diff --git a/.gitattributes b/.gitattributes
index cd586ea8ddc2e1444ad0a8efc9cedec21c2e76d1..2366375bd4d80f4d0b7a823227242d5beb3e55cc 100644
--- a/.gitattributes
+++ b/.gitattributes
@@ -214,3 +214,5 @@ data_all_eng_slimpj/shuffled/split/split_finalac/part-16.finalac filter=lfs diff
 data_all_eng_slimpj/shuffled/split/split_finalac/part-07.finalac filter=lfs diff=lfs merge=lfs -text
 data_all_eng_slimpj/shuffled/split/split_finalad/part-01.finalad filter=lfs diff=lfs merge=lfs -text
 data_all_eng_slimpj/shuffled/split/split_finalac/part-14.finalac filter=lfs diff=lfs merge=lfs -text
+data_all_eng_slimpj/shuffled/split/split_finalac/part-02.finalac filter=lfs diff=lfs merge=lfs -text
+data_all_eng_slimpj/shuffled/split/split_finalac/part-05.finalac filter=lfs diff=lfs merge=lfs -text
diff --git a/data_all_eng_slimpj/shuffled/split/split_finalac/part-02.finalac b/data_all_eng_slimpj/shuffled/split/split_finalac/part-02.finalac
new file mode 100644
index 0000000000000000000000000000000000000000..c35179b655e47d5a044f4ae7cdec578c0c115ce9
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split/split_finalac/part-02.finalac
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:f58035dc339d5280c7aaf72bee09b13670a24424b7e4819c233e9c7e9de06ae0
+size 12576660901
diff --git a/data_all_eng_slimpj/shuffled/split/split_finalac/part-05.finalac b/data_all_eng_slimpj/shuffled/split/split_finalac/part-05.finalac
new file mode 100644
index 0000000000000000000000000000000000000000..a95b37a5c21440fcb3690d03522038f65fe57864
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split/split_finalac/part-05.finalac
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:b257f509ba706c5612e817ec888930a1d07220f00e925c86190a4d54ab49b80b
+size 12576662770
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzwmu b/data_all_eng_slimpj/shuffled/split2/finalzwmu
new file mode 100644
index 0000000000000000000000000000000000000000..d87528cad3d3ea0c9778e924593740b95da8ae0f
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzwmu
@@ -0,0 +1,5 @@
+{"text":"\n\\chapter{Hardware}\n\\label{app:hardware}\n\nThis appendix gives an overview of the processors used throughout this work and\ntheir relevant properties.\n\nNote that, while the single-threaded peak performance is, where appropriate,\nbased on the processors' maximum turbo frequency, the multi-threaded peak\nperformance is instead computed from the base frequency.  Furthermore, we only\nlist the vector instructions that allow to reach a processor's theoretical peak\nperformance.\n\n\\section{\\hwstyle Harpertown E5450}\n\\label{hardware:E5450}\n\n\\href{http:\/\/ark.intel.com\/products\/33083\/Intel-Xeon-Processor-E5450-12M-Cache-3_00-GHz-1333-MHz-FSB}{\\nolinkurl{http:\/\/ark.intel.com\/products\/33083\/Intel-Xeon-Processor-E5450-}\\\\\\nolinkurl{12M-Cache-3_00-GHz-1333-MHz-FSB}}\n\nOur {\\namestyle Harpertown E5450}s were part of our compute cluster. Because\nthey were disposed of in mid~2016, they are only used in a part of this work's\nperformance analyses.\n\n\\begin{hwtable}\n    Name                &\\namestyle Intel\\textsuperscript\\textregistered{} Xeon\\textsuperscript\\textregistered{} Processor E5450 \\\\\n    Codename            &\\namestyle Harpertown \\\\\n    Lithography         &\\SI{45}{\\nano\\meter} \\\\\n    Release             &Q4 2007 \\\\\n    Cores \/ Threads     &4 \/ 4 \\\\\n    Base Frequency      &\\SI{3.00}{\\giga\\hertz} \\\\\n    Peak Performance    &\\SI{12}{\\giga\\flops\\per\\second} (single-threaded) \\\\\\nopagebreak\n                        &\\SI{48}{\\giga\\flops\\per\\second} (all cores) \\\\\n    Peak Bandwidth      &\\SI{10.6}{\\giga\\byte\\per\\second} \\\\\n    L2~cache            &\\SI6{\\mebi\\byte} {\\em per 2~cores}, 24-way set associative\\\\\n    L1d~cache           &\\SI{32}{\\kibi\\byte} per core, 8-way set-associative \\\\\n    Vector Instructions &1 SSE FMUL + 1 SSE FADD per cycle \\\\\\nopagebreak\n                        &$= \\SI4{\\flops\\per\\cycle}$ \\\\\n\\end{hwtable}\n\n\\section{\\hwstyle Sandy Bridge-EP E5-2670}\n\\label{hardware:E5-2670}\n\n\\href{http:\/\/ark.intel.com\/products\/64595\/Intel-Xeon-Processor-E5-2670-20M-Cache-2_60-GHz-8_00-GTs-Intel-QPI}{\\nolinkurl{http:\/\/ark.intel.com\/products\/64595\/Intel-Xeon-Processor-E5-2670-}\\\\\\nolinkurl{20M-Cache-2_60-GHz-8_00-GTs-Intel-QPI}}\n\nOur {\\namestyle Sandy Bridge E5-2680 v2}s are part of our compute cluster.\n\\intel{} \\turboboost is disabled on these machines unless otherwise stated.\n\n\\begin{hwtable}\n    Name                &\\namestyle Intel\\textsuperscript\\textregistered{} Xeon\\textsuperscript\\textregistered{} Processor E5-2670 \\\\\n    Codename            &\\namestyle Sandy Bridge-EP \\\\\n    Lithography         &\\SI{32}{\\nano\\meter} \\\\\n    Release             &Q1 2012 \\\\\n    Cores \/ Threads     &8 \/ 16 \\\\\n    Base Frequency      &\\SI{2.60}{\\giga\\hertz} \\\\\n    Max Turbo Frequency &\\SI{3.30}{\\giga\\hertz} ({\\em disabled unless otherwise stated})\\\\\n    Peak Performance    &\\SI{20.8}{\\giga\\flops\\per\\second} (single-threaded) \\\\\\nopagebreak\n                        &\\SI{166.4}{\\giga\\flops\\per\\second} (all cores) \\\\\n    Peak Bandwidth      &\\SI{51.2}{\\giga\\byte\\per\\second} \\\\\n    L3~cache            &\\SI{20}{\\mebi\\byte} shared, 20-way set associative \\\\\n    L2~cache            &\\SI{256}{\\kibi\\byte} per core, 8-way set associative \\\\\n    L1d~cache           &\\SI{32}{\\kibi\\byte} per core, 8-way set-associative \\\\\n    Vector Instructions &1 AVX FMUL + 1 AVX FADD per cycle \\\\\\nopagebreak\n                        &$= \\SI8{\\flops\\per\\cycle}$ \\\\\n\\end{hwtable}\n\n\n\\section{\\hwstyle Ivy Bridge-EP E5-2680 v2}\n\\label{hardware:E5-2680 v2}\n\n\\href{http:\/\/ark.intel.com\/products\/75277\/Intel-Xeon-Processor-E5-2680-v2-25M-Cache-2_80-GHz}{\\nolinkurl{http:\/\/ark.intel.com\/products\/75277\/Intel-Xeon-Processor-E5-2680-}\\\\\\nolinkurl{v2-25M-Cache-2_80-GHz}}\n\nOur {\\namestyle Ivy Bridge E5-2680 v3}s are part of our compute cluster.\n\n\\begin{hwtable}\n    Name                &\\namestyle Intel\\textsuperscript\\textregistered{} Xeon\\textsuperscript\\textregistered{} Processor E5-2680 v2\\\\\n    Codename            &\\namestyle Ivy Bridge-EP \\\\\n    Lithography         &\\SI{22}{\\nano\\meter} \\\\\n    Release             &Q3 2013 \\\\\n    Cores \/ Threads     &10 \/ 20 \\\\\n    Base Frequency      &\\SI{2.80}{\\giga\\hertz} \\\\\n    Max Turbo Frequency &\\SI{3.60}{\\giga\\hertz} \\\\\n    Peak Performance    &\\SI{28.8}{\\giga\\flops\\per\\second} (single-threaded) \\\\\\nopagebreak\n                        &\\SI{224}{\\giga\\flops\\per\\second} (all cores) \\\\\n    Peak Bandwidth      &\\SI{59.7}{\\giga\\byte\\per\\second} \\\\\n    L3~cache            &\\SI{25}{\\mebi\\byte} shared, 20-way set associative \\\\\n    L2~cache            &\\SI{256}{\\kibi\\byte} per core, 8-way set associative \\\\\n    L1d~cache           &\\SI{32}{\\kibi\\byte} per core, 8-way set-associative \\\\\n    Vector Instructions &1 AVX FMUL + 1 AVX FADD per cycle \\\\\\nopagebreak\n                        &$= \\SI8{\\flops\\per\\cycle}$ \\\\\n\\end{hwtable}\n\n\\section{\\hwstyle Haswell-EP E5-2680 v3}\n\\label{hardware:E5-2680 v3}\n\n\\href{http:\/\/ark.intel.com\/products\/81908\/Intel-Xeon-Processor-E5-2680-v3-30M-Cache-2_50-GHz}{\\nolinkurl{http:\/\/ark.intel.com\/products\/81908\/Intel-Xeon-Processor-E5-2680-}\\\\\\nolinkurl{v3-30M-Cache-2_50-GHz}}\n\nOur {\\namestyle Haswell-EP E5-2680 v3}s are part of our compute cluster.\n\n\\begin{hwtable}\n    Name                &\\namestyle Intel\\textsuperscript\\textregistered{} Xeon\\textsuperscript\\textregistered{} Processor E5-2680 v3\\\\\n    Codename            &\\namestyle Haswell-EP \\\\\n    Lithography         &\\SI{22}{\\nano\\meter} \\\\\n    Release             &Q3 2014 \\\\\n    Cores \/ Threads     &12 \/ 24 \\\\\n    Base Frequency      &\\SI{2.50}{\\giga\\hertz} \\\\\n    Max Turbo Frequency &\\SI{3.30}{\\giga\\hertz} \\\\\n    Peak Performance    &\\SI{52.8}{\\giga\\flops\\per\\second} (single-threaded) \\\\\\nopagebreak\n                        &\\SI{480}{\\giga\\flops\\per\\second} (all cores) \\\\\n    Peak Bandwidth      &\\SI{68}{\\giga\\byte\\per\\second} \\\\\n    L3~cache            &\\SI{30}{\\mebi\\byte} shared, 20-way set associative \\\\\n    L2~cache            &\\SI{256}{\\kibi\\byte} per core, 8-way set associative \\\\\n    L1d~cache           &\\SI{32}{\\kibi\\byte} per core, 8-way set-associative \\\\\n    Vector Instructions &2 AVX FMA per cycle \\\\\\nopagebreak\n                        &$= \\SI{16}{\\flops\\per\\cycle}$ \\\\\n\\end{hwtable}\n\n\\section{\\hwstyle Broadwell i7-5557U}\n\\label{hardware:i7-5557U}\n\n\\href{https:\/\/ark.intel.com\/products\/84993\/Intel-Core-i7-5557U-Processor-4M-Cache-up-to-3_40-GHz}{\\nolinkurl{https:\/\/ark.intel.com\/products\/84993\/Intel-Core-i7-5557U-}\\\\\\nolinkurl{Processor-4M-Cache-up-to-3_40-GHz}}\n\nOur {\\namestyle Broadwell i7-5557U} is part of a {\\namestyle MacBook Pro}.\n\n\\begin{hwtable}\n    Name                &\\namestyle Intel\\textsuperscript\\textregistered{} Core\\texttrademark{} i7-5557U Processor \\\\\n    Codename            &\\namestyle Broadwell-U \\\\\n    Lithography         &\\SI{14}{\\nano\\meter} \\\\\n    Release             &Q1 2015 \\\\\n    Cores \/ Threads     &2 \/ 4 \\\\\n    Base Frequency      &\\SI{3.10}{\\giga\\hertz} \\\\\n    Max Turbo Frequency &\\SI{3.40}{\\giga\\hertz} \\\\\n    Peak Performance    &\\SI{54.4}{\\giga\\flops\\per\\second} (single-threaded) \\\\\\nopagebreak\n                        &\\SI{99.2}{\\giga\\flops\\per\\second} (all cores) \\\\\n    Peak Bandwidth      &\\SI{25.6}{\\giga\\byte\\per\\second} \\\\\n    L3~cache            &\\SI4{\\mebi\\byte} shared, 16-way set associative \\\\\n    L2~cache            &\\SI{256}{\\kibi\\byte} per core, 8-way set associative \\\\\n    L1d~cache           &\\SI{32}{\\kibi\\byte} per core, 8-way set-associative \\\\\n    Vector Instructions &2 AVX FMA per cycle \\\\\\nopagebreak\n                        &$= \\SI{16}{\\flops\\per\\cycle}$ \\\\\n\\end{hwtable}\n\n\\subsection{\\blasl1}\n\n\\routinedoc{dcopy,\n    arguments={\n        n=dimension $n$,\n        x=vector $\\dv x \\in \\R^n$,\n        incx=increment for \\dv x,\n        y=vector $\\dv y \\in \\R^n$,\n        incy=increment for \\dv y\n    },\n    description={double-precision vector copy},\n    operations={$\\dv y \\coloneqq \\alpha \\dv x$},\n    flops=0,\n    datavol=$2 n$,\n    datamov=$2 n$,\n}\n\n\\routinedoc{dswap,\n    arguments={\n        n=dimension $n$,\n        x=vector $\\dv x \\in \\R^n$,\n        incx=increment for \\dv x,\n        y=vector $\\dv y \\in \\R^n$,\n        incy=increment for \\dv y\n    },\n    description={double-precision vector swap},\n    operations={${\\dv x, \\dv y \\coloneqq \\dv y, \\dv x}$},\n    flops=0,\n    datavol=$2 n$,\n    datamov=$4 n$,\n}\n\n\\routinedoc{daxpy,\n    arguments={\n        n=dimension $n$,\n        alpha=scalar $\\alpha$,\n        x=vector $\\dv x \\in \\R^n$,\n        incx=increment for \\dv x,\n        y=vector $\\dv y \\in \\R^n$,\n        incy=increment for \\dv y\n    },\n    description={double-precision scaled vector addition},\n    operations={$\\dv y \\coloneqq \\alpha \\dv x + \\dv y$},\n    flops=$2 n$,\n    datavol=$2 n$,\n    datamov=$3 n$,\n}\n\n\\routinedoc{ddot,\n    arguments={\n        n=dimension $n$,\n        x=vector $\\dv x \\in \\R^n$,\n        incx=increment for \\dv x,\n        y=vector $\\dv y \\in \\R^n$,\n        incy=increment for \\dv y\n    },\n    description={double-precision inner vector product},\n    operations={${\\alpha \\coloneqq \\dm[height=0, ']x \\dv x}$},\n    flops=$2 n$,\n    datavol=$2 n$,\n    datamov=$2 n$,\n}\n\n\n\\subsection{\\blasl2}\n\n\\routinedoc{dgemv,\n    arguments={\n        trans=\\dm A is transposed,\n        m=dimension $m$,\n        n=dimension $n$,\n        alpha=scalar $\\alpha$,\n        A=matrix $\\dm A \\in \\R^{m \\times n}$,\n        ldA=leading dimension for \\dm A,\n        x={vector $\\dv x \\in \\begin{cases}\n            \\R^n &\\text{if } \\code{trans} = \\code N\\\\\n            \\R^m &\\text{else}\n        \\end{cases}$},\n        incx=increment for \\dv x,\n        beta=scalar $\\beta$,\n        y={vector $\\dv y \\in \\begin{cases}\n            \\R^m &\\text{if } \\code{trans} = \\code N\\\\\n            \\R^n &\\text{else}\n        \\end{cases}$},\n        incy=increment for \\dv y\n    },\n    description={double-precision matrix-vector product},\n    operations={\n        {$\\dv y \\coloneqq \\alpha \\dm A \\matmatsep \\dv x + \\beta\\dv y$},\n        {$\\dv y \\coloneqq \\alpha \\dm[']A \\dv x + \\beta\\dv y$}\n    },\n    flops=$2 m n$,\n    datavol={$\\begin{array}{ll}\n        m n + m &\\text{if } \\code{trans} = \\code N\\\\\n        m n + n &\\text{else}\n    \\end{array}$},\n    datamov={$\\begin{array}{ll}\n        m n + 2 m &\\text{if } \\code{trans} = \\code N\\\\\n        m n + 2 n &\\text{else}\n    \\end{array}$},\n}\n\n\\routinedoc{dger,\n    arguments={\n        m=dimension $m$,\n        n=dimension $n$,\n        alpha=scalar $\\alpha$,\n        x=vector $\\dv x \\in \\R^m$,\n        incx=increment for \\dv x,\n        y=vector $\\dv y \\in \\R^n$,\n        incy=increment for \\dv y,\n        A=matrix $\\dm A \\in \\R^{m \\times n}$,\n        ldA=leading dimension for \\dm A\n    },\n    description={double-precision vector outer product},\n    operations={${\\dm A \\coloneqq \\alpha \\dv x \\dm[height=0, ']y + \\dm A}$},\n    flops=$2 m n$,\n    datavol=$m n + m + n$,\n    datamov=$2 m n + m + n$,\n}\n\n\\routinedoc{dtrsv,\n    arguments={\n        uplo=\\dm[lower]A is lower- or upper-triangular,\n        trans=\\dm[lower]A is transposed,\n        diag=\\dm[lower]A is unit triangular,\n        n=dimension $n$,\n        A=matrix $\\dm[lower]A \\in \\R^{n \\times n}$,\n        ldA=leading dimension for \\dm[lower]A,\n        x=vector $\\dv x \\in \\R^n$,\n        incX=increment for \\dv x\n    },\n    description={double-precision triangular linear system solve},\n    operations={\n        {$\\dv x \\coloneqq \\dm[lower, inv]A \\dv x$},\n        {$\\dv x \\coloneqq \\dm[lower, inv']A \\dv x$}\n    },\n    flops=$n^2$,\n    datavol={$\\frac12 n (n + 1) + n$},\n    datamov={$\\frac12 n (n + 1) + 2 n$}\n}\n\n\n\\subsection{\\blasl3}\n\n\\routinedoc{dgemm,\n    arguments={\n        transA=\\dm A is transposed,\n        transB=\\dm B is transposed,\n        m=dimension $m$,\n        n=dimension $n$,\n        k=dimension $k$,\n        alpha=scalar $\\alpha$,\n        A={matrix $\\dm A \\in \\begin{cases}\n            \\R^{m \\times k} &\\text{if } \\code{transA} = \\code N\\\\\n            \\R^{k \\times m} &\\text{else}\n        \\end{cases}$},\n        ldA=leading dimension for \\dm A,\n        B={matrix $\\dm B \\in \\begin{cases}\n            \\R^{k \\times n} &\\text{if } \\code{transB} = \\code N\\\\\n            \\R^{n \\times k} &\\text{else}\n        \\end{cases}$},\n        ldB=leading dimension for \\dm B,\n        beta=scalar $\\beta$,\n        C={matrix $\\dm C \\in \\R^{m \\times n}$},\n        ldC=leading dimension for \\dm C\n    },\n    description={double-precision matrix-matrix product},\n    operations={\n        {$\\dm C \\coloneqq \\alpha \\dm A \\matmatsep \\dm B + \\beta \\dm C$},\n        {$\\dm C \\coloneqq \\alpha \\dm A \\matmatsep \\dm[']B + \\beta \\dm C$},\n        {$\\dm C \\coloneqq \\alpha \\dm[']A \\matmatsep \\dm B + \\beta \\dm C$},\n        {$\\dm C \\coloneqq \\alpha \\dm[']A \\matmatsep \\dm[']B + \\beta \\dm C$}\n    },\n    flops=$2 m n k$,\n    datavol=$m k + k n + m n$,\n    datamov=$m k + k n + 2 m n$,\n}\n\n\\routinedoc{dsymm,\n    arguments={\n        side=\\dm A is on the left or right of \\dm B,\n        uplo=\\dm A is in lower- or upper-triangular storage,\n        m=dimension $m$,\n        n=dimension $n$,\n        alpha=scalar $\\alpha$,\n        A={matrix $\\dm A \\in \\begin{cases}\n            \\R^{m \\times m} &\\text{if } \\code{side} = \\code L\\\\\n            \\R^{n \\times n} &\\text{else}\n        \\end{cases}$},\n        ldA=leading dimension for \\dm A,\n        B={matrix $\\dm B \\in \\R^{m \\times n}$},\n        ldB=leading dimension for \\dm B,\n        beta=scalar $\\beta$,\n        C={matrix $\\dm C \\in \\R^{m \\times n}$},\n        ldC=leading dimension for \\dm C\n    },\n    description={double-precision symmetric matrix-matrix product},\n    operations={\n        {$\\dm C \\coloneqq \\alpha \\dm A \\matmatsep \\dm B + \\beta \\dm C$},\n        {$\\dm C \\coloneqq \\alpha \\dm B \\matmatsep \\dm A + \\beta \\dm C$}\n    },\n    flops={$\\begin{array}{ll}\n        2 m^2 n &\\text{if } \\code{side} = \\code L\\\\\n        2 m n^2 &\\text{else}\n    \\end{array}$},\n    datavol={$\\begin{array}{ll}\n        \\frac12 m (m + 1) + 2 m n &\\text{if } \\code{side} = \\code L\\\\\n        \\frac12 n (n + 1) + 2 m n &\\text{else}\n    \\end{array}$},\n    datamov={$\\begin{array}{ll}\n        \\frac12 m (m + 1) + 3 m n &\\text{if } \\code{side} = \\code L\\\\\n        \\frac12 n (n + 1) + 3 m n &\\text{else}\n    \\end{array}$},\n}\n\n\\routinedoc{dtrmm,\n    arguments={\n        side=\\dm[lower]A is on the left or right of \\dm B,\n        uplo=\\dm[lower]A is lower- or upper-triangular,\n        transA=\\dm[lower]A is transposed,\n        diag=\\dm[lower]A is unit triangular,\n        m=dimension $m$,\n        n=dimension $n$,\n        alpha=scalar $\\alpha$,\n        A={matrix $\\dm[lower]A \\in \\begin{cases}\n            \\R^{m \\times m} &\\text{if } \\code{side} = \\code L\\\\\n            \\R^{n \\times n} &\\text{else}\n        \\end{cases}$},\n        ldA=leading dimension for \\dm[lower]A,\n        B={matrix $\\dm B \\in \\R^{m \\times n}$},\n        ldB=leading dimension for \\dm B\n    },\n    description={double-precision triangular matrix-matrix product},\n    operations={\n        {$\\dm B \\coloneqq \\alpha \\dm[lower]A \\matmatsep \\dm B$},\n        {$\\dm B \\coloneqq \\alpha \\dm[lower, ']A \\dm B$},\n        {$\\dm B \\coloneqq \\alpha \\dm[upper]A \\matmatsep \\dm B$},\n        {$\\dm B \\coloneqq \\alpha \\dm[upper, ']A \\dm B$},\n        {$\\dm B \\coloneqq \\alpha \\dm B \\matmatsep \\dm[lower]A$},\n        {$\\dm B \\coloneqq \\alpha \\dm B \\matmatsep \\dm[lower, ']A$},\n        {$\\dm B \\coloneqq \\alpha \\dm B \\matmatsep \\dm[upper]A$},\n        {$\\dm B \\coloneqq \\alpha \\dm B \\matmatsep \\dm[upper, ']A$}\n    },\n    flops={$\\begin{array}{ll}\n        m^2 n &\\text{if } \\code{side} = \\code L\\\\\n        m n^2 &\\text{else}\n    \\end{array}$},\n    datavol={$\\begin{array}{ll}\n        \\frac12 m (m + 1) + m n &\\text{if } \\code{side} = \\code L\\\\\n        \\frac12 n (n + 1) + m n &\\text{else}\n    \\end{array}$},\n    datamov={$\\begin{array}{ll}\n        \\frac12 m (m + 1) + 2 m n &\\text{if } \\code{side} = \\code L\\\\\n        \\frac12 n (n + 1) + 2 m n &\\text{else}\n    \\end{array}$},\n}\n\n\\routinedocforward\\dsyrk{ssyrk,\n    arguments={uplo=, trans=, n=, k=, alpha=, A=, ldA=, beta=, C=, ldB=},\n    description={single-precision symmetric rank-k update},\n}\n\n\\routinedoc{dsyrk,\n    arguments={\n        uplo=\\dm C has lower- or upper-triangular storage,\n        trans=\\dm A is transposed,\n        n=dimension $n$,\n        k=dimension $k$,\n        alpha=scalar $\\alpha$,\n        A={matrix $\\dm A \\in \\begin{cases}\n            \\R^{n \\times k} &\\text{if } \\code{trans} = \\code N\\\\\n            \\R^{k \\times n} &\\text{else}\n        \\end{cases}$},\n        ldA=leading dimension for \\dm A,\n        beta=scalar $\\beta$,\n        C={symmetric matrix $\\dm C \\in \\R^{n \\times n}$},\n        ldB=leading dimension for \\dm C\n    },\n    description={double-precision symmetric rank-k update},\n    operations={\n        {$\\dm C \\coloneqq \\alpha \\dm A \\matmatsep \\dm[']A + \\dm C$},\n        {$\\dm C \\coloneqq \\alpha \\dm[']A \\dm A + \\dm C$},\n    },\n    flops={$n (n + 1) k$},\n    datavol={$\\frac12 n (n + 1) + n k$},\n    datamov={$n (n + 1) + n k$},\n}\n\n\\routinedocforward\\dsyrk{cherk,\n    arguments={uplo=, trans=, n=, k=, alpha=, A=, ldA=, beta=, C=, ldB=},\n    description={single-precision complex Hermitian rank-k update},\n}\n\n\\routinedocforward\\dsyrk{zherk,\n    arguments={uplo=, trans=, n=, k=, alpha=, A=, ldA=, beta=, C=, ldB=},\n    description={double-precision complex Hermitian rank-k update},\n}\n\n\\routinedoc{dsyr2k,\n    arguments={\n        uplo=\\dm C has lower- or upper-triangular storage,\n        trans=\\dm A is transposed,\n        n=dimension $n$,\n        k=dimension $k$,\n        alpha=scalar $\\alpha$,\n        A={matrix $\\dm A \\in \\begin{cases}\n            \\R^{n \\times k} &\\text{if } \\code{trans} = \\code N\\\\\n            \\R^{k \\times n} &\\text{else}\n        \\end{cases}$},\n        ldA=leading dimension for \\dm A,\n        B={matrix $\\dm B \\in \\begin{cases}\n            \\R^{n \\times k} &\\text{if } \\code{trans} = \\code N\\\\\n            \\R^{k \\times n} &\\text{else}\n        \\end{cases}$},\n        ldB=leading dimension for \\dm B,\n        beta=scalar $\\beta$,\n        C={symmetric matrix $\\dm C \\in \\R^{n \\times n}$},\n        ldC=leading dimension for \\dm C\n    },\n    description={double-precision symmetric rank-2k update},\n    operations={\n        {$\\dm C \\coloneqq \\alpha \\dm A \\matmatsep \\dm[']B + \\alpha \\dm B \\matmatsep \\dm[']A + \\dm C$},\n        {$\\dm C \\coloneqq \\alpha \\dm[']A \\dm B + \\alpha \\dm[']B \\dm A + \\dm C$}\n    },\n    flops={$2 n (n + 1) k$},\n    datavol={$\\frac12 n (n + 1) + 2 n k$},\n    datamov={$n (n + 1) + 2 n k$},\n}\n\n\\routinedocforward\\dtrsm{strsm,\n    arguments={side=, uplo=, transA=, diag=, m=, n=, alpha=, A=, ldA=, B=, ldB=},\n    description={single-precision triangular linear system solve with multiple\n    right hand sides},\n}\n\n\\routinedoc{dtrsm,\n    arguments={\n        side=\\dm[lower]A is on the left or right of \\dm B,\n        uplo=\\dm[lower]A is lower- or upper-triangular,\n        transA=\\dm[lower]A is transposed,\n        diag=\\dm[lower]A is unit triangular,\n        m=dimension $m$,\n        n=dimension $n$,\n        alpha=scalar $\\alpha$,\n        A={matrix $\\dm[lower]A \\in \\begin{cases}\n            \\R^{m \\times m} &\\text{if } \\code{side} = \\code L\\\\\n            \\R^{n \\times n} &\\text{else}\n        \\end{cases}$},\n        ldA=leading dimension for \\dm[lower]A,\n        B={matrix $\\dm B \\in \\R^{m \\times n}$},\n        ldB=leading dimension for \\dm B\n    },\n    description={double-precision triangular linear system solve with multiple\n    right hand sides},\n    operations={\n        {$\\dm B \\coloneqq \\alpha \\dm[lower, inv]A \\dm B$},\n        {$\\dm B \\coloneqq \\alpha \\dm[lower, inv']A \\dm B$},\n        {$\\dm B \\coloneqq \\alpha \\dm[upper, inv]A \\dm B$},\n        {$\\dm B \\coloneqq \\alpha \\dm[upper, inv']A \\dm B$},\n        {$\\dm B \\coloneqq \\alpha \\dm B \\matmatsep \\dm[lower, inv]A$},\n        {$\\dm B \\coloneqq \\alpha \\dm B \\matmatsep \\dm[lower, inv']A$},\n        {$\\dm B \\coloneqq \\alpha \\dm B \\matmatsep \\dm[upper, inv]A$},\n        {$\\dm B \\coloneqq \\alpha \\dm B \\matmatsep \\dm[upper, inv']A$}\n    },\n    flops={$\\begin{array}{ll}\n        m^2 n &\\text{if } \\code{side} = \\code L\\\\\n        m n^2 &\\text{else}\n    \\end{array}$},\n    datavol={$\\begin{array}{ll}\n        \\frac12 m (m + 1) + m n &\\text{if } \\code{side} = \\code L\\\\\n        \\frac12 n (n + 1) + m n &\\text{else}\n    \\end{array}$},\n    datamov={$\\begin{array}{ll}\n        \\frac12 m (m + 1) + 2 m n &\\text{if } \\code{side} = \\code L\\\\\n        \\frac12 n (n + 1) + 2 m n &\\text{else}\n    \\end{array}$},\n}\n\n\\routinedocforward\\dtrsm{ctrsm,\n    arguments={side=, uplo=, transA=, diag=, m=, n=, alpha=, A=, ldA=, B=, ldB=},\n    description={single-precision complex triangular linear system solve with\n    multiple right hand sides},\n}\n\n\\routinedocforward\\dtrsm{ztrsm,\n    arguments={side=, uplo=, transA=, diag=, m=, n=, alpha=, A=, ldA=, B=, ldB=},\n    description={double-precision complex triangular linear system solve with\n    multiple right hand sides},\n}\n\n\\subsection*{\\codestyle\\bf\\llap{\\routine(}\\arglist)}\n            \\label{routine:\\routine}\n            {\\it\\description}\n        ]\n        \\def\\empty{}\\small\\singlespacing\n        \\expandafter\\ifx\\note\\pgfkeysnovalue\\else\n            \\paragraph{Note\\strut}\n            \\note\n        \\fi\n\n        \\ifx\\operations\\empty\\else\n            \\paragraph{Operations\\strut}\n            \\operations\n        \\fi\n\n        {\n            \\raggedright\n            \\hbadness=10000\n            \\hangafter=1\n            \\renewcommand\\newline{\n                \\par\n                \\settowidth\\hangindent{\\hspace\\argwidth: }\n                \\makebox[\\hangindent]{}%\n            }\n            \\paragraph{Arguments\\strut}\n            \\arguments\n        }\n\n        \\expandafter\\ifx\\flops\\pgfkeysnovalue\\else\n            \\paragraph{Minimal FLOP-count\\strut}\n            \\flops\n        \\fi\n\n        \\expandafter\\ifx\\datavol\\pgfkeysnovalue\\else\n            \\paragraph{Data volume\\strut}\n            \\datavol\n        \\fi\n\n        \\expandafter\\ifx\\datamov\\pgfkeysnovalue\\else\n            \\paragraph{Minimal data movement\\strut}\n            \\datamov\n        \\fi\n    \\end{multicols}\n\n    \\filbreak\n}}\n\n\\newcommand\\routinedocforward[2]{\n    \\pgfkeys{\n        \/routine,\n        #2,\n        name\/.get=\\routine,\n        arglist\/.get=\\arglist,\n        description\/.get=\\description,\n    }\n    \\subsection*{\\codestyle\\bf\\routine(\\arglist)}\n    \\label{routine:\\routine}\n    {\\it\\description.}\n    See #1.\n\n    \\filbreak\n}\n\n\n\n\\subsection*{Reference Implementations}\n\nThe \\blas and \\lapack reference implementations~\\cite{blasweb, lapackweb} are\nfully functional and well-documented and thus of great value as references for\nroutine interfaces and semantics.  However, on their own they only attain poor\nperformance, and should therefore not be used in production codes.\n\nAll routines in the \\blas reference implementation are single-threaded\nand unoptimized.  The central kernel \\dgemm, for instance, is realized as a\nsimple triple loop that reaches around \\SI6{\\percent} of modern processors'\nsingle-threaded theoretical peak performance---optimized implementations are\ncommonly $15\\times$~faster on a single core and provide excellent multi-threaded\nscalability.\n\nSince \\lapack primarily relies on a tuned \\blas implementation for speed,\nthe reference implementation can in principle reach good performance.  However,\nas its documentation states, this requires careful tuning of its block sizes,\nwhose default values are generally too low on contemporary processors.\nOptimized implementations may further improve \\lapack's performance through\nfaster algorithms, tuned unblocked kernels (e.g, \\dtrti2, \\dpotf2), and\nalgorithm-level parallelism (e.g., task-based algorithms-by-blocks).\n\nThroughout this work, we use reference \\blas and \\lapack version~3.5.0.\n\n\n\\subsection*{\\namestyle OpenBLAS}\n\n{\\namestyle OpenBLAS}~\\cite{openblasweb} is a high-performance open-source \\blas\nand \\lapack implementation that is currently developed and maintained at the\n{\\namestyle Massachusetts Institute of Technology}.  It provides optimized and\nmulti-threaded \\blas kernels for a wide range of architectures, and offers tuned\nversion of core \\lapack routines, such as the \\dlauum, \\dtrtri, \\dpotrf, and\n\\dgetrf.  {\\namestyle OpenBLAS} is based on the discontinued {\\namestyle\nGotoBLAS2}, adopting its approach and much of its source-code; it includes\nassembly kernels for more recent architectures, such as \\sandybridgeshort and\n\\haswellshort, as well {\\namestyle AMD} processors.\n\nThroughout this work, we use {\\namestyle OpenBLAS} version~0.2.15.\n\n\n\\subsection*{\\namestyle BLIS}\n\nThe {\\namestyle BLAS-like Library Instantiation Software} ({\\namestyle\nBLIS})~\\cite{blis1, blis2, blis3, blisweb} is a fairly recent framework for\ndense linear algebra libraries that is actively developed at the {\\namestyle\nUniversity of Texas at Austin}.  While it comes with its own API, which is a\nsuperset, generalization, and extension of the \\blas, it contains a\ncompatibility layer offering the original de-factor standard \\blas interface.\n{\\namestyle BLIS} builds upon the {\\namestyle GotoBLAS} approach, yet\nrestructures and solidifies it to make all but a tiny ``micro-kernel''\narchitecture-independent.  While its performance is so far generally lower than\nthat of \\openblas (see examples in \\cref{sec:model:args}), its ambitious goal is\nto significantly speed up both the development of new application-specific\nkernels, and the adaptation to other architectures.\n\nAlthough multi-threading was introduced into {\\namestyle BLIS}~\\cite{blis3} soon\nafter its inception, its flexible threading model lacked a simple end-user\ninterface (such as following the environment variable \\code{OMP\\_NUM\\_THREADS})\nuntil November~2016 (commit\n\\href{https:\/\/github.com\/flame\/blis\/commit\/6b5a4032d2e3ed29a272c7f738b7e3ed6657e556}{\\sf\n6b5a403}).  As a result, we only presents single-threaded results for\n{\\namestyle BLIS}.\n\nThroughout this work we use {\\namestyle BLIS} version~0.2.0.\n\n\n\\subsection*{\\namestyle MKL}\n\n\\intel's {\\namestyle Math Kernel Library} ({\\namestyle MKL})~\\cite{mklweb} is a\nhigh-performance library for \\intel processors that covers \\blas and\n\\lapack, as well as other high-performance computations, such as for Fast\nFourier Transforms (FFT) and Deep Neural Networks (DNN).  While {\\namestyle MKL}\nis a closed-source library, it recently began offering free developer licenses.\nIn terms of performance, it is in most scenarios superior to open-source\nlibraries such as \\openblas and \\blis (see examples in \\cref{sec:model:args}).\n\nThroughout this work we use {\\namestyle MKL} version~11.3.\n\n\n\\subsection*{\\namestyle Accelerate}\n\n\\apple's framework {\\namestyle Accelerate}~\\cite{accelerateweb} is a\nhigh-performance library that ships with {\\namestyle macOS} and, among others,\nprovides full \\blas and \\lapack functionality.  Its performance is for many\ncases comparable to \\openblas or slightly better.\n\n\n\\subsection*{Other Implementations}\n\nThe following notable \\blas and \\lapack implementations are not used throughout\nthis work:\n\\begin{itemize}\n    \\item The {\\namestyle Automatically Tuned Linear Algebra Software}\n        ({\\namestyle ATLAS})~\\cite{atlas1, atlas2, atlas3, atlasweb} is a\n        high-performance \\blas implementation that relies on auto-tuning.  While\n        {\\namestyle ATLAS} kernels typically don not reach the performance of\n        hand-tuned implementations such as \\openblas, \\blis, and \\mkl, it\n        provides good performance for new and exotic architectures with little\n        effort.\n\n    \\item {\\namestyle GotoBLAS2}~\\cite{gotoblas1, gotoblas2, gotoblasweb} is a\n        high-performance \\blas implementation that was developed at the\n        {\\namestyle Texas Advanced Computing Center}.  Since its\n        discontinuation, much of its code-base was picked up by its successor\n        \\openblas in~2011, and its approach was refined and generalized in\n        \\blis.\n\n    \\item {\\namestyle IBM}'s {\\namestyle Engineering and Scientific Subroutine\n        Library} ({\\namestyle ESSL}) \\cite{esslweb} provides a high-performance\n        \\blas implementation and parts of \\lapack for {\\namestyle POWER}-based\n        systems, such as {\\namestyle Blue Gene} supercomputers.\n\\end{itemize}\n\n\\section{Storage Format}\n    \\label{app:libs:store}\n    \\input{applibs\/store}\n\n    \\section{\\namestyle Basic Linear Algebra Subprograms}\n    \\label{app:libs:blas}\n    \\input{applibs\/blas}\n\n    \\section{\\namestyle Linear Algebra PACKage}\n    \\label{app:libs:lapack}\n    \\input{applibs\/lapack}\n\n    \\section{Implementations}\n    \\label{app:libs:libs}\n    \\input{applibs\/libs}\n}\n\n\\subsection{Scalars}\nEach scalar operand (e.g., $\\alpha \\in \\R$) is passed as a single argument,\n (e.g., \\code{double *alpha}).  Complex scalars are stored as two consecutive\n elements of the basis data-type (\\code{float} or \\code{double}) that represent\n the real and imaginary parts.\n\n\n\\subsection{Vectors}\nEach vector operand (e.g., $\\dv x \\in \\R^n$) is specified by three arguments:\n\\begin{itemize}\n    \\item A size argument (e.g., \\code{int *n}) determines the length of the\n        vector.  One size argument can describe multiple vectors (and\/or\n        matrices) with the same size.\n\n    \\item A data argument (e.g., \\code{double *x}) points to the vector's first\n        element in memory.\n\n    \\item An increment argument (e.g., \\code{int *incx}) identifies the stride\n        between consecutive elements of the vector.  For instance, a\n        contiguously stored vector has an increment of~1.\n\n        Note that most routines allow negative increments.  In this case, the\n        vector is stored in reverse, and the data argument points to the\n        vector's last element---the first memory location.\n\\end{itemize}\nTo summarize, vector element~$x_i$ is stored at \\code{x[i * incx]} if\n\\code{incx} is positive and \\code{x[(i - n + 1) * incx]} otherwise.\n\n\n\\subsection{Matrices}\nEach matrix (e.g., $\\dm[width=.7]A \\in \\R^{m \\times n}$) is specified by four\narguments:\n\\begin{itemize}\n    \\item Two size arguments (e.g., \\code{int *m} and \\code{int *n}) determine\n        the matrix height~($m$) and width~($n$).  One size argument can describe\n        the dimensions of multiple matrices (and\/or vectors), or both dimensions\n        of a square matrix.\n\n    \\item A data argument (e.g., \\code{double *A}) points to the first matrix\n        element in memory (e.g., $a_{00}$).  The following elements of the\n        first column (e.g., $a_{i0}$) are stored consecutively in memory as\n        vector with increment~1.\n\n    \\item A leading dimension argument (e.g., \\code{int *ldA}) describes the\n        distance in memory between matrix columns.  It can hence be understood\n        and used as the increment argument for the matrix rows as vectors.  The\n        term ``leading dimension'' comes from the concept that a referenced\n        matrix is part of a larger, contiguously stored ``leading'' matrix. It\n        allows to operate on sub-matrices or tensor panels as shown throughout\n        this work.\n\n        Leading dimensions must be at least equal to the height of the matrix\n        (e.g., $m$).\n\\end{itemize}\nTo summarize, matrix element~$a_{ij}$ is stored at \\code{A[i + j * ldA]}.\n\n\n\\subsection{Compute-Bound Efficiency}\n\\label{sec:term:eff:compbound}\n\nA computation is compute-bound on a hardware platform if the memory operations\nto load and store the involved data can be amortized by floating-point\noperations, i.e., the available memory bandwidth is sufficient for all transfers\nand the speed at which the processor performs \\flops  is the bottleneck.  An\noperation is theoretically bandwidth bound when\n\\[\n    \\text{arithmetic intensity} \\geq \\frac\\pperf\\pbw \\enspace.\n\\]\nFurthermore, a computation's\n\\definition[(compute-bound)\\\\efficiency]{compute-bound efficiency} (or simply\n{\\em efficiency}) is given by\n\\begin{equation}\\label{eq:term:eff}\n    \\text{compute-bound efficiency}\n    \\defeqq \\frac{\\text{attained performance}}\\pperf \\enspace.\n\\end{equation}\nThis unit-less metric between 0 and~1 indicates how well the available hardware\nresources are utilized:  While a value close to~1 corresponds to near-optimal\nutilization, lower values indicate untapped resource potential.\n\n\\begin{example}{Compute-bound efficiency}{term:eff}\n    The matrix-matrix multiplication $\\dm C \\coloneqq \\dm A \\matmatsep \\dm B +\n    \\dm C$ (\\dgemm[NN]) with $\\dm A,\\allowbreak \\dm B,\\allowbreak \\dm C \\in\n    \\R^{1000 \\times 1000}$ has an arithmetic intensity of (see\n    \\cref{ex:term:ai})\n    \\[\n        \\SIvar{1000 \\times \\frac1{16}}{\\flops\\per\\Byte} \n        = \\SI{62.5}{\\flops\\per\\Byte} \\enspace.\n    \\]\n    On a single core of a \\sandybridge with a peak floating-point performance of\n    \\SI{20.8}{\\giga\\flops\\per\\second} (\\turboboost disabled) and peak bandwidth\n    of \\SI{51.2}{\\gibi\\byte\\per\\second} this operation is clearly compute bound:\n    \\[\n        \\frac\n            {\\SI{20.8}{\\giga\\flops\\per\\second}}\n            {\\SI{16.25}{\\gibi\\byte\\per\\second}}\n        \\approx \\SI{1.28}{\\flops\\per\\Byte}\n        < \\SI{62.5}{\\flops\\per\\Byte} \\enspace.\n    \\]\n    If the \\dgemm[NN] runs at \\SI{19.61}{\\giga\\flops\\per\\second}\n    (\\cref{ex:term:perf}), it reached an efficiency of\n    \\[\n        \\frac{\\text{attained performance}}\\pperf\n        = \\frac\n            {\\SI{19.61}{\\giga\\flops\\per\\second}}\n            {\\SI{20.8}{\\giga\\flops\\per\\second}}\n        \\approx \\SI{94.27}\\percent \\enspace.\n    \\]\n\\end{example}\n\nThere are many different ways to look at efficiency other than the ratio of\nattained performance to peak performance.  Rewriting the definition of\nefficiency as\n\\begin{align*}\n    \\text{efficiency}\n    &= \\frac{\\text{attained performance}}\\pperf \\\\\n    &= \\frac\n        {\\text{cost} \/ \\text{runtime}}\n        {\\text{cost} \/ \\text{optimal runtime}} \\\\\n    &= \\frac{\\text{optimal runtime}}{\\text{runtime}} \\enspace,\n\\end{align*}\nit is expressed as the ratio of the minimum time required to perform the\noperation's minimal \\flops on the given hardware to the computation's runtime.\nIf we reorganize it as\n\\begin{align*}\n    \\text{efficiency}\n    &= \\frac{\\text{attained performance}}\\pperf \\\\\n    &= \\frac{\\text{cost} \/ \\text{runtime}}\\pperf \\\\\n    &= \\frac{\\text{cost}}{\\text{runtime} \\times \\pperf} \\\\\n    &= \\frac{\\text{cost}}{\\text{available \\flops}} \\enspace,\n\\end{align*}\nit can be seen as the ratio of the operation's minimal \\flop-count to how many\n\\flops the processor could theoretically perform during the computation's\nruntime.\n\n\\begin{example}{Expressing compute-bound efficiency}{term:eff2}\n    In \\cref{ex:term:eff} the \\dgemm[NN] took \\SI{102}\\ms, while the\n    \\sandybridge with a peak performance of \\SI{20.8}{\\giga\\flops\\per\\second}\n    (\\turboboost disabled) could have performed the required $\\SIvar{2 \\times\n    1000^3}\\flops = \\SI{2e9}\\flops$ in\n    \\[\n        \\frac{\\SI{2e9}\\flops}{\\SI{20.8}{\\giga\\flops\\per\\second}}\n        \\approx \\SI{96.15}\\ms \\enspace .\n    \\]\n    Hence, the computation's efficiency can be computed as\n    \\[\n        \\frac{\\text{optimal runtime}}{\\text{runtime}}\n        = \\frac{\\SI{96.15}\\ms}{\\SI{102}\\ms} \\approx \\SI{94.26}\\percent \\enspace.\n    \\]\n\n    We can also consider that in the \\SI{102}{\\ms} that the \\dgemm[NN] took, the\n    \\sandybridgeshort core could have performed\n    \\[\n        \\SI{102}\\ms \\times \\SI{20.8}{\\giga\\flops\\per\\second}\n        \\approx \\SI{2.12e9}\\flops \\enspace.\n    \\]\n    Once again we obtain the same efficiency, as a \\flop-count ratio:\n    \\[\n        \\frac{\\text{cost}}{\\text{available \\flops}}\n        = \\frac{\\SI{2e9}\\flops}{\\SI{2.12e9}\\flops}\n        \\approx \\SI{94.26}\\percent\n        \\enspace.\n    \\]\n\\end{example}\n\n\n\\subsection{Bandwidth-Bound Efficiency}\n\\label{sec:term:eff:bwbound}\n\nA computation is bandwidth-bound on a hardware platform if the memory operations\ncannot load and store the involved data as fast as the processor's\nfloating-point units can process it, i.e., the memory bandwidth is the\nbottleneck and the compute units are partially idle.  An operation is\ntheoretically bandwidth-bound when\n\\[\n    \\text{arithmetic intensity} \\leq \\frac\\pperf\\pbw \\enspace.\n\\]\nFurthermore, a computation's \\definition{bandwidth-bound efficiency} is defined\nas\n\\begin{equation}\n    \\label{eq:term:eff:bwbound}\n    \\text{bandwidth-bound efficiency} \\defeqq\n    \\frac{\\text{attained bandwidth}}\\pbw \\enspace.\n\\end{equation}\nA bandwidth-bound efficiency close to~1 indicates a good utilization of the\nprocessor's main-memory bandwidth, while smaller values signal underutilization.\n\n\\begin{example}{Bandwidth-bound efficiency}{term:bwbeff}\n    The vector inner product $\\alpha \\coloneqq \\dm[height=0, ']x \\matvecsep \\dv\n    y$ (\\ddot) with $\\dv x, \\dv y \\in \\R^{\\num{100000}}$ has an arithmetic\n    intensity of \\SIvar{\\frac18}{\\flops\\per\\Byte} (\\cref{ex:term:ai}) and is\n    thus clearly bandwidth-bound.  If on one core of a \\sandybridge, it attains\n    a bandwidth of \\SI{11.49}{\\gibi\\byte\\per\\second} (\\cref{ex:term:bw}),\n    relative to the processor's empirical peak bandwidth of\n    \\SI{16.25}{\\gibi\\byte\\per\\second} (\\cref{ex:term:peakbw}), it performed at a\n    bandwidth-bound efficiency of\n    \\[\n        \\frac{\\text{attained bandwidth}}\\pbw\n        = \\frac\n            {\\SI{11.49}{\\gibi\\byte\\per\\second}}\n            {\\SI{16.25}{\\gibi\\byte\\per\\second}}\n        \\approx \\SI{70.71}\\percent \\enspace.\n    \\]\n\\end{example}\n\n\n\\subsection{The Roofline Model}\n\\label{sec:term:roofline}\n\nThe \\definition{Roofline model}~\\cite{roofline1} plots the performance of\ncomputations (in \\si{\\giga\\flops\\per\\second}) against their arithmetic intensity\n(in \\si{\\flops\\per\\Byte}).  In addition to data-points from measurements, two\nlines are added to such a plot to indicate the theoretically attainable\nperformance depending on the arithmetic intensity: The product of peak bandwidth\nand arithmetic intensity (in units: $\\si{\\gibi\\byte\\per\\second} \\times\n\\si{\\flops\\per\\Byte} = \\si{\\gibi\\flops\\per\\second} \\approx\n\\SI{.93}{\\giga\\flops\\per\\second}$) constitutes a straight line through the\norigin with the bandwidth as a gradient (visually: \\tikz\\draw[thick, darkred]\n(0, 0) -- (1.5ex, 1.5ex);) that represents the bandwidth-bound performance limit;\nand the peak floating-point performance is a constant line (\\tikz\\draw[thick,\ndarkred] (0,0) (0, 1.5ex) -- (3ex, 1.5ex);).  Together these two lines form the\nroofline-shaped performance limit (\\tikz\\draw[thick, darkred] (0, 0) -- (1.5ex,\n1.5ex) -- (4.5ex, 1.5ex);) that gives the visualization its name:\n\\begin{equation}\\label{eq:term:roofline}\n    \\text{performance limit} =\n    \\min\\left(\\begin{array}c\n            \\pbw \\times \\text{intensity},\\\\\n            \\pperf\n    \\end{array}\\right) \\enspace.\n\\end{equation}\nComparing the attained performance of a computation to this limit yields the\ncomputation's efficiency---bandwidth-bound below the left part of the ``roof''\nand compute-bound below the right part.\n\n\\input{appterm\/figures\/roofline}\n\n\\begin{example}{The roofline model}{term:roofline}\n    \\Cref{fig:term:roofline} presents the Roofline model for one core of a\n    \\sandybridge.  This processor has a  single-core peak performance of\n    \\SI{20.8}{\\giga\\flops\\per\\cycle} (\\turboboost disabled), and we use the\n    measured single-core peak bandwidth of \\SI{16.25}{\\gibi\\byte\\per\\second}\n    (\\cref{ex:term:peakbw}).  Together these two factors impose the performance\n    limit~(\\ref*{plt:term:roofline:peak})\n    \\[\n        \\min(\\SI{16.25}{\\gibi\\byte\\per\\second} \\times \\text{arithmetic\n        intensity}, \\SI{20.8}{\\giga\\flops\\per\\second})\n    \\]\n\n    \\cref{fig:term:roofline} also contains the measured performance of\n    representative \\blasl1, 2, and~3 operations, whose arithmetic intensity was\n    determined in \\cref{ex:term:ai}.\n    \\begin{itemize}\n        \\item The vector inner product $\\alpha \\coloneqq \\dm[height=0, ']x\n            \\matvecsep \\dv y$ (\\ddot) with $\\dv x, \\dv y \\in \\R^n$\n            (\\ref*{plt:term:roofline:ddot}) has a arithmetic intensity of\n            \\SIvar{\\frac18}{\\flops\\per\\Byte}, making it clearly bandwidth-bound\n            below the left part of the ``roofline''.  The attained\n            (bandwidth-bound) efficiency, which is given by the ratio of the\n            measured performance~(\\ref*{plt:term:roofline:ddot}) to the\n            attainable peak performance~(\\ref*{plt:term:roofline:peak}), is\n            quite high at~\\SI{87.93}\\percent.\n\n        \\item The matrix-vector multiplication $\\dv y \\coloneqq \\dm A \\matvecsep\n            \\dv x + \\dm y$ (\\dgemv) with $\\dm A \\in \\R^{n \\times n}$ and $\\dv x,\n            \\dv y \\in \\R^n$ (\\ref*{plt:term:roofline:dgemv}) has a computation\n            intensity of $\\approx \\SIvar{\\frac14}{\\flops\\per\\Byte}$, making it\n            also bandwidth-bound.  The (bandwidth-bound) efficiency\n            (\\ref*{plt:term:roofline:dgemv} divided by\n            \\ref*{plt:term:roofline:peak}) is between~\\SI{45.32}{\\percent} (for\n            $n = 100$) and \\SI{76.66}{\\percent} (for~$n = 2000$).\n\n        \\item The matrix-matrix multiplication $\\dm C \\coloneqq \\dm A \\matmatsep\n            \\dm B + \\dm C$ (\\dgemm[NN]) with $\\dm A, \\dm B, \\dm C \\in \\R^{n\n            \\times n}$ (\\ref*{plt:term:roofline:dgemm}) has a higher arithmetic\n            intensity of \\SIvar{\\frac n{16}}{\\flops\\per\\Byte}, which makes it\n            theoretically compute-bound on our system for~$n \\geq 21$.  In the\n            memory-bound domain it reaches its peak (memory-bound) efficiency\n            (\\ref*{plt:term:roofline:dgemm} divided\n            by~\\ref*{plt:term:roofline:peak}) of \\SI{50.15}{\\percent} at~$n =\n            20$.  Within the compute-bound domain, its (compute-bound)\n            efficiency grows towards \\SI{74.32}{\\percent} for our largest\n            problem size~$n = 100$.  Beyond this size the efficiency keeps\n            growing and converge to its peak of \\SI{93.70}{\\percent} for\n            matrices of size~$n = 2000$.\n    \\end{itemize}\n\\end{example}\n\n\n\n\n\\section{Workload}\n    \\label{sec:term:workload}\n    \\input{appterm\/workload}\n\n    \\section{Runtime}\n    \\label{sec:term:time}\n    \\input{appterm\/time}\n\n    \\section{Performance and Attained Bandwidth}\n    \\label{sec:term:perf}\n    \\input{appterm\/perf}\n\n    \\section{Hardware Constraints}\n    \\label{sec:term:hw}\n    \\input{appterm\/hardware}\n\n    \\section{Efficiency}\n    \\label{sec:term:eff}\n    \\input{appterm\/eff}\n\n    \\section{Other Metrics}\n    \\label{sec:term:other}\n    \\input{appterm\/othermetrics}\n}\n\n\n\n\n\\subsection{Floating-Point Operations}\n\\label{sec:term:flops}\n\nMost scientific computations, as complex as they may be, perform their work\nthrough a small set of elementary arithmetic operations on floating-point\nrepresentations of real numbers, such as scalar additions or\nmultiplications\\footnote{%\n    Exceptions that work on integer data or other structures include graph\n    algorithms and discrete optimization.\n}---These the so-called \\definition[\\flops: floating-point\noperations\\\\single- and double-precision]{floating-point operations}\n({\\em\\flops}).\\footnote{%\n    Not to be confused with floating-point operations {\\em per second}\n    (\\si{\\flops\\per\\second}).\n}\n\nContemporary hardware offers two floating-point precisions standardized in\nIEEE~754~\\cite{ieee754}: {\\em single-precision}, and {\\em double-precision}.\nThey differ in the range of representable numbers, their representation\naccuracy, and their implementation in hardware.  While we distinguish between\nsingle-precision \\flops and double-precision \\flops, throughout this work we are\nmostly concerned with double-precision computations. Hence we use ``\\flops''\nwithout a specification refers to double-precision floating-point operations,\nand \\R is used to denote double-precision numbers.\n\nAs commonly practiced in dense linear algebra, we assume that the multiplication\nof two $n \\times n$ matrices requires \\SIvar{2 n^3}\\flops{}---it has an\nasymptotic \\definition[matrix-matrix multiplication: $O(n^3)$]{complexity} of\n$O(n^3)$.  While algorithms with lower asymptotic complexities (such as the {\\em\nStrassen algorithm} with a complexity of $O(n^{2.807})$~\\cite{strassen} or the\n{\\em Coppersmith-Winograd algorithm} with a complexity of\n$O(n^{2.376})$~\\cite{coppersmith}) were already known in the 1970s, due to\nconsiderably higher constant factors they found little to no application in\nhigh-performance computing until recently~\\cite{blisstrassen}.\n\nThe \\flop-count of most dense liner algebra operations such as the matrix-matrix\nmultiplication is \\definition[data-independence]{data-independent}, i.e., the\noperand entries do not affect what arithmetic operations are\nperformed.\\footnote{%\n    Exceptions may be caused by corrupted input, such as \\code{NaN}s, or\n    floating-point exceptions, such as division by~0 or under-\/overflows.\n}  In particular, this means that all multiplications with 0's are explicitly\nperformed no matter how sparse an operand is (i.e., how few non-zero entries\nit has).  A notable exception to the data-independence are numerical\neigensolvers, whose FLOP-counts depend on the eigenspectrum of the input matrix;\nhowever, we do not study eigensolvers in further detail in this work.\n\nAssuming the cubic complexity of the matrix-matrix multiplication, the\ndata-independence allows us to compute the \\definition[cost = minimal\nFLOP-count]{minimal FLOP-count}---also referred to as {\\em cost}---for most\noperations solely based on their operands' sizes.\n\n\\begin{example}{Minimal \\flop-counts}{term:flops}\n    The vector inner product $\\alpha \\coloneqq \\dm[height=0, ']x \\matvecsep \\dv\n    y$ (\\ddot) with $\\dv x, \\dv y \\in \\R^n$ costs \\SIvar{2 n}\\flops: one\n    multiplication and one addition per vector entry.\n\n    The solution of a triangular linear system with multiple right-hand-sides\n    $\\dm[width=.4]B \\coloneqq \\dm[lower, inv]A \\dm[width=.4]B$ (\\dtrsm) with\n    $\\dm[lower]A\\lowerpostsep \\in \\R^{n \\times n}$ and $\\dm[width=.4]B \\in \\R^{n\n    \\times m}$ requires \\SIvar{n^2 m}\\flops.\n\n    The Cholesky decomposition of a symmetric positive definite (SPD) matrix\n    $\\dm[lower]L \\dm[upper, ']L \\coloneqq \\dm A$ (\\dpotrf) with $\\dm A \\in \\R^{n\n    \\times n}$ costs\n    \\[\n        \\SIvar{\\frac16 n (n + 1) (2 n + 1)}\\flops\n        \\approx \\SIvar{\\frac13 n^3}\\flops \\enspace.\n    \\]\n\\end{example}\n\nNote that an operation's minimal \\flop-count only provides a lower bound for\nroutines implementing it; reasons for exceeding this bound range from technical\nlimitations to cache-aware data movement patterns and algorithmic schemes that\nperform extra \\flops to use faster compute kernels.\n\n\n\\subsection{Data Volume and Movement}\n\\label{sec:term:datamovement}\n\nThe largest portion of a scientific computation's memory footprint is typically\noccupied by its numerical data consisting of floating-point numbers.  A real\nnumber in single- and double-precision requires, respectively, 4 and~\\SI8\\bytes,\nwhereas complex numbers are represented as two consecutive real numbers\nand thus require twice the space.  Since throughout this work we mostly use\ndouble-precision numbers---conventionally called ``\\definition[$\\SI1\\double =\n\\SI8\\bytes$]{doubles}''---we can proceed with the assumption that each number\ntakes up \\SI8\\bytes.\n\nIn dense linear algebra, the \\definition[data volume in \\bytes]{data volume}\n(in~\\bytes) involved in a computation is determined almost exclusively by the\ninvolved matrix operands.  For instance, a square matrix of size $1000 \\times\n1000$ consists of $\\SI{e6}\\doubles = \\SI{8e6}\\bytes \\approx\n\\SI{7.63}{\\mebi\\byte}$;\\footnote{%\n    We use the 1024-based binary prefixes for data volumes: $\\SI{1024}\\bytes =\n    \\SI1{\\kibi\\byte}$ (``kibibyte''), $\\SI{1024}{\\kibi\\byte} = \\SI1{\\mebi\\byte}$\n    (``mebibyte''), and $\\SI{1024}{\\mebi\\byte} = \\SI1{\\gibi\\byte}$\n    (``gibibyte'').\n} vector and scalar operands in comparison take up little space:  A vector of\nsize~1000  requires $\\SI{8000}\\bytes = \\SI{7.81}{\\kibi\\byte}$, and a scalar fits\nin just \\SI8\\bytes.\n\nWhile a computation's data volume describes how much data is involved in an\noperation, it says nothing about how often it is accessed.  For this purpose we\nintroduce the concept of \\definition{data movement} that quantifies how much\ndata is read from or written to memory.  A computation's data movement is\ncommonly higher than its data volume, because (parts of) the data are accessed\nmultiple times.\n\nWhile the actual data movement of any dense linear algebra operation is highly\nimplementation dependent, we can easily derive the \\definition{minimal data\nmovement} from the operation's mathematical formulation by summing the size of\nall input and output operands, counting the operands that are both input and\noutput twice.\n\n\\begin{example}{Data volume and movement}{term:datamov}\n    The vector inner product $\\alpha \\coloneqq \\dm[height=0, ']x \\matvecsep \\dv\n    y$ (\\ddot) with $\\dv x, \\dv y \\in \\R^n$ involves a data volume of $\\SIvar{2\n    n}\\doubles = \\SIvar{16 n}\\bytes$ (ignoring the scalar $\\alpha$); since both\n    \\dv x and \\dv y need only be read once the data movement is also \\SIvar{16\n    n}\\bytes.\n\n    The matrix-matrix product $\\dm C \\coloneqq \\dm A \\matmatsep \\dm B + \\dm C$\n    (\\dgemm[NN]) with $\\dm A,\\allowbreak \\dm B,\\allowbreak \\dm C \\in \\R^{n\n    \\times n}$ involves a data volume of $\\SIvar{3 n^2}\\doubles = \\SIvar{24\n    n^2}\\bytes$, however, since $\\dm C$ is updated, the minimal data movement is\n    $\\SIvar{4 n^2}\\doubles = \\SIvar{32 n^2}\\bytes$.\n\n    The Cholesky decomposition $\\dm[lower]L \\dm[upper, ']L \\coloneqq \\dm A$\n    (\\dpotrf) with $\\dm A \\in \\R^{n \\times n}$ uses only the lower-triangular\n    part of the symmetric matrix \\dm A,\\footnotemark{} and \\dm A is decomposed\n    in place, i.e., it is overwritten by \\dm[lower]L\\lowerpostsep upon\n    completion.  Hence the data volume is $\\SIvar{\\frac12 n (n + 1)}\\doubles\n    \\approx \\SIvar{4 n^2}\\bytes$, while the minimal data movement is at least\n    $\\SIvar{2 \\cdot \\frac12 n (n + 1)}\\doubles \\approx \\SIvar{8 n^2}\\bytes$.\n\\end{example}\n\\footnotetext{%\n    Space for the whole matrix is allocated, but the strictly upper-triangular\n    part is not accessed.\n}\n\nNote that the minimal data movement is a strict lower bound when none of the\ninvolved data is in any of the processor's caches.  Furthermore, depending on\nthe operation and the cache sizes, it may not be attainable in implementations.\n\n\n\\subsection{Arithmetic Intensity}\n\\label{sec:term:ai}\n\nDividing an operation's minimal flop count by its minimal data movement yields\nits \\definition{arithmetic intensity}:\n\\begin{equation}\n    \\label{eq:term:ai}\n    \\text{arithmetic intensity}\n    \\defeqq \\frac{\\text{minimal \\flop-count}}{\\text{minimal data movement}}\n    \\enspace.\n\\end{equation}\nA low arithmetic intensity means that few operations are performed per memory\naccess, thus making the data movement a likely bottleneck; a high arithmetic\nintensity on the other hand indicates that a lot of work is performed per data\nelement, thus making the floating-point computations the potential bottleneck.\nArithmetic intensity divides dense linear algebra operations into two groups:\nWhile for \\blasl1 (vector-vector) and~2 (matrix-vector) operations the intensity\nis quite small and independent of the problem size, it is considerably larger\nfor \\blasl3 (matrix-matrix) and dense \\lapack-level operations, for which\nincreases linearly with the problem size.\n\n\\begin{example}{Arithmetic intensity}{term:ai}\n    The vector inner product $\\alpha \\coloneqq \\dm[height=0, ']x \\dv y$ (\\ddot)\n    with $\\dv x, \\dv y \\in \\R^n$ is a \\blasl1 operation that performs \\SIvar{2\n    n}{\\flops} over \\SIvar{2 n}{\\doubles} of data movement.  Hence its\n    arithmetic intensity is\n    \\[\n        \\frac{\\text{minimal \\flop-count}}{\\text{minimal data movement}}\n        = \\frac{\\SIvar{2 n}\\flops}{\\SIvar{2 n}\\doubles}\n        = \\SIvar{\\frac18}{\\flops\\per\\Byte} \\enspace.\n    \\]\n\n    The matrix-vector multiplication $\\dv y \\coloneqq \\dm A \\matvecsep \\dv x +\n    \\dm y$ (\\dgemv[N]) with $\\dm A \\in \\R^{n \\times n}$ and $\\dv x, \\dv y \\in\n    \\R^n$ is a \\blasl2 operation that performs \\SIvar{2 n^2}{\\flops} over\n    \\SIvar{n^2 + 3 n}{\\doubles} of data movement ($\\dv y$ is both read and\n    written).  Therefore, its arithmetic intensity is\n    \\[\n        \\frac{\\text{minimal \\flop-count}}{\\text{minimal data movement}}\n        = \\frac{\\SIvar{2 n^2}\\flops}{\\SIvar{n^2 + 3 n}\\doubles}\n        \\approx \\SIvar{\\frac14}{\\flops\\per\\Byte} \\enspace.\n    \\]\n\n    The matrix-matrix multiplication $\\dm C \\coloneqq \\dm A \\matmatsep \\dm B +\n    \\dm C$ (\\dgemm[NN]) with $\\dm A,\\allowbreak \\dm B,\\allowbreak \\dm C \\in\n    \\R^{n \\times n}$ is a \\blasl3 that performs \\SIvar{2 n^3}{\\flops} over\n    \\SIvar{4 n^2}{\\doubles} of data movement ($\\dm C$ is both read and written).\n    Hence, its arithmetic intensity\n    \\[\n        \\frac{\\text{minimal \\flop-count}}{\\text{minimal data movement}}\n        = \\frac{\\SIvar{2 n^3}\\flops}{\\SIvar{4 n^2}\\doubles}\n        = \\SIvar{\\frac n{16}}{\\flops\\per\\Byte}\n    \\]\n    grows linearly with the problem size~$n$ and already exceeds the intensity\n    of \\dgemv for matrices as small as $5 \\times 5$.\n\\end{example}\n\nWe revisit the arithmetic intensity in \\cref{sec:term:eff}, where it determines\nwhether a computation's performance is limited by the processor's memory\nsubsystem or its floating-point units.\n\n\\section*{About This Document}\n\n\\def\\gettexliveversion#1, #2 (#3)#4\\relax{#2}\n\\newcommand\\pdftexver{\\expandafter\\gettexliveversion\\pdftexbanner\\relax\\xspace}\n\nThis document was written in \\href{https:\/\/www.latex-project.org\/}{\\LaTeXe} and\ntypeset with \\href{http:\/\/www.tug.org\/applications\/pdftex\/}{pdfTeX} \\pdftexver\non \\today.\n\nIt relies on the following packages:\n\\href{http:\/\/ctan.org\/pkg\/microtype}{\\code{microtype}} for micro-typography;\n\\href{http:\/\/ctan.org\/pkg\/listings}{\\code{listings}} and\n\\href{http:\/\/ctan.org\/pkg\/tcolorbox}{\\code{tcolorbox}} for algorithms, listings,\nand examples;  \\href{http:\/\/ctan.org\/pkg\/pgf}{\\code{tikz}} and\n\\href{http:\/\/ctan.org\/pkg\/pgfplots}{\\code{pgfplots}} for graphics and plots;\n\\href{http:\/\/ctan.org\/pkg\/drawmatrix}{\\code{drawmatrix}} for matrix\nvisualizations; \\href{http:\/\/ctan.org\/pkg\/cleveref}{\\code{cleveref}} and\n\\href{http:\/\/ctan.org\/pkg\/hyperref}{\\code{hyperref}} for references and\nhyperlinks; and \\href{http:\/\/ctan.org\/pkg\/biblatex}{\\code{biblatex}} for the\nbibliography.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Timing Kernels in \\lapack's \\texorpdfstring\\dgeqrf{dgeqrf}}\n        \\label{sec:cache:qr:alg}\n        \\input{cache\/alg}\n\n        \\subsection{Cache-Aware Timings}\n        \\label{sec:cache:qr:timings}\n        \\input{cache\/timings}\n\n        \\subsection{Modeling the Cache}\n        \\label{sec:cache:qr:cache}\n        \\input{cache\/cache}\n\n        \\subsection{Varying the Setup}\n        \\label{sec:cache:qr:res}\n        \\input{cache\/qrresults}\n\n    \\section{Application to Other Algorithms}\n    \\label{sec:cache:algs}\n    \\input{cache\/otheralgs}\n\n    \\section{Feasibility on Modern Hardware}\n    \\label{sec:cache:new}\n    \\input{cache\/new}\n\n    \\section{Summary}\n    \\label{sec:cache:conclusion}\n    \\input{cache\/conclusion}\n}\n\n\\subsection{In- and Out-of-Cache Timings}\n\\label{sec:cache:icoc}\n\n\\input{cache\/figures\/ooc}\n\nOut-of-core timings are hardware independent, and just as on the\n\\harpertownshort serve as an upper bound on the \\sandybridgeshort and\n\\haswellshort.  This is illustrated in \\cref{fig:cache:ooc} for the inversion of\na lower-triangular matrix $\\dm[lower]A \\in \\R^{3200 \\times 3200}$ with\n\\dtrtri[LN] (\\cref{alg:dtrtriLN2}) and block size~$b = 64$ on the \\haswellshort,\nand the QR~decomposition of $\\dm A \\in \\R^{2400 \\times 2400}$ with \\dgeqrf\n(\\cref{alg:dgeqrf}) and $b = 32$  on the \\sandybridgeshort{}---the chosen\nmatrices comprise around \\SI{40}{\\mebi\\byte} and thus exceed the\n\\sandybridgeshort's and \\haswellshort's last-level cache~(L3) of, respectively,\n\\SIlist{20,30}{\\mebi\\byte}.  The out-of-cache timings indeed consistently\noverestimate the in-algorithm timings---by up to~\\SI{347}{\\percent} for the last\ncall to \\refdtrmmRUNN in the QR~decomposition \\dgeqrf on the \\sandybridgeshort\n(\\cref{fig:cache:ooc:dgeqrf:sandybridge} is clipped at~\\SI{175}\\percent).  As\nsuch, these measurements serve well as an upper bound on the in-algorithm\ntimings.\n\n\\input{cache\/figures\/ic}\n\nFore the same scenarios \\cref{fig:cache:ic} presents the error of our previous\nin-cache setup with respect to the in-algorithm timings:  While we expect the\nour setup to yield faster kernel executions than the in-algorithm timings, on\nthe \\sandybridge (with \\turboboost disabled) the in-cache timings are still up to~\\SI{.51}{\\percent} slower than\nthe in-algorithm timings (not accounting for the small unblocked \\dgeqr2); on\nthe \\haswell (with \\turboboost enabled), the relative errors for \\dtrtri[LN] and\n\\dgeqrf reach, respectively, \\SIlist{1.67;3.44}\\percent.\n\n\\input{cache\/figures\/ictb}\n\nFurther investigation reveals that the processor's \\intel{} \\turboboost is a source of\ncomplication for out measurements:  As \\cref{fig:cache:ictb} shows, enabling\n\\turboboost on the \\sandybridge leads to overestimations of the \\dtrtri[LN]'s\nand \\dgeqrf's most compute-intensive operations (i.e., the \\refdtrmmLLNN and the\ntwo \\dgemm{}s (\\ref*{plt:dgemmTN}, \\ref*{plt:dgemmNT})), by up to, respectively,\n\\SIlist{3.20;2.79}\\percent.\n\nWhile \\turboboost increases the overestimation of individual kernels, this\nphenomenon's origin lies in the processor's cache hierarchy:  Within an\nalgorithm, each kernel is invoked with a distinct cache precondition, i.e., with\nonly portions of its operands in the processor's caches.  Since our\nalgorithm-independent measurements do clearly not match such preconditions, we\nattempted to construct conditions in which the kernel executes at its absolute\npeak performance with different cache setups:\n\\begin{itemize}\n    \\item First, we used simple repeated execution of the kernel without any\n        modification of the cache in between as before.\n\n    \\item Next, we accessed the kernel operands in various\n        orders prior to the invocation.  E.g., for a \\dgemm $\\dm[width=.25]C\n        \\coloneqq \\dm[width=.8]A \\matvecsep \\dm[width=.25, height=.8]B +\n        \\dm[width=.25]C$, we attempted all permutations of access orders, such\n        as \\dm[width=.8]A--\\dm[width=.25, height=.8]B--\\dm[width=.25]C and\n        \\dm[width=.25]C--\\dm[width=.8]A--\\dm[width=.25, height=.8]B.\n\n    \\item Finally, we refined the access granularity and attempted to bring\n        operands into cache not as a whole but only partially:  For a kernel\n        with one operand larger than the cache and the other operand(s) only a\n        fraction of that size (e.g., the \\dgemm[TN] (\\ref*{plt:dgemmTN}) in\n        \\dgeqrf: $\\dm[width=.25]C \\coloneqq \\dm[width=.8]A \\matvecsep \\dm\n        [width=.25, height=.8]B + \\dm[width=.25]C$ where \\dm[width=.25]B and\n        \\dm[width=.25]C are of width~$b$ and close to the problem size~$n$ in\n        height), we bring the entire small operand(s) into cache but only\n        portions of the large one.\n\n        \\input{cache\/figures\/acc}\n\n        \\Cref{fig:cache:acc} presents which operand portions we chose to load\n        into the cache.  These choices are based on the assumption that any\n        kernel implementation likely traverses the input matrix somehow form\n        from the top-left \\tsearrow to the bottom-right.\\footnote{%\n            Exceptions are, e.g., \\dtrsm[RLNN] ($B \\coloneqq B A^{-1}$) and\n            \\dtrsm[LUNN] ($B \\coloneqq A^{-1} B$), which must traverse the\n            triangular~$A$ from the bottom-right to the top-left---in these\n            cases the accessed matrix portions are mirrored accordingly.\n        }  Therefore, we bring a column panel of the operand, a row panel, a\n        square block, or any combination of these into the processor's caches.\n        While doing so, we varied the sizes~$s_1$, $s_2$, and~$s_3$ of the\n        accessed operand portions.\n\\end{itemize}\n\nWhile in some scenarios changing the in-cache setup for kernel invocations\nreduced the runtime overestimation, the effects were not consistent across\ndifferent algorithms, kernels, processors, and \\blas implementations.\nAltogether, it was not possible to determine general, algorithm-independent\nin-cache setups that yield a clear lower bound on the in-algorithm timings.\n\n\n\\subsection{Algorithm-Aware Timings}\n\\label{sec:cache:algaware}\n\nSince our above attempts at algorithm-independent in-cache timings did not yield\nthe required lower bound on in-algorithm timings, the only alternative is to\ntailor the timing setups to individual algorithms.  We might for instance setup\neach kernel timing with several preceding kernel invocations from within the\nalgorithms.  Such obtained \\definition{algorithm-aware timings} yield accurate\nestimates for the in-algorithm timings, and rid us of the need for combining in-\nand out-of-cache estimates.\n\n\\input{cache\/figures\/exact}\n\n\\begin{example}{Algorithm-aware timings}{cache:algaware}\n    \\Cref{fig:cache:exact} presents the accuracy of algorithm-aware timings as\n    estimates for in-algorithm timings for the inversion of a lower-triangular\n    matrix (\\dtrtri[LN]) and the QR~decomposition (\\dgeqrf) on a \\sandybridge\n    (with \\turboboost enabled) using single-threaded \\openblas.  The\n    algorithm-aware timings were created by preceding each measured kernel\n    invocation with the calls from the corresponding blocked algorithm that were\n    executed since that kernel's last invocation.\n\n    \\Cref{fig:cache:exact:dtrtri} shows that for \\dtrtri[LN] the algorithm-aware\n    timings are with few exceptions within~\\SI1{\\percent} of the in-algorithm\n    timings with an average absolute relative error (ARE) of~\\SI{.54}\\percent.\n    As seen in \\Cref{fig:cache:exact:dtrtri}, for the \\dgetrf the relative error\n    is overall larger yet similarly spread around~\\SI0{\\percent} with an average\n    ARE of~\\SI{.84}\\percent.\n\\end{example}\n\nWhile this approach yields accurate estimates, when the kernel invocations for\neach algorithm execution are timed separately and each measurement is preceded\nwith a setup of one or more kernels, the timing procedure takes effectively\nlonger than executing and measuring the target algorithm repeatedly.  As a\nresult, this method is at the same time highly accurate and impractical, which\nis why we do not further pursue it.\n\n\\subsection{Cholesky Decomposition: \\texorpdfstring{\\dpotrf[U]}{dpotrf}}\n\\label{sec:cache:dpotrfU}\n\n\\input{cache\/figures\/cholUalg}\n\nFirst, we consider \\lapack's upper triangular Cholesky decomposition \\dpotrf[U]\n\\[\n    \\dm[lower, ']U \\dm[upper]U \\coloneqq \\dm A\n\\]\nof a symmetric positive definite $\\dm A \\in \\R^{n \\times n}$ in upper triangular\nstorage.  \\Cref{alg:dpotrfU} presents the blocked algorithm employed in this\nroutine, which is the transpose of \\dpotrf's algorithm for lower-triangular\ncase (\\cref{alg:chol2} on \\cpageref{algs:chol}).  As the algorithm traverses\n\\dm A, both the size and shape of~\\dm[mat02, width=1.25]{A_{02}} (the largest\noperand) change noticeably:  It starts as row panel, then grows to a square\nmatrix and finally shrinks to a column panel.  \\dm[mat02, width=1.25]{A_{02}}'s\nsize determines the workload performed by the algorithm's large \\refdgemmTN,\nwhich is reflected in the in-algorithm timings in\n\\cref{fig:cache:dpotrfU:instr}.\n\n\\input{cache\/figures\/cholres}\n\nIn our experiments, we execute \\dpotrf[U] on a \\harpertown with single-threaded\n\\openblas, $\\dm A \\in \\R^{2400 \\times 2400}$,\\footnote{%\n    For $n = 2400$, the upper-triangular portion of~$A$ takes up about\n    \\SI{12}{\\mebi\\byte}---twice the size of the L2~cache.\n} and block size $b = 32$.  \\Cref{fig:cache:dpotrf:res} presents the relative\nperformance difference with respect to in-algorithm timings for both repeated\nexecution timings and our final estimates.  Our estimates yield improvements for\nthe \\refdsyrkUT and \\refdpotfU involving large matrices in the middle of \\dm A's\ntraversal.  In the beginning of the traversal, the estimates are generally too\npessimistic because some matrices are (partially) brought into cache by\nprefetching, which is not accounted for in our estimates.  On average the\nrelative error is reduced from~\\SIrange{11.11}{7.87}{\\percent}, i.e., by a\nfactor of~1.41.\n\nHowever, note that the improvement is only visible in the averaged per-kernel\nrelative error:  Since the runtime of large \\dgemm[TN]~(\\ref*{plt:dgemmTN}) is\noverestimated, the accumulated runtime estimate for the entire algorithm\nactually becomes less accurate.\n\n\n\\subsection{Inversion of a Triangular Matrix:\n\\texorpdfstring{\\dtrtri[LN]}{dtrtri}}\n\\label{sec:cache:dtrtriLN}\n\n\\input{cache\/figures\/trinvalg}\n\nWe now take a closer look at \\lapack's inversion of a lower-triangular matrix\n\\dtrtri[LN]\n\\[\n    \\dm[lower]A \\coloneqq \\dm[lower, inv]A\n\\]\nwith $\\dm A \\in \\R^{n \\times n}$, whose blocked algorithm is presented in\n\\cref{alg:dtrtriLN2}.  In contrast to the previous operations, this algorithm\ntraverses \\dm A \\tnwarrow from the bottom-right to the top-left, thereby\noperating on sub-matrices of increasing size.  \\Cref{fig:cache:dtrtriLN:instr}\nshows the in-algorithm timings for the algorithm, which are dominated by\n\\refdtrmmLLNN.\n\n\\input{cache\/figures\/trinvres}\n\nWe execute \\dtrtri[LN] on a \\harpertown with single-threaded \\openblas, $\\dm A\n\\in \\R^{2400 \\times 2400}$, and block size $b = 32$.\n\\Cref{fig:cache:dtrtriLN:res} compares the performance measurements from\nrepeated execution and our final estimates to in-algorithm timings:  The\nimprovements of our estimates are most significant in \\refdtrmmLLNN (which\nperforms the most computation) and \\refdtrtiLN; the error is reduced from an\naverage of~\\SIrange{6.70}{3.37}{\\percent}---a total improvement of~$1.99\\times$.\n\n\n\\subsection{Summary}\n\nWe have seen that, on a \\harpertown the accuracy of our runtime estimates for\nkernels within blocked algorithms is increased by taking the state of the\nL2~cache throughout the algorithm execution into consideration.  For different\nalgorithms, problem sizes, block sizes, \\blas implementations, and thread\ncounts, we have seen improvements between~$1.15\\times$ (with all 4~cores)\nand~$2.99\\times$.\n\n\n\n\n\n\n\n\\chapter{Conclusion}\n\\label{ch:conclusion}\n\nThis dissertation set out to predict the performance of dense linear algebra\nalgorithms.  It targeted two types of algorithms that require different\nprediction approaches: blocked algorithms and tensor contractions.\n\nFor blocked algorithms, we accomplished accurate performance predictions through\nautomatically generated performance models for compute kernels.  Our\npredictions both reliably identify the fastest blocked algorithm from\npotentially large numbers of available alternatives, and select a block size\nfor near-optimal algorithm performance.  Our approach's main advantage is its\nseparation of the model generation and the performance prediction:  While the\ngeneration may take several hours, thousands of algorithm executions are\nafterwards predicted within seconds.  A discussed downside to the approach,\nhowever, is that it does not account for algorithm-dependent caching effects.\n\nFor tensor contractions, we established performance predictions that identify\nthe fastest among potentially hundreds of alternative \\blas-based contraction\nalgorithms.  By using cache-aware micro-benchmarks instead of our performance\nmodels, our solution is highly accurate even for contractions with severely\nskewed dimensions.  Furthermore, since these micro-benchmarks only execute a\ntiny fraction of each tensor contraction, they provide performance predictions\norders of magnitude faster than empirical measurements.\n\nTogether, our model generation framework and micro-benchmarks form a solid\nfoundation for accurate and fast performance prediction for dense linear algebra\nalgorithms.\n\n\n\\section{Outlook}\nThe techniques presented in this dissertations offer numerous opportunities for\napplications and extensions:\n\\begin{itemize}\n    \\item Our methods can be applied to predict the performance various types of\n        algorithms and operations, such as recursive algorithms and\n        algorithms-by-blocks.\n\n    \\item For dense eigenvalue solvers, our models can predict the two most\n        computationally intensive stages:  The reduction to tridiagonal form and\n        the back-transformation.  By additionally estimating the data-dependent\n        performance of tridiagonal eigensolvers, one can predict the solution of\n        complete eigenproblems.\n\n    \\item Beyond individual operations, our predictions can be applied to\n        composite operations and algorithms, such as matrix chain\n        multiplications or least squares solvers.\n\n    \\item Our models were designed to provide estimates for configurable yet\n        limited ranges of problem sizes.  For extrapolations to larger problems\n        they should be revised to ensure that local performance phenomena do not\n        distort faraway estimates.\n\n    \\item Computations on distributed memory systems, accelerators, and graphics\n        cards can be predicted by combining our techniques with models for data\n        movement and communication.\n\\end{itemize}\n\n\\chapter*{Abstract}\n\n\\input{abstract\/abstract}\n\n\n\\chapter*{Acknowledgments}\n\nFirst and foremost, I would like to express my sincere gratitude to my advisor\nPaolo Bientinesi.  While guiding me through my studies, he always embraced my\nown ideas and helped me shape and develop them in countless discussions.  While\nhe granted me freedom in many aspects of my work, he always had time for\nanything between a quick exchange of thoughts and extensive brainstorming\nsessions.  Beyond our professional relationship, we enjoyed twisty puzzles and\nboard games in breaks from work, long game nights, and annual trips to SPIEL.  I\nconsider my self lucky to have spend my time as a doctoral student with him and\nhis research group.\n\nThe HPAC group proved to be much more than a collection of researchers working\non remotely associated projects; my colleagues were not only a source of\nincredibly valuable discussions and feedback regarding my work, we also indulged\nin various unrelated arguments and exchanges over lunch and at many other\noccasions.  My thanks go to Edoardo Di~Napoli, Diego Fabregat-Traver, Paul\nSpringer, Jan Winkelmann, Henrik Barthels, Markus H\u00f6hnerbach, Sebastian\nAchilles, William McDoniel, and Caterina Fenu, as well as our former group\nmembers Matthias Petschow, Roman Iakymchuk, Daniel Tameling, and Lucas Beyer.\n\nI am grateful for financial support from the {\\namestyle Deutsche\nForschungs\\-gemeinschaft} (DFG) through grant GSC~111 (the graduate school\nAICES) and the {\\namestyle Deutsche Telekomstiftung}.  Their programs not only\nfunded my work, but opened further opportunities in the form of seminars and\nworkshops, and connected me with like-minded students from various disciplines.\n\nThe {\\namestyle\\rwth IT Center} provided and maintained an extremely reliable\ninfrastructure central to my work: the {\\namestyle \\rwth Compute Cluster}.  I\nthank its staff not only for ensuring smooth operations but also for their\ncompetent and detailed responses to my many inquiries and requests regarding our\ninstitute's cluster partition.\n\nThe AICES service team did their best to shield me from the bureaucracy of\ncontracts, stipends, and reimbursements.  I am grateful they allowed me to focus\nsolely on my research.\n\nEven more important than a gratifying work environment is forgetting about it\nevery once in a while.  My friends played a bigger role in this effort than\nprobably most of them know, whether we were simply spending time hanging out or\nplaying games, went swimming, climbing or playing badminton, or taught swimming\nand worked as lifeguards.  You are too many to enumerate, but you know who you\nare.\n\nFinally, but most importantly, none of this would have been possible without the\nendless and uncompromising support of may parents.  You are the reason I grew\ninto the person I am today.  Danke!\n\n\\tableofcontents\n\n\\subsection{Motivation: Blocked Algorithms}\n\\label{sec:intro:blocked:algs}\n\n\\definitionp[blocked algorithm]{Blocked algorithms} are commonly used to exploit\nthe performance of optimized \\blasl3 kernels\\footnote{%\n    The {\\namestyle Basic Linear Algebra Subprograms} (BLAS) form the basis for\n    high-performance in dense linear algebra.  See \\cref{app:term,app:libs}.\n} in other matrix operations, such as decompositions, inversions, and\nreductions.  Every blocked algorithm traverses its input matrix (or matrices) in\nsteps of a fixed \\definition{block size}; in each step of this traversal, it\nexposes a set of \\definition[sub-matrices\\\\updates]{sub-matrices} to which it\napplies a series of {\\em updates}.  Through these updates, it progresses with\nthe computation and obtains a portion of the operation's result; once the matrix\ntraversal completes, the entire result is computed.\n\n\\input{intro\/figures\/blocked}\n\n\\footnotetextbefore{%\n    \\Cref{app:libs} gives an overview of the \\blas and \\lapack routines used\n    throughout this work.  When specified, the subscripts indicate the values of\n    the flag arguments, which identify the variant of the operation; e.g., in\n    \\dpotrf[L] the \\code L corresponds to the argument \\code{uplo} indicating\n    a lower-triangular decomposition.\n}\n\\begin{example}{Blocked algorithms for the Cholesky decomposition}{intro:chol}\n    \\newcommand\\Azz{\\dm[mat00, lower]{A_{00}}\\xspace}%\n    \\newcommand\\Aoz{\\dm[mat10, height=.5]{A_{10}}\\xspace}%\n    \\newcommand\\Aoo{\\dm[mat11, size=.5, lower]{A_{11}}\\xspace}%\n    \\newcommand\\Atz{\\dm[mat20, height=1.25]{A_{20}}\\xspace}%\n    \\Cref{algs:chol} illustrates blocked algorithms for a simple yet\n    representative operation: the lower-triangular Cholesky decomposition\n    \\[\n        \\dm[lower]L \\dm[upper, ']L \\coloneqq \\dm A\n    \\]\n    of a symmetric positive definite (SPD) matrix $\\dm A \\in \\R^{n \\times n}$ in\n    lower-triangular storage (\\lapack: \\dpotrf[L]\\footnotemark).  For this\n    operation there exist three different blocked algorithms.  Each algorithm\n    traverses \\dm A diagonally from the top-left to the bottom-right \\tsearrow\n    and computes the Cholesky factor~\\dm[lower]L in place.  At each step of the\n    traversal, the algorithm exposes the sub-matrices shown in\n    \\cref{algs:chol:traversal} and makes progress by applying the\n    algorithm-dependent updates in \\cref{alg:chol1,alg:chol2,alg:chol3}.  Before\n    these updates, the sub-matrix~\\Azz, which in the first step is of size $0\n    \\times 0$, already contains a portion of the Cholesky factor~\\dm[lower]L;\n    after the updates, the sub-matrices~\\Aoz and~\\Aoo also contain their\n    portions of~\\dm[lower]L, and in the next step become part of~\\Azz.  Once the\n    traversal reaches the bottom-right corner (i.e., \\Azz is now of size $n\n    \\times n$), the entire matrix is factorized.\n\\end{example}\n\nBlocked algorithms pose two \\definition[optimization challenges:\\\\alternative\nalgorithms]{optimization challenges}:\n\\begin{itemize}\n    \\item For each operation there typically exist several {\\em alternative\n        algorithms}, which are mathematically equivalent in exact arithmetic;\n        however, even if such algorithms perform the same number of floating\n        point operations, they may differ significantly in performance.\n\n    \\item For each algorithm, the \\definition{block size} influences the number\n        of traversal steps and the sizes and shapes of the exposed sub-matrices,\n        and thus the performance of the kernels applied to them.\n\\end{itemize}\nWhat makes matters more complicated is that the optimal choice depends on\nvarious factors, such as the hardware , the number of threads, the kernel\nimplementations, and the problem size.\n\n\\input{intro\/figures\/chol_vars}\n\n\\footnotetextbefore{%\n    \\Cref{app:hardware} provides an overview of the processors used throughout\n    this work.\n}\n\\begin{example}{Performance of alternative algorithms}{intro:chol:var}\n    \\Cref{fig:intro:chol:vars} shows the performance of the three blocked\n    Cholesky decompositions from \\cref{algs:chol} with block size~$b = 128$ and\n    increasing problem size~$n$ on a 12-core \\haswell\\footnotemark{} with\n    single- and multi-threaded \\openblas.\n\n    In both the single- and multi-threaded scenarios,\n    algorithm~3~(\\ref*{plt:chol3}) is the fastest among the three alternatives\n    for all problem sizes.  On a single core and for problem size $n = 4152$, it\n    is \\SIlist{27.40;12.89}{\\percent} faster than, respectively,\n    algorithms~1~(\\ref*{plt:chol1}) and~2~(\\ref*{plt:chol2}), and it reaches up\n    to \\SI{91.01}{\\percent} of the processor's theoretical peak performance (red\n    line \\legendline[very thick, darkred] at the top of the plot).  On all 12~of\n    the processor's cores, algorithm~3~(\\ref*{plt:chol3}) still reaches an\n    efficiency of~\\SI{69.70}\\percent, and outperforms\n    algorithms~1~(\\ref*{plt:chol1}) and~2~(\\ref*{plt:chol2}) by, respectively,\n    $5.21\\times$ and~$1.92\\times$.\n\n    Although algorithm~3~(\\ref*{plt:chol3}) is clearly the fastest in this and\n    many other scenarios, \\lapack's \\dpotrf[L] implements\n    algorithm~2~(\\ref*{plt:chol2}).\n\n    For other operations, the choice becomes more complicated, since no single\n    algorithm is the fastest for all problem sizes and scenarios.  For instance,\n    for the single-threaded inversion of a lower-triangular matrix $\\dm[lower]A\n    \\coloneqq \\dm[lower, inv]A$, two different algorithms are the fastest for\n    small and large matrices; with the performance differing by up\n    to~\\SI{13}{\\percent} in either direction (\\cref{sec:pred:var:trinv}).\n\\end{example}\n\n\\input{intro\/figures\/chol_b}\n\n\\begin{example}{Influence of the block size on performance}{intro:chol:b}\n    Let us consider the blocked Cholesky decomposition\n    algorithm~3~(\\ref*{plt:chol3} in \\cref{fig:intro:chol:vars}) with fixed\n    problem sizes~$n = 1000$, 2000, 3000, and~4000 and varying block size~$b$.\n    \\cref{fig:intro:chol:b} presents the performance of these algorithm\n    executions for 1 and 12~threads on the \\haswell using \\openblas:\n    Single-threaded, the optimal block size increases from~$b = 96$ for~$n =\n    1000$ to~$b = 184$ for~$n = 4000$.  On 12~cores, on the other hand, the\n    performance is less smooth and the optimal choices for~$b$ are between~56\n    and~112.\n\n    \\Cref{fig:intro:chol:b} demonstrates the importance of selecting the block\n    size dynamically:  If we use~$b = 184$, which is optimal for~$n = 4000$ on\n    one core, for~$n = 1000$ on 12~cores we only reach \\SI{77.62}{\\percent} of\n    the algorithm's optimal performance.  On the other hand, \\lapack's default\n    block size~$b = 64$ (which is close to the optimal~$b = 56$ for~$n = 1000$\n    on 12~cores) would reach \\SI{95.95}{\\percent} of the optimal single-threaded\n    performance for~$n = 4000$.\n\\end{example}\n\n\n\\subsection{Prediction through Performance Models}\n\\label{sec:intro:blocked:pred}\n\nNaturally, both the best algorithm and its optimal block size for a given\nscenario (operation, problem size, hardware, kernel library, multi-threading)\ncan be determined through exhaustive performance measurements; however, this is\nextremely time consuming and thus often impractical.  Instead we aim to\ndetermine the optimal configuration {\\em without executing} any of the\nalternative algorithms.  For this purpose, we use the hierarchical structure of\nblocked algorithms:  Their entire computation is performed in a series of calls\nto a few kernel routines; hence, by accurately estimating the runtime of these\nkernels, we can predict an entire algorithm's runtime and performance.\n\nIn order to estimate the kernel runtimes, let us study how these kernels are\nused:  In each algorithm execution, the same set of kernels is invoked\nrepeatedly---once for each step of the blocked matrix traversal.  Each\ninvocation, however, works on operands of different size depending on the\nprogress of the algorithms' traversal, the input problem size, and the block\nsize.  In short, we need to estimate the performance of only a few kernels, yet\nwith potentially wide ranges of operand sizes.\n\nOur solution is \\definition{performance modeling}, as detailed in\n\\cref{ch:model}:  Based on a detailed study of how a kernel's arguments (i.e.,\nflags, operand sizes, etc.) affect its performance, we design performance models\nin the form of piecewise multi-variate polynomials.  These models are generated\nautomatically once for each hardware and software setup and subsequently provide\naccurate performance estimates at a tiny fraction of the kernel's runtime.\n\nUsing such estimates, we \\definition[performance prediction]{predict} the {\\em\nperformance} of blocked algorithms, as presented in \\cref{ch:pred}.  These fast\npredictions prove to be highly accurate, and allow us to both rank the blocked\nalgorithms for a given operation according to their performance, and find\nnear-optimal values for the algorithmic block sizes.\n\nWhile our models yield accurate performance estimates for individual kernel\nexecutions, they do not capture the performance influence of\n\\definition{caching} between kernels.  Prior to the invocation of each compute\nkernel in an algorithm, typically only a portion of its operands are in cache,\nand loading operands from main memory increases the kernel runtime.\n\\cref{ch:cache} investigates how caching effects can be accounted for in blocked\nalgorithms, and attempts to combine pure in- and out-of-cache estimates into\nmore accurate prediction.  However, while the results look promising on a rather\nold \\harpertown, the analysis reveals that on modern processors the effect\ncaching on kernel performance is so complex that accounting for it in\nalgorithm-independent performance models to further improve our prediction\naccuracy is infeasible.\n\n\n\n\n\n\n\n\n\\chapter{Introduction}\n\\chapterlabel{intro}\n{\n    \\tikzsetexternalprefix{externalized\/intro-}\n\n    \\input{intro\/intro.tex}\n\n    \\section[Performance Modeling for Blocked Algorithms]\n            {Performance Modeling\\newline for Blocked Algorithms}\n    \\label{sec:intro:blocked}\n    \\input{intro\/blocked}\n\n    \\section{Micro-Benchmarks for Tensor Contractions}\n    \\label{sec:intro:tensor}\n    \\input{intro\/tensors}\n\n    \\section{Related Work}\n    \\label{sec:intro:relwork}\n    \\input{intro\/relwork}\n}\n\n\\subsection{Dense Linear Algebra Libraries and Algorithms}\n\\label{sec:relwork:libsalgs}\n\nWe begin with a brief history of the fundamental DLA libraries \\blas and \\lapack\nand prominent implementations in \\cref{sec:relwork:libs}.  We then focus on\nblocked algorithms and their tuning opportunities in \\cref{sec:relwork:blocked},\nand finally give an overview of alternative algorithms and libraries for\ndistributed-memory and accelerator hardware in, respectively,\n\\cref{sec:relwork:altalgs,sec:relwork:dist}.\n\n\n\\subsubsection{\\blas and \\lapack}\n\\label{sec:relwork:libs}\n\nThe development of standardized DLA libraries began in~1979 with the inception\nof the {\\namestyle Basic Linear Algebra Subprograms}\n(\\definition{\\blas})~\\cite{blasl1}, a \\fortran interface specification for,\ninitially, various ``Level~1'' scalar and vector operations.  It was\nsubsequently extended to kernels for ``Level~2'' matrix-vector~\\cite{blasl2} and\n``Level~3'' matrix-matrix~\\cite{blasl3} operations in, respectively, 1988\nand~1990.  The aim of the \\blas specification is to enable performance portable\napplications:  DLA codes reach high performance on different hardware by using\narchitecture-specific \\blas implementations.  Although computer architectures\nhave evolved dramatically in the last~40 years, this principle of performance\nportability is still at the core of all current DLA libraries.\n\nThe \\blas specification is accompanied by a reference\nimplementation~\\cite{blasweb} that, while fully functional and well documented,\nis deliberately simple and thus slow; to reach high performance, users instead\nlink with optimized \\definition[open-source implementations]{\\blas\nimplementations}.  The oldest {\\em open-source} implementation still in\nuse is the {\\namestyle Automatically Tuned Linear Algebra Software}\n(\\atlas)~\\cite{atlas1, atlas3, atlas2, atlasweb}, first released in 1997; this\nauto-tuning based library's main proficiency is to yield decent performance on a\nwide range of hardware platforms with little developer and user effort.  The\nfirst major open-source implementation hand-tuned for modern processors with\ncache hierarchies was {\\swstyle GotoBLAS}~\\cite{gotoblas1, gotoblas2,\ngotoblasweb}.  It reaches up to around \\SI{90}{\\percent} of a processor's peak\nfloating-point performance for both sequential and multi-threaded Level~3\nkernels and good bandwidth-bound performance for Level~1 and~2 operations.\nAfter {\\swstyle GotoBLAS}'s discontinuation in~2010, its code-base and approach\nwere picked up and extended to more recent processors in the \\openblas\nlibrary~\\cite{openblasweb}, which is currently the fastest open-source\nimplementation for many architectures.  Also inspired by {\\swstyle GotoBLAS}'s\napproach is the fairly recent {\\namestyle \\blas-like Library Instantiation\nSoftware} (\\blis)~\\cite{blis3, blis1, blis2, blisweb}, an open-source framework\nthat provides optimized kernels for basic DLA operations, such as the \\blas,\nbased on one hand-tuned micro-kernel per architecture.\n\nIn addition to open-source implementations, many hardware \\definition[vendor\nimplementations]{vendors} maintain and distribute their own high-performance\n{\\em\\blas}, e.g., \\intel's {\\namestyle Math Kernel Library}\n(\\mkl)~\\cite{mklweb}, \\apple's framework \\accelerate~\\cite{accelerateweb}, and\n{\\namestyle IBM}'s {\\namestyle Engineering and Scientific Subroutine Library}\n(\\essl)~\\cite{esslweb}.\n\n\\blas forms the basis for DLA libraries covering more advanced operations.  The\nearliest library built on top of first \\blasl1 and later Level~2 was {\\swstyle\nLINPACK}~\\cite{linpack, linpackweb}, a package of solvers for linear equations\nand least-squares problems from the~1970s and~1980s.  {\\swstyle LINPACK}\ntogether with {\\swstyle EISPACK}~\\cite{eispack, eispackweb}, a collection of\neigenvalue solvers, was superseded by the {\\namestyle Linear Algebra PACKage}\n(\\definition{\\lapack})~\\cite{lapack, lapackweb} in~1992.  \\lapack has since been\nextended with new features and algorithms, and is still under active\ndevelopment.  Just like \\blas, \\lapack functions as a de-facto standard\ninterface specification for many advanced DLA operations; libraries such as\n\\openblas and \\mkl adopt its interface and provide tuned implementations of\nvarious routines.\n\nFor more details on \\blas and \\lapack, and their kernels and implementations\nused throughout this work, see \\cref{app:libs}.\n\n\n\\subsubsection{Blocked Algorithms}\n\\label{sec:relwork:blocked}\n\n\\lapack uses \\definition{blocked algorithms} for most of its dense operations.\nThe core idea behind these algorithms is to leverage a processor's cache\nhierarchy by increasing the spacial and temporal locality of operands, as well\nas casting most of an operation's computation in terms of \\blasl3 kernels.  As a\nresult, complex operations can reach performance levels close to the hardware's\ntheoretical peak.\n\nHowever, for each operation, there typically exist multiple\n\\definition{alternative blocked algorithms}, of which \\lapack offers only one,\nbut not always the fastest.  The alternative algorithms for a given operation\ncan be derived from its mathematical formulation\nsystematically~\\cite{derivingbalgs} and automatically~\\cite{loopgen, pmegen}.\nBased on these principles, \\libflame~\\cite{libflameref, libflame, libflameweb}\noffers many alternative algorithms for each operation, and for several\noperations provides more efficient default algorithms than \\lapack.  In this\nwork we consider \\libflame's blocked algorithms for various operations, and aim\nto predict which of them is most efficient for given scenarios.\n\nAnother caveat of blocked algorithms is their \\definition[block size\ntuning]{block sizes}, which need to be carefully {\\em tuned} to maximize\nperformance.  Since this is a well-known aspect of blocked\nalgorithms~\\cite{rooflinedla, blocksizetuning}, \\lapack encapsulates and exposes\nall its tuning parameters in \\ilaenv, a central routine that is used to\nconfigure the library at compile time; for many operations the block\nsizes used by \\lapack's reference implementation of \\ilaenv (64~for most\nalgorithms) have been too small on recent hardware for quite some time.\nAlthough the necessity of optimizing block sizes is well understood and taken\ncare of by implementations such as \\mkl, it remains non-trivial, and in fact few\nend-users and application-developers are aware of it.  The automated model-based\noptimization of the block size for blocked algorithms is the second major goal\nof this work.\n\n\n\\subsubsection{Alternatives to Blocked Algorithms}\n\\label{sec:relwork:altalgs}\n\nAn alternative to blocked algorithms is \\definition{recursive algorithms},\nwhich avoid both the algorithm selection and block-size optimization.  They are\nalso known as ``cache oblivious'' algorithms~\\cite{cacheoblivious2,\ncacheoblivious1} since they minimize the data-movement between cache\nlevels~\\cite{dlarec}.  Recursion has been suggested for many DLA operations,\nsuch as the LU~decomposition~\\cite{lurec, lurec2}, the Cholesky\ndecomposition~\\cite{cholrec}, triangular matrix inversion~\\cite{trinvrec},\ntwo-sided linear systems~\\cite{sygstrec}, tall-and-skinny\nQR~factorization~\\cite{qrrec}, and Sylvester-type equation solvers~\\cite{recsy,\nrecsyweb}.\n\nHowever, since no readily-available recursion-based library comparable to\n\\lapack existed, we developed the {\\namestyle Recursive \\lapack collection}\n(\\definition{\\relapack})~\\cite{relapack, relapackweb}.  \\relapack provides\nrecursive implementations for 48~\\lapack routines, and outperforms not only the\nreference implementation but in many cases also optimized libraries such as\n\\openblas and \\mkl.\n\nA second alternative to blocked algorithms tailored to shared-memory systems are\ntask-based \\definition{algorithms-by-blocks}, also known as ``block algorithms''\nor ``tiled algorithms''.  However, these algorithms not only introduce a\nspecialized storage scheme of matrices ``by block'', but also require custom\ntask scheduling mechanisms.  Implementations of such schedulers include\n{\\namestyle QUARK}~\\cite{quark} as part of {\\namestyle PLASMA}~\\cite{plasma},\n{\\namestyle DAGuE}~\\cite{dague}, {\\namestyle SMPSs}~\\cite{smpssdla}, and\n{\\namestyle SuperMatrix}~\\cite{supermatrix}.\n\n\n\\subsubsection{Distributed-Memory and Accelerators}\n\\label{sec:relwork:dist}\n\n\\definition[distributed memory]{Distributed-memory} systems and super-computers\nare indispensable for large-scale DLA computations.  The first noteworthy\nextension of the \\blas and the \\lapack to this domain was the {\\namestyle\nScalable Linear Algebra PACKage} (\\scalapack)~\\cite{scalapack, scalapackweb},\nwritten in \\fortran and based on \\blas, \\lapack, and the {\\namestyle Message\nPassing Interface} (MPI).  However, {\\namestyle ScaLAPACK} is only sparingly\nupdated (last in~2012), and, instead, the state of the art for\ndistributed-memory DLA is {\\namestyle Elemental}~\\cite{elemental, elementalweb},\nan actively developed \\cpplang~library, based on \\libflame's methodology in and\nobject-oriented and templated programming techniques.\n\nSince \\definition{accelerators} such as {\\namestyle Xeon-Phi} coprocessors and\ngraphics processors lend themselves well to compute-intensive operations, they\nare a natural target for DLA codes.  While some classic \\blas implementations\nsuch as \\atlas, \\blis, and \\mkl, can be used on the x68-based {\\namestyle Xeon\nPhi}s, separate libraries are required for graphics processors: {\\namestyle\nNVIDIA}'s {\\namestyle cuBLAS}~\\cite{cublasweb} provides high-performance \\blas\nkernels for {\\langstyle CUDA}-enabled graphics cards, and {\\namestyle\nclBLAS}~\\cite{clblasweb} targets {\\langstyle OpenCL}-capable devices.\nFurthermore, {\\namestyle Matrix Algebra on GPU and Multicore Architectures}\n({\\namestyle MAGMA})~\\cite{magma, magmaweb} targets \\blas and \\lapack operations\non heterogeneous systems (e.g., CPU + GPU).\n\n\n\\subsection{Performance Measurements and Profiling}\n\\label{sec:relwork:meas}\n\nRuntime measurements of both application codes and algorithms are crucial in the\ninvestigation of performance behaviors, bottlenecks, as well as optimization\nand tuning in general; hence, numerous tools facilitate such\nmeasurements.  Simple timers are accessible in virtually any language and\nenvironment: e.g., \\code{time} in Unix, \\code{rdtsc} in x86~assembly,\n\\code{gettimeofday()} in~\\clang, \\code{omp\\_get\\_wtime()} in {\\namestyle\nOpenMP}, \\code{tic} and \\code{toc} in \\matlab, and \\code{timeit} in \\python.\nSeveral more advanced tools \\definition[profiling]{profile} executions of\nfunctions and communications in applications by tracing or sampling: e.g.,\n{\\namestyle gprof}~\\cite{gprof, gprofweb}, {\\namestyle VAMPIR}~\\cite{vampirweb},\n{\\namestyle TAU}~\\cite{tau, tauweb}, {\\namestyle Scalasca}~\\cite{scalasca,\nscalascaweb}, and \\intel's {\\namestyle VTune}~\\cite{vtuneweb}.  While such tools\nare invaluable in the performance analysis of application codes, their\ngenerality makes them somewhat unwieldy for our purposes of investigating DLA\nkernel performance.  Therefore, we designed {\\namestyle Experimental Linear\nAlgebra Performance Studies} (\\definition{\\elaps})~\\cite{elaps, elapsweb}, a\nframework for performance measurements and analysis of DLA routines and\nalgorithms, further detailed in \\cref{sec:meas:elaps}.\n\n\n\\subsection{Performance Modeling and Predictions}\n\\label{sec:relwork:model}\n\nPredicting and modeling application performance is an important aspect of\nhigh-performance computing, and the term ``performance modeling'' is used to\ndescribe many different techniques and approaches.  This section gives a brief\noverview of such approaches with focus on methods for DLA algorithms.\n\nThe well-established \\definition{Roofline model}~\\cite{roofline1} does not\npredict performance, but relates an algorithm's attained performance to the\nhardware's potential:  As detailed in \\ref{sec:term:roofline}, it allows to\nevaluate an execution's resource efficiency by relating its algorithm's\narithmetic intensity and int performance relative to the hardware's peak\nmain-memory bandwidth and floating-point performance.  It has been applied,\nimplemented, and extended in numerous publications, such as~\\cite{rooflinecache,\nrooflinetoolkit, roofline2}.  Notably, \\citeauthor*{rooflinedla} use the\nroofline model (the arithmetic intensity in particular) to optimize the block\nsize for a blocked matrix inversion algorithm~\\cite{rooflinedla}.\n\nModel-based performance tuning of \\blas implementations was suggested for both\n\\atlas~\\cite{atlasmodel} and \\blis~\\cite{blismodel}, showing that near-optimal\n\\blas performance can be reached without measurement-based autotuning:  Instead\nthey, e.g., select blocking sizes according to the \\blas implementation and the\ntarget processor's cache sizes.  Note that these approaches are used to tune\n\\blas kernels, and do not actually predict their performance; hence they cannot\nserve as a basis for our predictions.\n\nPrevious work in our research group by \\citeauthor*{roman1} constructed accurate\n\\definition[analytical models]{analytical performance models} for small DLA\nkernels~\\cite{romandis, roman1}.  These models target problems that fit within a\n\\harpertown's last-level cache (L2), and are based on the number of\nmemory-stalls and arithmetic operations as well as their overlap incurred by\nspecific kernel implementations.  As such, they require not only a deep\nunderstanding of the processor architecture, but also a detailed analysis of the\nkernel implementation.  While the resulting models yield accurate predictions\nwithin a few percent of reference measurements, they are not easily extended to\nlarger problems and other operations.  Therefore, this work instead\nconsiders automatically generated, measurement-based models.\n\n\\Citeauthor*{blis3model} construct \\definition[piecewise models]{piecewise}\nruntime and energy {\\em models}---somewhat similar to those presented in this\nwork---for the \\blis implementations of \\dgemm and \\dtrsm~\\cite{blis3model} on a\n{\\hwstyle Sandy Bridge-EP E5-2620}.  However, their approach is based on\nextensive knowledge of \\blis~\\cite{blismodel}, and their models only represent\none degree of freedom (by considering only square matrices or operations on\npanel matrices with fixed width\/height).  Their average runtime model accuracy\nfor \\dgemm and \\dtrsm is, respectively, \\SIlist{1.5;4.5}\\percent, with local\nerrors of up to, respectively, \\SIlist{4.5;7}\\percent.\n\\citeauthor*{blischolmodel} extend this work to multi-threaded \\dgemm, \\dtrsm,\nand \\dsyrk in order to predict the performance of a blocked Cholesky\ndecomposition algorithm with fixed block size~\\cite{blischolmodel}; their\naverage runtime prediction errors are \\SIlist{3.7;2.4}\\percent, depending on the\nparallelization within \\blis.  In contrast to these publications, the modeling\nframework presented in this work, which was developed around the same time, is\nfully automated, applicable to any \\blas- or \\lapack-like routine, not limited\nto one implementation and hardware, and offers models with multiple degrees of\nfreedom.\n\nIn a separate effort \\citeauthor*{tridiagmodel} constructs measurement-based,\nyet hardware- and \\definition{implementation-independent models} in the form of\na series of univariate polynomials (one kernel argument is represented by the\npolynomial, the other varied in the series) for several \\blasl3\nkernels~\\cite{tridiagmodel, qrmodel}.  These models are used to predict the\nperformance of both a blocked reduction to tridiagonal form~\\cite{tridiagmodel}\nand a blocked multishift QR~algorithm~\\cite{qrmodel}.  The resulting prediction\nerror on an unspecified {\\namestyle AMD Opteron} is reported to be\nbelow~\\SI{10}{\\percent} for the single-threaded tridiagonalization, and is on\naverage around~\\SI{10}{\\percent} for the QR~algorithm using multi-threaded\n\\blas.  In contrast, the more general piecewise models proposed in this work\nyield considerable smaller prediction errors for various blocked algorithms.\n\nSeveral research projects model the performance of \\definition[distributed\nmemory]{distributed-memory} applications.  A general purpose approach by\n\\citeauthor*{alex1} builds basic performance models for kernels in application\ncodes based on performance profiling~\\cite{alex2, alex1}, allowing to\ninvestigate the complexity and scalability of application components.  In the\nfield of distributed-memory DLA, most modeling efforts target \\scalapack using\ndomain-specific knowledge through, e.g., polynomial\nfitting~\\cite{scalapackpolfit} or hierarchical modeling of\nkernels~\\cite{scalapckhierarchmodel}.\n\n\n\\subsection{Tensor Contractions}\n\\label{sec:relwork:tensor}\n\nTensor contractions are at the core of scientific computations, such as machine\nlearning~\\cite{tensorml}, general relativity~\\cite{generalrelativity,\ngeneralrelativity2}, and quantum chemistry~\\cite{ccd2, ccd1}.  Since generally\nspeaking such contractions are high-dimensional matrix-matrix multiplications,\nthey are closely related to \\blasl3 operations, and in fact most contractions\ncan be cast in terms of one or more calls to \\dgemm, either by adding loops or\ntranspositions; this is implemented in many frameworks, such as the {\\namestyle\nTensor Contraction Engine} (TCE)~\\cite{tce, tceweb}, the {\\namestyle Cyclops\nTensor Framework} (CTF)~\\cite{cyclops, cyclopsweb}, the \\matlab{} {\\namestyle\nTensor Toolbox}~\\cite{matlabtt, matlabttweb}, and {\\namestyle\nlibtensor}~\\cite{libtensor, libtensorweb}.\n\nIn contrast to these implementations, which rely on a single algorithm for each\ncontraction (potentially selected through heuristics), previous work in our\ngroup by \\citeauthor*{tensorgen} investigated the automated generation of all\nalternative \\blas-based algorithms~\\cite{tensorgen}.  \\Cref{ch:tensor} picks up\nthis work and presents a performance prediction framework for such algorithms\nthat allow to automatically identify the fastest algorithm~\\cite{tensorpred}.\n\nMore recent and ongoing work in our group by \\citeauthor*{gett} attempts to go\nbreak the barrier between contraction algorithms and \\dgemm implementations.\nFollowing the structured design of \\blis~\\cite{blis1}, they propose code\ngenerators that provide high-performance algorithms tailored to specific\ncontraction problems that reach close to optimal performance~\\cite{gett}.  Their\ntools construct numerous alternative implementations, and identify the fastest\nthrough a combination of heuristics and micro-benchmarks.\n\n\n\\subsection{Motivation: Tensor Contraction Algorithms}\n\\label{sec:intro:tensor:algs}\n\nComputationally, tensor contractions are generalizations of matrix-vector and\nmatrix-matrix products to operands of higher dimensionality.  While\n\\blas covers contractions of up to two-dimensional operands (i.e., matrices),\nthere are no equivalently established and standardized high-performance\nlibraries for general tensor contractions.  Fortunately, just as a matrix-matrix\nproducts can be decomposed into sequences of matrix-vector products, higher\ndimensional tensor contractions can be cast in terms of matrix-matrix or\nmatrix-vector kernels.  (A broader overview of alternative approaches is given\nin \\cref{sec:relwork:tensor}.)\n\n\\input{intro\/figures\/tensor_algs}\n\n\\begin{example}{Tensor contraction algorithms}{intro:tensor:algs}\n    Let us consider the contraction $C_{abc} \\coloneqq A_{ai} B_{ibc}$ (in\n    Einstein notation), which is visualized as follows:\n    \\[\n        \\begin{tikzpicture}[baseline=(c.base)]\n            \\begin{drawcube}\n                \\node[anchor=east] at (-1, 0, 1) {$\\scriptstyle a$};\n                \\node[anchor=north] at (0, -1, 1) {$\\scriptstyle b$};\n                \\node[anchor=north west] at (1, -1, 0) {$\\scriptstyle c$};\n                \\node (c) {$C$};\n            \\end{drawcube}\n        \\end{tikzpicture}\n        \\coloneqq\n        \\begin{tikzpicture}[baseline=(c.base)]\n            \\begin{drawsquare}\n                \\node[anchor=east] at (-1, 0, 0) {$\\scriptstyle a$};\n                \\node[anchor=north] at (0, -1, 0) {$\\scriptstyle i$};\n                \\node {$A$};\n            \\end{drawsquare}\n        \\end{tikzpicture}\n        \\matmatsep\n        \\begin{tikzpicture}[baseline=(c.base)]\n            \\begin{drawcube}\n                \\node[anchor=east] at (-1, 0, 1) {$\\scriptstyle i$};\n                \\node[anchor=north] at (0, -1, 1) {$\\scriptstyle b$};\n                \\node[anchor=north west] at (1, -1, 0) {$\\scriptstyle c$};\n                \\node {$B$};\n            \\end{drawcube}\n        \\end{tikzpicture}\n        \\enspace.\n    \\]\n    The entries~$C$\\code{[a,b,c]} of the resulting three-dimensional tensor $C\n    \\in \\R^{a \\times b \\times c}$ are computed as\n    \\[\n        \\forall \\code a \\forall \\code b \\forall \\code c :\\\n        C\\text{\\code{[a,b,c]}} \\coloneqq \\sum_\\code i A\\text{\\code{[a,i]}}\n        B\\text{\\code{[i,b,c]}}\n        \\enspace.\n    \\]\n    As further described in \\cref{sec:tensor:alggen}, this contraction can be\n    performed by a total of 36~alternative algorithms, each consisting of one or\n    more \\code{\\bf for}-loops with a single \\blas kernel at its core.  Three\n    examples of such algorithms using \\blasl1, 2, and~3 kernels are shown in\n    \\cref{fig:intro:tensor:algs}.  These algorithms use \\matlab's ``\\code:''\n    slicing notation\\footnotemark{} to access matrices and vectors within the\n    tensors~$A$, $B$, and~$C$; the resulting operand shapes within the tensors\n    passed to the \\blas kernel are shown alongside the algorithms.\n\\end{example}\n\\footnotetext{%\n    The index ``\\code:'' in a tensor refers to all elements along that\n    dimension, e.g., $A$\\code{[a,:]} is the \\code a-th row of~$A$.\n}\n\nEach tensor contraction can be computed via \\blas kernels through many---even\nhundreds---of algorithms, each with its own performance behavior.  The\n\\definition[optimization challenge:\\\\alternative algorithms\\\\skewed\ndiensions]{optimization challenge} of identifying the fastest among such a set\nof {\\em alternative algorithms} is especially difficult due to the in practice\ncommonly encountered {\\em skewed dimensions} (i.e., one or more dimensions are\nextremely small) for which most \\blas implementations are typically not\noptimized.\n\n\\input{intro\/figures\/tensor_perf}\n\n\\begin{example}{Performance of contraction algorithms}{intro:tensor:perf}\n    Let us consider the tensor contraction $C_{abc} \\coloneqq A_{ai} B_{ibc}$\n    from \\cref{ex:intro:tensor:algs} with tensors $A \\in \\R^{n \\times 8}$, $B\n    \\in \\R^{8 \\times n \\times n}$, and thus $C \\in \\R^{n \\times n \\times n}$;\n    for~$n = 100$, this can be visualized as follows:\n    \\[\n        \\begin{tikzpicture}[baseline=(c.base)]\n            \\begin{drawcube}\n                \\node[anchor=east] at (-1, 0, 1) {$\\scriptstyle a$};\n                \\node[anchor=north] at (0, -1, 1) {$\\scriptstyle b$};\n                \\node[anchor=north west] at (1, -1, 0) {$\\scriptstyle c$};\n                \\node (c) {$C$};\n            \\end{drawcube}\n        \\end{tikzpicture}\n        \\coloneqq\n        \\begin{tikzpicture}[baseline=(a.base), x={(.08, 0)}]\n            \\begin{drawsquare}\n                \\node[anchor=east] at (-1, 0, 0) {$\\scriptstyle a$};\n                \\node[anchor=north] at (0, -1, 0) {$\\scriptstyle i$};\n                \\node (a) {$A$};\n            \\end{drawsquare}\n        \\end{tikzpicture}\n        \\begin{tikzpicture}[baseline=(a.base), y={(0, .08)}]\n            \\begin{drawcube}\n                \\node[anchor=east] at (-1, 0, 1) {$\\scriptstyle i$};\n                \\node[anchor=north] at (0, -1, 1) {$\\scriptstyle b$};\n                \\node[anchor=north west] at (1, -1, 0) {$\\scriptstyle c$};\n                \\node (b) {$B$};\n            \\end{drawcube}\n        \\end{tikzpicture}\n        \\enspace.\n    \\]\n\n    \\Cref{fig:intro:tensor:perf1} presents the performance of all 36~algorithms\n    for this contraction on a \\harpertown with single-threaded \\openblas.  While\n    the two \\dgemm-based algorithms~(\\ref*{plt:intro:tensor:dgemm}) are clearly\n    faster than the others, they differ in performance by up to\n    \\SI{23.32}\\percent; with other kernels the difference are even more extreme,\n    exceeding a factor of~60 for the \\daxpy-based\n    algorithms~(\\ref*{plt:intro:tensor:daxpy}).\n\n    \\Cref{fig:intro:tensor:perf2} showcases the performance of algorithms for\n    the more complex contraction $C_{abc} \\coloneqq A_{ija} B_{jbic}$ on all\n    10~cores of an \\ivybridge using multi-threaded \\openblas.  In this scenario,\n    the performance of the \\dgemm-based algorithms alone differs by up\n    to~$3\\times$.\n\\end{example}\n\nOne could argue that only \\dgemm-based algorithms are viable candidates to\nachieve the best performance; while for the most part this observation is true,\ndue to skewed dimensions, even the performance of only these algorithms can\ndiffer dramatically.  Furthermore, some contractions (e.g., $C_a \\coloneqq\nA_{iaj} B_{ji}$) cannot be implemented via \\dgemm in the first place.\nTherefore, we aim at the accurate prediction of any \\blas-based contraction,\nirrespective of which kernel is used.\n\n\n\\subsection{Prediction through Micro-Benchmarks}\n\\label{sec:intro:tensor:pred}\n\nAt first sight the situation seems similar to the selection of blocked\nalgorithms:  We want to avoid exhaustive performance measurements and select the\nbest algorithm {\\em without executing} any of the alternatives; our strategy is\nonce again to predict each algorithm's performance by estimating its invoked\nkernel's runtime.  However, while performance models accurately estimates the\nperformance of such kernels for many operand sizes, they perform rather poorly\nfor operations with skewed dimensions:  For extremely thin or small operands,\n\\blas kernels exhibit strong size-dependent performance fluctuations, which are\nimpractical to capture and represent in performance models.\n\nWhile we cannot rely on performance models, analyzing the structure of tensor\ncontraction algorithms suggests a different approach:  In contrast to blocked\nalgorithms, a contraction algorithm performs its entire computation in a series\nof calls to a \\definition[single kernel\\\\fixed size\\\\micro-benchmarks]{single\n\\blas kernel} of with operands of {\\em fixed size}.  Based on this observation,\nwe estimate the performance of such calls by constructing a small set of {\\em\nmicro-benchmarks} that executes the kernel only a few times, and thus performs\nonly a fraction of the algorithm's computation.  Since memory locality plays an\nespecially important role in contractions with skewed dimensions, we carefully\nrecreate the stat of the processor's caches within the micro-benchmarks to time\nthe kernel in conditions analogous to those in the actual algorithm.\n\nBased on such micro-benchmarks, we can predict the total runtime of contraction\nalgorithms for tensors of various shapes and sizes.  These predictions reliably\nsingle out the fastest algorithm from a set of alternatives several orders of\nmagnitude faster than a single algorithm execution.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsubsection{Background and System Noise}\n\\label{sec:meas:fluct:noise}\n\nThe potentially most disturbing, yet also quite easily avoidable source of\nfluctuations are other \\definition{background processes} competing for the\nprocessor's resources.\n\n\\input{meas\/figures\/fluct}\n\n\\begin{example}{Influence of background noise}{meas:fluct}\n    \\Cref{fig:meas:fluct} presents the runtime of 1000~repetitions of the\n    matrix-matrix multiplication $\\dm C \\coloneqq \\dm A \\matmatsep \\dm B + \\dm\n    C$ (\\dgemm[NN]) with $\\dm A, \\dm B, \\dm C \\in \\R^{100 \\times 100}$ on a\n    \\broadwell (as part of {\\namestyle MacBook Pro} with \\apple's framework\n    \\accelerate and a \\sandybridge (as part of \\rwth's computing cluster) with\n    \\mkl.\n\n    On the \\broadwell~(\\ref*{plt:ibacc:circ}) with various other applications\n    running in the background (e.g., browser and music player), the fluctuations\n    are enormous:  The measurement standard deviation is over $4\\times$~the mean\n    runtime.  On the \\sandybridge~(\\ref*{plt:sbmkl:circ}) with no other user\n    applications running during measurements, the fluctuations are already much\n    smaller at \\SI{2.36}{\\percent}~of the average time.  For larger problem\n    sizes, the fluctuations are considerably smaller, and quickly fall below\n    \\SI{.1}\\percent.\n\\end{example}\n\nWhile these type of fluctuations can be avoided to some extend by ensuring that\nno other applications run during measurements, they cannot be avoided altogether\neven with exclusive access to dedicated high-performance hardware---the\nremaining fluctuations are known as \\definition{system noise}.  Hence, for our\nexperiments, models, and micro-benchmarks all our measurements are repeated at\nleast five times and \\definition{summary statistics} of the runtime (or\nperformance) are presented, such as the minimum or median.\n\n\n\\subsubsection{\\intel{} \\turboboost}\n\\label{sec:meas:fluct:turbo}\n\nCompute-bound dense linear algebra computations, such as \\blasl3 and\n\\lapack-level routines, benefit directly from increased processing frequencies.\nTherefore, they usually trigger \\intel{} \\turboboost and constantly run at the\nmaximum turbo frequency if possible.  Since this frequency cannot be sustained\nindefinitely on most machines, the processor frequency is eventually lowered and\nhenceforth fluctuates to keep the hardware within its power and thermal limits.\n\n\\input{meas\/figures\/turbo}\n\n\\begin{example}{\\turboboost}{meas:turbo}\n    \\Cref{fig:meas:turbo} presents the runtime of repeated matrix-matrix\n    multiplications $\\dm C \\coloneqq \\dm A \\matmatsep \\dm B + \\dm C$\n    (\\dgemm[NN]) with $\\dm A, \\dm B, \\dm C \\in \\R^{1300 \\times 1300}$ alongside\n    the processor's temperature and frequency\\footnotemark{} on both cores of a\n    \\broadwell with multi-threaded \\accelerate; in this experiment, no other\n    resource intensive programs run in the background.\n\n    In the beginning, the processor is at a cool\n    \\SI{53}{\\celsius}~(\\ref*{plt:meas:turbo:temp}) and each \\dgemm[NN] takes\n    about \\SI{60}{\\ms}~(\\ref*{plt:meas:turbo:time}) at the maximum turbo\n    frequency of \\SI{3.4}{\\GHz}~(\\ref*{plt:meas:turbo:freq}).  The processor\n    temperature increases steadily up to \\SI{105}{\\celsius} around\n    repetition~200 (\\SI{12}{\\second} into the experiment); at this point the\n    frequency is reduced and continuously adjusted between \\SIlist{3;3.2}{\\GHz}\n    such that this temperature threshold is not exceeded.  This change in\n    frequency, as well as its fluctuations towards the end have a direct effect\n    on the \\dgemm[NN]'s runtime:  It increases by about~\\SI{10}{\\percent} to\n    roughly~\\SI{67}\\ms.\n\\end{example}\n\\footnotetext{%\n    Obtained through the \\intel {\\namestyle Power Gadget}.\n}\n\nThe behavior of \\turboboost depends enormously on the computation environment:\nWhile on a work-station or laptop system the processor temperature increases\nrapidly and the maximum turbo frequency is not sustained for long, on dedicated\nhigh-performance compute clusters, efficient cooling allows for the processor to\noperate at the maximum turbo frequency for much longer, if not indefinitely.\nHowever, even in our main computing facilities at the {\\namestyle\\rwth IT\nCenter}, we observed notable fluctuations of the frequency below its maximum\nwith negative impacts on our measurement quality and stability.\n\nThroughout this work, we consider processors with and without enabled\n\\turboboost.  While the performance of these two cases is not directly\ncomparable, we consider our methodologies for both scenarios.  In particular,\n\\turboboost is disabled on our \\sandybridge (unless otherwise stated) and\nenabled on our \\haswell---an overview of all hardware configurations is given in\n\\cref{app:hardware}.\n\n\n\\subsubsection{Distinct Long-Term Performance Levels}\n\\label{sec:meas:fluct:longterm}\n\nEven with \\turboboost disabled, a processor's speed is not always fixed to its\nbase frequency and we instead observed jumps between two or more\n\\definition{performance levels}.\n\n\\input{meas\/figures\/longterm}\n\n\\begin{example}{Performance levels}{meas:longterm}\n    \\Cref{fig:meas:longterm} presents the runtime of 1000~repetitions of the\n    matrix-matrix multiplication $\\dm[width=.05]C \\coloneqq \\dm A \\matvecsep\n    \\dm[width=.05]B + \\dm[width=.05]C$ (\\dgemm[NN]) with $\\dm A \\in \\R^{4000\n    \\times 4000}$ and $\\dm[width=.05]B, \\dm[width=.05]C \\in \\R^{4000 \\times\n    200}$ on a \\sandybridge and a \\haswell (both with \\turboboost disabled) with\n    single-threaded \\openblas.\n\n    On both systems, we can clearly make out two distinct runtime levels: on the\n    \\sandybridgeshort, the measurements jump between \\SIlist{354;359}\\ms, which\n    are \\SI{1.4}{\\percent}~apart, and on the \\haswellshort with twice the\n    floating-point performance per cycle, the two levels\n    at~\\SIlist{205;213}{\\ms} differ by~\\SI{3.9}\\percent.  There is no\n    discernible pattern to the jumps between these levels and the processors\n    commonly stay at the same level for~\\SI{10}{\\second} or longer\n    (50~repetitions at \\SI{200}{\\ms} each).\n\\end{example}\n\nSince we found no means to eradicate this type of fluctuations, we adopt our\nmeasurement setups to account for them:  Whenever we have more than one\nmeasurement point (e.g., varying the routines or problem sizes), we not only\nrepeat each measurement several times in isolation, but also shuffle the\nrepetitions.  As a result, the repetitions for each data point are spread across\nthe entire experiment duration and summary statistics such as the minimum and\nmedian yield a stable runtime estimate for only one performance level.\n\nIn summary, we can avoid or account for various types of fluctuations within our\nmeasurements.\n\n\n\n\\section{Performance Effects for Dense Linear Algebra Kernels}\n    \\label{sec:meas:effects}\n    \\input{meas\/effects}\n\n        \\subsection{Library Initialization Overhead}\n        \\label{sec:meas:effects:init}\n        \\input{meas\/init}\n\n        \\subsection{Fluctuations}\n        \\label{sec:meas:effects:fluct}\n        \\input{meas\/fluct}\n\n        \\subsection{Thread Pinning}\n        \\label{sec:meas:effects:pin}\n        \\input{meas\/pin}\n\n        \\subsection{Caching}\n        \\label{sec:meas:effects:caching}\n        \\input{meas\/caching}\n\n        \\subsection{Summary}\n        \\label{sec:meas:effects:sum}\n        \\input{meas\/effectssum}\n\n    \\section{Measurements and Experiments: \\elaps}\n    \\label{sec:meas:elaps}\n    \\input{meas\/elapsintro}\n\n        \\subsection{The \\sampler}\n        \\label{sec:meas:sampler}\n        \\input{meas\/sampler}\n\n        \\subsection{The \\elaps{} \\python Framework}\n        \\label{sec:meas:elapslib}\n        \\input{meas\/elaps}\n\n    \\section{Summary}\n    \\label{sec:meas:conclusion}\n    \\input{meas\/conclusion}\n}\n\n\n\n\n\n\n\n\n\n\\subsubsection{Alignment to Cache-Lines}\n\nData is moved through the memory hierarchy in blocks of \\SI{64}{\\bytes} ($=\n\\SI8\\doubles$) called \\definition{cache-lines}.\\footnote{%\n    The cache-line size is generally not fixed but for most processors it is\n    \\SI{64}{\\Byte}.\n}  Hence using multiples of the cache-lines size as memory access strides\ntypically shows a more regular and often better performance compared to other\nstrides.\n\n\\input{model\/figures\/ld8}\n\n\\footnotetextbefore{%\n    Since $A$ and~$B$ have 256~rows, the leading dimensions are at least~256.\n}\n\\begin{example}{Aligning leading dimensions to cache-lines}{model:args:ld:8}\n    \\Cref{fig:model:ld8} shows the runtime of\n    \\displaycall\\dtrsm{\n        \\arg{side}L, \\arg{uplo}L,\n        \\arg{transA}N, \\arg{diag}N,\n        \\arg m{256}, \\arg n{256},\n        \\arg{alpha}{1.0},\n        \\arg AA, \\arg{ldA}{\\it\\color{blue}ld},\n        \\arg BB, \\arg{ldB}{\\it\\color{blue}ld}\n    }\n    i.e., $\\dmB \\coloneqq \\dmAi \\dmB$ with $\\dmA\\lowerpostsep, \\dmB \\in \\R^{256\n    \\times 256}$, for leading dimensions\\footnotemark{} $ld = 256, \\ldots, 320$\n    in steps of~1 on a \\sandybridge and a \\haswell with single-threaded\n    \\openblas, \\blis, and \\mkl.\n\n    For all setups, the \\dtrsm[LLNN]'s runtime exhibits some regular pattern in\n    terms of the leading dimension arguments---with an average amplitude\n    of~\\SI{2.19}\\percent.  However the patterns are quite different:  While\n    \\openblas's runtime on the \\sandybridgeshort~(\\ref*{plt:sbopen}) drops\n    equally at every even leading dimension, \\mkl on the\n    \\haswellshort~(\\ref*{plt:hwmkl}) dips only at multiples of~4, and on the\n    \\sandybridgeshort~(\\ref*{plt:sbmkl}) it has stronger dips at multiples of~8.\n    \\blis on the other hand shows the exact opposite behavior:  On both\n    platforms~(\\ref*{plt:sbblis}, \\ref*{plt:hwblis}) its runtime spikes slightly\n    at multiples of~8.\n\n    Independent of the specific behavior of each setup, a smooth runtime curve\n    is obtained when only multiples of~8 are considered as leading dimensions.\n\\end{example}\n\nTo avoid small performance irregularities, we will generate our models using\n\\definition[use multiples of the cache-line size]{multiples of the cache-line\nsize} for leading dimensions---in double-precision: multiples of~8.\n\n\n\\subsubsection{Set-Associative Cache Conflicts}\n\\label{sec:model:args:ld512}\n\nThe Level~1 and~2 caches in our processors are \\definition{8-way\nset-associative}:  They are divided into sets of 8~cache-lines, and when a\ncache-line is loaded, its address's least significant bits determine which of the\nsets it is assigned to; within the set, an architecture-dependent cache\nreplacement policy determines in which of the 8~slots it is stored.  When the\naddress space is accessed contiguously, consecutive cache-lines are loaded into\nconsecutive sets, and the cache is filled evenly.  In the worst case, however,\nthe address space is accessed with a stride equal to the number of sets, and\nall loaded cache-lines are associated to the same set:  Only 8~cache-lines are\ncached, and each additional line results in a \\definition{cache conflict miss}\ncausing a recently loaded line to be evicted.  This effect should be avoided\nwhenever possible.\n\nOn recent \\intel{} {\\namestyle Xeon} processors, the Level~1 data cache~(L1d)\nfits \\SI{32}{\\kibi\\byte} organized as 64~sets of 8~cache-lines.  A memory\nlocation with address~$a$ is a part of cache-line~$\\lfloor a \/ 64 \\rfloor$ (due\nto the size of \\SI{64}{\\Byte} per line) and assigned to set $\\lfloor a \/ 64\n\\rfloor \\bmod 64$ (due to the capacity of 64~sets).  The Level~2 cache (L2) in\nturn fits \\SI{256}{\\kibi\\byte} in 1024~sets; here address~$a$ is assigned to set\n$\\lfloor a \/ 64 \\rfloor \\bmod 1024$.\n\nIn a double-precision matrix stored with leading dimension~$ld$, consecutive\nelements in each row are $8 ld$~\\bytes apart ($\\SI1\\double = \\SI8\\bytes$).\nHence, for $ld = 512$, the consecutive row elements starting at address~$a_0$\nare stored at~$a_i = a_0 + 8 ld \\cdot i  = a_0 + 4096 i$, and associated to the\nsame set in the L1d~cache:\n\\begin{align*}\n    \\left\\lfloor \\frac{a_i}{64} \\right\\rfloor \\bmod 64\n    &= \\left\\lfloor \\frac{a_0 + 4096 i}{64} \\right\\rfloor \\bmod 64 \\\\\n    &= \\left(\\left\\lfloor \\frac{a_0}{64} \\right\\rfloor + 64 i \\right) \\bmod 64 \\\\\n    &= \\left\\lfloor \\frac{a_0}{64} \\right\\rfloor \\bmod 64.\n\\end{align*}\nThe same problem occurs for leading dimensions that are multiples of~512, and\neven below~512 powers of~2 have a similar effect:  E.g., with $ld = 256$ the\nelements of a row are associated to only two of the cache's 64~sets.  Similarly,\nfor the L2~cache with 1024~sets, consecutive row-elements are associated to the\nsame cache set for leading dimensions that are multiples of~8192, and multiples\nof~4096 utilize only two sets.\n\n\\input{model\/figures\/ld512}\n\n\\begin{example}{Cache conflict misses caused by leading\n    dimensions}{model:args:ld:512}\n    \\Cref{fig:model:ld512} shows the runtime of\n    \\displaycall\\dtrsm{\n        \\arg{side}L, \\arg{uplo}L, \\arg{transA}N, \\arg{diag}N,\n        \\arg m{256}, \\arg n{256}, \\arg{alpha}{1.0},\n        \\arg AA, \\varg{ldA}{ld}, \\arg BB, \\varg{ldB}{ld}\n    }\n    i.e., $\\dmB \\coloneqq \\dmAi \\dmB$ with $\\dmA\\lowerpostsep, \\dmB \\in \\R^{256\n    \\times 256}$, for leading dimensions $ld = 256, \\ldots, 8320$ in steps\n    of~128 on a \\sandybridge and a \\haswell with single-threaded \\openblas,\n    \\blis, and \\mkl.\n\n    For most setups the runtime spikes above the baseline at multiples of~512.\n    However, the average magnitude of these spikes ranges\n    from~\\SI{.14}{\\percent} for \\blis on the\n    \\sandybridgeshort~(\\ref*{plt:sbblis}) to~\\SI{8.37}{\\percent} for \\openblas\n    on the \\haswellshort~(\\ref*{plt:hwopen}).  Especially for\n    \\openblas~(\\ref*{plt:sbopen}, \\ref*{plt:hwopen}), there are additional, yet\n    lower spikes of \\SI{1.40}{\\percent} at multiples of~256.  Furthermore, on\n    the \\haswellshort for both \\openblas~(\\ref*{plt:hwopen}) and\n    \\blis~(\\ref*{plt:hwblis}) the spikes are especially high at $ld = 4096$\n    and~8192, exceeding the baseline by, respectively,\n    \\SIlist{6.55;11.24}\\percent.\n\\end{example}\n\nTo prevent distortions from unfortunate leading dimensions in our model\ngeneration altogether, we will \\definition{avoid multiples of~256} for these\narguments.\n\nNote that by using leading dimensions that are multiples of~8, yet not of~256 in\nour measurements, our models will not yield accurate predictions for kernel\ninvocations that do not follow this pattern.  However, predicting the\nperformance of such unfortunate invocations, which can be systematically\navoided, is not part of our models' purpose and would exceed the scope of this\nwork.\n\n\n\\subsubsection{Smalls Scale Behavior}\n\\label{sec:model:args:size:small}\n\nOptimizations of compute kernels commonly involve vectorization and loop\nunrolling of length~4 or~8.  These optimizations typically have a direct\ninfluence on a kernel's runtime for small variations of the size arguments.\n\n\\input{model\/figures\/size8}\n\n\\begin{example}{Small variations of size arguments}{model:args:size:8}\n    \\Cref{fig:model:size8} shows the runtime of\n    \\displaycall\\dtrsm{\n        \\arg{side}L, \\arg{uplo}L, \\arg{transA}N, \\arg{diag}N,\n        \\varg mn, \\varg nn, \\arg{alpha}{1.0},\n        \\arg AA, \\arg{ldA}{400}, \\arg BB, \\arg{ldB}{400}\n    }\n    i.e., $\\dmB \\coloneqq \\dmAi \\dmB$ with $\\dmA\\lowerpostsep, \\dmB \\in \\R^{n\n    \\times n}$, for $n = 256, \\ldots, 320$ in steps of~1 on a \\sandybridge and a\n    \\haswell with single-threaded \\openblas, \\blis, and \\mkl.\n\n    All setups show periodic patterns in their runtimes.  While these patterns\n    differ between the implementations, most have local runtime minima at\n    multiples of~4, and all of them have minima at multiples of~8.\n\\end{example}\n\nTo avoid runtime artefacts introduced by vectorization and loop unrolling, we\nwill build our models on measurements that \\definition{use multiples of~8} for\nall size arguments.\n\n\n\\subsubsection{Piecewise Polynomial Behavior}\n\\label{sec:model:args:size:large}\n\nSince an operation's minimal \\flop-count is generally a (multivariate)\npolynomial function of the size arguments, one might expect that (for\ncompute-bound kernels) it translates directly into an equally polynomial\nruntime.  However, since a kernel's performance is generally not constant for\nvarying operand sizes, a single polynomial is often insufficient to accurately\nrepresent a kernel's runtime for large ranges of problem sizes.\n\n\\input{model\/figures\/size}\n\n\\begin{example}{Polynomial fitting for size arguments}{model:args:size}\n    \\Cref{fig:model:size} shows the runtime of\n    \\displaycall\\dtrsm{\n        \\arg{side}L, \\arg{uplo}L, \\arg{transA}N, \\arg{diag}N,\n        \\varg m{n}, \\varg n{n}, \\arg{alpha}{1.0},\n        \\arg AA, \\arg{ldA}{1000}, \\arg BB, \\arg{ldB}{1000}\n    }\n    i.e., $\\dmB \\coloneqq \\dmAi \\dmB$ with $\\dmA\\lowerpostsep, \\dmB \\in \\R^{n\n    \\times n}$, with $n = 24, \\ldots, 536$ in steps of~16 on a \\sandybridge and\n    a \\haswell with single-threaded \\openblas, \\blis, and \\mkl.\n\n    At first sight, the runtime for all setups follows a smooth cubic\n    behavior---perfectly in line with the operation's minimal cost of\n    \\SIvar{n^3}\\flops.  However, if for each setup we fit the measurements with\n    a single cubic polynomial that minimizes the least-squares relative error\n    (details in~\\cref{sec:model:fit}), we are left with the approximation error\n    shown in~\\cref{fig:model:size:err1}.  The absolute relative approximation\n    error\\footnotemark{} lies between \\SI{.86}{\\percent} for \\blis on the\n    \\sandybridgeshort~(\\ref*{plt:sbblis}) and \\SI{11.22}{\\percent} for \\openblas\n    on the \\haswellshort~(\\ref*{plt:hwopen}); on average it\n    is~\\SI{5.30}\\percent.\n\n    If we look closer at the approximation errors in\n    \\cref{fig:model:size:err1}---especially for \\openblas on the\n    \\haswellshort~(\\ref*{plt:hwopen})---we observe a piecewise smooth(er)\n    behavior.  Motivated by this observation, we now fit not one polynomial to\n    each data-set but two: one for the first half ($n \\leq 280$) and one for the\n    second half ($n \\geq 280$).  For this two-split polynomial fit the\n    approximation error is shown in~\\cref{fig:model:size:err2}: The largest\n    error is now reduced to~\\SI{5.25}{\\percent} for \\mkl on the\n    \\haswellshort~(\\ref*{plt:hwmkl}), and the average error\n    is~\\SI{2.55}{\\percent}---less than half of the original approximation error.\n    (Based on a more detailed analysis, a better splitting point than\n    $\\frac{24+536}2 = 280$ could have been chosen, but as\n    \\cref{fig:model:size:err1} shows such choices would be notably different for\n    each setup.)  Within the new approximation, the error for the second\n    polynomial ($n \\geq 280$) is already quite low---on\n    average~\\SI{.38}\\percent.  Hence, in a second step, we further subdivide\n    only the first half of the domain ($n \\leq 280$) at~$n = 152$, and generate\n    a new approximation consisting of three polynomials.  As\n    \\cref{fig:model:size:err3} shows, the error of this approximation is\n    below~\\SI{1.28}{\\percent}~(\\ref*{plt:hwmkl}) in all cases and on\n    average~\\SI{.71}\\percent.\n\\end{example}\n\\footnotetext{%\n    For a polynomial~$p(x)$ fit to measurements~$y_1, \\ldots, y_N$ in\n    points~$x_1, \\ldots, x_N$ we consider the error $1 \/ N \\sum_{i=1}^N \\lvert\n    y_i - p(x_i) \\rvert \/ y_i$.  Note that the least-squares fitting minimizes\n    not this sum of absolute relative errors but the sum of squared relative\n    errors.\n}\n\nTo account for the not purely polynomial influence of a kernel's size arguments\non its runtime, we will represent it in our models through \\definition{piecewise\npolynomials}.  Details on the such piecewise polynomial representations and\ntheir automated generation are given in\n\\cref{sec:model:fit,sec:model:adaptive,sec:model:config}.\n\n\n\n\n\\subsection{Configuration Parameters}\n\nThe adaptive refinement is controlled by a total of eight\n\\definition{configuration parameters}.  They allow to control the model\naccuracy, but also affect the time spent for the required measurements.  The\neight parameters regulate the model generation as follows:\n\\begin{itemize}\n    \\item To represent the runtime of a kernel, the monomial basis for the\n        fitted polynomials needs to at least cover the kernel's asymptotic\n        complexity (i.e., its minimal \\flop-count).  To better represent\n        performance variations, however, the maximum degree of the monomials can\n        be increased in each each dimension (i.e., size argument).  We refer to\n        this increase as \\definition[overfitting:\\\\between 0\n        and~2]{overfitting}; practical values are {\\em between 0 and~2}.\n\n    \\item To fit a polynomial to a routine's runtime, the number of sampling\n        points along each dimension needs to be at least one more than the\n        corresponding polynomial degree.  However, since this minimal number of\n        points yields a polynomial that fits the measurements perfectly, we\n        cannot use it to compute an approximation error.  We hence increase the\n        number of sampling points per dimension by at least one, and to further\n        improve the approximation accuracy, further points can be added; we\n        refer to the total number of points added as\n        \\definition[oversampling:\\\\between 1 and~10]{oversampling}; practical\n        values are values {\\em between 1 and~10}.\n\n    \\item We introduced two alternatives to \\definition[distribution\n        grid:\\\\Cartesian or Chebyshev]{distribute} sampling points on {\\em\n        grids} that cover the domains of problem sizes: a {\\em Cartesian} grid\n        and a {\\em Chebyshev} grid.\n\n    \\item For each sampling point, we perform several \\definition[measurement\n        repetitions:\\\\between 5 and~20]{measurement repetitions}; practical\n        values are {\\em between 5 and~20}.\n\n    \\item From the repetitions, we compute several runtime summary statistics:\n        minimum, median, maximum, average, and standard deviation.  One of these\n        is selected as the \\definition[reference statistic:\\\\minimum or\n        median]{reference statistic}; practical choices are the {\\em minimum and\n        median}.\n\n    \\item From the absolute relative errors in the reference statistic for all\n        sampling points, we compute the \\definition[error measure:\\\\average,\n        maximum, or 90th~percentile]{error measure} which is these relative\n        errors' {\\em average, maximum, or 90th~percentile}.\n\n    \\item The first termination criterion for the adaptive refinement process is\n        the approximation accuracy:  The refinement stops when the computed\n        error measure is below a \\definition[target error bound:\\\\between\n        {\\SIlist[detect-all=true]{1;5}\\percent}]{target error bound}; practical\n        values for this bound are {\\em between\n        \\SIlist[detect-all=true]{1;5}\\percent}.\n\n    \\item The second termination criterion is the size of the domains:  The\n        refinement stops when a new domain is smaller than a \\definition[minimum\n        width:\\\\32 or~64]{minimum width} along all dimensions; typical values\n        are {\\em 32 and~64}.\n\\end{itemize}\n\n\n\\subsection{Trade-Off and Configuration Selection}\n\nIn the following, we analyze the accuracy of our models and their generation\ncost, and select a configuration to generate the models for the performance\npredictions in the \\cref{ch:pred}.\n\nWe consider the model generation for\n\\displaycall\\dtrsm{\n    \\arg{side}L, \\arg{uplo}L, \\arg{transA}N, \\arg{diag}N,\n    \\arg m{\\it\\color{blue}m}, \\arg n{\\it\\color{blue}n}, \\arg{alpha}{1.0},\n    \\arg AA, \\arg{ldA}{5000}, \\arg BB, \\arg{ldB}{5000}\n}\ni.e., $\\dmB[height=.5] \\coloneqq \\dmAi[size=.5] \\dmB[height=.5]$ with\n$\\dmA[size=.5] \\in \\R^{m \\times m}$ and $\\dmB[height=.5] \\in \\R^{m \\times n}$,\nfor sizes $m \\in [24, 536]$ and $n \\in [24, 4152]$ on a \\sandybridge and a\n\\haswell using single-threaded \\openblas, \\blis, and \\mkl.\n\nFor each setup, our first step is to exhaustively measure the \\dtrsm[LLNN]'s\nruntime 15~times in all points $(m, n)$ in the domain $[24, 536] \\times [24,\n4152]$ at which both~$m$ and~$n$ are multiples of~8---a total of \\num{504075}\nmeasurements.   These measurements are used both as the basis for our model\ngeneration and to evaluate the model accuracy across the entire domain (contrary\nto the model generation, which can only evaluate the error in its sampling\npoints).\n\n\\input{model\/tables\/config}\n\nWe generate models for all 2880~configurations obtained from combining the\nparameter values shown in \\cref{tbl:model:config}.  These configurations result\nin a wide range of models with significantly different accuracies and generation\ncosts.  To evaluate them, we quantify the \\definition{model error} as the\naveraged relative error of the predicted minimum runtime~$p(\\x_i)$ relative to\nthe measured minimum~$y_i$ across all $N = \\num{33605}$ points~$\\x_i$ of the\ndomain:\n\\[\n    \\text{model error} \\defeqq\n    \\frac1N \\sum_{i=1}^N \\frac{\\lvert p(\\x_i) - y_i \\rvert}{y_i} \\enspace;\n\\]\nfurthermore, we define the \\definition{model cost} as the total runtime of the\nrequired measurements used as samples.\n\n\\input{model\/figures\/modelplots}\n\\input{model\/tables\/modelplots}\n\n\\begin{example}{Model accuracy}{model:acc}\n    \\Cref{fig:model:modelplots} shows the structure and point-wise accuracy of\n    the four models with minimum and maximum accuracy and cost for\n    single-threaded \\openblas on a \\sandybridge; \\cref{tbl:model:modelplots}\n    lists the corresponding configurations.  Both the cheapest and least\n    accurate model use only a single polynomial for the entire domain but also\n    offer only poor accuracy.  The expensive and accurate models on the other\n    hand subdivide the domain repetitively, and thus find a better fitting\n    piecewise polynomial.\n\\end{example}\n\n\\input{model\/figures\/tradeoff}\n\nThe accuracy and cost of all 2880~generated models for each setup are presented\nin \\cref{fig:model:tradeoff:full}; in this plot, the preferable models with low\nerror and cost are found close to the origin.  All setups share the same general\ntrend:  Models with low accuracy are quite cheap, while models with high\naccuracy are more expensive.  Hence we are faced with a\n\\definition[trade-off:\\\\accuracy vs.~cost]{trade-off between accuracy and cost}.\nHowever, the configuration selection is not straight-forward:  Models with\npractically identical accuracy are up to a factor of~16 apart in generation\ncost, and a cheap and accurate configuration for one setup may be neither for\nother setups.  In the following, we describe how we approach the search-space of\nall considered configurations, and identify a desirable default configuration\nthat we subsequently use to generate the models for all setups and kernels\nneeded for our performance predictions in \\cref{ch:pred}.\n\nBefore we begin to reduce our search space, we notice that on the \\haswellshort,\nthe models for both \\blis~(\\ref*{plt:hwblis:circ}) and\n\\mkl~(\\ref*{plt:hwmkl:circ}) are on average less than half as accurate than for\nthe other setups.  The cause is a rather jagged performance behavior, which is\ndifficult to represent accurately.  Hence, to identify a good default\nconfiguration, we consider only the \\sandybridgeshort~(\\ref*{plt:sbopen:circ},\n\\ref*{plt:sbblis:circ}, \\ref*{plt:sbmkl:circ}) and \\openblas on the\n\\haswellshort~(\\ref*{plt:hwopen:circ}).\n\nOur first step is to \\definition{prune by accuracy}:  We discard any\nconfiguration that for any of the considered setups yields a model error larger\nthan $1.5\\times$ the minimum error for that setup;  in other words, all\nremaining configurations generate models that are at most \\SI{50}{\\percent}\nless accurate than the most accurate model.  This step reduces the number of\npotential configurations from 2880 to~163; all remaining configurations use an\noversampling value of~3 or higher, and a target error bound of~\\SI1\\percent.\n\\cref{fig:model:tradeoff:within2err} shows the 163~remaining models' accuracy\nand cost.\n\n\\input{model\/tables\/tradeofffinal}\n\nOur second step is to similarly \\definition{prune by cost}:  We discard any\nconfiguration that for any considered setup takes longer than the first quartile\nin generation time for that setup; in other words, the remaining models are all\nwithin the \\SI{25}{\\percent} that are generated the fastest.  This step further\nreduces the number of potential configurations from 163 to~14, as shown in\n\\cref{fig:model:tradeoff:belowmedcost}.\n\nThe parameter values for the 14~remaining configurations are shown in\n\\cref{tbl:model:tradeoff:final}.  For each parameter, we can find one value that\nis common to at least 8~of the 14~configurations (highlighted in {\\bf bold}).\nWe choose our \\definition{default configuration} by selecting this most common\nvalue for each parameter.  It corresponds to line~(10) in\n\\cref{tbl:model:tradeoff:final} (highlighted in {\\bf\\color{blue}blue}), and is\nmarked for each setup in \\cref{fig:model:tradeoff:belowmedcost}.  Note that it\nalso serves as a good choice between accuracy and cost for\n\\blis~(\\ref*{plt:hwblis:circ}) and \\mkl~(\\ref*{plt:hwmkl:circ}) on the\n\\haswellshort, which were not included in the pruning process.\n\n\n\\subsection{Variations of the Default Configuration}\n\nWhile the configuration was found to yield good accuracies at reasonable costs\nfor almost all encountered kernels, it proves to be quite expensive for kernels\nwith \\definition[3D case (\\dgemm)]{three degrees of freedom}, which for the\npredictions in \\cref{ch:pred} only applies to {\\em\\dgemm} with its three size\narguments~\\code m, \\code n, and~\\code k.  To reduce the modeling cost for this\nkernel, we adjust the default configuration by reduce the overfitting from~2\nto~0, and increasing the minimum width from~32 to~64.\n\nFurthermore, the performance of \\blas kernels becomes less smooth when we bring\n\\definition{multi-threading} into the picture.  Hence, to avoid excessive\npartitioning as seen in \\cref{fig:model:modelplots:maxcost}, we increase the\nminimum width for all models to~64, and for \\dgemm to~256.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\chapter{Performance Modeling}\n\\chapterlabel{model}\n{\n    \\input{model\/commands}\n\n    \\input{model\/intro}\n\n    \\section{Kernel Argument Analysis}\n    \\label{sec:model:args}\n    \\input{model\/args}\n\n        \\subsection{Flag Arguments}\n        \\label{sec:model:args:flag}\n        \\input{model\/arg-flag}\n\n        \\subsection{Scalar Arguments}\n        \\label{sec:model:args:scalar}\n        \\input{model\/arg-scalar}\n\n        \\subsection{Leading Dimension Arguments}\n        \\label{sec:model:args:ld}\n        \\input{model\/arg-ld}\n\n        \\subsection{Increment Arguments}\n        \\label{sec:model:args:inc}\n        \\input{model\/arg-inc}\n\n        \\subsection{Size Arguments}\n        \\label{sec:model:args:size}\n        \\input{model\/arg-size}\n\n        \\subsection{Data Arguments}\n        \\label{sec:model:args:data}\n        \\input{model\/arg-data}\n\n        \\subsection{Summary}\n        \\label{sec:model:args:sum}\n        \\input{model\/arg-sum}\n\n    \\section{Model Generation}\n    \\label{sec:model:generation}\n    \\input{model\/generation}\n\n        \\subsection{Model Structure}\n        \\label{sec:model:structure}\n        \\input{model\/structure}\n\n        \\subsection{Sample Distribution}\n        \\label{sec:model:grids}\n        \\input{model\/grids}\n\n        \\subsection{Repeated Measurements and Summary Statistics}\n        \\label{sec:model:stat}\n        \\input{model\/stat}\n\n        \\subsection{Relative Least-Squares Polynomial Fitting}\n        \\label{sec:model:fit}\n        \\input{model\/fit}\n\n        \\subsection{Adaptive Refinement}\n        \\label{sec:model:adaptive}\n        \\input{model\/adaptive}\n\n    \\section{Model Generator Configuration}\n    \\label{sec:model:config}\n    \\input{model\/config}\n\n    \\section{Summary}\n    \\label{sec:model:sum}\n    \\input{model\/model-sum}\n}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Varying Problem Size}\n\\label{sec:pred:chol:n}\n\n\\input{pred\/figures\/cholperf}\n\nIn our first analysis, we use only one of the \\sandybridgeshort's 8~cores and\nvary the problem size between~$n = 56$ and~4152 in steps of~64 while keeping the\nblock size fixed at~$b = 128$.  \\Cref{fig:pred:chol:time_perf} shows the runtime\nand performance of predictions and measurements for this setup side-by-side.\n(Since the red line \\legendline[very thick, darkred] at the top of the\nperformance plots indicates the processor's theoretical peak performance, such\nplots can also be interpreted as compute-bound efficiencies with\n\\SI0{\\percent}~at the bottom and \\SI{100}{\\percent}~at the top.)  The\npredictions give a good idea of the algorithm behavior:  While the runtime\nincreases cubically with the problem size~$n$, the performance is low for small\nmatrices and increases steadily towards \\SI{18}{\\giga\\flops\\per\\second}.  At\nfirst sight, the predictions match the measurements well.\n\n\\input{pred\/figures\/cholerr}\n\nTo further study the accuracy of our predictions, the top half of\n\\cref{fig:pred:chol:err} presents the prediction errors.  As one might expect,\n\\cref{fig:pred:chol:err:time} indicates that with increasing problem size, the\nmagnitude of the runtime prediction error increases for all summary\nstatistics---most notably for the maximum~(\\ref*{plt:max}).  Since in contrast\nthe performance prediction error~(\\cref{fig:pred:chol:err:perf}) is not affected\nby the decomposition's cubic runtime, we instead observe the largest prediction\nerrors for the smallest problem size~$n = 56$.  Furthermore, we find that the\nminimum performance prediction error~(\\ref*{plt:min}) seems to alternate between\ntwo separate levels: one around \\SI0{\\mega\\flops\\per\\second} and one close to\n\\SI{200}{\\mega\\flops\\per\\second}.  This behavior, which is also already somewhat\nvisible in \\cref{fig:pred:chol:perf:meas,fig:pred:chol:err:time}, is caused by\nmeasurement fluctuations as discussed in \\cref{sec:meas:fluct:longterm}.\n\nWe gain more insights from the prediction errors when we compare it to the\npredicted quantities.  For this purpose, the bottom half of\n\\cref{fig:pred:chol:err} presents the relative runtime and performance\nprediction errors.  These relative errors for these metrics are almost identical\nup to a change in the sign---since the runtime is generally slightly\nunderestimated, the performance is somewhat overestimated.  Focusing on the\nruntime in \\cref{fig:pred:chol:re:time}, we notice that the average standard\ndeviation ARE is~\\SI{194.70}\\percent~(\\ref*{plt:std}), which, as in\n\\cref{ex:pred:err}, exceeds the error of the other prediction statistics by far.\nFurthermore, the previously addressed measurement fluctuations are also clearly\nvisible in the maximum~(\\ref*{plt:max}) as variations with a magnitude\nof~\\SI{1.5}\\percent.  The minimum~(\\ref*{plt:min}), median~(\\ref*{plt:med}), and\nmean~(\\ref*{plt:avg}) AREs on the other hand quickly fall below~\\SI2{\\percent}\nfor matrices larger than~$n = 200$ and further below below~\\SI1{\\percent}\nbeyond~$n \\approx 1000$;  across all chosen problem sizes, the average AREs for\nthe minimum, median and mean runtime are, respectively,\n\\SIlist{.78;.91;.90}\\percent.\n\nAmong the eight metrics presented in\n\\cref{fig:pred:chol:time_perf,fig:pred:chol:err}, we gained the most insight\nfrom 1)~the performance prediction (\\cref{fig:pred:chol:perf:pred}), which gives\na good idea of both the algorithm's performance and efficiency, and 2)~the\nrelative runtime prediction error (\\cref{fig:pred:chol:re:time}), which provides\nnot only an accuracy measure independent of the operation, the algorithm, and\nthe actual performance, but also indicates whether the runtime is under- or\noverestimated.  Hence,  we use these two types of plots in our following\nanalyses.\n\n\n\\subsection{Varying Block Size}\n\\label{sec:pred:chol:b}\n\n\\input{pred\/figures\/cholnb}\n\nIn our next analysis, we fix the problem size to~$n = 3000$ and vary the block\nsize between~$b = 24$ and~536 in steps of~8.  \\Cref{fig:pred:chol:b} presents\nthe performance prediction and the relative runtime prediction error for this\nscenario using single-threaded \\openblas on the \\sandybridgeshort.\n\nThe performance prediction (\\cref{fig:pred:chol:b:perf}) exhibits the typical\ntrade-off for any blocked algorithm:  While for both small and large block sizes\nthe algorithm attains rather poor performance, in between it reaches up to\n\\SI{17.91}{\\giga\\flops\\per\\second}, which corresponds to an efficiency\nof~\\SI{85.10}\\percent.  The cause for this trade-off and the selection of block\nsizes are addressed in detail in \\cref{sec:pred:b}.\n\nCompared to our previous performance predictions\n(\\cref{fig:pred:chol:perf:pred}), \\cref{fig:pred:chol:b:perf} exhibits a far\nwider spread of the summary statistics for large block sizes.  In particular,\nthe predicted minimum performance~(\\ref*{plt:min}) drops drastically, which\nimmediately causes the mean performance~(\\ref*{plt:avg}) to decrease and an\nenormous increase in the predicted standard deviation~(\\ref*{plt:stdf}).\n\nThe relative runtime prediction error (\\cref{fig:pred:chol:b:re}) indicates that\nthe predicted performance fluctuations are not present in the performance\nmeasurements:  The maximum and mean relative errors (\\ref*{plt:max} and\n\\ref*{plt:avg}) increase drastically for large problem size, suggesting that the\nmodel generation was influenced by large outlier measurements.  (A repetition of\nthe generation process would likely encounter different outliers and distort\nthese metrics statistics for other problem sizes.)  The minimum~(\\ref*{plt:min})\nand median~(\\ref*{plt:med}), on the other hand, are with few exceptions\npredicted within~\\SI1\\percent; their average prediction AREs are\n\\SI{.36}{\\percent} (minimum \\ref*{plt:min}) and \\SI{.42}{\\percent} (median\n\\ref*{plt:med}).\n\n\n\\subsection{Varying Problem Size and Block Size}\n\\label{sec:pred:chol:nb}\n\n\\input{pred\/figures\/cholheatmap}\n\nIf we vary both the problem size~$n$ and the block size~$b$, we can visualize\nthe runtime prediction ARE as a set of heat-maps as shown in\n\\cref{fig:pred:chol:heatmap}.  Note that these plots are based on a total of\n\\num{39690}~measurements of the algorithm's runtime (65~problem sizes, $\\approx\n65$~block sizes, 10~repetitions) that took over 4~hours.  The performance models\nfor the kernels needed for the predictions (\\dpotf[L]2, \\dtrsm[RLTN], and\n\\dsyrk[LN]), on the other hand, were generated in just under 10~minutes,\nproduced our predictions in under \\SI{20}\\second.\n\nThe standard deviation ARE is once again too large to fit the chosen scale and\nis hence not shown.  Furthermore, as already seen in \\cref{fig:pred:chol:b}, the\nmaximum prediction becomes rather inaccurate for large~$n$ and~$b$, which also\nhas a negative impact on the mean prediction.  On the other hand, both the\nminimum and median predictions are overall quite accurate with an average ARE of\nonly~\\SI{.45}\\percent.\n\nSince in the following we compare multiple alternative algorithms and\nhardware\/software setups, we limit our focus to a single statistic.\nWhile in the previous analysis the runtime minimum or median were predicted with\nequivalent accuracy, in practice the expected performance is better represented\nby the median runtime.\\footnote{%\n    In scenarios other than our considered single-node computations different\n    measures might be preferable; e.g., the 90th~percentile runtime.\n}  Hence, from now on we use the \\definition[accuracy\nmeasure: relative median runtime prediction error]{relative median runtime\nprediction error}~\\Q t{med}{RE} as our {\\em prediction accuracy measure}.\n\n\n\\subsection{Other Data-Types}\n\\label{sec:pred:chol:dt}\n\n\\input{pred\/tables\/cholfp}\n\\input{pred\/figures\/cholfp}\n\nSo far, we have considered the Cholesky decomposition of real double-precision\nmatrices; however, the same algorithm is also applicable to other data-types.\nFor the four de-facto standard numerical data-types (real and complex\\footnote{%\n    For the complex cases, the Cholesky decomposition is of the form $L L^H\n    \\coloneqq A$, where $A$~must be Hermitian positive definite (HPD).\n} floating-point numbers in single- and double-precision)\n\\cref{tab:pred:chol:fp} summarizes the algorithm's \\blas and \\lapack kernels,\nand \\Cref{fig:pred:chol:fp} presents our model's performance predictions and\ntheir accuracy.  (For each data-type, we generated a separate set of performance\nmodels.)\n\nIn the performance predictions (\\cref{fig:pred:chol:fp:perf}), we observe that\nthe real double-precision version~(\\ref*{plt:dt:d}) is most efficient (with\nrespect to its theoretical peak performance); this was to be expected because\n\\openblas is most optimized for this data-type.  In contrast, it is somewhat\nsurprising that, while single-precision complex~(\\ref*{plt:dt:c}) is noticeably\nmore performant than single-precision real~(\\ref*{plt:dt:s}), double-precision\ncomplex~(\\ref*{plt:dt:z}) does not exceed an efficiency of~\\SI{50}\\percent.\n\nAlthough the algorithm's performance for the four data-types differs\nsignificantly, \\cref{fig:pred:chol:fp:perf} reveals that our models predict the\nruntime for all of them equally well.  Moreover, for the in comparison\ninefficient double-precision complex variant~(\\ref*{plt:dt:z}), the prediction\nis already notably accurate small problem sizes below~$n = 1000$.\n\nWith equally accurate predictions demonstrated for four data-types, we will in\nthe following focus on real operations in double-precision.\n\n\n\\subsection{Multi-Threaded \\blas}\n\\label{sec:pred:chol:mt}\n\n\\input{pred\/figures\/cholp}\n\nFinally, we consider how multi-threading (through \\openblas) impacts the\nalgorithm's performance and our predictions' accuracy.  For this purpose,\n\\cref{fig:pred:cholp} presents the predicted performance of the Cholesky\ndecomposition and the prediction accuracy with 1, 2, 4, and 8~threads on the\n8-core \\sandybridgeshort.  (For each of these four levels of parallelism, a\nseparate set of performance models was generated.)\n\nThe predictions show that, while the performance grows with the number of\nthreads, the efficiency decreases from~\\SI{87.74}{\\percent} with one thread to a\nmaximum of~\\SI{70.78}{\\percent} with eight threads.  Furthermore, the\nperformance curves become less smooth with increased parallelism.\n\nConsidering our prediction's accuracy, we notice that for small problem sizes\nbelow~$n = 500$, the prediction ARE increases significantly when more threads\nare added.  Beyond this point however, the prediction for 1~(\\ref*{plt:nt:1})\nand 2~threads~(\\ref*{plt:nt:2}) are both highly accurate with an average ARE\nof~\\SI{.46}{\\percent}; the predictions for 4~(\\ref*{plt:nt:4}) and\n8~threads~(\\ref*{plt:nt:8}) are slightly less accurate and the AREs fluctuate\naround~\\SI1\\percent.  Note that the large fluctuations within the ARE for the\nmulti-threaded algorithms are caused by the combination of the block size~$b =\n128$ and the chosen problem sizes in steps of~64.  While with\n8~threads~(\\ref*{plt:nt:8}) these fluctuations are represented by our\npredictions to some degree, with 2~(\\ref*{plt:nt:2}) and\n4~threads~(\\ref*{plt:nt:4}), they are most striking for large problem sizes,\nwhere our models do not predict such fluctuations.\n\n\n\\subsection{Summary}\n\\label{sec:pred:chol:sum}\n\nWe studied the blocked Cholesky decomposition algorithm~3 on a \\sandybridge\nusing \\openblas with varying problem and block sizes, data-types, and kernel\nparallelism.  We analyzed this algorithm's measured and predicted runtime and\nperformance to evaluate the accuracy of our predictions, and selected the\nrelative median runtime prediction error~\\Q t{med}{RE} as our primary accuracy\nmeasure.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Single-Threaded \\blas}\n\\label{sec:pred:acc:st}\n\nWe begin with a study of the single-threaded prediction accuracy with \\lapack's\ndefault block size ($b = 64$, except for \\dgeqrf with~$b = 32$).  While these\nare generally sub-optimal configurations and often even sub-optimal algorithms\nfor the performed operations, this configuration is unfortunately still\nencountered frequently in application codes that use the reference \\lapack\nimplementation.  As such, it forms a quite canonical reference for the\nevaluation of our predictions.\n\n\\input{pred\/figures\/accst}\n\\input{pred\/tables\/accst}\n\n\\Cref{fig:pred:lapack:st} presents the relative runtime prediction error~\\Q\nt{med}{RE} for this scenario.  For all algorithms and setups, our\npredictions are mostly within \\SI5{\\percent}~of the measured runtime, and in\nmany situations considerably closer.  The runtime prediction ARE averaged across\nall problem sizes for each routine and setup is summarized in\n\\cref{tbl:pred:acc:st}: It ranges from~\\SIrange{.71}{3.93}\\percent, and its\naverage and median are, respectively, \\SIlist{1.91;1.69}\\percent.  Overall, the\npredictions are slightly more accurate on the \\sandybridge (average $\\Q\nt{med}{ARE} = \\SI{1.66}\\percent$) with the lowest average $\\Q t{med}{ARE} =\n\\SI{1.22}\\percent$ for \\openblas~(\\ref*{plt:sbopen}); on the \\haswell (average\n$\\Q t{med}{ARE} = \\SI{2.16}\\percent$), the predictions are least accurate for\n\\mkl~(\\ref*{plt:hwmkl}) with an average of $\\Q t{med}{ARE} = \\SI{2.26}\\percent$.\n\nMost routines are predicted equally well (with an average \\Q t{med}{ARE} around\n\\SI{1.5}\\percent) with two exceptions: \\dsygst[1L] (average $\\Q t{med}{ARE} =\n\\SI{2.63}\\percent$) and \\dgeqrf (average $\\Q t{med}{ARE} = \\SI{2.87}\\percent$).\n\\begin{itemize}\n    \\item For the two-sided linear system solver \\dsygst,\n        \\cref{fig:pred:accst:dsygst} reveals that for most setups, the\n        predictions consistently underestimate the algorithm runtime for large\n        problem sizes~$n$.\n\n        A quick calculation shows that this effect is related to the size of the\n        last-level cache~(L3):  On the \\haswellshort, the problem emerges\n        beyond~$n \\approx 2000$ at which point the two operands~\\dm A (symmetric\n        in lower-triangular storage) and~\\dm[lower]L\\lowerpostsep take up\n        $\\SIvar{2 \\times \\frac{2000^2}2}\\doubles \\approx\n        \\SI{30.52}{\\mebi\\byte}$---slightly more than the L3~cache of\n        \\SI{30}{\\mebi\\byte}.  On the \\sandybridgeshort with \\SI{20}{\\mebi\\byte}\n        of L3~cache, the effect is accordingly already visible beyond~$n \\approx\n        1600$.\n\n        The cause for the underestimation of large problems is as follows:  Our\n        models are based on repeated kernel measurements, which operate on\n        cached (``warm'') data as long as all of the kernel's arguments fit in\n        the cache.  However, each traversal step of \\dsygst[1L]\n        (\\cref{alg:dsygst}) uses two separate kernels (namely \\dsyrtk[LN] and\n        \\dtrsm[LLNN]) that operate on the trailing parts of \\dm A and\n        \\dm[lower]L\\lowerpostsep{}---since these do not fit in the cache\n        simultaneously, they are mutually evicted by these kernels, and hence\n        have to be loaded from main memory repeatedly (``cold'' data).  To\n        summarize, our models estimate fast operations on cached data, while in\n        the algorithm the operations are slower due to cache misses.\n\n        A more detailed study of caching effects within blocked algorithms and\n        attempts to account for them are presented in \\cref{ch:cache}.\n\n        Note that only \\dsygst is affected by caching effects on this scale\n        because all other routines involve only one dense operand.\n\n    \\item For the QR~decomposition \\dgeqrf, \\cref{fig:pred:accst:dgeqrf} reports\n        that the runtime for almost all setups is consistently\n        underestimated---especially for small problems.\n\n        The cause is the transposed matrix copy and addition (see\n        \\cref{alg:dgeqrf}), which account for about~\\SI4{\\percent} of the\n        runtime for small problems ($n \\approx 250$) and \\SI1{\\percent} for\n        large problems ($n \\approx 4000$):  The copy, performed by a sequence of\n        $b = 32$~\\dcopy{}s, is underestimated by~$2\\times$ to~$7\\times$ because\n        our models do not account for caching effects; the addition, which\n        inlined as two nested loops, is not accounted for at all.\n\\end{itemize}\n\n\n\\subsection{Multi-Threaded \\blas}\n\\label{sec:pred:acc:mt}\n\nWe study the multi-threaded prediction accuracy for the same six \\lapack\nalgorithms using all available cores of the processors, i.e., 8~threads on the\n\\sandybridge and 12~threads on the \\haswell.  In contrast to the single-threaded\npredictions, we use a block size of~$b = 128$ for all algorithms---while this\nconfiguration is certainly not optimal for all algorithms and problem sizes, it\ngenerally yields better performance than \\lapack's default values.\n\n\\input{pred\/figures\/accmt}\n\\input{pred\/tables\/accmt}\n\n\\Cref{fig:pred:lapack:mt} presents the relative runtime prediction errors~\\Q\nt{med}{RE} for this scenario, and \\cref{tbl:pred:acc:mt} summarizes their\naveraged AREs~\\Q t{med}{ARE}.  Compared to the single-threaded case, the\nprediction errors are across the board around $2.5\\times$~larger with a total\naverage of $\\Q t{med}{ARE} = \\SI{4.85}\\percent$.  The predictions are roughly\nequally accurate across the two architectures and the two \\blas implementations.\n\nConsidering \\cref{fig:pred:lapack:mt}, we note fluctuation patterns in the\nprediction errors by up to~\\SI{10}\\percent, most notably for \\dsygst[1L] and\n\\dtrtri[LN] using \\mkl on the \\haswellshort~(\\ref*{plt:hwmkl}).  As observed in\n\\cref{sec:pred:chol:mt}, these fluctuations are an artefact of the block size~$b\n= 128$ interacting with the considered problem sizes in steps of~64:  Between\nconsecutive problem sizes, the remaining matrix portions in the last step of the\nmatrix traversal alternate between widths~56 and~120.\n\nAs in the single-threaded case, the QR~decomposition's runtime is\nunderestimated by on average~\\SI{8.00}\\percent, due to the \\dcopy{}s and the\ninlined matrix addition.  Since especially the latter cannot make any use of the\nmulti-threaded parallelism, their impact increases significantly with the number\nof available cores.\n\nFurthermore, several individual algorithms and setups are consistently under- or\noverestimated:  e.g., \\openblas on the \\sandybridge~(\\ref*{plt:sbopen}) for\n\\dlauum[L] and \\dpotrf[L].  These problems arise from the multi-threaded\nimplementations of \\dgemm, whose irregular performance is not well represented\nin our models:  Since \\blas implementations distribute computations among\nthreads along a certain dimension of the operation, for small dimension (such as\nthe block size), only a subset of the available threads is used.  When the small\ndimension is increased, more threads are activated and the performance increases\nsuddenly.\n\n\n\\subsection{Summary}\n\\label{sec:pred:acc:sum}\n\nThis section has shown that across experiments on two processor architectures,\nthree \\blas implementations, and six blocked \\lapack algorithms, our models\nyield accurate predictions that are on average within~\\SI{1.91}{\\percent}\n(single-threaded) and \\SI{4.85}{\\percent} (multi-threaded) of reference\nmeasurements.  Encouraged by these accuracy results, the following sections use\nperformance predictions to target our main goals of algorithm selection and\nblock-size optimization.\n\n\\section{Performance Prediction}\n    \\label{sec:pred:pred}\n    \\input{pred\/pred}\n\n    \\section{Accuracy Quantification}\n    \\label{sec:pred:acc}\n    \\input{pred\/acc}\n\n    \\section[Accuracy Case Study: Cholesky Decomposition]\n            {Accuracy Case Study:\\newline Cholesky Decomposition}\n    \\label{sec:pred:chol}\n    \\input{pred\/chol}\n\n    \\section[Accuracy Study: Blocked \\lapack Algorithms]\n            {Accuracy Study:\\newline Blocked \\lapack Algorithms}\n    \\label{sec:pred:lapack}\n    \\input{pred\/lapack}\n\n    \\section{Algorithm Selection}\n    \\label{sec:pred:var}\n    \\input{pred\/var}\n\n        \\subsection{Cholesky Decomposition}\n        \\label{sec:pred:var:chol}\n        \\input{pred\/varchol}\n\n        \\subsection{Triangular Inversion}\n        \\label{sec:pred:var:trinv}\n        \\input{pred\/vartrinv}\n\n        \\subsection{Sylvester Equation Solver}\n        \\label{sec:pred:var:sylv}\n        \\input{pred\/varsylv}\n\n        \\subsection{Summary}\n        \\label{sec:pred:var:sum}\n        \\input{pred\/varsum}\n\n    \\section{Block Size Optimization}\n    \\label{sec:pred:b}\n    \\input{pred\/b}\n\n        \\subsection{Cholesky Decomposition}\n        \\label{sec:pred:b:chol}\n        \\input{pred\/bchol}\n\n        \\subsection{Triangular Inversion}\n        \\label{sec:pred:b:trinv}\n        \\input{pred\/btrinv}\n\n        \\subsection{\\lapack Algorithms}\n        \\label{sec:pred:b:lapack}\n        \\input{pred\/blapack}\n\n    \\section{Summary}\n    \\label{sec:pred:conclusion}\n    \\input{pred\/conclusion}\n}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsubsection{Algorithms}\nThe solution to the triangular Sylvester equation is computed by traversing \\dmC\nfrom the bottom left to the top right.  However, in contrast to the previous\noperations, this traversal does not need to follow \\dmC's diagonal; in fact \\dmC\ncan be traversed in various different ways:  Two algorithms traverse \\dmC\nvertically, two horizontally (using $3 \\times 1$ and $1 \\times 3$ partitions),\nand 14~diagonally (exposing $3 \\times 3$ sub-matrices), making a total of\n18~algorithms.  Furthermore, as detailed in the following, the Sylvester\nequation requires two layers of blocked algorithms, resulting in a total of\n\\definition[Sylvester equation:\\\\64~``complete'' algorithms]{64~``complete''\nalgorithms}.\n\n\\input{pred\/figures\/sylv1dalgs}\n\n\\Cref{algs:sylv1d} presents the four algorithms that traverse \\dmC vertically or\nhorizontally, thereby exposing $3 \\times 1$ or $1 \\times 3$ sub-matrices; each\nof these algorithms consists of one call to \\dgemm[NN] and the solution of a\nsub-problem (another triangular Sylvester equation).  To obtain a ``complete''\nalgorithm, two of these algorithms with orthogonal traversals are combined---the\nfirst traverses the full~\\dmC and invokes the second to solve sub-problem in\neach iteration; the second, in turn, solves its small $b \\times b$ sub-problem\nusing \\lapack's unblocked \\dtrsyl[NN1].  E.g., one can use algorithm~$m1$ to\ntraverse \\dmC vertically and in each step apply algorithm~$n2$ to traverse the\nmiddle panel~\\dm[mat11, height=.2, width=.8]{C_1} horizontally.  We call the\nresulting ``complete'' algorithm~$m1n2$, and see that eight such combinations\nare possible: $m1n1$, $m1n2$, $m2n1$, $m2n2$, $n1m1$, $n1m2$, $n2m1$,\nand~$n2m2$.  Note that in principle the block sizes for the two layered blocked\nalgorithms can be chosen independently; however, we limit our study to a single\nblock size for both layers.\n\n\\input{pred\/figures\/sylv2dalgs}\n\nBeyond the combination of the vertically and horizontally traversing algorithms\nabove, an additional 14~algorithms traverse the matrix diagonally (with\npotentially different block sizes~$b_m$ and~$b_n$ for dimensions~$m$ and~$n$),\nand operate on a set of $3 \\times 3$ sub-matrices in each iteration;\n\\cref{algs:sylv2d} presents a sample of two of these algorithms (all\n14~algorithms are found in \\libflame~\\cite{libflameweb}).  Each algorithm\nconsists of a sequence of \\dgemm[NN]{}s and three solutions of sub-problems that\nare also triangular Sylvester equations.  While the sub-problem involving\n\\dm[mat11, size=.5]{B_{11}} of size $b_m \\times b_n$ is directly solved by the\nunblocked \\dtrsyl[NN1], the other two involve potentially large yet thin panels\nof~\\dmC.  Complete algorithms are constructed by solving each of these sub\nproblems with an appropriate vertical or horizontal traversal\nalgorithm.\\footnote{%\n    Setting one of the block sizes of a diagonally traversing algorithm to the\n    corresponding matrix size results in one of the vertical or horizontal\n    traversal algorithms.\n}  Since each of the 14~algorithms has\ntwo such sub-problems, for each of which we can choose from two algorithms, we\nend up with a total of $14 \\cdot 2 \\cdot 2 = 56$~possible combinations.\nTogether with the eight combinations of only vertical and horizontal traversal\nalgorithms, this results in a grand total of 64~different ``complete'' blocked\nalgorithms.\n\n\n\\subsubsection{Algorithm Selection}\n\n\\input{pred\/figures\/varsylv}\n\n\\cref{fig:pred:var:sylv} presents performance predictions and measurements for\nthe Sylvester equation solver for problem sizes between~$n = 56$ and~4152 in\nsteps of~64 and block size~$b = 64$ on a \\haswell using \\openblas.  Since the\nexecutions for this setup take between 40~minutes and 2~hours for each\nalgorithm, we only measured the eight algorithms based exclusively on orthogonal\nmatrix traversals.  Our predictions, which are generated up to\n$1500\\times$~faster at roughly \\SI5{\\second}~per algorithm, indicate that in\nterms of performance these eight algorithms are evenly spread across the entire\n64~``complete'' algorithms.\n\nFor the single-threaded scenario, the predictions in\n\\cref{fig:pred:var:sylv:pred:1} suggest that\nalgorithms~$n2m2$~(\\ref*{plt:sylvn2m2}) and $m1n1$~(\\ref*{plt:sylvm1n1}) are,\nrespectively, the fastest and slowest, and differ in performance\nby~\\SI{9.99}\\percent.  The measurements in \\cref{fig:pred:var:sylv:meas:1}\nconfirm that, while algorithm~$n2m2$~(\\ref*{plt:sylvn2m2}) is indeed the\nfastest, algorithm~~$n1m1$~(\\ref*{plt:sylvn1m1}) is the slowest.  While the\nperformance of algorithms~$m1n1$~(\\ref*{plt:sylvm1n1}) and\n$n1m1$~(\\ref*{plt:sylvn1m2}) is predicted to be almost identical, the\nmeasurements show that $m1n1$~(\\ref*{plt:sylvm1n1}) is in fact up to\n\\SI{3.00}{\\percent} faster than $n1m1$~(\\ref*{plt:sylvn1m2}).  Furthermore,\nwhile the remaining algorithms are correctly placed between the fastest and the\nslowest, they are not accurately ranked.\n\nThe predictions and measurements for the multi-threaded scenario in\n\\cref{fig:pred:var:sylv:pred:12,fig:pred:var:sylv:meas:12} are at first sight\nsurprising:  Compared to the  single-threaded case the attained performance is\nconsiderably lower.  For matrices of size~$n = 4000$, the algorithms reach\nroughly \\SI8{\\giga\\flops\\per\\second}, which corresponds to\nmerely~\\SI{1.67}{\\percent} of the processor's 12-core peak performance of\n\\SI{480}{\\giga\\flops\\per\\second} (without \\turboboost).  An analysis revealed\nthat the source of the drastic increase in runtime is the \\blasl1 kernel \\dswap,\nwhich the unblocked \\dtrsyl\\footnote{%\n    Technically within \\code{dlasy2}, which is called from \\dtrsyl.\n} uses to swap two vectors of length~4:  Although the workload for this\noperation is tiny, with multiple threads \\openblas (version~0.2.15) activates\nits parallelisation, which for a copy operation on only~\\SI{64}{\\bytes}\nintroduces a overhead of over~$200\\times$ the kernel's single-threaded runtime.\n(The problem was subsequently fixed in \\openblas version~0.2.16 (March 2016) and\nis not present in \\mkl.)\n\nWhile the multi-threaded predictions for all 64~algorithms indicate virtually\nidentical performance and thus do not allow a meaningful performance ranking,\nthey support the crucial revelation that using \\openblas~0.2.15 the triangular\nSylvester equation is solved considerably faster on a single core than on\n12~cores without exception.\n}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Algorithm Generation}\n    \\label{sec:tensor:alggen}\n    \\input{tensor\/alggen}\n\n    \\section{Runtime Prediction}\n    \\label{sec:tensor:pred}\n    \\input{tensor\/pred}\n\n        \\subsection{Example Contraction: \\texorpdfstring{$C_{abc} \\coloneqq\n        A_{ai} B_{ibc}$}{C\\_abc := A\\_ai B\\_ibc}}\n        \\label{sec:tensor:extc}\n        \\input{tensor\/predex}\n\n        \\subsection{Repeated Execution}\n        \\label{sec:tensor:repeat}\n        \\input{tensor\/repeat}\n\n        \\subsection{Operand Access Distance}\n        \\label{sec:tensor:accdist}\n        \\input{tensor\/accdist}\n\n        \\subsection{Cache Prefetching}\n        \\label{sec:tensor:prefetch}\n        \\input{tensor\/prefetch}\n\n        \\subsection{Prefetching Failures}\n        \\label{sec:tensor:prefetchfail}\n        \\input{tensor\/prefetchfail}\n\n        \\subsection{First Loop Iterations}\n        \\label{sec:tensor:firstiter}\n        \\input{tensor\/firstiter}\n\n    \\section{Results}\n    \\label{sec:tensor:results}\n    \\input{tensor\/results}\n\n    \\section{Summary}\n    \\label{sec:tensor:conclusion}\n    \\input{tensor\/conclusion}\n}\n\n\n\n\n\n\n\\subsection{Changing the Setup for \\texorpdfstring{$C_{abc} \\coloneqq A_{ai}\nB_{ibc}$}{C\\_abc := A\\_ai B\\_ibc}}\n\\label{sec:ai_ibc2}\n\n\\input{tensor\/figures\/ai_ibc2}\n\nWe consider the previously studied contraction with an entirely different setup:\nWe use $a = b = c = 128$ and $i = 8, \\ldots, 1000$ in steps of~8 on an\n\\ivybridge with single-threaded \\mkl.  For this scenario,\n\\cref{fig:tensor:ai_ibc2} presents the performance predictions and measurements\nfor all 36~algorithms (see \\cref{sec:tensor:extc}).  Although everything,\nranging from the problem sizes to the machine and \\blas library was changed in\nthis setup, the predictions are of equivalent quality and our tool correctly\ndetermines that the \\dgemm-based algorithms (\\ref*{plt:ai_ibc:c_gemm}),\n\\ref*{plt:ai_ibc:b_gemm}) not only perform best and equally well but also reach\nover~\\SI{75}{\\percent} of the \\ivybridgeshort's theoretical peak performance of\n\\SI{28.8}{\\giga\\flops\\per\\second}.\n\n\n\\subsection{Vector Contraction: \\texorpdfstring{$C_a \\coloneqq A_{iaj}\nB_{ji}$}{C\\_a := A\\_iaj B\\_ji}}\n\\label{sec:noblas3}\n\n\\input{tensor\/algs\/iaj_ji}\n\\input{tensor\/figures\/iaj_ji}\n\nFor certain contractions (e.g., those involving vectors), \\dgemm cannot be\nused as a compute kernel, and algorithms can only be based on \\blasl1 or~2\nkernels.  One such scenario is encountered in the contraction $C_a \\coloneqq\nA_{iaj} B_{ji}$, for which our generator yields 8~algorithms:\n\\begin{itemize}\n    \\item 4 \\ddot-based:\n        \\tensoralgname{aj}{dot}~(\\ref*{plt:iaj_ji:aj_dot}),\n        \\tensoralgname{ja}{dot}~(\\ref*{plt:iaj_ji:ja_dot}),\\\\\n        \\tensoralgname{ai}{dot}~(\\ref*{plt:iaj_ji:ai_dot}),\n        \\tensoralgname{ia}{dot}~(\\ref*{plt:iaj_ji:ia_dot});\n    \\item 2 \\daxpy-based:\n        \\tensoralgname{ij}{axpy}~(\\ref*{plt:iaj_ji:ij_axpy}),\n        \\tensoralgname{ji}{axpy}~(\\ref*{plt:iaj_ji:ji_axpy}), and\n    \\item 2 \\dgemv-based (see \\cref{algs:iaj_ji}):\n        \\tensoralgname j{gemv}~(\\ref*{plt:iaj_ji:j_gemv}),\n        \\tensoralgname{i'}{gemv}~(\\ref*{plt:iaj_ji:i'_gemv}).\n\\end{itemize}\nNote that since last algorithm operates on slices \\tind A{i,:,:}, which do not\nhave contiguously-stored dimension, a \\code{copy} kernel (indicated by the\napostrophe in the algorithm name) is required before each \\dgemv[N]\n(\\cref{alg:iaj_ji:i'-gemv}).\n\n\\Cref{fig:tensor:iaj_ji} presents the predicted and measured performance for\nthese algorithms.  Our predictions clearly identify the fastest algorithm\n\\tensoralgname j{gemv}~(\\ref*{plt:iaj_ji:j_gemv}) across the board.\nFurthermore, the next group of four algorithms is also correctly recognized, and\nthe low performance of the second \\dgemv[N]-based algorithm\n\\tensoralgname{i'}{gemv}~(\\ref*{plt:iaj_ji:i'_gemv}) (due to the overhead of the\ninvolved copy operation) is correctly predicted as well.\n\n\n\\subsection{Challenging Contraction: \\texorpdfstring{$C_{abc} \\coloneqq A_{ija}\nB_{jbic}$}{C\\_abc := A\\_ija B\\_jbic}}\n\\label{sec:ijb_jcid}\n\n\\input{tensor\/algs\/ijb_jcid}\n\nWe now turn to a more complex example inspired by space-time continuum\ncomputations in the field general relativity~\\cite{generalrelativity}: $C_{abc}\n\\coloneqq A_{ija} B_{jbic}$.  For this contraction, we generated a total of\n176~different algorithms:\n\\begin{itemize}\n    \\item 48 \\ddot-based~(\\ref*{plt:ijb_jcid:dot}),\n    \\item 72 \\daxpy-based~(\\ref*{plt:ijb_jcid:axpy}),\n    \\item 36 \\dgemv-based~(\\ref*{plt:ijb_jcid:gemv}),\n    \\item 12 \\dger-based~(\\ref*{plt:ijb_jcid:ger}), and\n    \\item 8 \\dgemm-based:\\\\\n        \\tensoralgname{cj'}{gemm}~(\\ref*{plt:ijb_jcid:cj'_gemm}),\n        \\tensoralgname{jc'}{gemm}~(\\ref*{plt:ijb_jcid:jc'_gemm}),\n        \\tensoralgname{ci'}{gemm}~(\\ref*{plt:ijb_jcid:ci'_gemm}),\n        \\tensoralgname{i'c}{gemm}~(\\ref*{plt:ijb_jcid:i'c_gemm}),\\\\\n        \\tensoralgname{bj'}{gemm}~(\\ref*{plt:ijb_jcid:bj'_gemm}),\n        \\tensoralgname{jb'}{gemm}~(\\ref*{plt:ijb_jcid:jb'_gemm}),\n        \\tensoralgname{bi'}{gemm}~(\\ref*{plt:ijb_jcid:bi'_gemm}),\n        \\tensoralgname{i'b}{gemm}~(\\ref*{plt:ijb_jcid:i'b_gemm}).\n\\end{itemize}\nAll \\dgemm-based (see \\cref{algs:ijb_jcid}) and several of the \\dgemv-based\nalgorithms involve copy operations to ensure that each matrix has a\ncontiguously-stored dimension as required by the \\blas interface.  Once again,\nwe consider a challenging scenario where both contracted indices are of size $i\n= j = 8$ and the free indices $a = b = c$ vary between~8 and~1000.\n\n\\input{tensor\/figures\/ijb_jcid}\n\n\\Cref{fig:tensor:ijb_jcid:pred} presents the predicted performance of the\n176~algorithms, where algorithms based on \\blasl1 and~2 are grouped by kernel.\nEven with the copy operations, the \\dgemm-based algorithms are the fastest.\nHowever, within these 8~algorithms, the performance differs by more\nthan~\\SI{20}\\percent.  \\Cref{fig:tensor:ijb_jcid:meas} compares our predictions\nwith corresponding performance measurements\\footnote{%\n    Slow tensor contraction algorithms were stopped before reaching the largest\n    problem size by limiting the total measurement time per algorithm\n    to~\\SI{15}\\min.\n}:  Among the \\dgemm-based algorithms, our predictions clearly separate the bulk\nof fast algorithms from the slightly less efficient ones.\n\n\\input{tensor\/figures\/ijb_jcid10}\n\n\\paragraph{Multi-Threading}\nOur contraction algorithms can profit from shared memory parallelism through\nmulti-threaded \\blas kernels.  To focus on the impact of parallelism, we\nincrease the contracted tensor dimension sizes to~$i = j = 32$ and use all\n10~cores of the \\ivybridge with multi-threaded \\openblas.\n\\Cref{fig:tensor:ijb_jcid10} presents performance predictions and measurements\nfor this setup:  Our predictions accurately distinguish the three groups of\n\\dgemm-based implementations, and algorithms\n\\tensoralgname{i'c}{gemm}~(\\ref*{plt:ijb_jcid:i'c_gemm}) and\n\\tensoralgname{i'b}{gemm}~(\\ref*{plt:ijb_jcid:i'b_gemm}) (see\n\\cref{algs:ijb_jcid}), which reach \\SI{170}{\\giga\\flops\\per\\second}, are\ncorrectly identified as the fastest.\n\\tensoralgname{jb'}{gemm}~(\\ref*{plt:ijb_jcid:jb'_gemm}) on the other hand\nmerely reaches \\SI{60}{\\giga\\flops\\per\\second}.  This $3\\times$~difference in\nperformance among \\dgemm-based algorithms emphasizes the importance of selecting\nthe right algorithm.\n\n\n\\subsection{Efficiency Study}\n\n\\input{tensor\/figures\/eff}\n\nThe above study provided evidence that our automated approach successfully\nidentifies the most efficient algorithm(s).  In the following we show how much\nfaster this approach is compared to empirical measurements.  For this purpose, we\nonce more consider the contraction $C_{abc} \\coloneqq A_{ai} B_{ibc}$ with $i =\n8$ and varying $a = b = c$ on a \\harpertown with \\openblas.\n\\Cref{fig:tensor:eff} presents the speedup of our micro-benchmark over\ncorresponding algorithm measurements:  Generally our predictions are several\norders of magnitude faster than such algorithm executions.  For $a = b = c =\n1000$, this relative improvement is smallest for the \\dgemm-based\nalgorithms~(\\ref*{plt:eff:gemm}) at $1000\\times$, because each \\dgemm performs a\nsignificant portion of the computation; for the \\dger-based\nalgorithms~(\\ref*{plt:eff:ger}), it lies between 6000 and \\num{10000} and for\nthe \\dgemv-based algorithms~(\\ref*{plt:eff:gemv}) the gain is $\\num{5e5}\\times$\nto $\\num{e6}\\times$; finally, for the \\blasl1-based\nalgorithms~(\\ref*{plt:eff:axpy}, \\ref*{plt:eff:dot}), where each kernel\ninvocation only performs a tiny fraction of the contraction, our predictions are\n\\num{1e6} to \\num{1e9}~times faster than the algorithm executions.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Affine and non-affine deformation}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.9\\linewidth]{figure1.pdf}\n\\caption{(a) The deformation of infinitesimal spherical fluid elements after $\\Delta t=3\\tau_\\eta$ (size not to scale). (b) The time evolution of the mean curvature $\\langle\\kappa_1\\rangle$ (black solid curve); The black dashed line represents a linear relationship, and the cyan dashed line represents the prediction based on an Eulerian quantity $\\langle|\\boldsymbol{\\hat{e}}_1\\cdot\\boldsymbol{H}\\cdot\\boldsymbol{\\hat{e}}_1|\\rangle$ with $\\boldsymbol{\\hat{e}}_1$ being the eigenvector corresponding to the maximum eigenvalue of the rate-of-strain tensor. Inset: the same figure for $\\langle\\kappa_1\\rangle$ (black solid line) with a linear scale in time. The black dashed line represents an exponential growth over time.\\label{fig_element_deformation}}\n\\end{figure}\n\n\nTo build a framework that makes that connection, we consider the folding of infinitesimal fluid elements. Fig. \\ref{fig_element_deformation}(a) shows a number of infinitesimal spherical fluid elements being deformed after a time $3\\tau_\\eta$ ($\\tau_\\eta$ is the Kolmogorov time scale) in 3D homogeneous and isotropic turbulence \\cite{li2008public,perlman2007data} (details of the direct numerical simulation (DNS) of the turbulence can be found in Supplemental Material). It is clear that the deformed fluid elements show complex geometry involving both stretching and folding. To mathematically describe this high-order deformation, we consider each point $\\boldsymbol{X}$ at $t_0$ within an infinitesimal fluid element mapped to another point $\\boldsymbol{x}$ within the deformed element after a finite time $\\Delta t$, where $\\boldsymbol{x}$ and $\\boldsymbol{X}$ are the relative positions with respect to the center of the fluid elements. The non-linear mapping function between $\\boldsymbol{X}$ and $\\boldsymbol{x}$ with the leading orders follows\n\\begin{equation}\\label{eqn_mapping}\n\\boldsymbol{x}=\\boldsymbol{F}(t_0+\\Delta t)\\cdot\\boldsymbol{X}+\\boldsymbol{X}\\cdot\\boldsymbol{G}(t_0+\\Delta t)\\cdot\\boldsymbol{X},\n\\end{equation}\nwhere $F_{ij}=\\partial x_i\/\\partial X_j$ is the deformation gradient tensor and $G_{ijk}=\\partial^2 x_i\/\\partial X_j\\partial X_k$ is the deformation Hessian tensor. The tensors $F_{ij}$ and $G_{ijk}$ can be then determined by integrating  $dF_{ij}(t)\/dt=A_{im}F_{mj}(t)$ and  $dG_{ijk}(t)\/dt=A_{im}G_{mjk}(t)+H_{imn}F_{mj}(t)F_{nk}(t)\/2$ along the trajectories of fluid elements, with $A_{ij}=\\partial u_i\/\\partial x_j$ and $H_{ijk}=\\partial^2 u_i\/\\partial x_j \\partial x_k$ being the velocity gradient and velocity Hessian tensors, respectively. Details of these equations can be found in Supplemental Material. \n\nTo further simplify Eq. (\\ref{eqn_mapping}), we consider the deformation of an arbitrary straight material line passing through the center of a fluid element, represented by a set of positions $\\boldsymbol{X}$ represented parametrically according to $\\boldsymbol{X}(\\lambda)=\\boldsymbol{\\hat{e}}\\lambda$.  $\\boldsymbol{\\hat{e}}$ is a selected unit vector and the parameter $\\lambda\\rightarrow0$ indicates the distance from the center of the fluid element. Substituting $\\boldsymbol{X}(\\lambda)=\\boldsymbol{\\hat{e}}\\lambda$ into Eq. (\\ref{eqn_mapping}) yields the expression for the deformed material line at $t_0+\\Delta t$, \n\\begin{equation}\\label{eqn_mapping_expand}\n    \\boldsymbol{x}(\\lambda)=\\boldsymbol{F}\\cdot \\boldsymbol{\\hat{e}} \\lambda+\\boldsymbol{\\hat{e}}\\cdot\\boldsymbol{G}\\cdot\\boldsymbol{\\hat{e}}\\lambda^2= \\boldsymbol{r}^s \\lambda+\\boldsymbol{r}^b \\lambda^2,\n\\end{equation}\nwhere $\\boldsymbol{r}^s=\\boldsymbol{F}\\cdot \\boldsymbol{\\hat{e}}$ and  $\\boldsymbol{r}^b=\\boldsymbol{\\hat{e}}\\cdot\\boldsymbol{G}\\cdot\\boldsymbol{\\hat{e}}$ are defined as the stretching vector and the bending vector, respectively. \n\n\n\n\nA highly relevant material line is the one that gets stretched the most, written as $\\boldsymbol{X}(\\lambda)=\\boldsymbol{\\hat{e}}_{R1}\\lambda$. Here, $\\boldsymbol{\\hat{e}}_{R1}$ is the unit eigenvector associated with the greatest eigenvalue of right Cauchy-Green strain tensor $\\boldsymbol{C}^R=\\boldsymbol{F}^T\\boldsymbol{F}$. This special material line, as the \"skeleton\" of the fluid element, can be used to reflect the overall geometry of the fluid element. Substituting $\\boldsymbol{\\hat{e}}=\\boldsymbol{\\hat{e}}_{R1}$ in Eq. (\\ref{eqn_mapping_expand}) results in the quadratic equation $\\boldsymbol{x}(\\lambda)= \\boldsymbol{r}^s_1 \\lambda+\\boldsymbol{r}^b_1 \\lambda^2$ where $\\boldsymbol{r}^s_1=\\boldsymbol{F}\\cdot\\boldsymbol{\\hat{e}}_{R1}$ and $\\boldsymbol{r}^b_1=\\boldsymbol{\\hat{e}}_{R1}\\cdot\\boldsymbol{G}\\cdot\\boldsymbol{\\hat{e}}_{R1}$. An example of this material line is shown as the inset of Fig.  \\ref{fig_element_deformation}(a) (black dashed line). Given this quadratic equation, the curvature of the material line $\\kappa_1$ can be found using $\\kappa_1=2r^b_{1\\perp}\/(r_1^s)^2$, where $\\boldsymbol{r}^b_{1\\perp}$ represents the component of $\\boldsymbol{r}^b_1$ that is perpendicular to $\\boldsymbol{r}^s_1$. Although $\\kappa_1$ is not sufficient to describe the complete deformation, it does reflect the overall folding of the fluid element. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nThe curvature $\\kappa_1$ can therefore be obtained by computing $\\boldsymbol{F}$ and $\\boldsymbol{G}$ and their associated $\\boldsymbol{r}^b_1$ and $\\boldsymbol{r}^s_1$ along with each fluid trajectory. Fig. \\ref{fig_element_deformation}(b) shows the time evolution of the mean curvature $\\langle\\kappa_1\\rangle$, averaged over $10^5$ fluid elements, as a function of the integration time $\\Delta t$ using the DNS data. It is evident that, for the available simulation duration, the mean curvature of the fluid elements grows continuously, but the growth rate changes appreciably between two regimes. In early times, $\\langle\\kappa_1\\rangle$ increases linearly. The linear regime lasts until about the Kolmogorov timescale $\\tau_\\eta$ when the length scale $1\/\\langle\\kappa_1\\rangle$ is around 25$\\eta$ ($\\eta$ is the Kolmogorov length scale), and the growth of $\\langle\\kappa_1\\rangle$ slows down, marking the transition of the curvature dynamics. Soon after $\\tau_\\eta$, the growth of  $\\langle\\kappa_1\\rangle$ accelerates again, and this late stage behavior is better fitted with an exponential function, which is illustrated in a semi-logarithmic plot in the inset of Fig. \\ref{fig_element_deformation}(b). \n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.9\\linewidth]{curv_pdf.pdf}\n    \\caption{(a) The PDFs of the curvature $p(\\kappa_1)$ at different time instants in the early stage. Inset of (a): the same PDFs but for the normalized curvature $p(\\kappa_1\/\\langle\\kappa_1\\rangle)$. (b) The PDFs of the curvature $p(\\kappa_1)$ at different time instants in the late stage with the solid curves representing the data and the dashed curves representing the prediction by the model (Eq. (\\ref{eqn_pdf_evolution})). Inset of (b): the time evolution of the kurtosis of $\\kappa_1$. }\n    \\label{fig_curv_pdf}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\nThe transition from the linear to the exponential growth of $\\langle \\kappa_1 \\rangle$ indicates different mechanisms at play, which can be better understood using local curvature. Here, the probability density function (PDF) of $\\kappa_1$, i.e. $p(\\kappa_1)$, at different times  are shown in Fig. \\ref{fig_curv_pdf} for the early (a) and late (b) stages. In the early stage, the curvature grows systematically, but follows a self-similar behavior as indicated by the collapsed PDFs of the normalized curvature $p(\\kappa_1\/\\langle \\kappa_1 \\rangle)$ in the inset of Fig. \\ref{fig_curv_pdf}(a). In the late stage, the tail of the PDF still rises over time, whereas the peak location remains constant. This distinct behavior suggests that the curvature distribution becomes more intermittent over time, which is confirmed by the growing kurtosis as shown in Fig. \\ref{fig_curv_pdf}(b) inset. This result highlights the growing inhomogeneity of local mixing as locations with extreme curvature should reach a well-mixed stage much sooner than what is implied by the mean.\n\n\n\n\n\n\n\n\n\nTo model the multi-stage growth behavior of curvature, we consider an arbitrary deforming infinitesimal material line as in Eq. (\\ref{eqn_mapping_expand}). The equation for this material line can therefore be decomposed along two directions, $\\boldsymbol{\\hat{e}}_\\parallel=\\boldsymbol{r}^s\/r^s$ and $\\boldsymbol{\\hat{e}}_\\perp=\\boldsymbol{r}^b_\\perp\/r^b_\\perp$ , following:\n\\begin{equation}\\label{eqn_geometry}\n    \\boldsymbol{x}(\\lambda)=\\left(r^s\\lambda+r^b_\\parallel\\lambda^2\\right)\\boldsymbol{\\hat{e}}_\\parallel+r^b_\\perp\\lambda^2\\boldsymbol{\\hat{e}}_\\perp,\n\\end{equation}\nwhere $\\boldsymbol{r}^b_\\parallel=(\\boldsymbol{r}^b\\cdot\\boldsymbol{\\hat{e}}_\\parallel)\\boldsymbol{\\hat{e}}_\\parallel$ and $\\boldsymbol{r}^b_\\perp=\\boldsymbol{r}^b-\\boldsymbol{r}^b_\\parallel$.\n\nThe velocity of any arbitrary material point on the material line, $\\boldsymbol{u}(\\lambda)$, can then be expressed in the frame spanned by ($\\boldsymbol{\\hat{e}}_\\parallel$, $\\boldsymbol{\\hat{e}}_\\perp)$ in two different ways by taking either direct time derivative of Eq. (\\ref{eqn_geometry}) or the Taylor expansion based on the velocity information (see Supplemental Material). Comparing these two expressions for $\\boldsymbol{u}(\\lambda)$ leads to evolution equations for $r^s$ and $r^b_\\perp$, which then yields the evolution equation for curvature of the material line\n\\begin{equation}\\label{eqn_curv_evolution}\n\\begin{split}\n    \\frac{d\\kappa}{dt}=&\\left(\\boldsymbol{\\hat{e}}_\\parallel\\cdot\\boldsymbol{H}\\cdot\\boldsymbol{\\hat{e}}_\\parallel\\right)\\cdot\\boldsymbol{\\hat{e}}_\\perp\\\\\n    &+\\left(\\boldsymbol{\\hat{e}}_\\perp\\cdot\\boldsymbol{S}\\cdot\\boldsymbol{\\hat{e}}_\\perp-2\\boldsymbol{\\hat{e}}_\\parallel\\cdot\\boldsymbol{S}\\cdot\\boldsymbol{\\hat{e}}_\\parallel\\right)\\kappa.\n\\end{split}\n\\end{equation}\nHere $\\boldsymbol{S}$ and $\\boldsymbol{H}$ are the rate-of-strain tensor and the velocity Hessian tensor following the trajectories of fluid elements, respectively.\n\nEq. (\\ref{eqn_curv_evolution}) holds for an arbitrary material line, so it also works for the curvature along the largest stretching ($\\boldsymbol{\\hat{e}}_{R1}$) direction $\\kappa_1$. The first term on the right side of Eq. (\\ref{eqn_curv_evolution}) represents the contribution from the velocity Hessian, which can directly bend the fluid element as shown in Fig. \\ref{fig_alignment}(a). Here, the thick blue arrows indicate the primary velocity Hessian that bends the element (i.e., the velocity gradient that changes along the $\\boldsymbol{\\hat{e}}_\\parallel$ direction). In the short time limit, $\\kappa_1\\rightarrow0$, all the terms multiplied by $\\kappa_1$ in Eq. (\\ref{eqn_curv_evolution}) are negligible, \nso Eq. (\\ref{eqn_curv_evolution}) can be simplified to $d\\kappa_1\/dt=\\left(\\boldsymbol{\\hat{e}}_\\parallel\\cdot\\boldsymbol{H}\\cdot\\boldsymbol{\\hat{e}}_\\parallel\\right)\\cdot\\boldsymbol{\\hat{e}}_\\perp$, which corresponds to the linear growth in the early stage as in Fig. \\ref{fig_element_deformation}(b). At later times ($\\Delta t>\\tau_\\eta$), this contribution of the velocity Hessian approaches zero as shown in Fig. \\ref{fig_alignment}(d) (blue solid line) because  $\\left(\\boldsymbol{\\hat{e}}_\\parallel\\cdot\\boldsymbol{H}\\cdot\\boldsymbol{\\hat{e}}_\\parallel\\right)$ may not be perfectly aligned with $\\boldsymbol{\\hat{e}}_\\perp$. Since the velocity Hessian is a small-scale quantity, it is not surprising that the transition in Fig. \\ref{fig_element_deformation}(b) begins at a small $\\Delta t$ as the velocity Hessian decorrelates \\cite{schumacher2007asymptotic}.\n\n\n\n\n\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.9\\linewidth]{alignment.pdf}\n    \\caption{(a-c) Schematics illustrating how (a) velocity Hessian , (b) strain along $\\boldsymbol{\\hat{e}}_\\perp$, and (c) strain along $\\boldsymbol{\\hat{e}}_\\parallel$  contribute to the curvature change, respectively. For all cases, the black dashed curves represent the special material line (skeleton) while the gray dashed curves indicate the same material line at a later time deformed by the surrounding flows indicated by the thick arrows. (d) The time evolution of the contribution to the mean curvature growth by each term in Eq. (\\ref{eqn_curv_evolution}), conditioned on $\\kappa_1>3\\langle\\kappa_1\\rangle$. All the terms are normalized by the Kolmogorov scales. }\n    \\label{fig_alignment}\n\\end{figure}\n\n\n\n \nIn addition to the Hessian term, the other two terms in Eq. (\\ref{eqn_curv_evolution}), both proportional to $\\kappa_1$, represent how the strain affects the curvature of an already-bent fluid element. Here, $\\boldsymbol{\\hat{e}}_\\perp\\cdot\\boldsymbol{S}\\cdot\\boldsymbol{\\hat{e}}_\\perp$ represents the stretching along $\\boldsymbol{\\hat{e}}_\\perp$, which tends to increase the curvature (as shown in Fig. \\ref{fig_alignment}(b));  $\\boldsymbol{\\hat{e}}_\\parallel\\cdot\\boldsymbol{S}\\cdot\\boldsymbol{\\hat{e}}_\\parallel$ represents the stretching along $\\boldsymbol{\\hat{e}}_\\parallel$, which straightens an already-bent fluid element and reduces the curvature (as shown in Fig. \\ref{fig_alignment}(c)). At later times, the mean curvature $\\langle\\kappa_1\\rangle$ is large so both terms associated with $\\kappa_1$ become dominant, leading to $d\\kappa_1\/dt\\propto \\kappa_1$. As a result, the late stage growth of curvature exhibits exponential trend, consistent with the results in Fig. \\ref{fig_element_deformation}(b) inset.\n\nThe contributions from strain by each of the two terms (dashed line) and their combination (red solid line) are shown in Fig. \\ref{fig_alignment}(d). The statistics were collected by only using the fluid elements with  $\\kappa_1>3\\langle\\kappa_1\\rangle$ because the late stage is dominated by the large-curvature cases as indicated by Eq. (\\ref{eqn_curv_evolution}). It is evident that, as the velocity Hessian contribution approaches zero, the total contribution by the strain grows significantly, signaling the transition of the roles between these two mechanisms. This growing contribution by the strain is dominated by $(\\boldsymbol{\\hat{e}}_\\perp\\cdot\\boldsymbol{S}\\cdot\\boldsymbol{\\hat{e}}_\\perp)\\kappa$ which enhances the folding, whereas the other term  $(-\\boldsymbol{\\hat{e}}_\\parallel\\cdot\\boldsymbol{S}\\cdot\\boldsymbol{\\hat{e}}_\\parallel)\\kappa$ that reduces the curvature plateaus close to zero.\n\n\nTo understand the enhanced curvature intermittency at the late stage, the time evolution of the PDF of $\\kappa_1$, i.e. $p(\\kappa_1,t)$ as shown in Fig. \\ref{fig_curv_pdf}(b), is modelled by assuming that $p(\\kappa_1,t)d\\kappa_1=p(\\kappa_1',t+dt)d\\kappa_1'$, where $\\kappa_1'=\\kappa_1+(d\\kappa_1\/dt)dt$ is the curvature of the fluid elements with an initial curvature $\\kappa_1$ after $dt$. Substituting $\\kappa_1'$ into the equation for PDF leads to,\n\\begin{equation}\\label{eqn_pdf_evolution}\n    \\frac{\\partial p}{\\partial t}+(d\\kappa_1\/dt)\\cdot\\frac{\\partial p}{\\partial \\kappa_1}+p\\cdot\\frac{d(d\\kappa_1\/dt)}{d\\kappa_1}=0.\n\\end{equation}\nHere we approximate $d\\kappa_1\/dt\\approx\\langle\\boldsymbol{\\hat{e}}_\\perp\\cdot\\boldsymbol{S}\\cdot\\boldsymbol{\\hat{e}}_\\perp-2\\boldsymbol{\\hat{e}}_\\parallel\\cdot\\boldsymbol{S}\\cdot\\boldsymbol{\\hat{e}}_\\parallel\\rangle\\kappa_1$ because (i) the strain is the dominant mechanism in the late stage and (ii) the contribution by velocity Hessian will only result in a self-similar distribution of curvature as shown in Fig. \\ref{fig_curv_pdf}(a), whereas the PDFs in the late stage exhibit longer tails over time. Eq. (\\ref{eqn_pdf_evolution}) is then solved numerically with  $p(\\kappa_1)$ at $t\/\\tau_\\eta=3$ obtained from the DNS data serving as the initial condition. \n\n\n\n\n\n\n\n\n\nThe predicted PDFs at different times are shown as the dashed curves in Fig. \\ref{fig_curv_pdf}(b). An overall good agreement between the prediction and the data is achieved up to $t\\approx10\\tau_\\eta$, particulary in the tail region extended beyond $\\kappa_1\\eta\\approx 0.2$ in Fig. \\ref{fig_curv_pdf}(b), which correspond to a length scale smaller than 5$\\eta$. This suggests that the intermittency shown here is related to the curved elements being stretched even further by small-scale straining motions in the dissipative range. Note that the range of $\\kappa_1\\eta$ is limited because of the exceedingly low probability of finding fluid elements with $\\kappa_1\\eta$ greater than 0.25. We also note that the model following Eq. (\\ref{eqn_pdf_evolution}) is simplified and it only holds when $d\\kappa_1\/dt$ increases with $\\kappa_1$, i.e., more curved elements are being bent at a faster rate, which can only be satisfied at the late stage given the overall positive magnitude of $\\langle\\boldsymbol{\\hat{e}}_\\perp\\cdot\\boldsymbol{S}\\cdot\\boldsymbol{\\hat{e}}_\\perp-2\\boldsymbol{\\hat{e}}_\\parallel\\cdot\\boldsymbol{S}\\cdot\\boldsymbol{\\hat{e}}_\\parallel\\rangle$ in Eq. (\\ref{eqn_curv_evolution}). Furthermore, the model is intended only for the tail region because the peak region with smaller $\\kappa_1$ is dominated by the velocity Hessian. As a result, a mismatch between model predictions and simulation results is not unexpected for smaller $\\kappa_1\\eta$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.9\\linewidth]{joint_pdf.pdf}\n    \\caption{The joint PDF of the normalized curvature along $\\boldsymbol{\\hat{e}}_1$  and $\\boldsymbol{\\hat{e}}_2$ directions. Two schematics show an initially spherical fluid elements deforming to a bowl shape (top) and a saddle shape (bottom) after a short time,  respectively. }\n    \\label{fig_joint_pdf}\n\\end{figure}\n\nEq. (\\ref{eqn_curv_evolution}) also enables us to use simple Eulerian quantities to understand folding in the early stage. As $\\Delta t\\rightarrow0$, $\\boldsymbol{\\hat{e}}_\\parallel$ approaches $\\boldsymbol{\\hat{e}}_1$, which is the one of the three eigenvectors [$\\boldsymbol{\\hat{e}}_i$ ($i=1,2,3$)] corresponding to the maximum eigenvalue of the rate-of-strain tensor $\\boldsymbol{S}$. The early growth of the material curvature can therefore be determined by an Eulerian quantity $\\langle|\\boldsymbol{\\hat{e}}_1\\cdot\\boldsymbol{H}\\cdot\\boldsymbol{\\hat{e}}_1|\\rangle$ following $d\\langle\\kappa_1\\rangle\/dt\\approx\\langle\\left(\\boldsymbol{\\hat{e}}_\\parallel\\cdot\\boldsymbol{H}\\cdot\\boldsymbol{\\hat{e}}_\\parallel\\right)\\cdot\\boldsymbol{\\hat{e}}_\\perp\\rangle\\approx \\langle|\\boldsymbol{\\hat{e}}_1\\cdot\\boldsymbol{H}\\cdot\\boldsymbol{\\hat{e}}_1|\\rangle\\beta$, where $\\beta\\approx0.85$ is the mean cosine of the angle between $\\boldsymbol{\\hat{e}}_\\parallel\\cdot\\boldsymbol{H}\\cdot\\boldsymbol{\\hat{e}}_\\parallel$ and $\\boldsymbol{\\hat{e}}_\\perp$ obtained from the DNS data.  The predicted result is shown as the cyan dashed line in Fig. \\ref{fig_element_deformation}(b), and it overlaps with the DNS data perfectly. \n\n\n\n\nThis Eulerian quantity $\\langle|\\boldsymbol{\\hat{e}}_1\\cdot\\boldsymbol{H}\\cdot\\boldsymbol{\\hat{e}}_1|\\rangle$ also helps to establish a better physical picture of the deformed fluid elements in the short time limit beyond a simple flat sheet that extends along the $\\boldsymbol{\\hat{e}}_1$ and  $\\boldsymbol{\\hat{e}}_2$ directions considered in the classical framework \\cite{lund1994improved}. As illustrated in the schematics of Fig. \\ref{fig_joint_pdf}, such a sheet could be curved along $\\boldsymbol{\\hat{e}}_3$ direction, and its geometry can be described by two curvatures, whose growth are controlled by $(\\boldsymbol{\\hat{e}}_1\\cdot\\boldsymbol{H}\\cdot\\boldsymbol{\\hat{e}}_1)\\cdot\\boldsymbol{\\hat{e}}_3$ and $(\\boldsymbol{\\hat{e}}_2\\cdot\\boldsymbol{H}\\cdot\\boldsymbol{\\hat{e}}_2)\\cdot\\boldsymbol{\\hat{e}}_3$, respectively. \n\n\nThe joint PDF of $(\\boldsymbol{\\hat{e}}_1\\cdot\\boldsymbol{H}\\cdot\\boldsymbol{\\hat{e}}_1)\\cdot\\boldsymbol{\\hat{e}}_3$ and $(\\boldsymbol{\\hat{e}}_2\\cdot\\boldsymbol{H}\\cdot\\boldsymbol{\\hat{e}}_2)\\cdot\\boldsymbol{\\hat{e}}_3$ normalized by Kolmogorov scales is shown in Fig. \\ref{fig_joint_pdf}. Here, the direction of $\\boldsymbol{\\hat{e}}_3$ is chosen such that $(\\boldsymbol{\\hat{e}}_1\\cdot\\boldsymbol{H}\\cdot\\boldsymbol{\\hat{e}}_1)\\cdot\\boldsymbol{\\hat{e}}_3>0$, while $(\\boldsymbol{\\hat{e}}_2\\cdot\\boldsymbol{H}\\cdot\\boldsymbol{\\hat{e}}_2)\\cdot\\boldsymbol{\\hat{e}}_3$ can be either positive (bowl shape) or negative (saddle shape). The joint PDF suggests a nearly symmetric probability for either shape, skewing only slightly towards the bowl case.  Nevertheless, for a given curvature in one direction, the most likely curvature in the other direction is zero, so there appears to be some preference for cigar like shapes. This is confirmed in Fig. \\ref{fig_element_deformation}(a) where the bending occurs mostly in one direction (although various other bending configurations can be seen).  Note that large values of the velocity Hessian may be the result of local instabilities (e.g. shear instabilities that are responsible for rolling up the vortex sheets into tubes \\citep{vincent1994dynamics}). Connecting the dynamics\nof instabilities to velocity Hessian and curvature requires further investigations.\n\n\n\n\n\n\n\n\nIn sum, our work establishes a new framework to connect folding dynamics to the velocity Hessian and deformation Hessian tensors in a way similar to the connection between stretching to velocity gradient  and Cauchy-Green strain tensors. As the stretching can be well described by the Lyapunov exponents based on strain, such a relationship may inspire the development of new ways to formulate the dynamical system for folding. Our framework also provides new insights into the flow intermittency that the  sharp-turning points in flows become even more curved due to strain, which could help gain deeper insights into the intermittency and inhomogeneity of turbulent mixing. Future work can possibly extend our framework to finite-sized fluid elements considering the coarse-graining effect at the same length scale. This extension will help develop improved models for length-scale reduction in the energy cascade process.\n\n\nWe acknowledge the financial support from the National Science Foundation under the award number CAREER-1905103. This project was also partially supported by the ONR award: N00014-21-1-2083. \n \n\n\n\n\n\n\n\n\n\n\n\n\n\n \n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{${\\cal F}$-transversality}\\label{sF}\n\n\nIn this section, we study properties\nof morphisms of schemes\nwith respect to\ncomplexes on the \\'etale site\nof a scheme.\nThe transversality is defined \nas a condition for a canonical morphism\nfor extraordinary pull-back to be\nan isomorphism.\nIn Section \\ref{ssFtr},\nafter preparing some sorites on\nthe canonical morphism,\nwe establish basic properties\non the transversality.\nIn Section \\ref{ssla},\nafter recalling basic properties\nof local acyclicity,\nwe study the relation between\nthe local acyclicity and\nthe transversality.\n\nIn this section\nand Section \\ref{sms},\n$\\Lambda$ denotes\na finite field of characteristic $\\ell$\ninvertible on relevant \nnoetherian schemes.\nThe derived categories\n$D^+(-,\\Lambda)$ of \nbounded below complexes\nand \n$D^b_c(-,\\Lambda)$ of \nconstructible complexes\nare defined as usual.\n\n\\subsection{${\\cal F}$-transversality}\\label{ssFtr}\n\n\n\nLet $h\\colon W\\to X$\nbe a separated morphism\nof finite type of noetherian schemes\nand $\\Lambda$ be a finite field\nof characteristic $\\ell$ invertible on $X$.\nThe functor\n$Rh^!\\colon D^+(X,\\Lambda)\n\\to D^+(W,\\Lambda)$\nis defined as the adjoint\nof\n$Rh_!\\colon D(W,\\Lambda)\n\\to D(X,\\Lambda)$\nin \\cite[Th\\'eor\\`eme 3.1.4.]{DP}.\nIf $X$ is quasi-excellent,\nby the finiteness theorem\n\\cite[{\\sc Th\\'eor\\`eme} 1.1.1]{fini},\nwe have a functor\n$Rh^!\\colon D^b_c(X,\\Lambda)\n\\to D^b_c(W,\\Lambda)$\nsee also \\cite[Corollaire 1.5]{TF}.\nRecall that a scheme of\nfinite type over\na Dedekind domain with\nfraction field of characteristic 0\nis quasi-excellent\nby \\cite[Scholie (7.8.3)]{EGA4}.\n\n\nLet ${\\cal F}\n\\in D^+(X,\\Lambda)$ and \n${\\cal G}\\in D^+(W,\\Lambda)$ .\nThen, the adjoint of the morphism\n$h^*{\\cal F}\\otimes h^*Rh_*{\\cal G}\n\\to h^*{\\cal F}\\otimes {\\cal G}$\ninduced by the adjunction\n$h^*Rh_*{\\cal G} \\to{\\cal G}$\ndefines a canonical morphism\n\\begin{equation}\n{\\cal F}\\otimes Rh_*{\\cal G}\n\\to Rh_*(h^*{\\cal F}\\otimes {\\cal G}).\n\\label{eqpr0}\n\\end{equation}\nIf $h$ is an open immersion\nand if ${\\cal G}=h^*{\\cal G}_X$\nfor some extension of ${\\cal G}$\non $X$,\n(\\ref{eqpr0}) is identified with\nthe morphism\n${\\cal F}\\otimes R{\\cal H}om(h_!\\Lambda,\n{\\cal G}_X)\n\\to R{\\cal H}om(h_!\\Lambda,{\\cal F}\\otimes {\\cal G}_X)$ defined\nby the product.\n\nApplying the construction\n(\\ref{eqpr0})\nto a compactification of $h$\nand the extension by $0$,\na canonical isomorphism\n\\begin{equation}\n{\\cal F}\\otimes\nRh_!{\\cal G} \n\\to Rh_!(h^*{\\cal F}\\otimes{\\cal G} ),\n\\label{eqprj}\n\\end{equation}\nthe projection formula\n\\cite[(4.9.1)]{Rapport}\nis defined.\n\\if{This is defined as the adjoint\n$h^*{\\cal F}\\otimes^L_\\Lambda\nh^*Rh_!{\\cal G} \n\\to h^*{\\cal F}\\otimes^L_\\Lambda{\\cal G}$\nof the morphism induced\nby the adjunction\n$h^*Rh_!{\\cal G} \n\\to {\\cal G}$\nif $h$ is proper.\nIt is defined as the inverse of\nthe isomorphism\n$h^*{\\cal F}\\otimes^L_\\Lambda\nh^*Rh_!{\\cal G} \n\\gets h^*{\\cal F}\\otimes^L_\\Lambda{\\cal G}$\nif $h$ is an open immersion.}\\fi\n\n\\begin{df}\\label{dfAB}\nLet $h\\colon W\\to X$\nbe a separated morphism of finite\ntype of quasi-excellent noetherian schemes.\nLet ${\\cal F}\\in D^+(X,\\Lambda)$.\n\n{\\rm 1.}\nLet ${\\cal G}\\in D^+(X,\\Lambda)$. \nWe define a canonical morphism\n\\begin{equation}\nc_{{\\cal F},{\\cal G},h}\\colon\nh^*{\\cal F}\n\\otimes\nRh^!{\\cal G}\n\\to \nRh^!({\\cal F}\n\\otimes\n{\\cal G})\n\\label{eqAB}\n\\end{equation}\nto be the adjoint of the composition\n$$\nRh_!(h^*{\\cal F}\n\\otimes\nRh^!{\\cal G})\n\\to \n{\\cal F}\n\\otimes\nRh_!Rh^!{\\cal G}\n\\to {\\cal F}\n\\otimes\n{\\cal G}$$\nof the inverse \nof the isomorphism {\\rm (\\ref{eqprj})}\nand the morphism induced\nby the adjunction\n$Rh_!Rh^!{\\cal G}\n\\to\n{\\cal G}$.\nFor ${\\cal G}=\\Lambda$,\nwe define a canonical morphism\n\\begin{equation}\nc_{{\\cal F},h}\n\\colon \nh^*{\\cal F}\n\\otimes^L\nRh^!\\Lambda\n\\to Rh^!{\\cal F}\n\\label{eqcF}\n\\end{equation}\nto be \n$c_{{\\cal F},\\Lambda,h}$.\n\\end{df}\n\n\n\\begin{lm}\\label{lmcF}\nLet $h\\colon W\\to X$\nbe a separated morphism of finite\ntype of noetherian schemes.\nLet ${\\cal F}\\in D^+(X,\\Lambda)$.\n\n{\\rm 1.}\nLet ${\\cal G},{\\cal H}\\in D^+(X,\\Lambda)$.\nThen, the diagram \n\\begin{equation}\n\\begin{CD}\nh^*{\\cal F}\n\\otimes \nRh^!({\\cal G}\\otimes {\\cal H})\n@>{c_{{\\cal F},\n{\\cal G}\\otimes {\\cal H},h}}>>\nRh^!({\\cal F}\n\\otimes \n{\\cal G}\\otimes {\\cal H})\\\\\n@A{1\\otimes c_{{\\cal G},\n{\\cal H},h}}AA\n@AA{c_{{\\cal F},\n{\\cal G},h}\\otimes 1}A\\\\\nh^*{\\cal F}\n\\otimes \nRh^!{\\cal G}\\otimes h^*{\\cal H}\n@>{c_{{\\cal F},\n{\\cal G},h}\\otimes 1}>>\nRh^!({\\cal F}\\otimes {\\cal G})\n\\otimes \nh^*{\\cal H}\n\\end{CD}\n\\end{equation}\nis commutative.\n\n\n{\\rm 2.}\nLet $g\\colon V\\to W$\nbe a separated morphism of finite\ntype of schemes\nand \nlet ${\\cal G}\\in D^+(X,\\Lambda)$.\nThen, the diagram\n\\begin{equation}\n\\xymatrix{\n(hg)^*{\\cal F}\n\\otimes\nR(hg)^!{\\cal G}\n\\ar[r]^{c_{{\\cal F},{\\cal G},hg}}\n&\nR(hg)^!({\\cal F}\\otimes\n{\\cal G})\n\\\\\ng^*h^*{\\cal F}\n\\otimes\nRg^!Rh^!{\\cal G}\n\\ar[u]\n\\ar[rd]^{c_{h^*{\\cal F},\nRh^!{\\cal G},g}}\n&\nRg^!Rh^!({\\cal F}\\otimes\n{\\cal G})\n\\ar[u]\n\\\\\ng^*h^*{\\cal F}\n\\otimes\nRg^!\\Lambda\n\\otimes\ng^*Rh^!{\\cal G}\n\\ar[u]^{1\\otimes\nc_{Rh^!{\\cal G},g}}\n\\ar[dr]_{\nc_{h^*{\\cal F},g}\\otimes 1}\n&\nRg^!(h^*{\\cal F}\n\\otimes Rh^!{\\cal G})\n\\ar[u]_{Rg^!(c_{{\\cal F},{\\cal G},h})}\n\\\\\n&\nRg^!h^*{\\cal F}\n\\otimes g^*Rh^!{\\cal G}.\n\\ar[u]_{c_{h^*{\\cal F},Rh^!{\\cal G},g}}\n}\n\\label{eqcgh}\n\\end{equation}\nwhere the upper vertical arrows\nare canonical isomorphisms\n{\\rm \\cite[(3.1.13.1)]{DP}}\nis commutative.\n\n\n{\\rm 3.}\nLet $$\\begin{CD}\nX@<h<< W\\\\\n@VfVV@VV{f'}V\\\\\nY@<g<<V\n\\end{CD}$$\nbe a cartesian diagram of\nseparated morphisms\nof finite type.\nThen, the diagram\n\\begin{equation}\n\\begin{CD}\ng^*Rf_*{\\cal F}\n\\otimes\nRg^!\\Lambda\n@>{c_{Rf_*{\\cal F},g}}>>\nRg^!Rf_*{\\cal F}\n\\\\\n@VVV@VVV\\\\\nRf'_*h^*{\\cal F}\n\\otimes\nRg^!\\Lambda\n@.\nRf'_*Rh^!{\\cal F}\n\\\\\n@V\n{\\rm (\\ref{eqpr0})}VV\n@AA{Rf'_*(c_{{\\cal F},h})}A\\\\\nRf'_*(h^*{\\cal F}\n\\otimes\nf'^*Rg^!\\Lambda)\n@>>>\nRf'_*(h^*{\\cal F}\n\\otimes\nRh^!\\Lambda)\n\\end{CD}\n\\label{eqcfg}\n\\end{equation}\nwhere the arrows without\ntags are defined by\nbase change morphisms\nis commutative.\n\\end{lm}\n\n\\proof{\n1.\nThe diagram\n$$\\begin{CD}\nRh_!Rh^!({\\cal G}\\otimes {\\cal H})\n@>>>\n{\\cal G}\\otimes {\\cal H}\\\\\n@A\n{Rh_!(c_{{\\cal G},{\\cal H},h})}AA\n@AAA\\\\\nRh_!(Rh^!{\\cal G}\\otimes h^*{\\cal H})\n@<{{\\rm (\\ref{eqprj})}}<<\nRh_!Rh^!{\\cal G}\\otimes {\\cal H}\n\\end{CD}$$\nwhere the arrows without\ntags are defined by the adjunction\nis commutative\nby the definition of\n$c_{{\\cal G},{\\cal H},h}$.\nTaking the tensor products with ${\\cal F}$,\napplying the projection formula\n(\\ref{eqprj}) and\ntaking the adjoint,\nwe see that the upper triangle in\n\\begin{equation*}\n\\xymatrix{\nh^*{\\cal F}\n\\otimes \nRh^!({\\cal G}\\otimes {\\cal H})\n\\ar[r]^{c_{{\\cal F},\n{\\cal G}\\otimes {\\cal H},h}}&\nRh^!({\\cal F}\n\\otimes \n{\\cal G}\\otimes {\\cal H})\\\\\nh^*{\\cal F}\n\\otimes \nRh^!{\\cal G}\\otimes h^*{\\cal H}\n\\ar[u]^{1\\otimes c_{{\\cal G},\n{\\cal H},h}}\n\\ar[ru]^\n{c_{{\\cal F}\\otimes{\\cal H},\n{\\cal G},h}}\n\\ar[r]^{c_{{\\cal F},\n{\\cal G},h}\\otimes 1}\n&\nRh^!({\\cal F}\\otimes {\\cal G})\n\\otimes \nh^*{\\cal H}\n\\ar[u]_{c_{{\\cal F},\n{\\cal G},h}\\otimes 1}\n}\n\\end{equation*}\nis commutative.\nThe lower triangle is similarly\ncommutative\nand the assertion follows.\n\n2.\nThe lower quadrangle\nis commutative by 1.\nThe composition \n$g^*h^*{\\cal F}\n\\otimes\nRg^!Rh^!{\\cal G}\n\\to\nRg^!Rh^!{\\cal F}$\nthrough\n$\nRg^!(h^*{\\cal F}\n\\otimes Rh^!{\\cal G})$\nis the adjoint of\n$Rh_!Rg_!\n(g^*h^*{\\cal F}\n\\otimes\nRg^!Rh^!{\\cal G})\n\\to\n{\\cal F}\\otimes\nRh_!Rg_!\nRg^!Rh^!{\\cal G}$\ninduced by\nthe adjunction\n$Rh_!Rg_!\nRg^!Rh^!{\\cal G}\n\\to\nRh_!Rh^!{\\cal G}\n\\to {\\cal G}$.\nSince the last morphism\nis identified\nwith \nthe adjunction\n$R(hg)_!\nR(hg)^!{\\cal G}\n\\to {\\cal G}$,\nthe upper pentagon is also commutative.\n\n\n3.\nFor ${\\cal G}\\in D^+(V,\\Lambda)$,\nwe consider the diagram\n\\begin{equation}\n\\begin{CD}\nf^*Rg_!(g^*Rf_*{\\cal F}\n\\otimes\n{\\cal G})\n@<{f^*{\\rm (\\ref{eqprj})}}<<\nf^*Rf_*{\\cal F}\n\\otimes\nf^*Rg_!{\\cal G}\n@>>>\n{\\cal F}\n\\otimes\nf^*Rg_!{\\cal G}\n\\\\\n@VVV@.@VVV\\\\\nRh_!f'^*(Rf'_*h^*{\\cal F}\n\\otimes\n{\\cal G})\n@>>>\nRh_!(h^*{\\cal F}\n\\otimes\nf^*{\\cal G})\n@<{\\rm (\\ref{eqprj})}<<\n{\\cal F}\n\\otimes\nRh_!f'^*{\\cal G}\n\\end{CD}\n\\label{eqcfga}\n\\end{equation}\ndefined as follows.\nThe vertical arrows are\ndefined by the base change morphisms\nand the horizontal arrows\nwithout labels are\ndefined by adjunction.\nWe see that the diagram is commutative\nby reducing to the case\nwhere $g$ is proper and\ngoing back to the definition\nof (\\ref{eqprj}).\n\nWe apply (\\ref{eqcfga}) to\n${\\cal G}=Rg^!\\Lambda$.\nSince the composition\n$f^*Rg_!Rg^!\\Lambda\n\\to\nRh_!f'^*Rg^!\\Lambda\n\\to Rh_!Rh^!\\Lambda\n\\to \\Lambda$\nof the base change morphisms\nwith the adjunction\nis induced by the adjuncion\n$Rg_!Rg^!\\Lambda\\to \\Lambda$,\nwe obtain a commutative diagram\n\\begin{equation}\n\\begin{CD}\nf^*Rg_!(g^*Rf_*{\\cal F}\n\\otimes\nRg^!\\Lambda)\n@<{f^*{\\rm (\\ref{eqprj})}}<<\nf^*Rf_*{\\cal F}\n\\otimes\nf^*Rg_!Rg^!\\Lambda\n@>>>\n{\\cal F}\n\\\\\n@VVV@.@AAA\\\\\nRh_!f'^*(Rf'_*h^*{\\cal F}\n\\otimes\nRg^!\\Lambda)\n@>>>\nRh_!(h^*{\\cal F}\n\\otimes\nRh^!\\Lambda)\n@<{\\rm (\\ref{eqprj})}<<\n{\\cal F}\n\\otimes\nRh_!Rh^!\\Lambda\n\\end{CD}\n\\label{eqcfgb}\n\\end{equation}\nSince the canonical morphism\n(\\ref{eqcF}) is defined as\nthe adjoint of (\\ref{eqprj}),\nwe obtain (\\ref{eqcfg})\nby taking the adjoint of (\\ref{eqcfgb}).\n\\qed\n\n}\n\n\n\\begin{lm}\\label{lmij}\nLet $i\\colon Z\\to X$ be a closed\nimmersion of noetherian schemes\nand let ${\\cal F},{\\cal G}\n\\in D^+(X,\\Lambda)$.\n\n{\\rm 1.}\nWe define the slant arrow\nand the vertical arrow\nin the diagram\n\\begin{equation}\n\\xymatrix{\n{\\cal F}\\otimes\ni_*Ri^!{\\cal G}\n\\ar[r]^-{\\rm(\\ref{eqprj})}\n\\ar[rd]\n&\ni_*(i^*{\\cal F}\\otimes\nRi^!{\\cal G})\n\\ar[r]^-{i_*(c_{{\\cal F},{\\cal G},i})}\n&\ni_*Ri^!({\\cal F}\\otimes {\\cal G})\n\\ar[d]\n\\\\\n&\n{\\cal F}\\otimes\nR{\\cal H}om(i_*\\Lambda,{\\cal G})\n\\ar[r]\n&\nR{\\cal H}om(i_*\\Lambda,{\\cal F}\n\\otimes{\\cal G})\n}\n\\label{eqic}\n\\end{equation}\nby the canonical isomorphism\n$i_*Ri^!\\to R{\\cal H}om(i_*\\Lambda,-)$\nand the lower horizontal arrow\nby the product.\nThen, the diagram\n{\\rm (\\ref{eqic})}\nis commutative.\n\n{\\rm 2.}\nLet $j\\colon U=X\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} Z\\to X$\nbe the open immersion of the complement.\nThen, \nthe exact sequence\n$0\\to j_!\\Lambda\n\\to \\Lambda\\to i_*\\Lambda\\to 0$\ndefines a commutative diagram\n\\begin{equation}\n\\begin{CD}\n{\\cal F}\\otimes\ni_*Ri^!{\\cal G}\n@>>> {\\cal F}\\otimes\n{\\cal G}\n@>>>\n{\\cal F}\\otimes\nRj_*j^*{\\cal G}\n@>>>\\\\\n@V{c_{{\\cal F},{\\cal G},i}}VV@|\n@VV{\\rm (\\ref{eqpr0})}V@.\\\\\ni_*Ri^!({\\cal F}\\otimes\n{\\cal G})\n@>>> {\\cal F}\\otimes\n{\\cal G}\n@>>>\nRj_*j^*({\\cal F}\\otimes\n{\\cal G})\n@>>>\n\\end{CD}\n\\label{eqij}\n\\end{equation}\nof distinguished triangles.\n\\end{lm}\n\n\n\\proof{\n1.\nBy the definition of\n$c_{{\\cal F},{\\cal G},i}$,\nthe morphism\n$i_*(c_{{\\cal F},{\\cal G},i})\\colon\ni_*(i^*{\\cal F}\\otimes\nRi^!{\\cal G})\n\\to i_*Ri^!({\\cal F}\n\\otimes {\\cal G})$\nis the unique morphism\nsuch that the diagram\n$$\n\\begin{CD}\n{\\cal F}\\otimes\ni^*Ri^!{\\cal G}\n@>>> {\\cal F}\\otimes{\\cal G}\n\\\\\n@V{\\rm (\\ref{eqprj})}VV@AAA\\\\\ni_*(i^*{\\cal F}\\otimes\nRi^!{\\cal G})\n@>{i_*(c_{{\\cal F},{\\cal G},i})}>>\ni_*Ri^!({\\cal F}\\otimes{\\cal G})\n\\end{CD}$$\nis commutative.\nHere the arrows without tag\nare defined by the\nadjunction $i_*Ri^!\\to 1$.\nSimilarly, the lower horizontal arrow\n${\\cal F}\\otimes\nR{\\cal H}om(i_*\\Lambda,{\\cal G})\n\\to\nR{\\cal H}om(i_*\\Lambda,{\\cal F}\n\\otimes{\\cal G})$\nis the unique morphism\nsuch that the diagram\n$$\n\\begin{CD}\n{\\cal F}\\otimes\ni^*Ri^!{\\cal G}\n@>>> {\\cal F}\\otimes{\\cal G}\n\\\\\n@VVV@AAA\\\\\n{\\cal F}\\otimes\nR{\\cal H}om(i_*\\Lambda,{\\cal G})\n@>>>\nR{\\cal H}om(i_*\\Lambda,{\\cal F}\n\\otimes{\\cal G})\n\\end{CD}$$\nis commutative.\nHere the left vertical arrow\nis the slant arrow in (\\ref{eqic})\nand the right vertical arrow\nis induced by $\\Lambda\\to i_*\\Lambda$.\nHence\nthe assertion follows.\n\n2.\nThe exact sequence\n$0\\to j_!\\Lambda\n\\to \\Lambda\\to i_*\\Lambda\\to 0$\ndefines a commutative diagram\n\\begin{equation}\n\\begin{CD}\n{\\cal F}\\otimes\nR{\\cal H}om(i_*\\Lambda,{\\cal G})\n@>>> {\\cal F}\\otimes\n{\\cal G}\n@>>>\n{\\cal F}\\otimes\nR{\\cal H}om(j_!\\Lambda,{\\cal G})\n@>>>\\\\\n@VVV@VVV@VVV@.\\\\\nR{\\cal H}om(i_*\\Lambda,{\\cal F}\\otimes\n{\\cal G})\n@>>> {\\cal F}\\otimes\n{\\cal G}\n@>>>\nR{\\cal H}om(j_!\\Lambda,{\\cal F}\\otimes\n{\\cal G})\n@>>>\n\\end{CD}\n\\label{eqij2}\n\\end{equation}\nof distinguished triangles.\nBy 1.,\nthe left vertical arrow \nof (\\ref{eqij}) is\nidentified with\nthat of (\\ref{eqij2})\nand similarly for the\nright vertical arrows.\n\\qed\n\n}\n\n\\begin{lm}\\label{lmtrbc}\nLet\n$$\\begin{CD}\nX@<h<<W\\\\\n@VfVV@VV{f'}V\\\\\nY@<g<<V\n\\end{CD}$$\nbe a cartesian of \nmorphisms of finite type\nof regular noetherian schemes.\nIf $f$ and $g$ are transversal,\nthen the base change\nmorphism\n$f^*Rg^!\\Lambda\\to\nRh^!\\Lambda$\nis an isomorphism\nof locally constant complexes.\n\\end{lm}\n\n\nWe say that a complex\n${\\cal F}$ is locally constant,\nif its cohomology sheaf\n${\\cal H}^q{\\cal F}$ is locally constant\nfor every $q$\nand \nif ${\\cal H}^q{\\cal F}=0$\nexcept for finitely many $q$.\n\n\n\\proof{\nSince the assertion is local,\nwe may assume that\nthe morphism $g$ is a composition\nof a smooth morphism\nand a regular immersion.\nHence, it suffices to show\neach case.\n\nAssume that $g$ is a smooth\nof relative dimension $d$.\nThen, the adjoint\nof the trace morphism\n$Rg_!\\Lambda(d)[2d]\\to\n\\Lambda$ \n\\cite[Th\\'eor\\`eme 2.9]{DP} defines\nan isomorphism\n$\\Lambda(d)[2d]\\to\nRg^!\\Lambda$ by Poincar\\'e duality\n\\cite[Th\\'eor\\`eme 3.2.5]{DP}.\nSince the formation of\nthe trace morphism commutes\nwith base change,\nthe assertion follows in this case.\n\nAssume that $g$ is a regular immersion \nof codimension $c$.\nThen, by the absolute purity \\cite[{\\sc Th\\'eor\\`eme 3.1.1}]{purete},\nthe fundamental\nclass $[V]$ defines an isomorphism\n$\\Lambda\\to Rg^!\n\\Lambda(c)[2c]$.\nSince $f$ and $g$ \nare transversal,\nfurther by the absolute purity,\nthe fundamental\nclass $[W]=f^![V]$\ndefines an isomorphism\n$\\Lambda\\to Rh^!\n\\Lambda(c)[2c]$.\nHence the base change morphism\n$f^*Rg^!\n\\Lambda\n\\to Rh^!\\Lambda$ is an isomorphism.\n\\qed\n\n}\n\n\n\\begin{df}\\label{dfFtr}\nLet $h\\colon W\\to X$\nbe a separated morphism of finite\ntype of noetherian schemes\nand let ${\\cal F}\\in D^+(X,\\Lambda)$.\nWe say that $h$ is ${\\cal F}$-transversal\nif the canonical morphism\n{\\rm (\\ref{eqcF})} is an isomorphism.\n\\end{df}\n\n\nFor a closed immersion\n$i\\colon Z\\to X$ of\nregular noetherian schemes\nand a separated morphism\n$h\\colon W\\to X$\nof finite type of\nregular noetherian schemes,\nwe show that\n$h$ is $i_*\\Lambda$-transversal\nif $h$ and $i$ are transversal\nin Corollary \\ref{cortrC}.2.\nIf $h$ is also an immersion,\nif $Z$ and $W$ meets properly,\nif the reduced part\n$V$ of $Z\\times_XW$ is regular\nand if\nthe intersection multiplicity\n$\\mu(Z,W)$ \nis invertible in $\\Lambda$,\nthen \n$h$ is still $i_*\\Lambda$-transversal.\nHence the converse does not hold.\n\n\n\\begin{lm}\\label{lmPoi}\nLet $h\\colon W\\to X$\nbe a separated morphism of finite\ntype of noetherian schemes\nand let ${\\cal F}\\in D^+(X,\\Lambda)$.\n\n\n{\\rm 1.}\nIf $h\\colon W\\to X$\nis smooth,\nthen $h$ is ${\\cal F}$-transversal.\n\n{\\rm 2.}\nIf ${\\cal F}$ is locally constant,\nthen $h$ is ${\\cal F}$-transversal.\n\\end{lm}\n\n\\proof{\n1.\nThis is exactly the Poincar\\'e duality\n\\cite[Th\\'eor\\`eme 3.2.5]{DP}.\n\n2.\nSince the assertion is \\'etale local,\nthe assertion is reduced to the case\nwhere ${\\cal F}=\\Lambda$\nby devissage.\n\\qed\n\n}\n\n\\begin{lm}\\label{lmiZ}\nLet $i\\colon Z\\to X$ be a closed\nimmersion of noetherian schemes\nand let ${\\cal F}\\in D^+(X,\\Lambda)$.\n\n\n{\\rm 1.}\nAssume that\n$Z$ is the union of\nclosed subsets $Z_1,\\ldots,Z_n\n\\subset X$\nand that\nfor each subset $I\\subset\n\\{1,\\ldots,n\\}$,\nthe immersion\n$i_I\\colon Z_I=\\bigcap_{j\\in I}Z_j\\to X$\nis ${\\cal F}$-transversal.\nThen, $i\\colon Z\\to X$\nis ${\\cal F}$-transversal.\n\n{\\rm 2.}\nLet $j\\colon U=X\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} Z\\to X$\nbe the open immersion of the complement.\nThen, the following conditions\nare equivalent:\n\n{\\rm (1)}\n$i\\colon Z\\to X$\nis ${\\cal F}$-transversal.\n\n{\\rm (2)}\nThe canonical morphism\n${\\cal F}\n\\otimes Rj_*\\Lambda\n\\to\nRj_*j^*{\\cal F}$\nis an isomorphism.\n\\end{lm}\n\n\\proof{\n1.\nThe quasi-isomorphism\n$i_*\\Lambda\n\\to[\n\\bigoplus_{j\\in I}i_{j*}\\Lambda\\to\n \\cdots \\to\n\\bigoplus_{|I|=p}i_{I*}\\Lambda\n\\to \\cdots]$\ndefines a spectral sequence\n$E_1^{p,q}=\\bigoplus_{|I|=-p}\nR^q{\\cal H}om\n(i_{I*}\\Lambda,-)\n\\Rightarrow\nR^{p+q}{\\cal H}om\n(i_*\\Lambda,-)$.\nBy Lemma \\ref{lmij}.1,\nthe assumption\nimplies that the morphisms\n${\\cal F}\n\\otimes R^q{\\cal H}om\n(i_{I*}\\Lambda,\\Lambda)\n\\to\nR^q{\\cal H}om\n(i_{I*}\\Lambda,{\\cal F})$\non $E_1$-terms are isomorphisms.\nHence the assertion follows.\n\n2.\nThe assertion follows from\nLemma \\ref{lmij}.2 for ${\\cal G}=\\Lambda$.\n\\qed\n\n}\n\n\n\\begin{pr}\\label{prhF}\nLet $h\\colon W\\to X$\nbe a separated morphism of finite\ntype of noetherian schemes\nand let ${\\cal F}\\in D^+(X,\\Lambda)$.\nAssume that $h$ is ${\\cal F}$-transversal.\n\n\n{\\rm 1}.\nAssume that $Rh^!\\Lambda$ \nis locally constant on\na neighborhood $W_1$\nof ${\\rm supp}\\, h^*{\\cal F}$\nsuch that\n$W_1\\subset\n{\\rm supp}\\, Rh^!\\Lambda$.\nThen, for a separated morphism \n$g\\colon V\\to W$\nof finite type of noetherian schemes,\nthe following conditions are equivalent:\n\n{\\rm (1)}\n$g$ is $h^*{\\cal F}$-transversal.\n\n{\\rm (2)}\n$hg$ is ${\\cal F}$-transversal.\n\n\n\n\n{\\rm 2}.\nLet \n$$\\begin{CD}\nX@<h<< W\\\\\n@VfVV@VV{f'}V\\\\\nY@<g<<V\n\\end{CD}$$\nbe a cartesian diagram\nof morphisms of\nfinite type of noetherian schemes.\nAssume that $g$ is separated\nand that\n$Rg^!\\Lambda$ is locally constant\nof support $V$.\nFurther assume that \nthe base change\nmorphism\n\\begin{equation}\nf'^*Rg^!\\Lambda\\to\nRh^!\\Lambda\n\\label{eqbcg}\n\\end{equation} \nis an isomorphism on \na neighborhood of \n${\\rm supp}\\, h^*{\\cal F}$.\nThen the following conditions\nare equivalent:\n\n{\\rm (1)}\nThe morphism\n$g\\colon V\\to Y$\nis $Rf_*{\\cal F}$-transversal.\n\n{\\rm (2)}\nThe base change morphism\n\\begin{equation}\ng^*Rf_*{\\cal F}\n\\to Rf'_*h^*{\\cal F}\n\\label{eqbch}\n\\end{equation}\nis an isomorphism.\n\\end{pr}\n\nBy Lemma \\ref{lmPoi}.1,\nProposition \\ref{prhF}.2\n(1)$\\Rightarrow$(2)\ngives a generalization of\nthe smooth base change theorem\n\\cite[Corollaire 1.2]{smbc}.\n\n\\proof{\n1.\nWe consider the commutative diagram\n(\\ref{eqcgh}) for ${\\cal G}=\\Lambda$.\nSince $h$ is assumed to\nbe ${\\cal F}$-transversal,\n$c_{{\\cal F},h}$ is an isomorphism.\nSince $Rh^!\\Lambda$ is \nlocally constant on a neighborhood\nof the support of $h^*{\\cal F}$,\nthe morphisms\n$1\\otimes c_{Rh^!\\Lambda,g}$\nand\n$c_{h^*{\\cal F},Rh^!\\Lambda,g}$\nare isomorphisms.\nHence\n$c_{{\\cal F},hg}$ is an isomorphism\nif and only if\n$c_{h^*{\\cal F},g}\\otimes 1$\nis an isomorphisms.\nFurther by the assumption on\nthe support of $Rh^!\\Lambda$,\nthe latter condition is equivalent\nto the condition that\n$c_{h^*{\\cal F},g}$\nis an isomorphism.\n\n2.\nWe consider the commutative diagram\n(\\ref{eqcfg}).\nBy the proper base change theorem\nor \\cite[Corollaire 3.1.12.3]{DP},\nthe upper right vertical arrow\nis an isomorphism.\nSince $h$ is assumed ${\\cal F}$-transversal,\nthe lower right vertical arrow\n$Rf'_*(c_{{\\cal F},h})$\nis an isomorphism.\nBy the assumption on\n$Rg^!\\Lambda$,\nthe upper left vertical arrow\nis an isomorphism if and only if\n(\\ref{eqbch}) is an isomorphism.\nFurther the arrow labeled\n(\\ref{eqpr0}) is an isomorphism.\nSince (\\ref{eqbcg}) \nis assumed to be an isomorphism,\nthe bottom horizontal arrow is an\nisomorphism.\nHence the assertion follows\nfrom the commutative diagram\n(\\ref{eqcfg}).\n\\qed\n\n}\n\n\\begin{cor}\\label{cortrC}\n{\\rm 1.}\nLet the assumption\nbe the same as in Proposition\n{\\rm \\ref{prhF}.2.}\nAssume further that $f$ is proper\non the support of ${\\cal F}$.\nThen, $g$ is $Rf_*{\\cal F}$-transversal.\n\n{\\rm 2.}\nLet $p\\colon Z\\to X$ be \na proper morphisms of regular schemes\nand\nlet $h\\colon W\\to X$\nbe a separated morphism\nof finite type of regular schemes.\nIf $h$ and $p$ are transversal,\nthen $h$ is $Rp_*\\Lambda$-transversal.\n\\end{cor}\n\n\\proof{\n1.\nBy the assumption that $f$ is proper\non the support of ${\\cal F}$,\nthe base change morphism\n(\\ref{eqbch}) is an isomorphism\nby the proper base change theorem.\nHence the assertion follows\nfrom Proposition \\ref{prhF}.2\n(2)$\\Rightarrow$(1).\n\n\n2.\nLet \n$$\\begin{CD}\nZ@<g<< V\\\\\n@VpVV@VV{p'}V\\\\\nX@<h<< W\n\\end{CD}$$\nbe a cartesian diagram.\nBy Lemma \\ref{lmtrbc},\nthe base change morphism\n$p'^*Rh^!\n\\Lambda\n\\to Rg^!\\Lambda$ is an isomorphism.\nSince $X$ and $W$ are regular,\nthe assumption in Proposition  \\ref{prhF}.2\nthat\n$Rh^!\\Lambda$ is locally constant \nof support $W$ is satisfied\nby the absolute purity \\cite[{\\sc Th\\'eor\\`eme 3.1.1}]{purete}\nas in the proof of Lemma \\ref{lmtrbc}.\nSince $g$ is $\\Lambda$-transversal\nand $p$ is proper,\n$h$ is $Rp_*\\Lambda$-transversal\nby 1.\n\\qed\n\n}\n\n\n\n\\subsection{Local acyclicity\nand ${\\cal F}$-transversality}\\label{ssla}\n\nLet $f\\colon X\\to Y$\nbe a morphism of schemes\nand $x$ and $y$ be\ngeometric points of $X$ and $Y$.\nLet $f(x)$ denote the geometric point\nof $Y$ defined by the composition\n$x\\to X\\to Y$\nand let\n$X_{(x)}\\to Y_{(f(x))}$ be the induced\nmorphism of strict localizations.\nWe call a morphism $y\\to Y_{(f(x))}$\nof schemes over $Y$\na specialization $f(x)\\gets y$\nand call $X_{(x),y}\n=X_{(x)}\n\\times_{Y_{(f(x))}}y$\nthe Milnor fiber.\nFor a complex ${\\cal F}$ of \n$\\Lambda$-modules\non $X$,\nthe pull-back by $X_{(x),y}\n\\to X_{(x)}$\ndefines a canonical morphism\n\\begin{equation}\n{\\cal F}_x\n=R\\Gamma(X_{(x)},{\\cal F}|_{{X_{(x)}}})\n\\to\nR\\Gamma(X_{(x),y},{\\cal F}|_{{X_{(x),y}}}).\n\\label{eqMil}\n\\end{equation}\n\n\n\n\n\\begin{df}[{cf.~\\cite[D\\'efinition 2.12]{TF}}]\\label{dfla}\nLet $f\\colon X\\to Y$\nbe a morphism of schemes\nand $Z\\subset X$ be a closed\nsubset.\nLet ${\\cal F}$\nbe a complex of $\\Lambda$-modules \non $X$.\nWe say that $f$ is locally acyclic\nrelatively to ${\\cal F}$\nor ${\\cal F}$-acyclic \nfor short along $Z$ \nif for every geometric point\n$x$ of $Z$ and for every specialization\n$f(x)\\gets y$,\nthe canonical morphism\n${\\cal F}_x\\to R\\Gamma(X_{(x),y},\n{\\cal F}|_{X_{(x),y}})$ \n{\\rm (\\ref{eqMil})} is an isomorphism.\nIf $X=Z$, we drop along $Z$\nin the terminology.\n\nWe say that $f$ is universally\n${\\cal F}$-acyclic along $Z$, if\nfor every morphism $Y'\\to Y$,\nthe base change\n$X'\\to Y'$ is locally acyclic \nrelatively to the pull-back of\n${\\cal F}$ along the inverse image\n$Z'\\subset X'$ of $Z$.\n\\end{df}\n\n\\begin{lm}\\label{lmlac}\nLet $f\\colon X\\to Y$\nbe a morphism of schemes\nand $Z\\subset X$ be a closed\nsubset.\nLet ${\\cal F}\n\\in D^+(X,\\Lambda)$.\n\n{\\rm 1.}\nThe following conditions are\nequivalent.\n\n{\\rm (1)}\n$f$ is ${\\cal F}$-acyclic along $Z$.\n\n{\\rm (2)}\nLet $s\\gets t$ be a specialization\nof geometric points of $Y$\nsuch that $t$ is\nthe spectrum of an algebraic\nclosure of the residue field\nof the point of $Y$ below $t$.\nLet $Y_{(s)}$ denote the strict localization\nand let\n\\begin{equation}\n\\begin{CD}\nX_s@>{i'_s}>>X\\times_YY_{(s)}@<{j'_t}<< X_t\\\\\n@V{f_s}VV@V{f_{(s)}}VV @VV{f_t}V\\\\\ns@>{i_s}>>Y_{(s)}@<{j_t}<<t\n\\end{CD}\n\\label{eqXYtpi}\n\\end{equation}\nbe the cartesian diagram.\nThen, the canonical morphism \n\\begin{equation}\ni^{\\prime*}_s{\\cal F}\n\\to \ni^{\\prime*}_sRj'_{t*}j^{\\prime*}_t{\\cal F}\n\\label{eqijst}\n\\end{equation}\nis an isomorphism on \nthe inverse image of $Z$.\n\n{\\rm 2.}\nFor a proper morphism $p\\colon X\\to P$\nof schemes over $Y$,\nwe consider the following conditions:\n\n{\\rm (1)}\n$f$ is ${\\cal F}$-acyclic along $Z$.\n\n{\\rm (2)}\nThe morphism $g\\colon P\\to Y$ is \n$Rp_*{\\cal F}$-acyclic along $p(Z)$.\n\n\\noindent\nWe have {\\rm (1)}$\\Rightarrow${\\rm (2)}\n if $Z=p^{-1}(p(Z))$.\nIf $p$ is finite, we have\n{\\rm (2)}$\\Rightarrow${\\rm (1)}.\n\n{\\rm 3.}\nThe following conditions are\nequivalent:\n\n{\\rm (1)}\n$f$ is universally\n${\\cal F}$-acyclic along $Z$.\n\n{\\rm (2)}\nFor every \nsmooth morphism $Y'\\to Y$\nand \nfor the pull-back ${\\cal F}'$\nof ${\\cal F}$ on \n$X'=X\\times_YY'$,\nthe base change\n$f'\\colon X'\\to Y'$ is \n${\\cal F}'$-acyclic along \nthe inverse image $Z'\\subset X'$ of $Z$.\n\\end{lm}\n\nSince the local acyclicity is a local property,\nby locally taking an immersion\n$X\\to {\\mathbf A}^n_Y$,\nthe study of local acyclicity\nis reduced to the case\nwhere $f\\colon X\\to Y$\nis the projection\n${\\mathbf A}^n_Y\\to Y$\nby Lemma \\ref{lmlac}.2.\n\n\n\\proof{\n1.\nA morphism $y'\\to y$ of \ngeometric points of $Y$\nis the composition of a limit of \nsmooth morphisms and\na homeomorphism in \\'etale topology.\nHence for a geometric point $x$ of\n$X$ and a specialization $f(x)\\gets y$,\nthe pull-back\n$R\\Gamma(X_{(x),y},\n{\\cal F}|_{X_{(x),y}})\\to\nR\\Gamma(X_{(x),y'},\n{\\cal F}|_{X_{(x),y'}})$\nis an isomorphism by the smooth base\nchange theorem\n\\cite[Corollaire 1.2]{smbc}.\nThus, in the definition of\nlocal acyclicity,\nit suffices to consider \nspecializations $f(x)\\gets y$\nsuch that $y$ is\nthe spectrum of an algebraic\nclosure of the residue field\nof the point of $Y$ below $y$.\n\nIn the notation of (2),\nfor a geometric point $x$ of $X_s$,\nthe morphism induced by\n(\\ref{eqijst}) on the stalks \nat $x$ equals \n${\\cal F}_x\\to\nR\\Gamma(X_{(x),t},\n{\\cal F}|_{X_{(x),t}})$\n(\\ref{eqMil}).\nHence, the assertion follows.\n\n2.\n{\\rm (1)}$\\Rightarrow${\\rm (2)}:\nLet\n\\begin{equation}\n\\begin{CD}\nX_s@>{i'_s}>>X\\times_YY_{(s)}@<{j'_t}<< X_t\\\\\n@V{p_s}VV@V{p_{(s)}}VV @VV{p_t}V\\\\\nP_s@>{i''_s}>>P\\times_YY_{(s)}@<{j''_t}<<P_t\n\\end{CD}\n\\end{equation}\nbe the base change of $X\\to P$.\nThen, the isomorphism (\\ref{eqijst})\non the inverse image of $Z=p^{-1}(p(Z))$\nimplies an isomorphism\n$i^{\\prime\\prime*}_sRp_{(s)*}{\\cal F}\n\\to \ni^{\\prime\\prime*}_sRj''_{t*}j^{\\prime\\prime*}_tRp_{(s)*}{\\cal F}$\non the inverse image of $p(Z)$\nby proper base change theorem.\n\n{\\rm (2)}$\\Rightarrow${\\rm (1)}:\nLet $z$ be a geometric point\nof $Z$ and let $w=p(z)\\gets y$\nbe a specialization.\nThen the cospecialization morphism\n$p_*{\\cal F}_w\n\\to R\\Gamma(P_{(w),y},p_*{\\cal F}|_{P_{(w),y}})$\nis the direct sum of\n(\\ref{eqMil}) for\n$x\\in p^{-1}(w)$\nsince $p$ is finite.\nHence\nthe assertion follows.\n\n3.\nSince the local acyclicity\nis a local property\npreserved by base change\nby immersions\nand commutes with limits,\nthe assertion follows.\n\\qed\n\n}\n\n\n\\begin{lm}\\label{lmla}\nLet $X$ be a noetherian scheme\nand ${\\cal F}\\in D^b(X,\\Lambda)$.\n\n{\\rm 1.}\nIf ${\\cal F}$ is locally constant\nand if $f\\colon X\\to Y$\nis smooth,\nthen $f$ is ${\\cal F}$-acyclic.\n\n{\\rm 2.}\nIf $1_X\\colon X\\to X$\nis ${\\cal F}$-acyclic along $Z$\nand if ${\\cal F}$ is constructible,\nthen ${\\cal F}$ is locally constant\non a neighborhood of $Z$.\n\\end{lm}\n\n\\proof{1.\nBy devissage,\nthe assertion follows from\nthe local acyclicity of smooth morphism\n\\cite[Th\\'eor\\`eme 2.1]{Artin}.\n\n2.\nFor every geometric point  $s$ of\n$Z$ and\nevery specialization \n$s\\gets t$ of geometric points\nof $X$,\nthe cospecialization morphism\n${\\cal F}_s\\to {\\cal F}_t$\nis an isomorphism.\nHence the constructible sheaf\n${\\cal H}^q{\\cal F}$ is locally constant\non a neighborhood of \n every geometric point $s$ \nof $Z$ for every $q\\in {\\mathbf Z}$\nby \\cite[Proposition 2.11]{cst}.\nHence ${\\cal F}$\nis locally constant on a\nneighborhood of $Z$.\n\\qed\n\n}\n\n\\begin{pr}[{cf.~\\cite[Proposition 8.11]{CC}}]\\label{prla1}\nLet $f\\colon X\\to Y$\nbe a smooth morphism\nof regular schemes\nof finite type over \na discrete valuation ring\n${\\cal O}_K$\nand $Z\\subset X$\nbe a closed subset.\nLet\n${\\cal F}$ be a constructible\ncomplex of $\\Lambda$-modules\non $X$.\nAssume that \nfor every separated morphism\n$V\\to Y$ of\nregular schemes of finite\ntype over ${\\cal O}_K$,\nthe projection\n$h\\colon W=X\\times_YV\\to X$ is ${\\cal F}$-transversal\non a neighborhood of $h^{-1}(Z)$.\n\n{\\rm 1.}\nLet\n\\begin{equation}\n\\begin{CD}\nX@<{p'}<< X'@<{j'}<<W\\\\\n@VfVV @VV{f'}V@VV{f'_V}V\\\\\nY@<p<<Y'@<j<<V\n\\end{CD}\n\\label{eqXYUpi}\n\\end{equation}\nbe a cartesian diagram of\nregular schemes\nof finite type over ${\\cal O}_K$.\nAssume  that\n$p$ is proper\nand that $j\\colon V=Y'\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} D\n\\to Y'$ is the open immersion\nof the complement of\na divisor $D$ with simple normal crossings.\nThen,\nthe composition\n\\begin{equation}\n\\begin{CD}\n{\\cal F}\\otimes f^*R(pj)_*\\Lambda\n\\to \n{\\cal F}\\otimes R(p'j')_*\\Lambda\n@>{\\rm (\\ref{eqpr0})}>>\nR(p'j')_*(p'j')^*{\\cal F}\n\\end{CD}\n\\label{eqYU}\n\\end{equation}\nwhere the first morphism is\ninduced by the base change morphism\nis an isomorphism\non a neighborhood of $Z$.\n\n\n\n{\\rm 2.}\n$f$ is universally ${\\cal F}$-acyclic\nalong $Z$.\n\\end{pr}\n\nFor the sake of completeness,\nwe record the proof in \\cite{CC}\nwith more detail.\n\n\\proof{\n1. \nLet \n$D_1,\\ldots,D_n$\nbe the irreducible components\nof $D$. For a subset $I\\subset \\{1,\\ldots,n\\}$,\nlet $X'_I=X'\\times_{Y'}(\\bigcap_{i\\in I}D_i)$\nand let $i'_I\\colon X'_I\\to X'$\nbe the closed immersion.\nBy the assumption,\n$p'\\colon X'\\to X$\nand $p'i'_I\\colon X'_I\\to X$\nare ${\\cal F}$-transversal\non neighborhoods of\nthe inverse images of $Z$.\n\nLet ${\\cal F}'=p'^*{\\cal F}$.\nSince the assumption\non $Rh^!\\Lambda$\nin Proposition \\ref{prhF}.1\nis satisfied by the absolute\npurity \\cite[{\\sc Th\\'eor\\`eme 3.1.1}]{purete},\nthe immersions \n$i'_I\\colon X'_I\\to X'$\nare ${\\cal F}'$-transversal\non neighborhoods of\nthe inverse images of $Z$\nby Proposition \\ref{prhF}.1.\nHence by Lemma \\ref{lmiZ},\nthe canonical morphism\n${\\cal F}'\n\\otimes Rj'_*\\Lambda\n\\to Rj'_*j'^*{\\cal F}'$ (\\ref{eqpr0}) is an\nisomorphism \non a neighborhood of $p'^{-1}(Z)$.\nSince $p'$ is proper,\nwe obtain an isomorphism\n$Rp'_*({\\cal F}'\n\\otimes Rj'_*\\Lambda)\n\\to R(pj')_*(pj')^*{\\cal F}$\non a neighborhood of $Z$.\n\n\nBy the projection formula\n(\\ref{eqprj}),\nwe have a canonical isomorphism\n${\\cal F}\n\\otimes Rp'_*Rj'_*\\Lambda\n\\to Rp'_*({\\cal F}'\n\\otimes Rj'_*\\Lambda)$.\nThe base change morphism\n$f^*R(pj)_*\\Lambda\\to\nRp'_*Rj'_*\\Lambda$ is an isomorphism\nby the smooth base change\ntheorem\n\\cite[Corollaire 1.2]{smbc}.\nHence the morphism (\\ref{eqYU})\nis an isomorphism on a neighborhood of $Z$.\n\n{\\rm 2.}\nIt suffices to show that\nfor a smooth morphism\n$Y'\\to Y$,\nthe base change\n$X'\\to Y'$ of $f$\nis locally acyclic with respect to the \npull-back of ${\\cal F}$ by Lemma \\ref{lmlac}.3.\nSimilarly as in the proof of 1.,\nthe assumption is satisfied\nfor the pull-back $Y'\\to Y$.\nHence, \nby replacing $Y$ by $Y'$,\nit suffices to show\nthat\n$f$ is locally acyclic with respect to ${\\cal F}$.\n\n\nLet $s\\gets t$ be a specialization\nof geometric points of $Y$\nas in Lemma \\ref{lmlac}.1\nand let the notation be as loc.~cit.\nBy \\cite[Theorem 4.1,\nTheorem 8.2]{dJ},\nwe may write $t$ as a limit\n$\\varprojlim_\\lambda\nU_\\lambda$\nof the complements $U_\\lambda=Y_\\lambda\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} \nD_\\lambda$,\nin regular schemes $Y_\\lambda$\nendowed with a\nproper, surjective\nand generically finite\nmorphism $p_\\lambda\\colon Y_\\lambda\n\\to Y$ \nof divisors $D_\\lambda\\subset\nY_\\lambda$ with simple normal crossings.\nThen, as the limit of\n(\\ref{eqYU}), the canonical morphism\n\\begin{equation}\n{\\cal F}\\otimes f_{(s)}^*\nRj_{t*}j^*_t\\Lambda\n\\to \nRj'_{t*}j^{\\prime*}_t{\\cal F}\n\\label{eqijst2}\n\\end{equation}\nis an isomorphism\non the inverse image of $Z$.\nSince $Y$ is normal,\nthe canonical morphism\n$\\Lambda\\to\ni_s^*Rj_{t*}j^*_t\\Lambda$\nis an isomorphism.\nHence the isomorphism\n(\\ref{eqijst2})\ninduces an isomorphism\n(\\ref{eqijst})\non the inverse image of $Z$.\n\\qed\n\n}\n\n\\begin{cor}\\label{corlc}\nLet $X$\nbe a regular scheme\nof finite type over \na discrete valuation ring\n${\\cal O}_K$\nand\n$Z\\subset X$ be a closed subset.\nLet\n${\\cal F}$ be a constructible\ncomplex of $\\Lambda$-modules\non $X$.\nAssume that every separated morphism\n$h\\colon W\\to X$ of\nregular schemes\nof finite type over ${\\cal O}_K$\nis ${\\cal F}$-transversal\non a neighborhood of\nthe inverse image $h^{-1}(Z)$.\nThen ${\\cal F}$ is locally constant\non a neighborhood of $Z$.\n\\end{cor}\n\n\\proof{\nBy Proposition \\ref{prla1}\napplied to $1_X\\colon X\\to X$,\nthe identity $1_X\\colon X\\to X$\nis ${\\cal F}$-acyclic\nalong $Z$.\nHence ${\\cal F}$ is locally constant\non a neighborhood of $Z$\nby Lemma \\ref{lmla}.2.\n\\qed\n\n}\n\n\\medskip\n\nWe have a partial converse of\nProposition \\ref{prla1}\nnot used in the article.\n\n\\begin{pr}[{\\cite[Corollary 8.10]{CC}}]\\label{prla2}\nLet $f\\colon X\\to Y$\nbe a smooth morphism\nof noetherian schemes\nand let\n${\\cal F}$ be a constructible\ncomplex of $\\Lambda$-modules\non $X$.\nLet $i\\colon Z\\to Y$ be an immersion\nand let\n$$\\begin{CD}\nX@<h<< W\\\\\n@VfVV@VVV\\\\\nY@<i<< Z\n\\end{CD}$$\nbe a cartesian diagram.\nIf $f\\colon X\\to Y$ is ${\\cal F}$-acyclic,\nthen $h\\colon W\\to X$ is \n${\\cal F}$-transversal.\n\\end{pr}\n\nFor the sake of convenience,\nwe record the proof in \\cite{CC}.\n\n\\proof{\nWe may assume that $i\\colon Z\\to Y$\nis a closed immersion.\nLet $V=Y\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} Z$\nand consider the cartesian diagram\n\\begin{equation}\n\\begin{CD}\nW@>h>>X@<{j'}<< U\\\\\n@VgVV @VfVV @VV{f_V}V\\\\\nZ@>i>>Y@<j<<V.\n\\end{CD}\n\\label{eqZYV}\n\\end{equation}\nBy  \\cite[Proposition 2.10]{app} applied\nto the right square,\nwe obtain an isomorphism\n${\\cal F}\\otimes f^*Rj_*\\Lambda\n\\to \nRj'_*j^{\\prime*}{\\cal F}$.\nSince $f$ is smooth,\nthis induces an isomorphism\n${\\cal F}\\otimes Rj'_*\\Lambda\n\\to \nRj'_*j^{\\prime*}{\\cal F}$\nby smooth base change theorem\n\\cite[Corollaire 1.2]{smbc}.\nHence the assertion follows\nby Lemma \\ref{lmiZ}.2.\n\\qed\n\n}\n\n\n\\begin{cor}\\label{corlabc}\nLet $$\\begin{CD}\nV'@>{g'}>>X'@>{f'}>>Y'\\\\\n@V{h_V}VV@VhVV@VV{h'}V\\\\\nV@>g>> X@>f>>Y\n\\end{CD}$$\nbe a cartesian diagram of\nmorphisms of finite type of schemes\nsuch that \n$f\\colon X\\to Y$ is smooth\nand that the vertical arrows are\nseparated.\nAssume that\n$Rh^!\\Lambda$ is locally constant\nof support $X'$\nand that the base change\nmorphism\n$g'^*Rh^!\\Lambda\n\\to Rh_V^!\\Lambda$\nis an isomorphism.\n\nLet ${\\cal G}$\nbe a constructible complex\nof $\\Lambda$-modules on $V$\nand \nassume that $f$\nis $Rg_*{\\cal G}$-acyclic\nand that $fg$ is\n${\\cal G}$-acyclic.\nThen, \nthe base change morphism\n\\begin{equation}\nh^*Rg_*{\\cal G}\n\\to \nRg'_*h_V^*{\\cal G}\n\\label{eqbcj}\n\\end{equation}\nis an isomorphism.\n\\end{cor}\n\n\\proof{\nSince $f$ is $Rg_*{\\cal G}$-acyclic\nand $fg$ is ${\\cal G}$-acyclic,\nby Proposition \\ref{prla2},\n$h$ is $Rg_*{\\cal G}$-transversal\nand\n$h_V$ is ${\\cal G}$-transversal.\nHence the assertion follows\nfrom Proposition \\ref{prhF}.2.\n\\qed\n\n}\n\n\\section{$C$-transversality}\\label{sTX}\n\n\nIn this section,\nfirst we define \nthe FW-cotangent bundle\nof a regular scheme,\nas a vector bundle\non the closed subscheme \ndefined by $p=0$.\nThen, \nwe study properties\nof morphisms with respect to\nits closed conical subsets\ncorresponding to the transversality\nand the local\nacyclicity studied in Section \\ref{sF}.\n\nFirst in Section \\ref{ssFW},\nwe recall basic properties\nof the sheaf $F\\Omega^1_X$ \nof Frobenius-Witt differentials\nfrom \\cite{FW}.\nIn particular if $X$ is regular,\nunder a certain finiteness condition,\nthe sheaf $F\\Omega^1_X$\nis a locally free \n${\\cal O}_{X_{{\\mathbf F}_p}}$-module\nof rank $\\dim X$ on\n$X_{{\\mathbf F}_p}=\nX\\times_{{\\rm Spec}\\, {\\mathbf Z}}\n{\\rm Spec}\\, {\\mathbf F}_p$.\nUnder this condition,\nwe define the FW-cotangent bundle\n$FT^*X|_{X_{{\\mathbf F}_p}}$ on $X_{{\\mathbf F}_p}$\nas the vector bundle\nassociated to the locally free\n${\\cal O}_{X_{{\\mathbf F}_p}}$-module\n$F\\Omega^1_X$.\n\n\nWe study properties\nof morphisms with respect to \na given closed conical subset in\nSections \\ref{ssCtr} and \\ref{ssCac}.\nIn Section \\ref{ssCtr},\nwe study the transversality\nfor morphisms to $X$.\nIn Section \\ref{ssCac},\nwe study the acyclicity,\nwhich was also called transversality,\nfor morphisms from $X$.\n\n\n\n\n\\subsection{FW-cotangent bundle}\n\\label{ssFW}\n\n\n\\begin{df}[{\\rm \\cite[Definition 1.1]{FW}}]\\label{dfFW}\nLet $p$ be a prime number.\n\n{\\rm 1.}\nDefine a polynomial\n$P\\in {\\mathbf Z}[X,Y]$\nby\n\\begin{equation}\nP=\n\\sum_{i=1}^{p-1}\n\\dfrac{(p-1)!}{i!(p-i)!}\\cdot\nX^iY^{p-i}.\n\\label{eqP}\n\\end{equation}\n\n{\\rm 2.}\nLet $A$ be a ring\nand $M$ be an $A$-module.\nWe say that a mapping\n$w\\colon A\\to M$\nis an Frobenius-Witt derivation\nor FW-derivation for short\nif the following condition is\nsatisfied:\nFor any $a,b\\in A$, we have\n\\begin{align}\nw(a+b)\\, &=\nw(a)+\nw(b)\n-P(a,b)\n\\cdot w(p),\n\\label{eqadd}\\\\\nw(ab)\\, &=\nb^p\\cdot w(a)+\na^p\\cdot w(b).\n\\label{eqLb}\n\\end{align}\n\\end{df}\n\n\nDefinition \\ref{dfFW}.2\nis essentially the same\nas \\cite[Definition 2.1.1]{DKRZ}.\nWe recall some results from \\cite{FW}.\n\n\\begin{lm}\\label{lmOm}\nLet $p$ be a prime number and\n$A$ be a ring.\n\n{\\rm 1.\n(\\cite[Lemma 2.1.1]{FW})}\nThere exists a universal pair\nof an $A$-module\n$F\\Omega^1_A$\nand an FW-derivation\n$w\\colon A\n\\to F\\Omega^1_A$.\n\n{\\rm 2.\n(\\cite[Corollary 2.3.1]{FW})}\nIf $A$ is a ring over ${\\mathbf Z}_{(p)}$,\nwe have $p\\cdot F\\Omega^1_A=0$.\n\n{\\rm 3.\n(\\cite[Corollary 2.3.2]{FW})}\nIf $A$ is a ring over ${\\mathbf F}_{p}$,\nthen there exists a canonical\nisomorphism $F\\Omega^1_A\n\\to F^*\\Omega^1_A\n=\\Omega^1_A\\otimes_AA$\nto the tensor product\nwith respect to the absolute\nFrobenius morphism $A\\to A$.\n\\end{lm}\n\n\n\nWe call $F\\Omega^1_A$\nthe module of FW-differentials of $A$\nand $w(a)\\in F\\Omega^1_A$\nthe FW-differential of $a\\in A$.\nFor a morphism $A\\to B$ of rings,\nwe have a canonical $B$-linear morphism\n$F\\Omega^1_A\\otimes_AB\n\\to\nF\\Omega^1_B$.\n\nWe may sheafify the construction\nand define $F\\Omega^1$\nas a quasi-coherent ${\\cal O}_X$-module\nfor a scheme $X$. \nWe call $F\\Omega^1_X$\nthe sheaf of FW-differentials on $X$.\nIf $X$ is a scheme over\n${\\mathbf Z}_{(p)}$,\nthe ${\\cal O}_X$-module\n$F\\Omega^1_X$ is\nan ${\\cal O}_{X_{{\\mathbf F}_p}}$-module\nwhere\n$X_{{\\mathbf F}_p}\n=X\\times_{{\\rm Spec}\\, {\\mathbf Z}_{(p)}}\n{\\rm Spec}\\, {\\mathbf F}_p$.\nFurther if $X$ is noetherian\nand if $X_{{\\mathbf F}_p}$\nis of finite type over a field\nof finite $p$-basis,\nthen $F\\Omega^1_X$ is\na coherent \n${\\cal O}_{X_{{\\mathbf F}_p}}$-module\nby \\cite[Lemma 4.1.2]{FW}.\nIf $X$ is a scheme over\n${\\mathbf F}_p$,\nwe have a canonical isomorphism\n\\begin{equation}\nF\\Omega^1_X\n\\to F^*\\Omega^1_X\n\\label{eqFFX}\n\\end{equation}\nto the pull-back by\nthe absolute Frobenius morphism\n$F\\colon X\\to X$,\nsending $w(a)$ to $da$.\n\n\nFor a morphism \n$f\\colon X\\to Y$ of schemes,\nwe have a canonical morphism\n\\begin{equation}\nf^*F\\Omega^1_Y\\to\nF\\Omega^1_X\n\\label{eqFXY}\n\\end{equation}\n\n\n\n\n\n\n\n\\begin{pr}[{\\rm \\cite[Proposition 2.4]{FW}}]\\label{prdx}\nLet $X$ be a scheme\nand $x\\in X$\nbe a point such that\nthe residue field $k(x)={\\cal O}_{X,x}\/\n{\\mathfrak m}_{X,x}$\nis of characteristic $p$.\nFor a $k(x)$-vector space $M$,\nlet $F^*M$\ndenote the tensor product\n$M\\otimes_{k(x)}k(x)$\nwith respect to the Frobenius\n$F\\colon k(x)\\to k(x)$.\nThen,  we have an exact\nsequence \n\\begin{equation}\n\\begin{CD}\n0@>>>\nF^*({\\mathfrak m}_{X,x}\/\n{\\mathfrak m}_{X,x}^2)\n@>{w}>>\nF\\Omega^1_{X,x}\n\\otimes_{{\\cal O}_{X,x}} k(x)\n@>{\\rm (\\ref{eqFFX})}>>\nF^*\\Omega^1_{k(x)}\n@>>>0\n\\end{CD}\n\\label{eqdx}\n\\end{equation}\nof $k(x)$-vector spaces.\n\\end{pr}\n\n\n\n\\begin{pr}[{\\rm \\cite[Proposition 2.8]{FW}}]\\label{prsm}\nLet $f\\colon X\\to Y$\nbe a morphism of finite type of\nregular noetherian schemes\nover ${\\mathbf Z}_{(p)}$.\nThen the following conditions\nare equivalent:\n\n{\\rm (1)}\n$f\\colon X\\to Y$ is smooth\non a neighborhood of\n$X_{{\\mathbf F}_p}$.\n\n{\\rm (2)}\nThe sequence \n\\begin{equation}\n\\begin{CD}\n0@>>>\nf^*F\\Omega^1_Y\n@>{\\rm (\\ref{eqFXY})}>>\nF\\Omega^1_X\n@>{\\rm (\\ref{eqFFX})}>>F^*\\Omega^1_{\nX_{{\\mathbf F}_p}\/\nY_{{\\mathbf F}_p}}\n@>>>\n0\n\\end{CD}\n\\end{equation}\nof ${\\cal O}_{X_{{\\mathbf F}_p}}$-modules\nis a locally split exact sequence.\n\\end{pr}\n\n\n\\begin{thm}[{\\rm \\cite[Theorem 3.1]{FW}}]\n\\label{thmreg}\nLet $X$ be a noetherian scheme\nover ${\\mathbf Z}_{(p)}$\nand $X_{{\\mathbf F}_p}=\nX\\times_{{\\rm Spec}\\, \n{\\mathbf Z}_{(p)}}\n{\\rm Spec}\\, {\\mathbf F}_p$\nbe the closed subscheme.\nAssume that the reduced\npart $X_{{\\mathbf F}_p,{\\rm red}}$ is\na scheme of finite type over\na field $k$ with finite $p$-basis.\nIf $X$ is regular and \nis equi-dimensional\nof dimension $n$ and if\n$[k:k^p]=p^r$, then \nthe ${\\cal O}_{X_{{\\mathbf F}_p}}$-module\n$F\\Omega^1_X$\nis locally free of rank $n+r$.\n\\end{thm}\n\n\n\n\n\\begin{cor}[{\\rm \\cite[Corollary 2.6,\nCorollary 3.2]{FW}}]\n\\label{corXZ}\nLet $X$ be a regular noetherian scheme\nover ${\\mathbf Z}_{(p)}$\nsuch that the reduced\npart $X_{{\\mathbf F}_p,{\\rm red}}$ \nof\n$X_{{\\mathbf F}_p}=X\\times_{{\\rm Spec}\\, \n{\\mathbf Z}_{(p)}}\n{\\rm Spec}\\, {\\mathbf F}_p$\nis\na scheme of finite type over\na field $k$ of finite $p$-basis.\nLet $Z\\subset X$ be a closed subscheme.\n\nWe consider the following conditions:\n\n{\\rm (1)}\n$Z$ is regular on a neighborhood of\n$Z_{{\\mathbf F}_p}=Z\\times_{{\\rm Spec}\\, \n{\\mathbf Z}_{(p)}}\n{\\rm Spec}\\, {\\mathbf F}_p$.\n\n{\\rm (1$'$)}\nAt every point $x\\in Z\n_{{\\mathbf F}_p}$,\nthe local ring\n${\\cal O}_{Z,x}$ is regular.\n\n{\\rm (2)}\nThe sequence \n\\begin{equation}\n\\begin{CD}\n0@>>>\nF^*(N_{Z\/X}\\otimes_{{\\cal O}_Z}\n{\\cal O}_{Z_{{\\mathbf F}_p}})\n@>w>>\nF\\Omega^1_X\\otimes_{{\\cal O}_X}\n{\\cal O}_{Z_{{\\mathbf F}_p}}\n\\longrightarrow\nF\\Omega^1_Z\n@>>>\n0\n\\end{CD}\n\\label{eqXZ}\n\\end{equation}\nof \n${\\cal O}_{Z_{{\\mathbf F}_p}}$-modules\nis a locally splitting exact sequence.\n\nThen, we have\n{\\rm (1)}$\\Rightarrow${\\rm (2)}$\\Rightarrow${\\rm (1$'$)}.\nConsequently \nif the subset\n${\\rm Reg}(Z)\n\\subset Z$ consisting\nof regular points is an open subset,\nthe 3 conditions are equivalent.\n\\end{cor}\n\n\n\\proof{\nThe implications\n{\\rm (1)}$\\Rightarrow${\\rm (2)} and\n{\\rm (2)}$\\Rightarrow${\\rm (1$'$)}\nare proved in \n\\cite[Corollary 3.2]{FW} and in\n\\cite[Corollary 2.6.1]{FW}\nrespectively.\nSince (1$'$) means\n$Z_{{\\mathbf F}_p}\n\\subset {\\rm Reg}(Z)$,\nthe last assertion follows.\n\\qed\n\n}\n\n\n\n\n\n\\begin{df}\\label{dfFTX}\nLet $k$ be a perfect field \nof characteristic $p>0$\nand let $X$ be\na regular noetherian scheme\nsatisfying the following condition:\n\n{\\rm (F)}\n$X_{{\\mathbf F}_p}\n=X\\times_{{\\rm Spec}\\, {\\mathbf Z}}\n{\\rm Spec}\\, {\\mathbf F}_p$\nis a scheme of\nfinite type over $k$.\n\n\\noindent\nThen, we define\nthe FW-cotangent bundle\n$FT^*X|_{X_{{\\mathbf F}_p}}$\nof $X$ to be the vector bundle\non $X_{{\\mathbf F}_p}$\nassociated with the locally free\n${\\cal O}_{X_{{\\mathbf F}_p}}$-module\n$F\\Omega^1_X$\nof rank $\\dim X$.\n\\end{df}\n\nLet $x\\in X_{{\\mathbf F}_p}$\nbe a closed point \nand let\n$T^*_xX$\ndenote the cotangent space\nat $x$ defined as a scheme\n${\\rm Spec}\\,\nS_{k(x)}({\\mathfrak m}_x\/\n{\\mathfrak m}_x^2)^\\vee$\nassociated to the\n$k(x)$-vector space\n${\\mathfrak m}_x\/\n{\\mathfrak m}_x^2$.\nSince $k(x)$ is perfect,\nthe exact sequence\n(\\ref{eqdx}) defines a canonical\nisomorphism\n\\begin{equation}\nF^*T^*_xX\n\\to \nFT^*X|_x\n\\label{eqTx}\n\\end{equation}\nto the fiber of \nthe FW-cotangent bundle\nat $x$ \nfrom the pull-back by Frobenius\n$F\\colon x\\to x$ of \n$T^*_xX$.\nIf $X=X_{{\\mathbf F}_p}$,\nthen the FW-cotangent bundle\n$FT^*X|_{X_{{\\mathbf F}_p}}$\nis the pull-back of\nthe cotangent bundle\n$T^*X$ \nby the Frobenius morphism\n$F\\colon X\\to X$\nby {\\rm (\\ref{eqFFX})}.\n\n\nLet $X\\to Y$ be a morphism\nof finite type\nof regular noetherian schemes\nsatisfying the condition (F)\nin Definition \\ref{dfFTX}.\nThen, the morphism (\\ref{eqFXY})\ndefines morphisms\n\\begin{equation}\n\\begin{CD}\nFT^*X|_{X_{{\\mathbf F}_p}}\n@<{f^*}<<\nFT^*Y|_{Y_{{\\mathbf F}_p}}\n\\times_{Y_{{\\mathbf F}_p}}\nX_{{\\mathbf F}_p}\n@>>>\nFT^*Y|_{Y_{{\\mathbf F}_p}}\n\\end{CD}\n\\label{eqdXY}\n\\end{equation}\n of\nschemes.\n\nAssume that $X\\to Y$ is\nsmooth and let \n$F^*T^*X\/Y|_{\nX_{{\\mathbf F}_p}}$ denote the\npull-back by the Frobenius\n$F\\colon X_{{\\mathbf F}_p}\n\\to X_{{\\mathbf F}_p}$\nof the restriction to\n$X_{{\\mathbf F}_p}$ of \nthe vector\nbundle defined $T^*X\/Y$ by\nthe locally free ${\\cal O}_X$-module\n$\\Omega^1_{X\/Y}$.\nThen, by Proposition \\ref{prsm},\nwe have an exact sequence\n\\begin{equation}\n0\\to FT^*Y|_{Y_{{\\mathbf F}_p}}\n\\times_{Y_{{\\mathbf F}_p}}\nX_{{\\mathbf F}_p}\n\\to FT^*X|_{X_{{\\mathbf F}_p}}\n\\to F^*T^*X\/Y|_{\nX_{{\\mathbf F}_p}}\\to 0\n\\label{eqTEf}\n\\end{equation}\nof vector bundles on $X_{{\\mathbf F}_p}$.\n\nSimilarly,\nlet $Z\\to X$ be a closed immersion\nof regular noetherian schemes\nsatisfying the condition (F).\nLet ${\\cal I}_Z\\subset {\\cal O}_X$\nbe the ideal sheaf\nand let $T^*_ZX$ be\nthe conormal bundle\ndefined by\nthe locally free ${\\cal O}_Z$-module\n${\\cal I}_Z\/{\\cal I}_Z^2$.\nLet \n$F^*T^*_ZX|_{\nZ_{{\\mathbf F}_p}}$ denote the\npull-back by the Frobenius\n$F\\colon Z_{{\\mathbf F}_p}\n\\to Z_{{\\mathbf F}_p}$\nof the restriction to\n$Z_{{\\mathbf F}_p}$.\nThen, by Corollary \\ref{corXZ},\nwe have an exact sequence\n\\begin{equation}\n0\\to F^*T^*_ZX|_{\nZ_{{\\mathbf F}_p}}\\to \nFT^*X|_{Z_{{\\mathbf F}_p}}\n\\to \nFT^*Z|_{Z_{{\\mathbf F}_p}}\n\\to 0\n\\label{eqTEi}\n\\end{equation}\nof vector bundles on $Z_{{\\mathbf F}_p}$.\n\n\n\\subsection{$C$-transversality}\\label{ssCtr}\n\nIn the rest of this section,\nwe fix a perfect field $k$\nof characteristic $p>0$.\n\nWe fix some terminology\non closed conical subsets of\na vector bundle of a scheme.\nLet $V$ be a vector bundle\nover a scheme $Y$.\nWe say that a closed subset\nof $V$ is conical if it is\nstable under the action of\n${\\mathbf G}_{m,Y}$.\nFor a closed conical subset\n$C\\subset V$,\nthe intersection\n$B=C\\cap Y$ with the\n$0$-section $Y\\subset V$ regarded\nas a closed subset of $Y$\nis called the base of $C$.\nThe base $B$ equals the\nimage of $C$ by\nthe projection $V\\to Y$.\n\n\nWe say that a separated\nmorphism $f\\colon X\\to Y$\nof finite type of schemes\nis proper on a closed subset $Z\\subset X$\nif for every base change\n$f'\\colon X'\\to Y'$ of $f$\nits restriction to\nthe inverse image $Z'\n\\subset X'$ is a closed mapping.\nFor a morphism\n$V\\to V'$ of vector bundles\non a scheme $Y$\nand a closed conical subset\n$C$ of $V$,\nthe morphism $V\\to V'$\nis proper on $C$ if and only\nif the intersection\n$C\\cap {\\rm Ker}(V\\to V')$\nis a subset of the $0$-section of $V$\nby \\cite[Lemma 1.2(ii)]{Be}.\n\n\n\\begin{df}\\label{dfhC}\nLet $X$ be a \nregular noetherian scheme\nsatisfying the condition {\\rm (F)}\nin Definition {\\rm \\ref{dfFTX}}\nand let $C\\subset FT^*X|_{X_{{\\mathbf F}_p}}\n$ be a closed\nconical subset\nof the FW-cotangent bundle.\nLet $h\\colon W\\to X$\nbe a morphism of finite type\nof regular schemes.\n\n\n{\\rm 1.}\n{\\rm (\\cite[1.2]{Be}, \\cite[Definition 3.3]{CC})}\nWe say that \n$h\\colon W\\to X$ is $C$-transversal\nif \nthe intersection of\n$h^*C=C\\times_XW\n\\subset \nFT^*X|_{X_{{\\mathbf F}_p}}\n\\times_{X_{{\\mathbf F}_p}}\nW_{{\\mathbf F}_p}$\nwith the kernel \n${\\rm Ker}(\nFT^*X|_{X_{{\\mathbf F}_p}}\n\\times_{X_{{\\mathbf F}_p}}\nW_{{\\mathbf F}_p}\n\\to\nFT^*W|_{W_{{\\mathbf F}_p}})$\nis a subset of the $0$-section.\n\n{\\rm 2.}\nAssume that $h$ is\n$C$-transversal.\nThen we define\na closed conical subset \n$h^\\circ C\n\\subset FT^*W|_{W_{{\\mathbf F}_p}}$\nto be the image\nof $h^*C$ by\n$\nFT^*X|_{X_{{\\mathbf F}_p}}\n\\times_{X_{{\\mathbf F}_p}}\nW_{{\\mathbf F}_p}\n\\to\nFT^*W|_{W_{{\\mathbf F}_p}}$.\n\\end{df}\n\nExample.\nLet $Z\\subset X$ be a regular closed subscheme.\nThen a closed\nconical subset \n$C=F^*T^*_ZX|_{Z_{{\\mathbf F}_p}}\n\\subset \nFT^*X|_{X_{{\\mathbf F}_p}}$\nis defined by (\\ref{eqTEi}).\nIn particular,\nfor $Z=X$,\nthe $0$-section\n$F^*T^*_XX|_{X_{{\\mathbf F}_p}}\n=X_{{\\mathbf F}_p}$ is\na closed conical subset of\n$FT^*X|_{X_{{\\mathbf F}_p}}$.\n\n\n\n\\begin{lm}\\label{lmTXC}\nLet $X$ be a \nregular noetherian scheme\nsatisfying the condition {\\rm (F)}\nin Definition {\\rm \\ref{dfFTX}}\nand let $C\\subset FT^*X|_{X_{{\\mathbf F}_p}}\n$ be a closed\nconical subset.\nLet $h\\colon W\\to X$\nbe a morphism of finite type\nof regular schemes.\n\n{\\rm 1.}\nLet $C=FT^*X|_Z$\nbe the restriction to \na closed subset\n$Z\\subset X_{{\\mathbf F}_p}$\nof the closed fiber.\nIf $h$ is $C$-transversal,\nthen $h$ is smooth\non a neighborhood of\nthe inverse image $h^{-1}(Z)$.\n\n{\\rm 2.}\nIf $C$ is the $0$-section\nof $FT^*X|_{X_{{\\mathbf F}_p}}$,\nthen $h$ is $C$-transversal.\n\n{\\rm 3.}\nIf $h$ is smooth,\nfor any closed conical subset\n$C$ of $FT^*X|_{X_{{\\mathbf F}_p}}$,\nthe morphism\n$h$ is $C$-transversal.\n\\end{lm}\n\n\n\\proof{\n1.\nThe condition that the\nintersection of\n$h^*C=FT^*X|_Z\n\\times_{X_{{\\mathbf F}_p}}\nW_{{\\mathbf F}_p}\n=FT^*X\n\\times_{X_{{\\mathbf F}_p}}\nh^{-1}(Z)$\nwith the kernel\n${\\rm Ker}(FT^*X|_{X_{{\\mathbf F}_p}}\n\\times_{X_{{\\mathbf F}_p}}\nW_{{\\mathbf F}_p}\n\\to FT^*W|_{W_{{\\mathbf F}_p}})$\nis a subset of the $0$-section\nmeans that\n$F\\Omega^1_X\n\\otimes_{{\\cal O}_X}{\\cal O}_W\n\\to F\\Omega^1_W$\nis a locally splitting injection\non a neighborhood of $h^{-1}(Z)$.\nBy Proposition \\ref{prsm},\nthis means that\n$W\\to X$\nis smooth on a neighborhood of\nthe inverse image $h^{-1}(Z)$.\n\n\n2.\nIf $C$ is the $0$-section,\nits intersection with the\nkernel \n${\\rm Ker}(FT^*X|_{X_{{\\mathbf F}_p}}\n\\times_{X_{{\\mathbf F}_p}}\nW_{{\\mathbf F}_p}\n\\to FT^*W|_{W_{{\\mathbf F}_p}})$\nis also the $0$-section.\n\n\n\n3.\nIf $h$ is smooth,\nthe morphism $FT^*X|_{X_{{\\mathbf F}_p}}\n\\times_{X_{{\\mathbf F}_p}}\nW_{{\\mathbf F}_p}\n\\to FT^*W|_{W_{{\\mathbf F}_p}}$ is an injection\nby Proposition \\ref{prsm}.\nHence \nfor any subset\n$C\\subset FT^*X|_{X_{{\\mathbf F}_p}}$,\nits intersection with the\nkernel \n${\\rm Ker}(FT^*X|_{X_{{\\mathbf F}_p}}\n\\times_{X_{{\\mathbf F}_p}}\nW_{{\\mathbf F}_p}\n\\to FT^*W|_{W_{{\\mathbf F}_p}})$\nis a subset of the $0$-section.\n\\qed\n\n}\n\n\\begin{lm}\\label{lmhC}\nLet $h\\colon W\\to X$\nbe a morphism of\nfinite type of \nregular noetherian schemes\nsatisfying the condition {\\rm (F)}\nand let $C$ be a closed\nconical subset of $FT^*X|_{X_{{\\mathbf F}_p}}$.\nAssume that $h$ is\n$C$-transversal.\nThen, for a morphism \n$g\\colon V\\to W$ of finite type of\nregular noetherian schemes\nthe following conditions\nare equivalent:\n\n{\\rm (1)}\nThe morphism\n$g$ is $h^\\circ C$-transversal.\n\n{\\rm (2)}\nThe composition\n$hg$ is $C$-transversal.\n\n\\noindent\nIf these equivalent conditions\nare satisfied,\nwe have $(hg)^\\circ C=g^\\circ h^\\circ C$.\n\\end{lm}\n\n\\proof{\nThe condition (1)\nmeans that\nthe intersection\n$h^*C\\cap {\\rm Ker}(\nFT^*X|_{X_{{\\mathbf F}_p}}\n\\times_{X_{{\\mathbf F}_p}}\nW_{{\\mathbf F}_p}\n\\to FT^*W|_{W_{{\\mathbf F}_p}})$\nis a subset of the $0$-section\nand further that\nfor $h^\\circ C\n\\subset FT^*W|_{W_{{\\mathbf F}_p}}$,\nthe intersection\n$g^*h^\\circ C\\cap {\\rm Ker}(\nFT^*W|_{W_{{\\mathbf F}_p}}\n\\times_{W_{{\\mathbf F}_p}}\nV_{{\\mathbf F}_p}\n\\to FT^*V|_{V_{{\\mathbf F}_p}})$\nis a subset of the $0$-section.\nThis means that\n$(hg)^*C\\cap {\\rm Ker}(\nFT^*X|_{X_{{\\mathbf F}_p}}\n\\times_{X_{{\\mathbf F}_p}}\nV_{{\\mathbf F}_p}\n\\to FT^*V|_{V_{{\\mathbf F}_p}})$\nis a subset of the $0$-section,\nnamely the condition (2).\n\nThe image of \n$(hg)^*C$ by\n$FT^*X|_{X_{{\\mathbf F}_p}}\n\\times_{X_{{\\mathbf F}_p}}\nV_{{\\mathbf F}_p}\n\\to FT^*V|_{V_{{\\mathbf F}_p}}$\nequals \nthe image of $g^*h^\\circ C$\nby \n$FT^*W|_{W_{{\\mathbf F}_p}}\n\\times_{W_{{\\mathbf F}_p}}\nV_{{\\mathbf F}_p}\n\\to FT^*V|_{V_{{\\mathbf F}_p}}$.\n\\qed\n\n}\n\n\\medskip\nThe terminology transversality\nis related to the transversality\nof morphisms of regular\nschemes defined as follows.\n\n\n\n\n\n\\begin{df}\\label{dftrans}\nLet $f\\colon X\\to Y$\nand $g\\colon V\\to Y$\nbe morphisms of finite type of\nregular schemes\nand set $W\n=X\\times_YV$.\n\n\n\n{\\rm 1.}\nLet $w\\in W$\nand $x\\in X,y\\in Y,v\\in V$\nbe the images.\nWe say that \n$f$ and $g$ are transversal\nat $w$, if ${\\cal O}_{W,w}$\nis regular and if\n$Tor_q^{{\\cal O}_{Y,y}}\n({\\cal O}_{X,x},{\\cal O}_{V,v})=0$\nfor $q>0$.\n\n{\\rm 2.}\nLet $W_1\\subset W$\nbe an open subscheme.\nWe say that \n$f$ and $g$ are transversal on\n$W_1$ if \n$f$ and $g$ are transversal \nat every point of $W_1$.\n\\end{df}\n\nExample.\nLet $Z\\subset X$ be a regular closed subscheme\nand \n$C=F^*T^*_ZX|_{Z_{{\\mathbf F}_p}}\n\\subset \nFT^*X|_{Z_{{\\mathbf F}_p}}$\nbe the closed\nconical subset defined by the conormal bundle.\nThen, as we will see\nin Corollary \\ref{corfC},\na morphism\n$h\\colon W\\to X$ of\nfinite type \nof regular quasi-excellent\nnoetherian schemes\nis $C$-transversal\nif and only if\n$h\\colon W\\to X$\nis transversal to $Z\\subset X$\non a neighborhood of\nthe closed fiber $W_{{\\mathbf F}_p}$.\n\nIn particular, \nif $X$ is smooth over \na discrete valuation \nring ${\\cal O}_K$ of mixed\ncharacteristic with residue field $k$\nand  if $C=F^*T^*_{X_k}X|_{X_k}$\nfor the closed fiber $Z=X_k$,\nthen the condition that\n$h\\colon W\\to X$ is\n$C$-transversal\nmeans that\n$W$ is smooth over ${\\cal O}_K$ \non a neighborhood of\nthe closed fiber $W_k$.\n\n\n\n\n\\begin{lm}\\label{lmtrreg}\nLet $f\\colon X\\to Y$\nand $g\\colon V\\to Y$\nbe morphisms of finite type of\nregular schemes\nand set $W\n=X\\times_YV$.\nLet $w\\in W$\nand $x\\in X,y\\in Y,v\\in V$\nbe the images.\n\n{\\rm 1.}\nSuppose that $g\\colon V\\to Y$\nis an immersion.\nThen, the following conditions\nare equivalent:\n\n{\\rm (1)}\n$f$ and $g$ are transversal\nat $w$.\n\n{\\rm (2)}\nThe morphism\n$T^*_yY\\times_yx\\to T^*_xX$\non the cotangent space\ninduces an injection\non the subspace\n$T^*_VY\\times_Vy\n\\subset\nT^*_yY\\times_yx$.\n\n{\\rm 2.}\nSuppose that the\nsubset ${\\rm Reg}(W)\n\\subset W$\nconsisting of regular points\nis an open subset.\nIf\n$f$ and $g$ are transversal\nat $w\\in W$,\nthen\n$f$ and $g$ are transversal\non a neighborhood of $w$.\n\\end{lm}\n\n\n\nThe condition that\n${\\rm Reg}(W)\n\\subset W$\nis an open subset is satisfied\nif $W$ is of finite type\nover a Dedekind domain\nsuch that the fraction field\nis of characteristic $0$\nor a semi-local ring of dimension\nat most $1$\nby \\cite[Corollaire (6.12.6)]{EGA4}.\n\n\n\\proof{\n1.\nLet $a_1,\\ldots,a_r\\in {\\cal O}_{Y,y}$\nbe a minimal system of generators\nof ${\\rm Ker}({\\cal O}_{Y,y}\n\\to {\\cal O}_{V,y})$.\nThen, the both conditions are\nequivalent to the condition\nthat $a_1,\\ldots,a_r\\in {\\cal O}_{X,x}$\nis a part of a regular system of\nparameters.\n\n\n2.\nSince the ${\\cal O}_W$-modules\n${\\cal T}or_q^{{\\cal O}_Y}\n({\\cal O}_X,{\\cal O}_V)=0$\nare coherent\nand $w$ is an element\nof the open subset ${\\rm Reg}(W)$,\nthe assertion follows.\n\\qed\n\n}\n\n\n\\medskip\n\n\nLet $f\\colon X\\to Y$\nbe a morphism of finite type of\nregular noetherian schemes\nsuch that $Y_{{\\mathbf F}_p}$\nis of finite type\nover $k$\nand consider the\nmorphisms\n(\\ref{eqdXY}).\nLet $C$ be a closed conical subset\nof $FT^*X|_{X_{{\\mathbf F}_p}}$\nsuch that \n$f\\colon X\\to Y$ is proper\non the base $B(C)$.\nThen we define a closed conical subset\n$f_\\circ C$ of $FT^*Y|_{Y_{{\\mathbf F}_p}}$\nto be\nthe image by\n$FT^*X|_{X_{{\\mathbf F}_p}}\n\\times_{X_{{\\mathbf F}_p}}\nY_{{\\mathbf F}_p}\n\\to\nFT^*Y|_{Y_{{\\mathbf F}_p}}$\nof the inverse image of\n$C$ by\n$FT^*X|_{X_{{\\mathbf F}_p}}\n\\times_{X_{{\\mathbf F}_p}}\nY_{{\\mathbf F}_p}\n\\to\nFT^*X|_{X_{{\\mathbf F}_p}}$.\n\n\nFor a closed immersion\n$i\\colon Z\\to X$\nof regular noetherian schemes\nsuch that $X_{{\\mathbf F}_p}$\nis of finite type\nover $k$,\nthe closed conical subset\n$F^*T^*_ZX|_{X_{{\\mathbf F}_p}}$\ndefined by the conormal bundle\nequals $i_\\circ C$\nfor the $0$-section\n$C=FT^*_ZZ|_{Z_{{\\mathbf F}_p}}$\nof $FT^*Z|_{Z_{{\\mathbf F}_p}}$.\n\n\n\n\\begin{pr}\\label{prfC}\nLet $X,Y$ and $V$ be regular\nnoetherian schemes \nsatisfying the condition {\\rm (F)} and\n\\begin{equation}\n\\begin{CD}\nX@<h<< W\\\\\n@VfVV@VV{f'}V\\\\\nY@<g<< V\n\\end{CD}\n\\label{eqprfC}\n\\end{equation}\nbe a cartesian\ndiagram of morphisms of\nfinite type.\nAssume that the\nsubset ${\\rm Reg}(W)\n\\subset W$\nconsisting of regular points\nis an open subset.\nLet $C$ be a closed\nconical subset of $FT^*X|_{X_{{\\mathbf F}_p}}$\nsuch that $f$ is proper on the\nbase $B(C)$.\nThen,\nthe following conditions\nare equivalent:\n\n{\\rm (1)}\nThe morphism\n$g$ is $f_\\circ C$-transversal.\n\n{\\rm (2)}\nThere exists a\nregular neighborhood $W_1\n\\subset W$  \nof the inverse image of\nthe base $B(C)$\nsuch that \n$f$ and $g$ are transversal on\n$W_1$ and that the restriction\n$h_1\\colon W_1\\to X$ is $C$-transversal.\n\n\\noindent\nIf these equivalent conditions are satisfied,\nwe have $g^\\circ f_\\circ C=\nf'_{1\\circ} h_1^\\circ C$\nfor the restriction\n$f'_1\\colon W_1\\to V$ of $f'$.\n\\end{pr}\n\n\\proof{\n(1)$\\Rightarrow$(2):\nLet $x\\in B(C)\\subset X_{{\\mathbf F}_p}$\nbe a closed point and $y=f(x)\\in Y_{{\\mathbf F}_p}$.\nSince the assertion is \\'etale local,\nwe may assume that\nthe morphism $k(y)\\to k(x)$\nof residue fields is an isomorphism.\nThere exist\nan open neighborhood \n$U\\subset X$ of $x\\in X$\nand a cartesian diagram\n$$\\begin{CD}\nW@<{\\supset}<<W\\times_XU\n@>>> Q@>>> V\\\\\n@VVV@VVV@VVV@VVV\\\\\nX@<{\\supset}<<U@>>> P@>>>Y\n\\end{CD}$$\nsuch that $P\\to Y$\nis smooth and \n$U\\to P$ is a closed immersion.\nLet $w\\in W$ be a closed point\nabove $x$ and $v=f'(w)\\in V$.\nWe may also assume that\nthe morphisms \n$k(y)\\to k(v)$\nand hence $k(x)\\to k(w)$\nare isomorphisms.\nWe consider the cartesian\ndiagram\n$$\\begin{CD}\n@.\nT^*_wQ\n@<<<\nT^*_vV\n\\\\\n@.@AAA@AAA\\\\\nT^*_xX\n@<<<\nT^*_xP\n@<<<\nT^*_yY\n\\end{CD}$$\nof cotangent spaces\nand identify their Frobenius\npull-backs with the fibers\nof FW-cotangent bundles\nby the isomorphism (\\ref{eqTx}).\n\nLet $\\widetilde C_x\n\\subset F^*T^*_xP$\nand $A_x \\subset F^*T^*_yY$\nbe the inverse images of\n$C_x\\subset F^*T^*_xX$.\nThen, by the condition (1),\nthe intersection \n$A_x\\cap {\\rm Ker}(F^*T^*_yY \\to \nF^*T^*_vV)$\nis a subset of the $0$-section.\nSince\n$T^*_yY \\to T^*_xP$\ninduces an isomorphism\n${\\rm Ker}(T^*_yY\\to T^*_vV)\n\\to \n{\\rm Ker}(T^*_xP\\to T^*_wQ)$,\nthe intersection \n$\\widetilde C_x\\cap\n{\\rm Ker}(F^*T^*_xP\\to F^*T^*_wQ)$\nis a subset of the $0$-section.\n\nBy the exact sequence\n$0\\to T^*_XP|_x\n\\to T^*_xP\\to T^*_xX\\to 0$\nand $x\\in B(C)$,\nwe have\n$F^*T^*_XP|_x\\subset \\widetilde C_x$.\nHence\n$T^*_xP\\to T^*_wQ$\ninduces an injection on\n$T^*_XP|_x$.\nNamely, \nthe morphism\n$Q\\to P$ and the immersion\n$U\\to P$ are transversal\non a neighborhood of $w$\nby Lemma \\ref{lmtrreg}.\n\nHence\nthe horizontal arrows\nof the commutative diagram\n\\begin{equation}\n\\begin{CD}\nT^*_wW\n@<<<\nT^*_vV\n\\\\\n@AAA@AAA\\\\\nT^*_xX\n@<<<\nT^*_yY\n\\end{CD}\n\\label{eqUVW}\n\\end{equation}\ninduce isomorphisms\non the kernels and cokernels\nof the vertical arrows.\nSince the intersection of\nthe inverse image $A_x$ with \n${\\rm Ker}(F^*T^*_yY\n\\to F^*T^*_wV)$\nis a subset of the $0$-section,\nthe intersection of\n$C_x$ with \n${\\rm Ker}(F^*T^*_xX\n\\to F^*T^*_wW)$\nis also a subset of the $0$-section.\nNamely,\n$h$ is $C$-transversal\non a neighborhood of $w$.\nThus $h$ is $C$-transversal on\na neighborhood of \nthe inverse image of $B(C)$.\n\nFurther an elementary\ndiagram chasing shows\nthat the inverse image of\n$h^\\circ C|_w$\nby $F^*T^*_wW\\gets F^*T^*_vV$\nequals the image of\n$A_x$ by\n$F^*T^*_yY\\to F^*T^*_vV$.\nHence we have\n$g^\\circ f_\\circ C=\nf'_{1\\circ} h_1^\\circ C$.\n\n\n\n(2)$\\Rightarrow$(1):\nLet $w\\in B(h_1^\\circ C)$\nbe a closed point\nand let $v\\in V, x\\in X$\nand $y\\in Y$ be the image.\nThen, the commutative diagram\n(\\ref{eqUVW})\ninduces an isomorphism\n${\\rm Ker}(T^*_yY\\to T^*_vV)\n\\to\n{\\rm Ker}(T^*_xX\\to T^*_wW)$\non the kernels.\nIn the same notation,\nsince the intersection of\n$C_x$ with \n${\\rm Ker}(F^*T^*_xX\\to F^*T^*_wW)$\nis a subset of the $0$-section,\nthe intersection of\n$A_x$ with \n${\\rm Ker}(F^*T^*_yY\\to F^*T^*_vV)$\nis also a subset of the $0$-section.\n\\qed\n\n}\n\n\\begin{cor}\\label{corfC}\nLet $X,Y$ and $V$\nbe regular noetherian schemes \nsatisfying the condition {\\rm (F)}\nand let\n{\\rm (\\ref{eqprfC})}\nbe a cartesian diagram\nof morphisms of finite type.\nAssume that the\nsubset ${\\rm Reg}(W)\n\\subset W$\nconsisting of regular points\nis an open subset\nand that\n$f\\colon X\\to Y$ is an immersion.\nThen,\nthe following conditions\nare equivalent:\n\n{\\rm (1)}\nThe morphism\n$g$ is $F^*T^*_XY|_{Y_{{\\mathbf F}_p}}$-transversal.\n\n{\\rm (2)}\nThe morphism $g\\colon V\\to Y$\nis transversal with\nthe immersion\n$f\\colon X\\to Y$ on\na neighborhood of $W_{{\\mathbf F}_p}\n=V\\times_XX_{{\\mathbf F}_p}$.\n\\end{cor}\n\n\\proof{\nIt suffices to apply\nProposition \\ref{prfC} \ntogether with Lemma \\ref{lmTXC}.2\nto\nthe $0$-section $C$\nof $FT^*X|_{X_{{\\mathbf F}_p}}$.\n\\qed\n\n}\n\n\\begin{df}\\label{dfCet}\nLet $f\\colon U\\to X$ be an \\'etale morphism\nof regular noetherian schemes \nsatisfying the condition {\\rm (F)}\nand let $C'$ be a closed\nconical subset of $FT^*U$.\nWe identify $FT^*U$ with\nthe pull-back\n$FT^*X\\times_{X_{{\\mathbf F}_p}}\nU_{{\\mathbf F}_p}$ by the\ncanonical isomorphism\ninduced by\n$F\\Omega^1_X\\otimes_{{\\cal O}_X}\n{\\cal O}_U\\to\nF\\Omega^1_U$ \nand let\n${\\rm pr}_1\\colon \nFT^*X\\times_{X_{{\\mathbf F}_p}}\nU_{{\\mathbf F}_p} \\to \nFT^*X$ be the projection.\nThen, we define\na closed conical subset $f_*C'$ of \n$FT^*X$ to be the union of\nthe closure $\\overline{{\\rm pr}_1(C')}$\nand the restriction\n$FT^*X|_{X_{{\\mathbf F}_p}\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} \nf(U_{{\\mathbf F}_p})}$ \nto the image of the complement.\n\\end{df}\n\n\\begin{lm}\\label{lmCet}\nLet $$\n\\begin{CD}\nV@>>>W\\\\\n@VgVV@VVhV\\\\\nU@>f>>X\n\\end{CD}\n$$ be a cartesian diagram\nof regular noetherian schemes \nsatisfying the condition {\\rm (F)}\nsuch that $f$ is\nan \\'etale morphism of finite type.\nLet $C'$ be a closed\nconical subset of $FT^*U$\nand set $C=f_*C'\\subset FT^*X$\nas in Definition {\\rm \\ref{dfCet}}.\n\nIf $h$ is $C$-transversal,\nthen $h$ is smooth on a neighborhood\nof $h^{-1}(X_{{\\mathbf F}_p}\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} \nf(U_{{\\mathbf F}_p}))$\nand $g$ is $C'$-transversal.\n\\end{lm}\n\n\\proof{\nAssume that\n$h$ is $C$-transversal.\nSince \n$C\\supset \nFT^*X|_{X_{{\\mathbf F}_p}\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} \nh(U_{{\\mathbf F}_p})}$,\nthe morphism \n$h$ is smooth on a neighborhood\nof $h^{-1}(X_{{\\mathbf F}_p}\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} \nf(W_{{\\mathbf F}_p}))$\nby Lemma \\ref{lmTXC}.1.\nSince \n$f^\\circ C\\supset C'$,\nthe morphism \n$g$ is $C'$-transversal\nby Lemma \\ref{lmhC}.\n\\qed\n\n}\n\n\\subsection{$C$-acyclicity}\\label{ssCac}\n\nWe keep fixing a perfect field\n$k$ of characteristic $p>0$.\n\n\\begin{df}\\label{dffC}\nLet $f\\colon X\\to Y$\nbe a morphism of finite type of \nregular noetherian schemes\nsatisfying the condition {\\rm (F)}\nin Definition {\\rm \\ref{dfFTX}}\nand \nlet $C$\nbe a closed conical subset\nof the FW-cotangent bundle\n$FT^*X|_{X_{{\\mathbf F}_p}}$.\nWe say that $f$\nis $C$-acyclic if the inverse image of\n$C$ by the morphism\n$FT^*Y|_{Y_{{\\mathbf F}_p}}\n\\times_{Y_{{\\mathbf F}_p}}\nX_{{\\mathbf F}_p}\n\\to FT^*X|_{{\\mathbf F}_p}$\nis a subset of the $0$-section.\n\\end{df}\n\n\nThe corresponding notion is\ncalled $C$-transversality\nin \\cite[1.2]{Be} and\n\\cite[Definition 3.5]{CC}.\nHere to avoid confusion with\nthe $C$-transversality \nfor morphisms to $X$ in\nDefinition \\ref{dfhC}.1,\n\\cite[1.2]{Be} and\n\\cite[Definition 3.3]{CC},\nwe introduce another terminology.\nWe will show in Lemma \\ref{lmhf}.2\nthat\nfor a morphism\n$f\\colon X\\to Y$\nof regular schemes and\na closed immersion\n$i\\colon Z\\to X$\nof regular schemes,\nthe morphism\n$f$ is $F^*T^*_ZX|_{X_{{\\mathbf F}_p}}$-acyclic\nif and only if\nthe composition\n$fi$ is smooth on a neighborhood\nof $Z_{{\\mathbf F}_p}$.\n\n\n\\begin{lm}\\label{lmftr}\nLet $f\\colon X\\to Y$\nbe a morphism of finite type of \nregular noetherian schemes\nsatisfying the condition {\\rm (F)}\nand \nlet $C$\nbe a closed conical subset\nof $FT^*X|_{X_{{\\mathbf F}_p}}$.\n\n{\\rm 1.}\nThe following conditions\nare equivalent:\n\n{\\rm (1)}\n$f$ is $C$-acyclic.\n\n{\\rm (2)}\n$f$ is smooth\non a neighborhood of\nthe base $B(C)\\subset \nX_{{\\mathbf F}_p}$\nand \nthe intersection of\n$C\\subset FT^*X|_{X_{{\\mathbf F}_p}}$ \nwith the image of the morphism\n$FT^*Y|_{Y_{{\\mathbf F}_p}}\n\\times_{Y_{{\\mathbf F}_p}}X_{{\\mathbf F}_p}\n\\to FT^*X|_{X_{{\\mathbf F}_p}}$\nis a subset of the $0$-section.\n\n\n{\\rm 2.}\nIf $C$ is the $0$-section\n$F^*T^*_XX|_{X_{{\\mathbf F}_p}}$,\nthe following conditions\nare equivalent:\n\n\n{\\rm (1)}\n$f$ is $C$-acyclic.\n\n{\\rm (2)}\n$f$ is smooth\non the neighborhood of\n$X_{{\\mathbf F}_p}$.\n\\end{lm}\n\n\\proof{\n1.\nThe condition (1)\nis equivalent to the conjunction\nof the following (1$'$) and (1$''$):\n\n(1$'$)\nThe inverse image of\nthe $0$-section by\n$FT^*Y|_{Y_{{\\mathbf F}_p}}\n\\times_{Y_{{\\mathbf F}_p}}X_{{\\mathbf F}_p}\n\\to FT^*X|_{X_{{\\mathbf F}_p}}$\non the base $B(C)\\subset X_{{\\mathbf F}_p}$\nis a subset of the $0$-sections.\n\n(1$''$)\nThe intersection of\n$C\\subset FT^*X|_{X_{{\\mathbf F}_p}}$ \nwith the image of the morphism\n$FT^*Y|_{Y_{{\\mathbf F}_p}}\n\\times_{Y_{{\\mathbf F}_p}}X_{{\\mathbf F}_p}\n\\to FT^*X|_{X_{{\\mathbf F}_p}}$\nis a subset of the $0$-sections.\n\n\\noindent\nThe condition (1$'$)\nmeans that\nthe morphism\n$f^*F\\Omega^1_Y\n\\to\nF\\Omega^1_X$\nis a locally splitting injection\non a neighborhood\nof the base $B(C)\\subset X_{{\\mathbf F}_p}$.\nHence the assertion follows\nfrom Proposition \\ref{prsm}.\n\n2.\nFor the $0$-section\n$C=F^*T^*_XX|_{X_{{\\mathbf F}_p}}$,\nthe base\n$B(C)$ is $X_{{\\mathbf F}_p}$\nand the condition \n(1$''$) in the proof of\n1 is satisfied.\nHence the assertion follows from 1.\n\\qed\n\n}\n\n\n\\begin{pr}\\label{prfCX}\nLet $X,Y,V$\nbe regular noetherian schemes\nsatisfying the condition {\\rm (F)}\nand let\n$$\\begin{CD}\nX@<h<<W\\\\\n@VfVV@VV{f'}V\\\\\nY@<g<<V\n\\end{CD}$$\nbe a cartesian diagram\nof morphisms of\nfinite type.\nLet $C$\nbe a closed conical subset\nof $FT^*X|_{X_{{\\mathbf F}_p}}$.\nThen the following conditions are\nequivalent:\n\n\n{\\rm (1)}\n$f$ is $C$-acyclic on\na neighborhood of the image\n$h(W_{{\\mathbf F}_p})$.\n\n{\\rm (2)}\nThere exists\na regular neighborhood $W_1\\subset W$\nof the inverse image of the base\n$B(C)$ satisfying\nthe following conditions:\n$f$ and $g$ are transversal\non $W_1$,\nthe restriction $h_1\\colon W_1\n\\to X$ is $C$-transversal\nand the restriction\n$f'_1\\colon W_1\\to V$\nis $h_1^\\circ C$-acyclic.\n\\end{pr}\n\n\\proof{\nFirst, \nwe show that the both conditions\nimply that $f$ is smooth on\na neighborhood of\nthe intersection $B(C)\n\\cap h(W_{{\\mathbf F}_p})$.\nFor (1), this follows from\nLemma \\ref{lmftr}.1.\nFor (2), similarly,\n$f'_1$ is smooth on\na neighborhood of $B(h_1^\\circ C)\n=h_1^{-1}(B(C))$.\nThis implies that\n$f$ is smooth on\na neighborhood of $\nh(h_1^{-1}(B(C)))\n=B(C)\\cap h(W_{{\\mathbf F}_p})$\nsince $f$ and $g$ are transversal\non $W_1$.\n\nBy replacing $X$ by\na neighborhood of\n$B(C)\\cap h(W_{{\\mathbf F}_p})$\nsmooth over $Y$,\nwe may assume that \n$f$ is smooth.\nThen, $W$ is regular\nand $f$ and $g$ are transversal.\nBy Proposition \\ref{prsm},\nwe have a commutative diagram\n$$\\begin{CD}\n0\\to& FT^*V|_{V_{{\\mathbf F}_p}}\n\\times_{V_{{\\mathbf F}_p}}\nW_{{\\mathbf F}_p}\n@>>> FT^*W|_{W_{{\\mathbf F}_p}}\n@>>> F^*T^*W\/V|_{W_{{\\mathbf F}_p}}\n&\\to 0\\\\\n&@AAA@AAA@AA{\\cong}A&\\\\\n0\\to& FT^*Y|_{Y_{{\\mathbf F}_p}}\n\\times_{Y_{{\\mathbf F}_p}}\nW_{{\\mathbf F}_p}\n@>>> FT^*X|_{X_{{\\mathbf F}_p}}\n\\times_{X_{{\\mathbf F}_p}}\nW_{{\\mathbf F}_p}\n@>>> F^*T^*X\/Y|_{X_{{\\mathbf F}_p}}\n\\times_{X_{{\\mathbf F}_p}}\nW_{{\\mathbf F}_p}\n&\\to0\\\\\n\\end{CD}$$\nof exact sequences\nof vector bundles on $W_{{\\mathbf F}_p}$.\nLet $C'\\subset FT^*W|_{W_{{\\mathbf F}_p}}$\nbe the image\nof $h^*C=C\\times_{X_{{\\mathbf F}_p}}\nW_{{\\mathbf F}_p}\n\\subset FT^*X|_{X_{{\\mathbf F}_p}}\n\\times_{X_{{\\mathbf F}_p}}\nW_{{\\mathbf F}_p}$\nand let\n$A\\subset FT^*Y|_{Y_{{\\mathbf F}_p}}\n\\times_{Y_{{\\mathbf F}_p}}\nW_{{\\mathbf F}_p}$\nand $A'\\subset FT^*V|_{V_{{\\mathbf F}_p}}\n\\times_{V_{{\\mathbf F}_p}}\nW_{{\\mathbf F}_p}$\nbe their inverse images.\n\nSince the right vertical arrow is an\nisomorphism,\nthe lower left arrow induces\nan isomorphism\n${\\rm Ker}(FT^*Y|_{Y_{{\\mathbf F}_p}}\n\\times_{Y_{{\\mathbf F}_p}}\nW_{{\\mathbf F}_p}\n\\to \nFT^*V|_{V_{{\\mathbf F}_p}}\n\\times_{V_{{\\mathbf F}_p}}\nW_{{\\mathbf F}_p})\n\\to\n{\\rm Ker}(FT^*X|_{X_{{\\mathbf F}_p}}\n\\times_{X_{{\\mathbf F}_p}}\nW_{{\\mathbf F}_p}\n\\to FT^*W|_{W_{{\\mathbf F}_p}})$.\nHence $A\n\\subset FT^*Y|_{Y_{{\\mathbf F}_p}}\n\\times_{Y_{{\\mathbf F}_p}}\nW_{{\\mathbf F}_p}$ is a subset\nof the $0$-section\nif and only if\n$A'\\subset\nFT^*V|_{V_{{\\mathbf F}_p}}\n\\times_{V_{{\\mathbf F}_p}}\nW_{{\\mathbf F}_p}$ and\n$h^*C\\cap \n{\\rm Ker}(FT^*X|_{X_{{\\mathbf F}_p}}\n\\times_{X_{{\\mathbf F}_p}}\nW_{{\\mathbf F}_p}\n\\to FT^*W|_{W_{{\\mathbf F}_p}})$\nare subsets\nof the $0$-sections\nand the assertion follows.\n\\qed\n\n}\n\n\n\\begin{lm}\\label{lmhf}\nLet $f\\colon X\\to Y$\nbe a morphism of finite type of \nregular noetherian schemes\nsatisfying the condition {\\rm (F)}.\n\n{\\rm 1.}\nLet $C$ be a closed conical subset\nof $FT^*X|_{X_{{\\mathbf F}_p}}$\nand \nassume that $f$ is proper\non the base $B(C)$.\nLet $g\\colon Y\\to Z$\nbe a morphism of finite type of \nregular noetherian schemes\nsuch that $Z_{{\\mathbf F}_p}$\nis of finite type over $k$.\nThen the following conditions are\nequivalent:\n\n{\\rm (1)}\n$g$ is $f_\\circ C$-acyclic.\n\n{\\rm (2)}\n$gf$ is $C$-acyclic.\n\n{\\rm 2.}\nLet $p\\colon V\\to X$\nbe a proper morphism of\nregular schemes\nand let $C=p_\\circ F^*T^*_VV|_{V_{{\\mathbf F}_p}}\n\\subset FT^*X|_{X_{{\\mathbf F}_p}}$.\nThen, the following conditions\nare equivalent:\n\n{\\rm (1)}\n$f$ is $C$-acyclic.\n\n{\\rm (2)}\nThe composition\n$fp$ is smooth\non a neighborhood of $V_{{\\mathbf F}_p}$.\n\\end{lm}\n\n\\proof{\n1.\nLet $x\\in X_{{\\mathbf F}_p}$ be a closed\npoint and $y\\in Y_{{\\mathbf F}_p}$ and $z\\in Z_{{\\mathbf F}_p}$\nbe the images.\nSince the assertion is \\'etale local,\nwe may also assume that\nthe morphisms \n$k(z)\\to k(y)\\to k(x)$\nare isomorphisms.\n\nLet $A_x$ be the inverse image of\n$C_x$ by $F^*T^*_xX\\gets F^*T^*_yY$.\nThen, the inverse image $A'_x$\nof $C_x$ by $F^*T^*_xX\\gets F^*T^*_zZ$\nequals the inverse image $A''_x$\nof $A_x$\nby $F^*T^*_yY\\gets F^*T^*_zZ$.\nSince the condition (1) \n(resp.\\ (2)) is equivalent to\nthat $A'_x$ (resp.\\ $A''_x$)\nis a subset of the $0$-section\nfor any $x$,\nthe assertion follows.\n\n2.\nBy 1.~applied to\n$p_\\circ F^*T^*_VV|_{V_{{\\mathbf F}_p}}\n=F^*T^*_VX|_{X_{{\\mathbf F}_p}}$,\nthe condition (1)\nis equivalent to that\nthe composition $fp$\nis $F^*T^*_VV|_{V_{{\\mathbf F}_p}}$-acyclic.\nHence the assertion follows from\nLemma \\ref{lmftr}.2.\n\\qed\n\n}\n\n\n\\begin{df}\\label{dfhfC}\nLet $X$\nbe a regular noetherian scheme\nsatisfying the condition {\\rm (F)}\nand let $C$ be a closed conical subset\nof $FT^*X|_{X_{{\\mathbf F}_p}}$.\nWe say that a pair\n$(h,f)$ of morphisms\n$h\\colon W\\to X$,\n$f\\colon W\\to Y$\nof finite type\nof regular noetherian schemes\nsuch that $Y_{{\\mathbf F}_p}$\nis of finite type over $k$\nis $C$-acyclic\nif the intersection of\n$(C\\times_{X_{{\\mathbf F}_p}}\nW_{{\\mathbf F}_p})\n\\times_{W_{{\\mathbf F}_p}} \n(FT^*Y|_{Y_{{\\mathbf F}_p}}\n\\times_{Y_{{\\mathbf F}_p}}\nW_{{\\mathbf F}_p})\n\\subset \n(FT^*X|_{X_{{\\mathbf F}_p}}\n\\times_{X_{{\\mathbf F}_p}}\nW_{{\\mathbf F}_p})\n\\times_{W_{{\\mathbf F}_p}} \n(FT^*Y|_{Y_{{\\mathbf F}_p}}\n\\times_{Y_{{\\mathbf F}_p}}\nW_{{\\mathbf F}_p})$\nwith the kernel \n${\\rm Ker}((h^*,f^*)\\colon$\n$(FT^*X|_{X_{{\\mathbf F}_p}}\n\\times_{X_{{\\mathbf F}_p}}\nW_{{\\mathbf F}_p})\n\\times_{W_{{\\mathbf F}_p}} \n(FT^*Y|_{Y_{{\\mathbf F}_p}}\n\\times_{Y_{{\\mathbf F}_p}}\nW_{{\\mathbf F}_p})\n\\to\nFT^*W|_{W_{{\\mathbf F}_p}})$\nis a subset of the $0$-section.\n\\end{df}\n\n\n\\begin{lm}\\label{lmhfC}\nLet $X$\nbe a regular noetherian scheme\nsatisfying the condition {\\rm (F)}\nand let $C$ be a closed conical subset\nof $FT^*X|_{X_{{\\mathbf F}_p}}$.\n\n\n{\\rm 1.}\nLet $f\\colon X\\to Y$\nbe a morphism of finite type\nof regular noetherian schemes\nsatisfying the condition {\\rm (F)}.\nThen, the following conditions are\nequivalent:\n\n{\\rm (1)}\n$f$ is $C$-acyclic.\n\n{\\rm (2)}\n$(1_X,f)$ is $C$-acyclic.\n\n{\\rm 2.}\nLet $h\\colon W\\to X$\nand $f\\colon W\\to Y$\nbe morphisms of finite type\nof regular noetherian schemes\nsatisfying the condition {\\rm (F)}.\nThen the following conditions are\nequivalent:\n\n{\\rm (1)}\n$(h,f)$ is $C$-acyclic.\n\n{\\rm (2)}\n$h$ is $C$-transversal\nand \n$f$ is $h^\\circ C$-acyclic.\n\\end{lm}\n\n\\proof{1.\nIdentify the kernel of\n$(1,f^*)\\colon\nFT^*X|_{X_{{\\mathbf F}_p}}\n\\times_{X_{{\\mathbf F}_p}} \n(FT^*Y|_{Y_{{\\mathbf F}_p}}\n\\times_{Y_{{\\mathbf F}_p}}\nX_{{\\mathbf F}_p})\n\\to\nFT^*X|_{X_{{\\mathbf F}_p}}$\nwith the image of\nthe injection\n$(f^*,-1)\\colon \nFT^*Y|_{Y_{{\\mathbf F}_p}}\n\\times_{Y_{{\\mathbf F}_p}}\nX_{{\\mathbf F}_p}\n\\to\nFT^*X|_{X_{{\\mathbf F}_p}}\n\\times_{X_{{\\mathbf F}_p}}\n(FT^*Y|_{Y_{{\\mathbf F}_p}}\n\\times_{Y_{{\\mathbf F}_p}}\nX_{{\\mathbf F}_p})$.\nThen the inverse image\nin $\nFT^*Y|_{Y_{{\\mathbf F}_p}}\n\\times_{Y_{{\\mathbf F}_p}}\nX_{{\\mathbf F}_p}$ of\n$C\n\\times_{X_{{\\mathbf F}_p}}\n(FT^*Y|_{Y_{{\\mathbf F}_p}}\n\\times_{Y_{{\\mathbf F}_p}}\nX_{{\\mathbf F}_p})\n\\subset\nFT^*X|_{X_{{\\mathbf F}_p}}\n\\times_{X_{{\\mathbf F}_p}}\n(FT^*Y|_{Y_{{\\mathbf F}_p}}\n\\times_{Y_{{\\mathbf F}_p}}\nX_{{\\mathbf F}_p})$\nis the same as\nthe inverse image of\n$C\\subset T^*X$\nand the assertion follows.\n\n2.\nSince ${\\rm Ker}(h^*\nFT^*X|_{X_{{\\mathbf F}_p}}\n\\times_{X_{{\\mathbf F}_p}}\nW_{{\\mathbf F}_p}\n\\to\nFT^*W|_{W_{{\\mathbf F}_p}})\n\\times 0\n\\subset\n{\\rm Ker}((h^*,f^*)\\colon\n(FT^*X|_{X_{{\\mathbf F}_p}}\n\\times_{X_{{\\mathbf F}_p}}\nW_{{\\mathbf F}_p})\n\\times_{W_{{\\mathbf F}_p}} \n(FT^*Y|_{Y_{{\\mathbf F}_p}}\n\\times_{Y_{{\\mathbf F}_p}}\nW_{{\\mathbf F}_p})\n\\to\nFT^*W|_{W_{{\\mathbf F}_p}})$,\nthe $C$-acyclicity\nof $(h,f)$ implies\nthe $C$-transversality of\n$h$.\nBy 1.,\nthe $h^\\circ C$-acyclicity\nof $f$ is equivalent to the condition that\nthe intersection of\n$h^\\circ C\\times_{W_{{\\mathbf F}_p}} \n(FT^*Y|_{Y_{{\\mathbf F}_p}}\n\\times_{Y_{{\\mathbf F}_p}}\nW_{{\\mathbf F}_p})$\nwith \n${\\rm Ker}(\nFT^*W|_{W_{{\\mathbf F}_p}}\n\\times_{W_{{\\mathbf F}_p}}\n(FT^*Y|_{Y_{{\\mathbf F}_p}}\n\\times_{Y_{{\\mathbf F}_p}}\nW_{{\\mathbf F}_p})\n\\to \nFT^*W|_{W_{{\\mathbf F}_p}})$\nis a subset of the\n$0$-section.\nThis condition is equivalent to the\n$C$-acyclicity\nof $(h,f)$\nsince \n$h^\\circ C\\times_{W_{{\\mathbf F}_p}} \n(FT^*Y|_{Y_{{\\mathbf F}_p}}\n\\times_{Y_{{\\mathbf F}_p}}\nW_{{\\mathbf F}_p})$\nis the image of \n$h^*C\\times_{W_{{\\mathbf F}_p}} \n(FT^*Y|_{Y_{{\\mathbf F}_p}}\n\\times_{Y_{{\\mathbf F}_p}}\nW_{{\\mathbf F}_p})$.\n\\qed\n\n}\n\n\n\\section{Micro-support}\\label{sms}\n\nWe fix a perfect field $k$ \nof characteristic $p>0$\nand a finite field \n$\\Lambda$ of characteristic $\\ell\\neq p$.\nWe will assume\nthat a regular noetherian scheme $X$\nover ${\\mathbf Z}_{(p)}$\nsatisfies the condition {\\rm (F)}\nin Definition {\\rm \\ref{dfFTX}}.\n\n\\subsection{Micro-support}\n\\begin{df}\\label{dfms}\nLet $X$ be a regular noetherian scheme \nover ${\\mathbf Z}_{(p)}$\nsatisfying the condition {\\rm (F)}\nin Definition {\\rm \\ref{dfFTX}} and\nlet $C$ be a closed conical subset\nof the FW-cotangent bundle \n$FT^*X|_{X_{{\\mathbf F}_p}}$.\nLet ${\\cal F}$\nbe a constructible complex\nof $\\Lambda$-modules\non $X$.\nWe say that ${\\cal F}$\nis micro-supported on $C$\nif the following conditions\n{\\rm (1)} and {\\rm (2)}\nare satisfied:\n\n{\\rm (1)}\nThe intersection of\nthe support ${\\rm supp}\\, {\\cal F}$\nwith the closed fiber $X_{{\\mathbf F}_p}$ is \na subset of the base $B(C)$.\n\n{\\rm (2)}\nEvery $C$-transversal separated morphism\n$h\\colon W\\to X$ of finite type of\nregular schemes\nis ${\\cal F}$-transversal\non a neighborhood of the closed\nfiber $W_{{\\mathbf F}_p}$.\n\\end{df}\n\nThis definition of micro-support\nis related to\n\\cite[Proposition 8.13]{CC}\nbut is different from\n\\cite[1.3]{Be}.\nWe discuss this point in\nRemark after Proposition \\ref{prtrla}.\nIt is a property on a neighborhood\nof $X_{{\\mathbf F}_p}$.\nIf $X_{\\mathbf Q}\n=X\\times_{{\\rm Spec}\\, {\\mathbf Z}_{(p)}}\n{\\rm Spec}\\, {\\mathbf Q}$\nis smooth over a field $K$\nof characteristic $0$,\nto cover $X_{\\mathbf Q}$,\none can use the micro-support\nof the restriction of ${\\cal F}$ on $X_{\\mathbf Q}$\ndefined as closed conical subset\nof the cotangent bundle\n$T^*X_{\\mathbf Q}\/K$.\n\n\n\\begin{lm}\\label{lmTX}\nLet $X$ be a regular noetherian scheme \nover ${\\mathbf Z}_{(p)}$\nsatisfying the condition {\\rm (F)}\nand ${\\cal F}$\nbe a constructible complex\nof $\\Lambda$-modules.\n\n{\\rm 1.}\n${\\cal F}$ is micro-supported\non $FT^*X|_{X_{{\\mathbf F}_p}}$.\n\n{\\rm 2.}\nIf ${\\cal F}$ is locally constant\non a neighborhood of\nthe closed fiber $X_{{\\mathbf F}_p}$,\nthen \n${\\cal F}$ is micro-supported\non the $0$-section $F^*T^*_XX|_{X_{{\\mathbf F}_p}}$.\n\n{\\rm 3.}\nAssume that $X$ is a\nsmooth scheme over $k$.\nLet $C\\subset T^*X$ be\na closed conical subset\nand $F^*C\\subset F^*T^*X\n=FT^*X$ \nbe the pull-back of $C$\nThen, ${\\cal F}$ is micro-supported\non $C$ in the sense of\n{\\rm (\\cite[1.3]{Be}, \\cite[Definition 4.1]{CC})}\nif and only if \n${\\cal F}$ is micro-supported\non $F^*C$.\n\\end{lm}\n\nWe show the converse of\n2 in Corollary \\ref{cortrla}.\n\n\n\\proof{\n1.\nLet $h\\colon W\\to X$\nbe a separated\nmorphism of finite type of regular schemes.\nIf $h$ is $FT^*X|_{X_{{\\mathbf F}_p}}$-transversal,\nthen $h$ is smooth\non a neighborhood\nof $W_{{\\mathbf F}_p}$ by Lemma \\ref{lmTXC}.1.\nHence $h$ is ${\\cal F}$-transversal\non a neighborhood\nof $W_{{\\mathbf F}_p}$ by Lemma \\ref{lmPoi}.1.\n\n2.\nLet $h\\colon W\\to X$\nbe a separated\nmorphism of finite type \nof regular schemes.\nThen, since \n${\\cal F}$ is locally constant\non a neighborhood of\nthe closed fiber $X_{{\\mathbf F}_p}$,\n$h$ is ${\\cal F}$-transversal\non a neighborhood of $W_{{\\mathbf F}_p}$\nby Lemma \\ref{lmPoi}.2.\n\n3.\nLet $h\\colon W\\to X$\nbe a separated\nmorphism of finite type of regular schemes.\nThen, $h\\colon W\\to X$\nis a separated morphism \nof smooth schemes\nof finite type over $k$.\nThe morphism\n$h\\colon W\\to X$ is $F^*C$-transversal\nif and only if\n$h\\colon W\\to X$ is $C$-transversal.\nHence\nthe equivalence follows\nfrom \\cite[Proposition 8.13]{CC}.\n\\qed\n\n}\n\n\n\\begin{pr}\\label{prmcf}\nLet $X$ be a regular scheme \nover ${\\mathbf Z}_{(p)}$\nsatisfying the condition {\\rm (F)}\nand let\n${\\cal F}$ be a constructible\ncomplex of $\\Lambda$-modules.\nLet $C$ be a closed conical\nsubset of $FT^*X|_{X_{{\\mathbf F}_p}}$\nsuch that ${\\cal F}$\nis micro-supported on $C$.\n\n{\\rm 1.}\nLet $h\\colon W\\to X$\nbe a separated morphism\nof finite type of regular schemes.\nIf $h$ is $C$-transversal,\nthen $h$ is ${\\cal F}$-transversal\non a neighborhood of $W_{{\\mathbf F}_p}$\nand $h^*{\\cal F}$\nis micro-supported on $h^\\circ C$.\n\n{\\rm 2.}\nLet $f\\colon X\\to Y$\nbe a separated\nmorphism of finite type \nproper on the base $B(C)$\nof regular quasi-excellent\nnoetherian schemes\nsatisfying the condition {\\rm (F)}.\nThen $Rf_*{\\cal F}$\nis micro-supported on $f_\\circ C$.\n\\end{pr}\n\n\\proof{\n1.\nLet $g\\colon V\\to W$\nbe an $h^\\circ C$-transversal\nseparated morphism of finite type of\nregular noetherian schemes.\nThen, by Lemma \\ref{lmhC},\n$hg$ and $h$ are $C$-transversal.\nSince ${\\cal F}$ is\nmicro-supported on $C$,\n$hg$ and $h$ are \n${\\cal F}$-transversal\non neighborhoods of\n$V_{{\\mathbf F}_p}$ and of $W_{{\\mathbf F}_p}$ respectively.\nHence by Proposition \\ref{prhF}.1,\n$g$ is $h^*{\\cal F}$-transversal\non a neighborhood of\n$V_{{\\mathbf F}_p}$.\n\n2.\nLet $g\\colon V\\to Y$\nbe an $f_\\circ C$-transversal \nseparated morphism of finite type \nof regular noetherian schemes\nand let\n$$\\begin{CD}\nX@<h<< W\\\\\n@VfVV@VV{f'}V\\\\\nY@<g<<V\n\\end{CD}$$\nbe a cartesian diagram.\nThen, $f$ and $g$ are\ntransversal on a regular neighborhood\n$W_1\\subset W$\nof the inverse image of\n$B(C)$\nand $h_1=h|_{W_1}\\colon W_1\\to X$\nis $C$-transversal by Proposition \\ref{prfC}.\nSince ${\\cal F}$\nis micro-supported on $C$,\nthe restriction $h_1\\colon W_1\\to X$ is\n${\\cal F}$-transversal.\n\nSince the intersection of\n${\\rm supp}\\, {\\cal F}$\nwith $X_{{\\mathbf F}_p}$\nis a subset of $B(C)$,\nthe intersection of\n$A={\\rm supp}\\, h^*{\\cal F}$\nwith $W_{{\\mathbf F}_p}$ is a subset of $W_1$.\nSince the closed\nset $A\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} A\\cap W_1$\ndoes not intersect \nthe closed fiber $W_{{\\mathbf F}_p}$,\nthe complement $V_0=\nV\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} f'(A\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} A\\cap W_1)$\nis an open neighborhood of $V_{{\\mathbf F}_p}$.\nBy replacing $V$ by $V_0$,\nwe may assume $A={\\rm supp}\\, h^*{\\cal F}\\subset W_1$.\nThen, $h\\colon W\\to X$ is\n${\\cal F}$-transversal.\nSince $f$ and $g$ are transversal on $W_1$,\nthe base change morphism\n$f'^*Rg^!\\Lambda\n\\to Rh^!\\Lambda$\nis an isomorphism on $W_1$ by \nLemma \\ref{lmtrbc}.\nHence\n$g$ is $Rf_*{\\cal F}$-transversal\non $V$ by Corollary \\ref{cortrC}.1.\n\\qed\n\n}\n\n\nWe show that\nbeing micro-supported is\nan \\'etale local property.\n\n\\begin{lm}\\label{lmmset}\nLet $X$ be a regular noetherian scheme \nover ${\\mathbf Z}_{(p)}$\nsatisfying the condition {\\rm (F)}\nand let\n${\\cal F}$ be a constructible\ncomplex of $\\Lambda$-modules.\n\n{\\rm 1.}\nLet $C$ be a closed conical\nsubset of $FT^*X|_{X_{{\\mathbf F}_p}}$\nand let\n$(f_i\\colon U_i\\to X)_{i\\in I}$\nbe an \\'etale covering.\nThen, the following conditions\nare equivalent:\n\n{\\rm (1)}\n${\\cal F}$\nis micro-supported on $C$.\n\n{\\rm (2)}\nFor every $i\\in I$,\nthe pull-back\n${\\cal F}_i=f_i^*{\\cal F}$\nis micro-supported on \n$C_i=f_i^\\circ C$.\n\n{\\rm 2.}\nLet $f\\colon U\\to X$\nbe an \\'etale morphism\nof finite type\nand let \n${\\cal F}_U=f^*{\\cal F}$ be the restriction.\nLet\n$C'\\subset FT^*U$\nbe a closed conical subset\nand let $C=f_*C'$ be\nthe closed conical subset defined\nin Definition {\\rm \\ref{dfCet}}.\nIf ${\\cal F}_U$ is micro-supported\non $C'$,\nthen ${\\cal F}$ is micro-supported\non $C$.\n\\end{lm}\n\n\\proof{1.\nThe equivalence for the condition \n(1) in Definition \\ref{dfms}\non the support is verified easily.\nWe show the equivalence\nfor the condition \n(2) in Definition \\ref{dfms}\non transversality.\n\n(1)$\\Rightarrow$(2):\nLet $i\\in I$\nand let $h\\colon W\\to U_i$ be a\nseparated morphism of finite type\nof regular schemes.\nIf $h$ is $C_i$-tranversal,\nthen since $f_i$ is \\'etale,\nthe composition\n$f_i\\circ h$ is $C$-tranversal\nby Lemma \\ref{lmhC}.\nHence\n$f_i\\circ h$ is ${\\cal F}$-tranversal\non a neighborhood of the closed\nfiber\nand consequently\n$h$ is ${\\cal F}_i$-tranversal\non a neighborhood of the closed\nfiber.\n\n\n\n\n(2)$\\Rightarrow$(1):\nLet $h\\colon W\\to X$ be a\nseparated morphism of finite type\nof regular schemes.\nAssume $h$ is $C$-tranversal.\nThen, for every $i\\in I$,\nthe base change\n$h_i\\colon W_i=W\\times_XU_i\\to U_i$ \nis $C_i$-tranversal\nby Lemma \\ref{lmhC}.\nHence\n$h_i$ is ${\\cal F}_i$-tranversal\non a neighborhood of the closed\nfiber\nfor every $i\\in I$\nand consequently\n$h$ is ${\\cal F}$-tranversal\non a neighborhood of the closed\nfiber.\n\n2.\nThe condition \n(1) in Definition \\ref{dfms}\non the support is verified easily.\nWe show that the condition \n(2) in Definition \\ref{dfms}\non transversality is satisfied.\nLet $h\\colon W\\to X$\nbe a separated morphism\nof finite type of regular\nnoetherian schemes.\nAssume that $h\\colon W\\to X$\nis $C$-transversal.\nThen, by Lemma \\ref{lmCet},\nthe morphism\n$h\\colon W\\to X$\nis smooth on a neighborhood $W_1$\nof $h^{-1}(X_{{\\mathbf F}_p}\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} \nf(U_{{\\mathbf F}_p}))$.\nHence\n$h|_{W_1}\\colon W_1\\to X$\nis ${\\cal F}$-transversal.\nFurther, by Lemma \\ref{lmCet},\nthe base change\n$W\\times_XU\\to U$ of $h$\nis $C'$-transversal.\nSince ${\\cal F}_U$ is micro-supported\non $C'$,\nthe morphism\n$W\\times_XU\\to U$\nis ${\\cal F}_U$-transversal\non a neighborhood of the closed\nfiber.\nSince $W_1$ and\n$W\\times_XU$\nform an \\'etale covering of $W$,\nthe morphism\n$h$ is ${\\cal F}$-transversal\non a neighborhood of the closed\nfiber.\n\\qed\n\n}\n\n\n\\medskip\nIn the rest of this subsection,\nlet ${\\cal O}_K$ be\na discrete valuation ring\nwith residue field $k$\nsuch that the fraction field\n$K$ is of characteristic $0$.\nRecall that ${\\cal O}_K$ is excellent.\nFor a scheme $X$ over\n${\\cal O}_K$,\nthe closed fiber\n$X_k=X\\times_{{\\rm Spec}\\, \n{\\mathcal O}_K}{\\rm Spec}\\, k$\nhas the same underlying set\nas\n$X_{{\\mathbf F}_p}\n=X\\times_{{\\rm Spec}\\, \n{\\mathbf Z}}{\\rm Spec}\\, \n{\\mathbf F}_p$.\n\n\\begin{pr}\\label{prtrla}\nLet $h\\colon W\\to X$ and\n$f\\colon W\\to Y$\nbe morphisms of regular schemes of finite\ntype over ${\\cal O}_K$.\nLet ${\\cal F}$ be\na constructible complex\nof $\\Lambda$-modules\nand $C$ be a closed\nconical subset of $FT^*X|_{X_k}$.\nSuppose that ${\\cal F}$ is micro-supported\non $C$.\nIf the pair $(h,f)$ is\n$C$-acyclic,\nthen \n$f\\colon W\\to Y$ is\n$h^*{\\cal F}$-acyclic along $W_k$.\n\\end{pr}\n\n\\proof{\nBy Lemma \\ref{lmhfC}.2,\n$h\\colon W\\to X$\nis $C$-transversal\nand $f\\colon W\\to Y$\nis $h^\\circ C$-acyclic.\nSince \n${\\cal F}$ is micro-supported on $C$,\nthe pull-back\n$h^*{\\cal F}$ is micro-supported\non $h^\\circ C$ by Proposition \\ref{prmcf}.1.\nHence by replacing $X$ by $W$,\nwe may assume $W=X$.\n\nSince $f\\colon X\\to Y$ is\n$C$-acyclic,\nthe morphism $f$ is\nsmooth on a neighborhood\nof the intersection \n$B(C) \\cap X_k\n\\supset\n{\\rm supp}\\, {\\cal F}\n\\cap X_k$ by Lemma \\ref{lmftr}.1.\nHence, we may assume\n$f\\colon X\\to Y$ is smooth.\n\nLet $V\\to Y$ be a separated morphism\nof regular schemes\nof finite type over ${\\cal O}_K$.\nThen the projection $p\\colon U=V\\times_YX\\to X$\nis $C$-transversal\nby\nProposition \\ref{prfCX}.\nHence $p$ is ${\\cal F}$-transversal\non a neighborhood of $U_k$.\nThus by Proposition \\ref{prla1},\n$f$ is ${\\cal F}$-acyclic\nalong $X_k$.\n\\qed\n\n}\n\n\\medskip\n\n\\begin{rmk}\\label{rmk}\n{\\rm \nThe conclusion of Proposition\n\\ref{prtrla} is an analogue\nof the original condition defining\nthe micro-support in \\cite[1.3]{Be}.\nIn the geometric case,\nthis is shown to be equivalent\nin \\cite[Proposition 8.13]{CC}\nto the condition analogous\nto that in Definition \\ref{dfms}.\nHowever the following example\nshows that the condition is\ntoo weak in the setting of this article.\n\nLet $X$ be a smooth scheme over\n${\\cal O}_K$ \nand $C=F^*T^*_{X_k}X|_{X_k}$\nbe the conormal bundle\nof the closed fiber.\nLet $(h,f)$ be a $C$-acyclic pair\nof morphisms of regular schemes\nof finite type over ${\\cal O}_K$.\nThen, since $h\\colon W\\to X$\nis transversal to the immersion\n$X_k\\to X$,\nthe closed fiber $W_k$ is regular\nand $W$ is smooth over ${\\cal O}_K$\non a neighborhood of $W_k$\nby Lemma \\ref{lmhfC}.2\nand Corollary \\ref{corfC}.\nSince $f\\colon W\\to Y$\nis $F^*T^*_{W_k}W|_{W_k}$-acyclic,\nfurther by Lemma \\ref{lmhfC}.2\nand Lemma \\ref{lmhf}.2,\nthe morphism $W\\to Y$ is smooth\non a neighborhood of $W_k$\nand $W_k\\to Y$ is also smooth.\nThis means that $W_k$ is empty.\nThus any ${\\cal F}$ satisfies\nthe conclusion of\nProposition \\ref{prtrla}.}\n\\end{rmk}\n\n\\begin{cor}\\label{cortrla}\nLet $X$ be a regular scheme of finite\ntype over ${\\cal O}_K$\nand ${\\cal F}$\nbe a constructible complex\nof $\\Lambda$-modules.\nThen, the following conditions\nare equivalent.\n\n{\\rm (1)}\n${\\cal F}$ is locally constant\non a neighborhood of\nthe closed fiber $X_k$.\n\n{\\rm (2)}\n${\\cal F}$ is micro-supported\non the $0$-section $F^*T^*_XX|_{X_k}$.\n\\end{cor}\n\n\\proof{\n(1)$\\Rightarrow$(2)\nis proved in Lemma \\ref{lmTX}.2.\n\n(2)$\\Rightarrow$(1):\nBy Proposition \\ref{prtrla}\napplied to $(1_X,1_X)$,\nthe identity $1_X\\colon X\\to X$\nis ${\\cal F}$-acyclic\nalong $X_k$.\nHence ${\\cal F}$ is locally constant\non a neighborhood of $X_k$\nby Lemma \\ref{lmla}.2.\n\\qed\n\n}\n\n\n\\subsection{Singular support}\n\n\\begin{df}\\label{dfss}\nLet $X$ be a regular noetherian scheme \nover ${\\mathbf Z}_{(p)}$\nsatisfying the condition {\\rm (F)}\nin Definition {\\rm \\ref{dfFTX}}\nand ${\\cal F}$\nbe a constructible complex\nof $\\Lambda$-modules\non $X$.\nWe say that a closed\nconical subset $C\n\\subset FT^*X|_{X_{{\\mathbf F}_p}}$ \nof the FW-cotangent bundle is the singular\nsupport $SS{\\cal F}$ of ${\\cal F}$\nif the following condition is satisfied:\nFor any closed conical subset\n$C'\\subset FT^*X|_{X_{{\\mathbf F}_p}}$,\n${\\cal F}$ is micro-supported\non $C'$ if and only if $C\\subset C'$.\n\\end{df}\n\nIf the singular support\n$SS{\\cal F}$ exists,\nit is the intersection of\nclosed conical subsets $C\n\\subset FT^*X|_{X_{{\\mathbf F}_p}}$ on\nwhich\n${\\cal F}$ is micro-supported. \nThe author does not know how\nto prove the existence of\nthe singular support in general.\nWe compute \nthe singular supports\nin some cases.\n\n\\begin{lm}\\label{lmsslc}\nLet $X$ be a regular noetherian scheme \nover ${\\mathbf Z}_{(p)}$\nsuch that $X_{{\\mathbf F}_p}$\nis of finite type over $k$\nand ${\\cal F}$\nbe a constructible complex\nof $\\Lambda$-modules\non $X$.\nWe consider the following condition:\n\n{\\rm (DVR)}\n$X$ is of finite type over ${\\cal O}_K$\nfor a discrete valuation field $K$\nof characteristic $0$\nwith residue field $k$.\n\n\n{\\rm 1.}\nWe consider\nthe following conditions:\n\n{\\rm (1)}\n$SS{\\cal F}$ is the $0$-section\n$FT^*_XX|_{X_{{\\mathbf F}_p}}$.\n\n{\\rm (2)}\n${\\cal F}$ is locally constant\non a neighborhood of $X_{{\\mathbf F}_p}$\nand $X_{{\\mathbf F}_p}$\nis a subset of the support of ${\\cal F}$.\n\n\\noindent\nWe have\n{\\rm (2)}$\\Rightarrow${\\rm (1)}.\nIf {\\rm (DVR)} is satisfied,\nwe have\n{\\rm (1)}$\\Rightarrow${\\rm (2)}.\n\n\n{\\rm 2.}\nWe consider\nthe following conditions:\n\n{\\rm (1)}\n$SS{\\cal F}=\\varnothing$.\n\n{\\rm (2)}\n${\\cal F}=0$\non a neighborhood of $X_{{\\mathbf F}_p}$.\n\n\\noindent\nWe have\n{\\rm (2)}$\\Rightarrow${\\rm (1)}.\nIf {\\rm (DVR)} is satisfied,\nwe have\n{\\rm (1)}$\\Rightarrow${\\rm (2)}.\n\n{\\rm 3.}\nAssume that \n$X={\\rm Spec}\\, {\\cal O}_K$\nfor a discrete valuation ring\n${\\cal O}_K$ as in {\\rm (DVR)}.\nIf ${\\cal F}$\nis not locally constant,\nwe have\n$SS{\\cal F}=FT^*X|_{X_k}$.\n\\end{lm}\n\nRecall that a discrete valuation ring\n${\\cal O}_K$ as in {\\rm (DVR)}\nis excellent\nby \\cite[Scholie (7.8.3)]{EGA4}.\n\n\\proof{\n1.\n(1)$\\Rightarrow$(2):\nSince ${\\cal F}$ is micro-supported\non the $0$-section\n$F^*T^*_XX|_{X_{{\\mathbf F}_p}}$,\nby Corollary \\ref{cortrla},\n${\\cal F}$ is locally constant\non a neighborhood of $X_{{\\mathbf F}_p}$.\nAfter replacing $X$ by\na neighborhood of $X_{{\\mathbf F}_p}$,\nwe may assume that\n${\\cal F}$ is locally constant.\nThen, the support\n$Z={\\rm supp}\\, {\\cal F}$\nis an open and closed subset of $X$\nand ${\\cal F}$ is micro-supported\non the $0$-section $T^*_ZX|_{X_{{\\mathbf F}_p}}$\non $Z$.\nBy the minimality of the singular\nsupport, we have $T^*_ZX|_{X_{{\\mathbf F}_p}}\n=T^*_XX|_{X_{{\\mathbf F}_p}}$\nand $X_{{\\mathbf F}_p}\\subset Z$.\n\n\n(2)$\\Rightarrow$(1):\nSince ${\\cal F}$ is locally constant\non a neighborhood of $X_{{\\mathbf F}_p}$,\nby Lemma \\ref{lmTX}.2,\n${\\cal F}$ is micro-supported\non the $0$-section\n$F^*T^*_XX|_{X_{{\\mathbf F}_p}}$.\nSuppose ${\\cal F}$ is micro-supported\non a closed conical subset\n$C\\subset FT^*X|_{X_{{\\mathbf F}_p}}$.\nSince $X_{{\\mathbf F}_p}\\subset {\\rm supp}\\, {\\cal F}$,\nwe have\n$X_{{\\mathbf F}_p}\\subset B(C)$.\nThis is equivalent to\n$F^*T^*_XX|_{X_{{\\mathbf F}_p}}\\subset C$\nand we obtain\n$F^*T^*_XX|_{X_{{\\mathbf F}_p}}=SS{\\cal F}$.\n\n2.\n(1)$\\Rightarrow$(2):\nSince the intersection\n${\\rm supp}\\, {\\cal F}\\cap\nX_{{\\mathbf F}_p}$ is a subset\nof $SS{\\cal F}=\\varnothing$,\nwe have $X_{{\\mathbf F}_p}\\subset \nX\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} {\\rm supp}\\, {\\cal F}$\nand  the condition (2) holds.\n\n(2)$\\Rightarrow$(1):\nSince every separated\nmorphism $h\\colon W\\to X$\nis ${\\cal F}$-transversal\non a neighborhood of $W_{{\\mathbf F}_p}$\nand since the intersection\n${\\rm supp}\\, {\\cal F}\\cap\nX_{{\\mathbf F}_p}$ is empty,\n${\\cal F}$ is micro-supported\non $\\varnothing$.\n\n3.\nBy Lemma \\ref{lmTX},\n${\\cal F}$ is micro-supported\non $FT^*X|_{X_k}$.\nSuppose that ${\\cal F}$ \nis micro-supported on \na closed conical subset $C\n\\subset FT^*X|_{X_k}$.\nSince $FT^*X|_{X_k}$\nis a line bundle\nover the point\n${\\rm Spec}\\, k$,\n$C$ is either\n$\\varnothing$,\nthe $0$-section\n$F^*T^*_XX|_{X_k}$\nor\n$FT^*X|_{X_k}$ itself.\nSince ${\\cal F}$ is not locally\nconstant, \nby the contraposition of\nCorollary \\ref{cortrla} (2)$\\Rightarrow$(1),\n${\\cal F}$ is not micro-supported\non the $0$-section\n$F^*T^*_XX|_{X_k}$.\n\\qed\n\n}\n\n\\medskip\n\nWe show that\nbeing singular support is\na local property.\n\n\\begin{lm}\\label{lmSSet}\nLet $X$ be a regular noetherian scheme \nover ${\\mathbf Z}_{(p)}$\nsatisfying the condition {\\rm (F)}\nand let\n${\\cal F}$ be a constructible\ncomplex of $\\Lambda$-modules.\nLet $C$ be a closed conical\nsubset of $FT^*X|_{X_{{\\mathbf F}_p}}$\nand let\n$(f_i\\colon U_i\\to X)_{i\\in I}$\nbe an \\'etale covering.\nWe consider the following conditions:\n\n{\\rm (1)}\n$C=SS{\\cal F}$.\n\n{\\rm (2)}\nFor every $i\\in I$\nand\nthe pull-backs\n$C_i=f_i^\\circ C$\nand\n${\\cal F}_i=f_i^*{\\cal F}$,\nwe have\n$C_i=SS{\\cal F}_i$.\n\n\\noindent\nWe have\n{\\rm (2)}$\\Rightarrow${\\rm (1)}.\nIf \n$(f_i\\colon U_i\\to X)_{i\\in I}$\nis a Zariski covering,\nwe have\n{\\rm (1)}$\\Rightarrow${\\rm (2)}\nconversely.\n\\end{lm}\n\n\\proof{\nSince the equivalence\nof the condition \nto be micro-supported\nis proved in Lemma \\ref{lmmset},\nwe show the inplications\nfor the minimality.\n\n(1)$\\Rightarrow$(2):\nLet $i\\in I$\nand let $C'\\subset FT^*U_i$ be a\nclosed conical subset\non which ${\\cal F}_i$\nis micro-supported.\nThen, by Lemma \\ref{lmmset},\n${\\cal F}$ is micro-supported\non $f_{i*}C'$\nin the notation of Definition \\ref{dfCet}.\nHence, we have\n$f_{i*}C'\\supset SS{\\cal F}=C$.\nIf $f_i\\colon U_i\\to X$\nis an open immersion,\nwe have\n$C'\\supset f_i^\\circ C=C_i$.\n\n(2)$\\Rightarrow$(1):\nLet $C'\\subset FT^*X$ be a\nclosed conical subset\non which ${\\cal F}$\nis micro-supported.\nThen,\nfor every $i\\in I$,\nthe pull-back\n${\\cal F}_i$ is micro-supported\non $f_i^\\circ C'$\nby Lemma \\ref{lmmset}.\nHence, we have\n$f_i^\\circ C'\\supset f_i^\\circ C$.\nSince $(U_i\\to X)_{i\\in I}$\nis an \\'etale covering,\nwe have $C'\\supset C$.\n\\qed\n\n}\n\n\n\\medskip\n\nWe compute the singular supports\nof some sheaves\non regular schemes \nof finite type over a\ndiscrete valuation ring.\n\n\\begin{lm}\\label{lmXW}\nLet $K$ be a discrete\nvaluation field of characteristic $0$\nsuch that\nthe residue field $k$\nis a perfect field of characteristic $p>0$.\nLet $h\\colon W\\to X$ be a \nfinite surjective morphism of\nregular flat schemes\nof finite type over\n${\\cal O}_K$\nsuch that the morphism\n$W_K\\to X_K$\non the generic fiber is \\'etale.\nAssume that the reduced parts\n$D=X_{k,{\\rm red}}$\nand\n$E=W_{k,{\\rm red}}$\nof the closed fibers\nare irreducible and are smooth \nof dimension $\\geqq 1$\nover the residue field $k$.\n\nAssume that the following condition\nis satisfied:\n\n{\\rm (1)}\nThe cokernel of the canonical morphism\n$F\\Omega^1_X\n\\otimes_{{\\cal O}_X}\n{\\cal O}_E\n\\to\nF\\Omega^1_W\n\\otimes_{{\\cal O}_W}\n{\\cal O}_E$\nof locally free \n${\\cal O}_E$-modules\nis locally free of rank $1$.\n\n{\\rm 1.}\nThe direct image\n$C= \\pi_\\circ FT^*_WW|_E\n\\subset FT^*X|_D$\nof the $0$-section\nis the image of the sub line bundle\n${\\rm Ker}(FT^*X|_D\\times_DE\n\\to FT^*W|_E)$\nof\n$FT^*X|_D\\times_DE$.\n\n{\\rm 2.}\nFurther assume \nthat the following condition is satisfied:\n\n{\\rm (2)}\nThe finite morphism\n$E\\to D$ is purely inseparable\nof degree $\\geqq 1$.\n\n\\noindent\nThen, for each closed point $x\\in D$\nand for the point $w\\in E$\nabove $x$,\nthere exists\na regular subscheme\n$Z\\subset W$ \nof codimension $1$\ncontaining $w$ and\nflat over ${\\cal O}_K$\nsatisfying the following conditions:\n\nThe composition\n$Z\\to W\\to X$ is unramified.\nThe pull-back $C\\times_{X_{{\\mathbf F}_p}}w\n\\subset FT^*X\\times_{X_{{\\mathbf F}_p}}\nw$ \nof the fiber at $x$\nequals the fiber of the\nkernel of the surjection\n$FT^*X\\times_{X_{{\\mathbf F}_p}}Z\n\\to FT^*Z$.\n\\end{lm}\n\n\\proof{\n1. \nSince the ${\\cal O}_E$-linear morphism\n$F\\Omega^1_X\n\\otimes_{{\\cal O}_X}\n{\\cal O}_E\n\\to\nF\\Omega^1_W\n\\otimes_{{\\cal O}_W}\n{\\cal O}_E$\nof locally free \n${\\cal O}_E$-modules\nof the same rank has\nthe cokernel of rank 1,\nthe kernel is also locally free of\nrank 1.\nHence the assertion follows.\n\n\n2.\nLet $n=\\dim {\\cal O}_{X,x}$.\nSince $E\\to D$ is assumed\nto be purely inseparable,\nthe residue field\n$k(w)$ is a purely inseparable\nextension of a perfect field\n$k(x)$ and hence \nthe morphism $k(x)\\to k(w)$ is an isomorphism.\nBy the assumption on the\nrank of the cokernel\nand by Proposition \\ref{prdx},\nthe $k(x)$-linear mapping\n${\\mathfrak m}_x\/\n{\\mathfrak m}_x^2\n\\to\n{\\mathfrak m}_w\/\n{\\mathfrak m}_w^2$\ninduced by\n${\\cal O}_{X,x}\\to\n{\\cal O}_{W,w}$ is of rank $n-1$.\n\nTake an element of\n${\\mathfrak m}_w\/\n{\\mathfrak m}_w^2$\nnot contained in the image\nof ${\\mathfrak m}_x\/\n{\\mathfrak m}_x^2$\nand take its lifting\n$f\\in {\\mathfrak m}_w$\nnot divisible by\na prime element $t$ defining \nthe divisor $E\\subset W$.\nThen,\na regular closed subscheme $Z$\nof codimension $1$ \nof a neighborhood \nof $w$ is defined \nby $f$.\nLet $z$ denote $w\\in W$\nregarded as a point of $Z$.\nSince $f$ is not divisible by $t$,\nwe may assume that $Z$ is flat\nover ${\\cal O}_K$.\n\nSince $\\bar f\\in\n{\\mathfrak m}_w\/\n{\\mathfrak m}_w^2$ is \nnot contained in the image\nof ${\\mathfrak m}_x\/\n{\\mathfrak m}_x^2$,\nthe induced morphism\n${\\mathfrak m}_x\/\n{\\mathfrak m}_x^2\n\\to\n{\\mathfrak m}_w\/((f)+\n{\\mathfrak m}_w^2)\n=\n{\\mathfrak m}_z\/\n{\\mathfrak m}_z^2$\nis a surjection.\nHence further shrinking \n$Z$ if necessary,\nwe may assume that\n$Z\\to X$ is unramified.\nSince the kernel of the surjection\n${\\mathfrak m}_x\/\n{\\mathfrak m}_x^2\n\\to\n{\\mathfrak m}_z\/\n{\\mathfrak m}_z^2$\nequals the kernel of\n${\\mathfrak m}_x\/\n{\\mathfrak m}_x^2\n\\to\n{\\mathfrak m}_w\/\n{\\mathfrak m}_w^2$,\nthe last condition \non the fibers is satisfied.\n\\qed\n\n}\n\n\\medskip\nWe show that some concrete\nexamples of Kummer coverings\nsatisfy the assumptions\nin Lemma \\ref{lmXW}.\nLet $K$ be a discrete valuation\nfield as in Lemma \\ref{lmXW}\ncontaining a primitive\n$p$-th root of 1.\nLet $X$ be a regular flat\nscheme of finite type\nover ${\\cal O}_K$\nand assume that the reduced part\n$D=X_{k,{\\rm red}}$\nis smooth over the residue field $k$.\nLet $L$ be the local field \nat the generic point of $D$\nand let $e={\\rm ord}_Lp\\geqq p-1$\nbe the absolute ramification index.\n\n\\begin{lm}\\label{lmKum}\nLet $\\pi \\in \\Gamma(X,{\\cal O}_X)$\nbe a uniformizer of the divisor $D\n=X_{k,{\\rm red}}\\subset X$\nand let $u \\in \\Gamma(X,{\\cal O}_X^\\times)$\nbe a unit.\nLet $1\\leqq n< \\dfrac{pe}{p-1}$\nbe an integer\ncongruent to $0$ or $1$\nmodulo $p$\nand set $n=pm$ or $n=pm+1$\nrespectively.\nIn the case $n=pm$,\nassume that $du$ defines locally\na part of a basis of $\\Omega^1_D$.\nDefine a Kummer covering\n$V\\to U=X_K$\nby $v^p=1+u\\pi^n$.\n\n\n{\\rm 1.}\nThe normalization $\\pi\\colon\nW\\to X$\nin $V$ is regular.\nThe reduced closed fiber\n$E=W_{k,{\\rm red}}$\nis smooth over $k$\nand the finite morphism\n$E\\to D$ is purely inseparable.\n\n\n{\\rm 2.}\nThe cokernel\n${\\rm Coker}(F\\Omega^1_X\n\\otimes_{{\\cal O}_X}\n{\\cal O}_E\n\\to\nF\\Omega^1_W\n\\otimes_{{\\cal O}_W}\n{\\cal O}_E)$\nis an invertible ${\\cal O}_E$-module.\n\n{\\rm 3.}\nAssume $n=pm$.\nIf $e=m+1$,\nlet $\\pi'$ denote the uniformizer\n$p\/\\pi^m$.\nThen, the kernel of the canonical morphism\n$FT^*X|_D\\times_DE\\to\nFT^*W|_E$\nis a line bundle spanned\nby \n$$\\begin{cases}\nw(u)-u\\cdot w(\\pi')\n&\n\\text{ if $p=2$ and $e=m+1$},\n\\\\\nw(u)&\n\\text{ otherwise}.\n\\end{cases}$$\n\\end{lm}\n\n\\proof{\n{\\rm 1.}\nSince the assertion is local,\nwe may assume that\n$X={\\rm Spec}\\, A$ is affine.\nWe show that the normalization\n$B$ of $A$ is generated by \n$t=(v-1)\/\\pi^m$.\nBy the assumption\n$n<\\dfrac{ep}{p-1}$,\nwe have\n$e+m>pm$ and\nthe polynomial\n$(1+\\pi^mT)^p-1\n\\in A[T]$\nis divisible by $\\pi^{pm}$.\nDefine a monic polynomial\n$F\\in A[T]$\nby $1+\\pi^{pm}F=(1+\\pi^mT)^p$.\nSince\n$F\\equiv T^p\\bmod \n\\pi A$ and since $u$ is a unit,\nin the case $n=pm+1$,\nthe equation\n$F=\\pi u$ is an Eisenstein equation.\nIn the case $n=pm$,\nthe reduction of the equation\n$F=u$\nmodulo $\\pi A$\ngives $T^p=u$.\nIn this case $du$ is a part of\na basis of $\\Omega^1_D$\nby the assumption.\nHence \nby setting $v=1+\\pi^mt$\nwhere $t\\in B$ denotes the class of\n$T$,\nwe obtain\n$B=A[T]\/(F-u\\pi)$ \nor $B=A[T]\/(F-u)$ \nrespectively.\n\nThe reduced part $E$\nis defined by $t$ or $\\pi$\naccording to $n=pm+1$ or $n=pm$\nrespectively.\nHence $E$ is smooth over $k$\nand the finite morphism\n$E\\to D$ is purely inseparable\nof degree 1 or $p$\nrespectively.\n\n2.\nBy Corollary \\ref{corXZ},\nwe have a commutative diagram\n$$\\begin{CD}\n0@>>>\nF^*N_{D\/X}\n\\otimes_{{\\cal O}_D}\n{\\cal O}_E\n@>>>\nF\\Omega^1_X\n\\otimes_{{\\cal O}_X}\n{\\cal O}_E\n@>>>\nF^*\\Omega^1_D\n\\otimes_{{\\cal O}_D}\n{\\cal O}_E\n@>>>\n0\\\\\n@.@VVV@VVV@VVV@.\\\\\n0@>>>\nF^*N_{E\/W}\n@>>>\nF\\Omega^1_W\n\\otimes_{{\\cal O}_W}\n{\\cal O}_E\n@>>>\nF^*\\Omega^1_E\n@>>>\n0\n\\end{CD}$$\nof exact sequences of\nlocally free ${\\cal O}_E$-modules.\nIn the case $n=pm+1$,\nthe right vertical arrow\nis an isomorphism\nsince $E\\to D$ is an isomorphism.\nFurther\nthe left vertical arrow is $0$\nsince the ramification index is $p$.\nIn the case $n=pm$,\nthe left vertical arrow\nis an isomorphism\nsince the ramification index is $1$.\nFurther\nthe cokernel of the right vertical arrow is \nlocally free of rank 1\nsince $E\\to D$ is a purely inseparable\ncovering defined by $T^p=u$\nand $du$ is a part of a basis\nof $\\Omega^1_D$.\nHence the assertion follows.\n\n\n\n3.\nWe compute\nthe polynomial $F\n\\bmod  \\pi^2$.\nRecall that we have\n$e+m>pm$.\nSince $e$ is divisible by\n$p-1$, the equality\n$e+m=pm+1$ holds\nif and only if\n$p=2$ and $e=m+1$.\nHence \nthe coefficients of $T^i$ \nfor $i=1,\\ldots, p-1$ in \nthe polynomial $F$\nare divisible by $\\pi^2$\nexcept $F=T^2+\n2\/\\pi^m\\cdot T$\nin the exceptional case.\n\nThus, except the exceptional case,\nwe have a congruence\n$F\\equiv T^p\n\\bmod \\pi^2$\nand hence the kernel is\nspanned by \n$w(u)$.\nIn the exceptional case,\nwe have\n$t^2+\\pi't=u$ for\n$\\pi'=2\/\\pi^m$.\nHence $w(u)$\nis sent to $t^2\\cdot w(\\pi')\n=u\\cdot w(\\pi')$.\n\\qed\n\n}\n\n\n\\begin{pr}\\label{prKum}\nLet $K$ be a discrete\nvaluation field of characteristic $0$\nsuch that\nthe residue field $k$\nis a perfect field of characteristic $p>0$.\nLet $X$ be a regular flat scheme\nof finite type over\n${\\cal O}_K$\nsuch that the reduced part\n$D=X_{k,{\\rm red}}$\nis irreducible and is smooth\nover the residue field $k$.\n\nLet ${\\cal F}_U$ be a locally constant\nconstructible sheaf of\n$\\Lambda$-modules on\nthe generic fiber $U=X_K$\nand let ${\\cal F}=j_!{\\cal F}_U$\nbe the $0$-extension\nfor the open immersion\n$j\\colon U\\to X$.\nLet $V\\to U$ be a finite\n\\'etale Galois covering \nof Galois group $G$ such that\nthe pull-back ${\\cal F}_V$\nis constant \nand let $\\pi\\colon W\\to X$ be\nthe normalization \nof $X$ in $V$.\n\nAssume that $W$ is regular\nand that\nthe reduced part\n$E=W_{k,{\\rm red}}$\nis also irreducible and smooth\nover the residue field $k$.\nAssume that the order of $G$\nis invertible in $\\Lambda$\nand that ${\\cal F}_U$ corresponds\nto a non-trivial\nirreducible representation $M$ of $G$.\n\n{\\rm 1.}\nThe canonical morphism\n${\\cal F}=j_!{\\cal F}_U\n\\to Rj_*{\\cal F}_U$\nis an isomorphism.\n\n{\\rm 2.}\nAssume that conditions\n{\\rm (1)} and {\\rm (2)} \nin Lemma {\\rm \\ref{lmXW}}\nare satisfied.\nThen, the singular support $SS{\\cal F}$\nequals the direct image\n$C=\\pi_\\circ FT^*_WW|_{W_k}$\nof the $0$-section.\n\\end{pr}\n\n\n\\proof{\n1.\nBy the assumption \nthat the order of $G$\nis invertible in $\\Lambda$\nand that $M$ is an irreducible\nrepresentation,\nthe locally constant sheaf\n${\\cal F}_U$\nis isomorphic to a direct summand\nof $\\pi_{K*}\\Lambda$\nwhere $\\pi_K\\colon V=W_K\\to U=X_K$\nis the restriction of $\\pi$.\n\nLet $j_W\\colon W_K\\to W$\nbe the open immersion\nof the generic fiber.\nSince $W$ is regular\nand the reduced part\nof the closed fiber\n$W_k$ is a regular divisor,\nwe have isomorphisms\n$\\Lambda\\to j_{W*}\\Lambda$,\n$\\Lambda_E(-1)\\to R^1 j_{W*}\\Lambda$\nand $R^qj_{W*}\\Lambda=0$\nfor $q\\neq 0,1$\nby the absolute purity \n\\cite[{\\sc Th\\'eor\\`eme 3.1.1}]{purete}.\nSimilarly,\nwe have isomorphisms\n$\\Lambda\\to j_*\\Lambda$,\n$\\Lambda_D(-1)\\to R^1 j_*\\Lambda$\nand $R^qj_*\\Lambda=0$\nfor $q\\neq 0,1$.\nSince $E\\to D$ induces a homeomorphism\non the \\'etale site by the assumption,\nthe canonical morphism\n$\\Lambda_D\\to \\pi_*\\Lambda_E$\nis an isomorphism.\nHence, for the cokernel\n${\\cal G}={\\rm Coker}\n(\\Lambda_X\\to \\pi_*\\Lambda_W)$,\nthe canonical morphisms\n$j_!j^*{\\cal G}\\to\n{\\cal G}\\to Rj_*j^*{\\cal G}$\nare isomorphisms.\n\nSince\n$M$ is a non-trivial irreducible\nrepresentation\nof a semi-simple algebra\n$\\Lambda[G]$,\nthe corresponding sheaf\n${\\cal F}$ is a direct summand\nof $j^*{\\cal G}$.\nHence the canonical morphism\n${\\cal F}=j_!{\\cal F}_U\n\\to Rj_*{\\cal F}_U$\nis an isomorphism.\n\n2.\nSince ${\\cal F}$ is a direct summand\nof $\\pi_*\\Lambda_W=\\Lambda_X\n\\oplus {\\cal G}$,\nby Proposition \\ref{prmcf}.2,\nthe constructible sheaf\n${\\cal F}$ is micro-supported on\n$C=\\pi_\\circ FT^*_WW|_{W_k}$.\n\nSuppose ${\\cal F}$\nis micro-supported on \na closed conical subset $C'$.\nIt suffices to prove $C\\subset C'$.\nLet $x\\in X_{{\\mathbf F}_p}$\nbe a closed point,\nlet $h\\colon Z\\to X$\nbe an unramified morphism\nas in Lemma \\ref{lmXW}\nand let $z\\in Z$ be\nthe unique point above $x$.\nSince $Z\\to X$ factors through\n$Z\\to W$,\nthe restriction \n${\\cal F}_{Z\\cap U}$\nis constant.\nHence the morphism\n$h$ is not\n${\\cal F}$-transversal\nby the contraposition\nof Proposition \\ref{prhF}.2\n(1)$\\Rightarrow$(2)\nand 1.\nSince ${\\cal F}$\nis micro-supported on $C'$,\nthe morphism\n$h$ is not $C'$-transversal,\non any open neighborhood of $z\\in Z$.\n\nThe kernel $L={\\rm Ker}\n(FT^*X\\times_{X_{{\\mathbf F}_p}}\nZ_{{\\mathbf F}_p}\\to FT^*Z)$\nis a line bundle\non $Z_{{\\mathbf F}_p}$.\nThe intersection $C'_1=h^*C'\n\\cap L\n\\subset FT^*X\\times_{X_{{\\mathbf F}_p}}\nZ_{{\\mathbf F}_p}$\nis a closed conical subset\nof $L$.\nLet $Z_1\n=\\{y\\in Z_{{\\mathbf F}_p}\\mid\nC'_{1,y}=L_y\\}$\nbe the image by the projection \nof the complement\n$C'_1\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} (C'_1\\cap Z_{{\\mathbf F}_p})$\nof the $0$-section.\nSince $C'_1\\subset L$ is a closed\nconical subset,\nthe image\n$Z_1\\subset\nZ_{{\\mathbf F}_p}$ is a closed subset.\nSince the restriction \n$Z\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} Z_1\\to X$ of $h$\nis $C'$-transversal,\nthe complement\n$Z\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} Z_1$ is not\nan open neighborhood of $z$.\nNamely,\nwe have $z\\in Z_1$\nand hence \n$C'_{1,z}=L_z$ \nis a subset of $C'_z$.\n\nSince \n$L_z=C_z=C_x\\times_xz$\nby the condition on $Z$, we get\n$C_x\\subset C'_x$\nfor each closed point $x\\in X_k$.\nThus we have $C\\subset C'$\nas required.\n\\qed\n\n}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Preliminaries}\n\n\\subsection{Cayley algebra and compact Lie group of type ${\\rm G}_2$} \n\nLet $\\mathfrak{C}=\\{e_0 =1, e_1, e_2, e_3, e_4, e_5, e_6, e_7 \\}_{\\sR}$ be the division Cayley algebra. In $\\mathfrak{C}$, since the multiplication and the inner product are well known, these are omitted.\n\\vspace{1mm}\n\nThe connected compact Lie group of type ${\\rm G_2}$ is given by\n$$\nG_2 =\\{\\alpha \\in \\Iso_{\\sR}(\\mathfrak{C})\\,|\\, \\alpha(xy)=(\\alpha x) (\\alpha y) \\}\\vspace{1mm}.\n$$ \n\\subsection{Exceptional Jordan algebra and compact Lie group of type ${\\rm F}_4$} \n\nLet  \n$\\mathfrak{J}(3,\\mathfrak{C} ) = \\{ X \\in M(3, \\mathfrak{C}) \\, | \\, X^* = X \\}$ be the \nexceptional Jordan algebra. \nIn $\\mathfrak{J}(3,\\mathfrak{C} )$, the Jordan multiplication $X \\circ Y$, the \ninner product $(X,Y)$ and a cross multiplication $X \\times Y$, called the Freudenthal multiplication, are defined by\n$$\n\\begin{array}{c}\nX \\circ Y = \\dfrac{1}{2}(XY + YX), \\quad (X,Y) = \\tr(X \\circ Y),\n\\vspace{1mm}\\\\\nX \\times Y = \\dfrac{1}{2}(2X \\circ Y-\\tr(X)Y - \\tr(Y)X + (\\tr(X)\\tr(Y) \n- (X, Y))E), \n\\end{array}$$\nrespectively, where $E$ is the $3 \\times 3$ unit matrix. Moreover, we define the trilinear form $(X, Y, Z)$, the determinant $\\det \\,X$ by\n$$\n(X, Y, Z)=(X, Y \\times Z),\\quad \\det \\,X=\\dfrac{1}{3}(X, X, X),\n$$\nrespectively, and briefly denote $\\mathfrak{J}(3, \\mathfrak{C})$\nby $\\mathfrak{J}$.\n\\vspace{1mm}\n\nThe connected compact Lie group of type ${\\rm F_4}$ is given by\n\\begin{align*}\n\tF_4 &= \\{\\alpha \\in \\Iso_{\\sR}(\\mathfrak{J}) \\, | \\, \\alpha(X \\circ Y) = \\alpha X \\circ \\alpha Y \\}\n\t\\\\[1mm]\n\t&=  \\{\\alpha \\in \\Iso_{\\sR}(\\mathfrak{J}) \\, | \\, \\alpha(X \\times Y) = \\alpha X \\times \\alpha Y \\}. \n\\end{align*}\nThen we have naturally the inclusion $G_2 \\subset F_4$ as follows:\n\\begin{align*}\n\\varphi:G_2 \\to F_4,\\,\\,\\varphi(\\alpha)X=\\begin{pmatrix}\n\\xi_1 & \\alpha x_3 & \\ov{\\alpha x_2} \\\\\n\\ov{\\alpha x_3} & \\xi_2 & \\alpha x_1 \\\\ \n\\alpha x_2 & \\ov{\\alpha x_1} & \\xi_3\n\\end{pmatrix},\\,\\, X \\in \\mathfrak{J}.\n\\end{align*} \n\\subsection{Complex exceptional Jordan algebra and Compact Lie group of type ${\\rm E}_6$} \nLet $\\mathfrak{J}(3,\\mathfrak{C})^C = \\{ X \\in M(3, \\mathfrak{C})^C \\, | \\, X^* = X \\}$ be the complexification of the exceptional Jordan algebra $\\mathfrak{J}$. In $\\mathfrak{J}(3,\\mathfrak{C})^C$, as in $\\mathfrak{J}$, we can also define the multiplication $X \\circ Y, X \\times Y$, the inner product $(X, Y)$, the trilinear forms $(X, Y, Z)$ and the determinant $\\det \\, X$ in the same manner, and those have the same properties. The  $\\mathfrak{J}(3,\\mathfrak{C} )^C$ is called the complex exceptional Jordan algebra, and briefly denote $\\mathfrak{J}(3, \\mathfrak{C})^C$ by $\\mathfrak{J}^C$. \n\\vspace{1mm}\n\nThe connected compact Lie group of type ${\\rm E_6}$ is given by\n\\begin{align*}\n\t\tE_6 &= \\{\\alpha \\in \\Iso_C(\\mathfrak{J}^C) \\, | \\, \\det\\, \\alpha X = \\det\\, X, \\langle \\alpha X, \\alpha Y \\rangle = \\langle X, Y \\rangle \\}\n\t\t\\\\ \n\t\t &=\\{\\alpha \\in \\Iso_C(\\mathfrak{J}^C) \\, | \\,\\alpha X \\times \\alpha Y=\\tau\\alpha\\tau(X \\times Y) , \\langle \\alpha X, \\alpha Y \\rangle = \\langle X, Y \\rangle \\}\n\\end{align*}\nwhere $\\tau$ is a complex conjugation in $\\mathfrak{J}^C$: $\\tau(X+iY)=X-iY, \\,X, Y \\in \\mathfrak{J}$ and the Hermite inner product $\\langle X, Y \\rangle$ is defined by $(\\tau X, Y)$.\n\n\\noindent Then we have naturally the inclusion $F_4 \\subset E_6$ as follows:\n\\begin{align*}\n   \\varphi:F_4 \\to E_6,\\,\\,\\varphi(\\alpha)(X_1+iX_2)=(\\alpha X_1)+i(\\alpha X_2),\\,\\,X_1+iX_2 \\in \\mathfrak{J}^C, X_i \\in \\mathfrak{J}.\n\\end{align*}\n\n\n\\if\nIn the last of this section, we state useful lemma. \n\n\\begin{lemma}\\label{lemma 2.3.}\n\tFor Lie groups $G, G' $,  let a mapping $\\varphi : G \\to G'$ be a homomorphism of Lie groups. When $G'$ is connected, if $\\Ker\\,\\varphi$ is discrete and $\\dim(\\mathfrak{g})=\\dim(\\mathfrak{g}')$, $\\varphi$ is surjective.\n\\end{lemma}\n\\begin{proof}\n\tThe proof is omitted (see \\cite[Proposition 8.2 (1)]{iy4} in detail).\n\\end{proof}\n\n\\begin{lemma}[E. Cartan-Ra\\v{s}evskii]\\label{lemma 2.3.1}\n\tLet $G$ be a simply connected Lie group with a finite order automorphism $\\sigma$\n\tof $G$. Then $G^\\sigma$ is connected.\n\\end{lemma}\n\\begin{proof}\n\tThe proof is omitted (cf. \\cite[Lemma 0.7]{realization G_2}).\n\\end{proof}\n\\noindent Hereafter, using these lemmas without permission each times, we often prove lemma, proposition or theorem.\n\nWe almost use the same notation as \\cite{iy0}, in particular the complex fields $\\C, C$ are as follows.\n\\begin{align*}\n   \\C=\\{x+ye_1 \\,|\\, x,y \\in \\R \\},\\quad C=\\{x+yi \\,|\\, x,y \\in \\R \\}(=\\R^C).\n\\end{align*}\n\\f\n\n\\section{The inner automorphisms of order $3$ and the fixed points subgroups by them}\\label{section 3\n\nIn this section, we will rewrite the inner automorphisms of order $3$ on $G=G_2, F_4,E_6$ and the fixed points subgroups of $G$ by them which were realized and determined in \\cite{iy1}, in association with the involutive inner automorphisms. However, the detailed proofs are omitted.\n\n\\subsection{In $G_2$}\\label{subsection 3.1\n\nLet $\\mathfrak{C}=\\H \\oplus \\H e_4$ be Cayley devision algebra, where $\\H$ is the field of quaternion number. Since a multiplication, a conjugation and inner product in $\\mathfrak{C}=\\H \\oplus \\H e_4$ are well known, these are ommited. If necessary, refer to \\cite{miya1},\\cite{realization G_2} and \\cite{iy0}.\n\nWe define an $\\R$-linear transformation $\\gamma$ of $\\mathfrak{C}$ by \n\\begin{align*}\n\t\t\\gamma(m+ne_4)=m-ne_4, \\,\\, m+ne_4 \\in \\H \\oplus \\H e_4 = \\mathfrak{C}.\n\\end{align*}\nThen we have that $\\gamma \\in G_2$ and $\\gamma^2 =1$. Hence $\\gamma$ induces the involutive inner automorphism $\\tilde{\\gamma}$ on $G_2: \\tilde{\\gamma}(\\alpha)=\\gamma\\alpha\\gamma, \\alpha \\in G_2$, so we have the following well-known result.\n\n\\begin{proposition}\\label{proposition 3.1.1\n\tThe group $(G_2)^\\gamma$ is isomorphism to the group $(Sp(1) \\times Sp(1))\/\\Z_2${\\rm:}  $(G_2)^\\gamma \\cong  (Sp(1) \\times Sp(1))\/\\Z_2, $ $ \\Z_2=\\{ (1,1), (-1,-1) \\}$.\n\\end{proposition}\n\\begin{proof}\n\tWe define a mapping $\\varphi_{{}_{311}}: Sp(1) \\times Sp(1) \\to (G_2)^\\gamma$ by \n\t\\begin{align*}\n\t\\varphi_{{}_{G_2,\\gamma}}(p, q)(m+n e_4)=qm \\ov{q}+(pn \\ov{q}) e_4, \\,\\,\\,m+n e_4 \\in \\H \\oplus \\H e_4 =\\mathfrak{C}.\n\t\\end{align*}\n\tThis mapping induces the required isomorphism (see \\cite [Theorem 1.10.1]{iy0} in detail).\n\\end{proof}\n\nLet $\\bm{\\omega}=-(1\/2)+(\\sqrt{3}\/2)e_1 \\in U(1) \\subset \\C \\subset \\H \\subset \\mathfrak{C}$. We define an $\\R$-linear transformation $\\gamma_3$ of $\\mathfrak{C}$ by\n\\begin{align*}\n\t\t\\gamma_3(m+ne_4)=m+(\\bm{\\omega} n)e_4, \\,\\,m+ne_4 \\in \\H \\oplus \\H e_4=\\mathfrak{C}.\n\\end{align*}\nThen, using the mapping $\\varphi_{{}_{G_2, \\gamma}}$ above, since $\\gamma_3$ is expressed by $\\varphi_{{}_{G_2,\\gamma}}(\\bm{\\omega},1)$: $\\gamma_3=\\varphi_{{}_{G_2,\\gamma}}(\\bm{\\omega},1)$, it is clear that $\\gamma_3 \\in G_2$ and $(\\gamma_3)^3=1$. Hence $\\gamma_3$ induces the inner automorphism $\\tilde{\\gamma}_3$ of order $3$ on $G_2: \\tilde{\\gamma}_3(\\alpha)={\\gamma_3}^{-1}\\alpha\\gamma_3, \\alpha \\in G_2$. \n\\vspace{1mm}\n\nNow, we have the following theorem.\n\n\\begin{theorem}\\label{theorem 3.1.2\n\tThe group $(G_2)^{\\gamma_3}$ is isomorphism to the group $(U(1) \\times Sp(1))\/\\Z_2${\\rm:}  $(G_2)^{\\gamma_3} \\cong  (U(1) \\times Sp(1))\/\\Z_2,  \\Z_2=\\{ (1,1), (-1,-1) \\}$.\n\\end{theorem}\n\\begin{proof}\n\tLet $U(1)=\\{a \\in \\C \\,|\\,\\ov{a}a=1 \\} \\subset Sp(1)$, where $\\C=\\{x+ye_1\\,|\\, x,y \\in \\R \\}$. Then we define a mapping $\\varphi_{{}_{G_2,\\gamma_3}}:U(1) \\times Sp(1) \\to (G_2)^{\\gamma_3}$ by the restriction of the mapping $\\varphi_{{}_{G_2,\\gamma}}$ (Proposition \\ref{proposition 3.1.1}). This mapping induces the required isomorphism (see \\cite [Theorem 1.2]{iy1} in detail).\n\\end{proof}\n\nThus, since the group $(G_2)^{\\gamma_3}$ is connected, together with the result of Theorem \\ref{theorem 3.1.2}, we have an exceptional $\\varmathbb{Z}_3$-symmetric space $G_2\/(U(1) \\times Sp(1))\/\\Z_2)$.\n\\vspace{2mm}\n\nLet $x = m_0 + m_1e_2 + m_2e_4 + m_3e_6 \\in \\mathfrak{C}, m_i \\in \\C$. Then we associate such elements $x$ of $\\mathfrak{C}$ with the elements \n\\begin{align*}\n\t\t\tm_0 + \\begin{pmatrix}\n\t\t\tm_1 \\\\\n\t\t\tm_2 \\\\\n\t\t\tm_3\n\t\t\t\\end{pmatrix}(=:m_0+\\m)\n\\end{align*}\nof $\\C \\oplus \\C^3$ and we can define a multiplication, a conjugation and an inner product in $\\C \\oplus \\C^3$ corresponding to the same ones in $\\mathfrak{C}$ (see \\cite[Subsection 1.5]{iy0} in detail). Hence we have that $\\C \\oplus \\C^3$ is isomorphic to $\\mathfrak{C}$ as algebra. Hereafter, if necessary, we identify $\\mathfrak{C}$ with $\\C \\oplus \\C^3$: $\\mathfrak{C}=\\C \\oplus \\C^3$. \n\n\n\\if\nWe will rewrite alternative definition of Cayley algebra $\\mathfrak{C}$ according to \\cite[Subsection 1.5]{iy0}.\n\nAny element $x \\in \\mathfrak{C}$ is expressed by\n\\begin{align*}\n\tx &= x_0 + x_1e_1 + x_2e_2 + x_3e_3 + x_4e_4 + x_5e_5 + \n\tx_6e_6 + x_7e_7 \\quad (x_i \\in \\R) \n\t\\\\\n\t&= (x_0 + x _1e_1) + (x_2 + x_3e_1)e_2 + (x_4 + x_5e_1)e_4\n\t+ (x_6 + x_7e_1)e_6,\n\\end{align*}\nthat is,\n$$\nx = m_0 + m_1e_2 + m_2e_4 + m_3e_6, \\quad m_i \\in \\C.\n$$\n\nWe associate such element $x$ of $\\mathfrak{C}$ with the element $m_0 + \\begin{pmatrix}\nm_1 \\\\\nm_2 \\\\\nm_3\n\\end{pmatrix}$ of $\\C \\oplus \\C^3$. \n\n\\noindent In $\\C \\oplus \\C^3$, we define a multiplication, an inner product $(\\;\\;,\\;\\,)$ and a conjugation $\\overline{{\\;}^{\\;}\\;\\;}$ respectively by\n\\begin{align*}\n\t(m_0 + \\m)(n_0 + \\n) &= (m_0 n_0 - \\langle \\m, \\n \\rangle ) + \n\t(m_0\\n + \\ov{n_0}\\m - \\ov{\\m \\times \\n}), \n\t\\\\\n\t(m_0 + \\m, n_0 + \\n) &= (m_0, n_0) + (\\m, \\n), \n\t\\\\\n\t\\ov{m_0 + \\m} &= \\ov{m_0} - \\m, \n\\end{align*}\nwhere the real valued symmetric inner product $(\\m, \\n)$, the Hermitian inner \nproduct $\\langle \\m, \\n \\rangle$  and the exterior product $\\m \\times \\n$ are \nusually defined respectively by\n\\begin{align*}\n(\\m, \\n) = \\frac{1}{2}(\\m^{*}\\n + \\n^{*}\\m) = \\sum_{i=1}^3(m_i,n_i), \\,\\, \\langle \\m, \\n \\rangle = \\sum_{i=1}^3m_i\\ov{n}_i, \\,\\,\\m \\times \\n = \n\\begin{pmatrix} \nm_2n_3 - n_2m_3 \\\\\nm_3n_1 - n_3m_1 \\\\ \nm_1n_2 - n_1m_2\n\\end{pmatrix}\n\\end{align*}\n\\noindent for $\\m = \\begin{pmatrix}m_1 \\\\ m_2 \\\\ m_3\\end{pmatrix}$, $\\n = \\begin{pmatrix}n_1 \\\\ n_2 \\\\ n_3\\end{pmatrix} \n\\in \\C^3$. Since these operations correspond to their respective operations in \n$\\mathfrak{C}$. From now on, we also identify $\\C \\oplus \\C^3$ with $\\mathfrak{C}$ \n: $\\mathfrak{C}=\\C \\oplus \\C^3$.\n\\vspace{1mm}\n\\f\n\nAgain let $\\bm{\\omega}=-(1\/2)+(\\sqrt{3}\/2)e_1 \\in U(1) \\subset \\C \\subset \\H \\subset \\mathfrak{C}$. We define an $\\R$-linear transformation $w_3$ of $\\mathfrak{C}=\\C \\oplus \\C^3$ by\n\\begin{align*}\n\t\tw_3(m_0+\\m)=m_0+\\bm{\\omega} \\m, \\,\\,m_0+\\m \\in \\C \\oplus \\C^3=\\mathfrak{C}.\n\\end{align*}\nThen we have that $w_3 \\in G_2$ (\\cite[Proposition 1.4]{iy1}) and $(w_3)^3=1$. Hence $w_3$ induces the inner automorphism $\\tilde{w}_3$ of order $3$ on $G_2$: $\\tilde{w}_3(\\alpha)={w_3}^{-1}\\alpha w_3, \\alpha \\in G_2$.\n\\vspace{1mm}\n\nNow, we have the following theorem.\n\n\\begin{theorem}\\label{theorem 3.1.3\n\tThe group $(G_2)^{w_3}$ is isomorphic to the group $SU(3)${\\rm :} $(G_2)^{w_3} \\cong SU(3)$.\n\\end{theorem}\n\\begin{proof}\nWe define a mapping $\\varphi_{{}_{G_2,w_3}}: SU(3) \\to (G_2)^{w_3}$ by\n\t\\begin{align*}\n\t\t\t\\varphi_{{}_{G_2,w_3}}(A)(m_0+\\m)=m_0+A\\m, \\,\\,m_0+\\m \\in \\C \\oplus \\C^3=\\mathfrak{C}.\n\t\\end{align*}\n\tThis mapping induces the required isomorphism (see \\cite[Theorem 1.6]{iy1} in detail).\n\\end{proof}\n\nThus, since the group $(G_2)^{w_3}$ is connected, together with the result of Theorem \\ref{theorem 3.1.3}, we have an exceptional $\\varmathbb{Z}_3$-symmetric space $G_2\/SU(3)$. As is well known, this space is homeomorphic to a $6$-dimensional sphere $S^6$: $G_2\/SU(3) \\simeq S^6$.   \n\\vspace{2mm}\n\nThe following lemma are useful to determine the structure of groups $G^{\\sigma_3} \\cap G^{\\tau_3}$ in $G_2$.\n\n\\begin{lemma}\\label{lemma 3.1.4\n\t{\\rm (1)} The mapping $\\varphi_{{}_{G_2,\\gamma_3}}:U(1) \\times Sp(1) \\to (G_2)^{\\gamma_3}$ of \\,Theorem {\\rm \\ref{theorem 3.1.2}} satisfies the relational formulas \n\t\\begin{align*}\n \t\\gamma_3&=\\varphi_{{}_{G_2,\\gamma_3}}(\\bm{\\omega},1),\n \t\\\\\n \tw_3&=\\varphi_{{}_{G_2,\\gamma_3}}(1, \\ov{\\bm{\\omega}}),\n\t\\end{align*}\n \twhere $\\bm{\\omega}=-(1\/2)+(\\sqrt{3}\/2)e_1 \\in U(1)$.\n\\vspace{1mm}\n\n\t{\\rm (2)} The mapping $\\varphi_{{}_{G_2,w_3}}:SU(3) \\to (G_2)^{w_3}$ of \\,Theorem {\\rm \\ref{theorem 3.1.3}} satisfies the relational formulas\n\t\\begin{align*}\n\t\\gamma_3&=\\varphi_{{}_{G_2,w_3}}(\\diag(1,\\bm{\\omega},\\ov{\\bm{\\omega}})),\n\t\\\\\n\tw_3&=\\varphi_{{}_{G_2,w_3}}(\\bm{\\omega}E),\n\t\\end{align*}\n\twhere $\\bm{\\omega}=-(1\/2)+(\\sqrt{3}\/2)e_1 \\in U(1)$. \n\\end{lemma}\n\\begin{proof}\n\t(1), (2) By doing straightforward computation we obtain the results above. \n\\end{proof}\n\n\\subsection{In $F_4$}\\label{subsection 3.2\n\nLet $\\mathfrak{J}$ be the exceptional Jordan algebra. As is well known, the elements $X$ of $\\mathfrak{J}$ take the form \n$$\nX = \\begin{pmatrix}\n\\xi_1 & x_3 & \\ov{x_2} \\\\\n\\ov{x_3} & \\xi_2 & x_1 \\\\ \nx_2 & \\ov{x_1} & \\xi_3\n\\end{pmatrix},\\,\\, \\xi_i \\in \\R,\\, x_i \\in \\mathfrak{C},\\, i=1, 2, 3.\n$$\nHereafter, in $\\mathfrak{J}$, we use the following nations:\n\\begin{align*}\nE_1 &= \\left(\\begin{array}{ccc}\n1 & 0 & 0 \\\\\n0 & 0 & 0 \\\\\n0 & 0 & 0\n\\end{array}\n\\right),  \\,\\,\\,\\,\\,\\,\\,\\,\nE_2 = \\left(\\begin{array}{ccc}\n0 & 0 & 0 \\\\\n0 & 1 & 0 \\\\\n0 & 0 & 0\n\\end{array}\n\\right),  \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\nE_3 = \\left(\\begin{array}{ccc}\n0 & 0 & 0 \\\\\n0 & 0 & 0 \\\\\n0 & 0 & 1\n\\end{array}\n\\right),    \n\\\\[2mm]\nF_1 (x) &= \\left(\\begin{array}{ccc}\n0 &      0 & 0 \\\\\n0 &      0 & x \\\\\n0 & \\ov{x} & 0\n\\end{array}\n\\right),  \\,\\,\nF_2(x) = \\left(\\begin{array}{ccc}\n0 & 0 & \\ov{x} \\\\\n0 & 0 & 0 \\\\\nx & 0 & 0\n\\end{array}\n\\right),  \\,\\,\nF_3 (x) = \\left(\\begin{array}{ccc}\n0 & x & 0 \\\\\n\\ov{x} & 0 & 0 \\\\\n0 & 0 & 0\n\\end{array}\n\\right).\n\\end{align*}\n\n\n\\if\nThen $\\mathfrak{J}(3, \\H) \\oplus \\H^3$ has the Freudenthal multiplication and the inner product \n\\begin{align*}\n(M + \\a) \\times (N + \\b) &= \\Big(M \\times N - \\dfrac{1}{2} (\\a^*\\b + \\b^*\\a)\\Big) - \\dfrac{1}{2}(\\a N + \\b M), \n\\\\[1mm]\n(M + \\a, N + \\b) &= (M, N) + 2(\\a, \\b) \n\\end{align*}\ncorresponding those of $\\mathfrak{J}$, where $(\\a, \\b) = (1\/2)(\\a\\b^* + \\b\\a^*)$. Hence $\\mathfrak{J}$ is isomorphic to $\\mathfrak{J}(3, \\H) $ $\\oplus \\,\\H^3$ as algebra. From now on, we identify $\\mathfrak{J}$ with $\\mathfrak{J}(3, \\H) \\oplus \\H^3$: $\\mathfrak{J}=\\mathfrak{J}(3, \\H) \\oplus \\H^3$.\n\\f\n\\vspace{1mm}\n\nWe define an $\\R$-linear transformation $\\gamma$ of $\\mathfrak{J}$ by\n$$\n\\gamma X= \\begin{pmatrix} \\xi_1 & \\gamma x_3 & \\ov{\\gamma x_2} \\\\\n\\ov{\\gamma x_3} & \\xi_2 & \\gamma x_1 \\\\\n\\gamma x_2 & \\ov{\\gamma x_1} & \\xi_3   \\end{pmatrix}\n,\\,\\,X \\in \\mathfrak{J},\n$$\nwhere $\\gamma$ on right hand side is the same one as $\\gamma \\in G_2$. Then we have that $\\gamma \\in F_4$ and $\\gamma^2 =1$. Hence $\\gamma$ induce involutive inner automorphism $\\tilde{\\gamma}$ of $F_4{\\rm :}\\,\\tilde{\\gamma}(\\alpha)=\\gamma\\alpha\\gamma, \\alpha \\in F_4$.\n\\vspace{1mm}\n\nHere, we associate the elements $X$ of $\\mathfrak{J}$ with the elements \n\\begin{align*}\n\\begin{pmatrix}\n\\xi_1 & m_3 & \\ov{m_2} \\\\\n\\ov{m_3} & \\xi_2 & m_1 \\\\ \nm_2 & \\ov{m_1} & \\xi_3\n\\end{pmatrix}\n+ (\\a_1, \\a_2, \\a_3)(=:M + \\a) \n\\end{align*}\nof $\\mathfrak{J}(3, \\H) \\oplus \\H^3$ and we can define a multiplication, a conjugation and an inner product in $\\mathfrak{J}(3, \\H) \\oplus \\H^3$ corresponding to the same ones in $\\mathfrak{J}$ (see \\cite[Subsection 2.11]{iy0} in detail). \nHence we have that $\\mathfrak{J}(3, \\H) \\oplus \\H^3$ is isomorphic to the exceptional Jordan algebra $\\mathfrak{J}$ as algebra. From now on, if necessary we identify $\\mathfrak{J}$ with $\\mathfrak{J}(3, \\H) \\oplus \\H^3$: $\\mathfrak{J}=\\mathfrak{J}(3, \\H) \\oplus \\H^3$.\nNote that the action to $\\mathfrak{J}(3, \\H) \\oplus \\H^3$ of $\\gamma$ is as follows.\n\\begin{align*}\n\t\t\\gamma(M+\\a)=M-\\a,\\,\\,M+\\a \\in \\mathfrak{J}(3, \\H) \\oplus \\H^3=\\mathfrak{J}.\n\\end{align*}\n\nThen we have the following well-known result.\n\n\\begin{proposition}\\label{proposition 3.2.1\n\tThe group $(F_4)^\\gamma$ is isomorphic to the group $(Sp(1) \\times Sp(3))\/\\Z_2${\\rm:}  $(F_4)^\\gamma \\cong (Sp(1) \\times Sp(3))\/\\Z_2, \\,$ $\\Z_2 =\\{(1, E), (-1, -E) \\}$.\n\\end{proposition}\n\\begin{proof}\n\tWe define a mapping $\\varphi_{{}_{F_4,\\gamma}}: Sp(1) \\times Sp(3) \\to (F_4)^\\gamma$ by\n\t$$\n\t\\varphi_{{}_{F_4,\\gamma}}(p, A)(M+\\a)=AMA^* +p\\a A^*,\\,\\,\\, M+\\a \\in \\mathfrak{J}(3, \\H) \\oplus \\H^3=\\mathfrak{J}.\n\t$$\n\tThis mapping induces the required isomorphism (see \\cite[Theorem 2.11.2]{iy0} in detail).\n\\end{proof}\n\\vspace{1mm}\n\nLet $\\gamma_3 \\in G_2$ be the $\\R$-linear transformation of $\\mathfrak{C}$. Using the inclusion $G_2 \\subset F_4$, $\\gamma_3$ is naturally extended to the $\\R$-linear transformation of $\\mathfrak{J}$. The explicit form of \n$\\gamma_3$ as action to $\\mathfrak{J}$ is as follows.\n\\begin{align*}\n\t\t\\gamma_3 X=\n\t\t\\begin{pmatrix} \\xi_1 & \\gamma_3 x_3 & \\ov{\\gamma_3 x_2} \\\\\n\t\t\\ov{\\gamma_3 x_3} & \\xi_2 & \\gamma_3 x_1 \\\\\n\t\t\\gamma_3 x_2 & \\ov{\\gamma_3 x_1} & \\xi_3   \n\t\t\\end{pmatrix},\\,\\,X \\in \\mathfrak{J},\n\\end{align*}\nwhere $\\gamma_3$ on the right hand side is the same one as $\\gamma_3 \\in G_2$. Needless to say, $\\gamma_3 \\in F_4$ and $(\\gamma_3)^3=1$. Hence $\\gamma_3$ induces the automorphism $\\tilde{\\gamma}_3$ of order $3$ on $F_4$: $\\tilde{\\gamma}_3(\\alpha)={\\gamma_3}^{-1}\\alpha\\gamma_3, \\alpha \\in F_4$. Note that the action to $\\mathfrak{J}(3, \\H) \\oplus \\H^3$ of $\\gamma_3$ is as follows.\n\\begin{align*}\n\\gamma_3(M+\\a)=M+\\bm{\\omega}\\a,\\,\\,M+\\a \\in \\mathfrak{J}(3, \\H) \\oplus \\H^3=\\mathfrak{J}.\n\\end{align*}\n\nNow, we have the following theorem.\n\n\\begin{theorem}\\label{theorem 3.2.2\n\tThe group $(F_4)^{\\gamma_3}$ is isomorphic to the group $(U(1) \\times Sp(3))\/\\Z_2$ {\\rm :} $(F_4)^{\\gamma_3} \\cong (U(1) \\times Sp(3))\/\\Z_2, \\Z_2=\\{(1,E), \n\t(-1,-E) \\}$.\n\\end{theorem}\n\\begin{proof}\n\tAs in the proof of Theorem \\ref{theorem 3.1.2}, let $U(1)=\\{a \\in \\C \\,|\\,\\ov{a}a=1 \\} \\subset Sp(1)$. We define a mapping $\\varphi_{{}_{F_4,\\gamma_3}}:U(1) \\times Sp(3) \\to (F_4)^{\\gamma_3}$ by the restriction of the mapping $\\varphi_{{}_{F_4,\\gamma}}$ (Proposition \\ref{proposition 3.2.1}). This mapping induces the required isomorphism (see \\cite [Theorem 2.2]{iy1} in detail).\t\n\\end{proof}\n\nThus, since the group $(F_4)^{\\gamma_3}$ is connected, together with the result of Theorem \\ref{theorem 3.2.2}, we have an exceptional $\\varmathbb{Z}_3$-symmetric space $F_4\/((U(1) \\times Sp(3))\/\\Z_2)$.\n\\vspace{1mm}\n\nWe define an $\\R$-linear transformation $\\sigma$ of $\\mathfrak{J}$ by\n\\begin{align*}\n\\sigma X= \\begin{pmatrix} \\xi_1 & -x_3 & -\\ov{x_2} \\\\\n-\\ov{x_3} & \\xi_2 & x_1 \\\\\n-x_2 & \\ov{x_1} & \\xi_3   \\end{pmatrix}\n,\\,\\,X \\in \\mathfrak{J},\n\\end{align*}\nThen we have that $\\sigma \\in F_4$ and $\\sigma^2 =1$. Hence $\\sigma$ induce involutive inner automorphism $\\tilde{\\sigma}$ on $F_4{\\rm :}\\,\\tilde{\\sigma}(\\alpha)=\\sigma\\alpha\\sigma, \\alpha \\in F_4$.\n\\vspace{1mm}\n\nThen we have the following well-known result.\n\n\\begin{proposition}\\label{proposition 3.2.3\n\tThe group $(F_4)^\\sigma$ is isomorphic to the group $Spin(9)${\\rm:}$(F_4)^\\sigma \\!\\cong \\!Spin(9)$.\n\\end{proposition}\n\\begin{proof}\n\tFrom \\cite[Thorem 2.7.4]{iy0}\n\t, we have $(F_4)_{E_1} \\cong Spin(9)$, so by proving that $(F_4)^\\sigma \\cong (F_4)_{E_1}$ (\\cite[Thorem 2.9.1]{iy0}) we have the required isomorphism (see \\cite[Sections 2.7, 2.9 ]{iy0} in detail).\n\\end{proof}\n\\vspace{1mm}\n\nLet $U(1)=\\{a \\in \\C \\,|\\,\\ov{a}a=1 \\}$. For $a \\in U(1)$, we define an $\\R$-linear transformation $D_a$ of $\\mathfrak{J}$ by\n\\begin{align*}\n\t\tD_a X= \n\t\t\\begin{pmatrix} \\xi_1 & x_3 a & \\ov{ax_2} \\\\\n\t\t\\ov{x_3 a} & \\xi_2 & \\ov{a}x_1\\ov{a} \\\\\n\t\ta x_2 & a\\ov{x_1}a & \\xi_3   \n\t\t\\end{pmatrix},\\,\\, X \\in \\mathfrak{J}.\n\\end{align*}\nThen, since $D_a=\\varphi_{{}_{F_4,\\gamma}}(1,\\diag(1,\\ov{a},a))$, we have that $D_a \\in F_4$. Hence, by corresponding $a \\in U(1)$ to $D_a \\in F_4$, $U(1)$ is embedded into $F_4$.\nIn addition, we can express $\\sigma$ defined above by $D_{-1}$: $\\sigma=D_{-1}$.\n\nLet $\\bm{\\omega}=-(1\/2)+(\\sqrt{3}\/2)e_1 \\in U(1)$. Then we define an $\\R$-linear transformation $\\sigma_3$ of $\\mathfrak{J}$ by\n\\begin{align*}\n\t\t\\sigma_3X= \n\t\t\\begin{pmatrix} \\xi_1 & x_3 \\bm{\\omega} & \\ov{\\bm{\\omega} x_2} \\\\\n\t\t\\ov{x_3 \\bm{\\omega}} & \\xi_2 & \\ov{\\bm{\\omega}}x_1\\ov{\\bm{\\omega}} \\\\\n\t\t\\bm{\\omega} x_2 & \\bm{\\omega}\\ov{x_1}\\bm{\\omega} & \\xi_3   \n\t\t\\end{pmatrix},\\,\\, X \\in \\mathfrak{J}.\n\\end{align*}\nNeedless to say, since $\\sigma_3=D_\\omega=\\varphi_{{}_{F_4,\\gamma}}(1,\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega}))$, we have that $\\sigma_3 \\in F_4$. Hence $\\sigma_3$ induces the automorphism $\\tilde{\\sigma}_3$ of order $3$ on $F_4$: $\\tilde{\\sigma}_3(\\alpha)={\\sigma_3}^{-1}\\alpha\\sigma_3, \\alpha \\in F_4$.\n\\vspace{1mm}\n\nNow, we have the following theorem.\n\n\\begin{theorem}\\label{theorem 3.2.4\n\tThe group $(F_4)^{\\sigma_3}$ is isomorphic to the group $(Spin(2) \\times Spin(7))\/\\Z_2${\\rm:} $(F_4)^{\\sigma_3} \\cong (Spin(2) \\times Spin(7))\/\\Z_2, \\Z_2=\\{(1,1), (\\sigma,\\sigma)\\}$.\n\\end{theorem}\n\\begin{proof}\n\tLet $Spin(2)$ as the group $\\{D_a \\in F_4 \\,|\\,a \\in U(1) \\}$ defined above which is isomorphic to the group $U(1)$ and $Spin(7)$ as the subgroup $(F_4)_{E_1, F_1(1),F_1(e_1)}$ of $F_4$ (cf. \\cite[Propsition 2.9 (1)]{iy2}, \\cite[Subsection 2.2]{iy1}). We define a mapping $\\varphi_{{}_{F_4,\\sigma_3}}: Spin(2) \\times Spin(7) \\to (F_4)^{\\sigma_3}$ by\n\t\\begin{align*}\n\t\t\t\\varphi_{{}_{F_4,\\sigma_3}}(D_a, \\beta)=D_a \\beta.\n\t\\end{align*}\t\n\tThis mapping induces the required isomorphism (see \\cite[Lemmas 2.5, 2.6, Theorem 2.7]{iy1} in detail).\n\\end{proof}\n\nThus, since the group $(F_4)^{\\sigma_3}$ is connected, together with the result of Theorem \\ref{theorem 3.2.4}, we have an exceptional $\\varmathbb{Z}_3$-symmetric space $F_4\/((Spin(2) \\times Spin(7))\/\\Z_2)$.\n\\vspace{2mm}\t\n\n We define an $\\R$-linear transformation $w_3$ of $\\mathfrak{J}$ by\n\\begin{align*}\nw_3X= \n\\begin{pmatrix} \\xi_1 & w_3 x_3 & \\ov{w_3 x_2} \\\\\n\\ov{w_3 x_3} & \\xi_2 & w_3 x_1 \\\\\nw_3 x_2 & \\ov{w_3 x_1} & \\xi_3   \n\\end{pmatrix},\\,\\, X \\in \\mathfrak{J},\n\\end{align*}\nwhere $w_3$ on the right hand side is the same one as $w_3 \\in G_2$. Needless to say, $w_3 \\in F_4$ and $(w_3)^3=1$. Hence $w_3$ induces the automorphism $\\tilde{w}_3$ of order $3$ on $F_4$: $\\tilde{w}_3(\\alpha)={w_3}^{-1}\\alpha w_3, \\alpha \\in F_4$.\n\nWe associate the elements $X$ of $\\mathfrak{J}$ with the elements \n\\begin{align*}\n\t\t\\begin{pmatrix}\n\t\t\\xi_1 & c_3 & \\ov{c_2} \\\\\n\t\t\\ov{c_3} & \\xi_2 & c_1 \\\\ \n\t\tc_2 & \\ov{c_1} & \\xi_3\n\t\t\\end{pmatrix} +\n\t\t\\begin{pmatrix}\n\t\t   &  &  \\\\\n\t\t\\m_1 \\!\\!\\!& \\m_2 \\!\\!\\!& \\m_3 \\\\ \n\t\t   &  & \n\t\t\\end{pmatrix}(=:X_{\\bm{C}}+M)\n\\end{align*}\nof $\\mathfrak{J}(3,\\C) \\oplus M(3,\\C)$, where $\\m_i \\in \\C^3$, \nand we can define a multiplication, a conjugation and an inner product in \n$\\mathfrak{J}(3, \\C) \\oplus M(3,\\C)$ corresponding to the same ones in $\\mathfrak{J}$ (see \\cite[Subsection 2.12]{iy0} in detail). Hence we have that $\\mathfrak{J}(3, \\C) \\oplus M(3,\\C)$ is isomorphic to $\\mathfrak{J}$ as algebra. Hereafter, if necessary we identify $\\mathfrak{J}$ with $\\mathfrak{J}(3, \\C) \\oplus M(3,\\C)$: $\\mathfrak{J}=\\mathfrak{J}(3, \\C) \\oplus M(3,\\C)$. Note that using $\\bm{\\omega}=-(1\/2)+(\\sqrt{3}\/2)e_1 \\in \\C$, the action to $\\mathfrak{J}=\\mathfrak{J}(3, \\C) \\oplus M(3,\\C)$ of $w_3$ is as follows.\n\\begin{align*}\nw_3(X_{\\bm{C}}+M)=X_{\\bm{C}}+\\bm{\\omega} M,\\,\\,X_{\\bm{C}}+M \\in \\mathfrak{J}(3, \\C) \\oplus M(3,\\C)=\\mathfrak{J}.\n\\end{align*}\n\nNow, we have the following theorem.\n\n\\begin{theorem}\\label{theorem 3.2.5\n\t\tThe group $(F_4)^{w_3}$ is isomorphic to the group $(SU(3) \\times SU(3))\/\\Z_3${\\rm :} $(F_4)^{w_3} \\cong (SU(3) \\times SU(3))\/\\Z_3, \\Z_3=\\{(E,E),(\\bm{\\omega} E,\\bm{\\omega} E),({\\bm{\\omega}}^{-1}E,{\\bm{\\omega}}^{-1}E) \\}$.\n\\end{theorem}\n\\begin{proof}\n\tWe define a mapping $\\varphi_{F_4,w_3}:SU(3) \\times SU(3) \\to  (F_4)^{w_3}$ by\n\t\\begin{align*}\n\t\t\t\t\\varphi_{F_4,w_3}(B, A)(X_{\\bm{C}}+M)=AX_{\\bm{C}}A^* + BMA^*,\\,\\,X_{\\bm{C}}+M \\in \\mathfrak{J}(3, \\C) \\oplus M(3,\\C)=\\mathfrak{J}.\n\t\\end{align*}\n\tThis mapping induces the required isomorphism (see \\cite[Theorem 2.9]{iy1} in detail).\n\\end{proof}\n\nThus, since the group $(F_4)^{w_3}$ is connected, together with the result of Theorem \\ref{theorem 3.2.5}, we have an exceptional $\\varmathbb{Z}_3$-symmetric space $F_4\/((SU(3) \\times SU(3))\/\\Z_3)$.\n\\vspace{1mm}\n\nAs in Section 3.1, the following lemma are useful to determine the structure of a group $G^{\\sigma_3} \\cap G^{\\tau_3}$ in $F_4$.\n\n\\begin{lemma}\\label{lemma 3.2.6\n\t{\\rm (1)} The mapping $\\varphi_{{}_{F_4,\\gamma_3}}:U(1) \\times Sp(3) \\to (G_2)^{\\gamma_3}$ of \\,Theorem {\\rm \\ref{theorem 3.2.2}} satisfies the relational formulas \n\t\\begin{align*}\n\t\\gamma_3&=\\varphi_{{}_{F_4,\\gamma_3}}(\\bm{\\omega},E), \n\t\\\\\n\t\\sigma_3&=\\varphi_{{}_{F_4,\\gamma_3}}(1,\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega})),\n\t\\\\\n\tw_3&=\\varphi_{{}_{F_4,\\gamma_3}}(1, \\ov{\\bm{\\omega}}E), \n\t\\end{align*}\n where $\\bm{\\omega}=-(1\/2)+(\\sqrt{3}\/2)e_1 \\in U(1)$.\n\t\\vspace{1mm}\n\t\n\t{\\rm (2)} The mapping $\\varphi_{{}_{F_4,w_3}}:SU(3)\\times SU(3) \\to (F_4)^{w_3}$ of \\,Theorem {\\rm \\ref{theorem 3.2.5}} satisfies the relational formulas\n\t\\begin{align*}\n\t\\gamma_3&=\\varphi_{{}_{F_4,w_3}}(\\diag(1,\\bm{\\omega},\\ov{\\bm{\\omega}}),E), \n\t\\\\\n\t\\sigma_3&=\\varphi_{{}_{F_4,w_3}}(E,\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega}))\\\\\n\tw_3&=\\varphi_{{}_{F_4,w_3}}(\\bm{\\omega}E,E), \n\t\\end{align*}\n where $\\bm{\\omega}=-(1\/2)+(\\sqrt{3}\/2)e_1 \\in U(1)$. \t\n\\end{lemma}\n\\begin{proof}\n\t(1), (2) By doing straightforward computation we obtain the results above. \n\\end{proof}\n\n\\subsection{In $E_6$}\\label{subsection 3.3\n\nLet $\\gamma, \\gamma_3 \\in G_2 \\subset F_4$, and using the inclusion $F_4 \\subset E_6$, \n$\\gamma, \\gamma_3$ are naturally extended to an $C$-linear \ntransformation of $\\mathfrak{J}^C$. Needless to say, $\\gamma, \\gamma_3 \\in E_6$ and $\\gamma^2=(\\gamma_3)^3=1$. Hence $\\gamma, \\gamma_3$ induce the involutive automorphism $\\tilde{\\gamma}$, the automorphism $\\tilde{\\gamma}_3$ of order $3$ on $E_6$, respectively: $\\tilde{\\gamma}(\\alpha)=\\gamma\\alpha\\gamma, \\tilde{\\gamma}_3(\\alpha)={\\gamma_3}^{-1}\\alpha\\gamma_3, \\alpha \\in E_6$. \n\\vspace{1mm}\n\nThen we have the following proposition and theorem.\n\n\\begin{proposition}\\label{proposition 3.3.1}\n\tThe group $(E_6)^\\gamma$ isomorphic to the group $(Sp(1) \\times SU(6))\/\\Z_2${\\rm:}\n\t$(E_6)^\\gamma \\cong (Sp(1) \\times SU(6))\/\\Z_2,\\Z_2 =\\{(1, E), (-1, -E) \\}$.\n\\end{proposition}\n\\begin{proof}\n\tLet $SU(6)=\\{A \\in M(6, C)\\,|\\,(\\tau\\,{}^t A) A$ $=1, \\det\\, A=1) \\}$, where $\\tau$ is the complex conjugation of $C=\\{x+iy \\,|\\,x,y \\in \\R \\}$, that is, $\\tau(x+yi)=x-yi, x,y \\in \\R$.\n\tWe define a mapping $\\varphi_{{}_{E_6,\\gamma}}:Sp(1) \\times SU(6) \\to (E_6)^\\gamma $ by\n\t\\begin{align*}\n\t\\varphi_{{}_{E_6,\\gamma}}(p, A)(M+\\a)={k_J}^{-1}(A(k_J M){}^t\\!A)+p\\a k^{-1}(\\tau \\,{}^t\\!A), M+\\a \\in \\mathfrak{J}(3, \\H)^C \\oplus (\\H^3)^C=\\mathfrak{J}^C,\n\t\\end{align*}\n\twhere both of $k_J:\\mathfrak{J}(3, \\H)^C \\to \\mathfrak{S}(6, C)$ and $k:M(3, \\H)^C \\to M(6, C)$ are the $C$-linear isomorphisms.\n\tThis mapping induces the required isomorphism (see \\cite[Theorem 3.11.4 ]{iy0} in detail).\n\\end{proof}\n\n\\begin{theorem}\\label{theorem 3.3.2\n\tThe group $(E_6)^{\\gamma_3}$ is isomorphic to the group $(U(1) \\times SU(6))\/\\Z_2${\\rm :} $(E_6)^{\\gamma_3} \\cong (U(1) \\times SU(6))\/\\Z_2, \\Z_2=\\{(1,E),\n\t(-1,-E) \\}$.\t\n\\end{theorem}\n\\begin{proof}\n\tLet $U(1)=\\{a \\in \\C\\,|\\, \\ov{a}a=1 \\} \\subset Sp(1)$. We define a mapping $\\varphi_{{}_{E_6,\\gamma_3}}: U(1) \\times SU(6) \\to (E_6)^{\\gamma_3}$ by the restriction of the mapping $\\varphi_{{}_{E_6,\\gamma}}$ (Proposition \\ref{proposition 3.3.1}). This mapping induces the required isomorphism (see \\cite [Theorem 3.2]{iy1} in detail). \n\\end{proof}\n\nThus, since the group $(E_6)^{\\gamma_3}$ is connected, together with the result of Theorem \\ref{theorem 3.3.2}, we have an exceptional $\\varmathbb{Z}_3$-symmetric space $E_6\/((U(1) \\times SU(6))\/\\Z_2)$.\n\\vspace{2mm}\n\nLet $\\sigma, \\sigma_3 \\in F_4$. Then, as in the case above, using the inclusion $F_4 \\subset E_6$, $\\sigma, \\sigma_3$ are naturally extended to\ntransformations of $\\mathfrak{J}^C$. Needless to say, $\\sigma, \\sigma_3 \\in E_6$ and $\\sigma^2=(\\sigma_3)^3=1$. Hence $\\sigma$ and $\\sigma_3$ induce the involutive automorphism $\\tilde{\\sigma}$ and the automorphism $\\tilde{\\sigma}_3$ of order $3$ on $E_6$, respectively: $\\tilde{\\sigma}(\\alpha)=\\sigma\\alpha\\sigma, \\tilde{\\sigma}_3(\\alpha)={\\sigma_3}^{-1}\\alpha\\sigma_3, \\alpha \\in E_6$. \n\\vspace{1mm}\n\nThen we have the following proposition and theorem.\n\n\\begin{proposition}\\label{proposition 3.3.3\n\tThe group $(E_6)^\\sigma$ is isomorphic to the group $(U(1) \\times Spin(10))\/\\Z_4${\\rm:}\\,\n\t$(E_6)^\\sigma \\!\\cong (U(1) \\times Spin(10))\/\\Z_4,\\Z_4=\\{ (1, \\phi_{{}_{6,\\sigma}}(1)), (-1, \\phi_{{}_{6,\\sigma}}(-1)), (i, \\phi_{{}_{6,\\sigma}}(-i)), (-i, \\phi_{{}_{6,\\sigma}}(i)) \\}$.\n\\end{proposition}\n\\begin{proof}\n\tLet $Spin(10)$ as the group $(E_6)_{E_1}=\\{\\alpha \\in E_6\\,|\\,\\alpha E_1=E_1 \\}$ (\\cite[Theorem 3.10.4]{iy0}).\n\tWe define a mapping $\\varphi_{{}_{E_6,\\sigma}}:U(1) \\times Spin(10) \\to (E_6)^\\sigma $ by\n\t$$\n\t\\varphi_{{}_{E_6,\\sigma}}(\\theta, \\delta)=\\phi_{{}_{6,\\sigma}}(\\theta)\\delta,\n\t$$\n\twhere $\\phi_{{}_{6,\\sigma}}:U(1) \\to E_6$ is defined by\n\t\\begin{align*}\n\t\\phi_{{}_{6,\\sigma}}(\\theta)X=\\begin{pmatrix}\n\t\\theta^4 \\xi_1 & \\theta x_3 & \\theta \\ov{x_2} \\\\\n\t\\theta \\ov{x_3} & {\\theta}^{-2}\\xi_2 &  {\\theta}^{-2}x_1 \\\\ \n\t\\theta x_2 &  {\\theta}^{-2}\\ov{x_1} &  {\\theta}^{-2}\\xi_3\n\t\\end{pmatrix}, \\,\\, X \\in \\mathfrak{J}^C.\n\t\\end{align*}\n\tThis mapping induces the required isomorphism (see \\cite[Theorem 3.10.7 ]{iy0} in detail).\n\\end{proof}\n\n\\begin{theorem}\\label{theorem 3.3.4\n\tThe group $(E_6)^{\\sigma_3}$ is isomorphic to the group $(U(1) \\times Spin(2) \\times Spin(8))\/(\\Z_4 \\allowbreak \\times \\Z_2)${\\rm :} $(E_6)^{\\sigma_3} \\cong (U(1) \\times Spin(2) \\times Spin(8))\/(\\Z_2 \\times \\Z_4), \\Z_2=\\{(1,1,1),(1,\\sigma,\\sigma) \\}, \\Z_4=\\{(1,1,1),(i,D_{e_1},\\phi_{{}_{6,\\sigma}}(-i)D_{-e_1}),(-1,\\allowbreak\\sigma,1),(-i,D_{-e_1},\\phi_{{}_{6,\\sigma}}(i)D_{e_1}) \\} \\}$.\n\\end{theorem}\n\\begin{proof}\n\tLet $U(1)=\\{\\theta \\in C\\,|\\,(\\tau \\theta)\\theta=1 \\}$ and $Spin(2)$, which is isomorphic to the group $U(1)$, as the group $\\{D_a \\in F_4 \\,|\\,a \\in U(1) \\}$ defined in $F_4$, moreover let $Spin(8)$ as the group $(E_6)_{E_1, F_1(1),F_1(e_1)}=\\{ \\alpha \\in E_6 \\,|\\,\\alpha E_1=E_1, \\alpha F_1(1)=F_1(1), \\alpha F_1(e_1)=F_1(e_1)\\}$ (cf.\\cite[Proposition 3.22]{iy2}, \\cite[Subsection 3.2]{iy1}), respectively. We define a mapping $\\varphi_{{}_{E_6,\\sigma_3}}: U(1) \\times Spin(2) \\times Spin(8) \\to (E_6)^{\\sigma_3}$ by\n\t\\begin{align*}\n\t\t\t\\varphi_{{}_{E_6,\\sigma_3}}(\\theta, D_a, \\beta)=\\phi_{{}_{6,\\sigma}}(\\theta)D_a \\beta.\n\t\\end{align*}\n\tThis mapping induces the required isomorphism (see \\cite[Theorem 3.9]{iy1} in detail).\n\\end{proof}\n\nThus, since the group $(E_6)^{\\sigma_3}$ is connected, together with the result of Theorem \\ref{theorem 3.3.4}, we have an exceptional $\\varmathbb{Z}_3$-symmetric space $E_6\/((U(1) \\times Spin(2) \\times Spin(8))\/(\\Z_2 \\times \\Z_4))$.\n\\vspace{2mm}\n\nLet $\\nu=\\exp(2\\pi i\/9) \\in U(1)=\\{ \\theta \\in C \\,|\\, (\\tau \\theta)\\theta=1\\} \\subset C$. We consider the element $A_\\nu \\in SU(6) \\subset M(6, C)$ as follows.\n\\begin{align*}\n\t\tA_\\nu=\\diag(\\nu^5, \\nu^{-1}, \\nu^{-1}, \\nu^{-1}, \\nu^{-1},\\nu^{-1}),\n\\end{align*} \nand using this $A_\\nu$, set $\\nu_3=\\varphi_{{}_{E_6,\\gamma}}(1,A_\\nu)$. Then we have that $\\nu_3 \\in (E_6)^\\gamma \\subset E_6$ and $(\\nu_3)^9=1$. Since ${A_\\nu}^3= \\nu^6 E \\in z(SU(6))$ (the center of $SU(6)$) and $(\\nu_3)^3=\\varphi_{{}_{E_6,\\gamma}}(1, {A_\\nu}^3)=\\omega 1$, where $\\omega= -(1\/2)+(\\sqrt{3}\/2)i \\in C$, $\\nu_3$ induces the automorphism $\\tilde{\\nu}_3$ of order $3$ on $E_6$: $\\tilde{\\nu}_3(\\alpha)={\\nu_3}^{-1}\\alpha\\nu_3, \\alpha \\in E_6$.\n\\vspace{1mm}\n\nNow, we have the following theorem.\n\n\\begin{theorem}\\label{theorem 3.3.5\n\tThe group $(E_6)^{\\nu_3}$ is isomorphic to the group $(Sp(1) \\times S(U(1) \\times U(5)))\/\\Z_2${\\rm :} $(E_6)^{\\nu_3} \\cong (Sp(1) \\times S(U(1) \\times U(5)))\/\\Z_2, \\Z_2=\\{(1,E), (-1,-E) \\}$.\n\\end{theorem}\n\\begin{proof}\n\tLet $S(U(1) \\times U(5)) \\subset SU(6)$. We define a mapping $\\varphi_{{}_{E_6, \\nu_3}}:Sp(1) \\times S(U(1) \\times U(5)) \\to  (E_6)^{\\nu_3}$ by the restriction of the mapping $\\varphi_{{}_{E_6,\\gamma}}$. This mapping induces the required isomorphism (see \\cite[Theorem 3.4]{iy1} in detail).\n\\end{proof}\n\nThus, since the group $(E_6)^{\\nu_3}$ is connected, together with the result of Theorem \\ref{theorem 3.3.5}, we have an exceptional $\\varmathbb{Z}_3$-symmetric space $E_6\/((U(1) \\times S(U(1) \\times U(5)))\/ \\Z_2)$.\n\\vspace{2mm}\n\nLet $\\phi_{{}_{6,\\sigma}}:U(1) \\to E_6$ be the embedding defined in the proof of Proposition \\ref{proposition 3.3.3}, and again let $\\nu=\\exp(2\\pi i\/9) \\in U(1) \\subset C$. Set $\\mu_3=\\phi_{{}_{6,\\sigma}}(\\nu)$. Then, needless to say, $\\mu_3 \\in E_6$ and $\\nu^9=1$. \nHence, since $\\mu^3=\\omega 1 \\in z(E_6)$ (the center of $E_6$), $\\mu_3$ induces the automorphism $\\tilde{\\mu_3}$ of order $3$ on $E_6$: $\\tilde{\\mu_3}(\\alpha)={\\mu_3}^{-1}\\alpha\\mu_3, \\alpha \\in E_6$.\n\\vspace{1mm}\n\nNow, we have the following theorem.\n\n\\begin{theorem}\\label{theorem 3.3.6\n\tThe group $(E_6)^{\\mu_3}$ coincides with the group $(E_6)^\\sigma$, that is, this group is isomorphic to the group $(U(1) \\times Spin(10))\/\\Z_4${\\rm :} $(E_6)^{\\mu_3} \\cong (U(1) \\times Spin(10))\/\\Z_4, \\Z_4=\\{ (1, 1), (-1, \\sigma), (i, \\phi_{{}_{6,\\sigma}}(-i)), (-i, \\phi_{{}_{6,\\sigma}}(i)) \\}$\n\\end{theorem}\n\\begin{proof}\n\tWe have to prove that $(E_6)^{\\mu_3}=(E_6)^\\sigma$.\n\n\tHowever the details of proof is omitted (see \\cite[Theorem 3.11]{iy1} in detail).\n\\end{proof}\n\\vspace{2mm}\n\nLet $w_3 \\in G_2 \\subset F_4$. Then, as in the cases above, using the inclusion $F_4 \\subset E_6$, $w_3$ are naturally extended to\ntransformation of $\\mathfrak{J}^C$.\nNeedless to say, $w_3 \\in E_6$ by inclusion $F_4 \\subset E_6$ and $(w_3)^3=1$. Hence $w_3$ induces the automorphism $\\tilde{w}_3$ of order $3$ on $E_6$: $\\tilde{w}_3(\\alpha)={w_3}^{-1}\\alpha w_3, \\alpha \\in E_6$.\nNote that using $\\bm{\\omega}=-(1\/2)+(\\sqrt{3}\/2)e_1 \\in \\C$, the action to $\\mathfrak{J}^C=\\mathfrak{J}(3, \\C)^C \\oplus M(3,\\C)^C$ of $w_3$ is as follows.\n\\begin{align*}\n\t\tw_3(X_{\\bm{C}}+M)=X_{\\bm{C}}+\\bm{\\omega}M,\\,\\,X_{\\bm{C}}+M \\in \\mathfrak{J}(3,\\C)^C \\oplus M(3, \\C)^C=\\mathfrak{J}^C.\n\\end{align*}\n\nNow, we have the following theorem.\n\n\\begin{theorem}\\label{theorem 3.3.7\n\tThe group $(E_6)^{w_3}$ is isomorphic to the group $(SU(3) \\times SU(3) \\times SU(3))\/\\Z_3${\\rm:} $(E_6)^{w_3} \\cong (SU(3) \\times SU(3) \\times SU(3))\/\\Z_3, \\Z_3=\\{(E,E,E),(\\bm{\\omega}E,\\bm{\\omega}E,\\bm{\\omega}E),(\\bm{\\omega}^{-1}E,\\bm{\\omega}^{-1}E,\\allowbreak \\bm{\\omega}^{-1}E)  \\}$.\n\\end{theorem}\n\\begin{proof}\n\tWe define a mapping $\\varphi_{{}_{E_6,w_3}}:SU(3) \\times SU(3) \\times SU(3) \\to (E_6)^{w_3}$ by\n\t\\begin{align*}\n\t\t\t\\varphi_{{}_{E_6,w_3}}(L,A,B)(X_{C}+M)&=h(A,B)X_{C}h(A,B)^*+LM\\tau h(A,B)^*, \n\t\t\t\\\\\n\t\t\t&\\hspace*{20mm} X_{C}+M \\in \\mathfrak{J}(3, \\C)^C \\oplus \n\t\t\tM(3,\\C)^C=\\mathfrak{J}^C,\n\t\\end{align*}\n\twhere $h:M(3,\\C) \\times M(3,\\C) \\to M(3,\\C)^C$ is defined by \n\t\\begin{align*}\n\t\t\th(A,B)=\\dfrac{A+B}{2}+i\\dfrac{(B-A)e_1}{2}.\n\t\\end{align*}\n\tThis mapping induces the required isomorphism (see \\cite[Theorem 13]{iy0} in detail). Note that there is a mistake for the numbering of theorems in \\cite{iy0}, so Theorem 13 above is corresponding to the last theorem.\n\\end{proof}\n\nThus, since the group $(E_6)^{w_3}$ is connected, together with the result of Theorem \\ref{theorem 3.3.7}, we have an exceptional $\\varmathbb{Z}_3$-symmetric space $E_6\/((SU(3) \\times SU(3))\/ \\Z_3)$.\n\nAs in Subsections 3.1, 3.2, the following lemma are useful to determine the structure of groups $G^{\\sigma_3} \\cap G^{\\tau_3}$ in $E_6$.\n\n\\begin{lemma}\\label{lemma 3.3.8\n\t{\\rm (1)} The mapping $\\varphi_{{}_{E_6,\\gamma_3}}:U(1) \\times SU(6) \\to (E_6)^{\\gamma_3}$ of \\,Theorem {\\rm \\ref{theorem 3.3.2}} satisfies the relational formulas \n\t\\begin{align*}\n\t\\gamma_3&=\\varphi_{{}_{E_6,\\gamma_3}}(\\omega,E), \n\t\\\\\n\t\\sigma_3&=\\varphi_{{}_{E_6,\\gamma_3}}(1,\\diag(1,1,\\tau\\omega,\\omega,\\omega,\\tau\\omega)), \n\t\\\\\n\t\\nu_3&=\\varphi_{{}_{E_6,\\gamma_3}}(1,\\diag(\\nu^5,\\nu^{-1},\\nu^{-1},\\nu^{-1},\\nu^{-1},\\nu^{-1})),\n\t\\\\\n\t\\mu_3&=\\varphi_{{}_{E_6,\\gamma_3}}(1,\\diag(\\nu^{-2},\\nu^2,\\nu^{-1},\\nu,\\nu^{-1},\\nu)),\n\t\\\\\n\tw_3&=\\varphi_{{}_{E_6,\\gamma_3}}(1,\\diag(\\tau\\omega,\\omega,\\tau\\omega,\\omega,\\tau\\omega,\\omega)),\n\t\\end{align*}\n  where $\\bm{\\omega}=-(1\/2)+(\\sqrt{3}\/2)i \\in U(1), \\nu=\\exp(2\\pi i\/9)$.\n\t\\vspace{1mm}\n\t\n\t{\\rm (2)} The mapping $\\varphi_{{}_{E_6,w_3}}:SU(3)\\times SU(3) \\times SU(3) \\to (E_6)^{w_3}$ of \\,Theorem {\\rm \\ref{theorem 3.3.7}} satisfies the relational formulas\n\t\\begin{align*}\n\t\\gamma_3&=\\varphi_{{}_{E_6,w_3}}(\\diag(1,\\bm{\\omega},\\ov{\\bm{\\omega}}),E,E),\n\t\\\\\n\t\\sigma_3&=\\varphi_{{}_{E_6,w_3}}(E,\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega}),\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega})),\n\t\\\\\n\t\\mu_3&=\\varphi_{{}_{E_6,w_3}}(E,\\diag({\\bm{\\varepsilon}}^{-2},\\bm{\\varepsilon},\\bm{\\varepsilon}),\\diag({\\bm{\\varepsilon}}^2,{\\bm{\\varepsilon}}^{-1},{\\bm{\\varepsilon}}^{-1})),\n\t\\\\\n\tw_3&=\\varphi_{{}_{E_6,w_3}}(\\bm{\\omega}E,E,E),\n\t\\end{align*}\n\t where $\\bm{\\omega}=-(1\/2)+(\\sqrt{3}\/2)e_1 \\in U(1), \\bm{\\varepsilon}=\\exp(2\\pi e_1\/9)$. \t\n\\end{lemma}\n\\begin{proof}\n\t(1), (2) By doing straightforward computation we obtain the results above. \n\\end{proof}\n\n\\section{Globally exceptional $\\varmathbb{Z}_3 \\times \\varmathbb{Z}_3$-symmetric spaces}\n\nIn this section, we construct a finite abelian group $\\varGamma=\\varmathbb{Z}_3 \\times \\varmathbb{Z}_3$ by using the inner automorphisms $\\tilde{\\sigma}_3, \\tilde{\\tau}_3$ of order $3$ on $G=G_2, F_4,E_6$ as the Case 1 below and determine the structure of the group $G^{\\sigma_3} \\cap G^{\\tau_3}$.\n\n\\subsection{Case 1: $\\{1, \\tilde{\\gamma}_3,  \\tilde{\\gamma}_3{}^{-1}\\} \\times \\{1, \\tilde{w}_3,  \\tilde{w}_3{}^{-1}\\}$-symmetric space}\n\nLet the $\\R$-linear transformations $\\gamma_3, w_3$ of $\\mathfrak{C}$  defined in Subsection \\ref{subsection 3.1}. \n\n\\noindent From Lemma \\ref{lemma 3.1.4} (1), since we can easily confirm that $\\gamma_3$ and $w_3$ are commutative, $\\tilde{\\gamma}_3$ and $\\tilde{w}_3$ are commutative in $\\Aut(G_2)$: $\\tilde{\\gamma}_3\\tilde{w}_3=\\tilde{w}_3\\tilde{\\gamma}_3$.\n\\vspace{1mm}\n\nNow, we will determine the structure of the group $(G_2)^{\\gamma_3} \\cap (G_2)^{w_3}$.\n\n\\begin{theorem}\\label{theorem 4.1.1\n\tThe group  $(G_2)^{\\gamma_3} \\cap (G_2)^{w_3}$ is isomorphic to the group $(U(1) \\times U(1))\/\\Z_2${\\rm :} $(G_2)^{\\gamma_3} \\cap (G_2)^{w_3} \\cong (U(1) \\times U(1))\/\\Z_2, \\Z_2=\\{(1,1), (-1,-1) \\}$.\n\\end{theorem}\n\\begin{proof}\n\tLet $U(1) \\subset Sp(1)$. \n\tWe define a mapping $\\varphi_{{}_{G_2,\\gamma_3, w_3}}: U(1) \\times U(1) \\to (G_2)^{\\gamma_3} \\cap (G_2)^{w_3}$ by \n\t\\begin{align*}\n\t\t\t\t\\varphi_{{}_{G_2,\\gamma_3, w_3}}(s,t)(m+ne_4)=tm\\ov{t}+(sn\\ov{t})e_4,\\,\\,m+ne_4 \\in \\H \\oplus \\H e_4=\\mathfrak{C}.\n\t\\end{align*}\n\tNeedless to say, this mapping is the restriction of the mapping $\\varphi_{{}_{G_2,\\gamma_3}}$ (Theorem \\ref{theorem 3.1.2}).\n\t\n\tFirst, we will prove that $\\varphi_{{}_{G_2,\\gamma_3, w_3}}$ is well-defined. Since this mapping is also the restriction of the mapping $\\varphi_{{}_{G_2,\\gamma_3}}$, it is trivial that $\\varphi_{{}_{G_2,\\gamma_3, w_3}}(s,t) \\in (G_2)^{\\gamma_3}$, and from $w_3=\\varphi_{{}_{G_2,\\gamma_3}}(1,\\ov{\\bm{\\omega}})$ (Lemma \\ref{lemma 3.1.4} (1)), it is almost clear that  $\\varphi_{{}_{G_2,\\gamma_3, w_3}}(s,t) \\in (G_2)^{w_3}$. Hence $\\varphi_{{}_{G_2,\\gamma_3, w_3}}$ is well-defined. Subsequently, since $\\varphi_{{}_{G_2,\\gamma_3, w_3}}$ is the restriction of $\\varphi_{{}_{G_2,\\gamma_3}}$, we easily see that $\\varphi_{{}_{G_2,\\gamma_3, w_3}}$ is a homomorphism.\n\t\n\tNext, we will prove that $\\varphi_{{}_{G_2,\\gamma_3, w_3}}$ is surjective. Let $\\alpha \\in (G_2)^{\\gamma_3} \\cap (G_2)^{w_3} \\subset (G_2)^{\\gamma_3}$. There exist $s \\in U(1)$ and $q \\in Sp(1)$ such that $\\alpha=\\varphi_{{}_{G_2,\\gamma_3}}(s,q)$ (Theorem \\ref{theorem 3.1.2}). Moreover, since $\\alpha=\\varphi_{{}_{G_2,\\gamma_3}}(s,q)$ commutes with $w_3$, again using $w_3=\\varphi_{{}_{G_2,\\gamma_3}}(1,\\ov{\\bm{\\omega}})$, we have that \n\t\\begin{align*}\n\t\t\t\\left\\{ \\begin{array}{l}\n\t\t\ts=s \\\\\n\t\t\t\\bm{\\omega}q\\ov{\\bm{\\omega}}=q \n\t\t\t\\end{array} \\right. \n\t\t\t\\quad \\text{or}\\quad\n\t\t\t\\left\\{ \\begin{array}{l}\n\t\t\ts=-s \\\\\n\t\t\t\\bm{\\omega}q\\ov{\\bm{\\omega}}=-q.\n\t\t\t\\end{array} \\right.\n\t\\end{align*}\n\tThe latter case is impossible because $s \\not=0$. As for the former case, from the relational formula $\\bm{\\omega}q\\ov{\\bm{\\omega}}=q$ we easily see that $q \\in U(1)$, and needless to say, $s \\in U(1)$. Hence there exist $s,t \\in U(1)$ such that $\\alpha=\\varphi_{{}_{G_2,\\gamma_3}}(s,t)$. Namely, there exist $s,t \\in U(1)$ such that $\\alpha=\\varphi_{{}_{G_2,\\gamma_3,w_3}}(s,t)$. The proof of surjective is completed.\n\t\n\tFinally, we determine $\\Ker \\,\\varphi_{{}_{G_2,\\gamma_3, w_3}}$. However, since $\\varphi_{{}_{G_2,\\gamma_3, w_3}}$ is the restriction of $\\varphi_{{}_{G_2,\\gamma_3}}$, it is easily obtain that $\\Ker \\,\\varphi_{{}_{G_2,\\gamma_3, w_3}}=\\{(1,1),(-1,-1) \\} \\cong \\Z_2$.\n\t\n\tTherefore we have the required isomorphism\n\t\\begin{align*}\n\t\t\t(G_2)^{\\gamma_3} \\cap (G_2)^{w_3} \\cong (U(1) \\times U(1))\/\\Z_2.\n\t\\end{align*}\n\\end{proof}\n\nThus, since the group $(G_2)^{\\gamma_3} \\cap (G_2)^{w_3}$ is connected from Theorem \\ref{theorem 4.1.1}, we have an exceptional $\\varmathbb{Z}_3 \\times \\varmathbb{Z}_3$-symmetric space\n\\begin{align*}\n\t\t\t\t\t\t\tG_2\/((U(1) \\times U(1))\/\\Z_2).\n\\end{align*}\n\n\\subsection{Case 2: $\\{1, \\tilde{\\gamma}_3,  \\tilde{\\gamma}_3{}^{-1}\\} \\times \\{1, \\tilde{\\sigma}_3,  \\tilde{\\sigma}_3{}^{-1}\\}$-symmetric space}\n\nLet the $\\R$-linear transformations $\\gamma_3, \\sigma_3$ of $\\mathfrak{J}$  defined in Subsection \\ref{subsection 3.2}. \n\n\\noindent From Lemma \\ref{lemma 3.2.6} (1), since we can easily confirm that $\\gamma_3$ and $\\sigma_3$ are commutative, $\\tilde{\\gamma}_3$ and $\\tilde{\\sigma}_3$ are commutative in $\\Aut(F_4)$: $\\tilde{\\gamma}_3\\tilde{\\sigma}_3=\\tilde{\\sigma}_3\\tilde{\\gamma}_3$.\n\\vspace{1mm}\n\nBefore determining the structure of the group $(F_4)^{\\gamma_3} \\cap (F_4)^{\\sigma_3}$, we prove proposition needed in the proof of theorem below.\n\\vspace{1mm}\n\nWe define subgroups $G_{1,2}$ and $G'_{1,2}$ of the group $Sp(3)$ by\n\\begin{align*}\n\tG_{1,2}&=\\left\\{ A=\\begin{pmatrix}\n\t                         h & 0 & 0 \\\\\n\t                         0 & a & c \\\\\n\t                         0 & d & b\n\t                        \\end{pmatrix} \\in Sp(3)\\,\\left|\\,h \\in Sp(1), \\begin{pmatrix}\t                      \n\t                         a & c \\\\\n\t                         d & b\n\t                        \\end{pmatrix} \\in U(2) \\subset Sp(2) \n\t                       \\right. \\right\\}, \n\t\\\\\n\tG'_{1,2}&=\\left\\{ A'=\\begin{pmatrix}\n\th' & 0 & 0 \\\\\n\t0 & a' & c'e_2 \\\\\n\t0 & \\ov{e_2}d' & b'\n\t\\end{pmatrix} \\in Sp(3)\\,\\left|\\,h' \\in Sp(1), \n\t\\begin{array}{l}\n\t(c'e_2)(\\ov{c'e_2})+a'\\ov{a'}=1\\\\\n\tb'\\ov{b'}+(\\ov{e_2}d')(\\ov{\\ov{e_2}d'})=1\\\\\n\t(c'e_2)\\ov{b'}+a'(\\ov{\\ov{e_2}d'})=0\\\\\n\ta',b',c',d' \\in \\C\n\t\\end{array}\n\t\\right. \\right\\}, \n\\end{align*}\nwhere $e_2$ is one of basis in $\\mathfrak{C}$.\n\n It goes without saying that $\\begin{pmatrix}\n                       a & c \\\\\n                       d & b\n                     \\end{pmatrix} \\in U(2)$ is equivalent to the conditions\n\\begin{align*}\n\t\tc\\ov{c}+a\\ov{a}=1, \\,\\,b\\ov{b}+d\\ov{d}=1,\\,\\,c\\ov{b}+a\\ov{d}=0,\n\\end{align*}\nmoreover, that $(c'e_2)(\\ov{c'e_2})+a'\\ov{a'}=1$ above is same as $c'\\ov{c}+a'\\ov{a'}=1$, so is others.\n\\vspace{1mm}\n\n\\begin{proposition}\\label{proposition 4.2.1\n\tThe group $G'_{1,2}$ is isomorphic to the group $Sp(1) \\times U(2)${\\rm :} $G'_{1,2} \\cong Sp(1) \\times U(2)$.\n\\end{proposition}\n\\begin{proof}\n\tFirst, we will prove that the group $G'_{1,2}$ is isomorphic to the group $G_{1,2}$. \n\tWe define a mapping $g_{{}_{421}}: G_{1,2} \\to G'_{1,2}$ by\n\t\\begin{align*}\n\t\t\tg_{{}_{421}}(\\begin{pmatrix}\n\t\t\th & 0 & 0 \\\\\n\t\t\t0 & a & c \\\\\n\t\t\t0 & d & b\n\t\t\t\\end{pmatrix})\n\t\t\t&=\\begin{pmatrix}\n\t\t\t1 & 0 & 0 \\\\\n\t\t\t0 & 1 & 0 \\\\\n\t\t\t0 & 0 & \\ov{e_2}\n\t\t\t\\end{pmatrix}\\begin{pmatrix}\n\t\t\th & 0 & 0 \\\\\n\t\t\t0 & a & c \\\\\n\t\t\t0 & d & b\n\t\t\t\\end{pmatrix}\\begin{pmatrix}\n\t\t\t1 & 0 & 0 \\\\\n\t\t\t0 & 1 & 0 \\\\\n\t\t\t0 & 0 & e_2\n\t\t\t\\end{pmatrix}\\left(=\\begin{pmatrix}\n\t\t\th & 0 & 0 \\\\\n\t\t\t0 & a & ce_2 \\\\\n\t\t\t0 & \\ov{e_2}d & b\n\t\t\t\\end{pmatrix} \\right).\n\t\\end{align*}\n\tFirst, it is clear that $g_{{}_{421}}$ is well-defined and a homomorphism. Moreover, it is easy to verify that $g_{{}_{421}}$ is bijective. Thus we have the isomorphism $G'_{1,2} \\cong G_{1,2}$. \n\t\n\tHere, by defining a mapping $f_{{}_{421}}:Sp(1) \\times U(2) \\to G_{1,2}$ as follows:\n\t\\begin{align*}\n\t\tf_{{}_{421}}(p,U)=\\scalebox{0.8}{$\n\t\t\t\\left( \\begin{array}{cccccccc@{\\!}}\n\t\t\t&\\multicolumn{2}{c}{\\raisebox{-15pt}[0pt][0pt]{\\Large$p$}}&&&&\n\t\t\t\\\\\n\t\t\t&&&&\\multicolumn{2}{c}{\\raisebox{-5pt}[0pt]{\\Large $0$}}&\n\t\t\t\\\\\n\t\t\t&&&&&&&\n\t\t\t\\\\\n\t\t\t&&&&\\multicolumn{2}{c}{\\raisebox{-18pt}[0pt][0pt]{\\huge $U$}}&\n\t\t\t\\\\\n\t\t\t&\\multicolumn{2}{c}{\\raisebox{-5pt}[0pt]{\\Large $0$}}&&&&\n\t\t\t\\\\[-2mm]\n\t\t\t&&&&&&&\n\t\t\t\\end{array}\\right)$},\n\t\t\\end{align*}\n\twe have the isomorphism $G_{1,2} \\cong Sp(1) \\times U(2)$.\n\t\n\tTherefore, together with the result of $G'_{1,2} \\cong G_{1,2}$, we have the required isomorphism \n\t\\begin{align*}\n\t\t\tG'_{1,2} \\cong Sp(1) \\times U(2).\n\t\\end{align*}\n\\end{proof}\n\nNow, we will determine the structure of the group $(F_4)^{\\gamma_3} \\cap (F_4)^{\\sigma_3}$.\n\n\\begin{theorem} \\label{theorem 4.2.2\n\tThe group $(F_4)^{\\gamma_3} \\cap (F_4)^{\\sigma_3}$ is isomorphic to the group $(U(1) \\times Sp(1) \\times U(2))\/\\Z_2$ {\\rm: } $(F_4)^{\\gamma_3} \\cap (F_4)^{\\sigma_3} \\cong (U(1) \\times Sp(1) \\times U(2))\/\\Z_2, \\Z_2=\\{(1,1,E),(-1,-1,-E) \\}$.\n\\end{theorem}\n\\begin{proof}\n\tFirst, we denote the composition of $g_{{}_{421}}$ and $f_{{}_{421}}$ by $h$: $h=g_{{}_{421}}f_{{}_{421}}$ (in the proof of Proposition \\ref{proposition 4.2.1}). Then we define a mapping $\\varphi_{{}_{F_4,\\gamma_3,\\sigma_3}}:U(1) \\times Sp(1) \\times U(2) \\to (F_4)^{\\gamma_3} \\cap (F_4)^{\\sigma_3}$ by\n\t\t\\begin{align*}\n\t\t\\varphi_{{}_{F_4,\\gamma_3,\\sigma_3}}(s,p,U)(M+\\a)=h(p,U)Mh(p,U)^*+s\\a h(p,U)^*,\\,\n\t\tM+\\a \\in \\mathfrak{J}(3,\\H) \\oplus \\H^3=\\mathfrak{J}.\n\t\t\\end{align*}\n\t\tNeedless to say, this mapping is the restriction of the mapping $\\varphi_{{}_{F_4,\\gamma_3}}$, that is, $\\varphi_{{}_{F_4,\\gamma_3,\\sigma_3}}(s,p,U) \\allowbreak =\\varphi_{{}_{F_4,\\gamma_3}}(s,h(p,U))$ (Theorem \\ref{theorem 3.2.2}).\n\t\t\n\t\tFirst, we will prove that $\\varphi_{{}_{F_4,\\gamma_3,\\sigma_3}}$ is well-defined. It is clear that $\\varphi_{{}_{F_4,\\gamma_3,\\sigma_3}}(s,p,U) \\in (F_4)^{\\gamma_3}$, and using $\\sigma_3=\\varphi_{{}_{F_4,\\gamma_3}}(1,\\diag(1,\\ov{\\bm{\\omega}}, \\bm{\\omega}))$ (Lemma \\ref{lemma 3.2.6} (1)), it follows that\n\t\t\\begin{align*}\n\t\t{\\sigma_3}^{-1}\\varphi_{{}_{F_4,\\gamma_3,\\sigma_3}}(s,p,U)\\sigma_3\n\t\t&=\\varphi_{{}_{F_4,\\gamma_3}}(1,\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega}))^{-1}\\varphi_{{}_{F_4,\\gamma_3,\\sigma_3}}(s,p,U)\n\t\t\\varphi_{{}_{F_4,\\gamma_3}}(1,\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega}))\n\t\t\\\\\n\t\t&=\\varphi_{{}_{F_4,\\gamma_3}}(1,\\diag(1,\\bm{\\omega},\\ov{\\bm{\\omega}}))\n\t\t\\varphi_{{}_{F_4,\\gamma_3}}(s,h(p,U))\\varphi_{{}_{F_4,\\gamma_3}}(1,\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega}))\n\t\t\\\\\n\t\t&=\\varphi_{{}_{F_4,\\gamma_3}}(s,\\diag(1,\\bm{\\omega},\\ov{\\bm{\\omega}})h(p,U)\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega})), h(p,U)\\!=\n\t\t\\begin{pmatrix}\n\t\tp & 0 & 0 \\\\\n\t\t0 & a & c \\\\\n\t\t0 & d & b\n\t\t\\end{pmatrix}\n\t\t\\\\\n\t\t&=\\varphi_{{}_{F_4,\\gamma_3}}(s,\n\t\t\\begin{pmatrix}\n\t\tp & 0 & 0 \\\\\n\t\t0 & \\bm{\\omega}a\\ov{\\bm{\\omega}} & \\bm{\\omega}(ce_2)\\bm{\\omega} \\\\\n\t\t0 & \\ov{\\bm{\\omega}}(\\ov{e_2}d)\\ov{\\bm{\\omega}} & \\ov{\\bm{\\omega}}b\\bm{\\omega}\n\t\t\\end{pmatrix})\n\t\t\\\\\n\t\t&=\\varphi_{{}_{F_4,\\gamma_3}}(s,\n\t\t\\begin{pmatrix}\n\t\tp & 0 & 0 \\\\\n\t\t0 & a & c e_2\\\\\n\t\t0 & \\ov{e_2}d & b\n\t\t\\end{pmatrix})\n\t\t\\\\\n\t\t&=\\varphi_{{}_{F_4,\\gamma_3}}(s,h(p,U))\n\t\t\\\\\n\t\t&=\\varphi_{{}_{F_4,\\gamma_3,\\sigma_3}}(s,p,U).\n\t\t\\end{align*}\n\tHence we have that $\\varphi_{{}_{F_4,\\gamma_3,\\sigma_3}}(s,p,U) \\in (F_4)^{\\sigma_3}$. Thus $\\varphi_{{}_{F_4,\\gamma_3,\\sigma_3}}$ is well-defined.\n\tSubsequently, since $\\varphi_{{}_{F_4,\\gamma_3,\\sigma_3}}$ is the restriction of the mapping $\\varphi_{{}_{F_4,\\gamma_3}}$, we easily see  that $\\varphi_{{}_{F_4,\\gamma_3,\\sigma_3}}$ is a homomorphism. \n\n\tNext, we will prove that $\\varphi_{{}_{F_4,\\gamma_3,\\sigma_3}}$ is surjective. Let $\\alpha \\in (F_4)^{\\gamma_3} \\cap (F_4)^{\\sigma_3} \\subset (F_4)^{\\gamma_3}$. There exist  $s \\in U(1)$ and $A \\in Sp(3)$ such that $\\alpha=\\varphi_{{}_{F_4,\\gamma_3}}(s,A)$ (Theorem \\ref{theorem 3.2.2}). Moreover, from the condition $\\alpha\\in\t(F_4)^{\\sigma_3}$, that is, ${\\sigma_3}^{-1}\\varphi_{{}_{F_4,\\gamma_3}}(s,A)\\sigma_3=\\varphi_{{}_{F_4,\\gamma_3}}(s,A)$, and using ${\\sigma_3}^{-1}\\varphi_{{}_{F_4,\\gamma_3}}(s,A)\\sigma_3\\!=\\!\\varphi_{{}_{F_4,\\gamma_3}}(s,\\diag(1,\\bm{\\omega},\\ov{\\bm{\\omega}})A\\,\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega}))$ (Lemma \\ref{lemma 3.2.6} (1)), we have that\n\t\\begin{align*}\n\t\\left\\{ \n\t\\begin{array}{l}\n\ts=s \\\\\n\t\\diag(1,\\bm{\\omega},\\ov{\\bm{\\omega}})A\\,\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega})=A \n\t\\end{array} \\right.\n\t\\quad {\\text{or}}\\quad\n\t\\left\\{ \n\t\\begin{array}{l}\n\ts=-s \\\\\n\t\\diag(1,\\bm{\\omega},\\ov{\\bm{\\omega}})A\\,\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega})=-A. \n\t\\end{array} \\right.\n\t\\end{align*}\n\tThe latter case is impossible because of $s\\not=0$. As for the former case, from the second condition, by doing straightforward computation $A$ takes the following form $\\begin{pmatrix}\n\tp & 0 & 0 \\\\\n\t0 & a & c e_2\\\\\n\t0 & \\ov{e_2}d & b\n\t\\end{pmatrix} \\allowbreak \\in Sp(3)$, that is, $A \\in G'_{1,2}$. Hence there exist $s \\in U(1)$ and $h(p,U) \\in Sp(3)$ such that $\\alpha=\\varphi_{{}_{F_4,\\gamma_3}}(s,h(p,U))$. \n\tMoreover, from Lemma \\ref{lemma 3.2.6} (1) there exist $p \\in Sp(1)$ and $U \\in U(2)$ such that $A=h(p,U)$. Needless to say, $s \\in U(1)$.\n  Thus, there exist $s \\in U(1),p \\in Sp(1)$ and $U \\in U(2)$ such that $\\alpha=\\varphi_{{}_{F_4,\\gamma_3, \\sigma_3}}(s,p,U)$. The proof of surjective is completed.\n\t\n\tFinally, we will determine $\\Ker\\,\\varphi_{{}_{F_4,\\gamma_3,\\sigma_3}}$. However, from $\\Ker\\,\\varphi_{{}_{F_4,\\gamma_3}}=\\{(1,E), (-1,-E) \\}$ we easily obtain that $\\Ker\\,\\varphi_{{}_{F_4,\\gamma_3,\\sigma_3}}=\\{(1,1,E), (-1,-1,-E) \\} \\cong \\Z_2$. \n\n Therefore we have the required isomorphism\n \\begin{align*}\n (F_4)^{\\gamma_3} \\cap (F_4)^{\\sigma_3} \\cong (U(1) \\times Sp(1) \\times U(2))\/\\Z_2.\n  \\end{align*}\n\\end{proof}\n\\vspace{1mm}\n\nThus, since the group $(F_4)^{\\gamma_3} \\cap (F_4)^{\\sigma_3}$ is connected from Theorem \\ref{theorem 4.2.2}, we have an exceptional $\\varmathbb{Z}_3 \\times \\varmathbb{Z}_3$-symmetric space\n\\begin{align*}\n     F_4\/((U(1) \\times Sp(1) \\times U(2))\/\\Z_2).\n\\end{align*}\n\n\\subsection{Case 3: $\\{1, \\tilde{\\gamma}_3,  \\tilde{\\gamma}_3{}^{-1}\\} \\times \\{1, \\tilde{w}_3,  \\tilde{w}_3{}^{-1}\\}$-symmetric space}\n\nLet the $\\R$-linear transformations $\\gamma_3, w_3$ of $\\mathfrak{J}$  defined in Subsection \\ref{subsection 3.2}. \n\n\\noindent From Lemma \\ref{lemma 3.2.6} (2), since we can easily confirm that $\\gamma_3$ and $w_3$ are commutative, $\\tilde{\\gamma}_3$ and $\\tilde{w}_3$ are commutative in $\\Aut(F_4)$: $\\tilde{\\gamma}_3\\tilde{w}_3=\\tilde{w}_3\\tilde{\\gamma}_3$.\n\\vspace{1mm}\n\nBefore determining the structure of the group $(F_4)^{\\gamma_3} \\cap (F_4)^{w_3}$, we prove lemma needed in the proof of theorem below.\n\n\\begin{lemma}\\label{lemma 4.3.1\n\tThe group $S(U(1)\\times U(1)\\times U(1))$ is isomorphic to the group $U(1)\\times U(1)${\\rm :} $S(U(1)\\times U(1)\\times U(1)) \\cong U(1)\\times U(1)$.\n\\end{lemma}\n\\begin{proof}\n\tWe define a mapping $f_{{}_{431}}: U(1)\\times U(1) \\to S(U(1)\\times U(1)\\times U(1))$ by \n\t\\begin{align*}\n\tf_{{}_{431}}(a,b)=\\left( \n\t\\begin{array}{ccc}\n\ts & & {\\raisebox{-7pt}[0pt]{\\large $0$}}\n\t\\\\[2mm]\n\t& t & \n\t\\\\[2mm]\n\t{\\raisebox{1pt}[0pt]{\\large $0$}}&& (st)^{-1}\n\t\\end{array}\\right) \\in SU(3).\n\t\\end{align*}\n\tThen this mapping induces the required isomorphism.\n\\end{proof}\n\nNow, we will determine the structure of the group $(F_4)^{\\gamma_3} \\cap (F_4)^{w_3}$.\n\n\\begin{theorem}\\label{theorem 4.3.2\n\tThe group $(F_4)^{\\gamma_3} \\cap (F_4)^{w_3}$ is isomorphic to the group $(U(1) \\times U(1) \\times SU(3))\/\\Z_3${\\rm :} $(F_4)^{\\gamma_3} \\cap (F_4)^{w_3} \\cong (U(1) \\times U(1)\\times SU(3))\/\\Z_3, \\Z_3=\\{(1,1,E), (\\bm{\\omega},\\bm{\\omega}, \\bm{\\omega}E),(\\bm{\\omega}^{-1},\\allowbreak\\bm{\\omega}^{-1}, \\bm{\\omega}^{-1}E)\\}$.\n\\end{theorem}\n\\begin{proof}\n\tLet $S(U(1) \\times U(1) \\times U(1)) \\subset SU(3)$.\n\tWe define a mapping $\\varphi_{{}_{F_4,\\gamma_3, w_3}}: S(U(1) \\times U(1) \\times U(1)) \\times SU(3) \\to (F_4)^{\\gamma_3} \\cap (F_4)^{w_3}$ by\n\t\\begin{align*}\n\t\t\t\\varphi_{{}_{F_4,\\gamma_3, w_3}}(L,A)(X_{\\bm{C}}+M)=AX_{\\bm{C}}A^*+LMA^*,\\,\\,X_{\\bm{C}}+M \\in \\mathfrak{J}(3,\\C)\\oplus M(3,\\C)=\\mathfrak{J}.\n\t\\end{align*}\n\tNeedless to say, this mapping is the restriction of the mapping $\\varphi_{{}_{F_4,w_3}}$, that is, $\\varphi_{{}_{F_4,\\gamma_3, w_3}}(L,A)=\\varphi_{{}_{F_4,w_3}}(L,A)$ (Theorem \\ref{theorem 3.2.5}).\n\t\n\tAs usual, we will prove that $\\varphi_{{}_{F_4,\\gamma_3, w_3}}$ is well-defined. It is clear that $\\varphi_{{}_{F_4,\\gamma_3, w_3}}(L,A) \\in (F_4)^{w_3}$, and using $\\gamma_3=\\varphi_{{}_{F_4,w_3}}(\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega}), E)$ (Lemma \\ref{lemma 3.2.6} (2)), it follows that \n\t\\begin{align*}\n\t     {\\gamma_3}^{-1}\\varphi_{{}_{F_4,\\gamma_3, w_3}}(L,A)\\gamma_3\n\t     &=\\varphi_{{}_{F_4,w_3}}(\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega}), E)^{-1}\\varphi_{{}_{F_4,\\gamma_3, w_3}}(L,A)\\varphi_{{}_{F_4,w_3}}(\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega}), E)\n\t     \\\\\n\t     &=\\varphi_{{}_{F_4,w_3}}(\\diag(1,\\bm{\\omega},\\ov{\\bm{\\omega}}), E)\\varphi_{{}_{F_4,w_3}}(L,A)\\varphi_{{}_{F_4,w_3}}(\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega}), E)\n\t     \\\\\n\t     &=\\varphi_{{}_{F_4,w_3}}(\\diag(1,\\bm{\\omega},\\ov{\\bm{\\omega}})L\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega}),A),L=\\diag(a,b,c), abc=1\n\t     \\\\\n\t     &=\\varphi_{{}_{F_4,w_3}}(L,A)\n\t     \\\\\n\t     &=\\varphi_{{}_{F_4,\\gamma_3,w_3}}(L,A).\n\t\\end{align*}\n\tHence we have that $\\varphi_{{}_{F_4,\\gamma_3, w_3}}(L,A) \\in (F_4)^{\\gamma_3}$.  Thus $\\varphi_{{}_{F_4,\\gamma_3, w_3}}$ is well-defined. Subsequently, since $\\varphi_{{}_{F_4,\\gamma_3,w_3}}$ is the restriction of the mapping $\\varphi_{{}_{F_4,w_3}}$, we easily see that $\\varphi_{{}_{F_4,\\gamma_3,w_3}}$ is a homomorphism. \n\t\n\tNext, we will prove that $\\varphi_{{}_{F_4,\\gamma_3,w_3}}$ is surjective. Let $\\alpha \\in (F_4)^{\\gamma_3} \\cap (F_4)^{w_3} \\subset (F_4)^{w_3}$. There exist $P, A \\in SU(3)$ such that $\\alpha=\\varphi_{{}_{F_4,w_3}}(P,A)$ (Theorem \\ref{theorem 3.2.5}). Moreover, from the condition $\\alpha \\in (F_4)^{\\gamma_3}$, that is, ${\\gamma_3}^{-1}\\varphi_{{}_{F_4,w_3}}(P,A)\\gamma_3=\\varphi_{{}_{F_4,w_3}}(P,A)$, and using ${\\gamma_3}^{-1}\\varphi_{{}_{F_4,w_3}}(P,A)\\gamma_3=\\varphi_{{}_{F_4,w_3}}(\\diag(1,\\bm{\\omega},\\ov{\\bm{\\omega}})P\\,\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega}),A)$ (Lemma \\ref{lemma 3.2.6} (2)), we have that\n\t\\begin{align*}\n\t&\\,\\,\\,{\\rm(i)}\\,\\left\\{\n\t\\begin{array}{l}\n\t\\diag(1,\\bm{\\omega},\\ov{\\bm{\\omega}})P\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega})=P \\\\\n\tA=A,\n\t\\end{array} \\right.\n\t\\qquad\n\t {\\rm(ii)}\\,\\left\\{\n \t\\begin{array}{l}\n \t\\diag(1,\\bm{\\omega},\\ov{\\bm{\\omega}})P\\diag(1,\\ov{\\bm{\\omega}}, \\bm{\\omega})=\\bm{\\omega}P \\\\\n\tA=\\bm{\\omega}A,\n\t\\end{array} \\right.\n\t\\\\[2mm]\n\t&{\\rm(iii)}\\,\\left\\{\n\t\\begin{array}{l}\n\t\\diag(1,\\bm{\\omega},\\ov{\\bm{\\omega}})P\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega})=\\bm{\\omega}^{-1}P \\\\\n\tA=\\bm{\\omega}^{-1}A.\n\t\\end{array} \\right.\n\t\\end{align*}\n\tThe Cases (ii) and (iii) are impossible because of $A\\not=0$. As for the Case (i), from the first condition, by doing straightforward computation $P$ takes the form  $\\diag(a,b,c)\n\n\n\n\n\n  \\in SU(3)$, that is, $P \\in S(U(1)\\times U(1)\\times U(1))$. Needless to say, $A \\in SU(3)$. Hence there exist $L \\in S(U(1)\\times U(1) \\times U(1))$ and $A \\in SU(3)$ such that $\\alpha=\\varphi_{{}_{F_4,w_3}}(L,A)$. Namely, there exist $L \\in S(U(1)\\times U(1) \\times U(1))$ and $A \\in SU(3)$ such that $\\alpha=\\varphi_{{}_{F_4,\\gamma_3,w_3}}(L,A)$. With above, the proof of surjective is completed.\n\t\n\tFinally, we will determine $\\Ker\\,\\varphi_{{}_{F_4,\\gamma_3,w_3}}$. However, from $\\Ker\\,\\varphi_{{}_{F_4,w_3}}=\\{(E,E),(\\bm{\\omega}E,\\bm{\\omega}E), \\allowbreak (\\bm{\\omega}^{-1}E,\\bm{\\omega}^{-1}E)\\}$, we easily obtain that $\\Ker\\,\\varphi_{{}_{F_4,\\gamma_3,w_3}}=\\{(E,E),(\\bm{\\omega}E,\\bm{\\omega}E), (\\bm{\\omega}^{-1}E,\\bm{\\omega}^{-1}E)\\} \\cong \\Z_3$. Thus we have the isomorphism $(F_4)^{\\gamma_3} \\cap (F_4)^{w_3} \\cong (S(U(1) \\times U(1) \\times U(1))\\times SU(3))\/\\Z_3$.\n\t\n\tTherefore, by Lemma \\ref{lemma 4.3.1} we have the required isomorphism \n\t\\begin{align*}\n\t(F_4)^{\\gamma_3} \\cap (F_4)^{w_3} \\cong (U(1) \\times U(1)\\times SU(3))\/\\Z_3,\n\t\\end{align*}\n\twhere $\\Z_3=\\{(1,1,E), (\\bm{\\omega},\\bm{\\omega}, \\bm{\\omega}E),(\\bm{\\omega}^{-1},\\bm{\\omega}^{-1}, \\bm{\\omega}^{-1}E)\\}$.\n\\end{proof}\n\\vspace{1mm}\n\nThus, since the group $(F_4)^{\\gamma_3} \\cap (F_4)^{w_3}$ is connected  from Theorem \\ref{theorem 4.3.2}, we have an exceptional $\\varmathbb{Z}_3 \\times \\varmathbb{Z}_3$-symmetric space\n\\begin{align*}\nF_4\/((U(1) \\times U(1) \\times SU(3))\/\\Z_3).\n\\end{align*}\n\n\\subsection{Case 4: $\\{1, \\tilde{\\sigma}_3,  \\tilde{\\sigma}_3{}^{-1}\\} \\times \\{1, \\tilde{w}_3,  \\tilde{w}_3{}^{-1}\\}$-symmetric space}\\label{case 4}\n\nLet the $\\R$-linear transformations $\\sigma_3, w_3$ of $\\mathfrak{J}$  defined in Subsection \\ref{subsection 3.2}. \n\n\\noindent From Lemma \\ref{lemma 3.2.6} (1), since we can easily confirm that $\\gamma_3$ and $\\sigma_3$ are commutative, $\\tilde{\\sigma}_3$ and $\\tilde{w}_3$ are commutative in $\\Aut(F_4)$: $\\tilde{\\sigma}_3\\tilde{w}_3=\\tilde{w}_3\\tilde{\\sigma}_3$.\n\\vspace{1mm}\n\nNow, we will determine the structure of the group $(F_4)^{\\sigma_3} \\cap (F_4)^{w_3}$. Note that we can prove theorem below as in the proof of Theorem \\ref{theorem 4.3.2}, however we give the proof as detailed as possible.\n\n\\begin{theorem}\\label{theorem 4.4.1\n\tThe group $(F_4)^{\\sigma_3} \\cap (F_4)^{w_3}$ is isomorphic to the group $(SU(3)\\times U(1) \\times U(1))\/\\Z_3${\\rm :} $(F_4)^{\\sigma_3} \\cap (F_4)^{w_3} \\cong (SU(3)\\times U(1) \\times U(1))\/\\Z_3, \\Z_3=\\{(E,1,1), (\\bm{\\omega}E,\\bm{\\omega},\\bm{\\omega}),( \\bm{\\omega}^{-1}E,\\allowbreak \\bm{\\omega}^{-1},\\bm{\\omega}^{-1}\\}$.\n\\end{theorem}\n\\begin{proof}\n\tLet $S(U(1) \\times U(1) \\times U(1)) \\subset SU(3)$.\n\tWe define a mapping $\\varphi_{{}_{F_4,\\sigma_3, w_3}}: SU(3) \\times S(U(1) \\times U(1) \\times U(1)) \\to (F_4)^{\\sigma_3} \\cap (F_4)^{w_3}$ by\n\t\\begin{align*}\n\t\\varphi_{{}_{F_4,\\sigma_3, w_3}}(P,L)(X_{\\bm{C}}+M)=LX_{\\bm{C}}L^*+PML^*,\\,\\,X_{\\bm{C}}+M \\in \\mathfrak{J}(3,\\C)\\oplus M(3,\\C)=\\mathfrak{J}.\n\t\\end{align*}\n\tNeedless to say, this mapping is the restriction of the mapping $\\varphi_{{}_{F_4,w_3}}$, that is, $\\varphi_{{}_{F_4,\\gamma_3, w_3}}(P,L)=\\varphi_{{}_{F_4,w_3}}(P,L)$ (Theorem \\ref{theorem 3.2.5}).\n\t\n\tAs usual, we will prove that $\\varphi_{{}_{F_4,\\sigma_3, w_3}}$ is well-defined. It is clear that $\\varphi_{{}_{F_4,\\sigma_3, w_3}}(P,L) \\in (F_4)^{w_3}$, and using $\\sigma_3=\\varphi_{{}_{F_4,w_3}}(E,\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega}))$ (Lemma \\ref{lemma 3.2.6} (2)), it follows that \n\t\\begin{align*}\n\t{\\sigma_3}^{-1}\\varphi_{{}_{F_4,\\sigma_3, w_3}}(P,L)\\sigma_3\n\t&=\\varphi_{{}_{F_4,w_3}}(E,\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega}))^{-1}\\varphi_{{}_{F_4,\\gamma_3, w_3}}(P,L)\\varphi_{{}_{F_4,w_3}}(E,\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega}))\n\t\\\\\n\t&=\\varphi_{{}_{F_4,w_3}}(E,\\diag(1,\\bm{\\omega},\\ov{\\bm{\\omega}}))\\varphi_{{}_{F_4,w_3}}(P,L)\\varphi_{{}_{F_4,w_3}}(E,\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega}))\n\t\\\\\n\t&=\\varphi_{{}_{F_4,w_3}}(P,\\diag(1,\\bm{\\omega},\\ov{\\bm{\\omega}})L\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega})),\\,\\,L=\\diag(a,b,c)\n\t\\\\\n\t&=\\varphi_{{}_{F_4,w_3}}(P,L)\n\t\\\\\n\t&=\\varphi_{{}_{F_4,\\sigma_3,w_3}}(P,L).\n\t\\end{align*}\n\tHence we have that $\\varphi_{{}_{F_4,\\sigma_3, w_3}}(P,L) \\in (F_4)^{\\sigma_3}$.  Thus $\\varphi_{{}_{F_4,\\sigma_3, w_3}}$ is well-defined. Subsequently, since $\\varphi_{{}_{F_4,\\sigma_3,w_3}}$ is the restriction of the mapping $\\varphi_{{}_{F_4,w_3}}$, we easily see that $\\varphi_{{}_{F_4,\\sigma_3,w_3}}$ is a homomorphism. \n\t\n\tNext, we will prove that $\\varphi_{{}_{F_4,\\sigma_3,w_3}}$ is surjective. Let $\\alpha \\in (F_4)^{\\sigma_3} \\cap (F_4)^{w_3} \\subset (F_4)^{w_3}$.\n\t There exist $P, A \\in SU(3)$ such that $\\alpha=\\varphi_{{}_{F_4,w_3}}(P,A)$ (Theorem \\ref{theorem 3.2.5}). Moreover, from the condition $\\alpha \\in (F_4)^{\\sigma_3}$, that is, ${\\sigma_3}^{-1}\\varphi_{{}_{F_4,w_3}}(P,A)\\sigma_3=\\varphi_{{}_{F_4,w_3}}(P,A)$, and using ${\\sigma_3}^{-1}\\varphi_{{}_{F_4,w_3}}(P,A)\\sigma_3\\allowbreak=\\varphi_{{}_{F_4,w_3}}(P,\\diag(1,\\bm{\\omega},\\ov{\\bm{\\omega}})A\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega}))$ (Lemma \\ref{lemma 3.2.6} (2)), we have that\n\t\\begin{align*}\n\t&\\,\\,\\,{\\rm(i)}\\,\\left\\{\n\t\\begin{array}{l}\n\tP=P\\\\\n\t\\diag(1,\\bm{\\omega},\\ov{\\bm{\\omega}})A\\,\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega})=A, \n\t\\end{array} \\right.\n\t\\qquad\n\t{\\rm(ii)}\\,\\left\\{\n\t\\begin{array}{l}\n\tP=\\bm{\\omega}P\\\\\n\t\\diag(1,\\bm{\\omega},\\ov{\\bm{\\omega}})A\\,\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega})=\\bm{\\omega}A, \n\t\\end{array} \\right.\n\t\\\\[2mm]\n\t&{\\rm(iii)}\\,\\left\\{\n\t\\begin{array}{l}\n\tP=\\bm{\\omega}^{-1}P\\\\\n\t\\diag(1,\\bm{\\omega},\\ov{\\bm{\\omega}})A\\,\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega})=\\bm{\\omega}^{-1}A.\n\t\\end{array} \\right.\n\t\\end{align*}\n\tThe Cases (ii) and (iii) are impossible because of $P\\not=0$. As for the Case (i), from the first condition, by doing straightforward computation $A$ takes the following form  $\\diag(a,b,c), a,b,c \\in U(1), abc=1$, that is, $A \\in S(U(1)\\times U(1)\\times U(1))$. Needless to say, $P \\in SU(3)$.  Hence there exist $P \\in SU(3)$ and $A \\in S(U(1)\\times U(1) \\times U(1))$ such that $\\alpha=\\varphi_{{}_{F_4,w_3}}(P,A)$. Namely, there exist $P \\in SU(3)$ and $A \\in S(U(1)\\times U(1) \\times U(1))$ such that $\\alpha=\\varphi_{{}_{F_4,\\sigma_3,w_3}}(P,A)$. The proof of surjective is completed.\n\t\n\tFinally, we will determine $\\Ker\\,\\varphi_{{}_{F_4,\\sigma_3,w_3}}$. However, from $\\Ker\\,\\varphi_{{}_{F_4,w_3}}=\\{(E,E),(\\bm{\\omega}E,\\bm{\\omega}E), \\allowbreak (\\bm{\\omega}^{-1}E,\\bm{\\omega}^{-1}E)\\}$, we easily obtain that $\\Ker\\,\\varphi_{{}_{F_4,\\sigma_3,w_3}}=\\{(E,E),(\\bm{\\omega}E,\\bm{\\omega}E), (\\bm{\\omega}^{-1}E,\\bm{\\omega}^{-1}E)\\} \\cong \\Z_3$. Thus we have the isomorphism $(F_4)^{\\sigma_3} \\cap (F_4)^{w_3} \\cong (SU(3)\\times S(U(1) \\times U(1) \\times U(1)))\/\\Z_3$.\n\t\n\tHere, as in the proof of Theorem \\ref{theorem 4.3.2} we have the isomorphism $U(1) \\times U(1) \\cong S(U(1) \\times U(1) \\times U(1))$.\n\t\n\tTherefore we have the required isomorphism \n\t\\begin{align*}\n\t(F_4)^{\\sigma_3} \\cap (F_4)^{w_3} \\cong (SU(3)\\times U(1) \\times U(1))\/\\Z_3,\n\t\\end{align*}\n\twhere $\\Z_3=\\{(E,1,1), (\\bm{\\omega}E,\\bm{\\omega},\\bm{\\omega}),( \\bm{\\omega}^{-1}E,\\allowbreak \\bm{\\omega}^{-1},\\bm{\\omega}^{-1}\\}$.\n\\end{proof}\n\\vspace{1mm}\n\nThus, since the group $(F_4)^{\\sigma_3} \\cap (F_4)^{w_3}$ is connected from Theorem \\ref{theorem 4.4.1}, we have an exceptional $\\varmathbb{Z}_3 \\times \\varmathbb{Z}_3$-symmetric space\n\\begin{align*}\nF_4\/((SU(3)\\times U(1) \\times U(1))\/\\Z_3).\n\\end{align*} \n\n\\begin{assertion}\\label{assertion}\n\tOn Theorem \\ref{theorem 4.4.1} from a different view point. \n\\end{assertion}\n\n\tFirst, let $U(3) \\subset Sp(3)$. Then, we can embed $U(3)$ into $F_4$ using the mapping $\\varphi_{{}_{F_4,\\gamma_3}}$ as follows:\n\t\\begin{align*}\n\t\t\\varphi_{{}_{F_4,\\gamma_3}}(1,U)(M+\\a)=UMU^*+\\a U^*,\\,\\,M+\\a \\in \\mathfrak{J}(3,\\H) \\oplus \\H^3=\\mathfrak{J},\n\t\\end{align*}\n\tmore detail, since $w_3$ induces an automorphism of the group $(F_4)_{E_1, F_1(1),F_1(e_1)}$, it follows that $\\varphi_{{}_{F_4,\\gamma_3}}(1,U) \\in ((F_4)_{E_1, F_1(1),F_1(e_1)})^{w_3} \\cong (Spin(7))^{w_3}$ , where $Spin(7)$ is defined in Theorem \\ref{theorem 3.2.4}. Here, we denote $\\varphi_{{}_{F_4,\\gamma_3}}(1,U)$ by $\\varphi(U)$: $\\varphi(U)=\\varphi_{{}_{F_4,\\gamma_3}}(1,U)$, and we define a mapping $\\psi: U(1) \\times U(3) \\to (F_4)^{\\sigma_3} \\cap (F_4)^{w_3}$ by\n\t\\begin{align*}\n\t\t\\psi(a,U)=D_a\\varphi(U),\n\t\\end{align*}\n\twhere $D_a$ is defined in Subsection 3.2. Then the mapping $\\psi$ induces the isomorphism $(F_4)^{\\sigma_3} \\cap (F_4)^{w_3} \\cong (U(1)\\times U(3))\/\\Z_3$, where $\\Z_3=\\{(1,E), (\\bm{\\omega},\\bm{\\omega}^{-1}E), (\\bm{\\omega}^{-1}, \\bm{\\omega}E) \\}$.\n\n\\subsection{Case 5: $\\{1, \\tilde{\\gamma}_3,  \\tilde{\\gamma}_3{}^{-1}\\} \\times \\{1, \\tilde{\\sigma}_3,  \\tilde{\\sigma}_3{}^{-1}\\}$-symmetric space}\n\nLet the $C$-linear transformations $\\gamma_3, \\sigma_3$ of $\\mathfrak{J}^C$  defined in Subsection \\ref{subsection 3.3}.\n\n\\noindent From Lemma \\ref{lemma 3.3.8} (1), since we can easily confirm that $\\gamma_3$ and $\\sigma_3$ are commutative, $\\tilde{\\gamma}_3$ and $\\tilde{\\sigma}_3$ are commutative in $\\Aut(E_6)$: $\\tilde{\\gamma}_3\\tilde{\\sigma}_3=\\tilde{\\sigma}_3\\tilde{\\gamma}_3$.\n\\vspace{1mm}\n\nBefore determining the structure of the group $(E_6)^{\\gamma_3} \\cap (E_6)^{\\sigma_3}$, we prove proposition and lemma needed in the proof of theorem below.\n\\vspace{1mm}\n\nWe define a $C$-linear transformation $\\sigma'_3$ of $\\mathfrak{J}^C$ by\n\\begin{align*}\n\t\t\t\\sigma'_3=\\varphi_{{}_{E_6,\\gamma_3}}(1,\\diag(1,1,\\omega,\\omega,\\tau\\omega,\\tau\\omega)) \\in (E_6)^{\\gamma_3} \\subset E_6,\n\\end{align*}\nwhere $\\omega=-(1\/2)+(\\sqrt{3}\/2)i \\in C$.\n\nLet an element \n\\begin{align*}R:=\\scalebox{0.8}{$\\begin{pmatrix}\n\t1&&&&&\\\\\n\t&1&&&&\\\\\n\t&&&&1&\\\\\n\t&&&1&&\\\\\n\t&&-1&&&\\\\\n\t&&&&&1\n\t\\end{pmatrix}$} \\in SO(6) \\subset SU(6), \n\\end{align*}\nwhere the blanks are $0$, and we consider an element $\\varphi_{{}_{E_6,\\gamma_3}}(1,R) \\in (E_6)^{\\gamma_3} \\subset E_6$. Here, we denote this element by $\\delta_R$: $\\delta_R=\\varphi_{{}_{E_6,\\gamma_3}}(1,R)$.\nThen by doing straightforward computation, we have that $\\sigma_3\\delta_R=\\delta_R\\sigma'_3$, that is, $\\sigma_3$ is conjugate to $\\sigma'_3$ under $\\delta_R \\in (E_6)^{\\gamma_3} \\subset E_6$: $\\sigma_3 \\sim \\sigma'_3$. Moreover, $\\sigma'_3$ induces the automorphism $\\tilde{\\sigma'}_3$ of order $3$ on $E_6$: $\\tilde{\\sigma'}_3(\\alpha)={\\sigma'_3}^{-1}\\alpha\\sigma'_3, \\alpha \\in E_6$.\n\\vspace{1mm}\n\nThen we have the following proposition.\n\n\\begin{proposition}\\label{proposition 4.5.1\n\tThe group $(E_6)^{\\gamma_3} \\cap (E_6)^{\\sigma_3}$ is isomorphic to the group $(E_6)^{\\gamma_3} \\cap (E_6)^{\\sigma'_3}${\\rm :} $(E_6)^{\\gamma_3} \\cap (E_6)^{\\sigma_3} \\cong (E_6)^{\\gamma_3} \\cap (E_6)^{\\sigma'_3}$.\n\\end{proposition}\n\\begin{proof}\n\tWe define a mapping $g_{{}_{451}}: (E_6)^{\\gamma_3} \\cap (E_6)^{\\sigma_3} \\to (E_6)^{\\gamma_3} \\cap (E_6)^{\\sigma'_3}$ by\n\t\\begin{align*}\n\t\t\tg_{{}_{451}}(\\alpha)={\\delta_R}^{-1}\\alpha\\delta_R.\t\n\t\\end{align*}\n\tIn order to prove this isomorphism, it is sufficient to show that $g_{{}_{452}}$ is well-defined. \n\t\n\t\\noindent First, we will show that $g_{{}_{451}} \\in (E_6)^{\\gamma_3}$. Since it follows from $\\delta_R=\\varphi_{{}_{E_6,\\gamma_3}}(1,R)$ and $\\gamma_3=\\varphi_{{}_{E_6,\\gamma_3}}(\\omega,E)$ that $\\delta_R\\gamma_3=\\gamma_3\\delta_R$, we have that $g_{{}_{451}} \\in (E_6)^{\\gamma_3}$. Similarly, from $\\sigma_3\\delta_R=\\delta_R\\sigma'_3$ we have that $g_{{}_{451}} \\in (E_6)^{\\sigma'_3}$. Hence $g_{{}_{451}}$ is well-defined. With above, the proof of this proposition is completed.\t\n\\end{proof}\n\\vspace{1mm}\n\nSubsequently, we will prove the following lemma. \n\n\\begin{lemma}\\label{lemma 4.5.2\n\tThe group $S(U(2)\\times U(2)\\times U(2))$ is isomorphic to the group $(U(1) \\times U(1)\\times SU(2)\\times SU(2)\\times SU(2))\/(\\Z_2\\times\\Z_2)${\\rm :} $S(U(2)\\times U(2)\\times U(2)) \\cong (U(1) \\times U(1)\\times SU(2)\\times SU(2)\\times SU(2))\/(\\Z_2\\times\\Z_2), \\Z_2=\\!\\{(1,1,E,E,E), (1,-1,E,-E,E) \\}, \\Z_2=\\!\\{(1,1,E,E,E), (-1,1,-E,\\allowbreak E,E) \\}$.\n\\end{lemma}\n\\begin{proof}\n\tWe define a mapping $f_{{}_{452}}:U(1) \\times U(1)\\times SU(2)\\times SU(2)\\times SU(2) \\to S(U(2)\\times U(2)\\times U(2))$ by\n\t\\begin{align*}\n\t\t\tf_{{}_{452}}(a,b,A,B,C)=\\left( \n\t\t\t\\begin{array}{ccc}\n\t\t\t a\\mbox{\\large {$A$}} & & {\\raisebox{-7pt}[0pt]{\\large $0$}}\n\t\t\t \\\\[2mm]\n\t\t\t & b\\mbox{\\large {$B$}} & \n\t\t\t \\\\[2mm]\n\t\t\t {\\raisebox{1pt}[0pt]{\\large $0$}}&& (ab)^{-2}\\mbox{\\large {$C$}}\n\t\t\t\\end{array}\\right) \\in SU(6).\n\t\\end{align*}\n\tThen it is clear that $f_{{}_{452}}$ is well-defined and a homomorphism. \n\t\n\tWe will prove that $f_{{}_{452}}$ is surjective. Let $P \\in S(U(2)\\times U(2)\\times U(2))$. Then $P$ takes the form of $\\diag(P_1,P_2,P_3),P_j \\in U(2), (\\det\\,P_1)(\\det\\,P_2)(\\det\\,P_3)=1$. Here, since $P_1 \\in U(2)$, we see that $\\det\\,P_1 \\in U(1)$. We choose $a \\in U(1)$ such that $a^2=\\det\\,P_1$, and set $A=(1\/a)P_1$. Then we have that $ A \\in SU(2)$. Similarly, for $P_2 \\in U(2)$, there exist $b \\in U(1)$ and $B \\in SU(2)$ such that $P_2=bB, b^2=\\det\\,P_2$. From $(\\det\\,P_1)(\\det\\,P_2)(\\det\\,P_3)=1$, we have that $\\det\\,P_3=(ab)^{-2}$. Set $C=(ab)^2P_3$. Then we have that $C \\in SU(2)$. With above, the proof of surjective is completed.\n\t\n\tFinally, we will determine $\\Ker\\,f_{{}_{452}}$. It follows from the kernel of definition that\n\t\\begin{align*}\n\t\t\t\\Ker\\,f_{{}_{452}}&=\\{(a,b,A,B,C)\\in U(1)^{\\times 2}\\times SU(2)^{\\times 3} \\,|\\,f_{{}_{453}}(a,b,A,B,C)=E \\}\n\t\t\t\\\\\n\t\t\n\t\t\n\t\t\n\t\t\t&=\\{(a,b,a^{-1}E,b^{-1}E,(ab)^2E)\\in U(1)^{\\times 2}\\times SU(2)^{\\times 3} \\,|\\,a^2=b^2=1 \\}\n\t\t\t\\\\\n\t\t\t&=\\{(1,1,E,E,E), (1,-1,E,-E,E),(-1,1,-E,E,E), (-1,-1,-E,-E,E) \\}\n\t\t\t\\\\\n\t\t\t&=\\{(1,1,E,E,E), (1,-1,E,-E,E) \\} \\times \\{(1,1,E,E,E), (-1,1,-E,E,E) \\}\n\t\t\t\\\\\n\t\t\t& \\cong \\Z_2 \\times \\Z_2.\n\t\\end{align*}\n\t\n\tTherefore we have the required isomorphism \n\t\\begin{align*}\n\t\t\t\tS(U(2)\\times U(2)\\times U(2)) \\cong (U(1) \\times U(1)\\times SU(2)\\times SU(2)\\times SU(2))\/(\\Z_2\\times\\Z_2).\n\t\\end{align*}\n\\end{proof}\n\nNow, we will determine the structure of the group $(E_6)^{\\gamma_3} \\cap (E_6)^{\\sigma_3}$.\n\n\\begin{theorem}\\label{theorem 4.5.3\n\tThe group $(E_6)^{\\gamma_3} \\cap (E_6)^{\\sigma_3}$ is isomorphic the group $(U(1)\\times U(1) \\times U(1)\\allowbreak \\times SU(2) \\times SU(2)\\times SU(2))\/(\\Z_2\\times\\Z_2\\times\\Z_2\\times\\Z_2)${\\rm :} $(E_6)^{\\gamma_3} \\cap (E_6)^{\\sigma_3} \\cong (U(1)\\times U(1) \\times U(1)\\times SU(2) \\allowbreak \\times SU(2)\\times SU(2))\/(\\Z_2\\times\\Z_2\\times\\Z_2\\times\\Z_2), \\Z_2=\\{(1,1,1,E,E,E), (-1,1,1,-E,-E,E) \\},\\,\\Z_2=\\{(1,1,1,E,E,E), (-1,1,-1,-E,E,E) \\},\\Z_2\\!=\\!\\{(1,1,1,E,E,E), (-1,-1,1,-E,-E,E) \\},\\!\\Z_2\\allowbreak=\\{(1,1,1,E,E,E), (-1,-1,-1,E,E,E) \\}$.\n\\end{theorem}\n\\begin{proof}\n\tLet $S(U(2)\\times U(2)\\times U(2)) \\subset SU(6)$. \n\tWe define a mapping $\\varphi_{{}_{E_6,\\gamma_3,\\sigma'_3}}: U(1)\\times S(U(2)\\times U(2)\\times U(2)) \\to (E_6)^{\\gamma_3} \\cap (E_6)^{\\sigma'_3}$ by\n\t\\begin{align*}\n\t\\varphi_{{}_{E_6,\\gamma_3,\\sigma'_3}}(s, P)(M+\\a)&={k_J}^{-1}(P(k_J M){}^t\\!P)+s\\a k^{-1}(\\tau \\,{}^t\\!P), \n  \\\\\n\t&\\hspace*{40mm}M+\\a \\in \\mathfrak{J}(3, \\H)^C \\oplus (\\H^3)^C\\!\\!=\\!\\mathfrak{J}^C.\n\t\\end{align*}\n\tNeedless to say, this mapping is the restriction of the mapping $\\varphi_{{}_{E_6,\\gamma_3}}$, that is, $\\varphi_{{}_{E_6,\\gamma_3,\\sigma'_3}}(s, P)=\\varphi_{{}_{E_6,\\gamma_3}}(s,P)$ (Theorem \\ref{theorem 3.3.2}). \n\t\n\tFirst, we will prove that $\\varphi_{{}_{E_6,\\gamma_3,\\sigma'_3}}$ is well-defined. It is clear that $\\varphi_{{}_{E_6,\\gamma_3,\\sigma'_3}}(s,P) \\in (E_6)^{\\gamma_3}$, and it follows from $\\sigma'_3=\\varphi_{{}_{E_6,\\gamma_3}}(1,\\diag(1,1,\\omega,\\omega,\\tau\\omega,\\tau\\omega))$ that\n\t\\begin{align*}\n\t\t\t&\\quad {\\sigma'_3}^{-1}\\varphi_{{}_{E_6,\\gamma_3,\\sigma'_3}}(s,P)\\sigma'_3\n\t\t\t\\\\\n\t\t\t&=\\varphi_{{}_{E_6,\\gamma_3}}(1,\\diag(1,1,\\omega,\\omega,\\tau\\omega,\\tau\\omega))^{-1}\\varphi_{{}_{E_6,\\gamma_3,\\sigma'_3}}(s,P)\\varphi_{{}_{E_6,\\gamma_3}}(1,\\diag(1,1,\\omega,\\omega,\\tau\\omega,\\tau\\omega))\n\t\t\t\\\\\n\t\t\t&=\\varphi_{{}_{E_6,\\gamma_3}}(1,\\diag(1,1,\\tau\\omega,\\tau\\omega,\\omega,\\omega))\\varphi_{{}_{E_6,\\gamma_3}}(s,P)\\varphi_{{}_{E_6,\\gamma_3}}(1,\\diag(1,1,\\omega,\\omega,\\tau\\omega,\\tau\\omega))\n\t\t\t\\\\\n\t\t\t&=\\varphi_{{}_{E_6,\\gamma_3}}(s,\\diag(1,1,\\tau\\omega,\\tau\\omega,\\omega,\\omega)P\\diag(1,1,\\omega,\\omega,\\tau\\omega,\\tau\\omega)),P=\\diag(P_1,P_2,P_3)\n\t\t\t\\\\\n\t\t\t&=\\varphi_{{}_{E_6,\\gamma_3}}(s,\\diag(P_1,(\\tau\\omega E) P_2(\\omega E),(\\omega E) P_3(\\tau\\omega E)))\n\t\t\t\\\\\n\t\t\t&=\\varphi_{{}_{E_6,\\gamma_3}}(s,P)\n\t\t\t\\\\\n\t\t\t&=\\varphi_{{}_{E_6,\\gamma_3,\\sigma'_3}}(s,P).\n\t\\end{align*}\n\tHence we have that $\\varphi_{{}_{E_6,\\gamma_3,\\sigma'_3}}(s,P) \\in (E_6)^{\\sigma'_3}$. Thus $\\varphi_{{}_{E_6,\\gamma_3,\\sigma'_3}}$ is well-defined. Subsequently, since $\\varphi_{{}_{E_6,\\gamma_3,\\sigma'_3}}$ is the restriction of the mapping $\\varphi_{{}_{E_6,\\gamma_3}}$, we easily see that $\\varphi_{{}_{E_6,\\gamma_3,\\sigma'_3}}$ is a homomorphism.\n\t\n\tNext, we will prove that $\\varphi_{{}_{E_6,\\gamma_3,\\sigma'_3}}$ is surjective. Let $\\alpha \\in (E_6)^{\\gamma_3} \\cap (E_6)^{\\sigma'_3} \\subset (E_6)^{\\gamma_3}$. There exist $s \\in U(1)$ and $A \\in SU(6)$ such that $\\alpha=\\varphi_{{}_{E_6,\\gamma_3}}(s,A)$ (Theorem \\ref{theorem 3.3.2}). Moreover, from the condition $\\alpha \\in (E_6)^{\\sigma'_3}$, that is, ${\\sigma'_3}^{-1}\\varphi_{{}_{E_6,\\gamma_3}}(s,A)\\sigma'_3=\\varphi_{{}_{E_6,\\gamma_3}}(s,A)$, and using ${\\sigma'_3}^{-1}\\varphi_{{}_{E_6,\\gamma_3}}(s,A)\\sigma'_3=\\varphi_{{}_{E_6,\\gamma_3}}(s,\\diag(1,1,\\tau\\omega,\\tau\\omega,\\omega,\\omega)A\\,\\diag(1,1,\\omega,\\omega,\\tau\\omega,\\tau\\omega))$ (Lemma \\ref{lemma 3.3.8} (1)), we have that\n\t\\begin{align*}\n\t&\\left\\{\n\t    \\begin{array}{l}\n\t      s=s \\\\\n\t      \\diag(1,1,\\tau\\omega,\\tau\\omega,\\omega,\\omega)A\\,\\diag(1,1,\\omega,\\omega,\\tau\\omega,\\tau\\omega)=A   \n\t    \\end{array}\\right. \n\t \\\\\n\t&\\hspace*{45mm}{\\text{or}}\n\t \\\\\n\t&\\left\\{\n\t\\begin{array}{l}\n\ts=-s \\\\\n\t\\diag(1,1,\\tau\\omega,\\tau\\omega,\\omega,\\omega)A\\,\\diag(1,1,\\omega,\\omega,\\tau\\omega,\\tau\\omega)=-A.   \n\t\\end{array}\\right.     \n\t\\end{align*}\n\tThe latter case is impossible because of $s\\not=0$. As for the former case, from the second condition, by doing straightforward computation $A$ takes the following form $\\diag(A_1, A_2, A_3), A_j \\in U(2), (\\det\\,A_1)(\\det\\,A_2)(\\det\\,A_3)=1$, that is, $A \\in S(U(2)\\times U(2)\\times U(2))$.\n\tNeedless to say, $s \\in U(1)$. Hence there exist $s \\in U(1)$ and $P \\in S(U(2)\\times U(2) \\times U(2))$ such that $\\alpha=\\varphi_{{}_{E_6,\\gamma_3}}(s,P)$. Namely, there exist $s \\in U(1)$ and $P \\in S(U(2)\\times U(2) \\times U(2))$ such that $\\alpha=\\varphi_{{}_{E_6,\\gamma_3,\\sigma'_3}}(s,P)$. The proof of surjective is completed.\n\t\n\tFinally, we will determine $\\Ker\\,\\varphi_{{}_{E_6,\\gamma_3,\\sigma'_3}}$. However, from $\\Ker\\,\\varphi_{{}_{E_6,\\gamma_3}}=\\{(1,E),(-1,-E) \\}$, we easily obtain that $\\Ker\\,\\varphi_{{}_{E_6,\\gamma_3,\\sigma'_3}}=\\{(1,E),(-1,-E) \\} \\cong \\Z_2$. Thus we have the isomorphism $(E_6)^{\\gamma_3} \\cap (E_6)^{\\sigma'_3} \\cong (U(1)\\times S(U(2)\\times U(2)\\times U(2)))\/\\Z_2$. Here, from Proposition \\ref{proposition 4.5.1} we have the isomorphism $(E_6)^{\\gamma_3} \\cap (E_6)^{\\sigma_3} \\cong (U(1)\\times S(U(2)\\times U(2)\\times U(2)))\/\\Z_2$. Moreover, by Lemma \\ref{lemma 4.5.2} we have the required isomorphism \n\t\\begin{align*}\n\t(E_6)^{\\gamma_3} \\cap (E_6)^{\\sigma_3} \\!\\cong (U(1)\\times U(1) \\times U(1)\\times SU(2) \\allowbreak \\times SU(2)\\times SU(2))\/(\\Z_2\\times\\Z_2\\times\\Z_2\\times\\Z_2), \n\t\\end{align*}\n\twhere \n\t\\begin{align*}\n\t&\\Z_2=\\{(1,1,1,E,E,E), (-1,1,1,-E,-E,E) \\},\n\t\\\\\n\t&\\Z_2=\\{(1,1,1,E,E,E), (-1,1,-1,-E,E,E) \\},\n\t\\\\\n\t&\\Z_2=\\{(1,1,1,E,E,E), (-1,-1,1,-E,-E,E) \\},\n\t\\\\\n\t&\\Z_2\\allowbreak=\\{(1,1,1,E,E,E), (-1,-1,-1,E,E,E) \\}.\n\t\\end{align*}\n\\end{proof}\n\nThus, since the group $(E_6)^{\\gamma_3} \\cap (E_6)^{\\sigma_3}$ is connected from Theorem \\ref{theorem 4.5.3}, we have an exceptional $\\varmathbb{Z}_3 \\times \\varmathbb{Z}_3$-symmetric space\n\\begin{align*}\n\t\t\t\t\tE_6\/((U(1)\\times U(1) \\times U(1)\\times SU(2) \\allowbreak \\times SU(2)\\times SU(2))\/(\\Z_2\\times\\Z_2\\times\\Z_2\\times\\Z_2)).\n\\end{align*}\n\n\\subsection{Case 6: $\\{1, \\tilde{\\gamma}_3,  \\tilde{\\gamma}_3{}^{-1}\\} \\times \\{1, \\tilde{\\nu}_3,  \\tilde{\\nu}_3{}^{-1}\\}$-symmetric space}\n\nLet the $C$-linear transformations $\\gamma_3, \\nu_3$ of $\\mathfrak{J}^C$  defined in Subsection \\ref{subsection 3.3}.\n\n\\noindent From Lemma \\ref{lemma 3.3.8} (1), together with $\\gamma_3=\\varphi_{{}_{E_6,\\gamma_3}}(\\omega,E)$, since we can easily confirm that $\\gamma_3$ and $\\nu_3$ are commutative, $\\tilde{\\gamma}_3$ and $\\tilde{\\nu}_3$ are commutative in $\\Aut(E_6)$: $\\tilde{\\gamma}_3\\tilde{\\nu}_3=\\tilde{\\nu}_3\\tilde{\\gamma}_3$. \n\nBefore determining the structure of the group $(E_6)^{\\gamma_3} \\cap (E_6)^{\\nu_3}$, we prove lemma needed in the proof of theorem below.\n\n\\begin{lemma}\\label{lemma 4.6.1\n\tThe group $S(U(1)\\times U(5))$ is isomorphism the group $(U(1)\\times SU(5))\/\\Z_5${\\rm :} $S(U(1)\\times U(5))\\! \\cong\\! (U(1)\\times SU(5))\/\\Z_5, \\Z_5\\!=\\!\\{(\\varepsilon_k, {\\varepsilon_k}^{-1}E) | \\varepsilon_k\\!=\\!\\exp((2\\pi i\/5)k), k\\!=0,1,2,3,4\\}$.\n\\end{lemma}\n\\begin{proof}\n\tWe define a mapping $f_{{}_{461}}:U(1) \\times SU(5) \\to S(U(1)\\times U(5))$ by\n\t\\begin{align*}\n\tf_{{}_{461}}(t, T)=\\scalebox{0.7}{$\n\t\t\\left(\\begin{array}{cccccccc@{\\!}}\n\t\t&\\multicolumn{2}{c}{\\raisebox{-15pt}[0pt][0pt]{\\Large$t^{-5}$}}&&&&\n\t\t\\\\\n\t\t&&&&\\multicolumn{2}{c}{\\raisebox{-5pt}[0pt]{\\Large $0$}}&\n\t\t\\\\\n\t\t&&&&&&&\n\t\t\\\\\n\t\t&&&&\\multicolumn{2}{c}\n\t\t{\\raisebox{-15pt}[0pt][0pt]{\\Large $t$}\\,\\raisebox{-18pt}[0pt][0pt]{\\huge $T$}}&\n\t\t\\\\\n\t\t&\\multicolumn{2}{c}{\\raisebox{0pt}[0pt]{\\Large $0$}}&&&&\n\t\t\\\\[-2mm]\n\t\t&&&&&&&\n\t\t\\end{array}\\right)$}.\n\t\\end{align*}\n\tThen it is clear that $f_{{}_{461}}$ is well-defined and a homomorphism. \n\t\n\tNow, we will prove that $f_{{}_{461}}$ is surjective. Let $P \\in S(U(1) \\times U(5))$. Then $P$ takes the form of \n\t\\scalebox{0.6}\n\t{$\\left(\\begin{array}{cccccccc@{\\!}}\n\t\t\t&\\multicolumn{2}{c}{\\raisebox{-15pt}[0pt][0pt]{\\Large $s$}}&&&&\n\t\t\t\\\\\n\t\t\t&&&&\\multicolumn{2}{c}{\\raisebox{-5pt}[0pt]{\\Large $0$}}&\n\t\t\t\\\\\n\t\t\t&&&&&&&\n\t\t\t\\\\\n\t\t\t&&&&\\multicolumn{2}{c}{\\raisebox{-18pt}[0pt][0pt]{\\huge $S$}}&\n\t\t\t\\\\\n\t\t\t&\\multicolumn{2}{c}{\\raisebox{0pt}[0pt]{\\Large $0$}}&&&&\n\t\t\t\\\\[-2mm]\n\t\t\t&&&&&&&\n\t\t\\end{array}\\right)$},\\,\\,$s \\in U(1), S \\in U(5), s(\\det S)=1$.\n\tHere, since $S \\in U(5)$, we see that $\\det\\,S \\in U(1)$, and so we choose $t \\in U(1)$ such that $t^5=\\det\\,S$. Set $T=t^{-1}S$, then we have that $T \\in SU(5)$ and $s=t^{-5}$. With above, the proof of surjective is completed.\n\t\n\tFinally, we will determine $\\Ker\\,f_{{}_{461}}$. It follows from the definition of kernel that \n\t\\begin{align*}\n\t\\Ker\\,f_{{}_{461}}&=\\{(t,T) \\in U(1)\\times SU(5)\\,|\\, f_{{}_{461}}(t,T)=E \\}\n\t\\\\\n\t&=\\{(t,T) \\in U(1)\\times SU(5)\\,|\\,t^5=1, T=t^{-1}E \\}\n\t\\\\\n\t&=\\{(\\varepsilon_k, {\\varepsilon_k}^{-1}E) \\,|\\, \\varepsilon_k=\\exp((2\\pi i\/5)k), k=0,1,2,3,4\\}\n\t\\\\\n\t& \\cong \\Z_5.\n\t\\end{align*}\n\t\n\tTherefore we have the required isomorphism \n\t\\begin{align*}\n\tS(U(1) \\times U(5)) \\cong (U(1)\\times SU(5))\/\\Z_5.\n\t\\end{align*}\n\\end{proof}\n \nNow, we will determine the structure of the group $(E_6)^{\\gamma_3} \\cap (E_6)^{\\nu_3}$.\n\n\\begin{theorem}\\label{theorem 4.6.2\n\tThe group $(E_6)^{\\gamma_3} \\cap (E_6)^{\\nu_3}$ is isomorphic to the group $(U(1)\\times U(1)\\times SU(5))\/\\Z_2${\\rm :} $(E_6)^{\\gamma_3} \\cap (E_6)^{\\nu_3} \\cong (U(1)\\times U(1)\\times SU(5))\/(\\Z_2\\times \\Z_5), \\Z_2=\\{(1,1,E), (-1,-1,\\allowbreak -E) \\}, \\Z_5=\\{(1,\\varepsilon_i, {\\varepsilon_i}^{-1}E) \\,|\\, \\varepsilon_i=\\exp ((2\\pi i\/5)k), k=0,1,2,3,4\\}$.\n\\end{theorem}\n\\begin{proof}\n\tLet $S(U(1)\\times U(5)) \\subset SU(6)$. Then we define a mapping $\\varphi_{{}_{E_6, \\gamma, \\nu_3}}: U(1)\\times S(U(1)\\times U(5)) \\to (E_6)^{\\gamma_3} \\cap (E_6)^{\\nu_3}$ by\n\t\\begin{align*}\n\t\\varphi_{{}_{E_6,\\gamma_3,\\nu_3}}(s, P)(M+\\a)&={k_J}^{-1}(P(k_J M){}^t\\!P)+s\\a k^{-1}(\\tau \\,{}^t\\!P), \n\t\\\\\n\t&\\hspace*{40mm}M+\\a \\in \\mathfrak{J}(3, \\H)^C \\oplus (\\H^3)^C=\\mathfrak{J}^C.\n\t\\end{align*}\n\tNeedless to say, this mapping is the restriction of the mapping $\\varphi_{{}_{E_6,\\gamma_3}}$, that is, $\\varphi_{{}_{E_6,\\gamma_3,\\nu_3}}(s, P)=\\varphi_{{}_{E_6,\\gamma_3}}(s,P)$ (Theorem \\ref{theorem 3.3.2}).  \n\n\tFirst, we will prove that $\\varphi_{{}_{E_6,\\gamma_3,\\nu_3}}$ is well-defined. It is clear that $\\varphi_{{}_{E_6,\\gamma_3,\\nu_3}}(s, P) \\in (E_6)^{\\gamma_3}$, and using $\\nu_3=\\varphi_{{}_{E_6,\\gamma_3}}(1,\\diag(\\nu^5, \\nu^{-1},\\nu^{-1},\\nu^{-1},\\nu^{-1},\\nu^{-1}))$ (Lemma \\ref{lemma 3.3.8} (1)), it follows that \n\t\\begin{align*}\n\t\t\t&\\quad {\\nu_3}^{-1}\\varphi_{{}_{E_6, \\gamma_3, \\nu_3}}(s,P)\\nu_3\n\t\t\t\\\\\n\t\t\t&=\\varphi_{{}_{E_6,\\gamma_3}}(1,\\diag(\\nu^5, \\nu^{-1},\\ldots,\\nu^{-1}))^{-1}\\varphi_{{}_{E_6, \\gamma_3, \\nu_3}}(s,P)\\varphi_{{}_{E_6,\\gamma_3}}(1,\\diag(\\nu^5, \\nu^{-1},\\ldots,\\nu^{-1}))\n\t\t\t\\\\\n\t\t\t&=\\varphi_{{}_{E_6,\\gamma_3}}(1,\\diag(\\nu^{-5}, \\nu,\\ldots,\\nu))\\varphi_{{}_{E_6, \\gamma_3}}(s,P)\\varphi_{{}_{E_6,\\gamma_3}}(1,\\diag(\\nu^5, \\nu^{-1},\\ldots,\\nu^{-1}))\n\t\t\t\\\\\n\t\t\t&=\\varphi_{{}_{E_6, \\gamma_3}}(s,\\diag(\\nu^{-5}, \\nu,\\ldots,\\nu)P\\,\\diag(\\nu^5, \\nu^{-1},\\ldots,\\nu^{-1})), P=\\scalebox{0.6}{$\n\t\t\t\t\\left( \\begin{array}{cccccccc@{\\!}}\n\t\t\t\t&\\multicolumn{2}{c}{\\raisebox{-15pt}[0pt][0pt]{\\Large$t$}}&&&&\n\t\t\t\t\\\\\n\t\t\t\t&&&&\\multicolumn{2}{c}{\\raisebox{-5pt}[0pt]{\\Large $0$}}&\n\t\t\t\t\\\\\n\t\t\t\t&&&&&&&\n\t\t\t\t\\\\\n\t\t\t\t&&&&\\multicolumn{2}{c}{\\raisebox{-18pt}[0pt][0pt]{\\huge $U$}}&\n\t\t\t\t\\\\\n\t\t\t\t&\\multicolumn{2}{c}{\\raisebox{0pt}[0pt]{\\Large $0$}}&&&&\n\t\t\t\t\\\\[-2mm]\n\t\t\t\t&&&&&&&\n\t\t\t\t\\end{array}\\right)$}\n\t\t\t\\\\\n\t\t\t&=\\varphi_{{}_{E_6, \\gamma_3}}(s,P)\n\t\t\t\\\\\n\t\t\t&=\\varphi_{{}_{E_6, \\gamma_3, \\nu_3}}(s,P)\n\t\\end{align*}\n\tHence we have that $\\varphi_{{}_{E_6, \\gamma_3, \\nu_3}}(s,P) \\in (E_6)^{\\nu_3}$. Thus $\\varphi_{{}_{E_6,\\gamma_3,\\nu_3}}$ is well-defined. Subsequently, since $\\varphi_{{}_{E_6,\\gamma_3,\\nu_3}}$ is the restriction of the mapping $\\varphi_{{}_{E_6,\\gamma_3}}$, we easily see  that $\\varphi_{{}_{E_6,\\gamma_3,\\nu_3}}$ is a homomorphism.\n\t\n\tNext, we will prove that $\\varphi_{{}_{E_6, \\gamma_3, \\nu_3}}$ is surjective. Let $\\alpha \\in  (E_6)^{\\gamma_3} \\cap (E_6)^{\\nu_3} \\subset (E_6)^{\\nu_3}$. There exist $ q \\in Sp(1)$ and $P \\in S(U(1) \\times U(5))$ such that $\\alpha=\\varphi_{{}_{E_6, \\nu_3}}(q, P)$ (Theorem \\ref{theorem 3.3.5}). Moreover, from the condition $\\alpha \\in (E_6)^{\\gamma_3}$, that is, ${\\gamma_3}^{-1}\\varphi_{{}_{E_6, \\nu_3}}(q, P)\\gamma_3=\\varphi_{{}_{E_6, \\nu_3}}(q, P)$, and note that $\\gamma_3=\\varphi_{{}_{E_6, \\nu_3}}(\\omega,E)(=\\varphi_{{}_{E_6,\\gamma_3}}(\\omega,E))$ (Lemma \\ref{lemma 3.3.8} (1)), since it follows that\n\t${\\gamma_3}^{-1}\\varphi_{{}_{E_6, \\nu_3}}(q, P)\\gamma_3=\\varphi_{{}_{E_6,\\nu_3}}(\\omega^{-1}q\\omega, P)$, we have that\n\t\\begin{align*}\n\t\t\t\t\\left\\{\n\t\t\t\t\\begin{array}{l}\n\t\t\t\t\\omega^{-1}q\\omega=q \\\\\n\t\t\t\tP=P\n\t\t\t\t\\end{array}\\right.\n\t\t\t\t\\quad {\\text{or}}\\quad\n\t\t\t\t\\left\\{\n\t\t\t\t\\begin{array}{l}\n\t\t\t\t\\omega^{-1}q\\omega=-q \\\\\n\t\t\t\tP=-P.\n\t\t\t\t\\end{array}\\right.\n\t\\end{align*}\n\tThe latter case is impossible because of $P\\not=0$. As for the former case, from the first condition, we easily see that $q \\in U(1)$, and needless to say, $P \\in S(U(1)\\times U(5))$.  Hence there exist $s \\in U(1)$ and $P \\in S(U(1)\\times U(5))$ such that $\\alpha=\\varphi_{{}_{E_6,\\nu_3}}(s,P)$. Namely, there exist $s \\in U(1)$ and $P \\in S(U(1)\\times U(5))$ such that $\\alpha=\\varphi_{{}_{E_6,\\gamma_3,\\nu_3}}(s,P)$. The proof of surjective is completed.\n\t\n\tFinally, we will determine $\\Ker\\,\\varphi_{{}_{E_6, \\gamma_3, \\nu_3}}$. However, from $\\Ker\\,\\varphi_{{}_{E_6,\\gamma_3}}=\\{(1,E),(-1,-E) \\}$, we easily obtain that $\\Ker\\,\\varphi_{{}_{E_6, \\gamma_3, \\nu_3}}=\\{(1,(1,E)),(-1,(-1,-E)) \\} \\cong \\Z_2$. Thus we have the isomorphism $(E_6)^{\\gamma_3} \\cap (E_6)^{\\nu_3} \\cong (U(1)\\times S(U(1)\\times U(5)))\/\\Z_2$.\n\n\tTherefore, by Lemma \\ref{lemma 4.6.1} we have the required isomorphism\n\t\\begin{align*}\n\t\t\t\t(E_6)^{\\gamma_3} \\cap (E_6)^{\\nu_3} \\cong (U(1)\\times U(1)\\times SU(5))\/(\\Z_2\\times \\Z_5),\n\t\\end{align*}\n\twhere \n\t\\begin{align*}\n\t\t\t&\\Z_2=\\{(1,1,E), (-1,-1,-E) \\},\n\t\t\t\\\\\n\t\t\t&\\Z_5=\\{(1,\\varepsilon_i, {\\varepsilon_i}^{-1}E) \\,|\\, \\varepsilon_i=\\exp ((2\\pi i\/5)k), k=0,1,2,3,4\\}.\n\t\\end{align*}\n\\end{proof}\n\nThus, since the group $(E_6)^{\\gamma_3} \\cap (E_6)^{\\nu_3}$ is connected from Theorem \\ref{theorem 4.6.2}, we have an exceptional $\\varmathbb{Z}_3 \\times \\varmathbb{Z}_3$-symmetric space\n\\begin{align*}\nE_6\/((U(1)\\times U(1)\\times SU(5))\/(\\Z_2\\times \\Z_5)).\n\\end{align*}\n\n\\subsection{Case 7: $\\{1, \\tilde{\\gamma}_3,  \\tilde{\\gamma}_3{}^{-1}\\} \\times \\{1, \\tilde{\\mu}_3,  \\tilde{\\mu}_3{}^{-1}\\}$-symmetric space}\n\nLet the $C$-linear transformations $\\gamma_3, \\mu_3$ of $\\mathfrak{J}^C$  defined in Subsection \\ref{subsection 3.3}. \n\n\\noindent From Lemma \\ref{lemma 3.3.8} (1), since we can easily confirm that $\\gamma_3$ and $\\mu_3$ are commutative, $\\tilde{\\gamma}_3$ and $\\tilde{\\mu}_3$ are commutative in $\\Aut(E_6)$: $\\tilde{\\gamma}_3\\tilde{\\mu}_3=\\tilde{\\mu}_3\\tilde{\\gamma}_3$.\n\\vspace{1mm}\n\nBefore determining the structure of the group $(E_6)^{\\gamma_3} \\cap (E_6)^{\\mu_3}$, we prove proposition and lemma needed in the proof of theorem below.\n\\vspace{1mm}\n\nWe define a $C$-linear transformation $\\mu'_3$ of $\\mathfrak{J}^C$ by\n\\begin{align*}\n\\mu'_3=\\varphi_{{}_{E_6,\\gamma_3}}(1,\\diag(\\nu^{-2},\\nu^2,\\nu^{-1},\\nu^{-1},\\nu,\\nu))\\in (E_6)^{\\gamma_3} \\subset E_6,\n\\end{align*}\nwhere ${\\nu}=\\exp(2\\pi i\/9)\\in C$.\n\nLet an element \n\\begin{align*}Q:=\\scalebox{0.8}{$\\begin{pmatrix}\n\t1&&&&&\\\\\n\t&1&&&&\\\\\n\t&&1&&&\\\\\n\t&&&&1&\\\\\n\t&&&-1&&\\\\\n\t&&&&&1\n\t\\end{pmatrix}$} \\in SO(6) \\subset SU(6), \n\\end{align*}\nwhere the blanks are $0$, and we consider an element $\\varphi_{{}_{E_6,\\gamma_3}}(1,Q) \\in (E_6)^{\\gamma_3} \\subset E_6$. Here, we denote this element by $\\delta_Q$: $\\delta_Q=\\varphi_{{}_{E_6,\\gamma_3}}(1,Q)$.\nThen by doing straightforward computation, we have that $\\mu_3\\delta_Q=\\delta_Q\\mu'_3$, that is, $\\mu_3$ is conjugate to $\\mu'_3$ under $\\delta_Q \\in (E_6)^{\\gamma_3} \\subset E_6$: $\\mu_3 \\sim \\mu'_3$. Moreover, $\\mu'_3$ induces the automorphism $\\tilde{\\mu'}_3$ of order $3$ on $E_6$: $\\tilde{\\mu'}_3(\\alpha)={\\mu'_3}^{-1}\\alpha\\mu'_3, \\alpha \\in E_6$.\n\\vspace{1mm}\n\nThen we have the following proposition.\n\n\\begin{proposition}\\label{proposition 4.7.1\n\tThe group $(E_6)^{\\gamma_3} \\cap (E_6)^{\\mu_3}$ is isomorphic to the group $(E_6)^{\\gamma_3} \\cap (E_6)^{\\mu'_3}${\\rm :} $(E_6)^{\\gamma_3} \\cap (E_6)^{\\mu_3} \\cong (E_6)^{\\gamma_3} \\cap (E_6)^{\\mu'_3}$.\n\\end{proposition}\n\\begin{proof}\n\tWe define a mapping $g_{{}_{471}}: (E_6)^{\\gamma_3} \\cap (E_6)^{\\mu'_3} \\to (E_6)^{\\gamma_3} \\cap (E_6)^{\\mu_3}$ by\n\t\\begin{align*}\n\tg_{{}_{471}}(\\alpha)=\\delta_Q\\alpha{\\delta_Q}^{-1}.\t\n\t\\end{align*}\n\tIn order to prove this isomorphism, it is sufficient to show that $g_{{}_{471}}$ is well-defined. \n\t\n\t\\noindent First, we will show that $g_{{}_{471}} \\in (E_6)^{\\gamma_3}$. Since it follows from $\\delta_Q=\\varphi_{{}_{E_6,\\gamma_3}}(1,Q)$ and $\\gamma_3=\\varphi_{{}_{E_6,\\gamma_3}}(\\omega,E)$ that $\\delta_Q\\gamma_3=\\gamma_3\\delta_Q$, we have that $g_{{}_{471}} \\in (E_6)^{\\gamma_3}$. Similarly, from $\\mu_3\\delta_Q=\\delta_Q\\mu'_3$ we have that $g_{{}_{471}} \\in (E_6)^{\\sigma_3}$. Hence $g_{{}_{471}}$ is well-defined. With above, the proof of this proposition is completed.\t\n\\end{proof}\n\\vspace{1mm}\n\nSubsequently, we will prove the following lemma. \n\n\\begin{lemma}\\label{lemma 4.7.2\n\tThe group $S(U(1)\\times U(1)\\times U(2)\\times U(2))$ is isomorphic to the group $(U(1) \\times U(1)\\times  U(1)\\times SU(2)\\times SU(2))\/(\\Z_2\\times\\Z_2\\times \\Z_2)${\\rm :} $S(U(1)\\times U(1)\\times U(2)\\times U(2)) \\cong (U(1) \\times U(1)\\times U(1)\\times SU(2)\\times SU(2)\\times SU(2))\/(\\Z_2\\times\\Z_2\\times  \\Z_2), \\Z_2=\\{(1,1,1,E,E), (1,-1,1,E,-E) \\}, \\Z_2=\\{(1,1,1,E,E), (1,-1,-1,-E,E) \\}, \\Z_2=\\{ (1,1,1,E,E), (-1,1,1,E,-E)\\}$.\n\\end{lemma}\n\\begin{proof}\n\tWe define a mapping $f_{{}_{472}}:U(1)\\times U(1) \\times U(1)\\times SU(2)\\times SU(2) \\to S(U(1)\\times U(1)\\times U(2)\\times U(2))$ by\n\t\\begin{align*}\n\tf_{{}_{472}}(a,b,c,A,B)=\\left( \n\t\\begin{array}{cccc}\n\ta^{-2} && &{\\raisebox{-7pt}[0pt]{\\large $0$}}\n\t\\\\[2mm]\n\t& b^{-2} &&\n\t\\\\[2mm]\n\t&& c^{-1}\\mbox{\\large {$A$}}&\n\t\\\\[2mm]\n\t{\\raisebox{3pt}[0pt]{\\large $0$}}&&&(abc)\\mbox{\\large {$B$}}\n\t\\end{array}\\right) \\in SU(6).\n\t\\end{align*}\n\tThen it is clear that $f_{{}_{472}}$ is well-defined and a homomorphism. \n\t\n\tNow, we will prove that $f_{{}_{472}}$ is surjective. Let $P \\in S(U(1)\\times U(1)\\times U(2)\\times U(2))$. Then $P$ takes the form of $\\diag(s,t,P_1,P_2),s,t \\in U(1),P_j \\in U(2), (st)(\\det\\,P_1)(\\det\\,P_2)=1$. Here, first we choose $a \\in C$ such that $s=a^{-2}$. Then it is clear that $a \\in U(1)$, so is $b \\in C$ such that $t=b^{-2}$, that is, $b \\in U(1)$. \n\tMoreover, since $P_1 \\in U(2)$, we see that $\\det\\,P_1 \\in U(1)$, and so we choose $c \\in U(1)$ such that $c^2=\\det\\,P_1$. Set $A=c^{-1}P_1$, then we have that $A \\in SU(2)$. Similarly, for $P_2 \\in U(2)$, set $B=(stc)P_2$. Since $stc=(\\det\\,P_2)^{-1}$, we have that $B \\in SU(2)$. With above, the proof of surjective is completed.\n\t\n\tFinally, we will determine $\\Ker\\,f_{{}_{472}}$. It follows from the kernel of definition that\n\t\\begin{align*}\n\t\\Ker\\,f_{{}_{472}}&=\\{(a,b,c,A,B)\\in U(1)^{\\times 3}\\times SU(2)^{\\times 2} \\,|\\,f_{{}_{472}}(a,b,c,A,B)=E \\}\n\t\\\\\n\t&=\\{(a,b,c,A,B)\\in U(1)^{\\times 3}\\times SU(2)^{\\times 2} \\,|\\,a^2=b^2=1,A=cE, B=(abc)^{-1}E \\}\n\t\\\\\n\t&=\\{(1,1,1,E,E), (1,1,-1,-E,-E),(1,-1,1,E,-E), (1,-1,-1,-E,E) \\}\n\t\\\\\n\t& \\quad \\cup \\{ (-1,1,1,E,-E), (-1,1,-1,-E,E),(-1,-1,1,E,E), (-1,-1,-1,-E,-E)\\}\n\t\\\\\n\t&=\\{(1,1,1,E,E), (1,-1,1,E,-E) \\}\\times \\{(1,1,1,E,E), (1,-1,-1,-E,E) \\}\n\t\\\\\n\t&\\quad \\times \\{(1,1,1,E,E), (-1,1,1,E,-E) \\}\n\t\\\\\n\t& \\cong \\Z_2 \\times \\Z_2 \\times\\Z_2.\n\t\\end{align*}\n\t\n\tTherefore we have the required isomorphism \n\t\\begin{align*}\n\t&\\quad S(U(1)\\times U(1)\\times U(2)\\times U(2)) \n\t\\\\\n\t&\\cong (U(1)\\times U(1) \\times U(1)\\times SU(2)\\times SU(2))\/(\\Z_2\\times\\Z_2\\times \\Z_2).\n\t\\end{align*}\n\\end{proof}\n\nNow, we will determine the structure of the group $(E_6)^{\\gamma_3} \\cap (E_6)^{\\mu_3}$.\n\n\\begin{theorem}\\label{theorem 4.7.3\n\tThe group $(E_6)^{\\gamma_3} \\cap (E_6)^{\\mu_3}$ is isomorphic the group $(U(1)\\times U(1) \\times U(1)\\allowbreak \\times U(1) \\times SU(2)\\times SU(2))\/(\\Z_2\\times\\Z_2\\times\\Z_4)${\\rm :} $(E_6)^{\\gamma_3} \\cap (E_6)^{\\mu_3} \\cong (U(1)\\times U(1) \\times U(1)\\times U(1) \\allowbreak \\times SU(2)\\times SU(2))\/(\\Z_2\\times\\Z_2\\times\\Z_4), \\Z_2=\\{(1,1,1,1,E,E), (1,1,-1,1,E,-E) \\},\n\t\\Z_2=\\{(1,1,1,1,E,E), (1,1,-1,-1,-E,E) \\},\n\t\\Z_4=\\{(1,1,1,E,E,E), (1,-1,1-,1,E,E), (-1,i,i,\\allowbreak 1,-E,E), (-1,-i,-i,1,-E,E) \\}$.\n\\end{theorem}\n\\begin{proof}\n\tLet $S(U(1)\\times U(1)\\times U(2)\\times U(2)) \\subset SU(6)$. \n\tWe define a mapping $\\varphi_{{}_{E_6,\\gamma_3,\\mu'_3}}: U(1)\\times S(U(1)\\times U(1)\\times U(2)\\times U(2)) \\to (E_6)^{\\gamma_3} \\cap (E_6)^{\\mu'_3}$ by\n\t\\begin{align*}\n\t\\varphi_{{}_{E_6,\\gamma_3,\\mu'_3}}(s, P)(M+\\a)&={k_J}^{-1}(P(k_J M){}^t\\!P)+s\\a k^{-1}(\\tau \\,{}^t\\!P), \n\t\\\\\n\t&\\hspace*{40mm}M+\\a \\in \\mathfrak{J}(3, \\H)^C \\oplus (\\H^3)^C=\\mathfrak{J}^C.\n\t\\end{align*}\n\tNeedless to say, this mapping is the restriction of the mapping $\\varphi_{{}_{E_6,\\gamma_3}}$, that is, $\\varphi_{{}_{E_6,\\gamma_3,\\mu'_3}}(s, P)=\\varphi_{{}_{E_6,\\gamma_3}}(s,P)$ (Theorem \\ref{theorem 3.3.2}). \n\t\n\tAs usual, we will prove that $\\varphi_{{}_{E_6,\\gamma_3,\\mu'_3}}$ is well-defined. It is clear that $\\varphi_{{}_{E_6,\\gamma_3,\\mu'_3}}(s,P) \\in (E_6)^{\\gamma_3}$, and it follows from $\\mu'_3=\\varphi_{{}_{E_6,\\gamma_3}}(1,\\diag({\\nu}^{-2},\\nu^2,{\\nu}^{-1},{\\nu}^{-1},\\nu,\\nu))$ that\n\t\\begin{align*}\n\t&\\quad {\\mu'_3}^{-1}\\varphi_{{}_{E_6,\\gamma_3,\\nu'_3}}(s,P)\\mu'_3\n\t\\\\\n\t&=\\varphi_{{}_{E_6,\\gamma_3}}(1,\\diag({\\nu}^{-2},\\nu^2,{\\nu}^{-1},{\\nu}^{-1},\\nu,\\nu))^{-1}\\varphi_{{}_{E_6,\\gamma_3,\\mu'_3}}(s,P)\\varphi_{{}_{E_6,\\gamma_3}}(1,\\diag({\\nu}^{-2},\\nu^2,{\\nu}^{-1},{\\nu}^{-1},\\nu,\\nu))\n\t\\\\\n\t&=\\varphi_{{}_{E_6,\\gamma_3}}(1,\\diag({\\nu}^2,\\nu^{-2},\\nu,\\nu,{\\nu}^{-1},{\\nu}^{-1}))\\varphi_{{}_{E_6,\\gamma_3}}(s,P)\\varphi_{{}_{E_6,\\gamma_3}}(1,\\diag({\\nu}^{-2},\\nu^2,{\\nu}^{-1},{\\nu}^{-1},\\nu,\\nu))\n\t\\\\\n\t&=\\varphi_{{}_{E_6,\\gamma_3}}(s,\\diag({\\nu}^2,\\nu^{-2},\\nu,\\nu,{\\nu}^{-1},{\\nu}^{-1})P\\diag({\\nu}^{-2},\\nu^2,{\\nu}^{-1},{\\nu}^{-1},\\nu,\\nu)),P=\\diag(a,b,P_1,P_2)\n\t\\\\\n\t&=\\varphi_{{}_{E_6,\\gamma_3}}(s,\\diag(\\nu^2 a \\nu^{-2}, {\\nu}^{-2} b \\nu^2, (\\nu E) P_1(\\nu^{-1}E), ({\\nu}^{-1}E) P_2 (\\nu E) ))\n\t\\\\\n\t&=\\varphi_{{}_{E_6,\\gamma_3}}(s,P)\n\t\\\\\n\t&=\\varphi_{{}_{E_6,\\gamma_3,\\mu'_3}}(s,P).\n\t\\end{align*}\n\tHence we have that $\\varphi_{{}_{E_6,\\gamma_3,\\mu'_3}}(s,P) \\in (E_6)^{\\sigma'_3}$. Thus $\\varphi_{{}_{E_6,\\gamma_3,\\mu'_3}}$ is well-defined. Subsequently, since $\\varphi_{{}_{E_6,\\gamma_3,\\mu'_3}}$ is the restriction of the mapping $\\varphi_{{}_{E_6,\\gamma_3}}$, we easily see that $\\varphi_{{}_{E_6,\\gamma_3,\\mu'_3}}$ is a homomorphism.\n\t\n\tNext, we will prove that $\\varphi_{{}_{E_6,\\gamma_3,\\mu'_3}}$ is surjective. Let $\\alpha \\in (E_6)^{\\gamma_3} \\cap (E_6)^{\\mu'_3} \\subset (E_6)^{\\gamma_3}$. There exist $s \\in U(1)$ and $A \\in SU(6)$ such that $\\alpha=\\varphi_{{}_{E_6,\\gamma_3}}(s,A)$ (Theorem \\ref{theorem 3.3.2}). Moreover, from the condition $\\alpha \\in (E_6)^{\\mu'_3}$, that is, ${\\mu'_3}^{-1}\\varphi_{{}_{E_6,\\gamma_3}}(s,A)\\mu'_3=\\varphi_{{}_{E_6,\\gamma_3}}(s,A)$, and using ${\\mu'_3}^{-1}\\varphi_{{}_{E_6,\\gamma_3}}(s,A)\\mu'_3=\\varphi_{{}_{E_6,\\gamma_3}}(s,\\diag({\\nu}^2,\\nu^{-2},\\nu,\\nu,{\\nu}^{-1},{\\nu}^{-1}) A \\,\\diag({\\nu}^{-2},\\nu^2,{\\nu}^{-1},{\\nu}^{-1},\\nu,\\nu))$, we have that\n\t\\begin{align*}\n\t&\\left\\{\n\t\\begin{array}{l}\n\ts=s \\\\\n\t\\diag({\\nu}^2,\\nu^{-2},\\nu,\\nu,{\\nu}^{-1},{\\nu}^{-1}) A \\diag({\\nu}^{-2},\\nu^2,{\\nu}^{-1},{\\nu}^{-1},\\nu,\\nu)=A   \n\t\\end{array}\\right. \n\t\\\\\n\t&\\hspace*{50mm}{\\text{or}}\n\t\\\\\n\t&\\left\\{\n\t\\begin{array}{l}\n\ts=-s \\\\\n\t\\diag({\\nu}^2,\\nu^{-2},\\nu,\\nu,{\\nu}^{-1},{\\nu}^{-1}) A \\diag({\\nu}^{-2},\\nu^2,{\\nu}^{-1},{\\nu}^{-1},\\nu,\\nu)=-A.   \n\t\\end{array}\\right.     \n\t\\end{align*}\n\tThe latter case is impossible because of $s\\not=0$. As for the former case, from the second condition, by doing straightforward computation $A$ takes the following form $\\diag(a, b, C, D), a,b \\in U(1),C, D \\in U(2),  (ab)(\\det\\,C)(\\det\\,D)=1$, that is, $A \\in S(U(1)\\times U(1)\\times  U(2)\\times U(2))$. Needless to say, $s \\in U(1)$.\n\tHence there exist $s \\in U(1)$ and $P \\in S(U(1)\\times U(1)\\times U(2)\\times U(2))$ such that $\\alpha=\\varphi_{{}_{E_6,\\gamma_3}}(s,P)$. Namely, there exist $s \\in U(1)$ and $P \\in S(U(1)\\times U(1)\\times U(2)\\times U(2))$ such that $\\alpha=\\varphi_{{}_{E_6,\\gamma_3,\\mu'_3}}(s,P)$. With above, the proof of surjective is completed.\n\t\n\tFinally, we will determine $\\Ker\\,\\varphi_{{}_{E_6,\\gamma_3,\\mu'_3}}$. However, from $\\Ker\\,\\varphi_{{}_{E_6,\\gamma_3}}=\\{(1,E),(-1,-E) \\}$, we easily obtain that $\\Ker\\,\\varphi_{{}_{E_6,\\gamma_3,\\mu'_3}}=\\{(1,E),(-1,-E) \\} \\cong \\Z_2$. Thus we have the isomorphism $(E_6)^{\\gamma_3} \\cap (E_6)^{\\mu'_3} \\cong (U(1)\\times S(U(1)\\times U(1)\\times U(2)\\times U(2)))\/\\Z_2$. In addition, from Proposition \\ref{proposition 4.7.1} we have the isomorphism $(E_6)^{\\gamma_3} \\cap (E_6)^{\\mu_3} \\cong (U(1)\\times S(U(1)\\times U(1)\\times U(2)\\times U(2)))\/\\Z_2$. Here, using the mapping $f_{{}_{472}}$ in the proof of Lemma \\ref{lemma 4.7.2}, we define a homomorphism $h_{{}_{473}}:U(1)\\times (U(1)\\times U(1)\\times U(1)\\times SU(2)\\times SU(2)) \\to U(1)\\times S(U(1)\\times U(1)\\times U(2)\\times U(2))$ by\n\t\\begin{align*}\n\t\t\th_{{}_{473}}(s,(a,b,c,A,B))=(s,f_{{}_{472}}(a,b,c,A,B)).\n\t\\end{align*}\n\tThen, the elements $(s,(a,b,c,A,B))$ corresponding to the elements $(1,E), (-1,-E) \\in \\Ker\\,\\varphi_{{}_{E_6,\\gamma_3,\\mu'_3}}$ under the mapping $h_{{}_{473}}$ are as follows.\n\t\\begin{align*}\n\t&(1,(1,1,1,E,E)),(1,(1,1,-1,-E,-E)),(1,(1,-1,1,E,-E)),(1,(1,-1,-1,-E,E)),\n\t\\\\\n\t&(1,(-1,1,1,E,-E)),(1,(-1,1,-1,-E,E)),(1,(-1,-1,1,E,E)),(1,(-1,-1,-1,-E,-E)),\n\t\\\\\n\t&(-1,(i,i,1,-E,E)),(-1,(i,i,-1,E,-E)),(-1,(i,-i,1,-E,-E)),(-1,(i,i,-1,-E,E)),\n\t\\\\\n\t&(-1,(-i,i,1,-E,\\!E)),(-1,(-i,i,-1,E,\\!E)),(-1,(-i,-i,1,-E,E)),(-1,(-i,-i,-1,E,-E)).\n\t\\end{align*}\n\t\n\tTherefore we have the required isomorphism \n\t\\begin{align*}\n\t(E_6)^{\\gamma_3} \\cap (E_6)^{\\mu_3} \\!\\cong (U(1)\\times U(1) \\times U(1)\\times U(1) \\allowbreak \\times SU(2)\\times SU(2))\/(\\Z_2\\times\\Z_2\\times\\Z_4), \n\t\\end{align*}\n\twhere \n\t\\begin{align*}\n\t&\\Z_2=\\{(1,1,1,1,E,E), (1,1,-1,1,E,-E) \\},\n\t\\\\\n\t&\\Z_2=\\{(1,1,1,1,E,E), (1,1,-1,-1,-E,E) \\},\n\t\\\\\n\t&\\Z_4=\\{(1,1,1,E,E,E), (1,-1,1-,1,E,E), (-1,i,i,1,-E,E), (-1,-i,-i,1,-E,E) \\}.\n\t\\end{align*}\n\\end{proof}\n\nThus, since the group $(E_6)^{\\gamma_3} \\cap (E_6)^{\\mu_3}$ is connected from Theorem \\ref{theorem 4.7.3}, we have an exceptional $\\varmathbb{Z}_3 \\times \\varmathbb{Z}_3$-symmetric space\n\\begin{align*}\nE_6\/((U(1)\\times U(1) \\times U(1)\\times U(1) \\allowbreak \\times SU(2)\\times SU(2))\/(\\Z_2\\times\\Z_2\\times\\Z_4)).\n\\end{align*}\n\n\\subsection{Case 8: $\\{1, \\tilde{\\gamma}_3,  \\tilde{\\gamma}_3{}^{-1}\\} \\times \\{1, \\tilde{w}_3,  \\tilde{w}_3{}^{-1}\\}$-symmetric space}\n\nLet the $C$-linear transformations $\\gamma_3, w_3$ of $\\mathfrak{J}^C$  defined in Subsection \\ref{subsection 3.3}. \n\n\\noindent From Lemma \\ref{lemma 3.3.8} (1), since we can easily confirm that $\\gamma_3$ and $w_3$ are commutative, $\\tilde{\\gamma}_3$ and $\\tilde{w}_3$ are commutative in $\\Aut(E_6)$: $\\tilde{\\gamma}_3\\tilde{w}_3=\\tilde{w}_3\\tilde{\\gamma}_3$.\n\\vspace{1mm}\n\nBefore determining the structure of the group $(E_6)^{\\gamma_3} \\cap (E_6)^{w_3}$, we prove proposition and lemma needed in the proof of theorem below.\n\\vspace{1mm}\n\nWe define a $C$-linear transformation $w'_3$ of $\\mathfrak{J}^C$ by\n\\begin{align*}\n\t\tw'_3=\\varphi_{{}_{E_6,\\gamma_3}}(1,\\diag(\\tau\\omega,\\tau\\omega,\\tau\\omega,\\omega,\\omega,\\omega)) \\in (E_6)^{\\gamma_3} \\subset E_6.\n\\end{align*}\n\nLet an element \n\\begin{align*}N:=\\scalebox{0.8}{$\\begin{pmatrix}\n\t1&&&&&\\\\\n\t&&&&1&\\\\\n\t&&1&&&\\\\\n\t&&&1&&\\\\\n\t&-1&&&&\\\\\n\t&&&&&1\n\t\\end{pmatrix}$} \\in SO(6) \\subset SU(6), \n\\end{align*}\nwhere the blanks are $0$, and we consider an element $\\varphi_{{}_{E_6,\\gamma_3}}(1,N) \\in (E_6)^{\\gamma_3} \\subset E_6$. Here, we denote this element by $\\delta_N$: $\\delta_N=\\varphi_{{}_{E_6,\\gamma_3}}(1,N)$.\nThen by doing straightforward computation, we have that $w_3\\delta_Q=\\delta_Q w'_3$, that is, $w_3$ is conjugate to $w'_3$ under $\\delta_N \\in (E_6)^{\\gamma_3} \\subset E_6$: $w_3 \\sim w'_3$. Moreover, $w'_3$ induces the automorphism $\\tilde{w'}_3$ of order $3$ on $E_6$: $\\tilde{w'}_3(\\alpha)={w'_3}^{-1}\\alpha w'_3, \\alpha \\in E_6$.\n\\vspace{1mm}\n\nThen we have the following proposition.\n\n\\begin{proposition}\\label{proposition 4.8.1\n\tThe group $(E_6)^{\\gamma_3} \\cap (E_6)^{w_3}$ is isomorphic to the group $(E_6)^{\\gamma_3} \\cap (E_6)^{w'_3}${\\rm :} $(E_6)^{\\gamma_3} \\cap (E_6)^{w_3} \\cong (E_6)^{\\gamma_3} \\cap (E_6)^{w'_3}$.\n\\end{proposition}\n\\begin{proof}\n\tWe define a mapping $g_{{}_{481}}: (E_6)^{\\gamma_3} \\cap (E_6)^{w'_3} \\to (E_6)^{\\gamma_3} \\cap (E_6)^{w_3}$ by\n\t\\begin{align*}\n\tg_{{}_{481}}(\\alpha)=\\delta_N\\alpha{\\delta_N}^{-1}.\t\n\t\\end{align*}\n\tIn order to prove this isomorphism, it is sufficient to show that $g_{{}_{481}}$ is well-defined. \n\t\n\t\\noindent First, we will show that $g_{{}_{481}} \\in (E_6)^{\\gamma_3}$. Since it follows from $\\delta_N=\\varphi_{{}_{E_6,\\gamma_3}}(1,N)$ and $\\gamma_3=\\varphi_{{}_{E_6,\\gamma_3}}(\\omega,E)$ that $\\delta_N\\gamma_3=\\gamma_3\\delta_N$, we have that $g_{{}_{481}} \\in (E_6)^{\\gamma_3}$. Similarly, from $w_3\\delta_N=\\delta_N w'_3$ we have that $g_{{}_{481}} \\in (E_6)^{w_3}$. Hence $g_{{}_{481}}$ is well-defined. With above, the proof of this proposition is completed.\t\n\\end{proof}\n\nSubsequently, we will prove the following lemma. \n\n\\begin{lemma}\\label{lemma 4.8.2\n\tThe group $S(U(3)\\times U(3))$ is isomorphic to the group $(U(1) \\times SU(3)\\times SU(3))\/\\Z_3${\\rm :} $S(U(3)\\times U(3)) \\cong (U(1) \\times SU(3)\\times SU(3))\/\\Z_3, \\Z_3=\\{(1,E,E), (\\omega,{\\omega}^{-1} E,\\omega E),\\allowbreak (\\omega,\\omega E,{\\omega}^{-1} E)\\}$, where $\\omega=(-1\/2)+(\\sqrt{3}\/2)i \\in C$.\n\\end{lemma}\n\\begin{proof}\n\tWe define a mapping $f_{{}_{482}}:U(1)\\times SU(3)\\times SU(3) \\to S(U(3)\\times U(3))$ by\n\t\\begin{align*}\n\tf_{{}_{482}}(a,A,B)=\\left( \n\t\\begin{array}{cc}\n\taA &{\\raisebox{-5pt}[0pt]{\\large $0$}}\n\t\\\\[4mm]\n\t{\\raisebox{1pt}[0pt]{\\large $0$}}& a^{-1}B\n\t\\end{array}\\right) \\in SU(6).\n\t\\end{align*}\n\tThen it is clear that $f_{{}_{482}}$ is well-defined and a homomorphism. \n\t\n\tNow, we will prove that $f_{{}_{482}}$ is surjective. Let $P \\in S(U(3)\\times U(3))$. Then $P$ takes the form of $\\diag(P_1,P_2),P_j \\in U(3), (\\det\\,P_1)(\\det\\,P_2)=1$. Here, since $P_1 \\in U(3)$, we see that $\\det\\,P_1 \\in U(1)$, and so we choose $a \\in U(1)$ such that $a^3=\\det\\,P_1$. Set $A=a^{-1}P_1$, then we have that $A \\in SU(3)$. Similarly, for $P_2 \\in U(2)$, set $B=aP_2$, we have that $B \\in SU(3)$. With above, the proof of surjective is completed.\n\t\n\tFinally, we will determine $\\Ker\\,f_{{}_{482}}$. It follows from the kernel of definition that\n\t\\begin{align*}\n\t\\Ker\\,f_{{}_{482}}&=\\{(a,A,B)\\in U(1)\\times SU(3)\\times SU(3) \\,|\\,f_{{}_{482}}(a,A,B)=E \\}\n\t\\\\\n\t&=\\{(a,A,B)\\in U(1)\\times SU(3)\\times SU(3) \\,|\\,a^3=1,A=a^{-1}E, B=aE \\}\n\t\\\\\n\t&=\\{(1,E,E), (\\omega,{\\omega}^{-1}E,\\omega E),({\\omega}^{-1},\\omega E,{\\omega}^{-1}E) \\}\n\t\\\\\n\t& \\cong \\Z_3.\n\t\\end{align*}\n\t\n\tTherefore we have the required isomorphism \n\t\\begin{align*}\n\tS(U(3)\\times U(3)) \\cong (U(1) \\times SU(3)\\times SU(3))\/\\Z_3.\n\t\\end{align*}\n\\end{proof}\n\nNow, we will determine the structure of the group $(E_6)^{\\gamma_3} \\cap (E_6)^{w_3}$.\n\n\\begin{theorem}\\label{theorem 4.8.3\n\tThe group $(E_6)^{\\gamma_3} \\cap (E_6)^{w_3}$ is isomorphic the group $(U(1) \\times U(1) \\times SU(3)\\times SU(3)))\/(\\Z_2 \\times \\Z_3)${\\rm :} $(E_6)^{\\gamma_3} \\cap (E_6)^{w_3} \\cong ((U(1) \\times U(1) \\times SU(3)\\times SU(3)))\/(\\Z_2 \\times \\Z_3), \\Z_2=\\{(1,1,E,E), (-1,-1,E,E)\\},\n\t\\Z_3=\\{(1,1,E,E), (1,\\omega,{\\omega}^{-1}E,\\omega E),(1,{\\omega}^{-1},\\omega E,{\\omega}^{-1}E)\\}$.\n\\end{theorem}\n\\begin{proof}\n\tLet $S(U(3)\\times U(3)) \\subset SU(6)$. \n\tWe define a mapping $\\varphi_{{}_{E_6,\\gamma_3,w'_3}}: U(1)\\times S(U(3)\\times U(3)) \\to (E_6)^{\\gamma_3} \\cap (E_6)^{w'_3}$ by\n\t\\begin{align*}\n\t\\varphi_{{}_{E_6,\\gamma_3,w'_3}}(s, P)(M+\\a)&={k_J}^{-1}(P(k_J M){}^t\\!P)+s\\a k^{-1}(\\tau \\,{}^t\\!P), \n\t\\\\\n\t&\\hspace*{40mm}M+\\a \\in \\mathfrak{J}(3, \\H)^C \\oplus (\\H^3)^C=\\mathfrak{J}^C.\n\t\\end{align*}\n\tNeedless to say, this mapping is the restriction of the mapping $\\varphi_{{}_{E_6,\\gamma_3}}$, that is, $\\varphi_{{}_{E_6,\\gamma_3,w'_3}}(s, P)=\\varphi_{{}_{E_6,\\gamma_3}}(s,P)$ (Theorem \\ref{theorem 3.3.2}). \n\t\n\tFirst, we will prove that $\\varphi_{{}_{E_6,\\gamma_3,w'_3}}$ is well-defined. It is clear that $\\varphi_{{}_{E_6,\\gamma_3,w'_3}}(s,P) \\in (E_6)^{\\gamma_3}$, and it follows from $w'_3=\\varphi_{{}_{E_6,\\gamma_3}}(1,\\diag(\\tau\\omega,\\tau\\omega,\\tau\\omega,\\omega,\\omega,\\omega))$ that\n\t\\begin{align*}\n\t&\\quad {w'_3}^{-1}\\varphi_{{}_{E_6,\\gamma_3,\\nu'_3}}(s,P) w'_3\n\t\\\\\n\t&=\\varphi_{{}_{E_6,\\gamma_3}}(1,\\diag(\\tau\\omega,\\tau\\omega,\\tau\\omega,\\omega,\\omega,\\omega))^{-1}\\varphi_{{}_{E_6,\\gamma_3,w'_3}}(s,P)\\varphi_{{}_{E_6,\\gamma_3}}(1,\\diag(\\tau\\omega,\\tau\\omega,\\tau\\omega,\\omega,\\omega,\\omega))\n\t\\\\\n\t&=\\varphi_{{}_{E_6,\\gamma_3}}(1,\\diag(\\omega,\\omega,\\omega,\\tau\\omega,\\tau\\omega,\\tau\\omega))\\varphi_{{}_{E_6,\\gamma_3}}(s,P)\\varphi_{{}_{E_6,\\gamma_3}}(1,\\diag(\\tau\\omega,\\tau\\omega,\\tau\\omega,\\omega,\\omega,\\omega))\n\t\\\\\n\t&=\\varphi_{{}_{E_6,\\gamma_3}}(s,\\diag(\\omega,\\omega,\\omega,\\tau\\omega,\\tau\\omega,\\tau\\omega)P\\,\\diag(\\tau\\omega,\\tau\\omega,\\tau\\omega,\\omega,\\omega,\\omega)),P=\\diag(P_1,P_2)\n\t\\\\\n\t&=\\varphi_{{}_{E_6,\\gamma_3}}(s,\\diag((\\omega E)P_1(\\tau\\omega E), \\tau(\\omega E) P_2 (\\omega E) ))\n\t\\\\\n\t&=\\varphi_{{}_{E_6,\\gamma_3}}(s,P)\n\t\\\\\n\t&=\\varphi_{{}_{E_6,\\gamma_3,w'_3}}(s,P).\n\t\\end{align*}\n\tHence we have that $\\varphi_{{}_{E_6,\\gamma_3,w'_3}}(s,P) \\in (E_6)^{w'_3}$. Thus $\\varphi_{{}_{E_6,\\gamma_3,w'_3}}$ is well-defined. Subsequently, since $\\varphi_{{}_{E_6,\\gamma_3,w'_3}}$ is the restriction of the mapping $\\varphi_{{}_{E_6,\\gamma_3}}$, we easily see that $\\varphi_{{}_{E_6,\\gamma_3,w'_3}}$ is a homomorphism.\n\t\n\tNext, we will prove that $\\varphi_{{}_{E_6,\\gamma_3,w'_3}}$ is surjective. Let $\\alpha \\in (E_6)^{\\gamma_3} \\cap (E_6)^{w'_3} \\subset (E_6)^{\\gamma_3}$. There exist $s \\in U(1)$ and $A \\in SU(6)$ such that $\\alpha=\\varphi_{{}_{E_6,\\gamma_3}}(s,A)$ (Theorem \\ref{theorem 3.3.2}). Moreover, from the condition $\\alpha \\in (E_6)^{w'_3}$, that is, ${w'_3}^{-1}\\varphi_{{}_{E_6,\\gamma_3}}(s,A)w'_3=\\varphi_{{}_{E_6,\\gamma_3}}(s,A)$, and using ${w'_3}^{-1}\\varphi_{{}_{E_6,\\gamma_3}}(s,A)w'_3=\\varphi_{{}_{E_6,\\gamma_3}}(s,\\diag(\\omega,\\omega,\\omega,\\tau\\omega,\\tau\\omega,\\tau\\omega)A\\,\\diag(\\tau\\omega,\\tau\\omega,\\tau\\omega,\\omega,\\omega,\\omega))$, we have that\n\t\\begin{align*}\n\t&\\left\\{\n\t\\begin{array}{l}\n\ts=s \\\\\n\t\\diag(\\omega,\\omega,\\omega,\\tau\\omega,\\tau\\omega,\\tau\\omega)A\\,\\diag(\\tau\\omega,\\tau\\omega,\\tau\\omega,\\omega,\\omega,\\omega)=A   \n\t\\end{array}\\right. \n\t\\\\\n\t&\\hspace*{50mm}{\\text{or}}\n\t\\\\\n\t&\\left\\{\n\t\\begin{array}{l}\n\ts=-s \\\\\n\t\\diag(\\omega,\\omega,\\omega,\\tau\\omega,\\tau\\omega,\\tau\\omega)A\\,\\diag(\\tau\\omega,\\tau\\omega,\\tau\\omega,\\omega,\\omega,\\omega)=-A.   \n\t\\end{array}\\right.     \n\t\\end{align*}\n\tThe latter case is impossible because of $s\\not=0$. As for the former case, from the second condition, by doing straightforward computation $A$ takes the following form $\\diag(C, D), C, D \\in U(2), (\\det\\,C)(\\det\\,D)=1$, that is, $A \\in S(U(3)\\times U(3))$. Needless to say, $s \\in U(1)$.\n\tHence there exist $s \\in U(1)$ and $P \\in S(U(3)\\times U(3))$ such that $\\alpha=\\varphi_{{}_{E_6,\\gamma_3}}(s,P)$. Namely, there exist $s \\in U(1)$ and $P \\in S(U(3)\\times U(3))$ such that $\\alpha=\\varphi_{{}_{E_6,\\gamma_3,w'_3}}(s,P)$. The proof of surjective is completed.\n\t\n\tFinally, we will determine $\\Ker\\,\\varphi_{{}_{E_6,\\gamma_3,w'_3}}$. However, from $\\Ker\\,\\varphi_{{}_{E_6,\\gamma_3}}=\\{(1,E),(-1,-E) \\}$, we easily obtain that $\\Ker\\,\\varphi_{{}_{E_6,\\gamma_3,w'_3}}=\\{(1,E),(-1,-E) \\} \\cong \\Z_2$. Thus we have the isomorphism $(E_6)^{\\gamma_3} \\cap (E_6)^{w'_3} \\cong (U(1)\\times S(U(3)\\times U(3)))\/\\Z_2$. In addition, from Proposition \\ref{proposition 4.8.1} we have the isomorphism $(E_6)^{\\gamma_3} \\cap (E_6)^{w_3} \\cong (U(1)\\times S(U(3)\\times U(3)))\/\\Z_2$. Here, using the mapping $f_{{}_{482}}$ in the proof of Lemma \\ref{lemma 4.8.2}, we define a homomorphism $h_{{}_{484}}:U(1)\\times (U(1)\\times SU(3)\\times SU(3)) \\to U(1)\\times S(U(3)\\times U(3))$ by\n\t\\begin{align*}\n\th_{{}_{483}}(s,(a,A,B))=(s,f_{{}_{482}}(a,A,B)).\n\t\\end{align*}\n\tThen, the elements $(s,(a,A,B))$ corresponding to the elements $(1,E), (-1,-E) \\in \\\\\n\t\\Ker\\,\\varphi_{{}_{E_6,\\gamma_3,w'_3}}$ under the mapping $h_{{}_{483}}$ are as follows.\n\t\\begin{align*}\n\t& (1,1,E,E), (1,\\omega,{\\omega}^{-1}E,\\omega E),(1,{\\omega}^{-1},\\omega E,{\\omega}^{-1}E) \n\t\\\\\n\t& (-1,-1,E,E), (-1,-\\omega,{\\omega}^{-1}E,\\omega E),(-1,-{\\omega}^{-1},\\omega E,{\\omega}^{-1}E).\n\t\\end{align*}\n\tTherefore we have the required isomorphism\n\t\\begin{align*}\n\t  (E_6)^{\\gamma_3} \\cap (E_6)^{w_3} \\cong (U(1) \\times U(1) \\times SU(3)\\times SU(3))\/(\\Z_2 \\times \\Z_3), \n\t \\end{align*}\n\t where\n\t \\begin{align*}\n\t   &\\Z_2=\\{(1,1,E,E), (-1,-1,E,E)\\},\n\t   \\\\\n\t   &\\Z_3=\\{(1,1,E,E), (1,\\omega,{\\omega}^{-1}E,\\omega E),(1,{\\omega}^{-1},\\omega E,{\\omega}^{-1}E)\\}.\n\t \\end{align*}\n\\end{proof}\n\nThus, since the group $(E_6)^{\\gamma_3} \\cap (E_6)^{w_3}$ is connected from Theorem \\ref{theorem 4.8.3}, we have an exceptional $\\varmathbb{Z}_3 \\times \\varmathbb{Z}_3$-symmetric space\n\\begin{align*}\nE_6\/((U(1) \\times U(1) \\times SU(3)\\times SU(3))\/(\\Z_2 \\times \\Z_3)).\n\\end{align*}\n\n\\subsection{Case 9: $\\{1, \\tilde{\\sigma}_3,  \\tilde{\\sigma}_3{}^{-1}\\} \\times \\{1, \\tilde{\\nu}_3,  \\tilde{\\nu}_3{}^{-1}\\}$-symmetric space}\n\nLet the $C$-linear transformations $\\sigma_3, \\nu_3$ of $\\mathfrak{J}^C$  defined in Subsection \\ref{subsection 3.3}. \n\n\\noindent From Lemma \\ref{lemma 3.3.8} (1), since we can easily confirm that $\\sigma_3$ and $\\nu_3$ are commutative, $\\tilde{\\sigma}_3$ and $\\tilde{\\nu}_3$ are commutative in $\\Aut(E_6)$: $\\tilde{\\sigma}_3\\tilde{\\nu}_3=\\tilde{\\nu}_3\\tilde{\\sigma}_3$.\n\nBefore determining the structure of the group $(E_6)^{\\sigma_3} \\cap (E_6)^{\\nu_3}$, we confirm that useful lemma holds and prove proposition needed in the proof of theorem below.\n\n\\begin{lemma}\\label{lemma 4.9.1\n\tThe mapping $\\varphi_{{}_{E_6,\\nu_3}}:Sp(1) \\times S(U(1)\\times U(5)) \\to (E_6)^{\\nu_3}$ of \\,Theorem {\\rm \\ref{theorem 3.3.5}} satisfies the relational formulas \n\t\\begin{align*}\n\t\\sigma_3&=\\varphi_{{}_{E_6,\\nu_3}}(1, \\diag(1,1,\\tau\\omega,\\omega,\\omega,\\tau\\omega)),\n\t\\\\\n\t\\nu_3&=\\varphi_{{}_{E_6,\\nu_3}}(1,\\diag(\\nu^5,\\nu^{-1},\\nu^{-1},\\nu^{-1},\\nu^{-1},\\nu^{-1})),\n\t\\end{align*}\n\twhere ${\\omega}=-(1\/2)+(\\sqrt{3}\/2)i \\in U(1)$.\n\\end{lemma}\n\\begin{proof}\n\tFrom Lemma \\ref{lemma 3.3.8} (1), these results are trivial. \n\\end{proof}\n\nThe $C$-linear transformation $\\sigma'_3$ defined in the Case 5 is expressed by\n\\begin{align*}\n\\sigma'_3=\\varphi_{{}_{E_6,\\nu_3}}(1, \\diag(1,1,\\omega,\\omega,\\tau\\omega,\\tau\\omega)),\n\\end{align*}\nand note that $\\delta_R=\\varphi_{{}_{E_6, \\nu_3}}(1,R)(=\\varphi_{{}_{E_6, \\gamma_3}}(1,R))$, where $\\delta_R$ is also defined in the Case 5, moreover needless to say, $\\sigma_3$ is conjugate to $\\sigma'_3$ under $\\delta_R=\\varphi_{{}_{E_6, \\nu_3}}(1,R)$.\n\n\\begin{proposition}\\label{proposition 4.9.2\n\tThe group $(E_6)^{\\sigma_3} \\cap (E_6)^{\\nu_3}$ is isomorphic to the group $(E_6)^{\\sigma'_3} \\cap (E_6)^{\\nu_3}${\\rm :} $(E_6)^{\\sigma_3} \\cap (E_6)^{\\nu_3} \\cong (E_6)^{\\sigma'_3} \\cap (E_6)^{\\nu_3}$.\n\\end{proposition}\n\\begin{proof}\n\tWe define a mapping $g_{{}_{492}}: (E_6)^{\\sigma_3} \\cap (E_6)^{\\nu_3} \\to (E_6)^{\\sigma'_3} \\cap (E_6)^{\\nu_3}$ by\n\t\\begin{align*}\n\tg_{{}_{492}}(\\alpha)={\\delta_R}^{-1}\\alpha\\delta_R,\n\t\\end{align*}\n\twhere $\\delta_R$ is same one above. Since it is easy to verify that $\\delta_R\\nu_3=\\nu_3\\delta_R$ using $\\nu_3=\\varphi_{{}_{E_6,\\nu_3}}(1,\\diag(\\nu^5,\\nu^{-1},\\nu^{-1},\\nu^{-1},\\nu^{-1},\\nu^{-1}))$ (Lemma \\ref{lemma 4.9.1}), we can prove this proposition as in the proof of Proposition \\ref{proposition 4.5.1}\n\\end{proof}\n\nNow, we will determine the structure of the group $(E_6)^{\\sigma_3} \\cap (E_6)^{\\nu_3}$.\n\n\\begin{theorem}\\label{theorem 4.9.3\n\tThe group $(E_6)^{\\sigma_3} \\cap (E_6)^{\\nu_3}$ is isomorphic the group $(Sp(1)\\times U(1) \\times U(1)\\allowbreak \\times U(1) \\times SU(2)\\times SU(2))\/(\\Z_2\\times\\Z_2\\times\\Z_4)${\\rm :} $(E_6)^{\\sigma_3} \\cap (E_6)^{\\nu_3} \\cong (Sp(1)\\times U(1) \\times U(1)\\times U(1) \\allowbreak \\times SU(2)\\times SU(2))\/(\\Z_2\\times\\Z_2\\times\\Z_4), \\Z_2=\\{(1,1,1,1,E,E), (1,1,-1,1,E,-E) \\},\n\t\\Z_2=\\{(1,1,1,1,E,E), (1,1,-1,-1,-E,E) \\},\n\t\\Z_4=\\{(1,1,1,E,E,E), (1,-1,1-,1,E,E), (-1,i,i,\\allowbreak 1,-E,E), (-1,-i,-i,1,-E,E) \\}$.\n\\end{theorem}\n\\begin{proof}\n\tLet $S(U(1)\\times U(1)\\times U(2)\\times U(2)) \\subset S(U(1)\\times U(5))$ as in the proof of Theorem \\ref{theorem 4.7.3}. \n\tWe define a mapping $\\varphi_{{}_{E_6,\\sigma'_3,\\nu_3}}: Sp(1)\\times S(U(1)\\times U(1)\\times U(2)\\times U(2)) \\to (E_6)^{\\sigma'_3} \\cap (E_6)^{\\nu_3}$ by\n\t\\begin{align*}\n\t\\varphi_{{}_{E_6,\\sigma'_3,\\nu_3}}(q, P)(M+\\a)&={k_J}^{-1}(P(k_J M){}^t\\!P)+q\\a k^{-1}(\\tau \\,{}^t\\!P), \n\t\\\\\n\t&\\hspace*{40mm}M+\\a \\in \\mathfrak{J}(3, \\H)^C \\oplus (\\H^3)^C=\\mathfrak{J}^C.\n\t\\end{align*}\n\tNeedless to say, this mapping is the restriction of the mapping $\\varphi_{{}_{E_6,\\nu_3}}$, that is, $\\varphi_{{}_{E_6,\\sigma'_3,\\nu_3}}(q, P)=\\varphi_{{}_{E_6,\\nu_3}}(q,P)$ (Theorem \\ref{theorem 3.3.5}). \n\t\n\tAs usual, we will prove that $\\varphi_{{}_{E_6,\\sigma'_3,\\nu_3}}$ is well-defined. It is clear that $\\varphi_{{}_{E_6,\\sigma'_3,\\nu_3}}(q,P) \\in (E_6)^{\\nu_3}$, and it follows from $\\sigma'_3=\\varphi_{{}_{E_6,\\nu_3}}(1,\\diag(1,1,\\omega,\\omega,\\tau\\omega,\\tau\\omega))$ that\n\t\\begin{align*}\n\t&\\quad {\\sigma'_3}^{-1}\\varphi_{{}_{E_6,\\sigma'_3,\\nu_3}}(q,P)\\sigma'_3\n\t\\\\\n\t&=\\varphi_{{}_{E_6,\\nu_3}}(1,\\diag(1,1,\\omega,\\omega,\\tau\\omega,\\tau\\omega))^{-1}\\varphi_{{}_{E_6,\\sigma'_3,\\nu_3}}(q,P)\\varphi_{{}_{E_6,\\nu_3}}(1,\\diag(1,1,\\omega,\\omega,\\tau\\omega,\\tau\\omega))\n\t\\\\\n\t&=\\varphi_{{}_{E_6,\\nu_3}}(1,\\diag(1,1,\\tau\\omega,\\tau\\omega,\\omega,\\omega))\\varphi_{{}_{E_6,\\nu_3}}(q,P)\\varphi_{{}_{E_6,\\nu_3}}(1,\\diag(1,1,\\omega,\\omega,\\tau\\omega,\\tau\\omega))\n\t\\\\\n\t&=\\varphi_{{}_{E_6,\\nu_3}}(q,\\diag(1,1,\\tau\\omega,\\tau\\omega,\\omega,\\omega)P\\,\\diag(1,1,\\omega,\\omega,\\tau\\omega,\\tau\\omega)),P=\\diag(a,b,P_1,P_2)\n\t\\\\\n\t&=\\varphi_{{}_{E_6,\\nu_3}}(q,\\diag(a, b, (\\tau\\omega E)P_1(\\omega E), (\\omega E) P_2 (\\tau\\omega E) ))\n\t\\\\\n\t&=\\varphi_{{}_{E_6,\\nu_3}}(q,P)\n\t\\\\\n\t&=\\varphi_{{}_{E_6,\\sigma'_3,\\nu_3}}(q,P).\n\t\\end{align*}\n\tHence we have that $\\varphi_{{}_{E_6,\\sigma'_3,\\nu_3}}(q,P) \\in (E_6)^{\\sigma'_3}$. Thus $\\varphi_{{}_{E_6,\\sigma'_3,\\nu_3}}$ is well-defined. Subsequently, since $\\varphi_{{}_{E_6,\\sigma'_3,\\nu_3}}$ is the restriction of the mapping $\\varphi_{{}_{E_6,\\nu_3}}$, we easily see that $\\varphi_{{}_{E_6,\\sigma'_3,\\nu_3}}$ is a homomorphism.\n\t\n\tNext, we will prove that $\\varphi_{{}_{E_6,\\sigma'_3,\\nu_3}}$ is surjective. Let $\\alpha \\in (E_6)^{\\sigma'_3} \\cap (E_6)^{\\nu_3} \\subset (E_6)^{\\nu_3}$. There exist $q \\in Sp(1)$ and $A \\in S(U(1)\\times U(5))$ such that $\\alpha=\\varphi_{{}_{E_6,\\nu_3}}(q,A)$ (Theorem \\ref{theorem 3.3.5}). Moreover, from the condition $\\alpha \\in (E_6)^{\\sigma'_3}$, that is, ${\\sigma'_3}^{-1}\\varphi_{{}_{E_6,\\nu_3}}(q,A)\\sigma'_3=\\varphi_{{}_{E_6,\\nu_3}}(q,A)$, and using ${\\sigma'_3}^{-1}\\varphi_{{}_{E_6,\\nu_3}}(q,A)\\sigma'_3=\\varphi_{{}_{E_6,\\nu_3}}(q,\\diag(1,1,\\tau\\omega,\\tau\\omega,\\omega,\\omega)A\\,\\diag(1,1,\\omega,\\omega,\\tau\\omega,\\tau\\omega))$, we have that\n\t\\begin{align*}\n\t&\\left\\{\n\t\\begin{array}{l}\n\tq=q \\\\\n\t\\diag(1,1,\\tau\\omega,\\tau\\omega,\\omega,\\omega)A\\,\\diag(1,1,\\omega,\\omega,\\tau\\omega,\\tau\\omega)=A   \n\t\\end{array}\\right. \n\t\\\\\n\t&\\hspace*{50mm}{\\text{or}}\n\t\\\\\n\t&\\left\\{\n\t\\begin{array}{l}\n\tq=-q \\\\\n\t\\diag(1,1,\\tau\\omega,\\tau\\omega,\\omega,\\omega)A\\,\\diag(1,1,\\omega,\\omega,\\tau\\omega,\\tau\\omega)=-A.   \n\t\\end{array}\\right.     \n\t\\end{align*}\n\tThe latter case is impossible because of $q\\not=0$. As for the former case, from the second condition, by doing straightforward computation $A$ takes the following form $\\diag(a, b, C, D), a,b \\in U(1),C, D \\in U(2),  (ab)(\\det\\,C)(\\det\\,D)=1$, that is, $A \\in S(U(1)\\times U(1)\\times  U(2)\\times U(2))$. Needless to say, $q \\in Sp(1)$.\n\tHence there exist $q \\in Sp(1)$ and $P \\in S(U(1)\\times U(1)\\times U(2)\\times U(2))$ such that $\\alpha=\\varphi_{{}_{E_6,\\nu_3}}(s,P)$. Namely, there exist $q \\in Sp(1)$ and $P \\in S(U(1)\\times U(1)\\times U(2)\\times U(2))$ such that $\\alpha=\\varphi_{{}_{E_6,\\sigma'_3,\\nu_3}}(s,P)$. The proof of surjective is completed.\n\t\n\tFinally, we will determine $\\Ker\\,\\varphi_{{}_{E_6,\\sigma'_3,\\nu_3}}$. However, from $\\Ker\\,\\varphi_{{}_{E_6,\\nu_3}}=\\{(1,E),(-1,-E) \\}$, we easily obtain that $\\Ker\\,\\varphi_{{}_{E_6,\\sigma'_3,\\nu_3}}=\\{(1,E),(-1,-E) \\} \\cong \\Z_2$. Thus we have the isomorphism $(E_6)^{\\sigma'_3} \\cap (E_6)^{\\nu_3} \\cong (Sp(1)\\times S(U(1)\\times U(1)\\times U(2)\\times U(2)))\/\\Z_2$. In addition, from Proposition \\ref{proposition 4.8.1} we have the isomorphism $(E_6)^{\\sigma_3} \\cap (E_6)^{\\nu_3} \\cong (Sp(1)\\times S(U(1)\\times U(1)\\times U(2)\\times U(2)))\/\\Z_2$. \n\t\\if\n\tHere, \tusing the mapping $f_{{}_{473}}$ in the proof of Lemma \\ref{lemma 4.7.3}, we define a homomorphism $h_{{}_{474}}:U(1)\\times (U(1)\\times U(1)\\times U(1)\\times SU(2)\\times SU(2)) \\to U(1)\\times S(U(1)\\times U(1)\\times U(2)\\times U(2))$ by\n\t\\begin{align*}\n\th_{{}_{474}}(s,(a,b,c,A,B))=(s,f_{{}_{473}}(a,b,c,A,B)).\n\t\\end{align*}\n\tThen, the elements of $(s,(a,b,c,A,B))$ corresponding to $(1,E), (-1,-E) \\in \\Ker\\,\\varphi_{{}_{E_6,\\gamma_3,\\mu'_3}}$ under the mapping $h_{{}_{474}}$ are as follows.\n\t\\begin{align*}\n\t&(1,(1,1,1,E,E)),(1,(1,1,-1,-E,-E)),(1,(1,-1,1,E,-E)),(1,(1,-1,-1,-E,E)),\n\t\\\\\n\t&(1,(-1,1,1,E,-E)),(1,(-1,1,-1,-E,E)),(1,(-1,-1,1,E,E)),(1,(-1,-1,-1,-E,-E)),\n\t\\\\\n\t&(-1,(i,i,1,-E,E)),(-1,(i,i,-1,E,-E)),(-1,(i,-i,1,-E,-E)),(-1,(i,i,-1,-E,E)),\n\t\\\\\n\t&(-1,(-i,i,1,-E,\\!E)),(-1,(-i,i,-1,E,\\!E)),(-1,(-i,-i,1,-E,E)),(-1,(-i,-i,-1,E,-E)).\n\t\\end{align*}\n\t\\f\n\tTherefore, as in the proof of Theorem \\ref{theorem 4.7.3}, we have the required isomorphism \n\t\\begin{align*}\n\t(E_6)^{\\sigma_3} \\cap (E_6)^{\\nu_3} \\!\\cong (Sp(1)\\times U(1) \\times U(1)\\times U(1) \\allowbreak \\times SU(2)\\times SU(2))\/(\\Z_2\\times\\Z_2\\times\\Z_4), \n\t\\end{align*}\n\twhere \n\t\\begin{align*}\n\t&\\Z_2=\\{(1,1,1,1,E,E), (1,1,-1,1,E,-E) \\},\n\t\\\\\n\t&\\Z_2=\\{(1,1,1,1,E,E), (1,1,-1,-1,-E,E) \\},\n\t\\\\\n\t&\\Z_4=\\{(1,1,1,E,E,E), (1,-1,1-,1,E,E), (-1,i,i,1,-E,E), (-1,-i,-i,1,-E,E) \\}.\n\t\\end{align*}\n\\end{proof}\n\nThus, since the group $(E_6)^{\\sigma_3} \\cap (E_6)^{\\nu_3}$ is connected from Theorem \\ref{theorem 4.9.3}, we have an exceptional $\\varmathbb{Z}_3 \\times \\varmathbb{Z}_3$-symmetric space\n\\begin{align*}\nE_6\/((Sp(1)\\times U(1) \\times U(1)\\times U(1) \\times SU(2)\\times SU(2))\/(\\Z_2\\times\\Z_2\\times\\Z_4)).\n\\end{align*}\n\n\\subsection{Case 10: $\\{1, \\tilde{\\sigma}_3,  \\tilde{\\sigma}_3{}^{-1}\\} \\times \\{1, \\tilde{\\mu}_3,  \\tilde{\\mu}_3{}^{-1}\\}$-symmetric space}\n\nLet the $C$-linear transformations $\\sigma_3, \\mu_3$ of $\\mathfrak{J}^C$  defined in Subsection \\ref{subsection 3.3}. \n\n\\noindent From Lemma \\ref{lemma 3.3.8} (1), since we can easily confirm that $\\sigma_3$ and $\\mu_3$ are commutative, $\\tilde{\\sigma}_3$ and $\\tilde{\\mu}_3$ are commutative in $\\Aut(E_6)$: $\\tilde{\\sigma}_3\\tilde{\\mu}_3=\\tilde{\\mu}_3\\tilde{\\sigma}_3$.\n\nBefore determining the structure of the group $(E_6)^{\\sigma_3} \\cap (E_6)^{\\mu_3}$, we prove proposition needed in the proof of theorem below.\n\n\\begin{proposition}\\label{proposition 4.10.1\n\tThe group $(E_6)^{\\sigma_3}$ is a subgroup of the group $(E_6)^\\sigma${\\rm: } $(E_6)^{\\sigma_3} \\subset (E_6)^\\sigma$.\n\\end{proposition}\n\\begin{proof}\n\tLet $\\alpha \\in (E_6)^{\\sigma_3}$. Then, from Theorem \\ref{theorem 3.3.4}, there exist $\\theta \\in U(1), D_a \\in Spin(2)$ and $\\beta \\in Spin(8)$ such that $\\alpha=\\phi_{{}_{6,\\sigma}}(\\theta) D_a \\beta$. Here, note that $(E_6)_{E_1} \\subset (E_6)^\\sigma$ (\\cite[Theorem 3.10.2]{iy0}), and so since $Spin(8)$ as the group $(E_6)_{E_1,F_1(1),F_1(e_1)} \\subset (E_6)_{E_1} \\subset (E_6)^\\sigma$, it follows that\n\t\\begin{align*}\n\t\t\t\\sigma\\alpha=\\sigma(\\phi_{{}_{6,\\sigma}}(\\theta) D_a \\beta)=\\phi_{{}_{6,\\sigma}}(\\theta)\\sigma D_a\\beta=\\phi_{{}_{6,\\sigma}}(\\theta)D_a \\sigma\\beta=(\\phi_{{}_{6,\\sigma}}(\\theta) D_a \\beta)\\sigma=\\alpha\\sigma.\n\t\\end{align*}\n\tHence we have that $\\alpha \\in (E_6)^\\sigma$, that is, $(E_6)^{\\sigma_3} \\subset (E_6)^\\sigma$.\n\\end{proof}\n\nNow, we will determine the structure of the group $(E_6)^{\\sigma_3} \\cap (E_6)^{\\mu_3}$.\n\n\\begin{theorem}\\label{theorem 4.10.2\n\tThe group $(E_6)^{\\sigma_3} \\cap (E_6)^{\\mu_3}$ coincides with the group $(E_6)^{\\sigma_3}$, that is, the group $(E_6)^{\\sigma_3} \\cap (E_6)^{\\mu_3}$ is isomorphic to the group $(U(1)\\times Spin(2)\\times Spin(8))\/(\\Z_2 \\times \\Z_4),\\\\ \\Z_2=\\!\\{(1,1,1),(1,\\sigma,\\sigma) \\}, \\Z_4\\!=\\!\\{(1,1,1),(i,D_{e_1},\\phi_{{}_{6,\\sigma}}(-i)D_{-e_1}),(-1,\\allowbreak \\sigma,-1),(-i,D_{-e_1}, \\phi_{{}_{6,\\sigma}}(i) \\allowbreak D_{e_1}) \\}$. \n\\end{theorem}\n\\begin{proof}\n\tFrom Proposition \\ref{proposition 3.3.3} and Theorem \\ref{theorem 3.3.6}, we have that the group $(E_6)^{\\sigma_3} \\cap (E_6)^{\\mu_3}$ coincides with the group $(E_6)^{\\sigma_3} \\cap (E_6)^{\\sigma}$: $(E_6)^{\\sigma_3} \\cap (E_6)^{\\mu_3}=(E_6)^{\\sigma_3} \\cap (E_6)^{\\sigma}$. In addition, from Proposition \\ref{proposition 4.10.1} above, we have that \n\t\\begin{align*}\n\t      (E_6)^{\\sigma_3} \\cap (E_6)^{\\mu_3}=(E_6)^{\\sigma_3} \\cap (E_6)^{\\sigma}=(E_6)^{\\sigma_3}.\n\t\\end{align*}\n\tTherefore, by Theorem \\ref{theorem 3.3.4}, we have the required isomorphism \n\t\\begin{align*}\n\t \t\t   (E_6)^{\\sigma_3} \\cap (E_6)^{\\mu_3} \\cong (U(1)\\times Spin(2)\\times Spin(8))\/(\\Z_2\\times \\Z_4).\n\t\\end{align*}\n\\end{proof}\n\nThus, since the group $(E_6)^{\\sigma_3} \\cap (E_6)^{\\mu_3}$ is connected from Theorem \\ref{theorem 4.10.2}, we have an exceptional $\\varmathbb{Z}_3 \\times \\varmathbb{Z}_3$-symmetric space\n\\begin{align*}\nE_6\/((U(1)\\times Spin(2)\\times Spin(8))\/(\\Z_2\\times \\Z_4)).\n\\end{align*}\n\n\\subsection{Case 11: $\\{1, \\tilde{\\sigma}_3,  \\tilde{\\sigma}_3{}^{-1}\\} \\times \\{1, \\tilde{w}_3,  \\tilde{w}_3{}^{-1}\\}$-symmetric space}\n\nLet the $C$-linear transformations $\\sigma_3, w_3$ of $\\mathfrak{J}^C$  defined in Subsection \\ref{subsection 3.3}. \n\n\\noindent From Lemma \\ref{lemma 3.3.8} (2), since we can easily confirm that $\\sigma_3$ and $w_3$ are commutative, $\\tilde{\\sigma}_3$ and $\\tilde{w}_3$ are commutative in $\\Aut(E_6)$: $\\tilde{\\sigma}_3\\tilde{w}_3=\\tilde{w}_3\\tilde{\\sigma}_3$.\n\nNow, we will determine the structure of the group $(E_6)^{\\sigma_3}\\cap (E_6)^{w_3}$.\n\n\\begin{theorem}\\label{theorem 4.11.1\n\tThe group $(E_6)^{\\sigma_3}\\cap (E_6)^{w_3}$ is isomorphic to the group $(SU(3)\\times U(1)\\times U(1)\\times U(1)\\times U(1))\/\\Z_3${\\rm :} $(E_6)^{\\sigma_3}\\cap (E_6)^{w_3} \\cong (SU(3)\\times U(1)\\times U(1)\\times U(1)\\times U(1))\/\\Z_3, \\Z_3=\\{(E,1,1,1,1),(\\bm{\\omega}E,\\bm{\\omega},\\bm{\\omega},\\bm{\\omega},\\bm{\\omega}),(\\bm{\\omega}^{-1}E,\\bm{\\omega}^{-1},\\bm{\\omega}^{-1},\\bm{\\omega}^{-1},\\bm{\\omega}^{-1})\\}$.\n\\end{theorem}\n\\begin{proof}\n\tLet $S(U(1)\\times U(1)\\times U(1)) \\subset SU(3)$. We define a mapping $\\varphi_{{}_{E_6,\\sigma_3,w_3}}: SU(3)\\times S(U(1)\\times U(1)\\times U(1))\\times S(U(1)\\times U(1)\\times U(1)) \\to (E_6)^{\\sigma_3}\\cap (E_6)^{w_3}$ by\n\t\\begin{align*}\n\t\t\t\\varphi_{{}_{E_6,\\sigma_3,w_3}}(L,P,Q)(X_{C}+M)&=h(P,Q)X_{C}h(P,Q)^*+LM\\tau h(P,Q)^*, \n\t\t\t\\\\\n\t\t\t&\\hspace*{20mm} X_{C}+M \\in \\mathfrak{J}(3, \\C)^C \\oplus \n\t\t\tM(3,\\C)^C=\\mathfrak{J}^C.\n\t\\end{align*}\n\tNeedless to say, this mapping is the restriction of the mapping $\\varphi_{{}_{E_6,w_3}}$, that is, $\\varphi_{{}_{E_6,\\sigma_3,w_3}}(L,P,\\allowbreak Q)=\\varphi_{{}_{E_6,w_3}}(L,P,Q)$ (Theorem \\ref{theorem 3.3.7}). \n\t\n\tWe will prove that $\\varphi_{{}_{E_6,\\sigma_3,w_3}}$ is well-defined. It is clear that $\\varphi_{{}_{E_6,\\sigma_3,_3}}(L,P,Q) \\in (E_6)^{w_3}$, and it follows from $\\sigma_3=\\varphi_{{}_{E_6,w_3}}(E,\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega}), \\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega}))$ (Lemma \\ref{lemma 3.3.8} (2)) that \n\t\\begin{align*}\n\t&\\quad {\\sigma_3}^{-1}\\varphi_{{}_{E_6,\\sigma_3,w_3}}(L,P,Q)\\sigma_3\n\t\\\\\n\t&=\\varphi_{{}_{E_6,w_3}}(E,\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega}), \\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega}))^{-1}\\varphi_{{}_{E_6,\\sigma_3,w_3}}(L,P,Q)\n\t\\\\\n\t&\\hspace*{70mm}\\varphi_{{}_{E_6,w_3}}(E,\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega}), \\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega}))\n\t\\\\\n\t&=\\varphi_{{}_{E_6,w_3}}(E,\\diag(1,\\bm{\\omega},\\ov{\\bm{\\omega}}), \\diag(1,\\bm{\\omega},\\ov{\\bm{\\omega}}))\\varphi_{{}_{E_6,w_3}}(L,P,Q)\n\t\\\\\n\t&\\hspace*{70mm}\\varphi_{{}_{E_6,w_3}}(E,\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega}), \\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega}))\n\t\\\\\n\t&=\\varphi_{{}_{E_6,w_3}}(L,\\diag(1,\\bm{\\omega},\\ov{\\bm{\\omega}})P\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega}),\\diag(1,\\bm{\\omega},\\ov{\\bm{\\omega}})Q\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega})),\n\t\\\\\n\t&\\hspace*{85mm}P=\\diag(a,b,c), Q=\\diag(s,t,v)\n\t\\\\\n\t&=\\varphi_{{}_{E_6,w_3}}(L,P,Q)\n\t\\\\\n\t&=\\varphi_{{}_{E_6,\\sigma_3,w_3}}(L,P,Q).\n\t\\end{align*}\n\tHence we have that $\\varphi_{{}_{E_6,\\sigma_3,w_3}}(L,P,Q) \\in (E_6)^{\\sigma_3}$. Thus $\\varphi_{{}_{E_6,\\sigma_3,w_3}}$ is well-defined. Subsequently, since $\\varphi_{{}_{E_6,\\sigma_3,w_3}}$ is the restriction of the mapping $\\varphi_{{}_{E_6,w_3}}$, we easily see that $\\varphi_{{}_{E_6,\\sigma_3,w_3}}$ is a homomorphism.\n\t\n\tNext we will prove that $\\varphi_{{}_{E_6,\\sigma_3,w_3}}$ is surjective. Let $\\alpha \\in (E_6)^{\\sigma_3}\\cap (E_6)^{w_3} \\subset (E_6)^{w_3}$. There exist $L, A, B \\in SU(3)$ such that $\\alpha=\\varphi_{{}_{E_6,w_3}}(L,A,B)$ (Theorem \\ref{theorem 3.3.7}). Moreover, from the condition $\\alpha \\in (E_6)^{\\sigma_3}$, that is, ${\\sigma_3}^{-1}\\varphi_{{}_{E_6,w_3}}(L,A,B)\\sigma_3=\\varphi_{{}_{E_6,w_3}}(L,A,B)$, and using \n\t\\begin{align*}\n\t\t\t\t&\\quad {\\sigma_3}^{-1}\\varphi_{{}_{E_6,w_3}}(L,A,B)\\sigma_3\n\t\t\t\t\\\\\n\t\t\t\t&=\\varphi_{{}_{E_6,w_3}}(L,\\diag(1,\\bm{\\omega},\\ov{\\bm{\\omega}})A\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega}),\\diag(1,\\bm{\\omega},\\ov{\\bm{\\omega}})B\\diag(1,\\ov{\\bm{\\omega}}, \\bm{\\omega}))\n\t\\end{align*}\n\t(Lemma \\ref{lemma 3.3.8} (2)) we have that \n\t\\begin{align*}\n\t&\\,\\,\\,{\\rm(i)}\\,\\left\\{\n\t\\begin{array}{l}\n\tL=L\\\\\n\t\\diag(1,\\bm{\\omega},\\ov{\\bm{\\omega}})A\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega})=A \\\\\n\t\\diag(1,\\bm{\\omega},\\ov{\\bm{\\omega}})B\\diag(1,\\ov{\\bm{\\omega}}, \\bm{\\omega})=B,\n\t\\end{array} \\right.\n\t\\qquad\n\t{\\rm(ii)}\\,\\left\\{\n\t\\begin{array}{l}\n\tL=\\bm{\\omega}L\\\\\n\t\\diag(1,\\bm{\\omega},\\ov{\\bm{\\omega}})A\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega})=\\bm{\\omega}A \\\\\n\t\\diag(1,\\bm{\\omega},\\ov{\\bm{\\omega}})B\\diag(1,\\ov{\\bm{\\omega}}, \\bm{\\omega})=\\bm{\\omega}B,\n\t\\end{array} \\right.\n\t\\\\[2mm]\n\t&{\\rm(iii)}\\,\\left\\{\n\t\\begin{array}{l}\n\tL=\\bm{\\omega}^{-1}L\\\\\n\t\\diag(1,\\bm{\\omega},\\ov{\\bm{\\omega}})A\\diag(1,\\ov{\\bm{\\omega}},\\bm{\\omega})=\\bm{\\omega}^{-1}A \\\\\n\t\\diag(1,\\bm{\\omega},\\ov{\\bm{\\omega}})B\\diag(1,\\ov{\\bm{\\omega}}, \\bm{\\omega})=\\bm{\\omega}^{-1}B.\n\t\\end{array} \\right.\n\t\\end{align*}\n\tThe Cases (ii) and (iii) are impossible because $L\\not=0$. As for the Case (i), from the second and third conditions, it is easy to see that $A,B \\in S(U(1)\\times U(1) \\times U(1))$. Needless to say, $L \\in SU(3)$. \n\tHence there exist $L \\in SU(3)$ and $A,B \\in S(U(1)\\times U(1)\\times U(1))$ such that $\\alpha=\\varphi_{{}_{E_6,w_3}}(L,P,Q)$. Namely, there exist $L \\in SU(3)$ and $A,B \\in S(U(1)\\times U(1)\\times U(1))$ such that $\\alpha=\\varphi_{{}_{E_6,\\sigma_3,w_3}}(L,P,Q)$. The proof of surjective is completed.\n\t\n\tFinally, we will determine $\\Ker\\,\\varphi_{{}_{E_6,\\sigma_3,w_3}}$. However, from $\\Ker\\,\\varphi_{{}_{E_6,w_3}}=\\{(E,E,E),(\\bm{\\omega}E,\\allowbreak \\bm{\\omega}E,\\bm{\\omega}E),(\\bm{\\omega}^{-1}E,\\bm{\\omega}^{-1}E, \\bm{\\omega}^{-1}E) \\}$, we easily obtain that $\\Ker\\,\\varphi_{{}_{E_6,\\sigma_3,w_3}}=\\{(E,E,E),(\\bm{\\omega}E,\\allowbreak \\bm{\\omega}E,\\bm{\\omega}E),(\\bm{\\omega}^{-1}E,\\bm{\\omega}^{-1}E, \\bm{\\omega}^{-1}E) \\} \\cong \\Z_3$.\n\tThus we have the isomorphism $(E_6)^{\\sigma_3}\\cap (E_6)^{w_3} \\cong SU(3)\\times S(U(1)\\times U(1)\\times U(1))\\times S(U(1)\\times U(1)\\times U(1))\/\\Z_3$. \n\t\n\tTherefore, by Lemma \\ref{lemma 4.3.1} we have the required isomorphism \n\t\\begin{align*}\n\t(E_6)^{\\sigma_3}\\cap (E_6)^{w_3} \\cong (SU(3)\\times U(1)\\times U(1)\\times U(1)\\times U(1))\/\\Z_3,\n\t\\end{align*}\n\twhere $\\Z_3=\\{(E,1,1,1,1),(\\bm{\\omega}E,\\bm{\\omega},\\bm{\\omega},\\bm{\\omega},\\bm{\\omega}),(\\bm{\\omega}^{-1}E,\\bm{\\omega}^{-1},\\bm{\\omega}^{-1},\\bm{\\omega}^{-1},\\bm{\\omega}^{-1})\\}$.\n\\end{proof}\n\nThus, since the group $(E_6)^{\\sigma_3} \\cap (E_6)^{w_3}$ is connected from Theorem \\ref{theorem 4.11.1}, we have an exceptional $\\varmathbb{Z}_3 \\times \\varmathbb{Z}_3$-symmetric space\n\\begin{align*}\nE_6\/((SU(3)\\times U(1)\\times U(1)\\times U(1)\\times U(1))\/\\Z_3).\n\\end{align*}\n\n\\subsection{Case 12: $\\{1, \\tilde{\\nu}_3,  \\tilde{\\nu}_3{}^{-1}\\} \\times \\{1, \\tilde{\\mu}_3,  \\tilde{\\mu}_3{}^{-1}\\}$-symmetric space}\n\nLet the $C$-linear transformations $\\nu_3, \\mu_3$ of $\\mathfrak{J}^C$  defined in Subsection \\ref{subsection 3.3}. \n\n\\noindent From Lemma \\ref{lemma 3.3.8} (1), since we can easily confirm that $\\nu_3$ and $\\mu_3$ are commutative, $\\tilde{\\nu}_3$ and $\\tilde{\\mu}_3$ are commutative in $\\Aut(E_6)$: $\\tilde{\\nu}_3\\tilde{\\mu}_3=\\tilde{\\mu}_3\\tilde{\\nu}_3$.\n\nBefore determining the structure of the group $(E_6)^{\\nu_3} \\cap (E_6)^{\\mu_3}$, we confirm that useful lemma holds and prove proposition needed in the proof of theorem below.\n\n\\begin{lemma}\\label{lemma 4.12.1\n\tThe mapping $\\varphi_{{}_{E_6,\\nu_3}}:Sp(1) \\times S(U(1)\\times U(5)) \\to (E_6)^{\\nu_3}$ of \\,Theorem {\\rm \\ref{theorem 3.3.5}} satisfies the relational formulas \n\t\\begin{align*}\n\t\t\t\\nu_3&=\\varphi_{{}_{E_6,\\nu_3}}(1,\\diag(\\nu^5,\\nu^{-1},\\nu^{-1},\\nu^{-1},\\nu^{-1},\\nu^{-1})),\n\t\t\t\\\\\n\t\t\t\\mu_3&=\\varphi_{{}_{E_6,\\nu_3}}(1, \\diag(\\nu^{-2},\\nu^{2},\\nu^{-1},\\nu,\\nu^{-1},\\nu)),\n\t\\end{align*}\n\t where $\\nu=\\exp(2\\pi i\/9)\\in U(1)$.\n\\end{lemma}\n\\begin{proof}\n\t From Lemma \\ref{lemma 3.3.8} (1), these results are trivial. \n\\end{proof}\n\nIt goes with out saying that $\\delta_Q=\\varphi_{{}_{E_6, \\nu_3}}(1,Q)(=\\varphi_{{}_{E_6, \\gamma_3}}(1,Q))$, where $\\delta_Q$ is defined in the Case 7, and so from Lemma \\ref{lemma 3.3.8} (1) the $C$-linear transformation $\\mu'_3$ which is conjugate to $\\mu_3$ under $\\delta_Q \\in (E_6)^{\\nu_3}$ is also expressed by\n\\begin{align*}\n\\mu'_3=\\varphi_{{}_{E_6,\\nu_3}}(1, \\diag(\\nu^{-2},\\nu^{2},\\nu^{-1},\\nu^{-1},\\nu,\\nu)).\n\\end{align*}\n\nThen we have the following proposition.\n\n\\begin{proposition}\\label{proposition 4.12.2\n\tThe group $(E_6)^{\\nu_3} \\cap (E_6)^{\\mu_3}$ is isomorphic to the group $(E_6)^{\\nu_3} \\cap (E_6)^{\\mu'_3}${\\rm :} $(E_6)^{\\nu_3} \\cap (E_6)^{\\mu_3} \\cong (E_6)^{\\nu_3} \\cap (E_6)^{\\mu'_3}$.\n\\end{proposition}\n\\begin{proof}\n\tWe define a mapping $g_{{}_{4122}}: (E_6)^{\\nu_3} \\cap (E_6)^{\\mu'_3} \\to (E_6)^{\\gamma_3} \\cap (E_6)^{\\mu_3}$ by\n\t\\begin{align*}\n\tg_{{}_{4122}}(\\alpha)=\\delta_Q\\alpha{\\delta_Q}^{-1}.\t\n\t\\end{align*}\n\tSince it is easily to verify that $\\delta_Q\\nu_3=\\nu_3\\delta_Q$ using $\\nu_3=\\varphi_{{}_{E_6,\\nu_3}}(1,\\diag(\\nu^5,\\nu^{-1},\\nu^{-1},\\nu^{-1},\\nu^{-1}, \\allowbreak \\nu^{-1}))$ (Lemma \\ref{lemma 4.12.1}), we can prove this proposition as in the proof of Proposition \\ref{proposition 4.7.1}.\n\\end{proof}\n\\vspace{1mm}\n\nNow, we will determine the structure of the group $(E_6)^{\\nu_3} \\cap (E_6)^{\\mu_3}$.\n\n\\begin{theorem}\\label{theorem 4.12.3\n\tThe group $(E_6)^{\\nu_3} \\cap (E_6)^{\\mu_3}$ is isomorphic the group $(Sp(1)\\times U(1) \\times U(1)\\allowbreak \\times U(1) \\times SU(2)\\times SU(2))\/(\\Z_2\\times\\Z_2\\times\\Z_4)${\\rm :} $(E_6)^{\\nu_3} \\cap (E_6)^{\\mu_3} \\cong (Sp(1)\\times U(1) \\times U(1)\\times U(1) \\allowbreak \\times SU(2)\\times SU(2))\/(\\Z_2\\times\\Z_2\\times\\Z_4), \\Z_2=\\{(1,1,1,1,E,E), (1,1,-1,1,E,-E) \\},\n\t\\Z_2=\\{(1,1,1,1,E,E), (1,1,-1,-1,-E,E) \\},\n\t\\Z_4=\\{(1,1,1,E,E,E), (1,-1,1-,1,E,E), (-1,i,i,\\allowbreak 1,-E,E), (-1,-i,-i,1,-E,E) \\}$.\n\\end{theorem}\n\\begin{proof}\n\tLet $S(U(1)\\times U(1)\\times U(2)\\times U(2)) \\subset S(U(1)\\times U(5))$. \n\tWe define a mapping $\\varphi_{{}_{E_6,\\nu_3,\\mu'_3}}: Sp(1)\\times S(U(1)\\times U(1)\\times U(2)\\times U(2)) \\to (E_6)^{\\nu_3} \\cap (E_6)^{\\mu'_3}$ by\n\t\\begin{align*}\n\t\\varphi_{{}_{E_6,\\nu_3,\\mu'_3}}(q, P)(M+\\a)&={k_J}^{-1}(P(k_J M){}^t\\!P)+q\\a k^{-1}(\\tau \\,{}^t\\!P), \n\t\\\\\n\t&\\hspace*{40mm}M+\\a \\in \\mathfrak{J}(3, \\H)^C \\oplus (\\H^3)^C=\\mathfrak{J}^C.\n\t\\end{align*}\n\tNeedless to say, this mapping is the restriction of the mapping $\\varphi_{{}_{E_6,\\nu_3}}$, that is, $\\varphi_{{}_{E_6,\\nu_3,\\mu'_3}}(q, P)=\\varphi_{{}_{E_6,\\nu_3}}(q,P)$ (Theorem \\ref{theorem 3.3.5}). \n\t\n\tAs usual, we will prove that $\\varphi_{{}_{E_6,\\nu_3,\\mu'_3}}$ is well-defined. It is clear that $\\varphi_{{}_{E_6,\\nu_3,\\mu'_3}}(q,P) \\in (E_6)^{\\nu_3}$, and it follows from $\\mu'_3=\\varphi_{{}_{E_6,\\gamma_3}}(1,\\diag({\\nu}^{-2},\\nu^2,{\\nu}^{-1},{\\nu}^{-1},\\nu,\\nu))$ that\n\t\\begin{align*}\n\t&\\quad {\\mu'_3}^{-1}\\varphi_{{}_{E_6,\\nu_3,\\mu'_3}}(q,P)\\mu'_3\n\t\\\\\n\t&=\\varphi_{{}_{E_6,\\nu_3}}(1,\\diag({\\nu}^{-2},\\nu^2,{\\nu}^{-1},{\\nu}^{-1},\\nu,\\nu))^{-1}\\varphi_{{}_{E_6,\\nu_3,\\mu'_3}}(q,P)\\varphi_{{}_{E_6,\\nu_3}}(1,\\diag({\\nu}^{-2},\\nu^2,{\\nu}^{-1},{\\nu}^{-1},\\nu,\\nu))\n\t\\\\\n\t&=\\varphi_{{}_{E_6,\\nu_3}}(1,\\diag({\\nu}^2,\\nu^{-2},\\nu,\\nu,{\\nu}^{-1},{\\nu}^{-1}))\\varphi_{{}_{E_6,\\nu_3}}(q,P)\\varphi_{{}_{E_6,\\nu_3}}(1,\\diag({\\nu}^{-2},\\nu^2,{\\nu}^{-1},{\\nu}^{-1},\\nu,\\nu))\n\t\\\\\n\t&=\\varphi_{{}_{E_6,\\nu_3}}(q,\\diag({\\nu}^2,\\nu^{-2},\\nu,\\nu,{\\nu}^{-1},{\\nu}^{-1})P\\diag({\\nu}^{-2},\\nu^2,{\\nu}^{-1},{\\nu}^{-1},\\nu,\\nu)),P=\\diag(a,b,P_1,P_2)\n\t\\\\\n\t&=\\varphi_{{}_{E_6,\\nu_3}}(q,\\diag(\\nu^2 a \\nu^{-2}, {\\nu}^{-2} b \\nu^2, (\\nu E) P_1(\\nu^{-1}E), ({\\nu}^{-1}E) P_2 (\\nu E) ))\n\t\\\\\n\t&=\\varphi_{{}_{E_6,\\nu_3}}(q,P)\n\t\\\\\n\t&=\\varphi_{{}_{E_6,\\nu_3,\\mu'_3}}(q,P).\n\t\\end{align*}\n\tHence we have that $\\varphi_{{}_{E_6,\\nu_3,\\mu'_3}}(q,P) \\in (E_6)^{\\mu'_3}$. Thus $\\varphi_{{}_{E_6,\\nu_3,\\mu'_3}}$ is well-defined. Subsequently, since $\\varphi_{{}_{E_6,\\nu_3,\\mu'_3}}$ is the restriction of the mapping $\\varphi_{{}_{E_6,\\nu_3}}$, we easily see that $\\varphi_{{}_{E_6,\\nu_3,\\mu'_3}}$ is a homomorphism.\n\t\n\tNext, we will prove that $\\varphi_{{}_{E_6,\\nu_3,\\mu'_3}}$ is surjective. Let $\\alpha \\in (E_6)^{\\nu_3} \\cap (E_6)^{\\mu'_3} \\subset (E_6)^{\\nu_3}$. There exist $q \\in Sp(1)$ and $A \\in S(U(1)\\times U(5))$ such that $\\alpha=\\varphi_{{}_{E_6,\\nu_3}}(q,A)$ (Theorem \\ref{theorem 3.3.5}). Moreover, from the condition $\\alpha \\in (E_6)^{\\mu'_3}$, that is, ${\\mu'_3}^{-1}\\varphi_{{}_{E_6,\\nu_3}}(q,A)\\mu'_3=\\varphi_{{}_{E_6,\\nu_3}}(q,A)$, and using ${\\mu'_3}^{-1}\\varphi_{{}_{E_6,\\nu_3}}(q,A)\\mu'_3=\\varphi_{{}_{E_6,\\nu_3}}(q,\\diag({\\nu}^2,\\nu^{-2},\\nu,\\nu,{\\nu}^{-1},{\\nu}^{-1}) A \\,\\diag({\\nu}^{-2},\\nu^2,{\\nu}^{-1},{\\nu}^{-1},\\nu,\\nu))$, we have that\n\t\\begin{align*}\n\t&\\left\\{\n\t\\begin{array}{l}\n\tq=q \\\\\n\t\\diag({\\nu}^2,\\nu^{-2},\\nu,\\nu,{\\nu}^{-1},{\\nu}^{-1}) A\\, \\diag({\\nu}^{-2},\\nu^2,{\\nu}^{-1},{\\nu}^{-1},\\nu,\\nu)=A   \n\t\\end{array}\\right. \n\t\\\\\n\t&\\hspace*{50mm}{\\text{or}}\n\t\\\\\n\t&\\left\\{\n\t\\begin{array}{l}\n\tq=-q \\\\\n\t\\diag({\\nu}^2,\\nu^{-2},\\nu,\\nu,{\\nu}^{-1},{\\nu}^{-1}) A \\,\\diag({\\nu}^{-2},\\nu^2,{\\nu}^{-1},{\\nu}^{-1},\\nu,\\nu)=-A.   \n\t\\end{array}\\right.     \n\t\\end{align*}\n\tThe latter case is impossible because of $q\\not=0$. As for the former case, from the second condition, by doing straightforward computation $A$ takes the following form $\\diag(a, b, C, D), a,b \\in U(1),C, D \\in U(2),  (ab)(\\det\\,C)(\\det\\,D)=1$, that is, $A \\in S(U(1)\\times U(1)\\times  U(2)\\times U(2))$. Needless to say, $q \\in Sp(1)$.\n\tHence there exist $q \\in Sp(1)$ and $P \\in S(U(1)\\times U(1)\\times U(2)\\times U(2))$ such that $\\alpha=\\varphi_{{}_{E_6,\\nu_3}}(q,P)$. Namely, there exist $q \\in Sp(1)$ and $P \\in S(U(1)\\times U(1)\\times U(2)\\times U(2))$ such that $\\alpha=\\varphi_{{}_{E_6,\\nu_3,\\mu'_3}}(q,P)$. The proof of surjective is completed.\n\t\n\tFinally, we will determine $\\Ker\\,\\varphi_{{}_{E_6,\\nu_3,\\mu'_3}}$. However, from $\\Ker\\,\\varphi_{{}_{E_6,\\nu_3}}=\\{(1,E),(-1,-E) \\}$, we easily obtain that $\\Ker\\,\\varphi_{{}_{E_6,\\nu_3,\\mu'_3}}=\\{(1,E),(-1,-E) \\} \\cong \\Z_2$. Thus we have the isomorphism $(E_6)^{\\nu_3} \\cap (E_6)^{\\mu'_3} \\cong (Sp(1)\\times S(U(1)\\times U(1)\\times U(2)\\times U(2)))\/\\Z_2$. In addition, by Proposition \\ref{proposition 4.12.2} we have the isomorphism $(E_6)^{\\nu_3} \\cap (E_6)^{\\mu_3} \\cong (Sp(1)\\times S(U(1)\\times U(1)\\times U(2)\\times U(2)))\/\\Z_2$. \n\t\n\tTherefore, as in the proof of Theorem \\ref{theorem 4.7.3}, we have the required isomorphism \n\t\\begin{align*}\n\t(E_6)^{\\nu_3} \\cap (E_6)^{\\mu_3} \\!\\cong (Sp(1)\\times U(1) \\times U(1)\\times U(1) \\allowbreak \\times SU(2)\\times SU(2))\/(\\Z_2\\times\\Z_2\\times\\Z_4), \n\t\\end{align*}\n\twhere \n\t\\begin{align*}\n\t&\\Z_2=\\{(1,1,1,1,E,E), (1,1,-1,1,E,-E) \\},\n\t\\\\\n\t&\\Z_2=\\{(1,1,1,1,E,E), (1,1,-1,-1,-E,E) \\},\n\t\\\\\n\t&\\Z_4=\\{(1,1,1,E,E,E), (1,-1,1-,1,E,E), (-1,i,i,1,-E,E), (-1,-i,-i,1,-E,E) \\}.\n\t\\end{align*}\n\\end{proof}\n\nThus, since the group $(E_6)^{\\nu_3} \\cap (E_6)^{\\mu_3}$ is connected from Theorem \\ref{theorem 4.12.3}, we have an exceptional $\\varmathbb{Z}_3 \\times \\varmathbb{Z}_3$-symmetric space\n\\begin{align*}\nE_6\/((Sp(1)\\times U(1) \\times U(1)\\times U(1) \\allowbreak \\times SU(2)\\times SU(2))\/(\\Z_2\\times\\Z_2\\times\\Z_4)).\n\\end{align*}\n\n\\subsection{Case 13: $\\{1, \\tilde{\\nu}_3,  \\tilde{\\nu}_3{}^{-1}\\} \\times \\{1, \\tilde{w}_3,  \\tilde{w}_3{}^{-1}\\}$-symmetric space}\n\nLet the $C$-linear transformations $\\nu_3, w_3$ of $\\mathfrak{J}^C$  defined in Subsection \\ref{subsection 3.3}. \n\n\\noindent From Lemma \\ref{lemma 3.3.8} (1), since we can easily confirm that $\\nu_3$ and $w_3$ are commutative, $\\tilde{\\nu}_3$ and $\\tilde{w}_3$ are commutative in $\\Aut(E_6)$: $\\tilde{\\nu}_3\\tilde{w}_3=\\tilde{w}_3\\tilde{\\nu}_3$.\n\nBefore determining the structure of the group $(E_6)^{\\nu_3} \\cap (E_6)^{w_3}$, we confirm that useful lemma holds, and we prove proposition and lemma needed in the proof of theorem below.\n\n\\begin{lemma}\\label{lemma 4.13.1\n\tThe mapping $\\varphi_{{}_{E_6,\\nu_3}}:Sp(1) \\times S(U(1)\\times U(5)) \\to (E_6)^{\\nu_3}$ of \\,Theorem {\\rm \\ref{theorem 3.3.5}} satisfies the relational formula \n\t\\begin{align*}\n\tw_3=\\varphi_{{}_{E_6,\\nu_3}}(1, \\diag(\\tau\\omega,\\omega,\\tau\\omega,\\omega,\\tau\\omega,\\omega)),\n\t\\end{align*}\n\twhere $\\nu=\\exp(2\\pi i\/9)\\in U(1)$.\n\\end{lemma}\n\\begin{proof}\n\tFrom Lemma \\ref{lemma 3.3.8} (1), these results are trivial. \n\\end{proof}\n\nThe $C$-linear transformation $w'_3$ defined in the Case 8 is expressed by\n\\begin{align*}\nw'_3=\\varphi_{{}_{E_6,\\nu_3}}(1, \\diag(\\tau\\omega,\\tau\\omega,\\tau\\omega,\\omega,\\omega,\\omega)),\n\\end{align*}\nand note that $\\delta_N=\\varphi_{{}_{E_6, \\nu_3}}(1,N)(=\\varphi_{{}_{E_6, \\gamma_3}}(1,N))$, where $\\delta_N$ is also defined in the Case 8, needless to say, $w_3$ is conjugate to $w'_3$ under $\\delta_N=\\varphi_{{}_{E_6, \\nu_3}}(1,N)$.\n\\vspace{1mm}\n\nThen we have the following proposition.\n\n\\begin{proposition}\\label{proposition 4.13.2\n\tThe group $(E_6)^{\\nu_3} \\cap (E_6)^{w_3}$ is isomorphic to the group $(E_6)^{\\gamma_3} \\cap (E_6)^{w'_3}${\\rm :} $(E_6)^{\\nu_3} \\cap (E_6)^{w_3} \\cong (E_6)^{\\nu_3} \\cap (E_6)^{w'_3}$.\n\\end{proposition}\n\\begin{proof}\n\tWe define a mapping $g_{{}_{4132}}: (E_6)^{\\nu_3} \\cap (E_6)^{w'_3} \\to (E_6)^{\\nu_3} \\cap (E_6)^{w_3}$ by\n\t\\begin{align*}\n\tg_{{}_{4132}}(\\alpha)=\\delta_N\\alpha{\\delta_N}^{-1},\t\n\t\\end{align*}\n\twhere $\\delta_N$ is same one above. Since it is easy to verify that $\\delta_N \\nu_3=\\nu_3\\delta_N$ using $\\nu_3=\\varphi_{{}_{E_6,\\nu_3}}(1,\\diag(\\nu^5,\\nu^{-1},\\nu^{-1},\\nu^{-1},\\nu^{-1},\\nu^{-1}))$ (Lemma \\ref{lemma 4.9.1}) and $w_3\\delta_N=\\delta_N w'_3$ (Lemma \\ref{lemma 4.13.1}), we can prove this proposition as in the proof of Proposition \\ref{proposition 4.8.1}.\n\\end{proof}\n\nSubsequently, we will prove the following lemma.\n\n\\begin{lemma}\\label{lemma 4.13.3\n\tThe group $S(U(1)\\times U(2)\\times U(3))$ is isomorphic to the group $(U(1)\\times U(1)\\times SU(2)\\times SU(3))\/(\\Z_2\\times \\Z_3)${\\rm :} $S(U(1)\\times U(2)\\times U(3)) \\cong (U(1)\\times U(1)\\times SU(2)\\times SU(3))\/(\\Z_2\\times \\Z_3), \\Z_2\\!=\\{(1,1,E,E),(-1,1,-E,E) \\},\\Z_3\\!=\\!\\{(1,1,E,E),(1,\\omega,E,\\omega E),(1,\\omega^{-1},E,\\omega^{-1}E) \\}$.\t\n\\end{lemma}\n\\begin{proof}\n\tWe define a mapping $f_{{}_{4133}}:U(1) \\times U(1)\\times SU(2)\\times SU(3) \\to S(U(1)\\times U(2)\\times U(3))$ by\n\t\\begin{align*}\n\tf_{{}_{4133}}(a,b,A,B)=\\left( \n\t\\begin{array}{ccc}\n\ta^{-2}b^{-3} & & {\\raisebox{-7pt}[0pt]{\\large $0$}}\n\t\\\\[2mm]\n\t& a\\mbox{\\large {$A$}} & \n\t\\\\[2mm]\n\t{\\raisebox{1pt}[0pt]{\\large $0$}}&& b\\mbox{\\large {$B$}}\n\t\\end{array}\\right) \\in SU(6).\n\t\\end{align*}\n\tThen it is clear that $f_{{}_{4133}}$ is well-defined and a homomorphism. \n\t\n\tWe will prove that $f_{{}_{4133}}$ is surjective. Let $P \\in S(U(1)\\times U(2)\\times U(3))$. Then $P$ takes the form of $\\diag(s,P_1,P_2),s \\in U(1),P_1 \\in U(2), P_2 \\in U(3),s(\\det\\,P_1)(\\det\\,P_2)=1$. Here, since $P_1 \\in U(2), P_2 \\in U(3)$, we see that $\\det\\,P_1, \\det\\,P_2 \\in U(1)$. We choose $a,b \\in U(1)$ such that $a^2=\\det\\,P_1, b^3=\\det\\,P_2$, respectively, and set $A=(1\/a)P_1, B=(1\/b)P_2$. Then we have that $ A \\in SU(2), B \\in SU(3)$. With above, the proof of surjective is completed.\n\t\n\tFinally, we will determine $\\Ker\\,f_{{}_{4133}}$. It follows from the kernel of definition that\n\t\\begin{align*}\n\t\\Ker\\,f_{{}_{4133}}&=\\{(a,b,A,B)\\in U(1)\\times U(1)\\times SU(2) \\times SU(3) \\,|\\,f_{{}_{4133}}(a,b,A,B)=E \\}\n\t\\\\\n\t&=\\{(a,b,A,B)\\in U(1)\\times U(1)\\times SU(2) \\times SU(3)\\,|\\,a^2b^3=1,aA=bB=E \\}\n\t\\\\\n\t&=\\{(a,b,a^{-1}E,b^{-1}E)\\in U(1)\\times U(1)\\times SU(2) \\times SU(3) \\,|\\,a^2=b^3=1 \\}\n\t\\\\\n\t&=\\{(1,1,E,E), (1,\\omega,E,{\\omega}^{-1}E),(1,{\\omega}^{-1},E,\\omega E), \n\t\\\\\n\t&\\hspace*{20mm}(-1,1,-E,E), (-1,\\omega,-E,{\\omega}^{-1}E),(-1,{\\omega}^{-1},E,\\omega E)\\}\n\t\\\\\n\t&=\\{(1,1,E,E), (-1,1,-E,E) \\} \\times \\{(1,1,E,E), (1,\\omega,E,{\\omega}^{-1}E),(1,{\\omega}^{-1},E,\\omega E) \\}\n\t\\\\\n\t& \\cong \\Z_2 \\times \\Z_3.\n\t\\end{align*}\n\t\n\tTherefore we have the required isomorphism \n\t\\begin{align*}\n\tS(U(1)\\times U(2)\\times U(3)) \\cong (U(1) \\times U(1)\\times SU(2)\\times SU(3))\/(\\Z_2\\times\\Z_3).\n\t\\end{align*}\n\\end{proof} \n\nNow, we will determine the structure of the group $(E_6)^{\\nu_3} \\cap (E_6)^{w_3}$.\n\n\\begin{theorem}\\label{theorem 4.13.4\n\tThe group $(E_6)^{\\nu_3} \\cap (E_6)^{w_3}$ is isomorphic the group $(Sp(1)\\times U(1) \\times SU(2)\\times SU(3))\/(\\Z_2 \\times \\Z_3)${\\rm :} $(E_6)^{\\nu_3} \\cap (E_6)^{w_3} \\cong ((Sp(1)\\times U(1) \\times SU(2)\\times SU(3)))\/(\\Z_2 \\times \\Z_2 \\times \\Z_3), \\Z_2\\{(1,1,1,E,E), (1,-1,1,-E,E)\\},\\Z_2=\\{(1,1,1,E,E), (-1,-1,-1,E,E)\\}\n\t\\Z_3=\\{(1,\\allowbreak 1,1,E,E), (1,1,\\omega,E,{\\omega}^{-1}E),(1,1,{\\omega}^{-1},E,\\omega E)\\}$.\n\\end{theorem}\n\\begin{proof}\n\tLet $S(U(1)\\times U(2)\\times U(3)) \\subset S(U(1) \\times U(5))$. \n\tWe define a mapping $\\varphi_{{}_{E_6,\\nu_3,w'_3}}: Sp(1)\\times S(U(1)\\times U(2)\\times U(3)) \\to (E_6)^{\\nu_3} \\cap (E_6)^{w'_3}$ by\n\t\\begin{align*}\n\t\\varphi_{{}_{E_6,\\nu_3,w'_3}}(q, P)(M+\\a)&={k_J}^{-1}(P(k_J M){}^t\\!P)+q\\a k^{-1}(\\tau \\,{}^t\\!P), \n\t\\\\\n\t&\\hspace*{40mm}M+\\a \\in \\mathfrak{J}(3, \\H)^C \\oplus (\\H^3)^C=\\mathfrak{J}^C.\n\t\\end{align*}\n\tNeedless to say, this mapping is the restriction of the mapping $\\varphi_{{}_{E_6,\\nu_3}}$, that is, $\\varphi_{{}_{E_6,\\nu_3,w'_3}}(q, P)=\\varphi_{{}_{E_6,\\nu_3}}(q,P)$ (Theorem \\ref{theorem 3.3.5}). \n\t\n\tAs usual, we will prove that $\\varphi_{{}_{E_6,\\nu_3,w'_3}}$ is well-defined. It is clear that $\\varphi_{{}_{E_6,\\nu_3,w'_3}}(q,P) \\in (E_6)^{\\nu_3}$, and it follows from $w'_3=\\varphi_{{}_{E_6,\\nu_3}}(1,\\diag(\\tau\\omega,\\tau\\omega,\\tau\\omega,\\omega,\\omega,\\omega))$ that\n\t\\begin{align*}\n\t&\\quad {w'_3}^{-1}\\varphi_{{}_{E_6,\\nu_3,\\nu'_3}}(q,P) w'_3\n\t\\\\\n\t&=\\varphi_{{}_{E_6,\\nu_3}}(1,\\diag(\\tau\\omega,\\tau\\omega,\\tau\\omega,\\omega,\\omega,\\omega))^{-1}\\varphi_{{}_{E_6,\\nu_3,w'_3}}(q,P)\\varphi_{{}_{E_6,\\nu_3}}(1,\\diag(\\tau\\omega,\\tau\\omega,\\tau\\omega,\\omega,\\omega,\\omega))\n\t\\\\\n\t&=\\varphi_{{}_{E_6,\\nu_3}}(1,\\diag(\\omega,\\omega,\\omega,\\tau\\omega,\\tau\\omega,\\tau\\omega))\\varphi_{{}_{E_6,\\nu_3}}(q,P)\\varphi_{{}_{E_6,\\nu_3}}(1,\\diag(\\tau\\omega,\\tau\\omega,\\tau\\omega,\\omega,\\omega,\\omega))\n\t\\\\\n\t&=\\varphi_{{}_{E_6,\\nu_3}}(q,\\diag(\\omega,\\omega,\\omega,\\tau\\omega,\\tau\\omega,\\tau\\omega)P\\,\\diag(\\tau\\omega,\\tau\\omega,\\tau\\omega,\\omega,\\omega,\\omega)),P=\\diag(s,P_1,P_2)\n\t\\\\\n\t&=\\varphi_{{}_{E_6,\\nu_3}}(q,\\diag(\\omega s (\\tau\\omega),(\\omega E)P_1(\\tau\\omega E), \\tau(\\omega E) P_2 (\\omega E) ))\n\t\\\\\n\t&=\\varphi_{{}_{E_6,\\nu_3}}(q,P)\n\t\\\\\n\t&=\\varphi_{{}_{E_6,\\nu_3,w'_3}}(q,P).\n\t\\end{align*}\n\tHence we have that $\\varphi_{{}_{E_6,\\nu_3,w'_3}}(s,P) \\in (E_6)^{w'_3}$. Thus $\\varphi_{{}_{E_6,\\nu_3,w'_3}}$ is well-defined. Subsequently, since $\\varphi_{{}_{E_6,\\nu_3,w'_3}}$ is the restriction of the mapping $\\varphi_{{}_{E_6,\\nu_3}}$, we easily see that $\\varphi_{{}_{E_6,\\nu_3,w'_3}}$ is a homomorphism.\n\t\n\tNext, we will prove that $\\varphi_{{}_{E_6,\\nu_3,w'_3}}$ is surjective. Let $\\alpha \\in (E_6)^{\\nu_3} \\cap (E_6)^{w'_3} \\subset (E_6)^{\\nu_3}$. There exist $q \\in Sp(1)$ and $A \\in S(U(1)\\times U(5))$ such that $\\alpha=\\varphi_{{}_{E_6,\\nu_3}}(q,A)$ (Theorem \\ref{theorem 3.3.5}). Moreover, from the condition $\\alpha \\in (E_6)^{w'_3}$, that is, ${w'_3}^{-1}\\varphi_{{}_{E_6,\\nu_3}}(q,A)w'_3=\\varphi_{{}_{E_6,\\nu_3}}(q,A)$, and using ${w'_3}^{-1}\\varphi_{{}_{E_6,\\nu_3}}(q,A)w'_3=\\varphi_{{}_{E_6,\\nu_3}}(q,\\diag(\\omega,\\omega,\\omega,\\tau\\omega,\\tau\\omega,\\tau\\omega)A\\,\\diag(\\tau\\omega,\\tau\\omega,\\tau\\omega,\\omega,\\omega,\\omega))$, we have that\n\t\\begin{align*}\n\t&\\left\\{\n\t\\begin{array}{l}\n\tq=q \\\\\n\t\\diag(\\omega,\\omega,\\omega,\\tau\\omega,\\tau\\omega,\\tau\\omega)A\\,\\diag(\\tau\\omega,\\tau\\omega,\\tau\\omega,\\omega,\\omega,\\omega)=A   \n\t\\end{array}\\right. \n\t\\\\\n\t&\\hspace*{50mm}{\\text{or}}\n\t\\\\\n\t&\\left\\{\n\t\\begin{array}{l}\n\tq=-q \\\\\n\t\\diag(\\omega,\\omega,\\omega,\\tau\\omega,\\tau\\omega,\\tau\\omega)A\\,\\diag(\\tau\\omega,\\tau\\omega,\\tau\\omega,\\omega,\\omega,\\omega)=-A.   \n\t\\end{array}\\right.     \n\t\\end{align*}\n\tThe latter case is impossible because of $q\\not=0$. As for the former case, from the second condition, by doing straightforward computation $A$ takes the following form $\\diag(s,C, D), C \\in U(2),D \\in U(3), s(\\det\\,C)(\\det\\,D)=1$, that is, $A \\in S(U(1)\\times U(2)\\times U(3))$. Needless to say, $q \\in Sp(1)$.\n\tHence there exist $q \\in Sp(1)$ and $P \\in S(U(1)\\times U(2)\\times U(3))$ such that $\\alpha=\\varphi_{{}_{E_6,\\nu_3}}(q,P)$. Namely, there exist $q \\in Sp(1)$ and $P \\in S(U(1)\\times U(2)\\times U(3))$ such that $\\alpha=\\varphi_{{}_{E_6,\\nu_3,w'_3}}(q,P)$. The proof of surjective is completed.\n\t\n\tFinally, we will determine $\\Ker\\,\\varphi_{{}_{E_6,\\nu_3,w'_3}}$. However, from $\\Ker\\,\\varphi_{{}_{E_6,\\nu_3}}=\\{(1,E),(-1,-E) \\}$, we easily obtain that $\\Ker\\,\\varphi_{{}_{E_6,\\nu_3,w'_3}}=\\{(1,E),(-1,-E) \\} \\cong \\Z_2$. Thus we have the isomorphism $(E_6)^{\\nu_3} \\cap (E_6)^{w'_3} \\cong (Sp(1)\\times S(U(1)\\times U(2)\\times U(3)))\/\\Z_2$. In addition, by Proposition \\ref{proposition 4.13.2} we have the isomorphism $(E_6)^{\\nu_3} \\cap (E_6)^{w_3} \\cong (Sp(1)\\times S(U(1)\\times U(2)\\times U(3)))\/\\Z_2$. Here, using the mapping $f_{{}_{4133}}$ in the proof of Lemma \\ref{lemma 4.13.3}, we define a homomorphism $h_{{}_{4134}}:Sp(1)\\times (U(1)\\times U(1)\\times SU(2)\\times SU(3)) \\to Sp(1)\\times S(U(1)\\times U(2)\\times U(3)))$ by\n\t\\begin{align*}\n\th_{{}_{4134}}(q,(a,b,A,B))=(q,f_{{}_{4133}}(a,b,A,B)).\n\t\\end{align*}\n\tThen, the elements of $(q,(a,b,A,B))$ corresponding to the elements \n\t $(1,E), (-1,-E) \\in \\Ker\\,\\varphi_{{}_{E_6,\\nu_3,w'_3}}$ under the mapping $h_{{}_{4134}}$ are as follows.\n\t\\begin{align*}\n\t& (1,1,1,E,E), (1,1,\\omega,E,{\\omega}^{-1}E),(1,1,{\\omega}^{-1},E,\\omega E), (1,-1,1,-E,E),\n\t\\\\\n\t& (1,-1,\\omega,-E,{\\omega}^{-1}E),(1,-1,{\\omega}^{-1},-E,\\omega E),\n\t\\\\\n\t& (-1,1,-1,-E,E), (-1,1,-\\omega,-E,{\\omega}^{-1}E),(-1,1,-{\\omega}^{-1},-E,\\omega E), (-1,-1,-1,E,E),\n\t\\\\\n\t& (-1,-1,-\\omega,E,{\\omega}^{-1}E),(-1,-1,-{\\omega}^{-1},E,\\omega E).\n\t\\end{align*}\n\t\n\tTherefore we have the required isomorphism\n\t\\begin{align*}\n\t(E_6)^{\\nu_3} \\cap (E_6)^{w_3} \\cong (Sp(1) \\times U(1)\\times U(1) \\times SU(2)\\times SU(3))\/(\\Z_2\\times \\Z_2\\times \\Z_3), \n\t\\end{align*}\n\twhere\n\t\\begin{align*}\n\t&\\Z_2=\\{(1,1,1,E,E), (1,-1,1,-E,E)\\},\n\t\\\\\n\t&\\Z_2=\\{(1,1,1,E,E), (-1,-1,-1,E,E)\\},\n\t\\\\\n\t&\\Z_3=\\{(1,1,1,E,E), (1,1,\\omega,E,{\\omega}^{-1}E),(1,1,{\\omega}^{-1},E,\\omega E)\\}.\n\t\\end{align*}\n\\end{proof}\n\nThus, since the group $(E_6)^{\\nu_3} \\cap (E_6)^{w_3}$ is connected from Theorem \\ref{theorem 4.13.4}, we have an exceptional $\\varmathbb{Z}_3 \\times \\varmathbb{Z}_3$-symmetric space\n\\begin{align*}\nE_6\/((Sp(1) \\times U(1)\\times U(1) \\times SU(2)\\times SU(3))\/(\\Z_2\\times \\Z_2\\times \\Z_3)).\n\\end{align*}\n\n\\subsection{Case 14: $\\{1, \\tilde{\\mu}_3,  \\tilde{\\mu}_3{}^{-1}\\} \\times \\{1, \\tilde{w}_3,  \\tilde{w}_3{}^{-1}\\}$-symmetric space}\n\nLet the $C$-linear transformations $\\mu_3, w_3$ of $\\mathfrak{J}^C$  defined in Subsection \\ref{subsection 3.3}. \n\n\\noindent From Lemma \\ref{lemma 3.3.8} (2), since we can easily confirm that $\\mu_3$ and $w_3$ are commutative, $\\tilde{\\mu}_3$ and $\\tilde{w}_3$ are commutative in $\\Aut(E_6)$: $\\tilde{\\mu}_3\\tilde{w}_3=\\tilde{w}_3\\tilde{\\mu}_3$.\n\nNow, we will determine the structure of the group $(E_6)^{\\mu_3}\\cap (E_6)^{w_3}$.\n\n\\begin{theorem}\\label{theorem 4.14.1\n\tThe group $(E_6)^{\\mu_3}\\cap (E_6)^{w_3}$ is isomorphic to the group $(SU(3)\\times U(1)\\times U(1)\\times U(1)\\times U(1))\/\\Z_3${\\rm :} $(E_6)^{\\mu_3}\\cap (E_6)^{w_3} \\cong (SU(3)\\times U(1)\\times U(1)\\times U(1)\\times U(1))\/\\Z_3, \\Z_3=\\{(E,1,1,1,1),(\\bm{\\omega}E,\\bm{\\omega},\\bm{\\omega},\\bm{\\omega},\\bm{\\omega}),(\\bm{\\omega}^{-1}E,\\bm{\\omega}^{-1},\\bm{\\omega}^{-1},\\bm{\\omega}^{-1},\\bm{\\omega}^{-1})\\}$.\n\\end{theorem}\n\\begin{proof}\n\tLet $S(U(1)\\times U(1)\\times U(1)) \\subset SU(3)$. We define a mapping $\\varphi_{{}_{E_6,\\mu_3,w_3}}: SU(3)\\times S(U(1)\\times U(1)\\times U(1))\\times S(U(1)\\times U(1)\\times U(1)) \\to (E_6)^{\\nu_3}\\cap (E_6)^{w_3}$ by\n\t\\begin{align*}\n\t\\varphi_{{}_{E_6,\\mu_3,w_3}}(L,P,Q)(X_{C}+M)&=h(P,Q)X_{C}h(P,Q)^*+LM\\tau h(P,Q)^*, \n\t\\\\\n\t&\\hspace*{20mm} X_{C}+M \\in \\mathfrak{J}(3, \\C)^C \\oplus \n\tM(3,\\C)^C=\\mathfrak{J}^C.\n\t\\end{align*}\n\tNeedless to say, this mapping is the restriction of the mapping $\\varphi_{{}_{E_6,w_3}}$, that is, $\\varphi_{{}_{E_6,\\nu_3,w_3}}(L,P,\\allowbreak Q)\\allowbreak=\\varphi_{{}_{E_6,w_3}}(L,P,Q)$ (Theorem \\ref{theorem 3.3.7}). \n\t\n\tAs usual, we will prove that $\\varphi_{{}_{E_6,\\mu_3,w_3}}$ is well-defined. It is clear that $\\varphi_{{}_{E_6,\\mu_3,_3}}(L,P,Q) \\in (E_6)^{w_3}$, and it follows from $\\mu_3=\\varphi_{{}_{E_6,w_3}}(E,\\diag({\\bm{\\varepsilon}}^{-2},\\bm{\\varepsilon},\\bm{\\varepsilon}), \\diag({\\bm{\\varepsilon}}^{2},\\bm{\\varepsilon}^{-1},\\bm{\\varepsilon}^{-1}))$ (Lemma \\ref{lemma 3.3.8} (2)) that \n\t\\begin{align*}\n\t&\\quad {\\mu_3}^{-1}\\varphi_{{}_{E_6,\\sigma_3,w_3}}(L,P,Q)\\mu_3\n\t\\\\\n\t&=\\varphi_{{}_{E_6,w_3}}(E,\\diag({\\bm{\\varepsilon}}^{-2},\\bm{\\varepsilon},\\bm{\\varepsilon}), \\diag({\\bm{\\varepsilon}}^{2},\\bm{\\varepsilon}^{-1},\\bm{\\varepsilon}^{-1}))^{-1}\\varphi_{{}_{E_6,\\mu_3,w_3}}(L,P,Q)\n\t\\\\\n\t&\\hspace*{70mm}\\varphi_{{}_{E_6,w_3}}(E,\\diag({\\bm{\\varepsilon}}^{-2},\\bm{\\varepsilon},\\bm{\\varepsilon}), \\diag({\\bm{\\varepsilon}}^{2},\\bm{\\varepsilon}^{-1},\\bm{\\varepsilon}^{-1}))\n\t\\\\\n\t&=\\varphi_{{}_{E_6,w_3}}(E,\\diag({\\bm{\\varepsilon}}^{2},\\bm{\\varepsilon}^{-1},\\bm{\\varepsilon}^{-1}), \\diag({\\bm{\\varepsilon}}^{-2},\\bm{\\varepsilon},\\bm{\\varepsilon}))\\varphi_{{}_{E_6,\\mu_3,w_3}}(L,P,Q)\n\t\\\\\n\t&\\hspace*{70mm}\\varphi_{{}_{E_6,w_3}}(E,\\diag({\\bm{\\varepsilon}}^{-2},\\bm{\\varepsilon},\\bm{\\varepsilon}), \\diag({\\bm{\\varepsilon}}^{2},\\bm{\\varepsilon}^{-1},\\bm{\\varepsilon}^{-1}))\n\t\\\\\n\t&=\\varphi_{{}_{E_6,w_3}}(L,\\diag({\\bm{\\varepsilon}}^{2},\\bm{\\varepsilon}^{-1},\\bm{\\varepsilon}^{-1})P\\diag({\\bm{\\varepsilon}}^{-2},\\bm{\\varepsilon},\\bm{\\varepsilon}),\\diag({\\bm{\\varepsilon}}^{-2},\\bm{\\varepsilon},\\bm{\\varepsilon})Q \\diag({\\bm{\\varepsilon}}^{2},\\bm{\\varepsilon}^{-1},\\bm{\\varepsilon}^{-1})),\n\t\\\\\n\t&\\hspace*{90mm}P=\\diag(a,b,c), Q=\\diag(s,t,v)\n\t\\\\\n\t&=\\varphi_{{}_{E_6,w_3}}(L,P,Q)\n\t\\\\\n\t&=\\varphi_{{}_{E_6,\\mu_3,w_3}}(L,P,Q).\n\t\\end{align*}\n\tHence we have that $\\varphi_{{}_{E_6,\\mu_3,w_3}}(L,P,Q) \\in (E_6)^{\\mu_3}$. Thus $\\varphi_{{}_{E_6,\\mu_3,w_3}}$ is well-defined. Subsequently, since $\\varphi_{{}_{E_6,\\mu_3,w_3}}$ is the restriction of the mapping $\\varphi_{{}_{E_6,w_3}}$, we easily see that $\\varphi_{{}_{E_6,\\mu_3,w_3}}$ is a homomorphism.\n\t\n\tNext we will prove that $\\varphi_{{}_{E_6,\\mu_3,w_3}}$ is surjective. Let $\\alpha \\in (E_6)^{\\mu_3}\\cap (E_6)^{w_3} \\subset (E_6)^{w_3}$. There exist $L, A, B \\in SU(3)$ such that $\\alpha=\\varphi_{{}_{E_6,w_3}}(L,A,B)$ (Theorem \\ref{theorem 3.3.7}). Moreover, from the condition $\\alpha \\in (E_6)^{\\mu_3}$, that is, ${\\mu_3}^{-1}\\varphi_{{}_{E_6,w_3}}(L,A,B)\\mu_3=\\varphi_{{}_{E_6,w_3}}(L,A,B)$, and using \n\t\\begin{align*}\n\t&\\quad {\\mu_3}^{-1}\\varphi_{{}_{E_6,w_3}}(L,A,B)\\mu_3\n\t\\\\\n\t&=\\varphi_{{}_{E_6,w_3}}(L,\\diag({\\bm{\\varepsilon}}^{2},\\bm{\\varepsilon}^{-1},\\bm{\\varepsilon}^{-1})A\\,\\diag({\\bm{\\varepsilon}}^{-2},\\bm{\\varepsilon},\\bm{\\varepsilon}),\\diag({\\bm{\\varepsilon}}^{-2},\\bm{\\varepsilon},\\bm{\\varepsilon})B \\,\\diag({\\bm{\\varepsilon}}^{2},\\bm{\\varepsilon}^{-1},\\bm{\\varepsilon}^{-1}))\n\t\\end{align*}\n\t(Lemma \\ref{lemma 3.3.8} (2)) we have that \n\t\\begin{align*}\n\t&\\,\\,\\,{\\rm(i)}\\,\\left\\{\n\t\\begin{array}{l}\n\tL=L\n\t\\\\\n\t\\diag({\\bm{\\varepsilon}}^{2},\\bm{\\varepsilon}^{-1},\\bm{\\varepsilon}^{-1})A\\diag({\\bm{\\varepsilon}}^{-2},\\bm{\\varepsilon},\\bm{\\varepsilon})=A \n\t\\\\\n\t\\diag({\\bm{\\varepsilon}}^{-2},\\bm{\\varepsilon},\\bm{\\varepsilon})B \\diag({\\bm{\\varepsilon}}^{2},\\bm{\\varepsilon}^{-1},\\bm{\\varepsilon}^{-1})=B,\n\t\\end{array} \\right.\n\n\t\\\\[2mm]\n\t&{\\rm(ii)}\\,\\left\\{\n\t\\begin{array}{l}\n\tL=\\bm{\\omega}L\n\t\\\\\n\t\\diag({\\bm{\\varepsilon}}^{2},\\bm{\\varepsilon}^{-1},\\bm{\\varepsilon}^{-1})A\\diag({\\bm{\\varepsilon}}^{-2},\\bm{\\varepsilon},\\bm{\\varepsilon})=\\bm{\\omega}A \n\t\\\\\n\t\\diag({\\bm{\\varepsilon}}^{-2},\\bm{\\varepsilon},\\bm{\\varepsilon})B \\diag({\\bm{\\varepsilon}}^{2},\\bm{\\varepsilon}^{-1},\\bm{\\varepsilon}^{-1})=\\bm{\\omega}B,\n\t\\end{array} \\right.\n\t\\\\[2mm]\n\t&{\\rm(iii)}\\,\\left\\{\n\t\\begin{array}{l}\n\tL=\\bm{\\omega}^{-1}L\n\t\\\\\n\t\\diag({\\bm{\\varepsilon}}^{2},\\bm{\\varepsilon}^{-1},\\bm{\\varepsilon}^{-1})A\\diag({\\bm{\\varepsilon}}^{-2},\\bm{\\varepsilon},\\bm{\\varepsilon})=\\bm{\\omega}^{-1}A \n\t\\\\\n\t\\diag({\\bm{\\varepsilon}}^{-2},\\bm{\\varepsilon},\\bm{\\varepsilon})B \\diag({\\bm{\\varepsilon}}^{2},\\bm{\\varepsilon}^{-1},\\bm{\\varepsilon}^{-1})=\\bm{\\omega}^{-1}B.\n\t\\end{array} \\right.\n\t\\end{align*}\n\tThe Cases (ii) and (iii) are impossible because $L\\not=0$. As for the Case (i), from the second and third conditions, it is easy to see that $A,B \\in S(U(1)\\times U(1) \\times U(1))$. Needless to say, $L \\in SU(3)$. Hence there exist $L \\in SU(3)$ and $P,Q \\in S(U(1)\\times U(1) \\times U(1))$ such that $\\alpha=\\varphi_{{}_{E_6,w_3}}(L,P,Q)$. Namely, there exist $L \\in SU(3)$ and $P,Q \\in S(U(1)\\times U(1) \\times U(1))$ such that $\\alpha=\\varphi_{{}_{E_6,\\mu_3, w_3}}(L,P,Q)$. The proof of surjective is completed.\n\t\n\tFinally, we will determine $\\Ker\\,\\varphi_{{}_{E_6,\\mu_3,w_3}}$. However, from $\\Ker\\,\\varphi_{{}_{E_6,w_3}}=\\{(E,E,E),(\\bm{\\omega}E,\\allowbreak \\bm{\\omega}E,\\bm{\\omega}E),(\\bm{\\omega}^{-1}E,\\bm{\\omega}^{-1}E, \\bm{\\omega}^{-1}E) \\}$, we easily obtain that $\\Ker\\,\\varphi_{{}_{E_6,\\mu_3,w_3}}=\\{(E,E,E),(\\bm{\\omega}E,\\allowbreak \\bm{\\omega}E,\\bm{\\omega}E),(\\bm{\\omega}^{-1}E,\\bm{\\omega}^{-1}E, \\bm{\\omega}^{-1}E) \\} \\cong \\Z_3$.\n\tThus we have the isomorphism $(E_6)^{\\mu_3}\\cap (E_6)^{w_3} \\cong (SU(3)\\times S(U(1)\\times U(1)\\times U(1))\\times S(U(1)\\times U(1)\\times U(1)))\/\\Z_3$. \n\t\n\tTherefore, by Lemma \\ref{lemma 4.3.1} we have the required isomorphism \n\t\\begin{align*}\n\t(E_6)^{\\mu_3}\\cap (E_6)^{w_3} \\cong (SU(3)\\times U(1)\\times U(1)\\times U(1)\\times U(1))\/\\Z_3,\n\t\\end{align*}\n\twhere $\\Z_3=\\{(E,1,1,1,1),(\\bm{\\omega}E,\\bm{\\omega},\\bm{\\omega},\\bm{\\omega},\\bm{\\omega}),(\\bm{\\omega}^{-1}E,\\bm{\\omega}^{-1},\\bm{\\omega}^{-1},\\bm{\\omega}^{-1},\\bm{\\omega}^{-1})\\}$.\n\\end{proof}\n\nThus, since the group $(E_6)^{\\mu_3} \\cap (E_6)^{w_3}$ is connected from Theorem \\ref{theorem 4.14.1}, we have an exceptional $\\varmathbb{Z}_3 \\times \\varmathbb{Z}_3$-symmetric space\n\\begin{align*}\nE_6\/((SU(3)\\times U(1)\\times U(1)\\times U(1)\\times U(1))\/\\Z_3).\n\\end{align*}\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\nUpcoming 21~cm surveys are poised to make a first detection of redshifted\n21~cm fluctuations from the EoR within the next several years\n\\citep{DeBoer:2016tnn}. These measurements will provide a direct probe of the\ndistribution of neutral hydrogen in the IGM, revealing the spatial structure\nof the reionization process, and its redshift evolution. Along with these\nmeasurements, several other ``line-intensity'' mapping surveys are planned to\nmap out large-scale structure in the galaxy distribution using convenient\nemission lines with current targets including [C~\\textsc{ii}], CO, Ly-$\\alpha$, and\nH-$\\alpha$ \\citep[see e.g.][and references therein]{kovetz2017:im_review}.\nThese surveys study the spatial fluctuations in the collective emission from\nmany individually unresolved sources (e.g.\n\\citealt{Suginohara:1998ti,Righi:2008br,Visbal10}). These measurements should\nnicely complement 21~cm observations (e.g. \\citealt{Lidz11,gong11:probing}):\nwhile the 21~cm fluctuations trace-out remaining neutral hydrogen residing\nmostly in the low-density IGM, the galactic emission lines track the galaxies\nthemselves, which presumably lie within ``bubbles'' of mostly ionized hydrogen\n\\citep{Lidz:2008ry}.\n\nIn fact, recent work has led to detections in various lines at low redshift\n\\citep{2010Natur.466..463C,2001defi.conf..241D,Keating:2016pka,2018MNRAS.478.1911P,2016MNRAS.457.3541C,Croft:2018rwv},\nbolstering efforts to employ the line-intensity mapping technique at earlier\ntimes during the EoR. It is hence timely to explore the scientific benefits of\ncombining 21~cm observations of the EoR with line-intensity mapping surveys in\nother emission lines.\n\nHere we consider, for the first time, one potential advantage of combining\n21~cm surveys of the EoR with line-intensity mapping surveys in {\\em two\nadditional lines.} Specifically, we show that the linear bias factor of the\n21~cm field may be extracted solely from cross-power spectra between the 21~cm\nfluctuations and those in each of two separate lines. This can provide an\nimportant cross-check on inferences from the 21~cm auto-power spectrum since\ncross-power spectra should be less prone to bias from residual foregrounds\n\\citep[e.g.][]{Furlanetto:2006pg,Lidz:2008ry}; only shared foregrounds\ncontribute to the average cross spectrum signal. \n\nThe foreground problem is especially daunting in the case of redshifted\n21 cm surveys, where the expected foreground-to-signal strength is on the\norder of $\\sim 10^5$\n\\citep[e.g.][]{2009A&A...500..965B,2013ApJ...768L..36P,Dillon:2013rfa}. The\nbasic strategy for extracting the signal is to exploit the fact that the\nforegrounds should be smooth functions of frequency, while the reionization\nsignal has a great deal of spectral structure. In practice, this is\nchallenging because the instrument, calibration errors, and other effects may\nimprint artificial spectral variations. Cross-spectrum measurements should be\nless sensitive to such systematic effects and can therefore help confirm early\ndetections. For instance, \\citet{2015JCAP...03..034V} show that cross-spectra\ncan be robustly measured even in the presence of polarized synchrotron\nforegrounds; this is a troublesome case for auto-spectrum analyses because\nFaraday rotation leads to frequency structure.\n\nThe amplitude of the 21~cm power spectrum evolves with redshift in a\ndistinctive way as reionization proceeds \\citep[e.g.][]{Lidz08}, and recent\nwork has demonstrated that linear biasing describes the large-scale 21~cm\npower spectrum rather well\n\\citep{McQuinn:2018zwa,Hoffmann:2018clb,2018ApJ...867...26B}. Therefore, if\nour three-field method may be employed over a range of redshifts, it can be\nused to extract key and robust information regarding the reionization history\nof the Universe.\n\nIn recent related work we showed that the large-scale 21~cm bias factor may be\nrecovered using suitable cross-bispectra between the 21~cm fluctuations and\nthe [C~\\textsc{ii}] emission field \\citep{2018ApJ...867...26B}. While the\ncross-bispectra method requires only the 21~cm fluctuations and one additional\ntracer field, the technique we propose here should be vastly simpler to\nimplement in practice (provided two additional tracers are available with\ncommon sky and redshift coverage). This is the case because our present method\nrelies only on two-point statistics, and it therefore avoids practical\ndifficulties in carrying out cross-bispectrum analyses. For example, it is\nchallenging to estimate the bispectrum covariance as this involves computing a\nsix-point function. In addition, we will show that our present technique\nallows for a more faithful extraction of the 21~cm bias factor. Ultimately,\nboth analyses may be carried out for additional cross-checks.\n\nThere are a broad range of possible lines that may be combined with the 21~cm\nsurveys. Currently, there are projects -- either ramping-up or in the planning\nstages -- to perform EoR-era line-intensity surveys in: [C~\\textsc{ii}]~$158\\,\\mu\n\\text{m}$ \\citep{Crites14,Lagache:2018hmk,Vavagiakis:2018gen}, rotational\ntransitions from CO molecules \\citep{Chung:2017uot}, Ly-$\\alpha$\n\\citep{Dore16}, and H-$\\alpha$ \\citep{Cooray:2016hro}. Additional\nfine-structure lines such as [O~\\textsc{iii}]~$88\\,\\mu \\text{m}$ \\citep{Moriwaki18} and\n[N \\textsc{ii}]~$122\\,\\mu \\text{m}$ \\citep{Serra:2016jzs} may also be suitable\n--- in some cases, these lines will land in the proposed frequency bands of\nthe planned [C~\\textsc{ii}] surveys. The [O~\\textsc{iii}]~$88\\,\\mu \\text{m}$ line appears\nespecially promising since targeted ALMA observations around $z \\sim 7-9$\ngalaxies have found that this line is {\\em brighter} at high redshift than\nexpected based on local correlations between line-luminosity and\nstar-formation rate \\citep[e.g.][and references therein]{Moriwaki18}.\n\nIn principle, one could extract the 21~cm bias using the cross-spectrum with a\ntraditional galaxy survey, in which case the galaxy bias may be measured\nrobustly from the auto-power spectrum. In practice, this is extremely\nchallenging because one needs {\\em spectroscopic redshifts} for the galaxy\nsurvey over a huge sky area at $z \\sim 8$. If only photometric redshifts are\navailable, then one only accesses long-wavelength line-of-sight modes (with\nsmall or vanishing line-of-sight wavenumbers) in the galaxy survey but\nprecisely these modes are lost to foreground cleaning\/avoidance in the 21~cm\nsurveys (e.g. \\citealt{Lidz:2008ry}). Fortunately, multi-line intensity\nmapping provides a promising way forward here and our approach avoids\nmeasuring bias factors from auto-spectra.\n\nIn Section~\\ref{sec:approach}, we describe our three cross-spectra approach in\ndetail. In Section~\\ref{sec:simulations} we briefly discuss the radiative\ntransfer simulations of reionization \\citep{2007MNRAS.377.1043M,Lidz08} used\nin our analysis, the reionization model assumed, and our method for generating\nmock line-intensity mapping data cubes. We then quantify the accuracy of our\ntechnique in Section~\\ref{sec:results}. The survey specifications required to\nextract bias factors with this method are discussed briefly in\nSection~\\ref{sec:detectability}. We conclude in Section~\\ref{sec:conclusions}.\nWe assume a $\\Lambda$CDM cosmology, parameterized by $(\\Omega_m,\n\\Omega_{\\Lambda}, \\Omega_b, h, \\sigma_8, n_s) = (0.27, 0.73, 0.046, 0.7, 0.8,\n1)$ as in the simulations used in this work \\citep{McQuinn:2007dy}. While\nthese parameters differ slightly from presently favored values (e.g.\n\\citealt{2018arXiv180706209P}), this should not impact our conclusions.\n\n\\section{Approach}\\label{sec:approach}\nHere we define terms and describe our three cross-spectra approach. Ignoring\nredshift-space distortions and spin-temperature fluctuations, the 21~cm\nbrightness temperature contrast between neutral hydrogen gas and the cosmic\nmicrowave background is:\n\\begin{equation}\\label{eq:brightness_temp}\nT_{21}(\\bm{x}) = T_0 X_{\\text{HI}}(\\bm{x})[1+\\delta_\\rho(\\bm{x})]\\text{.}\n\\end{equation}\nHere $T_0 = 28\\,\\text{mK}[(1+z)\/10]^{1\/2}$ \\citep[e.g.][]{Zaldarriaga:2003du},\n$X_{\\text{HI}}(\\bm{x})$ is the neutral hydrogen fraction at position $\\bm{x}$, and\n$\\delta_\\rho(\\bm{x})$ is the gas density contrast, which is assumed to follow\nthe overall matter density field on the large scales of interest. Although\nionized regions imprint large-scale fluctuations in the 21~cm field, on scales\nmuch larger than the size of the ionized regions, the 21~cm fluctuations\nshould nevertheless follow a linear biasing relation\n\\begin{equation}\\label{eq:21cm_bias}\nT_{21}(\\bm{k}) = \\pm \\avg{T_{21}} b_{21} \\delta_{\\text{lin}}(\\bm{k})\\text{,}\n\\end{equation}\nwhere the $\\pm$ indicates that the fields are either correlated ($+$) or\nanti-correlated ($-$) --- during the bulk of the EoR, the 21~cm and density\nfields are anti-correlated on large scales in most models\n\\citep[e.g.][]{Lidz:2008ry}. Here $T_{21}(\\bm{k})$ is the Fourier transform of\nthe brightness temperature field (Equation~\\ref{eq:brightness_temp}) and\n$\\delta_{\\text{lin}}(\\bm{k})$ is the Fourier transform of the linear density\ncontrast.\\footnote{Our Fourier convention is: $T_{21}(\\bm{k}) = \\int\n\\text{d}^3x\\, T_{21}(\\bm{x}) e^{i \\bm{k} \\cdot \\bm{x}}$ and $T_{21}(\\bm{x}) =\n\\int \\frac{\\text{d}^3k}{(2\\pi)^3}\\, T_{21}(\\bm{k}) e^{-i \\bm{k} \\cdot\n\\bm{x}}$.} The quantity $b_{21}$ is the dimensionless, and scale-independent,\nlinear bias factor of the 21~cm fluctuation contrast, $\\delta_{21}(\\bm{x}) =\n\\left(T_{21}(\\bm{x}) - \\avg{T_{21}}\\right)\/\\avg{T_{21}}$, while the\n$\\avg{T_{21}}$ factor reverts to brightness temperature units (since the\naverage brightness temperature is not itself observable from interferometric\nmeasurements.) In this work when we refer to the ``bias'' we mean\n$\\avg{T_{21}}b_{21}$ (and likewise for the intensity mapping surveys.)\n\nLikewise, we can consider additional tracer lines, such as [C~\\textsc{ii}]. On large\nscales, the Fourier transform of the specific intensity of each of these lines\nshould be well-described by\n\\begin{equation}\\label{eq:linear_biasing}\nI_{i}(\\bm{k}) = \\avg{I_{i}} b_{i}  \\delta_{\\text{lin}}(\\bm{k})\\text{,}\n\\end{equation}\nwhere $\\avg{I_{i}}$ is the mean specific intensity of the emission\nline.\\footnote{We follow standard conventions in expressing 21~cm fluctuations\nin brightness temperature units, i.e. in $\\text{mK}$, while we use specific\nintensity units for the other tracer lines, i.e. $I_{i}$ is the specific\nintensity in $\\text{Jy\/str}$.} For the case of emission lines sourced by gas\nwithin galaxies, the relevant bias factor is the luminosity-weighted bias of\nthe line-emitting host halos (e.g. \\citealt{Lidz11}). To be completely general\nwe should also include a $\\pm$ here (as in Equation~\\ref{eq:21cm_bias}), but\nfor the galactic emission lines we generally expect brighter line emission in\noverdense regions.\n\nOn sufficiently large scales, the auto-power spectrum of the fluctuations in\neach tracer line (Equation~\\ref{eq:linear_biasing}) will be\n\\begin{equation}\\label{eq:bias_ps}\n\\begin{split}\nP_{i, i}(k, z) &\\equiv \\avg{I_{i}(k, z) I_{i}^{*}(k, z)} \\\\\n&= \\left[\\avg{I_{i}}(z) b_{i}(z)\\right]^2 P_{\\text{lin}}(k, z)\\text{,}\n\\end{split}\n\\end{equation}\nwhere $P_{\\text{lin}}(k,z)$ is the linear matter power spectrum. Similarly, on\nlarge scales the 21~cm auto-power spectrum should follow $P_{21,21}(k,z) =\n\\left[\\avg{T_{21}}(z) b_{21}(z)\\right]^2 P_{\\text{lin}}(k,z)$. In principle,\none can infer the bias factors $\\avg{I_i} b_i$ and $\\avg{T_{21}} b_{21}$ from\nauto-power spectrum measurements (assuming a model for the linear power\nspectrum). However, foreground cleaning\/avoidance present significant\nchallenges here\n\\citep[e.g.][]{2012MNRAS.419.3491L,2013ApJ...769..154M,2015ApJ...804...14T,2016ApJ...819....8P,Ewall-Wice:2016bhu}\nand residual foregrounds may bias such inferences.\n\nAnother approach is to measure the cross-power spectrum between two lines $i$\nand $j$. In this case, one measures\n\\begin{equation}\\label{eq:xps}\nP_{i,j} = r_{i, j} \\avg{I_{i}}  \\avg{I_{j}} b_{i} b_{j} P_{\\text{lin}}\\text{,}\n\\end{equation}\nwhere $r_{i,j}$ is the cross-correlation coefficient which ranges from $-1$ to\n$1$.\\footnote{Note that here we adopt the convention that the bias factors are\nalways positive and that the sign of the cross-spectrum is determined solely\nby that of the correlation coefficient. This convention differs from our\nprevious work \\citep{2018ApJ...867...26B}.} In the above equation and in what\nfollows, we generally suppress redshift and wavenumber labels for brevity. In\ngeneral, $r_{i,j}$ is scale-dependent, but asymptotes to $-1$ (for\nanticorrelated fields) or $1$ (for correlated fields) on large\nscales.\\footnote{Note that we neglect shot-noise contributions to the\nauto-spectrum in Equation~\\ref{eq:bias_ps}, as well as correlated shot-noise\nterms in the cross-power spectrum. This should be a very good approximation on\nthe scales of interest unless the line-emitting sources are quite rare (e.g.\n\\citealt{lidz2016:remove_interloper}). Even in the case of rare sources, the\nshot-noise term should be a white-noise contribution on scales much larger\nthan the size of the host halos. In this case, one can perform a joint fit for\nthe shot-noise along with the clustering terms.} If one of the lines is the\n21~cm field, we replace $\\avg{I_i}$ with $\\avg{T_{21}}$ in\nEquation~\\ref{eq:xps}.\n\nHowever, in the presence of a third line $k$, and with $P_{j,k}$ and $P_{k,i}$\ndefined analogously as in Equation~\\ref{eq:xps}, we can simply write\n\\begin{equation}\\label{eq:threefields}\n\\begin{split}\nP_{i,i}=(\\avg{I_i} b_i)^2 P_{\\text{lin}} &= \\frac{r_{j,k}}{r_{i,j} r_{k,i}} \\frac{P_{i,j} P_{k, i}}{P_{j,k}} \\\\\n&\\equiv R_{i,j,k} P_{i,j,k}\\text{,}\n\\end{split}\n\\end{equation}\nwhere we have defined $R_{i,j,k} \\equiv r_{j,k}\/(r_{i,j} r_{k,i})$ and\n$P_{i,j,k} \\equiv (P_{i,j}P_{k,i})\/P_{j,k}$. On sufficiently large scales,\n$R_{i,j,k} \\rightarrow 1$, but on intermediate scales $R_{i,j,k} > 1$ for most\nreasonable cases when the various $r$'s are close in magnitude.\nEquation~\\ref{eq:threefields} shows that (on sufficiently large scales where\nlinear biasing holds and $R_{i,j,k} \\sim 1$) we can recover the linear bias\nfactor of field $i$ from a suitable ratio of cross-spectra. Here we suppose\nthat the underlying density power spectrum is well known.\nEquation~\\ref{eq:threefields} is the main point of this paper; in the\nremainder of this work we consider an application to the EoR and quantify its\naccuracy. Specifically, we will test the range of validity -- in spatial scale\nand redshift\/ionization fraction -- of the assumption that $R_{i,j,k}=1$,\nalong with the linear biasing approximations of\nEquations~\\ref{eq:21cm_bias}~\\&~\\ref{eq:linear_biasing}. Note that testing the\nassumption that $R_{i,j,k} = 1$ directly from upcoming data will require\nreliable auto-spectra.\n\nWe turn now to the specific case of EoR surveys with the goal of extracting\nthe 21~cm bias factor using only cross-power spectra. For further specificity\nwe suppose that the two additional tracer lines are [C~\\textsc{ii}] and [O~\\textsc{iii}],\nalthough little of the analysis that follows depends on the choice of these\ntwo lines --- any of the lines mentioned in Section~\\ref{sec:intro} can be\nused instead of [C~\\textsc{ii}] or [O~\\textsc{iii}]. In this case,\nEquation~\\ref{eq:threefields} may be applied as\n\\begin{equation}\\label{eq:threefields_specific}\n\\begin{split}\nP_{21,21} &= (\\avg{T_{21}} b_{21})^2 P_{\\text{lin}}\\\\\n&= \\frac{P_{21,\\text{C~\\textsc{ii}}} P_{\\text{O~\\textsc{iii}}, 21}}{P_{\\text{C~\\textsc{ii}}, \\text{O~\\textsc{iii}}}}\\text{,}\n\\end{split}\n\\end{equation}\ni.e. assuming $R_{21,\\text{C~\\textsc{ii}},\\text{O~\\textsc{iii}}} = 1$.\n\nWe expect this approach to break down on small scales. First, the three fields\nwill be well-correlated (or anti-correlated) only on large scales, with the\n21~cm field and the [C~\\textsc{ii}], [O~\\textsc{iii}] fields decorrelating on scales smaller\nthan the size of the ionized regions \\citep{Lidz11}. Second, we assume linear\nbiasing which should break down on scales where second-order bias terms become\nsignificant \\citep{McQuinn:2018zwa}.\n\nOne caveat here is that we neglect redshift space distortions throughout.\nIncluding these effects will make the power spectra in\nEquation~\\ref{eq:threefields_specific} angle-dependent. Although these effects\nare well studied in the case of the 21~cm auto-spectrum (e.g.\n\\citealt{Mao12}), an extension of our three cross-spectra method may be needed\nto account for these distortions.\n\n\\section{Simulations}\\label{sec:simulations}\n\nIn order to investigate the accuracy of Equation~\\ref{eq:threefields_specific}\nwe turn to $(186\\,\\text{Mpc})^3$ radiative transfer simulations of the EoR\n\\citep{2007MNRAS.377.1043M,McQuinn:2007dy,Lidz08}. In these calculations,\nradiative transfer is post-processed onto a $(1024)^3$ dark matter only\nsimulation run with \\texttt{GADGET-2} \\citep{Springel:2005mi}. The dark matter\nsimulation resolves halos only down to $10^{10}\\,\\text{M}_\\odot$, however halos down to\n$10^8\\,\\text{M}_\\odot$ are added manually in post-processing with the correct\nstatistical properties \\citep{2007MNRAS.377.1043M}. Halos resolved directly in\nthe simulation (i.e. $>10^{10}\\,\\text{M}_\\odot$) are identified with a\nFriends-of-Friends algorithm with a linking length of 0.2.\n \n In what follows, we adopt the abundant mini-halo sink scenario\n \\citep{2007MNRAS.377.1043M,Lidz08} as our baseline reionization model.\n Although the detailed model for photon sinks implemented in these simulations\n may not be fully realistic, the smaller ionized regions in ``abundant sink''\n scenarios may, in fact, be more plausible than the other cases considered in\n this previous work \\citep{McQuinn:2018zwa}. In any case, the accuracy of our\n method does not depend strongly on the precise reionization model assumed.\n\nIn order to model the [C~\\textsc{ii}] and [O~\\textsc{iii}] emission fluctuations, we assume that\nthe luminosity in each line is correlated with the host halo mass.\nSpecifically, we adopt a power-law average relation between line-luminosity\nand halo mass:\n\\begin{equation}\\label{eq:im_form}\n\\avg{L_i}(M) = L_{i,0} \\left[\\frac{M}{M_0}\\right]^{\\alpha_i},\n\\end{equation}\nwhere $M$ is the mass of the halo, $\\avg{L_i}$ is the average luminosity, and\n$L_{i,0}$ is the luminosity at characteristic mass $M_0$. In order to account\nfor scatter in this relation, we add a random number so that each halo's\nluminosity is $L_i = \\avg{L_i}(1 + \\epsilon)$ where $\\epsilon$ is drawn from a\nzero-mean lognormal distribution of width 0.4 dex.\n\nIn what follows we assume that each host halo in the simulation hosts a [C~\\textsc{ii}]\nand [O~\\textsc{iii}] emitter. If only a random fraction $f$ of halos host active [C~\\textsc{ii}]\nand\/or [O~\\textsc{iii}] emitters while $L_{i,0}$ is boosted to fix the average\nspecific-intensity in each line, this does not change the 21~cm-[C~\\textsc{ii}] or\n21~cm-[O~\\textsc{iii}] cross-power spectra. This represents the case that\nstar-formation activity has a short duty-cycle, yet the total star-formation\nrate density is fixed to the observed value. If the same random fraction emit\nin both [C~\\textsc{ii}] and [O~\\textsc{iii}] this can boost the cross-shot noise contribution to\n$P_{\\text{C~\\textsc{ii}},\\text{O~\\textsc{iii}}}$, but this is highly sub-dominant on the scales of interest\n($k \\leq 0.4$ Mpc$^{-1}$) even for $f=10^{-2}$.\n\nIn order to estimate the specific intensity of the two fields, we use nearest\ngrid-point interpolation to estimate the emissivity on a $512^3$ Cartesian\ngrid, matching the resolution of the density and 21~cm fields from\n\\citet{Lidz08}. Note that we can test the accuracy of\nEquation~\\ref{eq:threefields_specific} without specifying the numerical value\nof $L_{i,0}$ or $M_0$ since they cancel in the ratio. The value of $\\alpha_i$,\non the other hand, controls which host-halos (and galactic star-formation\nrates) produce most of the specific intensity in line $i$.\\footnote{Note we\nassume that the minimum host halo mass of the [C~\\textsc{ii}] and [O~\\textsc{iii}] emitters is\n$10^8 M_\\odot$, comparable to the atomic cooling mass.  The true minimum host\nmass of the emitters may, in fact, be larger. However, note that the average\nspecific intensity may be fixed by the total star-formation rate density and\nthe line-luminosity star-formation rate correlation. Provided these quantities\nare fixed, then the main impact of boosting the minimum host halo mass will be\nto increase slightly the bias factors, $b_i$, and the signal strength. See\ne.g. \\citep{lidz2016:remove_interloper} for more details regarding\nline-intensity fluctuation models.} If the value of $\\alpha_i$ is the same for\n[C~\\textsc{ii}] and [O~\\textsc{iii}], then the two fields differ only by an overall\nmultiplicative factor and Equation~\\ref{eq:threefields_specific} reduces to a\nsimple ratio between a single cross-spectrum and an\nauto-spectrum.\\footnote{This assumes, as we do here, that the scatter in the\nluminosity-mass relation is perfectly correlated between [C~\\textsc{ii}] and [O~\\textsc{iii}] at\nfixed $\\alpha_i$.}\n\nWe consider three different values for $\\alpha_i$: $2\/3$, $1$, and $4\/3$. We\nrefer to these as L, M, and H since they provide most weight to low, medium,\nand high mass host-halos respectively. We allow for the case that the two\nlines have different values of $\\alpha_i$: i.e., we consider 21~cm-L-M,\n21~cm-M-H, and 21~cm-H-L, with L, M, or H standing in for [C~\\textsc{ii}] or [O~\\textsc{iii}] in\nEquation~\\ref{eq:threefields_specific}. We then measure the various\ncross-spectra using a slightly modified version of the power spectrum\ncalculator in \\texttt{21cmFAST} \\citep{Mesinger11,2018arXiv180908995P}.\n\n\\section{Results}\\label{sec:results}\n\nWe first investigate how well our three cross-spectra approach for measuring\nthe large-scale 21~cm bias agrees with the true bias. We measure the true bias\nas\n\\begin{equation}\\label{eq:truebias}\n\\avg{T_{21}} b_{21}(k) \\equiv \\sqrt{\\frac{P_{21,21}(k)}{P_{\\delta,\\delta}(k)}}\\text{,}\n\\end{equation}\nand also estimate the bias as\n\\begin{equation}\\label{eq:truebias_cross}\n\\avg{T_{21}} b_{21}(k) \\simeq\n\\left|\\frac{P_{21,\\delta}(k)}{P_{\\delta,\\delta}(k)} \\right| \\text{,}\n\\end{equation}\nwhere $P_{\\delta,\\delta}(k)$ is the auto-power spectrum of the simulated\ndensity field and $P_{21,\\delta}(k)$ is the 21~cm-density cross-power\nspectrum. Note that Equation~\\ref{eq:truebias_cross} assumes that the\ncorrelation coefficient $\\left|r_{21,\\delta}\\right| = 1$ and so will depart\nfrom Equation \\ref{eq:truebias} on small scales, but the two should converge\non large scales (see Section~\\ref{sec:intro}). The absolute value in\nEquation~\\ref{eq:truebias_cross} comes about from the convention adopted in\nSection~\\ref{sec:approach}. On large scales where the 21~cm, [C~\\textsc{ii}], and\n[O~\\textsc{iii}] fields are each well correlated or anti-correlated with the density\nfield and linear theory applies, we expect all estimates of $\\avg{T_{21}}\nb_{21}$ to agree. When we estimate the bias factors using our three\ncross-spectra method (Equation~\\ref{eq:threefields_specific}) we use the\nsimulated density power-spectrum, since this is extremely close to the linear\ntheory prediction on the relevant scales and redshifts.\n\nThe bias factors inferred from Equation~\\ref{eq:threefields_specific} are\nshown in Figure~\\ref{fig:b21_vs_k} for each of the three combinations of our\nluminosity-mass relation models (L-M, M-H, H-L) at $z=8.34$ when the model\nvolume-averaged ionization fraction is $\\avg{x_i}=0.36$. These are compared\nwith the bias inferred from the 21~cm auto-spectrum\n(Equation~\\ref{eq:truebias}) and the 21~cm-density cross-spectrum\n(Equation~\\ref{eq:truebias_cross}). On large scales ($k \\lesssim\n0.3\\,\\text{Mpc}^{-1}$), the methods converge to very nearly the same value. We find\nthat on a scale of $k=0.1\\,\\text{Mpc}^{-1}$ at $\\avg{x_i}=0.36$ the three methods\nagree with the true value to within $0.6\\%$. In the case of 21~cm-L-L,\n21~cm-M-M, or 21~cm-H-H models the agreement is slightly worse but still at\nthe percent-level. Note that another approach for estimating the 21~cm bias\nwould use only the 21~cm-[C~\\textsc{ii}] cross-spectrum and the [C~\\textsc{ii}] auto-spectrum.\nThis requires measuring the [C~\\textsc{ii}] auto-spectrum, which is subject to\ncontamination from interloping line emission, and so we pursue only the more\nrobust three-field technique here.\n\nThe success results because the ionized regions are sufficiently smaller than\nthis scale ($k=0.1\\,\\text{Mpc}^{-1}$), ensuring that the 21~cm and line-intensity\nfields are highly anti-correlated and that second-order biasing contributions\nare small. For example, the cross-correlation coefficient between the 21~cm\nfield and the density field is $r_{21,\\delta} = -0.99$ at $k=0.1\\,\\text{Mpc}^{-1}$\nfor $\\avg{x_i}=0.36$, $z=8.34$.\n\nOn smaller scales, our approach breaks down. At $\\avg{x_i}=0.36$, the\ndifferent bias factor estimates begin diverging at the $\\geq 10\\%$ level near\n$k \\sim 0.4\\,\\text{Mpc}^{-1}$. This occurs because the fields start to de-correlate\nand second order biasing terms become more important. As anticipated after\nEquation~\\ref{eq:threefields}, the three cross-spectra approach underestimates\nthe bias factor in this regime. This underestimation may allow one to place\nrobust lower limits on $P_{21,21}$ that are only $\\sim50\\%$ smaller than the\ntrue value down to $k\\sim2\\,\\text{Mpc}^{-1}$ at this stage of the EoR, although the\nmodel-dependence of such limits warrants further investigation.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{{b21_k_z8.34}.pdf}\n\\caption{{\\em Upper:} The simulated, dimensionless 21~cm auto-power spectrum\n(gray) compared to that inferred from our three-cross spectra approach\nassuming linear biasing at $\\avg{x_i}=0.36$, $z=8.34$. The different colors\ncorrespond to various possible line-luminosity mass relations (L, M, H), as\ndescribed in Section~\\ref{sec:simulations}. The shaded area shows the\n$1\\,\\sigma$ expected errors for the 21~cm-L-M survey described in\nSection~\\ref{sec:detectability}. {\\em Middle:} The 21~cm bias factor extracted\nfrom our three cross-spectra approach in the different line-luminosity models.\nThese are compared with that inferred from the 21~cm auto-spectrum (solid\ngray) and the 21~cm-density cross-spectrum (gray dashed). {\\em Bottom:} The\nrelative difference between the different bias-factor models. On large scales\nall inferences agree.}\n\\label{fig:b21_vs_k}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{{b21_z}.pdf}\n\\caption{{\\em Upper:} The inferred 21~cm bias factor as a function of\nredshift\/volume-average ionization fraction at $k=0.1\\,\\text{Mpc}^{-1}$. The\ndifferent colored lines show inferences from our three cross-spectra approach\nin the different line-luminosity models (see Figure~\\ref{fig:b21_vs_k} and\ntext). The gray line shows the true bias factor measured from the\n21~cm-$\\delta$ cross-spectrum. {\\em Lower:} The relative error in the three\ncross-spectra approach. We find better than $5\\%$ agreement for most of the\nEoR with sub-percent accuracy achieved near $\\avg{x_i}=0.36$ at $z=8.34$. At\n$\\avg{x_i} \\sim 0.15$ the fields decorrelate on large scales and so the\napproach breaks down (see text).}\n\\label{fig:b21_vs_z}\n\\end{figure}\n\nAt later times, the average ionization fraction and the bubble sizes increase\nand so the scale at which the linear biasing approximation breaks down moves\nto larger scales. For example, at $\\avg{x_i} = 0.7$ ($z = 7.32$), the approach\nbreaks down at the $\\sim 10\\%$ level at $k\\sim0.3\\,\\text{Mpc}^{-1}$, though an\naccuracy of only a few percent is achieved at the largest scales considered\nhere. We suspect better agreement on even larger scales than probed by our\nrelatively small simulation volume.\n\nIn Figure~\\ref{fig:b21_vs_z} we turn to consider the redshift evolution of the\n21 cm bias factor at $k=0.1\\,\\text{Mpc}^{-1}$. As emphasized earlier, the redshift\nevolution of the 21~cm bias factor encodes interesting information about how\nreionization proceeds. The three cross-spectra method generally recovers the\noverall evolution of the 21~cm bias factor with redshift and volume-averaged\nionization fraction quite accurately. This suggests that our technique may\nhelp in reconstructing the reionization history of the Universe, or in\nverifying the results from 21~cm auto-spectrum measurements.\n\nThe one exception is near $\\avg{x_i} \\sim 0.15$, where our technique is\nrelatively inaccurate. This occurs because large-scale overdense regions are\ninitially brighter in 21~cm than typical regions in our model and so the 21~cm\nfields are intially {\\em positively-correlated} with the density fluctuations.\nAs reionization begins, the large-scale overdense regions ionize first which\ncauses the correlation coefficient between the 21~cm and density fields to\nreverse signs. Consequently, there is an intermediate period (near $\\avg{x_i}\n\\sim 0.15$ in this model) where the two fields are roughly {\\em uncorrelated}\non large scales \\citep{Lidz08}. This causes our method to break down, although\nwe caution that incorporating spin-temperature fluctuations into the modeling\nmay modify this conclusion. Note also that it will be challenging to perform\nline-intensity mapping observations at very early times before, e.g.,\nsufficient metal enrichment occurs.\n\nWhile our baseline model assumes the abundant mini-halo sinks scenario we have\nalso investigated the fiducial model used in \\citet{Lidz08}. Although this\nlatter model has a different ionization history and bias factor evolution, the\naccuracy of our three cross-spectra method is broadly similar in this case.\nFor example, near the midpoint of reionization in this model ($z=7.32,\n\\avg{x_i}=0.54$), the 21~cm bias extraction also reaches sub-percent accuracy.\n\n\\section{Detectability}\\label{sec:detectability}\n\nEncouraged by the success of our approach in simulations, we briefly describe\nthe survey specifications required to infer 21~cm bias factors using this\ntechnique. Here we consider only rough estimates and defer an in depth\ntreatment of noise power spectra, variance from residual foregrounds, and a\nfull probabilistic, multi-field framework to future work.\n\nWe first describe the relevant variance and covariance formulae (for\nderivations, see e.g. \\citealt{2015JCAP...03..034V}):\n\\begin{equation}\\label{eq:var_covar}\n\\begin{split}\n\\Var{P_{i,j}} &= P_{i,j}^2 + P_{i,\\text{tot}}P_{j,\\text{tot}} \\\\\n\\Cov{P_{i,j}}{P_{i,k}} &= P_{i,\\text{tot}}P_{j,k} + P_{i,j}P_{i,k}\\text{,}\n\\end{split}\n\\end{equation}\nwhere $P_{i,\\text{tot}} = P_{i} + N_{i}$ and $N_{i}$ is the instrumental noise\npower spectrum of line $i$. For simplicity, we neglect the shot-noise\ncontribution to each field. We note that Equation~\\ref{eq:var_covar} is only\nvalid in the Gaussian approximation, but this is suitable for the large scales\nof interest in our approach.\n\nWe can now apply the standard propagation of errors formula to\nEquation~\\ref{eq:threefields_specific} and substitute\nEquation~\\ref{eq:var_covar}, yielding:\n\\begin{equation}\\label{eq:noise_P21}\n\\begin{split}\n&\\Var{P_{21}} = \\\\\n& \\left(\\frac{P_{21,\\text{C~\\textsc{ii}}}}{P_{21,\\text{O~\\textsc{iii}}}}\\right)^2\\left(P_{21,\\text{O~\\textsc{iii}}}^2 + P_{21,\\text{tot}}P_{\\text{O~\\textsc{iii}},\\text{tot}}\\right) \\\\\n&+ \\left(\\frac{P_{21,\\text{O~\\textsc{iii}}}}{P_{21,\\text{C~\\textsc{ii}}}}\\right)^2\\left(P_{21,\\text{C~\\textsc{ii}}}^2 + P_{21,\\text{tot}}P_{\\text{C~\\textsc{ii}},\\text{tot}}\\right) \\\\\n&+ \\left(\\frac{P_{21,\\text{C~\\textsc{ii}}}P_{21,\\text{O~\\textsc{iii}}}}{P_{\\text{C~\\textsc{ii}},\\text{O~\\textsc{iii}}}^2}\\right)^2\\left(P_{\\text{C~\\textsc{ii}},\\text{O~\\textsc{iii}}}^2 + P_{\\text{C~\\textsc{ii}},\\text{tot}}P_{\\text{O~\\textsc{iii}},\\text{tot}}\\right) \\\\\n&+ \\frac{P_{21,\\text{C~\\textsc{ii}}}P_{21,\\text{O~\\textsc{iii}}}}{P_{\\text{C~\\textsc{ii}},\\text{O~\\textsc{iii}}}^2}\\left(P_{21,\\text{tot}}P_{\\text{C~\\textsc{ii}},\\text{O~\\textsc{iii}}} + P_{21,\\text{C~\\textsc{ii}}}P_{21,\\text{O~\\textsc{iii}}} \\right) \\\\\n&- \\frac{P_{21,\\text{C~\\textsc{ii}}}^2P_{21,\\text{O~\\textsc{iii}}}}{P_{\\text{C~\\textsc{ii}},\\text{O~\\textsc{iii}}}^3}\\left(P_{\\text{O~\\textsc{iii}},\\text{tot}}P_{21,\\text{C~\\textsc{ii}}} + P_{21,\\text{O~\\textsc{iii}}}P_{\\text{C~\\textsc{ii}},\\text{O~\\textsc{iii}}} \\right) \\\\\n&- \\frac{P_{21,\\text{C~\\textsc{ii}}}P_{21,\\text{O~\\textsc{iii}}}^2}{P_{\\text{C~\\textsc{ii}},\\text{O~\\textsc{iii}}}^3}\\left(P_{\\text{C~\\textsc{ii}},\\text{tot}}P_{21,\\text{O~\\textsc{iii}}} + P_{21,\\text{C~\\textsc{ii}}}P_{\\text{C~\\textsc{ii}},\\text{O~\\textsc{iii}}} \\right)\\text{.}\n\\end{split}\n\\end{equation}\n\nThe number of modes in a bin of width $\\delta k$ centered on $k$ is,\n\\begin{equation}\\label{eq:num_modes}\nN_m = \\frac{4\\pi k^2 \\delta k}{V_{\\text{fund}}}\\text{,}\n\\end{equation}\nwhere $V_{\\text{fund}}$ is the volume of a fundamental mode. We assume a\nsquare survey area and therefore compute,\n\\begin{equation}\\label{eq:vfund}\nV_{\\text{fund}} = \\frac{(2\\pi)^3}{L_{\\bot}^2L_{\\parallel}}\\text{,}\n\\end{equation}\nwhere $L_{\\bot}$ is the side length of the survey area and $L_{\\parallel}$ is\nthe length of the redshift bin $\\Delta z$.\n\nWe assume a joint survey area of $100\\,\\text{deg}^2$ and bin widths of $\\delta\nk = 0.03 \\text{Mpc}^{-1}$ and $\\Delta z = 0.25$. In order to make a rough estimate,\nwe assume that each experiment reaches sample-variance limited sensitivity at\n$k=0.1\\,\\text{Mpc}^{-1}$, with $N_i=P_i$ at this wavenumber and adopt a pure,\nisotropic white-noise power spectrum. In the case of [C~\\textsc{ii}], the required\nnoise depends on the uncertain average specific intensity which determines, in\npart, the signal strength, $P_i$. A plausible value is $\\avg{I_{\\text{C~\\textsc{ii}}}}=5\n\\times 10^2\\,\\text{Jy\/str}$ at $z=8.34$ \\citep{2018ApJ...867...26B}. In this\ncase, $N_{\\text{C~\\textsc{ii}}} = 1.6 \\times 10^9$, $2.5 \\times 10^9$, $3.9 \\times\n10^9\\,(\\text{Jy}\/\\text{str})^2\\,\\text{Mpc}^3$ for the L, M, and H models of the\n[C~\\textsc{ii}] line, respectively at $z=8.34$. These noise requirements are comparable\nto the values forecasted for Stage-II [C~\\textsc{ii}] line-intensity mapping forecasts\nin \\citet{silva15:prospects,lidz2016:remove_interloper}. We expect broadly\nsimilar noise requirements for hypothetical future [O~\\textsc{iii}] surveys but defer\ndetailed forecasts to future work. As we discussed previously\n\\citep{2018ApJ...867...26B}, the 21~cm sensitivity requirement assumed here\nseems plausible considering HERA-350 will {\\em image} some large scale modes\n\\citep{DeBoer:2016tnn} --- although the white noise approximation is rather\ncrude and should be refined in future work.\n\nWe caution that the strength of the [C~\\textsc{ii}] signal at the redshifts of interest\nis quite uncertain. A broad range of estimates appear in the current\nliterature, depending on assumptions about: the correlation between [C~\\textsc{ii}]\nluminosity and SFR at high redshift, the total star-formation rate density\n(estimates from UV luminosity functions are sensitive to whether and how one\nextrapolates to faint luminosities beyond current detection limits), and the\nhost-halo masses of [C~\\textsc{ii}] emitters. For example, our model values for\n$\\avg{I_{C~\\textsc{ii}}}$ are similar to a number of recent forecasts\n\\citep{2018arXiv180204804D, 2015ApJ...806..209S}, but are more than an order\nof magnitude larger than some more pessimistic estimates in\n\\citet{2015ApJ...806..209S,2018arXiv181208135C}. In any case, at fixed\nluminosity-weighted bias, the required noise scales quadratically with the\naverage specific intensity and so the reader can rescale our results according\nto their preferred specific intensity model. For instance, in the case of\n$\\avg{I_{C~\\textsc{ii}}} = 20\\,\\text{Jy}\/\\text{sr}$ \\citep{2018arXiv181208135C}, one\nwould require that $N_{C~\\textsc{ii}} = 2.6 \\times 10^6$, $4 \\times 10^6$, $6.2 \\times\n10^6\\,(\\text{Jy}\/\\text{sr})^2\\,\\text{Mpc}^3$ for the L, M, and H models of the\n[C~\\textsc{ii}] line, respectively at $z=8.34$. On the other hand, a more moderate\nestimate of $\\avg{I_{\\text{C~\\textsc{ii}}}} = 100 \\,\\text{Jy}\/\\text{sr}$ \\citep{\n2015ApJ...806..209S,2018arXiv180204804D}, requires $N_{C~\\textsc{ii}} = 6.4\n\\times 10^7$, $1 \\times 10^8$, $1.6 \\times\n10^8\\,(\\text{Jy}\/\\text{sr})^2\\,\\text{Mpc}^3$ for the L, M, and H models of the\n[C~\\textsc{ii}] line, respectively at $z=8.34$.\n\n\\begin{deluxetable}{cCCC}\n\\tablecaption{The noise power-spectrum for upcoming [C~\\textsc{ii}] surveys at $z=7.4$.\n\\label{tab:noise}}\n\\tablehead{\\colhead{survey} & \\colhead{$A_{\\text{survey}}$} & \\colhead{$A_{\\text{pix}}$} & \\colhead{$N_{\\text{C~\\textsc{ii}}}$} \\\\ \n\\colhead{} & \\colhead{$(\\text{deg}^2)$} & \\colhead{$(\\text{deg}^2)$} & \\colhead{$((\\text{Jy}\/\\text{sr})^2\\,\\text{Mpc}^3)$} } \n\\startdata\nCCAT-p & 2 & 2.5\\times10^{-4} & 2.66\\times10^{9} \\\\\nCONCERTO & 1.4 & 6.7\\times10^{-5} & 2.04 \\times 10^{9} \\\\\nTIME & 1.3 \\times 0.0084 & 6.7\\times10^{-5} & 1.04\\times10^9 \\\\\n\\enddata\n\\tablerefs{See \\citet{2018arXiv181208135C} for more details.}\n\\end{deluxetable}\n\nWith the assumed noise and survey requirements for our fiducial model, we show\nthe resulting error bars in the {\\em Upper} and {\\em Middle} panels of\nFigure~\\ref{fig:b21_vs_k} for our particular choice of binning. At least for\nthe hypothetical surveys considered here, the 21~cm bias factor may be\nrecovered with good statistical precision. In other words, if sample-variance\nlimited sensitivity may be reached at $k=0.1\\,\\text{Mpc}^{-1}$ in each line over a\ncommon survey area of $\\sim 100\\,\\text{deg}^2$, then a strong detection\nappears feasible. Of course we have neglected sample variance contributions\nfrom residual foregrounds among other complications, and so this should be\ninterpreted as a best-case scenario. On the other hand, increasing the common\nsurvey area above $100\\,\\text{deg}^2$, for example, could help shrink the\nerror bars.\n\nWhile our fiducial [C~\\textsc{ii}] survey is somewhat futuristic, we can also consider\nthe prospects with current, shortly upcoming surveys, specifically\nCCAT-prime\\footnote{\\url{http:\/\/www.ccatobservatory.org}}\n\\citep{2018SPIE10700E..1MS},\nCONCERTO\\footnote{\\url{https:\/\/people.lam.fr\/lagache.guilaine\/CONCERTO.html}}\n\\citep{Lagache:2018hmk}, and\nTIME\\footnote{\\url{https:\/\/cosmology.caltech.edu\/projects\/TIME}}\n\\citep{Crites14}. We use the pixel noise values, $\\sigma_{\\text{pix}}\nt_{\\text{pix}}^{-1\/2}$, for each survey from \\citet{2018arXiv181208135C}. We\nreport the noise power spectrum at $z=7.4$ (assuming a pure white-noise\nspectrum) in Table~\\ref{tab:noise}. We generically find that\n$N\\sim2\\times10^9\\, (\\text{Jy}\/\\text{sr})^2\\,\\text{Mpc}^3$. If we assume a model\nwith $\\avg{I_{\\text{C~\\textsc{ii}}}}\\sim500\\,\\text{Jy}\/\\text{sr}$ then even the\nfirst-generation surveys reach our requisite noise. However, deeper surveys\nwill be needed in the case of the more pessimistic estimates of\n$\\avg{I_{\\text{C~\\textsc{ii}}}}\\sim100$ or $\\sim20\\,\\text{Jy}\/\\text{sr}$. That being said,\nour fiducial calculations also assume a larger survey area of\n$100\\,\\text{deg}^2$. At $z=8.34$ we find a $\\text{S}\/\\text{N}$ of $3.3$,\n$2.7$, and $2.9$ for the L-M, M-H, and H-L models, respectively at\n$k=0.1\\,\\text{Mpc}^{-1}$ and bin width of $\\Delta k=0.03\\,\\text{Mpc}^{-1}$. Since the\nnumber of modes scale with the square root of the survey area, we estimate\nthat CCAT-p might be able to recover a $\\text{S}\/\\text{N}$ of $0.5$, $0.4$,\nand $0.4$ for the L-M, M-H, and H-L models, respectively at\n$k=0.1\\,\\text{Mpc}^{-1}$. Including some higher $k$-modes, even this\nfirst-generation survey might be capable of a marginal detection (if [O~\\textsc{iii}]\ncan be surveyed as well), but this is only for our optimistic signal strength\nmodel.\n\nSince the strength of the [C~\\textsc{ii}] signal is likely a strong function of\nredshift, the survey requirements should be less stringent at $z \\sim 7$ than\nthe $z \\sim 8$ case considered above. The main effect here should be from\nredshift evolution in the average specific intensity; again, the noise\nrequirements scale with the average intensity squared. The required noise can\ntherefore be adjusted according to one's preferred model for redshift\nevolution in the signal strength.\n\n\\section{Conclusions}\\label{sec:conclusions}\n\nWe have shown that the amplitude of large-scale 21 cm fluctuations may be\ninferred from measuring cross-power spectra between the 21 cm fluctuations and\neach of two separate line-intensity maps, such as [C~\\textsc{ii}] or [O~\\textsc{iii}]. Although\nit has long been recognized that the cross-power spectrum between two fields\nis more robust to foreground contamination than the auto-power spectrum of\neither field alone, the amplitude of a single cross-power spectrum provides\nonly a product of two bias factors. We found that using a suitable combination\nof three cross-power spectra\n(Equations~\\ref{eq:threefields}~and~\\ref{eq:threefields_specific}) one can\ninstead infer the 21~cm bias alone to high accuracy.\n\nQuantitatively, in the reionization model we considered, the accuracy reaches\npercent-level on large scales ($k \\sim 0.1-0.3\\,\\text{Mpc}^{-1}$) during much of the\nEoR. The inferred bias factor evolution can then be compared to that extracted\nfrom the 21~cm auto spectrum. In principle, checking whether the 21~cm\nauto-power spectrum follows linear-biasing on large scales might itself be a\ngood systematics check. However, linear biasing holds only over a limited span\nof wavenumbers and early measurements may probe a small dynamic range in\nspatial scale. Hence we believe that our three cross-spectra approach might\nplay an important role in confirming initial detections. Since our method\nunderestimates $P_{21,21}$ on intermediate scales, it can place informative\nlower limits (i.e. $\\sim 50\\%$ of the true value) down to $k\\sim1\\,\\text{Mpc}^{-1}$,\ndepending on the stage of reionization. More work is necessary, however, to\nsee if there are some allowed reionization and line-intensity models where our\ntechnique actually overestimates $P_{21,21}$.\n\nAlthough we focused here on the case of 21~cm fluctuations during the EoR, the\nmethod has broader applicability. For example, one can also estimate the bias\nof the [C~\\textsc{ii}] and [O~\\textsc{iii}] fluctuations by using a similar ratio of\ncross-spectra. This should help circumvent the line-interloper problem that\npresents a challenge for such surveys \\citep[e.g.][]{kovetz2017:im_review}.\nSince the ionized bubbles lead to scale-dependent biasing in the 21~cm field\non large spatial scales, the 21~cm case is an especially demanding\napplication, and we expect even better performance for [C~\\textsc{ii}], [O~\\textsc{iii}], and\nrelated lines.\n\nIn order to implement the strategy proposed here, there must be a coordinated\neffort to probe the same regions on the sky over common redshifts in multiple\nlines of interest. Ultimately, we envision line-intensity mapping surveys in\n$N$ different lines, all probing the same cosmological volume. Among other\nbenefits, this will provide $N(N-1)\/2$ measurements of the bias factor in each\nline using the same basic technique outlined here.\n\n\\acknowledgments\nWe thank the anonymous referee for providing helpful comments. We thank Matt\nMcQuinn for the simulations used in this analysis. A.B. would like to thank\nTodd Phillips for helpful discussions. A.B. was supported in part by the Roy\n\\& Diana Vagelos Program in the Molecular Life Sciences and the Roy \\& Diana\nVagelos Challenge Award. The work of A.B. and F.V.-N. is supported by the\nSimons Foundation.\n\n\\software{\\texttt{colossus} \\citep{2018ApJS..239...35D}, \\texttt{matplotlib}\n\\citep{Hunter:2007}, \\texttt{numpy} \\citep{numpy:2011}, and \\texttt{scipy}\n\\citep{scipy:2001}.}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzbfyb b/data_all_eng_slimpj/shuffled/split2/finalzzbfyb
new file mode 100644
index 0000000000000000000000000000000000000000..b009920646978ae863c0cf8e46942d69bdd3173c
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzbfyb
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\n\nIt is known that at least half of the stars in the Galaxy are multiple systems containing two or more stars orbiting each other \\citep{2001ASPC..229...91K, 2017ApJ...836..139F}, thus in many surveys and large samples of stars, binaries are ubiquitous. This is in contrast with the Sun, which is a single star, and attempts to find a faint stellar companion orbiting it rendered no results thus far \\citep[e.g.,][]{2014ApJ...781....4L}. Many studies avoid contamination by binaries in their samples, the main reasons being because we do not understand well how binaries evolve and how the presence of a companion affects the primary star. However, with the development of instruments with higher spatial and spectral resolution and coronagraphs, it is now possible to better probe the secondary component of such systems.\n\n\\defcitealias{2014A&A...572A..48R}{Paper~I}\n\\defcitealias{2015A&A...581A..34B}{Paper~II}\n\\defcitealias{2016A&A...590A..32T}{Paper~III} \\defcitealias{2016A&A...592A.156D}{Paper~IV}\n\\defcitealias{2017A&A...597A..34M}{Paper~V}\n\nWe have been carrying out a radial velocity planet search focused on solar twins using HARPS \\citep[][hereafter Papers I, II, III, IV and V, respectively]{2014A&A...572A..48R, 2015A&A...581A..34B, 2016A&A...590A..32T, 2016A&A...592A.156D, 2017A&A...597A..34M}. The definition of solar twin we use is a star with stellar parameters inside the ranges $5777 \\pm 100$ K, $4.44 \\pm 0.10$ dex(cgs) and $0.0 \\pm 0.1$ dex, respectively, for $T_{\\mathrm{eff}}$, $\\log{g}$ and [Fe\/H]. In total, 81 solar twins\\footnote{\\footnotesize{Some of the stars in our sample do not fit the strict definition of solar twins because one or more parameters are off the definition intervals, but they are still very close solar analogues.}} were observed on HARPS. As part of our survey we previously identified 16 clear spectroscopic binaries (SB) \\citepalias{2016A&A...592A.156D}. We report here the identification of four additional SBs (HIP 14501, HIP 18844, HIP 65708 and HIP 83276) and the withdrawal of HIP 43297 and HIP 64673, which are unlikely to host stellar-mass companions, bringing the number of solar twin SBs to 18. Most of these SBs are single-lined -- they do not contain a second component in their spectral lines --, meaning that their companions are either faint stars or located outside the $\\sim$$1\\arcsec$ aperture of the HARPS spectrograph. We confirm that there are three solar twins with spectra contaminated by a relatively bright companion (see discussion in Section \\ref{peculiar}). In our sample there are an additional 18 visual binaries\\footnote{\\footnotesize{We define as visual companions those with separations larger than $1\\arcsec$}} or multiple systems, of which HIP 6407 and HIP 18844 have wide companions \\citep[see table 5 in][]{2014AJ....147...86T} as well as the spectroscopic companions reported here.\n\nIn \\citetalias{2016A&A...592A.156D} we saw that the single or visual binary solar twins display a rotational evolution that can described with a relation between stellar age $t$ and rotational velocity $v_{\\mathrm{rot}}$ in the form of a power law plus a constant: $v_{\\mathrm{rot}} = v_{\\mathrm{f}} + m\\ t^{-b}$, where $v_{\\mathrm{f}}$, $m$ and $b$ are free parameters to be fit with observations. This relation is explained by loss of angular momentum due to magnetized winds \\citep[e.g.,][]{1984RSPTA.313...19M, 1992ASPC...26..416C, 2003ApJ...586..464B, 2013A&A...556A..36G}, and the index $b$ reflects the geometry of the stellar magnetic field \\citep{1988ApJ...333..236K}. There are at least two solar twin binaries that display enhanced rotational velocities -- above $2\\sigma$ from the expected -- and activity for their ages: HIP 19911 and HIP 67620; if we consider the revised age for HIP 103983 (Spina et al., in preparation), it can also be considered a fast rotator for its age.\n\nBesides the enhanced rotational velocities and higher chromospheric activity (\\citetalias{2014A&A...572A..48R}; \\citealt{F16sub}), some of these binaries also display peculiar chemical abundances (\\citetalias{2016A&A...590A..32T}; \\citealt{2016A&A...587A..46D}). As pointed out by \\citeauthor{2016A&A...587A..46D}, the ultra-depletion of beryllium, which is observed on HIP 64150, could be explained by the interaction of the main star with the progenitor of the white dwarf companion. In addition to HIP 64150, the confirmed binaries HIP 19911 and HIP 67620 also display clearly enhanced $\\mathrm{[Y\/Mg]}$ abundances \\citepalias[see fig. 9 in][and the discussion in Section \\ref{peculiar} of this paper]{2016A&A...590A..32T}.\n\nOne interesting aspect about stars with enhanced activity and rotation is that these characteristics were hypothesized to be the result of dynamo action from close-in giant planets \\citep[see][and references therein]{2016ApJ...830L...7K}. In fact, some of our early results pointed out that the star HIP 68468, for which we inferred two exoplanets candidates \\citepalias{2017A&A...597A..34M}, had an enhanced rotational velocity when compared to other solar twins of the same age. However, a more careful analysis showed that the enhancement was instead a contribution of macroturbulence. Another explanation for these enhancements is due to magnetic interactions with either a close-in or an eccentric giant planet \\citep{2000ApJ...533L.151C}, but recent results obtained by, e.g., \\citet{2016A&A...592A.143F} and \\citet{2017MNRAS.465.2734M} show that they cannot explain such anomalies.\n\n\\defcitealias{2010exop.book...15M}{MC10}\n\nIn light of these intriguing results, we sought to better understand the nature of these solar twin multiple systems by studying their orbital parameters, and use them to search for explanations of the observed anomalies, especially stellar rotation. The orbital parameters can be estimated from the radial velocity data of the stars \\citep[see, e.g.,][hereafter MC10]{2010exop.book...15M}, with the quality of the results depending strongly on the time coverage of the data.\n\n\\section{Radial velocities}\n\nOur solar twins HARPS data\\footnote{\\footnotesize{Based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere under ESO programs 188.C-0265, 183.D-0729, 292.C-5004, 077.C-0364, 072.C-0488, 092.C-0721, 093.C-0409, 183.C-0972, 192.C-0852, 091.C-0936, 089.C-0732, 091.C-0034, 076.C-0155, 185.D-0056, 074.C-0364, 075.C-0332, 089.C-0415, 60.A-9036, 075.C-0202, 192.C-0224, 090.C-0421 and 088.C-0323.}} are completely described in \\citetalias{2014A&A...572A..48R}. Their radial velocities (RV) are automatically measured from the HARPS Data Reduction Software \\textbf{(see Table \\ref{HARPS_rvs})}, and the noise limit of the instrument generally remains around 1 m s$^{-1}$. In order to broaden the coverage of our RV data, we also obtained more datasets that were available in the literature and public databases, including the HARPS archival data for other programs.\n\nThe mass and other stellar parameters of the solar twins were estimated with high precision using the combined HARPS spectra and differential analysis owing to their similarity with the Sun \\citep[see, e.g.,][]{2014ApJ...795...23B, 2010A&A...519A..87B, 2016A&A...589A..17Y}. The ages for the solar twins were obtained using Yonsei-Yale isochrones \\citep{2001ApJS..136..417Y} and probability distribution functions as described in \\citet{2013ApJ...764...78R} and in \\citetalias{2014A&A...572A..48R}. The full description and discussion of the stellar parameters of the HARPS sample are going to be presented in a forthcoming publication (Spina et al., in preparation).\n\nThe additional radial velocities data obtained from online databases and the literature are summarized in Table \\ref{add_rvs}. These are necessary to increase the time span of the observations to include as many orbital phases as possible at the cost of additional parameters to optimize for (see Section \\ref{short}).\n\n\\begin{table}\n\\begin{center}\n\\caption{HARPS radial velocities for stars in the Solar Twin Planet Search program. This table is presented for guidance regarding the form and content of the online supplementary data, which is available in its entirety in machine-readable format.}\n\\begin{tabular}{ccc}\n\\toprule[\\heavyrulewidth]\nJulian Date & RV & $\\sigma_{\\mathrm{RV}}$ \\\\\n(d) & (km s$^{-1}$) & (km s$^{-1}$) \\\\\n\\bottomrule[\\heavyrulewidth]\n\\multicolumn{3}{c}{HIP 6407} \\\\\n2455846.750257 & 6.816873 & 0.001020 \\\\\n2455850.716200 & 6.811186 & 0.000997 \\\\\n2455851.710847 & 6.806352 & 0.000886 \\\\\n2455852.703837 & 6.801079 & 0.001140 \\\\\n2456164.849296 & 6.204053 & 0.001059 \\\\\n2456165.853256 & 6.203359 & 0.001049 \\\\\n2456298.564449 & 5.954045 & 0.001029 \\\\\n\\hline\n\\multicolumn{3}{c}{HIP 14501} \\\\\n2452937.683821 & 7.024814 & 0.000475 \\\\\n2452940.727854 & 7.025289 & 0.000516 \\\\\n2453001.575609 & 7.026322 & 0.000658 \\\\\n2453946.941856 & 7.023707 & 0.000708 \\\\\n\\midrule[\\heavyrulewidth]\n\\label{HARPS_rvs}\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{table*}\n\\begin{center}\n\\caption{Additional radial velocities from other programs and instruments.}\n\\begin{tabular}{lll}\n\\toprule[\\heavyrulewidth]\nInstrument\/Program & References & Data available for (HIP numbers) \\\\\n\\bottomrule[\\heavyrulewidth]\n CfA Digital Speedometers & \\citet{2002AJ....124.1144L} & 65708 \\\\\n ELODIE\\textsuperscript{a} & \\citet{1996AandAS..119..373B, 2004PASP..116..693M} & 43297, 54582, 62039, 64150, 72043, 87769 \\\\\n SOPHIE\\textsuperscript{b} & \\citet{2011SPIE.8151E..15P} & 6407, 43297, 54582, 62039, 64150, 87769 \\\\\n Lick Planet Search & \\citet{2014ApJS..210....5F} & 54582, 65708 \\\\\n AAT Planet Search & \\citet{2015MNRAS.453.1439J} & 18844, 67620, 73241, 79578, 81746 \\\\\n Various & \\citet{2016AJ....152...46W} & 67620 \\\\\n HIRES\/Keck RV Survey\\textsuperscript{c} & \\citet{2017AJ....153..208B} & 14501, 19911, 62039, 64150, 72043, 103983 \\\\\n\\bottomrule[\\heavyrulewidth]\n\\multicolumn{3}{l}{\\textsuperscript{a}\\footnotesize{Available at \\url{http:\/\/atlas.obs-hp.fr\/elodie\/}.}}\\\\\n\\multicolumn{3}{l}{\\textsuperscript{b}\\footnotesize{Available at \\url{http:\/\/www.obs-hp.fr\/guide\/sophie\/data_products.shtml}.}}\\\\\n\\multicolumn{3}{l}{\\textsuperscript{c}\\footnotesize{Available at \\url{http:\/\/home.dtm.ciw.edu\/ebps\/data\/}.}}\\\\\n\\label{add_rvs}\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\\section{Methods}\n\nThe variation of radial velocities of a star in binary or multiple system stems from the gravitational interaction between the observed star and its companions. For systems with stellar or substellar masses, the variation of radial velocities can be completely explained by the Keplerian laws of planetary motion. For the sake of consistency, we will use here the definitions of orbital parameters as presented in \\citetalias{2010exop.book...15M}.\n\nTo completely characterize the orbital motion of a binary system from the measured radial velocities of the main star, we need to obtain the following parameters: the semi-amplitude of the radial velocities $K$, the orbital period $T$, the time of periastron passage $t_0$, the argument of periapse $\\omega$ and the eccentricity $e$ of the orbit. In order to estimate the minimum mass $m \\sin{i}$ of the companion and the semi-major axis $a$ of the orbit, we need to know the mass $M$ of the main star.\n\nDue to their non-negative nature, the parameters $K$ and $T$ are usually estimated in logarithmic scale in order to eliminate the use of search bounds. Additionally, for orbits that are approximately circular, the value of $\\omega$ may become poorly defined. In these cases, a change of parametrization may be necessary to better constrain them. \\citet{2013PASP..125...83E}, for instance, suggest using $\\sqrt{e} \\cos{\\omega}$ and $\\sqrt{e} \\sin{\\omega}$ (which we refer to as the EXOFAST parametrization) instead of $\\omega$ and $e$ to circumvent this problem, which also can help improve convergence time.\n\nOne issue that affects the radial velocities method is the contamination by stellar activity \\citep[see, e.g.,][]{2016MNRAS.457.3637H}. This activity distorts the spectral lines \\citep{2005oasp.book.....G}, which in turn produces artificial RV variations that can mimic the presence of a massive companion orbiting the star. More active stars are expected to have RV variations with larger amplitudes and a shorter activity cycle period \\citep{2011arXiv1107.5325L}. For most binaries in our sample, the contamination by activity in the estimation of orbital parameters is negligible; the cases where this is not applicable are discussed in detail.\n\n\\subsection{Binaries with well-sampled orbits}\\label{short}\n\nFor binaries with orbital periods $T \\lesssim 15$ yr, usually there are enough RV data measured to observe a complete phase. In these cases, the natural logarithm of the likelihood of observing radial velocities $\\mathbf{y}$ on a specific instrument, given the Julian dates $\\mathbf{x}$ of the observations, their uncertainties $\\mathbf{\\sigma}$ and the orbital parameters $\\mathbf{p}_{\\mathrm{orb}}$ is defined as:\n\n\\begin{equation}\n \\ln{p \\left(\\mathbf{y} \\mid \\mathbf{x},\\mathbf{\\sigma}, \\mathbf{p}_{\\mathrm{orb}} \\right)} = -\\frac{1}{2} \\sum_n \\left[ \\frac{ \\left(y_\\mathrm{n} - y_{\\mathrm{model}} \\right)^2}{\\sigma_\\mathrm{n}^2} + \\ln{ \\left(2 \\pi \\sigma_\\mathrm{n}^2 \\right)} \\right] \\mathrm{,}\n\\label{likelihood}\n\\end{equation}where $y_\\mathrm{n}$ are the RV datapoints, $y_{\\mathrm{model}}$ are the model RV points for a given set of orbital parameters, and $\\sigma_\\mathrm{n}$ are the RV point-by-point uncertainties. The RV models are computed from Eq. 65 in \\citetalias{2010exop.book...15M}:\n\n\\begin{equation}\n v_\\mathrm{r} = \\gamma + K (\\cos{(\\omega + f)} + e \\cos{\\omega}) \\mathrm{,}\n\\end{equation}where $f$ is the true anomaly, and $\\gamma$ is the systemic velocity (usually including the instrumental offset). The true anomaly depends on $e$ and the eccentric anomaly $E$:\n\n\\begin{equation}\n \\cos{f} = \\frac{(1 - e^2)}{1 - e \\cos{E}} - 1 \\mathrm{;}\n\\end{equation}the eccentric anomaly, in turn, depends on $T$, $t_0$ and time $t$ in the form of the so called Kepler's equation:\n\n\\begin{equation}\n E - e \\sin{E} = \\frac{2 \\pi}{T} \\left(t - t_0 \\right) \\mathrm{.}\n\\end{equation}\nEq.~\\ref{likelihood} is minimized using the Nelder-Mead algorithm implementation from \\texttt{lmfit} \\citep[][version 0.9.5]{2016ascl.soft06014N} to obtain the best-fit orbital parameters to the observed data. Because different instruments have different instrumental offsets, the use of additional RV data from other programs require the estimation of an extra value of $\\gamma$ for each instrument.\n\nThe uncertainties of the orbital parameters are estimated using \\texttt{emcee}, an implementation of the Affine Invariant Markov chain Monte Carlo Ensemble sampler \\citep[][version 2.2.1]{2013PASP..125..306F} using flat priors for all parameters in both \\citetalias{2010exop.book...15M} and EXOFAST parametrizations. These routines were implemented in the Python package \\texttt{radial}\\footnote{\\footnotesize{Available at \\url{https:\/\/github.com\/RogueAstro\/radial}.}}, which is openly available online. The uncertainties in $m \\sin{i}$ and $a$ quoted in our results already take into account the uncertainties in the stellar masses of the solar twins.\n\n\\subsection{Binaries with partial orbits}\\label{methods_long_period}\n\nFor the binary systems with long periods (typically 20 years or more), it is possible that the time span of the observations does not allow for a full coverage of at least one phase of the orbital motion. In these cases, the estimation of the orbital parameters renders a number of possible solutions, which precludes us from firmly constraining the configuration of the system. Nevertheless, RV data containing a curvature or one inflection allows us to place lower limits on $K$, $T$ and $m \\sin{i}$, whilst leaving $e$ and $\\omega$ completely unconstrained. When the RV data are limited but comprise two inflections, it may be possible to use the methods from Section \\ref{short} to constrain the orbital parameters, albeit with large uncertainties.\n\nFor stars with very large orbital periods ($T \\gtrsim 100$ yr), the variation of radial velocities may be present in the form of a simple linear trend. In these cases, it is still possible to obtain an estimate of the mass of the companion -- a valuable piece of information about it: \\citet{1999PASP..111..169T} describes a statistical approach to extract the sub-stellar companion mass when the only information available from radial velocities is the inclination of the linear trend, provided information about the angular separation of the system is also available. In this approach, we need to adopt reasonable prior probability density functions (PDF) for the eccentricity $e$, the longitude of periastron $\\varpi$, phase $\\phi$ and the inclination $i$ of the orbital plane. As in \\citeauthor{1999PASP..111..169T}, we adopt the following PDFs: $p(i) = \\sin{i}$, $p(e) = 2e$ and flat distributions for $\\varpi$ and $\\phi$.\n\nWe sample the PDFs using \\texttt{emcee}, 20 walkers and 10000 steps; the first 500 burn-in steps are discarded. From these samples, we compute the corresponding companion masses and its posterior distribution. This posterior usually displays a very strong peak and long tails towards low and high masses which can be attributed to highly unlikely orbital parameters (see Fig. \\ref{64150_pdf} for an example). In our results, we consider that the best estimates for the companion masses are the central bin of the highest peak of the distribution in a histogram with log-space bin widths of about 0.145 dex(M$_\\odot$).\n\nWhen no adaptive optics (AO) imaging data are available for the stars with a linear trend in their RVs, the most conservative approach is to provide the minimum mass for the putative companion. In the case of a linear trend, the lowest mass is produced when $e = 0.5$, $\\omega = \\pi\/2$ and $\\sin{i} = 1$ \\citep{2015ApJ...800...22F}, yielding\n\n\\begin{equation}\\label{feng_eq}\n  m_{\\mathrm{min}} \\approx \\left( 0.0164\\ \\mathrm{M}_{\\mathrm{Jup}}\\right) \\left( \\frac{\\tau}{\\mathrm{yr}} \\right)^{4\/3} \\left| \\frac{dv\/dt}{\\mathrm{m\\ s}^{-1}\\ \\mathrm{yr}^{-1}} \\right| \\left( \\frac{M}{\\mathrm{M}_\\odot} \\right)^{2\/3} \\mathrm{,}\n\\end{equation}where $\\tau$ is 1.25 multiplied by the time span of the radial velocities and $dv\/dt$ is the inclination of the linear trend.\n\n\\section{Results}\n\nWe discovered new, short-period companions for the stars HIP 6407 and HIP 30037 (see Fig. \\ref{orbits_new}) and new long-period companions for HIP 54582 and HIP 62039, and updated or reproduced the parameters of several other known binaries that were observed in our program (see Figs. \\ref{orbits_updated} and \\ref{long_period_rvs}). We briefly discuss below each star, pointing out the most interesting results, inconsistencies and questions that are still open about each of them. The orbital parameters of the binaries with well-sampled orbits in their RV data are presented in Table \\ref{short_params} and the systems with partial orbits are reported in Tables \\ref{curvature_results} and \\ref{linear_trend_results}.\n\n\\subsection{Withdrawn binary candidates}\n\nIn \\citetalias{2016A&A...592A.156D}, we showed that HIP 43297 had a rotational velocity $v \\sin{i}$ higher than expected for its age. Moreover, its radial velocities had variations that hinted for one or more companions orbiting it. We carefully analyzed the RVs and concluded that the periodic ($T = 3.8$ yr) signal observed is highly correlated (Pearson $R = 0.893$) with the activity \\textit{S}-index of the star \\citep{F16sub}. In addition, we tentatively fitted a linear trend to the combined RVs from HARPS, ELODIE and SOPHIE, and obtained an inclination of $4.53 \\pm 0.04$ m s$^{-1}$ yr$^{-1}$, but further monitoring of the system is required to infer the presence of a long-period spectroscopic companion. The revised stellar age for HIP 43297 yields $1.85 \\pm 0.50$ Gyr (Spina et al., in preparation), which explains the high rotational velocity and activity.\n\nThe solar twin HIP 64673 displays significant fluctuations in its radial velocities, but they do not correlate with its activity index; the data covers approximately 5 years of RV monitoring and displays an amplitude $> 20$ m s$^{-1}$. If confirmed to be caused by massive companions, the RV variations of both HIP 43297 and HIP 64673 suggest substellar masses for the most likely orbital configurations. These stars are, thus, removed from the binaries sample of the Solar Twin Planet Search program.\n\n\\begin{figure*}\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[width=0.47\\textwidth]{images\/6407_rvs_lmfit.pdf} & \\includegraphics[width=0.47\\textwidth]{images\/30037_rvs_lmfit.pdf} \\\\\n\\end{tabular}\n\\caption{The radial velocities and orbital solutions of the newly discovered companions for solar twins with short orbital periods.}\n\\label{orbits_new}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[width=0.45\\textwidth]{images\/19911_rvs_folded.pdf} & \\includegraphics[width=0.45\\textwidth]{images\/65708_rvs_lmfit.pdf} \\\\\n\\includegraphics[width=0.45\\textwidth]{images\/67620_rvs_folded.pdf} & \\includegraphics[width=0.45\\textwidth]{images\/79578_rvs_lmfit.pdf} \\\\\n\\includegraphics[width=0.45\\textwidth]{images\/81746_rvs_lmfit.pdf} & \\includegraphics[width=0.45\\textwidth]{images\/103983_rvs_lmfit.pdf} \\\\\n\\end{tabular}\n\\caption{The radial velocities and updated orbital solutions of the known solar twin binaries with at least one complete or near-complete orbital phase. The acronyms CfADS, W16 and AATPS correspond to data from, respectively, the Center for Astrophysics Digital Speedometers, \\citet{2016AJ....152...46W} and the Anglo-Australian Telescope Planet Search program.}\n\\label{orbits_updated}\n\\end{figure*}\n\n\\begin{table*}\n\\begin{center}\n\\caption{Orbital parameters of the binaries with complete or near-complete orbital phases in their RV data.}\n\\begin{tabular}{llccccccc}\n\\toprule[\\heavyrulewidth]\n\\multirow{2}{*}{HIP} & \\multirow{2}{*}{HD} & $K$ & $T$ & $t_0$ & $\\omega$ & $e$ & $m\\sin{i}$ & $a$ \\\\\n& & (km s$^{-1}$) & (days) & (JD-$2.45E6$ days) & ($^o$) & & (M$_\\odot$) & (AU) \\\\\n\\bottomrule[\\heavyrulewidth]\n\\multirow{2}{*}{6407\\textsuperscript{$\\dag$}} & \\multirow{2}{*}{8291} & $2.614$ & $1852.3$ & $5076.7$ & $-57.1$ & $0.682$ & $0.119$ & $3.070$ \\\\\n& & $\\pm 0.084$ & $^{+3.3}_{-3.1}$ & $^{+1.1}_{-1.3}$ & $\\pm 0.9$ & $^{+0.009}_{-0.010}$ & $\\pm 0.002$ & $\\pm 0.005$ \\\\\n\\hline\n\\multirow{2}{*}{19911} & \\multirow{2}{*}{26990} & $7.707$ & $2074.15$ & $4627.4$ & $40.92$ & $0.8188$ & $0.313$ & $6.16$ \\\\\n& & $\\pm 0.007$ & $\\pm 0.09$ & $\\pm 0.1$ & $\\pm 0.06$ & $\\pm 0.0003$ & $\\pm 0.002$ & $\\pm 0.02$ \\\\\n\\hline\n\\multirow{2}{*}{30037} & \\multirow{2}{*}{45021} & $4.246$ & $31.61112$ & $5999.413$ & $-133.60$ & $0.30205$ & $0.0610$ & $0.1971$ \\\\\n& & $\\pm 0.003$ & $\\pm 0.00006$ & $\\pm 0.001$ & $\\pm 0.02$ & $\\pm 0.00008$ & $\\pm 0.0002$ & $\\pm 0.0003$ \\\\\n\\hline\n\\multirow{2}{*}{65708} & \\multirow{2}{*}{117126} & $5.754$ & $207.273$ & $-3675.8$ & $-137.9$ & $0.311$ & $0.170$ & $0.851$ \\\\\n& & $\\pm 0.007$ & $\\pm 0.004$ & $\\pm 0.3$ & $\\pm 0.2$ & $\\pm 0.002$ & $\\pm 0.001$ & $\\pm 0.001$ \\\\\n\\hline\n\\multirow{2}{*}{67620} & \\multirow{2}{*}{120690} & $6.311$ & $3803.3$ & $4945.7$ & $145.10$ & $0.3428$ & $0.578$ & $5.50$ \\\\\n& & $\\pm 0.002$ & $\\pm 0.4$ & $\\pm 0.7$ & $\\pm 0.07$ & $\\pm 0.0002$ & $\\pm 0.002$ & $\\pm 0.01$ \\\\\n\\hline\n\\multirow{2}{*}{79578} & \\multirow{2}{*}{145825} & $1.125$ & $6681.8$ & $879.2$ & $138.9$ & $0.3322$ & $0.1014$ & $7.216$ \\\\\n& & $\\pm 0.007$ & $\\pm 1.5$ & $\\pm 1.9$ & $\\pm 0.1$ & $\\pm 0.0003$ & $\\pm 0.0002$ & $\\pm 0.007$ \\\\\n\\hline\n\\multirow{2}{*}{81746} & \\multirow{2}{*}{150248} & $1.987$ & $3246.5$ & $5623.8$ & $-2.49$ & $0.6644$ & $0.1079$ & $4.387$ \\\\\n& & $\\pm 0.001$ & $\\pm 0.7$ & $\\pm 0.6$ & $\\pm 0.06$ & $\\pm 0.0005$ & $\\pm 0.0002$ & $\\pm 0.003$ \\\\\n\\hline\n\\multirow{2}{*}{103983} & \\multirow{2}{*}{200565} & $2.100$ & $10278$ & $6659$ & $-51.6$ & $0.50$ & $0.210$ & $9.8$ \\\\\n& & $\\pm 0.018$ & $^{+274}_{-247}$ & $\\pm 11$ & $\\pm 1.5$ & $\\pm 0.01$ & $\\pm 0.005$ & $\\pm 0.2$ \\\\\n\\midrule[\\heavyrulewidth]\n\\multicolumn{9}{l}{\\textsuperscript{$\\dag$}\\footnotesize{Triple or higher-order system. The orbital parameters correspond to the closer-in companion.}}\\\\\n\\end{tabular}\n\\label{short_params}\n\\end{center}\n\\end{table*}\n\n\\subsection{Solar twins with new companions}\n\n\\textbf{HIP 6407:} This is a known binary system located 58 pc away from the solar system \\citep{2007A&A...474..653V}, possessing a very low-mass (0.073 M$_\\odot$) L2-type companion separated by $44.8\\arcsec$ (2222 AU), as reported by \\citet[][and references therein]{2015ApJ...802...37B}. In this study, we report the detection of a new close-in low-mass companion with $m\\sin{i} = 0.12$ M$_\\odot$ on a very eccentric orbit ($e = 0.67$) with $a = 3$ AU and an orbital period of approximately 5 years. As expected, the long-period companion does not appear in the RV data as a linear trend.\n\n\\textbf{HIP 30037:} The most compact binary system in our sample, hosting a brown dwarf companion orbiting the main star with a period of 31 days. The high precision of its parameters owes to the wide time span of observations, which covered several orbits. This is one of the first detections of a close-in brown dwarf orbiting a confirmed solar twin\\footnote{\\footnotesize{There are at least 4 solar twin candidates with a close-in brown dwarf companion listed in table A.1 in \\citet{2016A&A...588A.144W}.}}. HIP 30037 is a very quiet star, displaying no excessive jitter noise in its radial velocities. We ran stellar evolution models with \\texttt{MESA}\\footnote{\\footnotesize{Modules for Experiments in Stellar Astrophysics, available at \\url{http:\/\/mesa.sourceforge.net}}} \\citep{2011ApJS..192....3P, 2015ApJS..220...15P} to test the hypothesis of the influence of tidal acceleration caused by the companion on a tight orbit, and found that, for the mass and period of the companion, we should expect no influence in the rotational velocity.\n\n\\textbf{HIP 54582:} \\textit{RV Curvature only}. There are no reports of binarity in the literature. The slight curvature in the RVs of this star is only visible when we combine the HARPS data and the Lick Planet Search archival data. Owing to the absence of an inflection point, the orbital parameters of this system are highly unconstrained. We found that an orbit with $e \\approx 0.2$ produces the least massive companion and shortest orbital period ($m \\sin{i} = 0.03$ M$_\\odot$ and $T = 102$ yr).\n\n\\textbf{HIP 62039:} \\textit{Linear trend}. There are no reports of visually detected close-in ($\\rho < 2\\arcsec$) companions around it. This can be attributed to: i) low luminosity companion, which is possible if it is a white dwarf or a giant planet, and ii) unfavorable longitude of periapse during the observation windows. By using Eq. \\ref{feng_eq}, we estimate that the minimum mass of the companion is 19 M$_{\\mathrm{Jup}}$.\n\n\\subsection{The peculiar binaries}\\label{peculiar}\n\n\\subsubsection{HIP 19911}\\label{19911_results}\n\nThis is one of the main outlier stars in the overall sample of solar twins in regards to its rotation and activity, which are visibly enhanced for both the previous and revised ages (\\citetalias{2016A&A...592A.156D}; \\citealt{F16sub}; Spina et al., in preparation). For the estimation of orbital parameters reported below, we used only the LCES HIRES\/Keck radial velocities, because there are too few HARPS data points to justify the introduction of an extra source of uncertainties (the HARPS points are, however, plotted in Fig. \\ref{orbits_updated} for reference). When using the HARPS data, although the solution changes slightly, our conclusions about the system remain the same.\n\nThe orbital solution of HIP 19911 renders a 0.31 M$_\\odot$ companion in a highly eccentric orbit ($e = 0.82$, the highest in our sample), with period $T = 5.7$ yr. Visual scrutiny reveals what seems to be another signal with large amplitude  in the residuals of this fit ($> 250$ m s$^{-1}$, see Fig. \\ref{orbits_updated}); the periodogram of the residuals shows a very clear peak near the orbital period of the stellar companion.\n\nThe cross-correlation function (CCF) plots for the HARPS spectra of HIP 19911 display a significant asymmetry -- longer tail in the blue side -- for the observations between October 2011 and February 2012, which suggests that the companion is contaminating the spectra. Upon visual inspection of the archival HIRES spectra\\footnote{\\footnotesize{Available at \\url{http:\/\/nexsci.caltech.edu\/archives\/koa\/}.}} taken on 17 January 2014, which is when we expect the largest RV difference between the main star and its companion, we saw a clear contamination of the spectrum by the companion (see Fig. \\ref{SBII}). This contamination could explain the large residuals of the orbital solution, as it introduces noise to the measured radial velocities. The double-lines also explain the inferred high rotational velocity of HIP 19911, since they introduce extra broadening to the spectral lines used to measure rotation. The presence of a bright companion may also affect estimates of chemical abundances, which elucidates the yttrium abundance anomaly \\citepalias{2016A&A...590A..32T}. The double-lined nature of this system is not observed on the HARPS spectra due to an unfavorable observation window.\n\nEven at the largest RV separation, we did not detect the Li I line at 6707.75 \\AA\\ in the HIRES spectrum of the companion. This is expected because M dwarf stars have deeper convection zones, which means they deplete lithium much faster than Sun-like stars. This leads us to conclude that estimates of Li abundance on solar twin binaries using this line do not suffer from strong contamination by their companions; consequently, age estimates with lithium abundances may be more reliable for such binaries than isochronal or gyro ages.\n\n\\begin{figure*}\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[width=0.47\\textwidth]{images\/19911_SBII.pdf} & \\includegraphics[width=0.47\\textwidth]{images\/103983_SBII.pdf} \\\\\n\\end{tabular}\n\\caption{The spectrum of HIP 19911 and HIP 103983 are contaminated by bright companions. The double-lined nature of these stars is only clear only in the maximum RV separation, which are the solid purple curves. The dashed green curves show spectra during minimum RV separation where there is no obvious contamination. The spectra in this figure are not Doppler-corrected for their systemic velocities or instrumental RV offset.}\n\\label{SBII}\n\\end{figure*}\n\n\\defcitealias{2014AJ....147...86T}{T14}\n\nAnother observation conundrum for this system is that \\citet{2015ApJ...799....4R}, using AO imaging without a coronagraph, reports the detection of a visual companion with orbital period $\\sim 12.4$ yr (roughly twice the one we estimated), a lower eccentricity ($e = 0.1677$) and similar semi-major axis of the orbit ($a = 6.17$ AU, if we consider a distance of 30.6 pc). Moreover, \\citet[][hereafter T14]{2014AJ....147...86T} reports that this visual companion has $m = 0.85$ M$_\\odot$. The most likely explanation is that the observations of \\citeauthor{2015ApJ...799....4R} did in fact detect the spectroscopic companion, but the coarse timing of the observations produced a larger period; the lower eccentricity could be explained by a strong covariance between $e$ and the inclination $i$. If $i$ is lower, that means the mass of the companion is significantly higher than $m \\sin{i} = 0.316$ M$_\\odot$, and that would explain the value obtained by \\citetalias{2014AJ....147...86T}. A companion with a mass as large as 0.85 M$_\\odot$ would likely pollute the spectra of HIP 19911, which agrees with our observation that this is a SB II system. If confirmed, this prominent $\\sim$0.85 M$_\\odot$ red dwarf companion could explain the observed activity levels for HIP 19911, since red dwarf stars are expected to be more active than Sun-like stars.\n\n\\defcitealias{2014ApJ...783L..25M}{M14}\n\n\\subsubsection{HIP 67620}\\label{67620_results}\n\nThis is a well-known binary and the target with the largest amount of RV data available (see Fig. \\ref{orbits_updated}). Its orbital parameters have been previously determined by \\citet{2006ApJS..162..207A} and more recently by \\citet{2015MNRAS.453.1439J} and \\citet{2016AJ....152...46W}. The orbital parameters we obtained are in good agreement with \\citet{2016AJ....152...46W}. It has one of the most peculiar rotation rates from our sample (2.77 km s$^{-1}$ for an age of 7.18 Gyr), an enhanced chromospheric activity \\citep{F16sub} and an anomalous [Y\/Mg] abundance \\citepalias{2016A&A...590A..32T}. The orbital period of the system is far too long for gravitational interaction to enhance the rotation of the main star through tidal acceleration, thus we should expect that they evolve similarly to single stars from this point of view.\n\nHigh-resolution imaging of HIP 67620 revealed a companion with $V_{\\mathrm{mag}} \\approx 10$ and separations which are consistent with the spectroscopic companion (\\citealt{2012AJ....143...42H}; \\citetalias{2014AJ....147...86T}). As explained by \\citet{2015MNRAS.453.1439J}, the presence of a companion with $m > 0.55$ M$_\\odot$ can produce contaminations to the spectra that introduce noise to the measured RVs; our estimate of $m \\sin{i}$ for this system is 0.58 M$_\\odot$. These results suggest that, similarly to HIP 19911 but to a lesser degree, the companion of HIP 67620 may be offsetting our estimates for rotational velocity, stellar activity, chemical abundances and isochronal age.\n\nWe were unable to discern double-lines in the HARPS spectra, likely resulting from  unfavorable Doppler separations (observations range from February 2012 to March 2013). However, an analysis of the CCF of this star shows slight asymmetries in the line profiles of the HARPS spectra, which indicates a possible contamination by the companion. \\citet{2017ApJ...836..139F} reported HIP 67620 as a double-lined binary using spectra taken at high-resolution ($R \\approx 60$$,$$000$) in February 2014 and July 2015. As expected due to the short time coverage of the HARPS spectra, we did not see any correlation between the bisector inverse slope \\citep[as defined in][]{2001A&A...379..279Q} and the radial velocities of HIP 67620.\n\n\\citet{2015MNRAS.453.1439J} found an additional signal on the periodogram of HIP 67620 at 532 d, which could be fit with a 1 M${_{\\mathrm{Jup}}}$ planet, bringing down the $rms$ of the fit by a factor of 2. However, we did not find any significant peak in the periodogram of the residuals of the radial velocities for HIP 67620.\n\n\\subsubsection{HIP 103983}\\label{103983_results}\n\nThe revised age for HIP 103983 ($4.9 \\pm 0.9$ Gyr; Spina et al., in preparation) renders this system as an abnormally fast rotator ($3.38$ km s$^{-1}$) for its age. However, upon a careful inspection of the HARPS data obtained at different dates, we identified that the spectrum from 2015 July 27 displays clearly visible double-lines, albeit not as well separated as those observed in the HIRES spectra of HIP 19911 (see Fig. \\ref{SBII}). No other anomalies besides enhanced rotation were inferred for this system. The CCF plots of the HARPS spectra show clear longer tails towards the blue side for most observations.\n\nIn \\citetalias{2016A&A...592A.156D} we reported distortions in the combined spectra of HIP 103983; this is likely a result from the combination of the spectra at orbital phases in which the Doppler separation between the binaries is large. Since the observing windows of the HARPS spectra of HIP 19911 and HIP 67620 do not cover large RV separations (see Fig. \\ref{orbits_updated}), the same effect is not seen in the combined spectra of these stars. This effect also explains why HIP 103983 is an outlier in fig. 4 of \\citetalias{2016A&A...592A.156D}.\n\nAlthough we have limited RV data, the \\texttt{emcee} simulations converge towards a well-defined solution instead of allowing longer periods, as these produce larger residuals. It is important, however, to keep monitoring the radial velocities of this system in order to confirm that the most recent data points are in fact a second inflection in the radial velocities. The residuals for the fit for the HIRES spectra are on the order of 100 m s$^{-1}$, which is likely a result from the contamination by a bright companion. \\citetalias{2014AJ....147...86T} reported a $0.91$ M$_\\odot$ visual companion at a separation of $0.093 \\arcsec$, which is consistent with the spectrocopic semi-major axis we estimated: $0.149 \\arcsec$ for a distance of 65.7 pc \\citep{2007A&A...474..653V}.\n\n\\subsection{Other binaries with updated orbital parameters}\n\nAmong the known binaries in the solar twins sample, five of them display curvature in their RV data which allows the estimation of limits for their orbital parameters (see Table \\ref{curvature_results} and Fig. \\ref{long_period_rvs}). Some of the linear trend binaries observed in our HARPS Solar Twin Planet Search program are targets with large potential for follow-up studies. For the companions with visual detection, we were able to estimate their most likely mass (see Table \\ref{linear_trend_results}).\n\n\\textbf{HIP 14501:} \\textit{Linear trend.} Its companion is reported by \\citet{2014ApJ...781...29C} as the first directly imaged T dwarf that produces a measurable doppler acceleration in the primary star. Using a low-resolution direct spectrum of the companion, \\citet{2015ApJ...798L..43C} estimated a model-dependent mass of 56.7 M$_{\\mathrm{Jup}}$. Using the HARPS and HIRES\/Keck RV data and the observed separation of $1.653\\arcsec$ \\citep{2014ApJ...781...29C}, we found that the most likely value of the companion mass is 0.043 M$_\\odot$ (45 M$_{\\mathrm{Jup}}$), which agrees with the mass obtained by \\citet{2015ApJ...798L..43C}. The most recent HARPS data hints of an inflection point in the orbit of HIP 14501 B (see Fig. \\ref{rv_14501}), but further RV monitoring of the system is necessary to confirm it.\n\n\\begin{figure*}\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[width=0.45\\textwidth]{images\/54102_rvs_lmfit.pdf} & \\includegraphics[width=0.45\\textwidth]{images\/54582_rvs_lmfit.pdf} \\\\\n\\includegraphics[width=0.45\\textwidth]{images\/72043_rvs_lmfit.pdf} & \\includegraphics[width=0.45\\textwidth]{images\/87769_rvs_lmfit.pdf} \\\\\n\\end{tabular}\n\\caption{The radial velocities and the solutions that produced the lower limits of orbital parameters of the solar twin binaries with RV curvature. A similar plot for HIP 73241 can be found in \\citet{2015MNRAS.453.1439J}. Time is given in $\\mathrm{JD} - 2.45 \\times 10^{6}$ d.}\n\\label{long_period_rvs}\n\\end{figure*}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.47\\textwidth]{images\/14501_rvs.pdf}\n\\caption{Radial velocities of HIP 14501. The RV shift in the $y$-axes is arbitrary. Time is given in $\\mathrm{JD} - 2.45 \\times 10^{6}$ d.}\n\\label{rv_14501}\n\\end{figure}\n\n\\textbf{HIP 18844:} \\textit{Linear trend.} It is listed as a multiple system containing a closer-in low-mass stellar companion (estimated 0.06 M$\\odot$, which agrees with our most likely mass) and orbital period $T = 6.5$ yr \\citepalias[][and references therein]{2014AJ....147...86T}. For the companion farther away, \\citet{2015MNRAS.453.1439J} reported a minimum orbital period of $\\sim 195$ yr and $m\\sin{i} = 0.33$ M$_\\odot$, with a separation of $29\\arcsec$ in 1941 ($\\sim$$750$ AU for a distance of 26 pc).\n\n\\textbf{HIP 54102:} \\textit{RV curvature only.} It is listed as a proper motion binary by \\citet{2005AJ....129.2420M}, but there are no other information about the companions in the literature. Its eccentricity is completely unconstrained due to lack of RV coverage. We estimate that its companion's minimum mass is $12.6$ M$_\\mathrm{Jup}$, with an orbital period larger than 14 years.\n\n\\textbf{HIP 64150:} \\textit{Linear trend.} The most likely companion mass obtained by the method explained in Section \\ref{methods_long_period} renders an estimate of 0.26 M$_\\odot$, as seen in Fig. \\ref{64150_pdf}. The higher mass (0.54 M$_\\odot$) obtained by \\citealt{2013ApJ...774....1C} and \\citetalias{2014ApJ...783L..25M} can be attributed to less likely orbital configurations, but it is still inside the 1-$\\sigma$ confidence interval of the RV+imaging mass estimate. The main star displays clear signals of atmosphere pollution caused by mass transfer from its companion during the red giant phase \\citep{2011PASJ...63..697T}, characterizing the only confirmed blue straggler of our sample. The measured projected separation of the binary system is 18.1 AU \\citepalias{2014ApJ...783L..25M}, which indicates that even for such a wide system the amount of mass transferred is still large enough to produce measurable differences in chemical abundances. It seems, however, that the amount of angular momentum transfer was not enough to produce significant enhancement in the rotation rate and activity of the solar twin. It is also important to note that the isochronal age measured for this system \\citepalias{2016A&A...590A..32T} has a better agreement with the white dwarf (WD) cooling age estimated by \\citetalias{2014ApJ...783L..25M} than previous estimates, illustrating the importance of studying these Sirius-like systems to test the various methods of age estimation.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.47\\textwidth]{images\/64150_torres.pdf}\n\\caption{Posterior probability distribution of the companion mass for HIP 64150. The mass obtained by \\citet{2014ApJ...783L..25M} using SED fitting for the spectra of the WD companion is shown as a red vertical line.}\n\\label{64150_pdf}\n\\end{figure}\n\n\\textbf{HIP 65708:} This star has previously been reported as a single-lined spectroscopic binary with an orbital solution \\citep{2002AJ....124.1144L}. Here we update this solution by leveraging the extremely precise radial velocities measured in the Lick Planet Search program and with the HARPS spectrograph. The minimum mass of the companion is 0.167 M$_\\odot$, indicating it is a red dwarf, orbiting at less than 1 AU with a slight eccentricity of 0.31. Our results agree with the previous orbital solution, which was based solely on data with uncertainties two orders of magnitude higher than the most recent data from HARPS and the Lick Planet Search.\n\n\\textbf{HIP 72043:} \\textit{RV curvature only.} Similarly to HIP 54102, it is listed as a proper motion binary and we could not constrain its eccentricity. A fairly massive ($> 0.5$ M$_\\odot$) companion is inferred at a very large period; this fit suggests that the longitude of periapse of the companion of HIP 72043 is currently at an unfavorable position for visual detection.\n\n\\textbf{HIP 73241:} \\textit{RV curvature only.} The companion's orbit is eccentric enough to allow an estimation of the minimum eccentricity; its companion has been previously been confirmed by \\citet{2010ApJS..190....1R} and visually detected by \\citetalias{2014AJ....147...86T} with a separation $0.318\\arcsec$. In \\citetalias{2016A&A...592A.156D} we listed this star as having an unusually high rotation, but here we revise this conclusion and list HIP 73241 as a candidate peculiar rotator because its $v \\sin{i}$ is less than 2$\\sigma$ above the expected value for its age. Similarly to HIP 67620, this peculiarity, if real,  could also be explained by contamination by a bright companion, since we determined that the minimum companion mass $m \\sin{i} > 0.49$ M$_\\odot$.\n\n\\textbf{HIP 79578:} The companion is a well-defined 0.10 M$_\\odot$ red dwarf orbiting the main star approximately every 18 years in a fairly eccentric orbit ($e = 0.33$). The orbital parameters we obtained differ significantly from the ones obtained by \\citet{2015MNRAS.453.1439J} by more than $10\\%$, except for the eccentricity; also in contrast, \\citeauthor{2015MNRAS.453.1439J} report it as a brown dwarf companion. The fit for this binary displays residuals of up to 30 m s$^{-1}$ for the AATPS radial velocities, and the periodogram of these residuals shows a peak near the period 725 days. When we fit an extra object with $m\\sin{i} = 0.70$ M$_{\\mathrm{Jup}}$ at this period ($a = 1.62$ AU and $e = 0.87$), it improves the general fit of the RVs by a factor of 7. It is important to mention, however, that there are only 17 data points for the AATPS dataset, and the HARPS dataset does not display large residuals for a single companion fit. We need thus more observations to securely infer the configuration of this binary system, and if it truly has an extra substellar companion at a shorter period.\n\n\\textbf{HIP 81746:} This is another high-eccentricity ($e = 0.7$) binary that does not display clear anomalies in its rotation and activity. Its companion is a 0.1 M$_\\odot$ red dwarf orbiting the main star every 9 years. The orbital parameters we obtained are in good agreement with the ones reported by \\citet{2015MNRAS.453.1439J}.\n\n\\textbf{HIP 83276:} \\textit{RV curvature only.} Although the HARPS radial velocities suggest the presence of a stellar mass companion, we do not have enough RV data points to infer any information about the orbital parameters of the system. Using radial velocities measured with the CORAVEL spectrograph, \\citet{1991A&A...248..485D} found the companion has $m\\sin{i}=0.24$ M$_\\odot$, $e=0.185$ and an orbital period of 386.72 days.\n\n\\textbf{HIP 87769:} \\textit{RV curvature only.} It is reported as a binary system by \\citetalias{2014AJ....147...86T} but, similarly to HIP 54102, lacks an inflection point in its RV data from HARPS, which spans 3.3 yr. There is a wide range of possible orbital solutions that suggest $m \\sin{i}$ varying from brown dwarf masses to $\\sim 1$ M$_\\odot$. Higher eccentricities ($e > 0.8$) can be ruled out as unlikely because they suggest a companion with $m \\sin{i} \\approx 1$ M$_\\odot$ at an orbital period of more than 500 yr and $a > 80$ AU.\n\n\\begin{table}\n\\begin{center}\n\\caption{Lower limits of the orbital parameters of the spectroscopic binaries with curvature in their RV data.}\n\\begin{tabular}{llcccc}\n\\toprule[\\heavyrulewidth]\n\\multirow{2}{*}{HIP} & \\multirow{2}{*}{HD} & $K$ & $T$ & $m \\sin{i}$ & $e$\\\\\n & & (km s$^{-1}$) & (yr) & (M$_\\odot$) & \\\\\n\\bottomrule[\\heavyrulewidth]\n 54102 & 96116 & $> 0.182$ & $> 14$ & $> 0.012$ & $\\dots$ \\\\\n 54582 & 97037 & $> 0.193$ & $> 102$ & $> 0.03$ & $\\dots$ \\\\\n 72043 & 129814 & $> 2.11$ & $> 104$ & $> 0.40$ & $\\dots$ \\\\\n 73241 & 131923 & $> 5.93$ & $> 21.0$ & $> 0.49$ & $> 0.72$ \\\\\n 87769 & 163441 & $> 1.90$ & $> 81.5$ & $> 0.30$ & $\\dots$ \\\\\n\\bottomrule[\\heavyrulewidth]\n\\label{curvature_results}\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{table}\n\\begin{center}\n\\caption{Measured RV slopes of the linear trend binaries. The most likely mass for the spectroscopic companion is estimated when their separation is available. Otherwise a minimum mass is provided.}\n\\begin{tabular}{llcccc}\n\\toprule[\\heavyrulewidth]\n\\multirow{2}{*}{HIP} & \\multirow{2}{*}{HD} & $dv_r \/ dt$ & $\\rho$ & Dist. & $m$  \\\\\n & & (m s$^{-1}$ yr$^{-1}$) & (arcsec) & (pc)\\textsuperscript{d} & (M$_{\\mathrm{Jup}}$) \\\\\n\\bottomrule[\\heavyrulewidth]\n14501 & 19467 & $-1.30 \\pm 0.01$ & 1.653\\textsuperscript{a} & 30.86 & 45 \\\\\n18844\\textsuperscript{$\\dag$} & 25874 & $424 \\pm 3$ & 0.140\\textsuperscript{b} & 25.91 & 79 \\\\\n62039 & 110537 & $7.25 \\pm 0.03$ & $\\dots$ & 42.68 & $> 19$ \\\\\n64150 & 114174 & $61.72 \\pm 0.02$ & 0.675\\textsuperscript{c} & 26.14 & 270 \\\\\n\\bottomrule[\\heavyrulewidth]\n\\multicolumn{6}{l}{\\textsuperscript{a}\\footnotesize{\\citet{2014ApJ...781...29C}.}}\\\\\n\\multicolumn{6}{l}{\\textsuperscript{b}\\footnotesize{\\citet{2014AJ....147...86T}.}}\\\\\n\\multicolumn{6}{l}{\\textsuperscript{c}\\footnotesize{\\citet{2014ApJ...783L..25M}.}}\\\\\n\\multicolumn{6}{l}{\\textsuperscript{d}\\footnotesize{\\citet{2007A&A...474..653V}.}}\n\\\\\n\\multicolumn{6}{l}{\\textsuperscript{$\\dag$}\\footnotesize{Triple or higher-order system.}}\\\\\n\\label{linear_trend_results}\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\subsection{Considerations on multiplicity statistics}\n\nAlthough planet search surveys are generally biased against the presence of binaries due to avoiding known compact multiple systems, the fraction of binary or higher-order systems in the whole sample of the Solar Twin Planet Search program is $42\\% \\pm 6\\%$\\footnote{\\footnotesize{Counting stellar and brown dwarf companions. The uncertainty is computed using a bootstrap resampling analysis with 10,000 iterations, similarly to \\citet{2010ApJS..190....1R}. In each iteration, a new set of 81 solar twins is randomly drawn from the original sample allowing stars to be selected more than once.}}. This value agrees with previous multiplicity fractions reported by, e.g., \\citet{2010ApJS..190....1R} and \\citetalias{2014AJ....147...86T}; however, it is signficantly lower than the $58\\%$ multiplicity factor for solar-type stars reported by \\citet{2017ApJ...836..139F}, who argues that previous results are subject to selection effects and are thus biased against the presence of multiple systems.\n\nThe orbital period vs. mass ratio plot of companions in the Solar Twin Planet Search is shown in Fig. \\ref{mratio}. A comparison with the sample of solar-type stars from \\citetalias{2014AJ....147...86T} reveals two important biases in our sample: i) Mass ratios are mostly below 0.3 because of selection of targets that do not show large radial velocity variations in previous studies; ii) Orbital periods are mostly lower than 30 yr because longer values cannot be constrained from the recent RV surveys targeting solar-type stars with low-mass companions. In such cases, further monitoring of linear trend and RV curvature-only binaries may prove useful to understand the origins of the brown dwarf desert \\citep{2006ApJ...640.1051G}. These targets are particularly appealing because the long periods mean that the separation from the main star is large enough to allow us to observe them directly using high-resolution imaging.\n\nPrevious studies on the period-eccentricity relation for binary stars found that systems with orbital periods below 10 days tend to have eccentricities near zero, while those between 10 and 1000 days follow a roughly flat distribution of eccentricities \\citep[][and references therein]{2016AJ....152..189K}, an effect that is due to the timescales for circularization of orbits. In relation to our sample, with the exception of HIP 30037, HIP 65708 and HIP 83276, all of the binaries we observed have periods longer than 1000 days and eccentricities higher than 0.3, which agrees with the aforementioned findings. According to \\citet{1991A&A...248..485D}, the distribution of eccentricities on systems with $T > 1000$ d is a function of energy only, and does not depend on $T$ (see fig. 5 in \\citeauthor{1991A&A...248..485D}). Interestingly, HIP 30037, which hosts a brown dwarf companion with $T = 31.6$ d, falls inside the 25--35 days interval of orbital periods found by \\citeauthor{2016AJ....152..189K} that corresponds to a short stage of evolution of binaries undergoing a fast change in their orbits.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.47\\textwidth]{images\/mratio_period.pdf}\n\\caption{Mass ratios in function of the orbital periods of binary stars or higher-order systems in the solar neighborhood. The purple circles are binaries in our sample with well defined period and $m \\sin{i}$; the blue triangles correspond to the binaries in our sample for which we only have lower limits for the periods and $m \\sin{i}$. The stars from \\citetalias{2014AJ....147...86T} are plotted as black dots (the darker ones are those with main star masses between 0.9 and 1.1 M$_\\odot$).}\n\\label{mratio}\n\\end{figure}\n\n\\section{Conclusions}\n\nThe Solar Twin Planet Search and several other programs observed 81 solar twins using the HARPS spectrograph. In total, 18 of these solar twins are spectroscopic binaries, 18 are visual binaries, and two intersect these categories. We found a multiplicity fraction of $42\\% \\pm 6\\%$ in the whole sample, which is lower than the expected fraction ($\\sim$$58\\%$) because of selection effects that are generally seen in exoplanet search surveys.\n\nWe updated or reproduced the solutions of several known binaries, and determined all the orbital parameters of HIP 19911, HIP 65708, HIP 67620, HIP 79578, HIP 81746 and HIP 103983. The stars HIP 43297 and HIP 64673, which we previously reported as binaries, are likely to host long-period giant planets instead of stellar companions. For binaries with partial orbits, we were able to place lower limits for some of their orbital parameters owing to the presence of curvature or an inflection point in their RV data. We estimated the most likely mass of the companions of the binaries that display only linear trends in their RV data. Future work is needed on studying the long-period binaries using photometry data and high-resolution imaging in order to constrain the nature of their companions. These wide solar twin binaries are prime targets for detailed physical characterization of their companions owing to the favorable separation for AO imaging and the precision with which we can measure the stellar parameters of the main star -- this is particularly important for fully convective red dwarf stars and very low-mass companions such as the T dwarf HIP 14501 B, whose evolution and structure is still poorly constrained.\n\nAdditionally, we reported the detailed discovery of new companions to the following solar twins: HIP 6407, HIP 30037, HIP 54582, and HIP 62039, for which we are able to determine an orbital solution for the first two using radial velocities. The latter two do not have enough RV data to obtain precise orbital parameters, but we can nonetheless estimate their minimum companions masses. We found that these new companions are likely very low-mass, ranging from 0.02 to 0.12 M$_\\odot$ (although stressing that these are lower limits), which should be useful in understanding the origins of the brown dwarf desert in future research.\n\nThe anomalies and RV residuals observed on HIP 19911, HIP 67620 and HIP 103983 are likely due to contamination by the companion on the spectra of the main star. Although the peculiar stars in our sample are no longer considered blue straggler candidates, it is important to note that the detection of WD companions is particularly important for the study of field Sun-like stars because they allow the estimation of their cooling ages; these are more reliable than isochronal and chromospheric ages in some cases, providing thus robust tests for other age estimate methods. We do not expect that the presence of M dwarf companions contaminate lithium spectral lines in Sun-like stars, thus stellar ages derived from Li abundances may be more reliable for double-lined solar twins. We recommend a revision of the stellar parameters of the peculiar binary stars by analyzing high-resolution spectra at the highest Doppler separations possible, or using Gaussian processes to disentangle the contaminated spectra \\citep[see, e.g.,][]{2017ApJ...840...49C}.\n\nWe conclude that single-lined solar twin binaries with orbital periods larger than several months and moderate to low eccentricities do not display signals of distinct rotational evolution when compared to single solar twins. The most compact system in our sample, HIP 30037, which hosts a 0.06 M$_\\odot$ brown dwarf companion at an orbital period of 31 days is, in fact, one of the quietest stars in the sample (in regards of its activity levels), and is thus a viable target for further efforts in detecting moderate- to long-period circumbinary planets.\n\n\\section*{Acknowledgements}\n\nLdS acknowledges the financial support from FAPESP grants no. 2016\/01684-9 and 2014\/26908-1. JM thanks FAPESP (2012\/24392-2) for support. LS acknowledges support by FAPESP (2014\/15706-9). This research made use of SciPy \\citep{scipy_ref}, Astropy \\citep{2013A&A...558A..33A}, Matplotlib \\citep{Hunter:2007}, and the SIMBAD and VizieR databases \\citep{2000A&AS..143....9W, 2000A&AS..143...23O}, operated at CDS, Strasbourg, France. We thank R. P. Butler, S. Vogt, G. Laughlin and J. Burt for allowing us to analyze the LCES HIRES\/Keck data prior to publication. LdS also thanks B. Montet, J. St\\\"urmer and A. Seifahrt for the fruitful discussions on the results and code implementation. We would also like to thank the anonymous referee for providing valuable suggestions to improve this manuscript.\n\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nPolarimetric imagery consists in forming an image of the state of polarization of the light backscattered by a scene. We consider in this paper that the scene is artificially illuminated with coherent light (laser). For example, this illumination is used in active imagery in order to combine night vision capability and to improve image resolution for a given aperture size. In practice, using a coherent illumination produces speckle noise that deteriorate the image \\cite{goo85}. However, the backscattered light gives information about the capability of the scene to polarize or depolarize the emitted light and thus allows one to determine the medium that compose the scene. These information can be described by a scalar parameter: the degree of polarization of light. This quantity is obtained in standard configurations of polarimetric systems using four pair of angular rotations of both a compensator and a polarizer. Four transmittance are thus recorded \\cite{bro98} that lead to the estimation of the degree of polarization. However, this system is complex and it is interesting to develop methods to estimate the degree of polarization that could reduce the number of images to register. In \\cite{gou01}, the authors proposed to estimate the degree of polarization with only two intensity images, however this method relies on the assumption that the measurements of the two components are uncorrelated which can be in some cases a too restrictive hypothesis. This paper extends the work of \\cite{gou01} by taking into account the correlation of the different components.\\\\\n Let us first introduce the context of the study.\\\\\n\n\n\\section{Background}\nThe electric field of the light at a point of coordinates { \\bf r} (vector of 3 component) in a 3D space and at time $t$ can be written, if we assume the light to propagate in a homogeneous and isotropic medium, as\n\\begin{equation}\n {\\bf E}({\\bf r},t) = \\left[ A^{X}({\\bf r},t) {\\bf e_x} + A^{Y}({ \\bf r},t) {\\bf e_y}\\right]. e^{- i 2 \\pi \\nu t}\n\\end{equation}\nwhere $\\nu$ is the central frequency of the field and $\\bf{e_x}, \\bf{e_y}$ are unitary orthogonal vectors (in the following bold letters represent vectors).\\\\\nThe terms $A^{X}({\\bf r},t)$ and $ A^{Y}({\\bf r},t)$ are complex and define the random vector called Jones vector\n\\begin{equation}\n\\bf{A} = \\left[\\begin{array}{l}\n   A^{X}({\\bf r},t) \\\\\n   A^{Y}({\\bf r},t)\n\\end{array}\\right].\n\\end{equation}\n\n\nThe state of polarization of light corresponds to the properties of ${\\bf{E}}({\\bf r},t)$ at a particular point of space. It can be described by the covariance matrix $\\Gamma$\n\n\\begin{equation}\n\\Gamma = \\left[\\begin{array}{ll}\n   <A^{X*}A^{X}> & <A^{X*}_{\\lambda}A^{Y}> \\\\\n   <A^{Y*} A^{X}>&<A^{Y*}_{\\lambda}A^{Y}>  \n   \\end{array}\\right]\n\\end{equation}\nwhere $<.>$ and $.^*$ define respectively the statistical average and the complex conjugate. For sake of brevity, the following notations for $\\Gamma$ are introduced\n\\begin{equation}\\label{cov}\n\\Gamma = \\left[\\begin{array}{ll}\n   a_1 & a_2 \\\\\n   a_2^* & a_4 \\\\\n   \\end{array}\\right] = \\frac{1}{c_1c_4-|c_2|^2} \\left[\\begin{array}{ll}\n   c_4 & -c_2 \\\\\n   -c_2^* & c_1 \\\\\n   \\end{array}\\right]; \n\\end{equation}\n\\begin{equation}\n\\hspace{0.6cm} \\Gamma^{-1} = \\left[\\begin{array}{ll}\n   c_1 & c_2  \\\\\n   c_2^* & c_4 \\\\\n   \\end{array}\\right] \n\\end{equation}\nLet us note that this matrix can be diagonalized since it is hermitic. \\\\\n In the case of coherent light, the electric field is represented by the complex Jones vector $\\bf{A}$ which follows a Gaussian circular law \\cite{goo85}\n\\begin{equation}\n p_{\\Gamma}({\\bf{A}}) = \\frac{1}{\\pi^2 det(\\Gamma)} e^{-{\\bf{A}}^{\\dagger}  \\Gamma^{-1}{\\bf{A}}} \n\\end{equation}\nwhere ${\\bf A}^{\\dagger}$ stands for the adjoint of the vector ${\\bf{A}}$.\\\\\n\nThe degree of polarization is defined by  \\cite{goo85}\n\n\\begin{equation}\nP^2= \\frac{\\mu_1 - \\mu_2}{\\mu_1 + \\mu_2}\n\\end{equation}\nwhere $\\mu_1$ and $\\mu_2$ are the eigenvalues of $\\Gamma$ ($\\mu_1 \\geq \\mu_2 \\geq 0$).\\\\\nThe degree of polarization is a scalar parameter that characterizes the state of polarization of the light: if $P = 0$, the light is said to be totally depolarized, and if  $P = 1$, the light is said to be totally polarized. In the intermediate cases, the light is partially polarized.\\\\\nWith the notations introduced in (\\ref{cov}) one can show that \\cite{goo85}\n\\begin{equation}\\label{deg_pol}\nP^2= 1 - \\frac{4 (a_1 a_4 - |a_2|^2)}{(a_1 + a_4)^2}.\n\\end{equation}\nThe knowledge of this quantity allows one to study the way the illuminated scene \npolarized or depolarized the emitted light. This degree gives information about the nature of the medium in the scene. In the standard configuration, four measurements are needed to estimate it. In the case of uncorrelated measurements, two intensity images gives a good estimation of this degree using the OSCI \\cite{gou01}. However, in some cases the uncorrelation of the measurements may be not valid.\\\\\nIn this paper, an original estimation method that both uses a pair of images and that accounts for correlation in the components is proposed. We recall in the following the method proposed in \\cite{gou01} and we extend it to correlated measurements. We then compare them through statistical measures using simulated data. The results are also presented when the standard estimation of $P^2$ is used (four images) as this case is expected to give the best results. Finally we conclude and give the perspectives of this work.\n\\section{The OSCI}\nIn the case of two uncorrelated images, the Orthogonal State Contrast Image (OSCI) is determined from \\cite{gou01}, \\cite{bre99}, \\cite{gou04b}\n\\begin{equation}\n\\rho(i) = \\frac{I_1(i) - I_2(i)}{I_1(i) + I_2(i)}\n\\end{equation}\nwhere $I_1(i)$ and $I_2(i)$ are intensity measurements at the pixel site $(i)$, assuming a lexicographic order for the pixels.\\\\\nThese two images are obtained with simple polarimetric systems. First, the scene is illuminated by coherent light with single elliptical polarization state. Then the backscattered light is analysed in the polarization state parallel (which leads to $I_1(i) = |A_X(i)|^2)$ and orthogonal (which leads to $I_2(i) = |A_Y(i)|^2) $ to the incident one. \\\\\nThe OSCI is an estimation of $P^2$ in each pixel provided that the materials of the scene  modify the degree of polarization of incident light without modifying its principal polarized state (\\footnote{State represented by the eigenvector associated to eigenvalue $\\mu_1$.}). Let us recall that this kind of material is called pure depolarizer. For such materials, the covariance matrix $\\Gamma$ is diagonal and, since the diagonal terms represent the intensity images, the OSCI gives an estimation of $P^2$ with \n\\begin{equation}\\label{posci}\n{\\hat P}_{OSCI}^2(i) = \\left(\\frac{I_1(i) - I_2(i)}{I_1(i) + I_2(i)}\\right)^2 = \\eta^2(i).\n\\end{equation}\nIn the case of non pure depolarizer objects (i.e. $\\Gamma$ is non diagonal), the OSCI still reveals interesting contrast image but no more defines $P^2$. This leads us to a new method that considers the cases of correlated  measurements. This is the object of the following part.\n\n\\section{Correlated measurements}\nIn the case of correlated measurements, the covariance matrix is non diagonal and of the form (\\ref{cov}). In its standard estimation, the degree of polarization needs four measurements, however, two images are sufficient to get an estimation of $P^2$ . Indeed, the coefficient $a_1$ is obtained from one measurement, as the coefficient $a_4$ and the squared modulus of $a_2$ can be estimated from the cross-correlation coefficient $\\delta_{12}$ between two measurements $I_1$ and $I_2$. We have\n\\begin{equation}\n\\delta_{12}= \\int \\int I_1 I_2 p(I_1,I_2) dI_1 dI_2 = <I_1 I_2>.\n\\end{equation}\n$\\delta_{12}$ can be calculated by using the joined density probability function $p(I_1,I_2)$ assuming that ${\\bf A}$ is Gaussian circular (i.e. the speckle is supposed to be fully developped).\nIt can be shown that the correlation coefficient is obtained with\n\\begin{equation}\n\\delta_{12}= \\frac{1}{det \\Gamma c_1^2 c_4^2}  \\left( \\frac{1 + \\frac{|c_2|^{2}}{c_1 c_4}}{\\left( 1 - \\frac{|c_2|^{2}}{c_1 c_4} \\right )^3 }\\right )\n\\end{equation}\nwhere $|c_2|$ stands for the modulus of $c_2$.\nCalculating the centered correlation coefficient defined by\n\\begin{equation}\\label{cross_coef0}\n\\Delta_{12}=  \\delta_{12} - <I_1> <I_2> \n\\end{equation}\nAfter some simple algebra, we get\n\\begin{equation}\\label{cross_coef}\n\\Delta_{12}=  |a_2|^2\n\\end{equation}\n\nThus the coefficient $|a_2|^2$ can be obtained from two measurements with $ <I_1 I_2>- <I_1> <I_2>$. This remark leads to write the following property.\\\\\n\n{\\bf Property A:} {\\it For fully developped speckle fluctuations, the degree of polarization can be obtained from only two intensity images.}\n\\\\\\\\\nOne can easily note that the degree of polarization can be written as a function of the OSCI with \n\\begin{equation}\\label{cor}\nP^2(i,j)=  \\eta^2(i,j) + 4 \\frac{\\Delta_{12}}{(<I_1> + <I_2>)^2}\n\\end{equation}\nFrom (\\ref{posci}), the degree of polarization estimated from the OSCI is clearly under-estimated. Thus, we can correct the OSCI in order to get an estimation of the degree of polarization using the correlation coefficient $\\Delta_{12}$. \\\\\nIn the following part, the different estimation of the degree of polarization are compared to the estimation with four images through simulated data.\n\n\\section{Comparison with numerical experiments}\nWe generated $R$ experiments of $N$ samples of complex Jones vectors which follow a Gaussian circular law. The covariance matrix is known and thus $P^2$ is also known. Under the assumption that the statistical average can be estimated by spatial averages in homogenous regions, the coefficients $a_1$  can be estimated from a single image ( $I_1 = |A_X|^2$) like the coefficient $a_4$ ($I_2 = |A_Y|^2$) since\n\\begin{equation}\\label{estima_1}\n{\\hat a}_1 = \\frac{1}{N}\\sum ^{N}_{i=1}|A_X(i)|^2\n\\end{equation}\n\\begin{equation}\\label{estima_2}\n{\\hat a}_4 = \\frac{1}{N}\\sum ^{N}_{i=1}|A_Y(i)|^2\n\\end{equation}\nwhere $\\{ A_X(i), A_Y(i) \\}$ represents the component of the Jones vector for the sample $i$.\\\\\nThe estimation of $P^2$ differs in the studied cases by the way $|a_2|^2$  is estimated. Three different methods are used:\n\\begin{itemize}\n\n\\item Case of four images\\\\\nIn this situation, we have both the real and the imaginary part of the coefficient $a_2$. The quantity $|a_2|^2$ is  estimated by\n\n\\begin{equation}\\label{estim_a2_A}\n{\\hat \\rho}_{A}(i)= \\left|\\frac{1}{N}\\sum ^{N}_{i=1}A_X(i)A^{*}_Y(i) \\right|^2 \n\\end{equation}\n\n\\item Case of two images with the OSCI\\\\\nIn this case $a_2$ is assumed to be equal to zero.\n\n\\item Case of two images with the proposed approach\\\\\nThe coefficient  $|a_2|^2$ is estimated using (\\ref{cross_coef0}) with\n\n\\begin{equation}\\label{estim_a2_I}\n\\begin{array}{lll}\n{\\hat \\rho}_{I}(i) & = & \\frac{1}{N}\\sum ^{N}_{i=1}|A_X(i)|^2|A_Y(i)|^2 \\\\\n& &\\\\\n&  &- \\left( \\frac{1}{N}\\sum ^{N}_{i=1}|A_X(i)|^2 \\right) \\left(\\frac{1}{N}\\sum ^{N}_{i=1}|A_Y(i)|^2\\right)\n\\end{array}\n\\end{equation}\n\n\\end{itemize}\n\nIn the three cases, $P^2$ was estimated from the relation (\\ref{deg_pol}) with the estimated parameters.\\\\\nIn order to characterize the precision of the estimation, one considers six examples of matrix $\\Gamma$. \n\n\\begin{equation}\n\\Gamma_1=\\left[ \\begin{array}{cc}\n15 & 0.2 + 0.5i\\\\\n0.2 - 0.5i & 6\n\\end{array}\\right];  \\Gamma_2=\\left[ \\begin{array}{cc}\n16 & 0\\\\\n0 & 3.6\n\\end{array}\\right];\n\\end{equation}\n\n\n\n\\begin{equation}\n \\Gamma_3=\\left[ \\begin{array}{cc}\n82 & 13i\\\\\n-13i & 17\n\\end{array}\\right];  \\Gamma_4=\\left[ \\begin{array}{cc}\n18 & 7 + 8i\\\\\n7 - 8i & 11\n\\end{array}\\right];\n\\end{equation}\n\n\\begin{equation}\n\\Gamma_5=\\left[ \\begin{array}{cc}\n30 & 16 - 8i\\\\\n16 + 8i & 14\n\\end{array}\\right];  \\Gamma_6=\\left[ \\begin{array}{cc}\n1.25 & 5.5i\\\\\n-5.5i & 26\n\\end{array}\\right];\n\\end{equation}\n These matrices were chosen such as the degree of polarization are approximatively in $\\{ 0.2, 0.4, 0.5, 0.6, 0.8, 1 \\}$.\\\\\n\nThe simulations are performed for $R = 1000$ realisations of $N = 10000$ samples for the six covariance matrices. For the two matrices $\\Gamma_1$ and $\\Gamma_5$, supplementary cases have been studied when $N$ $\\in$ $\\{ 100, 500, 1000, 5000, 10000 \\}$. The results are presented in figures \\ref{fig1}, \\ref{fig2},  \\ref{fig3}, \\ref{fig4}, \\ref{fig5}, \\ref{fig6}. Several points are important to notice.\nFirst of all, as expected, the best estimations of the degree of polarization, regarding all the cases tested, was achieved with four images. However the proposed approach that relies on two correlated images produces good estimations of the degree of polarization whatever the covariance matrix is used (fig.\\ref{fig1}) as soon as $N > 1000$ (fig.\\ref{fig3} and fig.\\ref{fig5} ). Note that the estimation with the OSCI gives results which cannot be used if the term $\\Delta_{12}\/((<I_1> + <I_2>)^2)$ is too high (for example if $|a_2|^2$ is non negligible). Fig.\\ref{fig2}, \\ref{fig4} and \\ref{fig6} show that the experimental variance using four measurements or the OSCI are comparable whatever $\\Gamma$ and $N$ are, whereas the variance obtained with the proposed approach is larger. However this precision should be sufficient for some practical applications. This point should be studied in details in a future work.\n\n\\begin{figure}\n\\centerline{\\epsfxsize=9cm\\epsfbox{P_gamma.eps}}\n\\caption{The degree of polarization is plotted as a function of the six covariance matrix for $R = 1000$ and $N = 10000$. $P^2$ is the true degree of polarization, $P^2$ - A is the degree estimated from four measurements, $P^2$ - I is the degree obtained with the numerical simulations using the proposed method for evaluating $|a_2|^2$, $P^2$ - OSCI is the degree estimed from the OSCI.}\n\\label{fig1}\n\\end{figure}\n\\begin{figure}\n\\centerline{\\epsfxsize=9cm\\epsfbox{sigma_gamma3.eps}}\n\\caption{The experimental variance of the degree of polarization is plotted as a function of the six covariance matrix for $R = 1000$ and $N = 10000$. $\\sigma$ - A is the standard deviation of the degree of polarization estimated from four measures, $\\sigma$ - I is  the standard deviation of $P^2$ obtained with the numerical simulations using the proposed method for evaluating $|a_2|^2$, $\\sigma$ - OSCI is the standard deviation of $P^2$ estimed from the OSCI.}\n\\label{fig2}\n\\end{figure}\n\\begin{figure}\n\\centerline{\\epsfxsize=9cm\\epsfbox{P_gamma1_samples.eps}}\n\\caption{The degree of polarization is plotted as a function of the number of sample $N$ for $R = 1000$ for the covariance matrix $\\Gamma_1$. $P^2$ is the true degree of polarization, $P^2$ - A is the degree estimated from four measures, $P^2$ - I is the degree obtained with the numerical simulations using the proposed method for evaluating $|a_2|^2$, $P^2$ - OSCI is the degree estimed from the OSCI.}\n\\label{fig3}\n\\end{figure}\n\\begin{figure}\n\\centerline{\\epsfxsize=9cm\\epsfbox{sigma_gamma1_samples4.eps}}\n\\caption{The product of N and the experimental variance of the degree of polarization ($\\sigma^2$) is plotted as a function of $N$ for $R = 1000$ for the covariance matrix $\\Gamma_1$.  $N \\sigma^2$ - A is the product obtained when $P^2$ is estimated from four measures, $N \\sigma^2$ - I  is the product obtained when $P^2$ is estimated with the numerical simulations using the proposed method for evaluating $|a_2|^2$, $N \\sigma^2$ - OSCI  is the product obtained when $P^2$ is estimated from the OSCI.}\n\\label{fig4}\n\\end{figure}\n\\begin{figure} \n\\centerline{\\epsfxsize=9cm\\epsfbox{P_gamma5_samples.eps}}\n\\caption{Idem as the Figure 3. using  the covariance matrix $\\Gamma_5$ instead of $\\Gamma_1$.}\n\\label{fig5}\n\\end{figure}\n\\begin{figure}\n\\centerline{\\epsfxsize=9cm\\epsfbox{sigmagamma5samples4.eps}}\n\\caption{Idem as the Figure 4. using  the covariance matrix $\\Gamma_5$ instead of $\\Gamma_1$.}\n \\label{fig6}\n\\end{figure}\n\n\\section{Conclusion}\n\nWe have proposed a new approach to estimate the degree of polarization on polarimetric images degraded by speckle noise. Assuming that the speckle is fully developped, this method allows one to estimate this degree with only two intensity images whereas four images are needed in a standard experimental setup. This presents a great interest in term of reduction of cost of the imagery system since the original setup can be simplified. The proposed approach has been tested on simulated data and compared to the standard estimation techniques that requires either 4 images or 2 independant images (OSCI).  The results show that the proposed method gives good approximation of the degree of polarization.\nThis study needs to be extended with a theoritical analyses in order to precise the conditions of validity of the proposed approach.  \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\nIt has been known that the observed number density of satellite galaxies in the Local Group and our own Milky Way is orders of magnitude lower than the predictions from the cosmological simulations, the so-called `missing satellite problem' \\citep{kly99,moo99}. One of the popular solutions to this problem is that low-mass halos fail to form stars efficiently such that  they are below the detection limit of most imaging surveys. Under this paradigm,  `dark' galaxies are gas-rich galaxies that do not emit sufficient optical light due to their low efficiency in forming\nstars, but are thought to be building blocks of normal star-forming galaxies. This type of galaxy is an\nideal laboratory to study early stages of star formation, which could lead to understanding how\nthe star formation is triggered. \n\nIt is in general very difficult to identify dark galaxies since they\nare extremely faint in the optical. Very few dark candidates have been identified and confirmed to date. The ALFALFA (Arecibo Legacy Fast ALFA) HI survey \\citep{gio05} has discovered on the order of $\\sim 200$ HI sources not associated with apparent optical counterparts, but most of them are likely to have tidal origins \\citep{hay11}. A pilot study suggested that none of remaining objects they have explored is a dark galaxy after cross-checking with data in other wavelengths \\citep{can15}. \n\n\nRecently, \\citet{van15a} have identified a new class of low surface brightness (LSB) dwarf galaxies in the Coma cluster, often referred to as `ultra-diffuse' galaxies (UDGs), using the Dragonfly Telescope\nArray \\citep{abr14}. These UDGs not only have a low surface brightness (24 -- 26 mag arcsec$^{-2}$), but also show extended structures with size comparable to $L^{*}$ galaxies. Although these UDGs are found in special environments and may not necessarily represent the global dwarf population, the discovery of this type of galaxy suggests we might have missed numerous faint dwarfs due to observation limitations in the past.\n\nIn addition to the aforementioned efforts, the optical Integral Field Unit (IFU) observations open a plausible window to probe the dwarf populations. Normally, emission lines ionized by star-forming regions, AGNs, or shocks are stronger than the stellar continuum and hence can be easily detected with reasonable integration time. With the large area covered by IFU, the structures of ionized gas can be probed out to several tens of kpc. Isolated ionized gases that are separate from a nearby galaxy or a quasar have been readily studied in IFU as well as spectroscopic observations \\citep{fu07a,fu07b,fu08,lin09,hus10,kee12,che16b}. Although the majority of those ionized gas are suggested or inferred to have external origins, such as the result of gas accretion or minor mergers, their nature remains unknown.\n\nHere we report the discovery of a giant H$\\alpha$~blob which does not have any optical counterpart in deep CFHT $gri$ images. However, the morphology and emission line analyses suggest that it could either be ejected gas due to past AGN activity or a special type of UDGs (or `dark' galaxies). \nIn \\S2, we describe the multi-wavelength data for this system. We present the main results in \\S3. Section 4 discusses the plausible origins of this system and the important implications of our results. Conclusions are given in \\S5. Throughout this paper we adopt the following cosmology: \\textit{H}$_0$ = 100$h$~${\\rm km~s^{-1}}$  Mpc$^{-1}$, $\\Omega_{\\rm m} =\n0.3$ and $\\Omega_{\\Lambda } = 0.7$. We use a Salpeter IMF when deriving the star formation rate from various observables.\nWe adopt the Hubble constant $h$ = 0.7 when calculating rest-frame magnitudes. All magnitudes are given in the AB system.\n\n\n\n\\section{DATA \\label{sec:data}}\n\\subsection{Optical integral field data}\nThis system, MaNGA 1-24145 ($z$ = 0.0322, RA = 258.84693, DEC =57.43288, M$_{*}$ $\\sim 10^{11}$ $\\rm M_{\\odot}$\\footnote{based on the NASA-Sloan\nAtlas catalog: http:\/\/www.nsatlas.org\n }), was observed in the first 1392 galaxies, as part of the on-going SDSS-IV\/MaNGA survey \\citep{bun15,dro15,law16,yan16a,yan16b,sdss16}. MaNGA is an IFU program to survey for 10k nearby galaxies with a spectral resolution varying from R $\\sim$ 1400 at 4000 \\AA~ to R $\\sim$ 2600 at 9000 \\AA. The survey uses the BOSS spectrographs (Smee et al. 2013) on the 2.5m Sloan Foundation Telescope (Gunn et al. 2006). The median full width at half maximum (FHWM) of the MaNGA point spread function (PSF) of the datacube is $\\sim$ 2.5\".\nThe MaNGA data used were reduced using the MPL-4 version of the MaNGA data reduction pipeline \\citep{law16}.\nThe spectral line fitting is carried out using Pipe3D pipeline\\citep{san16a}. The stellar continuum was first modelled with a linear combination of 12 single stellar population (SSP) templates that were extracted\nfrom the MILES project \\citep{san06,vaz10,fal11}. The best-fit stellar continuum is then subtracted from the reduced data spectrum for the emission line measurements. Details of the fitting procedures are described in \\citet{san16b}. To ensure reliable measurements, we restrict our analysis to spaxels where the error-to-flux ratio of the line fitting is less than 1 in subsequent analyses.\n\n\nTo correct for dust reddening, we follow the method described in the Appendix of \\citet{vog13} to compute the reddening correction by using the Balmer decrement at each spaxel of the IFU cube. An extinction law with $Rv = 4.5$ \\citep{fis05} is used. \nThe star formation rate (SFR) is then estimated based on this extinction corrected H$\\alpha$~flux. \n\n\n\n\\begin{figure*}\n\\includegraphics[angle=-90,width=17cm]{manga-7991-12702_SDSSimage_Ha_edit.eps}\n\\caption{Left: The SDSS $gri$ composite image of MaNGA 1-24145 with the MaNGA hexagonal bundle field of view (FoV) overlaid. This system was observed with the 127 fibre bundle of MaNGA, so this hexagon is $\\sim$32.5\" in diameter. The data extend to regions a bit outside the hexagon because of the dithering. Three distinct objects are visible within the bundle, including two elliptical galaxies (Satsuki and Mei) and one foreground star. Right: the observed H$\\alpha$~flux map from the MaNGA observations. \\label{fig:ha}.}\n\\end{figure*}\n\n\\begin{figure}\n\\includegraphics[angle=0,width=8.5cm]{SDSS_manga-7991-12702_colorcomposite_pipe3D_enhanced_crop.eps}\n\\caption{Left: The $ugi$ composite image reconstructed using the MaNGA continuum with the MaNGA hexagonal bundle field of view (FoV) overlaid. Right: The [OIII]+H$\\alpha$+[NII] composite image. The flux scales of the three lines are adjusted in order to highlight the H$\\alpha$~blob. \\label{fig:pipe3Dimage}}\n\\end{figure}\n\n\n\n\\begin{figure*}\n\\includegraphics[angle=-90,width=17cm]{manga-7991-12702_pipe3d_spectra.eps}\n\\caption{Upper panels: The black curves represent the MaNGA spectra of the central spaxel of the main galaxy `Satsuki' (left), the southern companion `Mei' (middle), and the H$\\alpha$~blob `Totoro' (right). The red curves are the best-fitted SSP model spectra for the stellar continuum. The blue curves show the residual spectra that are used for the emission line fitting. Lower panels: The zoom-in spectra around the H$\\alpha$~line. The five dashed line denote the observed wavelengths of the [NII] 6584, H$\\alpha$, [NII] 6584, [SII] 6717, and [SII] 6731 lines, respectively. \n\\label{fig:spectra}}\n\\end{figure*}\nFigure \\ref{fig:ha} shows the SDSS $gri$ composite image (left panel), and the H$\\alpha$~flux map (right panel) of this system (MaNGA 1-24145\n). The SDSS image indicates that MaNGA 1-24145 (nicknamed `Satsuki')\n has a companion galaxy (nicknamed `Mei') located in the lower left to MaNGA 1-24145. \n The stellar absorption lines suggest that these two galaxies are at similar redshifts with the line-of-sight velocities differed by $\\sim$ 200 ${\\rm km~s^{-1}}$ . In addition, both galaxies are round and red according to the SDSS images. Therefore, these two galaxies likely form a dry (gas-poor) merger system. The compact source in the upper-left corner of the image is a foreground star according to the MaNGA spectrum, and hence it is irrelevant to the dry merger system we are probing for this work.\n\nWhat is striking about this system is that in the upper-right corner of the H$\\alpha$~map, there exists a giant H$\\alpha$~blob (RA = 258.84314; DEC = 57.43529) with which however does not have any optical counterpart in the SDSS images. The size of the H$\\alpha$~blob is $\\sim3.2$ kpc in radius. This H$\\alpha$~blob (nicknamed `Totoro') is 7.7 kpc away from Satsuki, and is connected to the H$\\alpha$~emission of Satsuki through tail-like structures. To ensure that the H$\\alpha$~emission at the position of Totoro is not due to artifacts in the data, we check the MaNGA spectra at various spaxels in the region of Totoro. We found that multiple emission lines, including H$\\alpha$, [NII] 6584, [SII] 6717,6731, and [OIII] 5007~ lines are clearly detected in those regions, suggesting that the H$\\alpha$~blob is a real feature. Figure \\ref{fig:pipe3Dimage} displays the reconstructed $ugr$ image from the MaNGA continuum (left panel) and the [OIII] + H$\\alpha$ + [NII] composite image (right panel). It can be seen that the reconstructed continuum map from the MaNGA data is as smooth as SDSS. The three upper panels of Figure \\ref{fig:spectra} show the MaNGA spectra of the central pixels for Satsuki, Mei, and Totoro, respectively. The two elliptical galaxies are mainly composed of old stellar populations, consistent with the morphology classification of being early-type galaxies, whereas the blob is dominated by emission lines. \n\n\\subsection{Optical imaging data \\label{sec:optical}}\nThe imaging data come from two sources: the DR12 release of the SDSS photometric survey \\citep{yor00}, which reaches $r\\sim22$, and deeper observations with CFHT\/MegaCam in $g$, $r$ and $i$. The latter combines archival data downloaded from the CADC server and the data taken in 2015 summer (PI: Lihwai Lin) with the Director Discretionary time (DDT) program. All the MegaCam data were processed and stacked via MegaPipe \\citep{gwy08}. The final images have 5$\\sigma$ limiting mag of 25.7, 26.2, and 25.2 mag (1\" aperture in radius) and surface brightness 5$\\sigma$ limit of 26.4, 26.9, and 25.9 mag arcsec$^{-2}$ in $g$, $r$, and $i$, respectively.\n\n\\subsection{Radio Continuum}\n \nWe observed this system with the Karl G. Jansky Very Large Array\n(VLA) in the A configuration on 2015 August 20 using the C-band\nreceiver tuned to $4-6$ GHz ($\\lambda = 7.5-5.0$\\,cm). The on-source\ntime is 42 minutes; we observed 3C147 for flux and bandpass calibrations,\nand J0920+4441 for phase calibration. Data reduction was carried out\nwith CASA \\citep{McMullin07} using the following steps: (1) standard\ncalibration using the VLA Data Reduction Pipeline (Chandler et al., in\nprep); (2) removal of any portions of the data corrupted by strong\nradio frequency interference; and (3) imaging with the task {\\tt\nCLEAN}. The imaging parameters are the following: MT-MFS deconvolver\nwith nterms of 2, $0\\farcs06$ pixel size, and Briggs weighting with\nrobust parameter of 0.5. The final image has a $0\\farcs39 \\times\n0\\farcs35$ synthesized beam and rms noise at the pointing center of 7\n$\\mu$Jy beam$^{-1}$.\n\n\nThe radio flux of this source is about 37 $\\pm$ 13 $\\mu$Jy at 5 GHz.\nAssuming that this emission is synchrotron-dominated and so following\na power law $S \\propto \\nu^{\u2212\\alpha}$ with a spectral index $\\alpha$ = - 0.7, we derive the\nradio luminosity of Satsuki to be 2.2 $\\times$ 10$^{20}$ WHz$^{\u22121}$ at 1.4\nGHz. This luminosity implies a star-formation rate of 0.12 $\\rm M_{\\odot}$ yr$^{-1}$\nusing the \\citet{bel03} radio SFR indicator, converted to the Salpeter IMF, which is a factor of 2.5 greater than the limit implied by\nH$\\alpha$~emission at the location of the radio source (see Table \\ref{tab:property}). Therefore, it is\nlikely that this radio point source is a faint AGN.\n\n\\subsection{HI}\nThis source was observed as part of the HI-MaNGA programme at the Robert C. Byrd Green Bank Telescope (GBT), which is obtaining HI 21cm observations of a large sample of MaNGA galaxies (AGBT16A\\_095, PI: K. Masters). This target was observed on 2016 Feb. 5 for 3 sets of 5 min ON\/OFF pairs using the VEGAS spectrometer with a bandwidth of 23.44 MHz, centred on the frequency of 21cm emission redshifted to cz=9653 km~s$^{-1}$. At this frequency the FWHM of the GBT beam is 9\\arcmin. No HI emission was detected in this volume to a rms of 1.58 mJy (after smoothing to 5.15 km~s$^{-1}$ velocity resolution). Assuming a velocity spread of 100-400 ${\\rm km~s^{-1}}$  this non-detection sets an 1-$\\sigma$ upper limit of 8.9--9.2 $\\times 10^{8}$ $\\rm M_{\\odot}$~for the HI mass of this system \\footnote{It is intended that HI-MaNGA data will be released as an SDSS Value Added Catalogue in a future data release from SDSS. In addition the raw data will be publicly available via the NRAO Data Archive at https:\/\/archive.nrao.edu a year following observations.}.\n \n\n\\subsection{X-ray}\nSatsuki is located within the field of view $\\sim$ 47 ks \\textit{Chandra} ACIS observation OBSID 4194 (PI: Trevor Ponman), which occurred on 2003 September 17. This pointing of this observation was centered on nearby galaxy NGC 6338 (258.84256, +57.407) [J2000], $\\sim$ 2' South of the MaNGA source of interest. Before analysis, this dataset was reprocessed using the Chandra$\\_$repro task in CIAO v4.8 \\citep{fru06} using CALDB v4.6.3. \n\nThe exposure corrected image of this field (Figure \\ref{fig:JVLA}) was generated using the `fluximage' command, and indicates diffuse X-ray emission coincident with the dry merger system, as also shown in \\citet{pan12}, which performed a study on a nearby BCG based on the same X-ray dataset. The spectrum of the X-ray emission coincident with the H$\\alpha$~blob of interest was then extracted using the CIAO script specextract using a source region of a 22\" $\\times$ 18\" ellipse centered at 17:15:23.7, +57.26:05 which encompasses all of the emission. We fit this X-ray spectrum with a single absorbed APEC model -- the emission spectrum from a collisionally-ionized diffuse gas -- assuming the redshift $z$ = 0.032202 measured from optical spectroscopy using XSPEC v12.8.2e \\citep{arn96}. This fit results in a reasonable reduced $\\chi^{2}$ (1.45 for 88 d.o.f), though somewhat under-predicts the flux $>$ 5 keV. As we will mention in Section 3.5, this dry merger system is part of a small group. The derived X-ray temperature is 1.26$\\pm$0.06 keV, consistent with the temperature on group scale \\citep{ket13}. No point-like source is found within Satsuki or Totoro, indicating that there is no strong X-ray AGN present in this system.\n\n\n\n\n\\section{RESULTS}\n\n\n\\subsection{The optical morphology \\label{sec:obs}}\n\nTo ensure that the absence of the optical counterpart at the position of Totoro as shown in Figure \\ref{fig:ha} is not due to relatively shallow depth of SDSS imaging, we carried out a follow-up observation for this system with CFHT\/MegaCam and combined it with the archival data (see section \\ref{sec:optical}). Figure \\ref{fig:cfht} displays the $gri$ composite image of this galaxy. It is clear that there are extended stellar halos surrounding the two galaxies. Again at the position of the H$\\alpha$~blob, no apparent optical continuum is revealed (see the right panel of Figure \\ref{fig:cfht}). \n\n\\begin{figure}\n\\includegraphics[angle=0,width=8.5cm]{Halpha_Xray_VLA_SDSSr_crop_axistitle.eps}\n\\caption{The Chandra X-ray contours (magenta) and extinction-corrected MaNGA H$\\alpha$~contours (cyan) overlaid on the SDSS $r$-band image (background image) of MaNGA 1-24145. The red cross marks the position of the VLA point-source detection. The X-ray contours correspond to 44\\%, 30\\%, 21\\%, and 15\\% of the peak value, respectively, whereas the H$\\alpha$~contours correspond to 100\\%, 60\\%, 37\\%, 23\\%, 14\\%, 8\\%, 5\\%, 3\\%, and 2\\% of the peak value, respectively.\n \\label{fig:JVLA}}\n\\end{figure}\n\n\n\n\n\\begin{figure*}\n\\includegraphics[angle=0,width=17cm]{7991-12702_3color_region_combined.eps}\n\\caption{\n(A) CFHT $gri$ composite color image for MaNGA 1-24145 with the MaNGA hexagonal FoV overlaid. A bright BCG is $\\sim$ 43 kpc away to the South. (B) A zoom-in picture of (A).\nThe white circle marks the location of Totoro. In both panels, North is up and East is left. \\label{fig:cfht}}\n\\end{figure*}\n\n\n\n\n\n\n\\begin{figure}\n\\includegraphics[angle=0,width=8cm]{show_galfit_g_result_resize.eps}\n\\caption{\n(A) The original CFHT Megacam $g-$band image, (B) the $g-$band image after subtracting nearby satellite galaxies, (C) model image for (B), and (D) the residual images produced by GALFIT. The dashed while circles mark the position of Totoro. The white bars in the lower-right corner of each panel corresponds to a scale of 5\". \\label{fig:song}}\n\\end{figure}\n\n\\begin{figure*}\n\\includegraphics[angle=-90,width=17cm]{manga-7991-12702_kinematics.eps}\n\\caption{\nVelocity fields of MaNGA 1-24145\n based on H$\\alpha$~line (upper panels) and stellar components (lower panels). The left panels show the radial velocities and the right panels show the velocity dispersions. \\label{fig:velocity}}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[angle=-90,width=17cm]{manga-7991-12702_lineratio_elines.eps}\n\\caption{Left: The extinction-corrected H$\\alpha$~map of MaNGA 1-24145\n. Right: The [OIII] 5007\/H$\\beta$~ratio as a function of distance from the center of Satsuki\n. The letters (A,B,C,..etc) mark different locations of this system.\n\\label{fig:o3hb}}\n\\end{figure*}\n\n\n\n\nTo more clearly see the underlying surface brightness structure at the location\nof Totoro, we build detailed photometric models for this merging system\nusing \\texttt{GALFIT\\ v3.0.2} (Peng\\ et al.\\ 2002, 2010).The MecaCam $g$-band image is used here for this purpose because it may better reveal the continuum of Totoro if has had recent star formation (see \\S 4). Since we are not interested in\nthe overall structures of this complex merging system, we will only focus on a\n400$\\times$400 pixels region centered on the MaNGA bundle.  We\nconstruct a mask image to exclude isolated objects around the merging system from the\nfitting.  The PSF of the image was modelled using the \\texttt{SExtractor} and \n\\texttt{PSFex} routines, and the PSF image used in the modelling was extracted using the \ncentral coordinate of the MaNGA bundle.  Meanwhile, on top of the extended evelope, there\nare 11 objects (most are galaxies) that can not be easily masked out.   We model them \nseparately, then subtract them from the image.  All of these smaller objects can be \nwell modelled by a single- or double-{S\\'{e}rsic\\ } model locally with the help of an additional \n{S\\'{e}rsic\\ } and sky components to account for the envelope in the background.  As shown in the \npanel (B) of Figure \\ref{fig:song}, they have been removed smoothly from the input image without any \nsignificant residual pattern.  Using this ``cleaned'' image as input, we model the two \nmerging galaxies along with their ``common envelope'' together using different\ncombinations of {S\\'{e}rsic\\ } components.  We started with three {S\\'{e}rsic\\ } components (one for each \ngalaxy, additional one for the envelope), and gradually build up the complexity by adding \nmore {S\\'{e}rsic\\ } component.  As we simply want to achieve smooth residual map to study the \nunderlying structure, the number of the {S\\'{e}rsic\\ } components and the detailed parameters are\nnot a concern as long as each component behaves normally (e.g. Gu\\ et al.\\ 2013).  \nFor each object (including the envelope), all {S\\'{e}rsic\\ } components are constrained to have the\nsame center, and only symmetric {S\\'{e}rsic\\ } components are used here.  After visualizing the\nresidual of initial model, it becomes clear that, on the lower part of the image, there is\nan additional surface brightness enhancement (caused by the merging process) that are not well fit by the simple model.  Eventually, the best model we achieved include 7 {S\\'{e}rsic\\ } components plus a\ntilted-plane sky background component.  Both of the merging galaxies, and the extended \nenvelope are described by 2 {S\\'{e}rsic\\ } components; while the surface brightness in the south is \nmodelled using single {S\\'{e}rsic\\ } component.  All components behave regularly in term of size and \nshape, except for the central component for the brighter galaxy, no component has {S\\'{e}rsic\\ } \nindex larger than 2.0.  \n\nThe panels (C) and (D) of Figure \\ref{fig:song} show the model image and the residual maps of this\nmodel reconstruction, respectively. It clearly reveals a rich system of shells and tidal tails around the main MaNGA \ngalaxy, indicating an on-going interaction between these two galaxies.  Although the\nresidual map is not perfectly smooth, no optical counterpart is found at the position of\nTotoro.  \n\nReducing the number of components used, or changing the initial guess will not affect the\nabove conclusion.  Adding more components could not further improve the residual map,\nbut results in ill-behaved {S\\'{e}rsic\\ } components.  Given the complex nature of this merging\nsystem, we also try to invoke the asymmetric features in \\texttt{GALFIT}, especially the \n1st (global lopsidedness) and 4th (boxiness of the isophote) Fourier components (see \nPeng\\ et al.\\ 2010 for details).  The residual map shows improvements around the tidal\nfeatures, but does not change the conclusion that there is no apparent optical counterpart\nfor Totoro \\footnote{The results still hold if we repeat the analysis using the $i$-band image, which is more sensitive to the stellar mass distributions.}. \n\n\n\n\\subsection{Kinematics from the MaNGA Observations\\label{sec:velocity}}\nFigure \\ref{fig:velocity} displays the velocity and dispersion maps for this system. \nWhile the stellar velocity field indicates that the main galaxy Satsuki\n is primarily pressure supported, the gas component reveals more complex structures. The central galaxy shows a weak rotation structure, while there is strong variation in the line-of-sight velocity across Totoro region, redshifted in the left tail and blueshifted in the right tail.  The inconsistent velocity fields between stars and gas suggests that some part of the gas of Satsuki might have been either accreted or ejected recently. In the former case, it is similar to those early-type galaxies \nthat exhibit misaligned gas and stellar kinematics, the so-called early-type `counter rotators' \\citep{sar06,dav11,che16,jin16}, although counter rotators\nin general are defined for systems with a rotating stellar component.  The gas inflow scenario is also consistent with the H$\\alpha$~morphology which shows bridge (tail)-like structures that connect Totoro and Satsuki. On the other hand, the complicated velocity field can also be explained if the main galaxy Satsuki underwent a strong outburst phase, during which Totoro was expelled from Satuski.\n\n\n\n\n\n\n\n\n\n\\subsection{Excitation State \\label{sec:bpt}}\n\n\n\n\n\nIn addition to H$\\alpha$~and two [NII] 6584~lines, some other weak lines such as [SII] 6717,6731~ and [OIII] 5007~ are also detected in both Satsuki and Totoro, allowing us to \nprobe the ionization state for this system. Figure \\ref{fig:o3hb} shows the [OIII] 5007\/H$\\beta$~ ratio, one of the frequently used ionization parameters, as a function of the distance from the main galaxy. There exists a strong [OIII] 5007\/H$\\beta$~gradient, decreasing from the main galaxy Satsuki (location A) to the left bridge (locations B, C, and D) that connects to Totoro (location E, F, G, H, and I). On the other hand, the [OIII] 5007\/H$\\beta$~ ratio is nearly constant across Totoro. \n\n\nThe multiple line detections also allow us to classify the emission line regions into HII or AGN regimes using the standard Baldwin-Phillips-Terlevich (BPT Baldwin, Phillips \\& Terlevich 1981; Veilleux \\& Osterbrock 1987; Kauffmann et al. 2003; Kewley et al. 2006) excitation diagnostic diagrams. Here we apply three types of line diagnostics based on four line ratios, [OIII] 5007\/H$\\beta$, [NII] 6584\/H$\\alpha$~, [SII] 6717,6731\/H$\\alpha$, and [OI] 6300\/H$\\alpha$. Figures \\ref{fig:bpt} and \\ref{fig:bptmap} display the line ratio diagrams  and classification maps for this system, respectively. We adopt the dividing curves suggested in the literature \\citep[e.g.,][]{kew01,kau03,cid10} to separate various regions. All the three classifications indicate `LINER'-type excitations for Satsuki. As the LINER emission is extended across the entire main galaxy, this object falls in to the extended LIER (eLIER) category according to the classification scheme by \\citet{bel16a,bel16b}\n\nOn the other hand, in the regions of Totoro, the line ratios are consistent with the `composite', `HI'. and 'LINER' regions when using the [NII], [SII], and [OI] diagnostics, respectively. In the remaining part of this paper, we treat Totoro as `composite' regions based on the [NII], as it allows for the intermediate excitation state as opposed to the other two methods.\n\n\n\n\n\\begin{figure*}\n\\includegraphics[angle=-90,width=17cm]{manga-7991-12702_BPT_elines.eps}\n\\caption{Upper panels: BPT diagnostic diagrams for MaNGA 1-24145\n based on the [OIII] 5007\/H$\\beta$~ vs. [NII] 6584\/H$\\alpha$~ (left panels), [OIII] 5007\/H$\\beta$~ vs. [SII] 6717,6731\/H$\\alpha$~(middle panels), and [OIII] 5007\/H$\\beta$~ vs. [OI] 6300\/H$\\alpha$~(right panels). Various colors on the data points indicate the physical separation from the center of Satsuki (from near to far: red, origin, green, blue, purple). The solid, dashed, and dotted lines show the classification lines suggested by \\citet{cid10}, \\citet{kew01}, and \\citet{kau03}, respectively. The blue curves display the model predictions of the shock and photoionization mixing sequences by \\citet{ho14}. Each line corresponds to a certain shock fraction (from bottom to top: 20\\% to 100\\%). The shock velocity ranges from 100 to 300 ${\\rm km~s^{-1}}$  (from left to right). The MAPPINGS III shock+precursor models (gray grid) with $n$ = 1 cm$^{-3}$ from \\citet{all08} are also shown for comparison. The thick lines represent constant magnetic parameter while the thin lines display the constant shock velocity ranging from 200 to 1000 ${\\rm km~s^{-1}}$  (bottom to top; with 25 ${\\rm km~s^{-1}}$  intervals).\nBottom panels: A zoomed-in view of the upper panels. The letters (A,B,C,..etc) mark different locations of this system, following the definitions of Figure \\ref{fig:o3hb}.\n\\label{fig:bpt}}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\includegraphics[angle=-90,width=17cm]{manga-7991-12702_BPTmap_elines.eps}\n\\caption{BPT classification maps for MaNGA 1-24145\n based on the [OIII] 5007\/H$\\beta$~ vs. [NII] 6584\/H$\\alpha$~ (left panel) and [OIII] 5007\/H$\\beta$~ and [SII] 6717,6731\/H$\\alpha$~(right panel). The [OI] diagnostic map is not included here as all spaxels are classified as LINER.\n\\label{fig:bptmap}}\n\\end{figure*}\n\n\\begin{figure}\n\\includegraphics[angle=-90,width=11cm]{manga-7991-12702_gasmetal_N2S2Ha.eps}\n\\caption{The gas metallcity map of MaNGA 1-24145\n, computed using the method described in \\citet{dop16} based on the emission line ratios among [NII], [SII], and H$\\alpha$.\n\\label{fig:metal}}\n\\end{figure}\n\n\\subsection{Gas metallicity \\label{sec:metal}}\nTo further understand the properties of Totoro, we also measure the gas-phase metallicity (Z) for this system. Conventionally, there are many ways to estimate the gas metallicity through emission line ratios \\citep[see][]{kew08}. However, most of those tracers are calibrated against the HII regions, and may not be applicable to systems with ionization parameters or interstellar medium (ISM) pressure different from typical HII regions. Here, we adopt the `N2S2H$\\alpha$'~ calibrator that is suggested to be less sensitive to the ionization parameters \\citep{dop16} to estimate the metallicity. The N2S2H$\\alpha$~calibration can be expressed as the following:\n\n\\begin{equation}\\label{eq:N2S2ha}\n12 + \\mathrm{log (O\/H)} = 8.77 + \\mathrm{N2S2H}\\alpha\n\\end{equation}\nwhere N2S2H$\\alpha$ = Log([NII]\/[SII]) + 0.264Log([NII]\/H$\\alpha$). The [NII] to [SII] ratio is found to nearly independent of the AGN luminosity and hence provides a good metallicity indicator for even AGNs \\citep{ste13}. However, we note with caution that the derived metallicity may still be subject to large systematic uncertainty since the line ratios of the main galaxy and Totoro are located in the LINER and composite regions, respectively. \n\nFigure \\ref{fig:metal} shows the Z(N2S2H$\\alpha$) map for this system. The metallicity of the gas around Satsuki is close to solar and is $\\sim$ 0.3 dex lower than the expected value (Z = 9.1) for a massive galaxy with a stellar mass of $10^{11}$ $\\rm M_{\\odot}$~  based on the local mass--metallicity relation \\citep{tre04}. The offset we see here, however, may be easily accounted for by the different metallicity calibrators adopted. On the other hand, the averaged metallicity in the H$\\alpha$~ region is greater for Totoro than Satsuki by a factor of 0.3 dex and is more consistent with the gas metalltcity of high mass galaxies. Although the difference is significantly larger than the statistical uncertainty (0.03-0.1 dex) in the metallicity measurement, it is difficult to interpret this result given the systematic uncertainty in the metallicity measurement due to their different ionization states.\n\n\n\\subsection{Environment \\label{sec:environment}}\nThe environment of this system\n is rather complex. As already mentioned, it has an early-type companion (Mei) just 4 kpc away to the South, which makes them a possible dry merger candidate. Moreover, Satsuki\n is also part of the system MCG+10+24-117, a small group falling onto a galaxy cluster with NGC 6338 as a central Brightest Cluster Galaxy that is 43 kpc away to the south (see Figure \\ref{fig:cfht}; Pandge et al. 2012; Dupke et al. 2013). \n\n\n\n\n\n\n\n\n\n\\section{DISCUSSION}\\label{sec:discussion}\n\n\n\n\n\n\n\n\n\\subsection{The origin of the H$\\alpha$~ blob}\nUsing the first-year MaNGA data, we discover a giant H$\\alpha$~blob, Totoro, associated with a dry merger system. This object, however, does not have any optical counterparts down to 26.9 mag arcsec$^{-2}$ in deep CFHT\/MegaCam $g,r$, and $i$ images. \nThere are several possible scenarios to explain the origins of Totoro: \n\n\n\n\n\n\nScenario 1: Totoro is associated with gas tidally stripped from Satsuki during\nthe interaction between Satsuki and Mei. \n\nScenario 2: Totoro is associated with the gas ram-pressure stripped during the infall of\nSatsuki\n toward the center of NGC 6338 galaxy cluster, similar to NGC4569 located in the Virgo cluster (Boselli et al. 2016).\n \nScenario 3: Totoro is associated with gas ejected from Satsuki by an AGN outflow during\nthe mergers between Satsuki and Mei. If the central black hole of\nSatsuki is turned on during mergers, the energy can ionize the stripped gas, resulting in\nthe H$\\alpha$~emissions, similar to the known `Hanny's voorwerp' phenomenon (Lintott et al. 2009).\n\nScenario 4: Totoro is an UDG (or alternatively, a LSB galaxy), which falls\nbelow the detection limit (26.9 mag arcsec$^{-2}$ in $r$-band) of CFHT imaging data, making it a `dark' galaxy. The morphology of Totoro (Figure 1), especially the (tail) bridge-like structures that connect between Satsuki and Totoro, indicates that Satsuki\n is likely under interaction with the hosting galaxy of\nTotoro, which has a comparable physical size as Satsuki. \n\n\n\n\nTo estimate the mass of the warm gas component, we follow the approach adopted by \\citet{che16b}. We first estimate the electron density $n_{e}$ to be $\\sim$ 260 cm$^{-3}$ based on the median value of the [SII]6717\/[SII]6731 ratio in the region of Totoro following Equation 3 of \\citet{pro14}, which assumes the electron temperature $T_{e} = 10,000K$. Next, we calculate the extinction-corrected H$\\beta$~luminosity, $L_{\\mathrm{H}\\beta}$, to be 3.1$\\times 10^{39}$ erg s$^{-1}$. The ionized gas mass can then  be derived using the following equation \\citep{ost06,fu12}:\n\\begin{equation}\\label{eq:mass}\n\\frac{M_{\\rm HII}} {6.8\\times 10^{7}~ \\rm M_{\\odot}} = \\bigg(\\frac{L_{\\mathrm{H}\\beta}}{10^{40}~\\mathrm{erg s^{-1}}}\\bigg)\\bigg(\\frac{n_{e}}{1~  \\mathrm{cm}^{-3}}\\bigg)^{-1}.\n\\end{equation}\nWe obtain the ionized gas mass $\\sim$ 8.2$\\times$ 10$^{4}$ $\\rm M_{\\odot}$.\n\nThe HI observation of this system provides an upper limit of (8.9--9.2) $\\times 10^{8}$ $\\rm M_{\\odot}$~for the HI mass due to the null detection. We account for He gas by applying a factor of 1.33 to this upper limit. We can also infer the H$_{2}$ content of this Totoro using the H$\\alpha$~emission. The integrated H$\\alpha$~flux over spaxels within Totoro region is 3.3$\\times 10^{-15}$ erg s$^{-1}$ cm$^{-2}$, corresponding to a luminosity of 7.8$\\times 10^{39}$ erg s$^{-1}$. Assuming all the H$\\alpha$~flux results from star formation, we obtain the total star formation rate (SFR) of Totoro to be 0.059$\\rm M_{\\odot}$ yr$^{-1}$ using the conversion between the H$\\alpha$~luminosity and star formation rate from \\citet{ken98}.\nWe then estimate the total H$_2$ gas mass ( = SFR x t$_{dep}$) of the H$\\alpha$ blob to be $\\sim1.2\\times 10^8\\,M_\\odot$ assuming a typical gas depletion time (t$_{dep}$) of 2 Gyr. This value is also an upper limit since we have  assumed that all the H$\\alpha$~fluxes are contributed by the star formation. This implies an upper limit of the entire cold gas (HI + He + H$_{2}$) of $\\sim$ 1.3 $\\times 10^{9}$ $\\rm M_{\\odot}$.\n\nAlthough Satsuki and its companion Mei are consistent with being early-type galaxies (ETGs), the amount of the cold gas in Totoro is not totally unexpected if it is part of Satsuki. For example, recent works have found that a significant fraction of massive ETGs that show signs of star formation possess cold gas with gas mass comparable to this Totoro \\citep{osu15,dav16}. Therefore, it is reasonable to speculate that Totoro may  originally be part (if not all) of the cold gas contained Satsuki, and then get ejected during the galaxy-galaxy interaction or because of the ram-pressure stripping (scenarios 1 to 3, respectively).  \n\n\n\n\nTo test the galaxy interaction scenario (scenario 1), we have performed and examined a set of N-body\nsimulations of the interaction of two Elliptical galaxies (E-E) with the\ncode Gadget2 (Springel 2005). Different initial orbital configurations\n(e.g pericentric distance) have been considered in a similar way to Peirani\net al. (2010). Although we did not include any gas component in our dry merger simulations, the features in the stellar components can still be useful to understand the system as the stellar streams well trace the gas streams before 1 Gyr during galaxy interaction. \n\nAmong all the cases we have looked at, we did not see the formation of any clear stellar stream or a centrally concentrated blob-like structures during the interaction. This strongly suggested that Totoro is unlikely to be produced during an E-E interaction. In addition, based on other existing gas-rich merger simulations (see\nfor instance Barnes \\& Hernquist 1996; Di Matteo et al. 2007; Peirani et\nal. 2010), the prominent gas (or stellar) streams are expected to be\nformed when at least one of merger progenitors have extended disk structures, unlikely to be the case for this dry merger system even if these two ellipticals have small amount of gas. Furthermore, such latter simulations also suggest that it is very\ndifficult to produce such blob-like structure. Although tidal dwarf\ngalaxies can be formed during the interaction of disk galaxies (see for\ninstance Bournaud, Duc et Masset 2003; Duc, Bournaud et Masset 2004) in these cases, the\nblob-like structure is more extended with no clear star formation\nactivity. Nevertheless, we note that the simulations we have examined have the following caveats: 1) the possible parameter space is extensive, so it is impossible to fully explore, and 2) the stellar\/AGN feedback which could drive galactic scale outflows and these physics are not included. Such advanced simulations are beyond the scope of this work. We defer more detailed comparisons with simulations to a future work.\n\n\nOn the other hand, the centrally concentrated blob-like structure is not expected in the ram-pressure stripped gas (scenario 2), either, which often shows the tail and\/or `jellyfish' like structures (e.g., Boselli et al. 2016). Moreover, galaxies that show ram-pressure stripping phenomena are mostly gas-rich late-type galaxies, unlike Satsuki.\n\n\nAnother explanation for Totoro is the materials ejected by an AGN located in Satsuki (scenario 3), similar to the extended emission-line regions found around quasars \\citep{fu07a, fu07b}.  The outflow scenario is also consistent with the complicated velocity field seen in the gas component of this system. On the other hand,\nas discussed in \\S 2, we do not detect an X-ray point source or an extended radio jet in the main galaxy, suggesting that there is no on-going strong AGN activity. However, it has been known that the AGN brightness can vary over timescales of several to 10$^{5}$ years \\citep{den14,mce16}, either due to a change in the black-hole accretion rate or Tidal-Disruption Events \\citep[TDEs, e.g.][]{sax12,mer15}. Therefore,  we can not rule out the possibility of a recent past AGN outflow, similar to the notable phenomena of Henny's Voorwerp, in which the ionizing source has already diminished whereas a clump of ionized gas is found several kpc away. Nevertheless, it is unclear why the ejected gas would have a higher metallicity than than the gas remaining in Satsuki (although it is unclear whether the metallicity difference is real given the potential systematics in our metallicity estimate).\n\n \nAn alternative explanation of the H$\\alpha$~blob is a separate faint galaxy, interacting with the dry merger system (scenario 4). \nA few morphological features, for example, the centrally concentrated H$\\alpha$~and the bridges extended from Totoro toward Satsuki, are most consistent with Totoro being a separated gas component. According to numerical simulations, these features can indeed be explained by the interaction between the main galaxy and a faint disk galaxy (scenario 1) (see for instance, Peirani et al. 2010; Cheung\net al. 2016).\n\nRecently, a class of UDGs has been identified in the Coma cluster and several low-redshift clusters \\citep{van15a,van16}. These UDGs are surprising large in size ($r_{eff}$ = 1.5 -- 4.6 kpc) despite of their low stellar masses ($< 10^{8}$ $\\rm M_{\\odot}$). The majority of these UDGs are found to lie on the red-sequence, indicating old stellar populations and quenched star formation activities \\citep{van16}. Possible formation mechanisms include that gas is stripped by the ram pressure when falling into the cluster, which prevents subsequent star formation, or that gas is expelled due to strong stellar or supernova feedback for galaxies with halo mass between $10^{10}$--$10^{11}$ $\\rm M_{\\odot}$~\\citep{dicin16}. Nevertheless, it remains an open question regarding the origin of UDGs and whether these UDGs are already quenched before falling into the cluster environments. Totoro identified in this work has a comparable size ($\\sim$ 3.2 kpc) as the UDGs. However, the averaged surface brightness of Totoro has an upper limit of 26.9 mag arcsec$^{-2}$, at least 1-3 mag dimmer compared to that of the known UDGS ranging from 24 to 26 mag arcsec$^{-2}$. Another aspect of Totoro that is distinct from the known UDGs is that the latter are generally old in the stellar populations as indicated from their red colors, whereas Totoro shown in this work is likely to have small amount of on-going star formation based on the BPT diagnostics but with little old stellar populations.  Therefore, Totoro may represent a different category of LSB galaxy from the quiescent UDGs.\n\n\n\nThe non-detection of optical light associated with Totoro can provide an upper limit of the amount of star formation. Assuming that this H$\\alpha$~cloud is a young star-forming system with age $< 0.1$ Gyr, the flux in the optical regime is expected to be comparable to that in the UV (1500--2800 \\AA) in the case of no dust extinction. Following the conversion between the UV luminosity, H$\\alpha$~luminosity, and the star formation rate given by \\citep{ken98} and adopt the surface brightness limit (26.9 mag arcsec$^{-2}$) from our CFHT MegaCam $r-$band observation, we estimate that the corresponding H$\\alpha$~surface density of this Totoro is on the order of 1.1 $\\times10^{-17}$erg s$^{-1}$ cm$^{-1}$ arcsec$^{-2}$, which is one magnitude lower than the peak value of Totoro. By integrating it over the H$\\alpha$~region, we estimate that the star formation contributes less than 37\\% to the total H$\\alpha$~flux, otherwise we should be able to detect the optical counterpart.\n\nGiven the very low amount of star formation that can possibly occur in Totoro, we speculate that Totoro is  a gas cloud that fails to form stars efficiently, and thus a `dark' gas cloud. The origin of this gas cloud, whether it is associated with a satellite dark matter, or a pure gas cloud, is difficult to pin down. However, the latter scenario is unlikely since it would be difficult to maintain the kpc-scale gas cloud against gravity without invoking the existence of a dark matter halo.   If Totoro is indeed hosted by a subhalo, it would be a strong evidence of the existence of `dark' subhalos, which help to alleviate the missing satellite problem. Since there is no stellar component in the region of Totoro, we could roughly estimate the halo mass of Totoro based on the gas velocity dispersion using the virial theorem. Taking R = 3.2 kpc and $\\sigma_{gas} = 50$ ${\\rm km~s^{-1}}$ , we obtain the halo mass M$_{halo}\\sim 5.6\\times10^{9}$ $\\rm M_{\\odot}$.\n\nOn the other hand, the high metallicity of Totoro (see Section 3.4) is unexpected for a typical dwarf galaxy. One possible explanation is that the gas cloud has been enriched by the surrounding environment, and may has a different evolution process from the typical known dwarf galaxies. However, we caution that the metallicity measurement presented in this work is subject to large uncertainty since the ionization source is not well constrained. \n\n\n\n\\subsection{Sources of ionization}\n\n\\begin{deluxetable*}{llllllllll}\n\\tabletypesize{\\scriptsize}\n\\tablewidth{0pt}\n\\tablecaption{Properties of MaNGA 1-24145 (Satsuki), its southern companion (Mei), and the H$\\alpha$~blob (Totoro).\\label{tab:property}}\n\\tablehead{\n\t\\colhead{Object} &\n    \\colhead{$z$} &\n    \\colhead{RA} &\n    \\colhead{DEC} &\n    \\colhead{M$_{*}$ ($\\rm M_{\\odot}$)} &\n    \\colhead{M$_{\\rm HII}$ ($\\rm M_{\\odot}$)} &\n    \\colhead{M$_{H_{2}}$ ($\\rm M_{\\odot}$)} &\n    \\colhead{M$_{HI}$ ($\\rm M_{\\odot}$)} &\n    \\colhead{M$_{halo}$ ($\\rm M_{\\odot}$)} &\n    \\colhead{SFR (yr$^{-1}$ $\\rm M_{\\odot}$)} \n    }\n\n\\startdata\nSatsuki & 0.0322  & 258.84695 & 57.43288 & $1.2 \\times 10^{11}$ & \\nodata & \\nodata & $< 9.2 \\times 10^{8}$ & \\nodata & $<$ 0.049$^{b}$\\\\\nMei & 0.0322  & 258.84750 & 57.43133 & $4.0 \\times 10^{10~a}$ & \\nodata  & \\nodata  & $< 9.2 \\times 10^{8}$ & \\nodata & \\nodata\\\\\nTotoro & 0.0322  & 258.84314 & 57.43529 & \\nodata & $8.2 \\times 10^{4}$  & $< 1.2 \\times 10^{8}$ & $< 9.2 \\times 10^{8}$ & $5.6 \\times 10^{9}$ & $<$ 0.059$^{b}$\n\\enddata\n\n\\tablecomments{$^{(a)}$ This is scaled from the stellar mass of Satsuki by using the difference in their SDSS $r$-band magnitudes.; $^{(b)}$ This upper limit is derived assuming all the H$\\alpha$~fluxes come from the star formation.}\n\n\\end{deluxetable*}\n\nShocks can often lead to line ratios similar to those occupying the composite regions \\citep{ho14}. In the case where there are shocks, the gas velocity dispersion ($\\sigma_{gas}$) is expected to be as high as several hundreds of ${\\rm km~s^{-1}}$ . As revealed in Figure \\ref{fig:velocity}, $\\sigma_{gas}$ in the region of Totoro is $\\sim 50$ ${\\rm km~s^{-1}}$ . Although it is close to the lower end of velocity dispersion distribution that have been found in typical shocked regions, we can not ruled out the shock excitation of the emission lines. To gain insights to whether shock is responsible for producing the ionized ratios seen in this system, we show the shock and photoionization mixing models from \\citet{ho14} as blue curves in the upper panels of Figure \\ref{fig:bpt}. These models are produced based on the \\texttt{MAPPINGS IV code} \\citep{dop13} and span a wide range of shock fractions (from 20\\% to 100\\%) and shock velocities (from 100 to 300 ${\\rm km~s^{-1}}$ ). As it can be seen, the models predict a greater [OIII] 5007\/H$\\beta$~ratio than what is observed in the data and do not cover the majority of the regions occupied by the data points even at shock velocity up to $\\sim300$ ${\\rm km~s^{-1}}$ . In addition to the pure shock models, we also compare the data to the shock + precursor models of Allen et al. (2008) with $n$ = 1 cm$^{-3}$ and solar metallicity, shown as the gray grid in figure \\ref{fig:bpt}. Although the grid starts with the shock velocity of 200 ${\\rm km~s^{-1}}$ , it is expected by extrapolation that models with lower velocity values still can not reproduce the [NII]\/H$\\alpha$~ratios of Totoro. These comparisons suggest that shocks are unlikely to be the dominant mechanism that is responsible for ionizing the gas blob. \n\n\nThere have been studies showing that AGN is able to ionize gas clouds extending to several kpc \\citep{lin09,fu08}, and the effect from AGN  may persist even $\\sim10^{5}$yr after the central engines shut off \\citep[e.g., the `Hanny's Voorewerp';][]{lin09}). If Totoro is indeed a `dark' galaxy or a cloud interacting with Satsuki, the tidal field would cause a gas inflow that fuels the central black hole of Satsuki and possibly trigger the AGN activity. \nAlthough no X-ray point source is detected in the position of either Satsuki or Totoro, the detection of point-like radio source (see Sec. 2.3) in the center of Satsuki indicates the presence of a low-activity AGN. Therefore, an alternative explanation for the line ratios seen in Totoro is due to the star formation -- AGN mixing effect \\citep{dav14a,dav14b}. \n\nObservationally a starburst--AGN mixing sequence is often found in starbursting galaxy that hosts a central AGN, in which case the line ratio moves from Seyfert to composite to HII regions as the distance from the central AGN increases. The position of line ratios depends on the fractional contribution between AGN and star formation. According to the mixing model by \\citet{dav14b}, when the star-forming cloud is  photonionized by an AGN, the line ratios can fall into the `composite' region on BPT diagrams. In the case of 100\\% contribution from AGN, the emission line ratio falls into the `AGN' region, instead of `composite' region where Totoro lies. This implies that the emission line ratios seen in Totoro can not be fully attributed to pure AGN photonionization, and some level of star formation may be required. Unlike the typical star formation -- AGN mixing \\citep{dav14a,dav14b}, our case is analogous to the star formation -- LINER sequence, in which the LINER excitation is due to the low-activity AGN located in Satsuki.\n\n\n\n\n\n\\subsection{Comparison to similar objects in the literature}\nThere are several similar systems reported in the literature that show offset ionized gas components, and hence it is intriguing to compare this Totoro with previous cases. For example, the famous Hanny's Voorewerp \\citep{lin09}, also exhibits a separate warm gas component that is offset from the main galaxy by several kpc.  However, there are several different features between Hanny's Voorewerp and Totoro presented in this work. First, the ionized gas of Hanny's Voorewerp has much higher excitation state and its line ratios are consistent with Seyfert, as opposed to the `composite' region for Totoro. Moreover, the H$\\alpha$~morphology of the Hanny's Voorewerp is lumpy and irregular, unlike a disk-like structure seen in our case. Another well-known example is an H$\\alpha$~emission line component (often referred as the `cap') at a projected distance of 11 kpc northwest of M82 \\citep{dev99,leh99,ste03}. The cap has a shell-like structure and may possibly be bow shock formed by the starburst-driven superwind. In both cases, the nearby galaxy is a late-type star forming galaxy, different from Satsuki\n, which is an early-type galaxy. Totoro may thus represent a different category of offset ionized gas in nearby galaxies. \n\n\\section{CONCLUSION}\\label{sec:conclusion}\n\n\n\nHere we report a discovery of a puzzling giant H$\\alpha$~blob, Totoro, identified from the first-year MaNGA data. The data disfavor the scenario that Totoro is tidally-stripped  gas from MaNGA 1-24145 (Satsuki) that is interacting with the southern companion (Mei), or the ram-pressure stripped gas when Satsuki falls into the center of the cluster it is located. Despite there being no X-ray point source or radio jet detected in this system, we can not rule out the possibility that Totoro is ejected from a past AGN activity in Satsuki, which likely hosts a faint AGN given its radio luminosity. On the other hand, the H$\\alpha$~morphology and the lack of stellar tidal streams suggest that Totoro could also be a separate `dark' galaxy (or an extremely LSB galaxy) interacting with Satsuki. The non-detection of the stellar continuum suggests Totoro is different from known dwarf populations or UDGs: it is either completely `dark' or with a star formation rate that contributes $<37\\%$ of the H$\\alpha$~flux. \n\nAs for the source that powers the line excitation for Totoro, the `composite' line excitation can be explained either by a star-forming cloud being excited by a low-velocity shock or by the star formation -- LINER mixing effect. The shock scenario, however, is less favoured because of the low velocity dispersion observed in Totoro region. The decrease in the [OIII] 5007\/H$\\beta$~ratio away from Satsuki indicates that the ionizing source is possibly located inside Satsuki. Thus, the star formation -- LINER mixing effect seems to be the most probable ionizing mechanism. In this scenario, the hypothesis is that the ionizing source is the low-activity AGN residing in Satsuki, being triggered by the gas inflow induced during the interaction between Satsuki and the `dark' galaxy (or gas cloud). The AGN subsequently photoionizes the `dark' gas cloud, which then emits the H$\\alpha$~photons. \n\nHowever, we have not considered the case where Totoro is tidally-stripped from Satsuki while falling into the cluster center at the same time, which results in the non-typical tidally or ram-pressure stripped gas morphology of Totoro. More sophisticated modelling considering both the effects of tidal disrupting and the orbital motions of Satsuki relative to the cluster environments, as well as future resolved atomic and molecular gas observations, such as HI and CO, are required to further understand the origin of Totoro. \n\n\n\n\n\\acknowledgments\n\nWe thank the anonymous referee for constructive suggestions which significantly improve the clarity of this paper. L. Lin thank I-Ting Ho, Michal Michalowski, Yen-Ting Lin, You-Hua Chu, Lisa Kewley, Tomo Goto, Jorge Barrera-Ballesteros, and Christy Tremonti for useful discussions. The work is supported by the Ministry of Science \\& Technology of Taiwan\nunder the grant MOST 103-2112-M-001-031-MY3. H.F. acknowledges support from the NSF grant AST-1614326 and funds from the University of Iowa. S. Peirani acknowledges support from the Japan Society for the Promotion of\nScience (JSPS long-term invitation fellowship). J.G.F-T is currently supported by Centre National d'Etudes Spatiales (CNES) through PhD grant 0101973 and the R\\'egion de Franche-Comt\\'e and by the French Programme National de Cosmologie et Galaxies (PNCG). \n\nFunding for the Sloan Digital Sky Survey IV has been\nprovided by the Alfred P. Sloan Foundation, the U.S.\nDepartment of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges support\nand resources from the Center for High-Performance\nComputing at the University of Utah. The SDSS web\nsite is www.sdss.org. SDSS-IV is managed by the Astrophysical Research Consortium for the Participating\nInstitutions of the SDSS Collaboration including the\nBrazilian Participation Group, the Carnegie Institution\nfor Science, Carnegie Mellon University, the Chilean\nParticipation Group, the French Participation Group,\nHarvard-Smithsonian Center for Astrophysics, Instituto\nde Astrof\\'isica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) \/ University of Tokyo, Lawrence\nBerkeley National Laboratory, Leibniz Institut f\\\"ur Astrophysik Potsdam (AIP), Max-Planck-Institut f\\\"ur Astronomie (MPIA Heidelberg), Max-Planck-Institut f\\\"ur\nAstrophysik (MPA Garching), Max-Planck-Institut f\\\"ur\nExtraterrestrische Physik (MPE), National Astronomical Observatory of China, New Mexico State University,\nNew York University, University of Notre Dame, Observat\\'ario Nacional \/ MCTI, The Ohio State University,\nPennsylvania State University, Shanghai Astronomical\nObservatory, United Kingdom Participation Group, Universidad Nacional Aut\\'onoma de M\\'exico, University of\nArizona, University of Colorado Boulder, University of\nOxford, University of Portsmouth, University of Utah,\nUniversity of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University. \n\nThe National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.  This work used data from project AGBT16A\\_095: `HI-MaNGA: HI Followup of MaNGA galaxies, PI Karen L. Masters. \n\nThis work is partly based on observations obtained with MegaPrime\/MegaCam, a joint project of CFHT and CEA\/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Science de l\\'Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii.\n\nThis research has made use of data obtained from the Chandra Data Archive and the Chandra Source Catalog, and software provided by the Chandra X-ray Center (CXC) in the application packages CIAO, ChIPS, and Sherpa.\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsection{Review of Crater Detection}\nPlanetary rovers to date have all used stereo cameras for terrain imaging and 3D perception \\cite{goldberg2002stereo}. Future rovers (to Moon and Mars) might have LIDAR, either by itself or in addition to one or more cameras. Thus, any combination of these sensors could be used for crater detection. To cover this set of options and to create a foundation for identifying the best approach in the future, crater detection algorithms were developed for three classes of technical approach: (1) Using 3D point clouds from LIDAR, (2) Using 3D point clouds from stereo vision, and (3) Using deep-learning based pattern recognition with monocular images. \n\nThe original plan was to treat crater detection as a process done independently from any knowledge of the crater landmark map and rover position; however, work last year showed that more reliable crater detection results was achieved by assuming approximate prior knowledge of rover position, which is realistic in practice, and using that to allow the crater detection process to invoke 3D models of craters expected to be around the rover based on this prior knowledge. Such approximate prior knowledge was used for the LIDAR- based approach developed, but has not yet been carried over to the other approaches. Geometric analysis methods was appointed to the point clouds from LIDAR and stereo vision; machine learning with neural nets was used for monocular images.\n\nResults of quantitative performance evaluation with geometric analysis applied to simulated 3D point clouds from LIDAR show high reliability for detecting craters with a leading edge within about 15 m from the rover. The results also suggested that rover localization with an error less than 5 m is highly probable. Somewhat simpler geometric analysis methods were applied to simulated 3D point clouds from stereo vision, which are noisier than LIDAR-based point clouds. Stereo-based detection degraded at shorter range than for LIDAR and obtained significantly higher crater position estimation error; nevertheless, rover localization with error in the 5 m range still appears to be possible. Monocular appearance-based detection was done with a CNN-based machine learning algorithm; this produced detection results in image space, but did not produce 3D crater position and size estimates. Detection performance exceeded the other two methods, making this a very promising approach for future crater-based localization systems. See \\cite{matthies2022lunar} for details on the algorithms and performance characterization of crater detection.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/particle-filter-overview.pdf}\n    \\vspace{2mm}\n    \\caption{\\textbf{Overview of the Particle Filter framework. Steps associated with a particle filter approach. (1) The particles associated with the a posteriori function t-1 are (2) sampled; and (3) propagated leading to the a priori function P($X_t|Z_{(t-1)}$). Then, upon (4) observation in t, we have the (5) a posteriori function P($X_t|Z_t$)in t.}}\n    \\label{fig:partcile_filter}\n\\end{figure}\n\n\\begin{figure*}[!t]\n    \\centering\n    \\includegraphics[width=0.7\\linewidth]{figures\/figure2_annotated.png}\n    \\caption{\\textbf{Formulating rover and orbital observations as gaussian mixture models. Using optimization to find the best match between two closed-form terrain models for localization.}}\n    \\vspace{5mm}\n    \\label{fig:gmm}\n\\end{figure*}\n\n\\begin{figure*}[!t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/figure3.png}\n    \\caption{\\textbf{Parametric loss function landscape of the gaussian mixture models.}}\n    \\vspace{2mm}\n    \\label{fig:loss_landscape}\n\\end{figure*}\n\n\\subsection{Crater-based Localization using Particle Filters}\nFor non-parametric methods, we implemented a crater-based localization algorithm based on Particle Filters \\cite{thrun2002particle,fox2001particle}, where each particle represents a hypothesis for the rover location, and the strength of the hypothesis is represented by a measurement probability. Figure \\ref{fig:partcile_filter} illustrates a generalized overview of the particle filter's cycle in which we have, in the first layer, some particles represented by ellipsis of various sizes. The size denotes the weight of a particle. The second layer illustrates the result of the sampling process which can lead to repeated particles. Upon sampling, the weights of the particles lose their meaning and new measurements are necessary as the third layer shows. In this layer, the particles' weights are adjusted according to the used motion model and a noise model applied to the sorted particles. Upon adjustments, we have an estimation for the next frame in the sequence. The fourth layer shows the function representing the updated particles. In this function, the height denotes the weight of the measurement at a given point. The fifth layer shows the a posteriori probability function as the result of the measurement step. In this step, the particles are properly weighted and prepared for the next iteration of the localization cycle.\n\nIn our implementation, a large number of particles can be used to approximate the distribution of the rover location as the rover moves and detects the craters. To estimate the location, the particles are moved according to the motion model, then the measurement probabilities are updated by comparing the ground craters with the orbital craters. Lastly, the particles are re-sampled according to their measurement probabilities to represent the new location distribution after motion. \n\nThere are many possible formulations on how to update the measurement probability given certain observations; we reasoned that the geometric relationship between a set of crater observations can be used to improve the accuracy of localization, instead of modeling each crater as an independent observation. Therefore, we constructed a spatial formulation where the measurement probability of each particle is represented by the average area Intersection Over Union (IoU) distance \\cite{rezatofighi2019generalized} between the orbital and ground craters:\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/iou.pdf}\n\\end{figure}\n\nTo account for missed detections, we adjust for the probability of new crater detection by using prior probability of detection from previous data.\n\n\\begin{figure*}[!t]\n    \\centering\n    \\begin{tabular}{cc}\n    \\includegraphics[width=0.5\\linewidth]{figures\/exp1.pdf} &\n    \\includegraphics[width=0.5\\linewidth]{figures\/exp2.pdf}\n    \\end{tabular}\n    \\caption{\\textbf{(Left) Example of particle filter performance on a simulated scenario, and (Right) Example of parametric model performance on a simulated scenario.}}\n    \\label{fig:lidar_res1}\n\\end{figure*}\n\n\\begin{figure*}[!t]\n    \\centering\n    \\begin{tabular}{ccc}\n    \\includegraphics[width=0.31\\linewidth]{figures\/exp3.pdf} &\n    \\includegraphics[width=0.31\\linewidth]{figures\/exp4.pdf} &\n    \\includegraphics[width=0.31\\linewidth]{figures\/exp5.pdf}\n    \\end{tabular}\n    \\caption{\\textbf{(Left) Performance of particle filter and parametric model over different (n=50) simulated environments. The red line indicates the 2$\\%$ noise in motion model baseline, and the distribution of the relative error at the end of the traversal for particle filter (Middle) and parametric model (Right) with the 2$\\%$ motion model baseline in red.}}\n    \\label{fig:lidar_res2}\n\\end{figure*}\n\n\\subsection{Crater-based based Localization using Parametric Matching}\nFor parametric method-based localization, we assumed that each crater is represented by a gaussian distribution, with the location of the crater as the mean and the half of the radius as the standard deviation. For orbital craters, the set of craters can then be expressed using a gaussian mixture; and this can also be independently formulated for ground craters. Parametric matching based localization is formulated as the problem of finding the translation between the two gaussian mixtures that best matches these two distributions as shown in \\ref{fig:gmm}. The KL-divergence \\cite{goldberger2003efficient} measures the distance between two distributions and can be used as the loss function. We minimize the loss function with gradient-descent based methods to find the best translation, which represents the location (as illustrated in Figure \\ref{fig:loss_landscape}). Further, the Hessian around the optimal solution can be used to estimate the standard error, since KL-divergence is equivalent to the negative log-likelihood.\n\n\\subsection{Blender-based Lunar Scene Simulator}\nThe Blender-based image scene simulation takes a lunar digital elevation map (DEM), applies a custom texture map to it, and creates a world model with a simulated sun as a light source; within this model, a stereo camera pair is added and the simulation generates locations for the cameras and the sun. For each camera and sun location, Blender's render engine produces a synthetic image. This image is rendered using a custom implementation of the Hapke radiometric model \\cite{hapke1963theoretical,hapke1981bidirectional,hapke1993opposition} incorporated into the Blender's path tracing algorithm to simulate accurate lighting across the surface. The simulation has multiple parameters that can be controlled, including sun position, camera extrinsic and intrinsic parameters, image size, and a few others. Images were were generated for craters with diameters ranging from 5 to 20 meters, with four different approach angles for each crater. For each crater and approach angle, the stereo camera was positioned at distances between 5 and 20 meters from the crater near rim, at 1 meter spacing; for each location, an image was rendered with a varied set of sun angles from 0$^{\\circ}$ to 80$^{\\circ}$ from nadir. The final dataset consisted of 1,792 stereo pairs.\n\n\\begin{figure}[!tbh]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/mc-sim-environment.pdf}\n    \\caption{\\textbf{Illustration of the simulated environment used for development and testing of localization algorithms. Shown here is an example for a 600m x 600m region with known orbital craters}}\n    \\label{fig:mc_sim_env}\n\\end{figure}\n\n\\subsection{2.5D Simulation Environment for Long-range Traverse}\nTo facilitate development of algorithms for crater matching based localization, we constructed a 2.5D simulation environment that allows for independent unit testing of the different localization algorithms. This allowed us to quickly iterate through different versions of the localization algorithms and validate their effectiveness with offline Monte Carlo simulations. In the simulated environment, the localization algorithm receives two sets of craters as input: the known craters detected from orbit and the ones observed by the rover from the ground. The orbital craters are parameterized by their size and location (x, y, diameter). Their craters sizes are drawn from a truncated power-law distribution ($\\alpha$ = 1, diameter $<$ 20), whereas their locations are drawn from a uniform distribution. The craters observed on the surface by the rover are generated by perturbing the known orbital craters according to a similar distribution of noise from crater detection, with a zero-centered gaussian ($\\sigma$ = 3m) error in the crater location and zero-centered gaussian ($\\sigma$ = 1m) error in the crater size. Further, we also added the capability to mask a percentage of either orbital or ground craters to simulate false positives and false negatives in crater detection. Figure \\ref{fig:mc_sim_env} shows an illustration of the simulated environment that was used. Assuming the motion model with 2$\\%$ noise, the rover traverses within this simulated environment and observes craters within a field of view range ($<$ 40m). The localization algorithms then match the observed ground craters to the orbital craters for localization. Then, the estimated route is compared against the ground truth route for evaluation. The simulated environment can be extended to long-range traversals and can be used to validate the effectiveness in overcoming the baseline $2\\%$ of motion model noise.\n\n\\begin{figure}[!t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/dataset-fig2.pdf}\n    \\caption{\\textbf{Illustration of an instance from the LiDAR simulation from RSIM.}}\n    \\label{fig:lidar_sim}\n\\end{figure}\n\n\\begin{figure*}[!t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/dataset-fig3.pdf}\n    \\caption{\\textbf{Example Chang'e 3 Yutu rover images (left), an example stereo disparity map and 3D rendering (middle), and orbital image showing the rover traverse (right). Stereo disparity is inversely proportional to range; in the disparity map, bright pixels are close and dark are far.}}\n    \\label{fig:change-stereo}\n\\end{figure*}\n\n\\begin{figure*}[!t]\n    \\centering\n    \\includegraphics[width=0.95\\linewidth]{figures\/lcam.pdf}\n    \\caption{\\textbf{The Chang'e 4 EDL Flight trajectory was recovered between 1000m and 100m above ground, using TRN algorithm on the LCAM images.}}\n    \\label{fig:change-lcam}\n\\end{figure*}\n\n\\begin{figure*}[!t]\n    \\centering\n    \\includegraphics[width=0.95\\linewidth]{figures\/change-improved.pdf}\n    \\caption{\\textbf{Comparison of (Left) Original resolution - 1.4 m\/pixel vs (Right) Improved resolution \u2013 0.2 m\/pixel for the orbital maps for the Chang'e 4 landing site.}}\n    \\label{fig:change-improved}\n\\end{figure*}\n\n\\begin{figure*}[!t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/dataset-fig4.pdf}\n    \\caption{\\textbf{Crater landmark map generation combining LRO-NAC images with higher resolution Chang'e LCAM descent imagery. Large image is from LRO-NAC; smaller overlay and the expanded view on the right is from LCAM.}}\n    \\label{fig:change-crater}\n\\end{figure*}\n\n\\subsection{Long-range LiDAR simulation using RSIM}\nTo further simulate long-range traverses from real Lunar terrain, we added a LiDAR simulation capability in RSIM \\cite{allan2019planetary}, a ROS2 and Gazebo based high-fidelity simulation environment developed at NASA Ames to support the development of the VIPER Rover \\cite{colaprete2019overview}. RSIM models the VIPER lunar rover structure, with the full CAD including rover kinematics, dynamics, and 3D model. RSIM also contains a lunar terrain model of the Nobile landing site, with 8k resolution. The terrain includes rocks and crater distributions, shadowing, and sun lighting. The LiDAR sensor (also referred to as the Velodyne Simulator), is forked from the Toyota Research Institute Velodyne Simulator. This package contains a URDF description and Gazebo plugins to simulate the Velodyne laser scanners (our model is the HDL-32E). This Gazebo plugin is based on the gazebo plugins ROS block laser, which publishes PointCloud2 messages with the same structure (x, y, z, intensity, ring) and simulates Gaussian noise. To capture the point clouds from the LiDAR sensor, we created a package that provides a ROS2 node that can subscribe to the Velodyne lidar sensor mounted on the VIPER rover and saves a snapshot to the desk. In addition to capturing the point cloud, this node also outputs the position and orientation of the rover with respect to the world origin. It also contains scripts to visualize the point cloud and export it to other formats. The final datasets used were two discrete drives of the VIPER rover in RSIM with LiDAR on the Nobile landing site; both drives started the rover at the bottom-left corner of the map (320m x 320m) and traversed diagonally along the hypotenuse to the upper-right corner, took approximately 30-32 steps of 10m each, with a change in orientation chosen randomly between (-5 to 5 degrees) at each step. Due to the orientation change, the rover could not complete all the steps as it would fall off the map since it was not driving in a purely straight path with step-length: 10m, total steps: 35 and orientation change: 5 degrees at every waypoint. Figure \\ref{fig:lidar_sim} shows an illustration of an instance the LiDAR simulation from RSIM. \n\n\\subsection{Real Lunar data from Chang'e missions}\nFor real sensor data, the original plan for to acquire a large- scale dataset of LIDAR and stereo image data from an analog site at the Cinder Lake crater field near Flagstaff, Arizona. This had to be postponed due to covid travel restrictions and forest fires; in lieu of this data, we decided to use real lunar stereo images that were available from the \"panoramic\" cameras (PCAM) on the Chang'e 3 \\cite{li2015chang} and Chang'e 4 \\cite{li2021overview} lunar rover missions. Since this is actual lunar data, in important ways it is better than the analogue stereo images that were originally planned. The Chang'e missions' rover data has been publicly released \\footnote{https:\/\/moon.bao.ac.cn\/}. The cameras for both missions are identical; their FOV is 19.7$^{\\circ}$x14.5$^{\\circ}$, resolution are 2352 x 1728 pixels (color) and 1176 x 864 pixels (monochrome) and stereo baseline length is 27 cm. A total of 168 pairs of Chang'e 3 and 1174 pairs of Chang'e 4 PCAM images were processed. A stereo camera self-calibration applied to this data yielded good camera models for both data sets. Stereo depth maps were computed from these images with good results, as shown in Figures \\ref{fig:change-stereo}. \n\nGround truth labeling of craters in this imagery and registration of this imagery against orbital imagery was done to prepare a dataset that was used for performance evaluations. Towards this end, we studied the usefulness of the Chang'e lander camera (LCAM) to obtain a high resolution and high precision crater database. The LCAM image sequences (5000 images) were downloaded, and the Chang'e EDL flight trajectory was recovered between 1000m and 100m above ground, using terrain relative navigation (TRN) and structure from motion algorithms. Figure \\ref{fig:change-lcam} shows visualization of the recovered LCAM trajectory. Then, the LCAM images were ortho-rectified to a coarser resolution LRO-NAC image map (1 m\/pixel) to obtain an ortho-image with higher resolution of up to 20 cm\/pixel around the lander. This process was used to improve the resolution for a 100m x 100m region around the Chang'e 4 lander as shown in Figure \\ref{fig:change-improved}. The craters were detected from this high-resolution map and their geographic locations, diameters, depths were extracted into a crater database, as shown in Figure \\ref{fig:change-crater}. This database was used for rover localization algorithm development\n\n\n\n\n\\subsection{Remaining Gaps, Risks and Challenges to Flight Infusion}\nAs mentioned in Section \\ref{sec:intro}, the capability developed in this project is intended for use in lunar rovers. No such missions are currently in development, but robotic lunar science rover mission concepts were strongly recommended by the PSADS report; in particular, the Endurance-A robotic lunar science rover mission concept was recommended as the highest priority medium-sized mission for the Moon. This concept involves traverse of roughly 2,000 km in several Earth years, which requires onboard absolute position estimation; LunarNav capability would be enabling for this mission. The Lunar Terrain Vehicle (LTV) envisioned for transporting astronauts may also benefit from having an option for autonomous position estimation as developed by LunarNav. Some of the potential challenges to flight infusion falls into the following main categories:\n\\begin{itemize}\n    \\item The state of maturity of the LunarNav algorithms is at intermediate TRLs; more maturation is required before it is fully ready for infusion. In particular, the performance of stereo-based crater localization needs to be improved by adding robustness to a wide-range of operational scenarios (partial crater visibility, variable crater shapes) \\\\\n    \\item LunarNav requires stereo cameras and\/or a lidar onboard the rover. For night operation, either headlights for the stereo cameras or a lidar is required. These sensors are not fully developed at present. Furthermore, the majority of the work done in LunarNav focused on day-time driving; further work needs to be done to extend this capability to night-time and operations in permanently shadowed regions. \\\\\n    \\item The computing load associated with crater-based localization is expected to be less than that required for rover obstacle avoidance. Since any autonomous lunar rover would need obstacle avoidance, it is expected that the necessary computing capability for LunarNav would be available. Nevertheless, this aspect of system architecture must be verified as being sufficient. \\\\\n    \\item Validation and verification (V\\&V) of crater-based localization requires datasets with craters, either from lunar analog terrain on Earth or from high fidelity lunar terrain simulators. There is a very limited amount of suitable analog terrain available. The Cinder Lake Crater Field planned for use in the LunarNav project is the only location of any reasonable size in the U.S.; it is nevertheless fairly small for this purpose, it has not been maintained (i.e. it is partly overgrowth with vegetation), and access to it is limited to spring and fall seasons, due to snow in the winter, heat in the summer, and the possibility of nearby forest fires in the summer through early fall.\n\\end{itemize}\n\\subsection{LIDAR-based Localization on Simulation Environment:}\nWe evaluated the particle filter and parametric localization algorithms on a 400m x 400m simulated environment with 100 craters. An example of the traversal and localization results are shown below in Figures \\ref{fig:lidar_res1}-\\ref{fig:lidar_res3}. Over a large number of simulated scenarios (n=50), both localization methods were able to perform better than the relative localization baseline of 2\\% noise motion model, with the particle filter performing slightly better. Further, the particle filter converged at around 2m of error over long ranges, suggesting that crater landmarks are valid features for accurate localization. \n\n\\begin{figure*}\n    \\centering\n    \\begin{tabular}{cc}\n    \\includegraphics[width=0.65\\linewidth]{figures\/exp8.pdf} &\n    \\includegraphics[width=0.3\\linewidth]{figures\/exp9.pdf}\n    \\end{tabular}\n    \\caption{\\textbf{Illustration of the rover traverse for the Apollo 2 landing site showing the (left) simulated LiDAR point cloud and (middle) DEM overlayed with the rover traverse. Note: Point cloud is color coded using elevation data. (Right) An instance of particle filter illustration. Blue circles indicate craters, green arrow indicates estimated position, red arrow indicates GT position, and black dots represent the distribution of particles.}}\n    \\label{fig:lidar_apollo}\n\\end{figure*}\n\n\\subsection{LIDAR-based Localization from Apollo 2 Landing Site:}\nWe performed preliminarily tests of the particle filter and parametric localization on craters detected from simulated LIDAR point clouds. Here, we used the elevation map from the Apollo 2 landing site to simulate a traversal of 20 steps (with 1m\/step) for crater detection. In this scenario, two craters (with diameter 6.20m and 14.76m) were manually labeled as orbital craters. The crater detection algorithm matches observed candidate craters on the ground with the two orbital craters, and the localization algorithm used the location of the observed craters for localization, as shown in Figure \\ref{fig:lidar_apollo}.\n\n\\begin{figure*}\n    \\centering\n    \\begin{tabular}{cc}\n    \\includegraphics[width=0.5\\linewidth]{figures\/exp10.pdf} &\n    \\includegraphics[width=0.5\\linewidth]{figures\/exp11.pdf}\n    \\end{tabular}\n    \\caption{\\textbf{Rumker Dome site localization algorithm performance. The particle filter converged at around 0.75m of error, which is lower than the expected 2$\\%$ (1m) motion model noise. The parametric method has an average error of 1.6m.}}\n    \\label{fig:lidar_rumker}\n\\end{figure*}\n\n\\subsection{LIDAR-based Localization from Rumker Dome Landing Site:}\nFurther, we evaluated the particle filter and parametric localization on LIDAR crater detected from a traverse through the Rumker Dome region. This is a more complex scenario compared to the Apollo 2 landing site, with 4 manually labeled craters (with diameter 21.94m, 5.60m, 6.98m, 8.58m) and 50 traversal steps (1m\/step). According to the 2\\% motion model baseline, our model is estimated to deviate 1m from the ground truth. Our results (Figure \\ref{fig:lidar_rumker}) indicate an average distance of 0.75m throughout the traverse, indicating that crater landmarks are a reliable feature for more accurate localization. \n\n\\begin{figure*}\n    \\centering\n    \\begin{tabular}{cc}\n    \\includegraphics[width=0.45\\linewidth]{figures\/exp12.pdf} &\n    \\includegraphics[width=0.45\\linewidth]{figures\/exp13.pdf}\n    \\end{tabular}\n    \\caption{\\textbf{An example of a successful crater detection. (Left) original image; (b) disparity map overlayed with white ellipse showing crater detection}}\n    \\label{fig:stereo-change}\n\\end{figure*}\n\n\\begin{figure*}\n    \\centering\n    \\begin{tabular}{ccc}\n    \\includegraphics[width=0.31\\linewidth]{figures\/exp14.pdf} &\n    \\includegraphics[width=0.31\\linewidth]{figures\/exp15.pdf} &\n    \\includegraphics[width=0.31\\linewidth]{figures\/exp16.pdf}\n    \\end{tabular}\n    \\caption{\\textbf{Example images where the stereo-based crater detection algorithm fails.}}\n    \\label{fig:stereo-fail}\n\\end{figure*}\n\n\\subsection{Stereo-based Localization on Chang'e 4 Landing Site}\nWe integrated the particle-filter-based localization algorithm with the stereo-based crater detection from FY21, and tested it with the Chang'e rover stereo data. The stereo crater detection algorithm succeeds in detecting craters on the real data when the assumptions of the algorithm \\cite{matthies2022lunar} hold true; for example, as shown in Figure \\ref{fig:stereo-change} which demonstrated a successful case of stereo-based crater detection on real Lunar data. \n\nHowever, the original crater detection makes two critical assumptions which regularly do not hold in the Chang'e 4 dataset. It first assumes that the entire crater is visible in the image. Due to the small field of view of the PCAM, this is often not the case in the Chang'E imagery. The second assumption is that the crater composes a relatively small section of the image, since otherwise the ground plane will not be detected correctly. This is also not the case with nearby craters, in part due once again to the small camera field of view. \n\nFigure \\ref{fig:stereo-fail} shows three example images where crater detection currently fails. In the leftmost image, the algorithm successfully detects the as a crater candidate in the penultimate step of the algorithm. However, it is filtered out as not a crater because the entire crater is not visible in the image, violating an assumption behind the algorithm. In the middle image, the crater is not entirely within the image. This case fails in step three of the algorithm, earlier than the previous case, since the critical near rim edge is not visible. Furthermore, since the crater takes up almost the entirety of the image, the back wall of the crater is detected as the ground plane. In the rightmost image, the crater takes up a large section of the image, and the crater's back wall is detected as the ground plane, leading the algorithm to fail. Additionally, the left and right edges of the crater are not visible so this would likely fail in a later step as well. Thus, the performance of the crater detection did not generalize to the current dataset of craters visible in Chang'e imagery. As a result, we were unable to benchmark the performance of the particle filter on stereo modality.\n\n\n\\section{Introduction}\n\\input{intro.tex}\n\\vspace{2mm}\n\\label{sec:intro}\n\n\\section{Related Work}\n\\vspace{2mm}\n\\input{related.tex}\n\n\\section{System Overview}\n\\vspace{2mm}\n\\input{system.tex}\n\n\\section{Real and Simulated Datasets}\n\\vspace{2mm}\n\\input{datasets.tex}\n\\label{sec:data}\n\n\\section{Technical Approach}\n\\vspace{2mm}\n\\input{approach.tex}\n\n\\section{Performance Evaluation}\n\\input{experiments.tex}\n\\label{sec:exp}\n\n\\section{Discussion}\n\\vspace{2mm}\n\\input{discussion}\n\n\n\n\n\\acknowledgments\nThe research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration (80NM0018D0004).\n\n\n\n\\bibliographystyle{IEEEtran}\n\n\n\\subsection{System Concept}\nThe LunarNav system concept requires creating a database of crater landmarks from orbital images, as shown in Figure \\ref{fig:system_concept}. This database would contain the position, diameter, and estimated depth of each crater. Craters with diameters $> 5$ meters are mappable with LRO-NAC imagery, under most orbital imaging conditions. Even in the youngest terrain on the Moon, such craters occur with a frequency $> 10^3$ per $km^2$, or on average about 30 meters apart, and they occur more frequently in older terrain \\cite{hiesinger2012old,minton2019equilibrium}. This would provide frequent landmarks that should be detectable at distances of roughly 10 to 20 meters from the rover.\n\nRobotic lunar rovers will carry a sensor suite for relative localization and obstacle detection that includes wheel odometry, an IMU, either stereo cameras or a lidar, and either a sun sensor or a star camera for absolute heading measurement. Assuming the availability of a star camera, which is applicable for driving in sunlight and in shadow, gives 3-axis attitude knowledge to a small fraction of a degree. With this, the relative navigation sensor suite enables dead-reckoning with position error that typically can be $< 2\\%$ of distance traveled. This provides a prior estimate of position that at all times strongly constrains which crater(s) from the landmark database are expected to be near the rover. Craters can then be detected near the rover with a combination of 3D point cloud data from stereo cameras or a lidar, image data from a camera, or reflectance image data from a lidar. This enables detecting craters with diameters roughly between 5 to 20 meters whose near rims are roughly less than 20 meters from the rover.\n\nOverall, these methods should enable reliable absolute localization; given typical resolution characteristics of cameras, stereo vision, and lidar, we estimate that it should be possible to maintain a rover absolute position estimate with $3 \\sigma$ error $< 5$ meters at all times. Furthermore, in terms of computational feasibility, crater-based localization is less expensive than obstacle detection and needs to be done much less frequently than obstacle detection, so any onboard computing system that can do obstacle detection would also be able to do crater-based localization.\n\n\\begin{table*}[!tbh]\n    \\centering\n    \\caption{\\bf{Key Performance Parameters (KPP)}}\n    \\includegraphics[width=0.9\\linewidth]{figures\/kpp.pdf}\n    \\label{tab:kpp}\n\\end{table*}\n\n\\subsection{Key Performance Parameters}\nThorough performance evaluation of the LunarNav framework was a function of many parameters, and depended fundamentally on the ability to detect and localize individual craters and to estimate their positions and diameters. This depends on (1) crater size and distance from the rover, (2) rover camera\/lidar sensor parameters, including angular resolution, range resolution, field of view, and sensor height above the ground, (3) lighting conditions, and (4) other characteristics of terrain geometry, like slope. We distilled key performance parameters (KPPs) in terms of the crater distribution, as defined in Table \\ref{tab:kpp} and anticipated that craters with diameters $>$ 5 meters will be readily detectable from distances of at least 15 meters, in many but not necessarily all lighting conditions. For example, detection probability ($P_d$) of 0.5 for 5 m craters at 15 m range should be a conservative estimate. With a notional stereo camera system with angular resolution of 1 milliradian\/pixel and binocular camera baseline of 30 cm, 3 $\\sigma$ errors in estimating the position and diameter of such craters should be $<$ 2 m each.\n\nUpdates to rover position and heading will be obtained every time an onboard crater map is registered to the orbital map. The precision of these rover position and heading estimates will be a function of the precision of crater positions and diameters in the onboard map, as well as detection and false alarm probabilities ($P_d$ and $P_f$) for onboard crater recognition. Quality of the onboard map will also depend on accuracy and precision of heading estimation and dead reckoning between crater detections. Past experience suggests that dead reckoned position error can be better than $2\\%$ of distance traveled with visual odometry (2 m per 100 m) \\cite{rankin2021mars}. Rover heading error should be bounded by about $5\\degree$ by a sun sensor, $<1\\degree$ per 100 m using visual odometry, or $<0.01 \\degree$ at all times using a star camera. A star camera is the preferred heading estimation solution for performance, but a sun sensor may offer a lower cost solution with adequate performance for predominantly sunlit scenarios. Combining this with errors in crater center positions relative to the rover and crater position errors of $<$ 2 m in the orbital map should enable rover position and heading estimation error to be conservatively bounded by about 10 m and 5$\\degree$ at all times. Erroneous crater detections (false alarms) will be filtered out through a combination of several techniques applied at different stages of the estimation pipeline. Since the effect of false detections is ultimately captured in rover position estimation error, a separate key performance parameter is not specified for false alarms. Performance modeling and evaluation throughout the course of the project characterized how performance varies as a function of these sensor parameters (See Section \\ref{sec:exp}). \n\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nNew physics (NP), or physics beyond the standard model, involves various models\nthat extend the well verified standard model (SM) of particle physics by\nintroducing a number of new particles with novel properties and interactions.\nThough various aspects of many of these particles and interactions are\nconstrained by existing experimental data, we are yet to detect any definitive\nsignature of new physics in our experiments. Nevertheless, recent experimental\nstudies in $B$ meson decays, such as $B \\to K^{(*)} \\ell^-\\ell^+$\n\\cite{B2KorKstLL}, $B_s \\to \\phi \\ell^-\\ell^+$ \\cite{Aaij:2015esa}, $B \\to\nD^{(*)}\\ell\\nu$ \\cite{B2DorDstLN} and $B_c \\to J\/\\psi \\ell\\nu$\n\\cite{Aaij:2017tyk} (where $\\ell$ can be $e,\\mu$ or $\\tau$) have reported\nanomalous observations raising the expectation of discovery of new physics with\nmore statistical significance. In this context, model-independent studies of\nsuch semi-leptonic three-body meson decay processes become important as they can\nidentify generic signatures of new physics which can be probed experimentally.\nIn this paper, we have analyzed the effects of new physics, in a\nmodel-independent manner, on the angular distribution of a general semi-hadronic\nthree-body meson decay of the type $P_i \\to P_f f_1 f_2$, where $P_i$ and $P_f$\nare the initial and final pseudo-scalar mesons respectively, and $f_{1,2}$\ndenote fermions (which may or may not be leptons but not quarks) out of which at\nleast one gets detected experimentally. Presence of new interactions, or new\nparticles such as fermionic dark matter (DM) particles or heavy sterile\nneutrinos or long lived particles (LLP) would leave their signature in the\nangular distribution and we show by example how new physics contribution can be\nquantified from angular asymmetries. Our methodology can be used for detection\nof new physics in experimental study of various three-body pseudo-scalar meson\ndecays at various collider experiments such as LHCb and Belle~II.\n\nThe structure of our paper is as follows. In\nSec.~\\ref{sec:Lagrangian-and-amplitude} we discuss the most general Lagrangian\nand amplitude which include all probable NP contributions to our process under\nconsideration. The relevant details of kinematics is then described in\nSec.~\\ref{sec:kinematics}. This is followed by a discussion on the angular\ndistribution and the various angular asymmetries in\nSec.~\\ref{sec:ang-dist-asymmetries}. In Sec.~\\ref{sec:example} we present a few\nwell chosen examples illustrating the effects of new physics on the angular\ndistribution. In Sec.~\\ref{sec:conclusion} we conclude by summarizing the\nimportant aspects of our methodology and its possible experimental realization.\n\n\\section{Most general Lagrangian and Amplitude}\\label{sec:Lagrangian-and-amplitude}\n\nFollowing the model-independent analysis of the decay $B \\to D \\ell^-\\ell^+$ as\ngiven in Ref.~\\cite{Kim:2016zbg} and generalizing it for our process $P_i \\to\nP_f f_1 f_2$ where $P_{i,f}$ can be $B, B_s, B_c, D, K, \\pi$ etc.\\ as\nappropriate and $f_1 f_2$ can be $\\ell^- \\ell^+$, $\\ell \\bar{\\ell'}$, $\\ell\n\\nu_{\\ell}$, $\\ell \\nu_S$, $\\ell f^{DM}$, $\\nu_{\\ell}\\ensuremath{\\overline{\\nu}}_{\\ell}$,\n$\\nu_S\\ensuremath{\\overline{\\nu}}_{\\ell}$, $\\nu_{\\ell}\\ensuremath{\\overline{\\nu}}_S$, $\\nu_S \\ensuremath{\\overline{\\nu}}_S$, $f^{DM}\n\\bar{f}^{DM}$, $f_1^{DM} f_2^{DM}$, $f_1^{LLP} f_2^{LLP}$ (with\n$\\ell,\\ell'=e,\\mu,\\tau$ denoting leptons, $\\nu_S$ being sterile neutrino,\n$f^{DM}_{1,2}$ as fermionic dark matter and $f_{1,2}^{LLP}$ as long lived\nfermions)\\footnote{It is clear that we can not only analyze processes allowed in\n\tthe SM but also those NP contributions from fermionic dark matter in the final\n\tstate as well as including flavor violation. Our analysis as presented in this\n\tpaper is fully model-independent and general in nature.}, we can write down the\neffective Lagrangian facilitating the decay under consideration as follows,\n\\begin{eqnarray}\n\\mathcal{L}_{\\textrm{eff}} &=& J_S \\left( \\bar{f}_1 f_2 \\right)\n+ J_P \\left( \\bar{f}_1~\\gamma^5~f_2 \\right) + \\left(J_V\\right)_{\\alpha}\n\\left( \\bar{f}_1~\\gamma^{\\alpha}~f_2 \\right) \\nonumber\\\\%\n&& + \\left(J_A\\right)_{\\alpha} \\left( \\bar{f}_1~\\gamma^{\\alpha}\\gamma^5~f_2\n\\right) + \\left(J_{T_1}\\right)_{\\alpha\\beta} \\left(\n\\bar{f}_1~\\sigma^{\\alpha\\beta}~f_2 \\right) \\nonumber\\\\%\n&& + \\left(J_{T_2}\\right)_{\\alpha\\beta} \\left(\n\\bar{f}_1~\\sigma^{\\alpha\\beta}\\gamma^5~f_2 \\right) + \\text{h.c.},\n\\label{eq:Effective-Lagrangian}\n\\end{eqnarray}\nwhere $J_S$, $J_P$, $\\left(J_V\\right)_{\\alpha}$, $\\left(J_A\\right)_{\\alpha}$,\n$\\left(J_{T_1}\\right)_{\\alpha\\beta}$, $\\left(J_{T_2}\\right)_{\\alpha\\beta}$ are\nthe different hadronic currents which effectively describe the quark level\ntransitions from $P_i$ to $P_f$ meson. It should be noted that we have kept both\n$\\sigma^{\\alpha\\beta}$ and $\\sigma^{\\alpha\\beta}\\gamma^5$ terms. This is because\nof the fact that the currents $\\bar{f}_1 \\, \\sigma^{\\alpha\\beta} \\, f_2$ and\n$\\bar{f}_1 \\, \\sigma^{\\alpha\\beta}\\gamma^5 \\, f_2$ describe two different\nphysics aspects namely the magnetic dipole and electric dipole contributions\nrespectively. In the SM, vector and axial-vector currents (mediated by photon,\n$W^{\\pm}$ and $Z^0$ bosons) and the scalar current (mediated by Higgs boson)\ncontribute. So every other term in Eq.~\\eqref{eq:Effective-Lagrangian} except\nthe ones with $J_S$, $\\left(J_V\\right)_{\\alpha}$ and $\\left(J_A\\right)_{\\alpha}$\ncan appear in some specific NP model. Since, in this paper, we want to\nconcentrate on a fully model-independent analysis to get generic signatures of\nnew physics, we shall refrain from venturing into details of any specific NP\nmodel, which nevertheless are also useful. It is important to note that $J_S$,\n$\\left(J_V\\right)_{\\alpha}$ and $\\left(J_A\\right)_{\\alpha}$ can also get\nmodified due to NP contributions.\n\n\\begin{figure}[hbtp]\n\\centering%\n\\includegraphics[scale=1]{fig_Feynman_diagram.pdf} \\caption{Feynman diagram\n\tfor $P_i \\to P_f f_1 f_2$ considering $f_1$ as a particle and $f_2$ as an\n\tanti-particle. Here the blob denotes the effective vertex and includes\n\tcontributions from all the form factors defined in\n\tEq.~\\eqref{eq:form-factors}.}%\n\\label{fig:Feynman_diagram}\n\\end{figure}\n\nIn order to get the most general amplitude for our process under consideration,\nwe need to go from the effective quark-level description of\nEq.~\\eqref{eq:Effective-Lagrangian} to the meson level description by defining\nappropriate form factors. It is easy to write down the most general form of the\namplitude for the process $P_i \\to P_f f_1 f_2$ depicted in\nFig.~\\ref{fig:Feynman_diagram} as follows,\n\\begin{align}\n\\mathcal{M} \\left( P_i \\to P_f f_1 f_2 \\right) &= F_S \\left(\n\\bar{f}_1 f_2 \\right) + F_P \\left( \\bar{f}_1~\\gamma^5~f_2 \\right)\n\\nonumber\\\\*%\n&\\quad + \\left( F_V^+ p_{\\alpha} + F_V^- q_{\\alpha} \\right) \\left(\n\\bar{f}_1~\\gamma^{\\alpha}~f_2 \\right) \\nonumber\\\\* %\n&\\quad + \\left( F_A^+ p_{\\alpha} + F_A^- q_{\\alpha} \\right) \\left(\n\\bar{f}_1~\\gamma^{\\alpha}~\\gamma^5~f_2 \\right) \\nonumber\\\\* %\n&\\quad + F_{T_1}~p_{\\alpha}~q_{\\beta} \\left( \\bar{f}_1~\\sigma^{\\alpha\\beta}~f_2\n\\right) \\nonumber\\\\* %\n&\\quad + F_{T_2}~p_{\\alpha}~q_{\\beta} \\left(\n\\bar{f}_1~\\sigma^{\\alpha\\beta}~\\gamma^5~f_2 \\right), \\label{eq:amplitude}\n\\end{align}\nwhere $F_{S}$, $F_{P}$, $F_{V}^{\\pm}$, $F_{A}^{\\pm}$, $F_{T_1}$ and $F_{T_2}$\nare the relevant form factors, and are defined as follows,\n\\begin{subequations}\\label{eq:form-factors}\n\\begin{align}\n\\bracket{P_f}{J_S}{P_i} &= F_S,\\\\%\n\\bracket{P_f}{J_P}{P_i} &= F_P,\\\\%\n\\bracket{P_f}{\\left(J_V\\right)_{\\alpha}}{P_i} &= F_V^+ p_{\\alpha} + F_V^-\nq_{\\alpha},\\\\%\n\\bracket{P_f}{\\left(J_A\\right)_{\\alpha}}{P_i} &= F_A^+ p_{\\alpha} + F_A^-\nq_{\\alpha},\\\\%\n\\bracket{P_f}{\\left(J_{T_1}\\right)_{\\alpha\\beta}}{P_i} &=\nF_{T_1}~p_{\\alpha}~q_{\\beta},\\\\%\n\\bracket{P_f}{\\left(J_{T_2}\\right)_{\\alpha\\beta}}{P_i} &=\nF_{T_2}~p_{\\alpha}~q_{\\beta},\n\\end{align}\n\\end{subequations}\nwith $p \\equiv k + k_3$ and $q \\equiv k - k_3 = k_1 + k_2$, in which $k, k_1,\nk_2, k_3$ are the 4-momenta of the $P_i, f_1, f_2 $ and $P_f$ respectively (see\nFig.~\\ref{fig:Feynman_diagram}). All the form factors appearing in the amplitude\nin Eq.~\\eqref{eq:amplitude} and as defined in Eq.~\\eqref{eq:form-factors} are,\nin general, complex and contain all NP information. It should be noted that for\nsimplicity we have implicitly put all the relevant Cabibbo-Kobayashi-Maskawa\nmatrix elements as well as coupling constants and propagators inside the\ndefinitions of these form factors. In the SM only $F_V^{\\pm}$ and $F_A^{\\pm}$\nare present. Presence of NP can modify these as well as introduce other form\nfactors\\footnote{It should be noted that the form factors, especially the ones\n\tdescribing semi-leptonic $B$ meson decays, can be obtained by using the heavy\n\tquark effective theory \\cite{HQET}, the lattice QCD \\cite{Lattice}, QCD\n\tlight-cone sum rule \\cite{Light-cone} or the covariant confined quark model\n\t\\cite{CCQM} etc. In this paper we present a very general analysis which is\n\tapplicable to a diverse set of meson decays. Hence we do not discuss any\n\tspecifics of the form factors used in our analysis. Moreover, we shall show, by\n\tusing certain examples and in a few specific cases, that one can also probe new\n\tphysics without worrying about the details of the form factors. Nevertheless,\n\twhen one concentrates on a specific decay mode, considering the form factors in\n\tdetail is always useful.}. These various NP contributions would leave behind\ntheir signatures in the angular distribution for which we need to specify the\nkinematics in a chosen frame of reference.\n\n\n\\section{Decay Kinematics}\\label{sec:kinematics}\n\n\\begin{figure}[hbtp]\n\\centering%\n\\includegraphics[scale=1]{fig_GJframe.pdf}%\n\\caption{Decay of $P_i \\to P_f f_1 f_2$ in the Gottfried-Jackson frame.} %\n\\label{fig:GJ-frame}\n\\end{figure}\n\nWe shall consider the decay $P_i \\to P_f f_1 f_2$ in the Gottfried-Jackson\nframe, especially the center-of-momentum frame of the $f_1,f_2$ system, which is\nshown in Fig.~\\ref{fig:GJ-frame}. In this frame the parent meson $P_i$ flies\nalong the positive $z$-direction with 4-momentum $k = \\left(E, \\mathbf{k}\n\\right) = \\left(E,0,0,\\modulus{\\mathbf{k}}\\right)$ and decays to the daughter\nmeson $P_f$ which also flies along the positive $z$-direction with 4-momentum\n$k_3 = \\left( E_3, \\mathbf{k}_3 \\right) = \\left(E_3, 0, 0,\n\\modulus{\\mathbf{k}_3}\\right)$ and to $f_1$, $f_2$ which fly away back-to-back\nwith 4-momenta $k_1 = \\left( E_1, \\mathbf{k}_1 \\right)$ and $k_2 = \\left( E_2,\n\\mathbf{k}_2 \\right)$ respectively, such that by conservation of 4-momentum we\nget, $\\mathbf{k}_1 + \\mathbf{k}_2 = \\mathbf{0}$, $\\mathbf{k} = \\mathbf{k}_3$,\nand $E = E_1 + E_2 + E_3$. The fermion $f_1$ (which we assume can be observed\nexperimentally) flies out subtending an angle $\\theta$ with respect to the\ndirection of flight of the $P_i$ meson, in this Gottfried-Jackson frame. The\nthree invariant mass-squares involved in the decay under consideration are\ndefined as follows,\n\\begin{subequations}\\label{eq:stu}\n\\begin{align}\ns &= (k_1 + k_2)^2 = (k - k_3)^2,\\\\%\nt &= (k_1 + k_3)^2 = (k - k_2)^2,\\\\%\nu &= (k_2 + k_3)^2 = (k - k_1)^2.\n\\end{align}\n\\end{subequations}\nIt is easy to show that $s + t + u = m_i^2 + m_f^2 + m_1^2 + m_2^2$, where\n$m_i, m_f,m_1$ and $m_2$ denote the masses of particles $P_i,P_f,f_1$ and $f_2$\nrespectively. In the Gottfried-Jackson frame, the expressions for $t$ and $u$\nare given by\n\\begin{subequations}\\label{eq:tu}\n\\begin{align}\nt &= a_t - b \\cos\\theta,\\label{eq:t}\\\\%\nu &= a_u + b \\cos\\theta,\\label{eq:u}\n\\end{align}\n\\end{subequations}\nwhere\n\\begin{subequations}\\label{eq:ab}\n\\begin{align}\na_t &= m_1^2 + m_f^2 + \\frac{1}{2s} \\left( s + m_1^2 - m_2^2 \\right) \\left(\nm_i^2 - m_f^2 -s \\right),\\label{eq:at}\\\\%\na_u &= m_2^2 + m_f^2 + \\frac{1}{2s} \\left( s - m_1^2 + m_2^2 \\right) \\left(\nm_i^2 - m_f^2 -s \\right),\\label{eq:au}\\\\%\nb &= \\frac{1}{2s} \\sqrt{\\lambda\\left( s, m_1^2, m_2^2 \\right)~\\lambda \\left( s,\n\tm_i^2, m_f^2 \\right)},\\label{eq:b}\n\\end{align}\n\\end{subequations}\nwith the K\\\"{a}ll\\'{e}n function $\\lambda(x,y,z)$ defined as,\n\\begin{equation*}\n\\lambda\\left( x,y,z \\right) = x^2 + y^2 + z^2 - 2 \\left( xy + yz + zx \\right).\n\\end{equation*}\nIt is clear that $a_t$, $a_u$ and $b$ are functions of $s$ only. For the special\ncase of $m_1 = m_2 = m$ (say) we have $a_t = a_u = \\tfrac{1}{2} \\left(m_i^2 +\nm_f^2 + 2m^2 -s\\right)$ and $b = \\tfrac{1}{2} \\sqrt{\\left(1-4m^2\/s\\right)\n\t\\lambda\\left(s,m_i^2,m_f^2\\right)}$. It is important to note that we shall use\nthe angle $\\theta$ in our angular distribution.\n\n\\section{Most general angular distribution and angular asymmetries}\\label{sec:ang-dist-asymmetries}\n\nConsidering the amplitude as given in Eq.~\\eqref{eq:amplitude}, the most general\nangular distribution in the Gottfried-Jackson frame is given by,\n\\begin{equation}\\label{eq:gen-angular-dist}\n\\frac{d^2\\Gamma}{ds \\, d\\cos\\theta} = \\frac{b\\sqrt{s} \\left( C_0 + C_1\n\t\\cos\\theta + C_2 \\cos^2\\theta \\right)}{128 \\, \\pi^3 \\, m_i^2 \\left(m_i^2 - m_f^2\n\t+ s \\right)},\n\\end{equation}\nwhere $C_0$, $C_1$ and $C_2$ are functions of $s$ and are given by,\n\\begin{subequations}\\label{eq:gen-C012}\n\\begin{align}\nC_0 &= 2 \\Bigg(-\\modulus{F_{T_1}}^2 \\bigg(-\\Sigma m_{12}^2 s^2 + 2 \\Sigma\nm_{12}^2 \\left( \\Sigma m^2 \\right)_{if} s \\nonumber\\\\%\n& \\hspace{1.5cm} + \\left( \\Delta m^2 \\right)_{12}^2 s -\\Delta a_{tu}^2 s - 2\n\\left( \\Delta m^2 \\right)_{12}^2 \\left( \\Sigma m^2 \\right)_{if} \\nonumber\\\\%\n& \\hspace{1.5cm} - \\left( \\Delta m^2 \\right)_{if}^2 \\Sigma m_{12}^2 + 2 \\Delta\na_{tu} \\left( \\Delta m^2 \\right)_{12} \\left( \\Delta m^2 \\right)_{if} \\bigg)\n\\nonumber\\\\%\n& \\quad - 2 \\Im\\left( F_V^+ F_{T_1}^* \\right) \\bigg( -\\Sigma m_{12} s^2 + 2\n\\Sigma m_{12} \\left( \\Sigma m^2 \\right)_{if} s \\nonumber\\\\%\n& \\hspace{1.5cm} + \\Delta m_{12} \\left( \\Delta m^2 \\right)_{12} s - 2 \\Delta\nm_{12} \\left( \\Delta m^2 \\right)_{12} \\left( \\Sigma m^2 \\right)_{if}\n\\nonumber\\\\%\n& \\hspace{1.5cm} - \\left( \\Delta m^2 \\right)_{if}^2 \\Sigma m_{12} +\\Delta a_{tu}\n\\Delta m_{12} \\left( \\Delta m^2 \\right)_{if} \\bigg) \\nonumber\\\\%\n& \\quad + \\modulus{F_{T_2}}^2 \\bigg( \\Delta m_{12}^2 s^2 - 2 \\Delta m_{12}^2\n\\left( \\Sigma m^2 \\right)_{if} s - \\left( \\Delta m^2 \\right)_{12}^2 s\n\\nonumber\\\\%\n& \\hspace{1.5cm} + \\Delta a_{tu}^2 s + 2 \\left( \\Delta m^2 \\right)_{12}^2 \\left(\n\\Sigma m^2 \\right)_{if} + \\Delta m_{12}^2 \\left( \\Delta m^2 \\right)_{if}^2\n\\nonumber\\\\%\n& \\hspace{1.5cm} - 2 \\Delta a_{tu} \\left( \\Delta m^2 \\right)_{12} \\left( \\Delta\nm^2 \\right)_{if} \\bigg) \\nonumber\\\\%\n& \\quad - 2 \\Im\\left( F_A^+ F_{T_2}^* \\right) \\bigg(\\Delta m_{12} s^2 - 2 \\Delta\nm_{12} \\left( \\Sigma m^2 \\right)_{if} s \\nonumber\\\\%\n& \\hspace{1.5cm} - \\left( \\Delta m^2 \\right)_{12} \\Sigma m_{12} s + 2 \\left(\n\\Delta m^2 \\right)_{12} \\Sigma m_{12} \\left( \\Sigma m^2 \\right)_{if}\n\\nonumber\\\\%\n& \\hspace{1.5cm} - \\Delta a_{tu} \\left( \\Delta m^2 \\right)_{if} \\Sigma m_{12} +\n\\Delta m_{12} \\left( \\Delta m^2 \\right)_{if}^2 \\bigg) \\nonumber\\\\%\n& \\quad + \\modulus{F_A^+}^2 \\bigg( s^2 - 2\\left( \\Sigma m^2 \\right)_{if} s -\n\\Sigma m_{12}^2 s \\nonumber\\\\%\n& \\hspace{1.5cm} + 2 \\Sigma m_{12}^2 \\left( \\Sigma m^2 \\right)_{if} + \\left(\n\\Delta m^2 \\right)_{if}^2-\\Delta a_{tu}^2 \\bigg) \\nonumber\\\\%\n& \\quad + \\modulus{F_V^+}^2 \\bigg( s^2 - 2 \\left( \\Sigma m^2 \\right)_{if} s\n-\\Delta m_{12}^2 s \\nonumber\\\\%\n& \\hspace{1.5cm} + 2 \\Delta m_{12}^2 \\left( \\Sigma m^2 \\right)_{if} + \\left(\n\\Delta m^2 \\right)_{if}^2 - \\Delta a_{tu}^2 \\bigg) \\nonumber\\\\%\n& \\quad + \\modulus{F_A^-}^2 \\left(\\Sigma m_{12}^2 s - \\left( \\Delta m^2\n\\right)_{12}^2 \\right) \\nonumber\\\\%\n& \\quad - 2 \\Re\\left( F_P F_A^{-*} \\right) \\left(\\Sigma m_{12} s - \\Delta m_{12}\n\\left( \\Delta m^2 \\right)_{12} \\right) \\nonumber\\\\%\n& \\quad - \\modulus{F_V^-}^2 \\left( \\left( \\Delta m^2 \\right)_{12}^2-\\Delta\nm_{12}^2 s \\right) \\nonumber\\\\%\n& \\quad - 2\\Re\\left( F_S F_V^{-*} \\right) \\left(\\left( \\Delta m^2 \\right)_{12}\n\\Sigma m_{12}-\\Delta m_{12} s \\right) \\nonumber\\\\%\n& \\quad -\\modulus{F_S}^2 \\left( \\Sigma m_{12}^2-s \\right) -\\modulus{F_P}^2\n\\left( \\Delta m_{12}^2-s \\right) \\nonumber\\\\%\n& \\quad + 2 \\Re\\left( F_A^+ F_A^{-*} \\right) \\left( \\left( \\Delta m^2\n\\right)_{if} \\Sigma m_{12}^2 - \\Delta a_{tu} \\left( \\Delta m^2 \\right)_{12}\n\\right) \\nonumber\\\\%\n& \\quad - 2 \\Re\\left( F_P F_A^{+*} \\right) \\left( \\left( \\Delta m^2 \\right)_{if}\n\\Sigma m_{12} - \\Delta a_{tu} \\Delta m_{12} \\right) \\nonumber\\\\%\n& \\quad - 2 \\Re\\left( F_S F_V^{+*} \\right) \\left( \\Delta a_{tu} \\Sigma m_{12} -\n\\Delta m_{12} \\left( \\Delta m^2 \\right)_{if} \\right) \\nonumber\\\\%\n& \\quad + 2 \\Re\\left( F_V^+ F_V^{-*} \\right) \\left( \\Delta m_{12}^2 \\left(\n\\Delta m^2 \\right)_{if} - \\Delta a_{tu} \\left( \\Delta m^2 \\right)_{12} \\right)\n\\Bigg),\\\\%\nC_1 &= 8 b \\Bigg( \\Delta m_{12} \\left( \\Im\\left( F_V^- F_{T_1}^* \\right)\ns-\\Re\\left( F_P F_A^{+*} \\right) \\right) \\nonumber\\\\%\n& \\hspace{1.5cm} + \\Sigma m_{12} \\Big(-\\Im\\left( F_A^- F_{T_2}^* \\right) s +\n\\Re\\left( F_S F_V^{+*} \\right) \\nonumber\\\\%\n& \\hspace{4cm} - \\left( \\Delta m^2 \\right)_{if} \\Im\\left( F_A^+ F_{T_2}^*\n\\right) \\Big) \\nonumber\\\\%\n& \\hspace{1.5cm} + \\Delta a_{tu} \\left( \\modulus{F_V^+}^2 + \\modulus{F_A^+}^2\n\\right) \\nonumber\\\\%\n& \\hspace{1.5cm} + \\left( \\Im\\left( F_S F_{T_1}^* \\right) + \\Im\\left( F_P\nF_{T_2}^* \\right) \\right)s \\nonumber\\\\%\n& \\hspace{1.5cm} + \\left( \\Delta m^2 \\right)_{12} \\left( \\Re\\left( F_V^+\nF_V^{-*} \\right) + \\Re\\left( F_A^+ F_A^{-*} \\right) \\right) \\nonumber\\\\%\n& \\hspace{1.5cm} + \\left( \\Delta m^2 \\right)_{if} \\Delta m_{12} \\Im\\left( F_V^+\nF_{T_1}^* \\right) \\Bigg),\\\\%\nC_2 &= 8 b^2 \\left( \\left( \\modulus{F_{T_2}}^2 + \\modulus{F_{T_1}}^2 \\right) s -\n\\modulus{F_V^+}^2 - \\modulus{F_A^+}^2 \\right),\n\\end{align}\n\\end{subequations}\nwith\n\\begin{subequations}\n\\begin{align}\n\\Delta a_{tu} &= a_t - a_u,\\\\%\n\\Delta m_{12} &= m_1 - m_2,\\\\%\n\\Delta m_{if} &= m_i - m_f,\\\\%\n\\Sigma m_{12} &= m_1 + m_2,\\\\%\n\\Sigma m_{if} &= m_i + m_f,\\\\%\n\\left(\\Delta m^2\\right)_{12} &= \\Delta m_{12} \\Sigma m_{12} = m_1^2 - m_2^2,\\\\%\n\\left(\\Delta m^2\\right)_{if} &= \\Delta m_{if} \\Sigma m_{if} = m_i^2 - m_f^2,\\\\%\n\\left(\\Sigma m^2\\right)_{if} &= m_i^2 + m_f^2.\n\\end{align}\n\\end{subequations}\n\nIn the limit $m_1=m_2$, which happens when $f_1 f_2 = \\ell^-\\ell^+, \\nu\\ensuremath{\\overline{\\nu}},$\nor $f^{DM} \\bar{f}^{DM}$ etc., our expressions for the angular distribution\nmatches with the corresponding expression in Ref.~\\cite{Kim:2016zbg}. It is\nimportant to remember that in the SM we come across scalar, vector and axial\nvector currents only. Therefore, in the SM, $F_P^{\\text{SM}} =\nF_{T_1}^{\\text{SM}} = F_{T_2}^{\\text{SM}} = 0$, which implies that,\n\\begin{subequations}\\label{eq:SM-C012}\n\\begin{align}\nC_0^{\\text{SM}} =& 2 \\Bigg( \\modulus{\\left(F_A^+\\right)_{\\text{SM}}}^2 \\bigg(\ns^2 - 2\\left( \\Sigma m^2 \\right)_{if} s - \\Sigma m_{12}^2 s \\nonumber\\\\%\n& \\hspace{2cm} + 2 \\Sigma m_{12}^2 \\left( \\Sigma m^2 \\right)_{if} + \\left(\n\\Delta m^2 \\right)_{if}^2-\\Delta a_{tu}^2 \\bigg) \\nonumber\\\\%\n& \\quad + \\modulus{\\left(F_V^+\\right)_{\\text{SM}}}^2 \\bigg( s^2 - 2 \\left(\n\\Sigma m^2 \\right)_{if} s -\\Delta m_{12}^2 s \\nonumber\\\\%\n& \\hspace{2.25cm} + 2 \\Delta m_{12}^2 \\left( \\Sigma m^2 \\right)_{if} + \\left(\n\\Delta m^2 \\right)_{if}^2 - \\Delta a_{tu}^2 \\bigg) \\nonumber\\\\%\n& \\quad + \\modulus{\\left(F_A^-\\right)_{\\text{SM}}}^2 \\left(\\Sigma m_{12}^2 s -\n\\left( \\Delta m^2 \\right)_{12}^2 \\right) \\nonumber\\\\%\n& \\quad - \\modulus{\\left(F_V^-\\right)_{\\text{SM}}}^2 \\left( \\left( \\Delta m^2\n\\right)_{12}^2-\\Delta m_{12}^2 s \\right) \\nonumber\\\\%\n& \\quad - \\modulus{\\left(F_S\\right)_{\\text{SM}}}^2 \\left( \\Sigma m_{12}^2 -s\n\\right) \\nonumber\\\\%\n& \\quad + 2 \\Re\\left( \\left(F_A^+\\right)_{\\text{SM}}\n\\left(F_A^-\\right)_{\\text{SM}}^* \\right) \\bigg( \\left( \\Delta m^2 \\right)_{if}\n\\Sigma m_{12}^2 \\nonumber\\\\*%\n& \\hspace{4cm} - \\Delta a_{tu} \\left( \\Delta m^2 \\right)_{12} \\bigg)\n\\nonumber\\\\%\n& \\quad + 2 \\Re\\left( \\left(F_V^+\\right)_{\\text{SM}}\n\\left(F_V^-\\right)_{\\text{SM}}^* \\right) \\bigg( \\left( \\Delta m^2 \\right)_{if}\n\\Delta m_{12}^2 \\nonumber\\\\%\n& \\hspace{4cm} - \\Delta a_{tu} \\left( \\Delta m^2 \\right)_{12} \\bigg) \\Bigg),\\\\%\nC_1^{\\text{SM}} =& 8 b \\Bigg( \\Delta a_{tu} \\left(\n\\modulus{\\left(F_V^+\\right)_{\\text{SM}}}^2 +\n\\modulus{\\left(F_A^+\\right)_{\\text{SM}}}^2 \\right) \\nonumber\\\\%\n& \\qquad + \\left( \\Delta m^2 \\right)_{12} \\bigg( \\Re\\left(\n\\left(F_V^+\\right)_{\\text{SM}} \\left(F_V^-\\right)_{\\text{SM}}^* \\right)\n\\nonumber\\\\%\n& \\hspace{2.5cm} + \\Re\\left( \\left(F_A^+\\right)_{\\text{SM}}\n\\left(F_A^-\\right)_{\\text{SM}}^* \\right) \\bigg) \\Bigg),\\\\%\nC_2^{\\text{SM}} =& - 8 b^2 \\left( \\modulus{\\left(F_V^+\\right)_{\\text{SM}}}^2 +\n\\modulus{\\left(F_A^+\\right)_{\\text{SM}}}^2 \\right).\n\\end{align}\n\\end{subequations}\n\nIt is interesting to note that in the special case of $m_1 = m_2$, such as in\n$P_i \\to P_f \\ell^+ \\ell^-$, we always have $C_1^{\\text{SM}}=0$. For specific\nmeson decays of the form $P_i \\to P_f f_1 f_2$ allowed in the SM, one can write\ndown $\\left(F_S\\right)_{\\text{SM}}$ , $\\left(F_V^{\\pm}\\right)_{\\text{SM}}$ and\n$\\left(F_A^{\\pm}\\right)_{\\text{SM}}$, at least in principle. The SM prediction\nfor the angular distribution can thus be compared with corresponding\nexperimental measurement. In order to quantitatively compare the theoretical\nprediction with experimental measurement, we define the following three angular\nasymmetries which can precisely probe $C_0$, $C_1$ and $C_2$ individually,\n\\begin{subequations}\\label{eq:ang-asymmetries}\n\\begin{align}\nA_0 \\equiv A_0(s) &= \\frac{- \\frac{1}{6} \\left( \\int_{-1}^{-1\/2} - 7\n\t\\int_{-1\/2}^{+1\/2} + \\int_{+1\/2}^{+1} \\right) \\dfrac{d^2 \\Gamma}{ds \\,\n\t\td\\cos\\theta} d\\cos\\theta}{d\\Gamma\/ds} \\nonumber\\\\%\n&= 3C_0\/\\left(6C_0+2C_2\\right),\\\\%\nA_1 \\equiv A_1(s) &= \\frac{- \\left( \\int_{-1}^{0} - \\int_{0}^{+1} \\right)\n\t\\dfrac{d^2 \\Gamma}{ds \\, d\\cos\\theta} d\\cos\\theta}{d\\Gamma\/ds} \\nonumber\\\\%\n&= 3C_1\/\\left(6C_0+2C_2\\right),\\\\%\nA_2 \\equiv A_2(s) &= \\frac{2 \\left( \\int_{-1}^{-1\/2} - \\int_{-1\/2}^{+1\/2} +\n\t\\int_{+1\/2}^{+1} \\right) \\dfrac{d^2 \\Gamma}{ds \\, d\\cos\\theta}\n\td\\cos\\theta}{d\\Gamma\/ds} \\nonumber\\\\%\n&= 3C_2\/\\left(6C_0+2C_2\\right).\n\\end{align}\n\\end{subequations}\nThe angular asymmetries of Eq.~\\eqref{eq:ang-asymmetries} are functions of $s$\nand it is easy to show that $A_2 = 3 \\left(1\/2 - A_0 \\right)$. We can do the\nintegration over $s$ in Eq.~\\eqref{eq:gen-angular-dist} and define the following\nnormalized angular distribution,\n\\begin{equation}\\label{eq:Gen-ang-dist}\n\\frac{1}{\\Gamma} \\frac{d\\Gamma}{d\\cos\\theta} = T_0 + T_1 \\cos\\theta + T_2 \\cos^2\\theta,\n\\end{equation}\nwhere\n\\begin{equation}\\label{eq:Def-T012}\nT_j = 3 c_j\/\\left(6c_0 + 2 c_2\\right),\n\\end{equation}\nfor $j=0,1,2$ and with\n\\begin{equation}\\label{eq:cj}\nc_j = \\int_{(m_1+m_2)^2}^{(m_i-m_f)^2} \\frac{b\\sqrt{s} \\, C_j}{128 \\pi^3 m_i^2 \\left(m_i^2 - m_f^2 + s\\right)} ds.\n\\end{equation}\nFrom Eq.~\\eqref{eq:Def-T012} it is easy to show that $T_2 = 3 \\left(1\/2 -\nT_0\\right)$ which also ensures that integration over $\\cos\\theta$ on\nEq.~\\eqref{eq:Gen-ang-dist} is equal to $1$. It is interesting to note that the\nangular distribution of Eq.~\\eqref{eq:Gen-ang-dist} can be written in terms of\nthe orthogonal Legendre polynomials of $\\cos\\theta$ as well,\n\\begin{equation}\\label{eq:Gen-ang-dist-Legendre}\n\\frac{1}{\\Gamma} \\frac{d\\Gamma}{d\\cos\\theta} = \\sum_{i=0}^{2} \\langle G^{(i)} \\rangle P_i \\left(\\cos\\theta\\right).\n\\end{equation}\nHere we have followed the notation of Ref.~\\cite{Gratrex:2015hna} which also\nanalyzes decays of the type $P_i \\to P_f f_1 f_2$, with only leptons for\n$f_{1,2}$, in a model-independent manner but using a generalized helicity\namplitude method. The observables $\\langle G^{(i)} \\rangle$ of\nEq.~\\eqref{eq:Gen-ang-dist-Legendre} are related to $T_0$, $T_1$ and $T_2$ of\nEq.~\\eqref{eq:Gen-ang-dist} as follows,\n\\begin{subequations}\n\\begin{align}\n\\langle G^{(0)} \\rangle &= T_0 + T_2\/3 = 1\/2,\\\\%\n\\langle G^{(1)} \\rangle &= T_1,\\\\%\n\\langle G^{(2)} \\rangle &= 2 T_2\/3.\n\\end{align}\n\\end{subequations}\nThese angular observables $\\langle G^{(i)} \\rangle$'s can be obtained by using\nthe method of moments \\cite{Gratrex:2015hna, Beaujean:2015xea}. Another\nimportant way to describe the normalized angular distribution is by using a flat\nterm $F_H\/2$ and the forward-backward asymmetry $A_{FB}$ \\cite{AngDist:Hiller}\nas follows,\n\\begin{equation}\\label{eq:Gen-ang-dist-expt}\n\\frac{1}{\\Gamma} \\frac{d\\Gamma}{d\\cos\\theta} = \\frac{1}{2} F_H + A_{FB}\n\\cos\\theta + \\frac{3}{4} \\left(1-F_H\\right) \\left(1 - \\cos^2\\theta\\right).\n\\end{equation} \nThis form of the angular distribution has also been used in the experimental\ncommunity \\cite{AngDist:Expt} in the study of $B \\to K \\ell^+ \\ell^-$. The parameters $F_H$ and $A_{FB}$ are related to $T_0$, $T_1$ and $T_2$ as follows,\n\\begin{subequations}\n\\begin{align}\nF_H &= 2 \\left( T_0 + T_2 \\right) = 3 - 4 T_0,\\\\%\nA_{FB} &= T_1.\n\\end{align}\n\\end{subequations}\nThus we have shown that Eqs.~\\eqref{eq:Gen-ang-dist},\n\\eqref{eq:Gen-ang-dist-Legendre} and \\eqref{eq:Gen-ang-dist-expt} are equivalent\nto one another. In this paper, we choose to work using the normalized angular\ndistribution in terms of $T_0$, $T_1$ and $T_2$ as shown in\nEq.~\\eqref{eq:Gen-ang-dist}. This is because the terms $T_0$, $T_1$ and $T_2$\ncan be easily determined experimentally by using the $t$-vs-$u$ Dalitz plot\nwhich does not depend on any specific frame of reference. This Dalitz plot can\nbe easily divided into four segments $I$, $II$, $III$ and $IV$ as shown in\nFig.~\\ref{fig:Dalitz-plot-region}. The segments are decided as follows,\n\\begin{center}\n\\begin{tabular}{lcl}\nSegment $I$ & : & $-1 \\leqslant \\cos\\theta \\leqslant -0.5$,\\\\%\nSegment $II$ & : & $-0.5 < \\cos\\theta \\leqslant 0$,\\\\%\nSegment $II$ & : & $0 < \\cos\\theta \\leqslant 0.5$,\\\\%\nSegment $IV$ & : & $0.5 < \\cos\\theta \\leqslant 1$.%\n\\end{tabular}\n\\end{center}\n\\begin{figure}[hbtp]\n\\centering%\n\\includegraphics[scale=0.8]{fig_Dalitz_plot_region.pdf}%\n\\caption{Two examples depicting the variation of $\\cos\\theta$ in the interior\n\tregion of the $t$-vs-$u$ Dalitz plot. The interior of the Dalitz plot can be\n\tdivided into four segments, $I$, $II$, $III$ and $IV$, as shown here.}%\n\\label{fig:Dalitz-plot-region}\n\\end{figure}\nThe terms $T_0$, $T_1$ and $T_2$ can thus be expressed in terms of the following\nasymmetries,\n\\begin{subequations}\\label{eq:T012}\n\\begin{align}\nT_0 &= - \\frac{1}{6} \\left( \\frac{N_I - 7 \\left( N_{II} + N_{III}\\right) +\n\tN_{IV}}{N_I + N_{II} + N_{III} + N_{IV}} \\right),\\\\%\nT_1 &= \\frac{\\left( N_I + N_{II} \\right) - \\left( N_{III} + N_{IV} \\right)}{N_I\n\t+ N_{II} + N_{III} + N_{IV}},\\\\%\nT_2 &= 2 \\left( \\frac{N_I - \\left( N_{II} + N_{III}\\right) + N_{IV}}{N_I +\n\tN_{II} + N_{III} + N_{IV}} \\right),\n\\end{align}\n\\end{subequations}\nwhere $N_j$ denotes the number of events contained in the segment $j$. Since the\n$t$-vs-$u$ Dalitz plot does not depend on the frame of reference, we need not\nconstraint ourselves to the Gottfried-Jackson frame of Fig.~\\ref{fig:GJ-frame}\nand can work in the laboratory frame as well. Furthermore, we can use the\nexpressions in Eq.~\\eqref{eq:T012} to search for NP.\n\n\n\\section{Illustrating the effects of new physics on the angular distribution}\\label{sec:example}\n\n\\subsection{Classification of the \\texorpdfstring{$P_i \\to P_f f_1 f_2$}{Pi -> Pf + f1 + f2} decays}%\nIt should be emphasized that for our methodology to work, we need to know the\nangle $\\theta$ in the Gottfried-Jackson frame, or equivalently the $t$-vs-$u$\nDalitz plot, which demand that 4-momenta of the final particles be fully known.\nUsually, the 4-momenta of the initial and final pseudo-scalar mesons are\ndirectly measured experimentally. However, depending on the detection\npossibilities of $f_1$ and $f_2$ we can identify three distinct scenarios for\nour process $P_i \\to P_f f_1 f_2$. We introduce the notations $f_i^{\\textrm{\\ding{51}}}$\nand $f_i^{\\textrm{\\ding{55}}}$ to denote whether the fermion $f_i$ gets detected\n(\\textrm{\\ding{51}}) or not (\\textrm{\\ding{55}}) by the detector. Using this notation the three\nscenarios are described as follows.\n\\begin{enumerate}\n\\item[(S1)] $P_i \\to P_f + f_1^{\\textrm{\\ding{51}}} + f_2^{\\textrm{\\ding{51}}} \\equiv P_f +\n\\textrm{`visible'}$. Here both $f_1$ and $f_2$ are detected, e.g.\\ when $f_1 f_2\n= \\ell^-\\ell^+$ or $\\ell \\bar{\\ell'}$.%\n\\item[(S2)] $P_i \\to\n\\begin{Bmatrix}\nP_f + f_1^{\\textrm{\\ding{51}}} + f_2^{\\textrm{\\ding{55}}}\\\\%\nP_f + f_1^{\\textrm{\\ding{55}}} + f_2^{\\textrm{\\ding{51}}}\n\\end{Bmatrix} \\equiv P_f + \\textrm{`visible'} + \\text{`invisible'}$. Here either\n$f_1$ or $f_2$ gets detected, e.g.\\ when $f_1 f_2 = \\ell \\nu_{\\ell}$, $\\ell\n\\nu_S$, $\\ell f^{DM}$, $\\ell f^{LLP}$.%\n\\item[(S3)] $P_i \\to P_f + f_1^{\\textrm{\\ding{55}}} + f_2^{\\textrm{\\ding{55}}} \\equiv P_f +\n\\textrm{`invisible'}$. Here neither $f_1$ nor $f_2$ gets detected, e.g.\\ when\n$f_1 f_2 = \\nu_{\\ell}\\ensuremath{\\overline{\\nu}}_{\\ell}$, $\\nu_{\\ell}\\ensuremath{\\overline{\\nu}}_S$, $\\nu_S\\ensuremath{\\overline{\\nu}}_{\\ell}$,\n$\\nu_S\\ensuremath{\\overline{\\nu}}_S$, $f^{DM} \\bar{f}^{DM}$, $f_1^{DM} f_2^{DM}$, $f_1^{LLP}\nf_2^{LLP}$ etc.\n\\end{enumerate}\nIt should be noted that the above classification is based on our existing\nexperimental explorations. What is undetected today might get detected in future\nwith advanced detectors. In such a case we can imagine that, in future, the\nmodes grouped in S2 might migrate to S1 and those in S3 might be grouped under\nS2. Below we explore each of the above scenarios in more details.\n\n\\subsection{Exploration of new physics effects in each scenario}\n\nThe first scenario (S1) is an experimenter's delight as in this case all final\n4-momenta can be easily measured and the $t$-vs-$u$ Dalitz plot can be obtained.\nHere, our methodology can be used to look for the possible signature of new\nphysics in rare decays such as $B \\to D \\ell^- \\ell^+$ (which can be found in\n\\cite{Kim:2016zbg}) or study the nature of new physics contributing to\nlepton-flavor violating processes such as $B \\to P \\ell^{\\pm} \\ell^{\\prime\\mp}$\nwhere $P=\\pi,K,D$, $\\ell\\neq\\ell'$ and $\\ell,\\ell'=e,\\mu,\\tau$. Let us consider\na few NP possibilities mediating this lepton-flavor violating decay. There is no\ncontribution within the SM to such decays. Therefore, all contribution to these\ndecays comes from NP alone. It is very easy to note that for the decay $B \\to P\n\\ell^{-} \\ell^{\\prime+}$, from Eqs.~\\eqref{eq:gen-C012} and\n\\eqref{eq:Gen-ang-dist} we get,\n\\begin{equation}\\label{eq:NPinB2PLl}\n\\frac{1}{\\Gamma} \\frac{d\\Gamma}{d\\cos\\theta} = \n\\begin{cases}\n\\dfrac{1}{2}, & \\left(\\parbox{0.23\\linewidth}{\\textrm{\\centering only scalar or pseudo-scalar interaction}}\\right)\\\\[5mm]%\nT_0 + T_2 \\cos^2\\theta, & \\left(\\parbox{0.23\\linewidth}{\\textrm{\\centering only tensorial interaction}}\\right)\\\\[3mm]%\nT_0 + T_1 \\cos\\theta + T_2 \\cos^2\\theta, & \\left(\\parbox{0.23\\linewidth}{\\textrm{\\centering only vector or axial-vector interaction}}\\right)\n\\end{cases}\n\\end{equation}\nwhere $T_2 = 3\\left(1\/2 - T_0\\right)$ with the quantities $T_0$, $T_1$ and $T_2$\nbeing easily obtainable from the Dalitz plot distribution by using\nEq.~\\eqref{eq:T012}. It is clear from Eq.~\\eqref{eq:NPinB2PLl} that scalar or\npseudo-scalar interaction would give rise to a uniform (or constant) angular\ndistribution, while tensorial interaction gives a non-uniform distribution which\nis symmetric under $\\cos\\theta \\leftrightarrow -\\cos\\theta$ and for this $T_0\n\\leqslant 1\/2$. On the other hand vector or axial-vector interaction can only be\ndescribed by the most general form of the angular distribution, with its\nsignature being $T_1 \\neq 0$. Nevertheless, if vector or axial-vector\ninteraction contributes to the flavor violating processes $B \\to P \\ell^{-}\n\\ell^{\\prime+}$, it is important to note that $T_1 \\propto \\left(m_{\\ell}^2 -\nm_{\\ell'}^2\\right)$, where $m_{\\ell}$, $m_{\\ell'}$ denote the masses of the\ncharged leptons $\\ell^-$ and $\\ell^{\\prime+}$ respectively. Therefore, we should\nobserve an increase in the value of $T_1$ when going from $B \\to P \\mu^- e^+$ to\n$B \\to P \\tau^- \\mu^+$ to $B \\to P \\tau^- e^+$. This would nail down the vector\nor axial vector nature of the NP, if it is the only NP contributing to these\ndecays. Thus far we have analyzed the first scenario (S1) in which the relevant\ndecays can be easily probed with existing detectors.\n\nThe second scenario (S2) can also be studied experimentally with existing\ndetectors. In this case, the missing 4-momentum can be fully deduced using\nconservation of 4-momentum. Thus the $t$-vs-$u$ Dalitz plot can readily be\nobtained. Using our methodology the signatures of NP can then be extracted. One\npromising candidate for search for NP in this kind of scenario is in the decay\n$B \\to P \\ell N$ where $P=\\pi$, $K$ or $D$ and $N$ can be an active neutrino\n($\\nu_{\\ell}$) or sterile neutrino ($\\nu_S$) or a neutral dark fermion\n($f^{DM}$) or a long lived neutral fermion ($f^{LLP}$) which decays outside the\ndetector. These S2 decay modes offer an exciting opportunity for study of NP\neffects.\n\nThe third scenario (S3), which has the maximum number of NP possibilities, is\nalso the most challenging one for the current generation of experimental\nfacilities, due to lack of information about the individual 4-momentum of $f_1$\nand $f_2$. This implies that we can not do any angular analysis for these kind\nof decays unless by some technological advancement such as by using displaced\nvertex detectors\\footnote{There are many existing proposals for such displaced\n\tvertex studies from other theoretical and experimental considerations (see\n\tRefs.~\\cite{DV:Theory,DV:Experiments} and references contained therein for\n\tfurther information). } we can manage to make measurement of the 4-momentum or\nthe angular information of at least one of the final fermions. Getting 4-momenta\nof both the fermions would be ideal, but knowing 4-momentum of either one of\nthem would suffice for our purpose. We are optimistic that the advancement in\ndetector technology would push the current S3 decay modes to get labelled as S2\nmodes in the foreseeable future. It is important to note that once the current\nS3 modes enter the S2 category, we can cover the whole spectrum of NP\npossibilities in the $P_i \\to P_f f_1 f_2$ decays. Below we make a comprehensive\nexploration of NP possibilities in the generalized S2 decay modes, which\nincludes the current S2 and S3 modes together.\n\n\\subsection{Probing effects of new physics in the (S2)\\\\and generalized (S2)\tscenarios}\n\nIn the generalized S2 (GS2) scenario we have decays of the type $P_i \\to\n\\begin{Bmatrix}\nP_f + f_1^{\\textrm{\\ding{51}}} + f_2^{\\textrm{\\ding{55}}}\\\\%\nP_f + f_1^{\\textrm{\\ding{55}}} + f_2^{\\textrm{\\ding{51}}}\n\\end{Bmatrix} \\equiv P_f + \\textrm{`visible'} + \\text{`invisible'}$, where the detected (\\textrm{\\ding{51}}) or undetected (\\textrm{\\ding{55}}) nature is not constrained by our existing detector technology. In some cases, even with advanced detectors, either of the fermions $f_1$, $f_2$ might not get detected simply because its direction of flight lies outside the finite detector coverage, especially when the detector is located farther from the place of origin of the particle. Such possibilities are also included here. As noted before, measuring the 4-momentum of either of the final fermions would suffice to carry out the angular analysis following our approach. \n\nIn this context let us analyze the following decays.\n\\begin{enumerate}\n\\item[(i)] S2 decay: $B \\to P\\ell^- f^{\\textrm{\\ding{55}}}$ where $P$ can be $\\pi$ or\n$D$ and $f^{\\textrm{\\ding{55}}}$ is a neutral fermion. In the SM this process is\nmediated by $W^-$ boson and we have $f^{\\textrm{\\ding{55}}} = \\ensuremath{\\overline{\\nu}}_{\\ell}$. Presence\nof NP can imply $f^{\\textrm{\\ding{55}}}$ being a sterile neutrino $\\nu_S$ or a fermionic\ndark matter particle $f^{DM}$ or a long lived fermion $f^{LLP}$, with additional\nnon-SM interactions.%\n\\item[(ii)] GS2 decay: $B \\to K f_1^{\\textrm{\\ding{51}}} f_2^{\\textrm{\\ding{55}}}$ where\n$f_1^{\\textrm{\\ding{51}}}$ and $f_2^{\\textrm{\\ding{55}}}$ are both neutral fermions. In the SM\nthis process is mediated by $Z^0$ boson and we have $f_1 f_2 = \\nu_{\\ell}\n\\ensuremath{\\overline{\\nu}}_{\\ell}$. However, in case of NP contribution we can get pairs of sterile\nneutrinos or fermionic dark matter or fermionic long lived particles etc.\\ along\nwith nonstandard interactions as well. Here we are assuming that either of the\nfinal neutral fermions leaves a displaced vertex signature in an advanced\ndetector so that its 4-momentum or angular information could be obtained.%\n\\end{enumerate}\n\n\\subsubsection{New physics effects in the S2 decay \\texorpdfstring{$B \\to P\\ell^- f^{\\textrm{\\ding{55}}}$}{B -> P + l- + fX}}\nAnalyzing the $B \\to P\\ell^- f^{\\textrm{\\ding{55}}}$ decay in the SM we find that only\nvector and axial vector currents contribute and $F_A^{\\pm} = -F_V^{\\pm}$ while\nother form factors are zero. Also considering the anti-neutrino to be massless,\ni.e.\\ $m_2 =0$ we find that \n\\begin{align*}\na_t &= m_{\\ell}^2 + m_P^2 + \\left(s + m_{\\ell}^2\\right) \\left(m_B^2 - m_P^2 -\ns\\right)\/(2s),\\\\%\na_u &= m_P^2 + \\left(s - m_{\\ell}^2\\right) \\left(m_B^2 - m_P^2 -\ns\\right)\/(2s),\\\\%\nb &= \\left(s-m_{\\ell}^2\\right) \\sqrt{\\lambda\\left(s, m_B^2, m_P^2\\right)}\/(2s),\n\\end{align*}\nwhere $m_{\\ell}$, $m_P$ and $m_B$ denote the masses of the charged lepton\n$\\ell^-$, mesons $P$ and $B$ respectively. Substituting these information in\nEqs.~\\eqref{eq:SM-C012} and in Eq.~\\eqref{eq:gen-angular-dist} we get,\n\\begin{equation}\\label{eq:B2Plnu-dist-gen}\n\\frac{d^2\\Gamma^{\\textrm{SM}}}{ds \\, d\\cos\\theta} = \\frac{b\\sqrt{s} \\left(\n\tC_0^{\\textrm{SM}} + C_1^{\\textrm{SM}} \\cos\\theta + C_2^{\\textrm{SM}}\n\t\\cos^2\\theta \\right)}{128 \\, \\pi^3 \\, m_B^2 \\left(m_B^2 - m_P^2 + s \\right)},\n\\end{equation}\nwhere\n\\begin{subequations}\\label{eq:C012-in-B2Plnu}\n\\begin{align}\nC_0^{\\text{SM}} =& 4 \\Bigg( \\modulus{\\left(F_V^+\\right)_{\\text{SM}}}^2 \\bigg(\n\\lambda\\left(s, m_B^2, m_P^2\\right) - m_{\\ell}^2 \\left(s - 2 \\left(m_B^2 -\nm_P^2\\right) \\right) \\nonumber\\\\%\n& \\hspace{2cm} - m_{\\ell}^4 \\left(m_B^2 - m_P^2\\right)^2\/s^2 \\bigg)\n\\nonumber\\\\%\n& \\quad + \\modulus{\\left(F_V^-\\right)_{\\text{SM}}}^2 m_{\\ell}^2 \\left( s -\nm_{\\ell}^2 \\right) \\nonumber\\\\%\n& \\quad + 2 \\Re\\left( \\left(F_V^+\\right)_{\\text{SM}}\n\\left(F_V^-\\right)_{\\text{SM}}^* \\right) m_{\\ell}^2 \\left(m_B^2 - m_P^2\\right)\n\\left(1- \\frac{m_{\\ell}^2}{s}\\right) \\Bigg),\\\\%\nC_1^{\\text{SM}} =& 16 m_{\\ell}^2 b \\Bigg( \\left(\\frac{m_B^2 - m_P^2}{s}\\right)\n\\modulus{\\left(F_V^+\\right)_{\\text{SM}}}^2 + \\Re\\left(\n\\left(F_V^+\\right)_{\\text{SM}} \\left(F_V^-\\right)_{\\text{SM}}^* \\right)\n\\Bigg),\\\\%\nC_2^{\\text{SM}} =& - 16 b^2 \\modulus{\\left(F_V^+\\right)_{\\text{SM}}}^2 .\n\\end{align}\n\\end{subequations}\nIt is important to notice that in Eq.~\\eqref{eq:C012-in-B2Plnu} we have many\nterms in the expression for $C_0^{\\textrm{SM}}$ that are proportional to some\npower of the lepton mass, while the entire $C_1^{\\textrm{SM}}$ is directly\nproportional to $m_{\\ell}^2$. If we compare the $m_{\\ell}$ dependent and\n$m_{\\ell}$ independent contributions in $C_0^{\\textrm{SM}}$ we find that the\ndependent terms are suppressed by about a factor of\n$\\mathcal{O}\\left(2m_{\\ell}^2\/m_B^2\\right)$ which is roughly $8\\times 10^{-4}$\nfor muon and $2\\times 10^{-8}$ for electron. Thus we can neglect these\n$m_{\\ell}$ dependent terms in comparison with mass independent terms.\nEquivalently, we can consider the charged leptons such as electron and muon as\nmassless fermions, when compared with the $B$ meson mass scale. In the limit\n$m_{\\ell} \\to 0$ the expression for angular distribution as given in\nEq.~\\eqref{eq:B2Plnu-dist-gen} becomes much simpler,\n\\begin{equation}\n\\frac{d^2\\Gamma^{\\text{SM}}}{ds \\, d\\cos\\theta} = \\frac{b^3\\sqrt{s}}{8 \\, \\pi^3\n\t\\, m_B^2 \\left( m_B^2 - m_P^2 + s \\right)}\n\\modulus{\\left(F_V^+\\right)_{\\text{SM}}}^2 \\sin^2\\theta.\n\\end{equation}\nIndependent of the expression for $\\left(F_V^+\\right)_{\\text{SM}}$, it is easy\nto show that the normalized angular distribution is given by,\n\\begin{equation}\\label{eq:SM-Dist-B2Plnu-massless}\n\\frac{1}{\\Gamma^{\\text{SM}}} \\frac{d\\Gamma^{\\text{SM}}\n}{d\\cos\\theta} = \\frac{3}{4} \\sin^2\\theta,\n\\end{equation}\nwhich implies that $T_0 = 3\/4 = -T_2$, $T_1 = 0$. Since the distribution of\nevents in the Dalitz plot is symmetric under $\\cos\\theta \\leftrightarrow -\n\\cos\\theta$, we have $N_I = N_{IV}$ and $N_{II} = N_{III}$ which automatically\nsatisfies the condition $T_1 = 0$. If we solve $T_0 = 3\/4 = -T_2$, we find that\nthe number of events in the different segments of the Dalitz plot (equivalently\nthe number of events in the four distinct bins of $\\cos\\theta$) are related to\none another by\n\\begin{equation}\\label{eq:SM-bins-B2Plnu}\n\\frac{N_I}{N_{II}} = \\frac{5}{11} = \\frac{N_{IV}}{N_{III}}.\n\\end{equation}\nAny significant deviation from this would imply presence of NP effects. To\nillustrate the effects of NP on the angular distribution in these types of\ndecays, we consider two simple and specific NP possibilities. Here we assume the\ncharged lepton to be massless ($m_{\\ell}=0$) and the undetected fermion\n($f^{\\textrm{\\ding{55}}}$) to have mass $m\\neq 0$.\n\n\\paragraph{\\textbf{Scalar type new physics:}} Considering the simplest scalar type NP\nscenario, with $F_S \\neq 0$, $F_P = F_V^{\\pm} = F_A^{\\pm} = F_{T_1} = F_{T_2} =\n0$, we get\n\\begin{align*}\nC_0^{\\text{NP}} =& 2 \\left(s - m^2\\right) \\modulus{F_S}^2,\\\\%\nC_1^{\\text{NP}} =& 0 = C_2^{\\text{NP}}.\n\\end{align*}\nIn other words, there is no angular dependence at all here, i.e.\\\n\\begin{equation*}\n\\frac{d^2\\Gamma^{\\text{NP}}}{ds \\, d\\cos\\theta} = \\frac{b\\sqrt{s}}{64 \\, \\pi^3 \\,\n\tm_B^2 \\left(m_B^2 - m_P^2 + s \\right)} \\left(s - m^2\\right) \\modulus{F_S}^2,\n\\end{equation*}\nwhere $b = \\left(s-m^2\\right) \\sqrt{\\lambda\\left(s,m_B^2,m_P^2\\right)}\/(2s)$ and\n$m^2 \\leqslant s \\leqslant \\left(m_B-m_P\\right)^2$. If we do the integration\nover $s$, then the normalized angular distribution is given by,\n\\begin{equation*}\n\\frac{1}{\\Gamma^{\\text{NP}}} \\frac{d\\Gamma^{\\text{NP}}}{d\\cos\\theta} =\n\\frac{1}{2}.\n\\end{equation*}\nIn fact, if such a new physics were present, our observation of $B \\to P +\n\\ell^- + f^{\\textrm{\\ding{55}}}$ would have the following angular distribution,\n\\begin{equation*}\n\\frac{d\\Gamma}{d\\cos\\theta} = \\Gamma^{\\text{SM}} \\left(\\frac{3}{4} \\sin^2\\theta\n+ \\frac{1}{2} \\epsilon_0 \\right),\n\\end{equation*}\nwhere we have parametrized the new physics contribution in terms of $\\epsilon_0$,\n\\begin{equation*}\n\\epsilon_0 = \\Gamma^{\\text{NP}}\/\\Gamma^{\\text{SM}}.\n\\end{equation*}\nDoing integration over $\\cos\\theta$ we get,\n\\begin{equation*}\n\\Gamma = \\Gamma^{\\text{SM}} \\left(1+\\epsilon\\right) = \\Gamma^{\\text{SM}} + \\Gamma^{\\text{NP}}.\n\\end{equation*}\nThis implies \n\\begin{equation}\\label{eq:Scalar-NP-Dist-B2Plnu}\n\\frac{1}{\\Gamma} \\frac{d\\Gamma}{d\\cos\\theta} = \\frac{3\\sin^2\\theta + 2\n\t\\epsilon_0}{4 \\left(1+\\epsilon_0\\right)}.\n\\end{equation}\nThis angular distribution is shown in Fig.~\\ref{fig:Scalar-NP-B2Plnu} where we\nhave varied $\\epsilon_0$ in the range $[0,1]$, i.e.\\ we have allowed the\npossibility that the NP contribution might be as large as that of the SM. It is\ninteresting to find that in Fig.~\\ref{fig:Scalar-NP-B2Plnu} at two specific\nvalues of $\\cos\\theta$ there is no difference between the standard model\nprediction alone and the combination of standard model and new physics\ncontributions. These two points can be easily obtained by equating\nEqs.~\\eqref{eq:SM-Dist-B2Plnu-massless} and \\eqref{eq:Scalar-NP-Dist-B2Plnu},\nand then solving for $\\cos\\theta$ gives us\n\\begin{equation}\n\\cos\\theta = \\pm 1\/\\sqrt{3} \\approx \\pm 0.57735.\n\\end{equation}\nThis corresponds to the angle $\\theta \\approx 54.74^{\\circ}$. At these two\npoints in $\\cos\\theta$, the normalized uni-angular distribution always has the\nvalue $0.5$, even if there is some scalar new physics contributing to our\nprocess under consideration.\n\n\\begin{figure}[hbtp]\n\\centering%\n\\includegraphics[scale=0.8]{fig_Scalar-NP.pdf}%\n\\caption{Normalized uni-angular distribution showing the effect of a scalar new\n\tphysics contribution to $B \\to P \\ell^- f^{\\textrm{\\ding{55}}}$ where we have neglected\n\tthe mass of the charged lepton $\\ell =e,\\mu$. This also shows the normalized\n\tuni-angular distribution showing the effect of a scalar new physics contribution\n\tto $B \\to K f_1^{\\textrm{\\ding{51}}} f_2^{\\textrm{\\ding{55}}}$ considering the $m_1 = m_2$ case\n\tonly.}%\n\\label{fig:Scalar-NP-B2Plnu}\n\\end{figure}\n\nFrom Eq.~\\eqref{eq:Scalar-NP-Dist-B2Plnu} it is clear that despite the scalar NP\neffect, the distribution is still symmetric under $\\cos\\theta \\leftrightarrow\n-\\cos\\theta$, and solving for the number of events in the four segments of the\nDalitz plot (equivalently the four $\\cos\\theta$ bins) we get,\n\\begin{equation}\\label{eq:Scalar-NP}\n\\frac{N_I}{N_{II}} = \\frac{5+8\\epsilon_0}{11 + 8\\epsilon_0} =\n\\frac{N_{IV}}{N_{III}}.\n\\end{equation}\nIt is easy to see that when $\\epsilon=0$ we get back the SM prediction of\nEq.~\\eqref{eq:SM-bins-B2Plnu} as expected.\n\n\\paragraph{\\textbf{Tensor type new physics:}} \n\n\\begin{figure*}[hbtp]\n\\centering%\n\\includegraphics[scale=0.8]{fig_B2Plnu-Tensor-NP.pdf}%\n\\caption{Normalized uni-angular distribution showing the effect of a tensor new\n\tphysics contribution to $B \\to P \\ell^- f^{\\textrm{\\ding{55}}}$ where we have neglected\n\tthe mass of the charged lepton $\\ell=e,\\mu$. These set of plots can also\n\tdescribe the effect of a vector new physics contribution to $B \\to K\n\tf_1^{\\textrm{\\ding{51}}} f_2^{\\textrm{\\ding{55}}}$ when the final fermions are equally\n\tmassive.}%\n\\label{fig:Tensor-NP}\n\\end{figure*}\n\nLet us consider a tensor type of new physics\npossibility in which $F_{T_1} \\neq 0$ and all other form factors are zero. In\nsuch a case we get,\n\\begin{align*}\nC_0^{\\textrm{NP}} &=2 m^2 \\left(s-m^2\\right)\n\\frac{\\lambda\\left(s,m_B^2,m_P^2\\right)}{s} \\modulus{F_{T_1}}^2,\\\\%\nC_1^{\\textrm{NP}} &=0,\\\\%\nC_2^{\\textrm{NP}} &= 2 \\left(s-m^2\\right)^2 \\frac{\\lambda\\left(s, m_B^2,\n\tm_P^2\\right)}{s} \\modulus{F_{T_1}}^2.\n\\end{align*}\nIt is easy to notice that in the limit $m \\to 0$ we have $C_0 \\to 0$ but $C_2\n\\not\\to 0$. If we do the integration over $s$, then the normalized angular\ndistribution is given by,\n\\begin{equation}\n\\frac{1}{\\Gamma^{\\textrm{NP}}} \\frac{d\\Gamma^{\\textrm{NP}}}{d\\cos\\theta} =\nT_0^{\\textrm{NP}} + T_2^{\\textrm{NP}} \\cos^2\\theta,\n\\end{equation}\nwhere $T_2^{\\textrm{NP}} = 3\\left(1\/2-T_0^{\\textrm{NP}}\\right)$ and\n$T_0^{\\textrm{NP}} = 3c_0\/\\left(6c_0 + 2c_2\\right)$ with\n\\begin{equation*}\nc_j = \\int_{m^2}^{\\left(m_B-m_P\\right)^2} \\frac{b\\sqrt{s} \\\n\tC_j^{\\textrm{NP}}}{128 \\pi^3 m_B^2 \\left(m_B^2 - m_P^2 + s\\right)} ds.\n\\end{equation*}\nThus in the limit $m\\to 0$ we have $T_0 = 0$. If such a new physics were\npresent, our observation of $B \\to P \\ell^- f^{\\textrm{\\ding{55}}}$ would have the\nfollowing angular distribution,\n\\begin{equation}\n\\frac{d\\Gamma}{d\\cos\\theta} = \\Gamma^{\\text{SM}} \\left(\\frac{3}{4} \\sin^2\\theta + \\left(T_0^{\\textrm{NP}} + 3 \\left(\\frac{1}{2} - T_0^{\\textrm{NP}}\\right) \\cos^2\\theta\\right) \\epsilon \\right),\n\\end{equation}\nwhere $\\epsilon=\\Gamma^{\\textrm{NP}}\/\\Gamma^{\\textrm{SM}}$ is the NP parameter\nwhich can vary in the range $\\left[0,1\\right]$ denoting the possibility that the\nNP contribution can be as large as that of the SM, and $T_0^{\\textrm{NP}}$ acts\nas a free parameter here which can vary in the range $\\left[0,3\/4\\right]$ in\nwhich $d\\Gamma^{\\textrm{NP}}\/d\\cos\\theta \\geqslant 0$ for all values of\n$\\cos\\theta$. Doing integration over $\\cos\\theta$ we get $\\Gamma =\n\\Gamma^{\\textrm{SM}} \\left(1 + \\epsilon\\right) = \\Gamma^{\\textrm{SM}} +\n\\Gamma^{\\textrm{NP}}$. This implies\n\\begin{equation}\\label{eq:Tensor-NP-Dist-B2Plnu}\n\\frac{1}{\\Gamma} \\frac{d\\Gamma}{d\\cos\\theta} = \\frac{3 + 4 T_0^{\\textrm{NP}}\n\t\\epsilon - 3 \\left(4T_0^{\\textrm{NP}} \\epsilon -2\\epsilon +\n\t1\\right)\\cos^2\\theta}{4\\left(1+\\epsilon\\right)}.\n\\end{equation}\n\nThis angular distribution is shown in Fig.~\\ref{fig:Tensor-NP} in which we have\nconsidered nine values of $T_0^{\\textrm{NP}}$ and varied $\\epsilon$ in the range\n$[0,1]$. It is clearly evident in Fig.~\\ref{fig:Tensor-NP} that\n$T_0^{\\textrm{NP}} = 3\/4$ case is always indistinguishable from the SM case, as\nit should be. Just like the scalar-type new physics case, we observe that there\nare two values of $\\cos\\theta$ at which there is no difference between the SM\nprediction alone and the combination of SM and NP contributions. These two\npoints can be easily computed by equating\nEqs.~\\eqref{eq:SM-Dist-B2Plnu-massless} and \\eqref{eq:Tensor-NP-Dist-B2Plnu},\nand then solving for $\\cos\\theta$ we once again find that,\n\\begin{equation}\n\\cos\\theta = \\pm 1\/\\sqrt{3} \\approx \\pm 0.57735,\n\\end{equation}\nwhich corresponds to the angle $\\theta \\approx 54.74^{\\circ}$. At these two\npoints in $\\cos\\theta$, the normalized uni-angular distribution always has the\nvalue $0.5$, even if there is some tensor new physics contributing to our\nprocess under consideration. It should be noted that these are also the same\npoints where the scalar new physics contribution shows similar effect.\n\nIt is also easy to notice that the angular distribution as given in\nEq.~\\eqref{eq:Tensor-NP-Dist-B2Plnu} is symmetric under $\\cos\\theta\n\\leftrightarrow -\\cos\\theta$, and solving for the number of events in the four\nsegments of the Dalitz plot (equivalently the four $\\cos\\theta$ bins) we get,\n\\begin{equation}\n\\frac{N_{I}}{N_{II}} = \\frac{5 + 2 \\epsilon \\left(7 - 6\n\tT_0^{\\textrm{NP}}\\right)}{11 + 2 \\epsilon \\left(1 + 6 T_0^{\\textrm{NP}}\\right)}\n= \\frac{N_{IV}}{N_{III}}.\n\\end{equation}\nIt is easy to see that when $\\epsilon=0$ or $T_0^{\\textrm{NP}}=3\/4$ we get back\nthe SM prediction of Eq.~\\eqref{eq:SM-bins-B2Plnu} as expected.\n\nFinally we analyze new physics possibilities in the decays belonging to the GS2\ncategory. Due to the very nature of the GS2 decay modes, the following\ndiscussion of NP effects presumes usage of advanced detector technology to get\nangular information.\n\n\\subsubsection{New physics effects in the GS2 decay \\texorpdfstring{$B \\to K f_1^{\\textrm{\\ding{51}}} f_2^{\\textrm{\\ding{55}}}$}{B -> K + f1V + f2X}}\n\nAs mentioned before, the GS2 decay modes are originally part of S3, i.e.\\ it is\nextremely difficult to get angular distribution for these cases unless we\ninnovate on detector technology. Here we consider such a decay mode $B \\to K\nf_1^{\\textrm{\\ding{51}}} f_2^{\\textrm{\\ding{55}}}$ in which both $f_1$, $f_2$ are neutral\nfermions who have evaded, till now, all our attempts to detect them near their\nplace of origin. But probably with displaced vertex detectors or some other\nadvanced detector we could bring at least one of these fermions (say $f_1$)\nunder the purview of experimental study and measure its 4-momentum or angular\ninformation. The missing fermion (which is $f_2$ in our example here) might have\nflied in a direction along which there is no detector coverage. To increase the\nsample size we should include $B \\to K f_1^{\\textrm{\\ding{55}}} f_2^{\\textrm{\\ding{51}}}$ events\nalso, provided we know how to ascertain the particle or anti-particle nature of\n$f_1$ and $f_2$. To illustrate this point, let us consider the possibility $f_1\nf_2 = \\nu_S \\ensuremath{\\overline{\\nu}}_S$. In a displaced vertex detector if we see $\\pi^+ \\mu^-$\nevents, they can be attributed to the decay of $\\nu_S$ and similarly $\\pi^-\n\\mu^+$ events would appear from the decay of $\\ensuremath{\\overline{\\nu}}_S$. In this case, we can\ninfer the angle $\\theta$ by knowing the 4-momentum of either $f_1 = \\nu_S$ or\n$f_2 = \\ensuremath{\\overline{\\nu}}_S$ (see Fig.~\\ref{fig:GJ-frame}). If we find that both $f_1$ and\n$f_2$ leave behind their signature tracks in the detector (i.e.\\\n$f_1^{\\textrm{\\ding{51}}} f_2^{\\textrm{\\ding{51}}}$) it would be the most ideal situation. But\nas we have already stressed before, measuring 4-momenta of either of the\nfermions would suffice for our angular studies.\n\nIn the SM the only contribution to $B \\to K f_1^{\\textrm{\\ding{51}}} f_2^{\\textrm{\\ding{55}}}$\nand $B \\to K f_1^{\\textrm{\\ding{55}}} f_2^{\\textrm{\\ding{51}}}$ would come from $B \\to K\n\\nu_{\\ell} \\ensuremath{\\overline{\\nu}}_{\\ell}$ where as in the case of NP we have a number of\npossibilities that includes sterile neutrinos, dark matter particles, or some\nlong lived particles in the final state, $f_1 f_2 = \\nu_{\\ell} \\ensuremath{\\overline{\\nu}}_S$, $\\nu_S\n\\ensuremath{\\overline{\\nu}}_{\\ell}$, $\\nu_S \\ensuremath{\\overline{\\nu}}_S$, $f^{\\textrm{DM}} \\bar{f}^{\\textrm{DM}}$,\n$f_1^{\\textrm{DM}} f_2^{\\textrm{DM}}$, $f^{\\textrm{LLP}}\n\\bar{f}^{\\textrm{LLP}}$, $f_1^{\\textrm{LLP}} f_2^{\\textrm{LLP}}$\netc.\\footnote{In addition to the new physics possibilities considered here,\n\tthere can be additional contributions to the $B \\to K + \\text{`invisible'}$\n\tdecay, e.g.\\ from SM singlet scalars contributing to the `invisible' part as\n\tdiscussed in Ref.~\\cite{Kim:2009qc}. As is evident, our analysis is instead\n\tfocused on a pair of fermions contributing to the `invisible' part.} One can\nalso consider non-standard neutrino interactions also contributing in these\ncases. To demonstrate our methodology, we shall analyze only a subset of these\nvarious NP possibilities in which $f_1$ and $f_2$ have the same mass, i.e.\\ $m_1\n= m_2 = m$ (say), as this greatly simplifies the calculation. As we shall\nillustrate below we can not only detect the presence of NP but ascertain whether\nit is of scalar type or vector type, for example, by analyzing the angular\ndistribution.\n\nBefore, we go for new physics contributions, let us analyze the SM contribution\n$B \\to K \\nu_{\\ell} \\ensuremath{\\overline{\\nu}}_{\\ell}$. Here only vector and axial-vector currents\ncontributions, and $F_A^{\\pm} = - F_V^{\\pm}$. Also the neutrino and\nanti-neutrino are massless, i.e.\\ $m_1 = 0 = m_2$, which implies $a_t = a_u =\n\\tfrac{1}{2} \\left(m_B^2 + m_K^2 -s\\right)$ and $b = \\tfrac{1}{2}\n\\sqrt{\\lambda\\left(s,m_B^2,m_K^2\\right)}$, where $m_B$ and $m_K$ denote the\nmasses of $B$ and $K$ mesons respectively. Substituting these information in\nEqs.~\\eqref{eq:SM-C012} and in Eq.~\\eqref{eq:gen-angular-dist} we get,\n\\begin{equation}\n\\frac{d^2\\Gamma^{\\text{SM}}}{ds \\, d\\cos\\theta} = \\frac{b^3\\sqrt{s}}{8 \\, \\pi^3 \\, m_B^2\n\t\\left( m_B^2 - m_K^2 + s \\right)} \\modulus{\\left(F_V^+\\right)_{\\text{SM}}}^2\n\\sin^2\\theta.\n\\end{equation}\nIrrespective of the expression for $\\left(F_V^+\\right)_{\\text{SM}}$, it can be\neasily shown that the normalized angular distribution is given by,\n\\begin{equation}\\label{eq:SM-Dist-B2Knn}\n\\frac{1}{\\Gamma^{\\text{SM}}} \\frac{d\\Gamma^{\\text{SM}}\n}{d\\cos\\theta} = \\frac{3}{4} \\sin^2\\theta,\n\\end{equation}\nwhich implies that $T_0 = 3\/4 = -T_2$, $T_1 = 0$. Following the same logic as\nthe one given after Eq.~\\eqref{eq:SM-Dist-B2Plnu-massless}, we find that the\nnumber of events in the different segments of the Dalitz plot (equivalently the\nnumber of events in the four distinct bins of $\\cos\\theta$) are related to one\nanother by,\n\\begin{equation}\\label{eq:SM-bins}\n\\frac{N_I}{N_{II}} = \\frac{5}{11} = \\frac{N_{IV}}{N_{III}}.\n\\end{equation}\nThis sets the stage for us to explore (i) a scalar type and (ii) a vector type\nof NP possibility, with final fermions for which $m_1 = m_2 = m \\neq 0$. \n\n\\paragraph{\\textbf{Scalar type new physics:}}\n\nOnce again we consider the simplest scalar type NP scenario, with $F_S \\neq 0$,\nand other form factors being zero. This leads us to,\n\\begin{align*}\n\tC_0^{\\text{NP}} =& 2 \\left(s - 4m^2\\right) \\modulus{F_S}^2,\\\\%\n\tC_1^{\\text{NP}} =& 0 = C_2^{\\text{NP}}.\n\\end{align*}\nIn other words, there is no angular dependence at all here, i.e.\\\n\\begin{equation}\n\\frac{d^2\\Gamma^{\\text{NP}}}{ds \\, d\\cos\\theta} = \\frac{b\\sqrt{s}}{64 \\, \\pi^3 \\,\n\tm_B^2 \\left(m_B^2 - m_K^2 + s \\right)} \\left(s - 4m^2\\right) \\modulus{F_S}^2,\n\\end{equation}\nwhere $b = \\left(\\sqrt{\\left(s-4m^2\\right)} \\,\n\\sqrt{\\lambda\\left(s,m_B^2,m_K^2\\right)}\\right)\/(2\\sqrt{s})$ and $4m^2 \\leqslant\ns \\leqslant \\left(m_B-m_K\\right)^2$. If we do the integration over $s$, then for\nNP only the normalized angular distribution is given by,\n\\begin{equation*}\n\\frac{1}{\\Gamma^{\\text{NP}}} \\frac{d\\Gamma^{\\text{NP}}}{d\\cos\\theta} =\n\\frac{1}{2}.\n\\end{equation*}\nAssuming such a NP contributing in addition to the SM, the experimentally\nobserved angular distribution can be written as,\n\\begin{equation*}\n\\frac{d\\Gamma}{d\\cos\\theta} = \\Gamma^{\\text{SM}} \\left(\\frac{3}{4} \\sin^2\\theta + \\frac{1}{2} \\epsilon_0 \\right),\n\\end{equation*}\nwhere $\\epsilon_0 = \\Gamma^{\\text{NP}}\/\\Gamma^{\\text{SM}}$ is the new physics\nparameter which can vary in the range $\\left[0,1\\right]$ if we assume the NP\ncontribution to be as large as that from the SM. Doing integration over\n$\\cos\\theta$ we get, $\\Gamma = \\Gamma^{\\text{SM}} \\left(1+\\epsilon_0\\right) =\n\\Gamma^{\\text{SM}} + \\Gamma^{\\text{NP}}$. This implies\n\\begin{equation}\\label{eq:Scalar-NP-Dist-B2Knn}\n\\frac{1}{\\Gamma} \\frac{d\\Gamma}{d\\cos\\theta} = \\frac{3\\sin^2\\theta + 2\n\t\\epsilon_0}{4 \\left(1+\\epsilon_0\\right)}.\n\\end{equation}\nSince Eq.~\\eqref{eq:Scalar-NP-Dist-B2Knn} is identical to\nEq.~\\eqref{eq:Scalar-NP-Dist-B2Plnu}, the angular distribution for this case is\nalso as shown in Fig.~\\ref{fig:Scalar-NP-B2Plnu} where we have varied\n$\\epsilon_0$ in the range $[0,1]$. Once again at two specific values of\n$\\cos\\theta$, namely $\\cos\\theta = \\pm 1\/\\sqrt{3} \\approx \\pm 0.57735$\ncorresponding to the angle $\\theta \\approx 54.74^{\\circ}$, there is no\ndifference between the standard model prediction alone and the combination of\nstandard model and scalar new physics contribution. At these two points in\n$\\cos\\theta$, the normalized uni-angular distribution always has the value\n$0.5$, even if there is some scalar new physics contributing to our process\nunder consideration.\n\nSince the angular distribution as shown in Eq.~\\eqref{eq:Scalar-NP-Dist-B2Knn}\nis fully symmetric under $\\cos\\theta \\leftrightarrow -\\cos\\theta$, the number of\nevents in the four segments of the Dalitz plot (equivalently in the four\n$\\cos\\theta$ bins) satisfy the following relationship,\n\\begin{equation}\\label{eq:Scalar-NP-bins}\n\\frac{N_I}{N_{II}} = \\frac{5+8\\epsilon_0}{11 + 8\\epsilon_0} =\n\\frac{N_{IV}}{N_{III}}.\n\\end{equation}\nIt is easy to see that $\\epsilon_0=0$ gives the SM prediction of\nEq.~\\eqref{eq:SM-bins} as expected.\n\n\\paragraph{\\textbf{Vector type new physics:}}\n\nLet us now discuss another new physics scenario, such as the case of a\nflavor-changing $Z'$ or a dark photon $\\gamma_D$ giving rise to the final pair\nof fermions $f_1 f_2$. We assume that for this kind of new physics scenario,\n$F_V^+ = F_V^{\\text{NP}} \\neq 0$ and other form factors are zero. For this kind\nof new physics we get,\n\\begin{align*}\nC_0^{\\text{NP}} =& 2 \\modulus{F_V^{\\text{NP}}}^2\n\\lambda\\left(s,m_B^2,m_K^2\\right),\\\\%\nC_1^{\\text{NP}} =& 0,\\\\%\nC_2^{\\text{NP}} =& -8 b^2 \\modulus{F_V^{\\text{NP}}}^2,\n\\end{align*}\nwhere $b = \\left(\\sqrt{\\left(s-4m^2\\right)} \\,\n\\sqrt{\\lambda\\left(s,m_B^2,m_K^2\\right)}\\right)\/\\left(2\\sqrt{s}\\right)$ and\n$4m^2 \\leqslant s \\leqslant \\left(m_B-m_K\\right)^2$. The angular distribution\nfor the NP alone contribution can, therefore, be written in terms of\n$T_0^{\\textrm{NP}}$ and $T_2^{\\textrm{NP}}$ which are directly proportional to\n$C_0^{\\text{NP}}$ and $C_2^{\\text{NP}}$ respectively. It would lead us to\ndescribe the complete angular distribution in terms of $T_0^{\\text{NP}}$ and\n$\\epsilon=\\Gamma^{\\textrm{NP}}\/\\Gamma^{SM}$ using\nEq.~\\eqref{eq:Tensor-NP-Dist-B2Plnu} and the angular distribution would look\nlike the one shown in Fig.~\\ref{fig:Tensor-NP}. However, it is possible to\ndescribe the effects of NP on the angular distribution using a different set of\nparameters as well. For this we start a fresh with the angular distribution for\nthe NP contribution alone, which in our case is given by\n\\begin{equation*}\n\\frac{d^2\\Gamma^{\\text{NP}}}{ds \\, d\\cos\\theta} = \\frac{b \\modulus{F_V^{\\text{NP}}}^2 \\lambda\\left(s,m_B^2,m_K^2\\right) \\, \\left( s \\sin^2\\theta + 4m^2 \\cos^2\\theta \\right)}{64 \\, \\pi^3 \\, m_B^2 \\left(m_B^2 - m_K^2\n\t+ s \\right) \\sqrt{s}}.\n\\end{equation*}\nDoing integration over $\\cos\\theta$ we obtain,\n\\begin{equation*}\n\\frac{d\\Gamma^{\\text{NP}}}{ds} = \\frac{b \\modulus{F_V^{\\text{NP}}}^2\n\t\\lambda\\left(s,m_B^2,m_K^2\\right)}{64 \\, \\pi^3 \\, m_B^2 \\left(m_B^2 - m_K^2 + s\n\t\\right) \\sqrt{s}} \\left( \\frac{4s + 8m^2}{3} \\right).\n\\end{equation*}\nTherefore, the normalized uni-angular distribution is given by\n\\begin{equation}\\label{eq:Vector-NP-Dist-s-B2Knn}\n\\frac{1}{d\\Gamma^{\\text{NP}}\/ds} \\frac{d^2\\Gamma^{\\text{NP}}}{ds \\, d\\cos\\theta} = \\frac{3}{4} \\left(\\frac{s \\sin^2\\theta + 4m^2 \\cos^2\\theta}{s + 2m^2}\\right).\n\\end{equation}\nIt is interesting to compare this with the standard model expression,\n\\begin{equation}\\label{eq:SM-Dist-s-B2Knn}\n\\frac{1}{d\\Gamma^{\\text{SM}}\/ds} \\frac{d^2\\Gamma^{\\text{SM}}}{ds \\, d\\cos\\theta} = \\frac{3}{4} \\sin^2\\theta.\n\\end{equation}\n\n\\begin{figure*}[hbtp]\n\\centering%\n\\includegraphics[scale=0.8]{fig_B2Knunu-Vector-NP.pdf}%\n\\caption{Normalized uni-angular distribution showing the effect of a vector new\n\tphysics contribution to $B \\to K f_1^{\\textrm{\\ding{51}}} f_2^{\\textrm{\\ding{55}}}$. }%\n\\label{fig:Vector-NP}\n\\end{figure*}\n\nSince the range for $s$ is different in the SM and the NP scenarios, we can not\nadd Eqs.~\\eqref{eq:Vector-NP-Dist-s-B2Knn} and \\eqref{eq:SM-Dist-s-B2Knn}\ndirectly. Carrying out the integration over $s$ we get,\n\\begin{equation*}\n\\frac{d\\Gamma^{\\text{NP}}}{d\\cos\\theta} = \\frac{3}{4} \\Big( \\mathcal{S}\n\\sin^2\\theta + \\mathcal{C} \\cos^2\\theta \\Big),\n\\end{equation*}\nwhere\n\\begin{align*}\n\\mathcal{S} &= \\int_{4m^2}^{(m_B-m_K)^2} \\frac{d\\Gamma^{\\text{NP}}}{ds}\n\\left(\\frac{s}{s+2m^2}\\right) ds,\\\\%\n\\mathcal{C} &= \\int_{4m^2}^{(m_B-m_K)^2} \\frac{d\\Gamma^{\\text{NP}}}{ds}\n\\left(\\frac{4m^2}{s+2m^2}\\right) ds.\n\\end{align*}\nDoing integration over $\\cos\\theta$ we get,\n\\begin{equation*}\n\\Gamma^{\\text{NP}} = \\mathcal{S} + \\mathcal{C}\/2,\n\\end{equation*}\nand hence\n\\begin{equation*}\n\\frac{1}{\\Gamma^{\\text{NP}}} \\frac{d\\Gamma^{\\text{NP}}}{d\\cos\\theta} = \\frac{3\n\t\\left(\\mathcal{S} \\sin^2\\theta + \\mathcal{C} \\cos^2\\theta\\right)}{2\n\t(2\\mathcal{S} + \\mathcal{C})}.\n\\end{equation*}\nFor the SM contribution we know that\n\\begin{equation*}\n\\frac{1}{\\Gamma^{\\text{SM}}} \\frac{d\\Gamma^{\\text{SM}}}{d\\cos\\theta} = \\frac{3}{4} \\sin^2\\theta.\n\\end{equation*}\nNow the uni-angular distribution for the process $B \\to K f_1^{\\textrm{\\ding{51}}}\nf_2^{\\textrm{\\ding{55}}}$ is given by,\n\\begin{equation*}\n\\frac{d\\Gamma}{d\\cos\\theta} =\\frac{3}{4} \\Gamma^{\\text{SM}} \\left( \\left(1 +\\epsilon_s \\right) \\sin^2\\theta + \\epsilon_c \\cos^2\\theta \\right),\n\\end{equation*}\nwhere $\\epsilon_s = \\mathcal{S}\/\\Gamma^{\\text{SM}}$ and $\\epsilon_c =\n\\mathcal{C}\/\\Gamma^{\\text{SM}}$, are the two parameters which describe the\neffect of vector type NP. It is easy to check that,\n\\begin{equation*}\n\\Gamma = \\frac{3}{4} \\Gamma^{\\text{SM}} \\left( \\frac{4}{3} \\left(1+\\epsilon_s\\right) + \\frac{2\\epsilon_c}{3} \\right) = \\Gamma^{\\text{SM}} + \\Gamma^{\\text{NP}}.\n\\end{equation*}\nTherefore, the normalized angular distribution is given by,\n\\begin{equation}\\label{eq:Vector-NP-Dist-B2Knn}\n\\frac{1}{\\Gamma} \\frac{d\\Gamma}{d\\cos\\theta} = \\frac{3 \\left(1 + \\epsilon_s\\right) \\sin^2\\theta + 3\\epsilon_c \\cos^2\\theta}{4 \\left(1+\\epsilon_s\\right) + 2 \\epsilon_c}.\n\\end{equation}\nIt is important to note that, if we consider the mass of the fermion $f$ to be\nzero, i.e.\\ $m=0$, then $\\epsilon_c = 0$, since $\\mathcal{C} =0$. In such a case\nthe uni-angular distribution is given by,\n\\begin{equation*}\n\\frac{1}{\\Gamma} \\frac{d\\Gamma}{d\\cos\\theta} = \\frac{3}{4} \\sin^2\\theta, \\qquad \\left(\\text{here } \\epsilon_c=0\\right)\n\\end{equation*}\nwhich is same as that of the SM case. This is plausible, as in the SM case also\none has $m=0$ for the neutrino mass and only vector and axial-vector currents\ncontribute.\n\nAssuming that the NP contribution can be smaller than or as large as the SM\ncontribution, i.e.\\ $0 \\leqslant \\Gamma^{\\text{NP}} \\leqslant\n\\Gamma^{\\text{SM}}$, we get\n\\begin{equation*}\n0 \\leqslant \\epsilon_s + \\epsilon_c\/2 \\leqslant 1.\n\\end{equation*}\nThus $0 \\leqslant \\epsilon_s \\leqslant 1$ implies that $0 \\leqslant \\epsilon_c\n\\leqslant 2(1-\\epsilon_s)$.\n\nIn Fig.~\\ref{fig:Vector-NP} we have considered nine values of $\\epsilon_s$ and\nvaried $\\epsilon_c$ in the range $[0,2\\left(1-\\epsilon_s\\right)]$, to obtain the\nuni-angular distribution. It is clearly evident in Fig.~\\ref{fig:Vector-NP} that\n$\\epsilon_c=0$ case is always indistinguishable from the SM case, as it should\nbe. Just like the scalar-type new physics case, we observe that at $\\cos\\theta =\n\\pm 1\/\\sqrt{3} \\approx \\pm 0.57735$, there is no difference between the SM\nprediction alone and the combination of SM and NP contributions.\n\nIt is also easy to notice that the angular distribution as given in\nEq.~\\eqref{eq:Vector-NP-Dist-B2Knn} is symmetric under $\\cos\\theta\n\\leftrightarrow -\\cos\\theta$, and solving for the number of events in the four\nsegments of the Dalitz plot (equivalently the four $\\cos\\theta$ bins) we get,\n\\begin{equation}\n\\frac{N_{I}}{N_{II}} = \\frac{5 \\left(1+\\epsilon_s\\right) +\n\t7\\epsilon_c}{11\\left(1+\\epsilon_s\\right) + \\epsilon_c} = \\frac{N_{IV}}{N_{III}}.\n\\end{equation}\nIt is easy to see that when $\\epsilon_c = 0 = \\epsilon_s$ we get back the SM\nprediction of Eq.~\\eqref{eq:SM-bins} as expected.\n\n\\subsection{Discussion}\n\nIt should be noted that our discussions on the types of NP contributions to the\nS2 and GS2 modes, specifically $B \\to P \\ell^- f^{\\textrm{\\ding{55}}}$ and $B \\to K\nf_1^{\\textrm{\\ding{51}}} f_2^{\\textrm{\\ding{55}}}$ respectively, has been fully general. There\nis no complications arising out of hadronic form factors since we have\nconsidered normalized angular distribution. It should be noted that our analysis\nalso does not depend on how large or small the masses of the fermions\n$f,f_{1,2}$ are, as long as they are non-zero.\n\nIt is also very interesting to note that both the scalar and tensor type of NP\nfor the $B \\to P \\ell^- f^{\\textrm{\\ding{55}}}$ decays and both the scalar and vector\ntypes of NP for the $B \\to K f_1^{\\textrm{\\ding{51}}} f_2^{\\textrm{\\ding{55}}}$ decays, exhibit\nsimilar behaviour at $\\cos\\theta = \\pm 1\/\\sqrt{3}$. In order to know the real\nreason behind this we must do a very general analysis. Let us assume that the\nmost general angular distribution for the processes $B \\to P \\ell^-\nf^{\\textrm{\\ding{55}}}$ and $B \\to K f_1^{\\textrm{\\ding{51}}} f_2^{\\textrm{\\ding{55}}}$ is given by\nEq.~\\eqref{eq:Gen-ang-dist}. If we now equate this distribution to the SM\nprediction of Eq.~\\eqref{eq:SM-Dist-B2Plnu-massless} or\nEq.~\\eqref{eq:SM-Dist-B2Knn}, and solve for $\\cos\\theta$ after substituting\nEq.~\\eqref{eq:Def-T012} we find that,\n\\begin{equation}\\label{eq:costheta-gen-sol}\n\\cos\\theta = \\frac{-c_1 \\pm \\sqrt{c_1^2 + 3\n\t\t\\left(c_0+c_2\\right)^2}}{3\\left(c_0+c_2\\right)},\n\\end{equation}\nwhere the $c_j$'s (for $j=0,1,2$) are obtained from Eq.~\\eqref{eq:cj} with\nappropriate substitutions of masses and form factors. Thus\nEq.~\\eqref{eq:costheta-gen-sol} is the most general solution that we can get for\nthe two specific values of $\\cos\\theta$. However, let us look at the specific\ncase when $c_1=0$. Only in this situation do we get\n\\begin{equation}\n\\cos\\theta = \\pm 1\/\\sqrt{3}.\n\\end{equation}\nNow it is clear that since, in both the scalar and tensor type of NP\nconsiderations for the $B \\to P \\ell^- f^{\\textrm{\\ding{55}}}$ decays and in both the\nscalar and vector types of NP considerations for the $B \\to K f_1^{\\textrm{\\ding{51}}}\nf_2^{\\textrm{\\ding{55}}}$ decays, the angular distribution did not have any term\ndirectly proportional to $\\cos\\theta$ (i.e.\\ $c_1=0$), we obtained the same\n$\\cos\\theta = \\pm 1\/\\sqrt{3}$ result in both the cases. Therefore, if the\nobserved normalized uni-angular distribution does not have the value $0.5$ at\n$\\cos\\theta = \\pm 1\/\\sqrt{3}$, it implies that $c_1 \\neq 0$.\n\nAnother interesting aspect of the two specific NP contributions we have\nconsidered, is that from Figs.~\\ref{fig:Scalar-NP-B2Plnu}, \\ref{fig:Tensor-NP}\nand \\ref{fig:Vector-NP} one can clearly see that the vector and tensor types of\nNP can accommodate a much larger variation in the angular distribution than the\nscalar type NP. However, there is also a certain part of the angular\ndistribution for which both scalar and vector (or tensor) types of NP give\nidentical results. This happens when\n\\begin{equation}\n\\epsilon_0 = \\frac{3\\epsilon_c}{2\\left(1 + \\epsilon_s - \\epsilon_c\\right)} =  \\frac{\\epsilon \\left( 3 - 4T_0^{\\textrm{NP}} \\right)}{1 - \\epsilon \\left( 2 - 4T_0^{\\textrm{NP}} \\right)}.\n\\end{equation}\nIn order for $\\epsilon_0$ to vary in the range $[0,1]$ we find that (i) for\n$0\\leqslant \\epsilon_s \\leqslant 1$ we have $0 \\leqslant \\epsilon_c \\leqslant\n2\\left(1+\\epsilon_s\\right)\/5$ and (ii) for $0 \\leqslant \\epsilon \\leqslant 1$ we\nhave $\\frac{1}{2} \\leqslant T_0^{\\textrm{NP}} \\leqslant \\frac{3}{4}$. In these\nspecific regions, therefore, it would not be possible to clearly distinguish\nwhether scalar or vector or tensor type NP is contributing to our process under\nconsideration. Nevertheless, our approach can be used to constraint these NP\nhypothesis without any hadronic uncertainties.\n\n\\section{Conclusion}\\label{sec:conclusion}\n\nWe have shown that all NP contributions to three-body semi-hadronic decays of\nthe type $P_i \\to P_f f_1 f_2$, where $P_{i(f)}$ denotes appropriate initial\n(final) pseudo-scalar meson and $f_{1,2}$ are a pair of fermions, can be\ncodified into the most general Lagrangian which gives rise to a very general\nangular distribution. The relevant NP information can be obtained by using\nvarious angular asymmetries, provided at least one of the final pair of fermions\nhas some detectable signature, such as a displaced vertex, in the detector.\nDepending on the detection feasibility of the final fermions we have grouped the\n$P_i \\to P_f f_1 f_2$ decays into three distinct categories: (i) S1 where both\n$f_1$ and $f_2$ are detected, (ii) S2 where either $f_1$ or $f_2$ gets detected,\nand (ii) S3 where neither $f_1$ nor $f_2$ gets detected. We consider the\npossibility that with advancement in detector technology S3 decays could, in\nfuture, be grouped under S2 category. We analyze some specific NP scenarios in\neach of these categories to illustrate how NP affects the angular distribution.\nSpecifically we have analyzed (a) lepton-flavor violating S1 decay $B \\to P\n\\ell^- \\ell'^+$ (with $P = \\pi, K, D$ and $\\ell,\\ell'=e,\\mu,\\tau$) showing\nangular signatures of all generic NP possibilities, (b) S2 decays of the type $B\n\\to P \\ell^- f$ (where $f$ is not detected in the laboratory) showing the effect\nof a scalar type and a tensor type NP on the angular distribution, and finally\n(c) S3 decays (more correctly generalized S2 decays) of the type $B \\to K f\n\\bar{f}$ (where either $f$ or $\\bar{f}$ gets detected in an advanced detector)\nshowing the effects of a scalar type and a vector type NP on the angular\ndistribution. The effects on the angular distribution can be easily estimated\nfrom Dalitz plot asymmetries. The signatures of NP in angular distribution are\ndistinct once the process is chosen carefully. Moreover, as shown in our\nexamples it can be possible to do the identification and quantification of NP\neffects without worrying about hadronic uncertainties. We are optimistic that\nour methodology can be put to use in LHCb, Belle II in the study of appropriate\n$B$ meson decays furthering our search for NP.\n\n\n\n\\acknowledgments \n\nThis work was supported in part by the National Research Foundation of Korea\n(NRF) grant funded by the Korean government (MSIP) (No.2016R1A2B2016112) and\n(NRF-2018R1A4A1025334). This work of D.S. was also supported (in part) by the\nYonsei University Research Fund (Post Doc. Researcher Supporting Program) of\n2018 (project no.: 2018-12-0145).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzcgdw b/data_all_eng_slimpj/shuffled/split2/finalzzcgdw
new file mode 100644
index 0000000000000000000000000000000000000000..0d67e68a73bcc508846a8293360af7c4bfd6b10a
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzcgdw
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\nIn this somewhat long introduction,we start from general analogies between the theory of Artin Presentations (AP Theory) with modern physics, before giving more specific ones related to the actual Clay YM problem,\\cite {JW}, \\cite{D}, \\cite{V}, \\cite{F},in section 3. \n\nIn the following $X^4$ will denote a connected,compact,smooth,simply-connected 4-manifold with a connected boundary,$\\partial X^4$; if $\\partial X^4$ = $S^3$,we will also denote the so determined closed manifold by $X^4$.Although all the $X^4$ we discuss are so-called '2-handle bodies' ,(see, e.g., \\cite{GS})\nthey form a very large class of smooth 4-manifolds: every connected, closed,orientable 3-manifold \nappears as a $\\partial X^4$ and at least every complex elliptic surface (e.g.the Kummer surface) is \npart of this class, \\cite{CW}.(It is unknown whether every {\\it closed}, smooth,simply-connected 4-manifold can be so obtained, although it seems very likely that this is so).\\cite{GS},p.344.\n\nIn AP Theory (see ahead) such a $X^4$ is already determined by a certain type of (group) presentation, an Artin Presentation, $r$, on $n$ generators with $n$ relations, of the fundamental group of its boundary,which we denote by $\\pi(r)$ and \nwe write, $X^4$ = $W^4(r)$.\n\nEven at this early stage,this relation of pure discrete group theory with 4D smooth structures has physical relevance:\n\nIt is the most radical,mathematically rigorous,universal form of $3+1$ holography:\nDiscrete Group theory on the boundary of $X^4$ determines the whole smooth 4-manifold $X^4$,up to diffeomorphism.\n\nFor the meaning and importance in physics of holography ('t Hooft, Susskind, Maldacena,\\dots,) see \\cite{Bo}.\n\nThe discrete,purely group-theoretic Artin presentation $r$ is a {\\it hologram} of the {\\it  smooth} 4D spacetime universe $W^4(r)$.Compare to \\cite{M2},p.63.\n\n\nThis is the more far reaching AP-analogue of Witten's most general version of  holography in \nmodern physics:\n\n{\\it \"Gravity on the bulk is built from Gauge Theory on the boundary\"},see p.13\nof \\cite{Wi1}.\n\n{\\it This analogy of the rigorous, radical, universal AP-holography with,e.g.the more heuristic, restrictive, AdS\/CFT holography,(\\cite{M1},\\cite{M2}, \\cite{Re},\\cite{Wi2}, \\cite{SW}, \\cite{S1},\\dots), is the prototype of the analogies exhibited and studied in this paper.}\n\nWe can say: the compact,smooth $3+1$ spacetime universe $W^4(r)$ {\\it emerges} from the discrete group-theoretic 'vacuum', via the purely group theoretic Artin presentation $r$; in other words,4D gravity,in its most abstract Einsteinian sense,emerges purely group-theoretically,{\\it not} metrically,and it does so in a non-infinitesimal,non-local manner.Compare to \\cite{AJL},\\cite {Lo},\\cite{Se},\\cite{Wi1},p.5,\\cite{Sm}, \\cite{M4}.\n\nThe 4D smooth {\\it 'continuum picture arises from this fundamental discreteness'}, \\cite {As}, p.174.\n\nThis {\\it 'immense gauge symmetry'} (\\cite{G},p.A98) already makes AP Theory a candidate for\n a topological  {\\it 'quantum gauge theory in four dimensional space-time'}, \\cite{JW},p.3, although it is, apriori, 'exterior',not confined to the Differential Geometry of a single fixed  manifold,in the same sense\nthat Cobordism Theory is not so confined. In AP Theory,gauge theory has exterior,autonomous, {\\it intrinsic}, absolute meaning,not related to any particular Lie group ,just as Cobordism Theory has exterior,intrinsic,absolute homological meaning.We note here that,nevertheless, cobordism theory has led to the solutions of important 'internal' problems on a given fixed manifold (Hirzebruch formula,Atiyah-Singer theorems,\nSteenrod's conjecture,etc. \\cite {A1}, \\cite{Su}).\n\nAs has been pointed out,even the more classical AdS\/CFT holographic principle {\\it 'calls into question not only the status of field theory but the\nvery notion of locality'}, \\cite{Bo},p.2, \\cite{Wi3},p.1579, \\cite{S1},p.42, \\cite{S2}, p.10.  Thus infinitesimal, analytic, smooth continuum using differential geometric methods are not, apriori, used in AP Theory, {\\it although} the mathematical results of such classic analytic gauge theoretic methods,e.g. Donaldson\/Seiberg-Witten Theory,can still lead to purely group-theoretic analogs in AP theory, {\\it despite} the absence of moduli spaces in AP Theory.  See \\cite {W1}, p.240, \\cite{R}, p.621 and section 2 ahead.\n\nIn other words,the non-local radical AP-holography forces the substitution of Field Theories by discrete group theory. We do not consider this a bad thing, as, e.g. the ultraviolet catastrophe in QFT:to AP  discrete group theory it does not matter 'whether a field at a point is not well-defined' and has to be remedied 'by smearing over test functions to tame the UV divergences'....On the contrary, AP theory creates our intrinsic gauge theory,\nwithout obtaining the gauge theory analytically  from vector bundles\/fields,connections,on a particular given manifold. In the 4D AP Theory no field redefinition or renormalization is needed.\n(compare to \\cite{Wi2},p.2). Furthermore, our non-local construction of 4D smooth structures is still so subtle and metamathematically \"local\" so to speak, that, as mentioned above, an analogue of Donaldson\/Seiberg-Witten Theory is still obtained.  This AP-theoretic non-locality is evidently the conceptually  \nsimplest way to obtain universal, {\\it intrinsic quantum} gauge theory, not based on a particular Lie group.  Thus AP Theory takes the place of {\\it \"the large N limit of YM theory\"}, \\cite{M3}, \\cite{M4}.\n\n{\\it In AP Theory the UV problem and other quantum field-theoretic problems are bypassed rigorously with discrete group theory}. (Compare to \\cite{Fe}, \\cite{P}).\n\nIn a more abstract vein, our metamathematical maximalization of group theory, which does not use infinite-dimensional spaces (i.e. does not rely on SUSY), but which nevertheless still contains a very general, at each stage, {\\it infinitely generated} graded group of topology-changing transitions (the\nAP-analogue of Morse Theory) relates the symmetries of our 'particles',i.e. the discrete Artin Presentations $r$, with the exterior symmetries of the smooth 4D spacetime universes $W^4(r)$.\nThis is an analogue of the very desired property for unifying the symmetries of the actual Standard Model with those of classical General Relativity.  For a promising relation between braids and the Standard Model, see \\cite{BT}.\n\nOne can say, in AP Theory, SUSY is already 'broken' metamathematically by its canonical,natural grading by the positive \nintegers; SUSY reliance on an infinite number of dimensions is substituted by the infinite number of generators of our group of topology-changing transitions, whose existence is requiered for any sensible  Quantum Gravity theory, see \\cite{Wi2}, p.4.  There is no incompatibility problem between SUSY and the gauge transitions and other symmetries of AP Theory.(\\cite{Wi5}, p.5, \\cite{Wi6}, p.362, \\cite{Wi18}).  The 'vacuum' of AP Theory, i.e. the purely discrete part, gets enough 'rigidity' from its own graded group theory. A priori, SUSY is merely the stabilization theory of AP Theory,when $n$ approaches infinity. \n\nThese topology-changing transitions, which we call Torelli transitions, (and which,                                                                                                                                                                                                                                               for each $n$, form a group, isomorphic to the commutator subgroup of the pure braid group on $n$ strands, compare to \\cite{N}, p.43) are more universal, versatile and sharper than those caused by any classical Morse Theory or $3+1$ TQFT, \\cite {A2}, in the sense that, e.g. they can also just leave the underlying {\\it topological} structure of the manifold $X^4$ invariant and just change the smooth structure or in fact also leave it fixed; (see section 2 ahead). It is discrete pure group theory, that in a general, systematic, exterior manner (thus avoiding symmetry destroying 'skein' and\/or ad hoc 'by hand' internal surgery methods) generates new smooth 4D structures,i.e.structures that are at the foundations of modern 4D gravitational physics,\\cite{C1},\\cite{C2}.\n\nIn order to further explain and augment the above, it is instructive to analize more explicitly how the {\\it smooth} 4D manifold $W^4(r)$ is obtained holographically in a non-infinitesimal, non-local manner from the {\\it purely discrete} Artin presentation $r$ of the\nfundamental group of its boundary.\n\nLet $\\Omega_n$ denote the compact 2-disk with $n$ holes in the plane and $\\partial \\Omega_n$ its boundary.  To obtain $W^4(r)$ from $r$ proceed as follows:an Artin presentation $r$ on $n$ generators defines a unique, {\\it framed}, pure braid on $n$ strands (and conversely).This framing,(i.e. an assignement of an integer to each strand,sometimes also called a 'coloring' of the braid), is not obtained 'by hand', but is obtained canonically by representing the pure,framed, braid uniquely by an Artin presentation $r$, which then defines a smooth diffeomorphism $h(r):\\Omega_n\\to \\Omega_n$,\nwhich restricts to the identity on $\\partial \\Omega_n$. With $h(r)$,the smooth 4-manifold $W^4(r)$ is constructed by means of a relative open book construction (see \\cite{W1}, p.250,\\cite{CW}, \\cite{C1},\\cite{C2},and references therein, for the rigorous technical details).  This is a construction which is structurally similar to the fundamental Lefschetz Hyperplane Theorem for non-singular complex algebraic varieties and goes beyond the mere surgery prescriptions of the Kirby Calculus, see section 2 ahead.\n\n{\\it The diffeomorphism $h(r):\\Omega_n\\to \\Omega_n$ is determined by $r$ only up to isotopies keeping it fixed as the identity on the boundary $\\partial \\Omega_n$.}\n\nThus (the still well-defined) smooth 4D diffeomorphism class of the 4-manifold $W^4(r)$ is obtained in a non-infinitesimal,non-local manner from the 2D diffeomorphism $h(r)$, which up to isotopy is determined by the discrete Artin presentation $r$.\n\nThis is a {\\it sporadic} 4D smooth analogue, in a metamathematical sense, to the celebrated Hilbert Vth Problem, which cleared up Lie Group Theory's 'infinitesimal mess' (\\cite{H}) by showing that the existence of the {\\it smooth} structure of a Lie group need not be postulated a priori, since its existence already follows from non-infinitesimal, non-analytic concepts.\n\n{\\it This is the fundamental smooth 4D topological construction of AP Theory.}\n\nIt is indeed universal $3+1$ {\\it'gravitational holography'},very different from the very restricted field theoretic version of\nMaldacena ,Rehren,et al. and is definitely not {\\it 'a metaphoric illusion'}. (\\cite{S2},p.9).\n\nAs far as gauge theory is concerned,it is in the reverse order of the dimensional reduction of \\cite{Be},which is a cornerstone of so-called Geometric Langlands Theory,\\cite{GW}, \\cite{W4}.\n\nIt is natural to conjecture that,just as in the mathematically analogous case of Lie Groups,our\nsporadic holographic 4D gravitational Hilbert Vth-like construction,where 4D smooth structures emerge from pure discrete group theory,will help clear up the current conceptual mess in Quantum Gravity,(see \\cite{Sm}, \\cite{N}) beyond the results of this paper.\n\n\nIn order to place our main section  3 ahead in its proper metamathematical setting,we point out some more similarities of our fundamental construction to more heuristic ideas and concepts of modern physics in the  literature,(\\cite{Wi7}, \\cite{Wi8}, \\cite{A2}, \\cite{G1}, \\cite{G2}, \\cite{GW},\\cite{Wi9},\\dots )\n\ni) In the above construction of the 4D smooth manifold $W^4(r)$ from the 2D diffeomorphism  $h(r):\\Omega_n\\to \\Omega_n$,the boundary components of $\\partial \\Omega_n$ define $n+1$ knots in the 3D  boundary of $W^4(r)$ on which the,in general topology-changing and knot-changing, Torelli transitions\/interactions  act.\\cite{W1},p.226.\n\nThis is a  more direct, non-categorical analogue of {\\it 'surface operators...realizing knot invariants'} of the 4D topological gauge theories of Gukov,Witten, et al., \\cite{G1}, \\cite{GW}, \\cite{Wi9}, p.5, which lead to Geometric Langlands Theory.  The following  quote of p.9 of \\cite{G1} also seems relevant here:{\\it \"..every topological gauge theory,which admits surface operators is,in a sense,a factory that produces examples of braid group actions on branes..\"}.\n\nii) The Artin presentation $r$ on $n$ generators, which is obtained by infinite intersection via the {\\it rigorous} Cayley-von Dyck process (see section 2), instead of the infinite 'sum' (as in the non-rigorous Feynman process for achieving independence of the metric),defines a pure framed   (i.e. colored) braid on $n$ strands,i.e.intuitively a back-ground independent, {\\it macroscopic}  'string',{\\it which is immediately related (without relying on SUSY), to 4D 'gravity' via the 4D smooth structure of $W^4(r)$, by the fundamental Hilbert Vth-like construction of AP Theory}, compare to \\cite{Wi11}, p.25, \\cite{Wi16}. These are the strings in AP Theory, when it is considered as {\\it 'the infinite limit of $SU(N)$ YM Theory'},\\cite{M3},p.10, \\cite{M4}, \\cite{M5}.\n\n(A priori,they do not seem to be related to so-called 'string topology',\\cite{Su1}.)\n\nThus AP Theory can be considered to be a graded,  background independent, non-perturbative, parameter-free, macroscopic  'string theory' where holography and topology changing transistions and interactions are as strong as possible, (compare to p.411 of \\cite{Gr} and p.8 of \\cite{We}). The intrinsic 'world-sheet' of the string $r$ consists  simply of the iterates of the canonically associated planar diffeomorphisms $h(r):\\Omega_n\\to \\Omega_n$ .The incredibly rich Iteration and  Covering theories (Nielsen, Thurston,...) of these  $h(r):\\Omega_n\\to \\Omega_n$  makes this 'string theory' a very strong one indeed; in fact,in should also be relevant to \nLoop Quantum Gravity,\\cite{Sm}, \\cite{N}, \\cite{P},\\cite{BT}, thus uniting these two,a priori different approaches of modern physics to Quantum Gravity. The true LQG,after it accomodates universal holography and topology change, when freed from the graph-theoretic combinatorics of 'spin networks',etc., revealing itself as being the \nCovering Theory of String\/M Theory in AP Theory.Then some of the basic difficult problems of LQG (see,e.g. \\cite{DT}) will be by-passed or solved intrinsically {\\it ab initio}.\n\niii) Similarly AP-holography,our gauge-gravity correspondence,is so sharp and general,it can also be considered to be a particle-wave,particle-field 'duality': the discrete Artin presentation $r$,an 'extended' particle,i.e. a crystallic,non-topological 'quantum string',determines the smooth 4D spacetime universe $W^4(r)$. We can say:\n\n{\\it The 'particle'\/'quantum string' $r$ is a hologram of the 4D spacetime universe $W^4(r)$; compare again to \\cite{M2},p.63.}\n\n\n\nThis makes the fundamental AP construction of $W^4(r)$ from $r$,a rigorous background independent analogue of 'The Wave Function of the Universe',\n\\cite{HH},without using path integrals.\nThese analogies and the ones in section 3 already support many of the fundamental concepts of String\/M theory as well as QCD,(see \\cite{Wi3},p.1577,\\cite{Wil1}, \\cite{M4}), in a conceptually very simple,rigorous purely mathematical way:these concepts are {\\it \"here to stay\"},\\cite{L},p.9; the concepts of radical holography,universal topological change,etc. have to be present in any Quantum Geometry\/ Gravity theory.\n\niv) We consider $h(r):\\Omega_n\\to \\Omega_n$  to be the AP-analogue (where the rigorous Cayley-von Dyck intersection procedure is substituted for the non-rigorous Feynman summation procedure)  of 't Hooft's 'planar dominant Feynman diagrams' on the sphere $S^2$ (see \\cite{Wi3},p.1577,\\cite{M2},p.8, Ashtekar's remark on p.10 of \\cite{Au}).\nThus AP Theory in its abstract,but rigorous,universal way,realizes 't Hooft's 'bold' Conjecture of\nrelating 4D Quantum Gauge Theory to String Theory,\\cite{Wi3},p.1577, \\cite{Wi10},p.25,\\cite{Sm},p.44,\\cite{S3}.\n\nv) It is interesting to observe (compare to,e.g.,\\cite{Mo}) that the truth of the 3D Poincar\\'e  Conjecture shows that\nthere is no analogue of 4D  Black Hole singularities in AP Theory; although it is obvious from our Hilbert Vth-like construction,that 4D smooth singularities are avoided,they could have\n'perversely' re-appeared as follows: if the Poincar\\'e Conjecture were false,i.e.,if there existed an Artin Presentation $r$ presenting the trivial group,but such that the boundary\nof the corresponding 4D smooth manifold $W^4(r)$ were not homeomorphic to the 3-sphere $S^3$,then this fact would imply that this smooth $W^4(r)$ would have a serious,unremovable\nsingularity: one would not be able to 'close' this smooth 4-manifold {\\it smoothly}.See also \\cite{Wi15}.\n\nvi) AP Theory has rigorous analogues to all the features indicated in Witten's Fig.1 d) on p.25 of \\cite{Wi11}. \n\nAll the above make AP Theory a strong candidate for contributing to:{\\it  \"..the core geometrical ideas which underlie string theory, the way Riemannian geometry underlies general relativity\"},see \\cite{Wi12}.\n\n\nIn fact,perhaps it is not an exageration to call AP Theory,due to its discrete,group-theoretic conceptual simplicity,an \"Erlanger Program\" for Quantum Geometry and Quantum Gravity.It emerges as a new geometry {\\it \"..one that aligns with the new physics of string theory; \"}, see \\cite{Gr},pp.231,232.\n\nIn AP Theory, pure discrete group theory,i.e. 'symmetry' in its purest form, has been 'maximized' metamathematically; compare to \\cite{Wil5},\\cite{Wil4}.\n\n{\\it Metamathematically it is the most basic, simplest,'outermost' enveloping {\\it framing} of Quantum Gravity}.\n\nThis basic intrinsic, autonomous mathematical consistency should also, a fortiori, encompass the logic, if rigorous, of any {\\it empirical} physical evidence.\\cite{FRS},p.24.\n\nAP Theory in a way solves Atiyah's \"joker in the pack\" mystery of why low-dimensional manifold theory should be relevant to modern physics,\\cite{A3},p.15: the, for each $n$, {\\it infinitely generated} graded groups of topology changing transitions and interactions in AP Theory, take the place of the classical infinite dimensional Hilbert space of Quantum Theory.\n\nThe 4D metamathematical 'sporadicity' of AP Theory is much more universal than the important known ones: $Spin(4)$=$SU(2)\\times SU(2)$,\\cite{DK},p.7, or that the  braid group $B_n$ has a non-trivial amalgamation only when $n=3,4$.\\cite{KPS}.\n\n\nIn our main section 3 we point out more rigorous specific analogies with the Clay YM problem and how these might impede its solution as stated and\/or lessen its importance in modern physics.\n\nWe thank J.S.Calcut and C.E.Hough for very helpful conversations.\n\n\\section{The pure Mathematics of AP Theory}\n\nIn this section,we list,reference and explain some of the main,purely mathematical, but physically relevant, facts of\nAP Theory,that have been rigorously proven in the refereed literature, (\\cite{W1},\\cite{CW},\\cite{C2},..)\n\nIn a purely algebraic sense,AP Theory starts off as a sub-theory of the very basic,discrete\ntheory of Finitely Presented Groups,which begins with the concept of a presentation (see,e.g.,\\cite{KMS},chapter 1) of a discrete group $G$ ,whose meaning is the following:\n\nLet $F_n$ denote the free group on  the $n$ generators $x_1,\\dots,x_n$ and let $w_1,\\dots,w_m$  be $m$ words in $F_n$ ;let $N$ be the normal subgroup of $F_n$ ,which is the intersection of all normal subgroups of $F_n$  which contain all the $w_1\\dots, w_m$;\nthen one says $<x_1,\\dots,x_n\\vert w_1\\dots,w_m>$ presents $G$, is called a {\\it presentation}  of $G$, if the factor group $F_n\/N=G$. This is the Cayley-von Dyck process.[KMS,p.12]\n\n\nIt is important to notice here that a presentation of the trivial group,i.e. the case when it happens that $N =F_n$, can be,a priori, as complicated as a presentation of any arbitrary group  and that the concept of infinity is used here, when saying:\"intersection of {\\it all} normal subgroups\".\n\nEvidently in Group Theory,Presentation Theory is more basic, canonical and intrinsic than Representation Theory.\n\nIf $m=n$ a presentation $r=<x_1,\\dots,x_n\\vert r_1,\\dots,r_n>$ is called an {\\it Artin presentation},if,in $F_n$, the following group-theoretic equation holds:$$x_1\\dots x_n=r_1^{-1}x_1r_1\\dots r_n^{-1}x_nr_n$$.\n\nAs already realized by Artin himself, {\\it it is an equation in the free group $F_n$ that actually  defines and characterizes pure,framed,(colored) braids.} See also \\cite{MS}.\n\n\nSee \\cite{W1},\\cite{W2},\\cite{W3},\\cite{CW},\\cite{C1},\\cite{C2}, \\cite{C3}, \\cite{C4}, \\cite{Ar}, and \\cite{R},Appendix, for many examples.\n\nThe set of Artin presentations on $n$ generators is denoted by $R_n$, the group so presented by $\\pi(r)$; $A(r)$ denotes the $n\\times n$ integer matrix obtained by abelianizing the presentation $r$.\n\n$A(r)$ is always symmetric and  determines the integer quadratic form of the compact 4-manifold $W^4(r)$; every symmetric, integer $n\\times n$ matrix is an $A(r)$, where $r$ lies in $R_n$; $A(r)$ determines the $Z$-homology of both $W^4(r)$ and its boundary,$M^3(r)$.\n\nWe consider the integer, symmetric matrix $A(r)$ to be the analogue of the  Hilbert space binary forms (of QM) in AP Theory; it is all that remains of them under the radical reductivity of AP Theory,compare to \\cite{Wi17},p.9,\\cite{Wi2},p.4.\n\nWe say $r$ is 'a Torelli',if $A(r)$ is the zero matrix.\n\nThe $r$ in $R_n$ can be multiplied in a very non-trivial way,$r\\cdot r'$ again being an Artin presentation, and so that $R_n$ is canonically isomorphic to $P_n\\times Z^n$, where  $P_n$ denotes the pure braid group on $n$ strands (see \\cite{W1},p.227]).The Torelli form a subgroup isomorphic to the (infinitely generated if $n>2$) commutator subgroup of $P_n$ and it is indeed a subgroup of the classical Torelli group of a 2D closed,orientable surface of genus $n$,hence the name. Multiplication of $r$ by a Torelli does not change $A(r)$ and hence preserves the Z-homology of both $W^4(r)$ and $M^3(r)$.\n\nThe fact that AP Theory is discrete and is graded by the positive integers,i.e. is 'cone-like',allows one to use mathematical induction,to stabilize,and to use computer based methods.See \\cite{W1},\\cite{W2}, \\cite{W3}, \\cite{CW}, \\cite{C1}, \\cite{C2}, \\cite{C3}, \\cite{C4}, for many examples of such computations.\n\nRecall that a group is called perfect,if its abelianization is trivial.\nFrom work of Milnor,see\\cite{W1},p.227, we have the following 'Triality' fact:\n{\\it If the group $\\pi(r)$ is finite and perfect,then it is either trivial,or isomorphic to I(120),the binary icosahedral group,i.e. the fundamental group of Poincar\\'e's Z-homology 3-sphere}. \n\nThe smooth 4D topological part of AP Theory starts as explained in the Introduction\n(see \\cite{W1},\\cite{CW},\\cite{C1}, \\cite{C2},for the rigorous technical mathematical details) and we add the following in order to stress its topological 3D and 4D importance: \n\ni) Let $M^3(r)$ denote the boundary of  $W^4(r)$,the smooth 4-manifold determined by the Artin\npresentation $r$ via the 2D diffeomorphism $h(r):\\Omega_n\\to \\Omega_n$;it is always a connected,orientable,closed 3-manifold and every such 3-manifold can be so obtained.\\cite{W1}, \\cite{W3}, \\cite{R},Appendix.To posess an Artin presentation characterizes the fundamental groups of such $M^3$ and the Artin presentation {\\it actually determines the 3-manifold} up to diffeomorphism,not just its fundamental group. \n\nIn particular,the theory of closed,orientable 3-manifolds,is strictly speaking,{\\it not} an autonomous 3D theory,since at the same time of its definition,the smooth 4D manifold $W^4(r)$ is defined, amalgamating  this 3D theory with the much more physically relevant theory of 4D smooth manifolds.Thus the Hamilton-Thurston-Perelman program  aquires more physical importance,and hence perhaps a simpler solution.\n\nii)The $n+1$ boundary components of $\\Omega_n$ define a link $L(r)$ of $n+1$ knots, $k_i(r)$,($i$=0,1,2,..,$n$),in $M^3(r)$; Gonz\\'alez-Acu\\~na showed (see \\cite{C1}),{\\it given any link $L$ in any closed,orientable 3D manifold $M^3$,there exists an Artin presentation $r$ such that $M^3$ = $M^3(r)$ and $L$ is a sublink of $L(r)$. In particular,any knot in any $M^3$ can be obtained this way}.\n\niii) When  $A(r)$ is unimodular,then $M^3(r)$ is a Z-homology 3-sphere,and the groups and peripheral structures of the above knots,have very simple,computer friendly presentations, \\cite{W1}, p.226,  (where framings do not have to be 'put in by hand'),avoiding self-linking problems and heeding the admonitions of Penrose,Witten, et al.against symmetry-destroying 'skein methods'in Knot and Linking theory,when it is used in physics.\n\niv) The Torelli, denote one by $t$, by transforming $W^4(r)$,$M^3(r)$ into $W^4(t\\cdot r)$,$M^3(t\\cdot r)$, provide a very general,unpredictable and subtle theory of {\\it topology changing} transitions, and form the analogue of Morse Theory in AP Theory, which is much sharper than that of any known $3+1$ TQFT.\n\nMultiplying by a Torelli always preserves the $Z$-homology of $W^4(r)$ and $M^3(r)$ but usually changes the topology of $W^4(r)$ and $M^3(r)$,and the knots $k_i(r)$ in $M^3(r)$,but they can also just change certain things and leave others invariant:\n\n\n\n{\\bf Example:}Consider $s\\in R_4$ and the Torelli $t\\in R_4$ given by $s_1=(x_1x_3)^2x_2s_2$, $s_2=(x_1(x_2x_3)^2x_2^2)^{-1}$, $s_3=(x_2x_3x_2)^{-1}x_4s_4$,  $s_4=x_4^{-2}x_2x_3x_2(x_2x_3)^{-2}$, and $t_1=(x_4^{-1},x_1(x_1x_2x_3)^{-1})$,$t_2=(x_1,x_4)=t_3$, $t_4=(x_1^{-1},x_4x_1x_2x_3)t_3$; (here $(x,y)=x^{-1}y^{-1}xy$) and we use the computer algebra system MAGMA to do the group-theoretic computations).\n\nMAGMA shows that $\\pi(s) =1$, $M^3(s)=S^3$ and that  all the knot groups of the $k_i(s)$ are isomorphic to $Z$,except that of  $k_3(s)$,  which is isomorphic to that of the trefoil in $S^3$.\n\nMAGMA,and some simplification by hand, gives $r=t\\cdot s$ as:$$r_1=(x_4^{-1},x_1(x_1x_2x_3)^{-1})(x_4,x_1)(x_2x_3)^2x_2r_2,$$ $$r_2=(x_2x_3x_2^2)^{-1}((x_1^{-1},x_4x_1x_2x_3),x_4^{-1})(x_4^{-1},x_1)(x_1x_2x_3)^{-1},$$ $$r_3=(x_2x_3x_2)^{-1}(x_4x_1x_2x_3,x_1^{-1})x_4r_4,$$ $$r_4=x_4^{-2}(x_1^{-1},x_4x_1x_2x_3)x_2x_3x_2(x_2x_3)^{-2}(x_1,x_4);$$\n\nNow we again have $\\pi(r)=1$ and $M^3(r)=S^3$, however the (non-amphicheiral) trefoil $k_3(s)$ of $M^3(s)$ has been transformed by the Torelli $t$ to a (amphicheiral) figure-8 knot $k_3(r)$ in $M^3(r)$; all the other knots stay trivial.  \n\n\n\nv) Another very important,physically relevant property of the Torelli transitions is the following:\n\n{\\it One can change the smooth structure of a smooth 4-manifold, but leave the underlying topological structure intact. The discrete pure group theory of AP Theory has the energy and power to juggle different 4D smooth structures on the same underlying 4D topological manifold}.\\cite{C1},\\cite{C2}.\n\nThis phenomenon should be considered to be the last vestige (in the radical reductivity of AP Theory) of any hypothetical gravitational Schroedinger Wave Equation,where now 4D smooth structures, 'powered' by the Torelli transitions,  are the analogues of gravitational waves. This also seems to solve the so-called Hierarchy Problem (\"Why gravity is so weak\") in AP Theory.\n\nvi) The symmetric integer matrix $A(r)$ determines the quadratic form of $W^4(r)$,in particular its integer homology,as well as the integer homology of $M^3(r)$.Any integer quadratic form can be so obtained,which implies the very non-trivial fact that AP Theory has a discrete purely group-theoretic analogue of the fundamental,physically relevant Donaldson Theorem,\\cite{DK},\\cite{GS},\\cite{FM},\\cite{Wi4}, \\cite{Wi7}, \\cite{Wi8}, \\cite{Wi13}, {\\it despite the fact that there are no moduli spaces nor deRham theory} in AP Theory:\n\n{\\bf THEOREM \\cite{W1},p.240, \\cite{R},p.621}:{\\it If A(r) is a symmetric,integer,unimodular matrix,prevented by Donaldson's theorem from representing the quadratic form of closed,smooth,simply-connected 4-manifold,then the group $\\pi(r)$ is non-trivial; in fact,it has a non-trivial representation into the Lie Group SU(2).} \n\nThere exist (necessarilly non-Artin)  presentations $w$ of the trivial group, where $A(w)$ = $E_8$,(see p.11 of \\cite{C4}) and hence it is the Artin Equation above, in the free group $F_n$,  for the presentation $r$, that makes this theorem true.  \n\nThus, in particular, the discreteness of AP Theory is related to the complex numbers in a very non-trivial way.\n\nThe 'modularity' hinted at by this {\\it  purely group-theoretic} theorem should be related to that of Borcherds,\\cite{B} and that of Tomita-Takesaki theory as in Algebraic QFT, \\cite{S2},\\cite{Sum}.\n\nWe remark,that at this point in time, (although,e.g. the Casson invariant, can be described in purely AP-theoretic fashion,\\cite{Ar}, \\cite{C3},\\cite{C4}), we have no purely AP-theoretic proof of this 'Langlands-ian' theorem,which relates the  Number Theory of Integer Quadratic Forms with Group Representations of the $\\pi(r)$  into $SU(2)$; compare to \\cite{GW}, \\cite{W4}.  We still need the actual classic  analytic field-theoretic methods for this.\\cite{T},\\cite{DS}.\n\n\nOn the other hand, this also shows that the non-analytic,non-local smooth 4D Hilbert Vth Problem-like construction above,\nis so subtle , sharp and metamathematically 'local',that certain important analytic,{\\it field-theoretic}, physically relevant results,pertaining to a single, given smooth 4-manifold, are still present\nand are actually {\\it detected} by the discrete AP Theory.\n\nThis  goes well beyond the mere ad hoc Tietze-like methods of the Kirby Calculus.\n\n\nFurthermore,(see \\cite{CW}),any complex elliptic surface,e,g., the Kummer surface, can be obtained smoothly as a $W^4(r)$,with  boundary $S^3$,thus also proving the existence in AP Theory\nof a discrete, purely group-theoretic analogue of Donaldson\/ Seiberg-Witten {\\it invariants}.\\cite{FM},p.4, \\cite{DK},p.376,\n\\cite{Wi4},\\cite{Wi6},p.375.\n\n\nFinally,although in this paper we will not resort to it,we note that a very non-trivial Covering Theory exists in AP Theory: the covering and lifting theory of the 2D diffeomorphisms $h(r):\\Omega_n\\to \\Omega_n$,{\\it which is non-trivial even if the group $\\pi(r)$ is trivial}.In particular,the covering theory of the so-called 'class surface',i.e. the regular covering corresponding to the commutator subgroup of  the fundamental group of $\\Omega_n$, and that corresponding to the normal closure in $F_n$ of the $r_i$, should be specially relevant here for obtaining rigorous proofs in String theory and LQG.\n\nThis makes AP Theory also a very sophisticated mathematical theory indeed.\n\n\n\\section{Intrinsic 4D Quantum YM Existence and Mass Gap}\n\nIn sections 1,2,we exhibited the existence of a mathematically rigorous, sporadic,purely group-theoretic,smooth $3+1$ theory,which unlike the classical gauge-theoretic approach (which uses the space of connections on a particular,fixed manifold) is,a priori, exterior,(as cobordism theory is) ,autonomous,intrinsic,graded by the positive integers,and  which is not related to any particular Lie group in the usual gauge-theoretic sense.\n\nAs mentioned in the introduction, due to the radical,universal holography described above,this theory can not,a priori,be described by a conventional analytic,infinitesimal Field Theory.\n\nThis is the AP-analogue to {\\it 'producing a mathematically complete example of quantum gauge field theory in four dimensional space time'}, \\cite{JW},p.5, i.e. the first part of the Clay YM Existence and Mass Gap problem.\n\nDespite that our '4D Quantum Gauge Theory' is not {\\it quantitative},we can still ask the other fundamental question:\n\n{\\it Do there exist natural qualitative analogues to the quantitative mass gap condition of the Clay YM Millenium problem?}\n\nWe quote \\cite{JW},p.3:{\\it \"..it must have a \"mass gap\":namely there must be some constant $\\Delta > 0$ such that every excitation of the vacuum has energy at least $\\Delta$\".}\n\nFurthermore,if such qualitative analogues of this quantitative mass gap do exist,how do they affect the rigorous solution,if it exists,of the actual {\\it quantitative} Clay mass gap problem as stated?\n\nWe start with the questions of Ashtekar,\\cite{As},p.174,regarding Quantum Geometry:{\\it \"What are the atoms of geometry? What are the fundamental excitations?\"}.\n\nWe show that by considering the $h(r):\\Omega_n\\to \\Omega_n$ as the analogue of 'vacuum fluctuations\/excitations' and their generation of the 4D smooth manifolds $W^4(r)$ as a 'giving gravitational mass' Higgs-like phenomenon,we obtain some of the most desired and important hypothetical {\\it qualitative} consequences of the {\\it quantitative} Clay mass gap, if it were true.\n\nIn other words,the fundamental Hilbert Vth Problem-like construction, from the topology-less Artin presentation $r$ (i.e.$r$ is of the {\\it discrete purely group-theoretic} 'vacuum' with zero 'mass'), gives non-zero 'gravitational mass',i.e.4D smooth structures to the $W^4(r)$ in a universal Higgs-like way.(Compare to the heuristic quantitative arguments in \\cite{K} and references therein).\n\n{\\it In AP Theory,'mass gap' is just its radical,vacuum based,universal holography dressed in physical jargon}.\n\n{\\bf In AP Theory,holography and mass gap are defined in unison;they are the same mathematical phenomenon.}\n\nWe point out some AP-analogies with the most important {\\it qualitative} consequences of a positive solution to the actual quantitative Clay YM problem:\n\nI.Our mass gap immediately gives a sharper version of the 'clustering property' of (\\cite{JW},p.6,or \\cite{Wi10},p.125) {\\it \"of the principle of exponential decay of correlations at long distances that makes it possible to deduce global results about four manifolds from a knowledge how the theory behaves on $R^4$. ..the mass gap is closely related to the behaviour of the Donaldson invariants on algebraic surfaces\"}.\n(see also \\cite{Wi13},p.291,\\cite{Wi6}).\nIn AP Theory,global,non-local 4D  results are deduced not from $R^4$,{\\it but already,more holographically,from $R^2$},the plane and Pure Braid Theory,via the planar $h(r):\\Omega_n\\to \\Omega_n$.\nAs pointed out above,the results of \\cite{CW} show that there even  exists a non-trivial,purely\nAP-theoretic theory of Donaldson\/Seiberg-Witten invariants.\n\nII.The above AP-analogy (in iv) of the Introduction) with 't Hooft's 'bold' conjecture which according to \\cite{Wi10},p.25,{\\it \".. if valid,it might give an effective way to demonstrate the mass gap\"} and which {\\it \"..seems like much the most plausible known approach to the problem,but an answer along these lines is not yet in sight,even at a heuristic level\"}.\n\n\nIII.However the strongest analogy is with the  most important desideratum of the Clay problem: namely relating the mass gap with Quantum Chromodynamics (QCD),i.e. 4D $SU(N)$ Quantum Gauge Theory,\\cite{Wi3}, p.1577, ('YM Theory without SUSY') and its important properties such as {\\it confinement} and {\\it asymptotic freedom}. \\cite{Wil1},\\cite{Wil2},\\cite{H1},\\cite{H2}.\n\nFirst we note that the $h(r):\\Omega_n\\to \\Omega_n$,our 'vacuum fluctuations\/excitations' are not only sophisticated purely mathematically,but also {\\it physically}:\n\nSince $h(r)$ is determined only up to isotopies of $\\Omega_n$,keeping it fixed as the identity on the boundary,we obtain a {\\it topological} analogue of {\\it uncertainty} in AP Theory (instead of minimal length 'uncertainty')\nfor our vacuum fluctuations.Compare to \\cite{AY},\\cite{SW}.\n\nThus,unlike as usual in quantum physics,'uncertainty' in AP Theory is {\\it deduced} from the vacuum fluctuations,not {\\it used} to 'prove' their existence.\n\nIn AP Theory, holography and 'uncertainty' are related; compare to \\cite{SW}.\n\nIf we dare call the generic fixed points of $h(r):\\Omega_n\\to \\Omega_n$, 'quarks',(compare to Wilczek,\\cite{Wil3},Kondo,\\cite{K} and Susskind's 'partons' in \\cite{S}), we immediately obtain a topological analogue of 'confinement':\n\n{\\it Although generically these fixed points do not dissapear under isotopies,they can not be individually determined due to the uncertainty above}.\n\nThis is a topologically more sophisticated explanation of {\\it confinement} ('quarks can not be individually determined'),than the classical string-theoretic one:that an open string has to have two inseparable end points,i.e.quarks.\\cite{Wi3},p.1577.\n\nThis should also explain,why the phenomenon of confinement resists being proven analytically by field-theoretic methods.\\cite{Wil4},p.7,\\cite{S1},pp.40-41.\n\n\n\nSimilarly,if we also dare call our 4D smooth manifolds $W^4(r)$,'gluons',then we have a topological resemblance to {\\it asymptotic freedom},('that in very high energy reactions quarks and gluons interact very weakly'):\n\n\nIf we iterate $h(r)$ it is natural to suppose that the mathematical relations between the fixed point theory (Nielsen,Thurston,..) of $h(r)^m$ and the 4D smooth topology of the corresponding $W^4(r^m)$ will grow weaker.This hints at the existence of an abstract intrinsic 'non-linear Fourier transform',\\cite{A3},p.14,and should be compared to Taubes, \\cite{T},p.367,Wilczek, \\cite{Wil3}, Kondo,\\cite{K}, and \\cite{GW}, \\cite{W4}.\n\nRelating the the generic fixed points of $h(r):\\Omega_n\\to \\Omega_n$ (i.e.'quarks') to the smooth structures of the 4D manifolds $W^4(r)$ is a sporadic {\\it 4D smooth topological},i.e. gravitational, version of the original Yang-Mills program of relating particles to the differential geometry of connections.\n\n\nIt seems natural to conjecture that AP Theory and QCD ,due to their conceptual simplicity,are related ,in the sense that AP Theory has rigorous mathematical  features that the still hypothetical 'highest temperature QCD' should have.\\cite{Wil1},p.25; see also \\cite{We},pp.13,14.\n\nIn conclusion,we have exhibited the existence of an absolute,intrinsic '4D Quantum YM Theory' with a qualitative analogue of 'mass gap',with all the physical analogies above. Due to the universality and rigorous mathematical conceptual simplicity of this theory,perhaps the actual quantitative Clay YM problem should be substituted by the more general one of the existence of a $3+1$ axiomatic,'constructive' QFT. \\cite{J}, \\cite{Ri}, \\cite{FRS}, \\cite{S2},\\cite{Wi14},\\cite{We}.\n\nDue to the crucial mathematical fact that AP Theory is not a model (in particular, does not introduce any new axioms) and its radical holography and strong Torelli transitions and interactions, which bypass difficult UV problems with pure group theory and which,a priori,does not seem to mix well with,e.g., the Wightman axioms,(see \\cite{S2},p.10),the fate of a mathematically rigorous axiomatic $3+1$ QFT is a serious and  more pressing problem in physics,than the more particular Clay YM problem.It seems reasonable to conjecture that any such $3+1$ QFT would have to be 'modular' with respect to the groups $\\pi(r)$ and therefore its rigorous construction, if indeed it exists, would be difficult.\n\nFor the Geometric Langlands Theory corresponding to Intrinsic 4D Quantum YM Theory, instead of $N=4$ Super YM Theory,  see \\cite{W4}. \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nRecently, in a series\nof publications\\cite{neuberger,hasenfratz,hasenfratz2,hasenfratz3,\nluscher,chandra}, \nit has become clear that, if one would\nmodify the chiral transformation laws away from their canonical form in\nthe continuum, chiral symmetries can be preserved on the lattice without\nthe problems of fermion doubling.\nThe lattice Dirac operator $D$ for \nthese fermions satisfies the Ginsparg-Wilson \\cite{ginsparg}\nrelation\\footnote{They are therefore named Ginsparg-Wilson fermions.}, \n\\begin{equation}\n\\label{eq:ginsparg}\n\\gamma_5 D +D\\gamma_5 D =aD \\gamma_5 D  \\;\\;,\n\\end{equation}\nwhere $a$ is the lattice spacing. \nAs a consequence of the  \nGinsparg-Wilson relation~(\\ref{eq:ginsparg}), it is easy \nto show that the fermion action,\n\\begin{equation}\nS_F=\\sum_{x,y}\\bar{\\psi}(x) D_{x,y} \\psi(x) \\;\\;,\n\\end{equation}\nis invariant under lattice chiral transformations, and \nchiral symmetry will protect the quark masses away from the\nadditive renormalizations.\n\nThe chiral properties of the Ginsparg-Wilson fermions is\na direct result of the Ginsparg-Wilson relation. In particular, several\ntypes of fermion actions could be written down which all fulfill the\ncondition. For definiteness, one particular choice \\cite{neuberger2,luscher} \nis adopted in this paper, namely: \n\\begin{eqnarray}\n\\label{eq:ferm_matrix}\naD&=&1-H_W(H^{\\dagger}_W H_W)^{-1\/2} \\;\\;, \\nonumber \\\\\nH_W&=& 1+s-a\\sum_{\\mu}{1 \\over 2}\n[\\gamma_\\mu(\\nabla_\\mu+\\nabla^*_\\mu)-a\\nabla^*_\\mu\\nabla_\\mu] \\;\\;,\n \\nonumber \\\\\n&=&(-3+s)\\delta_{x,y}+\n{1 \\over 2}\\sum_{\\mu}\\left[(1-\\gamma_\\mu)U_{\\mu}(x)\\delta_{x+\\mu,y}\n+(1+\\gamma_\\mu)U^{\\dagger}_{\\mu}(x-\\mu)\\delta_{x-\\mu,y}\\right]\n\\end{eqnarray}\nwhere $s$ is a parameter satisfying $|s|<1$.  \nThe lattice covariant derivatives\n$\\nabla^*_\\mu$ and $\\nabla_\\mu$ are defined as usual,\n\\begin{eqnarray}\n\\nabla_\\mu \\psi&=& U_\\mu(x)\\psi(x+\\mu)-\\psi(x) \\;\\;, \\nonumber \\\\\n\\nabla^*_\\mu \\psi&=& \\psi(x)-U^{\\dagger}_{\\mu}(x-\\mu)\\psi(x) \\;\\;. \n\\end{eqnarray}\n\nDue to their decent chiral properties, it is quite tempting to investigate\nthe possibility of performing Monte Carlo simulations using \nGinsparg-Wilson fermions, replacing the conventional\nWilson fermions.  Albeit their\nseemingly non-local appearance, the fermion matrix~(\\ref{eq:ferm_matrix})\nis in fact local (with exponentially decaying tails), \nas long as the parameter $s$\nin the matrix is chosen appropriately\n\\cite{luscher2,chiu}. The locality property of the\n fermion matrix will enable us to use iterative methods in\nKrylov space whenever inversion of the matrix becomes necessary.\n\nWhen performing the inversion of the fermion matrix~(\\ref{eq:ferm_matrix}), \none would encounter the problem of yet another matrix \ninversion of a fractional power.\nRecently, proposals have been put forward \\cite{neuberger2,bunk} which\nmake  such inversions possible. \nSome numerical calculations in quenched QCD \\cite{edwards}\nfor Ginsparg-Wilson fermions already indicate that these fermions\nindeed  have anticipated chiral properties.\nHowever, it is also realized that quenched calculations with Ginsparg-Wilson\nfermions are more costly than with conventional Wilson fermions\nprimarily due to fractional inversion of \nthe matrix $H^{\\dagger}_WH_W$.\nTwo specific types of methods will be discussed in this paper. One is the\nmethod proposed by Bunk \\cite{bunk}, which will be called \nfractional inverter method, or FIM. The other method is the\noptimal rational approximation method, or ORAM,\nproposed in Ref.\\cite{edwards}.\nIt was reported in Ref.~\\cite{edwards} that ORAM converges faster\nthan FIM for a desired accuracy and a given condition number of\nthe matrix $H^{\\dagger}_WH_W$.\nNow, each multiplication with the fermion matrix $D$\nfor Ginsparg-Wilson fermions is\nequivalent to $2N+1$ multiplications with matrix $H^{\\dagger}_W$ or\n$H_W$, where $N$ is some integer.\nWith FIM, $N$ is the number of \nhighest order Legendre polynomial kept in the iteration procedure.\nWith ORAM, $N=N_{cg}$ is the number of conjugate gradient \\footnote{\nBy the phrase \"conjugate gradient\", we mean all possible iterative\nalgorithms in Krylov space: conjugate gradient, minimal residue,\nBi-conjugate gradient, etc.} \niterations needed to perform the multi-shift matrix inversions.\nThis number is determined by\nthe condition number of the matrix  $H^{\\dagger}_WH_W$ and\nthe accuracy desired.\nTherefore, the calculations with Ginsparg-Wilson fermions is\nat least more costly than conventional Wilson fermions\nby a factor of $2N+1$.\n\nAlthough it is already quite costly in the quenched case, it remains \nan tempting problem to simulate {\\em dynamical} \nGinsparg-Wilson fermions.  No algorithms including dynamical fermions\n have been tested on these newly proposed fermions.\nIn this paper, it is shown that a Hybrid Monte Carlo algorithm would do\nthe job, however, just as in the quenched case, the simulation is\nmore costly than simulating {\\em dynamical} Wilson fermions.\nAlso, in the calculation of \nthe fermionic force to gauge links, using\ntwo different methods for the fractional inversion\nresults in very different memory and CPU time consumptions.\nFor FIM, it seems that\n $O(N)$ psudofermion fields have\nto be stored in order to make the simulation tractable.\nFor ORAM, only a moderate number of\npsudofermion fields have to be stored, and the number of matrix\nmultiplications also increases slower than in FIM.\nIn this paper, we will concentrate on a Hybrid Monte Carlo\nalgorithm for simulation of dynamical Ginsparg-Wilson fermions.\nThe fermionic force for the gauge links, which is the crucial\npart for the dynamical fermion simulation, will be calculated \nfor both ORAM and FIM. General properties of the two methods \nare compared. Test runs on small lattices with gauge group\n$SU(3)$ are now being investigated \\cite{chuan2} and detailed\nresults will be reported later.\n\nThis paper is organized in the following manner. In Section 2, \nthe Hybrid Monte Carlo algorithm suitable for \nsimulating dynamical Ginsparg-Wilson fermions\nare described and the formula for the force of the gauge link is \nderived, in both ORAM and FIM. These two methods are compared\nin the dynamical simulation in terms of CPU time consumption\nand memory consumption. Possible improvement methods are\nalso addressed. Some concluding remarks are in Section 3.\n\n\\section{ The Hybrid Monte Carlo algorithm}\n\nThe basic formalism of Hybrid Monte Carlo\nalgorithm \\cite{kennedy} remains the same\nas in the conventional Wilson case \\cite{chuan}.\nOnly the fermionic force has to be re-derived for the\nGinsparg-Wilson case, which will be dealt with below in detail.\nThe effective action with the psudofermion contribution now reads:\n\\begin{equation}\nS_{eff}=S_g[U_{\\mu}(x)]+\\phi^{\\dagger}Q^{-2}\\phi \\;\\;,\n\\end{equation}\nwhere the fermion matrix $Q \\equiv \\gamma_5(D+m)$ is hermitian and\n$\\phi(x)$ being the psudofermion field generated at the beginning of\na Hybrid Monte Carlo trajectory from  Gaussian noise.\nWe have also assumed that two flavors of quarks have degenerate masses. \nAt each molecular dynamics step in a Hybrid Monte Carlo trajectory,\none has to find solution vectors \n$X_1 \\equiv Q^{-2}\\phi$ and  $X_2 \\equiv Q^{-1}\\phi=QX_1$ from \nan iterative algorithm (conjugate gradient, for example) in Krylov space.\nThen, the total force with respect to the gauge fields can be found\nby investigating the variation of the action under infinitesimal \nchanges of the gauge fields:\n\\begin{eqnarray}\n\\delta S_{eff}&=&\\sum_{x,\\mu} Tr[V_{\\mu}(x)\\delta U_{\\mu}(x)+h.c.]\n+  \\delta S_f \\;\\;, \n\\nonumber \\\\\n\\delta S_f&=& \\delta[\\phi^{\\dagger}Q^{-2}\\phi]\n=  Tr[F_{\\mu}(x)\\delta U_{\\mu}(x)+h.c.] \\;\\;. \n\\end{eqnarray}\nThe gauge staple $V_{\\mu}(x)$ comes solely from\nthe pure gauge part of the action and could be\nobtained with little cost. The fermionic forces $F_{\\mu}(x)$, \nhowever, is much more costly. Once the fermionic forces\nare obtained, the standard Hybrid Monte Carlo updating procedure\ncan be carried on just as in the conventional Wilson case.\n\nTo derive the formula for the fermionic force, we take the variation\nof the fermionic part of the action and get,\n\\begin{equation}\n\\delta S_f= X^{\\dagger}_1(-\\delta Q) X_2 + X^{\\dagger}_2(-\\delta Q) X_1 \n\\end{equation}\nThe variation of the matrix $Q$ contains two parts, one being \nsimple, namely \n\\begin{equation}\n-\\delta_1 Q = \\gamma_5 (\\delta H_W) (H^{\\dagger}_WH_W)^{-1\/2} \\;\\;,\n\\end{equation}\nthe other being quite complicated, i.e.\n\\begin{equation}\n-\\delta_2 Q = \\gamma_5 H_W \\delta (H^{\\dagger}_WH_W)^{-1\/2} \\;\\;,\n\\end{equation}\nwhich depends on the detailed implementation of the fractional\ninversion of the matrix. We now proceed to discuss the fermionic\nforces in ORAM and FIM respectively.\n\n\n\\subsection{Fermionic force in ORAM}\n\nWe first present the force in ORAM which is more straightforward.\nWe recall that, this approximation \namounts to approximate the function  $z(z^2)^{-1\/2}$ in the\ninterval $[0,1]$ by a ratio of two polynomials:\n\\begin{equation}\nz(z^2)^{-1\/2} = z\\left(c_0+\\sum^N_{k=1}{c_k \\over z^2+q_k}\\right) \\;\\;,\n\\end{equation}\nIt is an approximation similar to the Pad\\'e approximation\nused in Ref.\\cite{ying}. For details about this approximation\nand how to get coefficients $c_k$, consult~\\cite{edwards}\nand references therein.\nApplying this method to the hermitian matrix $\\gamma_5H_W$,\nwe immediately obtain the following expression for\nthe variation of the fermionic action:\n\\begin{eqnarray}\n\\delta S_f &=& c_0Tr(X_2 \\otimes X^{\\dagger}_1\\gamma_5\\delta H_W)\n+\\sum^N_{k=1}c_k Tr(\\zeta_{2,k} \\otimes X^{\\dagger}_1 \\gamma_5 \\delta H_W)\n\\nonumber \\\\\n&+&c_0Tr(X_1 \\otimes X^{\\dagger}_2\\gamma_5\\delta H_W)\n+\\sum^N_{k=1}c_k Tr(\\zeta_{1,k} \\otimes X^{\\dagger}_2 \\gamma_5 \\delta H_W)\n\\nonumber \\\\\n&-&\\sum^N_{k=1}c_k Tr\\left( \\zeta_{2,k} \\otimes \\xi^{\\dagger}_{1,k} H^{\\dagger}_W \\delta H_W\n+ \\xi_{2,k} \\otimes \\xi^{\\dagger}_{1,k} \\gamma_5 \\delta H_W\n\\right) \\;\\;,\n\\nonumber \\\\\n&-&\\sum^N_{k=1}c_k Tr\\left( \\zeta_{1,k} \\otimes \\xi^{\\dagger}_{2,k} H^{\\dagger}_W \\delta H_W\n+ \\xi_{1,k} \\otimes \\xi^{\\dagger}_{2,k} \\gamma_5 \\delta H_W\n\\right) \\;\\;,\n\\nonumber \\\\\n\\zeta_{i,k}&=& {1 \\over (\\gamma_5H_W)^2+q_k}X_i \\;\\;, \\;\\;\\;\\;\n\\xi_{i,k}= \\gamma_5H_W {1 \\over (\\gamma_5H_W)^2+q_k}X_i \\;\\;.\n\\end{eqnarray}\nIn the above formula, ``$Tr$'' stands for taking trace in both Dirac and color\nspace and a summation over the whole lattice points.\nThe symbol $\\otimes$ stands for direct product of two vectors in\ncolor space.  Therefore, the fermionic force is obtained as\n\\begin{eqnarray}\nF_{\\mu}(x)&=&{c_0 \\over 2} tr_{Dirac}[X_2(x+\\mu) \\otimes X^{\\dagger}_1(x)\\gamma_5(1-\\gamma_{\\mu}) \n+X_1(x+\\mu) \\otimes X^{\\dagger}_2(x)\\gamma_5(1-\\gamma_{\\mu})]\n\\nonumber \\\\\n&+&\\sum^N_{k=1}tr_{Dirac}{c_k \\over 2}[\\zeta_{2,k}(x+\\mu) \\otimes X^{\\dagger}_1(x)\n\\gamma_5(1-\\gamma_{\\mu})]\n\\nonumber \\\\\n&+&\\sum^N_{k=1}tr_{Dirac}{c_k \\over 2}[\\zeta_{1,k}(x+\\mu) \\otimes X^{\\dagger}_2(x)\n\\gamma_5(1-\\gamma_{\\mu})]\n\\nonumber \\\\\n&-&\\sum^N_{k=1}tr_{Dirac}{c_k \\over 2}[\\zeta_{2,k}(x+\\mu) \\otimes \n[H_W\\xi_{1,k}]^{\\dagger}(x)(1-\\gamma_{\\mu})]\n\\nonumber \\\\\n&-&\\sum^N_{k=1}tr_{Dirac}{c_k \\over 2}[\\xi_{2,k}(x+\\mu) \\otimes \\xi^{\\dagger}_{1,k}(x)\n\\gamma_5(1-\\gamma_{\\mu})]\n\\nonumber \\\\\n&-&\\sum^N_{k=1}tr_{Dirac}{c_k \\over 2}[\\zeta_{1,k}(x+\\mu) \\otimes \n[H_W\\xi_{2,k}]^{\\dagger}(x)(1-\\gamma_{\\mu})]\n\\nonumber \\\\\n&-&\\sum^N_{k=1}tr_{Dirac}{c_k \\over 2}[\\xi_{1,k}(x+\\mu) \\otimes \\xi^{\\dagger}_{2,k}(x)\n\\gamma_5(1-\\gamma_{\\mu})] \\;\\;,\n\\end{eqnarray}\nwhere the trace $tr_{Dirac}$ is only taken within the Dirac space.\n\n\n\\subsection{Fermionic force in FIM}\n\nIn FIM, we would like to solve for $\\xi$ satisfying:\n\\begin{equation}\n\\label{eq:solve}\nM^{1\/2}\\xi=X \\;\\;\\;\\;, {\\rm given} \\;\\;\\; X.\n\\end{equation}\nwhere the matrix $M \\equiv H^{\\dagger}_W H_W$ by setting:\n\\begin{equation}\nM = c (1+t^2-2tA) \\;\\;,\n\\end{equation}\nwith the parameters $c$ and $t$ chosen in such a way \nthat all eigenvalues of the matrix $A$ lie within $[-1,1]$. \nTo be more specific, we choose,\n\\begin{equation}\nt = {\\sqrt{\\kappa}-1 \\over\\sqrt{\\kappa}+1}  \\;\\;,\\;\\;\\;\\;\nc = {(\\sqrt{\\kappa}+1)^2 \\over 4 \\lambda_{min}} \\;\\;,\n\\end{equation}\nwhere $\\lambda_{min}$ ($\\lambda_{max}$) is the lowest\n(highest) eigenvalue of the matrix $H^{\\dagger}_WH_W$ and\n $\\kappa \\equiv \\lambda_{max}\/\\lambda_{min}$ is the condition number.\nThen, the solution to eq.~(\\ref{eq:solve}) may be written as:\n\\begin{equation}\n\\xi=c^{-1\/2} \\sum^{\\infty}_{k=0} t^k P_k(A) \\cdot X\n=\\sum^{\\infty}_{k=0} s_k \\;\\;,\n\\end{equation}\nwhere $P_k(z)$ are Legendre polynomials. Therefore, an approximant for the\nsolution at the $n$-th level is\n\\begin{equation}\n\\xi_n =\\sum^{n}_{k=0} s_k \\;\\;.\n\\end{equation}\nThe shifts \\footnote{A extra factor $t^k$ has been included\nin the definition of $s_k$ as compared with Ref.\\cite{bunk}}, \n$s_k$, defined as\n\\begin{equation}\ns_k =c^{-1\/2}t^kP_k(A)\\cdot X \\;\\;,\n\\end{equation}\nsatisfy the following recursion relations:\n\\begin{eqnarray}\ns_{-1}&=&0 \\;\\;, s_0=c^{-1\/2}X \\;\\;, \\nonumber \\\\\ns_{k}&=&(2-1\/k)tAs_{k-1}-(1-1\/k)t^2s_{k-2} \\;\\;.\n\\end{eqnarray}\nFor the case of Legendre polynomials, it is claimed that the following\nbound for the residue is obtained \\cite{bunk}:\n\\begin{equation}\n|\\xi-\\xi_n|\/|\\xi|=|R_n(A)| \\le t^{n+1}=\n ({\\sqrt{\\kappa}-1 \\over\\sqrt{\\kappa}+1})^{n+1} \\;\\;,\n\\end{equation}\nwhich asserts the exponential convergence of the iteration.\n\nFor the vectors  $\\delta (H^{\\dagger}_WH_W)^{-1\/2}X_i$,\nsimilar strategy could be applied, \n\\begin{equation}\n\\delta(M^{-1\/2})\\eta = c^{-1\/2}\n\\sum^{N_{cut}}_{n=0} t^n \\delta P_n(A)\\eta \\;\\;, \\;\\;\\;\\;\\; {\\rm given} \\;\\;\\; \\eta,\n\\end{equation}\nwhere $\\eta$ represents either $X_1$ or $X_2$ and $N_{cut}$ is\nthe highest order of Legendre polynomials kept in the approximation.\nIn an analogous manner, we define,\n\\begin{equation}\n\\delta_k \\equiv c^{-1\/2} t^k (\\delta P_k(A)) \\cdot \\eta \\;\\;,\n\\end{equation}\nwhich satisfy the following recursion relation:\n\\begin{eqnarray}\n&&(k+1) \\delta_{k+1}+kt^2 \\delta_{k-1}=(2k+1)t(\\delta As_k+A\\delta_k) \\;\\;.\n\\nonumber \\\\\n&& \\delta_{-1}=0 \\;\\;\\;\\;, \\delta_0=0\\;\\;,\n\\end{eqnarray}\nUsing this relation, $\\delta_k$ could be expressed as:\n\\begin{equation}\n\\delta_k = \\sum^{k-1}_{l=0} tL^l_k(A)\\delta A s_l \\;\\;.\n\\end{equation}\nThe coefficients $L^l_k(A)$ are polynomials in $A$ with degree\n$(k-l-1)$\nand can be expressed as Legendre polynomials,\n\\begin{equation}\nL^l_k(A)= {2l+1 \\over l+1}t^{k-l-1}P_{k-l-1}({2l+3 \\over l+2}A) \\;\\;.\n\\end{equation}\nAfter rearranging the double summation and some trivial\nalgebra, we get the following formula for the variation of\nthe fermionic action:\n\\begin{eqnarray}\n\\delta S_f&=&\nTr(Z_1 \\otimes X_2^{\\dagger} \\gamma_5 \\delta H_W)\n+Tr(Z_2 \\otimes X_1^{\\dagger} \\gamma_5 \\delta H_W)\n\\nonumber \\\\\n&-&{1 \\over 2c} \\sum^{N_{cut}}_{l=0} ({2l+1 \\over l+1})Tr[(H_Wx_{2,l} \\otimes \nT^{\\dagger}_{1,l})\\gamma_5\\delta H^{\\dagger}_W]\n\\nonumber \\\\\n&-&{1 \\over 2c} \\sum^{N_{cut}}_{l=0} ({ 2l+1 \\over l+1})Tr[x_{2,l} \\otimes \n(H_W\\gamma_5T_{1,l})^{\\dagger}\\delta H_W] \n\\nonumber \\\\\n&-&{1 \\over 2c} \\sum^{N_{cut}}_{l=0} ({2l+1 \\over l+1})Tr[(H_Wx_{1,l} \\otimes \nT^{\\dagger}_{2,l})\\gamma_5\\delta H^{\\dagger}_W]\n\\nonumber \\\\\n&-&{1 \\over 2c} \\sum^{N_{cut}}_{l=0} ({ 2l+1 \\over l+1})Tr[x_{1,l} \\otimes \n(H_W\\gamma_5T_{2,l})^{\\dagger}\\delta H_W] \\;\\;,\n\\nonumber \\\\\nZ_i&=& (H^{\\dagger}_WH_W)^{-1\/2}X_i \\;\\;,\\;\\;\\;\\;\nx_{i,l}= t^l(P_l(A))X_i \\;\\;,\n\\nonumber \\\\\nT_{i,l}&=& \\sum^{N-l-1}_{m=0} {\\cal S}^{(m)}_{i,l} \\;\\;,\\;\\;\\;\\;\n{\\cal S}^{(m)}_{i,l} =\nt^m P_{m}(({2l+3 \\over l+2})\\gamma_5A\\gamma_5)H_W X_i \\;\\;.\n\\end{eqnarray}\nTherefore, the following expression for the fermionic force is obtained:\n\\begin{eqnarray}\nF_{\\mu}(x)&=&\ntr_{Dirac}(Z_1(x+\\mu) \\otimes X_2^{\\dagger}(x) \\gamma_5(1-\\gamma_{\\mu})\n+(Z_2(x+\\mu) \\otimes X_1^{\\dagger}(x) \\gamma_5(1-\\gamma_{\\mu}))\n\\nonumber \\\\\n&-&{1 \\over 2c} \\sum^N_{l=0} ({ 2l+1 \\over l+1}) tr_{Dirac}[(H_Wx_{2,l}(x+\\mu) \n\\otimes T_{1,l}(x))\\gamma_5(1+\\gamma_{\\mu})]\n\\nonumber \\\\\n&-&{1 \\over 2c} \\sum^N_{l=0} ({ 2l+1 \\over l+1}) tr_{Dirac}[(x_{2,l}(x+\\mu) \n\\otimes [H^{\\dagger}_WT_{1,l}]^{\\dagger}(x))\\gamma_5(1-\\gamma_{\\mu})] \n\\nonumber \\\\\n&-&{1 \\over 2c} \\sum^N_{l=0} ({ 2l+1 \\over l+1}) tr_{Dirac}[(H_Wx_{1,l}(x+\\mu) \n\\otimes T_{2,l}(x))\\gamma_5(1+\\gamma_{\\mu})]\n\\nonumber \\\\\n&-&{1 \\over 2c} \\sum^N_{l=0} ({ 2l+1 \\over l+1}) tr_{Dirac}[(x_{1,l}(x+\\mu) \n\\otimes [H^{\\dagger}_WT_{2,l}]^{\\dagger}(x))\\gamma_5(1-\\gamma_{\\mu})] \n\\;\\;.\n\\end{eqnarray}\nSince the CPU cost of a simulation program with dynamical\nfermions is dominated by the fermion matrix times vector operations.\nIt becomes clear that \nthe above formula for the fermionic force is not very useful\nfrom a practical point of view.\nThe most CPU consuming part is the calculation of\nthe vectors $T_{i,l}(x)$, for all values of $l$,\neach requiring an iterative procedure, i.e. the calculation\nof the quantities ${\\cal S}^{(m)}_{i,l}$. This implies that, in order to\ncalculate the fermionic force, \nthe multiplication of the matrix $H^{\\dagger}_WH_W$ \nhas to be performed $O(N^2_{cut})$ times, where $N_{cut}$ \ncan become large.  This would make the calculation of \nthe fermionic force too costly.\n\nHowever, there is a way around this difficulty. \nThe price to pay will be some extra\nstorage. Note that Legendre polynomials $P_m(\\beta(l)z)$, $\\beta(l)$\nbeing the constant $(2l+1)\/(l+1)$, could be\nexpressed as a linear combination of Legendre polynomials \nof lower or equal degrees  with argument changed to $z$, i.e.\n\\begin{equation}\nP_m(\\beta z)= \\sum^m_{j=0} \\sigma_{m,j}(\\beta) P_j(z) \\;\\;.\n\\end{equation}\nWith this, we  could express the quantities $T_{i,l}$ in the following way:\n\\begin{eqnarray}\nT_{i,l}&=&\\sum^{N-l-1}_{m=0} f_m(l,t)t^mP_m(\\gamma_5A\\gamma_5)H_WX_i \\;\\;,\n\\nonumber \\\\\nf_m(t,l)&=&\\sum^{N-l-1}_{j=m} t^{j-m} \\sigma_{j,m}(\\beta(l)) \\;\\;.\n\\end{eqnarray}\nThe functions $f_m(t,l)$ are just c-numbers and can\nbe calculated at the beginning of the simulation.\nTherefore,after the vectors $X_i$ are obtained,\none can calculate the vectors $P_m(\\gamma_5A\\gamma_5)HX_i$ \nonce for all values of $m$\nand store the resulting vectors. \nThus, $T_{i,l}$ could be obtained easily without\nfurther iteration of matrix multiplications.\nThe coefficients $\\sigma_{m,j}(l)$ satisfy the following recursion relation:\n\\begin{equation}\n\\sigma_{m,j}=({2m-1 \\over m})\\beta(l)[({j \\over 2j-1})\\sigma_{m-1,j-1}\n+({j+1 \\over 2j+3})\\sigma_{m-1,j+1}]-({m-1 \\over m}) \\sigma_{m-2,j} \\;\\;,\n\\end{equation}\nwhere the subscripts $m$ $j$ should satisfy $0\\le j \\le m$ and \na zero value is understood whenever an out-of-range subscript is\nencountered. Together with $\\sigma_{0,0} \\equiv 1$, the above recursion\nrelation completely determines all coefficients $\\sigma_{m,j}(\\beta(l))$\nand therefore the function $f_m(t,l)$.\nNow the calculation of the quantity $T_{i,l}$ only requires a \nlinear combination of vectors which costs little CPU time.\n\n\\subsection{comparison of the two methods}\n\nWe now compare the CPU time consumption and memory consumptions of the\ntwo methods discussed so far for the simulation of dynamical \nGinsparg-Wilson fermions. As is well known, the CPU time \nconsumption is proportional to the  number of operations\nof the matrix $H_W$ on vectors. \nFor each molecular dynamics step in the Hybrid Monte Carlo,\nORAM requires $2N_{CG}(2N_{cg}+1)$ number of matrix multiplications\nto obtain the solution vector $X_1$ and $4N_{cg}+4N_r$ more matrix\nmultiplications to obtain the fermionic force. Here,\n$N_{CG}$ is the number of conjugate gradient iterations\nneeded to obtain the solution $X_1$ and $N_{cg}$ is the\nnumber of conjugate gradient iterations needed to obtain\nthe vector $(H^{\\dagger}_WH_W)+q_{min})^{-1}X_i$ for the\nsmallest shift $q_{min}$.\nParameter $N_r$ is the order of the polynomials in the optimal\nrational approximation.\nORAM also requires to store $N_r$ psudofermion field vectors.\nAs a comparison, \nFIM requires $2N_{CG}(2N_{cut}+1)$ number of matrix multiplications\nto obtain the solution vector $X_1$ and $12N_{cut}$ more matrix\nmultiplications to obtain the fermionic force, where $N_{cut}$\nis the highest order of Legendre polynomials kept in the series\nexpansion.  FIM also needs to store $2N_{cut}$ psudofermion field vectors.\n\nConcerning the CPU time consumption, both method are more costly\nthan dynamical simulations with Wilson fermions by a factor\nof $2N+1$, where $N=N_{cg}$ for ORAM and $N=N_{cut}$ for FIM.\nFrom the theoretical upper bound of the error, the two methods\nbehave in a similar manner, $N_{cg} \\sim N_{cut}$. Practically,\nhowever, according\nto the experience in \\cite{edwards}, $N_{cg}$ is usually less\nthan $N_{cut}$ because the theoretical bound is saturated for\nFIM  while it is not for ORAM. Therefore, ORAM is more favorable\ncompared with FIM when doing simulations \nwith dynamical Ginsparg-Wilson\nfermions. Concerning the memory consumption, $N_r$ is usually much\nless than $N_{cut}$ which would \nagain put ORAM in a more favorable place.\n\nIt is clear from the above discussion that, if one would like\nto accelerate the simulation with dynamical Ginsparg-Wilson fermions,\none has to find ways to decrease $N_{cg}$ in ORAM or $N_{cut}$ in FIM.\nThese two parameters are mainly determined by the condition number\nof the matrix $H^{\\dagger}_WH_W$. \nAny preconditioning methods that would decrease the condition\nnumber of the matrix (while still maintaining the\nshifted nature of the matrix in ORAM)\n will bring an improvement to the simulation of\ndynamical Ginsparg-Wilson fermions.\nIt should be pointed out that other improvements, for example using \nbetter integration schemes, would apply to both methods.\nTest runs on small lattices are now under \ninvestigation \\cite{chuan2} where these \nalgorithmic issues will be further studied.\n\n\\section{Conclusions}\n\nIn this paper, possibilities of simulating dynamical Ginsparg-Wilson fermions\nare discussed.  The formula for\nthe fermionic force is derived for two specific\nimplementations of the algorithm, the optimal rational\napproximation method (ORAM) and fractional inverter method (FIM).\nIt turns out that, in both methods, \nsimulating dynamical Ginsparg-Wilson fermions are more costly than\nsimulating dynamical Wilson fermions. The extra CPU time\nconsumption mainly comes from the\nthe fractional inversion of the matrix.\nIn quenched simulations, it has been reported \\cite{edwards} \nthat ORAM performs better than FIM. In dynamical simulations,\nthis conclusion remains, both on CPU time consumption and \nmemory consumption.\nIt should be emphasized that, though being more costly, \nthe advantage of simulating dynamical Ginsparg-Wilson\nfermions over dynamical Wilson fermions or\nquenched Ginsparg-Wilson fermions would be a much better\nbehavior towards the chiral limit.\nThe feasibility of such simulation has been demonstrated in\nthis paper using a Hybrid Monte Carlo algorithm. \n\n\n\\section*{Acknowledgments}\nThis work is supported by the Chinese National Science Foundation \nand by the Natural Science Fund from Peking University.\n\n\\input refs.tex \n\n\\end{document}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n \n \n In models extended from the standard model (SM) Higgs sector, strongly first order phase transition (1stOPT) can be realized.\n We emphasize that the nature of electroweak phase transition (EWPT) can be probed by exploring the Higgs sector at ongoing and future experiments.\n Models predicting significant deviations in various Higgs boson couplings can be tested at the LHC~\\cite{CMS:2013xfa} as well as at future lepton colliders including, the International Linear Collider (ILC)~\\cite{ILC}, the Compact Linear Collider (CLIC)~\\cite{CLIC} and the Future Circular Collider of electrons and positrons (FCC-ee)~\\cite{FCC-ee}.\n\n On the cosmological side, the strongly 1stOPT that occurs in the early Universe produces stochastic gravitational waves (GWs).\n  In future, planned space-based interferometers such as LISA~\\cite{Seoane:2013qna}, DECIGO~\\cite{Kawamura:2011zz} and BBO~\\cite{Corbin:2005ny} will survey GWs in the millihertz to decihertz range, which is the typical frequency of GWs from the 1stOPT at the electroweak scale.\n\n Among various extensions of the Higgs sector, we here focus on a model with gauged dark $U(1)_X^{}$ symmetry including the $U(1)$ gauge kinetic mixing term.\n In general, $U(1)$ extended models are also testable at various experiments for the dark photon search.\n\n\\section{Model with dark $U(1)_X$  gauge symmetry}\n\\label{sec:model}\n\n We consider a model with a dark sector where the $U(1)_X^{}$ Abelian gauge symmetry is spontaneously broken by the so-called dark Higgs mechanism. \n We introduce a complex scalar $S$ with $U(1)_X^{}$-charge $Q_X^{}$ and the $U(1)_X^{}$ gauge field (dark photon) $X_\\mu^0$. \n In generic, there appears the gauge kinetic mixing term between the $U(1)_X^{}$ gauge boson $X_\\mu^0$ and the hypercharge $U(1)_Y^{}$ gauge boson $B_\\mu^{}$~\\cite{Holdom:1985ag}, and the Lagrangian is given by (e.g. Ref.~\\cite{Addazi:2017gpt})\n\n\\begin{align}\n{\\cal L} = - \\frac{1}{4} X_{\\mu\\nu} X^{\\mu\\nu} - \\frac{\\epsilon}{2} X_{\\mu \\nu} B^{\\mu \\nu} + |D_\\mu S|^2 - V_0\n\\label{eq:lagrangian}\n\\end{align}\nwhere $X_{\\mu\\nu}=\\partial_\\mu X_\\nu^0 - \\partial_\\nu X_\\mu^0$ and\n$B_{\\mu\\nu}^0=\\partial_\\mu B_{\\nu}^0 - \\partial_\\nu B_{\\mu}^0$, \\\n and the covariant derivative is defined as $D_\\mu = \\partial_\\mu + i g_X Q_X X_\\mu^0$.\nHere, the Higgs potential is given by \n\\begin{align}\nV_0^{} =\n-\\mu_\\Phi^2|\\Phi|^2\n-\\mu_S^2 |S|^2\n+\\lambda_\\Phi^{} |\\Phi|^4\n+\\lambda_S^{} |S|^4\n+\\lambda_{\\Phi S}^{} |\\Phi|^2 |S|^2.\n\\label{eq:full_theory}\n\\end{align}\n We normalize the $U(1)_X^{}$ charge of $S$, $Q_S^{} \\equiv Q_X^{}(S)$; $Q_S^{}=1$.\n Since the viable parameter range for $\\epsilon$ is too small to affect PT~\\cite{Addazi:2017gpt}, we focus on the rest six parameters, i.e. $\\mu^2_\\Phi, \\mu^2_S, \\lambda_\\Phi^{}, \\lambda_S^{}$, $\\lambda_{\\Phi S}^{}$ and $g_X^{}$.\n\n After the EW symmetry breaking, the two Higgs multiplets can be expanded as\n$\\Phi=(\nw^+, \\frac{1}{\\sqrt{2}}(v_\\Phi+\\phi_\\Phi+i z^0)\n)^T, \nS=\\frac{1}{\\sqrt{2}}(v_S+\\phi_S+i x^0)$, \nwhere $v_\\Phi$ and $v_S$ are the corresponding vacuum expectation values (VEVs). \n The Nambu-Goldstone modes $w^\\pm$, $z^0$ and $x^0$ are absorbed by the gauge bosons $W^\\pm_\\mu$, $Z^0_\\mu$ and $X_\\mu^0$.\n The mass of $X_\\mu^0$ is $m_{X} = g_X |Q_S| v_S$ (see also Ref.~\\cite{Farzan:2012hh}).\n The interaction basis state ($\\phi_\\Phi^{}$, $\\phi_S^{}$) is diagonalized to their mass eigenstate ($h$, $H$) through the rotation matrix with $c_\\theta\\equiv\\cos\\theta$ and $s_\\theta\\equiv\\sin\\theta$.\n We take $v_\\Phi^{}(= 246~{\\rm GeV})$, $m_h^{}(=125~{\\rm GeV})$, $m_H^{}$, $\\theta$, $g_X$ and $m_X$ as input parameters.\n The tree-level interactions of $h$ and $H$ with the SM gauge bosons $V$(=$W_\\mu^\\pm$, $Z_\\mu^0$) and with the SM fermions  $F$ are given by\n${\\cal L}_{\\Phi V V, \\Phi FF} = (h c_\\theta-H s_\\theta) \\{(2 m_W^2\/v_\\Phi) W^+_\\mu W^{- \\mu}+ (m_Z^2\/v_\\Phi)  Z^0_\\mu Z^{0 \\mu} -\\sum_F (m_F\/v_\\Phi)  \\bar{F} F \\}$. \n The couplings of $h$ is normalized by the corresponding SM ones are universally given by\n\\begin{align}\n\\kappa \\equiv \n\\frac{g_{h VV}}{g_{h VV}^{\\rm SM}} =\\frac{g_{h FF}}{g_{h FF}^{\\rm SM}}=c_\\theta. \n\\end{align}\n\n\\section{Numerical results}\n\\label{sec:results}\n\n In Fig.~\\ref{fig:200+1_1}, various types of multi-step PT that predict first order EWPT are marked with colored plots.\n The gray plots are insensitive at future GW observations, LISA~\\cite{Caprini:2015zlo} and DECIGO~\\cite{Kawamura:2011zz}.\n We find that there are still detectable regions satisfying the current collider constraints shown below.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.57\\textwidth]{gx2+1mx200_1.eps}\n \\caption{\\label{fig:200+1_1}\n Types of multi-step PT on the $(m_H^{},\\theta)$ plane for the benchmark point $m_X=200~{\\rm GeV}$ and $g_X=2$.\n Parameter sets predicting one-step PT with 1st order are marked with blue closed square, one-step PT with 2nd order with blue open square, two-step PT where both transitions are 1st order with green closed star, two-step PT where the latter one is 1st order with green closed triangle, and two-step PT where the former one is 1st order with green open triangle.\n The gray plots are insensitive at future GW observations.\n The colored regions are excluded by perturbative unitarity (black), vacuum stability (brown).\n The black lines show the combined exclusion limit obtained by $\\kappa_Z^{}$ measurement (orange) and direct searches for $H$ (gray).\n The black dashed lines and dotted lines show the expected accuracy of $\\kappa$ measurements at HL-LHC (14~TeV, 3~${\\rm ab}^{-1}$) and ILC (250~GeV, 2~${\\rm ab}^{-1}$).\n }\n\\end{figure}\n\n The measurements of the Higgs boson decay into weak gauge bosons give constraints on the $hVV$ couplings as $\\kappa_Z^{}=1.03^{+0.11}_{-0.11}$ and $\\kappa_W^{}=0.91^{+0.10}_{-0.10}$ from the ATLAS and CMS combination of the LHC Run-I data (68\\% CL)~\\cite{TheATLASandCMSCollaborations:2015bln}.\n In our numerical analysis, we take the 68\\% CL bound $\\kappa_Z^{}>0.92$ as the lower bound on the mixing angle, namely $|\\theta| \\leq 23.1^\\circ$. \n The exclusion limits from the direct searches for the $H$ boson at the LEP and LHC Run-II are examined in Ref.~\\cite{Robens:2015gla}.  \n We will show that a large portion of the model parameter space where strongly 1stOPT and detectable GW signals are possible is excluded by the collider bounds on the Higgs bosons discussed above.\n\n The expected accuracy of the measurements of the Higgs boson couplings are also displayed in Fig.~\\ref{fig:200+1_1} as follows. \n The high-luminosity (HL)-LHC with $\\sqrt{s}=14$~TeV and $L=3~{\\rm ab}^{-1}$ can constrain $\\Delta \\kappa_V$ with an accuracy of $2\\%$~\\cite{CMS:2013xfa}.\n Future $e^+e^-$ colliders can considerably ameliorate the precision.\n The stage of the ILC with $\\sqrt{s}=250$~GeV and $L=2~{\\rm ab}^{-1}$ can limit $\\Delta \\kappa_W^{}$ to 1.8\\% and $\\Delta \\kappa_Z^{}$ to 0.38\\%~\\cite{Fujii:2017vwa}.\n In addition, the limit obtained from direct searches for the $H$-boson at future colliders are discussed in Ref.~\\cite{Chang:2017ynj} for the small mass region and in Ref.~\\cite{Carena:2018vpt} for the large mass region.\n\n\n\n In Fig.~\\ref{fig:benchmark}, our numerical results about the EWPT and GW signals for the six benchmark points are defined.\n Scanning the parameter region with the collider bounds on the Higgs boson properties into consideration, we have found that GW signals are detectable only for larger dark photon mass, say $m_X^{}\\gtrsim {\\cal O} (25-100)~{\\rm GeV}$.\n As shown in Ref.~\\cite{He:2017zzr}, the recent data from LHCb~\\cite{Aaij:2017rft} and LHC Run-II~\\cite{Aaboud:2017buh} give constraints on $\\epsilon$, which is roughly smaller than $10^{-2}$ at least, for the mass regions $10.6~{\\rm GeV} <m_X <70~{\\rm GeV}$ and $150~{\\rm GeV} <m_X <350~{\\rm GeV}$, respectively.\n It is also shown that the mass region $20~{\\rm GeV} <m_X <330~{\\rm GeV}$ can be constrained at future lepton colliders in Ref.~\\cite{He:2017zzr}.\n We expect that 1stOPT with such a heavy dark photon will be tested by synergy between future GW observations and dark photon searches.\n\n\\begin{figure}[t]\n\\centering\n  \\includegraphics[width=0.57\\textwidth]{search.eps}\n   \\caption{\\label{fig:benchmark}\n The six benchmark points are blobbed on the ($m_X, g_X$)-plane.\n The solid lines show the cases of $v_S$=50~GeV (blue), 100~GeV (orange), 200~GeV (green) from the top.\n The gray plots in the small $g_X$ region shows that any parameters on the $(m_H^{},\\theta)$ cannot reach to the sensitivity of future GW observations.\n  }\n\\end{figure}\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\n We have comprehensively explored models with a dark photon whose mass stems from spontaneous $U(1)_X^{}$ gauge symmetry breaking by the nonzero VEV of the dark Higgs boson $S$ in light of the patterns of PT and the detectability of GWs from strongly 1stOPT as well as various collider and theoretical bounds.\n After imposing these constraints on the model parameter space, we have found that GWs produced from multi-step PT can be detected at future observations such as LISA and DECIGO, if the dark photon mass is $m_X^{} \\gtrsim 25~{\\rm GeV}$ with the $U(1)_X^{}$ gauge coupling being $g_X^{} \\gtrsim 0.5$.\n We have found that the parameter regions predicting detectable GWs are covered by the measurement of $\\kappa^{}$ at future colliders including the HL-LHC and ILC.\n\n\\acknowledgments\n\n This work is based on the collaboration with Katsuya Hashino, Mitsuru Kakizaki, Shinya Kanemura and Pyungwon Ko. I would like to thank them for their support.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nA new neutral gauge boson, $Z'$, is the most well-studied of all exotic \nparticles and is the hallmark signature for extensions of the SM gauge group. \nIf such a particle is found at future colliders the next step will be to \nascertain its couplings to fermions. At hadron colliders, \na rather long list of observables has been proposed over the years \nto probe these \ncouplings--each with its own limitations{\\cite {rev}}. It has been shown, \nat least within the context of $E_6$-inspired \nmodels, that the LHC($\\sqrt s=14$ TeV, $100fb^{-1}$) will be able \nto extract useful \ninformation on all of the $Z'$ couplings for $M_{Z'}$ below $\\simeq 1-1.5$ \nTeV. At the NLC, when $\\sqrt s < M_{Z'}$, a $Z'$ manifests itself only \nindirectly as deviations in, {\\it e.g.}, cross sections and asymmetries from their SM \nexpectations. Fortunately the list \nof useful precision measurements that can be performed at the NLC is \nreasonably long and the expected large beam polarization($P$)  \nplays an important role. In the past, analyses of the ability of \nthe NLC to extract $Z'$ coupling information in this situation have taken \nfor granted that the \nvalue of $M_{Z'}$ is already known from elsewhere, {\\it e.g.}, the LHC{\\cite {rev}}. \nHere we address the more complex issue of whether it is possible for the NLC to \nobtain information on couplings of the $Z'$ if the mass were for some \nreason {\\it a priori} unknown. In this case we would not only want to \ndetermine couplings but the $Z'$ mass as well.\n\n\nIf the $Z'$ mass were unknown it would appear that the traditional NLC $Z'$ \ncoupling analyses would become problematic. Given \ndata at a fixed value of $\\sqrt s$ which shows deviations from the SM, one \nwould not be able to {\\it simultaneously} extract the value of $M_{Z'}$ as \nwell as the corresponding couplings. The reason is clear: to leading order in \n$s\/M_{Z'}^2$, rescaling \nall of the couplings and the value of $M_{Z'}$ by a common factor \nwould leave the observed deviations from the SM invariant. In this \napproximation, the $Z'$ exchange appears only as a contact interaction. \nThus as long as $\\sqrt s < M_{Z'}$, the only potential solution to this \nproblem lies in obtaining data on deviations from the \nSM at {\\it several}, distinct $\\sqrt s$ values and combining them into a \nsingle fit. Here we report on the first analysis \nof this kind, focussing on observables involving only leptons and\/or  \n$b$-quarks. In performing such an analysis, how many $\\sqrt s$ values are \nneeded? How do we distribute the luminosity($\\cal L$) to optimize the \nresults? Can such an analysis be done while maintaining model-independence? \nIn this {\\it initial} study we begin to address these and some related \nquestions. \n\n\\section{Analysis}\n\nIn order to proceed with this benchmark study, we will make a number of \nsimplifying assumptions and parameter choices. These can be modified at a \nlater stage to \nsee how they influence our results. In this analysis we consider the following \nten observables: $\\sigma_{\\ell,b}$, $A_{FB}^{\\ell,b}$, $A_{LR}^{\\ell,b}$, \n$A_{pol}^{FB}(\\ell,b)$, $<P_\\tau>$, and $P_\\tau^{FB}$. Other inputs and \nassumptions are as follows:\n\\begin{tabbing}\n   Anomalous Trilinear Gauge Couplings Fun \\= 8.Fun and games with Extended \nTechnicolor models \\kill\n   ~~~~~~~e,$\\mu$,$\\tau$ universality \\> ISR with $\\sqrt {s'}\/\\sqrt {s} >0.7$\\\\\n   ~~~~~~~$P=90\\%$, $\\delta P\/P=0.3\\%$  \\> $\\delta {\\cal L}\/ {\\cal L}=0.25\\%$\\\\\n   ~~~~~~~$\\epsilon_b=50\\%$, $\\Pi_b=100\\%$  \\>   $|\\theta|>10^\\circ$\\\\\n   ~~~~~~~$\\epsilon_{e,\\mu,\\tau}(\\sigma)=100\\%$, $\\epsilon_\\tau(P_\\tau)=50\\%$ \n   \\> Neglect $t$-channel exchange in $e^+e^-\\rightarrow e^+e^-$ \n\\end{tabbing}\nOf special note on this list are ($i$) a $b$-tagging \nefficiency($\\epsilon_b$) of $50\\%$ for a purity($\\Pi_b$) of 100$\\%$, ($ii$) \nthe efficiency for identifying all leptons is assumed to be 100$\\%$, although \nonly $50\\%$ of $\\tau$ decays are assumed to be polarization analyzed, ($iii$) \na $10^\\circ$ angle cut has been applied to all final state fermions, and \n($iv$) a strong cut to  events with an excess of initial state \nradiation(ISR) has also been made. \nIn addition to the above, final state QED as well as QCD corrections are \nincluded, the \n$b$-quark and $\\tau$ masses have been neglected, and the possibility of \n$Z-Z'$ mixing has been ignored. Since \nour results will generally be statistics limited, the role played by the \nsystematic uncertainties associated with the parameter choices above will \ngenerally be rather minimal. \n\nTo insure model-independence, the values of the $Z'$ couplings, {\\it i.e.}, \n$(v,a)_{\\ell,b}$, as well as $M_{Z'}$, are chosen {\\it randomly} and \n{\\it anonymously} from \nrather large ranges representative of a number of extended gauge models. \nMonte Carlo data representing the above observables \nis then generated for several different values of $\\sqrt s$. At this point, the \nvalues of the mass and couplings are not `known'  \n{\\it a priori}, but will later be compared with what is extracted \nfrom the Monte Carlo generated event sample. Following this approach \nthere is no particular relationship between any of the couplings and \nthere is no dependence upon any particular $Z'$ model. (We normalize our \ncouplings so that \nfor the SM $Z$, $a_\\ell=-1\/2$.) Performing this analysis for a wide range of \npossible mass and coupling choices then shows the power as well as the \nlimitations of this technique. \n\nTo get an understanding for how this procedure works in general we will make \ntwo case studies for the $Z'$ mass and couplings, labelled here by I and II. \nThere is nothing special about these two choices and several other parameter \nsets have been analyzed in comparable detail to show that the results that \nwe display below are rather typical. To begin \nwe generate Monte Carlo data at $\\sqrt {s}=$0.5, 0.75 and 1 TeV \nwith associated integrated luminosities of 70, 100, and 150 \n$fb^{-1}$, respectively, and subsequently \ndetermine the 5-dimensional $95\\%$ CL allowed region for the mass and \ncouplings from a simultaneous fit using the assumptions listed above. This \n5-dimensional region is then \nprojected into a series of 2-dimensional plots which we can examine in detail. \nFigs. 1 and 2 show the results of our analysis for these two case studies \ncompared \nwith the expectations of a number of well-known $Z'$ models{\\cite {rev}}. \nSeveral things are immediately apparent-the most obvious being that two \ndistinct allowed regions are obtained from the fit in both cases. (The input \nvalues are seen to lie nicely inside one of them.) This two-fold ambiguity \nresults from our inability to make a  \ndetermination of the overall sign of {\\it one} of the couplings, {\\it e.g.}, \n$a_\\ell$. If the sign \nof  $a_\\ell$ were known, only a single allowed region would appear in \nFigs. 1a-b and 2a-b and a unique coupling determination would be obtained. \nNote that this {\\it same} sign ambiguity arises in SLD\/LEP data for the \nSM $Z$ and is only removed through the examination of low-energy neutrino \nscattering. Secondly, we see that the leptonic couplings \nare somewhat better determined than are those of the $b$-quark, which is \ndue to the fact that the leptonic \nobservables involve only leptonic couplings, while \nthose for $b$-quarks involve both \ntypes. In addition, there is more statistical power available in the lepton \nchannels due to the assumption of universality and the \nleptonic results employ two additional observables related to $\\tau$ \npolarization. Thirdly, we see from \nFigs. 1a-b the importance in obtaining coupling information for a number of \ndifferent fermion species. If only the Fig. 1a results were available, one \nmight draw the hasty conclusion that an $E_6$-type $Z'$ had been found. Fig. 1b \nclearly shows us that this is not the case. Evidently {\\it neither} $Z'$ \ncorresponds to any well-known model. Lastly, as promised, the $Z'$ mass is \ndetermined in both cases, although with somewhat smaller uncertainties in \ncase II. We remind the reader that there is nothing special about these two \nparticular cases. \n\n\\vspace*{-0.5cm}\n\\noindent\n\\begin{figure}[htbp]\n\\centerline{\n\\psfig{figure=zdrzpfig1a.ps,height=9.1cm,width=9.1cm,angle=-90}\n\\hspace*{-5mm}\n\\psfig{figure=zdrzpfig1b.ps,height=9.1cm,width=9.1cm,angle=-90}}\n\\vspace*{-0.75cm}\n\\centerline{\n\\psfig{figure=zdrzpfig1c.ps,height=9.1cm,width=9.1cm,angle=-90}}\n\\vspace*{-1cm}\n\\caption{\\small $95\\%$ CL allowed regions for the extracted values of the \n(a) lepton and (b) $b$-quark couplings \nfor the $Z'$ of case I compared with the predictions of the $E_6$ \nmodel(dotted), the Left-Right Model(dashed), and the Un-unified \nModel(dash-dot), \nas well as the Sequential SM and Alternative LR Models(labeled by `S' and `A', \nrespectively.) (c) Extracted $Z'$ mass; only the $a_\\ell >0$ branch is shown. \nIn all cases the diamond represents the corresponding input values.}\n\\end{figure}\n\n\n\\vspace*{-0.5cm}\n\\noindent\n\\begin{figure}[htbp]\n\\centerline{\n\\psfig{figure=zdrzpfig2a.ps,height=9.1cm,width=9.1cm,angle=-90}\n\\hspace*{-5mm}\n\\psfig{figure=zdrzpfig2b.ps,height=9.1cm,width=9.1cm,angle=-90}}\n\\vspace*{-0.75cm}\n\\centerline{\n\\psfig{figure=zdrzpfig2c.ps,height=9.1cm,width=9.1cm,angle=-90}}\n\\vspace*{-1cm}\n\\caption{\\small Same as Fig. 1 but for a different choice of $Z'$ mass and \ncouplings referred to as case II in the text.}\n\\end{figure}\n\n\nOf course, the clever reader must now be asking the question `why use 3 \ndifferent \nvalues of $\\sqrt s$, why not 2 or 5?' This is a very important issue which we \ncan only begin to address here. Let us return to the mass and couplings of \ncase I and generate Monte Carlo `data' for \nonly $\\sqrt s$=0.5 and 1 TeV with $\\cal L$= 100 and 220 $fb^{-1}$, \nrespectively, thus keeping the total $\\cal L$ the {\\it same} as in the \ndiscussion above. Repeating our analysis we then \narrive at the `2-point' fit as shown in Fig. 3a; unlike Fig. 1a, the \nallowed region does not \nclose and extends outward to ever larger values of \n$v_\\ell,a_\\ell$. The corresponding \n$Z'$ mass contour also does not close, again extending outwards to ever larger \nvalues. We realize immediately that this is just what happens when data at \nonly a single $\\sqrt s$ is available. For our fixed $\\cal L$, distributed as we \nhave done, we see that there is not enough of a lever arm to simultaneously \ndisentangle the $Z'$ mass and \ncouplings. Of course the reverse situation can also be just as bad. We   \nnow generate Monte Carlo `data' for the case I mass and couplings in 100 GeV \nsteps in $\\sqrt s$ over the 0.5 to 1 TeV interval with the same total \n$\\cal L$ as above but now distributed as 30, 30, 50, 50, 60, and 100 $fb^{-1}$, \nrespectively. We then arrive at the `6-point' fit shown in Fig. 3b \nwhich suffers \na problem similar to Fig. 3a. What has happened now is that we have spread \nthe fixed $\\cal L$ too thinly over too many points for the \nanalysis to work. This brief study indicates that a proper balance is \nrequired to simultaneously achieve the desired statistics as well as an \neffective lever arm to obtain the $Z'$ mass and couplings. It is important \nto remember that we \nhave {\\it not} demonstrated that the `2-point' fit will never work. We note \nonly that it fails with our specific fixed luminosity distribution for the \nmasses and couplings associated with cases I and II. It is possible that for \n`lucky' combinations of masses and couplings a 2-point fit will suffice. \nClearly, more work is required to further address this issue.  \n\n\\vspace*{-0.5cm}\n\\noindent\n\\begin{figure}[htbp]\n\\centerline{\n\\psfig{figure=zdrzpfig3a.ps,height=9.1cm,width=9.1cm,angle=-90}\n\\hspace*{-5mm}\n\\psfig{figure=zdrzpfig3b.ps,height=9.1cm,width=9.1cm,angle=-90}}\n\\vspace*{-1cm}\n\\caption{\\small Failure of the method in case I when data is taken at \n(a) too few (`2-point' fit) or (b) too many (`6-point' fit) different \ncenter of mass energies for the same total integrated \nluminosity as in Figs. 1 and 2. The luminosities are distributed as discussed \nin the text.}\n\\end{figure}\n\\vspace*{0.4mm}\n\n\nHow do these results change if $M_{Z'}$ {\\it were} known or if our input \nassumptions were modified? Let us return to case I and concentrate on the \nallowed \ncoupling regions corresponding to a choice of negative values of \n$v_{\\ell,b}$; these are \nexpanded to the solid curves shown in Figs. 4a and 4c. The large dashed curve \nin Fig. 4a corresponds to a reduction of the polarization to $80\\%$ with the \nsame relative error as before. While the allowed region expands the \ndegradation is not severe. If the $Z'$ mass were known, the `large'  \nellipses shrink to \nthe small ovals in Fig. 4a; these are expanded in Fig. 4b. This is clearly a \nradical reduction in the size of the allowed region! We see that when the \nmass is known, varying the polarization or its uncertainty over a reasonable \nrange has very little influence on the resulting size of the allowed \nregions. From Fig. 4c we see that while knowing the $Z'$ mass significantly \nreduces the size of the allowed region for the $b$ couplings, the impact is \nfar less than in the leptonic case \nfor the reasons discussed above. Figs. 5a and 5b show that case I is not \nspecial in that similar results are seen to hold for case II.\n\n\\vspace*{-0.5cm}\n\\noindent\n\\begin{figure}[htbp]\n\\centerline{\n\\psfig{figure=zdrzpfig4a.ps,height=9.1cm,width=9.1cm,angle=-90}\n\\hspace*{-5mm}\n\\psfig{figure=zdrzpfig4b.ps,height=9.1cm,width=9.1cm,angle=-90}}\n\\vspace*{-0.75cm}\n\\centerline{\n\\psfig{figure=zdrzpfig4c.ps,height=9.1cm,width=9.1cm,angle=-90}}\n\\vspace*{-1cm}\n\\caption{\\small (a) Expanded lobe(solid) from Fig. 1a; the dashed curve shows \nthe same result but for $P=80\\%$. The smaller ovals, expanded in (b) apply \nwhen the $Z'$ mass is known. Here, in (b), $P=90(80)\\%$ corresponds to the \ndash-dot(dotted) curve while the case of $P=90\\%$ with $\\delta P\/P=5\\%$ \ncorresponds to the square-dotted curve. (c) Expanded lobe(solid) from Fig.1b; \nthe dotted curve corresponds to the case when $M_{Z'}$ is known.} \n\\end{figure}\n\nWhat happens for larger $Z'$ masses or when data at larger values of $\\sqrt s$ \nbecomes available? Let us assume that the `data' from the above three center \nof mass \nenergies is already existent, with the luminosities as given. We now imagine \nthat the NLC increases its center of mass energy to $\\sqrt s$= 1.5 TeV and \ncollects an additional \n200 $fb^{-1}$ of integrated luminosity. Clearly for $Z'$ masses near or \nbelow 1.5 TeV our \nproblems are solved since an on-shell $Z'$ can now be produced. Thus we \nshall concern ourselves with $Z'$ masses in excess of 2 TeV. \nFigs. 6a-d  show the result of extending our procedure--now using 4 \ndifferent $\\sqrt s$ values, again for two distinct choices of the \n$Z'$ mass and \ncouplings. These `4-point' results are a combined fit to the data at \nall four center of mass energies. \n(Only one of the allowed pair of ellipses resulting from the overall sign \nambiguity is shown for simplicity.) \nNote that the $Z'$ input masses we have chosen are well in excess of 2 TeV \nwhere the LHC may provide only very minimal information on the fermion \ncouplings{\\cite {rev}}. Clearly by using the additional data from a \nrun at $\\sqrt s$=1.5 TeV this technique can be extended to perform coupling \nextraction for $Z'$ masses in excess of 2.5 TeV. The maximum `reach' for the \ntype of coupling analysis we have done is not yet known. It seems likely, \nbased on these initial studies, that the extraction of interesting coupling \ninformation for $Z'$ masses in excess of 3 TeV seems possible for a reasonable \nrange of parameters. \n\n\n\\vspace*{-0.5cm}\n\\noindent\n\\begin{figure}[htbp]\n\\centerline{\n\\psfig{figure=zdrzpfig5a.ps,height=9.1cm,width=9.1cm,angle=-90}\n\\hspace*{-5mm}\n\\psfig{figure=zdrzpfig5b.ps,height=9.1cm,width=9.1cm,angle=-90}}\n\\vspace*{-1cm}\n\\caption{\\small (a) Expanded lobe(solid) from Fig. 2a; the dashed curve shows \nthe same result but for $P=80\\%$. The smaller dotted oval, applies  \nwhen the $Z'$ mass is known and $P=90\\%$. (b) Expanded lobe(solid) from \nFig. 2b; \nthe dotted curve corresponds to the case when $M_{Z'}$ is known. }\n\\end{figure}\n\\vspace*{0.4mm}\n\n\n\\section{Outlook and Conclusions}\n\nIn this paper we have shown that it is possible for the NLC to extract \ninformation on the $Z'$ couplings to leptons and $b$-quarks even when the $Z'$ \nmass is not {\\it a priori} known. The critical step for the success of the \nanalysis was to combine the data available from measurements performed \nat several different center of mass energies. For reasonable luminosities \nthe specific results we have obtained suggest, but do not prove, that data \nsets at at least 3 different energies are necessary for the procedure to be \nsuccessful. \n\n\n\n\\bigskip  \\bigskip \\begin{center} \\begin{large\n\nThe author would like to thank J.L. Hewett and S. Godfrey for discussions \nrelated to this work.\n\n\\def\\MPL #1 #2 #3 {Mod.~Phys.~Lett.~{\\bf#1},\\ #2 (#3)}\n\\def\\NPB #1 #2 #3 {Nucl.~Phys.~{\\bf#1},\\ #2 (#3)}\n\\def\\PLB #1 #2 #3 {Phys.~Lett.~{\\bf#1},\\ #2 (#3)}\n\\def\\PR #1 #2 #3 {Phys.~Rep.~{\\bf#1},\\ #2 (#3)}\n\\def\\PRD #1 #2 #3 {Phys.~Rev.~{\\bf#1},\\ #2 (#3)}\n\\def\\PRL #1 #2 #3 {Phys.~Rev.~Lett.~{\\bf#1},\\ #2 (#3)}\n\\def\\RMP #1 #2 #3 {Rev.~Mod.~Phys.~{\\bf#1},\\ #2 (#3)}\n\\def\\ZP #1 #2 #3 {Z.~Phys.~{\\bf#1},\\ #2 (#3)}\n\\def\\IJMP #1 #2 #3 {Int.~J.~Mod.~Phys.~{\\bf#1},\\ #2 (#3)}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nUnmanned Underwater Vehicles (UUVs) have the capability to perform various operations including surveillance, exploration, and object detection in underwater environments. While traditional propeller-based propulsion enables UUVs to conduct deep water diving and high-speed traversal, flapping fin propulsion--inspired by the high levels of controllability observed in aquatic animals--offers high maneuverability at low speeds \\cite{Blake1979}. Flapping fin propulsion provides a solution for UUVs to effectively navigate near-shore, obstacle-ridden terrain.\n\nFlapping fin UUV control systems regulate vehicle propulsion by modifying the vehicle gait, the specific set of fin kinematics applied to the current flapping cycle. UUV control systems have been extensively explored \\cite{He2020}; however, there is sparse literature on flapping fin control. While the effect of various kinematics on propulsion has been studied through experimental \\cite{Santo2017}, computational fluid dynamics \\cite{Liu2017}, and surrogate model \\cite{Viswanath2019} approaches, prior UUV flapping fin control systems do not embed a full understanding of how gait affects propulsion. For example, they focus on experimentally determining a small set of high-propulsion gaits \\cite{Shan2019}, restrict chosen gaits to a line in the kinematic space \\cite{Bi2014}, or incrementally change kinematics that have a known positive or negative correlation with thrust to eventually reach the target propulsion \\cite{Palmisano2008}. We use a neural network model to embed a more comprehensive understanding of the relationship between gait and propulsion within a gait selection model; as a result, the control system can generate gaits that not only meet a target trajectory, but also optimize for other measures of performance such as a smooth transition between gaits and energy efficiency.\n\nWe propose gait generation using a search-based inverse model that invokes a forward surrogate model. Our work focuses on a control system for thrust, which is the forward propulsion of the vehicle. The inverse model determines the subsequent gait from the desired thrust, current gait, and relative performance metric weights for thrust accuracy and kinematic smoothness. Inverse model gait evaluation uses a neural network forward model to predict the thrust of a given gait. Figure \\ref{fig:config} shows the integration of the inverse model within the control system.\n\nSearch-based methods are frequently used for offline inverse problems as they require invoking the forward model multiple times \\cite{Zhou2012,Hansen2017}. Unlike other aerial and underwater vehicle control systems, our flapping fin control system only needs to generate one gait per flapping cycle, allowing for a more flexible time constraint. Therefore, search-based methods are a viable onboard approach that additionally allows for the incorporation of modifiable optimization parameters without retraining the forward model.\n\nAs an alternative to a search-based methods, autonomous aerial vehicle literature often uses neural networks in control systems to develop direct inverse models \\cite{Muliadi2018,ElHamidi2020}. In this approach, the inverse model consists of a neural network that is directly trained from a forward model or plant. While producing fast predictions, this method does not allow for a flexible optimizer that can be changed cycle-to-cycle by the controller to prioritize different performance metrics.\n\nWe demonstrate that our forward gait-to-thrust model accurately interpolates gaits, and we show that our thrust-to-gait search-based inverse model generates high thrust accuracy gaits while embedding adjustable performance weights. These weights allow a control system to make cycle-to-cycle trade-offs between performance metrics based on current system objectives. We compare the performance of different sampling and search-based techniques to improve upon our inverse model performance. Through inverse model integration on the Raspberry Pi, we demonstrate that the inverse model fulfills the one prediction per cycle time constraints, even with the use of an expensive time-series forward model.\n\n\\begin{figure}[t]\n  \\centering\n\t\\includegraphics[width=1.0\\linewidth]{NN_Control_System_Design4.png}\n\t\\caption{\n    Integration of the inverse model (shown in light blue) within a control system for position. Orange arrows indicate one request per thrust cycle.\n\t}\n\t\\label{fig:config}\n\\end{figure}\n\n\n\\section{Methods}\n\nOur inverse model uses the input from the controller---desired thrust, current gait, and desired performance metric weights---to output a new gait with the goal of minimizing the inverse model loss function. Proposed gaits are generated using a sampling or direct search method. These gaits are then evaluated using a multi-objective loss function that evokes the forward gait-to-thrust neural network model.\n\n\\subsection{Loss Function}\n\nProposed gaits from our inverse model are evaluated with a loss function that is defined as the weighted sum of the loss functions for the different performance metrics. Equation \\ref{eq:loss} describes the overall loss function. The losses $L_{t}$, $L_{k}$, and $L_{e}$ correspond to the thrust accuracy, kinematic smoothness, and efficiency loss functions, while $w_{t}$, $w_{k}$, and $w_{e}$ serve as the corresponding performance metric weights.\n\\begin{equation}\n\\fontsize{10}{0} L = w_{t} * L_{t} + w_{k} * L_{k} + w_{e} * L_{e}\n\\label{eq:loss}\n\\end{equation}\n\nThe thrust accuracy loss is computed as the difference between $T_{target}$, the desired thrust, and $T_{pred}$, the thrust from the proposed gait of the inverse model (Eq. \\ref{eq:loss_t}). $T_{pred}$ is calculated using the gait-to-thrust neural network.\n\\begin{equation}\n  \\fontsize{10}{0} L_{t} = \\left|T_{target} - T_{pred}\\right|\n  \\label{eq:loss_t}\n\\end{equation}\n\nThe kinematic smoothness loss accounts for the detrimental effect of frequently moving between gaits with highly deviant kinematics between flapping cycles. Transitioning between similar gaits allows the UUV to undergo a smoother motion, promoting system stability. We define a user-selected equivalent step size such that a change of $s_i$ units for kinematic $i$ has the same kinematic smoothness loss to the system as a change of $s_j$ units for kinematic $j$. The kinematic space is normalized by scaling each dimension based on its equivalent step size; then, kinematic loss is calculated as the Euclidean distance between the current and proposed gait. Equation \\ref{eq:loss_k} defines the kinematic smoothness loss function where $n_{k}$ is the number of kinematics, and $x_{i}$ and $y_{i}$ are the values of kinematic $i$ for the current and proposed gait.\n  \\begin{equation}\n    \\fontsize{10}{0} L_{k} = \\left(\\sum_{i=1}^{n_{k}} \\left(\\frac{|y_{i} - x_{i}|}{s_{i}}\\right)^2\\right)^{\\frac{1}{2}}\n    \\label{eq:loss_k}\n  \\end{equation}\n\nOur efficiency loss will be based upon the propulsive efficiency of the gait, which is equal to the product of output thrust and current velocity divided by power. As we do not currently have experimental positive flow cases available for forward gait-to-thrust training, our results set the efficiency weight to 0 and evaluate the trade-offs between thrust accuracy and kinematic smoothness.\n\n\\subsection{The Forward Model}\n\nThe forward model predicts average UUV thrust for a flapping cycle from the gait. Reduced-order analytic models can produce fast predictions, but they struggle to maintain accuracy when generalized beyond a small parameter space \\cite{Muscutt2017}. A model supporting a higher-order input space will allow future forward gait-to-propulsion models to incorporate fluid dynamics-related parameters such as flow speed as well as multi-fin kinematic parameters such as the flapping phase offset between front and rear fins.\n\nNeural network surrogate models support higher degree input spaces, and prior flapping fin propulsion research on fin design has developed neural network surrogate models for thrust prediction \\cite{Viswanath2019,Lee2021}. Both works demonstrate that time series models can predict the time history of thrust generation for a flapping cycle. Therefore, we implement a similar long-term short memory (LSTM) network for our gait-to-thrust forward model using the inputs in Table \\ref{tab:parameters}. A single pectoral fin setup is shown in Figure \\ref{fig:single_fin_setup}. LSTM networks process sequential data by generating an output at each time step and using information from past outputs to inform subsequent results. Compared to traditional recurrent neural networks, LSTM networks include a cell state to retain a long-term memory accumulated from multiple past time steps that influences the next output.\n\n\\begin{figure}[t]\n  \\centering\n\t\\includegraphics[width=0.65\\linewidth]{Single_Fin_Setup.png}\n\t\\caption{\n    Example single pectoral fin setup.\n\t}\n\t\\label{fig:single_fin_setup}\n\\end{figure}\n\nOur control system requests one gait per flapping cycle, so only the average thrust over a cycle is used by our system. While the LSTM model computes average thrust as the mean of the output thrusts, we also test the viability of a simple dense neural network (DNN) model that uses the static kinematic values in Table \\ref{tab:parameters} to directly predict average thrust. DNN networks consist of layers of nodes such that each node in layer $l$ is connected to every node in layer $l-1$. Both models are discussed in our results.\n\n\\begin{table}[ht]\n\\footnotesize\n\\begin{center}\n \\caption{Parameters that compose a UUV gait}\n \\label{tab:parameters}\n \\begin{tabular}{p{3.1cm} p{4.5cm}}\n \\hline\n \\hline\n Kinematic & Description \\\\\n \\hline\n \\hline\n \\textit{Static kinematics} &  \\\\\n \\hline\n Stroke Amplitude (\u00b0) & Maximum gait stroke angle \\\\\n \\hline\n Pitch Amplitude (\u00b0) & Maximum gait pitch angle \\\\\n \\hline\n Flap Frequency (Hz) & Frequency of a stroke cycle \\\\\n \\hline\n Stroke-Pitch Offset & Phase offset of the pitch cycle relative to the stroke cycle, calculated as a fraction of one cycle  \\\\\n \\hline\n \\textit{Time-varying kinematics} &  \\\\\n \\hline\n Stroke Angle (\u00b0) & Flapping angle as a function of time \\\\\n \\hline\n Pitch Angle (\u00b0)  & Pitching angle as a function of time \\\\\n\\end{tabular}\n\\end{center}\n\\end{table}%\n\n\\subsection{Sampling and Direct Search Methods}\n\nMonte Carlo sampling and direct search methods are common derivative-free approaches for optimization based on an objective function \\cite{Kroese2014,Audet2014}. Monte-Carlo sampling involves randomly sampling the domain of the objective function based on a probability distribution, while direct search methods use a heuristic approach where past searches influence future attempts. A popular and effective class of direct search methods is the generalized pattern search (GPS) family of algorithms defined by Torczon \\cite{Torczon1997}. When applied to optimization problems, GPS often provides fast convergence and high accuracy solutions \\cite{Herrera2015,Javed2016}.\n\nFor our flapping fin inverse model, each static kinematic listed in Table \\ref{tab:parameters} serves as an input for the objective function, and the loss function in Equation \\ref{eq:loss} acts as the objective function output. We restrict our input space to the domain of provided experimental data as well as the approximate set of attainable gaits as described in the Experimental Data section. The input space is normalized through compressions in the direction of each static kinematic setting $i$ by the corresponding equivalent step size of $s_{i}$ units; in the normalized input space, any movement between two points of distance $d$ has the same kinematic loss. Both GPS-based and Monte Carlo-based inverse models are implemented as specified below.\n\n\\subsubsection{Monte Carlo Approach}\n\nThe Monte Carlo method evaluates points from a hyper-sphere centered around the current gait in the input space, where the hyper-sphere radius $a$ determines the search scale. The search scale is calculated as $a = d_t\/10$, where $d_t$ is the difference between the thrust of the current gait and the target thrust. A uniform random sample of $n$ points satisfying the inequality in Equation \\ref{eq:MCIneq} are evaluated, and the gait with the lowest loss is selected. The variables $x_{i}$ and $y_{i}$ are the value of static kinematic $i$ for the current and new gaits respectively.\n\\begin{equation}\n\\fontsize{10}{0}\n\\sum_{n=1}^{n_{k}} \\left(\\frac{|y_{i} - x_{i}|}{a}\\right)^2 < 1\n\\label{eq:MCIneq}\n\\end{equation}\n\n\\subsubsection{Generalized Pattern Search Approaches}\n\nGeneralized pattern search algorithms provide a fast, derivative-free method for optimization. The objective function is evaluated at points on a mesh generated from a positive spanning set. GPS consists of a search and poll step. During the search step, a finite number of mesh points are evaluated from the objective function using a user-defined procedure; if an improvement is found for any proposed point, then the solution is accepted. If no new point is found, GPS polls neighboring mesh points to the current solution $x_i$ and accepts new points with improved objective function values. If both steps are unable to generate an improved solution, the mesh size $m$ is divided by a mesh size divider $d_m$ and the process is repeated until a certain precision $p$ is obtained. An outline of GPS is provided.\n\n\\begin{algorithm}[tb]\n\\caption{Outline of GPS}\n\\begin{algorithmic}\n    \\State Set initial solution $x_{prev}$, initial mesh size $m$, precision $p$;\n    \\While{$k \\ge p$}\n         \\State $x_{new} \\gets x_{prev}$;\n         \\State SEARCH for new solution in mesh, update $x_{new}$;\n         \\If{$x_{new}$ == $x_{prev}$}\n             \\State POLL for new solution in mesh, update $x_{new}$;\n             \\If{$x_{new}$ == $x_{prev}$}\n                  \\State $m \/= d_m$;\n             \\EndIf\n         \\EndIf\n    \\EndWhile\n\\end{algorithmic}\n\\end{algorithm}\n\nHooke-Jeeves pattern search (HJPS) is a commonly used pattern search algorithm \\cite{Hooke1961}. The mesh is created from the positive spanning set consisting of the standard basis vectors. For each input dimension, the poll step evaluates a neighboring mesh point in both the positive and negative directions, and $x_{prev}$ is updated if an improved solution is found. From this step, the vector $x_{new} - x_{prev}$ offers a promising direction for a continued search. The search step leverages this tactic by attempting to move the current solution in the direction $x_{new} - x_{prev}$ until no further improvement can be made.\n\n\n\\begin{algorithm}[tb]\n\\caption{Search Upon Failure for Our GPS Algorithm}\n\\begin{algorithmic}\n    \\State Set current gait and loss as $\\vec{g}_{curr}, l_{curr}$\n    \\State Set direction $\\vec{u}_k$ and magnitude $i_k$ of minimum loss increase for each kinematic $k$ during polling\n    \\State Sort $\\vec{u}_k$ by $i_k$ in ascending order\n    \\State $\\vec{v} \\gets \\sum_{c=0}^{n_k-1} \\vec{u}_k$\n    \\State $j \\gets n_k - 1$\n    \\While{$j \\ge 1$}\n        \\While{$(l_{pred} \\gets loss(\\vec{g}_{curr} + \\vec{v})) < l_{curr}$}\n            \\State $\\vec{g}_{curr} \\gets \\vec{g}_{curr} + \\vec{v}$\n            \\State $l_{curr} \\gets l_{pred}$\n            \\State $j \\gets 0$\n        \\EndWhile\n        \\State $\\vec{v} \\gets \\vec{v} - \\vec{u}_j$\n        \\State $j \\gets j - 1$\n    \\EndWhile\n\\end{algorithmic}\n\\end{algorithm}\n\nIn cases with a high kinematic smoothness weight, movement in any one coordinate direction results in a larger increase to kinematic loss than reduction in thrust accuracy loss, leading to a failed poll step and a resulting failed search step by HJPS. In these situations, HJPS is unable to escape the local minimum until the kinematic smoothness weight is lowered. With the goal of escaping these minima, our proposed generalized pattern search algorithm modifies HJPS to strategically search upon polling failure. Pseudocode for this additional search step is provided. Our GPS algorithm records the direction $\\vec{u}_k$ and magnitude $i_k$ of minimum loss increase for each kinematic $k$ during the poll step. Until failure, searches are conducted in the direction of vector $\\vec{v}$, where $\\vec{v}$ is equal to the sum of all $\\vec{u}_k$ and spans all $n_k$ kinematic directions. If the search fails, the component $\\vec{u}_k$ associated with the highest loss increase $i_k$ is removed from vector $v$; this process repeats until a search succeeds, or until the vector $v$ only spans one dimension.\n\n\\subsection{Experimental Data}\n\nTo train and evaluate our forward model, experimental thrust data for various gaits was collected for a setup consisting of rigid rectangular-shaped pectoral fins, one on each side of the UUV. Experiments were run for 864 gaits, which are combinations of the kinematics listed in Table \\ref{tab:expData}. Time-series data for five stroke cycles was collected for each gait.\n\n\\begin{table}[ht]\n\\footnotesize\n\\begin{center}\n \\caption{Provided experimental gaits}\n \\label{tab:expData}\n \\begin{tabular}{p{3.2cm} p{4.1cm}}\n \\hline\n \\hline\n Kinematic & Provided Values \\\\\n \\hline\n \\hline\n Stroke Amplitude (\u00b0) & 0, 15, 25, 32, 40, 55 \\\\\n \\hline\n Pitch Amplitude (\u00b0) & 0, 15, 25, 32, 38, 55 \\\\\n \\hline\n Flap Frequency (Hz) & 0.75, 1, 1.25, 1.5, 1.75, 2 \\\\\n \\hline\n Stroke-Pitch Offset & -0.0625, 0, 0.0625, 0.125  \\\\\n\\end{tabular}\n\\end{center}\n\\end{table}%\n\nIn order to invoke the LSTM network on interpolated gaits, the expected motor stroke and pitch angle time histories must be generated as an input for the network. The motors are commanded 16 values per cycle, where the commanded stroke follows a sinusoidal curve and commanded pitch oscillates between $-p_a$ and $p_a$, where $p_a$ is the pitch amplitude. Time histories are generated using a simple model of motor dynamics accounting for maximum velocity and acceleration. Across the provided experimental data with attainable gaits, mean stroke and pitch angle difference between the generated and experimental time histories was 2.94\u00b0 and 4.02\u00b0 for stroke and pitch respectively. An example generated and experimental time history is shown in Figure \\ref{fig:strokePitchTimeHist}.\n\nDue to physical system limitations, certain high commanded amplitudes are unattainable at high flap frequencies, $ff$, as the motor is unable to realize the full commanded amplitude within the provided time frame. Therefore, our inverse model excludes all gaits where the stroke amplitude exceeds $97 - ff * 30$ or the pitch amplitude exceeds $75 - ff * 26$. The above inequalities were experimentally determined.\n\n\\begin{figure}[t]\n  \\centering\n\t\\includegraphics[width=0.6\\linewidth]{Stroke_Pitch_Time_Histories_2.png}\n\t\\caption{\n    Sample stroke and pitch time history.\n\t}\n\t\\label{fig:strokePitchTimeHist}\n\\end{figure}\n\n\\section{Results and Discussion}\n\n\n\\subsection{Forward Model Performance}\n\nBoth the DNN and LSTM were evaluated using the experimental data described in Table 2. Forward passes of the models during evaluation are run using the Tensorflow Lite library \\cite{Tensorflow} to increase prediction speed. Z-score normalization was applied to each kinematic input. The DNN directly outputs average thrust over one flapping cycle while the LSTM outputs the full thrust time history for a cycle. The input stroke and pitch time histories for LSTM training and evaluation were generated based on the motor dynamics as described in the Experimental Data section. The LSTM contains 100 hidden units, and time histories consisted of 50 points that were evenly spaced over a flapping cycle. The DNN contains 3 layers with 100 nodes per layer. The LSTM was trained for 150 epochs, and the DNN was trained for 500 epochs.\n\nWhen trained and evaluated on all experimental gaits, the LSTM and DNN reached mean average thrust errors of 0.0174N and 0.0364N respectively, where the average thrust error refers to the difference between the predicted and experimental average thrust for a specific gait. The DNN produces a significantly higher error, signifying it is unable to effectively learn the gait-thrust relationship across the full space of experimental gaits. The increase in error is concentrated in the high flapping frequency, high amplitude gaits: physically unattainable gaits described in our Experimental Data section have a mean average thrust error of 0.0915 for the DNN. The inability of the DNN to memorize the full gait-thrust relationship poses a concern since the modeling task will grow more complex in the future through the addition of inputs accounting for flow speed and multi-fin kinematic interactions. Therefore, our inverse model will implement the LSTM model.\n\nTo test LSTM gait interpolation, a holdout set of gaits was excluded from training. Our holdout set consisted of all experimental gaits fulfilling one or more of the following criteria: a flap frequency of 1.25 Hz, a stroke-pitch offset of 0, or a stroke or pitch amplitude of 25\u00b0. The LSTM successfully interpolated kinematics for the excluded gaits with a mean average thrust error of 0.0344N. The worst performing subset of excluded gaits--gaits with a stroke pitch offset of 0--still obtained a mean average thrust error of 0.0374N.\n\nFigure \\ref{fig:LSTMThrustProf} shows example thrust time histories generated by the LSTM for interpolated gaits. The LSTM embeds an understanding of how thrust changes over the course of a flapping cycle, capturing the peak and troughs of the thrust time history; this understanding offers an explanation for the high-accuracy LSTM average thrust predictions for interpolated kinematics.\n\n\\begin{figure}[t]\n  \\centering\n\t\\includegraphics[width=1\\linewidth]{LSTM_Thrust_Profs.png}\n\t\\caption{\n        Sample thrust time histories for interpolated kinematics. The left graph involves interpolation to an unseen stroke and pitch angle, while the right graph involves interpolation to an unseen flap frequency and stroke pitch offset.\n\t}\n\t\\label{fig:LSTMThrustProf}\n\\end{figure}\n\n\\subsection{Inverse Model Performance}\n\nThree search-based methods for the inverse model were evaluated--Monte Carlo, Hooke-Jeeves Pattern Search (HJPS), and our Generalized Pattern Search (GPS) algorithm that builds upon HJPS. Monte Carlo generates 50 trial gaits ($n = 50$). The mesh size $k$ and precision $p$ for both pattern search algorithms is set to 3 and 0.375 respectively. The inverse models implement the LSTM forward model for gait-to-thrust prediction.\n\nSynthetic and simulated thrust requests were used to evaluate the inverse models. Each synthetic data set consists of a sequential list of 100 pseudo randomly generated thrust requests with a difference between 0 and $\\Delta T_{max}$ for adjacent thrusts. Thrust requests were restricted to the range 0.2N to 1.2N, and the value of $\\Delta T_{max}$ was varied from 0.1N to 1.0N in increments of 0.1N to produce 10 data sets.\n\nInverse model performance on synthetically generated thrust requests is shown in Figure \\ref{fig:SynDataInvModelResults}. As the maximum step size increases, the average kinematic loss generally trends upwards for all three models: the inverse models can obtain gaits closer in the kinematic input space for smaller changes in thrust. An increase in thrust weight for our inverse models results in a lower thrust loss and a higher kinematic loss. Therefore, priorities on different objectives--thrust accuracy and kinematic smoothness--can be directly regulated by a control system through a change in weights.\n\n\\begin{figure}[t]\n  \\centering\n\t\\includegraphics[width=1\\linewidth]{Synthetic_Data_Inverse_Model_Results.png}\n\t\\caption{\n        Inverse model performance for synthetic data, measured by thrust accuracy loss (left) and kinematic smoothness loss (right). The dotted, dashed, and solid lines have thrust weights of 0.9, 0.95, and 1. Kinematic weights are set to $1 - w_{t}$.\n\t}\n\t\\label{fig:SynDataInvModelResults}\n\\end{figure}\n\nThe models do not optimize for kinematic smoothness in the cases where the models have a thrust weight of 1, yet Monte Carlo still exhibits a low kinematic smoothness loss across the tested input thrust data. Compared to HJPS and GPS, the MC model has a tendency to find closer gaits in the input space without implicitly embedding kinematic smoothness into the loss function. However, HJPS and GPS successfully obtain similar kinematic smoothness values to MC for higher kinematic smoothness weights.\n\nThe GPS and MC inverse models demonstrate a high thrust accuracy performance across all three weights, achieving an average thrust loss of less than 0.04N. While the HJPS inverse model performs similarly to the GPS model for higher thrust weights, this model shows a significant deterioration in thrust accuracy when a higher kinematic smoothness weight of 0.1 is applied. The higher kinematic smoothness weight increases the overall cost of moving to a new gait; in these cases, the HJPS inverse model reaches a gait where movement in any coordinate direction results in a higher increase in kinematic smoothness loss than reduction in thrust accuracy loss. At this point, the algorithm becomes trapped at a local minimum for all subsequent iterations unless the thrust accuracy weight is increased. Our GPS method strategically searches promising points in situations where the algorithm is potentially trapped in a local minimum, namely during polling failure. These additional searches enable the GPS inverse model to maintain a low thrust loss for lower thrust weights.\n\nTo evaluate our inverse models on more realistic thrust requests, a PID controller for vehicle position was used for thrust request generation. The PID controller outputs a target thrust to control a simulated thrust-to-position plant that models the UUV based on the translational equations of motion for a rigid body. We simulate UUV movement to 100 randomly generated positions between 0 and 10m. For each location, the controller commands a thrust between -1.2N and 1.2N for 15 flapping cycles to reach and then remain at the position. A sample of the change in positional values and corresponding thrust requests from the simulation is provided in Figure \\ref{fig:PID_Pos_Thrust}. The inverse model generated kinematics is able to accurately track the target trajectory in the given timeframe. This experiment uses small positional changes to simulate the start and stop conditions of the UUV: for intermediary movement, the UUV thrust is held constant at the maximum or minimum value.\n\n\\begin{figure}[ht]\n  \\centering\n\t\\includegraphics[width=1.0\\linewidth]{PID_Pos_Thrust.png}\n\t\\caption{\n     Sample position (left) and thrust request (right) time histories from the simulated PID controller for position.\n\t}\n\t\\label{fig:PID_Pos_Thrust}\n\\end{figure}\n\nAs provided experimental data does not cover the domain of negative-thrust gaits producing negative trusts, the inverse model temporarily assumes symmetry in the gait landscape. A gait with a negative stroke pitch offset, stroke amplitude, and pitch amplitude generates a thrust $-T$, where its positive counterpart generates a thrust $T$. For this symmetry assumption to hold true, it would physically require a fin rotation of 180\u00b0 along the pitch axis during every transition between a positive and negative gait such that the leading edge of the fin faces the direction of movement. This temporary solution allows for the inverse model to access negative-thrust gaits for simulated data testing, and the kinematic landscape will be extended to incorporate negative-thrust gaits in the future.\n\nThe inverse models are evaluated on the 1500 thrust requests generated by the simulation. To emulate the onboard system, inverse models were run on a Raspberry Pi 4 Model B. Model performance across various weight settings is shown in Figure \\ref{fig:SimDataInvModelResults}. Table \\ref{tab:simInvModelResults} summarizes the performance of MC, HJPS, and GPS across weight settings: losses are averaged across weight settings, and the worst-case run time across all calls to each algorithm is provided.\n\n\\begin{figure}[t]\n  \\centering\n\t\\includegraphics[width=1\\linewidth]{Simulated_Data_Inverse_Model_Results.png}\n\t\\caption{\n        Inverse model performance for simulated data, measured by thrust accuracy loss (left) and kinematic smoothness loss (right). The kinematic smoothness weight was set to $1-w_t$.\n\t}\n\t\\label{fig:SimDataInvModelResults}\n\\end{figure}\n\n\\begin{table}[ht]\n\\footnotesize\n\\begin{center}\n \\caption{Inverse Model Simulation Performance Summary}\n \\label{tab:simInvModelResults}\n \\begin{tabular}{p{2.8 cm} p{1.2cm} p{1.2cm} p{1.2cm}}\n \\hline\n \\hline\n  & MC & HJPS & GPS \\\\\n \\hline\n \\hline\n Thrust Loss (N) & 0.219 & 0.224 & 0.157 \\\\\n \\hline\n Kinematic Loss & 4.693 & 4.747 & 4.940 \\\\\n \\hline\n Overall Loss & 0.484 & 0.469 & 0.435 \\\\\n \\hline\n Maximum Time (s) & 0.300 & 0.393 & 0.467  \\\\\n\\end{tabular}\n\\end{center}\n\\end{table}%\n\nOnboard predictions must meet the time constraints of 1 generated gait per flapping cycle. As the onboard UUV will not exceed a flapping frequency of 2 Hz, the inverse model must generate a gait within 0.5 seconds. All inverse models consistently run within 0.5 seconds on the Pi: the GPS algorithm has the slowest maximum run time of 0.467 seconds. Therefore, these inverse models are all viable for an onboard flapping fin UUV control system.\n\nAll algorithms were responsive to changes to performance metric weights: higher thrust weights resulted in lower thrust losses and higher kinematic losses. GPS reached the lowest mean overall loss of 0.435, as well as a mean thrust accuracy of less than 0.01N when a thrust weight of 1 is applied. While the inverse models demonstrate similar performances when minimizing kinematic loss, GPS consistently outperforms MC and HJPS in terms of minimizing thrust loss as seen in Figure \\ref{fig:SimDataInvModelResults}. GPS obtains a mean thrust accuracy loss of 0.157N averaged across the tested weights, while MC and HJPS obtain thrust accuracy losses of 0.219N and 0.224N. The inverse models demonstrate similar performances when minimizing $L_k$, obtaining mean kinematic smoothness losses of 4.693, 4.747, and 4.940 respectively.\n\n\\section{Conclusion and Deployment Strategy}\n\nOur work uses neural networks to embed a deep kinematic-thrust relationship in a flapping fin UUV control system with the goal of multi-objective optimization: more specifically, we design a search-based inverse model that invokes a gait-to-thrust forward model to select gaits for the controller.\n\nWe demonstrate that our LSTM forward model effectively learned the full space of kinematic-thrust mappings and accurately interpolated to unseen gaits; the DNN model, which does not use time-series data, was unable to match the performance of the LSTM. We implemented inverse models incorporating the LSTM forward model using three approaches: Monte-Carlo Sampling, Hooke-Jeeves Pattern Search, and our new Generalized Pattern Search method, which is an extension of HJPS. Upon integration with onboard hardware, the inverse models consistently ran within the time constraint of 0.5 seconds per iteration. When evaluated with simulated PID controller thrust requests, our GPS algorithm yielded the best performance for minimizing both thrust loss and overall loss across weight settings.\n\nAll three inverse models successfully made trade-offs between thrust accuracy and kinematic smoothness based on the applied performance metric weights. Our flexible inverse model framework enables future UUV control systems to incrementally adjust the emphasis placed on different measures of performance based on the current task and vehicle status. For example, the emphasis on thrust accuracy can be dynamically changed by the controller based on the degree of precise maneuvering required for the task at hand. Our inverse model framework also allows for the incorporation of additional performance metrics such as efficiency.\n\nPrior to deployment, we will collect experimental data for gaits producing negative thrusts and subsequently retrain the forward model based on this new data. Since our inverse model is already integrated with onboard hardware, the Raspberry Pi 4, the final deployment step consists of establishing onboard communication between the Raspberry Pi containing our inverse model and a PID-based micro controller; at this point, physical experiments can be conducted using the inverse model. Following deployment, our inverse model will be extended to incorporate the propulsive efficiency performance metric, which is calculated as the product of output thrust and current velocity divided by power; inverse model performance for thrust accuracy and efficiency trade-offs will be evaluated once experimental data for positive flow speeds becomes available.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzcicp b/data_all_eng_slimpj/shuffled/split2/finalzzcicp
new file mode 100644
index 0000000000000000000000000000000000000000..cef8f673da896475467459d348e5f8d4373c51fe
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzcicp
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\n\\IEEEPARstart{F}{uzzy} logic addresses reasoning about vague perceptions of true or false \\cite{zadeh1965fuzzy,zadeh1968probability,zadeh1988fuzzy}. Moreover, it represents very well our linguistic description about everyday dynamic processes  \\cite{zadeh1976fuzzy, zadeh1975conceptI, zadeh1975conceptII, zadeh1975conceptIII, zadeh1996fuzzy,zadeh1983computational, efstathiou1979multiattribute}. Thus, the introduction of tense operators becomes a  necessity \\cite{chajda2015tense, thiele1993fuzzy, moon2004fuzzy, cardenas2006sound, mukherjee2013fuzzy} because they are useful to represent the temporal indeterminism contained in some propositions \\cite{prior2003time,macfarlane2003future}. For example, the hypothesis ``The weather is very hot'' has its  sense of truth revealed when we say ``The weather will always be very hot''. The  quantifier ``always'' communicates the belief about hypothesis confirmation in the future.\n\nThe task of developing a general theory for decision-making in a fuzzy environment was started  in  \\cite{bellman1970decision}, but a temporal approach still needs to be discussed.\nFor example, do you prefer to receive  7 dollars today or to receive it 5 years later?\nIf you prefer today,   then you made a  distinction of the same value at distant points  from time, even though you have used vague reasoning about value changes  because we do not access  our equity value in memory to perform calculations. \n\nIn decision-making, we do not only evaluate fuzzy goals, but also the time. The change caused by 7 dollars is strongly distinguishable between a  short time and a distant future. This suggests the existence of temporal arguments in fuzzy logic, where ``little change in wealth today'' is different from ``little change in wealth 5 years later''. The linguistic value ``little''  is irrelevant in this problem, but the arguments ``today'' and ``5 years later'' are decisive in the judgment.\n\n\n\nThe proposal here is to connect different fuzzy sets through time attenuators and  intensifiers. Hence, it is possible to simulate two figures of thought in hypotheses of  dynamic systems: meiosis to argue underestimated  changes and hyperbole to argue overestimated changes.\n\n\nThrough meiosis it is possible to formulate a concave expected change curve because it argues  minor changes for maximizing the  sense of truth. By this figure of thought within the fuzzy temporal logic it is noticeable that the hyperbolic discounting \\cite{benzion1989discount, chapman1996temporal, chapman1995valuing, pender1996discount, redelmeier1993time, frederick2002time} and its subadditivity \\cite{read2001time, read2003subadditive}, despite its numerous expected utility theory  anomalies \\cite{loewenstein1992anomalies,ainslie2016cardinal,loewenstein1989anomalies}, is an intuitive dynamic  evaluation of wealth, where ergodicity is relevant in the problem \\cite{peters2016evaluating, peters2011time}. \nSimilarly, concave expected  change curve in lotteries \\cite{bernoulli2011exposition} make certain outcomes more attractive than lotteries with uncertain outcomes.\n\nHyperbole  has an inverse reasoning to meiosis and it produces a convex expected change curve. Similarly, risky losses in the Prospect Theory have the same characteristic in its subjective value function \\cite{tversky1986rational,tversky1992advances, kahneman2013prospect}. Then, it is shown here that the  risk seeking, which is a preference for risky losses rather than certain losses \\cite{kahneman2013choices}, can be described in fuzzy environment by an imprecise perception for small losses. \nThus, the indistinguishability between small negative changes leads to preference for hopes when only these are perfectly distinguishable.\nOn the other hand, when the losses are high, the risk seeking disappears and a kind of ruin aversion prevails \\cite{taleb2018skin}, where it is better to lose a lot and stay with little than risk losing all means of survival after the lottery. \nIn addition, the loss aversion behavior, where people prefer not to lose a given amount than to win the same amount \\cite{kahneman2013prospect},  is interpreted by a disjunction between gains and losses hypotheses leading to the conclusion that such behavior is also amount dependent. \n\n\nIn essence, all the behaviors analyzed here are speculation examples   in dynamic systems, where we evaluate hypotheses and commit to outcomes before they emerge.\nThis paper shows, by modeling the Time Preference and the  Prospect Theory, that the fuzzy temporal logic allows to construct a rhetoric  for intertemporal and  probabilistic choices in fuzzy environment. The first problem takes us to focus on values and time and  the second focuses on values and probabilities. However, if the future is uncertain, there is no reason for time and uncertainty are studied in different matters \\cite{lu2010many}. \nIn addition, the feelings about judgments are amount dependents where the fuzziness can be decisive in some situations.  Therefore, time, uncertainty and fuzziness are concepts that can be properly studied by the fuzzy temporal logic in order to elaborate the decision-making rhetoric.\n\n\n\\section{Theoretical model}\nThis section provides a theoretical framework for the reader to understand as the figures of thought, meiosis and hyperbole, can be elaborated in the fuzzy temporal logic. In short, it is discussed the need of many-valued temporal logic, the existence of different temporal prepositions with  similar goals over time and, finally, it is shown as to perform the rhetorical development to make a judgment between two different hypotheses about the future. \n\n\\subsection{Temporal and many-valued logic}\n\\label{TIL}\n Usually we  make decisions about dynamic processes whose states are unknown in the future. An amusing example can be found in an Aesop's fable, where the Grasshopper and the Ant have different outlooks for an intertemporal decision dilemma \\cite{perry1965babrius}. In short, while the Ant is helping to lay up  food for the winter, the Grasshopper is enjoying the summer without worrying about the food stock. \n \nThe narrative teaches about hard work, collaboration, and planning by presenting temporal propositions. These propositions have invariant meaning in time, but  sometimes they are true and sometimes false,  yet  never simultaneously true and false \\cite{ohrstrom2007temporal}.  This property can be noted in the statement:\n\\[\n\\begin{array}{l}\nD_1 =  \\text{``We have got plenty of food at present'' \\cite{perry1965babrius}.} \n\\end{array}\n\\]\nAlthough  $D_1$  has constant meaning over  time, its logical value is not constant. According to the fable, this statement is true in the summer, but it is false in the winter, hence there is a need to stock food.\n \n     \nIf the logical value varies in time according to random circumstances, such as the weather of the seasons, how can we make inferences about the future? There seems to be a natural uncertainty that sets a haze over the vision of what will happen. For instance, consider the following statements about the Grasshopper: \n\\[\\centering\n\\begin{array}{rl}\nD_2 = \\text{``The Grasshopper stored a lot of food }\\\\ \n   \\text{during the summer'';}\\\\\n   D_3 = \\text{``The Grasshopper stores food for winter''.}\n\\end{array}\n\\]   \nAccording to the fable, we know that the statement  $D_2$ is false, \nbut at the fable end, when the winter came, \nthe Grasshopper says ``It is best to prepare for days of need''  \\cite{perry1965babrius}. In this way, the truth  value in $D_3$ is ambiguous. We can not say with certainty that it is false, but \nwe do not know how true it can be. Thus, we can only propose hypotheses to speculate the Grasshopper's behavior.\n\n\n A hypothesis is a proposition (or a group of propositions) provisionally anticipated as an explanation of facts, behaviors, or natural phenomena that must be later verified by deduction or experience. They should always be spoken or written in the present tense because they are referring to the research being conducted. In the scientific method, this is done independently whether they are true or false and, depending on rigor, no logical value is attributed to them. However, in everyday  language this rigor does not exist.  Hypotheses are guesses that argue for the decision-making before the facts are verified, so they have some  belief degree about the logical value that is a contingency sense performing a sound practical judgment concerning future events, actions or whatever is at stake.\n\n \n\nSince there is no rigor in everyday speculations, then different propositions may arise about the same fact. If they are analyzed by binary logic, then we may have unnecessary redundancy that can lead to contradictions.\n However, the redundancy of propositions is not a problem within the speculation process, what suggests a many-valued temporal logic.\n \n We can discuss the limitations of a binary temporal logic by speculating the Grasshopper's behavior. For example, is it possible to guarantee  that the two hypotheses below are completely true simultaneously?\n\\[\n\\begin{array}{l}\n\\Theta = \\text{``The Grasshopper stores a lot of food'';}\\\\\n\\theta = \\text{``The Grasshopper stores little food''.}\n\\end{array}\n\\]\nThe hypotheses $\\Theta$ and $\\theta$  propose changes to different states. If $S_0$ is the initial state for stored food, the new state after storing a lot of food $M$ is $S_{\\Theta}=S_0+M$. Analogously, the next state after storing little food is \n$S_{\\theta}=S_0+m$ for $m<M$. \nRepresenting by $\\Theta(t)$ the affirmation that $\\Theta$ is true at the moment $t$ and considering the same initial state for both, then by binary logic\n\\[\\Theta(t) \\neq \\theta(t) \\text{ because } S_\\Theta > S_\\theta. \\]\nTherefore, affirming both as true leads to a  contradiction because the same subject can not simultaneously produce two different results on the same object. However, in a  speculation of the Grasshopper's behavior, none of the propositions can be discarded.\n\n\n\n\n\nEvaluating by fuzzy logic, the linguistic variable ``stored food'' has values ``a lot of'' and ``little'' in the propositions  $\\Theta$   and $\\theta$. According to Bellman and Zadeh's model \\cite{bellman1970decision}, these linguistic values  are the fuzzy constraints for inputs  $M$ and $m$ about the food supply. Meanwhile, the target states  $S_\\theta$ and $S_\\Theta$ can be the fuzzy goals.\n\nAn alternative development, but similar,  can be done through changes, what is in accordance with   human psychophysical reality \\cite{kahneman2013choices}. Thus, in  this paper, the  goals  are the factors $ S_{\\Theta}\/S_0= 1+X$ and $S_{\\theta}\/S_0= 1+x$, where $X=M\/S_0$ and $x=m\/S_0$ are  changes. For example, in \\cite{mukherjee2017loss}  the respondents were asked how they would feel gaining (or losing) a certain amount. It was noted that the emotional effects gradually increased as the amounts had grown. Therefore, the intensity of gains and losses are easily ordered by our perception, $x<X$.\n\n\n\nThe non-fuzzy set $-1\\leq x<X$ for changes can have the fuzzifier effect  to transform it into a fuzzy set. Here, the general rule for fuzzy  goals  is simply  ``the bigger the better''. This rule can be represented by the membership function $\\mu : [-1,\\infty ) \\to [0,1)$, where: \n\\begin{enumerate}\n\\item $\\mu(-1)=0$  means losing all;\n\\item if $-1\\leq x<X$, then $\\mu(x)<\\mu(X)$, i.e., $\\mu(x)$ is always growing;\n\\item $\\lim\\limits_{x\\to \\infty} \\mu (x)=1$ means that the best gain goal (or change)  is undefined.\n\\end{enumerate}\n\nFigure \\ref{figt} shows a membership function example that satisfies the above descriptions.   The membership function $\\mu(x)$ is the general rule in decision-making because higher gains and lower losses, generally, cause better well-being. Therefore, others membership functions  will be a composition with the form $\\mu(f(x))$, where $f$ must be in accordance with the rhetorical figure necessary to make a decision.\n \\begin{figure}[!ht]\n\t\\centering\n\t\\includegraphics[scale=0.65]{Scurve}\n\t\t\\caption{Membership function example $\\mu(x)=1-[1-\\alpha\\beta(x+1)]^\\frac{1}{\\alpha}$ for $\\alpha=-1.001$ and $\\beta=1.3$. }\t\n\t\\label{figt}\n\\end{figure}\n\n\n\n\n\n\\subsection{Different hypotheses with similar future sentences} \n\\label{HDSFS}\nThe  sense of truth that we have about a hypothesis becomes evident when we express it in the form of future statement. Given that a future sentence indicates an event occurrence after the moment of speech, then it can emphasize the fact realization with an intuitive implicit probability.\nFor example, consider the following future statements for\n $\\Theta$ and $\\theta$:\n\\[\\centering\n\\begin{array}{rl}\nF\\Theta = \\text{``The Grasshopper  will sometime store  }\\\\ \n   \\text{a lot of food'';}\\\\\n   G\\theta = \\text{``The Grasshopper  will always store}\\\\\n     \\text{ little food''.}\n\\end{array}\n\\]\nThe statement  $G\\theta$ expresses  certainty with the modifier ``always'', while $F\\Theta$ expresses uncertainty about the instant that the action occurs through the modifier ``sometime''. In temporal logic  $F\\Theta$ is equivalent to saying which there is an instant  $t$ in the future where $\\Theta$ is true, i.e, $\\exists \\;t$ such that $ (\\text{now}<t) \\wedge \\Theta(t)$. \nMeanwhile,  $G\\theta$ affirms  that $\\theta$ is always true in the future,  $ \\theta(t) \\;\\; \\forall \\: t>\\text{now}$. \n\n\n\n\n\nAbout the  certain sense, $F\\Theta$ is known as the weak operator because  $\\Theta$ is true only once in the future, while  $G\\theta$ is known as the strong operator because it is true in all future periods.\nIf $\\Theta$  does not always come true, then we can investigate its  sense of truth through the affirmative: \n\\[\n\\begin{array}{rl}\nGF\\Theta = \\text{``The Grasshopper  will frequently  store}\\\\\n\\text{ a lot of food''.}\\\\\n\\end{array}\n\\]\nWhere the quantifier ``frequently'' better argues for the  sense of truth of $\\Theta$ because we have undefined repetitions in the same period in which $G\\theta$ is true.\n\n \n\nThe frequency in which the propositions $\\Theta$ and $\\theta$ are true and the changes proposed by them determine the outcomes over time. \nThe affirmative $GF\\Theta$ communicates that the Grasshopper will frequently produce a strong change in the stored food stock  (change factor  $1+X$). On the other hand, $G\\theta$ proposes a  small   change factor, $1+x$, but continuously over time. So what is the relation between $X$ and $x$  that generates a similarity of states between the two hypotheses over time? The relation that constructs this similarity can be obtained by the time average. Therefore, let us consider $\\tau(t)$ as the total time where $GF\\Theta$ is true and $t$ as the observation time. If $\\Theta$ is true with a frequency given by\n\\begin{equation}\n\\lim\\limits_{t\\to \\infty} \\frac{\\tau(t)}{t}=s,\n\\end{equation} \nthen, the relation between  change factors  $1+x =(1+X)^s$, estimated by time average \\cite{peters2016evaluating}, indicates that the sentences $GF\\Theta$  and $G\\theta$ have similar goals in the long run. This similarity is denoted in this work by\n\n\\begin{equation}\nGF\\Theta \\sim G {\\theta}.\n\\end{equation}  \n\n\nThe  sense of truth for the sentence $GF\\Theta$ is quantified in the parameter $s$. It can be a stationary probability when\n $t$ is big enough, but we do not have much time to calculate this probability in practice.\nIn this way, we assume that the  sense of truth is an imprecise suggestion (intuition) for the time probability.\nFigure \\ref{fig1} presents some adverbs of frequency   that can suggest the  sense of truth  in future sentences.\n\\begin{figure}[!ht]\n\t\\centering\n\t\\includegraphics[scale=0.8]{Gfrequency}\n\t\t\\caption{Adverbs of frequency   that can suggest the  sense of truth of $GF$.   The quantifier ``always'' indicates certainty,  while  ``never'' indicates impossibility.}\t\n\t\\label{fig1}\n\\end{figure}\n\n\n\n\nThe axiomatic system of temporal logic proposes that  \n $G {\\theta} \\Rightarrow GN {\\theta}$, where  $N\\theta$ stands for $\\theta$ is true at the next instant. Therefore, the similarity  $GF\\Theta \\sim G {\\theta}$ can be written by\n\\begin{equation}\n  F\\Theta \\sim N {\\theta},\n\\end{equation}\nwhen $1+x \\approx (1+X)^s$. \nThus, the statement ``the Grasshopper will sometime store a lot of food'',  which have a  change factor $1+X$, is similar to the statement ``the Grasshopper will store little food at the next moment'', which has a  change factor $(1+X)^s$.\n\n\n\n\\subsection{Rhetoric: Meiosis and Hyperbole}\n\\label{MH}\nIn general, different hypotheses do not have similar changes in the future. For this reason, it is required that a rhetorical development  make a judgment between them. In this section, two figures  of thought are presented,  meiosis and hyperbole, which  can be used to compare hypotheses with different changes and  senses of truth. \n\nStill using Aesop's fable, imagine that we want to compare Ant and Grasshopper's performance with the following hypotheses:\n\\[\n\\begin{array}{l}\n\\phi = \\text{``The Ant stores little food'';}\\\\\n\\Theta = \\text{``The Grasshopper stores a lot of food''.}\\\\\n\\end{array}\n\\]\nAssume that the Ant produces a  change   $y$ in its food stock while the Grasshopper produces a  change  $X$. If we think that the Ant is more diligent in its work than the Grasshopper, i.e., the  sense of truth for $\\phi$ is maximum while the  sense of truth for $\\Theta$ is ambiguous, so we can affirm $N\\phi$  and $F\\Theta$ in order to develop the following argumentation process:\n\\begin{enumerate}\n\\item elaborate a proposition $\\theta$,  similar to $\\Theta$,  which proposes a lower outcome, that has a  change  $x$ (for example, $\\theta $ = ``The Grasshopper stores little food'');\n\\item express $\\theta$ with maximum certainty in the future, $N\\theta$ = ``The Grasshopper will store little food at the next moment'';\n\\item  calculate the relation $1+x \\approx (1+X)^s$ to match the average changes between $N\\theta$ and $F\\Theta$ in order to obtain the similarity $F\\Theta \\sim N {\\theta}$, where $X>x$ and $s\\in [0,1]$;\n\\item finally, judge the affirmations $N\\phi$ and $N\\theta$ through fuzzy logic. In this specific problem we have\n\\[N\\theta \\text{ or } N\\phi = \\max\\left\\{\\mu \\left((1+X)^s-1\\right),\\mu(y)\\right\\}.\\]\n\\end{enumerate}\n\n\nThe above argument uses meiosis and the upper part of Figure \\ref{figMH}  summarizes this procedure. \nIn linguistic meiosis, the meaning of something is reduced to simultaneously increase something else in its place.\nIn the above mentioned case, proposing $\\theta$ means reducing the stored  food change  by the Grasshopper.\nAt the same time, this suggests greater certainty because it makes the process more feasible in the future.\nHowever, it is only a figure of thought to make an easy comparison because judging two sentences with the same  sense of truth, looking only at the change goals, is much simpler.\n\n The meiosis for Grasshopper's case  has a membership composite function given by\n\\[ \\mu_{\\text{ Grasshopper's goal}}= \\mu \\circ \\mu_{\\text{meiotic change}}(X)=\\mu \\left((1+X)^s-1\\right).\\] \nIn general, $\\mu_{\\text{ Grasshopper's goal}}$ refers to the fuzzy goal ``the bigger the better is the change $(1+X)^s-1$ at the next moment''.  Like this, we can evaluate it for decision-making in real time.  \n\n\\begin{figure}[!ht]\n\t\\centering\n\t\\begin{tikzpicture}[node distance=2cm and 1cm,>=stealth',auto, every place\/.style={draw}]\n         \n    \\node [state,initial text=,accepting by double] (1) [] {$\\begin{array}{l}\n F\\Theta \\\\\n \\\\\n{\\color{blue} F\\Phi}\n\\end{array}$};  \n    \\node [state,initial text=,accepting by double] (0) [right=of 1] {$\\begin{array}{l}\n{\\color{blue} N\\theta} \\\\\n\\\\\n N\\phi\n\\end{array}$};\n    \\path[->] (1) edge [bend left] node {$\\underbrace{(1+X)^s}_{\\text{Meiosis}}$} (0);\n    \\path[->] (0) edge [bend left] node {$\\overbrace{(1+y)^\\frac{1}{s}}^{\\text{Hyperbole}}$} (1);      \n\\end{tikzpicture}\n\t\t\\caption{Diagram representing the meiosis and hyperbole procedure   for the judgment of hypotheses. The blue sentences $F\\Phi$ and $N\\theta$ are the  figures of thought.}\t \n\t\\label{figMH}\n\\end{figure}\n\n\n \n\n\n\n\nOn the other hand, there is an inverse process to meiosis that is called hyperbole. Basically, it exaggerates a change in the outcome to reduce its  sense of truth. In everyday language, such statements are commonplace, as ``the bag weighed a ton''.\nIn this statement, we realize that  the bag is really heavy. However, is there a bag weighing really a ton? This is just a figure of speech.\n\n\nIn order to understand how to judge two hypotheses through a process of hyperbolic argumentation, consider again that we can make the future statements $F\\Theta$ and $N\\phi$ pass through the following steps:\n\\begin{enumerate}\n\\item elaborate a proposition $\\Phi$,  similar to $\\phi$,  which proposes a larger outcome, that has a change $Y$  (for instance, $\\Phi $ = ``The Ant stores a lot of food'');\n\\item affirm $\\Phi$ in the future with the same  sense of truth as the proposition $\\Theta$, that is, $F\\Phi$ = ``The Ant will sometime store a lot of food'';\n\\item  calculate the relation $(1+y)^{\\frac{1}{s}}\\approx 1+Y$ to match the average changes between $N\\phi$ and $F\\Phi$ in order to obtain the similarity $N\\phi \\sim F\\Phi$, where $Y>y$ and $s\\in [0,1]$; \n\\item finally,  judge the fuzzy  changes goals of the affirmative $F\\Phi$ and $F\\Theta$. In this specific problem we have\n\\[F\\Theta \\text{ or } F\\Phi = \\max\\left\\{\\mu(X), \\mu\\left((1+y)^\\frac{1}{s}-1\\right)\\right\\}.\\]\n\\end{enumerate}\n\n\n\nThe hyperbole for Ant's case  has a membership composite function given by\n\\[ \\mu_{\\text{ Ant's goal}}= \\mu \\circ \\mu_{\\text{hyperbolic change}}(y)=\\mu \\left((1+y)^\\frac{1}{s} -1\\right).\\] \nThus, $\\mu_{\\text{ Ant's goal}}$ refers to the fuzzy goal ``the bigger the better is the change $ (1+y)^\\frac{1}{s} -1 $ sometime in the future''.    \n\n\n The bottom part of Figure \\ref{figMH}  summarizes hyperbolic argumentation procedure. Note that the arguments by meiosis and hyperbole lead to the same conclusion. Therefore, they can be only two ways to solve the same problem. However, in section     \\ref{PT}, where the Prospect Theory is evaluated, there may be preference for one of the methods according to the frame in which the hypotheses are inserted.\n\n\n\n\n\\section{Time Preference}\n\\label{PTemp}\n\nTime preference is the valuation placed on receiving a good on a short date compared with receiving it on a farther date. A typical situation is  choosing to receive a monetary amount $m$ after a brief period (a day or an hour) or to receive $M>m$ in a distant time (after  some months or years).\n\n\nThe time   preference choice is a problem of logic about the future and in order to model it consider the following hypotheses:\n\n\\begin{itemize}\n\\item $\\Theta_m =$ ``to receive $m$''  represents the receipt of the amount $m$ in short period, $t_m=t_0+\\delta t$;\n\\item $\\Theta_M =$ ``to receive $M$'' represents the receipt of the amount $M$ in longer time horizons, $t_M=t_0+\\Delta t$. \n\\end{itemize}\nEach proposition has a  change factor for the individual's wealth. The proposition $\\Theta_M$ has the change factor $(1+M\/W_0)$, while $\\Theta_m$ has the change factor $(1+m\/W_0)$. Now, we perform the meiosis procedure for both hypotheses, reducing the changes and maximizing the  sense of truth:\n\\begin{itemize}\n\\item $N\\theta_m =$ ``to receive an amount less than $m$ at the next moment''. This affirmative proposes a  change factor  $1+x_m$ in the individual's wealth;\n\\item $N\\theta_M =$ ``to receive an amount less than $M$ at the next moment''. Similarly,  this affirmative proposes a  change factor  $1+x_M$.\n\\end{itemize}\n\nThe  senses of truth for the hypotheses $\\Theta_M$  and $\\Theta_m$ are  revealed when they are  affirmed in the future. Therefore, we have the following similarities:\n\\begin{itemize}\n\\item $F\\Theta_M \\sim N\\theta_M$, if  $(1+x_M)\\approx \\left(1+\\frac{M}{W_0}\\right)^{s_M}$ ;\n\\item $F\\Theta_m \\sim N\\theta_m$, if  $(1+x_m)\\approx \\left(1+\\frac{m}{W_0}\\right)^{s_m}$ .\n\\end{itemize}\nWhere $s_M$ and $s_m$ are the   senses of truth regarding the receipt of the values $M$ and $m$.\nIn this problem,  they cannot be time probabilities since the probabilistic investigation in this case is not convenient. However, intuitions about the realization of the hypotheses $\\Theta_M$ and $\\Theta_m$  are  feasible for individuals and they should be represented.\n\nNow suppose, without loss of generality, that $M$ is large enough so that the individual prefers to receive it in the distant future. If we consider $n=\\Delta t\/\\delta t$  periods in which $n$ attempts to receive $m$ until $M$'s receipt date are allowed, then we have\n\\begin{eqnarray}\n\\nonumber (1+x_M) &>& (1+x_m)^n \\\\\n\\Rightarrow\\left(1+\\frac{M}{W_0} \\right)^{s_M} &>& \\left(1+\\frac{m}{W_0} \\right)^{ns_m}.\n\\label{BhomBawerk}\n\\end{eqnarray}\nJudging by fuzzy logic the two hyprothesis in the future, the ``or'' operation between change goals is indicated to finalize the meiosis procedure, \n\\begin{eqnarray}\n\\nonumber \\mu(x_M) &=& \\max \\left\\{\\mu(x_M), \\mu \\left( (1+x_m)^n-1 \\right)\\right\\}\\\\\n&=& \\mu\\left(\\left(1+\\frac{M}{W_0} \\right)^{s_M}-1\\right).\n\\end{eqnarray}\n\nIn general the time preference solution  is presented through a discount function. In order to use this strategy it is necessary to develop the same form on both sides of the inequality \\ref{BhomBawerk}. For this, there is a value $\\kappa$ such that $\\kappa s_M > m$, where we can write\n\\begin{equation}\n\\left(1+\\frac{M}{W_0} \\right)^{s_M} = \\left(1+\\frac{\\kappa {s_M}}{W_0} \\right)^{ns_m}>\\left(1+\\frac{m}{W_0} \\right)^{ns_m}.\n\\label{Passagem}\n\\end{equation}\nThe discount function undoes exactly the change caused by the proposition  $\\Theta_M$, that is,\n\\begin{equation}\n \\frac{1}{{\\left(1+\\frac{M}{W_0} \\right)}} =\\left(1+\\frac{\\kappa s_M}{W_0} \\right)^{-\\frac{s_m}{s_M}n} .\n \\label{DescontoHiperbolico}\n\\end{equation}\n\n\nEquation \\ref{DescontoHiperbolico} describes the hyperbolic discount, the most well documented empirical observation of discounted utility \\cite{frederick2002time}. When mathematical functions are explicitly fitted to experiment data, a hyperbolic shape fits the data better than the exponential form \\cite{kirby1997bidding, kirby1995modeling, myerson1995discounting,rachlin1991subjective}. Among the functions proposed for the adjustment of experimental data, the discount function proposed by Benhabib, Bisin and Schotter \\cite{benhabib2004hyperbolic}\n\\[e_h^{-\\rho n}\\equiv \\left(1-h \\rho n \\right)^{\\frac{1}{h}}\\]\nallows for greater flexibility of fit for exponential, hyperbolic, and quasi-hyperbolic discounting \\cite{laibson1997golden} (see Figure \\ref{figDiscounting}). In order to obtain it, we must reparametrize  equation \\ref{DescontoHiperbolico} by doing \n\\begin{eqnarray}\n\\frac{1}{h} &=& -\\frac{s_m}{s_M}n,  \\label{eqh} \\\\\n\\rho &=& \\frac{\\kappa s_m}{W_0}.\n\\end{eqnarray}\n\nThe parameter $h$ denotes  hyperbolicity and it gives the curve shape over time. For instance,  $e_h^{-\\rho x}$ equals the exponential function $e^{-\\rho x}$ when $h \\to 0^-$. This means that there is plenty of time for possible trials with higher sense of truth until the date of the great reward. On the other hand,  $h\\ll 0$ indicates time shortage for trials.  In theses cases, only the first periods have strong declines in the discount function. In short, equation \\ref{eqh} shows that the senses of truth  and the time between rewards  determinate  the value of $h$.\n\n\n Furthermore, the discount rate $\\rho$ quantifies the preference for goods and it is influenced by individual states of scarcity and abundance of goods. For instance, let us consider an individual called Bob. If an object is scarce for him (small $W_0$), then he places a higher preference (great $\\rho$). Analogously, if $W_0$ represents an abundance state for him, he has a lower preference (small $\\rho$).  This may cause great variability in experiments because the wealth distribution  follows the power law  \\cite{levy1997new,druagulescu2001exponential,sinha2006evidence,klass2006forbes}. This means that $W_0$  can vary abruptly from one individual to another in the same intertemporal arbitrage experiment. \n\n\\begin{figure}[!ht]\n\t\\centering\n\t\\includegraphics[scale=0.65]{Discount}\n\t\t\\caption{Discount function $e_h^{-\\rho n}$ versus number of delayed periods: the dashed black curve is the exponential discounting for $\\rho = 0.005$; the $\\circ$-blue curve is quasi-hyperbolic discounting for $\\rho = 0.7$ and $h = -3$; \nand the $\\ast$-red and cyan curves are hyperbolic discounting for  $\\rho_1 = 0.0175$ and $h_1 = -3$, and $\\rho_2 = 0.05$ and $h_2 = -5$,  respectively.  }\t\n\t\\label{figDiscounting}\n\\end{figure} \n\n\\subsection{Discussion about time preference behaviors}\n\n\nDaniel Read pointed out that a common evidence in time preference is the ``subadditve discounting'', in others words, the discounting over a delay is greater when it is divided into subintervals than when it is left undivided \\cite{read2001time}. For example, in \\cite{takahashi2006time,takahashi2008psychophysics} it has been argued that abstinent drug addicts may more readily relapse into addiction if an abstinence period is presented as a series of shorter periods, rather than a undivided long period. This property is present in the function $e_h^{-x}$, because if $h<0$, then\n\\[e_h^{-x} \\; e_h^{-y}=e_h^{-x-y+h xy}<e_h^{-x-y} \\text{ for all } x,y>0.\\]\n\nThe hyperbolicity, due to intertemporal arbitrage, is always negative, $\\frac{1}{h} = -\\frac{s_m}{s_M}n$. Therefore, we have a subadditivity as a mandatory property in the time preference for positive awards. \n\n\n\nHowever, the main experimentally observed behavior, hereinafter referred to as ``time effect'', concerns the observation that the discount rate  will vary inversely with the time period to be waited. Here, we can verify  this effect  as a consequence of the subadditivity found in the function $e_h^{-\\rho n}$. After all, when we want to find the discount rate through a given discount $D$ after $n$ periods, then we calculate $-\\frac{1}{n}\\ln D $. For example, if we have an exponential discounting $e^{-\\rho n}$, then $\\rho= -\\frac{1}{n}\\ln e^{-\\rho n} $. Thus, using the subadditive property we can develop \n\\begin{eqnarray}\n\\nonumber \\left(e_h^{-\\rho}\\right)^n\\leq e_h^{-\\rho n} & \\Rightarrow & \n-\\frac{1}{n}\\ln \\left(e_h^{-\\rho}\\right)^n\\geq  -\\frac{1}{n}\\ln  e_h^{-\\rho n} \\\\\n\\nonumber & \\Rightarrow & \\left\\langle -\\ln e_h^{-\\rho} \\right\\rangle \\geq   \\left\\langle -\\frac{1}{n}\\ln  e_h^{-\\rho n} \\right\\rangle .\n\\end{eqnarray}\nTherefore, if $h$ does not tend to zero from the left side, then the average discount rate over shorter time is higher than the  average discount rate over longer time horizons. \nFor example, in \\cite{thaler1981some} it was  asked to respondents how much money they would require to make waiting  one month, one year and ten years just as attractive as getting the \\$ 250 now.  The median responses (US \\$ 300,  US \\$ 400 and US \\$ 1000) had an average (annual) discount rate of 219\\% over the one month, 120\\% over the one year and 19\\% over the ten years. Other experiments presented a similar pattern \\cite{benzion1989discount, chapman1996temporal, chapman1995valuing, redelmeier1993time, pender1996discount}. Therefore, the time effect is a consequence of subadditivity when there is not plenty of time for trials at one of the hypotheses. \n\n\n\n\nA second behavior,  referred to as  ``magnitude effect'', is also consequence of subadditivity. The reason for this is that magnitude effect and time effects are mathematically similar, because the discount rate is $\\rho=\\kappa s_m\/W_0$ and  $ \\kappa $ is growing for large values of    $ M $ (see equation \\ref{Passagem}). In order to understand the similarity, note that the function $e_h^{-\\rho n}$  varies with $ n \\rho $. If we set the value $ n $, for example $ n = 1 $, and we vary the only rate $ \\rho = r \\rho_0 $, where $ \\rho_0 $ is constant and $ r>1 $ is a multiplier which is growing for values of $ M $, then we have  the function $e_h^{-r\\rho_0}$  analogous to $e_h^{-\\rho n}$. Therefore, the magnitude effect made by $r$, similarly to the time effect,  results in\n\\[\\left\\langle -\\ln e_h^{-\\rho_0} \\right\\rangle \\geq   \\left\\langle -\\frac{1}{r}\\ln  e_h^{-\\rho_0 r} \\right\\rangle .\\]\n\nFor example, in Thaler's investigation \\cite{thaler1981some}, the respondents preferred, on average, \\$ 30 in 3 months rather than \\$ 15 now, \\$ 300 in 3 months rather than \\$ 250 now, and \\$ 3500 in 3 months rather than \\$ 3000 now, where the discount rates are 277\\%, 73\\% and 62\\%, respectively. Other experiments have found similar results \\cite{ainslie1983motives, benzion1989discount,green1994temporal,holcomb1992another,kirby1997bidding, kirby1995modeling, kirby1999heroin, loewenstein1987anticipation,raineri1993effect,shelley1993outcome, green1997rate}.\n\n\nAnother experimentally observed behavior is the ``preference reversal''. Initially, when individuals are asked to choose between one apple today and two apples tomorrow, then  they may be tempted to prefer only one apple today. However, when the same options are long delayed, for example, choosing between one apple in one year and two apples in one year plus one day, then to add one day to receive two apples becomes acceptable \\cite{thaler1981some}. \n\nThe preference reversal  shows how we evaluate value and time in the same hypothesis. For example, if $M_1<M_2$, then, in hypothesis, it is easier ``to receive  $M_1$ today'' than ``to receive $M_2$ today''. In this case, the sense of truth $s_1$ regarding the receipt of  $M_1$ is greater than the sense of truth $s_2$ regarding the receipt of $M_2$. For purposes of evaluation, consider that $ s_2 $ is not small enough to make $ M_1 $ preferable, that is\n\\begin{equation}\n\\left( 1+\\frac{M_1}{W_0}\\right)^{s_1} < \\left( 1+\\frac{M_2}{W_0}\\right)^{s_2}.\n\\label{today}\n\\end{equation}\n\n\n\nHowever, when one has to choose between the hypotheses ``to receive $M_1$ today'' and ``to receive $ M_2 $ tomorrow'', then it should be noted that sequential execution of these actions can give an advantage to receive $M_1$ in the future.\nFor simplicity, imagine that the proposal of each hypothesis happens  always  after receiving the reward. Then, in this case, the first hypothesis represents the attempt to receive $M_1$ every day, while the second hypothesis represents the attempt to receive $ M_2 $ every other day. If $M_1$ is not much less than $M_2$, then it will be more advantageous trying to receive $ M_1 $ twice  rather than expecting two days to receive $M_2$. Therefore, in dynamic situations, the waiting time between the hypotheses affects the frequency of attempts, resulting in\n\\begin{equation}\n\\left( 1+\\frac{M_1}{W_0}\\right)^{2s_1} > \\left( 1+\\frac{M_2}{W_0}\\right)^{s_2}.\n\\label{Twice}\n\\end{equation}\n\n\nOn the other hand, when one has to choose between the hypotheses $ H_1=$``to receive $M_1$ in $n$ days'' and $ H_2=$ ``to receive  $M_2$ in $n+1$ days'', then a similar judgment can be made by evaluating the proposed action execution   over and over again over time. Since the waiting time to receive $ M_1 $ is shorter, then we can realize that the number of attempts to receive $M_1$ will be greater in the future ($(n+1)\/n$ trials to receive $M_1$ for each trial to receive $M_2$). By fuzzy temporal logic, the choice between  $ H_1$ and $ H_2$ depends on the following result: \n\\[ \\max \\left\\{\\left( 1+\\frac{M_1}{W_0}\\right)^{\\frac{n+1}{n} s_1},\\left( 1+\\frac{M_2}{W_0}\\right)^{s_2}\\right\\}.\\]\nWhen $n=1$, then it will be preferable to receive the reward $M_1$ (see equation  \\ref{Twice}). This can also happen to other small values of $n$, for example, $n$ equals 2 or 3.  \nHowever, when $n$ is large enough, the relation $(n+1)\/n$ tends to  1 and  makes $M_2$ a   preferable reward (see equation \\ref{today}). Thus, the preference between the rewards are reversed when they are shifted in time. In similar experiments, this behavior can be observed in humans \\cite{kirby1995preference,green1994temporal,millar1984self,solnick1980experimental} and in pigeons \\cite{ainslie1981preference,green1981preference}.\n\n\n\nHence, the time effect and magnitude effect on the discount rates, preference reversal and subadditivity are strong empirical evidences for the application of fuzzy temporal logic in intertemporal choices. \n\n\n\n\n\n\n \n\n\\section{Lotteries}\n\\label{PT}\nIn a  more realistic human behavior descriptive analysis with the psychophysics, the subjective value of lottery must be related to the wealth change \\cite{kahneman2013choices}. However, changes after lotteries have incomplete information because we avoid calculating them using equity values, premiums and probabilities. Therefore, fuzzy sets are good candidates for representing these hypothetical changes. In addition, more realistic expectations should consider the evolution of outcomes over time \\cite{peters2016evaluating}. Thus, the changes may have their expected values attenuated or intensified by the sense of truth (intuitive time probability).\n\n \n\\subsection{Meiosis and risk aversion}\n In order to model lotteries consider the hypothesis $\\Theta_2$ = ``to win $M $'' and a probability $p \\in [0,1]$. If $M>0$, then  by fuzzy temporal logic we can have:\n\\begin{itemize}\n\\item $l_1$ = to win $Mp$ (at the next moment);\n\\item $l_2$ = to win $M$ (at the next moment) with probability $p$.\n\\end{itemize}\nThe expression ``at the next moment'' does not appear in the experiments, but they are implicit because the low waiting time for the two lotteries seems to be the same. Moreover,  note that both lotteries are equivalents in the ensemble average ($E=pM$ for the two lotteries). \n\nAgain, let us consider an individual called Bob who may repeat similar lotteries in the future. Therefore, this repetition can affect his decision \\cite{peters2016evaluating}. If the lottery $l_2$ is repeated several times until he wins $M$, then $l_2$ is similar to affirm\n\\[F\\Theta_2=\\text{ ``will sometime win } M \\text{'',}\\]\nwhere $p$ is the time probability (or sense of truth for lottery). Thus, if we perform the meiotic argumentation procedure  (see section \\ref{MH}), then  the similar sentence which have equivalent outcomes to $F\\Theta_2$ is\n\\[N\\theta_2=\\text{ ``will win } W_0\\left(1+\\frac{M}{W_0}\\right)^p - W_0 \\text{ at the next moment'',}\\]\nin which this sentence is a future affirmation of the hypothesis \\[\\theta_2= \\text{ ``to win } W_0\\left(1+\\frac{M}{W_0}\\right)^p - W_0 \\text{''}.\\]\n\nNow, the  difference between $N\\theta_2$ and $Nl_1$ consists only in the value of the award. Although values are reported in lotteries, variations on wealth are unknown because the cognitive effort to perform the division   $M\/W_0$  is avoided. Thus, taking $x\\geq 0$, such that   $x=M\/W_0$, we can only evaluate  changes, \n\\begin{eqnarray}\n\\nonumber N l_1  \\text{ or } N\\theta_2  &=& \\max \\{\\mu(px),\\mu\\left((1+x)^p-1\\right)\\}\\\\\n&=& \\mu (px)  \\text{ for all } x\\geq 0.\n\\label{MeioseComp}\n\\end{eqnarray}\n\n\n\n\\begin{figure}[!ht]\n\t\\centering\n\t\\includegraphics[scale=0.65]{PT1}\n\t\t\\caption{Function  ${\\cal M}_p^+(x)$ to represent meiosis.   The blue dashed line  $x\/2$ is tangent to the $\\circ$-blue curve   given by ${\\cal M}_{1\/2}^+(x)$. Analogously, the red dashed line  $x\/10$ is tangent to the   $\\ast$-red curve given by ${\\cal M}_{1\/10}^+(x)$. Note that in the vicinity of zero the curves are close, so this is a region of low distinguishability for the changes.}\t\n\t\\label{figPT1}\n\\end{figure} \n\nThe line $px$ is tangent to the concave curve $(1+x)^p-1$ at the point $x=0$ what results in  $px \\geq (1+x)^p-1$ for $x\\geq 0$.  An example can be seen in Figure \\ref{figPT1} where the dashed blue line   $x\/2$ is above the curve $\\circ$-blue    $(1+x)^\\frac{1}{2}-1$. Analogously, a similar illustration can be seen for $p=1\/10$.  Thus, we can note that the lottery $l_1$ is preferable for any values of $M$ and $p$, because the line $px$  is always above the curve $(1+x)^p-1$. This result is consistent with the respondents' choices in the Kahneman and Tversky experiments  \\cite{kahneman2013prospect}. Thus, the concave curve representing the expected positive change at the next moment can be described by\n\\begin{eqnarray}\n\\nonumber {\\cal M}^+_p(x)&=& (1+x)^p -1  \\\\\n&=& p\\ \\text{ln}_p (1+x) \\text{ for all } x\\geq 0.\n\\label{MeioseMais}\n\\end{eqnarray}\nThe function $\\text{ln}_p (x)\\equiv (x^p -1)\/p$ is defined here as in \\cite{nivanen2003generalized} and it is commonly used in nonextensive statistics \\cite{tsallis1988possible,tsallis1999nonextensive}.\n\n\n\n\n\n \n\n\n\n\n\\subsection{Hyperbole and risk seeking}\n Kahneman and Tversky also show that the subjective value function  is not always concave \\cite{kahneman2013choices}. They noted that in a loss scenario there is a convexity revealing a preference for uncertain loss rather than for certain loss.\n   If we replace the word ``win'' for ``lose'' in the lotteries $l_1$ and $l_2$, then we have the following lotteries that result in the wealth decrease:  \n\\begin{itemize}\n\\item $l_3$ = to lose $Mp$ (at the next moment);\n\\item $l_4$ = to lose $M$ (at the next moment) with probability $p$.\n\\end{itemize}\n\nNow consider the hypothesis $\\Theta_4$ = ``to lose $M$''. If $l_4$ is repeated until the individual loses $M$, then this lottery becomes similar to \n\\[F\\Theta_4=\\text{ ``will sometime lose } M \\text{'',}\\] \nwhere $p$ is the time probability   (sense of truth for $F\\Theta_4$). \n\nSimultaneously, the lottery $l_3$ proposes a certain loss. Then we can reduce its sense of truth for $p$ to compare with $F\\Theta_4$. For this, let us consider the following hyperbole\n \\[L_3 = \\text{``to lose } W_0-W_0\\left(1-\\frac{pM}{W_0}\\right)^\\frac{1}{p}\\text{'',}\\]\nin which the affirmation in the future\n\\[\\centering\n\\begin{array}{rl}\nFL_3  = \\text{``will sometime lose } W_0-W_0\\left(1-\\frac{pM}{W_0}\\right)^\\frac{1}{p}\\text{''}\n   \\end{array}\n\\]\narguments an expected change  $(1+px)^\\frac{1}{p}-1$ for $-1\\leq x<0$.\n\n\n\n The line $x$ is tangent to the convex curve $(1+px)^\\frac{1}{p}-1$ at the point $x=0$ for any  $p$, what results in  $(1+x)^\\frac{1}{p}-1 \\geq x$ for $-1\\leq x<0$. In Figure \\ref{figPT2}  the dashed black line   $x$  represents the proposed  change in $l_4$ and the curves  $\\ast$-red and $\\circ$-blue, \nbelonging  to the  family of curves $(1+px)^\\frac{1}{p}-1$,  represent the hyperbolic argumentation for $l_3$.\n \\begin{figure}[!ht]\n\t\\centering\n\t\\includegraphics[scale=0.65]{PT2}\n\t\t\\caption{The hyperbolic curves $(1+px)^{\\frac{1}{p}}$ for $p=1\/2$ and $p=1\/10$. Note that the curves tangentiate the black dashed line $x$ at the point zero. The interval $-0.1\\leq x <0$  is a region of low distinguishability. }\t\n\t\\label{figPT2}\n\\end{figure} \nTherefore, note that  the family  of curves $(1+px)^\\frac{1}{p}-1$   is very close to  line $x$ until 0.1. \nThis means that they have low distinguishability in this region, in other words, uncertain and certain losses can be imperceptible changes  when the losses are small.\nIn fuzzy logic is equivalent  to choose between ``small decrease in wealth with certainty'' and ``small decrease in wealth with probability $p$''. \nThe decreases in wealth are almost the same and undesirable, but the uncertainty argues hope for escaping losses and it is desirable. Thus, the uncertain option for losses will be more attractive in this situation. Then, in order to simulate risk seeking  in the losses region   we must insert a rate $\\rho$ into the hyperbolic argumentation process, so that\n\\begin{eqnarray}\n\\nonumber {\\cal H}^-_p(\\rho x)&\\equiv &(1+p\\rho x)^\\frac{1}{p}-1\\\\\n\\nonumber &=& e_p^{\\rho x}-1.\n\\end{eqnarray}\n\n\nThe rate $\\rho$ makes the curve ${\\cal H}^-_p(\\rho x) $ more convex. Thus, its first values pass  below the line $x$ to simulate the risk seeking.  In Figure \\ref{figPT3} the red curve   has  $\\rho=1.2$ and $p=1\/2$ to simulate this effect in the interval $-0.55<x<0$. The value $\\rho=1.2$ was exaggerated just to make the risk seeking visible in the figure, but it should be slightly larger than 1 for a more realistic representation.\n\nNow we can make a judgment between the sentences $FL_3$ and $F\\Theta_4$ because the difference between them is only the argued decrease  wealth, \n\\begin{eqnarray}\n FL_3 \\text{ or } F\\Theta_4 &=& \\max\\{ \\mu \\left( e_p^{\\rho x}-1\\right), \\; \\mu(x)\\}  \\label{CompHiper}\\\\\n\\nonumber &=& \\left\\{ \\begin{array}{l}\n \\mu(x)  \\text{ for small losses},\\\\\n\\mu \\left( e_p^{\\rho x}-1\\right) \\text{ for big losses}.\n\\end{array}\\right.\n\\end{eqnarray}\nThis means that the lottery $l_4$ is preferable when $M$ represents small losses, but when the losses are large, then $l_3$ is interpreted as the best option. Thus, the risk seeking disappears when the fuzzy sets are distinguishable.\n\n\n\n In order to understand the disappearance of  risk seeking, imagine all the goods necessary for your survival. After imagining them, then what do you prefer?  ``to lose 50\\% of all assets'' or ``a chance to lose all assets with probability 0.5''? If you chose the first option then you understood that a ruin aversion \\cite{taleb2018skin} can be dominant in this context. On this account, if   ``all assets'' is representing all the means of survival, then the second option may mean a death anticipation  after the lottery, while the former will allow the continuation of life with half of all assets.\n\nFinally, the function with expected changes at the next moment, convex for losses and concave for gains, exhibiting a S-shape, can be defined by\n\\[{\\cal{S}}_p(x)=\\left\\{ \\begin{array}{ll}\n p\\ \\text{ln}_p (1+x) & \\mbox{if}\\quad x\\geq 0,\\\\\ne_p^{\\rho x}-1& \\mbox{if}\\quad -1<x<0.\n\\end{array}\\right.\\]\nIn Figure \\ref{figPT3}, when  $p=1\/2$ and $\\rho=1.2$, the function ${\\cal{S}}_p(x)$  has risk seeking for $-0.55<x<0$, but has a kind of ruin aversion for $-1<x<-0.55$. The risk aversion is always present for all $x>0$.\n\n\\begin{figure}[!ht]\n\t\\centering\n\t\\includegraphics[scale=0.65]{PT3}\n\t\t\\caption{Function ${\\cal S}_{0.5} (x)$ for $\\rho=1.2$. When the red curve is below the dashed black line we have risk seeking (interval $-0.55<x<0$). On the other hand, we have a kind of ruin aversion when the red line is above the line $x$ (interval $-1<x<-0.55$). The risk aversion behavior is always present for all $x>0$ (dashed black line $x\/2$ above the red curve). }\t\n\t\\label{figPT3}\n\\end{figure} \n\\subsection{Loss aversion and disjunction between hypotheses}\n\\label{averisco}\nThe loss aversion principle, ${\\cal{S}}_p(x)<-{\\cal{S}}_p(-x)$, refers to the tendency to avoid losses rather than to acquire equivalent gains, in other words,  it is better not to lose \\$ 1,000  than to win \\$ 1,000.  \n\nIn order to understand why the curve is steeper on the losses side, consider that Bob has only \\$10,000 (all of his money). If now he loses \\$ 1,000, then the variation is -10\\% and his new wealth is \\$ 9,000. However, if he wins \\$ 1,000,  then the positive variation is 10\\%  and his new wealth is \\$ 11,000. So far this process seems fair, but we need to look at it dynamically.\n If he will have \\$ 9,000 at the next moment, then he will need to gain 11.11\\% to get back \\$ 10,000. On the other hand, if  he will have \\$ 11,000 at the next moment, then the required variation  is -9.09\\%  to get back \\$ 10,000. So which is the most difficult change to happen? Exactly, the 11.11\\% that restore the previous state after to lose 10\\%. Therefore, it is better not lose 10\\% than to gain 10\\% in the long-term gamble.\n\n\nThis behavior can be modeled in fuzzy temporal logic through the disjunction operator. In order to understand the details about the disjunction between loss and gain hypotheses, consider the lottery    \n\\begin{eqnarray}\n\\nonumber L_{wl} &=& \\text{``to win } M_1 \\text{ with probability } p\\\\\n\\nonumber &\\text{ }&\\text{ or to  lose }M_2\\text{ with probability } q\\text{''.}\n\\label{AversoRisco}\n\\end{eqnarray}\nIf winning  $M_1$ produces a gain  $x_1>0$ and losing $M_2$ produces a loss, $-1\\leq x_2 <0$, so we have the following atomic hypotheses\n\\[\n\\begin{array}{l}\nH_1 = \\text{``to win }M_1\\text{'' and } H_2 =  \\text{``to lose }M_2\\text{''.}\n\\end{array}\n\\]\nWhere the sense of truth for $H_1$ is $p$ and the sense of truth for $H_2$ is $q$. The future statement for disjunction is \n\\begin{eqnarray}\n\\nonumber F (H_1 \\vee H_2) &=& \\text{``to win } M_1 \\text{ or to  lose }M_2\\\\\\nonumber &\\text{ }&\\text{once in the future''.}\n\\label{LoteriaDisjunta}\n\\end{eqnarray}\nThe change average in this disjunction is $(1+x_1)^p(1+x_2)^q$  for $p+q<1$. This means that one of the hypotheses may be true at the next instant, or none, because only one will sometime be true  in the future.\n \n Another way of affirming a disjunction of losses and gains  is ensuring that one or the other will be true at the next instant, $N (H_1 \\vee H_2)$. The change average in this case is  $(1+x_1)^p(1+x_2)^q$  for $p+q=1$. This means that $H_1$ or $H_2$ will be true at the next moment with absolute certainty. Uncertainty is just ``which hypothesis is true?''. Therefore, the judgment preceding the decision whether or not  participating in this lottery, $N(H_1 \\vee H_2)$ or  nothing, is equal to\n\\begin{equation}\n\\nonumber  \\max \\{\\mu\\left((1+x_1)^p(1+x_2)^q-1\\right),\\mu(0)\\}.\n\\end{equation}\nThe lottery $L_{wl}$, which is a  loss and gain disjunction, will be considered fair if the parameters $x_1$, $x_2$, $p$ and $q$  guarantee $(1+x_1)^p(1+x_2)^q-1>0$. In the experiment described at \\cite{kahneman2013prospect}, the value $p=1\/2$ and $x_1=x_2=x$ generate the inequality $\\sqrt{1-x^2}-1<0$  that makes the lottery unfair. Thus, the respondent's choice for not betting seems to reveal a perception about the lottery dynamics. In addition, it can be noted  that  expected negative change has its intensity increased by $x$. Therefore,   the intensity of loss aversion is amount magnitude dependent. This means that the feeling of aversion to the lottery increases with the growth of the amount. In \\cite{mukherjee2017loss} is presented an empirical evidence of this behavior. \n\n\n \n\\section{Conclusion}\nHeuristics are cognitive processes that ignore part of the information and use simple rules to make a decision quicker and easier. In general, they are defined as strategies that seek alternatives, stop searches in a short time, and make a decision soon after.\n\n\nWithin heuristic processes, some decision-making requires the hypotheses judgment about dynamic processes before they take place in time. Time Preference Problem and Prospect Theory are famous examples. The first evaluates the goods receipt at different future dates  and the second requires lotteries valuation before their outcomes are known. The common characteristics  between these two problems noted here  are the magnitude dependence and the inseparability between time and uncertainty. On the magnitude dependence it can be concluded that:\n\\begin{itemize}\n\\item the magnitude effect in time preference is a consequence of subadditivity;\n\\item the risk seeking can disappear in the Prospect Theory if  high magnitude losses were considered. In addition, the aversion risk  increases with the growth of the amounts.\n\\end{itemize}\nOn the other hand, on the  inseparability between time and uncertainty it can be concluded that:\n\\begin{itemize}\n\\item in the time preference problem, the number of uncertain trials for the short-term hypotheses until verification of the long-term hypothesis produces the subadditive discounting, and consequently, higher annual average rates as the waiting time decreases. In addition, the preference reversal  occurs because the number of allowed trials is changed when the  hypothesis deadlines are shifted in time;\n\\item the probabilities of lotteries represent the temporal indeterminism about the future. Thus, the S-shaped curve in the Prospect Theory can be described by expected fuzzy changes of temporal hyptotheses.\n\\end{itemize}\n\nIf the future is uncertain, then time and uncertainty about changes can not mean two independent matters. For this reason, choice under uncertainty and intertemporal choice, traditionally treated as separate subjects, are unified in a same matter in this paper to elaborate the rhetoric for the decision-making. \n\nIn addition, it is shown that the fuzziness can changes to prospective judgments about  magnitude dependent gains and losses. This means that  a given problem may have different decisions simply by changing  the values of the rewards, even if time and uncertainty context are not changed. \nExactly in these situations, fuzzy environment modeling will be essential to represent the decision-making.\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nUplift modeling involves a set of methods for estimating the expected causal impact of taking an action or a treatment at an individual or subgroup level, which could lead to an increase in their conversion probability \\cite{Zhao2019UpliftMF}.\nTypically for financial services and commercial companies looking to provide additional value-added services and products to their customers, marketers may be interested in evaluating the effectiveness of numerous marketing techniques, such as sending promotional coupons.\nWith the change in customer conversion possibilities, marketers are able to efficiently target prospects.\nMore than marketing campaigns, uplift modeling can be applied to a variety of real-world scenarios related to personalization, such as online advertising, insurance, or healthcare, where patients with varying levels of response to a new drug are identified, including the discovery of adverse effects on specific subgroups \\cite{Jaskowski2012UpliftMF}.\n\nIn essence, uplift modeling is a problem that combines causal inference and machine learning. For the former, it is mutually exclusive to estimate the change between two outcomes for the same individual. To overcome this counterfactual problem, samples are randomly assigned to a treatment group (receiving online advertisement or marketing campaign) and a control group (not receiving online advertisement nor marketing campaign). For the latter, the task is to train a model that predicts the difference in the probability of belonging to a given class on the two groups.\nAt present, two major categories of estimation techniques have been proposed in the literature, namely meta-learners and tailored methods \\cite{Zhang2022AUS}. The first includes the Two-Model approach \\cite{Radcliffe2007UsingCG}, the X-learner \\cite{Knzel2017MetalearnersFE} and the transformed outcome methods \\cite{Athey2015MachineLM} which extend classical machine learning techniques. The second refers to direct uplift modeling such as uplift trees \\cite{5693998} and various neural network based methods \\cite{ Louizos2017CausalEI,Yoon2018GANITEEO}, which modify the existing machine learning algorithms to estimate treatment effects. \nAlso, uplift trees can be extended to more general ensemble tree models, such as causal forests \\cite{doi:10.1080\/01621459.2017.1319839,10.1214\/18-AOS1709}, at the cost of losing true interpretability.\n\nIn order to take advantage of decision trees and the ensemble approach, we propose causal inference based single-branch ensemble trees for uplift modeling (CIET) with two completely different partition criteria that directly maximizes the difference between outcome distributions of the treatment and control groups. When building a single-branch tree, we employ lift gain and lift gain ratio as loss functions or partition criteria for node splitting in a recursive manner. \nSince our proposed splitting criteria are highly related to the incremental impact, the performance of CIET is thus expected to be reflected in the uplift estimation. Meanwhile, \nthe splitting logic of all nodes along the path from root to leaf is combined to form a single rule to ensure the interpretability of CIET. Moreover, the dataset not covered by the rule is then censored and the above tree-building process continue to repeat on the censored data. \nDue to this divide and conquer learning strategy, the dependencies between the formed rules can be effectively avoided. It leads to the formation of single-branch ensemble trees and a set of decorrelated inference rules.  \n\nNote that our CIET is essentially different from decision trees for uplift modeling and causal forests. There are three major differences: (1) single-branch tree $vs$ standard binary tree; (2) lift gain and its ratio as loss function or splitting criterion $vs$ Kullback-Leibler divergence and squared Euclidean distance; and (3) decorrelated inference rules $vs$ correlated inference rules or even no inference rules. \nIt is demonstrated through empirical experiments that CIET can achieve better uplift estimation compared with the existing models. Extensive experimental results on synthetic data and the public credit card data show the success of CIET in uplift modeling. We also train an ensemble model and evaluate its performance on a large real-world online loan application dataset from a national financial holdings group in China. As expected, the corresponding results show a significant improvement in the evaluation metrics in terms of both AUUC and Qini coefficient.\n\n\n\n\nThe rest of this ms is organized as follows. First, causal inference based single-branch ensemble trees for uplift modeling is introduced. Next, full details of our experimental results on synthetic data, credit card data and real-world online loan application data are given. \nIt is demonstrated that CIET performs well in estimating causal effects compared to decision trees for uplift modeling. Finally, conclusions are presented.\n\n\n\n\\section{Causal Inference Based Single-Branch Ensemble Trees (CIET) for Uplift Modeling}\\label{sec:Alg}\nThis section consists of three parts. We first present two splitting criteria, single-branch ensemble approach and pruning strategy specially designed for the uplift estimation problem. Evaluation metrics for uplift modeling are then discoursed. Three key algorithms of CIET are further described in detail.\n\n\\subsection{Splitting Criteria, Single-Branch Ensemble Method and Pruning Strategy}\nTwo distinguishing characteristics of CIET are splitting criteria for tree generation and the single-branch ensemble method, respectively.\n\nAs for splitting criteria in estimating uplift, it is motivated by our expectation to achieve the maximum difference between the distributions of the treatment and control groups. \nGiven a class-labeled dataset with $N$ samples, $N^{T}$ and $N^{C}$ are sample size of the treatment and control groups (recall that $N = N^{T} + N^{C}$, $T$ and $C$ represent the treatment and control groups).\nFormally, in the case of a balanced randomized experiment, the estimator of the difference in sample average outcomes between the two groups is given by:\n\\begin{align}\n    \\label{tau} \\tau = (P^{T} - P^{C})(N^{T} + N^{C})\n\\end{align}\nwhere $P^{T}$ and $P^{C}$ are the probability distribution of the outcome for the two groups. Motivated by Eq. (\\ref{tau}), the divergence measures for uplift modeling we propose are lift gain and its ratio, namely LG and LGR for short. The corresponding mathematical forms of LG and LGR can be thus expressed as\n\\begin{align}\n    \\label{LG} LG &=(P_{R}^{T} - P_{R}^{C})N_{R} - \\tau_{0} = \\tau_{R} - \\tau_{0}\\\\\n    \\label{LGR} LGR  &=\\frac{(P_{R}^{T} - P_{R}^{C})}{(P_{0}^{T} - P_{0}^{C})} \\propto (P_{R}^{T} - P_{R}^{C}) = \\frac{\\tau_{R}}{N_{R}}\n\\end{align}\nwhere $P_{0}^{T}$ and $P_{0}^{C}$ are the initial probability distribution of the outcome for the two groups, $\\tau_{0} = (P_{0}^{T} - P_{0}^{C})N_{R}$. And $N_{R}$ and $Y_{R}$ for a node logic $R$ represent coverage and correction classification,\nwhile $P_{R}^{T}$ and $P_{R}^{C}$ are the corresponding probability distribution of the outcome for both groups, respectively. Evidently, both Eq. (\\ref{LG}) and Eq. (\\ref{LGR}) represent the estimator for uplift, which are proposed as two criteria in our ms. Compared to the standard binary tree with left and right branches, only one branch is created after each node splitting in this ms. It is characterized by the fact that both LG and LGR are calculated using the single-branch observations that present following a node split. Accordingly, subscript $k$ indicating binary branches doesn't exist in the above equation. Furthermore, the second term of LG makes every node partition better than randomization, while LGR has the identical advantages to information gain ratio. \n\n\n\nThe proposed splitting criterion for a test attribute A is then defined for any divergence measure $D$ as \n\\begin{equation}\\label{CRITERION}\n    \\Delta = D(P^T(Y):P^C(Y)|A) - D(P^T(Y):P^C(Y))  \n\\end{equation}\nwhere $D(P^T(Y):P^C(Y)|A)$ is the conditional divergence measure. Apparently, $\\Delta$ is the incremental gain in divergence following a node splitting. Substituting for $D$ the $LG$ and $LGR$, \nwe obtain our proposed splitting criteria $\\Delta_{LG}$ and $\\Delta_{LGR}$. \nThe intuition behind these splitting criteria is as follows: we want to build a single-branch tree such that the distribution divergence between the treatment and control groups before and after splitting an attribute differ as much as possible. \nThus, an attribute with the highest $\\Delta$ is chosen as the best splitting one. In order to achieve it, we need to calculate and find the best splitting point for each attribute. In particular, an attribute is sorted in descending order by value when it is numerical. \nFor categorical attributes, some encoding methods are adopted for numerical type conversion. The average of each pair of adjacent values in an attribute with $n$ value, forms $n-1$ splitting points or values. \nAs for this attribute, the point of the highest $\\Delta$ can be seen as the best partition one. \nFurthermore, the best splitting attribute with the highest $\\Delta$ can be achieved by traversing all attributes. \nAs for the best splitting attribute, the instances are thus divided into two subsets at the best splitting point. One feeds into a single-branch node, while the other is censored. \nNote that the top\u2013down, recursive partition will continue unless there is no attribute that explains the incremental estimation with statistical significance. Also, histogram-based method can be employed to select the best splitting for each feature, which can reduce the time complexity effectively.\n\nDue to noise and outliers in the dataset, a node may merely represent these abnormal points, resulting in model overfitting. Pruning can often effectively deal with this problem. That is, using statistics to cut off unreliable branches. Since none of the pruning methods is essentially better than others, we use a relatively simple pre-pruning strategy. If $\\Delta$ gain is less than a certain threshold, node partition would stop. Thus, a smaller and simpler tree is constructed after pruning. Naturally, decision-makers prefer less complex inference rules, since they are considered to be more comprehensible and robust from business perspective.\n\n\n\\subsection{Evaluation Metrics for Uplift Modeling}\nAs noted above, it is impossible to observe both the control and treatment outcomes for an individual, which makes it difficult to find measure of loss for each observation.\nIt leads that uplift evaluation should differ drastically from the traditional machine learning model evaluation. \nThat is, improving the predictive accuracy of the outcome \ndoes not necessarily indicate that the models will have better performance in identifying targets with higher uplift. In practice, most of the uplift literature resort to aggregated\nmeasures such as uplift bins or curves. Two key metrics involved are area under uplift curve (AUUC) and Qini coefficient \\cite{Gutierrez2016CausalIA}, respectively. In order to define AUUC, binned uplift predictions are sorted from largest to smallest. For each $t$, the cumulative sum of the observations statistic is formulated as below,\n\\begin{equation}\\label{AUUC}\n    f(t) = (\\frac{Y_t^{T}}{N_t^{T}} - \\frac{Y_t^{C}}{N_t^{C}})(N_t^{T} + N_t^{C}) \n\\end{equation}\nwhere the $t$ subscript implies that the quantity is calculated on the first or top $t$ observations.\nThe higher this value, the better the uplift model. The continuity of the uplift curves makes it \npossible to calculate AUUC, i.e. area under the real uplift curve, which can be used to evaluate and compare different models. As for Qini coefficient, it represents a natural generalization of \nGini coefficient to uplift modeling. Qini curve is introduced with the following equation,\n\\begin{equation}\\label{QINI_Curve}\n   g(t) = {Y_t^{T}} - \\frac{Y_t^{C}N_t^{T}}{N_t^{C}}\n\\end{equation}\nThere is an obvious parallelism with the uplift curve since $f(t)=g(t)(N_t^{T}+N_t^{C})\/N_t^{T}$.\nThe difference between the area under the actual Qini curve and that under the diagonal corresponding to random targeting can be obtained. \nIt is further normalized by the area between the random and the optimal targeting curves, which is defined as Qini coefficient.  \n\n\n\\subsection{Algorithm Modules}\nThe following representation of three algorithms includes: selecting the best split for each feature using the splitting criteria described above, learning a top-down induced single-branch tree and forming ensemble trees with each resulting tree progressively.\n\nAlgorithm \\ref{single1:algorithm} depicts how to find the best split of a single feature $F$ on a given dataset $D[group\\_key, feature,$ $ target]$ using a histogram-based method with the proposed two splitting criteria. Gain\\_left and Gain\\_right are the uplift gains for the child nodes after each node partition. If the maximum value of Gain\\_left is greater than that of Gain\\_right, the right branch is censored and vice versa. Thus, the best split with its corresponding splitting logic, threshold and uplift gain is found, which is denoted by Best$\\_$Direction, Best$\\_$Threshold and Best$\\_\\Delta$. Besides, there are several thresholds to be initialized before training a CIET model, including minimum number of samples at a inner node $min\\_samples$, minimum recall $min\\_recall$ and minimum uplift gain required for splitting $min\\_\\Delta$. The top-down process would continue only when the restrictions are satisfied.\n\n\\begin{algorithm}[htbp]\n\\caption{Selecting the Best Split for One Feature}\n\\label{single1:algorithm}\n\\textbf{Input}: D, the given class-labeled dataset, including the group key (treatment\/control);\\\\\n\\textbf{Parameter}: feature $F$, min$\\_$samples, min$\\_$recall, min$\\_\\Delta$\\\\\n\\textbf{Output}: the best split that maximizes the lift gain or lift gain ratio on a feature\n\n\\begin{algorithmic}[1]\n\\STATE \\textbf{Set} Best$\\_$Value = 0, Best$\\_$Direction = \"\", Best$\\_\\Delta$ = 0, Best$\\_$Threshold = None\n\\STATE calculate $Y^T$, $Y^C$, $N^T$, $N^C$ on D\n\\STATE For each feature value $v$ , calculate the values of $Y_{F\\leq v}^T$, $Y_{F\\leq v}^C$, $N_{F\\leq v}^T$, $N_{F\\leq v}^C$, and then the Gain\\_left($v$) and Gain\\_right($v$) with LG \\eqref{LG} or LGR \\eqref{LGR}.\n\\STATE set the Gain\\_left($v$) and Gain\\_right($v$) to their minimum value on the $v$, whose split does not satisfy the restrictions on number of samples\/recall rate\/divergence measure gain.  \n\\STATE $v_1$ = argmax(Gain\\_left($v$)), $v_2$ = argmax(Gain\\_right($v$))\n\\IF {max(Gain\\_left) $\\geq$ max(Gain\\_right)}\n\\STATE Best$\\_$Value = max(Gain\\_left)\n\\STATE Best$\\_$Direction = \"$\\leq$\"\n\\STATE Best$\\_$Threshold = $v_1$\n\\ELSE\n\\STATE Best$\\_$Value = max(Gain\\_right)\n\\STATE Best$\\_$Direction = \"$>$\"\n\\STATE Best$\\_$Threshold = $v_2$\n\\ENDIF\n\\STATE \\textbf{return} Best$\\_$Value, Best$\\_$Direction, Best$\\_$Threshold\n\\end{algorithmic} \n\\end{algorithm}\n\nAlgorithm \\ref{single:algorithm} presents a typical algorithmic framework for top\u2013down induction of a single-branch uplift tree, which is built in a recursive manner using a greedy depth-first strategy. \nThe parameter $max\\_depth$ represents the depth of the tree and $cost$ indicates the threshold of LG or LGR.\nAs the tree grows deep, more instances are censored since they are not covered by the node partition logic of each layer. \nAs a result, each child node subdivides the original dataset hierarchically into a smaller subset until the stopping criterion is satisfied. Tracing the splitting logics on the path from the root to leaf nodes in the tree, an \"IF-THEN\" inference rule is thus extracted. \n\nFinally, adopting a divide-and-conquer strategy, the above tree-building process is repeated on the censored samples to form ensemble trees, resulting in the formation of a set of inference rules as shown in Algorithm \\ref{multi:algorithm}.\n\n\n\\begin{algorithm}[htbp]\n\\caption{Learning An \"IF-THEN\" Uplift Rule of A Single-branch Tree}\n\\label{single:algorithm}\n\\textbf{Input}: D, the given class-labeled dataset, including the group key (treatment\/control);\\\\\n\\textbf{Parameter}: max$\\_$depth, cost, min$\\_$samples, min$\\_$recall, min$\\_\\Delta$\\\\\n\\textbf{Output}: an \"IF-THEN\" uplift rule\n\n\\begin{algorithmic}[1]\n\\STATE \\textbf{Set} Rule$\\_$Single = [], Max$\\_$Gain = 0.0\n\\STATE \\textbf{Set} Add\\_Rule = \\textbf{True}\n\\WHILE{depth $\\leq$ max$\\_$depth \\textbf{and} Add\\_Rule}\n\\IF {the treatment group or control group in D is empty}\n\\STATE break\n\\ENDIF\n\\STATE \\textbf{Set}  Keep = \\{ \\}, Best$\\_$Split = \\{ \\}\n\\STATE depth $\\leftarrow $ depth + 1\n\\STATE Add\\_Rule = \\textbf{False}\n\\FOR{feature in features}\n\\STATE Keep[feature] =  Best\\_Split\\_for\\_One\\_Feature(D, feature,  min$\\_$samples, min$\\_$recall, min$\\_\\Delta$) (Algorithm \\ref{single1:algorithm})\n\\ENDFOR\n\\FOR{feature in Keep}\n\\IF {feature's best gain $>$ Max\\_Gain + cost}\n\\STATE Max$\\_$Gain = feature's best gain\n\\STATE \\textbf{Add} Keep[feature] \\textbf{to} Best$\\_$Split\n\\STATE Add\\_Rule = \\textbf{True}\n\\ELSE\n\\STATE continue\n\\ENDIF\n\\ENDFOR\n\\STATE \\textbf{Add} Best$\\_$Split \\textbf{to} Rule$\\_$Single\n\\STATE D $\\leftarrow $ D $\\setminus$ \\{\nSamples covered by Rule$\\_$Single\\} \n\\ENDWHILE\n\\STATE \\textbf{return} Rule$\\_$Single\n\\end{algorithmic} \n\\end{algorithm}\n\n\n\\begin{algorithm}[htbp]\n\\caption{Learning A Set of \"IF-THEN\" Uplift Rules}\n\\label{multi:algorithm}\n\\textbf{Input}: D, the given class-labeled dataset, including the group key (treatment\/control);\\\\\n\\textbf{Parameter}: max$\\_$depth, rule$\\_$count, cost, min$\\_$samples, min$\\_$recall, min$\\_\\Delta$\\\\\n\\textbf{Output}: a set of \"IF-THEN\" uplift rules\n\n\\begin{algorithmic}[1]\n\\STATE \\textbf{Set} Rule$\\_$Set = \\{\\}, number = 0\n\\WHILE{number $\\leq$ rule$\\_$count}\n\\STATE rule = Single$\\_$Uplift$\\_$Rule(D, cost, max$\\_$depth, min$\\_$samples, min$\\_$recall, min$\\_\\Delta$) (Algorithm \\ref{single:algorithm})\\\\\n\\STATE \\textbf{Add} rule \\textbf{to} Rule$\\_$Set  \n\\STATE D $\\leftarrow $ D $\\setminus$ dataset covered by rule \n\\STATE number $\\leftarrow$ number + 1\n\\ENDWHILE\n\\STATE \\textbf{return} Rule$\\_$Set\n\\end{algorithmic} \n\\end{algorithm}\n\n\\begin{figure*}[htbp]\n\\centering\n\\includegraphics[width=0.95\\textwidth]{figs\/AUUC.png} \n\\caption{The uplift curves of four analyzed classifiers with four different colors for the synthetic dataset, while the dashed line corresponding to random targeting.}\n\\label{fig:AUUC_SSD} \n\\end{figure*}\n\n\\section{Experiments}\\label{sec:er}\nIn this section, the effectiveness of our CIET is evaluated on synthetic and real-world business datasets. Since CIET fundamentally stems from tree-based approaches, we implement it and compare it with uplift decision trees based on squared Euclidean distance and Kullback-Leibler divergence \\cite{5693998}, which are referred as baselines.\n\n\n\n\\begin{table*}[ht]\n\\centering\n\\begin{tabular}{llll}\n\\hline\nrule number  &\"1\"  &\"2\"  &\"3\"\\\\\n\\hline\nnode logic     &x9$\\_$uplift $\\leq$ 0.17                                  &x3$\\_$informative $>$ -1.04                               &x6$\\_$informative $>$ 0.95 \\\\\nnode logic     &x10$\\_$uplift $\\leq$ 2.59                                 &x1$\\_$informative $\\leq$ 2.71                             &x1$\\_$informative $\\leq$ 1.58\\\\\nnode logic     &null                                                      &x2$\\_$informative $\\leq$ 1.28                             &x9$\\_$uplift $\\leq$ 2.09     \\\\\n$N_{before}$     &3000     &2210     &675     \\\\\n$N_{before}^T$   &1500     &1117     &348     \\\\\n$N_{before}^C$   &1500     &1093     &327     \\\\\n$N_{rule}$        &790     &1535     &180     \\\\\n$N_{rule}^T$      &383      &769      &76     \\\\\n$N_{rule}^C$      &407      &766     &104     \\\\\nnet gain       &195.99    &87.14   &37.66     \\\\\n$recall_{treatment}$  &36.62$\\%$ &70.42$\\%$ &42.11$\\%$ \\\\\n$recall_{control}$    &28.00$\\%$ &63.70$\\%$ &44.90$\\%$ \\\\\n\\hline\n\\end{tabular}\n\\caption{A set of inference rules found by CIET and their corresponding statistical indicators with $criterion\\_type = $\"LG\", $rule\\_count = 3$ and $max\\_depth = 3$.}\n\\label{tab:RS_CIET}\n\\end{table*}\n\n\n\\subsection{Experiments on Synthetic Data}\\label{subsec: SD}\n\\textbf{Dataset} We can test the methodology with numerical simulations. That is, generating synthetic datasets with known causal and non-causal relationships between the outcome, action (treatment\/control) and some confounding variables. More specifically, both the outcome and the action\/treatment variables are binary. A synthetic dataset is generated with the $make\\_uplift\\_classification$ function in \"Causal ML\" package, based on the algorithm in \\cite{Guyon2003DesignOE}.\nThere are 3,000 instances for the treatment and control groups, with response rates of 0.6 and 0.5, respectively. The input consist of 11 features in three categories. 8 of them are used for base classification, which are composed of 6 informative and 2 irrelevant variables. 2 positive uplift variables are created to testify positive treatment effect. The remaining one is a mix variable, which is defined as a linear superposition of a randomly selected informative classification variable and a randomly selected positive uplift variable.\n\n\n\n\n\\textbf{Parameters and Results} There are four hyper-parameters in CIET: $criterion\\_type$, $max\\_depth$ and $rule\\_count$. $criterion\\_type$ includes two options, LG and LGR. \nMore precisely, two main factors of business complexity and difficulty in online deployment, determine parameter assignment.\nDue to the requirement of model generalization and its interpretability, $max\\_depth$ is set to 3. That is, the business logics of a single inference rule are always less than or equal to 3. And, $rule\\_count$ is given a value of 3, indicating that a set of no more than three rules is defined to model the causal effect of a treatment on the outcome. Meanwhile, the default values for $min\\_samples$, $min\\_recall$, $cost$ and $min\\_\\Delta$ are 50, 0.1, 0.01 and 0, respectively.\n\n\n\n\\begin{table}[htbp]\n\\centering\n\\begin{tabular}{lllll}\n\\hline\ndataset    &KL  &Euclid  &LG  &LGR\\\\\n\\hline\ntraining   &0.187  &0.189  &0.239  &0.235 \\\\\ntest       &0.176  &0.178  &0.210  &0.225 \\\\\n\\hline\n\\end{tabular}\n\\caption{Qini coefficients of four analyzed classifiers on the training and test sets of the synthetic data.}\n\\label{tab:QC_SSD}\n\\end{table}\n\n\nThe stratified sampling method is used to divide the synthetic dataset into training and test sets in a ratio of fifty percent to fifty percent. Figure \\ref{fig:AUUC_SSD} shows the uplift curves of the four analyzed classifiers. The AUUC of CIET with LG and LGR are 294 and 292 on the training set, which are significantly greater than 266 and 265 for the decision trees for uplift modeling with KL divergence and squared Euclidean distance. \nAt the 36th percentile of the population, the cumulative profit increase reach 303 and 354 for LG and LGR, resulting in a growth rate of more than 18$\\%$ and 37$\\%$ compared to baselines. Besides, AUUC shows little variation in the training and test datasets, indicating that the stability of CIET is also excellent. According to Table \\ref{tab:QC_SSD}, Qini coefficients of CIET are also obviously greater, with an increase of more than 24.5$\\%$ and 17.8$\\%$. Furthermore, all three rules are determined by uplift and informative variables as expected, which can be seen from Table \\ref{tab:RS_CIET}.   \n\n\n\n\n\n\\subsection{Experiments on Credit Card Data}\n\n\\textbf{Dataset} We use the publicly available dataset $Credit$ $Approval$ from the UCI repository as one of the real-world examples, which contains 690 credit card applications. All the 15 attributes and the outcome are encoded as nonsense symbols, where $A7 \\neq v$ \nis applied as the condition for dividing the dataset into treatment and control groups. \nThere are 291 and 399 observations in the two groups with response rates of 0.47 and 0.42, respectively.\nAttributes with more than 25$\\%$ difference in distribution between the two groups should be removed before any experiments are performed. This leads that 12 attributes are left as input variables. For further preprocessing, categorical features are binarized through one-hot encoding.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.95\\linewidth]{figs\/CC.png} \n\\caption{The uplift curves of four analyzed classifiers with four different colors for the $Credit\\ Approval$ dataset, while the dashed curve corresponding to the random targeting.}\n\\label{fig:FIG_CAD} \n\\end{figure}\n\n\\begin{table}[htbp]\n\\centering\n\\begin{tabular}{lllll}\n\\hline\nmetrics    &KL  &Euclid  &LG  &LGR\\\\\n\\hline\nAUUC   &37.337  &40.887  &42.893  &48.222 \\\\\nQini   &0.201  &0.236  &0.257  &0.310 \\\\\n\\hline\n\\end{tabular}\n\\caption{Model performance of four analyzed classifiers on the $Credit\\ Approval$ dataset.}\n\\label{tab:CAD}\n\\end{table}\n\n\n\\textbf{Parameters and Results} Based on the business decision-making perspective, the initial parameters are also the same as above.\n\n\n\n\\begin{figure*}[htbp]\n\\centering\n\\includegraphics[width=0.95\\textwidth]{figs\/ROP.png} \n\\caption{The uplift curves of four analyzed classifiers with four different colors for the real world online loan application dataset, while the dashed line corresponding to random targeting.}\n\\label{fig:AUUC_ROP} \n\\end{figure*}\n\nIn order to avoid the distribution bias caused by the division on such small sample size dataset, there is no need to divide $Credit\\ Approval$ into training and test parts. Figure \\ref{fig:FIG_CAD} shows the uplift curves for the four analyzed classifiers, from which we can see that CIET is able to obtain higher AUUC and Qini coefficients. As shown in Table \\ref{tab:CAD}, the former increases from 37$\\sim$40 at baselines to 42$\\sim$48 at CIET approximately, while the latter also improves significantly from 0.20$\\sim$0.23 to 0.25$\\sim$0.31. Especially when LGR serves as the splitting criterion, the cumulative profit has a distinguished peak of 74.5, while only 48.4$\\%$ of the samples are covered.\n\n\n\n\n\\subsection{Experiments on Online Loan Application Data}\n\n\n\n\\textbf{Dataset} We further extend our CIET to precision marketing for new customer application. A telephone marketing campaign is designed to promote customers to apply for personal credit loans at a national financial holdings group in China via its official mobile app. The target is 1\/0, indicating whether a customer would submit an application or not. \nThe data contains 53,629 individuals, consisting of a treated group of 32,984 (receiving marketing calls) and a control group of 20,645 (not receiving marketing calls). These two groups have 300 and 124 credit loan applications with response rates of 0.9\\% and 0.6\\%, which are typical values in real world marketing practice. There are 24 variables in all, which are characterized as credit card-related information, loan history, customer demographics et al.\n\n\n\n\n\n\\begin{table}[htbp]\n\\centering\n\\begin{tabular}{lllll}\n\\hline\ndataset    &KL  &Euclid  &LG  &LGR\\\\\n\\hline\ntraining   &0.173  &0.168  &0.414  &0.302 \\\\\ntest       &0.108  &0.124  &0.385  &0.319 \\\\\n\\hline\n\\end{tabular}\n\\caption{Qini coefficients of four analyzed classifiers on the training and test sets of the real world online loan application data.}\n\\label{tab:QC_ROP}\n\\end{table}\n\n\n\n\\textbf{Parameters and Results} All parameters are the same as in the above experiments. The dataset is first divided into training and test sets in a ratio of sixty percent to forty percent. The response rates are consistent across two sets for two groups. Figure \\ref{fig:AUUC_ROP} diplays the results graphically. As for the training dataset, CIET based on LG and LGR reach AUUC of about 104 and 89, while the decision trees based on KL divergence and squared Euclidean distance are 73 and 72. It can be seen that CIET achieves a significant improvement compared to baselines on this real-world dataset even with a very low response rate. Moreover, as can be seen in Table \\ref{tab:QC_ROP}, Qini coefficient based on our approaches increases to 0.30$\\sim$0.41 from 0.16$\\sim$0.17 \non the training dataset. Meanwhile, Qini coefficient changes little when crossing to test dataset, indicating a better stability. Consequently, classifier with our CIET for precision marketing is effectively improved as well as stabilized in terms of AUUC and Qini coefficient. At present, CIET has already been applied to personal credit telemarketing.  \n\n\n\n\\section{Conclusion}\\label{sec:con}\nIn this ms, we propose new methods for constructing causal inference based single-branch ensemble trees for uplift estimation, CIET for short. Our methods provide two partition criteria for node splitting and strategy for generating ensemble trees. The corresponding outputs are uplift gain between the two outcomes and a set of interpretable inference rules, respectively. Compared with the classical decision tree for uplift modeling, CIET can not only be able to avoid dependencies among inference rules, but also improve the model performance in terms of AUUC and Qini coefficient. It would be widely applicable to any randomized controlled trial, such as medical studies and precision marketing. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nQuantum teleportation experiments have shown that quantum coherence can be \nmaintained for ever increasing distances. Indeed, the factor that hinders \ncoherence (breaking the required entanglement for teleportation) is the \nloss of signal to the medium, mainly the atmosphere. This obstacle is no \nlonger present in space, which hints at the possibility of performing \nsuch experiments at interstellar distances, or even detecting quantum \nsignals from astrophysical sources. In this context, one of us showed \nrecently that the quantum state of a photon could indeed be maintained \nat galactic distances, at least for a range of the electromagnetic \nspectrum \\cite{Berera2020}. The reason for this is that the mean free \npaths associated to the different interactions the photon could have are \nmany order of magnitude larger than the galactic scales \n(or even the observable Universe). \nAs an outcome of this observation, one seminal suggestion that paper \nmade was the possibility\nfor interstellar quantum communication, due to the viability for\nmaintaining quantum coherence over these distances\nfor certain frequency bands. Another possibility suggested in\nthat paper was if there were any natural quantum coherent sources,\nsuch signals could maintain their coherence over interstellar\ndistances.   Extending on these ideas,\nthat paper also noted that this (lack of) effect most likely can be extrapolated to cosmological distances. \n\nThis work will explore that possibility.  Here we consider\na wider variety of decoherence factors, like the expansion of the Universe itself. However, even for this case we do not give up on the philosophy that decoherence takes place due to the interaction of the quantum state with some environment. To do so, we consider the environment to be constituted by particles produced by the expansion of the Universe at different epochs. The mechanism to achieve this is squeezing, which has been widely studied in quantum optics and, in cosmology, in the theory of inflationary perturbations. So, borrowing from this mechanism, we compute the number of scalar particles through squeezing, and argue that this effect is essentially absent for fermions and $U(1)$ gauge bosons. Moreover, we identify the scalar field (interacting with photons) to be that of axion-like particles (ALPs), as a natural extension of the Standard Model. With these considerations, we are able to look at different interactions of the photon with the ALPs (or their decay products) in order to estimate the probability of interactions, which we find to be basically null. Thus, in practice, the expansion is not a decoherence factor for photons (at the energies we shall consider). We also look at other potential sources of decoherence, like interaction with CMB radiation or with electrons after reionization. The latter is more likely to be a source of decoherence, although the probabilities remain low enough to consider that the quantum state could remain undisturbed after decoupling. This opens up a new window to look for quantum signals from certain astrophysical objects or even from cosmic strings. \n\n\\section{Expansion-induced decoherence}\n\n\\subsection{Scalar fields}\n\nIn order to learn how the expansion of the Universe can lead to decoherence, let us look at the theory of cosmic inflation for guidance. \nCosmological perturbations during inflation undergo a process known as {\\it squeezing}, where states of the type $|n_{\\bf k}, n_{-{\\bf k}}\\rangle$ are created at superhorizon scales. This is an effect purely due to expansion, whose basic principles can be grasped just by studying a massless scalar field minimally coupled to gravity, as follows:\n\\begin{align}\n\t{\\cal S} & = \\frac{1}{2} \\int dt\\ d^3 x\\ \\sqrt{-g} \\partial_{\\mu} \\phi \\partial^{\\mu} \\phi \\nonumber \\\\\n\t& = \\frac{1}{2} \\int d\\tau\\ d^3 x\\ a^2 \\left[(\\phi')^2 - (\\nabla \\phi)^2 \\right],\n\\end{align}\nwhere primes denote derivative w.r.t. the conformal time $\\tau$. It is convenient to introduce the change of variable $\\varphi \\equiv a \\phi$, such that\n\\begin{equation}\\label{act1}\n\t{\\cal S} = \\frac{1}{2}\\int d\\tau\\ d^3 x\\ \\left[ \\left(\\varphi' - \\frac{a'}{a}\\varphi\\right)^2 - (\\nabla \\varphi)^2 \\right].\n\\end{equation}\nUsing the Euler-Lagrange equations, and going to Fourier space, one gets the mode equations\n\\begin{equation}\\label{eom1}\n\t\\varphi_k'' + \\left(k^2 - \\frac{a''}{a}\\right) \\varphi_k = 0.\n\\end{equation} \nIn the case of a perfect de Sitter expansion, $a''\/a = 2\/\\tau^2$, the equation of motion becomes\n\\begin{equation}\n\t\\varphi_k'' + \\left(k^2 - \\frac{2}{\\tau^2} \\right) \\varphi_k = \\varphi_k'' + \\left(k^2 - 2 (aH)^2 \\right) \\varphi_k = 0\\,.\n\\end{equation}\nClearly, the solutions are oscillatory for $k^2 > 2 (aH)^2$, whereas for $k^2 < 2(aH)^2$ there is a growing and a decaying-mode solution. The question which then arises is what should be the right initial state for solving this equation. For inflation, one usually takes Bunch-Davies initial states\n\\begin{equation}\n\t\\varphi_k (\\tau) = \\frac{e^{-i k \\tau}}{\\sqrt{2k}} \\left(1 - \\frac{i}{k\\tau}\\right)\\,,\n\\end{equation}\nsuch that the time-dependent field operator and the canonical momentum are given by\n\\begin{widetext}\n\\begin{equation}\n\t\\hat{\\varphi} (\\tau,\\vt{x}) = \\int \\frac{d^3k}{(2\\pi)^3} \\frac{1}{\\sqrt{2k}} \\left[e^{-ik\\tau}\\left(1 - \\frac{i}{k\\tau}\\right) \\cm{k} (\\tau_0) + e^{ik\\tau} \\left(1 + \\frac{i}{k\\tau}\\right) \\cpp{k} (\\tau_0) \\right] e^{i \\dpr{k}{x}},\n\\end{equation}\n\\begin{equation}\n\\hat{\\pi}(\\tau, \\vt{x}) = \\varphi' - \\frac{a'}{a}\\varphi = -i \\int \\frac{d^3k}{(2\\pi)^3} \\sqrt{\\frac{k}{2}} \\left[e^{-ik\\tau} \\cm{k} (\\tau_0) - e^{ik\\tau} \\cpp{k} (\\tau_0)\\right] e^{i \\dpr{k}{x}}\\,.\n\\end{equation}\n\\end{widetext}\n\nThe creation and annihilation operators at later times can be found through a Bogolyubov transformation, such that\n\\begin{align}\n\t\\cm{k} (\\tau) & = \\alpha_k (\\tau) \\cm{k} (\\tau_0) + \\beta_k (\\tau) \\cpp{k}(\\tau_0)\\,, \\nonumber\\\\\n\t\\cpp{k} (\\tau) & =  \\alpha_k^* (\\tau) \\cpp{k} (\\tau_0) + \\beta_k^* (\\tau) \\cm{k} (\\tau_0)\\,,\n\\end{align}\nwhere $|\\alpha_k|^2 - |\\beta_k|^2 = 1$.\n\nConsidering this, one can parametrize these coefficients as\n\\begin{equation}\n\t\\alpha_k = \\cosh (r_k) e^{-i \\Theta_k},  \\quad \\beta_k = -\\sinh (r_k) e^{i (\\Theta_k + 2\\phi_k)}\\,,\n\\end{equation}\nwhich renders\n\\begin{widetext}\n\\begin{align}\\label{sqpar}\n\t\\hat{\\varphi}_k (\\tau) =& \\frac{1}{\\sqrt{2k}} \\left\\{\\left[ \\cosh (r_k) e^{-i \\Theta_k} - \\sinh(r_k) e^{-i(\\Theta_k + 2\\phi_k)}\\right] \\cm{k} + \\left[\\cosh(r_k) e^{i\\Theta_k} - \\sinh(r_k) e^{i(\\Theta_k + 2\\phi_k)}\\right] \\cpp{k} \\right\\}, \\nonumber \\\\\n\t\\hat{\\pi}_k (\\tau) = & -i \\sqrt{\\frac{k}{2}} \\left\\{\\left[ \\cosh (r_k) e^{-i \\Theta_k} + \\sinh(r_k) e^{-i(\\Theta_k + 2\\phi_k)}\\right] \\cm{k} - \\left[\\cosh(r_k) e^{i\\Theta_k} + \\sinh(r_k) e^{i(\\Theta_k + 2\\phi_k)}\\right] \\cpp{k} \\right\\}\\,.\n\\end{align}\n\\end{widetext}\n\nComparing with the equations above (depending on Bunch-Davies functions), one readily finds the parameters\n\\begin{align}\n\tr_k & = \\sinh^{-1} \\left(\\frac{1}{2k\\tau}\\right), \\qquad \\Theta_k = k\\tau + \\tan^{-1}  \\left(\\frac{1}{2k\\tau}\\right)\\,, \\nonumber \\\\\n\t\\phi_k & = -\\frac{\\pi}{4} - \\frac{1}{2} \\tan^{-1}  \\left(\\frac{1}{2k\\tau}\\right).\n\\end{align}\nThe vacuum expectation value of the number of particles for the new vacuum in the $k$ mode  is given by\n\\begin{equation}\n\t\\left\\langle N_k \\right\\rangle = |\\beta_k|^2 = \\sinh^2 (r_k) = \\left(\\frac{1}{2k\\tau}\\right)^2.\n\\end{equation}\nThus, for $k < 2\/(aH)$ the expectation number is bigger than $1$. In practice, this matches the region for which the equation of motion has the exponential solutions, and in particular, where squeezing takes place. \n\n\\subsubsection{Particle production during the standard cosmological expansion}\n\nIn order to find the density of particles created during the expansion history, it is convenient to have at hand the evolution of the scale factor as a function of the conformal time, starting from the inflationary era until the matter dominated era. We shall assume the transitions between epochs to be instantaneous, commonly known as the sudden approximation.\\footnote{Relaxing this assumption does not change our main findings.} Using the sudden approximation between the (quasi)-de Sitter expansion and the hot big bang phase, the scale factor is given by\n\n\\begin{equation*}\n\ta(\\tau)=\\left\\{\\begin{array}{l}\n(H_{\\rm Inf}|\\tau|)^{-1}, \\quad \\tau<\\tau_{e}<0 \\\\\n\\alpha_M (\\tau-\\tau_e)^2 + \\alpha_R (\\tau - \\tau_e) + \\alpha_I, \\quad \\tau>\\tau_{e}\n\\end{array}\\right.\\,,\n\\end{equation*}\n\\begin{align}\n\t\\alpha_M & = \\frac{\\pi G}{3} \\rho_{eq} a_{eq}^3, \\quad \\alpha_I = \\frac{1}{H_{\\rm Inf} |\\tau_e|}, \\nonumber \\\\\n\t \\alpha_R & = \\left[ \\frac{4\\pi G}{3} \\rho_{eq} a_{eq}^3 \\left(\\frac{1}{H_{\\rm Inf} |\\tau_e|} + a_{eq} \\right)\\right]^{1\/2}\\,,\n\\end{align}\nwhere  $\\tau_e$ denotes the conformal time at the end of inflation and ``$eq$'' refers to the time of matter-radiation equality. The quadratic term corresponds to the evolution during matter domination, whereas the linear term to radiation domination. \n\nThen, the equation of motion Eq. \\eqref{eom1} (which is for a completely general cosmological background) during this epoch(s) is given by\n\\begin{equation}\n\t\\varphi_k'' + \\left(k^2 - \\frac{2\\alpha_M}{\\alpha_M(\\tau-\\tau_e)^2 + \\alpha_R(\\tau-\\tau_e) + \\alpha_I} \\right) \\varphi_k = 0\\,.\n\\end{equation}\nNaturally, one can identify regions where the equation gets simplified. For the radiation-dominated era, the e.o.m. is\n\\begin{equation}\n\t\\varphi_k'' + k^2 \\varphi_k = 0\\,,\n\\end{equation}\nwhereas for the matter-dominated era, it is given by\n\\begin{equation}\n\t\\varphi_k'' + \\left(k^2 - \\frac{2}{\\tau^2} \\right) \\varphi_k = 0\\,,\n\\end{equation}\ni.e., the same equation as for the inflationary era. In principle, there can be small changes to the usual (positive-frequency) vacuum state coming from effects of gravitational phase transitions. However, the corrections to the positive-frequency vacuum are small or, in other words, the number of particles created due to these phase transitions quickly dilute. Therefore, it is reasonable to consider the Bunch Davies-like initial states, such that the solutions to these equations are, respectively, given by\n\\begin{eqnarray}\n{}^{\\rm Rad}\\varphi_k (\\tau) = \\frac{1}{\\sqrt{2k}} e^{-ik\/\\mathcal{H}}\\,,\n\\end{eqnarray}\nand the basis for the matter-dominated era as\n\\begin{eqnarray}\\label{solm}\n\t{}^{\\rm Mat}\\varphi_k (\\tau) = \\frac{1}{\\sqrt{2k}} \\left(1 - \\frac{i \\mathcal{H}}{2k}\\right) e^{-2ik\/\\mathcal{H}}\\,,\n\\end{eqnarray}\nwhere ${\\cal H}$ is the comoving rate of expansion.\\footnote{We here ignore any\n    squeezing which takes place before the epoch of radiation domination.} As mentioned, the matching of the solutions during the different epochs will lead to excited states that will increase the number of generated particles. However, for now it will be enough to concentrate on this simple form of solutions. Moreover, notice that Eq. \\eqref{solm} has the same functional form as the Bunch-Davies solution for de Sitter spacetime, and thus the squeezing formalism derived for that case also applies here. In particular, the vacuum expectation of the number of particles is \n\\begin{equation}\n\t\\langle N_k \\rangle = |\\beta_k|^2 = \\left(\\frac{1}{2k\\tau}\\right)^2\\,.\n\\end{equation}\nOn the contrary, during radiation domination there is no mass term in the e.o.m., so there is no squeezing and particle production during this era. Therefore, expansion induces particle excitations of a scalar field only during the de Sitter and matter-dominated eras. Before moving on, we will cover the case of massive scalar fields during inflation, and similar calculations can be done for standard expansion.\n\n\\subsubsection{The massive scalar case}\n\nHere we will cover (although somewhat superficially) the case of a massive scalar field. A priori, one would expect particle production for massive fields to be less efficient, so we need to quantify the required corrections to the functions displayed above.\n\nLet us start with a generalization of the action Eq. \\eqref{act1},\n\\begin{equation}\n\t{\\cal S} = \\frac{1}{2} \\int d\\tau d^3 x \\left[ \\left(\\varphi' - \\frac{a'}{a}\\varphi \\right)^2 - (\\nabla \\varphi)^2 - a^2 m^2 \\varphi^2 \\right]\\,,\n\\end{equation}\nwhich renders the following equation of motion for the field\n\\begin{equation}\n\t\\varphi_k'' + \\left[k^2 - \\left( \\frac{a''}{a} - a^2 m^2 \\right) \\right] \\varphi_k = 0\\,.\n\\end{equation}\nOnce again, this equation is completely general for any cosmological epoch. For inflation this becomes\n\\begin{equation}\n\t\\varphi_k'' + \\left[k^2 - \\frac{1}{\\tau^2} \\left(2 - \\frac{m^2}{H^2}\\right) \\right]\\varphi_k = 0\\,,\n\\end{equation}\nwhere the Bunch-Davies solution is\n\\begin{equation}\\label{fim}\n\t\\varphi_k = \\frac{e^{i (2\\nu+1)\\frac{\\pi}{4}}}{\\sqrt{2k}} \\sqrt{\\frac{\\pi}{2}} w^{1\/2} H_{\\nu}^{(1)} (w)\\,,\n\\end{equation}\nwhere $w = |k \\tau|$ and $\\nu^2 = 9\/4 - (m\/H)^2$. The conjugate momentum is\n\\begin{align}\\label{pim}\n\t\\pi_k = & -i \\sqrt{\\frac{k}{2}} \\bigg\\{-i \\sqrt{\\frac{\\pi}{2}} e^{i (2\\nu+1)\\frac{\\pi}{4}} \\bigg[ w^{1\/2} H_{\\nu-1}^{(1)} (w) \\nonumber \\\\\n\t& + w^{-1\/2} \\left(3\/2 - \\nu \\right) H_{\\nu}^{(1)} (w) \\bigg] \\bigg\\}\\,.\n\\end{align}\nNotice that negative values of $\\nu$ lead to exponentially suppressed solutions. Thus, as expected, there is no particle production for $m \\gtrsim H$. Then, assuming that the mass term is small enough so that $\\nu$ is safely larger than 0, one can compute the number of generated particles due to squeezing by comparing the equations above with Eq. \\eqref{sqpar}. In order to have analytical expressions, one can expand eqs.\\eqref{fim},\\eqref{pim} in powers of $m\/H$, which yields\n\\begin{align}\n\t|\\beta_k|^2 \\simeq \\left(\\frac{1}{2k\\tau}\\right)^2 \\left[1 + \\frac{2}{3} \\frac{m^2}{H^2} \\left( -1 + \\gamma_E + \\ln (2w) \\right)\\right]\\,,\n\\end{align}\nwhere $\\gamma_E$ denotes the Euler-Mascheroni constant. Naturally, during radiation domination this type of mass term does not enhance squeezing, whereas during matter dominance it is more subdominant than in the other eras ($\\tau^{-4}$ vs. $\\tau^{-2}$). \n\n\\subsection{Setting up an environment}\n\nWhat we have covered so far is valid for a scalar field, so the natural next step is to try and reproduce this for photons. However, in this case there is no induced time-dependent mass-term and thus no squeezing (similarly to the scalar case during radiation domination). Naturally, there can be particle production due to interactions with other fields, but such processes are not linked to the background dynamics. In fact, in some cases the expansion just dilutes whatever number of particles are produced through these couplings. Consequently, in order to grasp the effects of decoherence of photons due to expansion alone, the next best thing is to look at the interactions between the quantum state (of a test photon) and an environment encompassed by either pseudoscalar particles produced by the squeezing of super-horizon states, or by decay products of these scalars, in particular, into photons. Arguably, the preeminent example of a scalar field in such scenario is the axion, which has a well-known interaction with $U(1)$ fields. Moreover, the interactions between axions and photons through other means have been widely explored in the literature, where the search of this particle is largely based on this interaction. \nThe interaction between axions and $U(1)$ gauge fields is described by the Lagrangian\n\\begin{equation}\n\t{\\cal L}_{A\\gamma\\gamma} = - \\frac{g_{A \\gamma\\gamma}}{4} F_{\\mu\\nu} \\tilde{F}^{\\mu\\nu} \\phi_A = g_{A\\gamma\\gamma} \\vt{E}\\cdot \\vt{B}\\ \\phi_A\\,,\n\\end{equation}\nwith\n\\begin{align}\n\tg_{A\\gamma\\gamma} & = \\frac{\\alpha}{2\\pi f_A} \\left(\\frac{E}{N} - 1.92(4)\\right) \\nonumber \\\\\n\t& = \\left(0.203(3) \\frac{E}{N} - 0.39(1) \\right) \\frac{m_A}{{\\rm GeV}^2}\\,,\n\\end{align}\nwhere $E$ and $N$ are the electromagnetic and color anomalies of the axial current \\cite{RRR2018}. \n\n\\subsubsection{Number density}\n\nLet us estimate the number density of $\\phi_A$-particles created during inflation (just by squeezing). For this, we need to compute the total number of particles. Assuming the states are homogeneously distributed, the amount of states within a ``radius'' $k$ is \n\\begin{equation}\n\tG(k) = \\frac{V}{(2\\pi)^3} \\frac{4\\pi k^3}{3}\\,,\n\\end{equation}\nwhere $V$ stands for a comoving volume. In this way, the density of states is given by\n\\begin{equation}\n\tg(k) = \\frac{\\partial G}{\\partial k} = \\frac{V}{2\\pi^2} k^2\\,.\n\\end{equation}\nThus, the total number of particles is\n\\begin{widetext}\n\\begin{equation}\n\tN = \\sum_k N_k = \\int dk\\ g(k) f(k) \\nonumber  = \\frac{V}{2\\pi^2} \\int_{0}^{-1\/\\tau_e}  dk\\ k^2 \\left(\\frac{1}{2k\\tau_e}\\right)^2 \\left[1 + \\frac{2}{3} \\frac{m^2}{H_{\\rm Inf}^2} \\left( -1 + \\gamma_E + \\ln (-2k\\tau_e) \\right)\\right]\\,,\n\\end{equation}\n\\end{widetext}\nwhere we have used the formula for the average number of particles on a mode $k$ created due to squeezing, which we identified with $f(k)$. The integration limits correspond only to modes that have been superhorizon at some point during inflation, as those are the ones that undergo squeezing. Performing the integral we get\n\\begin{eqnarray}\n\tN & = & \\frac{V}{8\\pi^2} \\left. \\frac{k}{\\tau_e^2} \\left[1 + \\frac{2}{3}\\frac{m_a^2}{H_{\\rm Inf}^2} \\left( -2 + \\gamma_E + \\ln (-2k\\tau_e)\\right] \\right]\\right|_{0}^{-1\/\\tau_e} \\nonumber \\\\\n\t& = & -\\frac{V}{8\\pi^2} \\frac{1}{\\tau_e^3} \\left[1 + \\frac{2}{3}\\frac{m_a^2}{H_{\\rm Inf}^2} \\left( -2 + \\gamma_E + \\ln 2 \\right)  \\right]\\,.\n\\end{eqnarray}\nThen, one can obtain the density of these particles at any given time through\n\\begin{equation*}\n\tn = \\frac{N}{V_{\\rm phys}} = - \\frac{1}{8\\pi^2} \\left(\\frac{1}{a \\tau_e}\\right)^3 \\left[1 + \\frac{2}{3} \\frac{m_a^2}{H_{\\rm Inf}^2} \\left( -2 + \\gamma_E + \\ln 2 \\right) \\right]\\,.\n\\end{equation*}\n\nThis is a good point to make some estimates. First, one can get away (for now) with not choosing a value of $m_a$, as it will be subdominant. Thus, we are left to find $\\tau_e$. To do so, notice that \n\\begin{equation}\n\t\\frac{1}{k_0 |\\tau_e|} = \\frac{k_0|\\tau_*|}{k_0 |\\tau_e|} = \\frac{a_e}{a_*} \\sim e^{60}\\,,\n\\end{equation}\nwhere `$0$' and `$*$' stand for present-day and horizon-crossing magnitudes. In particular, $k_0$ can be identified with the current horizon length. As it is widely known, inflation had to last at least 60 $e-$folds after this mode crossed the horizon in order to solve the horizon problem.\\footnote{Actually, the number of $e-$folds needed to solve the horizon problem depends on the energy scale of inflation, but we are only interested in rough estimates here.} Then, the above is equivalent to \n\\begin{equation}\n\t\\frac{(aH)_0^{-1}}{|\\tau_e|} = \\frac{H_0^{-1}}{|\\tau_e|} \\sim e^{60}\\,,\n\\end{equation}\nrendering a conformal time at the end of inflation,\n\\begin{equation}\n\t\\tau_e \\sim - 4 \\times 10^{-9}\\ {\\rm sec} = - 1.465\\times 10^{34} \\Mp^{-1}\\,.\n\\end{equation}\nWith this, we have the necessary values to estimate the number density of squeezing-generated ALPs at any given era. The free parameters are the energy scale of inflation and the mass of the particles. However, if the latter is small in comparison to the former, the contribution from the ratio will be negligible and one can get away with working with the first term. \n\n\\subsection{$\\phi_A \\gamma_t \\rightarrow x \\overline{x}$}\n\nFermion production from the interaction of an ALP and a (test) photon $\\gamma_t$ is mediated by the Lagrangian\n\\begin{equation}\n\tq_x A_{\\mu} \\overline{\\psi} \\gamma^{\\mu} \\psi\\,.\n\\end{equation}\n\nLet us take the initial momenta of the particles to be\n\\begin{equation}\n\tk_a = E_a(1, \\cos \\theta, \\sin \\theta ,0), \\qquad k_{\\gamma} = E_{\\gamma}(1, 1, 0, 0)\\,.\n\\end{equation}\nWith a center of mass energy given by $E^2_{\\rm com} = 2 E_a E_{\\gamma} (1-\\cos \\theta)$, one can find the cross section of the interaction to be\n\\begin{equation}\n\t\\sigma = \\frac{1}{4E_a E_{\\gamma} |v_a - v_{\\gamma}|} \\frac{q_x^2 m_x^2}{2\\pi f_a^2} \\ln \\left(\\frac{E' + p'}{E'-p'}\\right)\\,,\n\\end{equation}\nwhere $E' = E_{\\rm com}\/2$, $p' = \\sqrt{(E')^2 - m_x^2}$ and $m_x$ denotes the mass of the fermions. Now, we introduce the variables $\\lambda = 2 m_x^2\/(E_a E_{\\gamma})$ and $y = \\cos \\theta$, such that the average over the initial axion momentum is \\cite{Conlon:2013isa}\n\\begin{widetext}\n\\begin{align}\n\t\\langle \\sigma v \\rangle & = \\frac{q_x^2 \\lambda}{16\\pi f_a^2} \\int_{-1}^{1-\\lambda} dy\\ \\ln \\left(\\frac{\\sqrt{1-y} + \\sqrt{1-y-\\lambda}}{\\sqrt{1-y} - \\sqrt{1-y-\\lambda}} \\right) \\nonumber \\\\\n\t& = \\frac{q_x^2 \\lambda}{16\\pi f_a^2} \\left[ - \\sqrt{4-2\\lambda} + (\\lambda - 2) \\ln (\\sqrt{2} - \\sqrt{2-\\lambda}) + 2 \\ln (\\sqrt{2-\\lambda} + \\sqrt{2}) - \\frac{1}{2} \\lambda \\ln \\lambda \\right] \\nonumber \\\\\n\t& = \\frac{q_x^2 \\lambda}{16\\pi f_a^2} \\left[ - \\sqrt{4-2\\lambda} + (4-\\lambda) \\ln (\\sqrt{2} + \\sqrt{2-\\lambda}) + \\left(\\frac{\\lambda}{2} - 2\\right) \\ln \\lambda \\right]\\,.\n\\end{align}\n\\end{widetext}\nNotice this expression tells us that $0 < \\lambda \\leq 2$. \n\nNext, we identify two contributions to the ALP number density during the matter dominated era: those produced during inflation and those produced during matter domination itself. \n\\begin{equation*}\n\tn = \\int dn = \\frac{1}{8\\pi^2} \\left[ \\int_{aH}^{-1\/\\tau_e} \\frac{dk}{a^3 \\tau_e^2} + \\int_{aH}^{(aH)_{eq}} \\frac{dk}{a^3 \\tau^2} \\right]\\,.\n\\end{equation*}\nThen, it is convenient to write every expression in terms of the variable $\\lambda$ introduced above, \n\\begin{equation}\nk = \\frac{2 a m_x^2}{\\lambda E_{\\gamma}} \\implies d k= - \\frac{2am_x^2}{E_{\\gamma}} \\frac{d\\lambda}{\\lambda^2}\\,,\n\\end{equation}\nsuch that the interaction rate is given by\n\\begin{widetext}\n\\begin{align}\n\t\\langle n \\sigma v \\rangle = & \\frac{q_x^2}{16\\pi f_a^2} \\frac{1}{8\\pi^2 a^3} \\frac{2am_x^2}{E_{\\gamma}} \\bigg\\{ \\int_{\\lambda_e}^{\\lambda_{\\tau}} \\frac{d\\lambda}{\\lambda \\tau_e^2} \\left[ - \\sqrt{4-2\\lambda} + (4-\\lambda) \\ln (\\sqrt{2} + \\sqrt{2-\\lambda}) + \\left(\\frac{\\lambda}{2} - 2\\right) \\ln \\lambda \\right] \\nonumber \\\\\n\t& + \\int_{\\lambda_{\\tau}}^{\\lambda_{eq}} \\frac{d\\lambda}{\\lambda \\tau^2} \\left[ - \\sqrt{4-2\\lambda} + (4-\\lambda) \\ln (\\sqrt{2} + \\sqrt{2-\\lambda}) + \\left(\\frac{\\lambda}{2} - 2\\right) \\ln \\lambda \\right] \\bigg\\}\\,,\n\\end{align}\n\\end{widetext}\nwhere \n\\begin{equation*}\n\t\\lambda_e = \\frac{2m_x^2}{E_{\\gamma}} a|\\tau_e|\\,, \\qquad \\lambda_{\\tau} = \\frac{2m_x^2}{H E_{\\gamma}}\\,, \\qquad \\lambda_{eq} = \\frac{2m_x^2}{H_{eq} E_{\\gamma}}\\,.\n\\end{equation*}\nThe kinematic constraints on $\\lambda$ place stringent bounds on the allowed values of the parameters of the model, in particular on the ratio $m_x^2\/E_{\\gamma}$. To see this, take the values at matter-radiation equality, where\n\\begin{equation*}\n\t\\lambda_e(a_{eq}) \\sim 10^{30} \\Mp^{-1} \\frac{2m_x^2}{E_{\\gamma}}\\,, \\quad \\lambda_{\\tau} (\\lambda_{eq}) \\sim 10^{55} \\Mp^{-1} \\frac{2m_x^2}{E_{\\gamma}}\\,.\n\\end{equation*}\nSo, taking the maximum allowed value of $\\lambda$, we conclude that $m_x^2\/E_{\\gamma} \\sim 10^{-55} \\Mp$. This could be satisfied only for extremely light fermions (even for not-so-realistic values of the photon energy). Assuming these rather implausible conditions are satisfied, we can notice that the first integral will dominate ($\\tau_e^{-2} \\gg \\tau^{-2}$), so we will just focus on this one (for now). Then, the interaction rate is\n\\begin{equation}\n\t\\langle n \\sigma v \\rangle \\approx \\frac{3356 q_x^2}{16\\pi f_a^2} \\frac{(1+z_{eq})^2}{8\\pi^2} \\frac{2\\times 10^{-55}\\Mp}{\\tau_e^2}\\,, \n\\end{equation}  \nwhere we have solved the integral numerically. Plugging the numerical values of $\\tau_e$ and $z_{eq}$, we have that \n\\begin{equation}\n\t\\langle n \\sigma v \\rangle \\sim 10^{-112} \\Mp^3 \\frac{q_x^2}{128\\pi^2 f_a^2}\\,.\n\\end{equation}\nClearly $f_a$ would need to be abnormally small in order to have a non negligible interaction rate. The only way to obtain non negligible values would be to suppress even more the ratio $m_x^2\/E_{\\gamma}$, such that the corresponding versions of $\\lambda$ approach to $0$, where the integral actually diverges. Needless to say, even considering very light fermions, the energy of the photon would be out of reach (and can even become trans-Planckian). Indeed, for axions coming from string theory, we generically expect $f_a>\\Mp$ from the Weak Gravity Conjecture (WGC) \\cite{Heidenreich:2015nta, Rudelius:2014wla, Bachlechner:2015qja}. Interestingly, the WGC also constrains the ratio of the charge-to-mass of fermions to be less than $q_x\/m_x < 1$ in Planck units. On excluding trans-Planckian photons on physical grounds, this means that $q_x$ gets naturally suppressed on considering very small values for $m_x^2\/E_{\\gamma}$. Therefore, it seems that the WGC highly disfavours having a non-negligible value for this interaction rate.\n\n\\subsection{$\\phi_A \\rightarrow \\gamma \\gamma \\implies \\gamma_t \\gamma \\rightarrow \\gamma \\gamma$}\n\nIn this case, we will check how likely it is for the photon to interact with an environment composed of photons which are produced from the decay of an ALP. For this, we need the decay width of the process, which is \n\\begin{equation}\n\t\\Gamma_{A\\rightarrow \\gamma\\gamma} = \\frac{g_{A\\gamma\\gamma}^2 m_A^3}{64\\pi}\\,,\n\\end{equation}\nand, assuming $E\/N = 0$, this becomes\n\\begin{equation}\n\t\\Gamma_{A\\rightarrow \\gamma\\gamma} =  1.1 \\times 10^{-24}\\ {\\rm s}^{-1} \\left(\\frac{m_A}{\\rm eV}\\right)^5\\,.\n\\end{equation}\nWithout any further calculations, one can see that for masses $m_A \\sim {\\cal O}(1)\\ {\\rm eV}$ or less, the decay width is too small even considering the age of the Universe ($\\sim 10^{17}\\ {\\rm s}$), and so no photons would be produced. Current bounds on the mass of the axion highly disfavour higher masses. This is why it is more appropriate to talk about ALPs, as they are more generic and well suited to be a test lab. \n\nNaturally, the photons resulting from the decaying of the ALP will not have the same momentum as it. We label the resulting photons as $1'$ and $2'$, with an angle $\\theta'$ between their momenta. Then, one can easily show that \n\\begin{equation}\n\t\\langle \\cos \\theta' \\rangle = - \\frac{m_A^2}{4 p_{1'} p_{2'}}\\,, \\qquad \\langle p_{1'} p_{2'} \\rangle = \\frac{m_A^2}{4}\\,,\n\\end{equation}\nso that\n\\begin{equation}\n\tp_{1'}^2 + p_{2'}^2 = p_A^2 + \\frac{m_A^2}{2}\\,,\n\\end{equation}\nleading to the following direction-averaged momenta\n\\begin{align}\n\tp_{1'}^2 &= \\frac{m_A^2}{4} + \\frac{p_A^2}{2} \\left[ 1 + \\sqrt{1+\\frac{m_A^2}{p_A^2}} \\right]\\,, \\nonumber \\\\\n\tp_{2'}^2 &= \\frac{m_A^2}{4} + \\frac{p_A^2}{2} \\left[ 1 - \\sqrt{1+\\frac{m_A^2}{p_A^2}} \\right]\\,.\n\\end{align}\nThis leads to a not-so-simple distribution of photons. However, considering the range of masses that render a photon population at matter domination, the distribution can be somewhat simplified. To see this, first notice that the comoving momentum is between $(aH)_{eq} \\lesssim k \\leq (aH)_{e}$, or plugging in numbers, $10^{-59}\\ \\Mp \\lesssim k \\lesssim 10^{-34}\\ \\Mp$. The physical momentum of massive particles varies with expansion the same way as for massless particles ($p \\propto a^{-1}$). Thus, the physical momentum of ALPs should be on the range $10^{-56}\\ \\Mp \\lesssim p \\lesssim 10^{-31}\\ \\Mp$ (or $ 10^{-38}\\ {\\rm GeV} \\lesssim p \\lesssim 10^{-13}\\ {\\rm GeV}$). Even for the upper limit, the physical momentum of ALPs is rather negligible in comparison with the rest mass required for it to decay by the matter dominated era (${\\cal O}(10^3)\\ {\\rm eV}$). Thus, it is a good approximation to treat the ALPs as non-relativistic. Then, the momentum of the resulting photons are roughly\n\\begin{equation}\n\tp_{1'} \\approx \\frac{m_A + p_A}{2}\\,, \\qquad p_{2'} \\approx \\frac{m_A - p_A}{2}\\,,\n\\end{equation}\nwhere for the sake of simplicity, we take $p_{1'} \\approx p_{2'} \\approx m_A\/2$.\n\nWith these considerations, one can compute the mean free path of a test photon interacting with an environment of photons decaying from ALPs. For starters, Euler and Kockel computed the cross section for photon-photon interactions \\cite{Euler:1935zz, Liang:2011sj}, \n\\begin{equation}\n\t\\sigma(\\gamma\\gamma \\rightarrow \\gamma\\gamma) = \\frac{937 \\alpha^4 \\omega^6}{10125 \\pi m^8},\n\\end{equation} \nwhere $\\alpha \\simeq 1\/137$ is the fine structure constant, $\\omega$ is the energy of the photons in the center-of-momentum frame, and $m$ is the mass of the electron. The momentum of each photon in the lab-frame can be written as\n\\begin{equation}\np_1^{\\mu} = E_{1} (1,1,0,0), \\qquad p_2^{\\mu} = E_2 (1, -\\cos \\theta, \\sin \\theta, 0),\n\\end{equation}\nsuch that\n\\begin{equation}\n\t\\omega = \\sqrt{E_1 E_2}\\; \\cos \\frac{\\theta}{2} .\n\\end{equation}\nNext, recalling the number density of ALPs (which translates into the number density of photons up to a factor of $2$), and considering that their mass is negligible in comparison to the energy scale of inflation, we have\n\\begin{equation}\\label{npeq}\n\tn = - \\frac{1}{4\\pi^2} \\left(\\frac{1}{a \\tau_e}\\right)^3\\,,\n\\end{equation}\nsuch that\n\\begin{equation}\n\t\\sigma n \\sim \\frac{937 \\alpha^4}{10125\\pi m^8} E_{\\gamma}^3 E_{1}^3 \\frac{2}{\\pi} \\frac{(1+z_{eq})^3}{4\\pi^2} |\\tau_e|^{-3}\\,,\n\\end{equation}\nwhere $E_{\\gamma}$ denotes the energy of the test photon (quantum state) and $E_{1}$ the energy of the environment photon. Then, taking $E_{\\gamma} = 10^{-17}\\ \\Mp$ and $E_{1} \\sim m_A = 10^{-24}\\ \\Mp\\ (1\\ {\\rm keV})$, the resulting mean free path is\n\\begin{equation}\n\t\\ell = (\\sigma n)^{-1} \t\\sim 10^{21}\\ {\\rm cm}\\,,\n\\end{equation}\nwhich should be compared to $H_{eq}^{-1} \\sim 10^{50}\\ {\\rm cm}$. Nevertheless, notice that we have taken a rather high energy for the test  photon, so much so that the cross section formula may be invalid due to other processes being predominant. A more sensible value would be $E_{\\gamma} = 10^{-24}\\ \\Mp$, which yields\n\\begin{equation}\n\t\\ell = (\\sigma n)^{-1} \t\\sim 10^{42}\\ {\\rm cm}\\,.\n\\end{equation}\nThus, in principle photons could interact with other photons emerging from the decay of ALPs (we will check this more carefully below). However, it is instructive to compare the possibility of these interactions to the interaction with CMB photons. According to our estimation for the number density of photons created through the process $\\phi_A \\rightarrow \\gamma\\gamma$, by the time of photon decoupling we have $ n \\sim 20 \\ {\\rm cm}^{-3}$ $(600\\ {\\rm cm}^{-3}$ by matter-radiation equality), whereas for CMB photons $n_{pd} \\approx n_{\\gamma,0} (1+z_{pd})^3 \\sim 4 \\times 10^{11}\\ {\\rm cm}^{-3}$. Thus, the number density of ALP photons is negligible in comparison to CMB photons, so the latter are in principle a more important source of decoherence than the former after $z \\sim 1000$. Let us compute next the mean free path due to this interaction. \n\n\\subsection*{Mean free path}\n\nIn order to compute the mean free path (or redshift in a cosmological setting), we will use the optical depth, defined as \n\\begin{equation}\n\t{\\cal T} = \\int \\sigma j_{\\mu} dx^{\\mu}\\,,\n\\end{equation}\nwhere $\\sigma$ is the cross section of the interaction and $j_{\\mu}$ is the four-current \\cite{Ruffini:2015oha}. The integral over the spatial dimensions are null due to isotropy and homogeneity. \nThis will be used to compute in a more robust manner the mean free path for the interaction of a photon with others produced by the decay of an ALP. Moreover, we will incorporate the time dependence from the decay width. With these considerations, the optical depth is written as \n\\begin{widetext}\n\\begin{equation}\\label{opd}\n\t{\\cal T} = \\int_{t}^{t_0} dt\\ (1-e^{-\\Gamma t}) \\frac{937\\alpha^4 E_{\\gamma}^3 m_A^3}{10125\\pi m^8} \\int_{-1}^{1} d(\\cos \\theta) \\cos^6 \\frac{\\theta}{2}\\ \\frac{1}{4\\pi^2} \\left[\\int_{aH}^{-1\/\\tau_e} \\frac{dk}{\\tau_e^2} + \\int_{aH}^{(aH)_{eq}} \\frac{dk}{\\tau^2} \\right]\\,.\n\\end{equation}\n\\end{widetext}\n\nNext, we shall assume a matter dominated Universe throughout the entire propagation of the photon. This will be convenient in order to deal with the explicit time dependence in the expression. Thus, we have that\n\\begin{equation}\n\tt = \\frac{2}{3H_0} (1+z)^{-3\/2}\\,.\n\\end{equation}\n\nWe shall focus on the first term inside the brackets of \\eqref{opd}, which is dominant (by many orders of magnitude). In doing so, the optical depth is written as\n\\begin{align}\n{\\cal T} & = \\frac{937\\alpha^4 E_{\\gamma,0}^3 m_A^3}{10125\\pi m^8} \\frac{1}{8\\pi^2} \\times \\nonumber \\\\ & \\int_0^z \\frac{dz'\\ (1+z')^3}{H_0 (1+z')^{5\/2}} (1-e^{-\\Gamma t})\\frac{1}{\\tau_e^2}\\left[-\\frac{1}{\\tau_e} - H_0 (1+z)^{1\/2}\\right]\\,. \\nonumber\n\\end{align}\nIn order to have numerical estimates we take $E_{\\gamma,0} = m_A = 10^{-24}\\ \\Mp$, such that\n\\begin{align}\n\t\\frac{937\\alpha^4 E_{\\gamma,0}^3 m_A^3}{10125\\pi m^8} \\frac{1}{8\\pi^2 H_0} \\simeq 10^{78}\\,.\n\\end{align}\nThe probability of the photon travelling without interacting with the environment is given by $P(z) = e^{- {\\cal T}(z)}$. For $z = 3400$, one gets ${\\cal T} \\sim 10^{-20}$, meaning that basically $P = 1$, and so there is no decoherence due to the interaction between the photon in some quantum state and the photons produced by the decay of expansion-generated ALPs. One could entertain the idea of going further into the past (higher redshift) in order to obtain non-trivial probabilities (even though the single-fluid approximation would break in the realistic setup). However, even for redshifts as high as $10^{20}$, the optical depth is just around $10^{-13}$, so that interactions remain highly unlikely. One could also argue that different input parameters could change this conclusion, however, smaller masses only lead to less efficient interactions and a slower decay, effectively increasing the mean free path. \n\nLet us emphasize that we have studied the potential interactions with particles that have been produced directly or indirectly due to the dynamics of the expansion of the Universe. In this sense, one could also ask if there can be interactions with a primordial population of ALPs (or their offspring). Such interactions can be potentially more important that the ones we have considered; however, it has been found that for realistic values of the parameters the growth of the photon field in particular is strongly suppressed \\cite{Garretson1992, Arza2020}, and thus by the time of decoupling this scenario should not be considered a source of decoherence. \n\nAn interesting thing to note is that the strength of the interaction, which we have considered in this work, has recently been constrained from the observation of the birefringence angle from the CMB data \\cite{Minami:2020odp}. It is also well-known that photons travelling significantly large distances, and interacting with magnetic fields, can lead to the production of ALPs (see, for instance, \\cite{DeAngelis:2008sk}). Conversions between photons and ALPs, in the presence of  primordial magnetic fields, can also leave observable signatures in the CMB \\cite{Mirizzi:2009nq}, which together with other cosmological considerations, has been used to constrain a considerable region of the parameter space \\cite{Irastorza:2018dyq}. In the future, we plan to combine the estimate coming from polarization data, and the requirement that ALPs from the early-universe do not decohere, to find new probes for the so-called cosmological axion background \\cite{Dror:2021nyr}.\n\n\n\n\\section{Decoherence through the cosmological medium}\n\nIn this section we will look at the potential sources of decoherence of a photon in some quantum state due to the interaction with other particles in the cosmological medium. Unlike for the estimates in the previous section, we know from observations the number density of the other species, with values that make interactions more likely. We already had a first glance at such interactions, like photon-photon scattering with CMB radiation. \n\n\\subsection{Abundance of particles}\\label{abpar}\n\nFirst, we shall compute the number density of photons. This is given by\n\\begin{align}\n& n_{\\gamma}=\\frac{8 \\pi}{c^{3}} \\int_{0}^{\\infty}\\left(\\frac{k T}{h}\\right)^{3} \\frac{x^{2} d x}{e^{x}-1} \\nonumber \\\\\n& \\implies n_\\gamma = 4.11 \\times 10^8 \\, (1+z)^3  \n\\, m^{-3}\\,,\n\\end{align}\nwhere the temperature of the CMB is $T_0 = 2.72548\\pm 0.00057$K. Other sources give far fewer photons. \n\nNext, we look at the abundance of baryons. The baryon-to-photon density is \n\\begin{equation}\n\\eta = \\frac{n_{\\rm b}}{n_\\gamma} = 2.75 \\times 10^{-8} \\Omega_b h^2\\,.\n\\end{equation}\nWith Planck's (2018) value of $\\Omega_{\\rm b} h^2 = 0.02237 \\pm 0.00015$ \\cite{Planck:2018vyg},  this gives an average baryon density today (if fully ionized) of\n\\begin{equation}\nn_{\\rm b,0} = 0.2526 \\, m^{-3}.\n\\end{equation}\nPrimordial nucleosynthesis and the CMB tell us that the Helium-4 mass fraction is about $Y_{\\rm P} = 0.246$.  To a good approximation, all the mass is in protons and Helium --- everything else is negligible in terms of number density.\n\nThe number density of Helium is given by $Y_{\\rm P} = 4n_{\\rm He}\/(4n_{\\rm He}+n_{\\rm p})$.   With $Y_{\\rm P}=0.246$, $n_{\\rm p}\/n_{\\rm He} = 12.26$.  This means that the fraction of baryonic nuclei that is Helium-4 is 0.0754.   We also have $n_{\\rm b} = n_{\\rm p} + 4 n_{\\rm He} = n_{\\rm p}(1+4\/12.26) = 1.33 n_{\\rm p}$.\n\nNext, the abundance of protons is related to that of baryons by\n\\begin{equation}\nn_{\\rm p,0} = \\frac{n_{\\rm b}}{1.33}  \\implies  n_{\\rm p} = 0.190 \\,(1+z)^3\\, m^{-3}\\,, z<z_{\\rm reion}\\,,\n\\label{np}\n\\end{equation}\nwhere $z_{\\rm reion} \\simeq 7.7\\pm 0.8$. Apart from protons, essentially all other baryons are Helium-4, which have a number density, after reionization, of\n\\begin{equation}\nn_{\\rm He,0} = (0.2526-0.190)\/4  \\implies  n_{\\rm  He} = 0.016\\,(1+z)^3 \\,m^{-3}\\,.\n\\label{nHe}\n\\end{equation}\nBefore reionization (and after recombination) the ionized fraction is about $10^{-4}$, so the proton number density is smaller than equation (\\ref{np}) by this factor.  Helium reionization is thought to occur at $z_{\\rm He, reion} \\simeq 3-4$, although the details remain uncertain. \n\nThe number density of electrons is related to that of protons and Helium-4. Indeed, there is one electron per proton and $2$ per Helium-4, which gives\n\\begin{equation}\nn_{\\rm e,0} = 0.190+2 \\times 0.016  \\implies  n_{\\rm e} = 0.222 \\,(1+z)^3\\,m^{-3}\\,,\n\\end{equation}\nafter $z_{\\rm He, reion}$.  For $z_{\\rm He,reion} < z < z_{\\rm reion}$, the electron number density would be  $n_{\\rm e} = n_{\\rm p}$.\n\n\\subsection{Interaction with CMB radiation}\n\nUsing the cross section for photon-photon scattering, the optical depth is computed as follows\n\\begin{align}\n\t{\\cal T} & = \\frac{1}{2}\\int_{t}^{t_0} dt \\frac{937\\alpha^4 E_{\\gamma}^3}{10125\\pi m^8} \\int_{0}^{\\infty} \\frac{dE}{2\\pi^2} \\frac{E^5}{\\exp(E\/T) - 1} \\nonumber \\\\\n\t& = \\frac{1}{2}\\int_{t}^{t_0} dt \\frac{937\\alpha^4 E_{\\gamma}^3}{10125\\pi m^8} \\frac{1}{2\\pi^2} \\left(\\frac{8\\pi^6}{63} T^6 \\right) \\nonumber \\\\\n\t& =  \\frac{1874 \\pi^3 \\alpha^4}{637875} \\frac{E_{\\gamma,0}^3}{m^8} \\int_0^{z} \\frac{dz'}{(1+z') H(z')} T_{0}^6 (1+z')^9\\,,\n\\end{align}\nwhere ${\\cal T} = 1$ determines the mean free path\/redshift of the test photon. The solutions to the equation depend strongly on $E_{\\gamma,0}$. For instance, for $E_{\\gamma,0} = 10^{-24}\\ \\Mp \\sim 1\\ {\\rm keV}$, we get{\\footnote{We have used $\\Omega_m h^2 = 0.1424$, $\\Omega_{\\Lambda} = 0.6889$ and $\\Omega_r h^2 = 4.2\\times10^{-5}$.}\n\\begin{align*}\n\t\\int_0^z dz' \\frac{(1+z')^8}{\\left[\\Omega_{m} (1+z)^3 + \\Omega_r (1+z)^4 + \\Omega_{\\Lambda}\\right]^{1\/2}} \\approx 1.13 \\times 10^{34}\\,,\n\\end{align*}\nwhich renders $z \\approx 50000$, whereas for $E_{\\gamma,0} = 10^{-21}\\ \\Mp \\sim 1\\ {\\rm MeV}$,\n\\begin{equation*}\n\t\\int_0^z dz' \\frac{(1+z')^8}{\\left[\\Omega_{m} (1+z)^3 + \\Omega_r (1+z)^4 + \\Omega_{\\Lambda}\\right]^{1\/2}} \\approx 1.13 \\times 10^{25}\\,,\n\\end{equation*}\nwhich yields $z \\approx 2700$. The conclusion is clear, that the test photon can propagate without interacting with CMB radiation for longer than the latter has been around ($z \\sim 1000$). Conversely, let us fix $z = z_{\\rm dec} = 1000$, which allows us to compute the probability of the photon travelling without interacting ($P = e^{- {\\cal T}(z_{\\rm dec})}$). For $E_{\\gamma,0} \\sim 1\\ {\\rm keV}$ the probability is essentially $1$, whereas for $E_{\\gamma,0}  \\sim 1\\ {\\rm MeV}$, we get $0.9994$, signalling that most likely the test photon would not have interacted from the decoupling era until today due to interaction with CMB radiation. Evidently, one can get less trivial values for higher photon energies. However, notice that at such energies other processes are predominant, which we will not consider for our purposes. \n\n\\subsection{Interactions through Thomson scattering}\n\nFollowing the same philosophy as before, we will compute the mean free path of a photons interacting through Thomson scattering. The cross section for this process is given by \n\\begin{equation}\n\t\\sigma_{\\rm th} = \\frac{8\\pi}{3} \\frac{\\alpha^2}{m^2}\\,,\n\\end{equation}\nwhere $m$ is the mass of the charged particle. Due to the dependence on this parameter, the interaction with electrons are predominant in comparison with interactions with protons. \nThen, the optical depth is given by\n\\begin{equation}\n\t{\\cal T} = \\frac{8\\pi}{3} \\frac{\\alpha^2}{m^2} \\int_0^z \\frac{dz'}{(1+z') H(z')} n_e (z')\\,,\n\\end{equation}\nwhere the number density of free electrons is given by\n\\begin{align}\nn_{e,1}(z) &= 0.222 (1+z)^3\\ m^{-3}, \\quad z<z_{\\rm He,reion} \\nonumber \\\\\nn_{e,2}(z) &= 0.19 (1+z)^3\\ m^{-3}, \\quad z_{\\rm He,reion} < z < z_{\\rm reion}\n\\end{align}\nas computed in the sub-section \\ref{abpar}. We are assuming that the ionization fraction is $1$ after reionization and $0$ before it (but after decoupling). With these considerations, the optical depth is\n\\begin{align}\n\t{\\cal T}  = \\frac{8\\pi}{3} \\frac{\\alpha^2}{m^2} & \\bigg\\{ \\int_0^{z_{\\rm He,reion}} \\frac{dz'\\ n_{e,1}(z')}{(1+z') H(z')} \\nonumber \\\\\n\t& + \\int_{z_{\\rm He,reion}}^{z_{\\rm reion}} \\frac{dz'\\ n_{e,2}(z')}{(1+z') H(z')} \\bigg\\}\\,.\n\\end{align}\nFor the sake of concreteness, we take $z_{\\rm He,reion} = 3.5$ and $z_{\\rm reion} = 7.8$, yielding\n\\begin{equation}\n\t{\\cal T} = 0.0529585 \\implies P = \\exp(-{\\cal T}) = 0.94819\\,,\n\\end{equation}\ni.e., there is roughly a $95\\%$ probability of photons travelling freely from the reionization epoch until present time. Notice that this analysis is basically the same as that for the optical depth for CMB radiation due to the same process. The value obtained by the Planck mission is ${\\cal T} = 0.0561 \\pm 0.0071$ \\cite{Planck:2018vyg}, in good agreement with the result estimated here. Notice that for higher energies one would have to use the Klein-Nishina formula for the cross section; however, it is always less or equal than the Thomson cross section, which renders larger mean free paths. \n\n\\subsection{Other processes}\n\nThere are other processes involving photons which could lead to decoherence (or the annihilation of the photon). One such process is the pion photoproduction\n\\begin{equation}\np + \\gamma \\longrightarrow\t\\left( \\begin{array}{l} \n\tp\\\\\n\tn\n\t\\end{array} \\right) + \\pi\\,.\n\\end{equation}\nHowever, the threshold energies for this kind of processes are very large for our purposes (see \\cite{Ruffini:2015oha} for an in-depth study of these processes). Indeed, one should take a closer look to them for energies of order $\\sim 10^{15}$ eV or higher. \n\nOn the other hand, as shown in \\cite{Berera2020}, x-rays would be more interesting for quantum communication purposes at present day (or low redshifts in general). For said range and due to the dominant constituents of the interstellar medium (photons, electrons and protons) and the weakness of QED, the interactions between photons and the background are negligible. Case in point, the mean free path for interactions with electrons in the interstellar medium was found to be of order $1$ Mpc, which is larger than the size of the Milky Way. Looking at dense regions of the HII gas, the mean free path reduces to $0.1$ kpc, which is a considerable distance within the galaxy. A more in depth discussion of these interactions and others, like with dust particles, galactic magnetic fields, etc., can be found in \\cite{Berera2020} and references within. The upshot is that for the (soft) x-ray region of the spectrum, the interactions of a test photon at such energies are rather negligible, as opposed of radio signals which can be affected by galactic magnetic fields, or UHE photons, where particle production and other processes dominate. These conclusions can be extrapolated to low redshifts, as demonstrated in \\cite{Mack:2008nv}. That work reports a considerable transparency window for photons in the same energy range ($E \\lesssim 10$ keV) for redshifts up to $z \\sim 100$. \n\n\\section{Discussion}\n\nIn this work we have looked at the possibility of photons maintaining their quantum state over cosmological distances. Naturally, this is an intriguing question in many respects, including quantum communication and quantum teleportation, even more so considering the success achieved in Earth-based experiments. An analysis of quantum coherence to interstellar distances was presented in \\cite{Berera2020}, where it was shown that photons in the x-ray range are the prime candidates for these purposes. This work reinforces that conclusion, only now extending to cosmological distances, by generalizing to include other potential sources of decoherence. \n\nWe have used the standard definition that decoherence takes place due to the interaction between a quantum state and an environment. Thus, for expansion-induced decoherence, the question we had to tackle was how gravity can produce an environment. For scalar fields it is well known that squeezing can do the job, where a large number of particles at super-horizon momenta scales are produced. That is not the case for EM fields, because the EM field is conformally invariant and the FLRW metric is conformally flat. The same argument can be applied for the free Dirac theory of fermions. In consequence, there are no excitations of the field owing to gravitational effects and thus no `extra' environment wrecking the quantum state. One obvious loophole consists in breaking the conformal invariance through a coupling with other fields. We have considered ALP-photon interactions, which have been widely studied in the literature. One of the options is a direct interaction which leads to the production of fermions, and the other is the decay of the ALP into two photons, which in turn can interact with the one in a coherent state. In both scenarios the clear conclusion is that interaction rates range from negligibly small to zero, depending on the parameters of the interaction. In some sense we played against our odds by taking large masses for ALPs so that they can decay by the matter-dominated era, or by taking abnormally large energies for the test photon. Regardless of whatever be the case, the conclusion remains the same. We should emphasize that the number densities for ALPs considered are only those produced by expansion, not some primordial population which may lead to an enhanced effect. In fact, axion production by other (standard) mechanisms can lead to a wider variety of more important processes, like the inverse-Primakoff effect, which future radio telescopes could exploit for ALP DM, although this is not expected for the QCD axion \\cite{Irastorza:2018dyq}. In consequence, as far as expansion-induced decoherence goes, the results from \\cite{Berera2020} can be confidently extrapolated to cosmological distances. \n\nIn order to search for stronger decoherence factors, one has to look at the population of different species in the cosmological medium. Interactions with the CMB radiation is one clear option, where for present-day energies of $~ 1$ keV the interaction rate is essentially 0, whereas for $1$ MeV, the probability of interactions is less than $0.1\\%$. Higher probabilities of interaction are associated to Thomson scattering after reionization, where there is roughly a $5\\%$ probability of interaction. Thus, for photon energies in the keV range the main decoherence factors lie at galactic scales, where they can maintain the state for considerable distances. Moreover, for the same sweet spot the conclusion holds for low redshifts, or even for $z \\sim 100$ \\cite{Mack:2008nv}.\n\nIn conclusion, the analysis in this paper has examined the free streaming requirements for photons, beyond just the classical condition that they maintain their initial momentum, to the stronger condition that the quantum coherence of the photons is also preserved. Notice for instance that groups of photons initially could also have some form of quantum coherence amongst them through their momentum or internal states, producing for example coherent or lasing states. So, even if the individual momentum of the photons was preserved it is still possible the more delicate quantum coherence amongst the photons could be destroyed. Here we have identified frequency regions in which photon quantum coherence can be maintained up to cosmological scales due to lack of interactions, extending on the work in \\cite{Berera2020} that only examined the galactic scale. Recently, the effect of a (curved) Schwarzschild background on the quantum state of coherent light, which can be verified by Earth-to-satellite signals, has been examined \\cite{Bruschi:2013sua, Bruschi:2021all, Exirifard:2020yuu}. Building on our present work, it will be natural to consider the effect of accelerating backgrounds on similar coherent wavepackets over cosmological scales and their consequences for quantum communication. \n\nA natural future direction to follow would be to check the conditions required for the axions to maintain a similar coherent state over cosmological distances. If one can find a similar result that axions also do not decohere due to the background expansion, this opens a new possibility for a complementary signal for axions coming from the very early universe. \n\n\\section*{Acknowledgements}\nAB is partially funded by STFC. SB is supported in part by the NSERC (funding reference CITA \\#490888-16) through a CITA National Fellowship and by a McGill Space Institute fellowship. RB is supported in part by funds from NSERC and from the Canada Research Chair program and is also grateful to the Institute for Theoretical Physics and the Institute for Particle Physics and Astrophysics of the ETH Zurich for hospitality. JCF is supported by the Secretary of Higher Education, Science, Technology and Innovation of Ecuador (SENESCYT).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nGiven a graph $G=(V,E)$, a matching $M$ of $G$ is a subset of edges such that no vertex is incident to two edges in $M$. Finding a maximum cardinality matching is a central problem in algorithmic graph theory. The most efficient algorithm for general graphs is that given by Micali and Vazirani  \\cite{MV} and runs in $O(|E||V|^{1\/2})$ time.\n\nIn a seminal paper, Karp and Sipser \\cite{KS} introduced two simple greedy algorithms for finding a large matching in the random graph $G_{n,m},m=cn\/2$ for some positive constant $c>0$. Let us call them  Algorithms 1 and 2 as they are in \\cite{KS}. Algorithm 2 is simpler than Algorithm 1 and has been intensely studied: see for example Aronson, Frieze and Pittel \\cite{AFP}, Bohman and Frieze \\cite{BF}, Balister and Gerke \\cite{BG} or Bordenave and Lelarge \\cite{BL}. In particular, \\cite{AFP} together with Frieze and Pittel \\cite{FP} shows that w.h.p. Algorithm 2 finds a matching that is within $\\tilde{\\Theta}(n^{1\/5})$  of the optimum, when applied to $G_{n,m}$. Subsequently, Chebolu, Frieze and Melsted \\cite{CFP} showed how to use Algorithm 2 as a basis for a linear expected time algorithm, when $c$ is sufficiently large.\n\nAlgorithm 2 proceeds as follows (a formal definition of Algorithm 1 is given in the next section). While there are isolated vertices  it deletes them. After which, while there are vertices of degree one in the graph, it chooses one at random and  adds the edge incident with it to the matching and deletes the endpoints of the edge. Otherwise, if the current graph has minimum degree at least two, then it adds a random edge to the matching and deletes the endpoints of the edge. \n\nIn the same paper Karp and Sipser proposed another algorithm for finding a matching that also runs in linear time. This was Algorithm 1. The algorithm sequentially reduces the graph until it reaches the empty graph. Then it unwinds some of the actions that it has taken and grows a matching which is then output. Even though it was shown empirically to outperform Algorithm 2, it has not been rigorously analyzed. In this paper we analyze Algorithm 2 in the special case where the graph is random with a fixed degree sequence $3\\leq d(i)\\leq 4$ for $i=1,2,\\ldots,n$. We prove the following:\n\\begin{thm}\\label{main}\nLet $G$ be a random graph with degree sequence $3\\leq d(i)\\leq 4$ for $i=1,2,\\ldots,n$. Then \n\\begin{enumerate}[(a)]\n\\item Algorithm 1 finds a matching of size $n\/2-O(\\log n)$, w.h.p.\n\\item Algorithm 1 can be modified to find a (near) perfect matching in $O(n)$ time w.h.p. and in expectation.\n\\end{enumerate}\n\\end{thm}\nA (near) perfect matching is one of size $\\rdown{n\/2}$. Note that in the case of cubic graphs, it is known that they have (near) perfect matchings w.h.p., see Bollob\\'as \\cite{Bol}. Note also that it was shown by Frieze, Radcliffe and Suen \\cite{FRS} that w.h.p. Algorithm 2 finds a matching of size $n\/2-\\tilde{\\Theta}(n^{1\/5})$. \\footnote{Recently, the junior author has extended Theorem \\ref{main} to random $r$-regular graphs for all $3\\leq r=O(1)$.}\n\\section{The Algorithm}\nThe algorithm that is given in \\cite{KS}  can be split into two parts. The first part sequentially reduces the graph until it reaches the empty graph. Then the second part reverses part of this reduction and grows a matching which is then output. \n\nTo reduce the graph, \n\\begin{enumerate}[(1)]\n\\item First, while there are vertices of degree 0 or degree 1 the algorithm removes them along with any edge incident to them. The edges removed at this stage will be part of the output matching. \n\\item Second, while there are vertices of degree 2 the algorithm contracts them along with their two neighbors. That is the induced path $(x,y,z)$ is replaced by a single contracted vertex $y_c$ whose neighbors are those of $x,z$ other than $y$. The description in \\cite{KS} does not explicitly say what to do with loops or multiple edges created by this process. In any case, such creations are very rare. We say a little more on this in Section \\ref{details}.\n\nIn the unwinding, if we have so far constructed a matching containing an edge $\\set{y_c,\\xi}$ incident with $y_c$ and $\\xi$ is a neighbor of $x$ then in our matching we replace this edge by $\\set{x,\\xi}$ and $\\set{y,z}$. If there is no matching edge so far chosen incident with $y_c$ then we add an arbitrary one of $\\set{x,y}$ or $\\set{y,z}$ to our matching.\n\\item Finally if the graph has minimum degree 3 then a random vertex is chosen among those of maximum degree and then a random edge incident to that vertex is deleted. These edges will not be used in the unwinding.\n\\end{enumerate}   \n\\subsection{Idea of proof:} No mistakes are made while handling vertices of degree 0,1 or 2. Each mistake decreases the size of the final matching produced by one from the maximum size. We will show that mistakes occur only at parts of the graph that have become denser than is likely. \n\nWe show that w.h.p. the maximum degree remains $O(\\log^{2}\\nu)$ where $\\nu$ is the number of vertices remaining and so as long as $\\nu\\log n$, say, then w.h.p. there will be no dense subgraphs and the algorithm will not make any mistakes. This explains the $O(\\log n)$ error term. Finally, we assert that removing an edge incident to a vertex of a maximum degree will help to control the maximum degree, explaining this choice of edge to delete.\n\n\\subsection{Details}\\label{details}\nThe precise algorithm that we analyze is called {\\sc reduce-construct} The algorithm description given in \\cite{KS} is not explicit  in how to deal with loops and multiple edges, as they arise. We will remove loops immediately, but keep the multiple edges until removed by other operations. \n\nWe assume that our input (multi-)graph $G = G([n],E)$ has degree sequence $\\bd$ and is generated by the configuration model of Bollob\\'as \\cite{Bol}. Let $W=[2\\nu]$, $2\\nu=\\sum_{i=1}^n d(i)$, be our set of {\\em configuration points} and let $\\Phi$ be the set of {\\em configurations} i.e. functions $\\phi:W \\mapsto [n]$ that such that $|\\f^{-1}(i)|=d(i)$ for every $i \\in [n]$. Given $\\phi \\in \\Phi$ we define the graph $G_\\phi=([n],E_\\phi)$ where $E_\\phi=\\{\\{\\phi(2j-1),\\phi(2j)\\}: j\\in [\\nu] \\}$. Choosing a function $\\phi \\in \\Phi$ uniformly at random yields a random (multi-)graph $G_\\phi$ with degree sequence $\\bd$. \n\nIt is known that conditional on $G_\\f$ being simple, i.e. having no loops or multiple edges, it is equally likely to be any graph that has degree sequence $\\bd$. Also, if the maximum degree is $O(1)$ then the probability that $G_\\f$ is simple is bounded below by a positive quantity that is independent of $n$. Thus results on this model can be translated immediately to random simple graphs.\n\nWe split the {\\sc reduce-construct} Algorithm into the {\\sc reduce} and {\\sc construct} algorithms which we present separately.\n\n\\textbf{Algorithm} \\textsc{Reduce}:\n\nThe input $G_0=G_\\f$ where we condition on there being no loops.\\\\ \n$i=\\hat{\\tau}=0$.\n\\\\ \\textbf{While} $G_i=(V_i,E_i) \\neq (\\emptyset,\\emptyset)$ do: \n\\begin{itemize}\n\\item[]\\textbf{If} $\\delta(G_i)=0$: Perform a {\\bf vertex-0 removal}: choose a  vertex of degree 0 and remove it from $V_i$.\n\\item[]\\textbf{Else if} $\\delta(G_i)=1$: Perform a {\\bf vertex-1 removal}: choose a random vertex $v$ of degree 1 and remove it along with its neighbor $w$ and any edge incident to  $w$. \n\\item[]\\textbf{Else if} $\\delta(G_i)=2$: Perform a {\\bf contraction}: choose a random vertex $v$ of degree 2. Then replace $\\{v\\} \\cup N(v)$ ($v$ and its neighbors $N(v)$) by a single vertex $v_c$. For $u\\in V \\setminus (\\{v\\} \\cup N(v))$, $u$ is joined to $v_c$ by as many edges as there are in $G_i$ from $u$ to $\\{v\\} \\cup N(v)$.  Remove any loops created.\n\\item[]\\textbf{Else if } $\\delta(G_i)\\geq 3$: Perform a {\\bf max-edge removal}: choose a random vertex of maximum degree and remove a random edge incident with it.\n\\\\ \\textbf{End if}\n\\item[]\\textbf{If} the last action was a max-edge removal, say the removal of edge $\\{u,v\\}$ and in the current graph we have $d(u)=2$ and $u$ is joined to a single vertex $w$ by a pair of parallel edges then perform an {\\bf auto correction contraction}: contract $u,v$ and $w$ into a single vertex. Remove any loops created.\n\\\\ \\textbf{End If}\n\\item[] Set $i=i+1$ and let $G_i$ be the current graph.\n\\end{itemize}\n\\textbf{End While}\n\\\\Set $\\hat{\\tau}=i$.\n\nObserve that we only reveal edges (pairs of the form $(\\phi(2j-1),\\phi(2j)): j\\in [\\nu]$) of $G_\\phi$ as the need arises in the algorithm. Moreover the algorithm removes any edges that are revealed. Thus if  we let $\\bd(i)$  be the degree sequence of $G_i$ then, given $\\bd(i)$ and the actions performed by {\\sc reduce} until it generates $G_i$ we have that $G_i$ is uniformly distributed among all configurations with degree sequence $\\bd(i)$ and no loops.\n\nCall a contraction that is performed by {\\sc reduce} and involves only 2 vertices \\emph{bad} i.e. one where we contract $u,v$ to a single vertex given that $G$ contains a parallel pair of the edge $\\set{u,v}$ and $u$ has degree 2. Otherwise call it \\emph{good}. Observe that a bad contraction can potentially be a mistake while a good contraction is never a mistake. By introducing the auto correction contraction we replace the bad contraction of the vertex set $\\{u,w\\}$, as presented in the description of {\\sc reduce}, with the good contraction of the vertex set $\\{v,u,w\\}$. Note that we do not claim that all bad contractions can be dealt with in this way. We only show later that other instance of bad contractions are very unlikely.\n\nWe now describe how we unwind the operations of {\\sc reduce} to provide us with a matching.\n\n\\textbf{Algorithm} {\\sc construct}:\n\n\\textbf{Input:} $G_0,G_1,...,G_{\\hat{\\tau}}$ - \nthe graph sequence produced by {\\sc reduce},\nan integer $j\\in \\{0,1,...,\\hat{\\tau}\\}$ and  a matching $M_{j}$ of $G_j$. (We allow the possibility of stopping {\\sc reduce} before it has finished.\nIf we do so when $|V(G_j)|=\\Theta(n^{2\/3})$ then, given that $G_j$ has a perfect matching w.h.p., we can use the $O(|E||V|^{1\/2})$ algorithm of \\cite{MV} applied to $G_j$ to find a perfect matching $M_j$ of $G_j$ in $O(n)$ time. Thereafter we can use {\\sc construct} to extend $M_j$ to a matching of $G_0$.)\n\\\\ \\textbf{For $i=1$ to $j$ do}: \n\\begin{itemize}\n\\item[]\\textbf{If} $\\delta(G_{j-i})=0$: Set $M_{j-i}=M_{j-i+1}$\n\\item[]\\textbf{Else if} $\\delta(G_{j-i})=1$: Let $v$ be the vertex of degree 1 chosen at the $(j-i)th$ step of {\\sc reduce} and let $e$ be the edge that is incident to $v$ in $G_{j-i}$. Then, \nSet $M_{j-i}=M_{j-i+1}\\cup\\{e\\}$.\n\\item[]\\textbf{Else if} $\\delta(G_{j-i})=2$: Let $v$ be the vertex of degree 2 selected in $G_{j-i}$. If $|N(v)|=1$ i.e. $v$ is joined to a single vertex by a double edge in $G_{j-i}$, set $M_{j-i}=M_{j-i+1}$. Else let $N(v)=\\{u,w\\}$ and $v_c$ be the new vertex resulting from the contraction of $\\{v,u,w\\}$. If $v_c$ is not covered by $M_{j-i+1}$ then set $M_{j-i}=M_{j-i+1}\\cup\\{v,u\\}$. Otherwise assume that $\\{v_c,z\\}\\in M_{j-i+1}$ for some $z\\in V(G_{j-i})$. Without loss of generality assume that in $G_{j-i}$, $z$ is connected to $u$. Set\n$M_{j-i}=(M_{j-i+1}\\cup \\{\\{v,w\\},\\{u,z\\}\\})\\setminus \\{ v_c,z\\}$.\n\\item[]\\textbf{Else if }: $\\delta(G_{j-i})\\geq 3$: Set $M_{j-i}=M_{j-i+1}$\n\\end{itemize}\n\\textbf{End For}\n\nFor a graph $G$ and $j\\in \\{0,1,...,\\hat{\\tau}\\}$ denote by $R_0(G,j)$ and $R_{2b}(G,j)$ the number of times that  {\\sc reduce} has performed a vertex-0 removal and a bad contraction respectively until it generates  $G_j$. For a graph $G$ and a matching $M$ denote by $\\kappa(G,M)$ the number of vertices that are not covered by $M$. The following Lemma determines the quality of the output of the {\\sc reduce-construct} algorithm.\n\\begin{lem}\\label{correction}\nLet $G$ be a graph and $M$ be the output of the Reduce-Backtrack algorithm applied to $G$. Then, for $j\\geq 0$,\n\\beq{true}{\n\\kappa(G,M)=R_0(G,j)+R_{2b}(G,j)+\\kappa(G_j,M_j).\n}\n\\end{lem}\n\\begin{proof}\nLet $G=G_0,G_1,...,G_{\\hat{\\tau}}$ be the sequence of graphs produced by {\\sc reduce} and let $M_j,M_{j-1},...,M_0=M$ be the sequence of matchings produced by {\\sc construct}. Let $R_0(G,j,i)$ and $R_{2b}(G,j,i)$ be the number of vertex-0 removals and bad contractions performed by {\\sc reduce} going from $G_{j-i}$ to $G_j$. We will prove that for every  $0\\leq i\\leq j$, \n\\beq{troo}{\n\\kappa(G_{j-i},M_{j-i})=R_0(G,j,i)+R_{2b}(G,j,i)+\\kappa(G_j,M_j).\n} \nTaking $i=j$ yields the desired result.\n\nFor $i=0$, equation \\eqref{true} holds as $R_0(G,j,0)=R_{2b}(G,j,0)=0$. Assume inductively that \\eqref{true} holds for $i=k-1$ where $k$ satisfies $0<k\\leq j$. For $i=k$, if a max-edge deletion was performed on $G_{j-k}$ then $|V_{j-k}|=|V_{j-k+1}|$. Furthermore, $R_0(G,j,k)= R_0(G,j,k-1)$ and $R_{2b}(G,j,k)=R_{2b}(G,j,k-1)$ and hence \\eqref{true} continues to hold. If a vertex-0 deletion or a bad contraction was performed on $G_{j-k}$ then $|V_{j-k}|=|V_{j-k+1}|+1$ and $M_{j-k}=M_{j-k+1}$. In the case of a vertex-0 deletion we have $R_0(G,j,k)= R_0(G,j,k-1)+1$ and $R_{2b}(G,j,k)=R_{2b}(G,j,k-1)$ and both sides of \\eqref{troo} increase by one. In the case of a bad contraction we have $R_0(G,j,k)= R_0(G,j,k-1)$ and $R_{2b}(G,j,k)=R_{2b}(G,j,k-1)+1$ and again  both sides of \\eqref{troo} increase by one. Finally if a good contraction or a vertex-1 removal was performed on $G_{j-k}$ then $R_0(G,j,k)= R_0(G,j,k-1)$ and $R_{2b}(G,j,k)=R_{2b}(G,j,k-1)$. At the same time we have that $\\kappa(G_{j-i},M_{j-i})=\\kappa(G_{j-i+1},M_{j-i+1})$, completing the induction.\n\\end{proof}\n\\subsection{Organizing the actions taken by {\\sc reduce}}\nWe do not analyze the effects of each action taken by {\\sc reduce} individually. Instead we group together sequences of actions, into what we call {\\em hyperactions}, and we analyze the effects of the individual hyperactions.\n\n\nWe construct a sub-sequence $\\Gamma_0=G,\\Gamma_1,\\ldots,\\Gamma_\\tau$ of $G_0,G_1,...,G_{\\hat{\\tau}}$. Every hyperaction, starts with a max-edge removal and it consists of all the actions taken until the next max-edge removal. We let $\\Gamma_{i}$ be the graph that results from performing the first $i$ hyperactions. Thus $\\Gamma_{i}$ is the $i$th graph in the sequence $G_0,G_1,...,G_{\\hat{\\tau}}$ that has minimum degree at least 3 and going from $\\Gamma_{i}$ to $\\Gamma_{i+1}$ {\\sc reduce} performs a max-edge removal followed by a sequence of vertex-0, vertex-1 removals and contractions. Thus $\\Gamma_0,\\Gamma_1,...,\\Gamma_{\\tau}$ consists of all the graphs in the sequence $G_0,G_1,...,G_{\\hat{\\tau}}$ with minimum degree at least 3.  \n\\subsection{Excess and Proof Mechanics}\\label{list}\nThe central quantity of this paper is the \\emph{excess} which we denote by $ex_\\ell(\\cdot)$. For a graph $G$ and a positive  integer $\\ell$ we let \n$$ ex_\\ell(G):=\\sum_{v \\in V(G)} [d(v)-\\ell]\\mathbb{I}(d(v)>\\ell).$$\n{\\bf Hyperactions of Interest}\\\\\nFor the analysis of {\\sc reduce} we consider 7 distinct hyperactions (sequences of actions) which we call hyperactions of Type 1,2,3,4,5,33 and 34 respectively. In the case that the maximum degree is larger than 3 we consider the following hyperactions: we have put some diagrams of these hyperactions at the end of the paper.\n\\begin{itemize}\n\\item[]\\textbf{ Type 1}: A single max-edge removal,\n\\item[]\\textbf{ Type 2}: A max edge-removal followed by an auto correction contraction.\n\\item[]\\textbf{ Type 3}: A single max-edge removal followed by a good contraction. \n\\item[]\\textbf{ Type 4}: A single max-edge removal followed by 2 good contractions.\nIn this case we add the restriction that there are exactly 6 distinct vertices $v,u,x_1,x_2,w_1,w_2$ involved in this hyperaction and they satisfy the following:\n(i) $v$ is a vertex of maximum degree, it is adjacent to $u$ and $\\{u,v\\}$ is removed during the max-edge removal, (ii) $d(u)=d(x_1)=d(x_2)=3$, (iii) $N(u)=\\{v,x_1,x_2\\}$, $N(x_1)=\\{u,x_2,w_1\\}$ and $N(x_2)=\\{u,x_1,w_2\\}$. (Thus $\\set{u,x_1,x_2}$ form a triangle.) The two contractions have the same effect as contracting $\\{u,x_1,x_2,w_1,w_2\\}$ into a single vertex.\n\\end{itemize}\n In the case that the maximum degree equals 3 we also consider the following hyperactions:\n\\begin{itemize}\n\\item[] \\textbf{Type 5}:  A max-edge removal followed by 2 good contractions that interact.\nIn this case the 5 vertices $u,v,x_1,x_2,z$ involved in the hyperaction satisfy the following:\n(i) $\\{u,v\\}$ is the edge removed by the max-edge removal, (ii) $N(v)=\\{u,x_1,x_2\\}$, $N(u)=\\{v,x_1,z\\}$, (so $\\set{u,v,x_1}$ form a triangle), (iii) $|(N(x_1) \\cup N(x_2) \\cup N(z)) \\setminus \\set{u,v,x_1,x_2,z}|\\geq 3$. This hyperaction has the same effect as contracting all of $\\set{u,v,x_1,x_2,z}$ into a single vertex.\n\\item[] \\textbf{Type 33}: A max-edge removal followed by 2 good contractions that do not interact. There are 6 distinct vertices involved $v,v_1,v_2,u,u_1,u_2$. During the max-edge removal $\\{u,v\\}$ is removed. Thereafter each of the 2 sets of vertices $\\{v,v_1,v_2\\}$ and $\\{u,u_1,u_2\\}$ is contracted to a single vertex.\n\\item[] \\textbf{Type 34}: A max-edge removal followed by 3 good contractions. There are 8 distinct vertices involved $v,v_1,v_2,v,u,u_1,u_2,w_1,w_2$. During the max-edge removal $\\{u,v\\}$ is removed. The conditions satisfied by $v,u,u_1,u_2,w_1,w_2$ and the actions that are performed on them are similar to the ones in a hyperaction of Type 4. The difference now is that $v$ has degree 3 before the hyperaction. In addition $\\{v,v_1,v_2\\}$ is contracted into a single vertex.\n\\end{itemize}\nWe divide Hyperactions of Type 3 into three classes. Assume that during a Hyperaction of Type 3 the set $\\{v,a,b\\}$ is contracted, $v$ is the contracted vertex and $v_c$ is the new vertex. We say that such a Hyperaction is of \\textbf{Type 3a} if $d(v_c)=d(a)+d(b)-2$, is of \\textbf{Type 3b} if $d(v_c)=d(a)+d(b)-4$ and is of \\textbf{Type 3c} if $d(v_c)<d(a)+d(b)-4$. Note that in general, $d(v_c)=d(a)+d(b)-2-2\\eta_{a,b}$, where $\\eta_{a,b}$ is the number of edges joining $a,b$ plus the number of edges joining $\\set{u,v}$ after the max-edge removal.\n\nWith the exception of a Hyperaction of Type 3c, where $\\eta_{a,b}\\geq 2$, we refer to the Hyperactions of interest as good Hyperactions. We call any Hyperaction that is not good, including a Hyperaction of Type 3c, bad. \n\nWe next state three lemmas, whose proofs will be deferred to Section \\ref{LemmasDegree}.\n\\begin{lem}\\label{hyper}\nLet $i\\geq 0$ and assume that $\\Gamma_i$  satisfies \n$ex_\\ell(\\Gamma_i)\\leq \\log^2 |V(\\Gamma_i)|$ for some $3\\leq \\ell =O(1)$.\n Then with probability $1-o(|V(\\Gamma_i)|^{-1.9})$ the hyperaction that {\\sc reduce} applies to $\\Gamma_i$ is good. In addition, it only applies a Hyperaction of Type 2, 3b or 4 with probability $o(|V(\\Gamma_i)|^{-0.9})$.\n\\end{lem}\nThe fact that w.h.p. we perform one hyperaction out of a small set of possibilities helps in controlling $ex_\\ell$. Namely we have  \n\\begin{lem}\\label{drift}\nSuppose that $\\Gamma_i$ satisfies $ex_4(\\Gamma)\\leq \\log^2 |V(\\Gamma_i)|$.\nThen either $ex_4(\\Gamma_i)=0$ or \n$$\\ex(ex_4(\\Gamma_{i+1})-ex_4(\\Gamma_i)|\\Gamma_i)\\leq -\\frac{1}{3}.$$\nMoreover with probability $1-o(|V(\\Gamma_i)|^{-1.9})$ we have that \n$$|ex_4(\\Gamma_i)-ex_4(\\Gamma_{i+1})|\\leq 2.$$\n\\end{lem}\n\\begin{lem}\\label{4inf}\nSuppose that $\\omega\\geq \\log n$. Then for every $i\\in \\{0,1,..,\\tau\\}$, $|V(\\Gamma_i)|\\geq \\omega$ implies that with probability $1-O(\\omega^{-0.8})$, we have $ex_4(\\Gamma_{i})\\leq \\log^2 |V(\\Gamma_i)|$. Furthermore with probability $1-O(\\omega^{-0.8})$ there exists $\\Gamma \\in \\{\\Gamma_{0},\\Gamma_{1},..,\\Gamma_{\\tau}\\}$ that satisfies $\\omega\\leq |V(\\Gamma)|\\leq 2\\omega$ and $ex_4(\\Gamma)=0$.\n\\end{lem}\nLemma  \\ref{hyper} states that as long as $ex_\\ell$ stays small then w.h.p. we perform only specific hyperactions. Lemmas \\ref{drift} and \\ref{4inf} imply that as long we perform only those hyperactions then w.h.p. $ex_4$ stays small. Furthermore, as none of those hyperactions consist of a vertex-0 removal or a bad contraction making we can appeal to Lemma \\ref{correction}. We use the above three lemmas to prove the following:\n\\begin{lem}\\label{reduction}\nLet $\\Gamma_h$ be as promised by Lemma \\ref{4inf} that is, $\\Gamma_h \\in \\{\\Gamma_{0},\\Gamma_{1},..,\\Gamma_{\\tau}\\}$ and it satisfies $\\log n\\leq \\omega \\leq V(\\Gamma_h)\\leq 2\\omega$ and $ex_4(\\Gamma)=0$. Then with probability $1-O(\\omega^{-0.8})$, for any matching $M_h$ of $\\Gamma_h$, the matching $M$ generated by {\\sc construct} with $j=h$, $M_h$ as an input satisfies\n\\begin{equation}\\label{kappa}\n\\kappa(G,M)=\\kappa(\\Gamma_h,M_h).\n\\end{equation} \n\\end{lem}\n\\begin{proof}\nFor $i<h$, Lemma \\ref{4inf} states that w.h.p. $ex_4(\\Gamma_{i})\\leq \\log^2 |V(\\Gamma_i)|$. Lemma \\ref{hyper} then implies that for $i<h$ with probability \n$$1-\\sum_{i=1}^{h}  o(|V(\\Gamma_i)|^{-1.9})\\geq 1-\\sum_{i=1}^{h}  o((2|E(\\Gamma_i)|\/3)^{-1.9}) \\geq 1- \\sum_{i=\\omega}^{\\infty}  o(2i^{-1.9}\/3) =1-o(\\omega^{-0.9})$$\nonly good hyperactions take place. For the first inequality we used the fact that $\\Gamma_i$ has minimum degree 3. For the second we used the fact that any Hyperaction decreases the number of edges, hence $|E(\\Gamma_i)|$ is decreasing with respect to $i$.\n\nHence w.h.p., until {\\sc reduce} generates $\\Gamma_h$ it performs no vertex-0 removals or bad contractions.\nLet $\\hat{h} \\in \\{0,1,,...,\\hat{\\tau}\\}$ be such that $G_{\\hat{h}}=\\Gamma_h$.\nThen from the above $R_0(G,\\hat{h})=R_{2b}(0,\\hat{h})=0$.\n Finally in this case, Lemma \\ref{correction} implies that \n$$\\kappa(\\Gamma,M_h)=\\kappa(G_{\\hat{h}},M_h)=\\kappa(\\Gamma_h,M_h).$$\n\\end{proof}\nObserve that by  a straightforward application of Lemma \\ref{reduction} with $\\omega=\\log n$ and $M_h$ being the matching produced by {\\sc construct} in the case $j=\\widehat{\\tau}$ we have that the matching $M$ output by the {\\sc reduce-construct} algorithm satisfies\n $$\\kappa(G,M)=\\kappa(\\Gamma_h,h) \\leq |V(\\Gamma_h)| \\leq 2\\log n.$$\nThis proves Part (a) of Theorem \\ref{main}.\n\\subsection{Finding a perfect matching in linear time} \nLemma \\ref{reduction} suggests that when we run {\\sc construct} we should finish our algorithm by finding a maximum matching $M'$ of $\\Gamma_h$. The final ingredient for the proof of Theorem \\ref{main}(b) is given by the next Theorem which implies that $\\Gamma_h$ has a parfect matching w.h.p.\n\\\\%Observe that the random graph in the statement of Lemma \\ref{34match} has even number of vertices. In the case that $|V(G_j)|$ is odd the we can remove a vertex, and the edges incident to it to obtain $G_j'$. Then we can apply {\\sc reduce} to $G_j'$ until we get a graph $G_j''$ that  satisfies $ex_4(G_j'')=0$. Lemmas 3 to 6 imply that w.h.p.\\@ $|V(G_j')|=(1+o(1))|V(G_j'')|$. Moreover w.h.p.\\@ any perfect matching of $G_j''$ can be extented to a perfect matching of $G_j'$ hence to a near perfect matching of $G_j$.\n\\begin{thm}\\label{34match}\nLet $\\bd$ be a degree sequence where $3\\leq d(i)\\leq 4$ for all $i$. Let $G$ be a random graph with degree sequence $\\bd$ and no loops. Then $G$ has a (near)-perfect matching with probability $1-O(n^{-3\/4})$. \n\\end{thm}\nWe defer the proof of Theorem \\ref{34match} to Section \\ref{Proofs Matchings}. To prove Part (b) of Theorem \\ref{main}, we run algorithm {\\em Reduce} until we find $\\Gamma_h$ as promised in Lemma \\ref{4inf}. Here we take $\\omega=n^{2\/3}$. We know from Lemma \\ref{reduction} and Lemma \\ref{34match} that $\\Gamma_h$ has a (near)-perfect matching with probability $1-O(\\omega^{-0.8})-O(\\omega^{-3\/4})=1-O(n^{-1\/2})$ that can be expanded to a (near)-perfect matching of the original graph $G$. Moreover since $|V(\\Gamma_h)|\\leq 2\\omega$, such a matching can be found in $O(\\omega^{3\/2})=O(n)$ time. The rest of the algorithm can be executed in $O(n)$ time and this completes the high probability part of the proof of Part (b). Also, if {\\sc reduce-construct} fails, then we can resort to an $O(n^{3\/2})$ time algorithm. This only happens with probability $O(n^{-1\/2})$ and this yields the claim in Part (b) about the expected running time. \n\\section{Proofs of Lemmas \\ref{hyper}, \\ref{drift} and \\ref{4inf}}\\label{LemmasDegree}\n\\begin{notn}\nWe sometimes write $A\\leq_O B$ in place of $A=O(B)$ for aesthetic purposes.\n\\end{notn}\n\\begin{notn}\nLet $K$ be an arbitrary positive integer and $b\\in \\{0,1\\}$. For a random graph $G$ and $v\\in V(G)$, let $\\cB_K(G,v,b)$ be the event that $G$ spans a subgraph that contains $v$, spans $a\\leq K$ vertices and $a+\\ell$ edges.\n\\end{notn}\nWe show that {\\sc reduce} either performs one of the good hyperactions given in Section \\ref{list} or $\\cB_K(G,v,1)$ occurs where $v$ is the vertex of maximum degree chosen by {\\sc reduce}. We will also use the following standard notation: For a graph $G$, $j,\\ell \\in \\mathbb{N}$ we let:\n\\begin{itemize}\n\\item $n(G):=|V(G)|$, $e(G):=|E(G)$,\n\\vspace{-2mm} \n\\item $\\delta(G)$ and $\\Delta(G)$ be the minimum and maximum degree of $G$ respectively, \n\\vspace{-2mm}\n\\item $n_j(G)$ be the number of vertices of $G$ of degree $j$,\n\\vspace{-2mm}\n\\item $p_j(G):=\\frac{jn_j(G)}{2|E(G)|}$ and $p_{>j}(G):=\\sum_{h>j} p_h(G)$,\n\\vspace{-2mm}\n\\item $ ex_\\ell(G):=\\sum_{v \\in V(G)} [d(v)-\\ell]\\mathbb{I}(d(v)>\\ell)$, \n\\end{itemize}\nWe denote by $n_i,e_i, n_{j,i},p_{j,i},p_{>j,i}$ and $ex_{\\ell,i}$ the corresponding quantities in relation to $\\Gamma_i$.  \n\\begin{lem}\\label{dence}\nLet $K$ be an arbitrary fixed positive integer. Let $\\bd$ be a degree sequence of length $n$ that satisfies $ex_\\ell(G)\\leq  \\log^2 n$ for some $3\\leq \\ell=O(1)$. Let $G$ be a random graph with degree sequence $\\bd$ and no loops. Let $b \\in \\{0,1\\}$, then $\\mathbf{Pr}(\\cB_K(G,v,b))=o(n^{-0.9-b})$.\n\\end{lem}\n\\begin{proof}\nLet $G$ be a random graph with degree sequence $\\bd$. The fact that $ex_\\ell(G)\\leq   \\log^2 n$ implies that $G$ has no loops with probability bounded below by a positive constant (see for example \\cite{FK}). Hence the condition of having no loops can be ignored in the proof that events have probability $o(1)$. Also it implies that $\\Delta=\\Delta(G)\\leq \\ell+ ex_\\ell(G) \\leq \\ell+\\log^2 n$.\n\nLet $2m=\\sum_{i=1}^n d(i)\\leq \\ell n+ex_{\\ell}(G) =\\Theta(n)$. Then for vertex $v$ and for $b=0,1$ the probability that $G$ spans a subgraph that covers $v$, spans $a\\leq K$ vertices and $a+b$ edges can be bounded above by \n\\begin{align*}\n&\\leq_O \\sum_{a=2}^{K} \\binom{n}{a-1} (\\Delta a)^{2(a+b)} \\frac{(2m-2(a+b))!}{2^{m-(a+b)}[m-(a+b)]!}\\times \\frac{2^{m}m!}{(2m)!}\\\\ \n&\\leq_O \\sum_{a=2}^{K} n^{a-1} \\Delta^{2(a+b)} \\frac{m(m-1)...(m-(a+b)+1)}{2m(2m-1)...(2m-2(a+b)+1)}\\\\ \n&\\leq_O \\sum_{a=2}^{K}  n^{a-1}  \\Delta^{2(a+b)}  m^{-(a+b)}\\\\\n&=o(n^{-0.9-b}).\n\\end{align*}\n\\end{proof}\nWe will drop the subscript $K$ from $\\cB$. Taking $K=20$ will easily suffice for the rest of the proof.  Thus $\\cB(G,v,b)=\\cB_{20}(G,v,b)$.\n\\subsection{Proof of Lemma \\ref{hyper}}\nLet $v$ be the vertex of maximum degree chosen by {\\sc reduce} and let $u$ be the vertex adjacent to $v$ such that $\\{u,v\\}$ is chosen for removal. We will show that if $\\cB(\\Gamma_i,v,1)$ does not occur then {\\sc reduce} performs one of the hyperactions given in Section \\ref{list}.\nAlso observe that if a Hyperaction of Type 2,3b,4,5 or 34 occurs then $\\cB(\\Gamma_i,v,0)$ occurs. Lemma \\ref{dence} states that $\\mathbf{Pr}(\\cB(\\Gamma_i,v,0))=o(|V(\\Gamma_i)|^{-0.9})$ thus proving the second part of Lemma \\ref{hyper}. Also note that if  a Hyperaction of Type 3c, occurs, corresponding to a bad Hyperaction, then $\\cB(\\Gamma_i,v,1)$ occurs. Lemma \\ref{dence} states that $\\mathbf{Pr}(\\cB(\\Gamma_i,v,1))=o(|V(\\Gamma_i)|^{-1.9})$.\n\n{\\bf Case A: $d(v) \\geq 4$.}\\\\\nIf $d(u)\\geq 4$ then a  hyperaction of Type 1 is performed.  \nThus assume  $d(u)=3$ and consider the cases where $|N(u)|=1,2,3$, (recall that we allow parallel edges but not self-loops). \n\n{\\textbf{Case A1: $|N(u)|=1$}}.\\\\\n$u$ is connected to $v$ by 3 parallel edges and so $\\cB(\\Gamma_i,v,1)$ occurs.\t\n\n{\\textbf{Case A2: $|N(u)|=2$}}.\\\\\nLet $N(u)=\\{v,u'\\}$ and $S=\\set{u,u',v}$. Let $T=(N(u')\\cup N(v) )\\setminus S$. If $|T|\\leq 2$ then we have $d(S)\\geq 10$ and either $S$ spans more than 3 edges or $S \\cup T$ spans at least 7 edges. In both cases  $\\cB(\\Gamma_i,v,1)$ occurs. Assume then that  $|T|\\geq 3$. Now exactly one of $\\set{u,u'}$, $\\set{u,v}$ is repeated, else $\\cB(\\Gamma_i,v,1)$ occurs.  If $\\set{u,u'}$ is repeated then we perform an auto correction contraction resulting to a Hyperaction of Type 2 . If $\\set{u,v}$ is repeated then we contract the remaining path $(u',u,v)$. Hence we have performed a Hyperaction of Type 3b.\n\n{\\textbf{Case A3: $|N(u)|=3$.} }\\\\\nLet $N(u)=\\set{v,x_1,x_2}$ and $T=(N(x_1)\\cup N(x_2))\\setminus\\{u,x_1,x_2\\}$. \n\n{\\textbf{Sub-case A3a: $|T|\\leq 1$}. \\\\\n$\\{u,x_1,x_2\\} cup T$ spans at least $(d(u)-1+d(x_1)+d(x_2)+|T|)\/2 \\geq 4+|T|\/2$ edges and the event $\\cB(\\Gamma_i,v,1)$ occurs.\n\n{\\textbf{Sub-case A3b: $T=\\{w_1,w_2\\}$}}. \\\\\nIf  $v$ is at distance less than 6 from $\\{u\\} \\cup N(u)$ in $\\Gamma\\setminus \\{ u,v\\}$ then  $\\cB(\\Gamma_i,v,1)$ occurs. To see this consider the subgraph $H$ spanned by $\\{u,v,x_1,x_2,w_1,w_2,y\\}$ and the vertices on the shortest path $P$ from $v$ to $u$ in $\\Gamma_i\\setminus \\{ u,v\\}$. Here $y$ is the neighbor of $v$ on $P$. It must contain at least two distinct cycles. One contained in $\\set{u,x_1,x_2,w_1,w_2}$ and $u,v,P$. If there is no edge from $x_1$ to $x_2$ then $\\{v,u,x_1,x_2,w_1,w_2\\}$ spans at least 7 edges and so $\\cB(\\Gamma_i,v,1)$ occurs.\n\nThus we may assume that $N(x_1)=\\{u,x_2,w_1\\}$, $N(x_2)=\\{u,x_1,w_2\\}$ and $v\\notin \\{w_1,w_2\\}\\cup N(w_1) \\cup N(w_2)$. We may also assume that $\\set{w_1,w_2}$ is not an edge of $\\Gamma$, for otherwise $\\{u,x_1,x_2,w_1,w_2\\}$ contains two distinct cycles and $\\cB(G_i,v,1)$ occurs. The algorithm {\\sc Reduce} proceeds by contracting $u,x_1,x_2$ into a single vertex $x'$.  $x'$ has degree 2 and the algorithm proceeds by performing a contraction of $x',w_1,w_2$ into a new vertex $w'$. Let $S=N\\{w_1,w_2\\}\\setminus \\{x_1,x_2\\}$. If $|S|\\leq 3$ then $\\cB(\\Gamma_i,u,1)$ occurs. To see this observe that $w_1,w_2$ must then have a common neighbor $w_3$ say. Consider the subgraph $H$ spanned by $\\{u,x_1,x_2,w_1,w_2,w_3\\}$. $H$ contains at least 7 edges and 6 vertices. If $|S|\\geq 4$ then the new vertex has degree 4 and the sequence of actions taken by {\\sc reduce} corresponds to a hyperaction of Type 4.\n\\vspace{3mm}\n\\\\{\\textbf{Sub-case A3c: $|T| \\geq 3$.}}\nAfter the removal of $\\{v,u\\}$ we contract $\\{u,x_1,x_2\\}$ into a single vertex of degree at least 3, hence a hyperaction of Type 3 is performed.\n\\vspace{3mm}\n\\\\\n{\\bf Case B: $d(v)=d(u)=3$.}\\\\\nLet $\\Gamma'=\\Gamma_i\\setminus \\{e\\}$ where $e=\\{v,u\\}$.\n\\vspace{3mm}\n\\\\{\\textbf{Case B1: In $\\Gamma'$, $u$ and $v$ are at distance at least 4.}}\\\\ \nIf $|N(N (u))|$ and $|N(N(v))| \\leq 3$ then $\\cB(\\Gamma_i,v,1)$ occurs. Thus we can assume that either $|N(N (u))|=4$ and\/or $|N(N(v))|=4$. If both $|N(N (u))|,|N(N(v))|=4$ then {\\em Reduce} will perform 2 good contractions and this amounts to a hyperaction of Type 33. Assume then that $|N(N (u))|=4$ and that $|N(N (v))|\\leq 3$. If $|N(N (v))|=3$ then again {\\em Reduce} will perform 2 good contractions amounting to a hyperaction of Type 33. If $|N(N (v))|=2$ and so $v$ is in a triangle then {\\em Reduce} will perform a hyperaction of Type 34. Finally, if $|N(N (v))|=1$ then $\\cB(\\Gamma_i,v,1)$ occurs.\n\n{\\textbf{Case B2: In $\\Gamma'$, $u$ and $v$ are at distance 3.}}\\\\\nIn $\\Gamma_i$ there is a cycle $C$ of length 4 containing $u,v$. If in $\\Gamma'$ we find that $|N(N (u))|\\leq 3$ or $ |N(N(v))| \\leq 3$ or $|N(u)\\cap N(N(v))|>1$ or $|N(v)\\cap N(N(u))|>1$ then $\\cB(\\Gamma_i,v,1)$ occurs. This is because the graph spanned by $\\{u,v\\} \\cup N(u)\\cup N(v) \\cup N(N(u) \\cup N(N(v))$ in $\\Gamma_i$ will contain a cycle distinct from $C$. Assume this is not the case. Then after the max-edge removal of $\\{u,v\\}$ we have a contraction of $\\{u\\} \\cup N(u)$ followed by a contraction of $\\{u\\} \\cup N(u)$. Observe that neither contraction reduces the size of $N(N(u))$ or $N(N(v))$. Thus {\\sc reduce} performs a hyperaction of Type 33.\n\\vspace{3mm}\n\\\\{\\textbf{Case B3: In $\\Gamma'$, $u$ and $v$ are at distance 2.}\\\\\nIn the case that $u,v$ have 2 common neighbors in $\\Gamma'$ we see that $\\cB(\\Gamma_i,v,1)$ occurs. Assume then that they have a single common neighbor $x_1$. Let $z,x_2$ be the other neighbors of $u,v$ respectively. Then either $\\cB(\\Gamma_i,v,1)$ occurs or {\\sc reduce} performs a hyperaction of Type 5.\n\\vspace{3mm}\n\\\\{\\textbf{Case B4: In $\\Gamma'$, $u$ and $v$ are at distance 1}.\\\\\nSo here we have that $\\set{u,v}$ is a double edge in $\\Gamma_i$. Let $x,y$ be the other neighbors of $u,v$ repectively in $\\Gamma$. Assuming that $\\cB(\\Gamma_i,v,1)$ does not occur, {\\sc reduce} performs a max-edge removal followed by a single good contraction and this will be equivalent to a hyperaction of Type 3, involving the contraction of one of $x,u,v$ or $u,v,y$.\n\\qed\n\\subsection{Proof of Lemma \\ref{drift}}\nThe inequality $ex_{4,i} \\leq \\log^2 n_i$ implies that $p_{j,i}\\leq \\log^2n_i\/2e_i =o(n_i^{-0.95})$ for $5\\leq j \\leq \\Delta$. It also implies that the maximum degree $\\Delta$ of $\\Gamma$ satisfies $\\Delta=O(\\log^2 n)$. \n\nIn the case that $\\cB(G,v,1)$ occurs, i.e. with probability $O(n_i^{-1.9})$ (see Lemma \\ref{hyper}) \n$$|ex_{4,i}-ex_{4,i+1}|\\leq 2e_i \\leq 4n_i+ex_{4,i} \\leq 5n_i.$$ \nObserve that if an action of Type 2,3b,4,5 or 34 takes place then $v$ lies on a subgraph with $a<12$ vertices and $a$ edges. Lemma \\ref{hyper} states that this occurs with probability $o(n^{-0.9})$. For all the above hyperactions we have $|ex_{4,i+1}-ex_{4,i}|\\leq 2$. This follows from the fact that performing a contraction can increase $ex_4$ by at most 2. This is because the initial vertices with degrees say $2,d_1,d_2$ contributed $\\max\\{ 0, (d_1-4)+(d_2-4) \\}$ to $ex_{4,i}$ while the new contracted vertex has degree $d_1+d_2-2$ and contributes $\\max\\{ 0, d_1+d_2-2-4) \\}$ to $ex_{4,i+1}$. Moreover observe that if a hyperaction of Type 5, 33 or 34 occurs then all the vertices involved have degree 3. If $\\cB(\\Gamma,v,1)$ does not occur then a hyperaction of Type 5 will increase $ex_{4,i}$ by 1 (since there will be one new vertex of degree 5). Hyperactions of Type 33 or Type 34 do not change $ex_4$. Thus it remains to examine the effects of a hyperaction of Type 1 or of Type 3a. \n\nIf  $ex_{4,i}=0$ then a hyperaction of Type 1 does not increase $ex_{4,i}$ while a hyperaction of Type 3 could increase it by 2. \n\nIf $ex_4(\\Gamma)>0$ then the $i^{th}$ hyperaction starts with a max-edge removal. \\\\\n{\\bf Case 1:} If the smaller degree vertex involved is of degree larger than 3, then this results in a hyperaction of Type 1. This happens with probability $(1+o(1))(1-p_{3,i})$. Furthermore in this case $ex_{4,i}-2\\leq ex_{4,i+1}\\leq ex_{4,i}-1$.\n (The $(1+o(1))$ factor arises because of $O(1)$ degree changes during the hyperaction.)\\\\\n{\\bf Case 2:} If the smaller degree vertex $v$ involved is of degree 3 then a contraction is performed. And this occurs with probability $(1+o(1))p_{3,i}$. If the contraction results in a vertex of degree at least 3 then we have a hyperaction of Type 3a, and not of Type 1. Also this is the only way for a hyperaction of Type 3a to occur. Let the other two vertices involved in the contraction be $a,b$ and  have degrees $d_a,d_b$ respectively. Now $d_a=d_b=3$ with probability $(1+o(1))p_{3,i}^2$, resulting in a new vertex that has degree at most 4. In this case, $ex_{4,i+1}-ex_{4,i}= -1$ (the -1 here originates from the max-edge removal). Else if $d_a=3,d_b=4$ then we have $-1\\leq ex_{4,i}-ex_{4,i}\\leq -1+1=0$. And this occurs with probability $(1+o(1))2p_{3,i}p_{4,i}$. Else if $d_a=d_b=4$ then we have $-1\\leq ex_{4,i+1}-ex_{4,i}\\leq -1+2= 1$ (this occurs with probability $p_{3,i}p_{4,i}^2$. Otherwise a vertex of degree at least 5 is involved and given our upper bound on $ex_{4,i}$, this happens with probability $o(1)$. If the event $\\cB(\\Gamma,v,1)$ does not occur then the new contracted vertex has degree   $d_a+d_b-2$. Hence $-1\\leq ex_{4,i} - ex_{4,i} \\leq -1+2=1.$ \n\nThus in all cases $|ex_{4,i} - ex_{4,i}|\\leq 2$ and if $ex_{4,i}>0$ then\n\\begin{align*}\n\\ex[ex_{4,i+1}-ex_{4,i}| \\Gamma_i] &\\leq -(5n_i)\\cdot o(n_i^{-1.9})+ 2n_i^{-0.95} - (1-p_{3,i}) -p_{3,i}^3 + p_{3,i}p_{4,i}^2+o(1)\n\\\\&\\leq - (1-p_{3,i}) -p_{3,i}^3 +p_{3,i}(1-p_{3,i})^2+o(1)\n \\leq -\\frac13\n\\end{align*} \n(The final expression in $p_{3,i}$ is maximized at $p_{3,i}=1\/2$, for $p_{3,i}\\in [0,1]$.)\n\\qed\n\\subsection{Proof of Lemma \\ref{4inf}} \nWe start by proving the following Lemma:\n\\begin{lem}\\label{intervals}\nLet $\\Gamma_h$ be such that $ex_{4,h}=0$. Then with probability $1-o(n_j^{-1.8})$ there exists $1\\leq j\\leq 0.25 \\log^2n_h$ satisfying $ex_{4,h+j}=0$. Furthermore\n$ex_{4,h+i}\\leq \\log^2n_{h+i}$ for $i\\leq j$.  \n\\end{lem}\n\\begin{proof}\nIf $ex_{4,h+1}=0$ then we are done. Otherwise Lemma \\ref{drift} implies that \n$ex_{4,h+1}\\in\\{1, 2\\}$ with probability $1-o(n_h^{-1.9})$.\n\nLet $\\cE_T$ be the event that for $j\\leq 0.25\\log^2 n_h$ {\\sc reduce} performs only hyperactions of Type 1,2,3,4,5,33 or 34. Such a hyperaction reduces the vertex set by at most 8. Lemma \\ref{hyper} implies that $\\cE_T$ occurs with probability $1-o(n^{-1.8}_h)$.  Moreover if $\\cE_T$ occurs for $j<0.25\\log^2 n_h$ then $|n_{h+j}| \\geq n_h-8\\cdot 0.25 \\log^2 n_h$. In addition from Lemma \\ref{drift} we have that with probability $1-o(n^{-1.8})$, $|ex_{4,h+j}-ex_{4,h+j-1}|<2$ for $j<0.25 \\log^2 n$ hence $ex_{4,h+j} \\leq 2\\cdot 0.25 \\log^2 n_h \\leq \\log^2 n_{h+j}$. Finally conditioned on $\\cE_T$ since $ex_{4,h+1}=1$ or 2, the probability that there is no $2\\leq j\\leq 0.25\\log^2 n_h$ satisfying $ex_{4,h+j}=0$ is bounded by the probability that the sum of $0.25 \\log^2 n-2$ independent random variables with magnitude at most 2 and expected value smaller than $-1\/3$ (see Lemma \\ref{drift}) is positive. From Hoeffding's inequality \\cite{Hoeff} we have that the  later probability is bounded by $\\exp\\set{-\\frac{2(\\frac13\\log^2n_h-3)^2}{\\log^2 n_h}}=o(n^{-2}_h)$.\n\\end{proof}\nNow let $\\Gamma_0,\\Gamma_{i_1},...,\\Gamma_{i_\\ell}$ be the subsequence of $\\Gamma_{0},\\Gamma_{1},..,\\Gamma_{\\tau}$ that includes all the graphs that have $ex_4=0$ and at least $\\omega$ vertices. Then since $\\Gamma_i$ has minimum degree 3 and $e_i$ is decreasing with respect to $i$ using  Lemma \\ref{intervals} we have that with probability \n$$1-\\sum_{i:n_i\\geq \\omega}o( n_i^{-1.8} )\\geq 1-\\sum_{i:n_i\\geq \\omega} 2e_i^{-1.8}\/3 \\geq  1-\\sum_{i=\\omega}^{\\infty} i^{-1.8}=1-o(\\omega^{-0.8})$$\nwe have that for $j<\\ell$  we have $n_{i_j} -n_{i_{j+1}} \\leq 8\\cdot 0.25\\log ^2 n_{i_j}=2\\log ^2 n_{i_j}$ and all the graphs $\\Gamma_i$ preceding $\\Gamma_{i_\\ell}$ in $\\Gamma_0,\\Gamma_1,...,\\Gamma_\\tau$ satisfy $ex_{4,i} \\leq \\log^2 n_i$.  Suppose now that $n_{i_\\ell}> 2\\omega$. Then the above argument implies that w.h.p. there is $j>i_\\ell$ such that $ex_{4,j}=0$ and $n_j\\geq n_{i_\\ell}-2\\log^2n_{i_\\ell}\\geq \\omega$ and this contradicts the definition of $i_\\ell$. Thus, w.h.p., $\\omega\\leq n_{i_\\ell}\\leq 2\\omega$ and hhis completes the proof of Lemma \\ref{4inf}.\n\\qed\n\\section{Existence of a Perfect Matching}\\label{Proofs Matchings}\nWe devote this section to the proof of  Lemma \\ref{34match}.  As discussed in the previous section it is enough to prove that given a degree sequence $\\bd=(d(1),...,d(n))$ consisting only of 3's and 4's, if we let $G$ be a random configuration (multi)-graph with degree sequence $\\bd$, then w.h.p. $G$ has a (near)-perfect matching (i.e we can lift the condition that $G$ has no loops). We will first assume that $n$ is even and verify Tutte's condition. That is for every $W\\subset V$ the number of odd components induced by $V \\setminus W$, $q(V\/W)$, is not larger that $|W|$. We split the verification of Tutte's condition into two lemmas. \n\\begin{lem}\\label{minimal}\nLet $W\\subset V$ be a set of minimum size that satisfies $q(V\/W)>|W|$. Then with probability $1-O(n^{-3\/4})$, $|W| > 10^{-5}n$.  \n\\end{lem}\n\\begin{lem}\\label{maximal}\nLet $W\\subset V$ be a set of maximum size that satisfies $q(V\/W)>|W|$. Then  with probability $1-O(n^{-3\/4})$, $|W| < 10^{-5}n$.  \n\\end{lem}\nLemmas \\ref{minimal} and \\ref{maximal} together imply Tutte's condition as there exists no set with size that is both strictly larger and strictly smaller than $10^{-5}n$. In the proof of these lemmas we use the following estimates.\n\\begin{lem}\\label{estimates}\nThe number of distinct partitions of a set of size $2r$ into 2-element subsets, denoted by $\\phi(2r)$, satisfies $\\phi(2r)=\\Theta\\left(\\bfrac{2r}{e}^r\\right)$. Also for $\\ell<r$ we have $\\phi(2r)\\leq 2r^{r-\\ell} \\bfrac{2\\ell}{e}^\\ell$.\n\\end{lem}\n\\begin{proof}\nTo generate a matching we first choose a permutation of the $2r$ items and then we pair the $(2i-1)$th item with the $2i$th item. Therefore, using Stirling's approximation we have \n$$\\phi(2r)=\\frac{{(2r)}!}{2^r r!} =\\frac{\\Theta(\\sqrt{2r} \\bfrac{2r}{e}^{2r})}{\\Theta(2^r \\sqrt{r} \\bfrac{r}{e}^r)}= \\Theta\\bigg( \\bfrac{2r}{e}^r\\bigg).$$\nAlso\n$$\\phi(2r)=\\frac{{(2r)}!}{2^r r!} \\leq \\frac{(2r)^{r-\\ell}{(2\\ell)}!}{2^{r-\\ell} 2^\\ell \\ell!}   \\leq 2r^{r-\\ell} \\bfrac{2\\ell}{e}^\\ell,$$\nwhere we have used $(2r)!\\leq (2r)^{2(r-\\ell)}(2\\ell)!$.\n\\end{proof}\n\\subsection{Proof of Lemma \\ref{minimal}:}\nLet $W$ be a  set satisfying $q(V \\setminus W)>|W|$ of minimum size and assume $2\\leq w=|W|\\leq 10^{-5}n$. We can rule out the case $w=1$ from the fact that with probability $1-O(1\/n)$, $G$ will be 3-connected, see e.g. the proof of Theorem 10.8 in \\cite{FK} . Let $C_z$ be a component spanned by $V\\setminus W$ of maximum size and let $r=|C_z|$. \n\n\\textbf{Case 1: $r=|C_z|\\leq 0.997n$.}} In this case we can partition $V\\setminus W$ into two parts $V_1,V_2$ such that (i) each $V_l,l=1,2$ is the union of components of $V\\setminus W$, (ii) $|V_1|\\geq |V_2|$, and (iii) $|V_2|\\geq (n-(r+w))\/2\\geq 10^{-3}n$. \n\nLet $d_2=d(V_2)$ and $d_W=d(W)$. Out of the $d_W$ endpoints in $W$ (i.e. configuration points that correspond to vertices in $W$), $\\ell\\leq d_W$ are matched with endpoints in $V_2$ and the rest with endpoints in $V_1$. \n\nFor fixed $i,w,d_W$ the probability that there are sets $V_1,V_2,W$ with $w=|W|, d(W)=d_W$ and $|V_2|=i$ satisfying $1\\leq w\\leq 10^{-5}n, 10^{-3}n\\leq i\\leq 0.5n$ and $d_W\\leq 4w\\leq 0.04i$, such that $V_1 \\times V_2$ spans no edges is bounded by \n\\begin{align*}\np_1 &\\leq  \\sum_{\\ell=0}^{d_W}\n\\binom{n}{i} \\binom{n-i}{w} \\binom{d_W}{\\ell} \\frac{\\phi(d_2+\\ell)\\cdot \\phi(2m-d_2-\\ell) }{\\phi(2m)}\\\\ \n&\\leq_O \\sum_{\\ell=0}^{d_W}\n\\bfrac{en}{i}^i \\bfrac{en}{w}^w 2^{d_W}  \\frac{\\bfrac{d_2+\\ell}{e}^{(d_2+\\ell)\/2} \n\\bfrac{2m-d_2-\\ell}{e}^{(2m-d_2-\\ell)\/2} }{\\bfrac{2m}{e}^m}\\\\ \n&\\leq\\sum_{\\ell=0}^{d_W}\n \\bfrac{en}{i}^i \\bfrac{100en}{i}^{i\/100} 2^{i\/25}  \\bfrac{d_2+\\ell}{2m}^{(d_2+\\ell)\/2} \n\\bigg(1-\\frac{d_2+\\ell}{2m}\\bigg)^{(2m-d_2-\\ell)\/2} \\\\  \n& \\leq_O \\sum_{\\ell=0}^{d_W}\n  \\bfrac{1600(en)^{101}}{i^{101}}^{i\/100} \\bfrac{d_2+\\ell}{2m}^{(d_2+\\ell)\/2} \\exp\\set{- \\frac{d_2+\\ell}{2}\\brac{1-\\frac{d_2+\\ell}{2m}}} \n\\end{align*}\nFor the third line we used the fact that $w\\leq i\/100$ and $d_W\\leq 4w \\leq i\/25$. \n\nLet $f(x)=x^xe^{-x(1-x)}$ and $L(x)=\\log f(x)$. Then $L''(x)=x^{-1}+2$ and so $L$ and hence $f$ is convex for $x>0$. Now $d_2+\\ell \\in J= [3i,4.04i]$ and since $\\bfrac{d_2+\\ell}{2m}^{(d_2+\\ell)\/2}\\exp\\set{- \\frac{d_2+\\ell}{2}\\brac{1-\\frac{d_2+\\ell}{2m}}}=f\\bfrac{d_2+\\ell}{2m}^m$ we see that its  maxima are at the endpoints of $J$. In general $3i\\leq 3n\/2 \\leq m$. However when $d_2+\\ell=4.04i$ we have that \n\\beq{2m}{\n2m\\geq 4.04i+3(n-i-w)\\geq 4.04i+ 3(n-1.01i)= 3n+1.01i.\n}\n{\\bf Case 1a: $d_2+\\ell=3i$.}\\\\\nWe have $\\frac{d_2+\\ell}{2m} \\leq \\frac{3i}{3n}\\leq \\frac{1}{2}$ and $(d_2+\\ell)(1-\\frac{d_2+\\ell}{2m})\\geq 3i\/2$.  Therefore,\n\\begin{align*}\np_1&\\leq_O w   \\bfrac{1600(en)^{101}}{i^{101}}^{i\/100}\\bfrac{i}{n}^{3i\/2} e^{-3i\/4} \\\\\n&= w  \\bigg[\\frac{1600e^{26}}{2^{49}}  \\bfrac{2i}{n}^{49}\\bigg]^{i\/100} \\\\ \n&\\leq w \\bigg[ e^{-1\/2}  \\bfrac{2i}{n}^{49}\\bigg]^{i\/100}\\\\\n&\\leq w e^{-i\/200}.\n\\end{align*}\n{\\bf Case 1b: $d_2+\\ell=4.04i$.}\\\\\nIt follows from \\eqref{2m} that $\\frac{d_2+\\ell}{2m} \\leq \\frac{4.04i}{3n+1.01i} \\leq 0.577$ where the second inequality uses $i\\leq n\/2$. It follows from this that $(d_2+\\ell)(1-\\frac{d_2+\\ell}{2m})\/2\\geq 0.85i$. Hence,\n\\begin{align*}\np_1&\\leq_O w   \\bfrac{1600(en)^{101}}{i^{101}}^{i\/100} \\bfrac{4.04i}{3n+1.01i}^{2.02i} e^{-0.85i}\\\\\n&\\leq_O w  \\bigg[1600e^{16} \\bigg(\\frac{n}{i}\\bigg)^{101} \\bigg( \\frac{4.04i}{3n}\\bigg)^{101} \\bigg(\\frac{4.04i}{3n+1.01i}\\bigg) ^{101}\\bigg]^{i\/100}\\\\ \n&\\leq_O w  \\bigg[1600e^{16} \\bigg(\\frac{4.04}{3} \\cdot 0.577\\bigg)^{101}\\bigg]^{i\/100}\\\\\n& \\leq_O w e^{-i\/100}.\n\\end{align*}\nTherefore the probability that Case 1 is satisfied is bounded by a constant times\n\\begin{align*}\n& \\sum_{w=1}^{10^{-5}n} \\sum_{i=10^{-3}n}^{0.5n} w e^{-i\/200}=O(n^{-3\/4}).\n\\end{align*}\n\\vspace{3mm}\n\\\\ \\textbf{Case 2:} $r=|C_z|\\geq 0.997n$. \nLet $V_1=V(C_z)$, \n$V_2=V\\setminus (V_1\\cup W)$. First note that $V_2$ spans at least $w$ components. Therefore $|V_2|\\geq w$. To lower bound $e(V_2:W)$ we use the following Claim.\n\\vspace{3mm}\n\\\\ \\textbf{Claim 1} Every vertex in $W$ is adjacent to at least 3 distinct components in $V \\setminus W$, and hence to at least 2 vertices in $V_2$.\n\\vspace{3mm}\n\\\\ \\textbf{Proof of Claim 1:} Let $v \\in W$ be such that it is adjacent to $t\\in \\{0,1,2\\}$ components in $V\\setminus W$. Consider $W'=W\\setminus\\{v\\}$. Thus $|W'|=|W|-1$ . If $t=0$ then $q(V\\setminus W')=q(V\\setminus W)+1$. If $t=1$ then $q(V\\setminus W')\\geq q(V\\setminus W)-1$. If $t=2$ then if the both of the corresponding components have odd size then the new component will also have odd side, while if only one of them has odd size then the new one has even size. Finally if both have even size the new one has odd size. In all three cases the inequality  $q(V\\setminus W')\\geq q(V\\setminus W)-1$ is satisfied. Therefore $q(V\\setminus W') \\geq q(V\\setminus W) -1 >|W|-1=|W'|  $ contradicting the minimality of $W$.\n\\qed \n\\vspace{3mm}\n\\\\ From Claim 1 it follows that $W:V_2$ spans  at least $2w$ edges. \nWe also have that $|V_2|\\leq n-r-w \\leq 0.003n$. For fixed  $2\\leq w\\leq 10^{-5}n$, $3w\\leq d_W \\leq 4w$ and $w\\leq i$  the probability that there exist such sets $V_1,V_2,W$, $|V_2|=i$, $w=|W|$, $d(W)=d_W$ and $2w\\leq  \\ell= e(V_2:W)\\leq 4w$ is bounded by  \n\\begin{align*}\n&\\sum_{\\ell=2w}^{d_W} \n \\binom{n}{i} \\binom{n-i}{w} \\binom{d_W}{\\ell} \\frac{\\phi(d_2+\\ell)\\cdot \\phi(2m-d_2-\\ell) }{\\phi(2m)}\\\\\n& \\leq_O  \\sum_{\\ell=2w}^{d_W} \\bfrac{en}{i}^i \\bfrac{en}{w}^w 2^{4w} \\frac{\\bfrac{d_2+\\ell}{2}^{(d_2+\\ell-2w)\/2}  \\bfrac{2w}{e}^{w} \\bfrac{2m-d_2-\\ell}{e}^{(2m-d_2-\\ell)\/2} }{\\bfrac{2m}{e}^m}\\\\ \n&= \\sum_{\\ell=2w}^{d_W} \\bfrac{en}{i}^i \\bfrac{en}{w}^w 2^{4w}   \\bfrac{2w}{2m}^{w} \\brac{\\frac{d_2+\\ell}{2m}\\cdot \\frac{e}2}^{(d_2+\\ell-2w)\/2} \n\\bigg(1-\\frac{d_2+\\ell}{2m}\\bigg)^{(2m-d_2-\\ell)\/2} \\\\  \n& \\leq_O w \\bfrac{en}{i}^i \\bfrac{16e}{3}^{w} \\bigg( \\frac{5i}{3n} \\cdot \\frac{e}{2} \\bigg)^{3i\/2}\\\\\n&\\leq_O w   \\brac{e^2 \\bfrac{16e}{3}^{2w\/i} \\frac{5^3i}{3^3n}}^{i\/2}.\n\\end{align*} \nFor the second line we used the second inequality of Lemma \\ref{estimates}. For the fourth line we used that $2w\\leq \\ell $, $d_2+\\ell \\leq 4|V_2|+4w \\leq 0.01204n$ and so $\\brac{\\frac{d_2+\\ell}{2m}\\cdot\\frac{e}2}^{(d_2+\\ell-2w)\/2}$ is maximized when $d_2,\\ell$ are as small as possible, that is $d_2=3i,\\ell=2w$. Furthermore note that $d_2+\\ell-2w\\geq d_2\\geq 3i$ and $i\\geq q(V\\setminus W)-1\\geq w$. Therefore the probability that Case 2 is satisfied is bounded by a constant times\n\\begin{align*}\n&\\sum_{w=2}^{10^{-5}n} \\sum_{i=w}^{0.003n}  \\brac{e^2\\bfrac{16e}{3}^{2w\/i} \\frac{5^3i}{3^3n}}^{i\/2}\\\\\n&\\leq_O  \\sum_{w=2}^{10^{-5}n} \\sum_{i=w}^{2w}\\bfrac{C_1i}{n}^{i\/2}+\\sum_{w=2}^{10^{-5}n} \\sum_{i=2w}^{0.003n}\\bfrac{C_2i}{n}^{i\/2}\\\\\n\\noalign{where $C_1=16^25^3e^4\/3^5,C_2=16\\cdot 5^3 e^3\/3^4$,}\n&\\leq \\sum_{i=2}^{n^{1\/4}}i\\brac{\\bfrac{C_1}{n^{3\/4}}^{i\/2}+\\bfrac{C_2}{n^{3\/4}}^{i\/2}} +\\sum_{i=n^{1\/4}}^{2\\cdot 10^{-5}n}i\\bfrac{2C_1}{10^5}^{i\/2}+\\sum_{i=n^{1\/4}}^{0.003n}i\\bfrac{6C_2}{10^3}^{i\/2}\\\\\n&=O(n^{-3\/4}).\n\\end{align*} \nFinally, since $G$ has an even number of vertices, for $W=\\emptyset$ we have $|W|=q(V\\setminus W)=0$.\n\\qed\n\\subsection{Proof of Lemma \\ref{maximal}:}\nLet $W$ be a set satisfying $q(V \\setminus W)>|W|$ of maximum size and assume $w=|W|\\geq 10^{-5}n$.\n\\vspace{3mm}\n\\\\\n{\\textbf{Claim 2}} No component induced by $V\\setminus W$ is a tree with more than one vertex.\n\\vspace{3mm}\n\\\\ \\textbf{Proof of Claim 2:} Indeed assume that $C_i$ is such a component. If $|C_i|$ is even then\nlet $v$ be a leaf of $C_i$ and define $W'=W \\cup \\{v\\}$. Then $C_i \\setminus \\{v\\}$ is an odd component in $V \\setminus W'$ and $q(V \\setminus W')=q(V \\setminus W)+1>|W|+1=|W'|$ contradicting the maximality of $W$.\n\nThus assume that $|C_i|$ is odd. Let $L_1$ be the set of leaves of $C_i$ and $L_2$ be the neighbors of $L_1$. Set $W'=W \\cup |L_1|$. Then $|L_1| \\geq |L_2|$. Furthermore every vertex in $L_1$ is an odd component in $V \\setminus W'$ and in the case $|L_1|=|L_2|$ then $C_i \\setminus (L_1\\cup L_2)$ is also an odd component in $V \\setminus W'$. Therefore,\n\\begin{align*}\nq(V\/W') &=q(V\/W) -1 +|L_1|+\\mathbb{I}(|L_1|=|L_2|) \n\\\\&\\geq q(V\/W) +|L_2|  +|L_1|-|L_2| +\\mathbb{I}(|L_1|=|L_2|)-1\n\\\\&> |W|+|L_2|=|W'|,\n\\end{align*}  \ncontradicting the maximality of $W$. \\qed\n\\vspace{3mm}\n\\\\\nWe partition $V \\setminus W$ into three sets, $W_1,W_2$ and $W_3$, as follows. With the underlying graph being the one spanned by $V \\setminus W$, $W_1$ consists of the isolated vertices in $V \\setminus W$, $W_2$ consists of the vertices spanned by components that contain a cycle and have  size $s\\leq \\frac{1}{10}\\log n$ and $W_3$ consists of the vertices that are spanned by a component of size at least $\\frac1{10}\\log n$. Finally let $W_4=W_2 \\cup W_3$. To lower bound $W_1$ we use the following  claim.\n\n\\textbf{{{Claim 3:}}} W.h.p.\\@  $W_4$ spans at most $\\frac{11w}{\\log n}$ components in $V\\setminus W$.\n\n\\textbf{Proof of Claim 3:} First observe that the number of components spanned by $W_2$ is smaller than the number of cycles of size at most $\\frac1{10} \\log n$ in $G$, which we denote by $r$. \n\\begin{align*}\n\\mathbf{Pr}(r\\geq n^{0.3}) &\\leq n^{-0.3} \\sum_{q=1}^{0.1 \\log n} \\binom{n}{q} 4^q q! \\frac{\\phi(2q) \\phi(2m-2q)}{\\phi(2m)}\\\\\n& \\leq_O  n^{-0.3} \\sum_{q=1}^{0.1 \\log n} \\bfrac{en}{q}^q 4^q \\bfrac{2q}{e}^q \\bfrac{e}{2m}^q\n\\\\& \\leq_O n^{-0.3} \\sum_{q=1}^{0.1 \\log n} \\bfrac{8e}{3}^q \n\\leq_O   n^{-0.3} (\\log n) 8^{0.1 \\log n}=o(1). \n\\end{align*}\nHence w.h.p.\\@ $W_2$ spans at most $n^{0.3}$ components. Moreover every component spanned by $W_3$ has size at least $\\frac{1}{10}\\log n$. Therefore $W_4$ spans  at most $n^{0.3}+\\frac{10w}{\\log n}= \\frac{(1+o(1))10w}{\\log n}$ components in $V\\setminus W$.\\qed\n\nSince $W_4$ spans at most $u=\\frac{11w}{\\log n}$ components in $V\\setminus W$ and no component is a tree it follows that the rest of the components consist of at least $q(V\\setminus W) -u > w-u$  isolated vertices that lie in $W_1$.\n\nFor convenience, we move $|W_1|-(w-u)$ vertices from $W_1$ to $W_4$. Therefore  $|W_1|= w-u$.\nLet $k_1$ be the number of vertices of degree 4 in $W_1$ and $d=d(W)-d(W_1)$.\nThen $0\\leq d\\leq 4w-(3(w-u)+k_1)=w+3u-k_1$. For fixed $10^{-5}n\\leq w\\leq 0.5n$ the probability that there exist such sets $W,W_1,W_4$ is bounded by \n\n\\begin{align}\np_2&\\leq \\sum_{k_1=0}^{w-u} \\sum_{d=0}^{w+3u-k_1}\n\\binom{n}{2w} \\binom{2w}{w}\\binom{w}{u} \\binom{4w}{d} \\mathbf{Pr}(d(W)-d(W_1)=d) \\label{2}\\\\ \n&\\times  (3(w-u)+k_1)! \\times \\frac{ [2m- [6(w-u)+2k_1]]!}{2^{m-[3(w-u)+k_1}[m-[3(w-u)+k_1]]!}  \\times \\frac{2^m m!}{(2m)!}.\\label{3}\n\\end{align}\n\\textbf{Explanation:} We first choose the sets $W,W_1$  and $W_4$ of size $w,w-u$ and $n-2w+u$ respectively. This can be done in $\\binom{n}{2w} \\binom{2w}{w}\\binom{w}{u} \\binom{n-2w+u}{u}^{-1} $ ways, but we ignore the final factor. \n\nFrom the at most $4w$ copies of vertices in $W$ we choose a set $W''\\subset W$ be of size $d$. \nWe let $W'=W\/W''$. These are the copies of vertices that will be matched with those in $W_1$.  \n\nIn the calculations that follow we let $a=w\/n\\geq 10^{-5}$. We also let $k_4$  be the number of vertices of degree 4 that lie in $W_4$. We first bound the binomial coefficients, found in the first line.\n\\begin{align}\n  &\\binom{n}{2w} \\binom{2w}{w}\\binom{w}{u} \\binom{4w}{d} \n= \\binom{n}{2an} \\binom{2an}{an}\\binom{an}{u} \\binom{4an}{d} \\nonumber\\\\\n&\\leq 2^{o(n)} \\bfrac{1}{2a}^{2an} \\bfrac{1}{1-2a}^{(1-2a)n} 2^{2an} \\bfrac{4ean}{d}^{d} \\nonumber\\\\\n& =2^{o(n)}\\bfrac{1}{a}^{2an} \\bfrac{1}{1-2a}^{(1-2a)n} \\bfrac{4ean}{d}^{d}.\\label{f1}\n\\end{align}\nFor the second line we used that $u\\leq u_0$ which implies that $\\binom{an}{u}=2^{o(n)}$. Observe that \n\\begin{align}\\label{2m2}\n2m=6(w-u)+2k_1+d+3(n-2w+u)+k_4=3n+d+2k_1+k_4-3u.\n\\end{align}\nLet $m_0=d+2k_1+k_4-3u$. For the terms in line \\eqref{3} we have\n\\begin{align*}\n\\frac{(2m)!}{2^m m!}&=\\frac{(3n)!}{2^{1.5n}(1.5n)!}   \n\\frac{\\prod_{i=1}^{m_0}(3n+i)}{2^{m_0\/2} \\prod_{i=1}^{m_0\/2}(1.5n+i)}\n\\geq_O\\bfrac{3n}{e}^{1.5n} \\prod_{i=1}^{m_0\/2}[3n+(2i-1)].\\\\\n& \\geq \\bfrac{3n}{e}^{1.5n} e^{-o(n)}(3n)^{-3u\/2} \\prod_{i=1}^{d\/2+k_1+k_4\/2}[3n+(2i-1)]\n\\end{align*}\nEquation \\eqref{2m2} implies that \n$$2m-[6(w-u)+2k_1]=3(1-2a)n+3u+k_4+d.$$ \nThus,  \n\\begin{align*}\n& \\frac{ [2m- [6(w-u)+2k_1]]!}{2^{m-[3(w-u)+2k_1]}[m-[3(w-u)+k_1]]!}=\\frac{ [3(1-2a)n]!} {2^{1.5(1-2a)n}[1.5(1-2a)n]!} \\cdot \\frac{\\prod_{i=1}^{d}3(1-2a)n+i}{2^{d\/2}\\prod_{j=1}^{\\frac{d}{2}} 1.5(1-2a)n+j} \n\\\\&\\hspace{5mm} \\times \\frac{\\prod_{i=1}^{k_4}3(1-2a)n+d+i}{2^{k_4\/2}\\prod_{j=1}^{k_4\/2} 1.5(1-2a)n+d\/2   +j } \\cdot \\frac{\\prod_{i=1}^{3u}3(1-2a)n+d+k_4+i}{2^{3u\/2}\\prod_{j=1}^{\\frac{3u}{2}} 1.5(1-2a)n+d\/2+k_4\/2 +j}\n\\\\ &\\hspace{5mm}\\leq_O  \\bfrac{3(1-2a)n}{e}^{1.5(1-2a)n} \\prod_{i=1}^{d\/2} [3(1-2a)n+(2i-1)]\n \\prod_{j=1}^{k_4\/2} [3(1-2a)n+d+(2j-1) ] \\cdot (2m)^{3u\/2}\n\\\\ &\\hspace{5mm} \\leq_O  \\bfrac{3(1-2a)n}{e}^{1.5(1-2a)n} [3(1-2a)n+an\/2]^{d\/2} (2m)^{3u\/2} \\prod_{j=1}^{k_4\/2} [3(1-2a)n+d+(2j-1) ]\n\\end{align*}\nFor the last inequality we used the Arithmetic Mean-Geometric Mean inequality and the fact that $d\/2\\leq an\/2+ o(n)$, which follows from $d\\leq w+3u-k_1$. \n\nFor the first term of \\eqref{3} we have\n\\begin{align*}\n[3(w-u)+k_1]! \\leq \\frac{3w!}{ (3(w-u))^{3u}} \\prod_{i=1}^{k_1}(3(w-u)+i)\\leq \\bfrac{3an}{e}^{3an} \\frac{2^{o(n)}}{n^{3u}} \\prod_{i=1}^{k_1}(3(w-u)+i).\n\\end{align*}\nThus the expression in \\eqref{3} is bounded by\n\\begin{align}\n&2^{o(n)} \\bfrac{3an}{e}^{3an} \\frac{1}{n^{3u}} \\prod_{i=1}^{k_1}(3(w-u)+i) \\nonumber\\\\ \n&\\times \\bfrac{3(1-2a)n}{e}^{1.5(1-2a)n} [3(1-2a)n+an\/2]^{d\/2} (2m)^{3u\/2} \\prod_{j=1}^{k_4\/2} [3(1-2a)n+d+2j-1 ]\\nonumber\\\\ \n&\\times \\bigg[ \\bfrac{3n}{e}^{1.5n} (3n)^{-3u\/2} \\prod_{i=1}^{d\/2+k_1+k_4\/2}[3n+(2i-1)] \\bigg]^{-1}\\nonumber\\\\\n&=2^{o(n)}  a^{3an}  [(1-2a)n]^{1.5(1-2a)n} \\bfrac{6m}{n}^{3u\/2}    \\prod_{i=1}^{d\/2}\\frac{ 3(1-2a)n+an\/2}{3n+(2i-1)}        \\nonumber\\\\\n&\\times \n\\prod_{i=1}^{k_1}\\frac{ 3(w-u)+i}{3n+d+(2i-1)} \n\\prod_{i=1}^{k_4\/2} \\frac{ 3(1-2a)n+d+2i-1}{3n+d+2k_1+2i-1}\\nonumber\\\\\n&\\leq_O 2^{o(n)} a^{3an}  [(1-2a)n]^{1.5(1-2a)n}\n\\prod_{i=1}^{d\/2}\\frac{ 3(1-2a)n+an\/2}{3n} \n\\prod_{i=1}^{k_1}\\frac{ 3(w-u)+i}{3n+d+(2i-1)} \\prod_{i=1}^{k_4\/2} 1 \\nonumber\\\\ \n&\\leq_O 2^{o(n)} a^{3an}  [(1-2a)n]^{1.5(1-2a)n}[(1-2a)+a\/6]^{d\/2} 2^{-k_1} \\label{f2\n\\end{align}\nFinally we consider the term $\\mathbf{Pr}(d(W)-d(W_1)=d) $ and assume that $h$ vertices of degree 4 were chosen to be included in $W\\cup W_1$, so that $d=h+3u-2k_1$. Then, because there are $\\binom{h}{k_1}\\binom{2w-u-h}{(w-u)-k_1}$ out of $\\binom{2w}{w-u}$ ways to  distribute the $k_1$ vertices of degree 4,\n\\begin{align*}\np_3&=\\mathbf{Pr}(d(W)-d(W_1)=d) =\\binom{h}{k_1}\\binom{2w-u-h}{(w-u)-k_1} \\bigg\/ \\binom{2w-u}{w-u}\\\\\n&\\leq \\binom{h}{k_1}\\binom{2w-u-h}{w-u}\\bigg\/\\binom{2w-u}{w-u} \\leq \\binom{h}{k_1} \\prod_{i=0}^{h-1} \\frac{w-i}{2w-i}\\\\\n&\\leq  2^{hH(k_1\/h) -h}=2^{k_1}2^{-k_1+h\\cdot H(k_1\/h)-h}.\n\\end{align*}\nHere $H(x)=-x\\log_2( x) -(1-x) \\log_2(1-x)$ is the entropy function. For fixed $d$ we have $h=d+2k_1+o(n)$. Thus \n$$p_3 \\leq 2^{o(n)+k_1+df(k_1\/d)},\\text{ where }f(x)= -x + (1+2x) H\\brac{\\frac{x}{1+2x}}-(1+2x).$$ \n$f(x)$ has a unique maximum at $x^*$, the solution to $8x(1+x)=(1+2x)^2$ and $f(x^*) \\leq -0.771$. Hence \n\\beq{f3}{\np_3\\leq 2^{-0.771d+k_1+o(n)}.\n} \nMultiplying the bounds in \\eqref{f1}, \\eqref{f2}, \\eqref{f3} together we have a bound\n\\begin{align*}\np_2 & \\leq 2^{o(n)-0.771d+k_1} \\bfrac{1}{a}^{2an} \\bfrac{1}{1-2a}^{(1-2a)n}  \\bfrac{4ean}{d}^{d} \n\\\\ &\\times a^{3an} (1-2a)^{1.5(1-2a)n} \\bigg(1-2a+ \\frac{a}{6}\\bigg)^{d\/2} 2^{-k_1}\n\\\\ & = 2^{o(n)}  \\bfrac{2^{1.229}ean}{d}^{d}  a^{an} (1-2a)^{0.5(1-2a)n} \\bigg(1-\\frac{11a}{6}\\bigg)^{d\/2}\n\\end{align*}\nThus $p_2=o(1)$ when $d=o(n)$. Let $d=ban$ for some $0<b\\leq 1$. Then,\n\\begin{align*}\np_2 & \\leq  \\bigg\\{ 2^{o(1)} \\bigg[ \\frac{2^{1.229}e}{b}\\bigg(1-\\frac{11a}{6} \\bigg)^{0.5} \\bigg]^b  a (1-2a)^{0.5(1-2a)\/a}  \\bigg\\}^{an}\n\\end{align*}\nLet $g(a)= {2^{1.229}e}\\bigg(1-\\frac{11a}{6} \\bigg)^{0.5}$. When $g(a)<e$ then $\\bfrac{g(a)}{b}^b$ is maximized when $b=\\frac{g(a)}{e}$ which yields\n$$ p_2 \\leq  \\bigg\\{ 2^{o(1)} \\ e^{ 2^{1.229} (1-\\frac{11a}{6})^{0.5}}  a (1-2a)^{0.5(1-2a)\/a}  \\bigg\\}^{an} \n\\leq\\bfrac{99}{100}^{an}.$$\nThe last inequality is most easily verified numerically.\n\nWhen $g(a)>e$ then $\\bfrac{g(a)}{b}^b$ is maximized at $b=1$. Hence\n$$ p_2 \\leq  \\bigg\\{ 2^{o(1)} \\  2^{1.229} e \\brac{1-\\frac{11a}{6} }^{0.5}   a (1-2a)^{0.5(1-2a)\/a}  \\bigg\\}^{an} \\leq \\bfrac{19}{20}^{an}.$$\nThe last inequality is most easily verified numerically. Thus the probability that there exists a set $W$ satisfying $q(V \\setminus W)>|W|$ of size $w=|W|\\leq 10^{-5}n$\nis bounded by \n\\begin{align*}\n\\sum_{w=10^{-5}n}^{0.5n}  \\bfrac{99}{100}^w =o(1). \n\\end{align*}\nThis only leaves the case of $n$ odd. The reader will notice that in none of the calculations above, did we use the fact that $n$ was even. The Tutte-Berge formula for the maximum size of a matching $\\nu(G)$ is\n$$\\nu(G)=\\min_{W\\subseteq V}\\frac12(|V|+|W|-q(V\\setminus W)).$$\nWe have shown that the above expression is at least $|V|\/2$ for $W\\neq\\emptyset$ and so the case of $n$ odd is handled by putting $W=\\emptyset$ and $q(W)=1$.\n\\section{Conclusions and open questions}\nThe paper of Karp and Sipser \\cite{KS} has been a springboard for research on matching algorithms in random graphs. Algorithm 1 of that paper has not been the subject of a great deal of analysis, mainly because of the way it disturbs the degree sequences of the graphs that it produces along the way. In this paper we have shown that if the original graph has small maximum degree then the maximum degree is controllable and the great efficiency of the algorithm can be verified.\n\nIt is natural to try to extend the analysis to random regular graphs with degree more than four and we are in the process of trying to overcome some technical problems. It would also be of interest to analyse the algorithm on $G_{n,p}$, as originally intended.\n\n\\section{Diagrams of Hyperactions of interest}\n{\\bf Type 2.}\n\n\\begin{center}\n\\pic{\n\\node at (0,0.2) {$w$};\n\\node at (1,0.2) {$u$};\n\\node at (2,0.2) {$v$};\n\\node at (3,1.2) {$x$};\n\\node at (3,0.2) {$y$};\n\\node at (3,-.8) {$z$};\n\\node at (-1,1.2) {$a$};\n\\node at (-1,-0.8) {$b$};\n\\draw (1,0) -- (2,0);\n\\draw (2,0) -- (3,1);\n\\draw (2,0) -- (3,-1);\n\\draw (2,0) -- (3,0);\n\\draw (0,0) -- (-1,1);\n\\draw (0,0) -- (-1,-1);\n\\draw (0,0) to [out=45,in=135] (1,0);\n\\draw (0,0) to [out=-45,in=-135] (1,0);\n\\draw [fill=black] (0,0) circle [radius=.05];\n\\draw [fill=black] (1,0) circle [radius=.05];\n\\draw [fill=black] (2,0) circle [radius=.05];\n\\draw [fill=black] (3,0) circle [radius=.05];\n\\draw [fill=black] (3,1) circle [radius=.05];\n\\draw [fill=black] (3,-1) circle [radius=.05];\n\\draw [fill=black] (-1,1) circle [radius=.05];\n\\draw [fill=black] (-1,-1) circle [radius=.05];\n\\draw [->] [ultra thick] (4,0) -- (5,0);\n\\draw (7,0) ellipse (.6 and .3);\n\\node at (7.05,0) {$wuv$};\n\\draw (7.6,0) -- (8.6,1);\n\\draw (7.6,0) -- (8.6,-1);\n\\draw (7.6,0) -- (8.6,0);\n\\draw (5.6,1) -- (6.5,0);\n\\draw (5.6,-1) -- (6.5,0);\n\\draw [fill=black] (8.6,1) circle [radius=.05];\n\\draw [fill=black] (8.6,-1) circle [radius=.05];\n\\draw [fill=black] (8.6,0) circle [radius=.05];\n\\draw [fill=black] (5.6,-1) circle [radius=.05];\n\\draw [fill=black] (5.6,1) circle [radius=.05];\n\\node at (5.6,1.2) {$a$};\n\\node at (5.6,-0.8) {$b$};\n\\node at (8.6,1.2) {$x$};\n\\node at (8.6,0.2) {$y$};\n\\node at (8.6,-.8) {$z$};\n\\node at (8.6,0.2) {$y$};\n\\node at (8.6,-.8) {$z$};\n}\n\\end{center}\n\n{\\bf Type 3.}\n\n\\begin{center}\n\\pic{\n\\node at (1,0.2) {$u$};\n\\node at (2,0.2) {$v$};\n\\node at (3,1.2) {$x$};\n\\node at (3,0.2) {$y$};\n\\node at (3,-0.8) {$z$};\n\\node at (0,1.2) {$a$};\n\\node at (0,-.8) {$b$};\n\\node at (-1,2.2) {$c$};\n\\node at (-1,1.2) {$d$};\n\\node at (-1,-.8) {$e$};\n\\node at (-1,-1.8) {$f$};\n\\draw (1,0) -- (2,0);\n\\draw (2,0) -- (3,1);\n\\draw (2,0) -- (3,-1);\n\\draw (2,0) -- (3,0);\n\\draw (0,1) -- (1,0);\n\\draw (0,1) -- (-1,2);\n\\draw (0,1) -- (-1,1);\n\\draw (0,-1) -- (-1,-1);\n\\draw (0,-1) -- (-1,-2);\n\\draw (0,-1) -- (1,0);\n\\draw [fill=black] (-1,2) circle [radius=.05];\n\\draw [fill=black] (-1,1) circle [radius=.05];\n\\draw [fill=black] (-1,-2) circle [radius=.05];\n\\draw [fill=black] (-1,-1) circle [radius=.05];\n\\draw [fill=black] (0,1) circle [radius=.05];\n\\draw [fill=black] (0,-1) circle [radius=.05];\n\\draw [fill=black] (1,0) circle [radius=.05];\n\\draw [fill=black] (2,0) circle [radius=.05];\n\\draw [fill=black] (3,0) circle [radius=.05];\n\\draw [fill=black] (3,1) circle [radius=.05];\n\\draw [fill=black] (3,-1) circle [radius=.05];\n\\draw [->] [ultra thick] (4,0) -- (5,0);\n\\draw (7.5,0) ellipse (.6 and .3);\n\\node at (7.55,0) {$avb$};\n\\draw (9.1,0) -- (10.1,1);\n\\draw (9.1,0) -- (10.1,-1);\n\\draw (9.1,0) -- (10.1,0);\n\\draw [fill=black] (9.1,0) circle [radius=.05];\n\\draw [fill=black] (10.1,1) circle [radius=.05];\n\\draw [fill=black] (10.1,-1) circle [radius=.05];\n\\draw [fill=black] (10.1,0) circle [radius=.05];\n\\node at (9.1,0.2) {$u$};\n\\node at (10.1,1.2) {$x$};\n\\node at (10.1,0.2) {$y$};\n\\node at (10.1,-.8) {$z$};\n\\node at (6,2.2) {$c$};\n\\node at (6,1.2) {$d$};\n\\node at (6,-.8) {$e$};\n\\node at (6,-1.8) {$f$};\n\\draw [fill=black] (6,2) circle [radius=.05];\n\\draw [fill=black] (6,1) circle [radius=.05];\n\\draw [fill=black] (6,-1) circle [radius=.05];\n\\draw [fill=black] (6,-2) circle [radius=.05];\n\\draw (6,2) -- (7,0);\n\\draw (6,1) -- (7,0);\n\\draw (6,-2) -- (7,0);\n\\draw (6,-1) -- (7,0);\n}\n\\end{center}\nWe allow the edge $\\set{a,b}$ to be a single edge in this construction. This gives us a Type 3b hyperaction.\n\n{\\bf Type 4.}\n\n\\begin{center}\n\\pic{\n\\node at (0,0.2) {$v$};\n\\draw [fill=black] (0,0) circle [radius=.05];\n\\node at (-1,1.2) {$a$};\n\\node at (-1,0.2) {$b$};\n\\node at (-1,-0.8) {$c$};\n\\draw [fill=black] (-1,1) circle [radius=.05];\n\\draw [fill=black] (-1,0) circle [radius=.05];\n\\draw [fill=black] (-1,-1) circle [radius=.05];\n\\draw (-1,1) -- (0,0);\n\\draw (-1,0) -- (0,0);\n\\draw (-1,-1) -- (0,0);\n\\draw [fill=black] (1,0) circle [radius=.05];\n\\node at (1,0.2) {$u$};\n\\draw (0,0) -- (1,0);\n\\node at (2,1.2) {$x_1$};\n\\node at (2,-1.2) {$x_2$};\n\\draw [fill=black] (2,1) circle [radius=.05];\n\\draw [fill=black] (2,-1) circle [radius=.05];\n\\draw (1,0) -- (2,1);\n\\draw (1,0) -- (2,-1);\n\\draw (2,-1) -- (2,1);\n\\node at (3,1.2) {$w_1$};\n\\node at (3,-1.2) {$w_2$};\n\\draw [fill=black] (3,1) circle [radius=.05];\n\\draw [fill=black] (3,-1) circle [radius=.05];\n\\draw (2,1) -- (3,1);\n\\draw (2,-1) -- (3,-1);\n\\node at (4,2.2) {$p$};\n\\node at (4,1.2) {$q$};\n\\node at (4,-0.8) {$r$};\n\\node at (4,-1.8) {$s$};\n\\draw [fill=black] (4,2) circle [radius=.05];\n\\draw [fill=black] (4,1) circle [radius=.05];\n\\draw [fill=black] (4,-1) circle [radius=.05];\n\\draw [fill=black] (4,-2) circle [radius=.05];\n\\draw (3,1) -- (4,2);\n\\draw (3,1) -- (4,1);\n\\draw (3,-1) -- (4,-2);\n\\draw (3,-1) -- (4,-1);\n\\draw [->] [ultra thick] (5,0) -- (6,0);\n\\node at (8,0.2) {$v$};\n\\draw [fill=black] (0,0) circle [radius=.05];\n\\node at (7,1.2) {$a$};\n\\node at (7,0.2) {$b$};\n\\node at (7,-0.8) {$c$};\n\\draw [fill=black] (7,1) circle [radius=.05];\n\\draw [fill=black] (7,0) circle [radius=.05];\n\\draw [fill=black] (7,-1) circle [radius=.05];\n\\draw [fill=black] (8,0) circle [radius=.05];\n\\draw (8,0) -- (7,1);\n\\draw (8,0) -- (7,0);\n\\draw (8,0) -- (7,-1);\n\\draw (10,0) ellipse (1.4 and .3);\n\\node at (10,0) {$u,x_1,x_2,w_1,w_2$};\n\\node at (13,2.2) {$p$};\n\\node at (13,1.2) {$q$};\n\\node at (13,-0.8) {$r$};\n\\node at (13,-1.8) {$s$};\n\\draw [fill=black] (13,2) circle [radius=.05];\n\\draw [fill=black] (13,1) circle [radius=.05];\n\\draw [fill=black] (13,-1) circle [radius=.05];\n\\draw [fill=black] (13,-2) circle [radius=.05];\n\\draw (11.3,0) -- (13,2);\n\\draw (11.3,0) -- (13,1);\n\\draw (11.3,0) -- (13,-1);\n\\draw (11.3,0) -- (13,-2);\n}\n\\end{center}\n\n{\\bf Type 5}.\n\n\\begin{center}\n\\pic{\n\\draw [fill=black] (0,1) circle [radius=.05];\n\\draw [fill=black] (0,-1) circle [radius=.05];\n\\node at (0,1.2) {$a$};\n\\node at (0,-.8) {$b$};\n\\draw [fill=black] (1,0) circle [radius=.05];\n\\node at (1,.2) {$z$};\n\\draw (0,1) -- (1,0);\n\\draw (0,-1) -- (1,0);\n\\draw [fill=black] (2,0) circle [radius=.05];\n\\draw (1,0) -- (2,0);\n\\node at (2,.2) {$u$};\n\\draw [fill=black] (3,0) circle [radius=.05];\n\\draw (3,0) -- (2,0);\n\\node at (3,-.2) {$v$};\n\\draw [fill=black] (3,1) circle [radius=.05];\n\\node at (3,1.2) {$x_1$};\n\\draw (3,1) -- (2,0);\n\\draw (3,1) -- (3,0);\n\\draw [fill=black] (4,0) circle [radius=.05];\n\\node at (4,.2) {$x_2$};\n\\draw (3,0) -- (4,0);\n\\draw [fill=black] (5,0) circle [radius=.05];\n\\draw [fill=black] (5,1) circle [radius=.05];\n\\draw [fill=black] (5,-1) circle [radius=.05];\n\\draw (4,0) -- (5,0);\n\\draw (3,1) -- (5,1);\n\\draw (4,0) -- (5,-1);\n\\node at (5,1.2) {$p$};\n\\node at (5,0.2) {$q$};\n\\node at (5,-.8) {$r$};\n\\draw [->] [ultra thick] (6,0) -- (7,0);\n\\draw (10,0) ellipse (1.4 and .3);\n\\node at (10,0) {$z,u,v,x_1,x_2$};\n\\draw [fill=black] (8,1) circle [radius=.05];\n\\draw [fill=black] (8,-1) circle [radius=.05];\n\\node at (8,-0.8) {$b$};\n\\node at (8,1.2) {$a$};\n\\draw [fill=black] (12.5,0) circle [radius=.05];\n\\draw [fill=black] (12.5,1) circle [radius=.05];\n\\draw [fill=black] (12.5,-1) circle [radius=.05];\n\\draw (11.4,0) -- (12.5,1);\n\\draw (11.4,0) -- (12.5,-1);\n\\draw (11.4,0) -- (12.5,0);\n\\node at (12.5,1.2) {$p$};\n\\node at (12.5,-0.8) {$r$};\n\\node at (12.5,0.2) {$q$};\n\\draw (8,1) -- (8.6,0);\n\\draw (8,-1) -- (8.6,0);\n}\n\\end{center}\n\n\n{\\bf Type 33.}\n\\begin{center}\n\\pic{\n\\draw [fill=black] (0,1) circle [radius=.05];\n\\draw [fill=black] (0,-1) circle [radius=.05];\n\\node at (0,1.2) {$u_1$};\n\\node at (0,-.8) {$u_2$};\n\\draw [fill=black] (1,0) circle [radius=.05];\n\\node at (1,.2) {$u$};\n\\node at (2,.2) {$v$};\n\\draw (0,1) -- (1,0);\n\\draw (0,-1) -- (1,0);\n\\node at (-1,2.2) {$a$};\n\\node at (-1,1.2) {$b$};\n\\node at (-1,-.8) {$c$};\n\\node at (-1,-1.8) {$d$};\n\\draw (1,0) -- (2,0);\n\\draw (2,0) -- (3,1);\n\\draw (2,0) -- (3,-1);\n\\draw (0,1) -- (1,0);\n\\draw (0,1) -- (-1,2);\n\\draw (0,1) -- (-1,1);\n\\draw (0,-1) -- (-1,-1);\n\\draw (0,-1) -- (-1,-2);\n\\draw (0,-1) -- (1,0);\n\\draw [fill=black] (-1,2) circle [radius=.05];\n\\draw [fill=black] (-1,1) circle [radius=.05];\n\\draw [fill=black] (-1,-2) circle [radius=.05];\n\\draw [fill=black] (-1,-1) circle [radius=.05];\n\\draw [fill=black] (0,1) circle [radius=.05];\n\\draw [fill=black] (0,-1) circle [radius=.05];\n\\draw [fill=black] (3,1) circle [radius=.05];\n\\draw [fill=black] (3,-1) circle [radius=.05];\n\\node at (3,-.8) {$v_2$};\n\\node at (3,1.2) {$v_2$};\n\\draw [fill=black] (4,1) circle [radius=.05];\n\\draw [fill=black] (4,-1) circle [radius=.05];\n\\node at (4,2.2) {$p$};\n\\node at (4,1.2) {$q$};\n\\node at (4,-.8) {$r$};\n\\node at (4,-1.8) {$s$};\n\\draw [fill=black] (4,2) circle [radius=.05];\n\\draw [fill=black] (4,-2) circle [radius=.05];\n\\draw (3,1) -- (4,1);\n\\draw (3,1) -- (4,2);\n\\draw (3,-1) -- (4,-1);\n\\draw (3,-1) -- (4,-2);\n\\draw [->] [ultra thick] (5,0) -- (6,0);\n\\draw (9,0) ellipse (1 and .2);\n\\draw (12,0) ellipse (1 and .2);\n\\node at (14,2.2) {$p$};\n\\node at (14,1.2) {$q$};\n\\node at (14,-.8) {$r$};\n\\node at (14,-1.8) {$s$};\n\\node at (7,2.2) {$a$};\n\\node at (7,1.2) {$b$};\n\\node at (7,-.8) {$c$};\n\\node at (7,-1.8) {$d$};\n\\draw [fill=black] (14,2) circle [radius=.05];\n\\draw [fill=black] (14,1) circle [radius=.05];\n\\draw [fill=black] (14,-2) circle [radius=.05];\n\\draw [fill=black] (14,-1) circle [radius=.05];\n\\draw [fill=black] (7,1) circle [radius=.05];\n\\draw [fill=black] (7,-1) circle [radius=.05];\n\\draw [fill=black] (7,2) circle [radius=.05];\n\\draw [fill=black] (7,-2) circle [radius=.05];\n\\draw (14,2) -- (13,0);\n\\draw (14,1) -- (13,0);\n\\draw (14,-1) -- (13,0);\n\\draw (14,-2) -- (13,0);\n\\draw (7,2) -- (8,0);\n\\draw (7,1) -- (8,0);\n\\draw (7,-2) -- (8,0);\n\\draw (7,-1) -- (8,0);\n\\node at (9,0) {$u,u_1,u_2$};\n\\node at (12,0) {$v,v_1,v_2$};\n}\n\\end{center}\n\n{\\bf Type 34.}\n\n\\hspace{1in}\n\\pic{\n\\draw [fill=black] (0,1) circle [radius=.05];\n\\draw [fill=black] (0,-1) circle [radius=.05];\n\\node at (0,1.2) {$v_1$};\n\\node at (0,-.8) {$v_2$};\n\\draw [fill=black] (1,0) circle [radius=.05];\n\\node at (1,.2) {$v$};\n\\node at (2,.2) {$u$};\n\\draw (0,1) -- (1,0);\n\\draw (0,-1) -- (1,0);\n\\node at (-1,2.2) {$a$};\n\\node at (-1,1.2) {$b$};\n\\node at (-1,-.8) {$c$};\n\\node at (-1,-1.8) {$d$};\n\\draw (1,0) -- (2,0);\n\\draw (2,0) -- (3,1);\n\\draw (2,0) -- (3,-1);\n\\draw (0,1) -- (1,0);\n\\draw (0,1) -- (-1,2);\n\\draw (0,1) -- (-1,1);\n\\draw (0,-1) -- (-1,-1);\n\\draw (0,-1) -- (-1,-2);\n\\draw (0,-1) -- (1,0);\n\\draw [fill=black] (-1,2) circle [radius=.05];\n\\draw [fill=black] (-1,1) circle [radius=.05];\n\\draw [fill=black] (-1,-2) circle [radius=.05];\n\\draw [fill=black] (-1,-1) circle [radius=.05];\n\\draw [fill=black] (0,1) circle [radius=.05];\n\\draw [fill=black] (0,-1) circle [radius=.05];\n\\draw [fill=black] (3,1) circle [radius=.05];\n\\draw [fill=black] (3,-1) circle [radius=.05];\n\\node at (3,-1.2) {$u_2$};\n\\node at (3,1.2) {$u_1$};\n\\draw (3,1) -- (3,-1);\n\\draw [fill=black] (4,1) circle [radius=.05];\n\\draw [fill=black] (4,-1) circle [radius=.05];\n\\node at (4,-1.2) {$w_2$};\n\\node at (4,1.2) {$w_1$};\n\\draw (3,1) -- (4,1);\n\\draw (3,-1) -- (4,-1);\n\\node at (5,2.2) {$p$};\n\\node at (5,1.2) {$q$};\n\\node at (5,-.8) {$r$};\n\\node at (5,-1.8) {$s$};\n\\draw [fill=black] (5,2) circle [radius=.05];\n\\draw [fill=black] (5,1) circle [radius=.05];\n\\draw [fill=black] (5,-2) circle [radius=.05];\n\\draw [fill=black] (5,-1) circle [radius=.05];\n\\draw (4,1) -- (5,1);\n\\draw (4,1) -- (5,2);\n\\draw (4,-1) -- (5,-1);\n\\draw (4,-1) -- (5,-2);\n\\draw [->] [ultra thick] (6,0) -- (7,0);\n}\n\n\\hspace{3in}\n\\pic{\n\\draw (10,0) ellipse (1 and .2);\n\\draw (13,0) ellipse (1.4 and .3);\n\\node at (10,0) {$v,v_1,v_2$};\n\\node at (13,0) {$u,u_1,u_2,w_1,w_2$};\n\\node at (8,2.2) {$a$};\n\\node at (8,1.2) {$b$};\n\\node at (8,-.8) {$c$};\n\\node at (8,-1.8) {$d$};\n\\draw [fill=black] (8,2) circle [radius=.05];\n\\draw [fill=black] (8,1) circle [radius=.05];\n\\draw [fill=black] (8,-2) circle [radius=.05];\n\\draw [fill=black] (8,-1) circle [radius=.05];\n\\draw (9,0) -- (8,1);\n\\draw (9,0) -- (8,2);\n\\draw (9,0) -- (8,-1);\n\\draw (9,0) -- (8,-2);\n\\node at (15.4,2.2) {$p$};\n\\node at (15.4,1.2) {$q$};\n\\node at (15.4,-.8) {$r$};\n\\node at (15.4,-1.8) {$s$};\n\\draw [fill=black] (15.4,2) circle [radius=.05];\n\\draw [fill=black] (15.4,1) circle [radius=.05];\n\\draw [fill=black] (15.4,-2) circle [radius=.05];\n\\draw [fill=black] (15.4,-1) circle [radius=.05];\n\\draw (14.4,0) -- (15.4,1);\n\\draw (14.4,0) -- (15.4,2);\n\\draw (14.4,0) -- (15.4,-1);\n\\draw (14.4,0) -- (15.4,-2);\n}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nIn the past years, {\\it Integral}--IBIS (Ubertini et al. 2003)\nand {\\it Swift}--BAT (Barthelmy et al. 2005) catalogs of hard X--ray selected \nextragalactic sources have been published (Sambruna et al. 2007; Bird et al. 2010), opening \na new window in the study of blazars.\n\nThe hard-X-ray selection produces a set of diverse sources, including extremely ``red'' \nflat spectrum radio quasars (FSRQs), with Inverse Compton (IC) peak at keV-MeV\nenergies, or ``blue'' BL Lacs with synchrotron peak at these frequencies,\nand provides a way to test the validity of the blazar sequence, \nlooking for outliers not predicted by the sequence itself (Giommi et al. 2007).\nIn particular, by selecting hard X--ray luminous FSRQ at high redshift, i.e. with the highest intrinsic \nbolometric luminosities, it is possible to collect samples of \nsupermassive black holes (SMBHs) in the early Universe, thus introducing important observational \nconstraint on the number density of \nheavy black holes at high redshift and hence on their formation processes and timing.\n\nIGR J22517+2217 was first reported by Krivonos et al.\n(2007) as an unidentified object detected by  {\\it Integral}--IBIS (150 ks of total IBIS exposure).\n{\\it Swift} follow-up observations were used to associate the source \nwith MG3 J225155+2217 (Bassani et al. 2007). MG3 J225155+2217 has been optically identified as a QSO by Falco et al. (1998) in a redshift survey of \n177 FSRQs, on the basis of  S$_{IV}$, Ly$\\alpha$, C$_{II}$ and C$_{IV}$ emission lines.\nIGR J22517+2217 is the highest redshift ($z=3.668$) blazar detected in the fourth {\\it Integral}--IBIS hard X--rays catalog (Bird et al. 2010).\nThe source has a {\\it Swift}--BAT counterpart (SWIFT J2252.0+2218) in the 3 years BAT survey (Ajello et al. 2009) and is present in the multifrequency ``Roma--BZCAT\" \ncatalog of blazars (Massaro et al. 2009).\n\nUsing XRT, IBIS and archival optical\/IR data, Bassani et al. (2007) constructed a non simultaneous SED \nof the source, showing an extremely bright X--ray emission with respect to the\noptical emission ($\\alpha_{OX}<0.75$), \nand suggested that IGR J22517+2217 could be a rare FSRQ with synchrotron peak at X--ray frequencies,\nor a more canonical FSRQ, i.e. with the synchrotron peak at radio--mm frequencies and IC peak at MeV-GeV energies, \nbut with an exceptionally strong Compton dominance.\n\nThis ``controversial\" blazar has been studied also by Maraschi et al. (2008). \nThey reanalyzed the existing {\\it Swift} (XRT and BAT) and {\\it Integral}--IBIS data, \nand propose a ''standard leptonic one-zone emission model'' (Ghisellini \\& Tavecchio 2009, see Sect. 3) with the peak of the synchrotron component at microwave\/radio frequencies, \nand a high luminosity external Compton (EC) component peaking in hard X-rays to reproduce the SED  of the source.\nThis model ruled out both a synchrotron and a synchrotron self--Compton (SSC) interpretation for the \nX--ray emission. \n\nGhisellini et al. (2010) included IGR J22517+2217 in their sample of 10 X--ray selected blazar at $z>2$:\nthe intent of the paper was to characterize the physical properties of these powerful sources, and to confirm\nthe capability of the hard X--ray selection in finding \nextreme blazars with massive SMBH, powerful jets and luminous accretion disks. \nIGR J22517+2217 is the highest redshift FSRQ in their sample and shows the highest total jet power (P$_{Jet}=1.5\\times10^{48}$ erg s$^{-1}$).\n\nAll these previous studies have been performed through the analysis of the ''average'' X-ray spectra obtained with the INTEGRAL and {\\it Swift}-BAT surveys, \nwithout taking into account any possible flux variation of the source during the period of monitoring (5 years for INTEGRAL and BAT).\nIn this paper we present the discovery of strong flaring activity, \nin X--ray (IBIS and BAT) archival data, of this extremely bright and peculiar\nFSRQ, and the modelization of both its flaring and quiescent SEDs.\nNew {\\it Suzaku} and {\\it Fermi} data are used for characterizing the  \nquiescent state.\nOur goal is to investigate the evolution \nof the SED and obtain information on the physical condition of the source in the two different states.\n\nThe paper is organized as follows. In \\S2 we report the multiwavelength data analysis of the instruments involved in the SED building. \nIn \\S3 we describe the model adopted to reproduce the broad band SED, while in \\S4 we discuss the SED fitting of both the flaring and quiescent state. \nThe summary of our results is presented in \\S5.\nThroughout the paper, a $\\Lambda-CDM$ cosmology with $H_0 = 71$ km s$^{-1}$ Mpc$^{-1}$, \n$\\Omega_\\Lambda = 0.73$, and $\\Omega_m = 0.27$ is adopted.\n\n\n\n\\section{Data reduction}\n\n\n\\subsection{ New {\\it Suzaku} data}\n\n{\\it Suzaku} observed the source on 2009 Nov 26 (ID 704060010, PI A. De Rosa), \nfor a net exposure of $\\sim$40 ks, with both the X--ray Imaging Spectrometer (XIS; Koyama et al. 2007) and the \nHard X--ray Detector (HXD; Takahashi et al. 2007).\nThe XIS instrument has 3 operating detectors at the time of the observation: the front illuminated XIS 0 and XIS 3, sensitive in the 0.5--10 keV band, and the \nback--illuminated XIS 1, extending the low energy range to 0.2 keV.\nThe HXD is instead composed of GSO scintillators and silicon PIN diodes.\nThe PIN detectors observe in the 12--60 keV energy band, while the GSO ones can observe up to 600 keV.\nData reduction and processing were performed using HEASOFT $v6.9$ and {\\it Suzaku ftools v16}.\nOnly XIS and PIN data have been used in this analysis, since the source is below the \nsensitivity limit of the GSO scintillators.\nThe cleaned event files produced by the data processing with the standard selection criteria were used.\n\nXIS source events were extracted from a region of radius 200 arcsec, and the \nbackground events from an annulus (external radius 400 arcsec) outside the source region.\nThe response matrix and effective area were calculated for each detector using {\\it Suzaku} \nftools tasks {\\it xisrmfgen} and {\\it xissimarfgen}.\nGiven that XIS 0 and XIS 3 have similar responses, their event files were summed.\nOnly data in the 0.5--8 keV band were considered, and the spectral counts were \nrebinned using at least 30 counts per bin, in order to allow the use of $\\chi^2$ statistic.\n\nPIN data were extracted from the HXD cleaned event files after standard screening. \nThe tuned background model supplied\nby the {\\it Suzaku} team (Fukazawa et al. 2009), was used for the ``Non X--ray Background\" (NXB) events. \nThe background light curve was corrected for the 10x oversampling rate. \nThe source spectra were corrected for deadtime using {\\it hxddtcor}. \nWe estimate the cosmic X--ray background (CXB) contribution to the\nPIN background using the model given in Gruber et al. (1999),\nwhich is folded with the PIN response to estimate the CXB rate.\n\nThe XIS and PIN data were fitted with a simple power-law, modified by neutral absorption at the source redshift, plus \ngalactic absorption, fixed to the value measured by Kalberla et al. (2005) at the source \ncoordinates\\footnote{http:\/\/heasarc.nasa.gov\/cgi-bin\/Tools\/w3nh\/w3nh.pl} ($N_{H, Gal.} = 5\\times10^{20} cm^{-2}$). \nThe best fit values are reported in Table 1.\nThe XIS and PIN data show a flat spectrum, with $\\Gamma=1.5\\pm0.1$ and $\\Gamma=1.5\\pm0.8$ \nrespectively, with the PIN power law normalization being $\\sim1.2$ times the XIS one, a known cross calibration issue\n\\footnote{http:\/\/heasarc.gsfs.nasa.gov\/docs\/suzaku\/analysis\/abc sec 5.4}.\nThe XIS data show some curvature below 1 keV, that can be reproduced either by an intrinsic \ncolumn density of $N_H=(1.5\\pm1.1)\\times10^{22}$ cm$^{-2}$ at the source redshift (compatible with the value found in Bassani et al. using XRT data)\nor with a broken power-law with break energy of $0.8\\pm0.1$ keV and $\\Delta\\Gamma=0.8$.\nBoth models give a comparable reduced $\\chi^2$ of 1.03 and 1.04, respectively, thus the \nquality of the data do not allow do disentangle the two possibilities.\nIn Sect. 4 we will attempt to disentangle these two different models using broadband data. \n\nThe hard X-ray flux obtained by fitting the PIN data is a factor of $\\sim$10 (4) lower that the one\nreported in literature from {\\it Integral}-IBIS ({\\it Swift}-BAT) survey data (Bassani et al. 2007; Maraschi et al. 2008; Ghisellini et al. 2010).\nThis suggests that the source was in a much less active state during the 2009 {\\it Suzaku} observation, compared to previous measurements.\nThis lead us to reanalyze the archival IBIS and BAT data in order to check for the presence of variability in the hard X-ray source flux.\nIn Figure 1 top panel we show the {\\it Suzaku} XIS (red, black) and PIN (green) unfolded spectra and model best fit found for IGR J22517+2217.  \nFor comparison also the IBIS spectrum (blue) is shown, to emphasize the different flux state.\n\n\n\\subsection{{\\it Integral}-IBIS and {\\it Swift}-BAT survey data}\n\n\\begin{figure}\n\\begin{center}\n\\psfig{figure=plot_xis_integral_def.ps,width=8cm,height=6.5cm}\\hspace{1cm}\\psfig{figure=igr22_lc.ps,width=8cm,height=7cm}\n\\caption{\n{\\it a) Top panel:} {\\it Suzaku} XIS (red, black) and PIN (green) unfolded spectrum and model of IGR J22517+2217. \nFor comparison the IBIS cat4 spectrum is shown (blue). \n{\\it b) Bottom panel:} Hard X--ray light curve of IGR J22517+2217. \nRed squares (black circles) represent BAT (IBIS) 15--55 keV flux. \nThe blue empty (green filled) triangles show the time of XRT\/UVOT ({\\it Suzaku}) observation. \nThe cyan solid lines represent the time intervals used in the BAT spectra extraction.\nThe orange dashed line represents the period of observation of {\\it Fermi}\/LAT.\n}\n\\end{center}\n\\label{fig:light}\n\\end{figure}\n\n\nIBIS data are taken from the Fourth IBIS\/ISGRI Soft $\\gamma$--ray Survey Catalog (Bird et al. 2010), \ncovering the period Feb 2003 -- Apr 2008.\nImages from the ISGRI detector (Lebrun et al. 2003) for each pointing have been generated\nusing scientific Analysis\nSoftware (Goldwurm et al. 2003) OSA version 7.0. \nFive primary energy bands (20--40, 30--60, 20--100, 17--30, and\n18--60 keV) were used to maximize the detection sensitivity\nfor sources with various energy spectra (for details see Bird et al. 2010).\n\nThe BAT analysis results presented in this paper were derived with all the available data \nduring the time interval 2005 Jan 19 -- 2010 Apr 14.\nThe 15--55 keV spectra and light curve were extracted following the recipes presented in Ajello \net al. (2008, 2009). \nThe spectra are constructed by weighted averaging of the source spectra extracted from short \nexposures (e.g., 300s) and are accurate to the mCrab level. \nThe reader is referred to Ajello et al. (2009) for more details. \n\nThe total IBIS and BAT spectra can be reproduced with a simple power law having $\\Gamma=1.6\\pm 0.6$ and \n$1.6\\pm0.5$, respectively;\nthe 15--55 keV flux is F$_{15-55 \\, {\\rm keV}}=(2.5\\pm0.9)\\times10^{-11}$ and  \nF$_{15-55 \\, {\\rm keV}}=(2.1\\pm0.8)\\times10^{-11}$ erg cm$^{-2}$ s$^{-1}$, respectively.\n\nThe bottom panel of Fig. 1 shows the BAT (red squares) and IBIS (black diamonds) historical light curve \nof IGR J22517+2217 from 2003 Dec 04 to 2010 Mar 23.\nThe 15--55 keV IBIS light curve was extracted from the ISDC Science Products Archive\\footnote{http:\/\/www.isdc.unige.ch\/heavens\\_webapp\/integral} \nadopting a nominal bin size of 100 days. \nWe converted the observed IBIS and BAT count rates into 15--55 keV observed flux, using the \nWebPimms HEASARC tool\\footnote{http:\/\/heasarc.nasa.gov\/Tools\/w3pimms.html}, \nassuming as underling model a power law with photon index $\\Gamma=1.6$, consistent with \nthe values observed in the BAT and IBIS spectra, and assuming a constant cross-calibration between IBIS and BAT, equal to one.\n\n\\noindent Fig.1 (bottom panel) shows that: \n\n-- the source displays quite strong long term variability in hard X--rays;\n\n-- a strong flare episode occurred around Jan 2005, and the source reached a 15--55 keV flux maximum of $(8\\pm2) \\times10^{-11}$ erg cm$^{-2}$ s$^{-1}$\n(a factor of 20 higher than the flux measured by {\\it Suzaku}-PIN in 2009); \n\n-- after the flare, the source faded into a quiescent state, reaching a flux that is at or below the detection limit of both BAT and IBIS instruments.\nAs can be seen, the IBIS light curve is completely dominated by the flare, \ni.e. the source flux is below the IBIS detection limit after \nMJD 53550 and the total spectrum extracted from the entire period can be considered representative of the flare.\n\nAs a result of its different pointing strategy, BAT has a much more regular and extended coverage of the source, \nand we were able to characterize the source in both states.\nWe extracted a BAT spectrum from the period around the flare of 2005 (2004 Dec 11--2005 Mar 21)\nand also one from the remaining quiescent period (2005 May 10--2009 Jun 18, \nsolid cyan lines in the bottom panel of Fig. 1). \n\n\nThe BAT flux relative to the flare state is the highest one measured in the hard X-ray energy range\n(F$_{15-55 \\, {\\rm keV}}=(3.7\\pm0.8)\\times10^{-11}$ erg cm$^{-2}$ s$^{-1}$).\nDuring this state the source is detected up to 200 keV, and the spectrum is characterized by photon index of 1.5$\\pm$0.5.\nIn the quiescent state the source is detected, with significance $\\sim3\\sigma$, only up to $\\sim75$ keV, and \nthe spectrum has a flux F$_{15-55 \\, {\\rm keV}}$ a factor of $\\sim15$ lower than the flaring one, while the photon index is 1.7$\\pm$1.1.\nConsidering the large uncertainties on $\\Gamma$ we can consider the spectra (in flaring and quiescent state) comparable (see\nTable 1 for IBIS and BAT spectral analysis results).\n\n\\begin{table*}\n\\begin{center}\n\\label{tab:xray}\n\\begin{tabular}{ccccccccccccccc}\n\\hline\\hline\\\\\n\\multicolumn{1}{c} {Date}&\n\\multicolumn{1}{c} {Inst.}&\n\\multicolumn{1}{c} {Exp.}&\n\\multicolumn{1}{c} {$N_{\\rm H}$}&\n\\multicolumn{1}{c} {$\\Gamma$}&\n\\multicolumn{1}{c} {F$_{2-10}$}&\n\\multicolumn{1}{c} {$\\log L_{2-10}$}&\n\\multicolumn{1}{c} {F$_{15-55}$}& \n\\multicolumn{1}{c} {$\\log L_{15-55}$}\\\\\n (1) & (2) &(3) & (4) & (5) & (6) & (7) & (8) & (9) \\\\\n\\hline\\\\  \n2003 Dec 04--2007 Nov 17    & IBIS       & 191 & -                & 1.6$\\pm$0.6  & -          & -    &25.1$\\pm$8.9 & 48.1 \\\\\n2004 Dec 11--2005 Mar 21    & BAT Flare  & 160 & -                & 1.5$\\pm$0.5  & -          & -    &36.9$\\pm$8.1 & 48.3 \\\\\n2005 May 10--2009 Jun 18    & BAT quiesc.&1610 & -                & 1.7$\\pm$1.1  & -          & -    &2.6$\\pm$1.6  & 47.3 \\\\\n2007 May 01                 & XRT        & 40 & $2.0\\pm1.5$       & 1.4$\\pm$0.1  & 2.4$\\pm$0.4& 47.1 & -           & -     \\\\\n2009 Nov 01                 & XIS        & 40 & $1.5\\pm1.1$       & 1.5$\\pm$0.1  & 1.2$\\pm$0.1& 46.7 & -           & -    \\\\\n2009 Nov 01                 & PIN        & 40 & -                 & 1.5$\\pm$0.8  & -          & -    &3.8$\\pm$1.8  &  47.4\\\\\n\\hline      \n\\hline                              \n\\end{tabular}\n\\end{center} \n\\caption{Best fit model parameters  for the different observations of IGR 22517+2217 analyzed in this work.\nThe broadband continuum is reproduced with a simple power-law, modified by intrinsic absorption at the source redshift, \nplus galactic absorption N$_{\\rm H, Gal}$ = 5$\\times$10$^{20}$ cm$^{-2}$.\nColumn: \n(1) Date;\n(2) Instrument; \n(3) Effective exposure (ks);\n(4) Column density (in $10^{22}$ cm$^{-2}$ units);\n(5) Photon index; \n(6) 2--10 keV observed flux (in $10^{-12}$ erg s$^{-1}$ cm$^{-2}$ units);       \n(7) Log 2--10 keV deabsorbed luminosity (erg s$^{-1}$);\n(8) 15--55 keV flux (in $10^{-12}$ erg s$^{-1}$ cm$^{-2}$ units);\n(9) Log 15--55 keV luminosity (erg s$^{-1}$).}\n\\end{table*}\n         \n\n\n\n\\subsection{{\\it Fermi}\/LAT data}\n\n{\\it Fermi} Large Area Telescope (LAT; Atwood et al. 2009) data were collected from Aug 2008 (MJD 54679) to Aug 2010 (MJD 55409). \nDuring this period, the {\\it Fermi}\/LAT instrument operated mostly in survey mode, \nscanning the entire $\\gamma$-ray sky every 3 hours.\nThe analysis was performed with the ScienceTools software package version {\\it v9r17p0}, which\nis available from the Fermi Science Support Center. \nOnly events having a high probability of being photons\n-- those in the ``diffuse class\" -- were used. \n\nThe energy range used was 100 MeV -- 100 GeV, and the maximum zenith angle value was $105^{\\circ}$.\nWe adopted the unbinned likelihood analysis, using the standard P6\\_V3\\_DIFFUSE response functions.\nThe diffuse isotropic background used is {\\it isotropic\\_iem\\_v02} and the galactic diffuse emission model used is \n{\\it gll\\_iem\\_v02}\\footnote{http:\/\/fermi.gsfc.nasa.gov\/ssc\/data\/access\/lat\/BackgroundModels.html}.\nWe considered a region of interest (RoI) of $15^{\\circ}$ from the IGR J22517+2217 position.\nAll sources from the 1FGL catalog (Abdo et al. 2010) within the RoI of the source were included in the fit, \nwith their photon indexes and the integral fluxes free to vary, \nplus a point source with power law spectrum at the IGR J22517+2217 position, having photon index fixed to 2.2 (a value typical of FSRQ in the GeV band).\nThe normalization of the background was also left free to vary.\nWe also repeated the analysis using the source list of the 2FGL catalog (Ackermann et al. 2011), obtaining consistent results.\n\nIGR J22517+2217 is located $\\sim6^{\\circ}$ from 3C 454.3, a bright, extremely variable source in \nthe $\\gamma$--ray sky (Ackermann et al. 2010). It contributes for more than 90\\% to the total counts in the RoI.\nWe tried to exclude the period of flaring activity of 3c 454.3 from the data set, in order to minimize the contamination by this source.\nThe results obtained for IGR J22517+2217, however, did not change significantly.\n\nIGR J22517+2217 is not detected in the 2 year observation, and the computed Test Statistic (TS, Mattox et al. 1996)\nis $TS \\simeq3.5$ in the full band.\nWe therefore calculated the 95\\% upper limits in 5 energy bands (i.e. 0.1--0.3, 0.3--1, 1--3, 3--10, 10--100 GeV), using the profile likelihood method.\nThe upper-limits were corrected for attenuation due to extragalactic background light (EBL) through $\\gamma-\\gamma$ interactions (Chen et al. 2004, Razzaque et al. 2009), \nalthough only the 10-100 GeV band was found to be affected by significant attenuation.\nAs a consequence of being close to 3C 454.3, the background around the position \nof IGR J22517+2217 is higher than in a typical extragalactic field, making it difficult to constrain more strongly the upper limits  at GeV energies.\nThe upper limits are plotted in Figure 2, with all other data discussed in this paper.\n\nGiven that {\\it Fermi} data were collected starting from Aug 2008, they completely fall\nin the quiescent period of the source, and so it is not surprising that IGR J22517+2217 was not detected.\nTherefore, they will be used to characterize the \nstate of the source in the quiescent SED only, while we do not have available $\\gamma$-ray data \nfor the flaring state SED.\n\n\n\\subsection{Archival Data}\n\nWe collected archival radio and optical data from NASA\/IPAC EXTRAGALACTIC DATABASE \\footnote{http:\/\/ned.ipac.caltech.edu\/} for the source.\nRadio data at 1.4 4.8 and 8.4 GHz comes from NRAO\/VLA.\nOptical data in J, H, and K bands are taken with the  UFTI instrument at UKIRT (Kuhn 2004).\nThe UV data are taken from Ghisellini et al. (2010), where they corrected the {\\it Swift}-UVOT observed magnitudes\nfor the absorption of neutral hydrogen in intervening Lyman$-\\alpha$ absorption systems. \nWe also reanalyzed the archival XRT spectrum (average of 4 contiguous observations), obtaining consistent results with those found in Bassani et al. (2007)\nusing the same set of data, and with the results reported in \\S2.1 for the XIS spectra.\nThe only difference is in the XRT observed 2-10 keV flux, that is a factor 2 higher than the flux measured by XIS.\n\n\\begin{figure*}\n\\begin{center}\n\\psfig{figure=sed_tot2.ps,width=16cm,height=12cm}\n\\hspace{0.3cm}\n\\vspace{-0.5cm}\n\\psfig{figure=zoom.ps,width=10cm,height=8cm}\n\\caption{\n{\\it a) Top panel:} \nSpectral energy distribution of IGR J22517+2217. \nGray circles and arrows represent archival radio\/optical\/UV data from NED. \nEmpty red triangles and magenta  \nsquares represent XIS 0 and XIS 3 data, \nempty green circles and orange pentagons represent PIN and BAT quiescent data respectively, \nwhile black arrows are {\\it Fermi} upper limits in 5 bands.\nFilled violet squares represent XRT data, filled cyan diamonds and blue pentagons represent \nIBIS and BAT flare data respectively.\nThe solid cyan and orange curves are the results of the modeling of the  quiescent \nand flaring states, respectively.\nWith gray lines we show the different components of the non--thermal emission:\nsynchrotron (dotted), synchrotron self--Compton (long dashed) and\nexternal Compton (dot--dashed).\nThe black dashed line corresponds to the thermal emission of the disk, the IR torus and the X--ray disk corona. \nThe model does not account for radio emission, produced from much larger regions of the jet.\n{\\it a) Bottom panel:} Zoom in the X--ray energy range for the two SEDs. Symbols as in top panel.\n}\n\\end{center}\n\\label{fig:sed2}\n\\end{figure*}\n\n\\section{The SED model}\n\nThe model adopted to fit the SED is a leptonic, one--zone synchrotron and inverse Compton model, \nfully discussed in Ghisellini \\& Tavecchio (2009).\nThe assumptions can be summarized as follows:\n\n\\begin{table*}\n\\label{tab:sed}\n\\begin{center}\n\\begin{tabular}{ccccccccc}\n\\hline\\hline\\\\\n\\multicolumn{1}{c} {State}&\n\\multicolumn{1}{c} {$R_{\\rm diss}$}&\n\\multicolumn{1}{c} {$P'_{\\rm inj}$}&\n\\multicolumn{1}{c} {$B$}&\n\\multicolumn{1}{c} {$\\Gamma$}&\n\\multicolumn{1}{c} {$\\gamma_{\\rm b}$}&\n\\multicolumn{1}{c} {$\\gamma_{\\rm max}$}&\n\\multicolumn{1}{c} {$s_1$}&\n\\multicolumn{1}{c} {$s_2$}\\\\\n (1) & (2) &(3) & (4) & (5) & (6) & (7) & (8) & (9)  \\\\\n\\hline\\\\ \nLow  & 570 (1900)  &   0.045   & 1.06  & 16 & 70 & 2e3 & -1  & 4  \\\\\nHigh & 990 (3300)  &   0.30    & 0.61  & 16* & 70* & 2e3* & -1*  & 4*  \\\\\n\\hline                                    \n\\end{tabular}\\end{center}     \n\\caption{Input parameters of the SED fitting for the low and high state of IGR J22517+2217.\nColumn: \n(1) State ; \n(2) dissipation radius in units of $10^{15}$ cm and, in parenthesis, in units of\nSchwarzschild radii;\n(3) intrinsic injected power ($10^{45}$ erg s$^{-1}$) in the form of relativistic electrons;\n(4) magnetic field intensity (Gauss);\n(5) bulk Lorentz factor at R$_{diss}$; \n(6) and (7) break and maximum random Lorentz factors of the injected electrons; \n(8) and (9) slopes of the injected electron distribution [Q($\\gamma$)] below and above $\\gamma_{\\rm b}$;\n* fixed values from low to high state SED.}\n\\end{table*}\n\n-- The emitting region is assumed to be spherical \n(with radius $R$) and at a distance $R_{\\rm diss}$ (dissipation radius) from the central black hole.\nThe emitting electrons are injected at a rate Q($\\gamma$) [cm$^{-3}$ s$^{-1}$] for a finite time equal to the\nlight crossing time $R\/c$. \nThe adopted function Q($\\gamma$) is a smoothly broken power law with a break at $\\gamma_{\\rm b}$\nand slopes $s_1$ and $s_2$ below and above $\\gamma_{\\rm b}$, respectively.\nThe emitting region is moving with a velocity $\\beta_{\\rm c}$ corresponding to a bulk Lorentz factor $\\Gamma$. \nWe observe the source at the viewing angle $\\theta_{\\rm v}$.\n\n-- The external radiation sources taken into account are:\nthe broad line region (BLR) photons, assumed to re--emit 10\\% of the \naccretion disk luminosity from a shell--like distribution of clouds located at a distance \n$R_{\\rm BLR}=10^{17}L_{\\rm d, 45}^{1\/2}$ cm (Kaspi et al. 2005), where $L_{\\rm d}$ is the disk luminosity; \nthe IR emission from a dusty torus located at a distance \n$R_{\\rm IR}=2.5\\times10^{18}L_{d, 45}^{1\/2}$ cm (Elitzur 2006); \nthe direct emission from the accretion disk, including its X--ray corona. \nAlso the starlight contribution from the inner region of the host galaxy and the cosmic\nbackground radiation are taken into account, but these photon sources are unimportant in\nthe case of IGR J22517+2217.\n\n-- The accretion disk is a standard ``Shakura \\& Syunyaev\" (1973) disk, emitting as a blackbody at each radius.\nThe maximum temperature ($T_{\\rm max}$), i.e. the peak of the disk luminosity, is assumed to occur at $\\sim5$ \nSchwarzschild radii ($R_S$).\nThus from the position of the peak of the disk luminosity, and the total luminosity of the accretion \ndisk ($L_d$) it is possible to derive $M_{\\rm BH}$ and $\\dot M$, once a value for the efficiency $\\eta$ is assumed ($\\eta=0.08$ for a Schwarzschild black hole). \nSee Ghisellini et al. (2010a) for a discussion on caveats of this black hole mass estimate.\n\nWe can estimate the black hole mass $M_{\\rm BH}$ and the accretion luminosity \n$L_{\\rm d}$ of IGR J22517+2217 using optical and UV data and upper-limits corrected for the absorption\nof neutral hydrogen in intervening Lyman $\\alpha$ systems along the line of sight \n(see Ghisellini et al. 2010a for details).\nGiven the uncertainties in the amount of intervening Ly$\\alpha$ systems \nand the paucity of data the results must be considered an approximation.\nWe find $M_{\\rm BH}=10^9 M_{\\sun}$ and $L_{\\rm d}=6.8\\times10^{46}$ erg s$^{-1}$.\nThese values correspond to a disk radiating at 45\\% of the Eddington level.\n\nThe BLR are located at $R_{\\rm BLR}= 8\\times 10^{17}$ cm and the \nIR emitting torus at $R_{\\rm IR}=2.5\\times10^{19}$ cm.\nThe total X--ray corona luminosity is assumed to emit 30\\% of $L_{\\rm d}$. \nIts spectral shape is assumed to be $\\propto \\nu^{-1} \\exp(-h\\nu\/150 \\, {\\rm keV})$. \n\n\n\\section{Discussion}\n\nOur analysis shows that the extremely Compton dominated FSRQ IGR J22517+2217,\nexperienced a strong flare in the high energy hump in Jan 2005, and then faded in a quiescent state.\nIn order to investigate the physical properties of the source we built two SEDs for the two different states, and fitted \nthe data with the leptonic, one--zone synchrotron and inverse Compton model described in \\S3.\n\nIn order to build the SED for the quiescent state of IGR J22517+2217, we used the  X-ray data from {\\it Suzaku} (XIS and PIN), \nthe BAT spectrum extracted from the quiescent period and {\\it Fermi}\/LAT 24 months upper limits.\nThe archival non--simultaneous optical\/UV data were added to our data.\n\nFor the flaring SED we used hard X-ray data from {\\it Integral}--IBIS and the {\\it Swift}--BAT \nspectrum extracted from the Jan 2005 flare.\nWe do not have soft X-ray data available for the flaring period. \nHowever, in order to put some constraint at soft X-ray frequencies we chose to include the XRT spectrum in \nthe flaring SED. This has a factor 2 higher normalization with respect to the XIS data.\nWe stress that, as observed in other samples of bright, red blazars (Ghisellini et al. 2010a), \nthe flux variability is larger at higher energies (i.e. in hard X--rays)\nbut modest at few keV. This support our choice of include the XRT data in the flaring SED.\n\nAll data points and SED model components of the flaring and quiescent SED are plotted in Fig. 2, top panel. Fig. 2, bottom\npanel, shows a zoom in the region of X-ray data.\nThe strong and very bright hard X-ray spectrum, together with the upper limits in the {\\it Fermi}\/LAT\nenergy range, constrain the peak of the high energy hump of the quiescent SED to be located \nat $\\sim10^{20}$--$10^{21}$ Hz, and thus the form of energy spectrum of radiating electrons.\nThe same energy spectrum is then assumed in the flaring SED, for which no MeV\/GeV data are available.\nThe corresponding synchrotron peak falls in a region of the spectrum localized around \n$10^{11}$ Hz for both SEDs.\n\nHowever, in our single--zone leptonic model, the synchrotron and the high \nenergy humps are produced by the same population of electrons.\nFurthermore, if the high energy emission is given by the external Compton process,\nthe energies of the electrons emitting at the $\\sim$MeV peak are rather modest,\nimplying a corresponding low frequency synchrotron peak.\nWe then require that the synchrotron component peaks at low energies,\nclose to the self--absorption frequency, and furthermore require that the\nthin synchrotron emission has a steep spectrum, whose flux is smaller\nthan the optical archival data (characterized instead by a rather hard slope,\nthat we interpret as emission from the accretion disk).\n\nGiven the relative paucity of the observational data, the choice of the model parameters is not unique,\nand some further assumption has to be made.\nWe indeed fix the viewing angle $\\theta_{\\rm v}$ to 3$^\\circ$, to be close to the\n$\\theta_{\\rm v}\\sim 1\/\\Gamma$ condition.\n\nNote that we model both states of\nthe source by changing the minimum number of parameters, given that fitting \ntwo data sets with the same model, allows to constraint better\nthe model parameters.\nIn particular, we assume that the accretion luminosity does not change \nfor the two states, and require also the bulk Lorentz factor and the parameters\nof the distribution of the injected electrons (break and maximum random Lorentz factors\nand slopes below and above $\\gamma_{\\rm b}$) to be the same for the two SEDs.\nThus, the parameters that are left free to vary from one state to the other are \nthe dissipation radius $R_{\\rm diss}$, the injected power $P'_{\\rm inj}$ and the \nmagnetic field $B$, that is proportional to $1\/R_{\\rm diss}$ (the Poynting flux is assumed to be constant).\n\nThe results of our modeling are shown in Fig. 2, where we show the total flux together\nwith the contributions of the non--thermal (synchrotron, self Compton, external Compton)\nand thermal (accretion disk, IR torus, X--ray corona) components.\nAs can be seen, the curvature around 1 keV observed in the XRT and XIS spectra\ncan be well reproduced by an EC component changing (softening) slope from \n$\\sim10^{17}$ to $\\sim10^{19}$ Hz, disfavoring the intrinsic obscuration scenario\nfor the shape of the X-ray emission of IGR J22517+2217.\n\nTable 2 reports the parameters of the SED fitting for the quiescent and flaring states. \nThe main difference between the two SEDs is the power $P'_{\\rm inj}$ \ninjected in the source in the form of relativistic electrons, that changes by a factor $\\sim$7.\nThe increase of $P'_{\\rm inj}$ accounts for the enhanced X--ray flux in the high state.\nThe other difference is the location of the emitting region $R_{\\rm diss}$, \nbecoming larger for the high state.\nThis is dictated by the detailed modeling of the slope of the soft to hard X--ray spectrum,\nrequiring an electron distribution with a break at low energies, and another break\nat somewhat larger energies.\nThis is accounted for, in our modeling, by requiring that electron of very low energies\n(corresponding to random Lorentz factors $\\gamma<5$) do not cool in one light crossing time.\nThis can be achieved if the location of the emitting region is slightly beyond the BLR,\nin a zone where the BLR radiation energy density is somewhat smaller.\nThis is the reason leading to a larger $R_{\\rm diss}$ in the high state.\nAs a consequence of assuming a larger region, the magnetic field is lower,\nfollowing the assumption of a constant Poynting flux ($\\propto B^2 R^2$).\nThe large Compton dominance constrains the value of the magnetic field,\nand in turn the relevance of the self Compton flux, found to be almost negligible.\n\nThis is also in agreement with the results pointed out in Sikora et al. (2009):\nbright blazars with very hard ($\\alpha_x<0.5$) X--ray spectra and high luminosity ratio \nbetween high and low frequency spectral components\nchallenge both the standard synchrotron self-Compton and \nhadronic models, while EC can easily account for these observed properties.\n\nIn the analysis described above, the bulk Lorentz factor is assumed to be constant. \nHowever, the change of bulk Lorentz factor is often invoked to explain the variability of FSRQs.\nAs a further check, we performed a new fit of both low and high states, \nleaving $\\Gamma$ as a free parameter, in addition to $R_{\\rm diss}$ and $P'_{\\rm inj}$, although the fit is limited by the poor number of data points\n\nBoth SEDs are well reproduced with these new parameters for the low (high) state: $\\Gamma$=15 (20); $R_{\\rm diss}$=1700 (3500) $R_S$ and \nLog($P'_{\\rm inj}$)= 43.48 (44.17) erg s$^{-1}$.\nIn this case the values of $P'_{\\rm inj}$ are slightly lower for both low and high states,\nand the difference is lower (a factor of $\\sim5$), while the difference \nin $R_{\\rm diss}$ is slightly larger: \nfrom 1700 to 3500 $R_S$ instead of 1900 and 3300 $R_S$.\nThus the change in $\\Gamma$ has the main effect of slightly decreasing the required variation of the total injected power\nto account for the observed variability, but no other substantial differences are introduced.\nThis new fit also gives an idea of the degree of degeneracy in the fit parameters, due to the incompleteness of the data set,\nespecially in the $\\gamma$-ray band.\n\n\n\nIn Table 3 we report the logarithm of the jet power in the form of radiation, \nPoynting flux and bulk motion of electrons and protons, \nin erg s$^{-1}$, calculated for both SEDs.\nThey have been calculated from\n\\begin{equation}\nP_{\\rm i} = \\pi R^2 \\Gamma^2 c U^\\prime_{\\rm i} \n\\end{equation}\nwhere $U^\\prime_{\\rm i}$ is the energy density of interest, calculated in the comoving frame\n(see e.g. Celotti \\& Ghisellini 2008).\nAs discussed in Ghisellini et al (2011), the power $P_{\\rm r}$ dissipated by the jet\nto produce the radiation we see is almost model--independent, since it depends\nonly on the observed luminosity and on the bulk Lorentz factor $\\Gamma$.\nThe power dissipated in other forms, on the other hand, depends on the amount of electrons ($P_{\\rm e}$),\nprotons ($P_{\\rm p}$), and magnetic field ($P_{\\rm B}$) carried by the jet,\nwhich have been estimated by applying our specific model.\nFurthermore the power carried in the bulk motion of protons requires \na knowledge of how many protons are there, per emitting lepton.\nThe values given in Table 3 assume one proton for one emitting lepton.\n\n\\begin{table}\n\\label{tab:sed}\n\\begin{center}\n\\begin{tabular}{cccccccccccccccc}\n\\hline\\hline\\\\\n\\multicolumn{1}{c} {State}&\n\\multicolumn{1}{c} {$\\log P_{\\rm r}$}&\n\\multicolumn{1}{c} {$\\log P_{\\rm B}$}&\n\\multicolumn{1}{c} {$\\log P_{\\rm e}$}&\n\\multicolumn{1}{c} {$\\log P_{\\rm p}$}&\\\\\n (1) & (2) &(3) & (4) & (5) \\\\\n\\hline\\\\ \nLow  & 46.04 & 45.54 & 45.00 & 47.56 \\\\\nHigh & 46.83 & 45.54 & 46.17 & 48.41 \\\\\n\\hline                                    \n\\end{tabular}\\end{center}   \n\\caption{Input parameters of the SED model fitting both the low and high state of IGR J22517+2217.      \nColumn: \n(1) State ; \n(2)--(5) logarithm of the jet power in the form of radiation ($P_{\\rm r}$), \nPoynting flux ($P_{\\rm B}$), bulk motion of electrons ($P_{\\rm e}$) and protons ($P_{\\rm p}$,\nassuming one proton per emitting electron). Powers are in erg s$^{-1}$.}\n\\end{table}\n\nFrom Table 3 we can see that the power\n$P_{\\rm r}$ changes from $\\sim0.15\\times L_{\\rm d}$ in the quiescent state, \nto $P_{\\rm r} \\sim L_{\\rm d}$ in the flaring state, i.e. the jet requires a power comparable \nto the disk luminosity to produce the radiation we see, in the latter case.\n\nTanaka et al. (2011) reported a similar behavior, based on {\\it Fermi}-LAT data, for the strong 2010 GeV flare of the blazar 4C+21.35:\nassuming similar efficiencies for the accretion disk and the jet, they estimated a jet intrinsic power $L_{jet}$\nchanging from $\\sim0.1 L_{acc}$ in the quiescent state to $\\sim1 L_{acc}$ (beeing $L_{acc}$ the intrinsic accretion power).\nThey argued that these results, combined with the findings of Abdo et al. (2010b) on several FSRQ detected by {\\it Fermi}-LAT,\nsuggest a scenario in which the observed $\\gamma$-ray variability of blazars is due to the different power of the jet,\nthat normally represents only a small fraction of the accretion power, while during major flares is capable of \ncarry away almost all the available accretion power.\nOur findings are quantitatively similar, and thus result in agreement with this view.\n\nThe $P_{\\rm r}$ however is a {\\it lower limit} to the total jet power.\nThe total jet power, dominated by the bulk motion of protons associated to emitting \nelectrons, is $P_{\\rm jet} = P_{\\rm B} + P_{\\rm e} + P_{\\rm p} = 3.6\\times10^{47}$ \nand $2.6\\times10^{48}$ erg s$^{-1}$, in the low and high state, respectively. \n$P_{\\rm jet}$ is dominated by $P_{\\rm p}$: if there is indeed one proton per\nemitting lepton, then the jet power is from 3 to 30 times more powerful than the \naccretion luminosity. \n\nIt has been proposed that jets in luminous blazars may well be numerically dominated by pairs,\nbut being still dynamically dominated by protons (Sikora \\& Madejski 2000, Kataoka et al. 2008).\nWe computed the jet power due to protons assuming an upper limit for this ratio of one proton per 20 pairs.\nAbove this limit the jet is too ``light'' and the Compton drag produces significant\ndeceleration (Ghisellini \\& Tavecchio 2010).\nThe one-to-one ratio can be assumed as a lower limit for this ratio.\n\nThe lower limits for $P_{\\rm p}$ obtained in this way are $1.8\\times10^{46}$ and $1.3\\times10^{47}$ erg s$^{-1}$ for the low and high state, respectively.\nTherefore with this assumption, the jet power in electron, protons and magnetic field, becomes comparable with the radiation power.\nThis translates into a total jet power of $P_{\\rm jet} = 2.2\\times10^{46}$  and $1.4\\times10^{47}$ erg s$^{-1}$, respectively.\n\nThe values obtained assuming one proton per lepton are extreme, even if compared with the distribution of $P_{\\rm jet}$ and $L_{\\rm d}$ \ncomputed, with the same assumptions, for a sample of high redshift {\\it Fermi}\/LAT and BAT blazars in Ghisellini et al. (2011, Fig. 9), \nthat are in the range $P_{\\rm jet}\\simeq10^{46}-2\\times10^{48}$ and $L_{\\rm d}\\simeq8\\times10^{45}-2\\times10^{47}$ erg s$^{-1}$. \nTo better clarify the remarkable behavior of IGR J22517+2217, \nwe note that similar values of jet power have been achieved during the exceptional flare \nof 3C 454.3 in December 2009 (Ackermann et al. 2010; Bonnoli et al. 2011). Therefore, IGR J22517+2217 represents one of the ''monsters'' of the high-z Universe.\n\n\n\n\n\\section{Summary and conclusion}\n\n\nThanks to a new {\\it Suzaku} observation in the X-ray energy band, the {\\it Fermi} upper limits in the 0.1-100 GeV band,\nthe flux selected spectra obtained through a re-analysis  of the IBIS and BAT hard X-ray data, and other optical and radio archival data sets, \nwe were able to identify a strong flare episode in the high redshift, hard X-ray selected blazar IGR J22517+2217, which occurred in Jan 2005,\nfollowed by a period of quiescence, extending up to the present days.\nTo model the overall SEDs of the source in the flare and quiescent states, we adopted a leptonic, one-zone synchrotron and inverse Compton model. \nThe optical\/UV emission is interpreted as thermal emission from the accretion disk, plus IC from the corona, and reprocessed emission from the BLR. \n\nThe curvature observed in the X-ray spectra was proposed to be due to intrinsic, moderate absorption (N$_H\\sim2\\times10^{22}$ cm$^{-2}$).\nHowever, in the context of the broad band SED modelization proposed in this paper, it appears to be inherently accounted for by an intrinsic softening of the \nEC component around $\\sim10^{18}$ Hz.   \n\nIn both states a very strong Compton dominance is observed, with the high energy hump (produced by EC), that is at least two orders of magnitudes \nhigher than the low energy (synchrotron) one.\nThe high energy peak flux varies by a factor 10 between the two states, while the high energy peak frequency remain almost constant, between $10^{20}-10^{22}$ Hz.\nThe observed large Compton dominance constrains the value of the magnetic field,\nand hence the relevance of the self--Compton component, that is found to be negligible in both states.\nThe model can explain the observed variability as a variation of the total number of emitting electrons (a variation of factor $\\sim7$) \nand as a change in the dissipation radius, moving from within to outside the broad line region as the luminosity increases.\n\nIn the flaring state, the jet power lower limit, represented by the radiative component $P_{\\rm r}$,\nrequires a power comparable to the disk luminosity to produce the observed radiation.\nThe total jet power upper limit, dominated by the bulk motion of protons, and estimated assuming one proton per electron, \nis more than $\\sim30$ times more powerful than the accretion luminosity ($2.6\\times10^{48}$ erg s$^{-1}$). \nSuch extreme values have been derived only recently for a handful of extreme, high redshift, hard-X\/soft-$\\gamma$ ray selected FSRQs, \nshowing similar strong Compton dominance,\nand comparable with the value achieved by 3C 454.3 during the 2009 exceptional flare.\n\n\n\n\\section*{acknowledgements}\n\nWe thank the referee for useful comments that improved the paper.\n\nPartial support from the Italian Space Agency (contracts ASI-INAF\nASI\/INAF\/I\/009\/10\/0) is acknowledged. \n\nThis research has made use of the NASA\/IPAC Extragalactic Database (NED) \nwhich is operated by the Jet Propulsion Laboratory, California Institute of Technology, \nunder contract with the National Aeronautics and Space Administration.\n\nThe \\textit{Fermi} LAT Collaboration acknowledges generous ongoing support\nfrom a number of agencies and institutes that have supported both the\ndevelopment and the operation of the LAT as well as scientific data analysis.\nThese include the National Aeronautics and Space Administration and the\nDepartment of Energy in the United States, the Commissariat \\`a l'Energie Atomique\nand the Centre National de la Recherche Scientifique \/ Institut National de Physique\nNucl\\'eaire et de Physique des Particules in France, the Agenzia Spaziale Italiana\nand the Istituto Nazionale di Fisica Nucleare in Italy, the Ministry of Education,\nCulture, Sports, Science and Technology (MEXT), High Energy Accelerator Research\nOrganization (KEK) and Japan Aerospace Exploration Agency (JAXA) in Japan, and\nthe K.~A.~Wallenberg Foundation, the Swedish Research Council and the\nSwedish National Space Board in Sweden.\n\nAdditional support for science analysis during the operations phase is gratefully\nacknowledged from the Istituto Nazionale di Astrofisica in Italy and the Centre National d'\\'Etudes Spatiales in France.\n\n\n\n\n\n\n\n\\begin{appendix}\n\n\n\\end{appendix}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzeapf b/data_all_eng_slimpj/shuffled/split2/finalzzeapf
new file mode 100644
index 0000000000000000000000000000000000000000..81b7acc2ee2a54b8b6ce8757dcb2dc18460ea7bb
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzeapf
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\nFor a conventional orthogonal class system, it is believed that an arbitrarily weakly uncorrelated diagonal disorder in one and two dimension \\cite{Economou} can result in the Anderson localization \\cite{Anderson1957}.\nIn three dimension, there exists mobility edge $E_c$ which separates the localized states from extended states \\cite{Economou1972}. When the eigenenergies approach the mobility edge $E_c$, the localization length of localized states  would diverge. Interestingly, in the presence of off-diagonal uncorrelated disorders, one-dimensional system can have a singular density of states near the zero energy \\cite{Dyson,Eggarter1978,Balents1997}, which also results in an anomalous localization \\cite{Theodorou1976,Antoaiou1977} that the localization length is proportional to the square root of system  size \\cite{Fleishman1977,Inui1992,Izrailev2012}.\nIf the energy deviates from zero, the eigenstates are usually localized states.\nIf the off-diagonal disorder is correlated, the system can have localized-extended transition \\cite{Cheraghchi2005}.\nIn the presence of both diagonal and correlated off-diagonal disorders, the one-dimensional system can also have extended states \\cite{Zhangwei2004}.\nRecently, a so-called mosaic lattice model with diagonal quasiperiodic disorder has been proposed \\cite{Wangyucheng2020}. It is found that this model  has mobility edges and the mobility edges can be exactly obtained with Avila's theory \\cite{Liu2021,Avila2015}.\n\n\n\n\nSince there exist above anomalous properties of  localizations in the  model with a pure off-diagonal uncorrelated disorder,\na natural question arises, how is it if there the off-diagonal hopping is quasiperiodic?\n One may wonder whether there exist mobility edges for off-diagonal quasiperiodic disorder (hopping). What are the localization properties of eigenstates?\n\n\n\n\n\n\n\n\n\n\n\n\nIn this work, we try to answer the above questions by exploring a quasiperiodic off-diagonal disorder model with mosaic modulation. The model is\n \\begin{align}\\label{2}\nV_{i,i+1}\\psi(i+1)+V_{i,i-1}\\psi(i-1)=E\\psi(i).\n\\end{align}\nwhere\n \\begin{align}\nV_{i,i+1}=V_{i+1,i}\n\\left\\{\\begin{array}{cc}\nt,  & for \\ i\\neq 0 \\ mod \\ \\kappa\\\\\n\\frac{2\\lambda \\cos(2\\pi \\beta i+\\phi)}{\\sqrt{1-\\tau \\cos^2(2\\pi \\beta i+\\phi)}},& for \\ i= 0 \\ mod \\ \\kappa\n  \\end{array}\\right.\n\\end{align}\nwhere $t>0$ is constant hopping strength,  $\\lambda$ describes the quasi-periodic hopping strength, positive integer $\\kappa$ is mosaic period, $\\beta$ is an irrational number, and the parameter $\\tau$ is a real number.\nIn this work,  we only consider the parameter $\\tau$ is not larger than $1$, i.e., $\\tau\\leq 1$.\nIn addition, we should remark that there is no extended state in the above model Eq.(1). This is because by \\cite{Barry1989} (or the mechanism in \\cite{xwzl}), the absolutely continuous spectrum which corresponds extended states is empty since there exists a sequence $\\{n_k\\}$ such that $V_{n_k,n_{k+1}}\\rightarrow0$. Thus the mobility edges (if any ) would separate localized states and critical states.\nIn the whole paper, we take $\\beta=(\\sqrt{5}-1)\/2$ and  use the units of $t=1$ for $\\kappa>1$ (or $\\lambda=1$ for $\\kappa=1$).\n\n\n\n\n\nIt is found that the parity of mosaic period $\\kappa$ has important influences on the localization of eigenstates near zero-energy. To be specific, if mosaic period $\\kappa$ is odd, there is no Anderson localization for arbitrarily strong hopping strength.  While for even mosaic period, the system undergoes Anderson localization as the quasiperiodic hopping increases.\nIn addition, the Lyapunov exponent $\\gamma(E)$ and mobility edges are also exactly obtained with the Avila's theory.\nWith Lyapunov exponent, we find that,  some critical regions in the parameter plane would appear.\n In comparison with the localized states, the spatial extensions of eigenstates and their fluctuations in the critical region are much larger.\nNear localized-critical transition points ($E_c$),  the localization length diverges, i.e.,\n\\begin{align}\\label{10}\n\\xi(E)\\equiv1\/\\gamma(E)\\propto|E-E_c|^{-\\nu}\\rightarrow\\infty, \\ \\ as \\ E\\rightarrow E_c,\n\\end{align}\nwhere the critical index \\cite{Huckestein1990} $\\nu=1$.\nFinally, we show that the systems with different parameter $E$ can be systematically classified  by Lyapunov exponent and Avila acceleration.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nThe work is organized as follows. First of all, we discuss the localization properties of zero-energy states for both odd and even number $\\kappa$ in Sec.\\textbf{II}.  In Sec.\\textbf{III}, the Lyapunov exponent is calculated.  Next, with the Lyapunov exponent, we  determine the mobility edges and critical region  in Sec.\\textbf{IV}. In addition, Avila's acceleration is also calculated.\n At the end, a summary is given in Sec.\\textbf{V}.\n\n\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.0\\columnwidth]{bu0.eps}\n\\end{center}\n\\caption{ The average growth\/decreasing ratio of zero-energy wave function for $\\kappa=2$, and $\\tau=-2$.  The critical hopping strength $\\lambda=\\lambda_c\\simeq\\pm1.366t$ where $f$ is exactly zero (indicated by black arrows in the figure).  }\n\\label{bu0}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.0\\columnwidth]{bu1.eps}\n\\end{center}\n\\caption{Several typical zero-energy wave functions of critical states and localized edge states. Panels (a) and (c) are critical zero-energy wave functions where $f=0$. Panel(b) [(d)] is  localized  right (left)-hand edge states where $f>0$ ($f<0$).}\n\\end{figure}\n\n\n\\section{localization of zero-energy state}\nIn this section, we discuss the influences of parity of integer $\\kappa$ on the localization properties of the zero energy states.\nFor the quasi-periodic model Eq.(1), we note that if one applies a transform $\\psi(n)\\rightarrow(-1)^n\\psi(n)$ in the Eq.(1), the energy would change a sign, i.e., $E\\rightarrow -E$.\nDue to the chiral (sublattice) symmetry, the energy $E_n$ and $-E_n$ appear in pairs \\cite{Cheraghchi2005}.\nIn addition, the number of eigenenergies is same with the lattice site number. If the total lattice site number $N$ is an odd number, then there would be one zero energy state at least.\nIn the following, we find that when $N$ is even, usually there is no zero-energy eigenstates. In this section, we assume the total lattice number $N$ is odd, then the zero-energy state always exists.\n\n\n\nFurthermore, we assume the lattice sites of system are labeled with number $i=1,2,3,...,2m, N=2m+1$, where $m$ is a positive integer.\nStarting from wave functions of left-hand end site $\\psi(i=1)=1$ [and $\\psi(i=0)=0$], by Eq.(1),\nthe wave function of zero-energy state can be written as\n\\begin{align}\n&\\psi(N=2m+1)=\\frac{V_{2m,2m-1}V_{2m-2,2m-3}...V_{4,3}V_{2,1}}{V_{2N,2N+1}V_{2m-2,2m-1}...V_{4,5}V_{2,3}}\\psi(i=1),\\notag\\\\\n&=\\frac{v(2m-1)v(2m-3)...v(3)v(1)}{v(2m)v(2m-2)...v(4)v(2)}.\n\\end{align}\nIn the above equation, we set $v(i)\\equiv v(i,\\phi)\\equiv V_{i,i+1}$ and use the relation $V_{i+1,i}=V_{i,i+1}$.\nThe average growth\/decreasing ratio of wave function  is\n\\begin{align}\n&f=\\lim_{m\\rightarrow\\infty}\\frac{1}{2m}ln(|\\frac{\\psi(N=2m+1)}{\\psi(i=1)}|)\\notag\\\\\n&=\\lim_{m\\rightarrow\\infty}\\frac{1}{2m}ln(|\\frac{v(2m-1)v(2m-3)...v(3)v(1)}{v(2m)v(2m-2)...v(4)v(2)}|).\n\\end{align}\nIf $f\\geq0$,  $f$ would be the Lyapunov exponent $\\gamma(E)$ (see Sec. \\textbf{III}).\n\n\n\n\n\n\n\n\n\n\\subsection{$\\kappa$ is a positive even integer}\n When $\\kappa$ is a positive even integer, we can assume $N=n\\kappa+1$, where $n$ is an integer.\nBased on  Eqs. (1) and (5), due to the ergodicity of the map $\\phi\\longrightarrow 2\\pi\\beta i+\\phi$, the average growth\/decreasing rate of wave function can be reduced into\n\\begin{align}\n&f=\\lim_{n\\rightarrow\\infty}\\frac{1}{n\\kappa}ln(\\frac{|v(\\kappa-1)v(2\\kappa-1)...v(n\\kappa-1)|}{|v(\\kappa)v(2\\kappa)...v(n\\kappa)|}),\\notag\\\\\n&=\\frac{-1}{\\kappa\\times 2\\pi}[\\int_{0}^{2\\pi}d\\phi ln(|v(\\kappa,\\phi)|)],\\notag\\\\\n&=\\frac{-1}{\\kappa}ln(\\frac{2|\\lambda\/t|}{1+\\sqrt{1-\\tau}}).\n\\end{align}\n\n\nIt is shown that when $\\kappa$ is a positive even integer, for a generic  $\\lambda$, $f$ is usually not zero. Then the zero-energy state would be localized states which may situate at right-hand edge ($f>0$) or left-hand edge ($f<0$) of the lattices (see Figs.1 and 2). So for general parameters, the zero-energy state would be a localized edge state.\nOnly  when $f$ is exactly vanishing, i.e., $f=0$,\n\\begin{align}\n&\\rightarrow f=\\frac{-1}{\\kappa}ln(\\frac{2|\\lambda\/t|}{1+\\sqrt{1-\\tau}})=0\\notag\\\\\n&\\rightarrow |\\lambda\/t|=|\\lambda_c\/t|\\equiv\\frac{1+\\sqrt{1-\\tau}}{2},\n\\end{align}\n the zero-energy state would be  a critical state (see Figs.1 and 2).\nThe average growth\/decreasing rate $f$ for $\\kappa=2$ and $\\tau=-2$ is reported in Fig.1. At the critical strength $\\lambda_c=\\pm 1.366t$, $f=0$.\nWhen $\\lambda$ approaches the critical $\\lambda_c$, the localization  length can be arbitrarily large, i.e.,  $\\xi\\equiv 1\/|f|\\propto 1\/|\\lambda-\\lambda_c|\\rightarrow\\infty$.\n\n\n\n\nIn order to investigate the properties of zero-energy states, we also numerically solve Eq.(1) for $\\kappa=2$, $\\tau=-2$, and lattice size $N=2\\times500+1$.\n  Several typical wave functions for localized states and critical states are reported in Fig.2.\n  We know that the wave function of extended state usually extends all over the whole lattices, while localized state only occupies finite lattice sites.\nThe critical state consists of several disconnected patches which interpolates between the localized and extended states \\cite{yicai,Liu2022}.\nFrom Fig.2, we see the zero energy wave function of $f=0$ is critical state. The wave functions with non-vanishing $f$ correspond to localized edge states.\nWhen $f>0$, the state is at right-hand end edge, while for $f<0$, it is at left-hand end edge.\n\n\n\n\n\\subsection{$\\kappa$ is a positive odd integer}\nWhen $\\kappa$ is a positive odd integer, we can assume $N=2n\\kappa+1$, where $n$ is an integer.\n  Similarly, $f$ can be written as\n\\begin{align}\n&f=\\lim_{n\\rightarrow\\infty}\\frac{1}{2n\\kappa}ln(\\frac{|v(\\kappa)v(3\\kappa)...v((2n-1)\\kappa)|}{|v(2\\kappa)v(4\\kappa)...v(2n\\kappa)|}),\\notag\\\\\n&=\\frac{1}{2\\kappa\\times 2\\pi}[\\int_{0}^{2\\pi}d\\phi ln(|v(\\kappa,\\phi)|)-\\int_{0}^{2\\pi}d\\phi ln(|v(2\\kappa,\\phi)|)],\\notag\\\\\n&=0.\n\\end{align}\nIt is shown that when $\\kappa$ is a positive odd integer, the average growth\/decreasing ratio of zero-energy wave function is exactly zero.\nSo if $\\kappa$ is odd, all the zero-energy states are critical states.\n\n\n Some interesting even-odd effects of lattice site number $N$ have been investigated in the random off-diagonal disorder models.\nIt is found that there  exists a delocalization transition only when lattice size $N$ is odd \\cite{Brouwer1998}. In addition, the localization length of the zero-energy state depends sensitively boundary conditions \\cite{Brouwer2002}, and it can be an arbitrarily large value.\n\n\n\n\n The above discussions show that the parity of integer $\\kappa$ has important influences on the localization of zero-energy states.\n An odd integer $\\kappa$ results in a critical zero-energy state, while an even $\\kappa$ usually gives the localized edge states (see Fig.2).\n We notice that the above discussion can be also applied to other forms of quasiperiodic hopping with mosaic modulations.\n\n\nIn the following text, we will show that the above influences of parity of $\\kappa$ are also transmitted to other eigenstates near the zero energy.\nTo be specific, when $\\kappa$ is a positive odd integer, for a given hopping strength $\\lambda$,  the eigenstates  near zero energy are always critical (see Sec.IV). So if energy is sufficiently near the zero-energy, there is no Anderson localization transition for odd $\\kappa$.\nWhen $\\kappa$ is a positive even integer, the eigenstates near zero-energy would undergo  Anderson localization transition as the quasiperiodic hopping strength increases.\nThen the system would have localized states near the zero-energy.\n\n\n\n\n\\section{The  Lyapunov exponent  }\n When $E\\neq0$, the  localization properties of eigenstates  can be characterized by the Lyapunov exponent. In this section,  we calculate the Lyapunov exponent with  the transfer matrix method \\cite{Sorets1991,Davids1995}.\n\nFirst of all,  we assume the system is a half-infinite lattice system with left-hand end sites $i=0$ and $i=1$.\nFurther using Eq.(1), starting with $\\psi(0)$ and $\\psi(1)$ of left-hand end sites, the wave function can be obtained with relation\n\\begin{align}\n\\Psi(i)=T(i)T(i-1)...T(2)T(1)\\Psi(0)\n\\end{align}\nwhere transfer matrix\n\\begin{align}\\label{V}\nT(n)\\equiv\\left[\\begin{array}{ccc}\n\\frac{E}{V_{n,n+1}} &-\\frac{V_{n,n-1}}{V_{n,n+1}}  \\\\\n1&0\\\\\n  \\end{array}\\right].\n\\end{align}\nand\n\\begin{align}\n\\Psi(n)\\equiv\\left[\\begin{array}{ccc}\n\\psi(n+1)  \\\\\n\\psi(n)\\\\\n  \\end{array}\\right].\n\\end{align}\n\nFor a given parameter  $E$, with the increasing of $n$, we can assume that the wave function grows roughly according to an exponential law \\cite{Ishii,Furstenberg}, i.e.,\n\\begin{align}\n\\psi(n)\\sim e^{\\gamma(E) n}, &\\ as \\ n\\rightarrow \\infty,\n\\end{align}\nwhere $\\gamma(E)\\geq0$ is Lyapunov exponent which measures the average growth rate of wave function. If the parameter $E$ is not an eigen-energy of $H$, the Lyapunov exponent would be positive, i.e., $\\gamma(E)>0$ \\cite{Jonhnson1986}.\nWhen $E$ is an eigen-energy of system, the Lyapunov exponent can be zero or positive \\cite{yicai}.\nFor  critical states, the Lyapunov exponent $\\gamma(E)\\equiv0$ . While for localized states, the Lyapunov exponent $\\gamma(E)>0$.\n\nConsequently the Lyapunov exponent can be written as\n\\begin{align}\n&\\gamma(E)=\\lim_{L \\rightarrow \\infty }\\frac{\\log(|\\Psi(L)|\/|\\Psi(0)|)}{L}\\notag\\\\\n&=\\lim_{L\\rightarrow \\infty}\\frac{\\log(|T(L)T(L-1)...T(2)T(1)\\Psi(0)|\/|\\Psi(0)|)}{L}\n\\end{align}\nwhere $L$ is a positive integer and\n\\begin{align}\n|\\Psi(n)|=\\sqrt{|\\psi(n+1)|^2+|\\psi(n)|^2}.\n\\end{align}\n\n\n\nIn the following, we view the adjacent $\\kappa$ lattice sites as a ``super  unit cell\". Next we assume $L=m\\kappa+1$ ($m$ is an integer) and $|\\Psi(0)|\/|\\Psi(1)|$ is a finite non-zero real number, the Lyapunov exponent can be reduced into\n\\begin{align}\n&\\gamma(E)=\\lim_{L \\rightarrow \\infty }\\frac{\\log(|\\Psi(L)|\/|\\Psi(0)|)}{m\\kappa+1}\\notag\\\\\n&=\\lim_{m\\rightarrow \\infty}\\frac{\\log(|T(m\\kappa+1)T(m\\kappa)...T(2)\\Psi(1)|\/|\\Psi(0)|)}{m \\kappa}\\notag\\\\\n&=\\lim_{m\\rightarrow \\infty}\\frac{\\log(|T(m\\kappa+1)T(m\\kappa)...T(2)\\Psi(0)|\/|\\Psi(0)|)}{m \\kappa}\\notag\\\\\n&=\\frac{1}{\\kappa}\\lim_{m\\rightarrow \\infty}\\frac{\\log(|CT_m.CT(m-1)...CT_1\\Psi(0)|\/|\\Psi(0)|)}{m}\\notag\\\\\n\\end{align}\nwhere cluster transfer matrix $CT_n$  for $n-th$ ``super unit cell\"  is defined  as\n\\begin{align}\n&CT_n\\equiv T(n\\kappa+1)T(n\\kappa)...T((n-1)\\kappa+3)T((n-1)\\kappa+2)\\notag\\\\\n&=\\left[\\begin{array}{ccc}\nE &-v(n\\kappa) \\\\\n1&0\\\\\n  \\end{array}\\right]\\left[\\begin{array}{ccc}\n\\frac{E}{v(n\\kappa)} &-\\frac{1}{v(n\\kappa)} \\\\\n1&0\\\\\n  \\end{array}\\right]\\left[\\begin{array}{ccc}\nE &-1 \\\\\n1&0\\\\\n  \\end{array}\\right]^{\\kappa-2}\\notag\\\\\n  &=\\frac{\\left[\\begin{array}{ccc}\nE^2-(E^2\\tau+4\\lambda^2)cos^2(\\theta_n) &-E(1-\\tau cos^2(\\theta_n) ) \\\\\nE(1-\\tau cos^2(\\theta_n) )&-1+\\tau cos^2(\\theta_n) \\\\\n  \\end{array}\\right]}{2\\lambda cos(\\theta_n)\\sqrt{1-\\tau cos^2(\\theta_n)}}\\notag\\\\\n  &\\times \\left[\\begin{array}{ccc}\nE &-1 \\\\\n1&0\\\\\n  \\end{array}\\right]^{\\kappa-2}\n\\end{align}\nwhere $\\theta_n=2\\pi\\beta n\\kappa +\\phi$.\n\n\n\nThe cluster transfer matrix Eq.(16) can be further\nwritten as a product of two parts, i.e., $CT_n= A_nB_n$, where\n   \\begin{align}\n&A_n=\\frac{1}{2\\lambda cos(\\theta_n)\\sqrt{1-\\tau cos^2(\\theta_n)}},\\notag\\\\\n&B_n=\\left[\\begin{array}{ccc}\nB_{11} &B_{12} \\\\\nB_{21}&B_{22}\\\\\n  \\end{array}\\right]\\left[\\begin{array}{ccc}\nE &-1 \\\\\n1&0\\\\\n  \\end{array}\\right]^{\\kappa-2},\n\\end{align}\nwith $B_{11}=E^2-(E^2\\tau+4\\lambda^2)cos^2(\\theta_n)$, $B_{21}=-B_{12}=E(1-\\tau cos^2(\\theta_n))$ and $B_{22}=-1+\\tau cos^2(\\theta_n)$.\nNow  the Lyapunov exponent is\n\\begin{align}\n\\gamma(E)=\\frac{1}{\\kappa}[\\gamma_A(E)+\\gamma_B(E)],\n\\end{align}\nwhere\n\\begin{align}\n&\\gamma_A(E)=\\lim_{m\\rightarrow \\infty}\\frac{\\log(|A(m)A(m-1)...A(2)A(1)|)}{m}.\n\\end{align}\nand $\\gamma_B(E)$ are given by\n\\begin{align}\n&\\gamma_B(E)=\\lim_{m\\rightarrow \\infty}\\frac{\\log(|B(m)B(m-1)...B(2)B(1)\\Psi(0)|\/|\\Psi(0)|)}{m}.\n\\end{align}\n\n\nIn the following,  we would use Avila's global theory \\cite{Avila2015} to get the Lyapunov exponent and the Avila's acceleration (see next section).\nFollowing Refs.\\cite{Liu2021,YONGJIAN1}, first of all, we  complexify the phase $\\phi\\rightarrow \\phi+i \\epsilon$ with $\\epsilon >0$ , e.g., $A_n=\\frac{1}{2\\lambda cos(2\\pi \\beta n\\kappa +\\phi+i\\epsilon)\\sqrt{1-\\tau cos^2(2\\pi\\beta n\\kappa +\\phi+i\\epsilon)}}$.\n In addition, due to the ergodicity of the map $\\phi\\longrightarrow 2\\pi\\beta n+\\phi$, we can write  $\\gamma_A(E)$ as an integral over phase $\\phi$ \\cite{Longhi2019}, consequently\n\\begin{align}\n&\\gamma_A(E,\\epsilon)\\notag\\\\\n&=\\frac{1}{2\\pi}\\int_{0}^{2\\pi} d\\phi \\ln(|\\frac{1}{2\\lambda cos(\\phi+i\\epsilon)\\sqrt{1-\\tau cos^2(\\phi+i\\epsilon)}}|)\\notag\\\\\n&=-\\epsilon+\\ln(|\\frac{2}{\\lambda(1+\\sqrt{1-\\tau})}|),\n\\end{align}\nfor $\\epsilon< \\ln |\\frac{2+2\\sqrt{1-\\tau}-\\tau}{\\tau}|$.\n\n\n\nNext we take $\\epsilon\\rightarrow\\infty$\n \\begin{align}\n&B_n=\\frac{e^{-i(4\\pi\\beta n \\kappa+\\phi)+2\\epsilon}}{4}\\left[\\begin{array}{ccc}\n-( E^2 \\tau+4\\lambda^2) &E\\tau \\\\\n-E\\tau & \\tau\\\\\n  \\end{array}\\right]\\left[\\begin{array}{ccc}\nE & -1 \\\\\n1& 0\\\\\n  \\end{array}\\right]^{\\kappa-2}\\notag\\\\\n  &+O(1).\n\\end{align}\nThen for large $\\epsilon$, i.e.,  $\\epsilon\\gg1$, $\\gamma_B(E,\\epsilon)$ is determined by the largest eigenvalue (in absolute value) of $B_n$, i.e.,\n\\begin{align}\n\\gamma_B(E,\\epsilon)=2\\epsilon+\\ln(|\\frac{|P|+\\sqrt{P^2+16\\lambda^2\\tau}}{8}|),\n\\end{align}\nwhere\n\\begin{align}\nP=(\\tau E^2+4\\lambda^2)a_{\\kappa}+\\tau a_{\\kappa-2}-2\\tau E a_{\\kappa-1}.\n\\end{align}\nand $a_{\\kappa}$, is given by\n\\begin{align}\na_\\kappa=\\frac{1}{\\sqrt{E^2-4}}[(\\frac{E+\\sqrt{E^2-4}}{2})^{\\kappa-1}-(\\frac{E-\\sqrt{E^2-4}}{2})^{\\kappa-1}].\n\\end{align}\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.0\\columnwidth]{F211.eps}\n\\end{center}\n\\caption{ Lyapunov exponents for $\\kappa=3$,  $\\tau=-2$, and $\\lambda\/t=0.5,1.5, 2.5$.  The discrete points are the numerical results for all the eigenenergies. The solid lines are given by Eq.(27). The mobility edges for $\\lambda\/t=2.5$ are indicated by black arrows.  Near mobility edges of the localized-critical transition, the Lyapunov exponent $\\gamma(E)\\propto |E-E_c|$ approaches zero (as $E\\rightarrow E_c$). The critical index of the localization length $\\nu=1$.  }\n\\end{figure}\n\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.0\\columnwidth]{F221.eps}\n\\end{center}\n\\caption{ Lyapunov exponents for $\\kappa=3$ and  $\\tau=1\/2$, and $\\lambda\/t=0.5,1.0, 1.5$.  The discrete points are the numerical results for all the eigenenergies. The solid lines are given by Eq.(27). The mobility edges for $\\lambda\/t=0.5$ are indicated by black arrows.  Near mobility edges of the localized-critical transition (e.g., $E_c\/t\\simeq\\pm \\sqrt{2}$ for $\\lambda\/t=0.5$), the Lyapunov exponent $\\gamma(E)\\propto |E-E_c|$ approaches zero (as $E\\rightarrow E_c$). The critical index of the localization length $\\nu=1$.}\n\\end{figure}\nWhen $\\epsilon$ is very small, using the facts that $\\gamma(E,\\epsilon)\\geq0$  and $\\gamma_B(E,\\epsilon)$ is  a convex and piecewise linear function of $\\epsilon$ \\cite{Avila2015,YONGJIAN1},  one can get\n\\begin{align}\n&\\gamma(E,\\epsilon)=Max\\{0,\\gamma_A(E,\\epsilon)+\\gamma_B(E,\\epsilon)\\},\\notag\\\\\n&=\n\\frac{1}{\\kappa}Max\\{0,\\epsilon+\\ln(|\\frac{|P|+\\sqrt{P^2+16\\lambda^2\\tau}}{4\\lambda(1+\\sqrt{1-\\tau})}|)\\}\n\\end{align}\nFurthermore, when $\\epsilon=0$, the Lyapunov exponent $\\gamma(E)\\equiv\\gamma(E,\\epsilon=0) $ is\n\\begin{align}\n\\gamma(E)=\\frac{1}{\\kappa}Max\\{\\ln|\\frac{|P(E)|+\\sqrt{P^2(E)+16\\lambda^2\\tau}}{4\\lambda(1+\\sqrt{1-\\tau})}|,0  \\}\n\\end{align}\nwhere\n\\begin{align}\nP(E)=(\\tau E^2+4\\lambda^2)a_{\\kappa}+\\tau a_{\\kappa-2}-2\\tau E a_{\\kappa-1}\n\\end{align}\nand\n\\begin{align}\na_\\kappa=\\frac{1}{\\sqrt{E^2-4}}[(\\frac{E+\\sqrt{E^2-4}}{2})^{\\kappa-1}-(\\frac{E-\\sqrt{E^2-4}}{2})^{\\kappa-1}].\n\\end{align}\n\nWhen $\\tau=0$, then $P(E)=4\\lambda^2a_{\\kappa}$, and\n\\begin{align}\n\\gamma(E)=\\frac{1}{\\kappa}Max\\{\\ln|\\lambda a_{\\kappa}|,0  \\}.\n\\end{align}\n\n\n\n\n\n\n\nThe above  formula Eq.(27)  has been  verified by our numerical results (see Figs.3 and 4).\nIn our numerical calculations, in order to get the correct Lyapunov exponents, on the one hand, the integer $L$ should be sufficiently large.\nOn the other hand, $L$ should be also much smaller than the system size $N$, i.e., $1\\ll L\\ll N$.\n\n To be specific, taking $\\kappa=3$, $\\tau=-2,1\/2$,  system size $N=3\\times1000$, we get the $N=3\\times1000$ eigenenergies and eigenstates.\n  Then we calculate the Lyapunov exponents numerically for all the eigenenergies [see the several sets of discrete points in  Figs.3 and 4].\n In our numerical calculation, we take $L=200$,  phase $\\phi=0$, $\\psi(0)=0$ and $\\psi(1)=1$ in Eq.(13).\n  The solid lines of Figs.3 and 4 are given by Eq.(27) with same parameters. It is shown that most of all discrete points fall onto the solid lines.\n\nHowever, we also note that there are some discrete points of localized states which are not on the solid lines. This is because these localized wave functions  are too near the left-hand boundary of system.\n\n\n\\section{mobility edge and critical region }\nIn this section, based on the Lyapunov exponent formula Eq.(27), we determine the mobility edges and the critical region.\n By the Eq.(27), the mobility edges $E_c$ which separate the localized states from the critical states, is determined by\n\\begin{align}\n\\gamma(E=E_c)=\\frac{1}{\\kappa}\\ln|\\frac{|P(E)|+\\sqrt{P^2(E)+16\\lambda^2\\tau}}{4\\lambda(1+\\sqrt{1-\\tau})}|=0\n\\end{align}\nthen\n\\begin{align}\n|P(E=E_c)|=4|\\lambda|\\sqrt{1-\\tau},\n\\end{align}\nThe critical regions which consist of critical states is given by\n\\begin{align}\n|P(E)|<4|\\lambda|\\sqrt{1-\\tau}.\n\\end{align}\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.0\\columnwidth]{F11.eps}\n\\end{center}\n\\caption{ Phase diagram in $(\\lambda, E)$ plane for $\\kappa=2$ and $\\tau=-2$. When $E$ is near zero, there exists localized-critical transitions. The blue solid lines are the phase boundaries (mobility edges $E_c$), which  are given by Eq.(32).\n Standard deviations are represented with different colors. }\n\\end{figure}\n\n\nBy expanding the Lyapunov exponent near the mobility edges $E_c$, we get\n\\begin{align}\n\\gamma(E)\\propto |E-E_c|\\rightarrow0, \\ as \\ E \\rightarrow E_c.\n\\end{align}\nThen the localization length is\n\\begin{align}\n\\xi(E)\\equiv1\/\\gamma(E)\\propto |E-E_c|^{-1}\\rightarrow\\infty,\\ as \\ E \\rightarrow E_c.\n\\end{align}\nIts critical index is $1$ [see the finite slopes of solid lines near $E_c$ in Figs.3 and 4].\n\n\n\nIn order to further distinguish the localized states from the  critical states, we also numerically calculate standard deviation of coordinates of eigenstates \\cite{Boers2007}\n\\begin{align}\\label{37}\n&\\sigma=\\sqrt{\\sum_{i}(i-\\bar{i})^2|\\psi(i)|^2},\n\\end{align}\nwhere the average value of coordinate $\\bar{i}$ is\n\\begin{align}\n\\bar{i}=\\sum_{i}i|\\psi(i)|^2.\n\\end{align}\nThe standard deviation $\\sigma$ describes the spatial extension of wave function in the lattices.\nThe phase diagram in $[\\lambda ( \\tau)- E]$ plane is reported in Figs.5,6,7 and 8.  In Figs.5,6,7 and 8, the standard deviations of coordinates are represented with different colors.\nFrom Figs.5,6,7 and 8, we can see that when the states are localized, standard deviations of coordinates are very small. For critical states, the standard deviations are very large.\n\n\n\nFrom Figs. 5, 7, and 8, we see that when $\\kappa>1$, there are $\\kappa-1$ loops for small hopping strength $\\lambda\/t$ in the phase diagram.\nWithin the loops, the Lyapunov exponent is positive, i.e., $\\gamma(E)>0$. Then if there is some eigenstates in the loops, these states would be localized states.\nHowever, numerical results show that there are no eigenenergies fallen into the loops.\n\nWhen $\\lambda\/t=0$, we see the system has $\\kappa$ eigenenergies  with multitude degeneracies. This is because  when  $\\lambda\/t=0$, the system in fact is composed by a lot of identical independent unit cells with $\\kappa$ lattice sites. Within a unit cell, there are exactly $\\kappa$ eigenenergies. With the increasing of $\\lambda\/t$, the degeneracies are removed and the system enters into the critical regions.\nIn addition, we also find that  the parity of $\\kappa$ also has important effects on the phase diagram.\n\n\\subsection{$\\kappa$ is an even number}\nWhen $\\kappa$ is even number and the energy $E$ is very near zero, there exist Anderson localizations if potential hopping strength $\\lambda$ is sufficiently large (see Fig.5 for $\\kappa=2$).\nEspecially when $E\\rightarrow0$, we get two critical hopping strengths\n\\begin{align}\n\\lambda_{c1}=\\pm \\frac{1-\\sqrt{1-\\tau}}{2} \\ \\ \\& \\ \\ \\lambda_{c2}=\\pm \\frac{1+\\sqrt{1-\\tau}}{2},\n\\end{align}\nwhich are independent of the $\\kappa$.\nHere $\\lambda_{c1}$ ($\\lambda_{c2}$) corresponds the point $A$ ($B$) in Fig.5.\nIt is noticed that $\\lambda_{c2}$ coincides the critical $\\lambda_c$ in Sec.II, where the average growth rate of zero-energy wave function is zero.\n\nFrom Fig.5, we can see that when the hopping $\\lambda$ is in the interval $|\\lambda_{c1}|<\\lambda<|\\lambda_{c2}|$, the eigenstates are critical states. Outside the interval, the eigenstates\nbecome localized states. So there exist Anderson localization transitions for even $\\kappa$.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.0\\columnwidth]{F1.eps}\n\\end{center}\n\\caption{Phase diagram for $\\kappa=1$. When energy is near zero-energy, the states are always critical states with vanishing Lyapunov exponent.}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.0\\columnwidth]{F21.eps}\n\\end{center}\n\\caption{ Phase diagram in $(\\lambda, E)$ plane for $\\kappa=3$ and $\\tau=-2$. When $E$ is near zero, there are no localized-critical transitions. The blue solid lines are the phase boundaries (mobility edges $E_c$), which  are given by Eq.(32).\n Standard deviations are represented with different colors. }\n\\end{figure}\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.0\\columnwidth]{F22.eps}\n\\end{center}\n\\caption{ Phase diagram in $(\\lambda, E)$ plane for $\\kappa=3$ and $\\tau=1\/2$. There exists localized-critical transitions for nonzero energy states. The blue solid lines are the phase boundaries (mobility edges $E_c$), which  are given by Eq.(32).\n Standard deviations are represented with different colors.}\n\\end{figure}\n\n\n\n\n\n\n\\subsection{$\\kappa$ is an odd number}\nWhen $\\kappa=1$, the system has no  the energy scale $t$ , then we would use the units of $\\lambda=1$. In Fig.6,  we report the phase diagram of  the $\\tau-E$ plane.\nFrom Fig.6, we see if $\\tau<0$, all the eigenstates are in the critical region. Only when $\\tau$ is positive and sufficiently large, the system has localized states.\n When $\\tau$ gets nearer and nearer to $1$, i.e., $\\tau\\rightarrow 1^{-}$, the range of  energy spectrum becomes larger and larger, and eventually diverges. This is  because when $\\tau=1$, the Hamiltonian defined by the Eq.(1) would be an unbounded operator, e.g, $\\frac{2\\lambda \\cos(2\\pi \\beta i+\\phi)}{\\sqrt{1-\\tau \\cos^2(2\\pi \\beta i+\\phi)}}=\\frac{2\\lambda \\cos(2\\pi \\beta i+\\phi)}{|\\sin(2\\pi \\beta i+\\phi)|}$ diverges for some lattice site index $i$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.0\\columnwidth]{F5.eps}\n\\end{center}\n\\caption{ Standard deviations of localized states and critical states for parameters $\\kappa=3$, $\\tau=-2$ and $\\lambda\/t=1.5$. The eigenenergy $E_n$ increases gradually as eigenstate index $n$ runs from $1$ to $3000$ (along the black dashed line of Fig.7).}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\nWhen $\\kappa>1$  and the energy $E$ is very near zero, there are no Anderson localizations for a given potential strength $\\lambda$ (see Figs.7, and 8).\nIf $\\lambda$ is very large, i.e, $\\lambda\\rightarrow\\pm\\infty$, the mobility  edge can be obtained by Eq.(32), i.e.,\n\\begin{align}\n&E_c=\\pm\\frac{2\\sqrt{1-\\tau}}{(\\kappa-1)|\\lambda|}, \\ as \\ \\lambda\\rightarrow\\pm\\infty .\n\\end{align}\nIt is shown that for an arbitrarily strong quasiperiodic hopping strength $\\lambda$, there always exists an energy window, i.e., $-\\frac{2\\sqrt{1-\\tau}}{(\\kappa-1)|\\lambda|}<E<\\frac{2\\sqrt{1-\\tau}}{(\\kappa-1)|\\lambda|}$, in which there exist critical states. With the increasing of $\\lambda$, the energy window of critical regions gets smaller and smaller.\n\n\n\nFor given parameters $\\kappa=3$, $\\tau=-2$, $\\lambda\/t=1.5$, we report the standard deviations of all the eigenstates in Fig.9. It is shown that in comparison with localized states, the critical states have much larger fluctuation of standard deviations.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.0\\columnwidth]{F7.eps}\n\\end{center}\n\\caption{ The inverse\nparticipation ratio $IPR_n$ of all the eigenstates for system size $N=3\\times250,3\\times500, 3\\times1000$ and $ N=3\\times2000$. The eigenenergy $E_n$ increases gradually as eigenstate index $n$ runs from $1$ to $N$ (along the black dashed line of Fig.8).}\n\\end{figure}\n\n\n\nIn order to investigate the properties of  the wave functions of critical states, we also numerically calculate the inverse participation ratio $IPR_n$ of  all eigenstates for different system sizes $N=3\\times 250,3\\times 500,3\\times1000$ and $N=3\\times2000$ \\cite{Xiaopeng,Deng}, i.e.,\n\\begin{align}\n&IPR_n=\\sum_{i}|\\psi_n(i)|^4.\n\\end{align}\nwhere $\\psi_n(i)$ is the normalized wave function for $n-th$ eigenstate.\nThe results are reported in Fig.10.\nWe find that the IPRs of localized states are basically same for different system sizes $N$,  while the IPRs of critical states have much larger fluctuation.\n\nOn the whole,   the IPR of critical states decreases with the increasing of system size $N$.\n The decreasing law can be captured by a function\n\\begin{align}\\label{H0}\n\\overline{IPR}\\propto 1\/N^x,\n\\end{align}\nwhere $\\overline{IPR}$ is an average value of the  $IPR$ within a typical energy interval.\n It is believed  that, for localized states, the scaling exponent $x=0$.  While for extended states (like plane wave states),  the scaling exponent $x=1$. For critical states, the exponent should be $0<x<1$.\nFor different system sizes $N$, due to randomness of $IPR$ of critical states (see Fig.10),  it is difficult to get a definite scaling exponent $x$.\nHere we find that for the critical states in the energy interval $-0.12<E\/t<-0.12$, its average value $\\bar{x}\\simeq0.47$.\n\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.1\\columnwidth]{F9.eps}\n\\end{center}\n\\caption{ Lyapunov exponents and Avila's accelerations. (a): Lyapunov exponents for $\\kappa=3$, $\\tau=1\/2$ and $\\lambda\/t=0.5$ (along the blue dashed line of Fig.8).   (b): Avila's acceleration  for for $\\kappa=3$, $\\tau=1\/2$ and $\\lambda\/t=0.5$. The mobility edges $E_c=\\pm\\sqrt{2}t$ are indicated by black arrows.  }\n\\end{figure}\n\n\n\\subsection{Avila acceleration}\nIn addition, for the bounded quasi-periodic potentials, Avila also defined the acceleration  $\\omega(E)$ by \\cite{Avila2015}\n\\begin{align}\n\\kappa\\omega(E)=lim_{\\epsilon\\rightarrow 0_+}\\frac{\\gamma(E,\\epsilon)-\\gamma(E,0)}{\\epsilon}.\n\\end{align}\nUsing Eqs.(26), when real number $E$ is an eigenvalue,  we get the Avila acceleration\n\\begin{align}\n&\\kappa \\omega(E)\n\\left\\{\\begin{array}{cccc}\n1, \\ for  \\ energy \\ of \\ localized \\ state \\\\\n0, \\ for  \\  energy \\ of \\ critical \\ state\n  \\end{array}\\right.\n\\end{align}\nThe above results are verified by our numerical calculations [see Fig.11].\nTo be specific, taking $\\tau=1\/2$, $\\lambda\/t=0.5$ and $\\epsilon=0,0.1,0.2$, we calculate the Lyapunov exponents  with Eq.(13) (taking the complexified phase $\\phi\\rightarrow \\phi+i\\epsilon=i\\epsilon$) for interval $-2.2\\leq E\\leq2.2$ [see the three solid lines in panel (a) of Fig.11]. In our calculation, we take $L=200$, $\\psi(0)=0$ and $\\psi(1)=1$ in Eq.(13).\nAt the same time, we calculate the Lyapunov exponents for all the eigenenergies with same parameters [see the three sets of discrete points in panel (a) of Fig.11].\nWe can find that\nif $E$ is an eigenenergy of critical state [$\\gamma(E)=0$], the Lyapunov exponents are the same for all three different  $\\epsilon=0,0.1,0.2$.\nWhile when $E$ is an eigenenergy of localized state [$\\gamma(E)>0$], the Lyapunov exponents are different for three different  $\\epsilon=0,0.1,0.2$. Their differences are linearly proportional to $\\Delta\\epsilon=0.1$ in Fig. 11.\n\nBy taking $\\epsilon=0.1$, we also  approximately calculate the Avila's  acceleration $\\omega(E)$ by\n\\begin{align}\n\\kappa\\omega(E)\\simeq\\frac{\\gamma(E,\\epsilon)-\\gamma(E,0)}{\\epsilon},\n\\end{align}\n[see panel (b) of Fig.11]. It shows that when $E$ is an eigenenergy of localized state [$\\gamma(E)>0$], the Avila's  acceleration is 1. When $E$ is  an eigenenergy of critical  state [$\\gamma(E)=0$],  the Avila's  acceleration is 0.\nWe also notice that if $E$  is not an eigenenergy, the Avila's  acceleration is $-1$.\n\n\n\n Further combining Eq.(27) and Eq.(43),\nthen one can classify systems with different real parameter $E$ (different phases) by Lyapunov exponent and the quantized acceleration, i.e.,\n\\begin{align}\n &(a): \\gamma(E)>0 \\ \\ \\& \\ \\kappa\\omega(E)=-1,  \\ if \\ E \\ is \\ not\\ an\\ eigenvalue \\notag\\\\\n &(b): \\gamma(E)>0 \\ \\ \\& \\ \\kappa\\omega(E)=1,  \\ for \\ localized \\ state\\notag\\\\\n &(c): \\gamma(E)=0 \\ \\ \\& \\ \\kappa\\omega(E)=0,  \\ for \\  critical \\ state.\n\n\\end{align}\n\n\n\n\n\n\n\n\n\n\n\\section{summary}\nIn conclusion, we investigate the localization properties of the one-dimensional lattice model with off-diagonal  mosaic quasiperiodic hopping.\n The parity of mosaic periodic has important effects on the localization of zero-energy states.\n When the mosaic period is odd, there always exists an energy window for critical states regardless hopping strength.\n While for even period, the states near zero-energy would become localized edge states for a sufficiently large hopping strength.\n It is found that there exist mobility edges which separate the localized states from critical states.\n Within the critical region, the spatial extensions of eigenstates have large fluctuation.\n\n The Lyapunov exponents and  mobility edges are  exactly obtained with Avila's theory.\n Furthermore, it is found that the critical index of localization length $\\nu=1$.\nFor $\\kappa=3$ and $\\tau=1\/2$, the numerical results show that the scaling exponent of inverse participation ratio (IPR) of critical states $x\\simeq0.47$. It is shown that these states indeed are critical states.\nIn addition,  it is shown that the Lyapunov exponent and Avila's acceleration can be used to classify the systems with different $E$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section*{Acknowledgements}\nThis work was supported by the NSFC under Grants Nos.\n11874127, 12061031, 12171039, 11871007, the Joint Fund with\nGuangzhou Municipality under No.\n202201020137, and the Starting Research Fund from\nGuangzhou University under Grant No.\nRQ 2020083.\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\\thispagestyle{empty}\nA common problem faced in machine learning is the lack of sufficient training data. For colonoscopy the majority of readily-available public image data is limited to individual frames or short sequences for benchmarking CAD-based polyp detection. Public colonoscopy videos of the entire colon structure are limited to rather low-quality capsule endoscopy video footage. The lack of ground truth camera poses further hampers the training of models for applications different from polyp detection such as: anatomical segment classification, visual place recognition (VPR), simultaneous localization and mapping (SLAM) and structure from motion (SfM). These applications require high-quality colonoscopy videos of the entire examinations covering all phases of the intervention.\n\\begin{center}\n    \\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{domain.jpeg}\n    \\caption{Examples of synthetic colonoscopy images.}\n    \\label{fig:synthetic_images}\n\\end{figure}\n\\end{center}\n\nA common solution to this is the rendering of virtual endoscopy (VE) videos based on CT colonography data. VE provides both, image sequences and ground truth poses of varying anatomy, but (without further investigation) differs substantially from the visual appearance of real colonoscopy images. This entails gaps that have to be addressed by proper domain adaptation methods as demonstrated in \\cite{mathew2020augmenting}. This, however, implies that synthesized images resemble colonoscopy images (and their anatomical locations) of small datasets which likely do not generalize well to unseen or less observed colon regions. \nDomain randomization, in contrast, utilizes a large amount of data which is randomly sampled over the entire configuration space with the variables being carefully predefined. It is important to note that domain randomization is practically applicable to only simulated data as some of the parameters such as textures, materials, occlusions and coat masks have to be properly controlled in a simulated environment have to be more elaborated than for generating VE images in order to enable visual appearance close to real colonoscopy images (see Fig. \\ref{fig:synthetic_images}). Powerful engines such as \\textit{Unity} have gained particular interest in the computer vision and robotics communities \\cite{borkman2021unity,tremblay2018training}, but have been rarely investigated in medical imaging \\cite{incetan2020vrcaps,billot2021synthseg}. \n\n\\begin{figure*}[ht]\n    \\centering\n    \\includegraphics[width=0.8\\textwidth]{Architecture.png}\n    \\caption{Overview of the utilized processing pipeline for generating synthetic images.}\n    \\label{fig:architecture}\n\\end{figure*}\n\nGiven sufficient capabilities of simulation, models can solely be trained on domain-randomized data while still achieving high generalization performance for inference on real-world test data.\n\nThis paper presents an exemplary implementation of domain randomization for colonoscopy with all required algorithmic components. It is built up on prior work \\cite{incetan2020vrcaps} and supplements the latter by automated domain-randomized video recording through following waypoints along the interior colon's centerline.\n\n\n\\section{Material and Methods} \n\n\\subsection{Colon segmentation}\nAt first, a CT colonography (CT with radiocontrast material) obtained from TCIA is imported in \\textit{3D Slicer} for semi-automatic colon segmentation which is carried out as follows. A ROI around the colon is set manually with its image content being  thresholded. Subsequently we apply region-based segmentation on the (thresholded) mask to further delineate the colon structure. The segmentation mask is manually curated to ensure optimal results for successive steps.\n\n\\subsection{Centerline extraction}\nFor automated image collection we require an appropriate camera path through the interior colon. For this purpose, we estimate the centerline within the colon structure based on the prior work of \\cite{wan2002automatic}. The key idea is to plan an obstacle-free (w.r.t colon wall) path from the anus (colon entry) to the caecum. Since the intuitive approach based on the shortest path estimation tends to get too close to corners in turns, Wan et al. propose to explicitly incorporate the inversed map of distances to the colon wall \\cite{wan2002automatic} which was demonstrated to achieve optimal results with paths being exactly centered. Subsequently we sample equidistant waypoints along the extracted centerline which will be utilized within the simulation. Currently, we manually pick start and end points of the centerline extraction which, however, could be replaced by automatic anatomical landmark prediction through heatmap regression.\n\n\\subsection{3D model preparation}\nNext, the colon segmentation is imported in \\textit{Blender} for UV editing. Generally, a mesh is created surrounding the organ that can be edited along the vertices of the object. This mesh allows \\textit{UV mapping} which is a method for projecting a 3D model surface onto a 2D plane for texture mapping. An UV editing tool as part of \\textit{Blender} offers the possibility of unwrapping the 3D object onto a 2D plane where textures can be applied seamlessly throughout the region of the colon. This texture gives a realistic pattern to the object. Default shaders in \\textit{Blender} enable to change material properties corresponding to colon such as surface IOR, secular tint and anisotropy to further enhance the realism. \n\n\\subsection{Photorealistic rendering}\nThe 3D model prepared in \\textit{Blender} is subsequently imported in \\textit{Unity} which provides high definition render pipelines for our simulation environment that can produce photorealistic colonoscopy images. This virtual engine is commonly used for game development and has drawn particular interest in computer vision research due to its powerful graphical simulation platform for generating synthetic images. Using \\textit{Unity} we are able to synthesize images where parameters such as lighting, materials, occlusions, transparency and coat mask are altered to give it a more realistic appearance. These parameters are carefully selected such that real-world characteristics are optimally mimicked.  As a starting base we utilize parts of the \\textit{VR-Caps} project simulating a capsule endoscopic camera within \\textit{Unity} \\cite{incetan2020vrcaps}. A 3D model of this capsule with predefined attributes of an attached camera is placed inside the colon which is used for data collection. Adjusting these parameters is crucial for both mimicking real endoscopy and augmenting the data. The table below shows the camera parameters and post-processing effects required to achieve a fully synthetic model of the colon. For potential navigation tasks it is possible to additionally store corresponding depth images. \n\n\\begin{table}[!h]\n\\label{tab:fonts}\n\\centering\n\\begin{tabular}{|c|c| }\n\\hline\nAttributes &  Values\\\\\n\\hline\nSurface Metallic \t\t\t\t& 0.3 \\\\\nSurface Smoothness\t\t\t& 0.7 \\\\\nLens Intensity \t\t\t& 0.1\\\\\nChromic Abberation\t\t\t& 0.5 \\\\\nCoat Mask\t\t\t& 0.435\t \\\\\nCamera's Field of View \t\t\t& 91.375\t \\\\\nFocal length\t\t\t\t& 159.45\\\\\nISO \t\t& 200 \\\\\nAperture \t\t\t\t& 16\\\\\nAnisotropy \t\t\t& 1 \\\\\n\\hline\n\\end{tabular}\n\\caption{Camera parameters and Post-processing Effects}\n\\end{table}\n\n  \n \n   \n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\textwidth]{DR-Samples_Colon.png}\n    \\caption{Synthesized, domain-randomized images captured at the same pose inside the colon. Textures are obtained as from random patterns as well as synthetic patterns mimicking mucosa appearances.}\n    \\label{fig:dr-samples}\n\\end{figure*}\n\n\\subsection{Automated Video Rendering}\nManually collecting data for endoscopy becomes highly time-consuming when creating synthetic datasets consisting of all the required variations and diversity. For domain randomization we need to record sequences of images each time with different textures and materials which entails substantial individual setup. Thus, an approach for automating the process of data collection is introduced, which allows us to collect numerous samples inside the colon with different parameters. For this purpose we make use of the \\textit{scripting API} offered by \\textit{Unity} which gives access to the simulation environment and interactive components via executable scripts. Firstly, the simulated capsule is introduced to the colon and then automatically steered along the waypoints of the centerline (see Fig. \\ref{fig:waypoints}). The \\textit{Unity engine} is setup appropriately such that it enables smooth camera motion while following the waypoints.  \nOur path following script consist of two parts:  an \\textit{initialization} function which runs all required initial setups (parameter setup, initial capsule pose) and an \\textit{update} function which constantly controls the movement of the capsule (along the waypoints) and triggers actions such as changing parameters (e.g. lightning, texture). All images captured by the camera of the capsule are recorded. The parameters can either be adjusted on the fly allowing to capture images at the same pose with varying conditions or alternatively it is possible to alter the parameter set only for entire traversals. \\textit{Unity} also enables to configure the capsule's speed and camera's field of view and targeted frame rate (FPS). \n\n\\section{Results and Discussion} \n\nWe evaluate our simulation qualitatively based on image renderings for varying parameters which becomes particularly apparent when randomizing surface material and textural patterns. This is illustrated by Fig.\\ref{fig:dr-samples} which shows different renderings from the same captured from the same camera pose inside the colon. \nFig. \\ref{fig:waypoints} shows an example of an extracted centerline and generated waypoints being followed for automated video recording. For comparison, Fig. \\ref{fig:real_samples} shows exemplary real and synthetic images respectively.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.5\\textwidth]{cenerline.png}\n    \\caption{Path following the centerline of the colon. The green line visualize the path and red circles waypoints being traced by the simulated capsule.}\n    \\label{fig:waypoints}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.5\\textwidth]{RE-VE-comparison.jpeg}\n    \\caption{Comparison of synthetic (top) and real (bottom) colonoscopy images.}\n    \\label{fig:real_samples}\n\\end{figure}\n\\section{Conclusion}\nThis paper presented a pipeline for generating synthetic colonoscopy videos that can be used to improve training of deep learning models. By controlling environment (e.g. texture, reflectance) as well as virtual camera (e.g. lightning) properties we are able to simulate conditions being observed in inference but hardly ever presented in training data which is particularly the case for small-scale datasets. Inspired by substantive improvements reported on computer vision and robotics applications and limited prior work (VR-Caps)  \\cite{incetan2020vrcaps} we motivate to utilize domain-randomized synthesization for video colonoscopy. In our future work, we will incorporate this additional data for training deep learning-based approaches to SfM, SLAM and 3D reconstruction. In order to further simplify the variation in patient anatomy, we will investigate (fully) automatic segmentation of the colon in CT scans as well as an alternative to the 3D model preparation in \\textit{Blender}. \n\\pagebreak\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and main results}\n\n\n\nConsider the one-dimensional Schr\\\"odinger equation\n\\begin{equation}\\label{eq_Schrodinger}\n {\\rm i}\\partial_t u=\\frac{\\nu(E)}{2}H_0u + W(E,\\omega t, x, -{\\rm i} \\partial_x)u,\\qquad x\\in{\\mathbb R} ,\n\\end{equation}\nwhere, we assume that\n\\begin{itemize}\n  \\item the frequencies $\\omega\\in {\\mathbb R}^d$, $d\\geq 1$, satisfy the {\\it Diophantine} condition (denoted by $\\omega\\in {\\rm DC}_d(\\gamma,\\tau)$ for $\\gamma>0$, $\\tau>d-1$):\n$$\\inf_{j\\in{\\mathbb Z}}|\\langle n,\\omega\\rangle-j|>\\frac{\\gamma}{|n|^\\tau},\\qquad \\forall \\ n\\in{\\mathbb Z}^d\\setminus\\{0\\},$$\n  \\item the parameter $E\\in {{\\mathcal I}}$, an interval $\\subset {\\mathbb R}$, and $\\nu\\in C^2({{\\mathcal I}},{\\mathbb R})$ satisfies\n  $$|\\nu'(E)|\\geq l_1,\\quad |\\nu''(E)|\\leq l_2,\\qquad \\forall \\  E\\in{{\\mathcal I}},$$\nfor some $l_1,l_2>0$,\n  \\item $H_0$ is the {\\it one-dimensional quantum harmonic operator}, i.e.\n$$(H_0u)(x):=-(\\partial_{x}^2 u)(x)+x^2\\cdot u(x),\\qquad \\forall \\ u\\in L^2({\\mathbb R}),$$\n  \\item $W(E,\\theta, x,\\xi)$ is a quadratic form of $(x,\\xi)$:\n$$W(E, \\theta, x,\\xi)=\\frac12 \\big(a(E,\\theta) x^2+2b(E,\\theta) x\\cdot\\xi+c(E,\\theta) \\xi^2\\big),$$\nwith $a,b,c:{{\\mathcal I}}\\times {\\mathbb T}^d\\to {\\mathbb R}$, all of which are $C^2$ w.r.t. $E\\in{{\\mathcal I}}$ and $C^\\omega$ w.r.t. $\\theta\\in{\\mathbb T}^d:=({\\mathbb R}\/{\\mathbb Z})^d$, and for every $E\\in{{\\mathcal I}}$, for $m=0,1,2$,\n$|\\partial^m_E a(E,\\cdot)|_r:=\\sup_{|\\Im z|<r}|\\partial^m_E a(E,z)|$, $|\\partial^m_E b(E,\\cdot)|_r$, $|\\partial^m_E c(E,\\cdot)|_r$ are small enough.\n\\end{itemize}\nWe will prove that, for almost every $E$ in the interval ${{\\mathcal I}}$, Eq. (\\ref{eq_Schrodinger}) is reducible, i.e., via a unitary transformation, Eq. (\\ref{eq_Schrodinger}) is conjugated to an equation which is independent of time (while the transformation depends on time in an analytic quasi-periodic way).\nAccording to the reducibility, we deduce the behavior of Sobolev norms for the solutions to Eq. (\\ref{eq_Schrodinger}).\n\n\n\\subsection{Reducibility for harmonic oscillators}\n\nOur main result is the following:\n\n\\begin{Theorem}\\label{thm_Schro}\nThere exists $\\varepsilon_*=\\varepsilon_*(\\gamma,\\tau,r,d,l_1,l_2)>0$ such that if\n$$\\max_{m=0,1,2}\\left\\{\\left|\\partial_E^m a\\right|_r, \\ \\left|\\partial_E^m b\\right|_r, \\ \\left|\\partial_E^m c\\right|_r\\right\\}=:\\varepsilon_0\\leq \\varepsilon_*,\\qquad \\forall \\ E\\in{{\\mathcal I}},$$\nthen for a.e. $E\\in{{\\mathcal I}}$, Eq. (\\ref{eq_Schrodinger}) is reducible, i.e., there exists a time quasi-periodic transformation $U(\\omega t)$, unitary in $L^2$ and analytically depending on $t$, such that\nEq. (\\ref{eq_Schrodinger}) is conjugated to\n${\\rm i}\\partial_t v= G v$ by the transformation $u=U(\\omega t) \\, v$, with $G$ a linear operator independent of $t$.\n\nMore precisely,\nthere exists a subset\n$${{\\mathcal O}}_{\\varepsilon_0}=\\bigcup_{j\\in{\\mathbb N}}\\Lambda_j\\subset \\overline{{\\mathcal I}}$$ with $\\Lambda_j$'s being closed intervals \\footnote{In this paper, the ``closed interval\" is interpreted in a more general sense, i.e., it can be degenerated to a point instead of a positive-measure subset of ${\\mathbb R}$.}\nand ${\\rm Leb}({{\\mathcal O}}_{\\varepsilon_0})<\\varepsilon_0^{\\frac{1}{40}}$,\nsuch that the following holds.\n\\begin{enumerate}\n\\item For a.e. $E\\in{{\\mathcal I}}\\setminus{{\\mathcal O}}_{\\varepsilon_0}$, $G$ is unitary equivalent to $\\varrho H_0$ for some $\\varrho=\\varrho_E\\geq 0$;\n\\item If ${\\rm Leb}(\\Lambda_j)>0$, then\n\\begin{itemize}\n  \\item for $E\\in{\\rm int}\\Lambda_j$, $G$ is unitary equivalent to $-\\frac{\\lambda {\\rm i}}{2}(x\\cdot \\partial_x+ \\partial_x \\cdot x)$ for some $\\lambda=\\lambda_E> 0$;\n  \\item for $E\\in\\partial\\Lambda_j\\setminus\\partial{{\\mathcal I}}$, $G$ is unitary equivalent to $-\\frac{\\kappa}2 x^2$ for some $\\kappa=\\kappa_E\\in{\\mathbb R}\\setminus\\{0\\}$.\n\\end{itemize}\nIf ${\\rm Leb}(\\Lambda_j)=0$, then $G=0$ for $E\\in\\Lambda_j$.\n\\end{enumerate}\n\\end{Theorem}\n\nBefore giving its application on the growth of Sobolev norm, let us first make a review on previous works about the reducibility on harmonic oscillators as well as the relative KAM theory.\n\nFor 1-d harmonic oscillators with time periodic smooth perturbations, Combescure \\cite{Com87} firstly showed the pure point nature of Floquet operator (see also \\cite{DLSV2002, EV83, Kuk1993}).\nFor 1-d harmonic oscillators with time quasi-periodic bounded perturbations, we can refer to \\cite{GreTho11, Wang08, WLiang17} for the reducibility and the pure point spectrum of Floquet operator.\nFor 1-d harmonic oscillators with unbounded time quasi-periodic perturbations, similar results can be found in \\cite{Bam2018, Bam2017, BM2018, Liangluo19}.\nIn investigating the reducibility problems, KAM theory for 1-d PDEs has been well developed by Bambusi-Graffi \\cite{BG2001} and Liu-Yuan \\cite{LY2010} in order to deal with unbounded perturbations.\n\nReducibility for PDEs in higher-dimensional case was initiated by Eliasson-Kuksin \\cite{EK2009}, based on their KAM theory \\cite{EK2010}.\nWe refer to \\cite{GrePat16} and \\cite{LiangWang19} for any dimensional harmonic oscillator with bounded potential. \nWe mention that some higher-dimensional results with unbounded perturbations have been recently obtained\n\\cite{BLM18, FGMP19, FG19, FGN19, Mon19}. \nHowever, a general KAM theorem for higher-dimensional PDEs with unbounded perturbations is far from success.\n\n\nRecently, Bambusi-Gr\\'ebert-Maspero-Robert \\cite{BGMR2018} built a reducibility result for the harmonic oscillators on ${\\mathbb R}^n$, ,$n\\geq 1$, in which the perturbation is a polynomial of degree at most two in $x$ and $-{\\rm i}\\partial_x$ with coefficients quasi-periodically depending on time.\nThe proof in \\cite{BGMR2018} exploits the fact that for polynomial Hamiltonians of degree at most $2$, there is an exact correspondence between classical and quantum mechanics, so that the result can be proved by exact quantization of the classical KAM theory which ensures reducibility of the classical Hamiltonian system.\nThe exact correspondence between classical and quantum dynamics of quadratic Hamiltonians was already exploited in the paper \\cite{HLS86} to prove stability and instability results for one degree of freedom time periodic quadratic Hamiltonians.\nTo prove our main result, we use the same strategy as \\cite{BGMR2018} and the reducibility result for the classical Hamiltonian by Eliasson \\cite{Eli1992}.\n\n\n\n\\subsection{Growth of Sobolev norms}\n\nBesides reducibility, the construction of unbounded solutions in Sobolev space for Schr\\\"odinger equations  attracts even more attentions.\n\nAs an application of Theorem \\ref{thm_Schro}, we can study the long time behaviour of its solution $u(t)$ to Eq. (\\ref{eq_Schrodinger}) in Sobolev space.\nFor $s\\geq 0$, we define Sobolev space\n$${{\\mathcal H}}^s:=\\left\\{\\psi\\in L^2({\\mathbb R}):H_0^{\\frac{s}2}\\psi \\in L^2({\\mathbb R})\\right\\}$$\nand Sobolev norm\n$\\|\\psi\\|_{s}:=\\|H_0^{\\frac{s}2} \\psi\\|_{L^2({\\mathbb R})}$.\nIt is well known that, for $s\\in {\\mathbb N}$, the above definition of norm is equivalent to\n$$\n\\sum\\limits_{\\alpha+\\beta\\leq s\\atop{\\alpha,\\beta\\in{\\mathbb N}} }\\|x^{\\alpha}\\cdot\\partial^{\\beta} \\psi\\|_{L^2({\\mathbb R})}.\n$$\n\\begin{remark}\\label{remark_norm_equiv}\n In view of Remark 2.2 of \\cite{BM2018}, we get that, for a given $\\psi\\in {{\\mathcal H}}^s$,\n\\begin{equation}\\label{norm_equiv}\n\\|\\psi\\|_{s}\\simeq \\|\\psi\\|_{H^s}+ \\|x^{s} \\psi\\|_{L^2},\n\\end{equation}\nreplacing $K_0=H_0$ in that remark by $K_0=H_0^{\\frac12}$, where $H^s$ means the standard Sobolev space and $\\|\\cdot\\|_{H^s}$ is the corresponding norm. Hence, to calculate the norm $\\|\\psi\\|_s$, $s\\geq 0$, it is sufficient to focus on $\\|x^{s} \\psi\\|_{L^2}$ for $s\\geq0$ and $\\|\\psi^{(s)}\\|_{L^2}$ for $s\\in{\\mathbb N}$.\n\\end{remark}\n\n\nFor different types of reduced systems, Sobolev norm of solution exhibits different behaviors.\n\n\n\n\\begin{Theorem}\\label{thm_Schro_sobolev} Under the assumption of Theorem \\ref{thm_Schro}, for any $s\\geq 0$, and any non-vanishing initial condition $u(0)\\in {{\\mathcal H}}^s$, the following holds true for the solution $u(t)$ to Eq. (\\ref{eq_Schrodinger}) for $t\\geq0$.\n\\begin{enumerate}\n\\item For a.e. $E\\in{{\\mathcal I}}\\setminus{{\\mathcal O}}_{\\varepsilon_0}$,\n$\nc \\leq \\|u(t)\\|_{s}\\leq C\n$.\n\\item If ${\\rm Leb}(\\Lambda_j)>0$, then\n\\begin{itemize}\n  \\item for $E\\in{\\rm int}\\Lambda_j$,\n$c  e^{\\lambda st}  \\leq \\|u(t)\\|_{s} \\leq C  e^{\\lambda st}$,\n  \\item for $E\\in\\partial \\Lambda_j\\setminus \\partial{{\\mathcal I}}$,\n$\nc |\\kappa|^s t^s \\leq \\|u(t)\\|_{s}\\leq C |\\kappa| (1+ t^2)^{\\frac{s}2}\n$.\n\\end{itemize}\nIf ${\\rm Leb}(\\Lambda_j)=0$, then for $E\\in \\Lambda_j$, $c  e^{\\lambda st}  \\leq \\|u(t)\\|_{s} \\leq C  e^{\\lambda st}$.\n\\end{enumerate}\nHere $\\lambda=\\lambda_E$ and $\\kappa=\\kappa_E$ are the same with Theorem \\ref{thm_Schro} and $c, \\, C>0$ are two constants depending on $s$, $E$ and $u(0)$.\n\\end{Theorem}\n\n\nLet us make more comments on constructing solutions growing with time in Schr\\\"odinger equations.\nBourgain \\cite{Bou99} built logarithmic lower and upper growth bounds for linear Schr\\\"odinger equation on ${\\mathbb T}$ by\nexploiting  resonance effects.  And the optimal polynomial growth example was given by Delort \\cite{Del2014} for 1-d harmonic oscillator with a time periodic order zero perturbation. Maspero \\cite{Mas2018} reproved the result of Delort by exploiting the idea in \\cite{GY00}. In \\cite{BGMR2018}, the authors also considered the higher-dimensional harmonic oscillator with a linear perturbation in $x$ and $-{\\rm i}\\partial_x$ with time quasi-periodic coefficients. Under the Diophantine condition of frequencies, the time-dependent equation can be reduced to a special ``normal form\" independent of time (see Theorem 3.3 of \\cite{BGMR2018}), which implies the polynomial growth of Sobolev norm.   There are also many literatures, e.g., \\cite{BGMR2019, MR2017}, which are relative to the upper growth bound of the solution in Sobolev space.\n\nFrom the above mentioned literatures, we can see that almost all the growth results of lower growth bound of the solution are closely related to the resonance phenomenon. However, it is not clear to us which kind of parameter set is connected to the growth of Sobolev norm.\nComparing with all the above results, we introduce the parameter set $\\bigcup_{j\\in{\\mathbb N}}\\Lambda_j$ following \\cite{Eli1992}, in which the solutions has exponential lower and upper growth bounds, while on the boundaries of this set the solutions has polynomial lower and upper growth bounds.  In the following, we will present several concrete examples to show that the set $\\bigcup_{j\\in{\\mathbb N}}\\Lambda_j$ is of positive measure.\n\n\n\\subsection{Examples with ${\\rm Leb}({{\\mathcal O}}_{\\varepsilon_0})>0$}\n\nIn view of Theorem \\ref{thm_Schro} and \\ref{thm_Schro_sobolev}, the growth of Sobolev norm can be obtained via the reducibility if ${\\rm Leb}({{\\mathcal O}}_{\\varepsilon_0})>0$.\nWe need to point that the time-dependent quadratic perturbation $W(E,\\omega t, x, -{\\rm i} \\partial_x)$ with ${\\rm Leb}({{\\mathcal O}}_{\\varepsilon_0})>0$ exists universally. In other words, it is a quite ``extreme\" case that\n$${\\rm Leb}(\\Lambda_j)=0, \\qquad \\forall \\ j\\in{\\mathbb N}.$$\nWe have the following concrete examples.\n\n\\\n\nFor ${{\\mathcal I}}={\\mathbb R}$, $\\nu(E)=E$, the equation\n\\begin{equation}\\label{example_1}\n {\\rm i}\\partial_t u=\\frac{E}{2}H_0u + \\left(\\frac{a(\\omega t)}{2} x^2-\\frac{b(\\omega t)}2\\left(x\\cdot{\\rm i}\\partial_x+{\\rm i}\\partial_x\\cdot x\\right)-\\frac{c(\\omega t)}2 \\partial_x^2 \\right) u,\n\\end{equation}\nsatisfies the assumptions of Theorem \\ref{thm_Schro} if $a,b,c\\in C^\\omega({\\mathbb T}^d,{\\mathbb R})$ are small enough. Hence, for Eq. (\\ref{example_1}), the reducibility and the behaviors of ${{\\mathcal H}}^s$ norm of solutions described in Theorem \\ref{thm_Schro_sobolev} can be obtained.\n\\begin{Theorem}\\label{thm_example_1}\nFor generic $a,b,c\\in C^\\omega({\\mathbb T}^d,{\\mathbb R})$ with $|a|_r, |b|_r, |c|_r$ small enough (depending on $r,\\gamma,\\tau,d$), the conclusions of Theorem \\ref{thm_Schro} and \\ref{thm_Schro_sobolev} hold for Eq. (\\ref{example_1}) for ${{\\mathcal I}}={\\mathbb R}$ with ${\\rm Leb}({{\\mathcal O}}_{\\varepsilon_0})>0$.\n\\end{Theorem}\n\n\\\n\n\n\nFor $\\nu(E)=\\sqrt{E}$, consider the equation\n\\begin{equation}\\label{eq_Schrodinger-example}\n {\\rm i}\\partial_t u=\\frac{\\sqrt{E}}{2} H_0 u -\\frac{q(\\omega t)}{2\\sqrt{E}}\\left(x^2-x\\cdot{\\rm i}\\partial_x-{\\rm i}\\partial_x\\cdot x-\\partial^2_x \\right)u.\n\\end{equation}\nwith $q\\in C_r^\\omega({\\mathbb T}^d,{\\mathbb R})$. The equation is important, since as we will show later, it is closely related to quasi-periodic Schr\\\"odinger operator.\n\\begin{Theorem}\\label{thm_example_schro}\nFor generic $q\\in C^\\omega({\\mathbb T}^d,{\\mathbb R})$, the conclusions of Theorem \\ref{thm_Schro} and \\ref{thm_Schro_sobolev} hold for Eq. (\\ref{eq_Schrodinger-example}) for ${{\\mathcal I}}=[E_0,E_1]$ with ${\\rm Leb}(\\Lambda_j)>0$ for infinitely many $j$'s, where $E_0>0$ is large enough (depending on $|q|_r$) and $E_1<\\infty$.\n\\end{Theorem}\n\n\\\n\n\nTheorem \\ref{thm_example_1} gives the example that ${\\rm Leb}(\\Lambda_j)>0$ for at least one $j$, while Theorem \\ref{thm_example_schro} gives the  example that  ${\\rm Leb}(\\Lambda_j)>0$ for infinitely many $j$'s. Indeed, if the dimension of the frequency $d=2$, we could even gives ${\\rm Leb}(\\Lambda_j)>0$ for every $j$'s.  To construct such an example, we consider\n \\begin{equation}\\label{eq_AMO}\n  {\\rm i}\\partial_t u=\\frac{\\nu(E)}{2} H_0 u+ \\left(\\frac{a(E,\\omega t)}{2} x^2-\\frac{b(E,\\omega t)}2\\left(x\\cdot{\\rm i}\\partial_x+{\\rm i}\\partial_x\\cdot x\\right)-\\frac{c(E,\\omega t)}2 \\partial_x^2 \\right) u.\n \\end{equation}\nwhere $\\nu(E)=\\cos^{-1}(-\\frac{E}{2})$,  ${{\\mathcal I}}\\subset[-2+\\delta,2-\\delta]$ with $\\delta$ a small numerical constant (e.g., $\\delta=10^{-6}$). Then our result is the following:\n\n\n\n\n \\begin{Theorem}\\label{thm_AMO} There exist a sub-interval ${{\\mathcal I}}\\subset[-2+\\delta,2-\\delta]$ and $a,b,c:{{\\mathcal I}}\\times {\\mathbb T}^2\\to{\\mathbb R}$ with $a(E,\\cdot), \\, b(E,\\cdot), \\, c(E,\\cdot)\\in C^{\\omega}({\\mathbb T}^2,{\\mathbb R})$ for every $E\\in{{\\mathcal I}}$,\n\n such that the conclusions of Theorem \\ref{thm_Schro} and \\ref{thm_Schro_sobolev} hold for Eq. (\\ref{eq_AMO}). Moreover,  ${\\rm Leb}(\\Lambda_j)> 0$ for every $j\\in{\\mathbb N}$.\n \\end{Theorem}\n\n\n\n\\begin{remark} One can even further get precise size of ${\\rm Leb}(\\Lambda_j)$ according to \\cite{LYZZ}.\n\\end{remark}\n\n\\\n\nThe rest of paper will be organised as follows. In Section \\ref{sec_weyl}, which serves as a preliminary section, we recall the definition of Weyl quantization and some known results on the relation between classical Hamiltonian to quantum Hamiltonian.\nWe give an abstract theorem in Section \\ref{sec_abstract} on the reducibility for quantum Hamiltonian, provided that the reducibility for the corresponding classical Hamiltonian is known.\nBy applying this abstract theorem, we exploit the connection between reducibility and property of Sobolev norm.\nThe abstract theorem is proved in Section \\ref{sec_reduc}.\nIn Section \\ref{sec_proof}, we prove the main result just by verifying the hypothesis of abstract theorem.\nIn Section \\ref{sec_pr_examples}, the proofs of Theorem \\ref{thm_example_1} -- \\ref{thm_AMO} are given.\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Classical Hamiltonian and quantum Hamiltonian}\\label{sec_weyl}\n\nTo give some preliminary knowledge,\nlet us recall the definition of Weyl quantization, which relates the classical and quantum mechanics, and its properties. The conclusions listed in this section can also be found in \\cite{BGMR2018}.\n\n\nThe Weyl quantization is the operator ${\\rm Op}^W:f\\mapsto f^W$ for any symbol $f=f(x,\\xi)$, with $x,\\xi\\in{\\mathbb R}^n$, where $f^{W}$ is the Weyl operator of $f$:\n$$\\left(f^{W} u\\right)(x)=\\frac{1}{(2\\pi)^n}\\int_{y, \\, \\xi\\in{\\mathbb R}^n} f\\left(\\frac{x+y}{2},\\xi\\right) u(y) \\, dy  \\, d\\xi,\\qquad \\forall \\ u\\in L^2({\\mathbb R}^n).$$\nIn particular, if $f$ is a polynomial of degree at most $2$ in $(x,\\xi)$, then $f^W$ is exactly $f(x,-{\\rm i}\\partial_x)$.\n\n\n\nFor the $1-$parameter family of Hamiltonian $\\chi(t, x, \\xi )$, with $t$ an external parameter, let $\\phi^\\tau(t,x,\\xi)$ be the time $\\tau-$flow it generates, precisely the\nsolution of\n$$\\frac{dx}{d\\tau}=\\frac{\\partial\\chi}{\\partial\\xi}(t, x, \\xi ),\\qquad \\frac{d\\xi}{d\\tau}=-\\frac{\\partial\\chi}{\\partial x}(t, x, \\xi).$$\nThe time-dependent coordinate transformation\n\\begin{equation}\\label{time1}\n(x,\\xi)=\\phi^1\\left(t,\\tilde x,{\\tilde\\xi}\\right)=\\left.\\phi^{\\tau}\\left(t,\\tilde x,{\\tilde\\xi}\\right)\\right|_{\\tau=1}\n\\end{equation}\ntransforms a Hamiltonian system with\nHamiltonian $h$ into a system with Hamiltonian $g$ given by\n$$g(t,\\tilde x,\\tilde\\xi)=h(\\phi^1(t,\\tilde x,\\tilde\\xi))-\\int_0^1 \\frac{\\partial\\chi}{\\partial t}(t,\\phi^{\\tau}(t,\\tilde x,\\tilde\\xi) ) d\\tau.$$\n\n\n\\begin{Lemma} [Remark 2.6 of \\cite{BGMR2018}] If the Weyl operator $\\chi^W(t, x, -{\\rm i}\\partial_x)$ is self-adjoint for any fixed $t$, then the transformation\n\\begin{equation}\\label{tran}\n\\psi=e^{{\\rm i}\\chi^W(t, x, -{\\rm i}\\partial_x)}\\tilde\\psi\n\\end{equation}\ntransforms the equation ${\\rm i}\\partial_t\\psi=H\\psi$ into ${\\rm i}\\partial_t\\tilde\\psi=G\\tilde\\psi$ with\n\\begin{eqnarray*}\nG&:=&e^{{\\rm i}\\chi^W(t, x, -{\\rm i}\\partial_x)}He^{-{\\rm i}\\chi^W(t, x, -{\\rm i}\\partial_x)}\\\\\n& & - \\, \\int_0^1 e^{{\\rm i}\\tau\\chi^W(t, x, -{\\rm i}\\partial_x)}\\left(\\partial_t \\chi^W(t, x, -{\\rm i}\\partial_x)\\right)e^{-{\\rm i}\\tau\\chi^W(t, x, -{\\rm i}\\partial_x)}d\\tau.\n\\end{eqnarray*}\n\n\\end{Lemma}\n\n\\begin{Proposition} [Proposition 2.9 of \\cite{BGMR2018}]\\label{Prop_hami} Let $\\chi(t, x, \\xi )$ be a polynomial of degree at most $2$ in $(x,\\xi)$ with smooth time-dependent coefficients.\nIf the transformation (\\ref{time1}) transforms a classical system with Hamiltonian $h$ into\na system with Hamiltonian $g$, then the transformation (\\ref{tran}) transforms the quantum Hamiltonian system\n$h^W$ into $g^W$.\n\\end{Proposition}\n\n\nNow, let us focus on the case $n=1$.\n\n\\begin{Lemma} [Lemma 2.8 of \\cite{BGMR2018}]\\label{lem_Sobolev}\nLet $\\chi(\\theta,x,\\xi)$ be a polynomial of degree at most $2$ in $(x,\\xi)$ with real coefficients depending in a $C^\\infty-$way on $\\theta\\in {\\mathbb T}^d$.\nFor every $\\theta\\in {\\mathbb T}^d$, the Weyl operator $\\chi^W(\\theta,x, -{\\rm i}\\partial_x)$ is self-adjoint in $L^2({\\mathbb R})$ and $e^{-{\\rm i}\\tau\\chi^W(\\theta,x, -{\\rm i}\\partial_x)}$ is unitary in $L^2({\\mathbb R}^n)$ for every $\\tau\\in{\\mathbb R}$.\nFurthermore, if the coefficients of $\\chi(\\theta,x,\\xi)$ are uniformly bounded w.r.t. $\\theta\\in {\\mathbb T}^d$, then for any $s\\geq 0$, there exist $c'$, $C' > 0$ depending on $\\|[H_0^s,\\chi^W(\\theta,x, -{\\rm i}\\partial_x)]H_0^{-s}\\|_{L^2\\mapsto L^2}$ and $s$, such that\n   \\begin{equation}\\label{change_Sobolevnorm}\n      c'\\|\\psi\\|_{s}\\leq \\|e^{-{\\rm i}\\tau\\chi^W(\\theta,x,-{\\rm i}\\partial_x)}\\psi\\|_{s}\\leq C'\\|\\psi\\|_{s},\\qquad \\tau\\in [0,1], \\quad \\theta\\in{\\mathbb T}^d.\n    \\end{equation}\n\\end{Lemma}\n\n\n\n\n\\section{Reducibility and growth of Sobolev norm}\\label{sec_abstract}\n\n\n\n\\subsection{An abstract theorem on reducibility}\n\n\nConsider the 1-d time-dependent equation\n\\begin{equation}\\label{eq_abs}\n {\\rm i}\\partial_t u=L^{W}(\\omega t, x, -{\\rm i} \\partial_x)u,\\qquad x\\in{\\mathbb R} ,\n\\end{equation}\nwhere $L^{W}(\\omega t, x, -{\\rm i} \\partial_x)$ is a linear differential operator, $\\omega\\in{\\mathbb T}^d$, $d\\geq 1$, and the symbol $L(\\theta, x,\\xi)$ is a quadratic form of $(x,\\xi)$ with coefficients\nanalytically depending on $\\theta\\in{\\mathbb T}^d$. More precisely, we assume that\n\\begin{equation}\\label{op_L}\nL(\\theta, x,\\xi)=\\frac12 \\big(a(\\theta) x^2+ b(\\theta) x\\cdot \\xi + b(\\theta) \\xi\\cdot  x + c(\\theta)  \\xi^2\\big),\n\\end{equation}\nwith coefficients $a,b,c\\in C^\\omega({\\mathbb T}^d,{\\mathbb R})$.\n\n\n\nThrough Weyl quantization, the reducibility for the time-dependent PDE can be related to the reducibility for the ${\\rm sl}(2,{\\mathbb R})-$linear system $(\\omega, \\, A(\\cdot))$:\n$$X'=A(\\omega t)X,\\qquad A\\in C^{\\omega}({\\mathbb T}^d,{\\rm sl}(2,{\\mathbb R})).$$\nGiven $A_1, A_2  \\in C^{\\omega}({\\mathbb T}^d,{\\rm sl}(2,{\\mathbb R}))$, if there exists $Y\\in C^{\\omega}(2{\\mathbb T}^d,{\\rm SL}(2,{\\mathbb R}))$ such that\n$$\\frac{d}{dt}Y(\\omega t)=A_1(\\omega t)Y(\\omega t)-Y(\\omega t)A_2(\\omega t),$$\n we say that $(\\omega, \\, A_1(\\cdot))$ is conjugated to $(\\omega, \\, A_2(\\cdot))$ by $Y$.\nIf $(\\omega, \\, A(\\cdot))$ can be conjugated to $(\\omega, \\, B)$ with $B\\in{\\rm sl}(2,{\\mathbb R})$, we say that $(\\omega, \\, A(\\cdot))$ is {\\it reducible}.\n\n\\smallskip\n\nNow let $A(\\cdot):=\\left(\\begin{array}{cc}\n            b(\\cdot) & c(\\cdot) \\\\[1mm]\n           -a(\\cdot) & -b(\\cdot)\n          \\end{array}\\right) \\in C^{\\omega}({\\mathbb T}^d,{\\rm sl}(2,{\\mathbb R})) $ with $a,b,c$ coefficients given in (\\ref{op_L}).\n\\begin{Theorem}\\label{thm_redu}\nAssume that there exist $B\\in{\\rm sl}(2,{\\mathbb R})$ and $Z_j\\in C^\\omega(2{\\mathbb T}^d, {\\rm sl}(2,{\\mathbb R}))$, $j=0, \\cdots,K$,\nsuch that $(\\omega, \\, A(\\cdot))$ is conjugated to $(\\omega, \\, B)$ by $\\prod_{j=0}^K e^{Z_j}$. Then Eq. (\\ref{eq_abs}) is reducible, i.e., there exists a time quasi-periodic map $U(\\omega t)$, unitary in $L^2$ and analytic on $t$, satisfying\n\\begin{equation}\\label{norm_U}\nc'\\|\\psi\\|_{s}\\leq \\|U(\\omega t)\\psi\\|_{s}\\leq C'\\|\\psi\\|_{s},  \\quad \\forall \\ \\psi\\in{{\\mathcal H}}^s,\n\\end{equation}\nfor constants $c', \\, C'>0$ depending on $s$ and $\\psi$, such that\nEq. (\\ref{eq_abs}) is conjugated to\n\\begin{equation}\\label{Ham_G}\n{\\rm i}\\partial_t v= G v\n\\end{equation}\nby the transformation $u=U(\\omega t) v$, with $G$ an operator independent of time.\n\n\nMore precisely,\n\\begin{itemize}\n\\item [(\\uppercase\\expandafter{\\romannumeral1})] $G$ is unitary equivalent to $\\frac{\\sqrt{{\\rm det} B}}{2} H_0$ if\n\\begin{equation}\\label{type1}\n{\\rm det}B>0 \\  or  \\ B=\\left(\\begin{array}{cc}\n           0 & 0 \\\\[1mm]\n           0 & 0\n          \\end{array}\n\\right).\n\\end{equation}\n\n\n\\item [(\\uppercase\\expandafter{\\romannumeral2})]\n$G$ is unitary equivalent to $-\\frac{{\\rm i}\\sqrt{-{\\rm det}B}}{2}(x\\cdot \\partial_x+ \\partial_x \\cdot x)$ if\n\\begin{equation}\\label{type2}\n{\\rm det}B<0.\\end{equation}\n\n\\item [(\\uppercase\\expandafter{\\romannumeral3})]\n$G$ is unitary equivalent to $-\\frac{\\kappa}{2} x^2$ if\n\\begin{equation}\\label{type3}\nB \\ is \\  similar  \\  to \\  \\left(\\begin{array}{cc}\n            0 & 0\\\\[1mm]\n            \\kappa & 0\n          \\end{array}\n\\right) \\  with \\ \\kappa\\neq 0.\n\\end{equation}\n\\end{itemize}\n\\end{Theorem}\n\n\n\n\n\n\\subsection{Growth of Sobolev norm via reducibility}\n\nAs an corollary of Theorem \\ref{thm_redu}, we have:\n\n\\begin{Theorem}\\label{thm_sobo} Under the assumption of Theorem \\ref{thm_redu}, we consider the solution $u(t)=u(t,\\cdot)$ to Eq. (\\ref{eq_abs}) with the non-vanishing initial condition $u(0)\\in {{\\mathcal H}}^s$, $s\\geq 0$. There exists $c, C>0$, depending on $s$ and $u(0)$, such that, for any $t\\geq 0$,\n\\begin{itemize}\n\\item If (\\ref{type1}) holds, then\n$c \\leq \\|u(t)\\|_{s}\\leq C$.\n\\item If (\\ref{type2}) holds, then\n$c  e^{\\sqrt{-{\\rm det}B}st}  \\leq \\|u(t)\\|_{s}\\leq C  e^{\\sqrt{-{\\rm det}B}st}$.\n\\item If (\\ref{type3}) holds, then\n$\nc |\\kappa|^s t^s \\leq \\|u(t)\\|_{s}\\leq C   |\\kappa|^s \\sqrt{1+t^2}^{\\frac{s}2}\n$.\n\\end{itemize}\n\\end{Theorem}\n\n\n\nAccording to (\\ref{norm_U}), to precise the growth of Sobolev norms for the solution to Eq. (\\ref{eq_abs}), it is sufficient to study the reduced quantum Hamiltonian $G(x, -{\\rm i}\\partial_x)$ obtained in (\\ref{Ham_G}), or more simply, the unitary equivalent forms of types (\\uppercase\\expandafter{\\romannumeral1})$-$(\\uppercase\\expandafter{\\romannumeral3}) listed in Theorem \\ref{thm_redu}.\n\n\\smallskip\n\nIf (\\ref{type1}) holds,  then $G$ is unitary equivalent to $\\frac{\\sqrt{{\\rm det}B}}{2} H_0$. Since the ${{\\mathcal H}}^s-$norm of $e^{-{\\rm i}t\\frac{\\sqrt{{\\rm det}B}}{2} H_0}\\psi_0$ is conserved for any $\\psi_0\\in{{\\mathcal H}}^s$, the boundedness of Sobolev norm is shown.\nWe focus on the cases where (\\ref{type2}) and (\\ref{type3}) hold, in which the growth of Sobolev norm occurs.\n\n\n\\begin{Proposition} \\label{prop6}\nFor the equation\n\\begin{equation}\\label{eq_hyper}\n\\partial_t v(t,x)=-\\frac\\lambda2 x\\cdot\\partial_x v (t,x)-\\frac\\lambda2 \\partial_x(x\\cdot v (t,x)), \\qquad \\lambda>0,\n\\end{equation}\n with non-vanishing initial condition $ v (0, \\cdot)= v_0\\in {{\\mathcal H}}^s$, $s\\geq 0$, there exist two constants $\\tilde c, \\, \\tilde C>0$, depending on $s$, $\\lambda$ and $ v _0$, such that the solution satisfies\n\\begin{equation}\\label{bounds_hyper}\n\\tilde c e^{\\lambda st} \\leq \\|\\psi(t,\\cdot )\\|_{s}\\leq   \\tilde C e^{\\lambda st}, \\qquad \\forall \\ t\\geq0.\n\\end{equation}\n\\end{Proposition}\n\n\\begin{remark}\nThis conclusion is also given in Remark 1.4 of \\cite{MR2017}.\n\\end{remark}\n\n\n\n\\proof Through a straightforward computation, we can verify that, for the initial condition $ v (0,\\cdot)= v _0(\\cdot)\\in {{\\mathcal H}}^s$, the solution to Eq. (\\ref{eq_hyper}) satisfies\n$$ v (t,x)=e^{-\\frac\\lambda2 t}  v _0(e^{-\\lambda t} x).$$\nFor any $s\\geq 0$,\n\\begin{eqnarray}\n\\int_{\\mathbb R} x^{2s } |  v (t,x)|^2 \\, dx &=& \\int_{\\mathbb R}  x^{2s }   | v _0(e^{-\\lambda t} x)|^2 \\, d (e^{-\\lambda t}x) \\nonumber\\\\\n&=& e^{2\\lambda s t}\\int_{\\mathbb R} (e^{-\\lambda t} x)^{2s}| v _0(e^{-\\lambda t} x)|^2 \\, d (e^{-\\lambda t}x)\\nonumber\\\\\n&=& e^{2\\lambda s t}\\int_{\\mathbb R} x^{2s} | v _0(x)|^2 \\, dx.\\label{sobolev_hyper}\n\\end{eqnarray}\nand for $s\\in{\\mathbb N}$,\n\\begin{equation}\\label{sobolev_hyper-ds}\n\\int_{\\mathbb R} |\\partial_x^s  v (t,x)|^2 \\, dx = e^{-2\\lambda s t} \\int_{\\mathbb R}  | v _0^{(s)}(e^{-\\lambda t} x)|^2 \\, d (e^{-\\lambda t}x)\n= e^{-2\\lambda s t}\\int_{\\mathbb R} | v _0^{(s)}(x)|^2 \\, dx.\n\\end{equation}\nIn view of the equivalent definition (\\ref{norm_equiv}) of the ${{\\mathcal H}}^s-$norm given in Remark \\ref{remark_norm_equiv}, we get (\\ref{bounds_hyper}) by combining (\\ref{sobolev_hyper}) and (\\ref{sobolev_hyper-ds}).\\qed\n\n\n\n\n\n\n\n\\begin{Proposition}\\label{prop_para}\nFor the equation\n\\begin{equation}\\label{eq_para}\n{\\rm i}\\partial_t v (t,x)=-\\frac{\\kappa}{2} x^2\\cdot v (t,x), \\qquad \\kappa\\in{\\mathbb R},\n\\end{equation}\nwith non-vanishing initial condition $ v _0\\in {{\\mathcal H}}^s$, $s\\geq 0$, there exists constants $\\tilde c, \\tilde C>0$, depending on $s$, $\\kappa$ and $ v _0$, such that the solution satisfies\n\\begin{equation}\\label{sobo_para}\n\\tilde c |\\kappa|^s |t|^s \\leq \\| v (t,\\cdot)\\|_{s}\\leq \\tilde C  |\\kappa|^s (1+ t^2)^\\frac{s}2,\\qquad \\forall \\  t\\in{\\mathbb R}.\n\\end{equation}\n\\end{Proposition}\n\\proof\nWith the initial condition $ v (0,\\cdot)= v _0(\\cdot)\\in {{\\mathcal H}}^s$, the solution to Eq. (\\ref{eq_para}) is\n$$ v (t,x)=e^{{\\rm i}\\frac{\\kappa}{2} x^2 t} v _0(x).$$\nFor any $s\\geq 0$,\n$$\\|x^s v (t,x)\\|_{L^2}=\\|x^s e^{{\\rm i}\\frac{\\kappa}{2} x^2 t} v _0(x)\\|_{L^2}=\\|x^s  v _0(x)\\|_{L^2},$$\nand for $s\\in{\\mathbb N}$,\n\\begin{eqnarray*}\n\\partial_{x}^{s}( v (t,x))\n&=&\\partial_{x}^{s}(e^{{\\rm i}\\frac{\\kappa}{2} x^2 t} v _0(x))\\\\\n&=& \\sum_{\\alpha=0}^{s} C_s^\\alpha (e^{{\\rm i}\\frac{\\kappa}{2} x^2 t})^{(\\alpha)} v _0^{(s-\\alpha)}(x)\\\\\n&=& e^{{\\rm i}\\frac{\\kappa}{2} x^2 t}  \\sum_{\\alpha=0}^{s} C_s^\\alpha \\left(({\\rm i}\\kappa t)^{\\alpha} x^{\\alpha}+P_{\\alpha}({\\rm i}\\kappa t,x)\\right)  v _0^{(s-\\alpha)}(x) \\\\\n&=&({\\rm i} \\kappa t)^{s}  x^{s}  e^{{\\rm i}\\frac{\\kappa}{2} x^2 t}\\cdot  v _0(x) +P_{s}({\\rm i}\\kappa t,x)e^{{\\rm i}\\frac{\\kappa}{2} x^2 t}\\cdot  v _0(x)\\\\\n& &+ \\, x^{\\alpha} e^{{\\rm i}\\frac{\\kappa}{2} x^2 t}  \\sum_{\\alpha=0}^{s-1} C_s^\\alpha \\left(({\\rm i}\\kappa t)^{\\alpha} x^{\\alpha}+P_{\\alpha}({\\rm i}\\kappa t,x)\\right)  v _0^{(s-\\alpha)}(x),\n\\end{eqnarray*}\nwhere, for $\\alpha\\geq 2$, $P_{\\alpha}({\\rm i}\\kappa t,x)$ is a polynomial of degree $\\alpha-2$ of $x$, with the coefficients being monomials of ${\\rm i}\\kappa t$ of degree $\\leq \\alpha-1$ and $P_{1}=P_0=0$. Then, there exists a constant $D>0$ such that\n$$\\left|\\|\\partial_{x}^{s}( v (t,x))\\|_{L^2}-|\\kappa t |^{s}\\|x^{s}   v _0(x)\\|_{L^2}\\right|\\leq D |\\kappa t |^{s-1} \\| v _0(x)\\|_s.$$\nIn view of the equivalent definition (\\ref{norm_equiv}) of norm in Remark \\ref{remark_norm_equiv},\nwe get (\\ref{sobo_para}). \\qed\n\n\\smallskip\n\n\\noindent{\\bf Proof of Theorem \\ref{thm_sobo}.}\nFrom Theorem \\ref{thm_redu}, we know that\nEq. (\\ref{eq_abs}) is conjugated to\n${\\rm i}\\partial_t v= G v$ by the transformation $u=U(\\omega t) v$, with $G=G(x,-{\\rm i}\\partial_x)$ the operator independent of $t$ given in (\\ref{Ham_G_pr}).\n\nRecall Proposition \\ref{prop6} and \\ref{prop_para}.\nGiven $s\\geq 0$, for any non-vanishing $v_0\\in{{\\mathcal H}}^s$, for the three types of unitary equivalence of $G$, there are three different behaviours of the solution to the equation ${\\rm i}\\partial_t v= G v$ as $t\\to  \\infty$.\n\\begin{itemize}\n\\item If $G$ is unitary equivalent to $\\frac{\\sqrt{{\\rm det} B}}{2} H_0$ (under (\\ref{type1})), then\n$\\|e^{-{\\rm i}Gt}v_0\\|_s=O(1)$.\n\\item If $G$ is unitary equivalent to $-\\frac{{\\rm i}\\sqrt{-{\\rm det} B}}{2} (x\\cdot \\partial_x +\\partial_x\\cdot x)$ (under (\\ref{type2})), then\n$\\|e^{-{\\rm i}Gt}v_0\\|_s=O(e^{\\sqrt{-{\\rm det} B} s t}).$\n\\item If $G$ is unitary equivalent to $-\\frac{\\kappa}{2} x^2$ (under (\\ref{type3})), then $\\|e^{-{\\rm i}Gt}v_0\\|_s=O(|\\kappa|^st^s)$.\n\\end{itemize}\nMoreover, according to (\\ref{norm_U}), for $s\\geq 0$, there exist constants $c', C'>0$ such that\n$$  c'\\|v\\|_{s}\\leq \\|U(\\omega t)v\\|_{s}\\leq C'\\|v\\|_{s},\\qquad  \\forall \\  v\\in {{\\mathcal H}}^s.$$\nHence Theorem \\ref{thm_sobo} is shown.\\qed\n\n\n\\section{Reducibility in classical Hamiltonian system and Proof of Theorem \\ref{thm_redu}}\\label{sec_reduc}\n\n\\subsection{Conjugation between classical hamiltonians}\n\nGiven two quadratic classical Hamiltonians\n$$h_j(\\omega t, x, \\xi)=\\frac12 \\big(a_j(\\omega t)x^2+ 2b_j(\\omega t) x\\cdot \\xi+ c_j(\\omega t)  \\xi^2 \\big), \\qquad j=1,2,$$\nwhich can be presented as\n$$h_j(\\omega t, x,\\xi)=\\frac12\\left(\n                             \\begin{array}{c}\n                               x \\\\\n                               \\xi \\\\\n                             \\end{array}\n                           \\right)^{\\top}J A_j(\\omega t)\\left(\n                             \\begin{array}{c}\n                               x \\\\\n                               \\xi \\\\\n                             \\end{array}\n                           \\right), \\qquad j=1,2$$\nwith $J:=\\left(\\begin{array}{cc}\n0 & -1 \\\\1 & 0 \\end{array}\\right)$ and\n$A_j(\\cdot)=\\left(\\begin{array}{cc}\nb_j(\\cdot) & c_j(\\cdot) \\\\ -a_j(\\cdot) & -b_j(\\cdot) \\end{array}\\right)\\in C^{\\omega}({\\mathbb T}^d, {\\rm sl}(2,{\\mathbb R}))$.\nThe corresponding equations of motion are  given by\n$$x'=\\frac{\\partial h_j}{\\partial\\xi},\\quad \\xi'=-\\frac{\\partial h_j}{\\partial x},\\qquad j=1,2,$$\nwhich are the linear systems $(\\omega, \\, A_j)$:\n$$\\left(\\begin{array}{c}\n          x(t) \\\\\n          \\xi(t)\n        \\end{array}\n\\right)'=A_j(\\omega t)\\left(\\begin{array}{c}\n          x(t) \\\\\n          \\xi(t)\n        \\end{array}\n\\right).$$\n\n\n\n\\begin{Proposition}\\label{prop_ham_cl}\nIf the linear system $(\\omega, \\, A_1(\\cdot))$ is conjugated to $(\\omega, \\, A_2(\\cdot))$ by a time quasi-periodic ${\\rm SL}(2,{\\mathbb R})-$transformation, i.e.,\n\\begin{equation}\\label{conj_ode}\n\\frac{d}{dt} e^{Z(\\omega t)}=A_1(\\omega t)e^{Z(\\omega t)}-e^{Z(\\omega t)} A_2(\\omega t),\\qquad Z \\in C^\\omega(2{\\mathbb T}^d, {\\rm sl}(2,{\\mathbb R})),\n\\end{equation}\nthen the classical Hamiltonian $h_1(\\omega t,x,\\xi)$ is conjugated to $h_2(\\omega t,x,\\xi)$ via the time$-1$ flow $\\phi_{\\chi}^1(t,x,\\xi)$\n generated by the Hamiltonian\n \\begin{equation}\\label{chi_eZ}\n \\chi(\\omega t,x,\\xi)=\\frac12\\left(\n                             \\begin{array}{c}\n                               x \\\\\n                               \\xi \\\\\n                             \\end{array}\n                           \\right)^{\\top}J Z(\\omega t)\\left(\n                             \\begin{array}{c}\n                               x \\\\\n                               \\xi \\\\\n                             \\end{array}\n                           \\right).\n \\end{equation}\n\\end{Proposition}\n\n\\proof Note that the equation of motion of the classical Hamiltonian $h_1$ is the linear system $(\\omega, \\, A_1(\\cdot))$:\n$$ \\left(\\begin{array}{c}\n   x \\\\\n    \\xi\n  \\end{array}\\right)'=A_1(\\omega t)\\left(\\begin{array}{c}\n   x \\\\\n    \\xi\n  \\end{array}\\right).$$\nIn view of (\\ref{conj_ode}), the transformation\n\\begin{equation}\\label{tramSL}\n  \\left(\\begin{array}{c}\n   x \\\\\n    \\xi\n  \\end{array}\\right)= e^{Z(\\omega t)}\\left(\\begin{array}{c}\n    \\tilde x \\\\\n   \\tilde \\xi\n  \\end{array}\\right),\\qquad Z\\in C^{\\omega}(2{\\mathbb T}^d, {\\rm sl}(2,{\\mathbb R})),\n\\end{equation}\nconjugates $(\\omega, \\, A_1(\\cdot))$ to $(\\omega, \\, A_2(\\cdot))$. More precisely,\n\\begin{eqnarray*}\n\\left(\\begin{array}{c}\n\\tilde x \\\\\n\\tilde \\xi\n\\end{array}\\right)'&=&e^{-Z(\\omega t)}A_1(\\omega t)\\left(\\begin{array}{c}\nx \\\\\n\\xi\n\\end{array}\\right)-e^{-Z(\\omega t)}\\frac{d}{dt}e^{Z(\\omega t)}\\left(\\begin{array}{c}\n\\tilde x\\\\\n\\tilde \\xi\n\\end{array}\\right)  \\\\\n&=& e^{-Z(\\omega t)}A_1(\\omega t)e^{Z(\\omega t)}\\left(\\begin{array}{c}\n{\\tilde x} \\\\\n{\\tilde \\xi}\n\\end{array}\\right)-e^{-Z(\\omega t)}\\frac{d}{dt}e^{Z(\\omega t)}\\left(\\begin{array}{c}\n\\tilde x\\\\\n\\tilde \\xi\n\\end{array}\\right)\\\\\n&=&A_2(\\omega t)\\left(\\begin{array}{c}\n\\tilde x\\\\\n\\tilde \\xi\n\\end{array}\\right),\n\\end{eqnarray*}\nfor which the corresponding Hamiltonian is $h_2(\\omega t,\\tilde x,\\tilde\\xi)$.\nAs in (3-35) of \\cite{BGMR2018}, the time$-1$ map between the two Hamiltonians is generated by (\\ref{chi_eZ}) since there is only quadratic terms in the Hamiltonian in our case.\\qed\n\n\n\n\n\\subsection{Proof of Theorem \\ref{thm_redu}}\n\n\nWe consider the classical Hamiltonian\n\\begin{eqnarray*}\nL(\\omega t,x,\\xi)&=&\\frac{a(\\omega t)}{2}x^2+\\frac{b(\\omega t)}{2}(x\\cdot\\xi+\\xi \\cdot x)+\\frac{c(\\omega t)}{2}\\xi^2 \\nonumber \\\\\n&=&\\frac12X^{\\top}J A(\\omega t)X,\\qquad X:= \\left(\n                             \\begin{array}{c}\n                               x \\\\\n                               \\xi \\\\\n                             \\end{array}\n                           \\right).\n\\end{eqnarray*}\nwith $a,b,c\\in C^{\\omega}({\\mathbb T}^d)$ given in Eq. (\\ref{eq_abs}), and $A:=\\left(\\begin{array}{cc}\n            b & c \\\\\n           -a & -b\n          \\end{array}\n\\right)\\in C^{\\omega}({\\mathbb T}^d, {\\rm sl}(2,{\\mathbb R}))$.\n\n\n\nBy the hypothesis of Theorem \\ref{thm_redu}, the linear system $(\\omega, \\, A(\\cdot))$ can be reduced to the constant system $(\\omega, \\, B)$, with $B=\\left(\\begin{array}{cc}\n            B_{11} & B_{12} \\\\\n            -B_{21} & -B_{11}\n          \\end{array}\n\\right)\\in{\\rm sl}(2,{\\mathbb R})$, via finitely many transformations $(e^{Z_j})_{j=0}^K$ with $Z_j\\in C^{\\omega}(2{\\mathbb T}^d, {\\rm sl}(2,{\\mathbb R}))$. Hence the reduced classical Hamiltonian is\n$$g(x,\\xi)=\\frac12X^{\\top}J B X= \\frac{B_{21}}2 x^2+\\frac{B_{11}}{2}(x\\cdot \\xi+ \\xi\\cdot x)+\\frac{B_{12}}2 \\xi^2.$$\nBy Proposition \\ref{Prop_hami}, we see that $L^W(\\omega t, x, -{\\rm i}\\partial_x)$ is conjugated to\n\\begin{equation}\\label{Ham_G_pr}\nG(x, -{\\rm i}\\partial_x):=g^W(x, -{\\rm i}\\partial_x)=\\frac{B_{21}}2 x^2-\\frac{B_{11}}{2}(x\\cdot{\\rm i}\\partial_x+{\\rm i}\\partial_x\\cdot x)-\\frac{B_{12}}2 \\partial_x^2\n\\end{equation}\nvia the product of unitary (in $L^2({\\mathbb R})$) transformations\n$$\nU(\\omega t):= \\prod_{j=0}^K e^{-{\\rm i}\\chi^W_j(\\omega t,x,-{\\rm i}\\partial_x)}\n$$\nwhere $\\chi^W_j$ is the Weyl quantization of\n$$\\chi_j(\\omega t,x,\\xi)=\\frac12X^{\\top} J Z_j(\\omega t)X.$$\nThen (\\ref{norm_U}) is deduced from (\\ref{change_Sobolevnorm}) in Lemma \\ref{lem_Sobolev}.\nThe following diagram gives a straightforward explanation for the above proof.\n$$\n\\begin{array}{rcccl}\n& X'=A(\\omega t)X &\\stackrel{\\prod_{j=0}^K e^{Z_j(\\omega t)}}{\\longrightarrow} & X'=BX &  \\  \\  Z_j\\in C^\\omega(2{\\mathbb T}^d, {\\rm sl}(2,{\\mathbb R}))  \\\\\n  &   &  & &  \\\\\n  &  \\big\\updownarrow &  &  \\big\\updownarrow & \\\\\n  &  &  &  & \\\\\n & L(\\omega t)=\\frac12X^{\\top} J A(\\omega t)X & \\stackrel{ \\Phi^1_{\\chi_0(\\omega t)}\\circ \\cdots \\circ  \\Phi^1_{\\chi_K(\\omega t)}}{\\longrightarrow} & g=\\frac12X^{\\top} J B X & \\ \\ \\chi_j =\\frac12X^{\\top}J Z_j X\\\\\n  &  &  & &  \\\\\n   &  \\big\\updownarrow &  &  \\big\\updownarrow & \\\\\n  &  &  & &  \\\\\n & {\\rm i}\\partial_t u=L^W(\\omega t)u &  \\stackrel{\\prod_{j=0}^{K} e^{-{\\rm i} \\chi_j^W(\\omega t)}}{\\longrightarrow} &  {\\rm i}\\partial_t u = g^W u &\n  \\end{array}\n$$\n\nIf (\\ref{type1}) holds, i.e., ${\\rm det}B>0$ or $B=\\left(\\begin{array}{cc}\n            0 & 0 \\\\\n            0 & 0\n          \\end{array}\\right)$,\nthen there exists $C_B\\in {\\rm sl}(2,{\\mathbb R})$ such that\n\\begin{equation}\\label{elliptic}\nB=e^{C_B}\\left(\\begin{array}{cc}\n            0 & \\sqrt{{\\rm det}B} \\\\\n            -\\sqrt{{\\rm det}B} & 0\n          \\end{array}\\right)e^{-C_B}.\n\\end{equation}\nIf (\\ref{type2}) holds,  i.e., ${\\rm det}B<0$, then there exists $C_B\\in {\\rm sl}(2,{\\mathbb R})$ such that\n\\begin{equation}\\label{hyerbolic}\nB=e^{C_B}\\left(\\begin{array}{cc}\n            \\sqrt{-{\\rm det}B} & 0 \\\\\n            0 & -\\sqrt{-{\\rm det}B}\n          \\end{array}\\right)e^{-C_B}.\n\\end{equation}\nIf (\\ref{type3}) holds,\nthen there exists $C_B\\in {\\rm sl}(2,{\\mathbb R})$ such that\n\\begin{equation}\\label{parapolic}\nB=e^{C_B}\\left(\\begin{array}{cc}\n            0 & 0 \\\\\n            \\kappa & 0\n          \\end{array}\\right)e^{-C_B}.\n\\end{equation}\nTherefore, for Eq. (\\ref{eq_abs}), the three types of unitary equivalence of $G=G(x,-{\\rm i}\\partial_x)$ are shown by (\\ref{elliptic})$-$(\\ref{parapolic}) respectively. \\qed\n\n\n\n\n\n\n\n\n\n\\section{Proof of Theorem \\ref{thm_Schro} and \\ref{thm_Schro_sobolev}}\\label{sec_proof}\n\n\nIn view of Theorem \\ref{thm_redu}, to show the reducibility of Eq. (\\ref{eq_Schrodinger}), it is sufficient to show the reducibility of the corresponding ${\\rm sl}(2,{\\mathbb R})-$linear system.\n\nFor $E\\in{{\\mathcal I}}$, the symbol of the quantum Hamiltonian (\\ref{eq_Schrodinger}) is\n$$h_E(\\omega t, x,\\xi)=\\frac{\\nu(E)}{2}(\\xi^2+x^2)+W(E,\\omega t,x,\\xi)$$\nwhich corresponds the quasi-periodic linear system $(\\omega, \\,  A_0+F_0)$\n\\begin{equation}\\label{linear_system_pr}\n\\left(\\begin{array}{c}\n          x \\\\\n          \\xi\n        \\end{array}\n\\right)'=\\left[\\left(\\begin{array}{cc}\n             0 & \\nu(E) \\\\\n             -\\nu(E) & 0\n           \\end{array}\\right)\n           +\\left(\\begin{array}{cc}\n             b(E,\\omega t) & c(E,\\omega t) \\\\\n             -a(E,\\omega t) & -b(E,\\omega t)\n           \\end{array}\\right)\\right] \\left(\\begin{array}{c}\n          x \\\\\n          \\xi\n        \\end{array}\n\\right),\n\\end{equation}\nwhere, for every $E\\in{{\\mathcal I}}$,\n\\begin{align*}\nA_0(E):= & \\left(\\begin{array}{cc}\n             0 & \\nu(E) \\\\\n             -\\nu(E) & 0\n           \\end{array}\\right)\\in {\\rm sl}(2,{\\mathbb R}), \\\\\n F_0(E,\\cdot):=& \\left(\\begin{array}{cc}\n             b(E,\\cdot) & c(E,\\cdot) \\\\\n             -a(E,\\cdot) & -b(E,\\cdot)\n           \\end{array}\\right)\\in C^{\\omega}_r({\\mathbb T}^d,{\\rm sl}(2,{\\mathbb R}))\n\\end{align*}\nwith $|\\partial_E^m F_0|_r<\\varepsilon_0$, $m=0,1,2$, sufficiently small.\n\n\nThe reducibility of linear system (\\ref{linear_system_pr}) is exploited by Eliasson \\cite{Eli1992} (see also \\cite{HA} for results about ${\\rm SL}(2,{\\mathbb R})$-cocycles). We summarise the needed results in the following proposition. To make the paper as self-contained as possible, we give a short proof without adding too many details on known facts.\nSince every quantity depends on $E$, we do not always write this dependence explicitly in the statement of proposition.\n\n\nBefore stating the precise result, we introduce the concept of rotation number. The {\\it rotation number}\nof quasi-periodic ${\\rm sl}(2,{\\mathbb R})-$linear system (\\ref{linear_system_pr})\n is defined as\n$$\\rho(E)=\\rho(\\omega, \\, A_0(E)+F(E,\\omega t))=\\lim_{t\\to\\infty}\\frac{\\arg(\\Phi_E^t X)}{t},\\qquad \\forall \\ X\\in \\mathbb{R}^2\\setminus\\{0\\}$$\nwhere $\\Phi_E^t$ is the basic\nmatrix solution and $\\arg$ denotes the angle. The rotation number\n$\\rho$ is well-defined and it does not depend on $X$\n\\cite{JM82}.\n\n\n\\begin{Proposition}\\label{prop_eliasson} There exists $\\varepsilon_*=\\varepsilon_*(r,\\gamma,\\tau,d,l_1,l_2)>0$ such that if\n\\begin{equation}\\label{small_F_0}\n\\max_{m=0,1,2}|\\partial_E^m F_0|_r=:\\varepsilon_0<\\varepsilon_*,\n\\end{equation}\nthen the following holds for the quasi-periodic linear system $(\\omega, \\, A_0+F_0)$.\n\\begin{enumerate}\n  \\item [(1)] For a.e. $E\\in{{\\mathcal I}}$, $(\\omega, \\, A_0+F_0(\\cdot))$ is reducible. More precisely,\n there exist $B\\in{\\rm sl}(2,{\\mathbb R})$ and  $Z_j\\in C^\\omega(2{\\mathbb T}^d, {\\rm sl}(2,{\\mathbb R}))$, $j=0,1,\\cdots,K$, such that\n \\begin{equation}\\label{reducibility_sl2R}\n   \\frac{d}{dt}\\left(\\prod_{j=0}^K e^{Z_j(\\omega t)}\\right)=\\left(A_0+F_0(\\omega t)\\right)\\left(\\prod_{j=0}^K e^{Z_j(\\omega t)}\\right)-\\left(\\prod_{j=0}^K e^{Z_j(\\omega t)}\\right)B.\n \\end{equation}\n  \\item [(2)] The rotation number $\\rho=\\rho(E)$ is monotonic on ${\\mathcal I}$. For any $k\\in{\\mathbb Z}^d$,\n  $$\\tilde\\Lambda_k:=\\left\\{E\\in\\overline{{\\mathcal I}}:\\rho(E)=\\frac{\\langle k,\\omega\\rangle}{2}\\right\\}  \\   \\footnote{$\\tilde\\Lambda_k$ can be empty for some $k\\in{\\mathbb Z}^d$ if the closed interval $\\rho^{-1}\\left(\\frac{\\langle k,\\omega\\rangle}{2}\\right)$ does not intersect ${{\\mathcal I}}$.} $$ is a closed interval, and we have\n\\begin{equation}\\label{measure_esti}\n\\sum_{k\\in{\\mathbb Z}^d}{\\rm Leb}(\\tilde\\Lambda_k)<\\varepsilon_0^{\\frac{1}{40}}.\n\\end{equation}\n  \\item [(3)] For every $E\\in \\tilde\\Lambda_k=:[a_k,b_k]$, $(\\omega, \\, A_0+F_0(\\cdot))$ is reducible and the matrix $B\\in {\\rm sl}(2,{\\mathbb R})$ in (\\ref{reducibility_sl2R}) satisfies\n \\begin{itemize}\n   \\item if $a_k=b_k$, then $B=\\left(\\begin{array}{cc}\n                                       0 & 0 \\\\\n                                       0 & 0\n                                     \\end{array}\\right)$;\n   \\item if $a_k<b_k$, then\n\\begin{itemize}\n  \\item ${\\rm det} B<0$ for $E\\in(a_k,b_k)$,\n  \\item ${\\rm det} B=0$ for $E=a_k,b_k$ and $E\\not\\in\\partial {{\\mathcal I}}$.\n\\end{itemize}\n \\end{itemize}\n   \\item [(4)] For a.e. $E\\in{{\\mathcal I}}\\setminus\\bigcup_k \\tilde\\Lambda_k$, $(\\omega, \\, A_0+F_0(\\cdot))$ is reducible and the matrix $B\\in {\\rm sl}(2,{\\mathbb R})$ in (\\ref{reducibility_sl2R}) satisfies ${\\rm det} B>0$.\n\\end{enumerate}\n\\end{Proposition}\n\n\\proof Since $\\nu$ is a strictly monotonic real-valued function of $E\\in{{\\mathcal I}}$ and $|\\nu'|\\geq l_1$, $|\\nu''|\\leq l_2$, (\\ref{small_F_0}) implies that\n$|\\partial^m_E F_0(\\nu^{-1}(E),\\cdot)|_r$, $m=0,1,2$, is also small enough.\n Hence, to prove the above arguments, we can simply consider the case where $\\nu(E)=E\\in {{\\mathcal I}}= {\\mathbb R}$ and then obtain Proposition \\ref{prop_eliasson} by replacing $E$ by $\\nu(E)$.\n\n\n\n\n\n\n\n\n\n\\smallskip\n\n\\noindent {\\it Proof of (1).}\nThe almost reducibility has already been shown by Eliasson \\cite{Eli1992} for every $E\\in{\\mathbb R}$.\nIndeed, if $\\max_{m=0,1,2}|\\partial_E^m F_0|_r$ is small enough (depending on $r,\\gamma,\\tau,d$), then there exists sequences $(Y_j)_{j\\in{\\mathbb N}}\\subset C^\\omega(2{\\mathbb T}^d, {\\rm SL}(2,{\\mathbb R}))$, $(A_j)_{j\\in{\\mathbb N}}\\subset {\\rm sl}(2,{\\mathbb R})$, and $(F_j)_{j\\in{\\mathbb N}}\\subset C^\\omega(2{\\mathbb T}^d, {\\rm sl}(2,{\\mathbb R}))$, all of which are piecewise $C^2$ w.r.t. $E$,\n with $\\max_{m=0,1,2}|\\partial^m_E F_j|_{{\\mathbb T}^d}<\\varepsilon_j:=\\varepsilon_0^{(1+\\sigma)^j}$ for $\\sigma=\\frac1{33}$, such that\n$$\\frac{d}{dt}Y_{j}(\\omega t)=\\left(A_j+F_j(\\omega t)\\right)Y_{j}(\\omega t)-Y_{j}(\\omega t) \\left(A_{j+1}+F_{j+1}(\\omega t)\\right).$$\nMore precisely, at the $j-$th step, for $\\pm{\\rm i}\\xi_j\\in{\\mathbb R}\\cup{\\rm i}{\\mathbb R}$, the two eigenvalues of $A_j$, and\n$$N_j:=\\frac{2\\sigma}{r_j-r_{j+1}}\\ln\\left(\\frac{1}{\\varepsilon_j}\\right)$$\nwith $(r_j)_{j\\in{\\mathbb N}}$ a decreasing sequence of positive numbers such that $r_j-r_{j+1}\\geq 2^{-(j+1)}r$ for each $j$,\n\\begin{itemize}\n  \\item (non-resonant case) if for every $n\\in{\\mathbb Z}^d$ with $0<|n|\\leq N_j$, we have\n  \\begin{equation}\\label{non_resonant}\n  \\left|2\\xi_j-\\langle n,\\omega\\rangle\\right|\\geq \\varepsilon_j^{\\sigma},\n  \\end{equation}\n  then $Y_{j}=e^{\\tilde Z_{j}}$ for some $\\tilde Z_{j}\\in C^{\\omega}(2{\\mathbb T}^d,{\\rm sl}(2,{\\mathbb R}))$ with $|\\tilde Z_{j}|_{2{\\mathbb T}^d}<\\varepsilon_j^{\\frac23}$, and $|A_{j+1}-A_j|<\\varepsilon_j^{\\frac23}$;\n  \\item (resonant) if for some $n_j\\in{\\mathbb Z}^d$ with $0<|n_j|\\leq N_j$, we have\n  \\begin{equation}\\label{resonant}\n  \\left|2\\xi_j-\\langle n_j,\\omega\\rangle\\right|< \\varepsilon_j^{\\sigma},\n  \\end{equation}\n  then $Y_{j+1}(\\cdot)=e^{\\frac{\\langle n_j ,\\cdot\\rangle}{2\\xi_j}A_j} e^{\\tilde Z_{j+1}}$ for some $\\tilde Z_{j}\\in C^{\\omega}(2{\\mathbb T}^d,{\\rm sl}(2,{\\mathbb R}))$ with $|\\tilde Z_{j}|_{2{\\mathbb T}^d}<\\varepsilon_j^{\\frac23}$ and $|A_{j+1}|<\\varepsilon_j^{\\frac{\\sigma}2}$.\n\\end{itemize}\nAs $j$ goes to $\\infty$, the time-dependent part $F_{j}$ tends to vanish. Hence $(\\omega,\\, A_0(E)+F_0)$ is almost reducible. For the detailed proof, we can refer to Lemma 2 of \\cite{Eli1992} and its proof.\n\nIn view of Lemma 3 b) of \\cite{Eli1992}, if the rotation number $\\rho(E)$ of $(\\omega, \\, A_0(E)+F_0)$ is Diophantine or rational w.r.t. $\\omega$, which corresponds to a.e. $E\\in{\\mathbb R}$, then the resonant case occurs for only finitely many times.\nTherefore, for a.e. $E\\in{\\mathbb R}$, there exists a large enough $J_*\\in{\\mathbb N}^*$, depending on $E$, such that\n\\begin{equation}\\label{J_large}\nY_{j}=e^{\\tilde Z_{j}} \\  \\ {\\rm with} \\   \\  |\\tilde Z_{j}|_{2{\\mathbb T}^d}<\\varepsilon^{\\frac23}_{j},\\qquad \\forall \\ j\\geq J_*.\n\\end{equation}\nThis implies that $\\prod_{j=0}^\\infty|Y_j|_{2{\\mathbb T}^d}$ is convergent.\nAs explained in the proof of Lemma 3.5 of \\cite{BGMR2018}, (\\ref{J_large}) also implies that there exists $S\\in C^{\\omega}(2{\\mathbb T}^d,{\\rm sl}(2,{\\mathbb R}))$ such that\n$\\prod_{j=J_*}^\\infty Y_{j}=e^S$,\nsince $\\varepsilon_0$ is sufficiently small.\nHence (\\ref{reducibility_sl2R}) is shown, i.e., the reducibility is realized via finitely many transformations of the form $e^{Z_j(\\omega t)}$ with $Z_j\\in C^{\\omega}(2{\\mathbb T}^d,{\\rm sl}(2,{\\mathbb R}))$.\n\n\n\n\n\n\n\n\\\n\n\\noindent {\\it Proof of (2).}\nFor $k\\in{\\mathbb Z}^d$, $\\tilde\\Lambda_k$ is obtained after several resonant KAM-steps, saying $j_1$, $\\cdots$, $j_L$, where $n_{j_i}\\in{\\mathbb Z}^d$ with $0<|n_{j_i}|\\leq N_{j_i}$, $i=1,\\cdots,L$, satisfies\n$$ \\left|2\\xi_{j_i}-\\langle n_{j_i},\\omega\\rangle\\right|< \\varepsilon_{j_i}^{\\sigma},$$\nand $k=n_{j_1}+\\cdots + n_{j_L}$. We will show that\n\\begin{equation}\\label{k_n_j_L}\n\\frac{10|n_{j_L}|}{11}\\leq |k|\\leq  \\frac{12|n_{j_L}|}{11}.\n\\end{equation}\n Assume that $L\\geq 2$ (otherwise we have already $k=n_{j_L}$). After the $(j_{i-1}+1)-$th step, $i=2,\\cdots,L$, the eigenvalues $\\pm{\\rm i}\\xi_{j_{i-1}+1}$ satisfies $|\\xi_{j_{i-1}+1}|<2\\varepsilon_{j_{i-1}}^{\\frac\\sigma2}$. On the other hand, before the $(j_{i}+1)-$th step, the resonant condition (\\ref{resonant}) implies that the eigenvalues $\\pm{\\rm i}\\xi_{j_{L}}$ satisfy that\n$$|2\\xi_{j_{i}}-\\langle n_{j_i},\\omega\\rangle|\\leq \\varepsilon_{j_{i}}^\\sigma.$$\nSince the steps between these two successive resonant steps are all non-resonant, and $\\omega\\in {\\rm DC}_{d}(\\gamma,\\tau)$,\nwe have that\n$$\\frac{\\gamma}{|n_{j_i}|^\\tau}\\leq|\\langle n_{j_i},\\omega\\rangle|\\leq 2|\\xi_{j_{i-1}+1}|+2\\varepsilon^{\\frac13}_{j_{i-1}+1}+\\varepsilon_{j_{i}}^\\sigma<3\\varepsilon_{j_{i-1}}^{\\frac\\sigma2},$$\nwhich implies that\n$$|n_{j_i}|>\\left(\\frac\\gamma3\\right)^{\\frac{1}{\\tau}}\\varepsilon^{-\\frac{\\sigma}{2\\tau}}_{j_{i-1}}>12|N_{j_{i-1}}|\\geq 12|n_{j_{i-1}}|.$$\nHence, we get (\\ref{k_n_j_L}).\n\n\n\n$\\tilde\\Lambda_k$ is firstly formed at the $j_L-$th step, with the initial measure smaller than $\\varepsilon_{j_L}^{2\\sigma}$.\nSince all the succedent steps are non-resonant, the measure of $\\tilde\\Lambda_k$ varies up to $\\varepsilon_{j_L}^{2\\sigma}$. Then, for $\\varsigma:=\\frac{\\ln(1+\\sigma)}{\\ln(8+8\\sigma)}$, we have\n$$\n{\\rm Leb}(\\tilde\\Lambda_k)< 2\\varepsilon_{j_L}^{2\\sigma}< 2\\varepsilon_0^{\\sigma} e^{-\\left(\\frac{12}{11}\\right)^\\varsigma N_{j_L}^\\varsigma}\\leq 2\\varepsilon_0^{\\sigma} e^{-\\left(\\frac{12}{11}\\right)^\\varsigma |n_{j_L}|^\\varsigma}.\n$$\nIndeed, recalling that $r_j-r_{j+1}\\geq 2^{-(j+1)}r$ for every $j$, we have\n\\begin{eqnarray*}\n\\varepsilon_{j_L} &=& \\exp\\{-|\\ln\\varepsilon_0|(1+\\sigma)^{j_L}\\} \\\\\n &=& \\exp\\left\\{- \\frac{|\\ln\\varepsilon_0|^{1-\\varsigma}(1+\\sigma)^{j_L(1- \\varsigma)}(r_{j_L}-r_{j_L+1})^{\\varsigma}}{(2\\sigma)^{\\varsigma}} N_{j_L}^\\varsigma\\right\\}\\\\\n &\\leq&\\exp\\left\\{- \\frac{|\\ln\\varepsilon_0|^{1-\\varsigma} r^\\varsigma}{(4\\sigma)^{\\varsigma}} \\left(\\frac{(1+\\sigma)^{1- \\varsigma}}{2^\\varsigma}\\right)^{j_L} N_{j_L}^\\varsigma\\right\\}\\\\\n &<&\\exp\\left\\{-\\left(\\frac{12}{11}\\right)^\\varsigma \\frac{N_{j_L}^\\varsigma}{\\sigma}\\right\\},\n\\end{eqnarray*}\nsince $\\varepsilon_0$ is small enough and\n$$\\frac{(1+\\sigma)^{1- \\varsigma}}{2^\\varsigma}=\\exp\\left\\{\\frac{\\ln(1+\\sigma)}{\\ln(8+8\\sigma)}\\left(\\ln 8-\\ln 2\\right)\\right\\}>1.$$\nTherefore, by (\\ref{k_n_j_L}), we get\n${\\rm Leb}(\\tilde\\Lambda_k)<2\\varepsilon_0^{\\sigma} e^{- |k|^\\varsigma}$,\nwhich implies (\\ref{measure_esti}). For detailed proof of the measure estimate of $\\tilde\\Lambda_k$, we can also refer to Corollary 1 of \\cite{HA}.\n\n\\\n\n\n\\noindent {\\it Proof of (3) and (4).} It can be deduced from Lemma 5 of \\cite{Eli1992}. \\qed\n\n\n\n\\\n\n\\noindent{\\bf Proof of Theorem \\ref{thm_Schro} and \\ref{thm_Schro_sobolev}.} Theorem \\ref{thm_Schro_sobolev} can be seen as a corollary of Theorem \\ref{thm_sobo}.\nAccording to Theorem \\ref{thm_redu}, the reducibility of Eq. (\\ref{eq_Schrodinger}) for a.e. $E\\in{{\\mathcal I}}$ is deduced from Proposition \\ref{prop_eliasson}-(1).\nLet $\\{\\Lambda_j\\}_{j\\in{\\mathbb N}}$ be the intervals $\\tilde\\Lambda_k$'s intersecting ${{\\mathcal I}}$ and let\n$${{\\mathcal O}}_{\\varepsilon_0}:=\\bigcup_{j\\in{\\mathbb N}}\\Lambda_j=\\bigcup_{k\\in{\\mathbb Z}^d}\\tilde\\Lambda_k.$$\nProposition \\ref{prop_eliasson}-(2) gives the measure estimate of ${{\\mathcal O}}_{\\varepsilon_0}$.\nThe unitary equivalences of the reduced quantum Hamiltonian follow from Proposition \\ref{prop_eliasson}-(3) and (4). Hence Theorem \\ref{thm_Schro} is shown. \\qed\n\n\\section{Proof of Theorem \\ref{thm_example_1} -- \\ref{thm_AMO}}\\label{sec_pr_examples}\n\n\nIn this section, we show that the measure of the subset ${{\\mathcal O}}_{\\varepsilon_0}$ is positive for the equations (\\ref{example_1}) -- (\\ref{eq_AMO}),\nwhich implies the growths of Sobolev norm.\n\n\\subsection{Proof of Theorem \\ref{thm_example_1}}\n\nFor Eq. (\\ref{example_1}), $E\\in{\\mathbb R}$, the corresponding linear system is\n$$\n\\left(\\begin{array}{c}\n          x \\\\\n          \\xi\n        \\end{array}\n\\right)'=\\left[\\left(\\begin{array}{cc}\n             0 & E \\\\\n             -E & 0\n           \\end{array}\\right)\n           +\\left(\\begin{array}{cc}\n             b(\\omega t) & c(\\omega t) \\\\\n             -a(\\omega t) & -b(\\omega t)\n           \\end{array}\\right)\\right] \\left(\\begin{array}{c}\n          x \\\\\n          \\xi\n        \\end{array}\n\\right).$$\nIn view of Lemma 5 of \\cite{Eli1992}, for ``generic\" $a,b,c\\in C^\\omega({\\mathbb T}^d,{\\mathbb R})$, there is at least one non-degenerate $\\tilde\\Lambda_k$, $k\\in{\\mathbb Z}^d$.\nMore precisely, at the resonant step of KAM scheme described in the proof of Proposition \\ref{prop_eliasson}-(1),\nthe condition (\\ref{resonant}) defines a resonant interval of $E$, on which the two eigenvalues $\\pm{\\rm i}\\xi_j$ of $A_j$ are purely imaginary since $\\xi_j$ is bounded frow below. After this resonant step, the two new eigenvalues $\\pm{\\rm i}\\xi_{j+1}$ of $A_{j+1}$ can be real or still purely imaginary for $E$ in this resonant interval, since $|\\xi_{j+1}|$ is close to zero.\nWe say that $a,b,c\\in C^\\omega({\\mathbb T}^d,{\\mathbb R})$ are {\\it generic} if, for at least one resonant step in the KAM scheme, the two new eigenvalues $\\pm{\\rm i}\\xi_{j+1}$ become real on a sub-interval of the resonant interval.\n\n\n\n\n\n\n\\subsection{Proof of Theorem \\ref{thm_example_schro}}\n\n\n\nFor Eq. (\\ref{eq_Schrodinger-example}) with $E\\in{{\\mathcal I}}=[E_0,E_1]$ with $E_0>0$ large enough, and $E_1<\\infty$, Theorem \\ref{thm_Schro} and \\ref{thm_Schro_sobolev} hold.\nThe corresponding linear system $(\\omega, \\, A_0+F_0)$ of Eq. (\\ref{eq_Schrodinger-example}) is\n$$\n\\left(\\begin{array}{c}\n          x \\\\\n          \\xi\n        \\end{array}\n\\right)'=\\left[\\left(\\begin{array}{cc}\n             0 & \\sqrt{E} \\\\\n             -\\sqrt{E} & 0\n           \\end{array}\\right)\n           +\\frac{q(\\omega t)}{2\\sqrt{E}}\\left(\\begin{array}{cc}\n             -1 & -1 \\\\\n             1 & 1\n           \\end{array}\\right)\\right] \\left(\\begin{array}{c}\n          x \\\\\n          \\xi\n        \\end{array}\n\\right).$$\nThen, through the change of variables\n$$\\left(\\begin{array}{c}\n          x \\\\\n          \\xi\n        \\end{array}\\right)=\\frac{1}{2\\sqrt{E}}\\left(\\begin{array}{cc}\n           \\sqrt{E} & -1 \\\\\n           \\sqrt{E} & 1\n           \\end{array}\\right)\\left(\\begin{array}{c}\n          \\tilde x \\\\\n          \\tilde\\xi\n        \\end{array}\\right),$$\n$(\\omega, \\, A_0+F_0)$ is conjugated to\n$$\n\\left(\\begin{array}{c}\n          \\tilde x \\\\\n          \\tilde\\xi\n        \\end{array}\\right)'=C^E_q(\\omega t)\\left(\\begin{array}{c}\n          \\tilde x \\\\\n          \\tilde\\xi\n        \\end{array}\\right):=\\left(\\begin{array}{cc}\n             0 & 1 \\\\\n             -E+q(\\omega t) & 0\n           \\end{array}\\right)\\left(\\begin{array}{c}\n          \\tilde x \\\\\n          \\tilde\\xi\n        \\end{array}\\right).\n$$\nThe quasi-periodic linear system $(\\omega, \\, C^E_q(\\cdot))$ corresponds exactly to the eigenvalue problem of the quasi-periodic continuous Schr\\\"odinger operator ${{\\mathcal L}}_{\\omega, q}$:\n $$({{\\mathcal L}}_{\\omega, q}y)(t)=-y''(t)+q(\\omega t) y(t).$$\nBy Gap labeling Theorem \\cite{JM82}, if $\\tilde\\Lambda_k$ is not empty for $k\\in{\\mathbb Z}^d$, then it is indeed a ``spectral gap\" of ${{\\mathcal L}}_{\\omega, q}$ intersecting $[E_0,E_1]$, i.e., a connected component of $[E_0,E_1]\\setminus\\Sigma_{\\omega, q}$ with $\\Sigma_{\\omega, q}$ denoting the spectrum of ${{\\mathcal L}}_{\\omega, q}$.\nIn view of Theorem C of \\cite{Eli1992}, for a generic potential $q$ (in the $|q|_r$-topology), for $E_0>0$ large enough, $[E_0,\\infty[ \\, \\cap \\, \\Sigma_{\\omega, q}$ is a Cantor set.\nHence there are infinitely many $\\tilde\\Lambda_k$'s satisfying ${\\rm Leb}(\\tilde\\Lambda_k)>0$.\n\n\n\\subsection{Proof of Theorem \\ref{thm_AMO}}\n\nFor Eq. (\\ref{eq_AMO}) with $\\nu(E)=\\cos^{-1}(-\\frac{E}{2})$, $E\\in[-2+\\delta,2-\\delta]$ with $\\delta>0$ a sufficiently small numerical constant (e.g. $\\delta:=10^{-6}$), we can apply Theorem \\ref{thm_Schro} and \\ref{thm_Schro_sobolev}.\n if $a,b,c:[-2+\\delta,2-\\delta]\\times{\\mathbb T}^2\\to{\\rm  sl}(2,{\\mathbb R})$ are small enough as assumed in Theorem \\ref{thm_Schro}.\n\n\n\nFor the quasi-periodic Schr\\\"odinger cocycle $(\\alpha, \\, S_E^\\lambda)$\n$$\nX_{n+1} =S_E^\\lambda(\\theta+n\\alpha) X_n= \\left[\\left(\\begin{array}{cc}\n             -E & -1 \\\\\n             1 & 0\n           \\end{array}\\right)+\\left(\\begin{array}{cc}\n             2\\lambda \\cos(\\theta+n\\alpha) & 0 \\\\\n             0 & 0\n           \\end{array}\\right)\\right] X_n,\n$$\nwith $\\alpha\\in{\\rm DC}_1(\\gamma,\\tau)$, $|\\lambda|$ small enough,\nit can be written as\n$$X_{n+1}=e^{B(E)}e^{G(E,\\theta+n\\alpha)} X_n,$$\nfor $e^{B(E)}:=\\left(\\begin{array}{cc}\n             -E & -1 \\\\\n             1 & 0\n           \\end{array}\\right)$ and some $G(E,\\cdot)\\in{\\rm sl}(2,{\\mathbb R})$.\nThis cocycle is related to the almost-Mathieu operator $H_{\\lambda,\\alpha,\\theta}$ on $\\ell^2({\\mathbb Z})$:\n$$(H_{\\lambda,\\alpha,\\theta}\\psi)_n=-(\\psi_{n+1}+\\psi_{n-1})+2\\lambda \\cos(\\theta+n\\alpha)\\psi_n,\\qquad n\\in{\\mathbb Z}.$$\nIt is known that its spectrum, denoted by $\\Sigma_{\\lambda,\\alpha}$, is a Cantor set \\cite{AvilaJito1},  which is well-known as Ten Martini Problem. In fact, Avila-Jitomirskaya\n \\cite{AvilaJito} further show that  all spectral gaps are ``open\" , which means that, for every $k\\in{\\mathbb Z}$,\n$$\\tilde\\Lambda_k:=\\left\\{E\\in{\\mathbb R}: \\tilde\\rho_{(\\alpha, \\, S_E^\\lambda)}=\\frac{k\\alpha}{2} \\mod {\\mathbb Z} \\right\\}$$\nhas positive measure. Indeed, the size of $\\tilde\\Lambda_k$ decays exponentially with respect to $|k|$, as was shown in \\cite{LYZZ}.\nHere, $\\tilde\\rho_{(\\alpha, \\, S_E^\\lambda)}$ is the fibered rotation number of cocycle $(\\alpha, \\, S_E^\\lambda)$.\nRecall that for any  $A:{\\mathbb T}^d \\to {\\rm SL}(2,{\\mathbb R})$ is continuous and homotopic to the identity,   \\textit{fibered rotation number} of\n$(\\alpha,A)$ is defined as\n\\begin{equation*}\n\\tilde\\rho(\\alpha,A)=\\int \\psi \\, d \\tilde{\\mu} \\mod {\\mathbb Z}\n\\end{equation*}\nwhere  $\\psi:{\\mathbb T}^{d+1} \\to {\\mathbb R}$ is  lift of $A$ such that\n$$\nA(x) \\cdot \\left (\\begin{matrix} \\cos 2 \\pi y \\\\ \\sin 2 \\pi y \\end{matrix} \\right )=u(x,y)\n\\left (\\begin{matrix} \\cos 2 \\pi (y+\\psi(x,y)) \\\\ \\sin 2 \\pi (y+\\psi(x,y)) \\end{matrix} \\right),\n$$\nand $\\tilde{\\mu}$ is\ninvariant probability  measure of $(x,y) \\mapsto (x+\\alpha,y+\\psi(x,y))$  (according to \\cite{Her}, it does not depend on the choices of $\\psi, \\tilde\\mu$).\n\n\nNote that $(\\alpha, \\, S_E^\\lambda)$ is a discrete dynamical system, however, with the help of Local Embedding Theorem (Theorem \\ref{localemb-sl}), we can embed the cocycle $(\\alpha, \\, S_E^\\lambda)$ into a quasi-periodic linear system $(\\omega, \\, B(E)+F(E,\\cdot))$. \nFor an individual cocycle, the Local Embedding Theorem was already shown in \\cite{YZ2013}.\nNevertheless, the crucial point here is that we really need a parameterized version of Local Embedding Theorem, that means the embedded system   $(\\omega, \\, B(E)+F(E,\\cdot))$ should have smooth dependence on $E$.\n\n\n\nTo show the parameterized version of Local Embedding Theorem, let us first introduce more notations.\n Given $f \\in C^2({\\mathcal I})$, define\n$$|f|_{*}= \\sum_\n{0\\leq m\\leq 2} \\sup_{E\\in{\\mathcal I}}|f^{(m)}|.$$\nFor any $ f(E,\\theta)=\\sum_{k\\in {\\mathbb Z}^d}\\widehat f_k(E)e^{2\\pi  {\\rm i}\n\\langle k,\\theta\\rangle}$ which is $C^2$ w.r.t. $E\\in{{\\mathcal I}}$, $C^\\omega$ w.r.t. $\\theta\\in{\\mathbb T}^d$, denote\n$$\\|f\\|_h:=\\sum_{k\\in {\\mathbb Z}^d}|\\widehat f_k(E)|_{*}e^{2\\pi |k|h}<\\infty,$$\nand we denote by  $C_h^\\omega( {\\mathcal I} \\times {\\mathbb T}^d,{\\mathbb C})$ all these functions with $\\|f\\|_h<\\infty$. Then our result is the following:\n\n\n\n\\begin{Theorem}\\label{localemb-sl}[Local Embedding Theorem]\nGiven $d\\geq 2$, $h>0$ and $G\\in C^\\omega_{h}({{\\mathcal I}} \\times {\\mathbb T}^{d-1}, {\\rm sl}(2,{\\mathbb R}))$, suppose that $\\mu\\in {\\mathbb T}^{d-1}$ such that $(1,\\mu)$ is rationally independent. Then, for any $\\nu\\in C^2({\\mathcal I})$ satisfying\n \\begin{equation}\\label{varition}\n \\sup_{E\\in{\\mathcal I}}|\\nu'(E)|\\cdot |{\\mathcal I}|< \\frac{1}{6},\n \\end{equation}\nthere exist $\\epsilon=\\epsilon(|\\nu|_{*},h,|\\mu|)>0,$\n$c=c(|\\nu|_*,h,|\\mu|)>0,$ and $F\\in\nC^\\omega_{\\frac{h}{1+|\\mu|}}(  {\\mathcal I} \\times {\\mathbb T}^d,{\\rm sl}(2,{\\mathbb R}))$ such that the cocycle $(\\mu,e^{2\\pi \\nu J}\ne^{G(\\cdot)})$ is the Poincar\\'e map of linear system\n\\begin{eqnarray}\\label{al-ref1}\n\\left(\\begin{array}{c}x\\\\ \\xi\n \\end{array}\n \\right)'=\\left(\\nu J+F(\\omega t)\\right)\\left(\\begin{array}{c}x\\\\ \\xi\n \\end{array}\n \\right) , \\qquad \\omega=(1,\\mu)\n\\end{eqnarray}\nprovided that $\\|G\\|_{h}<\\epsilon.$ Moreover, we have $\\|F\\|_{\\frac{h}{1+|\\mu|}}\\leq 2c \\|G\\|_{h}$.\n\\end{Theorem}\nWe postpone the proof of Theorem \\ref{localemb-sl} to Appendix \\ref{app_proof}.\n\n\\smallskip\n\nNow let us show how we can apply Theorem \\ref{localemb-sl} to finish the proof of Theorem \\ref{thm_AMO}. First note the constant matrix $e^{B}$ can be rewritten as\n\n$$e^{B}:=\\left(\\begin{array}{cc}\n             -E & -1 \\\\\n             1 & 0\n           \\end{array}\\right) = M\\left(\\begin{array}{cc}\n            \\cos(\\nu) & -\\sin(\\nu) \\\\\n             \\sin(\\nu) & \\cos(\\nu)\n           \\end{array}\\right)M^{-1} , $$\nwhere\n$$M:=\\frac{1}{\\sqrt{\\sin(\\nu)}}\\left(\\begin{array}{cc}\n             \\cos(\\nu) & -\\sin(\\nu) \\\\\n             1 & 0\n           \\end{array}\\right), $$\nrecalling that\n$$\\cos(\\nu(E))=-\\frac{E}{2},\\quad \\sin(\\nu(E))=\\frac{\\sqrt{4-E^2}}{2},\\qquad E\\in[-2+\\delta,\\ 2-\\delta].$$\nHence, by noting\n$$\\left(\\begin{array}{cc}\n            \\cos(\\nu) & -\\sin(\\nu) \\\\\n             \\sin(\\nu) & \\cos(\\nu)\n           \\end{array}\\right)=\\exp\\left\\{\\left(\\begin{array}{cc}\n            0 & -\\nu \\\\\n             \\nu & 0\n           \\end{array}\\right)\\right\\},$$\n          we see that $B$ can be written as\n$B=M  \\cdot ( \\nu J )  \\cdot  M^{-1}.$\n\n\n\n\\smallskip\n\nFor $\\nu(E)=\\cos^{-1}(-\\frac{E}{2})$, there exists ${{\\mathcal I}}\\subset [-2+\\delta, 2-\\delta]$ such that (\\ref{varition}) is satisfied.\nFor example, we can take ${{\\mathcal I}}= ]-\\frac{2}{\\sqrt{37}},\\frac{2}{\\sqrt{37}}[$.\nTherefore, according to Theorem \\ref{localemb-sl}, for $\\omega\\in (1,\\alpha)$, we have a quasi-periodic linear system $(\\omega, \\, B(E)+F(E,\\cdot))$ from  the quasi-periodic Schr\\\"odinger cocycle $(\\alpha, \\, S_E^\\lambda)$:\n\\begin{equation}\\label{Schrodinger_cocycle-conti}\n\\left(\\begin{array}{c}\n          x \\\\\n         \\xi\n        \\end{array}\n\\right)'=(B(E)+F(E,\\omega t)) \\left(\\begin{array}{c}\n          x \\\\\n          \\xi\n        \\end{array}\n\\right),\n\\end{equation}\nThrough the change of variables\n$$\\left(\\begin{array}{c}\n          x \\\\\n          \\xi\n        \\end{array}\\right)=M\\left(\\begin{array}{c}\n          \\tilde x \\\\\n          \\tilde\\xi\n        \\end{array}\\right),$$\n$(\\omega, \\, B(E)+F(E,\\cdot))$ is conjugated to\n$$\\left(\\begin{array}{c}\n          \\tilde x \\\\\n         \\tilde\\xi\n        \\end{array}\n\\right)'=\\left(\\left(\\begin{array}{cc}\n            0 & -\\nu \\\\\n             \\nu & 0\n           \\end{array}\\right)+MF(E,\\omega t)M^{-1}\\right)\\left(\\begin{array}{c}\n          \\tilde x \\\\\n         \\tilde\\xi\n        \\end{array}\n\\right).$$\nThen by Theorem \\ref{thm_Schro} and \\ref{thm_Schro_sobolev},  Theorem \\ref{thm_AMO} is shown with\n$$\\left(\\begin{array}{cc}\n            b(E,\\cdot) & c(E,\\cdot) \\\\\n            -a(E,\\cdot) & -b(E,\\cdot)\n           \\end{array} \\right)=MF(E,\\cdot)M^{-1}.$$\n\nFinally we point out that $\\rho_{(\\omega, \\, B(E)+F(E,\\cdot))}=\\tilde\\rho_{(\\alpha, \\, S_E^\\lambda(\\cdot))}$, since  $(\\alpha, \\, S_E^\\lambda)$ is the Poincar\\'e map of linear system\n$(\\omega, \\, B(E)+F(E,\\cdot))$.  Let\n$$\\tilde\\Lambda_{(-p, k)}:=\\left\\{E\\in\\overline{{\\mathcal I}}:  \\rho_{(\\omega, \\, B(E)+F(E,\\cdot))} =   \\frac{ k\\alpha-p}{2} =    \\min_{j\\in {\\mathbb Z}}  \\left| \\frac{ k\\alpha}{2} -j\\right| \\right\\} ,$$\nthen by well-known result of Avila-Jitomirskaya  \\cite{AvilaJito},  ${\\rm Leb}(\\tilde\\Lambda_{(-p, k)})>0$, for every $k\\in{\\mathbb Z}$ such that  $\\tilde\\Lambda_k$ intersect with ${{\\mathcal I}}$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Abstract}\nIn a world in which many pressing global issues require large scale cooperation, understanding the group size effect on cooperative behavior is a topic of central importance. Yet, the nature of this effect remains largely unknown, with lab experiments insisting that it is either positive or negative or null, and field experiments suggesting that it is instead curvilinear. Here we shed light on this apparent contradiction by considering a novel class of public goods games inspired to the realistic scenario in which the natural output limits of the public good imply that the benefit of cooperation increases fast for early contributions and then decelerates. We report on a large lab experiment providing evidence that, in this case, group size has a curvilinear effect on cooperation, according to which intermediate-size groups cooperate more than smaller groups and more than larger groups. In doing so, our findings help fill the gap between lab experiments and field experiments and suggest concrete ways to promote large scale cooperation among people.\n\n\n\n\n\\section*{Introduction}\n\nCooperation has played a fundamental role in the early evolution of our societies\\cite{KG,Tomasello14natural} and continues playing a major role still nowadays. From the individual level, where we cooperate with our romantic partner, friends, and co-workers in order to handle our individual problems, up to the global level where countries cooperate with other countries in order to handle global problems, our entire life is based on cooperation.\n\nGiven its importance, it is not surprising that cooperation has inspired an enormous amount of research across all biological and social sciences, spanning from theoretical accounts \\cite{Tr,Ax-Ha,FF03,nowak2006five,Perc10coevolutionary,press2012iterated,perc2013evolutionary,Ca,hilbe2013evolution,Ra-No,capraro2014translucent} to experimental studies \\cite{Andreoni1988why,Fischbacher2001people,milinski2002reputation,Frey2004social,Fischbacher2010social,traulsen2010human,apicella2012social,capraro2014heuristics,capraro2014benevolent,capraro2014good,hauser2014cooperating,gallo2015effects} and numerical simulations\\cite{Nowak92evolutionary,boyd2003evolution,santos2005scale,perc2008social,roca2009evolutionary,gardenes2012evolution,jiang2013spreading}.\n\nSince the resolution of many pressing global issues, such as global climate change and depletion of natural resources, requires cooperation among many actors, one of the most relevant questions about cooperation regards the effect of the size of the group on cooperative behavior. Indeed, since the influential work by Olson \\cite{olson1965logic}, scholars have recognized that the size of a group can have an effect on cooperative decision-making. However, the nature of this effect remains one of the most mysterious areas in the literature, with some scholars arguing that it is negative \\cite{olson1965logic,dawes1977behavior,komorita1982cooperative,baland1999ambiguous,ostrom2005understanding,grujic2012three,vilone2014partner,nosenzo2015cooperation}, others that it is positive \\cite{mcguire1974group,isaac1994group,haan2002free,agrawal2006explaining,masel2007bayesian,zhang2011group,szolnoki2011group}, and yet others that it is ambiguous \\cite{esteban2001collective,pecorino2008group,oliver1988paradox,chamberlin1974provision} or non-significant \\cite{todd1992collective,gautam2007group,rustagi2010conditional}. Interestingly, the majority of field experiments seem to agree on yet another possibility, that is, that group size has a curvilinear effect on cooperative behavior, according to which intermediate-size groups cooperate more than smaller groups and more than larger groups \\cite{poteete2004heterogeneity,agrawal2001group,agrawal2000small,yang2013nonlinear,cinner2013looking}.\nThe emergence of a curvilinear effect of the group size on cooperation in real life situations is also supported by data concerning academic research, which in fact support the hypothesis that research quality of a research group is optimized for medium-sized groups \\cite{kenna2011critical,kenna2011critical2,kenna2012managing}.\n\nHere we aim at shedding light on this debate, by providing evidence that a single parameter can be responsible for all the different and apparently contradictory effects that have been reported in the literature. Specifically, we show that the effect of the size of the group on cooperative decision-making depends critically on a parameter taking into account different ways in which the notion of cooperation itself can be defined when there are more than two agents.\n\nIndeed, while in case of only two agents a cooperator can be simply defined as a person willing to pay a cost $c$ to give a greater benefit $b$ to the other person \\cite{nowak2006five}, the same definition, when transferred to situations where there are more than two agents, is subject to multiple interpretations. If cooperation, from the point of view of the cooperator, means paying a cost $c$ to create a benefit $b$, what does it mean from the point of view of the \\emph{other} player\\emph{s}? Does $b$ get earned by each of the other players or does it get shared among all other players, or none of them? In other words, what is the marginal return for cooperation?\n\nOf course, there is no general answer and, in fact, previous studies have considered different possibilities. For instance, in the standard Public Goods game it is assumed that $b$ gets earned by each player (including the cooperator); instead, in the N-person Prisoner's dilemma (as defined in \\cite{barcelo2015group}) it is assumed that $b$ gets shared among all players; yet, the Volunteer's dilemma \\cite{diekmann1985volunteer} and its variants using critical mass \\cite{szolnoki2010impact} rest somehow in between: one or more cooperators are needed to generate a benefit that gets earned by each player, but, after the critical mass is reached, new cooperators do not generate any more benefit; finally, it has been pointed out \\cite{marwell1993critical,heckathorn1996dynamics} that a number of realistic situations can be characterized by a marginal return which increases linearly for early contributions and then decelerates, reflecting the natural decrease of marginal returns that occurs when output limits are approached.\n\nIn order to take into account this variety of possibilities, we consider a class of \\emph{social dilemmas} parametrized by a function $\\beta=\\beta(\\Gamma,N)$ describing the marginal return for cooperation when $\\Gamma$ people cooperate in a group of size $N$. More precisely, our \\emph{general Public Goods game} is the N-person game in which N people have to simultaneously decide whether to cooperate (C) or defect (D). In presence of a total of $\\Gamma$ cooperators, the payoff of a cooperator is defined as $\\beta(\\Gamma,N)-c$ ($c>0$ represents the cost of cooperation) and the payoff of a defector is defined as $\\beta(\\Gamma,N)$. In order to have a social dilemma (i.e., a tension between individual benefit and the benefit of the group as a whole) we require that:\n\\begin{itemize}\n\\item Full cooperation pays more than full defection, that is, $\\beta(N,N) - c > \\beta(0,N)$, for all $N$; \n\\item Defecting is individually optimal, regardless of the number of cooperators, that is, for all $\\Gamma < N$, one has $\\beta(\\Gamma,N)-c < \\beta(\\Gamma-1,N)$.\n\\end{itemize}\n\nThe aim of this paper is to provide further evidence that the function $\\beta$ might be responsible for the confusion in the literature about group size effect on cooperation. In particular, we focus on the situation, inspired from realistic scenarios, in which the natural output limits of the public good imply that $\\beta(\\Gamma,N)$ increases fast for small $\\Gamma$'s and then stabilizes. \n\nIndeed, in our previous work \\cite{barcelo2015group}, we have shown that the size of the group has a positive effect on cooperation in the standard Public Goods game and has a negative effect on cooperation in the N-person Prisoner's dilemma. A reinterpretation of these results is that, if $\\beta(N,N)$ increases linearly with $N$ (standard Public Goods game), then the size of the group has a positive effect on cooperation; and, if $\\beta(N,N)$ is constant with $N$ (N-person Prisoner's dilemma), then the size of the group has a negative effect on cooperation. This reinterpretation suggests that, in the more realistic situations in which the benefit for full cooperation increases fast for early contributions and then decelerates once the output limits of the public good are approached, we may observe a curvilinear effect of the group size, according to which intermediate-size groups cooperate more than smaller groups and more than larger groups.\n\nTo test this hypothesis, we have conducted a lab experiment using a general public goods game with a piecewise function $\\beta$, which increases linearly up to a certain number of cooperators, after which it remains constant. While it is likely that realistic scenarios would be better described by a smoother function, this is a good approximation of all those situations in which the natural output limits of a public good imply that the increase in the marginal return for cooperation tends to zero as the number of contributors grows very large. The upside of choosing a piecewise function $\\beta$ is that, in this way, we could present the instructions of the experiment in a very simple way, thus minimizing random noise due to participants not understanding the decision problem at hand (see Method).\n\nOur results support indeed the hypothesis of a curvilinear effect of the size of the group on cooperative decision-making. Taken together with our previous work \\cite{barcelo2015group}, our findings thus (i) shed light on the confusion regarding the group size effect on cooperation, by pointing out that different values of a single parameter might give rise to qualitatively different group size effects, including positive, negative, and even curvilinear; and (ii) they help fill the gap between lab experiments and field experiments. Indeed, while lab experiments use either the standard Public Goods game or the N-person Prisoner's dilemma, \\emph{real} public goods game are mostly characterized by a marginal return of cooperation that increases fast for early contributions and then approaches a constant function as the number of cooperators grows very large - and our results provide evidence that these three situations give rise to three different group size effects.\n\n\\section*{Method}\n\nWe have recruited participants through the online labour market Amazon Mechanical Turk (AMT) \\cite{paolacci2010running,horton2011online,mason2012conducting}. After entering their TurkID, participants were directed to the following instruction screen.\n\n\\emph{Welcome to this HIT.}\n \n\\emph{This HIT will take about 5 minutes and you will earn 20c for participating.} \n \n\\emph{This HIT consists of a decision problem followed by a few demographic questions.}  \n \n\\emph{You can earn an additional bonus depending on the decisions that you and the participants in your cohort will make.} \n\n\\emph{We will tell you the exact number of participants in your cohort later.} \n\n\\emph{Each one of you will have to decide to join either Group A or Group B.} \n \n\\emph{Your bonus depends on the group you decide to join and on the size of the two groups, A and B, as follows:}\n\\begin{itemize}\n\\item \\emph{If the size of Group A is 0 (that is, everybody chooses to join Group B), then everybody gets 10c}\n\\item \\emph{If the size of Group A is 1, then the person in Group A gets 5c and each person in Group B gets 15c}\n\\item \\emph{If the size of Group A is 2, then each person in Group A gets 10c and each person in Group B gets 20c}\n\\item \\emph{If the size of Group A is 3, then each person in Group A gets 15c and each person in Group B gets 25c}\n\\item \\emph{If the size of Group A is 4, then each person in Group A gets 20c and each person in Group B gets 30c}\n\\item \\emph{And so on, up to 10: If the size of Group A is 10, then each person in Group A gets 50c and each person in Group B gets 60c}\n\\item \\emph{However, if the size of Group A is larger than 10, then, independently of the size of the two groups, each person in group A will still get 50c and each person in group B will still get 60c.}\n\\end{itemize}\n\nAfter reading the instructions, participants were randomly assigned to one of 12 conditions, differing only on the size of the cohort ($N=3,5,10,15,20,25,30,40,50,60,80,100$). For instance, the decision screen for the participants in the condition where the size of the cohort is 3 was:\n\n\\emph{You are part of a cohort of 3 participants.}\n\n\\emph{Which group do you want to join?}\n\nBy using appropriate buttons, participants could select either Group A or Group B. \n\nWe opted for not asking any comprehension questions. We made this choice for two reasons. First, with the current design, it is impossible to ask general comprehension questions such as ``what is the strategy that benefits the group as a whole'', since this strategy depends on the strategy played by the other players. Second, we did not want to ask particular questions about the payoff structure since this may anchor the participants' reasoning on the examples presented. Of course, a downside of our choice is that we could not avoid random noise. However, as it will be discussed in the Results section, random noise cannot be responsible for our findings. Instead, our results would have been even cleaner, if we had not had random noise, since the initial increase of cooperation and its subsequent decline would have been more pronounced (see Results section for more details).\n\nAfter making their decisions, participants were asked to fill a standard demographic questionnaire (in which we asked for their age, gender, and level of education), after which they received the ``survey code'' needed to claim their payment. After collecting all the results, bonuses were computed and paid on top of the participation fee, that was \\$0.20. In case the number of participants in a particular condition was not divisible by the size of the cohort (it is virtually impossible, in AMT experiments, to decide the exact number of participants playing a particular condition), in order to compute the bonus of the remaining people we formed an additional cohort where these people where grouped with a random choice of people for which the bonus had been already computed.  Additionally, we anticipate that only 98 subjects participated in the condition with N=100. This does not generate deception in the computation of the bonuses since the payoff structure of the game does not depend on $N$ (as long as $N>10$). As a consequence of these observations, no deception was used in our experiment. \n\nAccording to the Dutch legislation, this is a non-WMO study, that is (i) it does not involve medical research and (ii) participants are not asked to follow rules of behavior. See http:\/\/www.ccmo.nl\/attachments\/files\/wmo- engelse-vertaling-29-7-2013-afkomstig-van-vws.pdf, Section 1, Article 1b, for an English translation of the Medical Research Act. Thus (see http:\/\/www.ccmo.nl\/en\/non-wmo- research) the only legislations which apply are the Agreement on Medical Treatment Act, from the Dutch Civil Code (Book 7, title 7, section 5), and the Personal Data Protection Act (a link to which can be found in the previous webpage). The current study conforms to both. In particular, anonymity was preserved because AMT ``requesters'' (i.e., the experimenters) have access only to the so-called TurkID of a participant, an anonymous ID that AMT assigns to a subject when he or she registers to AMT. Additionally, as demographic questions we only asked for age, gender, and level of education. \n\n\\section*{Results}\n\nA total of 1.195 \\emph{distinct} subjects located in the US participated in our experiment. \\emph{Distinct} subjects means that, in case two or more subjects were characterized by either the same TurkID or the same IP address, we kept only the first decision made by the corresponding participant and eliminated the rest. These multiple identities represent usually a minor problem in AMT experiments (only 2\\% of the participants in the current dataset). Participants were distributed across conditions as follows: 101 participants played with $N=3$, 99 with $N=5$, 102 with $N=10$, 101 with $N=15$, 98 with $N=20$, 103 with $N=25$, 97 with $N=30$, 99 with $N=40$, 97 with $N=50$, 101 with $N=60$, 99 with $N=80$, 98 with $N=100$.\n\nFig. 1 summarizes the main result. The rate of cooperation, that is the proportion of people opting for joining Group A, first increases as the size of the group increases from $N=3$ to $N=15$, then it starts decreasing. The figure suggests that the relation between the size of the group and the rate of cooperation is \\emph{not} quadratic: while the initial increase of cooperation is relatively fast, the subsequent decrease of cooperation seems extremely slow. This is confirmed by linear regression predicting rate of cooperation as a function of $N$ and $N^2$, which shows that neither the coefficient of $N$ nor that of $N^2$ are significant ($p=0.4692, p=0.2003$, resp.). For this reason we use a more flexible econometric model than the quadratic model, consisting of two linear regressions, one with a positive slope (for small $N$'s) and the other one with a negative slope (for large $N$'s). As a switching point, we use the $N=15$, corresponding to the size of the group which reached maximum cooperation. Doing so, we find that both the initial increase of cooperation and its subsequent decline are highly significant (from $N=3$ to $N=15$: coeff $= 0.0187553$, $p=0.00042$; from $N=15$ to $N=100$: coeff $= -0.00177618$, $p=0.00390$). \n\n\\begin{figure}\n   \\centering\n   \\includegraphics[scale=0.80]{Fig1.jpg} \n   \\label{fig:intermediate}\n   \\caption{\\emph{Proportion of cooperators (people choosing to join Group A) for each group size. Error bars represent the standard errors of the means. Group size has initially a positive effect on cooperation, which increases and reaches its maximum in groups of size 15, followed by a gradual decrease. Linear regression predicting cooperation using group size as independent variable confirms that both the initial increase of cooperation and its subsequent decline are highly significant (from $N=3$ to $N=15$: coeff $= 0.0187553$, $p=0.00042$; from $N=15$ to $N=100$: coeff $= -0.00177618$, $p=0.00390$).}}\n\\end{figure}\n\nWe conclude by observing that not only random noise cannot explain our results, but, without random noise, the effect would have been even stronger. Indeed, first we observe that there is no a priori worry that random noise would interact with any condition and so we can assume that it is randomly distributed across conditions. Then we observe that subtracting a binary distribution with average $0.5$ from a binary distribution with average $\\mu>0.5$, one would obtain a distribution with average $\\mu_0>\\mu$. Similarly, subtracting a binary distribution with average $0.5$ from a binary distribution with average $\\mu<0.5$ one would obtain a distribution with average $\\mu_0<\\mu$. Thus, if the $\\mu$'s are the averages that we have found (containing random noise) and the $\\mu_0$'s are the \\emph{true} averages (without random noise), the previous inequalities allow us to conclude that the initial increase of cooperation and its following decrease would have been stronger in absence of random noise.\n\n\\section*{Discussion}\n\nHere we have reported on a lab experiment providing evidence that the size of a group can have a curvilinear effect on cooperation in one-shot social dilemmas, with intermediate-size groups cooperating more than smaller groups and more than larger groups. Joining the current results with those of a previously published study of us \\cite{barcelo2015group}, we can conclude that group size can have qualitatively different effects on cooperation, ranging from positive, to negative and curvilinear, depending on the particular decision problem at hand. Interestingly, our findings suggest that different group size effects might be ultimately due to different values of a single parameter, the number $\\beta(N,N)$, describing the benefit for full cooperation. If $\\beta(N,N)$ is constant in $N$, then group size has a negative effect on cooperation; if $\\beta(N,N)$ increases linearly with $N$, then group size has a positive effect on cooperation; in the \\emph{middle}, all sorts of things may a priori happen. In particular, in the realistic situation in which $\\beta(N,N)$ is a piecewise function that increases linearly with $N$ up to a certain $N_0$ and then remains constant, then group size has a curvilinear effect, according to which intermediate-size groups cooperate more than smaller groups and more than larger groups. See Table 1.\n\n\\begin{center}\n\\begin{table}\n\\begin{tabular}{| l | c | c| }\n  \\hline                       \n  shape of $\\beta(N,N)$ & group size effect on cooperation & paper \\\\\n\\hline\n  linear & positive & Barcelo \\& Capraro (2015) \\\\\n  constant & negative & Barcelo \\& Capraro (2015) \\\\\n  linear-then-constant & curvilinear & this paper\\\\\n  \\hline  \n\\end{tabular}\n\\caption{Summary of the different group size effects on cooperation depending on how the benefit for full cooperation varies as a function of the group size.}\n\\end{table}\n\\end{center}\nTo the best of our knowledge, ours is the first study reporting a curvilinear effect of the group size on cooperation in an experiment conducted in the ideal setting of a lab, in which confounding factors are minimized. Previous studies reporting a qualitatively similar effect \\cite{poteete2004heterogeneity,agrawal2001group,agrawal2000small,yang2013nonlinear} used field experiments, in which it is difficult to isolate the effect of the group size from possibly confounding effects. In our case, the only possibly confounding factor is random noise due to a proportion of people that may have not understood the rules of the decision problem. As we have shown, our results cannot be driven by random noise and, in fact, the curvilinear effect would have been even stronger, without random noise. Moreover, since our experimental design was inspired by a tentative to mimic all those \\emph{real} public goods games in which the natural output limits of the public good imply that the increase of the marginal return for cooperation, when the number of cooperators diverges, tends to zero, our results might explain the apparent contradiction that field experiments tend to converge on the fact that the effect of the group size is curvilinear, while lab experiments tend to converge on either of the two linear effects.\n\nOur contribution is also conceptual, since we have provided evidence that a single parameter might be responsible for different group size effects: the parameter $\\beta(N,N)$, describing the way the benefit for full cooperation varies as a function of the size of the group. Of course, we do not pretend to say that this is the only ultimate explanation of why different group size effects have been reported in experimental studies. In particular, in real-life situations, which are typically repeated and in which communication among players is allowed, other factors, such as within-group enforcement, may favor the emergence of a curvilinear effect of the group size on cooperation, as highlighted in \\cite{yang2013nonlinear}. If anything, our results provide evidence that the curvilinear effect on cooperation goes beyond contingent factors and can be found also in the ideal setting of a lab experiment using one-shot anonymous games. We believe that this is a relevant contribution in light of possible applications of our work. Indeed, the difference between $\\beta(N,N)$ and the total cost of full cooperation $cN$ can be interpreted has the incentive that an institution needs to pay to the contributors in order to make them cooperate. Since institutions are interested in minimizing their costs and, at the same time, maximizing the number of cooperators, it is crucial to understand what is the ``lowest'' $\\beta$ such that the resulting effect of the group size on cooperation is positive. This seems to be an non-trivial question. For instance, does $\\beta(\\Gamma,N)=\\frac{\\Gamma}{N}\\log_2(N+1)$ give rise to a positive effect or is it still curvilinear or even negative? The technical difficulty here is that it is hard to design an experiment to test people's behavior in these situations, since one cannot expect that an average person would understand the rules of the game when presented using a logarithmic functions. \n\nIn terms of economic models, our results are consistent with utilitarian models such as the Charness \\& Rabin model \\cite{charness2002understanding} and the novel cooperative equilibrium model \\cite{Ca,capraro2013cooperative,barcelo2015group}. Both these models indeed predict that, in our experiment, cooperation initially (i.e., for $N\\leq10$) increases with $N$ (see \\cite{barcelo2015group} for the details), and then starts decreasing. This behavioral transition follows from the simple observation that free riding when there are more than 10 cooperators costs zero to each of the other players and benefits the free-rider. Thus, cooperation in larger groups is not supported by utilitarian models, which then predict a decrease in cooperative behavior whose speed depends on the particular parameters of the model, such as the extent to which people care about the group payoff versus their individual payoff, and people's beliefs about the behavior of the other players. Thus our results add to the growing body of literature showing that utilitarian models are qualitatively good descriptors of cooperative behavior in social dilemmas.\n\nHowever, we note that while theoretical models predict that the rate of cooperation should start decreasing at $N=10$, our results show that the rate of cooperation for $N=15$ is marginally significantly higher than the rate of cooperation for $N=10$ (Rank sum, $p=0.0588$). Although ours is a between-subjects experiment, this finding seems to hint at the fact that there is a proportion of subjects who would defect for $N=10$ and cooperate for $N=15$. This is not easy to explain: why should a subject cooperate with $N=15$ and defect with $N=10$? One possibility is that there is a proportion of ``inverse conditional cooperators'', who cooperate only if a small percentage of people cooperate: if these subjects believe that the rate of cooperation decreases quickly after $N=10$, they would be more motivated to cooperate for $N=15$ than for $N=10$. Another possibility, of course, is that this discrepancy is just a false positive. In any case, unfortunately our experiment is not powerful enough to detect the reason of this discrepancy between theoretical predictions and experimental results and thus we leave this interesting question for future research.\n\n\\section*{Acknowledgements}\n\nV.C. is supported by the Dutch Research Organization (NWO) Grant No. 612.001.352. This material is based upon work supported by the National Science Foundation under Grant No. 0932078000 while the first author was in residence at the Mathematical Science Research Institute in Berkeley, California, during the Spring 2015 semester.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n    In 1982, Feynman showed that a classical Turing machine would not be able to efficiently simulate quantum mechanical systems \\cite{feynman1982simulating}. Feynman went on to propose a model of computation based on quantum mechanics, which would not suffer the same limitations. Feynman's ideas were later refined by Deutsch who proposed a \\textit{universal quantum computer} \\cite{Deutsch97}. In this scheme, computation is performed by a series of \\textit{quantum gates}, which are the quantum analog to classical binary logic gates. A series of  gates is called a \\textit{quantum circuit} \\cite{nielsen-book}. Quantum gates act on \\textit{qubits} which is the quantum analog of a bit.\n    \\\\\n    \\\\\n    Lloyd later proved that a quantum computer would be able to simulate any quantum mechanical system efficiently \\cite{lloyd1996universal}. Equivalently, this can be stated as; given some special unitary operation \\( U \\in \\mathrm{SU}(2^n)\\), \\( U^{\\dagger} U = I\\), there exists some quantum circuit that  approximates \\(U\\), where \\(n\\) is the number of qubits.  One pertinent question that remains is how to find the circuit which implements this \\(U\\). In certain situations the circuit to implement \\(U\\) can be found exactly. However in general it is a difficult problem, and it is acceptable to approximate \\(U\\). Previously \\(U\\) has been found via expensive algebraic means \\cite{qcompiler,opt-qcompiler,cosine-sekigawa, Mottonen2004}.  Another novel attempt at finding an approximate \\(U\\) has been to use the tools of \\textit{Riemannian geometry}. \n    \\\\\n    \\\\\n    Nielsen originally proposed calculating special curves called \\textit{geodesics} between two points, \\(I\\) and \\(U\\) in \\(\\mathrm{SU}(2^n)\\). Geodesics are fixed points of the energy functional \\cite{wolfgang}. Nielsen claimed that when an energy minimising geodesic is discretised into a quantum circuit,  this would efficiently simulate \\(U\\) \\cite{nielsen-geom-1,nielsen-geom-2,nielsen-geom-3,nielsen-geom-4,nielsen-geom-5}. In practice however, finding the geodesics is a difficult task. Computing geodesics requires one to solve a boundary value problem in a high dimensional space. Furthermore, Nielsen originally formulated the problem on a Riemannian manifold equipped with a so called \\textit{penalty} metric, where the penalty was made large. This complicated solving the boundary value problem \\cite{brachistochrone}.\n    \\\\\n    \\\\\n    The Nielsen approach can be refined by considering subRiemannian geodesics. A subRiemannian geodesic is only allowed to evolve in directions from a \\textit{horizontal subspace} of the tangent space \\cite{montgomery}. This approach still involves solving a complicated boundary value problem. For a practical tool, a much faster methodology to synthesise a \\(U\\) is required. With recent advances in computing power, \\textit{neural networks} (NN) are an attractive option.\n    \\\\\n    \\\\\n    The problem is to find \\(U\\) approximately as a product of exponentials\n    \\begin{align} U \\approx \\,  &\\mathbf{E}(c) = \\exp(c^1_1 \\tau_{1}) \\dots \\exp( c^1_m \\tau_m ) \\nonumber \\\\\n                            &\\dots \\exp( c^N_1 \\tau_1 ) \\dots  \\exp( c^N_m \\tau_{m}), \\label{eqn:U} \\end{align}\n    where \\( \\mathbf{E} \\) we call the \\textit{embedding} function, \\( c = (c_1^1 ,\\dots, c^N_m )\\) and the \\(\\tau_i\\) are a basis for a \\textit{bracket generating} subset of the Lie algebra \\( \\Delta \\subset  \\mathfrak{su}(2^n)\\) of dimension \\(m\\). Bracket generating means that repeated Lie brackets of terms in \\(\\Delta \\) can generate any term in \\( \\mathfrak{su}(2^n)\\). Because products of matrix exponentials generate Lie bracket terms\n    \\[ \\exp(A) \\exp(B) = \\exp(A+ B+ \\frac{1}{2}[A,B] + \\dots ),\\]\nany \\(U \\in \\mathrm{SU}(2^n) \\) can be written as Equation (\\ref{eqn:U}) with sufficiently many products. We restrict ourselves to \\(U\\) which can be written as a product of a polynomial in \\(n\\) terms . An example of such a \\(\\Delta\\) could be the matrix logarithms of universal gates. For convenience it is easier to work with all permutations of Kronecker products of one and two Pauli matrices, so \n    \\[ \\Delta = \\mathrm{span} \\{\\frac{\\mathrm{i}}{\\sqrt{2^n}} \\sigma_i^j , \\frac{\\mathrm{i}}{\\sqrt{2^n}} \\sigma_i^k \\sigma_j^l \\} ,\\]\n    where \\( \\sigma^j_i \\) represents the \\(N\\) fold Kronecker product, \\( I \\otimes \\dots \\otimes \\sigma_i \\otimes \\dots \\otimes I\\), with a \\( \\sigma_i \\) inserted in the \\(j\\)-th slot and \\(I\\) representing the \\( 2 \\times 2 \\) identity matrix. Exponentials of these basis elements have very simple circuits, for more detail see Appendix A.\n    \\\\\n    \\\\\n    We propose that a neural network be trained to learn \\( \\mathbf{E}^{-1}\\). The neural network will try to find all the coefficients \\( c^k_{i} \\) so the product approximates \\(U\\). In this approach, the neural network takes a unitary matrix \\(U\\) as an input and returns a list \\(c\\) of \\( c^k_{i}\\). A segment is a product of \\( m \\) exponentials of each basis element. In total there are \\(N\\) segments. We only examine \\(U\\) which are implementable in a reasonable number of segments. We found that we required two neural networks to achieve this. The first is a Gated Recurrent Unit, GRU, network \\cite{gru_paper_1,gru_paper_2} which factors a \\(U\\) into a product of \\(U_j\\),\n    \\[U \\approx U_1 U_{2} \\dots U_j \\dots U_N, \\]\n    where each \\(U_j\\) is implementable in polynomially many gates, which we call \\textit{global decomposition}.\n    The second is simply several dense fully connected layers, which decomposes the \\(U_j\\) into products of exponentials\n    \\[ U_j \\approx \\exp( c^j_1 \\tau_1 ) \\dots \\exp( c^j_m \\tau_m ), \\]\n    which we term \\textit{local decomposition}. These procedures can be done with traditional optimisation methods. The lack of a good initial guess meant that it took an order of an hour in \\(\\mathrm{SU}(8)\\). While the output from the neural network may not implement \\(U\\) to a required tolerance, it does provide a good initial guess as the error will be small. The output from the neural network could be refined with another optimisation algorithm.\n\\section{Training data \\label{training_data}}\n    \\noindent To generate the training data, the \\(c\\) should not be chosen randomly. If there is no structure to how \\(c\\) is chosen, it will introduce extra redundancy. More seriously, \\(\\mathbf{E}^{-1} \\) will not be well defined. There are infinitely many ways to factor a \\(U\\) into some unordered product of matrix exponentials. Geometrically this could be visualised as taking any path from \\(I\\) to \\(U\\) on \\(\\mathrm{SU}(2^n)\\). Randomly generating data may give two different decompositions for a \\(U\\), and so \\(\\mathbf{E}\\) is not one to one. To ensure the training data is unique, we propose that these paths should be chosen to be, at least approximately, minimal normal subRiemannian geodesics. \n    \\\\\n    \\\\\n    The choice of using geodesics is not particularly special. Other types of curves could be used as long as it uniquely joins \\(I\\) and \\(U\\). This is so \\( \\mathbf{E}^{-1} \\) is well defined. Generating random geodesics can be done simply by generating random initial conditions. However the geodesics must also be minimal. The first way to try and ensure they are minimal is to bound the norms of the initial conditions. \n    \\\\\n    \\\\\n    The normal subRiemannian geodesics in \\(\\mathrm{SU}(2^n)\\) can be found via the Pontryagin Maximum Principle \\cite{pmp-intro,pmp-book} by minimising the energy functional\n    \\[ \\mathcal{E}[x] = \\int_0^1 dt \\langle \\dot{x}, \\dot{x} \\rangle, \\]\n    where \\( \\langle, \\rangle \\) is the restriction of the bi-invariant norm to \\(\\Delta \\subset \\mathfrak{su}(2^n)\\), and \\( x :[0,1] \\rightarrow \\mathrm{SU}(2^n) \\). See Chapter 7 of \\cite{opt-control-lie-group} for a review. The normal subRiemannian geodesic equations can be written as\n    \\begin{align*}\n        \\dot{x} &= u x, \\\\\n        \\dot{\\Lambda} &= [\\Lambda, u ],\\\\\n        u &= \\mathrm{proj}_{\\Delta}( \\Lambda),\n    \\end{align*}\n    where  \\( \\Lambda : [0,1] \\rightarrow \\mathfrak{su}(2^n) \\),  \\( u : [ 0,1] \\rightarrow \\Delta \\subset \\mathfrak{su}(2^n) \\) and \\( \\mathrm{proj}_{\\Delta} \\) is projection onto \\( \\Delta\\).\n    This can be re-written as the single equation\n    \\begin{equation} \\dot{x} = \\mathrm{proj}_{\\Delta}( x \\Lambda_0 x^{\\dagger} ) x, \\label{eqn:geod} \\end{equation}\n    where \\( \\Lambda_0 = \\Lambda(0) \\).\n    Choosing the \\( \\Lambda_0 \\) completely determines the geodesic. To generate the training data for the \\(U_j\\), first randomly choose a \\( \\Lambda_0\\).  The \\(U_j \\) are then matrices which forward solve the geodesic equations\n    \\[ x(t_{j+1} ) = U_j x(t_j),\\]\n    where \\( [0,1] \\) has been divided into \\(N\\) segments of width \\(h\\). For this paper we utilised the simple first order integrator\n    \\[ U_j = \\exp\\big( h \\, \\mathrm{proj}_{\\Delta}(x_j \\Lambda_0 x_j^{\\dagger}) \\big),  \\]\n    since approximating the geodesic is sufficient. There are infinitely many bi-invariant Riemannian geodesics joining \\(I \\) and \\(U\\), for the different branches of \\( \\log(U) \\). SubRiemannian geodesics are similarly behaved, but it varies on the norm of \\( \\Lambda_0\\). To generate the training data we bounded the norms by \\(  \\mathrm{dim}(\\Delta) = \\mathcal{O}(n^2) \\), to try and ensure the geodesics are unique. \n    \\\\\n    \\\\\n    Further, the norm \\( || \\mathrm{proj}_{\\Delta} (\\Lambda_0 ) || = || u_0 ||\\) determines the distance between \\(I \\) and a \\(U\\). Nielsen showed that the distance can be thought of as approximately the complexity to implement \\(U\\). Lemma (3) in \\cite{nielsen-geom-1} shows that a \\(U\\) further away from \\(I\\) requires more gates. The distance however is likely to scale exponentially. By bounding the norm by a polynomial, this ensures the training data only contains \\(U\\) which are reachable with a polynomial number of quantum gates.    \n\\section{Network Design - SU(8) }    \n    \\subsection{Global decomposition}\n    The neural network for the global decomposition takes an input of \\(U\\) and returns a list of \\(U_j\\). To do this \\(U\\) is decomposed into rows of length \\(2^n\\). This makes \\( 2^n\\) real vectors. Each row is treated as a single timestep in the GRU layer. The output \\(U_j\\) are also decomposed into their rows and these rows are treated as timesteps in the output. This gives \\( 2^n N \\) output vectors of length \\( 2^n\\). In particular we examined the \\(n=3\\) qubit case. For \\(\\mathrm{SU}(8) \\) we found \\(10\\) stacked GRU layers was sufficient to give reasonable results.  In \\(\\mathrm{SU}(8)\\) we chose \\(N=10\\) , so there were \\( 8 \\) input vectors of length \\( 8\\) and \\( 80 \\) output vectors of length \\( 8\\). The network was implemented in the Keras Python library \\cite{keras} with the TensorFlow backend, on a Nvidia GTX 1080.\n    \\subsection{Local decomposition}\n    For \\(\\mathrm{SU}(8)\\) a network with \\(2\\) fully connected dense hidden layers of \\(2000\\) neurons, with the ReLU activation function was found to be sufficient. The input layer took a vectorised \\(U_j\\), and outputted \\( \\dim(\\Delta)\\) values. The network was implemented in the Keras Python library with the TensorFlow backend, on a Nvidia GTX 1080.\n\\section{Results - SU(8)}\n   \\subsection{Global decomposition}\n    The global decomposition network was trained on \\(U_j\\) taken from \\(5000\\) randomly generated geodesics in \\(\\mathrm{SU}(8)\\). \\(500\\) were used for validation data. The loss function used was the standard Euclidean distance between the output vector and the desired output. After \\( 1500 \\) training epochs the validation loss reached \\( \\sim 0.9 \\) and did not decrease. This was found to be sufficient to generate \\(U_j\\) close to the training data. Figure (\\ref{fig:gruLoss}) shows the validation and training loss. Figure (\\ref{fig:outUi}) and figure (\\ref{fig:validUi}) shows a randomly chosen \\( U_j\\) from a list of \\(U_j\\) generated by the network, and from the training data respectively for some random \\(U\\). Most \\(U_j\\) appeared to be very similar. Figure (\\ref{fig:u34}) and figure (\\ref{fig:u25}) show the same entry in consecutive \\( U_i\\) for validation data. Again the network was able to output values very close to the values in the validation dataset. This similarity was typical. This shows the network is able to reasonably approximate the \\(U_j\\). \n    \\begin{figure}[h!]\n        \\centering\n        \\includegraphics[width=0.45\\textwidth]{GRU_loss.pdf}\n        \\caption{The loss and validation loss from training the global decomposition.}\n        \\label{fig:gruLoss}\n    \\end{figure}\n    \\begin{figure}[h!]\n        \\centering\n        \\begin{subfigure}{0.48\\textwidth}\n            \\centering\n            \\includegraphics[width=0.4\\linewidth]{out.pdf}\n            \\caption{Real components of a \\(U_j\\) generated by the NN.}\n            \\label{fig:outUi}\n        \\end{subfigure}\n        \\begin{subfigure}{0.48\\textwidth}\n            \\centering\n            \\includegraphics[width=0.4\\linewidth]{valid.pdf}\n            \\caption{The respective known real components of a \\(U_j\\) from the validation dataset.}\n            \\label{fig:validUi}\n        \\end{subfigure}\n        \\caption{A known \\(U_j\\) from the validation data and the \\(U_j\\) generated by the NN in \\(\\mathrm{SU}(8)\\) for global decomposition. Each \\(U_j\\) is close to the identity matrix. The shading from blue to orange represents \\( [-1,1] \\)}\n    \\end{figure}\n    \\begin{figure}[h!]\n        \\centering\n        \\begin{subfigure}{0.48\\textwidth}\n            \\centering\n            \\includegraphics[width=0.6\\linewidth]{u14_25.pdf}\n            \\caption{The same real entry from the \\(10\\) \\(U_i\\) from the validation data set (blue), vs the predicted output (red).}\n            \\label{fig:u34}\n        \\end{subfigure}\n        \\begin{subfigure}{0.48\\textwidth}\n            \\centering\n            \\includegraphics[width=0.6\\linewidth]{u34_40.pdf}\n            \\caption{The same real entry from the \\(10\\) \\(U_i\\) from the validation data set (blue), vs the predicted output (red). }\n            \\label{fig:u25}\n        \\end{subfigure}\n        \\caption{Real entries of validation \\(U_i\\) vs the \\(U_i\\) generated by the NN. Recall the \\(U_i\\) are not constant, and solve equation (\\ref{eqn:geod}). The behaviour displayed here was typical in other entries. }\n    \\end{figure}\n\n\n\\subsection{Local decomposition}\nThe network to implement the local decomposition was trained on \\(U_j\\) generated by choosing a random \\(m\\)-vector of the coefficients \\(c^j_i\\), where each \\(c^j_i \\) was order \\( 1\/N \\). In total there were \\(5000\\) pairs in the training set, and \\(500\\) in the validation set. Figure (\\ref{fig:denseLoss}) shows the validation and training loss. After \\( 500\\) epochs the network was able to sufficiently compute the local decomposition to reasonable error (on average 0.16).  Figures (\\ref{fig:localoutUi}) and (\\ref{fig:localvalidUi}) show a matrix generated by the neueral network and the target matrix.\n    \\begin{figure}[h!]\n        \\centering\n        \\includegraphics[width=0.45\\textwidth]{Dense_loss.pdf}\n        \\caption{The loss and the validation loss from training the local decomposition. There was no significant improvement after \\(500\\) epochs.}\n        \\label{fig:denseLoss}\n    \\end{figure}\n    \\begin{figure}[h!]\n        \\centering\n        \\begin{subfigure}{0.48\\textwidth}\n            \\centering\n            \\includegraphics[width=0.4\\linewidth]{outMat.pdf}\n            \\caption{Real components of a \\(U_j\\) generated by the NN.}\n            \\label{fig:localoutUi}\n        \\end{subfigure}\n        \\begin{subfigure}{0.48\\textwidth}\n            \\centering\n            \\includegraphics[width=0.4\\linewidth]{localValidM.pdf}\n            \\caption{The respective known real components of a \\(U_j\\) from the validation dataset.}\n            \\label{fig:localvalidUi}\n        \\end{subfigure}\n        \\caption{A known \\(U_j\\) from the validation data and the \\(U_j\\) generated by the NN in \\(\\mathrm{SU}(8)\\). These figures are for the local decomposition network. The shading from blue to orange represents \\( [-1,1] \\)}\n    \\end{figure}\n\n\\section{Conclusion}\nTraining two neural networks to together decompose \\(U\\) into \\(c^j_i \\) via a two-step approach (global decomposition followed by local decomposition) was found to be successful, when restricting the set of training data generated to paths which approximate minimal normal subRiemannian geodesics.  This restriction limited the training data pairs to ones which were one-to-one, eliminating redundancy.  For the global decomposition, using a neural network consisting of stacked GRU layers allowed for efficient training of the network, with the validation loss of the network approaching its minimum at 500 epochs for \\(\\mathrm{SU}(8)\\).  A simple dense network with two hidden layers proved sufficient for the local decomposition.  In \\(\\mathrm{SU}(8)\\), the networks were small enough that both networks were able to be trained on a desktop machine with a single NVidia GTX 1080 GPU.  The two stage decomposition proved more successful than single-stage attempts to form a solution, with the decomposition of a given \\(U\\) into \\(U_j\\) being crucial for this increase in effectiveness.  This approach to the solution for this problem demonstrates a novel use of neural networks.\n\\\\\n\\\\\nAlthough this approach works well for systems with small numbers of qubits (such as the \\(\\mathrm{SU}(8)\\) case used as an example), the approach does not scale well with increasing number of qubits. This is because the size of the network scales by the number of entries in matrices in \\(\\mathrm{SU}(2^n)\\). Although this is not a significant problem for currently realisable quantum computers, or those in the near future, it will increasingly become problematic as quantum computing continues to advance.  To somewhat counteract this, the complexity of the problem can be decreased by restricting the set of \\(U\\) on which the neural network is trained. For example if the \\(U\\) are sparse, some savings in the size of the network may be made. Investigating this will be increasingly significant, as it will increase the practical usefulness of this approach.  \n\\\\\n\\\\\nAs noted in section \\ref{training_data}, the choice of using geodesics to restrict the training data is fairly arbitrary, and as such, there may be different ways of restricting the training data which, while still ensuring the input\/output is one-to-one, may produce a better dataset, improving the accuracy of the networks. This is heavily related to the nature of \\( \\Lambda_0 \\) which is currently not fully understood. Exploring this problem is a possible future avenue of investigation, which may improve the effectiveness of the approach described in this paper.  \n\\\\\n\\\\\nFinally note that training the network is the most computationally expensive part of this approach. Once the network is trained, propagating an input through through the network is much more efficient than the conventional optimisation techniques for compiling \\(U\\). \n\\\\\n\\\\\nAll data and programs used to produce this work can be found at \\href{https:\/\/github.com\/Swaddle\/nnQcompiler}{\\url{https:\/\/github.com\/Swaddle\/nnQcompiler}}. This work was supported by resources provided by the Pawsey Supercomputing Centre with funding from the Australian Government and the Government of Western Australia \\footnote{\\url{https:\/\/www.pawsey.org.au\/}}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzewsi b/data_all_eng_slimpj/shuffled/split2/finalzzewsi
new file mode 100644
index 0000000000000000000000000000000000000000..abc9dedd117763fb54c2875b3a8187cff772296f
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzewsi
@@ -0,0 +1,5 @@
+{"text":"\\section{Nomenclature}\n\n{\\renewcommand\\arraystretch{1.0}\n\\noindent\\begin{longtable*}{@{}l @{\\quad=\\quad} l@{}}\n$altitude$ & height above WGS84 ellipsoid \\\\\n$ecef$  & Earth-centered, Earth-fixed coordinate system\\\\\n$ENU$  & East, North, Up coordinate system\\\\\n$\\Delta t_{imu}$  & time between IMU measurements, s\\\\\n$\\bm{\\theta}$ &    gyroscope measurements $[\\theta_x, \\theta_y, \\theta_z]^T$, $\\frac{rad}{s}$ \\\\\n$\\omega$& $\\| \\bm{\\theta} \\| $ \\\\\n$q_{1}^{2}$ & unit quaternion describing the rotation from frame 1 to frame 2 \\\\\n$P$ & camera projection matrix \\\\\n$\\pi_{WGS84}$ & projection of a pixel coordinate to a 3D point on the surface of the WGS84 model \\\\\n$\\alpha_{max}$ & max acceptable angle between camera boresight and normal of a landmark \\\\\n$\\delta x$ & amount to shift a point by in pixel space \\\\\n$surface\\_normal()$ & function that finds normal vector at a point on the WGS84 model \\\\\n$angle\\_between()$ & function that finds the angle between a camera boresight and a vector \\\\\n\\end{longtable*}} \\clearpage\n\n\\section{Introduction}\n\\label{sec:intro}\n\nTerrain Relative Navigation (TRN) is a method for absolute pose estimation in a GPS-denied environment using a prior map of the environment and onboard sensors such as a camera. TRN \nis commonly desired %\nfor applications requiring accurate pose estimation, such as planetary landings and airdrops, where GPS \nis either unavailable or cannot be relied upon. Due to the high altitude of planetary TRN missions, acquiring non-simulation test data oftentimes proves difficult, \nand thus many datasets used to test TRN systems are from lower altitudes than what the system would actually be used at during a mission. Additionally, \nfor vision-based TRN systems, the large distance between the camera and features on the ground can make position changes of the camera \ndifficult to accurately observe due to the high ratio of meters per pixel in the image plane.\n\nThis paper presents an experimental analysis on performing TRN using a camera-based approach aided by a gyroscope for high-altitude navigation by associating mapped landmarks from satellite\nimagery to camera images. We evaluate performance of both a sideways-tilted and downward-facing camera on data collected from a World View Enterprises high-altitude balloon (\\cref{fig:balloon_launch}) \nwith\ndata beginning at an altitude of 33 km and descending to ground level with almost 1.5 hours of flight time (\\cref{fig:overview}) and on data collected at speeds up to  \n880 km\/h (550 mph) from two sideways-tilted cameras mounted inside the capsule of Blue Origin's New Shepard rocket (\\cref{fig:rocket_all}), during payload mission NS-23. We also demonstrate the \nrobustness of the TRN system to rapid motions of the balloon which causes fast attitude changes (\\cref{fig:challenges_a})\nand can cause image blur (\\cref{fig:challenges_b}). Additionally, we demonstrate performance in the presence of dynamic camera obstructions \n caused by cords dangling below the balloon (\\cref{fig:challenges_c}), and clouds obstructing sections of \nthe image (\\cref{fig:challenges_d}). \n\nSideways-angled cameras are a common choice for TRN applications when mounting a downward camera is either infeasible due to vehicle constraints or \nwould be occluded by exhaust from an engine on vehicles such as a lander or a rocket. Additionally, for \nplanetary landings, a sideways-angled camera allows for a single camera to be used \nduring both the braking phase when the side of the lander faces the surface and during the \nfinal descent phase when the bottom of the lander faces the surface (\\cref{fig:landing}). We thus use both a \nsideways-angled camera and downward-facing camera during our high-altitude balloon flight \nto separately evaluate the performance of TRN using a camera from each orientation.\n\nWe use Draper's Image-Based Absolute Localization (IBAL) \\cite{Denver17aiaa-airdrop} software for our analysis. \nWhile our dataset has images at a rate of 20Hz, we subsample images by a factor of 10 and hence post-process images at 2Hz in real-time.\nIBAL could additionally be combined with a nonlinear estimator such as an Extended Kalman Filter (EKF) or a fixed-lag smoother through either a loosely coupled approach using IBAL's pose estimate or a tightly-coupled approach using landmark matches~\\cite{Forster17tro}. \nSince the quality of the feature matches generated by IBAL would affect all these methods, here we limit ourselves to evaluating IBAL as an independent system and also analyze the quality of the\nfeature matches. At the same time, we investigate the impact of using a gyroscope in conjunction with IBAL to aid with the challenges of our balloon dataset and show the advantage that \neven a simple sensor fusion method can provide. \nFinally, we extend IBAL to incorporate methods to \nefficiently process images when a camera views above the horizon.\n\n\\begin{figure}[hbt!]\n    \\centering\n    \\begin{subfigure}[t]{0.47\\textwidth}\n        \\centering\n        \\includegraphics[width=.5\\textwidth]{fig\/balloon2.png}\n        \\caption{Release of high-altitude balloon for data collection. \\\\\\ Image: courtesy of World View\\textregistered  Enterprises}\n        \\label{fig:balloon_launch}\n        \\end{subfigure}\n    \\hfil\n    \\begin{subfigure}[t]{0.418\\textwidth}\n        \\centering\n        \\includegraphics[width=.50\\textwidth]{fig\/rocket_all.png}\n        \\caption{Blue Origin's New Shepard rocket carrying Draper experimental payload in the capsule. Image: courtesy of Blue Origin}\n        \\label{fig:rocket_all}\n    \\end{subfigure}\n    \\caption{Data collection platforms used for experimental analysis.}\n\\end{figure}\n\n\\begin{figure}[hbt!]\n    \\centering\n    \\includegraphics[width=1.0\\textwidth]{fig\/image_overview.png}\n    \\caption{Example of images collected at different altitudes (32, 23, 14, and 4 km) from the balloon dataset with the downward-facing camera (top)\n    and sideways-facing camera (bottom).}\n    \\label{fig:overview}\n\\end{figure}\n\n\\begin{figure}[hbt!]\n    \\centering\n    \\begin{subfigure}[t]{0.24\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{fig\/challenge_b.png}\n        \\caption{Rapid rotations, here over $90^\\circ$ in 4 seconds. Red dots show ground reference points between top image and bottom image.}\n        \\label{fig:challenges_a}\n    \\end{subfigure}\n    \\unskip\\ \\vrule\\ \n    \\begin{subfigure}[t]{0.24\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{fig\/challenge_c.png}\n        \\caption{Image blur (top) due to rapid motion compared to crisp image (bottom).}\n        \\label{fig:challenges_b}\n    \\end{subfigure}\n    \\unskip\\ \\vrule\\ \n    \\begin{subfigure}[t]{0.24\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{fig\/challenge_a.png}\n        \\caption{Moving cords in the image. Top and bottom images showing example range of cord motion.}\n        \\label{fig:challenges_c}\n    \\end{subfigure}\n    \\unskip\\ \\vrule\\ \n    \\begin{subfigure}[t]{0.24\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{fig\/challenge_d.png}\n        \\caption{images partially occluded by clouds}\n        \\label{fig:challenges_d}\n    \\end{subfigure}\n    \\caption{Different types of TRN challenges in the balloon dataset.}\n\\end{figure}\n\n\\begin{figure}[hbt!]\n    \\centering\n    \\includegraphics[width=0.5\\textwidth]{fig\/landing.png}\n    \\caption{Demonstration of a sideways-angled camera viewing the terrain and being used \n    during the braking phase, pitch-up maneuver, and terminal descent phase.}\n    \\label{fig:landing}\n\\end{figure}\n %\n\\newpage\n\n\\section {Related Work}\n\nWe present an overview of existing Terrain Relative Navigation approaches and experiments, noting that our primary contribution are two experiments that\nallow us to perform indepth analysis of vision-based terrain relative navigation on challenging high-altitude data and on data from a high speed vehicle. \nTRN methods primarily use either cameras, radar, or lidar as an exteroceptive sensor. The majority of \nearly TRN methods such as the Mars Science Laboratory \\cite{Katake10-landingNav} and NASA's ALHAT Project (\\cite{Brady11gnc-alhat}, \\cite{Amzajerdian12ac-lidarTRN}) \nuse radar or lidar. However, due to the high power \nand weight budget of radar and lidar, cameras have been motivated as an active area of exploration for more recent TRN systems.\n\nThe seminal work of Mourikis \\textit{et al.} \\cite{Mourikis09tro-EdlSoundingRocket} describes a visual-inertial navigation method for \nEntry, Descent, and Landing (EDL) using an Extended Kalman Filter (EKF) with matched landmarks and\ntracked feature points in an image. They use inertial navigation results from their entire sounding rocket launch with an apogee of 123 km, and leverage visual methods after the vehicle reaches altitudes below 3800m. Johnson and Montgomery~\\cite{Johnson09ac-trnReview}\npresent a survey of TRN methods that use either image or lidar to detect the location of known landmarks.\n\nSingh and Lim~\\cite{Singh12aiaa-trnEKF} \ndemonstrate a visual TRN approach leverging an EKF for lunar navigation using known crater locations as landmarks. Recently, Downes \\textit{et al.} \\cite{Downes20aiaa-lunarTRN} \npresent a deep learning method for lunar crater detection to improve TRN landmark tracking. \nThe Lander Vision System (LVS) \\cite{Johnson17-lvs} used for the Mars 2020 mission uses vision-based landmark matching starting at an altitude of 4200m above the \nmartian surface with the objective of achieving less than 40m error with respect to the landing site. \nOur analysis focuses on higher altitudes and on a larger span on altitudes (4.5 km to 33 km for the \nballoon dataset).\n\nDever \\textit{et al.}\\cite{Denver17aiaa-airdrop} demonstrate visual navigation for guided parachute airdrops using IBAL and a \nMulti-State Constraint Kalman Filter (MSCKF). Additionally, the work incorporates\na lost robot approach to recover from a diverged pose estimate and to initialize the system if the pose is unknown. \nSteffes \\textit{et al.} \\cite{Steffes19aiaa-trnEDL} present a theoretical analysis of three types of visual terrain navigation \napproaches, namely template matching, SIFT \\cite{lowe2004ijcv-distinctive} descriptor matching, and crater matching. \nThe  work of Lorenz \\textit{et al.} \\cite{Lorenz17ac-osirisrex} demonstrates \nvision-based terrain relative navigation for a touch and go landing on an asteroid for the OSIRIS-REx mission. Due to extreme computation limits,\nthey used a maximum of five\n manually selected mapped template features per frame. Mario \\text{et al.} \\cite{Mario22psj-osirisRexTesting} provide additional discussion on ground tests \n used to prepare the TRN system for the OSIRIS-REx mission. Our balloon dataset has much faster rotional motion than \n what was present during the OSIRIS-REx mission along with camera obstructions.\n\nSteiner \\textit{et al.} \\cite{Steiner15ac-landmarkSelection} present a utility-based approach for optimal landmark selection and demonstrates performance \non a rocket testbed flight up to 500m. As shadows and variable lighting conditions are a well known challenge for TRN, \nSmith \\textit{et al.} \\cite{Smith22aiaa-blenderTRN} demonstrates the ability to use Blender to enhance a satellite database for different lighting conditions. %\n\n\\section{Data Collection}\n\nThe collection of both datasets used in this paper was supported by the NASA Flight Opportunities Program. The high-altitude balloon dataset \nwas designed to test TRN on a wide range of high-altitude data and occured in April of 2019. The New Shepard dataset was intended to \ntest TRN on a high speed vehicle with a flight profile similar to that of a precision landing and occured in August of 2022.\n\n\\subsection{Balloon Flight}\n\nWe captured downward and sideways camera images along with data from a GPS and an inertial measurement unit (IMU) on board a World View \nEnterprises high-altitude balloon shown in \\cref{fig:balloon_launch}, \nwith data recorded up to an altitude of 33 km.\nWe used FLIR Blackfly S Color 3.2 MP cameras for both downward and sideways facing views using 12 mm EFL lens and 4.5 mm EFL lens, respectively. \nThe field of view (FOV) for the downward and sideways camera with their respective lens is $32^{\\circ}$ and $76^{\\circ}$.\nBoth cameras, along with the IMU (Analog Devices ADIS16448) \nand data logging computer are self contained inside the Draper Multi-Environment Navigator (DMEN) package, shown in \\cref{fig:hardware}. \nBoth cameras generated images at 20 Hz with a resolution of $1024 \\times 768$. The IMU logged data at 820 Hz. \n\nAs mentioned in \\cref{sec:intro}, some TRN applications ---such as \nplanetary landing--- might prefer using a sideways-angled camera, while other applications \n---such as \nhigh-altitude drone flights--- may prefer a downward-facing camera. Therefore, we collect data from \nboth a downward and sideways angled camera to allow for IBAL to be evaluated at both these camera \nangles. Some planetary landings may also desire a downward-facing camera since it allows the boresight of the camera \nto be normal to the surface during the terminal descent phase, \nsuch as was done for OSIRIS-REx \\cite{Lorenz17ac-osirisrex}.\n\n\\begin{figure}[hbt!]\n    \\centering\n    \\includegraphics[width=.4\\textwidth]{fig\/hardware.png}\n    \\caption{Draper Multi-Environment Navigator (DMEN) package: data collection package containing sideways and downward facing cameras, IMU, and logging computer.}\n    \\label{fig:hardware}\n\\end{figure}\n\n\\subsection{Blue Origin New Shepard Flight}\n\nWe captured images from two sideways-angled cameras with 12.5 mm lens on opposite sides inside the New Shepard capsule which  \nlook out the capsule windows. Having two cameras was intended to allow us to study the effects of different cloud cover, terrain, and angle to the sun. \nWe will refer to these cameras as camera 1 and camera 2. \nWe additionally log IMU data from a Analog Devices ADIS16448, and \ntelemetry from the capsule which served as ground truth for our experiment. Data was logged with a NUC mounted inside a payload locker in the capsule.\nBoth cameras generated images at 20 Hz with a resolution of $1024 \\times 768$ and FOV of $31^{\\circ}$. \nThe IMU logged data at 820 Hz. The rocket reached speeds up to 880 km\/h and an altitude of 8.5 km before an anomaly occurred during the NS-23 flight \nwhich triggered the capsule escape system.\n\n\\Cref{fig:payload_blue} shows our payload locker containing the NUC, IMU, and a power converter which is \nmounted inside the New Shepard capsule. An ethernet cable and two USB cables transfer \ntelemetry data from the capsule and data from the cameras to the NUC, respectively.\n\n\\Cref{fig:cam_mount_a} shows camera 2 mounted inside the capsule with a sideways-angle and \n\\cref{fig:cam_mount_b} shows the location of both cameras inside the capsule on opposite sides while New Shepard \nis on the launch pad. Both cameras are mounted at the same tilt angle such that they can view the terrain while not \nhaving their FOVs obstructed by components on the rocket. Additionally, a mounting angle was selected to reduce \nthe effects of distortion caused by the windows, and to ensure the cameras did not come in direct \ncontact with the windows.\n\nDistortion effects from the windows were addressed by calibrating the instrinsic parameters \nof the camera while the camera was mounted in the capsule (i.e., a calibration board was positioned outside \nthe capsule window). We used the Brown-Conrady model \\cite{Brown66-brownConrady} which helps account for decentralized distortion caused by the window \nin addition to distortion from the camera lens. Further evaluation on the effects of distortion caused \nby the window of the capsule is left as a topic for future work.\n\n\\begin{figure}[hbt!]\n    \\centering\n    \\begin{subfigure}[t]{0.49\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{fig\/payload1.png}\n    \\end{subfigure}\n    \\hfil\n    \\begin{subfigure}[t]{0.49\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{fig\/payload2.JPG}\n    \\end{subfigure}\n    \\caption{Payload locker inside the New Shepard capsule containing a NUC, IMU, and DC\/DC Converter. Images courtesy of Blue Origin.}\n    \\label{fig:payload_blue}\n\\end{figure}\n\n\\begin{figure}[H]\n    \\centering\n    \\begin{subfigure}[t]{0.49\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{fig\/cam_capsule.png}\n        \\caption{}\n        \\label{fig:cam_mount_a}\n    \\end{subfigure}\n    \\hfil\n    \\begin{subfigure}[t]{0.49\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{fig\/rocket_cams.png}\n        \\caption{}\n        \\label{fig:cam_mount_b}\n    \\end{subfigure}\n    \\caption{Cameras 1 and 2 mounted inside the New Shepard capsule looking out the capsule windows. Images courtesy of Blue Origin.}\n    \\label{fig:cameras_in_window}\n\\end{figure} %\n\n\\section{Terrain Relative Navigation Method}\n\\label{sec:method}\n\nWe use Draper's IBAL  software~\\cite{Denver17aiaa-airdrop} to perform TRN \nfor our datasets. A database of image templates is created in advance from satellite imagery and stored using known \npixel correspondence with the world frame. Using satellite images and elevation maps from USGS~\\cite{usgs}, we automatically select patches of interest \nfrom the satellite images and create a collection of templates that serve as 3D landmarks. For each camera image processed by IBAL, IBAL uses an initial guess of \nthe camera pose to predict \nwhich templates from the database are in the field of view (FOV) of the camera using a projection from \nthe image plane to an ellipsoidal model of the planet. The templates are then matched to the camera \nimage using cross correlation. The resulting match locations are passed to a 3-point RANSAC \\cite{Fischler81} (using a Perspective-Three-Point method as a minimal solver) to reject outliers. \nThe output is a list of the inlier matches, their pixel location in the image, and their known location \nin the world frame that can be passed to a nonlinear estimator or fixed-lag smoother for tightly-coupled pose estimation. \nA secondary output of RANSAC is an absolute pose estimate found by using the Perspective-n-Point (PnP) \nalgorithm on the set of inliers. \n\nInstead of a tightly-coupled approach, we will use a simpler method to evaluate performance on the balloon and New Shepard datasets. \nFor the balloon dataset, we take the PnP absolute pose estimate directly from IBAL, \n forward propagate it with the gyroscope measurements, and use it at the next time step as a pose guess for IBAL. \n We do not use accelerometer data since \nin the image frame most scene changes for the balloon dataset \nover a short time span will be due to rotations. This\nis due to the high altitude and hence large distance between the camera and the Earth's surface. Using the gyroscope to propagate the rotation also allows for\nreduced computation since we are able to down-sample our camera data by a factor of 10 (2Hz image input to IBAL). \nAdditionally, the gyro allows for robust handling of rapid motions of the balloon and images that have large obstruction\nfrom cords which makes generating landmark matches unreliable. An ablation study on incorporating the gyroscope with IBAL is provided in \\cref{sec:gyro_ablation}.\n Since the New Shepard capsule does not experience rapid rotations like the balloon, we did not find it \nnecessary to use the gryoscope to forward propagate the pose estimate for the New Shepard dataset.\n\nWe propagate the rotation estimate of the vehicle, $q^{cam_T}_{ecef}$ (i.e., the orientation of the earth-centered, earth-fixed frame \nw.r.t. the camera frame at time $T$, represented as a unit quaternion), to the time of the next processed image ($T+1$) \nwith the gyro using second order strapdown quaternion expansion \\cite{Mckern68mit-transforms}.  \nUsing 3-axis gyro measurements $\\bm{\\theta}$ and their magnitude $\\omega = \\|\\bm{\\theta}\\|$, we compute the orientation $q_{IMU_{t+1}}^{IMU_t}$ between gyro measurements \nusing the following equation\n\\begin{equation}\n    \\label{eq:quat_gyro1}\n    q_{IMU_{t+1}}^{IMU_t}= [1 - \\frac{\\omega^2 \\Delta t_{IMU}^2}{8}, \\frac{\\bm{\\theta}^T \\Delta t_{IMU}}{2}]\n\\end{equation}\nwhere $t+1$ and $t$ represent the time of consecutive IMU measurements occuring $\\Delta t_{IMU}$ seconds apart.\n\n\nUsing the rotations $q_{IMU_{t+1}}^{IMU_t}$ between consecutive IMU timestamps, we \ncan compute the relative rotation $q_{cam_{T+1}}^{cam_T}$ between the camera pose between consecutive images collected at time $T$ and $T+1$:\n\\begin{equation}\n    \\label{eq:quat_gyro2}\n    q_{cam_{T+1}}^{cam_T} = \\prod_{t = T}^{T+1} q_{IMU}^{cam} \\otimes q_{IMU_{t+1}}^{IMU_t} \\otimes (q_{IMU}^{cam})^{-1}\n\\end{equation}\nwhere $\\otimes$ is the quaternion product and $q_{IMU}^{cam}$ is the static transform from the IMU frame to the camera frame:\n\nFinally, we can compute the rotation estimate $q^{cam_{T+1}}_{ecef}$ of the vehicle at time $T+1$:\n\\begin{equation}\n    \\label{eq:quat_gyro3}\n    q^{cam_{T+1}}_{ecef} = (q_{cam_{T+1}}^{cam_T})^{-1} \\otimes q^{cam_{T}}_{ecef}\n\\end{equation}\n\nWe use a simple yet effective logic for handling short segments in our datasets when PnP is unable to produce a reliable pose, which can be caused by image obstructions \nor blurry images caused by rapid vehicle motion. If PnP RANSAC selects a small set of inliers (i.e., less than 8) or if the pose is clearly infeasible (i.e., an altitude change between \nprocessed images greater than 450 m for the balloon dataset), we reject the \npose estimate, keep forward propagating the pose using gyroscope data, and run IBAL with the next available image, ignoring the down-sampling rate. %\n\n\\section{Addressing Challenges of High-Altitude Images}\n\nWe apply simple and effective methods to address two common challenges we encountered with high-altitude images, namely determining the projection\nto the ellipsoid when the camera views the horizon, and reducing the number of potential landmarks from the database that have a lower probability of \ngenerating good matches when there is a large number of landmarks in view of the camera. \n\nWhen the horizon is in view of the camera, as is true for the higher altitude images from the sideways camera for the balloon dataset \n(\\cref{fig:overview}), our \nbaseline method of determining the camera's viewing bounds of the planet's surface is insufficient. Our baseline method is to use an \ninitial estimate of the camera's pose to project each corner of the image to the ellipsoid model. From this, we can create a bounding box on the \nellipsoid defined by a minimum and maximum latitude and longitude. However, this is ill-defined if at least one corner of the image falls \nabove the horizon. \nTo resolve this case, if the projection of a corner point does not intersect the ellipsoid we incrementally \nmove the point (in the image space) towards the opposite corner of the image until it intersects the ellipsoid (\\cref{fig:works}). This process is summarized in \\cref{alg:horizon_detection}. \nThis process is shown to be effective for our dataset, despite the fact that the approach could fail (see line 15 in~\\cref{alg:horizon_detection}) when the projection of the ellipsoid does not intersect the main diagonals of the image (e.g., when the camera is too far away from Earth or has a large tilt angle).\n\n\n\n\\begin{figure}[H]\n    %\n        \\centering\n        \\includegraphics[width=.3\\textwidth]{fig\/works.png}\n        \\caption{Example of our horizon detection method finding the horizon of an ellipsoidal body. Each corner point of the image \n        is incremented towards the opposite corner until the ellipsoid body is intersected.}\n        \\label{fig:works}\n    %\n    %\n    %\n    %\n    %\n    %\n    %\n    %\n    %\n    %\n\\end{figure}\n\n\\begin{algorithm}\n    \\caption{Horizon Detection} \n    \\label{alg:horizon_detection}\n    \\small\n    \\begin{algorithmic}[1]\n        \\State \\textbf{Inputs:} \n            \\State \\indent \\indent P \\Comment{estimate of camera projection matrix (containing intrinsic and extrinsic parameters)}\n            \\State \\indent \\indent $\\pi_{WGS84}$  \\Comment{projection of a pixel coordinate to a 3D point on the surface of the WGS84 model}\n            \\State \\indent \\indent $\\delta x$ \\Comment{amount to shift a point by in pixel space (default 10 pixels)}\n\n        \\State \\textbf{Output:} $image\\_corners$ \\Comment{set of four pixel coordinates bounding image}\n        \n        \\For{$x_{corner}, \\in image\\_corners$}\n            \\While{True}\n                \\State X $\\gets \\pi_{WGS84}(P, x_{corner})$\n                \\If{X intersects ellipsoid} \n                    %\n                    \\State break \\Comment{found valid image boundary}\n                \\Else\n                    \\State increment $x_{corner}$ towards opposite corner by $\\delta x$\n                \\EndIf\n                \\If{$x_{corner}$ outside image}\n                    \\State \\textbf{return} error \\Comment{failed to find horizon boundary}\n                \\EndIf\n            \\EndWhile\n        \\EndFor\n        \\State \\textbf{return} $image\\_corners$\n    \\end{algorithmic}\n\\end{algorithm}\n\nSince we select a maximum number of landmarks based on the landmarks in our satellite database that are in view of the camera, we need additional logic to\navoid the possibility of selecting landmarks that mostly fall near the horizon, since these are unlikely to lead to good matches. \nThe ratio\nof meters per pixels grows rapidly as we approach the horizon, and image matching becomes difficult or impossible near\nthe horizon line due to glare or heavy warping needed to match a shallow surface angle. Additionally, there \nis significant atmospheric distortion. \nRemoving those landmarks helps avoid unnecessary computation and reduces the number of outliers we pass to RANSAC. Towards this goal,\n we set a maximum acceptable angle between the boresight of the \ncamera and the surface normal of \na landmark and reject landmarks that fail to meet this threshold. To increase the number of potential landmarks that \nmeet our angle requirement, we filter out sections of the camera's FOV projection to the ellipsoid that are unlikely \nto produce landmarks that meet the angle threshold. \nThis filtering method follows our prior method for intersecting the ellipsoid and uses \nsimilar logic. Starting at the first point near each image corner that views the ellipsoid, we find the surface normal by projecting \nfrom the image plane to the ellipsoid and move towards the oppostite corner of the image\nuntil the angle requirement is met. This process is summarized in \\cref{alg:landmark_angle} and a corresponding ablation \nis shown in \\cref{fig:angle_ablation}. Notice that \nwithout \\cref{alg:landmark_angle}, more landmarks are selected near the horizon (\\cref{fig:landmark_angle_no_angle}) \nwhere template matching is more difficult resulting in more outliers. Using \\cref{alg:landmark_angle} allows IBAL to target \nregions of the image with more distinguishable features for matching which results in a higher concentration of inliers \n(\\cref{fig:landmark_angle_with_angle}).\n\n\n\n\\begin{algorithm}\n    \\small\n    \\caption{Landmark Angle Filter} \n    \\label{alg:landmark_angle}\n    \\begin{algorithmic}[1]\n        \\State \\textbf{Inputs:} \n            \\State \\indent \\indent P \\Comment{estimate of camera projection matrix (containing intrinsic and extrinsic parameters)}\n            \\State \\indent \\indent $\\pi_{WGS84}$  \\Comment{projection of a pixel coordinate to a 3D point on the surface of the WGS84 model}\n            \\State \\indent \\indent $\\alpha_{max}$ \\Comment{max acceptable angle between camera boresight and normal of a landmark}\n            \\State \\indent \\indent $\\delta x$ \\Comment{amount to shift a point by in pixel space (default 10 pixels)}\n        \n        \\State \\textbf{Output:} $image\\_corners$ \\Comment{set of four pixel coordinates bounding image}\n\n        \\State surface\\_normal() $\\gets$ function that finds normal vector at a point on the WGS84 model\n        \\State angle\\_between() $\\gets$ function that finds the angle between a camera boresight and a vector \n        \\For{$x_{corner}, \\in image\\_corners$}\n            \\While{True}\n                \\State X $\\gets \\pi_{WGS84}(P, x_{corner})$\n                \\State $x_n \\gets surface\\_normal(X)$\n                \\State $\\alpha \\gets angle\\_between(P, x_n)$\n                \\If{$\\alpha \\leq \\alpha_{max}$}\n                    \\State break \\Comment{found valid image bounary}\n                \\Else\n                    \\State increment $x_{corner}$ towards opposite corner by $\\delta x$\n                \\EndIf\n                \\If{$x_{corner}$ outside image}\n                    \\State \\textbf{return} error \\Comment{failed to meet landmark angle requirement}\n                \\EndIf\n            \\EndWhile\n        \\EndFor\n        \\State \\textbf{return} $image\\_corners$\n    \\end{algorithmic}\n\\end{algorithm}\n\n\\begin{figure}[H]\n    \\centering\n    \\begin{subfigure}[t]{0.46\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{fig\/without_angle.png}\n        \\caption{Higher concentration of outliers near the horizon without using landmark angle filter. Ratio of inliers to outliers: 0.3}\n        \\label{fig:landmark_angle_no_angle}\n    \\end{subfigure}\n    \\hfil\n    \\begin{subfigure}[t]{0.46\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{fig\/with_angle.png}\n        \\caption{Higher concentration of inliers using landmark angle filter. Ratio of inliers to outliers: 1.3}\n        \\label{fig:landmark_angle_with_angle}\n    \\end{subfigure}\n    \\caption{Ablation study for \\cref{alg:landmark_angle}, which filters regions of the image for landmark matching \n    based on the angle between the surface and the camera boresight. This leads to a higher ratio of inliers to outliers, reducing computation and improving accuracy. \n    Inliers matches are shown in green and outlier are shown in red. \n    Blue shows initial estimate of landmark location based on initial pose estimate before utilizing cross correlation. Images are from sideways camera \n    from balloon dataset.}\n    \\label{fig:angle_ablation}\n\\end{figure} %\n\n\\section{Experiment Results}\n\n\\subsection{Balloon Flight}\n\nWe present results from running IBAL with both a sideways-tilted and downward-facing camera aided by gyroscope measurements on altitudes ranging from 33km to 4.5km. \nNote that we use the term altitude to mean height above the WGS84 ellipsoid. \nDuring this time, the system is descending under a parachute. \nWe split our data into \n7 segments, each about 15 minutes long, and evaluate our estimated TRN position by comparing with GPS. We manually reseed IBAL at the start of each segment. \nResults are defined with respect to an East North Up (ENU) frame centered at the landing site of the balloon. \n\\Cref{fig:all_trajectory} shows the ground truth trajectory from GPS compared to the trajectory estimates from IBAL with a downward and sideways \nfacing camera. The corresponding plot of absolute position \nerror is shown in \\cref{fig:all_trajectory_error} for each of the East, North, and Up axes. \nIBAL is able to achieve an average position error along the up axis of 78 m and 66 m for the entire trajectory with the downward-facing and sideways-tilted camera, \nrespectively, while the balloon travels almost 30 km in elevation.\nIBAL achieves 207 m and 124 m of average position error for the east and north axis across the entire trajectory of the \ndownward-facing camera, and likewise an average error of 177 m and 164 m along the east and north axis for the sideways camera \nwhile the balloon transverses well over 100 km laterally. \n\\Cref{fig:all_trajectory_total_error} shows total absolute error (defined as the Euclidean distance between the estimate and the GPS position) with respect to flight time and with respect to height above ground level. \nAverage absolute position error for the entire trajectory is 287 m and 284 m for the downward and sideways-tilted camera, respectively. \nSpikes in position estimates could be diminished \nusing filtering methods such as coupling with an accelerometer or with visual odometry as mentioned in \\cref{sec:method}. \nWe run IBAL in real-time on a laptop with an Intel Xeon 10885M CPU. \nWhile IBAL is designed to run in real-time \non flight hardware, we do not make showcasing run-time performance a focus of this paper.\n\n\\begin{figure}[hbt!]\n    \\centering\n    \\includegraphics[width=\\textwidth, height=0.40\\textheight]{fig\/traj_all.png}\n    \\caption{IBAL+gyro trajectory estimate vs. GPS for altitude range of 33 km to 4.5 km on balloon dataset. \n    Vertical lines show start of each new data segment.}\n    \\label{fig:all_trajectory}\n\\end{figure}\n\n\\newpage\n\n\\begin{figure}[hbt!]\n    \\centering\n    \\includegraphics[width=\\textwidth, height=0.40\\textheight]{fig\/error_all.png}\n    \\caption{IBAL+gyro absolute position error for altitude range of 33 km to 4.5 km on balloon dataset. Vertical lines show start of each new data segment.}\n    \\label{fig:all_trajectory_error}\n\\end{figure}\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=\\textwidth, height=0.37\\textheight]{fig\/total_error.png}\n    \\caption{IBAL+gyro total trajectory error vs. time and vs. height above ground level on balloon dataset. Error tends to \n    show slight decrease in magnitude at lower altitudes. Vertical lines show start of each new data segment.}\n    \\label{fig:all_trajectory_total_error}\n\\end{figure}\n\n\\newpage\n\n\nWe also provide an analysis of the match correlation for both cameras for the entire balloon dataset. \n\\Cref{fig:all_matches_down} and \\cref{fig:all_matches_side} show number of inliers and outliers \nfor the downward and sideways facing cameras. After estimating the location of a landmark in the image \nwith cross correlation and peak finding, inliers and outliers are labeled using PnP and RANSAC. \nThere are generally more inliers than outliers which shows the effectiveness of the correlation approach, and\nthat IBAL is able to perform well in the presence of outliers. \nWe observe a greater number of inliers with the \ndownward-facing camera than with the sideways-tilted camera.\n\nAdditionally, \\cref{fig:histograms} shows a histogram of the amount of pixel error for the inliers and outliers \ndetermined by PnP and RANSAC for both the downward and sideways-tilted cameras. Inlier pixel error is distributed such that \nmost inliers have between 0 and 1 pixel of error as determined by PnP and RANSAC which shows the effectiveness of IBAL's correlation approach. \nThat there is an increase in the ratio of outliers to inliers at lower altitudes. This is due in part to shadows, lack of distinct texture on the ground, and \nregions with a sparse amount of landmarks in our database. \nDepending on mission requirements, this issue can be greatly reduced \nduring the landmark database creation process such as by optimizing for landmark template size, ensuring sufficent landmark coverage at low altitudes \nfor all phases of a flight, and by baking shadows into the database as was demonstrated in \\cite{Smith22aiaa-blenderTRN}. \nHowever, for the purposes of the balloon experiment in this paper, we determined our database to be sufficient.\n\nLastly, we provide visual examples of IBAL matches on a selected subset of frames from the downward and sideways facing cameras. \n\\Cref{fig:down_match_135} shows landmark matches for the downward camera at 13.5 km with inliers shown in green and \noutliers shown in red. Blue dots show the inital estimate of the landmark locations in the image by using the pose estimated by IBAL's \nprior pose and the gyro before matching with cross correlation. \n\\Cref{fig:down_match_23} shows matches for the downward camera at 23 km. \nCords from the high-altitude balloon are partially in view, but incorrect matches caused by the cords are correctly \nrejected as outliers. \\Cref{fig:side_match_135} and \\cref{fig:side_match_23} show results for the sideways-tilted camera \nat 13.5 km and 23 km.\n\n\\begin{figure}[H]\n    \\centering\n    \\begin{subfigure}[t]{1.0\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth, height=0.195\\textheight]{fig\/matches_all_down.png}\n        \\caption{IBAL landmarking matching results for downward-facing camera}\n        \\label{fig:all_matches_down}\n    \\end{subfigure}\n\n    \n    \\begin{subfigure}[t]{1.0\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth, height=0.195\\textheight]{fig\/matches_all_side.png}\n        \\caption{IBAL landmarking matching results for sideways-tilted camera}\n        \\label{fig:all_matches_side}\n    \\end{subfigure}\n    \\caption{IBAL+gyro number of inliers and outliers for sideways-tilted and downward-facing cameras on balloon dataset for altitude range of 33 km to 4.5 km \n    as determined by PnP and RANSAC. Vertical lines show start of each new data segment. The downward camera tends to have more matches than the sideways-tilted camera.}\n\\end{figure}\n\n\\begin{figure}[H]\n    \\centering\n    \\begin{subfigure}[t]{0.48\\textwidth}\n        \\centering\n        \\textbf{Downward Camera}\\par\\medskip\n        \\includegraphics[width=\\textwidth, height=0.1\\textheight]{fig\/hist_down_1.png}\n        \\caption{altitude range: 33 km to 32.5 km}\n    \\end{subfigure}\n    \\hfil\n    \\begin{subfigure}[t]{0.48\\textwidth}\n        \\centering\n        \\textbf{Sideways Camera}\\par\\medskip\n        \\includegraphics[width=\\textwidth, height=0.1\\textheight]{fig\/hist_side_1.png}\n        \\caption{altitude range: 33 km to 32.5 km}\n    \\end{subfigure}\n\n    \\begin{subfigure}[t]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth, height=0.1\\textheight]{fig\/hist_down_2.png}\n        \\caption{altitude range: 32.5 km to 29 km}\n    \\end{subfigure}\n    \\hfil\n    \\begin{subfigure}[t]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth, height=0.1\\textheight]{fig\/hist_side_2.png}\n        \\caption{altitude range: 32.5 km to 29 km}\n    \\end{subfigure}\n\n\n    \\begin{subfigure}[t]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth, height=0.1\\textheight]{fig\/hist_down_3.png}\n        \\caption{altitude range: 29 km to 23 km}\n    \\end{subfigure}\n    \\hfil\n    \\begin{subfigure}[t]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth, height=0.1\\textheight]{fig\/hist_side_3.png}\n        \\caption{altitude range: 29 km to 23 km}\n    \\end{subfigure}\n\n\n    \\begin{subfigure}[t]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth, height=0.1\\textheight]{fig\/hist_down_4.png}\n        \\caption{altitude range: 23 km to 18 km}\n    \\end{subfigure}\n    \\hfil\n    \\begin{subfigure}[t]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth, height=0.1\\textheight]{fig\/hist_side_4.png}\n        \\caption{altitude range: 23 km to 18 km}\n    \\end{subfigure}\n\n\n    \\begin{subfigure}[t]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth, height=0.1\\textheight]{fig\/hist_down_5.png}\n        \\caption{altitude range: 18 km to 14 km}\n    \\end{subfigure}\n    \\hfil\n    \\begin{subfigure}[t]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth, height=0.1\\textheight]{fig\/hist_side_5.png}\n        \\caption{altitude range: 18 km to 14 km}\n    \\end{subfigure}\n\n    \n    \\begin{subfigure}[t]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth, height=0.1\\textheight]{fig\/hist_down_6.png}\n        \\caption{altitude range: 14 km to 9 km}\n    \\end{subfigure}\n    \\hfil\n    \\begin{subfigure}[t]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth, height=0.1\\textheight]{fig\/hist_side_6.png}\n        \\caption{altitude range: 14 km to 9 km}\n    \\end{subfigure}\n\n\n    \\begin{subfigure}[t]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth, height=0.1\\textheight]{fig\/hist_down_7.png}\n        \\caption{altitude range: 9 km to 4.5 km}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth, height=0.1\\textheight]{fig\/hist_side_7.png}\n        \\caption{altitude range: 9 km to 4.5 km}\n    \\end{subfigure}\n\n\\caption{Inlier and outlier pixel error for each segment of balloon dataset. Error is the reprojection error determined by PnP and RANSAC. \nLeft Column: downward camera, Right Column: sideways camera. \nRows correspond to different altitude ranges.\n}\n\\label{fig:histograms}\n\\end{figure}\n\n\\begin{figure}[H]\n    \\centering\n    \\begin{subfigure}[t]{0.49\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{fig\/chords_outlier_rejection_13_5.png}\n        \\caption{Downward Camera, altitude 13.5 km}\n        \\label{fig:down_match_135}\n    \\end{subfigure}\n    \\hfil\n    \\begin{subfigure}[t]{0.49\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{fig\/chord_outlier_rejection_23.png}\n        \\caption{Downward Camera, altitude 23 km}\n        \\label{fig:down_match_23}\n    \\end{subfigure}\n\n    \n    \\begin{subfigure}[t]{0.49\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{fig\/side_cam_matches_13_5.png}\n        \\caption{Sideways Camera, altitude 13.5 km}\n        \\label{fig:side_match_135}\n    \\end{subfigure}\n    \\hfil\n    \\begin{subfigure}[t]{0.49\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{fig\/side_cam_matches_23.png}\n        \\caption{Sideways Camera, altitude 23 km}\n        \\label{fig:side_match_23}\n    \\end{subfigure}\n\n    \\caption{IBAL landmark match analysis on balloon dataset. Inliers matches are shown in green and outlier are shown in red. \n    Points in blue show initial estimate of landmark location based on initial pose estimate before utilizing cross correlation. Lines connect blue estimate \n    to calculated match location. Landmarks locations covered \n    by the cords are correctly rejected as outliers (top row).}\n\\end{figure}\n\n\\subsection{Blue Origin New Shepard Flight}\n\nWe present results from running IBAL with two cameras (referred to as camera 1 and camera 2) mounted inside the Blue Origin New Shepard capsule. \nWe only show results up to an altitude of approximately 8.5 km \nsince there was an anomaly that occurred during flight NS-23 which triggered the capsule escape system. \nNevertheless, we are still able to show IBAL working while the rocket achieves \nnominal speeds up to 880 km\/h (550 mph). We seed the initial input image to IBAL using telemetry from New Shepard and then use the previous IBAL pose estimate \nas the initial pose guess for the next timestep. Unlike the balloon experiment, we do not incorporate the gyroscope measurement to forward propagate the \npose estimate since the capsule does not experience significant rotations during its ascent.\n\nWe show a similar series of analysis of trajectory error and landmark matches as was presented for the high-altitude balloon experiment. \nResults are defined with respect to a ENU frame centered at the launch pad. \n\\Cref{fig:blue_error} shows absolute error for each of the East, North, and Up axes by comparing the position estimate of IBAL with GPS. \n\\Cref{fig:blue_total_error} shows total absolute error with respect to flight time and with respect to height above ground level. IBAL's total position \nerror estimate is below 120 m for the duration of the dataset, and that error with camera 2 is as low as 10 m when the rocket is at an altitude of 3.5 km. \nAverage absolute position error for the entire trajectory is 54 m and 34 m for camera 1 and camera 2, respectively. Both cameras show similar performance with IBAL, and \nslight differences in performance can be explained by the cameras being located on opposite sides of the capsule (and thus viewing different terrain) \nand by potential unaccounted distortion effects in the camera calibration. \n\n\n\\begin{figure}[hbt!]\n    \\centering\n    \\includegraphics[width=1.0\\textwidth]{fig\/blue_error.png}\n    \\caption{IBAL absolute position error on New Shepard dataset: altitude range of 3.5 km to 8.5 km.}\n    \\label{fig:blue_error}\n\\end{figure}\n\n\\begin{figure}[hbt!]\n    \\centering\n    \\includegraphics[width=1.0\\textwidth]{fig\/blue_error_time_alt}\n    \\caption{IBAL total trajectory error vs. time and height above ground level on New Shepard dataset. Total error is less than 120 m while reaching \n    speeds up to 880 km\/h and a peak altitude of 8.5 km.}\n    \\label{fig:blue_total_error}\n\\end{figure}\n\nWe also provide  an analysis of match correlation for both cameras. Since each processed frame only had at most 2 matches identified as outliers \nby PnP and RANSAC, we do not include match analysis for outliers in our results. \n\\cref{fig:blue_inliers_outliers_1} and \\cref{fig:blue_inliers_outliers_2}\nshow number of inliers for both cameras. \n\\cref{fig:blue_histogram} shows a histogram of the amount of pixel error for the inliers determined by PnP\nRANSAC for both cameras. Similarly to the results from the balloon flight, pixel error for a majority of the inliers is less than two pixels.\n\nWe provide visual examples of IBAL matches on a frame from both cameras in \\cref{fig:blue_match_visualize}. \nMatches labeled as inliers are shown in green, while outliers are shown in red. There is only one outlier present in the processed image from \ncamera 1 (\\cref{fig:blue_match_visualize_1}) and no outliers in the image from camera 2 (\\cref{fig:blue_match_visualize_2}).\n\nLastly, we remark on one difficulty of the New Shepard dataset.\nA mountain range is in view of camera 2 which makes landmark matching more difficult near the latter portion of the dataset as the mountain comes into the camera's \nFOV (\\cref{fig:mountain}). This is due to the presence of shadows in the mountain that may not be consistent with shadows present in the \ntime of day the database imagery was collected. Additionally, the 2D-2D homography assumption which we use to warp landmark templates into the image for \ncorrelation begins to break down when 3D structures \nsuch as mountains are viewed from low altitudes. Work with database creation such as \\cite{Smith22aiaa-blenderTRN} along with advances in IBAL \nnot mentioned in the paper can be used to reduce these issue for low altitude navigation over mountains. \n\n\n\\begin{figure}[H]\n    \\centering\n    \\begin{subfigure}[t]{0.46\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{fig\/cam1_blue_matches.png}\n        \\caption{IBAL landmarking matching results for camera 1}\n        \\label{fig:blue_inliers_outliers_1}\n    \\end{subfigure}\n    \\hfil\n    \\begin{subfigure}[t]{0.46\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{fig\/cam2_blue_matches.png}\n        \\caption{IBAL landmarking matching results for camera 2}\n        \\label{fig:blue_inliers_outliers_2}\n    \\end{subfigure}\n    \\caption{IBAL number of inliers and outliers for cameras 1 and 2 on New Shepard dataset as determined by PnP and RANSAC. \n    The data corresponds to an altitude range between 3.5 km and 8.5 km.}\n\\end{figure}\n\n\\begin{figure}[H]\n    \\centering\n    \\begin{subfigure}[t]{0.45\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth, height=0.1\\textheight]{fig\/blue_hist_cam1.png}\n        \\caption{Camera 1}\n    \\end{subfigure}\n    \\hfil\n    \\begin{subfigure}[t]{0.45\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth, height=0.1\\textheight]{fig\/blue_hist_cam2.png}\n        \\caption{Camera 2}\n    \\end{subfigure}\n    \\caption{Inlier pixel error distribution for Cameras 1 and 2 on New Shepard dataset. \n    %\n    }\n    \\label{fig:blue_histogram}\n\\end{figure}\n\n\\begin{figure}[H]\n    \\centering\n    \\begin{subfigure}[t]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{fig\/cam1_54.png}\n        \\caption{IBAL inlier and outlier matches for camera 1 on New Shepard dataset at an altitude of 6.4 km}\n        \\label{fig:blue_match_visualize_1}\n    \\end{subfigure}\n    \\hfil\n    \\begin{subfigure}[t]{0.476\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{fig\/cam2_54.png}\n        \\caption{IBAL inlier and outlier matches for camera 2 on New Shepard dataset at an altitude of 6.4 km}\n        \\label{fig:blue_match_visualize_2}\n    \\end{subfigure}\n    \\caption{IBAL inlier and outlier matches for cameras 1 and 2 on New Shepard dataset. Inliers matches are shown in green and outlier are shown in red. \n    Blue shows initial estimate of landmark location based on initial pose estimate before utilizing cross correlation. Lines connect blue estimate \n    to calculated match location. Images have been rotated by \n    180 $^{\\circ}$ for visual appeal.}\n    \\label{fig:blue_match_visualize}\n\\end{figure}\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=0.46\\textwidth]{fig\/mountain.png}\n    \\caption{IBAL Camera 2 viewing a mountain range on New Shepard dataset. Inliers matches are shown in green. \n    Blue shows initial estimate of landmark location based on initial pose estimate before utilizing cross correlation. Lines connect blue estimate to calculated match location. \n    Image has been rotated by \n    180 $^{\\circ}$ for visual appeal.}\n    \\label{fig:mountain}\n\\end{figure} %\n\n\\section{Gyroscope Incorporation Ablation Study}\n\\label{sec:gyro_ablation}\n\nWe provide an ablation study of forward propagating the IBAL pose estimate with a gyroscope for the high-altitude balloon dataset \nas mentioned in \\cref{sec:method}. The benefits of incorporating the gyroscope data is two-fold. Firstly, since the balloon experiences \nrapid rotations, in some cases exceeding $20^\\circ$ per second, the gyro provides a more accurate initial guess of the balloon's pose for IBAL, which \nreduces the frequency at which images must be to used to estimate the pose, hence reducing computation. Additionally, if landmark match quality is temporarily insufficient \n(typically on the order of 1 to 3 seconds) for PnP and RANSAC, which can be caused for example by significant obstruction by the cords below the balloon, the gyro allows the pose estimate to be carried over until good landmark matches can be found.\n\n\\Cref{table:gyro_ablation} shows the benefits of using the gyro with our balloon dataset. Using the downward-facing camera, we show the percentage \nof each of the seven data segments IBAL is able to successfully complete with and without incorporating the gyroscope. We also test on two different rates of \nimage processing, noting that while one could partially compensate the lack of gyroscope measurements by increasing the rate of image processing,  that strategy is only effective at high altitudes in our dataset.\n\n\n\\begin{table}[H]\n    \\begin{tabular}{ | l | l | l | l | l | l | l | l |}\n        \\hline\n        & 33-32.5 km & 32.5-29 km & 29-23 km & 23-18 km & 18-14 km & 14-9 km & 9-4.5 km\\\\ \\hline\n        4 Hz w\/ gyro & 100 & 100 & 100 & 100 & 100 & 100 & 100  \\\\ \\hline\n        2 Hz w\/ gyro & 100 & 100 & 100 & 100 & 100 & 100 & 100  \\\\ \\hline\n        4 Hz w\/o gyro & 100 & 100 & 96 & 3 & 3 & 1 & 1  \\\\ \\hline\n        2 Hz w\/o gyro & 100 & 100 & 63 & 0 & 0 & 1 & 1  \\\\ \n        \\hline\n    \\end{tabular}\n    \\caption{Ablation study showing the benefit of incorporating gyroscope measurements with IBAL on each of the seven altitude segments of the balloon dataset \n    for different rates of image processing. \n    Results show the percent of each dataset segment IBAL successfully processes using images from the downward camera.}\n    \\label{table:gyro_ablation}\n\\end{table} %\n\n\\section{Conclusion}\n\nThis paper reports on the performance of a vision-based terrain relative navigation method on data ranging from 4.5 km to 33 km on a high-altitude \nballoon dataset and on data collected onboard Blue Origin's New Shepard rocket. We evaluate \nperformance \nof both a sideways-tilted and downward-facing camera for the balloon dataset and two sideways-tilted \ncameras on the New Shepard dataset. We observe less than 290 meters of \naverage position error on the balloon data over a trajectory of 150 kilometers and \nwith the presence of rapid motions and dynamic obstructions in the field of view of the camera. Additionally, we report less than 55 m of \naverage position error on the \nNew Shepard dataset while reaching an altitude of 8.5 km and a max nominal speed of 880 km\/h. As future work, we plan to fly again onboard the New Shepard \nrocket and capture camera data from ground level to an altitude of over 100 km. \n\\section*{Acknowledgments}\nWe would like to gratefully acknowledge Andrew Olguin, Carlos Cruz, Alanna Ferri, Laura Henderson, and\neveryone else at Draper who supported IBAL and data collection for the balloon flight and New Shepard flight. This\nwork was authored by employees of The Charles Stark Draper Laboratory, Inc. under Contract No. 80NSSC21K0348\nwith the National Aeronautics and Space Administration. The United States Government retains and the publisher, by\naccepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up,\nirrevocable, worldwide license to reproduce, prepare derivative works, distribute copies to the public, and perform\npublicly and display publicly, or allow others to do so, for United States Government purposes. All other rights are\nreserved by the copyright owner.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\nStochastic or simulation models are only approximations to the reality. A conjectured model may not align with the true system because of unobserved complexity. Moreover, some highly accurate models, even if formulable, may not be implementable due to computational barriers and time constraints, in which case a simpler, lower-fidelity model is adopted. In all these cases, there is a discrepancy between the model and the reality, which we call \\emph{model discrepancy}. This article describes a data-processing framework to integrate data from both a simulated response and the real system of interest, under the presence of model discrepancy, to reliably predict stochastic outputs of interest.\n\nOur objective is motivated from everyday practice of simulation analysis. For example, this article describes  a major manufacturer that is interested in assessing the impact of the staffing level of support workers on a production line via discrete-event simulation.  Twelve weeks were spent carefully designing and tuning the simulation model and the final report included seventy-five realizations of the simulation model at each potential staffing level.   The limited amount of realizations gives rise to a simulation error (also termed a Monte Carlo error).  In addition, when data at the current staffing level was compared to the simulation model realizations, it was clear the simulation model was inaccurate.  Yet, given the resources already invested, the manufacturer was interested if the simulation model could still be used to guide the staffing level decision.  An approach that can account for both sources of errors can save significant costs and improve the decisions in situations like these.\n\nDifferences between a simulation and real data is traditionally addressed during the important practice of \\emph{model validation} and \\emph{calibration} in the simulation literature, which refers to the joint task of checking whether a developed stochastic model sufficiently reflects the reality (validation), and if not, re-developing the model until it matches (calibration) (e.g., \\cite{Sargent2013}, \\cite{banks2000dm} Chapter 10, \\cite{kelton2000simulation} Chapter 5). Conventional validation methods compare relevant outputs from simulation models and real-world data via statistical or Turing tests (e.g. \\cite{schruben1980establishing} and \\cite{balci1982some}). In the case of a mismatch, guided expert opinions, together with possibly more data collection, are used to re-calibrate the model recursively until acceptable accuracy \\citep{sargent1998verification}.   While these tools are fundamentally critical to the practice of simulation, there can be two deficiencies when using calibration in an ad-hoc way:\n\\begin{enumerate}\n\\item  It necessitates building increasingly sophisticated models after unsatisfactory conclusions.  This process potentially places a heavy burden on a simulation modeler\/software, consumes time and, moreover, may end up in non-convergence to an acceptable ultimate model.\n\\item The recursive refinement of the model to align it with the real data along the development process involves hidden parameter choices and simultaneous estimations. These details, which are often overlooked and unaccounted for, complicate statistically justified uncertainty quantification alongside prediction.\n\\end{enumerate}\n\n\nOur goal is thus to investigate a framework that systematically offers predictive bounds using a simulation model without the traditionally encountered recursive efforts. Our framework is a stochastic version of model calibration that is similar in spirit to deterministic model calibration \\citep{kennedy2001bayesian}. The basic idea is to view potential model discrepancy as an object that can be inferred statistically, or plainly put, to ``model\" this potential error.  To conduct feasible inference, often the model discrepancy is assumed to have some structure decided a priori of observing data, and data are used to update the uncertainty on predictions of the true system. Since \\cite{kennedy2001bayesian}, this idea has been extended and widely applied in various scientific areas, e.g., \\cite{tuo2015efficient,higdon2004combining,plumlee2016bayesian}.  In the stochastic simulation literature, similar machinery has appeared under the heading of  stochastic kriging (\\cite{ankenman2010stochastic,Staum2009,chen2013enhancing,chen2012effects,chen2014stochastic,chen2016efficient}). In the stochastic kriging literature, the oracle benchmark is the simulation model and stochastic kriging is used to reduce simulation effort by borrowing information from the simulation outputs at a collection of design values. In the model discrepancy setting, the oracle benchmark is the real system's probabilistic generating mechanism and our goal is to improve the prediction accuracy and quantification of uncertainties associated with the simulation model.\n\nOne challenge in bringing the deterministic model discrepancy machinery to stochastic simulation is that in the latter case, the inference objects are themselves embedded in probability spaces. The stochastic simulation model and the real system are naturally represented as probability distributions (think of the output distributions of a queueing or a stochastic inventory model), which constitute the basis of calculation in many decision-making tasks (for example, computing the chance that the outcome is in some region that indicates poor performance). Consequently, the learning and the uncertainty quantification of the discrepancies need to take into account the resulting probabilistic constraints. This is beyond the scope of the established inference and computation tools in the deterministic model discrepancy literature.\n\nAs our main contribution, we develop a framework to infer stochastic model discrepancies that is statistically justified and computationally tractable under the constraints discussed above. On the statistical aspect, we build a Bayesian learning framework that operates on the space of likelihood ratios as the representation of model discrepancies between simulation and reality. We study how this representation satisfies the constraints necessarily imposed in capturing stochastic model discrepancies and leads to desirable asymptotic behavior. On the computational aspect, we propose an optimization approach to obtain prediction bounds. Though sampling techniques such as Markov chain Monte Carlo \\cite[Chapters 11 and 12]{gelman2014bayesian} are widely used in Bayesian computation, they encounter difficulties in our setting due to the constraints and high-dimensionality. Our approach, inspired from the recent literature in robust optimization (\\cite{ben2002robust,ben2009robust,bertsimas2011theory}), alleviates this issue via the imposition of suitable optimization formulations over posterior high probability regions. We study the statistical properties of these formulations and demonstrate that they are equally tight in terms of asymptotic guarantees to traditional Bayesian inference.\n\nWe close this introduction by briefly reviewing two other lines of related work. First, in stochastic simulation, the majority of work in handling model uncertainty focuses on input uncertainty; see, e.g. the surveys \\cite{barton2002panel,henderson2003input,chick2006bayesian,barton2012tutorial,song2014advanced,lam2016advancedtutorial}, \\cite{nelson2013foundations} Chapter 7. They quantify the impacts on simulation outputs due to the statistical uncertainty in specifying the input models (distributions, stochastic assumptions etc.), assuming input data are available. Approaches include the delta method (\\cite{cheng1997sensitivity}) and its variants such as the two-point method (\\cite{cheng1998two,cheng2004calculation}), the bootstrap (\\cite{barton1993uniform,barton2001resampling,cheng1997sensitivity}) which can be assisted with stochastic kriging-based meta-models (\\cite{barton2013quantifying,xie2014bayesian}), and Bayesian methods (\\cite{chick2001input,zouaoui2003accounting,zouaoui2004accounting,xie2014bayesian,biller2011accounting}). Added to these approaches are recent perspectives of model risks and robust optimization that do not necessarily directly utilize data (\\cite{glasserman2014robust,lam2013robust,lam2011sensitivity,ghosh2015computing}). The second line of related work is queueing inference that investigates the calibration of input processes and system performances from partially observed queueing outputs such as congestion or transaction data (e.g., the queue inference engine; \\cite{larson1990queue}). This literature utilizes specific queueing structures that can be approximated either analytically or via diffusion limits, and as such allow tractable inference. Techniques include maximum likelihood estimation (\\cite{basawa1996maximum,pickands1997estimation}), nonparametric approaches (\\cite{bingham1999non,hall2004nonparametric}) and point processes (\\cite{whitt1981approximating}).\nRecently, \\cite{goeva2014reconstructing} study calibration of input distributions under more general simulation models. Like the input uncertainty literature, however, these studies assume correctly specified system logics that imply perfect matches of the simulation models with real-world outputs.\n\n\n\\section{Stochastic Model Discrepancy: Setting and Notations} \\label{sec:setting}\nThis section describes our setting and notations throughout this paper. We consider a system of interest that outputs a discrete random response over the space $\\mathcal{Y}$ with cardinality $m$.  For notational simplicity, we will use the space $\\mathcal Y=\\{1,\\ldots,m\\}$.  This response depends on a vector of design variables, denoted $x$, which can be broadly defined to include input variables that are not necessarily controllable. We presume a finite set of design points or design values $x_j$, $j = 1,\\ldots,s$.  The probability mass function $\\pi_j=\\{\\pi_j(i)\\}_{i=1,\\ldots,m}$ describes the distribution of the response of the real system on $\\mathcal Y$ under $x_j$.   Examples of the response include the waiting times in call centers \\citep{brown2005statistical} and hospitals \\citep{helm2014design}.    In the first example, design variables could be the number of servers, the system capacity, and the arrival rate.  In  the second example, the design variable could be the rate of elective admissions.\n\nThe objective is to draw conclusions about $\\pi_j$ for several $j$'s. These distributions form the basis in evaluating quantities of interest used for decision-making.  When responses are independently observed from the real system (e.g., from a designed experiment \\citep{li2015value}), $n_j(i)\/n_j$ is a reasonable estimate of $\\pi_j$, where $n_j(i)$ counts the number of outcomes equal to $i$ and $n_j$ is the total number of recorded responses at $x_j$.  In the setting of simulation modeling, however, these empirical estimates are often inadequate because typical decision-making tasks, like feasibility or sensitivity tests, are applied on system configurations that are sparsely sampled or even never observed. This means that accurate empirical estimates for the $x_j$ values of interest are not available. In fact, for these $j$'s, $n_j$ can often times be $0$.\n\nIn contrast, using state-of-the-art understanding of the system, possibly simplified for computational concerns, an operations researcher builds a simulation model (typically based on discrete-event simulation) to estimate $\\tilde{\\pi}_{j}=\\{\\tilde\\pi_j(i)\\}_{i=1,\\ldots,m}$, the simulated distribution of the response at the design point $x_j$. In parallel to the real responses, we denote $\\tilde{n}_j(i)$ as the count of outcome $i$ and $\\tilde n_j$ as the total number of replications in a simulation experiment at $x_j$, and $\\tilde{n}_j(i)\/\\tilde{n}_j$ is hence an estimate of $\\tilde\\pi_j(i)$. Unlike the real responses, it is often affordable to generate a more abundant number of $\\tilde n_j$ and hence a more accurate estimate of $\\tilde\\pi_j$.  However, the difference between $\\tilde{n}_j(i)\/\\tilde{n}_j$ and $\\tilde\\pi_j(i)$ remains a source of uncertainty.\n\nOur premise is that the real response distribution $\\pi_j$ and the simulated distribution $\\tilde\\pi_j$ differ. Thus, in order to make conclusions about $\\pi_j$, we must conjecture about the potential gap between $\\pi_j$ and $\\tilde\\pi_j$ with the limited simulation and real-world data. The remainder of this section describes our framework for defining the discrepancy between $\\pi_j$ and $\\tilde\\pi_j$.\n\nFirst note that both $\\pi_j$ and $\\tilde\\pi_j$ obviously must satisfy the criteria of a probability distribution:\n\\begin{defination} \\label{def:valid_distribution}\nAny mapping $p: \\mathcal{Y} \\rightarrow \\mathbb{R}$ is a \\emph{valid distribution} if\n\\begin{enumerate}[(i)]\n\\item $p(i)\\geq0$ for all $i = 1,\\ldots,m$ and\n\\item $\\sum_{i=1}^m   p(i) =  1$.\n\\end{enumerate}\n\\end{defination}\n\nWe define the discrepancy between $\\pi_j$ and $\\tilde\\pi_j$ as $\\delta_j=\\{\\delta_j(i)\\}_{i=1,\\ldots,m}$  where\n\\begin{equation}\n\\delta_j(i)  =  \\frac{\\pi_j(i)}{\\tilde{\\pi}_j(i)}.  \\label{def discrepancy}\n\\end{equation}\nIn other words, $\\delta_j$ reflects the the ratio between the probabilities of the true responses and simulated responses. If $\\delta_j (i) = 1$ for all $i$, the simulation model is correctly specified. Definition \\ref{def discrepancy} is analogous to that of likelihood ratio in the context of importance sampling (e.g., \\cite{mcbook}, Chapter 9; \\cite{asmussen2007stochastic}, Chapter IV; \\cite{glasserman2003monte}, Chapter 4). In the model risk literature, similar object as Definition \\ref{def discrepancy} also appears as a decision variable in worst-case optimization problems used to bound performance measures subject to the uncertainty on the true model relative to a conjectured stochastic model (often known as the baseline model).   Examples include Gaussian models with mean and covariance uncertainty represented by linear matrix inequalities (\\cite{hu2012robust}), and nonparametric uncertainty measured by Kullback-Leibler divergence (e.g., \\cite{glasserman2014robust,lam2013robust}). Our definition \\ref{def discrepancy} is along a similar vein as these work, but  rather than using it as a tool to speed up simulation (in importance sampling) or an optimization decision variable (in model risk), our $\\delta_j$ is an object to be \\emph{inferred} from data.\n\nNote that \\eqref{def discrepancy} is not the only way to define stochastic model discrepancy. Another natural choice, which more closely mimics the established deterministic counterpart \\citep{kennedy2001bayesian}, is via\n\\[\\pi_j(i)-\\tilde{\\pi}_j(i)\\]\nThe choice of which version of discrepancy to use relates to the convenience in statistical modeling. We adopt the multiplicative version in \\eqref{def discrepancy} based on its analog with likelihood ratio, which facilitates our inference.\n\n\n\nSince $\\pi_j$ and $\\tilde\\pi_j$ are valid distributions, the model discrepancy $\\delta_j$ defined in \\eqref{def discrepancy} must satisfy the following criteria with respect to $\\tilde\\pi_j$:\n\\begin{defination} \\label{def:valid}\nSay $p$ is a valid distribution with $p(i)> 0$.  $d$ is a \\emph{valid discrepancy} with respect to $p$ if\n\\begin{enumerate}[(i)]\n\\item $d(i)\\geq0$ for all $i = 1,\\ldots,m$ and\n\\item $\\sum_{i=1}^m   d(i) p(i)  =  1$.\n\\end{enumerate}\n\\end{defination}\nClearly, if $d$ is a valid discrepancy and $p$ is a valid distribution then $\\{d(i) p(i)\\}_{i=1,\\ldots,m}$ will also be a valid distribution.\n\nDefinition \\ref{def:valid} plays a vital role in our subsequent analysis as they characterize the properties of our inference targets. Unlike deterministic model discrepancies, these conditions come from the probabilistic structure that arises uniquely in stochastic model discrepancies. Note that Definition \\ref{def:valid} coincides with that of a likelihood ratio (e.g., \\cite{asmussen2007stochastic}).\n\nLastly, in addition to model discrepancy, simulation noise and experimental noise also contribute to the uncertainty in estimating $\\pi_i$, i.e., the noise of the estimator $n_j(i)\/n_j$ for $\\pi_j(i)$ and $\\tilde n_j(i)\/\\tilde n_j$ for $\\tilde\\pi_j(i)$. Our analysis will also incorporate these sources of uncertainty.\n\\section{A Bayesian Framework} \\label{sec:learn}\nWe propose a Bayesian framework to infer the discrepancy $\\delta_j$. The framework has the capability to quantify uncertainty under limited data environments (common in our setting where observed responses from the real system may be sparse or absent for some design points), and to incorporate prior information that  anticipates similar discrepancies for similar design points, where the similarity is measured by the distance between the design values. We will also see how the framework can account for the notion of a valid discrepancy provided in Definition \\ref{def:valid}.\n\n\nThe term \\emph{data} substitutes for the collection of all observed responses from the real system and the simulation model, which is sufficiently represented as\n\\[\\text{data} = \\left\\{n_j(i),  i = 1,\\ldots,m, j = 1,\\ldots, s  \\text{ and } \\tilde{n}_j(i),  i = 1,\\ldots,m,  j = 1,\\ldots, s \\right\\}.\\]\nOur main inference procedure is the Bayes rule summarized as\n\\begin{equation}\n\\operatorname{post}\\left(d,\\tilde{p},\\text{data}\\right) \\propto \\operatorname{likelihood}(d,\\tilde{p},\\text{data} ) \\operatorname{prior}(d,\\tilde{p}), \\label{Bayesian update}\n\\end{equation}\nwhere $d$ and $\\tilde{p}$ are the locations at which the density is evaluated for $\\delta=(\\delta_j)_{j=1,\\ldots,s}$ and $\\tilde{\\pi}=(\\tilde\\pi_j)_{j=1,\\ldots,s}$. The notations ``$\\operatorname{post}$\", ``$\\operatorname{likelihood}$\" and ``$\\operatorname{prior}$\" stand for the posterior, likelihood and prior distribution of $(\\delta,\\tilde\\pi)$. Note that we have defined $\\tilde\\pi$ as an inference target in addition to the discrepancy $\\delta$, in order to handle the simulation noise (as we will describe momentarily). The relationship\n$$p_j(i) =  d_j(i) \\tilde{p}_j(i) $$\ncan be used to define the posterior distribution of $\\pi_j(i)$ at $p_j(i) $.\n\nThe likelihood for \\eqref{Bayesian update} is straightforward to compute as\n\\begin{align}\n\\operatorname{likelihood}(d,\\tilde{p},\\text{data})  \\propto \\exp &\\left( \\sum_{j=1}^{s} \\sum_{i=1}^{m} n_{j}(i) \\log\\left( d_{j}(i) \\tilde{p}_j(i) \\right)  + \\sum_{j=1}^{s} \\sum_{i=1}^{m} \\tilde{n}_j(i) \\log \\tilde{p}_j(i)\\right). \\label{eq:likelihood}\n \\end{align}\n\n\nWe now discuss the prior for \\eqref{Bayesian update}. We restrict ourselves to independent priors for the discrepancy and the simulation model. The prior on the simulation model needs to exhibit the properties of a valid distribution. These properties can be enforced by conditioning an arbitrary prior distribution on a vector which takes real values in a space $\\mathbb{R}^{sm}$ on the constrained region associated with Definition \\ref{def:valid_distribution}. Similarly, the properties of a valid discrepancy can be enforced by conditioning an arbitrary prior distribution on the constrained region associated with Definition \\ref{def:valid}. More precisely, let the logarithm of this arbitrary prior mass function for the simulation model be denoted with $f$ and the discrepancy with $g$.  Our construction leads to\n\\begin{equation}\n\\text{prior}(d,\\tilde{p}) \\propto \\begin{cases} \\exp\\left(f(\\tilde{p}) +  g(d) \\right) &\\text{ if }  \\begin{cases} \\tilde{p}_j(i) \\geq 0, &  1\\leq i \\leq m,1\\leq j \\leq s  \\\\\n d_j(i) \\geq 0,  &  1\\leq i \\leq m,1\\leq j \\leq s \\\\\n\\sum_{i=1}^m \\tilde{p}_j(i) =1,   & 1\\leq j \\leq s\\\\\n\\sum_{i=1}^m \\tilde{p}_j(i) d_j(i) =1, & 1\\leq j \\leq s\n\\end{cases} \\\\\n0 &\\text{otherwise}. \\end{cases}  \\label{eq:prior} \\end{equation}\n\nThe choices of $f(\\cdot)$ and $g(\\cdot)$ are open to the investigator. For computational reasons that will be detailed in Section \\ref{sec:optim}, we prefer $g(\\cdot)$ that is concave. One widely used option that exhibits this property will be a multivariate Gaussian with a mean $\\mu$ and correlation  matrix $R$ that borrows information across design points and observation points.  It is recommended that one uses a vector of $1$s as the prior mean for $\\delta$ and  $(1\/m)$s for $\\tilde{\\pi}$.   $R$ should be built with domain specific logic, e.g., similar design points and\/or similar responses should have similar discrepancies. For more detailed ideas toward constructing correlation structures for responses, see \\cite{ankenman2010stochastic} on the topic of stochastic kriging. In general, this approach leads to\n\\begin{align}\n \\exp\\left(f(\\tilde{p}) +  g(d) \\right) \\propto  \\exp & \\left( -\\lambda_{\\tilde{p}} (\\tilde{p} - 1\/m)^\\mathsf{T} R_{\\tilde{p}}^{-1} (\\tilde{p} - 1\/m) -\\lambda_d (d - 1)^\\mathsf{T} R_d^{-1} (d - 1) \\right), \\label{eq:Gaussian}\n\\end{align}\nwhere the $\\tilde{p}$ and the $d$ are understood to be vectorizations of the probability masses represented by themselves, and $\\lambda$s are positive constants that scale the correlation matrices $R$s.\n\nNote that, in the settings where simulation is cheap and $\\tilde\\pi$ is estimated with negligible error, one can drop the parameter $\\tilde p$ in the likelihood \\eqref{Bayesian update} and correspondingly the second terms in the likelihood \\eqref{eq:likelihood} and the prior \\eqref{eq:prior}.\n\n\\section{Optimization-based Procedure for Bayesian Inference} \\label{sec:optim}\nThis section presents our computation procedure to make conclusions about $\\pi_j$ based on \\eqref{Bayesian update}. In particular, we propose an optimization-based approach.  There are two reasons for considering this inference package in place of the more traditional Markov Chain Monte Carlo. First, a typical decision-making in simulation analysis often boils down to the estimation of expectation-type quantities of interest evaluated at $\\pi_j$. The optimization we study will provide efficiently computable bounds on these expectations. Second, because of the constrained structure of the prior distribution \\eqref{eq:prior}, standard sampling-based Bayesian computation tools are deemed to be inefficient, and optimization serves as a competitive alternative.\n\n\nTo elaborate the second rationale, note that common solution mechanisms in Bayesian inference consist of drawing samples from the posterior of the parameters of interest. However, because the posterior is often not a standard distribution like Normal (and that there is an unknown proportionality constant), direct Monte Carlo sampling is not possible.  Sophisticated Markov chain Monte Carlo samplers were designed explicitly for this purpose \\cite[Chapters 11 and 12]{gelman2014bayesian}.  Popular samplers include the classic Metropolis Hastings algorithm with a symmetric proposal \\cite[pp 278-280]{gelman2014bayesian}, and other useful methods such as Hamilton Monte Carlo \\citep{duane1987hybrid} and slice sampling \\citep{neal2003slice}.  The latter two methods are specifically designed to alleviate the problems faced by classical samplers.  But there are still many practical issues for these new samplers regarding their execution and choices of parameters in constrained and high dimensional spaces, which is the setting we encounter in the posterior induced from \\eqref{eq:prior} (probabilistically constrained and with dimension $sm$).  See, for example, \\cite{betancourt2017conceptual} for an intuitive history and theoretical summary of these conclusions.  It should be acknowledged that theoretical results do not always reveal these practical issues; see, for example, the positive results from \\cite{dyer1991random}. However, numerical tests in \\cite{plumlee2016learning} demonstrate these issues in a closely related setting.\n\n\nIn the following subsections, we will present our optimization formulation, the statistical guarantees, and discussion on computational tractability.  The summaries of the sections are:  1) We use an uncertainty set in place of a typical Bayesian integration; 2) The method is guaranteed to produce tight bounds that will contain the truth with the typical desired confidence; and 3) Given we simulate enough, the optimization problem can be reformulated into a convex problem.\n\n\\subsection{Optimization Formulation}\\label{sec:formulation}\nSuppose we are interested in estimating quantities of interest in the form $E[z(Y_j)]$ where $Y_j\\sim\\pi_j$ and $z:\\mathcal Y\\to\\mathbb R$  is some function. We can write this in terms of $\\delta$ and $\\tilde\\pi$ as $$\\zeta(\\delta,\\tilde\\pi)= \\sum_{i=1}^m z(i) \\pi_j(i) =\\sum_{i=1}^mz(i)\\delta_j(i)\\tilde\\pi_j(i).$$ Our procedure consists of solving the optimization pairs\n\\begin{equation}\\begin{array}{ll}\n\\max \\text{ or } \\min_{d,\\tilde{p}} & \\zeta(d,\\tilde{p}) =  \\sum_{i=1}^m z(i) d_j(i) \\tilde{p}_j(i), \\\\\n\\text{subject to}& \\operatorname{post}(d,\\tilde{p},\\text{data}) \\geq c \\end{array} \\label{obj:optim}\n\\end{equation}\nwhere $c$ is chosen such that\n\\begin{equation}\n  c = \\exp \\left(-\\frac{1}{2} \\Phi^{-1}(q)^2 + \\max_{d,\\tilde{p}} \\log \\text{post}(d,\\tilde{p},\\text{data})\\right),\\label{choice}\n\\end{equation}\nand $ \\Phi^{-1}(q)$ is the standard Normal quantile at level $q$. The optimal values of these optimization problems form an approximate confidence interval for $E[z(Y_j)]$ at a confidence level in the frequentist sense, as we will describe in Section \\ref{sec:theo}.\n\nOptimization problems  \\eqref{obj:optim} can be motivated from a robust optimization viewpoint. This  literature uses deterministic sets, the so-called ambiguity or uncertainty sets, to represent the probabilistic uncertainty in the parameters  (e.g., \\cite{ben2002robust,ben2009robust,bertsimas2011theory}). Typically, these sets are chosen as prediction sets that contains the truth with a prescribed confidence. The optimal values of the resulting robust optimizations then bound the true quantity of interest with at least the same confidence level. This approach has been applied in many contexts, such as approximating chance-constrained programs (e.g., \\cite{ben2002robust}, Chapter 2) and performance measures driven by complex stochastic models (e.g., \\cite{bandi2012tractable,bandi2014robust}). Here, we consider using a prediction set given by a posterior high probability region\n\\begin{equation}\n\\mathcal{U}(c) = \\left\\{d,\\tilde{p} \\left| \\operatorname{post}(d,\\tilde{p},\\text{data}) \\geq c \\right.\\right\\} \\label{uncertainty set}\n\\end{equation}\nas the set of points $(d,\\tilde p)$ with posterior probability higher than level $c$. From the view of robust optimization, if $c$ is chosen such that $\\mathcal U(c)$ contains $1-\\alpha$ posterior content of $(\\delta,\\tilde\\pi)$, the optimal values of \\eqref{obj:optim} will form an interval covering at least $1-\\alpha$ posterior content of $\\zeta(\\delta,\\tilde\\pi)$.\n\nInstead of looking for an exact $(1-\\alpha)$-content prediction set, we choose our $c$ based on asymptotic theory that guarantees an asymptotically exact coverage of the true value of $E[z(Y_j)]$, which in general can be different from the choice discussed above. Our result that justifies this approach has a similar spirit to some recent studies in calibrating uncertainty sets in distributionally robust optimization,  a setting in which the uncertainty is on the underlying distribution in a stochastic problem, via asymptotic analysis based on empirical likelihood (\\cite{lam2016recovering,duchi2016statistics,blanchet2016sample,lam2017empirical}) and Bayesian methods \\citep{gupta2015near}. Despite these connections, to our best knowledge, there has been no direct attempt in using robust optimization as a principled Bayesian computation tool.\n\nOur procedure essentially recovers the quantiles of the quantity of interest directly from the posterior distribution, which is the aforementioned goal of our Bayesian analysis and is conventionally obtained from sampling (e.g., Markov chain Monte Carlo). To intuitively explain the connection, consider the case when the posterior is normalized such that\n\\[\\int_{d,\\tilde p} \\text{post}(d,\\tilde p,\\text{data}) \\mathrm{d}  d \\mathrm{d}\\tilde p = 1.\\]\nThe described quantile is defined as\n\\begin{equation}\n\\min\\left\\{a\\in\\mathbb R: \\int_{\\zeta(d,\\tilde p)\\leq a}  \\text{post}(d,\\tilde p,\\text{data}) \\mathrm{d}  d \\mathrm{d}\\tilde p \\geq1-\\alpha\\right\\}\\label{quantile def}\n\\end{equation}\nAssume that for every $a$ in consideration, there exists $(d,\\tilde p)$ such that $\\zeta(d,\\tilde p)=a$. Then \\eqref{quantile def} is equal to\n\\begin{equation}\\begin{array}{ll}\n\\min_{a}&  \\left\\{\\begin{array}{ll}\\max_{d,\\tilde{p}}  & \\zeta(d,\\tilde{p}) \\\\\n\\text{subject to}&\\zeta(d,\\tilde{p}) \\leq a\\end{array}\\right\\}\\\\\n\\text{subject to}&  \\int_{\\zeta(d,\\tilde p)\\leq a}  \\text{post}(d,\\tilde p,\\text{data}) \\mathrm{d}  d \\mathrm{d}\\tilde p  \\geq1-\\alpha\n\\end{array} \\label{quantile reformulation}\n\\end{equation}\nDenote $\\mathcal U_a=\\left\\{d,\\tilde{p} \\left| \\zeta(d,\\tilde{p}) \\leq a \\right. \\right\\}$. We can further rewrite \\eqref{quantile reformulation} as an optimization over the collection of sets in the form $\\mathcal U_a$, given by\n\\begin{equation}\\begin{array}{ll}\n\\min_{\\mathcal U_a}&  \\left\\{\\begin{array}{ll}\\max_{d,\\tilde{p}}  & \\zeta(d,\\tilde{p}) \\\\\n\\text{subject to}&(d,\\tilde{p}) \\in\\mathcal U_a\\end{array}\\right\\}\\\\\n\\text{subject to}&  \\int_{\\mathcal U_a} \\text{post}(d,\\tilde p,\\text{data}) \\mathrm{d}  d \\mathrm{d}\\tilde p  \\geq1-\\alpha\n\\end{array} \\label{MC_sampler}\n\\end{equation}\nSuppose there exists an optimal solution $\\mathcal U^*$ to the outer optimization in \\eqref{MC_sampler}. We conclude that the $q$ quantile of $\\zeta(d,\\tilde p)$ under $\\text{post}(\\cdot,\\text{data})$ is equal to $\\max_{(d,\\tilde{p})\\in \\mathcal U^*} \\zeta(d,\\tilde{p}) $. Our chosen uncertainty set $\\mathcal U(c)$ turns out to bear a similar performance in bounding the quantity of interest as the set $\\mathcal U^*$, despite the potential vast difference in their geometries.\n\n\nTo illustrate graphically the difference between sampling quantiles and the optimization approach, suppose we are trying to find the $97.5\\%$ confidence level upper bounds for the sum of two probabilities in our system. Figure \\ref{fig:graphical_ROvSAMPLE} illustrates this with samples imposed on top of the projection of the uncertainty set in \\eqref{uncertainty set}, and it shows the similarity of the bounds provided by the two approaches.  Clearly, $\\mathcal{U}(c)$ is much smaller compared to $\\mathcal{U}^*$, yet the resulting bounds are quite similar. The next subsection investigates the properties of $\\mathcal U(c)$ and explains such a phenomenon.\n\\begin{figure}[htb]\n{\n\\centering\n\\includegraphics[width=5in]{graphical_ROvSAMPLE-eps-converted-to.pdf}\n\\caption{Graphical description of the differences between optimization- and sampling-based approaches where the objective is to bound the sum of the probabilities.  The $1000$ dots are samples from the posterior.  The upper limit of the $97.5 \\%$ quantile of the sum is indicated by the dashed line where $\\mathcal{U}^*$ is the solution to the outer optimization in (\\ref{MC_sampler}) given that these samples are the entirety of the posterior distribution.    The region labeled $\\mathcal{U}$ is the projection of $\\mathcal{U}$ onto this two-dimensional plane and the maximum of the optimization is determined by the solid line with $c=2$.   \\label{fig:graphical_ROvSAMPLE}}\n}\n\\end{figure}\n\\subsection{Theoretical Guarantees} \\label{sec:theo}\nWe first study the asymptotic behavior of the optimal values in \\eqref{obj:optim}.  We will consider a more general setting in which the objective function is $ \\sum_{j=1}^s\\sum_{i=1}^m z_j(i) p_j(i)$ for some functions $z_j:\\mathcal Y\\to\\mathbb R$, i.e., a linear combination of individual expectations at $x_j$. Evidently, $z_j(i)=0$ for all but one $j$ will reduce to the setting in \\eqref{obj:optim}. For ease of exposition, define\n\\[Z_j \\stackrel{\\text{dist}}{=}z_j(Y_j),\\]\nwhere $Y_j\\sim\\pi_j$. Let $\\mathcal{U}_n (c) $ be defined as in \\eqref{uncertainty set}, with the subscript $n=\\sum_{j=1}^sn_j$ indicating the total number of observed responses on the real system. Similarly, let $\\text{post}_n(d,\\tilde p)$ represent the posterior function when the data contains $n$ observations.\n\nWe have the following result (which is shown as Lemma \\ref{lem:op_consistency} in the appendix):\n\\begin{theorem} \\label{thm:op_consistency}\nSuppose that $\\tilde{\\pi}_j(i)>0$ and $\\pi_j(i)>0$ for all $i = 1,\\ldots,m$ and $j = 1,\\ldots,s$.  For each observation, the design point is an independent random variable with sample space $\\{x_1,\\ldots,x_m\\}$ and respective positive probabilities $\\xi_1,\\ldots, \\xi_m$.\n\nLet $\\text{post}_n^* = \\max_{d,\\tilde{p}} \\text{post}_n (d,\\tilde{p})$ and $\\hat{\\pi}_i^n (i) = n_j(i)\/n_j .$  Then for all $\\ell >0$,\n\\begin{equation}\n\\lim_{n \\rightarrow \\infty} \\sqrt{n}  \\left(\\max_{(d,\\tilde p)\\in\\mathcal{U}_n(\\text{post}_n^*-\\ell^2\/2)} \\sum_{j=1}^s\\sum_{i=1}^m z_j(i) d_j(i)\\tilde p_j(i) - \\sum_{j=1}^s \\sum_{i=1}^m z_j(i) \\hat{\\pi}_j^n(i)    \\right)  =    \\ell \\sqrt{\\sum_{j=1}^s \\xi_j^{-1} \\mathbb{V} Z_j} \\label{eq:optim1}\n\\end{equation}\nand\n\\begin{equation}\n\\lim_{n \\rightarrow \\infty} \\sqrt{n} \\left( \\min_{(d,\\tilde p)\\in\\mathcal{U}_n(\\text{post}_n^*-\\ell^2\/2)} \\sum_{j=1}^s\\sum_{i=1}^m z_j(i)d_j(i)\\tilde p_j(i) - \\sum_{j=1}^s \\sum_{i=1}^m z_j(i) \\hat{\\pi}_j^n(i)    \\right)  =    -\\ell \\sqrt{\\sum_{j=1}^s \\xi_j^{-1} \\mathbb{V} Z_j} \\label{eq:optim2}\n\\end{equation}\n\nalmost surely, where $\\mathbb{V} $ represents the variance.\n\\end{theorem}\n\nAn immediate observation of Theorem \\ref{thm:op_consistency} is that the simulation replication size $\\tilde n_j$ plays no role in the asymptotic behavior of the optimization output as $n$ gets large. Thus, with enough real data, the accuracy of the simulation runs is inconsequential, as the values of the real data dominate the results.\nThe same observation also holds for the prior choices made for  $f(\\cdot)$ and $g(\\cdot)$. This asymptotic independence of the prior resembles the classic Bernstein-von Mises theorem. In summary, our optimization approach generates bounds in tight asymptotic agreement with those obtained from the typical data-only inference approaches.\n\nIt is known that not every posterior distribution is guaranteed to have appropriate consistency properties; see the works of \\cite{freedman1963asymptotic} and \\cite{diaconis1986consistency}.    Bayesian credible sets resembling the form of \\eqref{uncertainty set} are not guaranteed to produce rational inference; for more information on the general properties of Bayesian credible sets, see   \\cite{cox1993analysis} or \\cite{szabo2015frequentist}.  In particular, two complications arise in proving Theorem \\ref{thm:op_consistency}.  First, the measure associated with the likelihood function only concentrates on a lower dimensional manifold (dimension $sm-s$) of the parameter space (dimension $sm$). This issue is by-and-large a technical one and is addressed in Lemmas \\ref{lemma:str_consis} and \\ref{lemma:weak_consis} proved in the appendix.    Second, the optimization problem requires a particular shape of the uncertainty set to yield the desired asymptotic properties. As a main observation, the uncertainty set $\\mathcal U_n(\\text{post}_n^*-\\ell^2\/2)$ can be shown to asymptotically become an ellipsoid, and optimization problem \\eqref{obj:optim} therefore reduces to a quadratic program with an elliptical constraint, which can be analyzed and elicits the convergence behavior in Theorem \\ref{thm:op_consistency}.\n\n\nThere are several implications of Theorem \\ref{thm:op_consistency}.  The first is that both the upper and lower limits provided by the optimization converge to the true value almost surely as the data gets large, as described below:\n\\begin{corollary}\nUnder the same assumptions in Theorem \\ref{thm:op_consistency}, we for all $\\ell>0$,\n$$\\max_{(d,\\tilde p)\\in\\mathcal{U}_n(\\text{post}_n^*-\\ell)} \\sum_{j=1}^s\\sum_{i=1}^m z_j(i)d_j(i)\\tilde p_j(i)\\to\\sum_{j=1}^s\\sum_{i=1}^m z_j(i)\\pi_j(i)$$\nand\n$$\\min_{(d,\\tilde p)\\in\\mathcal{U}_n(\\text{post}_n^*-\\ell)} \\sum_{j=1}^s\\sum_{i=1}^m z_j(i)d_j(i)\\tilde p_j(i)\\to\\sum_{j=1}^s\\sum_{i=1}^m z_j(i)\\pi_j(i)$$\nalmost surely as $n\\to\\infty$.\\label{consistency}\n\\end{corollary}\n\nCorollary \\ref{consistency} shows that  with enough data the proposed posterior estimate is a good representation of the truth. It is a basic property that is in line with Bayesian consistency results studied traditionally by statisticians \\citep{schwartz1965bayes}.\n\nFurthermore, Theorem \\ref{thm:op_consistency} also implies that, as $n$ gets large,\n\\begin{equation}\n\\max_{(d,\\tilde p)\\in\\mathcal{U}_n(\\text{post}_n^*-\\ell^2\/2)} \\sum_{j=1}^s\\sum_{i=1}^m z_j(i)d_j(i)\\tilde p_j(i) \\approx\\sum_{j=1}^s \\sum_{i=1}^m z_j(i) \\hat{\\pi}_j^n(i)    + \\ell \\sqrt{\\frac{\\sum_{j=1}^s \\xi_j^{-1} \\mathbb{V} Z_j}{n}} \\label{asymptotic max}\n\\end{equation}\nand\n\\begin{equation}\\min_{(d,\\tilde p)\\in\\mathcal{U}_n(\\text{post}_n^*-\\ell^2\/2)} \\sum_{j=1}^s\\sum_{i=1}^m z_j(i)d_j(i)\\tilde p_j(i) \\approx\\sum_{j=1}^s \\sum_{i=1}^m z_j(i) \\hat{\\pi}_j^n(i)    - \\ell \\sqrt{\\frac{\\sum_{j=1}^s \\xi_j^{-1} \\mathbb{V} Z_j}{n}} \\label{asymptotic min}\n\\end{equation}\nNote that the left hand sides of \\eqref{asymptotic max} and \\eqref{asymptotic min} are precisely the classical confidence bounds on $\\sum_{j=1}^s\\sum_{i=1}^m z_j(i) \\pi_j(i)$ generated from the central limit theorem with $n_j \\approx \\xi_j n$. This hints at a proper coverage in large samples at the level $1-\\alpha$. In fact, we have the following result:\n\\begin{corollary}\nUnder the same assumptions in Theorem \\ref{thm:op_consistency}, we have\n$$\\mathbb P\\left(\\max_{(d,\\tilde p)\\in\\mathcal{U}_n(\\text{post}_n^*-\\ell^2\/2)} \\sum_{j=1}^s\\sum_{i=1}^m z_j(i)d_j(i)\\tilde p_j(i)\\geq\\sum_{j=1}^s\\sum_{i=1}^m z_j(i)\\pi_j(i) \\right)\\to \\Phi(\\ell)$$\nand\n$$\\mathbb P\\left(\\min_{(d,\\tilde p)\\in\\mathcal{U}_n(\\text{post}_n^*-\\ell^2\/2)} \\sum_{j=1}^s\\sum_{i=1}^m z_j(i)d_j(i)\\tilde p_j(i)\\leq\\sum_{j=1}^s\\sum_{i=1}^m z_j(i)\\pi_j(i)\\right)\\to \\Phi(\\ell)$$\nas $n\\to\\infty$, where $\\mathbb P$ denotes the probability generated from a data set of size $n$.\\label{CI}\n\\end{corollary}\n\n\n\nThe above results reveal that the proposed inference differs from purely empirical estimates only when data is sparsely collected.   If data from the real system is abundant, our simulation models $\\tilde{\\pi}_1(\\cdot),\\ldots,\\tilde{\\pi}_s(\\cdot)$ will have very little impact on our resulting conclusions.  In a sense, the Bayesian approach automatically balances the influences from the empirical data versus the simulation model. Complement to our asymptotic result in this section, our numerical examples in Section \\ref{sec:illustration} will demonstrate that the difference in inference between our approach and one that ignores the simulation model can be sizable in sparse data environments.\n\n\nWe conclude this section by presenting a result on the consistency of a ``ranking and selection\" task:\n\\begin{corollary} \\label{corr:op_consistency}\nSuppose the conditions and definitions of Theorem \\ref{thm:op_consistency}.  For all $\\ell>0$, if $j$ and $k$ are such that  $\\mathbb{E} Z_j >  \\mathbb{E} Z_k $, then\n\\[\\lim_{n \\rightarrow \\infty} \\mathbb P \\left( \\min_{(\\tilde{p},d) \\in \\mathcal{U}(\\text{post}_n^*-\\ell)} \\sum_{i=1}^m z_j(i)d_j(i)\\tilde p_j(i) < \\max_{(\\tilde{p},d) \\in\\mathcal{U}(\\text{post}_n^*-\\ell)} \\sum_{i=1}^m z_k(i)d_k(i)\\tilde p_k(i) \\right)  = 0 .\\]\\label{rs}\n\\end{corollary}\nCorollary \\ref{rs} implies that the intervals for the quantities of interest at different design points do not overlap as the data gets large, if their values are truly different.  Thus, in practice, a user who notes that the two intervals generated from the optimization problems do not overlap can reasonably conclude there is a difference between the two values.\n\n\\subsection{Solvability of the optimization} \\label{sec:optim}\nThis subsection discusses the tractability of the imposed optimization problems in Section \\ref{sec:formulation}. We focus on the convexity of the problems which, in contrast to the previous section,  will  depend on the replication size from the simulation model.\n\n\n  To begin, we write optimization (\\ref{obj:optim}) in full (focusing only on the minimization problem) as\n\\begin{equation}\\begin{array}{ll}\n\\min_{d,\\tilde{p}} & \\zeta(d,\\tilde{p}) =  \\sum_{i=1}^m z_j(i) d_j(i) \\tilde{p}_j(i), \\\\\n\\text{subject to}& f(\\tilde{p}) + g(d)  + \\sum_{j=1}^{s} \\sum_{i=1}^{m} n_{j} (i) \\log( d_j(i) \\tilde{p}_j(i)) + \\sum_{j=1}^{s} \\sum_{i=1}^{m} \\tilde{n}_{j}(i) \\log \\tilde{p}_j(i) \\geq \\log(c)\\\\\n&\\tilde{p}_j(i) \\geq 0 , d_j(i) \\geq 0, \\text{ for all } i,j \\\\\n&\\sum_{i=1}^m \\tilde{p}_j(i) =1, \\sum_{i=1}^m \\tilde{p}_j(i) \\delta_j(i) =1, \\text{ for all } j\n\\end{array} \\label{obj:optim1}\n\\end{equation}\nThis formulation is generally non-convex because of the non-convex objective function and the non-convex constraint $\\sum_{i=1}^m \\tilde{p}_j(i) \\delta_j(i) =1$, regardless of the sample sizes and the priors $f(\\cdot)$ and $g(\\cdot)$. However, noting that the program is individually convex in $d$ and $\\tilde p$, one approach is to use alternating minimization, by sequentially optimizing $d$ fixing $\\tilde p$ and $\\tilde p$ fixing $d$ until no improvement is detected. Though it does not guarantee a global solution, this approach has been shown to be effective for certain chance-constrained programs (see, e.g., \\cite{chen2010cvar,zymler2013distributionally,jiang2016data}).\n\nOn the other hand, supposing that there is no simulation error in estimating $\\tilde\\pi$, then the prior  on $\\tilde p$ and the associated calculations can be removed, resulting in\n\\begin{equation}\\begin{array}{ll}\n\\max _{d} & \\sum_{i=1}^m z(i)\\tilde{\\pi}_j(i)d_j(i)   , \\\\\n\\text{subject to}& g(d)  + \\sum_{j=1}^{s} \\sum_{i=1}^{m} n_{j}(i) \\log d_{j}(i) \\geq \\log(c) \\\\\n&d_j(i) \\geq 0, \\text{ for all } i,j \\\\\n &\\sum_{i=1}^m \\tilde{\\pi}_j(i) d_j(i) =1, \\text{ for all } j\n\\end{array}  \\label{optimization no MC}\n\\end{equation}\nIf in addition the function $g(\\cdot)$ is a concave function, then \\eqref{optimization no MC} is a convex optimization problem. We summarize this as:\n\\begin{proposition}\nProblem (\\ref{optimization no MC}) is a convex program if $g(\\cdot)$ is a concave function on $\\mathbb{R}^{sm}$.\n\\end{proposition}\n\nRecalling our discussion in Section \\ref{sec:learn}, one example of a concave $g(\\cdot)$ corresponds to the multivariate Gaussian prior of (\\ref{eq:Gaussian}).\n\nFormulation \\eqref{optimization no MC} can be reasonably used in situations where simulation replications are abundant, so that the simulation outputs are very close to $\\tilde\\pi_j$. Our next result shows that, in the case that $\\tilde n_j$ is sufficiently large and $g(\\cdot)$ satisfies a slightly stronger condition, using \\eqref{obj:optim1} also leads to a convex problem. To prepare for this result, we rewrite the decision variables in \\eqref{obj:optim1} to get\n\\begin{equation}\\begin{array}{ll}\n\\max _{p,\\tilde{p}} & \\sum_{i=1}^j z(i) p_j(i), \\\\\n\\text{subject to}& f(\\tilde{p}) + h(p,\\tilde{p})  + \\sum_{j=1}^{s} \\sum_{i=1}^{m} n_{j} (i) \\log p_j(i) + \\sum_{j=1}^{s} \\sum_{i=1}^{m} \\tilde{n}_{j}(i) \\log \\tilde{p}_j(i) \\geq \\log(c)\\\\\n&\\tilde{p}_j(i) \\geq 0 , p_j(i) \\geq 0, \\text{ for all } i,j \\\\\n&\\sum_{i=1}^m \\tilde{p}_j(i) =1, \\sum_{i=1}^m p_j(i) =1, \\text{ for all } j\n\\end{array} \\label{optimization reformulation}\n\\end{equation}\nwhere $h(p,\\tilde{p}) = g(p_j\/\\tilde{p}_j)$ with the operation $p_j\/\\tilde p_j$ defined component-wise. We recall the definition that a function $r(\\cdot)$ is strongly concave if for all $a$ and $b$ and $0 \\leq \\lambda \\leq 1$,\n\\[r(\\lambda a+ (1-\\lambda) b ) \\geq \\lambda r(a)+ (1-\\lambda) r(b) + \\beta \\lambda (1-\\lambda) \\|a-b\\|^2,  \\]\nwhere $\\|a-b\\|$ is the Euclidean norm and  $\\beta$ is some positive constant \\cite[pp 60] {nesterov2003introductory}. Our result is:\n\n\\begin{theorem} \\label{thm:convex}\nAssume that $g(\\cdot)$ is strongly concave and differentiable on $\\mathbb R_+^{sm}$ and the derivative is bounded on all compact sets in $\\mathbb R_+^{sm}$, $f(\\cdot)$ is bounded from above, and $\\tilde\\pi_j(i)>0$ for all $i,j$.\n\nLet $\\mathcal{U}_{\\tilde{n}} (c_{\\tilde{n}})$ be the set of feasible solutions for (\\ref{optimization reformulation}) where\n\\[\\log c_{\\tilde{n}} = -\\frac{\\ell^2}{2} + \\max_{p,\\tilde{p}}\\left\\{  f(\\tilde{p}) + h(p,\\tilde{p}) + \\sum_{j=1}^{s} \\sum_{i=1}^{m} n_j(i) \\log p_j(i) + \\sum_{j=1}^{s} \\sum_{i=1}^{m} \\tilde{n}_{j}(i) \\log \\tilde{p}_j(i)\\right\\},\\]\nfor some constant $\\ell > 0$.  Then as $\\tilde{n} \\rightarrow \\infty$, $\\mathbb{P} \\left( \\mathcal{U}_{\\tilde{n}} (c_{\\tilde{n}}) \\text{ is convex}\\right) \\rightarrow 1. $\n\\end{theorem}\n\nThus, given access to sufficient computing resources and properly choosing $g(\\cdot)$, one can use a convex optimization solver to carry out our proposed approach, no matter how few or many data were collected from the real system. Note that this observation holds even when $f(\\cdot)$ is not concave. Theorem \\ref{thm:convex} hinges on a joint convexity argument with respect to $(d,\\tilde{p})$ in the asymptotic regime as $\\tilde n$ grows but $n$ is fixed.\n\n\\section{Numerical Illustrations } \\label{sec:illustration}\nWe demonstrate our approach with two real-data examples. First is a proof-of-concept investigation in modeling a call center. Second is on the support of staffing decision in a  manufacturing production line discussed in the introduction of this article.\n\n\\subsection*{Call center example}\\label{sec:call center}\nConsider the call center data originally analyzed in \\cite{brown2005statistical}.  This dataset is associated with a call center where a customer calls in and is placed a queue until one of $x$ servers is available. From these data, the sample mean of the waiting time (from entry to service for a customer) from 9:00 to 10:00 am is calculated.   In this narrow time period, the arrival rate and service rate, which is time inhomogenous according to \\cite{brown2005statistical}, should be approximately homogenous.  Here, we also account for the number of servers operating in the system at any given time, which appears to differ between days (see the appendix for details). To our reading,  this subset of the dataset was by-and-large ignored in \\cite{brown2005statistical}'s original analysis.\n\nOur model for this call center will be an $x$-server first-come-first-serve queue.  Following practice, both the interarrival and service times are modeled as exponentially distributed.  After a warm-up period, the sample average of the waiting time is measured over the course of a one-hour window.    This, in principle, agrees with \\cite{brown2005statistical}.    In the spirit of ad-hoc calibration, two additional features were added:  (i) the arrival rate is randomly generated each day from a log-normal distribution with associated mean $1.8$ and variance $0.4$, and (ii) a customer will abandon the queue if the waiting time is longer than an exponential random variable with mean $5$.  Adding both of these features resulted in a simulation model that was closer to the observed data.\n\nThe response is discretized into the four categories $<1$, $1-2$, $2-3$, and $> 3$ minutes ($m=4$) and we study $5-9$ servers ($s = 5$).    No data from the real system is observed at either $5$ or $9$ servers. The simulation model was evaluated 250 times at each design point.\n\\begin{figure}[htb]\n{\n\\centering\n\\includegraphics[width=\\textwidth]{orig_data_calibration-eps-converted-to.pdf}\n\\caption{The predictive intervals for $\\pi$ used for the call center study described in Section \\ref{sec:illustration}.  The subplots from left to right show the results with 5 to 9 servers, respectively.  The solid line is the observed frequency from the data when it is available at $6-8$ server levels.  The dashed line is the observed frequency from the simulation model.  The rectangles represent intervals predictions from either using the typical intervals from traditional sampling (left rectangles) or the optimization (right rectangles). } \\label{fig:orig_data_calibration}\n}\n\\end{figure}\n\nFigure \\ref{fig:orig_data_calibration} shows the intervals implied by the proposed posterior distribution using a sampling-based approach and our optimization approach using \\eqref{obj:optim} and \\eqref{choice}.  The functions $f$ and $g$ were of Gaussian form (\\ref{eq:Gaussian}), with correlation $0.75^{|x_i-x_j| } \\cdot 0.75^{|k-l| }$  between the $i$th and $j$th staffing levels and the $k$th and $l$th outputs with $\\lambda_d = 1\/4$ and $\\lambda_p = 1\/100$.  The key is the ability to answer question such as: how likely is it that the average waiting time when there are $9$ servers is between $2$ and $3$ minutes?  There is no data, but the simulation model combined with the observed responses and our prior information gives us an estimate of somewhere less than $15 \\%$.  This accounts for the discrepancy that we observed based on the recorded responses at $6-8$ servers as well as the potential Monte Carlo error from running a finite number of simulations.  Overall, the ranges at other staffing levels appear to agree with both the data and the simulation outputs.    We are not confident, for example, that staffing $5$ servers will produce the same results as the simulation model, which has average waiting times over $3$ minutes about $15 \\%$ of the time.  Based on the recorded responses, this could be  $30 \\%$, but it could also be as low as about $1\\%$.\n\nThe above discussion offers some preliminary validity check on the practical implementation. Next we illustrate the theoretical discussions in Section \\ref{sec:theo}. For this purpose, consider an example where the true model is specified by us, some data is generated from this model, and an inexact simulation model is specified.\n\nWe use the same simulation model. In the dataset we found that waiting times are under-estimated by the simulation model when many servers are present.  To replicate this, we add onto our ``true\" model an event (according to a Poisson process, average 5 min between events) in which if there are $5$ idle servers, all idle servers will take a break (average 30 minutes, exponentially distributed) and if there are more than $7$ idle servers, these additional severs will stop servicing for the remainder of the hour.  This will naturally inflate the waiting times.  While not exactly mimicking the real system, this reflects the general phenomena that all operators may not be working at all times in a call center.  Thus even though $8$ servers may be ``working'', because of miscellaneous personnel reasons, the queue behaves differently than the simulation model.  All other features of the true model are exactly the same as the simulation model, including the arrival rates and departure rates.\n\nIn this numerical experiment there are either $5$, $10$, $20$, $200$ or $2000$ total observations.  Two observation schemes are examined:  in the first, each observation comes from one of the staffing levels 6, 7 and 8 with equal probabilities of $1\/3$;  in the second, each observation comes from one of the staffing levels 5, 6, 7, 8 and 9 with equal probabilities of $1\/5$.\n\n Figure \\ref{fig:ROvSAMPLE_p} shows the prediction of the probability the average waiting time will be less than $1$ minute.   We compare to a data-only approach which consists of bounds based on the classic confidence interval with binomial responses (either less than one minute or not).  All approaches behave similarly when the amount of data is large, agreeing with Theorem \\ref{thm:op_consistency}.   But there are differences in the data-poor performances.  Consider the first observation scheme, where no data is collected at $5$ and $9$ servers.  The proposed approach correctly predicts the chance of a short average waiting time with $9$ servers to be large, while the data-only approach does not have access to the simulation model and thus predicts the chance of a short average waiting time with $9$ servers to be possibly small (the prediction covers all possibilities).   Moreover, the data-only approach can be quite poor when only a few data points exist. The conclusions reached from using the posterior with either traditional sampling or our optimization approach are comparable in the large data cases, but do differ in the small data cases.  The computational speed of the optimization approach was orders of magnitude smaller for this example compared to the sampling.\n\\begin{figure}[htb]\n{\n\\centering\n\\includegraphics[width=\\textwidth]{ROvSAMPLE_p-eps-converted-to.pdf}\n\\caption{The bounds produced in the call center example in Section \\ref{sec:illustration}.   The rectangles in the six panels are decided by the data-only approach (left), the sampling-based calibration approach with $2.5\\%$ and $97.5 \\%$ quantiles (middle), and  the proposed optimization-based calibration approach.   The set of $5$ rectangles for each number of servers  represent  $5$, $10$, $20$, $200$ and $2000$ observations, respectively.  The long horizontal line represents the true value.    The top set of panels refers to the case where data is observed only at  6, 7, and 8 and the bottom set of panels refers to the case where data is observed at 5, 6, 7, 8 and 9.  \\label{fig:ROvSAMPLE_p}}\n}\n\\end{figure}\n\n\n\\subsection*{Manufacturing line example}\nThis subsection uses our calibration framework to assist a decision process for staffing a real production line.   A major manufacturer of automobiles has two parallel production lines, labeled box and closure, that suffer from frequent failures.   These failures are predominately handled by a group of workers trained to quickly identify and resolve small issues.  Due to the time needed to traverse the line combined with the relative frequency of failures, four workers are currently staffed in this support position.   The manufacturer is interested in the impact of this staffing level on the throughput of the line, measured in units per hour.    The lines' behavior are classified into thirteen categories from 46 to 74 in $2$ units per hour increments.\n\nThe two lines have different criteria for ill-performance.   The box line will starve the next line if the throughput drops below 60.  The closure line will starve the next line if the throughput drops below 56.  The goal is thus to ensure that the chance of starving the next line remains near the current level when there are $4$ workers.  Since experiments on the real system would be extremely costly and potentially dangerous, an outside company was hired to design a discrete-event simulation model to investigate potential staffing reconfigurations for this group of workers.  Additionally, a two-person internal team was tasked with refining and adjusting the simulation model via ad-hoc calibration, including detailed input analysis that broke down failure rates by stations along the line.  Despite these extensive and costly efforts, the simulation model did not perfectly agree with the data collected in the current four worker configuration (see Figure \\ref{fig:illustration}) due to several assumptions made along the model development process. These included typical input assumptions like independent and exponentially distributed inter-failure times as well as more complicated structural assumptions such as workers returning to their station in between maintenance calls.  Roughly $75$ realizations from the simulation were completed at each design point, as decided was sufficient toward the end of the project.\n\n\\begin{figure}[t]\n{\n\\centering\n\\includegraphics[width=6.5in]{manufacturing_example-eps-converted-to.pdf}\n\\caption{Illustration for the manufacturing case study presented in Section \\ref{sec:illustration}.  Subplots far left and middle right are for the box and closure lines comparing the model histogram and the frequency histogram from the data.  Subplots middle left and far right show the predictive intervals from the case study presented in Section \\ref{sec:illustration} for the box and closure lines, respectively.  \\label{fig:illustration}}\n}\n\\end{figure}\n\nKnowing this simulation model is not perfect, what would be a reasonable estimate for the mean throughput of the line at each staffing level from $1$ worker to $6$ workers?  If we can define the priors $f$ and $g$, then this becomes an answerable question using the method described in this article.  Like the previous example, the functions $f$ and $g$ were of Gaussian form, of (\\ref{eq:Gaussian}), with correlation $0.75^{|x_i-x_j| } \\cdot 0.9^{|k-l| }$  between the $i$th and $j$th staffing levels and the $k$th and $l$th possible throughputs with $\\lambda_d = 1\/4$ and $\\lambda_p = 1\/100$.     This agreed, as best as possible, with the expectations of the builders of the simulation model, who think that there is a large correlation across outputs (i.e. a similar likelihood ratio at places close in the sample space) and smaller amounts of correlation across the inputs (i.e. the builders are unsure of the behavior of likelihood ratio across the input variables, but generally anticipate it is close for similar staffing levels).\n\nFigure \\ref{fig:illustration} displays the lower and upper bounds on the probabilities of low production for each line constructed from our method.  As we move away from our observations at a staffing level of $4$, the predictive bounds on the mean throughput get larger and become closer to the simulation model.  This expansion of predictive intervals and  the regression to  the simulation model mimic what is seen in calibration of deterministic models \\citep{kennedy2001bayesian} and stochastic kriging \\citep{ankenman2010stochastic}.   Around $4$ workers, the assumption is that that there is some correlation between staffing levels that decays as we move away from a staffing level of $4$, thus expanding our predictive intervals.\n\n In terms of comparison to a data-only alternative, there is clearly no ability to distinguish between different staffing levels using data alone.  In terms of an answer to the fundamental question posed by the manufacturer, a few things can be gleaned from these bounds.  For example, it becomes clear from this analysis that staffing a single worker would in high likelihood starve the next lines, which is the core problem the manufacturer would like to avoid.  The ultimate decision from the manufacturer was to do a field study of the three worker staffing level.  This was based on both feasibility assurance provided by the simulation model and the potential benefit of redeploying a worker into a different position.\n\n\n\\section*{Acknowledgements}\nWe thank Ilan Guedj for the data organization and for Avi Mandelbaum for continuing to place the data on the website \\href{http:\/\/ie.technion.ac.il\/serveng\/}{http:\/\/ie.technion.ac.il\/serveng\/}. Additional thanks are due to the\nTauber Institute for Global Operations  at the University of Michigan, Anthony Sciuto, Anusuya Ramdass and Brian Talbot. We also gratefully acknowledge support from the the National Science Foundation under grants CMMI-1542020, CMMI-1523453 and CAREER CMMI-1653339.\n\n\n\n\\bibliographystyle{informs2014}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Submission of conference papers to ICLR 2021}\n\nICLR requires electronic submissions, processed by\n\\url{https:\/\/openreview.net\/}. See ICLR's website for more instructions.\n\nIf your paper is ultimately accepted, the statement {{ {\\bm{\\theta}}_{t} }\n  {\\textbackslash}iclrfinalcopy} should be inserted to adjust the\nformat to the camera ready requirements.\n\nThe format for the submissions is a variant of the NeurIPS format.\nPlease read carefully the instructions below, and follow them\nfaithfully.\n\n\\subsection{Style}\n\nPapers to be submitted to ICLR 2021 must be prepared according to the\ninstructions presented here.\n\n\nAuthors are required to use the ICLR \\LaTeX{} style files obtainable at the\nICLR website. Please make sure you use the current files and\nnot previous versions. Tweaking the style files may be grounds for rejection.\n\n\\subsection{Retrieval of style files}\n\nThe style files for ICLR and other conference information are available online at:\n\\begin{center}\n   \\url{http:\/\/www.iclr.cc\/}\n\\end{center}\nThe file \\verb+iclr2021_conference.pdf+ contains these\ninstructions and illustrates the\nvarious formatting requirements your ICLR paper must satisfy.\nSubmissions must be made using \\LaTeX{} and the style files\n\\verb+iclr2021_conference.sty+ and \\verb+iclr2021_conference.bst+ (to be used with \\LaTeX{}2e). The file\n\\verb+iclr2021_conference.tex+ may be used as a ``shell'' for writing your paper. All you\nhave to do is replace the author, title, abstract, and text of the paper with\nyour own.\n\nThe formatting instructions contained in these style files are summarized in\nsections \\ref{gen_inst}, \\ref{headings}, and \\ref{others} below.\n\n\\section{General formatting instructions}\n\\label{gen_inst}\n\nThe text must be confined within a rectangle 5.5~inches (33~picas) wide and\n9~inches (54~picas) long. The left margin is 1.5~inch (9~picas).\nUse 10~point type with a vertical spacing of 11~points. Times New Roman is the\npreferred typeface throughout. Paragraphs are separated by 1\/2~line space,\nwith no indentation.\n\nPaper title is 17~point, in small caps and left-aligned.\nAll pages should start at 1~inch (6~picas) from the top of the page.\n\nAuthors' names are\nset in boldface, and each name is placed above its corresponding\naddress. The lead author's name is to be listed first, and\nthe co-authors' names are set to follow. Authors sharing the\nsame address can be on the same line.\n\nPlease pay special attention to the instructions in section \\ref{others}\nregarding figures, tables, acknowledgments, and references.\n\n\nThere will be a strict upper limit of 8 pages for the main text of the initial submission, with unlimited additional pages for citations. Note that the upper page limit differs from last year!Authors may use as many pages of appendices (after the bibliography) as they wish, but reviewers are not required to read these. During the rebuttal phase and for the camera ready version, authors are allowed one additional page for the main text, for a strict upper limit of 9 pages.\n\n\\section{Headings: first level}\n\\label{headings}\n\nFirst level headings are in small caps,\nflush left and in point size 12. One line space before the first level\nheading and 1\/2~line space after the first level heading.\n\n\\subsection{Headings: second level}\n\nSecond level headings are in small caps,\nflush left and in point size 10. One line space before the second level\nheading and 1\/2~line space after the second level heading.\n\n\\subsubsection{Headings: third level}\n\nThird level headings are in small caps,\nflush left and in point size 10. One line space before the third level\nheading and 1\/2~line space after the third level heading.\n\n\\section{Citations, figures, tables, references}\n\\label{others}\n\nThese instructions apply to everyone, regardless of the formatter being used.\n\n\\subsection{Citations within the text}\n\nCitations within the text should be based on the \\texttt{natbib} package\nand include the authors' last names and year (with the ``et~al.'' construct\nfor more than two authors). When the authors or the publication are\nincluded in the sentence, the citation should not be in parenthesis using \\verb|\\citet{}| (as\nin ``See \\citet{Hinton06} for more information.''). Otherwise, the citation\nshould be in parenthesis using \\verb|\\citep{}| (as in ``Deep learning shows promise to make progress\ntowards AI~\\citep{Bengio+chapter2007}.'').\n\nThe corresponding references are to be listed in alphabetical order of\nauthors, in the \\textsc{References} section. As to the format of the\nreferences themselves, any style is acceptable as long as it is used\nconsistently.\n\n\\subsection{Footnotes}\n\nIndicate footnotes with a number\\footnote{Sample of the first footnote} in the\ntext. Place the footnotes at the bottom of the page on which they appear.\nPrecede the footnote with a horizontal rule of 2~inches\n(12~picas).\\footnote{Sample of the second footnote}\n\n\\subsection{Figures}\n\nAll artwork must be neat, clean, and legible. Lines should be dark\nenough for purposes of reproduction; art work should not be\nhand-drawn. The figure number and caption always appear after the\nfigure. Place one line space before the figure caption, and one line\nspace after the figure. The figure caption is lower case (except for\nfirst word and proper nouns); figures are numbered consecutively.\n\nMake sure the figure caption does not get separated from the figure.\nLeave sufficient space to avoid splitting the figure and figure caption.\n\nYou may use color figures.\nHowever, it is best for the\nfigure captions and the paper body to make sense if the paper is printed\neither in black\/white or in color.\n\\begin{figure}[h]\n\\begin{center}\n\\fbox{\\rule[-.5cm]{0cm}{4cm} \\rule[-.5cm]{4cm}{0cm}}\n\\end{center}\n\\caption{Sample figure caption.}\n\\end{figure}\n\n\\subsection{Tables}\n\nAll tables must be centered, neat, clean and legible. Do not use hand-drawn\ntables. The table number and title always appear before the table. See\nTable~\\ref{sample-table}.\n\nPlace one line space before the table title, one line space after the table\ntitle, and one line space after the table. The table title must be lower case\n(except for first word and proper nouns); tables are numbered consecutively.\n\n\\begin{table}[t]\n\\caption{Sample table title}\n\\label{sample-table}\n\\begin{center}\n\\begin{tabular}{ll}\n\\multicolumn{1}{c}{\\bf PART}  &\\multicolumn{1}{c}{\\bf DESCRIPTION}\n\\\\ \\hline \\\\\nDendrite         &Input terminal \\\\\nAxon             &Output terminal \\\\\nSoma             &Cell body (contains cell nucleus) \\\\\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\section{Default Notation}\n\nIn an attempt to encourage standardized notation, we have included the\nnotation file from the textbook, \\textit{Deep Learning}\n\\cite{goodfellow2016deep} available at\n\\url{https:\/\/github.com\/goodfeli\/dlbook_notation\/}.  Use of this style\nis not required and can be disabled by commenting out\n\\texttt{math\\_commands.tex}.\n\n\n\\centerline{\\bf Numbers and Arrays}\n\\bgroup\n\\def1.5{1.5}\n\\begin{tabular}{p{1in}p{3.25in}}\n$\\displaystyle a$ & A scalar (integer or real)\\\\\n$\\displaystyle {\\bm{a}}$ & A vector\\\\\n$\\displaystyle {\\bm{A}}$ & A matrix\\\\\n$\\displaystyle {\\tens{A}}$ & A tensor\\\\\n$\\displaystyle {\\bm{I}}_n$ & Identity matrix with $n$ rows and $n$ columns\\\\\n$\\displaystyle {\\bm{I}}$ & Identity matrix with dimensionality implied by context\\\\\n$\\displaystyle {\\bm{e}}^{(i)}$ & Standard basis vector $[0,\\dots,0,1,0,\\dots,0]$ with a 1 at position $i$\\\\\n$\\displaystyle \\text{diag}({\\bm{a}})$ & A square, diagonal matrix with diagonal entries given by ${\\bm{a}}$\\\\\n$\\displaystyle {\\textnormal{a}}$ & A scalar random variable\\\\\n$\\displaystyle {\\mathbf{a}}$ & A vector-valued random variable\\\\\n$\\displaystyle {\\mathbf{A}}$ & A matrix-valued random variable\\\\\n\\end{tabular}\n\\egroup\n\\vspace{0.25cm}\n\n\\centerline{\\bf Sets and Graphs}\n\\bgroup\n\\def1.5{1.5}\n\n\\begin{tabular}{p{1.25in}p{3.25in}}\n$\\displaystyle {\\mathbb{A}}$ & A set\\\\\n$\\displaystyle \\mathbb{R}$ & The set of real numbers \\\\\n$\\displaystyle \\{0, 1\\}$ & The set containing 0 and 1 \\\\\n$\\displaystyle \\{0, 1, \\dots, n \\}$ & The set of all integers between $0$ and $n$\\\\\n$\\displaystyle [a, b]$ & The real interval including $a$ and $b$\\\\\n$\\displaystyle (a, b]$ & The real interval excluding $a$ but including $b$\\\\\n$\\displaystyle {\\mathbb{A}} \\backslash {\\mathbb{B}}$ & Set subtraction, i.e., the set containing the elements of ${\\mathbb{A}}$ that are not in ${\\mathbb{B}}$\\\\\n$\\displaystyle {\\mathcal{G}}$ & A graph\\\\\n$\\displaystyle \\parents_{\\mathcal{G}}({\\textnormal{x}}_i)$ & The parents of ${\\textnormal{x}}_i$ in ${\\mathcal{G}}$\n\\end{tabular}\n\\vspace{0.25cm}\n\n\n\\centerline{\\bf Indexing}\n\\bgroup\n\\def1.5{1.5}\n\n\\begin{tabular}{p{1.25in}p{3.25in}}\n$\\displaystyle {a}_i$ & Element $i$ of vector ${\\bm{a}}$, with indexing starting at 1 \\\\\n$\\displaystyle {a}_{-i}$ & All elements of vector ${\\bm{a}}$ except for element $i$ \\\\\n$\\displaystyle {A}_{i,j}$ & Element $i, j$ of matrix ${\\bm{A}}$ \\\\\n$\\displaystyle {\\bm{A}}_{i, :}$ & Row $i$ of matrix ${\\bm{A}}$ \\\\\n$\\displaystyle {\\bm{A}}_{:, i}$ & Column $i$ of matrix ${\\bm{A}}$ \\\\\n$\\displaystyle {\\etens{A}}_{i, j, k}$ & Element $(i, j, k)$ of a 3-D tensor ${\\tens{A}}$\\\\\n$\\displaystyle {\\tens{A}}_{:, :, i}$ & 2-D slice of a 3-D tensor\\\\\n$\\displaystyle {\\textnormal{a}}_i$ & Element $i$ of the random vector ${\\mathbf{a}}$ \\\\\n\\end{tabular}\n\\egroup\n\\vspace{0.25cm}\n\n\n\\centerline{\\bf Calculus}\n\\bgroup\n\\def1.5{1.5}\n\\begin{tabular}{p{1.25in}p{3.25in}}\n$\\displaystyle\\frac{d y} {d x}$ & Derivative of $y$ with respect to $x$\\\\ [2ex]\n$\\displaystyle \\frac{\\partial y} {\\partial x} $ & Partial derivative of $y$ with respect to $x$ \\\\\n$\\displaystyle \\nabla_{\\bm{x}} y $ & Gradient of $y$ with respect to ${\\bm{x}}$ \\\\\n$\\displaystyle \\nabla_{\\bm{X}} y $ & Matrix derivatives of $y$ with respect to ${\\bm{X}}$ \\\\\n$\\displaystyle \\nabla_{\\tens{X}} y $ & Tensor containing derivatives of $y$ with respect to ${\\tens{X}}$ \\\\\n$\\displaystyle \\frac{\\partial f}{\\partial {\\bm{x}}} $ & Jacobian matrix ${\\bm{J}} \\in \\mathbb{R}^{m\\times n}$ of $f: \\mathbb{R}^n \\rightarrow \\mathbb{R}^m$\\\\\n$\\displaystyle \\nabla_{\\bm{x}}^2 f({\\bm{x}})\\text{ or }{\\bm{H}}( f)({\\bm{x}})$ & The Hessian matrix of $f$ at input point ${\\bm{x}}$\\\\\n$\\displaystyle \\int f({\\bm{x}}) d{\\bm{x}} $ & Definite integral over the entire domain of ${\\bm{x}}$ \\\\\n$\\displaystyle \\int_{\\mathbb{S}} f({\\bm{x}}) d{\\bm{x}}$ & Definite integral with respect to ${\\bm{x}}$ over the set ${\\mathbb{S}}$ \\\\\n\\end{tabular}\n\\egroup\n\\vspace{0.25cm}\n\n\\centerline{\\bf Probability and Information Theory}\n\\bgroup\n\\def1.5{1.5}\n\\begin{tabular}{p{1.25in}p{3.25in}}\n$\\displaystyle P({\\textnormal{a}})$ & A probability distribution over a discrete variable\\\\\n$\\displaystyle p({\\textnormal{a}})$ & A probability distribution over a continuous variable, or over\na variable whose type has not been specified\\\\\n$\\displaystyle {\\textnormal{a}} \\sim P$ & Random variable ${\\textnormal{a}}$ has distribution $P$\\\\% so thing on left of \\sim should always be a random variable, with name beginning with \\r\n$\\displaystyle  \\mathbb{E}_{{\\textnormal{x}}\\sim P} [ f(x) ]\\text{ or } \\mathbb{E} f(x)$ & Expectation of $f(x)$ with respect to $P({\\textnormal{x}})$ \\\\\n$\\displaystyle \\mathrm{Var}(f(x)) $ &  Variance of $f(x)$ under $P({\\textnormal{x}})$ \\\\\n$\\displaystyle \\mathrm{Cov}(f(x),g(x)) $ & Covariance of $f(x)$ and $g(x)$ under $P({\\textnormal{x}})$\\\\\n$\\displaystyle H({\\textnormal{x}}) $ & Shannon entropy of the random variable ${\\textnormal{x}}$\\\\\n$\\displaystyle D_{\\mathrm{KL}} ( P \\Vert Q ) $ & Kullback-Leibler divergence of P and Q \\\\\n$\\displaystyle \\mathcal{N} ( {\\bm{x}} ; {\\bm{\\mu}} , {\\bm{\\Sigma}})$ & Gaussian distribution %\nover ${\\bm{x}}$ with mean ${\\bm{\\mu}}$ and covariance ${\\bm{\\Sigma}}$ \\\\\n\\end{tabular}\n\\egroup\n\\vspace{0.25cm}\n\n\\centerline{\\bf Functions}\n\\bgroup\n\\def1.5{1.5}\n\\begin{tabular}{p{1.25in}p{3.25in}}\n$\\displaystyle f: {\\mathbb{A}} \\rightarrow {\\mathbb{B}}$ & The function $f$ with domain ${\\mathbb{A}}$ and range ${\\mathbb{B}}$\\\\\n$\\displaystyle f \\circ g $ & Composition of the functions $f$ and $g$ \\\\\n  $\\displaystyle f({\\bm{x}} ; {\\bm{\\theta}}) $ & A function of ${\\bm{x}}$ parametrized by ${\\bm{\\theta}}$.\n  (Sometimes we write $f({\\bm{x}})$ and omit the argument ${\\bm{\\theta}}$ to lighten notation) \\\\\n$\\displaystyle \\log x$ & Natural logarithm of $x$ \\\\\n$\\displaystyle \\sigma(x)$ & Logistic sigmoid, $\\displaystyle \\frac{1} {1 + \\exp(-x)}$ \\\\\n$\\displaystyle \\zeta(x)$ & Softplus, $\\log(1 + \\exp(x))$ \\\\\n$\\displaystyle || {\\bm{x}} ||_p $ & $L^p$ norm of ${\\bm{x}}$ \\\\\n$\\displaystyle || {\\bm{x}} || $ & $L^2$ norm of ${\\bm{x}}$ \\\\\n$\\displaystyle x^+$ & Positive part of $x$, i.e., $\\max(0,x)$\\\\\n$\\displaystyle \\bm{1}_\\mathrm{condition}$ & is 1 if the condition is true, 0 otherwise\\\\\n\\end{tabular}\n\\egroup\n\\vspace{0.25cm}\n\n\n\n\\section{Final instructions}\nDo not change any aspects of the formatting parameters in the style files.\nIn particular, do not modify the width or length of the rectangle the text\nshould fit into, and do not change font sizes (except perhaps in the\n\\textsc{References} section; see below). Please note that pages should be\nnumbered.\n\n\\section{Preparing PostScript or PDF files}\n\nPlease prepare PostScript or PDF files with paper size ``US Letter'', and\nnot, for example, ``A4''. The -t\nletter option on dvips will produce US Letter files.\n\nConsider directly generating PDF files using \\verb+pdflatex+\n(especially if you are a MiKTeX user).\nPDF figures must be substituted for EPS figures, however.\n\nOtherwise, please generate your PostScript and PDF files with the following commands:\n\\begin{verbatim}\ndvips mypaper.dvi -t letter -Ppdf -G0 -o mypaper.ps\nps2pdf mypaper.ps mypaper.pdf\n\\end{verbatim}\n\n\\subsection{Margins in LaTeX}\n\nMost of the margin problems come from figures positioned by hand using\n\\verb+\\special+ or other commands. We suggest using the command\n\\verb+\\includegraphics+\nfrom the graphicx package. Always specify the figure width as a multiple of\nthe line width as in the example below using .eps graphics\n\\begin{verbatim}\n   \\usepackage[dvips]{graphicx} ...\n   \\includegraphics[width=0.8\\linewidth]{myfile.eps}\n\\end{verbatim}\nor %\n\\begin{verbatim}\n   \\usepackage[pdftex]{graphicx} ...\n   \\includegraphics[width=0.8\\linewidth]{myfile.pdf}\n\\end{verbatim}\nfor .pdf graphics.\nSee section~4.4 in the graphics bundle documentation (\\url{http:\/\/www.ctan.org\/tex-archive\/macros\/latex\/required\/graphics\/grfguide.ps})\n\nA number of width problems arise when LaTeX cannot properly hyphenate a\nline. Please give LaTeX hyphenation hints using the \\verb+\\-+ command.\n\n\\subsubsection*{Author Contributions}\nIf you'd like to, you may include  a section for author contributions as is done\nin many journals. This is optional and at the discretion of the authors.\n\n\\subsubsection*{Acknowledgments}\nUse unnumbered third level headings for the acknowledgments. All\nacknowledgments, including those to funding agencies, go at the end of the paper.\n\n\n\n\\section{Introduction}\n\\input{subtex\/intro.tex}\n\n\\section{Preliminaries} \\label{sec:prel}\n\\input{subtex\/preliminary.tex}\n\n\n\\section{Differential Dynamic Programming Neural Optimizer} \\label{sec:ddp-dnn}\n\\input{subtex\/ddp.tex}\n\n\n\\section{The Role of Feedback Policies} \\label{sec:dnn-trajopt}\n\\input{subtex\/role-of-feedback.tex}\n\n\\section{Experiments} \\label{sec:experiment}\n\\input{subtex\/experiment.tex}\n\n\\section{Conclusion} \\label{sec:conclusion}\nIn this work, we introduce DDPNOpt, a new class of optimizer arising from a novel perspective by bridging DNN training to optimal control and trajectory optimization.\nDDPNOpt features {layer-wise feedback policies} which improve convergence and robustness to hyper-parameters\nover existing optimizers.\nIt outperforms other OCP-inspired methods in both training performance and scalability.\nThis work provides a new algorithmic insight and bridges between deep learning and optimal control.\n\n\\newpage\n\n\n\\section*{Acknowledgments}\nThe authors would like to thank Chen-Hsuan Lin, Yunpeng Pan, Yen-Cheng Liu, and Chia-Wen Kuo for many helpful discussions on the paper.\nThis research was supported by NSF Award Number 1932288.\n\n\n\n\\subsection{Connection between Pontryagin Maximum Principle and DNNs Training} \\label{app:pmp-dev}\n\nDevelopment of the optimality conditions to OCP can be dated back to 1960s,\ncharacterized by both the Pontryagin\\textquotesingle s Maximum Principle (PMP)\nand the Dynamic Programming (DP).\nHere we review Theorem of PMP and its connection to training DNNs.\n\\begin{theorem}[Discrete-time PMP \\citep{pontryagin1962mathematical}] \\label{the:pmp} %\nLet $\\bar{{\\bm{u}}}^*$ be the optimal control trajectory for OCP and\n$\\bar{{\\bm{x}}}^*$ be the corresponding state trajectory.\nThen, there exists a co-state trajectory $\\bar{{\\bm{p}}}^* \\triangleq \\{{\\bm{p}}_t^*\\}_{t=1}^{T}$,\nsuch that\n\\begin{subequations} \\label{eq:mf-pmp}\n\\begin{align}\n{{\\bm{x}}}_{t+1}^{*}&= \\nabla_{{\\bm{p}}_{}} H_t\\left({\\bm{x}}_{t}^{*}, {\\bm{p}}_{t+1}^{*}, {\\bm{u}}_{t}^{*}\\right) { \\text{ ,} }\n\\text{ } {\\bm{x}}_{0}^{*}={\\bm{x}}_{0}\n{ \\text{ ,} } \\label{eq:pmp-forward} \\\\\n{{\\bm{p}}}_{t}^{*}&=\\nabla_{{\\bm{x}}} H_t\\left({\\bm{x}}_{t}^{*}, {\\bm{p}}_{t+1}^{*}, {\\bm{u}}_{t}^{*}\\right) { \\text{ ,} }\n\\text{ } {\\bm{p}}_{T}^{*}= \\nabla_{{\\bm{x}}} \\phi\\left({\\bm{x}}_{T}^{*}\\right)\n{ \\text{ ,} } \\label{eq:pmp-backward} \\\\\n{\\bm{u}}_{t}^{*} &= \\argmin_{v\\in {\\mathbb{R}^{m}}}\nH_t\\left({\\bm{x}}_{t}^{*}, {\\bm{p}}_{t+1}^{*}, {\\bm{v}} \\right)\n{ \\text{ .} } \\label{eq:pmp-max-h}\n\\end{align}\n\\end{subequations}\nwhere $H_t: {\\mathbb{R}^{n}} \\times {\\mathbb{R}^{n}} \\times {\\mathbb{R}^{m}} \\mapsto \\mathbb{R} $\nis the discrete-time Hamiltonian given by\n\\begin{align}\nH_t\\left({\\bm{x}}_t, {\\bm{p}}_{t+1}, {\\bm{u}}_t \\right) \\triangleq \\ell_t({\\bm{x}}_t, {\\bm{u}}_t) + {\\bm{p}}_{t+1}^{\\mathsf{T}} f_t({\\bm{x}}_t, {\\bm{u}}_t) { \\text{ ,} }\n\\end{align}\nand \\eq{\\ref{eq:pmp-backward}} is called the \\textit{adjoint equation}.\n\n\\end{theorem}\nThe discrete-time PMP theorem can be derived using KKT conditions,\nin which the co-state ${\\bm{p}}_t$ is equivalent to the Lagrange multiplier.\nNote that the solution to \\eq{\\ref{eq:pmp-max-h}} admits an open-loop process in the sense that it does not depend on state variables.\nThis is in contrast to the Dynamic Programming principle,\nin which a feedback policy is considered.\n\n\nIt is natural to ask whether the necessary condition in the PMP theorem relates to first-order optimization methods in DNN training.\nThis is indeed the case as pointed out in \\citet{li2017maximum}: %\n\\begin{lemma}[\\cite{li2017maximum}] \\label{lm:bp-gd}\nBack-propagation satisfies \\eq{\\ref{eq:pmp-backward}} and gradient descent iteratively solves \\eq{\\ref{eq:pmp-max-h}}.\n\\end{lemma}\nLemma \\ref{lm:bp-gd} follows by first expanding the derivative of Hamiltonian w.r.t. ${\\bm{x}}_t$,\n\\begin{align}\n    \\nabla_{{\\bm{x}}_t} H_t({\\bm{x}}_{t}, {\\bm{p}}_{t+1}, {\\bm{u}}_{t}) &= \\nabla_{{\\bm{x}}_t} \\ell_t({\\bm{x}}_{t}, {\\bm{u}}_{t}) + \\nabla_{{\\bm{x}}_t} f_t({\\bm{x}}_{t}, {\\bm{u}}_{t})^{\\mathsf{T}}  {\\bm{p}}_{t+1} \\text{ }  = \\nabla_{{\\bm{x}}_t} J({\\bar{{\\bm{u}}}}; {\\bm{x}}_0) { \\text{ .} }\n\\end{align}\nThus, \\eq{\\ref{eq:pmp-backward}} is simply the chain rule used in the Back-propagation.\nWhen $H_t$ is differentiable w.r.t. ${\\bm{u}}_t$, one can attempt to solve \\eq{\\ref{eq:pmp-max-h}} by iteratively taking the gradient descent.\nThis will lead to\n\\begin{align} %\n{\\bm{u}}^{(k+1)}_t\n= {\\bm{u}}^{(k)}_t - \\eta \\nabla_{{\\bm{u}}_t} H_t({\\bm{x}}_{t}, {\\bm{p}}_{t+1}, {\\bm{u}}_{t})\n= {\\bm{u}}^{(k)}_t - \\eta \\nabla_{{\\bm{u}}_t} J({\\bar{{\\bm{u}}}};{\\bm{x}}_0) { \\text{ ,} }\n\\end{align}\nwhere $k$ and $\\eta$ denote the update iteration and step size.\nThus, existing optimization methods can be interpreted as iterative processes to match the PMP optimality conditions.\n\n\nInspired from Lemma \\ref{lm:bp-gd}, \\citet{li2017maximum} proposed\na PMP-inspired method, named Extended Method of Successive Approximations (E-MSA),\nwhich solves the following augmented Hamiltonian\n\\begin{equation}\n\\begin{split}\n\\tilde{H}_t\\left({\\bm{x}}_t, {\\bm{p}}_{t+1}, {\\bm{u}}_t, {\\bm{x}}_{t+1}, {\\bm{p}}_{t} \\right)\n&\\triangleq\nH_t\\left({\\bm{x}}_t, {\\bm{p}}_{t+1}, {\\bm{u}}_t \\right) \\\\ &\\quad +\n\\frac{1}{2}\\rho \\norm{{\\bm{x}}_{t+1} - f_t({\\bm{x}}_t,{\\bm{u}}_t)} +\n\\frac{1}{2}\\rho \\norm{{\\bm{p}}_t - \\nabla_{{\\bm{x}}_t} H_t}\n{ \\text{ .} }\n\\end{split}  \\label{eq:emsa}\n\\end{equation}\n$\\tilde{H}_t$ is the original Hamiltonian augmented with the feasibility constraints on both forward states and backward co-states.\nE-MSA solves the minimization\n\\begin{align}\n    {\\bm{u}}_t^* = \\argmin_{{\\bm{u}}_t \\in \\mathbf{R}^{m_t}} \\tilde{H}_t\\left({\\bm{x}}_t, {\\bm{p}}_{t+1}, {\\bm{u}}_t, {\\bm{x}}_{t+1}, {\\bm{p}}_{t} \\right)\n\\end{align}\nwith L-BFGS per layer and per training iteration.\nAs a result, we consider E-MSA also as second-order method.\n\n\n\\subsection{Proof of Proposition \\ref{prop:bp2ddp}} \\label{app:c1}\n\\begin{proof}\nWe first prove the following lemma which connects the backward pass between two frameworks in the degenerate case.\n\\begin{lemma}\nAssume $Q_{{\\bm{u}} {\\bm{x}}}^t=\\mathbf{0}$ at all stages,\nthen we have\n\\begin{align} \\label{eq:v-dyn-degenerate}\nV_{\\bm{x}}^t = \\nabla_{{\\bm{x}}_t} J { \\text{ ,} } \\text{ and } \\quad\nV_{{\\bm{x}}\\vx}^t = \\nabla_{{\\bm{x}}_t}^2 J { \\text{ ,} } \\quad \\forall t\n{ \\text{ .} }\n\\end{align}\n\\end{lemma}\n\\begin{proof}\nIt is obvious to see that \\eq{\\ref{eq:v-dyn-degenerate}} holds at $t=T$.\nNow, assume the relation holds at $t+1$ and observe that at the time $t$, the backward passes take the form of\n\\begin{align*} %\n    V_{\\bm{x}}^t\n    &= Q_{\\bm{x}}^t - Q_{{\\bm{u}} {\\bm{x}}}^{t\\text{ }{\\mathsf{T}}} {({Q^{t}_{{\\bm{u}} {\\bm{u}}})}^{-1}} Q^t_{{\\bm{u}}}\n    = \\ell^t_{{\\bm{x}}} + {{{f}^t_{{\\bm{x}}}}^{\\mathsf{T}}} \\nabla_{{\\bm{x}}_{t+1}} J\n    = \\nabla_{{\\bm{x}}_t} J { \\text{ ,} } \\\\\n    V_{{\\bm{x}}\\vx}^t &= Q_{{\\bm{x}}\\vx}^t - Q_{{\\bm{u}} {\\bm{x}}}^{t\\text{ }{\\mathsf{T}}} {({Q^{t}_{{\\bm{u}} {\\bm{u}}})}^{-1}} Q^t_{{\\bm{u}} {\\bm{x}}}\n    = \\nabla_{{\\bm{x}}_t} \\{ \\ell^t_{{\\bm{x}}} + {{{f}^t_{{\\bm{x}}}}^{\\mathsf{T}}} \\nabla_{{\\bm{x}}_{t+1}} J \\}\n    = \\nabla_{{\\bm{x}}_{t}}^2 J\n{ \\text{ ,} }\n\\end{align*}\nwhere we recall $J_t = \\ell_t + J_{t+1}(f_t) $ in \\eq{\\ref{eq:Jt}}.\n\\end{proof}\nNow, \\eq{\\ref{eq:newton}} follows by substituting \\eq{\\ref{eq:v-dyn-degenerate}} to the definition of $Q_{{\\bm{u}}}^t$ and $Q_{{\\bm{u}}\\vu}^t$\n\\begin{align*} %\n    Q_{{\\bm{u}}}^t\n    &= \\ell^t_{{\\bm{u}}} + {{{f}^t_{{\\bm{u}}}}^{\\mathsf{T}}} V_{\\bm{x}}^{t+1}\n    = \\ell^t_{{\\bm{u}}} + {{{f}^t_{{\\bm{u}}}}^{\\mathsf{T}}} \\nabla_{{\\bm{x}}_{t+1}} J\n    = \\nabla_{{\\bm{u}}_t} J\n{ \\text{ ,} } \\\\\n    Q_{{\\bm{u}}\\vu}^t\n    &= \\ell^t_{{\\bm{u}}\\vu} + {{{f}^t_{{\\bm{u}}}}^{\\mathsf{T}}} V_{{\\bm{x}}\\vx}^{t+1} {f}_{{\\bm{u}}}^t + V_{\\bm{x}}^{t+1} \\cdot {f}_{{\\bm{u}}\\vu}^t \\\\\n    &= \\ell^t_{{\\bm{u}}\\vu} + {{{f}^t_{{\\bm{u}}}}^{\\mathsf{T}}} (\\nabla^2_{{\\bm{x}}_{t+1}} J) {f}_{{\\bm{u}}}^t + \\nabla_{{\\bm{x}}_{t+1}} J \\cdot {f}_{{\\bm{u}}\\vu}^t \\\\\n    &= \\nabla_{{\\bm{u}}_t} \\{ \\ell^t_{{\\bm{u}}} + {{{f}^t_{{\\bm{u}}}}^{\\mathsf{T}}} \\nabla_{{\\bm{x}}_{t+1}} J \\}\n    = \\nabla_{{\\bm{u}}_t}^2 J\n{ \\text{ .} }\n\\end{align*}\nConsequently, the DDP feedback policy degenerates to layer-wise Newton update.\n\\end{proof}\n\n\n\n\\subsection{Proof of Proposition \\ref{prop:gn-ddp}} \\label{app:prop:gn-ddp}\n\\begin{proof}\nWe will prove Proposition \\ref{prop:gn-ddp} by backward induction.\nSuppose at layer $t+1$, we have ${V^{t+1}_{{\\bm{x}}\\vx}} = {\\bm{z}}_{\\bm{x}}^{t+1} \\otimes {\\bm{z}}_{\\bm{x}}^{t+1}$ and $\\ell_t\\equiv\\ell_t({\\bm{u}}_t)$,\nthen \\eq{\\ref{eq:Qt}} becomes\n\\begin{align*} %\n{Q^t_{{\\bm{x}} {\\bm{x}}}} &= {{{f}^t_{{\\bm{x}}}}^{\\mathsf{T}}} {V^{t+1}_{{\\bm{x}}\\vx}} {{f}^t_{{\\bm{x}}}} = {{{f}^t_{{\\bm{x}}}}^{\\mathsf{T}}} ({\\bm{z}}_{\\bm{x}}^{t+1} \\otimes {\\bm{z}}_{\\bm{x}}^{t+1}) {{f}^t_{{\\bm{x}}}} = ({{{f}^t_{{\\bm{x}}}}^{\\mathsf{T}}}{\\bm{z}}_{\\bm{x}}^{t+1}) \\otimes ({{{f}^t_{{\\bm{x}}}}^{\\mathsf{T}}}{\\bm{z}}_{\\bm{x}}^{t+1}) \\\\\n{Q^t_{{\\bm{u}} {\\bm{x}}}} &= {{{f}^t_{{\\bm{u}}}}^{\\mathsf{T}}} {V^{t+1}_{{\\bm{x}}\\vx}} {{f}^t_{{\\bm{x}}}} = {{{f}^t_{{\\bm{u}}}}^{\\mathsf{T}}} ({\\bm{z}}_{\\bm{x}}^{t+1} \\otimes {\\bm{z}}_{\\bm{x}}^{t+1}) {{f}^t_{{\\bm{x}}}} = ({{{f}^t_{{\\bm{u}}}}^{\\mathsf{T}}}{\\bm{z}}_{\\bm{x}}^{t+1}) \\otimes ({{{f}^t_{{\\bm{x}}}}^{\\mathsf{T}}}{\\bm{z}}_{\\bm{x}}^{t+1})\n{ \\text{ .} }\n\\end{align*}\nSetting ${\\bm{q}}_{\\bm{x}}^t:={{{f}^t_{{\\bm{x}}}}^{\\mathsf{T}}}{\\bm{z}}_{\\bm{x}}^{t+1}$ and ${\\bm{q}}_{\\bm{u}}^t:={{{f}^t_{{\\bm{u}}}}^{\\mathsf{T}}}{\\bm{z}}_{\\bm{x}}^{t+1}$ will give the first part of Proposition \\ref{prop:gn-ddp}.\n\nNext, to show the same factorization structure preserves through the preceding layer, it is sufficient to show\n$V_{{\\bm{x}}\\vx}^t = {\\bm{z}}_{\\bm{x}}^{t} \\otimes {\\bm{z}}_{\\bm{x}}^{t}$ for some vector ${\\bm{z}}_{\\bm{x}}^{t}$. This is indeed the case.\n\\begin{align*} %\nV_{{\\bm{x}}\\vx}^t\n&= {Q^t_{{\\bm{x}} {\\bm{x}}}} - Q_{{\\bm{u}} {\\bm{x}}}^{t\\text{ }\\text{ } {\\mathsf{T}}} ({Q^t_{{\\bm{u}} {\\bm{u}}}})^{-1} {Q^t_{{\\bm{u}} {\\bm{x}}}} \\\\\n&= {\\bm{q}}_{\\bm{x}}^t \\otimes {\\bm{q}}_{\\bm{x}}^t - ({\\bm{q}}_{\\bm{u}}^t \\otimes {\\bm{q}}_{\\bm{x}}^t)^{\\mathsf{T}} ({Q^t_{{\\bm{u}} {\\bm{u}}}})^{-1} ({\\bm{q}}_{\\bm{u}}^t \\otimes {\\bm{q}}_{\\bm{x}}^t) \\\\\n&= {\\bm{q}}_{\\bm{x}}^t \\otimes {\\bm{q}}_{\\bm{x}}^t - ({\\bm{q}}_{\\bm{u}}^{t \\text{ } {\\mathsf{T}}} {({Q^{t}_{{\\bm{u}} {\\bm{u}}})}^{-1}} {\\bm{q}}_{\\bm{u}}^t) ({\\bm{q}}_{\\bm{x}}^t \\otimes {\\bm{q}}_{\\bm{x}}^t) { \\text{ ,} }\n\\end{align*}\nwhere the last equality follows by observing ${\\bm{q}}_{\\bm{u}}^{t \\text{ } {\\mathsf{T}}} {({Q^{t}_{{\\bm{u}} {\\bm{u}}})}^{-1}} {\\bm{q}}_{\\bm{u}}^t$ is a scalar.\n\nSet ${\\bm{z}}_{\\bm{x}}^t = \\mathpalette\\DHLhksqrt{1-{\\bm{q}}_{\\bm{u}}^{t \\text{ } {\\mathsf{T}}} {({Q^{t}_{{\\bm{u}} {\\bm{u}}})}^{-1}} {\\bm{q}}_{\\bm{u}}^t } \\text{ }{\\bm{q}}_{\\bm{x}}^t$ will give the desired factorization.\n\n\\end{proof}\n\n\n\n\\subsection{Derivation of \\eq{\\ref{eq:q-fc}}}\nFor notational simplicity, we drop the superscript $t$ and denote\n$V^{\\prime}_{{\\bm{x}}^{\\prime}} \\triangleq \\nabla_{{\\bm{x}}} V_{t+1}({\\bm{x}}_{t+1})$\nas the derivative of the value function at the next state.\n\\begin{align*}\nQ_{{\\bm{u}}}\n&=\\ell_{{\\bm{u}}}+f_{{\\bm{u}}}^{{\\mathsf{T}}}V_{{\\bm{x}}^{\\prime}}^{\\prime}\n=\\ell_{{\\bm{u}}}+g_{{\\bm{u}}}^{{\\mathsf{T}}}\\sigma_{{\\bm{h}}}^{{\\mathsf{T}}}V_{{\\bm{x}}^{\\prime}}^{\\prime} { \\text{ ,} } \\\\\n  Q_{{\\bm{u}} {\\bm{u}}} &=\\ell_{{\\bm{u}} {\\bm{u}}} + \\frac{\\partial}{\\partial {\\bm{u}}} \\{g_{{\\bm{u}}}^{{\\mathsf{T}}}\\sigma_{{\\bm{h}}}^{{\\mathsf{T}}}V_{{\\bm{x}}^\\prime}^{\\prime} \\} \\\\\n              &=\\ell_{{\\bm{u}} {\\bm{u}}} + g_{{\\bm{u}}}^{{\\mathsf{T}}}\\sigma_{{\\bm{h}}}^{{\\mathsf{T}}} \\frac{\\partial}{\\partial {\\bm{u}}} \\{V_{{\\bm{x}}^\\prime}^{\\prime} \\}\n                               + g_{{\\bm{u}}}^{{\\mathsf{T}}}(\\frac{\\partial}{\\partial {\\bm{u}}} \\{ \\sigma_{{\\bm{h}}} \\})^{{\\mathsf{T}}}V_{{\\bm{x}}^\\prime}^{\\prime} + (\\frac{\\partial}{\\partial {\\bm{u}}} \\{ g_{{\\bm{u}}}\\})^{{\\mathsf{T}}}\\sigma_{{\\bm{h}}}^{{\\mathsf{T}}}V_{{\\bm{x}}^\\prime}^{\\prime} \\\\\n              &=\\ell_{{\\bm{u}} {\\bm{u}}} + g_{{\\bm{u}}}^{{\\mathsf{T}}}\\sigma_{{\\bm{h}}}^{{\\mathsf{T}}} V^\\prime_{{\\bm{x}}^\\prime {\\bm{x}}^\\prime} \\sigma_{{\\bm{h}}} g_{{\\bm{u}}}\n                               + g_{{\\bm{u}}}^{{\\mathsf{T}}}(V_{{\\bm{x}}^\\prime}^{\\prime {\\mathsf{T}}} \\sigma_{{\\bm{h}} {\\bm{h}}} g_{{\\bm{u}}})\n                               + g_{{\\bm{u}} {\\bm{u}}}^{{\\mathsf{T}}}\\sigma_{{\\bm{h}}}^{{\\mathsf{T}}}V_{{\\bm{x}}^\\prime}^{\\prime} \\\\\n              &=\\ell_{{\\bm{u}} {\\bm{u}}}+g_{{\\bm{u}}}^{{\\mathsf{T}}} (V_{{\\bm{h}} {\\bm{h}}} + V_{{\\bm{x}}^{\\prime}}^{\\prime} \\cdot \\sigma_{{\\bm{h}} {\\bm{h}}}) g_{{\\bm{u}}}+V_{{\\bm{h}}} \\cdot g_{{\\bm{u}} {\\bm{u}}}\n\\end{align*}\nThe last equation follows by recalling\n$V_{{\\bm{h}}}           \\triangleq \\sigma_{{\\bm{h}}}^{{\\mathsf{T}}}V_{{\\bm{x}}^\\prime}^{\\prime}$ and\n$V_{{\\bm{h}}\\vh} \\triangleq \\sigma_{{\\bm{h}}}^{{\\mathsf{T}}} V^\\prime_{{\\bm{x}}^\\prime {\\bm{x}}^\\prime} \\sigma_{{\\bm{h}}}$.\nFollow similar derivation, we have\n\\begin{equation}\n\\begin{split} \\label{eq:q-fc2}\n Q_{{\\bm{x}}} &=\\ell_{{\\bm{x}}}+g_{{\\bm{x}}}^{{\\mathsf{T}}}V_{{\\bm{h}}}\\\\\n Q_{{\\bm{x}} {\\bm{x}}} &=\\ell_{{\\bm{x}} {\\bm{x}}}+g_{{\\bm{x}}}^{{\\mathsf{T}}} (V_{{\\bm{h}} {\\bm{h}}} + V_{{\\bm{x}}^{\\prime}}^{\\prime} \\cdot \\sigma_{{\\bm{h}} {\\bm{h}}}) g_{{\\bm{x}}}+V_{{\\bm{h}}} \\cdot g_{{\\bm{x}} {\\bm{x}}} \\\\\n Q_{{\\bm{u}} {\\bm{x}}} &=\\ell_{{\\bm{u}} {\\bm{x}}}+g_{{\\bm{u}}}^{{\\mathsf{T}}} (V_{{\\bm{h}} {\\bm{h}}} + V_{{\\bm{x}}^{\\prime}}^{\\prime} \\cdot \\sigma_{{\\bm{h}} {\\bm{h}}}) g_{{\\bm{x}}}+V_{{\\bm{h}}} \\cdot g_{{\\bm{u}} {\\bm{x}}}\n\\end{split}\n\\end{equation}\n\n\\textbf{Remarks.}\nFor feedforward networks, the computational overhead in Eq.~\\ref{eq:q-fc} and \\ref{eq:q-fc2} can be mitigated by leveraging its affine structure.\nSince $g$ is bilinear in ${\\bm{x}}_t$ and ${\\bm{u}}_t$, the terms ${{g}^t_{{\\bm{x}} {\\bm{x}}}}$ and ${{g}^t_{{\\bm{u}} {\\bm{u}}}}$ vanish.\nThe tensor ${{g}^t_{{\\bm{u}} {\\bm{x}}}}$ admits a sparse structure, whose computation can be simplified to\n\\begin{equation} \\begin{split}\n    [{{g}^t_{{\\bm{u}} {\\bm{x}}}}&]_{(i,j,k)} = 1 \\quad \\text{iff} \\quad j = (k-1)n_{t+1} + i { \\text{ ,} }\n\\\\  [{V^{t}_{{\\bm{h}}}} \\cdot {{g}^t_{{\\bm{u}} {\\bm{x}}}}&]_{((k-1)n_{t+1}:kn_{t+1},k)} = {V^{t}_{{\\bm{h}}}}\n{ \\text{ .} } \\label{eq:gux-compute}\n\\end{split}\n\\end{equation}\nFor the coordinate-wise nonlinear transform, $\\sigma^t_{{\\bm{h}}}$ and $\\sigma^t_{{\\bm{h}}\\vh}$ are diagonal matrix and tensor.\nIn most learning instances, stage-wise losses typically involved with weight decay alone; thus the terms ${{\\ell}^t_{{\\bm{x}}}}, {{\\ell}^t_{{\\bm{x}} {\\bm{x}}}}, {{\\ell}^t_{{\\bm{u}} {\\bm{x}}}}$ also vanish.\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Derivation of \\eq{\\ref{eq:qux-math}}} \\label{app:c2}\n\\eq{\\ref{eq:qux-math}} follows by an observation that the feedback policy $\\mathbf{K}_t {\\delta{\\bm{x}}}_t = -(Q^t_{{\\bm{u}} {\\bm{u}}})^{-1} Q^t_{{\\bm{u}} {\\bm{x}}} \\delta {\\bm{x}}_t$\nstands as the minimizer of the following objective\n\\begin{align} \\label{eq:Kdx-interpret}\n    \\mathbf{K}_t {\\delta{\\bm{x}}}_t =\n    \\argmin_{\\delta {\\bm{u}}_t({\\delta{\\bm{x}}}_t) \\in \\Gamma^\\prime(\\delta {\\bm{x}}_t )} \\norm{\\nabla_{{\\bm{u}}_t} Q({\\bm{x}}_t + \\delta {\\bm{x}}_t,{\\bm{u}}_t + \\delta {\\bm{u}}_t({\\delta{\\bm{x}}}_t)) - \\nabla_{{\\bm{u}}_t} Q({\\bm{x}}_t,{\\bm{u}}_t)}\n    { \\text{ ,} }\n\\end{align}\nwhere $\\Gamma^\\prime(\\delta {\\bm{x}}_t )$ denotes all affine mappings from $\\delta {\\bm{x}}_t$ to $\\delta {\\bm{u}}_t$ and\n$\\norm{\\cdot}$ can be any proper norm in the Euclidean space.\n\\eq{\\ref{eq:Kdx-interpret}}\nfollows by the Taylor expansion of $Q({\\bm{x}}_t + \\delta {\\bm{x}}_t,{\\bm{u}}_t + \\delta {\\bm{u}}_t)$ to its first order,\n\\begin{align*} %\n    \\nabla_{{\\bm{u}}_t} Q({\\bm{x}}_t + \\delta {\\bm{x}}_t,{\\bm{u}}_t + \\delta {\\bm{u}}_t)\n    =\n    \\nabla_{{\\bm{u}}_t} Q({\\bm{x}}_t,{\\bm{u}}_t) + Q^t_{{\\bm{u}} {\\bm{x}}} \\delta {\\bm{x}}_t + Q^t_{{\\bm{u}} {\\bm{u}}} \\delta {\\bm{u}}_t\n    { \\text{ .} }\n\\end{align*}\nWhen $Q = J$, we will arrive at \\eq{\\ref{eq:qux-math}}.\nFrom Proposition \\ref{prop:bp2ddp}, we know the equality holds when all $Q^s_{{\\bm{x}}{\\bm{u}}}$ vanish for $s>t$.\nIn other words, the approximation in \\eq{\\ref{eq:qux-math}} becomes equality\nwhen all aferward layer-wise objectives $s>t$ are expanded only w.r.t. ${\\bm{u}}_s$.\n\n\n\n\\subsection{Performance on Classification Dataset}\n\\textbf{Networks \\& Baselines Setup.}\nWe first validate the performance of training fully-connected (FCN) and convolution networks (CNN) using DDPNOpt on classification datasets.\n{\nFCN consists of $5$ fully-connected layers with the hidden dimension ranging from $10$ to $32$, depending on the size of the dataset.\nCNN consists of $4$ convolution layers (with $3{\\times}3$ kernel, $32$ channels), followed by $2$ fully-connected layers.\nWe use ReLU activation on all datasets except Tanh for WINE and DIGITS to better distinguish the differences between optimizers.\nThe batch size is set to $8$-$32$ for datasets trained with FCN, and $128$ for datasets trained with CNN.}\nAs DDPNOpt combines strengths from both standard training methods and OCP framework, we select baselines from both sides.\nThis includes first-order methods, \\emph{i.e.} SGD (with tuned momentum), RMSprop, Adam,\nand second-order method EKFAC \\citep{george2018fast}, which is a recent extension of the popular KFAC \\citep{martens2015optimizing}.\nFor OCP-inspired methods,\nwe compare DDPNOpt with vanilla DDP and E-MSA \\citep{li2017maximum},\nwhich is also a second-order method\nyet built upon the PMP framework.\nRegarding the curvature approximation used in DDPNOpt (${\\bm{M}}_t$ in Table \\ref{table:update-rule}),\nwe found that using adaptive diagonal and GN matrices respectively for FCNs and CNNs\ngive the best performance in practice.\nWe leave the complete experiment setup and additional results in Appendix \\ref{app:experiment}.\n\n\\input{subtex\/training_table2.tex}\n\n\\begin{figure}[t]\n\\vskip -0.1in\n\\centering\n\\begin{minipage}{0.42\\textwidth}\n\\centering\n\\includegraphics[width=\\linewidth]{fig\/complexity.pdf}\n\\vskip -0.15in\n\\caption{Runtime comparison on MNIST.}\\label{fig:runtime}\n\\end{minipage}\n\\begin{minipage}{0.57\\textwidth}\n\\centering\n  \\captionsetup{type=table}\n  \\captionsetup{justification=centering}\n  \\caption{Computational complexity {in backward pass}. \\\\ ($B$: batch size, $X$: hidden state dim., $L$: \\# of layers)}\n  \\vskip -0.1in\n  \\begin{small}\n  \\begin{tabular}{c|cc|c}\n    \\toprule\n     Method & Adam      & Vanilla DDP & \\textbf{DDPNOpt} \\\\\n    \\midrule\n    {\\small Memory}  & {\\small $\\mathcal{O}(X^2L)$} & {\\small $\\mathcal{O}(BX^3L)$} & {\\small$\\mathcal{O}(X^2L+BX)$} \\\\\n    {\\small Speed}   & {\\small $\\mathcal{O}(BX^2L)$} & {\\small $\\mathcal{O}(B^3X^3L)$} & {\\small$\\mathcal{O}(BX^2L)$} \\\\\n    \\bottomrule\n  \\end{tabular} \\label{table:complexity}\n  \\end{small}\n\\end{minipage}\n\\vskip -0.2in\n\\end{figure}\n\n\\textbf{Training Results.}\nTable \\ref{table:training} presents the results over $10$ random trials.\nIt is clear\nthat DDPNOpt outperforms two OCP baselines on \\emph{all datasets and network types}.\nIn practice, both baselines suffer from unstable training and require careful tuning on the hyper-parameters.\nIn fact, we are not able to obtain results for vanilla DDP with any reasonable amount of computational resources when the problem size goes beyond FC networks.\nThis is in contrast to DDPNOpt which adapts amortized curvature estimation from widely-used methods;\nthus exhibits much stabler training dynamics with superior convergence.\nIn Table~\\ref{table:complexity}, we provide the {analytic} runtime and memory complexity among different methods.\nWhile vanilla DDP grows cubic w.r.t. $BX$,\nDDPNOpt reduces the computation by orders of magnitude with efficient approximation presented in Sec.~\\ref{sec:ddp-dnn}.\nAs a result,\n{when measuring the actual computational performance with GPU parallelism,}\nDDPNOpt runs nearly as fast as standard methods and outperforms E-MSA by a large margin.\nThe additional memory complexity, when comparing DDP-inspired methods with Back-propagation methods,\ncomes from the layer-wise feedback policies.\nHowever, DDPNOpt is much memory-efficient compared with vanilla DDP by exploiting the factorization in Proposition~\\ref{prop:gn-ddp}.\n\n\n\\textbf{Ablation Analysis.}\nOn the other hand, the performance gain between DDPNOpt and standard methods appear comparatively small.\nWe conjecture this is due to the inevitable use of similar curvature adaptation,\nas the local geometry of the landscape directly affects the convergence behavior.\nTo identify scenarios where DDPNOpt best shows its effectiveness,\nwe conduct an ablation analysis on the feedback mechanism.\nThis is done by recalling Proposition~\\ref{prop:bp2ddp}: when ${Q^t_{{\\bm{u}} {\\bm{x}}}}$ vanishes,\nDDPNOpt degenerates to the method associated with each precondition matrix.\nFor instance, DDPNOpt with identity (\\emph{resp.} adaptive diagonal and GN) precondition ${\\bm{M}}_t$ will generate the same updates as SGD (\\emph{resp.} RMSprop and EKFAC) when all ${Q^t_{{\\bm{u}} {\\bm{x}}}}$ are zeroed out.\nIn other words, these DDPNOpt variants can be viewed as the \\emph{DDP-extension} to existing baselines.\n\n\nIn Fig.~\\ref{fig:exp-grid} we report the performance difference between each baseline and its associated DDPNOpt variant.\nEach grid corresponds to a distinct training configuration that is averaged over $10$ random trails,\nand we keep all hyper-parameters (\\emph{e.g.} learning rate and weight decay) the same between baselines and their DDPNOpt variants.\nThus, the performance gap only comes from the feedback policies,\nor equivalently the update directions in Table~\\ref{table:update-rule}.\nBlue (\\emph{resp.} red) indicates an improvement (\\emph{resp.} degradation) when the feedback policies are presented.\nClearly, the improvement over baselines\nremains consistent across most hyper-parameters setups, and\nthe performance gap tends to become obvious as the learning rate increases.\nThis aligns with the previous study on numerical stability \\citep{liao1992advantages},\n{\nwhich suggests the feedback can stabilize the optimization when \\emph{e.g.} larger control updates are taken.\nSince larger control corresponds to a further step size in the application of DNN training, one should expect DDPNOpt to show its robustness as the learning rate increases.\n}\nAs shown in Fig.~\\ref{fig:exp-comp},\nsuch a stabilization can also lead to smaller variance and faster convergence.\nThis sheds light on\nthe benefit gained by bridging two seemly disconnected methodologies between DNN training and trajectory optimization.\n\n\n\\begin{figure}[t]\n\\vskip -0.34in\n\\subfloat{\\includegraphics[width=0.78\\columnwidth]{fig\/grid-exp3.pdf} \\label{fig:exp-grid} }\n\\subfloat{\\includegraphics[width=0.21\\columnwidth]{fig\/comp2-fix.pdf}     \\label{fig:exp-comp}}\n\\vskip -0.1in\n\\caption{ %\n(a) Performance difference between DDPNOpt and baselines on DIGITS across hyper-parameter grid.\nBlue (\\emph{resp.} red) indicates an improvement (\\emph{resp.} degradation) over baselines.\nWe observe similar behaviors on other datasets.\n(b) Examples of the actual training dynamics.\n}\n\\label{fig:grid-exp}\n\\end{figure}\n\n\n\\begin{figure}[t]\n\\vskip -0.1in\n\\centering\n\\begin{minipage}{0.29\\textwidth}\n    \\centering\n    \\includegraphics[width=0.7\\linewidth]{fig\/K-vis.png}\n    \\vskip -0.1in\n    \\caption{Visualization of the feedback policies on MNIST.}\\label{fig:K-vis}\n\\end{minipage}\n\\begin{minipage}{0.05\\textwidth}\n\\end{minipage}\n\\begin{minipage}{0.65\\textwidth}\n    \\centering\n    \\subfloat{\n        \\includegraphics[width=0.95\\linewidth]{fig\/vg-update2.pdf}\n        \\label{fig:vanish-grad-a}%\n    }\n    \\subfloat{%\n        \\textcolor{white}{\\rule{1pt}{1pt}}\n        \\label{fig:vanish-grad-b}%\n    }%\n    \\vskip -0.1in\n    \\caption{Training a $9$-layer sigmoid-activated {FCN} on \\\\ DIGITS using MMC loss.\n    DDPNOpt2nd denotes when the layer dynamics is fully expanded to the second order.\n    }\n    \\label{fig:vanish-grad}\n\\end{minipage}\n\\vskip -0.2in\n\\end{figure}\n\n\n\\subsection{Discussion on Feedback Policies}\n\n\\textbf{Visualization of Feedback Policies.}\nTo understand the effect of feedback policies more perceptually,\n{\nin Fig.~\\ref{fig:K-vis} we visualize the feedback policy when training CNNs.\nThis is done by first conducting singular-value decomposition on the feedback matrices ${{\\bm{K}}_t}$,\nthen projecting the leading right-singular vector back to image space\n(see Alg.~\\ref{alg:K-vis} and Fig.~\\ref{fig:K-vis-app} in Appendix for the pseudo-code).\nThese feature maps, denoted $\\delta x_{\\max}$ in Fig.~\\ref{fig:K-vis}, correspond to the dominating differential image that the policy shall respond with during weight update.\nFig.~\\ref{fig:K-vis} shows that the feedback policies indeed capture\nnon-trivial visual features related to the pixel-wise difference between spatially similar classes, \\emph{e.g.} $(8,3)$ or $(7,1)$.\nThese differential maps differ from adversarial perturbation \\citep{goodfellow2014explaining}\nas the former directly links the parameter update to the change in activation;\nthus being more interpretable.\n\n\n\\textbf{Vanishing Gradient.}\nLastly, we present an interesting finding on how the feedback policies help mitigate vanishing gradient (VG),\na notorious effect when DNNs become impossible to train as gradients vanish along Back-propagation.\nFig.~\\ref{fig:vanish-grad-a} reports results on training a sigmoid-activated DNN on DIGITS.\nWe select SGD-VGR, which imposes a specific regularization to mitigate VG \\citep{pascanu2013difficulty}, and EKFAC as our baselines.\nWhile both baselines suffer to make any progress,\nDDPNOpt continues to generate non-trivial updates as the state-dependent feedback, \\emph{i.e.} $\\mathbf{K}_t {\\delta{\\bm{x}}}_t$, remains active.\nThe effect becomes significant when dynamics is fully expanded to the second order.\nAs shown in Fig.~\\ref{fig:vanish-grad-b}, the update norm from DDPNOpt is typically $5$-$10$ times larger.\nWe note that in this experiment, we replace the cross-entropy (CE) with Max-Mahalanobis center (MMC) loss,\na new classification objective that improves robustness on standard vision datasets \\citep{pang2019rethinking}.\nMMC casts classification to distributional regression, providing denser Hessian and making problems similar to original trajectory optimization.\nNone of the algorithms escape from VG using CE.\nWe highlight that while VG is typically mitigated on the \\textit{architecture} basis, by having either unbounded activation function or residual blocks,\nDDPNOpt provides an alternative from the \\textit{algorithmic} perspective.\n\n\n\n\n\n\n\n\n\n\n\\iffalse\n\nthe update signal remain active\nand, as we discussed in the previous experiment, become significant\n\nwhen the networks going ``deep''.\n\n\n\n\n\noptimizer basis\n\n\n\n gradient vanishes along the Back-propagation\n\nThe layer-wise feedback policy has an surprising effect on preventing \\textit{vanishing gradient},\na notorious effect\n\nWe find that in additional to\n\n\nIn Fig. X we report the training performance on MNIST for $20$-layers feedforward networks\n\n the training curves between different optimizers\n\n of a $20$-layers feedforward networks trained with on MNIST.\n\n\n\n\nknown to be notoriously hard to train\n\n\nIn Fig. X\n\n\nexplain\n\noptimizer\narchitectures\n\n\nfuture works.\n\n\\fi\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Experiment Detail} \\label{app:experiment}\n\\subsubsection{Setup} \\label{app:exp-set}\n\n\\begin{minipage}[t]{0.56\\textwidth}\n\\textbf{Clarification Dataset.}\nAll networks in the classification experiments are composed of $5$-$6$ layers.\nFor the intermediate layers, we use ReLU activation on all dataset, except Tanh on WINE and DIGITS.\nWe use identity mapping at the last prediction layer on all dataset except WINE,\nwhere we use sigmoid instead to help distinguish the performance among optimizers.\nFor feedforward networks, the dimension of the hidden state is set to $10$-$32$.\nOn the other hand, we use standard $3\\text{ }\\times\\text{ }3$ convolution\n\\end{minipage}\n\\begin{minipage}[t]{0.43\\textwidth}\n  \\captionof{table}{Hyper-parameter search}\n  \\vskip -0.5in\n  \\begin{center}\n  \\begin{small}\n    \\begin{tabular}{c|c}\n    \\toprule\n    Methods & Learning Rate \\\\\n    \\midrule\n    SGD             & $(7\\mathrm{e}\\text{-}2,5\\mathrm{e}\\text{-}1)$ \\\\\n    Adam \\& RMSprop & $(7\\mathrm{e}\\text{-}4,1\\mathrm{e}\\text{-}2)$ \\\\\n    EKFAC           & $(1\\mathrm{e}\\text{-}2,3\\mathrm{e}\\text{-}1)$ \\\\\n    \\bottomrule\n    \\end{tabular} \\label{table:hyper}\n  \\end{small}\n  \\end{center}\n  \\vskip 0.5in\n\\end{minipage}\nkernels for all CNNs.\nThe batch size is set $8$-$32$ for dataset trained with feedforward networks, and $128$ for dataset trained with convolution networks.\nFor each baseline we select its own hyper-parameter from an appropriate search space, which we detail in Table~\\ref{table:hyper}.\nWe use the implementation in \\url{https:\/\/github.com\/Thrandis\/EKFAC-pytorch} for EKFAC\nand implement our own E-MSA in PyTorch since the official code released from \\citet{li2017maximum} does not support GPU implementation.\nWe impose the GN factorization presented in Proposition \\ref{prop:gn-ddp} for all CNN training.\nRegarding the machine information,\nwe conduct our experiments on GTX 1080 TI, RTX TITAN, and four Tesla V100 SXM2 16GB.\n\n\n\\textbf{Procedure to Generate Fig.~\\ref{fig:K-vis}.}\nFirst, we perform standard DDPNOpt steps to compute layer-wise policies. Next, we conduct singular-value decomposition on the feedback matrix $({{\\bm{k}}_t},{{\\bm{K}}_t})$.\nIn this way, the leading right-singular vector corresponding to the dominating that the feedback policy shall respond with.\nSince this vector is with the same dimension as the hidden state, which is most likely not the same as the image space, we project the vector back to image space using the techniques proposed in \\citep{zeiler2014visualizing}. The pseudo code and computation diagram are included in Alg.~\\ref{alg:K-vis} and Fig.~\\ref{fig:K-vis-app}.\n\n\n\\vspace{-10pt}\n\n\\begin{minipage}[t]{0.52\\textwidth}\n\\vskip -0.in\n\\begin{algorithm}[H]\n\\small\n     \\caption{\\small Visualizing the Feedback Policies}\n     \\label{alg:K-vis}\n  \\begin{algorithmic}[1]\n   \\STATE {\\bfseries Input:}\n   Image ${{\\bm{x}}}$ (we drop the time subscript for notational simplicity, {\\emph{i.e.}} ${\\bm{x}} \\equiv {\\bm{x}}_0$)\n   \\STATE Perform backward pass of DDPNOpt. Compute $({{\\bm{k}}_t},{{\\bm{K}}_t})$ backward\n   \\STATE Perform SVD on ${{\\bm{K}}_t}$\n   \\STATE Extract the right-singular vector corresponding to the largest singular value, denoted $v_{\\max} \\in \\mathbb{R}^{n_t} $\n   \\STATE Project $v_{\\max}$ back to the image space using deconvolution procedures introduced in \\citep{zeiler2014visualizing}\n  \\end{algorithmic}\n\\end{algorithm}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[t]{0.42\\textwidth}\n\\vskip -0.1in\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=\\linewidth]{fig\/K-viz-procedure.png}\n\\caption{Pictorial illustration for Alg. \\ref{alg:K-vis}.}\n\\label{fig:K-vis-app}\n\\end{figure}\n\\end{minipage}\n\n\n\n\\subsubsection{Additional Experiment and Discussion} \\label{app:exp-more}\n\n\n\n\n\\textbf{Batch trajectory optimization on synthetic dataset.}\nOne of the difference between DNN training and trajectory optimization is that for the former,\nwe aim to find an ultimate control law that can drive every data point in the training set, or sampled batch, to its designed target.\nDespite seemly trivial from the ML perspective, this is a distinct formulation to OCP since the optimal policy typically varies at different initial state.\nAs such, we validate performance of DDPNOpt in batch trajectories optimization on a synthetic dataset,\nwhere we sample data from $k\\in\\{5,8,12,15\\}$ Gaussian clusters in $\\mathbb{R}^{30}$.\nSince conceptually\na DNN classifier can be thought of as a dynamical system guiding\ntrajectories of samples toward the target regions belong to their classes,\nwe hypothesize that\nfor the DDPNOpt to show its effectiveness on batch training,\nthe feedback policy must act as an ensemble policy that combines the locally optimal policy of each class.\nFig.~\\ref{fig:feedback-spectrum} shows the spectrum distribution, sorted in a descending order, of the feedback policy in the prediction layer.\nThe result shows that the number of nontrivial eigenvalues matches exactly the number of classes in each setup (indicated by the vertical dashed line).\nAs the distribution in the prediction layer\nconcentrates to $k$ bulks through training,\nthe eigenvalues also increase, providing stronger feedback to the weight update.\n\n\n\\begin{figure}[H]\n\\vskip -0.1in\n\\begin{center}\n\\centerline{\\includegraphics[width=\\columnwidth]{fig\/feedback-policy-spectrum.pdf}}\n\\vskip -0.1in\n\\caption{\nSpectrum distribution on synthetic dataset.\n}\n\\label{fig:feedback-spectrum}\n\\end{center}\n\\vskip -0.2in\n\\end{figure}\n\n\n\n\n\\textbf{Ablation analysis on Adam.}\nFig.~\\ref{fig:grid-exp-adam} reports the ablation analysis on Adam using the same setup as in Fig. \\ref{fig:exp-grid}, \\emph{i.e.}\nwe keep all hyper-parameters the same for each experiment so that the performance difference only comes from the existence of feedback policies.\nIt is clear that the improvements from the feedback policies remain consistent for Adam optimizer.\n\n\n\\begin{figure}[H]\n\\vskip -0.1in\n\\begin{center}\n\\centerline{\\includegraphics[width=0.6\\columnwidth]{fig\/grid-exp-adam.pdf}}\n\\vskip -0.1in\n\\caption{\nAdditional experiment for Fig.~\\ref{fig:exp-grid} where we compare the performance difference between DDPNOpt and Adam.\nAgain, all grids report values averaged over $10$ random seeds.\n}\n\\label{fig:grid-exp-adam}\n\\end{center}\n\\vskip -0.2in\n\\end{figure}\n\n\n\n\n\n\\textbf{Ablation analysis on DIGITS compared with best-tuned baselines.}\nFig.~\\ref{fig:grid-exp} reports the performance difference between baselines and DDPNOpt under different hyperparameter setupts.\nHere, we report the numerical values when each baseline uses its best-tuned learning rate (which is the values we report in Table 3) and compare with its DDPNOpt counterpart using the same learning rate. As shown in Tables \\ref{table:6}, \\ref{table:7}, and \\ref{table:8}, for most cases extending the baseline to accept the Bellman framework improves the performance.\n\n\\providecommand{\\e}[1]{\\ensuremath{\\times 10^{#1}}}\n\n\\bgroup\n\\setlength\\tabcolsep{0.07in}\n\\begin{table}[H]\n\\caption{ Learning rate $=0.1$}\n\\label{table:6}\n\\vspace{-3pt}\n\\begin{center}\n\\begin{small}\n\\vskip -0.1in\n\\begin{tabular}{c?cc}\n\\toprule\n& {SGD} & {DDPNOpt with {${\\bm{M}}_t = {\\bm{I}}_t$} } \\\\[2pt]\n\\midrule\nTrain Loss &\n$0.035$ &  $\\textbf{0.032}$ \\\\\nAccuracy (\\%) &\n$95.36$ &  $\\textbf{95.52}$ \\\\\n\\bottomrule\n\\end{tabular}\n\\end{small}\n\\end{center}\n\\end{table}\n\n\\vspace{-20pt}\n\n\\begin{table}[H]\n\\caption{ Learning rate $=0.001$}\n\\label{table:7}\n\\vspace{-3pt}\n\\begin{center}\n\\begin{small}\n\\vskip -0.1in\n\\begin{tabular}{c?cc}\n\\toprule\n& {RMSprop} & { DDPNOpt with {${\\bm{M}}_t = \\diag(\\mathpalette\\DHLhksqrt{\\mathbb{E}[{Q^t_{{\\bm{u}}}} \\odot {Q^t_{{\\bm{u}}}}]} +\\epsilon)$} } \\\\[2pt]\n\\midrule\nTrain Loss &\n$0.058$ &  $\\textbf{0.052}$ \\\\\nAccuracy (\\%) &\n$94.33$ &  $\\textbf{94.63}$ \\\\\n\\bottomrule\n\\end{tabular}\n\\end{small}\n\\end{center}\n\\end{table}\n\n\\vspace{-20pt}\n\n\\begin{table}[H]\n\\caption{ Learning rate $=0.03$}\n\\label{table:8}\n\\vspace{-3pt}\n\\begin{center}\n\\begin{small}\n\\vskip -0.1in\n\\begin{tabular}{c?cc}\n\\toprule\n& {EKFAC} & {DDPNOpt with {${\\bm{M}}_t = \\mathbb{E}{[{\\bm{x}}_t{\\bm{x}}_t^{\\mathsf{T}}]} \\otimes \\mathbb{E}{[V_{{\\bm{h}}}^tV_{{\\bm{h}}}^{t\\text{ }{\\mathsf{T}}}]}$} } \\\\[2pt]\n\\midrule\nTrain Loss &\n$0.074$ &  $\\textbf{0.067}$ \\\\\nAccuracy (\\%) &\n$\\textbf{95.24}$ &  ${95.19}$ \\\\\n\\bottomrule\n\\end{tabular}\n\\end{small}\n\\end{center}\n\\end{table}\n\n\\egroup\n\n\n\\input{subtex\/figure_4_absolute_value}\n\n\\textbf{More experiments on vanishing gradient.}\nRecall that Fig.~\\ref{fig:vanish-grad} reports the training performance using MMC loss on Sigmoid-activated networks.\nIn Fig.~\\ref{fig:vg-more1-a}, we report the result when training the same networks but using CE loss (notice the numerical differences in the $y$ axis for different objectives).\nNone of the presented optimizers were able to escape from vanishing gradient, as evidenced by the vanishing update magnitude.\nOn the other hands, changing the networks to ReLU-activated networks eliminates the vanishing gradient,\nas shown in Fig.~\\ref{fig:vg-more1-b}.\n\nFig.~\\ref{fig:vg-more1-c} reports the performance with other first-order adaptive optimizers including Adam and RMSprop.\nIn general, adaptive first-order optimizers are more likely to escape from vanishing gradient since the diagonal precondition matrix (recall ${\\bm{M}}_t = \\mathbb{E}[{J}_{{\\bm{u}}_t} \\odot {J}_{{\\bm{u}}_t}]$ in Table~\\ref{table:update-rule}) rescales the vanishing update to a fixed norm. However, as shown in Fig.~\\ref{fig:vg-more1-c}, DDPNOpt* (the variant of DDPNOpt that\nutilizes similar adaptive first-order precondition matrix) converges faster compared with these adaptive baselines.\n\nFig.~\\ref{fig:vg-more2} illustrates the selecting process on the learing-rate tuning when we report Fig.~\\ref{fig:vanish-grad}.\nThe training performance for both SGD-VGR and EKFAC remains unchanged when tuning the learning rate. In practice, we observe unstable training with SGD-VGR when the learing rate goes too large.\nOn the other hands, DDPNOpt and DDPNOpt2nd are able to escape from VG with all tested learning rates.\nHence, Fig.~\\ref{fig:vanish-grad} combines Fig.~\\ref{fig:vg-more2-a} (SGD-VGR-lr$0.1$) and Fig.~\\ref{fig:vg-more2-c} (EKFAC-lr$0.03$, DDPNOpt-lr$0.03$, DDPNOpt2nd-lr$0.03$) for best visualization.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{figure}[H]%\n  \\centering\n  \\vskip -0.1in\n  \\subfloat{\n    \\textcolor{white}{\\rule{1pt}{1pt}}\n    \\label{fig:vg-more1-a}%\n  }\n  \\subfloat{\n    \\includegraphics[width=0.85\\linewidth]{fig\/vg-1115.pdf}\n    \\label{fig:vg-more1-b}%\n  }\n  \\vskip -0.1in\n  \\caption{\n    Vanishing gradient experiment for different losses and nonlinear activation functions.\n  }\n  \\label{fig:weight-update}%\n  \\vskip -0.1in\n\\end{figure}\n\n\n\\begin{figure}[H]%\n  \\centering\n  \\subfloat{\n    \\includegraphics[width=0.8\\linewidth]{fig\/vg-1115-2.pdf}\n  }\n  \\vskip -0.1in\n  \\caption{\n    Vanishing gradient experiment for other optimizers.\n    The legend ``DDPNOpt*''\n    denotes DDPNOpt with adaptive diagonal matrix.\n  }\n  \\label{fig:vg-more1-c}%\n  \\vskip -0.1in\n\\end{figure}\n\n\\begin{figure}[H]%\n  \\centering\n  \\subfloat{\n    \\includegraphics[width=0.8\\linewidth]{fig\/vg-lrs-1115.pdf}\n    \\label{fig:vg-more2-a}%\n  }\n  \\subfloat{\n    \\textcolor{white}{\\rule{1pt}{1pt}}\n    \\label{fig:vg-more2-b}%\n  }\n  \\subfloat{\n    \\textcolor{white}{\\rule{1pt}{1pt}}\n    \\label{fig:vg-more2-c}%\n  }\n  \\subfloat{\n    \\textcolor{white}{\\rule{1pt}{1pt}}\n    \\label{fig:vg-more2-d}%\n  }\n  \\vskip -0.1in\n  \\caption{\n    Vanishing gradient experiment for different learning rate setups.\n  }\n  \\label{fig:vg-more2}%\n  \\vskip -0.1in\n\\end{figure}\n\n\n\n\\subsection{Training DNNs as Trajectory Optimization}\n\nRecall that DNNs can be interpreted as dynamical systems where each layer is viewed as a distinct time step.\nConsider \\emph{e.g.} the propagation rule in feedforward layers,\n\\begin{align} \\label{eq:dnn-dyn}\n{\\bm{x}}_{t+1} = \\sigma_t ({\\bm{h}}_t) { \\text{ ,} } \\quad\n{\\bm{h}}_t = g_t({\\bm{x}}_{t},{\\bm{u}}_t) = {\\bm{W}}_t {\\bm{x}}_t + {\\bm{b}}_t { \\text{ .} }\n\\end{align}\n${\\bm{x}}_t \\in \\mathbb{R}^{n_t}$ and ${\\bm{x}}_{t+1} \\in \\mathbb{R}^{n_{t+1}}$ represent the activation vector at layer $t$ and $t+1$, with\n${\\bm{h}}_t \\in \\mathbb{R}^{n_{t+1}}$ being the pre-activation vector.\n$\\sigma_t$ and $g_t$ respectively denote\nthe nonlinear activation function and\nthe affine transform parametrized by the vectorized weight ${\\bm{u}}_t \\triangleq [\\mathrm{vec}({\\bm{W}}_t), {\\bm{b}}_t]^{\\mathsf{T}}$.\n\\eq{\\ref{eq:dnn-dyn}} can be seen as a dynamical system (by setting $f_t \\equiv \\sigma_t \\circ g_t$ in OCP)\npropagating the activation vector ${\\bm{x}}_t$ using ${\\bm{u}}_t$.\n\n\nNext, notice that the gradient descent (GD) update, denoted\n$\\delta {\\bar{{\\bm{u}}}}^* \\equiv -\\eta \\nabla_{{\\bar{{\\bm{u}}}}} J$ with $\\eta$ being the learning rate,\ncan be break down into each layer, \\textit{i.e.}\n$\\delta {\\bar{{\\bm{u}}}}^* \\triangleq \\{\\delta {\\bm{u}}_t^*\\}_{t=0}^{T-1}$,\nand computed backward by\n\\begin{align}\n{\\delta{\\bm{u}}}_t^*\n&= \\argmin_{{\\delta{\\bm{u}}}_t \\in \\mathbb{R}^{m_t}}\\{\n  J_t +  \\nabla_{{\\bm{u}}_t} J_t^{\\mathsf{T}} {\\delta{\\bm{u}}}_t + \\textstyle \\frac{1}{2} {\\delta{\\bm{u}}}_t^{\\mathsf{T}} (\\textstyle \\frac{1}{\\eta}{\\bm{I}}_t) {\\delta{\\bm{u}}}_t\n\\} { \\text{ ,} } \\label{eq:du-star} \\\\\n\\text{where } J_t({\\bm{x}}_t,{\\bm{u}}_t) &\\triangleq \\ell_t({\\bm{u}}_t) + J_{t+1}(f_t({\\bm{x}}_t,{\\bm{u}}_t),{\\bm{u}}_{t+1})\n{ \\text{ ,} } \\quad J_T({\\bm{x}}_T)\\triangleq\\phi({\\bm{x}}_T)\n\\label{eq:Jt}\n\\end{align}\nis the per-layer objective\\footnote{\n  Hereafter we drop ${\\bm{x}}_t$ in all $\\ell_t(\\cdot)$\n  as the layer-wise loss typically involves weight regularization alone.\n} at layer $t$.\nIt can be readily verified that ${\\bm{p}}_t \\equiv \\nabla_{{\\bm{x}}_t}J_t$ gives the exact Back-propagation dynamics.\n\\eq{\\ref{eq:Jt}} suggests that\nGD minimizes the quadratic expansion of $J_t$ with the Hessian $\\nabla^2_{{\\bm{u}}_t}J_t$ replaced by $\\frac{1}{\\eta}{\\bm{I}}_t$.\nSimilarly, adaptive first-order methods, such as RMSprop and Adam,\napproximate the Hessian\nwith the diagonal of the covariance matrix.\nSecond-order methods, such as KFAC and EKFAC \\citep{martens2015optimizing,george2018fast},\ncompute full matrices using Gauss-Newton (GN) approximation:\n\\begin{align} \\label{eq:kfac}\n    \\nabla^2_{{\\bm{u}}}J_t \\approx\n    \\mathbb{E}{[J_{{\\bm{u}}_t} J_{{\\bm{u}}_t}^{\\mathsf{T}}]}\n   = \\mathbb{E}{[({\\bm{x}}_t \\otimes J_{{\\bm{h}}_t}) ({\\bm{x}}_t \\otimes J_{{\\bm{h}}_t})^{\\mathsf{T}}]}\n   \\approx \\mathbb{E}{[({\\bm{x}}_t {\\bm{x}}_t^{\\mathsf{T}})]} \\otimes \\mathbb{E}{[( J_{{\\bm{h}}_t} J_{{\\bm{h}}_t}^{\\mathsf{T}})]}\n    { \\text{ .} }\n\\end{align}\n\n\n\nWe now draw a novel connection between the training procedure of DNNs and DDP.\nLet us first summarize the Back-propagation (BP) {with gradient descent} in Alg. \\ref{alg:bp} and compare it with DDP (Alg. \\ref{alg:ddp}).\nAt each training iteration, we treat the current weight as the {control} ${\\bar{{\\bm{u}}}}$ that simulates the activation sequence ${\\bar{{\\bm{x}}}}$.\nStarting from this {nominal} trajectory $({\\bar{{\\bm{x}}}},{\\bar{{\\bm{u}}}})$,\nboth algorithms\nrecursively define some layer-wise objectives ($J_t$ in \\eq{\\ref{eq:Jt}} vs $V_t$ in \\eq{\\ref{eq:bellman}}),\ncompute the weight\/control update from the quadratic expansions (\\eq{\\ref{eq:du-star}} vs \\eq{\\ref{eq:du-star-ddp}}),\nand then carry certain information ($\\nabla_{{\\bm{x}}_t}J_t$ vs $(V_{\\bm{x}}^t,V_{{\\bm{x}}\\vx}^t)$)\nbackward to the preceding layer.\nThe computation graph between the two approaches is summarized in Fig.~\\ref{fig:1}.\nIn the following proposition,\nwe make this connection formally and provide conditions when the two algorithms become equivalent.\n\n\\begin{minipage}[t]{0.49\\textwidth}\n\\vskip -0.1in\n\\begin{proposition} \\label{prop:bp2ddp}\nAssume $Q_{{\\bm{u}} {\\bm{x}}}^t=\\mathbf{0}$ at all stages,\nthen the backward dynamics of the value derivative can be described by the Back-propagation,\n\\begin{align}\n\\begin{split}\n\\forall t { \\text{ ,} }\nV_{{\\bm{x}}}^t = \\nabla_{{\\bm{x}}_t} J { \\text{ ,} } \\text{ } Q_{{\\bm{u}}}^t = \\nabla_{{\\bm{u}}_t} J\n{ \\text{ ,} } \\text{ }\nQ_{{\\bm{u}}\\vu}^t = \\nabla^2_{{\\bm{u}}_t}J { \\text{ .} }\n\\end{split}\n\\end{align}\nIn this case, the DDP policy is equivalent to the stage-wise Newton,\nin which the gradient is preconditioned by the block-wise inverse Hessian at each layer:\n\\begin{align}\n{{\\bm{k}}_t} + {{\\bm{K}}_t} {\\delta{\\bm{x}}}_t\n= - (\\nabla_{{\\bm{u}}_t}^2 J)^{{-1}}\\nabla_{{\\bm{u}}_t}J { \\text{ .} } \\label{eq:newton}\n\\end{align}\nIf further we have ${Q_{{\\bm{u}} {\\bm{u}}}^{t}} \\approx \\frac{1}{\\eta}{\\mathbf{I}}$,\nthen DDP degenerates to Back-propagation with gradient descent.\n\\end{proposition}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[t]{0.48\\textwidth}\n  \\vskip -0.27in\n  \\begin{figure}[H]\n    \\vspace{-10pt}\n    \\includegraphics[width=\\linewidth]{fig\/compute-graph4.png}\n    \\caption{\n    DDP backward propagates the value derivatives $(V_{\\bm{x}},V_{{\\bm{x}}\\vx})$ instead of $\\nabla_{{\\bm{x}}_t}J$\n    and updates weight using layer-wise feedback policy, $\\delta {\\bm{u}}^{*}_t(\\delta {\\bm{x}}_t)$, with additional forward propagation.}\n  \\label{fig:1}\n  \\end{figure}\n\\end{minipage}\n\nProof is left in Appendix \\ref{app:c1}.\nProposition \\ref{prop:bp2ddp} states that the backward pass in DDP collapses to BP when $Q_{{\\bm{u}} {\\bm{x}}}$ vanishes at all stages.\nIn other words, existing training methods can be seen as special cases of DDP\nwhen the mixed derivatives (\\emph{i.e.} $\\nabla_{{\\bm{x}}_t{\\bm{u}}_t}$) of the layer-wise objective are discarded.\n\n\\subsection{Efficient Approximation and Factorization} \\label{sec:eff-approx}\n\nMotivated by Proposition \\ref{prop:bp2ddp},\nwe now present a new class of optimizer, DDP Neural Optimizer (DDPNOpt), on training feedforward and convolution networks.\nDDPNOpt follows the same procedure in vanilla DDP (Alg.~\\ref{alg:ddp})\nyet adapts several key traits arising from DNN training,\nwhich we highlight below.\n\n\\textbf{Evaluate derivatives of $Q_t$ with layer dynamics.}\nThe primary computation in DDPNOpt comes from constructing the derivatives of $Q_t$ at each layer.\nWhen the dynamics is represented by the layer propagation (recall \\eq{\\ref{eq:dnn-dyn}} where we set $f_t \\equiv \\sigma_t \\circ g_t$), we can rewrite \\eq{\\ref{eq:Qt}} as:\n\\begin{equation}\n\\begin{split}\n {Q^t_{{\\bm{x}}}}  = {{g}^{t \\text{ }{\\mathsf{T}}}_{{\\bm{x}}}} {V^{t}_{{\\bm{h}}}} { \\text{ ,} } \\quad\n {Q^t_{{\\bm{u}}}}  = {{\\ell}^t_{{\\bm{u}}}}  + {{g}^{t \\text{ }{\\mathsf{T}}}_{{\\bm{u}}}} {V^{t}_{{\\bm{h}}}} { \\text{ ,} } \\quad\n {Q^t_{{\\bm{x}} {\\bm{x}}}} = {{g}^{t \\text{ }{\\mathsf{T}}}_{{\\bm{x}}}} {V^{t}_{{\\bm{h}}\\vh}} g^t_{{\\bm{x}}} { \\text{ ,} } \\quad\n {Q^t_{{\\bm{u}} {\\bm{x}}}} =        {{g}^{t \\text{ }{\\mathsf{T}}}_{{\\bm{u}}}} {V^{t}_{{\\bm{h}}\\vh}} g^t_{{\\bm{x}}} { \\text{ ,} }\n  \\label{eq:q-fc}\n\\end{split}\n\\end{equation}\nwhere\n${V^{t}_{{\\bm{h}}}}  \\triangleq {\\sigma^{t\\text{ } {\\mathsf{T}}}_{{\\bm{h}}}} {V^{t+1}_{{\\bm{x}}}}$ and\n${V^{t}_{{\\bm{h}}\\vh}} \\triangleq {\\sigma^{t\\text{ } {\\mathsf{T}}}_{{\\bm{h}}}} V^{t+1}_{{\\bm{x}} {\\bm{x}}} \\sigma^t_{{\\bm{h}}}$\nabsorb the computation of the non-parametrized activation function $\\sigma$.\n{\n  Note that \\eq{\\ref{eq:q-fc}} expands the dynamics only up to first order,\n  \\emph{i.e.} we omitt the tensor products which involves second-order expansions on dynamics,\n  as the stability obtained by keeping only the linearized dynamics is thoroughly discussed and widely adapted in practical DDP usages \\citep{todorov2005generalized}.\n}\nThe matrix-vector product with the Jacobian of the affine transform (\\emph{i.e.} ${{g}^{t}_{{\\bm{u}}}},{{g}^{t}_{{\\bm{x}}}}$) can be evaluated efficiently for both feedforward (FF) and convolution (Conv) layers:\n\\begin{alignat}{4}\n{\\bm{h}}_t &\\FCeq {\\bm{W}}_t{\\bm{x}}_t+{\\bm{b}}_t\n&&\\Rightarrow\n{g^t_{{\\bm{x}}}}^{\\mathsf{T}} V^{t}_{{\\bm{h}}} &&= {\\bm{W}}_t^{\\mathsf{T}} V^{t}_{{\\bm{h}}}\n{ \\text{ ,} } \\quad\n{g^t_{{\\bm{u}}}}^{\\mathsf{T}} V^{t}_{{\\bm{h}}} &&= {\\bm{x}}_t \\otimes V^{t}_{{\\bm{h}}}\n{ \\text{ ,} } \\\\\n{\\bm{h}}_t &\\CONVeq \\omega_t * {\\bm{x}}_t\n&&\\Rightarrow\n{g^t_{{\\bm{x}}}}^{\\mathsf{T}} V^{t}_{{\\bm{h}}} &&= \\omega_t^{\\mathsf{T}} {\\text{ } \\hat{*} \\text{ }} V^{t}_{{\\bm{h}}}\n{ \\text{ ,} } \\quad\n{g^t_{{\\bm{u}}}}^{\\mathsf{T}} V^{t}_{{\\bm{h}}} &&= {\\bm{x}}_t {\\text{ } \\hat{*} \\text{ }} V^{t}_{{\\bm{h}}}\n{ \\text{ ,} }\n\\end{alignat}\nwhere $\\otimes$, ${\\text{ } \\hat{*} \\text{ }}$, and $*$ respectively denote the Kronecker product and (de-)convolution operator.\n\n\n\n\n\n\n\n\n\n\\input{subtex\/table-optimizer-relation}\n\n\n\\textbf{Curvature approximation.}\n{\n  Next, since DNNs are highly over-parametrized models, ${\\bm{u}}_t$ (\\emph{i.e.} the layer weight) will be in high-dimensional space.}\nThis makes ${Q^t_{{\\bm{u}} {\\bm{u}}}}$ and $({Q^t_{{\\bm{u}} {\\bm{u}}}})^{-1}$ computationally intractable to solve; thus requires approximation.\nRecall the interpretation we draw in \\eq{\\ref{eq:Jt}}\nwhere existing optimizers differ in approximating the Hessian $\\nabla_{{\\bm{u}}_t}^2J_t$.\nDDPNOpt adapts the same curvature approximation to ${Q^t_{{\\bm{u}} {\\bm{u}}}}$.\nFor instance, we can approximate ${Q^t_{{\\bm{u}} {\\bm{u}}}}$ simply with\nan identity matrix ${\\bm{I}}_t$, adaptive diagonal matrix $\\diag(\\mathpalette\\DHLhksqrt{\\mathbb{E}[{Q^t_{{\\bm{u}}}} \\odot {Q^t_{{\\bm{u}}}}]})$, or the GN matrix:\n\\begin{align} \\label{eq:ekfac-ddp}\n    {Q}^t_{{\\bm{u}} {\\bm{u}}} \\approx\n    \\mathbb{E}{[Q^t_{{\\bm{u}}} {Q^t_{{\\bm{u}}}}^{\\mathsf{T}}]}\n   =\\mathbb{E}{[({\\bm{x}}_t \\otimes {V^{t}_{{\\bm{h}}}}) ({\\bm{x}}_t \\otimes {V^{t}_{{\\bm{h}}}})^{\\mathsf{T}}]}\n   \\approx \\mathbb{E}{[{\\bm{x}}_t {\\bm{x}}_t^{\\mathsf{T}}]} \\otimes \\mathbb{E}{[{V^{t}_{{\\bm{h}}}} {V^{t}_{{\\bm{h}}}}^{\\mathsf{T}}]}\n    { \\text{ .} }\n\\end{align}\n\nTable \\ref{table:update-rule} summarizes the difference in curvature approximation\n(\\emph{i.e.} the precondition ${\\bm{M}}_t$ ) for different methods.\nNote that DDPNOpt constructs these approximations using $(V,Q)$ rather than $J$\nsince they consider different layer-wise objectives. %\nAs a direct implication from Proposition $\\ref{prop:bp2ddp}$,\nDDPNOpt will degenerate to the optimizer it adapts for curvature approximation\nwhenever all ${Q^t_{{\\bm{u}} {\\bm{x}}}}$ vanish.\n\n\n\\textbf{Outer-product factorization.}\nWhen the memory efficiency becomes nonnegligible as the problem scales,\nwe make GN approximation to $\\nabla^2\\phi$,\nsince the low-rank structure at the prediction layer has been observed\nfor problems concerned in this work \\citep{nar2019cross,lezama2018ole}.\nIn the following proposition, we show that\nfor a specific type of OCP, which happens to be the case of DNN training,\nsuch a low-rank structure preserves throughout the DDP backward pass.\n\\begin{proposition}[Outer-product factorization in DDPNOpt] \\label{prop:gn-ddp}\n  Consider the OCP where $\\ell_t \\equiv \\ell_t({\\bm{u}}_t)$ is independent of ${\\bm{x}}_t$,\n  If the terminal-stage Hessian can be expressed by the outer product of vector ${\\bm{z}}_{\\bm{x}}^T$,\n  $\\nabla^2\\phi({\\bm{x}}_{T}) = {\\bm{z}}_{\\bm{x}}^T \\otimes {\\bm{z}}_{\\bm{x}}^T$ (for instance, ${\\bm{z}}_{\\bm{x}}^T=\\nabla \\phi$ for GN), then we have the factorization for all $t$:\n  \\begin{equation}\n  \\begin{split}\n  {Q^t_{{\\bm{u}} {\\bm{x}}}} = {\\bm{q}}_{\\bm{u}}^t \\otimes {\\bm{q}}_{\\bm{x}}^t { \\text{ ,} } \\quad\n  {Q^t_{{\\bm{x}} {\\bm{x}}}} = {\\bm{q}}_{\\bm{x}}^t \\otimes {\\bm{q}}_{\\bm{x}}^t { \\text{ ,} } \\quad\n  V_{{\\bm{x}}\\vx}^t = {\\bm{z}}_{\\bm{x}}^t \\otimes {\\bm{z}}_{\\bm{x}}^t { \\text{ .} }\n  \\end{split} \\label{eq:qxqu}\n  \\end{equation}\n  ${\\bm{q}}_{\\bm{u}}^t$, ${\\bm{q}}_{\\bm{x}}^t$, and ${\\bm{z}}_{\\bm{x}}^t$\n  are outer-product vectors\n  which are also computed along the backward pass.\n  \\begin{align}\n  {\\bm{q}}_{\\bm{u}}^t = {{{f}^t_{{\\bm{u}}}}^{\\mathsf{T}}} {\\bm{z}}_{\\bm{x}}^{t+1} { \\text{ ,} } \\quad\n  {\\bm{q}}_{\\bm{x}}^t = {{{f}^t_{{\\bm{x}}}}^{\\mathsf{T}}} {\\bm{z}}_{\\bm{x}}^{t+1} { \\text{ ,} } \\quad\n  {\\bm{z}}_{\\bm{x}}^t = \\mathpalette\\DHLhksqrt{1-{\\bm{q}}_{\\bm{u}}^{t \\text{ } {\\mathsf{T}}} {({Q^{t}_{{\\bm{u}} {\\bm{u}}})}^{-1}} {\\bm{q}}_{\\bm{u}}^t } \\text{ }{\\bm{q}}_{\\bm{x}}^t\n  { \\text{ .} }\n  \\label{eq:vxvx}\n  \\end{align}\n\\end{proposition}\n\\vskip -0.08in\nThe derivation is left in Appendix \\ref{app:prop:gn-ddp}.\nIn other words,\nthe outer-product factorization at the final layer can be backward propagated to all proceeding layers.\nThus, large matrices, such as ${Q^t_{{\\bm{u}} {\\bm{x}}}}$, ${Q^t_{{\\bm{x}} {\\bm{x}}}}$, $V_{{\\bm{x}}\\vx}^t$, and even feedback policies ${{\\bm{K}}_t}$,\ncan be factorized accordingly, greatly reducing the complexity. %\n\n\n\n\n\\vspace{-8pt}\n\\input{subtex\/pseudo-code}\n\\vspace{-10pt}\n\n{\n  \\textbf{Regularization on $V_{{\\bm{x}}\\vx}$.}\n  Finally,\n  we apply Tikhonov regularization to the value Hessian $V^t_{{\\bm{x}}\\vx}$ (line $12$ in Alg.~\\ref{alg:ddpnopt}).\n  This can be seen as placing a quadratic state-cost and has been shown to improve stability on optimizing complex humanoid behavior \\citep{tassa2012synthesis}.\n  For the application of DNN where the dimension of the state ({\\emph{i.e.}} the vectorized activation) varies during forward\/backward pass,\n  the Tikhonov regularization prevents the value Hessian from low rank (throught ${{g}^{t \\text{ }{\\mathsf{T}}}_{{\\bm{u}}}} {V^{t}_{{\\bm{h}}\\vh}} g^t_{{\\bm{x}}}$);\n  hence we also observe similar stabilization effect in practice.\n}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nAlthough standard four-dimensional (4D) General Relativity (GR) is\nbelieved to be the correct description of gravity at the classical level,\nits quantization faces many well-known problems. Therefore,\nthree-dimensional (3D) gravity has gained much interest, since classically\nit is much simpler and thus one can investigate more efficiently its\nquantization. Amongst others, in 3D gravity one obtains the\nBanados-Teitelboim-Zanelli (BTZ) black hole~\\cite{BTZ}, which is a\nsolution to the Einstein equations with a negative cosmological constant.\nThis black-hole solution presents interesting properties at both classical\nand quantum levels, and it shares several features of the Kerr black hole\nof 4D GR~\\cite{Carlip}. \n\nFurthermore, remarkable attention was addressed recently to topologically\nmassive gravity, which is a generalization of 3D GR that amounts to\naugment the Einstein-Hilbert action by adding a Chern-Simons gravitational\nterm, and thus the propagating degree of freedom is a massive\ngraviton, which amongst others also admits BTZ black-hole as exact\nsolutions~\\cite{deser}. The renewed interest on  topologically\nmassive gravity relies on the possibility of\nconstructing a chiral theory of gravity at a special point of the\nparameter-space, as it was suggested in~\\cite{Li:2008dq}. This idea has\nbeen extensively analyzed in the last years~\\cite{Strominger:2008dp}, leading to a fruitful\ndiscussion that ultimately led to a significantly better understanding of\nthe model~\\cite{Maloney:2009ck}. Moreover, another\n3D massive gravity theory known as new massive gravity \\cite{Bergshoeff:2009hq, Bergshoeff:2009tb} (where the action is given by the\nEinstein-Hilbert term plus a specific\ncombination of square-curvature terms which gives rise to\nfield equations with a second order trace) have attracted considerable attention, this theory also admits interesting solutions, see for instance \\cite{Clement:2009gq, Clement:2009ka, Oliva:2009ip}. Furthermore, 3D gravity with torsion has been extensively studied in~\\cite{3dgravitywithtorsion} and references therein.\n\n\n\nOn the other hand, hairy black holes are interesting solutions of Einstein's Theory\nof Gravity and also of certain types of Modified Gravity Theories. The first attempts to couple a scalar field to gravity was done in\nan asymptotically flat spacetime. Then, hairy black hole solutions\nwere found \\cite{BBMB} but these\nsolutions were not examples of hairy black hole configurations\nviolating the no-hair theorems because they were not physically\nacceptable as the scalar field was divergent on the horizon and\nstability analysis showed that they were unstable\n\\cite{bronnikov}. To remedy this a regularization procedure has to\nbe used to make the scalar field finite on the horizon. Hairy black hole solutions have been extensively studied over the years mainly in connection\n with the no-hair theorems. The recent developments in string theory and\nspecially the application of the AdS\/CFT principle to condense\nmatter phenomena like superconductivity (for a review see\n\\cite{Hartnoll:2009sz}), triggered the interest of further  study\nof the behavior of matter fields outside the black hole horizon\n\\cite{Gubser:2005ih,Gubser:2008px}. There are also very\ninteresting recent developments in Observational Astronomy. High\nprecision astronomical observations of the supermassive black\nholes may pave the way to  experimentally test the no-hair\nconjecture\n \\cite{Sadeghian:2011ub}. Also, there are numerical investigations\n of single and binary black holes in the presence of scalar fields\n\\cite{Berti:2013gfa}. The aforementioned is a small part on the relevance that  has taken the study of hairy black holes currently in the field of physics,  for more details see for instance \\cite{Gonzalez:2013aca, Gonzalez:2014tga} and references therein. Also, we refer the reader to references \\cite{Martinez:1996gn, Henneaux:2002wm, Zhao:2013isa, Xu:2014uka, Cardenas:2014kaa} and references therein, where black holes solutions in three space-time dimensions with a scalar field (minimally and\/or confomally) coupled to gravity have been investigated.\n\n\n\n\n\nIn the present work we are interested in investigating the existence of 3D hairy black holes solutions for theories based on torsion. In particular, the so-called ``teleparallel\nequivalent of General Relativity\" (TEGR) \\cite{ein28,Hayashi79} is an\nequivalent formulation of gravity but instead of using the curvature\ndefined via the Levi-Civita connection, it uses the Weitzenb{\\\"o}ck\nconnection that has no curvature but only torsion. So, we consider a scalar field non-minimally coupled with the torsion scalar, with a self-interacting potential in TEGR, and we find three-dimensional asymptotically AdS, hairy black holes. It is worth  mentioning, that this kind of theory (known as scalar-torsion theory), has been studied in the cosmological context, where the dark energy sector is attributed to the scalar field. It was shown that the minimal case is equivalent to standard quintessence. However,  the nonminimal case has a richer structure, exhibiting quintessence-like or phantom-like behavior, or experiencing the phantom-divide crossing \\cite{Geng:2011aj, Geng:2011ka, Gu:2012ww}, see also \\cite{Horvat:2014xwa} for aplications of this theory (with a complex scalar field) to boson stars.\n\nIt is also worth to mention that a natural extension of TEGR is the so called $f(T)$ gravity, which is represented by a function of the scalar torsion $T$ as Lagrangian density \\cite{Ferraro:2006jd, Ferraro:2008ey, Bengochea:2008gz,Linder:2010py}.\nThe $f(T)$ theories picks up preferred referential frames which constitute the autoparallel curves of the given manifold. \nA genuine advantage of $f(T)$ gravity  compared with other deformed gravitational schemes is that the differential equations for the vielbein components are second order differential equations. However, the effects of the additional degrees of freedom that certainly exist in $f(T)$ theories is a consequence of breaking the local Lorentz invariance that  these theories exhibit. Despite this, it was found that on the flat FRW background with a scalar field,  up to second order linear perturbations  does not reveal any extra degree of freedom at all \\cite{Izumi:2012qj}. As such, it is fair to say  that the nature of these additional degrees of freedom remains unknown. Remarkably, it is possible to modify $f(T)$ theory in order to make it manifestly Lorentz invariant. However, it will generically have different dynamics and will reduce to $f(T)$ gravity in some local Lorentz frames \\cite{Li:2010cg, Weinberg, Arcos:2010gi}.\nClearly, in extending this\ngeometry sector, one of the goals is to solve the puzzle of dark energy and\ndark matter without asking for new material ingredients that have\nnot yet been detected  by  experiments \\cite{Capozziello:2007ec,Ghosh:2012pg}. For instance, a Born-Infeld $f(T)$ gravity Lagrangian was used to address the physically inadmissible divergencies occurring in the standard cosmological Big Bang model, rendering the spacetime geodesically complete and powering an inflationary stage without the introduction of an inflaton field \\cite{Ferraro:2008ey}. Also, it is believed that $f(T)$ gravity could be a reliable approach to address the shortcomings of general relativity at high energy scales \\cite{Capozziello:2011et}.  Furthermore, both inflation and the dark energy dominated stage can be realized in Kaluza-Klein and Randall-Sundrum models, respectively \\cite{Bamba:2013fta}.\nIn this way, $f(T)$ gravity has gained attention and\nhas been proven to exhibit interesting cosmological implications. On the other hand, the search for black hole solutions in $f(T)$ gravity is not a trivial problem, and there are only few exact solutions, see for instance \\cite{G1, solutions,Rodrigues:2013ifa}. Remarkably, it is possible to construct other generalizations, as Teleparallel Equivalent of Gauss-Bonnet Gravity \\cite{Kofinas:2014owa, Kofinas:2014daa}, Kaluza-Klein theory for teleparallel gravity \\cite{Geng:2014nfa} and scalar-torsion gravity theories \\cite{Geng:2011aj, Kofinas:2015hla}. \n\n\n\n\n\n\n\n\n\n\n\n\nThe paper is organized as follows. In Section II we give a brief review of three-dimensional Teleparallel Gravity. Then, in Section III we find asymptotically AdS black holes with scalar hair, and we conclude in Section IV with final remarks.\n\n\\section{3D Teleparallel  Gravity}\n\\label{Tel3D}\n\nIn 1928, Einstein proposed the idea of teleparallelism to unify gravity and electromagnetism into a unified field theory; this corresponds to an equivalent formulation of General Relativity (GR), nowadays known as Teleparallel Equivalent to General Relativity (TEGR) \\cite{ein28, Hayashi79}, where the Weitzenb\\\"{o}ck connection is used  to define the covariant derivative (instead of the Levi-Civita connection which is used to define the covariant derivative in the context of GR). The first \n investigations on teleparallel 3D gravity were\nperformed by Kawai almost twenty years ago \\cite{Kawai1,Kawai2,Kawai3}. The Weitzenb\\\"{o}ck connection mentioned above has not null torsion. However, it is curvatureless, which implies that this formulation of gravity exhibits only torsion. The Lagrangian density $T$ is constructed from the torsion tensor.\nTo clarify, the torsion scalar $T$ is the result of a very specific quadratic combination of irreducible representations of the torsion tensor under the Lorentz group $SO(1,3)$ \\cite{Hehl:1994ue}. In this way,  the torsion tensor in TEGR  \nincludes all the information concerning  the\ngravitational field.\nThe theory is called ``Teleparallel\nEquivalent to General Relativity'' since the field equations are exactly the same as those of GR for every geometry choice.\n\nThe Lagrangian of teleparallel 3D gravity corresponds to\nthe more general quadratic Lagrangian for torsion, under the\nassumption of zero spin-connection. So, the action can be written as \\cite{Muench:1998ay,Itin:1999wi}\n\\begin{equation}  \\label{action2}\nS=\\frac{1}{2 \\kappa}\\int \\left( \\rho_{0} \\mathcal{L}_{0}+ \\rho_{1}\n\\mathcal{L%\n}_{1}+ \\rho_{2} \\mathcal{L}_{2}+\\rho_{3} \\mathcal{L}_{3}+ \\rho_{4}\n\\mathcal{L%\n}_{4}\\right)~,\n\\end{equation}%\nwhere $\\kappa$ is the three-dimensional gravitational constant,\n$\\rho_i$ are parameters, and\n\\begin{equation}\n\\mathcal{L}_{0}= \\frac{1}{4}e^{a} \\wedge \\star e_a~,\\quad \\mathcal{L}_{1}=de^{a}\n\\wedge \\star de_{a}~,\\quad \\mathcal{L}_{2}= (de_{a} \\wedge \\star e^a)\n\\wedge \\star (de_b \\wedge e^b)~,\\nonumber\n\\end{equation}\n\\begin{equation}\n\\mathcal{L}_{3}=(de^{a} \\wedge e^{b}) \\wedge \\star (de_{a} \\wedge\ne_{b})~,\\quad \\mathcal{L}_{4}= (de_{a} \\wedge \\star e^b) \\wedge\n\\star (de_b \\wedge e^a)~,\n\\end{equation}\nwhere $e^a$ denotes the vielbein, $d$ is the exterior derivative, $\\star $ denotes the Hodge dual operator and $\\wedge$ the wedge\nproduct. The coupling constant $\\rho_{0}=-\\frac{8}{3} \\Lambda$ represents\nthe cosmological constant term.  Moreover, since $\\mathcal{L}_{3}$ can be\nwritten completely in terms of $\\mathcal{L}%\n_{1}$, in the following we set $\\rho_{3}=0$ \\cite{Muench:1998ay}.  Action (\\ref{action2}) can be written in a more convenient form as\n\\begin{equation}\n\\label{actiontele0}\nS=\\frac{1}{2\\kappa} \\int \\left (T -2\\Lambda   \\right )\\star 1~,\n\\end{equation}\nwhere $\\star1=e^{0} \\wedge e^{1} \\wedge e^{2}$, and the torsion\nscalar $T$ is given by\n\\begin{equation}  \\label{scalartorsion}\nT= \\star \\left[\\rho_{1}(de^{a} \\wedge \\star de_{a})+\\rho_{2}(de_{a} \\wedge\ne^a) \\wedge \\star (de_b \\wedge e^b)+\\rho_{4}(de_{a} \\wedge\ne^b) \\wedge \\star (de_b \\wedge e^a) \\right]~.\n\\end{equation}\nExpanding this expression in terms of its components, the torsion scalar yields\n\\begin{equation}\n\\label{scalartorsionrho}\nT=\\frac{1}{2} (\\rho_{1}+\\rho_{2}+\\rho_{4})T^{abc}T_{abc}+%\n\\rho_{2}T^{abc}T_{bca}-\\rho_{4}T_{a}^{ac}T^{b}_{bc}~,\n\\end{equation}\nnote that for TEGR $\\rho_{1}=0$, $\\rho_{2}=-\\frac{1}{2}$ and $\\rho_{4}=1$. \nA variation of  action (\\ref{actiontele0}) with respect to the\nvielbein provides the following field equations:\n\\begin{eqnarray}\\label{fieldequations}\n&&\\delta \\mathcal{L} =\\delta e^{a}\\wedge \\left\\{\\left\\{\\rho\n_{1}\\left[2d\\star de_{a}+i_{a}(de^{b}\\wedge \\star\nde_{b})-2i_{a}(de^{b})\\wedge\n\\star de_{b}\\right]\n\\right.\\right.\\nonumber\n\\\\ && \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n+\\rho _{2}\\left\\{-2e_{a}\\wedge d\\star (de^{b}\\wedge\ne_{b})\n+2de_{a}\\wedge \\star (de^{b}\\wedge e_{b})+i_{a}\\left[de^{c}\\wedge\ne_{c}\\wedge\n\\star (de^{b}\\wedge e_{b})\\right]\\right.\\nonumber\\\\\n&&\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\, \\ \\ \\ \\ \\ \\left.\n-2i_{a}(de^{b})\\wedge e_{b}\\wedge \\star\n(de^{c}\\wedge e_{c})\\right\\}\n \\nonumber\\\\\n&& \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ +\\rho_{4}\\left\\{-2e_{b}\\wedge\nd\\star (e_{a}\\wedge de^{b})+2de_{b}\\wedge \\star\n(e_{a}\\wedge de^{b})+i_{a}\\left[e_{c}\\wedge de^{b}\\wedge \\star\n(de^{c}\\wedge\ne_{b})\\right]\\right.\\nonumber \\\\\n&&\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\, \\ \\ \\ \\ \\ \\left.\\left.\n -2i_{a}(de^{b})\\wedge e_{c}\\wedge \\star (de^{c}\\wedge\ne_{b})\\right\\}\\right\\}\\nonumber\\\\\n&&\\ \\left. \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n-2\\Lambda \\star e_a  \\right\\}\n=0~,\n\\end{eqnarray}\nwhere $i_a$ is the interior product and for generality's sake we have kept the\ngeneral\ncoefficients $\\rho_i$, and we have used $\\epsilon^{012}=+1$. Through the following choice of the coefficients $\\rho_{1}=0$, $\\rho_{2}=-\\frac{1}{2}$ and $\\rho_{4}=1$ Teleparallel Gravity  coincides with the\nusual curvature-formulation of General Relativity and therefore the following BTZ metric is solution of TEGR  \n \\begin{equation}\n  \\label{metric}\nds^2=N^2dt^2-N^{-2}dr^2-r^2(d\\varphi+N_{\\varphi}dt)^2~,\n\\end{equation}\nwhere the lapse $N$ and shift $N_{\\varphi}$ functions are given by,\n\\begin{equation}\nN^2= -8GM+\\frac{r^2}{l^2}+\\frac{16G^2J^2}{r^2}~,\\quad\nN_{\\varphi}=-\\frac{4GJ}{r^2},\n\\label{BTZ0}\n\\end{equation}\nand the two constants of integration $M$ and $J$ are the usual conserved\ncharges associated with the asymptotic invariance under time displacements\n(mass) and rotational invariance (angular momentum)\nrespectively,  given by flux integrals\nthrough a large circle at spacelike infinity, and $\\Lambda=-1\/l^2$ is the\ncosmological constant \\cite{BTZ}.\nFinally, note\nthat \nthe torsion scalar can be  \ncalculated, leading to\nthe constant value\n\\begin{equation}\n\\label{Tteleresult}\nT=-2\\Lambda,\n\\end{equation}\nwhich is the cosmological constant as the sole source of torsion.\\\\\n\n\n\n\\section{3D Teleparallel Hairy Black Holes}\n\\label{Tel3DH}\n\\subsection{The Model}\n\nIn this section we will extend the above discussion considering a scalar field $\\phi$ non-minimally coupled with the torsion scalar with a self-interacting potential $V(\\phi)$, and then we will find hairy black hole solutions. \n So, the action can be written as \n\\begin{equation}\n  \\label{accionHT}\nS=\\int \\left( \\frac{1}{2 \\kappa} T \\star 1 - \\xi \\phi^2 T \\star 1 + \\frac{1}{2} d\\phi \\wedge \\star d\\phi -V(\\phi)\\star 1\\right)~,\n\\end{equation}\nwhere $T$ is given by (\\ref{scalartorsion}) and $\\xi$ is the non-minimal coupling parameter. \nThus, the variation with respect to the vielbein leads to\n the following field equations:\n\\begin{eqnarray}  \\label{fieldeq}\n\\delta_{e^{a}} \\mathcal{L} &=&\\delta e^{a}\\wedge\n\\left\\{\\left(\\frac{1}{2\\kappa}-\\xi\\phi^2\\right)\n\\left\\{\\rho\n_{1}\\left[2d\\star de_{a}+i_{a}(de^{b}\\wedge \\star\nde_{b})-2i_{a}(de^{b})\\wedge\n\\star de_{b}\\right]\\right.\\right.\\nonumber\n\\\\\n&&\\ \\ \\ \\ \\ \\ \\ \\ +\\rho _{2}\\left\\{-2e_{a}\\wedge d\\star (de^{b}\\wedge\ne_{b})+2de_{a}\\wedge \\star\n(de^{b}\\wedge e_{b})+i_{a}\\left[de^{c}\\wedge e_{c}\\wedge\n\\star (de^{b}\\wedge e_{b})\\right] \\right.\n\\nonumber\\\\\n&&\\left. \\ \\ \\ \\ \\ \\ \\ \\  \\ \\ \\ \\ \\ \\ \\ \\,  -2i_{a}(de^{b})\\wedge\ne_{b}\\wedge \\star\n(de^{c}\\wedge e_{c})\\right\\}\\nonumber\\\\\n&&\\ \\ \\ \\ \\ \\ \\ \\ \n+\\rho_{4}\\left\\{-2e_{b}\\wedge d\\star (e_{a}\\wedge\nde^{b})+2de_{b}\\wedge \\star\n(e_{a}\\wedge de^{b})\\right.\n\\nonumber \\\\\n&&\\left.\\left.\\ \\ \\ \\ \\ \\ \\ \\  \\ \\ \\ \\ \\ \\ \\ \\,\n+i_{a}\\left[e_{c}\\wedge de^{b}\\wedge \\star (de^{c}\\wedge\ne_{b})\\right] -2i_{a}(de^{b})\\wedge e_{c}\\wedge \\star (de^{c}\\wedge\ne_{b})\\right\\}\n\\right\\}\\nonumber\\\\\n&&\\ \\ \\ \\ \\ \\ \\ \\ -4\\xi\\left[\\rho_1\\phi d\\phi \\wedge \\star de_a+\\rho_2 \\phi d\\phi\\wedge\ne_a\\wedge\n\\star(de_b\\wedge e^b)+\\rho_4\\phi d\\phi \\wedge e_b\\wedge \\star(de^b \\wedge\ne_a)\\right]\\nonumber\\\\\n&&\\left. \\ \\ \\ \\ \\ \\ \\ \\ - V(\\phi) i_a(\\star 1)-\\frac{1}{2}d\\phi\\wedge i_a(\\star d\\phi)-\\frac{1}{2} i_a(d\\phi)\\wedge\\star d\\phi    \\right\\}=0~,\n\\label{fieldeq000}\n\\end{eqnarray}\nand the variation with respect to the scalar field leads to the Klein-Gordon equation \n\\begin{equation}\n\\label{fieldeq001}\n\\delta_{\\phi} \\mathcal{L}=\\delta \\phi \\left(-2\\xi\\phi T \\star 1-d\\star d\\phi - \\frac{dV}{d\\phi}\\star 1 \\right)=0~.\n\\end{equation}\n\n\n\\subsection{Circularly Symmetric Hairy Solutions}\n\\label{circsymmsol}\nLet us now investigate hairy black hole solutions of the theory. In order to\nanalyze static solutions we consider\nthe metric form as\n\\begin{equation}\\label{metric}\nds^{2}=A\\left( r\\right) ^{2}dt^{2}-\\frac{1}{B\\left( r\\right) ^{2}}\ndr^{2}-r^{2}d\\varphi^{2}~,\n\\end{equation}\nwhich arises from the triad \ndiagonal ansatz \n\\begin{equation}\\label{diagonal}\ne^{0}=A\\left( r\\right) dt~,\\text{ \\ }e^{1}=\\frac{1}{B\\left( r\\right) }dr~,\n\\text{ \\ }e^{2}=rd\\varphi~.\n\\end{equation}\nThen, inserting this vielbein in the field equations (\\ref{fieldeq000}), (\\ref{fieldeq001}) yields\n\\begin{equation}\n-\\frac{1}{r}{(\\frac{1}{2\\kappa}-\\xi\\phi(r)^2)\\frac{dB^2}{dr}}+\\frac{4}{r}\\xi B(r)^2\\phi(r)\\frac{d\\phi}{dr}-\\frac{1}{2}B(r)^2(\\frac{d\\phi}{dr})^2-V(\\phi)=0~,\n\\label{q1}\n\\end{equation}\n\\begin{equation}\n\\frac{B(r)^2}{rA(r)^2}{(\\frac{1}{2\\kappa}-\\xi\\phi(r)^2)\\frac{dA^2}{dr}}-\\frac{1}{2}B(r)^2(\\frac{d\\phi}{dr})^2+V(\\phi)=0~, \n\\label{q2}\n\\end{equation}\n\\begin{eqnarray}\n\\notag&& 2 \\xi\\phi(r) \\frac{d\\phi}{dr}\\frac{B(r)^2}{A(r)^2}\\frac{dA^2}{dr}-\\frac{1}{2}B(r)^2(\\frac{d\\phi}{dr})^2\\\\\n\\notag&&+\\frac{1}{2A(r)^4}(\\frac{1}{2\\kappa}-\\xi\\phi(r)^2)\\left(-A(r)^2\\frac{dA^2}{dr}\\frac{dB^2}{dr}+B(r)^2(\\frac{dA^2}{dr})^2-2A(r)^2 B(r)^2\\frac{d^2A^2}{dr^2}\\right)\\\\\n&&-V(\\phi)=0~,\n\\label{q3}\n\\end{eqnarray}\n\\begin{equation}\n-\\frac{2B(r)^2}{rA(r)^2}\\xi\\phi(r)\\frac{dA^2}{dr}+\\frac{1}{r}B(r)^2\\frac{d\\phi}{dr}+\\frac{1}{2}\\frac{dB(r)^2}{dr}\\frac{d\\phi}{dr}+\\frac{B(r)^2}{2A(r)^2}\\frac{dA(r)^2}{dr}\\frac{d\\phi}{dr}+B(r)^2\\frac{d^2\\phi}{dr^2}-\\frac{dV}{d\\phi}=0~.\n\\label{q4}\n\\end{equation}\n\nIt is worth mentioning that, in the case of a minimally coupled scalar field,  \n the above  simple, diagonal  relation \n between the metric and the vielbeins (\\ref{diagonal}) is\nalways allowed, due to in this case the theory is invariant under local Lorentz transformations of the vielbein. In contrast, in the extension of a non-minimally coupled scalar field with the torsion scalar, the theory is not local Lorentz invariant, therefore, one could have in general a more complicated relation connecting the vielbein with the metric, with the vielbeins being non-diagonal even for a diagonal\nmetric \\cite{fTLorinv0}. However, for the three-dimensional solutions considered here, using a preferred diagonal frame is allowed, in the sense that this frame defines a global set of basis covering the whole tangent bundle, i.e., they parallelize the spacetime \\cite{Fiorini:2013hva}, \\cite{Ferraro:2011us}.\n\nIn the following, and in order to solve the above system of equations, we will consider two cases:  first, we analyze the case $A(r)=B(r)$, and then we analyze the more general case $A(r) \\neq B(r)$.\n\\subsubsection{$A(r)=B(r)$}\nIn this case the field equations (\\ref{q1})-(\\ref{q4}) simplify to\n\\begin{equation}\n-\\frac{1}{r}{(\\frac{1}{2\\kappa}-\\xi\\phi(r)^2)\\frac{dA^2}{dr}}+\\frac{4}{r}\\xi A(r)^2\\phi(r)\\frac{d\\phi}{dr}-\\frac{1}{2}A(r)^2(\\frac{d\\phi}{dr})^2-V(\\phi)=0~, \n\\label{fieldequation1}\n\\end{equation}\n\\begin{equation}\n\\frac{1}{r}{(\\frac{1}{2\\kappa}-\\xi\\phi(r)^2)\\frac{dA^2}{dr}}-\\frac{1}{2}A(r)^2(\\frac{d\\phi}{dr})^2+V(\\phi)=0~, \n\\label{fieldequation2}\n\\end{equation}\n\\begin{equation}\n2\\xi\\phi(r) \\frac{d\\phi}{dr}\\frac{dA^2}{dr}-\\frac{1}{2}A(r)^2(\\frac{d\\phi}{dr})^2-(\\frac{1}{2\\kappa}-\\xi\\phi(r)^2)\\frac{d^2A^2}{dr^2}-V(\\phi)=0~,\n\\label{fieldequation3}\n\\end{equation}\n\\begin{equation}\n-\\frac{2}{r}\\xi\\phi(r)\\frac{dA^2}{dr}+\\frac{1}{r}A(r)^2\\frac{d\\phi}{dr}+\\frac{dA(r)^2}{dr}\\frac{d\\phi}{dr}+A(r)^2\\frac{d^2\\phi}{dr^2}-\\frac{dV}{d\\phi}=0~.\n\\label{fieldequation4}\n\\end{equation}\nNow, by adding equations (\\ref{fieldequation1}) and (\\ref{fieldequation2}) we obtain\n\\begin{equation}\nA(r)^2\\frac{d \\phi}{dr}\\left( \\frac{4 \\xi}{r} \\phi-\\frac{d \\phi}{dr}\\right)=0~.\n\\end{equation}\nTherefore, the nontrivial solution for the scalar field is given by\n\\begin{equation}\n\\phi(r)=Br^{4\\xi} ~,\n\\end{equation}\nand by using this profile for the scalar field in the remaining equations, we obtain the solution\n\\begin{equation}\n\\label{A}\nA(r)^2=Gr^{2}+H {}_{2}F_1(1,-\\frac{1}{4\\xi}, 1-\\frac{1}{4\\xi}, 2\\kappa B^2\\xi r^{8\\xi})~,\n\\end{equation}\n\\begin{eqnarray}\nV(\\phi) & = & \\frac{H}{\\kappa}\\left(\\frac{\\phi}{B}\\right)^{-\\frac{1}{2\\xi}}+2G\\left(-\\frac{1}{2\\kappa}+B^2\\xi(1+4\\xi)\\left(\\frac{\\phi}{B}\\right)^2\\right) \\\\ \\notag \n&& -2H\\left(\\frac{\\phi}{B}\\right)^{-\\frac{1}{2\\xi}}\\left(\\frac{1}{2\\kappa}-B^2\\xi(1+4\\xi)\\left(\\frac{\\phi}{B}\\right)^2\\right){}_{2}F_1(1,-\\frac{1}{4\\xi}, 1-\\frac{1}{4\\xi}, 2\\kappa \\xi \\phi^2)~,\n\\end{eqnarray}\nwhere $B$, $G$, and $H$ are integration constant and ${}_{2}F_{1}$ is the Gauss hypergeometric function. In the limits $\\xi \\rightarrow 0$ or $B \\rightarrow 0$ the theory reduces to TEGR, therefore, we must hope our solution reduces to the BTZ black hole, this is indeed the case, as we show below. For those limits we obtain:\n\\begin{eqnarray}\n\\lim_{\\xi \\rightarrow 0} \\,\\,{}_{2}F_1(1,-\\frac{1}{4\\xi}, 1-\\frac{1}{4\\xi}, 2\\kappa B^2\\xi r^{8\\xi})=1~,\\\\\n\\lim_{B \\rightarrow 0} \\,\\,{}_{2}F_1(1,-\\frac{1}{4\\xi}, 1-\\frac{1}{4\\xi}, 2\\kappa B^2\\xi r^{8\\xi})=1~,\n\\end{eqnarray}\ntherefore,\n\\begin{equation}\n\\lim_{\\xi \\rightarrow 0 \\,\\, or \\,\\, B \\rightarrow 0} \\,\\, A(r)^2=Gr^{2}+H~,\n\\end{equation}\nwhich is the non-rotating BTZ metric.\nIn order to see the asymptotic behavior of $A(r)^2$, we expand the hypergeometric function for large $r$ and $\\xi<0$:\n\\begin{equation}\n{}_{2}F_1(1,-\\frac{1}{4\\xi}, 1-\\frac{1}{4\\xi}, 2\\kappa B^2\\xi r^{8\\xi})\\approx 1-\\frac{\\kappa B^2 r^{8 \\xi}}{2 \\left( 1-\\frac{1}{4 \\xi}\\right)}+...~.\n\\end{equation}\nThis expansion shows that the hairy black hole is asymptotically AdS.\nOn other hand, in the limit $\\phi\\rightarrow 0$ the potential goes to a constant (the effective cosmological constant) $V(\\phi)\\rightarrow -\\frac{G}{\\kappa}=\\Lambda$. In Fig.~(\\ref{function}) we plot the  behavior of the metric function\n$A\\left( r\\right)^2 $ given by Eq. (\\ref{A}) \nfor a choice of parameters $H=-1$, $G=1$, $B=1$, $\\kappa=1$, \nand $\\xi=-0.25,-0.5,-1$. The metric function $A(r)^2$ changes\nsign for low values of $r$, signalling the presence of a horizon,\nwhile the scalar field is\nregular everywhere outside the event horizon (for $\\xi < 0$) and null at large\ndistances. In Fig.~(\\ref{Pot1}) we show the behavior of the potential, and we observe that it tends asymptotically ($\\phi \\rightarrow 0$)  to a negative constant (the effective cosmological constant). We also plot the\nbehavior of the Ricci scalar $R(r)$, the principal quadratic invariant of the Ricci tensor $R^{\\mu\\nu}R_{\\mu\\nu}(r)$, and the Kretschmann scalar\n$R^{\\mu\\nu\\lambda\\tau}R_{\\mu\\nu\\lambda\\tau}(r)$  in Fig.~(\\ref{figuraRR}) by using the Levi-Civita connection, and we observe that there is not a\nRiemann curvature singularity outside the horizon for  $\\xi=-0.25,-0.5,-1$. Also, we observe a Riemann curvature singularity at $r=0$ for $\\xi=-0.25$ and the torsion scalar is singular at $r=0$ for $\\xi=-0.25$, see Fig.~(\\ref{figuraR}).\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.6\\textwidth]{funcion.eps}\n\\end{center}\n\\caption{The behavior of $A(r)^2$, for $H=-1$, $G=1$, $B=1$, $\\kappa=1$, \nand $\\xi=-0.25,-0.5,-1$.} \\label{function}\n\\end{figure}\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.6\\textwidth]{Pot1.eps}\n\\end{center}\n\\caption{The Potencial $V(\\phi)$, for $H=-1$, $G=1$, $B=1$, $\\kappa=1$, \nand $\\xi=-0.25,-0.5,-1$.} \\label{Pot1}\n\\end{figure}\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.4\\textwidth]{epsilon1.eps}\n\\includegraphics[width=0.4\\textwidth]{epsilon2.eps}\n\\includegraphics[width=0.55\\textwidth]{epsilon3.eps}\n\\end{center}\n\\caption{The behavior of $R(r)$, $R^{\\mu\\nu}R_{\\mu\\nu}(r)$ and $R^{\\mu\\nu\\lambda\\tau}R_{\\mu\\nu\\lambda\\tau}(r)$ for $H=-1$, $G=1$, $B=1$, $\\kappa=1$, and $\\xi=-0.25$ (left figure), $\\xi=-0.5$ (right figure), and $\\xi=-1$ (bottom figure).} \\label{figuraRR}\n\\end{figure}\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.6\\textwidth]{epsilon.eps}\n\\end{center}\n\\caption{The behavior of torsion scalar $T$ as function of $r$ for $H=-1$, $G=1$, $B=1$, $\\kappa=1$, and $\\xi=-0.25, -0.5, -1$.} \\label{figuraR}\n\\end{figure}\n\n\n\\subsubsection{$A(r) \\neq B(r)$}\n\nNow, by considering the following ansatz for the scalar field\n\\begin{equation}\n\\phi(r)=Br^{\\gamma} ~,\n\\end{equation}\nwe find the following solution to the field equations\n\\begin{equation}\nA(r)^2=Gr^{2}+H {}_{2}F_1(-\\frac{1}{\\gamma},\\frac{\\gamma}{4\\xi}, 1-\\frac{1}{\\gamma}, 2\\kappa B^2\\xi r^{2\\gamma}) ~,\n\\end{equation}\n\\begin{equation}\nB(r)^2= \\left( \\frac{1}{2\\kappa} - r^{2 \\gamma} \\xi B^2\\right)^{-2+\\frac{\\gamma}{2 \\xi}} A(r)^2 ~,\n\\label{horizon}\n\\end{equation}\n\\begin{eqnarray}\n\\notag V(\\phi) & = & 2H \\left( \\frac{ \\phi}{B} \\right)^{-\\frac{2}{ \\gamma}} \\left( \\frac{1}{2\\kappa}-\\xi \\phi ^2 \\right)^{-1+\\frac{\\gamma}{2 \\xi}} \\left( 1-2\\kappa \\xi \\phi ^2\\right)^{-\\frac{\\gamma}{4 \\xi}}\\\\\n&& -\\frac{G}{2}\\left( \\frac{1}{2\\kappa}-\\xi \\phi ^2 \\right)^{-2+\\frac{\\gamma}{2 \\xi}}\n\\left( \\frac{2}{\\kappa}-(\\gamma^2+4\\xi) \\phi^2\\right)\\\\ \\notag \n&& -\\frac{H}{2} \\left( \\frac{ \\phi}{B} \\right)^{-\\frac{2}{ \\gamma}}\\left( \\frac{1}{2\\kappa}-\\xi \\phi ^2 \\right)^{-2+\\frac{\\gamma}{2 \\xi}}\n\\left( \\frac{2}{\\kappa}-(\\gamma^2+4\\xi) \\phi^2\\right) {}_{2}F_1(-\\frac{1}{\\gamma},\\frac{\\gamma}{4\\xi}, 1-\\frac{1}{\\gamma}, 2\\kappa\\xi \\phi ^2)~,\n\\end{eqnarray}\nwhere $B$, $G$ and $H$ are integration constants. \nThis solution is asymptotically AdS and generalizes the previous one, because if we take $\\gamma=4\\xi$ it reduces to the solution of the case $A(r)=B(r)$. Furthermore, for $\\gamma=0$ we recover the static BTZ black hole. On the other hand, in the limit $\\phi \\rightarrow 0$ the potential tends to a constant $V(\\phi) \\rightarrow -2G (2 \\kappa)^{1-\\frac{\\gamma}{2 \\xi}}=\\Lambda$. \n\nAs in the previous case, we plot the  behavior of the metric function\n$B\\left( r\\right)^2 $ given by (\\ref{horizon}), in Fig.~(\\ref{functionn}) \nfor a choice of parameters $H=-1$, $G=1$, $B=1$, $\\kappa=1$, $\\xi=-0.25$\nand $\\gamma=-0.25,-1,-2$. The metric function $B(r)^2$ changes\nsign for low values of $r$, signalling the presence of a horizon,\nwhile for $\\gamma < 0$ the scalar field is\nregular everywhere outside the event horizon and null at large\ndistances.  In Fig.~(\\ref{Pot2}) we show the behavior of the potential, asymptotically ($\\phi \\rightarrow 0$) it tends to a negative constant (the effective cosmological constant) as in the previous case. Also, we plot the\nbehavior of $R(r)$, $R^{\\mu\\nu}R_{\\mu\\nu}(r)$, and \n$R^{\\mu\\nu\\lambda\\tau}R_{\\mu\\nu\\lambda\\tau}(r)$  in Fig.~(\\ref{figureinvariant}) by using the Levi-Civita connection, and we observe that there is not a\nRiemann curvature singularity outside the horizon for  $\\gamma=-0.25,-1,-2$. Also, we observe a Riemann curvature singularity at $r=0$ and the torsion scalar is singular at $r=0$ for all the cases considered. Asymptotically, the torsion scalar goes to $-2\\Lambda$ since this spacetime is asymptotically AdS, see Fig.~(\\ref{torsionRR}). Therefore, we have shown that there are three-dimensional black hole solutions with scalar hair in Teleparallel Gravity.\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.6\\textwidth]{funcionn.eps}\n\\end{center}\n\\caption{The behavior of $B(r)^2$, for $H=-1$, $G=1$, $B=1$, $\\kappa=1$, $\\xi=-0.25$ \nand $\\gamma=-0.25,-1,-2$.} \\label{functionn}\n\\end{figure}\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.6\\textwidth]{Pot2.eps}\n\\end{center}\n\\caption{The Potencial $V(\\phi)$, for $H=-1$, $G=1$, $B=1$, $\\kappa=1$, $\\xi=-0.25$ \nand $\\gamma=-0.25,-1,-2$.} \\label{Pot2}\n\\end{figure}\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.4\\textwidth]{epsilonn1.eps}\n\\includegraphics[width=0.4\\textwidth]{epsilonn2.eps}\n\\includegraphics[width=0.55\\textwidth]{epsilonn3.eps}\n\\end{center}\n\\caption{The behavior of $R(r)$, $R^{\\mu\\nu}R_{\\mu\\nu}(r)$ and $R^{\\mu\\nu\\lambda\\tau}R_{\\mu\\nu\\lambda\\tau}(r)$ for $H=-1$, $G=1$, $B=1$, $\\kappa=1$, $\\xi=-0.25$ and $\\gamma=-0.25$ (left figure), $\\gamma=-1$ (right figure), and $\\gamma=-2$ (bottom figure).} \\label{figureinvariant}\n\\end{figure}\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.6\\textwidth]{epsilonn.eps}\n\\end{center}\n\\caption{The behavior of torsion scalar $T$ as function of $r$ for $H=-1$, $G=1$, $B=1$, $\\kappa=1$, $\\xi=-0.25$ \nand $\\gamma=-0.25,-1,-2$.} \\label{torsionRR}\n\\end{figure}\n\n\\section{Final Remarks}\n\\label{conclusions}\n\n\nMotivated by the search of hairy black holes solutions in theories based on torsion, we have considered an extension of three-dimensional TEGR with a scalar field non-minimally coupled to the torsion scalar along with a self-interacting potential, and we have found three-dimensional asymptotically AdS black holes with scalar hair. These hairy black holes are characterized by a scalar field with a power-law behavior and by a self-interacting potential, which tends to an effective cosmological constant at spatial infinity. We have considered two cases $A(r)=B(r)$ and $A(r)\\neq B(r)$. In the first case the scalar field depends on the non-minimal coupling parameter $\\xi$, and it is regular everywhere outside the event horizon and null at spatial infinity for $\\xi < 0$, while for $\\xi = 0$ we recover the non-rotating BTZ black hole. In the second case the scalar field depends on a parameter $\\gamma$, and it is regular everywhere outside the event horizon and null at spatial infinity for $\\gamma < 0$, this solution generalizes the solution of the first case, which is recovered for $\\gamma=4\\xi$. Furthermore, for $\\gamma = 0$ we recover the non-rotating BTZ black hole. Moreover, the analysis of the Riemann curvature invariants and the torsion scalar shows that they are all regular outside the event horizon. In furthering our understanding, it would be interesting to study the thermodynamics of these hairy black hole solutions in order to study the phase transitions. Work in this direction is in progress.\n\n\n\n\\acknowledgments \nThis work was funded by Comisi\\'{o}n\nNacional de Ciencias y Tecnolog\\'{i}a through FONDECYT Grants 11140674 (PAG),\n1110076 (JS) and 11121148 (YV) and by DI-PUCV Grant 123713\n(JS). P.A.G. acknowledge the hospitality of the\nUniversidad de La Serena.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\n\nIt is well known that $7$-dimensional area minimizing hypersurfaces can have isolated singularities. Using work of Hardt--Simon \\cite{HS}, Smale proved in \\cite{Smale} that in an $8$-dimensional manifold $M$ with $H_7(M; \\mathbb{Z}) \\neq 0$, there exists a smooth embedded area minimizing hypersurface for a generic choice of metric. In other words, he showed that isolated singularities of an area-minimizing $7$-dimensional hypersurface can generically be perturbed away. \n\nOne may thus seek to find a smooth embedded minimal hypersurface in all $8$-manifolds $M$ equipped with a generic metric $g$, even when $H_7(M;\\mathbb{Z})=0$. Here, we find such a hypersurface in the case of positive Ricci curvature, and give a partial answer in general. We let $\\Met^{2,\\alpha}(M)$ denote the space of Riemannian metrics of regularity $C^{2,\\alpha}$ on $M$ and $\\Met^{2,\\alpha}_{\\Ric>0}(M)\\subset\\Met^{2,\\alpha}(M)$ denote the open subset of Ricci positive metrics. \n\n\\begin{theorem}[Generic regularity with positive Ricci in dimension $8$]\\label{thm:generic_bound_ricci}\nLet $M^8$ be a compact smooth $8$-manifold. There is an open and dense set $\\mathcal G \\subset \\Met_{\\Ric>0}^{2,\\alpha}(M)$ so that for $g\\in \\mathcal G$, there exists a smooth embedded minimal hypersurface $\\Sigma \\subset M$. \n\\end{theorem}\n\nWithout the curvature condition, we have the following partial result. \n\n  \\begin{theorem}[Generic almost regularity in dimension $8$]\\label{thm:generic_bound}\n  \tLet $M^8$ be a compact smooth $8$-manifold. There exists a dense set $\\mathcal G \\subset \\Met^{2,\\alpha}(M)$ so that for $g \\in \\mathcal G$,  there exists a smooth embedded minimal hypersurface $\\Sigma \\subset M$ with at most one singular point. \n  \\end{theorem}\n \n We actually prove more general results valid in all dimensions, see Theorem \\ref{thm:generic_stratum} below. \n  \n  As mentioned above, the principal motivation for such results is to study generic regularity of non-minimizing, high-dimensional minimal submanifolds. This contrasts with previous works on generic regularity:\n\\begin{itemize}\n    \\item Hardt--Simon \\cite{HS} (resp.\\ Smale \\cite{Smale}), cf.\\ \\cite{Liu}, show that regular singularities of (one-sided) minimizing hypersurfaces can be perturbed away by perturbing the boundary (resp.\\ metric). \n    \\item White \\cite{White:85,white2019generic} shows that minimizing integral 2-cycles are smoothly embedded surfaces for a generic metric.  \n    \\item Moore \\cite{Moore,Moore:book} shows that parametrized minimal (2-dimensional) surfaces are free of branch points for a generic ambient metric. \n\\end{itemize}\n\n  \n \\medskip\n\nIn fact, our work proves that generically there exists a minimal hypersurface of optimal regularity avoiding \\emph{certain} singularities in ambient dimensions beyond the singular dimension. Indeed, Theorem \\ref{thm:generic_bound} is a consequence of a more general result stated below.\n\n\\begin{theorem}[Generic removability of isolated singularities]\\label{thm:generic_stratum}\nConsider a compact smooth $(n+1)$-manifold, for $n\\geq 7$. There is a dense set $\\mathcal G\\subset \\Met^{2,\\alpha}(M)$ with the following properties:\n\\begin{itemize}\n\\item If $g\\in\\mathcal G$ then there exists a minimal hypersurface $\\Sigma$, smooth away from a closed singular set of Hausdorff dimension at most $n-7$, so that for $\\mathcal S_0 \\subset \\sing(\\Sigma)$ the set of singular points with regular tangent cones, we have $\\cH^0(S_0) \\leq 1$. \n\\item If $g \\in \\mathcal G \\cap \\Met^{2,\\alpha}_{\\Ric>0}(M)$ then the same statement holds, except we can conclude that $\\cH^0(\\mathcal S_0) = 0$. \n\\end{itemize}\n\\end{theorem}\n\n\n \\medskip\n\nIn order to remove the topological condition $H_7(M; \\mathbb{Z}) \\neq 0$ of Smale, we will use the Almgren--Pitts min-max construction \\cite{Pitts}, which guarantees the existence of a minimal hypersurface $\\Sigma^n$ in a closed Riemannian manifold $(M^{n+1},g)$. As in the area-minimizing case, when the dimension $n$ satisfies $2\\leq n \\leq 6$, the Almgren--Pitts minimal hypersurface is smooth, but for larger values of $n$ there may be an at most $(n-7)$-dimensional singular set (this follows from work of Schoen--Simon \\cite{SS}). However tangent cones to min-max hypersurfaces are \\emph{a priori} only stable, while only area-minimizing cones have complements that are foliated by smooth minimal hypersurfaces (cf.\\ \\cite{BDG, Lawlor}) and it seems that such a foliation is needed (at least on one side) to perturb the singularity away by adjusting the metric  \\cite{HS}.\n\nThe key technical result of this paper is that (for one-parameter min-max) at all points---except possibly one---of the singular set with a regular tangent cone, the tangent cone is area minimizing on at least one side. Put another way, we show that tangent cones that are not area minimizing on either side ``contribute to the Morse index'' from the point of view of min-max (and these are precisely the cones that we are unable to perturb away using Hardt--Simon \\cite{HS}). \n\n\\subsection{Detailed description of results} \nLet $(M^{n+1},g)$ be a closed Riemannian manifold. By a \\emph{sweepout} of $M$ we will mean a family of (possibly singular) hypersurfaces $\\{ \\Phi(x)=\\partial \\Omega(x)\\}_{x \\in [0,1]}$, where each hypersurface $\\Phi(x)$ is the boundary of an open set\n$\\Omega(x)$ with $\\Omega(0) = \\emptyset$ and $\\Omega(1) = M$, and we denote the family of such sweepouts by $\\mathcal S$ (see Section \\ref{sec:min-max} for the precise definition).\nThe width, $W(M)$, is then defined by \n\\[\nW(M) = \\inf_{\\Phi\\in \\mathcal S} \\left\\{ \\sup_x \\mathbf{M}(\\Phi(x))\\right\\}\\,.\n\\]\n\n\n\n  Given a stationary integral varifold $V$, with $\\supp V$ regular outside of a set of $n-7$ Hausdorff dimension, we define \n \\[\n\\mathfrak{h}_\\textnormal{nm}(V):= \\left\\{ p\\in \\supp(V)\\,:\\,  \\begin{gathered}\n\\textrm{for all $r>0$ small, $\\supp V\\cap B_r(p)$ is not one-sided} \\\\\n\\text{homotopy area minimizing on either side (in $B_r(p)$).}\n\\end{gathered}\n\\right\\}\\,\n \\]\n  \nIn other words, $p\\in\\mathfrak{h}_\\textrm{nm}(V)$ implies that in any small ball there are one-sided homotopies on both sides of $\\supp V$ that strictly decrease area without ever increasing area. \nLet $\\cR$ denote the set of integral varifolds, \nwhose support is a complete embedded minimal hypersurface regular away from a closed singular set of Hausdorff dimension $n-7$.\nFinally, we let  $\\Index(V)$ denote the Morse index of the regular part of the support of $V$, that is\n$$\n\\Index (V)=\\Index (\\supp(\\reg(V)))\\,.\n$$\nThen the main technical estimate of this paper is the following result.\n\n\\begin{theorem}[Index plus non-area minimizing singularities bound]\\label{thm:nm+index_bound}\nFor $n\\geq 7$, let $(M^{n+1}, g)$ be a closed Riemannian manifold of class $C^2$. There exists a stationary integral varifold\n\t$V\\in \\cR$ such that $|V|(M)=W$, which satisfies \n\\begin{equation}\\label{e:bound1}\n \\cH^{0} (\\mathfrak{h}_\\textnormal{nm}(V)) +\\Index(V) \\leq 1\\,.\n\\end{equation}\nIf equality holds in \\eqref{e:bound1}, then for any point $p\\in \\supp V\\setminus \\mathfrak{h}_\\textnormal{nm}(V)$ there is $\\varepsilon>0$ so that $\\supp V$ is area-minimizing to one side in $B_\\varepsilon(p)$. Finally, we can write $V=\\sum_{i}\\kappa_i\\,|\\Sigma_i|$, where $\\Sigma_i$ are finitely many disjoint embeddded minimal hypersursufaces smooth away from finitely many points with  $\\kappa_i\\leq 2$ for every $i$; if $\\Sigma_i$ is one-sided then $\\kappa_i =2$ and if $\\kappa_j=2$ for some $j$ then each $\\Sigma_i$ is stable. \n\\end{theorem}\n\nThe above bound is valid in all dimensions and can be seen as a generalization of the work of Calabi--Cao concerning min-max on surfaces \\cite{CaCa}. Indeed if we define  $\\mathcal{S}_{\\textnormal{nm}}(V)$ by\\footnote{Here $\\omega$ is a modulus of continuity, and we could take it to be logarithmic, as suggested by the work of \\cite{Simon}.  Notice in fact that at all isolated singularities $\\mathcal S_0$, minimal surfaces have unique tangent cone and are locally $C^{1,\\log}$ deformation of the cone itself.}\n\\[\n\\mathcal{S}_{\\textnormal{nm}}(V):= \\left\\{ p\\in \\supp(V)\\,:\\, \\begin{gathered}\nV \\text{ is locally a $C^{1,\\omega}$ graph over its \\emph{unique} tangent cone $\\mathcal C$}\\\\\n\\text{at $p$ and both sides of $\\mathcal C$ are not one-sided minimizing} \n\\end{gathered}  \n\\right\\}\\,\n\\]\nthen we will see that $\\mathcal{S}_{\\textnormal{nm}}(V) \\subset \\mathfrak{h}_\\textrm{nm}(V)$ in Lemma \\ref{lem:snm-hnm}. In particular, \\eqref{e:bound1} implies that\n\\[\n\\cH^0(\\mathcal{S}_{\\textnormal{nm}}(V)) + \\Index(V) \\leq 1.\n\\]\nThus, if we are guaranteed to have $\\Index(V) =1$ (e.g., in positive curvature) we see that $\\mathcal{S}_{\\textnormal{nm}}(V) = \\emptyset$. This is precisely the higher dimensional analogue of the result of Calabi--Cao (cf. Figure \\ref{fig:starfish} and the remark below). \n\nSee also the more recent work of Mantoulidis \\cite{mantoulidis} which makes a more explicit connection with Morse index, using the Allen--Cahn approach (as developed by Guaraco and Gaspar \\cite{Guaraco,GasparGuaraco}) rather than Almgren--Pitts; it would be interesting to elucidate the relationship between Mantoulidis's Allen--Cahn techniques and our proof of Theorem \\ref{thm:nm+index_bound}. \n\n\n\\begin{figure}[h]\n\t\\centering\n\t\\includegraphics[scale=0.4]{starfish.png}\n\t\\caption{The figure eight geodesic $c$ is an \n\texample of a min-max closed geodesic\n\tthat is stable and has one singularity \n\twith non-area minimizing tangent cone.}\n\t\\label{fig:starfish}\n\\end{figure}\n\n\n\\begin{remark}\nBy the index bound in Theorem \\ref{thm:nm+index_bound}, any tangent cone to $V$ has stable regular part. Moreover, we note that the Simons cones \\cite{Simons} in $\\mathbb{R}^8$ (formed from products of two spheres) are all stable and area minimizing on (at least) one side (cf.\\ \\cite{Lawlor}). We particularly emphasize that the Simons cone \n\\[\n\\mathbf{C}^{1,5}: = \\{(x,y) \\in \\mathbb{R}^2\\times \\mathbb{R}^6 : 5|x|^2 = |y|^2\\}\n\\]\nis one-sided minimizing (and stable), but is not minimizing on the other side. It seems to be an open question whether or not there exists an $n$-dimensional stable cone that does not minimize area on either side, for $n\\geq 7$. \n\nEven assuming the existence of a stable minimal cone which is not area minimizing on either sides, it is hard to decide if the above bound is optimal. In dimension $n=1$, such an example is provided by the classical starfish example (cf.\\ Figure \\ref{fig:starfish}), whose tangent cone at the singular point (the union of two lines through the origin) is indeed stable non-area minimizing on either sides (and the starfish fails to be one-sided homotopy minimizing on either side).\n\nWe conjecture that if there is a regular stable minimal cone that is not area-minimizing on either side, then it can arise as the tangent cone to a min-max minimal hypersurface (possibly in a manifold geometrically similar to the starfish); note that were this to occur, Theorem \\ref{thm:nm+index_bound} would imply that the resulting hypersurface would necessarily be stable.\n\\end{remark}\n\nTheorem \\ref{thm:nm+index_bound} generalizes the index upper bound of Marques and Neves \\cite{MN16} for Riemannian manifolds $M^{n+1}$, $3\\leq n+1\\leq 7$ (see also \\cite{Zhou-reg-ind}).\nIn recent years there has been tremendous progress in the understanding of the geometry of minimal hypersurfaces constructed using min-max methods in\nthese dimensions\n(see \\cite{DL}, \\cite{MN19}, \\cite{CM}, \\cite{Zh19}, \\cite{So} and references therein).\n\nFor manifolds of dimension $n+1\\geq 8$ much less is known. \nWhen Ricci curvature is positive\nZhou obtained index and multiplicity bounds for one-parameter min-max minimal hypersurface \\cite{Z17} (see also the work of  Ram\\'irez-Luna \\cite{RL} and Bellettini \\cite{bellettini}). Upper Morse index bounds are known to hold in arbitrary manifolds of any dimensions for hypersurfaces constructed by Allen--Cahn, as proven by Hiesmayr and Gaspar \\cite{Hiesmayr,Gaspar} (see also the recent work of Dey showing that the Almgren--Pitts and Allen--Cahn approaches are equivalent \\cite{dey}). Li proved \\cite{li2019} existence \nof infinitely many distinct minimal hypersurfaces constructed \nvia min-max methods for a generic set of metrics, using the\nWeyl law of Liokumovich--Marques--Neves \\cite{LMN}.\n\n\n\\subsection{Overview of the proof}\nThe construction of a minimal hypersurface in Almgren-Pitts min-max theory proceeds by considering a sequence of sweepouts $\\{ \\Phi_i(x) \\}$\nwith the supremum of the mass $\\sup_x \\mathbf{M}(\\Phi_i(x)) \\rightarrow W(M)$ as $i \\rightarrow \\infty$.\nIt is then proved that we can find a subsequence $\\{i_k\\}$\nand $\\{\\Phi_{i_k}(x_k) \\}$ with mass tending\nto $W$, so that $|\\Phi_{i_k}|(x_k)$ converges to some $V \\in \\cR$.\n\nWe outline the proof of Theorem \\ref{thm:nm+index_bound}.\nFor the sake of simplicity, let's focus on the non-cancellation case, i.e., when all multiplicities of $V$ are one (in the case of cancellation we must argue slightly differently but the main strategy is the same).\nThe main geometric idea is to show that there cannot be two disjoint open sets $U_1,U_2$ so that $\\Sigma=\\supp V$ fails to be one-sided homotopy minimizing on \nthe same side in both $U_1$ and $U_2$. \nThis property is reminiscent of (but different from) \nalmost minimizing \nproperty introduced by Pitts to prove regularity\nof min-max minimal hypersurfaces.\n\nGranted this fact, it is easy to deduce the bound \\eqref{e:bound1}. For example, if $\\Index(\\Sigma) = 1$ and $\\hnm(\\Sigma) = \\{p\\}$, then we can localize the index in some $U$ disjoint from $p$. Because $\\Sigma$ is unstable in $U$, we can find area decreasing homotopies to both sides there, and we can also find $B_r(p)$ disjoint from $U$ with area decreasing homotopies (by definition). This contradicts the above fact. \n\nAs such, we want to show the one-sided homotopy minimizing property in pairs by using the fact that $V$ is a min-max minimal hypersurface. However, this leads us to a major difficulty. Indeed, the approximating currents $\\Phi_{i_k}(x_k)$ might cross $\\Sigma$ many times, making it difficult to glue in one-sided homotopies to push down the mass. \n\nAt a technical level, the main tool used in this paper is that it is possible to \n simplify the one-parameter case of min-max \n theory by constructing a nested optimal sweepout $\\Phi(x)$ with $\\sup \\mathbf{M}(\\Phi(x)) = W$.\nThis allows us to work with one\nsweepout $\\Phi(x)$\ninstead of a sequence of sweepouts. The nested property allows us to directly ``glue in'' the one-sided homotopies to push down the mass. \n\n\nThe existence of a nested optimal\nsweepout follows from a monotonization\ntechnique from \\cite{CL}.\nThere Chambers and Liokumovich proved that\neach sweepout\n$\\Phi_i(x)$ can be replaced by a nested sweepout \n$\\Psi_i(x)$ with $\\sup \\mathbf{M}(\\Psi_i(x)) \\leq \\sup\n\\mathbf{M}(\\Psi_i(x)) + \\frac{1}{i}$. ``Nested'' here means \nthat $\\Psi_i(x) = \\partial \\Omega(x)$ for a family of  open sets with $\\Omega(x) \\subset \\Omega(y)$ if $x<y$. The proof\nused ideas of Chambers and Rotman \\cite{CR} on existence of monotone homotopies of closed curves on surfaces.\n\nAfter we reparametrize $\\Psi_i(x)$\nby the volume swept out we obtain a sequence of families that is uniformly Lipschitz in flat topology. By Arzel\\`a--Ascoli a subsequence will converge to an optimal sweepout.\n\n\nIn the Almgren--Pitts theory, a ``pull-tight'' procedure is used to find a varifold achieving the width with good properties. We can apply this procedure to our sweepout to deduce that one of the critical varifolds in the sweepout is a smooth (up to small singular set) minimal hypersurface. We would then like to prove that this hypersurface satisfies \\eqref{e:bound1}. However, this poses another issue, namely that it could a priori be possible to push the mass of the sweepout near this critical value down, while not decreasing the global width. This could lead to an infinite sequence of ``pushing down'' operations that could create extra critical points not dealt with previously. \n\n\nAs such, our second main technical tool is that we construct an \noptimal sweepout $\\Phi(x)$ with the following special property. \nFor every point $x_0$ \nwith \n$\\sup_{x\\in U} \\mathbf{M}(\\Phi(x)) = W$\nfor every neighbourhood $U$ of $x_0$, \nthere \\emph{does not} exist an open interval $I \\ni x_0$\nand a family \n$\\{\\Phi'(x)\\}$ that coincides with $\\Phi$ for $x\\notin I$ and satisfying $\\sup_{x \\in I'} \\mathbf{M}(\\Phi'(x)) < W$ for every closed interval $I' \\subset I$.\n(In fact, we will prove a somewhat stronger property\nthat holds for open, half-open and closed intervals $I$).\nIn other words, we can not make a\nsmall ``dip'' in the graph of $\\mathbf{M}(x)$,\npushing it below $W$\nin the neighbourhood of $x_0$.\n\nThis property of $\\Phi$ allows us to easily prove the ``homotopy minimizing to one-side'' property of $V$ discussed above (and thus Theorem \\ref{thm:nm+index_bound}). A diagram of the procedure to prove this can be found in Figure \\ref{fig:two-homotopies-push-down}. \n\n\\begin{figure}\n\\begin{tikzpicture}\n\n\\begin{scope}[shift={(-2,0)}]\n\\draw (0,0) -- (0,.5);\n\\draw (0,-1) to [bend left = 80]  (0,0);\n\\draw (0,-2) -- (0,-1);\n\\draw (0,-3) to [bend left = 80]  (0,-2);\n\\draw (0,-3.5) -- (0,-3);\n\\end{scope}\n\n\\begin{scope}\n\\draw (0,0) -- (0,.5);\n\\draw (0,-1) to [bend left = 80]  (0,0);\n\\draw [opacity=.3] (0,-1) to [bend left = 40]  (0,0);\n\\draw [opacity=.3] (0,-1) to [bend left = 20]  (0,0);\n\\draw [opacity=.3]  (0,-1) to [bend left = 0]  (0,0);\n\\draw [opacity=.3] (0,-1) to [bend left = -20]  (0,0);\n\\draw [opacity=.3] (0,-1) to [bend left = -40]  (0,0);\n\\node at (0,-.5) {$\\Rightarrow$};\n\\draw (0,-1) to [bend left = -80]  (0,0);\n\\draw (0,-2) -- (0,-1);\n\\draw (0,-3) to [bend left = 80]  (0,-2);\n\\draw (0,-3.5) -- (0,-3);\n\\end{scope}\n\n\\begin{scope}[shift={(2,0)}]\n\\draw (0,0) -- (0,.5);\n\\draw (0,-1) to [bend left = -80]  (0,0);\n\\draw (0,-2) -- (0,-1);\n\\draw (0,-3) to [bend left = 80]  (0,-2);\n\\draw (0,-3.5) -- (0,-3);\n\\end{scope}\n\n\\begin{scope}[shift={(4,0)}]\n\\draw (0,0) -- (0,.5);\n\\draw (0,-1) to [bend left = -80]  (0,0);\n\\draw (0,-2) -- (0,-1);\n\\draw (0,-3) to [bend left = 80]  (0,-2);\n\\draw [opacity=.3] (0,-3) to [bend left = 40]  (0,-2);\n\\draw [opacity=.3] (0,-3) to [bend left = 20]  (0,-2);\n\\draw [opacity=.3] (0,-3) to [bend left = 0]  (0,-2);\n\\draw [opacity=.3] (0,-3) to [bend left = -20]  (0,-2);\n\\draw [opacity=.3] (0,-3) to [bend left = -40]  (0,-2);\n\\draw (0,-3) to [bend left = -80]  (0,-2);\n\\node at (0,-2.5) {$\\Rightarrow$};\n\\draw (0,-3.5) -- (0,-3);\n\\end{scope}\n\n\\begin{scope}[shift={(6,0)}]\n\\draw (0,0) -- (0,.5);\n\\draw (0,-1) to [bend left = -80]  (0,0);\n\\draw (0,-2) -- (0,-1);\n\\draw (0,-3) to [bend left = -80]  (0,-2);\n\\draw (0,-3.5) -- (0,-3);\n\\end{scope}\n\\end{tikzpicture}\n\\caption{A diagram of how to use two disjoint regions that are not one-sided homotopy minimizing to either side to push down the mass. The ``bumps'' have smaller mass than the original surface and are homotopic along one-sided homotopies that never increase mass. By doing one homotopy fully and then the other, we can see that the mass is always strictly decreased in this range. This can then be glued into a nested sweepout without increasing mass of the adjacent leaves. }\\label{fig:two-homotopies-push-down}\n\\end{figure}\n\n\n\\medskip\n\n\nIt thus remains to explain how we perturb remaining singularities. Hardt-Simon (cf. \\cite{HS}) proved that one sided perturbations of area minimizing cones with isolated singularities are smooth, and Liu extended this result to one sided stationary minimizers (cf. \\cite{Liu}). The proofs of the bounds in Theorems \\ref{thm:generic_bound_ricci}, \\ref{thm:generic_bound}, and \\ref{thm:generic_stratum} are then obtained by combining a weak generalization of White \\cite{W}, where ``stability'' is replaced by the stronger hypothesis that the surface is homotopic minimizing to one side, with a simple surgery procedure in an annulus around the singularity, to show that singular points with regular tangent cones in $\\sing(V) \\setminus \\hnm(V)$ are not generic. As such, the main results follow from \\eqref{e:bound1} when combined with a result of Simon on uniqueness of the blow up at certain singularities \\cite{Simon}.  The rest of the theorem follows from the index lower bound in manifolds with positive Ricci curvature.\n\n\n\\subsection{Organization of the paper} The paper is divided into four sections. The first section contains the basic definitions and some useful geometric tools. The second section introduces the notion of non-excessive optimal sweepouts, which is a key idea in the present work. The third section proves Theorem \\ref{thm:nm+index_bound}, while the last section is dedicated to the proofs of  Theorems \\ref{thm:generic_bound} and \\ref{thm:generic_stratum}.\n\n\\subsection{Remark} Since the first version of this paper appeared, Li--Wang have shown in a remarkable work \\cite{LiWang} that the results of this paper can be combined with \\cite{Wang2020} and some further geometric arguments to prove a version of Theorem \\ref{thm:generic_bound_ricci} for generic metrics without any curvature assumption.\n\n\\subsection{Acknowledgements} We are grateful to Zhihan Wang for pointing out an error in the original version of this paper and for pointing out an interesting alternative approach to prove Theorem \\ref{thm:generic_bound_ricci}.\nWe are also grateful to Zhichao Wang for carefully reading an earlier version of this paper and making several helpful suggestions.\nWe would like to thank the anonymous referees for numerous corrections. \n\nO.C. was partially supported by a Terman Fellowship, a Sloan Fellowship, and an\nNSF grant DMS-1811059\/2016403. Y. L. was partially supported by NSERC Discovery grant. L. S. acknowledges the support of the NSF grant DMS-$1951070$.  \n\n\\section{Geometric preliminaries}\n \nIn this section we introduce the main notations, we prove the existence of optimal nested sweepouts and we recall some useful geometric tools.\n\n\\subsection{Notations} Let $(M^{n+1},g)$ be a closed Riemannian manifold. By scaling, it suffices to consider $\\Vol(M,g) = 1$, which we will always assume below. We use $\\mathcal{Z}_{n}(M; \\mathbb{Z}_2)$ to denote\nthe space of mod 2 flat cycles in $M$. The topology on the space $\\mathcal{Z}_{n}(M; \\mathbb{Z}_2)$ is induced by the usual flat norm $\\mathcal F$.\n\nWe will make extensive use of the notion of \\emph{Caccioppoli set}. A measurable set $E\\subset M$ is Caccioppoli if \n\\[\n\\Per(E):=\\sup\\left\\{ \\int_M \\chi_E\\,{\\rm div} \\omega\\,:\\, \\omega\\in X(M)\\,,\\,\\,\\|\\omega\\|_{C^0}\\leq 1  \\right\\}<\\infty\\,,\n\\]\nwhere $\\chi_E$ denotes the indicator function of $E$. By De Giorgi's strutcure theorem we have that the distributional derivative $D\\chi_E$ (which is is a Radon measure) of a set of finite perimeter $E$ is given by $D\\chi_E=\\nu_E\\, \\cH^{n}\\res \\partial^* E$, where $\\partial^* E$ is the reduced boundary of $E$, which is a $n$-rectifiable set, and $\\nu_E$ is the normal direction to $\\partial^*E$ pointing outside $E$ defined $\\cH^{n}$-a.e.. This allows us to identify $D\\chi_E$ with an element of $\\mathcal{Z}_{n}(M; \\mathbb{Z}_2)$, which we will abuse notation and denote by\n\\[\n\\partial E := \\nu_E\\, \\cH^{n}\\res \\partial^* E\\,.\n\\]\n\nIn particular, with this identification we have\n$$\n\\mathbf{M}(\\partial E)=\\Per(E)\\qquad\\text{and}\\qquad \\mathcal F(\\partial E, \\partial F)=\\| \\chi_E -\\chi_F\\|_{L^1}=\\Vol(E\\Delta F)\\,,\n$$\nwhere $E\\Delta F$ denotes the symmetric difference between two sets. As usual, the perimeter of $E$ in an open set $U$, denoted by $\\Per(E\\,|\\,U)$, is the total variation of $D\\chi_E$ in the set $U$.\n\nWe let $\\mathcal V$ denote the set of varifolds of $(M^{n+1},g)$. Given a cycle $\\Gamma\\in \\mathcal{Z}_{n}(M; \\mathbb{Z}_2)$, we will write $|\\Gamma|$ for the associated integral varifold. In particular if $\\Gamma=\\partial \\Omega$, then \n$$\n|\\Gamma|=\\cH^{n}\\res \\partial^*\\Omega\\,\\otimes\\, \\delta_{T\\partial^*\\Omega}\\qquad \\text{and}\\qquad\n\\mu_{\\Gamma}:=\\cH^{n}\\res \\partial^*\\Omega\n$$ \nis the total variation measure of the measure $D\\chi_\\Omega$. Finally, given a set $\\Sigma$ regular outside a set of dimension $n-7$, we will denote with $|\\Sigma|$ the associated integral varifold.\n\n\\subsection{Optimal nested sweepouts} We start by recalling the notion of sweepouts that we will use in this paper.\n\n\\begin{definition}[Sweepout]\n\tA sweepout of $M$ is a map $\\Phi: [0,1] \\rightarrow \\mathcal{Z}_{n}(M; \\mathbb{Z}_2)$ continuous in $\\mathcal F$-topology such that $\\Phi(x) = \\partial \\Omega(x)$, where $\\{\\Omega(x) \\,:\\, x\\in [0,1]\\}$ is a family of Caccioppoli sets with $\\Omega(0)$ the $0$-cycle and $\\Omega(1) = M$.  We will denote with $\\mathcal S$ the collection of all such sweepouts.\n\tMoreover, we define the width $W$ to be\n\\[\nW = \\inf_{\\Phi \\in \\mathcal S} \\,\\sup_{x\\in [0,1]} \\mathbf{M}(\\Phi(x)) \\,.\n\\]\n\tIt is a consequence of the isoperimetric inequality that $W >0$ \\cite[Proposition 0.5]{DL}.\n\\end{definition}\n\nWe will switch freely between the equivalent notation $\\mathbf{M}(\\Phi(x))$ and $\\Per(\\Omega(x))$. We now  introduce the notion of optimal nested sweepouts and prove their existence.\n\n\n\\begin{definition}[Optimal nested volume parametrized (ONVP) sweepout]\n\tA sweepout $\\{ \\Phi(x) = \\partial \\Omega(x) \\,:\\, x\\in[0,1]\\}$ is called\n\t\\begin{itemize}\n\t\t\\item \\textit{optimal} if $\\sup_{x\\in [0,1]} \\mathbf{M}(\\Phi(x)) = W$;\n\t\t\\item \\textit{nested} if $\\Omega(x_1) \\subset \\Omega(x_2)$, for all $0\\leq x_1 \\leq x_2\\leq 1$;\n\t\t\\item \\textit{volume parametrized} if $\\Vol(\\Omega(x)) = x$, for every $x\\in [0,1]$ (recall that we have assumed $\\Vol(M,g) = 1$). \n\t\\end{itemize}\n\\end{definition}\n\nNested volume parametrized sweepouts enjoy nice compactness properties.\n\n\\begin{lemma}[Compactness for nested volume parametrized sweepouts] \\label{nested sequence}\n\tLet $\\left( \\Phi_i\\right)_i$ be a sequence of nested volume-parametrized sweepouts\n\twith mass uniformly bounded, that is\n\t\\begin{equation}\\label{e:mass_bounds}\n     \\sup_{i\\in \\mathbb N} \\sup_{x\\in [0, 1]} \\mathbf{M}(\\Phi_i(x))\\leq M<\\infty\\,.\n\t\\end{equation}\n\tThen there exists a subsequence $\\left(\\Phi_{i_k}\\right)_k$ converging uniformly to a nested volume parametrized sweepout $\\Psi$ such that\n\t\\begin{equation}\\label{e:sup_optimal}\n\t\t\\sup_x \\mathbf{M}(\\Psi(x)) \\leq \\liminf_k \\left(\\sup_x \\mathbf{M}( \\Phi_{i_k}(x) )\\right).\n\t\\end{equation}\n\\end{lemma}\n\n\\begin{proof} The sequence of continuous functions $\\Phi_i\\colon [0,1] \\to \\mathcal{Z}_{n-1}(M; \\mathbb{Z}_2)$ is uniformly Lispchitz continuous, since for every $0\\leq x<y\\leq 1$ we have\n\t$$\n\t\\mathcal F(\\Phi_i(x), \\Phi_i(y))\\leq \\Vol( \\Omega_i(y)\\setminus\\Omega_i(x)) = \\Vol(\\Omega_i(y))-\\Vol(\\Omega_i(x))=y-x\n\t$$\n\tand $\\Phi_i(0)=\\emptyset$ for every $i$, so by Arzel\\`a--Ascoli Theorem there exists a subsequence $\\Phi_{i_k}$ and a nested volume parametrized sweepout $\\Psi\\colon [0,1] \\to \\mathcal{Z}_{n}(M; \\mathbb{Z}_2)$ such that $\\Phi_{i_k}$ converges uniformly to $\\Psi$. Then \\eqref{e:sup_optimal} follows from \\eqref{e:mass_bounds} and the lower semi-continuity of $\\mathbf{M}$ with respect to the flat topology.\n\\end{proof}\n\n\nOptimal nested volume parametrized sweepouts exist.\n\n\n\\begin{theorem}[Existence of (ONVP) sweepouts] \\label{nested exists}\n\tFor any closed Riemannian manifold $(M,g)$ there exists an optimal \n\tnested volume-parametrized sweepout.\n\\end{theorem}\n\n\\begin{proof}\n\tLet $ \\{\\Psi_i\\}_i$ be a min-max sequence of sweepouts with $\\lim_{i \\rightarrow \\infty }\\sup_x \\mathbf{M}(\\Psi_i(x)) = W$.\n\tBy \\cite[Theorem 1.4]{CL}\n\twe can replace $\\{\\Psi_i(x)\\}_i$ by a  sequence of nested sweepouts $\\{\\Phi_i\\}_i$, such that $\\Phi_i(x) = f_i^{-1}(x)$ for\n\ta surjective Morse function $f_i: M \\rightarrow [0,1]$ and $\\lim_{i \\rightarrow \\infty }\\sup_x \\mathbf{M}(\\Phi_i(x)) = W$. \n\tLet $\\phi_i(x) = \\Vol (f_i^{-1}([0,x]))$.\n\tSince $f_i$ is a Morse function we have that $\\phi_i: [0,1] \\rightarrow [0,1]$ is a continuous strictly increasing function. Then $\\left( \\Phi_i\\circ \\phi_i^{-1}\\right)_i$ is a sequence\n\tof nested volume-parametrized sweepouts.\n\tBy Lemma \\ref{nested sequence}, a subsequence of $\\{ \\Phi_i\\circ \\phi_i^{-1}\\}_i$ converges \n\tto a (ONVP) sweepout.\n\\end{proof}\n\n\nFinally we recall the definition of critical set for a sweepout. \n\n\\begin{definition}[Critical set]\n\tGiven a sweepout $\\Phi$, we define\n\t$$\n\tM(x) = \\limsup_{r \\rightarrow 0} \\{\\mathbf{M}(\\Phi(y)): |y-x| < r\\}.\n\t$$\n\tIf $\\Phi$ is an optimal sweepout, we define the \\emph{critical domain of $\\Phi$} to be the set \n\t$$\n\t{\\bf m}(\\Phi)=\\{ x \\in [0,1]: M(x) =W \\}\\,.\n\t$$\n\tWe will say that a sequence $x_i \\rightarrow x \\in {\\bf m}(\\Phi)$ is a \\emph{min-max sequence} if $|\\Phi|(x_i)$ converges in the varifold sense to a varifold $V$ of mass $W$, i.e. $|V|(M)=W$. We denote the set of such varifolds by ${\\bf C}(\\Psi)$. \n\\end{definition}\n\nIn fact, it is convenient to refine this definition somewhat. \n\n\\begin{definition}[Left and right critical set]\nGiven a sweepout $\\Phi$, we say that $x\\in {\\bf m}_L(\\Phi)$ if there are $x_i\\nearrow x$ with \n\\[\n\\mathbf{M}(\\Psi(x_i)) \\to W\n\\]\nand similarly, $x\\in {\\bf m}_R(\\Phi)$ if there are $x_i\\searrow x$ with \n\\[\n\\mathbf{M}(\\Psi(x_i)) \\to W. \n\\]\n\\end{definition}\n\nNote that ${\\bf m}(\\Phi) = {\\bf m}_L(\\Phi) \\cup {\\bf m}_R(\\Phi)$ (and ${\\bf m}_L(\\Phi) \\cap {\\bf m}_R(\\Phi)$ need not necessarily be empty).\n\n\\begin{definition}[Varifolds of optimal regularity]\nFor an open set $U \\subset M$ we say that a varifold $V$ is in $\\cR(U)$ if \n\\[\nV\\res U = \\sum_{k=i}^K \\kappa_i \\,|\\Sigma_i|\\,,\n\\]\nfor $\\Sigma_i\\subset U$ embedded minimal hypersurfaces that are regular away from  a closed singular set of Hausdorff dimension $n-7$, and $\\kappa_i\\in\\mathbb{N}$ integer multiplicities. We set $\\cR = \\cR(M)$. \n\nAbusing notation, we will say that a (singular) hypersurface $\\Sigma$ is in $\\cR(U)$ if the associated varifold satisfies $|\\Sigma|\\in\\cR(U)$. \n\\end{definition}\n\nAs we will discuss later, Almgren--Pitts theory \\cite{Pitts} implies that given an optimal sweepout $\\Psi$, there is $V\\in{\\bf C}(\\Psi)$ with $V \\in \\cR$. \n\n\\subsection{(Homotopic) one sided minimizers} \nThe notions of one sided minimizers and homotopic minimizers will be useful when dealing with our deformations theorems, as we will see in Section \\ref{ss:deformation_thm}. \n\nLet $U \\subset M^{n+1}$ be an open set. Given two Caccioppoli sets $E\\subset \\Omega$ with $E \\Delta \\Omega \\subset U$ and $\\varepsilon \\geq 0$, we denote the inner families of deformations between $E$ and $\\Omega$ in $U$ which do not increase the volume more than $\\varepsilon$ by\n\\[\n\\mathcal I_\\varepsilon(\\Omega, E\\,|\\,U):=\\left\\{ \\{\\Omega(t)\\}_{t \\in [0,1]} :  \\begin{gathered} \\, \\Omega(0) = E,\\, \\Omega(1) = \\Omega, (\\Omega(t)\\Delta \\Omega) \\setminus U =\\emptyset,\\\\ \\Omega(t_1) \\subset \\Omega(t_2) \\text{ for } t_1 < t_2, \\\\\n\\Per(\\Omega_t)  \\leq \\Per(\\Omega) + \\varepsilon\\ \\end{gathered}   \\right\\}\\,, \n\\]\nand analogously for $\\Omega\\subset E$ with $E \\Delta \\Omega \\subset U$, we define the outer families of deformations between $\\Omega$ and $E$ in $U$ which do not increase the volume more than $\\varepsilon$ by\n\\[\n\\mathcal O_\\varepsilon(\\Omega, E\\,|\\, U):=\\left\\{\\{\\Omega(t)\\}_{t \\in [0,1]}:\\begin{gathered} \\Omega(0) =\\Omega,\\, E = \\Omega(1), (\\Omega(t)\\Delta \\Omega) \\setminus U =\\emptyset,\\\\ \\Omega(t_1) \\subset \\Omega(t_2) \\text{ for } t_1 < t_2  \\\\ \\Per(\\Omega_t) \\leq \\Per(\\Omega) + \\varepsilon\\end{gathered} \\right\\}\\,. \n\\]\nMoreover, given $\\Omega$ and $\\varepsilon\\geq 0$, we denote the collections of inner and outer Caccioppoli sets that can be reached by an inner or outer family of deformations by\n\\begin{gather*}\n    \\mathcal I_\\varepsilon(\\Omega\\,|\\, U)=\\{E\\subset \\Omega\\,: E \\Delta \\Omega \\subset U,\\, \\mathcal I_\\varepsilon(\\Omega, E\\,|\\,U)\\neq \\emptyset\\},\\\\\n    \\mathcal O_\\varepsilon(\\Omega\\,|\\, U)=\\{E\\supset \\Omega\\,: E \\Delta \\Omega \\subset U, \\, \\mathcal O_\\varepsilon(\\Omega, E\\,|\\,U)\\neq \\emptyset\\}\\,.\n\\end{gather*}\nIn both definitions, if we do not include an $\\varepsilon$-subscript, it should be understood that we are taking $\\varepsilon=0$; this will happen most of the time below, but we will crucially rely on the definition with $\\varepsilon>0$ to obtain \\emph{regular} homotopic minimizers in certain situations. \n\n\n\\begin{definition}[Homotopic inner and outer minimizers] \\label{homotopy minimizer}\n\tGiven a Caccioppoli set $\\Omega$ we say that a Caccioppoli set $L(\\Omega\\,|\\, U)\\in \\mathcal I(\\Omega\\,|\\, U)$ is a \\emph{homotopic inner minimizer for $\\Omega$ in $U$}, if\n\t\\begin{enumerate}\n\t\t\\item $\\Per(L(\\Omega\\,|\\, U)\\,|\\, U)\\leq \\Per(\\Omega'\\,|\\,U)$, for every $\\Omega'\\in \\mathcal I(\\Omega\\,|\\, U)$ and\n        \\item if $E\\in \\mathcal I(\\Omega\\,|\\, U)$ satisfies (1) and $ L(\\Omega\\,|\\, U) \\subset E$ then $E= L(\\Omega\\,|\\, U)$.\n\t\\end{enumerate}\n\tSimilarly, define $R(\\Omega\\,|\\, U)\\in \\mathcal O(\\Omega\\,|\\, U)$ to be a \\emph{homotopic outer minimizer for $\\Omega$ in $U$}, if\n\t\\begin{enumerate}\n\t\t\\item $\\Per( R(\\Omega\\,|\\, U)\\,|\\,U)\\leq \\Per(\\Omega'\\,|\\,U)$, for every $\\Omega'\\in \\mathcal I(\\Omega\\,|\\, U)$;\n\\item if $E\\in \\mathcal I(\\Omega\\,|\\, U)$ satisfies (1) and $E \\subset R(\\Omega\\,|\\, U)$ then $E= R(\\Omega\\,|\\, U)$.\n\t\\end{enumerate}\n\tWe say that a Caccioppoli set $\\Omega$ is an \\emph{inner (resp. outer) homotopic minimizer in $U$} if $\\Omega$ is a homotopic inner (resp.\\ outer) minimizer relative to itself. \n\\end{definition}\n\n\nIt is easy to see that inner and outer homotopic minimizers for a fixed set $\\Omega$ always exist.\n\n\\begin{lemma}[Existence of homotopic minimizers] \\label{l:existence_homotopic}\n\tFor any Caccioppoli set $\\Omega$ and open set $U$\twe can find a homotopic inner (resp.\\ outer) minimizer $L(\\Omega\\,|\\, U)$ (resp.\\ $R(\\Omega\\,|\\, U)$) for $\\Omega$ in $U$. \n\\end{lemma}\n\n\\begin{proof} We consider only the case of inner minimizers as the outer minimizers are handled identically. \n\nThis is once again an application of Arzel\\`a--Ascoli theorem. Indeed, notice that $\\mathcal I(\\Omega\\,|\\, U) \\not = \\emptyset$, since $\\Omega \\in \\mathcal I(\\Omega\\,|\\, U)$, so we can consider a minimizing sequence $(E_j)_j$, that is \n\\[\n\t\\lim_{j}\\Per( E_j\\,|\\,U)=\\inf\\{ \\Per(E\\,|\\,U)\\,:\\, E\\in  \\mathcal I(\\Omega\\,|\\, U)\n\t\\}\n\\]\n\tand let $\\{E_j(x): x\\in [0,1]\\} \\in \\mathcal I(\\Omega,E_j \\,|\\, U)$ be the corresponding inner volume non increasing sweepout between $E_j$ and $\\Omega$. We can assume that it is volume parametrized (being nested). Moreover $\\Per(E_j(x)\\,|\\,U)$ is uniformly bounded by $\\Per(\\Omega\\,|\\,U)$, so by Arzel\\`a--Ascoli there is a subsequence converging to $\\{E_\\infty(x)\\} \\in \\mathcal I(\\Omega, E_\\infty\\,|\\, U)$, with $E_\\infty$ satisfying the desired minimality property by lower semi-continuity of the perimeter. \n\t\n\tFinally, again by Arzel\\`a--Ascoli, we can find $L(\\Omega\\,|\\, U)\\subset \\Omega$ in the set of minimizers, which infimizes the flat distance to $\\partial \\Omega$, and so satisfies condition (2) (otherwise there would be a competitor closer to $\\Omega$ in flat norm).\n\\end{proof}\n\nWe recall the definition of one-sided minimizers, which will be useful in the sequel when we perform cut and paste arguments. \n\n\\begin{definition}[One sided minimizers]\n\tLet $E$ be a Caccioppoli set. We say that $E$ is \\emph{locally one-sided inner (resp. outer) area-minimizing} in $U$ if for every $A\\Subset U$ and $V$ with $V\\Delta E \\subset A$, we have\n\\[\n\t\\Per( E\\,|\\, A) \\leq \\Per (V \\,|\\, A)\n\\]\n\twhenever $V\\subset E$ (resp.\\ $E\\subset V$). We say that $E$ is \\emph{strictly locally one-sided inner (resp.\\ outer) area-minimizing} if the inequality holds strictly except when $E=V$ as Caccioppoli sets. \n\\end{definition}\n\nWe show that homotopic minimizers are in fact strict one sided minimizers into the region they sweep out. \n\n\\begin{lemma}[Homotopic minimizers are one sided minimizers in the swept out region] \\label{strict minimizer}\n\tSuppose $L(\\Omega\\,|\\, U)$ is an homotopic inner (resp. outer) minimizer for $\\Omega$ in $U$. Then $L(\\Omega\\,|\\, U)$ (resp. $R(\\Omega\\,|\\, U)$) is strict locally outer (resp. inner) one-sided minimizing in $U\\cap \\Omega$ (resp. $U\\setminus \\Omega$) .\n\\end{lemma}\n\n\\begin{proof}\n\tWe consider homotopic inner minimizers; the case of outer minimizers is similar. \n\t\n\tIf $L(\\Omega\\,|\\, U)$ is not a strict outer minimizer in $U\\cap \\Omega$ then there is $V'$ with $L(\\Omega\\,|\\, U)\\subset V'$ and $L(\\Omega\\,|\\, U) \\Delta V' \\subset A \\Subset U$ and\n\t\\[\n\t\\Per(V'\\,|\\,A) \\leq P(L(\\Omega\\,|\\, U)\\,|\\,A).\n\t\\]\n\tWe can minimize perimeter in $A$ among all such $V'$ to find $V$. Namely,\n\t\\begin{equation}\\label{e:confusing1}\n\t\\Per( V \\,|\\,A )\\leq \\Per(W\\,|\\,A)  \n\t\\end{equation} \n\t\tfor all $W$ with $W\\Delta V\\subset A\\setminus L(\\Omega\\,|\\, U)$. Since $L(\\Omega\\,|\\, U) \\in \\mathcal I(\\Omega\\,|\\, U)$, there is $\\{U(x)\\,:\\,x \\in [0,1]\\} \\in \\mathcal I(\\Omega, L(\\Omega\\,|\\,U)\\,|\\,U)$. Set $\\Omega(x) = U(x) \\cup V$. Since $V$ satisfies \\eqref{e:confusing1},\nwe have that \n\t$$\n\t\\Per(\\Omega_t\\,|\\,A) \\leq \\Per(U_t\\,|\\,A).\n\t$$\n\tThis implies that $\\Omega(1)=V$ satisfies (1) of Definition \\ref{homotopy minimizer} and $V \\Delta L(\\Omega\\,|\\,U)\\subset A \\setminus L(\\Omega\\,|\\, U)$, therefore by (2) of Definition \\ref{homotopy minimizer}, it follows that $V= L(\\Omega\\,|\\,U)$. This completes the proof. \n\\end{proof}\n\n\nWe have the following lemma that will allow us to find bounded mass homotopies in certain situations. \n\\begin{lemma}[Interpolation lemma] \\label{l:close in flat}\n\tFix $L>0$. For every $\\varepsilon>0$ there exists $\\delta>0$, such that the following holds. If $\\Omega_0,\\Omega_1 $ are two sets of finite perimeter, such that $\\Omega_0 \\subset \\Omega_1$, $\\Per(\\Omega_i) \\leq L$, $i=0,1$, and $\\Vol(\\Omega_1 \\setminus \\Omega_0)\\leq \\delta$, then there exists a nested $\\mathcal F$-continuous family $\\{\\partial\\Omega_t \\}_{t \\in [0,1]}$ with \n\\[\n\\Per(\\Omega_t)\\leq \\max\\{\\Per(\\Omega_0),\\Per(\\Omega_1) \\}+\\varepsilon\n\\]\nfor all $t\\in[0,1]$\n\\end{lemma}\n\\begin{proof}\nLet $\\Omega$ be a Caccioppoli set that minimizes\nperimeter among sets $\\Omega'$\nwith $\\Omega_0 \\subset \\Omega' \\subset \\Omega_1$.\n\nFix $r>0$ such that for every $x \\in M$ the ball $B(x,2r)$\nis $2$-bi-Lipschitz diffeomorphic to\nthe Euclidean ball of radius $2r$.\nLet $\\{B(x_i,r) \\}_{i=1}^N$ be a collection of balls covering $M$. By coarea inequality\nwe can find a radius $r_i \\in [r,2r]$,\nso that $\\mathbf{M}(\\partial B(x_i,r_i) \\cap \\Omega \\setminus \\Omega_0) \\leq \\frac{\\delta}{r}$.\n\nLet $U_1 = B(x_1,r_1) \\cap\n\\Omega \\setminus \\Omega_0$.\nBy a result of Falconer (see \\cite{Falconer},\n\\cite[Appendix 6]{Guth}) there exists a family of \nhypersurfaces sweeping out $U_1$ of area bounded by\n$c(n) \\delta^{\\frac{n}{n+1}}$. It follows (see\n\\cite[Lemma 5.3]{CL}) that\nthere exists a nested family \n$\\{ \\Xi^1(t)\\} $ of Caccioppoli sets with\n$\\Xi^1(0) = \\Omega_0$ and $\\Xi^1(1) = \\Omega_0 \\cup U_1$\nand satisfying\n$$\\Per(\\Xi^1(t)) \\leq \\Per(\\Omega_0)+2c(n)\\delta^{\\frac{n}{n+1}} \\, . $$\nLet $\\Omega^1 =  \\Omega_0 \\cup U_1$. Observe, that\nthe minimality of $\\Omega$ implies that\n$$\\Per(\\Omega^1) \\leq \\Per(\\Omega_0) + \\frac{2\\delta}{r} $$\n\nInductively, we define $\\Omega^k= \\Omega^{k-1} \\cup U_k$ and\n$U_k = B(x_k,r_k) \\cap\n\\Omega \\setminus \\Omega^{k-1}$. As above we can construct a nested homotopy of Caccioppoli sets $\\Xi^k(t)$ from $\\Omega^{k-1}$ to $\\Omega^k$, satisfying\n$$\\Per(\\Xi^k(t)) \\leq \\Per(\\Omega_0)+2c(n)\\delta^{\\frac{n}{n+1}} \n+ \\frac{2N\\delta}{r}$$\n\nWe choose $\\delta>0$ so small that $\\Per(\\Xi^k(t))<  \\Per(\\Omega_0) + \\varepsilon$.\nIt follows then that we have obtained a homotopy from $\\Omega_0$ to $\\Omega$ satisfying the\ndesired perimeter bound. Similarly,\nwe construct a homotopy from $\\Omega$ to $\\Omega_1$.\n\\end{proof}\n\n\n\n\n\n\n\nFinally, we have the following result. Recall that White \\cite{W} proves that strictly stable  \\emph{smooth} minimal hypersurfaces are locally area-minimizing. A generalization of such a result to the case of hypersurfaces with singularities (i.e., elements of $\\cR$) would be very interesting. The following (weaker) result will suffice for our needs; it can be seen as a result along these lines, except ``stability'' is replaced by a stronger hypothesis: the surface is homotopic minimizing to one side.\\footnote{Note that one certainly needs a condition on the singularities rather than just a condition on the regular part like strict stability, since as we show in Proposition \\ref{p:def_thm}, the existence of (regular) non-minimizing tangent cones implies that the hypersurface is not homotopic minimizing irrespective of any stability condition that might hold on the regular part.}\n\n \\begin{proposition}[Comparing the notions of minimizing vs.\\ homotopic minimizing for minimal surfaces]\\label{prop:min-vs-htpy-min}\nSuppose that $\\Omega$ is a Caccioppoli set and for some strictly convex open set $U \\subset M$ with smooth boundary, the associated varifold $V = |\\partial \\Omega|$ satisfies $V \\in \\cR(U)$. Assume that $\\supp V \\cap U$ is connected. \n\nSuppose that $\\Omega$ is inner (resp.\\ outer) homotopy minimizing in $U$. Then, at least one of the following two situations holds:\n\\begin{enumerate}\n\\item for all $p \\in \\supp V \\cap U$, there is $\\rho_0>0$ so that for $\\rho<\\rho_0$, $B_\\rho(p) \\subset U$ and $\\Omega$ is inner (resp.\\ outer) minimizing in $B_\\rho(p)$, or\n\\item there exists a sequence of Caccioppoli sets $E_i\n\\neq \\Omega$\nwith $|\\partial E_i| \\in \\cR(U)$ so that $E \\Delta \\Omega \\subset \\Omega \\cap U$ (resp.\\ $\\Omega^c \\cap U$), $|\\partial E_i|$ has stable regular part, and $\\partial E_i \\to \\partial \\Omega$ in the flat norm.  \n\\end{enumerate}\n\\end{proposition}\n\n\\begin{remark}\nIt is interesting to ask if the second possibility occurs; it seems possible that one could rule this out in the case where $V$ has regular tangent cones that are all strictly minimizing in the sense of Hardt--Simon \\cite[\\S 3]{HS}. \n\\end{remark}\n\n\n\\begin{proof}[Proof of Proposition \\ref{prop:min-vs-htpy-min}]\nWe consider the ``inner'' case, as the ``outer'' case is similar. Let $E^\\varepsilon \\in \\mathcal I_\\varepsilon(\\Omega\\, |\\, U)$ minimize perimeter among all sets in $\\mathcal I_\\varepsilon(\\Omega\\,|\\, U)$ (as usual, the existence of $E^\\varepsilon$ follows from Arzel\\`a--Ascoli). We claim that $E^\\varepsilon$ is area-minimizing to the inside of $\\Omega$ in sufficiently small balls. \n\nMore precisely, for $r>0$ sufficiently small, suppose there was a Caccioppoli set $E'$ so that $E'\\Delta E^\\varepsilon \\subset B_r(p) \\cap U \\cap \\Omega$ and $\\Per(E'\\,|\\,U) < \\Per(E^\\varepsilon\\,|\\,U)$. As long as $r$ was chosen sufficiently small, Lemma \\ref{l:close in flat} guarantees\nthat $E' \\in \\mathcal I_\\varepsilon(\\Omega\\, |\\,U)$. This is a contradiction. \n\nNow, consider $p \\in \\reg V \\cap U$. We note that $E^\\varepsilon$ is almost minimizing (with no constraint coming from $\\Omega$) in the sense of \\cite{Tam}, and thus has $C^{1,\\alpha}$ boundary in $B_r(p) \\cap U$,  thanks to standard results on the obstacle problem; see \\cite[\\S 1.9, \\S1.14(iv)]{Tam}. As such, away from $\\sing V$ (which has Hausdorff dimension at most $n-7$) we can thus conclude that $\\partial^*E^\\varepsilon$ is regular, stationary and stable.\\footnote{Cf.\\ the proof of \\cite[Proposition 2.1]{Liu} for the proof of stability.} A capacity argument then implies that $|\\partial E^\\varepsilon| \\in \\cR(U)$ and $\\partial^*E^\\varepsilon$ is stable. Therefore, the maximum principle for (possibly singular) hypersurfaces \\cite{Ilm} implies that either $E^\\varepsilon = \\Omega$ or $\\partial^*E^\\varepsilon \\cap \\supp V = \\emptyset$. In the first case, we can conclude that $\\Omega$ is inner minimizing in small balls (since $E^\\varepsilon$ is). \n\nWe can thus assume that the latter possibility holds for all $\\varepsilon>0$ sufficiently small. Taking $\\varepsilon_j\\to 0$, there is $E \\in \\mathcal I_0(\\Omega | U)$ so that $E^{\\varepsilon_j}\\to E$ with respect to the flat norm. If $E=\\Omega$, then the second possibility in the conclusion of the proposition holds for $E_j = E^{\\varepsilon_j}$. \n\nThe final case to consider is $E\\neq \\Omega$. By curvature estimates for stable minimal hypersurfaces \\cite{SS}, $|\\partial E| \\in \\cR(\\Omega)$ and thus $\\partial^*E \\cap \\supp V = \\emptyset$ again by the maximum principle.\n\nWe know that $\\Per(E_i| U) \\leq \\Per(\\Omega| U)$, so in the limit we get\n$\\Per(E| U) \\leq \\Per(\\Omega| U)$\nBy Arzel\\`a-Ascoli nested homotopies from $\\Omega$ to $E_i$ will converge to\na nested homotopy $E(t)$ from $\\Omega$ to $E$ that does not increase volume.\nBy the inner homotopy minimizing property of $\\Omega$ we have\n\\[\n\\Per(E\\,|\\,U) = \\Per(\\Omega\\, |\\, U).\n\\]\n\nSuppose we minimize perimeter among Caccioppoli sets $A'$ sandwiched between $E$ and $\\Omega$, $E \\subset A' \\subset \\Omega$. We claim that the\nminimizer $A$ has perimeter equal to that of $\\Omega$. Indeed, if  $A$ has strictly\nsmaller perimeter, then family $E(t) \\cup A$\nis an area decreasing nested homotopy between $\\Omega$ and $A$, contradicting that\n$\\Omega$ is inner homotopic minimizing.\n\nWe thus see that $\\Omega$ is minimizing in $\\Omega \\cap E^c \\cap U$, which implies that it is inner minimizing in small balls, as asserted. \n\\end{proof}\n\n\n\\section{Non-excessive sweepouts}\\label{sec:min-max}\n\nIn this section we introduce the concept of excessive intervals and excessive points for a sweepout and prove that there is a sweepout, such that every point in the critical domain is not excessive.\n\n\\begin{definition}[Excessive points and intervals] \\label{def:excessive_interval}\n\tSuppose $\\{\\Phi(x)=\\partial \\Omega(x)\\}$ is a sweepout. Given a connected interval $I$ (we allow $I$ to be open, closed, or half-open)\n\twe will say that $\\{\\Phi^I(x) = \\partial \\Omega^I(x)\\}_{x\\in \\bar I}$ is an \\emph{$I$-replacement family for $\\Phi$} if\n\t$\\Omega^I(a) = \\Omega(a)$, $\\Omega^I(b)=\\Omega(b)$ and for all $x \\in I$, \n\\[\n\\limsup_{I \\ni y\\to x} \\mathbf{M}(\\Phi^I(y)) < W.\n\\]\nWe say that a connected interval $I$ is an \\emph{excessive interval for $\\Phi$} if there is an $I$-replacement family for  $\\Phi$. We say that a point $x$ is \\emph{left (resp.\\ right) excessive for $\\Phi$} if there is an excessive interval $I$ for $\\Phi$ so that $(x-\\varepsilon,x]\\subset I$ (resp.\\ $[x,x+\\varepsilon)\\subset I$) for some $\\varepsilon>0$. \n\\end{definition}\n\n \n The goal of this section is to prove the following result.\n\n\\begin{theorem}[Existence of non-excessive min-max hypersurface]\\label{c:non-excessive_minmax}\nThere exists a (ONVP) sweepout $\\Psi$ such that every $x\\in {\\bf m}_L(\\Psi)$ is not left excessive and every $x \\in {\\bf m}_R(\\Psi)$ is not right excessive. \n\\end{theorem}\n\n\\subsection{Preliminary results} We establish several results that will be used in the proof of Theorem \\ref{c:non-excessive_minmax}. \n\n\n\n\n\n\n\\begin{lemma}[Extension lemma I]\\label{l:union-excessive}\nIf $I,J$ are excessive for $\\Phi$ and $I \\cap J \\not = \\emptyset$, then $I \\cup J$ is excessive for $\\Phi$. \n\\end{lemma}\n\\begin{proof}\nLet $\\{\\partial \\Omega^I(x) \\}_{x \\in I}$\nand $\\{\\partial \\Omega^J(x) \\}_{x \\in J}$\nbe $I$ and $J$ replacement families\nfor $\\Phi$.\n\nLet $a_1 = \\inf\\{x \\in I\\} $, $a_2 = \\inf\\{x \\in J\\}$ and $b_1 = \\sup\\{x \\in I\\} $, $b_2 = \\sup\\{x \\in J\\}$.\nAssume without any loss of generality that\n$a_1\\leq a_2$ and $b_1 \\leq b_2$\nand at least one of the two inequalities\nis strict.\n\nLet $K = I \\cap J$; let $a,b$ denote, respectively, left and right\nboundary points of $K$ and $c = \\frac{a+b}{2}\\in K$.\nLet $\\tilde{\\Omega}$ be a Caccioppoli set minimizing perimeter among all $\\Omega'$\nwith $\\Omega(a) \\subset \\Omega' \\subset \\Omega(b)$.\nDefine $\\phi_1:[a_1,c] \\rightarrow [a_1,b_1]$\nand $\\phi_2:[c,b_2] \\rightarrow [a_2,b_2]$\ngiven by $\\phi_1(x)=a_1+\\frac{b_1-a_1}{c-a_1}(x-a_1)$\nand $\\phi_2(x)=a_2 + \\frac{b_2-a_2}{b_2-c}(x-c)$.\nWe define an $I\\cup J$ replacement family for\n$\\Phi$ by setting\n\\[\n\\Phi^{I\\cup J}(x) = \\begin{cases} \n\\partial (\\Omega^I(\\phi_1(x)) \\cap \\tilde{\\Omega}) & x \\in [a_1,c]\\\\\n\\partial (\\Omega^J(\\phi_2(x)) \\cup \\tilde{\\Omega}) & x \\in [c,b_2]\n\\end{cases}\n\\]\nObserve that $\\Phi^{I\\cup J}$\nis continuous since $\\Phi^{I\\cup J}(c) = \\partial \\tilde{\\Omega}$.\nIt follows from our choice of $\\tilde{\\Omega}$ that\n$\\mathbf{M}(\\Phi^{I\\cup J}(x))\\leq \\mathbf{M}(\\Phi^I(\\phi_1^{-1}(x)))<W$\nfor $x \\in I \\cap (-\\infty, c] $ and \n$\\mathbf{M}(\\Phi^{I\\cup J}(x))\\leq \\mathbf{M}(\\Phi^J(\\phi_2^{-1}(x)))<W$\nfor $x \\in J \\cap [c, \\infty) $. \n\\end{proof}\n\n\\begin{lemma}[Extension lemma II]\\label{l:union-excessive-repeated}\nIf $I$ is excessive for $\\Phi$ and $J$ has $J\\cap I \\neq \\emptyset$ and is excessive for\n\\[\n\\Psi(x) : = \\begin{cases} \\Phi^I(x) & x \\in I \\\\ \\Phi(x) & x \\not \\in I \\end{cases}\n\\]\nthen $J \\cup I$ is excessive for $\\Phi$. \n\\end{lemma}\n\\begin{proof}\nDefine an $I\\cup J$-replacement family $\\Phi^{I\\cup J}$ for $\\Phi$ by\n\\[\n\\Phi^{I\\cup J}(x) = \\begin{cases} \n\\Phi(x) & x \\in \n[0,1] \\setminus (I \\cup J) \\\\\n\\Phi^I(x) & x\\in I \\setminus J \\\\\n\\Psi^J(x) & x \\in J\n\\end{cases}\n\\]\nwhere $\\Psi^J$ is a $J$-replacement family for $\\Psi$. \n\\end{proof}\n\nThe following is the technical core of the proof of Theorem \\ref{c:non-excessive_minmax}. \n\n\\begin{proposition}[Existence of maximal excessive intervals]\\label{p:max_int}\nGiven an (ONVP) sweepout $\\Phi$, if $\\hat J$ is excessive for $\\{\\Phi(x)=\\partial\\Omega(x)\\}$, then there exists an excessive interval $J \\supset \\hat J$ so that $J$ is maximal in the sense that if $\\tilde J$ is excessive with $\\tilde J \\cap J \\not = \\emptyset$, then $\\tilde J \\subset J$. \n\\end{proposition}\n\n\\begin{proof}\nLet \n\\[\n\\alpha : = \\sup\\{|\\tilde J| : \\tilde J \\textrm{ excessive }, \\hat J \\cap \\tilde J \\not = \\emptyset\\}. \n\\]\nChoose excessive intervals $\\tilde J_n$ with $\\tilde J_n \\cap \\hat J \\neq \\emptyset$ and $|\\tilde J_n|\\to \\alpha$. By Lemma \\ref{l:union-excessive}, we can replace $\\tilde J_n$ by $\\tilde J_n \\cup \\hat J$, and thus assume that $\\hat J \\subset \\tilde J_n$. In particular $\\tilde J_n \\cap \\tilde J_m\\neq \\emptyset$ for all $m,n$. Using Lemma \\ref{l:union-excessive} again, we can replace $\\tilde J_n$ by\n\\[\n\\bigcup_{m=1}^n\\tilde J_m\n\\]\nso that the $\\tilde J_n$ form an increasing sequence of excessive intervals (still with $|\\tilde J_n| \\to\\alpha$). Note that the interior of an excessive interval is still excessive, so we can consider $J_n : = (\\tilde J_n)^\\circ$. Note that $|J_n| \\to \\alpha$ and the $J_n$ are increasing. \n\nWe will show below that \n\\[\nJ'  : = \\bigcup_n J_n\n\\]\nis excessive. Write $J' = (a,b)$. Granted the fact that $J'$ is excessive, we claim that one of the intervals $(a,b),(a,b],[a,b)$, or $[a,b]$ is the desired maximal excessive interval. Note that by Lemma \\ref{l:union-excessive}, if $[a,b)$ and $(a,b]$ are excessive, then so is $[a,b]$, so we can choose the largest excessive interval out of these four choices and call it $J$. Suppose that $\\tilde J$ is excessive with $\\tilde J \\cap J\\not = \\emptyset$. Then, $J \\cup \\tilde J$ is excessive by Lemma \\ref{l:union-excessive} and $\\hat J \\subset J \\cup\\tilde J$. Thus, \n\\[\n|J \\cup \\tilde J| \\leq \\alpha,\n\\]\nso $\\tilde J \\subset \\bar J$ (where $\\bar J$ is the closure of $J$). Now, $J \\cup \\tilde J$ is excessive, but strictly larger than $\\tilde J$ (by assumption). This contradicts the choice of $J$ as the largest excessive interval out of $(a,b),(a,b],[a,b)$, and $[a,b]$. This shows that $J$ is maximal, as desired.\n\nIt thus remains to prove that $J' = \\cup_n J_n$ is excessive for a nested sequence of open excessive intervals $J_n$. Write $J_n = (a_n,b_n)$ and set $a'_n = a_n + \\frac 1n,b_n'=b_n - \\frac 1n$. \n\n\n\n\n\nFix $i =0,1,\\dots$ and assume we have real numbers $0<A_1,\\dots,A_i < W$ and integers $n_i\\geq i$ (with $n_1< n_2<\\dots<n_i$) so that for $n\\geq n_i$, there is a $J_n$-replacement $\\{\\Phi^n_i(x)=\\partial \\Omega^n_i(x)\\}$ so that \n\\[\n\\Per(\\Omega^n_i(x)) \\leq A_j\n\\]\nfor $x \\in [a_j',b_j']$ and $1 \\leq j \\leq i$. (Note that for $i=0$, we can find such objects because the $J_n$ are excessive.) \n\nWe will choose $0 < A_{i+1} < W$, and $n_{i+1}>\\max\\{n_i,i+1\\}$ so that we can construct $J_n$-replacements $\\{\\Phi^n_{i+1}(x)=\\partial \\Omega^n_{i+1}(x)\\}$ for $n\\geq n_{i+1}$ with \n\\[\n\\Per(\\Omega^n_{i+1}(x)) \\leq A_j\n\\]\nfor $x \\in [a_j',b_j']$ and $1 \\leq j \\leq i+1$. Granted this, we can easily (inductively) complete the proof by passing $\\Phi^{n_{i+1}}_{i+1}$ to a subsequential limit (using Arzel\\`a--Ascoli).\n\n It is useful to introduce the following notation, used in the construction of $\\Phi^n_{i+1}$. Given two nested sets of finite perimeter $V \\subset W$, we\n\tlet \n\t\\begin{itemize}\n\t\t\\item $\\mathcal M_{V,W}$ an outermost Caccioppoli set minimizing perimeter among all the Caccioppoli sets $\\Omega$ with $V \\subset \\Omega \\subset W$;\n\t\t\\item $\\{\\mathcal V_{(V,W)}(x)\\}_x$ the optimal nested homotopy from $V$ to $W$.\n\t\\end{itemize} \nFor $n \\geq n_i$, we set\n\\[\nL_n : = \\mathcal M_{\\Omega(a_n),\\Omega^{n_i}_i(a_{i+1}')}, \\qquad U_n: = \\mathcal M_{\\Omega^{n_i}_i(b_{i+1}'),\\Omega(b_n)}\n\\]\nNote that for $n \\leq m$, $ L_m \\subset L_n$ and $U_n \\subset U_m$. Hence, $L_n$ and $U_n$ have $\\mathcal F$-limits as $n\\to\\infty$. For $\\varepsilon > 0$ fixed so that \n\\[\n\\max\\left\\{\\Per\\left(\\Omega^{n_{i}}(a_{i+1}')\\right),\\Per\\left(\\Omega^{n_i}_i(b_{i+1}')\\right)\\right\\}+\\varepsilon < W,\n\\]\nLemma \\ref{l:close in flat} thus guarantees that there is $n_{i+1}\\geq i+1$ sufficiently large so that for $n\\geq n_{i+1}$, \n\\[\n\\sup_t \\Per\\left(\\mathcal V_{(L_n,L_{n_{i+1}})}(t)\\right) < W, \\qquad \\sup_t \\Per\\left(\\mathcal V_{(U_{n_{i+1}},U_n)}(t)\\right) < W. \n\\]\nFor $n\\geq n_{i+1}$, we define\n\\[\n\\tilde \\Phi^n_{i+1}(x) = \\begin{cases}\n\\partial\\left(\\Omega^n_i(x+1) \\cap L_n\\right) & x \\in [a_n-1,b_n-1]\\\\\n\\partial \\tilde \\mathcal V_{L_n,L_{n_{i+1}}}(x) & x \\in [b_n-1,a_{n_{i+1}}]\\\\\n\\partial\\left(\\Omega^{n_{i+1}}_i(x) \\cup L_{n_{i+1}} \\cap U_{n_{i+1}} \\right)& x \\in [a_{n_{i+1}},b_{n_{i+1}}]\\\\\n\\partial \\tilde \\mathcal V_{U_{n_{i+1}},U_n}(x) & x \\in [b_{n_{i+1}},a_n+1]\\\\\n\\partial\\left(\\Omega^n_i(x-1) \\cup U_n \\right)& x \\in[a_n+1,b_n+1]. \n\\end{cases}\n\\]\nHere, the $\\tilde \\mathcal V$ are the homotopies $\\mathcal V$ reparametrized to be defined on the given intervals (the exact parametrization is immaterial). It is easy to check that $\\tilde\\Phi^n_{i+1}$ is continuous. \n\nLet $\\Phi_{i+1}^n$ denote the reparametrization of $\\tilde\\Phi_{i+1}^n$ by volume. We have arranged that $\\Phi_{i+1}^n$ is a $J_{n}$-replacement. Moreover, for $x \\in [a_{i+1}',b_{i+1}']$, we have that $\\Phi^n_{i+1}(x) = \\Phi^{n_{i+1}}_i(x)$, so \n\\[\n\\mathbf{M}(\\Phi^n_{i+1}(x)) \\leq A_j\n\\]\nfor $x \\in [a_j',b_j']$ and $1\\leq j\\leq i$. Finally, we can set \n\\[\nA_{i+1} : = \\sup_{x\\in[a_{i+1}',b_{i+1}']} \\mathbf{M}(\\Phi^{n_{i+1}}_{i}(x)) < W\n\\]\n(which is independent of $n$). This completes the proof. \n\\end{proof}\n\n\n\\subsection{Proof Theorem \\ref{c:non-excessive_minmax}}\nWe are now able to complete proof of Theorem \\ref{c:non-excessive_minmax}\n\n\t\tLet $\\Phi$ be a nested optimal sweepout. Consider the collection $\\mathcal A$ of the maximal (with respect to inclusion) excessive intervals for $\\Phi$, that is $I\\in \\mathcal A$ if for every excessive interval $I'$ such that $I'\\cap I\\neq \\emptyset$, we have $I\\supset I'$. The existence of maximal intervals follows from Proposition \\ref{p:max_int} proven above. \n\t\n\t    Notice that by definition $I \\neq J\\in \\mathcal A$ implies that $I\\cap J=\\emptyset$, so we can define a new sweepout $\\Psi$ in the following way\n\\[\n\t    \\Psi(x)=\\begin{cases}\n\t    \\Phi^I(x) & \\quad \\text{if }x \\in I \\in \\mathcal A\\\\\n\t    \\Phi(x) & \\quad \\text{otherwise}\\,.\n\t    \\end{cases}\n\\]\nNote that $\\Psi$ is a nested optimal sweepout, so up to reparametrization we can assume it is (ONVP), and moreover by construction ${\\bf m}(\\Psi)\\subset {\\bf m}(\\Phi)$. Suppose that $x \\in {\\bf m}_L(\\Psi)$ is left excessive. Then, there is a $\\Psi$-excessive interval $J$ with $(x-\\varepsilon,x]\\subset J$. We claim that there is $I \\in \\mathcal A$ with $J \\subset I$. Indeed, if $J \\cap I = \\emptyset$ for all $I \\in \\mathcal A$, then $J$ is a $\\Phi$-excessive interval, contradicting the definition of $\\mathcal A$. On the other hand, if there is $I \\in \\mathcal A$ with $J \\cap I \\not =\\emptyset$, then $J \\cup I$ is excessive by Lemma \\ref{l:union-excessive-repeated}. Thus, $J \\subset I$ by definition of $\\mathcal A$ again. Thus, for $y \\in (x-\\varepsilon,x]\\subset I$, $\\Psi(y) = \\Psi^I(y)$. By the definition of replacement family, we know that if $x_i \\in (x-\\varepsilon,x]$ has $x_i\\to x$, then\n\\[\n\\limsup_{i\\to\\infty} \\mathbf{M}(\\Psi^I(x_i)) < W. \n\\]\nHowever, this contradicts the assumption that $x \\in{\\bf m}_L(\\Psi)$. The same proof works to prove that $x \\in {\\bf m}_R(\\Psi)$ is not right excessive. This finishes the proof. \\qed\n\n\n\\section{Deformation Theorems and Proof of Theorem \\ref{thm:nm+index_bound}}\\label{ss:deformation_thm}\nIn this section we conclude the proof of Theorem \\ref{thm:nm+index_bound}. By Theorem \\ref{c:non-excessive_minmax}, there exists an (ONVP) sweepout $\\Phi$ so that every $x \\in {\\bf m}_L(\\Phi)$ is not left excessive and every $x\\in{\\bf m}_R(\\Phi)$ is not right excessive. By Almgren--Pitts pull-tight and regularity theory \\cite{Pitts}, we find that for some $x_0\\in{\\bf m}(\\Phi)$, there is a min-max sequence $x_i\\to x_0$ so that $|\\Phi(x_i)|$ converges to some $V \\in \\cR$. Indeed, we can pull-tight $\\Phi$ to find a sweepout (in the sense of Almgren--Pitts, not in the (ONVP) sense considered in this paper) $\\tilde\\Phi$; we have that ${\\bf C}(\\tilde\\Phi) \\subset {\\bf C}(\\Phi)$ and some $V\\in{\\bf C}(\\tilde\\Phi)$ is in $\\cR$. By replacing\n$\\Phi(x)$ by $\\Phi(1-x)$ if necessary,\nwe can then assume for the rest of this section that:\n\\begin{equation}\\label{e:no_can}\n\\begin{gathered}\n\\text{there is a (ONVP) sweepout $\\{\\Phi(x)=\\partial \\Omega(x)\\}$ and $x_i \\nearrow x_0 \\in {\\bf m}_L(\\Phi)$, so that}\\\\\n\\text{$|\\Phi(x_i)| \\to V\\in\\cR$ and $\\Phi$ is not left excessive at $x_0$}\n\\end{gathered}\n\\end{equation}\n\nWe then consider two cases: $\\mathbf{M}(\\Phi(x_0)) = W$ (no cancellation) and $\\mathbf{M}(\\Phi(x_0)) < W$ (cancellation). We analyze the geometric properties of $V$ in both cases separately,\nproving deformation theorems reminiscent of those in \\cite{MN16}. \n\t\n\\subsection{No cancellation} \nThroughout this subsection we will assume the no cancellation condition\n\\[\n\\mathbf{M}(\\Phi(x_0)) = W \\,.\n\\] \nIn this case we have that $|\\Phi(x_i)| \\to |\\partial \\Omega|$, see for instance \\cite[Proposition A.1]{DL}, so we can rephrase our assumption \\eqref{e:no_can} as\n\\begin{equation}\\label{e:no_canc2}\n\\begin{gathered}\n\\text{there is a (ONVP) sweepout $\\{\\Phi(x)=\\partial \\Omega(x)\\}$ and $x_i \\nearrow x_0 \\in {\\bf m}_L(\\Phi)$, so that} \\\\\n\\text{$|\\Phi(x_i)| \\to |\\Sigma|:=|\\partial \\Omega|\\in\\cR$ and $\\Phi$ is not left excessive at $x_0$.}\n\\end{gathered}\n\\end{equation}\nIn particular, in this case the multiplicity bound of Theorem \\ref{thm:nm+index_bound} follows immediately.\n\n\n\n\n\n\t\n\n\n\\begin{proposition}\n\\label{p:pairs}\nLet $\\Sigma$ be as in \\eqref{e:no_canc2}. \nSuppose $\\Sigma$ is not homotopic minimizing to either side\nin some  open set $U$. Then the following holds:\n\\begin{enumerate}\n    \\item for every $x \\not\\in \\overline{U}$ there exists $r>0$, such that\n$\\Sigma$ is minimizing to one side in $B_r(x)$;\n    \\item for every open set $U'$ disjoint from $U$,\n    we have that $\\Sigma$ is homotopic minimizing\n    to one side in $U'$.\n\\end{enumerate}\n\n\\end{proposition}\t\n\t\\begin{proof} \nWe prove statement (1).\nThere is $\\delta>0$ and Caccioppoli sets $E^-_1 \\in \\mathcal I(\\Omega\\,|\\,U)$ and $ E^+_1\\in \\mathcal O(\\Omega\\,|\\,U)$ with \n\t\\begin{equation}\\label{e:vol_dec}\n\t\\Per(E^\\pm_1\\,|\\, U)\\leq \\Per(\\Omega\\,|\\, U)-\\delta\\,,\n\t\\end{equation}\n\tand nested families $\\{\\Omega^-_1(x)\\,:x\\in [0,1]\\} \\in \\mathcal I(\\Omega, E_1^-\\,|\\,U)$ and $\\{\\Omega^+_1(x)\\,:x\\in [0,1]\\} \\in \\mathcal O(\\Omega, E_1^+\\,|\\,U)$. Furthermore, by Lemma \\ref{l:existence_homotopic}, we can assume that $E_1^+$ are inner and $E_1^-$ are outer homotopic minimizers in $U$.\n\t\n    Let $x \\in \\Sigma \\setminus \\overline{U}$ and assume, for contradiction,\n\tthat $\\Sigma$ is not area minimizing on both sides \n\tin every ball $B_r(x)$, $r< {\\rm dist}(x, U)$. Let $E^-_2 \\subset \\Omega$, \n\twith $\\Omega \\setminus  E^-_2 \\subset B_r(x)$, denote a Caccioppoli set\n\tthat is a strict outer minimizer in $\\Omega \\cap B_r(x)$.\n\tSimilarly, let $ \\Omega \\subset E^+_2$, with\n\t$ E^+_2 \\setminus \\Omega \\subset B_r(x)$, denote a Caccioppoli set\n\tthat is a strict inner minimizer in $\\Omega \\cap B_r(x)$.\n    We have $$\\Per(\\Omega)> \\max\\{\\Per(E^\\pm_2) \\}.$$\n\tIf we choose $r>0$ sufficiently small,\n\tthen, by Lemma \\ref{l:close in flat}, there exist nested families $\\{\\Omega^-_2(x)\\,:x\\in [0,1]\\}$ and $\\{\\Omega^+_2(x)\\,:x\\in [0,1]\\} $ that interpolate between $E^-_2$ and \n\t$\\Omega$ and between $\\Omega$ and $E^+_2$ and satisfying\n\\begin{equation}\\label{e:vol_dec2}\n\\Per(\\Omega^\\pm_2(x))\\leq \\Per(\\Omega) + \\frac{\\delta}{2}\n\\end{equation}\n\n\tLet $(x_l,x_r)\\neq \\emptyset$ be the interval (since $\\Phi$ is nested) such that\n\t$$\n\t\\Phi(x) \\cap (\\cup_i E_i^+\\setminus \\cup_i E_i^-)\\neq \\emptyset \\, .\n\t$$\n\tThen we define a family $\\bar \\Psi\\colon [x_l-2, x_r+2] \\to \\mathcal{Z}_{n}(M; \\mathbb{Z}_2)$ by setting\n\t$$\n\t\\bar \\Psi(x):=\n\t\\begin{cases}\n\t\\partial\\left(\\Omega(x+2)\\cap E_1^-\\cap E_2^-\\right) & \\text{if } x\\in (x_l-2,x_0-2]\\\\\n\t\\partial \\left(\\Omega_1^-(x-x_0+2)\\cap E_2^- \\right) & \\text{if } x\\in [x_0-2,x_0-1]\\\\\n\t\\partial \\left(\\Omega_1^+(x-x_0+1)\\cap E_2^-\\right) & \\text{if } x\\in [x_0-1,x_0]\\\\\n\t\\partial \\left(\\Omega_2^-(x-x_0)\\cup E_1^+\\right) & \\text{if } x\\in [x_0,x_0+1]\\\\\n\t\\partial \\left(\\Omega_2^+(x-x_0-1) \\cup E_1^+\\right) & \\text{if } x\\in [x_0+1,x_0+2]\\\\\n\t\\partial\\left(\\Omega(x-2)\\cup E_1^+\\cup E_2^+\\right) & \\text{if } x\\in [x_0+2,x_r+2)\n\t\\end{cases}\n\t$$\n\tIt is easy to see that $\\bar \\Psi$ is continuous, and moreover notice that, since by Lemma \\ref{strict minimizer} $E_1^+$ is a strict inner minimizer in $U$ and $E_{1}^-$ strict outer minimizers in $U$, we have that\n\t$$\n\t\\limsup_{y\\to x} \\mathbf{M} (\\bar \\Psi(y)) < \\limsup_{y\\to x} \\mathbf{M}(\\bar\\Phi(y)) \\leq W \n\t$$\n\tfor $x\\in(x_l-2,x_0-2] \\cup [x_0-2,x_r-2)$.\n\tSince the families $\\Omega_1^\\pm(x)$ do not increase the volume of $\\Sigma$ in $U_i$ and using \\eqref{e:vol_dec} and \\eqref{e:vol_dec2}, we also have\n\t$$\n\t\\mathbf{M} (\\bar \\Psi(x)) \\leq W -\\frac{\\delta}{2} \\qquad \\forall \\, x\\in [x_0-2, x_0+2]\\,.\n\t$$\n\tWe let $\\Psi$ be the volume reparametrization of the nested sweepout $\\bar \\Psi$, then $\\Psi$ is a $(x_l,x_r)$-replacement for $\\Phi$, thus giving a contradiction with the fact that $x_0\\in (x_l,x_r)$ and $x_0\\in {\\bf m}_L(\\Phi)$.\n\t\n\tThe proof of statement (2) is completely analogous.\n\\end{proof}\n\n\n\n\n\\begin{proposition}\n\\label{p:def_thm}\nLet $\\Sigma$ be as in \\eqref{e:no_canc2}, then the following holds\n\\begin{enumerate}\n    \\item $\\Index(\\Sigma)\\leq 1$;\n    \\item If $\\Index(\\Sigma)=1$, then \n for every point $x \\in \\Sigma$ there exists $r>0$, such that\n$\\Sigma$ is minimizing to one side in $B_r(x)$;\n    \\item If $\\hnm(\\Sigma)$ is non-empty, then $\\Sigma$ is stable,\n    $\\cH^0(\\hnm(\\Sigma))=1$ and for every point $x \\in \\Sigma\\setminus \\hnm(\\Sigma)$ there exists $r>0$, such that $\\Sigma$ is minimizing to one side in $B_r(x)$.\n\\end{enumerate}\nIn particular, Theorem \\ref{thm:nm+index_bound} holds in the case of no cancellations.\n\\end{proposition}\n\t\n\\begin{proof} \nNote that if $U \\cap \\Sigma$ is smooth and unstable, it is easy to see that $\\Sigma$ is not homotopic minimizing to either side in $U$ (just consider the normal flow generated by a compactly supported unstable variation of fixed sign). Statements (2) and (3)\nof the Proposition now immediately follow from Proposition \\ref{p:pairs}. \nThe upper bound on the index (1) follows from (2) of Proposition \\ref{p:pairs} and Lemma \\ref{l:localizing-index-two-sided} below. \n\\end{proof}\t\n\\begin{lemma}[Localizing the index]\\label{l:localizing-index-two-sided}\nSuppose that $\\Sigma \\in \\cR$ is two-sided and has $\\Index(\\Sigma) \\geq 2$. Then, there is $\\Sigma^*_1,\\Sigma^*_2 \\subset \\Sigma$ smooth hypersurfaces with boundary so that the $\\Sigma^*_i$ are both unstable (for variations fixing the boundary). \n\\end{lemma}\n\\begin{proof}\nA standard capacity argument implies that there is a subset $\\Sigma' \\subset \\Sigma$ where $\\Sigma'$ is a smooth minimal surface with smooth boundary and $\\Index(\\Sigma') \\geq 2$ (with Dirichlet boundary conditions). Let $u$ denote the second (Dirichlet) eigenfunction (with eigenvalue $\\lambda <0$) for the stability operator for $\\Sigma$. Because $u$ must change sign, there are at least two nodal domains $\\Sigma_1,\\Sigma_2\\subset \\Sigma$. One can find subsets with smooth boundary $\\Sigma^*_i\\subset \\Sigma_i$ so that $\\Sigma^*_i$ are unstable. This follows from the argument in \\cite[p.\\ 21]{Chavel} (namely, by considering $(u|_{\\Sigma_i} - \\varepsilon)_+$ in the stability operator for $\\varepsilon\\to 0$ chosen so that $\\{u|_{\\Sigma_i} > \\varepsilon\\}$ has smooth boundary). \n\\end{proof}\n\n\\begin{lemma}\\label{lem:snm-hnm}\n$\\mathcal{S}_{\\textnormal{nm}}(\\Sigma)\\subset \\mathfrak{h}_\\textrm{nm}(\\Sigma)$.\n\\end{lemma}\n\\begin{proof}\nSuppose that $p \\in \\mathcal{S}_{\\textnormal{nm}}(V)$, we claim that $\\Sigma$ is not homotopic minimizing to either side in $B_\\varepsilon(p)$ for any $\\varepsilon>0$ sufficiently small. Indeed, by assumption, the unique tangent cone $\\mathbf{C} = \\partial\\Omega_\\mathbf{C}$ to $\\Sigma$ at $p$ is not minimizing to either side. This implies that there are Caccioppoli sets $E_\\mathbf{C}^- \\subset \\Omega_\\mathbf{C} \\subset E_\\mathbf{C}^+$ so that $E_\\mathbf{C}^\\pm \\Delta\\Omega_\\mathbf{C} \\subset B_1 \\subset \\mathbb{R}^{n+1}$ and so that\n\\[\n\\Per_{\\mathbb{R}^{n+1}}(E_\\mathbf{C}^\\pm\\,|\\,B_1) \\leq \\Per_{\\mathbb{R}^{n+1}}(\\Omega_\\mathbf{C}\\,|\\,B_1) - \\delta.\n\\]\nChoose $C^{1,\\omega}$ coordinates on $M$ around $p$ so that $\\Omega = \\Omega_\\mathbf{C}$ in $B_\\varepsilon(p)$ and so that $g_{ij}(p) = \\delta_{ij}$, which we can do since $g\\in C^2$ and $\\Sigma$ is a $C^{1,\\omega}$ deformation of $\\mathbf{C}$ near $p$ by assumption. Then, set \n\\[\nE(x) := \\begin{cases}\n (\\Omega\\setminus B_\\varepsilon) \\cup (|x| E_\\mathbf{C}^- \\cap B_\\varepsilon) & x < 0\\\\\n \\Omega & x = 0\\\\\n (\\Omega\\setminus B_\\varepsilon) \\cup (|x| E_\\mathbf{C}^+ \\cap B_\\varepsilon) & x > 0\\\\\n \\end{cases}\n\\]\nWe have that \n\\[\n\\Per_g(E(x)) - \\Per_g(\\Omega) = - |x|^n \\delta (1+o(1))\n\\]\nas $x\\to 0$ (since the metric $g_{ij}$ converges to the flat metric $\\delta_{ij}$ after rescaling $|x| \\to 1$, by the $C^{1,\\omega}$ regularity of the chart). This shows that $\\Sigma$ is not homotopic minimizing to either side in $B_\\varepsilon(p)$, so $p\\in\\mathfrak{h}_\\textrm{nm}(\\Sigma)$ as claimed. \n\\end{proof}\n\n\\subsection{Cancellation} We will assume the cancellation condition\n\\[\n\\mathbf{M}(\\Phi(x_0)) < W\n\\]\nthroughout this subsection. In particular, we can find $q \\in \\reg V$ so that for all $\\varepsilon>0$ sufficiently small,\n\\[\n\\Per(\\Omega\\,|\\,B_\\varepsilon(q)) < | V |(B_\\varepsilon(q)) \n\\]\nwhere $\\partial \\Omega = \\Phi(x_0)$. Like in the previous section we set $\\Sigma := \\supp V$. \n\nFurthermore we set $V=\\sum_{i}\\kappa_i\\,|\\Sigma_i|$, where each $\\Sigma_i$ is a minimal hypersurface with optimal regularity and $\\kappa_i\\in \\mathbb{N}$ are constant multiplicities, by the constancy theorem \\cite[Theorem 41.1]{Sim}. So \\eqref{e:no_can} becomes\n\n\\begin{equation}\n\\label{e:canc}\n\\begin{gathered}\n\\text{there is a (ONVP) sweepout $\\{\\Phi(x)=\\partial \\Omega(x)\\}$ and $x_i \\nearrow x_0 \\in {\\bf m}_L(\\Phi)$, so that} \\\\\n\\text{$|\\Phi(x_i)| \\to V=\\sum_{i} \\kappa_i\\,|\\Sigma_i|\\in\\cR$, $\\Phi$ is not left excessive at $x_0$ and} \\\\\n\\text{there is $q\\in \\Sigma$ such that }\\, \\Per(\\Omega\\,|\\,B_\\varepsilon(q)) \\leq | V |(B_\\varepsilon(q))-\\delta(\\varepsilon) \\quad \\text{for all } \\varepsilon>0\\,.\n\\end{gathered}\n\\end{equation}\n\nWe write $\\Omega=\\Omega(x_0)$ and observe that $\\Sigma\\subset \\overline\\Omega$. We would like to claim that $\\Sigma$ is homotopically minimizing, but this condition might not make sense if $\\Sigma$ is one-sided. However, thanks to the cancellation we can actually prove \nthat $\\Sigma$ is area-minimizing in its neighborhood in $\\Omega$\naway from a small ball around $q$.\n\n\n\\begin{definition} \\label{def:nbhd minimizing}\nWe will call a set $\\Omega'$ a $(q,\\varepsilon, \\tau, \\Sigma, \\Omega)$-competitor\nif $$\\big(\\Omega \\setminus B_\\tau(\\Sigma)\\big)  \\cup \\big(B_\\varepsilon(q) \\setminus \\Sigma \\big) \n\\subset \\Omega' \\subsetneqq \n\\Omega \\setminus \\Sigma \\, .$$\nAn $(q,\\varepsilon, \\tau, \\Sigma, \\Omega)$-competitor $\\Omega'$ will be called a\nminimizing competitor if \nits perimeter is strictly less than perimeter of any \n$(q,\\varepsilon, \\tau, \\Sigma, \\Omega)$-competitor $\\Omega''$ \nwith $\\Omega' \\subset \\Omega''$.\n(Note that we do not require $\\Per (\\Omega')$ to \nbe less that the perimeter of all competitors, but\nonly those that contain $\\Omega'$).\n\\end{definition}\n\n\n\n\\begin{proposition} \\label{thm:no_competitor}\n    Suppose \\eqref{e:canc} holds, then \n    for every $\\varepsilon>0$ there is $\\tau>0$,\n    such that \n    minimizing $(q,\\varepsilon, \\tau, \\Sigma, \\Omega)$-competitor\n    does not exist.\n\\end{proposition}\n\n\\begin{proof} \nFor contradiction suppose \nthere exists a minimizing $(q,\\varepsilon, \\tau, \\Sigma, \\Omega)$-competitor\n$U$. Observe that by the cancellation\nassumption for every $\\eta>0$ \nwe can find $(q,\\varepsilon, \\tau, \\Sigma, \\Omega)$-competitors \n$\\Omega'$\nwith $\\Per(\\Omega') \\leq W+ \\eta - \\delta(\\epsilon)$,\nwhere $\\delta(\\epsilon)$ is from (\\ref{e:canc}). \nIt follows that \n$$\\Per(U) \\leq \\Per(\\Omega')<W \\, .$$\nIf we choose $\\tau>0$ sufficiently small,\nthen by Lemma \\ref{l:close in flat}\nthere exists a nested family $\\{E(x)\\,:\\,x\\in [0,1]\\}$\nwith $E(0)= U$, $E(1) = \\Omega$ and\n$$\\Per(E(x))< W \\, .$$\n\nLet $(x_l,x_0]$ be the connected interval such that $\\Omega(x)\\setminus U\\neq \\emptyset$, where $\\{\\Phi(x)=\\partial \\Omega(x)\\}$, and define \nfamily $\\Psi \\colon (x_l, x_0+1] \\to \\mathcal Z_n(M, \\mathbb Z_2)$ by\n$$\n\\Psi(x):=\n\\begin{cases}\n\\partial(\\Omega(x)\\cap U) & \\text{if }x\\in (x_l,x_0] \\\\\n\\partial  E(x-x_0)\n& \\text{if }x\\in [x_0, x_0+1]\n\\end{cases}\n$$\nClearly $\\Psi$ is continuous, since $\\Omega=\\Omega(x_0)$ and moreover we have that\n$$\n\\limsup_{y\\to x} \\mathbf{M}(\\Psi(y)) < \\limsup_{y\\to x} \\mathbf{M}(\\Phi(x)) \\leq W\n$$\nfor every $x\\in (x_l,x_0)$ by\nstrict minimality condition in Definition \\ref{def:nbhd minimizing}. For every $x\\in [x_0, x_0+1]$\nwe also have $\\mathbf{M}(\\Psi(x)) = \\mathbf{M}(\\partial E(x))<W$. This implies that $x_0$ is left excessive for $\\Phi$ which is a contradiction.\n\\end{proof}\n\n\n \n\n\n\n\n\n\n\n\n\\begin{proposition}\\label{p:def-thm-cancel}\nSuppose $V=\\sum_i\\kappa_i \\,|\\Sigma_i|$ is as in \\eqref{e:canc}, then each $\\Sigma_i$ has stable regular part and $\\hnm(V) = \\emptyset$.\nMoreover, for every point \n$x \\in support(V)$\nthere exists $r>0$, such that \nthe support of $V$ is minimizing to one side in $B_r(x)$\n\\end{proposition}\n\n\n\n\\begin{proof} \nFirst we observe that we can find two points $q_1$ and $q_2$ in $\\reg V$,\nsuch that for all $\\varepsilon>0$ sufficiently small,\n\\[\n\\Per(\\Omega\\,|\\,B_\\varepsilon(q_j)) < | V |(B_\\varepsilon(q)) \\, .\n\\]\nBy Proposition \\ref{thm:no_competitor} we have non-existence\nof minimizing $(q_j,\\varepsilon, \\tau, \\Sigma, \\Omega)$-competitors\nfor $j=1,2$. This implies that \n $\\Sigma_i$ is area minimizing to one side\nin a small ball around every point of $V$. In particular,\nwe have $\\cH^{0}(\\hnm(V)))=0$.\n\nThe stability of the regular part of each $\\Sigma_i$ also follows from the\nnon-existence of minimizing $(q,\\varepsilon, \\tau, \\Sigma, \\Omega)$-competitors.\nIndeed, if a component $\\Sigma_i$ has index $\\geq 1$,\nthen for $\\varepsilon>0$ sufficiently small, the  minimal\nhypersurface $\\Sigma_i \\setminus B_{\\varepsilon}(q)$ with fixed boundary\nwill be unstable by a standard capacity argument.\nIf $\\Sigma_i$ is two-sided, then by considering a minimization problem\nto one side of $\\Sigma_i$ \nin $B_\\tau(\\Sigma_i) \\setminus B_\\varepsilon(q)$\nwe can find open set $U \\subset \\Omega$,\nsuch that $\\Omega \\setminus U$ is a minimizing \n$(q,\\varepsilon, \\tau, \\Sigma, \\Omega)$-competitor.\n\nSuppose $\\Sigma_i$ is one-sided.\nSince $\\Sigma_i \\subset \\overline{\\Omega}$\nwe have that $B_\\tau(\\Sigma_i)\\setminus \\Sigma_i \\subset \\Omega$\nfor all sufficiently small $\\tau>0$.\nIn particular, for small $\\tau< \\varepsilon$ \nwe can minimize in the class of hypersurfaces\n $\\{S\\subset B_\\tau(\\Sigma_i) : S \\cap B_\\varepsilon(q) = \\Sigma_i \\cap B_\\varepsilon(q)\\}$\n to obtain a minimizer $\\Sigma_i'$ in the same homology class\n and open set $U \\subset \\Omega$ with $\\partial U = \\Sigma_i \\cup \\Sigma_i'$.\n Then $\\Omega \\setminus U$ is a minimizing \n$(q,\\varepsilon, \\tau, \\Sigma, \\Omega)$-competitor.\n\\end{proof}\n\n\\subsection{Multiplicity $2$ bound} In this subsection we show that if $\\kappa_i> 2$ for some $i$, then $x_0$ is excessive, by using simple comparisons with disks. Notice that if any multiplicity satisfies $\\kappa_i\\geq 2$ then we must be in the cancellation case considered above. \n\n\\begin{lemma}[Multiplicity $2$ bound]\\label{l:mult_bound}\nLet $V=\\sum_{i}\\kappa_i\\,| \\Sigma_i|$ be as in \\eqref{e:no_can}. Then $\\kappa_i\\leq 2$ for every $i$.\n\\end{lemma}\n\t\n\\begin{proof}  Suppose by contradiction $\\kappa_i\\geq 3$ for some $i$. Then let $p\\in \\reg(\\Sigma_i)$, $p\\neq q$ (where $q$ is the cancellation point considered above). Consider a ball  $B_r(p)$, $r < \\frac{1}{2}dist(p,q)$, sufficiently small so that $\\Sigma_i\\cap B_r(p)$ is two-sided. Let $\\tau(r)>0$ be a small constant\nto be chosen later\nand set $U = B_r(p) \\cap B_\\tau(\\Sigma_i)$.\n\nConsider sequence $x_j \\nearrow x_0$\nwith $|\\partial \\Omega(x_j)| \\rightarrow V$.\nWe can assume that the radius $r$ was\n chosen sufficiently small, so that\n\\begin{equation}\\label{e:mult_bound1}\n    \\mathbf{M}(\\partial \\Omega(x_j) \\cap U)\\geq \\left(\\kappa_i - \\frac{1}{10}\\right) \\omega_n  r^n\\,,\n\\end{equation}\nfor all $j$ large enough, where $\\omega_n$ denotes the measure of the $n$-dimensional ball of radius one.\n\nLet $\\Omega_j' \\subset \\Omega(x_j)$, $\\Omega_j' \\setminus U = \\Omega(x_j)\n\\setminus U$, be a strict one-sided outer area minimizer\nin $\\Omega(x_j)\\cap U$.\nObserve that if $\\Omega'_j$ does not converge to $\\Omega(x_0)$,\nthen $\\lim \\Omega_j'$ is a $(q,\\frac{1}{2}dist(p,q), \\tau, \\Sigma, \\Omega(x_0))$-competitor,\nwhich contradicts Proposition \\ref{thm:no_competitor}.\n\nWe conclude that $\\lim \\Omega_j' = \\Omega(x_0)$.\nOn the other hand, \nby comparing $\\Omega(x_j) \\setminus U$ to $\\Omega_j'$ and\nassuming that $\\tau(r)$\nwas chosen sufficiently small, we have that one-sided \narea minimizing property of $\\Omega_j'$ implies \n\\begin{equation*}\n    \\mathbf{M}(\\partial \\Omega_j' \\cap U)\\leq \\Per(U) \\leq \\left(2 + \\frac{1}{10}\\right) \\omega_n  r^n\\,,\n\\end{equation*}\nFor $\\tau(r)$\n sufficiently small and $j$ large we can apply\nLemma \\ref{l:close in flat} to find a nested family $E(x)$\ninterpolating between $\\Omega_j'$ and $\\Omega$, such that\n\\begin{align*}\\Per( E(x) ) &\\leq \\max \\{\\mathbf{M}(\\partial \\Omega_j' \\setminus U),\n\\mathbf{M}(\\partial \\Omega(x_0) \\setminus U) \\} + \\left(2 + \\frac{2}{10}\\right) \\omega_n  r^n\\\\\n& \\leq W - \\left( 1-\\frac{3}{10} \\right) \\omega_n r^n.\n\\end{align*}\nBy combining families $\\Omega(x) \\cap \\Omega_j'$\nand $E(x)$ we obtain that $x_0$ is left-excessive.\n\\end{proof}\n\t\n\t\n\t\n\t\n\t\n\t\\subsection{Proof of Theorem \\ref{thm:nm+index_bound}} The result follow immediately by combining Corollary \\ref{c:non-excessive_minmax} with Propositions \\ref{p:pairs}, \\ref{p:def_thm}, \\ref{p:def-thm-cancel} and Lemma \\ref{l:mult_bound}. \\qed\n\t\t\n\t\\section{Proof of Theorems \\ref{thm:generic_bound_ricci},  \\ref{thm:generic_bound}, and \\ref{thm:generic_stratum}}\n    In this section we prove Theorem \\ref{thm:generic_stratum} (Theorems \\ref{thm:generic_bound_ricci} and \\ref{thm:generic_bound} follow immediately from Theorem \\ref{thm:generic_stratum} when combined with the facts that when $n=8$ all singularities are regular and that the set of bumpy metrics is open and dense \\cite{White:bumpy,White:bumpy2}). Theorem \\ref{thm:generic_stratum} will follow from Theorem \\ref{thm:nm+index_bound} and Proposition \\ref{prop:min-vs-htpy-min}, together with a simple surgery procedure.\n    \n    \n    \\subsection{Surgery procedure} We show here how to regularize minimal hypersurfaces with regular singularities under the assumption that the hypersurface minimizes area in a small ball around each singularity. \n    \\begin{proposition}[Perturbing away regular singularities of locally area minimizing surfaces]\\label{p:surgery}\n\tFor $(M^{n+1},g)$ a compact $C^{2,\\alpha}$-Riemannian metric and $\\Sigma\\in\\cR$ a minimal hypersurface, recall that $\\mathcal S_0(\\Sigma)\\subset \\sing \\Sigma$ is defined to be the set of singular points with a regular tangent cone. There is $\\tilde g\\in \\Met^{2,\\alpha}(M)$ arbitrarily close to $g$ and $\\tilde \\Sigma$ arbitrarily close in the Hausdorff sense to $\\Sigma$ so that $\\tilde\\Sigma$ is minimal with respect to $\\Sigma$ and $\\mathcal S_0(\\tilde\\Sigma) \\subset \\hnm(\\tilde\\Sigma) = \\hnm(\\Sigma)$. \n\\end{proposition} \n\\begin{proof}\nFor every $p \\in \\mathcal S_0(\\Sigma)\\setminus \\hnm(\\Sigma)$, and $\\varepsilon_0=\\varepsilon_0(p)$ so that $\\Sigma \\cap (B_{\\varepsilon_0}(p)\\setminus p)$ is regular, we will show how to perturb $g$ and $\\Sigma$ so that $p$ becomes regular. We will do this by making an arbitrarily small change to $g$, $\\Sigma$ supported in $B_{\\varepsilon_0}(p)$. Because $\\mathcal S_0$ is discrete (but not necessarily closed when $n\\geq 9$) it is easy to enumerate the elements of $\\mathcal S_0(\\Sigma)\\setminus \\hnm(\\Sigma)$ and make a summably small change around each point. As such, it suffices to consider just the perturbation near $p$.\n\nBy definition, taking $\\varepsilon<\\varepsilon_0$ sufficiently small, $\\Sigma \\cap B_\\varepsilon(p)$ is one-sided homotopy area-minimizing. For concreteness write $\\Sigma \\cap B_\\varepsilon(p) = \\partial \\Omega$ in $B_\\varepsilon(p)$ and assume that $\\Omega$ is inner homotopy minimizing. By Lemma \\ref{lem:snm-hnm}, the tangent cone at $p$ is area-minimizing (to the same side). \n\nWe claim that (after taking $\\varepsilon>0$ smaller if necessary) there is a sequence of $\\Sigma_i\\in\\cR(B_\\varepsilon(p))$ with stable regular part, with $\\Sigma_i \\subset \\Omega$, $\\Sigma_i$ disjoint from $\\Sigma$, and $\\Sigma_i\\to \\Sigma$. Indeed, we can apply Proposition \\ref{prop:min-vs-htpy-min} to conclude that either (after shrinking $\\varepsilon>0$), $\\Omega$ is area-minimizing to the inside, or there are $\\Sigma_i$ as asserted.\n\nIn the case that $\\Omega$ is area-minimizing to the inside, we can still construct the $\\Sigma_i$ by shrinking $\\varepsilon>0$ even further so that $\\Omega$ is strictly area-minimizing to the inside and then minimizing area with respect to a boundary\n$\\Sigma \\cap \\partial B_\\varepsilon(p) + \\delta_i$, for a sequence $\\delta_i\\to 0$;\ni.e., the boundary of $\\Sigma\\cap B_\\varepsilon(p)$ pushed slightly into $\\Omega$. By the unique minimizing property, the minimizers will converge back to $\\Sigma$ in $B_\\varepsilon(p)$. \n\nFor $i$ sufficiently large we can write the intersection of $\\Sigma_i$\nwith the annulus $A(p,\\varepsilon\/5,\\varepsilon)$ as a graph of function $u_i$\nover $\\Sigma$.\n\nReasoning as in Hardt--Simon \\cite[Theorem 5.6]{HS} (cf. \\cite[Theorem 3.1]{Liu}), for $i$ sufficiently large, $\\Sigma_i$ will be regular in $B_{\\varepsilon\/2}(p)$. We now set \n\\[\n\\tilde\\Sigma_i =(\\Sigma_i \\cap B_{\\varepsilon\/5}) \\cup (\\Sigma\\setminus B_\\varepsilon(p)) \\cup ((\\Sigma + \\chi u_i)\\cap A(p,\\varepsilon\/5,\\varepsilon))\n\\]\nwhere $\\chi$ is a smooth cutoff function with $\\chi\\equiv1$ on $B_{\\varepsilon\/5}$ and $\\chi\\equiv 0$ on $B_{3\\varepsilon\/5}$. Note that\n\\[\nH_g(\\tilde\\Sigma_i) \\quad \\text{is supported in $B_{4\\varepsilon\/5}(p) \\setminus B_{\\varepsilon\/5}(p)$}\n\\]\nand $\\Vert H_g(\\tilde \\Sigma_i) \\Vert_{C^{2,\\alpha}} = o(1)$ as \n$i \\to \\infty$.\n\n\tNow, define $\\tilde g = e^{2f} g$, in this new metric, since $\\tilde \\Sigma$ is smooth, we have the transformation\n\t$$\n\tH_{\\tilde g}(\\tilde \\Sigma)= e^{-f}\\left(H_g(\\tilde\\Sigma)+\\frac{\\partial f}{\\partial \\nu} \\right)\\,,\n\t$$\n\twhere $\\nu$ is the normal direction to $\\Sigma$. \n\tSetting $H_{\\tilde g}(\\tilde \\Sigma)=0$, this reduces to the equation\n\t$$\n\tH_g(\\tilde\\Sigma)+\\frac{\\partial f}{\\partial \\nu}=0\n\t$$\n\twhich implies that $f=-H_g(\\tilde \\Sigma) \\zeta(\\nu)$, for a function $\\zeta(t)$ such that $\\zeta'(0)=1$ and $\\zeta\\equiv0$ for $|t|\\geq \\varepsilon\/100$ is a solution. Since, as observed, $H_g(\\tilde\\Sigma)$ is supported in $A(p,\\varepsilon\/5,4\\varepsilon\/5)$, so is the metric change, and since $\\|u\\|_{C^{4,\\alpha}}\\leq o(1)$ and $\\chi$ is smooth, we have\n\t$$\n\t\\|g-\\tilde g\\|_{2,\\alpha}=\\|e^f-1\\|_{C^{2,\\alpha}} \\|g\\|_{2,\\alpha} \\leq C\\, \\|u\\|_{C^{4,\\alpha}}\\,   \\|g\\|_{2,\\alpha} = o(1)\n\t$$ \n\tas $i\\to \\infty$. This completes the proof. \n\\end{proof}\n\t\n\t\\subsection{Proof of Theorem \\ref{thm:generic_stratum}} For $g \\in \\Met^{2,\\alpha}(M)$, apply Theorem \\ref{thm:nm+index_bound} to find $V\\in\\cR$ with\n\t\\[\n\t\\cH^0(\\hnm(V)) + \\Index(V) \\leq 1.\n\t\\]\n\tWe can apply Proposition \\ref{p:surgery} to $\\Sigma = \\supp V$ to find a metric $\\tilde g$ that is arbitrarily $C^{2,\\alpha}$-close to $g$ and a $\\tilde g$ minimal hypersurface $\\tilde\\Sigma \\in \\cR$ so that $\\mathcal S_0(\\Sigma) \\subset \\hnm(\\Sigma)$. (Note that if $\\Index(V) = 1$, then $ \\hnm(\\Sigma) =\\emptyset$, so $\\mathcal S_0(\\Sigma) = \\emptyset$.) This completes the first part of the proof. \n\t\nWe now consider $g \\in \\Met^{2,\\alpha}_{\\Ric>0}(M)$.\\footnote{The idea is that positive Ricci curvature rules out stable hypersurfaces but this requires the hypersurface to be two-sided. As such, we must consider two cases, depending on whether $\\Sigma$ is one or two-sided.} If $\\Sigma$ is two-sided, then $\\Index(\\Sigma)\\geq1$, so we can argue as above.  On the other hand if $\\Sigma$ is one-sided, then $[\\Sigma]\\neq 0 \\in H_{n}(M,\\mathbb{Z}_{2})$. We can then find $\\hat\\Sigma \\in [\\Sigma]$ by minimizing area in the homology class. The surface $\\hat\\Sigma$ may have singularities, but they are all locally area minimizing. Thus, we can apply Proposition \\ref{p:surgery} to $\\Sigma$ yielding $\\tilde\\Sigma$ and $\\tilde g$ with $\\mathcal S_0(\\tilde\\Sigma) = \\emptyset$. \n\\qed\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzfoqm b/data_all_eng_slimpj/shuffled/split2/finalzzfoqm
new file mode 100644
index 0000000000000000000000000000000000000000..24a1e9607cc6f283a0390d6603856bd43051418e
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzfoqm
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\n\n\n\n\n\nIn the series of articles \\cite{FKS, FJK1, FJK2}, degenerate elliptic equations in divergence form with real symmetric coefficients are studied. There, the degeneracy is given in terms of  an $A_{2}$ weight or a power of the jacobian of a quasi-conformal map. The first article gives interior estimates, the second article deals with the Wiener test and the third one study  boundary behavior and  harmonic measure.\n Further work along these lines, for example \\cite{FS, BM}, has been done. However, little work on the fundamental $L_p$ Dirichlet and Neumann\nproblems in the degenerate setting seems to have been done.\nHere, we want to initiate such study of boundary value problems, with $L_2$ boundary data, for a large class of weights, in the case of domains which are Lipschitz\ndiffeomorphic to the upper half space ${\\mathbb R}^{1+n}_+ := \\sett{(t,x)\\in{\\mathbb R}\\times {\\mathbb R}^n}{t>0}$, $n\\ge 1$. Thus, our work includes the case of special Lipschitz domains.\nAnother difference to earlier work that we want to stress is that we consider general elliptic divergence form systems, and not only scalar equations, as this is the natural setting for the\nmethods used. In this generality, interior pointwise regularity estimates fail in general, even\nin the uniform elliptic case.\nHowever, we emphasize that the methods used in this paper do not require such pointwise estimates.\n\n\nWe consider\ndivergence form second order, real and complex, \\textbf{degenerate} elliptic systems\n\\begin{equation}  \\label{eq:divform}\n   \\sum_{i,j=0}^n\\sum_{\\beta= 1}^m \\partial_i\\Big( A_{i,j}^{\\alpha, \\beta}(t,x) \\partial_j u^{\\beta}(t,x)\\Big) =0,\\ \\alpha=1,\\ldots, m\n\\end{equation}\nin ${\\mathbb R}^{1+n}_+$,\nwhere $\\partial_0= \\tdd{}{t}$ and $\\partial_i= \\tdd{}{x_i}$, $1\\le i\\le n$, which we abbreviate as\n${\\text{{\\rm div}}} A \\nabla u=0$, where\n\\begin{equation}   \\label{eq:boundedmatrix}\n  A=(A_{i,j}^{\\alpha,\\beta}(t, x))_{i,j=0,\\ldots,n}^{\\alpha,\\beta= 1,\\ldots,m}.  \\end{equation}\nWe assume $A$ to be degenerate   in the sense that for some $w\\in A_{2}({\\mathbb R}^n)$ and $C<\\infty$,\n  \\begin{equation}   \\label{eq:bounded}\n  |A(t,x)| \\le Cw(x),  \\ \\mathrm{for \\ a.e.}\\ (t,x)\\in {\\mathbb R}^{1+n}_{+}\n  \\end{equation}\nand elliptic degenerate in the sense that $w^{-1}A$ is  accretive on a space ${\\mathcal H}^0$ that we define below. This ellipticity condition   means that\nthere exists $\\kappa>0$ such that\n\\begin{equation}   \\label{eq:accrassumption}\n  \\re \\int_{{\\mathbb R}^n}  (Af(x),f(x)) dx\\ge   \\kappa\n   \\sum_{i=0}^n\\sum_{\\alpha=1}^m \\int_{{\\mathbb R}^n} |f_i^\\alpha(x)|^2 w(x)\\, dx,\n\\end{equation}\nfor all $f\\in {\\mathcal H}^0$ and a.e. $t>0$.  We have set $$(A\\xi,\\xi)= \\sum_{i,j=0}^n\\sum_{\\alpha,\\beta=1}^m A_{i,j}^{\\alpha,\\beta}(t,x)\\xi_j^\\beta\\, \\conj{\\xi_i^\\alpha}.\n$$\nThe space ${\\mathcal H}^0$ is the closed subspace of $L^2({\\mathbb R}^n,w;{\\mathbb C}^{m(1+n)})$ consisting of those functions with ${\\text{{\\rm curl}}}_{x}(f_{i}^\\alpha)_{i=1,\\ldots,n}=0$ for all $\\alpha$.  The case of equations is when $m=1$ or, equivalently, when $A_{i,j}^{\\alpha,\\beta}=A_{i,j}\\delta _{\\alpha,\\beta}$. In this case, the accretivity condition becomes the usual pointwise accretivity\n\\begin{equation}   \\label{eq:poinwiseaccr}\n \\re  \\sum_{i,j=0}^n A_{i,j}\\xi_j \\conj{\\xi_i} \\ge\n \\kappa  \\sum_{i=0}^n |\\xi_i|^2 w(x),\n\\end{equation}\nfor all $\\xi\\in{\\mathbb C}^{1+n}$ and a.e. $(t,x)\\in {\\mathbb R}^{1+n}_{+}$.\nObserve that the function $(t,x)\\mapsto w(x)$ is an $A_{2}$ weight in  ${\\mathbb R}^{1+n}$  if $w$ is an  $A_{2}$ weight in ${\\mathbb R}^n$. So, the degeneracy is a special case of that considered in the works mentioned above. However, for the boundary value problems we wish to consider, this seems a natural class. To our knowledge,  this has not been considered before.\n\n A natural question is whether weights could depend on both variables or only on the $t$-variable. Already in the non-degenerate case there are regularity conditions without which the Dirichlet problem is ill-posed.  As for the degenerate case, the well known example from \\cite{CaS}  when $A= t^{1-2s}I$, $0<s<1$, leading to representation of the fractional Laplacian $(-\\Delta)^s$ as the Dirichlet to Neumann operator, is of  a  nature  that our theory cannot cover.  In fact, our weights need to have a trace at the boundary in some sense. Another option is to assume the weight depends on the $x$-variable only and put the $t$-perturbations in the coefficients. See below in this introduction and Section \\ref{sec:rep}.  \n\nThe equation \\eqref{eq:divform} must be properly interpreted.\nSolutions are taken with $u$ and $\\nabla u$ locally in $L^2({\\mathbb R}^{1+n}_{+}, w(x)dxdt)$, and the equation is taken in the sense of distributions. Note that we allow complex coefficients and systems, so most of the theory developed for real symmetric equations does not apply, and even for real coefficients we want to develop methods regardless of regularity theory.\n\nThe boundary value problems can be formulated as follows:   find weak solutions $u$ with appropriate interior estimates of $\\nabla_{t,x} u$\nsatisfying one of the following three natural boundary conditions.\n\\begin{itemize}\n\\item The Dirichlet condition $u= \\varphi$ on ${\\mathbb R}^n$,\ngiven the Dirichlet datum $\\varphi\\in L^2({\\mathbb R}^n,w;{\\mathbb C}^m)$.\n\\item The Dirichlet regularity condition $\\nabla_x u= \\varphi$ on ${\\mathbb R}^n$, given the regularity datum\n$\\varphi\\in L^2({\\mathbb R}^n,w;{\\mathbb C}^{mn})$ satisfying ${\\text{{\\rm curl}}}_x \\varphi=0$. Alternately, $\\varphi$ is the tangential gradient of the Dirichlet datum.\n\\item The Neumann condition $\\partial_{\\nu_{A}}u= ( A\\nabla_{t,x} u, e_{0})= \\varphi$ on ${\\mathbb R}^n$, given\n$\\varphi\\in L^2({\\mathbb R}^n,w^{-1};{\\mathbb C}^{m})$. Here $e_{0}$ is the upward unit  vector in the $t$-direction.\n\\end{itemize}\nObserve that  the natural space for  the gradient at the boundary is  $L^2({\\mathbb R}^n,w;{\\mathbb C}^{m(1+n)})$, and since $A$ is of the size $w$, multiplication by $A$ maps into $L^2(w^{-1})$.  Thus  the weight $w^{-1}$ is  natural  for the conormal derivative. In order   to work with  the same weighted space for all three problems, we shall  consider the $w$-normalized conormal derivative $\\partial_{\\nu_{w^{-1}A}}u|_{t=0}=w^{-1}\\partial_{\\nu_{A}}u|_{t=0}\\in L^2({\\mathbb R}^n, w;{\\mathbb C}^m)$.\n\nBoundary value problems can be formulated  for $L^p$ data with $p\\ne 2$. This is for later work and here we restrict our attention to $p=2$.\n We mention that there are also ``intermediate'' boundary value problems for regularity\/Neumann data in some negative Sobolev spaces based on $L^2(w)$ using  fractional powers of the Laplacian $-\\Delta_{w}$ (defined later) that can be treated by the same methods.  One important case is the treatment of variational solutions in this context, \\textit{i.e.} those with $\\iint_{\\mathbb{R}^{1+n}_+} |\\nabla u |^2\\, dw(x) dt<\\infty$, in which these problems are always well-posed. In the case $w=1$, the methods have been worked out completely in \\cite{R}.\n\n\nLet us also comment on the corresponding degenerate inhomogeneous problem ${\\text{{\\rm div}}} A \\nabla u=f$, with ${\\text{{\\rm div}}} A \\nabla u$ denoting the left hand side in \\eqref{eq:divform}. A study of such equations would be of interest in its own right, but could also prove useful in the study of boundary value problems of the type we study here. Such applications were implemented in \\cite{DinR,DPR} in the non-degenerate setting where the coefficients of the operator were assumed to satisfy a Carleson measure condition in place of $t$-independence. In \\cite{DinR} a duality argument reduced the desired estimate for solutions to the homogeneous equation to an estimate on a solution to an inhomogeneous equation, which could subsequently be proved. While such investigations in the degenerate setting would certainly make the theory more complete, studying the inhomogeneous equation is not our goal here.\n\n\nWe shall obtain \\textit{a priori} representations of solutions and existence of boundary  traces (in the almost everywhere sense) for appropriate solution spaces together with various estimates involving non-tangential maximal functions, a characterization of well-posedness, a duality principle between regularity and Dirichlet problems, perturbation results for well-posedness on perturbing the coefficients,  and well-posedness for hermitian systems and some block-triangular coefficients.\n\n\nTo this end, we follow the two step strategy developed in \\cite{AA1} and \\cite{AA2} and we assume the reader has these references handy.  The first step is to obtain a  priori representations and boundary traces (in weighted $L^2$ and almost everywhere) for solutions in appropriate classes. Actually, we shall even improve upon known results on almost everywhere convergence  when $w=1$. The starting point is to transform the second order equation to a first order system in the $w$-normalized conormal gradient\n$$\\nabla_{w^{-1}A} u= \\begin{bmatrix} \\partial_{\\nu_{w^{-1}A}}u \\\\ \\nabla_x u \\end{bmatrix},$$\nand then prove a weighted  quadratic estimate  and a non-tangential maximal estimate for functions of  a bisectorial operator $DB$ when the coefficients are $t$-independent.  Then one can develop a semigroup representation of those $w$-normalized  conormal gradients. This semigroup method  for $t$-independent $A$ and their $t$-dependent perturbations is presented in  such an abstract way in \\cite{AA1}  that it applies in the weighted space without any change once  the initial quadratic estimate is established. We shall basically define the necessary objects and prove only what is not \\textit{mutatis mutandi} the same. Here, the discrepancy function should be renormalized by the weight, so it reads $w^{-1}(x)(A(t,x) -A(0,x))$, and it is measured by a weighted Carleson-Dahlberg condition.  Also some estimates such as  weighted Carleson embedding require the fact that $dw$ is doubling, which follows from the $A_{2}$ condition.\n\nThe quadratic estimate for bisectorial $DB$ is also a fundamental ingredient in the proof of non-tangential maximal estimates and almost everywhere convergence. As in \\cite{AA2}, the almost everywhere convergence is not in a pointwise sense but in some averaged sense on Whitney regions approaching the boundary.  We obtain convergence results for solutions in spaces relative to the Dirichlet problem, and for solutions,  their gradients and $w$-normalized conormal derivatives  for solutions in spaces relative to the regularity and Neumann problems. We stress that even when $w=1$, this convergence at the level of gradients is new in this generality.\n\n\n\n Once representations and boundary traces of solutions are obtained, the second step is to  formulate the boundary value problems and show that solvability amounts to inverting boundary operators. As for positive results, we shall obtain\nwell-posedness results for $t$-independent $A$ which, in addition, are hermitian, block-diagonal,  block-triangular    or  merely satisfy a divergence free condition on the triangular block. Let us remark that in the non-degenerate case $w=1$, all these BVPs are well posed with constant matrices. Here, constant would mean that  $A$ is  a constant multiple of $w$, or that  $DB$ comes with constant $B$. However, we do not know well-posedness  for  such $B$, in particular  for  the example below with $\\varepsilon \\sim 2$.\nAs usual, all positive well-posedness results are stable under perturbation, that is, when $w^{-1}(A'-A)$ is small in $L^\\infty$. We also obtain perturbation results for well-posedness with $t$-dependent coefficients $A$ under  smallness of the discrepancy function $w^{-1}(A(t,x) -A(0,x))$ in a weighted Carleson-Dahlberg sense.\n\n The key result is therefore the quadratic estimates for bisectorial operators $DB$ in $L_2({\\mathbb R}^n,w)$, stated in Theorem \\ref{th:main}. The differential operator $D$, although not explicit in our notation, depends on the weight $w$ as well. The structure of the operator dictates a different\nreduction to a Carleson measure estimate than in the uniformly elliptic case. In particular there\nappears a dyadic averaging\/expectation operator $E_t$ which acts on vector valued functions by\nunweighted averages of some components and weighted averages  of some other components.\nIt seems therefore from the proof that the $A_2$ condition on the weight is sharp for this\nquadratic estimate, since the $L_2({\\mathbb R}^n,w)$ bound of $E_t$ requires the $A_2$ condition.\nIn particular, our method does not to apply to the quasiconformal Jacobians mentioned above.\n\n\nThe quadratic estimate from Theorem \\ref{th:main} is the central estimate without which the method breaks down. We summarize the harvest  of results using in an essential way this quadratic estimate:\n\n1. The estimates of the functional calculus in Proposition \\ref{prop:equi};\n\n2. The non-tangential maximal estimates in $L^2$ and  almost\neverywhere convergence in Theorem \\ref{thm:NTmaxandaeCV};\n\n3. The estimates, through operational calculus, of the singular integral\nwith operator-valued kernel $S_{\\mathcal E}$ in Theorems \\ref{thm:estSA} and \\ref{thm:esttSA};\n\n4. The perturbation results for boundary value problems in Theorems \\ref{thm:Nellie} and \\ref{thm: DirLip}.\n\nThe proof of the quadratic estimate is not an easy matter\nand will be the hard part of the article. Let us comment on this. First, it does not follow from available extensions of the proof of the Kato square root problem such as the ones in \\cite{AKMc}, \\cite{elAAM} or \\cite{Ban}. This is because $D$ will no longer be a constant coefficient operator.  Let us give  an example to illustrate the differences. When $w=1$, the basic example is $$\nD=\\begin{bmatrix}\n  0     & - \\frac{d}{dx}   \\\\\n \\frac{d}{dx}     & 0\n\\end{bmatrix}.$$\nOn ${\\mathbb R}$, if $B$ is any constant elliptic matrix, the domain of $DB$ is the Sobolev space $H^1({\\mathbb R};{\\mathbb C}^2)$. This can be seen using a Fourier transform argument. If $w\\ne 1$, then $D$ becomes\n$$\nD=\\begin{bmatrix}\n  0     & - \\frac{1}{w}\\frac{d}{dx}w   \\\\\n \\frac{d}{dx}     & 0\n\\end{bmatrix} $$\nand (the column vector) $(f,g)$ is in the domain of $D$ if and only if $f\\in H^1({\\mathbb R},w)$ and $wg\\in  H^1({\\mathbb R},w^{-1})$ (the weighted Sobolev spaces).   Note that elements in these spaces are  absolutely continuous functions when $w\\in A_{2}$. Let $B$ be  the constant matrix\n$$\nB=\\begin{bmatrix}\n  1     & 0   \\\\\n\\varepsilon     & 1\n\\end{bmatrix}.$$\nThen for any $\\varepsilon\\ne 0$, the domain of $DB$ cannot be the same as the domain of $D$. Indeed, the conditions to be in the domain of $DB$ are $w(g+\\varepsilon f) \\in H^1({\\mathbb R},w^{-1})$ and $f\\in H^1({\\mathbb R},w)$.   If $wg$ were in $H^1({\\mathbb R},w^{-1})$, then so would $wf$, and  both $f$ and $wf$ would  be continuous on ${\\mathbb R}$, which is a restriction on $f$.\n\nMoreover, $D$ cannot be coercive on its range. In the previous example,  that would lead to an estimate like $g'$ controlled by $\\frac 1 w (gw)'$, which is absurd.\n\n Nevertheless,  Cruz-Uribe and Rios \\cite{CR} recently proved  the Kato conjecture for square roots of degenerate elliptic operators $-{\\text{{\\rm div}}}_{{\\mathbb R}^n} A \\nabla_{{\\mathbb R}^n}$ on ${\\mathbb R}^n$, which  corresponds here to the special case of  block diagonal, bounded, measurable and accretive $B=\\begin{bmatrix}\n  1     & 0   \\\\\n0     & w^{-1}A\n\\end{bmatrix}.$  The preceding example indicates that non-block terms  introduce a difficulty that was not in the above work; indeed, it makes the argument  harder. Let us explain where this difficulty appears  in the proof and how we overcome it.   As is customary in this area, we look for  a $Tb$ argument to prove a Carleson estimate which proceeds via a stopping time argument on the averages on dyadic cubes of  some test functions. In our situation, averages are taken   with respect to both the unweighted and weighted measure $dx$ and $w(x)dx$. But if the weight varies too much on a given cube $Q$,  the stopping time does\n  not give  useful information.   We are forced to organize  the collection of all subcubes of $Q$ in subclasses  on which the $dx$-averages of $w$  do not vary too much from a parent cube. The ideal situation is that all cubes fall into one of these subclasses. This is not the case, but the next best thing happens, namely that  the those subcubes which are left out satisfy a packing condition with uniform, possibly large, constant. This organization of cubes is intrinsic to the given weight and is in fact a weighted version of a result found in Garnett's book \\cite{Gar}. Here, it allows us to run the stopping time on the test functions for the Carleson estimate.  In the situation of \\cite{CR},  one can split things so as to run two separate stopping time arguments each concerned with one measure so this step is not necessary.\n\n\n\n\n\nTo conclude this introduction,  we discuss two different boundary geometries.  First, all of the results here are invariant under bilipschitz changes of variables. Thus,  whenever one can pull back a boundary value problem on a special Lipschitz domain  to one that fits our  hypotheses, one obtains a result for the initial problem. As an example, this shows that  if $w\\in A_{2}({\\mathbb R}^n)$,  the three boundary value problems for $-{\\text{{\\rm div}}}_{x,t} (w(x)\\nabla_{x,t}u)(x,t)=0$ with appropriate interior estimates (see Section \\ref{sec:solvability}) are  well-posed on any special Lipschitz domain $\\Omega=\\{t>\\varphi(x)\\}$ when the corresponding datum is in $L^2(\\partial\\Omega; \\tilde wd\\sigma)$ where $\\sigma$ is the surface measure and $\\tilde w(\\varphi(x)):=w(x)$. Applying the standard pullback $(x,t) \\mapsto (y, t-\\varphi(y))=(y,s)$, we obtain an equation of the form $ {\\text{{\\rm div}}}_{y,s} (A(y)\\nabla_{y,s}v)(y,s)=0$ that is degenerate elliptic.\nSecondly, it is natural to expect results on bounded domains. Bilipschitz invariance implies that one can look at the case of the unit  ball as the non-smoothness is carried by the coefficients. There, the setup of \\cite{AA2} applies to  radially independent weights and degenerate coefficients, and perturbations of the latter. It will be clear from the present article that it all depends on the weighted quadratic estimate on the boundary. This requires a proof that is left to further work.\n\n\n\n\nThe first  author was partially supported by the ANR project ``Harmonic analysis at its boundaries'' ANR-12-BS01-0013-01. The second one was supported by the Grant 621-2011-3744 from the Swedish Research Council, VR. The first author wants to thank Yannick Sire for bringing his attention to this problem.\n\n\n\\section{Preliminaries on weights}\n\n\\subsection{Muckenhoupt weights}\n\nRecall that for a weight $w$ on ${\\mathbb R}^n$ and $p>1$, the  $A_{p}$ condition reads\n$$\n\\bigg(\\barint_{\\hspace{-6pt}Q} w\\, dx\\bigg)\\,\n\\bigg(\\barint_{\\hspace{-6pt}Q} w^{1-p'} \\, dx\\bigg)^{p\/p'}\\le C,\n$$\n for all cubes $Q$ with $p'$ the conjugate exponent to $p$.  The smallest possible  $C$ is denoted by $[w]_{A_{p}}$.\nThe notation $\\barint_{\\hspace{-2pt}E}$ means the average with respect to the indicated measure on $E$.\n\nWe identify  $w$ with the measure $dw=w(x)\\, dx$ and write $w(E)$ for $\\int_{E} dw$ while $|E|= \\int_{E} dx$.\nRecall that $w\\in A_{p}$ implies $w\\in A_{q}$ for all $q>p-\\varepsilon$ where $\\varepsilon>0$ depends on  $[w]_{A_{p}}.$  Every $w\\in A_{p}$ is an $A_{\\infty}$ weight:  there exist constants $0< \\sigma \\le 1 \\le \\tau<\\infty$ such that\n\\begin{equation}\n\\label{eq:ainfty}\n\\left(\\frac {|E|}{|Q|}\\right)^\\tau \\lesssim \\frac{w(E)}{w(Q)} \\lesssim  \\left(\\frac {|E|}{|Q|}\\right)^\\sigma\n\\end{equation}\nfor all cubes $Q$ and measurable subsets $E$ of $Q$ (actually, $\\tau=p$ if $w\\in A_{p}$ and $(1\/\\sigma)'$ is the reverse H\\\"older exponent of $w$, which can be arbitrary in $(1,\\infty]$). In particular, $dw$ is a doubling measure. There also exists a constant $c_0 > 0$ such that\n\\begin{equation} \\label{reversejensen}\n\\barint_{\\hspace{-6pt}Q} \\ln w(x) dx \\leq \\ln\\bigg(\\barint_{\\hspace{-6pt}Q} w(x) dx\\bigg) \\leq \\barint_{\\hspace{-6pt}Q}\\ln w(x) dx + c_0,\n\\end{equation}\nthe first inequality being Jensen's inequality and the second being a reverse form of it.\n\nDenote $ L^p(w; {\\mathbb C}^d)=L^p({\\mathbb R}^n, w; {\\mathbb C}^d)$ for $d\\ge 1$, and $L^p(w)=L^p(w;{\\mathbb C})$.\nIf $w\\in A_{p}$,\n\\begin{equation}\n\\label{eq:dxdwAp}\n\\barint_{\\hspace{-6pt}Q} \\left|f(y)\\right|\\,  dy \\le [w]_{A_{p}}^{1\/p}  \\left( \\barint_{\\hspace{-6pt}Q} |f(y)|^p\\,  dw(y)\\right)^{1\/p}.\n\\end{equation}\nIn particular, for $w\\in A_{2}$, since $w\\in A_{2\/p}$ for some $p>1$, this implies Muckenhoupt's theorem: the Hardy-Littlewood maximal operator $M$ with respect to $dx$ is bounded on $L^2(w)$.\n\n\\subsection{A corona decomposition for $A_{2}$ weights} \\label{coronasec}\n\n\n\n\n\nWe use the following dyadic decomposition of ${\\mathbb R}^n$. Let\n$\\triangle= \\bigcup_{j=-\\infty}^\\infty\\triangle_{2^j}$ where\n$\\triangle_{2^j}:=\\{ 2^j(k+(0,1]^n) :k\\in{\\mathbb Z}^n \\}$. For a dyadic cube $Q\\in\\triangle_{2^j}$, denote by $\\ell(Q)=2^j$\nits \\emph{sidelength} and  by $|Q|= 2^{nj}$ its {\\em Lebesgue volume}.  We set $\\triangle_t=\\triangle_{2^j}$  if $2^{j-1}<t\\le\n2^j$.\n\nHere we describe a decomposition of a dyadic cube $Q \\in \\triangle$ with respect to a Muckenhoupt weight $w \\in A_2$.\n\n\nWe follow a construction of Garnett \\cite{Gar}. For a fixed $\\sigma_w > 0$,  to be chosen later,  we consider  $B^w(Q)$ the collection of  those  (``bad for $\\ln w$'')  maximal sub-cubes of $Q$ for which\n\\begin{equation*} \\label{st1}\n|(\\ln w)_R - (\\ln w)_Q| > \\sigma_w.\n\\end{equation*}\n In this section, we use the notation $f_{Q}= (f)_{Q}:= \\barint_{\\hspace{-2pt}Q} f\\, dx$.\nWe can then define $B^w_j(Q)$ inductively  as   $B^w_{1}(Q)=B^w(Q)$  and for $j = 2,3,\\dots$ by\n\\[\nB^w_j(Q) = \\bigcup_{R \\in B^w_{j-1}(Q)} B^w(R), \\quad \\mbox{and set} \\quad B^w_*(Q) = \\bigcup_{j=1}^\\infty B^w_j(R).\n\\]\nThe following proposition shows that the ``number'' of  cubes {on which the oscillation of $\\ln w$  differs too much from the one on some ancestor} can be controlled.\n\n\\begin{prop} \\label{prop4}\nFor a cube $R \\in \\triangle$,  if $R\\subseteq Q$ is not contained in a  cube of $B^w(Q)$  then $|(\\ln w)_R - (\\ln w)_Q|  \\leq   \\sigma_w$. We also have that\n\\begin{equation} \\label{eqsq}\n\\sum_{R \\in B^w_*(Q)} w(R) \\leq \\frac{C}{\\sigma_w^2} w(Q)\n\\end{equation}\nfor some $C < \\infty$ which depends only on $[w]_{A_2}$.\n\\end{prop}\n\n\\begin{proof}\nThe first statement of the Proposition is clear from the definition of $B^w(Q)$, so we only need to prove \\eqref{eqsq}.\n\n\n\nFor each $R \\in B^w_*(Q)$, there is a unique $R'\\in B^w_*(Q) \\cup \\{Q\\}$ such that $R \\in B^w_1(R')$, that is, $R'$ is the stopping-time parent of $R$. Thus\n\\[\nw(R) \\leq \\int_{R} \\frac{|(\\ln w)_R - (\\ln w)_{R'}|^2}{\\sigma_w^2}\\,  dw\n\\]\nand so\n\\begin{align*}\n\\sum_{R \\in B^w_*(Q)} w(R) & \\lesssim \\frac{1}{\\sigma_w^2} \\sum_{R \\in B^w_*(Q)} \\int_{R} |(\\ln w)_R - (\\ln w)_{R'}|^2\\,  dw \\\\\n& = \\frac{1}{\\sigma_w^2} \\int_{Q} \\bigg( \\, \\sum_{R \\in B^w_*(Q)} 1_{R} |(\\ln w)_R - (\\ln w)_{R'}|^2 \\bigg)\\,  dw,\n\\end{align*}\nwhere $1_R$ is the characteristic function of the set $R$.   For any $b\\in L^1_{loc}({\\mathbb R}^n, dx)$ consider\n the square function\n\\[\nS(b)(x) := \\bigg(\\, \\sum_{R \\in B^w_*(Q)} 1_{R}(x) |b_R - b_{R'}|^2 \\bigg)^{1\/2}.\n\\]\nFor any $w \\in A_\\infty$ we have (see  Lemma 6.4 in \\cite{Gar} for the unweighted estimate and   \\cite{HR2}  for the weighted one) the estimate\n\\[\n\\int_{{\\mathbb R}^n} |S(b)|^2 dw \\lesssim \\int_{{\\mathbb R}^n} |M(b)|^2 dw,\n\\]\nwhere $M$ is the Hardy-Littlewood maximal operator with respect to  $dx$. Applying this to $b(x) := (\\ln w(x) - (\\ln w)_Q)1_Q(x)$ and using Muckenhoupt's  theorem,  it suffices to show that\n\\begin{equation} \\label{eq:rem}\n\\int_Q |\\ln w - (\\ln w)_Q|^2\\, dw \\lesssim w(Q).\n\\end{equation}\n\n This inequality follows from the fact that  $\\ln w \\in BMO({\\mathbb R}^n, dx)$ when $w\\in A_{2}$ and the fact that    $ BMO({\\mathbb R}^n, dx)= BMO({\\mathbb R}^n, dw)$ with equivalent norms  for $w\\in A_{\\infty}$. The latter statement is a consequence of the John-Nirenberg inequality and reverse H\\\"older inequality for $w$. However, we present a direct proof.  It follows from Jensen's inequality and the $A_2$ condition on $w$ that\n\\[\n\\int_Q e^{|\\ln w(x) - (\\ln w)_Q|}\\, dx \\lesssim |Q|.\n\\]\n Setting $M_\\lambda = \\{x \\in Q \\, ; \\, |\\ln w(x) - (\\ln w)_Q| > \\lambda  \\}$, we can write\n\\[\n\\int_{0 }^\\infty e^\\lambda |M_\\lambda| d\\lambda=  \\int_Q \\int_{ 0  }^{|\\ln w(x) - (\\ln w)_Q|} e^{\\lambda} d\\lambda dx \\le \\int_Q e^{|\\ln w(x) - (\\ln w)_Q|} \\, dx ,\n\\]\nso\n\\[\n\\int_{0  }^\\infty e^\\lambda |M_\\lambda| d\\lambda \\lesssim |Q|.\n\\]\nSimilarly,\n\\[\n\\int_Q |\\ln w - (\\ln w)_Q|^2\\,  dw  = \\int_{0}^\\infty 2\\lambda w(M_\\lambda) d\\lambda.\n\\]\nCondition \\eqref{eq:ainfty}, in which we take $\\sigma<1$ as one can always do, ensures that\n\\begin{align*}\n\\int_{0}^\\infty \\lambda w(M_\\lambda) d\\lambda & \\lesssim \\int_{0}^\\infty \\lambda w(Q)\\left(\\frac{|M_\\lambda|}{|Q|}\\right)^\\sigma d\\lambda \\\\\n& = w(Q) \\int_{0}^\\infty \\left(\\frac{|M_\\lambda|}{|Q|}e^\\lambda \\right)^\\sigma (\\lambda e^{-\\lambda\\sigma})d\\lambda \\\\\n& \\lesssim  w(Q) \\left(\\int_{0}^\\infty \\frac{|M_\\lambda|}{|Q|}e^\\lambda d\\lambda \\right)^\\sigma \\left(\\int_{0}^\\infty (\\lambda e^{-\\lambda\\sigma})^{1\/(1-\\sigma)}d\\lambda \\right)^{1-\\sigma} \\\\\n& \\lesssim w(Q),\n\\end{align*}\nwhich proves \\eqref{eq:rem} and with it the proof of the Proposition.\n\\end{proof}\n\n\\subsection{Review of weighted Littlewood-Paley inequalities}\n\nWe recall here a few facts of Littlewood-Paley theory. The treatment in Wilson's book used in \\cite{CR} is not completely adapted for our needs. In particular, we use Calder\\'on's reproducing formula in a different way. Here $w$ denotes a weight in $A_{2}$. We begin with approximation issues.\n\n\n\\begin{lem}\\label{lem:approx} Let $\\varphi\\in L^1({\\mathbb R}^n)$ with a  radially decreasing integrable majorant $\\phi$. Let $\\varphi_{\\varepsilon}(x) =\\varepsilon^{-n} \\varphi(x\/\\varepsilon)$ for $\\varepsilon>0$.\n\\begin{itemize}\n  \\item Convolution with $\\varphi_{\\varepsilon}$ is a bounded operator on $L^2(w)$, uniformly with respect to $\\varepsilon$.\n  \\item For every $f\\in L^2(w)$, $\\varphi_{\\varepsilon}\\star f\\to  cf$ and $ \\varphi_{1\/\\varepsilon}\\star f \\to 0$ in $L^2(w)$ as $\\varepsilon\\to 0$, where $c=\\int_{{\\mathbb R}^n}\\varphi(x)\\, dx$.\n\n\\end{itemize}\n\n\n\\end{lem}\n\n\n\\begin{proof} For the first point, we observe that $\\sup_{\\varepsilon>0}|\\varphi_{\\varepsilon}\\star f| \\le \\|\\phi\\|_{1} Mf$ almost everywhere (See \\cite[Corollary 2.1.12]{Gra}) and  recall that $M$ is bounded on  $L^2(w)$.\nFor the second point,\n it is easy to see that  $L^\\infty_{c}({\\mathbb R}^n)$, the set of bounded functions with bounded support, is dense in $L^2(w)$.  Thus, it is enough to assume $f\\in L^\\infty_{c}({\\mathbb R}^n)$. Then   $\\varphi_{\\varepsilon}\\star f- cf$ and  $\\varphi_{1\/\\varepsilon}\\star f$  converge  almost everywhere (for $dx$, thus for $dw$) to 0  as $\\varepsilon\\to 0$ (See \\cite[Corollary 2.1.19]{Gra}). The conclusion follows using the dominated convergence theorem in $L^2(w)$ as  $Mf\\in L^2(w)$.\n\\end{proof}\n\n\n\n\n\n\\begin{cor}\\label{lem:density}  $C_{0}^\\infty({\\mathbb R}^n)$ is dense in $L^2(w)$.  Also the space ${\\mathcal E}$ of\nfunctions of the form $\\int_{\\varepsilon}^R \\psi_{t}\\star f\\, \\frac{dt}t$ where $f\\in C^\\infty_{0}({\\mathbb R}^n)$, $\\psi_{t}(x)=t^{-n}\\psi(x\/t)$  with  $\\psi\\in S({\\mathbb R}^n)$ with  Fourier transform supported away from 0 and  $\\infty$,  $\\varepsilon>0, R<\\infty$ and $\\int_{0}^\\infty \\hat \\psi(t\\xi) \\, \\frac {dt} t=1$ for all $\\xi\\ne 0$, form a dense subspace of $L^2(w)$. [Here, $\\hat g$ is the Fourier transform of $g$ in ${\\mathbb R}^n$.]\n\\end{cor}\n\n\\begin{proof} As  $L^\\infty_{c}({\\mathbb R}^n)$ is dense in $L^2(w)$,\nchoosing $\\varphi\\in C^\\infty_{0}({\\mathbb R}^n)$ in the previous lemma proves the first density. Next, with $\\psi$ as in the statement, it is easy to see  that $\\hat \\varphi(\\xi): = \\int_{1}^\\infty  \\hat\\psi(t\\xi)\\, \\, \\frac {dt} t= 1- \\int_{0}^1 \\hat\\psi(t\\xi) \\, \\frac {dt} t$ for $\\xi\\ne 0$ and $\\hat \\varphi(0)=1$ is a Schwartz class function and that $\\int_{\\varepsilon}^{R} \\psi_{t}\\star f\\, \\frac{dt}t= \\varphi_{\\varepsilon}\\star f - \\varphi_{R}\\star f$. So the lemma follows.\n\\end{proof}\n\n\\begin{cor}\\label{cor:Calderon}  Assume $\\psi$ and $\\varphi$ are integrable functions, that  $\\varphi$ is as in Lemma \\ref{lem:approx} with $\\int \\varphi =1$ and   that   $\\int_{\\varepsilon}^{R} \\hat\\psi(t\\xi)\\, \\frac{dt}t= \\hat\\varphi(\\varepsilon\\xi)- \\hat\\varphi({R}\\xi)$ for all $\\xi\\in {\\mathbb R}^n, 0<\\varepsilon<R$. Set $Q_{t}f= \\psi_{t}\\star f$. Then the Calder\\'on representation formula $\\int_{0}^\\infty Q_{t}f\\, \\frac{dt}t= f$ holds for all $f\\in L^2(w)$ in the sense that\n$\\int_{\\varepsilon}^R Q_{t}f\\, \\frac{dt}t$ converges to $f$ in $L^2(w)$ as $\\varepsilon\\to 0$ and $R\\to \\infty$.\n\\end{cor}\n\n\\begin{proof} The lemma follows immediately from the formula\n$\\int_{\\varepsilon}^{R} \\psi_{t}\\star f\\, \\frac{dt}t= \\varphi_{\\varepsilon}\\star f - \\varphi_{R}\\star f$ and Lemma \\ref{lem:approx}.\n\\end{proof}\n\nNext, we recall that any Calder\\'on-Zygmund operator in the sense of  Meyer's book \\cite{Me}  is bounded on $L^2(w)$ (the sharp dependance of the norm with respect to $[w]_{A_{2}}$ has been obtained by T. Hyt\\\"onen \\cite{H} but we do not need such a refined estimate). This result extends to vector-valued Calder\\'on-Zygmund operators, and consequently\nwe have the following useful corollary.\n\n\n\n\n\n\n\\begin{lem}\\label{lem:LPw}  Let $\\psi$ be an integrable function in ${\\mathbb R}^n$ with $\\int_{{\\mathbb R}^n} \\psi(x)\\, dx=0$ and such that $(\\psi_{t}(x-y))_{t>0}$ is an $ L^2({{\\mathbb R}_+}, \\frac{dt}t)$-valued Calder\\'on-Zygmund kernel and set  $Q_{t}f= \\psi_{t}\\star f$. Then for all $f\\in L^2(w)$,\n$$\\int_{0}^\\infty \\|Q_{t}f\\|^2_{L^2(w)}\\, \\frac{dt}t \\le C_{w,\\psi}\\|f\\|^2_{L^2(w)}.$$\n\\end{lem}\n\n\\begin{proof}\nNote that $T: f\\mapsto (Q_{t}f)_{t>0}$ is a Calder\\'on-Zygmund operator, bounded from $L^2({\\mathbb R}^n)$ to $L^2({\\mathbb R}^n, L^2({{\\mathbb R}_+}, \\frac{dt}t))$. Thus weighted $L^2$ theory for $A_{2}$ weights applies (see \\cite{GCRF}, Chapter V).\n\\end{proof}\n\n\n\n\nTo conclude this section, we  recall a  trick proved by  Cruz-Uribe and Rios \\cite{CR} originating from the proof of \\cite[Corollary 4.2]{DR}.\n\n\\begin{lem}\\label{lem:DRF} If $T$ is a linear operator that is bounded on $L^2(w)$ for any $w\\in A_{2}$ with norm depending only on $[w]_{A_{2}}$, then for any fixed $w\\in A_{2}$, there exists $\\theta\\in (0,1)$ and $C>0$ such that\n$\\|T\\|_{{\\mathcal L}(L^2(w))}\\le C\\|T\\|_{{\\mathcal L}(L^2(dx))}^\\theta$.\n\\end{lem}\n\n\n\n\\subsection{Weighted Sobolev spaces and the gradient operator}\n\n\nFor $1<p<\\infty$,  $W^{1,p}(w)$ is the  closure of $ C^\\infty_{0}({\\mathbb R}^n)$ in the norm\n$$\n\\left(\\int_{{\\mathbb R}^n} \\big( |f|^p + |\\nabla f|^p \\big)\\, dw\\right)^{1\/p}.\n$$ For $p=2$,  we set $H^1(w)=W^{1,2}(w).$\n\nIt follows from \\cite[Theorem 1.5]{FKS} that $w\\in A_{p}$ satisfies a Poincar\\'e inequality:\n\\begin{equation}\n\\label{eq:Poincare}\n\\left(\\barint_{\\hspace{-6pt}Q} |\\psi - c |^p \\, dw\\right)^{1\/p} \\lesssim \\ell(Q) \\left(\\barint_{\\hspace{-6pt}Q} |\\nabla \\psi|^p \\, dw\\right)^{1\/p}\n\\end{equation}\nwhere $\\psi \\in C^\\infty_{0}({\\mathbb R}^n)$, $c$ is   the mean of $\\psi$ with respect to either  $dw$ or $dx$   and the implicit constant depends only on $[w]_{A_{p}}$, $n$ and $p$.  In particular, this implies that $W^{1,p}(w)$ can be identified with a space of measurable functions imbedded in $L^p(w)$.\nThe gradient (taken in the sense of distributions) extends to an operator on $W^{1,p}(w)$, still denoted by $\\nabla$, and the Poincar\\'e inequality extends to all $W^{1,p}(w)$ functions. The space of Lemma \\ref{lem:density} is also dense in $W^{1,2}(w)$ when $w\\in A_{2}$.\n\n\n\n\n We shall need the following result.\n\n\\begin{prop} \\label{prop:interpolation} If $w\\in A_{2}$, the spaces $W^{1,p}(w)$ interpolate by the real or complex methods for $r(w)< p<\\infty$ where $r(w)<2$ is the infimum of  exponents $p$ for which $w\\in A_{p}$.\n\\end{prop}\n\n\n\\begin{proof}\nThis follows from N. Badr's interpolation theorem \\cite[Theorem 8.4]{Badr}, using the Poincar\\'e inequality and doubling of $dw$.\n\\end{proof}\n\n\nWe denote   the Riesz transforms by $R_{j}$, $j=1,\\ldots, n$.  Set   ${\\mathcal R} f= (R_{1}f, \\ldots, R_{n}f)$ and ${\\mathcal R}^*g= R_{j}^*g_{1}+\\cdots + R_{n}^*g_{n}$ where $R_{j}^*=-R_{j}$, the adjoint being for the $L^2(dx)$ duality. The Riesz transforms  are bounded on $L^2(w)$.\n\n\nIn this paper, if $A$ is an unbounded linear operator on a Banach space,  then $\\textsf{D}(A), \\textsf{N}(A), \\textsf{R}(A)$ denote respectively, its domain, null space and range. Closure of the range is denoted by $\\clos{\\textsf{R}(A)}$.\n\n\n\\begin{lem}\\label{lem:gradient} Let $w\\in A_{2}$.  The following statements hold.\n\n\\begin{enumerate}\n  \\item  The operator $\\nabla$ on $L^2(w)$ into $L^2(w;{\\mathbb C}^n)$ with domain $\\textsf{D}(\\nabla)=H^1(w)$ is densely defined, closed and injective.\n   \\item  Its adjoint $\\nabla^*:=-{\\text{{\\rm div}}}_{w}:= - \\frac 1 w {\\text{{\\rm div}}} w$ is also closed, densely defined  and $\\clos{\\textsf{R}(\\nabla^*)}=L^2(w)$.   Moreover,  $ \\frac 1 w C^\\infty_{0}({\\mathbb R}^n; {\\mathbb C}^n)$ is  a dense subspace of $\\textsf{D}({\\text{{\\rm div}}}_{w})$.\n \\item The operator $\\Delta_{w}=-  \\nabla^*\\nabla=  {\\text{{\\rm div}}}_{w}\\nabla$ constructed by the method of sesquilinear forms from $ \\int_{{\\mathbb R}^n} \\nabla u \\cdot \\overline{\\nabla v}\\, dw$ on $H^1(w)$ is a  negative  self-adjoint, injective operator on $L^2(w)$.\n \\item  The operator ${\\mathcal R}_{w}:= \\nabla (-\\Delta_{w})^{-1\/2}$ is bounded from $L^2(w)$ into $L^2(w;{\\mathbb C}^n)$ and its range is $\\clos{\\textsf{R}(\\nabla)}$.\n  \\item   $\\clos{\\textsf{R}(\\nabla)}= {\\mathcal R}(L^2(w))$.\n   \\item Moreover $\\clos{\\textsf{R}(\\nabla)}= \\clos{\\nabla (C_{0}^\\infty({\\mathbb R}^n))}=\\{g\\in L^2(w;{\\mathbb C}^n)\\, ; \\, curl g=0\\,\\}$.\n\n\\end{enumerate}   \\end{lem}\n\n\nAs we can see, the closure of the range of $\\nabla$ has several characterizations: the one in (4) comes from spectral theory for the weighted Laplacian, the one in (5) comes from weighted theory  for the usual Laplacian.  Both will  be useful.\n\n\\begin{proof}\nTo prove  (1), observe that $\\nabla$ is densely defined  by construction and the fact that it is closed is proved in \\cite{FKS}. Injectivity follows from Poincar\\'e inequalities: indeed if $u\\in H^1(w)$ such that $\\nabla u=0$ implies $u$ constant. As $w$ is doubling and ${\\mathbb R}^n$ is unbounded, we have that $w({\\mathbb R}^n)=+\\infty$, so that the constant must be 0.\n\nThe  first part of   (2) follows upon standard functional analysis from (1).  To show the density of $ \\frac 1 w C^\\infty_{0}({\\mathbb R}^n; {\\mathbb C}^n)$ in $\\textsf{D}({\\text{{\\rm div}}}_{w})$, we note that this is equivalent to the density of $ C^\\infty_{0}({\\mathbb R}^n; {\\mathbb C}^n)$ in $\\textsf{D}({\\text{{\\rm div}}}):=\\{ f\\in L^2(w^{-1};{\\mathbb C}^n)\\, ; \\, {\\text{{\\rm div}}} f\\in L^2(w^{-1})\\}$ (equipped with the graph norm). This equivalent statement follows using regularisation by convolution as in Lemma \\ref{lem:approx}  and smooth truncations.\n\nThe construction in (3) is classical (see for example, Kato's book \\cite{Kato}). The injectivity follows from that of the gradient.   Next, the boundedness of ${\\mathcal R}_{w}$ in (4) follows from the equality  $\\|\\nabla f\\|=\\|(-\\Delta_{w})^{1\/2}f\\|$, which itself is obtained using self-adjointness and the form. The identification of  its range follows immediately.\n\n\nLet us turn to (5).\nAs the  Riesz transforms are bounded on $L^2(w)$ when $w\\in A_{2}$ and ${\\mathcal R}^*{\\mathcal R}=I$, ${\\mathcal R}(L^2(w))$ is a closed subspace of $L^2(w; {\\mathbb C}^n)$.  For $f$ in the  dense subspace ${\\mathcal E}$ of Corollary \\ref{lem:density}, it is easy to see that ${\\mathcal R} f=\\nabla g$ with $g=(-\\Delta)^{-1\/2}f \\in {\\mathcal S}({\\mathbb R}^n)$ using the properties of the function $\\psi$. Thus,   ${\\mathcal R}({\\mathcal E}) \\subseteq  {\\textsf{R}(\\nabla)}$ so that ${\\mathcal R}(L^2(w)) \\subseteq  \\clos{\\textsf{R}(\\nabla)}$. Conversely, if $g$ belongs to  this dense subspace  ${\\mathcal E}$ then $\\nabla g = {\\mathcal R} f$ for $f=(-\\Delta)^{1\/2}g\\in {\\mathcal S}({\\mathbb R}^n).$ Thus $\\nabla ({\\mathcal E}) \\subseteq {\\mathcal R}(L^2(w))$ and, by density, $\\clos{\\textsf{R}(\\nabla)}\\subseteq {\\mathcal R}(L^2(w))$.\n\n We now prove (6). We introduce $\\dot H^1(w)$ as the closure of $C_{0}^\\infty({\\mathbb R}^n)$ for the semi norm $\\| \\nabla f\\|_{L^2(w;{\\mathbb C}^n)}$. We claim that this space coincides with  the space of $f\\in L^2_{\\text{{\\rm loc}}}(w)$ such that $ \\nabla f\\in L^2(w; {\\mathbb C}^n)$. In one direction, if one has a Cauchy sequence $\\nabla g_{k}$ in $L^2(w)$  with $g_{k}\\in C_{0}^\\infty({\\mathbb R}^n)$, then using the Poincar\\'e inequality, if $c_{k}$ is the mean of $g_{k}$ on the unit ball, one can see that $g_{k}-c_{k}$ converges in $L^2_{\\text{{\\rm loc}}}(w)$\nto some $g$ and thus $\\nabla g_{k}$ converges to the distributional gradient of $g$ in $L^2(w; {\\mathbb C}^n)$.\nConversely, if $g\\in L^2_{loc}(w)$ and $\\nabla g\\in L^2(w; {\\mathbb C}^n)$, then  set\n$g_{k}= (g-c_{k})\\varphi_{k}$  where $c_{k}$ is the mean of $g$ on the ball $B(0,2^{k+1})$ and $\\varphi_{k}(x)= \\varphi(2^{-k}x)$ with $\\varphi$ is a smooth function that vanishes outside $B(0,2)$ and that is 1 on $B(0,1)$. It follows from Poincar\\'e's inequality that $\\nabla g_{k}$ converges weakly to $\\nabla g$ in $L^2(w)$. By Mazur's theorem, this implies that a sequence of  convex combinations converges strongly to $\\nabla g$. Noticing that $g_{k}\\in H^1(w)$ in which $C_{0}^\\infty({\\mathbb R}^n)$ is dense  concludes the proof of the claim.\nClearly, $\\clos{\\textsf{R}( \\nabla)}= \\nabla (\\dot H^1(w))$, so this proves the first equality.\n\nNext, we have $\\nabla (C_{0}^\\infty({\\mathbb R}^n)) \\subseteq \\{g\\in L^2(w;{\\mathbb C}^n)\\, ; \\, {\\text{{\\rm curl}}} g=0\\,\\}$. This inclusion is preserved by taking the closure in $L^2(w;{\\mathbb C}^n)$ as the second set is clearly closed. Conversely,  using the boundedness of the Riesz transforms, the idendity\n$g= (I-{\\mathcal R}\\mR^*)g+ {\\mathcal R}\\mR^*g$ holds for $g\\in L^2(w;{\\mathbb C}^n)$. Moreover ${\\text{{\\rm div}}} ((I-{\\mathcal R}\\mR^*)g)=0$ and ${\\text{{\\rm curl}}}  ({\\mathcal R}\\mR^*g)=0$ in the distributions sense. Indeed, it is certainly true for $g\\in C_{0}^\\infty({\\mathbb R}^n;{\\mathbb C}^n)$ and  one can use a limiting argument. If one assumes   ${\\text{{\\rm curl}}} g=0$ then one  also  has ${\\text{{\\rm curl}}} ((I-{\\mathcal R}\\mR^*)g)=0$. Hence $(I-{\\mathcal R}\\mR^*)g$ is constant as a distribution, and this equals to  0 since it is in $L^2(w; {\\mathbb C}^n)$. Thus $g={\\mathcal R}\\mR^*g \\in {\\mathcal R}(L^2(w))$ and we conclude using (5).\n\\end{proof}\n\n\n\n\n\\section{The main quadratic estimate}\n\n\n\n\\subsection{Review of the $DB$ formalism}\n\nLet ${\\mathcal H}$ be a separable complex Hilbert space.  One uses $(\\, , \\, )$ and $\\|\\, \\|$ for the hermitian product and norm on ${\\mathcal H}$.  Let  $D$ be an unbounded self-adjoint operator on ${\\mathcal H}$. There is an orthogonal  splitting\n\\begin{equation}\\label{eq:splitting}\n{\\mathcal H}= \\textsf{N}(D) \\oplus \\clos{\\textsf{R}(D)}.\n\\end{equation}\n\nDefine closed  double sectors in the complex plane by\n$$\nS_{\\mu} := \\{z\\in{\\mathbb C} : {| \\arg (\\pm z)|\\le\\mu}\\}\\cup\\{0\\}.\n$$\n Consider $B\\in {\\mathcal L}({\\mathcal H})$   and  assume that\n $B$ is (strictly) accretive on $\\clos{\\textsf{R}(D)}$, that is, there exists $\\kappa>0$ such that\n\\begin{equation}\n\\label{accretive}\n\\re (Bv,v) \\ge\\kappa \\|v\\|^2, \\ \\forall v\\in \\clos{\\textsf{R}(D)}.\n\\end{equation}\nIn this case, let\n$$\n   \\mu(B):= \\sup_{v\\in \\textsf{R}(D), v\\not = 0} |\\arg(Bv,v)|  <\\pi\/2\n$$\ndenote the  {\\em angle of accretivity} of $B$ on $\\textsf{R}(D)$.\nNote that $B$ may not be invertible on ${\\mathcal H}$. Still for $X$ a subspace of ${\\mathcal H}$, we set\n$B^{-1}X= \\{u \\in {\\mathcal H}\\, ; \\,  Bu \\in X\\}$.\n\n\\begin{prop}   \\label{prop:typeomega} With the above assumptions, we have the following facts.\n\\begin{itemize}\n\\item[{\\rm (i)}]\nThe operator $DB$, with domain $B^{-1}\\textsf{D}(D)$,  is $\\mu(B)$-bisectorial, i.e. $\\sigma(DB)\\subseteq S_{\\mu(B)}$ and there are resolvent bounds\n$\\|(\\lambda I - DB)^{-1}\\| \\lesssim 1\/ \\text{{\\rm dist}}\\,(\\lambda, S_\\mu)$ when $\\lambda\\notin S_\\mu$, $\\mu(B) <\\mu<\\pi\/2$.\n\\item[{\\rm (ii)}]\nThe operator $DB$ has range $\\textsf{R}(DB)=\\textsf{R}(D)$ and null space $\\textsf{N}(DB)=B^{-1}\\textsf{N}(D)$ such that topologically (but  not necessarily orthogonally) one has\n$$\n{\\mathcal H} = \\clos{\\textsf{R}(DB)} \\oplus \\textsf{N}(DB).\n$$\n\\item[{\\rm (iii)}]\nThe restriction of $DB$ to $\\clos{\\textsf{R}(D)}$\nis a closed, injective operator with dense range in\n$\\clos{\\textsf{R}(D)}$.  Moreover, the same statements  on spectrum and resolvents as in (i) hold.\n\n\\item[{\\rm (iv)}]   Statements similar to (i) and (ii) hold for $BD$ with $\\textsf{D}(BD)=\\textsf{D}(D)$, defined as the adjoint of $DB^*$ or equivalently by $BD= B(DB)B^{-1}$ on $\\textsf{R}(BD)\\cap \\textsf{D}(D)$, with $\\textsf{R}(BD):=B\\textsf{R}(D)$ and  $BD=0$ on the null space $\\textsf{N}(BD):=\\textsf{N}(D)$.\n\\end{itemize}\n\\end{prop}\n\nFor a proof, see \\cite{ADMc}. Note that the accretivity is only needed on $\\textsf{R}(D)$.\nFor $t\\in {\\mathbb R}$,  we shall\n\\begin{align*}R_t^B&=(I+itDB)^{-1},\\\\\nQ_t^B&= \\frac{1}{2i} (R_{-t}^B -R_{t}^B)= t DB(I+t^{2}DBDB)^{-1},\\\\\nP_{t}^B&= \\frac{1}{2} (R_{-t}^B +R_{t}^B)= (I+t^{2} DBDB)^{-1}.\n\\end{align*}  It follows from the previous result that $R_t^B$,  $P_t^B$ and  $Q_{t}^B$\nare uniformly bounded operators on ${\\mathcal H}$.\n\n\n\n\n\\subsection{The main theorem}\\label{sec:main}\n\n\n\n\nLet us specify in this section the operator $D$ we consider throughout.   In ${\\mathbb C}^{m(n+1)}={\\mathbb C}^m\\oplus {\\mathbb C}^{mn}= {\\mathbb C}^m \\oplus ({\\mathbb C}^m\\otimes {\\mathbb C}^n) $, we use the  notation $v=\\begin{bmatrix}\n        v_{{\\scriptscriptstyle\\perp}} \\\\\n      v_{{\\scriptscriptstyle \\parallel}}\n     \\end{bmatrix}$, with $v_{{\\scriptscriptstyle\\perp}}\\in {\\mathbb C}^m$ and $v_{{\\scriptscriptstyle \\parallel}}\\in {\\mathbb C}^{mn}={\\mathbb C}^m\\otimes {\\mathbb C}^n$ which we call respectively the normal (or scalar) and tangential parts of $v$. To simplify the exposition, we carry out the detailed proof only in the ``scalar '' case $m=1$ as it carries  in the vector-valued case $m>1$ without change. See Section \\ref{sec:vv} for the notation.\n\n\n\n\\begin{prop}\\label{prop:D} Let $w\\in A_{2}({\\mathbb R}^n)$ and set ${\\mathcal H}=L^2(w;{\\mathbb C}^{n+1})$ with norm $\\|f \\|= (\\int_{{\\mathbb R}^n} |f|^2\\, dw)^{1\/2}$.\n\\begin{enumerate}\n  \\item The operator $D:=\n     \\begin{bmatrix}\n        0 & {\\text{{\\rm div}}}_{w} \\\\\n       - \\nabla  & 0\n     \\end{bmatrix}$ with domain  $ \\begin{bmatrix}\n        \\textsf{D}(\\nabla) \\\\\n      \\textsf{D}({\\text{{\\rm div}}}_{w})\\end{bmatrix}$ is self-adjoint.\n       \\item  $\\begin{bmatrix}\n     C^\\infty_{0}({\\mathbb R}^n) \\\\\n     \\frac 1 w C^\\infty_{0}({\\mathbb R}^n; {\\mathbb C}^n)\n     \\end{bmatrix}$  is a dense subspace of $\\textsf{D} (D)$.\n       \\item  $ \\clos{\\textsf{R}(D)}  =  \\begin{bmatrix}\n       L^2(w) \\\\\n     {\\mathcal R}_{w}(L^2(w))\n     \\end{bmatrix}.$\n \\item  $ \\clos{\\textsf{R}(D)}  =  \\begin{bmatrix}\n       L^2(w) \\\\\n     {\\mathcal R}(L^2(w))\n     \\end{bmatrix}=\\{g\\in L^2({\\mathbb R}^n,w;{\\mathbb C}^{1+n}); {\\text{{\\rm curl}}}_x(g_{\\scriptscriptstyle \\parallel})=0\\}$, where the closure is taken in  ${\\mathcal H}$  and\n     $\\clos{\\textsf{R}(D)} \\cap\n      C^\\infty_{0}({\\mathbb R}^n; {\\mathbb C}^{n+1})$ is dense in $\\clos{\\textsf{R}(D)}$.\n     \\end{enumerate}\n   \\end{prop}\n\n   The proof is a direct consequence of   Lemma \\ref{lem:gradient}. We omit the  details.\n\n\\bigskip\n\n Let us specify the required assumption on $B$: this is the operator of multiplication by  an\n$(n+1)\\times(n+1)$ matrix $B(x)$ which has  bounded entries and is  accretive   on $\\clos{\\textsf{R}(D)}$.  Associated to $B$ are the constants $\\|B\\|_{\\infty}$ and (the best) $\\kappa>0$ in \\eqref{accretive}.\n\n\n In this section,  $\\|f\\|$ systematically designates the  weighted  $L^2(w; {\\mathbb C}^d)$ norm with $d=1,n$ or $n+1$ depending on the context.\n\n\n\n\n\n\n\n\n\\begin{thm}\\label{th:main}  With the preceding assumptions, one has\n\\begin{equation}  \\label{eq:sfBD}\n  \\int_0^\\infty\\| Q_{t}^B v \\|^2 \\, \\frac{dt}t \\lesssim \\|v\\|^2, \\qquad  \\text{for all }\\ v \\in {\\mathcal H}.\n\\end{equation}\n \\end{thm}\n\n\n\n Theorem \\ref{th:main} differs from previous results in the field. First, $D$ still being  a first order  differential operator,  no longer has constant coefficients. Secondly, the coercivity assumption\n $\\|\\nabla u\\| \\lesssim \\|Du\\|$ for $u\\in \\textsf{R}(D) \\cap \\textsf{D}(D) $ which is of utmost importance in other proofs cannot be true here, thus we cannot apply the metric measure space (here ${\\mathbb R}^n$ with Euclidean distance and measure $dw$) generalisation made by Bandara  \\cite{Ban} of the Euclidean  results  in \\cite{AKMc} or \\cite{elAAM}.\n\n Note that\n the block-matrix case  $B=\\begin{bmatrix}\n  1    & 0   \\\\\n    0  &  d\n\\end{bmatrix}$ in the splitting ${\\mathbb C}^{n+1}={\\mathbb C} \\oplus {\\mathbb C}^n$   gives a proof of  the Kato conjecture for degenerate operators, first proved by Cruz-Uribe and Rios \\cite{CR}.  See Section \\ref{sec:consequences}.\n\n\n\nOur strategy to prove Theorem \\ref{th:main} is to follow the line of argument in \\cite{elAAM} but with some necessary twists. In particular, we will use different spectral decompositions on the scalar and  tangential parts. Also the   stopping-time argument requires the use of the Corona decomposition for $w$ (Proposition \\ref{prop4}).\n\n\n\n\\subsection{Reduction to a Carleson measure estimate}\\label{sec:reduction}\n\n\\begin{lem} [Off-diagonal decay] \\label{lem:odd}\n For every integer $N$ there exists $C_N>0$\nsuch that\n\\begin{equation} \\label{odn}\n\\|1_{E}\\,R_{t}^B u\\|+ \\|1_{E}\\,Q_{t}^B u\\| \\le C_N \\brac{\\text{{\\rm dist}}\\, (E,F)\/|t|}^{-N}\\|u\\|\n\\end{equation}\nfor all $t\\ne 0$,\nwhenever $E,F \\subseteq {\\mathbb R}^n$ are closed sets,  $u \\in {\\mathcal H}$\nis such that $\\text{{\\rm supp}}\\, u\\subseteq F$.  We have set $\\brac x:=1+|x|$ and\n$\\text{{\\rm dist}}\\,(E,F) :=\\inf\\{|x-y|\\, ; \\, x\\in E,y\\in F\\}$.\n\n\\end{lem}\n\n\\begin{proof}\nRepeat the argument in \\cite[Proposition 5.1]{elAAM}, which only uses that $D$ is of order 1 and that the commutator $[\\chi, D]$ of $D$ with a Lipschitz function $\\chi$  is bounded on the  space ${\\mathcal H}$. More precisely,  it is the pointwise multiplication by\n \\begin{equation}\n\\label{eq:comm}\n\\begin{bmatrix}\n        0 &  -(\\nabla \\chi)^t  \\\\\n        \\nabla \\chi  & 0\n     \\end{bmatrix}.\n\\end{equation}\n We omit further details.\n\\end{proof}\n\n\n Let  $A_t$ be the   dyadic averaging operator  with respect to Lebesgue measure and $A_{t}^w$ the one with respect  to $dw$.  Set\n$$E_{t}u(x):= \\begin{bmatrix}\n      A_{t}^w u_{{\\scriptscriptstyle\\perp}}(x)    \\\\\n       A_{t}u_{{\\scriptscriptstyle \\parallel}}(x)     \\end{bmatrix} =\\begin{bmatrix}\n      \\barint_{\\hspace{-2pt}Q} u_{{\\scriptscriptstyle\\perp}}(y)\\,  dw(y)    \\\\\n       \\barint_{\\hspace{-2pt}Q} u_{{\\scriptscriptstyle \\parallel}}(y)\\,  dy\n     \\end{bmatrix},\n     $$\n     where $Q$ is the   unique dyadic cube  $Q \\in \\triangle_t$ that contains $x$.  The notation for dyadic cubes is the same as in Section \\ref{coronasec}.  Recall that    $\\barint_{\\hspace{-2pt}Q}$ means the average on $Q$ with respect to the indicated measure. We also set\n     $$\n     E_{Q}u= \\begin{bmatrix}\n      \\barint_{\\hspace{-2pt}Q} u_{{\\scriptscriptstyle\\perp}}(y)\\,  dw(y)    \\\\\n       \\barint_{\\hspace{-2pt}Q} u_{{\\scriptscriptstyle \\parallel}}(y)\\,  dy\n     \\end{bmatrix}.\n     $$\nObserve  that $E_{t}$ acts  as a linear operator componentwise.  Using the inequality\n    \\eqref{eq:dxdwAp} with $p=2$,\nwe have\n   that $E_{t}$ is a bounded operator on ${\\mathcal H}$, uniformly in $t$.\n   We also have the  pointwise estimate\n   $$\n\\sup_{t>0}    |E_{t}u| \\le   M_{d,w}|u_{{\\scriptscriptstyle\\perp}}| + M_{d}|u_{{\\scriptscriptstyle \\parallel}}|\n   $$\n   where $M_{d,w}$ is the dyadic maximal function with respect to\n   $dw$ and $M_{d}$ the dyadic maximal function with respect to\n   $dx$. Both are bounded on $L^2(w)$ by the Hardy-Littlewood  theorem for the doubling measure $dw$ and\n   by Muckenhoupt's theorem since $w\\in A_{2}$. Thus, $u\\mapsto \\sup_{t>0}    |E_{t}u|$ is bounded from ${\\mathcal H}$ into $L^2(w)$.\n\n\n\n\n\\begin{defn}  \\label{defn:princpart}\nBy the {\\em principal part} of $(Q_t^B)_{t>0}$,\nwe mean the  function   $(x,t) \\mapsto \\gamma_t(x)$ defined from ${\\mathbb R}^{n+1}_+$  to $ {\\mathcal L}({\\mathbb C}^{n+1}) $ by\n$$\n   \\gamma_t(x)z:= (Q_t^B z)(x)\n$$\nfor every $z\\in {\\mathbb C}^{n+1}$. We view $z$ on the right-hand side\nof the above equation as the constant function valued in ${\\mathbb C}^{n+1}$ defined on ${\\mathbb R}^n$\nby $z(x):=z$.  We denote by $|\\gamma_t(x)|$ its norm in ${\\mathcal L}({\\mathbb C}^{n+1})$ subordinated to the hermitian structure on ${\\mathbb C}^{n+1}$.\nWe identify $\\gamma_t(x)$ with the (possibly unbounded) multiplication\noperator $\\gamma_t: u(x)\\mapsto \\gamma_t(x)u(x)$, $u\\in {\\mathcal H}$.\n\\end{defn}\n\\begin{lem}\\label{lem:gammat}\nThe operator $Q_t^B$ extends to a bounded operator from\n$L^\\infty(w; {\\mathbb C}^{n+1})$ into $ {\\mathcal H}_{\\text{loc}}=L^2_{\\text{loc}}(w; {\\mathbb C}^{n+1})$.\nIn particular we have\n$\\gamma_t\\in L^2_{\\text{loc}}(w; {\\mathcal L}({\\mathbb C}^{n+1}))$ with bounds\n$$\n   \\barint_{\\hspace{-6pt}Q} |\\gamma_t(y)|^2 \\, dw(y) \\lesssim\n  1\n$$\nfor all $Q\\in\\triangle_t$.\nMoreover, $\\gamma_t E_{t}$ are bounded on ${\\mathcal H}$ with $\\|\\gamma_t E_{t}u\\|\\lesssim \\|E_{t}u\\|$ uniformly for all $t>0$ and $u\\in {\\mathcal H}$.\n\\end{lem}\n\n\\begin{proof} Fix a cube $Q \\in \\triangle_t$ and $u \\in L^\\infty(w;{\\mathbb C}^{n+1})$ with $\\|u\\|_\\infty=1$. Then\nwrite $u= u_0+ u_1+u_2+\\ldots$ where $u_0=u$ on $2Q$ and $0$ elsewhere and if $j\\ge 1$, $u_j=u$ on $2^{j+1}Q \\setminus 2^{j}Q$ and $0$ elsewhere. Then apply $Q_t^B$  and use Lemma \\ref{lem:odd} for each term $Q_t^B u_j$ with $N$ large enough  and sum  to obtain\n$$\n   \\barint_{\\hspace{-6pt}Q} |(Q_t^Bu)(y)|^2 \\, dw(y) \\le\n  C.\n$$\nIf we do this for  the constant functions with values describing an orthonormal basis of ${\\mathbb C}^{n+1}$ and sum, we obtain an upper bound for the desired average of $\\gamma_t$.\nNext,  for a function $u \\in {\\mathcal H}$ and $Q\\in \\Delta_{t}$, as $E_{t}u$ is constant on $Q$,\n\\begin{align*}\n\\label{}\n   \\int_Q \\left|\\gamma_t (y)E_{t}u(y)\\right|^2 \\, dw(y)&\\le \\int_Q \\left|\\gamma_t (y)\\right|^2 \\, dw(y) \\times  \\barint_{\\hspace{-6pt}Q} \\left|E_{t}u(y)\\right|^2 \\, dw(y) \\\\\n   &\\lesssim    \\int_Q \\left|E_{t}u(y)\\right|^2 \\, dw(y).\n   \\end{align*}  Thus\n$$\n\\|\\gamma_t E_{t}u\\|^2  \\lesssim  \\sum_{Q\\in \\triangle_t}  \\int_Q \\left|E_{t}u(y)\\right|^2 \\, dw(y) = \\|E_{t}u\\|^2 \\lesssim \\|u\\|^2.\n$$\n\\end{proof}\n\nAs $Q_t^B$ vanishes on $ \\textsf{N}(DB)$, it is enough to prove the quadratic estimate (\\ref{eq:sfBD}) for $v \\in \\clos{\\textsf{R}(DB)}=\\clos{\\textsf{R}(D)}$.\n Our principal part approximation reads as follows.\n\n\\begin{prop}  \\label{lem:ppa} We have\n \\begin{equation}\\label{eq:ppa1}\n  \\qe{Q_t^B v-\\gamma_t E_{t} v}\n  \\lesssim \\|v\\|^2,  \\quad v\\in \\clos{\\textsf{R}(D)}.\n\\end{equation}\n\n\n\\end{prop}\n\nThe function  from ${\\mathbb R}^{n+1}_+$ to $ {\\mathbb R}_+$ defined by $ (x,t) \\mapsto |\\gamma_t(x)|^2$, is\na {\\em weighted dyadic Carleson function} if there exists $C<\\infty$\nsuch that\n$$\n  \\iint_{\\carl{Q}} |\\gamma_t(x)|^2 \\, \\frac{dw(x)dt}t \\le C^2 w(Q)\n$$\nfor all dyadic cubes $Q\\subseteq{\\mathbb R}^n$.\nHere $\\carl{Q}:= Q\\times (0,l(Q)]$ is the Carleson box over $Q$. We define the dyadic Carleson norm $\\|\\gamma_t\\|_C$ to be the smallest\nconstant $C$ for which this inequality holds.  The form of Carleson's lemma that we need and will apply componentwise is  the following (see \\cite{AT}, p.168 and references therein).\n\\begin{lem}  \\label{lem:Carleson} For all $u\\in {\\mathcal H}$,\n$$\n  \\qe{\\gamma_t E_{t}u} \\lesssim \\|\\gamma_t\\|_C^2 \\| \\sup_{t>0}|E_{t}u|\\|^2 \\lesssim \\|\\gamma_t\\|_C^2 \\| u\\|^2.\n$$\n\\end{lem}\n\n\\begin{cor}\\label{cor:cor1} If $|\\gamma_{t}(x)|^2$ is a weighted dyadic Carleson function,  then Theorem \\ref{th:main} holds.\n\\end{cor}\n\nThis corollary clearly  follows from the above lemma and \\eqref{eq:ppa1}.\n\n\n\n\n\\subsection{Proof of the principal part approximation}\n\nWe begin the proof of the principal part approximation \\eqref{eq:ppa1} with some further notation.\nDefine\n$$P_{t}=\\begin{bmatrix}\n        (I-t^2\\Delta_{w})^{-1} &  0  \\\\\n        0  & (I-t^2\\Delta)^{-1} I_{{\\mathbb C}^n}\n     \\end{bmatrix}.\n     $$\n\n           Here $\\Delta_{w}={\\text{{\\rm div}}}_{w}\\nabla$ is the  negative   self-adjoint operator  on $L^2(w)$ defined in Lemma \\ref{lem:gradient} while $\\Delta$ is the usual  negative  Laplacian  on $L^2(dx)$.  From Lemma \\ref{lem:approx},\n       the convolution operator  $(I-t^2\\Delta)^{-1}$ is bounded on $L^2(w)$ uniformly with respect to $t>0$: It is indeed classical that $(I-t^2\\Delta)^{-1}$ is the convolution with $t^{-n}G_{2}(x\/t)$ where the Bessel potential $G_{2}$ is integrable and radially decreasing (see \\cite[Chapter 6]{Gra}).  Thus, $P_{t}$ is uniformly bounded on ${\\mathcal H}$.\n\n\n\n\n\n\n\n\\begin{lem}\\label{whatisneeded}   For all $ v \\in \\clos{\\textsf{R}(D)}$, one has\n\\begin{equation}  \\label{firsttermprop}\n     \\qe{Q_t^B(I-P_t) v}\\lesssim \\|v\\|^2.\n\\end{equation}\n\\end{lem}\n\n\\begin{proof}\\label{proof:thm}   The key  point is that $ \\clos{\\textsf{R}(D)}$ is preserved by $P_{t}$. This follows from the characterization of  $\\clos{\\textsf{R}(D)}$ in Proposition \\ref{prop:D}  as $\\begin{bmatrix}\n       L^2(w) \\\\\n     {\\mathcal R}(L^2(w))\n     \\end{bmatrix}$, since  the Riesz transforms and $(I-t^2\\Delta)^{-1}$ commute and ${\\mathcal R}(- \\Delta)^{1\/2}=\\nabla$.\n     Write $v=\\begin{bmatrix}\n       f \\\\\n     {\\mathcal R} g\n     \\end{bmatrix}$, with $f, g\\in L^2(w)$.  Then,\n     \\begin{align*}\n     (I-P_{t})v&=\\begin{bmatrix}\n       -t^2\\Delta_{w} (I-t^2\\Delta_{w})^{-1}f   \\\\\n        -t^2\\Delta (I-t^2\\Delta)^{-1} {\\mathcal R} g\n     \\end{bmatrix}\\\\\n     & = \\begin{bmatrix}\n       -t^2{\\text{{\\rm div}}}_{w} \\nabla (I-t^2\\Delta_{w})^{-1}f   \\\\\n        t^2\\nabla (- \\Delta)^{1\/2} (I-t^2\\Delta)^{-1}  g\n     \\end{bmatrix}\\\\\n     &  =  - tD  \\begin{bmatrix}\n               t (- \\Delta)^{1\/2} (I-t^2\\Delta)^{-1}  g\n               \\\\\n               t \\nabla (I-t^2\\Delta_{w})^{-1}f\n\\end{bmatrix} .\n\\end{align*}\nThus using that $tQ_{t}^BD$ is uniformly bounded on ${\\mathcal H}$, it suffices to prove\n$$\n\\qe{t \\nabla (I-t^2\\Delta_{w})^{-1}f}\\lesssim \\|f\\|^2\n$$\nand\n$$\\qe{t (- \\Delta)^{1\/2} (I-t^2\\Delta)^{-1}  g} \\lesssim \\|g\\|^2.\n$$\nThe first estimate is a simple consequence of the construction of $-\\Delta_{w}$ as the self-adjoint operator $\\nabla^*\\nabla$ on $L^2(w)$.  Indeed, $\\|\\nabla (I-t^2\\Delta_{w})^{-1}f\\|= \\| (-\\Delta_{w})^{1\/2} (I-t^2\\Delta_{w})^{-1}f\\|$ and one concludes using the spectral theorem for the self-adjoint operator $\\Delta_{w}$ that $\\qe{t \\nabla (I-t^2\\Delta_{w})^{-1}f }= c \\|f\\|^2$ with $c= \\int_{0}^\\infty t(1+t^2)^{-1} \\frac{dt}{t}$.\nThe second estimate is a consequence of Lemma \\ref{lem:LPw} as soon as the conditions for the function $\\psi$ defined on the Fourier transform side by $\\hat\\psi(\\xi)= |\\xi|(1+|\\xi|^2)^{-1}$ have been checked.  \\end{proof}\n\n\n \\begin{lem}\\label{lem:ppalem2} For all $v \\in \\clos{\\textsf{R}(D)}$, one has\n $$\n \\qe{Q_t^BP_tv -\\gamma_tE_{t}P_tv } \\lesssim  \\|v\\|^2.\n $$\n \\end{lem}\n\n  \\begin{proof}\n  We remark that for $t>0$ fixed and $x\\in {\\mathbb R}^n$, we have\n$$(Q_t^BP_tv -\\gamma_tE_{t}P_t v)(x)= Q_t^B (u- u_Q ) (x),\n$$\nwhere $u=P_tv$,  $Q$ is the unique dyadic cube in $\\triangle_t$ containing $x$ and\n$u_{Q}$ is the value of $E_{t}u$ on $Q$.\nDefine $C_0(Q)=2Q$ and $C_j(Q)=2^{j+1}Q\\setminus 2^jQ$ if $j\\in {\\mathbb N}^*$. Then,\n\\begin{align*}\n\\|Q_t^BP_t v-\\gamma_tE_{t}P_t v  \\|^2 &=  \\sum_{Q\\in \\triangle_t} \\int_Q | Q_t^B(u- u_{Q})|^2 \\, dw\n\\\\\n& \\le \\sum_{Q\\in \\triangle_t} \\left(\\sum_{j\\ge 0} \\bigg(\\int_Q | Q_t^B ({\\bf 1}_{C_j(Q)} (u -u_{Q}))|^2\\, dw \\bigg)^{1\/2}\\right)^{2}\n\\\\\n&\n\\lesssim  \\sum_{Q\\in \\triangle_t} \\left(\\sum_{j\\ge 0} 2^{-jN} \\bigg(\\int_{C_j(Q)}|u-u_{Q} |^2\\, dw \\bigg)^{1\/2}\\right)^{2}\n\\\\\n&\n\\lesssim  \\sum_{Q\\in \\triangle_t} \\sum_{j\\ge 0}  2^{-jN} \\int_{C_j(Q)}  |u - u_{Q} |^2\\, dw\n\\\\\n&\n\\lesssim  \\sum_{Q\\in \\triangle_t} \\sum_{j\\ge 0}  2^{-jN} 2^{2j} \\ell(Q)^2 2^{jd}\\int_{2^{j+1}Q} |\\nabla  u |^2\n\\, dw \\\\\n&\n\\lesssim   t^2 \\sum_{j\\ge 0}  2^{-jN} 2^{2j} 2^{jd} 2^{jn} \\int_{{\\mathbb R}^n} |\\nabla u   |^2\\, dw.\n\\\\\n&\n\\lesssim   t^2 \\|\\nabla  u \\|^2.\n\\end{align*}\nWe  used the Minkowski inequality on the second line, off-diagonal decay  on the third,  Cauchy--Schwarz inequality on the fourth,\nPoincar\\'e inequality on the fifth  (recalling that one can take  the average  with respect to either  $dx$ or $dw$), and a telescoping argument which produces the doubling exponent $d$ of $w$,\nthe covering inequality $ \\sum_{Q\\in \\triangle_t}  {\\bf 1}_{2^{j+1}Q} \\lesssim 2^{jn}$ and $\\ell(Q)\\sim t$  on the sixth. Finally, we choose $N> n+d+ 2$ in the last.\n  Hence\n$$\\qe{Q_t^BP_tv-\\gamma_t E_{t} P_t v}\n  \\lesssim \\qe{t \\nabla  P_t v}.\n  $$\n\n\n For the scalar part of $v$, we have to control the weighted quadratic estimate for $t\\nabla (I-t^2\\Delta_{w})^{-1}v_{{\\scriptscriptstyle\\perp}}$ which we have seen already. Using  $v\\in \\clos{\\textsf{R}(D)}$,  the tangential part $v_{{\\scriptscriptstyle \\parallel}}$ is of the form ${\\mathcal R} g$ for some $g\\in L^2(w)$. Hence we have to control the  quadratic estimate of $t\\nabla  (I-t^2\\Delta)^{-1}R_{j} g=R_{j} {\\mathcal R} t  (-\\Delta)^{1\/2}(I-t^2\\Delta)^{-1}g$ for $j=1, \\ldots, n$. We can eliminate $R_{j}$ and ${\\mathcal R}$ as the Riesz transforms are bounded on $L^2(w)$ and the weighted Littlewood-Paley estimate  for $t  (-\\Delta)^{1\/2}(I-t^2\\Delta)^{-1}$ has been already seen.   \\end{proof}\n\n  \\begin{lem}\\label{lem:mean1} There are constants $C<\\infty$ and $\\tau_{1}\\in(0,1)$ such that for all $f\\in \\textsf{D}({\\text{{\\rm div}}}_{w})$ and all  dyadic cubes $Q$,\n  $$\n\\left| \\barint_{\\hspace{-6pt}Q} {\\text{{\\rm div}}}_{w}f \\, dw \\right| \\le   \\frac{C}{\\ell(Q)^{\\tau_{1}}}\n\\left( \\barint_{\\hspace{-6pt}Q} |{\\text{{\\rm div}}}_{w}f|^2 \\, dw \\right)^{\\frac{1-\\tau_{1}}2}\\left( \\barint_{\\hspace{-6pt}Q} |f|^2 \\, dw \\right)^{\\frac{\\tau_{1}}2}.\n$$\n  \\end{lem}\n\n\\begin{proof} Observe that if $f$ has support contained in $Q$, then $\\int_{Q}  {\\text{{\\rm div}}}_{w}f \\, dw=0$.   Thus this lemma  follow from \\cite{Ban}. Here is a simple proof in our situation.   Let $A=\\left( \\barint_{\\hspace{-2pt}Q} |{\\text{{\\rm div}}}_{w}f|^2 \\, dw \\right)^{1\/2}$ and $B=\\left( \\barint_{\\hspace{-2pt}Q} |f|^2 \\, dw \\right)^{1\/2}$. If $B\\ge A\\ell$ a simple application of the Cauchy-Schwarz inequality gives the result.\n\nAssume next that $B<A\\ell$.\nLet  $\\varphi\\colon {\\mathbb R}^n\\to [0,1]$ be a smooth function which equals to  1 on $(1-t)Q$, 0 on $Q^c$\nwith $\\|\\nabla\\varphi\\|_\\infty \\le C(t\\ell)^{-1}$ for some $t\\in (0,1)$ to be chosen. We can write\n$$\n\\barint_{\\hspace{-6pt}Q} {\\text{{\\rm div}}}_{w}f\\, dw\n= \\barint_{\\hspace{-6pt}Q} \\varphi\\, {\\text{{\\rm div}}}_{w}f \\, dw + \\barint_{\\hspace{-6pt}Q} (1-\\varphi){\\text{{\\rm div}}}_{w}f \\, dw\n=I+II.\n$$\nUsing that $-{\\text{{\\rm div}}}_{w}$ is the adjoint of $\\nabla$ (writing the integral as an integral on ${\\mathbb R}^n$ thanks to the support of $\\varphi$),\n$$| I | = \\left|\\barint_{\\hspace{-6pt}Q} \\nabla \\varphi f\\, dw\\right| \\le C(t\\ell)^{-1} B.$$\nFor $II$,   using Cauchy-Schwarz inequality  and  the right hand side of  \\eqref{eq:ainfty} with $E$ being  the support of $1-\\varphi$ in $Q$,\n$$\n| II |  \\le A  \\frac{w(Q\\setminus (1-t)Q)^{1\/2}}{w(Q)^{1\/2}} \\le C_{w}  A t^{\\sigma\/2}.\n$$\nHence, choosing $t^{1+\\sigma\/2}=B\/A\\ell$, we obtain the inequality with $\\tau_{1}=\\frac{\\sigma\/2}{1+\\sigma\/2}$.\n\\end{proof}\n\n \\begin{lem}\\label{lem:mean2} There are constants $C<\\infty$ and $\\tau_{2}\\in(0,1)$ such that for all $g\\in \\textsf{D}(\\nabla)$ and all dyadic cubes $Q$,\n  $$\n\\left| \\barint_{\\hspace{-6pt}Q} \\nabla g \\, dx \\right| \\le   \\frac{C}{\\ell(Q)^{\\tau_{2}}}\n\\left( \\barint_{\\hspace{-6pt}Q} |\\nabla g|^2 \\, dw \\right)^{\\frac{1-\\tau_{2}}2}\\left( \\barint_{\\hspace{-6pt}Q} |g|^2 \\, dw \\right)^{\\frac{\\tau_{2}}2}.\n$$\n  \\end{lem}\n\n\\begin{proof} The proof follows a similar pattern to that of Lemma \\ref{lem:mean1}.     Let $A=\\left( \\barint_{\\hspace{-2pt}Q} |\\nabla g|^2 \\, dw \\right)^{1\/2}$ and $B=\\left( \\barint_{\\hspace{-2pt}Q} |g|^2 \\, dw \\right)^{1\/2}$. If $B\\ge A\\ell$ a simple application of \\eqref{eq:dxdwAp} with $p=2$ gives the result.\n\nAssume next that $B<A\\ell$.\nLet  $\\varphi\\colon {\\mathbb R}^n\\to [0,1]$ be a smooth function which equals to  1 on $(1-t)Q$, 0 on $Q^c$\nwith $\\|\\nabla\\varphi\\|_\\infty \\le C(t\\ell)^{-1}$ for some $t\\in (0,1)$ to be chosen. We can write\n$$\n\\barint_{\\hspace{-6pt}Q}\\nabla g \\, dx\n= \\barint_{\\hspace{-6pt}Q} \\varphi \\nabla g\\, dx + \\barint_{\\hspace{-6pt}Q} (1-\\varphi)\\nabla g \\, dx\n=I+II.\n$$\nInterpreting $I$ as a distribution-test function pairing, we may integrate by parts coordinatewise,  and using that $w\\in A_{2}$ we find\n$$| I | = \\left|\\barint_{\\hspace{-6pt}Q}  g \\nabla \\varphi \\, dx\\right| \\le C(t\\ell)^{-1} \\barint_{\\hspace{-6pt}Q}  |g|\\, dx \\le C[w]_{A_{2}}(t\\ell)^{-1} B.$$\nFor $II$, pick $p>1$ such that  $w\\in A_{2\/p}$ and use H\\\"older's inequality with conjugate exponents $p$ and $p'$, the support  properties of $1-\\varphi$, and the fact that  $w\\in A_{2\/p}$, to conclude that\n$$\n| II |  \\lesssim \\left( \\barint_{\\hspace{-6pt}Q} |\\nabla g|^p \\, dx \\right)^{1\/p} t^{1\/p'} \\le [w]_{A_{2\/p}}^{1\/p}  A t^{1\/p'}.\n$$\nHence, choosing $t^{1+1\/p'}=B\/A\\ell$, we obtain the inequality with $\\tau_{2}=\\frac{1\/p'}{1+1\/p'}$.\n\\end{proof}\n\n\n\\begin{lem}\\label{lem:ppalem3} For all $u\\in {\\mathcal H}=L^2(w;{\\mathbb C}^{n+1})$,\n$$\\qe{\\gamma_tE_{t}(P_t-I)u} \\lesssim \\|u\\|^2.$$\n\\end{lem}\n\n\n \\begin{proof} {It follows from Lemma \\ref{lem:gammat} that\n $\\|\\gamma_tE_{t}(P_t-I)u\\| \\lesssim \\|E_{t}(P_t-I)u\\|$. Given the definitions of $E_{t}$ and $P_{t}$, the lemma reduces to  the scalar inequalities\n $$\\qe{A_t^w((I-t^2\\Delta_{w})^{-1}-I)f} \\lesssim \\|f\\|^2$$\n and\n  $$\\qe{A_t((I-t^2\\Delta)^{-1}-I)f} \\lesssim \\|f\\|^2.$$\n\n For the first one, we follow \\cite{AKMc} with a minor simplification. Let $Q_{s}^w= s^2\\Delta_{w}e^{s^2\\Delta_{w}}$. Then for $f\\in L^2(w)$ we have by the spectral theorem,  $f= 8  \\int_{0}^\\infty (Q_{s}^w)^2f\\, \\frac {ds}s$\n and also $\\|f\\|^2= 8 \\int_{0}^\\infty \\|Q_{s}^w f\\|^2 \\, \\frac {ds}s$.\n By Schur's lemma,  it is enough to show that  the operator norm of $A_t^w((I-t^2\\Delta_{w})^{-1}-I)Q_{s}^w$ in $L^2(w)$ is bounded by $h(s\/t)$ with $h\\ge 0$ and $\\int_{0}^\\infty h(u)\\, \\frac{du}u<\\infty$. We shall find $h(u)= C \\inf (u^{\\tau_{1}}, u^{-2})$.\n\n If $t<s$,\n \\begin{align*}\n\\label{}\n   \\|A_{t}^w((I-t^2\\Delta_{w})^{-1}-I)Q_{s}^wf\\| &\\lesssim \\|((I-t^2\\Delta_{w})^{-1}-I)s^2\\Delta_{w}e^{s^2\\Delta_{w}}f\\| \\\\\n   &\\lesssim C(t\/s)^2\\|f\\|.\n    \\end{align*}\n\n\n If $s<t$,\n \\begin{align*}\n\\label{}\n   \\|A_{t}^w(I-t^2\\Delta_{w})^{-1}Q_{s}^wf\\|&\\lesssim \\|(I-t^2\\Delta_{w})^{-1}s^2\\Delta_{w}e^{s^2\\Delta_{w}}f\\|\n   \\\\\n   &\n   \\lesssim C(s\/t)^2\\|f\\|. \\end{align*}\n\n Thus, it remains to study the operator norm of $A_{t}^wQ_{s}^w$ for $s<t$. For this, we remark\n that $A_{t}^wQ_{s}^wf(x) $ is an average with respect to $dw$ of $ {\\text{{\\rm div}}}_{w}(sg)$ with $g=-s \\nabla e^{s^2\\Delta_{w}}f$ so that we can use Lemma \\ref{lem:mean1}.\n Thus,\n \\begin{align*}\n \\|A_{t}^wQ_{s}^wf\\|^2&= \\sum_{Q\\in \\Delta_{t}}  w(Q) \\left| \\barint_{\\hspace{-6pt}Q} {\\text{{\\rm div}}}_{w}(sg)\\, dw \\right|^2\n \\\\\n &\n \\lesssim  \\frac {1}{t^{2\\tau_{1}}} \\sum_{Q\\in \\Delta_{t}}  w(Q)  \\left( \\barint_{\\hspace{-6pt}Q} |{\\text{{\\rm div}}}_{w}(sg)|^2 \\, dw \\right)^{{1-\\tau_{1}}}\\left( \\barint_{\\hspace{-6pt}Q} |sg|^2 \\, dw \\right)^{{\\tau_{1}}}\n \\\\\n& = \\frac {s^{2\\tau_{1}}}{t^{2\\tau_{1}}} \\sum_{Q\\in \\Delta_{t}}    \\left( \\int_{Q} |Q_{s}^wf|^2 \\, dw \\right)^{{1-\\tau_{1}}}\\left( \\int_{Q} |s \\nabla e^{s^2\\Delta_{w}}f|^2 \\, dw \\right)^{{\\tau_{1}}}\n\\\\\n &\\le  \\frac {s^{2\\tau_{1}}}{t^{2\\tau_{1}}} \\|Q_{s}^wf\\|^{2(1-\\tau_{1})}  \\|s \\nabla e^{s^2\\Delta_{w}}f\\|^{2\\tau_{1}}\n \\\\\n &\n \\lesssim  \\frac {s^{2\\tau_{1}}}{t^{2\\tau_{1}}} \\|f\\|^2\n \\end{align*}\n where we used H\\\"older's inequality for the sum.\n\n The proof of the second inequality is as follows. Setting $Q_{s}= s^2\\Delta e^{s^2\\Delta }$, as before  we have that  the operator norm  of\n $A_t((I-t^2\\Delta)^{-1}-I)Q_{s}$ on $L^2(dx)$ is bounded by $h(s\/t)$ with $h$ similar to  above. As this operator is also bounded on $L^2(w)$ uniformly in $s$ and $t$ for all $w\\in A_{2}$ (for the convolution operators,  it follows from Lemma \\ref{lem:approx}  and for  $A_{t}$ this has been seen before), its operator norm on $L^2(w)$ is bounded by $h(s\/t)^\\theta$ for some power $\\theta>0$ depending only on  $w$  using Lemma \\ref{lem:DRF}. Thus, we can use  the fact that  the integral $f=  8  \\int_{0}^\\infty (Q_{s})^2f\\, \\frac {ds}s$ converges in $L^2(w)$ from Corollary \\ref{cor:Calderon}, Schur's Lemma and that $ \\int_{0}^\\infty \\|Q_{s} f\\|^2 \\, \\frac {ds}s \\lesssim \\|f\\|^2$ from Lemma \\ref{lem:LPw}.\n }\n \\end{proof}\n\n\n\n\n\n\\begin{proof}[Proof of  Proposition \\ref{lem:ppa}]\nIt is enough to write\n$$\nQ_t^B v-\\gamma_t E_{t} v= Q_t^B(I-P_{t}) v +  (Q_t^BP_{t}v -\\gamma_t E_{t}P_{t} v) +  \\gamma_t E_{t}(P_{t}-I) v\n$$\nand to use successively Lemma \\ref{whatisneeded}, Lemma \\ref{lem:ppalem2} and Lemma \\ref{lem:ppalem3}.\n\\end{proof}\n\n\\subsection{Preamble to the Carleson measure estimate}\n\nWe are now ready to prove that $|\\gamma_t(x)|^2$ is a weighted dyadic Carleson function and so, via Proposition \\ref{lem:ppa}, complete the proof of Theorem \\ref{th:main}. The first step towards this is a compactness argument. As was seen in the solution of the Kato square root problem (\\cite{AHLMcT}), the application of a stopping-time argument was made possible by restricting $\\gamma_t(x)$ so that, once normalised, it is close to a fixed element in the unit sphere of $\\mathcal{L}({\\mathbb C}^{1+n})$, the set of bounded linear transformations on ${\\mathbb C}^{1+n}$. We will make use of the same stopping-time argument, but also require a second stopping-time related to the oscillation of the weight, and with it comes a second compactness argument which restricts our attention to Whitney boxes on which the average of the weight is close to that of the top cube.\n\n A convenient way to define the  Whitney box $W_{Q}$ associated to a given  dyadic cube $Q$ is\n$W_{Q}=\\{ (x,t)\\, ; \\, x\\in Q, Q\\in \\Delta_{t}\\} $. With our definition $\\widehat Q$ is the union all $W_{Q'}$ for which $Q'$ is a dyadic subcube of $Q$.\n\nConsider the compact unit sphere in $\\mathcal{L}({\\mathbb C}^{1+n})$ and the compact interval $[0,c_0]$, where $c_0$ is as in \\eqref{reversejensen}. For each $\\nu \\in \\mathcal{L}({\\mathbb C}^{1+n})$ such that $|\\nu| = 1$, $\\tau \\in [0,c_0]$ and $\\sigma_{1},\\sigma_2 > 0$, define   $G_{\\tau,\\sigma_2}$ as the union of those Whitney boxes $W_{Q}$ for which $|\\ln(w_Q) - (\\ln w)_Q - \\tau| < \\sigma_2$,\n and\n\\begin{equation}\\label{gammatilde}\n\\widetilde{\\gamma}_t(x)=\n\\begin{cases} \\gamma_t(x) & \\text{if $\\gamma_t(x)\\ne 0$,  $\\left|\\frac{\\gamma_t(x)}{|\\gamma_t(x)|} - \\nu \\right| \\leq \\sigma_1$ and $(x,t) \\in G_{\\tau,\\sigma_2}$,}\n\\\\\n0 &\\text{otherwise.}\n\\end{cases}\n\\end{equation}\n We recall the notation $B^w(Q)$ from Section \\ref{coronasec} and set\n$$\\Omega^w(Q) = \\carl{Q}   \\setminus \\cup_{R \\in B^w(Q)} \\carl{R}.\n$$\n\n\n\n\n\\begin{lem} \\label{lemma5}\nSuppose that we can show\n\\[\n K=  \\sup_{\\nu, \\tau}\\sup_{Q \\in \\triangle} \\frac{1}{w(Q)} \\iint_{\\Omega^w(Q)} |\\widetilde{\\gamma}_t(x)|^2\\,  \\frac{dw(x)dt}{t} < \\infty\n\\]\nfor some choice of parameters $\\sigma_1$ and $\\sigma_2$ depending only on $\\|B\\|_{\\infty}$, $\\kappa$, $[w]_{A_2}$ and $n$. Then $|\\gamma_t(x)|^2$ is a weighted dyadic Carleson function.\n\\end{lem}\n\n\\begin{proof}  Fix $\\sigma_{1}$ and $\\sigma_{2}$ so that the hypothesis applies.  Let $Q\\in \\Delta$.  Observe  that the sets $\\Omega^w(R)$ form a partition of $\\carl{Q}$ when $R$ runs over elements of $ B^w_*(Q) \\cup \\{Q\\}$.  Thus,\nby the hypothesis and Proposition \\ref{prop4},\n\\begin{align*}\n\\iint_{\\carl{Q}} |\\widetilde{\\gamma}_t(x)|^2\\,  \\frac{dw(x)dt}{t} & = \\sum_{R \\in B^w_*(Q) \\cup \\{Q\\}} \\iint_{\\Omega^w(R)} |\\widetilde{\\gamma}_t(x)|^2\\,  \\frac{dw(x)dt}{t} \\\\\n& \\leq  K   \\sum_{R \\in B^w_*(Q) \\cup \\{Q\\}}  w(R) \\leq  \\frac{KC}{\\sigma^2_{w}}  w(Q)  + K w(Q).\n\\end{align*}\n By the compactness of the unit sphere in $\\mathcal{L}({\\mathbb C}^{1+n})$ and the interval $[0,c_0]$, there exist a finite index set $A \\subseteq {\\mathbb N}$ and, for each $j \\in A$, choices of $\\nu_j$ and $\\tau_j$ such that $|\\gamma_t(x)|^2 \\leq \\sum_{j \\in A} |\\widetilde{\\gamma}^j_t(x)|^2$, where $\\widetilde{\\gamma}^j_t(x) = \\widetilde{\\gamma}_t(x)$ with the choice $\\nu = \\nu_j$ and $\\tau = \\tau_j$.\nThis completes the proof.\n\\end{proof}\n\n\\subsection{Stopping-time arguments for test functions}\n\nWe fix an arbitrary vector $\\xi$ in the unit sphere of ${\\mathbb C}^{1+n}$.  For any  $Q_1 \\in \\triangle$ and   $\\sigma_{3}>0$ to be chosen, define a test function\n\\begin{equation}\n\\label{testf}\nf^\\xi_{Q_1} := \\Big(I + \\big(\\sigma_3\\ell(Q_{1})DB\\big)^2\\Big)^{-1}(1_{Q_1}\\xi)  = P^B_{\\sigma_3\\ell(Q_{1})}(1_{Q_1}\\xi),\n\\end{equation}\nwhere   $1_{Q_1}$ is the indicator of $Q_{1}$.\n    Note that $\\|f^\\xi_{Q_1}\\|^2 \\lesssim w(Q_{1})$ and $\\|\\sigma_{3}\\ell(Q_{1})DBf^\\xi_{Q_1}\\|^2\\lesssim w(Q_{1})$ with uniform implicit constants with respect to  $|\\xi|=1$, $\\sigma_{3}>0$ and $Q_{1}$,   as can be seen using the uniform  boundedness in $t>0$ of $Q^B_{t}$ and $P^B_{t}$.\n\n\\begin{lem} \\label{lemma6}\nThere exist a constant $c$ depending only on $\\|B\\|_{\\infty}$, $\\kappa$, $[w]_{A_2}$ and $n$, and  a constant $\\delta > 0$ depending only on $[w]_{A_2}$, such that for all such $\\xi$, $Q_{1}$ and $\\sigma_{3}$,\n\\[\n|E_{Q_1}(f^\\xi_{Q_1}) - \\xi| \\leq c\\sigma_3^{\\delta}.\n\\]\n\\end{lem}\n\n\\begin{proof}\nWe have  $E_{Q_1}(f^\\xi_{Q_1}) - \\xi = E_{Q_1}D u$ with\n\\[\nu: = - (\\sigma_3 \\ell(Q_1))^2BDB \\Big(I + \\big(\\sigma_3\\ell(Q_{1})DB\\big)^2\\Big)^{-1}(1_{Q_1}\\xi)\n\\]\n and notice that $E_{Q_1}D$ acts on  $u =  \\begin{bmatrix}\nu_1 \\\\\nu_2\n\\end{bmatrix}$ componentwise by averaging  ${\\text{{\\rm div}}}_w u_2$ with respect to $dw$ and $\\nabla u_1$ with respect to $dx$.\nLemma \\ref{lem:mean1} says\n\\begin{align*}\n& \\left| \\barint_{\\hspace{-6pt}Q_1} {\\text{{\\rm div}}}_{w}u_2 \\, dw \\right| \\le   \\frac{C}{\\ell(Q_1)^{\\tau_{1}}}\n\\left( \\barint_{\\hspace{-6pt}Q_1} |{\\text{{\\rm div}}}_{w}u_2|^2 \\, dw \\right)^{\\frac{1-\\tau_{1}}2}\\left( \\barint_{\\hspace{-6pt}Q_1} |u_2|^2 \\, dw \\right)^{\\frac{\\tau_{1}}2} \\\\\n& \\leq   C\\sigma_3^{\\tau_1}\n\\left( \\barint_{\\hspace{-6pt}Q_1} | \\xi - f^\\xi_{Q_1}  |^2 \\, dw \\right)^{\\frac{1-\\tau_{1}}2}\\left( \\barint_{\\hspace{-6pt}Q_1} | \\sigma_{3}\\ell(Q_{1})DBf^\\xi_{Q_1}|^2 \\, dw \\right)^{\\frac{\\tau_{1}}2} \\\\\n& \\leq   C\\sigma_3^{\\tau_1}\n\\end{align*}\nand Lemma \\ref{lem:mean2} says\n\\begin{align*}\n& \\left| \\barint_{\\hspace{-6pt}Q} \\nabla u_1 \\, dx \\right| \\le   \\frac{C}{\\ell(Q)^{\\tau_{2}}}\n\\left( \\barint_{\\hspace{-6pt}Q} |\\nabla u_1|^2 \\, dw \\right)^{\\frac{1-\\tau_{2}}2}\\left( \\barint_{\\hspace{-6pt}Q} |u_1|^2 \\, dw \\right)^{\\frac{\\tau_{2}}2} \\\\\n& \\leq   C\\sigma_3^{\\tau_2}\n\\left( \\barint_{\\hspace{-6pt}Q_1} |\\xi - f^\\xi_{Q_1} |^2 \\, dw \\right)^{\\frac{1-\\tau_{2}}2}\\left( \\barint_{\\hspace{-6pt}Q_1} | \\sigma_{3}\\ell(Q_{1})DBf^\\xi_{Q_1}  |^2 \\, dw \\right)^{\\frac{\\tau_{2}}2} \\\\\n& \\leq   C\\sigma_3^{\\tau_2}.\n\\end{align*}\nSo taking $\\delta = \\min(\\tau_1,\\tau_2)$ completes the proof.\n\\end{proof}\n\nRecall that  $w_{Q}= \\barint_{\\hspace{-2pt}Q} w\\, dx$ and similarly for $(\\ln w)_{Q}$.\n\n\\begin{lem} \\label{lemma7}\nFix $\\tau\\in [0,c_{0}]$ and $\\xi$ in the unit sphere of ${\\mathbb C}^{1+n}$. Let\n\\[\nS_{Q_1}^\\tau=\\begin{bmatrix}\n        w_{Q_1} e^{-\\tau - (\\ln w)_{Q_1}} &  0  \\\\\n        0  & I\n     \\end{bmatrix},\n\\]\nand define the collection of `bad' cubes $B^{\\tau,\\xi}(Q_1)$ to be the set of maximal $Q' \\in \\triangle$ such that $Q' \\subseteq Q_1$ and\n\\[\n\\mbox{either $|E_{Q'}(f^\\xi_{Q_1})| > \\frac{1}{\\sigma_4}$ or $ \\re  \\left(S_{Q_1}^\\tau(\\xi),\\begin{bmatrix}\n        w_{Q'}\/w_{Q_1} &  0  \\\\\n        0  & I\n     \\end{bmatrix}E_{Q'}(f^\\xi_{Q_1})\\right) < \\sigma_5$.}\n\\]\nWe can then choose positive $\\sigma_3$, $\\sigma_4$ and $\\sigma_5$  depending only on $\\|B\\|_{\\infty}$, $\\kappa$, $[w]_{A_2}$ and $n$, in particular independently on $\\tau,\\xi, Q_{1}$, so that\n\\begin{equation} \\label{geometric}\n\\sum_{R \\in B^{\\tau,\\xi}(Q_1)} w(R) \\leq (1-\\sigma_6)w(Q_1),\n\\end{equation}\nwith $0< \\sigma_{6}\\le 1$.\n\\end{lem}\n\n\\begin{proof}\nThere are two sets of cubes to consider. The first is the set of those   maximal  $Q'$ for which\n\\begin{equation} \\label{stopone}\n  \\re  \\left(S_{Q_1}^\\tau(\\xi),\\begin{bmatrix}\n        w_{Q'}\/w_{Q_1} &  0  \\\\\n        0  & I\n     \\end{bmatrix}E_{Q'}(f^\\xi_{Q_1})\\right) <  \\sigma_5.\n\\end{equation}\nBy \\eqref{reversejensen}, we know that\n\\[\n-c_0 \\leq \\ln(w_{Q_1}) - \\tau - (\\ln w)_{Q_1} \\leq c_0\n\\]\nso $S_{Q_1}^\\tau$ is a constant  self-adjoint  matrix with  $e^{-c_{0}} I \\le S_{Q_1}^\\tau  \\le e^{c_{0}} I$. Applying Lemma \\ref{lemma6},\n\\begin{align}\n \\re   \\big(S_{Q_1}^\\tau(\\xi),E_{Q_1}(f^\\xi_{Q_1})\\big) & = \\big(S_{Q_1}^\\tau(\\xi),\\xi\\big) +   \\re  \\big(S_{Q_1}^\\tau(\\xi), E_{Q_1}(f^\\xi_{Q_1})-\\xi\\big) \\nonumber \\\\\n& \\geq e^{-c_{0}} - ce^{c_{0}}\\sigma_3^{\\delta} \\geq \\frac 1 2 e^{-c_{0}}, \\label{sigma3}\n\\end{align}\non choosing $\\sigma_3$ so that $2c\\sigma_{3}^\\delta \\le e^{-2c_{0}}$. Consequently, setting $G = Q_1 \\setminus (\\cup Q')$   and $f^\\xi_{Q_1}= \\begin{bmatrix}\n       f_{1}   \\\\ f_{2}\n        \\end{bmatrix}\n        $,\n we have\n \\begin{align*}\n\\label{}\n    E_{Q_1}(f^\\xi_{Q_1})&= \\begin{bmatrix}\n        \\frac{1}{w(Q_1)} \\int_{Q_{1}} f_1\\, dw  \\\\\n        \\frac{1}{|Q_1|} \\int_{Q_{1}} f_2\\,  dx\n     \\end{bmatrix}  \\\\\n    &\n    =\\sum_{Q'} \\frac{|Q'|}{|Q_1|}  \\begin{bmatrix}\n        w_{Q'}\/w_{Q_{1}} & 0 \\\\\n        0 & I\n     \\end{bmatrix}E_{Q'}(f^\\xi_{Q_1}) +   \\begin{bmatrix}\n        \\frac{1}{w(Q_1)} \\int_G f_1\\, dw  \\\\\n        \\frac{1}{|Q_1|} \\int_G f_2\\,  dx\n     \\end{bmatrix} ,\n     \\end{align*}\nwhere the subcubes $Q'$ are those of \\eqref{stopone}.     Using \\eqref{stopone} and \\eqref{sigma3}, we obtain\n        \\begin{align*}\n\\frac 1 2 e^{-c_{0}}\n&\n \\leq   \\re  \\big(S_{Q_1}^\\tau(\\xi),E_{Q_1}(f^\\xi_{Q_1})\\big) \\\\\n& \\leq \\sigma_5 \\sum_{Q'} \\frac{|Q'|}{|Q_1|} +   \\re  \\left(S_{Q_1}^\\tau(\\xi),\\begin{bmatrix}\n        \\frac{1}{w(Q_1)} \\int_G f_1\\,  dw  \\\\\n        \\frac{1}{|Q_1|} \\int_G f_2\\,  dx\n     \\end{bmatrix}\\right) \\\\\n& \\leq \\sigma_5 +  \\re  \\left(S_{Q_1}^\\tau(\\xi),\\begin{bmatrix}\n        \\frac{1}{w(Q_1)} \\int_G f_1\\,  dw  \\\\\n        \\frac{1}{|Q_1|} \\int_G f_2\\,  dx\n     \\end{bmatrix}\\right)\n\\end{align*}\nand,  using the estimate $\\int_{Q_{1}} |f^\\xi_{Q_1}|^2\\, dw \\lesssim w(Q_{1}) $ and again the $A_{2}$ condition for $w$,\n\\begin{align*}\n& \\left|\\left(S_{Q_1}^\\tau(\\xi),\\begin{bmatrix}\n        \\frac{1}{w(Q_1)} \\int_G f_1\\, dw  \\\\\n        \\frac{1}{|Q_1|} \\int_G f_2 \\, dx\n     \\end{bmatrix}\\right)\\right| \\\\\n& \\leq e^{c_{0}}\\left(\\frac{w(G)}{w(Q_1)^2}\\left(\\int_G |f_1|^2 dw\\right) + \\frac{w^{-1}(G)}{|Q_1|^2}\\left(\\int_G |f_2|^2 dw\\right)\\right)^{1\/2} \\\\\n& \\lesssim \\left(\\frac{w(G)}{w(Q_1)}\\right)^{1\/2} + \\frac{(w^{-1}(G))^{1\/2}w(Q_1)^{1\/2}(w^{-1}(Q_1))^{1\/2}}{(w^{-1}(Q_1))^{1\/2}|Q_1|} \\\\\n& \\lesssim \\left(\\frac{w(G)}{w(Q_1)}\\right)^{1\/2} + \\left(\\frac{w^{-1}(G)}{w^{-1}(Q_1)}\\right)^{1\/2} \\lesssim \\left(\\frac{w(G)}{w(Q_1)}\\right)^{\\theta}.\n\\end{align*}\nfor some $\\theta>0$ by \\eqref{eq:ainfty} applied to $w^{-1}$ and $w$.\n Therefore, for a small enough choice of $\\sigma_5$, we have that\n\\[\n\\left(\\frac{w(G)}{w(Q_1)}\\right)^{\\theta} \\gtrsim 1\n\\]\nwhich implies that $w(G) \\geq 2\\sigma_6 w(Q_1)$ for some small $\\sigma_6 > 0$ and so\n\\begin{equation} \\label{firsthalf}\n\\sum_{Q'} w(Q') \\leq (1-2\\sigma_6) w(Q_1),\n\\end{equation}\nwhere the sum is taken over those cubes $Q'$ which satisfy \\eqref{stopone}.\n\nNow we consider the set of maximal dyadic subcubes $Q'$ of $Q_{1}$ for which\n\\begin{equation} \\label{stoptwo}\n|E_{Q'}(f^\\xi_{Q_1})| > \\frac{1}{\\sigma_4}\n.\\end{equation}\n Then $w(Q') \\leq  \\sigma_4^2  \\int_{Q'} |f^\\xi_{Q_1}|^2 dw$ and\n\\[\n\\sum_{Q'} w(Q') \\leq \\sum_{Q'} \\sigma_4^2 \\int_{Q'} |f^\\xi_{Q_1}|^2 dw \\leq \\sigma_4^2  \\int_{Q_1} |f^\\xi_{Q_1}|^2 dw \\lesssim \\sigma_4^2\\,   w(Q_1)\n\\]\nwhere the sum is now over those cubes $Q'$ which satisfy \\eqref{stoptwo}.\nSo we can choose $\\sigma_4$ so small that\n\\begin{equation} \\label{secondhalf}\n\\sum_{Q'} w(Q') \\leq \\sigma_6\\,  w(Q_1).\n\\end{equation}\nCombining \\eqref{firsthalf} and \\eqref{secondhalf} proves the lemma.\n\\end{proof}\n\n\\subsection{Conclusion of the Carleson measure estimate} Consider $\\tilde{\\gamma}_t(x)$ depending on  $\\nu$ and $ \\tau$ as defined in \\eqref{gammatilde}.\n Associate to $\\nu$ a vector  $\\xi \\in {\\mathbb C}^{1+n}$  such that $|\\nu(\\xi)|=1$ and $|\\xi|=1$. Such a $\\xi$ may not be uniquely  defined but we pick one.\nFor a cube $Q_1 \\in \\triangle$, consider the test function $f_{Q_{1}}^\\xi$ of   \\eqref{testf} and  set $\\Omega^{\\tau,\\xi}(Q_1) = \\carl{Q_1} \\setminus \\cup_{R \\in B^{\\tau,\\xi}(Q_1)} \\carl{R}$ with $B^{\\tau,\\xi}(Q_1)$ defined in Lemma \\ref{lemma7}.\n\n\n\\begin{lem}  \\label{lem:parameters}\nSuppose that $\\sigma_3$, $\\sigma_4$ and $\\sigma_5$  are chosen as in Lemma \\ref{lemma7} so that \\eqref{geometric} holds.\nThen there exists a choice of $\\sigma_w$, $\\sigma_1$ and $\\sigma_2$  so that for all  $Q_{0}, Q_{1}\\in \\triangle$ with $Q_{1}\\subseteq Q_{0}$, and all  $\\nu$ and $ \\tau$,\n\\begin{equation} \\label{43}\n|\\tilde{\\gamma}_t(x)| \\leq C|\\gamma_tE_t(f^\\xi_{Q_1})(x)|, \\quad \\mathrm{for}\\,(x,t) \\in \\Omega^w(Q_0) \\cap \\Omega^{\\tau,\\xi}(Q_1),\n\\end{equation}\n where $C > 0$ depends only on the choice of $\\sigma_w$, $\\sigma_1$, $\\sigma_2$, $\\sigma_3$, $\\sigma_4$ and $\\sigma_5$.\n\\end{lem}\n\n\\begin{proof}  We assume that $\\Omega^w(Q_0) \\cap \\Omega^{\\tau,\\xi}(Q_1)$  is non-empty, otherwise, there is nothing to prove. Recall that  it is a union of Whitney boxes $W_{Q'}$ and \\eqref{43} follows from $\\left| \\frac{\\tilde{\\gamma}_t(x)}{|\\tilde{\\gamma}_t(x)|} \\big( E_{Q'}(f^\\xi_{Q_1})\\big) \\right|\\gtrsim 1$ for $(x,t)\\in  W_{Q'}$ with $\\tilde{\\gamma}_t(x)\\ne 0$.\nFor  a Whitney box $W_{Q'} \\subseteq\\Omega^w(Q_0) \\cap \\Omega^{\\tau,\\xi}(Q_1)$ we have that\n\\begin{align*}\n|(\\ln w)_{Q'} - (\\ln w)_{Q_0}| & \\leq \\sigma_w, \\\\\n|E_{Q'}(f^\\xi_{Q_1})| & \\leq \\frac{1}{\\sigma_4} \\,\\,\\, \\mbox{and} \\\\\n \\re  \\left(S_{Q_1}^\\tau(\\xi),\\begin{bmatrix}\n        w_{Q'}\/w_{Q_1} &  0  \\\\\n        0  & I\n     \\end{bmatrix}E_{Q'}(f^\\xi_{Q_1})\\right) & \\geq \\sigma_5.\n\\end{align*}\n The last two inequalities are the definition of $\\Omega^{\\tau,\\xi}(Q_{1})$.  The first  comes from the fact  $Q'$ is not contained in a  cube of $B^w(Q_{0})$ by Proposition \\ref{prop4}. As $Q'\\subseteq Q_{1}\\subseteq Q_{0}$, $Q_{1}$ is also not contained in a cube of  $B^w(Q_{0})$ and    we also  have\n\\[ |(\\ln w)_{Q_{1}} - (\\ln w)_{Q_0}|  \\leq \\sigma_w.\n\\]\n Moreover, recall  that  if $\\tilde{\\gamma}_t(x)\\ne 0$,  then\n\\[\n\\left|\\frac{\\tilde{\\gamma}_t(x)}{|\\tilde{\\gamma}_t(x)|} - \\nu\\right| \\le \\sigma_{1}.\n\\]\nFinally, if $\\tilde{\\gamma}_t(x)\\ne 0$ and $(x,t)\\in W_{Q'}$ then\n\\[\n|\\ln(w_{Q'}) - (\\ln w)_{Q'} - \\tau| < \\sigma_2.\n\\]\nClearly then,   we may assume the  six  inequalities above.\n\nWe begin by observing that\n\\[\n \\begin{bmatrix}\n        w_{Q'}\/w_{Q_1} &  0  \\\\\n        0  & I\n     \\end{bmatrix} S_{Q_1}^\\tau = \\begin{bmatrix}\n        e^{\\ln (w_{Q'})-\\tau - (\\ln w)_{Q_1}} &  0  \\\\\n        0  & I\n     \\end{bmatrix}\n\\]\nand\n\\begin{align}\n |\\ln(w_{Q'}) - \\tau - (\\ln w)_{Q_1}|\n& \\leq |\\ln(w_{Q'}) - (\\ln w)_{Q'} - \\tau| \\nonumber \\\\\n& + |(\\ln w)_{Q'} - (\\ln w)_{Q_0}| + |(\\ln w)_{Q_1} - (\\ln w)_{Q_0}|  \\nonumber\\\\\n&  \\leq \\sigma_2 + \\sigma_w + \\sigma_w. \\label{1}\n\\end{align}\nRecall that we chose $\\sigma_3$ in \\eqref{sigma3} so that $0 < c\\sigma_3^{\\delta} \\leq e^{-2c_{0}}\/2  \\leq 1\/2$. Therefore, using Lemma \\ref{lemma6},\n\\[\n\\left| \\nu \\left(E_{Q'}(f^\\xi_{Q_1})\\right) \\right| \\geq \\left| \\nu \\left(\\xi\\right) \\right| - \\left| \\nu \\left(E_{Q'}(f^\\xi_{Q_1}) - \\xi\\right) \\right| \\geq 1 - c\\sigma_3^{\\delta} \\geq 1\/2\n\\]\nand\n\\[\n \\re  \\left( \\xi, E_{Q'}(f^\\xi_{Q_1})\\right) = \\left( \\xi, \\xi\\right) +  \\re  \\left( \\xi,  E_{Q'}(f^\\xi_{Q_1})-\\xi\\right) \\leq 1 + c\\sigma_3^{\\delta} \\leq 2,\n\\]\nso\n\\[\n\\left| \\nu \\left(E_{Q'}(f^\\xi_{Q_1})\\right) \\right| \\geq \\frac{1}{4}  \\re  \\left( \\xi, E_{Q'}(f^\\xi_{Q_1})\\right).\n\\]\nIt then follows that for $(x,t)\\in W_{Q'}$ with $\\tilde{\\gamma}_t(x)\\ne 0$,\n\\begin{align*}\n& \\left| \\frac{\\tilde{\\gamma}_t(x)}{|\\tilde{\\gamma}_t(x)|} \\left( E_{Q'}(f^\\xi_{Q_1})\\right) \\right| \\\\\n& \\geq \\left| \\nu \\left(E_{Q'}(f^\\xi_{Q_1})\\right) \\right| - \\left|\\left(\\frac{\\tilde{\\gamma}_t(x)}{|\\tilde{\\gamma}_t(x)|} - \\nu\\right)\\left( E_{Q'}(f^\\xi_{Q_1})\\right) \\right| \\\\\n& \\geq \\frac{1}{4} \\re   \\left( \\xi, E_{Q'}(f^\\xi_{Q_1})\\right) - \\frac{\\sigma_1}{\\sigma_4} \\\\\n& = \\frac{1}{4}  \\re  \\left( \\begin{bmatrix}\n        e^{\\ln(w_{Q'} ) -\\tau - (\\ln w)_{Q_1}}&  0  \\\\\n        0  & I\n     \\end{bmatrix}\\xi, E_{Q'}(f^\\xi_{Q_1})\\right) \\\\\n& \\quad + \\frac{1}{4} \\re  \\left(\\begin{bmatrix}\n        1-e^{\\ln(w_{Q'})-\\tau - (\\ln w)_{Q_1}} &  0  \\\\\n        0  &  0\n     \\end{bmatrix}\\xi, E_{Q'}(f^\\xi_{Q_1})\\right) - \\frac{\\sigma_1}{\\sigma_4} \\\\\n& \\geq \\frac{1}{4} \\re  \\left( S_{Q_1}^\\tau(\\xi), \\begin{bmatrix}\n        w_{Q'}\/w_{Q_1} &  0  \\\\\n        0  & I\n     \\end{bmatrix}E_{Q'}(f^\\xi_{Q_1})\\right) - \\frac{e}{\\sigma_4} (\\sigma_2 + 2\\sigma_w) - \\frac{\\sigma_1}{\\sigma_4} \\\\\n& \\geq \\frac{1}{4}\\sigma_5 - \\frac{e}{\\sigma_4} (\\sigma_2 + 2\\sigma_w + \\sigma_1).\n\\end{align*}\nWe have used \\eqref{1} and  $|1-e^u|\\le e u$ if $u$ is a real with $|u|\\le 1$, assuming $\\sigma_2 + 2\\sigma_w\\le 1$.\nWe have already chosen $\\sigma_4$ and $\\sigma_5$, but we are still free to choose $\\sigma_w$, $\\sigma_1$ and $\\sigma_2$ small  so that \\eqref{43} holds.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Theorem \\ref{th:main}]  By Corollary \\ref{cor:cor1} and  Lemma \\ref{lemma5}, we know it is enough to show, for fixed $\\nu$ and $\\tau$,  and   for any cube $Q_0$, that\n\\begin{equation} \\label{indeed}\n\\iint_{\\Omega^w(Q_0)} |\\widetilde{\\gamma}_t(x)|^2\\,  \\frac{dw(x)dt}{t} \\leq Kw(Q_0).\n\\end{equation}\n   Fix $Q_{1}\\in \\Delta$ with $Q_{1}\\subseteq Q_{0}$.  Having fixed the parameters in Lemma \\ref{lem:parameters}, we apply  \\eqref{43} in the first inequality to obtain\n\\begin{align*}\n \\iint_{\\Omega^w(Q_0)\\cap\\Omega^{\\tau,\\xi}(Q_1)}& |\\widetilde{\\gamma}_t(x)|^2\\,  \\frac{dw(x)dt}{t} \\\\\n& \\leq \\iint_{\\carl{Q_1}} |\\gamma_tE_t(f^\\xi_{Q_1})(x)|^2\\,  \\frac{dw(x)dt}{t} \\\\\n& \\lesssim \\iint_{\\carl{Q_1}} |(Q^B_tf^\\xi_{Q_1})(x)|^2\\,  \\frac{dw(x)dt}{t} \\\\\n& \\quad + \\iint_{\\carl{Q_1}} |(Q^B_t - \\gamma_tE_t)(f^\\xi_{Q_1} - 1_{Q_1}\\xi)(x)|^2\\,  \\frac{dw(x)dt}{t} \\\\\n& \\quad + \\iint_{\\carl{Q_1}} |(Q^B_t - \\gamma_tE_t)(1_{Q_1}\\xi)(x)|^2 \\, \\frac{dw(x)dt}{t}.\n\\end{align*}\nSince $Q_{t}^Bf^\\xi_{Q_1}= \\frac t{\\sigma_{3}\\ell(Q_{1})}P_{t}^B (\\sigma_{3}\\ell(Q_{1})DBf^\\xi_{Q_1} ) $, one has  that\n\\[\n\\iint_{\\carl{Q_1}} |(Q^B_tf^\\xi_{Q_1})(x)|^2\\,  \\frac{dw(x)dt}{t} \\lesssim \\int_{0}^{\\ell(Q_{1})} \\frac {t^2\\|\\sigma_{3}\\ell(Q_{1})DBf^\\xi_{Q_1}\\|^2}{(\\sigma_{3}\\ell(Q_{1}))^2} \\,  \\frac {dt}t \\lesssim w(Q_1),\n\\]\nand, by Proposition \\ref{lem:ppa} because $f^\\xi_{Q_1} - 1_{Q_1}\\xi \\in \\textsf{R}(D)$,\n\\[\n\\iint_{\\carl{Q_1}} |(Q^B_t - \\gamma_tE_t)(f^\\xi_{Q_1} - 1_{Q_1}\\xi)(x)|^2\\,  \\frac{dw(x)dt}{t} \\lesssim \\|f^\\xi_{Q_1} - 1_{Q_1}\\xi\\|^2 \\lesssim w(Q_1).\n\\]\n For the last term, using that by definition $Q^B_t - \\gamma_t E_t$ annihilates constants and $E_t((1_{Q_1} -1)\\xi)(x)=0$ when $(x,t) \\in \\carl{Q_{1}}$, we can rewrite\n\\[(Q^B_t - \\gamma_t E_t) (1_{Q_1} \\xi)(x)= (Q^B_t - \\gamma_t E_t) ((1_{Q_1} -1) \\xi)(x) =  Q^B_t ((1_{Q_1} -1)\\xi)(x).\\]\nUsing off-diagonal estimates for $Q^B_t$ as in Lemma \\ref{lem:gammat}, one can easily show $\\iint_{\\carl{Q_1}} |Q^B_t ((1_{2Q_{1}}-1)\\xi)(x)|^2  \\frac{dw(x)dt}{t} \\lesssim w(Q_{1})$. Next, decompose $2Q_{1}\\setminus Q_{1} = \\partial_{a(t)} \\cup (2Q_{1}\\setminus \\partial_{a(t)})$ where $a(t)= \\sqrt {t\/\\ell(Q_{1})} \\ell(Q_{1})$ and $\\partial_{a}=\\{y\\notin Q_{1}\\, ; \\, d(y,Q_{1})\\le a\\}$.  Again,  using the off-diagonal estimates for each $t$, the function $1_{2Q_{1}\\setminus \\partial_{a(t)}}$ contributes  $w(Q_{1})$. It  remains to control  the integral corresponding to $1_{\\partial_{a(t)}}$. From the uniform boundedness of $Q^B_{t}$, one has\n\\begin{align*}\n\\iint_{\\carl{Q_1}} |Q^B_t (1_{\\partial_{a(t)}}\\xi)(x)|^2  \\frac{dw(x)dt}{t}    &  \\lesssim  \\int_{0}^{\\ell(Q_{1})} w(\\partial_{a(t)})  \\frac{dt}{t}  \\\\\n    &  \\lesssim  \\int_{0}^{\\ell(Q_{1})} \\bigg(\\frac{|\\partial_{a(t)}|}{|Q_{1}|}\\bigg)^\\sigma   \\frac{dt}{t} \\, w(Q_{1})\\\\\n    &   \\lesssim w(Q_{1})\n\\end{align*}\nusing  \\eqref{eq:ainfty} and $\\frac{|\\partial_{a(t)}|}{|Q_{1}|} \\lesssim \\big(\\frac t{\\ell(Q_{1})}\\big)^{1\/2}$ obtained from elementary observations. Summarizing the estimates above, we have proved that\n$$\n\\iint_{\\Omega^w(Q_0)\\cap\\Omega^{\\tau,\\xi}(Q_1)} |\\widetilde{\\gamma}_t(x)|^2\\,  \\frac{dw(x)dt}{t}\n\\lesssim w(Q_{1}).\n$$\n\n\n We can now prove \\eqref{indeed}.\nDefine\n\\begin{align*}\nB^{\\tau,\\xi}_0(Q_0) & = \\{Q_0\\}, \\quad B^{\\tau,\\xi}_1(Q_0) = B^{\\tau,\\xi}(Q_0), \\\\\nB^{\\tau,\\xi}_{j+1}(Q_0) & = \\cup_{R \\in B^{\\tau,\\xi}_j(Q_0)}B^{\\tau,\\xi}(R) \\,\\,\\, \\mbox{for} \\,\\,\\, j = 1,2,\\dots, \\\\\n\\mbox{and} \\,\\,\\, B^{\\tau,\\xi}_*(Q_0) & = \\cup_{j=0}^\\infty B^{\\tau,\\xi}_j(Q_0).\n\\end{align*}\nUsing\n\\[\n\\iint_{\\Omega^w(Q_0)} |\\widetilde{\\gamma}_t(x)|^2\\,  \\frac{dw(x)dt}{t} = \\sum_{Q_1 \\in B^{\\tau,\\xi}_*(Q_0)} \\iint_{\\Omega^w(Q_0) \\cap \\Omega^{\\tau,\\xi}(Q_1)} |\\widetilde{\\gamma}_t(x)|^2 \\, \\frac{dw(x)dt}{t},\n\\]\n and summing the estimate above together with an iteration of  Lemma \\ref{lemma7} imply\n\\begin{align*}\n& \\iint_{\\Omega^w(Q_0)} |\\widetilde{\\gamma}_t(x)|^2\\,   \\frac{dw(x)dt}{t} \\lesssim \\sum_{Q_1 \\in B^{\\tau,\\xi}_*(Q_0)} w(Q_1) \\\\\n& \\leq \\sum_{j=0}^\\infty (1-\\sigma_6)^j w(Q_0) \\lesssim w(Q_0),\n\\end{align*}\nwhich proves \\eqref{indeed} and with it Theorem \\ref{th:main}.\n\\end{proof}\n\n\\subsection{The case of block matrices}\nWe show how to simplify the argument in this case.\nRecall that $B$ is a $(n+1) \\times (n+1)$ matrix. Assume here that it is block diagonal, namely\n$$\nB(x)= \\begin{bmatrix} a(x) & 0 \\\\ 0 & d(x)\n\\end{bmatrix}\n$$\nwith $a(x) $ scalar-valued and $d(x)$ $n\\times n$ matrix-valued. Define the normal and tangential spaces\n$${\\mathcal H}_{{\\scriptscriptstyle\\perp}}=\\begin{bmatrix} L^2({\\mathbb R}^n, w; {\\mathbb C}) \\\\ 0\n\\end{bmatrix} \\quad \\mathrm{and} \\quad  {\\mathcal H}_{{\\scriptscriptstyle \\parallel}}=\\begin{bmatrix} 0\\\\ L^2({\\mathbb R}^n, w; {\\mathbb C}^n)\n\\end{bmatrix}.$$\nIn this case, both operators $BD$ and $DB$  swap the normal and tangential spaces. So do $Q_{t}^B$ and  multiplication by $\\gamma_{t}$. This means that\n$$\n\\gamma_{t}(x)= \\begin{bmatrix} 0 & \\alpha_{t}(x) \\\\ \\beta_{t}(x) & 0\n\\end{bmatrix}\n$$\nso that   the Carleson function norms\nfor $|\\alpha_{t}(x)|^2$ and $|\\beta_{t}(x)|^2$ can be estimated separately. The normal and tangential parts of our test functions can be used in two separate  stopping-time arguments,  which do not require the Corona decomposition (Proposition \\ref{prop4}),   following the usual proof in the unweighted case,  since for each stopping-time we use the average against \\emph{one} measure: either $dx$ or $dw$.\n\n\n\n\\subsection{A vector-valued extension}\\label{sec:vv}\n\nThe proof of the main quadratic estimate carries straightforwardly to the case of systems where\n\\begin{itemize}\n  \\item ${\\mathcal H}= L_{2}({\\mathbb R}^n, w; {\\mathbb C}^{m(1+n)})$,  where $ {\\mathbb C}^{m(1+n)}= ({\\mathbb C}^m)^{1+n}$,\n  \\item $D$ acts componentwise on ${\\mathcal H}$ by $(Du)^\\alpha= D(u^\\alpha)$ for $\\alpha=1, \\ldots, m$  (in other words, the new $D$ is $D\\otimes I_{{\\mathbb C}^m}$ but we shall not use this notation),  and\n  \\item  $B(x)$ is an $(n+1)\\times (n+1)$ matrix whose entries are $m\\times m $ matrices and the multiplication by $B(x)$ is assumed to be bounded on ${\\mathcal H}$ and accretive on $\\clos{\\textsf{R}(D)}$.\n\n\n\\end{itemize}\n\n\n\\subsection{Consequences}\\label{sec:consequences}\n\nWe gather  here  some consequences  on the functional calculus for the convenience of the reader.\n\n\n\n\\begin{prop}\\label{prop:equi} Let  $w,D$ and $B$ be as above. If $T=DB$ or $T=BD$, then one has the equivalence\n\\begin{equation}\\label{eq:sfT}\n  \\int_0^\\infty\\|tT(1+t^2T^2)^{-1} u \\|^2 \\, \\frac{dt}t \\sim \\|u\\|^2, \\qquad  \\text{for all }\\ u\\in \\clos{\\textsf{R}(T)}.\n\\end{equation}\n\\end{prop}\n\n\\begin{proof} First, the square function estimate for $BD$  follows from that for $DB$. Indeed, on $\\textsf{N}(BD)$,\n$tBD(1+t^2(BD)^2)^{-1} u =0$. On $\\clos{\\textsf{R}(BD)}$, $BD$ is similar to $DB$ so the square function inequality  for $BD$   follows. We  conclude  using the splitting ${\\mathcal H}= \\textsf{N}(BD) \\oplus  \\clos{\\textsf{R}(BD)}$. Now, if one changes $B$ to  $B^*$, this means we have proved\n\\eqref{eq:sfBD} for both $T=DB$ (resp. $T=BD$) and its adjoint. It is classical (\\cite{ADMc}) that this implies the equivalence on the range.\n\\end{proof}\n\n\nThe next result summarizes consequences of quadratic estimates that are needed.\n\n\\begin{prop}\\label{prop:SFimpliesFC} Let  $T$ be an $\\omega$-bisectorial operator on  a separable Hilbert space ${\\mathcal H}$ with $0\\le \\omega<\\pi\/2$.  Assume that the quadratic estimate\n\\begin{equation}\n  \\int_0^\\infty\\|tT(1+t^2T^2)^{-1} u \\|^2 \\, \\frac{dt}t \\sim \\|u\\|^2 \\   \\text{holds for all }\\ u\\in \\clos{\\textsf{R}(T)}.\n\\end{equation}Then, the following statements hold.\n\\begin{itemize}\n  \\item $T$ has a bounded holomorphic functional calculus on $\\clos{\\textsf{R}(T)}$ on any bisector $|\\arg (\\pm z)| <\\mu$ for any $\\omega<\\mu<\\pi\/2$, which can be extended to all ${\\mathcal H}$ by setting $f(T)=f(0)I$ on $\\textsf{N}(T)$ whenever $f$ is also defined at 0.\n\\item The comparison  \\begin{equation} \\label{eq:psiT}\n  \\int_0^\\infty\\|\\psi(tT) u \\|^2 \\, \\frac{dt}t \\sim \\|u\\|^2 \\  \\text{holds for all }\\ u\\in \\clos{\\textsf{R}(T)}.\n\\end{equation}\nfor any $\\omega<\\mu<\\pi\/2$ and  for any  holomorphic function $\\psi$ in the bisector $|\\arg (\\pm z)| <\\mu$, which is not identically zero on each connected component of the bisector and which satisfies\n$|\\psi(z)|\\le C\\inf (|z|^\\alpha, |z|^{-\\alpha})$ for some $C<\\infty$ and $\\alpha>0$.\n  \\item The operator $\\text{{\\rm sgn}}(T)$ is a bounded involution on $\\clos{\\textsf{R}(T)}$.\n  \\item  $\\clos{\\textsf{R}(T)}$ splits topologically into two spectral subspaces\n\\begin{equation}     \\label{eq:hardysplit}\\clos{\\textsf{R}(T)}={\\mathcal H}^+_{T}\\oplus {\\mathcal H}^-_{T}\n\\end{equation}\n  with ${\\mathcal H}^\\pm_{T}=E_{T}^\\pm(\\clos{\\textsf{R}(T)})$ and $E_{T}^+=\\chi^\\pm(T)$ are projections with $\\chi^\\pm(z)=1$ if $\\pm \\re z>0$ and $\\chi^\\pm(z)=0$ otherwise.\n\\item The operator $|T|=sgn(T)T = \\sqrt {T^2}$ with $\\textsf{D}(|T|)=\\textsf{D}(T)$ is a $\\omega$-sectorial operator and\n$-|T|$ generates an analytic semigroup of operators $(e^{-z|T|})_{|\\arg z| <\\pi\/2 - \\omega}$.\n\\item For $h\\in \\textsf{D}(T)$, $h\\in {\\mathcal H}^\\pm_{T}$ if and only if $|T|h=\\pm Th.$ As a consequence\n$e^{\\mp zT}$ are well-defined operators on ${\\mathcal H}^\\pm_{T}$ respectively,  and $e^{-zT}E_{T}^+$ and $e^{+zT}E_{T}^-$ are well-defined operators on ${\\mathcal H}$   for $|\\arg z | <\\pi\/2 - \\omega$.\n\\end{itemize}\\end{prop}\n\n\nAs announced in Section \\ref{sec:main}, we recall here why this implies the Kato conjecture for block diagonal\n$$\nB= \\begin{bmatrix} a & 0 \\\\ 0 & d\n\\end{bmatrix}\n$$\nidentifying the functions $a$ and $d$ with the corresponding multiplication operators.\nWe have\n$$BD= \\begin{bmatrix}  0 & a {\\text{{\\rm div}}}_{w}  \\\\ - d\\nabla  & 0\n\\end{bmatrix} \\ , \\ (BD)^2= \\begin{bmatrix} -a{\\text{{\\rm div}}}_{w} d\\nabla & 0 \\\\ 0 & - d\\nabla a  {\\text{{\\rm div}}}_{w}\n\\end{bmatrix}, $$ so that for $u\\in H^1({\\mathbb R}^n, w;{\\mathbb C}^m)$, $v=\\begin{bmatrix}\n      u    \\\\\n      0\n\\end{bmatrix} \\in \\textsf{D}(BD)= \\textsf{D}  (|BD|)$ and\n$$\\|\\sqrt {-a{\\text{{\\rm div}}}_{w} d\\nabla} u\\| \\sim \\| |BD|v \\| \\sim \\|BD v\\|  \\sim \\|d\\nabla u \\| \\sim \\|\\nabla u\\|.\n$$\n\n\n\n\n\n\\section{Representations for solutions of degenerate elliptic systems}\\label{sec:rep}\n\n\nFrom now on, we write  points in the upper half-space  ${\\mathbb R}^{1+n}_+$ as ${\\bf x}=(t,x)$, $t>0, x\\in {\\mathbb R}^n$.\n\n\n\n\n\n\\subsection{From second order to first order}\n\nWe shall now follow closely \\cite{AA1}, and its extension \\cite{R}, but in the weighted setting. It is necessary to have these references handy.  The estimates  of these two articles obtained in  abstract Hilbert spaces  evidently apply here.  Some other estimates use harmonic analysis (tent spaces, maximal functions). Thus we shall try to extract the relevant information and give proofs only when the argument uses a particular feature of the weighted situation.\n\nWe recall the notation ${\\mathcal H}=L^{2}({\\mathbb R}^n,w;{\\mathbb C}^{m(1+n)})$ and  use ${\\mathcal H}^{0}=\\clos{\\textsf{R}(D)}$ where $D$ was defined in Section  \\ref{sec:vv}. Beware that in \\cite{AA1}, ${\\mathcal H}$ was taken as $\\clos{\\textsf{R}(D)}$.\nWe continue to use $\\|\\ \\|$ to denote the norm in ${\\mathcal H}$, and occasionally use other notation when needed.\n\nWe construct solutions $u$ to the divergence form system (\\ref{eq:divform}),\nby  solving the equivalent  vector-valued ODE (\\ref{eq:firstorderODE}) below for the $w$-normalized conormal gradient\n$$\n  f=\\nabla_{w^{-1}A} u= \\begin{bmatrix} \\partial_{\\nu_{w^{-1}A}}u \\\\ \\nabla_x u \\end{bmatrix},$$\nand\n$\\partial_{\\nu_{w^{-1}A}}u$ denotes the upward  (hence inward for $\\mathbb{R}^{1+n}_+$) $w$-normalized conormal derivative of $u$.\n\nUsing the normal\/tangential  decomposition for  ${\\mathbb C}^{m(1+n)}= {\\mathbb C}^m \\oplus {\\mathbb C}^{mn}= {\\mathbb C}^m \\oplus ({\\mathbb C}^m \\otimes {\\mathbb C}^n)$ (see Section \\ref{sec:main}),  we  write matrices acting on ${\\mathbb C}^{m(1+n)}$   as\n$$\n  M= \\begin{bmatrix} M_{{\\scriptscriptstyle\\perp}\\no} & M_{{\\scriptscriptstyle\\perp}{\\scriptscriptstyle \\parallel}} \\\\ M_{{\\scriptscriptstyle \\parallel}{\\scriptscriptstyle\\perp}} & M_{{\\scriptscriptstyle \\parallel}\\ta} \\end{bmatrix},\n$$\nthe entries being matrices acting from and into the various spaces in the splitting.\n\n\n\n\\begin{prop}\\label{prop:hat}\n  The transformation\n$$\n   C\\mapsto \\widehat C:=   \\begin{bmatrix} I & 0  \\\\\n     C_{{\\scriptscriptstyle \\parallel}{\\scriptscriptstyle\\perp}} & C_{{\\scriptscriptstyle \\parallel}\\ta} \\end{bmatrix} \\begin{bmatrix} C_{{\\scriptscriptstyle\\perp}\\no} &  C_{{\\scriptscriptstyle\\perp}{\\scriptscriptstyle \\parallel}}  \\\\\n     0 & I \\end{bmatrix}^{-1} =  \\begin{bmatrix} C_{{\\scriptscriptstyle\\perp}\\no}^{-1} & -C_{{\\scriptscriptstyle\\perp}\\no}^{-1} C_{{\\scriptscriptstyle\\perp}{\\scriptscriptstyle \\parallel}}  \\\\\n     C_{{\\scriptscriptstyle \\parallel}{\\scriptscriptstyle\\perp}}C_{{\\scriptscriptstyle\\perp}\\no}^{-1} & C_{{\\scriptscriptstyle \\parallel}\\ta}-C_{{\\scriptscriptstyle \\parallel}{\\scriptscriptstyle\\perp}}C_{{\\scriptscriptstyle\\perp}\\no}^{-1}C_{{\\scriptscriptstyle\\perp}{\\scriptscriptstyle \\parallel}} \\end{bmatrix}\n$$\nis a self-inverse bijective transformation of the set of operator-valued matrices which are bounded on ${\\mathcal H}$ and   accretive on $\\clos{\\textsf{R}(D)}$.\n\\end{prop}\n\nThe proof is analogous to that of \\cite{AA1}.\n\n  We set\n  $$\n  \\widehat {w^{-1}A}= B$$ in what follows.\n  Our assumption is that as a multiplication operator,  $w^{-1}A(t,\\cdot)$ is bounded on ${\\mathcal H}$  and  accretive on $\\clos{\\textsf{R}(D)}$ for a.e.~$t>0$ with uniform bounds with respect to $t$. In particular,  the matrix $w^{-1}A_{{\\scriptscriptstyle\\perp}\\no}(t,\\cdot)$ is invertible as an operator acting on $L^2({\\mathbb R}^n; w; {\\mathbb C}^m)$, hence it is also invertible in $L^\\infty({\\mathbb R}^n, {\\mathcal L}({\\mathbb C}^{m}))$, with uniform bounds a.e.~in $t>0$.  Thus, $B(t,\\cdot)$ is also a multiplication operator.\n\nWe now introduce some notation.\nLet\n$\n   \n    {\\mathcal D}_{w}=\\begin{bmatrix} C_{0}^\\infty({\\mathbb R}^{1+n}_{+}; {\\mathbb C}^m)  \\\\\n    w^{-1} C_{0}^\\infty({\\mathbb R}^{1+n}_{+}; {\\mathbb C}^{mn})\\end{bmatrix}.\n    $$\n\n    Let ${\\mathcal C} url_{{\\scriptscriptstyle \\parallel},0}= \\{f \\in {\\mathcal D}'({\\mathbb R}^n; {\\mathbb C}^{m(1+n)})\\, ;\\, {\\text{{\\rm curl}}}_{x} f_{{\\scriptscriptstyle \\parallel}}=0\\}$, where the curl operator is computed componentwise.\n    Let ${\\mathcal H}_\\text{{\\rm loc}}=L^2_\\text{{\\rm loc}}({\\mathbb R}^n, w;{\\mathbb C}^{m(1+n)})$.\n\n\\begin{prop}  \\label{prop:divformasODE}\nFor a pair of coefficient matrices $A$ and $B$ related by  $A= w\\widehat B$, or equivalently $B= \\widehat {w^{-1}A}$,\nthe pointwise map $g\\mapsto f= \\begin{bmatrix}  (w^{-1} A g)_{\\scriptscriptstyle\\perp} \\\\ g_{\\scriptscriptstyle \\parallel} \\end{bmatrix} $ gives\na one-to-one correspondence, with inverse $g= \\begin{bmatrix}  (B f)_{\\scriptscriptstyle\\perp} \\\\ f_{\\scriptscriptstyle \\parallel} \\end{bmatrix} $,\nbetween solutions $g$ to the equations\n\\begin{equation}  \\label{eq:firstorderdiv}\n     \\begin{cases}\n     g\\in L^2_\\text{{\\rm loc}}({\\mathbb R}_+;{\\mathcal H}_\\text{{\\rm loc}}), \\\\\n     {\\text{{\\rm div}}}_{t,x} (Ag)=0, \\\\\n     {\\text{{\\rm curl}}}_{t,x} g=0,\n     \\end{cases}\n\\end{equation}  in the sense of distributions on ${\\mathbb R}^{1+n}_{+}$\nand solutions $f$  to the generalized Cauchy--Riemann equations\n\\begin{equation}  \\label{eq:firstorderODE}\n  \\begin{cases}\n  f\\in L^2_\\text{{\\rm loc}}({\\mathbb R}_+;{\\mathcal H}_\\text{{\\rm loc}}\\cap {\\mathcal C} url_{{\\scriptscriptstyle \\parallel},0}), \\\\\n  \\partial_t f+ DB f=0,\n   \\end{cases}\n\\end{equation}\n in the weak sense\n\\begin{equation}\n\\label{firstorderweak}\n\\int_{0}^\\infty -(f, \\partial_{t}\\varphi) + (Bf,D\\varphi)\\, dt =0 \\quad \\forall \\varphi\\in {\\mathcal D}_{w},\n\\end{equation}\nwhere $(\\ ,\\ )$ is the complex inner product with respect to $dw$.  \\end{prop}\n\nThe proof is almost completely identical to the one in \\cite{AA1}.\n\n\n\n\\begin{proof}\nThe transformation $g\\mapsto f= \\begin{bmatrix}  (w^{-1} A g)_{\\scriptscriptstyle\\perp} \\\\ g_{\\scriptscriptstyle \\parallel} \\end{bmatrix} $ is easily seen to be invertible on $L^2_\\text{{\\rm loc}}({\\mathbb R}_+;{\\mathcal H}^\\text{{\\rm loc}})$.  Consider a pair of functions $g$ and $f$ in $L^2_\\text{{\\rm loc}}({\\mathbb R}_+;{\\mathcal H}^\\text{{\\rm loc}})$ related in this fashion.\n  Equations (\\ref{eq:firstorderdiv}) for $g$ are equivalent to\n\\begin{equation}\n\\begin{cases}\n  \\partial_t (Ag)_{\\scriptscriptstyle\\perp} + {\\text{{\\rm div}}}_x( A_{{\\scriptscriptstyle \\parallel}{\\scriptscriptstyle\\perp}} g_{\\scriptscriptstyle\\perp} + A_{{\\scriptscriptstyle \\parallel}\\ta}g_{\\scriptscriptstyle \\parallel}) =0, \\\\\n  \\partial_t g_{\\scriptscriptstyle \\parallel} - \\nabla_x g_{\\scriptscriptstyle\\perp} =0, \\\\\n  {\\text{{\\rm curl}}}_x g_{\\scriptscriptstyle \\parallel} =0,\n\\end{cases}\n\\end{equation}\neach in the sense of distributions on ${\\mathbb R}^{1+n}_{+}$.\n  The last equation is equivalent to $f_t=f(t,\\cdot)\\in  {\\mathcal C} url_{{\\scriptscriptstyle \\parallel},0}$.\n  Moreover, using that $(w^{-1}Ag)_{\\scriptscriptstyle\\perp}= f_{\\scriptscriptstyle\\perp}$, $g_{\\scriptscriptstyle \\parallel}= f_{\\scriptscriptstyle \\parallel}$ and\n  $g_{\\scriptscriptstyle\\perp}= (Bf)_{\\scriptscriptstyle\\perp}= A_{{\\scriptscriptstyle\\perp}\\no}^{-1}(wf_{\\scriptscriptstyle\\perp}- A_{{\\scriptscriptstyle\\perp}{\\scriptscriptstyle \\parallel}}f_{\\scriptscriptstyle \\parallel})$,\n  the first two equations are seen to be equivalent to the equation $\\partial_t f+ DB f=0$ in the prescribed sense.\n  \\end{proof}\n\n\n\n\n  Next, the strategy in \\cite{AA1} is to integrate the weak differential equation  \\eqref{firstorderweak}\n  to obtain an equivalent formulation in the Duhamel sense. Again,  this  can be followed almost line by line, once we have the following density lemma.\n\n  \\begin{lem} The space ${\\mathcal D}_{w}$ is dense in  $H^{1}_\\text{{\\rm c}}({\\mathbb R}_+;{\\mathcal H}) \\cap L^2_\\text{{\\rm c}}({\\mathbb R}_+;\\textsf{D}(D))$, where the subscript $\\text{{\\rm c}}$ means that elements have compact support in ${\\mathbb R}_{+}$. Thus, if $f\\in L^2_\\text{{\\rm loc}}({\\mathbb R}_+;{\\mathcal H}^{0})$, \\eqref{firstorderweak} holds for any $\\varphi\\in H^{1}_\\text{{\\rm c}}({\\mathbb R}_+;{\\mathcal H}) \\cap L^2_\\text{{\\rm c}}({\\mathbb R}_+;\\textsf{D}(D))$ if it does for any $\\varphi\\in{\\mathcal D}_{w}$.\n\\end{lem}\n\n\n \\begin{proof} The density of ${\\mathcal D}_{w}$ in  $H^{1}_\\text{{\\rm c}}({\\mathbb R}_+;{\\mathcal H}) \\cap L^2_\\text{{\\rm c}}({\\mathbb R}_+;\\textsf{D}(D))$ can be easily established using (2) in Proposition \\ref{prop:D} and standard truncation and regularization in the $t$-variable.  If  $f\\in L^2_\\text{{\\rm loc}}({\\mathbb R}_+;{\\mathcal H}^{0})$ and $\\varphi\\in H^{1}_\\text{{\\rm c}}({\\mathbb R}_+;{\\mathcal H}) \\cap L^2_\\text{{\\rm c}}({\\mathbb R}_+;\\textsf{D}(D))$ then the integral  in \\eqref{firstorderweak} makes sense and vanishes by approximating $\\varphi$ by elements in ${\\mathcal D}_{w}$.\n\\end{proof}\n\nIn the above proposition, we are mostly interested in  having $g\\in L^2_\\text{{\\rm loc}}({\\mathbb R}_+;{\\mathcal H})$. Recall that\n    $${\\mathcal H}^{0}=\\clos{\\textsf{R}(D)}= L^{2}({\\mathbb R}^n,w;  {\\mathbb C}^{m(1+n)})\\cap {\\mathcal C} url_{{\\scriptscriptstyle \\parallel},0}={\\mathcal H}\\cap {\\mathcal C} url_{{\\scriptscriptstyle \\parallel},0}.$$\n    In particular, $g\\in L^2_\\text{{\\rm loc}}({\\mathbb R}_+;{\\mathcal H})$ if and only if $f\\in L^2_\\text{{\\rm loc}}({\\mathbb R}_+;{\\mathcal H}^{0})$ and we can apply the above lemma.\nFormally writing $\\partial_{t}f+ DB_{0}f= D({\\mathcal E} f)$, ${\\mathcal E}=B_{0}-B$, where $B_{0}$ is now  multiplication by a $t$-independent matrix,\nthe integration of \\eqref{firstorderweak} leads to the following equation\n\\begin{equation}   \\label{eq:inteqroadmap}\n  f_{t}= e^{-tDB_{0}} E_{0}^+ h^+ + (S_Af)_{t},\n\\end{equation}\nfor a unique $h^+\\in {\\mathcal H}^+_{DB_{0}}$\nand where $S_A$ is the vector-valued singular integral operator given\n\\begin{equation}   \\label{eq:firstformalSAdefn}\n    (S_A f)_t :=  \\int_0^t e^{-(t-s)DB_0} E_0^+ D {\\mathcal E}_s f_s \\, ds - \\int_t^\\infty e^{(s-t)DB_0} E_0^- D{\\mathcal E}_s f_s\\,  ds.\n\\end{equation}\nHere $E_{0}^\\pm=\\chi^\\pm(DB_{0})$ are the projections defined in Section \\ref{sec:consequences}\nand $ {\\mathcal H}^+_{DB_{0}}:= E_{0}^\\pm{\\mathcal H} $ are the ranges of  the respective projections. We also use  the notation $g_{t}=g(t,\\cdot)$.\nThis operator can be rigorously defined using  the maximal regularity operator for $|DB_0|$ viewed from the operational calculus point of view as $F(|DB_0|)$ with $F(z)$ being the operator-valued analytic function given by\n$$\n(F(z)g)_t:= \\int_0^t ze^{-(t-s) z}g_s\\,  ds,  \\ \\re z>0,\n$$\nso  \\eqref{eq:firstformalSAdefn} becomes\n\\begin{equation}    \\label{eq:inteqopcalcroadmap}\n   S_A= F(|DB_0|) \\widehat E_0^+{\\mathcal E} + F^*(|DB_0|) \\widehat E_0^- {\\mathcal E},\n\\end{equation}\nwhere $\\widehat E_0^\\pm$ are bounded operators on ${\\mathcal H}$ such that $E_0^\\pm D= (DB_0)\\widehat E_0^\\pm$. These two representations and Proposition \\ref{prop:equi} allow us to prove most relevant boundedness results concerning the regularity and Neumann problems. For the Dirichlet problem, they can be used as well  in an appropriate sense (see \\cite{R}) but there is  another useful representation using the operator \\begin{equation}    \\label{eq:tildeSA}\n  (\\widetilde S_A f)_t:=  \\int_0^t e^{-(t-s)B_0D} \\widetilde E_0^+ {\\mathcal E}_s f_s \\, ds - \\int_t^\\infty e^{(s-t)B_0D} \\widetilde E_0^- {\\mathcal E}_s f_s \\, ds,\n\\end{equation} where $\\widetilde E_0^\\pm=\\chi^\\pm(B_{0}D)$, and the vector field defined by\n$$\n   v_{t}  := e^{-tB_{0}D} \\widetilde E_{0}^+  \\tilde h^+ + (\\widetilde S_A f)_{t},\n   $$\n   for some $\\tilde h^+$. From the  intertwining property $b(DB_0)D= Db(B_0D)$ of the functional\ncalculi of $DB_0$ and $B_0D$, one has $D\\widetilde S_A =S_{A}$, so that the relation $D\\tilde h^+=h^+$, which uniquely determines a choice $\\tilde h^+\\in {\\mathcal H}^+_{B_{0}D}$, shows  that $Dv=f$. Solutions $u$ to the second order equations are related to $v$ in the sense that   there exists a constant $c\\in {\\mathbb C}^m$ such that\n   $\n   u =c  -v_{\\scriptscriptstyle\\perp}.\n$\nThis means that the tangential part $-v_{{\\scriptscriptstyle \\parallel}}$ encodes a conjugate to the solution $u$. This notion of conjugate was further developed in \\cite{AA2}.\n\nThese representations are justified provided one has  the operator bounds in the next section.\n\n\\subsection{Functions spaces and operator estimates}\n\n\nHere, we give  the definition of the functions spaces associated to the BVPs. What changes compared to \\cite{AA1} is that\nthe Lebesgue measure $dx$ on ${\\mathbb R}^n$ is replaced by $dw$ and Lebesgue measure $d{\\bf x}=dtdx$ on ${\\mathbb R}^{1+n}$  by $d{\\underline w}=dtdw$ where ${\\underline w}$ is the $A_{2}$ weight on ${\\mathbb R}^{1+n}$ defined by ${\\underline w}(t,x)=w(x)$. The only property required for $w$  in this section is the doubling property of $dw$, except when we use the quadratic estimate which uses the $A_{2}$ property.  Also we incorporate the posterior duality and multiplier results of\n \\cite{HR} (for $dx$), which can be extended to the weighted setting too. See below.\n\n\n\n\n\\begin{defn}    \\label{defn:NTandC} For an $L^q_{loc}$ function $f$, $1\\le q\\le \\infty$, define\n$W_{q}f(t,x)= \\left(\\barint_{\\hspace{-2pt}W(t,x)} |f|^q \\, d{\\underline w}\\right)^{1\/q}$ with the usual essential supremum definition if $q=\\infty$ and where $W(t,x):= (c_0^{-1}t,c_0t)\\times B(x;c_1t)$ is a Whitney region, for some fixed constants $c_0>1$, $c_1>0$.\nThe {\\em weighted non-tangential maximal function} of an $L^2_{\\text{{\\rm loc}}}$ function $f$ in ${\\mathbb R}^{1+n}_+$ is\n$$\n  \\widetilde N_*(f)(x):= \\sup_{t>0}  W_{2}f(t,x), \\qquad x\\in {\\mathbb R}^n,\n \n$$\nThe {\\em weighted Carleson functional} of an $L^1_{\\text{{\\rm loc}}}$ function $f$ is\n$$\n  Cf(x) := \\sup_{Q\\ni x} \\frac 1{w(Q)} \\int_{(0, \\ell(Q))\\times Q} |f(t,y)| \\, d{\\underline w}(t,y),\\qquad x\\in{\\mathbb R}^n,\n$$\nwhere the supremum is taken over all cubes $Q$ in ${\\mathbb R}^n$ containing $x$, with $\\ell(Q)$ denoting their sidelengths.\nThe {\\em modified weighted Carleson norm} of a measurable function $g$ in ${\\mathbb R}^{1+n}_+$ is\n$$\n  \\|g\\|_* := \\| C(W_{\\infty}(|g|^2\/t))\\|_\\infty^{1\/2}.\n$$\n\\end{defn}\n\n\nWe will use the modified Carleson norm to measure the size of perturbations\nof $t$-independent coefficients $A_0$.\n  The proof of Lemma 2.2 in \\cite{AA1} adapts to show that if there exists $A_{0}(x)$ with\n  $\\|w^{-1}(A-A_0)\\|_* < \\infty$, then it is unique and $w^{-1}A_{0}$ is bounded, and accretive on $\\clos{\\textsf{R}(D)}$, so that we may call $A_{0}$ the trace of $A$.\n\n\n  \\begin{defn}    \\label{defn:XY}\n   Define the Banach\/Hilbert spaces\n\\begin{align*}\n  {\\mathcal X} & := \\sett{ f\\in L^2_{loc}({\\mathbb R}^{1+n}_+; {\\mathbb C}^{m(1+n)})}{ \\|\\widetilde N_*(f)\\|<\\infty }, \\\\\n  {\\mathcal C} & :=   \\sett{f\\in L^2_{loc}({\\mathbb R}^{1+n}_+; {\\mathbb C}^{m(1+n)}) }{ \\|C(W_{2}f)\\|<\\infty },\\\\\n  {\\mathcal Y} &:= \\sett{f\\in L^2_{loc}({\\mathbb R}^{1+n}_+; {\\mathbb C}^{m(1+n)})}{\\int_0^\\infty \\| f_t \\|^2 \\, tdt < \\infty}, \\\\\n  {\\mathcal Y}^* &:= \\sett{f\\in L^2_{loc}({\\mathbb R}^{1+n}_+; {\\mathbb C}^{m(1+n)}) }{\\int_0^\\infty \\| f_t \\|^2\\,  \\frac{dt}t < \\infty},\n\\end{align*}\nwith the obvious norms.\n\\end{defn}\n\nWe use the same notation as in \\cite{AA1}, but of course, here all norms are weighted. Note that ${\\mathcal Y}^*$ is the dual space of\n${\\mathcal Y}$ with respect to  the inner product $\\langle\\ , \\ \\rangle_{{\\underline w}}$ of ${\\mathcal H}=L^2({\\mathbb R}^{1+n}_+, d{\\underline w} ;{\\mathbb C}^{m(1+n)})$.\n\n\n\n\n\\begin{lem}      \\label{lem:XlocL2}\n  There are estimates\n$$\n  \\sup_{t>0} \\frac 1t\\int_t^{2t} \\| f_s \\|^2\\,  ds \\lesssim \\| \\widetilde N_*(f) \\|^2 \\lesssim \\int_0^\\infty \\| f_s \\|^2\\, \\frac {ds}s.\n$$\nIn particular ${\\mathcal Y}^*\\subseteq {\\mathcal X}$.\n\\end{lem}\n\n A fundamental quantity is the norm  of multiplication operators mapping ${\\mathcal X}$ into ${\\mathcal Y}^*$ or ${\\mathcal Y}$ into ${\\mathcal C}$.\n\n\n\n\n\n\\begin{lem}   \\label{lem:Carleson} The dual of ${\\mathcal X}$  with respect to the pairing   $\\langle\\ , \\ \\rangle_{{\\underline w}}$ is ${\\mathcal C}$.\n  For  functions ${\\mathcal E}: {\\mathbb R}^{1+n}_+\\to {\\mathcal L}({\\mathbb C}^{m(1+n)})$, we have estimates\n$$\n  \\|{\\mathcal E}\\|_\\infty \\lesssim  \\|{\\mathcal E}\\|_* \\sim \\sup_{\\|f\\|_{\\mathcal X}=1}\\|{\\mathcal E} f\\|_{{\\mathcal Y}^*} \\sim \\sup_{\\|f\\|_{\\mathcal Y}=1}\\|{\\mathcal E} f\\|_{{\\mathcal C}}.\n$$\n\\end{lem}\n\n\n\\begin{proof} When $w=1$, the duality was established in \\cite{HR} and recently  another more direct proof  was given in \\cite{Huang}. This second proof passes to the  doubling measure setting (personal communication  of Amenta and Huang). Next,\nthe first inequality is proved in a similar way than in  \\cite{AA1}.  The equivalences for the pointwise multiplier operator norms were also established in \\cite{HR}, and reproved in \\cite{Huang} when $w=1$, and the latter proof extends in a doubling measure context as well  (personal communication  of Amenta and Huang).\n  \\end{proof}\n\n\n\n  \\begin{prop}\\label{prop:X}\n  Let $u\\in W^{1,2}_{\\text{{\\rm loc}}}({\\mathbb R}^{1+n}_+,w) $  be such that $\\|\\widetilde N_*(\\nabla_{t,x}u)\\|<\\infty$. Then there exists $u_{0}\\in \\dot H^1({\\mathbb R}^n,w)$ (as defined in the proof of  Lemma \\ref{lem:gradient}) such that $\\|u_{t}-u_{0}\\| \\lesssim t$,  $\\|\\nabla_{x}u_{0}\\| \\lesssim \\|\\widetilde N_*(\\nabla_{t,x}u)\\|$,  and for $dw$ almost every $x_{0} \\in {\\mathbb R}^n$,\n\\begin{equation}\n\\label{eq:pointwiset}\n \\barint_{\\hspace{-6pt}W(t,x_{0})} |u-u_{0}(x_{0})|^2\\, d{\\underline w} \\leq t^2g(x_{0}).\n\\end{equation}\nwith $g\\in L^1({\\mathbb R}^n,w)$.  Conversely, if $u_{0}\\in \\dot H^1({\\mathbb R}^n,w)$, then $u=e^{t^2\\Delta_{w}}u_{0}$ satisfies $\\|\\widetilde N_*(\\nabla_{t,x}u)\\| \\lesssim \\|\\nabla_{x} u_{0}\\|$.\n \\end{prop}\n\n\\begin{proof} The first part is the weighted version of a result in \\cite{KP}.\nFirst, it is easy to show   $\\|u_{t}-u_{t'}\\| \\lesssim |t-t'|$ by using $u_{t}-u_{t'}= \\int_{t'}^t \\partial_{s}u_{s}\\, ds$, Cauchy-Schwarz inequality and   the left hand  inequality in Lemma \\ref{lem:XlocL2}. This gives the existence of  $u_{0}\\in L^2_{\\text{{\\rm loc}}}({\\mathbb R}^n,w)$ with $\\|u_{t}-u_{0}\\| \\lesssim t$  (observe that only the difference is in $L^2({\\mathbb R}^n,w)$).\nNext, the Poincar\\'e inequality and a telescopic  sum argument implies\n$$\n\\bigg |\\barint_{\\hspace{-6pt}W(t,x_{0})} u \\,  d{\\underline w} - \\barint_{\\hspace{-6pt}W(t',x_{0})} u \\,  d{\\underline w}\\bigg| \\le C\\tau  \\widetilde N_*^1(\\nabla_{t,x}u)(x_{0})\n$$\nwhenever $t,t'\\le \\tau$ up to using a non-tangential maximal function with appropriately large Whitney regions and $\\widetilde N_*^1$ is the analogue of $\\widetilde N_*$ with $L^1$-averages. Thus for  every $x_{0}$ where  $u_{0}(x_{0})$ exists\n$$\n\\bigg|\\barint_{\\hspace{-6pt}W(t,x_{0})} u \\,  d{\\underline w}  -u_{0}(x_{0})\\bigg| \\lesssim t \\widetilde N_*^1(\\nabla_{t,x}u)(x_{0})\n$$\nand $\\widetilde N_*^1(\\nabla_{t,x}u)\\in L^2({\\mathbb R}^n,w)$ by equivalences of norms if we change the Whitney regions.  By the Poincar\\'e inequality again,\n$$\n\\barint_{\\hspace{-6pt}W(t,x_{0})} \\bigg|u-\\barint_{\\hspace{-6pt}W(t,x_{0})} u\\bigg|^2\\, d{\\underline w} \\le Ct^2\\widetilde N_*(\\nabla_{t,x}u)(x_{0})\n$$\nand we deduce  \\eqref{eq:pointwiset} on combining the last two inequalities.\n Finally, note that  if $x_{0}, y_{0}$ are different points and $t =  10 (c_{0}+c_{1}) |x_{0}-y_{0}|$, then\n$$\n\\bigg |\\barint_{\\hspace{-6pt}W(t,x_{0})} u \\,  d{\\underline w} - \\barint_{\\hspace{-6pt}W(t',y_{0})} u \\,  d{\\underline w}\\bigg| \\le C |x_{0}-y_{0}|  \\widetilde N_*^1(\\nabla_{t,x}u)(x_{0})\n$$\nagain with  slightly larger Whitney regions in the definition of $\\widetilde N_*^1$, so that combining with\n  inequalities above, we obtain\n$$\n|u_{0}(x_{0})-u_{0}(y_{0})| \\le C |x_{0}-y_{0}|(\\widetilde N_*^1(\\nabla_{t,x}u)(x_{0})+ \\widetilde N_*^1(\\nabla_{t,x}u)(y_{0})).\n$$\nUsing the theory of Sobolev spaces on the complete doubling  metric-measure space $({\\mathbb R}^n, |\\ |, w)$, it follows that $u_{0}\\in \\dot H^1({\\mathbb R}^n,w)$ (identified with the Haj\\l ash space), see \\cite{HajlashKoskela}.\n\nThe converse will be proved after Theorem \\ref{thm:NTmaxandaeCV}.\n\\end{proof}\n\nAt this stage, we do not know if $\\nabla_{t,x}u$ has  almost everywhere limits or even strong $L^2(w)$ limits in the above averaged sense (although weak $L^2(w)$ convergence can be shown as in \\cite{KP}). This will be the case, however, when $u$ is a solution of our systems.\nWe remark that in comparison, the space defined by $\\int_0^\\infty \\|\\nabla_{t,x}u_t\\|^2\\, tdt<\\infty$ does not have a trace on ${\\mathbb R}^n$.\n\nWith the above notation, we can state our main theorem for  $t$-independent $B_{0}$, thus for semigroups only.\n\n\\begin{thm}\\label{thm:NTmaxandaeCV} Let $T=DB_{0}$ or $B_{0}D$.\nThen one has the estimate\n\\begin{equation}\n\\label{eq:Ntmax}\n\\|e^{-t|T|}h\\|_{{\\mathcal X}}  \\sim \\|h\\| \\sim \\|\\partial_{t}e^{-t|T|}h\\|_{{\\mathcal Y}} , \\ \\forall h\\in \\clos{\\textsf{R}(T)}.\n\\end{equation}\nFurthermore, for any $h\\in {\\mathcal H}$ (not just $\\clos{\\textsf{R}(T)}$), we have that the Whitney averages of $e^{-t|T|}h$ converge to $h$ in $L^2$ sense, that is for $dw$ almost every $x_{0}\\in {\\mathbb R}^n$,\n\\begin{equation}\n\\label{eq:CVae}\n\\lim_{t\\to 0}\\ \\barint_{\\hspace{-6pt}W(t,x_{0})} |e^{-s|T|}h-h(x_{0})|^2\\, d{\\underline w}=0.\n\\end{equation}\nIn particular, this implies the $dw$ almost everywhere convergence of Whitney averages\n\\begin{equation}\n\\label{eq:CVaew}\n\\lim_{t\\to 0}\\ \\barint_{\\hspace{-6pt}W(t,x_{0})} e^{-s|T|}h\\,  d{\\underline w}=h(x_{0}).\n\\end{equation}\nMore generally, one can replace $e^{-s|T|}$ by any $\\varphi(sT)$ where $\\varphi$ is holomophic and bounded in some bisector containing $\\sigma(T)$ and satisfies $|\\varphi(z)| \\lesssim |z|^{-\\alpha}$ and $|\\varphi(z)-a| \\lesssim  |z|^\\alpha$ for some $\\alpha>0$, $a\\in {\\mathbb C}$. In this case,  convergence is towards $ah$, and only the upper bound $\\|\\varphi({tT})h\\|_{{\\mathcal X}}  \\lesssim  \\|h\\| $ holds if $a=0$.\n\\end{thm}\n\nThe proof  of this theorem will be given in Section \\ref{sec:NTmax}.\nThe last equivalence in \\eqref{eq:Ntmax}  is nothing but \\eqref{eq:psiT}, we put it here for completeness.\n\nIf $B_{0}=I$, then $T^2=D^2= \\begin{bmatrix} -\\Delta_{w} & 0 \\\\ 0 & -\\nabla {\\text{{\\rm div}}}_{w} \\end{bmatrix}$, so that $$\\nabla_{x} e^{t^2\\Delta _{w}}u_{0}= -\\bigg(De^{-t^2D^2}\\begin{bmatrix} u_{0}  \\\\ 0  \\end{bmatrix}\\bigg)_{{\\scriptscriptstyle \\parallel}}= \\bigg(e^{-t^2D^2}\\begin{bmatrix} 0  \\\\ \\nabla_{x }u_{0}  \\end{bmatrix}\\bigg)_{{\\scriptscriptstyle \\parallel}}\n$$\nand\n$$\\nabla_{t} e^{t^2\\Delta _{w}}u_{0}= \\bigg(2tDe^{-t^2D^2}\\begin{bmatrix} 0  \\\\ \\nabla_{x }u_{0}  \\end{bmatrix}\\bigg)_{{\\scriptscriptstyle\\perp}}\n$$\nThus, $\\|\\widetilde N_*(\\nabla_{t,x} e^{t^2\\Delta _{w}}u_{0})\\| \\lesssim \\|\\nabla_{x }u_{0}\\|$ follows from this result, proving the converse statement in Proposition \\ref{prop:X}.\n\nWe observe that  only the weak type bound  $\\|\\widetilde N_*(e^{-t|T|}h) \\|_{L^{2,\\infty}(w)} \\lesssim \\|h\\|$ holds if $h\\in \\textsf{N}(T)$. Concerning  the convergence \\eqref{eq:CVae}, this is new even when $w=1$    for $T=DB_{0}$ in this generality.\nWhat was proved in \\cite{AA2} is \\eqref{eq:CVae} for $|B_{0}e^{-t|DB_{0}|}h-(B_{0}h)(x_{0})|^2$  (which is also true in this situation), and the removal of $B_{0}$ was done only when $B_{0}^{-1}$ is given by pointwise multiplication. It turns out this is not necessary.   This will yield the almost everywhere limits in full generality in Theorem \\ref{apriori_HSNeumann} as compared to \\cite{AA2}.\n\n\\begin{rem}\\label{rem} the almost everywhere limit  \\eqref{eq:CVaew} is stated with respect to $d{\\underline w}$, which is natural. However, as they are derived from the weighted $L^2({\\underline w})$ limits \\eqref{eq:CVae}, using that ${\\underline w}\\in A_{2}$, we also have unweighted $L^1$ averages that converge to 0 almost everywhere. This means that \\eqref{eq:CVaew} holds also with Lebesgue measure replacing $d{\\underline w}$.\n\\end{rem}\n\n The next two theorems are for the $t$-dependent $S_{A}$ and $\\widetilde S_{A}$. Note that we may rewrite $S_{A}f=S({\\mathcal E} f):=S_{{\\mathcal E}}f$ and $\\widetilde S_{A}f= \\widetilde S({\\mathcal E} f):=\\widetilde S_{{\\mathcal E}}f$ where ${\\mathcal E}$ may not be related to $A$. We use this notation in what follows.\n\n\\begin{thm}\\label{thm:estSA} Assume that $ \\|{\\mathcal E}\\|_*<\\infty$. Then we have the following estimates for arbitrary $f\\in {\\mathcal X}$.\n\\begin{equation}\n\\|S_{{\\mathcal E}}f\\|_{{\\mathcal X}}\\lesssim \\|{\\mathcal E}\\|_*\\|f\\|_{{\\mathcal X}}.\n\\end{equation}\nThe function $h^-:= -\\int_0^\\infty  e^{sDB_{0}}E_0^-D {\\mathcal E}_s f_s ds$ belongs to $ E_0^-{\\mathcal H}={\\mathcal H}^-_{DB_{0}}$ and\n\\begin{equation}\n\\|h^-\\|\\lesssim \\|{\\mathcal E}\\|_*\\|f\\|_{{\\mathcal X}},\n\\end{equation}\n\\begin{equation}\n\\|S_{{\\mathcal E}}f-e^{tDB_{0}} E_0^- h^-\\|_{{\\mathcal Y}^*}\\lesssim \\|{\\mathcal E}\\|_*\\|f\\|_{{\\mathcal X}},\n\\end{equation}\n\\begin{equation}  \\label{eq:SAavlim}\n\\lim_{t\\to 0} t^{-1} \\int_t^{2t} \\| (S_{{\\mathcal E}}f)_{s} -h^- \\|^2 \\,ds =0=\n\\lim_{t\\to \\infty} t^{-1} \\int_t^{2t} \\| (S_{{\\mathcal E}}f)_s \\|^2 ds,\n\\end{equation}\nand\n\\begin{equation}\n\\label{eq:CVaeSA}\n\\lim_{t\\to 0}\\ \\barint_{\\hspace{-6pt}W(t,x_{0})} |S_{{\\mathcal E}}f-h^-(x_{0})|^2\\, d{\\underline w}=0, \\ \\mathrm{for \\ a.e.} \\ x_{0}\\in {\\mathbb R}^n.\n\\end{equation}\nMoreover, $\\tilde h^-:= -\\int_0^\\infty  e^{sB_{0}D}\\widetilde E_0^- {\\mathcal E}_s f_s \\, ds$ satisfies $D\\tilde h^-=h^-\\in E_0^-{\\mathcal H}$,\n\\begin{equation}  \\label{eq:SAavlimint}\n \\| (\\widetilde S_{{\\mathcal E}}f)_{t} -\\tilde h^- \\|  \\lesssim t \\|{\\mathcal E}\\|_*\\|f\\|_{{\\mathcal X}}.\n\\end{equation}\nIn addition, if $\\|{\\mathcal E}\\|_*$ is sufficiently small\n and  ${\\mathcal E}$ satisfies the $t$-regularity condition\n$\n  \\|t\\partial_{t}{\\mathcal E}\\|_{*}<\\infty,\n$\n  then\n  \\begin{equation}\n\\label{ }\n \\|\\partial_t (S_{{\\mathcal E}} f)\\|_{\\mathcal Y}\\lesssim (\\|{\\mathcal E}\\|_* + \\|t\\partial_t {\\mathcal E}\\|_*) \\|f\\|_{\\mathcal X} + \\|{\\mathcal E}\\|_\\infty \\|\\partial_t f\\|_{{\\mathcal Y}},\n\\end{equation}\nand one has $t\\mapsto (S_{{\\mathcal E}}f)_{t}$ is continuous  into  ${\\mathcal H}$ if $\\|f\\|_{\\mathcal X} + \\|\\partial_t f\\|_{{\\mathcal Y}}<\\infty$ with improved limits\n\\begin{equation}  \\label{eq:SAavlimimp}\n\\lim_{t\\to 0}  \\| (S_{{\\mathcal E}}f)_{t} -h^- \\|  =0=\n\\lim_{t\\to \\infty}  \\| (S_{{\\mathcal E}}f)_t \\| .\n\\end{equation}\n \\end{thm}\n\n\n\n\n\n\\begin{thm}\\label{thm:esttSA} Assume that $ \\|{\\mathcal E}\\|_*<\\infty$. Then we have the following estimates for arbitrary $f\\in {\\mathcal Y}$.\n\\begin{equation}\n\\|S_{{\\mathcal E}}f\\|_{{\\mathcal Y}}\\lesssim \\|{\\mathcal E}\\|_*\\|f\\|_{{\\mathcal Y}}.\n\\end{equation}\nThe operator $\\widetilde S_{{\\mathcal E}}$ maps ${\\mathcal Y}$ into $C([0,\\infty); {\\mathcal H})$ with\n\\begin{equation}\n\\label{ }\n\\sup_{t\\ge 0}\\|(\\widetilde S_{{\\mathcal E}}f)_{t}\\| \\lesssim   \\|{\\mathcal E}\\|_*\\|f\\|_{{\\mathcal Y}}.\n\\end{equation}\nMoreover, $\\tilde h^-:= -\\int_0^\\infty  e^{sB_{0}D}\\widetilde E_0^- {\\mathcal E}_s f_s\\,  ds \\in \\widetilde E_0^-{\\mathcal H}= {\\mathcal H}^-_{B_{0}D}$,\n\\begin{equation}  \\label{eq:tSAavlimint}\n \\lim_{t\\to 0}\\| (\\widetilde S_{{\\mathcal E}}f)_{t} -\\tilde h^- \\|=0 =\\lim_{t\\to\\infty}  \\| (\\widetilde S_{{\\mathcal E}}f)_{t} \\|.\n\\end{equation}\nFurthermore, if $p<2$ and $\\widetilde N_*^p$ is the $p$-modified version of $\\widetilde N_*$ by taking $W_{p}$ functionals on Whitney regions,\n\\begin{equation}\n\\label{ }\n \\|\\widetilde N_*^p(\\widetilde S_{{\\mathcal E}}f)\\| \\lesssim  \\|{\\mathcal E}\\|_*\\|f\\|_{{\\mathcal Y}}\n \\end{equation}\nand\n\\begin{equation}\n\\label{eq:CVaetSAint}\n \\barint_{\\hspace{-6pt}W(t,x_{0})} |\\widetilde S_{{\\mathcal E}}f-\\tilde h^-(x_{0})|^p\\, d{\\underline w} =0 , \\ \\mathrm{for \\ a.e.} \\ x_{0}\\in {\\mathbb R}^n.\n\\end{equation}\n\n\\end{thm}\n\n\n\nThese two theorems can be proved following the corresponding results in  Sections 6,~7,~8,~9 and 10 of \\cite{AA1} (some of the arguments were simplified in \\cite{R})  and, concerning the almost everywhere convergence limits \\eqref{eq:CVaeSA} and  \\eqref{eq:CVaetSAint},  in Section 15 of \\cite{AA2} : they  hold in  a doubling weighted context when coming to use maximal functions and Carleson estimates.  The inequality \\eqref{eq:SAavlimint} is not proved in \\cite{AA1} and merely sketched in \\cite[Section 13]{AA2}, but is easy to prove following the same decompositions as there. We shall not give details.  We just mention that $\\tilde h^-$ in Theorem \\ref{thm:estSA} is not an element of ${\\mathcal H}$: it is only defined as a limit of the integrals  truncated away from 0 and $\\infty$ in the sense that $-\\int_\\varepsilon^R  De^{sB_{0}D}\\widetilde E_0^- {\\mathcal E}_s f_s \\, ds$ converges to $h^-$ in ${\\mathcal H}$. Thus, only the difference $(\\widetilde S_{{\\mathcal E}}f)_{t} -\\tilde h^- $ in  \\eqref{eq:SAavlimint} makes sense in ${\\mathcal H}\n $. The scalar part of $\\tilde h^-$ belongs to the homogeneous Sobolev space $\\dot H^1({\\mathbb R}^n,w;{\\mathbb C}^m)$ (as defined in the proof of Lemma \\ref{lem:gradient}) and as such is also an $L^2_{loc}(w)$ function.\n\n\n\n\n\n\\subsection{{\\em A priori} estimates}\\label{sec:apriori}\n\nIn this subsection, we derive a priori estimates for solutions of ${\\text{{\\rm div}}} A \\nabla u=0$ with $\\nabla u \\in {\\mathcal X}$ or ${\\mathcal Y}$.  Again, these are obtained as in \\cite{AA1}, together with \\cite{AA2} for the almost everywhere statements and the improvements noticed in Theorem \\ref{thm:NTmaxandaeCV}.\n\n\\begin{thm}   \\label{apriori_HSNeumann}\n   Consider coefficients $w^{-1}A\\in L^\\infty({\\mathbb R}^{1+n}_+; {\\mathcal L}({\\mathbb C}^{m(1+n)}))$ such that $w^{-1}A$ is  accretive on ${\\mathcal H}^{0}$ and assume there exists $t$-independent measurable coefficients $A_0$ such that\n $\\|w^{-1} (A-A_0) \\|_* <\\infty$ or equivalently that $ \\|{\\mathcal E}\\|_*\\sim \\| w^{-1}(A-A_{0})\\|_{*}<\\infty$ where ${\\mathcal E}=B_{0}-B$ and $B=\\widehat {w^{-1}A}, B_{0}=\\widehat{w^{-1}A_{0}}$.\n\n  Let  $u$\n    be a weak solution of\n   ${\\text{{\\rm div}}} A\\nabla u=0$ in $\\mathbb{R}^{1+n}_+$ with  $\\|\\widetilde N_*(\\nabla_{t,x}u)\\| <\\infty$.\n  Then\n$$\n  \\lim_{t\\to 0} t^{-1} \\int_t^{2t} \\| \\nabla_{s,x} u_s - g_0 \\|^2 ds =0=\n   \\lim_{t\\to \\infty} t^{-1} \\int_t^{2t} \\| \\nabla_{s,x} u_s \\|^2 ds,\n$$\nfor some $g_0 \\in L^2({\\mathbb R}^n ,w;{\\mathbb C}^{m(1+n)})$, with estimate\n$\\|g_0\\|\\lesssim\\|\\widetilde N_*(\\nabla_{t,x}u)\\|$, which we call the gradient of $u$ at the boundary and we set\n$\\nabla_{t,x}u|_{t=0}:=g_{0}$.\nFurthermore, one has that for $dw$ almost every $x_{0}\\in {\\mathbb R}^n$,\n\\begin{equation}\n\\label{eq:CVaewgradsol}\n\\lim_{t\\to 0}\\ \\barint_{\\hspace{-6pt}W(t,x_{0})} \\nabla_{s,x} u  \\, d{\\underline w}=g_{0}(x_{0}).\n\\end{equation}\nAll three limits hold with $\\nabla u, g_{0}$ replaced by  the $w$-normalized conormal gradient $f= \\nabla_{w^{-1}A}u$ and $f_{0}=\\begin{bmatrix}\n  (w^{-1}A_{0}g_{0})_{\\perp}        \\\\\n     ( g_{0})_{{\\scriptscriptstyle \\parallel}}\n\\end{bmatrix}:=\\nabla_{w^{-1}A}u|_{t=0}$ (in particular, they hold for the $w$-normalised conormal derivative $\\partial_{\\nu_{w^{-1}A}}u$). Moroever,   one has the\nrepresentation\n\\begin{equation}\n\\label{eq:representationRegNeu}\n\\nabla_{w^{-1}A}u = e^{- t  DB_{0}}h^+ + S_{A}(\\nabla_{w^{-1}A}u).\n\\end{equation}\nfor a unique $h^+\\in {\\mathcal H}^+_{DB_{0}}$ and\n\\begin{equation}\n\\label{eq:representationRegNeu1}\n\\nabla_{w^{-1}A}u|_{t=0} = h^+ + h^-, \\quad h^-=-\\int_0^\\infty  e^{sDB_{0}}E_0^-D {\\mathcal E}_s (\\nabla_{w^{-1}A}u)_s ds.\n\\end{equation}\nFinally, there exists $u_{0}\\in \\dot H^1(w)$ (as defined in Lemma \\ref{lem:gradient}) such that  $\\nabla_{x}u_{0}= (g_{0})_{{\\scriptscriptstyle \\parallel}}$ and one has $\\|u_{t}-u_{0}\\|\\lesssim t$   and for $dw$ almost every $x_{0}\\in {\\mathbb R}^n$\n\\begin{equation}\n\\label{eq:CVaewsolreg}\n\\lim_{t\\to 0}\\ \\barint_{\\hspace{-6pt}W(t,x_{0})} u  \\, d{\\underline w}=u_{0}(x_{0}).\n\\end{equation}\nRemark \\ref{rem} about replacing $d{\\underline w}$ by Lebesgue measure in the almost everywhere limit applies here too.\n\n\\end{thm}\n\n\n\n\n\\begin{thm}   \\label{apriori_HSDir}\n   Consider coefficients $w^{-1}A\\in L^\\infty({\\mathbb R}^{1+n}_+; {\\mathcal L}({\\mathbb C}^{m(1+n)}))$ such that $w^{-1}A$ is  accretive on ${\\mathcal H}^{0}$ and assume there exists $t$-independent measurable coefficients $A_0$\n such that $\\| w^{-1}(A-A_0) \\|_* <\\infty$, or equivalently that $ \\|{\\mathcal E}\\|_*\\sim \\| w^{-1}(A-A_{0})\\|_{*}<\\infty$ where ${\\mathcal E}=B_{0}-B$ and $B=\\widehat {w^{-1}A}, B_{0}=\\widehat{w^{-1}A_{0}}$.\n\n  Let  $u$     be a weak solution of\n   ${\\text{{\\rm div}}} A\\nabla u=0$ in $\\mathbb{R}^{1+n}_+$ and assume that\n  $\\int_0^\\infty \\|\\nabla_{t,x}u\\|^2\\, tdt<\\infty$.\n Then $u= \\hat u+c$ almost everywhere,\n for a unique constant $c\\in {\\mathbb C}^m$ and  $\\hat u\\in C([0,\\infty); L^2({\\mathbb R}^n,w;{\\mathbb C}^m))$ given by\n $\\hat u= -v_{\\perp}$ with\n \\begin{equation}\n\\label{eq:representationDir}\nv= e^{- t B_{0}D } \\tilde h^+ + \\widetilde S_{A}(\\nabla_{w^{-1}A}u),\n\\end{equation}\nfor a unique $ \\tilde h^+\\in \\widetilde E_{0}^+{\\mathcal H}$. Moreover,\n\\begin{equation}\n\\label{eq:representationDir1}\nv_{0}=  \\tilde h^+ + \\tilde h^-\n\\ \\mathrm{with}\\  \\tilde h^-= -\\int_0^\\infty  e^{sB_{0}D}\\widetilde E_0^- {\\mathcal E}_s (\\nabla_{w^{-1}A}u)_{s}\\,  ds,\n\\end{equation}\nand we call $-v$ the conjugate system associated to $u$.  In addition, we have $Dv=\\nabla_{w^{-1}A}u$.\n\n Identifying the functions $u$ and $\\hat u+c$, we have\n  limits\n$$\n  \\lim_{t\\to 0} \\| u_t -\\hat u_0-c \\| =0=\n   \\lim_{t\\to \\infty} \\| u_t -c\\|,\n$$\nfor  $\\hat u_0= -(\\tilde h^+)_{\\perp}\\in L^2({\\mathbb R}^n, w;{\\mathbb C}^m)$, and we have the estimates\n$$\n\\|\\hat u_{0}\\|\\lesssim \\max(\\|\\widetilde N_*(\\hat u)\\|, \\sup_{t>0}\\|\\hat u_t \\|)\\lesssim \\bigg(\\int_0^\\infty \\|\\nabla_{t,x}u\\|^2\\,  tdt\\bigg)^{1\/2}.\n$$\nFinally, for $dw$ almost every $x_{0}\\in {\\mathbb R}^n$,\n\\begin{equation}\n\\label{eq:CVaewsoldir}\n\\lim_{t\\to 0}\\ \\barint_{\\hspace{-6pt}W(t,x_{0})} u  \\, d{\\underline w}=u_{0}(x_{0}).\n\\end{equation}\n Remark \\ref{rem} about replacing $d{\\underline w}$ by Lebesgue measure in the almost everywhere limit applies here too.\n\n\n\\end{thm}\n\nThe representation formula suggests a possible construction of solution given $\\tilde h^+$ provided $\\|{\\mathcal E}\\|_{*}$ is sufficiently small. This is what leads to well-posedness results.\n\n\\begin{rem}\nWe also have the representation \\eqref{eq:representationRegNeu} with both $e^{-t|DB_{0}|}$ and $h^+$ interpreted in a suitable  sense with Sobolev spaces of order $s=-1$. This point of view is developed more systematically in \\cite{R} when $w=1$ and with Sobolev regularity $-1\\le s \\le0$. We refer the reader there to make the straightforward adaptation, as it is again an abstract argument.   We just warn the reader that the ${\\mathcal E}$ in \\cite{R} is not exactly the same as ours because the author assumed pointwise accretivity to simplify matters. One should use  representation \\eqref{eq:firstformalSAdefn}  for $S_{A}$ instead.\n\\end{rem}\n\n\n\\begin{cor}\\label{cor:aentcv}\nAssume that $A$ satisfies $\\| w^{-1}(A-A_0) \\|_* <\\infty$ for some $t$-independent $A_{0}$ and is such that all weak solutions $u$ to the system ${\\text{{\\rm div}}} A \\nabla u=0$ in a ball $B\\subseteq \\mathbb{R}^{1+n}_+$ satisfy\n the local boundedness property\n$$\n\\sup_{\\alpha B}|u| \\le C \\left( \\,\\, \\barint_{\\hspace{-6pt}\\beta B} |u|^2 d{\\underline w}\\right)^{1\/2},\n$$\nfor any fixed constant $\\alpha<\\beta<1$,  with  $C$ independent of $u$ and  $B$.  Then any\nweak solution $u$ with $\\int_0^\\infty \\|\\nabla_{t,x}u\\|^2\\, tdt<\\infty$ or $\\|\\widetilde N_*(\\nabla_{t,x}u)\\|<\\infty$ converges non-tangentially almost everywhere to its boundary trace.\n\\end{cor}\n\nThe proof is a straightforward consequence of the more precise almost everywhere convergences\nwe stated in the previous section. We skip the details.\nThis result applies in particular to real equations as a consequence of \\cite{FKS}.\n\n\n\n\\section{Well-posedness}\\label{sec:solvability}\n\n\\subsection{Formulation and general results}\n\n\n\n\\begin{defn} Fix $w\\in A_{2}({\\mathbb R}^n)$. Consider degenerate coefficients $A$ with $w^{-1}A\\in L^\\infty({\\mathbb R}^{1+n}_+; {\\mathcal L}({\\mathbb C}^{m(1+n)}))$ such that $w^{-1}A$ is  accretive on ${\\mathcal H}^{0}$.\n \\begin{itemize}\n  \\item By the Dirichlet problem with coefficients $A$  being well-posed, we mean that\n  given $\\varphi\\in L^2({\\mathbb R}^n,w;{\\mathbb C}^m)$, there is a unique weak solution $u$\n   solving (\\ref{eq:divform}),  with $ \\int_0^\\infty \\|\\nabla_{t,x} u\\|^2\\,  tdt<\\infty$\n and trace $u_0= \\varphi$.\n  \\item By the regularity problem with coefficients $A$  being well-posed, we mean that\n  given $\\varphi\\in L^2({\\mathbb R}^n,w; {\\mathbb C}^{mn})$, where $\\varphi$ satisfies ${\\text{{\\rm curl}}}_x \\varphi=0$, there is a  weak solution $u$,\n  unique modulo constants,\n  solving  (\\ref{eq:divform}),    with $\\|\\widetilde N_*(\\nabla_{t,x} u)\\|<\\infty$\n  and  such that $\\nabla_{x}u|_{t=0}= \\varphi$.\n   \\item By the Neumann problem with coefficients $A$  being well-posed, we mean that\n  given $\\varphi\\in L^2({\\mathbb R}^n,w;{\\mathbb C}^m)$, there is a weak solution $u$,\n  unique modulo constants,\n  solving  (\\ref{eq:divform}),  with $\\|\\widetilde N_*(\\nabla_{t,x} u)\\|<\\infty$\n  and  such that $\\partial_{\\nu_{w^{-1}A}}u|_{t=0}= \\varphi$.\n\\end{itemize}\nWe  write $A \\in$ WP(BVP), if the corresponding boundary value problem (BVP) is well-posed with coefficients  $A$.\n\\end{defn}\n\n\n\n\n\nWe remark that the definition does not include  almost everywhere requirements. For the regularity and Neumann problems, one can make sense of the trace in a weak sense, but for the Dirichlet problem, the trace may not even make sense. However, as soon as we assume $\\|w^{-1}(A-A_{0})\\|_{*}<\\infty$, which will be the case here, we know exactly the meaning of the trace from the results in Section \\ref{sec:apriori}.\n\nThe most important observation following the \\textit{a priori}  estimates in Theorems \\ref{apriori_HSNeumann} and \\ref{apriori_HSDir} is the fact that in the $t$-independent coefficient case, we completely identify the trace spaces: $ {\\mathcal H}^+_{DB}$ is the trace space of $w$-normalized conormal gradients for solutions with $\\|\\widetilde N_*(\\nabla_{t,x} u)\\|<\\infty$;\n$ {\\mathcal H}^+_{BD}$ is the trace space of  conjugate systems $v$  for solutions with $ \\int_0^\\infty \\|\\nabla_{t,x} u\\|^2\\,  tdt<\\infty$. In each case this is an isomorphism.\n\nThis leads to the following characterisation of well-posedness.\n\n\\begin{thm} Consider coefficients $w^{-1}A\\in L^\\infty({\\mathbb R}^{1+n}_+; {\\mathcal L}({\\mathbb C}^{m(1+n)}))$ such that $w^{-1}A$ is  accretive on ${\\mathcal H}_{0}$. Assume that $A$ has $t$-independent coefficients.  Let $B=\\widehat {w^{-1}A}$.\nThen $A\\in$ WP(Reg)\/WP(Neu)\/WP(Dir)   if and only if\n\\begin{align*}\n  {\\mathcal H}^+_{DB} \\longrightarrow \\sett{g\\in L^2({\\mathbb R}^n,w;{\\mathbb C}^{mn})}{{\\text{{\\rm curl}}}_x g=0} &: f\\longmapsto f_{\\scriptscriptstyle \\parallel}, \\\\\n   {\\mathcal H}^+_{DB} \\longrightarrow L^2({\\mathbb R}^n,w;{\\mathbb C}^m) &: f\\longmapsto f_{\\perp}, \\\\\n   {\\mathcal H}^+_{BD} \\longrightarrow L^2({\\mathbb R}^n,w;{\\mathbb C}^m) &: f\\longmapsto f_{\\perp},\n\\end{align*}\nare isomorphisms respectively.\n\\end{thm}\n\n\nObserve the change of space in the third line.\n\nLet us mention a connection to so-called Rellich estimates. The isomorphisms imply the Rellich estimates\n\\begin{align*}\n  \\|f_{\\perp}\\|\\lesssim \\|f_{{\\scriptscriptstyle \\parallel}}\\|, \\quad \\forall f \\in {\\mathcal H}^+_{DB}, \\\\\\\n  \\|f_{{\\scriptscriptstyle \\parallel}}\\|\\lesssim \\|f_{\\perp}\\|, \\quad \\forall f \\in {\\mathcal H}^+_{DB}, \\\\\n  \\|f_{{\\scriptscriptstyle \\parallel}}\\|\\lesssim \\|f_{\\perp}\\|, \\quad \\forall f \\in  {\\mathcal H}^+_{BD},\n\\end{align*}\nrespectively. Assuming the Rellich estimates is not enough to conclude well-posedness because this only gives injectivity with closed range. The surjectivity  usually follows from  a continuity argument starting with a situation where one knows surjectivity. Thus, if, in a connected component of validity of a Rellich estimate, there is one $B$   for which surjectivity holds, then surjectivity holds for all $B$ in  this connected component and the corresponding BVP is well-posed for all $B$ in this connected component. Usually one considers $B=I$ so that, here,  $A=wI$. See \\cite{AAM} for a discussion which applies \\textit{in extenso}.  We remark that this depends on the continuous dependence on $B\\in L^\\infty$ of the projections $E_{0}^+$ and $\\widetilde E_{0}^+$, which follows from Theorem \\ref{th:main}.\nLet us mention also the duality principle between Dirichlet and Regularity, whose proof is the same as that of Proposition 17.6 in \\cite{AA2}.\n\n\\begin{thm}\\label{th:dualityDirReg} Let $w^{-1}A\\in L^\\infty({\\mathbb R}^{1+n}_+; {\\mathcal L}({\\mathbb C}^{m(1+n)}))$ such that $w^{-1}A$ is  accretive on ${\\mathcal H}^{0}$. Assume  there exists $t$-independent measurable coefficients $A_0$\n such that $\\| w^{-1}(A-A_0) \\|_* <\\varepsilon$. If $\\varepsilon$ is small enough, then $A\\in$ WP(Dir) if and only if $A^*\\in$ WP(Reg).\n\\end{thm}\n\n\n\n\nWe now turn to perturbation results for both $t$-dependent and $t$-independent coefficients. Adapting \\cite{AAH, AAM}, see especially Lemma 4.3 in \\cite{AAM}, one obtains that\neach WP(BVP) is open under perturbation of $t$-independent coefficients in $wL^\\infty$.\nWe refer to \\cite{AA1}  for the proofs of the $t$-dependent perturbations, which carry over without change to our setting. We gather these observations together in the following statement.\n\n\n\\begin{thm}   \\label{thm:Nellie} Assume the Neumann problem with $t$-independent $A_0$ is well-posed.\n Then there exist $\\varepsilon_{0}>0$ and $ \\varepsilon_{1}>0$ such that if $\\| w^{-1}(A-A_1) \\|_* <\\varepsilon_{1}$ and $A_{1}$ has $t$-independent coefficients with $\\| w^{-1}(A_{1}-A_0) \\|_\\infty <\\varepsilon_{0}$, then the Neumann problem  with coefficients $A$ is well-posed.\n\nThe corresponding result holds when the Neumann problem is replaced by the regularity problem.\n\nMoreover, for all such $A$, the solutions  $u$ of the BVP satisfy $$\n   \\|\\widetilde N_*(\\nabla_{t,x} u)\\| \\approx \\|g_0\\| \\approx \\|\\varphi\\|,\n $$\n with $\\varphi$ the $w$-normalized Neumann data or  the regularity data, and one has the limits and regularity estimates as described in Theorem \\ref{apriori_HSNeumann}.\n\n\n\n\\end{thm}\n\n\nWith the duality principle above, we obtain.\n\n\\begin{thm}\\label{thm: DirLip}\n  Assume the Dirichlet problem with $t$-independent $A_0$ is well-posed.\n Then there exist $\\varepsilon_{0}>0$ and $\\varepsilon_{1}>0$ such that if $\\| w^{-1}(A-A_1) \\|_* <\\varepsilon_{1}$ and $A_{1}$ has $t$-independent coefficients with $\\| w^{-1}(A_{1}-A_0) \\|_\\infty <\\varepsilon_{0}$, then the Dirichlet\n  problem  with coefficients $A$ is well-posed.\nMoreover, one has $$\n   \\|\\widetilde N_*(u)\\| \\approx \\sup_{t>0}\\|u_t \\| \\approx\n   \\bigg(\\int_0^\\infty \\|\\nabla_{t,x} u\\|^2\\,  tdt \\bigg)^{1\/2}\\approx \\|\\varphi\\|,\n $$\n  if $\\varphi$ is the Dirichlet data,\nand one has the limits and regularity estimates described in Theorem \\ref{apriori_HSDir}.\n\\end{thm}\n\n\n\n\n\n\n\n\n\\subsection{Well-posedness for $t$-independent  hermitian coefficients}\n\n\n\n\n\\begin{prop} Assume that  $A=A^*$ and that $A$ is $t$-independent and satisfies the usual degenerate boundedness and accretivity on ${\\mathcal H}^{0}=\\clos{\\textsf{R}(D)}$ conditions. Then the regularity, Neumann and Dirichlet problems with coefficients $A$ are well-posed.\n \\end{prop}\n\n\\begin{proof} Let $B=\\widehat {w^{-1}A}$ and\n  $f\\in E^+_0{\\mathcal H}={\\mathcal H}^+_{DB}$.\nTheorem \\ref{apriori_HSNeumann} in the case of  $t$-independent coefficients implies that the vector field $F_t=e^{-tDB}f$ in ${\\mathbb R}^{1+n}_+$ is such that $\\partial_t F_t=-DB F_t$,\n$\\lim_{t\\rightarrow\\infty}F_t=0$ and $\\lim_{t\\rightarrow 0}F_t=f$.\nLet $N:= \\begin{bmatrix} -I & 0 \\\\ 0 & I\\end{bmatrix}$\nand note that $D N +N D=0$.\nNow,    the definition of $B=\\widehat {w^{-1}A}$ and   the Hermitian\ncondition  $A^*=A$ imply $B^*N=NB$.  Using the hermitian inner product  $(\\ , \\ )$  for $dw$, we have\n\\begin{multline*}\n    \\partial_t (N F_t,  BF_t)\n  = (NDB F_t,  BF_t)+ (N F_t, BDB F_t) \\\\\n  =  (NDB F_t, B F_t)+ (DB^*N F_t, B F_t)\n  = ((N D+ D N)B F_t, B F_t) =0.\n\\end{multline*}\nHence, integrating in $t$ and taking into account the  limit at $\\infty$ gives us $(Nf,Bf)=0$.  Thus, separating scalar and tangential parts, we obtain the Rellich equality:\n \\begin{equation}   \\label{eq:rellich}\n(f,Bf)= 2(f_{\\perp}, (Bf)_{\\perp})=2 (f_{{\\scriptscriptstyle \\parallel}}, (Bf)_{{\\scriptscriptstyle \\parallel}}).\n\\end{equation}\nConsider first the Neumann problem. It follows  from  (\\ref{eq:rellich}) and the accretivity of $B$ on ${\\mathcal H}^{0}$ that\n$$\n \\kappa \\|f\\|^2 \\le \\re(f,Bf)= 2\\re(f_{\\perp}, (Bf)_{\\perp})\\lesssim 2\\|B\\|_{\\infty} \\|f_{\\perp}\\| \\|f\\|.\n$$\nThis shows that $\\|f\\|\\lesssim \\|f_{\\perp}\\|$  for the Neumann map for any hermitian $A$, which implies that this map is injective with closed range. The continuity argument explained above implies that $A\\in $ WP(Neu) provided that  $I\\in$  WP(Neu). That\n$I\\in$ WP(Neu) can be seen from the equality $\\|\\nabla (-\\Delta_{w})^{-1\/2}u\\|=\\|u\\|$ and\n$$\n\\text{{\\rm sgn}}(D)= \\begin{bmatrix} 0 & (-\\Delta_{w})^{-1\/2}{\\text{{\\rm div}}}_{w} \\\\ -\\nabla (-\\Delta_{w})^{-1\/2}  & 0\\end{bmatrix}.\n$$\nThus, for $f\\in {\\mathcal H}^{0}$, $f\\in {\\mathcal H}^+_{D}$ if and only if $f_{{\\scriptscriptstyle \\parallel}}= -\\nabla (-\\Delta_{w})^{-1\/2}f_{\\perp}\n$, which in turn holds  if and only if $f_{\\perp}= (-\\Delta_{w})^{-1\/2}{\\text{{\\rm div}}}_{w}f_{{\\scriptscriptstyle \\parallel}}$. This implies that the map used for solving the Neumann problem are invertible.\n\nThat $A\\in$ WP(Reg) is proved in the similar way. Then, by Theorem \\ref{th:dualityDirReg}, it follows that $A=A^* \\in$ WP(Dir).\n\\end{proof}\n\n\\subsection{Well-posedness with algebraic structure and $t$-independent coefficients}\n\n Recall that we write our coefficients $A$ as a $2 \\times 2$ block matrix. We say that it is {\\em block lower-triangular} if the upper off-diagonal block $ A_{\\perp{\\scriptscriptstyle \\parallel}}$ is 0,  and {\\em block upper-triangular} if the lower block  $A_{{\\scriptscriptstyle \\parallel}\\perp}$ is 0.\n\n\n\\begin{thm}\\label{thm:triangular} We assume that $A$ is $t$-independent and satisfies the usual degenerate boundedness and accretivity conditions.\n\\begin{itemize}\n  \\item The Neumann problem  with block lower-triangular   coefficients $A$ is well-posed.\n    \\item The regularity problem  with block upper-triangular   coefficients $A$ is well-posed. More generally, it suffices for the off-diagonal lower block of $A$ to be divergence free and have real entries.\n  \\item  The Dirichlet problem  with block lower-triangular   coefficients $A$ is well-posed. More generally, it suffices for the off-diagonal upper block to be divergence free and have real entries.\\end{itemize}\n\n\\end{thm}\n\nLet us clarify  the statements above. The off-diagonal lower block is $ A_{{\\scriptscriptstyle \\parallel}{\\scriptscriptstyle\\perp}}= (A^{\\alpha,\\beta}_{i,0})_{i=1,\\ldots, n}^{\\alpha,\\beta= 1,\\ldots, m}$. Real entries means that all these coefficients are real: it guarantees that $A$ and $$A'=A-\\begin{bmatrix} 0 & -A_{{\\scriptscriptstyle \\parallel}{\\scriptscriptstyle\\perp}}^t \\\\ A_{{\\scriptscriptstyle \\parallel}{\\scriptscriptstyle\\perp}}  & 0\\end{bmatrix}=\\begin{bmatrix} A_{{\\scriptscriptstyle\\perp}\\no} & A_{{\\scriptscriptstyle\\perp}{\\scriptscriptstyle \\parallel}}+A_{{\\scriptscriptstyle \\parallel}{\\scriptscriptstyle\\perp}}^t \\\\ 0  & A_{{\\scriptscriptstyle \\parallel}\\ta}\\end{bmatrix} $$\nhave the same accretivity  bounds. The divergence free condition is $\\sum_{i=1}^n\\partial_{i}A^{\\alpha,\\beta}_{i,0}=0$\nfor all $\\alpha,\\beta$. It implies that weak solutions with coefficients $A$ or $A'$ are the same as can be seen by integrating by parts. In other words, we can reduce matters the special case where the  off-diagonal lower block is zero\nThis possibility does not appear to be available for the Neumann problem because the conormal derivative depends on the coefficients.\n\n  The proof of this theorem is obtained by a line by line adaptation of  \\cite{AMM} to the weighted setting using well-posedness  of the three problems (modulo constants) in the class  of energy solutions, that is,  having finite energy $\\int_{\\mathbb{R}^{1+n}_+} |\\nabla u|^2 d{\\underline w}<\\infty$.  We  mention that to carry out the algebra  there, one should replace the standard Riesz transforms by the Riesz transforms ${\\mathcal R}_{w}$ defined in Lemma \\ref{lem:gradient}. We leave details to the interested reader.\n\n\n\n\n\\section{Non-tangential maximal estimates and Fatou type results}\\label{sec:NTmax}\n\n\n\nRecall that ${\\underline w}$ is the $A_{2}$ weight on ${\\mathbb R}^{1+n}$ defined by ${\\underline w}(t,x)=w(x)$ and that   ${\\underline w}$ and $w$ have identical $A_{2}$ constants.\nWriting equations ${\\text{{\\rm div}}} A \\nabla u =0$  as ${\\text{{\\rm div}}}_{{\\underline w}} (w^{-1}A\\nabla u)= 0$ allows one to carry  some proofs to the degenerate case  without much change from the non-degenerate case. We quote two results we will be using. The first is the usual Caccioppoli inequality with a completely analogous proof:\nall weak solutions $u$ in a ball  $ B=B({\\bf x},r) \\subseteq \\mathbb{R}^{1+n}_+$ of  ${\\text{{\\rm div}}} A \\nabla u=0$ enjoy the Caccioppoli inequality \\begin{equation}\n\\label{eq:caccio}\n{}\\int_{\\alpha B} |\\nabla u|^2 \\, d{\\underline w} \\le Cr^{-2} {}\\int_{\\beta B}  |u|^2 \\, d{\\underline w}\n\\end{equation}\nfor any $0<\\alpha<\\beta<1$,\n the implicit constants depending only on $\\|w^{-1}A\\|_{\\infty}$ the accretivity constant of $w^{-1}A$, $n$, $m$, $\\alpha$ and $\\beta$.\n\nThe second one is a corollary of this and Poincar\\'e inequalities: there exists $1<p_{1}<2$ such that for any $p_{1}<p<2$, all weak solutions $u$ in a ball $B=B({\\bf x},r) \\subseteq \\mathbb{R}^{1+n}_+$ of  ${\\text{{\\rm div}}} A \\nabla u=0$ enjoy the reverse H\\\"older inequality  \\begin{equation}\n\\label{eq:meyers}\n\\bigg(\\barint_{\\hspace{-6pt}\\alpha B} |\\nabla u|^2 \\, d{\\underline w}\\bigg)^{1\/2} \\lesssim \\bigg(\\barint_{\\hspace{-6pt}\\beta B}  |\\nabla u|^p \\, d{\\underline w}\\bigg)^{1\/p}\n\\end{equation}\nfor any $0<\\alpha<\\beta<1$,\n the implicit constants depending only on $\\|w^{-1}A\\|_{\\infty}$ the accretivity constant of $w^{-1}A$, $n$, $m$, $p$, $\\alpha,\\beta$ and the $A_{2}$ constant of $w$. The usual proof follows from Caccioppoli's inequality applied to $u-u_{\\beta B}$ and using the Poincar\\'e inequality with  the gradient\nin $L^p$. Here, to be able to do that we need ${\\underline w} \\in A_{2-\\varepsilon}$ for $\\varepsilon>0$ but this can be done  for some $\\varepsilon>0$ depending only in the size  of the $A_{2}$ constant of ${\\underline w}$. Then, one can use \\cite[Theorem 1.5]{FKS} which asserts that the gain of exponent in the Poincar\\'e inequality for an $A_{p}$ weight on ${\\mathbb R}^{1+n}$ is at least $p \\frac{1+n}n$. So for $p> \\frac {2n}{1+n}$  (and $p>2-\\varepsilon$) we are done.\n\n\nWe continue with the analogue of Lemma 10.3 in \\cite{AA1}.\n\n\\begin{lem}  \\label{lem:lqoffdiag}\n  Let $B$ be $t$-independent, bounded on ${\\mathcal H}$ and accretive on ${\\mathcal H}^{0}= \\clos{\\textsf{R}(D)}$. Let $T=DB$ or $BD$. Then there exists $1<p_{0}<2$, such that for $p_{0}<q<p_{0}'$,\n$$\n  \\| (I+it T)^{-1} f \\|_{q}\\le  {C_N}\\bigg(1+\\frac{\\text{{\\rm dist}}\\,(E,F)}{t}\\bigg)^{-N} \\|f\\|_{q}\n$$\nfor all   integer $N$,   $t>0$ and sets $E,F\\subseteq {\\mathbb R}^n$ such that $\\text{{\\rm supp}}\\, f\\subseteq F$, with $C_{N}$ independent of $f, t, E, F$.  Here $\\text{{\\rm dist}}\\,(E,F):= \\inf\\sett{|x-y|}{x\\in E, y\\in F}$ and $\\|\\ \\|_{q}$ are the weighted $L^q$ norms.\n\\end{lem}\n\n\\begin{proof} It suffices to prove the lemma for $T=DB$ as then it holds for $T^*=B^*D$, and hence for $BD$ upon changing $B^*$ to $B$.\n\n  For $q=2$, this is contained in  Lemma \\ref{lem:odd}.  By interpolation, it suffices to estimate the operator norm of $ (I+it DB)^{-1}$ on $L^q({\\mathbb R}^n,w;{\\mathbb C}^{m(1+n)})$, uniformly for $t$.\n\n  To this end, assume that $(I+ it DB)\\tilde f= f$.\n  As in Proposition~\\ref{prop:divformasODE}, but replacing $\\partial_t$ by $(it)^{-1}$, this equation is equivalent to\n$$\n\\begin{cases}\n  (w^{-1} A\\tilde g)_{\\scriptscriptstyle\\perp} + it{\\text{{\\rm div}}}_w(w^{-1} A\\tilde g)_{\\scriptscriptstyle \\parallel} = (w^{-1} A g)_{\\scriptscriptstyle\\perp}, \\\\\n  \\tilde g_{\\scriptscriptstyle \\parallel} - it \\nabla_x\\tilde g_{\\scriptscriptstyle\\perp} = g_{\\scriptscriptstyle \\parallel},\n\\end{cases}\n$$\nwhere $A, g$ and $ \\tilde g$ are related to $B, f $ and $ \\tilde f$ respectively, as in Proposition~\\ref{prop:divformasODE}.\nUsing the second equation to eliminate $\\tilde g_{\\scriptscriptstyle \\parallel}$ in the first, shows that $\\tilde g_{\\scriptscriptstyle\\perp}$ satisfies the divergence form\nequation\n$$\n  L_{t}\\tilde g_{\\scriptscriptstyle\\perp}:= \\begin{bmatrix} 1 & it{\\text{{\\rm div}}}_w \\end{bmatrix}\n   (w^{-1}A)\n    \\begin{bmatrix} 1 \\\\ it\\nabla_x \\end{bmatrix}\n    \\tilde g_{\\scriptscriptstyle\\perp} =\n     \\begin{bmatrix} 1 & it{\\text{{\\rm div}}}_w \\end{bmatrix}\n      \\begin{bmatrix} w^{-1}A_{{\\scriptscriptstyle\\perp}\\no} g_{\\scriptscriptstyle\\perp} \\\\ - w^{-1}A_{{\\scriptscriptstyle \\parallel}\\ta} g_{\\scriptscriptstyle \\parallel} \\end{bmatrix}.\n$$\nLet $r(w)<2$ be the infimum of those exponents $q$ for which $w\\in A_{q}$. For $r(w)<q<r(w)\n'$, $L_{t}$ is bounded from $W^{1,q}({\\mathbb R}^n,w; {\\mathbb C}^{m})$ equipped with the norm\n$\\|u\\|_{q}+ t\\|\\nabla u \\|_{q}$ into $W^{-1, q}({\\mathbb R}^n,w;{\\mathbb C}^m)=(W^{1, q'}({\\mathbb R}^n, w;{\\mathbb C}^m))'$ for the duality $\\langle \\ , \\ \\rangle_{w}$, uniformly in $t$. (Observe that the $A_{2}$ constant is unchanged by scaling, so we can change variable to reduce to $t=1$ up to changing $w(x)$ to $w(tx)$.) The accretivity condition on ${\\mathcal H}^{0}$ tells us  that\n$$\n\\re \\langle   L_{t}u,u \\rangle_{w} \\ge \\kappa (\\|u\\|_{2}^2 + t^2\\|\\nabla u \\|_{2}^2).$$\n Hence $L_{t}$ is an isomorphism for $p=2$. Using Proposition \\ref{prop:interpolation} and\nthe stability result of {\\v{S}}ne{\\u\\i}berg~\\cite{sneiberg}, it follows that\n $L_{t}$ is an isomorphism for $p_0<q<p_{0}'$ for some $r(w)<p_{0}<2$. This gives us the desired estimate in the weighted $L^q(w)$ norms,\n$$\n  \\|\\tilde f\\|_q\\approx \\|\\tilde g\\|_q \\lesssim \\|\\tilde g_{\\scriptscriptstyle\\perp}\\|_q+ t\\|\\nabla_x \\tilde g_{\\scriptscriptstyle\\perp}\\|_q + \\|g_{\\scriptscriptstyle \\parallel}\\|_q\n  \\lesssim \\|g\\|_q\\approx \\|f\\|_q.\n$$\n\\end{proof}\n\nWe state a simple but useful lemma.\n\n\\begin{lem}\\label{lem:SFEimplies0limit} Let $f$ be a function\n satisfying the square function estimate $\\int_{0}^\\infty \\|f_{t}\\|^2\\, \\frac{dt}t <\\infty$. Then the Whitney averages of $f$ converge $dw$ almost everywhere in $L^2$ sense to 0.\n\\end{lem}\n\n\\begin{proof} Recall that $c_{0},c_{1}$ are the parameters for the Whitney box $W(t,x)$. Let\n$$\nh(t,x):= \\int_{|x-y|\\le c_{1}c_{0}s, \\, s<c_{0}t}  |f(s,y)|^2\\frac{d{\\underline w}(s,y)}{s w(B(y,c_{1}s))}.\n$$\nIt is an increasing function of $t$ with\n$$\n\\int_{{\\mathbb R}^n} h(t,x) \\, dw(x) \\lesssim\n\\int_{0}^{c_{0}t} \\|f_{s}\\|^2\\, \\frac{ds}s \\to 0\n$$\nas $t\\to 0$.  Thus $h(t,x)$ converges to 0 as $t\\to 0$ for $dw$ almost every $x$. We conclude observing that\n$W_{2}f(t,x)^2 \\lesssim h(t,x)$.\n\\end{proof}\n\nWe now turn to the proof of Theorem \\ref{thm:NTmaxandaeCV}.\n\n\\begin{proof}[Proof of \\eqref{eq:Ntmax}.]  Here it remains to show $\\|\\widetilde N_*(e^{-t|T|}h) \\| \\sim \\|h\\|$ whenever $h\\in \\clos{\\textsf{R}(T)}$ and $T=DB$ or $BD$ with $B$ being $t$-independent (we drop the  subscript 0).  The bound from below is easy:\n$$\n\\|h\\|^2 = \\lim_{t\\to 0}  \\frac 1t\\int_t^{2t} \\| e^{-s|T|}h \\|^2\\,  ds \\lesssim \\| \\widetilde N_*(e^{-s|T|}h) \\|^2\n$$\nfrom Lemma \\ref{lem:XlocL2}.\n\nLet us see the converse when $T=DB$.  Split $h\\in \\clos{\\textsf{R}(DB)}={\\mathcal H}^{0}$ as $h=h^++h^-$ according to $h^\\pm= \\chi^\\pm(DB)h$. Then $e^{-t|DB|}h^\\pm$ is the $w$-normalized conormal gradient of a weak solution $u^\\pm$ in the upper-(resp. lower-)half space. Thus one can use\n\\eqref{eq:meyers} and estimate $\\widetilde N_*(e^{-t|DB|}h^\\pm)$ by $\\widetilde N_*^p(e^{-t|DB|}h^\\pm)$ for some $p<2$ which we can take larger than $p_{0}$ of the Lemma \\ref{lem:lqoffdiag}. From here follow the proof of Lemma 2.52 in \\cite{AAH}: decompose $  e^{-t|DB|}=\\psi(tDB) + (I+itDB)^{-1}$ and use  the quadratic estimate \\eqref{eq:psiT} and Lemma \\ref{lem:XlocL2} to obtain $$\n\\| \\widetilde N_*^p(\\psi(tDB)h^\\pm)\\|^2 \\lesssim \\int_{0}^\\infty \\|\\psi(tDB)h^\\pm\\|^2 \\, \\frac{dt}{t} \\lesssim \\|h^\\pm\\|^2\n$$\nand   Lemma \\ref{lem:lqoffdiag} to obtain the pointwise bound\n$$\n\\widetilde N_*^p((I+itDB)^{-1}h^\\pm) \\le C M_{w}(|h^\\pm|^p)^{1\/p}, $$\nwhere $M_{w}$ is the Hardy-Littlewood maximal operator with respect to $dw$.\n\n\n\nThe result for $T=BD$ follows from that of $DB$:  If $g\\in \\clos{\\textsf{R}(BD)}$, then $B^{-1}g=h\\in \\clos{\\textsf{R}(DB)}$  with $\\|h\\| \\sim \\|g\\|$ and  $e^{-t|BD|}g=B e^{-t|DB|}h$. Thus\n$$\n \\| \\widetilde N_*(e^{-t|BD|}g) \\| =  \\| \\widetilde N_*(B e^{-t|DB|}h) \\| \\leq \\|B\\|_{\\infty}  \\| \\widetilde N_*( e^{-t|DB|}h) \\|  \\sim \\|h\\|.\n$$\n\\end{proof}\n\n\n\\begin{proof}[Proof of \\eqref{eq:CVae}.] This time we begin with $T=BD$.  We can split $h=h_{N}+ h_{R}$ where $h_{N}\\in \\textsf{N}(BD) $ and $h_{R}\\in \\clos{\\textsf{R}(BD)}. $ As $e^{-s|BD|}h_{N}=h_{N}$,  the almost everywhere limit\n$\\lim_{t\\to 0}\\ \\barint_{\\hspace{-2pt}W(t,x_{0})} |e^{-s|BD|}h_{N}-h_{N}(x_{0})|^2\\, d{\\underline w}=0$ follows from the Lebesgue differentiation theorem.  We can thus assume that $h\\in \\clos{\\textsf{R}(BD)}$.\n\n Pick  a Lebesgue point  $x_{0}$ for the conditions\n\\begin{equation}\n\\label{eq:leb2}\n\\lim_{t\\to 0}\\ \\barint_{\\hspace{-6pt}B(x_{0},t)} |h-h(x_{0})|^2\\, dw=0= \\lim_{t\\to 0}\\ \\barint_{\\hspace{-6pt}B(x_{0},t)} |w^{-1}-w^{-1}(x_{0})|^2\\, dw.\n\\end{equation}\nThe second equality is possible  since $w^{-1} \\in L^2_{\\text{{\\rm loc}}}(w)$ as $w\\in A_{2}$.\nAs a consequence of \\eqref{eq:leb2} and \\eqref{eq:dxdwAp} with $p=2$, we have\n\\begin{equation}\n\\label{eq:leb1}\n\\lim_{t\\to 0}\\ \\barint_{\\hspace{-6pt}B(x_{0},t)} w \\, dx=w(x_{0}), \\qquad  \\lim_{t\\to 0}\\ \\barint_{\\hspace{-6pt}B(x_{0},t)} w^{-1} \\, dx = w^{-1}(x_{0})\n\\end{equation}\nand\n$$\\lim_{t\\to 0}\\ \\barint_{\\hspace{-6pt}B(x_{0},t)} |h-{(hw)(x_{0})}w^{-1}|^2\\, dw=0.\n$$\nWrite as above, $e^{-s|BD|}h=\\psi(sBD)h + (I+isBD)^{-1}h$. The quadratic estimate  \\eqref{eq:psiT}  and Lemma \\ref{lem:SFEimplies0limit} imply that\n$$\n\\lim_{t\\to 0}\\ \\barint_{\\hspace{-6pt}W(t,x_{0})} | \\psi(sBD)h|^2 \\, d{\\underline w} =0\n$$\nfor $dw$ almost every $x_{0}\\in {\\mathbb R}^n$. Now the key point is that $D\\begin{bmatrix}\n      c_{\\perp}    \\\\\n      c_{{\\scriptscriptstyle \\parallel}}w^{-1}\n\\end{bmatrix}= 0$ if $c$ is a constant, thus $(I+isBD)^{-1} \\begin{bmatrix}\n      h_{\\perp}(x_{0})    \\\\\n      (h_{{\\scriptscriptstyle \\parallel}}w)(x_{0})w^{-1}\n\\end{bmatrix}= \\begin{bmatrix}\n      h_{\\perp}(x_{0})    \\\\\n      (h_{{\\scriptscriptstyle \\parallel}}w)(x_{0})w^{-1}\n\\end{bmatrix}.$  It follows that\n\\begin{align*}\n& (I+isBD)^{-1}h -h(x_{0}) \\\\\n&\n=(I+isBD)^{-1}h- \\begin{bmatrix}\n      h_{\\perp}(x_{0})    \\\\\n      (h_{{\\scriptscriptstyle \\parallel}}w)(x_{0})w^{-1}\n\\end{bmatrix} + \\begin{bmatrix}\n      0    \\\\\n      (h_{{\\scriptscriptstyle \\parallel}}w)(x_{0})w^{-1} -h_{{\\scriptscriptstyle \\parallel}}(x_{0})\n\\end{bmatrix}\\\\\n&= (I+isBD)^{-1} \\begin{bmatrix}\n      h_{\\perp}- h_{\\perp}(x_{0})      \\\\\n     h_{{\\scriptscriptstyle \\parallel}}-  (h_{{\\scriptscriptstyle \\parallel}}w)(x_{0})w^{-1}\n\\end{bmatrix}  + (h_{{\\scriptscriptstyle \\parallel}}w)(x_{0})\\begin{bmatrix}\n      0    \\\\\n      w^{-1} -w^{-1}(x_{0})\n\\end{bmatrix}.\n\\end{align*}\nBy the assumption on  $x_{0}$,\n$$\n\\lim_{t\\to 0}\\ \\barint_{\\hspace{-6pt}W(t,x_{0})} \\bigg| (h_{{\\scriptscriptstyle \\parallel}}w)(x_{0})\\begin{bmatrix}\n      0    \\\\\n      w^{-1} -w^{-1}(x_{0})\n\\end{bmatrix} \\bigg|^2\\, d{\\underline w}=0.\n$$\nDecomposing the inner function of the other term using annuli centred around $B(x_{0},t)$ and using Lemma \\ref{lem:odd}, we have that\n$$\n \\barint_{\\hspace{-6pt}W(t,x_{0})} \\bigg | (I+isBD)^{-1} \\begin{bmatrix}\n      h_{\\perp}- h_{\\perp}(x_{0})    \\\\\n     h_{{\\scriptscriptstyle \\parallel}}-  (h_{{\\scriptscriptstyle \\parallel}}w)(x_{0})w^{-1}\n\\end{bmatrix} \\bigg|^2 \\, d{\\underline w}\n$$\nis bounded by\n\\begin{equation}\n\\label{eq:NTbound}\n \\sum_{j\\ge 1} 2^{-jN}  (w(B(x_{0},t)))^{-1} \\int_{B(x_{0}, 2^j t)} \\bigg |  \\begin{bmatrix}\n      h_{\\perp}- h_{\\perp}(x_{0})    \\\\\n     h_{{\\scriptscriptstyle \\parallel}}-  (h_{{\\scriptscriptstyle \\parallel}}w)(x_{0})w^{-1}\n\\end{bmatrix} \\bigg|^2 \\, dw.\n\\end{equation}\nThe scalar term is bounded by\n\\begin{align*}\n&\\sum_{j\\ge 1} 2^{-j(N-d_{w})}   \\barint_{\\hspace{-6pt}B(x_{0}, 2^j t)}  |\n      h_{\\perp}- h_{\\perp}(x_{0})     |^2 \\, dw\n      \\\\\n      \\lesssim &\n      \\sup_{\\tau\\le \\sqrt t }\\  \\barint_{\\hspace{-6pt}B(x_{0}, \\tau)}  |        h_{\\perp}- h_{\\perp}(x_{0})\n     |^2 \\, dw + \\sqrt t M_{w}(|h|^2)(x_{0})\n \\end{align*}\nas can be seen using the doubling condition on $w$ ($d_{w}$ being its homogeneous dimension) and\nbreaking the sum up at $j_{0}$ with $2^{j_{0}}\\sim 1\/ \\sqrt t$ if $N\\ge d_{w}+1$,\nwhere $M_{w}$ is the Hardy-Littlewood maximal operator with respect to $dw$. The weak type (1,1) estimate of $M_{w}$ implies that  $M_{w}(|h|^2)(x_{0})<\\infty$ for almost every $x_{0}\\in {\\mathbb R}^n$.  Hence,  the scalar term  goes to 0  as $t\\to 0$ at those $x_{0}$ which meet all these requirements.\nThe tangential term  in \\eqref{eq:NTbound} is bounded by\n\\begin{align*}\n&\\sum_{j\\ge 1} 2^{-j(N-d_{w^{-1}})}   \\barint_{\\hspace{-6pt}B(x_{0}, 2^j t)}  |\n      h_{{\\scriptscriptstyle \\parallel}}- h_{{\\scriptscriptstyle \\parallel}}(x_{0})     |^2 \\, dw\n      \\\\ + &  \\sum_{j\\ge 1} 2^{-jN} |h_{{\\scriptscriptstyle \\parallel}}(x_{0})|^2   (w(B(x_{0},t)))^{-1} \\int_{B(x_{0}, 2^j t)} |w^{-1}(x_{0})-w^{-1}|^2\\,dw.\n \\end{align*}\n The first sum can be treated as above. For the second when  $j\\le j_{0}$, we also do as above. For $j\\ge j_{0}$, we write $|w^{-1}(x_{0})-w^{-1}|^2 \\le 2 w^{-2}(x_{0}) + 2w^{-2}$. The first term leads to a bound $\\sqrt t  |h_{{\\scriptscriptstyle \\parallel}}(x_{0})|^2 w^{-2}(x_{0})$ if $N\\ge d_{w}+1$. For the second term,\nobserve that\n $$\\int_{B(x_{0}, 2^j t)} w^{-2}dw= w^{-1}(B(x_{0}, 2^j t)) \\lesssim 2^{jd_{w^{-1}}} w^{-1}(B(x_{0},  t))$$ and  that  by \\eqref{eq:leb1}$$(w(B(x_{0},t)))^{-1}w^{-1}(B(x_{0},  t)) \\to 1\/w^2(x_{0}), \\quad t\\to 0.$$\n Thus, if also $N\\ge d_{w^{-1}}+1$, then the tangential term in \\eqref{eq:NTbound} tends to 0.\n\nWe now turn to the proof for $T=DB$.  If $g\\in \\textsf{N}(DB)$, \\eqref{eq:CVae} is a consequence of the Lebesgue differentiation theorem for the measure $dw$ on ${\\mathbb R}^n$ as $e^{-s|DB|}g=g$ for all $s$.\nNow consider $g\\in \\clos{\\textsf{R}(DB)}$. As\n$$\n\\lim_{t\\to 0}\\ \\barint_{\\hspace{-6pt}W(t,x_{0})} |g-g(x_{0})|^2\\, d{\\underline w}= \\lim_{t\\to 0}\\ \\barint_{\\hspace{-6pt}B(x_{0},c_{1}t_{0})} |g-g(x_{0})|^2\\, dw= 0$$\nfor almost every $x_{0}\\in {\\mathbb R}^n$,\n it is enough to show the almost everywhere limit\n$$\n\\lim_{t\\to 0}\\ \\barint_{\\hspace{-6pt}W(t,x_{0})} |e^{-s|DB|}g-g|^2\\, d{\\underline w}=0.\n$$\nWrite\n$e^{-s|DB|}g-g= \\psi(sDB)g + (I+isDB)^{-1}g-g$. The quadratic estimate  \\eqref{eq:psiT} and Lemma \\ref{lem:SFEimplies0limit} imply that\n$$\n\\lim_{t\\to 0}\\ \\barint_{\\hspace{-6pt}W(t,x_{0})} | \\psi(sDB)g|^2 \\, d{\\underline w} =0\n$$\nfor almost every $x_{0}\\in {\\mathbb R}^n$. Now $(I+isDB)^{-1}g-g= -isDh_{s}$ with $ h_{s}=B(I+isDB)^{-1}g= (I+isBD)^{-1}(Bg)$ and $Bg\\in {\\mathcal H}$. Let\n $$\\tilde h_{s}:= (I+isBD)^{-1}(Bg)- \\begin{bmatrix}\n      (Bg)_{\\perp}(x_{0})    \\\\\n      ((Bg)_{{\\scriptscriptstyle \\parallel}}w)(x_{0})w^{-1}\n\\end{bmatrix}.$$ As above we have that $Dh_{s}=D\\tilde h_{s}$. We now apply the following local coercivity inequality on ${\\mathbb R}^n$: for any $u\\in {\\mathcal H}_{\\text{{\\rm loc}}}$ with $Du\\in {\\mathcal H}_{loc}$ and any ball $B(x,r)$ in ${\\mathbb R}^n$,\n\\begin{equation}\n\\label{eq:localcoerc}\n\\int_{B(x,r)} |Du|^2\\, dw \\lesssim \\int_{B(x,2r)} |BDu|^2\\, dw + r^{-2} \\int_{B(x,2r)} | u|^2\\, dw,\n\\end{equation}\nwhere the implicit constant depends only on the ellipticity constants of $B$, the dimension and $m$. (Of course, if $B^{-1}$ is a multiplication operator, this inequality improves considerably.)\n\nApplying this inequality  to $u=\\tilde h_{s}$, using $Dh_{s}=D\\tilde h_{s}$ and integrating with respect to $s$ implies\n\\begin{align*}\n\\label{}\n\\barint_{\\hspace{-6pt}W(t,x_{0})} |isDh_{s}|^2\\, d{\\underline w}    & \\lesssim  \\barint_{\\hspace{-6pt}\\widetilde W(t,x_{0})} |isBDh_{s}|^2\\, d{\\underline w}  +  \\barint_{\\hspace{-6pt}\\widetilde W(t,x_{0})} |\\tilde h_{s}|^2\\, d{\\underline w}  \\\\\n    & \\lesssim  \\barint_{\\hspace{-6pt}\\widetilde W(t,x_{0})} |(I+isBD)^{-1}(Bg) -Bg|^2\\, d{\\underline w}\n    \\\\\n  &  \\qquad +  \\barint_{\\hspace{-6pt}\\widetilde W(t,x_{0})} \\bigg|Bg- \\begin{bmatrix}\n      (Bg)_{\\perp}(x_{0})    \\\\\n      ((Bg)_{{\\scriptscriptstyle \\parallel}}w)(x_{0})w^{-1}\n\\end{bmatrix}\\bigg|^2\\, d{\\underline w},\n\\end{align*}\nwhere $\\widetilde W(t,x_{0})$ is a slightly expanded version of $W(t,x_{0})$ and, in the last inequality, we have written\n$$\\tilde h_{s}= (I+isBD)^{-1}(Bg) -Bg + Bg- \\begin{bmatrix}\n      (Bg)_{\\perp}(x_{0})    \\\\\n      ((Bg)_{{\\scriptscriptstyle \\parallel}}w)(x_{0})w^{-1}\n\\end{bmatrix}.\n$$\nThe  last two integrals have been shown to converge to $0$ as $t\\to 0$ for almost every $x_{0}\\in {\\mathbb R}^n$ in the argument for $BD$.  It only remains to prove   \\eqref{eq:localcoerc}.\n\nFor this inequality, we let $\\chi$ be a smooth scalar-valued function with $\\chi=1$ on $B(x,r)$, $\\chi$ supported in $B(x,2r)$  and  $|\\nabla \\chi|\\lesssim r^{-1}$.  Using the commutator identity \\eqref{eq:comm} between $\\chi$ and $D$, we have\n$$\n\\int_{B(x,r)} |Du|^2\\,  dw  \\leq \\int  |\\chi Du|^2\\,  dw  \\lesssim  \\int  |D(\\chi u)|^2\\,  dw + \\int  |\\nabla\\chi |^2| u|^2\\,  dw.\n$$\nSince $B$ is accretive on $\\textsf{R}(D)$ and  $\\chi u \\in \\textsf{D}(D)$, we have $ \\int  |D(\\chi u)|^2\\,  dw \\lesssim  \\int  |BD(\\chi u)|^2\\,  dw$. Now, we use again the commutation between $\\chi$ and $D$ together with $\\|B\\|_{\\infty}$. This proves \\eqref{eq:localcoerc}.\\end{proof}\n\n\n\n\\bibliographystyle{acm}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nIntersession Network Coding (NC) enables the local processing and mixing of independent traffic flows. \n Combining such flows\nat wireless routers  can increase the available capacity \\cite{ahlswede, proutiere, cope}.   \nHowever, \n such increase is \nevident only when:   \n(a) routers (which perform the encoding operations) are able to quickly identify efficient coding opportunities that increase the NC gain;   \n(b) packet decoders are able to correctly decipher the encoded packets and acknowledge the decoded packets that they receive in diverse channel conditions;  \n and \n(c) the overheads imposed due to the inclusion of additional packet headers\nas well as packet processing operations \\cite{memorycopy} are kept minimal. \n While NC can increase the router throughput in random-access networks \\cite{ieee80211},  \nprior studies have shown that \n  when the packets are scheduled carelessly, the benefit is reduced \\cite{proutiere, nococo}. \n  With multi-rate links, and when\n  decisions are made based on statistical information,  \n  scheduling is necessary to avoid \n  packet\n  losses. \nAll these factors should be taken into account when designing and developing practical \nNC algorithms and systems. \n\n{\\bf Prior  NC systems\n do not consider such effects:}\nAlthough intersession NC can theoretically offer unprecedented wireless router capacity benefits, realizing these benefits in practical systems is by no means an easy task.\nPrior implemented efforts\ntowards practical NC \\cite{cope,er,clone,more}, have demonstrated\n throughput benefits at low transmission rates but have also \ndiscovered a series of complexity issues arising in such implementations. \n Our measurements across different  testbeds suggest that \n it is very difficult for such NC\n implementations\n\n to deliver the expected throughput gains\n  in practical multi-rate\n deployments.\n This is due to \n  two reasons, which motivate our study. We explain them below. \n\n{\\bf \\em a. Overhead intensive design:} \n  With NC,  routers \n need to be\n aware of the packets that have been successfully overheard by each neighbor,\nin order to \ndecide\nwhich packets to encode together\nand when.   \nA method that has been commonly adopted\nfor addressing this requirement,\n is by enforcing every neighbor into  explicitly acknowledging \n   overheard packets. \n However, in certain practical settings this approach may perform poorly. \nThis is because:\n (a) \nthe timeliness of information can be severely impaired, as\nit depends on random medium access;\n and \n (b) \n additional packet processing needs to be performed,  \n \n which intrusively increases the already imposed processing overhead by NC.\n Due to such operations, routers become overloaded. \n Thus,\n although the channel may be conducive to the use of high bit rates, routers may be incapable of transmitting as many packets in order to meet those rates. \nOur experiments across two different \ntestbed\n (with various approaches such as \\cite{cope} and  \\cite{er} running on 1 GHz CPU, 1 GB RAM devices \\cite{orbit}) suggest that in the simple scenario where Alice and Bob exchange packets through an \nintermediate relay \nwith all links having a PDR (Packet Delivery Ratio) close to 1, the observed CPU utilization was at 100\\% in 802.11g when bit rates of 36 Mbps or above were used.  \nMoreover, the maximum\nrouter throughput for Alice's and Bob's flows  \nwas apprx. 6.5 Mbps on average, compared to apprx. 8 Mbps of the pure IEEE 802.11 protocol \\cite{ieee80211}. \n These\n measurements suggest that  \n  with such\n\n design choices,\n  the benefits due to NC \ncannot outweigh the performance degradation due to\nexcessive imposed overheads. \n\n{\\bf \\em b. Absence of scheduling techniques:} \n Previous studies have demonstrated the benefits of jointly applying NC and link scheduling  \\cite{proutiere, nococo}.\n Prior practical implementations  have not incorporated any such techniques, \n while they have not been designed to \n\n host scheduling ideas. \nThis necessitates the design of a broad, although lightweight framework, which can facilitate the efficient coexistence of NC and scheduling. \n\n{\\bf Our contributions:} \n We present the design and implementation of  NCRAWL, our Network Coding framework for Rate Adaptive Wireless Links.  \n  NCRAWL   %\n has been optimized at each stage of NC operations \n %\n It is a modular tool, which can easily host the  \n\n implementation of\n intersession NC schemes that are either standalone, \nor tightly integrated with scheduling algorithms.\nMore specifically, \n our framework differs from other related systems in the following aspects:\n\\begin{itemize}\n\\item {\\bf  \\em NCRAWL is modular:}\n\n Algorithms can be easily developed as extensions to        \n  NCRAWL modules.   \n These modules manage all the  NC operations, such as encoding, decoding, storage and routing\n\n in a lightweight manner, which allows for overhead-limited\n network operations. \n\\item {\\bf  \\em NCRAWL realizes joint NC and scheduling:} \nOur framework is the first to facilitate the practical coexistence of NC and scheduling. \nTheoretically shown throughput benefits can be easily assessed on NCRAWL and adapted for operating on real networks with limited effort. \n\\item {\\bf  \\em NCRAWL uses solely stochastic information for overhearing:} \nWith this we\navoid many\noverhead intensive\nprocessing operations. \nThe practical integration of NC with scheduling\n provides a well performing \n lightweight solution. \n\\item {\\bf  \\em NCRAWL is channel aware:} \n Routers are aware of all the potential NC opportunities that can take place within their neighborhoods at all times, as well as \n\n the maximum transmission rate of the encoded packets that allows for decoding.\nRouters can also quickly determine which packets should be encoded together to offer the highest performance benefit. \n\n\\end{itemize}\nWe implement NCRAWL on\n {\\em Click}\n as a Linux kernel module \\cite{clicksite}. \nWe evaluate NCRAWL on a wireless testbed \n through measurements with various indoor and outdoor topological settings.\nOur experiments demonstrate that NCRAWL \nidentifies efficient\nNC opportunities;\nit offers significant  throughput improvements even at high bit rate regimes, where\n prior schemes are\n unable to operate, due to the imposed overheads.\n\n\n\\textbf{The scope of our work:}\n Our focus is not on  \n proposing optimal scheduling policies for\nNC, but on developing an accurate, lightweight and easy-to-deploy system that can host NC and\/or scheduling schemes. %\nNCRAWL can be applied on \n routers,\n keeping their functionality simple and fast,\n given that they need to process and route many thousands of packets per second. \nWe provide a practical and realistic design, by considering cases where encoded packets are decoded at the next hop.  \n\nThe rest of the paper is structured as follows.\nIn section \\ref{sec:background} we discuss relevant previous studies. \n In section  \\ref{sec:nc_scheme} we provide a high level overview of the considered \n NC scheme.\nIn section \\ref{sec:arch} we present the modular design and implementation of NCRAWL.\nIn section \\ref{sec:case} we demonstrate the strengths of our\n design through a scheduling-driven case study. \nIn section \\ref{sec:measurements}  we assess the performance of  NCRAWL  \n \n  via extensive measurements. %\n Our conclusions form  \n section \\ref{sec:conclusions}. \n\\section{Related work}\n\\label{sec:background}\n\\setcounter{paragraph}{0}\nIn this section we discuss previous related NC studies and differentiate our work. \n\n{\\bf Experimental studies on wireless coding:} \nKatti et al. \\cite{cope} propose COPE, the seminal implementation of wireless NC.\nWith COPE, \n\n  routers are fully aware of packets that have been overheard\n  by every neighbor. \nFor this, each node is required to inform the router about overheard\npackets.\nExperiments with COPE on a wireless testbed show that even with very simple encoding operations, intersession NC\ncan provide significant capacity gains. \nThey also study the interactions of network coding with the routing and the higher layer protocols. \nWe provide more details about COPE later in this paper, where we differentiate our design and implementation.\nRozner et al. in \\cite{er} present ER, a scheme that adopts the design of COPE and employs NC to perform efficient packet retransmissions. \nWith ER, packets that need to be retransmitted are coded together, such that one retransmission can recover multiple packet losses. The authors show that the problem of selecting the optimal set of packets to code together is NP-hard; \nthey propose a set of heuristics that can be followed to make coding decisions. \nKim et al. \\cite{tskim} extend the design of COPE to include NC-aware bit rate control and clever selection of nodes that acknowledge the reception of encoded packets. \nRayanchu et al. \\cite{clone} propose CLONE, a suit of algorithms for NC that take into account    losses on wireless links. However,  \\cite{er}, \\cite{tskim} and \\cite{clone} all follow COPE's logic regarding the dissemination of information about which packets have been stored as keys. Moreover, these studies do not make online decisions about whether to enable coding or not, based on the link quality. \nMORE \\cite{more} is a routing protocol\nwhere routers perform random mixing of packets\nbefore forwarding them. \n Routers that overhear the same transmission may decide not to forward the same packets. \n Sources keep sending linear combinations of a batch of packets, until receiving an ACK from the destination.\nMIXIT \\cite{mixit} encodes symbols rather than packets. \nSimilarly to MORE, batches of packets are coded together. However, since a packet is a sequence of symbols, \n Intermediate relays use hints from the PHY layer in order to infer which symbols within a packet are\ncorrectly received with high probability.\nRelays choose a vector of coefficients at random and encode packets symbol by symbol, using only the {\\em clean} symbols at a certain position. \n\nThe above experimental approaches differ from ours in that we use solely stochastic information for overhearing instead of acknowledging each particular packet.\nThis allows for efficient implementation\nand avoids\ncomputationally expensive packet processing operations. \nOur work offers a valuable\ntool for studying problems regarding joint NC and scheduling with feedback, and \n\n  can potentially be intertwined with an optimal algorithm to provide the best solution within the class of implicit ACK schemes. \n  %\n  %\n  %\n We showcase the operation of NCRAWL with various heuristic lightweight  algorithms.\nAs we show  in  section \\ref{sec:measurements}, there are cases of implicit ACK schemes where NC incurs throughput loss, unless careful scheduling is used.   Making decisions based on multiple rates is another important innovation of our framework. \n To the best of our knowledge, our work is the first to provide a coherent, lightweight framework for practically assessing joint NC and scheduling schemes. \n\n{\\bf Analytical and simulation NC studies:} \nChaporkar and Proutiere \\cite{proutiere} show that systems with NC may actually have smaller throughputs than if coding is not applied. \nThey show that unless appropriate scheduling is applied, NC may lead to performance degradation.\n We support this claim by identifying an additional example when implicit ACKs are used. In the same work, \n Chaporkar and Proutiere propose a generic framework that characterizes the throughput region with NC and enables the design of adaptive joint NC and scheduling schemes. \nFinally, \nthey propose XOR-Sym, a computationally simple \nNC scheme that can be applied to symmetric routes. \nWith XOR-Sym, packets are decoded at their destinations only and not at intermediate nodes along a path. This protocol\nconsiders only symmetric flows disregarding opportunistic listening. \nOn the contrary we focus on exploiting opportunistic listening. \nLiu and Xue in \\cite{chliu} consider NC for two-way relaying in a three-node network. They analytically characterize the achievable rate regions for the traditional Alice-Relay-Bob topology, and they find the theoretically optimal end-to-end sum rates. \nScheuermann et al. \\cite{nococo} propose noCoCo, a deterministic packet scheduling scheme for NC within two-way multihop traffic flows. \nTheir scheme involves per-hop packet scheduling, NC and congestion control. \nSeferoglu and Markopoulou in \\cite{athina_distr_video}  provide an understanding of the interplay between application data rate control and NC. \nFinally, Vieira et al. \\cite{mario_globe07}\nprovide observations on\thow the combination of NC and bit rate diversity affects the performance of practical broadcasting protocols.\nThey show that it is possible for multi-rate link layer broadcasts and NC to jointly increase the network throughput in multicast applications. \n\nMore theoretical results can be found in \\cite{ncpage} with the list being non-exhaustive.\n\\section{Network coding scheme}\n\\label{sec:nc_scheme}\n\nWe study the generalized $N$--wheel topology having $N+1$ nodes as shown in figure \\ref{figure:wheel}. The central node, called the \\emph{relay}\n (or router),  \nis connected to all other $N$ nodes, called \\emph{neighbors}. Links between neighbors may exist as well. Each link connecting nodes $i$ and $j$ is characterized by two channel rates $r_{ij}$, $r_{ji}$ and two probabilities $q_{ij}$, $q_{ji}$ which correspond to the packet delivery ratios in each direction. \n\nIn the above topology we focus only on 2--hop flows having neighbors as source and destination and the relay as the intermediate node. In the uplink part (the first hop of these flows), the packets are transmitted without NC towards the relay. In the downlink part, the relay selects a number of packets, applies the XOR operator and transmits the encoded packet. If a receiver recognizes its address in the header, it attempts to decode the packet in order to obtain the intended packet. To achieve this, it should have all the other packets in its buffer, in order to apply the XOR operator again. These packets are known to the receiver either because they have been generated by it (in case of symmetric flows) or they have been obtained by means of opportunistic listening, as explained above. Whenever the packet is successfully decoded, an acknowledgment message is sent back to the relay at layer-3.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=2in]{figures\/wheel.pdf}\n\\caption{The $\\frac{x}{2}$--wheel setting, with $x+1$ nodes and $\\frac{x}{2}$ activated flows.}\n\\label{figure:wheel}\n\\end{figure}\n\nComing back to the $N$--wheel topology, in order to experiment on high gain topologies, we can impose the extra constraint that $N$ symmetric flows are defined by splitting the neighbors in two equal sets, the source and the destination set, selecting a matching of these two sets which corresponds to $\\frac{N}{2}$ flows, and finally create another  $\\frac{N}{2}$ flows by inverting the roles of source and destination. If only the initial $\\frac{N}{2}$ flows are enabled, we refer to $\\frac{x}{2}$--wheel setting else if all $N$ flows are enabled, we refer to $x$--wheel setting. Evidently, $2$--wheel corresponds to the well known Alice-Relay-Bob topology. Throughout the paper, we also use $\\frac{4}{2}$--wheel referred to as half-cross and $4$--wheel referred to as cross, as well as a \n$6$--wheel where we activate the flows one after the other. It should be noted that NCRAWL supports any random subgraph of the $N$--wheel topology with any possible set of flows activated on top of it. In addition, it supports settings with any possible combination of link qualities and\/or channel rates. \nThe wheel topology is the most general topology to be considered around a single node. Any actual network topology can be reduced to a wheel topology if nodes and links irrelevant to NC on a relay are removed. Since our scheme runs on all nodes in the network, this implies that our scheme works with any arbitrary network topology.\nThe encoding opportunities at each node are automatically discovered by the combination of NCRAWL and SRCR. Thus NCRAWL operates under any assumed graph providing opportunistically throughput gains. We study the wheel setting where the maximum such gain arises in order to showcase that NCRAWL can achieve it in many cases.\n\nWe have incorporated the \n\n following features in order to enhance the practicality of NCRAWL. \n As a tradeoff,  such features may limit the performance of our framework against the maximum theoretical performance of intersession coding. \n\n\\begin{itemize}\n\\item The XOR operator is used instead of linear coding. It is known that the capacity region of XOR NC schemes is a subset of the one achieved with linear coding.\n\t\\item We\n  enforce the decoding of encoded packets at the next hop, since this is practically the most possible case in today's wireless access deployments. \n\t\\item Only native packets are allowed to be stored as keys. This might reduce the capacity region as well.\n\t\t\\item We use implicit ACKs of overheard packets. The capacity region is reduced in comparison to explicit ACKs whenever the overhearing probabilities are small.\n\t\n\\end{itemize}\nThe decision on imposing these features is  justified by the necessity to keep the NC scheme simple, practical, implementable and efficient in terms of processing overhead.\n\\section{Architectural Blueprint}\n\\label{sec:arch}\n\nIn this section, we describe the modular design and implementation of NCRAWL. \n\n{\\bf Employing Click as the basis of our framework:} \nThe main NCRAWL system has been developed in the \\emph{Click} modular\nrouter framework \\cite{clicksite}. Click can be used to develop primarily OSI layer 3 \npacket processors, which can be directly deployed inside the standard Linux network stack. \nA {Click} processor is mainly comprised of (a) processing stages which are called \n\\emph{elements} and (b) an \nelement interconnection configuration that indicates the processing flow. \nExecution in {Click} is event-driven with 4 different types of asynchronous events, namely the {\\em incoming packet} event, the {\\em ready-to-forward packet} event, the {\\em timer expire} event and the {\\em external read or write} events to {Click} memory. The first two events require  \n\n  some handling code \nto deal with network packets;\n this is not always necessary for serving the rest of the event types. \nSince all {Click} events are asynchronous, a {Click} packet processor typically\n features internal queues to temporarily store\n incoming \n packets.\n\nIn what follows,\n we describe the NCRAWL system design\nand implementation. \nWe also present the NCRAWL\ninterface that can be used to\n develop new algorithms, as well as for\n deploying and managing\n experiments on   wireless testbeds. \n\n\\subsection {Design preliminaries}\n\nNCRAWL realizes\nan OSI layer 2.5 protocol that lies immediately under\nthe\nrouting layer. More specifically, it can be considered \nas an extension to the {Click} modular router implementation of the\nSRCR   protocol \\cite{roofnet_wiki}, \nwhich  is the heart of the {MIT Roofnet}\n wireless\n network. \nSince NCRAWL operates below the routing layer, encoded packets\nare not forwarded by a\n node-relay; they are\ndecoded at the next hop.\nThis\nsimplifies the format of the NCRAWL packet headers:  they are now used only\nto encapsulate encoded packets and transfer the \nrespective acknowledgments for\n successful decoding\nback to the sender.\n Both\ntypes\nof the NCRAWL header format are depicted in figure~\\ref{NCRAWL:headers}.\nNetwork wide unique 32-bit packet identifiers are made by applying the \n\\emph{sdbm} \\cite{sdbm} hashing algorithm on data tuples, comprised of \npacket source IP, the IP header sequence number and the respective offset. \n\n\\begin{figure}[t]\n\\centerline{ \\includegraphics[width=4in]{figures\/WiNCS_headers.pdf}}\n\\caption{NCRAWL header format. The Ethernet header magic number distinguishes\nbetween encoded packets and ACK headers which are otherwise identical.}\n\\label{NCRAWL:headers}\n\\end{figure}\n\n\\subsection {NCRAWL System Description}\n\n Next, we discuss the modular design of NCRAWL in detail. \n\n{\\bf The big picture:} \nThe main NCRAWL system is a Click network packet processor that includes the {SRCR}\nrouting protocol implementation for wireless networks \\cite{roofnet_wiki}.   We have included two additional\nprocessing stages:\n\n the NCRAWL decoder and the NCRAWL encoder.\n We have developed these stages\n as individual {Click} {elements}, and we have placed them\nbefore the beginning and after the end of the {SRCR} processing\nflow, respectively, as depicted in figure \\ref{NCRAWL:system}. \n\n\\begin{figure}[t]\n\\centerline{ \\includegraphics[width=3.7in]{figures\/system2.pdf}}\n\\caption{NCRAWL framework system components and their interaction}\n\\label{NCRAWL:system}\n\\end{figure}\n\n{\\bf a. The packet decoder:} \n The main tasks of the decoder module are the following: \n \\begin{itemize} \n\\item To use the available (from overhearing or ownership) key packets in order to decode the received encoded packet. \n\\item To schedule the transmission of layer-3 acknowledgments (ACKs) for the correctly retrieved native packets,\n derived by the decoding operation.\n\\item To determine any potential pending acknowledgments, as well as to verify any received acknowledgments.\n\\item To tag and  store all the correctly overheard data packets as potential keys; as discussed above, these will be potentially used in the near future for decoding received encoded packets. Moreover the\nkey repository is used for packet retransmissions,  in case an\n expected \n acknowledgment never arrives.\n\\end{itemize}\nThe   decoder\n resides at the packet receiving side of the system and\n is invoked by the corresponding packet arrival event. \n\n{\\bf b. The NC packet encoder:} \n The NCRAWL encoder element resides at the sending side of the system and is more\ncomplicated, since\n it maintains\n and manages the processor packet queues.\n A  part of the element handles incoming packet events, another part deals with outgoing\npacket events, and there is also code that gets invoked upon timer expiry \nas well as read and write\n{Click} configuration events (figure \\ref{NCRAWL:system}). It is this element that exports the framework {API} \nwhich can be used to develop NC algorithms.\nSpecifically, the main assigned tasks for this module are the following: \n\\begin{itemize}\n\\item \nTo process and place\nincoming native packets (keys) into particular\n\n  maintained queues.\nOur system supports a plurality of queueing operations, which can be configured \nas per the requirements of the {NC} algorithm under development.\n\\item\nTo identify and combine packets together, towards forming encoded packets.\nThe selection of the appropriate packet set follows the directions of the NC algorithm under consideration, supported by NCRAWL. \n\\item \nTo piggyback any acknowledgments  \n(through the use of scheduled, upcoming data packet transmissions) \nthat have been scheduled by the decoder element.\n\\item\nTo generate   potentially \nexpected acknowledgment tokens for each   of the packets of an encoded combination.\n\\end{itemize}\n\n{\\bf c. Maintaining up-to-date topological information:}\nThe \\emph{link metrics updater} is responsible for collecting information about the\n current neighbors \n\n  as well as the corresponding link transmission rates and PDR values.\n  This  information is gathered and passed to the rest of the system via the {Click} memory write event mechanism. Furthermore, the code that configures the encoding combination policies is invoked \n as needed. \n\n{\\bf \\em Gathering link quality information:} \nThe {NCRAWL} updater relies on the existing {SRCR} protocol component, which maintains \n\n link connectivity information and performs periodic measurements on all links. {SRCR} sends probe packets at all rates to determine the PDR for each link and chooses the highest rate that performs well. PDR information is then used by {SRCR} to calculate the {ETX}  or ETT metric \\cite{etx, ett}, which\n provides information about entire routing paths (not just 1-hop links).  \n  This information is kept in the {SRCR} link table, and is accessible by our\n   {Click} components. The {SRCR} measurement period can be set as desired  (the default value, also used in our work, is 3 sec). \n\n{\\bf \\em Managing neighbor information:} \nBased on the information stored in the {SRCR} link table, the\n link updater maintains\n  its own so-called {Neighbor Table} (NT), which includes information for its\n  neighbors. Initially, the NT is empty. The updater periodically reads the {SRCR} link table \n \n  and updates NT as needed.\n   The  NT contents are updated\n   whenever (i) a new neighbor appears,   (ii) an existing neighbor disappears, or (iii) a certain link quality changes. \n   In such cases, the {NCRAWL} updater broadcasts a packet with the new NT contents and sets a timer. When such a NT packet is received (overheard), the updater replies by broadcasting its NT, provided it has not done so recently. The reply suppression threshold is set equal to the {SRCR} period.\nThe NT packets are used by the {NCRAWL} updater to maintain the so-called {Received NT Table} (RNTT). This table complements the NT, holding information about the link quality as  \n\n   measured by the neighbor nodes \\emph{themselves}. \n\n  When an NT packet arrives, the corresponding RNTT entry is updated. \n   Packets from nodes that are not in the NT are ignored; a node must be ``officially'' reported as a direct neighbor by {SRCR} in order to be considered by {NCRAWL}.\n\n{\\bf \\em Feeding NC algorithms with updated topological information:}  \nEach time the updater modifies the contents of the NT or RNTT (i.e., each time it proactively sends   \n\n or receives a NT packet which leads to an update of the RNTT) a timer is set. \n \n  Upon timer expiry,   \n  the new link qualities are passed to the main {NCRAWL} system,  \nwhere they will potentially drive adaptive NC decisions, based on the NC designer's needs. \nThis timeout is (generously) set to 1 sec, providing ample time for any NT reply packets to arrive. \n\n{\\bf \\em Keeping overheads low:} \nThe NCRAWL updater employs its own threads of execution to perform these information maintenance tasks (the current implementation uses 2 threads), but these remain suspended most of the time, making this component quite unintrusive in terms of CPU occupancy. Moreover, only a small fraction of the wireless bandwidth is typically used to collect the required link quality information from the neighboring nodes. Finally, the ``reactiveness'' of the updater is a function of the {SRCR} period. If a smaller period is used, link changes can be tracked faster (and more accurately) but the processing and communication overhead will increase too. \nNote that the implementation of the {NCRAWL} updater \n \n \n  can be trivially adjusted to cooperate with other link information gathering protocols as well,  \n  i.e., it is not tied to SRCR. \n\n{\\bf d. NCRAWL logger:}  \n The  {\\em read} events are used by another application, the\nNCRAWL {\\em logger}, which gathers\nvarious statistics that are generated online by both the encoder and decoder. \n\n\n{\\bf e. NCRAWL acknowledgments:} \n\n NCRAWL acknowledges individual (native) data\npackets, but also groups packet acknowledgments per encoded combination. \n With this, if the same encoded packet has been successfully decoded at one recipient but failed at\nanother, the sender can figure out which of the undelivered packets can be reused in encoded\ncombinations, based on whether they have been logged successfully as keys by fellow recipient\nnodes.\n  Note that NCRAWL provides this support; however,  \n it expects that the user algorithm will make the final scheduling decisions. \nNCRAWL uses by default a user defined timeout threshold to resend \npackets that have not been acknowledged. Note also that \n\na timer-expire event triggers the transmission of acknowledgments in separate packets when there is not\nenough outgoing traffic to piggyback them (figure \\ref{NCRAWL:system}). NCRAWL also\nreschedules packets from the key repository for which acknowledgments have not arrived.\n\n\n{\\bf \\em Utilizing resources effectively:} \n Efficient resource utilization \n\n is an inherent property of NCRAWL.  \n\n  The repository that stores copies of packets\nuses a FIFO queue as the main indexing mechanism and can host up to a user defined quantity.\nAfter the storage limit is reached, the oldest packet is removed in order for a new one\nto get stored. The same packets are also indexed in a hash table based on their network-wide\nunique identifiers, as\n  we \n  previously discussed. The hash table is used to quickly\nretrieve packets either as keys for decoding, or for resending them in case an expected\nACK token expires. The same indexing approach has been used for the\nACKs and expected ACK tokens as well.\n\n\\subsection{Implementing {NC} algorithms} \n\n  NCRAWL exports an API (Application Programming Interface) that can be used to implement scheduling algorithms for intersession NC.\n  This API is\n  a library of functions that can\nbe used to carry out NCRAWL\ncommon tasks and mandatory function extensions to the handling code of each event. Points of extensibility and\/or programmability are denoted in figure \\ref{NCRAWL:system} with shadowed   boxes.\n\n{\\bf Implementing packet handling operations:} \n Regarding \nthe incoming packet event,\n the designer should account for placing arriving packets \ninto  proper queues. \nIn particular, \n each flow is associated with a queue and the scheduler checks all available controls, i.e. activates  a set of queues by combining one packet from each queue. It maintains a list with the expected score (reward) of all controls and selects one of those controls at each time instance. The controls that activate only one queue correspond to the case of transmitting non coded packets. It is always possible to deactivate NC by imposing the use of only those latter controls. \n With this,  \n the developer may implement logic that disables NCRAWL \nwhen needed. Furthermore, after the placement of the incoming packet, the developer may: \n (a) invoke the function that chooses the next encoded packet queue combination according \nto the NC algorithm, \n (b) retrieve the packets from the respective queues, \n (c) encode them and schedule the encoded packet for transmission by placing it into the outgoing queue. \n This operation may be repeated until the outgoing queue contains a user\ndefined number of packets. Apart from the queue combination retrieval \nfunction, which needs to be implemented by the algorithm developer, the rest \nof the required functionality is already seamlessly provided by NCRAWL.\n\n{\\bf Implementing the core NC logic:} \nThe main algorithm implementation takes place in the context of the {Click} memory \nwrite event, generated by the NCRAWL updater. The latter provides \nthe user with a table of single direction links with entries denoted by the corresponding  \nsource and destination IP combinations. Each entry holds the link direction \n{PDR} and transmission rate. \nThe provided information can be used by the network \n coding algorithm to decide the valid NC combinations  \n by selecting the queues to activate together.\n\n Since this part of the code runs periodically, the developer \nis encouraged to implement any complex algorithm steps here and thoroughly\nindex the NC available combinations.  With such an approach, the overhead of choosing \nthe most beneficial packet combination during the incoming or outgoing packet events \nwill be minimized.\n\n{\\bf Sending data and ACK packets:} \nThe outgoing packet event checks the size\n of the outgoing packet queue. \n If  this is\n below the   defined threshold, the functions that choose and encode \ncombinations are called \n in the exact say way as for \n\n  the   incoming packet event. Then the\nnext packet on the outgoing queue gets transmitted. \nThe developer may also add logic for the handling of ACKs. By default, \nNCRAWL resends packets that have not been acknowledged, by directly placing \nthem on the outgoing queue. \n It is possible, however, that an algorithm deals with the failed packets (see section 5 for an example). \n Since the ACK scheme groups ACKs \nthat belong to the same encoded packet, the developer knows which packets have been decoded\nsuccessfully at which destination, and may extend the NC algorithm to decide\nwhether a packet should be resent directly, or reconsidered for encoding combinations.\n\n{\\bf System monitoring:} \n\n\n Our framework  \n allows\n  for \n user defined timer events. \nStatistics for incoming--outgoing packet activity as well as queue lengths are all logged using counters.\nThe NCRAWL logger periodically retrieves\nstatistics\nand \nnotifies the user at runtime about the flow stability and the corresponding queue lengths. The latter are \nalso available for use in the NC algorithm if needed. Finally, the developer may also \nimplement additional debugging support for inspecting the algorithm configuration at runtime, using \nthe standard Click support for the read handler. \n\n\\subsection{Deploying NCRAWL Experiments}\n\\label{subsec:deploy_NCRAWL_experiments}\n\nWe have integrated NCRAWL deployment scripts with the OMF framework \\cite{orbit} for wireless testbeds. {OMF} is a Control, Management and Measurement Framework that provides\n users with   tools to describe, execute and collect experimental results\n in a straightforward manner. There are three main components that comprise OMF; we   describe them below:\n\\\\\n$\\bullet$ {\\bf \\em Gridservices:} This is a set of web services that are used by {OMF} to fetch information and perform actions remotely on the nodes. These services can be used for loading the system image to nodes, executing   experiments and   collecting   results.\n\\\\\n$\\bullet$ {\\bf \\em Nodehandler:} This component resides on the central server that interacts with the user for the experiment submission. Moreover, it provides the necessary applications for node system image loading, experiment execution, image saving and node status check. This component communicates with both the gridservices and the nodeagent to get the required information and perform actions. Regarding the experiment deployment, nodehandler contains a set of prototypes that can be used for experiment definition. It also\nnotifies the user for any problems that may arise.\n\\\\\n$\\bullet$ {\\bf \\em Nodeagent:}\nThe task of this constantly active component is to wait for information  that contains instructions for the experiment deployment,  arriving  from the nodehandler.\n\n NCRAWL extensions have been written for both nodehandler and nodeagent. The former   \nperforms transfers of the Click executable along with user defined parameters. \n  \n The latter retrieves local node information (e.g. the network interface name and MAC address)  and then   parametrizes a generic NCRAWL deployment script that gets immediately invoked to start the local Click NCRAWL instance. \nFinally, the nodehandler is notified if the deployment was successful; Upon success,\n experiments can start. We have written nodeagent scripts to deploy \\emph{iperf} instances and collect results at the nodehandler. \nWith\nOMF,\nNCRAWL experiments can be deployed\n with minimal user effort.\n\n\\section{Case study}\n\\label{sec:case}\n\nAs discussed above, the relay maintains a queue for each pair of source-destination in the neighborhood that lacks direct connection (2--hop flows traversing the relay). At each transmission slot, the scheduler should select a number of packets, encode them and send the encoded packet to the MAC layer for transmission. The problem is then of scheduling nature; to select a number of packets for transmission. Note that selecting only one packet corresponds to transmission without NC. This section demonstrates by example how the NCRAWL framework allows for easy implementation of scheduling algorithms and presents a case study to be used for the performance comparison of the next section.\n\nA well known family of optimal algorithms for scheduling is the maximum weighted matching algorithms, applied in the stability theory of stochastic networks and input queues switches, see for example \\cite{TW92,MMAW99}. \nIn these algorithms, the available control actions are chosen to maximize a reward which depends on link rates and queue lengths. \nThe application of such algorithms for the solution of the joint NC and scheduling problem with arbitrary rates is then promising (see \\cite{Eryilmaz07controlfor} and papers in which it appears as a reference).\n\nFor the case of the implicit ACKing scheme, used by our framework, one observes that the rate of service for packets of a particular source-destination pair is actually random. The randomness comes from the fact that some packets needed for decoding may be missing because of an overhearing failure. Nevertheless, a NCRAWL equipped relay owns the probabilistic information of overhearing and is then in position to determine the probability of decoding and thus the average service rate. A control action consists of selecting a number of queues to serve at a single decision instance. The reward of each control is the sum of queue length times the average service rate for each queue activated by the control. The average service rate is the expected number of successfully serviced packets times the transmission rate for which the encoded packet can be received by all intended receivers. This rate is the minimum of the reception rates of all intended receivers. We write: \n\\begin{equation}\n\\mu_i(C)=\\prod_{j\\in C}q_{s_j,d_i}\\min\\{\\mathbf{r}_C\\},\n\\label{equation:service_rate}\n\\end{equation}\nwhere $C$ is the control, $q_{s_j,d_i}$ is the probability that the destination of flow $i$ overhears the source of flow $j$ at the uplink phase, $\\min\\{\\mathbf{r}_C\\}$ is the transmission rate of the control and $\\mu_i(C)$ is the expected service rate of flow $i$ when the control $C$ is selected. Here we have assumed that the overhearing events are independent. This is a realistic assumption since the time and space for each overhearing event is different. Collisions and Rayleigh fading may be the causes for this randomness.\n\n\\subsection{Algorithm 1}\n\nLet $Q_i$ be the queue backlog at the relay for the flow $i$. We are then in position to design our first algorithm.\n\\begin{algorithm}[ht]\n\\small{ \n\\KwIn\n{$Q_i, \\mu_i(C)$}\n\\KwOut\n{$C^*$}\n\n$w_{\\max}:=0$\\;\n\n\\For\n{$C\\in \\mathcal{C}$}{\n\n  $w(C):=\\sum_{i\\in C} Q_i \\mu_i(C)$\\;\n  \n\t\\If\n\t{$w(C)>w_{\\max}$}{ $C^*$ :=\n    $C$\\; } \n\n}\n\\caption{MaxWeight Algorithm without feedback}\n\\label{alg:1}\n} \n\\end{algorithm}\n\nAn issue raised in this algorithm is the fact that the number of possible encoding combinations to be examined is exponential in nature. If, for example, we assume that overhearing is possible for all receivers except the destinations, then the number of combinations is actually $2^N-1$ where $N$ is the number of source-destination pairs (or 2--hop flows). The question is then whether the computational overhead for the weights is prohibitively high. In subsection \\ref{sec:case-implement}, we explain how the list of weights is maintained in order to reduce the number of calculations per slot.\n\nThe algorithm 1 is throughput optimal under the condition that the knowledge of the aforementioned probabilities of overhearing cannot be altered during transmissions. This happens when (i) the probabilities are 0 or 1, as in Alice-Relay-Bob topology (and any other symmetric flow setting) or (ii) upon a decoding failure we reschedule the uplink transmission for the failed flow. The latter may arise in a TCP scenario. In the general case, however, whenever a particular encoded packet is not correctly decoded, the packet remains in the queue at the relay but extra feedback information is obtained. If for example $P_1 \\oplus P_2$ is not decoded by both receivers, the relay knows that these two packets are not overheard by receiver 2 and 1 respectively, and the proper action is to correct the overhearing probabilities to zero and never encode these two specific packets again. The impact of feedback clearly biases the probabilities of decoding.\n The knowledge state of each packet evolves in a such a way that future states depend on the control action selected at present and\n as a result, in the general case, not only algorithm 1 is not optimal but it might perform quite badly when the overhearing probabilities are quite small. \n\n\n\\subsection{Algorithm 2: the case of two queues}\n\nAnother idea is to propose an algorithm which is not necessarily optimal, but manages to handle the feedback information successfully. In general, an algorithm should be able to predict the future effects of current control actions. Here we restrict our search to the category of the so-called myopic algorithms, trying to solve the problem given only the current state and disregarding the future. We  consider the\nproblem of mixing only two flows. \n\nIn order to cope with feedback, we add two more knowledge states. Apart from newly arrived (unknown) packets whose behavior is captured by known probabilities, we have a state for ``good'' packets (overheard by the other receiver) and one for ``bad'' packets (those not overheard by the other receiver). Thus, the system maintains the queues $Q_i^s$ where $i\\in\\{1,2\\}$ signifies the flow and $s\\in\\{u,g,b\\}$ signifies the state. The set of controls $\\mathcal{C}$ contains all controls that activate one or two queues with the constraint that no two queues from the same flow can be activated.\n\\begin{algorithm}[h]\n\\small{ \n\\KwIn\n{$Q_i^s, \\mu_i^s(C)$}\n\\KwOut\n{$C^*$}\n\nAt feedback time:\n\t\\begin{itemize}\n\t\t\\item For each packet that was not correctly decoded define whether it is good or bad.\n\t\t\\item Bad packets are directly sent to the MAC layer for transmission without coding.\n\t\t\\item Good packets are sent to the corresponding queue at the good state.\n\t\\end{itemize}\nAt decision time:\n\n$w_{\\max}:=0$\\;\n\n\\For\n{$C\\in \\mathcal{C}$}{\n\n  $w(C):=\\sum_{C} Q_i^s \\mu_i^s(C)$\\;\n  \n\t\\If\n\t{$w(C)>w_{\\max}$}{ $C^*$ :=\n    $C$\\; } \n\n}\n\\caption{Myopic Algorithm with feedback}\n\\label{alg:2}\n}\n\\end{algorithm}\nThe packets are initially injected in the queues at the unknown state. Once a packet is not decoded properly, the relay classifies it as either good or bad based on feedback information. If bad, it is retransmitted without encoding (thus the queues $Q_i^{b}$ are not needed actually). If it is deemed as good, it is transferred to the corresponding queue at the good state ($Q_1^{g}$ or $Q_2^{g}$ depending on the flow it belongs to). When calculating average service rates, the packets at the good state have probability of overhearing equal to one. Apart from these alterations, algorithm 2 works in the same way as algorithm 1.\n\nIn \\cite{rawnet} it is shown that an enhanced queue length based algorithm solves optimally the joint NC and scheduling problem arising in intersession coding at the relay node. This solution might be costly in terms of resources, and therefore suboptimal algorithms might be preferred. For this reason, our framework serves as an ideal substrate for performing measurements of such algorithms.\n\n\\subsection{Algorithm 3: fixed threshold policy}\n\nFor reasons of performance comparison we define a third algorithm. This algorithm operates only with implicit ACKs and makes decisions based on principles used in the COPE framework. In this sense, it emulates COPE in its probabilistic mode. The important differences to our algorithm are that instead of calculating average service rates, the $\\delta$--Fixed Threshold Policy ($\\delta$--FTP) simply marks the incoming packets with information about decoding opportunities. In order to do so, overhearing probabilities $q_{i,j}$ are compared with a fixed threshold $\\delta\\in[0,1]$ and set to $1$ if they exceed the threshold or zero otherwise. \nThe algorithm selects at each decision instance the control that maximizes the number of transmitted packets.\n\n\\subsection{ NCRAWL algorithm implementation}\n\\label{sec:case-implement}\nNext, we demonstrate how to implement the three above algorithms on NCRAWL. \nFor all three cases, \n\n we configure NCRAWL at each node to maintain  \n\n one queue per flow for incoming packets\n\n\n{\\bf Implementing algorithm 1:} We first describe how one may organize queues in an efficient manner. Subsequently, we show how to utilize the queue information to apply NC. \n\\\\\n$\\bullet$ {\\bf \\em Organizing packet queues:} \nTo begin with,\nwe dedicate one vector per control which contains\nthe identity of the involved queues (e.g. the flow it belongs to and\/or the state) and the identities of the packets enqueued at the involved queues. \n\nThe formed vectors are stored in\n a double linked list. \n Each vector is assigned a weight (or reward); the higher this weight, the higher the   \n preference of the encoder for using the combination.  \nThis weight is recalculated every time the backlog size of \n a\n member queue\n changes. \n    The linked list is formed such that the head of the list contains always the \n current maximum\n weight.\n For the sake of  low processing overhead, \n vectors\n are also directly indexed by their member\n queues; with this,\nthe weight update process is fast. \n As one may expect,  vectors as well as their linked list \n are all constructed during the NCRAWL updater write event. \n\\\\\n$\\bullet$ {\\bf \\em Applying NC operations:}  \n Given the construction of the control list, \nthe encoder event examines the head of the list, and further: \n(a) retrieves packets from their respective queues, \n (b) updates the\n vector weights (since the respective backlogs are decremented), \nand \n(c)  sets the vector with the highest weighted combination as the head of the list. \nThe latter is actually\na process with slowly scaling complexity with the number of vectors-combinations, \nsince each updated vector weight is just compared against the weight of the current head,  \n and only takes its place if it is higher.\nRetrieved packets are subsequently combined using the NCRAWL \\emph{encode} \nlibrary call, and the resulting encoded packet is scheduled for transmission.\n \n{\\bf Implementation considerations for algorithm 2:} \n This algorithm is similar to algorithm 1, however it involves an additional \nacknowledgment scheme logic. \nTherefore, for each flow NCRAWL now maintains two queues: \n (a) one with new incoming packets, and \n (b) one with packets that have been successfully \nlogged as keys by fellow nodes, but have not reached their ultimate \ndestinations\\footnote{For example this could be due to the fact that the destination failed to decode properly a previously sent encoded packet.}. \nAlgorithm  2  exploits the \nNCRAWL acknowledgment scheme facility; this process groups the packet \nacknowledgment tokens, which have been created for \noutgoing packets combined together in the same encoded packet. \nThis information is provided by NCRAWL to the   \ndeveloper. Algorithm 2  directly sends packets that have not \nyet reached their destinations;\n  such packets are not reconsidered for encoding. \n\n However, the algorithm considers favorable queues and ``unknown\" queues for the same flow separately, when forming vectors. \n Note that the vectors formed with this algorithm scale intrusively, \ncompared to the simple maxweight algorithm described previously. Throughout our measurements we only consider the scenario of two flows and thus avoid the arising complexity. This issue is expected to be resolved in the future using the NCRAWL framework.\n\n{\\bf Algorithm 3 in NCRAWL:} \n For the implementation of the third algorithm we simply need to create vectors, (i.e. controls or queue combinations) for which the decoding\nprobability is nonzero, according to the user-defined threshold $\\delta$ and the channel quality. \nAs soon as packets are \navailable in all\nqueues that constitute a vector, they are combined\nand transmitted at once, without considering or updating the queue backlogs. This\nalgorithm\nselects controls that mix the largest possible number of packets each time.\n\n We should note here that NCRAWL does not use any time-threshold policy \n towards increasing the backlog size of the incoming packet queues, before \n\n deciding to send outgoing packets. On the other hand, COPE adopts such a design decision. \n\n With NCRAWL,\nqueue backlogs will increase when the relay's outgoing packet rate is smaller than the incoming packet rate. In such cases, NC proves to be a panacea for the router stability; if the NC algorithmic operations are supported by a lightweight implementation, the router capacity can be truly increased, as our measurements suggest. \n\\vspace{-0.1in}\n\\section{Evaluating our Framework}\n\\label{sec:measurements}\n\nIn this section, we\n evaluate \nNCRAWL\nin conjunction with scheduling algorithms (NCRAWL + alg1, NCRAWL + alg2 and $\\delta$--FTP) described in section \\ref{sec:case},  \nin terms of both throughput and resource utilization. We begin by describing \nthe wireless testbed infrastructure\\footnote{Our motivating experiments discussed in section \\ref{sec:introduction} were performed on 2 different testbeds. We have evaluated NCRAWL on both testbeds; here we present results for one of them.} \nand the configurations that we \nused to deploy experiments. Next, we quantify the \n{CPU} overheard that is introduced by each NCRAWL \nprocessing stage, under maximum traffic loads, and we compare \ntotal {CPU} utilization to: (i) the public COPE implementation \nthat uses an explicit acknowledgment scheme and (ii) legacy IEEE 802.11b-g.\nFollowing, we demonstrate that NCRAWL can support theoretical \ngains even when coding opportunities lead to more than 2-packet \ncombinations. Finally, we deploy    experiments that \ndemonstrate how the proposed algorithms perform in cases with variable link qualities and different rates.\n\n\\subsection{Experimental setup}\n\nOur testbed is comprised of 20 ORBIT-like nodes, deployed both indoors and outdoors. \nEach node consists  of one 1{GHz} 386 processor, 512MB of {RAM}, two ethernet ports and two miniPCI slots which are used to host two  AR5212 Atheros 802.11a\/b\/g WiFi \ncards.\nAll the nodes are connected through wired Ethernet with the testbed's \nserver -  console. On console, we have all the required testbed \nservices running as well as the NCRAWL deployment scripts that \nwe described in section~\\ref{subsec:deploy_NCRAWL_experiments}. \n For conducting  throughput measurements we use the {iperf} bandwidth meter tool, \\cite{iperf}. \nFor {CPU} occupancy measurements  we appropriately instrument  \nNCRAWL\n with the Linux \\emph{getrusage} system call, which  \naccurately estimates {CPU} usage time. We   place  several {getrusage} \ncalls at the borders of each processing stage, we record  the average usage time \nof each stage and we compare  it to the whole NCRAWL system usage time. \n We have repeatedly performed all of our experiments late at night, in order to avoid interference from collocated networks. \n\n\n\n\\subsection{CPU occupancy measurements}\n\nIn order to measure the efficiency of our framework in terms of CPU occupancy, we compare it to the case of running COPE \\cite{cope}, as well as\nthe legacy IEEE 802.11 protocol. \n\n\n\n\\begin{figure*}[h]\n\\begin{center} \\hspace{-0.12in}\n\\parbox{1.5in} {\n    \\centerline{\\subfigure\n    ]{\\includegraphics[scale=.19]{figures\/cpub.pdf}}}\n\n\n\n}\n\\makebox[.15in] {}\n\\parbox{1.5in} {\n    \\centerline{\\subfigure\n    ]{\\includegraphics[scale=.19]{figures\/cpug.pdf}}}\n  \n\n\n}\n\\makebox[.15in] {}\n\\parbox{1.8in} {\n\\vspace{0.1in}\n    \\centerline{\\subfigure\n    ]{\\includegraphics[scale=.17]{figures\/origin\/fig_CPU_pie}}}\n   \n\n \n}\n\\makebox[.15in] {}\n\\parbox{1.2in} {\n\\hspace{-0.2in}\n    \\centerline{\\subfigure\n    ]{\\includegraphics[scale=.19]{figures\/cpustage.pdf}}}\n   \n\n \n}\n\\end{center}\n\n\\end{figure*}\n\n\n\\begin{figure*}[h] \\vspace{-0.12in}\n\\begin{center} \\hspace{-0.12in}\n\\parbox{1.3in} {\n    \\centerline{\\subfigure\n    ]{ \\includegraphics[scale=.19]{figures\/tputrateb.pdf}}}\n\n\n\n}\n\\makebox[.34in] {}\n\\parbox{1.3in} {\n    \\centerline{\\subfigure\n    ]{ \\includegraphics[scale=.19]{figures\/gainb.pdf}}}\n\n\n\n}\n\\makebox[.34in] {}\n\\parbox{1.3in} {\n    \\centerline{\\subfigure\n    ]{ \\includegraphics[scale=.19]{figures\/tputrateg.pdf}}}\n\n\n\n}\n\\makebox[.34in] {}\n\\parbox{1.3in} {\n    \\centerline{\\subfigure\n    ]{ \\includegraphics[scale=.19]{figures\/gaing.pdf}}}\n\n\n\n}\n\\caption{Results in Alice-Relay-Bob topology.}\n\\label{fig:alice}\n\\end{center}\n\\end{figure*}\n\n\n\\begin{figure*}[t!]\n\\begin{center} \\hspace{-0.12in}\n\\parbox{1.3in} {\n    \\centerline{\\subfigure\n    ]{ \\includegraphics[scale=.19]{figures\/flowtput_flows.pdf}}}\n\n\n\n}\n\\makebox[.35in] {}\n\\parbox{1.3in} {\n    \\centerline{\\subfigure\n    ]{\\includegraphics[scale=.19]{figures\/gainflows.pdf}}}\n\n\n}\n\\makebox[.35in] {}\n\\parbox{1.3in} {\n    \\centerline{\\subfigure\n    ]{\\includegraphics[scale=.19]{figures\/flowtput_prob.pdf}}}\n\n\n}\n\\makebox[.35in] {}\n\\parbox{1.3in} {\n    \\centerline{\\subfigure\n    ]{\\includegraphics[scale=.19]{figures\/flowtput_rate.pdf}}}\n \n \n}\n\\caption{Results in wheel topologies. }\n\\label{fig:wheel}\n\\end{center}\n\\end{figure*}\n\n{\\bf NCRAWL is much more CPU friendly than COPE-based approaches:} \n We  invoke the Alice-Relay-Bob setting (see section \\ref{sec:nc_scheme}) and we inject fully saturated traffic in both flows. \nWe compare NCRAWL + alg1, NCRAWL + alg2,\nCOPE and the plain 802.11, for the case of 802.11b;   figure \\ref{fig:alice}-a depicts the results.   \nNote that COPE can support at most the IEEE 802.11b rate set as discussed in section \\ref{sec:introduction}; for the sake of a fair comparison here, we use this mode of operation for NCRAWL as well.\nWe observe that NCRAWL makes use of the CPU resources in a very efficient manner: it reduces the CPU utilization by at least 2 and as much as 7 times compared to COPE  \n (we have validated these observations for the case of ER \\cite{er} as well, which is based on COPE). \nFurthermore, we test NCRAWL for the case of 802.11g. Our measurements  (figure  \\ref{fig:alice}-b) \n suggest that NCRAWL does not need to occupy more than 37\\% of the CPU resources for NC operations at 54 Mbps, with fully saturated UDP traffic! This implies that the design of NCRAWL  includes low additional overhead functions (as opposed to legacy 802.11). \n\n{\\bf Evaluating  individual operations of NCRAWL:} \nNext, we deploy {getrusage} calls and measure the breakdown of CPU occupancy per processing stage (figure \\ref{fig:alice}-c). \nThe most CPU intensive operation is the SRCR stage (it contains legacy IEEE 802.11 operations as well).\n\n The most computationally heavy pieces of NCRAWL are the encode stage and the key house-keeping. Note  here  that these two lie at the heart of any NC system and in a way represent unavoidable costs. It should also be noted that the   processing stage of the scheduler remains at very low values and there is a certain percentage dedicated to dealing with ACKs. \n Furthermore, as depicted in figure \\ref{fig:alice}-d,   by increasing the channel rate (and thus the number of packets into the system per unit time), the coding stage increases in complexity disproportionally with the SRCR stage. This implies that the coding complexity increases faster than SRCR as the rate increases. \n Nevertheless,\n for\n  high channel rates  the differences  are reduced. \n  This suggests that NCRAWL could potentially operate efficiently at much higher channel rates, such as with 802.11n systems. We plan to test NCRAWL on MIMO networks in our future work. \n   \n\\subsection{Throughput measurements with UDP}\nNext, we assess the ability of NCRAWL to approach the theoretically expected benefits of NC. \n\n{\\bf Experiments with the simple Alice-Relay-Bob topology:}\n\n We calculate and measure the maximum throughput for both symmetric   flows, such that the system remains stable (i.e. the queues do not rise more than a large permissible number). \nFigures \\ref{fig:alice}-e, \\ref{fig:alice}-f, \\ref{fig:alice}-g and \\ref{fig:alice}-h show the results. \n\n Note that since the receivers   always have the proper keys (these are the keys from their own transmitted packets \\cite{proutiere}), decoding is always possible and thus algorithm 1, algorithm 2 and COPE are optimal in this setting. \n In each case, a gain in throughput of $\\frac{4}{3}$ is identified, which matches the theoretical for this topology. \n %\n %\n %\n %\n Our measurements suggest that \n\nCOPE achieves the theoretical throughput for small rates, but\n it fails to do so in higher rates. \n Note that the public COPE code was initially available for 802.11b only; \n while we carefully\n  modified COPE to operate at 802.11g rates, \n  we observed that such modifications lead to a very unstable system when rates higher than 18 Mbps are used. A closer look at certain individual components of the COPE implementation revealed that the reason for this instability is the  excessive overhead induced by the NC system operations (as discussed earlier). \nFor this reason we do not explicitly\ncompare COPE here at these high rates. Nevertheless, from these measurements  \n one can realize\n that COPE cannot provide benefits at \n rates higher than 18 Mbps, due to the tremendous CPU processing overheads that its design  incurs. \n In contrast, NCRAWL manages to\n reach the theoretical gain at high channel rates (e.g. at 54 Mbps), as shown in figures \\ref{fig:alice}-f and \\ref{fig:alice}-h.\n\n{\\bf The case for wheel topologies:} \nFurthermore, we scale the number of flows (see figures \\ref{fig:wheel}-a and \\ref{fig:wheel}-b); the topology is an $\\frac{x}{2}$--wheel.\nThe theoretical gain in this case is $\\frac{2x}{x+1}$ where $x$ is the number of flows combined at the downlink. Our measurements support the theoretically predicted gain at the channel rate of 54 Mbps. We observe the per flow throughput naturally drops, as the number of flows increases, but the aggregate throughput increases. The gain (figure \\ref{fig:wheel}-b)) is an increasing function of flows and approaches asymptotically 2; note that this is perfectly aligned   with the findings in \\cite{proutiere} as well.  \n\n  Note also that in $\\frac{x}{2}$--wheel topologies, piggybacking is not available since there is no return flow from the receivers. NCRAWL is able to select the appropriate ACKing method and the results show that the overhead incurred is negligible.\n\n{\\bf Experiments with cross topologies:} \nWe now present two more cases of interest that can appear in realistic environments. \nWe setup\n various {\\em cross} topologies with nodes in different locations across our testbed; we activate the flows Alice-Relay-Chloe and Bob-Relay-David. The arrivals are again chosen in a symmetric way, i.e. the arrival rate of the one flow is equal to the other.\n\n$\\bullet$ In the first case (figure \\ref{fig:wheel}-c),  David overhears Alice's uplink transmissions with probability 1 and Chloe hears Bob with probability $q$. The rates of all links are equally set to 12Mbps (the channel rate is not important in this experiment). We measure the highest throughput that guarantees queue stability while varying the probability $q$, by considering different node locations. \nWe compare NCRAWL+alg1, NCRAWL+alg2 and IEEE 802.11g as well as $\\delta$--FTP for $\\delta=\\{0.7,0.8,0.9\\}$ (see section \\ref{sec:case} for description). The results demonstrate the superiority of NCRAWL+alg2, which is able to deliver the maximum throughput in each case. Evidently, our framework in combination with the proposed scheduling algorithms is able to effectively handle\nthe several link quality conditions. \n\n$\\bullet$  In the second case (figure \\ref{fig:wheel}-d),\n the overhearing probability from Bob to Chloe is set to $q=0.7$. All channel rates are set to 24Mbps with the exception of the link Relay-Chloe which is varied. Our measurements demonstrate the inefficiency of policies oblivious to rates like the $\\delta$--FTP. In this case, the choice of a small value for $\\delta$ is penalized when the Relay-Chloe link is slow enough. Instead NCRAWL+alg2 is able to handle in an effective way the several rate and link conditions and deliver important throughput gains. \nFrom figures \\ref{fig:wheel}-c and \\ref{fig:wheel}-d  \nwe also observe that \n\n given that  \n  overhearing links are not perfect in terms of PDR, NCRAWL+alg2 always outperforms NCRAWL+alg1, since it is able to use feedback information.\n\n\\subsection{Performance with TCP traffic}\n\nFinally, we assess the efficacy of NCRAWL in scenarios with TCP traffic. \nIn \\cite{cope}, experiments with TCP have demonstrated a loss in efficiency due to packet losses and reordering. \n\nFirst, throughout our experiments with the Alice-Relay-Bob topology, where no losses or delays are incurred, the throughput is reduced due to the additional TCP  overheads. We observe that when the 54 Mbps rate is used, the per flow throughput rate is 7 Mbps for plain 802.11 and 8.5 Mbps for NCRAWL+alg1. \n A slight loss in NC gain is observed; this   is the result of mixing TCP ACKs with data packets. The same gain is obtained for all the other available bit rates.\n\nFurthermore, \n\n we perform experiments with {\\em half-cross} topologies, where flows are unidirectional (from Alice to Chloe and from Bob to Dave),   \n with probabilities of overhearing $q_{AD}=q_{BC}=0.7$ and several channel rates. In this case, NCRAWL+alg1 achieves a slightly lower throughput than IEEE 802.11. This is due to the fact that some packets are not correctly decoded at the destination and therefore they arrive delayed and out of order. This causes abrupt reactions from TCP and leads to throughput reduction. When adding the reordering module of COPE \\cite{cope}, the packets arrive always in order, however this module increases the delay for each packet. This in turn is interpreted by TCP as congestion; it ends up in TCP window increments, and thereby decreases performance. \nNCRAWL is not optimized to cooperate with TCP at this point and thus, it faces the common problems of TCP in wireless networks. Improving this component is the main goal of our future work.\n\\vspace{-0.1in}\n\\section{Conclusions}\n\\label{sec:conclusions}\n\n We design and develop  NCRAWL. Our  framework \n is an extended, generic NC framework that can be used to quickly develop networking systems in order to evaluate intersession NC and\/or scheduling  algorithms,   entirely based on the implicit (probabilistic) acknowledgment that a packet can get decoded at \nthe destination. The design of NCRAWL involves all the common processing steps that are always \nneeded to implement such algorithms; these steps  have been abstracted such \nthat designers\nneed to simply focus only on the  implementation of their algorithms. \nOur measurements\ndemonstrate that NCRAWL is a powerful NC development system. \nIt offers significant throughput benefits \neven at high channel rates. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\t\n\t\n\tThe original Kaluza-Klein idea \\cite{Kaluza}\\cite{Klein1}\\cite{Klein2} consists in a $5D$ space-time formulation having the aim to include in a geometrical picture also the electromagnetic interaction. \n\t\n\t\\noindent The surprising formal success in providing a metric representation of the vector potential suggested, in the Seventies, to attempt for a geometrical unification \\cite{ModernKKtheories}, able to assess all the fundamental interactions into a multi-dimensional space-time, with particular attention to the Electroweak Model. \n\tThe suggestive idea at the ground of these approach consists of the possibility to reproduce the Lie algebra, characterizing the elementary particle symmetries by the isometries of the extra-dimensional space. The non-trivial result obtained by the extra-dimensional Kaluza-Klein theories relies on the emergence from the multi-dimensional Einstein-Hilbert Lagrangian of the correct Yang-Mills action for the vector bosons which are the interaction carriers. \n\t\n\tHowever, many non-trivial problems affected this fascinating attempt for a geometrization of Nature. One of the main questions came out from the difficulty to provide a geometrical version of the chirality singled out by the electroweak interaction \\cite{Wetterich}, as well as the impossibility to represent the Standard Model of elementary particles in a Kaluza-Klein scenario \\cite{Witten}.\n\tFor alternative non-Riemannian approaches to solve the chirality problem of the Electroweak model see \\cite{Cianfrani-Montani1}\\cite{Cianfrani-Montani2}. \n\t\n\t\n\t\\noindent Finally, we observe that a full geometrical picture of Nature would involve the geometrical formulation of the fermionic field, a really non-trivial perspective if supersymmetry is not considered \\cite{Ferrara}. \n\t\n\t\\medskip\n\t\n\tEven the $5D$ Kaluza-Klein theory presents some important difficulties, see for a review \\cite{Cianfrani-Marrocco-Montani}, which leaves open the question concerning the viability of this approach as a geometrization of the electromagnetic interaction. \n\t\n\t\\noindent First of all, the $5D$ metric tensor contains an additional degree of freedom besides the $4D$ metric and the vector potential, namely the fifth diagonal component. \n\tUnder the necessary restriction of the coordinate transformation in order to deal with the $U(1)$ symmetry, this quantity behaves as an additional scalar field, which presence non-trivially affects basic features of the electromagnetism, for instance, the charge conservation itself \\cite{ModernKKtheories}\\cite{Lacquaniti-Montani}\\cite{Lacquaniti-Montani-Vietri}. \n\tBut, even fixing this scalar field to unity in the Lagrangian for the model (with the right sign of a space-like component), nonetheless, the ratio between the charge and the mass of an elementary particle is constrained to remain too small in order to reproduce the Standard Model spectrum of masses (for a proposal to solve the charge to mass ratio problem see \\cite{Lacquaniti-Montani}).\n\t\n\t\\noindent Finally, studying the morphology of a five-dimensional D'A\\-lam\\-ber\\-tian\\- operator, it is immediate to recognize the emergence of huge massive modes of a boson field, as result of the compactified scale of the fifth dimension \\cite{Chodos-Detweiler}.\n\t\n\t\\medskip \n\t\n\tIn the present analysis, we approach the formulation of the $5D$ Kaluza-Klein theory within the semi-classical and quantum framework of the so-called Polymer Quantum Mechanics \\cite{Corichi1}\\cite{Corichi2}.\n\tThis revised formulation of quantum physics has the aim to introduce a discrete nature in the generalized coordinate (a real coordinate of a generic degree of freedom), as an effect of the emergence of cut-off physics. \n\t\n\tIndeed, the fifth compactified dimension, being in the standard approach about two order greater than the Planck size, it is in the natural condition to be approached via the continuum limit of Polymer Quantum Mechanics as referred to a point particle living in this dimension.\n\tFurthermore, also the corresponding diagonal metric component (namely the additional Universe scale factor) in such a dynamical regime is to \n\tbe interested - as expected - by cut-off physics effects. \n\t\n\tThe present analysis follows the scenario proposed in \\cite{Chodos-Detweiler} but revised in view of the polymer formulation. \n\t\n\t\\noindent We first show that a five-dimensional Kasner solution \\cite{Kasner} \\cite{Landau}\\cite{Montani} (characterizing the Bianchi I Universe) admits a configuration in which three spatial directions isotropically expand, while the fourth remains static. \n\tThis result is of impact in the implementation of Kaluza-Klein theory, since it removes some of the non-trivial inconvenient features of a collapsing dimension, closely to a Planckian size. \n\tFor a previous attempt to deal with a static compactified dimension, on the base of a physical phenomenon, see \\cite{Salam}. \n\t\n\t\\noindent Then, we analyse the geodesic motion on a generic $5D$ space-time, having a fifth steady dimension and we outline a natural solution to the charge to mass ratio problem. \n\tThis result comes out from the details of the semi-classical polymer formulation, adopted for the Hamiltonian dynamics of the free-falling particle. \n\tIn particular, the modified expression taken by the fifth momentum of the particle leads to a modified constitutive relation, that is - when passing from the momenta to the velocities - the previous constraint on the charge to mass ratio allows considering values which are natural in the Standard Model particles. \n\t\n\t\\noindent Finally, we study a five-dimensional Klein-Gordon equation and we clarify that, addressing the fifth coordinate via the quantum polymer prescription, the spectrum of emerging masses can fit some values of the Standard Model one and no tachyonic mode emerges, differently from the case discussed in \\cite{Chodos-Detweiler}. \n\t\n\tHowever, it should be noticed that, in this quantum field approach, a problem with the definition of a correct $q\/m$ ratio for a Standard Model particle still survives.\n\t\n\t\\medskip \n\t\n\tThe present study suggests that, when cut-off physics is included in the Kaluza-Klein formulation, some of the puzzling features of this approach are restated into a form that can give new physical insight for their understanding and overcoming. \n\t\n\t\\medskip \n\t\n\tThe manuscript presentation is structured as follows:\n\tin Section \\ref{Kaluza-Klein theory} we review the main features of ordinary Kaluza-Klein theory, from the metric tensor construction and the resulting field equations to the geodesic motion of a point-like particle, which analysis, in particular, leads to the ordinary quantisation law for the electric charge, an estimate for the size $L$ of the fifth dimension and the aforementioned shortcoming\n\tof the charge to mass ratio of a particle. \n\t\n\t\\noindent In Section \\ref{Polymer quantum mechanics} we review polymer quantum mechanics, summarizing the construction of the relative kinematics Hilbert space, via the introduction of a Weyl-Heisenberg algebra and under the assumption of the existence of a discrete spatial coordinate, and the implementation of the proper dynamics both on a quantum and semi-classical level, with particular regard to the p-polarization.\n\t\n\t\\noindent In Section \\ref{Polymer Kasner solution} we analyse the polymer-modified Kasner solution obtained from the introduction of the polymer framework on a semi-classical level in a $5D$ Bianchi I model, focusing on the behaviour of the fifth dimension.\n\t\n\t\\noindent Finally, in Section \\ref{Kaluza-Klein theory in polymer quantum mechanics framework}, based on the result of the previous section, first, we analyse, in a semi-classical formulation of Polymer Quantum Mechanics, the geodesic motion of a point-like particle and all its features, comparing all the results with the ones from the ordinary theory, and then we carry out the study of the polymer quantum dynamics of a complex Klein-Gordon field, along the lines of \\cite{Chodos-Detweiler}, discussing with particular attention the resulting electric charge distribution and mass spectrum. \n\t\n\t\\noindent In Section \\ref{Conlusion} brief concluding remarks follow.\n\t\n\t\\section{Kaluza-Klein theory} \n\t\\label{Kaluza-Klein theory}\n\t\n\tKaluza-Klein theory is a $5D$ extension of Einstein's General Relativity which aim is to provide a unified description of gravitational and electromagnetic interaction in a purely geometric fashion. \n\t\n\t\\noindent In the original theory \\cite{Kaluza}\\cite{Klein1}\\cite{Klein2} the space-time is described by a $5D$ smooth manifold $V^5$, which is assumed to be the direct product $V^4\\otimes S^1$ between a generic $4D$ manifold and a circus of length $L$, that is a compact space.\n\t\n\t\\noindent A crucial assumption relies on the fact that all the observable physical quantities do not depend on the fifth coordinate $x^5$.\n\tThis hypothesis can be further motivated by noticing that, due to the compactness of the fifth dimension, all the observable physical quantities are periodic in $x^5$; hence the independence on the fifth coordinate can be regarded as zero-order cut-off of a Fourier expansion of these quantities themselves, dubbed as cylinder condition.\n\t\n\t\\noindent Once restricted the $5D$ general relativity principle to the following coordinate transformations (and their inverse):\n\t\\begin{equation}   \\label{KK_group_trans_1}\n\t\t\\begin{cases}\n\t\t\t\n\t\t\tx^{\\mu'}=\\Psi(x^{\\mu}) \\\\\n\t\t\tx^{5'}=x^5 + k\\Lambda(x^{\\mu}) \\\\\n\t\t\t\n\t\t\\end{cases}\n\t\\end{equation} \n\tthe $5D$ metric tensor of the expanded theory can be written as follows:\n\t\\begin{equation}\\label{KK_metric_tensor}\n\t\t\\tilde{g}_{ab}=\n\t\t\\left(\n\t\t\\begin{array}{c|c}\n\t\t\tg_{\\mu \\nu} + k^2 \\phi^2 A_\\mu A_\\nu & k \\phi^2 A_\\mu \\\\\n\t\t\t\\hline\n\t\t\tk \\phi^2 A_\\nu & \\phi^2\n\t\t\\end{array}\n\t\t\\right) ,\n\t\\end{equation}\n\twhere $g_{\\mu \\nu}$ is the $4D$ metric tensor of the ordinary theory, $A_{\\mu}$ is the electromagnetic four-potential, $\\phi$ is a scalar field and $k$ is a constant to be properly determined. \n\t\n\t\\subsection{Kaluza-Klein field equations}\n\tThe field equations of the theory can be obtained from a $5D$ Einstein-Hilbert action:\n\t\\begin{equation}\n\t\t^{(5)}S:=\\tilde{S}= - \\frac{1}{16\\pi\\tilde{G}}\\int_{V^{4}\\otimes S^1}  dx^{0}dx^{1}dx^{2}dx^{3}dx^{5}   \\sqrt{-\\tilde{g}} \\tilde{R} ,\n\t\\end{equation}\n\twhere $\\tilde{G}$, $\\tilde{g}$ and $\\tilde{R}$ are respectively the $5D$ gravitational constant, the metric tensor $\\tilde{g}_{ab}$ determinant and the $5D$ scalar curvature.\n\t\n\tBy performing a 4+1 dimensional reduction the ordinary $4D$ Einstein-Maxwell action is surprisingly obtained:\n\t\\begin{equation}\n\t\t\\begin{split}\n\t\t\t\\tilde{S}=& - \\frac{c^3}{16\\pi{G}}\\int_{V^{4}}  d^4x \\sqrt{{-g}}\\phi \\biggl( R + \\frac{1}{4}\\phi^2 k^2 F_{\\mu \\nu}F^{\\mu \\nu}\\\\\n\t\t\t&+\\frac{2}{\\phi}\\nabla_{\\mu} \\partial^{\\mu} \\phi \\biggr).\n\t\t\\end{split}\n\t\\end{equation}\n\tBy setting $\\phi=1$ in the action - as in the original work of Kaluza \\cite{Kaluza} and Klein \\cite{Klein1}\\cite{Klein2} - and by using the variational principle ordinary, Einstein-Maxwell field equations can be correctly recovered, once $k$ is setted equal to $2\\sqrt{G}\/c^2$.\n\t\n\t\n\t\\subsection{Geodesic motion}\n\t\n\tA free point-like particle in this theory will move along a $5D$ geodesic, hence the respective action, with signature (-,+,+,+,+), will be:\n\t\\begin{equation}\n\t\t\\tilde{S}=-mc\\int d\\tilde{s}=-mc \\int \\sqrt{-\\tilde{g}_{a b}\\frac{d x^a}{d \\tilde{s}}\n\t\t\t\\frac{d x^b}{d \\tilde{s}}} d\\tilde{s},\n\t\\end{equation}\n\twhere $d\\tilde{s}$ is the $5D$ line element, to be distinguished from the $4D$ line element $ds$.\n\t\n\t\\noindent Once setted - here and in the further developments - $\\phi=1$ in the metric \\eqref{KK_metric_tensor}, from the variational principle $5D$ geodesic equation is immediately restored:\n\t\\begin{equation} \\label{geodesic}\n\t\t\\tilde{u}^a \\tilde{\\nabla}_{a} \\tilde{u}^b=0.\n\t\\end{equation}\n\t\n\tIt is essential to point out that $5D$ velocity $\\tilde{u}^a$ is different from $4D$ velocity $u^a$; indeed they are related as follows:\n\t\\begin{equation} \\label{KK_five-four_velocity_rel}\n\t\t\\tilde{u}^a=\\frac{1}{\\sqrt{1-u_5^2}} u^a.\n\t\\end{equation}\n\t\n\t\\noindent From relations \\eqref{geodesic} and \\eqref{KK_five-four_velocity_rel} it can be easily shown that $u_5$ is a constant of motion. \n\t\n\tIn order to achieve $4D$ equation of motion, the geodesic equation \\eqref{geodesic} has to be evaluated for the usual space-time variables only, which we indicate with Greek letters. \n\t\n\t\\noindent By making use of relation \\eqref{KK_five-four_velocity_rel}, the following result is attained:\n\t\\begin{equation}\n\t\tu^\\nu\\nabla_{\\nu}u^{\\mu}=\\frac{2\\sqrt{G}}{c^2}u_5u^{\\nu}g^{\\mu \\lambda}F_{ \\nu \\lambda},\n\t\\end{equation}\n\twhere $F_{ \\nu \\lambda}$ is the antisymmetric electromagnetic tensor. \n\t\n\t\\noindent By comparison with the ordinary classical equation:\n\t\\begin{equation} \\label{KK_ordinary_4d_geodesic}\n\t\tu^\\nu\\nabla_{\\nu}u^{\\mu}=\\frac{q}{mc^2}u^{\\nu}g^{\\mu \\lambda}F_{\\nu\\lambda},\n\t\\end{equation}\n\tthe following fundamental identification is achieved:\n\t\\begin{equation} \\label{KK_u5_rel_q}\n\t\tu_{5}=\\frac{q}{2m\\sqrt{G}}.\n\t\\end{equation}\n\t\\noindent Being $p_5=mcu_5$ it can then be written:\n\t\\begin{equation} \\label{KK_p5_rel_q}\n\t\tp_5=\\frac{qc}{2 \\sqrt{G}},\n\t\\end{equation}\n\twhich establishes a fundamental relation between the particle fifth component of momentum and its electric charge. \n\t\n\tThe compactness of the fifth dimension implies a quantisation of momentum along the fifth direction:\n\t\\begin{equation} \\label{KK_p_quantised}\n\t\tp_5=\\frac{2\\pi \\hbar}{L}n \\qquad n \\in \\mathbb{Z},\n\t\\end{equation}\n\twhere we remind that $L$ is the length of the circus describing the fifth dimension.\n\t\\noindent By a direct comparison between relations \\eqref{KK_p5_rel_q} and \\eqref{KK_p_quantised} a natural quantisation law for the electric charge and an estimate of the size $L$ of the fifth dimension are obtained:\n\t\\begin{equation} \\label{KK_ordinary_L}\n\t\t{L}=4\\pi \\frac{\\hbar\\sqrt{G}}{e c}\\approx 2.37\\cdot10^{-31}\\>cm \\qquad q=ne,\n\t\\end{equation}\n\twhere $e$ is the electron charge.\n\t\n\t\\noindent Coherently the size of the fifth dimension is in agreement with its non-observability and with its impossibility to be currently detected. \n\t\n\t\\medskip \n\t\n\tNevertheless, despite these remarkable results, the relation \\eqref{KK_five-four_velocity_rel} sets the constraint $\\abs{u_5}<1$; \n\tby virtue of relation \\eqref{KK_u5_rel_q}, this implies the following condition on the charge\/mass ratio of a particle:\n\t\\begin{equation} \\label{KK_q_over_m}\n\t\t\\frac{\\abs{q}}{m}<2\\sqrt{G}\\approx 5.16\\cdot10^{-4} \\> e.s.u.\/g,\n\t\\end{equation}\n\twhich, unfortunately, has not phenomenological confirmation, neither for elementary particle nor for macroscopic object, hence representing one of the puzzling shortcomings of the theory.\n\t\n\t\\section{Polymer quantum mechanics}\n\t\\label{Polymer quantum mechanics}\n\t\n\tPolymer quantum mechanics is a non-standard representation of the non-relativistic quantum theory, unitarily inequivalent to the Schr\u00f6dinger one \\cite{Corichi1}\\cite{Corichi2}.\n\tIts developments are due mainly to the exploration of background-independent theories, such as quantum gravity, of which mimics several structures \\cite{Ashtekar}. \n\t\n\t\\smallskip\n\t\n\tGiven a discrete orthonormal basis $\\ket{\\mu_i}$ for a space $\\mathcal{H'}$, such that $\\braket{\\mu_i}{\\mu_j}=\\delta_{ij}$, where $\\mu_i \\in \\mathbb{R}$ and $i=1,2...,n$, the kinematic polymer Hilbert space $\\mathcal{H}_{poly}$ is obtained as a Cauchy completion of $\\mathcal{H'}$.\n\t\n\t\\noindent On this space two abstract operators can be defined:\n\t\\begin{align}\n\t\t& \\hat{\\epsilon}\\ket{\\mu}:=\\mu \\ket{\\mu} \\\\\n\t\t& \\hat{s}(\\lambda)\\ket{\\mu}:=\\ket{\\mu +\\lambda}.\n\t\\end{align}\n\t\\noindent The operator $\\hat{\\epsilon}$ is a symmetric operator and $\\hat{s}(\\lambda)$ defines a one-parameter family of unitary operators.\n\tIn spite of this $\\hat{s}(\\lambda)$ is discontinuous with respect to $\\lambda$;\n\tthis means that no self-adjoint operator exists that can generate $\\hat{s}(\\lambda)$ by exponentiation. \n\tTaking now in exam a physical system with configuration space spanned by the coordinate $q$, which is assumed to have a discrete character, and its conjugate momentum $p$, the previous abstract representation can be projected and studied with respect to p-polarization.\n\tIn this polarization the basis states will be:\n\t\\begin{equation} \\label{basic_states_p}\n\t\t\\psi_\\mu(p)=\\braket{p}{\\mu}=e^{i\\mu p\/\\hbar}.\n\t\\end{equation}\n\tFollowing the algebraic construction method, a Weyl-Hei\\-sen\\-berg\\- algebra is introduced on $\\mathcal{H}_{poly}$ and the action of its generators on the basis states is defined as follows:\n\t\\begin{align}\n\t\t&\\hat{\\mathcal{U}}(\\nu)\\psi_\\mu(p)=\\psi_\\mu(p+\\nu)=e^{i\\mu(p+\\nu)\/\\hbar}=e^{i\\mu\\nu \/\\hbar}e^{i\\mu p\/\\hbar} \\\\\n\t\t&\\hat{\\mathcal{V}}(\\lambda)\\psi_\\mu(p)=e^{i\\lambda p\/\\hbar}e^{i\\mu p\/\\hbar}=e^{i(\\lambda + \\mu)p\/\\hbar}=\\psi_{\\mu+\\lambda}(p).\n\t\\end{align}\n\tFrom this it can be inferred that the shifting operator $\\hat{s}(\\lambda)$ can be identified with the operator $\\hat{\\mathcal{V}}(\\lambda)$, which is then discontinuous in $\\lambda$; this means that the spatial translations generator, that is the momentum operator $\\hat{p}$, does not exist. \n\tOn the other hand, the operator $\\hat{\\mathcal{U}}(\\nu)$ is continuous, so that the translations generator in the momentum space, i.e. the position operator $\\hat{q}$, exists and it can be identified with the abstract operator $\\hat{\\epsilon}$.\n\t\n\t\\noindent Indeed:\n\t\\begin{equation} \\label{PQ_q_op}\n\t\t\\hat{q}\\psi_{\\mu}(p)=-i\\hbar \\partial_p\\psi_{\\mu}(p)=\\mu\\psi_{\\mu}(p).\n\t\\end{equation}\n\tIt can be proved \\cite{Corichi2} that the kinematic polymer Hilbert space in this polarization is explicitly given by $\\mathcal{H}_{poly,p}=L^2\\left(\\mathbb{R}_{B},d\\mu_{H}\\right)$, where $\\mathbb{R}_{B}$  is the so-called Bohr compactification of the real line and $d\\mu_{H}$ is the Haar measure. \n\t\n\t\\medskip \n\t\n\tA similar picture is obtained in the q-polarization: the momentum operator cannot still be defined, while it is possible to show that the fundamental wave functions are Kronecker deltas and that the kinematic polymer Hilbert space is explicitly given by $\\mathcal{H}_{poly,x}=L^2\\left(\\mathbb{R}_{d},d\\mu_{c}\\right)$, where $\\mathbb{R}_{d}$  is the real line equipped with a discrete topology and $d\\mu_{c}$ is the counting measure. \n\t\n\t\\medskip \n\t\n\tIn order to build the dynamics a Hamiltonian operator $\\hat{H}$ has to be defined on $\\mathcal{H}_{poly}$, but since $\\hat{p}$ do not exist, a direct implementation is not possible.\n\tTo overcome this problem the momentum operator can be approximated by defining on the configuration space of the system a regular graph $\\gamma_{\\mu}=\\{q \\in \\mathbb{R} \\> | \\> q=n\\mu, n \\in \\mathbb{Z}\\}$, where $\\mu$ is the fundamental scale introduced by the polymer representation. \n\tThe basis kets $\\ket{\\mu}$ can now be indicated as $\\ket{\\mu_n}$, where $\\mu_n=n\\mu$ are the points belonging to the graph $\\gamma_{\\mu_0}$.\n\tConsequently the generic states will be:\n\t\\begin{equation}\n\t\t\\ket{\\psi}_{\\gamma_{\\mu}}=\\sum_n a_n \\ket{\\mu_n},\n\t\\end{equation}\n\tand they will belong to the new Hilbert space $\\mathcal{H}_{\\gamma_{\\mu}} \\subset \\mathcal{H}_{poly}$, posed that they satisfy the condition $\\sum_n \\abs{a_n}^2 < \\infty$.\n\tSince the dynamics has to be closed in $\\mathcal{H}_{\\gamma_{\\mu}}$, the shift parameter $\\lambda$ has to be fixed equal to $\\mu$, hence the action of $\\hat{\\mathcal{V}}(\\lambda)$ will be:\n\t\\begin{equation}\n\t\t\\hat{\\mathcal{V}}(\\lambda)\\ket{\\mu_n}=\\hat{\\mathcal{V}}(\\mu)\\ket{\\mu_n}=\\ket{\\mu_{n+1}}.\n\t\\end{equation}\n\t\n\t\\noindent On a general ground, the variable $p$ it can be written as:\n\t\\begin{equation} \\label{PQ_p_var_approx}\n\t\tp \\approx \\frac{\\hbar}{\\mu} \\sin(\\frac{\\mu}{\\hbar}p)=\\frac{\\hbar}{2i\\mu}\\left(e^{i\\frac{\\mu}{\\hbar}p}-e^{-i\\frac{\\mu}{\\hbar}p}\\right),\n\t\\end{equation}\n\twhen the condition $p<<\\hbar\/\\mu$ holds.\n\t\n\t\\noindent Based on this approximation and visualizing the action of  $\\hat{\\mathcal{V}}(\\mu)$ in the p-polarization, it is clear that the operator $\\hat{p}$ and its action can be approximated as:\n\t\\begin{equation} \\label{PQ_p_op_approx}\n\t\t\\hat{p}_{\\mu}\\ket{\\mu_n}\\approx \\frac{\\hbar}{2i\\mu} \\left[\\hat{\\mathcal{V}}(\\mu)-\\hat{\\mathcal{V}}(-\\mu)\\right]\\ket{\\mu_n},\n\t\\end{equation}\n\twhere $\\mu$ acts as a regulator.\n\t\n\t\\noindent To approximate the operator $\\hat{p}^2$ two paths are possible:\n\t\\begin{equation} \\label{PQ_p^2_op_approx_1}\n\t\t\\hat{p}^2_{\\mu}\\approx \\frac{\\hbar^2}{4\\mu^2} \\left[ 2-\\hat{\\mathcal{V}}(2\\mu)-\\hat{\\mathcal{V}}(-2\\mu)\\right],\n\t\\end{equation} \n\tbased on the approximation \n\t\\begin{equation} \\label{PQ_p^2_var_approx_1}\n\t\tp^2 \\approx \\frac{\\hbar^2}{\\mu^2} \\sin[2](\\frac{\\mu}{\\hbar}p)\n\t\\end{equation}\n\tand hence defined by iterating the action of $\\hat{p}$ according \\eqref{PQ_p_op_approx}, or\n\t\\begin{equation} \\label{PQ_p^2_op_approx_2}\n\t\t\\hat{p}^2_{\\mu}\\approx \\frac{\\hbar^2}{{\\mu}^2} \\left[2-\\hat{\\mathcal{V}}(\\mu)-\\hat{\\mathcal{V}}(-\\mu)\\right],\n\t\\end{equation}\n\texploiting the approximation \n\t\\begin{equation}\\label{PQ_p^2_var_approx_2}\n\t\tp^2 \\approx \\frac{2\\hbar^2}{{\\mu}^2}\\left(1- \\cos(\\frac{\\mu}{\\hbar}p)\\right),\n\t\\end{equation}\n\tvalid as long as $p<<\\hbar\/\\mu$.\n\t\n\t\n\t\\noindent Hence, the well-defined, symmetric Hamiltonian operator will be:\n\t\\begin{equation}\n\t\t\\hat{H}_{\\mu}:=\\frac{\\hat{p}^2_{\\mu}}{2m}+\\hat{V}(\\hat{q}),\n\t\\end{equation}\n\twhere $\\hat{V}(\\hat{q})$ is the potential operator. \n\t\n\tTherefore, quantising a system according to the polymer representation, in the p-polarization, implies the use of the approximation \\eqref{PQ_p^2_op_approx_1} or \\eqref{PQ_p^2_op_approx_2} for the momentum operator, while the position operator will be the natural differential operator, which action is expressed in \\eqref{PQ_p_op_approx}.\n\t\n\tIn a semi-classical approach, this procedure corresponds to the proper introduction of the approximations \\eqref{PQ_p_var_approx}, \\eqref{PQ_p^2_var_approx_1} and \\eqref{PQ_p^2_var_approx_2} on the variable $p$ in the dynamics of the system of interest. \n\tHence, on this level, the whole procedure can be thought of as a prescription to provide physical insight into the behaviour of the quantum expectations values, according to the so-called Ehrenfest theorem.\n\t\n\t\n\t\\section{Polymer Kasner solution}\n\t\\label{Polymer Kasner solution}\n\t\n\tWe will now apply polymer formalism in a semi-classical framework to the study of a $5D$ Bianchi I model, which solution in the vacuum is the well-known Kasner metric \\cite{Kasner}\\cite{Landau}, focusing on the kinematics and dynamics of the fifth dimension. \n\t\n\t\\medskip\n\t\n\t\\noindent In order to obtain the polymer Kasner cosmological solution, we need a minisuperspace Hamiltonian formulation, extended to the $5D$ case.\n\t\n\t\\noindent The $5D$ Bianchi I line element, written in the ADM formalism \\cite{ADM}, is a straightforward generalisation of the $4D$ one\\footnote{As it should be clear from the context, the metric coefficient $c(t)$ is not to be confused with the velocity of light $c$.}:\n\t\\begin{equation}\n\t\t\\begin{split}\n\t\t\tds^2=&-N^2(t)c^2dt^2+^{(4)}h_{ij}dx^idx^j=\\\\\n\t\t\t&-N^2(t)c^2dt^2+a^2(t)(dx^1)^2+b^2(t)(dx^2)^2\\\\\n\t\t\t& +c^2(t)(dx^3)^2+d^2(t)(dx^5)^2, \\qquad (\\text{i,j=1,2,3,5})\n\t\t\\end{split}\n\t\\end{equation}\n\twhere $N(t)$ is the lapse function of the ADM formalism, $^{(4)}h_{ij}$ is the metric tensor of the $4D$ manifold, which coordinate are all space-like.\n\t\n\t\\noindent Having the general structure of this metric as starting point, we can build the Hamiltonian of the system:\n\t\\begin{equation} \\label{PBM_hamiltonian_bianchi_non_diag}\n\t\t\\begin{split}\n\t\t\tH_{Bianchi \\> I}:=H_{B}=Ne^{-\\sum_a q^a\/2}\\biggl\\{\\sum_a p_a^2-\\frac{1}{3}\\biggl[\\sum_b p_b\\biggr]^2\\biggr\\},\n\t\t\\end{split}\n\t\\end{equation}\n\twhich, by varying with respect to $N(t)$ turns out to be a constraint for the dynamics, namely $H_B=0$.\n\t\n\t\\noindent The couple $(q^a,p_a)$  in \\eqref{PBM_hamiltonian_bianchi_non_diag} are the conjugate variables spanning an highly symmetric phase space, the so-called \\textit{minisuperspace}, and the relation between the metric coefficients and the q-variables is the usual one of the literature \\cite{Montani}, extended to the $5D$ case:\n\t\\begin{equation} \\label{PBM_metric_coeff_to_q}\n\t\t\\begin{split}\n\t\t\t&a(t)=e^{q^1(t)\/2} \\quad b(t)=e^{q^2(t)\/2} \\\\\n\t\t\t&c(t)=e^{q^3(t)\/2} \\quad d(t)=e^{q^5(t)\/2}.\n\t\t\\end{split}\n\t\\end{equation}\n\t\n\tAs is well-known, it is more convenient to express the obtained Hamiltonian in its diagonal form:\n\t\\begin{equation} \\label{PBM_hamilt_bianchi_diagonal}\n\t\tH'_{B}=Ne^{-\\alpha}\\left[-\\frac{1}{3}p^2_\\alpha+p^2_{+}+p^2_{-}+p^2_{\\gamma}\\right],\n\t\\end{equation}\n\twhich is the canonical form of the quadratic form associated to $H_{B}$. \n\t\n\t\\noindent The p-variables in the Hamiltonian \\eqref{PBM_hamilt_bianchi_diagonal} are the conjugate momenta of a set of variables $\\alpha,\\beta_{+},\\beta_{-},\\gamma$, which represent the generalisation of the Misner variables \\cite{Misner}. The relation between these variables and the previous q-variables is defined through the following linear transformation:\n\t\\begin{equation} \\label{q_to_misner_var}\n\t\t\\begin{cases}\n\t\t\tq^1=\\frac{1}{2}\\alpha-\\frac{1}{2\\sqrt{3}}\\beta_{+}-\\frac{1}{\\sqrt{6}}\\beta_{-}-\\frac{1}{\\sqrt{2}}\\gamma\\\\ q^2=\\frac{1}{2}\\alpha-\\frac{1}{2\\sqrt{3}}\\beta_{+}-\\frac{1}{\\sqrt{6}}\\beta_{-}+\\frac{1}{\\sqrt{2}}\\gamma \\\\\n\t\t\tq^3=\\frac{1}{2}\\alpha-\\frac{1}{2\\sqrt{3}}\\beta_{+}+\\sqrt{\\frac{2}{3}}\\beta_{-} \\\\\n\t\t\tq^5=\\frac{1}{2}\\alpha+\\frac{\\sqrt{3}}{2}\\beta_{+}.\n\t\t\\end{cases}\n\t\\end{equation}\n\t\n\t\n\tBy using the Hamilton-Jacobi method and the Hamilton equation for the variable $\\alpha$ - which represents the universe volume - in the synchronous reference frame, the standard classical Kasner solution for the $5D$ case can be recovered:\n\t\\begin{equation} \\label{PBM_kasner_metric}\n\t\t\\begin{split}\n\t\t\tds^2&=-c^2dt^2+(t\/t_0)^{2k_1}(dx^1)^2+(t\/t_0)^{2k_2}(dx^2)^2\\\\\n\t\t\t&+(t\/t_0)^{2k_3}(dx^3)^2+(t\/t_0)^{2k_5}(dx^5)^2.\n\t\t\\end{split}\n\t\\end{equation}\n\t\n\t\\noindent The k parameters are the so-called Kasner exponents and they satisfy the following conditions:\n\t\\begin{equation} \\label{PBM_kasner_buondaries_standard}\n\t\t\\begin{cases}\n\t\t\tk_1+k_2+k_3+k_5=1 \\\\\n\t\t\tk_1^2+k_2^2+k_3^2+k_5^2=1.\n\t\t\\end{cases}\n\t\\end{equation}\n\t\\noindent In particular, if we assume isotropy in the three usual spatial dimensions, as observations suggest, that is if we set $k_1=k_2=k_3$, solution of the previous system becomes:\n\t\\begin{equation}\n\t\tk_1=k_2=k_3=\\frac{1}{2} \\quad k_5=-\\frac{1}{2}.\n\t\\end{equation}\n\t\\noindent This means that while the three usual spatial dimensions expand, the fifth one collapses indefinitely. \n\t\n\t\\medskip \n\t\n\tWe want now to introduce polymer formalism in Bianchi I dynamics.\n\tIn order to do this we choose to operate the substitutions \\eqref{PQ_p_var_approx} and \\eqref{PQ_p^2_var_approx_1} on the conjugate momentum $p_\\gamma$ of the Misner variable $\\gamma$, connected with the metric coefficient of the fifth dimension.\n\tThe new \\textit{polymerized} Hamiltonian will be:\n\t\\begin{equation} \\label{PBM_hamlit_poly_Bianchi}\n\t\tH_{B}^{poly}=e^{-\\alpha}\\biggl[-\\frac{1}{3}p^2_\\alpha+p^2_{+}+p^2_-+\\frac{\\hbar^2}{\\mu^2}\\sin[2](\\frac{\\mu}{\\hbar}p_\\gamma)\\biggr].\n\t\\end{equation}\n\t\n\t\\noindent By the exactly above procedure we find that the solution is still a Kasner solution, where the metric coefficients have a coordinate time power trend as in \\eqref{PBM_kasner_metric}, but their exponents, that is the Kasner indices, due to the quantum polymer modifications, satisfy different constraints:\n\t\n\t\\begin{equation}\n\t\t\\begin{cases}\n\t\t\tk_1+k_2+k_3+k_5=1 \\\\\n\t\t\tk_1^2+k_2^2+k_3^2+k_5^2=1-\\frac{3}{4}\\frac{\\hbar^2}{\\mu^2}\\frac{\\sin[4](\\frac{\\mu}{\\hbar}p_{\\gamma})}{\\sqrt{p^2_++p^2_-+\\frac{\\hbar^2}{\\mu^2}\\sin[2](\\frac{\\mu}{\\hbar}p_\\gamma)}}.\n\t\t\\end{cases}\n\t\\end{equation}\n\t\n\t\\noindent The second term of the right-hand side of the second condition is non-negative, so that we can restate the system as follows:\n\t\\begin{equation}\n\t\t\\begin{cases} \\label{PBM_kasner_ind_poly_conditions}\n\t\t\tk_1+k_2+k_3+k_5=1 \\\\\n\t\t\tk_1^2+k_2^2+k_3^2+k_5^2\\leq 1.\n\t\t\\end{cases}\n\t\\end{equation}\n\t\n\t\\noindent Assuming isotropy in the three usual spatial dimensions and introducing an order between exponents, in particular setting $k_5<k$, the system \\eqref{PBM_kasner_ind_poly_conditions} has the following solution:\n\t\\begin{equation} \\label{PBM_constraint_polymer_Kasner}\n\t\t\\begin{cases}\n\t\t\t1\/4 < k\\leq1\/2 \\\\\n\t\t\t-1\/2 \\leq k_5 < 1\/4.\n\t\t\\end{cases}\n\t\\end{equation}\n\t\n\tWe can notice that the Kasner exponents relative to the three usual spatial dimensions are bound to be positive and they can take value in a precise interval, while the exponent relative to the fifth dimension has value in an interval in which it can take both positive and negative values, which is a new feature of the Kasner solution entirely due to the polymer physics.\n\t\n\t\\noindent The latter statement grows in importance when we draw our attention to the permitted values of $k_5$.\n\t\n\t\\medskip\n\t\n\tIndeed, even if the power trend with respect to the coordinate time variable of fifth dimension metric coefficient - which is ultimately responsible for the singularity at an infinite time - is not removed in this scenario, the introduction of the polymer formalism leads to a modification on the constraint of the fifth Kasner exponent $k_5$, which interval of definition allows the value $k_5=0$.\n\t\n\t\n\t\n\t\\noindent For this particular choice of $k_5$, the other Kasner indices will be equal to $k=1\/3$, correctly reproducing the observed isotropic expansion in the three usual spatial dimensions, while the metric coefficient relative to the fifth dimension will be of course equal to one. \n\t\n\t\\noindent This means that the fifth dimension has no dynamics since the time-dependence of the relative metric coefficient disappears. \n\tWe, therefore, have obtained a static solution, which somehow solves the singularity problem by remarkably removing the indefinite collapse of the dimension itself.\n\t\n\t\\noindent The developments of the next sections will be based on this important result.\n\t\n\t\\section{Kaluza-Klein theory in polymer quantum mechanics framework}\n\t\\label{Kaluza-Klein theory in polymer quantum mechanics framework}\n\t\n\tIn this section we face the analysis of the Kaluza-Klein paradigm in the framework of Polymer Quantum Mechanics, both studying the behaviour of the geodesic motion and the quantum dynamics of a Klein-Gordon field. \n\tFrom the point of view of the geodesic discussion, the polymer framework is addressed on a semi-classical level, in the spirit of the Ehrenfest theorem. \n\tInstead, the study of the quantum scalar field is performed in a full quantum picture, as restricted to the fifth coordinate. \n\t\n\t\\subsection{Geodesic motion}\n\t\n\tIn this section polymer formalism will be now applied to the Hamiltonian formulation of the $5D$ geodesic motion, at a semi-classical level, that is introducing quantum modifications to the classical dynamics. \n\t\n\t\\medskip\n\t\n\tThe $5D$ Hamiltonian of a free particle in a general Kaluza-Klein background, with $\\tilde{g}_{55}=1$ - see previous section - reads as:\n\t\\begin{equation}\n\t\t\\tilde{H}=\\frac{1}{2mc}\\left[\\tilde{p}_{\\mu} \\tilde{p}_{\\nu} {g}^{\\mu\\nu}-\\frac{4\\sqrt{G}}{c^2}\\tilde{p}_\\mu \\tilde{p}_5A^{\\mu}+\\tilde{p}^2_5\\left(1+\\frac{4G}{c^4}A_{\\mu}A^{\\mu} \\right)\\right].\n\t\\end{equation}\n\twhere, as before, the quantities with a tilde are the $5D$ ones, that is those defined with respect to the $5D$ line element.\n\t\n\tWe want now introduce polymer formalism only with respect to the canonical couple $(x^5,\\tilde{p}_5)$, that is we assume that the coordinate $x^5$ has an essential discrete nature and we redefine the respective conjugate momentum $\\tilde{p}_5$ by introducing a regular graph structure on $S^1_{x^5}$, i.e. on the fifth dimension, which is now equipped with a discrete topology, according to the discussion in Section \\ref{Polymer quantum mechanics}.\n\tFollowing the polymer prescription \\eqref{PQ_p_var_approx}, we can made the substitution:\n\t\\begin{equation}\n\t\t\\tilde{p}_5 \\rightarrow \\frac{\\hbar}{\\mu}\\sin(\\frac{\\mu}{\\hbar}\\tilde{p}_5).\n\t\\end{equation}\n\t\n\t\\noindent The new Hamiltonian will be rewritten as:\n\t\\begin{equation}\n\t\t\\begin{split}\n\t\t\t\\tilde{H}_{poly}= & \\frac{1}{2mc}\\bigg[\\tilde{p}_{\\mu} \\tilde{p}_{\\nu} {g}^{\\mu\\nu}-\\frac{4\\sqrt{G}}{c^2}\\frac{\\hbar}{\\mu}\\tilde{p}_\\mu \\sin(\\frac{\\mu}{\\hbar}\\tilde{p}_5)A^{\\mu}\\\\\n\t\t\t&+\\frac{\\hbar^2}{\\mu^2}\\sin[2](\\frac{\\mu}{\\hbar}\\tilde{p}_5)\\left(1+\\frac{4G}{c^4}A_{\\mu}A^{\\mu} \\right)\\bigg],\n\t\t\\end{split}\n\t\\end{equation}\n\t\n\tfrom which the equations of motion (Hamilton equations) can be obtained:\n\t\\begin{equation} \\label{PG_polymer_equation_of_motion_4_index}\n\t\t\\begin{cases}\n\t\t\t\\tilde{u}^\\mu=\\frac{1}{mc}\\left[\\tilde{p}_{\\nu}{g}^{\\mu \\nu}-\\frac{2\\sqrt{G}}{c^2}\\frac{\\hbar}{\\mu}\\sin(\\frac{\\mu}{\\hbar}\\tilde{p}_5)A^{\\mu}\\right] \\\\\n\t\t\t\\begin{split}\n\t\t\t\t\\dot{\\tilde{p}}_{\\mu}=&-\\frac{1}{2mc}\\bigg[\\tilde{p}_{\\rho}\\tilde{p}_{\\sigma}\\partial_{\\mu}{g}^{\\rho \\sigma}-\\frac{4\\sqrt{G}}{c^2}\\tilde{p}_{\\rho}\\frac{\\hbar}{\\mu}\\sin(\\frac{\\mu}{\\hbar}\\tilde{p}_5)\\partial_{\\mu}A^{\\rho}\\\\\n\t\t\t\t&+\\frac{4G}{c^4}\\frac{\\hbar^2}{\\mu^2}\\sin[2](\\frac{\\mu}{\\hbar}\\tilde{p}_5)\\partial_{\\mu}\\left(A_\\nu A^\\nu\\right)\\bigg], \\\\\n\t\t\t\\end{split}\n\t\t\\end{cases}\n\t\\end{equation}\n\t\n\t\\begin{equation} \\label{PG_polymer_equation_of_motion_5}\n\t\t\\begin{cases}\n\t\t\t\\begin{split}\n\t\t\t\t\\tilde{u}^5=&\\frac{1}{mc}\\bigg[-\\frac{2\\sqrt{G}}{c^2}\\tilde{p}_\\mu A^{\\mu}\\cos(\\frac{\\mu}{\\hbar}\\tilde{p}_5)\\\\\n\t\t\t\t&+\\frac{\\hbar}{\\mu}\\sin(\\frac{\\mu}{\\hbar}\\tilde{p}_5)\\cos(\\frac{\\mu}{\\hbar}\\tilde{p}_5)\\left(1+\\frac{4G}{c^4}A_\\nu A^\\nu\\right)\\bigg] \n\t\t\t\\end{split} \\\\\t \n\t\t\t\\dot{\\tilde{p}}_5=0,\n\t\t\\end{cases}\n\t\\end{equation}\n\twhere the dot refers to the derivative with respect to the five-dimensional line element $d\\tilde{s}$. \n\t\n\t\\noindent We outline that in the last equation we made use of the \\textit{cylinder condition}, which implies that $\\tilde{p}_5$ is a constant of motion, as expected.\n\t\n\t\\noindent As we know from the standard case, in order to make a comparison with the $4D$ equations of motion, we need to know the relation between $4D$ and $5D$ quantities, in particular the relation between five-velocities and four-velocities.\n\tThese are formally the same of equation \\eqref{KK_five-four_velocity_rel} also in our framework.\n\t\n\tNevertheless, while in the ordinary theory, with the scalar field of the Kaluza Klein metric set to be constant, $u_5$ is a constant of motion, in the polymer framework this is not the case. \n\t\n\t\\noindent Indeed, from the explicit expression of $\\tilde{u}^5$ \\eqref{PG_polymer_equation_of_motion_5} we can obtain the corresponding expression of $\\tilde{u}_5$:\n\t\n\t\\begin{equation}\n\t\t\\tilde{u}_5=\\frac{2\\sqrt{G}}{c^2} \\tilde{u}^\\mu A_\\mu \\cos(\\frac{\\mu}{\\hbar}\\tilde{p}_5)+\\frac{\\hbar}{\\mu mc}\\sin(\\frac{\\mu}{\\hbar}\\tilde{p}_5)\\cos(\\frac{\\mu}{\\hbar}\\tilde{p}_5),\n\t\\end{equation}\n\t\n\twhere we have made use of the first Hamilton equation for $\\tilde{u}^{\\mu}$.\n\t\n\t\\noindent The latter can be solved in function of $u_5$ and two solutions are obtained:\n\t\\begin{equation} \\label{PG_u_5 exact expression}\n\t\t\\begin{split}\n\t\t\tu_5=&\\frac{1}{D(\\mu,\\tilde{p}_5)}\\bigg\\{16\\sqrt{G}  \\mu ^2 m^2   \\left(1-\\cos(\\frac{\\mu  }{\\hbar}\\tilde{p}_5)\\right)A_\\mu u^\\mu\\\\\n\t\t\t&\\pm \\sqrt{2}\\hbar\\abs{\\sin(2\\frac{\\mu }{\\hbar}\\tilde{p}_5)} \\sqrt{D(\\mu, \\tilde{p}_5)+ K(\\mu,\\tilde{p}_5)A_\\mu A^\\mu u_\\nu u^\\nu}\\bigg\\},\n\t\t\\end{split}\n\t\\end{equation}\n\twhere:\n\t\\begin{gather}\n\t\tD(\\mu, \\tilde{p}_5):={\\hbar^2+8\\mu ^2 m^2 c^2-\\hbar^2\\cos (4\\frac{\\mu  }{\\hbar}\\tilde{p}_5)} \\\\\n\t\tK(\\mu,\\tilde{p}_5):=-128 \\frac{G}{c^2} \\mu ^2 m^2\\sin[4](\\frac{\\mu}{2\\hbar}\\tilde{p}_5).\n\t\\end{gather}\n\t\n\t\\noindent These are rather complicated non-constant expressions, due to the dependence from the electromagnetic and the four-velocity fields, which consequently heavily modify the $\\frac{ds}{d\\tilde{s}}$ factor. \n\t\n\t\\noindent From the Hamilton equations \\eqref{PG_polymer_equation_of_motion_4_index}, by exploiting the formal relation \\eqref{KK_five-four_velocity_rel}, we can write down the part relative to the $4D$ indices of the new generalised $5D$ geodesic, as a function of the $4D$ quantities:\n\t\n\t\\begin{equation} \\label{PG_4d_polymer_geodesic_equation}\n\t\tu^{\\nu}\\nabla_{\\nu}u^{\\mu}=\\frac{2\\sqrt{G}\\hbar}{mc^3\\mu} \\sqrt{1-u_5^2}\\sin(\\frac{\\mu}{\\hbar}\\tilde{p}_5)u^{\\nu}F^{\\mu}_{\\nu}-\\frac{u_5u^{\\mu}}{1-u^2_5}\\frac{du_5}{ds},\n\t\\end{equation}\n\tin which it is possible to distinguish the polymer-modified part of the standard $4D$ geodesic \\eqref{KK_ordinary_4d_geodesic} and an extra term, entirely due to the variation of $u_5$. \n\tIt is worth pointing out that eq.\\eqref{PG_4d_polymer_geodesic_equation} correctly reduces to the standard one in the $\\mu \\to 0$ limit.\n\t\n\t\\noindent At this point, in order to compare the last expression with the equation of motion \\eqref{KK_ordinary_4d_geodesic} we need to make a series expansion of $u_5$ by assuming $A_\\mu << 1$.\n\tThis can be perfectly legitimate in a cosmological setting since there is no \n\trelevant coherent electromagnetic field which permeates the Universe.\n\t\n\tThis leads to the following expression, to the first order in $A_\\mu$:\n\t\\begin{equation}\n\t\t\\begin{split}\n\t\t\tu_5\\approx &\\pm\\bigg\\{\\frac{\\sqrt{2} \\hbar\\abs{\\sin(2\\frac{\\mu}{\\hbar}\\tilde{p}_5) }}{D(\\mu,\\tilde{p}_5)^{1\/2}}+\\frac{16\\sqrt{G}\\mu^2m^2 \\left(\\cos(\\frac{\\mu }{\\hbar}\\tilde{p}_5) -1\\right)}{D(\\mu,\\tilde{p}_5)} A_{\\mu}u^{\\mu}\\\\ \n\t\t\t&+O\\left(A^2\\right)\\bigg\\}.\n\t\t\\end{split}\n\t\\end{equation}\n\t\\noindent Hence, by inserting this expansion in the geodesic equation \\eqref{PG_4d_polymer_geodesic_equation} and ignoring the extra term - which meaning it is beyond the purpose of this article - we obtain, keeping only the overall first order term in $A_\\mu$:\n\t\\begin{equation}\n\t\tu^{\\nu}\\nabla_{\\nu}u^{\\mu}\\approx\\frac{2\\sqrt{G}\\hbar}{\\mu mc^3} \\frac{2\\sqrt{2}\\mu mc \\sin(\\frac{\\mu}{\\hbar}\\tilde{p}_5)}{\\sqrt{\\hbar^2+8\\mu^2m^2c^2-\\hbar^2 \\cos(4\\frac{\\mu}{\\hbar}\\tilde{p}_5)}}u^{\\nu}F^{\\mu}_{\\nu}\n\t\\end{equation}\n\tfrom which we can read off the perturbative polymer relation between the electric charge $q$ and $\\tilde{p}_5$ - which we remind it is a constant of motion:\n\t\\begin{equation} \\label{PG_polymer_q_expression}\n\t\tq\\approx\\frac{2\\sqrt{G}}{c} \\frac{2\\sqrt{2}\\hbar mc }{\\sqrt{\\hbar^2+8\\mu^2m^2c^2-\\hbar^2 \\cos(4\\frac{\\mu}{\\hbar}\\tilde{p}_5)}}\\sin(\\frac{\\mu}{\\hbar}\\tilde{p}_5).\n\t\\end{equation}\n\t\n\tTo correctly interpret this formula we can refer to the ordinary relation between $q$ and $\\tilde{p}_5$, which can be obtained from the expression \\eqref{KK_p5_rel_q}.\n\tIn fact, since in the standard case the following simple statement holds:\n\t\\begin{equation} \\label{PG_ordinary_rel_between_p_5}\n\t\t\\tilde{p}_5=\\frac{mc}{\\sqrt{m^2c^2-p^2_5}}p_5 \\qquad \n\t\tp_5=\\frac{mc}{\\sqrt{m^2c^2+\\tilde{p}^2_5}}\\tilde{p}_5,\n\t\\end{equation}\n\tthe relation \\eqref{KK_p5_rel_q} can be rewritten as:\n\t\\begin{equation} \\label{PG_rel_q_p5_standard}\n\t\tq=\\frac{2\\sqrt{G}}{c}\\frac{mc}{\\sqrt{m^2c^2+\\tilde{p}^2_5}}\\tilde{p}_5.\n\t\\end{equation}\n\t\\noindent This suggests that the expression \\eqref{PG_polymer_q_expression} is the polymer generalisation to the first perturbative order in the electromagnetic field of the latter relation.\n\t\n\t\\noindent This interpretation is confirmed by the $\\mu\\to0$ limit of \\eqref{PG_polymer_q_expression}, from which the standard expression \\eqref{PG_rel_q_p5_standard} is recovered. \n\t\n\t\\noindent Therefore, based on the relations \\eqref{PG_ordinary_rel_between_p_5}, we can impose:\n\t\\begin{equation}\n\t\t\\frac{\\hbar}{\\mu}\\sin(\\frac{\\mu}{\\hbar}p_5)\\approx\\frac{2\\sqrt{2}\\hbar mc }{\\sqrt{\\hbar^2+8\\mu^2m^2c^2-\\hbar^2 \\cos(4\\frac{\\mu}{\\hbar}\\tilde{p}_5)}}\\sin(\\frac{\\mu}{\\hbar}\\tilde{p}_5)\n\t\\end{equation}\n\tand, in the end:\n\t\\begin{equation} \\label{PG_poly_charge_p5}\n\t\tq\\approx\\frac{2\\hbar\\sqrt{G}}{\\mu c}\\sin(\\frac{\\mu}{\\hbar}p_5).\n\t\\end{equation}\n\t\\noindent This is a periodic and bounded function of $p_5$ with period $2\\pi$, defined in the generic periodic interval $[\\pi(2k-1)\\hbar\/\\mu,\\pi(2k+1)\\hbar\/\\mu]$, with $k\\in\\mathbb{Z}$. \n\tSince we want to preserve the Cauchy problem symmetry with respect to the initial values of $p_5$, that is the symmetry of left-handed and right-handed particles in the fifth direction, we are led to choose the interval $[-\\pi\\hbar\/\\mu,\\pi\\hbar\/\\mu]$, i.e. the one for k=0, as the natural periodic interval of definition.\n\tIn the $\\mu\\to0$ limit, the obtained function correctly reproduces the ordinary relation inferable from \\eqref{KK_p5_rel_q}.\n\t\n\tAs in the ordinary case, being the fifth dimension a compact space, we have to assume that $p_5$ is quantized; in order to determine the quantisation law, we need to represent the free particle state in the fifth coordinate as described by the wave function of the polymer free particle. \n\tAs comprehensively discussed in \\cite{Corichi2}, in the position representation, this will be:\n\t\\begin{equation} \\label{PG_free_poly_particle_normalized}\n\t\t\\psi_{\\mu}(x^5)=\\frac{1}{\\sqrt{L}}e^{ ix^5p_{5}^{(\\mu)}\/\\hbar},\n\t\\end{equation}\n\tnormalized on $S^1_{x_5}$, where $L$ is the length of the circus which characterizes the fifth dimension.\n\tHence, by imposing $\\psi_{\\mu}(x^5)$ to be periodic in $x^5$, we achieve the following quantisation law:\n\t\\begin{equation} \\label{PG_polymer_momentum_quantisation}\n\t\tp_{5}^{(\\mu)}= \\frac{2\\pi n}{L}\\hbar  \\qquad n \\in\\mathbb{Z},\n\t\\end{equation}\n\twhich immediately leads, accordingly to \\eqref{PG_poly_charge_p5}, to a new quantisation law for the electric charge:\n\t\\begin{equation} \\label{PG_charge_quantisation}\n\t\tq_{n}\\approx\\frac{2\\hbar\\sqrt{G}}{\\mu c}\\sin(\\frac{2\\pi \\mu}{L}n),\n\t\\end{equation}\n\tdefined in the aforementioned periodic interval.\n\t\n\t\\noindent We can choose $q_n$ to be equal to the electron electric charge $e$ for $n=1$, as in the ordinary case, and in this way we are able to find an expression for $L$:\n\t\\begin{equation} \\label{PG_L_function}\n\t\tL\\approx\\frac{2\\pi\\mu}{\\arcsin(\\frac{e\\mu c}{2\\hbar\\sqrt{G}})},\n\t\\end{equation}\n\t\n\tfrom which we deduce a constraint for the polymer scale $\\mu$:\n\t\\begin{equation} \\label{PG_scale_constraint}\n\t\t0 < \\mu \\leq \\frac{2\\hbar\\sqrt{G}}{\\abs{e}c} \\approx 3.78 \\cdot 10^{-32} \\> cm.\n\t\\end{equation}\n\tAt this point we choose $\\mu$ equal to the Planck length and obtain: \n\t\\begin{equation} \\label{PG_poly_geo_L}\n\t\tL\\approx 2.377 \\cdot 10^{-31} \\> cm,\n\t\\end{equation}\n\twhich almost coincide with the result of the standard theory and therefore it can account for the non observability of the fifth dimension.\n\t\n\tThere are basically two reasons behind this particular choice of the polymer scale: \n\t\\begin{itemize}\n\t\t\\item it is a scale with a strong physical meaning;\n\t\t\\item the $L\/\\mu$ ratio, for this value of $\\mu$, is large enough to ensure some calculations to be carried out in the polymer continuum limit, through the assumptions discussed in \\cite{Corichi1} and \\cite{Corichi2}.\n\t\\end{itemize}\n\t\n\t\\noindent A further discussion about the function \\eqref{PG_L_function} is postponed to the next subsection.\n\t\n\t\\medskip \n\t\n\tThe charge function \\eqref{PG_poly_charge_p5}, instead, can be rewritten, by means of the function $L(\\mu)$ \\eqref{PG_L_function}, as follows:\n\t\\begin{equation} \\label{PG_charge_distr_func_poly}\n\t\tq(\\mu;n)=\\frac{2\\hbar\\sqrt{G}}{\\mu c}\\sin\\bigg(n\\arcsin(\\frac{\\mu ec}{2\\hbar\\sqrt{G}})\\bigg)\n\t\\end{equation}\n\t\n\tand by setting the polymer scale $\\mu$ equal to the Planck length, we obtain the symmetric distribution of positive and negative charges reported in Figure \\ref{PG_planckian_charge_distribution_plt}.\n\t\n\t\\noindent The number of modes $n$ in the considered interval is limited by the periodic condition of the function itself and it clearly depends on $\\mu$.\n\tFor our choice of $\\mu$ we find that $-73\\leq n \\leq 73$ (see again Figure \\ref{PG_planckian_charge_distribution_plt}). \n\t\n\t\\noindent It is worth noticing that for any fixed value of $n=n^*$ it is always possible to expand the sine function in correspondence to a suitable small cut-off parameter $\\mu$, accordingly to the inequality $n^*\\mu<<2\\hbar\\sqrt{G}\/ec$, where we also have expanded the function $L(\\mu)$.\n\t\n\t\\noindent In this limit we recover the standard expression for the charge as multiple of the elementary electron charge, at least for $n<n*$.\n\t\n\t\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\\includegraphics[width=%\n\t\t0.5\\textwidth]{poly_charge_geo}\t\n\t\t\\caption{Plot of charges distribution for a value of $\\mu$ equal to the Planck length. It is possible to appreciate the oscillating profile of the function and the negative and positive symmetric branches.}\n\t\t\\label{PG_planckian_charge_distribution_plt}\n\t\\end{figure}\n\t\n\t\\medskip\n\t\n\t\n\tIn the end, we want to evaluate the consequence of the condition $\\abs{u_5}<1$ in our framework.  \n\t\n\t\\noindent The relation between $u_5$ and $q$ can be found employing the expression \\eqref{PG_polymer_q_expression}.\n\t\n\t\\noindent Hence, by retaining only the zero-order term in $u_5$ - coherently with the overall expansion in $A_{\\mu}$ and its derivatives we have done in the equation of motion \\eqref{PG_4d_polymer_geodesic_equation} - we can write:\n\t\\begin{equation}\n\t\t\\abs{u_5}\\approx \\frac{\\abs{q}}{2m\\sqrt{G}}\\sqrt{\\frac{1}{2}+\\frac{2 G}{q^2m^2}\n\t\t\t\\pm \\frac{1}{q^2\\hbar}F(\\mu)},\n\t\\end{equation}\n\twhere:\n\t\\begin{equation}\n\t\tF(\\mu):=\\sqrt{4\\mu ^2 m^2c^2 q^4+16G^2\\hbar^2m^4-8G \\hbar^2 m^2 q^2+h^2 q^4}.\n\t\\end{equation}\n\t\n\t\\noindent By imposing the condition $\\abs{u_5}<1$, the solution with plus sign can be ruled out, since it always violates the constraint, while for the solution with minus sign - which is the meaningful one - we obtain:\n\t\\begin{equation} \\label{PG_poly_q\/m_constraint}\n\t\t\\frac{\\abs{q}}{m}\\leq \\frac{2\\hbar\\sqrt{G}}{\\mu m c},\n\t\\end{equation}\n\twhich is in agreement with the quantisation law \\eqref{PG_poly_charge_p5}.\n\t\n\t\\medskip\n\t\n\t\\noindent This constraint - that we stress is valid only in a perturbative regime - is different from the ordinary one \\eqref{KK_q_over_m} and it introduces a dependence in the $q\/m$ ratio from the polymer scale $\\mu$ - as expected - and from the mass itself.\n\tRemarkably the empirical $q\/m$ ratio of any known particle always respects the bound for every value of $\\mu$ in the interval \\eqref{PG_scale_constraint}.\n\tThis means that the new constraint does not set an unphysical condition and at least does not contradict the experimental evidence, hence resolving one of the shortcoming aspects of the standard theory. \n\tIn other words, we are now able, in principle, to reproduce the $q\/m$ ratio of any Standard Model particle.\n\t\n\t\n\t\\subsection{Polymer complex scalar field coupled with Kaluza-Klein metric}\n\t\n\tIn this subsection we will carry out the study of a complex scalar field polymer dynamics, on a Kaluza-Klein background, obtained through a proper perturbation of the previous polymer Kasner metric.\n\t\n\t\\medskip \n\t\n\tFollowing the literature \\cite{Chodos-Detweiler}\\cite{Montani}\\cite{Landau}, under the isotropy assumption in the three usual spatial dimensions, we rewrite the polymer Kasner solution as:\n\t\\begin{equation}\n\t\t\\begin{split}\n\t\t\tds^2=&-c^2dt^2+\\left(\\frac{t}{\\tau}\\right)^{2k}(d\\vec{x})^2+\\left(\\frac{t}{\\tau}\\right)^{2k_5}(dx^5)^2,\n\t\t\\end{split}\n\t\\end{equation}\n\twhere $\\tau$ is a time characteristic of the present age of the Universe and we have rescaled the coordinates by the same factor.\n\t\n\t\\noindent Since we are interested in the static case with respect to the fifth dimension, we choose $k_5=0$ and hence, as we have seen in the previous section, $k=1\/3$:\n\t\\begin{equation} \\label{PCSF_Kasner_metric}\n\t\tds^2=-c^2dt^2+\\left(\\frac{t}{\\tau}\\right)^{2\/3}(d\\vec{x})^2+(dx^5)^2.\n\t\\end{equation}\n\tTaken into account the rescaling, the coordinates will now satisfy the following conditions:\n\t\\begin{equation} \\label{x5_values}\n\t\t0\\leq \\abs{\\vec{x}}<L(\\tau\/t_0)^{k} \\qquad  0\\leq x^5<L.\n\t\\end{equation}\n\t\n\t\\noindent In order to include a perturbative electromagnetic field, a small perturbation $h_{\\mu 5}$ ($\\mu$=0,1,2,3) is added to the polymer-modified Kasner metric, properly proportional to the electromagnetic field itself:\n\t\\begin{equation}\n\t\th_{\\mu 5}=\\frac{2\\sqrt{G}}{c^2}A_{\\mu}.\n\t\\end{equation}\n\t\\noindent As stated in the previous section this is naturally justified in a cosmological setting.\n\t\n\tCoupled to this background we introduce a quantum complex scalar field $\\Phi$, which ordinary dynamics is described by a $5D$ Klein-Gordon equation and we set the $5D$ \"mass\" term equal to zero:\n\t\\begin{equation} \\label{PCSF_KG_5d_eq}\n\t\t^{(5)}\\Box \\Phi(x^a)=0 \\qquad a=0,1,2,3,5.\n\t\\end{equation}\n\t\n\t\\noindent As in the standard case, our analysis relies on the assumption of the cylinder hypothesis for observable physical quantities, still providing the scalar field $\\Phi$ with a phase factor depending on $x^5$, the latter described by the polymer free particle wave function, with periodicity condition on the fifth coordinate due to the topology of the space:\n\t\\begin{equation}\n\t\t\\Phi(x^{\\mu},x^5)=\\frac{1}{\\sqrt{L}}\\phi(x^\\mu) e^{ix^52\\pi n\/L}.\n\t\\end{equation}\n\twhere the function $\\phi(x^\\mu)$ depends only on the variables of the $4D$ space-time. \n\t\n\t\\noindent A 4+1 dimensional splitting of eq.\\eqref{PCSF_KG_5d_eq} results in the following expression:\n\t\\begin{equation} \\label{PCSF_KG_5d_eq_splitted}\n\t\t\\begin{split}\n\t\t\t&\\partial_\\mu\\partial^\\mu\\Phi-\\frac{2\\sqrt{G}}{c^2}\\partial_\\mu A^\\mu\\partial_5\\Phi-\\frac{4\\sqrt{G}}{c^2}A^\\mu\\partial_\\mu\\partial_5\\Phi\\\\\n\t\t\t&+\\left[1+\\frac{4G}{c^4}A_\\nu A^\\nu\\right]\\partial_5^2\\Phi=0,\n\t\t\\end{split}\n\t\\end{equation}\n\tWe want now to study eq.\\eqref{PCSF_KG_5d_eq_splitted} by introducing polymer quantum framework only with respect to the fifth dimension; therefore we switch to momentum representation only on the fifth coordinate and then let the operator $\\hat{p}_5$ act according to the polymer prescription \\eqref{PQ_p_op_approx}.\n\t\n\t\\noindent It is worth noticing that, acting in this way, we are working in a mixed representation of position and momentum, which would not be possible universally speaking, nevertheless, since the coupling term in \\eqref{PCSF_KG_5d_eq_splitted} between $x^{\\mu}$ and $x^5$ is a small perturbation, we are legitimate, under this assumption, to proceed along this path, according to the diagonal form of the background metric.\n\t\n\t\\noindent Finally, the following equation for the complex scalar field is achieved:\n\t\\begin{equation} \\label{PCSF_poly_KG_field_eq}\n\t\t\\begin{split}\n\t\t\t&{}^{(4)}\\Box\\phi(x^\\mu)-i\\frac{2\\sqrt{G}}{c^2\\mu}\\phi(x^\\mu)\\partial_{\\mu}A^{\\mu}\\frac{1}{\\mu}\\sin({\\mu}\\frac{2\\pi n}{L})\\\\\n\t\t\t&-i\\frac{4\\sqrt{G}}{c^2\\mu}\\sin({\\mu}\\frac{2\\pi n}{L})A^{\\mu}\\partial_\\mu\\phi(x^\\mu)\\\\\n\t\t\t&-\\left[1+\\frac{4G}{c^4}A_\\nu A^\\nu\\right]\\frac{1}{\\mu^2}\\sin[2]({\\mu}\\frac{2\\pi n}{L})\\phi(x^\\mu)=0.\n\t\t\\end{split}\n\t\\end{equation}\n\t\n\t\\noindent By comparing term by term the latter with the $4D$ equation of a massive complex scalar field, coupled to an electromagnetic one, on a curved space-time, we attain the following identifications:\n\t\\begin{equation} \\label{PCSF_polymer_law_charge_and_mass}\n\t\t\\begin{split}\n\t\t\t&q_n=\\frac{2\\hbar \\sqrt{G}}{\\mu c}\\sin(\\frac{2\\pi \\mu}{L}n) \\\\\n\t\t\t&m_n=\\frac{\\hbar}{\\mu c}\\abs{\\sin(\\frac{2\\pi \\mu}{L}n)}.\n\t\t\\end{split}\n\t\\end{equation}\n\t\\noindent The first one is again a quantisation law for the electric charge and it is clearly the same we have obtained in \\eqref{PG_charge_quantisation}, in the modified geodesic study.\n\t\n\t\\noindent Hence, by imposing once again $q_n=e$ for $n=1$, we recover for the size $L$ of the fifth dimension the function \\eqref{PG_L_function}, with the constraint \\eqref{PG_scale_constraint}.\n\t\n\t\\noindent This function is a monotonically decreasing function of $\\mu$, which reaches its minimum value at the right endpoint of the interval in which is defined and it tends exactly to the value of the ordinary theory \\eqref{KK_ordinary_L} as $\\mu$ approaches zero (see Figure \\ref{L_function_plt}).\n\t\n\t\\noindent Clearly, choosing again $\\mu$ around the Planck length, we find for the value of $L$ the result \\eqref {PG_poly_geo_L}, discussed above.\n\t\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\\includegraphics[width=%\n\t\t0.5\\textwidth]{L_function}\t\n\t\t\\caption{Monotonically decreasing trend of the size $L$ of the fifth dimension as a function of the polymer scale $\\mu$. The constraint \\eqref{PG_scale_constraint} for $\\mu$ establishes a precise domain and codomain for the function $L(\\mu)$.}\n\t\t\\label{L_function_plt}\n\t\\end{figure}\n\t\n\t\\noindent On the other hand, the second relation \\eqref{PCSF_polymer_law_charge_and_mass} represents a mass distribution law for the scalar field, which, by means of the expression \\eqref{PG_L_function}, can be rewritten as a function\n\tof the integer parameter $n$ and of the continuous variable $\\mu$:\n\t\\begin{equation} \\label{mass_distr_func_poly}\n\t\tm(\\mu;n)=\\frac{\\hbar}{\\mu c}\\abs{\\sin\\bigg(n\\arcsin(\\frac{\\mu ec}{2\\hbar\\sqrt{G}})\\bigg)}.\n\t\\end{equation}\n\t\n\t\\noindent This function - for the same reasons stated in the previous subsection regarding the charge function - can be defined in the periodic interval $[-\\pi \\hbar\/\\mu,\\pi \\hbar\/\\mu]$, but, because of the presence of the modulus, we can restrict our attention only on the positive segment of the interval itself, that is $[0, \\pi \\hbar\/\\mu]$.\n\tFor a fixed value of the polymer scale $\\mu$, the resulting sequence of masses, due to the periodicity condition, will include only a finite number of mass modes $n$ and it clearly will result bounded and oscillating, as $n$ changes. \n\t\n\t\\noindent This freedom in the choice of the scale leads to a crucial feature of such distribution: picked up a particular mode $n$, through a fine-tuning of $\\mu$, it is possible to fit any desired value of the mass, therefore also Standard Model-comparable masses. \n\tIn particular, the spectrum can be explored in correspondence to a Planckian-like value of the parameter $\\mu$. \n\t\n\t\\noindent The specific pattern of the distribution then will depend on the chosen reference mass to be fitted to a certain mode $n$, for a fixed value of $\\mu$; nevertheless, it can be shown that a qualitative general trend can be recovered, which provides the presence of the assigned mass in the minimum of the sequence as a ground level of the Kaluza-Klein tower.\n\tWe stress that these regular values appear accompanied by Planckian masses, as sketched in Figure \\ref{planckian_masses_distribution_plt}.\n\t\n\t\\noindent This distribution deeply differs from the one obtained in the standard theory, reported in \\cite{Chodos-Detweiler}.\n\tIn fact, the latter is a linear and increasing sequence of only Planckian masses, without an upper limit.\n\t\n\t\\noindent Actually, the possibility of fitting arbitrary masses and from this obtaining various distributions can be achieved also in the ordinary theory, as shown in \\cite{Chodos-Detweiler}, by introducing a $5D$ mass term $a$ in the Klein-Gordon equation \\eqref{PCSF_KG_5d_eq}, which role is that of an additive parameter to be fine-tuned.\n\t\n\t\\noindent Nevertheless, this way of proceeding reveals several issues, which consequently are all addressed in our framework, not being necessary the introduction of such an extra quantity $a$.\n\t\n\t\\noindent First of all, in general, the fine-tuning procedure cannot be applied to every mode, since the linear and increasing trend of the distribution is not modified at all; in the second place, the addition of this parameter eventually generates the rise of tachyonic masses in the past.\n\tThis happens because of the presence of the metric coefficient $(t\/\\tau)^{\\abs{k_5}}$ in the standard momentum expression, as discussed in \\cite{Chodos-Detweiler}, which cannot be removed, since in the ordinary theory the Kasner solution does not admit the case $k_5=0$. \n\t\n\t\\noindent This means that in our framework the tachyonic masses are ruled out from the polymer-modified distribution. \n\t\n\t\\smallskip\n\t\n\tWe outline that - in our analysis - even in the case $k_5\\neq 0$, i.e. in the non-static case, the tachyonic problem would be solved since ultimately it is the chance to set $a=0$ which removes the unphysical masses.\n\t\n\t\\smallskip\n\t\n\tFinally, the $a$ parameter has to be introduced on purpose in the theory and its physical meaning remains ambiguous, while in our framework the fine-tuning parameter is an internal degree of freedom of the theory.\n\t\n\t\\medskip\n\t\n\tHence, even if polymer quantum mechanics fails in removing completely the Planckian masses from the spectrum, it succeeds first in introducing a cut-off since the masses distribution ref is bounded.\n\tFurthermore, in the considered natural interval a finite number of variable mass values take place. \n\tin the masses distribution in the considered periodic interval of definition of the masses function itself and secondly in accounting for Standard Model-comparable masses, under a proper fine-tuned choice of the scale variable $\\mu$.\n\t\n\t\\medskip \n\t\n\tSince we are dealing with a complex scalar field, the charged pions $\\pi^{\\pm}$ seem to be relevant (phenomenologically) candidates in Nature for the reference masses.\n\t\n\t\\noindent It results that the pion mass can be fitted at the mode $n=73$ for a fine-tuned value of $\\mu$ (up to twenty decimal figures) almost equal to the Planck length, which we will denote from now on as $\\mu_{Planck}$.\n\t\n\t\\noindent Nevertheless, for this procedure to be consistent, it is necessary to verify that the resulting mode $n=73$ belongs to the range of $n$ admitted in the periodic interval of definition of the masses function.\n\t\n\t\\noindent It is easy to show that for $\\mu=\\mu_{Planck}$ the condition for positive $n$ for belonging - in which we are interested - is $n\\leq73$.\n\t\n\t\\noindent Therefore, the results coming from the fit procedure are valid and legitimate in this regard.\n\t\n\tIn Figure \\ref{planckian_masses_distribution_plt} it is reported the whole sequence of masses resulting for $\\mu=\\mu_{Planck}$, where it is possible to appreciate the general behaviour discussed above.\n\t\n\t\\noindent In particular, we observe that the pion mass is in the minimum of the distribution and that the maximum available mass is always: \\[m_{max}=\\frac{\\hbar}{\\mu c}\\]\n\twhich is inversely proportional to $\\mu$, while the first mode of the sequence ($n=1$) is: \n\t\\begin{equation}\n\t\tm(\\mu;1)=\\frac{e}{2\\sqrt{G}}\\approx 9.29\\cdot 10^{-7} \\> g.\n\t\\end{equation}\n\t\n\t\\noindent It does not depend on $\\mu$ and it is almost a Planckian mass, defined only by fundamental constants, equal to the one obtained in the standard framework (without the introduction of the \\textit{ad hoc} parameter $a$).\n\t\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\\includegraphics[width=%\n\t\t0.5\\textwidth]{mass_poly}\t\n\t\t\\caption{Plot of masses distribution for the complex scalar field, for the value $\\mu_{Planck}\\approx 1.627\\cdot10^{-33} \\> cm$. It is possible to observe the oscillating profile of the function and the fitted Pion mass, placed in the minimum, as ground level of the Kaluza-Klein tower.}\n\t\t\\label{planckian_masses_distribution_plt}\n\t\\end{figure}\n\t\n\tIn Figure \\ref{KG_planckian_charge_distribution_plt} instead the corresponding charge distribution - which again can be put in the form \\eqref{PG_charge_distr_func_poly} - for $\\mu=\\mu_{Planck}$ is represented. \n\t\n\t\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\\includegraphics[width=%\n\t\t0.5\\textwidth]{poly_charge_KG}\t\n\t\t\\caption{Plot of charges distribution for value $\\mu=\\mu_{Planck}$. Again, it is possible to observe the oscillating profile of the sequence and the negative and positive symmetric branches.}\n\t\t\\label{KG_planckian_charge_distribution_plt}\n\t\\end{figure}\n\t\n\t\\noindent We observe, however, that only the electron charge ($n=1$) has a phenomenological correspondence, while the remaining points of the sequence have not a clear interpretation.\n\tIn particular according to these mass and charge distributions the fundamental charge $e$ has to be associated with a particle of Planckian mass, while the corresponding charge of the pion would be several orders of magnitude smaller than the electron charge.\n\t\n\t\n\t\\noindent Clearly, this peculiar charge-mass configuration is not phenomenological consistent.\n\t\n\t\\noindent Indeed, calculating the $q\/m$ ratio for the modes of the scalar field, we obtain:\n\t\\begin{equation}\n\t\t\\frac{q(\\mu;n)}{m(\\mu;n)}=\\pm 2\\sqrt{G}\\approx 5.16\\cdot 10^{-4} e.s.u.\/g,\n\t\\end{equation}\n\twhere the $\\pm$ sign is due to the sign of the electric charge.\n\t\n\t\\noindent This value, which coincides with the upper limit of the $q\/m$ ratio \\eqref{KK_q_over_m} of the standard classical case, does not depend on $\\mu$ nor $n$, rather it is constant for every polymer scale and every mode and no known particle satisfies such a relation. \n\t\n\t\n\t\\section{Conclusion}\n\t\\label{Conlusion}\n\t\n\tWe investigated the formulation of a five-dimensional Kaluza-Klein theory in the framework of Polymer Quantum Mechanics, viewed both in a semi-classical and quantum approach. \n\tThe polymer modifications have been implemented to the fifth coordinate only, on a semi-classical level in the spirit of the Ehrenfest theorem (the modification provides the dynamics of the quantum expectation values) and \n\tin a full quantum approach when a Klein-Gordon equation has been investigated. \n\t\n\t\\medskip \n\t\n\tWe started by applying the semi-classical polymer formulation to the evolution of the Bianchi I model, by showing that the corresponding Kasner solution can be taken in a form in which three scale factors isotropically expand and the remaining one is static and, in the considered model, it coincides with the compactified extra-dimension. \n\t\n\t\\noindent Then we studied the geodesic motion of a particle, starting with a Hamiltonian formulation (the only one in which the polymer formulation is viable) and then turning to a formalism based on the particle velocities.\n\tThis procedure allows, in analogy to the standard literature on this same subject, to identify the expression for the electric charge via the fifth momentum component of the particle. \n\tThe important consequence of this revised formulation consists of the overcoming of the problem of a too small charge to mass ratio to account in the model for any known elementary particle. \n\tIn fact, the revised constraint, due to the polymer relation between the fifth momentum component and the corresponding velocity, is in principle compatible with all the elementary particle predicted by the Standard Model. \n\t\n\t\\noindent Finally, we implemented a quantum polymer modification in the Klein-Gordon equation, by adopting a mixed representation of quantum mechanics (based on the coordinates for the usual four dimensions and the momentum for the extra one). \n\tThis study has the aim to revise the analysis in \\cite{Chodos-Detweiler} for a static (now available) extra-dimension, under a polymer prescription for the \n\tcompactified dimension physics. \n\t\n\tWe got the fundamental result that the tachyon mode, present in \\cite{Chodos-Detweiler} is now removed from the mass spectrum and that the obtained values for the boson mass can fit the values spanned in the Standard Model. \n\tActually, we arrived at a deformed morphology of the so-called Kaluza-Klein tower (the steps are no longer equispaced), but this revised structure allows us to avoid the only Planckian mode naturally present in the standard Kaluza-Klein formulation. \n\t\n\t\\medskip \n\t\n\tAll these results suggest that some of the puzzling question affecting the viability of the Kaluza-Klein idea must be reanalysed phenomenologically including the notion of a cut-off physics. \n\tIn fact, in the case of small dimensions, living about two orders of magnitude over the Planck scale, it should be unavoidable to feel the effects of the nearby cut-off and when its presence is made manifest a new paradigm can be assessed.\n\tIn other words, we argue that some limits of the geometrical unification theories are possibly due to the ultraviolet divergence that the gravitational field possesses and when they are somehow attenuated, like by the polymer scenario adopted here, the compactified dimension takes a more regular behaviour, which is reflected into the solution of some inconsistencies of the underlying model.\n\t\n\tThe emergence of a static dimension in the $5D$ Kasner solution - which prevents the necessity to deal with unphysical tachyonic modes - undoubtedly represents the simplest elucidation of this point of view. \n\t\n\t\n\t\n\\section{Introduction}\n\t\n\t\n\tThe original Kaluza-Klein idea \\cite{Kaluza}\\cite{Klein1}\\cite{Klein2} consists in a $5D$ space-time formulation having the aim to include in a geometrical picture also the electromagnetic interaction. \n\t\n\t\\noindent The surprising formal success in providing a metric representation of the vector potential suggested, in the Seventies, to attempt for a geometrical unification \\cite{ModernKKtheories}, able to assess all the fundamental interactions into a multi-dimensional space-time, with particular attention to the Electroweak Model. \n\tThe suggestive idea at the ground of these approach consists of the possibility to reproduce the Lie algebra, characterizing the elementary particle symmetries by the isometries of the extra-dimensional space. The non-trivial result obtained by the extra-dimensional Kaluza-Klein theories relies on the emergence from the multi-dimensional Einstein-Hilbert Lagrangian of the correct Yang-Mills action for the vector bosons which are the interaction carriers. \n\t\n\tHowever, many non-trivial problems affected this fascinating attempt for a geometrization of Nature. One of the main questions came out from the difficulty to provide a geometrical version of the chirality singled out by the electroweak interaction \\cite{Wetterich}, as well as the impossibility to represent the Standard Model of elementary particles in a Kaluza-Klein scenario \\cite{Witten}.\n\tFor alternative non-Riemannian approaches to solve the chirality problem of the Electroweak model see \\cite{Cianfrani-Montani1}\\cite{Cianfrani-Montani2}. \n\t\n\t\n\t\\noindent Finally, we observe that a full geometrical picture of Nature would involve the geometrical formulation of the fermionic field, a really non-trivial perspective if supersymmetry is not considered \\cite{Ferrara}. \n\t\n\t\\medskip\n\t\n\tEven the $5D$ Kaluza-Klein theory presents some important difficulties, see for a review \\cite{Cianfrani-Marrocco-Montani}, which leaves open the question concerning the viability of this approach as a geometrization of the electromagnetic interaction. \n\t\n\t\\noindent First of all, the $5D$ metric tensor contains an additional degree of freedom besides the $4D$ metric and the vector potential, namely the fifth diagonal component. \n\tUnder the necessary restriction of the coordinate transformation in order to deal with the $U(1)$ symmetry, this quantity behaves as an additional scalar field, which presence non-trivially affects basic features of the electromagnetism, for instance, the charge conservation itself \\cite{ModernKKtheories}\\cite{Lacquaniti-Montani}\\cite{Lacquaniti-Montani-Vietri}. \n\tBut, even fixing this scalar field to unity in the Lagrangian for the model (with the right sign of a space-like component), nonetheless, the ratio between the charge and the mass of an elementary particle is constrained to remain too small in order to reproduce the Standard Model spectrum of masses (for a proposal to solve the charge to mass ratio problem see \\cite{Lacquaniti-Montani}).\n\t\n\t\\noindent Finally, studying the morphology of a five-dimensional D'A\\-lam\\-ber\\-tian\\- operator, it is immediate to recognize the emergence of huge massive modes of a boson field, as result of the compactified scale of the fifth dimension \\cite{Chodos-Detweiler}.\n\t\n\t\\medskip \n\t\n\tIn the present analysis, we approach the formulation of the $5D$ Kaluza-Klein theory within the semi-classical and quantum framework of the so-called Polymer Quantum Mechanics \\cite{Corichi1}\\cite{Corichi2}.\n\tThis revised formulation of quantum physics has the aim to introduce a discrete nature in the generalized coordinate (a real coordinate of a generic degree of freedom), as an effect of the emergence of cut-off physics. \n\t\n\tIndeed, the fifth compactified dimension, being in the standard approach about two order greater than the Planck size, it is in the natural condition to be approached via the continuum limit of Polymer Quantum Mechanics as referred to a point particle living in this dimension.\n\tFurthermore, also the corresponding diagonal metric component (namely the additional Universe scale factor) in such a dynamical regime is to \n\tbe interested - as expected - by cut-off physics effects. \n\t\n\tThe present analysis follows the scenario proposed in \\cite{Chodos-Detweiler} but revised in view of the polymer formulation. \n\t\n\t\\noindent We first show that a five-dimensional Kasner solution \\cite{Kasner} \\cite{Landau}\\cite{Montani} (characterizing the Bianchi I Universe) admits a configuration in which three spatial directions isotropically expand, while the fourth remains static. \n\tThis result is of impact in the implementation of Kaluza-Klein theory, since it removes some of the non-trivial inconvenient features of a collapsing dimension, closely to a Planckian size. \n\tFor a previous attempt to deal with a static compactified dimension, on the base of a physical phenomenon, see \\cite{Salam}. \n\t\n\t\\noindent Then, we analyse the geodesic motion on a generic $5D$ space-time, having a fifth steady dimension and we outline a natural solution to the charge to mass ratio problem. \n\tThis result comes out from the details of the semi-classical polymer formulation, adopted for the Hamiltonian dynamics of the free-falling particle. \n\tIn particular, the modified expression taken by the fifth momentum of the particle leads to a modified constitutive relation, that is - when passing from the momenta to the velocities - the previous constraint on the charge to mass ratio allows considering values which are natural in the Standard Model particles. \n\t\n\t\\noindent Finally, we study a five-dimensional Klein-Gordon equation and we clarify that, addressing the fifth coordinate via the quantum polymer prescription, the spectrum of emerging masses can fit some values of the Standard Model one and no tachyonic mode emerges, differently from the case discussed in \\cite{Chodos-Detweiler}. \n\t\n\tHowever, it should be noticed that, in this quantum field approach, a problem with the definition of a correct $q\/m$ ratio for a Standard Model particle still survives.\n\t\n\t\\medskip \n\t\n\tThe present study suggests that, when cut-off physics is included in the Kaluza-Klein formulation, some of the puzzling features of this approach are restated into a form that can give new physical insight for their understanding and overcoming. \n\t\n\t\\medskip \n\t\n\tThe manuscript presentation is structured as follows:\n\tin Section \\ref{Kaluza-Klein theory} we review the main features of ordinary Kaluza-Klein theory, from the metric tensor construction and the resulting field equations to the geodesic motion of a point-like particle, which analysis, in particular, leads to the ordinary quantisation law for the electric charge, an estimate for the size $L$ of the fifth dimension and the aforementioned shortcoming\n\tof the charge to mass ratio of a particle. \n\t\n\t\\noindent In Section \\ref{Polymer quantum mechanics} we review polymer quantum mechanics, summarizing the construction of the relative kinematics Hilbert space, via the introduction of a Weyl-Heisenberg algebra and under the assumption of the existence of a discrete spatial coordinate, and the implementation of the proper dynamics both on a quantum and semi-classical level, with particular regard to the p-polarization.\n\t\n\t\\noindent In Section \\ref{Polymer Kasner solution} we analyse the polymer-modified Kasner solution obtained from the introduction of the polymer framework on a semi-classical level in a $5D$ Bianchi I model, focusing on the behaviour of the fifth dimension.\n\t\n\t\\noindent Finally, in Section \\ref{Kaluza-Klein theory in polymer quantum mechanics framework}, based on the result of the previous section, first, we analyse, in a semi-classical formulation of Polymer Quantum Mechanics, the geodesic motion of a point-like particle and all its features, comparing all the results with the ones from the ordinary theory, and then we carry out the study of the polymer quantum dynamics of a complex Klein-Gordon field, along the lines of \\cite{Chodos-Detweiler}, discussing with particular attention the resulting electric charge distribution and mass spectrum. \n\t\n\t\\noindent In Section \\ref{Conlusion} brief concluding remarks follow.\n\t\n\t\\section{Kaluza-Klein theory} \n\t\\label{Kaluza-Klein theory}\n\t\n\tKaluza-Klein theory is a $5D$ extension of Einstein's General Relativity which aim is to provide a unified description of gravitational and electromagnetic interaction in a purely geometric fashion. \n\t\n\t\\noindent In the original theory \\cite{Kaluza}\\cite{Klein1}\\cite{Klein2} the space-time is described by a $5D$ smooth manifold $V^5$, which is assumed to be the direct product $V^4\\otimes S^1$ between a generic $4D$ manifold and a circus of length $L$, that is a compact space.\n\t\n\t\\noindent A crucial assumption relies on the fact that all the observable physical quantities do not depend on the fifth coordinate $x^5$.\n\tThis hypothesis can be further motivated by noticing that, due to the compactness of the fifth dimension, all the observable physical quantities are periodic in $x^5$; hence the independence on the fifth coordinate can be regarded as zero-order cut-off of a Fourier expansion of these quantities themselves, dubbed as cylinder condition.\n\t\n\t\\noindent Once restricted the $5D$ general relativity principle to the following coordinate transformations (and their inverse):\n\t\\begin{equation}   \\label{KK_group_trans_1}\n\t\t\\begin{cases}\n\t\t\t\n\t\t\tx^{\\mu'}=\\Psi(x^{\\mu}) \\\\\n\t\t\tx^{5'}=x^5 + k\\Lambda(x^{\\mu}) \\\\\n\t\t\t\n\t\t\\end{cases}\n\t\\end{equation} \n\tthe $5D$ metric tensor of the expanded theory can be written as follows:\n\t\\begin{equation}\\label{KK_metric_tensor}\n\t\t\\tilde{g}_{ab}=\n\t\t\\left(\n\t\t\\begin{array}{c|c}\n\t\t\tg_{\\mu \\nu} + k^2 \\phi^2 A_\\mu A_\\nu & k \\phi^2 A_\\mu \\\\\n\t\t\t\\hline\n\t\t\tk \\phi^2 A_\\nu & \\phi^2\n\t\t\\end{array}\n\t\t\\right) ,\n\t\\end{equation}\n\twhere $g_{\\mu \\nu}$ is the $4D$ metric tensor of the ordinary theory, $A_{\\mu}$ is the electromagnetic four-potential, $\\phi$ is a scalar field and $k$ is a constant to be properly determined. \n\t\n\t\\subsection{Kaluza-Klein field equations}\n\tThe field equations of the theory can be obtained from a $5D$ Einstein-Hilbert action:\n\t\\begin{equation}\n\t\t^{(5)}S:=\\tilde{S}= - \\frac{1}{16\\pi\\tilde{G}}\\int_{V^{4}\\otimes S^1}  dx^{0}dx^{1}dx^{2}dx^{3}dx^{5}   \\sqrt{-\\tilde{g}} \\tilde{R} ,\n\t\\end{equation}\n\twhere $\\tilde{G}$, $\\tilde{g}$ and $\\tilde{R}$ are respectively the $5D$ gravitational constant, the metric tensor $\\tilde{g}_{ab}$ determinant and the $5D$ scalar curvature.\n\t\n\tBy performing a 4+1 dimensional reduction the ordinary $4D$ Einstein-Maxwell action is surprisingly obtained:\n\t\\begin{equation}\n\t\t\\begin{split}\n\t\t\t\\tilde{S}=& - \\frac{c^3}{16\\pi{G}}\\int_{V^{4}}  d^4x \\sqrt{{-g}}\\phi \\biggl( R + \\frac{1}{4}\\phi^2 k^2 F_{\\mu \\nu}F^{\\mu \\nu}\\\\\n\t\t\t&+\\frac{2}{\\phi}\\nabla_{\\mu} \\partial^{\\mu} \\phi \\biggr).\n\t\t\\end{split}\n\t\\end{equation}\n\tBy setting $\\phi=1$ in the action - as in the original work of Kaluza \\cite{Kaluza} and Klein \\cite{Klein1}\\cite{Klein2} - and by using the variational principle ordinary, Einstein-Maxwell field equations can be correctly recovered, once $k$ is setted equal to $2\\sqrt{G}\/c^2$.\n\t\n\t\n\t\\subsection{Geodesic motion}\n\t\n\tA free point-like particle in this theory will move along a $5D$ geodesic, hence the respective action, with signature (-,+,+,+,+), will be:\n\t\\begin{equation}\n\t\t\\tilde{S}=-mc\\int d\\tilde{s}=-mc \\int \\sqrt{-\\tilde{g}_{a b}\\frac{d x^a}{d \\tilde{s}}\n\t\t\t\\frac{d x^b}{d \\tilde{s}}} d\\tilde{s},\n\t\\end{equation}\n\twhere $d\\tilde{s}$ is the $5D$ line element, to be distinguished from the $4D$ line element $ds$.\n\t\n\t\\noindent Once setted - here and in the further developments - $\\phi=1$ in the metric \\eqref{KK_metric_tensor}, from the variational principle $5D$ geodesic equation is immediately restored:\n\t\\begin{equation} \\label{geodesic}\n\t\t\\tilde{u}^a \\tilde{\\nabla}_{a} \\tilde{u}^b=0.\n\t\\end{equation}\n\t\n\tIt is essential to point out that $5D$ velocity $\\tilde{u}^a$ is different from $4D$ velocity $u^a$; indeed they are related as follows:\n\t\\begin{equation} \\label{KK_five-four_velocity_rel}\n\t\t\\tilde{u}^a=\\frac{1}{\\sqrt{1-u_5^2}} u^a.\n\t\\end{equation}\n\t\n\t\\noindent From relations \\eqref{geodesic} and \\eqref{KK_five-four_velocity_rel} it can be easily shown that $u_5$ is a constant of motion. \n\t\n\tIn order to achieve $4D$ equation of motion, the geodesic equation \\eqref{geodesic} has to be evaluated for the usual space-time variables only, which we indicate with Greek letters. \n\t\n\t\\noindent By making use of relation \\eqref{KK_five-four_velocity_rel}, the following result is attained:\n\t\\begin{equation}\n\t\tu^\\nu\\nabla_{\\nu}u^{\\mu}=\\frac{2\\sqrt{G}}{c^2}u_5u^{\\nu}g^{\\mu \\lambda}F_{ \\nu \\lambda},\n\t\\end{equation}\n\twhere $F_{ \\nu \\lambda}$ is the antisymmetric electromagnetic tensor. \n\t\n\t\\noindent By comparison with the ordinary classical equation:\n\t\\begin{equation} \\label{KK_ordinary_4d_geodesic}\n\t\tu^\\nu\\nabla_{\\nu}u^{\\mu}=\\frac{q}{mc^2}u^{\\nu}g^{\\mu \\lambda}F_{\\nu\\lambda},\n\t\\end{equation}\n\tthe following fundamental identification is achieved:\n\t\\begin{equation} \\label{KK_u5_rel_q}\n\t\tu_{5}=\\frac{q}{2m\\sqrt{G}}.\n\t\\end{equation}\n\t\\noindent Being $p_5=mcu_5$ it can then be written:\n\t\\begin{equation} \\label{KK_p5_rel_q}\n\t\tp_5=\\frac{qc}{2 \\sqrt{G}},\n\t\\end{equation}\n\twhich establishes a fundamental relation between the particle fifth component of momentum and its electric charge. \n\t\n\tThe compactness of the fifth dimension implies a quantisation of momentum along the fifth direction:\n\t\\begin{equation} \\label{KK_p_quantised}\n\t\tp_5=\\frac{2\\pi \\hbar}{L}n \\qquad n \\in \\mathbb{Z},\n\t\\end{equation}\n\twhere we remind that $L$ is the length of the circus describing the fifth dimension.\n\t\\noindent By a direct comparison between relations \\eqref{KK_p5_rel_q} and \\eqref{KK_p_quantised} a natural quantisation law for the electric charge and an estimate of the size $L$ of the fifth dimension are obtained:\n\t\\begin{equation} \\label{KK_ordinary_L}\n\t\t{L}=4\\pi \\frac{\\hbar\\sqrt{G}}{e c}\\approx 2.37\\cdot10^{-31}\\>cm \\qquad q=ne,\n\t\\end{equation}\n\twhere $e$ is the electron charge.\n\t\n\t\\noindent Coherently the size of the fifth dimension is in agreement with its non-observability and with its impossibility to be currently detected. \n\t\n\t\\medskip \n\t\n\tNevertheless, despite these remarkable results, the relation \\eqref{KK_five-four_velocity_rel} sets the constraint $\\abs{u_5}<1$; \n\tby virtue of relation \\eqref{KK_u5_rel_q}, this implies the following condition on the charge\/mass ratio of a particle:\n\t\\begin{equation} \\label{KK_q_over_m}\n\t\t\\frac{\\abs{q}}{m}<2\\sqrt{G}\\approx 5.16\\cdot10^{-4} \\> e.s.u.\/g,\n\t\\end{equation}\n\twhich, unfortunately, has not phenomenological confirmation, neither for elementary particle nor for macroscopic object, hence representing one of the puzzling shortcomings of the theory.\n\t\n\t\\section{Polymer quantum mechanics}\n\t\\label{Polymer quantum mechanics}\n\t\n\tPolymer quantum mechanics is a non-standard representation of the non-relativistic quantum theory, unitarily inequivalent to the Schr\u00f6dinger one \\cite{Corichi1}\\cite{Corichi2}.\n\tIts developments are due mainly to the exploration of background-independent theories, such as quantum gravity, of which mimics several structures \\cite{Ashtekar}. \n\t\n\t\\smallskip\n\t\n\tGiven a discrete orthonormal basis $\\ket{\\mu_i}$ for a space $\\mathcal{H'}$, such that $\\braket{\\mu_i}{\\mu_j}=\\delta_{ij}$, where $\\mu_i \\in \\mathbb{R}$ and $i=1,2...,n$, the kinematic polymer Hilbert space $\\mathcal{H}_{poly}$ is obtained as a Cauchy completion of $\\mathcal{H'}$.\n\t\n\t\\noindent On this space two abstract operators can be defined:\n\t\\begin{align}\n\t\t& \\hat{\\epsilon}\\ket{\\mu}:=\\mu \\ket{\\mu} \\\\\n\t\t& \\hat{s}(\\lambda)\\ket{\\mu}:=\\ket{\\mu +\\lambda}.\n\t\\end{align}\n\t\\noindent The operator $\\hat{\\epsilon}$ is a symmetric operator and $\\hat{s}(\\lambda)$ defines a one-parameter family of unitary operators.\n\tIn spite of this $\\hat{s}(\\lambda)$ is discontinuous with respect to $\\lambda$;\n\tthis means that no self-adjoint operator exists that can generate $\\hat{s}(\\lambda)$ by exponentiation. \n\tTaking now in exam a physical system with configuration space spanned by the coordinate $q$, which is assumed to have a discrete character, and its conjugate momentum $p$, the previous abstract representation can be projected and studied with respect to p-polarization.\n\tIn this polarization the basis states will be:\n\t\\begin{equation} \\label{basic_states_p}\n\t\t\\psi_\\mu(p)=\\braket{p}{\\mu}=e^{i\\mu p\/\\hbar}.\n\t\\end{equation}\n\tFollowing the algebraic construction method, a Weyl-Hei\\-sen\\-berg\\- algebra is introduced on $\\mathcal{H}_{poly}$ and the action of its generators on the basis states is defined as follows:\n\t\\begin{align}\n\t\t&\\hat{\\mathcal{U}}(\\nu)\\psi_\\mu(p)=\\psi_\\mu(p+\\nu)=e^{i\\mu(p+\\nu)\/\\hbar}=e^{i\\mu\\nu \/\\hbar}e^{i\\mu p\/\\hbar} \\\\\n\t\t&\\hat{\\mathcal{V}}(\\lambda)\\psi_\\mu(p)=e^{i\\lambda p\/\\hbar}e^{i\\mu p\/\\hbar}=e^{i(\\lambda + \\mu)p\/\\hbar}=\\psi_{\\mu+\\lambda}(p).\n\t\\end{align}\n\tFrom this it can be inferred that the shifting operator $\\hat{s}(\\lambda)$ can be identified with the operator $\\hat{\\mathcal{V}}(\\lambda)$, which is then discontinuous in $\\lambda$; this means that the spatial translations generator, that is the momentum operator $\\hat{p}$, does not exist. \n\tOn the other hand, the operator $\\hat{\\mathcal{U}}(\\nu)$ is continuous, so that the translations generator in the momentum space, i.e. the position operator $\\hat{q}$, exists and it can be identified with the abstract operator $\\hat{\\epsilon}$.\n\t\n\t\\noindent Indeed:\n\t\\begin{equation} \\label{PQ_q_op}\n\t\t\\hat{q}\\psi_{\\mu}(p)=-i\\hbar \\partial_p\\psi_{\\mu}(p)=\\mu\\psi_{\\mu}(p).\n\t\\end{equation}\n\tIt can be proved \\cite{Corichi2} that the kinematic polymer Hilbert space in this polarization is explicitly given by $\\mathcal{H}_{poly,p}=L^2\\left(\\mathbb{R}_{B},d\\mu_{H}\\right)$, where $\\mathbb{R}_{B}$  is the so-called Bohr compactification of the real line and $d\\mu_{H}$ is the Haar measure. \n\t\n\t\\medskip \n\t\n\tA similar picture is obtained in the q-polarization: the momentum operator cannot still be defined, while it is possible to show that the fundamental wave functions are Kronecker deltas and that the kinematic polymer Hilbert space is explicitly given by $\\mathcal{H}_{poly,x}=L^2\\left(\\mathbb{R}_{d},d\\mu_{c}\\right)$, where $\\mathbb{R}_{d}$  is the real line equipped with a discrete topology and $d\\mu_{c}$ is the counting measure. \n\t\n\t\\medskip \n\t\n\tIn order to build the dynamics a Hamiltonian operator $\\hat{H}$ has to be defined on $\\mathcal{H}_{poly}$, but since $\\hat{p}$ do not exist, a direct implementation is not possible.\n\tTo overcome this problem the momentum operator can be approximated by defining on the configuration space of the system a regular graph $\\gamma_{\\mu}=\\{q \\in \\mathbb{R} \\> | \\> q=n\\mu, n \\in \\mathbb{Z}\\}$, where $\\mu$ is the fundamental scale introduced by the polymer representation. \n\tThe basis kets $\\ket{\\mu}$ can now be indicated as $\\ket{\\mu_n}$, where $\\mu_n=n\\mu$ are the points belonging to the graph $\\gamma_{\\mu_0}$.\n\tConsequently the generic states will be:\n\t\\begin{equation}\n\t\t\\ket{\\psi}_{\\gamma_{\\mu}}=\\sum_n a_n \\ket{\\mu_n},\n\t\\end{equation}\n\tand they will belong to the new Hilbert space $\\mathcal{H}_{\\gamma_{\\mu}} \\subset \\mathcal{H}_{poly}$, posed that they satisfy the condition $\\sum_n \\abs{a_n}^2 < \\infty$.\n\tSince the dynamics has to be closed in $\\mathcal{H}_{\\gamma_{\\mu}}$, the shift parameter $\\lambda$ has to be fixed equal to $\\mu$, hence the action of $\\hat{\\mathcal{V}}(\\lambda)$ will be:\n\t\\begin{equation}\n\t\t\\hat{\\mathcal{V}}(\\lambda)\\ket{\\mu_n}=\\hat{\\mathcal{V}}(\\mu)\\ket{\\mu_n}=\\ket{\\mu_{n+1}}.\n\t\\end{equation}\n\t\n\t\\noindent On a general ground, the variable $p$ it can be written as:\n\t\\begin{equation} \\label{PQ_p_var_approx}\n\t\tp \\approx \\frac{\\hbar}{\\mu} \\sin(\\frac{\\mu}{\\hbar}p)=\\frac{\\hbar}{2i\\mu}\\left(e^{i\\frac{\\mu}{\\hbar}p}-e^{-i\\frac{\\mu}{\\hbar}p}\\right),\n\t\\end{equation}\n\twhen the condition $p<<\\hbar\/\\mu$ holds.\n\t\n\t\\noindent Based on this approximation and visualizing the action of  $\\hat{\\mathcal{V}}(\\mu)$ in the p-polarization, it is clear that the operator $\\hat{p}$ and its action can be approximated as:\n\t\\begin{equation} \\label{PQ_p_op_approx}\n\t\t\\hat{p}_{\\mu}\\ket{\\mu_n}\\approx \\frac{\\hbar}{2i\\mu} \\left[\\hat{\\mathcal{V}}(\\mu)-\\hat{\\mathcal{V}}(-\\mu)\\right]\\ket{\\mu_n},\n\t\\end{equation}\n\twhere $\\mu$ acts as a regulator.\n\t\n\t\\noindent To approximate the operator $\\hat{p}^2$ two paths are possible:\n\t\\begin{equation} \\label{PQ_p^2_op_approx_1}\n\t\t\\hat{p}^2_{\\mu}\\approx \\frac{\\hbar^2}{4\\mu^2} \\left[ 2-\\hat{\\mathcal{V}}(2\\mu)-\\hat{\\mathcal{V}}(-2\\mu)\\right],\n\t\\end{equation} \n\tbased on the approximation \n\t\\begin{equation} \\label{PQ_p^2_var_approx_1}\n\t\tp^2 \\approx \\frac{\\hbar^2}{\\mu^2} \\sin[2](\\frac{\\mu}{\\hbar}p)\n\t\\end{equation}\n\tand hence defined by iterating the action of $\\hat{p}$ according \\eqref{PQ_p_op_approx}, or\n\t\\begin{equation} \\label{PQ_p^2_op_approx_2}\n\t\t\\hat{p}^2_{\\mu}\\approx \\frac{\\hbar^2}{{\\mu}^2} \\left[2-\\hat{\\mathcal{V}}(\\mu)-\\hat{\\mathcal{V}}(-\\mu)\\right],\n\t\\end{equation}\n\texploiting the approximation \n\t\\begin{equation}\\label{PQ_p^2_var_approx_2}\n\t\tp^2 \\approx \\frac{2\\hbar^2}{{\\mu}^2}\\left(1- \\cos(\\frac{\\mu}{\\hbar}p)\\right),\n\t\\end{equation}\n\tvalid as long as $p<<\\hbar\/\\mu$.\n\t\n\t\n\t\\noindent Hence, the well-defined, symmetric Hamiltonian operator will be:\n\t\\begin{equation}\n\t\t\\hat{H}_{\\mu}:=\\frac{\\hat{p}^2_{\\mu}}{2m}+\\hat{V}(\\hat{q}),\n\t\\end{equation}\n\twhere $\\hat{V}(\\hat{q})$ is the potential operator. \n\t\n\tTherefore, quantising a system according to the polymer representation, in the p-polarization, implies the use of the approximation \\eqref{PQ_p^2_op_approx_1} or \\eqref{PQ_p^2_op_approx_2} for the momentum operator, while the position operator will be the natural differential operator, which action is expressed in \\eqref{PQ_p_op_approx}.\n\t\n\tIn a semi-classical approach, this procedure corresponds to the proper introduction of the approximations \\eqref{PQ_p_var_approx}, \\eqref{PQ_p^2_var_approx_1} and \\eqref{PQ_p^2_var_approx_2} on the variable $p$ in the dynamics of the system of interest. \n\tHence, on this level, the whole procedure can be thought of as a prescription to provide physical insight into the behaviour of the quantum expectations values, according to the so-called Ehrenfest theorem.\n\t\n\t\n\t\\section{Polymer Kasner solution}\n\t\\label{Polymer Kasner solution}\n\t\n\tWe will now apply polymer formalism in a semi-classical framework to the study of a $5D$ Bianchi I model, which solution in the vacuum is the well-known Kasner metric \\cite{Kasner}\\cite{Landau}, focusing on the kinematics and dynamics of the fifth dimension. \n\t\n\t\\medskip\n\t\n\t\\noindent In order to obtain the polymer Kasner cosmological solution, we need a minisuperspace Hamiltonian formulation, extended to the $5D$ case.\n\t\n\t\\noindent The $5D$ Bianchi I line element, written in the ADM formalism \\cite{ADM}, is a straightforward generalisation of the $4D$ one\\footnote{As it should be clear from the context, the metric coefficient $c(t)$ is not to be confused with the velocity of light $c$.}:\n\t\\begin{equation}\n\t\t\\begin{split}\n\t\t\tds^2=&-N^2(t)c^2dt^2+^{(4)}h_{ij}dx^idx^j=\\\\\n\t\t\t&-N^2(t)c^2dt^2+a^2(t)(dx^1)^2+b^2(t)(dx^2)^2\\\\\n\t\t\t& +c^2(t)(dx^3)^2+d^2(t)(dx^5)^2, \\qquad (\\text{i,j=1,2,3,5})\n\t\t\\end{split}\n\t\\end{equation}\n\twhere $N(t)$ is the lapse function of the ADM formalism, $^{(4)}h_{ij}$ is the metric tensor of the $4D$ manifold, which coordinate are all space-like.\n\t\n\t\\noindent Having the general structure of this metric as starting point, we can build the Hamiltonian of the system:\n\t\\begin{equation} \\label{PBM_hamiltonian_bianchi_non_diag}\n\t\t\\begin{split}\n\t\t\tH_{Bianchi \\> I}:=H_{B}=Ne^{-\\sum_a q^a\/2}\\biggl\\{\\sum_a p_a^2-\\frac{1}{3}\\biggl[\\sum_b p_b\\biggr]^2\\biggr\\},\n\t\t\\end{split}\n\t\\end{equation}\n\twhich, by varying with respect to $N(t)$ turns out to be a constraint for the dynamics, namely $H_B=0$.\n\t\n\t\\noindent The couple $(q^a,p_a)$  in \\eqref{PBM_hamiltonian_bianchi_non_diag} are the conjugate variables spanning an highly symmetric phase space, the so-called \\textit{minisuperspace}, and the relation between the metric coefficients and the q-variables is the usual one of the literature \\cite{Montani}, extended to the $5D$ case:\n\t\\begin{equation} \\label{PBM_metric_coeff_to_q}\n\t\t\\begin{split}\n\t\t\t&a(t)=e^{q^1(t)\/2} \\quad b(t)=e^{q^2(t)\/2} \\\\\n\t\t\t&c(t)=e^{q^3(t)\/2} \\quad d(t)=e^{q^5(t)\/2}.\n\t\t\\end{split}\n\t\\end{equation}\n\t\n\tAs is well-known, it is more convenient to express the obtained Hamiltonian in its diagonal form:\n\t\\begin{equation} \\label{PBM_hamilt_bianchi_diagonal}\n\t\tH'_{B}=Ne^{-\\alpha}\\left[-\\frac{1}{3}p^2_\\alpha+p^2_{+}+p^2_{-}+p^2_{\\gamma}\\right],\n\t\\end{equation}\n\twhich is the canonical form of the quadratic form associated to $H_{B}$. \n\t\n\t\\noindent The p-variables in the Hamiltonian \\eqref{PBM_hamilt_bianchi_diagonal} are the conjugate momenta of a set of variables $\\alpha,\\beta_{+},\\beta_{-},\\gamma$, which represent the generalisation of the Misner variables \\cite{Misner}. The relation between these variables and the previous q-variables is defined through the following linear transformation:\n\t\\begin{equation} \\label{q_to_misner_var}\n\t\t\\begin{cases}\n\t\t\tq^1=\\frac{1}{2}\\alpha-\\frac{1}{2\\sqrt{3}}\\beta_{+}-\\frac{1}{\\sqrt{6}}\\beta_{-}-\\frac{1}{\\sqrt{2}}\\gamma\\\\ q^2=\\frac{1}{2}\\alpha-\\frac{1}{2\\sqrt{3}}\\beta_{+}-\\frac{1}{\\sqrt{6}}\\beta_{-}+\\frac{1}{\\sqrt{2}}\\gamma \\\\\n\t\t\tq^3=\\frac{1}{2}\\alpha-\\frac{1}{2\\sqrt{3}}\\beta_{+}+\\sqrt{\\frac{2}{3}}\\beta_{-} \\\\\n\t\t\tq^5=\\frac{1}{2}\\alpha+\\frac{\\sqrt{3}}{2}\\beta_{+}.\n\t\t\\end{cases}\n\t\\end{equation}\n\t\n\t\n\tBy using the Hamilton-Jacobi method and the Hamilton equation for the variable $\\alpha$ - which represents the universe volume - in the synchronous reference frame, the standard classical Kasner solution for the $5D$ case can be recovered:\n\t\\begin{equation} \\label{PBM_kasner_metric}\n\t\t\\begin{split}\n\t\t\tds^2&=-c^2dt^2+(t\/t_0)^{2k_1}(dx^1)^2+(t\/t_0)^{2k_2}(dx^2)^2\\\\\n\t\t\t&+(t\/t_0)^{2k_3}(dx^3)^2+(t\/t_0)^{2k_5}(dx^5)^2.\n\t\t\\end{split}\n\t\\end{equation}\n\t\n\t\\noindent The k parameters are the so-called Kasner exponents and they satisfy the following conditions:\n\t\\begin{equation} \\label{PBM_kasner_buondaries_standard}\n\t\t\\begin{cases}\n\t\t\tk_1+k_2+k_3+k_5=1 \\\\\n\t\t\tk_1^2+k_2^2+k_3^2+k_5^2=1.\n\t\t\\end{cases}\n\t\\end{equation}\n\t\\noindent In particular, if we assume isotropy in the three usual spatial dimensions, as observations suggest, that is if we set $k_1=k_2=k_3$, solution of the previous system becomes:\n\t\\begin{equation}\n\t\tk_1=k_2=k_3=\\frac{1}{2} \\quad k_5=-\\frac{1}{2}.\n\t\\end{equation}\n\t\\noindent This means that while the three usual spatial dimensions expand, the fifth one collapses indefinitely. \n\t\n\t\\medskip \n\t\n\tWe want now to introduce polymer formalism in Bianchi I dynamics.\n\tIn order to do this we choose to operate the substitutions \\eqref{PQ_p_var_approx} and \\eqref{PQ_p^2_var_approx_1} on the conjugate momentum $p_\\gamma$ of the Misner variable $\\gamma$, connected with the metric coefficient of the fifth dimension.\n\tThe new \\textit{polymerized} Hamiltonian will be:\n\t\\begin{equation} \\label{PBM_hamlit_poly_Bianchi}\n\t\tH_{B}^{poly}=e^{-\\alpha}\\biggl[-\\frac{1}{3}p^2_\\alpha+p^2_{+}+p^2_-+\\frac{\\hbar^2}{\\mu^2}\\sin[2](\\frac{\\mu}{\\hbar}p_\\gamma)\\biggr].\n\t\\end{equation}\n\t\n\t\\noindent By the exactly above procedure we find that the solution is still a Kasner solution, where the metric coefficients have a coordinate time power trend as in \\eqref{PBM_kasner_metric}, but their exponents, that is the Kasner indices, due to the quantum polymer modifications, satisfy different constraints:\n\t\n\t\\begin{equation}\n\t\t\\begin{cases}\n\t\t\tk_1+k_2+k_3+k_5=1 \\\\\n\t\t\tk_1^2+k_2^2+k_3^2+k_5^2=1-\\frac{3}{4}\\frac{\\hbar^2}{\\mu^2}\\frac{\\sin[4](\\frac{\\mu}{\\hbar}p_{\\gamma})}{\\sqrt{p^2_++p^2_-+\\frac{\\hbar^2}{\\mu^2}\\sin[2](\\frac{\\mu}{\\hbar}p_\\gamma)}}.\n\t\t\\end{cases}\n\t\\end{equation}\n\t\n\t\\noindent The second term of the right-hand side of the second condition is non-negative, so that we can restate the system as follows:\n\t\\begin{equation}\n\t\t\\begin{cases} \\label{PBM_kasner_ind_poly_conditions}\n\t\t\tk_1+k_2+k_3+k_5=1 \\\\\n\t\t\tk_1^2+k_2^2+k_3^2+k_5^2\\leq 1.\n\t\t\\end{cases}\n\t\\end{equation}\n\t\n\t\\noindent Assuming isotropy in the three usual spatial dimensions and introducing an order between exponents, in particular setting $k_5<k$, the system \\eqref{PBM_kasner_ind_poly_conditions} has the following solution:\n\t\\begin{equation} \\label{PBM_constraint_polymer_Kasner}\n\t\t\\begin{cases}\n\t\t\t1\/4 < k\\leq1\/2 \\\\\n\t\t\t-1\/2 \\leq k_5 < 1\/4.\n\t\t\\end{cases}\n\t\\end{equation}\n\t\n\tWe can notice that the Kasner exponents relative to the three usual spatial dimensions are bound to be positive and they can take value in a precise interval, while the exponent relative to the fifth dimension has value in an interval in which it can take both positive and negative values, which is a new feature of the Kasner solution entirely due to the polymer physics.\n\t\n\t\\noindent The latter statement grows in importance when we draw our attention to the permitted values of $k_5$.\n\t\n\t\\medskip\n\t\n\tIndeed, even if the power trend with respect to the coordinate time variable of fifth dimension metric coefficient - which is ultimately responsible for the singularity at an infinite time - is not removed in this scenario, the introduction of the polymer formalism leads to a modification on the constraint of the fifth Kasner exponent $k_5$, which interval of definition allows the value $k_5=0$.\n\t\n\t\n\t\n\t\\noindent For this particular choice of $k_5$, the other Kasner indices will be equal to $k=1\/3$, correctly reproducing the observed isotropic expansion in the three usual spatial dimensions, while the metric coefficient relative to the fifth dimension will be of course equal to one. \n\t\n\t\\noindent This means that the fifth dimension has no dynamics since the time-dependence of the relative metric coefficient disappears. \n\tWe, therefore, have obtained a static solution, which somehow solves the singularity problem by remarkably removing the indefinite collapse of the dimension itself.\n\t\n\t\\noindent The developments of the next sections will be based on this important result.\n\t\n\t\\section{Kaluza-Klein theory in polymer quantum mechanics framework}\n\t\\label{Kaluza-Klein theory in polymer quantum mechanics framework}\n\t\n\tIn this section we face the analysis of the Kaluza-Klein paradigm in the framework of Polymer Quantum Mechanics, both studying the behaviour of the geodesic motion and the quantum dynamics of a Klein-Gordon field. \n\tFrom the point of view of the geodesic discussion, the polymer framework is addressed on a semi-classical level, in the spirit of the Ehrenfest theorem. \n\tInstead, the study of the quantum scalar field is performed in a full quantum picture, as restricted to the fifth coordinate. \n\t\n\t\\subsection{Geodesic motion}\n\t\n\tIn this section polymer formalism will be now applied to the Hamiltonian formulation of the $5D$ geodesic motion, at a semi-classical level, that is introducing quantum modifications to the classical dynamics. \n\t\n\t\\medskip\n\t\n\tThe $5D$ Hamiltonian of a free particle in a general Kaluza-Klein background, with $\\tilde{g}_{55}=1$ - see previous section - reads as:\n\t\\begin{equation}\n\t\t\\tilde{H}=\\frac{1}{2mc}\\left[\\tilde{p}_{\\mu} \\tilde{p}_{\\nu} {g}^{\\mu\\nu}-\\frac{4\\sqrt{G}}{c^2}\\tilde{p}_\\mu \\tilde{p}_5A^{\\mu}+\\tilde{p}^2_5\\left(1+\\frac{4G}{c^4}A_{\\mu}A^{\\mu} \\right)\\right].\n\t\\end{equation}\n\twhere, as before, the quantities with a tilde are the $5D$ ones, that is those defined with respect to the $5D$ line element.\n\t\n\tWe want now introduce polymer formalism only with respect to the canonical couple $(x^5,\\tilde{p}_5)$, that is we assume that the coordinate $x^5$ has an essential discrete nature and we redefine the respective conjugate momentum $\\tilde{p}_5$ by introducing a regular graph structure on $S^1_{x^5}$, i.e. on the fifth dimension, which is now equipped with a discrete topology, according to the discussion in Section \\ref{Polymer quantum mechanics}.\n\tFollowing the polymer prescription \\eqref{PQ_p_var_approx}, we can made the substitution:\n\t\\begin{equation}\n\t\t\\tilde{p}_5 \\rightarrow \\frac{\\hbar}{\\mu}\\sin(\\frac{\\mu}{\\hbar}\\tilde{p}_5).\n\t\\end{equation}\n\t\n\t\\noindent The new Hamiltonian will be rewritten as:\n\t\\begin{equation}\n\t\t\\begin{split}\n\t\t\t\\tilde{H}_{poly}= & \\frac{1}{2mc}\\bigg[\\tilde{p}_{\\mu} \\tilde{p}_{\\nu} {g}^{\\mu\\nu}-\\frac{4\\sqrt{G}}{c^2}\\frac{\\hbar}{\\mu}\\tilde{p}_\\mu \\sin(\\frac{\\mu}{\\hbar}\\tilde{p}_5)A^{\\mu}\\\\\n\t\t\t&+\\frac{\\hbar^2}{\\mu^2}\\sin[2](\\frac{\\mu}{\\hbar}\\tilde{p}_5)\\left(1+\\frac{4G}{c^4}A_{\\mu}A^{\\mu} \\right)\\bigg],\n\t\t\\end{split}\n\t\\end{equation}\n\t\n\tfrom which the equations of motion (Hamilton equations) can be obtained:\n\t\\begin{equation} \\label{PG_polymer_equation_of_motion_4_index}\n\t\t\\begin{cases}\n\t\t\t\\tilde{u}^\\mu=\\frac{1}{mc}\\left[\\tilde{p}_{\\nu}{g}^{\\mu \\nu}-\\frac{2\\sqrt{G}}{c^2}\\frac{\\hbar}{\\mu}\\sin(\\frac{\\mu}{\\hbar}\\tilde{p}_5)A^{\\mu}\\right] \\\\\n\t\t\t\\begin{split}\n\t\t\t\t\\dot{\\tilde{p}}_{\\mu}=&-\\frac{1}{2mc}\\bigg[\\tilde{p}_{\\rho}\\tilde{p}_{\\sigma}\\partial_{\\mu}{g}^{\\rho \\sigma}-\\frac{4\\sqrt{G}}{c^2}\\tilde{p}_{\\rho}\\frac{\\hbar}{\\mu}\\sin(\\frac{\\mu}{\\hbar}\\tilde{p}_5)\\partial_{\\mu}A^{\\rho}\\\\\n\t\t\t\t&+\\frac{4G}{c^4}\\frac{\\hbar^2}{\\mu^2}\\sin[2](\\frac{\\mu}{\\hbar}\\tilde{p}_5)\\partial_{\\mu}\\left(A_\\nu A^\\nu\\right)\\bigg], \\\\\n\t\t\t\\end{split}\n\t\t\\end{cases}\n\t\\end{equation}\n\t\n\t\\begin{equation} \\label{PG_polymer_equation_of_motion_5}\n\t\t\\begin{cases}\n\t\t\t\\begin{split}\n\t\t\t\t\\tilde{u}^5=&\\frac{1}{mc}\\bigg[-\\frac{2\\sqrt{G}}{c^2}\\tilde{p}_\\mu A^{\\mu}\\cos(\\frac{\\mu}{\\hbar}\\tilde{p}_5)\\\\\n\t\t\t\t&+\\frac{\\hbar}{\\mu}\\sin(\\frac{\\mu}{\\hbar}\\tilde{p}_5)\\cos(\\frac{\\mu}{\\hbar}\\tilde{p}_5)\\left(1+\\frac{4G}{c^4}A_\\nu A^\\nu\\right)\\bigg] \n\t\t\t\\end{split} \\\\\t \n\t\t\t\\dot{\\tilde{p}}_5=0,\n\t\t\\end{cases}\n\t\\end{equation}\n\twhere the dot refers to the derivative with respect to the five-dimensional line element $d\\tilde{s}$. \n\t\n\t\\noindent We outline that in the last equation we made use of the \\textit{cylinder condition}, which implies that $\\tilde{p}_5$ is a constant of motion, as expected.\n\t\n\t\\noindent As we know from the standard case, in order to make a comparison with the $4D$ equations of motion, we need to know the relation between $4D$ and $5D$ quantities, in particular the relation between five-velocities and four-velocities.\n\tThese are formally the same of equation \\eqref{KK_five-four_velocity_rel} also in our framework.\n\t\n\tNevertheless, while in the ordinary theory, with the scalar field of the Kaluza Klein metric set to be constant, $u_5$ is a constant of motion, in the polymer framework this is not the case. \n\t\n\t\\noindent Indeed, from the explicit expression of $\\tilde{u}^5$ \\eqref{PG_polymer_equation_of_motion_5} we can obtain the corresponding expression of $\\tilde{u}_5$:\n\t\n\t\\begin{equation}\n\t\t\\tilde{u}_5=\\frac{2\\sqrt{G}}{c^2} \\tilde{u}^\\mu A_\\mu \\cos(\\frac{\\mu}{\\hbar}\\tilde{p}_5)+\\frac{\\hbar}{\\mu mc}\\sin(\\frac{\\mu}{\\hbar}\\tilde{p}_5)\\cos(\\frac{\\mu}{\\hbar}\\tilde{p}_5),\n\t\\end{equation}\n\t\n\twhere we have made use of the first Hamilton equation for $\\tilde{u}^{\\mu}$.\n\t\n\t\\noindent The latter can be solved in function of $u_5$ and two solutions are obtained:\n\t\\begin{equation} \\label{PG_u_5 exact expression}\n\t\t\\begin{split}\n\t\t\tu_5=&\\frac{1}{D(\\mu,\\tilde{p}_5)}\\bigg\\{16\\sqrt{G}  \\mu ^2 m^2   \\left(1-\\cos(\\frac{\\mu  }{\\hbar}\\tilde{p}_5)\\right)A_\\mu u^\\mu\\\\\n\t\t\t&\\pm \\sqrt{2}\\hbar\\abs{\\sin(2\\frac{\\mu }{\\hbar}\\tilde{p}_5)} \\sqrt{D(\\mu, \\tilde{p}_5)+ K(\\mu,\\tilde{p}_5)A_\\mu A^\\mu u_\\nu u^\\nu}\\bigg\\},\n\t\t\\end{split}\n\t\\end{equation}\n\twhere:\n\t\\begin{gather}\n\t\tD(\\mu, \\tilde{p}_5):={\\hbar^2+8\\mu ^2 m^2 c^2-\\hbar^2\\cos (4\\frac{\\mu  }{\\hbar}\\tilde{p}_5)} \\\\\n\t\tK(\\mu,\\tilde{p}_5):=-128 \\frac{G}{c^2} \\mu ^2 m^2\\sin[4](\\frac{\\mu}{2\\hbar}\\tilde{p}_5).\n\t\\end{gather}\n\t\n\t\\noindent These are rather complicated non-constant expressions, due to the dependence from the electromagnetic and the four-velocity fields, which consequently heavily modify the $\\frac{ds}{d\\tilde{s}}$ factor. \n\t\n\t\\noindent From the Hamilton equations \\eqref{PG_polymer_equation_of_motion_4_index}, by exploiting the formal relation \\eqref{KK_five-four_velocity_rel}, we can write down the part relative to the $4D$ indices of the new generalised $5D$ geodesic, as a function of the $4D$ quantities:\n\t\n\t\\begin{equation} \\label{PG_4d_polymer_geodesic_equation}\n\t\tu^{\\nu}\\nabla_{\\nu}u^{\\mu}=\\frac{2\\sqrt{G}\\hbar}{mc^3\\mu} \\sqrt{1-u_5^2}\\sin(\\frac{\\mu}{\\hbar}\\tilde{p}_5)u^{\\nu}F^{\\mu}_{\\nu}-\\frac{u_5u^{\\mu}}{1-u^2_5}\\frac{du_5}{ds},\n\t\\end{equation}\n\tin which it is possible to distinguish the polymer-modified part of the standard $4D$ geodesic \\eqref{KK_ordinary_4d_geodesic} and an extra term, entirely due to the variation of $u_5$. \n\tIt is worth pointing out that eq.\\eqref{PG_4d_polymer_geodesic_equation} correctly reduces to the standard one in the $\\mu \\to 0$ limit.\n\t\n\t\\noindent At this point, in order to compare the last expression with the equation of motion \\eqref{KK_ordinary_4d_geodesic} we need to make a series expansion of $u_5$ by assuming $A_\\mu << 1$.\n\tThis can be perfectly legitimate in a cosmological setting since there is no \n\trelevant coherent electromagnetic field which permeates the Universe.\n\t\n\tThis leads to the following expression, to the first order in $A_\\mu$:\n\t\\begin{equation}\n\t\t\\begin{split}\n\t\t\tu_5\\approx &\\pm\\bigg\\{\\frac{\\sqrt{2} \\hbar\\abs{\\sin(2\\frac{\\mu}{\\hbar}\\tilde{p}_5) }}{D(\\mu,\\tilde{p}_5)^{1\/2}}+\\frac{16\\sqrt{G}\\mu^2m^2 \\left(\\cos(\\frac{\\mu }{\\hbar}\\tilde{p}_5) -1\\right)}{D(\\mu,\\tilde{p}_5)} A_{\\mu}u^{\\mu}\\\\ \n\t\t\t&+O\\left(A^2\\right)\\bigg\\}.\n\t\t\\end{split}\n\t\\end{equation}\n\t\\noindent Hence, by inserting this expansion in the geodesic equation \\eqref{PG_4d_polymer_geodesic_equation} and ignoring the extra term - which meaning it is beyond the purpose of this article - we obtain, keeping only the overall first order term in $A_\\mu$:\n\t\\begin{equation}\n\t\tu^{\\nu}\\nabla_{\\nu}u^{\\mu}\\approx\\frac{2\\sqrt{G}\\hbar}{\\mu mc^3} \\frac{2\\sqrt{2}\\mu mc \\sin(\\frac{\\mu}{\\hbar}\\tilde{p}_5)}{\\sqrt{\\hbar^2+8\\mu^2m^2c^2-\\hbar^2 \\cos(4\\frac{\\mu}{\\hbar}\\tilde{p}_5)}}u^{\\nu}F^{\\mu}_{\\nu}\n\t\\end{equation}\n\tfrom which we can read off the perturbative polymer relation between the electric charge $q$ and $\\tilde{p}_5$ - which we remind it is a constant of motion:\n\t\\begin{equation} \\label{PG_polymer_q_expression}\n\t\tq\\approx\\frac{2\\sqrt{G}}{c} \\frac{2\\sqrt{2}\\hbar mc }{\\sqrt{\\hbar^2+8\\mu^2m^2c^2-\\hbar^2 \\cos(4\\frac{\\mu}{\\hbar}\\tilde{p}_5)}}\\sin(\\frac{\\mu}{\\hbar}\\tilde{p}_5).\n\t\\end{equation}\n\t\n\tTo correctly interpret this formula we can refer to the ordinary relation between $q$ and $\\tilde{p}_5$, which can be obtained from the expression \\eqref{KK_p5_rel_q}.\n\tIn fact, since in the standard case the following simple statement holds:\n\t\\begin{equation} \\label{PG_ordinary_rel_between_p_5}\n\t\t\\tilde{p}_5=\\frac{mc}{\\sqrt{m^2c^2-p^2_5}}p_5 \\qquad \n\t\tp_5=\\frac{mc}{\\sqrt{m^2c^2+\\tilde{p}^2_5}}\\tilde{p}_5,\n\t\\end{equation}\n\tthe relation \\eqref{KK_p5_rel_q} can be rewritten as:\n\t\\begin{equation} \\label{PG_rel_q_p5_standard}\n\t\tq=\\frac{2\\sqrt{G}}{c}\\frac{mc}{\\sqrt{m^2c^2+\\tilde{p}^2_5}}\\tilde{p}_5.\n\t\\end{equation}\n\t\\noindent This suggests that the expression \\eqref{PG_polymer_q_expression} is the polymer generalisation to the first perturbative order in the electromagnetic field of the latter relation.\n\t\n\t\\noindent This interpretation is confirmed by the $\\mu\\to0$ limit of \\eqref{PG_polymer_q_expression}, from which the standard expression \\eqref{PG_rel_q_p5_standard} is recovered. \n\t\n\t\\noindent Therefore, based on the relations \\eqref{PG_ordinary_rel_between_p_5}, we can impose:\n\t\\begin{equation}\n\t\t\\frac{\\hbar}{\\mu}\\sin(\\frac{\\mu}{\\hbar}p_5)\\approx\\frac{2\\sqrt{2}\\hbar mc }{\\sqrt{\\hbar^2+8\\mu^2m^2c^2-\\hbar^2 \\cos(4\\frac{\\mu}{\\hbar}\\tilde{p}_5)}}\\sin(\\frac{\\mu}{\\hbar}\\tilde{p}_5)\n\t\\end{equation}\n\tand, in the end:\n\t\\begin{equation} \\label{PG_poly_charge_p5}\n\t\tq\\approx\\frac{2\\hbar\\sqrt{G}}{\\mu c}\\sin(\\frac{\\mu}{\\hbar}p_5).\n\t\\end{equation}\n\t\\noindent This is a periodic and bounded function of $p_5$ with period $2\\pi$, defined in the generic periodic interval $[\\pi(2k-1)\\hbar\/\\mu,\\pi(2k+1)\\hbar\/\\mu]$, with $k\\in\\mathbb{Z}$. \n\tSince we want to preserve the Cauchy problem symmetry with respect to the initial values of $p_5$, that is the symmetry of left-handed and right-handed particles in the fifth direction, we are led to choose the interval $[-\\pi\\hbar\/\\mu,\\pi\\hbar\/\\mu]$, i.e. the one for k=0, as the natural periodic interval of definition.\n\tIn the $\\mu\\to0$ limit, the obtained function correctly reproduces the ordinary relation inferable from \\eqref{KK_p5_rel_q}.\n\t\n\tAs in the ordinary case, being the fifth dimension a compact space, we have to assume that $p_5$ is quantized; in order to determine the quantisation law, we need to represent the free particle state in the fifth coordinate as described by the wave function of the polymer free particle. \n\tAs comprehensively discussed in \\cite{Corichi2}, in the position representation, this will be:\n\t\\begin{equation} \\label{PG_free_poly_particle_normalized}\n\t\t\\psi_{\\mu}(x^5)=\\frac{1}{\\sqrt{L}}e^{ ix^5p_{5}^{(\\mu)}\/\\hbar},\n\t\\end{equation}\n\tnormalized on $S^1_{x_5}$, where $L$ is the length of the circus which characterizes the fifth dimension.\n\tHence, by imposing $\\psi_{\\mu}(x^5)$ to be periodic in $x^5$, we achieve the following quantisation law:\n\t\\begin{equation} \\label{PG_polymer_momentum_quantisation}\n\t\tp_{5}^{(\\mu)}= \\frac{2\\pi n}{L}\\hbar  \\qquad n \\in\\mathbb{Z},\n\t\\end{equation}\n\twhich immediately leads, accordingly to \\eqref{PG_poly_charge_p5}, to a new quantisation law for the electric charge:\n\t\\begin{equation} \\label{PG_charge_quantisation}\n\t\tq_{n}\\approx\\frac{2\\hbar\\sqrt{G}}{\\mu c}\\sin(\\frac{2\\pi \\mu}{L}n),\n\t\\end{equation}\n\tdefined in the aforementioned periodic interval.\n\t\n\t\\noindent We can choose $q_n$ to be equal to the electron electric charge $e$ for $n=1$, as in the ordinary case, and in this way we are able to find an expression for $L$:\n\t\\begin{equation} \\label{PG_L_function}\n\t\tL\\approx\\frac{2\\pi\\mu}{\\arcsin(\\frac{e\\mu c}{2\\hbar\\sqrt{G}})},\n\t\\end{equation}\n\t\n\tfrom which we deduce a constraint for the polymer scale $\\mu$:\n\t\\begin{equation} \\label{PG_scale_constraint}\n\t\t0 < \\mu \\leq \\frac{2\\hbar\\sqrt{G}}{\\abs{e}c} \\approx 3.78 \\cdot 10^{-32} \\> cm.\n\t\\end{equation}\n\tAt this point we choose $\\mu$ equal to the Planck length and obtain: \n\t\\begin{equation} \\label{PG_poly_geo_L}\n\t\tL\\approx 2.377 \\cdot 10^{-31} \\> cm,\n\t\\end{equation}\n\twhich almost coincide with the result of the standard theory and therefore it can account for the non observability of the fifth dimension.\n\t\n\tThere are basically two reasons behind this particular choice of the polymer scale: \n\t\\begin{itemize}\n\t\t\\item it is a scale with a strong physical meaning;\n\t\t\\item the $L\/\\mu$ ratio, for this value of $\\mu$, is large enough to ensure some calculations to be carried out in the polymer continuum limit, through the assumptions discussed in \\cite{Corichi1} and \\cite{Corichi2}.\n\t\\end{itemize}\n\t\n\t\\noindent A further discussion about the function \\eqref{PG_L_function} is postponed to the next subsection.\n\t\n\t\\medskip \n\t\n\tThe charge function \\eqref{PG_poly_charge_p5}, instead, can be rewritten, by means of the function $L(\\mu)$ \\eqref{PG_L_function}, as follows:\n\t\\begin{equation} \\label{PG_charge_distr_func_poly}\n\t\tq(\\mu;n)=\\frac{2\\hbar\\sqrt{G}}{\\mu c}\\sin\\bigg(n\\arcsin(\\frac{\\mu ec}{2\\hbar\\sqrt{G}})\\bigg)\n\t\\end{equation}\n\t\n\tand by setting the polymer scale $\\mu$ equal to the Planck length, we obtain the symmetric distribution of positive and negative charges reported in Figure \\ref{PG_planckian_charge_distribution_plt}.\n\t\n\t\\noindent The number of modes $n$ in the considered interval is limited by the periodic condition of the function itself and it clearly depends on $\\mu$.\n\tFor our choice of $\\mu$ we find that $-73\\leq n \\leq 73$ (see again Figure \\ref{PG_planckian_charge_distribution_plt}). \n\t\n\t\\noindent It is worth noticing that for any fixed value of $n=n^*$ it is always possible to expand the sine function in correspondence to a suitable small cut-off parameter $\\mu$, accordingly to the inequality $n^*\\mu<<2\\hbar\\sqrt{G}\/ec$, where we also have expanded the function $L(\\mu)$.\n\t\n\t\\noindent In this limit we recover the standard expression for the charge as multiple of the elementary electron charge, at least for $n<n*$.\n\t\n\t\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\\includegraphics[width=%\n\t\t0.5\\textwidth]{poly_charge_geo}\t\n\t\t\\caption{Plot of charges distribution for a value of $\\mu$ equal to the Planck length. It is possible to appreciate the oscillating profile of the function and the negative and positive symmetric branches.}\n\t\t\\label{PG_planckian_charge_distribution_plt}\n\t\\end{figure}\n\t\n\t\\medskip\n\t\n\t\n\tIn the end, we want to evaluate the consequence of the condition $\\abs{u_5}<1$ in our framework.  \n\t\n\t\\noindent The relation between $u_5$ and $q$ can be found employing the expression \\eqref{PG_polymer_q_expression}.\n\t\n\t\\noindent Hence, by retaining only the zero-order term in $u_5$ - coherently with the overall expansion in $A_{\\mu}$ and its derivatives we have done in the equation of motion \\eqref{PG_4d_polymer_geodesic_equation} - we can write:\n\t\\begin{equation}\n\t\t\\abs{u_5}\\approx \\frac{\\abs{q}}{2m\\sqrt{G}}\\sqrt{\\frac{1}{2}+\\frac{2 G}{q^2m^2}\n\t\t\t\\pm \\frac{1}{q^2\\hbar}F(\\mu)},\n\t\\end{equation}\n\twhere:\n\t\\begin{equation}\n\t\tF(\\mu):=\\sqrt{4\\mu ^2 m^2c^2 q^4+16G^2\\hbar^2m^4-8G \\hbar^2 m^2 q^2+h^2 q^4}.\n\t\\end{equation}\n\t\n\t\\noindent By imposing the condition $\\abs{u_5}<1$, the solution with plus sign can be ruled out, since it always violates the constraint, while for the solution with minus sign - which is the meaningful one - we obtain:\n\t\\begin{equation} \\label{PG_poly_q\/m_constraint}\n\t\t\\frac{\\abs{q}}{m}\\leq \\frac{2\\hbar\\sqrt{G}}{\\mu m c},\n\t\\end{equation}\n\twhich is in agreement with the quantisation law \\eqref{PG_poly_charge_p5}.\n\t\n\t\\medskip\n\t\n\t\\noindent This constraint - that we stress is valid only in a perturbative regime - is different from the ordinary one \\eqref{KK_q_over_m} and it introduces a dependence in the $q\/m$ ratio from the polymer scale $\\mu$ - as expected - and from the mass itself.\n\tRemarkably the empirical $q\/m$ ratio of any known particle always respects the bound for every value of $\\mu$ in the interval \\eqref{PG_scale_constraint}.\n\tThis means that the new constraint does not set an unphysical condition and at least does not contradict the experimental evidence, hence resolving one of the shortcoming aspects of the standard theory. \n\tIn other words, we are now able, in principle, to reproduce the $q\/m$ ratio of any Standard Model particle.\n\t\n\t\n\t\\subsection{Polymer complex scalar field coupled with Kaluza-Klein metric}\n\t\n\tIn this subsection we will carry out the study of a complex scalar field polymer dynamics, on a Kaluza-Klein background, obtained through a proper perturbation of the previous polymer Kasner metric.\n\t\n\t\\medskip \n\t\n\tFollowing the literature \\cite{Chodos-Detweiler}\\cite{Montani}\\cite{Landau}, under the isotropy assumption in the three usual spatial dimensions, we rewrite the polymer Kasner solution as:\n\t\\begin{equation}\n\t\t\\begin{split}\n\t\t\tds^2=&-c^2dt^2+\\left(\\frac{t}{\\tau}\\right)^{2k}(d\\vec{x})^2+\\left(\\frac{t}{\\tau}\\right)^{2k_5}(dx^5)^2,\n\t\t\\end{split}\n\t\\end{equation}\n\twhere $\\tau$ is a time characteristic of the present age of the Universe and we have rescaled the coordinates by the same factor.\n\t\n\t\\noindent Since we are interested in the static case with respect to the fifth dimension, we choose $k_5=0$ and hence, as we have seen in the previous section, $k=1\/3$:\n\t\\begin{equation} \\label{PCSF_Kasner_metric}\n\t\tds^2=-c^2dt^2+\\left(\\frac{t}{\\tau}\\right)^{2\/3}(d\\vec{x})^2+(dx^5)^2.\n\t\\end{equation}\n\tTaken into account the rescaling, the coordinates will now satisfy the following conditions:\n\t\\begin{equation} \\label{x5_values}\n\t\t0\\leq \\abs{\\vec{x}}<L(\\tau\/t_0)^{k} \\qquad  0\\leq x^5<L.\n\t\\end{equation}\n\t\n\t\\noindent In order to include a perturbative electromagnetic field, a small perturbation $h_{\\mu 5}$ ($\\mu$=0,1,2,3) is added to the polymer-modified Kasner metric, properly proportional to the electromagnetic field itself:\n\t\\begin{equation}\n\t\th_{\\mu 5}=\\frac{2\\sqrt{G}}{c^2}A_{\\mu}.\n\t\\end{equation}\n\t\\noindent As stated in the previous section this is naturally justified in a cosmological setting.\n\t\n\tCoupled to this background we introduce a quantum complex scalar field $\\Phi$, which ordinary dynamics is described by a $5D$ Klein-Gordon equation and we set the $5D$ \"mass\" term equal to zero:\n\t\\begin{equation} \\label{PCSF_KG_5d_eq}\n\t\t^{(5)}\\Box \\Phi(x^a)=0 \\qquad a=0,1,2,3,5.\n\t\\end{equation}\n\t\n\t\\noindent As in the standard case, our analysis relies on the assumption of the cylinder hypothesis for observable physical quantities, still providing the scalar field $\\Phi$ with a phase factor depending on $x^5$, the latter described by the polymer free particle wave function, with periodicity condition on the fifth coordinate due to the topology of the space:\n\t\\begin{equation}\n\t\t\\Phi(x^{\\mu},x^5)=\\frac{1}{\\sqrt{L}}\\phi(x^\\mu) e^{ix^52\\pi n\/L}.\n\t\\end{equation}\n\twhere the function $\\phi(x^\\mu)$ depends only on the variables of the $4D$ space-time. \n\t\n\t\\noindent A 4+1 dimensional splitting of eq.\\eqref{PCSF_KG_5d_eq} results in the following expression:\n\t\\begin{equation} \\label{PCSF_KG_5d_eq_splitted}\n\t\t\\begin{split}\n\t\t\t&\\partial_\\mu\\partial^\\mu\\Phi-\\frac{2\\sqrt{G}}{c^2}\\partial_\\mu A^\\mu\\partial_5\\Phi-\\frac{4\\sqrt{G}}{c^2}A^\\mu\\partial_\\mu\\partial_5\\Phi\\\\\n\t\t\t&+\\left[1+\\frac{4G}{c^4}A_\\nu A^\\nu\\right]\\partial_5^2\\Phi=0,\n\t\t\\end{split}\n\t\\end{equation}\n\tWe want now to study eq.\\eqref{PCSF_KG_5d_eq_splitted} by introducing polymer quantum framework only with respect to the fifth dimension; therefore we switch to momentum representation only on the fifth coordinate and then let the operator $\\hat{p}_5$ act according to the polymer prescription \\eqref{PQ_p_op_approx}.\n\t\n\t\\noindent It is worth noticing that, acting in this way, we are working in a mixed representation of position and momentum, which would not be possible universally speaking, nevertheless, since the coupling term in \\eqref{PCSF_KG_5d_eq_splitted} between $x^{\\mu}$ and $x^5$ is a small perturbation, we are legitimate, under this assumption, to proceed along this path, according to the diagonal form of the background metric.\n\t\n\t\\noindent Finally, the following equation for the complex scalar field is achieved:\n\t\\begin{equation} \\label{PCSF_poly_KG_field_eq}\n\t\t\\begin{split}\n\t\t\t&{}^{(4)}\\Box\\phi(x^\\mu)-i\\frac{2\\sqrt{G}}{c^2\\mu}\\phi(x^\\mu)\\partial_{\\mu}A^{\\mu}\\frac{1}{\\mu}\\sin({\\mu}\\frac{2\\pi n}{L})\\\\\n\t\t\t&-i\\frac{4\\sqrt{G}}{c^2\\mu}\\sin({\\mu}\\frac{2\\pi n}{L})A^{\\mu}\\partial_\\mu\\phi(x^\\mu)\\\\\n\t\t\t&-\\left[1+\\frac{4G}{c^4}A_\\nu A^\\nu\\right]\\frac{1}{\\mu^2}\\sin[2]({\\mu}\\frac{2\\pi n}{L})\\phi(x^\\mu)=0.\n\t\t\\end{split}\n\t\\end{equation}\n\t\n\t\\noindent By comparing term by term the latter with the $4D$ equation of a massive complex scalar field, coupled to an electromagnetic one, on a curved space-time, we attain the following identifications:\n\t\\begin{equation} \\label{PCSF_polymer_law_charge_and_mass}\n\t\t\\begin{split}\n\t\t\t&q_n=\\frac{2\\hbar \\sqrt{G}}{\\mu c}\\sin(\\frac{2\\pi \\mu}{L}n) \\\\\n\t\t\t&m_n=\\frac{\\hbar}{\\mu c}\\abs{\\sin(\\frac{2\\pi \\mu}{L}n)}.\n\t\t\\end{split}\n\t\\end{equation}\n\t\\noindent The first one is again a quantisation law for the electric charge and it is clearly the same we have obtained in \\eqref{PG_charge_quantisation}, in the modified geodesic study.\n\t\n\t\\noindent Hence, by imposing once again $q_n=e$ for $n=1$, we recover for the size $L$ of the fifth dimension the function \\eqref{PG_L_function}, with the constraint \\eqref{PG_scale_constraint}.\n\t\n\t\\noindent This function is a monotonically decreasing function of $\\mu$, which reaches its minimum value at the right endpoint of the interval in which is defined and it tends exactly to the value of the ordinary theory \\eqref{KK_ordinary_L} as $\\mu$ approaches zero (see Figure \\ref{L_function_plt}).\n\t\n\t\\noindent Clearly, choosing again $\\mu$ around the Planck length, we find for the value of $L$ the result \\eqref {PG_poly_geo_L}, discussed above.\n\t\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\\includegraphics[width=%\n\t\t0.5\\textwidth]{L_function}\t\n\t\t\\caption{Monotonically decreasing trend of the size $L$ of the fifth dimension as a function of the polymer scale $\\mu$. The constraint \\eqref{PG_scale_constraint} for $\\mu$ establishes a precise domain and codomain for the function $L(\\mu)$.}\n\t\t\\label{L_function_plt}\n\t\\end{figure}\n\t\n\t\\noindent On the other hand, the second relation \\eqref{PCSF_polymer_law_charge_and_mass} represents a mass distribution law for the scalar field, which, by means of the expression \\eqref{PG_L_function}, can be rewritten as a function\n\tof the integer parameter $n$ and of the continuous variable $\\mu$:\n\t\\begin{equation} \\label{mass_distr_func_poly}\n\t\tm(\\mu;n)=\\frac{\\hbar}{\\mu c}\\abs{\\sin\\bigg(n\\arcsin(\\frac{\\mu ec}{2\\hbar\\sqrt{G}})\\bigg)}.\n\t\\end{equation}\n\t\n\t\\noindent This function - for the same reasons stated in the previous subsection regarding the charge function - can be defined in the periodic interval $[-\\pi \\hbar\/\\mu,\\pi \\hbar\/\\mu]$, but, because of the presence of the modulus, we can restrict our attention only on the positive segment of the interval itself, that is $[0, \\pi \\hbar\/\\mu]$.\n\tFor a fixed value of the polymer scale $\\mu$, the resulting sequence of masses, due to the periodicity condition, will include only a finite number of mass modes $n$ and it clearly will result bounded and oscillating, as $n$ changes. \n\t\n\t\\noindent This freedom in the choice of the scale leads to a crucial feature of such distribution: picked up a particular mode $n$, through a fine-tuning of $\\mu$, it is possible to fit any desired value of the mass, therefore also Standard Model-comparable masses. \n\tIn particular, the spectrum can be explored in correspondence to a Planckian-like value of the parameter $\\mu$. \n\t\n\t\\noindent The specific pattern of the distribution then will depend on the chosen reference mass to be fitted to a certain mode $n$, for a fixed value of $\\mu$; nevertheless, it can be shown that a qualitative general trend can be recovered, which provides the presence of the assigned mass in the minimum of the sequence as a ground level of the Kaluza-Klein tower.\n\tWe stress that these regular values appear accompanied by Planckian masses, as sketched in Figure \\ref{planckian_masses_distribution_plt}.\n\t\n\t\\noindent This distribution deeply differs from the one obtained in the standard theory, reported in \\cite{Chodos-Detweiler}.\n\tIn fact, the latter is a linear and increasing sequence of only Planckian masses, without an upper limit.\n\t\n\t\\noindent Actually, the possibility of fitting arbitrary masses and from this obtaining various distributions can be achieved also in the ordinary theory, as shown in \\cite{Chodos-Detweiler}, by introducing a $5D$ mass term $a$ in the Klein-Gordon equation \\eqref{PCSF_KG_5d_eq}, which role is that of an additive parameter to be fine-tuned.\n\t\n\t\\noindent Nevertheless, this way of proceeding reveals several issues, which consequently are all addressed in our framework, not being necessary the introduction of such an extra quantity $a$.\n\t\n\t\\noindent First of all, in general, the fine-tuning procedure cannot be applied to every mode, since the linear and increasing trend of the distribution is not modified at all; in the second place, the addition of this parameter eventually generates the rise of tachyonic masses in the past.\n\tThis happens because of the presence of the metric coefficient $(t\/\\tau)^{\\abs{k_5}}$ in the standard momentum expression, as discussed in \\cite{Chodos-Detweiler}, which cannot be removed, since in the ordinary theory the Kasner solution does not admit the case $k_5=0$. \n\t\n\t\\noindent This means that in our framework the tachyonic masses are ruled out from the polymer-modified distribution. \n\t\n\t\\smallskip\n\t\n\tWe outline that - in our analysis - even in the case $k_5\\neq 0$, i.e. in the non-static case, the tachyonic problem would be solved since ultimately it is the chance to set $a=0$ which removes the unphysical masses.\n\t\n\t\\smallskip\n\t\n\tFinally, the $a$ parameter has to be introduced on purpose in the theory and its physical meaning remains ambiguous, while in our framework the fine-tuning parameter is an internal degree of freedom of the theory.\n\t\n\t\\medskip\n\t\n\tHence, even if polymer quantum mechanics fails in removing completely the Planckian masses from the spectrum, it succeeds first in introducing a cut-off since the masses distribution ref is bounded.\n\tFurthermore, in the considered natural interval a finite number of variable mass values take place. \n\tin the masses distribution in the considered periodic interval of definition of the masses function itself and secondly in accounting for Standard Model-comparable masses, under a proper fine-tuned choice of the scale variable $\\mu$.\n\t\n\t\\medskip \n\t\n\tSince we are dealing with a complex scalar field, the charged pions $\\pi^{\\pm}$ seem to be relevant (phenomenologically) candidates in Nature for the reference masses.\n\t\n\t\\noindent It results that the pion mass can be fitted at the mode $n=73$ for a fine-tuned value of $\\mu$ (up to twenty decimal figures) almost equal to the Planck length, which we will denote from now on as $\\mu_{Planck}$.\n\t\n\t\\noindent Nevertheless, for this procedure to be consistent, it is necessary to verify that the resulting mode $n=73$ belongs to the range of $n$ admitted in the periodic interval of definition of the masses function.\n\t\n\t\\noindent It is easy to show that for $\\mu=\\mu_{Planck}$ the condition for positive $n$ for belonging - in which we are interested - is $n\\leq73$.\n\t\n\t\\noindent Therefore, the results coming from the fit procedure are valid and legitimate in this regard.\n\t\n\tIn Figure \\ref{planckian_masses_distribution_plt} it is reported the whole sequence of masses resulting for $\\mu=\\mu_{Planck}$, where it is possible to appreciate the general behaviour discussed above.\n\t\n\t\\noindent In particular, we observe that the pion mass is in the minimum of the distribution and that the maximum available mass is always: \\[m_{max}=\\frac{\\hbar}{\\mu c}\\]\n\twhich is inversely proportional to $\\mu$, while the first mode of the sequence ($n=1$) is: \n\t\\begin{equation}\n\t\tm(\\mu;1)=\\frac{e}{2\\sqrt{G}}\\approx 9.29\\cdot 10^{-7} \\> g.\n\t\\end{equation}\n\t\n\t\\noindent It does not depend on $\\mu$ and it is almost a Planckian mass, defined only by fundamental constants, equal to the one obtained in the standard framework (without the introduction of the \\textit{ad hoc} parameter $a$).\n\t\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\\includegraphics[width=%\n\t\t0.5\\textwidth]{mass_poly}\t\n\t\t\\caption{Plot of masses distribution for the complex scalar field, for the value $\\mu_{Planck}\\approx 1.627\\cdot10^{-33} \\> cm$. It is possible to observe the oscillating profile of the function and the fitted Pion mass, placed in the minimum, as ground level of the Kaluza-Klein tower.}\n\t\t\\label{planckian_masses_distribution_plt}\n\t\\end{figure}\n\t\n\tIn Figure \\ref{KG_planckian_charge_distribution_plt} instead the corresponding charge distribution - which again can be put in the form \\eqref{PG_charge_distr_func_poly} - for $\\mu=\\mu_{Planck}$ is represented. \n\t\n\t\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\\includegraphics[width=%\n\t\t0.5\\textwidth]{poly_charge_KG}\t\n\t\t\\caption{Plot of charges distribution for value $\\mu=\\mu_{Planck}$. Again, it is possible to observe the oscillating profile of the sequence and the negative and positive symmetric branches.}\n\t\t\\label{KG_planckian_charge_distribution_plt}\n\t\\end{figure}\n\t\n\t\\noindent We observe, however, that only the electron charge ($n=1$) has a phenomenological correspondence, while the remaining points of the sequence have not a clear interpretation.\n\tIn particular according to these mass and charge distributions the fundamental charge $e$ has to be associated with a particle of Planckian mass, while the corresponding charge of the pion would be several orders of magnitude smaller than the electron charge.\n\t\n\t\n\t\\noindent Clearly, this peculiar charge-mass configuration is not phenomenological consistent.\n\t\n\t\\noindent Indeed, calculating the $q\/m$ ratio for the modes of the scalar field, we obtain:\n\t\\begin{equation}\n\t\t\\frac{q(\\mu;n)}{m(\\mu;n)}=\\pm 2\\sqrt{G}\\approx 5.16\\cdot 10^{-4} e.s.u.\/g,\n\t\\end{equation}\n\twhere the $\\pm$ sign is due to the sign of the electric charge.\n\t\n\t\\noindent This value, which coincides with the upper limit of the $q\/m$ ratio \\eqref{KK_q_over_m} of the standard classical case, does not depend on $\\mu$ nor $n$, rather it is constant for every polymer scale and every mode and no known particle satisfies such a relation. \n\t\n\t\n\t\\section{Conclusion}\n\t\\label{Conlusion}\n\t\n\tWe investigated the formulation of a five-dimensional Kaluza-Klein theory in the framework of Polymer Quantum Mechanics, viewed both in a semi-classical and quantum approach. \n\tThe polymer modifications have been implemented to the fifth coordinate only, on a semi-classical level in the spirit of the Ehrenfest theorem (the modification provides the dynamics of the quantum expectation values) and \n\tin a full quantum approach when a Klein-Gordon equation has been investigated. \n\t\n\t\\medskip \n\t\n\tWe started by applying the semi-classical polymer formulation to the evolution of the Bianchi I model, by showing that the corresponding Kasner solution can be taken in a form in which three scale factors isotropically expand and the remaining one is static and, in the considered model, it coincides with the compactified extra-dimension. \n\t\n\t\\noindent Then we studied the geodesic motion of a particle, starting with a Hamiltonian formulation (the only one in which the polymer formulation is viable) and then turning to a formalism based on the particle velocities.\n\tThis procedure allows, in analogy to the standard literature on this same subject, to identify the expression for the electric charge via the fifth momentum component of the particle. \n\tThe important consequence of this revised formulation consists of the overcoming of the problem of a too small charge to mass ratio to account in the model for any known elementary particle. \n\tIn fact, the revised constraint, due to the polymer relation between the fifth momentum component and the corresponding velocity, is in principle compatible with all the elementary particle predicted by the Standard Model. \n\t\n\t\\noindent Finally, we implemented a quantum polymer modification in the Klein-Gordon equation, by adopting a mixed representation of quantum mechanics (based on the coordinates for the usual four dimensions and the momentum for the extra one). \n\tThis study has the aim to revise the analysis in \\cite{Chodos-Detweiler} for a static (now available) extra-dimension, under a polymer prescription for the \n\tcompactified dimension physics. \n\t\n\tWe got the fundamental result that the tachyon mode, present in \\cite{Chodos-Detweiler} is now removed from the mass spectrum and that the obtained values for the boson mass can fit the values spanned in the Standard Model. \n\tActually, we arrived at a deformed morphology of the so-called Kaluza-Klein tower (the steps are no longer equispaced), but this revised structure allows us to avoid the only Planckian mode naturally present in the standard Kaluza-Klein formulation. \n\t\n\t\\medskip \n\t\n\tAll these results suggest that some of the puzzling question affecting the viability of the Kaluza-Klein idea must be reanalysed phenomenologically including the notion of a cut-off physics. \n\tIn fact, in the case of small dimensions, living about two orders of magnitude over the Planck scale, it should be unavoidable to feel the effects of the nearby cut-off and when its presence is made manifest a new paradigm can be assessed.\n\tIn other words, we argue that some limits of the geometrical unification theories are possibly due to the ultraviolet divergence that the gravitational field possesses and when they are somehow attenuated, like by the polymer scenario adopted here, the compactified dimension takes a more regular behaviour, which is reflected into the solution of some inconsistencies of the underlying model.\n\t\n\tThe emergence of a static dimension in the $5D$ Kasner solution - which prevents the necessity to deal with unphysical tachyonic modes - undoubtedly represents the simplest elucidation of this point of view. \n\t\n\t\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn this paper, we are devoted to the orbital stability of the two-component peakon solutions  and the corresponding configuration of train-profiles. The system we are concerned with is the following integrable two-component Novikov system \\cite{Li}\n\\begin{equation}\\label{tcNK}\n\\left\\{\n\\begin{aligned}\n&m_t+uvm_x+(2vu_x+uv_x)m=0,\\quad m=u-u_{xx},\\\\\n&n_t+uvn_x+(2uv_x+vu_x)n=0, \\quad n=v-v_{xx}.\n\\end{aligned}\n\\right.\n\\end{equation}\nNote that this system reduces to the well-studied integrable Novikov equation \\cite{HW, Nov}\n\\begin{equation}\\label{NOV}\nm_t+u^2m_x+3uu_{x}m=0, \\quad m=u-u_{xx},\n\\end{equation}\nwhen $v=u$.\n\n\nSince the celebrated work \\cite{CH} by Camassa and Holm, in which they first discovered the non-smooth peaked soliton solutions (called peakons) to the Camassa-Holm (CH) equation, the existence of peakons and multi-peakons is one of the significant properties of the integrable CH-type equations. The basic wave profile of a peakon takes a quite compact form\n\\begin{equation}\\label{profile}\n\\varphi_c(x-ct)=a(c)\\,e^{-|x-ct|},\n\\end{equation}\nwhere $c\\in \\R$ is the wave speed and the amplitude $a(c)$ is a function related to $c$. Remarkably, the seemingly simple form of the peakon displays the deep relationship with some important phenomena of wave propagation in shallow water waves. Indeed, due to the discussion in \\cite{cons, conesc, To}, the feature of the peakons that their profile \\eqref{profile} is smooth, except at the crest where it is continuous but the lateral tangents differ, is similar to that of the so-called Stokes waves of greatest height, i.e. traveling waves of largest possible amplitude which are solutions to the governing equations for irrotational water waves. There is no closed forms available for these waves, and the peakons can capture these described essential features. It is well-understood that the CH equation\n\\begin{equation*}\nm_t+um_x+2u_{x}m=0, \\quad m=u-u_{xx},\n\\end{equation*}\nand the Degasperis-Procesi (DP) equation\n\\begin{equation*}\nm_t+um_x+3u_{x}m=0, \\quad m=u-u_{xx},\n\\end{equation*}\nboth arise as the appropriate asymptotic approximations of the Euler equations for the free-surface shallow water waves in the moderately nonlinear regime \\cite{CL}, and admit the following peakon solutions ($c>0$, and anti-peakon for $c<0$) in the line \\cite{CH, CHH, CHT, DHK, Len1, Len2, LS}\n\\begin{equation}\\label{CHpeakons}\nu(t, x)=\\varphi_c(x-ct)=ce^{-\\left| x-ct\\right|}, \\qquad c\\neq 0.\n\\end{equation}\nOn the other hand, the peakons and the corresponding multi-peakons admit a rich mathematical structure related to the underlying integrable features. Indeed, both the CH and DP equations are integrable equations with Lax-pair formulations, and the inverse scattering approach can be used to derive explicitly these peakons \\eqref{CHpeakons} and the related multi-peakon solutions \\cite{BSS, ET, LS-0}. In the past ten years, two typical CH-type integrable equations with cubic nonlinearity that support peakon dynamics attracted much attention. One is the Novikov equation \\eqref{NOV}, whose peakon solutions take the form of \\cite{HW}\n\\begin{equation*}\nu(t, x)=\\varphi_c(x-ct)=\\sqrt{c}e^{-\\left| x-ct\\right|}, \\qquad c> 0.\n\\end{equation*}\nAnother one is the modified Camassa-Holm (mCH) equation \\cite{OR}\n\\begin{equation*}\nm_t+\\left( (u^2-u_x^2)m\\right)_x=0, \\quad m=u-u_{xx},\n\\end{equation*}\nwhich has the following peakon structure \\cite{gloq}\n\\begin{equation*}\nu(t, x)=\\varphi_c(x-ct)=\\sqrt{\\frac{3c}{2}}\\,e^{-\\left| x-ct\\right|}, \\qquad c> 0.\n\\end{equation*}\nAlthough the physical background of the Novikov and mCH equations is not so clear, their peakon and multi-peakon dynamics are demonstrated to have several non-trivial properties in the framework of Lax integrability (see the discussion in \\cite{CSz, HLS}, etc).\n\nIn this paper, special concern for these peakon solutions is the issue of their stability, which lies in the fact that they are the explicit weak solutions in the sense of distribution to the corresponding equations. The peakon equations exhibit different features in contrast with the classical integrable equations such as the KdV equation, the modified KdV equation and the Schr\\\"{o}dinger equation that admit the smooth solitions, especially in the study of qualitative properties related to stability and instability. There are huge number of papers to study the stability and instability of solitons for classical integrable systems. We don't attempt to exhaust all the literatures, one can refer to \\cite{BEN, BON, CaLi, GSS-1, GSS-2, MM, MMT, pava, PW} and the references therein. Note that the peakons don't admit the classical second-order derivatives and the linearized operators at peakons appear to be degenerate. So the classical methods based on the spectral analysis are not available in the case of peakons. In an intriguing paper due to Constantin and Strauss \\cite{CS}, they proved the orbital stability of peakons for the CH equation by discovering several precise optimal inequalities relating to the maximum value of the approximate solutions and the conserved quantity of quadratic form and higher-order conserved quantity (see also \\cite{CM} for a variational argument and \\cite{Len} for the periodic case). This approach in \\cite{CS} was further developed by Dika and Molinet \\cite{DM} to study orbital stability of the train of peakons to the CH equation. Orbital stability of peakons and the train of peakons of other CH-type integrable equations were investigated in \\cite{Kab, LL, LLQ-1, LLQ, LLOQ, QLL}. More recently, the issue of instability of peakons for the CH equation and the Novikov equation are addressed in \\cite{NP} and \\cite{CP}, respectively.\n\nCompared with the rich results of the stability or instability properties of peakons for the CH-type integrable equations of scalar form, the corresponding work for the multi-component peakon profiles is quite a few. It is worth noticing that multi-component CH-type integrable systems, that are verified to admit the peakon structures as the distributional solutions, are very rare up to now. A celebrated integrable multi-component extension to the CH equation is the so-called two-component CH system \\cite{CLZ, CI, OR}\n\\begin{eqnarray}\\label{tcCH}\n\\left\\{\n\\begin{aligned}\n&\\; m_t+um_x+2u_xm+\\rho\\rho_x=0, \\quad m=u-u_{xx},\\\\\n&\\rho_t+(\\rho u)_x=0.\n\\end{aligned}\n\\right.\n\\end{eqnarray}\nHowever, system \\eqref{tcCH} does not admit the peaked solitons \\cite{HNT}. The orbital stability of a kind of reduced peakon profile was studied via variational method \\cite{CLLQ}. The Novikov equation \\eqref{NOV} admits the following two-component integrable extension, called the Geng-Xue system \\cite{GX},\n\\begin{equation}\\label{GX}\n\\left\\{\n\\begin{aligned}\n&m_t+uvm_x+3vu_xm=0,\\quad m=u-u_{xx},\\\\\n&n_t+uvn_x+3uv_xn=0, \\quad n=v-v_{xx},\n\\end{aligned}\n\\right.\n\\end{equation}\nwhich has been paid much attention recently \\cite{LiLiu, LS-1, LS-2}. Although system \\eqref{GX} supports the multi-peakon structures, they are derived in the framework of Lax-pair formulation of \\eqref{GX} (see \\cite{LS-2}), which are not weak solutions in the sense of distribution. To the best of our  knowledge, there is no work to study orbital stability of multi-component peakons for integrable multi-component CH-type equations.\n\nRecently, another kind of two-component integrable generalization \\eqref{tcNK} of the Novikov equation \\eqref{NOV} was introduced in \\cite{Li} and we find that this system admits the two-component peakon structure, which are given by\n\\begin{equation}\\label{solitons}\n\\big(u(t, x), v(t, x)\\big)=\\big(\\varphi_c(x-ct), \\psi_c(x-ct)\\big)=\\big(a\\varphi(x-ct), b\\psi(x-ct)\\big)=\\big(ae^{-\\left| x-ct\\right|}, be^{-\\left| x-ct\\right|}\\big),\n\\end{equation}\ntraveling at constant speed $c=ab\\neq 0$. It is demonstrated that these peakons \\eqref{solitons} are indeed the weak solutions of \\eqref{tcNK} in the distribution form\n\\begin{equation}\\label{weakform}\n\\left\\{\n\\begin{aligned}\n&u_t+uvu_x+P_x*\\left( \\frac{1}{2}u_x^2v+uu_xv_x+u^2v\\right)+\\frac 12 P*(u_x^2v_x)=0,\\\\\n&v_t+uvv_x+P_x*\\left( \\frac{1}{2}v_x^2u+vv_xu_x+v^2u\\right)+\\frac 12 P*(v_x^2u_x)=0,\n \\end{aligned}\n\\right.\n\\end{equation}\nwhere $P(x)=e^{-\\left| x\\right|}\/2$ and $*$ stands for convolution with respect to the spatial variable $x\\in \\mathbb R$. Here a question arises: are these multi-component peakons and the corresponding train-profiles for system \\eqref{tcNK} stable in the energy space?\n\nSystem \\eqref{tcNK} adapts the following conserved densities\n\\begin{equation*}\nE_{0}[u,v]=\\int_{\\mathbb R}(mn)^{\\frac 13}\\,dx,\n\\end{equation*}\n\\begin{equation*}\nE_{u}[u]=\\int_{\\mathbb{R}}\\left( u^2+u_x^2\\right) \\, dx, \\quad E_{v}[v]=\\int_{\\mathbb{R}}\\left( v^2+v_x^2\\right) \\, dx, \\quad H[u,v]=\\int_{\\mathbb{R}}\\big(uv+u_xv_x\\big) \\, dx\n\\end{equation*}\nand\n\\begin{equation*}\nF[u,v]=\\int_{\\mathbb{R}}\\left( u^2v^2+\\frac{1}{3}u^2v_x^2+\\frac{1}{3}v^2u_x^2+\\frac{4}{3}uvu_xv_x-\\frac{1}{3}u_x^2v_x^2\\right) \\, dx,\n\\end{equation*}\nwhich will play prominent role in proving stability of peakons, while the corresponding three conserved quantities of Novikov equation \\eqref{NOV} are\n\\begin{equation*}\nH_0[u]=\\int_{\\mathbb R}m^{\\frac 23}\\,dx, \\;\\; E[u]=\\int_{\\mathbb{R}}{(u^2+u_x^2)} \\, dx,\\;\\;\nF[u]=\\int_{\\mathbb{R}}{\\left (u^4+2u^2u_x^2-\\frac{1}{3}u_x^4 \\right )}\\, dx.\n\\end{equation*}\nIf we choose $u=v$, then we have $H_0=E_0[u,u]$, $E[u]=E_{u}[u]=E_{v}[v]=H[u,v]$ and $F[u,v]=F[u]$. Due to the existence of the $H^1$ conservation law respectively for the $u$ and $v$-components as well as the mutual interaction conservation laws $H[u, v]$, it is expected to prove stability for the two-component Novikov system in the sense of the energy space of $H^{1}\\times H^{1}$-norm. In general, a small perturbation of a solitary wave can yield another one with a different speed and phase shift. It is appropriate to define the orbit of traveling-wave solutions $(\\varphi_{c}, \\, \\psi_{c})$ to be the set $U(\\varphi, \\psi)=\\{(a\\varphi(\\cdot+x_{1}), b\\psi(\\cdot+x_{2})), x_{1} \\in \\mathbb{R}, x_{2} \\in \\mathbb{R}\\}$. However, if $x_{1}\\neq x_{2}$, the functionals $F[\\varphi_{c}, \\, \\psi_{c}]$ and $H[\\varphi_{c}, \\, \\psi_{c}]$ are not conserved in the time evolution for $(\\varphi_{c}, \\, \\psi_{c})$ in this set $U(\\varphi, \\psi)$. Thus, we consider here a suitable orbit of the traveling-wave solutions $\\varphi_{c}$ and $\\psi_{c}$ to be the set $U_0(\\varphi, \\psi)=\\{(a\\varphi(\\cdot+x_{0}), b\\psi(\\cdot+x_{0})), x_{0} \\in \\mathbb{R}\\}$ and the peakon solutions of the two-component Novikov equation are called orbitally stable if a wave standing close to the peakon remains close to the orbit $U_0(\\varphi, \\psi)$ at all the later existence time.\n\nThe first main result is stated as follows. Here, we only consider the case of peakons traveling to the right, i.e. the case of $a>0$, $b>0$ and then $c=ab>0$ in \\eqref{solitons}.\n\n\\begin{theorem}\\label{thm1.1}\n\\, Let $(\\varphi_c, \\, \\psi_c)$ be the peaked solitons in \\eqref{solitons}, traveling with speed $c=ab>0$. Then $(\\varphi_c, \\, \\psi_c)$ are orbitally stable in the following sense. Assume that $u_0,v_0 \\in H^s(\\mathbb{R})$ for some $s \\geq 3$, $(1-\\partial_x^2)u_0(x)$ and $(1-\\partial_x^2)v_0(x)$ are nonnegative, and there is a $\\delta>0$ such that\n\\begin{align*}\n\\left \\| (u_0,v_0)-(\\varphi_c,\\psi_c) \\right \\|_{H^1(\\mathbb{R}) \\times H^1(\\mathbb{R})}\\leq \\left \\| u_0-\\varphi_c \\right \\|_{H^1(\\mathbb{R})} +\\left \\| v_0-\\psi_c \\right \\|_{H^1(\\mathbb{R})} < \\delta.\n\\end{align*}\nThen the corresponding solution $(u(t, x), v(t, x))$ of the Cauchy problem for the two-component Novikov system \\eqref{tcNK} with the initial data $u(0, x)=u_0(x)$ and $v(0, x)=v_0(x)$ satisfies\n\\begin{eqnarray*}\n\\begin{aligned}\n&\\sup_{t \\in [0,T)}\\left \\| \\big{(}u(t, \\cdot), v(t, \\cdot)\\big{)}-\\big{(}\\varphi_c(\\cdot-\\xi(t)), \\psi_c(\\cdot-\\xi(t))\\big{)} \\right \\|_{H^1(\\mathbb{R}) \\times H^1(\\mathbb{R})}\\\\\n&\\leq \\sup_{t \\in [0,T)}{\\left \\| u(t, \\cdot)-\\varphi_c(\\cdot-\\xi(t)) \\right \\|_{H^1(\\mathbb{R})}+\\left \\| v(t, \\cdot)-\\psi_c(\\cdot-\\xi(t)) \\right \\|_{H^1(\\mathbb{R})}} < A\\delta^\\frac14,\n\\end{aligned}\n\\end{eqnarray*}\nwhere $T>0$ is the maximal existence time, $\\xi(t) \\in \\mathbb{R}$ is the maximum point of the function $u(t, x)v(t, x)$, the constant $A$ depends only on $a$, $b$ as well as the norms $\\left \\| u_0 \\right \\|_{H^s(\\mathbb{R})}$ and $\\left \\| v_0 \\right \\|_{H^s(\\mathbb{R})}$.\n\\end{theorem}\n\nTo prove orbital stability of the two-component peakons $(\\varphi_c, \\, \\psi_c)$, some new insights are developed. We aim to obtain for each component $u$ and $v$ the dynamical estimates $|u(t, \\xi(t))-a|$ and $|v(t, \\xi(t))-b|$ along some trajectory $t \\mapsto \\xi(t)$, where $a$ and $b$ are the maximal value of the component $\\varphi_c$ and $\\psi_c$, respectively. Here, due to the nonlinear interaction between $u$ and $v$ involved in system \\eqref{tcNK}, the key obstacle is how to find the suitable location of $\\xi(t)$ in order to derive the precise estimates for $|u(t, \\xi(t))-a|$ and $|v(t, \\xi(t))-b|$ (note that in the case of scalar peakons such $\\xi(t)$ is always chosen to locate at the maximal point of perturbed solution, which is no longer valid in the multi-component case considered here and $\\xi(t)$ must change according to the appearance of characteristic speed of nonlinear interaction $uv$). Moreover, the dynamical energy identities and energy inequalities should involve the nonlinear interaction of the two components. In addition, the conservation law $F[u,v]$ is much more complicated than $F[u]$ of the Novikov equation. Therefore, the stability issue of the two-component peakon solutions is more subtle. To overcome the difficulties, two observations will be crucial. System \\eqref{tcNK} not only has the separated $H^1$ conserved quantities $\\int (u^2+u_x^2) dx$ and $\\int(v^2+v_x^2) dx$ with which one can derive the pointwise estimates separately for each component $u$ and $v$, but also the second-order interacting conserved quantity $H[u,v]=\\int (uv+u_xv_x) dx$ with which we are motivated to find the exact location of $\\xi(t)$ as the maximal point of the multiplication function $u(t, \\cdot)v(t, \\cdot)$ of two components. This argument is quite different from the case for the scalar CH-type equations. On the other hand, new optimal energy identities for $H[u, v]$ and $F[u, v]$ are established. The new insight is that the precise control of two components is involved in one optimal energy identity. This point is also different with the scalar case. Based on these observations together with corresponding refined analysis, we are able to prove orbital stability of the two-component peakons $(\\varphi_c, \\, \\psi_c)$ on the line ${\\mathbb R}$.\n\n\\begin{remark}\nFor the Geng-Xue system \\eqref{GX}, even though the peakon solutions in the Lax-pair sense are considered, system \\eqref{GX} does not admit sufficient conserved quantities to establish the corresponding estimates for the stability.\n\\end{remark}\n\nFor the issue of orbital stability of train-profiles of these two-component peakons, we have the following result.\n\n\\begin{theorem}\\label{trainsstable}\nLet be given $N$ velocities $c_{1},c_{2},\\cdots, c_{N}$ such that $0<a_{1}<a_{2}<...<a_{N}$, $0<b_{1}<b_{2}<...<b_{N}$ and $c_{i}=a_{i}b_{i}$ for any $i\\in \\{1,...,N\\}$. There exist $A>0$, $L_0>0$ and $\\epsilon_{0}>0$ such that if the initial data $(u_0,v_0) \\in H^s(\\mathbb{R})\\times H^s(\\mathbb{R})$ for some $s \\geq 3$ with $(1-\\partial_x^2)u_0(x)$ and $(1-\\partial_x^2)v_0(x)$ being nonnegative, satisfy\n\\begin{align}\\label{initialdata-1}\n{\\left \\| u_0- \\sum_{i=1}^N{\\varphi_c(\\cdot - z_i^0)} \\right \\|_{H^1}}+{\\left \\| v_0- \\sum_{i=1}^N{\\psi_c(\\cdot - z_i^0)} \\right \\|_{H^1}}\\leq {\\epsilon}\n\\end{align}\nfor some $0<\\epsilon <\\epsilon_0$ and $ z_i^0-z_{i-1}^0 \\geq L$ with $L>L_{0}$, then there exist ${x}_1(t),...,{x}_N(t)$ such that the corresponding strong solution $(u(t, x), v(t, x))$ satisfies\n\\begin{align*}\n{\\left \\| u(t, \\cdot)- \\sum_{i=1}^N{\\varphi_c(\\cdot - {x}_i(t))} \\right \\|_{H^1}}+{\\left \\| v(t, \\cdot)- \\sum_{i=1}^N{\\psi_c(\\cdot - {x}_i(t))} \\right \\|_{H^1}} \\leq A\\left( \\epsilon^{ \\frac{1}{4}}+L^{- \\frac{1}{8}}\\right),\n\\end{align*}\n$\\forall t \\in [0,T)$, where $x_{j}(t)-x_{j-1}(t)>L\/2$.\n\\end{theorem}\n\nIn general, two main ingredients in the proof of orbital stability for the train-proflies of peakons are involved \\cite{DM,LLQ,MMT}. One is orbital stability of the single peakons, and the another one is the property of almost monotonicity of the local energy on the right hand side of the peakons. For the two-component peakons, more difficulties come from the interaction of the two components $u(t, x)$ and $v(t, x)$. The first one is how to establish inequalities among the localized conserved quantities to verify the orbital stability of the two-component peakons separately. For the train-profiles of peakons, we need to apply $N$ inequalities to control $2N$ estimates. The second one is to establish monotonicity result of the functionals $\\mathcal{J}^{u,v}_{j,k}(t)$ since we can not identify the sign for the term $u_xv_x$ in the conserved density $H[u,v]$. To overcome the difficulties, we use \nthe conserved densities $E_u[u]$, $E_v[v]$, $H[u,v]$ and $F[u,v]$, and establish the delicate inequalities relating to the conserved densities and the maximal value of the two components $u(t,x)$ and $v(t,x)$. And in the case of train-profile of two-component peakons, we apply the proof of the single peakons, the modulation theory, the accurate estimates on the conserved densities $E_u[u]$, $E_v[v]$, $H[u, v]$ and $F[u,v]$ and the induction method to obtain the desired result.\n\nThe remainder of the paper is organized as follows. In Section 2, we provide a brief discussion on the integrability, conservation laws, the sign invariant property of $m(t, x)$ and $n(t, x)$ of system \\eqref{tcNK},  and local well-posdness result on Cauchy problem of system \\eqref{tcNK}. In Section 3, we prove orbital stability of single peakons on the line. Finally in Section 4, we verify orbital stability of the train of peakons.\n\n\n\n\n\n\n\\section{Preliminaries}\n\nIn the present section, the issue of well-posedness is discussed. First of all, we call that functions $(u,v)\\in C([0,T);H^1(\\mathbb R))\\times C([0,T);H^1(\\mathbb R))$ is a solution of system \\eqref{tcNK}, if $(u,v)$ is a solution of \\eqref{weakform} in the sense of distributions and $E_{u}[u]$, $E_{v}[v]$, $H[u, v]$ and $F[u, v]$ are conserved quantities.\n\nConsider the following Cauchy problem of system \\eqref{tcNK} in the whole line $\\R$\n\\begin{equation}\\label{tcNK-1}\n\\left\\{\n\\begin{aligned}\n&m_t+uvm_x+(2vu_x+uv_x)m=0,\\quad m=u-u_{xx},\\\\\n&n_t+uvn_x+(2uv_x+vu_x)n=0, \\quad n=v-v_{xx}, \\quad t>0, \\; x\\in \\R,\\\\\n&u(0,x)=u_0(x), \\;\\;v(0,x)=v_0(x), \\quad x\\in \\R.\n\\end{aligned}\n\\right.\n\\end{equation}\n\nFirst, similar to the results for the two-component CH system given in \\cite{FQ}, one can establish the following local well-posedness result.\n\\begin{theorem}\\label{th-2.1}\nGiven $z_0=(u_0, v_0)^T\\in H^s(\\R)\\times H^s(\\R) (s>3\/2)$, there exists a maximal $T=T(\\| z_0\\|_{H^s(\\R)\\times H^s(\\R)})>0$ and a unique strong solution $z=(u, v)^T$ to \\eqref{tcNK-1} such that\n\\begin{equation*}\nz=z(\\cdot, z_0)\\in C([0, T);H^s(\\R)\\times H^s(\\R))\\cap C^1([0,T); H^{s-1}(\\R)\\times H^{s-1}(\\R)).\n\\end{equation*}\nIn addition, the solution depends continuously on the initial data, i.e. the mapping\n\\begin{equation*}\nz_0\\rightarrow z(\\cdot, z_0): H^s(\\R)\\times H^s(\\R) \\rightarrow C([0, T); H^s(\\R)\\times H^s(\\R))\\cap C^1([0, T); H^{s-1}(\\R)\\times H^{s-1}(\\R))\n\\end{equation*}\nis continuous. Furthermore, the quantities $E_{u}[u]$, $E_{v}[v]$, $H[u, v]$ and $F[u, v]$ are all conserved along the solution $z=(u, v)^T$.\n\\end{theorem}\n\nConsider the flow governed by $(uv)(t, x)$\n\\begin{equation}\\label{firstflow}\n\\left\\{\n\\begin{aligned}\n&\\frac{d}{dt}q(t, x)=(uv)(t, q(t, x)),\\quad x\\in \\R, \\quad t\\in [0, T),\\\\\n&q(0, x)=x,\\quad x\\in \\R.\n\\end{aligned}\n\\right.\n\\end{equation}\nJust as the case for the Novikov equation \\eqref{NOV}, the following lemma can be proved.\n\\begin{lemma}\\label{Flow}\nAssume that $(u_{0},v_0)\\in H^s(\\mathbb R)\\times H^s(\\mathbb R)$ with $s>3\/2$ and $T>0$ be the maximal existence time of the corresponding strong solution $(u, v)$ to the Cauchy problem of system \\eqref{tcNK-1}. Then the problem \\eqref{firstflow} has a unique solution $q(t, x)\\in C^1([0, T)\\times \\R)$. Furthermore, the map $q(t,\\cdot)$ is an increasing diffeomorphism over $\\R$ with\n\\begin{eqnarray}\\label{diffflow}\n\\begin{aligned}\nq_{x}(t,x)=exp\\left(\\int^t_{0}(uv)_x(s,q(s,x))ds\\right),\\quad (t,x)\\in [0,T)\\times \\mathbb R.\n\\end{aligned}\n\\end{eqnarray}\n\\end{lemma}\n\n\\begin{lemma}\\label{nonnegative}\nLet $u_0, v_0\\in H^s(\\mathbb R)$, $s\\geq 3$. Assume that $m_0=u_0-u_{0xx}$ and $n_0=v_0-v_{0xx}$ are nonnegative in the line.\nThen for the corresponding strong solution $u, v\\in C([0,T); H^s(\\R))\\cap C^1([0,T); H^{s-1}(\\R))$ of the Cauchy problem of the two-component Novikov system \\eqref{tcNK-1} with the initial data $u_0,\\,v_0$, we have for all $t\\in [0, T)$, $m(t, x)$ and $n(t, x)$ are both nonnegative functions. In addition, $u(t, \\cdot) \\geq 0$, $v(t, \\cdot) \\geq 0$ and $|u_{x}(t, \\cdot)|\\leq u(t, \\cdot)$, $|v_{x}(t, \\cdot,)|\\leq v(t, \\cdot)$ on the line.\n\\end{lemma}\n\\begin{proof}\nIt follows from the Cauchy problem \\eqref{tcNK-1} of the two-component Novikov system that along the flow \\eqref{firstflow}, $m$ and $n$ satisfy\n\\begin{equation}\\label{weakformFLOW}\n\\left\\{\n\\begin{aligned}\n&m'+(2vu_x+uv_x)(t,q(t,x))m(t,q(t,x))=0,\\\\\n&n'+(2uv_x+vu_x)(t,q(t,x))n(t,q(t,x))=0,\n\\end{aligned}\n\\right.\n\\end{equation}\nwhere $'$ denotes the derivative with respect to $t$ along the flow \\eqref{firstflow}. Denote\n\\begin{eqnarray*}\n\\gamma_1(t,x)=exp\\left(\\int^t_{0}(2vu_x+uv_x)(s,q(s,x))ds\\right), \\;  \\gamma_2(t,x)=exp\\left(\\int^t_{0}(2uv_x+vu_x)(s,q(s,x))ds\\right).\n\\end{eqnarray*}\nThen they satisfy\n\\begin{eqnarray*}\n\\gamma'_{1}(t,x)=(2vu_x+uv_x)(t,q(t,x))\\gamma_1,\\quad \\gamma'_{2}(t,x)=(2uv_x+vu_x)(t,q(t,x))\\gamma_2.\n\\end{eqnarray*}\nLet\n\\begin{eqnarray*}\n\\begin{aligned}\n\\tilde{m}(t,x)=\\gamma_{1}(t,x)m(t,q(t,x)), \\quad \\tilde{n}(t,x)=\\gamma_{2}(t,x)n(t,q(t,x)).\n\\end{aligned}\n\\end{eqnarray*}\n\\eqref{weakformFLOW} become\n\\begin{eqnarray}\n\\begin{aligned}\n\\tilde{m}'(t,x)=0, \\quad \\tilde{n}'(t,x)=0.\n\\end{aligned}\n\\end{eqnarray}\nThe equations\n\\begin{eqnarray*}\n\\begin{aligned}\n\\tilde{m}(t,x)=m_0 \\quad and \\quad \\tilde{n}(t,x)=n_0\n\\end{aligned}\n\\end{eqnarray*}\nlead to\n\\begin{eqnarray*}\n\\begin{aligned}\n&m(t,q(t,x))=exp\\left(-\\int^t_{0}(2vu_x+uv_x)(s,q(s,x))ds\\right)m_0,\\\\\n&n(t,q(t,x))=exp\\left(-\\int^t_{0}(2uv_x+vu_x)(s,q(s,x))ds\\right)n_0.\n\\end{aligned}\n\\end{eqnarray*}\nThus, for all $t\\in [0, T)$, we have $m(t, \\cdot) \\geq 0$, $n(t, \\cdot) \\geq 0$. And then $u(t, \\cdot) \\geq 0$, $v(t, \\cdot) \\geq 0$.\n\nFormally regarding $m(x)=u(x)-u_{xx}(x)$, it holds that for all $x\\in \\R$,\n\\begin{eqnarray*}\n\\begin{aligned}\nu(x)=\\frac{e^{-x}}{2}\\int^{x}_{-\\infty}e^{y}m(y)dy+\\frac{e^{x}}{2}\\int^{\\infty}_{x}e^{-y}m(y)dy\n\\end{aligned}\n\\end{eqnarray*}\nand\n\\begin{eqnarray*}\n\\begin{aligned}\nu_{x}(x)=-\\frac{e^{-x}}{2}\\int^{x}_{-\\infty}e^{y}m(y)dy+\\frac{e^{x}}{2}\\int^{\\infty}_{x}e^{-y}m(y)dy.\n\\end{aligned}\n\\end{eqnarray*}\nThen we infer that\n\\begin{eqnarray*}\n\\begin{aligned}\nu(x)\\geq |u_{x}(x)|, \\; \\forall x\\in \\mathbb R.\n\\end{aligned}\n\\end{eqnarray*}\nSimilarly, we find\n\\begin{eqnarray*}\n\\begin{aligned}\nv(x)\\geq |v_{x}(x)|, \\; \\forall x\\in \\mathbb R.\n\\end{aligned}\n\\end{eqnarray*}\nThis completes the proof of the lemma.\n\\end{proof}\n\n\n\n\n\n\n\\section{Stability of two-component peakons}\n\nIn this section, we prove Theorem 1.1, which will be based on a series of lemmas. Note that the assumptions on the initial profile guarantee the existence of a unique positive solution for the Cauchy problem \\eqref{tcNK-1} of the two-component Novikov system. In general, for $a>0$ and $b>0$, the profile functions of peakon solutions $\\varphi_c(x)=ae^{-\\left| x\\right|}$ and $\\psi_c(x)=be^{-\\left| x\\right|}$ are in $H^1(\\R)$, which have peaks at $x=0$, and thus\n\\begin{equation*}\n\\max_{x \\in \\mathbb{R}}{\\varphi_c(x)}=\\varphi_c(0)=a \\quad \\mathrm{and} \\quad \\max_{x \\in \\mathbb{R}}{\\psi_c(x)}=\\psi_c(0)=b.\n\\end{equation*}\nA direct calculation gives\n\\begin{equation*}\nE_{u}[\\varphi_c(x)]=\\left \\| \\varphi_c \\right \\|^2_{H^1}=2a^2, \\qquad E_{v}[\\psi_c(x)]=\\left \\| \\psi_c \\right \\|^2_{H^1}=2b^2\n\\end{equation*}\nand\n\\begin{equation*}\nH[\\varphi_c(x),\\psi_c(x)]=2ab, \\qquad F[\\varphi_c(x),\\psi_c(x)]=\\frac43 \\, a^2b^2.\n\\end{equation*}\n\nDue to the conservation of the $H^1$-norm of each component $u$ and $v$, the following pointwise identities still hold for the two-component Novikov system as in the scalar CH and Novikov cases.\n\\begin{lemma}\\label{lem1.1}\nFor any $u,v \\in H^1(\\mathbb{R})$ and $\\xi \\in \\mathbb{R}$, we have\n\\begin{eqnarray}\\label{energyESI}\n\\begin{aligned}\n&E_{u}[u]-E_{u}[\\varphi_c]=\\left \\| u- \\varphi_c(\\cdot - \\xi) \\right \\|^2_{H^1(\\mathbb{R})}+4a(u(\\xi)-a),\\\\\n&E_{v}[v]-E_{v}[\\psi_c]=\\left \\| v- \\psi_c(\\cdot - \\xi) \\right \\|^2_{H^1(\\mathbb{R})}+4b(v(\\xi)-b).\n\\end{aligned}\n\\end{eqnarray}\n\\end{lemma}\n\n\nIn the following two lemmas, two energy identities relating some kind of critical values of $u$ and $v$ to the invariants $H[u, v]$ and $F[u, v]$ are established. Consider $0\\not\\equiv u,\\,v\\in H^{s}(\\R)$, $s\\geq 3$, and $u, v\\geq 0$. Then $u,v\\in C^2$ due to the Sobolev imbedding theory. Since $u$ and $v$ decay at infinity, it must have a point with global maximal value of the multiplication function $u(x)v(x)$. Thus, in the following, set for some $\\xi\\in \\R$\n\\begin{eqnarray}\\label{maxpoint}\nM=\\max_{x\\in \\R}\\{u(x)v(x)\\}=u(\\xi)v(\\xi).\n\\end{eqnarray}\n\n\\begin{lemma}\\label{lem1.2}\nLet $0\\not\\equiv u,\\,v\\in H^{s}(\\R)$, $s\\geq 3$ and $u, v\\geq 0$. Define the functions $g_1(x)$ and $g_2(x)$ by\n\\begin{equation}\\label{functiong1}\ng_1(x)=\n\\left\\{\n\\begin{aligned}\n&u(x)-u_{x}(x), \\quad x<\\xi,\\\\\n&u(x)+u_{x}(x), \\quad x>\\xi,\n \\end{aligned}\n\\right.\n\\end{equation}\nand\n\\begin{equation}\\label{functiong2}\ng_2(x)=\n\\left\\{\n\\begin{aligned}\n&v(x)-v_{x}(x), \\quad x<\\xi,\\\\\n&v(x)+v_{x}(x), \\quad x>\\xi.\n \\end{aligned}\n\\right.\n\\end{equation}\nThen\n\\begin{eqnarray}\\label{g1g2ESI}\n\\begin{aligned}\n\\int_{\\mathbb{R}}{g_{1}(x)g_{2}(x)}dx=H[u,v]-2M.\n\\end{aligned}\n\\end{eqnarray}\n\\end{lemma}\n\n\\begin{proof}\nTo show \\eqref{g1g2ESI}, we evaluate the integral of $g_1(x)g_2(x)$ on $\\mathbb R$. Thus,\n\\begin{eqnarray*}\n\\begin{aligned}\n&\\int_{\\mathbb{R}}{g_{1}(x)g_{2}(x)}dx\\\\\n&=\\int_{-\\infty}^{\\xi}{\\left(u(x)-u_{x}(x)\\right)\\left(v(x)-v_{x}(x)\\right)}dx+\\int_{\\xi}^{\\infty}{\\left(u(x)+u_{x}(x)\\right)\\left(v(x)+v_{x}(x)\\right)}\\,dx\\\\\n&=\\int_{-\\infty}^{\\xi}{\\left(uv-(uv)_x+u_xv_x\\right)}dx+\\int_{\\xi}^{\\infty}{\\left(uv+(uv)_x+u_xv_x\\right)}\\,dx\\\\\n&=H[u,v]-2M.\n\\end{aligned}\n\\end{eqnarray*}\n\\end{proof}\n\nThe construction of the auxiliary function $h(x)$ in the following lemma is crucial in the proof of stability of peakons, which is different from the scalar cases of CH and DP equations. This new defined function is a nontrivial refinement of the case in the Novikov equation.\n\n\\begin{lemma}\\label{lem1.3}\nWith the same assumptions and notation as in Lemma \\ref{lem1.2}, define the function $h$ by\n\\begin{equation}\\label{functionalh}\nh(x)=\n\\left\\{\n\\begin{aligned}\n&uv-\\frac{1}{3}(uv)_{x}-\\frac{1}{3}u_{x}v_{x}, \\quad x<\\xi,\\\\\n&uv+\\frac{1}{3}(uv)_{x}-\\frac{1}{3}u_{x}v_{x}, \\quad x>\\xi.\n \\end{aligned}\n\\right.\n\\end{equation}\nThen\n\\begin{eqnarray}\\label{Finalesi}\n\\begin{aligned}\n\\int_{\\mathbb{R}}{h(x)g_{1}(x)g_{2}(x)}dx=F[u,v]-\\frac{4}{3}\\,M^2.\n\\end{aligned}\n\\end{eqnarray}\n\\end{lemma}\n\n\\begin{proof}\nTo show \\eqref{Finalesi}, we evaluate the integral of $h(x)g_{1}(x)g_{2}(x)$ on $\\mathbb R$. Thus,\n\\begin{eqnarray*}\n\\begin{aligned}\n&\\int_{\\mathbb{R}}{h(x)g_{1}(x)g_{2}(x)}dx\\\\\n&=\\int_{-\\infty}^{\\xi}\\left[uv-\\frac{1}{3}(uv)_{x}-\\frac{1}{3}u_{x}v_{x}\\right]\\left[uv-(uv)_{x}+u_{x}v_{x}\\right]\\,dx\\\\\n&\\qquad+\\int_{\\xi}^{\\infty}\\left[uv+\\frac{1}{3}(uv)_{x}-\\frac{1}{3}u_{x}v_{x}\\right]\\left[uv+(uv)_{x}+u_{x}v_{x}\\right]\\,dx\\triangleq I+\\Pi.\n\\end{aligned}\n\\end{eqnarray*}\nWe do computation:\n\\begin{eqnarray*}\n\\begin{aligned}\nI&=\\int_{-\\infty}^{\\xi}\\left[uv-\\frac{1}{3}(uv)_{x}-\\frac{1}{3}u_{x}v_{x}\\right]\\left[uv-(uv)_{x}+u_{x}v_{x}\\right]\\,dx\\\\\n&=\\int_{-\\infty}^{\\xi}\\left(u^2v^2+\\frac{1}{3}u^2v_x^2+\\frac{1}{3}v^2u_x^2+\\frac{4}{3}uvu_xv_x-\\frac{1}{3}u_x^2v_x^2\\right)\\,dx-\\frac{4}{3}\\,\\int_{-\\infty}^{\\xi}(uv)(uv)_{x}\\,dx\\\\\n&=\\int_{-\\infty}^{\\xi}\\left(u^2v^2+\\frac{1}{3}u^2v_x^2+\\frac{1}{3}v^2u_x^2+\\frac{4}{3}uvu_xv_x-\\frac{1}{3}u_x^2v_x^2\\right)\\,dx-\\frac{2}{3}M^2.\n\\end{aligned}\n\\end{eqnarray*}\nSimilarly,\n\\begin{eqnarray*}\n\\begin{aligned}\n\\Pi=\\int_{\\xi}^{\\infty}\\left(u^2v^2+\\frac{1}{3}u^2v_x^2+\\frac{1}{3}v^2u_x^2+\\frac{4}{3}uvu_xv_x-\\frac{1}{3}u_x^2v_x^2\\right)\\,dx-\\frac{2}{3}M^2.\n\\end{aligned}\n\\end{eqnarray*}\nCombining above, we have\n\\begin{eqnarray*}\n\\begin{aligned}\n&\\int_{\\mathbb{R}}{h(x)g_{1}(x)g_{2}(x)}dx\\\\\n&=\\int_{-\\infty}^{\\infty}\\left(u^2v^2+\\frac{1}{3}u^2v_x^2+\\frac{1}{3}v^2u_x^2+\\frac{4}{3}uvu_xv_x-\\frac{1}{3}u_x^2v_x^2\\right)\\,dx-\\frac{4}{3}\\,M^2=F[u,v]-\\frac{4}{3}\\,M^2.\n\\end{aligned}\n\\end{eqnarray*}\n\\end{proof}\n\nWith the two energy identities \\eqref{g1g2ESI} and \\eqref{Finalesi} in hand, one can derive the following delicate relation between the second order conserved quantity $H[u, v]$ and the higher-order conserved quantity $F[u, v]$ for the strong solution $(u, v)$.\n\n\\begin{lemma}\\label{lem1.4}\nAssume that $0\\not\\equiv u_0,\\,v_0\\in H^s$, $s\\geq 3$ and $m_0=u_0-u_{0xx}\\geq 0$, $n_0=v_0-v_{0xx}\\geq 0$. For the corresponding strong solution $(u(t, x),\\,v(t, x))$ with initial data $(u_0,\\,v_0)$ in the lifespan $[0,\\,T)$, there holds\n\\begin{eqnarray}\\label{functionalESI}\n\\begin{aligned}\nF[u, v]-\\frac{4}{3}M(t)H[u, v]+\\frac{4}{3}M(t)^2 \\leq 0, \\quad \\forall t\\in [0,\\,T),\n\\end{aligned}\n\\end{eqnarray}\nwhere $M(t)=\\max_{x\\in R}\\{u(t, x)v(t, x)\\}=u(t, \\xi(t))v(t, \\xi(t))$ for some trajectory $\\xi(t)\\in \\R$ in $[0,\\,T)$.\n\\end{lemma}\n\n\\begin{proof}\nFirst, by the sign-invariant property, the solution $(u(t, x),\\,v(t, x))$ satisfies $m(t, x)=u(t, x)-\\partial^2_{x}u(t, x)\\geq 0$ and $n(t, x)=v(t, x)-\\partial^2_{x}v(t, x)\\geq 0$ for all $(t, x)\\in [0,\\,T)\\times \\R$. It follows that $(u(t, x),\\,v(t, x))$ is positive solution and fulfills all the conditions assumed in Lemmas 3.2 and 3.3. Hence, we have the following energy identities of the dynamical forms in $[0,\\,T)$\n\\begin{equation}\\label{H(t)}\n\\int_{\\mathbb{R}}{g_{1}(t, x)g_{2}(t, x)}dx=H[u, v]-2M(t)\n\\end{equation}\nand\n\\begin{equation}\\label{F(t)}\n\\int_{\\mathbb{R}}{h(t, x)g_{1}(t, x)g_{2}(t, x)}dx=F[u, v]-\\frac{4}{3}\\,M(t)^2.\n\\end{equation}\n\nWe now claim that for any $(t, x)\\in [0,\\,T)\\times \\R$\n\\begin{equation*}\nh(t, x)=\n\\left\\{\n\\begin{aligned}\n&\\left( uv-\\frac{1}{3} (uv)_{x}-\\frac{1}{3}u_{x}v_{x}\\right)(t, x), \\quad x<\\xi(t),\\\\\n&\\left( uv+\\frac{1}{3} (uv)_x -\\frac{1}{3}u_{x}v_{x}\\right)(t, x), \\quad x>\\xi(t)\n \\end{aligned}\n\\right.\n\\leq \\frac{4}{3}u(t, x)v(t, x).\n\\end{equation*}\nIn fact, due to the definition of $h(t, x)$, it follows from the fact $u(t, x)\\geq |u_{x}(t, x)|$ and $v(t, x)\\geq |v_{x}(t, x)|$ that\n\\begin{equation*}\nh(t, x)= \\frac{4}{3}u(t, x)v(t, x)-\\frac{1}{3}(u\\pm u_{x}(t, x))(v\\pm v_{x}(t, x))\\leq \\frac{4}{3}u(t, x)v(t, x).\n\\end{equation*}\nThus, the combination of the above inequalities yields\n\\begin{eqnarray*}\nh(t, x)\\leq \\frac{4}{3}u(t, x)v(t, x)\\leq \\frac{4}{3}\\max_{x\\in \\R}\\{u(t, x)v(t, x)\\}=\\frac{4}{3}M(t), \\quad \\forall (t, x)\\in [0,\\,T)\\times \\R.\n\\end{eqnarray*}\n\nNow, using estimates $|u_x|\\leq u$ and $|v_x|\\leq v$ again in the expressions \\eqref{functiong1} and \\eqref{functiong2}, we obtain \\eqref{functionalESI} from \\eqref{H(t)} and \\eqref{F(t)}.\n\\end{proof}\n\nIn the following lemma, we study the perturbation of the conserved quantities around the profile functions $\\varphi_c(x)$ and $\\psi_c(x)$ of peakon solutions, under the case of time independence.\n\n\\begin{lemma}\\label{lem1.5}\nFor $u, v \\in H^{s}(\\R)$, $s\\geq 3$, if $\\|u-\\varphi_{c}\\|_{H^{1}(\\mathbb{R})} < \\delta$ and $\\|v-\\psi_{c}\\|_{H^{1}(\\mathbb{R})} < \\delta$ with $0<\\delta<1\/2$, then\n\\begin{equation*}\n\\left |E_{u}[u]-E_{u}[\\varphi_{c}]\\right |<2\\sqrt{2}a\\delta+\\delta^2, \\quad \\left |E_{v}[v]-E_{u}[\\psi_{c}]\\right|<2\\sqrt{2}b\\delta+\\delta^2\n\\end{equation*}\nand\n\\begin{equation*}\n\\left|H[u,v]-H[\\varphi_{c},\\psi_{c}]\\right|<2\\sqrt{2}(a+b)\\delta +6\\delta^{2}, \\quad \\left|F[u,v]-F[\\varphi_{c},\\psi_{c}]\\right|<C\\delta+O(\\delta^{2}),\n\\end{equation*}\nwhere $C>0$ is a constant depending on $a$, $b$, $\\|u\\|_{H^s}$ and $\\|v\\|_{H^s}$.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\tilde{u}=u-\\varphi_{c}$ and $\\tilde{v}=v-\\psi_{c}$, for convenience. Since $\\|\\tilde{u}\\|_{H^{1}(\\mathbb{R})} < \\delta$ and $\\|\\tilde{v}\\|_{H^{1}(\\mathbb{R})} < \\delta$, it follows that\n\\begin{eqnarray*}\n\\begin{aligned}\n\\left|E_{u}[u]-E_{u}[\\varphi_{c}]\\right|=&\\left|\\|u_{0}\\|^{2}_{H^{1}(\\mathbb R)}-\\|\\varphi_{c}\\|^{2}_{H^{1}(\\mathbb R)}\\right|\\\\\n=&\\left|\\|u_{0}\\|_{H^{1}(\\mathbb R)}-\\|\\varphi_{c}\\|_{H^{1}(\\mathbb R)}\\right|\\left|\\|u_{0}\\|_{H^{1}(\\mathbb R)}+\\|\\varphi_{c}\\|_{H^{1}(\\mathbb R)}\\right|\\\\\n\\leq &\\|\\tilde{u}_{0}\\|_{H^{1}(\\mathbb{R})} \\left(\\|\\tilde{u}_{0}\\|_{H^{1}(\\mathbb R)}+2\\|\\varphi_{c}\\|_{H^{1}(\\mathbb R)}\\right)\\leq 2\\sqrt{2}\\,a\\delta+\\delta^2.\n\\end{aligned}\n\\end{eqnarray*}\nSimilarly, we have\n\\begin{align*}\n\\left|E_{v}[v]-E_{v}[\\psi_{c}]\\right|<2\\sqrt{2}\\,b\\delta+\\delta^2.\n\\end{align*}\n\nNext, we now estimate\n\\begin{eqnarray*}\n\\begin{aligned}\n\\Big|H[u, v]-&H[\\varphi_{c},\\psi_{c}]\\Big|=\\left|\\int_{\\mathbb R}\\left(uv+u_xv_x\\right)\\,dx-\\int_{\\mathbb R}\\left(\\varphi_{c}\\psi_{c}+\\varphi'_{c}\\psi'_{c}\\right)dx\\right|\\\\\n=&\\left|\\int_{\\mathbb R}\\left(\\tilde{u}\\tilde{v}+u\\tilde{v}+v\\tilde{u}\\right)\\,dx+\\int_{\\mathbb R}\\left[(u_x-\\varphi'_{c})(v_x-\\psi'_{c})+u_{x}(v_{x}-\\psi'_{c})+v_{x}(u_{x}-\\varphi'_{c})\\right]\\,dx\\right|\\\\\n\\leq & \\int_{\\mathbb R}\\left|\\tilde{u}\\tilde{v}\\right|\\,dx+\\int_{\\mathbb R}\\left|\\tilde{u}v\\right|\\,dx+\\int_{\\mathbb R}\\left|(v-\\varphi_{c})u\\right|\\,dx\\\\\n&\\quad +\\int_{\\mathbb R}\\left|\\tilde{u}'\\tilde{v}'\\right|dx+\\int_{\\mathbb R}\\left|\\tilde{u}'v_{x}\\right|\\,dx+\\int_{\\mathbb R}\\left|\\tilde{v}'u_{x}\\right|\\,dx.\n\\end{aligned}\n\\end{eqnarray*}\nUsing the H$\\mathrm{\\ddot{o}}$lder inequality\n\\begin{eqnarray*}\n\\int_{\\mathbb R}\\left|\\tilde{u}v\\right|\\,dx \\leq \\left(\\int_{\\mathbb R}\\tilde{u}^{2}\\,dx\\right)^{\\frac12}\\left(\\int_{\\mathbb R}v^2\\,dx\\right)^{\\frac12}\\leq \\|\\tilde{u}\\|_{H^{1}(\\mathbb{R})}\\left(\\|\\tilde{v}\\|_{H^{1}(\\mathbb{R})}+\\|\\varphi_{c}\\|_{H^{1}(\\mathbb{R})}\\right) \\leq \\delta^{2}+\\sqrt{2}\\,a \\delta,\n\\end{eqnarray*}\nwe deduce that\n\\begin{align*}\n\\left|H[u, v]-H[\\varphi_{c},\\psi_{c}]\\right|<2\\sqrt{2}(a+b)\\delta +6\\delta^{2}.\n\\end{align*}\n\nFinally, we estimate\n\\begin{eqnarray*}\n\\begin{aligned}\n\\Big|F&[u, v]-F[\\varphi_{c},\\psi_{c}]\\Big|=\\Bigg|\\int_{\\mathbb{R}}\\left(u^2v^2+\\frac{1}{3}u^2v_{x}^2+\\frac{1}{3}v^2u_{x}^2+\\frac{4}{3}uvu_{x}v_{x}-\\frac{1}{3}u_{x}^2v_{x}^2\\right)\\,dx\\\\\n&\\qquad\\qquad\\qquad\\qquad\\quad\\; -\\int_{\\mathbb{R}}\\left[\\varphi_{c}^{2}\\psi_{c}^{2}+\\frac{1}{3}\\varphi_{c}^{2}(\\psi'_{c})^{2}+\\frac{1}{3}(\\varphi'_{c})^{2}\\psi_{c}^{2}+\\frac{4}{3}\\varphi_{c}\\psi_{c}\\varphi'_{c}\\psi'_{c}-\\frac{1}{3}(\\varphi'_{c})^{2}(\\psi'_{c})^{2}\\right]\\,dx\\Bigg|\\\\\n&\\leq \\int_{\\mathbb{R}}\\left|u^2v^2-\\varphi_{c}^{2}\\psi_{c}^{2}\\right|\\,dx+\\frac{1}{3}\\,\\int_{\\mathbb{R}}\\left|u^2v_{x}^2-\\varphi_{c}^{2}(\\psi'_{c})^{2}\\right|\\,dx+\\frac{1}{3}\\,\\int_{\\mathbb{R}}\\left|v^2u_{x}^2-(\\varphi'_{c})^{2}\\psi_{c}^{2}\\right|\\,dx\\\\\n&\\quad\\quad+\\frac{4}{3}\\,\\left|\\int_{\\mathbb{R}}\\big(uvu_{x}v_{x}-\\varphi_{c}\\psi_{c}\\varphi'_{c}\\psi'_{c}\\big)dx\\right|+\\frac{1}{3}\\,\\left|\\int_{\\mathbb{R}}\\left[u_{x}^2v_{x}^2-(\\varphi'_{c})^{2}(\\psi'_{c})^{2}\\right]dx\\right|\\\\\n&\\triangleq I_{1}+\\frac{1}{3}I_{2}+\\frac{1}{3}I_{3}+\\frac{4}{3}I_{4}+\\frac{1}{3}I_{5}.\n\\end{aligned}\n\\end{eqnarray*}\nFor the first term $I_{1}$, we obtain\n\\begin{eqnarray*}\n\\begin{aligned}\nI_{1}& \\leq  \\int_{\\mathbb R}\\left|\\left(u^2-\\varphi^{2}_{c}\\right)v^2\\right|\\,dx+\\int_{\\mathbb R}\\left|\\left(v^2-\\psi^{2}_{c}\\right)\\varphi^{2}_{c}\\right|\\,dx\\\\\n&\\leq  \\|v\\|^{2}_{L^\\infty}\\left(\\int_{\\mathbb R}\\tilde{u}^2\\,dx\\right)^{\\frac 12}\\left(\\int_{\\mathbb R}(u+\\varphi_{c})^2\\,dx\\right)^{\\frac 12}+\\|\\varphi_{c}\\|^{2}_{L^\\infty}\\left(\\int_{\\mathbb R}\\tilde{v}^2\\,dx\\right)^{\\frac 12}\\left(\\int_{\\mathbb R}(v+\\psi_{c})^2\\,dx\\right)^{\\frac 12}\\\\\n&\\leq \\|v\\|^{2}_{L^\\infty}\\|\\tilde{u}\\|_{H^1}\\left(\\|\\tilde{u}\\|_{H^1}+2\\|\\varphi_{c}\\|_{H^1}\\right)+\\|\\varphi_{c}\\|^{2}_{L^\\infty}\\|\\tilde{v}\\|_{H^1}\\left(\\|\\tilde{v}\\|_{H^1}+2\\|\\psi_{c}\\|_{H^1}\\right)\\\\\n&\\leq \\frac{1}{2}\\left(\\delta+\\sqrt{2}b\\right)^2\\delta\\left(\\delta+2\\sqrt{2}a\\right)+b^2\\delta\\left(\\delta+2\\sqrt{2}b\\right)\\leq 2\\sqrt{2}b^2(a+b)\\delta+O(\\delta^2).\n\\end{aligned}\n\\end{eqnarray*}\nFor the second term $I_{2}$, we have\n\\begin{align*}\nI_{2}&\\leq \\int_{\\mathbb R}\\left|\\left(u^2-\\varphi^{2}_{c}\\right)v_{x}^2\\right|\\,dx+\\int_{\\mathbb R}\\left|\\left(v_{x}^2-\\psi'^{2}_{c}\\right)\\varphi^{2}_{c}\\right|\\,dx\\\\\n&\\leq \\|v_{x}\\|^{2}_{L^\\infty}\\left(\\int_{\\mathbb R}(\\tilde{u})^2\\,dx\\right)^{\\frac 12}\\left(\\int_{\\mathbb R}(u+\\varphi_{c})^2\\,dx\\right)^{\\frac 12}\n+\\|\\varphi_{c}\\|^{2}_{L^\\infty}\\left(\\int_{\\mathbb R}(\\tilde{v}_{x})^2\\,dx\\right)^{\\frac 12}\\left(\\int_{\\mathbb R}(v_{x}+\\psi'_{c})^2\\,dx\\right)^{\\frac 12}\\\\\n&\\leq  \\|v_{x}\\|^{2}_{L^\\infty}\\|\\tilde{u}\\|_{H^1}\\left(\\|\\tilde{u}\\|_{H^1}+2\\|\\varphi_{c}\\|_{H^1}\\right)+\\|\\varphi_{c}\\|^{2}_{L^\\infty}\\|\\tilde{v}\\|_{H^1}\\left(\\|\\tilde{v}\\|_{H^1}+2\\|\\psi_{c}\\|_{H^1}\\right)\\\\\n&\\leq \\frac{1}{2}\\left(\\delta+\\sqrt{2}b\\right)^2\\delta\\left(\\delta+2\\sqrt{2}a\\right)+b^2\\delta\\left(\\delta+2\\sqrt{2}b\\right)\\leq 2\\sqrt{2}b^2(a+b)\\delta+O(\\delta^2).\n\\end{align*}\nSimilarly, for the term $I_{3}$, we get\n\\begin{align*}\nI_{3}< 2\\sqrt{2}a^2(a+b)\\delta+O(\\delta^{2}).\n\\end{align*}\nFor the fourth term $I_{4}$, we estimate\n\\begin{align*}\nI_{4} & \\leq \\int_{\\mathbb{R}}\\left|\\tilde{u}vu_{x}v_{x}\\right|\\,dx+\\int_{\\mathbb{R}}\\left|\\varphi_{c}\\tilde{v}vu_{x}v_{x}\\right|\\,dx+\\int_{\\mathbb{R}}\\left|\\varphi_{c}\\psi_{c}\\tilde{u}_{x}v_{x}\\right|\\,dx+\\int_{\\mathbb{R}}\\left|\\varphi_{c}\\psi_{c}\\varphi'_{c}\\tilde{v}_{x}\\right|\\,dx\\\\\n& \\leq \\frac{1}{2}\\|\\tilde{u}\\|_{L^{\\infty}}\\|v\\|_{L^{\\infty}}\\int_{\\mathbb{R}}(u^{2}_{x}+v^{2}_{x})\\,dx+\\frac{1}{2}\\|\\tilde{v}\\|_{L^{\\infty}}\\|u\\|_{L^{\\infty}}\\int_{\\mathbb{R}}(u^{2}_{x}+v^{2}_{x})\\,dx\\\\\n&\\;\\;\\; +\\|\\varphi_{c}\\|_{L^{\\infty}}\\|\\psi_{c}\\|_{L^{\\infty}}\\left(\\int_{\\mathbb{R}}\\tilde{u}_{x}^{2}\\,dx\\right)^{\\frac{1}{2}}\\left(\\int_{\\mathbb{R}}v^{2}_{x}\\,dx\\right)^{\\frac{1}{2}}+\\|\\varphi_{c}\\|_{L^{\\infty}}\\|\\psi_{c}\\|_{L^{\\infty}}\\left(\\int_{\\mathbb{R}}\\tilde{v}_{x}^{2}\\,dx\\right)^{\\frac{1}{2}}\\left(\\int_{\\mathbb{R}}u^{2}_{x}\\,dx\\right)^{\\frac{1}{2}}\\\\\n& \\leq \\frac{1}{2}\\left(\\|\\tilde{u}\\|_{H^{1}(\\mathbb{R})}\\|v\\|_{H^{1}(\\mathbb{R})}+\\|\\tilde{v}\\|_{H^{1}(\\mathbb{R})}\\|u\\|_{H^{1}(\\mathbb{R})}\\right)\\left(\\|v\\|^{2}_{H^{1}(\\mathbb{R})}+\\|u\\|^{2}_{H^{1}(\\mathbb{R})}\\right)\\\\\n&\\quad +\\frac{1}{2}\\|\\varphi_{c}\\|_{H^{1}(\\mathbb{R})}\\|\\psi_{c}\\|_{H^{1}(\\mathbb{R})}\\left(\\|\\tilde{u}\\|_{H^{1}(\\mathbb{R})}\\|v\\|_{H^{1}(\\mathbb{R})}+\\|\\tilde{v}\\|_{H^{1}(\\mathbb{R})}\\|u\\|_{H^{1}(\\mathbb{R})}\\right)\\\\\n& \\leq \\sqrt{2}(a+b)(a^{2}+b^{2}+ab)\\delta+O(\\delta^{2}).\n\\end{align*}\nFor the fifth term $I_{5}$, we have\n\\begin{align*}\nI_{5} & \\leq \\left|\\int_{\\mathbb{R}}u_{x}^{2}\\left(v_{x}^2-(\\psi'_{c})^{2}\\right)\\,dx\\right|+\\left|\\int_{\\mathbb{R}}(\\psi'_{c})^{2}(u^{2}_{x}-(\\varphi'_{c})^{2})\\,dx\\right|\\\\\n& \\leq \\|u\\|^{2}_{L^{\\infty}}\\left(\\int_{\\mathbb{R}}\\tilde{v}_{x}^{2}\\,dx\\right)^{\\frac{1}{2}}\\left(\\int_{\\mathbb{R}}(v_{x}+\\psi'_{c})^{2}\\,dx\\right)^{\\frac{1}{2}}+\\|\\psi'_{c}\\|^{2}_{L^{\\infty}}\\left(\\int_{\\mathbb{R}}\\tilde{u}_{x}^{2}dx\\right)^{\\frac{1}{2}}\\left(\\int_{\\mathbb{R}}(u_{x}+\\varphi'_{c})^{2}dx\\right)^{\\frac{1}{2}}\\\\\n& \\leq \\frac{1}{2}\\left[\\|u\\|^{2}_{H^{1}(\\mathbb{R})}\\|\\tilde{v}\\|_{H^{1}(\\mathbb{R})}\\left(\\|\\tilde{v}\\|_{H^{1}(\\mathbb{R})}+2\\|\\psi_{c}\\|_{H^{1}(\\mathbb{R})}\\right)+\\|\\psi'_{c}\\|^{2}_{H^{1}(\\mathbb{R})}\\|\\tilde{u}\\|_{H^{1}(\\mathbb{R})}\\left(\\|\\tilde{u}\\|_{H^{1}(\\mathbb{R})}+2\\|\\varphi_{c}\\|_{H^{1}(\\mathbb{R})}\\right)\\right]\\\\\n& \\leq 2\\sqrt{2}ab(a+b)\\delta+O(\\delta^{2}).\n\\end{align*}\nAccordingly, for $0<\\delta<1\/2$, we have\n\\begin{align*}\n\\left|F(u,v)-F(\\varphi_{c},\\psi_{c})\\right|\\leq C\\delta +O(\\delta^{2}),\n\\end{align*}\nwhere $C$ is a constant depending on $a$, $b$, $\\|u\\|_{H^s}$ and $\\|v\\|_{H^s}$, which completes the proof of this lemma.\n\\end{proof}\n\nNow, we are in the position to prove that the strong solution satisfies the novel error estimates at some kind of critical point under the assumption of small perturbation of initial data around the profile of peakon solutions.\n\n\\begin{lemma}\nAssume that $(u(t, x),\\,v(t, x))$, $t\\in [0,\\,T)$, is the corresponding strong solution of the Cauchy problem \\eqref{tcNK-1} with initial data $(u_0(x),\\,v_0(x))$ satisfying $0\\not\\equiv u_0,\\,v_0\\in H^s(\\R)$, $s\\geq 3$ and $m_0=u_0-u_{0xx}\\geq 0$, $n_0=v_0-v_{0xx}\\geq 0$. If $(u_0(x),\\,v_0(x))$ satisfies\n\\begin{equation*}\n\\|u_0-\\varphi_{c}\\|_{H^{1}(\\mathbb{R})} < \\delta \\quad \\mathrm{and} \\quad \\|v_0-\\psi_{c}\\|_{H^{1}(\\mathbb{R})} < \\delta,\n\\end{equation*}\nwith $0<\\delta<1\/2$, then there exists a constant $C>0$ such that\n\\begin{eqnarray*}\n|u(t, \\xi(t))-a|<C\\delta^{\\frac{1}{2}} \\quad \\mathrm{and} \\quad |v(t, \\xi(t))-b|<C\\delta^{\\frac{1}{2}}, \\quad \\forall t\\in [0,\\,T),\n\\end{eqnarray*}\nwhere $\\xi(t)\\in \\R$ is located such that $u(t, \\xi(t))v(t, \\xi(t))=\\max_{x\\in \\mathbb R}\\{u(t, x)v(t, x)\\}=M(t)$.\n\\end{lemma}\n\n\n\\begin{proof}\nIn view of the conservation of the functionals $H[u, v]$ and $F[u, v]$, as well as the invariant property established in Lemma \\ref{nonnegative} of the strong solution $(u(t, x),\\,v(t, x))$, we deduce from Lemma 3.5 that for any $t\\in [0,\\,T)$,\n\\begin{equation}\\label{H_t}\n\\Big|H[u(t, \\cdot), v(t, \\cdot)]-H[\\varphi_{c},\\psi_{c}]\\Big|<2\\sqrt{2}(a+b)\\delta +6\\delta^{2}\n\\end{equation}\nand\n\\begin{equation}\\label{F_t}\n\\Big|F[u(t, \\cdot), v(t, \\cdot)]-F[\\varphi_{c},\\psi_{c}]\\Big|<C\\delta+O(\\delta^{2}),\n\\end{equation}\nwhere the constants involved depend on $a$, $b$, $\\|u_0\\|_{H^s}$ and $\\|v_0\\|_{H^s}$.\n\nFurthermore, we have obtained in Lemma \\ref{lem1.4} the following dynamical inequality\n\\begin{align*}\nF[u, v]-\\frac{4}{3}M(t)H[u, v]+\\frac{4}{3}M(t)^2 \\leq 0.\n\\end{align*}\nThen, define the polynomial $P(y)$ to be\n\\begin{align*}\nP(y)=\\frac{4}{3}y^{2}-\\frac{4}{3}H[u, v]y+F[u, v].\n\\end{align*}\nFor the peakons, we have $H[\\varphi_{c}, \\psi_{c}]=2ab$ and $F[\\varphi_{c}, \\psi_{c}]=4a^{2}b^{2}\/3$, thus the above polynomial with respect to peakons is defined by\n\\begin{align*}\nP_{0}(y)=\\frac{4}{3}y^{2}-\\frac{4}{3}H[\\varphi_{c}, \\psi_{c}]y+F[\\varphi_{c}, \\psi_{c}]=\\frac{4}{3}(y-ab)^{2}.\n\\end{align*}\nNote that\n\\begin{align*}\nP(y)=P_0(y)-\\frac{4}{3}\\big(H[u, v]-H[\\varphi_{c}, \\psi_{c}]\\big)\\,y+\\big(F[u, v]-F[\\varphi_{c}, \\psi_{c}]\\big).\n\\end{align*}\nIt follows that for any $t\\in [0,\\,T)$\n\\begin{align*}\n\\frac{4}{3}\\big(M(t)-ab\\big)^{2}=P_{0}(M(t))\\leq \\frac{4}{3}\\big|H(u, v)-H(\\varphi_{c}, \\psi_{c})\\big|M(t)+\\big|F(u, v)-F(\\varphi_{c}, \\psi_{c})\\big|,\n\\end{align*}\nwhich together with \\eqref{H_t} and \\eqref{F_t} leads to\n\\begin{align}\\label{M-ab}\n\\big|M(t)-ab\\big|\\leq C\\delta^{\\frac{1}{2}}.\n\\end{align}\nOn the other hand, since\n\\begin{align*}\n2u^{2}(t, \\xi(t))\\leq \\|u_0\\|^{2}_{H^{1}(\\mathbb{R})}, \\quad  2v^{2}(t, \\xi(t))\\leq \\|v_0\\|^{2}_{H^{1}(\\mathbb{R})},\n\\end{align*}\nwe have\n\\begin{align*}\n0<u(t, \\xi(t))\\leq a+\\tilde{C}\\delta^{\\frac{1}{2}}, \\quad  0<v(t, \\xi(t))\\leq b+\\tilde{C}\\delta^{\\frac{1}{2}}.\n\\end{align*}\nIt follows from \\eqref{M-ab} and the above estimate that\n\\begin{align*}\nab-\\tilde{C}\\delta^{\\frac 12}\\leq u(\\xi)v(\\xi)\\leq (a+\\tilde{C}\\delta^{\\frac 12})v(\\xi),\n\\end{align*}\nwhich implies\n\\begin{align*}\nv(t, \\xi(t))\\geq b-C\\delta^{\\frac{1}{2}}.\n\\end{align*}\nSimilarly, we have\n\\begin{align*}\nu(t, \\xi(t))\\geq a-C\\delta^{\\frac{1}{2}}.\n\\end{align*}\nThis finishes the proof of the lemma.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm1.1}]\nLet $u \\in C([0,T); H^{s})$ and $v \\in C([0, T); H^{s})$, $s \\geq 3$, be the unique strong solution for Cauchy problem \\eqref{tcNK-1} of the two-component Novikov system \\eqref{tcNK} with initial data $u(0, x)=u_{0}(x)$ and $v(0, x)=v_{0}(x)$. Since $E_u[u]$ and $E_v[v]$ are both conserved, we deduce that\n\\begin{align}\\label{conserved}\nE_u[u(t, \\cdot)]=E_u[u_0] \\quad and \\quad E_v[v(t, \\cdot)]=E_v[v_0], \\quad \\forall t\\in (0,\\,T).\n\\end{align}\n\nBy \\eqref{conserved} and Lemma \\ref{lem1.1}, we infer that for any $t\\in [0,\\,T)$\n\\begin{align*}\n\\|u(t, \\cdot)-\\varphi_{c}(\\cdot-\\xi(t))\\|^{2}_{H^{1}(\\mathbb{R})}=&E_{u}[u_{0}]-E_{u}[\\varphi_{c}]-4a(u(t, \\xi(t))-a),\\\\\n\\|v(t, \\cdot)-\\psi_{c}(\\cdot-\\xi(t))\\|^{2}_{H^{1}(\\mathbb{R})}=&E_{v}[v_{0}]-E_{v}[\\psi_{c}]-4b(v(t, \\xi(t))-b).\n\\end{align*}\nCombining the above two dynamical pointwise identities with Lemmas 3.5 and 3.6 , we conclude that for any $t\\in [0,\\,T)$\n\\begin{align*}\n\\|u-\\varphi_{c}(\\cdot-\\xi(t))\\|_{H^{1}(\\mathbb{R})}<C\\delta^{\\frac{1}{4}} \\quad \\mathrm{and} \\quad \\|v-\\psi_{c}(\\cdot-\\xi(t))\\|_{H^{1}(\\mathbb{R})}<C\\delta^{\\frac{1}{4}},\n\\end{align*}\nwhere as in Lemma 3.6, $C$ is a constant depending on $a$, $b$, $\\|u\\|_{H^s}$ and $\\|v\\|_{H^s}$. This thus completes the proof of Theorem \\ref{thm1.1}.\n\\end{proof}\n\n\n\n\n\n\n\\section{Stability of train-profiles}\n\nIn this section, we are devoted to proving orbital stability of train-profiles of two-component peakons $(\\varphi_c, \\, \\psi_c)$. For $\\alpha>0$ and $L>0$, we define the following neighborhood of all the sums of N peakons of speed $c_1,...,c_N$ with spatial shifts $x_i$ that satisfy $x_i-x_{i-1} \\geq L$,\n\\begin{align*}\nU( \\alpha , L) = \\Big\\{(u , v) & \\in H^{1}(\\mathbb{R}) \\times H^{1}(\\mathbb{R}), \\\\\n& \\inf_{x_i-x_{i-1} \\geq L}{{\\Big\\| u- \\sum_{i=1}^N{\\varphi_{c_i}(\\cdot - {x}_i)} \\Big\\|_{H^1}}+{\\Big\\| v- \\sum_{i=1}^N{\\psi_{c_i}(\\cdot - {x}_i)} \\Big\\|_{H^1}}} \\leq \\alpha \\Big\\}.\n\\end{align*}\n\nBy the continuity of the map $t \\mapsto (u(t),v(t))$ from $[0,T[$ into $H^1(\\mathbb{R})\\times H^1(\\mathbb{R})$, to prove Theorem 1.2 it suffices to verify that there exist $ A>0$, $\\epsilon_0>0$ and $L_0>0$ such that for any $L>L_0$ and $0<\\epsilon<\\epsilon_0$, if $(u_{0},v_{0})$ satisfies \\eqref{initialdata-1} and for some $0<t_0<T$, we have\n\\begin{align}\\label{uvinitial}\n(u(t),v(t)) \\in U\\left( A({\\epsilon}^{ \\frac{1}{4}}+L^{- \\frac{1}{8}}),\\frac{L}{2}\\right) \\quad {\\forall}t \\in [0, t_0],\n\\end{align}\nthen\n\\begin{align}\\label{uvending}\n(u(t_{0}),v(t_{0})) \\in U\\left( \\frac{A}{2}({\\epsilon}^{ \\frac{1}{4}}+L^{- \\frac{1}{8}}),\\frac{2L}{3}\\right).\n\\end{align}\nTherefore, in the sequel of this section we will assume \\eqref{uvinitial} for some $0< \\epsilon < \\epsilon_{0}$ and $L>L_{0}$, with $A$, $\\epsilon_{0}$ and $L_{0}$ to be specified later.\n\\subsection{Control of the distance between the peakons}\nIn this subsection we shall prove that the different bumps of $u$ and $v$ that are individually close to their own peakons and get away from each others as time is increasing. This is crucial in our analysis since we do not know how to manage strong interactions.\n\n\\begin{prop}\\label{implicitfunction}\n Assume $(u_{0}, v_{0})$ satisfies \\eqref{initialdata-1}, and there exist $\\alpha_{0} > 0$, $L_{0} > 0$ and $C_{0} > 0$ such that for all $0 < \\alpha < \\alpha_{0}$, $L >L_{0} >0$. If $(u(t),v(t)) \\in U(\\alpha, \\frac {L}{2})$ on $[0, t_{0}]$ for some $0< t_{0} <T$, then there exist $C^{1} functions$ $\\tilde{x}_{1} ,..., \\tilde{x}_{N}$ defined on $[0, t_{0}]$ such that\n\\begin{align}\\label{initial-3-1}\n\\frac{d}{dt}\\tilde{x}_{i}=c_{i}+O(\\sqrt{\\alpha})+O(L^{-1}),\\;\\; i=1,...,N,\n\\end{align}\n\\begin{align}\\label{initial-3-2}\n{\\Big\\| u(t)- \\sum_{i=1}^N{\\varphi_{c_i}(\\cdot - \\tilde{x}_i(t))} \\Big\\|_{H^1}}+{\\Big\\| v(t)- \\sum_{i=1}^N{\\psi_{c_i}(\\cdot - \\tilde{x}_i(t))} \\Big\\|_{H^1}}=O(\\sqrt{\\alpha}),\n\\end{align}\n\\begin{align}\\label{initial-3-3}\n\\tilde{x}_{i}(t)-\\tilde{x}_{i-1}(t) \\geq \\frac{3L}{4} + \\frac{c_{i}-c_{i-1}}{2}t, \\;\\;i=2,...,N.\n\\end{align}\nMoreover, setting $J_{i}:=[y_{i}(t),y_{i-1}(t)]$,\\;\\; $i=1,...,N$, with\n\\begin{align}\\label{initial-3-4}\ny_{1}=-\\infty,\\quad y_{N+1}=+\\infty \\quad and \\quad y_{i}(t)=\\frac{\\tilde{x}_{i-1}(t)+\\tilde{x}_{i}(t)}{2},\\;\\; i=2,...,N,\n\\end{align}\nit holds\n\\begin{align}\\label{initial-3-5}\n|\\tilde{x}_{i}(t)-x_{i}(t)| \\leq \\frac{L}{12} ,\\;\\; i=1,...,N,\n\\end{align}\nwhere $x_{1}(t),...,x_{N}(t)$ are any point such that\n\\begin{align}\\label{initial-3-6}\nu(t,x_{i}(t))v(t,x_{i}(t))= \\max_{x \\in J_{i}(t)}u(t)v(t),\\;\\; i=1,...,N.\n\\end{align}\n\\end{prop}\n\nTo prove this proposition we use a modulation argument. The strategy is to construct $C^{1}$ functions $\\tilde{x}_{1} ,..., \\tilde{x}_{N}$ on $[0,t_{0}]$ satisfying a suitable orthogonality condition, see \\eqref{orthogonality}. Thanks to this orthogonality condition, we will be able to prove that the speed of the $\\tilde{x}_{i}$ stays close to $c_{i}$ on $[0,t_{0}]$.\n\n\\begin{proof}\nFor $Z=(z_{1},...,z_{N}) \\in \\mathbb{R}^{N}$ satisfying $z_{i}-z_{i-1}>L\/2$, we set\n\\begin{align}\\label{sumofsolitons}\nR_{Z}(\\cdot)=\\sum_{i=1}^{N}\\varphi_{c_{i}}(\\cdot - z_{i}) \\quad and \\quad S_{Z}(\\cdot)=\\sum_{i=1}^{N}\\psi_{c_{i}}(\\cdot - z_{i}).\n\\end{align}\nFor $\\alpha_{0} > 0$, $L_{0} > 0$, we define the function\n\\begin{align*}\nY:(-\\alpha ,\\alpha )^{N} \\times B_{H^{1} \\times H^{1}}((R_{Z},S_{Z}),\\alpha) \\mapsto {\\mathbb{R}}^{N}, \\\\\n(y_{1},...,y_{N},u,v) \\mapsto (Y^{1}(y_{1},...,y_{N},u,v),...,Y^{N}(y_{1},...,y_{N},u,v))\n\\end{align*}\nwith\n\\begin{align*}\nY^{i}(y_{1},...,y_{N},u,v)= &\\int_{\\mathbb{R}}\\bigg{(}(u-\\sum_{i=1}^{N}\\varphi_{c_{j}}(\\cdot - z_{j} -y_{j})) \\partial_{x}\\varphi_{c_{i}}(\\cdot - z_{i} -y_{i}) \\\\\n&\\qquad \\quad+ (v-\\sum_{i=1}^{N}\\psi_{c_{j}}(\\cdot - z_{j} -y_{j})) \\partial_{x}\\psi_{c_{i}}(\\cdot - z_{i} -y_{i})\\bigg{)},\n\\end{align*}\n$Y$ is clearly of class $C^{1}$. For $i=1,...,N,$\n\\begin{align*}\n\\frac{\\partial{Y}^{i}}{\\partial y_{i}}(y_{1},...,y_{N},u,v)=&\\int_{\\mathbb{R}}\\bigg{(}(u_{x}-\\sum_{i \\ne j}^{N}\\partial_{x} \\varphi_{c_{j}}(\\cdot - z_{j} -y_{j})) \\partial_{x}\\varphi_{c_{i}}(\\cdot - z_{i} -y_{i}) \\\\\n&\\qquad\\qquad + (v_{x}-\\sum_{i \\ne j}^{N}\\partial_{x} \\psi_{c_{j}}(\\cdot - z_{j} -y_{j})) \\partial_{x}\\psi_{c_{i}}(\\cdot - z_{i} -y_{i})\\bigg{)},\n\\end{align*}\nand $\\forall j\\ne i,$\n\\begin{align*}\n\\frac{\\partial{Y}^{i}}{\\partial y_{j}}(y_{1},...,y_{N},u,v)=&\\int_{\\mathbb{R}}\\bigg{(}\\partial_{x} \\varphi_{c_{j}}(\\cdot - z_{j} -y_{j}) \\partial_{x}\\varphi_{c_{i}}(\\cdot - z_{i} -y_{i}) \\\\\n&\\qquad\\qquad + \\partial_{x} \\psi_{c_{j}}(\\cdot - z_{j} -y_{j}) \\partial_{x}\\psi_{c_{i}}(\\cdot - z_{i} -y_{i})\\bigg{)}.\n\\end{align*}\nHence\n\\begin{align*}\n\\frac{\\partial{Y}^{i}}{\\partial y_{i}}(0,...,0,R_{Z},S_{Z})=\\| \\partial_{x}\\varphi_{c_{i}} \\|_{L^2}^2+\\| \\partial_{x}\\psi_{c_{i}} \\|_{L^2}^2 \\geq a_{1}^2+b_{1}^{2},\n\\end{align*}\nand, for $\\forall j\\ne i$, using the exponential decay of $\\varphi_{c}$ and that $z_{i}-z_{i-1} >L $ we infer  for $L_{0}$ large enough that (recall that $L>L_{0}$),\n\\begin{equation*}\n\\begin{aligned}\n\\frac{\\partial{Y}^{i}}{\\partial y_{j}}(0,...,0,R_{Z},S_{Z})=&\\int_{\\mathbb{R}}\\bigg{(}\\partial_{x} \\varphi_{c_{j}}(\\cdot - z_{j} ) \\partial_{x}\\varphi_{c_{i}}(\\cdot - z_{i} ) + \\partial_{x} \\psi_{c_{j}}(\\cdot - z_{j}) \\partial_{x}\\psi_{c_{i}}(\\cdot - z_{i})\\bigg{)}\\\\\n&\\leq O(e^{-\\frac{L}{4}}).\n\\end{aligned}\n\\end{equation*}\nWe conclude that, for $L>0$ large enough, $D_{(y_{1},...,y_{N})}Y(0,...,0,R_{Z},S_{Z})=D+P$ where $D$ is an invertible diagonal matrix with $\\|D^{-1}\\|\\leq (a^{2}_{1}+b^{2}_{1})^{-n}$ and $\\|P\\|\\leq O(e^{-L\/4})$. Hence there exists $L_{0}>0$ such that for $L>L_{0}$, $D_{(y_{1},...,y_{N})}Y(0,...,0,R_{Z},S_{Z})$ is invertible with an inverse matrix of norm smaller than $2(a^{2}_{1}+b^{2}_{1})^{-n}$. The implicit function theorem implies that there exists $\\beta_{0}>0$ and $C^{1}$ functions $y_{1},y_{2},...,y_{N}$ from $B_{H^1 \\times H^1}((R_{Z},S_{Z}),\\beta_{0})$ to a neighborhood of $(0,0,...,0)$ which uniquely determined such that\n\\begin{align*}\nY(y_{1},...,y_{N},u,v)=0 \\quad {\\rm for}\\;\\;  {\\rm  all} \\quad (u,v) \\in B((R_{Z},S_{Z}),\\beta_{0}).\n\\end{align*}\nIn particular, there exists $C_{0}>0$ such that if $(u,v) \\in B((R_{Z},S_{Z}),\\beta)$, with $0<\\beta \\leq \\beta_{0}$, then\n\\begin{align}\\label{implicitfunc}\n\\sum^{N}_{i=1}{\\big{|}y_{i}(u,v)\\big{|}}\\leq C_{0}\\beta.\n\\end{align}\nNote that $\\beta_{0}$ and $C_{0}$ depend on only $a_{1}$, $b_{1}$ and $L_{0}$ and not on the point $(z_{1},...,z_{N})$. For $(u,v) \\in B((R_{Z},S_{Z}),\\beta_{0})$ we set $\\tilde{x}_{i}(u,v)=z_{i}+y_{i}(u,v)$. Assuming that $\\beta_{0}\\leq L_{0}\/(8C_{0})$, $\\tilde{x}_{1},...,\\tilde{x}_{N}$ are thus $C^{1}$ functions on $B((R_{Z},S_{Z}),\\beta)$ satisfying\n\\begin{align}\\label{dist-1}\n\\tilde{x}_{j}(u,v)-\\tilde{x}_{j-1}(u,v)>\\frac{L}{2}-2C_{0}\\beta>\\frac{L}{4}.\n\\end{align}\nFor $L>L_{0}$ and $0<\\alpha<\\alpha_{0}<\\beta_{0}\/2$ to be chosen later, we define the modulation of $(u,v) \\in U(\\alpha,L\/2)$ in the following way, the trajectory of $(u,v)$ is covered by a finite number of open balls:\n\\begin{align*}\n\\Big{\\{}\\big{(}u(t),v(t)\\big{)}, t \\in [0,t_{0}]\\Big{\\}} \\subset \\bigcup_{k=1,...,M}B\\big{(}(R_{Z^{k}},S_{Z^{k}}),2\\alpha\\big{)}.\n\\end{align*}\nIt is worth noticing that, since $0<\\alpha<\\alpha_{0}<\\beta_{0}\/2$, the functions $\\tilde{x}_{i}(u,v)$ are uniquely determined for $(u,v)\\in B((R_{Z^{k}},S_{Z^{k}}),2\\alpha)\\bigcap B((R_{Z^{k'}},S_{Z^{k'}}),2\\alpha)$. We can thus define the functions $t\\mapsto \\tilde{x}_{i}(t)$ on $[0,t_{0}]$ by setting $\\tilde{x}_{i}(t)=\\tilde{x}_{i}(u(t),v(t))$. By construction\n\\begin{eqnarray}\\label{orthogonality}\n\\begin{aligned}\n\\int_{\\mathbb{R}} &\\Big{(}\\Big{(}u(t,\\cdot)-\\sum^{N}_{j=1}\\varphi_{c_{j}}(\\cdot-\\tilde{x}_{j}(t))\\Big{)}\\partial_{x}\\varphi_{c_{i}}(\\cdot-\\tilde{x}_{i}(t))\\\\\n&\\qquad \\qquad\\qquad +\\Big{(}v(t,\\cdot)-\\sum^{N}_{j=1}\\psi_{c_{j}}(\\cdot-\\tilde{x}_{j}(t))\\Big{)}\\partial_{x}\\psi_{c_{i}}(\\cdot-\\tilde{x}_{i}(t))\\Big{)}dx=0.\n\\end{aligned}\n\\end{eqnarray}\nMoreover, on account of \\eqref{implicitfunc} and the fact that $\\varphi''_{c}$ and $\\psi''_{c}$ are the sum of an $L^{1}$ function and a Dirac mass, we claim\n\\begin{align}\\label{preestimate}\n\\Big{\\|}\\Big{(}u(t),v(t)\\Big{)}-\\Big{(}R_{\\tilde{X}(t)},S_{\\tilde{X}(t)}\\Big{)}\\Big{\\|}_{H^{1}\\times H^{1}}\\leq O(\\sqrt{\\alpha}), \\quad \\forall t \\in [0,t_{0}].\n\\end{align}\nIndeed, one can calculate\n\\begin{align*}\n&\\Big{\\|}\\Big{(}u(t),v(t)\\Big{)}-\\Big{(}R_{\\tilde{X}(t)},S_{\\tilde{X}(t)}\\Big{)}\\Big{\\|}_{H^{1}\\times H^{1}} \\triangleq \\Big{\\|}u(t)-R_{\\tilde{X}(t)}\\Big{\\|}_{H^{1}}+\\Big{\\|}v(t)-S_{\\tilde{X}(t)}\\Big{\\|}_{H^{1}}\\\\\n&\\leq \\Big{\\|}u(t)-R_{{Z^{k}}(t)}\\Big{\\|}_{H^{1}}+\\Big{\\|}R_{{Z^{k}}(t)}-R_{\\tilde{X}(t)}\\Big{\\|}_{H^{1}}+\\Big{\\|}v(t)-S_{{Z^{k}}(t)}\\Big{\\|}_{H^{1}}+\\Big{\\|}S_{{Z^{k}}(t)}-S_{\\tilde{X}(t)}\\Big{\\|}_{H^{1}}\\\\\n&\\leq\n\\alpha+\\hat{C}\\sum^{N}_{i=1}\\Big{\\|}\\varphi(\\cdot-z^{k}_{i})-\\varphi(\\cdot-z^{k}_{i}-y_{i}(u,v))\\Big{\\|}_{H^{1}}\\\\\n&\\leq \\alpha+\\hat{C}\\sum^{N}_{i=1}\\Big{(}2(1-e^{-y_{i}})^{2}+2y_{i}+O(y^{2}_{i})\\Big{)}^{\\frac{1}{2}}\\leq O(\\sqrt{\\alpha}).\n\\end{align*}\nLet us now prove that the speed of $\\tilde{x}_{i}$ stays close to $c_{i}$. We set\n\\begin{align*}\n&R_{j}(t)=\\varphi_{c_{j}}(\\cdot-\\tilde{x}_{j}(t)) \\quad \\mathrm{and} \\quad \\tilde{u}(t)=u(t)-\\sum^{N}_{j=1}R_{j}(t)=u(t,\\cdot)-R_{\\tilde{X}(t)},\\\\\n&S_{j}(t)=\\psi_{c_{j}}(\\cdot-\\tilde{x}_{j}(t)) \\quad \\mathrm{and} \\quad \\tilde{v}(t)=v(t)-\\sum^{N}_{j=1}S_{j}(t)=u(t,\\cdot)-S_{\\tilde{X}(t)}.\n\\end{align*}\nDifferentiating \\eqref{orthogonality} with respect to $t$ we get\n\\begin{align*}\n\\int_{\\mathbb{R}}\\Big{(}\\tilde{u}_{t}\\partial_{x}R_{i}+\\tilde{v}_{t}\\partial_{x}S_{i}\\Big{)}=\\dot{\\tilde{x}}_{i}\\big{(}<\\partial^{2}_{x}R_{i},\\tilde{u}>_{H^{-1},H^{1}}+<\\partial^{2}_{x}S_{i},\\tilde{v}>_{H^{-1},H^{1}}\\big{)},\n\\end{align*}\nand thus\n\\begin{align}\\label{speedestimate}\n\\bigg{|}\\int_{\\mathbb{R}}\\Big{(}\\tilde{u}_{t}\\partial_{x}R_{i}+\\tilde{v}_{t}\\partial_{x}S_{i}\\Big{)}\\bigg{|}\\leq |\\dot{\\tilde{x}}_{i}-c_{i}|\\Big(O\\big{(}\\|\\tilde{u}\\|_{H^{1}}\\big{)}+O\\big{(}\\|\\tilde{v}\\|_{H^{1}}\\big{)}\\Big)+\\Big(O\\big{(}\\|\\tilde{u}\\|_{H^{1}}\\big{)}+O\\big{(}\\|\\tilde{v}\\|_{H^{1}}\\big{)}\\Big).\n\\end{align}\nSubstituting $u$ by $\\tilde{u}+\\sum^{N}_{j=1}R_{j}(t)$ and $v$ by $\\tilde{v}+\\sum^{N}_{j=1}S_{j}(t)$ into  \\eqref{weakform} and using\n\\begin{equation}\n\\left\\{\n\\begin{aligned}\n&\\partial_{t}R_{i}+\\big{(}\\dot{\\tilde{x}}_{i}(t)-c_{i}\\big{)}\\partial_{x}R_{i}+R_{i}S_{i}\\partial_{x}R_{i}+P_{x}\\ast\\Big{(}\\frac{1}{2}(\\partial_{x}R_{i})^{2}S_{i}+R_{i}(\\partial_{x}R_{i})(\\partial_{x}S_{i})+R^{2}_{i}S_{i}\\Big{)}\\\\\n&+P\\ast\\Big{(}\\frac{1}{2}(\\partial_{x}R_{i})^{2}\\partial_{x}S_{i}\\Big{)}=0,\\\\ \n&\\partial_{t}S_{i}+\\big{(}\\dot{\\tilde{x}}_{i}(t)-c_{i}\\big{)}\\partial_{x}S_{i}+S_{i}R_{i}\\partial_{x}S_{i}+P_{x}\\ast\\Big{(}\\frac{1}{2}(\\partial_{x}S_{i})^{2}R_{i}+S_{i}(\\partial_{x}S_{i})(\\partial_{x}R_{i})+S^{2}_{i}R_{i}\\Big{)}\\\\\n&+P\\ast\\Big{(}\\frac{1}{2}(\\partial_{x}S_{i})^{2}\\partial_{x}R_{i}\\Big{)}=0,\n\\end{aligned}\n\\right.\n\\end{equation}\nwe infer that $(\\tilde{u},\\tilde{v})$ satisfies on $[0,t_{0}]$ that\n\\begin{equation*}\\label{speedequation}\n\\begin{aligned}\n&\\tilde{u}_{t}-\\sum^{N}_{i=1}(\\dot{\\tilde{x}}_{i}(t)-c_{i})\\partial_{x}R_{i}+\\big{(}\\tilde{u}+\\sum^{N}_{j=1}R_{j}\\big{)}\\big{(}\\tilde{v}+\\sum^{N}_{j=1}S_{j}\\big{)}\\Big{(}\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx}\\Big{)}-\\sum^{N}_{j=1}(R_{j}S_{j}\\partial_{x}R_{j})\\\\\n&\\quad +P_{x}\\ast\\bigg{(}\\frac{1}{2}\\Big{(}\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx}\\Big{)}^{2}(\\tilde{v}+\\sum^{N}_{j=1}S_{j})\n+(\\tilde{u}+\\sum^{N}_{j=1}R_{j})\\Big{(}\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx}\\Big{)}\\Big{(}\\tilde{v}_{x}+\\sum^{N}_{j=1}S_{jx}\\Big{)}\\\\\n&\\qquad\\quad+(\\tilde{u}+\\sum^{N}_{j=1}R_{j})^{2}(\\tilde{v}+\\sum^{N}_{j=1}S_{j})-\\frac{1}{2}\\sum^{N}_{j=1}R^{2}_{jx}S_{j}-\\sum^{N}_{j=1}R_{j}R_{jx}S_{jx}-\\sum^{N}_{j=1}R^{2}_{j}S_{j}\\bigg{)}\\\\\n&\\qquad\\qquad+P\\ast \\bigg{(}\\frac{1}{2}\\big{(}\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx}\\big{)}^{2}\\big{(}\\tilde{v}_{x}+\\sum^{N}_{j=1}S_{jx}\\big{)}-\\frac{1}{2}\\sum^{N}_{j=1}R^{2}_{jx}S_{jx}\\bigg{)}=0,\\\\\n\\end{aligned}\n\\end{equation*}\n\\begin{eqnarray}\\label{speedequation}\n\\begin{aligned}\n&\\tilde{v}_{t}-\\sum^{N}_{i=1}(\\dot{\\tilde{x}}_{i}(t)-c_{i})\\partial_{x}S_{i}+\\big{(}\\tilde{v}+\\sum^{N}_{j=1}S_{j}\\big{)}\\big{(}\\tilde{u}+\\sum^{N}_{j=1}R_{j}\\big{)}\\Big{(}\\partial_{x}\\big{(}\\tilde{v}+\\sum^{N}_{j=1}S_{j}\\big{)}\\Big{)}-\\sum^{N}_{j=1}(S_{j}R_{j}\\partial_{x}S_{j})\\\\\n&\\quad+P_{x}\\ast\\bigg{(}\\frac{1}{2}\\Big{(}\\tilde{v}_{x}+\\sum^{N}_{j=1}S_{jx}\\Big{)}^{2}(\\tilde{u}+\\sum^{N}_{j=1}R_{j})\n+(\\tilde{v}+\\sum^{N}_{j=1}S_{j})\\Big{(}\\tilde{v}_{x}+\\sum^{N}_{j=1}S_{jx}\\Big{)}\\Big{(}\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx}\\Big{)}\\\\\n&\\qquad+(\\tilde{v}+\\sum^{N}_{j=1}S_{j})^{2}(\\tilde{u}+\\sum^{N}_{j=1}R_{j})-\\frac{1}{2}\\sum^{N}_{j=1}S^{2}_{jx}R_{j}-\\sum^{N}_{j=1}S_{j}S_{jx}R_{jx}-\\sum^{N}_{j=1}S^{2}_{j}R_{j}\\bigg{)}\\\\\n&\\qquad\\qquad+P\\ast \\bigg{(}\\frac{1}{2}\\big{(}\\tilde{v}_{x}+\\sum^{N}_{j=1}S_{jx}\\big{)}^{2}\\big{(}\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx}\\big{)}-\\frac{1}{2}\\sum^{N}_{j=1}S^{2}_{jx}R_{jx}\\bigg{)}=0.\n\\end{aligned}\n\\end{eqnarray}\nTaking the $L^{2}$ scalar product with $\\partial_{x}R_{i}$ in the first equation of \\eqref{speedequation} and $\\partial_{x}S_{i}$ in the second equation of \\eqref{speedequation}, summing up the resulting equations, integrating by parts, and using the decay of $R_{j}$, $S_{j}$ and their first derivative, we claim\n\\begin{align}\\label{speedfinestimate}\n|\\dot{\\tilde{x}}_{i}(t)-c_{i}|\\leq O(\\sqrt{\\alpha})+O(e^{-\\frac{L}{8}}).\n\\end{align}\nIndeed, according to the above, we obtain that\n\\begin{align*}\n&(\\dot{\\tilde{x}}_{i}-c_{i})\\big{(}<\\partial^{2}_{x}R_{i},\\tilde{u}>_{H^{-1},H^{1}}+<\\partial^{2}_{x}S_{i},\\tilde{v}>_{H^{-1},H^{1}}\\big{)}\\\\\n&+c_{i}\\big{(}<\\partial^{2}_{x}R_{i},\\tilde{u}>_{H^{-1},H^{1}}+<\\partial^{2}_{x}S_{i},\\tilde{v}>_{H^{-1},H^{1}}\\big{)}-(\\dot{\\tilde{x}}_{i}-c_{i})\\int_{\\mathbb{R}}\\bigg{(}(\\partial_{x}R_{i})^{2}+(\\partial_{x}S_{i})^{2}\\bigg{)}dx\\\\\n&=\\sum_{i\\ne j}(\\dot{\\tilde{x}}_{j}-c_{j})\\Big{(}\\int_{\\mathbb{R}}\\big{(}\\partial_{x}R_{i}\\partial_{x}R_{j}+\\partial_{x}S_{i}\\partial_{x}S_{j}\\big{)}dx\\Big{)}\\\\\n&\\quad -\\int_{\\mathbb{R}}\\Bigg{(}\\bigg{(}\\big{(}\\tilde{u}+\\sum^{N}_{j=1}R_{j}\\big{)}\\big{(}\\tilde{v}+\\sum^{N}_{j=1}S_{j}\\big{)}\\Big{(}\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx}\\Big{)}-\\sum^{N}_{j=1}(R_{j}S_{j}\\partial_{x}R_{j})\\bigg{)}\\partial_{x}R_{i}\\Bigg{)}dx\\\\\n&\\quad -\\int_{\\mathbb{R}}\\Bigg{(}\\bigg{(}\\big{(}\\tilde{v}+\\sum^{N}_{j=1}S_{j}\\big{)}\\big{(}\\tilde{u}+\\sum^{N}_{j=1}R_{j}\\big{)}\\Big{(}\\partial_{x}\\big{(}\\tilde{v}+\\sum^{N}_{j=1}S_{j}\\big{)}\\Big{)}-\\sum^{N}_{j=1}(S_{j}R_{j}\\partial_{x}S_{j})\\bigg{)}\\partial_{x}S_{i}\\Bigg{)}dx\\\\\n&\\quad +\\int_{\\mathbb{R}}\\Bigg{(}P\\ast\\bigg{(}\\frac{1}{2}\\Big{(}\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx}\\Big{)}^2(\\tilde{v}+\\sum^{N}_{j=1}S_{j})\n+(\\tilde{u}+\\sum^{N}_{j=1}R_{j})\\Big{(}\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx}\\Big{)}\\Big{(}\\tilde{v}_{x}+\\sum^{N}_{j=1}S_{jx}\\Big{)}\\\\\n&\\quad +(\\tilde{u}+\\sum^{N}_{j=1}R_{j})^{2}(\\tilde{v}+\\sum^{N}_{j=1}S_{j})-\\frac{1}{2}\\sum^{N}_{j=1}R^{2}_{jx}S_{j}-\\sum^{N}_{j=1}R_{j}R_{jx}S_{jx}-\\sum^{N}_{j=1}R^{2}_{j}S_{j}\\bigg{)}\\partial^{2}_{x}R_{i}\\Bigg{)}dx\\\\\n&\\quad +\\int_{\\mathbb{R}}\\Bigg{(}P\\ast\\bigg{(}\\frac{1}{2}\\Big{(}\\tilde{v}_{x}+\\sum^{N}_{j=1}S_{jx}\\Big{)}^2(\\tilde{u}+\\sum^{N}_{j=1}R_{j})\n+(\\tilde{v}+\\sum^{N}_{j=1}S_{j})\\Big{(}\\tilde{v}_{x}+\\sum^{N}_{j=1}S_{jx}\\Big{)}\\Big{(}\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx}\\Big{)}\\\\\n&\\quad +(\\tilde{v}+\\sum^{N}_{j=1}S_{j})^{2}(\\tilde{u}+\\sum^{N}_{j=1}R_{j})-\\frac{1}{2}\\sum^{N}_{j=1}S^{2}_{jx}R_{j}-\\sum^{N}_{j=1}S_{j}S_{jx}R_{jx}-\\sum^{N}_{j=1}S^{2}_{j}R_{j}\\bigg{)}\\partial^{2}_{x}S_{i}\\Bigg{)}dx\\\\\n&\\quad -\\int_{\\mathbb{R}}\\Bigg{(}P\\ast\\bigg{(}\\frac{1}{2}\\big{(}\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx}\\big{)}^{2}\\big{(}\\tilde{v}_{x}+\\sum^{N}_{j=1}S_{jx}\\big{)}-\\frac{1}{2}\\sum^{N}_{j=1}R^{2}_{jx}S_{jx}\\bigg{)}\\partial_{x}R_{i}\\Bigg{)}dx\\\\\n&\\quad -\\int_{\\mathbb{R}}\\Bigg{(}P\\ast\\bigg{(}\\frac{1}{2}\\big{(}\\tilde{v}_{x}+\\sum^{N}_{j=1}S_{jx}\\big{)}^{2}\\big{(}\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx}\\big{)}-\\frac{1}{2}\\sum^{N}_{j=1}S^{2}_{jx}R_{jx}\\bigg{)}\\partial_{x}S_{i}\\Bigg{)}dx.\n\\end{align*}\nFor every term, we have the following estimates\n\\begin{eqnarray*}\n\\begin{aligned}\n<\\partial^{2}_{x}R_{i},\\tilde{u}>_{H^{-1},H^{1}}&=\\int_{\\mathbb{R}}\\partial^{2}_{x}R_{i}\\tilde{u}\\,dx=\\int_{\\mathbb{R}}\\left[R_{i}-2a_i\\delta\\left(x-\\tilde{x}_{i}(t)\\right))\\right]\\tilde{u}\\,dx\\\\\n&=\\int_{\\mathbb{R}}R_{i}\\tilde{u}\\,dx-2a_{i}\\tilde{u}(\\tilde{x}_{i}(t))\\leq \\|\\tilde{u}\\|_{L^\\infty}\\Big(\\Big|\\int_{\\mathbb{R}}R_{i}\\,dx\\Big|+2a_i\\Big)\\leq O(\\sqrt{\\alpha}).\n\\end{aligned}\n\\end{eqnarray*}\nSimilarly,\n\\begin{align*}\n<\\partial^{2}_{x}S_{i},\\tilde{v}>_{H^{-1},H^{1}}\\leq O(\\sqrt{\\alpha}).\n\\end{align*}\nFor the term, $\\int_{\\mathbb{R}}\\partial_{x}R_{i}\\partial_{x}R_{j}\\,dx$, we find\n\\begin{eqnarray*}\n\\begin{aligned}\n\\int_{\\mathbb{R}}\\partial_{x}R_{i}\\partial_{x}R_{j}\\,dx&=-\\int_{\\mathbb{R}}\\partial_x^2 R_{i}R_{j}\\,dx=-\\int_{\\mathbb{R}}\\left(R_{i}-2a_{i}\\delta(x-\\tilde{x}_{i}(t))\\right)R_{j}\\,dx\\\\\n&=-\\int_{\\mathbb{R}}R_{i}R_{j}\\,dx+2a_{i}R_{j}(\\tilde{x}_{i}(t)) \\leq O(e^{-\\frac L4}).\n\\end{aligned}\n\\end{eqnarray*}\nSimilarly,\n\\begin{align*}\n\\int_{\\mathbb{R}}\\partial_{x}S_{i}\\partial_{x}S_{j}\\,dx \\leq O(e^{-\\frac L4}).\n\\end{align*}\n\n\\begin{lemma}\\label{implicit-1}\nAssume $(\\tilde{u},\\tilde{v})$ satisfies \\eqref{preestimate}, then we have\n\\begin{equation*}\n\\int_{\\mathbb{R}}\\Big[(\\tilde{u}+\\sum^{N}_{j=1}R_{j})(\\tilde{v}+\\sum^{N}_{j=1}S_{j})(\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx})-\\sum^{N}_{j=1}R_{j}S_{j}R_{jx}\\Big]R_{ix}\\,dx \\leq O(\\sqrt{\\alpha})+O(e^{-\\frac L4})\n\\end{equation*}\nand\n\\begin{equation*}\n\\int_{\\mathbb{R}}\\Big[(\\tilde{u}+\\sum^{N}_{j=1}R_{j})(\\tilde{v}+\\sum^{N}_{j=1}S_{j})(\\tilde{v}_{x}+\\sum^{N}_{j=1}S_{jx})-\\sum^{N}_{j=1}R_{j}S_{j}S_{jx}\\Big]S_{ix}\\,dx \\leq O(\\sqrt{\\alpha})+O(e^{-\\frac L4}).\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nWe calculate\n\\begin{align*}\n&\\int_{\\mathbb{R}}\\Big[(\\tilde{u}+\\sum^{N}_{j=1}R_{j})(\\tilde{v}+\\sum^{N}_{j=1}S_{j})(\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx})-\\sum^{N}_{j=1}R_{j}S_{j}R_{jx}\\Big]R_{ix}\\,dx\\\\\n=&\\int_{\\mathbb{R}}\\Big[\\tilde{u}(\\tilde{v}+\\sum^{N}_{j=1}S_{j})(\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx})+\\sum^{N}_{j=1}R_{j}\\tilde{v}(\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx})+\\sum^{N}_{j=1}S_{j}\\sum^{N}_{j=1}R_{j}\\tilde{u}_{x}\\\\\n&\\qquad\\qquad+ \\sum^{N}_{j=1}S_{j}\\sum^{N}_{j=1}R_{j}\\sum^{N}_{j=1}R_{jx}-\\sum^{N}_{j=1}R_{j}S_{j}R_{jx}\\Big]R_{ix}\\,dx\\\\\n\\leq &\\Big|\\int_{\\mathbb{R}}\\tilde{u}(\\tilde{v}+\\sum^{N}_{j=1}S_{j})(\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx})R_{ix}\\,dx\\Big|+\\Big|\\int_{\\mathbb{R}}\\sum^{N}_{j=1}R_{j}\\tilde{v}(\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx})R_{ix}\\,dx\\Big|\\\\\n&\\quad+\\Big|\\int_{\\mathbb{R}}\\sum^{N}_{j=1}S_{j}\\sum^{N}_{j=1}R_{j}\\tilde{u}_{x}R_{ix}\\,dx\\Big|+\\Big|\\int_{\\mathbb{R}}\\Big[\\sum^{N}_{j=1}S_{j}\\sum^{N}_{j=1}R_{j}\\sum^{N}_{j=1}R_{jx}-\\sum^{N}_{j=1}R_{j}S_{j}R_{jx}\\Big]R_{ix}\\,dx\\Big|\\\\\n\\leq &\\|\\tilde{u}\\|_{L^{\\infty}}\\|u\\|_{L^{\\infty}}\\|v\\|_{L^{\\infty}}\\Big|\\int_{\\mathbb{R}}R_{ix}\\,dx\\Big|+\\sum^{N}_{j=1}\\|R_{j}\\|_{L^{\\infty}}\\|\\tilde{v}\\|_{L^{\\infty}}\\|u\\|_{L^{\\infty}}\\Big|\\int_{\\mathbb{R}}R_{ix}\\,dx\\Big|\\\\\n&\\quad+\\Big(\\sum^{N}_{j=1}\\|R_{j}\\|_{L^{\\infty}}\\Big)\\Big(\\sum^{N}_{j=1}\\|S_{j}\\|_{L^{\\infty}}\\Big)\\Big(\\int_{\\mathbb{R}}(\\tilde{u}_{x})^2\\,dx\\Big)^{\\frac 12}\\Big(\\int_{\\mathbb{R}}R_{ix}^2\\,dx\\Big)^{\\frac 12}\\\\\n&\\quad\\quad +\\Big|\\int_{\\mathbb{R}}\\Big[\\sum_{i\\ne j\\,or\\,i\\ne k\\,or\\,j\\ne k}R_{i}S_{j}R_{kx}\\Big]R_{ix}\\,dx\\Big|\\leq  O(\\sqrt{\\alpha})+O(e^{-\\frac L4}).\n\\end{align*}\nSimilarly, we have\n\\begin{align*}\n\\int_{\\mathbb{R}}\\Big[(\\tilde{u}+\\sum^{N}_{j=1}R_{j})(\\tilde{v}+\\sum^{N}_{j=1}S_{j})(\\tilde{v}_{x}+\\sum^{N}_{j=1}S_{jx})-\\sum^{N}_{j=1}R_{j}S_{j}S_{jx}\\Big]S_{ix}\\,dx \\leq O(\\sqrt{\\alpha})+O(e^{-\\frac L4}).\n\\end{align*}\n\\end{proof}\n\nIn order to estimate the next terms, we need the following lemma.\n\n\\begin{lemma}\\label{implicit-2}\nUnder the same assumptions as in Lemma \\eqref{implicit-1}, we have\n\\begin{equation*}\n\\begin{aligned}\n\\Big{\\|} &P\\ast \\Big{(}\\frac{1}{2}\\Big{(}\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx}\\Big{)}^2(\\tilde{v}+\\sum^{N}_{j=1}S_{j})\n+(\\tilde{u}+\\sum^{N}_{j=1}R_{j})\\Big{(}\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx}\\Big{)}\\Big{(}\\tilde{v}_{x}+\\sum^{N}_{j=1}S_{jx}\\Big{)}\\\\\n&+(\\tilde{u}+\\sum^{N}_{j=1}R_{j})^{2}(\\tilde{v}+\\sum^{N}_{j=1}S_{j})-\\frac{1}{2}\\sum^{N}_{j=1}R^{2}_{jx}S_{j}-\\sum^{N}_{j=1}R_{j}R_{jx}S_{jx}-\\sum^{N}_{j=1}R^{2}_{j}S_{j}\\bigg{)}\\Big{\\|}_{L^\\infty}\\\\\n\\leq & O(\\sqrt{\\alpha})+O(e^{-\\frac L4})\n\\end{aligned}\n\\end{equation*}\nand\n\\begin{equation*}\n\\begin{aligned}\n\\Big{\\|} &P\\ast \\Big{(}\\frac{1}{2}\\Big{(}\\tilde{v}_{x}+\\sum^{N}_{j=1}S_{jx}\\Big{)}^2(\\tilde{u}+\\sum^{N}_{j=1}R_{j})\n+(\\tilde{v}+\\sum^{N}_{j=1}S_{j})\\Big{(}\\tilde{v}_{x}+\\sum^{N}_{j=1}S_{jx}\\Big{)}\\Big{(}\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx}\\Big{)}\\\\\n&+(\\tilde{v}+\\sum^{N}_{j=1}S_{j})^{2}(\\tilde{u}+\\sum^{N}_{j=1}R_{j})-\\frac{1}{2}\\sum^{N}_{j=1}S^{2}_{jx}R_{j}-\\sum^{N}_{j=1}S_{j}S_{jx}R_{jx}-\\sum^{N}_{j=1}S^{2}_{j}R_{j}\\Big{)}\\Big{\\|}_{L^\\infty}\\\\\n\\leq & O(\\sqrt{\\alpha})+O(e^{-\\frac L4}).\n\\end{aligned}\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nBy using H$\\rm{\\ddot{o}}$lder inequality and triangle inequality, we get\n\\begin{align*}\n\\Big{\\|} &P\\ast \\Big{(}\\frac{1}{2}\\Big{(}\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx}\\Big{)}^2(\\tilde{v}+\\sum^{N}_{j=1}S_{j})\n+(\\tilde{u}+\\sum^{N}_{j=1}R_{j})\\Big{(}\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx}\\Big{)}\\Big{(}\\tilde{v}_{x}+\\sum^{N}_{j=1}S_{jx}\\Big{)}\\\\\n&\\;\\;+(\\tilde{u}+\\sum^{N}_{j=1}R_{j})^{2}(\\tilde{v}+\\sum^{N}_{j=1}S_{j})-\\frac{1}{2}\\sum^{N}_{j=1}R^{2}_{jx}S_{j}-\\sum^{N}_{j=1}R_{j}R_{jx}S_{jx}-\\sum^{N}_{j=1}R^{2}_{j}S_{j}\\Big{)}\\Big{\\|}_{L^\\infty}\\\\\n\\leq &\\frac{1}{2}\\,\\int_{\\mathbb{R}}\\Big|\\Big{(}\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx}\\Big{)}^2(\\tilde{v}+\\sum^{N}_{j=1}S_{j})-\\sum^{N}_{j=1}R^{2}_{jx}S_{j}\\Big|\\,dx\\\\\n&\\;\\;+\\int_{\\mathbb{R}}\\Big| (\\tilde{u}+\\sum^{N}_{j=1}R_{j})\\Big{(}\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx}\\Big{)}\\Big{(}\\tilde{v}_{x}+\\sum^{N}_{j=1}S_{jx}\\Big{)}-\\sum^{N}_{j=1}R_{j}R_{jx}S_{jx}\\Big|\\,dx\\\\\n&\\;\\; +\\int_{\\mathbb{R}}\\Big| (\\tilde{u}+\\sum^{N}_{j=1}R_{j})^{2}(\\tilde{v}+\\sum^{N}_{j=1}S_{j})-\\sum^{N}_{j=1}R^{2}_{j}S_{j}\\Big|\\,dx\\\\\n&=\\frac{1}{2}\\,I_{2,1}+I_{2,2}+I_{2,3}.\n\\end{align*}\nFor the term $I_{2,1}$, we have\n\\begin{align*}\nI_{2,1}=&\\int_{\\mathbb{R}}\\Big|\\Big{(}\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx}\\Big{)}^2(\\tilde{v}+\\sum^{N}_{j=1}S_{j})-\\sum^{N}_{j=1}R^{2}_{jx}S_{j}\\Big|\\,dx\\\\\n\\leq &\\int_{\\mathbb{R}}\\Big|\\Big[\\Big{(}\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx}\\Big{)}^2-\\sum^{N}_{j=1}R^{2}_{jx}\\Big](\\tilde{v}+\\sum^{N}_{j=1}S_{j})\\Big|\\,dx+\\int_{\\mathbb{R}}\\Big|\\Big(\\sum^{N}_{j=1}R^{2}_{jx}\\Big)\\tilde{v}\\Big|\\,dx\\\\\n&\\;\\;+\\int_{\\mathbb{R}}\\Big|\\sum^{N}_{j=1}R^{2}_{jx}\\sum^{N}_{i=1}S_{i}-\\sum^{N}_{j=1}R^{2}_{jx}S_{j}\\Big|\\,dx\\\\\n\\leq &\\|\\tilde{v}\\|_{L^\\infty}\\int_{\\mathbb{R}}\\Big( \\Big|\\tilde{u}^2_{x}\\Big|+\\Big|2\\tilde{u}_{x}\\sum^{N}_{j=1}R_{jx}\\Big|+\\Big|\\Big(\\sum^{N}_{j=1}R_{jx}\\Big)^2-\\sum^{N}_{j=1}R^2_{jx}\\Big|\\Big)\\,dx\\\\\n&\\;\\;+\\|\\tilde{v}\\|_{L^\\infty}\\int_{\\mathbb{R}}\\sum^{N}_{j=1}R^{2}_{jx}\\,dx+\\int_{\\mathbb{R}}\\sum^{N}_{i\\ne j}\\sum^{N}_{j=1}R^{2}_{jx}S_{i}\\,dx\\leq O(\\sqrt{\\alpha})+O(e^{\\frac L4}).\n\\end{align*}\nFor the term $I_{2,2}$, we have\n\\begin{align*}\nI_{2,2}=&\\int_{\\mathbb{R}}\\Big|(\\tilde{u}+\\sum^{N}_{j=1}R_{j})\\Big{(}\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx}\\Big{)}\\Big{(}\\tilde{v}_{x}+\\sum^{N}_{j=1}S_{jx}\\Big{)}-\\sum^{N}_{j=1}R_{j}R_{jx}S_{jx}\\Big|\\,dx\\\\\n\\leq &\\int_{\\mathbb{R}}\\Big|\\tilde{u}\\Big{(}\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx}\\Big{)}\\Big{(}\\tilde{v}_{x}+\\sum^{N}_{j=1}S_{jx}\\Big{)}\\Big|\\,dx+\\int_{\\mathbb{R}}\\Big|\\Big(\\sum^{N}_{j=1}R_{j}\\Big)\\tilde{u}_{x}\\Big{(}\\tilde{v}_{x}+\\sum^{N}_{j=1}S_{jx}\\Big{)}\\Big|\\,dx\\\\\n&\\;+\\int_{\\mathbb{R}}\\Big|\\Big(\\sum^{N}_{j=1}R_{j}\\Big)\\Big(\\sum^{N}_{j=1}S_{jx}\\Big)\\tilde{u}_{x}\\Big|\\,dx+\\int_{\\mathbb{R}}\\Big|\\Big(\\sum^{N}_{j=1}R_{j}\\Big)\\Big(\\sum^{N}_{j=1}S_{jx}\\Big)\\Big(\\sum^{N}_{j=1}R_{jx}\\Big)-\\sum^{N}_{j=1}R_{j}R_{jx}S_{jx}\\Big|\\,dx\\\\\n\\leq\n&\\|\\tilde{u}\\|_{L^\\infty}\\int_{\\mathbb{R}}\\left|u_{x}v_{x}\\right|\\,dx+\\|v\\|_{L^\\infty}\\Big(\\int_{\\mathbb{R}}\\Big(\\sum^{N}_{j=1}R_{j}\\Big)^2\\,dx\\Big)^{\\frac{1}{2}}\\Big(\\int_{\\mathbb{R}}\\tilde{u}^2_{x}\\,dx\\Big)^{\\frac 12}\\\\\n&\\;\\;+\\sum^{N}_{j=1}\\|S_{j}\\|_{L^\\infty}\\Big(\\int_{\\mathbb{R}}\\Big(\\sum^{N}_{j=1}R_{j}\\Big)^2\\,dx\\Big)^{\\frac 12}\\Big(\\int_{\\mathbb{R}}\\tilde{u}^2_{x}\\,dx\\Big)^{\\frac 12}+\\int_{\\mathbb{R}}\\Big|\\sum_{i\\ne j\\,or\\,j\\ne k\\,or\\,i\\ne k}R_{i}S_{jx}R_{jx}\\Big|\\,dx\\\\\n\\leq & O(\\sqrt{\\alpha})+O(e^{\\frac L4}).\n\\end{align*}\nFor this term $I_{2,3}$, we obtain\n\\begin{align*}\nI_{2,3}=&\\int_{\\mathbb{R}}\\Big(\\tilde{u}+\\sum^{N}_{j=1}R_{j}\\Big)^{2}\\Big(\\tilde{v}+\\sum^{N}_{j=1}S_{j}\\Big)-\\sum^{N}_{j=1}R^{2}_{j}S_{j}\\,dx\\\\\n\\leq & \\int_{\\mathbb{R}}\\Big|\\Big((\\tilde{u}+\\sum^{N}_{j=1}R_{j})^{2}-\\sum^{N}_{j=1}R^{2}_{j}\\Big)\\Big(\\tilde{v}+\\sum^{N}_{j=1}S_{j}\\Big)\\Big|\\,dx+\\int_{\\mathbb{R}}\\Big|\\sum^{N}_{j=1}R^2_{j}\\tilde{v}\\Big|\\,dx\\\\\n&\\qquad\\qquad +\\int_{\\mathbb{R}}\\Big|\\sum^{N}_{j=1}R^{2}_{j}\\sum^{N}_{i=1}S_{i}-\\sum^{N}_{j=1}R^{2}_{j}S_{j}\\Big|\\,dx\\\\\n\\leq &\\|v\\|_{L^\\infty}\\int_{\\mathbb{R}}|\\tilde{u}|^2+\\Big|2\\tilde{u}\\sum^{N}_{j=1}R_{j}\\Big|+\\Big|\\Big(\\sum^{N}_{j=1}R_{j}\\Big)^2-\\sum^{N}_{j=1}R^{2}_{j}\\Big|\\,dx+\\|\\tilde{v}\\|_{L^\\infty}\\int_{\\mathbb{R}}\\sum^{N}_{j=1}R^{2}_{j}\\,dx\\\\\n&\\qquad\\qquad+\\int_{\\mathbb{R}}\\sum^{N}_{i\\ne j}\\sum^{N}_{j=1}R^{2}_{j}S_{i}\\,dx\\leq O(\\sqrt{\\alpha})+O(e^{\\frac L4}).\n\\end{align*}\nAccordingly, we get\n\\begin{align*}\n\\Big{\\|} &P\\ast \\Big{(}\\frac{1}{2}\\Big{(}\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx}\\Big{)}^2(\\tilde{v}+\\sum^{N}_{j=1}S_{j})\n+(\\tilde{u}+\\sum^{N}_{j=1}R_{j})\\Big{(}\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx}\\Big{)}\\Big{(}\\tilde{v}_{x}+\\sum^{N}_{j=1}S_{jx}\\Big{)}\\\\\n&\\qquad\\qquad+(\\tilde{u}+\\sum^{N}_{j=1}R_{j})^{2}(\\tilde{v}+\\sum^{N}_{j=1}S_{j})-\\frac{1}{2}\\sum^{N}_{j=1}R^{2}_{jx}S_{j}-\\sum^{N}_{j=1}R_{j}R_{jx}S_{jx}-\\sum^{N}_{j=1}R^{2}_{j}S_{j}\\Big{)}\\Big{\\|}_{L^\\infty}\\\\\n\\leq & O(\\sqrt{\\alpha})+O(e^{-\\frac L4}).\n\\end{align*}\nSimilarly, we can prove\n\\begin{align*}\n\\Big{\\|} &P\\ast \\Big{(}\\frac{1}{2}\\Big{(}\\tilde{v}_{x}+\\sum^{N}_{j=1}S_{jx}\\Big{)}^2(\\tilde{u}+\\sum^{N}_{j=1}R_{j})\n+(\\tilde{v}+\\sum^{N}_{j=1}S_{j})\\Big{(}\\tilde{v}_{x}+\\sum^{N}_{j=1}S_{jx}\\Big{)}\\Big{(}\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx}\\Big{)}\\\\\n&\\qquad\\qquad\\qquad+(\\tilde{v}+\\sum^{N}_{j=1}S_{j})^{2}(\\tilde{u}+\\sum^{N}_{j=1}R_{j})-\\frac{1}{2}\\sum^{N}_{j=1}S^{2}_{jx}R_{j}-\\sum^{N}_{j=1}S_{j}S_{jx}R_{jx}-\\sum^{N}_{j=1}S^{2}_{j}R_{j}\\Big{)}\\Big{\\|}_{L^\\infty}\\\\\n\\leq & O(\\sqrt{\\alpha})+O(e^{-\\frac L4}).\n\\end{align*}\nThis completes the proof of this lemma.\n\\end{proof}\n\nThanks to the lemma \\ref{implicit-2}, we have\n\\begin{align*}\n\\int_{\\mathbb{R}}&P\\ast\\Big{(}\\frac{1}{2}\\Big{(}\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx}\\Big{)}^2(\\tilde{v}+\\sum^{N}_{j=1}S_{j})\n+(\\tilde{u}+\\sum^{N}_{j=1}R_{j})\\Big{(}\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx}\\Big{)}\\Big{(}\\tilde{v}_{x}+\\sum^{N}_{j=1}S_{jx}\\Big{)}\\\\\n&+(\\tilde{u}+\\sum^{N}_{j=1}R_{j})^{2}(\\tilde{v}+\\sum^{N}_{j=1}S_{j})-\\frac{1}{2}\\sum^{N}_{j=1}R^{2}_{jx}S_{j}-\\sum^{N}_{j=1}R_{j}R_{jx}S_{jx}-\\sum^{N}_{j=1}R^{2}_{j}S_{j}\\Big{)}\\partial^{2}_{x}R_{i}\\,dx\\\\\n& \\triangleq \\int_{\\mathbb{R}}A(x)\\partial^{2}_{x}R_{i}\\,dx=\\int_{\\mathbb{R}}A(x)\\left(R_{i}-2a_{i}\\delta(x-\\tilde{x}_{i}(t))\\right)\\,dx\\\\\n=&\\int_{\\mathbb{R}}A(x)R_{i}\\,dx-2a_{i}A(\\tilde{x}_{i}(t))\\leq O\\left(\\|A\\|_{L^\\infty}\\right) \\leq O(\\sqrt{\\alpha})+O(e^{-\\frac L4}).\n\\end{align*}\nSimilarly, we get\n\\begin{align*}\n\\int_{\\mathbb{R}}&P\\ast\\Big{(}\\frac{1}{2}\\Big{(}\\tilde{v}_{x}+\\sum^{N}_{j=1}S_{jx}\\Big{)}^2(\\tilde{u}+\\sum^{N}_{j=1}R_{j})\n+(\\tilde{v}+\\sum^{N}_{j=1}S_{j})\\Big{(}\\tilde{v}_{x}+\\sum^{N}_{j=1}S_{jx}\\Big{)}\\Big{(}\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx}\\Big{)}\\\\\n&+(\\tilde{v}+\\sum^{N}_{j=1}S_{j})^{2}(\\tilde{u}+\\sum^{N}_{j=1}R_{j})-\\frac{1}{2}\\sum^{N}_{j=1}S^{2}_{jx}R_{j}-\\sum^{N}_{j=1}S_{j}S_{jx}R_{jx}-\\sum^{N}_{j=1}S^{2}_{j}R_{j}\\Big{)}\\partial^{2}_{x}S_{i}\\,dx\\\\\n\\leq & O(\\sqrt{\\alpha})+O(e^{-\\frac L4}).\n\\end{align*}\n\n\\begin{lemma}\\label{implicit-3}\nUnder the same assumption as \\eqref{implicit-1}, we have\n\\begin{align*}\n&\\Big\\|P\\ast\\Big{(}\\frac{1}{2}\\big{(}\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx}\\big{)}^{2}\\big{(}\\tilde{v}_{x}+\\sum^{N}_{j=1}S_{jx}\\big{)}-\\frac{1}{2}\\sum^{N}_{j=1}R^{2}_{jx}S_{jx}\\Big{)}\\Big\\|_{L^\\infty}\\leq O(\\sqrt{\\alpha})+O(e^{-\\frac L4}),\\\\\n&\\Big\\|P\\ast\\Big{(}\\frac{1}{2}\\big{(}\\tilde{v}_{x}+\\sum^{N}_{j=1}S_{jx}\\big{)}^{2}\\big{(}\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx}\\big{)}-\\frac{1}{2}\\sum^{N}_{j=1}S^{2}_{jx}R_{jx}\\Big{)}\\Big\\|_{L^\\infty}\\leq O(\\sqrt{\\alpha})+O(e^{-\\frac L4}).\n\\end{align*}\n\\end{lemma}\n\\begin{proof}\nWe estimate\n\\begin{align*}\n&\\Big\\|P\\ast\\Big{(}\\frac{1}{2}\\big{(}\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx}\\big{)}^{2}\\big{(}\\tilde{v}_{x}+\\sum^{N}_{j=1}S_{jx}\\big{)}-\\frac{1}{2}\\sum^{N}_{j=1}R^{2}_{jx}S_{jx}\\Big{)}\\Big\\|\\\\\n\\leq & \\frac{1}{2}\\,\\int_{\\mathbb{R}}\\Big|(\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx})^{2}(\\tilde{v}_{x}+\\sum^{N}_{j=1}S_{jx})-\\sum^{N}_{j=1}R^{2}_{jx}S_{jx}\\Big|\\,dx\\\\\n\\leq &\n\\frac{1}{2}\\,\\Big(\\int_{\\mathbb R}\\Big|\\Big((\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx})^{2}-\\sum^{N}_{j=1}R^{2}_{jx}\\Big)(\\tilde{v}_{x}+\\sum^{N}_{j=1}S_{jx})\\Big|\\,dx+\\int_{\\mathbb R}\\Big|\\tilde{v}_{x}\\sum^{N}_{j=1}R^{2}_{jx}\\Big|\\,dx\\Big)\\\\\n&\\qquad\\qquad +\\frac{1}{2}\\int_{\\mathbb R}\\Big|\\sum^{N}_{j=1}R^{2}_{jx}\\sum^{N}_{j=1}S_{jx}-\\sum^{N}_{j=1}R^{2}_{jx}S_{jx}\\Big|\\,dx=\\frac{1}{2}\\,(I_{3,1}+I_{3,2}+I_{3,3}).\n\\end{align*}\nFor the term $I_{3,1}$, we obtain\n\\begin{align*}\nI_{3,1}=&\\int_{\\mathbb{R}}\\Big|\\Big{(}\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx}\\Big{)}^2(\\tilde{v}_{x}+\\sum^{N}_{j=1}S_{jx})-\\sum^{N}_{j=1}R^{2}_{jx}S_{jx}\\Big|\\,dx\\\\\n\\leq &\\int_{\\mathbb{R}}\\Big|\\Big[\\Big{(}\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx}\\Big{)}^2-\\sum^{N}_{j=1}R^{2}_{jx}\\Big](\\tilde{v}_{x}+\\sum^{N}_{j=1}S_{jx})\\Big|\\,dx+\\int_{\\mathbb{R}}\\Big|\\Big(\\sum^{N}_{j=1}R^{2}_{jx}\\Big)\\tilde{v}_{x}\\Big|\\,dx\\\\\n&\\qquad\\qquad+\\int_{\\mathbb{R}}\\Big|\\sum^{N}_{j=1}R^{2}_{jx}\\sum^{N}_{i=1}S_{ix}-\\sum^{N}_{j=1}R^{2}_{jx}S_{jx}\\Big|\\,dx\\\\\n\\leq &\\|v\\|_{L^\\infty}\\int_{\\mathbb{R}}\\Big|\\tilde{u}^2_{x}\\Big|+\\Big|2\\tilde{u}_{x}\\sum^{N}_{j=1}R_{jx}\\Big|+\\Big|\\Big(\\sum^{N}_{j=1}R_{jx}\\Big)^2-\\sum^{N}_{j=1}R^2_{jx}\\Big|\\,dx\\\\\n&\\qquad\\qquad+\\|\\tilde{v}\\|_{L^\\infty}\\int_{\\mathbb{R}}\\sum^{N}_{j=1}R^{2}_{jx}\\,dx+\\int_{\\mathbb{R}}\\sum^{N}_{i\\ne j}\\sum^{N}_{j=1}\\left|R^{2}_{jx}S_{ix}\\right|\\,dx\\leq O(\\sqrt{\\alpha})+O(e^{-\\frac L4}).\n\\end{align*}\nFor the term $I_{3,2}$, we get\n\\begin{align*}\nI_{3,2}=\\int_{\\mathbb R}\\sum^{N}_{j=1}R^2_{jx}|\\tilde{v}_{x}|\\,dx= \\sum^{N}_{j=1}\\int_{\\mathbb R}R^2_{jx}|\\tilde{v}_{x}|\\,dx\\leq \\sum^{N}_{j=1}\\Big(\\int_{\\mathbb R}\\tilde{v}^2_{x}\\,dx\\Big)^{\\frac 12}\\Big(\\int_{\\mathbb R}R^4_{jx}\\,dx\\Big)^{\\frac 12}\\leq O(\\sqrt{\\alpha}).\n\\end{align*}\nFor the term $I_{3,3}$, we have\n\\begin{align*}\n \\int_{\\mathbb R}\\Big|\\sum^{N}_{j=1}R^{2}_{jx}\\sum^{N}_{j=1}S_{jx}-\\sum^{N}_{j=1}R^{2}_{jx}S_{jx}\\Big|\\,dx \\leq \\int_{\\mathbb R}\\sum^{N}_{i\\ne j}\\sum^{N}_{j=1}\\left|R^{2}_{jx}S_{ix}\\right|\\,dx \\leq  O(e^{-\\frac L4}).\n\\end{align*}\nThus, we deduce that\n\\begin{align*}\n\\Big\\|P\\ast\\Big{(}\\frac{1}{2}\\big{(}\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx}\\big{)}^{2}\\big{(}\\tilde{v}_{x}+\\sum^{N}_{j=1}S_{jx}\\big{)}-\\frac{1}{2}\\sum^{N}_{j=1}R^{2}_{jx}S_{jx}\\Big{)}\\Big\\|_{L^\\infty}\\leq O(\\sqrt{\\alpha})+O(e^{-\\frac L4})\n\\end{align*}\nand\n\\begin{align*}\n&\\Big\\|P\\ast\\Big{(}\\frac{1}{2}\\big{(}\\tilde{v}_{x}+\\sum^{N}_{j=1}S_{jx}\\big{)}^{2}\\big{(}\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx}\\big{)}-\\frac{1}{2}\\sum^{N}_{j=1}S^{2}_{jx}R_{jx}\\Big{)}\\Big\\|_{L^\\infty}\\leq O(\\sqrt{\\alpha})+O(e^{-\\frac L4}).\n\\end{align*}\nThis completes the proof of this lemma.\n\\end{proof}\n\nOn account of Lemma \\eqref{implicit-3}, we obtain\n\\begin{align*}\n&\\int_{\\mathbb{R}}P\\ast\\Big{(}\\frac{1}{2}\\big{(}\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx}\\big{)}^{2}\\big{(}\\tilde{v}_{x}+\\sum^{N}_{j=1}S_{jx}\\big{)}-\\frac{1}{2}\\sum^{N}_{j=1}R^{2}_{jx}S_{jx}\\Big{)}\\partial_{x}R_{i}\\,dx\\\\\n\\leq &\\Big( O(\\sqrt{\\alpha})+O(e^{\\frac L4})\\Big) \\int_{\\mathbb R}|R_{ix}|\\,dx \\leq O(\\sqrt{\\alpha})+O(e^{-\\frac L4}).\n\\end{align*}\nSimilarly, we find\n\\begin{align*}\n\\int_{\\mathbb{R}}P\\ast\\Big{(}\\frac{1}{2}\\big{(}\\tilde{v}_{x}+\\sum^{N}_{j=1}S_{jx}\\big{)}^{2}\\big{(}\\tilde{u}_{x}+\\sum^{N}_{j=1}R_{jx}\\big{)}-\\frac{1}{2}\\sum^{N}_{j=1}S^{2}_{jx}R_{jx}\\Big{)}\\partial_{x}S_{i}\\,dx\\leq O(\\sqrt{\\alpha})+O(e^{-\\frac L4}).\n\\end{align*}\nThanks to the lemmas \\ref{implicit-1}, \\ref{implicit-2} and \\ref{implicit-3}, we arrive at\n\\begin{align*}\n|\\dot{\\tilde{x}}_{i}(t)-c_{i}|\\Big{(}\\|R_{ix}\\|^{2}_{L^{2}}+\\|S_{ix}\\|^{2}_{L^{2}}+O(\\sqrt{\\alpha})\\Big{)}\\leq O(\\sqrt{\\alpha})+O(e^{-\\frac{L}{8}}).\n\\end{align*}\nSince $\\|R_{ix}\\|_{L^2}>a_1$ and $\\|S_{ix}\\|^{2}_{L^2}>b_1$, then we have\n\\begin{align*}\n|\\dot{\\tilde{x}}_{i}(t)-c_{i}|\\leq O(\\sqrt{\\alpha})+O(e^{-\\frac{L}{8}}).\n\\end{align*}\n\nFinally, we claim\n\\begin{align*}\n\\left|x_{i}-\\tilde{x}_{i}\\right|\\leq \\frac{L}{12}.\n\\end{align*}\nIndeed, if $x\\notin [\\tilde{x}_{i}-L\/12,\\tilde{x}_{i}+L\/12]$, then\n\\begin{align*}\nu(x,t)v(x,t)\\leq c_{i}e^{-\\frac{L}{6}}+O(\\sqrt{\\alpha})+O(e^{-\\frac{L}{4}}) \\leq c_{i}-O(\\sqrt{\\alpha})-O(e^{-\\frac{L}{4}}),\n\\end{align*}\nby choosing $\\alpha$ small enough and $L$ large enough.\nHowever,\n\\begin{align*}\nu(t,x_{i})v(t,x_{i})=\\max_{x\\in J_{i}(t)}u(x)v(x)\\geq u(\\tilde{x}_{i})v(\\tilde{x}_{i})=c_{i}+O(\\sqrt{\\alpha})+O(e^{-\\frac{L}{4}}),\n\\end{align*}\nwhich is a contradiction.\nTherefore, we have $\\left|x_{i}-\\tilde{x}_{i}\\right|\\leq L\/12$.\n\\end{proof}\n\n\n\\subsection{Monotonicity property}\nThanks to the preceding proposition, for $\\epsilon_{0}>0$ small enough and $L_{0}>0$ large enough, one can construct $C^1$ functions $\\tilde{x}_{1},...,\\tilde{x}_{N}$ defined on $[0,t_{0}]$ such that \\eqref{initial-3-1}-\\eqref{initial-3-5} are satisfied. In this subsection we investigate the almost monotonicity of functionals that are very close to the energy at the right of $i$th bump, $i=1,...,N-1$ of $(u, v)$. Let $\\Psi$ be a $C^{\\infty}$-function such that\n\\begin{equation*}\n\\left\\{\n\\begin{aligned}\n&0<\\Psi(x)<1,\\,\\Psi'(x)>0,\\quad\\quad\\quad & x\\in{\\mathbb R},\\\\\n&|\\Psi'''|\\leq 10|\\Psi'|,&x\\in [-1,1],\\\\\n\\end{aligned}\n\\right.\n\\end{equation*}\nand\n\\begin{equation*}\n\\Psi(x)=\\left\\{\n\\begin{aligned}\n&e^{-|x|},\\quad &x<-1,\\\\\n&1-e^{-|x|},&x>1.\n\\end{aligned}\n\\right.\n\\end{equation*}\nSetting $\\Psi_{K}=\\Psi(\\cdot\/K)$, we introduce for $i=2,...,N$,\n\\begin{align*}\n&\\mathcal{J}^{u}_{j,K}(t)=\\int_{\\mathbb R}\\left(u^2(t)+u^2_{x}(t)\\right)\\Psi_{j,K}(t)\\,dx,\\quad \\mathcal{J}^{v}_{j,K}(t)=\\int_{\\mathbb R}\\left(v^2(t)+v^2_{x}(t)\\right)\\Psi_{j,K}(t)\\,dx,\\\\\n&\\mathcal{J}^{u,v}_{j,K}(t)=\\int_{\\mathbb R}\\left(u(t)v(t)+u_{x}(t)v_{x}(t)\\right)\\Psi_{j,K}(t)\\,dx,\n\\end{align*}\nwhere $\\Psi_{j,K}(t,x)=\\Psi_K(x-y_{j}(t))$ with $y_{j}(t)$, $j=2,...,N$, defined in \\eqref{initial-3-4}. Note that $\\mathcal{J}^{u}_{j,K}(t)$ is close to $\\left\\|u(t)\\right\\|_{H^{1}(x>y_{j}(t))}$ and thus measures the energy at the right of the $(j-1)th$ bump of $u$, $\\mathcal{J}^{v}_{j,K}(t)$ is close to $\\left\\|v(t)\\right\\|_{H^{1}(x>y_{j}(t))}$ and thus measures the energy at the right of the $(j-1)$th bump of $v$, and $\\mathcal{J}^{u,v}_{j,K}(t)$ is close to $<u(t),v(t)>_{H^{1}\\times H^{1}(x>y_{j}(t))}$ and thus measures the energy at the right of the $(j-1)th$ bump of $(u, v)$. Finally, we set\n\\begin{align*}\n\\sigma_{0}=\\frac{1}{4}\\,\\min(c_{1},c_{2}-c_{1},...,c_{N}-c_{N-1}).\n\\end{align*}\n\nWe have the following monotonicity result.\n\\begin{prop}\\label{monotonicity}\n(Exponential decay of the functionals $\\mathcal{J}^{u}_{j,K}(t)$, $\\mathcal{J}^{v}_{j,K}(t)$ and $\\mathcal{J}^{u,v}_{j,K}(t)$). Let $(u,v)\\in Y([0,T[)$ be a solution of two component Novikov equations satisfying \\eqref{initial-3-2} on $[0,t_{0}]$. There exist $\\alpha_{0}>0$ and $L_{0}>0$ only depending on $c_{1}$ such that if $0<\\alpha<\\alpha_{0}$ and $L\\geq L_{0}$ then for any $4 \\leq K \\lesssim \\sqrt{L}$,\n\\begin{align*}\n&\\mathcal{J}^{u}_{j,K}(t)-\\mathcal{J}^{u}_{j,K}(0)\\leq O(e^{-\\frac{\\sigma_{0}L}{8K}}), \\quad \\mathcal{J}^{v}_{j,K}(t)-\\mathcal{J}^{v}_{j,K}(0)\\leq O(e^{-\\frac{\\sigma_{0}L}{8K}}),\\\\\n&\\mathcal{J}^{u,v}_{j,K}(t)-\\mathcal{J}^{u,v}_{j,K}(0)\\leq O(e^{-\\frac{\\sigma_{0}L}{8K}}),\\quad \\forall j\\in \\{2,...,N\\},\\;\\forall t\\in[0,t_{0}].\n\\end{align*}\n\\end{prop}\n\nThe proof of this Proposition relies on the following Virial type identity.\n\n\\begin{lemma}\\label{virial}\n(Virial type identity). Let $(u,v)\\in Y([0,T[)$, with $0<T\\leq +\\infty$, be a solution of equation \\eqref{tcNK} that satisfies \\eqref{initialdata-1}. For any smooth space function $g:\\mathbb{R}\\rightarrow\\mathbb{R}$, there holds\n\\begin{align*}\n\\frac{d}{dt}\\,&\\int_{\\mathbb R}\\left(u^{2}+u^{2}_{x}\\right)g\\,dx\\\\\n=&\\int_{\\mathbb R}u\\Big[u^2_{x}v+2P\\ast\\Big(\\frac{1}{2}u^{2}_{x}v+uu_{x}v_{x}+u^{2}v\\Big)+2P_{x}\\ast\\Big(\\frac{1}{2}u^{2}_{x}v_{x}\\Big)\\Big]g'\\,dx+\\int_{\\mathbb R}\\left(u^{2}+u^{2}_{x}\\right)g'\\,dx,\\\\\n\\frac{d}{dt}\\,&\\int_{\\mathbb R}\\left(v^{2}+v^{2}_{x}\\right)g\\,dx\\\\\n=&\\int_{\\mathbb R}v\\Big[v^2_{x}u+2P\\ast\\Big(\\frac{1}{2}v^{2}_{x}u+vv_{x}u_{x}+v^{2}u\\Big)+2P_{x}\\ast\\Big(\\frac{1}{2}v^{2}_{x}u_{x}\\Big)\\Big]g'\\,dx+\\int_{\\mathbb R}\\left(v^{2}+v^{2}_{x}\\right)g'\\,dx,\\\\\n{\\rm and}\\quad &\\frac{d}{dt}\\,\\int_{\\mathbb R}\\left(uv+u_{x}v_{x}\\right)g\\,dx\\\\\n=&\\int_{\\mathbb R}u\\Big[vu_{x}v_{x}+P\\ast\\Big(\\frac{1}{2}v^{2}_{x}u+vv_{x}u_{x}+v^{2}u\\Big)+P_{x}\\ast\\left(\\frac{1}{2}v^{2}_{x}u_{x}\\right)\\Big]g'\\,dx\\\\\n&\\;\\;+\\int_{\\mathbb R}v\\Big[P\\ast\\Big(\\frac{1}{2}u^{2}_{x}v+uu_{x}v_{x}+u^{2}v\\Big)+P_{x}\\ast\\Big(\\frac{1}{2}u^{2}_{x}v_{x}\\Big)\\Big]g'\\,dx+\\int_{\\mathbb R}\\left(uv+u_{x}v_{x}\\right)g'\\,dx.\n\\end{align*}\n\\end{lemma}\n\\begin{proof}\nBy using the weak form of $u$, integrating by parts, we calculate\n\\begin{align*}\n\\frac{d}{dt}\\,&\\int_{\\mathbb R}\\left(u^{2}+u^{2}_{x}\\right)g\\,dx=2\\int_{\\mathbb R}\\left(uu_{t}+u_{x}u_{xt}\\right)g\\,dx+\\int_{\\mathbb R}\\left(u^{2}+u^{2}_{x}\\right)g'\\,dx\\\\\n=&2\\int_{\\mathbb R}\\left(uu_{t}-uu_{xxt}-uu_{xt}\\right)g'\\,dx+\\int_{\\mathbb R}\\left(u^{2}+u^{2}_{x}\\right)g'\\,dx\\\\\n=&\\int_{\\mathbb R}\\Big(2u^2vm-2u\\Big[u^2v-\\frac{1}{2}u^2_{x}v-uvu_{xx}-P\\ast\\Big(\\frac{1}{2}\\,u^2_{x}v+uu_{x}v_{x}+u^2v\\Big)-P_{x}\\ast\\Big(\\frac{1}{2}\\,u^2_{x}v_{x}\\Big)\\Big]\\Big)g'\\,dx\\\\\n&\\;\\;+\\int_{\\mathbb R}\\left(u^{2}+u^{2}_{x}\\right)g'\\,dx\\\\\n=&\\int_{\\mathbb R}\\Big[uu^2_{x}v+2uP\\ast\\Big(\\frac{1}{2}u^{2}_{x}v+uu_{x}v_{x}+u^{2}v\\Big)+2uP_{x}\\ast\\Big(\\frac{1}{2}u^{2}_{x}v_{x}\\Big)\\Big]g'\\,dx+\\int_{\\mathbb R}\\left(u^{2}+u^{2}_{x}\\right)g'\\,dx.\n\\end{align*}\nSimilarly, we have\n\\begin{eqnarray*}\n\\begin{aligned}\n\\frac{d}{dt}\\,&\\int_{\\mathbb R}\\left(v^{2}+v^{2}_{x}\\right)g\\,dx=\\int_{\\mathbb R}\\Big[vv^2_{x}u+2vP\\ast\\Big(\\frac{1}{2}v^{2}_{x}u+vv_{x}u_{x}+v^{2}u\\Big)+2vP_{x}\\ast\\Big(\\frac{1}{2}v^{2}_{x}u_{x}\\Big)\\Big]g'\\,dx\\\\\n&\\qquad\\quad\\qquad\\quad\\qquad\\quad+\\int_{\\mathbb R}\\left(v^{2}+v^{2}_{x}\\right)g'\\,dx.\\\\\n\\end{aligned}\n\\end{eqnarray*}\nUsing the weak form of $(u,v)$ and its first derivative, integrating by parts, we obtain\n\\begin{align*}\n\\frac{d}{dt}\\,&\\int_{\\mathbb R}\\left(uv+u_{x}v_{x}\\right)g\\,dx\\\\\n=&\\int_{\\mathbb R}\\left(uv_{t}+u_{t}v-u_{xxt}v-uv_{xxt}-u_{xt}v-uv_{xt}\\right)g\\,dx+\\int_{\\mathbb R}\\left(uv+u_{x}v_{x}\\right)g'\\,dx\\\\\n=&\\int_{\\mathbb R}\\left(un+mv-u_{xt}v-uv_{xt}\\right)g\\,dx+\\int_{\\mathbb R}\\left(uv+u_{x}v_{x}\\right)g'\\,dx\\\\\n=&\\int_{\\mathbb R}\\Big[uvu_{x}v_{x}+uP\\ast\\Big(\\frac{1}{2}v^{2}_{x}u+vv_{x}u_{x}+v^{2}u\\Big)+uP_{x}\\ast\\Big(\\frac{1}{2}v^{2}_{x}u_{x}\\Big)\\Big]g'\\,dx\\\\\n&\\;\\;+\\int_{\\mathbb R}\\Big[vP\\ast\\Big(\\frac{1}{2}u^{2}_{x}v+uu_{x}v_{x}+u^{2}v\\Big)+vP_{x}\\ast\\Big(\\frac{1}{2}u^{2}_{x}v_{x}\\Big)\\Big]g'\\,dx+\\int_{\\mathbb R}\\left(uv+u_{x}v_{x}\\right)g'\\,dx.\n\\end{align*}\n\\end{proof}\n\n\\begin{proof} [Proof of Proposition 4.2.]\nWe first note that, combining \\eqref{initial-3-1} and \\eqref{initial-3-4}, it holds for $i=2,..,N$,\n\\begin{eqnarray}\\label{infspeed}\n\\begin{aligned}\n\\dot{y}_{i}(t)=\\frac{\\dot{\\tilde{x}}_{i-1}(t)+\\dot{\\tilde{x}}_{i}(t)}{2}=\\frac{c_{i-1}+c_{i}}{2}+O(\\sqrt{\\alpha})+O(L^{-1})\\geq \\frac{c_1}{2}.\n\\end{aligned}\n\\end{eqnarray}\nRecall that the assumption ensures that $u\\geq 0$ and $v\\geq 0$ on $\\mathbb R$. Now, applying the Virial type identity with $g=\\Psi_{i,K}$ and using \\eqref{infspeed}, we get\n\\begin{align*}\n\\frac{d}{dt}\\,\\mathcal{J}^{u}_{i,K}(t)=&-\\dot{y}_{i}\\,\\int_{\\mathbb R}\\left(u^{2}+u^{2}_{x}\\right)\\Psi'_{i,K}\\,dx\\\\\n&\\;\\;+\\int_{\\mathbb R}\\Big[uu^2_{x}v+2uP\\ast\\Big(\\frac{1}{2}u^{2}_{x}v+uu_{x}v_{x}+u^{2}v\\Big)+2uP_{x}\\ast\\Big(\\frac{1}{2}u^{2}_{x}v_{x}\\Big)\\Big]\\Psi'_{i,K}\\,dx\\\\\n\\leq &-\\frac{c_1}{2}\\int_{\\mathbb R}\\left(u^{2}+u^{2}_{x}\\right)\\Psi'_{i,K}\\,dx+\\int_{\\mathbb R}uu^2_{x}v\\Psi'_{i,K}\\,dx\\\\\n&\\;\\;+\\int_{\\mathbb R}\\Big[2uP\\ast\\Big(\\frac{1}{2}u^{2}_{x}v+uu_{x}v_{x}+u^{2}v\\Big)\\Big]\\Psi'_{i,K}\\,dx+\\int_{\\mathbb R}\\Big[2uP_{x}\\ast\\Big(\\frac{1}{2}u^{2}_{x}v_{x}\\Big)\\Big]\\Psi'_{i,K}\\,dx\\\\\n\\triangleq &-\\frac{c_1}{2}\\int_{\\mathbb R}\\left(u^{2}+u^{2}_{x}\\right)\\Psi'_{i,K}\\,dx+\\widetilde{I}_{1,1}+\\widetilde{I}_{1,2}+\\widetilde{I}_{1,3}.\n\\end{align*}\nWe claim that for the terms $\\tilde{I}_{1,i}$, $i=1,2,3$, it holds\n\\begin{align*}\n\\widetilde{I}_{1,i}\\leq \\frac{c_1}{20}\\int_{\\mathbb R}\\left(u^{2}+u^{2}_{x}\\right)\\Psi'_{i,K}\\,dx+\\frac{C}{K}\\|u_{0}\\|^{3}_{H^{1}(\\mathbb R)}\\|v_{0}\\|_{H^{1}(\\mathbb R)}e^{-\\frac{1}{K}(\\sigma_{0}t+\\frac{L}{8})}.\n\\end{align*}\nIndeed, we divide $\\mathbb R$ into two regions $D_{i}$ and $D^{c}_{i}$ with\n\\begin{align*}\nD_{i}=\\left[\\tilde{x}_{i-1}(t)+\\frac{L}{4},\\tilde{x}_{i}(t)-\\frac{L}{4}\\right],\\;i=2,...,N.\n\\end{align*}\nCombining \\eqref{initial-3-3} and \\eqref{initial-3-4}, one can check that for $x\\in D^c_{i}$,\n\\begin{eqnarray}\\label{Estimate}\n\\begin{aligned}\n\\left|x-y_{i}(t)\\right|\\geq \\frac{\\tilde{x}_{i}(t)-\\tilde{x}_{i-1}(t)}{2}-\\frac{L}{4}\\geq \\frac{c_{i}-c_{i-1}}{4}\\,t+\\frac{L}{8}\\geq \\sigma_{0}t+\\frac{L}{8}.\n\\end{aligned}\n\\end{eqnarray}\nLet us begin by an estimate of $\\widetilde{I}_{1,1}$. Using \\eqref{Estimate}, Sobolev imbedding and the exponential decay of $\\Psi'_{i,K}$ on $D^c_{i}$, we get\n\\begin{align*}\n\\widetilde{I}_{1,1}=&\\int_{D_{i}}uu^2_{x}v\\Psi'_{i,K}\\,dx+\\int_{D^c_{i}}uu^2_{x}v\\Psi'_{i,K}\\,dx\\\\\n\\leq &\\|u\\|_{L^\\infty(D_{i})}\\|v\\|_{L^\\infty(D_{i})}\\int_{D_{i}}\\left(u^2_{x}+u^2\\right)\\Psi'_{i,K}\\,dx+\\|\\Psi'_{i,K}\\|_{L^\\infty(D^{c}_{i})}\\|u\\|_{L^\\infty(\\mathbb R)}\\|v\\|_{L^\\infty(\\mathbb R)}\\|u\\|^2_{H^1}.\n\\end{align*}\nNow, using the exponential decay of $\\varphi_{c}$ and $\\psi_{c}$ on $D_{i}$, it holds\n\\begin{eqnarray}\\label{predict-1}\n\\begin{aligned}\n&\\|u\\|_{L^\\infty(D_{i})}\\leq \\Big\\|u-\\sum^{N}_{j=1}\\varphi(\\cdot-\\tilde{x}_{j}(t))\\Big\\|_{L^\\infty(D_{i})}+\\sum^{N}_{j=1}\\left\\|\\varphi(\\cdot-\\tilde{x}_{j}(t))\\right\\|_{L^\\infty(D_{i})}\\leq  O(\\sqrt{\\alpha})+O(e^{-\\frac{L}{8}}),\\\\\n& \\|v\\|_{L^\\infty(D_{i})}\\leq \\Big\\|v-\\sum^{N}_{j=1}\\psi(\\cdot-\\tilde{x}_{j}(t))\\Big\\|_{L^\\infty(D_{i})}+\\sum^{N}_{j=1}\\left\\|\\psi(\\cdot-\\tilde{x}_{j}(t))\\right\\|_{L^\\infty(D_{i})}\\leq  O(\\sqrt{\\alpha})+O(e^{-\\frac{L}{8}}).\n\\end{aligned}\n\\end{eqnarray}\nTherefore, for $0<\\alpha<\\alpha_{0}$ and $L>L_{0}>0$, with $\\alpha_{0}\\ll 1$ and $L_{0}\\gg 1$, we obtain\n\\begin{align*}\n\\widetilde{I}_{1,1} \\leq \\frac{c_1}{20}\\int_{\\mathbb R}\\left(u^{2}+u^{2}_{x}\\right)\\Psi'_{i,K}\\,dx+\\frac{C}{K}\\|u_{0}\\|^{3}_{H^{1}(\\mathbb R)}\\|v_{0}\\|_{H^{1}(\\mathbb R)}e^{-\\frac{1}{K}(\\sigma_{0}t+\\frac{L}{8})}.\n\\end{align*}\nBefore estimating the term $\\widetilde{I}_{1,2}$, using that $|\\Psi'''_{i,K}|\\leq 10K^{-2}\\Psi'_{i,K}$, we have\n\\begin{align*}\n\\left(1-\\partial^2_{x}\\right)\\Psi'_{i,K}(x)=\\Psi'_{i,K}(x)-\\frac{1}{K^2}\\Psi'''_{i,K}(x)\\geq \\left(1-\\frac{10}{K^2}\\right)\\Psi'_{i,K}(x),\\; \\forall x\\in \\mathbb R,\n\\end{align*}\nand since $K>4$, it holds\n\\begin{eqnarray}\\label{convolutionESI}\n\\begin{aligned}\n\\left(1-\\partial^2_{x}\\right)^{-1}\\Psi'_{i,K}(x)\\leq \\Big(1-\\frac{10}{K^2}\\Big)^{-1}\\Psi'_{i,K}(x),\\; \\forall x\\in \\mathbb R.\n\\end{aligned}\n\\end{eqnarray}\nNext, the estimate of $\\widetilde{I}_{1,2}$ gives us\n\\begin{align*}\n\\widetilde{I}_{1,2}=&\\int_{D_{i}}\\Big[2uP\\ast\\Big(\\frac{1}{2}u^{2}_{x}v+uu_{x}v_{x}+u^{2}v\\Big)\\Big]\\Psi'_{i,K}\\,dx+\\int_{D^c_{i}}\\Big[2uP\\ast\\Big(\\frac{1}{2}u^{2}_{x}v+uu_{x}v_{x}+u^{2}v\\Big)\\Big]\\Psi'_{i,K}\\,dx\\\\\n\\leq\\; & \\|u\\|_{L^\\infty(D_{i})}\\int_{\\mathbb R}P\\ast\\left|u^{2}_{x}v+2uu_{x}v_{x}+2u^{2}v\\right|\\Psi'_{i,K}\\,dx\\\\\n&\\quad +\\|\\Psi'_{i,K}\\|_{L^\\infty(D^c_{i})}\\|u\\|_{L^\\infty(\\mathbb R)}\\int_{\\mathbb R}\\left|u^{2}_{x}v+2uu_{x}v_{x}+2u^{2}v\\right|\\,dx\\\\\n\\leq\\; & 3\\|u\\|_{L^\\infty(D_{i})}\\|v\\|_{L^\\infty(D_{i})}\\int_{\\mathbb R}\\left|u^{2}_{x}+u^{2}\\right|\\Psi'_{i,K}\\,dx+\\frac{C}{K}\\|u_{0}\\|^{3}_{H^{1}(\\mathbb R)}\\|v_{0}\\|_{H^{1}(\\mathbb R)}e^{-\\frac{1}{K}(\\sigma_{0}t+\\frac{L}{8})}\\\\\n\\leq\\; & \\frac{c_1}{20}\\int_{\\mathbb R}\\left(u^{2}+u^{2}_{x}\\right)\\Psi'_{i,K}\\,dx+\\frac{C}{K}\\|u_{0}\\|^{3}_{H^{1}(\\mathbb R)}\\|v_{0}\\|_{H^{1}(\\mathbb R)}e^{-\\frac{1}{K}(\\sigma_{0}t+\\frac{L}{8})},\n\\end{align*}\nwhere the Young's inequality is used, the exponential decay of $\\Psi'_{i,K}$ on $D^c_{i}$, \\eqref{predict-1} and \\eqref{convolutionESI}.\nLet us tackle now the estimate of $\\widetilde{I}_{1,3}$. On $D^c_{i}$ we have\n\\begin{align*}\n&\\int_{D^c_{i}}\\left[uP_{x}\\ast(u^2_{x}v_{x})\\right]\\Psi'_{i,K}(t)\\,dx \\leq  \\|\\Psi'_{i,K}\\|_{L^\\infty(D^c_{i})}\\|v\\|_{L^\\infty(D^c_{i})}\\int_{\\mathbb R}u^2_{x}\\left[P\\ast u\\right]\\,dx\\\\\n&\\quad \\leq \\;  \\|\\Psi'_{i,K}\\|_{L^\\infty(D^c_{i})}\\|v\\|_{L^\\infty(D^c_{i})}\\|P\\ast u\\|_{L^\\infty(\\mathbb R)}\\int_{\\mathbb R}u^2_{x}\\,dx.\n\\end{align*}\nApplying the Holder inequality, we have for all $x\\in \\mathbb R$,\n\\begin{eqnarray}\\label{convolutionESI-1}\n\\begin{aligned}\nP\\ast u= \\frac{1}{2}\\int_{\\mathbb R}e^{-|x-y|}u(y)\\,dy\\leq \\frac{1}{2}\\left(\\int_{\\mathbb R}e^{-2|x-y|}\\,dy\\right)^{\\frac{1}{2}}\\left(\\int_{\\mathbb R}u^2(y)\\,dy\\right)^{\\frac{1}{2}}\\leq \\frac{1}{2}\\|u\\|_{L^2(\\mathbb R)},\n\\end{aligned}\n\\end{eqnarray}\nand then using \\eqref{convolutionESI-1} and the exponential decay of $\\Psi'_{i,K}$ on $D^c_{i}$, it holds\n\\begin{align*}\n\\int_{D^c_{i}}\\left[uP_{x}\\ast(u^2_{x}v_{x})\\right]\\Psi'_{i,K}(t)\\,dx\\leq \\frac{C}{K}\\|u_{0}\\|^{3}_{H^{1}(\\mathbb R)}\\|v_{0}\\|_{H^{1}(\\mathbb R)}e^{-\\frac{1}{K}(\\sigma_{0}t+\\frac{L}{8})}.\n\\end{align*}\nThe estimate of $\\widetilde{I}_{1,3}$ on $D_{i}$ leads to\n\\begin{align*}\n\\int_{D_{i}}\\left[uP_{x}\\ast(u^2_{x}v_{x})\\right]\\Psi'_{i,K}(t)\\,dx \\leq & \\|u\\|_{L^\\infty(D_{i})}\\|v\\|_{L^\\infty(D_{i})}\\int_{\\mathbb R}\\left[P\\ast\\Psi'_{i,K}(t)\\right]u^{2}_{x}\\,dx\\\\\n\\leq&  \\frac{c_1}{20}\\int_{\\mathbb R}\\left(u^{2}+u^{2}_{x}\\right)\\Psi'_{i,K}\\,dx.\n\\end{align*}\nTherefore, for $0<\\alpha<\\alpha_{0}$ and $L>L_{0}>0$, with $\\alpha_{0}\\ll 1$ and $L_{0}\\gg 1$, it holds\n\\begin{align*}\n\\frac{d}{dt}\\mathcal{J}^{u}_{j,K}(t)\\leq \\frac{C}{K}\\|u_{0}\\|^{3}_{H^{1}(\\mathbb R)}\\|v_{0}\\|_{H^{1}(\\mathbb R)}e^{-\\frac{1}{K}(\\sigma_{0}t+\\frac{L}{8})}.\n\\end{align*}\nIntegrating between $0$ and $t$, we obtain\n\\begin{align*}\n\\mathcal{J}^{u}_{j,K}(t)-\\mathcal{J}^{u}_{j,K}(0)\\leq O(e^{-\\frac{\\sigma_{0}L}{8K}}).\n\\end{align*}\nSimilarly, we also obtain\n\\begin{align*}\n\\mathcal{J}^{v}_{j,K}(t)-\\mathcal{J}^{v}_{j,K}(0)\\leq O(e^{-\\frac{\\sigma_{0}L}{8K}}).\n\\end{align*}\nNow, we need to prove the third inequality of this proposition. Applying the Virial type identity with $g=\\Psi_{i,K}$ and using \\eqref{infspeed}, we get\n\\begin{eqnarray*}\n\\begin{aligned}\n\\frac{d}{dt}\\mathcal{J}^{u,v}_{j,K}(t)\n=&\\int_{\\mathbb R}\\Big[uvu_{x}v_{x}+uP\\ast\\Big(\\frac{1}{2}v^{2}_{x}u+vv_{x}u_{x}+v^{2}u\\Big)+uP_{x}\\ast\\Big(\\frac{1}{2}v^{2}_{x}u_{x}\\Big)\\Big]\\Psi'_{i,K}\\,dx\\\\\n&\\;\\;+\\int_{\\mathbb R}\\Big[vP\\ast\\Big(\\frac{1}{2}u^{2}_{x}v+uu_{x}v_{x}+u^{2}v\\Big)+uP_{x}\\ast\\Big(\\frac{1}{2}u^{2}_{x}v_{x}\\Big)\\Big]\\Psi'_{i,K}\\,dx\\\\\n&\\;\\;-\\dot{y}_{i}\\int_{\\mathbb R}\\left(uv+u_{x}v_{x}\\right)\\Psi'_{i,K}\\,dx\\\\\n=&\\int_{\\mathbb R}\\left[uv(uv+u_{x}v_{x})+uP\\ast\\left(v(uv+u_{x}v_{x})\\right)+vP\\ast\\left(u(uv+u_{x}v_{x})\\right)\\right]\\Psi'_{i,K}\\,dx\\\\\n&\\;\\;+\\int_{\\mathbb R}\\Big[-\\frac{1}{2}u^2v^2+vP\\ast\\Big(\\frac{1}{2}u^2_{x}v\\Big)+vP_{x}\\ast\\Big(\\frac{1}{2}u^2_{x}v_{x}\\Big)\\Big]\\Psi'_{i,K}\\,dx\\\\\n&\\;\\;+\\int_{\\mathbb R}\\Big[-\\frac{1}{2}u^2v^2+uP\\ast\\Big(\\frac{1}{2}v^2_{x}u\\Big)+uP_{x}\\ast\\Big(\\frac{1}{2}v^2_{x}u_{x}\\Big)\\Big]\\Psi'_{i,K}\\,dx\\\\\n&\\;\\;-\\dot{y}_{i}\\int_{\\mathbb R}\\left(uv+u_{x}v_{x}\\right)\\Psi'_{i,K}\\,dx\\\\\n\\triangleq &\\widetilde{I}_{2,1}+\\widetilde{I}_{2,2}+\\widetilde{I}_{2,3}-\\dot{y}_{i}\\int_{\\mathbb R}\\left(uv+u_{x}v_{x}\\right)\\Psi'_{i,K}\\,dx.\n\\end{aligned}\n\\end{eqnarray*}\nIn the same way as the proof of the first inequality, we obtain\n\\begin{align*}\n\\widetilde{I}_{2,1} \\leq \\frac{c_1}{20}\\int_{\\mathbb R}\\left(uv+u_{x}v_{x}\\right)\\Psi'_{i,K}\\,dx+\\frac{C}{K}\\|u_{0}\\|^{2}_{H^{1}(\\mathbb R)}\\|v_{0}\\|^{2}_{H^{1}(\\mathbb R)}e^{-\\frac{1}{K}(\\sigma_{0}t+\\frac{L}{8})}.\n\\end{align*}\nFor this term $\\widetilde{I}_{2,2}$, we note\n\\begin{align*}\n|u_x|\\leq u \\quad {\\rm and} \\quad |v_x|\\leq v.\n\\end{align*}\nThus,\n\\begin{align*}\n(u+u_{x})(v+v_{x})\\geq 0\\quad {\\rm and} \\quad (u-u_{x})(v-v_{x})\\geq 0,\n\\end{align*}\nwhich is\n\\begin{eqnarray}\\label{oneESI}\n\\begin{aligned}\n|uv_{x}+u_{x}v| \\leq |uv+u_{x}v_{x}|.\n\\end{aligned}\n\\end{eqnarray}\nThanks to this estimate \\eqref{oneESI}, we have\n\\begin{align*}\n\\widetilde{I}_{2,2}=&\\frac{1}{2}\\int_{\\mathbb R}\\left[(P_{xx}-P)\\ast u^2v+P\\ast\\left(u^2_{x}v\\right)+P_{x}\\ast\\left(u^2_{x}v_{x}\\right)\\right]v\\Psi'_{i,K}\\,dx\\\\\n=&\\frac{1}{2}\\int_{\\mathbb R}\\left[P\\ast \\left((u^2_{x}-u^2)v\\right)+P_{x}\\ast(2uu_{x}v+u^2v_{x}+u^2_{x}v_{x})\\right]v\\Psi'_{i,K}\\,dx\\\\\n\\leq &\\frac{1}{2}\\int_{\\mathbb R}\\left[P_{x}\\ast\\left((uv+u_{x}v_{x})u_{x}\\right)+P_{x}\\ast\\left((uv_{x}+u_{x}v)u\\right)\\right]v\\Psi'_{i,K}\\,dx\\\\\n\\leq &\\frac{1}{2}\\int_{\\mathbb R}\\left[P_{x}\\ast\\left((uv+u_{x}v_{x})u_{x}\\right)+P\\ast\\left((uv+u_{x}v_{x})u\\right)\\right]v\\Psi'_{i,K}\\,dx.\n\\end{align*}\nIn the same way as proof the inequality of the first inequality, we find\n\\begin{align*}\n\\widetilde{I}_{2,2}\\leq \\frac{c_1}{20}\\int_{\\mathbb R}\\left(uv+u_{x}v_{x}\\right)\\Psi'_{i,K}\\,dx+\\frac{C}{K}\\|u_{0}\\|^{2}_{H^{1}(\\mathbb R)}\\|v_{0}\\|^{2}_{H^{1}(\\mathbb R)}e^{-\\frac{1}{K}(\\sigma_{0}t+\\frac{L}{8})}.\n\\end{align*}\nSimilarly, we have\n\\begin{align*}\n\\widetilde{I}_{2,3}\\leq \\frac{c_1}{20}\\int_{\\mathbb R}\\left(uv+u_{x}v_{x}\\right)\\Psi'_{i,K}\\,dx+\\frac{C}{K}\\|u_{0}\\|^{2}_{H^{1}(\\mathbb R)}\\|v_{0}\\|^{2}_{H^{1}(\\mathbb R)}e^{-\\frac{1}{K}(\\sigma_{0}t+\\frac{L}{8})}.\n\\end{align*}\nTherefore, for $0<\\alpha<\\alpha_{0}$ and $L>L_{0}>0$, with $\\alpha_{0}\\ll 1$ and $L_{0}\\gg 1$, it holds\n\\begin{align*}\n\\frac{d}{dt}\\mathcal{J}^{u,v}_{j,K}(t)\\leq \\frac{C}{K}\\|u_{0}\\|^{3}_{H^{1}(\\mathbb R)}\\|v_{0}\\|_{H^{1}(\\mathbb R)}e^{-\\frac{1}{K}(\\sigma_{0}t+\\frac{L}{8})}.\n\\end{align*}\nIntegrating between $0$ and $t$, we obtain\n\\begin{align*}\n\\mathcal{J}^{u,v}_{j,K}(t)-\\mathcal{J}^{u,v}_{j,K}(0)\\leq O(e^{-\\frac{\\sigma_{0}L}{8K}}).\n\\end{align*}\nThis completes the proof of this proposition.\n\\end{proof}\n\n\n\\subsection{A localized and a global estimate}\nWe define the function $\\Phi_{i}=\\Phi_{i}(t,x)$ by $\\Phi_{1}=1-\\Psi_{2,K}=1-\\Psi_{K}(\\cdot-y_{2}(t))$, $\\Phi_{N}=\\Psi_{N,K}=\\Psi_{K}(\\cdot-y_{N}(t))$ and for $i=2,...,N-1$,\n\\begin{eqnarray*}\n\\begin{aligned}\n\\Phi_{i}=\\Psi_{i,K}-\\Psi_{i+1,K}=\\Psi_{K}(\\cdot-y_{i}(t))-\\Psi_{K}(\\cdot-y_{i+1}(t)),\n\\end{aligned}\n\\end{eqnarray*}\nwhere $\\Psi_{K}$ and the $y_{i}$ are defined in the previous section. It is easy to check that $\\sum^{N}_{i=1}\\Phi_{i,K}\\equiv1$. We take $L>0$ and $L\/K>0$ large enough so that $\\Phi_{i}$ satisfies\n\\begin{eqnarray}\\label{weightesi-1}\n\\begin{aligned}\n\\left|1-\\Phi_{i,K}\\right|\\leq 4e^{-\\frac{L}{8K}} \\quad {\\rm on}\\quad \\Big[\\tilde{x}_{i}-\\frac{L}{4},\\tilde{x}_{i}+\\frac{L}{4}\\Big],\n\\end{aligned}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\\label{weightesi-2}\n\\begin{aligned}\n\\left|\\Phi_i\\right|\\leq 4e^{-\\frac{L}{8K}} \\quad {\\rm on}\\quad \\Big[\\tilde{x}_{j}-\\frac{L}{4},\\tilde{x}_{j}+\\frac{L}{4}\\Big]\\,\\; {\\rm whenever}\\;\\; j\\ne i.\n\\end{aligned}\n\\end{eqnarray}\nWe now use the following localized version of $E_{u}$, $E_{v}$, $H$ and $F$ defined for $i\\in \\{1,...,N\\}$, by\n\\begin{eqnarray}\\label{weightesi-3}\n\\begin{aligned}\n&E_{ui}(u)=\\int_{\\mathbb R}\\left(u^2+u^2_{x}\\right)\\Phi_{i}\\,dx, \\quad E_{vi}(v)=\\int_{\\mathbb R}\\left(v^2+v^2_{x}\\right)\\Phi_{i}\\,dx, \\\\\n& H_{i}(u,v)=\\int_{\\mathbb R}\\left(uv+u_{x}v_{x}\\right)\\Phi_{i}\\,dx\\\\\n{\\rm and}\\;\\; &F_{i}(u,v)=\\int_{\\mathbb R}\\Big(u^2v^2+\\frac{1}{3}u^2v_x^2+\\frac{1}{3}v^2u_x^2+\\frac{4}{3}uvu_xv_x-\\frac{1}{3}u_x^2v_x^2\\Big)\\Phi_{i}\\,dx.\n\\end{aligned}\n\\end{eqnarray}\nPlease note that henceforth we take $K=\\sqrt{L}\/8$.\n\nThe following lemma gives a localized version of \\eqref{functionalESI}. Note that the functionals $H_{i}$ and $F_{i}$ do not depend on time in the statement below since we fix $\\tilde{x}_{1}<...<\\tilde{x}_{N}$.\n\\begin{lemma}\\label{localizedESI}\nLet be given $N$ real numbers $\\tilde{x}_{1}<...<\\tilde{x}_{N}$ with $\\tilde{x}_{i}-\\tilde{x}_{i-1}\\geq 2L\/3$. Define the $J_{i}$ as in \\eqref{initial-3-4} and assume that, for $i=1,...,N$, there exists $x_{i}\\in J_{i}$ such that $|x_{i}-\\tilde{x}_{i}|\\leq L\/12$ and $u(x_{i})v(x_{i})=\\max_{x\\in J_{i}}uv:=M_{i}$. Then for any $(u,v)\\in H^{1}(\\mathbb R)\\times H^{1}(\\mathbb R)$, it holds\n\\begin{eqnarray}\\label{localizedESI-1}\n\\begin{aligned}\n\\frac{4}{3}M^2_{i}-\\frac{4}{3}M_{i}H_{i}(u,v)+F_{i}(u,v)\\leq O(L^{-\\frac{1}{2}}),\\quad i\\in\\{1,...,N\\}.\n\\end{aligned}\n\\end{eqnarray}\n\\end{lemma}\n\\begin{proof}\nLet $i\\in \\{1,...,N\\}$ be fixed. We introduce the functions $g_{u}$, $g_{v}$ and $h$ defined by\n\\begin{equation*}\ng_{u}(x)=\\left\\{\n\\begin{aligned}\n&u-u_{x},\\quad &x<x_{i},\\\\\n&u+u_{x},&x>x_{i},\n\\end{aligned}\n\\right.\n\\end{equation*}\n\\begin{equation*}\ng_{v}(x)=\\left\\{\n\\begin{aligned}\n&v-v_{x},\\quad &x<x_{i},\\\\\n&v+v_{x},&x>x_{i},\n\\end{aligned}\n\\right.\n\\end{equation*}\nand\\\\\n\\begin{equation*}\nh(x)=\\left\\{\n\\begin{aligned}\n&uv-\\frac13 (uv)_x-\\frac{1}{3}u_{x}v_{x},\\quad &x<x_{i},\\\\\n&uv+\\frac 13 (uv)_x+\\frac{1}{3}u_{x}v_{x},&x>x_{i}.\n\\end{aligned}\n\\right.\n\\end{equation*}\nIntegrating by parts we compute\n\\begin{eqnarray}\\label{localizedfuncESI-1}\n\\begin{aligned}\n&\\int_{\\mathbb R}h(x)g_{u}(x)g_{v}(x)\\Phi_{i}dx\\\\\n=&\\int_{-\\infty}^{x_i}{\\Big(uv-\\frac{1}{3}(uv)_x-\\frac{1}{3}u_{x}v_{x}\\Big)\\Big(uv-(uv)_x+u_{x}v_{x}\\Big)}\\Phi_{i}\\,dx\\\\\n&\\qquad +\\int_{x_i}^{\\infty}{\\Big(uv+\\frac 13 (uv)_x-\\frac{1}{3}u_{x}v_{x}\\Big)\\Big(uv+(uv)_x+u_xv_x\\Big)}\\Phi_{i}\\,dx\\\\\n=&\\int_{-\\infty}^{\\infty}{\\Big(u^2v^2+\\frac{1}{3}u^2v_x^2+\\frac{1}{3}v^2u_x^2+\\frac{4}{3}uvu_xv_x-\\frac{1}{3}u_x^2v_x^2\\Big)}\\,dx-\\frac{4}{3}\\,M_{i}^2\\Phi_{i}(x_{i})\\\\\n&\\qquad+\\frac{2}{3}\\int^{x_{i}}_{\\-\\infty}u^2v^2\\Phi'_{i}\\,dx-\\frac{2}{3}\\int^{\\-\\infty}_{x_{i}}u^2v^2\\Phi'_{i}\\,dx.\\\\\n\\end{aligned}\n\\end{eqnarray}\nRecall that we take $K=\\sqrt{L}\/8$ and thus $|\\Phi'|\\leq C\/K=O(L^{-1\/2})$. Moreover, since $|x_{i}-\\tilde{x}_{i}|\\leq L\/12$, it follows from \\eqref{weightesi-1} that $\\Phi_{i}(x_{i})=1+O(e^{-\\sqrt{L}})$ and thus\n\\begin{eqnarray}\\label{localizedfuncESI-2}\n\\begin{aligned}\n\\int_{\\mathbb R}h(x)g_{u}(x)g_{v}(x)\\Phi_{i}dx=&F_{i}(u,v)-\\frac{4}{3}M^2_{i}+\\|u_{0}\\|^2_{H^{1}(\\mathbb R)}\\|v_{0}\\|^2_{H^{1}(\\mathbb R)}O(L^{-\\frac{1}{2}})+O(e^{-\\sqrt{L}}).\n\\end{aligned}\n\\end{eqnarray}\nOn the other hand, we firstly claim that $x_{1}$ is the maximum point of the function $u(x,t)v(x,t)$ on $(-\\infty,y_{2}(t)+aL]$, $x_{N}$ is the maximum point of the function $u(x,t)v(x,t)$ on $[y_{N}(t)-aL,+\\infty)$ and $x_{i}$ is the maximum point of the function $u(x,t)v(x,t)$ on $[y_{i}(t)-aL,y_{i+1}(t)+aL]$, where $a$ is a constant chosen later and $i\\in \\{2,...,N-1\\}$.\n\nWe need to show  $x_{i}$ is the maximum point of the function $u(x,t)v(x,t)$ on $[y_{i}(t)-aL,y_{i}(t)]\\,\\cup\\, [y_{i+1}(t),y_{i+1}(t)+aL]$, where $i\\in \\{2,...,N-1\\}$.\nIf $x\\in [y_{i}(t)-aL,y_{i}(t)]\\,\\cup\\, [y_{i+1}(t),y_{i+1}(t)+aL]$, then\n\\begin{eqnarray*}\n\\begin{aligned}\nu(x,t)v(x,t)\\leq c_{i+1}e^{-\\left(\\frac{3}{8}-a\\right)L}+c_{i}e^{-\\frac{3}{8}L}+O(\\sqrt{\\alpha})+O(e^{-\\frac{L}{4}}).\n\\end{aligned}\n\\end{eqnarray*}\nChoosing $a$ with\n\\begin{eqnarray*}\n\\begin{aligned}\nc_{i+1}e^{-\\left(\\frac{3}{8}-a\\right)L}+c_{i}e^{-\\frac{3}{8}L}+O(\\sqrt{\\alpha})+O(e^{-\\frac{L}{4}})\\leq c_{i}-O(\\sqrt{\\alpha})-O(e^{-\\frac{L}{4}}),\n\\end{aligned}\n\\end{eqnarray*}\nand using the same estimate as above, we have proved similar conclusion of $x_{1}$ and $x_{N}$. \n\nNext, we define the intervals $\\hat{J}_{1}=(-\\infty,y_{2}(t)+aL]$, $\\hat{J}_{N}=[y_{N}(t)-aL,+\\infty)$ and for $i=2,...,N-1$, $\\hat{J}_{i}=[y_{i}(t)-aL,y_{i+1}(t)+aL]$, where $a$ is chosen above.\n\\begin{eqnarray}\n\\begin{aligned}\n\\int_{\\mathbb R}h(x)g_{u}(x)g_{v}(x)\\Phi_{i}dx=&\\int_{\\hat{J}_{i}}h(x)g_{u}(x)g_{v}(x)\\Phi_{i}dx+\\int_{\\hat{J}^c_{i}}h(x)g_{u}(x)g_{v}(x)\\Phi_{i}dx\\\\\n\\leq & \\frac{4}{3}M_{i}\\int_{\\mathbb R}g_{u}(x)g_{v}(x)\\Phi_{i}dx+\\int_{\\hat{J}^c_{i}}h(x)g_{u}(x)g_{v}(x)\\Phi_{i}dx,\n\\end{aligned}\n\\end{eqnarray}\nwhere we use the fact $h(x)\\leq 4u(x)v(x)\/3\\leq 4M_{i}\/3$ on the $\\hat{J}_{i}$. One chooses $L$ large enough satisfying $aL>10\\sqrt{L}$, which leads to\n\\begin{eqnarray*}\n\\begin{aligned}\n\\int_{\\hat{J}^c_{i}}h(x)g_{u}(x)g_{v}(x)\\Phi_{i}dx\\leq C\\|u_{0}\\|^2_{H^{1}(\\mathbb R)}\\|v_{0}\\|^2_{H^{1}(\\mathbb R)}e^{-\\tilde{a}L}\\leq O(L^{-1}),\n\\end{aligned}\n\\end{eqnarray*}\nwhere $C$ and $\\tilde{a}$ are constants. Therefore,\n\\begin{eqnarray*}\n\\begin{aligned}\n\\frac{4}{3}M^2_{i}-\\frac{4}{3}M_{i}H_{i}(u,v)+F_{i}(u,v)\\leq O(L^{-\\frac{1}{2}}),\\quad i\\in\\{1,...,N\\}.\n\\end{aligned}\n\\end{eqnarray*}\nThis completes the proof.\n\\end{proof}\nNow let us state a global identity related to \\eqref{energyESI}.\n\\begin{lemma}\\label{globalIDE}\nFor any $Z\\in \\mathbb R^N$ satisfying $|z_{i}-z_{i-1}|\\geq L\/2$ and any $(u,v)\\in H^{1}(\\mathbb R)\\times H^{1}(\\mathbb R)$, there holds\n\\begin{eqnarray*}\n\\begin{aligned}\n&E_{u}(u)-\\sum^{N}_{i=1}E_{u}(\\varphi_{c_{i}})=\\|u-R_{Z}\\|^2_{H^1}+4\\sum^N_{i=1}a_{i}(u(z_{i})-a_{i})+O\\left(e^{-\\frac{L}{4}}\\right),\\\\\n&E_{v}(v)-\\sum^{N}_{i=1}E_{v}(\\psi_{c_{i}})=\\|v-S_{Z}\\|^2_{H^1}+4\\sum^N_{i=1}b_{i}(v(z_{i})-b_{i})+O\\left(e^{-\\frac{L}{4}}\\right).\n\\end{aligned}\n\\end{eqnarray*}\n\\end{lemma}\n\\begin{proof}\nUsing the relation between $\\varphi$ and its derivative, integrating by parts, we get\n\\begin{eqnarray*}\n\\begin{aligned}\n&E_{u}(u-R_{Z})=E_{u}(u)+E_{u}(R_{Z})-2\\sum^{N}_{i=1}\\int_{\\mathbb R}u\\varphi_{c_{i}}(\\cdot-z_{i})+u_{x}\\partial_{x}\\varphi_{c_{i}}(\\cdot-z_{i})\\\\\n&=E_{u}(u)+E_{u}(R_{Z})-2\\sum^{N}_{i=1}\\int_{\\mathbb R}u\\varphi_{c_{i}}(\\cdot-z_{i})+2\\sum^{N}_{i=1}\\int^{\\infty}_{z_{i}}u_{x}\\varphi_{c_{i}}(\\cdot-z_{i})-2\\sum^{N}_{i=1}\\int^{z_{i}}_{-\\infty}u_{x}\\varphi_{c_{i}}(\\cdot-z_{i})\\\\\n&=E_{u}(u)+E_{u}(R_{Z})-4\\sum^{N}_{i=1}a_{i}u(z_{i}).\n\\end{aligned}\n\\end{eqnarray*}\nOn the other hand, since $|z_{i}-z_{i-1}|\\geq L\/2$, it is easy to check that\n\\begin{eqnarray*}\n\\begin{aligned}\nE(R_{Z})=\\sum^{N}_{i=1}E(\\varphi_{c_{i}})+O\\left(e^{-\\frac{L}{4}}\\right).\n\\end{aligned}\n\\end{eqnarray*}\nCombining these two identities, the desired result follows, and similarly, we obtain the second equality of this lemma.\n\\end{proof}\n\n\n\n\n\n\\subsection{End of the proof of Theorem 1.2}\n\n\nBefore we give the final proof, we need to prove the following lemmas.\n\\begin{lemma}\\label{FinalESI}\nAssume $\\|u_{0}-R_{Z^{0}}\\|_{H^{1}}+\\|v_{0}-S_{Z^{0}}\\|_{H^{1}}< \\epsilon $. Then for any $i=\\{1,...,N\\}$, we have\n\\begin{eqnarray*}\n\\begin{aligned}\n&\\left|H_{i}(u_{0},v_{0})-H_{i}(\\varphi_{c_{i}},\\psi_{c_{i}})\\right|<O(\\epsilon)+O\\left(e^{-\\sqrt{L}}\\right), \\\\\n&\\left|E_{ui}(u_{0})-E_{ui}(\\varphi_{c_{i}})\\right|<O(\\epsilon)+O\\left(e^{-\\sqrt{L}}\\right),\\\\\n&\\left|E_{vi}(v_{0})-E_{vi}(\\psi_{c_{i}})\\right|<O(\\epsilon)+O\\left(e^{-\\sqrt{L}}\\right).\n\\end{aligned}\n\\end{eqnarray*}\n\\end{lemma}\n\\begin{proof}\nDefine the interval $\\bar{J}_{i}=\\left[z^0_{i}-L\/4, \\,z^0_{i}+L\/4\\right]$. Using \\eqref{weightesi-1}, \\eqref{weightesi-2} and the exponential decay of $\\varphi_{c_{i}}$'s and the $\\psi_{c_{i}}$'s, we have\n\\begin{eqnarray*}\n\\begin{aligned}\n&\\left|H_{i}(\\varphi_{c_{i}},\\psi_{c_{i}})-H_{i}(u_{0},v_{0})\\right|=\\left|\\int_{\\mathbb R}\\varphi_{c_{i}}\\psi_{c_{i}}+\\varphi'_{c_{i}}\\psi'_{c_{i}}\\,dx-\\int_{\\mathbb R}\\left(u_{0}v_{0}+u_{0x}v_{0x}\\right)\\Phi_{i}\\,dx\\right|\\\\\n\\leq &\\left|\\int_{\\bar{J}_{i}}\\varphi_{c_{i}}\\psi_{c_{i}}+\\varphi'_{c_{i}}\\psi'_{c_{i}}\\,dx-\\int_{\\bar{J}_{i}}\\left(u_{0}v_{0}+u_{0x}v_{0x}\\right)\\,dx\\right|+O\\left(e^{-\\sqrt{L}}\\right)+O\\left(e^{-\\frac{L}{8}}\\right).\n\\end{aligned}\n\\end{eqnarray*}\nSince $\\|u_{0}-R_{Z}\\|_{H^1}+\\|v_{0}-S_{Z}\\|_{H^1}<\\epsilon$, then we obtain\n\\begin{eqnarray*}\n\\begin{aligned}\n&\\|u_{0}-\\varphi_{c_{i}}\\|_{H^1(\\bar{J}_{i})}+\\|v_{0}-\\psi_{c_{i}}\\|_{H^1(\\bar{J}_{i})}\\\\\n&\\leq |u_{0}-R_{Z}\\|_{H^1(\\bar{J}_{i})}+\\|v_{0}-S_{Z}\\|_{H^1(\\bar{J}_{i})}+\\sum_{i\\ne j}\\left(\\|\\varphi_{c_{j}}\\|_{H^1(\\bar{J}_{i})}+\\|\\psi_{c_{j}}\\|_{H^1(\\bar{J}_{i})}\\right)\\\\\n&\\leq |u_{0}-R_{Z}\\|_{H^1(\\mathbb R)}+\\|v_{0}-S_{Z}\\|_{H^1(\\mathbb R)}+O\\left(e^{-\\frac{L}{8}}\\right)<  \\epsilon +O\\left(e^{-\\frac{L}{8}}\\right).\n\\end{aligned}\n\\end{eqnarray*}\nUsing the same way in proving the third inequality of \\eqref{lem1.3}, we obtain\n\\begin{eqnarray*}\n\\begin{aligned}\n\\left|H_{i}(\\varphi_{c_{i}},\\psi_{c_{i}})-H_{i}(u_{0},v_{0})\\right|<O(\\epsilon)+O\\left(e^{-\\sqrt{L}}\\right).\n\\end{aligned}\n\\end{eqnarray*}\nSimilarly,\n\\begin{eqnarray*}\n\\left|E_{ui}(\\varphi_{c_{i}})-E_{ui}(u_{0})\\right|<O(\\epsilon)+O\\left(e^{-\\sqrt{L}}\\right)\\; {\\rm and} \\; \\left|E_{vi}(\\psi_{c_{i}})-E_{vi}(v_{0})\\right|<O(\\epsilon)+O\\left(e^{-\\sqrt{L}}\\right).\n\\end{eqnarray*}\n\\end{proof}\n\\begin{lemma}\\label{productESI}\nAssume $u\\in H^s$,$v\\in H^s$, $s>3$ and $u\\geq 0$, $v\\geq 0$. Let\n\\begin{align*}\nM_{i}=u(x_{i})v(x_{i})=\\max_{x\\in J_{i}}u(x)v(x),\n\\end{align*}\nand set $\\alpha_{0}=A({\\epsilon_0}^{1\/4}+L_{0}^{-1\/8})$. Then we have\n\\begin{align*}\n|M_{i}-c_{i}|\\leq O\\big(L^{-\\frac{1}{4}}\\big)+O\\big(\\epsilon^{\\frac{1}{2}}\\big),\\quad  i=1,...,N.\n\\end{align*}\n\\end{lemma}\n\\begin{proof}\nDefine the second-order polynomials $\\bar{P}^{i}$ and $\\hat{P}^{i}$ by:\n\\begin{align*}\n\\bar{P}^{i}(y)=y^2-H_{i}(u,v)y+\\frac{3}{4}F_{i}(u,v) \\;\\; \\mathrm{and} \\;\\; \\hat{P}^{i}(y)=y^2-H_{i}(u_0,v_0)y+\\frac{3}{4}F_{i}(u_{0},v_{0}).\n\\end{align*}\nFor the peaked solution, $H(\\varphi_{c_{i}},\\psi_{c_{i}})=2c_{i}$ and $F(\\varphi_{c_{i}},\\psi_{c_{i}})=4c^2_{i}\/3$, the above polynomial becomes\n\\begin{eqnarray}\\label{finalESI-1}\n\\begin{aligned}\n&\\hat{P}^{i}_{0}(y)=y^2-2c_{i}y+c^2_{i}=(y-c_{i})^2.\n\\end{aligned}\n\\end{eqnarray}\nSince\n\\begin{align*}\n\\bar{P}^{i}(y)=&\\hat{P}^{i}_{0}(y)+\\left(H(\\varphi_{c_{i}},\\psi_{c_{i}})-H_{i}(u_0,v_0)\\right)y+\\left(H_{i}(u_0,v_0)-H_{i}(u,v)\\right)y\\\\\n&\\qquad +\\frac{3}{4}\\left[\\left(F_{i}(u_{0},v_{0})-F(\\varphi_{c_{i}},\\psi_{c_{i}})\\right)+\\left(F_{i}(u,v)-F_{i}(u_{0},v_{0})\\right)\\right],\n\\end{align*}\nit follows that\n\\begin{eqnarray*}\n\\begin{aligned}\n\\sum^{N}_{i=1}\\bar{P}^{i}&(M_{i})=\\sum^{N}_{i=1}\\hat{P}^{i}_{0}(M_{i})+\\sum^{N}_{i=1}M_{i}\\left(H(\\varphi_{c_{i}},\\psi_{c_{i}})-H_{i}(u_0,v_0)\\right)\\\\\n&+\\frac{4}{3}\\Big(F(u_{0},v_{0})-\\sum^{N}_{i=1}F(\\varphi_{c_{i}},\\psi_{c_{i}})\\Big)+\\sum^N_{i=1}M_{i}\\Big((H_{i}(u_0,v_0)-H_{i}(u,v)\\Big)\\leq O(L^{-\\frac{1}{2}}).\n\\end{aligned}\n\\end{eqnarray*}\nUsing the Abel transformation of the last term and the monotonicity property and combining Lemma \\eqref{FinalESI}, we thus get\n\\begin{align*}\n\\sum^{N}_{i=1}(M_{i}-c_{i})^2\\leq &\\sum^{N}_{i=1}M_{i}\\left|H(\\varphi_{c_{i}},\\psi_{c_{i}})-H_{i}(u_0,v_0)\\right|+\\frac{3}{4}\\Big|F(u_{0},v_{0})-F(R_{Z},S_{Z})\\Big|+O\\big(L^{-\\frac{1}{2}}\\big)+O(\\epsilon)\\\\\n\\leq & O\\big(L^{-\\frac{1}{2}}\\big)+O(\\epsilon),\n\\end{align*}\nwhich completes the proof of the lemma.\n\\end{proof}\n\n\n\\begin{lemma}\\label{FLGJ}\nAssume $|M_{i}-c_{i}|\\leq O\\left(L^{-1\/4}\\right)+O\\left(\\epsilon^{1\/2}\\right),\\quad i=1,...,N$. Then for any $i=\\{1,...,N\\}$, we have\n\\begin{align*}\n|u(x_{i})-a_{i}|\\leq O\\left(L^{-1\/4}\\right)+O\\Big(\\epsilon^{1\/2}\\Big)\\quad {\\rm and} \\quad |v(x_{i})-b_{i}|\\leq O\\left(L^{-1\/4}\\right)+O\\left(\\epsilon^{1\/2}\\right).\n\\end{align*}\n\\end{lemma}\nNote the monotonicity property that $\\mathcal{J}^{u}_{j,K}(t)$ is close to $\\left\\|u(t)\\right\\|_{H^{1}(x>y_{j}(t))}$ and thus measures the energy at the right of the $(j-1)th$ bump of $u$ and  $\\mathcal{J}^{v}_{j,K}(t)$ is close to $\\left\\|v(t)\\right\\|_{H^{1}(x>y_{j}(t))}$ and thus measures the energy at the right of the $(j-1)th$ bump of $v$. We thus use the induction argument to prove this lemma.\n\\begin{proof}\n{Step 1.}\nDefine\n\\begin{equation}\ng_{uN}(x)=\n\\left\\{\n\\begin{aligned}\n&u(x)-u_{x}(x), \\quad x<x_N,\\\\\n&u(x)+u_{x}(x), \\quad x>x_N.\n \\end{aligned}\n\\right.\n\\end{equation}\nThen we have\n\\begin{eqnarray*}\n\\begin{aligned}\n0\\leq \\int_{\\mathbb R}g^2_{uN}(x)\\Phi_{N}(x)\\,dx=&\\int_{\\mathbb R}\\left(u^2+u^2_{x}\\right)\\Phi_{N}(x)\\,dx-2u^{2}(x_{N})\\Phi_{N}(x_N)\\\\\n&+\\int^{x_{N}}_{-\\infty}u^2\\Phi'_{N}\\,dx-\\int^{+\\infty}_{x_{N}}u^2\\Phi'_{N}\\,dx.\n\\end{aligned}\n\\end{eqnarray*}\nUsing $|x_{i}-\\tilde{x}_{i}|\\leq L\/12$, it follows from \\eqref{weightesi-1} that $\\Phi_{i}(x_{i})=1+O(e^{-\\sqrt{L}})$ and thus\n\\begin{align*}\nE_{uN}(u(t))-2u^2(x_{N})+O\\left(L^{-\\frac{1}{2}}\\right)\\geq 0.\n\\end{align*}\nNote that\n\\begin{eqnarray*}\nE_{uN}(u(t))=\\mathcal{J}^u_{N,K}(t)\\quad {\\rm and} \\quad E_{uN}(u(t))-E_{uN}(u(0))\\leq O\\big(e^{-\\sigma_{0}\\sqrt{L}}\\big).\n\\end{eqnarray*}\nWe thus get\n\\begin{eqnarray*}\n\\begin{aligned}\nu^2(x_{N})\\leq O(L^{-\\frac{1}{2}})+O(\\epsilon)+a^2_{N}.\n\\end{aligned}\n\\end{eqnarray*}\nSimilarly,\n\\begin{eqnarray*}\n\\begin{aligned}\nv^2(x_{N})\\leq O(L^{-\\frac{1}{2}})+O(\\epsilon)+b^2_{N}.\n\\end{aligned}\n\\end{eqnarray*}\nCombining the identity $|M_{N}-c_{N}|\\leq O\\left(L^{-1\/4}\\right)+O\\left(\\epsilon^{1\/2}\\right)$, we arrive at\n\\begin{eqnarray*}\n\\begin{aligned}\n|u(x_{N})-a_{N}|\\leq O\\big(L^{-\\frac{1}{4}}\\big)+O\\big(\\epsilon^{\\frac{1}{2}}\\big) \\quad {\\rm and} \\quad |v(x_{N})-b_{N}|\\leq O\\big(L^{-\\frac{1}{4}}\\big)+O\\big(\\epsilon^{\\frac{1}{2}}\\big).\n\\end{aligned}\n\\end{eqnarray*}\n{Step 2.}  Assume for any $k\\leq i<N$, we have\n\\begin{align*}\n|u(x_{i})-a_{i}|\\leq O\\big(L^{-\\frac{1}{4}}\\big)+O\\big(\\epsilon^{\\frac{1}{2}}\\big) \\quad {\\rm and} \\quad |v(x_{i})-b_{i}|\\leq O\\big(L^{-\\frac{1}{4}}\\big)+O\\big(\\epsilon^{\\frac{1}{2}}\\big).\n\\end{align*}\nWe need to prove\n\\begin{align*}\n|u(x_{i-1})-a_{i-1}|\\leq O\\big(L^{-\\frac{1}{4}}\\big)+O\\big(\\epsilon^{\\frac{1}{2}}\\big) \\quad {\\rm and} \\quad |v(x_{i-1})-b_{i-1}|\\leq O\\big(L^{-\\frac{1}{4}}\\big)+O\\big(\\epsilon^{\\frac{1}{2}}\\big).\n\\end{align*}\nFirstly, we have\n\\begin{align*}\n\\sum^{N}_{i=k-1}u^2(x_{i})\\leq &\\;\\frac{1}{2}\\sum^{N}_{i=k-1}E_{ui}(u(t))+O(L^{-\\frac{1}{2}})\\\\\n\\leq &\\; \\frac{1}{2}\\left(\\mathcal{J}^u_{k-1,K}(t)-\\mathcal{J}^u_{k-1,K}(0)\\right)+\\frac{1}{2}\\Big(\\mathcal{J}^u_{k-1,K}(0)-\\sum^{N}_{i=k-1}E_{u}(\\varphi_{c_{i}})\\Big)\\\\\n&\\quad + \\frac{1}{2}\\sum^{N}_{i=k-1}E_{u}(\\varphi_{c_{i}})+O\\Big(L^{-\\frac{1}{2}}\\Big).\n\\end{align*}\nIt follows that\n\\begin{align*}\n\\sum^{N}_{i=k-1}u^2(x_{i})\\leq \\frac{1}{2}\\sum^{N}_{i=k-1}E_{u}(\\varphi_{c_{i}})+O\\Big(L^{-\\frac{1}{2}}\\Big)+O(\\epsilon).\n\\end{align*}\nUsing the assumption, we find\n\\begin{align*}\nu^2(x_{k-1})\\leq a^2_{k-1}+O\\big(L^{-\\frac{1}{4}}\\big)+O\\left(\\sqrt{\\epsilon}\\right).\n\\end{align*}\nSimilarly\n\\begin{align*}\nv^2(x_{k-1})\\leq b^2_{k-1}+O\\big(L^{-\\frac{1}{4}}\\big)+O\\left(\\sqrt{\\epsilon}\\right).\n\\end{align*}\nThose together with the identity \\eqref{productESI} imply that for any $i=1,\\cdots ,N$\n\\begin{align*}\n|u(x_{i})-a_{i}|\\leq O\\big(L^{-\\frac{1}{4}}\\big)+O\\big(\\epsilon^{\\frac{1}{2}}\\big) \\quad {\\rm and} \\quad |v(x_{i})-b_{i}|\\leq O\\big(L^{-\\frac{1}{4}}\\big)+O\\big(\\epsilon^{\\frac{1}{2}}\\big). \n\\end{align*}\n\\end{proof}\n\n\\begin{proof} [End of the Proof of Theorem 1.2]\nTo conclude the proof, from the \\eqref{globalIDE}, it thus suffices to prove that there exists $C>0$ which does not depend on $A$ such that\n\\begin{eqnarray*}\n\\begin{aligned}\n|u(x_{N})-a_{N}|\\leq C\\Big(L^{-\\frac{1}{4}}+\\epsilon^{\\frac{1}{2}}\\Big) \\quad {\\rm and} \\quad |v(x_{N})-b_{N}|\\leq C\\Big(L^{-\\frac{1}{4}}+\\epsilon^{\\frac{1}{2}}\\Big),\n\\end{aligned}\n\\end{eqnarray*}\nwhich has been verified by \\eqref{FLGJ}.\nFrom \\eqref{initial-3-3} and \\eqref{initial-3-5}, we know that for $i=2,...,N$,\n\\begin{eqnarray*}\n\\begin{aligned}\nx_{i}-x_{i-1}\\geq \\frac{2}{3}L.\n\\end{aligned}\n\\end{eqnarray*}\nThis completes the proof of the theorem.\n\\end{proof}\n\n\n\\vglue .5cm\n\n\\vskip 0.2cm\n\\noindent {\\bf Acknowledgments.} The work of He is partially supported by the NSF-China grant-11971251.  The work of Liu is partially supported by the NSF-China grant-11722111 and  grant-11871395. The work of Qu is partially supported by the NSF-China grant-11631007 and grant-11971251.\n\n\n\n\\vskip 1cm\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nProtostellar jets appear intimately linked to the process of mass accretion onto the growing star; their strikingly similar properties across protostellar age,  mass, and accretion rate all point to universal ejection and collimation mechanisms\n\\citep{2002EAS.....3..147C,2007A&A...468L..29C, 2013A&A...551A...5E}. Yet, jets from the youngest protostars --- so-called Class 0 --- are much brighter in molecules \\citep[\\BT{e.g.,}][]{2000A&A...359..967T} than jets from more evolved protostars and pre-main sequence stars which are mainly atomic; Molecules have been traced as close as 20-100 au from the source \\citep[\\BT{e.g.,}][]{2017NatAs...1E.152L,2014ApJ...794..169H}. \nThe origin of this selective molecular richness remains an important issue for models of the jet origin.\nThree broad scenarios have been considered, with no fully validated answer so far. \n\nIn models of ejection from the stellar magnetosphere or the inner disk edge \\citep[\\BT{e.g.,}][]{1994ApJ...429..781S,2002ApJ...578..420R,2005ApJ...632L.135M,2013A&A...550A..99Z}, the jet\nwould be expected to be dust-free (the grain sublimation radius around a typical solar-mass protostar is $R_{\\rm sub}\\sim 0.3$~au,  see \\BT{for example} \\citet{2016A&A...585A..74Y}).\nThe lack of dust screening then makes the wind extremely sensitive to photodissociation by the accretion shock.\nChemical models of dust-free winds by \\citet{1991ApJ...373..254G} found that CO, \\BT{SiO,} and H$_2$O could no longer form at the wind base in the presence of a typical expected level of FUV excess\\footnote{the flat UV flux in their UV1-UV2 models is $\\sim$ 30-500 that in BP Tau \\citep{2003ApJ...591L.159B}, for a wind-mass flux corresponding to a 1000 times larger accretion rate ($\\sim 3 \\times 10^{-5} M_\\odot$ yr$^{-1}$)}. \\citet{2005RMxAA..41..137R} showed that H$_2$ could form further out behind internal shocks. However, the key ions involved are also easily destroyed by FUV photons. \nHence, molecule formation in a dust-free jet within 20-100 au of protostars remains an open issue. \n\n\nA second proposed explanation \nis that the molecular component of jets may be tracing dusty MHD disk winds launched beyond $R_{\\rm sub}$, where dust can shield molecules against the FUV field and allow faster H$_2$ reformation. Detailed models are successful at reproducing the higher molecule richness of Class 0 jets \\citep{2012A&A...538A...2P}\nthe broad water line components revealed by \\textit{Herschel}\/HIFI  \\citep{2016A&A...585A..74Y}, and the rotation signatures \\BT{recently resolved by ALMA in the HH212 jet and in the slow wider angle wind surrounding it} \\citep{2017NatAs...1E.152L, 2017A&A...607L...6T}. However, the same disk wind models predict that the fastest, SiO-rich streamlines in  HH212 (flowing at $\\sim 100$ km~s$^{-1}$)  would be launched\nfrom 0.05-0.2 au, within the dust sublimation radius \\citep{2017A&A...607L...6T}. Hence, this scenario still partly faces the unsolved question of molecule survival in a dust-free wind.\n\nA third scenario is that molecules could be somehow \"entrained\" from the surroundings into the jet, assumed initially atomic. In a time-dependent jet, travelling internal shocks will squeeze out high-pressure jet material, which then sweeps up the surrounding gas into a curved bowshock. If the surrounding material is molecular, a partly molecular bowshock will result, with a more tenuous \"wake\" of shocked molecular gas trailing behind it \\citep{1993A&A...278..267R,2005RMxAA..41..137R}. As the next \"internal working surface\" (IWS) propagates into this wake, it may again produce a molecular jet bowshock. However,  after the passage of many such IWS, the wake will be so shock-processed and tenuous that not enough molecules may be left to produce molecular bowshocks close to the jet axis.\n\nIn the present paper, we revisit this last scenario in a new light by investigating whether a slower molecular \"disk wind\" surrounding the jet could help refill the wake and re-inject\nfresh (unprocessed) molecules into the jet path. This new outlook is prompted by the discovery of a potential molecular disk wind wrapped around the dense axial jet in HH212 \\citep{2017A&A...607L...6T}. We explore this possibility by studying analytically and numerically the propagation of bow shocks driven by a time-variable, inner jet into a surrounding slower disk wind. This scenario may be seen as an extension of the recent modeling work of \\citet{2016MNRAS.455.2042W} who studied the turbulent mixing layer between a jet and disk wind, with the novel addition of internal working surfaces in the jet to produce a stronger coupling between the two outflow components. \n\nBesides our main goal of exploring the impact of a DW on the chemical richness of Class 0 jets, our study has two other important motivations. \\BT{First, we aim to identify specific signatures in the morphology and kinematics of jet bowshocks that could reveal the presence of a surrounding DW. Secondly, we aim to\nidentify in which regions of space the pristine DW material would remain unperturbed, for comparison to theoretical DW models.}  \n\n\nIn the present exploratory study, \\BT{we have limited} ourselves to purely hydrodynamical and cylindrical flows, which allow us to develop an analytical model that greatly helps to capture the main effects of the two-flow interaction, and to understand the numerical results. Also, this is expected to be an optimal case for interaction between the two flows, as magnetic tension would tend to oppose mixing. \n\nThe paper is organized as follows.\nIn \\BT{Section 2}, we build an analytical model (in the thin shell approximation) for the propagation of a bow shock driven by an IWS into\na surrounding disk wind. The model is extended to the case of two or\nmore successive IWS in \\BT{Section 3}. In \\BT{Section 4} we compare the analytic model with axisymmetric\nsimulations of a variable jet+surrounding disk wind configuration, and compare the results with a\n\"reference simulation\" in which the same variable jet propagates into a stationary environment.\nFinally, the results are discussed in \\BT{Section 5}.\n\n\\section{Analytical approach}\n\\subsection{Basic equations}\n\nWe \\BT{considered} the \"disk wind+jet\" configuration shown in \\BT{Fig.}~\\ref{fig:shockframe}.\nwhere a \ncylindrical jet of radius $r_j$ and time-variable velocity v$_j$ directed along the $z$-axis\nis immersed in a plane-parallel \"disk wind''\nwith uniform density $\\rho_w$ and time-independent velocity v$_w$ parallel to v$_j$.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=8cm]{fig2-eps-converted-to.pdf}\n\\caption{Schematic diagram of the flow around an internal working surface (IWS)\nin the frame of reference co-moving with the IWS\nat velocity v$_z$ = v$_j$ (a similar\nconfiguration would apply for the leading working surface of a jet).\nThe thick, horizontal line at the bottom of the graph is the jet\n(with a gap showing the position of the IWS).\nThe working surface ejects jet material sideways at an initial velocity\nv$_0$ into the slower disk wind, which in this frame of reference moves\ntowards the outflow source at velocity v$_j-$v$_w$. The distance $x$\nis measured towards the outflow source\nThe shape of the thin shell bow shock \nis given by $r_b(x)$ (see Equ.~\\ref{rx}), and terminates \nat the cylindrical radius $r_{b,f}(t)$ with $t$ the time elapsed since formation of the IWS\n(see Equ.~\\ref{rf}).\n}\n\\label{fig:shockframe}\n\\end{figure}\n\n\\begin{center}\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[width=12cm]{fig1-new2-eps-converted-to.pdf}\n\\caption{Schematic diagram showing the flow around a working surface of a jet\n(in this case the\nleading working surface, but the diagram also applies for an internal\nworking surface). \nThe jet is the horizontal, red rectangle at the bottom of the graph, with the source located at $z$=0. \nThe working surface in yellow is located at a distance $z_j$ from the source and travels\nat a velocity v$_j$.  It ejects material away from the axis at an initial velocity v$_0$. The jet is\nsurrounded by a ``disk wind'', which travels along the outflow\naxis at a velocity v$_w$.  The shape of the thin-shell bow shock  (thick cyan line) is given by\n$z_b$ as a function of $r$ and ends at the edge of the bow wing (cyan point) (z$_{bf}$,r$_{bf}$). \nThe bow shock leaves behind a ``cavity'' (black region), which is partially refilled by the disk wind (brown region). \nThe boundaries of the initial swept up cavity (in black dashed line) \nand of the refilled region (cyan dash-dotted line) are given by $z_f$ an $z_c$ (respectively) as\na function of cylindrical radius $r$ (see \\BT{Eqs.}~\\ref{zfrf2} and \\ref{zc}).\n}\n\\label{fig:restframe}\n\\end{figure*}\n\\end{center}\n\nThe jet velocity variation is such that\nan internal working surface is produced within the jet beam. \nIn the following derivations, we assume\nthat the working surface is formed at $t=0$ at the position of the \nsource (\\BT{i.e.,} $z=0$),\nand that it then travels at a constant\nvelocity v$_j$ (for $t>0$). Such a working surface could be produced,\n\\BT{for example}, by an outflow velocity with a constant value v$_1<$v$_j$ for $t<0$,\njumping to a constant value v$_2>$v$_j$ for $t\\geq 0$.\nNote that if the shock is produced at distance $z_s>0$ at a time $t_s >0$, \nthe equations below remain valid with the transformation $z\\rightarrow z-z_s$ and $t\\rightarrow t-t_s$. \n\nIn a frame of reference moving with the internal working surface  \\SC{(see \\BT{Fig.}~\\ref{fig:shockframe})}\nthe over-pressured shocked jet material which is ejected sideways from the\nworking surface interacts with the slower moving, surrounding disk wind.\nIn the strong radiative cooling limit, this sideways ejection leads to\nthe formation of a thin-shell bow shock, which sweeps up material of the\nsurrounding disk wind, \nflowing towards the source at a relative velocity (v$_j$-v$_w$).\n\nAssuming full mixing between jet and disk-wind material, we can write the mass, $r$-\nand $x$-momentum conservation equations \nat any point of radius $r_b$ along this thin-shell ($r$,  $x$ and $r_b$ being defined in \\BT{Fig.}~\\ref{fig:shockframe})\nas:\n\\begin{equation}\n{\\dot m}={\\dot m}_0+\\int_{r_j}^{r_b} 2\\pi r'\\rho_w(\\textnormal{v}_j-\\textnormal{v}_w)dr'\\,,\n\\label{mass}\n\\end{equation}\n\\begin{equation}\n{\\dot \\Pi}_r={\\dot m}_0 \\textnormal{v}_0={\\dot m}\\textnormal{v}_r\\,,\n\\label{rmom}\n\\end{equation}\n\\begin{equation}\n{\\dot \\Pi}_x=\\int_{r_j}^{r_b} 2\\pi r'\\rho_w(\\textnormal{v}_j-\\textnormal{v}_w)^2dr'={\\dot m} \\textnormal{v}_x\\,,\n\\label{xmom}\n\\end{equation}\nwhere\n${\\dot m}$ is the mass rate, ${\\dot \\Pi}_r$ the $r$-momentum rate and\n${\\dot \\Pi}_x$ the $x$-momentum rate of the \\SC{mixed jet+disk-wind} material flowing along the thin-shell\nbow shock, and ${\\dot m}_0$ and $v_0$ are the mass rate and velocity (respectively)\n\\SC{at which jet material is initially} ejected sideways by the working surface. These equations have\nstraightforward interpretations. As an illustration, we point out that \\BT{Eq.}~(\\ref{rmom})\nstates that the radial momentum of the material flowing along the thin shell remains constant over time (which is due to the fact that the disk wind material adds no $r$-momentum), so that\nits radial velocity v$_r$ decreases as \\SC{${\\dot m}$ increases,} the $r$-momentum being shared with \\SC{a larger amount of}\nzero $r$-momentum material from the disk wind.\n\n\nThe mass rate ${\\dot m}_0$ and velocity v$_0$ of the sideways\nejected material are determined \\SC{only} by the properties of the working surface. For\na highly radiative working surface, we would expect the \\SC{post-shock jet}\nmaterial to cool to $\\sim 10^4$~K before exiting sonically into the disk\nwind. Therefore, we would expect v$_0\\sim 10$~km~s$^{-1}$.\nThe mass rate  ${\\dot m}_0$\nwill have values of the order of the (time-dependent) mass loss rate ${\\dot M}_j$\n\\SC{in} the jet beam.\n\n\\SC{We note that although our basic equations are similar to those of  \\citet{2001ApJ...557..443O} our approaches and derived equations will differ.\nThey considered only the case of the leading jet bowshock propagating in a medium at rest (v$_w = 0$), so that\nthe injected mass and momentum rates ${\\dot m}_0$ and ${\\dot m}_0$v$_0$ were\nexpressed as a function of the velocity of the shock and the jet radius. Here, \nwe keep ${\\dot m}_0$ and v$_0$ as explicit parameters, so that we can consider a moving surrounding disk wind of arbitrary velocity v$_w$,\nand an arbitrarily small $r_j$}.\n\n\\subsection{Shape of the bow shock shell}\n\nFor a disk-wind with position-independent density $\\rho_w$ and velocity v$_w$, the\nintegrals in \\BT{Eqs.}~(\\ref{mass}-\\ref{xmom}) can be trivially performed, and from\nthe ratio of \\BT{Eqs.}~(\\ref{rmom}-\\ref{xmom}) one obtains the differential\nequation of $r_b(x)$ :\n\\begin{equation}\n\\frac{dr_b}{dx}\n=\\frac{{\\dot m}_0 \\textnormal{v}_0}{\\pi\\rho_w(r_b^2-r_j^2)(\\textnormal{v}_j-\\textnormal{v}_w)^2}\\,,\n\\label{drdx}\n\\end{equation}\nwhich can be integrated to obtain the shape of the thin shell bow shock as a function of $x$ \\SC{in the IWS reference frame}:\n\\begin{equation}\nr_b(r_b^2-3r_j^2)+2r_j^3=L_0^2x\\,,\n\\label{rx}\n\\end{equation}\nwhere we defined the characteristic scale\\footnote{\n\\SC{Noting that} $L_0$ is the radius where the swept-up \\textit{x}-momentum is equal to 3\ntimes the injected $r$-momentum ${\\dot m}_0 \\textnormal{v}_0$, \\BT{Eq.}~(\\ref{rx}) is equivalent to \\BT{Eq.}~(22) in \n\\citet{2001ApJ...557..443O}.}\n\\begin{equation}\nL_0\\equiv \\sqrt{\\frac{3{\\dot m}_0 \\textnormal{v}_0}{\\pi \\rho_w(\\textnormal{v}_j-\\textnormal{v}_w)^2}}\\,.\n\\label{l0}\n\\end{equation}\n\nClearly, as the thin shell bow shock began to grow at $t=0$, the solution\ngiven by \\BT{Eq.}~(\\ref{rx}) \\SC{must} terminate at a finite maximum radius $r_{b,f}$ (see \\BT{Fig.}~\\ref{fig:shockframe}). \n\\SC{The growth of this outer radius with time}\ncan be calculated combining \\BT{Eqs.}~(\\ref{mass}-\\ref{rmom}) to\nobtain:\n\\begin{equation}\n\\frac{dr_{b,f}}{dt}=\\textnormal{v}_r=\\frac{{\\dot m}_0 \\textnormal{v}_0}{{\\dot m}_0+\\pi\\rho_w(r_{b,f}^2-r_j^2)(\\textnormal{v}_j-\\textnormal{v}_w)}\\,,\n\\label{drfdt}\n\\end{equation}\nwhich can be integrated with the boundary condition $r_{b,f}(t=0)=r_j$ to\nobtain \\SC{$r_{b,f}(t)$ at the current time $t$}:\n\\begin{equation}\n\\frac{1}{\\gamma L_0^2}\\left[r_{b,f}^3-r_j^3+3r_j^2(r_j-r_{b,f})\\right]+r_{b,f}-r_j=\\textnormal{v}_0t\\,,\n\\label{rf}\n\\end{equation}\nwith\n\\begin{equation}\n\\gamma\\equiv \\frac{\\textnormal{v}_j-\\textnormal{v}_w}{\\textnormal{v}_0}\\,.\n\\label{beta}\n\\end{equation}\n\nNow, in order to obtain the shape of the bowshock shell in the \\SC{source frame} $(z,r)$  (see \\BT{Fig.}~\\ref{fig:restframe}, cyan curve),\nwhen the IWS is located at distance $z_j$ from the source, we simply \ninsert the relation $x = (z_j-z_b)$  into \\BT{Eq.}~(\\ref{rx}) and $t = t_j\\equiv z_j\/$v$_j$ in \\BT{Eq.}~(\\ref{rf}). \nIn the \\textit{``narrow jet'} limit where $r_j \\to 0$, the thin shell bow shock has the \\SC{simple cubic} shape\ngiven by equation:\n\\begin{equation}\n\\frac{r_b}{L_0}=\\left(\\frac{z_j-z_b}{L_0}\\right)^{1\/3}\\,,\n\\label{rz}\n\\end{equation}\nending at the maximum ``outer edge\" radius $r_{b,f}$ \n(see the cyan dot in \\BT{Fig.}~\\ref{fig:restframe}) given by Eq.~\\ref{rf} evaluated at $t=t_j$:\n\\begin{equation}\n\\frac{1}{\\gamma}\\left(\\frac{r_{b,f}}{L_0}\\right)^3+\\frac{r_{b,f}}{L_0}=\\left(\\frac{\\textnormal{v}_0}{L_0}\\right)\\,t_j = \\left(\\frac{\\textnormal{v}_0}{L_0}\\right)\\left(\\frac{z_j}{\\textnormal{v}_j}\\right).\n\\label{rfz}\n\\end{equation}\nFor a \\textit{\"wide jet''} where $r_j$ is no longer negligible, the corresponding equations can also straightforwardly be obtained\nfrom \\BT{Eqs.}~(\\ref{rx}) and (\\ref{rf}), and are given in appendix \\ref{appendixA}. In the following, we will consider the\n\\textit{\"narrow jet\"} regime, which leads to simpler equations. \n\n\\subsection{The post-bow shock cavity}\n\nLet us now consider the trajectory \\SC{$r_f(z_f)$ described in the $z,r$ plane by} the outer edge of the thin\nshell bow shock \\SC{at earlier times, when the IWS travelled from its formation point $z=0$ at $t=0$ \nto its current location $z_j$ at time $t_j$}.\nThis trajectory will define the shape of the volume\nswept out by the travelling and expanding bowshock into the slower disk wind\n(see Fig.~\\ref{fig:restframe}, dashed black line).\n\nAt an earlier time $t_f$  ($0\\leq t_f\\leq t_j$), the bowshock terminated at an outer radius $r_f \\le r_{b,f}$ \ngiven by Eq.~\\ref{rfz} with $t_j=t_f$:\n\\begin{equation}\n\\frac{r_f^3}{\\gamma L_0^2}+r_f=\\textnormal{v}_0t_f.\n\\label{rftf}\n\\end{equation}\n\nThe distance $z_f$ from the source where this radius $r_f$ \\SC{was reached}\nis obtained from \\BT{Eq.}~(\\ref{rz}) by setting $z_b = z_f$, $r_b = r_f$ and $z_j=$v$_j \\, t_f$.\n\\begin{equation}\n\\frac{r_f^3}{L_0^2} = \\textnormal{v}_j t_f - z_f.\n\\label{zftf}\n\\end{equation}\n\nCombining \\BT{Eqs.}~(\\ref{rftf}-\\ref{zftf}) to eliminate $t_f$, and recalling \nthat $\\gamma = (\\textnormal{v}_j-\\textnormal{v}_w\/){\\textnormal{v}_0}$\\, we then obtain\nthe shape $r_f(z_f)$ of the cavity swept by the (growing) edge of the bow shock wing associated\nwith the travelling internal working surface (see dashed black curve in Fig.~\\ref{fig:restframe}) :\n\\begin{equation}\n\\frac{\\textnormal{v}_w}{\\textnormal{v}_j-\\textnormal{v}_w}\\left(\\frac{r_f}{L_0}\\right)^3+\\frac{\\textnormal{v}_j}{\\textnormal{v}_0}\\left(\\frac{r_f}{L_0}\\right)\n= \\frac{z_f}{L_0}\\,.\n\\label{zfrf2}\n\\end{equation}\n\n\\subsection{Refilling of the cavity by the disk-wind}\n\nOf course, as soon as the bow shock wing has passed by, the disk wind (travelling in the $z$-direction\nat a velocity v$_w$, see \\BT{Fig.}~\\ref{fig:restframe}) immediately starts to refill the swept-up cavity. \nFor a given radius $r_f(z_f)$ along the boundary of the swept-up volume, the refilling by the disk-wind will thus start at the \ntime $t_f$ (given by Equ.~\\ref{zftf}) when the bowshock edge reached this position; at \nthe present time $t_j$ the disk wind will have refilled a region of length $(t_j-t_f)$v$_w$ along the $z$-axis.\nThe boundary between the \\SC{wind-refilled} region and the emptied cavity thus\nhas a locus $z_c(r_c)$ (see the cyan dash-dotted line in Fig.~\\ref{fig:restframe}) given by:\n\\begin{equation}\nz_c=z_f+(t_j-t_f)\\textnormal{v}_w=\\gamma r_c+\\textnormal{v}_w t_j\\,,\n\\label{zc}\n\\end{equation}\nwhere for the second equality we have used \\BT{Eqs.}~(\\ref{rftf}-\\ref{zftf}) and set $r_c=r_f$.\n\nTherefore, the slower disk wind refills the cavity \\SC{swept} by the bow shock except\nfor an inner, conical ``hole'' with half-opening angle $\\alpha=\\arctan \\gamma^{-1}$ =\narctan[v$_0$ \/ (v$_j$-v$_w$)].\nThe conical cavity is attached to the wings of the bow shock at $(z_{bf},r_{bf})$, \nand its vertex along the jet axis is located at a distance from the source $z_a$= v$_w\\,t_j$ = $z_j$ (v$_w$\/v$_j$)\n(see \\BT{Eq.}~\\ref{zc} with $r_c=0$ and cyan asterisk in Fig.~\\ref{fig:restframe}).\n\n\\BT{Fig.}~\\ref{fig:analytic-timevol} shows the analytical flow configurations obtained at three different evolutionary times\n(corresponding to $t=2L_0\/$v$_j$, $4L_0\/$v$_j$ and $8L_0\/$v$_j$), and for\ntwo choices of the wind velocity (v$_w=0$ and v$_w=0.4$v$_j$). In the two models,\nwe have set v$_0=0.2$v$_j$. The model with v$_w=0$ (\\BT{left} frames of \\BT{Fig.~\\ref{fig:analytic-timevol}})\nproduces a cavity which does not fill up. For v$_w=0.4$v$_j$  (right frames\nof \\BT{Fig.~\\ref{fig:analytic-timevol}}), the bow shock has a more stubby shape compared to the v$_w=0$\nbow shock ($L_0$ is larger) and the cavity which it leaves behind is partially refilled by\nthe disk wind (brown region).\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[width=17cm]{fig3-new2-eps-converted-to.pdf}\n\\centering\n\\caption{The time evolution of the bow shock + cavity flow predicted by\nthe analytic model for two choices of the wind velocity: v$_w=0$ (left \nframe) and v$_w=0.4$v$_j$  (right frame).  The dark region is\nthe empty part of the cavity (swept-up in the thin shell bow shock) and\nthe brown region is the part of the cavity that has been refilled\nby the disk wind (this region being of course absent in the v$_w=0$ model\nof the left frame). \nFor both models, we show snapshots \ncorresponding to $t=2L_0\/$v$_j$, $4L_0\/$v$_j$ and $8L_0\/$v$_j$), which result\nin positions $z_j=2L_0$, $4L_0$ and $8L_0$ for the working surface (see the\nlabels on the top left of each frame). Both models have v$_0=0.2 $v$_j$.}\n\\label{fig:analytic-timevol}\n\\end{figure*}\n\n\\subsection{Kinematics along the shell}\n\nFrom \\BT{Eqs.}~(\\ref{mass}-\\ref{xmom}), it is straightforward to show that for a narrow\njet surrounded by a homogeneous disk-wind, the radial and axial velocities \n\\SC{(in the source rest frame) \nof the well-mixed thin shell material as a function of cylindrical radius $r_b$ are }:\n\\begin{equation}\n\\textnormal{v}_r= \\textnormal{v}_0 \\left(1+\\frac{3r_b^2}{\\gamma L_0^2}\\right)^{-1}\\,,\n\\label{vrv0}\n\\end{equation}\n\\begin{equation}\n\\textnormal{v}_z=\\textnormal{v}_w+(\\textnormal{v}_j -\\textnormal{v}_w) \\left(1+\\frac{3r_b^2}{\\gamma L_0^2}\\right)^{-1}\\,,\n\\label{vzv0}\n\\end{equation}\nwhere for the second equation we have also considered that v$_z=$v$_j-$v$_x$ (see\n\\BT{Figs.} \\ref{fig:shockframe} and \\ref{fig:restframe}). In evaluating the radial velocities, one should keep in\nmind that the radius $r_b$ is always smaller than the $r_{b,f}$ value given by\n\\BT{Eq.}~(\\ref{rfz}).\n\nAs expected, \\SC{we find the following asymptotic limits }:\n\\begin{itemize}\n\\item $\\textnormal{v}_r$\nhas an initial value v$_0$ for $r_b\\to 0$ (i.e., as it\nleaves the jet working surface) and goes to 0 at large radii (as the radial\nmomentum of the thin shell bow shock is shared with an increasing mass of\ndisk wind),\n\\item $\\textnormal{v}_z$\nhas an initial value v$_j$ when it leaves the working surface ($r_b \\rightarrow 0$),\nand for large radii tends to the disk wind velocity v$_w$.\n\\end{itemize}\n\n\\BT{Eqs.}~(\\ref{vrv0}-\\ref{vzv0}) give the velocity of the well-mixed material\nwithin the thin shell bow shock. These velocities correspond to the Doppler\nvelocities observed in an astronomical observation provided that the emission\ndoes indeed come from fully-mixed material.\n\nAnother extreme limit is if the emission is actually \\SC{dominated by} the gas that has\njust gone through the bow shock and which is not yet mixed with the thin shell\nflow material. In this case, the axial and radial velocities of the emitting\nmaterial would correspond to the velocity directly behind a highly compressive\nradiative shock. For such a shock, the velocity of the post shock flow\n(measured in the reference system moving with the bow shock)\nis basically equal to the \\SC{projection of the incoming flow velocity} parallel to the shock front. \nIt is straightforward\nto show that in this case the immediate post shock radial and axial velocities of the emitting\nmaterial \\SC{in the source rest frame} are given - in the \\textit{\"narrow jet\"} limit - by:\n\\begin{equation}\n\\textnormal{v}_{r,ps}= (\\textnormal{v}_j-\\textnormal{v}_w) \\frac{3 (r_b\/L_0)^2}{1+9  (r_b\/L_0)^4},\n\\label{vrps}\n\\end{equation}\n\\begin{equation}\n\\textnormal{v}_{z,ps}= \\frac{\\textnormal{v}_j+9~\\textnormal{v}_w~(r_b\/L_0)^4}{1+9(r_b\/L_0)^4}.\n\\label{vzps}\n\\end{equation}\nWe note that while v$_{z,ps}$ has the same asymptotic limits as v$_z$ in the full mixing case (see \\BT{Eqs.}~\\ref{vzv0} and \\ref{vzps}), the radial post-shock velocity v$_{r,ps}$ tends to zero both for $r_b\\to 0$ \nand for $r_b\\to \\infty$ (see \\BT{Eqs.}~\\ref{vrps}), reaching a maximum value of (v$_j-$v$_w)\/2$ for a radius equal to $L_0 \/\\sqrt{3}$. This peak value for\nthe radial velocity is a general result of bow shock kinematics, valid regardless\nof the shape of the bow shock, which was first derived by \\citet{1987ApJ...316..323H}.\n\n\nBy combining \\BT{Eqs.}~(\\ref{vrv0}-\\ref{vzps}) with \\BT{Eq.}~(\\ref{rz}) it is\nstraightforward to obtain the axial and radial velocities as a function of\nthe distance $z$ along the symmetry axis. Examples of these dependencies are\nshown in the following section.\n\n\\subsection{A dimensional example}\n\n\\BT{We now consider a particular model of an internal working surface\nmoving at a velocity v$_j=100$~km~s$^{-1}$, located at $z_j=10^{16}$~cm along the $z$-axis and ejecting side way material at a rate ${\\dot m}_0=10^{-8}$M$_\\odot$yr$^{-1}$ with a lateral ejection velocity v$_0=10$~km~s$^{-1}$.} \n\nFor the surrounding disk wind, we assume a number density of atomic nuclei\n$n_w$=$\\rho_w\/1.4m_H$ =$10^4$~cm$^{-3}$ and\nvelocities v$_w=0$ and v$_w=40$~km~s$^{-1}$. With these parameters,\nwe obtain $L_0=5.2\\times 10^{14}$~cm (for v$_w=0$) and $L_0=8.7\\times 10^{14}$~cm \n(for v$_w=40$~km~s$^{-1}$). Note that ${\\dot m}_0$ and $\\rho_w$ are only involved in the shape and kinematic equations through\n$L_0 \\propto ({\\dot m}_0\/\\rho_w)^{1\/2}$, so that only their ratio actually matters in defining the flow properties.\n\nFor these two working surfaces, we obtain the shapes, and the radial and axial velocities\n(as a function of $z$) shown in \\BT{Fig.}~\\ref{fig4}. From this figure, it is clear that for the v$_w=40$~km~s$^{-1}$ model\nwe obtain a flatter working surface than for the v$_w=0$ case, \\SC{because $L_0$ is larger.}\n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[width=8cm]{fig4-new5-eps-converted-to.pdf}\n\\caption{Shape of the bow shock and the cavity (top), radial\nvelocities v$_r$ (center) and axial velocities v$_z$ (bottom) for the\ntwo models discussed in the text. The solid curves show the\nvelocities of the well-mixed material within the thin shell flow,\nand the dashed curves show the immediate\npost-bow shock velocities. The dotted red line shows v$_z=$ v$_w$.}\n\\label{fig4}\n\\end{figure}\n\nThe velocities of the fully mixed thin shell material (shown with solid lines\nin \\BT{Fig.}~\\ref{fig4}) have the following behaviors:\n\\begin{itemize}\n\\item v$_r$ has a value of v$_0=10$~km~s$^{-1}$ at\n$z=z_j$, and monotonically decreases toward  (but not reaching) \\BT{zero} for decreasing values of $z$,\n\\item v$_z$  (lower plots of \\BT{Fig.}~\\ref{fig4})\nhas a value of v$_j=100$~km~s$^{-1}$ at\n$z_j$, and monotonically decreases for lower values of $z$,\ntowards (but not reaching) an asymptotic limit of v$_w$.\n\\end{itemize}\n\n The immediate post-bow shock velocities (shown with dashed lines in \\BT{Fig.}~\\ref{fig4}) have the following behaviors:\n\\begin{itemize}\n\\item v$_{r,ps}$ (central plots of \\BT{Fig.}~\\ref{fig4}) is \\BT{zero} at the apex of the bow shock surface at $z=z_j$, and rapidly grows to a \nmaximum value of\n50~km~s$^{-1}$ (for v$_w=0$) and 30~km~s$^{-1}$ (for v$_w=40$~km~s$^{-1}$), \nthis value corresponding to (v$_j-$v$_w$)$\/2$, as\ndiscussed in the previous section is reached at $z=z_j-L_0\/3\\sqrt{3}$. \nThe radial velocity then decreases again at smaller $z$ until the end of the bowshock wings,\n\\item the axial velocity v$_{z,ps}$ has the same qualitative behavior\nas the well-mixed v$_z$ (see above), but with\na different functional form that approaches faster its limit v$_w$ in the bowshock wings.\n\\end{itemize}\n\nWe expect that in reality, due to incomplete mixing, the emitting material will have axial\nand radial velocities between the fully-mixed layer and immediate post-bow\nshock velocities shown in \\BT{Fig.}~\\ref{fig4}. The difference between\nthese two velocities is particularly important for the\nradial component of the velocity of the emitting material.\n\n\\subsection{Successive bow shocks}\n\nIn the previous section, we assumed that the bow shock associated with an internal\nworking surface travels through undisturbed disk wind material. However, we saw that the cavity formed\nbehind it is only partially refilled by fresh disk wind. Therefore, a second bow shock will travel\ninto a disk wind structure containing an empty, conical cavity left behind by the first bow shock.\n\nWe now assume that the variable ejection velocity of the jet produces a second working surface \nat $z = 0$\nat a time $\\tau_{j}$, which also travels along the jet axis with\nthe same velocity v$_j$ as the first working surface.\n\\BT{Fig.}~\\ref{fig-twoiws} illustrates three steps of the propagation of this second working surface.\n\n\\begin{figure*}[!t]\n\\includegraphics[width=14cm]{fig5-new-eps-converted-to.pdf}\n\\centering\n\\caption{The time evolution of two successive bow shocks and the cavities predicted by the analytical model.\n  The first bow shock is ejected at $t=0$, the second shock is formed at time $t= \\tau_j$ when the first bow shock\n  is at $z=$v$_j \\tau_j$ taken here as $6 L_0$ (top). At a time $t=\\tau_j+2 L_0\/$v$_j$, the second bow shock is still\n  propagating in the undisturbed disk wind material (center). At a time $t_c$ the second bow shock catches up the emptied cavity of the first bow shock at its vertex (bottom).}\n  \\label{fig-twoiws}\n\\end{figure*}\n\nAt $t=\\tau_j$ (\\BT{Fig.}~\\ref{fig-twoiws}, first panel) the first working surface is at a distance $z_{j1} =$v$_j \\tau_j$ from the outflow source \nand its cavity is partially filled with fresh disk wind material, while the second working surface \nhas not yet expanded.\nAt a time $\\tau_j < t < t_c$ (\\BT{Fig.}~\\ref{fig-twoiws}, center) the second bow shock travels in unperturbed, pristine disk wind material  \n\\SC{that refilled the cavity behind the first bowshock};\n\\SC{hence its shape and kinematics are still given by the same equations derived above for the leading internal working surface,\nand any molecules present in the disk-wind can enter the second bowshock.}\n\\SC{At a time $t = t_c$ (\\BT{Fig.}~\\ref{fig-twoiws}, bottom panel)  the apex of the second bowshock just catches up with}\nthe vertex of the conical cavity emptied by the first bow shock and not refilled by the disk wind. \n\nTo obtain the time $t_c$,\nwe note that at any time $t$ ($\\tau_j < t < t_c$), the position of the apex of the second bow shock is $z_{j2} =$v$_j (t-\\tau_j)$ and the position of the vertex of the empty cavity behind the first bow shock is $z_{a1} = $v$_w t$. \nBy equating these two quantities we get:\n\\begin{equation}\n  t_c = \\frac{\\textnormal{v}_j}{\\textnormal{v}_j-\\textnormal{v}_w} \\tau_j.\n  \\label{criticaltime}\n\\end{equation}\nThis interaction occurs at a distance $l_c$ from the source:\n\\begin{equation}      \n  l_c = z_{a1}(t_c) = \\textnormal{v}_w t_c =\\frac{\\textnormal{v}_j}{\\textnormal{v}_j\/\\textnormal{v}_w-1} \\tau_j = \\frac{\\Delta z}{\\textnormal{v}_j\/\\textnormal{v}_w - 1 \n\\label{criticaldist}\n\\end{equation}\nwhere $\\Delta z = \\tau_j $v$_j$ denotes the distance between two successive IWS.\nUnless v$_w$ is very close to v$_j$, we find that $l_c$ is of the order of a few times the typical IWS spacing.\n\nOur model thus predicts that no more pristine unperturbed disk wind material can remain close to the jet axis beyond $z = l_c$. \nWhen the second IWS reaches $z > l_c$, the central region of its bow shock shell propagates into the emptied\ncavity left behind by the previous IWS. This second bow shock shell will in general become less curved than the first one, \nbecause its central region travels into a low density cavity instead of unperturbed disk wind. \n\n\\section{Numerical simulations}\nIn the previous section, we proposed a simple analytical \"thin-shell\" model that describes the morphology and the kinematics of a bow shock produced by a pulsating jet travelling in a surrounding disk wind. We especially show that the disk wind refills up part of the cavity carved by the bow shock, allowing us to observe pristine disk wind close the source. For successive bow shocks, bow shocks travels in an undisturbed disk wind up to a critical distance $l_c$. Above this altitude, bow shock shells interact with each other and analytical models can only be heuristic.\n\nIn this section, we present numerical simulations that start with the simple configuration adopted above, first to determine to what extent the analytical model can be used to describe a realistic situation -e.g., with a partial mixing - and secondly to study briefly the long term evolution of the interacting bow shock shells. \n\n\\subsection{Numerical method and setup}\n\nWe carry out numerical simulations of a variable ejection jet surrounded by a wide \"disk wind\" outflow.\nWe have implemented the new HD numerical code \\textit{Coyotl} which solves the \"2.5D\"\nEuler ideal fluid equations in cylindrical coordinates:\n\n\\begin{equation}\n\\frac{\\partial \\textbf{U} }{\\partial t} + \\frac{1}{r} \\frac{\\partial r \\textbf{F}}{\\partial r} + \\frac{\\partial }{\\partial z} \\textbf{G} = \\textbf{S}, \n\\label{euler2D1}\n\\end{equation}\nwhere $\\textbf{U}$ is the vector of conserved quantities\n\\begin{equation}\n\\textbf{U} = (\\rho, \\rho \\textnormal{v}_r, \\rho \\textnormal{v}_z, e, n_i)\n\\label{euler2D2}\n\\end{equation}\nwith fluxes in the $r$- and $z$- directions given, respectively, by\n\\begin{equation}\n\\textbf{F} = (\\rho \\textnormal{v}_r, \\rho \\textnormal{v}_r^2+p, \\rho \\textnormal{v}_z \\textnormal{v}_r, \\textnormal{v}_r(e+p), \\textnormal{v}_r n_i),\n\\label{eulerU2D3}\n\\end{equation}\n\\begin{equation}\n\\textbf{G} = (\\rho \\textnormal{v}_z, \\rho \\textnormal{v}_r \\textnormal{v}_z, \\rho \\textnormal{v}_z^2 +p, \\textnormal{v}_z (e+p), \\textnormal{v}_z n_i).\n\\label{eulerU2D4}\n\\end{equation}\n$n_i$ are passive scalars used to \\SC{separate} the jet from the \\SC{disk}-wind material in the flow.\nAssuming an ideal equation of state, the total energy density $e$ is\n\\begin{equation}\ne = \\frac{p}{\\rho (\\gamma -1)} + \\frac{1}{2} \\rho (\\textnormal{v}_r^2 + \\textnormal{v}_z^2).\n\\label{eulerU2D4}\n\\end{equation}\nand the source term is\n\\begin{equation}\n\\textbf{S} = (0, \\frac{p}{r}, 0, -\\rho^2 \\Lambda(T), 0).\n\\label{eulerU2D4bis}\n\\end{equation}\nwhere the cooling function $\\Lambda(T)$ is the parametrized atomic\/ionic cooling term\nof \\citet{1989ApJ...344..404R}, which approximates the cooling curve of \\citet{1976ApJ...204..290R} for temperatures above $10^4$K.\n\nThe numerical scheme is based on a second order Godunov method with an HLLC Riemann solver (Toro 1999).\nThe calculation of the fluxes and data reconstruction uses the second order scheme\ndescribed by \\citet{1991MNRAS.250..581F}. This algorithm solves Euler equations in a true cylindrical coordinate\nsystem as written in \\BT{Eq.}~(\\ref{euler2D1}) and calculates the cell gradients through\nthe center of gravity of the cylindrical cells.\n\nWe ran two simulations: a reference simulation called \\textit{no-DW} model, with v$_w$ = 0 (i.e., a jet in a stationary\nambient medium) and a simulation with v$_w = 0.4 $v$_j$ called \\textit{DW} model. \nTo follow the refilling of the cavity close to the source and the interaction between various shells, \nwe integrate equations on a $2000$ au $\\times~350$ au domain, with a resolution of 1~au per cell.\nAll  jet and wind parameters except v$_w$ are kept equal between the two simulations, and \nare summarized in Table 1.\n\n\\begin{table}\n\\caption{Model parameters}             \n\\label{table:1}     \n\\centering                                     \n\\begin{tabular}{c c }         \n\\hline\\hline                       \nParameter & Value  \\\\   \n\\hline                                  \nresolution &  $1$ au per cell  \\\\\nsimulation domain $z \\times r$ & $2000$ au $ \\times 350$ au\\\\     \n\\hline \n Jet  \\\\\n\\hline     \naverage jet velocity, $\\textnormal{v}_j$ & $96$ km~s$^{-1}$ \\\\\nvariability amplitude, $\\delta $v$_j$  & $48$ km~s$^{-1}$ \\\\\nvariability period, $\\Delta \\tau_j$  & 33 yr \\\\\ntime of velocity increase $\\eta \\Delta \\tau_j$  & 0.1~$\\Delta \\tau_j$ \\\\\njet density & $9 \\times 10^{-22}$g~cm$^{-3}$ \\\\\njet temperature  & $28$K \\\\\n\\SC{jet radius} & 20 au \\\\\n\\hline \n Disk wind  \\\\\n\\hline   \ndisk wind velocity, v$_W$ & $0$ (\\textit{no-DW} reference model)  \\\\ \n\t\t\t\t\t\t\t & $0.4$ v$_j$ (\\textit{DW} model) \\\\\ndisk wind density & $3 \\times 10^{-23}$g~cm$^{-3}$  \\\\\ndisk wind temperature & 800K \\\\\n\n \n\\hline                                            \n\\end{tabular}\n\\end{table}\n\nOur initial conditions have an inner, constant velocity jet filling the $r<r_j$ region \\SC{at all $z$}, and the disk wind (or \nexternal stationary medium) filling the rest of the computational domain. This setup differs from the standard\njet initialization in which \\SC{the jet is introduced only in a small region\naround $z \\simeq 0$} and then propagates through the domain, producing a transient with\na leading jet bow shock that sweeps aside the ambient medium \\citep{1990ApJ...360..370B,1993ApJ...410..686D}.\nBy initializing our simulations with a jet that \\SC{already} extends across \\SC{the whole range of $z$, we do not \nperturb the surrounding medium with the transient leading jet} bow shock and we can directly follow the interaction\nbetween an IWS and the unperturbed disk wind close to the outflow source.\n\nIn order to form internal working surfaces in the jet, we impose a saw-tooth ejection variability with a mean velocity $\\textnormal{v}_j$, a velocity\njump $\\delta $v$_j$ and a period $\\Delta\\tau_j$. The ejection velocity is assumed to rise linearly from\n$\\textnormal{v}_j-\\delta $v$_j\/2$ to $\\textnormal{v}_j+\\delta $v$_{j}\/{2}$ during a time-lapse\n$\\eta \\Delta \\tau_j$ and to linearly decrease over a time $(1-\\eta) \\Delta \\tau_j$ back to a velocity\n$\\textnormal{v}_j-\\delta $v$_j\/2$. Using the small amplitude velocity variability\napproximation of \\citep{1990ApJ...364..601R}, we estimate that \\SC{for our chosen parameters in Table~1} this variability will produce an internal\nworking surface in the jet at time $t_S$=5yr and distance $z_S = {(\\textnormal{v}_j-\\delta \\textnormal{v}_j\/2)^2}\\eta \\Delta \\tau_j\/{\\delta \\textnormal{v}_j} = 75$ au from the central source, and that the working surface will travel in the jet at a velocity v$_j$ =  96~km~s$^{-1}$.\n\nWe adopt a jet radius $r_j = 20$~au consistent with the width of the HH~212 molecular jet obtained from\nVLBI measurements by \\citet{1998ApJ...507L..79C} and with the widths \\SC{of atomic jets} estimated by \\citet{2000A&A...357L..61D} close to\nthe source. The density contrast between the jet and the wind is chosen to be sufficiently high ($\\rho_j\/\\rho_w$ = 29) to produce wide bow shock shell. Temperatures are chosen to insure a transverse pressure equilibrium between the jet and the wind. Note that simulations are not very sensitive to the jet temperature since strong internal shocks cooled down by atomic lines set the temperature of the IWS at T$\\sim10^4$ K.\n\nThe boundary conditions are reflecting on the symmetry axis ($r=0$) and outflowing in the outer radial and\naxial cells. On the $z=0$ boundary, we introduce the jet by imposing fixed constant physical conditions for $r<r_j$.\nFor $r>r_j$ we impose either the disk wind physical conditions, or a reflecting condition (for the reference simulation\nwith v$_w$ = 0). In order to avoid numerical problems due to the $z$-velocity shear \\SC{between the jet and the surrounding \ndisk-wind} we put a velocity gradient on \\BT{three} cells (i.e., 3~au) at the outer edge of the jet inflow. \n\n\\subsection{Single bow shock propagation}\n\n\\begin{figure*}[!h]\n\\includegraphics[width=15cm]{simu_nov_compressed_v3.pdf}\n\\centering\n\\caption{\n  Maps of density and pressure for the reference no-DW simulation with v$_w$ = 0 (left) and \n  the  \\textit{DW} simulation with v$_w$ = 0.4v$_j$ (right) at a time $t=48$yr. \n \n  Color scales on top are in g cm$^{-3}$ for density and in dyn cm$^{-2}$  for pressure.\n  White contours show the locus of 50\\% mixing ratio between jet and disk-wind\/ambient material.\n  The cyan curve shows a fit (to the numerical results) by the analytic shell shape in \\BT{Eq.}~\\ref{rfz}, with $L_0 = 65$ au (left) and\n  $108$ au (right).\n  The cyan dot indicates the \\SC{maximum radius} of\n  the shell,\n \n the cyan asterix indicates the predicted vertex of the empty conical cavity left behind the shell, \n and the cyan dash-dotted line is the analytical predicted boundary between the emptied cavity  and \n  the region refilled from below by fresh disk wind (see Fig.~\\ref{fig:restframe} and \\BT{Eq.}~\\ref{zc}). }\n\\label{nodw_simu}\n\\end{figure*}\n\n\\BT{Fig.~\\ref{nodw_simu}} shows snapshots of the  \\textit{no-DW}  simulation (two frames on the left) and of the  \\textit{DW}  simulation (two frames on the right)\nafter a $t=48$~yr time integration, which is larger than\nthe ejection variability period of 33~yr (see Table 1). The first internal working surface (IWS) has travelled to a distance\nof 995~au from the source, and a second IWS to 355~au. \nIn this subsection, we study successively the shape of the first bow shock shell, the refilling of the cavity behind it, and the kinematics of the shell, comparing each of them with our analytical predictions.\n\n\\subsubsection{The shape of the bow shock shell}\n\nThe cyan curves in Fig. 6 show that the bow shock shells in the two simulations can be well fitted with the cubic analytic solution for the thin-shell (\\ref{rx}), with values for the characteristic scale $L_0 = 65$~au for the reference \\textit{no-DW} simulation and $L_0= 108$~au for the \\textit{DW} simulation. \nIn the simulations, the sideways ejection velocity v$_0$ and mass-flux  $\\dot{m}_0$ (see Section 2)  are a result of the IWS shock configuration, which compresses\nthe jet material and ejects it sideways.\nSince the two simulations only differ in the presence or lack of a surrounding disk wind, the jet IWS in the two simulations have similar characteristics. We then expect that $\\dot{m}_0$v$_0$ is the same, and that $L_0$ should vary with the wind velocity as $L_0 \\propto ($v$_j-$v$_W)^{-1}$ (see \\BT{Eq.}~\\ref{l0}).  \nThe values of $L_0$ found above by fitting the shell shape are indeed consistent with this expectation.\n\n\\begin{figure}[!h]\n\\includegraphics[width=8.5cm]{fig7-new-june.pdf}\n\\caption{Transverse cut across the flow at the IWS location ($z=993$~au) in the \\textit{no-DW} time-frame shown in \\BT{Fig.} \\ref{nodw_simu}. This cut shows the radial velocity\n  as a function of distance from the jet axis in solid line. We also plot the radial velocity weighted by the abundance of the jet tracer with a dashed line. The radial velocity first grows outwards, reaches a maximum velocity of $\\approx 14$~km~s$^{-1}$\n  at a radius of $\\sim 25$~au (somewhat larger than the 20~au initial jet radius), and then remains with values $>10$~km~s$^{-1}$ until it drops\nsharply to 0 at $r\\sim 50$~au. The velocity maximum at $r \\sim 25$~au corresponds to the shock against the jet material. The second maximum at r$\\sim 50$~au is the shock that propagates in the disk-wind and the zero radial velocity material at larger radii is the undisturbed disk wind.}\n\\label{v0est}\n\\end{figure}\n\nThe cyan dots on the leading bow shock wings in \\BT{Fig.} \\ref{nodw_simu} indicate the maximum radius of the bow shock shell as observed in the numerical simulations,  $r_{\\rm max} = $ 133 au in the \\textit{no-DW} and $r_{\\rm max} = $ 137 au in the \\textit{DW} cases. Assuming that it corresponds to \nthe current position of the edge of the thin shell $(r_{bf},z_{bf})$, as defined in \\BT{Fig.}~\\ref{fig:restframe},  our analytic model  predicts  that $r_{bf}$ depends on $L_0$ and v$_0$ through \\BT{Eq.}~(\\ref{rfz}) or (\\ref{rf-wide}). With $L_0=65$, $108$~au, we would deduce v$_0 = 27$, $19$ km~s$^{-1}$ for\nthe \\textit{no-DW} and  \\textit{DW} simulations, respectively.\n\nTo obtain a direct measurement of v$_0$, \nwe plot in \\BT{Fig.} \\ref{v0est} a transverse cut of the radial velocity v$_r$ at the position of the leading working\nsurface ($z=993$~au) in the \\textit{DW} simulation. Inside the IWS, because of both the adiabatic expansion and the mass flux across the IWS,\n$u_r$ increases from zero to a maximum velocity of\n$14$ km~s$^{-1}$. This direct measurement of v$_0$\nis smaller than the values of 27, 19~km~s$^{-1}$ inferred from the maximum radius of the shell in our simulations using \\BT{Eq.}~\\ref{rfz} (see above). \nHowever, we note that taking v$_0$ = 14 km~s$^{-1}$ (its real value), the predicted $r_{bf}$ would be $r_{bf} = 111$~au in the no-DW case and $r_{bf} = 121$~au in the DW case, only slightly\nsmaller than the $r_{\\rm max}$ found in our simulations.\n\nThis slight difference in outer radius \nbetween the analytic model and the numerical simulations could be a result of several effects:\n\\begin{itemize}\n\\item The analytic model assumes a working surface with a time-independent, sideways ejection,\n  while the numerical simulation has an IWS with time-dependent sideways ejection that depends on the evolution\n  of the IWS shocks. The IWS in the simulations produces a higher sideways velocity at early times (v$_0 \\sim 18$~km~s$^{-1}$)\\footnote{Note that following \\citet{2001ApJ...557..443O} the maximum velocity that an atomic gas at T$=10^4$K can reach through adiabatic expansion is $\\sqrt{3} c_s = 18$~km~s$^{-1}$, where $c_s$ is the adiabatic sound speed.}, closer to the values deduced from the analytic cavity shapes,\n\\item in the numerical simulation, the sideways ejection from the IWS is not highly super-sonic. The thermal gas pressure is therefore expected\n  to be an additional source of sideways momentum for the shell (an effect not included in our momentum conserving analytic model); \\SC{this will act to\n  produce a higher ``effective\" v$_0$.}\n\\item similarly, the thermal pressure in the head of the bow shock driven into the surrounding environment will result in\n  a sideways push which is not present in the momentum conserving analytic model. \n\\item the numerical simulations do not have instant mixing between the sideways ejected jet material and the shocked\n  environment (or disk wind), as assumed in the analytic model. Since the immediate post shock velocity in the radial direction is generally greater that the radial mean shell velocity (see example in \\BT{Fig.}~\\ref{fig4}), the growth rate of the bow shock can be enhanced. \\SC{In the reference frame of the IWS (see Fig.~\\ref{fig:shockframe}), the non-mixed material will \"slide\" along the shell surface, extending $r_{b,f}$ to larger values.}\n\\end{itemize}\n\\citet{2001ApJ...557..429L} found in their simulations similar disagreements between direct measurements of the sideways momentum ejected by the IWS\nand the momentum estimated from the fitted shape of their analytic shell model. \n\n\n\\subsubsection{Cavity refilling}\n\nThe asterisk in cyan in each panel of \\BT{Fig. \\ref{nodw_simu}} indicates the location of the vertex of the emptied cavity as predicted from the analytic model\n(see \\BT{Fig.}~\\ref{fig:restframe}). For the \\textit{no-DW} simulation, this point is located at the shock formation position ($z_s=75$~au)\nwhereas for the \\textit{DW} simulation this point is located at $z_a = z_{sf}$ + $\\frac{v_w}{v_j} (z_j - z_{sf}) = 440$~au. \nWe also plot in cyan dash-dotted \\SC{the line connecting this vertex to $r_{\\rm max}$,} which traces the \nboundary predicted by the analytical model between the emptied swept-out conical cavity and the unperturbed surrounding medium\/refilled disk wind\n(see black conical region in Fig.~\\ref{fig:restframe}). Three important features can be seen.\n\n\n\\textit{In both numerical simulations, the emptied cavity predicted by the analytical thin-shell model,\ni.e. the conical volume inside the dash-dotted cyan line,\nis not really empty, but partially filled with a cocoon of low\ndensity and pressure material.} \nNo unperturbed ambient gas or disk wind can be left inside this volume (in black in Fig. \\ref{fig:analytic-timevol}), which was entirely swept-out by the growing shell \nduring the IWS propagation.\nHence this cocoon is made of shocked material that did not fully mix in the shell, and re-expanded in the low-pressure cavity behind it,\n refilling it ``from above''.\nThe white contour, which denotes the surface of 50\\%\\  jet\/environment mixing fraction (obtained following a passive scalar) \n\\SC{shows that} the cocoon is mainly filled with jet material close to the axis, where the shell mass is dominated by \n\\SC{gas ejected from the IWS}.\n\\SC{Further from the axis and closer to the theoretical boundary (cyan dash-dot line)} \nit is filled by ambient material that was swept up by the bowshock and re-expanded behind it.\n\n\\textit{The boundary with unperturbed ambient \\BT{or} disk wind material}\ncloses back to the axis at the predicted vertex position (see cyan asterisk \\BT{Fig.}~\\ref{nodw_simu}), \nbut is delimited by a weak shock that extends slightly outside from the predicted analytical boundary (dash-dotted cyan line \\BT{Fig.}~\\ref{nodw_simu}) .\nIn the \\textit{no-DW} model, the analytical boundary\nrepresents the trajectory of the edge of the bow shock ($z_c = z_f$ in the case v$_w=0$).\nHence, this weak shock is produced by the supersonic motion of the high-pressure edge of the bow shock ($r_{b,f},z_{b,f}$) in the static surrounding medium.\nThis launches a weak outward shock that repels the boundary of the unperturbed \nambient\nmaterial slightly outside the predicted cavity boundary (in cyan dash-dotted line). \nIn the presence of a supersonic disk wind, the weak shock front is advected away from the source so that it still closes back \non-axis at the predicted vertex position $z_a$. \nHence, \\BT{Eq.}~(\\ref{zc}) gives a strong limit on the boundary between perturbed and unperturbed material. \n\n\n\n\\textit{In the presence of a disk-wind, the region between the predicted cavity boundary (cyan dash-dotted line) and the weak shock front outside it\nis refilled by fresh disk wind material coming from below.}\n\\begin{figure}[!h]\n\\includegraphics[width=9cm]{dw-zoom-5_compressed-v2.pdf}\n\\caption{Zoom of the leading IWS of the simulation with a surrounding disk wind at time $t=48$~yr. Left: density\nstratification (with the logarithmic color scale given by the top bar in g~cm$^{-3}$), center: radial velocity\n(with the linear scale of top bar in km~s$^{-1}$) and right: axial velocity structure\n(with the linear scale of the top bar in km~s$^{-1}$). The white contours show the surfaces of\n50\\% (solid line), 10\\%, 0.1\\% and 0.001\\% (outer contour) jet material mixing fractions. \nThe cyan asterisk is the predicted vertex of the cavity, the cyan dash-dotted line in the predicted boundary \nof the cavity, and the cyan curve is the fitted shape of the bow shock.}\n\\label{refilling}\n\\end{figure}\nTo analyse this process\nwe show in Fig. \\ref{refilling} density and velocity maps of the region around the leading bow shock of the \\textit{DW} simulation.\nThe dashed white contours show  \n10\\%, 0.1\\% and 0.001\\% jet material mixing fractions. \nMaterial that went through the bowshock and re-expanded in the cocoon has also been partially mixed with jet material. As a consequence, regions where no jet material is observed are regions where the disk wind has refilled the cocoon from below.\nThe location of the last, outer contour (corresponding to a $10^{-5}$ jet material mixing fraction)\nshows that the weak outer shock front propagates into un-mixed, fresh DW material. This material manages to cross the weak \nshock to refill ``from below\" the bottom part of the swept-out cavity.\n\nThe weak shock provides a slight push outwards to the refilling DW, with radial velocities \nthat vary from $+6$~km s$^{-1}$ to $+3$~km s$^{-1}$ along the shock front (middle panel of Fig.~\\ref{refilling}), \nsimilar to the adiabatic sound speed of the disk wind ($c_{s w}\\approx2.8~$km~s$^{-1}$). \nThe weak shock also reduces the DW inflow velocity v$_z$ to values slightly below v$_w$ = 0.4v$_j$ = 38.4~km~s$^{-1}$ (right panel of Fig.~\\ref{refilling}).\nHowever,  refilling remains efficient up to the locus predicted by our analytical model (dash-dotted cyan line), \nas the jet mixing fraction there remains very small ($\\simeq$ 0.1\\%). \nThe presence of the weak shock does not appear to significantly modify the extent of DW refilling compared to analytical expectations.\n\n\nIn summary, we can therefore distinguish in our simulations three refilling regions behind the bowshock:\n  \\begin{itemize}\n  \\item a \\SC{low density cocoon} trailing the bow shock, that is refilled ``from above\" by shell material re-expanding into the emptied cavity.\n  This region is mainly composed of jet material close to the apex of the bow shock, and of shocked swept-up disk wind material \n  behind the wings of the bowshock,\n  \\item an intermediate region (outside the cyan dash-dotted line and inside the weak shock closing the cavity) refilled ``from below''  \n  by weakly shocked disk wind material, \n    \\item a region upstream of the weak shock closing the cavity, that is refilled by unperturbed fresh disk wind keeping its initial physical conditions.\n  \\end{itemize}\n\n\n\\subsubsection{Kinematics}\n\nWe now compare the kinematics in both simulations with our analytical predictions.\n\\BT{Fig.}~\\ref{PV} shows \"position-velocity\" (PV) diagrams for v$_r$ and v$_z$ as a function of distance $z$ along the flow axis.\nIn order to enhance the contribution from the material that has just been shocked, each pixel in a snap shot has been weighted by the cube of the pressure $p^3$ times the elementary volume $2\\pi r \\Delta r \\Delta z$.\nUsing this\nweighting, the maximum intensity (in yellow and orange shades) at each position in the PV diagrams then traces the velocity in the shell.\nThe separation between material originating mainly from the jet or mainly from the surroundings\/disk wind is done using a passive scalar.\nThe predicted mixed shell velocities (\\BT{Eqs.}~\\ref{vrv0} and \\ref{vzv0}) are shown in blue, and the predicted immediate post-bowshock velocities (\\BT{Eqs.}~\\ref{vrps} and \\ref{vzps}) are shown in magenta. Following the discussion of \\BT{Fig. \\ref{v0est}}, we take v$_0= 14$ km s$^{-1}$. \n\\begin{figure}[!h]\n\\centering\n\\includegraphics[width=9.5cm]{kinematics-nodw4-june-compressed.pdf}  \\\\\n\\includegraphics[width=9.5cm]{kinematics-dw4-june-compressed.pdf}\n\\caption{Longitudinal position-velocity (PV) diagrams for the \\textit{no DW} simulation (top) and the \\textit{DW} simulation (bottom).  From left to right: v$_r$ for the jet material\nonly, v$_r$ for the surrounding material only, v$_z$ for the jet material only, v$_z$ for the surrounding material only, and density stratification. The ordinate of all frames\nis position along the outflow axis (in au). The color scale in the PVs is scaled by volume $\\times$ cube of pressure so as to be maximum (in red and yellow shades) \nfor shocked material in the shell, while the \\SC{color scale} for density is in g~cm$^{-3}$. \nBlue curves are predicted velocities in the full mixing hypothesis (\\BT{Eqs.}~\\ref{vrv0} and \\ref{vzv0}), while\nmagenta curves are the predicted immediate-post shock velocities (\\BT{Eqs.}~\\ref{vrps} and \\ref{vzps}).}\n\\label{PV}\n\\end{figure}\n\nIn the v$_r$ PV diagrams of the surrounding\nmaterial, the \\SC{expansion velocities of shocked material in the shell (orange shading) are always larger than predicted by the (blue) full-mixing curve}\n(except very close to the bow shock apex where the shear is maximum). The \nsimulation more closely follows the immediate post-bow shock velocity curve (magenta), \\SC{indicating that high-pressure shocked material in the shell\nis not fully mixed in our simulations.} \\SC{Conversely,} the v$_r$ PV diagram of the jet material \ndecreases monotonically along the bow shock wing\nwith velocities always \\SC{slightly smaller than} the full mixing velocity curve (in blue).  \n\nConcerning the velocity along the jet $z$-axis,\nthe v$_z$ values for jet material lie close to, or slightly above the full mixing curve in blue.\nThe v$_z$ PV diagrams for the surrounding material \n\\SC{generally show smaller v$_z$ than predicted by the full mixing curves.}\n\\SC{The high-pressure swept-up shell material (in orange)} \nlies close to the immediate post-shock velocities (magenta curve).\n\n \nThe relatively small v$_r$ velocities and large v$_z$ velocities \nobserved in the jet dominated material indicate that even if the full mixing hypothesis does not hold, the momentum is still conserved: if the velocities of the swept-up surrounding material are greater than expected from the full mixing hypothesis, then the velocities of the jet material (in the IWS rest-frame) must be smaller than the predicted full mixing velocities (and vice-versa). \n \nAs predicted, the most striking difference between the disk wind model and the reference no-DW model is\nthe saturation of the v$_z$ velocity in the bowshock wings \\SC{to a non-zero value of} v$_z \\approx $v$_w$. Even if this asymptotic limit does\nnot depend on any mixing, the \\SC{incomplete}\nmixing obtained in the simulations produces a \\SC{more} rapid convergence to v$_w$\n\\SC{than predicted in the case of full mixing (blue curve)}.\n \n \\subsection{Long-term evolution}\n\n\\begin{figure*}[!h]\n\\includegraphics[width=6cm]{longterm-nodw-4-compressed.pdf}\n\\includegraphics[width=7.97cm]{longterm-dw-5-compressed.pdf}\n\\centering\n\\caption{Density maps for the \\SC{no-DW} reference simulation (three frames on the left) and the  \\SC{DW simulation} \n(three frames on the right) at $t=71$, 119 and 167~yr. The white contours indicate the surface of $50\\%$ (solid line) and 90 \\% (dashed line) \njet material mixing fractions. The black lines in the disk-wind simulation show\na cone of $\\alpha = 11^\\circ$ opening half-angle, which circumscribes the\nboundary the region perturbed by the jet and its internal working surfaces. The black dashed lines show the predicted trajectory of the edge of the bow shock (see eq. \\ref{zfrf2}). The density color scale is given by the right bar (in g~cm$^{-3}$).}\n\\label{long_term}\n\\end{figure*}\n\n\\BT{Fig.} \\ref{long_term} shows the longer-term evolution of the \nreference no-DW simulation (\\BT{three} left frames) and of the disk-wind simulation (\\BT{three} right frames)  at times $t=71$, $119$ and $167$~yr.\nFrom this figure,\nwe see that the morphologies of the regions perturbed by the jet after the passage of several IWS\nare very different in the two cases.\n\nIn the \\textit{no-DW} simulation, the region perturbed by the jet behind the leading bowshock\nexpands into a roughly\ncylindrical shape, which tapers off close to the position of the outflow source (where it becomes a weak, radially expanding shock). This\nis the standard shape of the perturbed region in simulations of variable, radiative jets propagating into a uniform static medium,\nseen since the early\nwork of \\citet{1993ApJ...413..198S} and \\citet{1994ApJ...434..221B}.\n\nIn the disk wind simulation, in contrast, the\nregion perturbed by the jet behind the leading bowshock takes\na conical shape, tapering off at large distances from the outflow\nsource.  \nFor the parameters of our DW simulation, the half-opening angle of the perturbed region \nis $\\alpha\\approx 11^\\circ$ (see Fig.~\\ref{long_term}). \nThis cone is located outside the predicted trajectory of the edge of the bow shock (drawn in black dashed line) given by \\BT{Eq.}~(\\ref{zfrf2}). This broadening occurs because, as discussed in item 3 of Section 3.2.2, the edge of the bow shock drives a weak outer shock into the undisturbed DW, which propagates away at a speed close to $c_s\\sim3.8$ km~s$^{-1}$. Taking into account the advection of the weak shock by the DW, one predicts that this will broaden the disturbed region by an angle  $\\beta = \\arctan{c_s\/\\textnormal{v}_w} = 4\\degr$, in agreement with the observed cone opening. Obviously, in the no-DW simulations, this weak outer shock travels laterally without being advected, and no limiting cone forms.\n\n\nIn this surprisingly simple configuration\nadopted by our jet+disk wind simulation, the overall long-term effect of the disk wind is\nto stop the perturbations from travelling beyond this ``opening cone'' of the sideways ejection from the IWS. \n\nAnother important effect of the DW is to push the locus of 50\\% ambient material (white contour) \ncloser to the jet axis than in the no-DW case, due to the disk wind partial refilling behind each bowshock.\nHence, the first few IWS close to the source can still sweep up (possibly molecular) DW material.\nThe internal IWS are also more curved than in the no-DW case, where material ejected sideways meets a very low-density cocoon,\nproducing flat-topped internal bowshocks (see \\BT{Fig.} \\ref{long_term}).\n\n\\section{Summary}\n\nIn this paper we have presented a first exploration of an hydrodynamical flow composed by an inner,\nvariable jet surrounded by a slower, steady, cylindrical disk wind. The jet variability\nproduces internal working surfaces (IWS) which drive bow shocks into the disk wind, producing\na strong coupling between the two components of the flow.\n\nWe have developed a standard thin shell model for the bow shock driven into the disk wind\nby a single IWS, for a jet of arbitrarily small radius  (see Section 2), \nderiving the shape of the bow shock and the refilling by the continuing disk wind\nof the cavity left behind by the bow shock. The model was extended\nto give a qualitative description of the flow resulting from two or more successive IWS bow shocks\nplowing through the disk wind (Section 3).\n\nThe appropriateness and limitations of the predictions of bow shock shapes and kinematics from this analytic model \nhave been checked with axisymmetric numerical simulations:\none of a variable jet+disk wind configuration,\nand a second reference simulation with the same variable jet surrounded by a stationary\nenvironment. We compared the analytic model with the numerical simulations, and we found\na relatively good agreement, giving us an understanding of the main features of\nthe simulated flows. These features are:\n\\begin{itemize}\n\\item the bow shocks of the numerical IWS have cubic morphologies which can be reproduced\nquite convincingly with the thin-shell, momentum conserving analytic\nmodel (see \\BT{Eqs.}~\\ref{rz}-\\ref{rfz} and Fig. \\ref{nodw_simu}),\n\\item the kinematics in the simulated bow shocks has a behavior\nwhich approximately follows the kinematics predicted from the analytic model\nfor the fully-mixed layer (for jet-dominated material) or \nthe immediate post-bow shock gas (for high-pressure swept-up ambient gas)  (see Figs. 4. and 8).\n\\item these bow shocks leave behind cavities which are partially refilled\nby the slower disk wind (see Figs.~3, 5 and 8).\n\\item thanks to this refilling, subsequent IWS will propagate into fresh disk wind material \nup to a distance from the source $l_c = {\\Delta z}$\/(v$_j$\/v$_w$-1) (see Fig.~7).\n\\end{itemize}\n\nThe main contribution of this paper is thus to provide a numerically validated, simple analytic model which can\nbe used to model bow-like shapes of knots observed close to the outflow sources in high velocity,\ncollimated optical and molecular outflows \\citep{2000A&A...356L..41L,2015A&A...581A..85P}.\nAs shown by our simulations, this shape modeling (in the narrow jet limit) allows one to estimate the sideways ejection velocity \nfrom the IWS and the length scale of the bowshock. From this, constraints could be inferred on the \nmass-loss rate from the IWS and the surrounding flow properties (see \\BT{Eq.}~\\ref{l0}).\n\nAnother important contribution of this paper is to predict the regions\nwhere a surrounding disk-wind would remain unperturbed.\nA quite dramatic result of our jet+disk wind simulation is that the perturbations of the disk wind by the IWS bow shocks \nare confined inside a cone.\nTherefore, all of the gas outside this confinement cone is unperturbed disk wind material. Also,\nthere are pockets of undisturbed disk wind material within this cone, in the refilled region\nbetween the source and the last IWS, and also ahead of the latest\nIWS when it is at $z < l_c$ (see the three right hand frames of Fig. 10). \nThese are the regions in which one still \nfinds a record of the undisturbed characteristics of the disk wind, \nwhich could be useful for comparisons with disk wind models.\n\nFinally, another result of observational interest is that we identify several \ndistinctive signs of a cylindrical DW around a time-variable jet:\n(i) bow shocks that close upon the axis at a finite distance from the source (at a fraction v$_w$\/v$_j$ of the distance to the bow apex), \n(ii) a non-zero (= v$_w$) asymptotic value of longitudinal velocity in the far bowshock wings,\n(iii) internal bowshocks that are curved rather than \"flat-topped\", \n(iv) a predominance of DW material ahead of the first few IWS, which (if the DW is chemically richer and\/or dustier than the jet) \nshould produce different emission signatures compared to the more distant IWS.\n\nExtensions of the analytic model to more complex jet+disk wind flows do not\nappear very attractive (as, e.g., relaxing the assumption of a cylindrical uniform disk\nwind) as quite complex expressions are obtained, and are therefore not straightforwardly\napplicable to model observed structures. On the other hand, the numerical simulations\npresented here can be extended in many directions:\n\\begin{itemize}\n\\item including a more realistic disk wind model (e.g., with a radial dependence of the density and velocity, and a velocity not aligned with\nthe outflow axis),\n\\item studying the effect of a non-top hat jet cross section,\n\\item going from the HD to the MHD equations,\n\\item including a chemical\/ionic network and the associated cooling functions.\n\\end{itemize}\nIf future comparisons between jet+disk wind models and observations are sufficiently\npromising, the items listed above (as well as other easily imagined possibilities)\nwill become worthy of exploration.\n\n\\begin{acknowledgements}\n\\BT{We thank the anonymous referee for useful comments.} This work has been done within the LABEX PLAS@PAR project, and received financial state aid managed by the Agence Nationale de la Recherche, as part of the ''Programme d'Investissements d'Avenir'' under the reference ANR-11-IDEX-0004-02. AR acknowledges support from the DGAPA (UNAM) grant IG100218. This research has made use of NASA's Astrophysics Data System.  \n\\end{acknowledgements}\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzfpxm b/data_all_eng_slimpj/shuffled/split2/finalzzfpxm
new file mode 100644
index 0000000000000000000000000000000000000000..be41f662434e2ba0683e70a26775ff6b61e86054
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzfpxm
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\n\\ \\ We have recently developed  methods for obtaining exact two-point resistance of certain circulant graph namely, the complete graph minus $N$ edges \\cite {Chair1}. In this paper, using  similar techniques and ideas, we    consider  trigonometrical sums that arise in the computation of the two-point resistance of the finite  resistor networks \\cite{Wu}, in  the work of  McCoy  and Orrick  on the chiral Potts model \\cite {McCoy}, and in the Verlinde dimension formula of the  twisted \/untwisted space of  conformal blocks of the $SO(3)$\/$SU(2)$ WZW model \\cite{Verlinde}.\n\n\\ \\  Before considering these trigonometrical sums, we test the techniques  used in \\cite{Chair1}, by first deriving the Green's function of the one-dimensional lattice graphs with free boundaries, and the two-point resistance of the $N$-cycle graph \\cite{Wu}. The same techniques is then used to  evaluate  a trigonometrical sum that played a crucial role to prove  R. F Scott's  conjecture  on the permanent of the Cauchy matrix \\cite{Minc,Todd}. Having tested these techniques, an  alternative derivation is then given for certain trigonometrical sum that appeared in the perturbative  chiral Potts model \\cite {McCoy}, \\cite {Mehta}.\n\n\\ \\  We have also considered the general case studied by  Gervois and Mehta \\cite {Mehta},  Berndt and Yeap \\cite {Berndt}, here, our results agree with those in \\cite {Mehta}.  It turns out that the Verlinde's dimension formulas for the untwisted space of conformal blocks, may be obtained simply by summing over  certain parameter of a trigonometrical sum  considered in \\cite{Mehta}.  For the twisted space of conformal blocks, however,  the parameter is restricted to take some value. It is shown that the dimension of the conformal blocks on a genus $g\\geq 2$ Riemann surface may be obtained  through a recursion formula that relates  different genera. Mathematically speaking, the dimension of the space of conformal blocks is obtained by expanding  certain generating function order by order, or using the Hirzebruch-Rlemann-Roch theorem \\cite{Zagier}.\n\n\\ \\   By using the method given in \\cite {Chair1}, we are able to obtain closed form formula for the  two-point resistance of a $2\\times N $ resistor network \\cite {Chair2}. In this paper, an exact computation of the corner-to-corner resistance as well as the total effective resistance of a $2\\times N$ will be given. The total effective effective resistance, also called the Kirchhoff index \\cite{Randic}, this  is an invariant quantity of the resistor network or graph.\n\n\\ \\   The exact  two-point resistance of an $M\\times N$ resistor network is given in terms of a double sum and not in a  closed form \\cite {Wu}. Therefore, our computation carried out in this paper, represents the first non-trivial exact results for the two-point resistance of a two-dimensional resistor network.\n\n\\ \\ This method is then used  to evaluate variant of trigonometrical sums, some of which are related to number theory, we hope that these trigonometrical sums will have some physical applications.  It is  interesting to point out that  all the computations of the trigonometrical sums  in this  paper  are based on a  formula by Schwatt \\cite{ Schwatt} on trigonometrical power sums, and  the representation  of the binomial coefficients  by the  residue operator. The  Schwatt's formula is modified slightly,  only in the case of the corner-to-corner resistance, the Kirchhoff index  and  trigonometrical sum given by $ F_{1}(N,l,2) $, and $ F_{1}(N,l,2)$, see Section 6, this was also the case in our previous paper  \\cite{Chair1}.\n\n\\ \\ This paper is organized as follows; in section 2, we give an explicit computations of the two-point resistance of the $N$-cycle graph and the Green's function of the one-dimensional lattice, and in Section 3,  we give a simple  derivation of a trigonometrical sum connected with Scott's conjecture on the permanent of the Cauchy matrix. In section 4, we consider trigonometrical sums  arising in the chiral Potts model, and in the Verlinde formula of the dimension of the conformal blocks. Exact computations of the corner-to-corner and the Kirchhoff index of $2\\times N$ resistor network will be given  in section 5. In Section 6, we consider other class of trigonometrical sums  some of which are related to number theory, and  finally, in Section 7,  our conclusions are given.\n\\section{ The two-point resistance of one-dimensional lattice using the residue operator }\n\n\\ \\ In this Section, we first  start with the two-point resistance of the $N$-cycle graph computations, then, move to the trigonometrical sum related to the two-point resistance of the one-dimensional lattice with free boundaries, that is, the  path graph. The two-point resistance of the $N$-cycle graph  between any  two nodes $\\alpha$ and $\\beta$ is  given by the following simple closed formula \\cite{Wu},\n\\begin{equation} \n\\label{t1}\nR(l)=\\frac{1}{N}\\sum_{n=1}^{N-1}\\frac{\\sin^2(nl\\pi\/N)}{\\sin^2(n\\pi\/N)}=\\frac{l(N-l)}{N},\n\\end{equation}\nwhere $l=|\\alpha-\\beta|$, and $1\\leq \\alpha,\\beta \\leq N. $\nOur derivation for the two-point resistance starts with the following trigonometrical identity  $$\\cos(2ln\\pi\/N)=\\sum_{s=0}^{l}(-1)^s \\frac{l}{l+s}\\binom {l+s} {l-s}2^{2s}\\sin^{2s}(n\\pi\/N), $$ from which the  above trigonometrical sum may be rewritten as\n\\begin{equation} \n\\label{t2}\nR(l)=\\frac{1}{2N}\\sum_{s=1}^{l}(-1)^{s +1}\\frac{l}{l+s}\\binom {l+s} {l-s}2^{2s}\\sum_{n=1}^{N-1} \\sin^{2(s-1)}(n\\pi\/N).\n\\end{equation}  \nOn the other hand, Schwatt's formula for trigonometrical power sums \\cite{Schwatt}, gives\n\\begin{equation} \n\\label{t3}\n\\sum_{n=1}^{N-1} \\sin^{2(s-1)}(n\\pi\/N)=\\frac{1}{2^{2(s-1)-1}}\\sum_{t=1}^{s-1}(-1)^{t +1}\\binom {2(s-1)}{s-1-t} +\\frac{N-1}{2^{2(s-1)}}\\binom {2(s-1)} {s-1}.\n\\end{equation}\n\n\\ \\ \n Therefore,  the two-point resistance may be obtained by evaluating  the binomial sums in the expression of $R(l)$, based on the  residue operator. This operator played a crucial  role in evaluating combinatorial sums and proving combinatorial identities \\cite{Egorychev}.  First, let us recall the definition of the residue operator $\\hbox{res}$. To that end, let  $G(w)= \\sum_{k=0}^{\\infty}a_{k}w^{k}$ be a generating function for a sequence $\\{a_{k}\\}$. Then the k-th coefficient of $G(w)$ may be represented by the formal residue as follows\n$$a_{k}= \\hbox{res}_w G(w){w^{-k-1}}.$$\nThis is equivalent  to the Cauchy integral representation of $a_k$,\n\\[a_k=\\frac{1}{2\\pi i}\\oint _{|z|=\\rho}\\frac{G(w)}{w^{k+1}}dw ,\\] for coefficients of the Taylor series in a punctured neighborhood of zero. In particular, the generating function of the binomial coefficient sequence  $\\binom {n}{k}$ for a fixed $n$ is given by $$G(w)= \\sum_{k=0}^{n}\\binom {n}{k} w^{k} =(1+w)^{n},$$ \nand hence $$\\binom {n}{k}=\\hbox{res}_w (1+w){^n}{w^{-k-1}}.$$ The other binomial coefficient that we need in this paper is the following, $$\\binom {2n}{n}=\\hbox{res}_w (1-4w){^{-1\/2}}{w^{-n-1}}.\n$$\n\n\\ \\  Before finishing this brief summary, we should mention  one important property of the  residue operator $\\hbox{res}$, namely linearity, this is a  crucial in doing  computations,  linearity states that;  given some contants $\\alpha$ and $\\beta$, then $$\\alpha \\hbox{res}_w G_{1}(w){w^{-k-1}}+ \\beta\\hbox{res}_w G_{2}(w){w^{-k-1}}=\\hbox{res}_w(\\alpha G_{1}(w)+ \\beta G_{2}(w)) {w^{-k-1}}.$$ \nLet us now evaluate the first term in Eq.(\\ref{t2}), using the residue operator, namely the following term\n\\begin{eqnarray}\n\\label{t4}\nR_{1}(l):&=&\\frac{2}{N}\\sum_{s=1}^{l}(-1)^{s }\\frac{2l}{l+s}\\binom {l+s} {l-s}\\sum_{t=1}^{s-1}(-1)^{t }\\binom {2(s-1)}{s-1-t}\\nonumber\\\\&=&\\frac{2}{N}\\sum_{s=1}^{l}(-1)^{s }\\frac{2l}{l+s}\\binom {l+s} {l-s}\\sum_{t=1}^{s-1}(-1)^{t }\\hbox{res} \\frac{(1+w)^{2(s-1)}}{w^{s-t}}\\nonumber\\\\&=&\\frac{2}{N}\\sum_{s=1}^{l}(-1)^{s }\\frac{2l}{l+s}\\binom {l+s} {l-s}\\hbox{res}_{w=0}\\frac{(1+w)^{2(s)}}{(1+w)^{3}w^{s-1}}.\n\\end{eqnarray}\nIn obtaining Eq.(\\ref{t4}), we  discarded an analytic term at the pole $w=0$ of order $s-1$. By making a change of variable $l-s=k $, then, Eq. (\\ref{t4}) may be rewritten as \n\\begin{eqnarray}\n\\label{t5}\nR_{1}(l)&=&(-1)^{l+1}\\frac{2}{N}\\hbox{res}_{w=0}\\frac{w}{(1+w)^{3}}\\sum_{k=1}^{l-1}(-1)^{ k}\\frac{2l}{2l-k}\\binom {2l-k} {k}\\bigg(\\frac{1+w}{\\sqrt{w}}\\bigg)^{2(l-k)}\\nonumber\\\\&=& (-1)^{l+1}\\frac{2}{N}\\hbox{res}_{w=0}\\frac{w}{(1+w)^{3}}\\bigg(C_{2l}\\big(\\frac{1+w}{\\sqrt{w}}\\big)-(-1)^{l}\\bigg),\n\\end{eqnarray}\nwhere $$ C_{2l}(x)= 2T_{2l}(x\/2)=\\sum_{k=0}^{l}(-1)^{k} \\frac{2l}{2l-k}\\binom {2l-k} {k}x^{2l-2k},$$\nis the normalized Chebyshev polynomial of the first kind\\cite{Rivlin}, and\n$$T_{2l}(x\/2)=\\frac{1}{2}\\Bigg\\lbrack\\Bigg(\\frac{x}{2}+\\sqrt{(x\/2)^2-1}\\bigg)^{2l} +\\Bigg(\\frac{x}{2}-\\sqrt{(x\/2)^2-1}\\bigg)^{2l} \\Bigg\\rbrack.$$\nUsing the fact that $C_{2l}\\big(\\frac{1+w}{\\sqrt{w}}\\big)= \\frac{1}{w^{l}}+w^{l}$, then the final expression for the first term $R_{1}(l) $, reads\n\\begin{eqnarray}\n\\label{t6}\nR_{1}(l)&=&(-1)^{l+1}\\frac{2}{N}\\hbox{res}_{w=0}\\frac{1}{(1+w)^{3}w^{l-1}}\\nonumber\\\\&=&-\\frac{l(l-1)}{N}\n\\end{eqnarray}\nSimilarly, the second term may written as \n\\begin{eqnarray}\n\\label{t7}\nR_{2}(l):&=&\\frac{(N-1)}{N}\\sum_{s=1}^{l}(-1)^{s }\\frac{2l}{l+s}\\binom {l+s} {l-s}\\binom {2(s-1)} {s-1}\\nonumber\\\\&=&(-1)^{l+1}\\frac{(N-1)}{N}\\hbox{res}_{w=0}\\frac{1}{(1+w)^{2}w^{l}}\\nonumber\\\\&=& \\frac{l(N-1)}{N}.\n\\end{eqnarray}\n Adding the contributions given by Eqs. (\\ref{t6}), and  (\\ref{t7}), then, we get  \n\\begin{equation}\n\\label{t8}\nR(l)=\\frac{1}{N}\\sum_{n=1}^{N-1}\\frac{\\sin^2(nl\\pi\/N)}{\\sin^2(n\\pi\/N)}=\\frac{l(N-l)}{N}.\n\\end{equation}\n\n\\ \\   Now, we want to evaluate the following trigonometrical sum $$  F_N(l ) = \\frac {1} {N}\\sum_{n=1}^{N-1} \\frac {1-\\cos nl\\pi\/N} {1-\\cos n\\pi\/N},$$ this sum arises in connection with the two-point resistance of a path graph \\cite{Wu}. The evaluation of this term may be done as follows; \n\\begin{eqnarray}\n\\label{t9}\nF_N(l ) &= &\\frac {1} {N}\\sum_{n=1}^{N-1} \\frac {1-\\cos nl\\pi\/N} {1-\\cos n\\pi\/N}\\nonumber\\\\&=&\\frac {1} {N}\\sum_{n=1}^\\frac{N}{2} \\frac {1-\\cos(2n-1)l\\pi\/N} {1-\\cos(2n-1)\\pi\/N}+\\frac {1} {N}\\sum_{n=1}^{\\frac{N}{2}-1} \\frac {1-\\cos 2nl\\pi\/N} {1-\\cos2n\\pi\/N},\n\\end{eqnarray}\nhere,  $N$ is  assumed to be even, similar steps may be used for  $N$ odd. It is interesting to note that in evaluating $F_N(l )$, we only need to compute the first term since the second term is related to the two-point resistance of the $N$-cycle graph given by Eq. (\\ref{t8}). Then, the first term may be written as \n\\begin{eqnarray}\n\\label{t10}\n\\frac {1} {N}\\sum_{n=1}^\\frac{N}{2} \\frac {1-\\cos(2n-1)l\\pi\/N} {1-\\cos(2n-1)l\\pi\/N}&=&\\frac {1} {N} \\sum_{n=1}^\\frac{N}{2} \\frac {\\sin^{2}(2n-1)l\\pi\/2N} {\\sin^{2}(2n-1)l\\pi\/2N}\\nonumber\\\\&=&\\frac{1}{2N}\\sum_{s=1}^{l}(-1)^{s +1}\\frac{l}{l+s}\\binom {l+s} {l-s}2^{2s}\\sum_{n=1}^\\frac{N}{2}\\sin^{2(s-1)}\\frac{(2n-1)\\pi}{2N}.\\nonumber\\\\.\n \\end{eqnarray}\n By using the identity $$\\sum_{n=1}^\\frac{N}{2}\\sin^{2(s-1)}(2n-1)\\pi\/2N= \\frac{1}{2}\\bigg(\\sum_{n=1}^{N-1}\\sin^{2(s-1)}n\\pi\/2N+\\sum_{n=1}^{N-1}(-1)^{n-1}\\sin^{2(s-1)}n\\pi\/2N\\bigg),  $$ \n and the formulas for trigonometrical power sums given in \\cite{Schwatt}, then, one can show\n\\begin{eqnarray}\n\\label{t11}\n\\sum_{n=1}^\\frac{N}{2}\\sin^{2(s-1)}(2n-1)\\pi\/2N=\\frac{2N}{2^{2s}}\\binom {2(s-1)} {s-1},\n\\end{eqnarray}\nwhich in turn, implies that the formula for the first term  should be\n \\begin{eqnarray}\n\\label{t12}\n\\frac {1} {N}\\sum_{n=1}^\\frac{N}{2} \\frac {1-\\cos(2n-1)l\\pi\/N} {1-\\cos(2n-1)\\pi\/N}&=&\\frac{1}{2}\\sum_{s=1}^{l}(-1)^{s +1}\\frac{2l}{l+s}\\binom {l+s} {l-s}\\binom {2(s-1)} {s-1}\\nonumber\\\\&=&\\frac{l}{2}.\n\\end{eqnarray}\nTo compute the second term given in Eq. (\\ref{t9}), we use the following symmetry \nenjoyed by the two-point resistance of the $N$-cycle graph,   $N$ even\n\\begin{eqnarray}\n\\label{t13}\n\\frac {1} {N}\\sum_{n=1}^{N-1} \\frac {1-\\cos 2nl\\pi\/N} {1-\\cos2n\\pi\/N}&=&\\frac {2} {N}\\sum_{n=1}^{\\frac{N}{2}-1} \\frac {1-\\cos 2nl\\pi\/N} {1-\\cos2n\\pi\/N}+\\frac {1} {2N}\\big(1-(-1)^{l}\\big).\n\\end{eqnarray}\nTherefore, the second term may be obtained to give the following closed formula for $F_N(l ) $\n\\begin{equation}\n\\label{t14}\nF_N(l ) = \\frac {1} {N}\\sum_{n=1}^{N-1} \\frac {1-\\cos nl\\pi\/N} {1-\\cos n\\pi\/N}= l-\\frac{1}{N}\\bigg(\\frac{l^2}{2}+\\frac{1}{4}(1-(-1)^{l})\\bigg)\n\\end{equation}\nThis is in a complete agreement with the formula for the Green's function for the path graph in \\cite{Wu}\n\\section{Trigonometrical sum connected with  Scott's conjecture}\n\n\\ \\ \n In  proving  R. F Scott's conjecture on the permanent  of the Cauchy matrix,  Minc in \\cite{Minc} needed to evaluate the following trigonometrical sum;\n  \\begin{equation}\n\\label{sc0}\n \\sum_{n=1}^{N} \\frac {\\cos (2n-1)l\\pi\/N} {1-\\cos (2n-1)\\pi\/N}.\n \\end{equation}\n He obtained a closed-form formula for this sum using induction,\n and the sum turns out to  be equal to  $\\frac{N}{2}(N-2l)$.  A short  time  later, Stembridge and  Todd \\cite{Todd}, gave a proof for the evaluation for this  sum, based on linear algebra. Here, we give a short derivation for this sum using  our formula given by Eq. (\\ref{t12}), and the well-known identity \n \\begin{equation}\n\\label{sc1}\n \\sum_{n=1}^{N-1}\\frac{1}{\\sin^2(n\\pi\/N)}=\\frac{N^{2}-1}{3}.\n\\end{equation}\nOur derivation follows easily by realizing that  Eq. (\\ref{t12}) is symmetric under the shift $n\\rightarrow N-n$, and as a consequence one  gets \n\\begin{equation}\n\\label{sc2}\n\\sum_{n=1}^{N} \\frac {1-\\cos(2n-1)l\\pi\/N} {1-\\cos(2n-1)\\pi\/N}=N l.\n\\end{equation}\nIn order to evaluate the sum in  Eq. (\\ref{sc0}), we need a formula for the sum $$\\sum_{n=1}^{N} \\frac {1} {1-\\cos(2n-1)\\pi\/N} =\\frac{1}{2}\\sum_{n=1}^{N} \\frac {1} {\\sin^{2}(2n-1)\\pi\/2N }.$$\nThe latter may be evaluated as follows\n\\begin{eqnarray}\n\\label{sc3}\n\\frac{1}{2}\\sum_{n=1}^{N} \\frac {1} {\\sin^{2}(2n-1)\\pi\/2N}&=&\\frac{1}{2}\\sum_{n=1}^{2N-1}\\frac{1}{\\sin^2(n\\pi\/2N)}-\\frac{1}{2}\\sum_{n=1}^{N-1}\\frac{1}{\\sin^2(2n\\pi\/2N)}\\nonumber\\\\&=&\\frac{N^{2}}{2}.\n\\end{eqnarray}\nIn obtainnig Eq. (\\ref{sc3}), we used the identity given in Eq. (\\ref{sc1}), thus, using Eq. (\\ref{sc2}) and  Eq. (\\ref{sc3}), we may write\n\\begin{equation}\n\\label{sc4}\n \\sum_{n=1}^{N} \\frac {\\cos (2n-1)l\\pi\/N} {1-\\cos (2n-1)\\pi\/N}=\\frac{N^2}{2}-Nl.\n\\end{equation}\nThis is exactly the result obtained by Minc, Stembridge and  Todd \\cite{Minc,Todd}.\n\\section{Trigonometrical sums arising in the chiral Potts model and in the Verlinde's formula}\n\n\\ \\ In this section  the  trigonometrical sum $T_{4}(l):=\\sum_{n=1}^{N-1}\\frac{\\sin^2(nl\\pi\/N)}{\\sin^4(n\\pi\/N)} $ is evaluated in a closed form using the residue operator. We also give an almost closed formula for the general case $T_{2m}(l):=\\sum_{n=1}^{N-1}\\frac{\\sin^{2}(nl\\pi\/N)}{\\sin^{2m}(n\\pi\/N)} $, for $m\\geq1$. The first  trigonometrical sum arises  in the work of  McCoy  and Orrick  on the chiral Potts model \\cite {McCoy},  this sum including other trigonometrical identities  were proved  by Gervois and Mehta \\cite {Mehta}. The second sum namely the sum  $T_{2m}(l)$,  was considered by  Gervois and Mehta \\cite {Mehta} using a recursion formula. Here, we will obtain   recursion formulas for both  $T_{2m}(l)$ and $$T_{2m}:=\\sum_{n=1}^{N-1}\\frac{1}{\\sin^{2m}(n\\pi\/N)}.$$\n\n\\ \\  If, we set $N=k+2$ and $m =g-1$, $k,g$ being the level of the $su(2)$ Kac-Moody algebra, and the genus of the Riemann surface respectively. Then, the  sum $ T_{2m}$ up to to some normalization factor  is nothing but the dimension of the space of the conformal blocks of the $SU(2)$ WZW model. As a consequence, the recursion formula derived for  $ T_{2m}$, may be used to obtain the expression for the dimension of the space of  the conformal blocks for a given genus $g$. Similar computations are carried out for the twisted trigonometrical sum $$T_{2m}^{t}:=\\sum_{n=1}^{N-1}(-1)^{n+1}\\frac{1}{\\sin^{2m}(n\\pi\/N)}.$$ This is related to the dimension of the space of the conformal blocks of the $SO(3)$ WZW model.\n \\subsection {Trigonometrical sums and the perturbative chiral Potts model}\n \n \\ \\  Let us first start with the trigonometrical sums arising in the perturbative treatment  of the the chiral Potts model \\cite {McCoy}. Techniques of the previous section, may be used to evaluate the  sum  $T_{4}(l)$, as follows\n\\begin{eqnarray}\n\\label{t15}\nT_{4}(l)&=&\\sum_{n=1}^{N-1}\\frac{\\sin^2(nl\\pi\/N)}{\\sin^4(n\\pi\/N)}\\nonumber\\\\&=&l^{2}\\sum_{n=1}^{N-1}\\frac{1}{\\sin^2(n\\pi\/N)}+\\frac{1}{2}\\sum_{s=2}^{l}(-1)^{s +1}\\frac{l}{l+s}\\binom {l+s} {l-s}2^{2s}\\sum_{n=1}^{N-1} \\sin^{2(s-2)}(n\\pi\/N)\\nonumber\\\\&=&\\frac{l^2}{3}(N^{2}-1)+8\\sum_{s=2}^{l}(-1)^{s }\\frac{2l}{l+s}\\binom {l+s} {l-s}\\sum_{t=1}^{s-2}(-1)^{t }\\binom {2(s-2)}{s-2-t}\\nonumber\\\\&+& 4(N-1)\\sum_{s=2}^{l}(-1)^{s +1}\\frac{2l}{l+s}\\binom {l+s} {l-s}\\binom {2(s-2)}{s-2},\n\\end{eqnarray}\nthe first term in the above equation follows from the well-known identity $$\\sum_{n=1}^{N-1}\\frac{1}{\\sin^2(n\\pi\/N)}=\\frac{N^{2}-1}{3},$$ while the second and the third terms may computed using the residue operator as in the previous section to give,\n\\begin{eqnarray}\n\\label{t16}\n\\sum_{s=2}^{l}(-1)^{s +1}\\frac{2l}{l+s}\\binom {l+s} {l-s}\\sum_{t=1}^{s-2}(-1)^{t }\\binom {2(s-2)}{s-2-t}&=&(-1)^{l+1} \\hbox{res}_{w=0}\\frac{1}{(1+w)^{5}w^{l-2}}\\nonumber\\\\&=&\\frac{1}{4!}(l+1)l(l-1)(l-2),\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\label{t17}\n\\sum_{s=2}^{l}(-1)^{s +1}\\frac{2l}{l+s}\\binom {l+s} {l-s}\\binom {2(s-2)}{s-2}&=& (-1)^{l+1}\\hbox{res}_{w=0}\\frac{1}{(1+w)^{4}w^{l-1}}\\nonumber\\\\&=&-\\frac{1}{3!}(l+1)l(l-1).\n\\end{eqnarray}\nTherefore, the closed formula for the sum given in Eq. (\\ref{t15}), reads\n\\begin{eqnarray}\n\\label{t18}\n\\sum_{n=1}^{N-1}\\frac{\\sin^2(nl\\pi\/N)}{\\sin^4(n\\pi\/N)}&=&\\frac{l^2}{3}(N^{2}-1)+\\frac{1}{3}(l+1)l(l-1)(l-2)-\\frac{2(N-1)}{3}(l+1)l(l-1)\\nonumber\\\\&=&\\frac{l^2}{3}(N-l)(N-l)+\\frac{2}{3}(N-l).\n\\end{eqnarray}\nThis is exactly the result obtained by  Gervois and Mehta using a recursion formula satisfied by  $T_{2m}(l)$ \\cite {Mehta}. Next, we will  give another recursion formula for the  sum  $T_{2m}(l)$. Now, $T_{2m}(l)$, may be written as \n\\begin{eqnarray}\n\\label{t19}\nT_{2m}(l)&=&\\sum_{n=1}^{N-1}\\frac{\\sin^2(nl\\pi\/N)}{\\sin^{2m}(n\\pi\/N)}\\nonumber\\\\&=&\\frac{1}{2}\\sum_{n=1}^{N-1}\\sum_{s=1}^{m-1}(-1)^{s+1 }\\frac{l}{l+s}\\binom {l+s} {l-s}2^{2s}\\frac{1}{\\sin^{2(m-s)}(n\\pi\/N)}\\nonumber\\\\&+&\\frac{1}{2}\\sum_{s=m}^{l}(-1)^{s +1}\\frac{l}{l+s}\\binom {l+s} {l-s}2^{2s}\\sum_{n=1}^{N-1} \\sin^{2(s-m)}(n\\pi\/N).  \n\\end{eqnarray}\nThe first term on the right-hand side, is written in terms of the sum $$T_{2k}=:\\sum_{n=1}^{N-1}\\frac{1}{\\sin^{2k}(n\\pi\/N)}. $$ This may be  computed  \\cite {Mehta}, using $$T_{2k}=\\sum_{n=1}^{N-1}\\Big(\\cot^{2}(\\frac{n\\pi}{N})+1\\Big)^{k}=\\sum_{l=1}^{k} \\binom{k}{l}S_{l}, $$ where $S_{l}=\\sum_{n=1}^{N-1}\\Big(\\cot^{2}(\\frac{n\\pi}{N}) \\Big)^{2l}$  and a recurrence  relation satisfied by the power sums $S_{l} $. It turns out that  $T_{2k}$,   may also be obtained  using a recursion formula, this will be shown shortly. Now, the second term may be written as follows \n\\begin{eqnarray}\n\\label{t20}\n\\tilde{T}_{2m}(l):&=&\\frac{1}{2}\\sum_{s=m}^{l}(-1)^{s +1}\\frac{l}{l+s}\\binom {l+s} {l-s}2^{2s}\\sum_{n=1}^{N-1} \\sin^{2(s-m)}(n\\pi\/N)\\nonumber\\\\&=&2^{2m-1}\\sum_{s=m}^{l}(-1)^{s }\\frac{2l}{l+s}\\binom {l+s} {l-s}\\sum_{t=1}^{s-m}(-1)^{t }\\binom {2(s-m)}{s-m-t}\\nonumber\\\\&+&2^{2m-2}(N-1)\\sum_{s=m}^{l}(-1)^{s +1}\\frac{2l}{l+s}\\binom {l+s} {l-s}\\binom {2(s-m)}{s-m}\\nonumber\\\\&=&(-1)^{l+1}2^{2m-1}\\hbox{res}_{w=0}\\frac{1}{(1+w)^{2m+1}w^{l-m}}\\nonumber\\\\&+&(-1)^{l+1}2^{2m-2}(N-1)\\hbox{res}_{w=0}\\frac{1}{(1+w)^{2m}w^{l+1-m}}\\nonumber\\\\&=&(-1)^{m}2^{2m-1}\\frac{(l+m-1)!}{(l-m-1)!(2m)!}+ (-1)^{m+1}(N-1)2^{2m-2}\\frac{(l+m-1)!}{(l-m)!(2m-1)!}\\nonumber\\\\.\n\\end{eqnarray}\nTherefore, we succeeded  in writing $\\tilde{T}_{2m}(l)$  in a closed form  formula,   One can check easily that our results agree with those given in  \\cite {Mehta}, and so the formula for  $T_{2m}(l)$ becomes\n\\begin{eqnarray}\n\\label{t21}\nT_{2m}(l)&=&\\sum_{n=1}^{N-1}\\frac{\\sin^2(nl\\pi\/N)}{\\sin^{2m}(n\\pi\/N)}\\nonumber\\\\&=&\\frac{1}{2}\\sum_{s=1}^{m-1}(-1)^{s+1 }\\frac{l}{l+s}\\binom {l+s} {l-s}2^{2s}T_{2(m-s)}\\nonumber\\\\&+&(-1)^{m+1}2^{2m-1}\\frac{(l+m-1)!}{(l-m)!(2m)!}(mN-l).\n\\end{eqnarray}\nsetting   $m=1,2$, then, our previous  results given by Eq's. (\\ref{t8}) and (\\ref{t18}) respectively  are recovered. From Eq. (\\ref{t21}), it is clear that  in order to have a closed formula for  $T_{2m}(l)$, one  needs also,  the exact  expressions  for $T_{2k}$, $ k=1\\cdots, m-1$. Next, we will  show that $ T_{2k}$ satisfies a recursion formula that involves  the $T_{2k}$'s.  The expression for $T_{2m}$  may be obtained from  $T_{2m}(l)$ from the following simple formula;\n\\begin{eqnarray}\n\\label{t22}\n\\sum_{l=1}^{N-1}T_{2m}(l)&=&\\sum_{l=1}^{N-1}\\sum_{n=1}^{N-1}\\frac{\\sin^2(nl\\pi\/N)}{\\sin^{2m}(n\\pi\/N)}\\nonumber\\\\&=&\\frac{N}{2}\\sum_{n=1}^{N-1}\\frac{1}{\\sin^{2m}(n\\pi\/N)},\n\\end{eqnarray}\nTherefore, we may write\n\\begin{eqnarray}\n\\label{t23}\nT_{2m}&=&\\sum_{n=1}^{N-1}\\frac{1}{\\sin^{2m}(n\\pi\/N)}\\nonumber\\\\&=&\\frac{1}{N}\\sum_{l=1}^{N-1}\\sum_{s=1}^{m-1}(-1)^{s+1 }\\frac{l}{l+s}\\binom {l+s} {l-s}2^{2s}T_{2(m-s)}\\nonumber\\\\&+&\\frac{2}{N}\\sum_{l=1}^{N-1}(-1)^{l+1}2^{2m-1}\\hbox{res}_{w=0}\\frac{1}{(1+w)^{2m+1}w^{l-m}}\\nonumber\\\\&+&\\frac{2}{N}\\sum_{l=1}^{N-1}(-1)^{l+1}2^{2m-2}(N-1)\\hbox{res}_{w=0}\\frac{1}{(1+w)^{2m}w^{l+1-m}}\\nonumber\\\\&=&\\frac{1}{N}\\sum_{l=1}^{N-1}\\sum_{s=1}^{m-1}(-1)^{s+1 }\\frac{l}{l+s}\\binom {l+s} {l-s}2^{2s}T_{2(m-s)}\\nonumber\\\\&+&(-1)^{m+1}2^{2m-1}\\frac{(N+m-1)!}{N (N-m-1)!(2m+1)!}(2mN+1-N).\n\\end{eqnarray}\nAs a result, from our recursion formula,  the different $T_{2m}$'s may be obtained directly, we do not have to  use  the recurrence relation satisfied by the power sum $S_{l}$ \\cite {Mehta}. For $m=1$ the first term in Eq. (\\ref{t23}) does not contribute and the second term gives the well known formula $ T_{2}=\\frac{N^{2}-1}{3}$. Now, for $m=2$, the first term may be computed to give  $ \\frac{{(N^2-1)(N-1)}(2N-1)}{9}$, while the second term gives $ -\\frac{{(N^2-1)(N-2)}(3N+1)}{15}$, and hence,   $T_{4}= \\frac{{(N^2-1)(N^2+11)}}{45}$ in a full agreement with  \\cite {Mehta}, \\cite { Berndt}.\n\\subsection{The Verlinde dimension formula}\n\n\\ \\ The verlinde dimension formula may be obtained simply by setting $m=g-1$,  $N=k+2$ in the expression of  $T_{2m}$, where $g\\ge2$, $ k$ are the genus of the Riemann surface, and the level of the lie algebra $SU(2)$, respectively. Then,   $T_{2(g-1)}$ up to some normalization factor, is  the dimension of the space  of conformal blocks $V_{g}$ of the $SU(2)$ WZW model \\cite{Verlinde},\n \\begin{eqnarray}\n\\label{t24}\ndimV_{g,k}&=&\\Big(\\frac{k+2}{2}\\Big)^{g-1}\\sum_{n=1}^{k+1}\\frac{1}{\\sin^{2g-2}n\\pi\/(k+2)}\\nonumber\\\\&=&\\Big(\\frac{k+2}{2}\\Big)^{g-1} \\frac{1}{(k+2)}\\sum_{l=1}^{k+1}\\sum_{s=1}^{g-2}(-1)^{s+1 }\\frac{l}{l+s}\\binom {l+s} {l-s}2^{2s}T_{2g-2 -2s} \\nonumber\\\\&+&(-1)^{g}2^{2g-3}\\Big(\\frac{k+2}{2}\\Big)^{g-1} \\frac{1}{(k+2)}\\frac{1}{(2g-1)}\\binom{k+g}{2g-2}\\big((k+2)(2g-3)+1\\big).\n\\end{eqnarray}\n As a consequence, the dimension of the space of the conformal blocks of the $SU(2)$ WZW model,  may be  computed using our recursion formula for $T_{2k}$. Our formula Eq. (\\ref{t24}) may be used to give \n\\begin{eqnarray}\n\\label{t25}\ndimV_{2,k}&=&\\frac{(k+1)(k+2)(k+3)}{6}\\nonumber\\\\dimV_{3,k}&=&\\frac{1}{5}\\frac{(k+1)(k+2)(k+3)}{6}\\Big[\\frac{(k+1)(k+2)(k+3)}{6}+2(k+2)\\Big]\\nonumber\\\\dimV_{4,k}&=&\\frac{1}{7}.\\frac{1}{5}\\frac{(k+1)(k+2)(k+3)}{6}\\nonumber\\\\&.&\\Big[\\frac{(k+1)(k+2)(k+3)}{6}\\Big[\\frac{2(k+1)(k+2)(k+3)+27(k+2)}{6}\\Big]+6(k+2)^{2}\\Big].\\nonumber\\\\\n\\end{eqnarray}\nOur first two expressions for dimension of the space of conformal blocks agree with those computed using conformal field theory \\footnote{The last term $k+2$ in the expression of  $dimV_{3,k}$  should be corrected  in \\cite{Piunikhin} in order to make it a positive integer.} \\cite{Piunikhin}. For $g=4$, our formula  for $dimV_{4,k} $ is identical to the formula given by Zagier \\cite{Zagier} provided  the shift $k\\rightarrow k+2$ is taken. This shift is natural, since Zagier defined the dimension of the space conformal  blocks as $dimV_{g,k-2}$.\n\n\\ \\  For the WZW model based on $SO(3)$, the level $k$ must be even \\cite {Thaddeus}, and the formula for the  dimension of the twisted space of the conformal blocks $V_{g,k}^{t}$,  may be written as \n$$dimV_{g,k}^{t}=\\Big(\\frac{k+2}{2}\\Big)^{g-1}\\sum_{n=1}^{k+1}(-1)^{n+1}\\frac{1}{\\sin^{2g-2}n\\pi\/(k+2)}.$$ In order to  derive a recursion formula for the  dimension $ dimV_{g,k}^{t}$, we first note that the expression for the twisted version of the  trigonometrical sum $ T_{2m}(l)$ given by Eq. (\\ref{t19}) is\n\\begin{eqnarray}\n\\label{t26}\nT_{2m}^{t}(l)&=&\\sum_{n=1}^{N-1}(-1)^{n+1} \\frac{\\sin^2(nl\\pi\/N)}{\\sin^{2m}(n\\pi\/N)}\\nonumber\\\\&=&\\frac{1}{2}\\sum_{s=1}^{m-1}(-1)^{s+1 }\\frac{l}{l+s}\\binom {l+s} {l-s}2^{2s}T_{2(m-g)}^{t}\\nonumber\\\\&+&\\frac{1}{2}\\sum_{s=m}^{l}(-1)^{s +1}\\frac{l}{l+s}\\binom {l+s} {l-s}2^{2s}\\sum_{n=1}^{N-1} (-1)^{n+1}\\sin^{2(s-m)}(n\\pi\/N),  \n\\end{eqnarray}\nwhere $$T_{2m}^{t}=\\sum_{n=1}^{N-1}(-1)^{n+1}\\frac{1}{\\sin^{2m}(n\\pi\/N)} $$\nThe trigonometrical sum $$ \\sum_{n=1}^{N-1} (-1)^{n+1}\\sin^{2(s-m)}(n\\pi\/N),$$ is non-vanishing only if $ N$ is even\\cite{Schwatt} and is given by\n \\begin{eqnarray}\n\\label{t27}\n \\sum_{n=1}^{N-1} (-1)^{n+1}\\sin^{2(s-m)}(n\\pi\/N)&=&2^{2(m-s)+1}\\sum_{t=1}^{s-m}(-1)^{t }\\binom {2(s-m)}{s-m-t}\\nonumber\\\\&+&2^{2(m-s)} \\binom {2(s-m)}{s-m}\n\\end{eqnarray}\nThis formula shows clearly that the dimension of the twisted space of the conformal blocks is is non-vanishing only if $k$ is even, $N=k+2$ in agreement with the algebro-geometrical argument \\cite {Thaddeus}. Then, the expression for $T_{2m}^{t}$ may be written as\n\\begin{eqnarray}\n\\label{t28}\nT_{2m}(l)^{t}&=&\\sum_{n=1}^{N-1}(-1)^{n+1}\\frac{\\sin^2(nl\\pi\/N)}{\\sin^{2m}(n\\pi\/N)}\\nonumber\\\\&=&\\frac{1}{2}\\sum_{n=1}^{N-1}(-1)^{n+1}\\sum_{s=1}^{m-1}(-1)^{s+1 }\\frac{l}{l+s}\\binom {l+s} {l-s}2^{2s}\\frac{1}{\\sin^{2(m-s)}(n\\pi\/N)}\\nonumber\\\\&+&2^{2m-1}\\sum_{s=m}^{l}(-1)^{s +1}\\frac{2l}{l+s}\\binom {l+s} {l-s}\\sum_{t=1}^{s-m}(-1)^{t }\\binom {2(s-m)}{s-m-t}\\nonumber\\\\&+&2^{2m-2}\\sum_{s=m}^{l}(-1)^{s +1}\\frac{2l}{l+s}\\binom {l+s} {l-s}\\binom {2(s-m)}{s-m}\\nonumber\\\\&=&\\frac{1}{2}\\sum_{s=1}^{m-1}(-1)^{s+1 }\\frac{l}{l+s}\\binom {l+s} {l-s}2^{2s}T_{2(m-s)}^{t}\\nonumber\\\\&+&(-1)^{m+1}2^{2m-1}\\frac{(l+m-1)!}{(l-m)!(2m)!}l.  \n\\end{eqnarray}\nThe recursion formula for the  twisted trigonometrical sum $ T_{2m}^{t}$, may be derived by setting $ l= N\/2$ in Eq. (\\ref{t28})\n\\begin{eqnarray}\n\\label{t29}\nT_{2m}^{t}&=&\\sum_{n=1}^{N-1}(-1)^{n+1}\\frac{1}{\\sin^{2m}(n\\pi\/N)}\\nonumber\\\\&=&\\sum_{s=1}^{m-1}(-1)^{s+1 }\\frac{N\/2}{N\/2+s}\\binom {N\/2+s} {N\/2-s}2^{2s}T_{2(m-s)}^{t}\\nonumber\\\\&+&(-1)^{m+1}2^{2m}\\frac{(N\/2+m-1)!}{(N\/2-m)!(2m)!}N\/2 \\nonumber\\\\&-&\\sum_{n=1}^{N-1}\\frac{1}{\\sin^{2m}(n\\pi\/N)}.\n\\end{eqnarray}\nFor $m=1$ and $m=2$ the twisted trigonometrical sums are\n\\begin{equation}\n\\label{30}\nT_{2}^{t}=\\frac{N^{2}+2}{6}\n\\end{equation}\nand\n\\begin{equation}\n\\label{31}\nT_{4}^{t}=\\frac{7N^{4}+40N^{2}+88}{360},\n\\end{equation}\nrespectively. In obtaining these results we used the expressions for $T_{2}$ and $T_{4}$. These twisted trigonometrical sums appeared  earlier as coefficients of certain generating function  \\cite{ Zagier}. Using the recursion formula Eq. (\\ref{t29}), the twisted trigonometrical sum $ T_{6}^{t}$ is\n\\begin{equation}\n\\label{32}\nT_{4}^{t}=\\frac{31N^{6}+294N^{4}+1344N^{2}+3056}{15120}. \n\\end{equation}\nThe twisted trigonometrical sum formula given by  Eq. (\\ref{t29}), implies that the dimension of the twisted space of the conformal blocks is may be deduced for any genus $g\\geq2$, through the following formula \n\\begin{eqnarray}\n\\label{t33}\ndimV_{g,k}^{t}&=&\\Big(\\frac{k+2}{2}\\Big)^{g-1}\\sum_{n=1}^{k+1}(-1)^{n+1}\\frac{1}{\\sin^{2g-2}n\\pi\/(k+2)}\\nonumber\\\\&=&\\Big(\\frac{k+2}{2}\\Big)^{g-1}\\sum_{s=1}^{g-2}(-1)^{s+1 }\\frac{(k+2)\/2}{(k+2)\/2+s}\\binom {(k+2)\/2+s} {(k+2)\/2-s}2^{2s}T_{2g-2-2s}^{t}\\nonumber\\\\&+&(-1)^{g}2^{2g-2}\\Big(\\frac{k+2}{2}\\Big)^{g-1}\\frac{((k+2)\/2+g-2)!}{((k+2)\/2-g+1)!(2g-2)!}(k+2)\/2 \\nonumber\\\\&-&dimV_{g,k}.\n\\end{eqnarray}\nNote that, the relation  between $dimV_{g,k}^{t} $ and $ dimV_{g,k}$ is expected from the simple identity $$dimV_{g,k-2}^{t}= dimV_{g,k-2}-2^{g}dimV_{g,k\/2-1},$$ where $k$ even. The  formula  by Zagier \\cite{Zagier}   for $dimV_{g,k-2}$,  may be obtained using the following generating function \n$$ \\sum_{g=1}^{\\infty}dimV_{g,k-2}\\Big(\\frac{2}{k}\\sin^{2}x\\Big)^{g-1}=\\frac{k\\sin (k-1)x}{\\sin kx \\cos x }.$$\n\\section{The corner-to-corner resistance  and the Kirchhoff index of a $ 2\\times N $ resistor network}\n\n\\ \\ In general, it is hard to have a closed-form expression for the two-point resistance of a resistor network, however, if the latter has certain symmetries like circulant resistor network, then this may be possible \\cite{Chair1}. The situation gets more and more complicated in two and three dimensional resistor networks \\cite{Wu}, as the exact  two-point resistance are expressed in terms of the double and triple summations.  It turns out that the  recently developed techniques by the author \\cite{Chair1},  may be used  to obtain exact formula for the two-point resistance of the first non-trivial  $ 2\\times N $ resistor network \\cite{Chair2}. In this section, we derive an exact formula for the corner-to-corner resistance and the total effective  resistance of a $ 2\\times N $ resistor network. The total effective resistance  is also known as the  Kirchhoff index, this index  was introduced in chemistry  as a molecular structure descriptor, it is used  for discriminating among different molecules with similar shapes and structures \\cite{Randic}. At the moment, the only   exact two-point resistance not written as a double summation of an $ M\\times N $ resistor network is the asymptotic expansion of the corner-to-corner resistance \\cite{Essam, Huang}.  It is known that the value of  the asymptotic expansion of the corner-to-corner resistance of a rectangular resistor network  provides a lower bound to the resistance of compact percolation clusters \\cite{ Domany}.\n\\subsection{The exact evaluation of the corner-to-corner resistance }\n\n\\ \\  The exact expression for the resistance between two nodes of a rectangular network of resistors with\nfree boundary conditions was given by Wu \\cite{Wu}. Suppose that  the resistances in the two spatial directions\nare r = s = 1, then the resistance  ${R}_{\\,\\rm free}$ between two nodes ${\\bf r}_1=(x_1, y_1)$\nand ${\\bf r}_2=(x_2, y_2)$  is \n\\begin{eqnarray}\n&&R_{\\{M\\times N\\}}^{\\,\\rm free}({\\bf r}_1,{\\bf r}_2) = \\frac {1} {N} \\Big| x_1 -x_2 \\Big|  + \\frac 1 {M} \\Big| y_1 - y_2 \\Big| +\\frac 2 {MN} \\nonumber\\\\\n&&\\times{\\sum_{m=1}^{M-1}\\sum_{n=1}^{N-1}\n\\frac {\\Big[\\cos\\Big(x_1+\\frac 1 2\\Big)\\theta_m \\cos\\Big(y_1+\\frac 1 2\\Big)\\phi_n\n - \\cos\\Big(x_2+\\frac 1 2\\Big)\\theta_m \\cos\\Big(y_2+\\frac 1 2\\Big)\\phi_n\n\\Big]^2 } \n{ (1-\\cos \\theta_m )  +(1-\\cos \\phi_n )  } } ,\\nonumber \\\\\n\\label{cc1}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\theta_m= \\frac{m\\pi} M, \\hskip1cm \\phi_n= \\frac{n\\pi} N.\\nonumber\n\\end{eqnarray} \nIn order to compute the corner-to-corner resistance of a $2 \\times N$  resistor network, we set $ M = 2$, ${\\bf r}_1=(0, 0)$\nand ${\\bf r}_2=(1, N-1)$ into Eq. (\\ref{cc1}), and so\nthe double sum of the above equation is reduced to a single sum, and the corner-to-corner\nresistance may be written as\n\\begin{eqnarray}\nR_{\\{2\\times N\\}}^{\\,\\rm free}((0,0),(1,N-1))&=&\\frac{1}{N}+\\frac{N-1}{2} \\nonumber\\\\&+& \\frac{1}{3N}\\sum_{n=1}^{N-1} \\frac{(1+(-1)^{n})(1+\\cos n\\pi\/N)}{2(1-2\/3\\cos^{2}n\\pi\/2N)}.\n\\label{cc2}\n\\end{eqnarray}\nFor $N$ even, the sum over  $n$ may be reduced to\n\\begin{eqnarray}\n\\frac{2}{3N}\\sum_{n=1}^{N\/2-1} \\frac{\\cos^{2} n\\pi\/N}{(1-2\/3\\cos^{2}n\\pi\/N)}&=&\\frac{1}{3N}\\sum_{n=1}^{N-1} \\frac{\\cos^{2} n\\pi\/N}{(1-2\/3\\cos^{2}n\\pi\/N)}\\nonumber\\\\&=&\\frac{1}{3N}\\sum_{j=0}^{\\infty}(2\/3)^{j}\\sum_{n=1}^{N-1}\\cos^{2(j+1)} n\\pi\/N,\n\\label{cc3}\n\\end{eqnarray}\nto evaluate this sum, we follow closely the method developed by the author in \\cite{Chair1}.  As explained in \\cite{Chair1},\nthe formula for the sum $\\sum_{n=1}^{N-1} \\cos^{2J} n\\pi\/N $,  given by Schwatt \\cite{Schwatt} is not the right one to use, we use instead the formula \n\\begin{eqnarray}\n\\sum_{n=1}^{N-1} \\cos^{2J} n\\pi\/N =-1+\\frac{N}{2^{2j-1}}\\sum_{p=1}^{[J\/N]}\\binom {2J}{J-pN} +\\frac{1}{2^{2j}}\\binom {2J}{J},\n\\label{cc4}\n\\end{eqnarray}\nthus, the sum contribution to the corner-to-corner resistance using the residue representation of binomials is\n\\begin{eqnarray}\n\\frac{1}{3N}\\sum_{j=0}^{\\infty}(2\/3)^{j}\\sum_{n=1}^{N-1}\\cos^{2(j+1)} n\\pi\/N&=&-\\frac{1}{N}+\\sum_{j=0}^{\\infty}\\hbox{res}_{w}\\frac{(1+w)^{2j}}{(6w)^{j}}\\frac{w^{N}}{w(1-w^{N})}\\nonumber\\\\&+&\\frac{1}{2}\\Big[\\hbox{res}_w (1-4w){^{-1\/2}}\\sum_{j=0}^{\\infty}(1\/6w)^{j}{w^{-1}}-1\\Big]\\nonumber\\\\&=&-\\frac{1}{N}+\\sqrt{3}\\frac{(2-\\sqrt{3})^{N}}{1-(2-\\sqrt{3})^{N}}+\\frac{1}{2}(\\sqrt{3}-1).\n\\label{cc5}\n\\end{eqnarray}\nFinally, the corner-to-corner resistance of  $ 2\\times N $ resistor network becomes,\n\\begin{eqnarray}\nR_{\\{2\\times N\\}}^{\\,\\rm free}((0,0),(1,N-1))&=&\\frac{N-1}{2} +\\sqrt{3}\\frac{(2-\\sqrt{3})^{N}}{1-(2-\\sqrt{3})^{N}}+\\frac{1}{2}(\\sqrt{3}-1).\n\\label{cc6}\n\\end{eqnarray}\n It is not difficult to see that this formula is also valid for $N$ odd.\n  \n  \\ Examples. For $N=2,3, 4$ our formula  Eq. (\\ref{cc6}), gives\n \\begin{eqnarray}\n R_{\\{2\\times 2\\}}^{\\,\\rm free}((0,0),(1,1))&=&1\\nonumber\\\\R_{\\{2\\times 3\\}}^{\\,\\rm free}((0,0),(1,2))&=&1.4\\nonumber\\\\R_{\\{2\\times 4\\}}^{\\,\\rm free}((0,0),(1,3))&=&1.875,\n \\label{cc7}\n \\end{eqnarray} \nthese results are in a full agreement with Eq. (\\ref{cc2}).\n\\subsection{The Kirchhoff index}\n\n\\ \\ The  computation of  the total effective resistance of a  $ 2\\times N $ resistor network, that is, the  Kirchhoff index, may be computed in two ways.  It may be evaluated  by summing over all effective resistances between nodes of a given resistor  network, or alternatively by summing over all eigenvalues of a Laplacian associated with resistor  network \\cite{Gutman}. So,  we do not have to know the effective resistance between each node to compute the total effective resistance of a resistor network. The formula that gives  the Kirchhoff index of a resistor network in terms of  the eigenvalues is\n $$ Kf(G)=N\\sum_{n=1}^{N-1}\\frac{1}{\\lambda_n},$$  where $\\lambda_{n}$ are the eigenvalues of the Laplacian of the network, or the graph $G$ made of nodes and edges considered as  unit resistors.  Our network is given by the cartesian product $ 2\\times N$, that is, made of two path lines with $N$ nodes, and $N$ path lines with two nodes. Now,  the Kirchhoff index of a path line is    $$Kf(P_{n})=N\\sum_{n=1}^{N-1}\\frac{1}{4\\sin^2(n\\pi\/2N)}=\\frac{N}{8}\\Big[ \\sum_{n=1}^{2N-1}\\frac{1}{\\sin^2(n\\pi\/2N)}-1\\Big]=\\frac{N^{3}-N}{6}.$$   \n  Thus, the contribution from these path lines is $N+\\frac{N^{3}-N}{3}$, by connecting the system together, then the corresponding  eigenvalues of the laplacian are $\\lambda_{1,n}= 3(1-2\/3\\cos^{2}n\\pi\/2N)$. As a consequence, the Kirchhoff index  of a  $ 2\\times N$, resistor network can be written as\n\\begin{eqnarray} \nKf(2\\times N)=N+\\frac{N^{3}-N}{3}+N\\sum_{n=1}^{N-1}\\frac{1}{3(1-2\/3\\cos^{2}n\\pi\/2N)},\n\\label{Kf1}\n\\end{eqnarray}\nNote that, our simple deduction of this expression gives the same value of the Kirchhoff index given by theorem 4.1 in \\cite{Yang}. The above sum seems  difficult to  evaluate, however,  using a simple trick, we will be able to get a closed form for the Kirchhoff index. To that end, let us write\n\\begin{eqnarray}\n\\label{Kf2}\n\\sum_{n=1}^{N-1}\\frac{1}{(1-2\/3\\cos^{2}n\\pi\/2N)}&=&\\sum_{n=1}^{N\/2-1}\\frac{1}{(1-2\/3\\cos^{2}n\\pi\/N)}\\nonumber\\\\&+&\\sum_{n=1}^{N\/2}\\frac{1}{(1-2\/3\\cos^{2}(2n-1)\\pi\/2N)},\n\\end{eqnarray}\nwhere $N$ is assumed to be even. The first sum may be \ncarried out using the following trick;\n\\begin{eqnarray}\n\\sum_{j=0}^{\\infty}(2\/3)^{j}\\sum_{n=1}^{N-1}\\cos^{2(j+1)} n\\pi\/N&=&\\frac{3}{2}\\Big[\\sum_{j=0}^{\\infty}(2\/3)^{j}\\sum_{n=1}^{N-1}\\cos^{2j} n\\pi\/N-(N-1)\\Big].\n\\label{Kf3}\n\\end{eqnarray}\nNow, the sum on the left-hand side was computed before, see Eq. (\\ref{cc5}), then one may deduce\n\\begin{eqnarray}\n\\sum_{n=1}^{N-1}\\frac{1}{(1-2\/3\\cos^{2}n\\pi\/N)}&=&\\sum_{j=0}^{\\infty}(2\/3)^{j}\\sum_{n=1}^{N-1}\\cos^{2j} n\\pi\/N \\nonumber\\\\&=&-3+\\frac{6N(2-\\sqrt{3})^{N}}{\\sqrt{3}(1-(2-\\sqrt{3})^{N})}+\\sqrt{3}N\n\\label{Kf5}.\n\\end{eqnarray}\nand so,\n\\begin{eqnarray}\n\\sum_{n=1}^{N\/2-1}\\frac{1}{(1-2\/3\\cos^{2}n\\pi\/N)}&=&\\frac{1}{2}\\Big[\\sum_{n=1}^{N-1}\\frac{1}{(1-2\/3\\cos^{2}n\\pi\/N)}-1\\Big]\\nonumber\\\\&=&-2+\\frac{3N(2-\\sqrt{3})^{N}}{\\sqrt{3}(1-(2-\\sqrt{3})^{N})}+\\frac{\\sqrt{3}}{2}N.\n\\label{Kf6}\n\\end{eqnarray}\nUsing the identity \\cite{Chair1},\n\\begin{eqnarray}\n\\label{Kf7}\n\\sum_{n=1}^{N\/2}\\cos^{2j}(2n-1)l\\pi\/N&=&\n\\frac{N}{2^{2j+1}}\\binom {2j}{j}+\\frac{N}{2^{2j}}\\sum_{p=1}^{[j\/2N]}\\binom {2j}{j-2pN}\\nonumber\\\\&-&\\frac{N}{2^{2j}}\\sum_{p=1}^{[j\/2N]}\\binom {2j}{j-(2p-1)N},\n\\end{eqnarray}\n and by following similar steps as in the  above computations, then, one may show\n \\begin{eqnarray}\n \\sum_{n=1}^{N\/2}\\frac{1}{(1-2\/3\\cos^{2}(2n-1)\\pi\/2N)}&=&\\frac{3N(2-\\sqrt{3})^{2N}}{\\sqrt{3}(1-(2-\\sqrt{3})^{2N})}-\\frac{3N(2-\\sqrt{3})^{N}}{\\sqrt{3}(1-(2-\\sqrt{3})^{2N})}\\nonumber\\\\&+&\\frac{\\sqrt{3}}{2}N.\n \\label{Kf8}\n \\end{eqnarray}\n Finally, the exact expression of the Kirchhoff index of a $2\\times N$ reeds\n\\begin{eqnarray}\nKf(2\\times N)=N+\\frac{N^{3}-N}{3}+\\frac{N}{3}\\Big[-2+\\frac{6N(2-\\sqrt{3})^{2N}}{\\sqrt{3}(1-(2-\\sqrt{3})^{2N})}+\\sqrt{3}N\\Big].\n\\label{Kf9}\n\\end{eqnarray} \nOne can show that, the above formula for the Kirchhoff formula is valid for $N$ odd as well.\n\nExample, For $N=1,2,3,4,5$, the Kirchhoff indices are respectively, \n \\begin{eqnarray}\nKf(2\\times 2)&=&5\\nonumber\\\\Kf(2\\times 3)&=&14.2 \\nonumber\\\\Kf(2\\times 4)&=&30.57142857\n\\nonumber\\\\Kf(2\\times 5)&=&56.10047847\n\\end{eqnarray} \n These results are in complete agreement with those obtained using formula given by Eq. (\\ref{Kf1}), or theorem 4.1 of reference \\cite{Yang}.\n  \\section{Some  trigonometrical sums related to number theory}\n  \n  \\ \\  In this section other class of  trigonometrical sums  will be evaluated using similar techniques as in the previous sections. Some of these trigonometrical sums are related to number theory. We  will start with the following sum\n$$S(l):=\\sum_{n=1}^{N-1}(-1)^n\\frac{\\sin^2(nl\\pi\/N)}{\\sin^2(n\\pi\/N)}, $$ which is the alternating  sum associated with the sum  $R(l)$ given in Eq. (\\ref{t1}). This sum has the following closed formula\n\\begin{proposition}\n\\begin{equation}\n\\label{s}\nS(l)=\\sum_{n=1}^{N-1}(-1)^n\\frac{\\sin^2(nl\\pi\/N)}{\\sin^2(n\\pi\/N)}=-l^2\n\\end{equation}\n\\end{proposition}\nTo derive the above formula, we follow  similar computations carried out for $R(l)$, except that this time the sum over $n$ is  non-vanishing only if $N$ is even\n\\begin{equation} \n\\label{a1}\n\\sum_{n=1}^{N-1}(-1)^n\\sin^{2(s-1)}(n\\pi\/N)=\\frac{1}{2^{2(s-1)-1}}\\sum_{t=1}^{s-1}(-1)^{t +1}\\binom {2(s-1)}{s-1-t} +\\frac{-1}{2^{2(s-1)}}\\binom {2(s-1)} {s-1},\n\\end{equation}\n Comparing Eq. (\\ref{a1}) and Eq. (\\ref{t3}), and using the previous results, then,  without any further computations,  the formula for the trigonometrical sum $S(l)$ is obtained. \n Due to the the symmetry enjoyed by $ S(l)$, $S(l)=S(N-l)$, the right hand side of equation (\\ref{s}) should be read with this constraint, that is  for  both $l$, and $N-l$,  $ S(l)=-l^2$ . Next, let us consider the sums $ S_{1}(l):=\\sum_{n=1}^{N-1}\\frac{\\sin(nl\\pi\/N)}{\\sin(n\\pi\/N)}$ and $S_{2}(l):=\\sum_{n=1}^{N-1}(-1)^n\\frac{\\sin(nl\\pi\/N)}{\\sin(n\\pi\/N)}$ that are closely related. We will prove that  closed formulas for the sums  $ S_{1}(l)$ and $ S_{2}(l)$ are given by\n \\begin{theorem}\n\\begin{equation} \n\\label{s1}\nS_{1}(l)=\\sum_{n=1}^{N-1}\\frac{\\sin(nl\\pi\/N)}{\\sin(n\\pi\/N)}=\\left\\{\\begin{array}{cl}N-l & \\text{for } l  \\text{odd }\\\\\n0 &  \\text{for } l  \\text{even },\n\\end{array} \\right.\n\\end{equation}\n\\begin{eqnarray} \n\\label{s2}\nS_{2}(l)=\\sum_{n=1}^{N-1}(-1)^n\\frac{\\sin(nl\\pi\/N)}{\\sin(n\\pi\/N)}=\\left\\{\\begin{array}{cl}-(2l-1) & \\text{for } l  \\text{odd } \\text{and } N \\text{even} \\\\\n0 &  \\text{for } l  \\text{odd } \\text{and } N \\text{odd}\\\\\n-2l& \\text{for } l  \\text{even } \\text{and } N \\text{odd}\\\\\n0& \\text{for } l  \\text{even } \\text{and } N \\text{even}.\n\\end{array} \\right.\n\\end{eqnarray}\n\\end{theorem}\n\nTo prove the first formula, we use the following trigonometrical identity \\cite{Hobson},\n \\begin{equation} \n\\label{Hobson}\n\\frac{\\sin(nl\\pi\/N)}{\\sin(n\\pi\/N)}=\\sum_{s\\geq0}(-1)^{s}\\binom {l-s-1}{s}2^{l-2s-1}\\cos^{l-2s-1}(n\\pi\/N), \n \\end{equation}\n thus, for $l$ odd, one has\n \\begin{eqnarray} \n\\label{n1}\nS_{1}(l)=\\sum_{n=1}^{N-1}\\frac{\\sin(2l-1)n\\pi\/N}{\\sin n\\pi\/N}&=&\\sum_{s\\geq0}(-1)^{s}\\binom {2l-2-s}{s}2^{2l-2-2s}\\sum_{n=1}^{N-1}\\cos^{2l-2-2s}(n\\pi\/N)\\nonumber\\\\.\n\\end{eqnarray}\nThe sum over $n$, may be computed from Schwatt's book \\cite{Schwatt}, see Eq. (107), page $221$ to give \n\\begin{eqnarray} \n\\label{n2}\n\\sum_{n=1}^{N-1}\\cos^{2l-2-2s}(n\\pi\/N)= \\frac{-2}{2^{2l-2-2s}}\\sum_{t=1}^{l-1-s}\\binom {2l-2-2s}{l-1-s-t} +\\frac{N-1}{2^{2l-2-2s}}\\binom {2l-2-2s} {l-1-s},\n\\end{eqnarray}\nthen, the first contribution to the sum given in Eq. (\\ref{n1}) is\n\\begin{eqnarray} \n\\label{n3}\nS_{1}(l)^{'}=-2\\sum_{s=0}^{l-1}(-1)^{s}\\binom {2l-2-s}{s}\\sum_{t=1}^{l-1-s}\\binom {2l-2-2s}{l-1-s-t}=\\nonumber\\\\-2\\hbox{res}_{w=0}\\sum_{s=0}^{l-1}(-1)^{s}\\binom {2l-2-s}{s}\\Big(\\frac{1+w}{\\sqrt {w}}\\Big)^{2l-2-2s}\\frac{1}{1-w}\\nonumber\\\\=-2\\hbox{res}_{w=0}U_{2l-2}\\Big(\\frac{1+w}{2\\sqrt {w}}\\Big)\\frac{1}{1-w},\n\\end{eqnarray}\nin obtaining the last line of the above equation we used the expresion for the normalized Chebyshev polynomial of the second kind $U_{n}(\\frac{x}{2})= \\sum_{k=0}^{[n\/2]}(-1)^{k}\\binom {n-k}{k}x^{n-2k}$. The residue may be evaluated using $$U_{n}(\\frac{x}{2})= \\frac{(x+\\sqrt{x^2-1})^{n+1}-(x-\\sqrt{x^2-1})^{n+1}}{2\\sqrt{x^2-1}}, $$  to obtain \n\\begin{eqnarray} \n\\label{n4}\nS_{1}(l)^{'}=-2\\hbox{res}_{w=0} \\frac{1}{w^{l-1}}\\frac{1}{(1-w)^2}=-2(l-1).\n\\end{eqnarray}\nSimilarly, the second contribution reads\n\\begin{eqnarray} \n\\label{ns3}\nS_{1}(l)^{''}&=&(N-1)\\hbox{res}_{w=0}U_{2l-2}\\Big(\\ \\frac{1+w}{2\\sqrt {w}}\\Big)\\frac{1}{w}\\nonumber\\\\&=& N-1.\n\\end{eqnarray}\nTherefore, combining these contributions, the closed formula for $ S_{1}(l)$ is \n\\begin{eqnarray} \n\\label{n00}\nS_{1}(l)=\\sum_{n=1}^{N-1}\\frac{\\sin(2l-1)n\\pi\/N}{\\sin n\\pi\/N}=N-(2l-1).\n\\end{eqnarray}\nIt is not difficult to show that there is no contribution to the sum $ S_{1}(l)$ for $l$ even. In proving the second  formula for $S_{2}(l)$ Eq. (\\ref{s2}), one notes that in evaluting the sum over $n$ in $$S_{2}(l)=\\sum_{s\\geq0}(-1)^{s}\\binom {l-s-1}{s}2^{l-2s-1}\\sum_{n=1}^{N-1}(-1)^n\\cos^{l-2s-1}(n\\pi\/N),$$  turns out to depend on both $l$, $N$ unlike the previous case. By using formulas  given by Eq (113), and  Eq (114) in \\cite{ Schwatt}, we have\n\\begin{eqnarray} \n\\label{n5}\n\\sum_{n=1}^{N-1}(-1)^n\\cos^{l-2s-1}(n\\pi\/N)&=&\\frac{-2}{2^{2l-2-2s}}\\sum_{t=1}^{l-1-s}\\binom {2l-2-2s}{l-1-s-t} -\\frac{1}{2^{2l-2-2s}}\\binom {2l-2-2s} {s-l-1},\\nonumber\\\\\n\\end{eqnarray}\nfor $l$ odd, $ N$ even, and the sum vanishes for $l$ odd, $ N$ odd. If $l$ is even, then, the above sum is non-vanishing only  for $N$ odd\n\nTherefore, for $l$ odd it follows from  Eq. (\\ref{n5}) that we have  \n\\begin{eqnarray} \n\\label{n7} \nS_{2}(l)&=&\\sum_{n=1}^{N-1}(-1)^n\\frac{\\sin (2l-1)n\\pi\/N}{\\sin(n\\pi\/N)}\\nonumber\\\\&=&-2\\hbox{res}_{w=0} \\frac{1}{w^{l-1}}\\frac{1}{(1-w)^2}-\\hbox{res}_{w=0} \\frac{1}{w^l}\\frac{1}{(1-w)}=-(2l-1),\n\\end{eqnarray}\n for $l$ even and  $N$ odd, one has the following identity\n\\begin{eqnarray} \n\\label{n6}\n\\sum_{n=1}^{N-1}(-1)^n\\cos^{l-2s-1}(n\\pi\/N)&=&\\frac{-2}{2^{2l-1-2s}}\\sum_{t=0}^{l-1-s}\\binom {2l-1-2s}{l-1-s-t},\n\\end{eqnarray}\nfrom which\n\\begin{eqnarray} \n\\label{n0} \nS_{2}(l)=\\sum_{n=1}^{N-1}(-1)^n\\frac{\\sin(2nl\\pi\/N)}{\\sin(n\\pi\/N)}=-2\\hbox{res}_{w=0} \\frac{1}{w^{l}}\\frac{1}{(1-w)^2}=-2l,\n\\end{eqnarray}\n these results were recently verified by simulations without proof in connection with number theory  \\footnote{Anonymous author  working on characters of a finite field and the Polya-Vinogradov inequality.}. It is interesting to note that the closed formula for $ S_{2}(l)$  may be expected  since if we   let  $l$ go to $ N-l$ in  $ S_{1}(l)$, then, $S_{2}(l)=- l$ . Now, we will consider the following non-trivial and interesting trigonometrical sums, $$ F_{1}(N,l,2):= \\sum_{n=1}^{N-1}\\frac{\\sin(nl\\pi\/N)}{\\sin(n\\pi\/N)}\\frac{\\sin(2nl\\pi\/N)}{\\sin(2n\\pi\/N)},$$ and \n $$ F_{2}(N,l,2):= \\sum_{n=1}^{N-1}(-1)^{n}\\frac{\\sin(nl\\pi\/N)}{\\sin(n\\pi\/N)}\\frac{\\sin(2nl\\pi\/N)}{\\sin(2n\\pi\/N)},$$\n where $ F_{1}(N,N-l,2)= F_{2}(N,l,2)$, and $ N$ is assumed to be odd. So, if $ F_{2}(N,l,2)$ is known, then,  $ F_{2}(N,l,2)$ may be obtained and vice-versa. In the  rest of this paper, we show that both the trigonometrical sums may be evaluated to give the following closed formulas\n\\begin{theorem}\n\\begin{eqnarray} \n\\label{f1}\nF_{1}(N,l,2)& = &\\sum_{n=1}^{N-1}\\frac{\\sin(nl\\pi\/N)}{\\sin(n\\pi\/N)}\\frac{\\sin(2nl\\pi\/N)}{\\sin(2n\\pi\/N)}\\nonumber\\\\&=&-\\frac{1}{2}(3l-2)(3l-3)+\\frac{1}{2}(l-1)(l-2)-l+\\frac{1}{2}(1-(-1)^l)\\nonumber\\\\&+&\\frac{N-1}{2}\\Big(2l-1-(-1)^l\\Big)+N\\Big(3l-2-N+\\frac{1}{2}(1-(-1)^{l-N})\\Big)\\nonumber\\\\\n\\end{eqnarray}\nwhere  $l$  is odd  and  the sum vanishes for even l, also, note that the last term namely the coefficient of $N$ is different from zero only if $ 3l-2>N$. The closed formula for $ F_{2}(N,l,2)$ reads\n\\begin{eqnarray} \n\\label{f2}\n F_{2}(N,l,2)&=& \\sum_{n=1}^{N-1}(-1)^{n}\\frac{\\sin(nl\\pi\/N)}{\\sin(n\\pi\/N)}\\frac{\\sin(2nl\\pi\/N)}{\\sin(2n\\pi\/N)}\\nonumber\\\\&=&-\\frac{1}{2}3l(3l-1)+\\frac{1}{2}l(l-1)-l+N\\Big(3l-\\frac{(N+1)}{2}+\\frac{1}{2}(1-(-1)^{l-\\frac{(N+1)}{2}})\\Big),\\nonumber\\\\,\n\\end{eqnarray}\nwhere  $l$  is even  and the sum vanishes for odd l, also, note that the last term whose  coefficient is $N$ is different from zero only if $ 3l>\\frac{N+1}{2}$\n\\end{theorem} \n   To prove the first formula, we note  that $  F_{1}(N,2l,2)=0$, and hence the only sum  to consider is the sum $  F_{1}(N,2l-1,2)$. The latter may be written as \n \\begin{eqnarray} \n\\label{n8}\n F_{1}(N,2l-1,2)&=&\\sum_{n=1}^{N-1}\\frac{\\sin(n(2l-1)\\pi\/N)}{\\sin(n\\pi\/N)}\\frac{\\sin(2n(2l-1)\\pi\/N)}{\\sin(2n\\pi\/N)}\\nonumber\\\\&=&\\sum_{s,k\\geq0}^{l-1}(-1)^{s+k}\\binom {2l-2-s}{s}\\binom {2l-2-k}{k}2^{2(2l-2)-2(s+k)}\\nonumber\\\\&\\times&\\sum_{j=0}^{2l-2-2k}(-1)^{j}2^j\\binom {2l-2-2k}{k}\\sum_{n=1}^{N-1}\\cos^{2l-2-2(s-j)}(n\\pi\/N).\n\\end{eqnarray}\nThe sum over $n$, formally looks like that given in Eq. (\\ref{n2}), however, the variable $t$, may be a multiple of $N$ and in that case the Schwatt's formula given by Eq (107), does not work it has to be modified slightly. The formula that takes into account this fact may be shown to be given by\n\\begin{eqnarray} \n\\label{n9}\n\\sum_{n=1}^{N-1}\\cos^{2l-2-2(s-j)}(n\\pi\/N)&= &\\frac{-2}{2^{2l-2-2(s-j)}}\\sum_{t=1}^{l-1-s}\\binom {2l-2-2(s-j)}{l-1-(s-j)-t}\\nonumber\\\\& +&\\frac{N-1}{2^{2l-2-2(s-j)}}\\binom {2l-2-2(s-j)} {l-1-(s-j)}\\nonumber\\\\&+&\\frac{2N}{2^{2l-2-2(s-j)}}\\sum_{p=1}^{[l-1-(s-j)\/N]}\\binom {2l-2-2(s-j)} {l-1-(s-j)-pN},\n\\end{eqnarray}\nwhere the first two terms in the above formula are those expected from Eq.  (\\ref{n2}), while the third term is precisely the correction to the formula Eq.  (\\ref{n2}) for  $t$  congruent to $N$. Therefore, there are three contributions to the sum given in Eq. (\\ref{n8}), the first of which  reads\n\\begin{eqnarray} \n\\label{n10}\nF_{1}^{'}(N,2l-1,2)&=&-2\\sum_{s,k\\geq0}^{l-1}(-1)^{s+k}\\binom {2l-2-s}{s}\\binom {2l-2-k}{k}2^{2l-2-2k}\\nonumber\\\\&\\times&\\sum_{j=0}^{2l-2-2k}(-1)^{j}\\frac{1}{2^j}\\binom {2l-2-2k}{k}\\sum_{t=1}^{l-1-s}\\binom {2l-2-2(s-j)}{l-1-(s-j)-t}\\nonumber\\\\&=&-2\\hbox{res}_{w=0}U_{2l-2}\\Big(\\frac{1+w}{2\\sqrt {w}}\\Big)U_{2l-2}\\Big(\\frac{1+w^2}{2w}\\Big)\\frac{1}{1-w}\\nonumber\\\\&=&-2\\hbox{res}_{w=0}\\Big(\\frac{1}{w^{3l-3}}-\\frac{1}{w^{l-2}}\\Big)\\frac{1}{(1-w)^2}\\frac{1}{1-w^2}\\nonumber\\\\&=&-\\frac{1}{2}(3l-2)(3l-3)+\\frac{1}{2}(l-1)(l-2)-l+\\frac{1}{2}(1-(-1)^l)\n\\end{eqnarray}\nwhile the second contribution is\n\\begin{eqnarray} \n\\label{n11}\nF_{1}^{''}(N,2l-1,2)&=&(N-1)\\sum_{s,k\\geq0}^{l-1}(-1)^{s+k}\\binom {2l-2-s}{s}\\binom {2l-2-k}{k}2^{2l-2-2k}\\nonumber\\\\&\\times&\\sum_{j=0}^{2l-2-2k}(-1)^{j}\\frac{1}{2^j}\\binom {2l-2-2k}{k}\\binom {2l-2-2(s-j)}{l-1-(s-j)}\\nonumber\\\\&=&(N-1)\\hbox{res}_{w=0}U_{2l-2}\\Big(\\frac{1+w}{2\\sqrt {w}}\\Big)U_{2l-2}\\Big(\\frac{1+w^2}{2w}\\Big)\\frac{1}{w}\\nonumber\\\\&=& (N-1)\\hbox{res}_{w=0}\\Big(\\frac{1}{w^{3l-2}}-\\frac{1}{w^{l-1}}\\Big)\\frac{1}{1-w^2}\\frac{1}{1-w}\\nonumber\\\\&=&\\frac{N-1}{2}\\Big(2l-1-(-1)^l\\Big).\n\\end{eqnarray}\n To obtain the last contribution we write the sum over $p$, in Eq. (\\ref{n9}), as $$ \\sum_{p=1}^{[l-1-(s-j)\/N]}\\binom {2l-2-2(s-j)} {l-1-(s-j)-pN}=\\hbox{res}_{w=0} \\Big( \\frac{(1+w)^{2l-2-2(s-j)}w^N}{w^{l-(s-j)}(1-w^{N})}\\Big),$$ and using the fact that $l\\leq N-1$, then, the third contribution may be computed to give\n \\begin{eqnarray} \n\\label{n12}\nF_{1}^{'''}(N,2l-1,2)&=&2N\\sum_{s,k\\geq0}^{l-1}(-1)^{s+k}\\binom {2l-2-s}{s}\\binom {2l-2-k}{k}2^{2l-2-2k}\\nonumber\\\\&\\times&\\sum_{j=0}^{2l-2-2k}(-1)^{j}\\frac{1}{2^j}\\binom {2l-2-2k}{k}\\hbox{res}_{w=0} \\Big( \\frac{(1+w)^{2l-2-2(s-j)}w^N}{w^{l-(s-j)}(1-w^{N})}\\Big)\\nonumber\\\\&=&2N\\hbox{res}_{w=0}\\Big(\\frac{1}{1-w^N}U_{2l-2}\\Big(\\frac{1+w}{2\\sqrt {w}}\\Big)U_{2l-2}\\Big(\\frac{1+w^2}{2w}\\Big)w^{N-1}\\Big)\\nonumber\\\\&=&2N\\hbox{res}_{w=0}\\frac{1}{1-w^N}\\frac{1}{w^{3l-2-N}}\\frac{1}{(1-w)(1-w^2)}\\nonumber\\\\&=&N\\big(3l-2-N+\\frac{1}{2}(1-(-1)^{l-N}\\big).\n\\end{eqnarray}\nNote that this will contribute only for $3l-2>N $, and as a result the formula for the sum $ F_{1}(N,2l-1,2)$ is\n\\begin{eqnarray} \n\\label{n13}\nF_{1}(N,2l-1,2)&=&\\sum_{n=1}^{N-1}\\frac{\\sin(n(2l-1)\\pi\/N)}{\\sin(n\\pi\/N)}\\frac{\\sin(2n(2l-1)\\pi\/N)}{\\sin(2n\\pi\/N)}\\nonumber\\\\&=&-\\frac{1}{2}(3l-2)(3l-3)+\\frac{1}{2}(l-1)(l-2)-l+\\frac{1}{2}(1-(-1)^l)\\nonumber\\\\&+&\\frac{N-1}{2}\\Big(2l-1-(-1)^l\\Big)+N\\Big(3l-2-N+\\frac{1}{2}(1-(-1)^{l-N})\\Big).\n\\end{eqnarray}\nHaving obtained a closed formula for the sum  $ F_{1}(N,2l-1,2)$, we now wish  to prove the formula for the   alternating sum  $ F_{2}(N,l,2) $. \nFirst, we note that for $N$ odd, the sum is non-vanishing only for $l$ is even. Therefore, the formula for  $ F_{2}(N,l,2) $ becomes\n\\begin{eqnarray} \n\\label{n14}\n F_{2}(N,2l,2&)=&\\sum_{n=1}^{N-1}(-1)^{n}\\frac{\\sin((2l)n\\pi\/N)}{\\sin(n\\pi\/N)}\\frac{\\sin((2l)2n\\pi\/N)}{\\sin(2n\\pi\/N)}\\nonumber\\\\&=&\\sum_{s,k\\geq0}^{[(2l-1)\/2]}(-1)^{s+k}\\binom {2l-1-s}{s}\\binom {2l-1-k}{k}2^{2(2l-1)-2(s+k)}\\nonumber\\\\&\\times&\\sum_{j=0}^{2l-1-2k}(-1)^{j}2^j\\binom {2l-1-2k}{k}\\sum_{n=1}^{N-1}(-1)^n\\cos^{2l-1-2(s-j)}(n\\pi\/N).\n\\end{eqnarray}\nThe sum over $n$  may carried out using Eq. (114), in \\cite{ Schwatt} with the slight modification as explained before, then, it is not difficult to show\n\\begin{eqnarray} \n\\label{n15}\n\\sum_{n=1}^{N-1}(-1)^n\\cos^{2l-1-2(s-j)-1}(n\\pi\/N)&=&\\frac{-2}{2^{2l-1-2(s-j)}}\\sum_{t=0}^{l-1-s}\\binom {2l-1-2(s-j)}{l-1-(s-j)-t}\\nonumber\\\\&+&\\frac{2N}{2^{2l-1-2(s-j)}}\\sum_{p\\geq1}\\binom {2l-1-2(s-j)} {l-1-(s-j)-\\frac{(2p-1)N-1}{2}},\\nonumber\\\\\n\\end{eqnarray}\nwhere the sum over $p$, may be written as\n$$\\sum_{p\\geq1}\\binom {2l-1-2(s-j)} {l-1-(s-j)-\\frac{(2p-1)N-1}{2}}=\\hbox{res}_{w=0} \\Big( \\frac{(1+w)^{2l-1-2(s-j)}w^{N\/2}}{w^{l-(s-j)+1\/2}(1-w^{N})}\\Big).$$\nBy using Eq. (\\ref{n14}), computations show that the closed formula for the sum $F_{2}(N,2l,2)$ is\n\\begin{eqnarray} \n\\label{n16}\n F_{2}(N,2l,2&)=&\\sum_{n=1}^{N-1}(-1)^{n}\\frac{\\sin((2l)n\\pi\/N)}{\\sin(n\\pi\/N)}\\frac{\\sin((2l)2n\\pi\/N)}{\\sin(2n\\pi\/N)}\\nonumber\\\\&=&-2\\hbox{res}_{w=0}\\Big(U_{2l-2}\\Big(\\frac{1+w}{2\\sqrt {w}}\\Big)U_{2l-2}\\Big(\\frac{1+w^2}{2w}\\Big)\\frac{1}{\\sqrt{w}(1-w)}\\Big)\\nonumber\\\\&+&2N\\hbox{res}_{w=0}\\frac{1}{1-w^N}\\Big(U_{2l-1}\\Big(\\frac{1+w}{2\\sqrt {w}}\\Big)U_{2l-1}\\Big(\\frac{1+w^2}{2w}\\Big)\\frac{w^{N\/2}}{w}\\Big)\\nonumber\\\\&=&-2\\hbox{res}_{w=0}\\Big(\\frac{1}{w^{3l-1}}-\\frac{1}{w^{l-1}}\\Big)\\frac{1}{(1-w)^2}\\frac{1}{1-w^2}\\nonumber\\\\&+&2N\\hbox{res}_{w=0}\\frac{1}{1-w^N}\\frac{1}{w^{3l-(N+1)\/2}}\\frac{1}{(1-w)(1-w^2)}\\nonumber\\\\&=&-\\frac{1}{2}3l(3l-1)+\\frac{1}{2}l(l-1)-l+N\\Big(3l-\\frac{(N+1)}{2}+\\frac{1}{2}(1-(-1)^{l-\\frac{(N+1)}{2}})\\Big),\\nonumber\\\\\n \\end{eqnarray}\n where the last term whose coefficient is $N$, contributes only for $3l> \\frac{(N+1)}{2} $. Let us now, check  that the  formulas   $  F_{1}(N,2l-1,2)$, $F_{2}(N,2l,2)$ are consistent with  symmetry   discussed earlier, that is,  $  F_{1}(N,N-l,2)= F_{2}(N,l,2)$, this in turns implies that the  correctness  of the formulas. To do so, we will give some explicit examples, from the expression of $ F_{1}(N,2l-1,2) $ given in Eq. (\\ref{n13}), it is clear that the sum should be $N-1$, for $l=1$ and to check this, one has to take into account that when substituting $l=1$ in the formula,  the last term of  Eq. (\\ref{n13}) does not contribute. From the symmetry that relates the two sums, we should have $ F_{2}(N,N-1,2)=N-1$. Indeed, this is the case, we simply let $l=\\frac{N-1}{2} $ into Eq. (\\ref{n16}), this time, however, the last term of this equation does contribute. An explicit computation shows that  $  F_{2}(N,2,2)= -4$, for $N> 3$, and  $  F_{2}(N,2,2)= 2$, for $N= 3$, it is interesting to note that these two cases for $l=1$ are contained in the last term of Eq. (\\ref{n15}), since   for $N> 3$, the last term is equal to $0$, and hence $  F_{2}(N,2,2)= -4$, while for $N= 3$, the last term is equal to $6$, that is, our formula gives the right answer. Using the symmetry, we obtain $ F_{1}(N,N-2,2)=-4$, this can be easily checked using our formula given by Eq. (\\ref{n13}), and $l=\\frac{N-1}{2} $.\n \\section{Conclusion}\n \n \\ \\ To conclude,  in this paper we used our method in \\cite{ Chair1}, to give alternative derivations to  closed formulas for trigonometrical sums that appear in one-dimensional lattice,  and in the proof of the  conjecture of F.  R Scott on Permanent of the Cauchy matrix.  A new derivation of certain trigonometrical sum of the perturbative chiral Potts model is given as well as new recursion formulas of certain trigonometrical sums \\cite{Mehta}. By  using these recursion formulas, then, one is able  deduce the Verlinde dimension formulas for the untwisted (twisted) space of conformal blocks of  $SU(2)$ ($SO(3)$)WZW. In this paper, we reported closed-form  formulas for the corner-to-corner resistance and the Kirchhoff index of the first non-trivial two-dimensional resistor network, $2\\times N$.  We have also, considered other class of trigonometrical sums, some of which appear in  number theory. Here, we followed  similar formalism as in  \\cite{ Chair1}, as a consequence  the non-trivial circulant electrical networks (the cycle and complete graphs are not included) are related to non-trivial trigonometrical sums in number theory. For example in\\cite{ Chair1}, we had to introduce certain numbers that we called the Bejaia and Pisa numbers with well known properties so that the trigonometrical sums that arise in the computation of the two-point resistance are written in terms of these numbers nicely. By using the  well known connection between the electrical networks and  the random walks \\cite{Doyle}, one may hope to give  interpretations  to some of the trigonometrical sums in number theory other than those associated with the two-point resistance of a given electrical network, since the latter  provides an alternative way to compute the basic quantity relevant to random walks known as  the first passage time,  the expected time to hit a target node for the first time  for a walker starting from a source node \\cite{Tetali}.\n \\vspace{7mm}\n\n{\\bf Acknowledgment:}\n\nI would like to thank Professor Bruce Berndt for reading and making comments on the manuscript. Also, I would like to thank the Abdus Salam centre for Theoretical Physics for supports and\nhospitality throughout these years \n\\newpage\n\\bibliographystyle{phaip}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction\\label{intro}}\nWe consider the Cauchy problem of the fourth order nonlinear Schr\\\"odinger type equations:\n\\begin{equation}\\label{D4NLS}\n\\begin{cases}\n\\displaystyle (i\\partial_{t}+\\Delta ^2)u=\\partial P_{m}(u,\\overline{u}),\\hspace{2ex}(t,x)\\in (0,\\infty )\\times \\R^{d} \\\\\nu(0,x)=u_{0}(x),\\hspace{2ex}x\\in \\R^{d}\n\\end{cases}\n\\end{equation}\nwhere $m\\in \\N$, $m\\geq 2$, $P_{m}$ is a polynomial which is written by\n\\[\nP_{m}(f,g)=\\sum_{\\substack{\\alpha ,\\beta \\in \\Z_{\\geq 0}\\\\ \\alpha +\\beta=m}}f^{\\alpha}g^{\\beta}, \n\\]\n$\\partial$ is a first order derivative with respect to the spatial variable, for example a linear combination of  \n$\\frac{\\partial}{\\partial x_1} , \\, \\dots , \\, \\frac{\\partial}{\\partial x_d}$ or $|\\nabla |= \\mathcal{F}^{-1}[|\\xi | \\mathcal{F}]$\nand the unknown function $u$ is $\\C$-valued. \nThe fourth order Schr\\\"{o}dinger equation with $P_{m}(u,\\overline{u})=|u|^{m-1}u$ appears in the study of deep water wave dynamics \\cite{Dysthe}, solitary waves \\cite{Karpman}, \\cite{KS}, vortex filaments \\cite{Fukumoto}, and so on.\nThe equation (\\ref{D4NLS}) is invariant under the following scaling transformation:\n\\[\nu_{\\lambda}(t,x)=\\lambda^{-3\/(m-1)}u(\\lambda^{-4}t,\\lambda^{-1}x), \n\\]\nand the scaling critical regularity is $s_{c}=d\/2-3\/(m-1)$. \nThe aim of this paper is to prove the well-posedness and the scattering for the solution of (\\ref{D4NLS}) \nin the scaling critical Sobolev space. \n\n\nThere are many results for the fourth order nonlinear Schr\\\"{o}dinger equation \nwith derivative nonlinearities (see \\cite{S1}, \\cite{S2}, \\cite{HJ1}, \\cite{HHW}, \\cite{HHW2}, \\cite{HJ3}, \\cite{S3}, \\cite{HJ2}, \\cite{Y12}, \\cite{HN15_1}, \\cite{HN15_2}, and references cited therein).\nEspecially, the one dimensional case is well studied.\nWang (\\cite{Y12}) considered (\\ref{D4NLS}) for the case $d=1$, $m=2l+1$, $l\\ge 2$, $P_{2l+1}(u,\\overline{u})=|u|^{2l}u$ \nand proved the small data global in time well-posedness for $s=s_{c}$ by using Kato type smoothing effect. \nBut he did not treat the cubic case.\nActually, a technical difficulty appears in this case (see Theorem \\ref{notC3} below).\n\n\nHayashi and Naumkin (\\cite{HN15_1}) considered (\\ref{D4NLS}) for $d=1$ with the power type nonlineality $\\partial_{x}(|u|^{\\rho -1}u)$ ($\\rho >4$) \nand proved the global existence of the solution and the scattering in the weighted Sobolev space.\nMoreover, they (\\cite{HN15_2}) also proved that the large time asymptotics is determined by the self similar solution in the case $\\rho =4$.\nTherefore, derivative quartic nonlinearity in the one spatial dimension is the critical in the sense of the asymptotic behavior of the solution.\n\nWe firstly focus on the quartic nonlinearity $\\partial _x (\\overline{u}^4)$ in one space dimension.\nSince this nonlinearity has some good structure, the global solution scatters to a free solution in the scaling critical Sobolev space.\nOur argument does not apply to \\eqref{D4NLS} with $P (u,\\overline{u}) = |u|^3 u$ because we rely on the Fourier restriction norm method.\nNow, we give the first results in this paper. \nFor a Banach space $H$ and $r>0$, we define $B_r(H):=\\{ f\\in H \\,|\\, \\|f\\|_H \\le r \\}$. \n\\begin{thm}\\label{wellposed_1}\nLet $d=1$, $m=4$ and $P_{4}(u,\\overline{u})=\\overline{u}^{4}$. Then the equation {\\rm (\\ref{D4NLS})} is globally well-posed for small data in $\\dot{H}^{-1\/2}$. \nMore precisely, there exists $r>0$ such that for any $T>0$ and all initial data $u_{0}\\in B_{r}(\\dot{H}^{-1\/2})$, there exists a solution\n\\[\nu\\in \\dot{Z}_{r}^{-1\/2}([0,T))\\subset C([0,T );\\dot{H}^{-1\/2})\n\\]\nof {\\rm (\\ref{D4NLS})} on $(0, T )$. \nSuch solution is unique in $\\dot{Z}_{r}^{-1\/2}([0,T))$ which is a closed subset of $\\dot{Z}^{-1\/2}([0,T))$ {\\rm (see Definition~\\ref{YZ_space} and (\\ref{Zr_norm}))}. \nMoreover, the flow map\n\\[\nS^{+}_{T}:B_{r}(\\dot{H}^{-1\/2})\\ni u_{0}\\mapsto u\\in \\dot{Z}^{-1\/2}([0,T))\n\\]\nis Lipschitz continuous. \n\\end{thm}\n\\begin{rem}\nWe note that $s=-1\/2$ is the scaling critical exponent of (\\ref{D4NLS}) for $d=1$, $m=4$. \n\\end{rem}\n\\begin{cor}\\label{sccat}\nLet $r>0$ be as in Theorem~\\ref{wellposed_1}. \nFor all $u_{0}\\in B_{r}(\\dot{H}^{-1\/2})$, there exists a solution \n$u\\in C([0,\\infty );\\dot{H}^{s_{c}})$ of (\\ref{D4NLS}) on $(0,\\infty )$ and the solution scatters in $\\dot{H}^{-1\/2}$. \nMore precisely, there exists \n$u^{+}\\in \\dot{H}^{-1\/2}$ \nsuch that \n\\[\nu(t)-e^{it\\Delta^2}u^{+}\n\\rightarrow 0\n\\ {\\rm in}\\ \\dot{H}^{-1\/2}\\ {\\rm as}\\ t\\rightarrow + \\infty. \n\\]\n\\end{cor}\n\nMoreover, we obtain the large data local in time well-posedness in the scaling critical Sobolev space.\nTo state the result, we put\n\\[\nB_{\\delta ,R} (H^s) := \\{ u_0 \\in H^s | \\ u_0=v_0+w_0 , \\, \\| v_0 \\| _{\\dot{H}^{-1\/2}} < \\delta, \\, \\| w_0 \\| _{L^2} <R \\}\n\\]\nfor $s<0$.\n\n\\begin{thm} \\label{large-wp}\nLet $d=1$, $m=4$ and $P_{4}(u,\\overline{u})=\\overline{u}^{4}$.  \nThen the equation {\\rm (\\ref{D4NLS})} is locally in time well-posed in $H^{-1\/2}$. \nMore precisely,\nthere exists $\\delta >0$ such that for all $R \\ge \\delta$ and $u_0 \\in B_{\\delta ,R} (H^{-1\/2})$ there exists a solution\n\\[\nu \\in Z^{-1\/2}([0,T]) \\subset C([0,T); H^{-1\/2})\n\\]\nfor $T=\\delta ^{8} R^{-8}$ of \\eqref{D4NLS}.\n\nFurthermore, the same statement remains valid if we replace $H^{-1\/2}$ by $\\dot{H}^{-1\/2}$ as well as $Z^{-1\/2}([0,T])$ by $\\dot{Z}^{-1\/2}([0,T])$.\n\\end{thm}\n\n\\begin{rem}\nFor $s>-1\/2$, the local in time well-posedness in $H^s$ follows from the usual Fourier restriction norm method, which covers for all initial data in $H^s$.\nIt however is not of very much interest.\nOn the other hand, since we focus on the scaling critical cases, which is the negative regularity, we have to impose that the $\\dot{H}^{-1\/2}$ part of  initial data is small.\nBut, Theorem \\ref{large-wp} is a large data result because the $L^2$ part is not restricted.\n\\end{rem}\n\n\nThe main tools of the proof are the $U^{p}$ space and $V^{p}$ space which are applied to prove \nthe well-posedness and the scattering for KP-II equation at the scaling critical regularity by Hadac, Herr and Koch (\\cite{HHK09}, \\cite{HHK10}).\n\nWe also consider the one dimensional cubic case and the high dimensional cases. \nThe second result in this paper is as follows.\n\\begin{thm}\\label{wellposed_2}\n{\\rm (i)}\\ Let $d=1$ and $m=3$. Then the equation {\\rm (\\ref{D4NLS})} is locally well-posed  in $H^{s}$ for $s\\ge 0$. \\\\\n{\\rm (ii)}\\ Let $d\\geq 2$ and $(m-1)d\\geq 4$. Then the equation {\\rm (\\ref{D4NLS})}\n is globally well-posed for small data in $\\dot{H}^{s_{c}}$ (or $H^{s}$ for $s\\ge s_{c}$)\n and the solution scatters in $\\dot{H}^{s_{c}}$ (or $H^{s}$ for $s\\ge s_{c}$).\n\\end{thm}\n\nThe smoothing effect of the linear part recovers derivative in higher dimensional case.\nTherefore, we do not use the $U^p$ and $V^p$ type spaces.\nMore precisely, to establish Theorem \\ref{wellposed_2}, we only use the Strichartz estimates\nand get the solution in $C([0,T);H^{s_c})\\cap L^{p_m}([0,T); W^{q_m,s_{c}+1\/(m-1)})$ \nwith $p_m =2(m-1)$, $q_m =2(m-1)d\/\\{ (m-1)d-2\\}$.\nAccordingly, the scattering follows from a standard argument.\nSince the condition $(m-1)d\\geq 4$ is equivalent to $s_{c}+1\/(m-1)\\ge 0$, \nthe solution space $L^{p_m}([0,T); W^{q_m,s_{c}+1\/(m-1)})$ has nonnegative regularity even if the data belongs to $H^{s_{c}}$ with $-1\/(m-1)\\le s_c <0$. \nOur proof of Thorem~\\ref{wellposed_2} {\\rm (ii)} cannot applied for $d=1$ \nsince the Schr\\\"odingier admissible $(a,b)$ in {\\rm (\\ref{admissible_ab})} does not exist. \n\\begin{rem}\nFor the case $d=1$, $m=4$ and $P_{4}(u,\\overline{u})\\ne \\overline{u}^{4}$, \nwe can obtain the local in time well-posedness of {\\rm (\\ref{D4NLS})} in $H^{s}$ for $s\\ge 0$ \nby the same way of the proof of Theorem~\\ref{wellposed_2}. \nActually, we can get the solution in $C([0,T];H^s)\\cap L^4 ([0,T];W^{s+1\/2,\\infty })$ \nfor $s\\ge 0$ by using the iteration argument  \nsince the fractional Leibnitz rule (see \\cite{CW91}) and the H\\\"older inequality imply\n\\[\n\\left\\| |\\nabla |^{s+\\frac{1}{2}}\\prod_{j=1}^{4}u_j \\right\\|_{L^{4\/3}_{t}([0,T);L_{x}^{1})}\n\\lesssim T^{1\/4}\\| |\\nabla |^{s+\\frac{1}{2}}u_1 \\|_{L^{4}_{t}L_{x}^{\\infty}}\\| u_2 \\|_{L^{4}_{t}L_{x}^{\\infty}}\n\\| u_3 \\|_{L^{\\infty}_{t}L_{x}^{2}}\\| u_4 \\|_{L^{\\infty}_{t}L_{x}^{2}}.\n\\]\n\\end{rem}\n\nWe give a remark on our problem, which shows that the standard iteration argument does not work.\n\\begin{thm}\\label{notC3}\n{\\rm (i)}\\ Let $d=1$, $m=3$, $s<0$ and $P_{3}(u,\\overline{u})=|u|^{2}u$. Then the flow map of {\\rm (\\ref{D4NLS})} from $H^s$ to $C(\\R ; H^s)$ is not smooth. \\\\\n{\\rm (ii)}\\ Let $m\\ge 2$, $s<s_c$ and $\\partial =|\\nabla |$ or $\\frac{\\partial}{\\partial x_k}$ for some $1\\le k \\le d$. \nThen the flow map of {\\rm (\\ref{D4NLS})} from $H^s$ to $C(\\R ; H^s)$ is not smooth.\n\\end{thm}\n\nMore precisely, we prove that the flow map is not $C^3$ if $d=1$, $m=3$, $s<0$ and $P_{3}(u,\\overline{u})=|u|^{2}u$ or $C^m$ if $d \\ge 1$, $m \\ge 2$, and $s<s_c$.\nIt leads that the standard iteration argument fails, because the flow map is smooth if it works.\nOf course, there is a gap between ill-posedness and absence of a smooth flow map.\n\nSince the resonance appears in the case $d=1$, $m=3$ and $P_{3}(u,\\overline{u})=|u|^{2}u$, there exists an irregular flow map even for the subcritical Sobolev regularity.\n\n\n\\kuuhaku \\\\\n\\noindent {\\bf Notation.} \nWe denote the spatial Fourier transform by\\ \\ $\\widehat{\\cdot}$\\ \\ or $\\F_{x}$, \nthe Fourier transform in time by $\\F_{t}$ and the Fourier transform in all variables by\\ \\ $\\widetilde{\\cdot}$\\ \\ or $\\F_{tx}$. \nThe free evolution $S(t):=e^{it\\Delta^{2}}$ is given as a Fourier multiplier\n\\[\n\\F_{x}[S(t)f](\\xi )=e^{-it|\\xi |^{4}}\\widehat{f}(\\xi ). \n\\]\nWe will use $A\\lesssim B$ to denote an estimate of the form $A \\le CB$ for some constant $C$ and write $A \\sim B$ to mean $A \\lesssim B$ and $B \\lesssim A$. \nWe will use the convention that capital letters denote dyadic numbers, e.g. $N=2^{n}$ for $n\\in \\Z$ and for a dyadic summation we write\n$\\sum_{N}a_{N}:=\\sum_{n\\in \\Z}a_{2^{n}}$ and $\\sum_{N\\geq M}a_{N}:=\\sum_{n\\in \\Z, 2^{n}\\geq M}a_{2^{n}}$ for brevity. \nLet $\\chi \\in C^{\\infty}_{0}((-2,2))$ be an even, non-negative function such that $\\chi (t)=1$ for $|t|\\leq 1$. \nWe define $\\psi (t):=\\chi (t)-\\chi (2t)$ and $\\psi_{N}(t):=\\psi (N^{-1}t)$. Then, $\\sum_{N}\\psi_{N}(t)=1$ whenever $t\\neq 0$. \nWe define frequency and modulation projections\n\\[\n\\widehat{P_{N}u}(\\xi ):=\\psi_{N}(\\xi )\\widehat{u}(\\xi ),\\ \n\\widetilde{Q_{M}^{S}u}(\\tau ,\\xi ):=\\psi_{M}(\\tau -|\\xi|^{4})\\widetilde{u}(\\tau ,\\xi ).\n\\]\nFurthermore, we define $Q_{\\geq M}^{S}:=\\sum_{N\\geq M}Q_{N}^{S}$ and $Q_{<M}^{S}:=Id -Q_{\\geq M}^{S}$. \n\nThe rest of this paper is planned as follows.\nIn Section 2, we will give the definition and properties of the $U^{p}$ space and $V^{p}$ space. \nIn Section 3, we will give the multilinear estimates which are main estimates to prove Theorems~\\ref{wellposed_1} and \\ref{large-wp}. \nIn Section 4, we will give the proof of the well-posedness and the scattering (Theorem~\\ref{wellposed_1}, Corollary~\\ref{sccat}, and Theorem \\ref{large-wp}). \nIn Section 5, we will give the proof of Theorem~\\ref{wellposed_2}. \nIn Section 6, we will give the proof of Theorem~\\ref{notC3}. \n\n\\section{The $U^{p}$, $V^{p}$ spaces  and their properties \\label{func_sp}}\nIn this section, we define the $U^{p}$ space and the $V^{p}$ space, \nand introduce the properties of these spaces which are proved by Hadac, Herr and Koch (\\cite{HHK09}, \\cite{HHK10}). \n\nWe define the set of finite partitions $\\ZZ$ as\n\\[\n\\ZZ :=\\left\\{ \\{t_{k}\\}_{k=0}^{K}|K\\in \\N , -\\infty <t_{0}<t_{1}<\\cdots <t_{K}\\leq \\infty \\right\\}\n\\]\nand if $t_{K}=\\infty$, we put $v(t_{K}):=0$ for all functions $v:\\R \\rightarrow L^{2}$. \n\\begin{defn}\\label{upsp}\nLet $1\\leq p <\\infty$. For $\\{t_{k}\\}_{k=0}^{K}\\in \\ZZ$ and $\\{\\phi_{k}\\}_{k=0}^{K-1}\\subset L^{2}$ with \n$\\sum_{k=0}^{K-1} \\| \\phi_{k} \\| _{L^{2}}^{p}=1$ we call the function $a:\\R\\rightarrow L^{2}$ \ngiven by\n\\[\na(t)=\\sum_{k=1}^{K}\\ee_{[t_{k-1},t_{k})}(t)\\phi_{k-1}\n\\]\na ``$U^{p}${\\rm -atom}''. \nFurthermore, we define the atomic space \n\\[\nU^{p}:=\\left\\{ \\left. u=\\sum_{j=1}^{\\infty}\\lambda_{j}a_{j}\n\\right| a_{j}:U^{p}{\\rm -atom},\\ \\lambda_{j}\\in \\C \\ {\\rm such\\ that}\\  \\sum_{j=1}^{\\infty}|\\lambda_{j}|<\\infty \\right\\}\n\\]\nwith the norm\n\\[\n \\| u \\| _{U^{p}}:=\\inf \\left\\{\\sum_{j=1}^{\\infty}|\\lambda_{j}|\\left|u=\\sum_{j=1}^{\\infty}\\lambda_{j}a_{j},\\ \na_{j}:U^{p}{\\rm -atom},\\ \\lambda_{j}\\in \\C\\right.\\right\\}.\n\\]\n\\end{defn}\n\\begin{defn}\\label{vpsp}\nLet $1\\leq p <\\infty$. We define the space of the bounded $p$-variation \n\\[\nV^{p}:=\\{ v:\\R\\rightarrow L^{2}|\\  \\| v \\| _{V^{p}}<\\infty \\}\n\\]\nwith the norm\n\\[\n \\| v \\| _{V^{p}}:=\\sup_{\\{t_{k}\\}_{k=0}^{K}\\in \\ZZ}\\left(\\sum_{k=1}^{K} \\| v(t_{k})-v(t_{k-1}) \\| _{L^{2}}^{p}\\right)^{1\/p}.\n\\]\nLikewise, let $V^{p}_{-, rc}$ denote the closed subspace of all right-continuous functions $v\\in V^{p}$ with \n$\\lim_{t\\rightarrow -\\infty}v(t)=0$, endowed with the same norm  $ \\| \\cdot  \\| _{V^{p}}$.\n\\end{defn}\n\\begin{prop}[\\cite{HHK09} Proposition\\ 2.2,\\ 2.4,\\ Corollary\\ 2.6]\\label{upvpprop}\nLet $1\\leq p<q<\\infty$. \\\\\n{\\rm (i)} $U^{p}$, $V^{p}$ and $V^{p}_{-, rc}$ are Banach spaces. \\\\ \n{\\rm (ii)} For every $v\\in V^{p}$, $\\lim_{t\\rightarrow -\\infty}v(t)$ and $\\lim_{t\\rightarrow \\infty}v(t)$ exist in $L^{2}$. \\\\\n{\\rm (iii)} The embeddings $U^{p}\\hookrightarrow V^{p}_{-,rc}\\hookrightarrow U^{q}\\hookrightarrow L^{\\infty}_{t}(\\R ;L^{2}_{x}(\\R^{d}))$ are continuous. \n\\end{prop}\n\\begin{thm}[\\cite{HHK09} Proposition\\ 2,10,\\ Remark\\ 2.12]\\label{duality}\nLet $1<p<\\infty$ and $1\/p+1\/p'=1$. \nIf $u\\in V^{1}_{-,rc}$ be absolutely continuous on every compact intervals, then\n\\[\n \\| u \\| _{U^{p}}=\\sup_{v\\in V^{p'},  \\| v \\| _{V^{p'}}=1}\\left|\\int_{-\\infty}^{\\infty}(u'(t),v(t))_{L^{2}(\\R^{d})}dt\\right|.\n\\]\n\\end{thm}\n\\begin{defn}\nLet $1\\leq p<\\infty$. We define\n\\[\nU^{p}_{S}:=\\{ u:\\R\\rightarrow L^{2}|\\ S(-\\cdot )u\\in U^{p}\\}\n\\]\nwith the norm $ \\| u \\| _{U^{p}_{S}}:= \\| S(-\\cdot )u \\| _{U^{p}}$, \n\\[\nV^{p}_{S}:=\\{ v:\\R\\rightarrow L^{2}|\\ S(-\\cdot )v\\in V^{p}_{-,rc}\\}\n\\]\nwith the norm $ \\| v \\| _{V^{p}_{S}}:= \\| S(-\\cdot )v \\| _{V^{p}}$.\n\\end{defn}\n\\begin{rem}\nThe embeddings $U^{p}_{S}\\hookrightarrow V^{p}_{S}\\hookrightarrow U^{q}_{S}\\hookrightarrow L^{\\infty}(\\R;L^{2})$ hold for $1\\leq p<q<\\infty$\nby {\\rm Proposition~\\ref{upvpprop}}. \n\\end{rem}\n\\begin{prop}[\\cite{HHK09} Corollary\\ 2.18]\\label{projest}\nLet $1< p<\\infty$. We have\n\\begin{align}\n& \\| Q_{\\geq M}^{S}u \\| _{L_{tx}^{2}}\\lesssim M^{-1\/2} \\| u \\| _{V^{2}_{S}},\\label{highMproj}\\\\\n& \\| Q_{<M}^{S}u \\| _{V^{p}_{S}}\\lesssim  \\| u \\| _{V^{p}_{S}},\\ \\  \\| Q_{\\geq M}^{S}u \\| _{V^{p}_{S}}\\lesssim  \\| u \\| _{V^{p}_{S}},\\label{Vproj}\n\\end{align}\n\\end{prop}\n\\begin{prop}[\\cite{HHK09} Proposition\\ 2.19]\\label{multiest}\nLet \n\\[\nT_{0}:L^{2}(\\R^{d})\\times \\cdots \\times L^{2}(\\R^{d})\\rightarrow L^{1}_{loc}(\\R^{d})\n\\]\nbe a $m$-linear operator. Assume that for some $1\\leq p, q< \\infty$\n\\[\n \\| T_{0}(S(\\cdot )\\phi_{1},\\cdots ,S(\\cdot )\\phi_{m}) \\| _{L^{p}_{t}(\\R :L^{q}_{x}(\\R^{d}))}\\lesssim \\prod_{i=1}^{m} \\| \\phi_{i} \\| _{L^{2}(\\R^{d})}.\n\\]\nThen, there exists $T:U^{p}_{S}\\times \\cdots \\times U^{p}_{S}\\rightarrow L^{p}_{t}(\\R ;L^{q}_{x}(\\R^{d}))$ satisfying\n\\[\n \\| T(u_{1},\\cdots ,u_{m}) \\| _{L^{p}_{t}(\\R ;L^{q}_{x}(\\R^{d}))}\\lesssim \\prod_{i=1}^{m} \\| u_{i} \\| _{U^{p}_{S}}\n\\]\nsuch that $T(u_{1},\\cdots ,u_{m})(t)(x)=T_{0}(u_{1}(t),\\cdots ,u_{m}(t))(x)$ a.e.\n\\end{prop}\nNow we refer the Strichartz estimate for the fourth order Schr\\\"odinger equation proved by Pausader.\nWe say that a pair $(p,q)$ is admissible if $2 \\le p,q \\le \\infty$, $(p,q,d) \\neq (2, \\infty ,2)$, and\n\\[\n\\frac{2}{p} + \\frac{d}{q} = \\frac{d}{2}.\n\\]\n\\begin{prop}[\\cite{P07} Proposition\\ 3.1]\\label{Stri_est}\nLet $(p,q)$ and $(a,b)$ be admissible pairs.\nThen, we have\n\\[\n\\begin{split}\n\\| S(\\cdot )\\varphi  \\| _{L_{t}^{p}L_{x}^{q}}&\\lesssim  \\| |\\nabla|^{-2\/p}\\varphi  \\| _{L^{2}_{x}},\\\\\n\\left\\| \\int_{0}^{t}S(t-t' )F(t')dt'\\varphi  \\right\\| _{L_{t}^{p}L_{x}^{q}}&\\lesssim  \\| |\\nabla|^{-2\/p-2\/a}F \\| _{L^{a'}_{t}L^{b'}_{x}},\n\\end{split}\n\\]\nwhere $a'$ and $b'$ are conjugate exponents of $a$ and $b$ respectively.\n\\end{prop}\nPropositions \\ref{multiest} and ~\\ref{Stri_est} imply the following.\n\\begin{cor}\\label{Up_Stri}\nLet $(p,q)$ be an admissible pair.\n\\begin{equation}\\label{U_Stri}\n \\| u \\| _{L_{t}^{p}L_{x}^{q}}\\lesssim  \\| |\\nabla|^{-2\/p}u \\| _{U_{S}^{p}},\\ \\ u\\in U^{p}_{S}.\n\\end{equation}\n\\end{cor}\nNext, we define the function spaces which will be used to construct the solution. \nWe define the projections $P_{>1}$ and $P_{<1}$ as\n\\[\nP_{>1}:=\\sum_{N\\ge 1}P_N,\\ P_{<1}:=Id-P_{>1}. \n\\]\n\n\\begin{defn}\\label{YZ_space}\nLet $s <0$.\\\\\n{\\rm (i)} We define $\\dot{Z}^{s}:=\\{u\\in C(\\R ; \\dot{H}^{s}(\\R^{d}))\\cap U^{2}_{S}|\\  \\| u \\| _{\\dot{Z}^{s}}<\\infty\\}$ with the norm\n\\[\n \\| u \\| _{\\dot{Z}^{s}}:=\\left(\\sum_{N}N^{2s} \\| P_{N}u \\| ^{2}_{U^{2}_{S}}\\right)^{1\/2}.\n\\]\n{\\rm (ii)} We define $Z^{s}:=\\{u\\in C(\\R ; H^{s}(\\R^{d})) |\\  \\| u \\| _{Z^{s}}<\\infty\\}$ with the norm\n\\[\n \\| u \\| _{Z^{s}}:= \\| P_{<1} u \\| _{\\dot{Z}^{0}}+ \\| P_{>1} u \\| _{\\dot{Z}^{s}}. \n\\]\n{\\rm (iii)} We define $\\dot{Y}^{s}:=\\{u\\in C(\\R ; \\dot{H}^{s}(\\R^{d}))\\cap V^{2}_{S}|\\  \\| u \\| _{\\dot{Y}^{s}}<\\infty\\}$ with the norm\n\\[\n \\| u \\| _{\\dot{Y}^{s}}:=\\left(\\sum_{N}N^{2s} \\| P_{N}u \\| ^{2}_{V^{2}_{S}}\\right)^{1\/2}.\n\\]\n{\\rm (iv)} We define $Y^{s}:=\\{u\\in C(\\R ; H^{s}(\\R^{d})) |\\  \\| u \\| _{Y^{s}}<\\infty\\}$ with the norm\n\\[\n \\| u \\| _{Y^{s}}:= \\| P_{<1} u \\| _{\\dot{Y}^{0}}+ \\| P_{>1 }u \\| _{\\dot{Y}^{s}}.\n\\]\n\\end{defn}\n\\section{Multilinear estimate for $P_{4}(u,\\overline{u})=\\overline{u}^{4}$ in $1d$ \\label{Multi_est}}\nIn this section, we prove multilinear estimates for the nonlinearity $\\partial_{x}(\\overline{u}^{4})$ in $1d$, which plays a crucial role in the proof of Theorem \\ref{wellposed_1}.\n\\begin{lemm}\\label{modul_est}\nWe assume that $(\\tau_{0},\\xi_{0})$, $(\\tau_{1}, \\xi_{1})$, $\\cdots$, $(\\tau_{4}, \\xi_{4})\\in \\R\\times \\R^{d}$ satisfy \n$\\sum_{j=0}^{4}\\tau_{j}=0$ and $\\sum_{j=0}^{4}\\xi_{j}=0$. Then, we have \n\\begin{equation}\\label{modulation_est}\n\\max_{0\\leq j\\leq 4}|\\tau_{j}-|\\xi_{j}|^{4}|\n\\geq \\frac{1}{5}\\max_{0\\leq j\\leq 4}|\\xi_{j}|^{4}. \n\\end{equation}\n\\end{lemm}\n\\begin{proof}\nBy the triangle inequality, we obtain (\\ref{modulation_est}).  \n\\end{proof}\n\n\\subsection{The homogeneous case}\n\n\\begin{prop}\\label{HL_est_n}\nLet $d=1$ and $0<T\\leq \\infty$. \nFor a dyadic number $N_{1}\\in 2^{\\Z}$, we define the set $A_{1}(N_{1})$ as\n\\[\nA_{1}(N_{1}):=\\{ (N_{2},N_{3},N_{4})\\in (2^{\\Z})^{3}|N_{1}\\gg N_{2}\\geq N_{3} \\geq N_{4}\\}. \n\\]\nIf $N_{0}\\sim N_{1}$, then we have\n\\begin{equation}\\label{hl}\n\\begin{split}\n&\\left|\\sum_{A_{1}(N_{1})}\\int_{0}^{T}\\int_{\\R}\\left(N_{0}\\prod_{j=0}^{4}P_{N_{j}}u_{j}\\right)dxdt\\right|\\\\\n&\\lesssim \n \\| P_{N_{0}}u_{0} \\| _{V^{2}_{S}} \\| P_{N_{1}}u_{1} \\| _{V^{2}_{S}}\\prod_{j=2}^{4} \\| u_{j} \\| _{\\dot{Y}^{-1\/2}}. \n\\end{split}\n\\end{equation}\n\\end{prop}\n\\begin{proof}\nWe define $u_{j,N_{j},T}:=\\ee_{[0,T)}P_{N_{j}}u_{j}$\\ $(j=1,\\cdots ,4)$ and put \n$M:=N_{0}^{4}\/5$. We decompose $Id=Q^{S}_{<M}+Q^{S}_{\\geq M}$. \nWe divide the integrals on the left-hand side of (\\ref{hl}) into $10$ pieces of the form \n\\begin{equation}\\label{piece_form_hl}\n\\int_{\\R}\\int_{\\R}\\left(N_{0}\\prod_{j=0}^{4}Q_{j}^{S}u_{j,N_{j},T}\\right) dxdt\n\\end{equation}\nwith $Q_{j}^{S}\\in \\{Q_{\\geq M}^{S}, Q_{<M}^{S}\\}$\\ $(j=0,\\cdots ,4)$. \nBy the Plancherel's theorem, we have\n\\[\n(\\ref{piece_form_hl})\n= c\\int_{\\sum_{j=0}^{4}\\tau_{j}=0}\\int_{\\sum_{j=0}^{4}\\xi_{j}=0}N_{0}\\prod_{j=0}^{4}\\F[Q_{j}^{S}u_{j,N_{j},T}](\\tau_{j},\\xi_{j}),\n\\]\nwhere $c$ is a constant. Therefore, Lemma~\\ref{modul_est} implies that\n\\[\n\\int_{\\R}\\int_{\\R}\\left(N_{0}\\prod_{j=0}^{m}Q_{<M}^{S}u_{j,N_{j},T}\\right) dxdt=0.\n\\]\nSo, let us now consider the case that $Q_{j}^{S}=Q_{\\geq M}^{S}$ for some $0\\leq j\\leq 4$. \n\nFirst, we consider the case $Q_{0}^{S}=Q_{\\geq M}^{S}$.  \nBy the Cauchy-Schwartz inequality, we have\n\\[\n\\begin{split}\n&\\left|\\sum_{A_{1}(N_{1})}\\int_{\\R}\\int_{\\R}\\left(N_{0}Q_{\\geq M}^{S}u_{0,N_{0},T}\\prod_{j=1}^{4}Q_{j}^{S}u_{j,N_{j},T}\\right)dxdt\\right|\\\\\n&\\leq N_{0} \\| Q_{\\geq M}^{S}u_{0,N_{0},T} \\| _{L^{2}_{tx}} \\| Q_{1}^{S}u_{1,N_{1},T} \\| _{L^{4}_{t}L^{\\infty}_{x}}\\prod_{j=2}^{4}\\left\\|\\sum_{N_{j}\\lesssim N_{1}}Q_{j}^{S}u_{j,N_{j},T}\\right\\|_{L^{12}_{t}L^{6}_{x}}. \n\\end{split}\n\\]\nFurthermore by (\\ref{highMproj}) and $M\\sim N_{0}^{4}$, we have\n\\[\n \\| Q_{\\geq M}^{S}u_{0,N_{0},T} \\| _{L^{2}_{tx}}\n\\lesssim N_{0}^{-2} \\| u_{0,N_{0},T} \\| _{V^{2}_{S}}\n\\]\nand by (\\ref{U_Stri}) and $V^{2}_{S}\\hookrightarrow U^{4}_{S}$, \nwe have\n\\[\n\\begin{split}\n \\| Q_{1}^{S}u_{1,N_{1},T} \\| _{L_{t}^{4}L_{x}^{\\infty}}\n&\\lesssim N_{1}^{-1\/2} \\| Q_{1}^{S}u_{1,N_{1},T} \\| _{U^{4}_{S}}\n\\lesssim N_{1}^{-1\/2} \\| Q_{1}^{S}u_{1,N_{1},T} \\| _{V^{2}_{S}}. \n\\end{split}\n\\]\nWhile by the Sobolev inequality, (\\ref{U_Stri}), $V^{2}_{S}\\hookrightarrow U^{12}_{S}$ and the Cauchy-Schwartz inequality for the dyadic sum\n, we have\n\\begin{equation}\\label{L12L6_est}\n\\begin{split}\n\\left\\|\\sum_{N_{j}\\lesssim N_{1}}Q_{j}^{S}u_{j,N_{j},T}\\right\\|_{L^{12}_{t}L^{6}_{x}}\n&\\lesssim \\left\\| |\\nabla |^{1\/6}\\sum_{N_{j}\\lesssim N_{1}}Q_{j}^{S}u_{j,N_{j},T} \\right\\| _{L^{12}_{t}L^{3}_{x}}\n\\lesssim \\left\\| \\sum_{N_{j}\\lesssim N_{1}}Q_{j}^{S}u_{j,N_{j},T} \\right\\| _{V^{2}_{S}}\\\\\n& \\lesssim N_1^{1\/2} \\left( \\sum _{N_j \\lesssim N_1} N_j^{-1} \\| u_{j,N_j,T} \\|_{V^2_S}^2 \\right) ^{1\/2}\n\\lesssim N_{1}^{1\/2} \\| \\ee_{[0,T)}u_{j} \\| _{\\dot{Y}^{-1\/2}}\n\\end{split}\n\\end{equation}\nfor $2\\leq j\\leq 4$. Therefore, we obtain\n\\[\n\\begin{split}\n&\\left|\\sum_{A_{1}(N_{1})}\\int_{\\R}\\int_{\\R}\\left(N_{0}Q_{\\geq M}^{S}u_{0,N_{0},T}\\prod_{j=1}^{m}Q_{j}^{S}u_{j,N_{j},T}\\right)dxdt\\right|\\\\\n&\\lesssim \n \\| P_{N_{0}}u_{0} \\| _{V^{2}_{S}} \\| P_{N_{1}}u_{1} \\| _{V^{2}_{S}}\\prod_{j=2}^{4} \\| u_{j} \\| _{\\dot{Y}^{-1\/2}}\n\\end{split}\n\\]\nby (\\ref{Vproj}) since $ \\| \\ee_{[0,T)}u \\| _{V^{2}_{S}}\\lesssim  \\| u \\| _{V^{2}_{S}}$ for any $T\\in (0,\\infty]$. \nFor the case $Q_{1}^{S}=Q_{\\geq M}^{S}$ is proved in same way. \n\nNext, we consider the case $Q_{i}^{S}=Q_{\\geq M}^{S}$ for some $2\\le i \\le 4$. \nBy the H\\\"older inequality, we have\n\\[\n\\begin{split}\n&\\left|\\sum_{A_{1}(N_{1})}\\int_{\\R}\\int_{\\R}\\left(N_{0}Q_{\\geq M}^{S}u_{i,N_{i},T}\\prod_{\\substack{0\\le j\\le 4\\\\ j\\neq i}}Q_{j}^{S}u_{j,N_{j},T}\\right)dxdt\\right|\\\\\n&\\lesssim N_{0} \\| Q_{0}^{S}u_{0,N_{0},T} \\| _{L_{t}^{12}L_{x}^{6}} \\| Q_{1}^{S}u_{1,N_{1},T} \\| _{L_{t}^{4}L_{x}^{\\infty}}\\\\\n&\\ \\ \\ \\ \\times \\left\\| \\sum_{N_{i}\\lesssim N_{1}}Q_{\\geq M}^{S}u_{i,N_{i},T} \\right\\|_{L_{tx}^{2}}\n\\prod_{\\substack{2\\le j\\le 4 \\\\ j\\neq i}}\\left\\| \\sum_{N_{j}\\lesssim N_{1}}Q_{j}^{S}u_{j,N_{j},T} \\right\\| _{L_{t}^{12}L_{x}^{6}}. \n\\end{split}\n\\]\nBy $L^{2}$ orthogonality and (\\ref{highMproj}), we have\n\\begin{equation}\\label{hi_mod_234}\n\\begin{split}\n\\left\\| \\sum_{N_{i}\\lesssim N_{1}}Q_{\\geq M}^{S}u_{i,N_{i},T}\\right\\| _{L_{tx}^{2}}\n&\\lesssim \\left(\\sum_{N_{2}} \\| Q_{\\geq M}^{S}u_{i,N_{i},T} \\| _{L_{tx}^{2}}^{2}\\right)^{1\/2}\\\\\n&\\lesssim N_{1}^{-3\/2} \\| \\ee_{[0,T)}u_{i} \\| _{\\dot{Y}^{-1\/2}}\n\\end{split}\n\\end{equation}\nsince $M\\sim N_{0}^{4}$. While, by the calculation way as the case $Q_{0}^{S}=Q_{\\geq M}^{S}$, we have\n\\[\n \\| Q_{0}^{S}u_{0,N_{0},T} \\| _{L_{t}^{12}L_{x}^{6}}\\lesssim  \\| Q_{0}^{S}u_{0,N_{0},T} \\| _{V^{2}_{S}},\n\\]\n\\[\n \\| Q_{1}^{S}u_{1,N_{1},T} \\| _{L_{t}^{4}L_{x}^{\\infty}}\\lesssim N_{1}^{-1\/2} \\| Q_{1}^{S}u_{1,N_{1},T} \\| _{V^{2}_{S}}\n\\]\nand\n\\[\n\\left\\| \\sum_{N_{j}\\lesssim N_{1}}Q_{j}^{S}u_{j,N_{j},T} \\right\\|_{L_{t}^{12}L_{x}^{6}}\n\\lesssim N_{1}^{1\/2} \\| \\ee_{[0,T)}u_{j} \\| _{\\dot{Y}^{-1\/2}}. \n\\]\nTherefore, we obtain\n\\[\n\\begin{split}\n&\\left|\\sum_{A_{1}(N_{1})}\\int_{\\R}\\int_{\\R}\\left(N_{0}Q_{\\geq M}^{S}u_{i,N_{i},T}\\prod_{\\substack{0\\le j\\le 4\\\\ j\\neq i}}Q_{j}^{S}u_{j,N_{j},T}\\right)dxdt\\right|\\\\\n&\\lesssim \n \\| P_{N_{0}}u_{0} \\| _{V^{2}_{S}} \\| P_{N_{1}}u_{1} \\| _{V^{2}_{S}}\\prod_{j=2}^{4} \\| u_{j} \\| _{\\dot{Y}^{-1\/2}}\n\\end{split}\n\\]\nby (\\ref{Vproj}) since $ \\| \\ee_{[0,T)}u \\| _{V^{2}_{S}}\\lesssim  \\| u \\| _{V^{2}_{S}}$ for any $T\\in (0,\\infty]$. \n\\end{proof}\n\\begin{prop}\\label{HH_est}\nLet $d=1$ and $0<T\\leq \\infty$. \nFor a dyadic number $N_{2}\\in 2^{\\Z}$, we define the set $A_{2}(N_{2})$ as\n\\[\nA_{2}(N_{2}):=\\{ (N_{3}, N_{4})\\in (2^{\\Z})^{4}|N_{2}\\geq N_{3}\\geq N_{4}\\}. \n\\]\nIf $N_{0}\\lesssim N_{1}\\sim N_{2}$, then we have\n\\begin{equation}\\label{hh}\n\\begin{split}\n&\\left|\\sum_{A_{2}(N_{2})}\\int_{0}^{T}\\int_{\\R}\\left(N_{0}\\prod_{j=0}^{4}P_{N_{j}}u_{j}\\right)dxdt\\right|\\\\\n&\\lesssim \n\\frac{N_{0}}{N_{1}} \\| P_{N_{0}}u_{0} \\| _{V^{2}_{S}} \\| P_{N_{1}}u_{1} \\| _{V^{2}_{S}}N_{2}^{-1\/2} \\| P_{N_{2}}u_{2} \\| _{V^{2}_{S}} \\| u_{3} \\| _{\\dot{Y}^{-1\/2}} \\| u_{4} \\| _{\\dot{Y}^{-1\/2}}. \n\\end{split}\n\\end{equation}\n\\end{prop}\nThe proof of Proposition~\\ref{HH_est} is quite similar as the proof of Proposition~\\ref{HL_est_n}. \n\n\n\\subsection{The inhomogeneous case}\n\\begin{prop}\\label{HL_est_n-inh}\nLet $d=1$ and $0<T\\leq 1$. \nFor a dyadic number $N_{1}\\in 2^{\\Z}$, we define the set $A_{1}'(N_{1})$ as\n\\[\nA_{1}'(N_{1}):=\\{ (N_{2},N_{3},N_{4})\\in (2^{\\Z})^{3}|N_{1}\\gg N_{2}\\geq N_{3} \\ge N_{4}, \\, N_4 \\le 1 \\}. \n\\]\nIf $N_{0}\\sim N_{1}$, then we have\n\\begin{equation}\\label{hl-inh}\n\\begin{split}\n\\left|\\sum_{A_{1}'(N_{1})}\\int_{0}^{T}\\int_{\\R}\\left(N_{0}\\prod_{j=0}^{4}P_{N_{j}}u_{j}\\right)dxdt\\right|\n\\lesssim T^{\\frac{1}{6}} \\| P_{N_{0}}u_{0} \\| _{V^{2}_{S}} \\| P_{N_{1}}u_{1} \\| _{V^{2}_{S}} \\prod_{j=2}^{4} \\| u_{j} \\| _{Y^{-1\/2}}. \n\\end{split}\n\\end{equation}\n\\end{prop}\n\\begin{proof}\nWe further divide $A_1'(N_1)$ into three pieces:\n\\begin{align*}\nA_1'(N_1) & = \\bigcup _{j=1}^3 A_{1,j}'(N_1), \\\\\nA_{1,1}'(N_1) &:= \\{ (N_{2},N_{3},N_{4}) \\in  A_1'(N_1) : N_3 \\ge 1 \\} ,\\\\\nA_{1,2}'(N_2) &:= \\{ (N_{2},N_{3},N_{4}) \\in  A_1'(N_1) : N_2 \\ge 1 \\ge N_3 \\} ,\\\\\nA_{1,3}'(N_2) &:= \\{ (N_{2},N_{3},N_{4}) \\in  A_1'(N_1) :  1 \\ge N_2 \\} .\n\\end{align*}\nWe define $u_{j,N_{j}}:=P_{N_{j}}u_{j}$, $u_{j,T}:=\\ee_{[0,T)}u_{j}$ and $u_{j,N_{j},T}:=\\ee_{[0,T)}P_{N_{j}}u_{j}$\\ $(j=1,\\cdots ,4)$.\nWe firstly consider the case $A_{1,1}'(N_1)$\nIn the case $T \\le N_0^{-3}$, the H\\\"older inequality implies\n\\begin{align*}\n& \\left|\\sum_{A_{1,1}'(N_{1})} \\int_{0}^{T}\\int_{\\R}\\left(N_{0}\\prod_{j=0}^{4}P_{N_{j}}u_{j}\\right)dxdt\\right| \\\\\n& \\le N_0 \\| \\ee_{[0,T)}\\|_{L^{2}_{t}}\n\\| u_{0,N_0} \\| _{L_t^4 L_x^{\\infty}} \\| u_{1,N_1} \\| _{L_t^4 L_x^{\\infty}} \n\\prod _{j=2}^3 \\left\\| \\sum_{1\\le N_j \\le N_{1}} u_{j,N_j} \\right\\| _{L_t^{\\infty} L_x^2} \\| P_{<1} u_{4} \\| _{L_t^{\\infty} L_x^{\\infty}\n\\end{align*}\nFurthermore by (\\ref{U_Stri}) and $V^{2}_{S}\\hookrightarrow U^{4}_{S}$, \nwe have\n\\[\n\\begin{split}\n\\| u_{0,N_0} \\| _{L_t^4 L_x^{\\infty}} \\| u_{1,N_1} \\| _{L_t^4 L_x^{\\infty}} \n&\\lesssim N_{0}^{-1\/2}\\| u_{0,N_0} \\| _{U^{4}_{S}}N_{1}^{-1\/2} \\| Q_{1}^{S}u_{1,N_{1}} \\| _{U^{4}_{S}}\\\\\n&\\lesssim N_{0}^{-1} \\| u_{0,N_{0}} \\| _{V^{2}_{S}} \\| u_{1,N_{1}} \\| _{V^{2}_{S}}\n\\end{split}\n\\]\nand by the Sobolev inequality, $V^{2}_{S}\\hookrightarrow L^{\\infty}_{t}L^{2}_{x}$ and the Cauchy-Schwartz inequality , we have\n\\[\n\\| P_{<1} u_{4} \\| _{L_t^{\\infty} L_x^{\\infty}}\\lesssim \\| P_{<1} u_{4} \\| _{L_t^{\\infty} L_x^{2}}\n\\lesssim \\left(\\sum_{N\\le 2}\\|P_{N}P_{<1}u_{4}\\|_{V^{2}_{S}}^{2}\\right)^{1\/2}\n\\le \\|P_{<1}u_4\\|_{\\dot{Y}^{0}}\n\\]\nWhile by $L^{2}$ orthogonality and $V^{2}_{S}\\hookrightarrow L^{\\infty}_{t}L^{2}_{x}$, we have\n\\[\n\\begin{split}\n\\left\\| \\sum_{1\\le N_j \\le N_{1}} u_{j,N_j} \\right\\| _{L_t^{\\infty} L_x^2}\n&\\lesssim \\left(\\sum_{1\\le N_j \\le N_{1}} \\| u_{j,N_{j}} \\| _{V^{2}_{S}}^{2}\\right)^{1\/2}\n\\lesssim N_{0}^{1\/2} \\| P_{>1}u_{j} \\| _{\\dot{Y}^{-1\/2}}\n\\end{split}\n\\]\nTherefore, we obtain\n\\[\n\\begin{split}\n&\\left|\\sum_{A_{1,1}'(N_{1})} \\int_{0}^{T}\\int_{\\R}\\left(N_{0}\\prod_{j=0}^{4}P_{N_{j}}u_{j}\\right)dxdt\\right| \\\\\n&\\lesssim T^{1\/2}N_0 \\| u_{0,N_{0}} \\| _{V^{2}_{S}} \\| u_{1,N_{1}} \\| _{V^{2}_{S}}\\prod_{j=2}^{3}\\| P_{>1}u_{j} \\| _{\\dot{Y}^{-1\/2}}\\|P_{<1}u_4\\|_{\\dot{Y}^{0}}\n\\end{split}\n\\]\nand note that $T^{1\/2}N_0\\le T^{1\/6}$.\n\nIn the case $T \\ge N_0^{-3}$, we divide the integrals on the left-hand side of (\\ref{hl}) into $10$ pieces of the form \\eqref{piece_form_hl} in the proof of Proposition \\ref{HL_est_n}.\nThanks to Lemma~\\ref{modul_est}, let us consider the case that $Q_{j}^{S}=Q_{\\geq M}^{S}$ for some $0\\leq j\\leq 4$.\nFirst, we consider the case $Q_{0}^{S}=Q_{\\geq M}^{S}$.  \nBy the same way as in the proof of Proposition \\ref{HL_est_n} and using\n\\[\n\\|Q_{4}^{S}P_{<1}u_{4,T}\\|_{L^{12}_{t}L^{6}_{x}}\\lesssim \\|Q_{4}^{S}P_{<1}u_{4,T}\\|_{V^{2}_{S}}\\lesssim \\|P_{<1}u_{4,T}\\|_{\\dot{Y}^{0}}\n\\]\ninstead of (\\ref{L12L6_est}), we obtain\n\\[\n\\begin{split}\n&\\left|\\sum_{A_{1,1}'(N_{1})}\\int_{\\R}\\int_{\\R}\\left(N_{0}Q_{\\geq M}^{S}u_{0,N_{0},T}\\prod_{j=1}^{4}Q_{j}^{S}u_{j,N_{j},T}\\right)dxdt\\right|\\\\\n&\\leq N_{0} \\| Q_{\\geq M}^{S}u_{0,N_{0},T} \\| _{L^{2}_{tx}} \\| Q_{1}^{S}u_{1,N_{1},T} \\| _{L^{4}_{t}L^{\\infty}_{x}}\n\\prod_{j=2}^{3} \\left\\|\\sum_{1 \\le N_{j}\\lesssim N_{1}}Q_{j}^{S}u_{j,N_{j},T}\\right\\|_{L^{12}_{t}L^{6}_{x}} \\|Q_{4}^{S}P_{<1}u_{4,T}\\|_{L^{12}_{t}L^{6}_{x}}\\\\\n& \\lesssim N_0^{-\\frac{1}{2}} \\| P_{N_0} u_0 \\| _{V^2_S} \\| P_{N_1} u_1 \\| _{V^2_S} \\prod_{j=2}^{3} \\left\\| P_{>1} u_j \\right\\| _{\\dot{Y}^{-1\/2}} \\| P_{<1} u_{4} \\| _{\\dot{Y}^0}\n\\end{split}\n\\]\nand note that $N_0^{-1\/2}\\le T^{1\/6}$. \nSince the cases $Q_j^S = Q_{\\ge M}^S$ ($j=1,2,3$) are similarly handled, we omit the details here.\n\n\nWe focus on the case $Q_4^S = Q_{\\ge M}^S$.\nBy the same way as in the proof of Proposition \\ref{HL_est_n} and using\n\\[\n\\|Q_{\\ge M}^{S}P_{<1}u_{4,T}\\|_{L^{2}_{tx}}\\lesssim N_{0}^{-2} \\|P_{<1}u_{4,T}\\|_{V^{2}_{S}}\\lesssim N_{0}^{-2}\\|P_{<1}u_{4,T}\\|_{\\dot{Y}^{0}}\n\\]\ninstead of (\\ref{hi_mod_234}) with $j=4$, we obtain\n\\[\n\\begin{split}\n&\\left|\\sum_{A_{1,1}'(N_{1})}\\int_{\\R}\\int_{\\R}\\left(N_{0}Q_{\\geq M}^{S}u_{4,N_{4},T}\\prod_{j=0}^{3}Q_{j}^{S}u_{j,N_{j},T}\\right)dxdt\\right|\\\\\n&\\leq N_{0} \\| u_{0,N_{0},T} \\| _{L^{12}_{t}L_x^6} \\| Q_{1}^{S}u_{1,N_{1},T} \\| _{L^{4}_{t}L^{\\infty}_{x}}\n\\prod_{j=2}^{3} \\left\\|\\sum_{1 \\le N_{j}\\lesssim N_{1}}Q_{j}^{S}u_{j,N_{j},T}\\right\\|_{L^{12}_{t}L^{6}_{x}} \n\\|Q_{\\geq M}^{S} P_{<1}u_{4,T}\\|_{L^{2}_{tx}}\\\\\n& \\lesssim N_{0}^{-1\/2}\\| P_{N_0} u_0 \\| _{V^2_S}  \\| P_{N_1} u_1 \\| _{V^2_S} \\prod_{j=2}^{3} \\left\\| P_{>1} u_j \\right\\| _{\\dot{Y}^{-1\/2}} \\| P_{<1} u_4 \\| _{\\dot{Y}^0}\n\\end{split}\n\\]\nand note that $N_0^{-1\/2}\\le T^{1\/6}$. \n\n\nWe secondly consider the case $A_{1,2}'(N_1)$.\nIn the case $T \\le N_0^{-3}$, the H\\\"older inequality implies\n\\[\n\\begin{split}\n& \\left|\\sum_{A_{1,2}'(N_{1})} \\int_{0}^{T}\\int_{\\R}\\left(N_{0}\\prod_{j=0}^{4}P_{N_{j}}u_{j}\\right)dxdt\\right| \\\\\n& \\le N_0 \\| \\ee_{[0,T)}\\|_{L^{2}_{t}}\n\\| u_{0,N_0} \\| _{L_t^4 L_x^{\\infty}} \\| u_{1,N_1} \\| _{L_t^4 L_x^{\\infty}} \\left\\| \\sum _{1 \\le N_2 \\lesssim N_1} u_{2,N_2} \\right\\| _{L_t^{\\infty} L_x^2}\n\\prod_{j=3}^{4}\\| P_{<1} u_{j} \\| _{L_t^{\\infty} L_x^4} .\n\\end{split}\n\\]\nBy the same estimates as in the proof for the case $A_{1,1}'(N_1)$ and\n\\[\n\\| P_{<1} u_{j} \\| _{L_t^{\\infty} L_x^4}\\lesssim \\| P_{<1} u_{j} \\| _{L_t^{\\infty} L_x^{2}}\n\\lesssim \\left(\\sum_{N\\le 2}\\|P_{N}P_{<1}u_{j}\\|_{V^{2}_{S}}^{2}\\right)^{1\/2}\n\\le \\|P_{<1}u_j\\|_{\\dot{Y}^{0}}\n\\]\nfor $j=3,4$, we obtain\n\\[\n\\begin{split}\n&\\left|\\sum_{A_{1,2}'(N_{1})} \\int_{0}^{T}\\int_{\\R}\\left(N_{0}\\prod_{j=0}^{4}P_{N_{j}}u_{j}\\right)dxdt\\right| \\\\\n&\\lesssim T^{1\/2}N_0^{1\/2} \\| u_{0,N_{0}} \\| _{V^{2}_{S}} \\| u_{1,N_{1}} \\| _{V^{2}_{S}}\\| P_{>1}u_{2} \\| _{\\dot{Y}^{-1\/2}}\\prod_{j=3}^{4}\\|P_{<1}u_j\\|_{\\dot{Y}^{0}}\n\\end{split}\n\\]\nand note that $T^{1\/2}N_0^{1\/2}\\le T^{1\/3}$. \n\n\nIn the case $T \\ge N_0^{-3}$, we divide the integrals on the left-hand side of (\\ref{hl}) into $10$ pieces of the form \\eqref{piece_form_hl} in the proof of Proposition \\ref{HL_est_n}.\nThanks to Lemma~\\ref{modul_est}, let us consider the case that $Q_{j}^{S}=Q_{\\geq M}^{S}$ for some $0\\leq j\\leq 4$.\nBy the same argument as in the proof for the case $A_{1,1}'(N_1)$, we obtain \n\\[\n\\begin{split}\n&\\left|\\sum_{A_{1,2}'(N_{1})}\\int_{\\R}\\int_{\\R}\\left(N_{0}Q_{\\geq M}^{S}u_{0,N_{0},T}\\prod_{j=1}^{4}Q_{j}^{S}u_{j,N_{j},T}\\right)dxdt\\right|\\\\\n&\\leq N_{0} \\| Q_{\\geq M}^{S}u_{0,N_{0},T} \\| _{L^{2}_{tx}} \\| Q_{1}^{S}u_{1,N_{1},T} \\| _{L^{4}_{t}L^{\\infty}_{x}} \\left\\|\\sum_{1 \\le N_{2}\\lesssim N_{1}}Q_{2}^{S}u_{2,N_{2},T}\\right\\|_{L^{12}_{t}L^{6}_{x}} \\prod_{j=3}^{4} \\| Q_{j}^{S}P_{<1}u_{j,T}\\|_{L^{12}_{t}L^{6}_{x}}\\\\\n& \\lesssim N_0^{-1} \\| P_{N_0} u_0 \\| _{V^2_S} | P_{N_1} u_1 \\| _{V^2_S} \\left\\| P_{>1} u_2 \\right\\| _{\\dot{Y}^{-1\/2}} \\prod _{j=3}^4 \\| P_{<1} v_j \\| _{\\dot{Y}^0}\n\\end{split}\n\\]\nif $Q_0 = Q_{\\ge M}^S$ and \n\\[\n\\begin{split}\n&\\left|\\sum_{A_{1,2}'(N_{1})}\\int_{\\R}\\int_{\\R}\\left(N_{4}Q_{\\geq M}^{S}u_{4,N_{4},T}\\prod_{j=0}^{3}Q_{j}^{S}u_{j,N_{j},T}\\right)dxdt\\right|\\\\\n&\\leq N_{0} \\| u_{0,N_{0},T} \\| _{L^{12}_{t}L_x^6} \\| Q_{1}^{S}u_{1,N_{1},T} \\| _{L^{4}_{t}L^{\\infty}_{x}} \\left\\|\\sum_{1 \\le N_{2}\\lesssim N_{1}}Q_{2}^{S}u_{2,N_{2},T}\\right\\|_{L^{12}_{t}L^{6}_{x}} \\\\\n&\\hspace{21ex}\\times \\|Q_{3}^{S} P_{<1}u_{3,T}\\|_{L^{12}_{t}L^{6}_{x}} \\| Q_{\\geq M}^{S} P_{<1}u_{4,T}\\|_{L^{2}_{tx}}\\\\\n& \\lesssim N_0^{-1} \\| P_{N_0} u_0 \\| _{V^2_S} \\| P_{N_1} u_1 \\| _{V^2_S} \\left\\| P_{>1} u_2 \\right\\| _{\\dot{Y}^{\\frac{1}{2}}} \\prod_{j=3}^{4}\\| P_{<1} u_j \\| _{\\dot{Y}^0}\n\\end{split}\n\\]\nif $Q_4 = Q_{\\ge M}^S$\nNote that $N_0^{-1}\\le T^{1\/3}$. \nThe remaining cases follow from the same argument as above.\n\n\nWe thirdly consider the case $A_{1,3}'(N_1)$.\nIn the case $T \\le N_0^{-3}$, the H\\\"older inequality implies\n\\[\n\\begin{split}\n& \\left|\\sum_{A_{1,3}'(N_{1})} \\int_{0}^{T}\\int_{\\R}\\left(N_{0}\\prod_{j=0}^{4}P_{N_{j}}u_{j}\\right)dxdt\\right| \\\\\n& \\le N_0 \\| \\ee_{[0,T)}\\|_{L^{2}_{t}}\\| u_{0,N_0} \\| _{L_t^4 L_x^{\\infty}} \\| u_{1,N_1} \\| _{L_t^4 L_x^{\\infty}}\n\\prod_{j=2}^{4} \\| P_{<1}u_{2} \\| _{L_t^{\\infty} L_x^3}.\n\\end{split}\n\\]\nBy the same estimates as in the proof for the case $A_{1,1}'(N_1)$ and\n\\[\n\\| P_{<1} u_{j} \\| _{L_t^{\\infty} L_x^3}\\lesssim \\| P_{<1} u_{j} \\| _{L_t^{\\infty} L_x^{2}}\n\\lesssim \\left(\\sum_{N\\le 2}\\|P_{N}P_{<1}u_{j}\\|_{V^{2}_{S}}^{2}\\right)^{1\/2}\n\\le \\|P_{<1}u_j\\|_{\\dot{Y}^{0}}\n\\]\nfor $j=2, 3,4$, we obtain\n\\[\n\\begin{split}\n&\\left|\\sum_{A_{1,3}'(N_{1})} \\int_{0}^{T}\\int_{\\R}\\left(N_{0}\\prod_{j=0}^{4}P_{N_{j}}u_{j}\\right)dxdt\\right| \n\\lesssim T^{1\/2}\\| u_{0,N_{0}} \\| _{V^{2}_{S}} \\| u_{1,N_{1}} \\| _{V^{2}_{S}}\\prod_{j=2}^{4}\\| P_{<1}u_{j} \\| _{\\dot{Y}^{0}}. \n\\end{split}\n\\]\n\n\nIn the case $T \\ge N_0^{-3}$, we divide the integrals on the left-hand side of (\\ref{hl}) into $10$ pieces of the form \\eqref{piece_form_hl} in the proof of Proposition \\ref{HL_est_n}.\nThanks to Lemma~\\ref{modul_est}, let us consider the case that $Q_{j}^{S}=Q_{\\geq M}^{S}$ for some $0\\leq j\\leq 4$.\nBy the same argument as in the proof for the case $A_{1,1}'(N_1)$, we obtain \n\\[\n\\begin{split}\n&\\left|\\sum_{A_{1,3}'(N_{1})}\\int_{\\R}\\int_{\\R}\\left(N_{0}Q_{\\geq M}^{S}u_{0,N_{0},T}\\prod_{j=1}^{4}Q_{j}^{S}u_{j,N_{j},T}\\right)dxdt\\right|\\\\\n&\\leq N_{0} \\| Q_{\\geq M}^{S}u_{0,N_{0},T} \\| _{L^{2}_{tx}} \\| Q_{1}^{S}u_{1,N_{1},T} \\| _{L^{4}_{t}L^{\\infty}_{x}} \n\\prod_{j=2}^{4} \\|Q_{j}^{S}P_{<1}u_{j,T}\\|_{L^{12}_{t}L^{6}_{x}}\\\\\n& \\lesssim N_0^{-3\/2} \\| P_{N_0} u_0 \\| _{V^2_S} \\| P_{N_1} u_1 \\| _{V^2_S} \\left\\| P_{<1} u_2 \\right\\| _{Y^{-1\/2}} \\prod _{j=3}^4 \\| P_{<1} v_j \\| _{\\dot{Y}^0} \n\\end{split}\n\\]\nif $Q_0 = Q_{\\ge M}^S$ and\n\\[\n\\begin{split}\n&\\left|\\sum_{A_{1,3}'(N_{1})}\\int_{\\R}\\int_{\\R}\\left(N_{4}Q_{\\geq M}^{S}u_{4,N_{4},T}\\prod_{j=0}^{3}Q_{j}^{S}u_{j,N_{j},T}\\right)dxdt\\right|\\\\\n&\\leq N_{0} \\| u_{0,N_{0},T} \\| _{L^{12}_{t}L_x^6} \\| Q_{1}^{S}u_{1,N_{1},T} \\| _{L^{4}_{t}L^{\\infty}_{x}} \\prod _{j=2}^3 \\|Q_{j}^{S} P_{<1}u_{j,T}\\|_{L^{12}_{t}L^{6}_{x}} \n\\|Q_{\\geq M}^{S} P_{<1}u_{4,T}\\|_{L^2_{tx}}\\\\\n& \\lesssim N_0^{-3\/2} \\| P_{N_0} u_0 \\| _{V^2_S} \\| P_{N_1} u_1 \\| _{V^2_S} \\prod _{j=2}^4 \\left\\| P_{<1} u_j \\right\\| _{Y^{0}}\n\\end{split}\n\\]\nif $Q_4 = Q_{\\ge M}^S$.\nNote that $N_0^{-3\/2}\\le T^{1\/2}$. \nThe cases  $Q_j^S = Q_{\\ge M}^S$ ($j=1,2,3$) are the same argument as above. \n\n\\end{proof}\n\n\nFurthermore, we obtain the following estimate.\n\n\\begin{prop}\\label{HH_est-inh}\nLet $d=1$ and $0<T\\leq 1$. \nFor a dyadic number $N_{2}\\in 2^{\\Z}$, we define the set $A_{2}'(N_{2})$ as\n\\[\nA_{2}'(N_{2}):=\\{ (N_{3}, N_{4})\\in (2^{\\Z})^{4}|N_{2}\\geq N_{3}\\ge N_{4} , \\, N_4 \\le 1 \\}. \n\\]\nIf $N_{0}\\lesssim N_{1}\\sim N_{2}$, then we have\n\\begin{equation}\\label{hh-inh}\n\\begin{split}\n&\\left|\\sum_{A_{2}'(N_{2})}\\int_{0}^{T}\\int_{\\R}\\left(N_{0}\\prod_{j=0}^{4}P_{N_{j}}u_{j}\\right)dxdt\\right|\\\\\n&\\lesssim T^{\\frac{1}{6}} \\frac{N_{0}}{N_{1}} \\| P_{N_{0}}u_{0} \\| _{V^{2}_{S}} \\| P_{N_{1}}u_{1} \\| _{V^{2}_{S}}N_{2}^{-1\/2} \\| P_{N_{2}}u_{2} \\| _{V^{2}_{S}} \\| u_{3} \\| _{Y^{-1\/2}} \\| u_{4} \\| _{Y^{-1\/2}}. \n\\end{split}\n\\end{equation}\n\\end{prop}\n\nBecause the proof is similar as above, we skip the proof.\n\n\\section{Proof of well-posedness \\label{pf_wellposed_1}}\n\n\\subsection{The small data case}\n\nIn this section, we prove Theorem~\\ref{wellposed_1} and Corollary~\\ref{sccat}. \nWe define the map $\\Phi_{T, \\varphi}$ as\n\\[\n\\Phi_{T, \\varphi}(u)(t):=S(t)\\varphi -iI_{T}(u,\\cdots, u)(t),\n\\] \nwhere\n\\[\nI_{T}(u_{1},\\cdots u_{4})(t):=\\int_{0}^{t}\\ee_{[0,T)}(t')S(t-t')\\partial_{x}\\left(\\prod_{j=1}^{4}\\overline{u_{j}(t')}\\right)dt'.\n\\]\nTo prove the well-posedness of (\\ref{D4NLS}) in $\\dot{H}^{-1\/2}$, we prove that $\\Phi_{T, \\varphi}$ is a contraction map \non a closed subset of $\\dot{Z}^{-1\/2}([0,T))$. \nKey estimate is the following:\n\\begin{prop}\\label{Duam_est}\nLet $d=1$. For any $0<T<\\infty$, we have\n\\begin{equation}\\label{Duam_est_1}\n \\| I_{T}(u_{1},\\cdots u_{4}) \\| _{\\dot{Z}^{-1\/2}}\\lesssim \\prod_{j=1}^{4} \\| u_{j} \\| _{\\dot{Y}^{-1\/2}}.\n\\end{equation}\n\\end{prop}\n\\begin{proof}\nWe decompose\n\\[\nI_{T}(u_{1},\\cdots u_{m})=\\sum_{N_{1},\\cdots ,N_{4}}I_{T}(P_{N_{1}}u_{1},\\cdots P_{N_{4}}u_{4}).\n\\]\nBy symmetry, it is enough to consider the summation for $N_{1}\\geq N_{2}\\geq N_{3} \\geq N_{4}$. We put\n\\[\n\\begin{split}\nS_{1}&:=\\{ (N_{1},\\cdots ,N_{m})\\in (2^{\\Z})^{m}|N_{1}\\gg N_{2}\\geq N_{3} \\geq N_{4}\\}\\\\\nS_{2}&:=\\{ (N_{1},\\cdots ,N_{m})\\in (2^{\\Z})^{m}|N_{1}\\sim N_{2}\\geq N_{3} \\geq N_{4}\\}\n\\end{split}\n\\]\nand\n\\[\nJ_{k}:=\\left\\| \\sum_{S_{k}}I_{T}(P_{N_{1}}u_{1},\\cdots P_{N_{4}}u_{4}) \\right\\| _{\\dot{Z}^{-1\/2}}\\ (k=1,2).\n\\]\n\nFirst, we prove the estimate for $J_{1}$. By Theorem~\\ref{duality} and the Plancherel's theorem, we have\n\\[\n\\begin{split}\nJ_{1}&\\leq \\left\\{ \\sum_{N_{0}}N_{0}^{-1}\\left\\| S(-\\cdot )P_{N_{0}}\\sum_{S_{1}}I_{T}(P_{N_{1}}u_{1},\\cdots P_{N_{4}}u_{4})\\right\\|_{U^{2}}^{2}\\right\\}^{1\/2}\\\\\n&\\lesssim \\left\\{\\sum_{N_{0}}N_{0}^{-1}\\sum_{N_{1}\\sim N_{0}}\n\\left( \\sup_{ \\| u_{0} \\| _{V^{2}_{S}}=1}\\left|\\sum_{A_{1}(N_{1})}\\int_{0}^{T}\\int_{\\R}\\left(N_{0}\\prod_{j=0}^{4}P_{N_{j}}u_{j}\\right)dxdt\\right|\\right)^{2}\\right\\}^{1\/2}, \n\\end{split}\n\\]\nwhere $A_{1}(N_{1})$ is defined in Proposition~\\ref{HL_est_n}. \nTherefore by Proposition~\\ref{HL_est_n}, we have\n\\[\n\\begin{split}\nJ_{1}&\\lesssim \\left\\{\\sum_{N_{0}}N_{0}^{-1}\\sum_{N_{1}\\sim N_{0}}\n\\left( \\sup_{ \\| u_{0} \\| _{V^{2}_{S}}=1} \\| P_{N_{0}}u_{0} \\| _{V^{2}_{S}} \\| P_{N_{1}}u_{1} \\| _{V^{2}_{S}}\\prod_{j=2}^{4} \\| u_{j} \\| _{\\dot{Y}^{-1\/2}}\\right)^{2}\\right\\}^{1\/2}\\\\\n&\\lesssim \n\\left(\\sum_{N_{1}}N_{1}^{-1} \\| P_{N_{1}}u_{1} \\| _{V^{2}_{\\Delta}}^{2}\\right)^{1\/2}\n\\prod_{j=2}^{4} \\| u_{j} \\| _{\\dot{Y}^{-1\/2}}\\\\\n&=\\prod_{j=1}^{4} \\| u_{j} \\| _{\\dot{Y}^{-1\/2}}.\n\\end{split}\n\\]\n\nNext, we prove the estimate for $J_{2}$. By Theorem~\\ref{duality} and the Plancherel's theorem, we have\n\\[\n\\begin{split}\nJ_{2}&\\leq\n\\sum_{N_{1}}\\sum_{N_{2}\\sim N_{1}}\\left(\\sum_{N_{0}}N_{0}^{-1}\\left\\|S(-\\cdot )P_{N_{0}}\\sum_{A_{2}(N_{2})}I_{T}(P_{N_{1}}u_{1},\\cdots P_{N_{4}}u_{4})\\right\\|_{U^{2}}^{2}\\right)^{1\/2}\\\\\n&=\\sum_{N_{1}}\\sum_{N_{2}\\sim N_{1}}\\left(\\sum_{N_{0}\\lesssim N_{1}}N_{0}^{-1}\n\\sup_{ \\| u_{0} \\| _{V^{2}_{S}}=1}\\left| \\sum_{A_{2}(N_{2})}\\int_{0}^{T}\\int_{\\R}\\left(N_{0}\\prod_{j=0}^{4}P_{N_{j}}u_{j}\\right)dxdt\\right|^{2}\\right)^{1\/2}, \n\\end{split}\n\\]\nwhere $A_{2}(N_{2})$ is defined in Proposition~\\ref{HH_est}. \nTherefore by {\\rm Proposition~\\ref{HH_est}} and Cauchy-Schwartz inequality for the dyadic sum, we have\n\\[\n\\begin{split}\nJ_{2}&\\lesssim\n\\sum_{N_{1}}\\sum_{N_{2}\\sim N_{1}}\\left(\\sum_{N_{0}\\lesssim N_{1}}N_{0}^{-1}\n\\left(\\frac{N_{0}}{N_{1}} \\| P_{N_{1}}u_{1} \\| _{V^{2}_{S}}N_{2}^{-1\/2} \\| P_{N_{2}}u_{2} \\| _{V^{2}_{S}} \\| u_{3} \\| _{\\dot{Y}^{-1\/2}} \\| u_{4} \\| _{\\dot{Y}^{-1\/2}}\\right)^{2}\\right)^{1\/2}\\\\\n&\\lesssim \\left(\\sum_{N_{1}}N_{1}^{-1} \\| P_{N_{1}}u_{1} \\| _{V^{2}_{S}}^{2}\\right)^{1\/2}\n\\left(\\sum_{N_{2}}N_{2}^{-1} \\| P_{N_{2}}u_{2} \\| _{V^{2}_{S}}^{2}\\right)^{1\/2} \\| u_{3} \\| _{\\dot{Y}^{-1\/2}} \\| u_{4} \\| _{\\dot{Y}^{-1\/2}}\\\\\n&= \\prod_{j=1}^{4} \\| u_{j} \\| _{\\dot{Y}^{s_{c}}}.\n\\end{split}\n\\]\n\\end{proof}\n\\begin{proof}[\\rm{\\bf{Proof of Theorem~\\ref{wellposed_1}.}}]\nFor $r>0$, we define \n\\begin{equation}\\label{Zr_norm}\n\\dot{Z}^{s}_{r}(I)\n:=\\left\\{u\\in \\dot{Z}^{s}(I)\\left|\\  \\| u \\| _{\\dot{Z}^{s}(I)}\\leq 2r \\right.\\right\\}\n\\end{equation}\nwhich is a closed subset of $\\dot{Z}^{s}(I)$. \nLet $T>0$ and $u_{0}\\in B_{r}(\\dot{H}^{-1\/2})$ are given. For $u\\in \\dot{Z}^{-1\/2}_{r}([0,T))$, \nwe have\n\\[\n \\| \\Phi_{T,u_{0}}(u) \\| _{\\dot{Z}^{-1\/2}([0,T))}\\leq  \\| u_{0} \\| _{\\dot{H}^{-1\/2}} +C \\| u \\| _{\\dot{Z}^{-1\/2}([0,T))}^{4}\\leq r(1+ 16 Cr^{3})\n\\]\nand\n\\[\n\\begin{split}\n \\| \\Phi_{T,u_{0}}(u)-\\Phi_{T,u_{0}}(v) \\| _{\\dot{Z}^{-1\/2}([0,T))}\n&\\leq C( \\| u \\| _{\\dot{Z}^{-1\/2}([0,T))}+ \\| v \\| _{\\dot{Z}^{-1\/2}([0,T))})^{3} \\| u-v \\| _{\\dot{Z}^{-1\/2}([0,T))}\\\\\n&\\leq 64Cr^{3} \\| u-v \\| _{\\dot{Z}^{-1\/2}([0,T))}\n\\end{split}\n\\]\nby Proposition~\\ref{Duam_est} and\n\\[\n \\| S(\\cdot )u_{0} \\| _{\\dot{Z}^{-1\/2}([0,T))}\\leq  \\| \\ee_{[0,T)}S(\\cdot )u_{0} \\| _{\\dot{Z}^{-1\/2}}\\leq  \\| u_{0} \\| _{\\dot{H}^{-1\/2}}, \n\\] \nwhere $C$ is an implicit constant in (\\ref{Duam_est_1}). Therefore if we choose $r$ satisfying\n\\[\nr <(64C)^{-1\/3},\n\\]\nthen $\\Phi_{T,u_{0}}$ is a contraction map on $\\dot{Z}^{-1\/2}_{r}([0,T))$. \nThis implies the existence of the solution of (\\ref{D4NLS}) and the uniqueness in the ball $\\dot{Z}^{-1\/2}_{r}([0,T))$. \nThe Lipschitz continuously of the flow map is also proved by similar argument. \n\\end{proof} \nCorollary~\\ref{sccat} is obtained by the same way as the proof of Corollaty\\ 1.2 in \\cite{Hi}. \n\n\\subsection{The large data case}\n\nIn this subsection, we prove Theorem \\ref{large-wp}.\nThe following is the key estimate.\n\n\\begin{prop}\\label{Duam_est-inh}\nLet $d=1$. We have\n\\begin{equation}\\label{Duam_est_1-inh}\n \\| I_{1}(u_{1},\\cdots u_{4}) \\| _{\\dot{Z}^{-1\/2}} \\lesssim  \\prod_{j=1}^{4} \\| u_{j} \\| _{Y^{-1\/2}}.\n\\end{equation}\n\\end{prop}\n\n\\begin{proof}\nWe decompose $u_j = v_j +w_j$ with $v_j = P_{>1}u_j \\in \\dot{Y}^{-1\/2}$ and $w_j  = P_{<1} u_j \\in \\dot{Y}^0$. \n>From Propositions \\ref{HL_est_n-inh}, \\ref{HH_est-inh}, and the same way as in the proof of Proposition~\\ref{Duam_est}, \nit remains to prove that\n\\[\n\\| I_{1}(w_{1},w_2,w_3,w_{4}) \\| _{\\dot{Z}^{-1\/2}} \\lesssim \\prod_{j=1}^{4} \\| u_{j} \\| _{\\dot{Y}^0}.\n\\]\nBy Theorem \\ref{duality}, the Cauchy-Schwartz inequality, the H\\\"older inequality and the Sobolev inequality, we have\n\\[\n\\| I_{1}(w_{1},w_2,w_3,w_{4}) \\| _{\\dot{Z}^{-1\/2}}\n\\lesssim \\left\\| \\prod_{j=1}^{4}\\overline{w_{j}} \\right\\|_{L^1([0,1];L^2)}\n\\lesssim \\prod _{j=1}^4 \\| w_j \\| _{L_t^{\\infty} L_x^2}\n\\lesssim \\prod_{j=1}^{4} \\| u_{j} \\| _{\\dot{Y}^{0}},\n\\]\nwhich completes the proof.\n\\end{proof}\n\n\\begin{proof}[\\rm{\\bf{Proof of Theorem \\ref{large-wp}}}]\nLet $u_0 \\in B_{\\delta ,R}(H^{-1\/2})$ with $u_0=v_0+w_0$, $v_0 \\in \\dot{H}^{-1\/2}$, $w_0 \\in L^2$.\nA direct calculation yields\n\\[\n\\| S(t) u_0 \\| _{Z^{-1\/2}([0,1))} \\le \\delta +R.\n\\]\nWe start with the case $R=\\delta = (4C+4)^{-4}$, where $C$ is the implicit constant in \\eqref{Duam_est_1-inh}.\nProposition \\ref{Duam_est-inh} implies that for $u \\in Z^{-1\/2}_r([0,1])$ with $r=1\/(4C+4)$\n\\begin{align*}\n\\| \\Phi_{1,u_{0}}(u) \\| _{Z^{-1\/2}([0,1))} &  \\leq  \\| S(t) u_0 \\| _{Z^{-1\/2}([0,1))} +C \\| u \\| _{Z^{-1\/2}([0,1))}^{4} \\\\\n& \\leq 2r^4 + 16C r^4\n= r^4 (16C+2)\n\\le r\n\\end{align*}\nand\n\\begin{align*}\n\\| \\Phi_{1,u_{0}}(u)-\\Phi_{1,u_{0}}(v) \\| _{Z^{-1\/2}([0,1))}\n&\\leq C( \\| u \\| _{Z^{-1\/2}([0,1))}+ \\| v \\| _{Z^{-1\/2}([0,1))})^{3} \\| u-v \\| _{Z^{-1\/2}([0,1))}\\\\\n&\\leq 64Cr^{3} \\| u-v \\| _{Z^{-1\/2}([0,1))}\n< \\| u-v \\| _{Z^{-1\/2}([0,1))}\n\\end{align*}\nif we choose $C$ large enough (namely, $r$ is small enough).\nAccordingly, $\\Phi_{1,u_{0}}$ is a contraction map on $\\dot{Z}^{-1\/2}_{r}([0,1))$.\n\nWe note that \nall of the above remains valid if we exchange $Z^{-1\/2}([0,1))$ by the smaller space $\\dot{Z}^{-1\/2}([0,1))$ since $\\dot{Z}^{-1\/2}([0,1)) \\hookrightarrow Z^{-1\/2}([0,1))$ and the left hand side of \\eqref{Duam_est_1-inh} is the homogeneous norm.\n\nWe now assume that $u_0 \\in B_{\\delta ,R}(H^{-1\/2})$ for $R \\ge \\delta = (4C+4)^{-4}$.\nWe define $u_{0, \\lambda}(x) = \\lambda ^{-1} u_0 (\\lambda ^{-1}x)$.\nFor $\\lambda = \\delta ^{-2} R^{2}$, we observe that $u_{0,\\lambda} \\in B_{\\delta ,\\delta}(H^{-1\/2})$.\nWe therefore find a solution $u_{\\lambda} \\in Z^{-1\/2}([0,1))$ with $u_{\\lambda}(0,x) = u_{0,\\lambda}(x)$.\nBy the scaling, we find a solution $u \\in Z^{-1\/2}([0, \\delta ^8 R^{-8}))$.\n\nThanks to Propositions \\ref{HL_est_n-inh} and \\ref{HH_est-inh}, the uniqueness follows from the same argument as in \\cite{HHK10}.\n\\end{proof}\n\n\\section{Proof of Theorem~\\ref{wellposed_2}}\\label{pf_wellposed_2}\\kuuhaku\nIn this section, we prove Theorem~\\ref{wellposed_2}. \nWe only prove for the homogeneous case since the proof for the inhomogeneous case is similar. \nWe define the map $\\Phi_{T, \\varphi}^{m}$ as \n\\[\n\\Phi_{T, \\varphi}^{m}(u)(t):=S(t)\\varphi -iI_{T}^{m}(u,\\cdots, u)(t),\n\\] \nwhere\n\\[\nI_{T}^{m}(u_{1},\\cdots u_{m})(t):=\\int_{0}^{t}\\ee_{[0,T)}(t')S(t-t')\\partial \\left(\\prod_{j=1}^{m}u_{j}(t')\\right)dt'.\n\\]\nand the solution space $\\dot{X}^{s}$ as\n\\[\n\\dot{X}^{s}:=C(\\R;\\dot{H}^{s})\\cap L^{p_{m}}(\\R;\\dot{W}^{s+1\/(m-1),q_{m}}),\n\\] \nwhere $p_{m}=2(m-1)$, $q_{m}=2(m-1)d\/\\{(m-1)d-2\\}$ for $d \\ge 2$ and $p_3=4$, $q_3=\\infty$ for $d=1$. \nTo prove the well-posedness of (\\ref{D4NLS}) in $L^{2}(\\R )$ or $H^{s_{c}}(\\R^{d})$, we prove that $\\Phi_{T, \\varphi}$ is a contraction map \non a closed subset of $\\dot{X}^{s}$. \nThe key estimate is the following:\n\\begin{prop}\\label{Duam_est_g}\n{\\rm (i)}\\ Let $d=1$ and $m=3$. For any $0<T<\\infty$, we have\n\\begin{equation}\\label{Duam_est_1d}\n \\| I_{T}^{3}(u_{1},u_{2}, u_{3}) \\| _{\\dot{X^{0}}}\\lesssim T^{1\/2}\\prod_{j=1}^{3} \\| u_{j} \\| _{\\dot{X}^{0}}.\n\\end{equation}\n{\\rm (ii)}\\ Let $d\\ge 2$, $(m-1)d\\ge 4$ and $s_c=d\/2-3\/(m-1)$ For any $0<T\\le \\infty$, we have\n\\begin{equation}\\label{Duam_est_2}\n \\| I_{T}^{m}(u_{1},\\cdots, u_{m}) \\| _{\\dot{X^{s_c}}}\\lesssim \\prod_{j=1}^{m} \\| u_{j} \\| _{\\dot{X}^{s_c}}.\n\\end{equation}\n\\end{prop}\n\\begin{proof}\n{\\rm (i)}\\ By Proposition~\\ref{Stri_est} with $(a,b)=\\left( 4, \\infty \\right)$,\nwe get\n\\[\n \\| I_{T}^{3}(u_{1},u_{2}, u_{3}) \\| _{L^{\\infty}_{t}L^{2}_{x}}\n\\lesssim \\left\\|\\ee_{[0,T)} |\\nabla |^{-1\/2}\\partial \\left(\\prod_{j=1}^{3}u_{j}\\right)\\right\\|_{L^{4\/3}_{t}L^{1}_{x}}\n\\]\nand\n\\[\n \\| |\\nabla |^{1\/2}I_{T}^{3}(u_{1},u_{2}, u_{3}) \\| _{L^{4}_{t}L^{\\infty}_{x}}\n\\lesssim \\left\\| \\ee_{[0,T)}|\\nabla |^{1\/2-1\/2-1\/2}\\partial \\left(\\prod_{j=1}^{3}u_{j}\\right)\\right\\|_{L^{4\/3}_{t}L^{1}_{x}}.\n\\]\nTherefore, thanks to the fractional Leibniz rule (see \\cite{CW91}), we have\n\\[\n\\begin{split}\n\\| I_{T}^{3}(u_{1},\\cdots, u_{3}) \\| _{\\dot{X^{0}}}\n& \\lesssim \\left\\| \\ee_{[0,T)}|\\nabla |^{1\/2}\\prod_{j=1}^{3}u_{j}\\right\\|_{L^{4\/3}_{t}L^{1}_{x}} \\\\\n& \\lesssim  \\| \\ee_{[0,T)}\\|_{L^{2}_{t}}\\| |\\nabla |^{1\/2}u_{i} \\| _{L^{4}_{t}L^{\\infty}_{x}}\\prod_{\\substack{1\\le j\\le 3\\\\ j\\neq i}} \\| u_{j} \\| _{L^{\\infty}_{t}L^{2}_{x}}\\\\\n&\\lesssim T^{1\/2}\\prod_{j=1}^{3} \\| u_{j} \\| _{\\dot{X}^{0}}\n\\end{split}\n\\]\nby the H\\\"older inequality. \n\\\\\n{\\rm (ii)}\\ By Proposition~\\ref{Stri_est} with \n\\begin{equation}\\label{admissible_ab}\n(a,b)=\\left( \\frac{2(m-1)}{m-2}, \\frac{2(m-1)d}{(m-1)d-2(m-2)}\\right),\n\\end{equation}\nwe get\n\\[\n \\| |\\nabla |^{s_c}I_{T}^{m}(u_{1},\\cdots u_{m}) \\| _{L^{\\infty}_{t}L^{2}_{x}}\n\\lesssim \\left\\| |\\nabla |^{s_c-2\/a}\\partial \\left(\\prod_{j=1}^{m}u_{j}\\right)\\right\\|_{L^{a'}_{t}L^{b'}_{x}}\n\\]\nand\n\\[\n \\| |\\nabla |^{s_c+1\/(m-1)}I_{T}^{m}(u_{1},\\cdots u_{m}) \\| _{L^{p_m}_{t}L^{q_m}_{x}}\n\\lesssim \\left\\| |\\nabla |^{s_c+1\/(m-1)-2\/p_m-2\/a}\\partial \\left(\\prod_{j=1}^{m}u_{j}\\right)\\right\\|_{L^{a'}_{t}L^{b'}_{x}}.\n\\]\nTherefore, thanks to the fractional Leibniz rule (see \\cite{CW91}), we have\n\\[\n\\begin{split}\n\\| I_{T}^{m}(u_{1},\\cdots u_{m}) \\| _{\\dot{X^{s_c}}}\n& \\lesssim \\left\\| |\\nabla |^{s_c+1\/(m-1)}\\prod_{j=1}^{m}u_{j}\\right\\|_{L^{a'}_{t}L^{b'}_{x}} \\\\\n& \\lesssim  \\sum_{i=1}^{m} \\| |\\nabla |^{s_c+1\/(m-1)}u_{i} \\| _{L^{p_{m}}_{t}L^{q_{m}}_{x}}\\prod_{\\substack{1\\le j\\le m\\\\ j\\neq i}} \\| u_{j} \\| _{L^{p_{m}}_{t}L^{(m-1)d}_{x}}\\\\\n&\\lesssim \\sum_{i=1}^{m} \\| |\\nabla |^{s_c+1\/(m-1)}u_{i} \\| _{L^{p_{m}}_{t}L^{q_{m}}_{x}}\\prod_{\\substack{1\\le j\\le m\\\\ j\\neq i}} \\| |\\nabla |^{s_{c}+1\/(m-1)}u_{j} \\| _{L^{p_{m}}_{t}L^{q_{m}}_{x}}\\\\\n&\\lesssim \\prod_{j=1}^{m} \\| u_{j} \\| _{\\dot{X}^{s_c}}\n\\end{split}\n\\]\nby the H\\\"older inequality and the Sobolev inequality, where we used the condition $(m-1)d\\ge 4$ which is equivalent to $s_{c}+1\/(m-1)\\ge 0$. \n\\end{proof}\nThe well-posedness can be proved by the same way as the proof of Theorem~\\ref{wellposed_1} and the scattering follows from \nthat the Strichartz estimate because the $\\dot{X}^{s_c}$ norm of the nonlinear part is bounded by the norm of the $L^{p_m}L^{q_m}$ space (see for example \\cite[Section 9]{P07}).\n\\section{Proof of Theorem~\\ref{notC3}}\\label{pf_notC3\n\nIn this section we prove the flow of (\\ref{D4NLS}) is not smooth.\nLet $u^{(m)}[u_0]$ be the $m$-th iteration of \\eqref{D4NLS} with initial data $u_0$:\n\\[\nu^{(m)}[u_0] (t,x) := -i \\int _0^t e^{i(t-t') \\Delta ^2} \\partial  P_m( S(t') u_0, S(-t') \\overline{u_0}) dt' .\n\\]\n\n\nFirstly we consider the case $d=1$, $m=3$, $P_{3}(u,\\overline{u})=|u|^{2}u$. \nFor $N\\gg 1$, we put\n\\[\nf_{N} = N^{-s+1\/2} \\mathcal{F}^{-1}[ \\ee _{[N-N^{-1}, N+N^{-1}]}]\n\\]\nLet $u^{(3)}_{N}$ be the third iteration of (\\ref{D4NLS}) for the data $f_{N}$.\nNamely, \n\\[\nu^{(3)}_{N}(t,x) = u^{(3)}[f_N] (t,x)= -i \\int _0^t e^{i(t-t') \\partial _x ^4} \\partial _x  \\left( |e^{it' \\partial _x^4} f_{N}| ^2 e^{it' \\partial _x^4} f_{N} \\right)(x) dt'.\n\\]\nNote that $ \\| f_{N} \\| _{H^s} \\sim 1$. \nThorem~\\ref{notC3} is implied by the following propositions.\n\\begin{prop}\nIf $s<0$, then for any $N\\gg 1$, we have\n\\[\n \\| u^{(3)}_{N} \\| _{L^{\\infty}([0,1]; H^s)} \\rightarrow \\infty\n\\]\nas $N\\rightarrow \\infty$. \n\\end{prop}\n\\begin{proof}\nA direct calculation implies\n\\[\n\\widehat{u^{(3)}_{N}} (t, \\xi ) =  e^{it \\xi ^4} \\xi \\int _{\\xi _1-\\xi _2+\\xi _3 =\\xi} \\int _0^t e^{it'(-\\xi ^4 +\\xi _1^4-\\xi _2^4+\\xi _3^4)} d t' \\widehat{f_{N}}(\\xi _1) \\overline{\\widehat{f_{N}}}(\\xi _2) \\widehat{f_{N}}(\\xi _3) \n\\]\nand\n\\begin{equation} \\label{modulation}\n\\begin{split}\n&-(\\xi _1-\\xi _2+\\xi _3)^4+\\xi _1^4-\\xi _2^4+\\xi _3^4\\\\\n&= 2 (\\xi _1- \\xi _2)(\\xi _2-\\xi _3) ( 2 \\xi _1^2 +\\xi _2^2+2\\xi _3^2 -\\xi _1 \\xi _2 -\\xi _2\\xi _3 +3 \\xi _3 \\xi _1) .\n\\end{split}\n\\end{equation}\n>From $\\xi _j \\in [N-N^{-1}, N+N^{-1}]$ for $j=1,2,3$, we get\n\\[\n|-(\\xi _1-\\xi _2+\\xi _3)^4+\\xi _1^4-\\xi _2^4+\\xi _3^4|\n\\lesssim 1.\n\\]\nWe therefore obtain for sufficiently small $t>0$\n\\begin{align*}\n|\\widehat{u^{(3)}_{N}} (t,\\xi ) |\n& \\gtrsim t N^{-3s+5\/2} \\left| \\int _{\\xi _1-\\xi _2+\\xi _3 =\\xi} \\ee _{[N-N^{-1}, N+N^{-1}]} (\\xi _1) \\ee _{[N-N^{-1}, N+N^{-1}]} (\\xi _2) \\ee _{[N-N^{-1}, N+N^{-1}]} (\\xi _3) \\right| \\\\\n& \\gtrsim t N^{-3s+1\/2} \\ee _{[N-N^{-1},N+N^{-1} ]} (\\xi ) .\n\\end{align*}\nHence,\n\\[\n\\| u^{(3)}_{N} \\| _{L^{\\infty}([0,1]; H^s)} \\gtrsim N^{-2s}.\n\\]\nThis lower bound goes to infinity as $N$ tends to infinity if $s<0$, which concludes the proof.\n\\end{proof}\n\n\nSecondly, we show that absence of a smooth flow map for $d \\ge 1$ and $m \\ge 2$.\nPutting\n\\[\ng_N := N^{-s-d\/2} \\mathcal{F}^{-1}[ \\ee _{[-N,N]^d}] ,\n\\]\nwe set $u_N^{(m)} := u^{(m)} [g_N]$.\nNote that $\\| g_N \\| _{H^s} \\sim 1$.\nAs above, we show the following.\n\n\\begin{prop}\nIf $s<s_c := d\/2-3\/(m-1)$ and $\\partial =|\\nabla |$ or $\\frac{\\partial}{\\partial x_k}$ for some $1\\le k\\le d$, then for any $N \\gg 1$, we have\n\\[\n\\| u_N^{(m)} \\| _{L^{\\infty}([0,1];H^s)} \\rightarrow \\infty\n\\]\nas $N \\rightarrow \\infty$.\n\\end{prop}\n\\begin{proof}\nWe only prove for the case $\\partial =|\\nabla |$ since the proof for the case $\\frac{\\partial}{\\partial x_k}$ is same.\nLet\n\\[\n\\mathcal{A} := \\{ (\\pm _1, \\dots , \\pm _m) : \\pm _j \\in \\{ +, - \\} \\, (j=1, \\dots ,m) \\} .\n\\]\nSince $\\mathcal{A}$ consists of $2^m$ elements, we write\n\\[\n\\mathcal{A} = \\bigcup _{\\alpha}^{2^m} \\{ \\pm ^{(\\alpha )} \\} ,\n\\]\nwhere $\\pm ^{( \\alpha )}$ is a $m$-ple of signs $+$ and $-$.\nWe denote by $\\pm _{j}^{(\\alpha )}$ the $j$-th component of $\\pm ^{(\\alpha )}$.\nA simple calculation shows that\n\\[\n\\widehat{u_N^{(m)}} (t,\\xi) = |\\xi | \\sum _{\\alpha =0}^{2^m} e^{it |\\xi |^4} \\int _{\\xi = \\sum _{j=1}^m \\pm _j^{(\\alpha)} \\xi _j} \\int _0^t e^{it' (-|\\xi|^4 + \\sum _{j=1}^m \\pm _j^{(\\alpha )} |\\xi _j|^4)} dt' \\prod _{j=1}^m \\widehat{g_N} (\\xi _j) .\n\\]\nFrom\n\\[\n\\left| -|\\xi|^4 + \\sum _{j=1}^m \\pm _j^{(\\alpha )} |\\xi _j|^4 \\right| \\lesssim N^4\n\\]\nfor $|\\xi _j| \\le N$ ($j= 1, \\dots , m$), we have\n\\[\n|\\widehat{u_N^{(m)}} (t, \\xi )|\n\\gtrsim |\\xi | N^{-4} N^{-m(s+d\/2)} N^{(m-1)d} \\ee _{[-N.N]^d} (\\xi )\n\\gtrsim N^{-3} N^{-m(s+d\/2)} N^{(m-1)d} \\ee _{[N\/2.N]^d} (\\xi )\n\\]\nprovided that $t \\sim N^{-4}$.\nAccordingly, we obtain\n\\[\n\\| u_N^{(m)} (N^{-4}) \\| _{H^s} \\gtrsim N^{-3} N^{-m(s+d\/2)} N^{(m-1)d} N^{s+d\/2}\n\\sim N^{-(m-1)s+(m-1)d\/2-3} ,\n\\]\nwhich conclude that $\\limsup _{t \\rightarrow 0} \\| u^{(m)}_N(t) \\| _{H^s} = \\infty$ if $s<s_c$.\n\\end{proof}\n\n\n\\section*{Acknowledgment}\n\nThe work of the second author was partially supported by JSPS KAKENHI Grant number 26887017.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Architecture}\n\\vspace*{-.2cm}\nThe ConvNet architecture is composed of repeatedly stacked feature stages.\nEach stage contains a convolution module, followed by a pooling\/subsampling module and a normalization module. While traditional pooling modules in ConvNet are either average or max poolings, we use an Lp pooling here. The normalization module is subtractive only as opposed to subtractive and divisive, i.e. the mean value of each neighborhood is subtracted to the output of each stage (but not divided by the standard deviation as it decreases performance with this dataset).\nFinally, multi-stage features are also used as opposed to single-stage features.\n\n\\subsection{Lp-Pooling}\n\\vspace*{-.5cm}\n\n\\begin{figure}[h]\n  \\begin{center}\n    \\includegraphics[width=.5\\textwidth]{ l2pool2}\n  \\end{center}\n  \\caption{\\small{L2-pooling applied to a 9x9 feature map with a 3x3 Gaussian kernel and 2x2 stride}}\n  \\label{fig:fig2}\n\\end{figure}\n\\vspace*{-.5cm}\n\nLp pooling is a biologically inspired pooling layer modelled on complex cells~\\cite{Simoncelli97amodel,Hyvarinen05complexcell} who's operation can be summarized in equation (1), where $G$ is a Gaussian kernel, $I$ is the input feature map and $O$ is the output feature map. It can be imagined as giving an increased weight to stronger features and suppressing weaker features. Two special cases of Lp pooling are notable. $P = 1$ corresponds to a simple Gaussian averaging, whereas $P = \\infty$ corresponds to max-pooling (i.e only the strongest signal is activated). Lp-pooling has been used previously in~\\cite{koray-cvpr-09,Yang09linearspatial} and a theoretical analysis of this method is described in~\\cite{boureau-icml-10}.\n\\begin{equation}\nO = (\\sum \\sum I(i,j)^P \\times G(i,j))^{1\/P} \n\\end{equation}\n\nFigure~\\ref{fig:fig2} demonstrates a simple example of L2-pooling.\n\n\n\\begin{table*}[htb] \n  \\small{\n    \\begin{center}\n      \\begin{tabular}{ | c | c | c | c |}\n        \\hline\n        Task & Single-Stage features & Multi-Stage features & Improvement \\% \\\\\n        \\hline \\hline\n        Pedestrians detection (INRIA)~\\cite{sermanet-snowbird-11}& 14.26\\% & 9.85\\% & 31\\%\\\\\n        Traffic Signs classification (GTSRB)~\\cite{sermanet-ijcnn-11}& 1.80\\% & 0.83\\% & 54\\%\\\\\n        House Numbers classification (SVHN) & 5.72\\% & 5.67\\% & 0.9\\%\\\\\n        \\hline\n      \\end{tabular}\n    \\end{center}\n    \\caption{\\small{Error rates improvements of multi-stage features over\n        single-stage features for different types of objects detection and\n        classification. Improvements are significant for multi-scale and\n        textured objects such as traffic signs and pedestrians but minimal for\n        house numbers.}}\n  }\n  \\label{table:ms}\n\\vspace*{-1.5cm}\n\\end{table*}\n\n\\begin{figure}[h]\n  \\begin{center}\n    \\begin{minipage}[t]{0.5\\textwidth}\n    \\vspace{0pt}\n      \\includegraphics[width=1\\textwidth]{ net2}\n  \\end{minipage}\n  \\end{center} \n  \\caption{\\small{A 2-stage ConvNet architecture where Multi-Stage features\n      (MS) are fed to a 2-layer classifier. The 1st stage features are branched\t\n      out, subsampled again and then concatenated to 2nd stage features.}}\n  \\label{fig:arch}\n\\vspace*{-.5cm}\n\\end{figure}\n\n\\subsection{Multi-Stage Features}\n\nMulti-Stage features (MS) are obtained by branching out outputs of all stages into the classifier (Figure~\\ref{fig:arch}). They provide richer representations compared to Single-Stage features (SS) by adding complementary information such as local textures and fine details lost by higher levels. MS features have consistently improved performance in other work~\\cite{fan-nn-10,sermanet-ijcnn-11,sermanet-snowbird-11} and in this work as well (Figure~\\ref{fig:ms}). However we observe minimal gains on this dataset compared to other types of objects such as pedestrians and traffic signs (Table 1). The likely explanation for this observation is that gains are correlated to the amount of texture and multi-scale characteristics of the objects of interest.\n\n\\vspace*{-.2cm}\n\\Section{Experiments}\n\\vspace*{-.2cm}\n\n\\SubSection{Data Preparation}\nThe SVHN classification dataset~\\cite{netzer-11} contains 32x32 images\nwith 3 color channels. The dataset is divided into three subsets: train set, extra set and test set. The extra set is a large set of easy samples and train set is a smaller set of more difficult samples. Since we are given no information about how the sampling of these images was done, we assume a random order to construct our validation set. We compose our validation set with $2\/3$ from training samples (400 per class) and $1\/3$ from extra samples (200 per class), yielding a total of 6000 samples. This distribution allows to measure success on easy samples but puts more emphasis on difficult ones.\n\nSamples are pre-processed with a local contrast normalization (with a 7x7 kernel) on the Y channel of the YUV space followed by a global contrast normalization over each channel. No sample distortions were used to improve invariance.\n\n\n\\begin{figure}[h]\n  \\begin{center}\n    \\begin{minipage}[t]{0.5\\textwidth}\n    \\vspace{0pt}\n      \\includegraphics[width=1\\textwidth]{ test_overall_errors}\n  \\end{minipage}\n  \\end{center} \n  \\caption{\\small{Improvement of Multi-Stage features (MS) over Single-Stage features (SS) in error rate on the validation set. MS features provide a slight\n      error improvement over SS features.}}\n  \\label{fig:ms}\n\\end{figure} \n \n \n\\subsection{Architecture Details} \nThe ConvNet has 2 stages of feature extraction and a two-layer non-linear classifier. The first convolution layer produces 16 features with 5x5 convolution filters while the second convolution layer outputs 512 features with 7x7 filters. The output to the classifier also includes inputs from the first layer, which provides local features\/motifs to reinforce the global features. The classifier is a 2-layer non-linear classifier with 20 hidden units. \nHyper-parameters such as learning rate, regularization constant and learning rate decay were tuned on the validation set.\nWe use stochastic gradient descent as our optimization method and shuffle our dataset after each training iteration.\n\n\\begin{figure}[h]\n  \\begin{center}\n    \\begin{minipage}[t]{0.5\\textwidth}\n    \\vspace{0pt}\n      \\includegraphics[width=1\\textwidth]{ pooling_compare}\n  \\end{minipage}\n  \\end{center} \n  \\caption{\\small{Error rate of Lp-pooling on the validation set for\n      $p = 1,2,4,8,12,16,32,\\infty$ ($p=\\infty$} is represented as $p=100$ for\n    convenience). These validation errors are reported after 1000 training\n    epochs. $p = 12$ performs best with an error rate of $5.61\\%$.}\n  \\label{fig:fig3}\n\\vspace*{-.5cm}\n\\end{figure}\n\nFor the pooling layers, we compare Lp-pooling for the value $p = 1,2,4,8,12,16,32,\\infty$ on the validation set and use the best performing pooling on the final testing. The performance of different pooling methods on the validation set can be seen in Figure~\\ref{fig:fig3}. Insights from~\\cite{boureau-icml-10} tell us that the optimal value of $p$ varies for different input spaces and there is no single globally optimal value for $p$. For our validation data, we observe that $p = 2,4,12$ give the best performance ($5.62\\%, 5.64\\%$ and $5.61\\%$ respectively). Max-pooling, which corresponds to $p = \\infty$ yielded a validation error rate of $7.57\\%$.\n\n\\begin{table*}[htb]\n  \\small{\n  \\begin{center}\n    \\begin{tabular}{ | c | c | }\n      \\hline\n      Algorithm & SVHN-Test Accuracy \\\\\n      \\hline \\hline\n      Binary Features (WDCH) & 63.3\\% \\\\\n      HOG & 85.0\\% \\\\\n      Stacked Sparse Auto-Encoders & 89.7 \\% \\\\\n      K-Means & 90.6\\% \\\\\n      {\\bf ConvNet \/ MS \/ Average} & {\\bf 90.75\\%} \\\\\n      {\\bf ConvNet \/ MS \/ L2 \/ Smaller training} & {\\bf 91.55\\%} \\\\\n      {\\bf ConvNet \/ SS \/ L2} & {\\bf 94.28\\%} \\\\\n      {\\bf ConvNet \/ MS \/ L2} & {\\bf 94.33\\%} \\\\\n      {\\bf ConvNet \/ MS \/ L12} & {\\bf 94.76\\%} \\\\\n      {\\bf ConvNet \/ MS \/ L4} & {\\bf 94.85\\%} \\\\\n     \n      \\hline\n      Human Performance &98.0\\% \\\\\n      \\hline\n    \\end{tabular}\n  \\label{table:res}\n  \\end{center}\n\\vspace*{-.2cm}\n  \\caption{\\small{Performance reported by~\\cite{netzer-11} with the additional\n      Supervised ConvNet with state-of-the-art accuracy of 94.85\\%.}}\n  }\n\\end{table*}\n\n\n\\section{Results \\& Future Work}\n\nOur experiments demonstrate a clear advantage of Lp pooling with $1 < p < \\infty$ on this dataset in validation (Figure~\\ref{fig:fig3}) and test (Average pooling is 3.58 points inferior to L2 pooling in Table 2). With L4 pooling, we obtain a state-of-the-art performance on the test set with an accuracy of 94.85\\% compared to the previous best of 90.6\\% (Table 2). We also show that using multi-stage features gives only a slight increase in performance, compared to the performance increase seen in other vision applications.\n\nAdditionally, it is important to note that our approach is trained fully supervised only, whereas the best previous methods are unsupervised learning methods (k-means, auto-encoders). We shall, in the future, run experiments with unsupervised learning, to compare the accuracy improvement that can be attributed to supervision. Figure~\\ref{fig:highest} shows the validation samples with highest energy. Many of these seem to exhibit large scale variations, future work could address this problem by introducing artificial scale deformations during training.\n\n\\begin{figure*}[htb]\n  \\begin{center}\n      \\includegraphics[width=.9\\textwidth]{ highest-y}\n  \\end{center} \n\\vspace*{-.5cm}\n  \\caption{\\small{Preprocessed Y channel of validation samples with\n      highest energy (i.e. highest error)\n      with the 94.33\\% accuracy L2-pool based multi-stage ConvNet.}}\n  \\label{fig:highest}\n\\vspace*{-.2cm}\n\\end{figure*}\n\n\n\\balance\n\n\\bibliographystyle{latex12}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\n\\textit{Fibroblasts}, \\textit{keratocytes}, \\textit{cancer cells}, and other types \\change{}{of} fast moving cells exhibit a particular crawling-like motion in which the \\textit{lamellipodium} of the cells plays a pivotal role, \\cite{Small2002,Svitkina1997,Yam2007,Postlethwaite1987,Gerisch1981,Iijima2002,Zigmond1973}. \n\nThe lamellipodium is a sheet-like dense network \\change{of linear \\textit{biopolymers} of the protein \\textit{actin} ---termed \\textit{actin-filaments} or simply \\textit{filaments}--- that can be found in the propagating front of the cells}{that can be found in the propagating front of the cells and is comprised of linear biopolymers of the protein \\textit{actin}  ---termed \\textit{actin-filaments} or simply \\textit{filaments}.}. These {actin-filaments} are highly dynamic; they continuously \\textit{polymerize}, \\textit{adhere} to the substrate, and are subject to numerous other processes like  \\textit{nucleation}, \\textit{fragmentation}, \\textit{capping}, and more \\cite{Blanchoin_Plastino2014,Lauffenburger1996, Gittes1993, Tojkander2012,Mitchison1996,Jay1995,Chen1981}.\n\nThese processes affect the structure and \\change{}{the} functionality of the lamellipodium and the motility of the cell, \\cite{Small2002}. They are \\change {greatly} influenced\\change{}{, to a large extent, } by the extracellular environment, \\change{e.g.}{} its chemical composition and the \\textit{architecture} of the \\textit{Extracellular Matrix} (ECM). The response of the cell to gradients of extracellular \\textit{chemical signals} and ECM-bound \\textit{adhesion sites} is termed \\textit{chemotaxis} and \\textit{haptotaxis} respectively.\n\nThere have been several approaches in the literature to model and simulate lamellipodium driven cell motility. In the current paper we do not aim to develop \\change{and}{} a new model or to perform biologically relevant numerical experiments, hence we do not discuss the literature in detail\\change{s}{} or compare the different model\\change{}{s}. We merely refer the interested reader to some of \\change{}{the} existing works \\cite{Schwarz2010, Stevens2015, Voigt2015, Mogilner2009, Alt2009, DeSimone2011,Preziosi2013, Merkel2012,Madzvamuse2013,Ambrosi2016}. \n\n\\change{In this work we follow}{The model we follow is} the \\textit{Filament Based Lamellipodium Model} (FBLM), a two dimensional, two phase model that describes the lamellipodium at the level of actin-filaments. \\change{This model}{It} was first introduced in \\cite{Oelz2008,Oelz2010a} and was further extended in \\cite{MOSS-model}. When endowed with a particular problem specific \\textit{Finite Element Method} (FEM), the \\change{}{the resulting } FBLM-FEM is able to reproduce a realistic, crawling-like lamellipodium driven motility \\cite{MOSS-numeric,Brunk2016}.\n\nAlthough the FBLM describes the dynamics of \\change{}{the} actin-filaments and the lamellipodium, the deduced motility is understood as the motility of the full cell. This is \\change{mostly}{} due to the fact that \\change{in the case}{the role of the lamellipodium in the motility of the} model biological cell (\\textit{fish keratocyte}) \\change{that we consider, the role of the lamellipodium in the motility}{} is predominant, \\cite{Small2002}. We henceforth will not distinguish between the two, and for convenience we will use the term \\textit{cell motility} for both cases.\n\nThe FBLM and the corresponding FEM have been used so far in several works to simulate \\change{some}{various} cases of cell motility, e.g. \\cite{MOSS-model,MOSS-numeric,Brunk2016}. To date though no numerical \\change{analysis}{} investigations have been presented in the literature to verify that the FBLM-FEM combination satisfies some (at least minimum) numerical prerequisites and justify thusly its further use in biological relevant situations. What moreover is missing from the relevant literature is a study of the corresponding \\change{parameter set,}{parameters} and of the effect \\change{that}{} they have \\change{in}{on} the dynamics of the FBLM. Such parameter study would facilitate the \\textit{parameter identification} procedure\\change{(-s)}{} in the reproduction\/simulation of realistic biological experiments. The current work aims to \\change{\\textit{partially fill}}{partially fill} these gaps and is split in two main parts\\change{:}{.}\n\nIn the first part of the paper, we identify appropriate timestep \\textit{stability} conditions in chemotaxis and haptotaxis experiments where we propose a relation between the time and the (two-dimensional) \\change{space}{``space''} discretization steps. We investigate the dependence of the stability constant on the gradient of the ECM and propose an \\change{\\textit{Adaptive Time Control}}{\\textit{adaptive time control}} (ATC) method for the \\change{automated identification}{automatic adaptation} of the timestep of the \\change{}{numerical} method. \n\nWe also exhibit the \\textit{convergence} of the FEM. For that we consider three generic and representative experiments and a \\change{}{discretization} grid that is refined with respect to both \\change{the}{} ``spatial'' variables. \\change{The benefit of this method stems} \n\nAs the model and the \\change{}{numerical} method are quite complex, they do not allow for a rigorous numerical analysis study\\change{. As}{; as} a result we restrict our work here to the experimental and case dependent study of these properties\\change{,}{} and postpone the more rigorous numerical and analytical investigations for a separate work. Nevertheless, the benefit that stems from the first part of our work is \\change{that we have verified}{mainly the verification} that the FBLM-FEM can be further employed in biologically relevant numerical simulations. Most notably, that the refinement of the mesh and the adaptation of the timestep will \\change{}{be able to} reveal the dynamics of the model.\n\nIn the second part of the paper, we make one more step towards biological realism and embed the FBLM in a complex and adaptive chemical and {haptic} extracellular environment. Such extended FBLM-environment \\change{models}{model combination} will allow us to better reproduce \\textit{in vitro} biological experiments. The model for the environment we propose here is minimal. It includes only some basic components \\change{and}{of the extracellular environment and we} use it primarily to present the \\change{}{FBLM-environment} coupling\\change{ between the FBLM and the environment}{}. \n\nBased on the \\change{combination of the FBLM with the environment,}{FBLM-environment combination,} we perform a sensitivity analysis \\change{of the FBLM}{} in a generic experimental case, where we identify the most influential parameters and address the effect they have in the dynamics of the model. We get this way a critical insight \\change{in}{of} the different components and dynamics of the FBLM and pave the way for \\change{}{more detailed and problem specific} parameter identification \\change{works}{investigations} and the simulation of biologically relevant experimental scenarios. \n\nThe structure of the paper is as follows: in Section \\ref{sec:model} we describe briefly the FBLM and give some details on extensions of the model that we consider for the first time. In Sections \\ref{sec:stbility} and \\ref{sec:converge} we study the \\textit{stability} and the \\textit{convergence} of the the FEM.\nIn Section \\ref{sec:env} we introduce the model for the extracellular environment, and in Section \\ref{sec:sensit}, we study the sensitivity of the FBLM-FEM on a series of its controlling parameters.\nIn the Appendix we provide some basic information on the FEM and the FV methods we use to solve the FBLM and the environment.\n\\section{The FBLM} \\label{sec:model}\nWe present here some information on the FBLM and the new components that we include, and refer to \\cite{Oelz2008,Oelz2010a,Schmeiser2010, MOSS-model,MOSS-numeric, Brunk2016} for more details.\n\nThe FBLM is a two-dimensional, two-phase continuum model that describes the dynamics of the lamellipodium by retaining key biological processes of the actin-filaments\\change{ and the their}{, the} interactions\\change{. The model distinguishes between two families (phases) of filaments and includes the interactions between them}{with each other}, as well as \\change{between the network of the filaments and}{their interactions with} the extracellular environment. \\change{In more details:}{}\n\nThe main assumptions behind the FBLM are \\change{that}{a)} the lamellipodium is a two dimensional structure, and \\change{that}{b)} the actin filaments are organized in two locally parallel families (denoted here by the superscripts $+$ or $-$). Each family covers a region with the filaments connecting the membrane with the inside of the cell. \n\nThe filaments of the $\\pm$ family are indexed by $\\alpha\\in [0,2\\pi)$ and have a maximal length $L^\\pm(\\alpha,t)$ at time $t$. The two families are parametrized with respect to their arclength as\n\\begin{equation} \\label{eq:arclength}\n\t\\left\\{\\vec{F}^\\pm(\\alpha,s,t): -L^\\pm(\\alpha,t)\\le s \\le 0\\right\\}\\subset {\\mathbb R}^2,\n\\end{equation}\nand coincide at their outer boundaries ($s=0$) with the membrane of the cell\n\\begin{equation}\\label{eq:tether}\n\t\\left\\{\\vec{F}^+(\\alpha,0,t): 0\\le \\alpha < 2\\pi\\right\\} = \\left\\{\\vec{F}^-(\\alpha,0,t): 0\\le \\alpha < 2\\pi\\right\\},  \\quad \\forall\\, t\\geq 0 \\,.\n\\end{equation}\nThey moreover satisfy the constraint \n\\begin{equation}\\label{eq:inext}\n\\left|\\partial_s \\vec{F}^\\pm(\\alpha,s,t)\\right| = 1 \\quad\\forall\\, (\\alpha,s,t) \\;,\n\\end{equation}\nthat is understood as an \\textit{inextensibility} condition between their monomers.\n\nWe assume that filaments of the same family do not intersect each other\n\\begin{equation}\n\\det \\(\\partial_\\a \\vec F^\\pm, \\partial_s \\vec F^\\pm\\)>0\n\\end{equation}\nand that filaments of different families cross at most once\n\\begin{equation}\n\t\\Big\\{ \\forall (\\a^+,\\a^-)\\ \\exists \\text{ at most one } (s^+,s^-) \\text{ such that }\\, \\vec{F}^+(\\alpha^+,s^+,t) = \\vec{F}^-(\\alpha^-,s^-,t)\\Big\\}.\n\\end{equation}\n\n\n\n\\begin{figure}[t]\n\t\\begin{center}\n\t\n\t\t\\begin{picture}(220,88)\n\t\t\\put(0,0){\\includegraphics[width=22em]{final.png}}\n\t\n\t\n\t\n\t\t\\put(117,65){\\scriptsize$F^+$}\n\t\t\\put(117,12){\\scriptsize$F^-$}\n\t\t\\put(175,37){\\scriptsize$B_0$}\n\t\t\\put(147,41){\\scriptsize$0$}\n\t\t\\put(141,28){\\scriptsize$-1$}\n\t\t\\put(198,48){\\scriptsize$2\\pi$}\n\t\t\\put(212,37){\\scriptsize$\\a$}\n\t\t\\put(157,53){\\scriptsize$s$}\n\t\t\\end{picture}\n\t\\end{center}\n\t\\caption{Graphical representation of the $\\vec F^\\pm$ that map the domain of dependence $B_0$ \\eqref{eq:rescaled.B} to the lamellipodium. The $s=0$ boundary of $B_0$ is mapped to the membrane of the cell and the $s=-1$ to the minus-ends of the filaments inside the cell. The filaments and the other functions of $\\a$ are periodic with respect to $\\a$. The ``filaments'' plotted in the lamellipodium correspond to the discretization interfaces of $B_0$ along the $\\a$ direction.}\\label{fig:domains}\n\\end{figure}\n\nIn the heart of the FBLM is found the system of equations\n\\begin{align} \n0=\\underbrace{\\mu^B \\partial_s^2\\left(\\eta \\partial_s^2 \\vec{F}\\right)}_\\subtext{bending} &-\\underbrace{\\partial_s\\left(\\eta \\lambda_{\\rm inext} \\partial_s \\vec{F}\\right)}_\\subtext{in-extensibility} +\\underbrace{\\mu^A \\eta D_t \\vec{F}}_\\subtext{adhesion} \\nonumber + \\underbrace{\\partial_s\\left( p(\\rho) \\partial_\\a \\vec{F}^{\\perp}\\right)\n\t-\\partial_\\a \\left( p(\\rho) \\partial_s \\vec{F}^{\\perp}\\right)}_\\subtext{pressure} \\nonumber\\\\\n&\\pm\\underbrace{\\partial_s\\left(\\eta\\eta^* \\widehat{\\mu^T} (\\phi-\\phi_0)\\partial_s \\vec{F}^{\\perp}\\right)}_\\subtext{twisting}  \n+ \\underbrace{\\eta\\eta^* \\widehat{\\mu^S}\\left(D_t \\vec{F} - D_t^* \\vec{F}^*\\right)}_\\subtext{stretching} \\;,\\label{eq:strong}\n\\end{align}\nwhere $\\vec{F}^\\bot = (F_1,F_2)^\\bot = (-F_2,F_1)$ and where we have dropped the $\\pm$ notation and focus on one of the two families\/equations. The other family is indicated by the superscript $^*$ for which a similar equation holds.\n\nThe function  $\\eta(\\alpha,s,t)$ represents the (number) density of filaments of length at least $-s$ at time $t$ with respect to $\\alpha$. The corresponding submodels used to derive the evolution of $\\eta$ (and $L(\\alpha,t)$) incorporate the effects of polymerization, depolymerization, branching, and capping, see \\cite{MOSS-model}.\n\nThe first term on the right hand side of (\\ref{eq:strong}) describes the resistance of the filaments against bending, the second term is a tangential tension force, stemming from the inextensibility constraint (\\ref{eq:inext}) with the \\textit{Lagrange multiplier} $\\lambda_{\\rm inext}(\\alpha,s,t)$. The third term describes the friction between the filament network and the substrate.\n\nThe filaments polymerize at the leading edge with rate $v(\\alpha,t)\\ge 0$. The \\textit{material derivative} operator\n\\[\n\tD_t  := \\partial_t  - v\\partial_s \n\\]\ndescribes the velocity of the actin-material relative to the substrate. In a similar way we set $D_t^\\ast  := \\partial_t  - v^\\ast \\partial_s$. \n\n\nThe pressure effect in (\\ref{eq:strong}) is caused by Coulomb repulsion between neighbouring filaments of the same family with \\textit{pressure} $p(\\rho)$ given by the density of actin as\n\\begin{equation} \\label{eq:rho}\n\t\\rho = \\frac{\\eta}{\\left|\\det(\\partial_\\a \\vec{F}, \\partial_s \\vec{F})\\right|}  \\;.\n\\end{equation}\n\nThe last two terms in (\\ref{eq:strong}) model the interaction between the two families caused by \\textit{elastic cross-links} and\/or \\textit{branch junctions}. The first  one describes the resistance against changing the angle \n\\[\n\t\\phi=\\arccos (\\partial_s \\vec{F}\\cdot\\partial_s \\vec{F}^*)\n\\]\nbetween crossing filaments away from the equilibrium angle $\\phi_0$ of the cross-linking molecule. The second one describes the friction between the two families analogously to the friction with the substrate.\n\n\nThe system (\\ref{eq:strong}) is subject to the boundary conditions \n\\begin{align} \\label{eq:newBC}\n\t- \\mu^B\\partial_s\\left(\\eta\\partial_s^2 \\vec{F}\\right) -&~ p(\\rho)\\partial_\\a \\vec{F}^\\perp + \\eta \\lambda_{\\rm inext} \\partial_s \\vec{F}  \n\t\t\\mp\\eta\\eta^* \\widehat{\\mu^T}(\\phi-\\phi_0)\\partial_s \\vec{F}^\\perp \\\\\n\t\t=&~\\left\\{\n\t\t\t\\begin{array}{l l}\n\t\t\t\t\\eta \\left(f_{\\rm tan}(\\alpha)\\partial_s \\vec{F} + f_{\\rm inn}(\\alpha) \\vec{V}(\\alpha)\\right), & \\quad \\mbox{for \t} s=-L \\;,\\\\\n\t\t\t\t\\pm\\lambda_{\\rm tether} \\nu, & \\quad  \\mbox{for }  s=0 \\;,\n\t\t\t\\end{array}\\right.\\nonumber\\\\\n\t\t\\eta \\partial_s^2 \\vec{F} =&~ 0, \\qquad \\mbox{for } s=-L,0 \\;.\\nonumber\n\\end{align}\nThe terms in the second line, describe forces applied to the filament ends. The force in the  direction $\\nu$ orthogonal to the leading edge at $s=0$ arises from the constraint (\\ref{eq:tether}) with the Lagrange parameter $\\lambda_{\\rm tether}$. The forces at the inner boundary $s=-L$ model the contraction effect of actin-myosin  interaction in the interior region, refer to \\cite{MOSS-model} for details.\n\nFundamental to the motility of the cell, is the breaking of the symmetry in the thickness of the lamellipodium. This way, the effective pulling force becomes stronger in the direction of the wider lamellipodium \\cite{Yam2007}. The maximal length of the filaments and width of the lamellipodium $L(\\a,t)$,  depends on the polymerization rate. Based on the capping, severing, and filament nucleation procedures, we have deduced in \\cite{MOSS-model} the relation\n\\begin{equation}\\label{eq:width_and_v}\n\tL(\\a,t)= \\frac{\\kappa_\\text{cap}}{\\kappa_\\text{sev}} + \\sqrt{\\frac{\\kappa_\\text{cap}^2}{\\kappa_\\text{sev}^2} + \\frac{2v(\\a,t)}{\\kappa_\\text{sev}}\\log{\\frac{\\eta(0,t)}{\\eta_\\text{min}}}},\n\\end{equation}\nwhich reveals the direct dependence of the width of the lamellipodium $L(\\a,t)$ to the polymerization rate of $v(\\a,t)$ of the local filaments.\n\nFor more details on the FBLM we refer to \\cite{Oelz2008,Oelz2010a,Schmeiser2010, MOSS-model,MOSS-numeric}.\n\n\\subsection*{Adjusting the polymerization rate}\nWe consider in this work two mechanisms to control the polymerization rate $v$ of the filaments. The first is the direct response of the intracellular polymerization mechanism to extracellular chemical signals as they are perceived by the cell through transmembrane receptors. The second mechanism represents (unspecified in this paper) intracellular processes that destabilize, cut off, or even enhance the response of the polymerization mechanism.\n\nFor the first mechanism we assume that the polymerization rate $v^\\pm_\\text{ext}(\\a,t)$ of the filament $\\a$ is adjusted between the minimum and the maximum value $v_{\\min}$, $v_{\\max}$ of the cell polymerization mechanism according to the density of the extracellular chemical signal $c$ (that serves as a chemo-attractant) by the formula\n\\begin{equation}\\label{eq:chem.pol}\n\tv^\\pm_\\text{ext} (\\a,t)= v_{\\max} - (v_{\\max} -v_{\\min}) e^{-\\lambda_\\text{res} c^\\pm (\\a,t)},\n\\end{equation}\nwhere $c^\\pm (\\a,t)$ is the density of the extracellular chemical $c$ at the barbed end of the filament $\\a$ at time $t$, i.e\n$$c^\\pm (\\a,t) = c\\( \\vec F^\\pm (\\a,0,t),t\\).$$\nThe coefficient $\\lambda_\\text{res}$ represents the response of the cell and in particular of the polymerization mechanism to changes of the extracellular chemical. Larger $\\lambda_\\text{res}$ values lead to more pronounced changes of the polymerization rate and to more polarized cells. \n\nThe exponential function in \\eqref{eq:chem.pol} has no biological justification; it is used merely to provide a smooth transition from the minimum $v_{\\min}$ to the maximum $v_{\\max}$ polymerization rate in a continuous and controlled manner. Other functions with the same attributes could be used in its place. \n\nThe second mechanism that we consider describes primarily intracellular processes. For biological reasons that are not specified in this work, the polymerization mechanism can be hampered or otherwise destabilized, leading to an assortment of phenomena like persistent very high or low polymerization rates, abrupt changes of the polymerization rate, etc. This part of the model was previously proposed in \\cite{MOSS-model} where it was used to prescribe the polymerization rate directly on the membrane of the cell.\n\nThe conditions that destabilize the polymerization mechanism are important in a assortment of phenomena (pathological or not) which are beyond the scope of this paper, so we will not comment on them any more. We understand though the biological significance of both mechanisms as well as their distinctive functionality and use both \\change{}{of} them in this work. \n\nOverall, the polymerization rate $v^\\pm$ that we consider is given as \n\\begin{equation}\\label{eq:int.pol}\n\tv^\\pm(\\a,t) = \\mathcal D_\\text{stb}\\(v^\\pm_\\text{ext}(\\a,t)\\),\n\\end{equation}\nwhere $\\mathcal D_\\text{stb}$ describes the internal controlling mechanism that can potentially depend on a large number of cellular processes. Nevertheless, unless otherwise stated, we assume throughout this paper that $\\mathcal D_\\text{stb}=\\text{id}$ and hence\n\\begin{equation}\\label{eq:int.pol.II}\n\tv^\\pm(\\a,t) = v^\\pm_\\text{ext}(\\a,t).\n\\end{equation}\n\n\n\nFor the numerical solution of the FBLM we employ a problem specific FEM that we briefly describe in Appendix \\ref{sec:FEM}. It was previously developed in \\cite{MOSS-numeric, Brunk2016}, where we refer for more details. \n\n\\begin{figure}[t]\n\t\\centering\n\t\\begin{tabular}{cc}\n\t\t\\hskip -1em\n\t\t{\\includegraphics[width=0.45\\linewidth]{{18x72x0.0002x1.6202}.png}}\n\t\t&{\\includegraphics[width=0.45\\linewidth]{{18x72x0.001x1681}.png}}\\\\\n\t\t\\footnotesize{(a) $\\Delta t = 0.0002$} & \\footnotesize{(b) $\\Delta t=0.001$}\n\t\\end{tabular}\n\t\\caption{Final time conformations for the Experiment \\ref{exp:dt.chemo} with $s_\\text{nodes}=18$, $\\alpha_\\text{nodes}=72$. Showing in the large panel the full lamellipodium as comprised by the two families of ``discrete'' filaments (corresponding to the discretization lines of the domain), and in the smaller panels, the polymerization rates of the individual ``discrete'' filaments and the corresponding number density of the filaments at the leading edge with respect to their index. In both, the conformation of the lamellipodium and its discrete filaments, as well as the polymerization rates and filament densities are smooth.}\\label{fig:18x72}\n\\end{figure}\n\n\\section{Timestep stability}\\label{sec:stbility} \n\nThe complexity of the FBLM \\eqref{eq:strong} and the FEM \\eqref{eq:FEM} do not allow for a rigorous stability analysis. Instead, we perform here an experimental\/numerical investigation where we exhibit the existence of regions of stability for the timestep $\\Delta t$ in terms of the space discretizaiton steps $\\Delta s$ and $\\Delta \\a$.\n\nTo this end, we first note that the structure of the FEM \\eqref{eq:FE:V}--\\eqref{eq:I2D} indicates a particular relation between $\\Delta t$, $\\Delta s$, and $\\Delta \\a$, of the form:\n\\begin{equation}\\label{eq:CFL}\n\t\\Delta t \\leq C\\,(\\Delta s)^3\\,\\Delta \\a\\,.\n\\end{equation}\nTo identify the stability constant $C$, we numerically solve indicative experiments for different combinations of $\\Delta t$, $\\Delta s$, $\\Delta a$. For each combination, we characterize the resulting conformation as smooth (or not) and accordingly accept (or not) the corresponding combination. The largest $\\Delta t$ to result to smooth solutions gives rise to $C$. As we see later, this procedure can be used to set the timestep of the method in an automated way.\n\nThe experiments that we consider are particular; one chemotaxis and one haptotaxis. In the fist we exhibit the approach we follow to identify the stability constraint, and discuss a computational approach to set the timestep automatically. In the second we go one step further and identify the relation between the stability constraint as the gradient of the ECM. \n\nThe first experiment we consider is a chemotaxis driven cell migration.\n\\begin{experiment}[Stability -- Chemotaxis]\\label{exp:dt.chemo}\n\tAn initially rotational symmetric cell migrates under the influence of a chemical signal. The direction and strength of the signal and the final simulation time are chosen in a way that the deformation and migration of the cell is small, while at the same time the width of the lamellipodium (and hence the effective pulling force) becomes significantly asymmetric around the lamellipodium.\n\n\tThe parameters for this experiment are given in Table \\ref{tbl:parameters}. The polymerization rate varies smoothly from the minimum value at the posterior side of the cell, to the maximum value at the anterior, see \\cite{MOSS-model} for more details.\n\\end{experiment}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\begin{tabular}{cc}\n\t\t\\hskip -0.7em\n\t\t{\\includegraphics[width=0.45\\linewidth]{{9x72x0.002x1.68}.png}}\n\t\t&{\\includegraphics[width=0.45\\linewidth]{{9x72x0.01x1.62}.png}}\\\\\n\t\t\\footnotesize{(a) $\\Delta t=0.002$} & \\footnotesize{(b) $\\Delta t = 0.01$}\n\t\\end{tabular}\n\t\\caption{Experiment \\ref{exp:dt.chemo} with $s_\\text{nodes}=9$, $\\alpha_\\text{nodes}=72$ and $\\Delta t =0.002$ in (a) and $\\Delta t =0.01$ in (b). Final time conformation. In both cases, the filaments in the lamellipodium are smooth but the polymerization rates in (b) are clearly not.} \\label{fig:9x72}\n\\end{figure}\n\nAs a first step for the computation of $C$, we set the resolution of $B_0$ in \\eqref{eq:cell} to be $\\alpha_\\text{nodes}=72$ and $s_\\text{nodes}=18$, along the $\\a$ and $s$ directions respectively. Accordingly, \\eqref{eq:CFL} recasts to $\\Delta t \\leq  0.017\\,C$.  Using Experiment \\ref{exp:dt.chemo} and varying $\\Delta t$ by small increments  we identify two smooth conformations: one for $\\Delta t=0.001$ and one for $\\Delta t=0.0002$, cf. Figure \\ref{fig:18x72}. When these timesteps are combined with \\eqref{eq:CFL} they yield respectively\n\\begin{equation}\\label{eq:C.choice}\n\tC=\\frac{1}{15}\\text{ and }C=\\frac{1}{75}.\n\\end{equation}\n\n\n\n\nTo distinguish between these two values of $C$, we consider the coarser grid with $s_\\text{nodes}=9$ and $\\alpha_\\text{nodes}=72$ and we deduce from \\eqref{eq:CFL} and \\eqref{eq:C.choice} the timesteps $\\Delta t = 0.01$ and $\\Delta t = 0.002$ respectively.  We test these $\\Delta t$ values in Experiment \\ref{exp:dt.chemo} and note that in both cases the conformation of the lamellipodium is ``smooth'', cf. Figure \\ref{fig:9x72}. In the case $\\Delta t = 0.01$ though, the corresponding polymerization rate functions are non smooth; this implies timestep driven numerical instabilities in the solution. In effect, the value $C=\\frac{1}{15}$ is too large, and hence we promote the value $C=\\frac{1}{75}$.\n\n\nWe verify this stability constant with an even coarser grid with $s_\\text{nodes} = 5$ and $\\alpha_\\text{nodes} = 36$.  This time, we deduce from \\eqref{eq:C.choice} and \\eqref{eq:CFL}, the timesteps $\\Delta t = 0.08$ and $\\Delta t=0.02$ respectively. The simulation results are shown in Figure \\ref{fig:5x36}, where we note that when $\\Delta t = 0.08$ the filaments in the lamellipodium and the polymerization rates are not smooth. Hence, we promote once again the stability constant \n\\begin{equation}\\label{eq:C.chemo}\n\tC=\\frac{1}{75}. \n\\end{equation}\n\n\n\\begin{figure}[t]\n    \\centering\n\t\\begin{tabular}{cc}\n\t\t{\\includegraphics[width=0.45\\linewidth]{{5x36x0.02x1.62}.png}}\n\t   &{\\includegraphics[width=0.45\\linewidth]{{5x36x0.08x1.68}.png}}\\\\\n\t   \\footnotesize{(a) $\\Delta t = 0.02$} & \\footnotesize{(b) $\\Delta t=0.08$}\n\t\\end{tabular}\n\t\\caption{Experiment \\ref{exp:dt.chemo} with $s_\\text{nodes} = 5$, $\\alpha_\\text{nodes} = 36$. Final time. Both the filaments and the polymerization rate function in (b)  are not smooth; an indication that the corresponding timestep is too large.}\\label{fig:5x36}\n\\end{figure}\n\nIt is understood that this stability constant depends on a number of environmental parameters and variables, most notably on the extracellular chemical and matrix. Nevertheless, it is important to note that the FEM clearly exhibits stability regions with respect to the time and space steps $\\Delta t$ and $\\Delta \\a$, $\\Delta s$. \n\n\\subsection*{Choosing the timestep}\nWhat we have seen by the previous analysis is that the stability of the method can be identified by the smoothness of the polymerization rate and the filament density functions. Using this information we can adjust the timestep of the method in an automated way, similar to the \\textit{adaptive timestep control} (ATC) methods, \\cite{hundsdorfer2003numerical}. For that, we propose to use the \\textit{divided difference smoothness measure} \\cite{LeVeque2002,Sfakianakis.2016} or the $\\beta_0$ \\textit{smoothness indicators} used in \\textit{weighted essentially non-oscillatory} (WENO) schemes, see e.g. \\cite{Shu2009}. \n\nIn some more detail, we consider the discrete polymerization rate (and\/or the filament density) function $\\left\\{v_i,\\, i=1\\dots\\alpha_\\text{nodes}\\right\\}$ of the filaments at the end of one timestep of the method, see e.g. upper right panel in Figures \\ref{fig:18x72}, \\ref{fig:9x72}, \\ref{fig:5x36}. \n\t\n\t\nThe $\\beta_i^0$ smoothness indicator is computed for $i\\in {1\\dots\\alpha_\\text{nodes}}$ as\n\\begin{equation}\\label{eq:smooth}\n\t\\beta_i^0 = \\frac13 \\(4v_{i-1}^2 - 13v_{i-1}v_i + 13v_i^2 + 5v_{i-1}v_{i+1} - 13v_iv_{i+1} + 4v_{i+1}^2\\),\n\\end{equation}\nwhere the ``boundary'' terms $\\beta_1^0$ and $\\beta_{\\alpha_\\text{nodes}}^0$ are computed periodically with respect to $i$, and where $\\beta_i^0\\geq 0$ (by construction). The formula \\eqref{eq:smooth} is derived from a fourth order polynomial fitted to the numerical values $v_i$. This ``assumption'' is made to meet the smoothness of the FEM shape functions, i.e. third order along the $s$- and first order along the $\\a$-direction, see \\eqref{eq:H}, \\eqref{eq:LG}.\n\nThe results ought to be understood as follows: the larger $\\beta_i^0$ is, the ``less smooth'' the discrete function $\\left\\{v_i,\\, i=1\\dots\\alpha_\\text{nodes}\\right\\}$ is, see also \\cite{Shu2009} for details. \n\nAccordingly, we update the timestep $\\Delta t$ of the method using the rule \n\\begin{equation}\n\t\\Delta t^{\\text{upd}} =\\left\\{\\begin{aligned}\n\t&1.1\\,\\Delta t,&& \\max_{i} \\beta_i^0\\leq \\frac{2}{3}\\,\\beta_\\text{thr}\\\\\n\t&0.9\\,\\Delta t,&& \\max_{i} \\beta_i^0> \\frac{4}{3}\\beta_\\text{thr}\\\\\n\t&\\Delta t,&& \\text{else}\n\\end{aligned}\\right.\\,,\n\\end{equation}\nwhere the threshold value $\\beta_\\text{thr}>0$ is set experimentally. In the first case, the updated $\\Delta t^{\\text{upd}}$ is employed in the next step of the method, whereas in the second case the same step is repeated with the updated $\\Delta t^{\\text{upd}}$ value. Subsequent repetitions of the same step of the method might be needed in the second case.\n\n\\subsection*{Haptotaxis and dependence on the gradient of the ECM}\nThe second experiment that we consider in the time-step stability study is a haptotaxis one. This time though  we make one more step and identify the relation between the stability constant $C$ and the gradient of the ECM. We do so in the following experiment:\n\n\\begin{experiment}[Stability -- Haptotaxis]\\label{exp:dt.hapto}\n\tAn initially rotational symmetric cell lies over a non-uniform adhesion substrate. We consider three different cases that are incorporated in \\eqref{eq:strong} by the adhesion coefficients:\n\t\\begin{subequations}\n\t\\begin{align}\n\t\t\\mu^A_1(\\vec x) &=\\begin{cases}  0.4 - 0.01\\,x, & x< 0\\\\ 0.4, & x\\geq0 \\end{cases},\\label{eq:mu_A_1}\\\\\n\t\t\\mu^A_2(\\vec x) &=\\begin{cases}  0.4 - 0.05\\,x, & x< 0\\\\ 0.4, & x\\geq0 \\end{cases},\\label{eq:mu_A_2}\\\\\n\t\t\\mu^A_3(\\vec x) &=\\begin{cases}  0.4 - 0.1\\,x, & x< 0\\\\ 0.4, & x\\geq0 \\end{cases}.\\label{eq:mu_A_3}\n\t\\end{align}\t\n\t\\end{subequations}\n\twhere $\\vec x=(x,y)\\in \\Omega=[-40,20]\\times[-30,30]$. We assume that the chemical environment is uniform to a level that the polymerization rate is approximately\\footnote{Small variations might emanate from the variable curvature of the membrane, see Section \\ref{sec:model} and \\cite{MOSS-model} for details.} $v^\\pm(\\a,t)=1$ for all filaments.  The domain is discretized with $s_\\text{nodes} = 7$ and $\\alpha_\\text{nodes}=36$, and the rest of the parameters are given in Table \\ref{tbl:parameters} and the simulation results are shown in Figure \\ref{fig:dt.hapto}. \n\\end{experiment}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\begin{tabular}{ccc}\n\t\t~\\includegraphics[height=9em]{hapto-04-1-slope-00-1-dt-0-003.png}~\n\t\t&~\\includegraphics[height=9em]{hapto-04-3-4-slope-0-05-dt-0-002.png}~\n\t\t&\\includegraphics[height=9em]{hapto-04-6-4-slope-0-1-dt-0-0015.png}\n\t\t~\\includegraphics[height=9em]{hapto-colorbar.png}\n\t\n\t\t\\\\\n\t\t\\footnotesize{(a) $\\mu^A_1$}\n\t\t&\\footnotesize{(b) $\\mu^A_2$}\n\t\t&\\footnotesize{(c) $\\mu^A_3$}\n\t\\end{tabular}\t\t\n\t\\caption{Experiment \\ref{exp:dt.hapto} for the three different adhesion coefficients \\eqref{eq:mu_A_1}--\\eqref{eq:mu_A_3}. Showing here the final time simulations where it is clearly exhibited that the stronger the gradient of the ECM is, the more pronounced the deformation of the cell is. The colorbar on the right refers to the ECM and is common for all three figures.}\\label{fig:dt.hapto}\n\\end{figure}\n\t\n\tAs with Experiment \\ref{exp:dt.chemo}, we identify for each $\\mu^A$ case  \\eqref{eq:mu_A_1}--\\eqref{eq:mu_A_3}, the corresponding stability constant $C$ in \\eqref{eq:CFL}. We do so by varying the timestep of the method by small increments and find the largest $\\Delta t$ that still delivers smooth results.\n\t\n\tThis way, we identify the stability constants \n\t\\[\n\t\tC_1=0.0041\\,,\\quad  C_2= 0.0033\\,,\\quad  C_3=0.0022.\n\t\\] \n\trespectively for $\\mu^A_1$, $\\mu^A_2$, $\\mu^A_3$. As the motility of the cell is haptotaxis driven, we correlate $C_1$ through $C_3$ with the corresponding (norm of the) gradients of the ECM\n\t\\[\n\t\t\\lambda_1=0.01\\,,\\quad \\lambda_2=0.05\\,,\\quad \\lambda_3=0.1.\n\t\\]\t\n\tThe relation between the stability constant $C$ and the gradient $\\lambda$ of the ECM is approximately linear \n\t\\begin{equation}\\label{eq:C.hapt}\n\t\tC \\sim -0.0225 \\lambda, \n\t\\end{equation}\n\tsee also Figure \\ref{fig:dependence}. Clearly, the coefficient $-0.0225$ might depend on several cellular and extracellular parameters. Nevertheless, it is clearly exhibited by this experiment that the timestep stability constant depends in a linear decreasing way on the gradient of the ECM.\n\n\n\t\\begin{figure}[t]\n\t\\centering\n\t\\footnotesize\n\t\\begin{tabular}{c}\n\t\t{\\includegraphics[width=0.4\\linewidth]{lambda-dependence-II}}\n\t\\end{tabular}\t\t\n\t\\caption{Experiment \\ref{exp:dt.hapto}. Showing the linear dependence of the stability constant $C$ on the ``gradient'' of the ECM $\\lambda$, cf.  \\eqref{eq:mu_A_1}--\\eqref{eq:mu_A_3} and Figure \\ref{fig:dt.hapto}.} \\label{fig:dependence}\n\t\\end{figure}\n\n\\section{Convergence of the FEM}\\label{sec:converge}\nAs is typically done in complex models and methods, we exhibit here the convergence of the FEM in characteristic numerical experiments. In particular, we consider \\change{a}{} chemotaxis, \\change{a}{} haptotaxis, and \\change{a}{} chemo-haptotaxis experiments. \n\nIn each of these experiments we vary $\\Delta s$ and\/or $\\Delta \\a$ and compute cascades of numerical solutions over nested refined grids. As the exact solutions are not known, we compare every numerical solution to the one of the finest grid. \n\nThe experiments that we consider are the following:\n\n\\begin{experiment}[Convergence -- Chemotaxis]\\label{exp:conv:chemo}\n\tThe cell migrates over a uniform adhesion substrate, under the influence of a chemical stimulus. The effect of chemotaxis is incorporated as a variable polymerization rate at the membrane. It varies smoothly $v_{\\min}=1.5$ at the posterior side of the lamellipodium, to $v_{\\max}=8$ at the anterior, see \\cite{MOSS-model} for more details. The rest of the parameters are given in Table \\ref{tbl:parameters}. \n\\end{experiment}\n\n\\begin{experiment}[Convergence -- Haptotaxis]\\label{exp:conv:hapt}\n\tThe cell migrates within a chemically uniform environment over a non-uniform adhesion substrate. The variable ECM density is included in \\eqref{eq:strong} by the adhesion coefficient,\n\t\\[\n\t\t\\mu^A(\\vec x) = 0.4101\\, \\begin{cases} 0.1, & x<0\\\\ 0.1 + x\/30, & x\\geq 0 \\end{cases},\\quad \\vec x =(x,y)\\in \\Omega \n\t\\]\n\tThe polymerization rate is set at the same value\\footnote{Small changes might occur due to variations in the curvature of the membrane, see Section \\ref{sec:model} and \\cite{MOSS-model} for details.} $v=8$ for all the filaments. The rest of the parameters are set as in Table \\ref{tbl:parameters}.\n\\end{experiment}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\footnotesize\n\t\\begin{tabular}{cc}\n\t\t{\\includegraphics[width=0.45\\linewidth]{conv-chem-wrt-anodes-snodes-9-new}}\n\t\t\t&{\\includegraphics[width=0.45\\linewidth]{conv-chem-wrt-snodes-anodes-36-new}}\\\\[-0.7em]\n\t\t$\\alpha_\\text{nodes}$ & $s_\\text{nodes}$\\\\[0.3em]\n\t\t(a) $s_\\text{nodes}=9$ & (b) $\\alpha_\\text{nodes}=36$\n\t\\end{tabular}\t\t\n\t\\caption{Convergence (log-log) plots for the Experiment \\ref{exp:conv:chemo} (chemotaxis) using the ``norm'' $|\\cdot|_A$. The ``error'' is computed as the difference against the numerical solution of the finest grid.  In {\\rm(a)} we fix the value $s_\\text{nodes}=9$ and study the convergence with respect to $\\alpha_\\text{nodes}$ (horizontal axis). We compare the resulting convergence curve with the slope $-2$. In {\\rm(b)} we fix $\\alpha_\\text{nodes}=36$ and study the convergence with respect to $s_\\text{nodes}$ (horizontal axis). The convergence rate in this experiment is comparable to 3.} \\label{fig:conv1}\n\\end{figure}\n\n\\begin{experiment} [Convergence -- Chemo-haptotaxis]\\label{exp:conv:mixed}\n\tThe cell migrates under the influence of a chemical signal and over a non-uniform adhesion substrate. The polymerization rates and the adhesion coefficients are set as in Experiments \\ref{exp:conv:chemo} and \\ref{exp:conv:hapt}, i.e.\n\t\\[\n\t\t\\mu^A(\\vec x) = 0.4101\\, \\begin{cases} 0.1, & x<0\\\\ 0.1 + x\/30, & x\\geq 0 \\end{cases},\\quad \\vec x =(x,y)\\in \\Omega \n\t\\]\n\tand $v$ varies smoothly from $v_{\\min}=1.5$ at the posterior side of the lamellipodium, to $v_{\\max}=8$ at the anterior.\n\\end{experiment}\n\nFor each of the these experiments, the comparison between the results takes place in physical space. We denote by $\\mathcal C\\subset \\mathbb{R}^2$ the full \\textit{cell} (area enclosed by the outer border of the lamellipodium) consider the following ``norms'': \n\\begin{subequations}\n\\begin{description}\n\t\\item[Invasiveness:] The maximum $x$-coordinate of the cell: \n\t\\begin{equation}\\label{eq:norm.invad}\n\t\t|\\mathcal C|_A=\\max_{(x,y)\\in \\mathcal C} x.\n\t\\end{equation}\n\tThis ``norm'' is more suited for experiments where the cell migrates to the right.\n\t\n\\item[Size:] The area of the minimum quadrilateral that the cell occupies:\n\t\\begin{equation}\\label{eq:norm.size}\n\t\t|\\mathcal C|_B= \\(\\max_{(x,y)\\in \\mathcal C} x - \\min_{(x,y)\\in \\mathcal C} x \\) \\(\\max_{(x,y)\\in \\mathcal C} y - \\min_{(x,y)\\in \\mathcal C} y \\).\n\t\\end{equation}\n\t\n\\item[Perimeter:] The length of the membrane of the cell, which coincides with the outer boundary of the lamellipodium:\n\t\\begin{equation}\\label{eq:norm.perim}\n\t\t|\\mathcal C|_C= \\int_{\\partial \\mathcal C} 1 .\n\t\\end{equation}\n\tWe compute this ``norm'' as the length of the closed \\textit{piecewise-linear} curve defined by the outer ends of the discretization filaments. \n\t\n\\item[Elongation:] The ratio of the sides of the minimum quadrilateral that the cell $\\mathcal C$ occupies:\n\t\\begin{equation}\\label{eq:norm.elong}\n\t\t|\\mathcal C|_D= \\frac{\\max_{(x,y)\\in \\mathcal C} x - \\min_{(x,y)\\in \\mathcal C} x } { \\max_{(x,y)\\in \\mathcal C} y - \\min_{(x,y)\\in \\mathcal C} y}.\n\t\\end{equation}\n\\end{description}\n\\end{subequations}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\footnotesize\n\t\\begin{tabular}{cc}\n\t\t{\\includegraphics[width=0.45\\linewidth]{conv-hapt-snodes-anodes-36-new}}\n\t\t&{\\includegraphics[width=0.46\\linewidth]{conv-chemhapt-snodes-anodes-144-new}}\\\\[-0.7em]\n\t\t$s_\\text{nodes}$ & $s_\\text{nodes}$\\\\[0.3em]\n\t\t(a) haptotaxis, $\\alpha_\\text{nodes}=36$ & (b) chemo-haptotaxis, $\\alpha_\\text{nodes}=144$\n\t\\end{tabular}\t\t\n\t\\caption{Convergence (log-log) plots in the haptotaxis and the combined chemo-haptotaxis Experiments \\ref{exp:conv:hapt}, \\ref{exp:conv:mixed} with respect to the the $|\\cdot|_A$  ``norm''. In {\\rm(a)} we consider the haptotaxis Experiment \\ref{exp:conv:hapt} and study the convergence with respect to $s_\\text{nodes}$ (horizontal axis) for fixed $\\alpha_\\text{nodes}=36$. In {\\rm(b)} we consider the chemo-haptotaxis Experiment \\ref{exp:conv:mixed} and study the convergence with respect to $s_\\text{nodes}$ (horizontal axis) for fixed $\\alpha_\\text{nodes}=144$.}\\label{fig:conv2}\n\\end{figure}\n\nWe first define the \\textit{distance} between two cells $\\mathcal C_1,\\ \\mathcal C_2$ as \n\\begin{equation}\\label{eq:diff.cells}\n\td_X(\\mathcal C_1,\\mathcal C_2) = \\big| |\\mathcal C_1|_X - |\\mathcal C_2|_X \\big|,\n\\end{equation}\nwhere $|\\cdot|_X$, $X=A,\\,B,\\,C$, or $D$, represents the ``norm'' of choice. The convergence results with respect to the $|\\cdot |_A$ and $|\\cdot |_B$ ``norms'' can be found in Figures \\ref{fig:conv1} -- \\ref{fig:conv3}; similar results are obtain for the $|\\cdot |_C$ and $|\\cdot |_D$ ``norms''.\n\nFigure \\ref{fig:conv1} refers to the chemotaxis Experiment \\ref{exp:conv:chemo}. We set the fixed value $s_\\text{nodes}=9$ and allow for $\\alpha_\\text{nodes}\\in\\left\\{36,54,72,108\\right\\}$ to vary. For every instance of $\\alpha_\\text{nodes}$ we compute the ``error'' as the distance of the corresponding $\\mathcal C_{\\alpha_\\text{nodes}}$ with the one with the finest grid, i.e. \n$$d_X(\\mathcal C_{\\alpha_\\text{nodes}}, \\mathcal C_{108}).$$\nIn this case, we deduce a convergence rate of about two. In a similar way, we consider the fixed value $\\alpha_\\text{nodes}=36$ and vary $s_\\text{nodes}\\in\\left\\{5,7,\\dots,15\\right\\}$ to deduce a convergence rate of approximately three. \n\nIn Figure \\ref{fig:conv2} we study the convergence of the haptotaxis Experiment \\ref{exp:conv:hapt} and the chemo-haptotaxis Experiment \\ref{exp:conv:mixed}. We set the fixed values $\\alpha_\\text{nodes}=36$ and $\\alpha_\\text{nodes}=144$ respectively, and allow for $s_\\text{nodes}\\in\\{5,7,9,11\\}$ to vary. We once again deduce the convergence of the numerical method. When comparing with the convergence in the chemotaxis Experiment \\ref{exp:conv:chemo} (shown in Figure \\ref{fig:conv1}), the results obtained here indicate that the influence of the non-uniform ECM on the numerical error dissipates fast with respect to the resolution of the grid. In other words, coarse discretization grids are suited better for haptotaxis rather than for chemotaxis experiments.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\footnotesize\n\t\\begin{tabular}{cc}\n\t\t\\includegraphics[width=0.45\\linewidth]{mixed-convergence-A}\n\t\t& \\includegraphics[width=0.45\\linewidth]{mixed-convergence-B}\\\\[-0.4em]\n\t\t~~~~~inscribed circle diameter &~~~~~inscribed circle diameter\\\\[0.3em]\n\t\t(a) norm $|\\cdot |_A$ & (b) norm $|\\cdot |_B$\\\\\n\t\\end{tabular}\t\t\n\t\\caption{Convergence (log-log) plots for the chemo-haptotaxis Experiment \\ref{exp:conv:mixed} using the $|\\cdot|_A$ and the $|\\cdot|_B$ norms. In both (a) and (b),  $\\alpha_\\text{nodes}$ and $s_\\text{nodes}$ vary at the same time. The convergence is studied with respect to the (decreasing) radius of the inscribed circle (horizontal axis) of the discretization cells $C_{i,j}$ \\eqref{eq:cell}. The convergence rates are comparable to 3.}\\label{fig:conv3}\n\\end{figure}\n\nIn Figure \\ref{fig:conv3} we consider again the mixed chemo-haptotaxis Experiment \\ref{exp:conv:mixed}. This time though, the discretization varies with respect to both $s_\\text{nodes}$ and $\\alpha_\\text{nodes}$ at the same time. As the grid is refined, we consider the \\textit{diameter of the inscribed circle of the discretization cell} $C_{i,j}$ as the controlling parameter, and study the convergence of the method with respect to it. We use the norms $|\\cdot|_A$ and $|\\cdot |_B$, and compare every numerical solution with the one of the finest grid. The detailed results for the $|\\cdot|_A$ ``norm'' can be found in Table \\ref{tbl:conv.mixed}, where the convergence rates are computed by the slope of piecewise linear curve.\n\n\n\n\\section{Embedding the FBLM in the extracellular environment}\\label{sec:env}\nIn this section we model and simulate the interactions between the FBLM-FEM and the extracellular environment. We start with a model that describes the extracellular environment and the cell, and with a short description of the numerical method that we use to solve it. We conclude this section with two particular numerical experiments that exhibit the combination of the FBLM-FEM with the environment.\n\n\\subsection{A model for the environment}\nWe assume that the extracellular environment is comprised of the ECM, a chemical ingredient that serves as \\textit{chemoattractant} for the cell, and \\textit{matrix metalloproteinases} (MMPs) that are secreted by the cell and are responsible for the degradation of the matrix. These environmental components participate in our study via the density of the corresponding (macro-)molecules.\n\nThe extracellular chemical, denoted by $c$, is injected in the environment by one or more \\textit{micro-pipettes} that are modelled here as source terms. The chemical is assumed to diffuse freely in the environment, to decay with time (chemical degradation), and to be degraded by the cell upon attachment. The MMPs, denoted by $m$, are produced by the cell, they diffuse freely in the environment, and decay with time. The ECM, denoted by $v$, is assumed to be an immovable component of the system that decays upon attachment of the MMPs, and is not remodelled. Overall the model of the environment reads\n\\begin{equation}\\label{eq:env}\n\\left\\{\\begin{aligned}\n\\frac{\\partial c}{\\partial t}(\\vec x,t) &= D_{c} \\Delta c(\\vec x,t)  + \\sum_{i=1}^{N^\\text{pip}}\\a_i\\, \\mathcal X_{\\mathcal P_i(t)}(\\vec x)  - \\gamma_1 c(\\vec x,t) - \\delta_1\\, \\mathcal X_{\\mathcal C(t)}(\\vec x)\\\\\n\\frac{\\partial m}{\\partial t}(\\vec x,t) &= D_m \\Delta m(\\vec x,t) + \\beta \\mathcal X_{\\mathcal C(t)}(\\vec x) - \\gamma_2 m(\\vec x,t)\\\\\n\\frac{\\partial v}{\\partial t}(\\vec x,t) &= -\\delta_2 m(\\vec x,t) v(\\vec x,t)\n\\end{aligned}\\right.\n\\end{equation}\nwith $\\vec x\\in \\Omega\\subset \\mathbb R^2$, $t\\geq 0$, and $D_c,\\, D_m, \\a_i, \\beta, \\gamma_i, \\delta_i \\geq 0$. The number of pipettes is $N^\\text{pip}$ and the corresponding domain\/source has support  $\\mathcal P_i$, $i=1\\dots N^\\text{pip}$. The feedback of the FBLM-FEM to the environment takes place \\change{at}{through} the term $\\mathcal X_{\\mathcal C(t)}(\\vec x)$, where $\\mathcal C(t)\\subset \\mathbb R^2$ represents the \\textit{full cell} (lamellipodium and internal structures). \n\n\\begin{table}[t]\n\t\\footnotesize\n\t\\begin{center}\n\t\t\\begin{tabular}{c|c|c}\n\t\t\tinscribed circle & $|\\cdot|_A$ & convergence \\\\\n\t\t\tdiameter&``error''&rate\\\\\n\t\t\t\\hline\n\t\t\t\\\\[-0.7em]\n\t\t\t0.27925 & 0.46966 & ---\\\\\n\t\t\t0.23271 & 0.32897 & 1.9529\\\\\n\t\t\t0.19947 & 0.22370 & 2.5022\\\\\n\t\t\t0.17453 & 0.18117 & 1.5788\\\\\n\t\t\t0.15514 & 0.13964 & 2.2108\\\\\n\t\t\t0.13963 & 0.10468 & 2.7357\\\\\n\t\t\t0.12693 & 0.08957 & 1.6343\\\\\n\t\t\t0.11636 & 0.06306 & 4.0358\\\\\n\t\t\t0.10740 & 0.04702 & 3.6627\\\\\n\t\t\t0.09973 & 0.03610 & 3.5346\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{The data corresponding to the convergence Figure \\ref{fig:conv3} (a) and the Experiment \\ref{exp:conv:mixed} for the $|\\cdot|_A$ ``norm''. The ``error'' of each numerical solution is computed by its difference against the numerical solution of the finest grid using \\eqref{eq:diff.cells}. The convergence rates are computed by the slope of the corresponding segment of the curve.}\\label{tbl:conv.mixed}\n\\end{table}\n\nClearly, \\change{the}{} model \\eqref{eq:env} is simple and accounts only for some of the basic extracellular processes and interactions between the FBLM and the environment. It can easily be extended to incorporate further and more precise biological properties\/phenomena. As it is not though the main aim of this paper, we refrain from such generalizations here and postpone this study for a follow-up work.\n\nAs it is easier for the presentation of the numerical method, we write the system \\eqref{eq:env} in an alternative operator form:\n\\begin{equation}\\label{eq:env.oper}\n\\partial_t \\vec w= R(\\vec w) + D(\\vec w)\\;,\n\\end{equation}\nwhere $\\vec w=(c, m, v)^T$ and where $R$, $D$ are the reaction and diffusion operators respectively:\n\\begin{align}\nD(\\vec w)&= \\(D_c \\Delta c,\\ D_m \\Delta m,\\ 0\\)^T,\\\\\nR(\\vec w)& = \\(\\sum_{i=1}^{N^\\text{pip}}\\a_i \\mathcal X_{\\mathcal P_i}  - \\gamma_1 c - \\delta_1 \\mathcal X_{\\mathcal C},\\ \\beta \\mathcal X_{\\mathcal C} - \\gamma_2 m,\\ -\\delta_2 m v\\)^T.\n\\end{align}\n\nThe system \\eqref{eq:env} (or \\eqref{eq:env.oper}) is equipped with initial and boundary conditions, and parameters that are experiment specific.\n\nFor the numerical treatment of \\eqref{eq:env.oper} we use a second order \\textit{Implicit-Explicit Runge-Kutta} (IMEX-RK) \\textit{Finite Volume} (FV) numerical method that was previously developed in \\cite{Sfakianakis.2016, Sfakianakis.2016b} where we refer for more details. Here, in Appendix \\ref{sec:FV}, we give some details.\n\n\\subsection{Coupling the FBLM with the environment}\n\\begin{figure}[t]\n\t\\centering\n\t\\begin{tabular}{ccc}\n\t\t\\raisebox{-0.21em}{\\includegraphics[height=9.5em]{embed-cbar-ecm-2}}~\n\t\t\\includegraphics[height=9.3em]{embed-21-A}\n\t\t&\\includegraphics[height=9.3em]{embed-10021-A.png}\n\t\t&\\includegraphics[height=9.3em]{embed-20021-A.png}\n\t\n\t\t~\\raisebox{-0.21em}{\\includegraphics[height=9.48em]{embed-cbar-chem-2}}\\\\\n\t\t\\footnotesize{(a) $t=0.021$}\n\t\t&\\footnotesize{(b) $t=10.021$}\n\t\t&\\footnotesize{(c) $t=20.021$}\n\t\\end{tabular}\t\t\n\t\\caption{Experiment \\ref{exp:embedding} (In the environment): A cell (closed curve) migrates on a non-uniform ECM (isolines) under the influence of an extracellular chemical attractor injected in the environment by a circular pipette (upper right). {\\rm (a):} As the chemical diffuses in the environment and decays, it creates a chemical gradient. {\\rm (b)-(c):} As the cell identifies the gradients of the chemical and of the ECM, it responds by adjusting its motility and shape. The size of the cell increases with the intensity of the surrounding chemical. At the same time, the MMPs produced by the cell degrade the ECM. The colorbars refer to the densities of the ECM (left) and the chemical (right).}\\label{fig:embedding}\n\\end{figure}\nThe FBLM-FEM \\eqref{eq:strong} and the environment \\eqref{eq:env} are coupled at three different places \\change{}{through}: a) \\change{in}{}the characteristic function $\\mathcal X_{\\mathcal C}$ in the model of the environment \\eqref{eq:env}, where the cell $\\mathcal C$ produces MMPs and degrades the chemical, b) \\change{in}{}the adhesion coefficient $\\mu^A$ of the FBLM in \\eqref{eq:strong} that reflects the density of the ECM, see e.g. Experiment \\ref{exp:conv:hapt}, and c) \\change{in}{}the polymerization rate $v^\\pm_\\text{ext}$ in  \\eqref{eq:chem.pol} which is primarily controlled by the intensity of the extracellular chemical $c$ at the membrane of the cell.\n\nNumerically, the FEM and FV methods that solve the FBLM \\eqref{eq:strong} and environment \\eqref{eq:env} model, are combined in a modular way. During the time period $[t^n,t^{n+1}]$, $t^{n+1}=t^n+\\Delta t^n$, $n=0, \\dots$ the following hold\n\\begin{itemize}\n\t\\item[---] The timestep $\\Delta t^n$ is common in the FEM and the FV and is dictated by the stability of both methods, see also Section \\eqref{sec:stbility}. \n\t\\item[---] The FV takes into account the position of the cell at $t^n$, i.e. the FV is explicit with respect to the cell.\n\t\\item[---] Similarly, the FEM is explicit with respect to the density of the ECM and the chemical.\n\\end{itemize}\n\nWe exhibit the coupling of the cell with the environment, and their interactions with two particular experiments:\n\\begin{experiment}[In the environment]\\label{exp:embedding}\n\tWe consider an initial non-uniform adhesion substrate, a circular pipette that injects chemical in the environment, and a rotational symmetric cell in some distance from the pipette.\n\tThe ECM is given by\n\t\\begin{equation}\n\t\t\\mu^A(\\vec x) = 0.5\\( \\sin\\( \\(2y-x^3\\)\\pi \\)^2 + 1\\)\\,, \t\n\t\\end{equation}\n\twhere $\\vec x=(x,y)\\in \\Omega = [-50,60]\\times [-50,70]$.\n\t\n\tThe simulation results are shown in Figure \\ref{fig:embedding}. The parameters for the FBLM and the environment models are given in Tables \\ref{tbl:parameters} and \\ref{tbl:env.parameters}.\n\t\\end{experiment}\nThe chemical diffuses in the environment and is identified by the cell, which responds with a combined motion to its gradient and to the gradient of the ECM. While the cell migrates, it secretes MMPs that diffuse in the environment, decay, and degrade the ECM.\n\t\n\\change{The }{}Experiment \\ref{exp:embedding} exhibits the relation between the size of the cell, the width of the lamellipodium, the polymerization rates of the filaments, and the density of the extracellular chemical attractant. We clearly see how the size of the cell increases as it approaches the pipette and the higher density of the chemical, cf. \\eqref{eq:width_and_v} and \\cite{MOSS-model,Brunk2016}.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\begin{tabular}{cccc}\n\t\t\\hspace{-0.2em}\\includegraphics[height=7.2em]{soft-571-B}\n\t\t&\\hspace{-0.3em}\\includegraphics[height=7.4em]{soft-16201-B.png}\n\t\t&\\hspace{-0.3em}\\includegraphics[height=7.4em]{soft-20611-B.png}\n\t\t&\\hspace{-0.2em}\\includegraphics[height=7.4em]{soft-28441-B.png}\n\t\t\\includegraphics[height=7.1em]{soft-cbar-chem.png}\\\\\n\t\t\\footnotesize{(a) $t=0.571$}\n\t\t&\\footnotesize{(b) $t=16.201$}\n\t\t&\\footnotesize{(e) $t=20.611$}\n\t\t&\\footnotesize{(e) $t=28.441$}\n\t\n\t\\end{tabular}\t\t\n\t\\caption{Time evolution of the Experiment \\ref{exp:softstiff} (Adhesion wall): A cell faces a discontinuity in the ECM with density that is lower in the left side of the domain and higher density of adhesion sites perturbed by a small amount of noise. The colorbar on the right refers to the intensity of the chemical. {\\rm(a):} Initially the cell is rotational symmetric and resides on the higher density part of the ECM. The three pipettes are located in the lower density part of the ECM and create a complex chemical environment that is attractive for the cell. {\\rm(b-c):} The cell identifies the gradient of the chemical and migrates towards the pipettes. When it reaches the discontinuity of the ECM, it ceases further migration as it cannot exert sufficient adhesions. The part though that still resides on the higher density ECM, keeps on migrating and as an effect the cell elongates parallel \\change{}{to} the discontinuity of the ECM.}\\label{fig:softstiff}\n\\end{figure}\n\nFor the next experiment we are motivated by \\cite{Chaplain2012, Lo2000}, and in particular by the opposing effects that chemical attraction and the lack of sufficient adhesion can have on the migration of the cell. As the aim of this paper is not to reproduce experimental scenarios, we postpone this more detailed work for a follow-up work. Nevertheless we present it here as an indication of the combination of the FBLM \\eqref{eq:strong} and the environment \\eqref{eq:env} with multiple chemical sources.\n\n\n\\begin{experiment}[Adhesion wall]\\label{exp:softstiff}\n\tWe consider three sources that inject the same chemical in the environment in a constant rate. The chemical diffuses and decays and a complex chemical landscape is formed in the environment. The ECM exhibits a jump-discontinuity between a higher and lower density that separates the domain in two parts. \n\t\\begin{equation}\n\t\t\\mu^A(\\vec x) = 0.5\\,\\begin{cases} 1, & x<-30\\\\ 5, & x\\geq -30 \\end{cases},\\quad \\vec x =(x,y)\\in \\Omega\n\t\\end{equation}\n\tThe three pipettes reside on the lower-ECM-density part of the domain ($x< -30$) whereas an (initially rotational symmetric) cell on the higher-ECM part ($x\\geq -30$). \n\t\n\tThe simulation results of this experiment are given in Figure \\ref{fig:softstiff} and the parameters \\change{follow}{are as shown in} the Tables \\ref{tbl:parameters} and \\ref{tbl:env.parameters}.\n\\end{experiment}\t\n\n\nAs the chemical gradient is formed, it is identified by the cell which starts migrating towards the pipettes. The propagating front of the cell ceases further migration when the cell arrives at the discontinuity of the ECM, as it cannot create sufficient adhesions to the lower-density ECM and transfer its momentum. The rest of the cell, that resides on the higher-density ECM, keeps on migrating and the cell effectively elongates in a direction parallel to the discontinuity.\n\n\\section{Sensitivity analysis}\\label{sec:sensit}\n\nWe investigate here the sensitivity of the FBLM-FEM on several of its parameters. We consider a particular experiment and a reference parameter set and compute with them a reference numerical solution.  We then vary the parameter set, compute the new results, compare them with the reference solution, and quantify the effect the varied parameters have on the numerical solution.\n\nAs opposed to the study of the stability and the convergence in Sections \\ref{sec:stbility} and \\ref{sec:converge}, the sensitivity includes more biological information and meaning. For that reason, we employ now the FBLM augmented with the model for the environment \\eqref{eq:env}.\n\nThe control experiment that we consider is a chemotaxis scenario with two sources of chemical.\n \n\\begin{experiment}[Sensitivity -- Chemotaxis]\\label{exp:sensit}\n\tAn initially rotational symmetric cell resides over an adhesively uniform substrate \n\t\\begin{equation}\n\t\t\\mu^A(\\vec x) = 0.75\\,.\n\t\\end{equation}\n\tTwo sources of \\change{}{the} chemical are found in the vicinity of the cell as seen Figure \\ref{fig:sensit} (a). They represent two \\textit{pipettes} of the same circular shape that inject \\change{}{the} chemical in the environment with the same rate, cf. Tables \\ref{tbl:parameters} and  \\ref{tbl:env.parameters} for the relevant parameters.\n\\end{experiment}\n\tIn Figure \\ref{fig:sensit} we reproduce the time evolution of the Experiment \\ref{exp:sensit}. As the chemical diffuses in the environment, it decays with time (chemical degradation), and is degraded upon attachment with the cell. The cell responds to the gradient of the chemical by adjusting its polymerization rate, breaking \\change{of}{} its symmetry, and moving towards the direction of the pipettes. \n\t\nThe parameters we consider in the sensitivity analysis of the FBLM-FEM are\n\\begin{equation}\\label{eq:sens.set}\n\t\\mathcal P^{\\text{sens}}=\\Big\\{\\mu^B, \\mu^A, \\mu^T, \\mu^S, \\phi_0, \\mu^{IP}, v_{\\min}, v_{\\max}, \\lambda_\\text{inext}, \\mu^P, \\lambda_\\text{tether}, A_0\\Big\\}\n\\end{equation}\nthat we index by $i= 1\\dots 12$ and set the \\textit{reference parameter set} to be  \n\\begin{equation}\\label{eq:ref.set}\n\t\\mathcal P^\\text{ref}=\\Big\\{ p_i^\\text{ref},\\ i=1\\dots 12\\Big\\}\n\\end{equation}\nwith values given in Table \\ref{tbl:parameters}. For this parameter set, the final time conformation of the cell is denoted by $\\mathcal C^\\text{ref}$ and is depicted in Figure \\ref{fig:sensit} (b).\n\n\\begin{figure}[t]\n\t\\centering\n\t\\begin{tabular}{ccl}\n\t\t\\includegraphics[height=10em]{sensitivity-initial.png}~~~\n\t\t& \\includegraphics[height=10em]{sensitivity-final.png}\n\t\t& \\includegraphics[height=10em]{sensitivity-cbar.png}\n\t\t\\\\\n\t\n\t\t\\footnotesize{(a) Initial time}\n\t\t& \\footnotesize{(b) Final time}\n\t\\end{tabular}\t\t\n\t\\caption{Initial and final time conformation of the Experiment \\ref{exp:sensit} (sensitivity analysis). Showing here the cell (solid line) and the two pipettes as circular sources of chemical. (a): The initial condition of the cell and the chemical. (b): In the final time the cell has responded to the chemical gradient by breaking its symmetry and protruding towards both sources of the chemical. The colorbar in the right refers to the density of the chemical and is common to both figures.}\\label{fig:sensit}\n\\end{figure}\n\nWe perturb one after the other the reference parameters $p_i^\\text{ref}\\in \\mathcal P^\\text{ref}$ to new values $p_i^\\text{per}$, while maintaining the rest to their reference values. For each perturbation, the new parameter set $\\mathcal P^\\text{per}_i$ differs from the reference set $\\mathcal P^\\text{ref}$ only at the parameter $i$. \n\nFor each perturbation $\\mathcal P^\\text{per}_i$ of the parameter set, we compute the final time conformation of the cell $\\mathcal C^\\text{per}_i$ and compare it with the reference $\\mathcal C^\\text{ref}$ as:\n\\begin{equation}\\label{eq:sensit}\n\t\\mathcal S^{|\\cdot|_X}_{i}=\\frac{|\\mathcal C^\\text{per}_i|_X - |\\mathcal C^\\text{ref}|_x }{p_i^\\text{per}-p_i^\\text{ref}}.\n\\end{equation}\nwhere the ``norm'' $|\\cdot|_X$, $X=A,B,C,D$ is one of the ``norms'' introduced in Section \\ref{sec:converge}. The perturbations of the parameters are small, hence the divided differences in \\eqref{eq:sensit} can also be viewed as approximations to the corresponding derivatives, around the reference state $\\mathcal P^\\text{ref}$.\n\nIn essence, $\\mathcal S_i^{|\\cdot|_X}$ represents the rate at which the cell changes, in the sense of the ``norm'' $|\\cdot|_X$, with respect to the parameter $i$. Accounting for all the parameters of $\\mathcal P^\\text{sens}$, the \\textit{local sensitivity} follows. We refer to the Tables and \\ref{tbl:parameters} for the full list of the reference parameters and to Table \\ref{tbl:sensit} and Figure \\ref{fig:sensit_graph} for a concise description of the sensitivity analysis results. \n\nWe note that these results are not global in the sense that they are influenced by the experiment under investigation, the initial state of the cell, the environment, the reference parameters, and more.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\footnotesize\n\t\\begin{tabular}{>{\\centering}m{0.9\\linewidth}}\n\t\t\\includegraphics[width=1\\linewidth]{sensitivity-all.png}\\\\\n\t\t(a)\tSensitivity results for all the parameters in $\\mathcal P^\\text{sens}$ \\eqref{eq:sens.set}.\\\\\n\t\t\\includegraphics[width=1\\linewidth]{sensitivity-withoutmax.png}\\\\\n\t\t(b) Subtracting the most influential parameters $\\mu^T$ and $\\lambda_\\text{tether}$ from (a), reveals the importance of the adhesion coefficient $\\mu^A$, the inner pulling force $f_\\text{inn}$, the pressure $\\mu^P$.\n\t\\end{tabular}\n\t\\caption{Graphical representation of the sensitivity analysis Experiment \\ref{exp:sensit}. The effect of all the parameters \\eqref{eq:sens.set} is shown in the panel (a). Clearly, the membrane tethering $\\lambda_\\text{texther}$ and the twisting coefficient $\\mu^T$ are the more influential parameters. In (b) we exclude these two parameters and can hence identify the stretching $\\mu^S$, inner-pulling $\\mu^\\text{IP}$, and the pressure $\\mu^P$ as the (next) more influential parameters, cf.  Table \\ref{tbl:sensit}. The ``norms'' $|\\cdot|_A$, $|\\cdot|_B$, $|\\cdot|_C$, and $|\\cdot|_D$ represent: \\change{I}{i}nvasiveness, \\change{S}{s}ize, \\change{P}{p}erimeter, and \\change{E}{e}longation \\change{}{of the cell as defined in \\eqref{eq:norm.invad}-\\eqref{eq:norm.elong}}.}\\label{fig:sensit_graph}\n\\end{figure}\n\nNevertheless, we present here some characteristic remarks that will assist in further investigations:\n\\begin{itemize}\n\t\\item[--] The membrane tethering parameter $\\lambda_\\text{tether}$, is the most influential of the parameters \\eqref{eq:sens.set} in all the ``norms''. The fact in particular that it has a negative effect in the \\textit{invasiveness} $|\\cdot|_A$ ``norm'' and a positive in the \\textit{perimeter} $|\\cdot|_C$ and \\textit{elongation} $|\\cdot|_D$ ``norms'', implies that its primer effect is retractive not on the protruding but rather on the side parts of the cell.\n\t\n\t\\item[--] The twisting parameter $\\mu^T$ is equally important. This is in contrast to the lesser influence of the $\\mu^S$ parameter (the other biological component of the crosslink protein). Moreover, the negative influence in \\textit{area} $|\\cdot|_B$ and positive in \\textit{perimeter} $|\\cdot|_C$ ``norms'' is understood by the fact that the decrease of $\\mu^T$ leads to more linear and radial filaments, and to rotational symmetric and circular cells; hence to (relative) increase of the area and decrease of the perimeter. The decrease of the preferred angle $\\phi_0$ has also the same effect.\n\n\t\\item[--] All the parameters, except for $A_0$, have opposite effects in the \\textit{area} $|\\cdot|_B$ and \\textit{perimeter} $|\\cdot|_C$ ``norms''. As in the case of $\\mu^T$, this is understood by the fact that each term either leads towards to or away from a more circular conformation of the cell, which in turn maximizes the area and minimizes the perimeter. In contrast, \\change{the} increase of $A_0$ leads to a larger cell by increasing the inner area of the cell (behind the lamellipodium) and which leads to increase of the total area of the cell and its perimeter.\n\t\n\t\\item[--] Increasing the polymerization rates $v_{\\min}$ and $v_{\\max}$ has a positive effect in all the ``norms'' except for the \\textit{perimeter} $|\\cdot|_C$. This is so since the polymerization of filaments opposes the retracting effects of the other components of the model which in turn are responsible for the deformation of the initial rotational symmetric cell; in short: increasing the polymerization rates, leads to more circular cells. \n\t\n\t\\item[--] The myosin-actin inner pulling parameter $\\mu^\\text{IP}$ is also very influential in all the ``norms''. Note also that $\\mu^\\text{IP}$ and $\\phi_0$ are the only parameters with negative impact in the \\textit{elongation} ``norm'' $|\\cdot|_D$. That is, the higher the $\\mu^\\text{IP}$ or the $\\phi_0$ are, the less elongated the cell becomes, see also Figure \\ref{fig:softstiff}.\n\t\n\t\\item[--] We note the effect of $\\mu^A$ and $\\mu^S$ is almost identical in the \\textit{invasiveness} and \\textit{elongation} ``norms'' $|\\cdot|_A$, $|\\cdot|_D$. On the other hand, in the \\textit{size} and \\textit{elongation} ``norms'' $|\\cdot|_B$, $|\\cdot|_C$, their effects are of similar magnitude but opposite sign. This is understood as follows: \\change{the} increase of the adhesion parameter $\\mu^A$ leads to an increase of the size of the cell (see also \\cite{Brunk2016}) whereas \\change{the} increase of the stretching parameter $\\mu^S$ leads to its decrease.\n\\end{itemize}\n\n\n\\begin{table}[t]\n\t\\footnotesize\n\t\\centering\n\t\\begin{tabular}{c|p{7em}|p{7em}|p{7em}|p{7em}}\n\t\t&\\centering$|\\cdot|_A$&\\centering$|\\cdot|_B$&\\centering$|\\cdot|_C$&~\\hfill$|\\cdot|_D$\\hfill~\\\\ \n\t\tvariable&\\centering{Invasiveness}&\\centering{Area}&\\centering{Perimeter}&~\\hfill{Elongation}\\hfill~\\\\\n\t\t\\hline\n\t\t&&&&\\\\[-0.7em]\n\t\t$\\mu^B$&$+4.1903$&$+5.4005\\times 10^2$&$-1.8422$&$+8.4974\\times 10^{-2}$\\\\\n\t\t$\\mu^A$&$+1.1291\\times 10^{1}$&$+1.2492\\times 10^{3}$&$-6.0469\\times 10^{-1}$&$+1.1648\\times 10^{-1}$\\\\\n\t\t$\\mu^T$&$+6.8379\\times 10^{1}$&$-6.3180\\times 10^{3}$&$+1.9945\\times 10^{1}$&$+2.2043$\\\\\n\t\t$\\mu^S$&$+3.1246$&$-1.5439\\times 10^{3}$&$+7.6924$&$+6.5057\\times 10^{-2}$\\\\\n\t\t$\\phi_0$&$-1.6577\\times 10^{-2}$&$-9.6335$&$+1.6478\\times 10^{-2}$&$-2.8298\\times 10^{-4}$\\\\\n\t\t$\\mu^\\text{IP}$&$-1.6111\\times 10^{1}$&$-2.7006\\times 10^{3}$&$+5.6310$&$-1.3288\\times 10^{-1}$\\\\\n\t\t$v_\\text{min}$&$+6.3210\\times 10^{-1}$&$+1.5715\\times 10^{2}$&$-5.2289\\times 10^{-1}$&$+1.2388\\times 10^{-2}$\\\\\n\t\t$v_\\text{max}$&$+5.2297\\times 10^{-1}$&$+1.6010\\times 10^{2}$&$-3.9309\\times 10^{-1}$&$+1.7660\\times 10^{-2}$\\\\\n\t\t$\\lambda_\\text{inext}$&$-1.9399\\times 10^{-2}$&$-4.5358$&$+8.7613\\times 10^{-3}$&$+9.6935\\times 10^{-6}$\\\\\n\t\t$\\mu^P $&$+5.1689$&$+1.5652\\times 10^{3}$&$-3.0199$&$+1.4201\\times 10^{-1}$\\\\\n\t\t$\\lambda_\\text{tether}$&$-2.3896\\times 10^{2}$&$-9.5477\\times 10^{4}$&$+9.2927\\times 10^{1}$&$+1.1029\\times 10^{1}$\\\\\n\t\t$A_0$&$+1.7537\\times 10^{-2}$&$+2.1346$&$+1.5987\\times 10^{-4}$&$+3.1229\\times 10^{-4}$\\\\\t\t\n\n\t\n\t\\end{tabular}\n\t\\caption{Sensitivity of the FBLM-FEM with respect to the parameters \\eqref{eq:sens.set} in Experiment \\ref{exp:sensit}. Larger absolute values imply stronger influence of the corresponding parameters (rows) to the different ``norms''(columns). See Section \\ref{sec:sensit} for the computation and discussion of these results.}\n\t\\label{tbl:sensit}\n\\end{table}    \n\n\n\\section{Discussion}\n\nThe aim of this work was twofold: to investigate two fundamental numerical properties (stability and convergence) of the FEM solving the FBLM, and to embed the FBLM in a complex and adaptive extracellular environment and study its sensitivity to several of its controlling parameters. \\change{In some more detail:}{}\n\nWe have showed in Section \\ref{sec:stbility} that the FEM exhibits the expected timestep stability behaviour \\eqref{eq:CFL} in both chemotaxis and haptotaxis experiments. We have verified this assertion in a particular chemotaxis experiment where we have identified the stability constant $C$ and have proposed an automated way of computing proper timesteps for the method. The technique proposed is inspired by the well known ATC methods used in many cases of scientific computing, and is based on the smoothness of the filament polymerization rate and the filament density functions. \n\nWe have proceeded further and identified with a series of haptotaxis experiments the dependence between the timestep stability and the gradient of the ECM. We have seen that this relation \\eqref{eq:C.hapt} is linear and we expect it to hold globally, although with different coefficients.\n\nWe have exhibited in Section \\ref{sec:converge} the convergence of the FEM as the discretization grid is refined with respect to $\\Delta \\alpha$ and\/or $\\Delta s$. As is commonly done in complex models\/methods, we have considered characteristic\/representative experimental cases. Since the exact solutions are unknown we deduce the convergence by comparison against the numerical solutions of the finest discretization grid. The comparisons themselves are conducted in terms of particular ``norms'' that we define in  \\eqref{eq:norm.invad}--\\eqref{eq:norm.elong}.\n\nIn Section \\ref{sec:env} we have dealt with the embedding of the FBLM-FEM in a complex and adaptive extracellular environment. The model for the environment that we propose in \\eqref{eq:env} is an reaction-diffusion system of the densities of the involved quantities. The proposed model is relatively simple and is used here mostly to exhibit the coupling between the FBLM and the environment. \n\nWe close this work in Section \\ref{sec:sensit} with a first study of the sensitivity of the FBLM-FEM on several of its controlling parameters. This sensitivity analysis is performed around predefined parameter values, most of which are biologically relevant and that have been previously proposed in the literature. We identify the most significant parameters and get an insight on the magnitude of the effect of the different model components.\n\nAlthough the current work is based on particular experimental cases and is not escorted by rigorous numerical (or other type of) analysis, the benefit is twofold: \n\nOn the one hand, the combination FBLM-FEM can be further used to model and simulate biologically relevant experimental situations. We know now that the refinement of the discretization grid, augmented by the automated adaptation of the timestep, provides with a stable and converging method that will reveal the inherent dynamics of the model. \n\nOn the other hand, having verified that the behaviour of the FBLM-FEM is the expected one in terms of convergence and stability, the rigorous numerical analysis is well warranted. It is expected that the remarks of this work will serve also as a guide in this effort.\n\nMoreover, the coupling of the FBLM with the extracellular environment  that we propose here will serve as springboard for the further modelling and simulation of more biologically relevant settings and \\textit{in-vitro} experiments. It can easily be extended to include e.g. the description of the ECM as a fibrous component of the environment, more than one chemical ingredients (attractants or repellents), as well as more than one cells interacting with each other. \n\nThe sensitivity analysis results have served here for the deeper understanding of the effect of different terms on the model. The insight we have gained will serve in the further refinement of the FBLM-FEM and the parameter estimation procedures, when reproducing and simulating realistic experimental scenarios.\n\n\n\\begin{table}[t]\n\t\\footnotesize\n\t\\centering\n\t\\begin{tabular}{ c p{0.29\\linewidth} |p{0.24\\linewidth}| p{0.27\\linewidth}}\n\n\t\tsymb.&\\hfill~description\\hfill~&\\hfill~value\\hfill~&\\hfill~comment\\hfill~\\\\ \n\t\t\\hline\n\t\t& & & \\\\\n\t\t$\\mu^B$&bending elasticity& $\\rm 0.07\\,pN\\,\\upmu m^2$ &\\cite{Gittes1993}  \\\\\t\t\n\t\t$\\mu^A$&adhesion& $\\rm 0.4101\\,pN\\, min\\, \\upmu m^{-2}$& \\cite{Li2003,Oberhauser2002} \\& \\cite{Oelz2008,Oelz2010a,Schmeiser2010}\\\\\n\t\t$\\mu^T$&cross-link twisting& $\\rm 7.1\\times 10^{-3}\\, \\upmu m$ &\\\\\n\t\t$\\mu^S$&cross-link stretching& $\\rm 7.1 \\times 10^{-3}\\, pN\\, min\\, \\upmu m^{-1} $& \\\\\n\t\t$\\phi_0$&crosslinker equil. angle& $70^o$ & \\cite{Schmeiser2010}\\\\\n\t\t$\\mu^\\text{IP}$&actin-myosin strength&$\\rm 0.1\\, pN\\, \\upmu m^{-2}$& \\\\\n\t\t$v_\\text{min}$&minimal polymerization&$\\rm 1.5\\,\\upmu m\\, min^{-1}$&in biological range\\\\\n\t\t$v_\\text{max}$&maximal polymerization&$\\rm 8\\,\\upmu m\\, min^{-1}$&in biological range\\\\\n\t\t$\\mu^P $ &pressure constant & $\\rm 0.05\\, pN\\, \\upmu m $& \\\\\n\t\t$A_0$ & equilibrium inner area & $\\rm 450\\, \\upmu m^2$&\\cite{Verkhovsky1999,Small1978}\\\\\n\t\t\\hline\n\t\t$\\lambda_\\text{inext}$&inextensibility&$20$&\\\\\n\t\t$\\lambda_\\text{tether}$&membrane tethering&$1\\times 10^{-3}$&\\\\\n\t\n\t\\end{tabular}\n\t\\caption{Basic set of parameter values used in the numerical simulations of the FBLM. Variants of these parameters are discussed in each experiment separately. These parameters (except for the $\\lambda_\\text{inext}$ and $\\lambda_\\text{tether}$) have been adopted from \\cite{MOSS-numeric}.}\n\t\\label{tbl:parameters}      \n\\end{table}\n \n\\begin{table}[t]\n\t\\footnotesize\n\t\\centering\n\t\\begin{tabular}{c p{12em} |p{8em}|c|c}\n\t\tsymb.&\\hfill~description\\hfill~ &\\hfill~Experiment \\ref{exp:embedding}\\hfill~& Experiment \\ref{exp:softstiff}  & Experiment \\ref{exp:sensit}\\\\ \n\t\t&&\\hfill~and Figure \\ref{fig:embedding}\\hfill~&and Figure \\ref{fig:softstiff}&and Figure \\ref{fig:sensit}\\\\\n\t\t\\hline\n\t\t& & & \\\\[-0.7em]\n\t\t$D_c$  &diffusion of the chemical& $2\\times10^3\\, {\\rm cm^2min^{-1}}$ & $2\\times10^3$& $5\\times10^3$\\\\          \n\t\t$D_m$ &diffusion of the MMPs& $1\\, {\\rm cm^2min^{-1}}$ & $2\\times10^3$ & $5\\times10^3$\\\\\n\t\t$\\alpha_1$ &production rate of chemical& $10^2\\, {\\rm mol\\,min^{-1}}$ & $10^2$ & $4\\times10^2$\\\\     \n\t\t$\\beta$ & production of MMPs& $10^{-1}\\, {\\rm mol\\,min^{-1}}$ & $0$ & $0$\\\\     \n\t\t$\\gamma_1$ &decay of the chemical& $10\\, {\\rm mol\\,min^{-1}}$ & $10$& $7\\times 10^1$\\\\      \n\t\t$\\gamma_2$ &decay of the MMPs& $10\\, {\\rm mol\\,min^{-1}}$& $0$& $0$\\\\      \n\t\t$\\delta_1$ &degr. chemical by the cell& $0$ & $0$ & $0$\\\\     \n\t\t$\\delta_2$ &degr. of the ECM by the MMPs&$2\\,{\\rm cm^2mol^{-1}min^{-1}}$& $5\\times10^{-1}$& $0$   \n\t\\end{tabular}\n\t\n\t\\caption{Parameter sets used for the simulation of the environment \\eqref{eq:env} in the Experiments \\ref{exp:embedding}, \\ref{exp:softstiff}, \\ref{exp:sensit}, see also Figures \\ref{fig:embedding}, \\ref{fig:softstiff}, \\ref{fig:sensit}. There is no biological justification of these values.}\n\t\\label{tbl:env.parameters}      \n\\end{table}\n \n\\section*{Acknowledgement}\nThe authors would like to thank Christian Schmeiser, Anna Marciniak-Czochra, and Mark Chaplain for the fruitful discussions and suggestions during the preparation of this manuscript.\n\n\n\t\\bibliographystyle{unsrtnat\n{\t\\footnotesize\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn this paper we\nshow the following\n\\begin{thm}\\label{thm1.1}\nLet $C$ be an $n$-dimensional complex torus embedded in a complex manifold $M$ of dimensional $n+d$.  Assume that $T_CM$, the restriction of $TM$ on $C$,   splits as $TC\\oplus N_C$. Suppose that the normal bundle of $C$ in $M$ admits transition functions that are\n Hermitian matrices and satisfy a {\\it non-resonant Diophantine} condition $($see Definition~$\\ref{dioph})$. Then a neighborhood of  $C$ in $M$ is biholomorphic to a neighborhood of   the zero section in the normal bundle.\n\\end{thm}\n\nWe first describe the organization of the proof of our main theorem.\n\nA complex torus $C$ can be identified with the quotient of ${\\mathbb C}^n$ by a lattice $\\Lambda$ spanned by the standard unit vectors $e_1,\\dots, e_n$ in ${\\mathbb C}^n$ and $n$ additional vectors $e'_1,\\dots, e'_n$ in ${\\mathbb C}^n$, where $\\operatorname{Im} e'_1,\\dots, \\operatorname{Im} e'_n$ are linearly independent vectors in ${\\mathbb R}^n$. Let $\\Lambda'$ be the lattice in the cylinder ${\\mathcal C}:={\\mathbb R}^n\/{\\mathbb Z}^n+i{\\mathbb R}^n$ spanned by $  e_1',\\dots,   e'_n\\mod {\\mathbb Z}^n$.   There are two coverings for the  torus $C={\\mathbb C}^n\/\\Lambda={\\mathcal C}\/{\\Lambda'}$: the universal covering $\\pi\\colon{\\mathbb C}^n\\to C$ and the covering by cylinder, $\\pi_{\\mathcal C}\\colon{\\mathcal C}\\to C$  that extends to a covering $\\mathcal M$ over $M$.  In section two we recall some facts about factors of automorphy  for   vector bundles on $C$ via the covering by ${\\mathbb C}^n$. In section three, we study the flat vector bundles on $C$.    The pull back of the flat vector bundle $N_C$  to  the cylinder ${\\mathcal C}$ is the normal bundle $N_{{\\mathcal C}}$ of ${\\mathcal C}$ in $\\mathcal M$. We show that $N_{{\\mathcal C}}$ is always the holomorphically trivial vector bundle ${\\mathcal C}\\times{\\mathbb C}^d$. By ``vertical coordinates\", we mean ``coordinates on ${\\mathbb C}^d$\", the normal component of the normal bundle $N_C$, while ``horizontal coordinates\" mean the tangential components of  $N_C$.\n\nSince ${\\mathcal C}$ is a Stein manifold, a theorem of   Siu \\cite{siu-stein} says that  a neighborhood of ${\\mathcal C}$ in $\\mathcal M$ is biholomorphic to a neighborhood of the zero section in its normal bundle, which is trivial as mentioned above.\nWe show that the holomorphic classification of neighborhoods $M$ of $C$ with flat $N_C$ is equivalent to\nthe holomorphic classification of the family of the deck transformations of coverings $\\mathcal M$ of $M$ in a neighborhood of $\\mathcal C$. These deck transformations are\n``higher-order\" (in the vertical coordinates) perturbations $\\tau_1,\\dots, \\tau_n$ of $\\hat\\tau_1,\\dots, \\hat\\tau_n$, where the latter are the deck transformations of the covering of $\\widetilde N_C$ over $N_C$. In order to find a biholomorphism between a neighborhood of $C$ in $M$ and a neighborhood of its\n  zero\n section in $N_C$, it is sufficient to find a biholomorphism that conjugates $\\{\\tau_1,\\dots, \\tau_n\\}$ to $\\{\\hat\\tau_1,\\dots, \\hat\\tau_n\\}$.\n\nThere are two useful features. First, since the fundamental group of $C$ is abelian, the deck transformations $\\tau_1,\\dots,\\tau_n$ commute pairwise. Second, we can also introduce suitable coordinates on ${\\mathcal C}$ so that the \"horizontal\" components of deck transformations have diagonal linear parts. In such a way the classification of neighborhoods of $C$ is reduced to a more attainable  classification of deck transformations. While the full theory for this classification is out the scope of this paper,  we study the case when $N_C$ admits  Hermitian transition functions. Since a Hermitian transition matrix must be locally constant, we call such an $N_C$ {\\it Hermitian flat}.  The convergence proof for \\rt{thm1.1} is given in section four.\n It relies on a Newton rapid convergence scheme adapted to our situation  based on an appropriate Diophantine condition among the lattice and the normal bundle.\n At step $k$ of the iteration scheme, let  $\\delta_k$ be the error of the deck transformations  $\\{\\tau_1^{(k)},\\dots, \\tau_n^{(k)}$\\} defined on domain $D^{(k)}$ to $\\hat\\tau_1,\\dots, \\hat\\tau_n$ in suitable norms. By an appropriate transformation $\\Phi^{(k)}$, we conjugate to a new set of deck transformations $\\{\\tau_1^{(k+1)},\\dots, \\tau_n^{(k+1)}\\}$ of which the error to the linear ones is now $\\delta_{k+1}$ on a slightly smaller domain $D^{(k+1)}$. Using our Diophantine conditions, related to the lattice $\\Lambda$ and the normal bundle, we show that the sequence $\\Phi^{(k)}\\circ \\cdots \\circ \\Phi^{(1)}$ converges to a holomorphic transformation $\\Phi$ on an open domain $D^{(\\infty)}$ where we linearize $\\{\\tau_1,\\dots,\\tau_n\\}$.\n\nWe now describe\nclosely related previous results.\nOur work is motivated by work of Arnol'd and Ilyashnko-Pyartli. Our main theorem was proved in \\cite{arnold-embed} when $C$ is an elliptic curve ($n=1$) and $N_C$ has rank one ($d=1$). Il'yashenko-Pyartli~\\cite{ilyashenko-pyartly-embed} extended Arnol'd's result to the case when the torus is the product of elliptic curves  together with a normal bundle which is a direct sum of line bundles, while \\rt{thm1.1} deals with general complex tori. We also assume that $N_C$ is {\\it non-resonant}, a condition that is weaker\n than the non-resonant condition used\nby Il'yashenko-Pyartli. Our small-divisor condition is also weaker. Of course, the study of neighborhood of embedded compact complex manifolds has a long history. Also, see some recent work \\cite{hwang-annals,koike-fourier,loray-moscou}.  We refer to \\cite{MR4392029} for some references and a different approach to this range of questions.\n\n{\\bf Acknowledgments.} This work benefits from helpful discussions with Jean-Pierre Demailly. Part of work was finished when X.~G. was supported by CNRS and  UCA for a visiting position at UCA.\n\n\\setcounter{thm}{0}\\setcounter{equation}{0}\n\t\\section{Vector bundles on Tori and factors of automorphy}\nIn this section, we identify vector bundles on a complex torus with factors of automorphy. The latter gives us a useful alternative definition of vector bundles on a higher dimensional complex torus $C$ and the isomorphisms of two vector bundles. \tGeneral references for line bundle on complex tori are \\cite{birkenhake-lange,debarre-book,iena-05} and \\cite[p.~307]{GH-book}.\n\t\n   Let $\\Lambda$ be a $2n$-dimensional lattice in $\\mathbb{C}^n$.\n  We may assume that  $\\Lambda$ is defined by $2n$  vectors $e_1,\\dots, e_n, e'_1,\\dots,e'_n$  of $\\mathbb{C}^n$, where $e_i=(0,\\ldots, 0,1,0,\\ldots, 0)$ with $1$ being at the $i$-th place, $e'_i=(e'_{i,1},\\ldots,e'_{i,n})$ and the matrix\n  $$\\operatorname{Im} \\tau:=(\\operatorname{Im}{e'_{i,j}})_{1\\leq i,j\\leq n}=:({e''_{i,j}})_{1\\leq i,j\\leq n}$$ is invertible \\cite[exerc. 2, p.~21]{birkenhake-lange}.\n  The compact complex manifold $C:=\\mathbb{C}^n\/\\Lambda$ is called an ($n$-dimensional) complex torus. Unless the lattice is equivalent to another one defined by a diagonal matrix $e'$, $C$ is not biholomorphic to a product of one-dimensional tori.\n  Let $\\pi: {\\mathbb C}^n\\rightarrow C$ be the universal cover of $C$. Its group $\\Gamma$ of deck transformations consists of translations\n  $$\n T_\\lambda\\colon z\\to z+\\lambda, \\quad \\lambda\\in\\Lambda.\n  $$\n  Note that $\\Gamma$ is abelian and is isomorphic to ${\\mathbb Z}^{2n}$. $\\Gamma$ is also isomorphic to $\\pi_1(C,0)$ since $\\mathbb C^n$ is a universal covering of $C$.\n\n Next, we consider equivalence relations for holomorphic vectors bundles on $C$ and $\\mathbb C^n$, following the realization proof of Theorem 3.2 in \\cite{iena-05}.\n Let $E$ be a vector bundle of rank $d$ over $C$. The pull-back bundle $\\pi^*E$ on $\\mathbb C^n$ is  trivial and  has global coordinates $\\hat\\xi$. Let $\\{U_j\\}$ be an open covering of $C$ so that coordinates $\\xi_j=(\\xi_{j,1},\\dots, \\xi_{j,d})^t$ of $E$ are well-defined (injective) on $U_j$. Then we have\n \\eq{hxixi}\n \\hat\\xi=h_j\\xi_j(\\pi),\n \\end{equation}\n  where $h_j$ is a non-singular holomorphic matrix on $\\pi^{-1}(U_j)$. The transition functions $g_{kj}$ satisfy\n \\eq{gkjh}\n g_{kj}(\\pi)=h_k^{-1}h_j, \\quad \\text{on $\\pi^{-1}(U_k)\\cap\\pi^{-1}(U_j)$}.\n \\end{equation}\n For any $z,\\lambda$, we know that both $\\pi(z+\\lambda)$ and  $\\pi(z)$ are in the same $U_j$  for some $j$.  Then we have\n$$\n\\hat\\xi(z+\\lambda)=h_j(z+\\lambda)\\xi_j(\\pi(z))=h_j(z+\\lambda)h_j(z)^{-1}\\hat\\xi(z).\n$$\nWe can define\n\\eq{rholaz}\n\\rho(\\lambda,z)=h_j(z+\\lambda)h_j(z)^{-1}, \\quad z\\in\\pi^{-1}(U_j)\n\\end{equation}\nas the latter is independent of the choice of $j$\nby\n\\re{gkjh}.\nTherefore, \n\n \\begin{gather}\\label{global-coord}\n \\hat\\xi(\\lambda+z)=\\rho(\\lambda,z)\\hat\\xi(z), \\quad \\rho(\\lambda,z)\\in GL(d,\\mathbb C),\\\\\n \\rho\\colon\\Lambda\\times \\mathbb C^n\\to GL(d,\\mathbb C).\n \\end{gather}\nHere $\\rho$ is called a  {\\it factor of automorphy}.\nWe can verify that\n\\eq{abelP}\n\\rho(\\lambda+\\mu,z)=\\rho(\\lambda,\\mu+z)\\rho(\\mu,z).\n\\end{equation}\nIn particular, if all $\\rho(\\lambda,z)=\\rho(\\lambda)$ are independent of $z$, then\n $\\rho(\\Lambda)$ is an abelian group.\n\n\nThe above construction from a vector bundle $E$ on $C$ to a factor of automorphy can be reversed. Namely, given\n\\re{global-coord}-\\re{abelP},\ndefine the vector bundle $E$\non $C$\nas the quotient vector space of ${\\mathbb C}^n\\times{\\mathbb C}^d$ via the equivalence relation\n\\eq{eqrel}\n(z,\\xi)\\sim (z+\\lambda,\\rho(\\lambda,z)\\xi), \\quad z\\in{\\mathbb C}^n,\\ \\xi\\in{\\mathbb C}^d,\\ \\lambda\\in\\Lambda.\n\\end{equation}\nWe denote the projection from the cylinder ${\\mathcal C}:={\\mathbb R}^n\/{\\mathbb Z}^n+i{\\mathbb R}^n={\\mathbb C}^n\/{\\mathbb Z}^n$ onto $C$ by $\\pi_{{\\mathcal C}}$.\nTherefore, we can define $\\pi_{\\mathcal C}^*E$ on the cylinder ${\\mathcal C}$ by the equivalence relation\n\\eq{eqrelS}\n(z,\\xi)\\sim (z+\\lambda,\\rho(\\lambda,z)\\xi), \\quad z\\in{\\mathbb C}^n,\\ \\xi\\in{\\mathbb C}^d,\\ \\lambda\\in{\\mathbb Z}^n.\n\\end{equation}\n\n\nOf course,  global coordinates $\\hat\\xi$, $\\rho$, and\n$g_{jk}$ are not uniquely determined by $E$. However, their equivalence classes are determined.  Two vector bundles $E,\\tilde E$ are isomorphic if their corresponding transitions $g_{jk},\\tilde g_{jk}$ satisfy $\\tilde g_{jk}=h_j^{-1}g_{jk}h_k$ where $h_j$ are non-singular holomorphic matrices.   Replacing global coordinates $\\hat\\xi$ by $\\nu\\hat\\xi$ where $\\nu\\colon\\mathbb C^n\\to GL(n,d)$ is holomorphic, we can verify that\n\\eq{nuLaz}\n\\nu(\\lambda+z)\\rho(\\lambda, z)\\nu(z)^{-1}=:\\tilde\\rho(\\lambda,z)\n\\end{equation}\nis also a factor of automorphy.   Define two factors of automorphy  $\\rho,\\tilde\\rho$ to be {\\it equivalent} if \\re{nuLaz} holds. Therefore,  the classification of holomorphic vector bundles is identified with the classification of factors of automorphy.\n\n\\setcounter{thm}{0}\\setcounter{equation}{0}\n\\section{Flat vector bundles}\nIn this section, we will show that the pull-back of a flat vector bundle $E$ on $C$ to the cylinder  ${\\mathcal C}={\\mathbb C}^n\/{\\mathbb Z}^n$ is always trivial.\n\n When  $E$ is flat, we can choose global coordinates as follows. We know that  $\\pi^*E$ is also flat and   we can choose its global flat basis, or global flat coordinates  $\\hat\\xi$ by using analytic continuation on ${\\mathbb C}^n$ and pulling back flat local coordinates of $E$. In other words,  in \\re{hxixi} $h_j$ are locally constants while $\\xi_j$ are locally flat coordinates.  Then $\\rho(\\lambda,z)$ depend only on $\\lambda$, in which case we write $\\rho(\\lambda)$ for $\\rho(\\lambda,z)$. As remarked above, $\\rho(\\Lambda)$ is abelian.\nWhen $E$ is unitary (flat), by the same reasoning  $\\pi^*E$ is unitary and we can choose\n$h_j$ and $\\rho(\\lambda)$ to be unitary.\n\n\n\n\n\n\n  A $d\\times d$ Jordan block $J_d(\\lambda)$  is a matrix of the form $\\lambda I_d+N_d$, where $I_d$ is the $d\\times d$ identity matrix and $N_d$ is the $d\\times d$ matrix with all entries being $0$, except all\n the $(i,i+1)$-th entries being $1$. A  matrix $T$  commutes with $J$ if and only if\n $$\nT=T_d(a):=a_0I_d+\\sum_{i>0} a_i N_d^i.\n $$\n Note that $N_d^i=0$ for $i\\geq d$. Following \\cite[p.~218]{gantmacher1},\n we call the above $T$ as well as the  following two types of matrices, {\\it regular upper triangular} matrices  w.r.t. $J$:\n   $$\n A=(0,T_d(a)), \\quad\\text{or} \\quad B=\\binom{T_{d'}(a')}{0},\n $$\n  where $0$ denotes in $A$ (resp. $B$) a $0$ matrix of $d$ rows (resp. $d'$ columns).\n\n Given a Jordan matrix\n $$\n \\tilde J=\\operatorname{diag}(J_{d_1}(\\lambda_1),\\dots, J_{d_k}(\\lambda_k)). $$\n the matrices that commute with $\\tilde J$ are precisely the block matrices\n $$\nX= (X_{\\alpha\\beta})_{m\\times m}\n $$\n where $X_{\\alpha\\beta}=0$ if $\\lambda_\\alpha\\neq\\lambda_\\beta$, while $X_{\\alpha\\beta}$ is a regular upper triangular $(d_\\alpha\\times d_\\beta)$ matrix if $\\lambda_\\alpha=\\lambda_\\beta$.\n Such a matrix $X$ is said to be a {\\it regular upper triangular matrix }\n w.r.t. $J$\n\n\nFrom the structure of matrices commuting with a Jordan matrix, we can verify the following two results.\n\\begin{prop}Let $A_1,\\dots, A_m$ be $2\\times 2$ matrices commuting pairwise. Then there is a non-singular matrix $S$ such that all $S^{-1}A_jS$\nare Jordan matrices.\n\\end{prop}\n\\begin{exmp} The $3\\times 3$ matrices $\\lambda I_3+N_3,\n\\mu I_3+N_3^2$ commute,  but they cannot be transformed into the Jordan normal forms simultaneously.\n\\end{exmp}\nThe following results on logarithms are likely classical. However, we cannot find a reference. Therefore, we give  proofs emphasizing  commutativity of logarithms  of matrices.\n\nWe start with the following.\n\\begin{lemma}\\label{up-t}\nLet $A_1,\\dots, A_m$ be pairwise commuting matrices. Then there is a non-singular matrix $S$ such that\n$$\nS^{-1}A_jS=(\\hat A_{\\alpha\\beta})_{1\\leq\\alpha,\\beta\\leq s}=:\\hat A_j, \\quad 1\\leq j\\leq m\n$$\nwhere  $\\hat A_1$ is a Jordan matrix and all $\\hat A_j$ are\nupper triangular matrices.\n\\end{lemma}\n \\begin{proof} We may assume that  $A_1$ is a Jordan matrix $J=\\operatorname{diag}(J_{d_1}(\\lambda_1),\\dots, J_{d_s}(\\lambda_s))$. Then\n $$\n A_j=(X^j_{\\alpha\\beta})_{1\\leq\\alpha,\\beta\\leq s}\n $$\n are regular w.r.t $J$. Note that pairwise commuting non-singular matrices have  non-trivial common eigenspaces.  The eigenspace of $J$ are spanned $e_{d_1'}, \\dots, e_{d'_s}$ with $d'_1=1$ and $d_j'=d_1+\\cdots d_{j-1}+1$.\n To simplify the indices, we may assume that $e_1$ is an eigenvector of all $A_j$. Then the first column of $A_j$ is $a_je_1$ with $a_j\\neq0$.  The new matrices $\\tilde A_j$,  obtained by removing all first rows and first columns,\n  still commute pairwise.  In particular all $\\tilde A_j$ are regular to the new Jordan matrix $\\tilde J=\\tilde A_1$. By induction on $d$,   we can find  a non-singular matrix  $\\tilde S$ which is regular to $\\tilde J$ so that all $\\tilde S^{-1}\\tilde A_j\\tilde S$ are\nupper-triangular.  Let $S=\\operatorname{diag}(1,\\tilde S)$. Now  $S^{-1}=\\operatorname{diag}(1,\\tilde S^{-1})$. We can check that\nall $\\hat A_j:=S^{-1}A_jS$ are upper triangular. Then $ \\hat A_1$, $J$ have the same entries, with only one possible exception\n$$\\hat A_{1;12}=s_{11}J_{12}.\n$$\nIf $\\hat A_{1;12}\\neq 0$, dilating the first coordinate can transform $\\hat A_{1}$ into the original $J$, while  $\\hat A_j$ remain upper-triangular. If $s_{12}J_{12}=0$, then $J_{12}$ must be $0$, i.e. $\\hat A_1=J$,  because one cannot transform a Jordan matrix, $A_1=J$, into a new Jordan matrix,\n$\\hat A_1$, by reducing  an entry $1$ to $0$ and keeping other entries unchanged.\n \\end{proof}\n The above simultaneous normalization of  upper-triangular matrices allows us to define the logarithms. The construction of logarithms of non-singular matrices can be found in \\cite[p.~239]{gantmacher1}. Here we need to find a definition that is suitable to determine the commutativity of the logarithm of pairwise commuting non-singular matrices.\n\nRecall that for a  $d\\times d$ matrix $A$, the generalized eigenspace $E_\\lambda (A)$ with eigenvalue $\\lambda$ is the kernel of $(A-\\lambda I)^d$, while ${\\mathbb C}^d$\nis the direct sum of all $E_\\lambda (A)$. A\nmatrix $B$\nthat commutes with $A$ leaves each $E_\\lambda (A)$ invariant, i.e. $B(E_{\\lambda}(A))\\subset E_{\\lambda}(A)$. Thus if $A_1,\\dots, A_m$ commute pairwise, we can decompose ${\\mathbb C}^d$ as a direct sum of liner subspaces $V_j$ such that each $V_j$ is invariant by $A_i$ and admits exactly one eigenvalue of $A_i$. Thus to define $\\ln A_j$, we will assume that each $A_j$ has a single eigenvalue on ${\\mathbb C}^d$ if we wish.\n\n Given a non-singular matrix $A$, a logarithm of $A$ is a matrix $\\ln A$  satisfying\n $$\n e^{\\ln A}=A\n $$\n where the exponential matrix $e^B=\\sum\\frac{B^n}{n!}$ is always well-defined.  However,  $\\ln A$\nis not  unique.\n\n\n  For a non-singular upper triangular matrix\n \\begin{equation}\\label{defLn0}\n A=\\lambda I_d+ a, \\quad a:=(a_{ij})_{1\\leq i,j\\leq d}, a_{ij=0}\\quad  \\forall i\\geq j,\n \\end{equation}\n we have $a^k=0$ for $k\\geq d$.\n  Using the identity $e^{B+C}=e^{B}e^C$ for two commuting matrices $B,C$, we  see that $e^{\\ln A}=A$ for\n \\begin{equation}\\label{defLn0+}\n \\ln A:=(\\ln \\lambda)I_d-\\sum_{k>0}\\f{(-\\lambda^{-1} a)^k}{k}\n \\end{equation}\nwith $0\\leq \\operatorname{Im}  \\ln(.) <2\\pi$.\nFor a non-singular Jordan matrix, we can define\n$$\n\\ln\\operatorname{diag}(J_{d_1}(\\lambda_1),\\dots, J_{d_m}(\\lambda_m))=\\operatorname{diag}(\\ln J_{d_1}(\\lambda_1),\\dots, \\ln J_{d_m}(\\lambda_m)).$$\n Note that $\\ln\\lambda_\\alpha=\\ln\\lambda_\\beta$ if and only if $\\lambda_\\alpha=\\lambda_\\beta$.\n Since matrices that commute with a fixed matrix is closed under multiplication by a scalar, addition and multiplication.\n It is thus clear that if $A$ is an upper triangular matrix that is regular to a non-singular Jordan matrix $J=\\operatorname{diag}(J_{d_1}(\\lambda_1),\\dots, J_{d_s}(\\lambda_s))$, then $\\ln A$ remains regular to $J$. Equivalently and more importantly,  $\\ln A$ is regular to the Jordan normal form of $\\ln J$, which is\n $$\n \\operatorname{diag}(J_{d_1}(\\ln\\lambda_1),\\dots, J_{d_s}(\\ln\\lambda_s)).\n $$\n\n\n \\begin{prop}\\label{defLnA}\n Let $A_1,\\dots, A_m$ be pairwise commuting\n $d\\times d$ matrices. Then there is a non-singular matrix $S$ such that\n $\\hat A_j:=S^{-1}A_jS$ are block diagonal matrices\n  of the form \\begin{equation}\\label{hataj}\n \\hat A_j=\\operatorname{diag}(\\hat A_{j,d_1},\\dots, \\hat A_{j,d_k})\n \\end{equation}\n  where all $\\hat A_{j,d_i}$\n  are   upper triangular $d_i\\times d_i$ matrices, and each $\\hat A_{j,d_i}$ has only one eigenvalue $\\lambda_{j,i}$. Assume further that all $A_j$ are non-singular. Then\n  \\begin{equation}\\label{defLn1}\n  \\ln A_j:=S\\operatorname{diag}(\\ln \\hat A_{j,d_1}, \\dots, \\ln \\hat A_{j,d_k})S^{-1}, \\quad 1\\leq j\\leq m\n  \\end{equation}\n  commute pairwise and $e^{\\ln A_j}=A_j$, where $\\ln\\hat A_{j,d_i}$ are defined by  \\rea{defLn0}-\\rea{defLn0+}.\n \\end{prop}\n \\begin{proof}Note that for pairwise commuting   matrices $A_1,\\dots, A_m$, we have a decomposition\n $\n  {\\mathbb C}^d=\\bigoplus_{i=1}^tV_i\n $\n where each $A_j$ preserves $V_i$ and\n has only one eigenvalue $\\lambda_{j,i}$. Their restrictions of $A_1,\\dots, A_m$\n  on $V_i$ remain commutative pairwise.\n Let $\\dim V_i=d_i, d_0'=0, d_{i+1}'-d_i'=d_i$.\n By \\rl{up-t}, we can find a basis\n $$e^*_{d'_{i-1}+1}, \\dots, e^*_{d_{i-1}'+d_{i}}$$\n for $V_i$ such that $A_1|_{V_i}, \\dots, A_m|_{V_i}$ are upper triangular matrices. Using the new basis $e_1^*,\\dots, e_d^*$ we can find the matrix $S$ for the decomposition \\re{hataj}. Therefore,\n  $ \\hat A_{1, d_i},\\dots, \\hat A_{m,d_i}$\n  commute pairwise.\n\nAssume now that all $\\lambda_{j,i}$ are non-zero.\n It remains to show that $\\ln \\hat A_{1, d_i},\\dots, \\ln \\hat A_{m,d_i}$ commute pairwise.\n Write\n $$\n \\hat A_{j,i}=\\lambda_{j,i}I_{d_i}+W_{j,i}\n $$\n where $W_{j,i}$ are upper triangular  matrices and  $W_{j,i}^{d_i}=0$.\n By a straightforward computation, we have\n $$\n [W_{j,i}, W_{j',i}]=[\\hat A_{j,i},\\hat A_{j',i}]=0.\n $$\n Therefore, $W_{j,i}^k$ commutes with $W_{j',i}^\\ell$ for any $k,\\ell$. Consequently,  the finite sum\n $$\n \\ln \\hat A_{j,i}=\\ln\\hat\\lambda_{j,i}I_{d_i}-\\sum_{k>0}\\frac{(-\\lambda_{j,i}^{-1}W_{j,i})^k}{k}\n $$\n commutes with $\\ln\\hat A_{j',i}$. Therefore,  $\\ln A_1,\\dots, \\ln A_m$ in \\re{defLn1}\n commute pairwise.\n  \\end{proof}\n\n\n    \\begin{lemma}\\label{muit-flows}\n Let $A_1,\\dots, A_n$ be   non-singular  upper triangular   $d\\times d$ matrices.\n Suppose that $A_1,\\dots, A_n$ commute pairwise.\n There exists a linear mapping $w\\to\\tilde v^z(w):=v(z)w$ in  ${\\mathbb C}^d$,\n\n  entire in $z\\in{\\mathbb C}^n$  such that\n $v(z)\\in GL_d({\\mathbb C})$, $v(0)=Id$ and $v(e_j)=A_j$ for $j=1,\\ldots,n$.  Furthermore,  $v({z+z'})=v(z) v({z'})$ for all $z,z'\\in{\\mathbb C}^n$\n \\end{lemma}\n \\proof{By \\rp{defLnA}, we  define pairwise commuting matrices $\\ln A_1,\\dots, \\ln A_n$ such that $e^{\\ln A_j}=A_j$.\n  Then the Lie brackets of the vector fields for $\\dot w=\\ln A_jw, j=1,\\dots,n$ vanish and their flows $\\varphi_j^t(w)$  commute pairwise.\n Note that\n $$\n \\varphi_j^0(w)=w, \\quad\n \\varphi_j^1(w)=e^{\\ln A_j}w=A_jw.\n $$\n Define\n $$\n \\tilde v^z(w)=\\varphi_1^{z_1}\\cdots\\varphi_n^{z_n}(w).\n $$\n  We conclude $\\tilde v^{z}\\tilde v^{z'}(w)=\\tilde v^{z+z'}(w)$, that is $v(z+z')=v(z)v({z'})$.\n\\qedhere }\n\n\nBy \\rp{defLnA}, we define $\\ln\\rho(e_1), \\dots, \\ln\\rho(e_{2n})$ and they commute pairwise.\nWe now define $\\ln\\rho(\\lambda)$ for all $\\lambda=\\sum_{j=1}^{2n}m_je_j\\in \\Lambda$ as follows\n$$\n\\ln\\rho\\left(\\sum_{j=1}^{2n}m_je_j\\right):=\\sum_{j=1}^{2n} m_j\\ln\\rho(e_j).\n$$\nThus the matrices $\\ln\\rho(\\lambda)$ for $\\lambda\\in\\Lambda$ commute pairwise.\n\n\\begin{prop}\\label{pi_SEtrivial}\nLet $E$ be a flat vector bundle on $C$.\n  Then  $\\pi_{{\\mathcal C}}^*E$ admits a factor of automorphy $\\rho$ satisfying $\\rho(e_j)=Id$ for $j=1,\\dots, n$; in particularly, $\\pi_{\\mathcal C}^*E$ is  holomorphically trivial.\n\\end{prop}\n\\proof{  Let $d$ be the rank of $E$.\nLet $A_j=\\rho(e_j)^{-1}$ for $j=1,\\dots, n$. With $v(e_j)=A_j$ and $v(0)=Id_d$, we first see that\n$$\n\\tilde\\rho(\\lambda,z):=v(z+\\lambda)\\rho(\\lambda)v(z)^{-1}\n$$\nsatisfies $\\tilde\\rho(e_j,0)=Id_d$.  We want to show that $\\tilde\\rho(\\lambda,z)$ depends only on $\\lambda$. Fix  $\\lambda=\\sum_{j=1}^{2n}m_je_j\\in \\Lambda$. By definition,\nthe matrix $\\ln\\rho(\\lambda)$ commutes with each $\\ln\\rho(e_j)$, $j=1,\\ldots, 2n$. Thus the flow $\\varphi_\\lambda^t$ of $\\dot w=\n\\ln\\rho(\\lambda)w$ commutes with the flows of\n$\\dot w=\n\\ln\\rho(e_j)w$, $j=1,\\ldots, 2n$.  As in the proof of the previous lemma, we know that $\\varphi_\\lambda^t(w)$ is linear in $w$ and entire in $t\\in{\\mathbb C}$. For $z\\in {\\mathbb C}^n$ and $w\\in {\\mathbb C}^d$, let $\\tilde v^z(w)$ be as defined in the previous lemma.   Thus we have\n$$\n\\varphi_\\lambda^t \\tilde v^z(w)=\\tilde v^z\\varphi_\\lambda^t(w).\n$$\nTaking derivatives in $w$ and plugging in $t=1$, we get $$\n\\exp(\\ln\\rho(\\lambda))v(z)=v(z)\\exp(\\ln\\rho(\\lambda)).\n$$ Since $\\exp(\\ln\\rho(\\lambda))=\\rho(\\lambda)$, we have  $\\rho(\\lambda)v(z)=v(z)\\rho(\\lambda)$ for all $\\lambda\\in\\Lambda$, $z\\in {\\mathbb C}^n$.\nConsidering $z=z_1e_1+\\cdots +z_ne_n\\in \\Lambda$, we have $v(z+\\lambda)=v(z)v(\\lambda)$. Hence,    $\\tilde\\rho(\\lambda, z)$ is independent of $z$.\n\nWe have achieved $\\tilde\\rho(\\lambda)=\\nu(z+\\lambda)\\rho(z)\\nu(z)^{-1}$ and $\\tilde\\rho(e_j)=Id_d$ for $j=1,\\dots, n$. Therefore, $\\pi|_{\\mathcal C}^*E$ is trivial, by the equivalence relation \\re{eqrelS}.\n\\qedhere}\n\n\nIt is known that there are Stein manifolds with non-trivial vector bundle \\cite{forster-rammspott}. Furthermore, we conclude the section by emphasizing that the triviality $\\pi^*|_{\\mathcal C} E$ relies on the extra assumption that it is a pull-back bundle. The following result is likely known, but we include a short proof for completeness.\n\\begin{prop}The set of holomorphic equivalence classes of flat holomorphic line bundles on ${\\mathcal C}$ can be identified with $H^1({\\mathcal C},\\mathbb C^*)$. The latter is non-trivial.\n\\end{prop}\n\\begin{proof} Each element $\\{c_{jk}\\}$ in $H^1({\\mathcal C},\\mathbb C^*)$ is clearly an element in $H^1({\\mathcal C},\\mathcal O^*)$.\nWe want to show that if $\\{c_{jk}\\},\\{\\tilde c_{jk}\\}\\in H^1({\\mathcal C},\\mathbb C^*)$ represent the same element in   $H^1({\\mathcal C}, \\mathcal O^*)$, then they are also the same in $H^1({\\mathcal C},\\mathbb C^*)$. Indeed, we can cover ${\\mathcal C}$ by convex open sets $U_1,U_2,U_3,U_4$ such that $U_1\\cap U_2\\cap U_3\\cap U_3$ is non empty. Thus $\\{U_i\\}$ is a Leray covering. If $\\tilde c_{jk}=h_jc_{jk}h_k^{-1}$, where each $h_j$ is a non-vanishing holomorphic function on $U_j$. Take $p$ in all $U_j$. We get $\\tilde c_{jk}=c_jc_{jk}c_k^{-1}$ for $c_j=h_j(p)$.\n\n  Note that\n  $H^1({\\mathcal C},\\mathbb C^*)$ is non-trivial.\n  Otherwise,   the  exact sequences $0\\to\\mathbb Z\\to\\mathbb C\\to\\mathbb C^*\\to0$ and\n  $0\\to  H^0({\\mathcal C},\\mathbb Z)\\to  H^0({\\mathcal C},\\mathbb C)\\to  H^0({\\mathcal C},\\mathbb C^*)\\to 0$ would imply that\n  $H^1({\\mathcal C}, {\\mathbb Z}) \\cong H^1({\\mathcal C},{\\mathbb C})$, a contradiction.\n \\end{proof}\n\\setcounter{thm}{0}\\setcounter{equation}{0}\n\n\n\\setcounter{thm}{0}\\setcounter{equation}{0}\n\n\n\n\\section{Equivalence  of neighborhoods and commuting deck transformations}\n\n\n In this section, we will discuss how the classification of neighhorhoods $U$ of a compact complex manifold $C$ is related to the classification of deck transformations of a holomorphic covering $\\tilde U\\to U$, where $U,\\tilde U$ are chosen carefully and $\\tilde U$ contain $C^*$ that covers $C$. When $C^*$ is additionally Stein, we can choose $\\tilde U$ to be a neighborhood of $C^*$ in its normal bundle $N_{C^*}(\\tilde U)$\n  by applying a result of Siu.  After  preliminary results in \\rl{deck-tran} and \\rl{CinN}, we will return to our previous study where $C$ is a complex torus, and its covering of $C$ is the Stein manifold  $C^*=\\tilde C$, and $N_C(M)$ is Hermitian flat. We then prove the main result of this paper by using a KAM rapid iteration scheme.\n\nLet us start with $\\iota: C\\hookrightarrow M$, a holomorphic embedding of a compact complex manifold $C$. We shall still denote $\\iota(C)$ by $C$.  Let $U$ be a neighborhood of $C$ in $M$ such that $U$ admits a smooth, possibly non-holomorphic,  \\emph{deformation or strong retract} $C$~~\\cite[p.~361]{MR3728284}; namely there is a smooth mapping $R\\colon U\\times [0,1]\\to U$ such that $R(\\cdot,0)=Id$ on $U$, $R(\\cdot,t)=Id$ on $C$, and $R(\\cdot,1)(U)=C$. Thus, $\\pi_1(U,x_0)=\\pi_1(C,x_0)$ for $x_0\\in C$ (see~~\\cite[p.~361]{MR3728284}).   When $M$ is $N_C$, we can find a {\\it holomorphic}  deformation retraction from a suitable neighborhood of its the zero section onto $C$,   by using  a  Hermitian metric on $N_C$.\n\nLet $X$ be a complex manifold and  $\\mathcal X$ be a universal covering of $X$. Then the group of deck transformations of the covering is identified with $\\pi_1(X, x_0)$. The set of equivalence classes of coverings of $X$ is identified with the conjugate classes of subgroups of $\\pi_1(X, x_0)$; see~~\\cite[Thm.~79.4, p.~492]{MR3728284}. Furthermore, $\\pi_1(X, x_0)$ acts transitively and freely on each fiber of the covering and $X$ is the quotient of $\\mathcal X$ by $\\pi_1(X, x_0)$.\n\\begin{lemma}\\label{deck-tran}Let $C$ be a compact complex manifold. Let $\\pi\\colon C^*\\to C$ be a holomorphic covering and $\\pi(x_0^*)=x_0$. Suppose that $(M,C)$ (resp. $(M', C)$) is a holomorphic neighborhood of $C$. There is a neighborhood $U$ in $M$ (resp. $U'$ in $M'$) of $C$\n and  a holomorphic neighborhood  $\\tilde U$ $($resp. $\\widetilde{U'})$ of $C^*$ such that $p\\colon\\tilde U\\to U$ $($resp $p\\colon\\widetilde{U'}\\to U')$ is an extended covering of the covering $\\pi\\colon C^*\\to C$ and $C$\n  $($resp. $C^*)$   is a smooth strong retract of $U,U'$ $($resp. $\\tilde U,\\widetilde{U'})$. Consequently, $$\n  \\pi_1(\\tilde U,x_0^*))=\\pi_1(C^*,x_0^*), \\quad \\pi_1(U,x_0)=\\pi_1(C,x_0).\n$$\nSuppose that $(M,C)$ is biholomorphic to $(M',C)$. Then $U,U', \\tilde U,\\widetilde{U'}$\ncan be so chosen that there is covering transformation sending $\\tilde U$ onto $\\widetilde{U'}$ and fixing $C^*$ pointwise. Conversely, if there is a convering transformation sending $\\tilde U$ onto $\\widetilde{U'}$ fixing $C^*$ pointwise, then $(U,C),(U',C)$ are holomorphically equivalent.\n\\end{lemma}\n\\begin{proof}\n Since $\\pi\\colon C^*\\to C$ is a covering map, \naccording to \\cite[Thm.~4.9]{vick-book}, it extends to a covering map $p:\\tilde U \\to U$  such that $\\tilde U$ contains $C^*$ and $p|_{C^*}=\\pi$. Suppose that $R$ is a strong retraction of $U$ onto $C$. We can lift $R(z,\\cdot)\\colon[0,1]\\to U$ to a continuous mapping $\\tilde R(\\tilde z,\\cdot)\\colon[0,1] \\to \\tilde U$ such that $\\tilde R(\\tilde z,0)=\\tilde z$ and $p\\tilde R(\\tilde z,.)=R(p(\\tilde z),.)$ for all $\\tilde z\\in\\tilde U$. One can verify that $\\tilde R$  is a strong retraction  of $\\tilde U$ onto $C^*$.\n\nSuppose that a biholomorphic map $f$ sends  $(M,C)$ onto $(M',C)$ fixing $C$ pointwise. We may assume that $f$ is a biholomorphic mapping from $U$ onto $U'$. Then we can lift the mapping $f\\pi\\colon \\tilde U\\to { U'}$ to obtain a desired covering biholomorphism $F$, since $(f\\pi)_*\\pi_1(\\tilde U,x_0^*)=\\pi_*\\pi_1(C^*,x_0^*)$. Conversely, a covering biholomorphism from $\\tilde{U}$ onto $\\tilde{ U'}$ fixing $C^*$ pointwise clearly induces a biholomorphism from $U$ onto $U'$ fixing $C$ pointwise.\n\\end{proof}\n\n\nWith the covering, we can identity $(M,C)$ with $(\\tilde M, C^*)\/{\\,\\sim}$ where $\\tilde p\\sim p$ if and only if $p,\\tilde p$ are in the same stack of the covering $\\pi$.\n\n\nApplying the above to $(N_C,C)$ and a covering $\\pi|_{C^*}\\colon C^*\\to C$, we have a covering $\\hat\\pi\\colon \\widetilde{N_C}\\to N_C$ such that\n$$\nC^*\\subset\\widetilde{N_C}, \\quad \\pi_1(\\widetilde{N_C},x_0^*)=\\pi_1(C^*,x_0^*), \\quad \\pi_1(N_C,x_0)=\\pi_1(C,x_0).\n$$\n\nTo simplify the notation, we denote $U,\\tilde U$ by $M,\\tilde M$ respectively.\nThus we have commuting diagrams for the coverings:\n$$\n\\begin{matrix}\n\t\\widetilde{N_C} & \\hookleftarrow& C^* & \\hookrightarrow & {\\tilde M}\\\\\n\t\\hat \\pi\\downarrow & \t&\\pi_{C^*}\\downarrow& & p\\downarrow \\\\\n\tN_C& \\hookleftarrow &\tC & \\hookrightarrow & M\n\\end{matrix}.\n$$\nThe set of deck transformations of $p$ (resp. $\\hat\\pi$) will be denoted by $\\{\\tau_1,\\dots, \\tau_n\\}$ (resp. $\\{\\hat\\tau_1,\\dots, \\hat\\tau_n\\}$).\nIf $\\pi\\colon \\tilde U\\to U$ is a covering map, $Deck(\\tilde U)$ denotes the set of deck transformations.\n\n\\begin{lemma}\\label{CinN}\nLet $C, C^*, M$ be as in \\rla{deck-tran}. Suppose that $C^*$ is a Stein manifold. Let $\\omega_0^*$ be an open set of $\\widetilde{N_C}$ such that $\\hat\\pi(\\omega^*)$ contains $C$. Then there is an open subset $\\omega^*$ of $\\omega^*_0$ such that $\\hat\\pi\\omega^*$ contains $C$ and  $(M,C)$ is holomorphically equivalent to the quotient space of $\\omega^*$ by $Deck( \\widetilde{N_C})$.\n\\end{lemma}\n\\begin{proof}\n By a result of Siu \\cite[Cor.~1]{siu-stein}, we find a biholomorphism $L$ from a holomorphic strong retraction neighborhood of $C^*$ in $\\tilde M$, still denoted by $\\tilde M$ into  $N_{C^*}(\\tilde M)$ and  a biholomorphism $L'$ from a   strong retraction neighborhood of $C^*$ in $\\widetilde{N_C}$, still denoted by  $\\widetilde{ N_C}$ into  $N_{C^*}( \\widetilde{N_C)}$. Furthermore, $L,L'$ fix $C^*$ pointwise.\n  We have\n \\begin{align*}\n p_*\\pi_1(N_{C^*}(\\tilde M),x_0^*)&=p_*L^{-1}_*\\pi_1(\\tilde M,x_0^*)=\\pi_1(C,x_0)=\\pi_1(L^{-1}\\tilde M,x_0^*)\\\\\n &=\\hat\\pi_*(L')^{-1}_*(\\pi_1(\\widetilde{N_C},x_0^*)).\n \\end{align*}\n Both $\\hat\\pi\\circ L'^{-1}\\colon N_{C^*}( \\widetilde{N_C)})\\to M$ and $p\\circ L^{-1}\\colon N_{C^*}( \\tilde M)\\to M$ are  coverings   and the above identifications show that the lifts of the two coverings yield a biholomorphism between neighborhoods of $C^*$ in $\\tilde M$ and $N_{C^*}(\\widetilde{N_C})$ fixing $C^*$ pointwise.\n \\end{proof}\n\n\nHere, $z=(z_1,\\ldots,z_n)$ belongs to the fundamental domain\n$$\n\\omega_0=\\left\\{\\sum_{j=1}^{2n} t_je_j\\in{\\mathbb C}^n\\colon t\\in[0,1)^{2n}\\right\\},\\quad e_{n+i}:=\\tau_i,\\quad i=1,\\ldots,n.\n$$\nThus $h$ belongs to the fundamental domain $\\Omega_0$ defined\n$$\n{\\Omega\n}_0:=\\{(e^{2\\pi i\\zeta_1}, \\dots e^{2\\pi i\\zeta_n})\\colon\\zeta\\in \\omega_0\\},\\quad {\n\\Omega\n}_0^+=\\{(|z_1|,\\dots, |z_n|)\\colon z\\in {\\Omega\n}_0\\}.\n$$\nThus ${\\Omega\n}_0$ is a Reinhardt domain, being $\\{(\\nu_1R_1,\\dots, \\nu_nR_n)\\colon |\\nu_j|=1\\colon R\\in \\Omega\n_0^+\\}$.\nWe have\n$$\n{\\Omega\n}_0^+=\\left\\{(e^{-2\\pi R_1},\\dots, e^{-2\\pi R_n})\\colon R=\\sum_{i=1}^n t_i\\operatorname{Im}\\tau_i,t\\in[0,1)^n\\right\\}.\n$$\nFor $\\epsilon>0$,  define a (Reinhardt) neighborhood ${\\Omega\n}_\\epsilon$ of $\\overline{{\\Omega\n}_0}$ by\n\\begin{gather*}\n\\omega_\\epsilon:=\\left\\{\\sum_{j=1}^{2n} t_je_j\\colon t\\in[0,1)^n\\times (-\\epsilon,1+\\epsilon)^{n}\\right\\},\n\\quad {\\Omega\n}_\\epsilon:=\\{(e^{2\\pi i\\zeta_1}, \\dots e^{2\\pi i\\zeta_n})\\colon\\zeta\\in \\omega_\\epsilon\\}.\n\\end{gather*}\nFor any $\\ell$-tuple of indices in $\\{n+1,\\ldots,2n\\}$, we set\n\\begin{gather}{}\n\\omega_\\epsilon^{j_1\\dots j_\\ell}:=\n\\left\\{\\sum_{j=1}^{2n} t_je_j\\colon t_{k}\\in(-\\epsilon,\\epsilon), k-n\\in \\{j_1,\\dots, j_\\ell\\};   t_{k}\\in(-\\epsilon,1+\\epsilon), k-n\\not\\in \\{j_1,\\dots, j_\\ell\\}\n  \\right\\},\n \\\\\n\\tilde \\omega_\\epsilon^{j_1\\dots j_\\ell}:=\n\\left\\{\\sum_{j=1}^{2n} t_je_j\\colon\nt_{k}-1\\in(-\\epsilon,\\epsilon),  k-n\\in \\{j_1,\\dots, j_\\ell\\}; t_{k}\\in(-\\epsilon,\\epsilon),  k-n\\not\\in \\{j_1,\\dots, j_\\ell\\}\n \\right\\}.\n\\end{gather}\nNote that $\\tilde \\omega_\\epsilon^{j_1\\dots j_\\ell}$ and $ \\omega_\\epsilon^{j_1\\dots j_\\ell}$ are subset of $\\omega_\\epsilon$, and\n   $\\omega_\\epsilon^{1\\dots n}=\n\\{\\sum_{j=1}^{2n} t_je_j\\in\\omega_\\epsilon\\colon t\\in[0,1)^n\\times(-\\epsilon,\\epsilon)^n\\}$.  Then\n\\begin{gather}\n\\Omega_\\epsilon^{j_1\\dots j_\\ell}:=\\Omega_\\epsilon\\cap\\bigcap_{k=1}^\\ell T_{j_k}^{-1}\\Omega_\\epsilon=\\{(e^{2\\pi i\\zeta_1}, \\dots e^{2\\pi i\\zeta_n})\\colon\\zeta\\in \\omega^{j_1\\dots j_\\ell}\n_\\epsilon\\},\n\\\\\n\\tilde\\Omega_\\epsilon^{j_1\\dots j_\\ell}:=\\Omega_\\epsilon\\cap\\bigcap_{k=1}^\\ell T_{j_k}\\Omega_\\epsilon=\\{(e^{2\\pi i\\zeta_1}, \\dots e^{2\\pi i\\zeta_n})\\colon\\zeta\\in \\tilde\\omega^{j_1\\dots j_\\ell}\\}\n\\end{gather}\nare connected non-empty Reinhardt domains. Moreover, $\\Omega_0^{1\\cdots n}:=\\cap_{\\epsilon>0} \\Omega_0^{1\\cdots n}$ and $ \\tilde\\Omega_0^{1\\cdots n}=\\cap_{\\epsilon>0}\\Omega_\\epsilon^{1\\cdots n} $ are diffeomorphic to the real torus $(S^1)^n$.\nWe remark that\n$\nT_iT_j$ maps $\\Omega^{ij}_\\epsilon$ into $\\Omega_\\epsilon$ for $i\\neq j$, while $T_i\\circ T_i$ does not map $\\Omega_{\\epsilon,r}^{ii}$ into   $\\Omega_\\epsilon$ .\n\n\n\n\nWith $\\Delta_r=\\{z\\in{\\mathbb C}\\colon|z|<r\\}$, we also define\n\\begin{equation}\\label{domains}\n\\omega_{\\epsilon,r}:= \\omega_\\epsilon\\times  \\Delta^d_r,\n\\quad \\Omega_{\\epsilon,r}:=\\Omega\n_\\epsilon\\times  \\Delta^d_r.\n\\end{equation}\nThroughout the paper, a mapping $(z',v')=\\psi^0(z,v)$ from $\\omega_{\\epsilon,r}$ into ${\\mathbb C}^{n+d}$ that commutes with $z_j\\to z_j+1$ for $j=1,\\dots, n$ will be identified with a well-defined mapping $(h',v')=\\psi(h,v)$ from $\\Omega_{\\epsilon,r}$ into ${\\mathbb C}^{n+d}$, where $z,h$ and $z',h'$ are related   as in \\re{hTj}. A function on $\\omega_{\\epsilon,r}$ that has period $1$ in all $z_j$ is identified with a function on $\\Omega_{\\epsilon,r}$. We shall use these identifications as we wish.\n\\begin{prop}Let $C$ be the complex torus and $\\pi_{{\\mathcal C}}\\colon{\\mathcal C}={\\mathbb C}^n\/{{\\mathbb Z}^n}\\to C$ be the covering.\nLet $(M,C)$ be a neighborhood of $C$. Assume that $N_C$ is flat.\n\\begin{list}[align=left]{$(\\roman{ppp})$}{\\usecounter{ppp}}\n\\item Then one can take $\\omega_{\\epsilon_0,r_0}=\\omega_{\\epsilon_0}\\times \\Delta_{r_0}^d$ such that $(M,C)$ is biholomorphic to the quotient of $\\omega_{\\epsilon_0,r_0}$ by  $\\tau^0_1,\\dots,\\tau^0_n$. Let $\\tau_j$ be the mapping defined on $\\Omega_{\\epsilon_0,r_0}$ corresponding to $\\tau_j^0$.\nThen $\\tau_1,\\dots, \\tau_n$ commute pairwise   wherever they are defined, i.e.\n$$\n\\tau_i\\tau_j(h,v)=\\tau_j\\tau_i(h,v)\\quad  \\forall  i\\neq j\n$$\nfor $(h,v)\\in \\Omega_{\\epsilon_0, r_0}\\cap \\tau_i^{-1}\\Omega_{\\epsilon_0,r_0}\\cap \\tau_j^{-1}\\Omega_{\\epsilon_0,r_0}$.\n\\item\n Let $(\\tilde M,C)$ be another  such neighborhood having the corresponding generators $\\tilde\\tau_1,\\dots,\\tilde\\tau_n$  of deck transformations defined on $\\Omega_{\\tilde\\epsilon_0,\\tilde r_0}$. Then $(M,C)$ and $(\\tilde M,C)$ are holomorphically equivalent if and only if there is a  biholomorphic mapping $F$  from $\\Omega_{\\epsilon,r}$ into $\\Omega_{\\tilde \\epsilon,\\tilde r}$ for some positive $\\epsilon,r,\\tilde e,\\tilde r$  such that\n$$\nF \\tilde\\tau_j(h,v)=\\tau_jF(h,v),\\ j=1,\\ldots, n,\n$$\nwherever   both sides are defined, i.e. $(h,v)\\in \\Omega_{\\tilde\\epsilon,\\tilde r}\\cap \\tilde\\tau_j^{-1}\\Omega_{\\epsilon,r}\\cap \\Omega_{\\epsilon,r}\\cap F^{-1}\\Omega_{\\epsilon,r}.$\n\\end{list}\n\\end{prop}\n\\begin{proof}\nWe now apply \\rl{CinN}, in which $C^*$ is replaced by\n  ${\\mathcal C}={\\mathbb R}^n\/{\\mathbb Z}^n+i{\\mathbb R}^n$ is a Stein manifold.\n  Assume that $N_C$ is flat. Then according to \\rp{pi_SEtrivial},  $N_{{\\mathcal C}}( \\widetilde{N_C)}=N_{{\\mathcal C}}(\\tilde M)=\\pi_{{\\mathcal C}}^*(N_C)$  is the trivial vector bundle  ${\\mathcal C}\\times {\\mathbb C}^d$ with coordinates $(h,v)$, while ${\\mathcal C}\\times\\{0\\}$ is defined by $v=0$.\n\\end{proof}\n\n\n\nSet\n\\begin{gather*}\n{\\mathcal P}_\\epsilon^+:=\\left\\{ \\sum_{i=1}^n t_i\\operatorname{Im}\\tau_i\\colon t\\in (-\\epsilon,1+\\epsilon)^{n}\\right\\},\\\\\n{\\Omega\n}_\\epsilon^+:=\\left\\{(e^{-2\\pi R_1},\\dots, e^{-2\\pi R_n})\\colon R\n\\in {\\mathcal P}_\\epsilon^+\n\\right\\}.\n\\end{gather*}\nNote that  ${\\mathcal P}_\\epsilon^+, {\\mathcal P}_0^+$ are $n$-dimensional parallelotopes,\n and $\\Omega_\\epsilon^+$ contains  $(1,\\ldots,1)$, the image of $0\\in{\\mathcal P}_\\epsilon^+$, corresponding to the real torus $(S^1)^n$.\n\n Since $\\Omega\n _\\epsilon$ is   Reinhardt, we have\n \\eq{boundary-of-R}\n \\Omega\n _\\epsilon\\supset\\Omega\n _\\epsilon^+,\\quad (\\partial \\Omega\n _\\epsilon)^+=\\partial(\\Omega\n _e^+), \\quad (\\partial \\Omega\n _\\epsilon)^+:=\\{(|h_1|,\\dots,|h_n|)\\colon h\\in \\partial \\Omega\n _\\epsilon\\}.\n \\end{equation}\n We now apply the above general results to the case where $C$ is a complex torus,  $C^*=\\tilde C$ and $N_C(M)$ is Hermitian flat.\n\n As in \\cite{ilyashenko-pyartly-embed}, the deck transformations of $(\\tilde N_C,{\\mathcal C})$ are   generated by $n$ biholomorphisms $\\hat\\tau_1,\\dots, \\hat\\tau_n$ that preserve ${\\mathcal C}$.\\begin{gather}\n \t\\hat \\tau_j(\n \th,v\n \t=\n \t(T_jh,  M_jv),\\quad M_j:=\\operatorname{diag}(\\mu_{j,1},\\dots, \\mu_{j,d})\\label{tauhat}\n \n \\end{gather}\n\n\n with $h,T_j$ being defined by~\n  \\eq{hTj} h=(e^{2\\pi i z_1}, \\cdots, e^{2\\pi iz_n}), \\quad\n T_j:= \\operatorname{diag}(\\lambda_{j,1},\\dots, \\lambda_{j,n}),\\quad \\lambda_{j,k}:=e^{2\\pi i\\tau_{jk}}.\n \\end{equation}\nHere the invertible $(d\\times d)$-matrix $M_j$ is the factor of automorphy $\\rho(e_{n+j})$ of $\\pi_{{\\mathcal C}}^*N_C$.\n\nRecall that each deck transformation\n $\\tau^0_j(z,v)$, $j=1,\\ldots, n$, is a holomorphic map defined on $\\overline \\omega_{\\epsilon,r}$.\n   In coordinates $(h,v)$, $\\tau_j^0$ becomes $\\tau_j$ defined on $\\overline\\Omega_{\\epsilon,r}$.\n      Since $T_CM$ splits, it is a higher-order perturbation of\n      $\\hat\\tau_j$\n      with $$\n      \\tau_j(h,0)=( T_jh,0)\n      $$\n\n\n\n The above computation is based on the assumption that $N_C$ is flat. We now assume that $N_C$ is {\\it Hermitian and flat}, i.e. the $N_C$ admits locally constant Hermitian transition matrices.\nThen all $\\rho(e_{n+j})$ are Hermitian and constant matrices. Since they pairwise commute, they  are simultaneously diagonalizable.\nHence, we can assume that $M_j=\\operatorname{diag}(\\mu_{j,1},\\ldots, \\mu_{j,d})$.\n   We  recall from \\re{hTj} that $T_j=\\operatorname{diag}(\\lambda_{j,1},\\dots, \\lambda_{j,n})$ are already diagonal.\n\n\\begin{defn}\\label{def-nr}\n\tThe\n\tnormal bundle $N_C$ is said to be\n\tnon-resonant if, for each $(Q,P)\\in \\mathbb{N}^d\\times {\\mathbb Z}^n$ with $|Q|>1$,  each $i=1,\\ldots, n$, and each $j=1,\\ldots ,d$, there exist $i_h:=i_h(Q,P,i)$  and\n $i_v:=i_v(Q,P,j)$ that are in $ \\{1,\\ldots, n\\}$ such that\n\t$$\n\n\t\\lambda_{i_h}^P\n\\mu_{i_h}^Q-\\lambda_{i_h,i\n}\\neq 0\\quad\\text{and}\n\\quad\n\\lambda_{i_v}^P\n\\mu_{i_v}^Q-\\mu_{{i_v},j}\\neq 0.\n$$\n\\end{defn}\n\n\\begin{defn}\\label{dioph}\nThe pullback\nnormal bundle $N_C$, or $N_C$,  is said to be  {Diophantine} if for all $(Q,P)\\in \\mathbb{N}^d\\times {\\mathbb Z}^n$,   $|Q|>1$ and all $i=1,\\ldots, n$, and  $j=1,\\ldots ,d$,\n\\begin{gather}\n\\max_{\\ell\\in \\{1,\\ldots, n\\}}\\left |\n\\lambda_{\\ell}^P\n\\mu_{\\ell}^Q\n-\\lambda_{\\ell,i}\n\\right|   >  \\frac{D}{(|P|+|Q|)^{\\tau}},\\label{dh}\\\\\n\\max_{\\ell\\in \\{1,\\ldots, n\\}}  \\left |\\lambda_\\ell^P\n\\mu_{\\ell}^Q-\\mu_{{\\ell},j}\\right |   >  \\frac{D}{(|P|+|Q|)^{\\tau}}.\\label{dv}\n\\end{gather}\n We shall choose $i_h$ (resp. $i_v$) to be the index that realizes the maximum of \\rea{dh} (resp. \\rea{dv}).\n\\end{defn}\n\\begin{remark}\nIf  the right-hand sides are replaced by $0$,\n then \n$N_C$ is non-resonant.\n\\end{remark}\n\n\\begin{prop}\n\tThe properties of being non-resonant and Diophantine is a property of the (abelian) group and not the choice of the generators.\n\\end{prop}\n\\begin{proof}\nRecall that $$\n\t\\lambda_{\\ell}=(\\lambda_{\\ell,1},\\ldots,\\lambda_{\\ell,n})=(e^{2\\pi i\\tau_{\\ell,1}},\\ldots, e^{2\\pi i\\tau_{\\ell,n}}).\n\t$$\nLet $G$ be the group generated by the $\\hat \\tau_{\\ell}$'s. Then, $\\{\\tilde \\tau_{\\ell}\\}_{\\ell}$ defines another set of generators of $G$ if $\\tilde \\tau_{\\ell}=\\hat\\tau_{1}^{a_{\\ell,1}}\\cdots \\hat\\tau_{n}^{a_{\\ell,n}}$, $\\ell=1,\\ldots, n$ where $A=(a_{i,j})_{1\\leq i,j\\leq n}\\in GL_n({\\mathbb Z})$ with $\\det A=\\pm1$. Then, the eigenvalues of $\\tilde \\tau_{\\ell}$ are\n$$\n\\tilde \\lambda_{\\ell,i}=\\prod_{k=1}^n \\lambda_{k,i}^{a_{\\ell,k}},\\quad \\tilde \\mu_{\\ell,j}=\\prod_{k=1}^{ d} \\mu_{k,j}^{a_{\\ell,k}}.\n$$\nHence, we have\n$$\n\\tilde \\lambda_{\\ell}^P\\tilde\\mu_{\\ell}^Q\\tilde \\lambda_{\\ell,i}^{-1}= (\\lambda_{1}^P\\mu_1^Q\\lambda_{1,i}^{-1})^{a_{\\ell,1}}\\cdots (\\lambda_{n}^P\\mu_n^Q\\lambda_{n,i}^{-1})^{a_{\\ell,n}}.\n$$\nFix $P,Q$ and $i$. Taking the logarithm, we have as $n$-vectors\n\\begin{equation}\\label{log}\n\\left(\\ln \\tilde \\lambda_{\\ell}^P\\tilde\\mu_{\\ell}^Q\\tilde \\lambda_{\\ell,i}^{-1}\\right)_{\\ell=1,\\dots, n}= A \\left(\\ln \\lambda_{\\ell}^P\\mu_{\\ell}^Q\\lambda_{\\ell,i}^{-1}\\right)_{\\ell=1,\\dots, n},\\mod 2\\pi i.\n\\end{equation}\nSince $A,A^{-1}\n\\in GL_n({\\mathbb Z})$, given $P, Q$ and $i$, $$\\left(\\ln \\tilde \\lambda_{\\ell}^P\\tilde\\mu_{\\ell}^Q\\tilde \\lambda_{\\ell,i}^{-1}\\right)_{\\ell=1,\\dots, n}=0\\mod 2\\pi i$$\n iff $\\left(\\ln \\lambda_{\\ell}^P\\mu_{\\ell}^Q\\lambda_{\\ell,i}^{-1}\\right)_{\\ell=1,\\dots, n}=0\\mod 2\\pi i$. Similarly, by considering $\\ln \\tilde\\lambda_{\\ell}^P\\tilde\\mu_{\\ell}^Q\\tilde \\mu_{\\ell,i}^{-1}$, we obtain that the non-resonant condition does not depend on the choice of generators.\nGiven $P,Q,i$,  if one of the $\\lambda_{\\ell}^P\\mu_{\\ell}^Q\\lambda_{\\ell,i}^{-1}$'s is not close to $1$, then $\\left\\|(\\ln \\tilde \\lambda_{\\ell}^P\\tilde \\mu_{\\ell}^Q\\tilde\\lambda_{\\ell,i}^{-1})_{\\ell}\\right\\|$ is bounded way from $0$. On the other hand, if all $\\lambda_{\\ell}^P\\mu_{\\ell}^Q\\lambda_{\\ell,i}^{-1}$'s are close to $1$, then\n$|\\ln \\lambda_{\\ell}^P\\mu_{\\ell}^Q\\lambda_{\\ell,i}^{-1}|$\n(with $\\operatorname{Im}\\ln$ in $(-\\pi,\\pi]$) is comparable to\n$|\\lambda_{\\ell}^P\\mu_{\\ell}^Q\\lambda_{\\ell,i}^{-1}-1|$.\nFurthermore, taking the module of \\re{log}, we obtain\n$$\n\\|A^{-1}\\|^{-1}\\left\\|(\\ln \\lambda_{\\ell}^P\\mu_{\\ell}^Q\\lambda_{\\ell,i}^{-1})_{\\ell}\\right\\|\\leq \\left\\|(\\ln \\tilde \\lambda_{\\ell}^P\\tilde \\mu_{\\ell}^Q\\tilde\\lambda_{\\ell,i}^{-1})_{\\ell}\\right\\|\\leq \\|A\\|\\left\\|(\\ln \\lambda_{\\ell}^P\\mu_{\\ell}^Q\\lambda_{\\ell,i}^{-1})_{\\ell}\\right\\|,\n$$\nwhere $\\|(a_{\\ell})_{\\ell}\\|=\\max_{\\ell}|a_{\\ell}|$.\nIf the latter is bounded  below by $\\frac{C}{(|P|+|Q|)^{\\tau}}$, so is  $\\left\\|(\\ln \\tilde \\lambda_{\\ell}^P\\tilde \\mu_{\\ell}^Q\\tilde\\lambda_{\\ell,i}^{-1})_{\\ell}\\right\\|$.\n\\end{proof}\n\\begin{thm}\n\tLet $C_n$ be an $n$-dimensional complex torus, holomorphically embedded into a complex manifold $M_{n+d}$.   Assume that $T_CM$ splits. Assume the normal bundle $N_C$ has (locally constant) Hermitian transition functions. Assume that  $N_C$ is   Diophantine. Then some neighborhood of $C$ is biholomorphic to a neighborhood of the zero section in the normal bundle.\n\\end{thm}\n\\begin{remark}\n\tWhen $C$ is a product of $1$-dimensional tori with normal bundle which is a direct sum of line bundles, the above result is due to Il'yashenko-Pyartli~\\cite{ilyashenko-pyartly-embed}.\n\\end{remark}\n\nWe have\n$\\tau_j(h,v) =\\hat\\tau_j(h,v)+(\\tau^h_j(h,v),\\tau^v_j(h,v) )$. Here, functions\n$$\n\\tau^{\\bullet}_j(h,v)=\\sum_{Q\\in \\mathbb{N}^d, |Q|\\geq 2}\\tau^{\\bullet}_{j,Q}(h)v^Q\n$$\nare holomorphic in $(h,v)$ in a neighborhood of $\\Omega_{\\epsilon,r}$ with values in ${\\mathbb C}^{n+d}$.\n\n\\begin{defn}\n\t\nSet\n $\\Omega_{\\epsilon,r}:=\\Omega_\\epsilon\\times\\Delta_r^d$,\n$\n\\tilde \\Omega_{\\epsilon,r}:=\\overline\\Omega_{\\epsilon,r}\\cup\\bigcup_{i=1}^n \\hat\\tau_i(\\overline\\Omega_{\\epsilon,r})$.\nDenote by $\\mathcal A_{\\epsilon,r}$ $($resp. $\\tilde{\\mathcal A}_{\\epsilon, r})$ the set of holomorphic functions on $\\overline{\\Omega_{\\epsilon,r}}$ $($resp. $\\overline{\\tilde \\Omega_{\\epsilon,r}})$.\nIf $f\\in \\mathcal{A}_{\\epsilon, r}$ $($resp. $\\tilde f\\in\\tilde{\\mathcal A}_{\\epsilon, r})$, we set\n$$\n\\|f\\|_{\\epsilon,r}:=\\sup_{(h,v)\\in\\Omega_{\\epsilon,r}}|  f(h,v)|, \\quad |||\\tilde f|||_{\\epsilon,r}:=\\sup_{(h,v)\\in\\tilde\\Omega_{\\epsilon,r}}| \\tilde f(h,v)|.\n$$\n\\end{defn}\n\nAs such, each\n  $f\\in \\mathcal{A}_{\\epsilon, r}$\ncan be expressed as a convergent Taylor-Laurent series\n$$\nf(h,v)= \\sum_{P\\in {\\mathbb Z}^n}f_{Q,P}h^Pv^Q\n$$\nfor $(h,v)\\in \\Omega_{\\epsilon,r}=\\Omega_\\epsilon\\times\\Delta_r^d$.\nRecall that each holomorphic function on $E_j$ in \\rl{connected} admits a unique Taylor-Laurent series expansion on $\\tilde\\Omega_{\\epsilon'}^{1\\cdots n}\\times \\Delta_{\\epsilon'}^d$ when $\\epsilon'>0$ is sufficiently small.\n\nWe can state the following, for later use~:\n\\begin{lemma}\\label{connected}\n\tFor  $i\\neq j$, the set $\n\\hat\\tau_j\\overline\\Omega_{\\epsilon,r}\\cap\\hat\\tau_i\n(\\overline\\Omega_{\\epsilon,r})$ is a connected Reinhardt domain containing  $\\tilde\n\\Omega_{\\epsilon}^{1\\dots n}\\times \\Delta_{r'}^d$ in ${\\mathbb C}^{n+d}$ when $r'>0$ is sufficiently small.\n\\end{lemma}\n\\subsection{Holomorphic functions on $\\Omega_{\\epsilon,r}$}\nIn this section, we study elementary properties and estimate  holomorphic functions   $f$ on $\\overline\\Omega_{\\epsilon,r}$.\n\\begin{lemma}\\label{laurent}\n\tAn element  $f(h)$ of ${\\mathcal A}_{\\epsilon}$, that is a holomorphic function in a neighborhood of $\\overline \\Omega_{\\epsilon}$,  admits a Laurent series expansion in $h$\n\t\\eq{L-exp}\n\t  f(h)=\\sum_{P\\in{\\mathbb Z}^n} c_Ph^P.\n\t\\end{equation}\n\tThe series converges normally on $\\Omega_\\epsilon$. Moreover, the Laurent coefficients\n\t\\eq{L-coef}\n\tc_P=\\f{1}{(2\\pi i)^n}\\int_{|\\zeta_1|=s_1,\\dots, |\\zeta_n|=s_n}   f(\\zeta)\\zeta^{-P-(1,\\dots, 1)}\\, d\\zeta_1\\wedge\\cdots \\wedge d\\zeta_n\n\t\\end{equation}\n\tare independent of $s\\in \\Omega_\\epsilon^+$ and\n\t\\eq{L-est}\n\t|c_P|\\leq\\sup_{\\Omega_\\epsilon}| f|\\inf_{s\\in \\Omega_\\epsilon^+} s^{-P}.\n\t\\end{equation}\n\\end{lemma}\n\\begin{proof}Obviously, estimate \\re{L-est} follows from \\re{L-coef}.  Define $A(a,b)=\\{w\\in {\\mathbb C}:a<|w|<b\\}$. Fix $h\\in \\Omega_\\epsilon$. Then $|h|:=(|h_1|,\\dots, |h_n|)\\in \\Omega_\\epsilon^+$.  The latter is an open set and we have for a small positive number $\\epsilon=\\epsilon(h)$,\n\t$$\n\t  f(h)=\\f{1}{(2\\pi i)^n}\n\t\\int_{\\partial A(|h_1|-\\epsilon,|h_1|+\\epsilon)}\n\t\\cdots \\int_{\\partial A(|h_n|-\\epsilon,|h_n|+\\epsilon)}\\f{  f(\\zeta)\\, d\\zeta_1\\dots d\\zeta_n}{(\\zeta_1-h_1)\\cdots(\\zeta_n-h_n)}.\n\t$$\n By  Laurent expansion in one-variable, we get the expansion \\re{L-exp} in which $c_P$ are given by \\re{L-coef} if we take $r_j=|h_j|\\pm\\epsilon$ according to the sign of $p_j\\in {\\mathbb Z}^{\\mp}$.\n\t\n\tWe want to show that $c_P$ is independent of $s\\in\\Omega_\\epsilon^+$. Note that $\\Omega_\\epsilon^+$ is a connected open set. For any two points $s,\\tilde s$ can be connected by a union of line segments\nin $\\Omega_\\epsilon^+$  which are parallel to coordinate axes in ${\\mathbb R}^n$.\n Using such line segments,  say a line segment $[a,b]\\times (s_2,\\dots, s_n)$ in $\\Omega_\\epsilon^+$, we know that $ f(\\zeta)\\zeta^{-P-(1,\\dots, 1)}$ is holomorphic in $\\zeta_1$ in the closure of $A(a,b)$ when $|\\zeta_2|=s_2,\\dots, |\\zeta_n|=s_n$. By Cauchy theorem, the integrals are independent of $s_1\\in[a,b]$ for these $s_2,\\dots, s_n$. This shows that \\re{L-coef} is independent of $s$.\n\t\n\tFinally the series converges uniformly on each compact subset of $\\Omega_\\epsilon$. Indeed, for a small perturbation of $h$, we can choose $\\epsilon(h)$ to be independent of $h$. Then we can see easily that the series converges locally uniform in $h$.\n\\end{proof}\nRecall that\n\\begin{gather*}\n{\\mathcal P}_\\epsilon^+=\\left\\{\\sum_{i=1}^n t_i\\operatorname{Im}\\tau_i\\in{\\mathbb R}^n\\colon t\\in (-\\epsilon,1+\\epsilon)^{n}\\right\\},\\\\\n{\\Omega}_\\epsilon^+=\\left\\{(e^{-2\\pi R_1},\\dots, e^{-2\\pi R_n})\\colon R\n\\in {\\mathcal P}_\\epsilon^+\n\\right\\}.\n\\end{gather*}\n\\begin{lemma}\\label{dist-lemma} There is a constant $\\kappa_0>0$ that depends only on $\\operatorname{Im}\\tau_1,\\dots,\\operatorname{Im}\\tau_n$ such that if  $P\\in{\\mathbb R}^n$ and $\\epsilon>\\epsilon'$, there exists $R\\in {\\mathcal P}_{\\epsilon}^+$ such that for all $R'  \\in {\\mathcal P}_{\\epsilon'}^+$ we have\n\\eq{RR'}\n  (R'-R)\\cdot P\\leq  -\\kappa_0(\\epsilon-\\epsilon')|P|.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}Without loss of generality, we may assume that $P$ is a unit vector.\nLet $\\pi(x)P$ be the orthogonal projection from $x\\in{\\mathbb R}^n$ onto the line spanned by $P$. Choose $R\\in\\overline {\\mathcal P}_\\epsilon^+$ so that $\\pi(R)$ has the largest value for $R\\in  \\overline {\\mathcal{P}_\\epsilon^+}$. Note that $R$ must be on the boundary of ${\\mathcal P}_\\epsilon^+$ and latter is contained in the half-space $H$ defined by $\\pi(y)\\leq\\pi(R)$ for $y\\in{\\mathbb R}^n$. Hence, $\\partial H$ is orthogonal to $P$.    Then for any $R'\\in \\mathcal {P}_{\\epsilon'}^+$,\n$$\n\\pi(R)-\\pi(R')=\\operatorname{dist}(R',\\partial H)\\geq\\operatorname{dist} ( \\partial {\\mathcal P}_{\\epsilon}^+,{\\mathcal P}_{\\epsilon'}^+)\\geq (\\epsilon-\\epsilon')\/C.\n$$\nTherefore,   we obtain $(R'-R)\\cdot P=-(\\pi(R)-\\pi(R'))\\leq -(\\epsilon-\\epsilon')\/C.$\n\\end{proof}\n\\begin{remark}Since $ {\\mathcal P}_{\\epsilon}^+$ is a   parallelotope, we can choose $R$ to be a vertex of ${\\mathcal P}_{\\epsilon}^+$.\n\\end{remark}\n In what follows, we denote the fixed constant~:\n\\begin{equation}\\label{kappa}\n\\kappa:=2\\pi\\kappa_0.\n\\end{equation}\n\\begin{lemma}[Cauchy estimates]\\label{cauchy}\n\tIf $f\\in\\mathcal A_{\\epsilon, r}$, then for all $(P,Q)\\in {\\mathbb Z}^n\\times \\mathbb {N}^d$,\n\t\\begin{equation}\\label{Reps}\n\t|f_{Q,P}|\\leq \\frac{M}{r^{|Q|}\\max_{s\\in \\Omega_\\epsilon^+} s^{P}}\n\t\\end{equation}\n\tFurthermore,\n if $f=\\sum_{Q\\in {\\bf N}^d, P\\in {\\mathbb Z}^n}f_{QP}h^Pv^Q\\in \\mathcal A_{\\epsilon , r}$  \n and  $0<\\delta<{\\kappa}\\epsilon$,\n\tthen\n\t$$\n\t\\|f\\|_{\\epsilon\n  {-{\\delta}\/{\\kappa}}, re^{-\\delta}}\\leq \\frac{C\\sup_{\\Omega_{\\epsilon,r}}|f|}{\\delta^{\\nu}},\n\t$$\n\twhere $C$ and $\\nu$ depends only on $n$ and $d$.\n\\end{lemma}\n\\begin{proof}\n\tAccording to \\rl{laurent}\n\tand Cauchy estimates on polydiscs, we have for any $s\\in \\Omega_\\epsilon^+$,\n$$\n\t|f_{Q,P}|\\leq \\frac{\\sup_{\\Omega_\\epsilon}|f_Q(h)|}{s^{P}}\\leq \\frac{\\sup_{\\Omega_{\\epsilon,r}}|f|}{s^{P}r^{|Q|}}\n$$\n\tAccording to \\rl{laurent} and Cauchy estimates for polydiscs, we have if $(h,v)\\in \\Omega_{\\epsilon e^{-\\delta'},re^{-\\delta}}$, then for all $s\\in \\Omega_{\\epsilon}^+$,\n\\begin{align*}\n|f_Q(h)|&\\leq \\frac{\\sup_{v\\in  \\Delta^d_r\n}|f(h,v)|}{r^{|Q|}},\\\\\n|f_{Q,P}h^P|&\\leq \\left|\\f{1}{(2\\pi i)^n}\\int_{|\\zeta_1|=s_1,\\dots, |\\zeta_n|=s_n} f_Q(\\zeta)\\frac{h^P}{\\zeta^{P}}\\, \\frac{d\\zeta_1\\wedge\\cdots \\wedge d\\zeta_n}{\\zeta_1\\cdots\\zeta_n}\\right|.\n\\end{align*}\nSet $s_j=e^{-2\\pi R_j}$, $|h_j|=e^{-2\\pi R_j'}$, $R=(R_1,\\dots, R_n)$ and $R'=(R_1',\\dots, R_n')$.  By \\rl{dist-lemma},\n\\begin{equation}\\label{fourier-type}\n\\inf_{(|\\zeta_1|,\\dots,|\\zeta_n|)=s\\in\\Omega_{\\epsilon}^+}\\sup_{h\\in \\Omega_{\\epsilon  {-\\delta'}}}\n\\left|\\frac{h^P}{\\zeta^P}\\right|=\\inf_{R\\in \\mathcal P_{\\epsilon}^+}\\sup_{R'\\in\\mathcal P_{\\epsilon  {-\\delta'}}^+} e^{-2\\pi <R-R',P>}\\leq e^{-\\kappa\\delta' |P|},\n\\end{equation}\nwhere the positive constant $\\kappa$, defined by \\rl{dist-lemma} and \\re{kappa}, is independent of $P,\\zeta_1,\\zeta_2$.  Thus\n\\begin{equation}\\label{est-small}\n|f_{Q,P}h^Pv^Q|\\leq  \\frac{\\sup_{\\Omega_{\\epsilon,r}}|f| e^{-\\kappa\\delta'|P|}r^{|Q|}e^{-\\delta|Q|}}{r^{|Q|}}\\leq \\sup_{\\Omega_{\\epsilon,r}}|f| e^{-\\delta'\\kappa |P|}e^{-\\delta|Q|}.\n\\end{equation}\n\tHence, setting $\\delta':=\\delta\/\\kappa$, we have\n\t$$\n\t\\|f\\|_{\\epsilon\n  {-{\\delta}\/{\\kappa}}, re^{-\\delta}}\\leq \\sum_{Q\\in {\\bf N}^d, P\\in {\\mathbb Z}^n}|f_{Q,P}h^Pv^Q|\\leq \\frac{C\\sup_{\\Omega_{\\epsilon,r}}|f|}{\\delta^{\\nu}},\n\t$$\n\twhere $C$ and $\\nu$ depends only on $n$ and $d$.\n\\end{proof}\n\\subsection{Conjugacy  of the deck transformations}\nLet us show that there is a biholomorphism $\\Phi=(h,v)+\\phi(h,v)$ of some neighborhood $\\Omega_{\\tilde\\epsilon,\\tilde r}$, fixing ${\\mathcal C}$ pointwise (i.e. $\\Phi(h,0)=(h,0)$) such that\n$$\n\\Phi\\circ \\hat \\tau_i = \\tau_i\\circ \\Phi,\\quad i=1,\\ldots, n.\n$$\nThis reads $\\hat\\tau_i+\\phi(\\hat\\tau_i)=\\hat \\tau_i(Id+\\phi)+\\tau^{\\bullet}_i(Id+\\phi)$, that is, for all $i=1,\\ldots n$,\n\\begin{equation}\n\\mathcal L_i(\\phi)=\\tau^{\\bullet}_i(Id+\\phi)+\\left(\\hat \\tau_i(Id+\\phi)- \\hat \\tau_i-D\\hat \\tau_i.\\phi\\right)=\\tau^{\\bullet}_i(Id+\\phi).\n\\end{equation}\nHere we define $\\mathcal L_i(\\phi):=\\phi(\\hat\\tau_i)-D\\hat \\tau_i.\\phi$ and\n$ (\\mathcal L^h_i(\\phi^h),\\mathcal L^v_i(\\phi^v)):=\\mathcal L_i(\\phi)$. We have\n$$\n(\\mathcal L^h_i(\\phi^h),\\mathcal L^v_i(\\phi^v))=\\left(\\phi^h(T_ih, M_iv)-T_i\\phi^h(h,v),  \\phi^v(T_ih, M_iv)-M_i\\phi^v(h,v)\\right).\n$$\n  Since $\\hat\\tau_j$ are linear, then $D\\hat\\tau_j=\\hat\\tau_j$.\nExpand the latter in Taylor-Laurent expansions as\n\\begin{eqnarray*}\n\\mathcal L_i^h(\\phi^h)&=&\\sum_{Q\\in{\\mathbb N}^d_2}\\left(\\sum_{P\\in {\\mathbb Z}^n}\\left(\\lambda^P_i\\mu_i^Q\\times Id_n-T_i\\right)\\phi^h_{Q,P}h^P\\right)v^Q, \\quad \\phi^h_{Q,P}\\in {\\mathbb C}^n,\\\\\n\\mathcal L_i^v(\\phi^v)&=&\\sum_{Q\\in{\\mathbb N}^d_2}\\left(\\sum_{P\\in {\\mathbb Z}^n}\\left(\\lambda^P_i\\mu_i^Q\\times Id_d-M_i\\right)\\phi^v_{Q,P}h^P\\right)v^Q, \\quad \\phi^v_{Q,P}\\in {\\mathbb C}^d.\\\\\n\\end{eqnarray*}\nRecall the notations\n$\n\\lambda_{\\ell}=(\\lambda_{\\ell,1},\\ldots,\\lambda_{\\ell,n})$ and $ \\mu_\\ell=(\\mu_{\\ell,1},\\ldots,\\mu_{\\ell,d}).\n$\nWith $P=(p_1,\\ldots, p_n)\\in{\\mathbb Z}^n$ and $Q=(q_1,\\ldots, q_d)\\in \\mathbb{N}^d$, we have\n$$\n\\lambda_{\\ell}^P\\mu_{\\ell}^Q:=\\prod_{i=1}^n\\lambda_{\\ell,i}^{p_i}\\prod_{j=1}^n\\mu_{\\ell,j}^{q_j}.\n$$\n\n\\begin{lemma}\\label{computeFij}Let $\\tau_j\\in\\mathcal A_{\\epsilon,r}^{n+d}$ with\n$\n\\tau_j=\\hat\\tau_j-F_j$ and $ F_j(h,v)=O(|v|^{q+1})\n$\n  with $q\\geq1$.\nSuppose that $\\tau_i\\tau_j=\\tau_j\\tau_i$ in a neighborhood of $\\Omega_{0}^{1\\cdots n}\\times\\{0\\}$ in ${\\mathbb C}^{n+d}$.   Then\n$\n\\mathcal L_iF_j-\\mathcal L_iF_i=O(|v|^{2q+1}).\n$\\end{lemma}\n\\begin{proof\n Recall that $\\hat\\tau_i(h,v)$ are linear maps in $h,v$.\nAlso,  $\\hat\\tau_i\\hat\\tau_j$ sends $\\Omega_{\\epsilon}^j\\times\\Delta_\\epsilon^d$ into ${\\mathbb C}^{n+d}$. Since  $\\tau_j\\in\\mathcal A_{\\epsilon,r}^{n+d}$ and $\\tau_j=\\hat\\tau_j+O(|v|^2)$, the continuity implies that $\\tau_i\\tau_j$ is well-defined on the product domain $\\Omega^{j}_{\\epsilon'}\\times\\Delta_{r'}^d$ when $\\epsilon', r'$ are sufficiently small. Fix $h\\in\\Omega^{j}_{\\epsilon'}$. By Taylor expansions in $v$, we obtain\n$$\n\\tau_i\\tau_j(h,v)=\\hat\\tau_i\\hat\\tau_j(h,v)+\n\\hat\\tau_jF_j (h,v)+F_i\\circ\\hat\\tau_j(h,v)+O(|v|^{2q+1}).\n$$\nSince $\\hat\\tau_i\\hat\\tau_j=\\hat\\tau_j\\hat\\tau_i$ and $\\tau_i\\tau_j=\\tau_j\\tau_i$ in a neighborhood of $\\Omega_{0}^{1\\cdots n}\\times\\{0\\}$, we get $\\mathcal L_iF_j=\\mathcal L_jF_i=O(|v|^{2q+1})$ in a possibly smaller neighborhood of $\\Omega_{0}^{1\\cdots n}\\times\\{0\\}$.\n\\end{proof}\n\n\n\n  We will apply the following result to   $F_j=\\hat\\tau_j-J^{2q}(\\tau_j)$,   where $J^{2q}(\\tau_j)$ denotes the $2q$-jet at $0$ in the variable $v$. These are holomorphic on $\\Omega_{\\epsilon,r}$.\n  Recall that\n  $$\n  \\Omega^{ij}_{\\epsilon',r'}:= \\Omega_{\\epsilon',r'}\\cap\\hat\\tau_i^{-1}(\\Omega_{\\epsilon',r'})\\cap\\hat\\tau_j^{-1}(\\Omega_{\\epsilon',r'})\n  $$\n\\begin{prop}\\label{cohomo}\n\t\tAssume   $N_C$\n\t\tis  \n\t\tDiophantine.   Fix $\\epsilon_0,r_0,\\delta_0$  in $(0,1)$. Let $0<\\epsilon'<\\epsilon<\\epsilon_0$, $0<r'<r<r_0$, $0<\\delta<\\delta_0$, and $\\frac{\\delta}{\\kappa}<\\epsilon$. Suppose that $F_i\\in {\\mathcal A}_{\\epsilon, r}$, $i=1,\\ldots, n$,\n satisfy\n\t\t\\begin{equation}\n\\label{almost-com}  \\mathcal L_i(F_j)-\\mathcal L_j(F_i)=0\\quad\n\\text{on  }  \\Omega^{ij}_{\\epsilon',r'}.\n\t\t\\end{equation}\nThere exist functions  $G \\in  \\tilde{\\mathcal A}_{\\epsilon\n {-{\\delta}\/{\\kappa}},re^{-\\delta}}$\n such that\n\t\t \\begin{align}\\label{formal2q+1}\n\t\t \\mathcal L_i(G)&=F_i \\ \\text{on   } \\Omega_{\\epsilon  {-{\\delta}\/{\\kappa}},re^{-\\delta}}.\n\t\t\t\t \\end{align}\n\t Furthermore, $G$ satisfies\n\t\t\\begin{align}\\label{estim-sol}\n\t\t\\|G\\|_{\\epsilon  {-{\\delta}\/{\\kappa}},re^{-\\delta}}&\\leq  \\max_i\\|F_{i}\\|_{\\epsilon,r}\\frac{C'}{\\delta^{\\tau+\\nu}},\\\\\n\\label{estim-solcompo}\n\t\t\\|G\\circ\\hat\\tau_i\\|_{\\epsilon  {-{\\delta}\/{\\kappa}},re^{-\\delta}}&\\leq  \\max_i\\|F_{i}\\|_{\\epsilon,r}\\frac{  C'}{\\delta^{\\tau+\\nu}}.\n\t\t\\end{align}\n\t\tfor some constant $\\kappa, C'$  that are independent of $F,q,\\delta, r,\\epsilon$ and $\\nu$ that depends only on $n$ and $d$.\nFurthermore, if $F_j(h,v)=O(|v|^{q+1})$ for all $j$, then\n\\eq{}\nG(h,v)=O(|v|^{q+1}), \\quad G(h,v)=J^{2q}G(h,v).\n\\end{equation}\n\\end{prop}\n\n\\begin{proof}\nSince $F_i\\in {\\mathcal A}_{\\epsilon, r}$, we can write\n$$\nF_i(h,v)=\\sum_{Q\\in \\mathbb{N}^d, |Q|\\geq2}\\sum_{P\\in{\\mathbb Z}^n}F_{i,Q,P}h^Pv^Q,\n$$\nwhich converges normally   for $(h,v)\\in\\Omega_{\\epsilon,r}$. We emphasize that $F_{i,Q,P}$\nare vectors, and its $k$th component\n is denoted  by $F_{i,k,Q,P}$.\nFor each $(Q,P)\\in \\mathbb{N}^d\\times {\\mathbb Z}^n$,   each $i=1,\\ldots, n$, and each $j=1,\\ldots ,d$, let $i_h:=i_h(Q,P,i), i_v:=i_v(Q,P,j)$ be in $ \\{1,\\ldots, n\\}$ as in \\rd{def-nr}.\nLet us set\n\\begin{align}\n\tG^h_i&:=\\sum_{Q\\in \\mathbb{N}^d, 2\\leq|Q|\\leq 2q}\\sum_{P\\in{\\mathbb Z}}\\frac{F^h_{i_h,i,Q,P}}{\\lambda_{i_h}^P\\mu_{i_h}^Q-\\lambda_{i_h,i}}h^Pv^Q,\\quad i=1,\\ldots, n\\label{sol-coh-h}\\\\\n\tG^v_j&:=\\sum_{Q\\in \\mathbb{N}^d, 2\\leq |Q|\\leq 2q}\\sum_{P\\in{\\mathbb Z}}\\frac{F^v_{i_v,j,Q,P}}{\\lambda_{i_v}^P\\lambda_{i_v}^Q-\\mu_{i_v,j}}h^Pv^Q,\\quad j=1,\\ldots. d\\label{sol-coh-v}.\n\\end{align}\nAccording to \\re{almost-com}, we have\n\\begin{equation}\\label{compat-coeff}\n(\\lambda_{i_h}^P\\mu_{i_h}^Q-\\lambda_{i_h,i})F_{m,i,Q,P}^h=(\\lambda_{m}^P\\mu_{m}^Q-\\lambda_{m,i}) F_{i_h,i,Q,P}^h, \\quad 2\\leq|Q|\\leq 2q.\n\\end{equation}\nTherefore, using \\re{compat-coeff}, the $i$th-component of $\\mathcal L_m(G)$ reads\n\\begin{align*}\n\t\\mathcal L_m(G^h_i) = &\\sum_{Q\\in \\mathbb{N}^d, 2\\leq |Q|\\leq 2q }\\sum_{P\\in {\\mathbb Z}^n}(\\lambda_{m}^P\\mu_{m}^Q-\\lambda_{m,i})\\frac{F^h_{i_h,i,Q,P}}{(\\lambda_{i_h}^P\\mu_{i_h}^Q-\\lambda_{i_h,i})}h^Pv^Q\\\\\n\t= &\\sum_{Q\\in \\mathbb{N}^d,2\\leq|Q|\\leq 2q}\\sum_{P\\in {\\mathbb Z}^n}F_{m,i,Q,P}^hh^Pv^Q.\n\\end{align*}\nProceeding similarly for the vertical component, we have obtained, the formal equality~:\n\\begin{equation}\\label{formal-lin}\n \\mathcal L_m(G) = F_m\n,\\quad m=1,\\ldots, n\n\\end{equation}\nLet us estimate these solutions.\nAccording to \\rd{dioph} and formulas \\re{sol-coh-h}-\\re{sol-coh-v}, we have\n\\eq{Ghv}\n\\max_{i,j}\\left (|G^h_{i,Q,P}|,|G^v_{j,Q,P}|\\right )\\leq \\max_i|F_{i,Q,P}|\\frac{(|P|+|Q|)^{\\tau}}{D}.\n\\end{equation}\nLet $(h,v)\\in \\Omega_{\\epsilon  {-{\\delta}\/{\\kappa}},re^{-\\delta}}$. According to \\re{est-small},\nwe have\n\\begin{align*}\n\\|G^h_{Q,P}h^Pv^Q\\|&\\leq \\max_i\\|F_{i}\\|_{\\epsilon,r}e^{-\\delta(|P|+|Q|)}\\frac{(|P|+|Q|)^{\\tau}}{D}\\\\\n&\\leq \\max_i\\|F_{i}\\|_{\\epsilon,r}e^{-\\delta\/2(|P|+|Q|)}\\frac{(2\\tau e)^{\\tau}}{D\\delta^{\\tau}}.\n\\end{align*}\nSumming over $P$ and $Q$, we obtain\n$$\n\\|G\\|_{\\epsilon  {-{\\delta}\/{\\kappa}},re^{-\\delta}}\\leq \\max_i\\|F_{i}\\|_{\\epsilon,r}\\frac{C'}{\\delta^{\\tau+\\nu}},\n$$\nfor some constants $C',\\nu$ that are independent of $F,\\epsilon,\\delta$.\n Hence, $G\\in\n \\mathcal A_{\\epsilon  {-{\\delta}\/{\\kappa}},re^{-\\delta}}$.\n\nLet us prove \\re{estim-solcompo}.   Let $B:=2\\max_{k,i,j}(|\\lambda_{k,i}|, |\\mu_{k,j}|)$. Then, there is a constant $D'$ such that\n\\begin{gather}\\label{sd-ehanced}\n\\max_{\\ell\\in \\{1,\\ldots, n\\}}\\left |\\lambda_{\\ell}^P\\mu_{\\ell}^Q-\\lambda_{\\ell,i}\\right|\n\\geq  \\frac{D'\\max_k|\\lambda_{k}^P\\mu_{k}^Q|}{(|P|+[Q|)^{\\tau}},\\\\\n\\label{sd-ehanced+}\n\\max_{\\ell\\in \\{1,\\ldots, n\\}}  \\left |\\lambda_{\\ell}^P\\mu_{\\ell}^Q-\\mu_{{\\ell},j}\\right | \\geq  \\frac{D'\\max_k|\\lambda_{k}^P\\mu_{k}^Q|}{(|P|+[Q|)^{\\tau}}.\n\\end{gather}\nIndeed, if $\\max_k|\\lambda_k^P\\mu_k^Q|<B$, then \\rd{dioph} gives \\re{sd-ehanced} with $D':= \\frac{D}{B}$. Otherwise, if $$|\\lambda_{k_0}^P\\mu_{k_0}^Q|:=\\max_k|\\lambda_{k}^P\\mu_{k}^Q|\\geq B,$$\nthen $|\\lambda_{k_0,i}|\\leq \\frac{B}{2}\\leq \\frac{|\\lambda_{k_0}^P\\mu_{k_0}^Q|}{2}$. Hence, we have\n$$\n\\left |\\lambda_{k_0}^P\\mu_{k_0}^Q-\\lambda_{k_0,i}\\right|\\geq \\left||\\lambda_{k_0}^P\\mu_{k_0}^Q|- |\\lambda_{k_0,i}|\\right|\\geq \\frac{|\\lambda_{k_0}^P\\mu_{k_0}^Q|}{2}.\n$$\nWe have verified \\re{sd-ehanced}. Similarly, we can verified \\re{sd-ehanced+}.\nFinally, combining all cases gives us, for $m=1,\\ldots, n$,\n\\begin{align*}\n|[G\\circ\\hat \\tau_m]_{QP}| &= \\left| G_{Q,P}\\lambda_{m}^P\\mu_{m}^Q\n\\right|\\leq\\max_{\\ell}|F_{\\ell,Q,P}| \\frac{|\\lambda_{m}^P\\mu_{m}^Q|}{|\\lambda_{i_h}^P\\mu_{i_h}^Q-\\lambda_{i_h,i}|}\\\\\n&\\leq \\max_{\\ell}|F_{\\ell,Q,P}|\\frac{|\\lambda_{m}^P\\mu_{m}^Q|(|P|+|Q|)^{\\tau}}{D'\\max_k|\\lambda_{k}^P\\mu_{k}^Q|}\\\\\n&\\leq  \\max_{\\ell}|F_{\\ell,Q,P}|\\frac{(|P|+|Q|)^{\\tau}}{D'}.\\end{align*}\n Hence,  $\\tilde G_m:=G\\circ \\hat \\tau_m\\in \\mathcal A_{\\epsilon  {-{\\delta}\/{\\kappa}},re^{-\\delta}}$.   We can define  $\\tilde G\\in \\tilde{\\mathcal A}_{\\epsilon  {-{\\delta}\/{\\kappa}},re^{-\\delta}}$ such that $\\tilde G=\\tilde G_m\\hat\\tau_m^{-1}$ on $\\hat\\tau_m\\Omega_{\\epsilon,r}$.  We verify that $\\tilde G$ extends to a single-valued holomorphic function of class $\\tilde {\\mathcal A}_{\\epsilon,r}$. Indeed,\n  $\\tilde G_i\\hat\\tau_i^{-1}=\\tilde G_j\\hat\\tau_j^{-1}$ on $\\hat\\tau_i\\Omega_{\\epsilon,r}\\cap \\hat\\tau_j\\Omega_{\\epsilon,r}$, since the latter is connected by \\rl{connected} and the two functions agree with $G$ on $\\hat\\tau_i\\Omega_{\\epsilon,r}\\cap \\hat\\tau_j\\Omega_{\\epsilon,r}\\cap \\Omega_{\\epsilon,r}$ that contains a neighborhood of $\\tilde\\Omega_\\epsilon\\times\\{0\\}$ in ${\\mathbb C}^{n+d}$.\n\\end{proof}\n\n\n  In what follows, we shall set $\\hat\\tau_0=Id$ and for any\n   $f \\in (\\tilde A_{\\epsilon,r})^{\n   n+d}$ and $F\\in (\\tilde{\\mathcal  A}_{\\epsilon,r})^{n+d}$, we set\n\\begin{align*}\n|||f|||_{\\epsilon,r}&:=\\|f\\|_{\\epsilon,r}+\\sum_{i=1}^n\\| \\hat\\tau_i^{-1}\nf\\circ \\hat\\tau_i\\|_{\\epsilon,r}= \\sum_{i=0}^n\\|  \\hat\\tau_i^{-1}\nf\\circ \\hat\\tau_i\\|_{\\epsilon,r},\\\\\nF_{(i)}&:=\\hat\\tau_i^{-1}F\\circ\\hat\\tau_i\\in \\mathcal A_{\\epsilon,r},\\quad F_{(0)}:=F,\\quad i=1,\\ldots, n.\n\\end{align*}\n\n\n\n\\subsection{Iteration scheme}\nWe shall prove the main result through a Newton scheme.\nLet us define sequences of positive real numbers\n$$\n\\delta_{k}:=\n  \\frac{\\delta_0}{(k+1)^2},\n\\quad r_{k+1}:=r_ke^{-5\\delta_k}, \\quad \\epsilon_{k+1}:= \\epsilon_k\n{ {-\\frac{5\\delta_k}{\\kappa}}}\n$$\nsuch that\n\\begin{equation}\\label{sigma}\n\\sigma:=\\sum_{k\\geq 0}\\delta_k <2\\delta_0\n\\end{equation}\nWe assume the following conditions hold~:\n\\begin{gather}\n\t \\label{cond1}\\delta_0<\\frac{\\kappa}{20}\\epsilon_0,\\\\\n\t\\label{cond2}\\delta_0<\\frac{\\ln 2}{10}.\n\\end{gather}\nCondition \\re{cond1} ensures that\n$$\n\\frac{5}{\\kappa}\\sigma<\\frac{10}{\\kappa}\\delta_0<\\frac{\\epsilon_0}{2},\\quad \\text{so that}\\quad \\epsilon_{k}>\\frac{\\epsilon_0}{2},\\; k\\geq 0.\n$$\nCondition \\re{cond2} ensures that\n$$\ne^{-5\\sigma}>e^{-10\\delta_0}>\\frac{1}{2},\\quad \\text{so that}\\quad r_k>\\frac{r_0}{2},\\;  k\\geq 0.\n$$\nLet  $m=5$ be  fixed. We define $\\epsilon_{k+1}<\\epsilon_{k}^{(\\ell)}<\\epsilon_{k}$ and $r_{k+1}<r_{k}^{(\\ell)}<r_{k}$\n, $\\ell=1,\\ldots m$  as follows~:\n\\begin{align*}\n\\epsilon_{k}^{(\\ell)}=\\epsilon_{k} {-\\frac{\\ell\\delta_k}{\\kappa}},&\\quad\\epsilon_{k+1}:=\\epsilon_{k}^{(m)},\\\\\nr_k^{(\\ell)}=r_ke^{-\\ell\\delta_k},&\\quad r_{k+1}:=r_k^{(m)} .\n\\end{align*}\n We emphasize that condition \\re{cond1} ensures $\\epsilon_{k}^{(\\ell)}>0$.\n Let us assume that for each $i=1,\\ldots, n$, $\\tau_i^{(k)}=\\hat \\tau_i+ \\tau^{\\bullet (k)}_i,$ is holomorphic and on $\\Omega_{\\epsilon_{k},r_k}$ it satisfies\n \\eq{4.32-15f}\n  ||\\tau^{\\bullet(k)}||_{\\epsilon_{k},r_k}<\\delta_k^{\\mu}, \\quad \\tau^{\\bullet(k)}(h,v)=O(|v|^{q_k+1}).\n  \\end{equation}\n We further assume that $\\tau_i^{(k)},\\tau_j^{(k)}$ commute on\n $\\Omega^{ij}_{\\epsilon',r'}\n\n \\subset \\Omega_{\\epsilon_{k},r_k}$\n for some positive\n $\\epsilon'<\\epsilon_k,r'<r_k$.\n   We take\n $$\n  q_{k+1}=2q_k+1\\geq q_02^{k}\n $$\n for $q_0\\geq\n 1$ to be determined.\n\nWe will define a sequence $\\Phi^{(k)}$ with   $\\Phi^{(k)}(h,0)=(h,0)$. Let us write on appropriate domains\n\\begin{gather*}\n\\Phi^{(k)}=Id+\\phi^{(k)},\\quad (\\Phi^{(k)})^{-1}:=Id-\\psi^{(k)},\\\\\n\\quad \\tau_i^{(k+1)} = \\Phi^{(k)}\\circ \\tau_i^{(k)}\\circ (\\Phi^{(k)})^{-1},\\quad i=1,\\ldots, n.\n\\end{gather*}\n\n\n\n\n\nNote that there is a constant $C>0$ (depending only on the $\\mu_{i,j}$'s) such that if   $\\epsilon''<\\epsilon'$, $Cr''<r'$ then\n$$\n\\Omega^{ij}_{\\epsilon'',r''}\\subset\\Omega_{\\epsilon'}\\times\\Delta_{r'}^d,\\quad \\Omega_{\\epsilon''}\\times\\Delta_{r''}^d\\subset \\Omega^{ij}_{\\epsilon',r'}.\n$$\nSince the $\\tau_i^{(k)}, \\tau_j^{(k)}$   commute   on $\\Omega_{\\epsilon'}\\times\\Delta_{\\epsilon'}^d$ for some $\\epsilon'>0$ and $\\Phi^{(k)}(h,v)=(h,v)+O(|v|^2)$, then   $\\tau_i^{(k+1)}, \\tau_j^{(k+1)}$ still commute on the same kind of domains for a possibly smaller $\\epsilon'$.\n  By \\rl{computeFij}, we obtain :\n\\begin{align}\\label{compatible}\n\\mathcal L_j(\\tau^{\\bullet (k)}_i)-\\mathcal L_i(\\tau^{\\bullet (k)}_j)=\nO(|v|^{2q_k+1}).\n\\end{align}\n\n\nWe want to find $\\phi^{(k)}$ so that\n\\begin{align}\n\t\\label{tik+1}\n\t\\tau_i^{(k+1)}:=\\hat\\tau_i+\\tau^{\\bullet (k+1)}_i&=\\Phi^{(k)}\\circ\\tau_i^{(k)}\\circ (\\Phi^{(k)})^{-1}\\\\\n\t&=\\hat \\tau_i(Id-\\psi^{(k)})+\\tau^{\\bullet(k)}_i(Id-\\psi^{(k)})\\nonumber\\\\\n\t&\\quad +\\phi^{(k)}\\left(\\hat \\tau_i(Id-\\psi^{(k)})+\\tau^{\\bullet(k)}_i(Id-\\psi^{(k)})\\right)\n\t\\nonumber\\end{align}\nis defined and bounded on $ \\Omega_{\\epsilon_{k+1},r_{k+1}}$.\n\n We now define $\\Phi^{(k)}$ by applying \\rp{cohomo} with these  $F_i:=-J^{2q_k}(\\tau^{\\bullet (k)}_i)$. Let $\\phi^{(k)}$ stands for $G$.  \n Therefore, given $0<\\delta<\\kappa \\epsilon_k^{(1)}$, $\\phi^{(k)}$ is holomorphic and bounded on $\\tilde\\Omega_{\\epsilon_{k} {-\\frac{\\delta}{\\kappa}},r_ke^{-\\delta}}$\n and it satisfies on $\\Omega_{\\epsilon_{k}^{(1)} {-\\frac{\\delta}{\\kappa}},r_k^{(1)}e^{-\\delta}}$,\n\n\\begin{equation}\\label{sol-cohom-approx}\n  \\mathcal L_i(\\phi^{(k)}):=\\phi^{(k)}(\\hat\\tau_i)-D\\hat \\tau_i.\\phi^{(k)}= -J^{2q_k}(\\tau^{\\bullet (k)}_i),\\quad i=1,\\ldots,n.\n\\end{equation}\nWriting formally $\\Phi^{(k)}\\circ \\Psi^{(k)}=Id$ and using linearity of $\\hat\\tau_i$, we obtain\n\\begin{align}\n\\psi^{(k)}_{(i)} &= \\phi^{(k)}_{(i)}(Id-\\psi^{(k)}_{(i)}), \\quad i=0,\\ldots, n,\\label{contr-i}\n\\end{align}\nrecalling the notation $\\phi^{(k)}_{(i)}=\\hat\\tau_i^{-1}\\phi^{(k)}\\circ \\hat\\tau_i$.\nAccording to \\rp{cohomo}, we have\n\\begin{equation}\\label{phik}\n|||\\phi^{(k)}|||_{\\epsilon_k  {-\\frac{\\delta}{\\kappa}},r_ke^{-\\delta}}\\leq ||\\tau^{\\bullet(k)}||_{\\epsilon_k,r_k}\\frac{C'}{\\delta^{\\tau+\\nu}}\\leq C'\\delta_k^{\\mu}\\delta^{-\\tau-\\nu}.\n\\end{equation}\n We recall that the constatn $C'$ does not depend on $k$, nor on $\\tau^{\\bullet(k)}$.\n\\begin{lemma}\\label{l-inclusion}   There is constant $\\tilde D>0$ (independent of $k$) such that, given a positive number $\\mu$,\nso that  $\\delta< \\min\\{\\tilde D,\\ln 2,\\frac{r_0}{2}\\}$  satisfies\n \\eq{2C'}\n 2^{m+2}C'\\delta_k^{\\mu}\\delta^{-\\tau-\\nu-3}<1,\n  \\end{equation}\nwhere  $m=5\n. Then, for all $0\\leq \\ell \\leq m$ and $i=0,\\ldots,n$, the maps $\\Phi^{(k)}_{(i)}:=Id+\\phi^{(k)}_{(i)}$ are biholomorphisms\n\t$$\n\t\\Phi^{(k)}_{(i)}\\colon \\Omega_{\\epsilon_k {-\\frac{(\\ell+1)\\delta}{\\kappa}},r_ke^{-(\\ell+1)\\delta}}\\to \\Omega_{\\epsilon_k  {-\\frac{\\ell\\delta}{2\\kappa}},r_ke^{-\\frac{\\ell\\delta}{2}}\n\t$$\nwith inverse $\\Psi^{(k)}_{(i)}:=Id-\\psi^{(k)}_{(i)} ~:\\Omega_{\\epsilon_k {-\\frac{(\\ell+2)\\delta}{\\kappa}},r_ke^{- (\\ell+2)\\delta}}\\to \\Omega_{\\epsilon_k {-\\frac{(\\ell+1)\\delta}{\\kappa}},r_ke^{-(\\ell+1)\\delta}}$ satisfying\n\t$$\\Phi^{(k)}_{(i)}\\circ \\Psi^{(k)}_{(i)}=\\hat\\tau_i^{-1}(\\Phi^{(k)}\\circ\\Psi^{(k)})\\circ\\hat\\tau_i=Id$$ on $\\Omega_{\\epsilon_k {-\\frac{(\\ell+2)\\delta}{\\kappa}},r_ke^{-(\\ell+2)\\delta}}$,\nprovided $q_0>C(\\delta_0,\\mu)$. \\end{lemma}\n\\begin{proof}\n\t\n\nWe have $1-e^{-\\delta}> \\delta\/2$   and $\\frac{1}{2}<e^{-\\delta}$ for $0<\\delta<\\ln 2$. Assuming  $\\delta<\\frac{r_0}{2}$, we have $\\delta<r_k$\n\nso that\n$$\\operatorname{dist}(\\Delta^d\n_{r_{k}e^{-( 1\n+\\ell)\\delta}},\\partial \\Delta^d\n_{r_ke^{-\\ell\\delta}})=r_ke^{-\\ell\\delta}(1-e^{- \n\\delta})>\\frac{\n\\delta^2}{2^{\\ell+1}}.$$\nOn the other hand,\nby \\re{boundary-of-R}, we have\n$$\\operatorname{dist}(\\Omega_{\\epsilon_{k} {-\\frac{(\\ell+  1\n)\\delta}{\\kappa}}},\\partial \\Omega_{\\epsilon_{k} {-\\frac{\\ell\\delta}{\\kappa}}})=\\operatorname{dist}(\\Omega_{\\epsilon_{k}\n {-\\frac{(\\ell+  1\n )\\delta}{\\kappa}}}^+,\\partial \\Omega_{\\epsilon_{k} {-\\frac{\\ell\\delta}{\\kappa}}}^+).$$\nLet $(e^{-2\\pi R},e^{-2\\pi R'})\\in \\Omega_{\\epsilon_{k} {-\\frac{(\\ell+1)\\delta}{\\kappa}}}^+\\times \\partial\\Omega_{\\epsilon_{k} {-\\frac{\\ell\\delta}{\\kappa}}}^+$. \nSince the matrix $(\\operatorname{Im}\\tau_i^j)_{1\\leq i,j\\leq n}$ is invertible, there is a constant $\\tilde C$ (independent of $k$) such that\n$$\n|e^{-2\\pi R}-e^{-2\\pi R'}|\\geq \\tilde C^{-1}\\left| 1-e^{-2\\pi (R- R')}\\right|>2\\pi\\tilde C^{-1}|R-R'|\\geq  2\\pi\\tilde C^{-1}\\frac{\\delta}{\\kappa}.\n$$\n Let us set $\\tilde D:= 2\\frac{2\\pi}{\\tilde C\\kappa}$.\nAssuming $\\delta<\\tilde D$, we have $\\delta<\\tilde D<2^{\\ell+2}\\tilde D\\leq 2^{m+2}\\tilde D$. Assume $2^{m+2}C'\\delta_k^{\\mu}\\delta^{-\\tau-\\nu-3}<1$. Then according to \\re{phik}, we have, for  $(h,v)\\in \\Omega_{\\epsilon_k {-\\frac{\\ell\\delta}{\\kappa}},r_ke^{-\\ell\\delta}}$,\n\\begin{align}{}\\label{dist-min}\n|\\phi^{(k)}_{(i)}(h,v)|&\\leq C'\\delta_k^{\\mu}\\delta^{-\\tau-\\nu}< \\frac{\\delta^3}{2^{m+2}}<\\frac{\\delta}{2}\\frac{\\delta^2}{2^{\\ell+1}} \\\\\n&< \\frac{\\delta}{2}\\operatorname{dist}(\\Omega_{\\epsilon_k {-\\frac{(\\ell+ 1\n)\\delta}{\\kappa}},r_ke^{-(\\ell+ 1\n)\\delta}},\\partial \\Omega_{\\epsilon_k {-\\frac{\\ell\\delta}{\\kappa}},r_ke^{-\\ell\\delta}}).\\nonumber\n\\end{align}\nBy the Cauchy inequality, we have, for $(h,v)\\in \\Omega_{\\epsilon_k {- \\frac{(\\ell+ 2\n)\\delta}{\\kappa}},r_ke^{-(\\ell+ 2\n)\\delta}}$\n\\eq{Dphi}\n|D\\phi^{(k)}_{(i)}(h,v)|\\leq \\frac{C'\\delta_k^{\\mu}\\delta^{-\\tau-\\nu}}{\\operatorname{dist}(\\Omega_{\\epsilon_k {-\\frac{(\\ell+ 2\n)\\delta}{\\kappa}},r_ke^{-(\\ell+  2\n)\\delta}},\\partial \\Omega_{\\epsilon_k {-\\frac{ (\\ell+1)\\delta}{\\kappa}},r_ke^{- (\\ell+1)\\delta}})}\\leq  \\frac{\\delta}{2\n<1\/2.\n\\end{equation}\nWe can apply the contraction mapping theorem to \\re{contr-i}  together with the last inequality of \\re{dist-min}. We find a holomorphic solution $\\psi^{(k)}_{(i)}$ such that  for $(h,v)\\in \\Omega_{\\epsilon_k {- \\frac{(\\ell+ 3\n)\\delta}{\\kappa}},r_ke^{-(\\ell+  \n)\\delta}}$\n\\begin{align}\n|\\psi^{(k)}_{(i)}(h,v)|&\\leq  ||\\phi^{(k)}_{(i)}||_{\\epsilon_k  {-\\frac{\\delta{({\\bf Corrected})}\n(\\ell+\n2)}{\\kappa}},r_ke^{- (\\ell+2)\n\\delta}}\n\\leq  C'\n \\delta_k^{\\mu}\\delta^{-\\tau-\\nu}\n \\leq  \\frac{\\delta^3}{2^{m+2}}\\label{estimpsi}\\\\\n&<\\frac{\\delta}{2^{m-\\ell-1}}\\operatorname{dist}(\\Omega_{\\epsilon_k {-\\frac{(\\ell+3)\\delta}{\\kappa}},r_ke^{-(\\ell+3)\\delta}},\\partial \\Omega_{\\epsilon_k {-\\frac{(\\ell+2)\\delta}{\\kappa}},r_ke^{-(\\ell+2)\\delta}}).\\nonumber\n\\end{align}\nHence, we have found a mapping $\\Psi^{(k)}_{(i)}:=Id-\\psi^{(k)}_{(i)}$ such that\n$$\n\\hat\\Omega_{\\epsilon_k {-\\frac{(\\ell+3)\\delta}{\\kappa}},r_ke^{-(\\ell+3)\\delta}}:=\\Psi^{(k)}_{(i)}(\\Omega_{\\epsilon_k {-\\frac{(\\ell+3)\\delta}{\\kappa}},r_ke^{-(\\ell+3)\\delta}})\\subset \\Omega_{\\epsilon_k {-\\frac{(\\ell+2)\\delta}{\\kappa}},r_ke^{-(\\ell+2)\\delta}}.\n$$\nAlso, $\\Phi^{(k)}_{(i)}\\circ\\Psi^{(k)}_{(i)}=Id$\n  on   $\\Omega_{\\epsilon_k {-\\frac{(\\ell+3)\\delta}{\\kappa}},r_ke^{-(\\ell+3)\\delta}}$.\nTherefore,\n$$\n\\Phi^{(k)}_{(i)}\\colon\\hat \\Omega_{\\epsilon_k {-\\frac{(\\ell+3)\\delta}{\\kappa}},r_ke^{- (\\ell+3)\\delta}}\\to  \\Omega_{\\epsilon_k {-\\frac{(\\ell\n+1)\\delta}{\\kappa}},r_ke^{-(\\ell\n+1)\\delta}\n$$\nis an (onto) biholomorphism such that $(\\Phi^{(k)}_{(i)})^{-1}=\\Psi^{(k)}_{(i)}$.\n\\end{proof}\n\\begin{prop}\\label{prop4.20}\nKeep conditions on $\\delta,\\mu$ in \\rla{l-inclusion} as well as conditions \\rea{cond1},\\rea{cond2}.\n\tIf $\\delta_0$ small enough   there is possibly larger\n $\\mu>0$  such that if for all $i=1,\\ldots, n$, $\\tau_i^{(0)}\\in  ({\\mathcal A}_{\\epsilon_{0},r_{0}})^{n+d}$ with $|||\\tau^{\\bullet(0)}_i|||_{\\epsilon_{0},r_{0}}\\leq \\delta_{0}^{\\mu}$, then for  all $k\\geq 0$  we have the following~:\n\n\\begin{align}\n\\tau_i^{\\bullet(k+1)}\\in ({\\mathcal A}_{\\epsilon_{k+1},r_{k+1}})^{n+d}, \\quad &\\tau_i^{\\bullet(k+1)}=O(|v|^{2q_k+1}),\\nonumber\\\\\n||\\tau^{\\bullet(k+1)}_i||_{\\epsilon_{k+1},r_{k+1}}&\\leq \\delta_{k+1}^{\\mu}.\\label{tauDotk}\n\\end{align}\n\\end{prop}\n\\begin{proof}\nLet us first show that $\\tau^{(k+1)}_{i}:=\\Phi^{(k)}\\circ \\tau^{(k)}_{i}\\circ(\\Phi^{(k)})^{-1}$\nis well defined on $\\Omega_{\\epsilon_k {- \\frac{\\ell\\delta}{\\kappa}},r_ke^{-\\ell\\delta}}$ for $m\\geq\\ell\\geq 5$ and all $i=1,\\ldots, n$.  Here $m$ is fixed from \\rl{l-inclusion}.\nIndeed, we have\n$$\n\\tau^{(k+1)}_{i}= \\hat\\tau_i(I+\\phi^{(k)}_{(i)})\\circ (I+\\hat\\tau_i^{-1}\\tau_{i}^{\\bullet(k)})\\circ (\\Phi^{(k)})^{-1}.\n$$\n Since $\\tau_i^{\\bullet(k)}$ is of order $\\geq 2q_{k-1}+1$, Schwarz inequality gives\n$$\n\\|\\hat\\tau_i^{-1}\\tau^{\\bullet(k)}_i\\|_{\\epsilon_k  {- \\frac{(\\ell-1)\\delta}{\\kappa}},r_ke^{-(\\ell-1)\\delta}}\\leq \\max_i \\|\\hat\\tau_i^{-1}\\|_{\\epsilon_0,r_0}C e^{-(2q_{k-1}+1)\\delta}\\|\\tau^{\\bullet(k)}_i\\|_{\\epsilon_k  {- \\frac{(\\ell-2)\\delta}{\\kappa}},r_ke^{-(\\ell-2)\\delta}}.\n$$\nWe recall that $\\delta_0$ satisfies \\re{cond1} and \\re{cond2}. Setting $\\delta:=\\delta_k$ and if $q_0$ large enough, we have\n$$\n\\max_i \\|\\hat\\tau_i^{-1}\\|_{\\epsilon_0,r_0}C e^{-(2q_{k-1}+1)\\delta}\\leq \\max_i \\|\\hat\\tau_i^{-1}\\|_{\\epsilon_0,r_0}C e^{-\\frac{2^{k-1}}{(k+1)^2}q_0\\delta_0}<1.\n$$\nAccording to \\rl{l-inclusion},  \\re{2C'} and the distance estimate in \\re{dist-min}, we have\n$$\n(I+\\hat\\tau_i^{-1}\\tau_i^{\\bullet(k)})\\circ (\\Phi^{(k)})^{-1}(\\Omega_{\\epsilon_k {- \\frac{\\ell\\delta}{\\kappa}},r_ke^{-\\ell\\delta}})\\subset \\Omega_{\\epsilon_k {- \\frac{(\\ell-3)\\delta}{\\kappa}},r_ke^{-(\\ell-3)\\delta}}\n$$\nso that $\\tau^{(k+1)}_i$ is defined on $\\Omega_{\\epsilon_k  {- \\frac{\\ell\\delta}{\\kappa}},r_ke^{-\\ell\\delta}}$ for  $\\ell\n=4\\leq m$ since  $\\phi^{(k)}_{(i)}$ is\ndefined on $\\Omega_{\\epsilon_k-\\delta\/\\kappa,r_ke^{-\\delta\/\\kappa}}$.\nFor the rest of the proof, we fix  $\\ell=4$.\n From the argument above, we have\n$$\n\\tau_i^{(k+1)}(\\Omega_{\\epsilon_k-4\\delta\/\\kappa,r_ke^{-4\\delta\/\\kappa}})\\subset \\hat\\tau_i(\\Phi_{(i)}^{k}(\\Omega_{\\epsilon_k-\\delta\/\\kappa,r_ke^{-\\delta\/\\kappa}}))\\subset \\hat\\tau_i(\\Omega_{\\epsilon_k,r_k})\\subset \\hat\\tau_i(\\Omega_{\\epsilon_0,r_0}).\n$$\nHence, $\\tau_i^{\\bullet(k+1)}$ is uniformly bounded on $\\Omega_{\\epsilon_k-4\\delta\/\\kappa,r_ke^{-4\\delta\/\\kappa}}$ w.r.t $k$~:\n$$\n\\|\\tau^{\\bullet (k+1)}_i\\|_{\\epsilon_k {-4\\delta\/\\kappa,r_ke^{-4\\delta}}\n\\leq C.\n$$\n\n\n\nOn the other hand, on $\\Omega_{\\epsilon_k-4\\delta\/\\kappa,r_ke^{-4\\delta\/\\kappa}}$, we have\n\\begin{align}\n\t\\tau^{\\bullet (k+1)}_i&=\\hat \\tau_i(\\phi^{(k)}-\\psi^{(k)})+\\left(\\tau^{\\bullet(k)}_i(Id-\\psi^{(k)})-\\tau^{\\bullet(k)}_i\\right)\\label{remainder1}\\\\\n\t&\\quad+\\left(\\phi^{(k)}\\left(\\hat \\tau_i-\\hat \\tau_i\\psi^{(k)}+\\tau^{\\bullet(k)}_i(Id-\\psi^{(k)})\\right)-\\phi^{(k)}(\\hat \\tau_i)\\right)\n\\nonumber\n\\\\\n\t&\\quad+(\\phi^{(k)}(\\hat \\tau_i)-\\hat \\tau_i\\phi^{(k)}+\\tau^{\\bullet(k)}_i).\\nonumber\n\\end{align}\n  The last term is equal to $\\tau^{\\bullet(k)}_i-J^{2q_k}(\\tau^{\\bullet(k)}_i)=O(|v|^{2q_k+1})$.\nWe also have $\\phi^{(k)}-\\psi^{(k)}=O(|v|^{2q_k+1})$, $\\phi^{(k)}=O(|v|^{q_k+1})$. Thus\n\\eq{tauk+1=2q+1}\n\\tau^{\\bullet (k+1)}_i=O(|v|^{2q_k+1}).\n\\end{equation}\n\nImproving the estimate by the Schwarz inequality, we obtain\n$$\n\\|\\tau^{\\bullet(k+1)}_i\\|_{\\epsilon_k  {- \\frac{5\\delta}{\\kappa}},r_ke^{-5\\delta}}\\leq C e^{-q_{k+1}\\delta}.\n$$\nWe have $e^{-x}<1\/x$ for $x>0$. For   $\\delta:=\\delta_k< \\min\\{\\tilde D,\\ln 2,\\frac{r_0}{2}\\}$,   let $\\mu$ satisfy $2^{m+2}C'\\delta_k^{\\mu-\\tau-\\nu-3}<1$, in which case assumptions of \\rl{l-inclusion} are satisfied. We then obtain\n$$\nC\\delta_{k+1}^{-\\mu}e^{-q_{k+1}\\delta} \\leq C \\left(\\f{(k+2)^2}{\\delta_0}\\right) ^{\\mu +1}\\f{1}{ q_02^{k}}<1\n$$\nprovided $q_0>C(\\delta_0,\\mu)$.\n\\end{proof}\nFinally, quite classically, since \\re{Dphi} and \\re{sigma}, the sequence of diffeomorphisms $\\{\\Phi_k\\circ\\Phi_{k-1}\\circ\\cdots\\circ \\Phi_1\\}_k$ converges uniformly on the open set $\\Omega_{\\frac{\\epsilon_{0}}{2},r_{0}e^{-\\sigma}}$  to a diffeomorphism $\\Phi$ which satisfies\n$$\n\\Phi\\circ \\hat \\tau_i = \\tau_i\\circ \\Phi,\\quad i=1,\\ldots, n\n$$\n  provided $q_0\\geq C(\\delta_0,\\mu)$ and    $ ||\\tau^{\\bullet(0)}_i||_{\\epsilon_{0},r_{0}}\\leq \\delta_{0}^{\\mu}$ for some $\\delta_0,\\mu$ are fixed. The condition $q_0>C(\\delta_0,\\mu,\\tau,\\nu)$ can be achieved by using finitely many $\\Phi_0,\\dots, \\Phi_{m}$.  Then initial condition $ ||\\tau^{\\bullet(0)}_i||_{\\epsilon_{0},r_{0}}\\leq \\delta_{0}^{\\mu}$   can be achieved easily by a dilation in the $v$ variable. Indeed, we apply the dilation in $v$ to the original $\\tau_1,\\dots,\\tau_n$ with $q_0=1$. This allows us to construct $\\Phi_0$ in \\rp{prop4.20} with $k=0$ and define $\\Phi_0\\tau_j\\Phi_0^{-1}$ to achieve  $q_1\\geq 2$. Then $\\Phi_0\\tau_j\\Phi_0^{-1}$, $j=1,\\dots, n$ still commute pairwise on $\\Omega_{\\epsilon'}\\times\\Delta_{\\epsilon'}^d$ for some $\\epsilon'>0$. Applying the procedure again, this allows us to find $\\Phi_1,\\dots, \\Phi_{k-1}$ to achieve $q_k\\geq 2^{k}>  C(\\delta_0,\\mu)$. Finally using dilation we can apply the full version of \\rp{prop4.20} for all $k$ to construction a new sequence of desired mapping $\\Phi_0,\\Phi_1,\\dots.$\nHence, the torus $C$ has a neighborhood in $M$ biholomorphic to a neighborhood of the zero section in its normal bundle $N_C$  since there is a biholomorphism fixing ${\\mathcal C}$ that conjugate the deck transformations of the covering of the latter to those of the former.\n\n \\begin{remark}\n\tThe assumption \"$T_CM$ splits\" can be replaced by the condition that for all $P\\in {\\mathbb Z}^n$, for all $Q\\in \\mathbb{N}^d$, $|Q|=1$ and for all $i=1,\\ldots, n$\n\t$$\n\t\\max_{\\ell\\in \\{1,\\ldots, n\\}}\\left |\\lambda_\\ell^P\n\\mu_{\\ell}^Q-\\lambda_{\\ell,i}\n\\right|   >  \\frac{D}{(|P|+1)^{\\tau}}.\n\t$$\n\\end{remark}\n\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzfsez b/data_all_eng_slimpj/shuffled/split2/finalzzfsez
new file mode 100644
index 0000000000000000000000000000000000000000..0250fb9437d6ece2ee43332bc6779acb076d838c
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzfsez
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\\label{sec1}\n\n\n\nLet $K$ be a smooth immersed curve in the plane.  Fabricius-Bjerre \n\\cite{FB1} found the following relation among the double tangent \nlines, crossings, and inflections points for a generic $K$:\n$$\nT_1-T_2=C+(1\/2)I\n$$\nwhere $T_1$ and $T_2$ are the number of exterior and interior double \ntangent lines of $K$, $C$ is the number of crossings, and $I$ is the \nnumber of inflection points.  Here ``generic'' means roughly that the\ninteresting attributes of the curve are invariant under small smooth perturbations.    Fabricius-Bjerre remarks on an example \ndue to Juel which shows that the theorem cannot be straightforwardly \ngeneralized to the projective plane.  A series of papers followed. \nHalpern \\cite{H} re-proved the theorem and obtained some additional \nformulas using analytic techniques.   Banchoff \n\\cite{B} proved an analogue of the theorem for piecewise linear \nplanar curves, using deformation methods.   Fabricius-Bjerre  \ngave a variant of the theorem for curves with cusps \\cite {FB2}. \nWeiner \\cite{W} generalized the formula to closed curves lying on a \n2--sphere.  Finally Pignoni \\cite{P} generalized the formula to curves \nin real projective space, but his formula depends, both in the \nstatement and in the proof, on the selection of a base point for the \ncurve.  Ferrand \\cite{F} relates the Fabricius-Bjerre and Weiner formulas\nto Arnold's invariants for plane curves.  Note that any formula for curves\nin $\\mathbb{R}P^2$ is more general than one for curves in $\\mathbb{R}^2$, since one\ncan specialize to curves in $\\mathbb{R}^2$ by considering curves lying inside a\nsmall disk in $\\mathbb{R}P^2$.     \n\nThere are two main results in this paper.  The first is a \ngeneralization of the theorem in \\cite {FB2} to $\\mathbb{R}P^2$, with no \nreference to a basepoint on the curve.  The original theorem is transparently a \nspecial case of this result, which is not surprising as the \ntechniques used to prove it are a combination of those found in \\cite{FB1} and \nin \\cite{W}.  The difficulties encountered in the generalization \nare due to the problems in distinguishing between two ``sides'' of a\nclosed geodesic in $\\mathbb{R}P^2$.  These are overcome by a careful attention to the natural \nmetric on the space inherited from the round 2--sphere of radius one.   \n\nThe main results are tied together by the observation that, in the \nversion of the original formula which includes cusps \\cite {FB2}, the quantities in the formula \nare naturally dual to each other in $\\mathbb{R}P^2$.  This leads to the second, more surprising,  \nmain result, which is a dual formula for generic curves in $\\mathbb{R}P^2$. \nThis specializes to a new formula for generic smooth curves in the plane. \nThis new formula has the interesting property that it reveals delicate \ngeometric distinctions between topologically similar planar \ncurves, for example quantifying some of the differences between the \ntwo curves shown in \\fullref{twocurves}.\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[scale=0.50]{\\figdir\/fbfig1}\\label{twocurves}\n\\caption{}\n\\end{figure}\n\nThe outline of the paper is as follows: in \\fullref{sec2} we state and \nprove the generalization of \\cite{FB2} to curves in $\\mathbb{R}P^2$.  In \n\\fullref{sec3} we describe the duality between terms of the formula.  In \n\\fullref{sec4} we state and prove the dual formulation, and give its \ncorollaries for planar curves.\n\\vspace{-2pt}\n\n\\section{A Fabricius-Bjerre formula for curves in $\\mathbb{R}P^2$}\\label{sec2}\n\\vspace{-2pt}\n\nLet $\\mathbb{R}P^2$ be endowed with the spherical metric, inherited from its double cover, the \nround 2--sphere of radius one.  With this metric, a simple closed \ngeodesic (or projective line)  in $\\mathbb{R}P^2$ has length $\\pi$.  The figures will use a \nstandard disk model for $\\mathbb{R}P^2$, in which the boundary of the disk \ntwice covers a closed geodesic.\nLet $K$ be a generic oriented closed curve in $\\mathbb{R}P^2$, which is smoothly immersed \nexcept for cusps of type 1, that is, cusps at which locally the two \nbranches of $K$ coming into the cusp are on opposite sides of the \ntangent geodesic.  We postpone the definition of {\\em generic} until the end of section 3. \nWe will need some definitions.\n\n\n\n{\\bf Definitions}\\par\nLet {\\em $\\tau_p$} be the geodesic tangent to $K$ at $p$, with \norientation induced by $K$.\n\nLet {\\em $a_p$}, the {\\em antipodal point to $p$}, be the point on \n$\\tau_p$ a distance $\\pi\/2$ from $p$.\n\n\n$\\tau_p$ is divided by $p$ and $a_p$ into two pieces.\nLet {\\em ${\\tau_p}^+$} be the segment from $p$ to $a_p$ and  {\\em \n${\\tau_p}^-$} the segment from $a_p$ to $p$.\nAt cusp points ${\\tau_p}^+$ and  ${\\tau_p}^-$ are not well-defined.\n\nLet {\\em $\\nu_p$} be the normal geodesic to $K$ at $p$.\n\nLet {\\em $c_p$} (which lies on $\\nu_p$) be the center of curvature of \n$K$ at $p$, that is, the center of the osculating circle to $K$ at $p$.\n\nWe orient {\\em $\\nu_p$} so that the length of the (oriented) segment \nfrom $p$ to $c_p$ is less than the length of the segment from $c_p$ \nto $p$.  This orientation is well-defined except at cusps and \ninflection points.\n\nThere is a natural duality from $\\mathbb{R}P^2$ to itself.  Under this duality \nsimple closed geodesics, or projective lines,  in $\\mathbb{R}P^2$ are sent to points and vice versa. \nThis duality is most easily described by passing to the 2--sphere $S$ \nwhich is the double cover of $\\mathbb{R}P^2$; in this view a simple closed \ngeodesic in $\\mathbb{R}P^2$ lifts to a great circle on $S$.  If this great \ncircle is called the equator,  the dual point in $\\mathbb{R}P^2$ is the image \nof the north (or south) pole.\n\nUnder this duality the image of $K$ is a {\\em dual curve} $K'$.   To \ndescribe $K'$ we need only observe that a point on $K$ comes equipped \nwith a tangent geodesic, $\\tau_p$.  The {\\em dual point to $p$}, \ncalled $p'$,  is the point dual to the tangent geodesic $\\tau_p$.\n\nAnother useful description is that $p'$ is the point a distance \n$\\pi\/2$ along the normal  geodesic to $K$ at $p$.  Notice that\n$\\nu_p=\\nu_{p'}$ and $c_{p}=c_{p'}$.\n\n\nAn ordered pair of points $(p,q)$ on $K$ is an {\\em antipodal pair} if $q=a_p$.\n\n\nLet {\\em $Y_p$} be the geodesic dual to the point $c_p$.\n\n\nLet $(p,q)$ be an antipodal pair.  Then\n $Y_p$ and $\\tau_p$\nintersect at $q$ and divide $\\mathbb{R}P^2$ into two regions, $R_1$ and $R_2$.\nOne of the regions, say $R_1$, contains $c_p$.  The geodesic $\\tau_q$\nlies in one of the two regions.     An antipodal pair $(p,q)$ is of\ntype 1 if  $\\tau_q$ lies in $R_1$,  type 2 if  $\\tau_q$ lies in\n$R_2$.  Let $A_1$ be the number of type 1 antipodal pairs of $K$,\n$A_2$ the number of type 2.\n\n\n$T$ is a {\\em double-supporting geodesic} of $K$ if $T$  is either a \ndouble tangent geodesic, a tangent geodesic through a cusp or a \ngeodesic through two cusps.  The two tangent or cusp points of $K$ \ndivide $T$ into two segments, one of which has length less than \n$\\pi\/2$.  We distinguish two types of double supporting geodesics, \ndepending on whether the two points of $K$ lie on the same side of \nthis segment  (type 1)  or opposite sides (type 2).  Let\n{\\em $T_1$} be the number of double supporting geodesics of $K$ of \ntype 1, {\\em $T_2$} the number of type 2 (see \\fullref{fig2}).\n\n\\begin{figure}[ht!]\\small\n\\centering\n\\includegraphics[scale=0.60]{\\figdir\/fbfig2a}\\qquad\\qquad\n\\includegraphics[scale=0.60]{\\figdir\/fbfig2b}\n\\cl{Type 1 double supporting geodesic \\qquad\\qquad\nType 2 double supporting geodesic}\n\\caption{}\\label{double tangent types}\n\\label{fig2}\n\\end{figure}\n\nThe tangent geodesics at a crossing of $K$ define four angles, two of \nwhich, $\\alpha$ and $\\beta$, are less than $\\pi\/2$.  In a small \nneighborhood of a crossing there are four segments of $K$.   The \ncrossing is of type 1 if one of these segments lies in $\\alpha$ and \nanother in $\\beta$, type 2 if two lie in $\\alpha$ or two lie in \n$\\beta$.  Let $C_1$ be the number of type 1 crossings of $K$, $C_2$ \nthe number of type 2 (see \\fullref{fig3}). \n\\vspace{2pt}\n\n\\begin{figure}[ht!]\\small\n\\centering\n\\includegraphics[scale=0.70]{\\figdir\/fbfig3}\\label{crossing types}\n\\cl{type 1 crossing\\qquad\\qquad type 2 crossing}\n\\caption{}\n\\label{fig3}\n\\end{figure}\n\\vspace{2pt}\n\nLet {\\em $I$} be the number of inflection points of $K$.\n\\vspace{2pt}\n\nLet {\\em $U$} be the number of (type 1) cusps of $K$.\n\\vspace{2pt}\n\nWe are now ready to state the first main theorem, which is a \ngeneralization of the main theorem of \\cite {FB1} to the projective \nplane.  We note that (unlike \\cite{P}) we do not need to \nchoose a base-point for $K$.\n\\vspace{2pt}\n\n\\begin{theorem}\\label{main1}\nLet $K$ be a generic singular curve in $\\mathbb{R}P^2$ with type 1 cusps.  Then\n$$\nT_1-T_2=C_1+C_2+(1\/2)I+U-(1\/2)A_1+(1\/2)A_2\n$$\n\\end{theorem}\n\n\\begin{proof}\nThe proof proceeds as in \\cite {FB2}, with some caution being \nrequired at antipodal pairs and at cusp points.  We choose a starting \npoint $p$ on $K$. Let ${M_p}^+$ be the number of times $K$ intersects \n${\\tau_p}^+$,  ${M_p}^-$ be the number of times $K$ intersects \n${\\tau_p}^-$, and $M_p={M_p}^+- {M_p}^-$.  We keep track of how $M_p$ \nchanges as we traverse the knot.   Double-supporting geodesics, \ncrossings, cusps and inflection points all behave as in \\cite{FB2}. \nSuppose $p$ is a point of an antipodal pair $(p,q)$. Let $p_1$ be a \npoint immediately before $p$ on $K$, $p_2$ a point immediately after. \nIf $(p,q)$ is of type 1 then $\\tau_{p_1}$ intersects the arc of $K$ \ncontaining $q$ on ${\\tau_ {p_1}}^-$ and  $\\tau_ {p_2}$ intersects the \narc of $K$ containing $q$ on ${\\tau_ {p_2}}^+$, hence $M_p$ increases \nby 2 as we pass through $p$. If $(p,q)$ is of type 2 then $\\tau_ \n{p_1}$ intersects the arc of $K$ containing $q$ on ${\\tau_ {p_1}}^+$ \nand  $\\tau_ {p_2}$ intersects the arc of $K$ containing $q$ on \n${\\tau_{p_2}}^-$, hence $M_p$ decreases by 2 as we pass through $p$. \nThis is easiest to see by approximating $K$ near $p$ by a circle \ncentered at $c_p$.  As we traverse a piece of this circle from $p_1$ \nthrough $p$ to $p_2$, the antipodal points to $p_1$ and $p_2$ lie on \nthe geodesic $ Y_p$.  Hence the critical distinction to be made at $q$ \nis where the tangent geodesic at $q$ lies in relation to $Y_p$ and \n$\\tau_p$.   This is the exactly the distinction between type 1 and \ntype 2 antipodal pairs.\n\\end{proof}\n\n\\section{Duality in $\\mathbb{R}P^2$}\\label{sec3}\n\nWe describe the dual relations between crossings and double \ntangencies, cusps and inflection points, and antipodal points and \nnormal-tangent pairs (defined below).\n\n\\medskip\n{\\bf Definitions}\\qua\nThe points $p$ and $c_p$ divide $\\nu_p$ into two pieces, {\\em \n${\\nu_p}^+$} from $p$ to $c_p$ and  {\\em ${\\nu_p}^-$} from $c_p$ to \n$p$.  An ordered pair of points $(p,q)$ on $K$ is a {\\em \nnormal-tangent pair} if $\\tau_q=\\nu_p$.\nA normal-tangent pair $(p,q)$ is of type 1 if $q$ lies on ${\\nu_p}^-$, \ntype 2 if $q$ lies on ${\\nu_p}^+$ (\\fullref{fig4}).  Let $N_1$ be the number of type \n1 normal-tangent pairs of $K$, $N_2$ the number of type 2 .\n\n\\begin{figure}[ht!]\n\\centering\n\\labellist\\small\n\\pinlabel $q$ [t] <0pt, -2pt> at 92 332\n\\pinlabel $c(p)$ [t] at 155 332\n\\pinlabel $p$ [tl] <0pt, -2pt> at 212 332\n\\pinlabel {$(p,q)$ is a type 1 normal-tangent pair} at 146 254\n\\pinlabel $q$ [t] <0pt, -2pt> at 185 156\n\\pinlabel $c(p)$ [t] at 155 156\n\\pinlabel $p$ [tl] <0pt, -2pt> at 212 156\n\\pinlabel {$(p,q)$ is a type 2 normal-tangent pair} at 146 87\n\\endlabellist\n\\includegraphics[scale=0.70]{\\figdir\/fbfig4}\\label{normal-tangent}\n\\caption{}\n\\label{fig4}\n\\end{figure}\n\n\\begin{prop} \\label{dualcor}\nLet $K$ be a generic curve in $\\mathbb{R}P^2$, with dual curve $K'$.  Let $i=1,2$.\nThen:\n\\begin{enumerate}\n\\item A crossing of type $i$ in $K$ is dual to a double supporting \ngeodesic of type $i$ in $K'$.\n\\item A cusp in $K$ is dual to an inflection point in $K'$.\n\\item An antipodal pair of type $i$ in $K$ is dual to a normal-tangent \npair  of type $i$ in $K'$.\n\\end{enumerate}\n\nAs the dual of $K'$ is again $K$, these correspondences work in both \ndirections.\n\\end{prop}\n\n\\begin{proof}\nThe proof is by construction in $\\mathbb{R}P^2$.\n\\end{proof}\n\n\nThis correspondence breaks down slightly when we consider double \nsupporting geod\\-esics between cusps and tangents, or cusps and cusps. \nFabricius-Bjerre suggests a small local alteration of $K$ to \nunderstand his argument at a cusp point, replacing the cusp point by \na small ``bump\".  Just as the dual to a small round circle in $\\mathbb{R}P^2$ \nis a (long) curve that is close to a geodesic, his local change at \ncusps induces a more global change at inflection points, and so in \norder to incorporate curves with inflection points we need to add \n{\\em inflection geodesics} to our picture of $K$.\n\nLet $p$ be an inflection point of $K$, with tangent geodesic \n$\\tau_p$.  Endow $\\tau_p-p$ with a normal direction at each point \n(except the inflection point) by the convention shown in \\fullref{fig5}.\n\n\\begin{figure}[ht!]\\small\n\\centering\n\\includegraphics[scale=0.70]{\\figdir\/fbfig5}\\label{inflectiongeo}\n\\cl{inflection geodesic with normal direction}\n\\caption{}\n\\label{fig5}\n\\end{figure}\n\n\\medskip\n{\\bf Definition}\\qua\nCall this the {\\em inflection geodesic to $K$ at $p$}.\n\nFor crossings between $K$ and an inflection geodesic $\\tau_p$, or \nbetween two inflection geodesics, the piece of $\\tau_p$ in the \nneighborhood of the crossing should be construed as bending slightly \ntowards its normal direction for the purposes of classifying the \ncrossing type.   This convention preserves the correct duality between crossing type in $K$ and \ndouble supporting geodesic type in $K'$. A point on the inflection geodesic has center of \ncurvature a distance $\\pi\/2$ in the normal direction, at $p'$. For \n$\\alpha$ a point on an inflection geodesic $\\tau_p$, $\\nu_{\\alpha}$ \nis the geodesic through $\\alpha$ and $p'$.\n\n\\medskip\n{\\bf Definition}\\qua\nDenote by  $\\wbar{K}$,  $K$ together with all its inflection \ngeodesics.    Crossings and normal tangencies are counted as described \nabove. The inflection points of $K$ (where the inflection geodesic\nintersects the curve) will still be counted as simply inflection points in\n$\\wbar{K}$, not as new crossing points.\n\nIf $K$ is a generic curve with dual $K'$, then double supporting geodesics in $K$ involving cusp points correspond to crossings in\n$\\wbar{K}'$ involving inflection geodesics,  and an antipodal pair \n$(p,q)$ with $p$ a cusp point will correspond to a normal-tangent \npair $(p',q')$ with $p'$ a point on an inflection geodesic in \n$\\wbar{K}'$.   \n\nWe end this section with the definition of what it means for $K$ to be generic:\n\n\\medskip\n{\\bf Definition}\\qua\n$K$ is {\\em generic} if:\n\\begin{itemize}\n\\item $K$ has a finite number of crossings, double tangent lines, cusps, inflection points, antipodal pairs, and normal-tangent pairs.\n\\item  Tangent geodesics at self-intersections of $K$ are neither \nparallel nor perpendicular.\n\\item The tangent geodesic through an \ninflection point or at a cusp is everywhere else transverse to $K$.\n\\item A geodesic goes through at most two tangent points or cusps of \n$K$.\n\\item No crossings occur at inflection points.\n\\item A geodesic normal to $K$ at one point is tangent to $K$ at at most one point and everywhere else transverse to $K$.\n\\item The distance between two points on a double-supporting geodesic is not $\\pi\/2$.\n\\item If $(p,q)$ is an antipodal pair, let $ Y_p$ be the geodesic dual \nto $c_p$.  Then $\\tau_q$ should be neither $\\tau_p$ nor $ Y_p$.\n\\item If $(p,q)$ on $K$ is a  normal-tangent pair, $q$ is not $c_p$.\n\\end{itemize}\n\n\\section{A dual formula, with applications}\\label{sec4}\n\nThe simplest version of the dual theorem applies to curves with no \ninflection points.\n\n\n\\begin{theorem}\\label{dualthm}\nLet $K$ be a generic singular curve in $\\mathbb{R}P^2$ with type 1 cusps and \nno inflection points.  Then\n$$C_1-C_2=T_1+T_2+(1\/2)U-(1\/2)N_1+(1\/2)N_2$$\n\\end{theorem}\n\nSince inflection points are dual to cusps, we also have:\n\\begin{cor}\nLet $K$ be the dual in $\\mathbb{R}P^2$ of a smooth singular curve.  Then\n\\fullref{dualthm} holds for $K$.\n\\end{cor}\n\n\\proof[Proof of \\fullref{dualthm}]\nThe theorem follows directly from duality on $\\mathbb{R}P^2$, but \nit is illuminating to consider the dual of the proof of Theorem 1, \nas it provides a direct proof for curves in the plane.   In Theorem 1 \nwe count the number of intersections between the curve $K$ and \n$\\tau_p$, with appropriate signs, as the curve is traversed once. \nHence the main technical point is to understand the dual to $M_p$. \n\nAssign an orientation to $K$.  The geodesics $\\tau_p$ and $\\nu_p$ \nintersect in a single point (at $p$) and divide $\\mathbb{R}P^2$ into two \nregions.  We first define the {\\em tangent-normal frame $F_p$} to $K$ \nat p as follows: $F_p$ is the union of $\\tau_p$ and $\\nu_p$ together \nwith a black-and-white coloring of the two regions of $\\mathbb{R}P^2$.  We \ncolor them by the convention that if we think of $\\tau_p$ and $\\nu_p$ \nat $P$ as being analogous to the standard $x- $ and $y-$ axes, the \nregion corresponding to the quadrants where $x$ and $y$ have the same \nsign is colored white, the complementary region black (see \\fullref{fig6}). \nThe frame and its coloring are well-defined at points that are \nneither cusps nor inflection points.  At cusps, the orientations of \n$\\tau_p$ and $\\nu_p$ {\\em both} reverse as we traverse $K$, with the \nhappy effect that the coloring of the normal-tangent frame is \nwell-defined as we pass through a cusp point (notice that this is not \ntrue if we allow type 2 cusps).\n\n\\begin{figure}[ht!]\n\\centering\n\\labellist\\small\\hair 5pt\n\\pinlabel $p$ [tl] at 109 428\n\\pinlabel $p$ [tl] <0pt,-1pt> at 104 143\n\\endlabellist\n\\includegraphics[scale=0.50]{\\figdir\/fbfig6}\n\\caption{}\n\\label{fig6}\n\\end{figure}\n\n\nWe now describe how the tangent-normal frame for the dual curve $K'$ is related to $M_p$  for the curve $K$.\n\nLet $p$ be a point on $K$ with \ntangent geodesic $\\tau_p$.  Let $r$ be a point of intersection \nbetween $K$ and $\\tau_p$.  $r$ contributes either $+1$ or $-1$ to \n$M_p$, depending on where it lies relative to the antipodal point \n$a_p$.  What does $r$ correspond to in the dual picture?  Under \nduality $p$ is sent to the point $p'$, and $\\nu_p=\\nu_p'$.  The point \n$r$ is mapped to a geodesic $g_r$ through $p'$.  A small neighborhood \nof $r$ in $K$ is mapped to an arc of $K'$ tangent to $g_r$. If $r$ \ncontributes $+1$ to $M_p$, $g_r$ lies in the white region of the \ntangent-normal frame at $p'$.  If $r$ contributes $-1$ to $M_p$, \n$g_r$ lies in the black region of the tangent-normal frame at $p'$. \nThis leads us to the following definition.\n\n\\medskip\n{\\bf Definition}\\qua\nAt a given point $p$ on $K$, we define $W_p$ to be the number of \ngeodesics through $p$ and tangent to $K$ which lie in the white \nregion and $B_p$ to be the number of geodesics through $p$ and \ntangent to $K$ which lie in the black region as defined by the \ntangent-normal frame at $p$. Let $V_p=W_p-B_p$.  The proof consists \nof tracking how $V_p$ changes as we traverse $K$ once; each type of \nsingularity contributes to $V_p$ according to the following table:\n\n\\def\\vrule width 0pt depth 5pt height 12pt{\\vrule width 0pt depth 5pt height 12pt}\n\\hfill\\begin{tabular}{||c|c||} \\hline\\hline\n\\vrule width 0pt depth 5pt height 12pt{\\em Singularity} & {\\em Contribution}\\\\ \\hline\n\\vrule width 0pt depth 5pt height 12pt$C_1$ & $+4$ \\\\ \\hline\n\\vrule width 0pt depth 5pt height 12pt$C_2$ & $-4$ \\\\ \\hline\n\\vrule width 0pt depth 5pt height 12pt$T_i$ & $-4$ \\\\ \\hline\n\\vrule width 0pt depth 5pt height 12pt$U$ & $-2$ \\\\ \\hline\n\\vrule width 0pt depth 5pt height 12pt$N_1$ & $+2$ \\\\ \\hline\n\\vrule width 0pt depth 5pt height 12pt$N_2$ & $-2$ \\\\ \\hline\n\\end{tabular}\\hfill\\lower60pt\\hbox{$\\square$}\n\n\n\\medskip\nWe can use the natural duality directly for the general case:\n\n\\begin {theorem}\nLet $K$ be a generic singular curve in $\\mathbb{R}P^2$ with type 1 cusps. \nThen for $\\wbar{K}$,\n$$\nC_1-C_2=T_1+T_2+(1\/2)U+I-(1\/2)N_1+(1\/2)N_2\n$$\n\\end {theorem}\n\\medskip\n\n\nIf $K$ is a curve with no cusps, inflection points, or antipodal pairs\n(or for a smooth immersed curve in $\\mathbb{R}^2$ with no inflection points),\nthen the pair of formulas:\n\\begin{eqnarray*}\nT_1-T_2 & = & C_1+C_2\\\\\nC_1-C_2 & = & T_1+T_2-(1\/2)N_1+(1\/2)N_2\n\\end{eqnarray*}\napplies, and combining them we can obtain:\n\\medskip\n \n\n\\begin {cor}\\label{tcurves}\n\nFor $K$ a curve with no cusps, inflection points, or \nantipodal pairs (or for a smooth immersed curve in $\\mathbb{R}^2$ with no \ninflection points):\n\\begin{eqnarray*}\n4T_1-4C_1=N_1-N_2\\\\\n4T_2+4C_2=N_1-N_2\n\\end{eqnarray*}\n\\end {cor}\n\\medskip\n\n\\begin{figure}[ht!]\\small\n\\centering\n\\includegraphics[scale=0.55]{\\figdir\/fbfig7}\\label{twocurves*}\n\\cl{$C_1=0$, $C_2=1$, $N_1=4$\\qquad\\qquad $C_1=1$, $C_2=0$, $N_1=N_2=0$}\n\\caption{}\\label{fig7}\n\\end{figure}\n\\medskip\n\n\n\nNote that for the two curves shown in \\fullref{twocurves} (redrawn \nin \\fullref{fig7}) , we obviously have the values $T_1=1$, $T_2=0$.  For \nthe right-hand curve, $C_1=1$ and $C_2=0$, while for the left, \n$C_1=0$ and $C_2=1$.   By observation, the right curve has no \nnormal-tangent pairs, and the two equations in \\fullref{tcurves} are easily seen to be satisfied.  Applying \\fullref{tcurves} to the left-hand curve, however, we obtain\n$$\n4=N_1-N_2\n$$\nand we can locate four normal-tangent pairs of type 1 (\\fullref{fig7}).\n\\newpage\n\n{\\bf Acknowledgement}\\qua Research supported in part by an NSF grant\nand by the von Neumann Fund and the Weyl Fund through the Institute\nfor Advanced Study.\n\n\\bibliographystyle{gtart}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nSince 1995, the Bose-Einstein condensation (BEC) of ultracold atomic\nand molecular gases has attracted considerable interests both\ntheoretically and experimentally. These trapped quantum gases are\nvery dilute and  most of their properties are governed by the\ninteractions between particles in the condensate \\cite{Pitaevskii}.\nIn the last several years, there has been a quest for realizing a\nnovel kind of quantum gases with the dipolar interaction, acting\nbetween particles having a permanent magnetic or electric dipole\nmoment. A major breakthrough has been very recently performed at\nStuttgart University, where a BEC of ${}^{52}$Cr atoms has been\nrealized in experiment and it allows the experimental investigations\nof the unique properties of dipolar quantum gases \\cite{Griesmaier}.\nIn addition, recent experimental developments on cooling and\ntrapping of molecules \\cite{Ellio}, on photoassociation \\cite{Wang},\nand on Feshbach resonances of binary mixtures  open much more\nexciting perspectives towards a degenerate quantum gas of polar\nmolecules \\cite{Sage}.  These success of experiments have spurred\ngreat excitement in the atomic physics community and renewed\ninterests in  studying the ground states\n\\cite{Santos,Yi,Goral,Goral1,Jiang,Ronen} and dynamics\n\\cite{Lahaye1,Parker, Pedri,Yi1} of dipolar BECs.\n\n\nAt temperature $T$ much smaller than the critical temperature $T_c$,\na dipolar BEC is well described by the macroscopic wave function\n$\\psi=\\psi({\\bf x} ,t)$ whose evolution is governed by the\nthree-dimensional (3D) Gross-Pitaevskii equation (GPE)\n\\cite{Yi,Santos} \\begin{equation} \\label{ngpe} i\\hbar \\partial_t\n\\psi({\\bf x} ,t)=\\left[-\\fl{\\hbar^2}{2m}\\nabla^2+V({\\bf x} )+U_0|\\psi|^2+\n\\left(V_{\\rm dip}\\ast |\\psi|^2\\right)\\right]\\psi, \\quad {\\bf x} \\in{\\mathcal\nR}^3, \\ t>0, \\end{equation} where $t$ is time, ${\\bf x} =(x,y,z)^T\\in {\\mathcal R^3}$ is\nthe Cartesian coordinates, $\\hbar$ is the Planck constant, $m$ is\nthe mass of a dipolar particle and $V({\\bf x} )$ is an external trapping\npotential. When a harmonic trap potential is considered,\n$V({\\bf x} )=\\fl{m}{2}(\\omega_{x}^2x^2+ \\omega_y^2y^2+\\omega_{z}^2 z^2)$\nwith $\\omega_x$, $\\omega_y$ and $\\omega_z$ being the trap\nfrequencies in $x$-, $y$- and $z$-directions, respectively.\n$U_0=\\frac{4\\pi \\hbar^2 a_s}{m}$ describes local (or short-range)\ninteraction between dipoles in the condensate with $a_s$ the\n$s$-wave scattering length (positive for repulsive interaction and\nnegative for attractive interaction). The long-range dipolar\ninteraction potential between two dipoles is given by\n\\begin{equation}\\label{kel0} V_{\\rm dip}({\\bf x} )= \\frac{\\mu_0\\mu_{\\rm\ndip}^2}{4\\pi}\\,\\fl{1-3({\\bf x} \\cdot \\bf\nn)^2\/|{\\bf x} |^2}{|{\\bf x} |^3}=\\frac{\\mu_0\\mu_{\\rm\ndip}^2}{4\\pi}\\,\\fl{1-3\\cos^2(\\theta)}{|{\\bf x} |^3}, \\qquad {\\bf x} \\in{\\mathcal\nR}^3,\\end{equation}  where $\\mu_0$ is the vacuum magnetic permeability,\n$\\mu_{\\rm dip}$ is permanent magnetic dipole moment (e.g. $\\mu_{\\rm\ndip}=6\\mu_{_B}$ for $^{52}$C$_{\\rm r}$ with $\\mu_{_B}$ being the\nBohr magneton), ${\\bf n}=(n_1,n_2,n_3)^T\\in {\\mathcal R}^3$ is the\ndipole axis (or dipole moment) which is a given unit vector, i.e.\n$|{\\bf n}|=\\sqrt{n_1^2+n_2^2+n_3^3}=1$, and $\\theta$ is the angle\nbetween the dipole axis ${\\bf n}$ and the vector ${\\bf x} $. The wave\nfunction is normalized according to \\begin{equation}\\label{norm00}\n\\|\\psi\\|^2:=\\int_{{\\mathcal R}^d} |\\psi({\\bf x} ,t)|^2\\;d{\\bf x}  =N,\\end{equation} where $N$\nis the total number of dipolar particles in the dipolar BEC.\n\n\nBy introducing the dimensionless variables, $t\\to \\frac{t}{\\omega_{0}}$\nwith $\\omega_0=\\min\\{\\omega_x,\\omega_y,\\omega_z\\}$,  ${\\bf x}  \\to a_0{\\bf x} $ with $\na_0=\\sqrt{\\fl{\\hbar}{m\\omega_{0}}}$, $\\psi\\to \\frac{\\sqrt{N}\n\\psi}{a_0^{3\/2}}$,\n we obtain the dimensionless GPE  in 3D from (\\ref{ngpe})\n as \\cite{Yi,Yi2,Pitaevskii,Bao_Jaksch_Markowich}:\n \\begin{equation} \\label{ngpe1} i\\partial_t \\psi({\\bf x} ,t)=\\left[-\\fl{1}{2}\\nabla^2+V({\\bf x} )+\\beta\n|\\psi|^2+\\lambda \\left(U_{\\rm dip}\\ast|\\psi|^2\\right)\\right]\\psi,\n\\qquad {\\bf x} \\in{\\mathcal R}^3, \\quad t>0,\\end{equation} where $\\beta\n=\\frac{NU_0}{\\hbar\\omega_0 a_0^3}= \\fl{4\\pi a_sN}{a_0}$, $\\lambda\n=\\fl{mN\\mu_0\\mu_{\\rm dip}^2}{3\\hbar^2 a_0}$,\n$V({\\bf x} )=\\fl{1}{2}(\\gamma_{x}^2x^2+ \\gamma_y^2y^2+\\gamma_{z}^2 z^2)$\nis the dimensionless harmonic trapping potential with\n$\\gamma_x=\\frac{\\omega_x}{\\omega_0}$, $\\gamma_y=\\frac{\\omega_y}{\\omega_0}$ and\n$\\gamma_z=\\frac{\\omega_z}{\\omega_0}$, and the dimensionless long-range\ndipolar interaction potential $U_{\\rm dip}({\\bf x} )$ is given as\n\\begin{equation}\\label{kel} U_{\\rm dip}({\\bf x} )= \\frac{3}{4\\pi}\\,\\fl{1-3({\\bf x} \\cdot\n\\bf n)^2\/|{\\bf x} |^2}{|{\\bf x} |^3}=\\frac{3}{4\\pi}\\,\n\\fl{1-3\\cos^2(\\theta)}{|{\\bf x} |^3}, \\qquad {\\bf x} \\in{\\mathcal R}^3.\\end{equation} From\nnow on, we will treat $\\beta$ and $\\lambda$ as two dimensionless\nreal parameters. We understand that it may not physical meaningful\nwhen $\\lambda<0$ for modeling dipolar BEC. However, it is an\ninteresting problem to consider the case when $\\lambda<0$ at least\nin mathematics and it may make sense for modeling other physical\nsystem. In fact, the above nondimensionlization is obtained by\nadopting a unit system where the units for length, time and energy\nare given by $a_0$, $1\/\\omega_0$ and $\\hbar \\omega_0$, respectively. Two\nimportant invariants of (\\ref{ngpe1}) are the {\\sl mass} (or\nnormalization) of the wave function\n \\begin{equation}\\label{norm}\nN(\\psi(\\cdot,t)):=\\|\\psi(\\cdot,t)\\|^2= \\int_{{\\mathcal R}^3}\n|\\psi({\\bf x} ,t)|^2\\;d {\\bf x} \\equiv \\int_{{\\mathcal R}^3} |\\psi({\\bf x} ,0)|^2\\;d\n{\\bf x} =1, \\qquad t\\ge0, \\end{equation} and the {\\sl energy} per particle \\begin{eqnarray}\n E(\\psi(\\cdot,t))&:=&\\int_{{\\mathcal R}^3}\\left[\n\\frac{1}{2}|\\nabla\\psi |^2 +V({\\bf x} ) |\\psi|^2 +\\fl{\\beta}{2}|\\psi|^4 +\n\\fl{\\lambda}{2}\\left(U_{\\rm dip}\\ast|\\psi|^2\\right) |\\psi|^2\n\\right]d {\\bf x} \\nonumber\\\\\n&\\equiv& E(\\psi(\\cdot,0)), \\qquad t\\ge0. \\label{energy} \\end{eqnarray}\n\n\nTo find the stationary states including ground and excited states of\na dipolar BEC, we take the ansatz \\begin{equation} \\label{stat}\n\\psi({\\bf x} ,t)=e^{-i\\mu t}\\phi ({\\bf x} ), \\qquad {\\bf x} \\in{\\mathcal R}^3, \\quad\nt\\ge0,\\end{equation} where $\\mu\\in{\\mathcal R}$ is the chemical potential and\n$\\phi:=\\phi({\\bf x} )$ is a time-independent function. Plugging\n(\\ref{stat}) into (\\ref{ngpe1}), we get the time-independent GPE (or\na nonlinear eigenvalue problem) \\begin{eqnarray}\n \\label{gpe22dstat} \\mu\\, \\phi({\\bf x} )=\\left[-\\fl{1}{2 }\\nabla^2\n+V({\\bf x} )+\\beta|\\phi|^2+\\lambda\\left(U_{\\rm\ndip}\\ast|\\phi|^2\\right)\\right] \\phi({\\bf x} ),\\qquad {\\bf x} \\in{\\mathcal R}^3,\n\\end{eqnarray} under the constraint \\begin{equation}\\label{const} \\|\\phi\\|^2:=\\int_{{\\mathcal\nR}^3}|\\phi({\\bf x} )|^2\\;d{\\bf x} =1.\\end{equation}\n The ground state of a dipolar BEC\nis usually defined as\nthe minimizer of the following nonconvex minimization problem:\\\\\nFind $\\phi_g \\in S $ and $\\mu^g\\in {\\mathcal R}$   such that \\begin{equation}\n\\label{groundstate} E^g:=E(\\phi_g)=\\min_{\\phi\\in S}\\ E (\\phi),\\qquad\n\\mu^g:=\\mu(\\phi_g), \\end{equation} where the nonconvex set $S$ is defined as\n\\begin{equation} S:=\\left\\{\\phi \\ |\\  \\|\\phi\\|^2=1, \\, E(\\phi)<\\infty \\right\\}\n\\end{equation} and the chemical potential (or eigenvalue of (\\ref{gpe22dstat}))\nis defined as \\begin{eqnarray}\n \\mu(\\phi)&:=&\\int_{{\\mathcal R}^3}\\left[\n\\frac{1}{2}|\\nabla\\phi |^2 +V({\\bf x} ) |\\phi|^2 +\\beta|\\phi|^4 +\n\\lambda\\left(U_{\\rm dip}\\ast|\\phi|^2\\right) |\\phi|^2\n\\right]d {\\bf x} \\nonumber\\\\\n&\\equiv& E(\\phi)+\\frac{1}{2}\\int_{{\\mathcal R}^3}\\left[ \\beta|\\phi|^4 +\n\\lambda\\left(U_{\\rm dip}\\ast|\\phi|^2\\right) |\\phi|^2 \\right]d {\\bf x} .\n\\label{chem00} \\end{eqnarray} In fact, the nonlinear eigenvalue problem\n(\\ref{gpe22dstat}) under the constraint (\\ref{const}) can be viewed\nas the Euler-Lagrangian equation of the nonconvex minimization\nproblem (\\ref{groundstate}). Any eigenfunction of the nonlinear\neigenvalue problem (\\ref{gpe22dstat}) under the constraint\n(\\ref{const}) whose energy is larger than that of the ground state\nis usually called as an excited state in the physics literatures.\n\n\nThe theoretical study of dipolar BECs including ground states and\ndynamics as well as quantized vortices has been carried out in\nrecent years based on the GPE (\\ref{ngpe}). For the study in\nphysics, we refer to\n\\cite{Eberlein,Giovanazzi,Klawunn,Recati,Abad,Glaum,Klawunn,Nath,Odell0,\nWilson,Wilson1,Yi2,Zhang} and references therein. For the study in\nmathematics, existence and uniqueness as well as the possible\nblow-up of solutions were studied in \\cite{Carles}, and existence of\nsolitary waves was proven in \\cite{Ant}. In most of the numerical\nmethods used in the literatures for theoretically and\/or numerically\nstudying the ground states and dynamics of dipolar BECs, the way to\ndeal with the convolution in (\\ref{ngpe1}) is usually to use the\nFourier transform \\cite{Lahaye1,Goral,Ronen,Xiong,Blakie,Tick,Yi3}.\nHowever, due to the high singularity in the dipolar interaction\npotential (\\ref{kel}), there are two drawbacks in these numerical\nmethods: (i) the Fourier transforms of the dipolar interaction\npotential (\\ref{kel}) and the density function $|\\psi|^2$ are\nusually carried out in the continuous level on the whole space\n${\\mathcal R}^3$ (see (\\ref{four11}) for details) and in the discrete\nlevel on a bounded computational domain $\\Omega$, respectively, and due\nto this mismatch, there is a locking phenomena in practical\ncomputation as observed in \\cite{Ronen}; (ii) the second term in the\nFourier transform of the dipolar interaction potential is\n$\\frac{0}{0}$-type for $0$-mode, i.e when $\\xi=0$ (see\n(\\ref{four11}) for details), and it is artificially omitted when\n$\\xi=0$ in practical computation\n\\cite{Ronen,Goral1,ODell,Yi1,Yi2,Xiong,Blakie} thus this may cause\nsome numerical problems too. The main aim of this paper is to\npropose new numerical methods for computing ground states and\ndynamics of dipolar BECs which can avoid the above two drawbacks and\nthus they are more accurate than those currently used in the\nliteratures. The key step is to decouple the dipolar interaction\npotential into a short-range and a long-range interaction (see\n(\\ref{Udip0}) for details) and thus we can reformulate the GPE\n(\\ref{ngpe1}) into a Gross-Pitaevskii-Poisson type system. In\naddition, based on the new mathematical formulation, we can prove\nexistence and uniqueness as well as nonexistence of the ground\nstates and discuss mathematically the dynamical properties of\ndipolar BECs in different parameter regimes.\n\nThe paper is organized as follows. In section 2, we reformulate the\nGPE for a dipolar BEC into a Gross-Pitaevskii-Poisson type system\nand study analytically the ground states and dynamics of dipolar\nBECs. In section 3, a backward Euler sine pseudospectral method is\nproposed for computing ground states of dipolar BECs; and in section\n4, a time-splitting sine pseudospectral (TSSP) method is presented\nfor computing the dynamics. Extensive numerical results are reported\nin section 5 to demonstrate the efficiency and accuracy of our new\nnumerical methods.  Finally, some conclusions are drawn in section\n6. Throughout this paper, we adapt the standard Sobolev spaces and\ntheir corresponding norms.\n\n\n\\section{Analytical results for ground sates and dynamics}\n\\setcounter{equation}{0}\n\nLet $r=|{\\bf x} |=\\sqrt{x^2+y^2+z^2}$ and denote \\begin{equation}\\label{pbn0}\n\\partial_{\\bf n} ={\\bf n}\\cdot \\nabla=n_1\\partial_x+n_2\\partial_y+n_3\\partial_z, \\qquad\n\\partial_{{\\bf n} \\bn}=\\partial_{\\bf n} (\\partial_{\\bf n} ).\\end{equation} Using the equality (see \\cite{Parker}\nand a mathematical proof in the Appendix)  \\begin{equation} \\label{decop1}\n\\fl{1}{r^3}\\left(1-\\frac{3({\\bf x} \\cdot {\\bf n})^2}{r^2}\\right) =\n-\\fl{4\\pi}{3} \\delta ({\\bf x} )-\\partial_{{\\bf n} \\bn}\\left(\n\\frac{1}{r}\\right),\\qquad {\\bf x} \\in {\\mathcal R}^3, \\label{formula}\\end{equation} with\n$\\delta({\\bf x} )$ being the Dirac distribution function and  introducing\na new function \\begin{equation} \\label{varp0} \\varphi({\\bf x} ,t):=\\left(\\frac{1}{4\\pi\n|{\\bf x} |}\\right) \\ast |\\psi(\\cdot,t)|^2 = \\fl{1}{4\\pi} \\int_{{\\mathcal\nR}^3} \\fl{1}{|{\\bf x} -{\\bf x} '|}|\\psi({\\bf x} ',t)|^2\\;d{\\bf x} ', \\qquad {\\bf x} \\in{\\mathcal\nR}^3, \\quad t\\ge0,\\end{equation} we obtain \\begin{equation}\\label{Udip0} U_{\\rm\ndip}\\ast|\\psi(\\cdot,t)|^2 = -|\\psi({\\bf x} ,t)|^2-3\\partial_{{\\bf n} \\bn}\n\\left(\\varphi ({\\bf x} , t)\\right), \\qquad {\\bf x} \\in {\\mathcal R}^3, \\quad\nt\\ge0. \\label{integral}\\end{equation} In fact, the above equality decouples the\ndipolar interaction potential into a short-range and a long-range\ninteraction which correspond to the first and second terms in the\nright hand side of (\\ref{Udip0}), respectively. In fact, from\n(\\ref{pbn0})-(\\ref{Udip0}), it is straightforward to get the Fourier\ntransform of $U_{\\rm dip}({\\bf x} )$ as \\begin{equation}\\label{four11}\n\\widehat{(U_{\\rm dip})}(\\xi)=-1+\\frac{3\\left({\\bf n} \\cdot\n\\xi\\right)^2}{|\\xi|^2}, \\qquad \\xi\\in {\\mathcal R}^3. \\end{equation} Plugging\n(\\ref{Udip0}) into (\\ref{ngpe1}) and noticing (\\ref{varp0}), we can\nreformulate the GPE (\\ref{ngpe1}) into  a Gross-Pitaevskii-Poisson\ntype system \\begin{eqnarray} \\label{gpe} &&i \\partial_t\n\\psi({\\bf x} ,t)=\\left[-\\fl{1}{2}\\nabla^2+V({\\bf x} )+(\\beta-\\lambda)\n|\\psi({\\bf x} ,t)|^2-3\\lambda \\partial_{{\\bf n} \\bn} \\varphi({\\bf x} ,t)\n\\right]\\psi({\\bf x} ,t),  \\\\\n\\label{poisson}&&\\qquad \\nabla^2 \\varphi({\\bf x} ,t) =\n-|\\psi({\\bf x} ,t)|^2,\\qquad\n\\lim\\limits_{|{\\bf x} |\\to\\infty}\\varphi({\\bf x} ,t)=0\\qquad {\\bf x} \\in{\\mathcal R}^3,\n\\quad t>0.\n \\end{eqnarray}\nNote that the far-field condition in (\\ref{poisson}) makes the\nPoisson equation uniquely solvable. Using (\\ref{poisson}) and\nintegration by parts, we can reformulate the energy functional\n$E(\\cdot)$ in (\\ref{energy}) as \\begin{equation}\\label{newener}\nE(\\psi)=\\int_{\\mathcal R^3 }\\left[\\frac 12|\\nabla\n\\psi|^2+V({\\bf x} )|\\psi|^2+\\frac{1}{2}(\\beta-\\lambda\n)|\\psi|^4+\\frac{3\\lambda}{2}\\left|\\partial_{\\bf n}\\nabla\n\\varphi\\right|^2\\right]\\,d{\\bf x} \\,,\\end{equation} where $\\varphi$ is defined\nthrough (\\ref{poisson}). This immediately shows that the decoupled\nshort-range and long-range interactions of the dipolar interaction\npotential are attractive and repulsive, respectively,  when\n$\\lambda>0$; and are repulsive and attractive, respectively, when\n$\\lambda<0$. Similarly, the nonlinear eigenvalue problem\n(\\ref{gpe22dstat}) can be reformulated as \\begin{eqnarray}\n \\label{gpe22dstat1} &&\\mu\\, \\phi({\\bf x} )=\\left[-\\fl{1}{2 }\\nabla^2\n+V({\\bf x} )+\\left(\\beta-\\lambda\\right)|\\phi|^2-3\\lambda \\partial_{{\\bf n} \\bn}\n\\varphi({\\bf x} )\\right] \\phi({\\bf x} ), \\\\\n&&\\qquad  \\nabla^2 \\varphi({\\bf x} ) =-|\\phi({\\bf x} )|^2, \\quad {\\bf x} \\in{\\mathcal\nR}^3,\\qquad\n\\lim\\limits_{|{\\bf x} |\\to\\infty}\\varphi({\\bf x} )=0.\\label{poi101} \\end{eqnarray}\n\n\n\n\n\n\n\\subsection{Existence and uniqueness for ground states}\nUnder the  new formulation for the energy functional $E(\\cdot)$ in\n(\\ref{newener}), we have\n\n\n\n\\begin{lemma}\\label{lem1} For the energy  $E(\\cdot)$ in (\\ref{newener}), we have\n\n(i) For any $\\phi\\in S$, denote $\\rho({\\bf x} )=|\\phi({\\bf x} )|^2$ for\n${\\bf x} \\in{\\mathcal R}^3$, then we have \\begin{equation} E(\\phi)\\geq\nE(|\\phi|)=E\\left(\\sqrt{\\rho}\\right), \\qquad \\forall \\phi\\in S,\\end{equation} so\nthe minimizer $\\phi_g$ of (\\ref{groundstate}) is of the form\n$e^{i\\theta_0}|\\phi_g|$ for some constant $\\theta_0\\in \\mathcal R$.\n\n(ii) When $\\beta\\ge0$ and $-\\frac12\\beta\\leq \\lambda\\leq \\beta$, the\nenergy $E(\\sqrt{\\rho})$ is strictly convex in $\\rho$.\n\\end{lemma}\n\n\\noindent {\\bf Proof:} For any $\\phi\\in S$, denote $\\rho=|\\phi|^2$\nand consider the Poisson equation \\begin{equation}\n\\label{poi11}\\nabla^2\\varphi({\\bf x} )=-|\\phi({\\bf x} )|^2:=-\\rho({\\bf x} ), \\quad\n{\\bf x} \\in{\\mathcal R}^3, \\qquad\n\\lim\\limits_{|{\\bf x} |\\to\\infty}\\varphi({\\bf x} )=0.\\end{equation} Noticing (\\ref{pbn0})\nwith $|{\\bf n} |=1$,  we have the estimate \\begin{equation} \\label{pbnf1}\\|\\partial_{\\bf\nn}\\nabla \\varphi\\|_2\\leq\\|D^2\\varphi\\|_2=\\|\\nabla^2\\varphi\\|_2=\n\\|\\rho\\|_2=\\|\\phi\\|_4^2,\\qquad \\hbox{with} \\quad D^2=\\nabla\\nabla.\n\\end{equation}\n\n\n\n(i) Write $\\phi({\\bf x} )=e^{i\\theta({\\bf x} )}|\\phi({\\bf x} )|$, noticing\n(\\ref{newener}) with $\\psi=\\phi$ and (\\ref{poi11}), we get\n\\begin{eqnarray}\nE(\\phi)&=&\\int_{\\mathcal R^3}\\left[|\\,\\nabla\n|\\phi|\\,|^2+|\\phi|^2|\\nabla\\theta({\\bf x} )|^2+V({\\bf x} )|\\phi|^2+\\frac{1}{2}(\\beta-\\lambda\n)|\\phi|^4+\\frac{3\\lambda}{2}|\\partial_{\\bf n}\\nabla\n\\varphi|^2\\right]\\,d{\\bf x} \\nonumber\\\\\n&\\geq&\\int_{\\mathcal R^3}\\left[|\\,\\nabla\n|\\phi|\\,|^2+V({\\bf x} )|\\phi|^2+\\frac{1}{2}(\\beta-\\lambda\n)|\\phi|^4+\\frac{3\\lambda}{2}|\\partial_{\\bf n}\\nabla\n\\varphi|^2\\right]\\,d{\\bf x} \\nonumber\\\\\n&=&E(|\\phi|)=E\\left(\\sqrt{\\rho}\\right), \\qquad \\forall \\phi\\in S,\n\\end{eqnarray}\nand the equality holds iff $\\nabla\\theta({\\bf x} )=0$ for ${\\bf x} \\in {\\mathcal\nR}^3$, which means $\\theta({\\bf x} )\\equiv \\theta_0$ is a constant.\n\n(ii) From (\\ref{newener}) with $\\psi=\\phi$ and noticing\n(\\ref{poi11}), we can split the energy $E\\left(\\sqrt{\\rho}\\right)$\ninto two parts, i.e.\n\\begin{eqnarray}E(\\sqrt{\\rho})\n=E_1(\\sqrt{\\rho})+E_2(\\sqrt{\\rho}), \\end{eqnarray} where \\begin{eqnarray}\\label{e11}\n&&E_1(\\sqrt{\\rho})=\\int_{\\mathcal R^3}\\left[|\\nabla\n\\sqrt{\\rho}|^2+V({\\bf x} )\\rho\\right]d{\\bf x} , \\\\ \\label{e12}\n&&E_2(\\sqrt{\\rho})=\\int_{\\mathcal R^3}\\left[\\frac{1}{2}(\\beta-\\lambda\n)|\\rho|^2+\\frac{3\\lambda}{2}|\\partial_{\\bf n}\\nabla\n\\varphi|^2\\right]\\,d{\\bf x} . \\label{e1e2} \\end{eqnarray} As shown in \\cite{Lie},\n$E_1\\left(\\sqrt{\\rho}\\right)$ is convex (strictly) in $\\rho$. Thus\nwe need only prove $E_2\\left(\\sqrt{\\rho}\\right)$ is convex too. In\norder to do so, consider $\\sqrt{\\rho_1}\\in S$, $\\sqrt{\\rho_2}\\in S$,\nand let $\\varphi_1$ and $\\varphi_2$ be the solutions of the Poisson\nequation (\\ref{poi11}) with $\\rho=\\rho_1$ and $\\rho=\\rho_2$,\nrespectively. For any $\\alpha\\in[0,1]$, we have\n$\\sqrt{\\alpha\\rho_1+(1-\\alpha)\\rho_2}\\in S$, and\n\\begin{eqnarray}\n\\lefteqn{\\alpha E_2(\\sqrt{\\rho_1})+(1-\\alpha)E_2(\\sqrt{\\rho_2})-E_2\n\\left(\\sqrt{\\alpha\\rho_1+(1-\\alpha)\\rho_2}\\right)\\nonumber}\\\\[2mm]\n&=&\\alpha(1-\\alpha)\\int_{\\mathcal R^3}\\left[\\frac{1}{2}(\\beta-\\lambda\n)(\\rho_1-\\rho_2)^2+\\frac{3\\lambda}{2}|\\partial_{\\bf n}\\nabla\n(\\varphi_1-\\varphi_2)|^2\\right]\\,d{\\bf x} ,\\label{poi00}\n\\end{eqnarray}\nwhich immediately implies that $E_2(\\sqrt{\\rho})$ is convex if\n$\\beta\\ge0$ and $0\\leq\\lambda\\leq \\beta$.  If $\\beta\\ge0$ and\n$-\\frac{1}{2}\\beta\\leq \\lambda<0$, noticing that\n$\\alpha\\varphi_1+(1-\\alpha)\\varphi_2$ is the solution of the Poisson\nequation (\\ref{poi11}) with $\\rho=\\alpha\\rho_1+(1-\\alpha)\\rho_2$,\ncombining (\\ref{pbnf1}) with $\\varphi=\\varphi_1-\\varphi_2$ and\n(\\ref{poi00}), we obtain $E_2(\\sqrt{\\rho})$ is convex again.\nCombining all the results above together, the conclusion follows.\n$\\Box$\n\nNow, we are able to prove the existence and uniqueness as well as\nnonexistence results for the ground state of a dipolar BEC in\ndifferent parameter regimes.\n\n\\begin{theorem} Assume $V({\\bf x} )\\ge0$ for ${\\bf x} \\in{\\mathcal R}^3$ and\n$\\lim\\limits_{|{\\bf x} |\\to\\infty}V({\\bf x} )=\\infty$ (i.e., confining\npotential), then we have:\n\n(i) If $\\beta\\ge0$ and $-\\frac{1}{2}\\beta\\leq \\lambda\\leq \\beta$,\nthere exists a ground state $\\phi_g\\in S$, and the positive ground\nstate $|\\phi_g|$ is unique. Moreover, $\\phi_g=e^{i\\theta_0}|\\phi_g|$\nfor some constant $\\theta_0\\in\\mathcal R$.\n\n(ii) If $\\beta<0$, or $\\beta\\ge0$ and $\\lambda<-\\frac{1}{2}\\beta$ or\n$\\lambda>\\beta$, there exists no ground state, i.e.,\n$\\inf\\limits_{\\phi\\in S}E(\\phi)=-\\infty$.\n\\end{theorem}\n\n\\noindent {\\bf Proof:} (i) Assume $\\beta\\ge0$ and\n$-\\frac{1}{2}\\beta\\leq \\lambda\\leq \\beta$, we first show $E(\\phi)$\nis nonnegative in $S$, i.e. \\begin{equation}\\label{Ephi} E(\\phi)=\\int_{\\mathcal\nR^3}\\left[|\\nabla \\phi|^2+V({\\bf x} )|\\phi|^2+\\frac{1}{2}(\\beta-\\lambda\n)|\\phi|^4+\\frac{3\\lambda}{2}|\\partial_{\\bf n}\\nabla\n\\varphi|^2\\right]\\,d{\\bf x} \\ge0, \\qquad \\forall \\phi\\in S.\\end{equation} In fact,\nwhen $\\beta\\ge0$ and $0\\leq\\lambda\\leq \\beta$, noticing\n(\\ref{newener}) with $\\psi=\\phi$, it is obvious that (\\ref{Ephi}) is\nvalid. When $\\beta\\ge0$ and  $-\\frac 12\\beta\\leq \\lambda<0$,\ncombining  (\\ref{newener}) with $\\psi=\\phi$, (\\ref{poi11}) and\n(\\ref{pbnf1}), we obtain (\\ref{Ephi}) again as \\begin{eqnarray}\nE(\\phi)&\\ge&\\int_{\\mathcal R^3}\\left[|\\nabla\n\\phi|^2+V({\\bf x} )|\\phi|^2+\\frac{1}{2}(\\beta-\\lambda)|\\phi|^4\n+\\frac{3\\lambda}{2}|\\phi|^4\\right]\\,d{\\bf x} \\nonumber\\\\\n&=&\\int_{\\mathcal R^3}\\left[|\\nabla\n\\phi|^2+V({\\bf x} )|\\phi|^2+\\frac{1}{2}\\left(\\beta+2\\lambda\\right)|\\phi|^4\n\\right]\\,d{\\bf x} \\ge0. \\end{eqnarray} Now, let $\\{\\phi^n\\}_{n=0}^\\infty\\subset S$\nbe a minimizing sequence of the minimization problem\n(\\ref{groundstate}). Then there exists a constant $C$ such that \\begin{equation}\n\\|\\nabla\\phi^n\\|_2\\le C, \\qquad \\|\\phi^n\\|_4\\le C, \\qquad \\int_{\\mathcal\nR^3} V({\\bf x} )|\\phi^n({\\bf x} )|^2d{\\bf x} \\le C, \\qquad n\\ge0.\\end{equation} Therefore\n$\\phi^n$ belongs to a weakly compact set in $L^4$, $H^1=\\{\\phi\\ |\\\n\\|\\phi\\|_2+\\|\\nabla \\phi\\|_2<\\infty\\}$, and $L^2_V=\\{\\phi\\ |\\\n\\int_{{\\mathcal R}^3} V({\\bf x} ) |\\phi({\\bf x} )|^2\\; d{\\bf x} <\\infty\\}$ with a\nweighted $L^2$-norm given by $\\|\\phi\\|_V=[\\int_{{\\mathcal\nR}^3}|\\phi({\\bf x} )|^2V({\\bf x} )d{\\bf x} ]^{1\/2}$. Thus, there exists a\n$\\phi^\\infty\\in H^1\\bigcap L^2_V\\bigcap L^4$ and a subsequence\n(which we denote as the original sequence for simplicity), such that\n\\begin{equation} \\label{conveg0}\n \\phi^n\\rightharpoonup\\phi^\\infty,\\quad  \\mbox{in } L^2\\cap L^4\\cap\n L^2_V,\\qquad\\quad\n\\nabla \\phi^n\\rightharpoonup\\nabla\\phi^\\infty,\\quad \\mbox{in } L^2.\n\\end{equation} Also, we can suppose that $\\phi^n$ is nonnegative, since we can\nreplace them with $|\\phi^n|$ , which also minimize the functional\n$E$.  Similar as in \\cite{Lie}, we can obtain $\\|\\phi^\\infty\\|_2=1$\ndue to the confining property of the potential $V({\\bf x} )$. So,\n$\\phi^\\infty\\in S$. Moreover, the $L^2$-norm convergence of $\\phi^n$\nand weak convergence in (\\ref{conveg0}) would imply the strong\nconvergence $\\phi^n\\to\\phi^\\infty\\in L^2$. Thus, employing\nH\\\"{o}lder inequality and Sobolev inequality, we obtain\n\\begin{eqnarray}\\lefteqn{\\|(\\phi^n)^2-(\\phi^\\infty)^2\\|_2\\leq\nC_1\\|\\phi^n-\\phi^\\infty\\|_2^{1\/2}(\\|\\phi^n\\|_6^{1\/2}+\\|\\phi^\\infty\\|_6^{1\/2})\\nonumber}\n\\\\[2mm]\n&\\leq&\nC_2(\\|\\nabla\\phi^n\\|_2^{1\/2}+\\|\\nabla\\phi^\\infty\\|_2^{1\/2})\\|\\phi^n-\\phi^\\infty\\|_2\\to\n0,\\qquad n\\to \\infty,\\end{eqnarray} which shows\n$\\rho^n=(\\phi^n)^2\\to \\rho^\\infty=(\\phi^\\infty)^2 \\in L^2$. Since\n$E_2(\\sqrt{\\rho})$ in (\\ref{e12}) is convex and  lower\nsemi-continuous in $\\rho$, thus\n$E_2(\\phi^\\infty)\\leq\\lim\\limits_{n\\to\\infty}E_2(\\phi^n)$. For $E_1$\nin  (\\ref{e11}),\n$E_1(\\phi^\\infty)\\leq\\lim\\limits_{n\\to\\infty}E_1(\\phi^n)$ because of\nthe lower semi-continuity of the $H^1$- and $L^2_V$-norm. Combining\nthe results together, we know\n$E(\\phi^\\infty)\\leq\\lim\\limits_{n\\to\\infty}E(\\phi^n)$, which proves\nthat $\\phi^\\infty$ is indeed a minimizer of the minimization problem\n(\\ref{groundstate}). The uniqueness follows from the strictly\nconvexity of $E(\\sqrt{\\rho})$ as shown in Lemma \\ref{lem1}.\n\n(ii) Assume $\\beta<0$, or $\\beta\\ge0$ and\n$\\lambda<-\\frac{1}{2}\\beta$ or $\\lambda>\\beta$. Without loss of\ngenerality,  we assume ${\\bf n}=(0,0,1)^T$ and choose the function\n\\begin{equation}\\label{phiv11}\\phi_{\\varepsilon_1,\\varepsilon_2}({\\bf x} )=\\frac{1}{(2\\pi\\varepsilon_1)^{1\/2}}\n\\cdot\\frac{1}{(2\\pi\\varepsilon_2)^{1\/4}}\\exp\\left(-\\frac{x^2+y^2}\n{2\\varepsilon_1}\\right)\\exp\\left(-\\frac{z^2}{2\\varepsilon_2}\\right), \\qquad\n{\\bf x} \\in{\\mathcal R}^3, \\end{equation} with $\\varepsilon_1$ and $\\varepsilon_2$ two small positive\nparameters (in fact, for general ${\\bf n}\\in {\\mathcal R}^3$ satisfies\n$|{\\bf n} |=1$, we can always choose $0\\ne {\\bf n}_1\\in{\\mathcal R}^3$ and\n$0\\ne {\\bf n}_2\\in{\\mathcal R}^3$ such that  $\\{{\\bf n}_1,\\, {\\bf\nn}_2,\\,{\\bf n} \\}$ forms an orthonormal basis of $\\mathcal R^3$ and do the\nchange of variables ${\\bf x} =(x,y,z)^T$ to ${\\bf y}=({\\bf x} \\cdot{\\bf\nn}_1,\\,{\\bf x} \\cdot{\\bf n}_2,\\,{\\bf x} \\cdot{\\bf n})^T$ on the right hand\nside of (\\ref{newener}), the following computation is still valid).\nTaking the standard Fourier transform at both sides of the Poisson\nequation \\begin{equation} -\\nabla^2\n\\varphi_{\\varepsilon_1,\\varepsilon_2}({\\bf x} )=|\\phi_{\\varepsilon_1,\\varepsilon_2}({\\bf x} )|^2=\\rho_{\\varepsilon_1,\\varepsilon_2}({\\bf x} ),\n\\quad {\\bf x} \\in{\\mathcal R}^3, \\qquad\n\\lim\\limits_{|{\\bf x} |\\to\\infty}\\varphi_{\\varepsilon_1,\\varepsilon_2}({\\bf x} )=0,\\end{equation} we\nget \\begin{equation}\n|\\xi|^2\\widehat{\\varphi_{\\varepsilon_1,\\varepsilon_2}}(\\xi)=\\widehat{\\rho_{\\varepsilon_1,\\varepsilon_2}}(\\xi),\n\\qquad \\xi\\in{\\mathcal R}^3. \\end{equation} Using the Plancherel formula and\nchanging of variables, we obtain\n\\begin{eqnarray} \\label{pbfn3}\n\\|\\partial_{\\bf n}\\nabla\n\\varphi_{\\varepsilon_1,\\varepsilon_2}\\|_2^2&=&\\frac{1}{(2\\pi)^3}\\|({\\bf n}\\cdot {\n\\xi})\\widehat{\\varphi_{\\varepsilon_1,\\varepsilon_2}}(\\xi)\\|_2^2\n=\\frac{1}{(2\\pi)^3}\\int_{\\mathcal\nR^3}\\frac{|\\xi_3|^2}{|\\xi|^2}\\left(\\widehat{\\rho_{\\varepsilon_1,\\varepsilon_2}}(\\xi)\\right)^2d\\xi\\nonumber\\\\\n&=&\\frac{1}{(2\\pi)^3\\varepsilon_1\\sqrt{\\varepsilon_2}}\\int_{\\mathcal\nR^3}\\frac{|\\xi_3|^2}{(|\\xi_1|^2+|\\xi_2|^2)\\cdot\\frac{\\varepsilon_2}\n{\\varepsilon_1}+|\\xi_3|^2}\\left(\\widehat{\\rho_{1,1}}(\\xi)\\right)^2\\,d\\xi,\n\\quad \\varepsilon_1,\\varepsilon_2>0.\\qquad\\quad\n\\end{eqnarray}\nBy the dominated convergence theorem, we get \\begin{eqnarray}\n \\|\\partial_{\\bf n}\\nabla\n\\varphi_{\\varepsilon_1,\\varepsilon_2}\\|_2^2\\to\\left\\{\\begin{array}{ll} 0,\n&\\varepsilon_2\/\\varepsilon_1\\to+\\infty, \\\\\n\\frac{1}{(2\\pi)^3\\varepsilon_1\\sqrt{\\varepsilon_2}}\\displaystyle\\int_{\\mathcal\nR^3}\\left(\\widehat{\\rho_{1,1}}(\\xi)\\right)^2\\,d\\xi\n=\\|\\rho_{\\varepsilon_1,\\varepsilon_2}\\|_2^2=\\|\\phi_{\\varepsilon_1,\\varepsilon_2}\\|_4^4,\n&\\varepsilon_2\/\\varepsilon_1\\to0^+. \\end{array}\\right.\\qquad \\end{eqnarray} When fixed\n$\\varepsilon_1\\sqrt{\\varepsilon_2}$, the last integral in (\\ref{pbfn3}) is\ncontinuous in $\\varepsilon_2\/\\varepsilon_1>0$. Thus, for any $\\alpha\\in(0,1)$, by\nadjusting $\\varepsilon_2\/\\varepsilon_1:=C_\\alpha>0$, we could have $\\|\\partial_{\\bf\nn}\\nabla\n\\varphi_{\\varepsilon_1,\\varepsilon_2}\\|_2^2=\\alpha\\|\\phi_{\\varepsilon_1,\\varepsilon_2}\\|_4^4$.\nSubstituting (\\ref{phiv11}) into (\\ref{e11}) and (\\ref{e12}) with\n$\\sqrt{\\rho}=\\phi_{\\varepsilon_1,\\varepsilon_2}$ under fixed $\\varepsilon_2\/\\varepsilon_1>0$, we\nget \\begin{eqnarray} E_1(\\phi_{\\varepsilon_1,\\varepsilon_2})&=&\\int_{\\mathcal R^3}\\left[|\\nabla\n\\phi_{\\varepsilon_1,\\varepsilon_2}|^2+V({\\bf x} )|\\phi_{\\varepsilon_1,\\varepsilon_2}|^2\\right]\\,\nd{\\bf x} =\\fl{C_1}{\\varepsilon_1}+\n\\fl{C_2}{\\varepsilon_2}+{\\mathcal{O}}(1), \\qquad \\\\\nE_2(\\phi_{\\varepsilon_1,\\varepsilon_2})&=&\\frac12\\int_{\\mathcal\nR^3}(\\beta-\\lambda+3\\alpha\\lambda)\n)|\\phi_{\\varepsilon_1,\\varepsilon_2}|^4\\,d{\\bf x} =\\frac{\\beta-\\lambda+3\\alpha\n\\lambda}{2}\\cdot\\frac{C_3}{\\varepsilon_1\\sqrt{\\varepsilon_2}}, \\end{eqnarray} with some\nconstants $C_1$, $C_2$, $C_3>0$ independent of $\\varepsilon_1$ and\n$\\varepsilon_2$. Thus, if $\\beta<0$, choose $\\alpha=1\/3$; if $\\beta\\ge0$\nand $\\lambda< -\\frac12\\beta$, choose\n$1\/3-\\frac{\\beta}{3\\lambda}<\\alpha<1$; and if $\\beta\\ge0$ and\n$\\lambda>\\beta$, choose\n$0<\\alpha<\\frac{1}{3}\\left(1-\\frac{\\beta}{\\lambda}\\right)$; as\n$\\varepsilon_1$, $\\varepsilon_2\\to 0^+$, we can get $\\inf\\limits_{\\phi\\in\nS}E(\\phi)=\\lim\\limits_{\\varepsilon_1,\\varepsilon_2\\to 0^+}\nE_1(\\phi_{\\varepsilon_1,\\varepsilon_2}) +E_2(\\phi_{\\varepsilon_1,\\varepsilon_2})=-\\infty$, which\nimplies that there exists no ground state of the minimization\nproblem (\\ref{groundstate}). $\\Box$\n\nBy splitting the total energy $E(\\cdot)$ in (\\ref{newener}) into\nkinetic, potential, interaction and dipolar energies, i.e. \\begin{equation}\nE(\\phi)=E_{\\rm kin}(\\phi)+E_{\\rm pot}(\\phi)+E_{\\rm int}(\\phi)+E_{\\rm\ndip}(\\phi),\\end{equation} where \\begin{eqnarray}  E_{\\rm\nkin}(\\phi)&=&\\frac{1}{2}\\int_{{\\mathcal R}^3} |\\nabla\\phi({\\bf x} ) |^2 d\n{\\bf x} ,\\   E_{\\rm pot}(\\phi) = \\int_{{\\mathcal R}^3} V({\\bf x} )|\\phi({\\bf x} )|^2 d\n{\\bf x} , \\ E_{\\rm int}(\\phi) =\n\\frac{\\beta}{2}\\int_{{\\mathcal R}^3} |\\phi({\\bf x} ) |^4 d {\\bf x} ,\\nonumber \\\\\nE_{\\rm dip} (\\phi)&=&\\frac{\\lambda}{2}\\int_{{\\mathcal R}^3} \\left(U_{\\rm\ndip}\\ast|\\phi|^2\\right) |\\phi({\\bf x} )|^2 d {\\bf x} =\n\\frac{\\lambda}{2}\\int_{{\\mathcal R}^3} |\\phi({\\bf x} )|^2\\left[3\n \\left(\\partial_{\\bf n{\\bf n} } \\varphi\\right)^2-|\\phi({\\bf x} )|^2\\right]d {\\bf x} \\nonumber\\\\\n&=&\\frac{\\lambda}{2}\\int_{{\\mathcal R}^3} \\left[-|\\phi({\\bf x} )|^4+3\n\\left|\\partial_{\\bf n}\\nabla \\varphi\\right|^2\\right]d {\\bf x} , \\label{dipp03}\n \\end{eqnarray}\nwith $\\varphi$ defined in (\\ref{poi101}), we have the following\nViral identity:\n\n\\begin{proposition}\nSuppose $\\phi_e$ is a stationary state of a dipolar BEC, i.e. an\neigenfunction of the nonlinear eigenvalue problem (\\ref{gpe22dstat})\nunder the constraint (\\ref{const}), then we have \\begin{equation} \\label{virial}\n2E_{\\rm kin}(\\phi_e)- 2E_{\\rm trap}(\\phi_e)+3E_{\\rm\nint}(\\phi_e)+3E_{\\rm dip}(\\phi_e)=0.\\end{equation}\n\\end{proposition}\n\n\\noindent {\\bf{Proof:}} Follow the analogous proof for a BEC without\ndipolar interaction  \\cite{Pitaevskii} and we omit the details here\nfor brevity. $\\Box$\n\n\n\n\\subsection{Analytical results for dynamics}\n\nThe well-posedness of the Cauchy problem of (\\ref{ngpe}) was\ndiscussed in \\cite{Carles} by analyzing the convolution kernel\n$U_{\\rm dip}({\\bf x} )$ with detailed Fourier transform. Under the new\nformulation (\\ref{gpe})-(\\ref{poisson}), here we present a simpler\nproof for the well-posedness and show finite time blow-up for the\nCauchy problem of a dipolar BEC in different parameter regimes.\nDenote\n$$ X=\\left\\{u\\in\nH^1({\\mathcal R}^3)\\ \\big|\\ \\|u\\|_X^2=\\|u\\|_{L^2}^2+\\|\\nabla\nu\\|_{L^2}^2+\\int_{\\mathcal R^3}V({\\bf x} )|u({\\bf x} )|^2\\,d{\\bf x} <\\infty\\right\\}.$$\n\n\n\n\\begin{theorem} (Well-posedness) Suppose the real-valued trap\npotential $V({\\bf x} )\\in C^\\infty(\\mathcal R^3)$ such that $V({\\bf x} )\\ge0$ for\n${\\bf x} \\in{\\mathcal R}^3$ and $D^\\alpha V({\\bf x} )\\in L^\\infty(\\mathcal R^3)$ for\nall $\\alpha\\in{\\mathcal N}_0^3$ with $|\\alpha|\\ge 2$. For any initial\ndata $\\psi({\\bf x} ,t=0)=\\psi_0({\\bf x} )\\in X$,\n there exists\n$T_{\\mbox{\\rm max}}\\in(0,+\\infty]$ such that the problem\n(\\ref{gpe})-(\\ref{poisson})\n has a unique maximal solution\n$\\psi\\in C\\left([0,T_{\\mbox{max}}),X\\right)$. It is maximal in the\nsense that if $T_{\\mbox{\\rm max}}<\\infty$, then\n$\\|\\psi(\\cdot,t)\\|_X\\to\\infty$ when  $t\\to T^-_{\\mbox{\\rm max}}$.\nMoreover, the {\\sl mass} $N(\\psi(\\cdot,t))$ and {\\sl energy}\n$E(\\psi(\\cdot,t))$ defined in (\\ref{norm}) and (\\ref{energy}),\nrespectively, are conserved for $t\\in[0,T_{\\rm max})$. Specifically,\nif $\\beta\\ge0$ and $-\\frac12\\beta\\leq\\lambda\\leq\\beta$, the solution\nto (\\ref{gpe})-(\\ref{poisson}) is global in time, i.e.,\n  $T_{\\mbox{max}}=\\infty$.\n\\end{theorem}\n\n\n\\noindent {\\bf{Proof: }} For any $\\phi\\in X$, let $\\varphi$ be the\nsolution of the Poisson equation (\\ref{poi11}), denote\n$\\rho=|\\phi|^2$ and define \\begin{equation}\nG(\\phi,\\bar{\\phi}):=G(\\rho)=\\frac12\\int_{\\mathcal\nR^3}|\\phi({\\bf x} )|^2\\partial_{\\bf nn}\\varphi({\\bf x} )\\,d{\\bf x} , \\qquad\ng(\\phi)=\\frac{\\delta G(\\phi,\\bar{\\phi})}{\\delta\n\\bar{\\phi}}=\\phi\\;\\partial_{\\bf nn}\\varphi,\\qquad \\end{equation} where $\\bar{f}$\ndenotes the conjugate of $f$.\n Noticing (\\ref{pbnf1}), it is easy to show that\n $G(\\phi)\\in C^1(X,\\mathcal\nR)$, $g(\\phi) \\in C(X,L^p)$ for some $p\\in(6\/5,2]$, and\n\\begin{equation}\\label{con3}\\|g(u)-g(v)\\|_{L^p}\\leq\nC(\\|u\\|_X+\\|v\\|_X)\\|u-v\\|_{L^r},\\quad \\hbox{for some }r\\in[2,6),\n\\qquad \\forall u,v\\in X.\\end{equation} Applying the standard Theorems 9.2.1,\n4.12.1 and 5.7.1 in \\cite{Cazen,Sulem} for the well-posedness of the\nnonlinear Schr\\\"{o}dinger equation, we can obtain the results\nimmediately. $\\Box$\n\n\n\n\n\n\n\\begin{theorem}(Finite time blow-up) If $\\beta<0$, or $\\beta\\ge0$ and\n$\\lambda<-\\frac{1}{2}\\beta$ or $\\lambda>\\beta$,\n and assume $V({\\bf x} )$ satisfies $3V({\\bf x} )+ {\\bf x} \\cdot \\nabla V({\\bf x} )\\ge0$ for\n${\\bf x} \\in{\\mathcal R}^3$. For any initial data\n$\\psi({\\bf x} ,t=0)=\\psi_0({\\bf x} )\\in X$ to the problem\n(\\ref{gpe})-(\\ref{poisson}), there exists finite time blow-up, i.e.,\n$T_{\\mbox{max}}<\\infty$, if one of the following holds:\n\n(i) $E(\\psi_0)<0$;\n\n(ii) $E(\\psi_0)=0$ and ${\\rm Im}\\left(\\int_{\\mathcal\nR^3}\\bar{\\psi}_0({\\bf x} )\\ ({\\bf x} \\cdot\\nabla\\psi_0({\\bf x} ))\\,d{\\bf x} \\right)<0$;\n\n(iii) $E(\\psi_0)>0$ and ${\\rm Im}\\left(\\int_{\\mathcal R^3}\n\\bar{\\psi}_0({\\bf x} )\\ ({\\bf x} \\cdot\\nabla\\psi_0({\\bf x} ))\\,d{\\bf x} \\right)\n<-\\sqrt{3E(\\psi_0)}\\|{\\bf x} \\psi_0\\|_{L^2}$;\n\n\\noindent where  {\\rm Im}(f) denotes the  imaginary part of $f$.\n\\end{theorem}\n\n\n\n\\noindent {\\bf{ Proof :}} Define the variance \\begin{equation}\\label{dtv001}\n\\sigma_V(t):=\\sigma_V(\\psi(\\cdot,t))=\\int_{\\mathcal\nR^3}|{\\bf x} |^2|\\psi({\\bf x} ,t)|^2\\,d{\\bf x} =\\delta_x(t)+\\delta_y(t)+\\delta_z(t),\\qquad\nt\\ge0,\\end{equation} where \\begin{equation}\n\\label{dtap01}\\sigma_\\alpha(t):=\\sigma_\\alpha(\\psi(\\cdot,t))=\\int_{\\mathcal\nR^3}\\alpha^2|\\psi({\\bf x} ,t)|^2\\,d{\\bf x} , \\qquad \\alpha=x,\\ y,\\ z.\\end{equation} For\n$\\alpha=x$, or $y$ or $z$, differentiating (\\ref{dtap01}) with\nrespect to $t$, noticing (\\ref{gpe}) and (\\ref{poisson}),\nintegrating by parts, we get \\begin{equation}\n\\frac{d}{dt}\\sigma_\\alpha(t)=-i\\int_{\\mathcal\nR^3}\\left[\\alpha\\bar{\\psi}({\\bf x} ,t)\\partial_{\\alpha}\\psi({\\bf x} ,t)-\n\\alpha\\psi({\\bf x} ,t)\\partial_{\\alpha}\\bar{\\psi}({\\bf x} ,t)\\right]\\,d{\\bf x} , \\qquad\nt\\ge0.\\end{equation} Similarly, we have \\begin{equation} \\label{d2ap22}\n\\frac{d^2}{dt^2}\\sigma_\\alpha(t)=\\int_{\\mathcal\nR^3}\\left[2|\\partial_{\\alpha}\\psi|^2+(\\beta-\\lambda)\n|\\psi|^4+6\\lambda|\\psi|^2\\alpha\\partial_{\\alpha}\\partial_{\\bf{nn}}\n\\varphi-2\\alpha|\\psi|^2\\partial_{\\alpha}V({\\bf x} )\\right]\\,d{\\bf x} . \\end{equation} Noticing\n(\\ref{poisson}) and\n\\[\n-\\int_{\\mathcal R^3}\\nabla^2\\varphi\\left( {\\bf x} \\cdot \\nabla\n\\partial_{\\bf{nn}}\\varphi\\right)\\,d{\\bf x} =\\frac 32\\int_{\\mathcal\nR^3}|\\partial_{\\bf{n}}\\nabla \\varphi|^2\\,d{\\bf x} ,\n\\]\nsumming (\\ref{d2ap22}) for $\\alpha=x$, $y$ and $z$, using\n(\\ref{dtv001}) and (\\ref{energy}), we get\n\\begin{eqnarray}\n\\frac{d^2}{dt^2}\\sigma_V(t)&=&2\\int_{\\mathcal\nR^3}\\left(|\\nabla\\psi|^2+\\frac 32(\\beta-\\lambda)|\\psi|^4+\\frac\n92\\lambda|\\partial_{\\bf{n}}\\nabla\\psi|^2-|\\psi|^2({\\bf x} \\cdot\\nabla\nV({\\bf x} ))\\right)\\,d{\\bf x} \\nonumber\\\\\n&=&6E(\\psi)-\\int_{\\mathcal R^3}|\\nabla\\psi({\\bf x} ,t)|^2-2\\int_{\\mathcal\nR^3}|\\psi({\\bf x} ,t)|^2\\left(3V({\\bf x} )+{\\bf x} \\cdot\\nabla V({\\bf x} )\\right)\\,d{\\bf x} \\nonumber\\\\\n&\\leq&6E(\\psi)\\equiv 6E(\\psi_0), \\qquad t\\ge0.\n\\end{eqnarray}\nThus,\n$$\n\\sigma_V(t)\\leq 3E(\\psi_0)t^2+\\sigma_V^\\prime(0)t+\\sigma_V(0), \\qquad t\\ge0,\n$$\nand the conclusion follows in the same manner as those in\n\\cite{Sulem,Cazen} for the standard nonlinear Schr\\\"{o}dinger\nequation. $\\Box$\n\n\n\\section{A numerical method for computing ground states} \\label{0tssp}\n \\setcounter{equation}{0}\n\nBased on the new mathematical formulation for the energy in\n(\\ref{newener}), we will present an efficient and accurate backward\nEuler sine pseudospectral method for computing the ground states of\na dipolar BEC.\n\nIn practice, the whole space problem is usually truncated into a\nbounded computational domain $\\Omega=[a,b]\\times[c,d]\\times[e,f]$ with\nhomogeneous  Dirichlet boundary condition. Various numerical methods\nhave been proposed in the literatures for computing the ground\nstates of BEC (see \\cite{Schnerder,Tosi,Bao1,Bao3,Bao5,Chang1,Ca1}\nand references therein). One of the popular and efficient techniques\nfor dealing with the constraint (\\ref{const}) is through the\nfollowing construction \\cite{Bao1,Bao2,Bao3}: Choose a time step\n$\\Delta t>0$ and set $t_n=n\\; \\Delta t$ for $n=0,1,\\ldots$  Applying\nthe steepest decent method to the energy functional $E(\\phi)$ in\n(\\ref{newener}) without the constraint (\\ref{const}), and then\nprojecting the solution back to the unit sphere $S$ at the end of\neach time interval $[t_n,t_{n+1}]$ in order to satisfy the\nconstraint (\\ref{const}). This procedure leads to the function\n$\\phi({\\bf x} ,t)$ is the solution of the following gradient flow with\ndiscrete normalization: \\begin{eqnarray} \\label{ngf1} &&\\partial_t\n\\phi({\\bf x} ,t)=\\left[\\fl{1}{2 }\\nabla^2\n-V({\\bf x} )-(\\beta-\\lambda)|\\phi({\\bf x} ,t)|^2+\n3\\lambda \\partial_{\\bf{nn}}\\varphi({\\bf x} ,t)\\right]\\phi({\\bf x} ,t),    \\\\\n\\label{ngf21}&&\\nabla^2 \\varphi({\\bf x} ,t) = -|\\phi({\\bf x} ,t)|^2, \\qquad\n\\qquad {\\bf x} \\in\\Omega,\\quad  t_n \\leq t < t_{n+1}, \\\\\n\\label{ngf2} &&\\phi({\\bf x} ,t_{n+1}):=\n\\phi({\\bf x} ,t_{n+1}^+)=\\fl{\\phi({\\bf x} ,t_{n+1}^-)}{\\|\\phi(\\cdot,t_{n+1}^-)\\|},\n\\qquad {\\bf x} \\in \\Omega, \\quad n\\ge 0,\\\\\n&&\\left.\\phi({\\bf x} ,t)\\right|_{{\\bf x} \\in\\partial\\Omega}=\\left.\\varphi({\\bf x} ,t)\\right|_{{\\bf x} \\in\\partial\\Omega}=0,\n\\qquad t\\ge0,\\\\\n \\label{ngf3}\n &&\\phi({\\bf x} ,0)=\\phi_0({\\bf x} ), \\qquad\n\\qquad \\hbox{with}\\quad \\|\\phi_0\\|=1; \\end{eqnarray} where\n $\\phi({\\bf x} ,\nt_n^\\pm)=\\lim\\limits_{t\\to t_n^\\pm} \\phi({\\bf x} ,t)$.\n\nLet $M$, $K$ and $L$ be even positive integers and define the index\nsets \\begin{eqnarray*} &&{\\cal T}_{MKL}=\\{(j,k,l)\\ |\\ j=1,2,\\ldots,M-1,\\\nk=1,2,\\ldots, K-1, \\ l=1,2,\\ldots,L-1\\}, \\\\\n&&{\\cal T}_{MKL}^0=\\{(j,k,l)\\ |\\ j=0,1,\\ldots,M,\\ k=0,1,\\ldots, K, \\\nl=0,1,\\ldots,L\\}. \\end{eqnarray*} Choose the spatial mesh sizes as\n$h_x=\\frac{b-a}{M}$, $h_y=\\frac{d-c}{K}$ and $h_z=\\frac{f-e}{L}$ and\ndefine\n\\[x_j:=a+j\\;h_x,\\qquad  y_k = c+ k\\; h_y,\\qquad\nz_l = e+ l\\; h_z, \\qquad (j,k,l)\\in {\\cal T}^0_{MKL}.\\] Denote the\nspace\n\\[Y_{MKL}={\\rm\nspan}\\{\\Phi_{jkl}({\\bf x} ), \\quad (j,k,l)\\in{\\cal T}_{MKL}\\},\\] with\n\\[\n\\Phi_{jkl}({\\bf x} )=\\sin\\left(\\mu_j^x(x-a)\\right)\\sin\\left(\\mu_k^y(y-c)\\right)\n\\sin\\left(\\mu_l^z(z-e)\\right), \\quad {\\bf x} \\in \\Omega,\\qquad\n(j,k,l)\\in{\\cal T}_{MKL}, \\]\n\\[\\mu_j^x =\n\\fl{\\pi j}{b-a}, \\qquad \\mu_k^y = \\fl{\\pi k}{d-c}, \\qquad \\mu_l^z =\n\\fl{\\pi l}{f-e}, \\qquad (j,k,l)\\in{\\cal T}_{MKL}; \\] and  $P_{MKL}:\nY=\\{\\varphi\\in C(\\Omega)\\ |\\ \\varphi({\\bf x} )|_{{\\bf x} \\in\\partial\\Omega}=0\\}\\to\nY_{MKL}$ be the standard project operator \\cite{ST}, i.e.\n\\[(P_{MKL}v)({\\bf x} )=\\sum_{p=1}^{M-1}\\sum_{q=1}^{K-1}\\sum_{s=1}^{L-1}\n\\widehat{v}_{pqs}\\; \\Phi_{pqs}({\\bf x} ), \\quad {\\bf x} \\in\\Omega,\\qquad \\forall\nv\\in Y,\n\\]\nwith \\begin{equation}\\label{FST} \\widehat{v}_{pqs}=\\int_{\\Omega} v({\\bf x} )\\;\n\\Phi_{pqs}({\\bf x} )\\;d{\\bf x} , \\qquad (p,q,s)\\in{\\cal T}_{MKL}. \\end{equation} Then a\nbackward Euler sine spectral discretization\nfor (\\ref{ngf1})-(\\ref{ngf3}) reads:\\\\\n Find\n$\\phi^{n+1}({\\bf x} )\\in Y_{MKL}$ (i.e. $\\phi^{+}({\\bf x} )\\in Y_{MKL}$) and\n$\\varphi^{n}({\\bf x} )\\in Y_{MKL}$ such that {\\small \\begin{eqnarray}\n&&\\frac{\\phi^{+}({\\bf x} )-\\phi^n({\\bf x} )}{\\Delta t}=\\frac{1}{2}\\nabla^2\n\\phi^{+}({\\bf x} )\n-P_{MKL}\\left\\{\\left[V({\\bf x} )+(\\beta-\\lambda)|\\phi^n({\\bf x} )|^2+ 3\\lambda\n\\partial_{\\bf{nn}}\\varphi^n({\\bf x} )\\right]\\phi^{+}({\\bf x} )\\right\\},\\qquad  \\\\\n&&\\nabla^2\\varphi^n({\\bf x} )=-P_{MKL}\\left(|\\phi^n({\\bf x} )|^2\\right),\\qquad\n\\phi^{n+1}({\\bf x} )=\\frac{\\phi^{+}({\\bf x} )}{\\|\\phi^{+}({\\bf x} )\\|_2}, \\qquad\n{\\bf x} \\in\\Omega,\\quad  n\\ge0; \\end{eqnarray}} where\n$\\phi^0({\\bf x} )=P_{MKL}\\left(\\phi_0({\\bf x} )\\right)$ is given.\n\n\nThe above discretization can be solved in phase space and it is not\nsuitable in practice due to the difficulty of computing the\nintegrals in (\\ref{FST}). We now present an efficient implementation\nby choosing $\\phi^0({\\bf x} )$ as the interpolation of $\\phi_0({\\bf x} )$ on\nthe grid points $\\{(x_j,y_k,z_l), \\ (j,k,l)\\in{\\cal T}_{MKL}^0\\}$,\ni.e $\\phi^0(x_j,y_k,z_l) =\\phi_0(x_j,y_k,z_l)$ for $(j,k,l)\\in{\\cal\nT}_{MKL}^0$, and approximating the integrals in (\\ref{FST}) by a\nquadrature rule on the grid points. Let $\\phi_{jkl}^n$ and\n$\\varphi_{jkl}^n$ be the approximations of $\\phi(x_j,y_k,z_l,t_n)$\nand $\\varphi(x_j,y_k,z_l,t_n)$, respectively, which are the solution\nof (\\ref{ngf1})-(\\ref{ngf3}); denote $\\rho_{jkl}^n=|\\phi^n_{jkl}|^2$\nand  choose $\\phi_{jkl}^0=\\phi_0(x_j,y_k,z_l)$ for $(j,k,l)\\in {\\cal\nT}_{MKL}^0$. For $n=0,1,\\ldots$, a backward Euler sine\npseduospectral discretization for (\\ref{ngf1})-(\\ref{ngf3}) reads:\n{\\small \\begin{eqnarray} &&\\fl{\\phi_{jkl}^+-\\phi_{jkl}^n}{\\triangle t}=\n\\fl{1}{2} \\left.\\left(\\nabla_s^2\n\\phi^+\\right)\\right|_{jkl}-\\left[V(x_j,y_k,z_l)+(\\beta-\\lambda)\n\\left|\\phi_{jkl}^n\\right|^2 +3\\lambda\\left.\\left(\\partial_{{\\bf n} \\bn}^s\n\\varphi^n\\right)\\right|_{jkl}\\right] \\phi^+_{jkl}, \\qquad\n\\label{discretized1} \\\\\n&&-\\left.\\left(\\nabla_s^2 \\varphi^n\\right)\\right|_{jkl}=\n|\\phi_{j,k,l}^n|^2=\\rho_{jkl}^n, \\qquad\n\\phi_{jkl}^{n+1}=\\fl{\\phi_{jkl}^+}{\\|\\phi^+\\|_h}, \\qquad (j,k,l)\\in\n{\\cal T}_{MKL}, \\\\\n&&\\phi_{0kl}^{n+1}=\\phi_{Mkl}^{n+1}=\\phi_{j0l}^{n+1}=\n\\phi_{jKl}^{n+1}=\\phi_{jk0}^{n+1}=\\phi_{jkL}^{n+1}=0,\\qquad\n(j,k,l)\\in {\\cal T}_{MKL}^0,\\\\\n&&\\varphi_{0kl}^{n}\n=\\varphi_{Mkl}^{n}=\\varphi_{j0l}^{n}=\\varphi_{jKl}^{n}=\n\\varphi_{jk0}^{n}=\\varphi_{jkL}^{n}=0, \\qquad (j,k,l)\\in {\\cal\nT}_{MKL}^0;\\label{discretized2} \\end{eqnarray}}  where $\\nabla_s^2$ and\n$\\partial_{{\\bf n} \\bn}^s$ are sine pseudospectral\n approximations of $\\nabla^2$ and $\\partial_{{\\bf n} \\bn}$, respectively,\n defined as\n{\\small \\begin{eqnarray}&&\\left.\\left(\\nabla_s^2 \\phi^n\\right)\\right|_{jkl} =\n-\\sum_{p=1}^{M-1}\\sum_{q=1}^{K-1}\\sum_{s=1}^{L-1}\n\\left[(\\mu_p^x)^2+(\\mu_q^y)^2+(\\mu_s^z)^2\\right]\\widetilde{(\\phi^n)}_{pqs}\n\\sin\\left(\\frac{jp\\pi}{M}\\right)\\sin\\left(\\frac{kq\\pi}{K}\\right)\n\\sin\\left(\\frac{ls\\pi}{L}\\right),\\qquad \\nonumber \\\\\n&&\\left.\\left(\\partial_{{\\bf n} \\bn}^s\n\\varphi^n\\right)\\right|_{jkl}=\\sum_{p=1}^{M-1}\\sum_{q=1}^{K-1}\\sum_{s=1}^{L-1}\n\\frac{\\widetilde{(\\rho^n)}_{pqs}}{(\\mu_p^x)^2+(\\mu_q^y)^2+(\\mu_s^z)^2}\n\\left.\\left(\\partial_{{\\bf n} \\bn}\\Phi_{pqs}({\\bf x} )\\right)\\right|_{(x_j,y_k,z_l)},\n\\ (j,k,l)\\in {\\cal T}_{MKL}, \\label{dstpp}\n \\end{eqnarray}}\nwith $\\widetilde{(\\phi^n)}_{pqs}$ ($(p,q,s)\\in{\\cal T}_{MKL})$ the\ndiscrete sine transform coefficients of the vector $\\phi^n$ as\n{\\small \\begin{equation}\\label{dst11}\\widetilde{(\\phi^n)}_{pqs}=\n\\frac{8}{MKL}\\sum_{j=1}^{M-1}\\sum_{k=1}^{K-1}\\sum_{l=1}^{L-1}\n\\phi^n_{jkl}\n\\sin\\left(\\frac{jp\\pi}{M}\\right)\\sin\\left(\\frac{kq\\pi}{K}\\right)\n\\sin\\left(\\frac{ls\\pi}{L}\\right), \\quad (p,q,s)\\in {\\cal\nT}_{MKl},\\end{equation} } and the discrete $h$-norm is defined as\n\\[ \\|\\phi^+\\|_h^2 = h_xh_yh_z\\sum_{j=1}^{M-1}\\sum_{k=1}^{N-1}\\sum_{l=1}^{L-1}\n|\\phi_{jkl}^+|^2.\\] Similar as those in \\cite{Bao6}, the linear\nsystem (\\ref{discretized1})-(\\ref{discretized2}) can be iteratively\nsolved in phase space very efficiently via discrete sine transform\nand we omitted the details here for brevity.\n\n\n\n\n\\section{A time-splitting sine pseudospectral method for dynamics}\\label{1tssp}\n\nSimilarly, based on the new Gross-Pitaevskii-Poisson type system\n(\\ref{gpe})-(\\ref{poisson}), we will present an efficient and\naccurate time-splitting sine pseudospectral (TSSP) method for\ncomputing the dynamics of a dipolar BEC.\n\nAgain, in practice, the whole space problem is truncated into a\nbounded computational domain $\\Omega=[a,b]\\times[c,d]\\times[e,f]$ with\nhomogeneous  Dirichlet boundary condition. From time $t=t_n$ to time\n$t=t_{n+1}$, the Gross-Pitaevskii-Poisson type system\n(\\ref{gpe})-(\\ref{poisson}) is solved in two steps.  One solves\nfirst \\begin{equation}\\label{fgpe}i \\partial_t\n\\psi({\\bf x} ,t)=-\\fl{1}{2}\\nabla^2\\psi({\\bf x} ,t), \\quad {\\bf x} \\in\\Omega, \\qquad\n\\left.\\psi({\\bf x} ,t)\\right|_{{\\bf x} \\in\\partial\\Omega}=0, \\qquad t_n\\le t\\le\nt_{n+1},\\end{equation} for the time step of length $\\Delta t$, followed by\nsolving \\begin{eqnarray}\\label{ode11} &&i \\partial_t\n\\psi({\\bf x} ,t)=\\left[V({\\bf x} )+(\\beta-\\lambda) |\\psi({\\bf x} ,t)|^2-3\\lambda\n\\partial_{{\\bf n} \\bn} \\varphi({\\bf x} ,t)\n\\right]\\psi({\\bf x} ,t),  \\\\\n\\label{poisson11}&&\\nabla^2 \\varphi({\\bf x} ,t) = -|\\psi({\\bf x} ,t)|^2,\\qquad\n {\\bf x} \\in\\Omega, \\qquad t_n\\le t \\le t_{n+1}; \\\\ \\label{bond123}\n &&\\left.\\varphi({\\bf x} ,t)\\right|_{{\\bf x} \\in\\partial\\Omega}=0, \\qquad \\left.\\psi({\\bf x} ,t)\\right|_{{\\bf x} \\in\\partial\\Omega}=0,\n \\qquad t_n\\le t \\le t_{n+1};\\end{eqnarray}\nfor the same time step. Equation (\\ref{fgpe}) will be discretized in\nspace by sine pseudospectral method and integrated in time {\\sl\nexactly} \\cite{Bao8}. For $t\\in[t_n,t_{n+1}]$, the equations\n(\\ref{ode11})-(\\ref{bond123}) leave $|\\psi|$ and $\\varphi$ invariant\nin $t$ \\cite{Bao_Jaksch_Markowich,Bao8} and therefore they collapses\nto {\\small \\begin{eqnarray}\\label{ode111} &&i \\partial_t\n\\psi({\\bf x} ,t)=\\left[V({\\bf x} )+(\\beta-\\lambda) |\\psi({\\bf x} ,t_n)|^2-3\\lambda\n\\partial_{{\\bf n} \\bn} \\varphi({\\bf x} ,t_n)\n\\right]\\psi({\\bf x} ,t),  \\quad {\\bf x} \\in\\Omega,\\ t_n\\le t \\le t_{n+1},\\qquad \\quad  \\\\\n\\label{poisson113}&&\\nabla^2 \\varphi({\\bf x} ,t_n) =\n-|\\psi({\\bf x} ,t_n)|^2,\\qquad\n {\\bf x} \\in\\Omega.\n\\end{eqnarray}} Again, equation (\\ref{poisson113}) will be discretized in\nspace by sine pseudospectral method \\cite{Bao8,ST} and the linear\nODE (\\ref{ode111}) can be integrated in time {\\sl exactly}\n\\cite{Bao_Jaksch_Markowich,Bao8}.\n\nLet $\\psi_{jkl}^n$ and $\\varphi_{jkl}^n$ be the approximations of\n$\\psi(x_j,y_k,z_l,t_n)$ and $\\varphi(x_j,y_k,z_l,t_n)$,\nrespectively, which are the solution of (\\ref{gpe})-(\\ref{poisson});\nand choose $\\psi^0_{jkl}=\\psi_0(x_j,y_k,z_l)$ for $(j,k,l)\\in {\\cal\nT}_{MKL}^0$. For $n=0,1,\\ldots$,  a second-order TSSP method for\nsolving (\\ref{gpe})-(\\ref{poisson}) via the standard Strang\nsplitting is \\cite{str,Bao_Jaksch_Markowich,Bao8} {\\small\n\\begin{eqnarray}\\label{tssp1}\n &&\\psi^{(1)}_{jkl}=\\sum_{p=1}^{M-1}\\sum_{q=1}^{K-1}\\sum_{s=1}^{L-1}\ne^{-i\\triangle t\\left[(\\mu_p^x)^2+(\\mu_q^y)^2+(\\mu_r^z)^2 \\right]\n\/4}\\;\\widetilde{(\\psi^n)}_{pqr}\\sin\\left(\\frac{jp\\pi}{M}\n\\right)\\sin\\left(\\frac{kq\\pi}{K}\\right)\n\\sin\\left(\\frac{ls\\pi}{L}\\right), \\nonumber \\\\\n &&\\psi^{(2)}_{jkl}=\n e^{-i\\triangle t\\left[V(x_j,y_k,z_l)+(\\beta-\\lambda)|\\psi_{jkl}^{(1)}|^2-3\\lambda\n\\left.\\left(\\partial_{{\\bf n} \\bn}^s\\varphi^{(1)}\\right)\\right|_{jkl}\\right]\n}\\; \\psi_{jkl}^{(1)},\n\\qquad (j,k,l)\\in{\\cal T}_{MKL}^0, \\\\\n&&\\psi^{n+1}_{jkl}=\\sum_{p=1}^{M-1}\\sum_{q=1}^{K-1}\\sum_{s=1}^{L-1}\ne^{-i\\triangle t\\left[(\\mu_p^x)^2+(\\mu_q^y)^2+(\\mu_r^z)^2 \\right]\n\/4}\\;\\widetilde{(\\psi^{(2)})}_{pqr}\\sin\\left(\\frac{jp\\pi}{M}\n\\right)\\sin\\left(\\frac{kq\\pi}{K}\\right)\n\\sin\\left(\\frac{ls\\pi}{L}\\right); \\nonumber\n \\end{eqnarray}}\nwhere $\\widetilde{(\\psi^n)}_{pqr}$ and\n$\\widetilde{(\\psi^{(2)})}_{pqr}$ ($(p,q,s)\\in{\\cal T}_{MKL}$) are\nthe discrete sine transform coefficients of the vectors $\\psi^n$ and\n$\\psi^{(2)}$, respectively (defined similar as those in\n(\\ref{dst11})); and\n$\\left.\\left(\\partial_{{\\bf n} \\bn}^s\\varphi^{(1)}\\right)\\right|_{jkl}$\n can be computed as in (\\ref{dstpp}) with\n$\\rho^n_{jkl}=\\rho^{(1)}_{jkl}:=|\\psi^{(1)}_{jkl}|^2$  for\n$(j,k,l)\\in {\\cal T}_{MKL}^0$.\n\nThe above method is explicit, unconditionally stable, the memory\ncost is $O(MKL)$ and the computational cost per time step is\n$O\\left(MKL\\ln(MKL)\\right)$. In fact, for the stability, we have\n\n\\begin{lemma}\nThe TSSP method (\\ref{tssp1}) is normalization conservation, i.e.\n\\begin{eqnarray}\n\\|\\psi^n\\|_h^2:=h_xh_yh_z\\sum_{j=1}^{M-1}\\sum_{k=1}^{K-1}\\sum_{l=1}^{L-1}|\\psi^n_{jkl}|^2\\equiv\nh_xh_yh_z\\sum_{j=1}^{M-1}\\sum_{k=1}^{K-1}\\sum_{l=1}^{L-1}|\\psi^0_{jkl}|^2=\\|\\psi^0\\|_h^2,\n\\ n\\ge0. \\quad \\end{eqnarray}\n\n\\end{lemma}\n\n\n\\noindent {\\bf Proof:} Follow the analogous proof in\n\\cite{Bao_Jaksch_Markowich,Bao8} and we omit the details here for\nbrevity. $\\Box$\n\n\n\n\n\\section{Numerical results}\n\\setcounter{equation}{0} In this section, we first compare our new\nmethods and the standard method used in the literatures\n\\cite{Yi2,Xiong,Tick,Blakie} to evaluate numerically the dipolar\nenergy and then report ground states and dynamics of dipolar BECs by\nusing our new numerical methods.\n\n\n\\subsection{Comparison for evaluating the dipolar energy}\n\nLet \\begin{equation} \\phi:=\\phi({\\bf x} ) = \\pi^{-3\/4} \\gamma_{x}^{1\/2} \\gamma_z^{1\/4}\ne^{-\\fl{1}{2}\\left( \\gamma_x (x^2+y^2)+\\gamma_z z^2\\right)},\\qquad\n{\\bf x} \\in{\\mathcal R}^3.\\end{equation} Then the\n dipolar energy $E_{\\rm dip}(\\phi)$ in (\\ref{dipp03})\n can be  evaluated analytically as \\cite{Tikhonenkov}\n\\begin{equation} E_{\\rm dip}(\\phi)= -\\fl{\\lambda\\gamma_x \\sqrt{\\gamma_z}}{4\\pi\n\\sqrt{2\\pi} }\\left\\{\n\\begin{array}{ll}\n  \\fl{1+2\\kappa^2}{1-\\kappa^2}-\\fl{3\\kappa^2 \\rm{arctan} \\left(\n\\sqrt{\\kappa^2-1}\\right)}{(1-\\kappa^2)\\sqrt{\\kappa^2-1}}, & \\kappa>1, \\\\\n  0, & \\kappa =1, \\\\\n  \\fl{1+2\\kappa^2}{1-\\kappa^2}-\\fl{1.5\\kappa^2\n}{(1-\\kappa^2)\\sqrt{1-\\kappa^2}} \\rm{ln} \\left(\n\\fl{1+\\sqrt{1-\\kappa^2}}{1-\\sqrt{1-\\kappa^2}}\\right), & \\kappa <1, \\\\\n\\end{array}\n\\right. \\end{equation}\n with $\\kappa =\\sqrt{ \\fl{\\gamma_z}{\\gamma_x}}$. This provides a\n perfect example to test the efficiency of different numerical\n methods to deal with the dipolar potential. Based on our new\n formulation (\\ref{dipp03}), the dipolar energy can be evaluated via\n discrete sine transform (DST) as\n\\begin{eqnarray} \\label{dstdip1}E_{\\rm dip}(\\phi)\\approx \\frac{\\lambda\nh_xh_yh_z}{2}\n\\sum_{j=1}^{M-1}\\sum_{k=1}^{K-1}\\sum_{l=1}^{L-1}|\\phi(x_j,y_k,z_l)|^2\\left[\n3\\left(\\left.\\left(\\partial_{{\\bf n} \\bn}^s\\varphi^n\n\\right)\\right|_{jkl}\\right)^2-|\\phi(x_j,y_k,z_l)|^2\\right], \\nonumber\\end{eqnarray}\nwhere $\\left.\\left(\\partial_{{\\bf n} \\bn}^s\\varphi^n \\right)\\right|_{jkl}$ is\ncomputed as in (\\ref{dstpp}) with\n$\\rho^n_{jkl}=|\\phi(x_j,y_k,z_l)|^2$ for $(j,k,l)\\in {\\cal \\cal\nT}_{MKL}^0$. In the literatures \\cite{Yi2,Tick,Xiong,Blakie}, this\ndipolar energy is usually calculated via discrete Fourier transform\n(DFT) as \\begin{eqnarray} \\label{dftdip1}E_{\\rm dip}(\\phi)\\approx \\frac{\\lambda\nh_xh_yh_z}{2}\n\\sum_{j=1}^{M-1}\\sum_{k=1}^{K-1}\\sum_{l=1}^{L-1}|\\phi(x_j,y_k,z_l)|^2\\left[\n{\\cal F}^{-1}_{jkl}\\left(\\widehat{(U_{\\rm\ndip})}(2\\mu_p^x,2\\mu_q^y,2\\mu_s^z)\\cdot {\\cal\nF}_{pqs}(|\\phi|^2)\\right) \\right], \\nonumber\\end{eqnarray} where ${\\cal F}$ and\n${\\cal F}^{-1}$ are the discrete Fourier and inverse Fourier\ntransforms over the grid points $\\{(x_j,y_k,z_l), \\ (j,k,l)\\in {\\cal\nT}_{MKL}^0\\}$, respectively \\cite{Xiong}. We take $\\lambda=24\\pi$,\nthe bounded computational domain  $\\Omega=[-16,16]^3$, $M=K=L$ and thus\n$h=h_x=h_y=h_z=\\frac{32}{M}$. Table \\ref{tab1} lists the errors\n$e:=\\left|E_{\\rm dip}(\\phi)-E_{\\rm dip}^h\\right|$ with $E_{\\rm\ndip}^h$ computed numerically via either (\\ref{dstdip1}) or\n(\\ref{dftdip1}) with mesh size $h$ for three cases:\n\n\\begin{itemize}\n\\item Case I. $\\gamma_x=0.25$ and $\\gamma_z=1$ which implies $\\kappa\n=2.0$ and $E_{\\rm dip}(\\phi) = 0.0386708614$;\n\n\\item Case II. $\\gamma_x=\\gamma_z=1$ which implies\n$\\kappa =1.0$ and  $E_{\\rm dip}(\\phi)= 0$;\n\n\\item Case III. $\\gamma_x=2$ and $\\gamma_z=1$ which implies\n$\\kappa =\\sqrt{0.5}$ and  $E_{\\rm dip}(\\phi)=-0.1386449741$.\n\\end{itemize}\n\n\n\\begin{table}[htb]\n\\begin{center}\n\\begin{tabular}{c|cc|cc|cc}\n  \\hline\n  &\\multicolumn{2}{c|}{Case I}&\\multicolumn{2}{c|}{Case II}&\\multicolumn{2}{c}{Case III}\\\\ \\cline{2-7}\n  &DST &DFT &DST &DFT &DST &DFT\\\\\\hline\n $M=32 \\& h=1$ &2.756E-2 &2.756E-2 &3.555E-18 &1.279E-4 &0.1018\n &0.1020\\\\\n $M=64 \\& h=0.5$ &1.629E-3 &1.614E-3  &9.154E-18 &1.278E-4\n &9.788E-5\n &2.269E-4\\\\\n $M=128 \\& h=0.25$ &1.243E-7  &1.588E-5 &7.454E-17 &1.278E-4\n&6.406E-7 &1.284E-4  \\\\\n  \\hline\n\\end{tabular}\n\\end{center}\n\\caption{Comparison for evaluating dipolar energy under different\nmesh sizes $h$.} \\label{tab1}\n\\end{table}\n\nFrom Tab. \\ref{tab1} and our extensive numerical results not shown\nhere for brevity, we can conclude that our new method via discrete\nsine transform based on a new formulation  is much more accurate\nthan that of the standard method via discrete Fourier transform in\nthe literatures for evaluating the dipolar energy.\n\n\n\n\\subsection{Ground states of dipolar BECs}\n\nBy using our new numerical method\n(\\ref{discretized1})-(\\ref{discretized2}), here we report the ground\nstates of a dipolar BEC (e.g., ${}^{52}$Cr \\cite{Parker}) with\ndifferent parameters and trapping potentials. In our computation and\nresults, we always use the dimensionless quantities. We take\n$M=K=L=128$, time step $\\Delta t=0.01$, dipolar direction\n${\\bf n} =(0,0,1)^T$ and the bounded computational domain $\\Omega=[-8,8]^3$\nfor all cases except $\\Omega=[-16,16]^3$ for the cases\n$\\frac{N}{10000}=1,\\;5,\\;10$ and $\\Omega=[-20,20]^3$ for the cases\n$\\frac{N}{10000}=50,\\;100$ in Table \\ref{tab2}. The ground state\n$\\phi_g$ is reached numerically when\n$\\|\\phi^{n+1}-\\phi^n\\|_\\infty:=\\max\\limits_{0\\le j\\le M,\\ 0\\le k\\le\nK,\\ 0\\le l\\le L} |\\phi^{n+1}_{jkl}-\\phi^n_{jkl}|\\le \\varepsilon:=10^{-6}$\nin (\\ref{discretized1})-(\\ref{discretized2}). Table \\ref{tab2} shows\nthe energy $E^g:=E(\\phi_g)$, chemical potential\n$\\mu^g:=\\mu(\\phi_g)$, kinetic energy $E_{\\rm kin}^g:=E_{\\rm\nkin}(\\phi_g)$, potential energy $E_{\\rm pot}^g:=E_{\\rm\npot}(\\phi_g)$, interaction energy $E_{\\rm int}^g:=E_{\\rm\nint}(\\phi_g)$, dipolar energy $E_{\\rm dip}^g:=E_{\\rm dip}(\\phi_g)$,\ncondensate widths $\\sigma_x^g:=\\sigma_x(\\phi_g)$ and\n$\\sigma_z^g:=\\sigma_z(\\phi_g)$ in (\\ref{dtap01}) and central density\n$\\rho_g({\\bf 0}):=|\\phi_g(0,0,0)|^2$ with harmonic potential\n$V(x,y,z)= \\fl{1}{2}\\left(x^2+y^2+0.25z^2\\right)$ for different\n$\\beta=0.20716N$ and $\\lambda=0.033146N$ with $N$ the total number\nof particles in the condensate; and Table \\ref{tab3} lists similar\nresults  with $\\beta=207.16$ for different values of $-0.5\\le\n\\frac{\\lambda}{\\beta}\\le 1$. In addition, Figure \\ref{fig1} depicts\nthe ground state $\\phi_g({\\bf x} )$, e.g. surface plots of\n$|\\phi_g(x,0,z)|^2$ and isosurface plots of  $|\\phi_g({\\bf x} )|=0.01$,\n of a dipolar BEC with $\\beta = 401.432$ and $\\lambda\n=0.16\\beta$ for harmonic potential $V({\\bf x} )=\n\\fl{1}{2}\\left(x^2+y^2+z^2\\right)$, double-well potential\n$V({\\bf x} )=\\fl{1}{2}\\left(x^2+y^2+z^2\\right)+4e^{-z^2\/2}$ and optical\nlattice potential\n$V({\\bf x} )=\\fl{1}{2}\\left(x^2+y^2+z^2\\right)+100\\left[\\sin^2\\left(\\fl{\\pi}{2}x\\right)\n+\\sin^2\\left(\\fl{\\pi}{2}y\\right)+\\sin^2\\left(\\fl{\\pi}{2}z\\right)\n\\right]$; and Figure \\ref{fig6} depicts  the ground state\n$\\phi_g({\\bf x} )$, e.g. isosurface plots of $|\\phi_g({\\bf x} )|=0.08$, of a\ndipolar BEC with the harmonic potential $V({\\bf x} )=\n\\fl{1}{2}\\left(x^2+y^2+z^2\\right)$ and $\\beta=207.16$ for different\nvalues of $-0.5\\le \\frac{\\lambda}{\\beta}\\le 1$.\n\n\n\n\\begin{table}[htb]\n\\begin{center}\n\\begin{tabular}{cccccccccc}\n\\hline \\\\\n$\\frac{N}{10000}$ &$E^g$ &$\\mu^g$  &$E_{\\rm kin}^g$ &$E_{\\rm pot}^g$\n&$E_{\\rm int}^g$ &$E_{\\rm dip}^g$\n&$\\sigma_x^g$ &$\\sigma_z^g$& $\\rho_g({\\bf 0})$\\\\\n\\hline\n\n\n0.1 &1.567 &1.813 &0.477 &0.844 &0.262 &-0.015 &0.796 &1.299 &0.06139 \\\\\n0.5 &2.225 &2.837 &0.349 &1.264 &0.659 &-0.047 &0.940 &1.745 &0.02675 \\\\\n1   &2.728 &3.583 &0.296 &1.577 &0.925 &-0.070 &1.035 &2.009 &0.01779\\\\\n5   &4.745 &6.488 &0.195 &2.806 &1.894 &-0.151 &1.354 &2.790 &0.00673 \\\\\n10  &6.147 &8.479 &0.161 &3.654 &2.536 &-0.204 &1.538 &3.212 &0.00442     \\\\\n50  &11.47 &15.98 &0.101 &6.853 &4.909 &-0.398 &2.095 &4.441 &0.00168  \\\\\n100 &15.07 &21.04 &0.082 &9.017 &6.498  &-0.526 &2.400 &5.103 &0.00111  \\\\\n\n\\hline\n\\end{tabular}\n \\end{center}\n\\caption{Different quantities of the ground states of a dipolar BEC\n for $\\beta=0.20716N$ and $\\lambda=0.033146N$ with different number of particles\n$N$.} \\label{tab2}\n \\end{table}\n\n\n\n\\begin{table}[htb]\n\\begin{center}\n\\begin{tabular}{cccccccccc}\n\\hline \\\\\n $\\fl{\\lambda}{\\beta}$  &  $E^g$  & $\\mu^g$ & $E_{kin}^g$ &   $E_{pot}^g$ &  $E_{int}^g$\n &  $E_{dip}^g$ &   $\\sigma_x^g$ & $\\sigma_z^g$ & $\\rho_g({\\bf 0})$\n \\\\ \\hline\n\n-0.5 &2.957 &3.927 &0.265 &1.721 &0.839 &0.131 &1.153 &1.770 &0.01575 \\\\\n-0.25 &2.883 &3.817 &0.274 &1.675 &0.853 &0.081 &1.111 &1.879 &0.01605 \\\\\n0 &2.794 &3.684 &0.286 &1.618 &0.890 &0.000 &1.066 &1.962 &0.01693 \\\\\n0.25 &2.689 &3.525 &0.303 &1.550 &0.950 &-0.114 &1.017 &2.030 &0.01842 \\\\\n0.5 &2.563 &3.332 &0.327 &1.468 &1.047 &-0.278 &0.960 &2.089 &0.02087 \\\\\n0.75 &2.406 &3.084 &0.364 &1.363 &1.212 &-0.534 &0.889 &2.141 &0.02536\\\\\n1.0 &2.193 &2.726 &0.443 &1.217 &1.575 &-1.041 &0.786  &2.189 &0.03630 \\\\\n \\hline\n\n\n\\end{tabular}\n \\end{center}\n \\caption{ Different quantities of the ground states of a dipolar BEC with different\n values of $\\frac{\\lambda}{\\beta}$ with $\\beta=207.16$.}\\label{tab3}\n \\end{table}\n\n\n\\begin{figure}[h!]\n\\centerline{\n\\psfig{figure=harmonic.eps,height=6cm,width=6cm,angle=0}\\qquad\n\\psfig{figure=harmonic-iso.eps,height=6cm,width=6cm,angle=0}\n }\n \\centerline{\n\\psfig{figure=double_well.eps,height=6cm,width=6cm,angle=0}\\qquad\n\\psfig{figure=double_well2-iso.eps,height=6cm,width=6cm,angle=0}\n }\n \\centerline{\n\\psfig{figure=optical_lattice.eps,height=6cm,width=6cm,angle=0}\\qquad\n\\psfig{figure=optical_lattice2-iso.eps,height=6cm,width=6cm,angle=0}\n }\n  \\caption{\nSurface plots of $|\\phi_g(x,0,z)|^2$ (left column) and isosurface\nplots of $|\\phi_g(x,y,z)|=0.01$ (right column) for the ground state\nof a dipolar BEC with $\\beta= 401.432$ and $\\lambda = 0.16 \\beta$\nfor harmonic potential (top row), double-well potential (middle row)\nand optical lattice potential (bottom row).\n  } \\label{fig1}\n\\end{figure}\n\n\n\n\n\n\n\n\nFrom Tabs. \\ref{tab2}\\&\\ref{tab3} and Figs. \\ref{fig1}\\&\\ref{fig6},\nwe can draw the following conclusions: (i) For fixed trapping\npotential $V({\\bf x} )$ and dipolar direction ${\\bf n} =(0,0,1)^T$, when\n$\\beta$ and $\\lambda$ increase with the ratio\n$\\frac{\\lambda}{\\beta}$ fixed, the energy $E^g$, chemical potential\n$\\mu^g$,  potential energy $E_{\\rm pot}^g$, interaction energy\n$E_{\\rm int}^g$,  condensate widths $\\sigma_x^g$ and $\\sigma_z^g$ of the\nground states increase; and resp., the  kinetic energy $E_{\\rm\nkin}^g$, dipolar energy $E_{\\rm dip}^g$ and central density\n$\\rho_g({\\bf 0})$ decrease (cf. Tab. \\ref{tab2}). (ii) For fixed\ntrapping potential $V({\\bf x} )$, dipolar direction ${\\bf n} =(0,0,1)^T$ and\n$\\beta$, when the ratio $\\frac{\\lambda}{\\beta}$ increases from\n$-0.5$ to $1$, the kinetic energy $E_{\\rm kin}^g$, interaction\nenergy $E_{\\rm int}^g$, condensate widths $\\sigma_z^g$ and central\ndensity $\\rho_g({\\bf 0})$ of the ground states increase; and resp.,\nthe energy $E^g$, chemical potential $\\mu^g$, potential energy\n$E_{\\rm pot}^g$, dipolar energy $E_{\\rm dip}^g$ and condensate\nwidths $\\sigma_x^g$\n decrease (cf. Tab. \\ref{tab3}). (iii) Our new numerical method can\n compute the ground states accurately and efficiently (cf. Figs.\n \\ref{fig1}\\&\\ref{fig6}).\n\n\n\\subsection{Dynamics of dipolar BECs}\n\nSimilarly, by using our new numerical method (\\ref{tssp1}), here we\nreport the dynamics of a dipolar BEC (e.g., ${}^{52}$Cr\n\\cite{Parker}) under different setups. Again, in our computation and\nresults, we always use the dimensionless quantities. We take the\nbounded computational domain $\\Omega=[-8,8]^2\\times[-4,4]$,\n$M=K=L=128$, i.e. $h=h_x=h_y=1\/8,h_z=1\/16$,  time step $\\Delta\nt=0.001$. The initial data $\\psi({\\bf x} ,0)=\\psi_0({\\bf x} )$ is chosen as\nthe ground state of a dipolar BEC computed numerically by our\nnumerical method with ${\\bf n} =(0,0,1)^T$,\n$V({\\bf x} )=\\fl{1}{2}(x^2+y^2+25z^2)$, $\\beta=103.58$ and\n$\\lambda=0.8\\beta=82.864$.\n\nThe first case  to study numerically is the dynamics of suddenly\nchanging the dipolar direction from ${\\bf n} =(0,0,1)^T$ to\n${\\bf n} =(1,0,0)^T$ at $t=0$ and keeping all other quantities unchanged.\nFigure \\ref{fig2} depicts time evolution of the energy\n$E(t):=E(\\psi(\\cdot,t))$, chemical potential\n$\\mu(t)=\\mu(\\psi(\\cdot,t)$, kinetic energy $E_{\\rm kin}(t):=E_{\\rm\nkin}(\\psi(\\cdot,t))$, potential energy $E_{\\rm pot}(t):=E_{\\rm\npot}(\\psi(\\cdot,t))$, interaction energy $E_{\\rm int}(t):=E_{\\rm\nint}(\\psi(\\cdot,t))$, dipolar energy $E_{\\rm dip}(t):=E_{\\rm\ndip}(\\psi(\\cdot,t))$, condensate widths\n$\\sigma_x(t):=\\sigma_x(\\psi(\\cdot,t))$,\n$\\sigma_z(t):=\\sigma_z(\\psi(\\cdot,t))$, and central density\n$\\rho(t):=|\\psi({\\bf 0},t)|^2$, as well as the isosurface of the\ndensity function $\\rho({\\bf x} ,t):=|\\psi({\\bf x} ,t)|^2=0.01$ for different\ntimes. In addition, Figure \\ref{fig3} show similar results for the\ncase of\n suddenly\nchanging the trapping potential   from\n$V({\\bf x} )=\\fl{1}{2}(x^2+y^2+25z^2)$ to\n$V({\\bf x} )=\\fl{1}{2}(x^2+y^2+\\fl{25}{4}z^2)$ at $t=0$, i.e. decreasing\nthe trapping frequency in z-direction from $5$ to $\\fl{5}{2}$,  and\nkeeping all other quantities unchanged;  Figure \\ref{fig4}  show the\nresults for the case of\n suddenly\nchanging the dipolar interaction from $\\lambda=0.8\\beta=82.864$ to\n$\\lambda=4\\beta=414.32$ at $t=0$ while keeping all other quantities\nunchanged, i.e. collapse of a dipolar BEC; and Figure \\ref{fig7}\nshow the results for the case of\n suddenly\nchanging the interaction constant $\\beta$ from $\\beta=103.58$ to\n$\\beta=-569.69$ at $t=0$ while keeping all other quantities\nunchanged, i.e. another collapse of a dipolar BEC.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nFrom Figs. \\ref{fig2}, \\ref{fig3}, \\ref{fig4} and \\ref{fig7}, we can\nconclude that the dynamics of dipolar BEC can be very interesting\nand complicated. In fact, global existence of the solution is\nobserved in the first two cases (cf. Figs. \\ref{fig2}\\&\\ref{fig3})\nand finite time blow-up is observed in the last two cases (cf. Figs.\n\\ref{fig4}\\&\\ref{fig7}). The total energy is numerically conserved\nvery well in our computation when there is no blow-up (cf. Figs.\n\\ref{fig2}\\&\\ref{fig3}) and before blow-up happens (cf. Figs.\n\\ref{fig4}\\&\\ref{fig7}). Of course, it is not conserved numerically\nnear or after blow-up happens because the mesh size and time step\nare fixed which cannot resolve the solution. In addition, our new\nnumerical method can compute the dynamics of dipolar BEC accurately\nand efficiently.\n\n\\begin{remark} Due to size limit at ariv, to read the full figures,\nyou can download this paper from:\\\\\nhttp:\/\/www.math.nus.edu.sg\/\\~{ }bao\/PS\/dipolar-bec.pdf\n\\end{remark}\n\n\n\n\\section{Conclusions}\n\\setcounter{equation}{0}\n\nEfficient and accurate numerical methods were proposed for computing\nground states and dynamics of dipolar Bose-Einstein condensates\nbased on the three-dimensional Gross-Pitaevskii equation (GPE)  with\na nonlocal dipolar interaction potential. By decoupling the dipolar\ninteraction potential into a short-range and a long-range part, the\nGPE for a dipolar BEC is re-formulated to a Gross-Pitaevskii-Poisson\ntype system. Based on this new mathematical formulation, we proved\nrigorously the existence and uniqueness as well as nonexistence of\nthe ground states, and discussed the dynamical properties of dipolar\nBEC in different parameter regimes. In addition, the backward Euler\nsine pseudospectral method and time-splitting sine pseudospectral\nmethod were proposed for computing the ground states and dynamics of\na dipolar BEC, respectively. Our new numerical methods avoided\ntaking the Fourier transform of the nonlocal dipolar interaction\npotential which is highly singular and causes some numerical\ndifficulties in practical computation. Comparison between our new\nnumerical methods and existing numerical methods in the literatures\nshowed that our numerical methods perform better. Applications of\nour new numerical methods for computing the ground states and\ndynamics of dipolar BECs were reported. In the future, we will use\nour new numerical methods to simulate the ground states and dynamics\nof dipolar BEC with experimental relevant setups and extend our\nmethods for rotating dipolar BECs.\n\n\n\n\\bigskip\n\n\\begin{center}\n {\\bf Acknowledgements}\n\\end{center}\n\n\nThis work was supported in part by the Academic Research Fund of\nMinistry of Education of Singapore grant R-146-000-120-112 (W.B.,\nY.C. and H.W.)\nand the National Natural Science Foundation of China grant 10901134\n(H.W.). We acknowledge very stimulating and helpful discussions with\nProfessor Peter A. Markowich on the topic.  This work was partially\ndone while the authors were visiting the Institute for Mathematical\nSciences, National University of Singapore, in 2009.\n\n\n\\bigskip\n\n\\begin{center}\n {\\bf Appendix  \\qquad Proof of the equality (\\ref{decop1})}\n\\end{center}\n\n\\renewcommand{\\theequation}{\\Alph{section}.\\arabic{equation}}\n\\setcounter{equation}{0} \\setcounter{section}{1}\n\n\nLet \\begin{equation} \\phi({\\bf x} )=\\frac{1}{r^3}\\left(1-\\frac{3({\\bf x} \\cdot {\\bf\nn})^2}{r^2}\\right), \\qquad r=|{\\bf x} |, \\qquad {\\bf x} \\in{\\mathcal R}^3.\\end{equation} For\nany ${\\bf n} \\in{\\mathcal R}^3$ satisfies $|{\\bf n} |=1$, in order to prove\n(\\ref{decop1}) holds in the distribution sense, it is equivalent to\nprove the following:\n  \\begin{equation} \\label{tt87}\n\\int_{{\\mathcal R}^3} \\phi({\\bf x} ) f({\\bf x} ) d{\\bf x}  =-\\frac{4\\pi}{3}f({\\bf 0}) -\n\\int_{{\\mathcal R}^3}f({\\bf x} )\\; \\partial_{{\\bf n} \\bn} \\left(\\fl{1}{r}\\right)  d{\\bf x} ,\n\\qquad \\forall f({\\bf x} ) \\in C_0^\\infty ({\\mathcal R}^3). \\label{theorem1}\n\\end{equation} For any fixed $\\varepsilon>0$, let $B_\\varepsilon=\\{{\\bf x} \\in\\mathcal R^3\\ |\\\n|{\\bf x} |<\\varepsilon\\}$ and $B_\\varepsilon^c=\\{{\\bf x} \\in\\mathcal R^3 \\ |\\\n|{\\bf x} |\\ge \\varepsilon\\}$. It is straightforward to check that \\begin{equation}\n\\label{phi1234}\\phi({\\bf x} )=-\\partial_{{\\bf n} \\bn}\\left(\\frac 1r\\right), \\qquad\n0\\ne {\\bf x} \\in {\\mathcal R}^3.\\end{equation} Using integration by parts and noticing\n(\\ref{phi1234}), we get \\begin{eqnarray}\\label{vkder} \\int_{B_\\varepsilon^c }\n\\phi({\\bf x} ) f({\\bf x} ) d{\\bf x} &=&- \\int_{B_\\varepsilon^c}f({\\bf x} )\\; \\partial_{{\\bf n} \\bn}\n\\left(\\fl{1}{r}\\right)  \\; d{\\bf x}  \\nonumber \\\\\n&=&\\int_{B_\\varepsilon^c}\\partial_{\\bf n}  \\left(\\fl{1}{r}\\right)\\; \\partial_{\\bf n} \n(f({\\bf x} ))\\; d{\\bf x}  +\\int_{\\partial B_\\varepsilon}f({\\bf x} )\\;\\frac{{\\bf\nn}\\cdot{\\bf x} }{r}\\;\\partial_{\\bf n} \n\\left(\\fl{1}{r}\\right)\\,  dS\\nonumber\\\\\n&=&-\\int_{B_\\varepsilon^c}\\frac{1}{r}\\;\\partial_{{\\bf n} \\bn}(f({\\bf x} ))\\;d{\\bf x} +I^\\varepsilon_1+I^\\varepsilon_2,\n    \\end{eqnarray}\nwhere \\begin{equation} \\label{I1I2}I^\\varepsilon_1:=\\int_{\\partial\nB_\\varepsilon}f({\\bf x} )\\;\\frac{{\\bf n}\\cdot{\\bf x} }{r}\\;\\partial_{\\bf n} \n\\left(\\fl{1}{r}\\right)\\, dS, \\qquad I^\\varepsilon_2:=-\\int_{\\partial\nB_\\varepsilon}\\frac{{\\bf n}\\cdot{\\bf x} }{r^2}\\;\\partial_{\\bf n} \n\\left(f({\\bf x} )\\right)\\,dS. \\end{equation} From (\\ref{I1I2}), changing of\nvariables, we get\n\\begin{eqnarray} \\label{I145}I^\\varepsilon_1&=&-\\int_{\\partial\nB_\\varepsilon}\\frac{({\\bf n}\\cdot {\\bf x} )^2}{r^4}f({\\bf x} )\\,dS=-\\int_{\\partial\nB_1}\\frac{({\\bf n}\\cdot\n{\\bf x} )^2}{\\varepsilon^2}f(\\varepsilon{\\bf x} )\\,\\varepsilon^2dS\\nonumber\\\\\n&=&-\\int_{\\partial B_1}({\\bf n}\\cdot {\\bf x} )^2f({\\bf 0})\\,dS-\\int_{\\partial\nB_1}({\\bf n}\\cdot {\\bf x} )^2\\left[f(\\varepsilon{\\bf x} )-f({\\bf 0})\\right]\\,dS.\n\\end{eqnarray} Choosing $0\\ne{\\bf n} _1\\in{\\mathcal R}^3$ and $0\\ne {\\bf n} _2\\in{\\mathcal R}^3$\nsuch that $\\{{{\\bf{n}_1},\\,{\\bf{n}_2}\\,\\bf{n}}\\}$ forms an\northornormal basis of ${\\mathcal R}^3$, by symmetry, we obtain\n\\begin{eqnarray}\\label{etd11} &&A:=\\int_{\\partial B_1}({\\bf n}\\cdot\n{\\bf x} )^2\\,dS=\\frac{1}{3}\\int_{\\partial\nB_1}\\left[({\\bf{n}}\\cdot{\\bf x} )^2+({\\bf{n}_1}\\cdot{\\bf x} )^2+\n({\\bf{n}_2}\\cdot{\\bf x} )^2\\right]\\,dS\\nonumber\\\\\n&&\\quad =\\frac{1}{3}\\int_{\\partial B_1} |{\\bf x} |^2dS=\\frac{1}{3}\\int_{\\partial B_1}dS=\\frac{4\\pi}{3},\\\\\n\\label{etd12} &&\\left|\\int_{\\partial B_1}({\\bf n}\\cdot\n{\\bf x} )^2\\left(f(\\varepsilon{\\bf x} )-f({\\bf 0})\\right)\\,dS\\right|=\\left|\\int_{\\partial\nB_1}({\\bf\nn}\\cdot {\\bf x} )^2\\varepsilon\\; \\left[{\\bf x} \\cdot \\nabla f(\\theta\\varepsilon{\\bf x} )\\right]\\,dS\\right|\\nonumber\\\\\n&&\\quad \\leq\\varepsilon \\,\\|\\nabla f\\|_{L^\\infty(B_\\varepsilon)} \\int_{\\partial\nB_1}\\,dS\\leq 4\\pi\\varepsilon\\,\\|\\nabla f\\|_{L^\\infty(B_\\varepsilon)},\n\\end{eqnarray}\nwhere $0\\le \\theta\\le 1$. Plugging (\\ref{etd11}) and (\\ref{etd12})\ninto (\\ref{I145}), we have \\begin{equation} \\label{I167} I^\\varepsilon_1\\to\n-\\frac{4\\pi}{3}f({\\bf 0}), \\qquad \\varepsilon \\to 0^+. \\end{equation} Similarly, for\n$\\varepsilon\\to0^+$, we get \\begin{eqnarray}\\label{I267} &&|I^\\varepsilon_2|\\leq \\|\\nabla\nf\\|_{L^\\infty(B_\\varepsilon)}\\int_{\\partial B_\\varepsilon} \\frac{1}{\\varepsilon}\\,dS=\n4\\pi\\varepsilon\\, \\|\\nabla\nf\\|_{L^\\infty(B_\\varepsilon)}\\to0, \\\\\n \\label{chat34} &&\\left|\\int_{B_\\varepsilon}\n\\frac{1}{r}\\;\\partial_{{\\bf n} \\bn}(f({\\bf x} ))\\,d{\\bf x} \\right|\\le\n\\|D^2f\\|_{L^\\infty(B_\\varepsilon)}\\, \\int_{B_\\varepsilon} \\frac{1}{r}\\,d{\\bf x}  \\leq\n2\\pi\\varepsilon^2\\,\\|D^2f\\|_{L^\\infty(B_\\varepsilon)} \\to 0. \\qquad \\qquad \\end{eqnarray}\nCombining (\\ref{I167}), (\\ref{I267}) and (\\ref{chat34}), taking\n$\\varepsilon\\to0^+$ in (\\ref{vkder}), we obtain\n  \\begin{equation} \\label {tt98}\\int_{{\\mathcal R}^3}\n\\phi({\\bf x} ) f({\\bf x} ) d{\\bf x}  =-\\fl{4\\pi}{3} f({\\bf 0}) - \\int_{{\\mathcal R}^3}\n\\fl{1}{r}\\;\\partial_{{\\bf n} \\bn} (f({\\bf x} )) \\,d{\\bf x} , \\qquad \\forall f({\\bf x} ) \\in\nC_0^\\infty ({\\mathcal R}^3). \\label{theorem3} \\end{equation} Thus (\\ref{tt87})\nfollows from (\\ref{tt98}) and the definition of the derivative in\nthe distribution sense, i.e. \\begin{equation} \\int_{{\\mathcal R}^3} f({\\bf x} )\\;\n\\partial_{{\\bf n} \\bn} \\left(\\fl{1}{r}\\right)\n      d{\\bf x}  = \\int_{{\\mathcal R}^3}\\fl{1}{r}\\;\\partial_{{\\bf n} \\bn}\n      (f({\\bf x} ))\\,d{\\bf x} , \\qquad \\forall f({\\bf x} ) \\in\nC_0^\\infty ({\\mathcal R}^3),\\end{equation} and the equality (\\ref{decop1}) is\nproven. $\\Box$\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \nSince the discovery of the 125 GeV Higgs boson in 2012 at the\nLHC \\cite{discovery},\nwe have been looking for a clear signal or \neven a hint of new physics beyond the Standard Model (SM) but without much success.\nMoreover, after completing the Runs I and II at the LHC, \nit turns out that the 125 GeV Higgs boson is best described as \nthe SM Higgs boson \\cite{higgcision},\nalthough there is an upward trend in the overall signal\nstrength \\cite{Cheung:2018ave}.\nUnder this situation,  one of the most solid avenues to explore for new physics\nis to measure the Higgs potential which could be significantly different from\nthat of the SM.\n\nHiggs-boson pair production at the high-luminosity and\/or high-energy\nhadron colliders provides a very useful way to probe the Higgs potential\nvia the investigation of the trilinear Higgs self-coupling \n(THSC)~\\cite{dihiggs1,dihiggs2,dihiggs3}.\nThe specific decay modes considered are: $b\\bar b b\\bar b$~\\cite{bbbb},\n$b\\bar b\\gamma\\gamma$~\\cite{bbaa,bbaa_atlas17},\n$b\\bar b\\tau^+\\tau^-$~\\cite{bbtata},\n$b\\bar b W^+W^-$~\\cite{bbWW}, and some combinations\nof these channels~\\cite{decaycombined,comb_atlas18}.\nHiggs-boson pair production also has been vastly studied in \nmodels beyond the SM~\\cite{dihiggs_np}.\n\nThe current limits on the THSC in units of $\\lambda_{3H}$, which takes\nthe value of 1 in the SM, are\n$-5.0 < \\lambda_{3H} < 12$  from ATLAS~\\cite{Aad:2019uzh}\nand $-11.8 < \\lambda_{3H} < 18.8$  from CMS~\\cite{Sirunyan:2018two}\nat 95\\% confidence level (CL). \nAt the high-luminosity option of the LHC running at 14 TeV (HL-LHC)\nwith an integrated luminosity of 3 ab$^{-1}$, \na combined ATLAS and CMS projection of the 68\\% CL interval \nis $0.57 < \\lambda_{3H} < 1.5$ \nwithout including systematic uncertainties~\\cite{Cepeda:2019klc}.\nOn the other hand, at the International Linear Collider (ILC) operated at\n1 TeV can reach the precision of 10\\% at 68\\%  CL with \nan integrated luminosity of 8 ab$^{-1}$~\\cite{Fujii:2015jha,Braathen:2019zoh}.\n\n\\bigskip\n\nIn this work, we perform a multivariate analysis of Higgs-pair production \nin $HH \\to b\\bar b \\gamma\\gamma$ channel at the \n100 TeV hadron collider.\nIn our previous work,   \nbased on the conventional cut-and-count analysis,\nit was shown that the THSC can be measured with about 20\\% accuracy\nat the SM value with a luminosity of 3 ab$^{-1}$~\\cite{Chang:2018uwu}.\nIn this Letter, with the use of the BDT method \nclosely following Ref.~\\cite{Chang:2019ncg},\nwe show that the THSC can be measured with \na precision of $7.5\\%$ at 68\\% CL at the 100 TeV hadron collider\nassuming 3 ab$^{-1}$ luminosity,\nwhich is superior to the accuracy expected\nat the 1 TeV ILC even with 8 ab$^{-1}$.\n\n\\section{Event generation and TMVA analysis}\nThe Higgs bosons in the signal event samples\nare generated on-shell with a zero width by\n\\texttt{POWHEG-BOX-V2}~\\cite{Heinrich:2017kxx,Heinrich:2019bkc} with\nthe damping factor $\\mathtt{hdamp}$ set to the default value of 250\nto limit the amount of hard radiation. \nThis code provides NLO distributions matched to a parton shower \ntaking account of the full top-quark mass dependence. \nThe signal cross section at NNLO order in QCD is calculated according to\n$\\sigma^{\\rm NNLO} ( \\lambda_{3H} )=\nK^{\\rm NNLO \/ NLO}_{\\rm SM}\\, \\sigma^{\\rm NLO}  (\\lambda_{3H}) \\;$\nusing $\\sigma^{\\rm NLO}(\\lambda_{3H})$ from {\\tt POWHEG-BOX-V2}\nand  $K^{\\rm NNLO\/NLO}_{\\rm SM}=1.067$~\\cite{Grazzini:2018bsd}\n\\footnote{\nAccording to the recent N3LO calculations~\\cite{n3lo}, \nthe signal cross section is further enhanced by the amount of $2.7$\\% which \nwould hardly affect our conclusion, or rather strengthen our results.}\nin the FT\napproximation in which the full top-quark mass dependence is considered only\nin the real radiation while the Born improved Higgs Effective Field Theory\nis taken in the virtual part.\nAnd then, \nthe \\texttt{MadSpin} code \\cite{Artoisenet:2012st} is used for the decay of\nboth Higgs bosons into two bottom quarks and two photons.\n\n\n\nFor generation and simulation of backgrounds,\nwe closely follow Ref.~\\cite{Chang:2018uwu}\n\\footnote{Specifically, the multi-variate MV1 $b$-tagging algorithm \nwith $\\epsilon_b=0.75$ is taken together with $P_{c\\to b}=0.1$, \n$P_{j\\to b}=0.01$, and $P_{j\\to\\gamma}=1.35\\times 10^{-3}$~\\cite{Contino:2016spe}.}, \nexcept for the use\nof the post-LHC PDF set of {\\tt CT14LO}~\\cite{Dulat:2015mca} \nfor non-resonant backgrounds.\nFurthermore,\nfor the two main non-resonant backgrounds of $b\\bar b\\gamma\\gamma$ and $c\\bar\nc\\gamma\\gamma$,\nwe use the merged cross sections and distributions\nby MLM matching~\\cite{Mangano:2006rw,Alwall:2007fs}\nwith {\\bf xqcut} and $Q_{\\rm cut}$ set to 20 GeV and 30 GeV, respectively.\nFor the remaining non-resonant backgrounds,\nwe are using the cross sections and distributions obtained by applying\nthe generator-level cuts as adopted in Ref.~\\cite{bbaa_atlas17,comb_atlas18}\nwhich might provide more reliable and conservative estimation of the non-resonant\nbackgrounds\ncontaining light jets~\\cite{Chang:2018uwu}.\n\nFor parton showering and hadronization,\n\\texttt{PYTHIA8}~\\cite{Sjostrand:2014zea} is used both for\nsignal and backgrounds.\nFinally, fast-detector simulation and analysis are\nperformed using \\texttt{Delphes3}~\\cite{deFavereau:2013fsa}\nwith the \\texttt{Delphes-FCC} template.\n\nAll the signal and backgrounds are summarized in Table~\\ref{tab:ParticleList_100TeV},\ntogether with information of the corresponding event\ngenerator, the cross section times the branching ratio and \nthe order in QCD, and the Parton Distribution Function (PDF) used.\n\n\\begin{table}[th!]\n\\centering\n\\caption{\\small \nMonte Carlo samples used in Higgs-pair production\nanalysis $H(\\rightarrow b\\bar{b})H(\\rightarrow \\gamma\\gamma)$, and\nthe corresponding codes for the matrix-element generation. \n$\\tt{PYTHIA8}$ is used for parton showering and hadronization. \nWe refer to Ref.~\\cite{Alwall:2014hca} for $\\mathtt{MG5\\_aMC@NLO}$.\n}\n\\vspace{3mm}\n\\label{tab:ParticleList_100TeV}\n\\begin{tabular}{  c  c  c  c  c  c }\n\\hline\n\\multicolumn{6}{c}{Signal} \\\\\n\\hline\n\\multicolumn{2}{c}{Signal process} & Generator &\n$\\sigma \\cdot BR$ [fb] & Order  & PDF used  \\\\\n&&&& in QCD &   \\\\\n\\hline\n\\multicolumn{2}{c}{$gg \\to HH \\to b\\bar b \\gamma\\gamma$} &\n {\\tt POWHEG-BOX-V2} & 3.25\n & NNLO &  PDF4LHC15$\\_$nlo \\\\\n\\hline\n\\hline\n\\multicolumn{6}{c}{Backgrounds} \\\\\n\\hline\nBackground(BG)  & Process  & Generator & $\\sigma\\cdot BR$~[fb] &  Order  &\nPDF used\\\\\n&&&& in QCD&   \\\\\n\\hline\n\\multirow{4}{*}{}\n& $ggH(\\rightarrow \\gamma\\gamma)$  &  $\\mathtt{POWHEG-BOX}$\n  & $1.82 \\times 10^{3}$ & $\\mathrm{NNNLO}$ & $\\mathtt{CT10}$ \\\\ \\cline{2-5}\nSingle-Higgs  & $t \\bar{t} H(\\rightarrow \\gamma\\gamma)$  &\n$\\mathtt{PYTHIA8}$  & $7.29\\times 10^1$ & NLO & \\\\ \\cline{2-5}\n associated BG             &  $ZH(\\rightarrow \\gamma\\gamma)$ &\n$\\mathtt{PYTHIA8}$  & $2.54\\times 10^1$ & NNLO &\\\\ \\cline{2-5}\n                  & $b\\bar{b}H(\\rightarrow \\gamma\\gamma)$ &\n$\\mathtt{PYTHIA8}$ & $1.96\\times 10^1$ & NNLO(5FS)  &\\\\ \\hline\n\\multirow{7}{*}{Non-resonant BG} \n & $b\\bar{b} \\gamma\\gamma$ & $\\mathtt{MG5\\_aMC@NLO}$ & $2.28 \\times 10^3$  &  LO & CT14LO \\\\  \\cline{2-5}\n &  $c\\bar{c} \\gamma\\gamma$ & $\\mathtt{MG5\\_aMC@NLO}$ &  $1.92 \\times 10^4$  & LO &  MLM~\\cite{Mangano:2006rw,Alwall:2007fs}  \\\\ \\cline{2-6}\n &  $jj\\gamma\\gamma$ & $\\mathtt{MG5\\_aMC@NLO}$& $4.20 \\times 10^5$  & LO &  \\\\ \\cline{2-5}\n &  $b\\bar{b}j\\gamma$ & $\\mathtt{MG5\\_aMC@NLO}$ & $0.96 \\times 10^7$ & LO & \\\\ \\cline{2-5}\n &  $c\\bar{c}j\\gamma$ & $\\mathtt{MG5\\_aMC@NLO}$& $3.19 \\times 10^7$ & LO &  CT14LO \\\\ \\cline{2-5}\n &  $b\\bar{b}jj$  & $\\mathtt{MG5\\_aMC@NLO}$ & $1.00 \\times 10^{10}$ & LO &  \nRefs.~\\cite{bbaa_atlas17,comb_atlas18,Chang:2018uwu}  \\\\ \\cline{2-5}\n & $Z(\\rightarrow b\\bar{b})\\gamma\\gamma$ &\n$\\mathtt{MG5\\_aMC@NLO}$ &  $7.87 \\times 10^1)$ & LO & \\\\ \\hline\n\\multirow{2}{*}{$t\\bar{t}$ and $t\\bar{t}\\gamma$ BG} \n& $t\\bar{t}$ & $\\mathtt{MG5\\_aMC@NLO}$  & $1.76 \\times 10^7$ & NLO &  $\\mathtt{CT10}$  \\\\ \n\\cline{2-6}\n  ($\\geq 1$ lepton)   & $t\\bar{t}\\gamma$\n & $\\mathtt{MG5\\_aMC@NLO}$  &\n$4.18 \\times 10^4$ & NLO & $\\mathtt{CTEQ6L1}$ \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}[th!]\n\\caption{\nSequence of event selection criteria applied in this analysis.}\n\\label{tab:event_selection}\n\\vspace{3mm}\n  \\begin{tabular}{|c|l| }\n  \\hline\n    Sequence &~ Event Selection Criteria at the 100 TeV hadron collider \\\\\n    \\hline\n    \\hline\n    1 &~ Di-photon trigger condition,\n$\\geq $ 2 isolated photons with $P_T > 30$ GeV, $|\\eta| < 5$\n\\\\\n    \\hline\n    2 &~ $\\geq $ 2 isolated photons with $P_T > 40$ GeV,\n$|\\eta| < 3$, $\\Delta R_{j\\gamma\\,,\\gamma\\gamma} > 0.4$ \\\\\n    \\hline\n    3 &~ $\\geq$ 2 jets identified as b-jets with leading(subleading) $P_T > 50(40)$ GeV, \n$|\\eta|<3$, $\\Delta R_{bb} > 0.4$ \\\\\n    \\hline\n    4 &~ Events are required to contain $\\le 5$ jets with\n $P_T >40$ GeV within $|\\eta|<5$ \\\\\n\\hline\n    5 &~ No isolated leptons with $P_T > 40$ GeV, $|\\eta| <3$ \\\\\n    \\hline\n    6 &~ TMVA analysis  \\\\\n    \\hline\n    \\end{tabular}\n\\end{table}\nA multivariate analysis is performed using TMVA~\\cite{TMVA2007}\nwith {\\tt ROOTv6.18}~\\cite{ROOT}.\nAfter applying a sequence of event selections\nas in Table~\\ref{tab:event_selection},\nwe choose the following 8 kinematic variables for TMVA:\n\\begin{equation}\nM_{bb}\\,, \\ \\ P_{T}^{bb}\\,, \\ \\ \\Delta R_{bb}\\,; \\ \\\nM_{\\gamma\\gamma}\\,, \\ \\ P_{T}^{\\gamma\\gamma}\\,, \\ \\ \\Delta R_{\\gamma\\gamma}\\,; \\ \\\nM_{\\gamma\\gamma bb}\\,, \\ \\ \\Delta R_{\\gamma b}\\,. \\nonumber\n\\end{equation}\nThe judicious choice of the two photons or two $b$ quarks for the above TMVA variables\nhas been made as in ~\\cite{Chang:2019ncg}.\nWe also refer to Ref.~\\cite{Chang:2019ncg} for the details of our TMVA setup and\nanalysis. And we choose  BDT for our analysis since\nthe BDT-related methods show higher performance\nwith better signal efficiency and stronger background rejection.\n\n\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=3.2in,height=2.9in]{overtrain_BDTL1_p.png}\n\\includegraphics[width=3.2in,height=2.9in]{BDT_Z_eseb_p.png}\n\\caption{\n(Left) \nNormalized SM BDT responses for test (histogram) and training (dots with error bars) \nsamples. BDT responses for signal (blue) and background (red) samples,\nwhich mostly populate\nin the regions with positive and negative BDT response, respectively.\n(Right)\nSignal and background efficiencies (inset) and\nsignificance $Z$ as functions of BDT response cut.\nBDT$_{\\rm SM}$ is used.\nThe vertical lines show the position of \nthe optimal cut on the BDT response which maximizes the significance.\n}\n\\label{fig:BDT_over}\n\\end{figure}\n\\section{Results}\nIn the left panel of Fig.~\\ref{fig:BDT_over}, we show the BDT responses obtained using\nBDT trained for $\\lambda_{3H}=1$ which is dubbed as BDT$_{\\rm SM}$.\nBy validating the BDT distributions for the training sample (dots with error bars)\nwith those for the test sample (histogram), we check that BDT$_{\\rm SM}$ is not \novertrained. \nIn the right panel of Fig.~\\ref{fig:BDT_over}, using BDT$_{\\rm SM}$,\nwe show the behavior of signal and background efficiencies (inset)\nand significance \n$ Z = \\sqrt{ 2 \\cdot \\left[ \\left( (s+b) \\cdot \\ln( 1+ s\/b) - s \\right)  \\right ] }$\nwith $s$ and $b$ being the numbers of signal and background events\nas functions of the cut value on BDT response.\nThe significance can reach up to $20.50$ when the BDT response is cut\nat $0.216$, at which, the signal and background efficiencies are\n$0.48$ and $1.58\\times 10^{-4}$, respectively.\nWe denote by vertical lines the positions of the optimal cut on \nthe BDT response which maximizes the significance.\n\n\n\\begin{table}[t!]\n\\caption{\\small\nExpected number of signal and background events at the 100 TeV hadron collider \nassuming 3 ab$^{-1}$ using BDT$_{\\rm SM}$ with the BDT response cut of $0.216$.\nSee text for explanation.\n}\n\\vspace{-1mm}\n\\label{tab:l3h1_opt}\n\\begin{center}\n\\begin{tabular}{| l || r | r || r || r |}\n\\hline\n&\\multicolumn{3}{c||}{Expected yields $(3~ \\mathrm{ab}^{-1})$}\n& \\\\ \\hline\nSignal and Backgrounds & ~Pre-Selection & ~BDT$_{\\rm SM}$ &  ~Cut-and & Eff. Lumi.  \\\\ \n &  & &  -Count~~ & (ab$^{-1}$)~~  \\\\ \n\\hline\n$H(b\\,\\bar{b})\\,H(\\gamma\\,\\gamma)$, $\\lambda_{3H} = -3$ & 7253.98\t&2408.37 &\t3400.08&10.7 \\\\\n$H(b\\,\\bar{b})\\,H(\\gamma\\,\\gamma)$, $\\lambda_{3H} = 0$ & 2072.09\t&\t902.49&\t1146.21&44.5 \\\\\n$\\mathbf{H(b\\,\\bar{b})\\,H(\\gamma\\,\\gamma)}$, $\\mathbf{\\lambda_{3H}= 1}$ & \\textbf{1124.48} & \\textbf{548.02} & \\textbf{673.29}  &\\textbf{615} \\\\\n$H(b\\,\\bar{b})\\,H(\\gamma\\,\\gamma)$, $\\lambda_{3H} = 5$ & 1480.24\t&251.13 &\t439.29&40.9 \\\\ \\hline\n$gg\\,H(\\gamma\\,\\gamma)$ &5827.41\t&255.86 &\t875.71&17.0 \\\\\n$t\\,\\bar{t}\\,H(\\gamma\\,\\gamma)$ &11371.21\t&145.88 &\t868.73&13.2 \\\\\n$Z\\,H(\\gamma\\,\\gamma)$ & 593.29\t&38.88 &\t168.86&39.4 \\\\\n$b\\,\\bar{b}\\,H(\\gamma\\,\\gamma)$ & 205.45\t&2.59 &\t9.82&51.0 \\\\ \\hline\n$b\\,\\bar{b}\\,\\gamma\\,\\gamma$&183493.56\t&\t55.01&\t336.49 &19.2 \\\\\n$c\\,\\bar{c}\\,\\gamma\\,\\gamma$ & 66600.78\t&0.00 &\t54.66&0.11 \\\\\n$j\\,j\\,\\gamma\\,\\gamma$ &14182.56\t&2.52 &\t25.20&2.38 \\\\\n$b\\,\\bar{b}\\,j\\,\\gamma$ & 1228956.91\t&\t38.53&\t1176.93&3.74 \\\\\n$c\\,\\bar{c}\\,j\\,\\gamma$ &208285.83\t&0.00 &\t187.92&0.26 \\\\\n$b\\,\\bar{b}\\,j\\,j$ & 1622778.23\t& 0.00 &\t2231.08&0.19 \\\\\n$Z(b\\,\\bar{b})\\,\\gamma\\gamma$ & 4540.20\t&4.72 &\t45.33&12.7 \\\\ \\hline\n$t\\,\\bar{t}$~($\\geq$ 1 leptons)& 78490.03 &0.00&56.93&~$11.5+3.68$ \\\\\n$t\\,\\bar{t}\\,\\gamma$~($\\geq$ 1 leptons)& 74885.54 &  9.09&105.16&~$8.69+2.07$ \\\\ \\hline\nTotal Background&3500211.00 &\t553.09&\t6142.83& \\\\ \\hline\\hline\nSignificance $Z$, $\\lambda_{3H} = 1$ && \\textbf{20.50} & \\textbf{8.44}& \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\nIn Table~\\ref{tab:l3h1_opt}, we present the\nexpected number of signal and background events at the 100 TeV hadron \ncollider assuming 3 ab$^{-1}$ using BDT$_{\\rm SM}$ with the BDT response cut of $0.216$.\nWe show the four representative values of $\\lambda_{3H}$ for signal\nand the backgrounds are separated into three categories.\nFor comparisons, we also show the results obtained using the cut-and-count\nanalysis~\\cite{Chang:2018uwu}.\nIn the last column, we additionally present the effective luminosity (Eff. Lumi.) for each\nof signal and background samples.\nIn the $t\\bar t$ and $t\\bar t\\gamma$ backgrounds,\nthe first (second) number is the effective luminosity\nwhen the two top quarks decay fully (semi-) leptonically.\nWe find about $550$ signal and $550$\nbackground events for $\\lambda_{3H}=1$. Comparing to the results using\nthe cut-and-count analysis~\\cite{Chang:2018uwu},\nthe number of signal events\ndecreases by only $19\\%$ while the number of backgrounds by almost $90\\%$, resulting in \nan increase in significance from $8.44$ to $20.50$.\nNote that the composition of backgrounds changes drastically by the use of BDT.\nIn the cut-and-count analysis, the non-resonant background is about two times \nlarger than the single-Higgs associated background. \nWhile, in the BDT analysis,\nthe single-Higgs associated background is more than four times  \nlarger than the non-resonant one\nand $t\\bar t$ associated background becomes negligible.\nNote that we generate relatively smaller number of events for\nthe $c\\bar c\\gamma\\gamma$, $c\\bar cj\\gamma$, and\n$b\\bar bjj$ backgrounds since we observe that they quickly decrease when\nthe BDT response cut approaches to the point $Z_{\\rm max}$ of $0.216$\n\\footnote{In fact,\nthere are some differences in kinematic distributions among \nthe non-resonant backgrounds. For example, \nthe $c\\bar c\\gamma\\gamma$ background is more populated in the\nregion of $\\Delta R_{bb}>3$ compared to the $b\\bar b\\gamma\\gamma$ one.}.\nSpecifically, the $b\\bar b j j$ background vanishes\nfor the BDT response cut larger than $0.2$.\nOtherwise, we generate enough number of events considering the assumed luminosity of\n3 ab$^{-1}$.\n\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=3.12in,height=2.4in]{uncertainty_1.pdf}\n\\includegraphics[width=3.12in,height=2.4in]{LogLHratio_BDTL1_100.pdf}\n\\caption{\n(Left) The total number $N =s+b$ of signal ($s$) and background ($b$) events\nversus $\\lambda_{3H}$ with 3 $\\mathrm{ab}^{-1}$.\nThe horizontal solid line denotes the total number of events\nobtained using the SM value of $\\lambda_{3H}=1$\nand the dashed lines for the statistical $1$-$\\sigma$ error.\n(Right) \nThe relative log likelihood distribution \nfor the nominal value of $\\lambda_{3H}=1$\nat the 100 TeV hadron collider assuming 3 ab$^{-1}$ and using\nBDT$_{\\rm SM}$ with the BDT response cut of $0.216$.\nThe distribution has been obtained by a likelihood\nfitting of $M_{\\gamma\\gamma b b}$ distribution for each value of \n$\\lambda_{3H}$.\nThe black solid line shows the result of a polynomial fitting\nand the horizontal \nsolid (red) line at $-\\ln(L_{\\lambda_{3H}}\/L_{\\lambda_{3H}=1})=32$ indicates the value\ncorresponding to the $8\\sigma$ level.\n}\n\\label{fig:mhh_BDTL1}\n\\end{figure}\nFirst, we try to determine the THSC considering the total number of events.\nAs shown in the left panel of Fig.~\\ref{fig:mhh_BDTL1},\nwe find that the THSC can be measured with about $11\\%$ accuracy\nat the SM value which is about two times better than the result\nbased on the conventional cut-and-count analysis~\\cite{Chang:2018uwu}.\nHowever, there is a second solution around $\\lambda_{3H}=6.5$.\nTo lift up the two-fold ambiguity,\nwe implement a likelihood fitting of the signal-plus-background\n$M_{\\gamma\\gamma bb}$ distribution\nand find the second solution is ruled out \n by more than $8\\sigma$ confidence,\nsee the right panel of Fig.~\\ref{fig:mhh_BDTL1}.\n\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=3.12in,height=2.4in]{LogLHratio_BDTL1_100_ZU.pdf}\n\\includegraphics[width=3.12in,height=2.4in]{mhh_BDTL1_100_sup.pdf}\n\\caption{\n(Left)\nThe relative log likelihood distribution \nfor the nominal value of $\\lambda_{3H}=1$\nat the 100 TeV hadron collider with 3 ab$^{-1}$.\nThe black circles are the values obtained by a likelihood\nfitting of $M_{\\gamma\\gamma b b}$ distributions \nusing BDT$_{\\rm SM}$ with the BDT response cut of $0.216$.\nThe black solid line shows the result of a polynomial fitting\nand the thin dashed line at $0.5\\,(2.0)$ indicates the value corresponding to\na $1\\sigma\\,(2\\sigma)$ CI.\nThe shaded region shows the $1\\sigma$ CI expected at the ILC\nat 1 TeV with 8 ab$^{-1}$.\n(Right)\nThe SM $M_{\\gamma\\gamma bb}$ distribution (solid line with dots with $1\\sigma$ error bars)\nand those for $\\lambda_{3H}=0.92$ and $1.08$ (dashed lines).\n}\n\\label{fig:result}\n\\end{figure}\n\n  To improve the sensitivity of the THSC around the SM value \n  and to tame the statistical fluctuation due to the limited\n  size of the MC samples, we repeat the likelihood fitting of\n  $M_{\\gamma\\gamma bb}$ distribution by optimizing the bin size\n  between $1\/20\\,$GeV and $1\/60\\,$GeV.\nFinally, we find that\nthe THSC can be determined with a precision of $7.5\\%$ at 68\\% CL\nas shown in the left panel of Fig.~\\ref{fig:result}. \nIn the right panel of Fig.~\\ref{fig:result}, $M_{\\gamma\\gamma bb}$ distributions\nare shown for\nthe THSC at the SM value and for the two values\ndeviated by $1\\sigma$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=4.12in,height=3.17in]{LogLHratio_BDTL1_100_sb.pdf}\n\\caption{\nThe same as in the left panel of Fig.~\\ref{fig:result} while taking\n$\\sigma_b\/b = 0$ (solid), $0.02$ (red dotted)\n$0.05$ (black dotted), $0.1$ (dashed), and $0.2$ (dash-dotted).\n}\n\\label{fig:sys}\n\\end{figure}\n\n\nBy now, we have considered only the statistical uncertainties which\nmay eventually dominate the total uncertainties.\nBefore concluding, we would like to discuss the effects of \nsystematic uncertainties which could be important\nat the early stage of 100 TeV hadron collider.\nThe systematic uncertainties might be taken into account by \nconsidering the variance of background $\\sigma_b^2$~\\cite{Cowan:2010js}.\nIn this case, the error due to systematic uncertainties is proportional to the number\nof background or $\\sigma_b \\propto b$.\nWe find that the THSC precision of $7.5$\\%$-18$\\%  at 68\\% CL\nwhile varying $\\sigma_b\/b$ between $0$\nand $0.2$, see Fig.~\\ref{fig:sys}\n\\footnote{Incidentally, by measuring only the total number of events,\nthe precision becomes worse to $11$\\%$-30$\\%.}.\n\n\nFinally, before we end this section, in Table~\\ref{ranking},\nwe show the relative importance of the variables\nthat we employed in this BDT analysis.\nWe observe that\nthe two most important variables are $\\Delta R_{bb}$ and $\\Delta R_{\\gamma\\gamma}$,\nwhich is consistent with our previous cut-and-count analysis\n\\cite{Chang:2018uwu}.\n\n\\begin{table}[h]\n  \\caption{\\small \\label{ranking}\n    The ranking of the variables that we employed in this BDT analysis\n    in the descending order of importance.}\n\\label{tab:ranking}\n\\vspace{0.1cm}\n  \\begin{ruledtabular}\n    \\begin{tabular}{cccccccc}\n    $\\Delta R_{bb}$ & $\\Delta R_{\\gamma\\gamma}$ & $M_{\\gamma\\gamma}$ &\n    $\\Delta R_{\\gamma b}$ & $P_T^{\\gamma\\gamma}$ & $M_{\\gamma\\gamma b b}$\n      & $P_T^{bb}$ & $M_{bb}$ \\\\\n      \\hline\n    0.163 & 0.152 & 0.150 & 0.133 & 0.110 & 0.102 & 0.096 & 0.095\n    \\end{tabular}\n    \\end{ruledtabular}\n    \\end{table}\n\n\n\n\\section{Conclusions:}\nHiggs-pair production is one of the most useful avenue to probe the EWSB\nsector. We have studied in great details, with the help of\nmachine learning, the sensitivity of measuring the THSC $\\lambda_{3H}$\nthat one can expect at the 100 TeV $pp$ collider with an integrated\nluminosity 3 ab$^{-1}$. With TMVA one can improve\nthe signal-to-background ratio for $\\lambda_{3H}=1$\nto $1:1$ compared with the ratio $1:10$ obtained in the conventional\ncut-and-count approach.\nFurthermore, the significance of such a signal jumps to $20$.\n\nOther than determining the THSC by measuring the total number of events,\none can also improve the sensitivity and lift the two-fold degeneracy\nby implementing a likelihood fitting of the signal-plus-background\n$M_{\\gamma\\gamma b b}$ distribution with optimized bin sizes. The THSC\ncan be determined with a precision of 7.5\\% at 68\\% CL\nwith 3 ab$^{-1}$, which is indeed\nbetter than the ILC running at 1 TeV with 8 ab$^{-1}$.\nExtrapolating our result\nconservatively, we expect that one can achieve the precision better than\n$\\sim 2$\\% with 30 ab$^{-1}$.\n\n\\bigskip\n\n\\noindent\n{\\it Note added}: After the completion of our work, we learned \na similar analysis performed considering various systematic \nuncertainties rigorously~\\cite{Mangano:2020sao}.\nThey found the combined\nprecision of $2.9$\\%$-5.5$\\% with 30 ab$^{-1}$ at 68\\% CL\nwhich is in a good quantitative agreement with our results.\n\n\\newpage\n\\section*{Appendix}\n\\defA.\\arabic{equation}{\\Alph{section}.\\arabic{equation}}\n\\begin{appendix}\n\n\\setcounter{equation}{0}\n\\section{More on the $c\\bar c\\gamma\\gamma$, $c\\bar cj\\gamma$, and\n$b\\bar bjj$ backgrounds}\nFor this work,\nwe generate relatively smaller number of events for\nthe $c\\bar c\\gamma\\gamma$, $c\\bar cj\\gamma$, and\n$b\\bar bjj$ backgrounds which may lead to underestimation\nof the relevant backgrounds.\n\nThe $c\\bar c\\gamma\\gamma$ and $c\\bar cj\\gamma$ backgrounds might be\nnegligible since, taking account of the fake rates $P_{c\\to b}$ and\n$P_{j\\to\\gamma}$, the cross sections are smaller than\nthat of the $b\\bar b\\gamma\\gamma$ background by about an order of magnitude.\nOn the other hand, our estimation of the $b\\bar bjj$ background\ncould be unreliable due to the limited size of the MC sample.\nHere we try to estimate the background yield based on the current sample.\n\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=3.18in]{AppFigs\/Exp_Y_BBJJ.pdf}\n\\includegraphics[width=3.0in]{AppFigs\/LogY_BBJJ.pdf}\n\\includegraphics[width=3.0in]{AppFigs\/ExpY_All_BBJJ.pdf}\n\\caption{Behavior of the $b\\bar b jj$ background yield $Y_{b\\bar bjj}$ versus\nthe BDT response cut.\nIn the upper panels, the MC data points are denoted by bullets and\nthe solid line in the upper-right panel shows the result of the linear\nfitting to $\\log Y_{b\\bar bjj}$.\nIn the lower panel, we show the result of extrapolation of the solid line\nto the region with BDT Cut $>0.19$, where no data points exist, together with\n$1$- and $2$-$\\sigma$ errors.\nThe vertical lines locate the BDT response cut of $0.216$\ntaken for BDT$_{\\rm SM}$.\n}\n\\label{fig:bbjj}\n\\end{figure}\nPrecisely, we study the behavior of the $b\\bar b jj$ background yield $Y_{b\\bar bjj}$ versus\nthe BDT response cut.\nFirst we observe that, based on the current $b\\bar b jj$ MC sample,\nour estimation of the background results in $0$ when BDT Cut $>0.19$, see\nthe upper-left panel of Fig.~\\ref{fig:bbjj}.\nTo extrapolate to the region with BDT Cut $>0.19$, we implement\na linear fitting to $\\log Y_{b\\bar bjj}$, see the solid line\nin the upper-right panel of Fig.~\\ref{fig:bbjj}.\nAnd we find that\n\\begin{equation}\nY_{b\\bar bjj}=13.7^{+2.2\\,(5.2)}_{-1.9\\,(3.7)}\n\\end{equation}\nat $68(95)$\\% CL as shown in the lower panel of Fig.~\\ref{fig:bbjj}.\nTaking the $1\\sigma$ upper value of $15.9$,\nthe number of total background increases by\nthe amount of about $3$\\% which hardly affect\nour main results significantly.\n\nIncidentally, we note that  the\n$jj\\gamma\\gamma$ background survives though its cross section\nis smaller than that of the $b\\bar b\\gamma\\gamma$ one\nby about {\\it two} orders of magnitude taking account of\nthe fake rate $P_{j\\to b}$. This is because\nits kinematical distributions quite resemble to those of the signal.\nFor example, compared to other non-resonant backgrounds,\nwe find that it is quite populated\nin the region of $\\Delta R_{bb}\\raisebox{-0.13cm}{~\\shortstack{$<$ \\\\[-0.07cm] $\\sim$}}~ 2$ where\nmost signal events are located.\n\n\n\\setcounter{equation}{0}\n\\section{Supplemental materials}\nIn this appendix, we present the\nnormalized distributions of the eight kinematic variables\nfor the SM signal with $\\lambda_{3H}=1$ (black solid) and \nthe six non-resonant backgrounds \nafter applying the event preselection cuts 1-5 in Table~\\ref{tab:event_selection},\nsee Fig.~\\ref{fig:app1}.\nFor $M_{bb\\,,\\gamma\\gamma}$ and $P_T^{bb\\,,\\gamma\\gamma}$, in terms of $P_T$,\nwe choose the least energetic two photons or two $b$ quarks\nwhile the most energetic ones are chosen for $\\Delta R_{bb,\\gamma\\gamma}$ and\n$M_{\\gamma\\gamma bb}$. For $\\Delta R_{\\gamma b}$, on the other hand, we choose the least\nenergetic $b$ and the next-to-the-least energetic photon.\n\n\n\n\\begin{figure}[t!]\n\\vspace{-1.0cm}\n\\centering\n\\includegraphics[width=3.2in]{AppFigs\/sig_nbkg_drbb.png}\n\\includegraphics[width=3.2in]{AppFigs\/sig_nbkg_draa.png}\n\\includegraphics[width=3.2in]{AppFigs\/sig_nbkg_maa.png}\n\\includegraphics[width=3.2in]{AppFigs\/sig_nbkg_drab.png}\n\\includegraphics[width=3.2in]{AppFigs\/sig_nbkg_ptaa.png}\n\\includegraphics[width=3.2in]{AppFigs\/sig_nbkg_maabb.png}\n\\includegraphics[width=3.2in]{AppFigs\/sig_nbkg_ptbb.png}\n\\includegraphics[width=3.2in]{AppFigs\/sig_nbkg_mbb.png}\n\\caption{Normalized distributions of the eight kinematic variables\nfor the SM signal and six non-resonant backgrounds after applying the event preselection\ncuts 1-5 in Table~\\ref{tab:event_selection}.\nPanels are in the descending order of importance \n(see Table~\\ref{tab:ranking}) from upper-left to lower-right.}\n\\label{fig:app1}\n\\end{figure}\n\n\\end{appendix}\n\n\n\\bigskip\n\n\\newpage\n\n\\section*{Acknowledgment}\nThis work was supported by the National Research Foundation of Korea \nGrant No. NRF-2016R1E1A1A01943297 (J.C., J.S.L., J.P.), \nNo. NRF-2018R1D1A1B07051126 (J. P.), \nand by the MoST of Taiwan under Grant No. 107-2112-M-007-029-MY3 (K. C.).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:Introduction}\nThe texture of the landscape and fluvial basins is the product of thousands of\nyears of tectonic movement coupled with erosion and weathering caused by water\nflow and climatic processes.  To gain insight into the time evolution of the\ntopography, a model has to include the essential processes responsible for the\nchanges of the landscape.  In Geology the formation of river deltas and braided\nriver streams has been studied since a long time describing the schematic\nprocesses for the formation of deltaic distributaries and inter-levee- basins\n\\cite{Coleman75, Allen81, Coleman88, Bridge93, Bristow93}.\nExperimental investigation of erosion and deposition has a long tradition in\nGeology \\cite{Jaggar08}. Field studies have been carried out for the Mississippi\nDelta \\cite{Fisk47, Kolb58, Coleman64, Could70}, the Niger Delta\n\\cite{Allen64,Allen65, Allen70}, or for the Brahmaputra Delta \\cite{Coleman69}.\nLaboratory experiments have also been set up in the last decades for\nquantitative measurements \\cite{Czirok93, Ashmore82, Ashmore85, Wright05,\n  Parker05, Parker03}.  For instance, in the eXperimental EarthScape (XES)\nproject the formation of river deltas is studied on laboratory scale and\ndifferent measurements have been carried out \\cite{Kim06,Swendson05,Lague03}.\n\nNevertheless modeling has proved to be very difficult as the system is\nhighly complex and a large range of time scales is involved. To\nsimulate geological time scales the computation power is immense and\nclassical hydrodynamical models cannot be applied.  Typically these\nmodels are based on a continuous \\emph{ansatz} (e.g., shallow water\nequations) which describes the interaction of the physical laws for\nerosion, deposition and water flow \\cite{Giacometti95, Willgoose91,\n  Howard94, Kooi96, Densmore97, Beaumont00}. The resulting set of\npartial differential equations are then solved with boundary and\ninitial conditions using classical finite element or finite volume\nschemes.  Unfortunately none of these continuum models is able to\nsimulate realistic land-forms as the computational effort is much too\nhigh to reproduce the necessary resolution over realistic time scales.\nTherefore in the last years discrete models based on the idea of\ncellular automata have been proposed \\cite{Wolfram02, Murray94,\n  Murray97, Davy00, Coulthard05, Coulthard06}.  These models consider\nwater input on some nodes of the lattice and look for the steepest\npath in the landscape to distribute the flow. The sediment flow is\ndefined as a nonlinear function of the water flow and the erosion and\ndeposition are obtained by the difference of the sediment inflow and\noutflow. This process is iterated to obtain the time evolution.  In\ncontrast to the former models, these models are fast and several\npromising results have been obtained, but as they are only based on\nflow, a well defined water level cannot be obtained with this\n\\emph{ansatz}.\n\nHere we introduce a new kind of model where the water level and the landscape\nare described on a lattice grid coupled by an erosion and sedimentation law.\nThe time evolution of the sediment and water flow is governed by conservation\nequations.  The paper is organized as follows. After an overview on the\ndifferent types of deltas and their classification the model is introduced and\ndiscussed in details.  The analysis of the model results and a comparison with\nreal landforms are provided.  According to different parameter combinations\ndifferent delta types can be reproduced and interesting phenomena in the time\nevolution of a delta such as the switching of the delta lobe can be observed.\nFinally the scaling structure of the delta pattern is analyzed and compared with\nthat obtained from satellite images.\n\n\\section{Classification}\\label{sec:Classification}\nThe word ``delta'' comes from the Greek capital letter $\\Delta$ and\ncan be defined as a coastal sedimentary deposit with both subaerial\nand subaqueous parts. It is formed by riverborne sediment which is deposited\nat the edge of a standing water. This is in most cases an ocean but\ncan also be a lake.  The morphology and sedimentary sequences of a\ndelta depend on the discharge regime and on the sediment load of the\nriver as well as on the relative magnitudes of tides, waves and\ncurrents \\cite{Coleman76}.  Also the sediment grain size and the water\ndepth at the depositional site are important for the shape of the\ndeltaic deposition patterns \\cite{Coleman75, Coleman76,\n  Bhattacharya92, Orton93}.  This complex interaction of different\nprocesses and conditions results in a large variety of different\npatterns according to the local situations.  Wright and Coleman\n\\cite{Coleman75, Coleman76, Wright73, Coleman73} described\ndepositional facies in deltaic sediments and concluded that they\nresult from a large variety of interacting dynamic processes (climate,\nhydrologic characteristics, wave energy, tidal action, etc.)  which\nmodify and disperse the sediment transported by the river.  By\ncomparing sixteen deltas they found that the\nMississippi Delta is dominated by the sediment supply of the river\nwhile the Senegal Delta or the S\\~{a}o Francisco River Delta are mainly\ndominated by the reworking wave activities. High tides and strong\ntidal currents are the dominant forces at the Fly River Delta.\n\nGalloway \\cite{Galloway75} introduced a classification scheme where\nthree main types of deltas are distinguished according to the dominant\nforces on the formation process: \\emph{river-}, \\emph{wave-} and\n\\emph{tide}-dominated deltas.  This simple classification scheme was later\nextended \\cite{Bhattacharya92, Orton93, Wright85} including also grain\nsize and other effects.\n\nAt the river-dominated end of the spectrum, deltas are indented and have more\ndistributaries with marshes, bays, or tidal flats in the interdistributary\nregions.  They occur when the stream of the river and the resulting sediment\ntransport is strong and other effects such as reworking by waves or by tides are\nminor \\cite{Coleman76,Wright73}. These deltas tend to form big delta lobes into\nthe sea which may have little more than the distributary channel and its levee\nis exposed above the sea level. Due to their similarity with a bird's foot, they\nare often referred in the literature as ``bird foot delta'' like in the case of\nthe Mississippi River Delta \\cite{Coleman76}.  When more of the flood plain\nbetween the individual distributary channels is exposed above the sea level, the\ndelta displays lobate shape.  Wave-dominated delta shorelines are more regular,\nassuming the form of gentle, arcuate protrusions, and beach ridges are more\ncommon (e.g., like for the Nile Delta or Niger Delta \\cite{Allen65, Oomkens74}).\nHere the breaking waves cause an immediate mixing of fresh and salt water. Thus\nthe stream immediately loses its energy and deposits all its load along the\ncost.  Tide-dominated deltas occur in locations of large tidal ranges or high\ntidal current speeds. Such a delta often looks like a estuarine bay filled with\nmany stretched islands parallel to the main tidal flow and perpendicular to the\nshore line (like e.g., the Brahmaputra Delta).  Using the classification of\nGalloway \\cite{Galloway75} the different delta types of deltas can be arranged\nin a triangle where the extremes are put in the edges (see\nFig.~\\ref{fig:Classification}).\n\n\\section{The Model}\\label{sec:Model}\nThe model discretizes the landscape on an rectangular grid where the surface\nelevation $H_i$ and the water level $V_i$ are assigned to the nodes. Both $H_i$\nand $V_i$ are measured from a common base point, which is defined by the sea\nlevel.  On the bonds between two neighboring nodes $i$ and $j$, a hydraulic\nconductivity for the water flow from node $i$ to node $j$ is defined as\n\\begin{equation}\n\\label{eq:Sigma}\n  \\sigma_{ij}=c_\\sigma\\left\\{\n\\begin{array}{cl}\n\\displaystyle{V_i+V_j\\over2}-{H_i+H_j\\over 2}&\\mbox{if $>0$}\\\\\n0 &\\mbox{else}.\n\\end{array}\n\\right.\n\\end{equation}\nAs only surface water flow is considered, $\\sigma_{ij}$ is set larger\nthan zero only if the water level of the source node is larger than\nthe topography, which means that water can only flow out of a node\nwhere the water level is above the surface.  The relation between the\nflux $I_{ij}$ along a bond and the water level is given by\n\\begin{equation}\n\\label{eq:Current}\n  I_{ij}=\\sigma_{ij}(V_i-V_j).\n\\end{equation}\nFurthermore water is routed downhill using the continuity equation for\neach node\n\\begin{equation}\n  \\label{eq:Continuity}\n  \\frac{V_i-V'_i}{\\Delta t}=\\sum_{N.N.} I_{ij},\n\\end{equation}\nwhere the sum runs over all currents that enter or leave node $i$ and $V_i'$ is\nthe new water level.  The boundaries of the system are chosen as follows: On the\nsea side the water level on the boundary is set equally to zero and water just\ncan flow out of the system domain.  On the land the water is retained in the\nsystem by high walls or choosing the computational domain for the terrain such\nthat the flow never reaches the boundary.  Water is injected into the system by\ndefining an input current $I_0$ at the entrance node.\n\nThe landscape is initialized with a given ground water table.  Runoff\nis produced when the water level exceeds the surface.  The sediment\ntransport is coupled to the water flow by the rule, that all sediment\nthat enters a node has to be distributed to the outflows according to\nthe strength of the corresponding water outflow. Thus the sediment\noutflow currents for node $i$ are determined via\n\\begin{equation}\n\\label{eq:Sediment_flow}\n  J^{out}_{ij}={\\sum_k J^{in}_{ik}\\over \\sum_k |I^{out}_{ik}|}I^{out}_{ij},\n\\end{equation}\nwhere the upper sum runs over all inflowing sediment and the lower one\nover the water outflow currents.  A sediment input current $s_0$ is\ndefined in the initial bond.\n\nThe sedimentation and erosion process is modeled by a phenomenological\nrelation which is based on the flow strength $I_{ij}$ and the local\npressure gradient imposed by the difference in the water levels in the\ntwo nodes $V_i$ and $V_j$. The sedimentation\/erosion rate $dS_{ij}$\nis defined through\n\\begin{equation}\n  \\label{eq:Erosion}\n  dS_{ij}=c_1(I^\\star-|I_{ij}|)+c_2(V^\\star-|V_i-V_j|),\n\\end{equation}\nwhere the parameters $I^\\star$ and $V^\\star$ are erosion thresholds and the\ncoefficients $c_1$ resp. $c_2$ determine the strength of the corresponding\nprocess.  The first term $c_1(I^\\star-|I_{ij}|)$ describes the dependency on the\nflow strength $I_{ij}$ \\cite{Wipple99} and is widely used in geomorphology,\nwhile the second term $c_2(V^\\star-|V_i-V_j|)$ relates sedimentation and erosion\nto the flow velocity, which in the model can be described by\n$I_{ij}\/\\sigma_{ij}\\sim|V_i-V_j|$.  The two terms of \\Ref{eq:Erosion} are not\nlinearly dependant on each other as one may think first by looking at Eq.\n\\Ref{eq:Current}.  In fact due to Eq.\\Ref{eq:Sigma} there is a nonlinear\nrelation between $V$ and $I$ which leads to different thresholds in the pressure\ngradient and the current.\n\n\nThe sedimentation\nrate $dS_{ij}$ is limited by the sediment supply through $J_{ij}$,\nthus in the case $dS_{ij}>J_{ij}$ the whole sediment is deposited on\nthe ground and $J_{ij}$ is set to zero. In the other cases $J_{ij}$ is\nreduced by the sedimentation rate or increased if we have erosion. The\nerosion process is also supply limited which means that the erosion\nrate is not allowed to exceed a certain threshold $T$; so $\\mbox{if }\ndS_{ij}<T, \\mbox{ then } dS'_{ij}=T$. Note that in the case of erosion\n$dS_{ij}$ is negative.  Due to erosion or deposition, the landscape is\nmodified according to\n\\begin{eqnarray}\n  \\label{eq:Landscape_Modification1}\n   H'_i&=&H_i+\\frac{\\Delta t}{2}dS_{ij}\\\\\n    \\label{eq:Landscape_Modification2}\n   H'_j&=&H_j+\\frac{\\Delta t}{2}dS_{ij},\n\\end{eqnarray}\nwhere the sediment deposits equally on both ends of the bond.  The new\ntopography is marked with $H'_i$. The same formulae (Eqs.\n\\Ref{eq:Landscape_Modification1}, \\Ref{eq:Landscape_Modification2}) also hold in\nthe case of erosion when $dS_{ij}$ is negative. \n\nIterating Eqns. \\Ref{eq:Sigma} to \\Ref{eq:Landscape_Modification2} determines\nthe time evolution of the system. Finally in a real system subaquateous water\ncurrents lead to a smoothening of the bottom which is modeled by the following\nexpression\n\\begin{equation}\n  \\label{eq:smooth}\n  H'_i= (1-\\epsilon)H_i+\\frac{\\epsilon}{4}\\sum_{N.N.}H_j,\n\\end{equation}\nwhere $\\epsilon$ is a smoothening constant determining the strength of\nthe smoothening process. The sum runs over all nearest neighbors of\nnode $i$.\n\n\\section{Simulation}\\label{sec:Simulation}\n\nThe simulation is initialized with a valley on a rectangular $N\\times N$ lattice\nwith equal spacing grid as shown in Fig.~\\ref{fig:Initialization}. The valley\nruns downhill with slope $S$ along the diagonal of the lattice and the\nhillslopes of the valley increase from the bottom of the valley sidewards\naccording to a power law with exponent $\\alpha$.  In the simulation shown in\nFig.~\\ref{fig:Birdfoot_simu1} the value of $\\alpha$ was chosen to be 2.0.  Under\nthe sea the landscape is flat with a constant slope downhill.  Furthermore we\nassume the initial landscape to have a disordered topography by assigning\nuniformly distributed random numbers to $H_i$.  This variable is then smoothed\nout according to Eq.~\\Ref{eq:smooth}.  The water level $V_i$ of the system is\ninitialized with a given ground water table.  In reality the distance of the\nground water to the surface is minimal on the bottom of the valley and increases\nuphill.  This is obtained in the simulation by choosing the water level $V_i$ in\nan incline plane $\\delta$ below the bottom of the valley.  The slope of the\nplane is the same as the slope of the valley $S$.\nThis also keeps the river close to the bottom of the valley.  As we are only\ninterested in studying the pattern formation at the mouth of the river, the\nbraiding conditions of the upper river only determine the width of the delta\nfront.  On the seaside when $H_i\\leq 0$ the water level is a constant and set to\nzero.  A sketch of the initial landscape is shown in\nFig.~\\ref{fig:Initialization}.\n\nAn initial channel network is created by running the algorithm without\nsedimentation and erosion until the water flow reaches a steady state.  Then the\nsedimentation and erosion procedure is switched on and the pattern formation at\nthe mouth of the river is studied.\n\nAccording to the dominance of the different processes, completely different\ncoastline shapes can be observed. The smoothening procedure Eq.~\\Ref{eq:smooth}\nleads to the formation of an estuary by reworking the coastline at the river\nmouth, while the stream dominant erosion term $c_1(I^\\star-|I_{ij}|)$ in\nEq.~\\Ref{eq:Erosion} favors the formation of river-dominated birdfoot shaped\ndelta.  In contrast to this, the second term $c_2(V^\\star-|V_i-V_j|)$ in\nEq.~\\Ref{eq:Erosion}, which depends on the pressure gradient represented by the\nheight difference of the water levels in the nodes $i$ and $j$, produces more\nclassical shaped deltas with several islands and channels. These patterns are\nsimilar to the distributary structure of the Lena or the Mahakam river delta,\nwhich are more sea or wave-dominated.  This difference can be explained by the\nfact that the first term sediments along the main current stream, whereas the\nsecond term distributes the sediment more equally to the neighboring nodes.\n\nFigures \\ref{fig:Birdfoot_simu1}(a-c) show some snapshots of the time evolution\nof the simulation of a birdfoot delta ($c_2=0$). A map of the Mississippi River\nis given in Fig.~\\ref{fig:Mississippi} for comparison.  In both cases one can\nsee how the main channel penetrated into the ocean depositing sediment mainly on\nits levee sides.  When the strength of the main channel decreases, side channels\nstart to appear breaking through the sidebars as can be seen in the snapshots of\nFigs.~\\ref{fig:Birdfoot_simu1}(b) and ~\\ref{fig:Birdfoot_simu1}(c). At the\nbeginning of the delta formation process the sediment transport is equally\ndistributed among the different channels and leads to a broader growth of the\ndelta front along the coast.  With time, the side channels are gradually\nabandoned and the sediment is primarily routed through the main channel, thus\nthis dominant channel is growing faster than the others forming the typical\nbirdfoot shaped deposits.\n\n\n\nFigure \\ref{fig:Wave_simu}(a) shows another type of delta where the smoothening\nof the waves reworks the deposits at the river mouth and distributes it along\nthe coast. Here the river could built up only a slight protrusion in the\nimmediate vicinity of the river mouth. The same happens in areas where the wave\ncurrents are dominant and lead to the formation of wave-dominated deltas like\nthe S\\~{a}o Francisco River in Brazil or the Nile Delta. A map of the S\\~{a}o\nFrancisco River Delta is also given in Fig.~\\ref{fig:Wave_simu}(b) for\ncomparison.  Here the coast line has been straightened by the wave activities\nand consists almost completely of beach ridges which have the typical triangular\nshape inland.  This flattened deposit can also be found in our simulation\nresults.  As there is no evaporation included in the simulation, small ponds and\nabandoned channels remain in the sedimented zone instead of disappearing with\ntime.\n\nFinally, if the term $c_2(V^\\star-|V_i-V_j|)$ dominates the\nsedimentation\/erosion process a half moon shaped delta with many small islands\nand channels appear.  This delta type shows more activity in the channel network\nthan the others.  The channels split and come together, and when the main\nchannel blocks its way due to sedimentation, the whole delta lobe switches to\nanother place.  This phenomenon is called delta switching.  During the\nsimulation, the switching of the delta occurred several times.\n\nThe best studied delta in the world is that of the Mississippi river where the\nswitching of the delta lobes was studied in detail \\cite{Kolb58}. The switching\nof the Mississippi Delta during the last 4000 years is well documented\n\\cite{Fisk47,Kolb58, Fisk52}.\nThe rich dynamics due to the switching phenomenon that is observed in the\nMississippi can be also identified in our simulations.  In the literature\n\\cite{Coleman76} three types of switching mechanisms are distinguished. The\nfirst type referred as switching type I, consists of a lobe switching in which\nthe delta propagates in a series of distributary channels. After a certain time,\nthe stream abandons the entire system close to the head of the delta and forms a\nnew lobe in an adjacent region.  Very often this lobe occupies an indentation in\nthe coastline between previous existing lobes so that with time the sediment\nlayers overlap each other.  One can find this type of delta switching in areas\nwhere the offshore slope is extremely low and the tidal and wave forces are too\nsmall for reworking the lobe \\cite{Coleman76, Wright73, Galloway75}.  In many\ncases the delta lobes merge with each other forming major sheet-type sand banks.\nThis phenomena can be nicely observed when comparing the two images of the\nsimulation in Figs.~\\ref{fig:SwitchingTypeI}(a) and \\ref{fig:SwitchingTypeI}(b).\nIn the Mississippi Delta this also happened several times in the past and the\ndifferent lobes have today different names.  A type I shift of the Mississippi\ndelta occurred for example, 4600 years B.C. between the Sal{\\'e}-Cypremort and\nthe Cocodrie (4600-3500) Lobe or when the St.  Bernard Lobe switched to which\ntoday is called Lafourche at about 1000 BC.\n\nAt about 3500 BC the Mississippi river switched far upstream from the Cocodrie\nto the Teche stream tailing a completely new course for the river itself and its\ndelta.  This type of switching is referred as type II switching \\cite{Coleman76}\nand can also be found in the simulation. When comparing\nFigs.~\\ref{fig:SwitchingTypeI}(b) and \\ref{fig:SwitchingTypeI}(c) one can see a\nmajor shift of the channel far upstream in the deltaic plain so the river takes\na completely different course and forms a new delta beside.\n\nThe type III of delta switching is referred in the literature as alternate\nchannel extension \\cite{Coleman76}. In this case not the complete channel but\nthe dominance of sediment flux in one or more distributaries is changing with\ntime.  This can be described as follows: two or more major channels split into\nseveral distributaries nearly at the same point at the head of the delta.\nCommonly one of the distributaries is dominant, so it will carry most of the\nsediment and water discharge at any time.  As a result, this active channel will\nrapidly propagate seaward, while the other channel will shrivel with time.  At\nsome point, the slope of the main active channel will decrease and the discharge\nwill seek one of the shorter distributaries. With the increased sediment flux\ndownstream, the new channel will rapidly propagate into the sea.  This switching\nprocess will repeat several times forming a deltaic plain characterized by a\nseries of multiple beach ridges.  This type of switching can be best observed in\nthe simulation of the birdfoot delta in Fig.~\\ref{fig:Birdfoot_simu1}(a-c),\nwhere the main path of the sediment flow is marked by the red arrows. One can\nsee how side channels emerge and are abandoned after a certain time.\nNevertheless, a major switching of the main channel could not be observed in the\nsimulations.  The average time between two lobe switchings was found to be\naround 1000 years for the Mississippi river \\cite{ Kolb58, Coleman76, Fisk52}.\n\nAt this point, we show that the river delta patterns generated from our\nsimulations display geometric features that are statistically similar to real\nriver delta structures.  More precisely, we analyze the self similar behavior of\nthe real and simulated river deltas using the box counting algorithm\n\\cite{Feder89}.  The box counting dimension is a quite common measure in\ngeomorphological pattern analysis and has been used by many authors to\ncharacterize river basin patterns and coastlines \n\\cite{Maritan96, Rodriguez-Iturbe98, Sapoval04}.\n\n\nFor the real satellite picture as well as for\nthe simulated river delta, we show in Figs.~\\ref{fig:Fractal} that the variation\nwith the cell size $s$ of the number of cells $N$ covering the land follows\ntypical power laws over more than 3 decades\n\\begin{equation}\n  \\label{eq:fractal}\n  N\\sim s^{-D},\n\\end{equation}\nwhere the exponent $D$ is the fractal dimension.  Moreover, the least square fit\nof this scaling function to the data gives exponents which are strikingly close\nto each other, namely $D=1.81\\pm 0.01$ for the real Lena river delta and\n$D=1.85\\pm 0.1$ for the simulation.\n\n\n\\section{Conclusion}\\label{sec:Conclusion}\nA new model for simulating the formation process of river deltas has been\npresented. It is based on simple conservation laws for water and sediment on a\nlattice grid, coupled by a phenomenological sedimentation\/erosion law.  Several\ninteresting features of river deltas like the different delta switching\nprocesses could be found with the model and compared with real landforms.\n\nDifferent delta shapes in the classification scheme of Galloway\n\\cite{Galloway75} could be reproduced by varying the model parameters and\ninitial conditions. The pattern structure of the simulation has been analyzed\nand good agreement with real deltas have been found.  Furthermore the delta\nshifting phenomena could be observed in the simulation and different types of\ndelta shifting could be distinguished.\n\n\\begin{acknowledgments}\n  This work was funded by the Swiss National Fond and the Max Planck Prize as\n  well as by the Landesstiftung BW, CNPq, CAPES and FUNCAP.\n\\end{acknowledgments}\n\\bibliographystyle{pnas}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nMRI is a widely used non-invasive technique for studying and characterizing diseases of the optic pathway such as optic neuritis, multiple sclerosis, and optic pathway glioma (OPG) \\cite{chan2007optic}. OPGs are low grade astrocytomas inherent to the AVP (i.e., optic nerve, chiasm and tracts). OPGs occur in $20\\%$ of children with neurofibromatosis type 1 (NF1), a very common genetic disorder {that carries increased risk of tumors in the nervous system}. The disease course is variable, as these tumors may demonstrate several distinct periods of growth, stability or regression. {{Currently, no quantitative imaging criteria exist to define OPGs secondary to NF1}}. Non-invasive computer-aided quantification of these changes can not only eliminate excessive physician's effort to segment these regions but also increases the precision of volume measures. However, automatic segmentation of cranial nerve pathways including AVP from MRI is challenging due to their thin-long shape and varying appearances. A few non-invasive automated methods to segment AVP from radiological images have been reported in the literature previously with modest success. Bekes et al. \\cite{bekes2008735} proposed a geometrical model based approach; however, their approach's reproducibility is found to be less than $50\\%$. Noble et al. \\cite{noble2011877} presented a hybrid approach using a deformable model with level set method to segment the optic nerves and the chiasm; {{however, the method was tested only on healthy cases}. Recently, Yang et al. \\cite{Xue2014109} developed a partitioned approach to healthy AVP segmentation by dividing the pathway into various shape homogenous segments and modeling each segment independently. The local appearance information in their approach was encoded using the normalized derivatives, three class fuzzy c-means, and spherical flux}. {{The approach was the first attempt to accommodate local shape and appearance variation for healthy AVP segmentation}}; the method, although promising, did not provide any objective criteria on the optimal number of partitions. Moreover, the approach did not accommodate local appearance characteristics along the nerve boundary that are particularly important in pathological cases. \n\nDepending on severity, pathological AVPs can have a drastically different local shape and appearance characteristics than healthy ones, thus failing the shape model based segmentation methods in cranial nerve pathways. To illustrate, Fig \\ref{fig:introduction}(a) demonstrates a healthy optic nerve along with a contralateral optic nerve having OPG. Fig \\ref{fig:introduction} (b)-(c) show the renderings of cases with OPG in optic nerve region. In this paper, we propose, \\emph{PAScAL}, an optimally partitioned statistical shape model with sparse appearance learning for the segmentation of AVPs for both healthy and pathological cases. The challenge of segmenting larger anatomical structures with pathologies have been addressed numerously in the literature \\cite{mansoor2014generic}. However, development of similar approaches for {{smaller vascular structures, such as the AVP,}} have traditionally been ignored. {{By illustrating the robustness of PAScAL to segment AVP with OPG, we demonstrate the applicability of the proposed method in segmenting other anatomical structures of similar characteristics.}}\n   \n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{introduction06.eps}\n\\caption{\\small{(a) MRI scan with a healthy (\\emph{left}) and a gliomic (\\emph{right}) optic nerve. The maximum diameter of OPG nerve is $9.54$ mm and $1.15$ mm for the healthy nerve of the same patient. (b)-(c) renderings of typical OPG cases in the optic nerve. {{(b) shows OPG in the distal region of left optic nerve, (c) shows one in the proximal region. (d) Shape consistent partitioning of a healthy AVP produced by PAScAL.}}}}\n\\label{fig:introduction}\n\\vspace{-.2in}\n\\end{figure}\n\n\\section{Methods}\nWe propose a hierarchical joint partitioned shape model and sparse appearance learning to automatically segment the AVP from MRI scans of the head. During \\textbf{the training stage} automatically selected landmarks from healthy cases are first clustered into various shape-consistent overlapping partitions thus creating individual simplistic shape and appearance models for each partition. The individually learned models are used to produce the initial segmentation of AVP using the partitioned active shape model (ASM\\textsubscript{p}) described in Section \\ref{sec:asmp}. In \\textbf{the testing stage}, the learned ASM\\textsubscript{p} is iteratively fitted to new data using the appearance guided model. A refinement stage follows to accommodate local appearance features particularly important in cases with pathologies (e.g. OPG): a sparse local appearance dictionary is learned \\emph{on-the-fly} from the testing image for each partition using the initial segmentation as training data acquired from the test image in real-time. Through these steps, PAScAL is adapting to each testing set to compensate for the difficulties with off-line training for pathological cases due to the unpredictable location, shape, and appearance of OPG. PAScAL is summarized in Fig. \\ref{fig:flowdiagram}. Details of the proposed method are provided in the subsequent sections.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{blockdiagram05.eps}\n\\caption{\\small{Flow diagram of the PAScAL approach to optic nerve segmentation.}}\n\\label{fig:flowdiagram}\n\\end{figure}\n\n\\subsection{Shape consistent agglomerative hierarchical landmark partitioning}\nIn the beginning, the annotated landmarks are grouped by using a modification of the agglomerative hierarchical clustering method proposed by Cerrolaza et al. \\cite{cerrolaza2015automatic}, minimizing the following objective function:\n\\begin{equation}\nJ\\left( \\Omega  \\right) = \\underbrace {\\alpha \\int\\limits_\\Omega  {{{\\left( {\\frac{{\\left| {{\\mathbf{V}_\\Omega } \\times {\\mathbf{V}_l}} \\right|}}{{\\left| {{\\mathbf{V}_l}} \\right|}}} \\right)}^2}\\frac{{{L_{\\max }}}}{{\\left| {{\\mathbf{V}_l}} \\right|}}dl} }_{{\\text{Colinearity term}}} + \\underbrace {\\left( {1 - \\alpha } \\right)\\left( {1 - \\frac{{\\int\\limits_\\Omega  {dl} }}{{\\int\\limits_\\mathbf{S} {dl}}}} \\right)}_{{\\text{Maximum area constraint}}}\n\\end{equation}\nwhere $\\mathbf{S}$ is the set of all landmarks over the AVP and $\\Omega\\subset\\mathbf{S}$ denotes the local shape to be sub-partitioned into optimal set of clusters. $\\mathbf{V}_\\Omega$ denotes the dominant direction in $\\Omega$. $\\mathbf{V}_l$ is the deformation vector for landmark $l$ obtained through well known point distribution model by Cootes et. al \\cite{cootes2001236} over $\\mathbf{S}$. $\\alpha\\in[0,1]$ is the coefficient that controls the relative weights ($\\alpha$ is set to $0.8$ in our experiments) and ${L_{\\max }} = \\mathop {\\max }\\limits_\\mathbf{S} \\left\\{ {\\left\\| {{V_l}} \\right\\|} \\right\\}$. We define the optimal number of partitions based on shape similarities calculated using a tailored Silhouette coefficient score. Specifically, let $\\Omega_p$ denotes the set of landmarks for the shape partition $p$ containing the landmark $l$ and $\\Omega_{p-l}$ denotes the set of landmarks for the same shape $p$ with landmark excluded then the contribution of the landmark $l$ in partition $p$ is defined as $a_{p,l}=J(\\Omega_{p})-J(\\Omega_{p-l}) \\in \\{0,1\\}$. A large $a_{p,l}$ denotes higher dissimilarity between the landmark $l$ and the shape $\\Omega_p$. The cost of including landmark $l$ to a partition $p$ is similarly defined as $b_{p,l}=J(\\Omega_{p+l})-J(\\Omega_{p})$. Then the optimal number of partitions $p_\\text{opt}$ are found by maximizing: \n$\n\\begin{aligned}\n & \\underset{\\Omega}{\\textit{maximize}}\n & \\frac{1}{|l|}\\sum_{p=1}^{|l|}\\frac{f_p(b_l)-f_p(a_l)}{\\text{max}(f_p(a_l),f_p(b_l))} \\\\\n \\end{aligned}\n$, where $f(.)$ is the logistic sigmoid function, $|l|$ is the total number of landmarks. To ensure that adjacent partitions are connected, an overlapping region is introduced by sharing the boundary landmarks of these partitions. During the shape model fitting, the shape parameters of the overlapping landmarks are calculated using the parameters of the overlapping shapes. {{Fig. \\ref{fig:introduction}(d) demonstrates the proposed agglomerative hierarchical landmark partitioning approach.}}\n\n\\subsection{Landmark weighted partitioned active shape model fitting}\n\\label{sec:asmp}\nOnce the shape partitions are generated, ASM\\textsubscript{p} is performed on the individual shapes in the partitioned hyperspace. In order to adapt to local appearance characteristics, following set of appearance features are used to create overlapping partitioned statistical appearance models for each partition: (i) the intensities of neighboring voxels of each landmark, (ii) the three-class fuzzy c-means filter to robustly delineate both tissues in dark as well as bright foregrounds (as explained before, the AVP passes through neighboring tissues of varying contrasts), and (iii) spherical flux to exploit the \\emph{vessel-like} characteristics. AVP has varying contrast in different regions (i.e, fatty regions has better contrast appearance with optic nerve than gray matter) thus we assigned different levels of confidence for the reliability of landmarks. Specifically, for each landmark in the training set, the covariance $\\Sigma$ of these features is calculated across the training examples under the assumption that the lower the variance of the appearance profile of a landmark, the higher would be our confidence in the landmark. The weight $w_l$ of a landmark $l$ can therefore be calculated as: ${w_l} = \\frac{1}{{\\left( {1 + tr\\left( {{\\Sigma_l}} \\right)} \\right)}}$, where $tr()$ denotes the trace of a matrix. The shape parameters for a partition $p$ can be computed as ${b_p} = {\\left( {\\varphi _p^T{W_p}\\varphi _p^T} \\right)^{ - 1}}\\varphi _p^T{W_p}\\left( {{x_p} - \\overline {{x_p}} } \\right)$, where $\\varphi_p$ is the eigenvector matrix, $x_p$ is the aligned training shape vector, $\\overline {{x_p}}$ is the mean shape vector, and $W_p$ is the diagonal weight matrix of landmarks belonging partition $p$. \n\n\\subsection{\\emph{On-the-fly} sparse appearance learning}\n\\label{sec:fly}\nPathologies can result in changes in shape and appearance of AVP at unpredictable locations (Fig. \\ref{fig:introduction}). {{Statistical shape models have been very successful in segmenting healthy organs; however, they struggle to accommodate cases where the shape of the target structure cannot be predicted through training, such as in the cases of OPG. Feature-based approaches have demonstrated superior performance in segmentation of pathological tissues \\cite{mansoor2014generic}; however, off-line feature-based training of pathological cases mostly fails due to large variations, in both shape and appearance, for pathological cases}}. To address these challenges, we present a novel \\emph{on-the-fly} learning approach by using the initial delineation of the test image obtained in the previous section as training to learn an appearance dictionary in real-time. Specifically, let $R_v(p)$ be a $m\\times m\\times k$ image patch extracted from within the initial partition $p$ centered at voxel $v\\in\\mathbb{R}^3$. Equal number of patches are extracted from each partition. The 2D co-occurrence matrix on every slice of the patch is then calculated from $R_{l_{p,i}}(p)$ and the following gray-level features are extracted: (1) autocorrelation, (2) contrast, (3) cluster shade, (4) dissimilarity, (5) energy, (6) entropy, (7) variance, (8) homogeneity, (9) correlation, (10) cluster prominence, and (11) inverse difference. To reduce the redundancy in the features, we use k-SVD dictionary learning \\cite{aharon20064311}. A dictionary $D^p$ for every partition $p\\in P$ is learned. Specifically, we begin by extracting the centerline of the initial ASM\\textsubscript{p} segmentation using the shortest path graph. Afterwards, we choose the point $c_{p,i}$ on the centerline that is closest to the landmark $l_{p,i}$ in $l^2$-\\textit{norm} sense. Subsequently, co-occurrence features are extracted from the patch $R_{c_{p,i}}(p)$. The likelihood of voxels belonging to the optic nerve is determined by using sparse representation classification (SRC) \\cite{wright2009210}. In SRC framework, the classification problem is formulated as:\n$\\mathop {{\\rm{argmin}}}\\limits_\\beta \\left\\| {f' - {D^p}\\beta} \\right\\|_2^2 + \\lambda {\\left\\| \\beta \\right\\|_1}$, where $f'$ is the discriminative feature representation of the testing voxel, $\\beta$ is the sparse code for the testing voxel, $\\lambda$  is the coefficient of sparsity, and $r^p= f' - {D^p}{\\beta^p}$ is the reconstruction residue of the sparse reconstruction. The likelihood $h$ of a testing voxel $y$ is calculated with the indicator function $h(\\nu)$ with $h(\\nu)=1$ if $r^p_y\\le r^p_{y+1}$ and $-1$ otherwise, $r^p_{y}$ is the reconstruction residue at testing voxel $y$ and $r^p_{y+1}$ is the reconstruction residue at the neighboring next voxel to $y$ in the normal direction outwards from the centerline. To move landmark $l_{p,i}$ on the surface of the segmentation, we search in the normal direction. A position with the most similar profile pattern to the boundary pattern is chosen as the new position of the landmark using the following objective function,\n$\n\\mathop {{\\rm{arg max}}}\\limits_h {\\rm{ }}\\left( {\\mathop {{\\rm{arg min}}}\\limits_\\delta  \\left( {{{\\left\\| {P_{\\left\\{ { - 1,1} \\right\\}}^h\\left( {{c_{p,i}} + \\delta .\\overrightarrow {{N_{{c_{p,i}}}}} } \\right) - \\overline P _{\\left\\{ { - 1,1} \\right\\}}^h} \\right\\|}_2}} \\right){\\rm{ + }}\\frac{1}{|h|}{\\rm{ }}} \\right)| \\delta  \\in \\left[ {0,A} \\right],\n$ where $\\overline P _{\\left\\{ { - 1,1} \\right\\}}^h = \\left[ {\\underbrace { - 1, - 1,..., - 1}_{\\left| h \\right|},\\underbrace {1,1,...,1}_{\\left| h \\right|}} \\right]$ is the boundary pattern, $A$ is the search range, $N_{c_{p,i}}$ is outward normal direction at point $c_{p,i}$, $\\delta$ is the position off-set to be optimized and $\\overline P _{\\left\\{ { - 1,1} \\right\\}}^h$ is the desired boundary pattern. The length of the boundary pattern $|h|$ is desirable to be maximized to mitigate the effects of noise and false positives in the pattern.\n  \n\\subsection{Hierarchical segmentation}\nIn order to enhance the robustness of the proposed method, we adopted a hierarchical segmentation approach by incorporating scale dependent information. The idea is that the coarser levels handles robustness while the finer-scale concentrates on the accuracy of the boundary. The segmentation at a coarser scale is subsequently used to initialize the finer scale. To achieve the hierarchical joint segmentation the following steps are adopted: (1) The number of shape partitions are dyadically increased from the coarsest to the finest scale. The number of partitions $n_j$ at the coarser scales $j$ are calculated as: $n_j=\\lceil 2^{-j}G^J\\rceil$, where $G^J$ is the number of partitions at the finest scale $J$. (2) The patch size used to calculate the appearance features (Section \\ref{sec:fly}) are dyadically decreased from coarser to finer scales.\n\n\\section{Results}\nAfter Institutional Review Board approval, 15 pediatric MRI scans with healthy AVPs and 6 with OPG were acquired for this study. The acquired data were T1 weighted cube with Gadolinium contrast enhancement having spatial resolution between $0.39\\times 0.39\\times 0.6\\text{mm}^3$ to $0.47\\times 0.47\\times 0.6\\text{mm}^3$. The manual ground truth for optic pathway segmentation was created by an expert neuro-radiologist and an expert neuro-ophthalmologist. During \\textbf{the training stage}, the dataset was affinely registered to a randomly chosen reference image using a two-stage hierarchical approach: first by optimizing the registration parameters for the entire brain and later by optimizing over the region of interest around the optic nerve. The surfaces for each training instance were computed using the tetrahedral mesh generation approach followed by point set registration to the reference surface. Based on our training set, optimal number of partitions were found to be 12. Three hierarchical scales for shape model and appearance were used. The refinement model was learned \\emph{on-the-fly} from the initial segmentation using a patch of size $11\\times 11\\times 11$ voxels at the coarsest level. The normalized derivative, the tissue intensity probability, and the tubular structure probability were used together as a unified feature set of size 33 to train the refinement model. To learn the sparse dictionary, co-occurrence features were extracted with an offset of 1 and four directions ($0$, $\\frac{\\pi}{4}$, $\\frac{\\pi}{2}$, $\\frac{3\\pi}{4}$). The co-occurrence features presented in Section \\ref{sec:fly} are then calculated for each direction. During \\textbf{the testing stage}, the test image was first registered to the randomly selected reference set followed by automatic overlapping partitioning. The mean shape of the training set was used to initialize the shape model. Fig. \\ref{fig:results} shows the qualitative results of PAScAL against the ground truth manual segmentation.\n\\begin{figure}[!t]\n\\vspace{-0.2in}\n\\centering\n\\includegraphics[width=0.85\\linewidth]{OpticNerveforMICCAI_fig_02.eps}\n\\caption{\\small{Segmentation results for a representative healthy (\\emph{left}) and OPG case (\\emph{right}). \\emph{Blue} label shows overlap area of manual and automated segmentation, \\emph{red} label shows the manual label while the \\emph{green} label shows the automated segmentation.}}\n\\label{fig:results}\n\\vspace{-0.2in}\n\\end{figure}\n\nFor quantitative evaluation, the Dice similarity coefficient (DSC) and Hausdorff distance (HD) were calculated between the segmentation obtained using PAScAL and the expert generated ground truth. The quantitative results based on the leave-one-out evaluation are reported in Fig. \\ref{fig:quantitative}. An average DSC of $0.32$ for ASM, $0.53$ for Yang et al.'s approach \\cite{Xue2014109}, and $0.68$ for PAScAL is obtained, showing significant improvement by PAScAL over both methods (p-value (\\emph{Wilcoxon signed rank test}): ASM=$<0.001$, Yang's partitioned ASM=$0.015$).\n\\begin{figure*}[!b]\n\\vspace{-0.2in}\n\\centering\n\\includegraphics[width=0.43\\linewidth]{OpticNerveforMICCAI_DSC_7.eps}\n\\includegraphics[width=0.43\\linewidth]{OpticNerveforMICCAI_HD_7.eps}\n\\caption{\\small{Quantitative comparison of PAScAL with traditional ASM and partitioned ASM method presented by Yang et al. \\cite{Xue2014109}.}}\n\\label{fig:quantitative}\n\\vspace{-0.3in}\n\\end{figure*}\n\n\\subsection{Automatic optic pathway glioma detection}\nThe demonstrated of the AVP is used to establish the clinical biomarker of the OPG based on the radius profile of the optic nerve. Specifically, the average radius of the optic nerve only (ref. Fig. \\ref{fig:introduction} (c)) is calculated along the center-line of the training data set for healthy and OPG cases. A statistically significant difference between the average radii of the two classes was found based on the ground truth data (healthy optic nerve ($0.401\\pm 0.050$mm), optic nerve with OPG ($0.800\\pm 0.293$mm), p-value$<0.001$). No significant correlation between the average radius and the patient age, head circumference, and brain volume was found. To date, no established nomogram exist for the assessment of OPG; however, according to the World Health Organization osteopenia is diagnosed if the T score is $<1$ standard deviation ($\\sigma$) from the mean of healthy population, osteoprosis is defined as $<2.5\\sigma$ from the mean \\cite{linguraru2012588}. Adopting similar approach, we define the detection of OPG in the optic nerve if the mean radius $>2.5\\sigma$ from the mean of healthy population. Based on the adopted criteria, all 21 cases (15 healthy + 6 OPG cases) were classified with accuracy demonstrating the PAScAL to automatically detect pathologies of the optic nerve.      \n\n\\section{Conclusion}\n{{We presented an automated technique, PAScAL, for the segmentation of anterior visual pathway from MRI scans of the brain based on partitioned shape models with sparse appearance learning. Our work addresses the challenge of segmenting cranial nerve pathways with shape and appearance variations due to unpredictable pathological changes.}} Experiments conducted using 21 T1 MRI scans, containing instances of both healthy and pathological cases, demonstrated superior performance of PAScAL over existing approaches. {{The application of PAScAL in segmenting anterior visual pathway shows its potential in analyzing other long and thin anatomical structures with pathologies.}} \n\n\\bibliographystyle{splncs}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzfuzu b/data_all_eng_slimpj/shuffled/split2/finalzzfuzu
new file mode 100644
index 0000000000000000000000000000000000000000..230d2a0a2bdd61cd22d3e2b96dcbfa5e75e9977e
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzfuzu
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\n\nThe mathematical challenge associated with the position-dependent mass (PDM)\nvon Roos Hamiltonian \\cite{1}, and the feasible applicability of the PDM\nsettings in different fields of physics, has inspired a relatively intensive\nrecent research attention on the quantum mechanical (see the sample of\nreferences \\cite{2,3,4,5,6,7,8,9,10,11,12}), classical mechanical and\nmathematical (see the sample of references \\cit\n{12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37\n) domains in general. The position-dependent mass is, in principle, a\nposition-dependent deformation in the standard constant mass settings that\nintroduces it's own PDM-byproducted reaction-type force $R_{PDM}\\left( x\n\\dot{x}\\right) =m^{^{\\prime }}\\left( x\\right) \\dot{x}^{2}\/2$, and manifests\ndeformation in the potential force field that may inspire nonlocal\nspace-time point transformations. That is, if a PDM-particle is moving in a\nharmonic oscillator potential force field $V\\left( x\\right) =m\\left(\nx\\right) \\omega ^{2}x^{2}\/2$, for example, then one may use $q=x\\sqrt\nm\\left( x\\right) }\\Longrightarrow V\\left( q\\right) =\\omega ^{2}q^{2}\/2$ to\nretain the standard constant mass settings. In the process, some\nposition-dependent deformation in time may be deemed vital (see \\cite{22,38}\nfor more details on this issue). For some comprehensive discussions on the\nquantum mechanical PDM related ordering ambiguity and on the\nclassical-quantum mechanical correspondence, the reader may refer to (c.f.,,\ne.g., \\cite{12,17}). However, on the issue of the classical mechanical\nequivalence between the Euler-Lagrange's and Newton's equations of motion\none may refer to \\cite{22}.\n\nIn a very recent study, Mustafa \\cite{38} has introduced a general nonlocal\npoint transformation for one-dimensional PDM Lagrangians and provided their\nmappings into a constant \\emph{\"unit-mass\"} Lagrangians in the generalized\ncoordinates. Therein, it has been shown that the applicability of such\nmappings not only results in the linearization of some nonlinear oscillators\nbut also extends into the extraction of exact solutions of more complicated\ndynamical systems. Hereby, the exactly solvable Lagrangians (labeled as \n\\emph{\"reference\/target-Lagrangians\"}) are mapped along with their exact\nsolutions into PDM-Lagrangians (labeled as \\emph\n\"target\/reference-Lagrangians\"}). It would be natural and interesting to\nextend\/generalize Mustafa's methodical proposal \\cite{38} to deal with\nLagrangians in more than one-dimension. Therefore, the current methodical\nproposal is a parallel extension to \\cite{38} and deals with PDM Lagrangians\nin two-dimensions.\n\nHowever, in handling such higher dimensional Lagrangians the notion\/concept\nof \\emph{superintegrability} (c.f., e.g., \\cit\n{39,40,41,42,43,44,45,46,47,48,49,50,51} and related references cited\ntherein) is unavoidable. A Lagrangian system is said to be \\emph\nsuperintegrable} if it admits the Liouville-Arnold sense of integrability\nand introduces more constants of motion (also called integrals of motion)\nthan the degrees of freedom the system is moving within (c.f., e.g., \\cit\n{45}). The set of the two-dimensional isotonic oscillator potential\n\\begin{equation}\nV(x_{1},x_{2})=\\frac{1}{2}\\left( \\omega _{1}^{2}x_{1}^{2}+\\omega\n_{2}^{2}x_{2}^{2}+\\frac{\\beta _{1}}{x_{1}^{2}}+\\frac{\\beta _{2}}{x_{2}^{2}\n\\right) ,\n\\end{equation\nfor example (a more general extension of the Smorodinsky-Winternitz isotonic\noscillator potentials where $\\ \\omega _{1}=\\omega _{2}=\\omega $) is known to\nbe \\emph{superintegrable} (c.f., e.g., \\cite{39,40,41,42,46,47}).\\ The\ndetails on their \\emph{superintegrability} (c.f.,e.g., \\cite{39,41}) or\ntheir \\emph{maximal superintegrability} (c.f., e.g., \\cite{40})\nclassification criteria lay far beyond the scope of our study here. They are\nconsidered as \\emph{superintegrable} potentials throughout the current\nmethodical proposal, though we verify their \\emph{superintegrability} in\nbrief to make the current methodical proposal self-contained. On the other\nhand, under some nonlocal space-time transformation (see (10) below) the\ntwo-dimensional \\emph{superintegrable} isotonic oscillator potential\n\\begin{equation}\nV(q_{1},q_{2})=\\frac{1}{2}\\left( \\omega _{1}^{2}q_{1}^{2}+\\omega\n_{2}^{2}q_{2}^{2}+\\frac{\\beta _{1}}{q_{1}^{2}}+\\frac{\\beta _{2}}{q_{2}^{2}\n\\right) ,\n\\end{equation\n(where $q_{j}\\equiv q_{j}\\left( x_{j}\\right) \\,;j=1,2$ are some invertible, \n\\partial x_{j}\/\\partial q_{j}\\neq 0\\neq \\partial q_{j}\/\\partial x_{j}$,\ngeneralized coordinates) may yield a PDM-deformed isotonic oscillator\npotentials. Hereby,\\ we argue that, if the exactly solvable \\emph\n\"superintegrable reference-Lagrangians\"} are mapped into\\ \\emph{\"PDM\ntarget-Lagrangians\"}, then the set of \\emph{\"PDM target-Lagrangians\"} is a\nset of \\emph{\"sub-superintegrable PDM-Lagrangians\"} (as shall be so labeled\nhereinafter). In addition to our main objective to extend\/generalize\nMustafa's methodical proposal \\cite{38} to deal with two-dimensional\nLagrangians, we anticipate that the introduction of the terminology of \\emph\n\"sub-superintegrability\"} \\ would\\ yet add a new flavour\/concept to \\emph\n\"superintegrability\"}. The organization of our methodical proposal is in\norder.\n\nThe two-dimensional nonlocal space-time PDM transformation and the\nEuler-Lagrange equations' invariance are discussed in section 2. Therein,\none would observe that the obvious non-separability of the two-dimensional \n\\emph{\"PDM target-Lagrangians\"} $L\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x\n_{2};t\\right) $ is accompanied by the separability of the two-dimensional \n\\emph{\"reference-Lagrangians\"} $L\\left( q_{1},q_{2},\\tilde{q}_{1},\\tilde{q\n_{2};\\tau \\right) $. Mandating in effect \\emph{\"sub-separability\"} of the\ntwo-dimensional \\emph{\"PDM target-Lagrangians\"} $L\\left( x_{1},x_{2},\\dot{x\n_{1},\\dot{x}_{2};t\\right) $, as a result of the two-dimensional nonlocal PDM\ntransformations. In section 3, the mapping from the \\emph{superintegrability}\nof some linear oscillators into \\emph{sub-superintegrability} of nonlinear\nPDM-oscillators is introduced. To make the current methodical proposal\nself-contained, we discuss in short the superintegrability of the \\emph\nreference-Lagrangians} (although similar discussions are available in the\nliterature). Three two-dimensional \\emph{sub-superintegrable}\nMathews-Lakshmanan type PDM-oscillators are used for clarification. They\nare, (i) a \\emph{sub-superintegrable} Mathews-Lakshmanan type-I\nPDM-oscillator for a PDM-particle moving in a harmonic oscillator potential,\n(ii) a \\emph{sub-superintegrable} Mathews-Lakshmanan type-II PDM-oscillator\nfor a PDM-particle moving in a constant potential, (iii) a \\emph\nsub-superintegrable} shifted Mathews-Lakshmanan type-III PDM-oscillator for\na PDM-particle moving in a a shifted harmonic oscillator potential. In the\nsame section, we discuss (in short) the superintegrability of some shifted\nlinear oscillators (a new superintegrable model to the best of our\nknowledge) and report their mapping into sub-superintegrable nonlinear\nPDM-Oscillators. We use, moreover, a \\emph{superintegrable} isotonic\noscillator and map it into a \\emph{sub-superintegrable} PDM isotonic\noscillator. Of course the mappings also include exact solvability of the \n\\emph{\"reference-Lagrangians\"} at hand as well. Our concluding remarks are\ngiven in section 4.\n\n\\section{Two-dimensional nonlocal PDM-point transformations and\nEuler-Lagrange's invariance}\n\nThe Lagrangian of a particle with a constant \\emph{\"unit mass\"} moving in\nthe generalized coordinates $\\left( q_{1},q_{2}\\right) \\equiv \\left(\nq_{1}\\left( x_{1}\\right) ,q_{2}\\left( x_{2}\\right) \\right) $, under the\ninfluence of a potential force field $V(q_{1},q_{2}),$ and a\ndeformed\/re-scaled time $\\tau $, is given b\n\\begin{equation}\nL\\left( q_{1},q_{2},\\tilde{q}_{1},\\tilde{q}_{2};\\tau \\right) =\\frac{1}{2\n\\left( \\tilde{q}_{1}^{2}+\\tilde{q}_{2}^{2}\\right) -V(q_{1},q_{2})\\text{ }\n\\text{ \\ }\\tilde{q}_{j}=\\frac{dq_{j}}{d\\tau };\\,j=1,2.\n\\end{equation\nThe\\ corresponding Euler-Lagrange equation\n\\begin{equation}\n\\frac{d}{d\\tau }\\left( \\frac{\\partial L}{\\partial \\tilde{q}_{j}}\\right) \n\\frac{\\partial L}{\\partial q_{j}}=0;\\,\\,j=1,2,\n\\end{equation\ntherefore, read \n\\begin{equation}\n\\text{\\ }\\frac{d}{d\\tau }\\tilde{q}_{j}+\\frac{\\partial }{\\partial q_{j}\nV(q_{1},q_{2})=0;\\,j=1,2.\n\\end{equation\nOne should notice that for a \\emph{\"unit mass\"} particle moving in a \\emph\nfree} force field $V(q_{1},q_{2})=0\\Longrightarrow d\\tilde{q}_{j}\\left(\nx_{j}\\right) \/d\\tau =0$, hence the linear momenta $\\tilde{q}_{1}\\left(\nx_{1}\\right) $ and $\\tilde{q}_{2}\\left( x_{2}\\right) $ are conserved\nquantities ( in this particular case) and serve as fundamental integrals\n(i.e., constants of motion). However, for the set of potentials in the from\nof \n\\begin{equation}\nV(q_{1},q_{2})=V_{1}\\left( q_{1}\\right) +V_{2}\\left( q_{2}\\right) \\neq\n0;\\,V_{j}\\left( q_{j}\\right) \\neq 0;\\,j=1,2\n\\end{equation\n(which is the set of potential force fields of our interest in the current\nstudy) one may recast (5) a\n\\begin{equation}\n\\text{\\ }\\frac{d}{d\\tau }\\tilde{q}_{j}\\left( x_{j}\\right) +\\frac{\\partial }\n\\partial q_{j}}V_{j}(q_{j})=0\\Longrightarrow \\tilde{q}_{j}\\left( x\\right) \n\\text{\\ }\\frac{d}{d\\tau }\\tilde{q}_{j}\\left( x\\right) +\\tilde{q}_{j}\\left(\nx\\right) \\frac{\\partial }{\\partial q_{j}}V_{j}(q_{j})=0.\n\\end{equation\nto obtain two integrals of motion $I_{1}=E_{1}$ and $I_{2}=E_{2}$ via \n\\begin{equation}\n\\frac{d}{d\\tau }\\left[ \\frac{1}{2}\\tilde{q}_{j}^{2}+V_{j}(q_{j})\\right]\n=0\\Longrightarrow \\frac{dE_{j}}{d\\tau }=\\frac{dI_{j}}{d\\tau }=0;\\,j=1,2.\n\\end{equation\nYet, the conservation of the total energy $E_{tot}$ is a natural and an\nimmediate consequence of such settings. That is\n\\begin{equation}\n\\frac{d}{d\\tau }\\left[ E_{1}+E_{2}\\right] =\\frac{d}{d\\tau }\\left[ \\frac{1}{2\n\\left( \\tilde{q}_{1}^{2}+\\tilde{q}_{2}^{2}\\right) +V_{1}\\left( q_{1}\\right)\n+V_{2}\\left( q_{2}\\right) \\right] =0\\Longrightarrow \\frac{dE_{tot}}{d\\tau }=0\n\\end{equation\nThe separability and\/or integrability of the above system is obvious,\ntherefore.\n\nNext, let us introduce the nonlocal point transformation of the for\n\\begin{equation}\nq_{j}\\equiv q_{j}\\left( x_{j}\\right) =\\int \\sqrt{g\\left( \\bar{x}\\right) \ndx_{j}\\text{, \\ }\\tau =\\int f\\left( \\bar{x}\\right) dt\\Longrightarrow \\frac\nd\\tau }{dt}=f\\left( \\bar{x}\\right) \\neq 0\\text{ };\\text{ \\ }x_{j}\\equiv\nx_{j}\\left( t\\right) ,\\,\\bar{x}=x_{1},x_{2}.\n\\end{equation\nConsequently, with an overhead dot to identify total time $t$ derivative, \n\\begin{equation}\n\\frac{dq_{j}}{d\\tau }=\\tilde{q}_{j}=\\frac{\\dot{x}_{j}\\sqrt{g\\left( \\bar{x\n\\right) }}{f\\left( \\bar{x}\\right) }\\text{ },\\text{ \\ \\ \\ }\\frac{d}{d\\tau \n\\tilde{q}_{j}\\left( x_{j}\\right) =\\frac{\\sqrt{g\\left( \\bar{x}\\right) }}\nf\\left( \\bar{x}\\right) ^{2}}\\left[ \\ddot{x}_{j}+\\left( \\frac{1}{2}\\frac{\\dot\ng}\\left( \\bar{x}\\right) }{g\\left( \\bar{x}\\right) }-\\frac{\\dot{f}\\left( \\bar{\n}\\right) }{f\\left( \\bar{x}\\right) }\\right) \\dot{x}_{j}\\right] .\n\\end{equation\nWhich when substituted in (7) would , in a straightforward manner, result in \n\\begin{equation}\n\\left[ \\dot{x}_{1}\\,\\ddot{x}_{1}+\\dot{x}_{2}\\,\\ddot{x}_{2}+\\left( \\frac{1}{2\n\\frac{\\dot{g}\\left( \\bar{x}\\right) }{g\\left( \\bar{x}\\right) }-\\frac{\\dot{f\n\\left( \\bar{x}\\right) }{f\\left( \\bar{x}\\right) }\\right) \\left( \\dot{x\n_{1}^{2}+\\dot{x}_{2}^{2}\\right) \\right] +\\frac{f\\left( \\bar{x}\\right) ^{2}}\ng\\left( \\bar{x}\\right) }\\frac{d}{dt}V\\left( \\bar{x}\\right) =0;\\,x_{j}\\equiv\nx_{j}\\left( q_{j}\\right) ,\n\\end{equation\nwher\n\\begin{equation*}\n\\frac{d}{dt}V\\left( \\bar{x}\\right) =\\left( \\dot{x}_{1}\\frac{\\partial }\n\\partial x_{1}}V_{1}\\left( x_{1}\\right) +\\dot{x}_{2}\\frac{\\partial }\n\\partial x_{2}}V_{2}\\left( x_{2}\\right) \\right) .\n\\end{equation*}\n\nOn the other hand, for a two-dimensional position-dependent mass particle\nmoving in a force field $V\\left( x_{1},x_{2}\\right) =V_{1}\\left(\nx_{1}\\right) +V_{2}\\left( x_{2}\\right) $ the Lagrangia\n\\begin{equation}\nL\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x}_{2};t\\right) =\\frac{1}{2}m\\left( \n\\bar{x}\\right) \\left( \\dot{x}_{1}^{2}+\\dot{x}_{2}^{2}\\right) -\\left[\nV_{1}\\left( x_{1}\\right) +V_{2}\\left( x_{2}\\right) \\right] ;,\n\\end{equation\nin the Cartesian coordinates, would yield two Euler-Lagrange equations \n\\begin{equation}\nm\\left( \\bar{x}\\right) \\ddot{x}_{j}+\\dot{m}\\left( \\bar{x}\\right) \\dot{x}_{j}\n\\frac{1}{2}\\frac{\\partial m\\left( \\bar{x}\\right) }{\\partial x_{j}}\\left( \n\\dot{x}_{1}^{2}+\\dot{x}_{2}^{2}\\right) +\\frac{\\partial }{\\partial x_{j}\nV_{j}\\left( x_{j}\\right) =0;\\,j=1,2.\n\\end{equation\nThe non-separability of this system is obviously manifested by the\nposition-dependent mass term. However, when multiplied, from the left, by \n\\dot{x}_{j}$ it reads \n\\begin{equation}\nm\\left( \\bar{x}\\right) \\dot{x}_{j}\\ddot{x}_{j}+\\dot{m}\\left( \\bar{x}\\right) \n\\dot{x}_{j}^{2}-\\frac{1}{2}\\left( \\dot{x}_{j}\\frac{\\partial m\\left( \\bar{x\n\\right) }{\\partial x_{j}}\\right) \\left[ \\dot{x}_{1}^{2}+\\dot{x}_{2}^{2\n\\right] +\\left( \\dot{x}_{j}\\frac{\\partial }{\\partial x_{j}}\\right)\nV_{j}\\left( x_{j}\\right) =0;\\,j=1,2.\n\\end{equation\nConsequently, the addition of the two equations of (15) yield\n\\begin{equation}\n\\dot{x}_{1}\\ddot{x}_{1}+\\dot{x}_{2}\\ddot{x}_{2}+\\frac{1}{2}\\frac{\\dot{m\n\\left( \\bar{x}\\right) }{m\\left( \\bar{x}\\right) }\\left( \\dot{x}_{1}^{2}+\\dot{\n}_{2}^{2}\\right) +\\frac{1}{m\\left( \\bar{x}\\right) }\\frac{d}{dt}V\\left( \\bar{\n}\\right) =0,\n\\end{equation\nand henc\n\\begin{equation}\n\\frac{d}{dt}\\left[ \\frac{1}{2}m\\left( \\bar{x}\\right) \\,\\left( \\dot{x\n_{1}^{2}+\\dot{x}_{2}^{2}\\right) +V_{1}\\left( x_{1}\\right) +V_{2}\\left(\nx_{2}\\right) \\right] =0\\Longrightarrow \\frac{dE_{tot}}{dt}=0.\n\\end{equation\nHereabout, we emphasize that the conservation of the total energy $E_{tot}$\nin (17) can \\emph{never} be considered as an immediate consequence of the\nsum of two fundamental integrals of motion $E_{x_{1}}$ and $E_{x_{2}}$. The\ntime evolutions of $E_{x_{j}j}$'s do not satisfy (15). i.e., one may easily\nshow that \n\\begin{equation}\n\\frac{d}{dt}E_{x_{j}}=\\frac{d}{dt}\\left[ \\frac{1}{2}m\\left( \\bar{x}\\right) \n\\dot{x}_{j}^{2}+V_{j}\\left( x_{j}\\right) \\right] =m\\left( \\bar{x}\\right) \n\\dot{x}_{j}\\ddot{x}_{j}+\\frac{1}{2}\\dot{m}\\left( \\bar{x}\\right) \\dot{x\n_{j}^{2}+\\dot{x}_{j}\\frac{\\partial }{\\partial x_{j}}V_{j}\\left( x_{j}\\right)\n\\neq 0\n\\end{equation\ncompared to (15). The third term in (15) is missing in (18). Moreover, the\ncomparison between (12) and (16) obviously suggests that the Euler-Lagrange\nequations of motion (12) and (16) are identical if and only if $f\\left( \\bar\nx}\\right) $ and $g\\left( \\bar{x}\\right) $ satisfy the conditions \n\\begin{equation}\ng\\left( \\bar{x}\\right) =m\\left( \\bar{x}\\right) \\,f\\left( \\bar{x}\\right)\n^{2}\\Longleftrightarrow \\frac{1}{2}\\frac{\\dot{g}\\left( \\bar{x}\\right) }\ng\\left( \\bar{x}\\right) }-\\frac{\\dot{f}\\left( \\bar{x}\\right) }{f\\left( \\bar{x\n\\right) }=\\frac{1}{2}\\frac{\\dot{m}\\left( \\bar{x}\\right) }{m\\left( \\bar{x\n\\right) }\\Longleftrightarrow q_{j}=\\int \\sqrt{m\\left( \\bar{x}\\right) \nf\\left( \\bar{x}\\right) dx_{j}.\n\\end{equation\nAs such, it is clear that the functional nature\/structure of the\nposition-dependent mass $m\\left( \\bar{x}\\right) $ determines the\nnature\/structure of the nonlocal transformation functions $g\\left( \\bar{x\n\\right) $ and $\\,f\\left( \\bar{x}\\right) .$ For the sake of simplicity,\nhowever, we shall work with $m\\left( \\bar{x}\\right) \\equiv m\\left( r\\right)\n\\Longrightarrow g\\left( \\bar{x}\\right) \\equiv g\\left( r\\right) $ and \n\\,f\\left( \\bar{x}\\right) \\equiv $ $f\\left( r\\right) $. Hence, $m\\left(\nr\\right) $, $g\\left( r\\right) $ and $f\\left( r\\right) $ are well behaved\nfunctions of explicit dependence on $r=\\sqrt{x_{1}^{2}+x_{2}^{2}}$, if not\notherwise mentioned.\n\nAt this point, one may safely conclude that under such invertible (i.e., the\nJacobian determinant $det\\left( \\partial x_{i}\/\\partial q_{i}\\right) \\neq 0\n) nonlocal transformation, (10) and (19), the two-dimensional Euler-Lagrange\nequations, (12) and (16), remain invariant. That is, the Lagrangian $L\\left(\nq_{1},q_{2},\\tilde{q}_{1},\\tilde{q}_{2};\\tau \\right) ,$ in (3), non-locally\ntransforms (via (10) and (19)) into the PDM Lagrangian $L\\left( x_{1},x_{2}\n\\dot{x}_{1},\\dot{x}_{2};t\\right) $, in (13), and leaves in the process the\ncorresponding two-dimensional Euler-Lagrange equations of motions, (12) and\n(16), invariant. The mappin\n\\begin{equation}\nL\\left( q_{1},q_{2},\\tilde{q}_{1},\\tilde{q}_{2};\\tau \\right)\n\\Longleftrightarrow \\left\\{ \n\\begin{array}{c}\ng\\left( r\\right) =m\\left( r\\right) f\\left( r\\right) ^{2}\\medskip \\medskip \\\\ \nq_{j}=\\int \\sqrt{m\\left( r\\right) }f\\left( r\\right)\ndx_{j}\\,;\\,j=1,2\\smallskip \\medskip \\\\ \n\\tau =\\int f\\left( r\\right) dt\\medskip \\\\ \n\\tilde{q}_{j}=\\dot{x}_{j}\\sqrt{m\\left( r\\right) };\\,j=1,\n\\end{array\n\\right\\} \\Longleftrightarrow L\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x\n_{2};t\\right) ,\n\\end{equation\nbetween the \\emph{\"unit mass\" } Lagrangian $L\\left( q_{1},q_{2},\\tilde{q\n_{1},\\tilde{q}_{2};\\tau \\right) $ and PDM Lagrangian $L\\left( x_{1},x_{2}\n\\dot{x}_{1},\\dot{x}_{2};t\\right) $ is clear, therefore.\n\nNevertheless, the obvious non-separability of the two-dimensional\nEuler-Lagrange equations associated with the PDM Lagrangian $L\\left(\nx_{1},x_{2},\\dot{x}_{1},\\dot{x}_{2};t\\right) $ (hence, $L\\left( x_{1},x_{2}\n\\dot{x}_{1},\\dot{x}_{2};t\\right) $ is non-separable) is accompanied by the\nseparability of the two-dimensional Euler-Lagrange equations associated with\nunit mass Lagrangian $L\\left( q_{1},q_{2},\\tilde{q}_{1},\\tilde{q}_{2};\\tau\n\\right) $ (hence, $L\\left( q_{1},q_{2},\\tilde{q}_{1},\\tilde{q}_{2};\\tau\n\\right) $ is separable). This should, in turn, mandate the notion of\n\"sub-separability\" of the two-dimensional Lagrangian $L\\left( x_{1},x_{2}\n\\dot{x}_{1},\\dot{x}_{2};t\\right) $ as a result of the two-dimensional PDM\nnonlocal transformations in (20). Likewise, if the two-dimensional unit mass\nLagrangian $L\\left( q_{1},q_{2},\\tilde{q}_{1},\\tilde{q}_{2};\\tau \\right) $\nadmits superseparability and\/or superintegrability then the two-dimensional\nPDM Lagrangian $L\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x}_{2};t\\right) $ may\nvery well be labeled as sub-superseparable and\/or sub-superintegrable. The\nlatter forms the focal point of our study here and shall be clarified in the\nforthcoming experimental examples.\n\n\\section{From superintegrability to sub-superintegrability; two-dimensional\nPDM-oscillators}\n\nAlthough the superintegrability of some of the \\emph{reference\nsuperintegrable} oscillators, we use here, is verified in the literature, we\nrecycle them in such a way that serves our methodical proposal and keeps it\nself-contained. The\\emph{\\ superintegrable linear oscillators}, the \\emph\nsuperintegrable shifted-oscillators} (is a new \\emph{superintegrable}\noscillator model, to the best of our knowledge), and the \\emph\nsuperintegrable isotonic oscillators} are used here as illustrative\nexamples. They are mapped along with their exact solutions into \\emph\nsub-superintegrable PDM-oscillators}.\n\n\\subsection{Superintegrable linear oscillators into sub-superintegrable\nnonlinear PDM-oscillators}\n\nConsider a \\emph{unit mass }particle moving under the influence of the\ntwo-dimensional oscillators potential \n\\begin{equation}\nV(q_{1},q_{2})=\\frac{1}{2}\\left( \\omega _{1}^{2}q_{1}^{2}+\\omega\n_{2}^{2}q_{2}^{2}\\right) \\Longrightarrow V_{j}\\left( q_{j}\\right) =\\frac{1}{\n}\\omega _{j}^{2}q_{j}^{2},\\,\\omega _{j}=n_{j}\\omega _{\\circ };\\,j=1,2,\n\\end{equation\nin the generalized coordinates. Then, the corresponding two-dimensional\nLagrangia\n\\begin{equation}\nL\\left( q_{1},q_{2},\\tilde{q}_{1},\\tilde{q}_{2};\\tau \\right) =\\frac{1}{2\n\\left( \\tilde{q}_{1}^{2}+\\tilde{q}_{2}^{2}\\right) -\\frac{1}{2}\\left( \\omega\n_{1}^{2}q_{1}^{2}+\\omega _{2}^{2}q_{2}^{2}\\right) ,\n\\end{equation\nleads to Euler-Lagrange equation\n\\begin{equation}\n\\frac{d}{d\\tau }\\tilde{q}_{j}+\\omega _{j}^{2}q_{j}=0\\Longrightarrow\nq_{j}\\left( \\tau \\right) =A_{j}\\cos \\left( \\omega _{j}\\tau +\\varphi \\right)\n,\\,\\,j=1,2,\n\\end{equation\nsubjected to the boundary conditions $\\,q_{j}\\left( 0\\right) =A_{j},\\,\\tilde\nq}_{j}\\left( 0\\right) =0$, say. It obviously admits separability and is\nknown to satisfy the superintegrability conditions (c.f. e.g., \\cit\n{39,40,41,42}) via the use of \\ a complex factorization technique (c.f.,\ne.g., \\cite{43,44}). That is, if we introduce the two complex functions \n\\begin{equation}\nQ_{j}=\\tilde{q}_{j}+i\\omega _{j}q_{j}\\Longrightarrow \\frac{d}{d\\tau \nQ_{j}=i\\omega _{j}Q_{j};\\,\\,j=1,2,\n\\end{equation\nthen the function\n\\begin{equation}\nQ_{jk}=Q_{j}^{\\omega _{k}}\\left( Q_{k}^{\\ast }\\right) ^{\\omega\n_{j}};\\,\\,j,k=1,2,\n\\end{equation\nrepresent \\emph{complex }constants of motion with vanishing\ndeformed\/rescaled-time evolution $\\tau $\n\\begin{equation}\n\\frac{d}{d\\tau }Q_{jk}=i\\left( \\omega _{j}\\omega _{k}-\\omega _{k}\\omega\n_{j}\\right) Q_{jk}=0.\n\\end{equation\nMoreover, one can, in a straightforward manner, verify that \n\\begin{equation}\nQ_{jj}=\\tilde{q}_{j}^{2}+\\omega _{j}^{2}q_{j}^{2}\\Longrightarrow\n2E_{1}=I_{1}=Q_{11},\\,2E_{2}=I_{2}=Q_{22},\n\\end{equation\nwhere $I_{1}$ and $I_{2}$ are two fundamental integrals of motion. Yet, for\nthe isotropic oscillator $\\omega _{1}=\\omega _{2}=\\omega _{\\circ }$, for\nexample\n\\begin{equation}\nQ_{12}=\\left( \\tilde{q}_{1}\\tilde{q}_{2}+\\omega _{\\circ\n}^{2}q_{1}q_{2}\\right) +i\\omega _{\\circ }\\left( q_{1}\\tilde{q}_{2}-q_{2\n\\tilde{q}_{1}\\right) =I_{3}+iI_{4}\n\\end{equation\nidentifies two more integrals of motion \n\\begin{equation}\nI_{3}=\\func{Re}Q_{12}=\\tilde{q}_{1}\\tilde{q}_{2}+\\omega _{\\circ\n}^{2}q_{1}q_{2}\\text{ \\ ; \\ \\ }I_{4}=\\func{Im}Q_{12}=\\omega _{\\circ }\\left(\nq_{1}\\tilde{q}_{2}-q_{2}\\tilde{q}_{1}\\right) .\n\\end{equation\nwhich are, in the general case $\\omega _{1}\\neq \\omega _{2}$, polynomials in\nthe momenta. Therefore, our two-dimensional Lagrangian (22) admits\nsuperintegrability in the generalized coordinates $\\left( q_{1},q_{2}\\right) \n$ and in the deformed\/rescaled time $\\tau $. The details on such\nsuperintegrability are far beyond the scope of our the current methodical\nproposal, though can be traced through the comprehensive article of Ranada \n\\cite{42} and related references cited therein.\n\n\\subsubsection{Sub-superintegrable Mathews-Lakshmanan type-I PDM-oscillators}\n\nLet us now consider a PDM particle $m\\left( r\\right) $\\ moving in the\nharmonic oscillator force field \n\\begin{equation*}\nV\\left( x_{1},x_{2}\\right) =\\frac{1}{2}m\\left( r\\right) \\omega ^{2}\\left(\nx_{1}^{2}+x_{2}^{2}\\right) ,\n\\end{equation*\nwith the corresponding PDM Lagrangia\n\\begin{equation}\nL\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x}_{2};t\\right) =\\frac{1}{2}m\\left(\nr\\right) \\left[ \\dot{x}_{1}^{2}+\\dot{x}_{2}^{2}-\\omega ^{2}\\left(\nx_{1}^{2}+x_{2}^{2}\\right) \\right] .\n\\end{equation\nThis Lagrangian indulges one and only one integral offered by the total\nenergy $E_{tot}$ as in (17), and the corresponding Euler-Lagrange equations\nare non-separable. However, with the substitutio\n\\begin{equation}\nq_{j}=x_{j}\\sqrt{m\\left( r\\right) }\\,;\\smallskip \\medskip \\,j=1,2,\n\\end{equation\none can non-locally transform $L\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x\n_{2};t\\right) $ of (30) into $L\\left( q_{1},q_{2},\\tilde{q}_{1},\\tilde{q\n_{2};\\tau \\right) $ of (22). Moreover, if this substitution is used along\nwith the transformation in (20) we ge\n\\begin{equation}\nq_{j}=x_{j}\\sqrt{m\\left( r\\right) }\\Longrightarrow \\frac{dq_{j}}{dx_{j}}\n\\sqrt{m\\left( r\\right) }\\left[ 1+\\frac{m^{\\prime }\\left( r\\right) }{2m\\left(\nr\\right) }\\left( \\frac{x_{j}^{2}}{r}\\right) \\right] ,\\medskip\n\\end{equation\nan\n\\begin{equation*}\nq_{j}=\\int \\sqrt{m\\left( r\\right) }f\\left( r\\right) dx_{j}\\Longrightarrow \n\\frac{dq_{1}}{dx_{1}}+\\frac{dq_{2}}{dx_{2}}=2\\sqrt{m\\left( r\\right) }f\\left(\nr\\right) .\n\\end{equation*\nhenc\n\\begin{equation}\nf\\left( r\\right) =1+\\frac{1}{4}\\frac{m^{^{\\prime }}\\left( r\\right) }{m\\left(\nr\\right) }r.\n\\end{equation\nAt this point, one should be aware that we are interested in $m\\left(\nr\\right) $ and $f\\left( r\\right) $ that are only explicit function in $r\n\\sqrt{x_{1}^{2}+x_{2}^{2}}$. Obviously, moreover, the choice of $f\\left(\nr\\right) $ would determine the position-dependent mass function $m\\left(\nr\\right) $ (of course the other way around works as well). This is clarified\nin the following assumption. Let us assume tha\n\\begin{equation}\nf\\left( r\\right) =m\\left( r\\right) -\\frac{1}{4}\\frac{m^{^{\\prime }}\\left(\nr\\right) }{m\\left( r\\right) }r\n\\end{equation\nto obtain a position-dependent mass of the for\n\\begin{equation}\nf\\left( r\\right) =m\\left( r\\right) -\\frac{1}{4}\\frac{m^{^{\\prime }}\\left(\nr\\right) }{m\\left( r\\right) }r=1+\\frac{1}{4}\\frac{m^{^{\\prime }}\\left(\nr\\right) }{m\\left( r\\right) }r\\Longleftrightarrow m\\left( r\\right) =\\frac{1}\n1\\pm \\beta r^{2}};\\,\\beta \\geq 0.\n\\end{equation\nThen the corresponding two-dimensional PDM Lagrangian of (30) reads a\ntwo-dimensional Mathews-Lakshmanan type-I oscillato\n\\begin{equation}\nL\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x}_{2};t\\right) =\\frac{1}{2}\\frac{\\left[\n\\dot{x}_{1}^{2}+\\dot{x}_{2}^{2}-\\omega ^{2}\\left( x_{1}^{2}+x_{2}^{2}\\right)\n\\right] }{1\\pm \\beta \\left( x_{1}^{2}+x_{2}^{2}\\right) }.\n\\end{equation}\n\nA Lagrangian of this type is neither separable nor superintegrable.\nNevertheless, our two-dimensional PDM Lagrangian $L\\left( x_{1},x_{2},\\dot{x\n_{1},\\dot{x}_{2};t\\right) $ (36) nonlocaly transforms into a \\emph\nsuperintegrable} two-dimensional Lagrangian $L\\left( q_{1},q_{2},\\tilde{q\n_{1},\\tilde{q}_{2};\\tau \\right) $ (22). Hence the two-dimensional PDM\\\nLagrangian $L\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x}_{2};t\\right) $ of (36)\nis a \\emph{sub-superintegrable \\ }PDM Lagrangian and the corresponding\nEuler-Lagrange equations (16) admit exact solution\n\\begin{equation}\nx_{j}\\left( t\\right) =A_{j}\\cos \\left( \\Omega t+\\varphi \\right) ;\\,\\Omega\n^{2}=\\frac{\\omega ^{2}}{1\\pm \\beta \\left( A_{1}^{2}+A_{2}^{2}\\right) }.\n\\end{equation\nLikewise, the reversed process is equally valid. That is, the relation \n\\begin{equation}\n\\left\\{ \n\\begin{tabular}{c}\n\\emph{\\ Superintegrable}\\medskip \\\\ \n$L\\left( q_{1},q_{2},\\tilde{q}_{1},\\tilde{q}_{2};\\tau \\right) \\medskip $ \\\\ \nof \\ (22) with\\medskip \\\\ \n$q_{j}\\left( \\tau \\right) =A_{j}\\cos \\left( \\omega \\tau +\\varphi \\right) \n\\end{tabular\n\\right\\} \\Longleftrightarrow \\left\\{ \n\\begin{tabular}{l}\n$\\text{Nonlocal transformation\\medskip }$ \\\\ \n$q_{j}=x_{j}\\sqrt{m\\left( r\\right) }\\,;\\smallskip \\medskip \\,j=1,2,$ \\\\ \n$\\tilde{q}_{j}=\\dot{x}_{j}\\sqrt{m\\left( r\\right) }\\medskip $ \\\\ \n$f\\left( r\\right) =m\\left( r\\right) -\\frac{1}{4}\\frac{m^{^{\\prime }}\\left(\nr\\right) }{m\\left( r\\right) }r\\medskip $ \\\\ \n$m\\left( r\\right) =1\/\\left( 1\\pm \\beta r^{2}\\right) \n\\end{tabular\n\\right\\} \\Longleftrightarrow \\left\\{ \n\\begin{tabular}{c}\n\\emph{Sub-superintegrable}\\medskip \\\\ \n$L\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x}_{2};t\\right) \\medskip $ \\\\ \nof \\ (36) with\\medskip \\\\ \n$x_{j}\\left( t\\right) =A_{j}\\cos \\left( \\Omega t+\\varphi \\right) \\,\\medskip $\n\\\\ \n$\\text{ \\ }\\Omega ^{2}=\\frac{\\omega ^{2}}{1\\pm \\beta \\left(\nA_{1}^{2}+A_{2}^{2}\\right) }\\medskip \n\\end{tabular\n\\right\\} .\n\\end{equation\nprovides the exact mapping from the \\emph{superintegrability} and exact\nsolvability of (22) into the \\emph{sub-superintegrability} and exact\nsolvability of the PDM Mathews-Lakshmanan type-I Lagrangian (36).\n\n\\subsubsection{Sub-superintegrable Mathews-Lakshmanan type-II PDM-oscillator\n}\n\nConsider a PDM particle $m\\left( r\\right) $ moving in a constant potential\nforce field of the for\n\\begin{equation}\nV\\left( x_{1},x_{2}\\right) =\\frac{1}{2}m\\left( r\\right) \\omega ^{2}\\left(\n\\xi _{1}^{2}+\\xi _{2}^{2}\\right)\n\\end{equation\nwith the corresponding Lagrangia\n\\begin{equation}\nL\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x}_{2};t\\right) =\\frac{1}{2}m\\left(\nr\\right) \\left[ \\dot{x}_{1}^{2}+\\dot{x}_{2}^{2}-\\omega ^{2}\\left( \\xi\n_{1}^{2}+\\xi _{2}^{2}\\right) \\right] .\n\\end{equation\nwhere $\\xi _{1},$ $\\xi _{2}\\in \n\\mathbb{R}\n$ are constants. This Lagrangian has the total energy $E_{tot}$ of (17) as\nthe only integral of motion and the corresponding Euler-Lagrange equations\nare non-separable. However, the substitution of $q_{j}=\\xi _{j}\\sqrt{m\\left(\nr\\right) }$ would nonlocaly transform it into the \\emph{superintegrable}\nLagrangian $L\\left( q_{1},q_{2},\\tilde{q}_{1},\\tilde{q}_{2};\\tau \\right) $\nof (22) for $\\omega _{1}=\\omega _{2}=\\omega $. Under such settings\n\\begin{equation}\nq_{j}=\\int \\sqrt{m\\left( r\\right) }f\\left( r\\right) dx_{j}=\\xi _{j}\\sqrt\nm\\left( r\\right) }\\Longrightarrow f\\left( r\\right) =\\frac{\\xi _{j}}{2}\\frac\nm^{^{\\prime }}\\left( r\\right) }{m\\left( r\\right) }\\left( \\frac{x_{j}}{r\n\\right) ,\n\\end{equation\nand for $\\xi _{1}=\\xi _{2}=\\xi \/\\sqrt{2}$, this would imply tha\n\\begin{equation}\n2f\\left( r\\right) ^{2}=\\left( \\frac{m^{^{\\prime }}\\left( r\\right) }{2m\\left(\nr\\right) }\\right) ^{2}\\left( \\frac{\\xi _{1}^{2}x_{1}^{2}+\\xi\n_{2}^{2}x_{2}^{2}}{r^{2}}\\right) \\Longrightarrow f\\left( r\\right) =\\frac{\\xi\nm^{^{\\prime }}\\left( r\\right) }{4m\\left( r\\right) }.\n\\end{equation\nConsequently, the corresponding two-dimensional PDM Euler-Lagrange equation\n(17), fo\n\\begin{equation*}\nm\\left( r\\right) =\\frac{1}{1\\pm \\beta r^{2}},\n\\end{equation*\nread\n\\begin{equation}\n\\frac{d}{dt}\\left[ \\frac{\\dot{x}_{1}^{2}+\\dot{x}_{2}^{2}+\\omega ^{2}\\xi ^{2\n}{2(1\\pm \\beta r^{2})}\\right] =0,\n\\end{equation\nand admits solutions of the form\n\\begin{equation}\nx_{j}\\left( t\\right) =A_{j}\\cos \\left( \\Omega t+\\varphi \\right) \\,;\\,\\Omega\n^{2}=\\frac{\\mp \\omega ^{2}\\beta \\xi ^{2}}{1\\pm \\beta \\left(\nA_{1}^{2}+A_{2}^{2}\\right) }\\,;\\,\\beta \\geq 0;\\,j=1,2\\medskip \\medskip\n\\end{equation\nUnder such settings, one can show that for $\\beta =\\mp 1\/\\xi ^{2}$ the\nLagrangian of (36) and the Lagrangian of (40) indulge the very same\ndynamical properties as documented in the corresponding Euler-Lagrange\nequation (16). Hence, the Lagrangian at hand here is a Mathews-Lakshmanan\ntype-II PDM-oscillators Lagrangian.\n\nObviously, moreover, our non-separable and non-superintegrable Lagrangian\n(40) non-locally transforms into a separable and \\emph{superintegrable}\nLagrangian (22). Our PDM Lagrangian (40) is a \\emph{sub-superintegrable}\nMathews-Lakshmanan type-II PDM-oscillators Lagrangian, therefore. In short,\nthe relatio\n\\begin{equation}\n\\left\\{ \n\\begin{tabular}{c}\n\\ \\emph{Superintegrable}\\medskip \\\\ \n$L\\left( q_{1},q_{2},\\tilde{q}_{1},\\tilde{q}_{2};\\tau \\right) \\medskip $ \\\\ \nof \\ (22) with\\medskip \\\\ \n$q_{j}\\left( \\tau \\right) =A_{j}\\cos \\left( \\omega \\tau +\\varphi \\right) \n\\end{tabular\n\\right\\} \\Longleftrightarrow \\left\\{ \n\\begin{tabular}{l}\n$\\text{Nonlocal transformation\\medskip }$ \\\\ \n$q_{j}=\\xi _{j}\\sqrt{m\\left( r\\right) }\\medskip \\,;\\,j=1,2$ \\\\ \n$\\tilde{q}_{j}=\\dot{x}_{j}\\sqrt{m\\left( r\\right) }\\medskip $ \\\\ \n$f\\left( r\\right) =\\frac{\\xi m^{^{\\prime }}\\left( r\\right) }{4m\\left(\nr\\right) }\\medskip ;\\,\\xi _{j}=\\frac{\\xi }{\\sqrt{2}}$ \\\\ \n$m\\left( r\\right) =1\/\\left( 1\\pm \\beta r^{2}\\right) ;\\beta =\\mp 1\/\\xi\n^{2}\\medskip \n\\end{tabular\n\\right\\} \\Longleftrightarrow \\left\\{ \n\\begin{tabular}{c}\n\\emph{Sub-superintegrable}\\medskip \\\\ \n$L\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x}_{2};t\\right) \\medskip $ \\\\ \nof \\ (39) with\\medskip \\\\ \n$x_{j}\\left( t\\right) =A_{j}\\cos \\left( \\Omega t+\\varphi \\right) \\,\\medskip\n\\medskip $ \\\\ \n$\\text{ \\ }\\Omega ^{2}=\\frac{\\omega ^{2}}{1\\pm \\beta \\left(\nA_{1}^{2}+A_{2}^{2}\\right) }\\medskip \n\\end{tabular\n\\right\\} .\n\\end{equation\nrepresents the sought after mapping from\/to \\emph{superintegrable}\nLagrangian $L\\left( q_{1},q_{2},\\tilde{q}_{1},\\tilde{q}_{2};\\tau \\right) $\nof \\ (22)\\ to\/from the \\emph{Sub-superintegrable}\\medskip\\ Lagrangian \nL\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x}_{2};t\\right) $ of \\ (39).\n\n\\subsubsection{Sub-superintegrable Mathews-Lakshmanan type-III PDM\nshifted-oscillators}\n\nLet us now use a position-dependent mass with a different functional\nstructure moving in a shifted-oscillator force field and described by the\nLagrangia\n\\begin{equation}\nL\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x}_{2};t\\right) =\\frac{1}{2}m\\left(\nr_{s}\\right) \\left\\{ \\dot{x}_{1}^{2}+\\dot{x}_{2}^{2}-\\omega ^{2}\\left[\n\\left( x_{1}+\\gamma _{1}\\right) ^{2}+\\left( x_{2}+\\gamma _{2}\\right) ^{2\n\\right] \\right\\} .\n\\end{equation\nwhere $r_{s}=\\sqrt{\\left( x_{1}+\\gamma _{1}\\right) ^{2}+\\left( x_{2}+\\gamma\n_{2}\\right) ^{2}}$ is introduced for convenience. The form of the shifted\noscillator potential is clear here. At this point, we may recollect that the\nfunctional nature\/structure of the position-dependent mass $m\\left( \\bar{x\n\\right) $ determines the nature\/structure of $g\\left( \\bar{x}\\right) $ and \n\\,f\\left( \\bar{x}\\right) $ in our nonlocal point transformation as suggested\nby equation (19). Therefore, $f\\left( r\\right) \\longrightarrow f\\left(\nr_{s}\\right) $ and $g\\left( r\\right) \\longrightarrow g\\left( r_{s}\\right) $\nfor our Lagrangian (46) at hand. Moreover, it is a straightforward manner,\nand in parallel with (32)-(35), one may show that the substitution o\n\\begin{equation}\nq_{j}=\\left( x_{j}+\\xi _{j}\\right) \\sqrt{m\\left( r_{s}\\right) \n\\Longrightarrow f\\left( r_{s}\\right) =1+\\frac{1}{4}\\frac{m^{^{\\prime\n}}\\left( r_{s}\\right) }{m\\left( r_{s}\\right) }r_{s}\n\\end{equation\nwould consequently, for the choice \n\\begin{equation}\nf\\left( r_{s}\\right) =m\\left( r_{s}\\right) -\\frac{1}{4}\\frac{m^{^{\\prime\n}}\\left( r_{s}\\right) }{m\\left( r_{s}\\right) }r_{s}=1+\\frac{1}{4}\\frac\nm^{^{\\prime }}\\left( r_{s}\\right) }{m\\left( r_{s}\\right) }r_{s},\n\\end{equation\nyiel\n\\begin{equation}\nm\\left( r_{s}\\right) =\\frac{1}{1\\pm \\beta r_{s}^{2}}=\\frac{1}{1\\pm \\beta\n\\left[ \\left( x_{1}+\\gamma _{1}\\right) ^{2}+\\left( x_{2}+\\gamma _{2}\\right)\n^{2}\\right] },\n\\end{equation\nThen the corresponding two-dimensional PDM Lagrangian of (46) reads a\ntwo-dimensional Mathews-Lakshmanan type-III PDM shifted-oscillators\nLagrangia\n\\begin{equation}\nL\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x}_{2};t\\right) =\\frac{1}{2}\\frac{\\left[\n\\dot{x}_{1}^{2}+\\dot{x}_{2}^{2}-\\omega ^{2}r_{s}^{2}\\right] }{1\\pm \\beta\nr_{s}^{2}}.\n\\end{equation\nOur two-dimensional PDM Lagrangian $L\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x\n_{2};t\\right) $ (50) nonlocaly transforms into a \\emph{superintegrable}\ntwo-dimensional Lagrangian $L\\left( q_{1},q_{2},\\tilde{q}_{1},\\tilde{q\n_{2};\\tau \\right) $ of (22). Hence our PDM-Lagrangian $L\\left( x_{1},x_{2}\n\\dot{x}_{1},\\dot{x}_{2};t\\right) $ (50) is a \\emph{sub-superintegrable \nLagrangian and the corresponding Euler-Lagrange equation (16) admits exact\nsolutions of the for\n\\begin{equation}\nx_{j}\\left( t\\right) =A_{j}\\cos \\left( \\Omega t+\\varphi \\right) -\\gamma\n_{j};\\,\\Omega ^{2}=\\frac{\\omega ^{2}}{1\\pm \\beta \\left(\nA_{1}^{2}+A_{2}^{2}\\right) };\\,j=1,2.\n\\end{equation\nThe process is summed up a\n\\begin{equation}\n\\left\\{ \n\\begin{tabular}{c}\n\\ Superintegrable\\medskip \\\\ \n$L\\left( q_{1},q_{2},\\tilde{q}_{1},\\tilde{q}_{2};\\tau \\right) \\medskip $ \\\\ \nof \\ (22) with\\medskip \\\\ \n$q_{j}\\left( \\tau \\right) =A_{j}\\cos \\left( \\omega \\tau +\\varphi \\right) \n\\end{tabular\n\\right\\} \\Longleftrightarrow \\left\\{ \n\\begin{tabular}{l}\n$\\text{Nonlocal transformation\\medskip }$ \\\\ \n$q_{j}=x_{j}\\sqrt{m\\left( r_{s}\\right) }\\medskip \\,;\\,j=1,2$ \\\\ \n$\\tilde{q}_{j}=\\dot{x}_{j}\\sqrt{m\\left( r_{s}\\right) }\\medskip $ \\\\ \n$f\\left( r_{s}\\right) =m\\left( r_{s}\\right) -\\frac{1}{4}\\frac{m^{^{\\prime\n}}\\left( r_{s}\\right) }{m\\left( r_{s}\\right) }r_{s}\\medskip \\medskip $ \\\\ \n$m\\left( r_{s}\\right) =1\/\\left( 1\\pm \\beta r_{s}^{2}\\right) \\medskip \n\\end{tabular\n\\right\\} \\Longleftrightarrow \\left\\{ \n\\begin{tabular}{c}\nSub-superintegrable\\medskip \\\\ \n$L\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x}_{2};t\\right) \\medskip $ \\\\ \nof \\ (50) with\\medskip \\\\ \n$x_{j}\\left( t\\right) =A_{j}\\cos \\left( \\Omega t+\\varphi \\right) -\\gamma\n_{j}\\,\\medskip $ \\\\ \n$\\text{ \\ }\\Omega ^{2}=\\frac{\\omega ^{2}}{1\\pm \\beta \\left(\nA_{1}^{2}+A_{2}^{2}\\right) }\\medskip \n\\end{tabular\n\\right\\} ,\n\\end{equation\nto represent the mapping from the \\emph{superintegrability} harmonic\noscillators (22) into \\emph{sub-superintegrability} of the above\nMathews-Lakshmanan type-III PDM shifted-oscillators (50).\n\n\\subsection{Superintegrable shifted-linear oscillators into\nsub-superintegrable nonlinear PDM-oscillators}\n\nConsider a \\emph{unit mass }particle moving in the two-dimensional\nshifted-oscillators potential \n\\begin{equation}\nV(q_{1},q_{2})=\\frac{1}{2}\\left[ \\alpha _{1}^{2}\\left( q_{1}+\\eta\n_{1}\\right) ^{2}+\\alpha _{2}^{2}\\left( q_{2}+\\eta _{2}\\right) ^{2}\\right]\n\\Longrightarrow V_{j}\\left( q_{j}\\right) =\\frac{1}{2}\\alpha _{j}^{2}\\left(\nq_{j}+\\eta _{j}\\right) ^{2};\\,j=1,2,\n\\end{equation\nin the generalized coordinates, with the constant shifts $\\eta _{1},$ $\\eta\n_{2}\\in \n\\mathbb{R}\n$. Then, the corresponding two-dimensional Lagrangia\n\\begin{equation}\nL\\left( q_{1},q_{2},\\tilde{q}_{1},\\tilde{q}_{2};\\tau \\right) =\\frac{1}{2\n\\left( \\tilde{q}_{1}^{2}+\\tilde{q}_{2}^{2}\\right) -\\frac{1}{2}\\left[ \\alpha\n_{1}^{2}\\left( q_{1}+\\eta _{1}\\right) ^{2}+\\alpha _{2}^{2}\\left( q_{2}+\\eta\n_{2}\\right) ^{2}\\right] ,\n\\end{equation\nyields the Euler-Lagrange equation\n\\begin{equation}\n\\frac{d}{d\\tau }\\tilde{q}_{j}+\\alpha _{j}^{2}\\left( q_{j}+\\eta _{j}\\right)\n=0\\Longrightarrow q_{j}\\left( \\tau \\right) =A_{j}\\cos \\left( \\alpha _{j}\\tau\n+\\varphi \\right) -\\eta _{j};\\,j=1,2,\n\\end{equation\nwith the initial conditions that $\\,q_{j}\\left( 0\\right) =A_{j}-\\eta _{j},\\\n\\tilde{q}_{j}\\left( 0\\right) =0$, say. The separability of this Lagrangian\nis obvious. However, the verification of the superintegrability of such a\nLagrangian follows (step-by-step) from the complex factorization recipe\n(c.f., e.g., \\cite{42,43,44}) by introducing the complex function\n\\begin{equation}\nQ_{j}=\\tilde{q}_{j}+i\\alpha _{j}\\left( q_{j}+\\eta _{j}\\right)\n\\Longrightarrow \\frac{d}{d\\tau }Q_{j}=i\\alpha _{j}Q_{j};\\,\\,j=1,2.\n\\end{equation\nWhich would, in turn, suggest that \n\\begin{equation}\nQ_{jk}=Q_{j}^{\\alpha _{k}}\\left( Q_{k}^{\\ast }\\right) ^{\\alpha\n_{j}}\\Longrightarrow \\frac{d}{d\\tau }Q_{jk}=i\\left( \\alpha _{j}\\alpha\n_{k}-\\alpha _{k}\\alpha _{j}\\right) Q_{jk}=0.;\\,\\,j,k=1,2.\n\\end{equation\nThat is, the complex function $Q_{jk}$ represent \\emph{complex }constants of\nmotion with vanishing deformed\/rescaled-time evolution $\\tau $, Yet, it is\nan easy task to show that the two fundamental integrals $I_{1}$ and $I_{2}$\nare given through the relation \n\\begin{equation}\nQ_{jj}=\\tilde{q}_{j}^{2}+\\alpha _{j}^{2}\\left( q_{j}+\\eta _{j}\\right)\n^{2}\\Longrightarrow 2E_{1}=I_{1}=Q_{11},\\,2E_{2}=I_{2}=Q_{22},\n\\end{equation\nWhereas, for $\\alpha _{1}=\\alpha _{2}=\\alpha _{\\circ }$, for example\n\\begin{equation}\nQ_{12}=\\left[ \\tilde{q}_{1}\\tilde{q}_{2}+\\alpha _{\\circ }^{2}\\left(\nq_{1}+\\eta _{1}\\right) \\left( q_{2}+\\eta _{2}\\right) \\right] +i\\alpha\n_{\\circ }\\left[ \\left( q_{1}+\\eta _{1}\\right) \\tilde{q}_{2}-\\left(\nq_{2}+\\eta _{2}\\right) \\tilde{q}_{1}\\right] ,\n\\end{equation\nwhich identifies two real integrals of motion $I_{3}$ and $I_{4}$ such that \n\\begin{equation}\nI_{3}=\\func{Re}Q_{12}=\\tilde{q}_{1}\\tilde{q}_{2}+\\alpha _{\\circ }^{2}\\left(\nq_{1}+\\eta _{1}\\right) \\left( q_{2}+\\eta _{2}\\right) \\text{ \\ ; \\ \\ }I_{4}\n\\func{Im}Q_{12}=\\alpha _{\\circ }\\left[ \\left( q_{1}+\\eta _{1}\\right) \\tilde{\n}_{2}-\\left( q_{2}+\\eta _{2}\\right) \\tilde{q}_{1}\\right] .\n\\end{equation\nTherefore, our two-dimensional Lagrangian (54) admits \\emph\nsuperintegrability} in the generalized coordinates $\\left(\nq_{1},q_{2}\\right) $ and in the deformed\/rescaled time $\\tau $.\n\nNext, under the nonlocal transformation (20), along with the substitution\n\\begin{equation}\nq_{j}=x_{j}\\sqrt{m\\left( r\\right) }-\\eta _{j}\\Longrightarrow \\frac{dq_{j}}\ndx_{j}}=\\sqrt{m\\left( r\\right) }\\left[ 1+\\frac{m^{\\prime }\\left( r\\right) }\n2m\\left( r\\right) }\\left( \\frac{x_{j}^{2}}{r}\\right) \\right] \\Longrightarrow\nf\\left( r\\right) =1+\\frac{1}{4}\\frac{m^{^{\\prime }}\\left( r\\right) }{m\\left(\nr\\right) }r.\n\\end{equation\nThis result looks very much the same as that of $f\\left( r\\right) $ used in\n(33). Thus it would, again, with the assumption tha\n\\begin{equation*}\nf\\left( r\\right) =m\\left( r\\right) -\\frac{1}{4}\\frac{m^{^{\\prime }}\\left(\nr\\right) }{m\\left( r\\right) }r=1+\\frac{1}{4}\\frac{m^{^{\\prime }}\\left(\nr\\right) }{m\\left( r\\right) }r\\Longrightarrow m\\left( r\\right) =\\frac{1}\n1\\pm \\beta r^{2}},\n\\end{equation*\nlead to the \\emph{sub-superintegrable }Mathews-Lakshmanan type-I\nPDM-oscillator (36). The corresponding Euler-Lagrange equation (16) admits\nexact solutions as those in (37). Then, the \\emph{sub-superintegrability} of\nour Mathews-Lakshmanan type-I PDM-Lagrangian (36) turns out to be a\nconsequence of the \\emph{superintegrability} of the linear oscillators (22)\nand\/or the \\emph{superintegrability} of the shifted-oscillators (54).\nLikewise, the \\emph{sub-superintegrable} and exact solvable\nMathews-Lakshmanan type-I PDM-Lagrangian (36) may very well be nonlocally\ntransformed into two \\emph{superintegrable} Lagrangians, a \\emph\nsuperintegrable} linear oscillator (22) and a \\emph{superintegrable}\nshifted-oscillator (54). That is, the relation \n\\begin{equation}\n\\left\\{ \n\\begin{tabular}{c}\n\\emph{\\ Superintegrable}\\medskip \\\\ \n$L\\left( q_{1},q_{2},\\tilde{q}_{1},\\tilde{q}_{2};\\tau \\right) \\medskip $ \\\\ \nof (54), with\\medskip \\\\ \n$q_{j}\\left( \\tau \\right) =A_{j}\\cos \\left( \\alpha _{j}\\tau +\\varphi \\right)\n-\\eta _{j}\n\\end{tabular\n\\right\\} \\Longleftrightarrow \\left\\{ \n\\begin{tabular}{l}\n$\\text{Nonlocal transformation\\medskip }$ \\\\ \n$q_{j}=x_{j}\\sqrt{m\\left( r\\right) }-\\eta _{j}\\,;\\smallskip \\medskip\n\\,j=1,2, $ \\\\ \n$\\tilde{q}_{j}=\\dot{x}_{j}\\sqrt{m\\left( r\\right) }\\medskip $ \\\\ \n$f\\left( r\\right) =m\\left( r\\right) -\\frac{1}{4}\\frac{m^{^{\\prime }}\\left(\nr\\right) }{m\\left( r\\right) }r\\medskip $ \\\\ \n$m\\left( r\\right) =1\/\\left( 1\\pm \\beta r^{2}\\right) \n\\end{tabular\n\\right\\} \\Longleftrightarrow \\left\\{ \n\\begin{tabular}{c}\n\\emph{Sub-superintegrable}\\medskip \\\\ \n$L\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x}_{2};t\\right) \\medskip $ \\\\ \nof \\ (36), with\\medskip \\\\ \n$x_{j}\\left( t\\right) =A_{j}\\cos \\left( \\Omega t+\\varphi \\right) \\,\\medskip $\n\\\\ \n$\\text{ \\ }\\Omega ^{2}=\\frac{\\omega ^{2}}{1\\pm \\beta \\left(\nA_{1}^{2}+A_{2}^{2}\\right) }\\medskip \n\\end{tabular\n\\right\\} .\n\\end{equation\nwould describe the mapping from the \\emph{superintegrable} Lagrangians (54)\ninto the \\emph{sub-superintegrable} ones of (36).\n\n\\subsection{Superintegrable Isotonic Oscillator into sub-superintegrable\nPDM-deformed Isotonic oscillator}\n\nA \\emph{\"unit mass\"} particle moving in a two-dimensional isotonic\noscillator potential fiel\n\\begin{equation}\nV(q_{1},q_{2})=\\frac{1}{2}\\left( \\omega _{1}^{2}q_{1}^{2}+\\omega\n_{2}^{2}q_{2}^{2}+\\frac{\\beta _{1}}{q_{1}^{2}}+\\frac{\\beta _{2}}{q_{2}^{2}\n\\right) \\Longrightarrow V_{j}\\left( q_{j}\\right) =\\frac{1}{2}\\left( \\omega\n_{j}^{2}q_{j}^{2}+\\frac{\\beta _{j}}{q_{j}^{2}}\\right) ,\n\\end{equation\nwhere $\\,\\omega _{j}=n_{j}\\omega _{\\circ };\\,\\,j=1,2$, is described by the\nLagrangia\n\\begin{equation}\nL\\left( q_{1},q_{2},\\tilde{q}_{1},\\tilde{q}_{2};\\tau \\right) =\\frac{1}{2\n\\left( \\tilde{q}_{1}^{2}+\\tilde{q}_{2}^{2}\\right) -\\frac{1}{2}\\left( \\omega\n_{1}^{2}q_{1}^{2}+\\omega _{2}^{2}q_{2}^{2}+\\frac{\\beta _{1}}{q_{1}^{2}}\n\\frac{\\beta _{2}}{q_{2}^{2}}\\right) ,\n\\end{equation\nwhich is known to be the \\emph{superintegrable} Smorodisky-Winternitz type\nLagrangian (c.f., e.g., \\cite{29,38,40,42,47}). Its \\emph{superintegrability}\ncan be verified through the two complex substitution\n\\begin{equation}\nQ_{j}=\\left( \\tilde{q}_{i}^{2}-\\omega _{j}^{2}q_{j}^{2}+\\frac{\\beta _{j}}\nq_{j}^{2}}\\right) +2i\\omega _{j}q_{j}\\tilde{q}_{j}\\Longrightarrow \\frac{d}\nd\\tau }Q_{j}=2i\\omega _{j}Q_{j},\\smallskip \\medskip \\,\\,j=1,2,\n\\end{equation\nthat satisfy (25) with $Q_{jk}$ representing complex constants of motion and\nleads to more than two integrals of motion that manifest \\emph\nsuperintegrability}. \\ Moreover, the corresponding Euler-Lagrange equations\nof which read two Ermakov-Pinney's like equation\n\\begin{equation}\n\\frac{d}{d\\tau }\\tilde{q}_{j}=-\\omega _{j}^{2}q_{j}+\\frac{\\beta _{j}}\nq_{j}^{3}},\n\\end{equation\nwith the corresponding exact solution\n\\begin{equation}\nq_{j}=\\sqrt{\\frac{A_{j}}{\\omega _{j}}\\sin \\left( \\omega _{j}\\tau +\\delta\n_{j}\\right) }\\Longrightarrow \\beta _{j}=-A_{j}^{2};\\,j=1,2.\n\\end{equation}\n\nYet, under the nonlocal transformation setting in (38) our \\emph\nsuperintegrable} Lagrangian $L\\left( q_{1},q_{2},\\tilde{q}_{1},\\tilde{q\n_{2};\\tau \\right) $ in (64) nonlocaly transforms into a \\emph\nsub-superintegrable} Smorodisky-Winternitz like PDM-oscillators Lagrangia\n\\begin{equation}\nL\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x}_{2};t\\right) =\\frac{1}{2}\\left\\{ \n\\frac{\\dot{x}_{1}^{2}+\\dot{x}_{2}^{2}-\\left( \\omega _{1}^{2}x_{1}^{2}+\\omega\n_{2}^{2}x_{2}^{2}\\right) }{1\\pm \\lambda \\left( x_{1}^{2}+x_{2}^{2}\\right) }\n\\left[ 1\\pm \\lambda \\left( x_{1}^{2}+x_{2}^{2}\\right) \\right] \\left( \\frac\n\\beta _{1}}{x_{1}^{2}}+\\frac{\\beta _{2}}{x_{2}^{2}}\\right) \\right\\} .\n\\end{equation\nThe Euler-Lagrange equation (16) for which admits exact solutions of the for\n\\begin{equation}\nx_{j}=\\sqrt{\\frac{A_{j}}{\\Omega }\\sin \\left( \\Omega \\tau +\\delta _{j}\\right) \n};\\,j=1,2\\medskip \\medskip ,\n\\end{equation\nwher\n\\begin{equation}\n\\Omega ^{2}=\\left\\{ \n\\begin{tabular}{ll}\n$\\omega ^{2}+\\lambda ^{2}\\left( A_{1}+A_{2}\\right) ^{2}$ & $;$ \\medskip\\ \n\\omega _{1}=\\omega _{2}=\\omega ,\\beta _{1}\\neq \\beta _{2}$ \\\\ \n$\\frac{A_{1}\\omega _{1}^{2}+A_{2}\\omega _{2}^{2}+\\lambda ^{2}\\left(\nA_{1}+A_{2}\\right) ^{3}}{A_{1}+A_{2}}\\medskip $ & $;$ $\\ \\omega _{1}\\neq\n\\omega _{2},\\beta _{1}\\neq \\beta _{2}$ \\\\ \n$\\left( \\omega _{1}^{2}+\\omega _{2}^{2}\\right) \/\\left( 16A^{2}\\lambda\n^{2}\\right) \\medskip $ & $;$ \\ $\\omega _{1}\\neq \\omega _{2},\\beta _{1}=\\beta\n_{2}\\Leftrightarrow A_{1}=A_{2}=A\n\\end{tabular\n\\right. .\n\\end{equation}\n\n\\section{Concluding Remarks}\n\nIn this article, and in parallel with our recent methodical proposal in \\cit\n{38}, we have introduced the two-dimensional extension of the\none-dimensional PDM-Lagrangians and their nonlocal transformation mappings'\nrecipes into constant \\emph{unit-mass} exactly solvable Lagrangians. Hereby,\nthe two-dimensional nonlocal point transformations (10) and the related\nEuler-Lagrange equations invariance conditions (19) are reported. However,\ndealing with Lagrangians in more than one-dimension renders \\emph\nsuperintegrability} to be unavoidably in the process. We have, therefore,\nasserted that, if a set of \\emph{\"superintegrable\nreference\/target-Lagrangians\"} $L\\left( q_{1},q_{2},\\tilde{q}_{1},\\tilde{q\n_{2};\\tau \\right) $ is mapped into a set of \\emph{\"PDM\ntarget\/reference-Lagrangians\"} $L\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x\n_{2};t\\right) $, then the set of \\emph{\"PDM target\/reference-Lagrangians\"} \nL\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x}_{2};t\\right) $\\ is a set of \\emph\n\"sub-superintegrable PDM-Lagrangians\"} (as shall be so labelled\nhereinafter). Two sets of illustrative examples are used. The first set is\ndevoted to the mappings from \\emph{superintegrable} linear oscillators into \n\\emph{sub-superintegrable} nonlinear PDM-oscillators. Where, three\ntwo-dimensional \\emph{sub-superintegrable} Mathews-Lakshmanan type\nPDM-oscillators are used: (i) a \\emph{sub-superintegrable}\nMathews-Lakshmanan type-I PDM-oscillator for a PDM-particle moving in a\nharmonic oscillator potential, (ii) a \\emph{sub-superintegrable}\nMathews-Lakshmanan type-II PDM-oscillator for a PDM-particle moving in a\nconstant potential, and (iii) a \\emph{sub-superintegrable} shifted\nMathews-Lakshmanan type-III PDM-oscillator for a PDM-particle moving in a\nshifted harmonic oscillator potential. The second set, nevertheless,\nincludes some \\emph{superintegrable} shifted linear oscillators (new to the\nbest of our knowledge) and isotonic oscillators that are mapped into \\emph\nsub-superintegrable} PDM-nonlinear and \\emph{sub-superintegrable}\nPDM-isotonic oscillators, respectively. The mappings included exact\nsolvability as well. Our observations are in order.\n\nWhilst the two-dimensional PDM Mathews-Lakshmanan type-I and type-III, and\nthe PDM shifted nonlinear oscillators share the same total energ\n\\begin{equation}\nE_{tot}=\\frac{1}{2}\\omega ^{2}\\frac{\\left( A_{1}^{2}+A_{2}^{2}\\right) }{1\\pm\n\\beta \\left( A_{1}^{2}+A_{2}^{2}\\right) },\n\\end{equation\nthe two-dimensional PDM Mathews-Lakshmanan type-II oscillators admit total\nenerg\n\\begin{equation}\nE_{tot}=\\frac{\\omega ^{2}\\xi ^{2}}{1-\\xi ^{2}\\left(\nA_{1}^{2}+A_{2}^{2}\\right) },\n\\end{equation\nand the two-dimensional PDM-deformed isotonic oscillators indulge total\nenerg\n\\begin{equation}\nE_{tot}=\\frac{1}{2}\\left\\{ \\frac{\\omega _{1}^{2}A_{1}+\\omega _{2}^{2}A_{2}}\n\\Omega \\pm \\lambda \\left( A_{1}+A_{2}\\right) }-\\left[ \\Omega \\pm \\lambda\n\\left( A_{1}+A_{2}\\right) \\right] \\left( A_{1}+A_{2}\\right) \\right\\} .\n\\end{equation\nYet, the two-dimensional PDM Mathews-Lakshmanan type-I and type-III inherit\nthe dynamical properties and trajectories of each other. On the other hand,\nthe \\emph{sub-superintegrability} of the PDM Mathews-Lakshmanan type-I\noscillators (36) may very well be attributed to the \\emph{superintegrability}\nof the linear oscillator (22) and\/or the shifted-oscillators (54).\n\nFinally, the generalization of the current methodical proposal into more\nthan two-dimensional recipes looks eminent and feasible. This is very\nobviously documented in the description of our equations (6) to (12), where \nj=1,2$ is used. Strictly speaking, for a three-dimensional case \nj=1,2\\rightarrow $ $j=1,2,3$ and $\\bar{x}=x_{1},x_{2}\\rightarrow \\bar{x\n=x_{1},x_{2},x_{3}$ in (10) and so on so forth. It would be interesting to\nstudy and explore the consequences of such generalization.\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:auto-hsp}\nThe main result of this paper is the Theorem ~\\ref{theo:cycle-graph-auto-weak-fails} and the Corollary ~\\ref{cor:cycle-graph-auto-strong-fails} where it is shown that both the weak and strong quantum Fourier samplings are guaranteed to always fail for the classically trivial cycle graph automorphism problem. It is also shown how to systematically determine the non-trivial classes of graphs for which quantum Fourier transform (QFT) always fails to construct the automorphism groups. Here, the term `non-trivial' refers to the classes of graph automorphism problems which can be solved trivially on a classical computer. This result puts an end to the decades long effort of finding a hidden subgroup algorithm for the graph isomorphism problem. Previously, the researchers have been proving increasing negative results which indicated that the probability of successfully deciding a graph isomorphism problem is exponentially small. This paper gives an algorithm to create arbitrarily large classes of graph automorphism problems for which the quantum hidden subgroup approach will always fail to compute the automorphism group no matter how big the computer or how small the size of the problem is. As the graph automorphism problem is Karp-reducible \\cite{kobler2012graph} to the graph isomorphism problem, it can be inferred that there are classes of graph isomorphism problem for which hidden subgroup approach will always fail irrespective of the size of the problem. The linear representation theory of the dihedral groups plays a very important role in proving this result.\n\nThe general framework for the hidden subgroup problems was first formulated in \\cite{brassard1997exact, hoyer2000quantum, mosca1999hidden}. The hidden subgroup problem can be defined as follows \\footnote{This pedagogically convenient version of HSP has been borrowed from the presentation titled 'Graph isomorphism, the hidden subgroup problem and identifying quantum states' by Pranab Sen.}.\n\n\\begin{mydef}[Hidden subgroup problem]\n\n{\\bf Given:} $G:$ group, $S:$ set, $f : G \\to S$ via an oracle.\\\\\n{\\bf Promise:} Subgroup $H \\le G$ such that $f$ is constant on the left\ncosets of $H$ and distinct on different cosets.\\\\\n{\\bf Task:} Find the hidden subgroup $H$ by querying $f$.\n\\end{mydef}\n\nThe hidden subgroup version of the graph isomorphism problem was first defined in \\cite{jozsa1998quantum}.The $n$-vertex graph isomorphism problem for rigid graphs of  $n$ vertices can be expressed as a hidden subgroup problem over the ambient symmetric group $S_{2n}$ or more specifically the wreath product $S_n \\wr \\mathbb{Z}_2$ where the hidden subgroup is promised to be either trivial or of order two \\cite{moore2010impossibility}.  The scheme and notation of the following definitions of the hidden subgroup problem, used in this paper, are borrowed from \\cite{lomonacopers2002, van2012quantum}. Erd{\\H{o}}s et al \\cite{erdHos1963asymmetric} have shown that the automorphism groups of the most of the graphs are trivial.  So, although, the problem was defined for all simple undirected graphs in \\cite{jozsa1998quantum}, this paper follows the example of \\cite{grigni2001quantum} and limit the discussion to the rigid graphs with trivial automorphism groups. The rigidity of the graphs in this definition will be temporarily relaxed in Section ~\\ref{sec:iso-auto} to prove reducibility.\n\n\\begin{mydef} [Graph isomorphism as a hidden subgroup problem ($\\text{{\\bf GI}}_{\\text{HSP}}$)]\nLet the $2 n$ vertex graph $\\Gamma = \\Gamma_1 \\sqcup \\Gamma_2$ be the disjoint union of the two rigid graphs $\\Gamma_1$ and $\\Gamma_2$ such that $Aut \\left(\\Gamma_1\\right) = Aut \\left(\\Gamma_2\\right) = \\left\\{e\\right\\}$. A map $\\varphi : S_{2n} \\to \\text{Mat}\\left(\\mathbb{C}, N \\right)$ \\footnote{$\\text{Mat}\\left(\\mathbb{C}, N \\right)$ is the algebra of all $N \\times N$ matrices over the complex numbers $\\mathbb{C}$.} from the group $S_{2n}$ is said to have hidden subgroup structure if there exists a subgroup $H_\\varphi$ of $S_{2n}$, called a hidden subgroup, an injection $\\ell_\\varphi : S_{2n}\/H \\to \\text{Mat}\\left(\\mathbb{C}, N \\right)$, called a hidden injection, such that the diagram\n\\[\n\\begin{tikzcd}\nS_{2n} \\arrow{r}{\\varphi} \\arrow{d}{\\nu} & \\text{Mat}\\left(\\mathbb{C}, N \\right)\\\\\nS_{2n}\/H \\arrow{ur}{\\ell_\\varphi}&\n\\end{tikzcd}\n\\]\nis a commutative diagram, where $S_{2n}\/H_{\\varphi}$ denotes the collection of right cosets of $H_\\varphi$ in $S_{2n}$, and where $\\nu : S_{2n}\/H_\\varphi$ is the natural map of $S_{2n}$ onto $S_{2n}\/H_\\varphi$. The group $S_{2n}$ is called the ambient group and  the set $\\text{Mat}\\left(\\mathbb{C}, N \\right)$ is called the target set.\n\nThe hidden subgroup version of the graph isomorphism problem is to determine a hidden subgroup $H$ of $S_{2n}$ with the promise that $H$ is either trivial or $|H| = 2$.\n\\end{mydef}\n\nThis section also gives a formal definition for the hidden subgroup representation of the graph automorphism problem.\n\n\\begin{mydef} [Graph automorphism as a hidden subgroup problem ($\\text{{\\bf GA}}_{\\text{HSP}}$)]\nFor a graph $\\Gamma$ with $n$ vertices, a map $\\varphi : S_{n} \\to \\text{Mat}\\left(\\mathbb{C}, N \\right)$ \\footnote{$\\text{Mat}\\left(\\mathbb{C}, N \\right)$ is the algebra of all $N \\times N$ matrices over the complex numbers $\\mathbb{C}$.} from the group $S_{n}$ is said to have hidden subgroup structure if there exists a subgroup $\\text{Aut}\\left(\\Gamma\\right)$ of $S_{n}$, called a hidden subgroup, an injection $\\ell_\\varphi : S_{n}\/\\text{Aut}\\left(\\Gamma\\right) \\to \\text{Mat}\\left(\\mathbb{C}, N \\right)$, called a hidden injection, such that for each $g \\in \\text{Aut}\\left(\\Gamma\\right)$, $g \\left(\\Gamma\\right) = \\Gamma$ and, the diagram\n\\[\n\\begin{tikzcd}\nS_{n} \\arrow{r}{\\varphi} \\arrow{d}{\\nu} & \\text{Mat}\\left(\\mathbb{C}, N \\right)\\\\\nS_{n}\/\\text{Aut}\\left(\\Gamma\\right) \\arrow{ur}{\\ell_\\varphi}&\n\\end{tikzcd}\n\\]\nis commutative, where $S_{n}\/\\text{Aut}\\left(\\Gamma\\right)$ denotes the collection of right cosets of $\\text{Aut}\\left(\\Gamma\\right)$ in $S_{n}$, and where $\\nu : S_{n}\/\\text{Aut}\\left(\\Gamma\\right)$ is the natural map of $S_{n}$ onto $S_{n}\/\\text{Aut}\\left(\\Gamma\\right)$.  $S_{n}$ is called  the ambient group and  $\\text{Mat}\\left(\\mathbb{C}, N \\right)$ is called  the target set.\n\nThe hidden subgroup version of the graph automorphism problem is to determine a hidden subgroup $\\text{Aut}\\left(\\Gamma\\right)$ of $S_{n}$ with the promise that $\\text{Aut}\\left(\\Gamma\\right)$ is either of trivial or non-trivial order depending on the type of $\\Gamma$.\n\\end{mydef}\n\n\\subsection{Outline of the paper}\nFor the convenience of the readers, a brief outline of the paper is given here. The Section ~\\ref{sec:Historical-context} gives an overview of the related literature, Section  ~\\ref{sec:prem} provides the preliminary background needed to follow the discussion used in this paper, Section ~\\ref{sec:qft-gi} provides the already known results on the hidden subgroup approach for graph isomorphism, Section ~\\ref{sec:weak-cycle} presents the original result that the hidden subgroup approach is guaranteed to fail for an easy class of graph automorphism problem, and Section ~\\ref{sec:qft-ga-plus} presents another original result which is a systematic way to build arbitrarily  large classes of graph automorphism problems for which hidden subgroup algorithms are guaranteed to fail. Finally, it has been discussed whether the hidden subgroup algorithms are the most appropriate ways to attempt combinatorial problems in quantum computation.\n\n\\subsection{Key technical ideas}\nThis section summarizes the key technical ideas used to prove the main results of this paper.  The automorphism group of a cycle graph is the dihedral group $D_n$ of order $2 n$. It has been shown in  Lemma ~\\ref{lem:d-n-1-d-rep-prob-zero} that the probability of measuring the labels of one dimensional irreducible representations of $D_n$ is zero for non-trivial representations. Then, in Lemma ~\\ref{lem:d-n-2-d-rep-prob-zero},  it has been shown that the probability of measuring the labels of two dimensional irreducible representations of $D_n$ is always zero. Combining these two lemmas, it has been proved in Theorem ~\\ref{theo:cycle-graph-auto-weak-fails} that Weak quantum Fourier sampling always fails to solve the cycle graph automorphism problem irrespective of its size. Finally, this paper gives Algorithm ~\\ref{algo:arb-fail} to create arbitrarily large class of graph automorphism problems (with rotational symmetries) for which quantum Fourier transform is guaranteed to always fail irrespective of the size of the graphs.\n\nIt was already proved in \\cite{grigni2001quantum} that single coset state can provide only exponentially less information for the graph isomorphism problem. Hence, later works, for example  \\cite{hallgren2010limitations, moore2010impossibility}, investigated the possibility of using multiple copies of coset states. The current work gives a stronger result on the graph automorphism problem for the single coset state. Moreover, all the multi-coset state algorithms are conditioned on the successful execution of weak sampling, which has been proved in this paper to fail with guarantee for the problem of interest. So, the case of failure can also be inferred for multi-coset approaches, e.g., the sieve algorithms. To summarize, if there are rotational symmetries in the graph, we will not get exponentially less information with a single copy coset state rather, but, even worse, we will get exactly zero amount of information no matter how large the quantum computer is. Same would be true for k-copy coset states.\n\n\\section{Historical context}\n\\label{sec:Historical-context}\nRead et al. \\cite{Read1977} have named the tendency of incessant but unsuccessful attempts at the graph isomorphism problem as the {\\it graph isomorphism disease}. This indicates the amount of interest about the problem among the researchers. For almost three decades, until $2015$, the  best known algorithm for the general graph isomorphism problem has been due to Babai et al. \\cite{babai1983canonical}. The algorithm exploits graph canonization techniques through label reordering in exponential time (\\ensuremath{exp \\left(n^{\\frac{1}{2} + o\\left(1\\right)}\\right)}), where \\ensuremath{n = |V|}. Faster algorithms have been proposed for graph sub classes with special properties. In \\cite{babai1983canonical}, Babai et al. also proved the bound for tournament graphs is \\ensuremath{n^{\\left(\\frac{1}{2} + o\\left(1\\right)\\right) \\log n}}. In \\cite{luks1982isomorphism}, Luks reduced the bounded valence graph isomorphism problem to the color automorphism problem, and gave a polynomial time algorithm. In another paper \\cite{babai1982isomorphism}, Babai et al. created two polynomial algorithms using two different approaches, i.e., the tower of groups method, and the recursion through systems of imprimitivity respectively, for the bounded eigenvalue multiplicity graph isomorphism problem. The isomorphism problem for planar graphs is known to be in polynomial time due to Hopcroft et al. \\cite{hopcroft1974linear}. In their paper, the authors used a reduction approach to eventually tranform the graphs into five regular polyhedral graphs and check the isomorphism by exhaustive matching in a fixed finite time. Miller \\cite{miller1980isomorphism} used a different approach by finding minimal embeddings of the graphs of bounded genus and checking their isomorphism by generating codes. Babai et al. in \\cite{babai1980random} and Czajka et al. in \\cite{czajka2008improved} showed that the isomorphism of almost all the graphs in a class of random graphs can be tested in linear time. Both of their approaches exploit the properties of the degree sequence of a random graph. Babai et al. \\cite{babai2013faster} proved that while the graph isomorphism problem for strongly regularly graphs may be solved faster than the general version it is still an exponential time algorithm. A series of dramatic events took place recently between $2015$ and $2017$ in the field of graph isomorphism. In December, $2015$ \\cite{babai2015graph}, Babai posted a pre-print claiming that the general graph isomorphism problem can be solved in quasipolynomial time. One of the authors of this papers was fortunate enough to witness a live proof session of the algorithm by Babai in Discrete Mathematics $2016$. Two years later, Helfgott \\cite{helfgott2017isomorphismes} pointed out a serious error in that proof. Babai immediately fixed the proof and graph isomorphism still remains in quasipolynomial time.\n\nAlthough there is a quasi-polynomial time algorithm for the general graph isomorphism problem, it is not proven to be optimal. So, the complexity class of the graph isomorphism problem is yet undecided. While it is known that the problem is in {\\bf NP} \\cite{garey2002computers}, it is not known whether the problem is in {\\bf P} or {\\bf NP}-complete. This is why the graph isomorphism problem is called an {\\bf NP}-intermediate problem. Sch{\\\"o}ning \\cite{schoning1988graph} has shown that graph isomorphism is in \\ensuremath{L^P_2} and not \\ensuremath{\\gamma}-complete under the assumption that the polynomial hierarchy does not collapse to \\ensuremath{L^P_2}. Given this information, many researchers believe that the graph isomorphism problem is not {\\bf NP}-complete.\n\nWhile the efforts towards finding an efficient solution for the general graph isomorphism problems have been unsuccessful, the researchers have attempted practically feasible methods to solve the problem in reasonable time frame.\n\nThe hidden subgroup approach for both the graph isomorphism and automorphism problems require the computing of the quantum Fourier sampling of the ambient symmetric group. This has been an active area of research since Peter Shor invented the famous Shor's algorithm, a quantum hidden subgroup algorithm for the abelian groups,  to solve prime factorization \\cite{shor1999polynomial}. While at this moment, there is no known efficient quantum hidden subgroup algorithm for symmetric groups, researchers have shed some light on why it had been so difficult to find them.\n\nWhile surveys like \\cite{RevModPhys.82.1}, summarizes the advances made so far in the area of hidden subgroup algorithms, it would always be helpful to review the negative results in this section to illustrate why this is a difficult problem. It is noteworthy that all the positive results, so far, have been demonstrated for the synthetically created product groups. While this approach may not have immediate practical application, this idea of creating synthetic groups has been used in this paper to generalize results. One of the first results for the non-abelian hidden subgroup problems was presented by Roetteler et al \\cite{roetteler1998polynomial}. In that paper, the authors proved an efficient hidden subgroup algorithm for the wreath product $\\mathbb{Z}^k_2 \\wr \\mathbb{Z}_2$ which is a non-abelian group. Similarly, Ivanyos et al \\cite{ivanyos2003efficient} proved the existence of an efficient hidden subgroup algorithm for a more general non-abelian nil-$2$ groups. Later Friedl et al  \\cite{friedl2003hidden} generalized the result such that there are efficient hidden subgroup algorithms for the groups whose\nderived series have constant length and whose Abelian factor groups are each the\ndirect product of an Abelian group of bounded exponent and one of polynomial\nsize. Ettinger et al \\cite{ettinger2000quantum} showed that it is possible to reconstruct a subgroup hidden inside the dihedral group using finite number of queries. This result was later generalized by Ettinger et al \\cite{ettinger1999hidden} that arbitrary groups may be reconstructed using finite queries but they did not give any specific set of measurement.\n\nIn \\cite{moore2002hidden}, Moore et al proved that although weak quantum Fourier sampling fails to determine the hidden subgroups of the non-abelian groups of the form $\\mathbb{Z}_q \\ltimes \\mathbb{Z}_p$, where $q \\mid \\left(p-1\\right)$ and $q = p \/ \\text{polylog}\\left(p\\right)$, strong Fourier sampling is able to do that. Later on, Moore et al  \\cite{moore2006generic, moore2005explicit}  proved the existence of $polylog \\left(|G|\\right)$ sized quantum Fourier circuits for the groups like $S_n$, $H \\wr S_n$, where $|H| = \\text{poly}\\left(n\\right)$, and the Clifford groups. The authors also gave the circuits of subexponential size for standard groups like $\\text{GL}_n \\left(q\\right)$, $\\text{SL}_n \\left(q\\right)$, $\\text{PGL}_n \\left(q\\right)$, and $\\text{PSL}_n \\left(q\\right)$, where $q$ is a fixed prime power. Moore et al  \\cite{moore2008symmetric} have also presented a stronger result where they have shown that it is not possible to reconstruct a subgroup hidden inside the symmetric group with strong Fourier sampling and both arbitrary POVM and entangled measurement. At the same time, the authors did not rule out the possibility of success using other possible measurements which is still an open question. Bacon et al \\cite{bacon2005optimaldi} proved that the so called {\\it pretty good measurement} is optimal for the dihedral hidden subgroup problem. Moore et al  \\cite{moore2005distinguishing} extended this result for the case where the hidden subgroup is a uniformly random conjugate of a given subgroup. Moore et al \\cite{moore2007power} eventually proved a more general results that strong quantum Fourier sampling can reconstruct $q$-hedral groups. Alagic et al \\cite{alagic2005strong} proved a general result that strong Fourier sampling fails to distinguish the subgroup of the power of a given non-abelian simple group. Moore et al \\cite{moore2005tight} later proved that arbitrary entangled measurement on $\\Omega \\left(n \\log n\\right)$ coset states is necessary and sufficient to extract non-negligible information. Similar result was also proved in \\cite{hallgren2010limitations} separately. Few years later, Moore et al \\cite{moore2010impossibility} proved a negative result that the {\\it quantum sieve algorithm}, i.e. highly entangled measurements across $\\Omega(n \\log n)$ coset states, cannot solve the graph isomorphism problem.\n\nIt is important to point out that all the groups used in the previously mentioned results are conveniently chosen and synthetically created. Moreover, they are sporadic so it is not clear how the knowledge can be extrapolated to the symmetric groups. As the graph automorphism problem is Karp-reducible to the graph isomorphism problem, it is believed to be sufficient to investigate the hidden subgroup representation of the graph isomorphism problem. With all these unsuccessful attempts for the last couple of decades presented above, one may ask whether the hidden subgroup approach is the right way to attempt the graph isomorphism problem. If it is, there would have been a Karp-reduction from the hidden subgroup representation of the graph automorphism problem to the hidden subgroup representation of the graph isomorphism problem. This paper gives one such reduction in Section ~\\ref{sec:iso-auto}. So, another way of looking at the problem is to understand the hidden subgroup  complexity of the graph automorphism problem and reduce the results to graph isomorphism.\n\n\\section{Preliminaries}\n\\label{sec:prem}\n\\subsection{Graph Theory}\nMost of the work presented in this paper involves the graph isomorphism and automorphism problems. So, it would be appropriate to start the background section with a few concepts of graph theory.  The materials in this section are reproduced from the very well written book by Bollob\\'{a}s \\cite{bollobas2013modern}. The section does not contain a comprehensive coverage on graph theory, rather they are only related to the discussion of this paper.\n\n\\begin{mydef} [Graph]\nA graph $\\Gamma$ is an ordered pair of disjoint sets $\\left(V, E\\right)$ such that $E$ is a subset of the set $V^{\\left(2\\right)}$ of unordered pairs of $V$.\n\\end{mydef}\n\n$V$ is known as the set of vertices, and $E$ is known as the set of edges. Each element of $E$ connects two elements of $V$. A graph is directed if the edge $\\left(v_i, v_j\\right)$ is an element of $E$ but $\\left(v_j, v_i\\right)$ is not for all $i$ and $j$. A simple graph does not have loops or multi-edges. This paper only focuses on questions defined on simple undirected graphs.\n\n\\subsubsection{Graph isomorphism and automorphism}\n\\label{sec:iso-auto}\nThe graph isomorphism and automorphism problems are the two of the oldest problems in combinatorics. The formal statement of the graph isomorphism problem goes as follows as mentioned in \\cite{fortin1996graph}. \n\n\\begin{mydef} [Graph isomorphism ({\\bf GI})]\n\\label{prob:gi}\nGiven two graphs, \\ensuremath{\\Gamma_1 = \\left(V_1, E_1\\right)} and \\ensuremath{\\Gamma_2 = \\left(V_2, E_2\\right)},  does there exist a bijection \\ensuremath{f : V_1 \\to V_2} such that \\ensuremath{\\forall a, b \\in V_1, \\left(a, b\\right) \\in E_1 \\iff \\left(f\\left(a\\right), f\\left(b\\right)\\right) \\in E_2}?\n\\end{mydef}\nHere, \\ensuremath{V_1} and \\ensuremath{V_2} are the sets of vertices and \\ensuremath{E_1} and \\ensuremath{E_2} are the sets of edges of \\ensuremath{\\Gamma_1} and \\ensuremath{\\Gamma_2} respectively.\n\nThe graph automorphism problem is a special version of Definition \\ref{prob:gi} when $\\Gamma_1 = \\Gamma_2$.  \n\n\\begin{mydef} [Graph automorphism ({\\bf GA})]\n\\label{prob:ga}\nGiven a graph $\\Gamma = \\left(V, E\\right)$, compute the automorphism groups which are $\\Gamma \\to \\Gamma$ isomorphisms; and form the subgroup $\\text{Aut}\\left(\\Gamma\\right)$ of the symmetric group $S_{|V|}$.\n\\end{mydef}\n\nThe reducibility from {\\bf GA} to {\\bf GI} is discussed in the rest of this section based on a few theorems proven in \\cite{kobler2012graph}.  The outline is as follows. First, it has been shown that {\\bf GA} is Turing-reducible to {\\bf GI} i.e. $\\text{\\bf GA} \\le^p_T \\text{\\bf GI}$ using Algorithm ~\\ref{algo:ga-gi-turing-reduction} which is the Example 1.10 of \\cite{kobler2012graph}. \n\n\\begin{algorithm}[H]\n\\caption{$GA \\le^p_T GI$}\n     \\label{algo:ga-gi-turing-reduction}\n\\begin{algorithmic}[1]\n\\Procedure {GA-GI-Turing-Reduction}{$\\Gamma, n$} \\Comment{graph $\\Gamma$ with $n$ nodes}\n\n\\For{$i\\gets 1, n-1 $}\n\\For{$j\\gets i+1, n $}\n\\If{$\\left(\\Gamma_{[i]}, \\Gamma_{[j]}\\right) \\in GI$} \\Comment{$\\Gamma_{[i]}$ denotes a copy of the graph $\\Gamma$ with a label attached with node $i$}\n\\State {\\bf accept}\n\\EndIf\n\\EndFor\n\\EndFor\n\\State {\\bf reject};\n\\EndProcedure\n\\end{algorithmic}\n\\end{algorithm}\n\nThen, it has been proven that {\\bf GA} is Karp-reducible to {\\bf GI}. Instead of reproducing the detailed proof from \\cite{kobler2012graph}, this section provides a sketch of it. First, it needs to be shown that {\\bf GA} has a polynomial time computable {\\it or}-function. Then, it has to be shown that {\\bf GI} has both polynomial time computable {\\it and-} and {\\it or}-functions. Combining these results, it can be argued that {\\bf GA} is Karp-reducible to {\\bf GI} i.e. $\\text{\\bf GA} \\le^p_m \\text{\\bf GI}$.\n\nAt this point, it is natural to ask whether $\\text{\\bf GA}_{HSP} \\le^p_T \\text{\\bf GI}_{HSP}$ or ${\\bf GA}_{HSP} \\le^p_m {\\bf GI}_{HSP}$.\n\nIt can be trivially shown that the hidden subgroup representation of the graph automorphism problem is Turing-reducible to the graph isomorphism problem by giving two input graphs as the original graph and the candidate automorphism of the original graph. The technique to prove the Karp-reducibility from $\\text{\\bf GA}_{HSP}$ to $\\text{\\bf GI}_{HSP}$ was kindly shown to the authors in a public forum by Grochow \\cite{Grochow2017}. The sketch of the algorithm is given below.\n\nFirst, the condition on the the rigidity of the input graphs is relaxed. This makes the case harder.  Instead, the (non-rigid) {\\bf GI} as an HSP is described in the same way, but now the goal is to determine the size of the hidden subgroup, or a generating set. The difference between the isomorphic and non-isomorphic cases will be a factor of $2$ in the order of the hidden subgroup. If  the problem is expressed as finding generators of the hidden subgroup, then the question is whether any generator switches $\\Gamma_1$ and $\\Gamma_2$.\n\nNow, an instance of $\\text{\\bf GA}_{HSP}$ corresponding to a graph $\\Gamma$ is given by the function from $S_n \\to M_n(\\mathbb{C})$ defined by $f(\\pi) = A(\\pi(\\Gamma))$ where $A(\\cdot)$ denotes the adjacency matrix. In particular, $f(e) = A(\\Gamma)$. Then the usual Karp reduction is applied from {\\bf GA} to {\\bf GI} to get a pair of graphs $\\Gamma_1, \\Gamma_2$. Then an instance of $\\text{\\bf GI}_{HSP}$ instance, of the type described in the preceding paragraph, can be created corresponding to the pair $\\Gamma_1, \\Gamma_2$ (that is, the disjoint union $\\Gamma_1 \\cup \\Gamma_2$). Thus ${\\bf GA}_{HSP} \\le^p_m {\\bf GI}_{HSP}$.\n\n\\subsection{Representation Theory}\nA few concepts of the representation theory are discussed in the current section which are relevant to this paper. The discussion is limited to the representation theory of symmetric groups. A more detailed introduction may be found in \\cite{fulton1991representation, curtis1966representation, sagan2013symmetric}. The paper has borrowed the notations and definitions of representation theory generously from the above mentioned standard resources.\n\n\\begin{mydef} [Matrix representations \\cite{sagan2013symmetric}]\nA matrix representation of a group $G$ is a group homomorphism\n\\begin{align}\nX : G \\to GL_d.\n\\end{align}\nEquivalently, to each $g \\in G$ is assigned $X \\left(g\\right) \\in \\text{Mat}_d$ such that\n\\begin{itemize}\n\\item $X \\left(e\\right) = I$ the identity matrix, and\n\\item $X \\left(g h\\right) = X \\left(g\\right) X \\left(h\\right)$ for all $g, h \\in G$.\n\\end{itemize}\nThe parameter $d$ is called the degree, or dimension, of the representation and is denoted by $\\text{deg } X$.\n\\end{mydef}\n\n\\begin{mydef} [Young diagram]\nFor any partition $\\lambda_1, \\ldots, \\lambda_k$ of an integer $\\lambda$, there is a diagram associated called the Young diagram where there are $\\lambda_i$ cells in the $i$-th row. The cells are lined up on the left.\n\\end{mydef}\n\n\\begin{mydef} [Restricted and induced representations]\nIf $H \\subset G$ is a subgroup, any representation $\\rho_1$ of $G$ restricts to a representation of $H$, denoted $Res^G_H \\rho_1$ or simple $Res \\rho_1$. Let $\\rho_2 \\subset \\rho_1$ be a subspace which is $H$-invariant. For any $g$ in $G$, the subspace $g . \\rho_2 = \\left\\{g . w : w \\in \\rho_2\\right\\}$ depends only on the left coset of $g H$ of $g$ modulo $H$, since $g h . W = g . \\left(h . \\rho_2\\right) = g. \\rho_2$; for a coset $c$ in $G\/H$,  $c . \\rho_2$ is the subspace of $\\rho_1$ subspace of $\\rho_1$. $\\rho_1$ is induced by $\\rho_2$ if every element in $\\rho_1$ can be written uniquely as a sum of elements in such translates of $\\rho_2$, i.e. \n\\begin{align}\n\\rho_1 &= \\bigoplus_{c \\in G\/H} c . \\rho_2\n\\end{align}\nIn this case, the induced representation is $\\rho_1 = Ind^G_H \\rho_2$  = Ind $\\rho_2$.\n\\end{mydef}\n\nA common representation to be seen in later sections of this report is the regular representation \\cite{fulton1991representation}.\n\n\\begin{mydef} [Regular representation]\nIf $X$ is any finite set and $G$ acts on the left on $X$, i.e., $G \\to Aut \\left(X\\right)$ is a homomorphism to the permutation group of $X$, there is a associated permutation representation: let $V$ be the vector space with basis $\\left\\{e_x: x \\in X\\right\\}$, and let $G$ act on $V$ by\n\\begin{align}\ng \\cdot \\sum a_x e_x = \\sum a_x e_{g x}.\n\\end{align}\nThe regular representation, denoted $R_G$ or $R$, corresponds to the left action of $G$ on itself.\n\\end{mydef}\n\nThe character of a group element is defined as follows \\cite{fulton1991representation}.\n\n\\begin{mydef} [Character]\nIf $\\rho$ is a representation of a group $G$, its character $\\chi_\\rho$ is the complex-valued function on the group defined by \n\\begin{align}\n\\chi_\\rho \\left(g\\right) &= Tr \\left(g|_\\rho\\right),\n\\end{align}\nthe trace of $g$ on $\\rho$.\n\\end{mydef}\n\nIt is also useful to define the inner product of characters \\cite{sagan2013symmetric}.\n\n\\begin{mydef}[Inner product of characters]\n\\label{def:inner-prod-char}\nLet $\\chi$ and $\\psi$ be the characters of a group $G$. The {\\it inner product} of $\\chi$ and $\\Psi$ is\n\\begin{align}\n\\langle \\chi, \\Psi\\rangle &= \\frac{1}{|G|} \\sum_{g\\in G} \\chi \\left(g\\right) \\Psi^\\dagger \\left(g\\right)\n\\end{align}\n\\end{mydef}\n\n\n\nThe character table of a finite group  is defined as follows \\cite{sagan2013symmetric}.\n\n\\begin{mydef} [Character table]\nLet $G$ be a group. The {\\it character table} of $G$ is an array with rows indexed by the inequivalent irreducible characters of $G$  and columns indexed by the conjugacy classes. The table entry in row $\\chi$ and column $K$ is $\\chi_K$:\n\n\\begin{tabular}{ c | c c c }\n   & \\ldots & $K$ & \\ldots \\\\\n  \\hline\n  \\vdots &   & \\vdots & \\\\\n$\\chi$ & \\ldots   & $\\chi_K$ & \\\\\n\\vdots &    &  & \\\\\n\\end{tabular}\n\nBy convention, the first row corresponds to the trivial character, and the first column corresponds to the class of the identity, $K = \\left\\{e\\right\\}$.\n\\end{mydef}\n\nTwo equivalent procedures are provided for computing the character table of any symmetric group $S_n$ improvising from \\cite{Gillespie2012}.\n\nThe most straight forward way \\cite{sagan2013symmetric} to compute the character table is given in Algorithm ~\\ref{algo:char-tab-sagan}.\n\n\\begin{algorithm}[H]\n\\caption{{\\bf CHARACTER-TABLE-SAGAN}}\n     \\label{algo:char-tab-sagan}\n\\begin{algorithmic}[1]\n\\Procedure {CHARACTER-TABLE-SAGAN}{$S_n$}\n\\State Determine all the partitions of $n$ which will also infer the conjugacy classes.\n\n\\State Enumerate all group elements and cluster them based on their cycle types. These clusters will coincide with the conjugacy classes.\n\n\\State For each class, compute the irreducible representation for each group element.\n\n\\State For each class, determine the character of the irreducible representation. All group elements of the same cycle type will have the same character.\n\n\\State Populate the table with the characters following the prescribed order of the partitions for both column and rows.\n\\EndProcedure\n\\end{algorithmic}\n\\end{algorithm}\n\nEnumeration of conjugacy classes becomes tedious when  groups larger than $S_5$ are being considered. By using the {\\it Murnaghan-Nakayama rule} , The process can be simplified even for larger groups \\cite{stanley1986enumerative} as shown by \\cite{Gillespie2012} in Algorithm ~\\ref{algo:char-tab-mn-rule}.\n\n\\begin{algorithm}[H]\n\\caption{{\\bf CHARACTER-TABLE-GILLESPIE}}\n     \\label{algo:char-tab-mn-rule}\n\\begin{algorithmic}[1]\n\\Procedure {CHARACTER-TABLE-GILLESPIE}{$S_n$}\n\\State   The  conjugacy classes of $S_n$ are the permutations having a fixed number of cycles of each length, corresponding to a partition of $n$ called the shape of the permutation. Since, the characters of a group are constant on its conjugacy classes,  index the columns of the character table by these partitions. The partitions are arranged in an increasing order.\n\n\\State There are precisely as many irreducible characters as conjugacy classes, so the irreducible characters can be indexed by the partitions of $n$. Represent each partition as a Young diagram and write them, or the characters directly down, the left of the table in a decreasing order of the partitions.\n\n\\Comment{The Murnaghan-Nakayama Rule}\n\\State  Calculate the entry in row $\\lambda$  and column $\\mu$. Define a filling of $\\lambda$ with content $\\mu$ to be a way of writing a number in each square of $\\lambda$ such that the numbers are weakly increasing along each row and column and there are exactly $\\mu_i$ squares labeled $i$ for each $i$.\n\n\\State Consider all fillings of $\\lambda$ with content $\\mu$ such that for each label $i$, the squares labeled $i$ form a connected skew tableaux that does not contain a $2 \\times 2$ square. Such a tableaux is called a {\\it border-strip tableaux}.\n\n\\State For each label in the tableau, define the height of the corresponding border strip to be one less than the number of rows of the border strip. Weight the tableau by $\\left(-1\\right)^s$ where $s$ is the sum of the heights of the border strips that compose the tableau.\n\n\\State The entry in the character table is simply the sum of these weights.\n\\EndProcedure\n\\end{algorithmic}\n\\end{algorithm}\n\nNow, the concepts of hook is introduced which is used in computing characters of group representations.\n\n\\begin{mydef} [Hook]\nFor a cell $\\left(i, j\\right)$ of a Young tableau $\\lambda$, the $\\left(i, j\\right)$-hook $h_{i, j}$ is the collection of all cells of $\\lambda$ which are beneath $\\left(i, j\\right)$ (but in the same column) or t the right of $\\left(i, j\\right)$ (but in the same row), including the cell $\\left(i, j\\right)$. The {\\it length} of the hook is the number of cells appearing in the hook.\n\\end{mydef}\n\n\\begin{mydef} [Skew hook]\nA skew hook $s$ of a Young diagram $\\lambda$ is a connection collection of boundary boxes such that their removal from $\\lambda$ results in a (smaller) diagram.\n\\end{mydef}\n\nThe Murnaghan-Nakayama rule \\cite{stanley1986enumerative} is given below.\n\n\\begin{mytheo}[The Murnaghan-Nakayama rule]\n\\label{theo:mnrule}\nLet $c$ be a permutation with cycle structure $\\left(c_1, \\ldots, c_t\\right)$, $c_1 \\ge \\ldots \\ge c_t$. Then\n\n\\begin{align}\n\\chi_\\lambda \\left(c\\right) &= \\sum_{s_1, \\ldots, s_t} \\left(-1\\right)^{v\\left(s_1\\right)} \\ldots \\left(-1\\right)^{v\\left(s_t\\right)},\n\\end{align}\n\nwhere each $s_i$ is a skew hook of length $c_i$ of the diagram or partition $\\lambda$ after $s_1, \\ldots, s_{i - 1}$ have been removed, and $v \\left(s_i\\right)$ denotes the number of vertical steps in $s_i$.\n\\end{mytheo}\n\nThe semidirect product is defined as follows \\cite{dummit2004abstract}.\n\n\\begin{mydef} [Semidirect product]\nLet $H$ and $K$ be groups, with $K$ acting on $H$ via an action $\\phi : K \\to Aut \\left(H\\right)$. The multiplication operation is defined as follows.\n\\begin{align}\n\\left(h_1, k_1\\right) * \\left(h_2, k_2\\right) &= \\left(h_1 \\phi_{k_1} \\left(h_2\\right), k_1 k_2 \\right) =  \\left(h_1 \\left(k_1 \\cdot h_2\\right), k_1 k_2\\right)\n\\end{align}.\nThen the group $G$ from is called the semidirect product of $H$ by $K$, and is denoted by $H \\rtimes_\\phi K$.\n\\end{mydef}\n\n\nThe wreath product is defined as follows \\cite{dummit2004abstract}.\n\n\\begin{mydef} [Wreath product]\nLet $K$ and $L$ be groups, let $n$ be a positive integer, let $\\phi : K \\to S_n$  be a homomorphism and let $H$ be the direct product of $n$ copies of $L$. Let $\\psi$ be an injective homomorphism from $S_n$ into $Auto \\left(H\\right)$  constructed by letting the elements of $S_n$ permute the $n$ factors of $H$. The composition $\\psi \\circ \\phi$ is a homomorphism from $G$ into $Aut \\left(H\\right)$. The wreath product of $L$ by $K$ is the semidirect product $H \\rtimes K$ with respect to this homomorphism and is denoted by $L \\wr K$. \n\\end{mydef}\n\nThe dimension of an irreducible representation of the symmetric group is defined as follows.\n\n\\begin{mydef} [Dimension of an irreducible representation]\nThe dimension $dim_{\\rho_\\lambda}$ of an irreducible representation $\\rho$ for a partition $\\lambda = \\left(\\lambda_1 + \\ldots + \\lambda_i + \\ldots + \\lambda_k\\right)$ of a symmetric group $S_n$ is given as follows \\cite{fulton1991representation}.\n\n\\begin{align}\ndim_{\\rho_\\lambda} &= \\frac{n!}{l_1 \\cdot \\ldots \\cdot l_k!} \\Pi_{i < j} \\left(l_i - l_j\\right),\n\\end{align}\nwith $l_i = \\lambda_i + k - i$.\n\\end{mydef}\n\nIt is suitable to mention the following theorem on the multiplicity of an irreducible representation in the regular representation \\cite{fulton1991representation}.\n\n\\begin{mytheo}\n\\label{theo:irrep-multiplicity-in-regular}\nEvery irreducible representation $\\rho$ occurs $\\text{dim}\\left(\\rho\\right)$ times in the regular representation.\n\\end{mytheo}\n\n\\begin{proof}[Proof of Theorem ~\\ref{theo:irrep-multiplicity-in-regular}]\nLet $\\chi$ be the character of the regular representation. Then\n\\begin{align*}\n\\chi \\left(g\\right) &=\\begin{cases} \n    n       & \\quad \\text{if } g = 1, \\text{ and}\\\\\n   0  & \\quad \\text{ otherwise.}\\\\\n  \\end{cases}\n\\end{align*}\nBecause, each group elements acts by a permutation matrix, and the trace of a permutation matrix is simply the number of fixed points of the permutation. Thus,\n\\begin{align*}\n\\langle \\chi_\\rho, \\chi \\rangle &= \\frac{1}{n} \\bar{\\chi_\\rho \\left(1\\right) \\chi \\left(1\\right)}\n\\\\\n&= \\frac{1}{n} \\text{dim} \\left(\\rho\\right) n\n\\\\\n&= \\text{dim} \\left(\\rho\\right)\n\\end{align*}\n\\qedhere\n\\end{proof}\n\nAnother two important concepts in representation theory are {\\it restriction} and {\\it induction} \\cite{sagan2013symmetric}.\n\n\\begin{mydef}[Restriction]\nLet $H$ be a subgroup of $G$ and $X$ be a matrix representation of $G$. The restriction of $X$ to $H$, $X \\downarrow^G_H$, is given by\n\\begin{align*}\nX\\downarrow^G_H \\left(h\\right) = X \\left(h\\right)\n\\end{align*}\nfor all $h \\in H$.\n\\end{mydef}\n\n\\begin{mydef}[Induction]\n\\label{def:induction}\nLet $H \\le G$ and $t_1, \\ldots, t_l$ be a fixed transversal for the left cosets of $H$, i.e., $G = t_1 H \\sqcup \\ldots \\sqcup t_l H$. If $Y$ is a representation of $H$, then the corresponding induced representation $Y\\uparrow^G_H$ assigns to each $g \\in G$ the block matrix\n\\begin{align*}\nY\\uparrow^G_H \\left(g\\right) &= Y \\left(t^{-1}_i g t_j\\right)\n\\nonumber\\\\\n&= \\begin{pmatrix}\n  Y \\left(t^{-1}_1 g t_1\\right) & Y \\left(t^{-1}_1 g t_2\\right) & \\cdots & Y \\left(t^{-1}_1 g t_l\\right) \\\\\n  Y \\left(t^{-1}_2 g t_1\\right) & Y \\left(t^{-1}_2 g t_2\\right) & \\cdots & Y \\left(t^{-1}_2 g t_l\\right) \\\\\n  \\vdots  & \\vdots  & \\ddots & \\vdots  \\\\\n  Y \\left(t^{-1}_l g t_1\\right) & Y \\left(t^{-1}_l g t_2\\right) & \\cdots & Y \\left(t^{-1}_l g t_l\\right)\n \\end{pmatrix}\n\\end{align*}\nwhere $Y \\left(g\\right)$ is the zero matrix if $g \\notin H$.\n\\end{mydef}\n\nIt is natural to define the characters for the restricted and induced representations \\cite{serre2012linear}.\n\n\\begin{mydef}[Character of restricted representation]\nLet $X$ be a matrix representation of a group $G$, and let $H \\le G$ be a subgroup. Then, the character of the restricted representation $\\chi \\downarrow^G_H \\left(h\\right)$ is the character of the original representation $\\chi \\left(h\\right)$ for all $h \\in H$.\n\\end{mydef}\n\nThe definition of the character of induced representation is reproduced from \\cite{JohnArmstrongCharInducRepre}.\n\n\\begin{mydef}[Character of induced representation]\n\\label{def:char-ind-rep}\nLet $Y$ be a matrix representation of a group $H$ such that $H \\le G$. A transversal of $H$ in $G$ is now picked. Using the previously mentioned formula for the induced representation, it is found that,\n\\begin{align*}\n\\chi \\uparrow^G_H \\left(g\\right) &= Tr \\left( Y \\left(t^{-1}_i g t_j\\right)\\right)\n\\nonumber\\\\\n&= Tr \\begin{pmatrix}\n  Y \\left(t^{-1}_1 g t_1\\right) & Y \\left(t^{-1}_1 g t_2\\right) & \\cdots & Y \\left(t^{-1}_1 g t_l\\right) \\\\\n  Y \\left(t^{-1}_2 g t_1\\right) & Y \\left(t^{-1}_2 g t_2\\right) & \\cdots & Y \\left(t^{-1}_2 g t_l\\right) \\\\\n  \\vdots  & \\vdots  & \\ddots & \\vdots  \\\\\n  Y \\left(t^{-1}_l g t_1\\right) & Y \\left(t^{-1}_l g t_2\\right) & \\cdots & Y \\left(t^{-1}_l g t_l\\right)\n \\end{pmatrix}\n \\nonumber\\\\\n &= \\sum^n_{i=1} Tr \\left(Y \\left(t^{-1}_i g t_i\\right)\\right)\n \\nonumber\\\\\n &= \\sum^n_{i=1} \\chi  \\left(t^{-1}_i g t_i\\right)\n\\end{align*}\nwhere $\\chi \\left(g\\right)$ is the zero matrix if $g \\notin H$.\n\\end{mydef}\n\nSince, $\\chi$ is a class function on $H$, conjugation by any element $h \\in H$ leaves it the same. So, $\\chi \\left(h^{-1} g h\\right) = \\chi \\left(g\\right)$ for all $g \\in G$ and $h \\in H$.\n\nThe same computation is performed for each element of $H$. Then, all the results are added together and divided by the number of elements of $H$. In other words, the above function is written out in $|H|$ different ways, added  all together, and divided by $|H|$ to get exactly what the section started with started with:\n\n\\begin{align}\n\\chi \\uparrow^G_H \\left(g\\right) &= \\frac{1}{|H|} \\sum_{h \\in H} \\sum^n_{i = 1} \\chi \\left(h^{-1} t^{-1}_i g t_i h\\right)\n\\nonumber\\\\\n&= \\frac{1}{|H|} \\sum_{h \\in H} \\sum^n_{i=1} \\chi \\left(\\left(t_i h\\right)^{-1} g \\left(t_i h\\right)\\right)\n\\end{align}\nBut now as $t_i$ varies over the transversal, and as $h$ varies over $H$, their product $t_i h$ varies exactly once over $G$. That is, every $x \\in G$ can be written in exactly one way in the form $t_ h$ for some transversal element $t_i$ and subgroup element $h$. Thus, the following relation can be stated.\n\n\\begin{align}\n\\chi \\uparrow^G_H \\left(g\\right) &= \\frac{1}{|H|} \\sum_{x \\in G} \\chi \\left(x^{-1} g x\\right)\n\\end{align}\n\nIt would be appropriate if the following theorem on the {\\it Frobenius reciprocity} is also mentioned in this section \\cite{sagan2013symmetric}.\n\n\\begin{mytheo}[Frobenius reciprocity]\nLet $H \\le G$ and suppose that $\\psi$ and $\\chi$ are characters of     $H$ and $G$, respectively. Then\n\\begin{align}\n\\langle \\psi \\uparrow^G_H, \\chi \\rangle_G &= \\langle \\psi, \\chi \\downarrow^G_H \\rangle_H\n\\end{align}\nwhere the left inner product is calculated in $G$ and the right one in $H$.\n\\end{mytheo}\n\nA special case of {\\it Frobenius reciprocity} is relevant to the discussion of this paper where the representation of $H$ is the trivial representation $\\text{\\bf 1}_H$ \\cite{hallgren2000normal}. The case is described as follows.\n\n\\begin{mylem}[Special case of Frobenius reciprocity]\n\\label{lem:frob-recip-special}\nLet $H \\le G$ and suppose that $\\chi_\\rho$ is the character of the irreducible representation $\\rho$ of $G$. Then\n\\begin{align}\n\\langle {\\chi \\uparrow^G_H}_{\\text{\\bf 1}_H}, \\chi_\\rho \\rangle_G &= \\langle \\chi_{\\text{\\bf 1}_H}, \\chi_\\rho \\downarrow^G_H \\rangle_H\n\\end{align}\nwhere the left inner product is calculated in $G$ and the right one in $H$.\n\\end{mylem}\n\nThe Example $3.13$ from  \\cite{fulton1991representation} can be reproduced here in relevance to the ongoing discussion.\n\n\\begin{myrem}\n\\label{rem:triv-rep-induc}\nThe permutation representation associated to the left action of $G$ on $G\/H$ is induced from the trivial one-dimensional representation $W$ of $H$. Here, the representation of $G$, $V$ has basis $\\left\\{e_\\sigma: \\sigma \\in G\/H\\right\\}$, and $W = \\mathbb{C}\\cdot e_H$, with     $H$the trivial coset.\n\\end{myrem}\n\nFollowing Remark ~\\ref{rem:triv-rep-induc}, it can be said that $\\text{\\bf 1}_H\\uparrow^G_H $ is the permutation representation of $G$. So, according to the Theorem ~\\ref{theo:irrep-multiplicity-in-regular}, the multiplicity of $\\rho$ in $\\text{\\bf 1}_H\\uparrow^G_H $ is $d_\\rho$.\n\n\\subsection{Quantum Fourier Sampling}\nQuantum Fourier Sampling is a class of quantum algorithms which uses quantum Fourier transformation as a subroutine.\nThe definition of quantum Fourier transform of a map from a finite group to its representation is defined as follows where the notations are borrowed from \\cite{hallgren2000normal}. Later in this section, the algorithm is also presented.\n\n\\begin{mydef}[Fourier transformation of a finite group]\n\\label{def:qft}\nLet $f: G \\to \\mathbb{C}$. The Fourier transform of $f$ at the irreducible representation $\\rho$ is the $d_\\rho \\times d_\\rho$ matrix\n\\begin{align}\n\\hat{f} \\left(\\rho\\right) &= \\sqrt{\\frac{d_\\rho}{|G|}} \\sum_{g\\in G} f\\left(g\\right) \\rho \\left(g\\right)\n\\end{align}\n\\end{mydef}\n\nIn quantum Fourier transform, the superposition $\\sum_{g\\in G} f_g |g\\rangle$ is identified  with the function $f: G \\to \\mathbb{C}$ defined by $f \\left(f \\left(g\\right)\\right) = f_g$. Using this notation, $\\sum_{g\\in G} f\\left(g\\right) |g\\rangle$ is mapped under the Fourier transform to $\\sum_{\\rho \\in \\hat{G}, 1 \\le i, j\\le d_\\rho} \\hat{f}\\left(\\rho\\right)_{i, j}|\\rho, i, j\\rangle$. Here, $\\hat{G}$ is the set of all irreducible representations of $G$ and $\\hat{f}\\left(\\rho\\right)_{i,j}$ is a complex number. The probability of measuring the register $|\\rho \\rangle$ is\n\\begin{align}\n\\sum_{1\\le i, j\\le d_\\rho} |\\hat{f}\\left(\\rho\\right)_{i,j}|^2 &= \\|\\hat{f}\\left(\\rho\\right)\\|^2\n\\end{align}\nwhere $\\|A\\|$ is the natural norm (also known as Frobenius norm) given by $\\|A\\|^2 = Tr \\left(A^\\dagger A\\right)$.\n\nThe Frobenius norm can be calculated from the characters of the group associated which is demonstrated in the following theorem reproduced from \\cite{hallgren2000normal}.\n\n\\begin{mytheo}\n\\label{theo:frob-char}\nIf, $f$ is an indicator function of a left closet of $H$ in $G$, i.e. for some $c \\in G$,\n\\begin{align}\nf \\left(g\\right) &= \\begin{cases}\n    \\frac{1}{\\sqrt{|H|}}       & \\quad \\text{ if } g \\in c H, \\text{ and }\\\\\n    0  & \\quad 0 \\text{ otherwise}\\\\\n  \\end{cases}\n\\end{align}, then,\n\n\\begin{align}\n\\|\\hat{f} \\left(\\rho\\right)\\|^2 &= \\frac{|H|}{|G|} d_\\rho \\langle \\chi_\\rho, \\chi_{\\text{\\bf 1}_H} \\rangle_H\n\\end{align}\n\\end{mytheo}\n\n\\begin{proof}[Proof of Theorem ~\\ref{theo:frob-char}]\nFrom Definition ~\\ref{def:qft}, it is known that,\n\\begin{align}\n\\|\\hat{f} \\left(\\rho\\right)\\|^2 &= \\sum_{1\\le i, j\\le d_\\rho} |\\hat{f}\\left(\\rho\\right)_{i,j}|^2\n\\end{align}\nOnly  $\\rho$ is to be measured. \n\nFollowing relation is assumed in the theorem.\n\\begin{align}\n\\|\\hat{f} \\left(\\rho\\right)\\|^2 &= \\| \\rho \\left(c\\right) \\sum_{h\\in H} \\rho \\left(h\\right) \\|^2\n\\end{align}\n$\\rho \\left(c\\right)$ is a unitary matrix. So, as a multiplier it does not change the norm \\cite{meyer2000matrix}.\n\\begin{align}\n\\|\\hat{f} \\left(\\rho\\right)\\|^2 &= \\| \\sum_{h\\in H} \\rho \\left(h\\right) \\|^2\n\\end{align}\n\nSo, the probability of measuring $\\rho$ is determined by $\\sum_{h\\in H} \\rho \\left(h\\right)$. If correctly normalized, $\\frac{1}{|H|}\\sum_{h\\in H}\\rho(h)$ is a projection.\n\n\\begin{align}\n\\left(\\frac{1}{|H|}\\sum_{h\\in H}\\rho(h)\\right)^2&=\\frac{1}{|H|^2}\\sum_{h_1,h_2\\in H} \\rho(h_1 h_2)\n\\nonumber\\\\\n&=\\frac{1}{|H|}\\sum_{h\\in H}\\rho(h)\n\\end{align}\nbecause $h_1h_2=h$ has $|H|$ solutions $(h_1,h_2)\\in H\\times H$.\n\n With the right choice of basis,   $\\hat{f} \\left(\\rho\\right)$ will be diagonal and consist of ones and zeros. The probability of measuring $\\rho$ will then be the sum of ones in the diagonal. As $\\rho$ is an irreducible representation of $G$,  the sum of the matrices $\\rho \\left(h\\right)$ for all $h \\in H$ needs to be taken into account. Based of the assumption of the current theorem, one may only consider to evaluate $\\rho$ on $H$. According to the assumption, the probability of measuring $\\rho$ when $g \\notin cH$ is zero. So, one may consider consider $\\rho \\downarrow^G_H$ instead of $G$.\n\nThen, the Fourier transform of $f$ at $\\rho$ is comprised of blocks, each corresponding to a representation in the decomposition of $\\rho\\downarrow^G_H$. Such as,\n\\begin{align}\n\\sum_{h \\in H} \\rho \\left(h\\right) &= U  \\begin{bmatrix}\n\\sum_{h \\in H} \\sigma_1 \\left(h\\right) & 0 & \\cdots & 0 \\\\\n  0 & \\sum_{h \\in H} \\sigma_2 \\left(h\\right) & \\cdots & 0 \\\\\n  \\vdots  & \\vdots  & \\ddots & \\vdots  \\\\\n 0 & 0 & \\cdots & \\sum_{h \\in H} \\sigma_t \\left(h\\right)\n \\end{bmatrix} U^\\dagger\n\\end{align}\nHere, $U$ is an arbitrary unitary transformation, $\\sigma_i$ is an irreducible representation of $H$ with possible repetition. Now, as a special case of the orthogonality relation among group characters,  $\\sum_{h \\in H} \\rho \\left(h\\right)$ is nonzero only when the irreducible representation is trivial, in which case, it is $|H|$.\n\nSo, the probability of measuring $\\rho$ is:\n\\begin{align}\n\\norm{\\hat{f} \\left(\\rho\\right)}^2 &= \\norm{ \\sqrt{\\frac{d_\\rho}{|G|}} \\sum_{h\\in H} \\rho \\left(h\\right) }^2\n\\nonumber\\\\\n&= \\frac{d_\\rho}{|G|}  \\norm{ \\sum_{h\\in H} \\rho \\left(h\\right) }^2\n\\nonumber\\\\\n&= \\frac{d_\\rho}{|G|} \\mathrm{tr}\\left(\\sum_{h_1\\in H}\\rho(h_1)\\right)\\left(\\sum_{h_2\\in H}\\rho(h_2)\\right)^\\dagger\n\\nonumber\\\\\n&= \\frac{d_\\rho}{|G|} \\mathrm{tr}\\sum_{h_1,h_2\\in H}\\rho(h_1h_2^{-1})\n\\nonumber\\\\\n&= \\frac{d_\\rho}{|G|} \\frac{1}{|H|} |H|^2 \\langle \\chi_\\rho, \\chi_{\\text{\\bf 1}_H}\\rangle_H\n\\nonumber\\\\\n&= \\frac{|H|}{|G|} d_\\rho \\langle \\chi_\\rho, \\chi_{\\text{\\bf 1}_H} \\rangle_H\n\\end{align}\nIt should be mentioned that, by definition, $\\rho$ appears  $\\langle \\chi_\\rho, \\chi_{1_H} \\rangle_H$ times in the decomposition of $\\text{\\bf 1}_H$.\n\\qedhere\n\\end{proof}\n\nInterested readers are encouraged to refer to \\cite{diaconis1990efficient} for a review on the classical complexity of Fourier transformation of the symmetric groups. The goal of quantum Fourier sampling algorithm is to sample the labels and elements of the irreducible representations available after quantum Fourier transformation. Sampling only the labels of representations is called weak sampling. On the other hand, sampling also the indices of the elements of the matrix is called strong Fourier sampling. The quantum Fourier sampling algorithm is reproduced from \\cite{hallgren2000normal}.\n\n\\begin{algorithm}[H]\n\\caption{{\\bf QUANTUM-FOURIER-SAMPLING}}\n \\label{algo:weak}\n\\begin{algorithmic}[1]\n\\Procedure {QUANTUM-FOURIER-SAMPLING}{$f : G \\to S$}\n\\State Compute $\\sum_{g \\in G} |g, f \\left(g\\right) \\rangle$ and measure the second register $f \\left(g\\right)$. The resulting super-position is $\\sum_{h \\in H} |c h \\rangle \\otimes | f \\left(c h\\right)\\rangle$ for some coset $c H$ of $H$. Furthermore, $c$ is uniformly distributed over $G$.\\;\n\n\\State  Compute the Fourier transform of the coset state which is $\\sum_{\\rho \\in \\hat{G}} \\sqrt{\\frac{d_\\rho}{|G|}} \\sqrt{\\frac{1}{|H|}} \\left(\\sum_{h \\in H} \\rho \\left(c h\\right)\\right)_{i, j} | \\rho, i, j\\rangle$, where $\\hat{G}$ denotes the set of irreducible representations of $G$.\\;\n\n\\State  Measure the first register and observe a representation $\\rho$ (weak) or $\\rho, i, \\text{ and } j$ (strong). \\;\n\\EndProcedure\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\\section{Quantum Fourier sampling for graph isomorphism}\n\\label{sec:qft-gi}\nThe goal of this section is to give the readers a detailed exposition of the standard approach of hidden subgroup algorithms for graph isomorphism problems. In this section, the quantum Fourier sampling algorithm for graph isomorphism is described. Although it is reproduced verbatim from \\cite{hallgren2000normal}, the notation is changed to match it to this paper's original discussion of quantum Fourier sampling for graph automorphism. Most of the algorithms for the non-Abelian hidden subgroup problem use a black box for $\\varphi$ in the same way as in the Abelian hidden subgroup problem \\cite{lomonacopers2002}. This has come to be known as the {\\it standard method}. The standard method begins by preparing a uniform superposition over group elements \\cite{childs2016lecture}:\n\n\\begin{align}\n|S_n \\rangle := \\frac{1}{\\sqrt{|S_n|}} \\sum_{g\\in S_n} |g\\rangle\n\\end{align}\n\nThe value of $\\varphi \\left(g\\right)$ is then computed in an ancilla register which creates the following state.\n\n\\begin{align}\n \\frac{1}{\\sqrt{|S_n|}} \\sum_{g\\in S_n} |g, \\varphi \\left(g\\right) \\rangle\n\\end{align}\n\nThen  the second register is discarded by just being traced it out. If the outcome of the second register is $s$  then the state is projected onto the uniform superposition of those $g \\in S_n$ such that $\\varphi \\left(g\\right) = s$. By definition of $\\varphi$, it is some left coset of the hidden subgroup $H$. Since every coset contains the same number of elements, each left coset occurs with equal probability. Thus, the standard method produces the following coset state.\n\n\\begin{align}\n|g H\\rangle := \\frac{1}{\\sqrt{|H|}} \\sum_{h \\in H} |g h \\rangle\n\\end{align}\n\nor equivalently as the following mixed {\\it hidden subgroup state}.\n\n\\begin{align}\n\\rho_H := \\frac{1}{|S_n|} \\sum_{g \\in S_n} |g h \\rangle \\langle g h |\n\\end{align}\n\nIt has been previously mentioned that $\\varphi$ maps the group elements of $S_n$ to $\\text{Mat}\\left(\\mathbb{C}, N \\right)$. Here,  more information is presented about the space $\\text{Mat}\\left(\\mathbb{C}, N \\right)$. Let the complete set of  irreducible representations of $S_n$ (which are unique up to isomorphism) be $\\hat{S_n}$. The Fourier transform is a unitary transformation from the group algebra, $\\mathbb{C} S_n$, to a complex vector space whose basis vectors correspond to matrix elements of the irreducible representations of $S_n$, $\\oplus_{\\rho \\in \\hat{S_n}} \\left(\\mathbb{C}^{d_\\rho} \\otimes \\mathbb{C}^{d_\\rho}\\right)$. Here, $d_\\rho$ is the dimension of the irreducible representation $\\rho$. \n\n$|g \\rangle$ is the basis vector chosen for the group element $g \\in S_n$. There will be $n !$ such basis vectors of dimension $n! \\times 1$. For a given group element $g$, there are a particular number of matrices  one for each irreducible representation $\\rho$. $|g h\\rangle$ is expressed as $|\\rho, j, k\\rangle$ which is the basis vector labeled by the $\\left( j, k \\right)$-th element of the irreducible representation $\\rho$ of $g$.\n\nAs it is mentioned earlier, when only $\\rho$ is measured  from $|\\rho, j, k\\rangle$, it is called {\\it weak Fourier sampling}. In {\\it strong Fourier sampling}, $j$ and $k$ are also measured.\n\n\\subsection{Weak Fourier sampling for $\\text{{\\bf GI}}_{\\text{HSP}}$}\n\\label{sec:wfs-gi}\nThis section summarizes what already is known about the weak Fourier sampling when applied to the graph isomorphism problem. The weak Fourier sampling for $\\text{{\\bf GI}}_{\\text{HSP}}$ attempts to measure the labels of irreducible representations of the symmetric group $S_{2 n}$ when the input graphs $\\Gamma_1$ and $\\Gamma_2$ are of $n$ vertices. It is assumed that $Aut \\left(\\Gamma_1\\right) = Aut \\left(\\Gamma_2\\right) = \\left\\{e\\right\\}$. If $\\Gamma = \\Gamma_1 \\sqcup \\Gamma_2$, one of the following two claims is true \\cite{hallgren2000normal}. \n\n\\begin{itemize}\n\\item If $\\Gamma_1 \\not\\approx \\Gamma_2$, then $\\text{Aut}\\left(\\Gamma \\right) = \\left\\{e\\right\\}$.\n\\item If $\\Gamma_1 \\approx \\Gamma_2$, then $\\text{Aut}\\left(\\Gamma \\right) = \\left\\{e, \\sigma\\right\\} = \\mathbb{Z}_2$, where $\\sigma \\in S_{2n}$ is a permutation with $n$ disjoint $2$-cycles.\n\\end{itemize}\n\nIt should be mentioned that in \\cite{hallgren2000normal}, the authors derived the success probability of measuring the label of the irreducible representations for $S_n$. In this paper, the same probability will be derived for $S_{2 n}$ to keep consistency with the definition of $\\text{{\\bf GI}}_{\\text{HSP}}$.\n\nThe weak Fourier sampling algorithm for finding $\\text{Aut}(\\Gamma)$ in $S_{2 n}$ is described below.\n\n\\begin{algorithm}[H]\n\\caption{{\\bf WEAK-QUANTUM-FOURIER-SAMPLING-}$S_{2 n}$}\n \\label{algo:weak-gi}\n\\begin{algorithmic}[1]\n\\Procedure {WEAK-QUANTUM-FOURIER-SAMPLING-$S_{2 n}$}{a graph $\\Gamma$ such that either $\\text{Aut} \\left(\\Gamma\\right) = \\left\\{e\\right\\}$ or $\\text{Aut} \\left(\\Gamma\\right) = \\left\\{e, \\sigma\\right\\}$}\n\\State    Compute $ \\frac{1}{\\sqrt{\\left(2 n\\right)!}} \\sum_{g\\in S_{2 n}} |g, \\varphi \\left(g\\right) \\rangle$\\;\n\n\\State  Compute $\\sum_{\\rho \\in \\hat{S_{2 n}}} \\sqrt{\\frac{d_\\rho}{\\left(2 n\\right)!}} \\sqrt{\\frac{1}{|\\text{Aut}\\left(\\Gamma \\right)|}} \\left(\\sum_{h \\in \\text{Aut}\\left(\\Gamma \\right)} \\rho \\left(c h\\right)\\right)_{i, j} |\\rho, i, j \\rangle $\\;\n\n\\State  Measure $\\rho$ as in tracing it out \\;\n\\EndProcedure\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\n\nLet $p_\\rho$ be the probability of sampling $\\rho$ in Algorithm ~\\ref{algo:weak-gi} when $\\Gamma_1 \\not\\approx \\Gamma_2$, and $q_\\rho$ when $\\Gamma_1 \\approx \\Gamma_2$. So,  the induced representation of $\\text{Aut} \\left(\\Gamma\\right)$ to $S_{2 n}$, $\\text{Ind}^{S_{2 n}}_{\\text{Aut} \\left(\\Gamma\\right)} 1$, is the regular representation. So, $\\langle \\chi_\\rho | \\chi_{\\text{Ind}^{S_{2 n}}_{\\text{Aut} \\left(\\Gamma\\right)} 1} \\rangle$, the multiplicity of $\\rho$ in the regular representation, is $d_\\rho$. Hence, $p_\\rho = \\frac{d^2_\\rho}{\\left(2 n\\right)!}$.\n\nWhen $\\Gamma_1 \\approx \\Gamma_2$, $\\text{Aut} \\left(\\Gamma\\right) = \\left\\{e, \\sigma\\right\\}$. In this case, the probability of measuring $\\rho$, \n\n\\begin{align}\nq_\\rho &= \\frac{|\\text{Aut}\\left(\\Gamma \\right)|}{\\left(2 n\\right)!} d_\\rho \\langle \\chi_1 | \\chi_\\rho \\rangle_{\\text{Aut}\\left(\\Gamma \\right)}\n\\end{align}\n\n$\\text{Aut}\\left(\\Gamma \\right)$ has only two elements, $e$ and $\\sigma$, hence\n\n\\begin{align}\n \\langle \\chi_1 | \\chi_\\rho \\rangle_{\\text{Aut}\\left(\\Gamma \\right)} &= \\frac{1}{2} \\left(\\chi_\\rho \\left(e\\right) + \\chi_\\rho \\left(\\sigma\\right)\\right)\n \\nonumber\\\\\n &= \\frac{1}{2} \\left(d_\\rho + \\chi_\\rho \\left(\\sigma\\right)\\right)\n\\end{align}\n\nSo, \n\n\\begin{align}\nq_\\rho &= \\frac{|\\text{Aut}\\left(\\Gamma \\right)|}{\\left(2 n\\right)!} d_\\rho \\frac{1}{2} \\left(d_\\rho + \\chi_\\rho \\left(\\sigma\\right)\\right)\n \\nonumber\\\\\n &= \\frac{2}{\\left(2 n\\right)!} d_\\rho \\frac{1}{2} \\left(d_\\rho + \\chi_\\rho \\left(\\sigma\\right)\\right)\n \\nonumber\\\\\n &=\\frac{d_\\rho}{\\left(2 n\\right)!}  \\left(d_\\rho + \\chi_\\rho \\left(\\sigma\\right)\\right)\n\\end{align}\n\nSo, \n\n\\begin{align}\n| p_\\rho - q_\\rho| &=| \\frac{d^2_\\rho}{\\left(2 n\\right)!} - \\frac{d_\\rho}{\\left(2 n\\right)!}  \\left(d_\\rho + \\chi_\\rho \\left(\\sigma\\right)\\right)|\n \\nonumber\\\\\n &=  \\frac{d_\\rho}{\\left(2 n\\right)!}   |\\chi_\\rho \\left(\\sigma\\right)|\n\\end{align}\n\nNow, the Murnaghan-Nakayama rule (Theorem ~\\ref{theo:mnrule}) is used to approximate  $|\\chi_\\rho \\left(\\sigma\\right)|$. \n\nThe number of unordered decompositions for the diagram $\\lambda$ with $2 n$ cells is $2^{4 n} = 16^n$. For each unordered decomposition, the number of ordered decomposition is at most $\\left(2 \\sqrt{2 n}\\right)^{\\frac{2 n}{2}} = \\left(2 \\sqrt{2 n}\\right)^{n}$ . So, by the Murnaghan-Nakayama rule, $\\chi_\\rho \\left(\\sigma\\right) \\le 16^n \\left(2 \\sqrt{2 n}\\right)^{n}$ .\n\nSo,\n\n\\begin{align}\n|\\chi_\\rho \\left(\\sigma\\right)| & \\le 16^n \\left(2 \\sqrt{2 n}\\right)^{n}\n\\nonumber\\\\\n &\\le 2^{4 n} 2^{n} \\sqrt{2}^n \\left( \\sqrt{ n}\\right)^{n}\n \\nonumber\\\\\n &\\le 2^{O\\left( n\\right)}  n^{\\frac{n}{2}}\n\\end{align}\n\nNow $| p_\\rho - q_\\rho|$ is computed for all irreducible representations. So,\n\n\\begin{align}\n| p - q|_1 &= \\sum_\\rho | p_\\rho - q_\\rho|\n \\nonumber\\\\\n & \\le \\sum_\\rho \\frac{d_\\rho}{\\left(2 n\\right)!} 2^{O\\left( n\\right)}  n^{\\frac{n}{2}}\n  \\nonumber\\\\\n  & \\le \\sum_\\rho \\frac{\\sqrt{\\left(2n\\right)!}}{\\left(2 n\\right)!} 2^{O\\left( n\\right)}  n^{\\frac{n}{2}}\n\\nonumber\\\\\n&= \\frac{2^{O\\left( n\\right)}  }{  \\frac{n}{2}^{\\frac{n}{2}}}\n\\nonumber\\\\\n&\\le \\frac{2^{O\\left( n\\right)}  }{  \\left(\\frac{n}{2}\\right)!}\n\\nonumber\\\\\n&\\lll 2^{-\\Omega \\left(n\\right)}\n\\end{align}\nSo, the probability of successfully measuring the labels of the irreducible representations is exponentially low in the size of the graphs.\n\n\\section{Quantum Fourier sampling for cycle graph automorphism}\n\\label{sec:weak-cycle}\nIn this section, it is shown that quantum Fourier sampling fails to compute the automorphism group of a cycle graph. The result is original to this paper. The scheme of the proof is as follows. First,  the automorphism group of the graph is computed which is trivial for this case. Then  its irreducible representation is computed. Then the general expressions for group characters are derived. And, finally, using the characters, the probabilities of measuring the labels of irreducible representations are computed. A more granular representation of the previously mentioned steps is given below.\n\n            \\begin{tikzpicture}[node distance=5cm, scale=0.8, every node\/.append style={transform shape}]\n\n\\node[text width=4cm] (tf) [startstop] {\\scriptsize Determine the symmetric group in which\\\\ the automorphism group is hidden in};\\\\\n\n\\node[text width=4cm] (ln) [startstop , right of = tf] {\\scriptsize Determine the presentation of the\\\\ automorphism group};\n\n\\node[text width=4cm] (mv) [startstop , below of = ln] {\\scriptsize Determine the order of the\\\\ automorphism group};\n\n\\node[text width=4cm] (qp) [startstop , left of = mv] {\\scriptsize Determine the irreducible representations\\\\ of the automorphism group};\n\n\\node[text width=4cm] (bq) [startstop , below of = qp] {\\scriptsize Determine the characters\\\\ of the representations};\n\n\\node[text width=5cm] (bq1) [startstop , right of = bq] {\\scriptsize Compute the inner product of the characters of the trivial representation of the automorphism group and the irreducible representations of the symmetric group restricted to the automorphism group};\n\n\\node[text width=5cm] (bq2) [startstop , below of = bq1] {\\scriptsize To compute previous quantity, compute the inner product of the characters of the trivial representation of the automorphism group induced up to the symmetric group and the irreducible representation of the symmetric group};\n\n\\node[text width=4cm] (bq3) [startstop , left of = bq2] {\\scriptsize Using the previous quantity compute the probability of sampling the labels of the irreducible representations of the automorphism group};\n\n\\draw [arrow] (tf) -- (ln);\n\\draw [arrow] (ln) -- (mv);\n\\draw [arrow] (mv) -- (qp);\n\\draw [arrow] (qp) -- (bq);\n\\draw [arrow] (bq) -- (bq1);\n\\draw [arrow] (bq1) -- (bq2);\n\\draw [arrow] (bq2) -- (bq3);\n\n\\end{tikzpicture}\n\n\\subsection{Automorphism group of cycle graph}\nThe automorphism group of an $n$-cycle graphs is the dihedral group $D_n$ of order $2 n$. This section intends to study the weak Fourier sampling of cycle graphs. To provide the background, this section discusses the irreducible representations of the dihedral group $D_n$. \n\n\\begin{mydef}[Cycle Graph]\nAn $n$-cycle graph is a single cycle with $n$ vertices.\n\\end{mydef}\n\nThe automorphism group of an $n$-cycle graph is the dihedral group $D_n$ which is of order $2 n$.  If $D_n$ is even, the group can be generated as $\\langle(2\\quad n)(3 \\quad n-1) \\ldots (\\frac{n}{2}-1 \\quad \\frac{n}{2}+1),\\, (1 \\ldots n)\\rangle$. If $D_n$ is odd, the group can be generated as $\\langle(2 \\quad n) (3 \\quad n-1) \\ldots (\\frac{n-1}{2} \\quad \\frac{n+1}{2}),\\, (1 \\ldots n)\\rangle$. This is a manifestation of the presentation \n$D_n = \\langle x, y \\mid x^n = y^2 = (xy)^2 = 1, y x y=x^{-1}\\rangle$. The correspondence consists of $x = (1 \\ldots n)$ and  $y = (2 \\quad n) (3 \\quad n-1) \\ldots (\\frac{n}{2}-1\\quad\\frac{n}{2}+1)$ if $n$ is even, and  $y = (2 \\quad n) (3 \\hspace{0.5cm} n-1) \\ldots (\\frac{n-1}{2} \\quad\\frac{n+1}{2})$ if $n$ is odd. The orders of $x$ and $y$ are $n$ and $2$ respectively.\n\nThe order of the symmetric group $S_n$ is $n!$. The order of a dihedral group $D_n$ is $2 n$.  So, the index of $D_n$ in $S_n$ is $\\frac{n!}{2n} \\approx \\frac{\\sqrt{2 \\pi n} \\left(\\frac{n}\n{e}\\right)^n}{2n}$.\n\nIt would be relevant if the following  important theorem proved in \\cite{conrad2009dihedral} is mentioned here.\n\n\\begin{mytheo}\n\\label{theo:d-n-subgroups}\nEvery subgroup of $D_n$ is cyclic or dihedral. A complete listing of the subgroups is as follows:\n\\begin{itemize}\n\\item $\\langle x^d \\rangle$, where $d \\mid n$, with index $2 d$,\n\\item $\\langle x^d, x^i y \\rangle$, where $d \\mid n$ and $0 \\le i \\le d-1$, with index $d$.\n\\end{itemize}\nEvery subgroup of $D_n$ occurs exactly once in this listing.\n\nIn this theorem, subgroups of the first type are cyclic and subgroups of the second type are dihedral: $\\langle x^d \\rangle \\cong \\text{\\bf Z}\/\\left(n\/d\\right)$ and $\\langle x^d, x^i y  \\rangle \\cong D_{n\/d}$.\n\\end{mytheo}\n\nBased on Theorem ~\\ref{theo:d-n-subgroups}, following remark can be made.\n\n\\begin{myrem}\n\\label{rem:d-n-subgroup-order}\nThe order of the subgroups $\\langle x^d \\rangle$ and $\\langle x^d, x^i y \\rangle$ are $\\frac{n}{d}$ and $\\frac{2 n}{d}$ respectively.\n\\end{myrem}\n\nEvery element of $D_n$ is either $x^i$ or $y x^i$ for $0 \\le i < n$. The conjugacy classes are small enough in number to be enumerated.\n\n$\\begin{array}{rlclcl}\n\\text{Conjugate} &x^i &\\text{by}&x^j &:&(x^j) x^i(x^{-j})=x^i\\\\\n                 &x^i &\\text{by}&yx^j&:&(yx^j)x^i(x^{-j}y)= yx^iy=x^{-i}\\\\\n                 &yr^i&\\text{by}&r^j &:&(r^j) yr^i(r^{-j}) = yr^{-j}r^ir^{-j} = yr^{i-2j}\\\\\n                 &yx^i&\\text{by}&yx^j&:&(yx^j)yx^i(x^{-j}y) = x^{i-2j}y=yx^{2j-i}\\\\\n\\end{array}$\n\nThe set of rotations decomposes into inverse pairs, $\\left\\{x^i, \\left(x^i\\right)^{-1}\\right\\}$. So, the classes are $\\left\\{1\\right\\}, \\left\\{x, x^{n-1}\\right\\}$, $\\left\\{x^2, x^{n-2}\\right\\}, \\ldots$. When $n$ is even, there are $\\frac{n}{2}+1$, and when $n$ is odd, there are $\\frac{n+1}{2}$ conjugacy classes.\n\n$y x$ is conjugate to $y x^3, y x^5, \\ldots$ while $y$ is conjugate to $y x^2, y x^4, \\ldots$. If $n$ is even, these two sets are disjoint. However, $y x$ is conjugate to $y x^{n-1}$ (via $x$), so if $n$ is odd, all the non trivial reflections are in one conjugacy class.\n\nSo, the total number of conjugacy classes are as follows. If $n$ is even, the total number of conjugacy classes is $\\left(\\frac{n}{2}+1\\right) + 2 = \\frac{n}{2}+3$. If $n$ is odd, the total number of conjugacy classes is $\\frac{n+1}{2} + 1 = \\frac{n+3}{2} $.\n\nThe commutators of $D_n$,\n\n\\begin{align}\n\\left[x^i, y x^j\\right] &=  x^{-i}\\left(y x^j\\right)x^i\\left(y x^j\\right)\n\\nonumber\\\\\n& = y x^{2i+j}y x^j\n\\nonumber\\\\\n& = \\left(x^i\\right)^2\n\\end{align}\n\n\\subsection{Irreducible representations}\n\\label{sec:irrepdn}\nThe commutators generate the subgroup of squares of rotations. When $n$ is even, only half the rotations are squares, hence $G\/\\left[G, G\\right]$ is of order four. When $n$ is odd, all rotations are squares, hence $G\/\\left[G, G\\right]$ is of order two. The number of one dimensional irreducible representations is the order of $G\/\\left[G, G\\right]$. So,  when $n$ is even, there are four one dimensional representations and when $n$ is odd, there are two one dimensional representations.\n\nThe representations can be enumerated as follows.\n\n\\begin{itemize}\n\\item When $n$ is even:\n\\begin{itemize}\n\\item The trivial representation, sending all group elements to the $1 \\times 1$ matrix $\\begin{pmatrix}1\\end{pmatrix}$.\n\\item The representation, sending all elements in $\\langle x \\rangle$ to $\\begin{pmatrix}1\\end{pmatrix}$ and all elements outside $\\langle x \\rangle$ to $\\begin{pmatrix}-1\\end{pmatrix}$.\n\\item The representation, sending all elements in $\\langle x^2, y \\rangle$ to $\\begin{pmatrix}1\\end{pmatrix}$ and $x$ to $\\begin{pmatrix}-1\\end{pmatrix}$.\n\\item The representation, sending all elements in $\\langle x^2, x y \\rangle$ to $\\begin{pmatrix}1\\end{pmatrix}$ and $x$ to $\\begin{pmatrix}-1\\end{pmatrix}$.\n\\end{itemize}\n\\item When $n$ is odd:\n\\begin{itemize}\n\\item The trivial representation, sending all group elements to the $1 \\times 1$ matrix $\\begin{pmatrix}1\\end{pmatrix}$.\n\\item The representation, sending all elements in $\\langle x \\rangle$ to $\\begin{pmatrix}1\\end{pmatrix}$ and all elements outside $\\langle x \\rangle$ to $\\begin{pmatrix}-1\\end{pmatrix}$.\n\\end{itemize}\n\\end{itemize}\n\nThe two dimensional irreducible representations are described as follows. There is an obvious subgroup $\\{1, x, \\ldots, x^{n-1}\\}$ which is a cyclic group of order $n$. It can be defined as $C_n < D_{n}$. Since $C_n$ is abelian, it has $n$ irreducible 1-dimensional representations over $\\mathbb{C}$, namely\n\n\\begin{align}\nx\\mapsto e^{2\\pi ki\/n},\\qquad 0 &\\leq k < n\n\\end{align}\n\nwhich captures the idea of rotating by an angle of $2\\pi k\/n$. These easily-described representations are induced to $D_{n}$ in order to find some possibly new representations.\n\nFor the representation $W$ of a subgroup $H\\leq G$ (i.e. an H-linear action on $W$), the induced representation of $W$ is \n\n\\begin{align}\n\\bigoplus_{g\\in G\/H}g\\cdot W\n\\end{align}\n\nwhere $g$ ranges over a set of representatives of $G\/H$.\n\nThe induced representation of $C_n$ to $D_{n}$ for fixed $k$ is straight forward since $D_{n}\/C_n$ has representatives $\\{1, y\\}$. So, one just need to describe the $D_{n}$-vector space $\\mathbb{C}\\oplus y \\cdot\\mathbb{C}$ where $\\mathbb{C}$ has basis consisting only of $w_1$. Now, the $D_{n}$ action turns into an actual matrix representation.\n\nSpecifically, it can be found out how $x$ acts on each summand using the representation of $C_n$: \n\n\\begin{align}\nx\\cdot w_1 &= e^{2\\pi ki\/n}w_1, \\text{ and}\n\\end{align}\n\n\\begin{align}\nx \\cdot(y \\cdot w_1) &= x y \\cdot w_1 \n\\nonumber\\\\\n&= y x^{-1}\\cdot w_1 = e^{-2\\pi ki\/n} y \\cdot w_1\n\\end{align}\n\n which means $x$ acts by the matrix \n $\\begin{pmatrix}\n e^{2\\pi ki\/n}&0\\\\\n 0&e^{-2\\pi ki\/n}\n \\end{pmatrix}\n  $.\n\nIt can also be figured out how $y$ acts. $y$ obviously takes $w_1$ to $y \\cdot w_1$, and $y$ takes $y \\cdot w_1$ to $y^2 w_1=w_1$, so $y$ simply interchanges the two summands. This entails  that $y$ acts by the matrix \n$\\begin{pmatrix}\n0&1\\\\\n1&0\n\\end{pmatrix}\n$.\n\nHere,  the $k$-th two dimensional irreducible representations are listed for the general group elements.\n\n\\begin{align}\nx \\mapsto \\begin{pmatrix}\ne^{\\frac{2 \\pi i k}{n}}&0\\\\\n0&e^{-\\frac{2 \\pi i k}{n}}\n\\end{pmatrix}\n\\nonumber\\\\\nx^l \\mapsto \\begin{pmatrix}\ne^{\\frac{2 \\pi i k l}{n}}&0\\\\\n0&e^{-\\frac{2 \\pi i k l}{n}}\n\\end{pmatrix}\n\\nonumber\\\\\ny \\mapsto \\begin{pmatrix}\n0&1\\\\\n1&0\n\\end{pmatrix}\n\\nonumber\\\\\nx^l y \\mapsto \\begin{pmatrix}\n0&e^{\\frac{2 \\pi i k l}{n}}\\\\\ne^{-\\frac{2 \\pi i k l}{n}}&0\n\\end{pmatrix}\n\\end{align}\n\nIt is observed that, $0 \\le l \\le n-1$. Both $l = 0$ and $l = n$ determine the identity matrix to which the identity element, $e$, is mapped. When $n$ is even, the $k$-th and $\\left(n-k\\right)$-th representations are equivalent, hence the distinct representations are found only for $k = 1, 2, \\ldots, \\frac{n-2}{2}$. The representations for $k = 0$ and $k = \\frac{n}{2}$ are not irreducible and they decompose into one dimensional representations. On the other hand, when, $n$ is odd, there are $\\frac{n-1}{2}$ irreducible representations.\n\nUsing the previous calculations,  the total number of irreducible representations for $D_n$ can be computed. When $n$ is even, the total number is $\\frac{n-2}{2} + 4 = \\frac{n}{2} + 3$. When $n$ is odd, it is $\\frac{n-1}{2} + 2 = \\frac{n+3}{2}$. \n\nAt this point, the following remark regarding the characters of the irreducible two dimensional representations can be made. \n\n\\begin{myrem}\n\\label{rem:char-dn}\nThe characters of the representations of the elements of type $y$ and $x^l y$ are both zeros. The representations of the elements of type $x$ have the same character, $2 \\cos \\left(\\frac{2 \\pi k}{n}\\right)$. Finally, the representations of the elements of type $x^l$ have the same character, $2 \\cos \\left(\\frac{2 \\pi k l}{n}\\right)$.\n\\end{myrem}\n\n\\subsection{Sampling the representations}\nThe interest of this paper with the dihedral group $D_n$, of order $2 n$, is based on the fact that it is the automorphism group of the $n$-cycle graph. A $p$-group is a group where the order of every group element is a power of the prime $p$.  So, $D_n$ can be a $p$-group only when $n = 2^m$, when $m \\in \\mathbb{Z}$, because that is when $x^{2^m} = y^{2^1} = 1$ for $p = 2$. \n\nThe restrictions from the irreducible representations of $S_n$ to $D_n$ is straight forward. For any irreducible representation $\\rho$ of $S_n$, its restriction to $D_n$ is $\\rho \\left(h\\right)$ for all $h \\in D_n$. This new representation may not be necessarily irreducible.\n\nThe inductions from the irreducible representations of $D_n$ to $S_n$ can be computed following the Definition ~\\ref{def:induction}. This new representation may also not be necessarily irreducible. \n\nFollowing \\cite{hallgren2000normal}, this paper seeks to compute the induced representation of  $\\text{\\bf 1}_{D_n}$ which is the trivial representation of the dihedral group $D_n$. As a prerequisite,  a transversal for the left cosets of $D_n$ needs to be computed which can be done by any of the two classical algorithms presented in Section $4.6.7$ of \\cite{holt2005handbook}. The computation of  a transversal of a group requires the computation of the base of a group which can be computed in polynomial time using the Schreier-Sims algorithm \\cite{seress2003permutation, sims1970computational, knuth1991efficient}.\n\nAs it has been mentioned previously in the current section, there are $l = \\frac{n!}{2n} \\approx \\frac{\\sqrt{2 \\pi n} \\left(\\frac{n}{e}\\right)^n}{2n}$ cosets for $D_n$ in $S_n$. This will also be the number of elements in a transversal for the left cosets of $D_n$ in $S_n$. Let the transversal be $t_1, \\ldots, t_l$. So, $S_n = t_1 D_n \\sqcup \\ldots \\sqcup t_l D_n$.  $\\text{\\bf 1}_{D_n} \\uparrow^{S_n}_{D_n}$ can be computed following Definition ~\\ref{def:induction}. \n\nIt would be instructive to discuss the character table of $S_n$ and $D_n$ here.\n\nComputing the character table of $S_n$ starts from computing the partitions $\\lambda_1 \\ge \\lambda_2 \\ge \\ldots \\ge \\lambda_r$ given $\\sum_i \\lambda = n$. These partitions can be partially ordered as follows. If there are two partitions $\\lambda = \\left(\\lambda_1 \\ge \\lambda_2 \\ge \\ldots\\right)$ and $\\mu = \\left(\\mu_1 \\ge \\mu_2 \\ge \\ldots\\right)$, $\\lambda \\ge \\mu$ if,\n\\begin{align}\n\\lambda_1 &\\ge \\mu_1\n\\nonumber\\\\\n\\lambda_1 + \\lambda_2 &\\ge \\mu_1 + \\mu_2\n\\nonumber\\\\\n\\lambda_1 + \\lambda_2 + \\lambda_3 &\\ge \\mu_1 + \\mu_2 + \\mu_3\n\\nonumber\\\\\n&\\vdots\n\\end{align}\n\nThe columns of the character table are indexed by the conjugacy classes such that the partitions are arranged in increasing order. On the other hand, the rows are indexed by the characters in the decreasing order of the partitions. Each cell in the table then contains the corresponding character.\n\nThe Lemma ~\\ref{lem:frob-recip-special} can be applied to obtain the Frobenius reciprocity for $D_n < S_n$.\n\n\\begin{align}\n\\langle \\chi_{\\text{\\bf 1}_{D_n}}, \\chi_\\rho \\downarrow^{S_n}_{D_n} \\rangle_{D_n} &= \\langle {\\chi \\uparrow^{S_n}_{D_n}}_{\\text{\\bf 1}_{D_n}} , \\chi_\\rho \\rangle_{S_n}\n\\end{align}\nwhere the left inner product is calculated in $S_n$ and the right one in $D_n$. So, if  $\\langle \\chi_{\\text{\\bf 1}_{D_n}}, \\chi_\\rho \\downarrow^{S_n}_{D_n} \\rangle_{D_n}$ needs to be determined, it would be sufficient to determine $\\langle {\\chi \\uparrow^{S_n}_{D_n}}_{\\text{\\bf 1}_{D_n}} , \\chi_\\rho \\rangle_{S_n}$.\n\nFollowing the Definition ~\\ref{def:inner-prod-char}, \n\n\\begin{align}\n\\langle {\\chi \\uparrow^{S_n}_{D_n}}_{\\mathbf{ 1}_{D_n}} , \\chi_\\rho \\rangle_{S_n} &= \\sum_{g_i \\in S_n}  {\\chi \\uparrow^{S_n}_{D_n}}_{\\mathbf{ 1}_{D_n}} (g_i) \\chi^\\dagger_\\rho (g_i)  \n\\nonumber\\\\\n&=\\sum_{g_i \\in S_n} \\left(\\sum^n_{i=1} \\delta_{{{ \\chi \\uparrow^{S_n}_{D_n}}_{\\mathbf{ 1}_{D_n}} (g_i)}_{i,i}}\\right)   \\chi^\\dagger_\\rho (g_i)  \n\\end{align}\n\n\nThe Theorem ~\\ref{theo:frob-char} is applied to determine the probabilities of measuring the irreducible representations of $D_n$ through proving a few lemmas. \n\n\\begin{mylem}\n\\label{lem:d-n-1-d-rep-prob-zero}\nThe probability of measuring the labels of one dimensional irreducible representations of $D_n$ is zero for non-trivial representations.\n\\end{mylem}\n\n\\begin{proof}[Proof of Lemma ~\\ref{lem:d-n-1-d-rep-prob-zero}]\nFirst, it is assumed that $n$ is even. So, there are four one dimensional representations and $\\frac{n-2}{2}$ two dimensional representations. The probability of measuring the one dimensional representations is then computed. Let the representations be denoted as $\\rho_1$, $\\rho_2$, $\\rho_3$, and $\\rho_4$. So, their dimensions are all the same i.e. $d_{\\rho_1} = d_{\\rho_2} = d_{\\rho_3} = d_{\\rho_4} = 1$. Let the probability of measuring $\\rho_i$ be $p_{\\rho_i}$. So, following Theorem ~\\ref{theo:frob-char},\n\n\\begin{align}\np_{\\rho_i} &= \\frac{|D_n|}{|S_n|} d_{\\rho_i} \\langle \\chi_{\\rho_i}, \\chi_{\\text{\\bf 1}_{D_n}} \\rangle_{D_n}\n\\nonumber\\\\\n&= \\frac{2 n}{n!}  \\langle \\chi_{\\rho_i}, \\chi_{\\text{\\bf 1}_{D_n}} \\rangle_{D_n}\n\\end{align}\n\n$D_n$ has $2 n$ elements. So, \n\n\\begin{align}\n\\langle \\chi_{\\rho_i}, \\chi_{\\text{\\bf 1}_{D_n}} \\rangle_{D_n} &= \\frac{1}{|D_n|} \\sum_{g\\in D_n} \\chi_{\\rho_i} \\left(g\\right) \\chi^\\dagger_{\\text{\\bf 1}_{D_n}} \\left(g\\right)\n\\nonumber\\\\\n&= \\frac{1}{2 n} \\sum_{g\\in D_n} \\chi_{\\rho_i} \\left(g\\right) \n\\end{align}\n\nWhen $i = 1$, $\\rho_1$ is the trivial representation which sends all group elements to the $1 \\times 1$ matrix $\\begin{pmatrix}1\\end{pmatrix}$. The probability of measuring this representation is given below.\n\n\\begin{align}\n\\langle \\chi_{\\rho_1}, \\chi_{\\text{\\bf 1}_{D_n}} \\rangle_{D_n} &= \\frac{1}{2 n} \\sum_{g\\in D_n} \\chi_{\\rho_1} \\left(g\\right) \n\\nonumber\\\\\n&= 1\n\\end{align}\n\nSo, \n\\begin{align}\np_{\\rho_1} &=  \\frac{2 n}{n!}  \\langle \\chi_{\\rho_1}, \\chi_{\\text{\\bf 1}_{D_n}} \\rangle_{D_n}\n\\nonumber\\\\\n&=  \\frac{2 n}{n!}  \n\\end{align}\n\nWhen $i = 2$, $\\rho_2$ is the representation which sends all elements in $\\langle x \\rangle$ to $\\begin{pmatrix}1\\end{pmatrix}$ and all elements outside $\\langle x \\rangle$ to $\\begin{pmatrix}-1\\end{pmatrix}$. The probability of measuring this representation is given below.\n\n\\begin{align}\n\\langle \\chi_{\\rho_2}, \\chi_{\\text{\\bf 1}_{D_n}} \\rangle_{D_n} &= \\frac{1}{2 n} \\sum_{g\\in D_n} \\chi_{\\rho_2} \\left(g\\right) \n\\end{align}\n\nAs observed from the presentation of $D_n$, the number of elements in $\\langle x \\rangle$ is $n$. So, $n$ elements will be mapped to $1$ and $n$ elements will e mapped to $-1$. So,\n\n\\begin{align}\n\\langle \\chi_{\\rho_2}, \\chi_{\\text{\\bf 1}_{D_n}} \\rangle_{D_n} &= \\frac{1}{2 n} \\left(n \\left(1\\right) + n \\left(-1\\right)\\right) \n\\nonumber\\\\\n&= 0\n\\end{align}\n\nSo, \n\\begin{align}\np_{\\rho_2} &=  \\frac{2 n}{n!}  \\langle \\chi_{\\rho_2}, \\chi_{\\text{\\bf 1}_{D_n}} \\rangle_{D_n}\n\\nonumber\\\\\n&= 0\n\\end{align}\n\nSo, the weak Fourier sampling will not be able to determine the labels of the sign representation which sends all elements in $\\langle x \\rangle$ to $\\begin{pmatrix}1\\end{pmatrix}$ and all elements outside $\\langle x \\rangle$ to $\\begin{pmatrix}-1\\end{pmatrix}$.\n\nWhen $i = 3$, $\\rho_3$ is the representation which sends all elements in $\\langle x^2, y \\rangle$ to $\\begin{pmatrix}1\\end{pmatrix}$ and $x$ to $\\begin{pmatrix}-1\\end{pmatrix}$. The probability of measuring this representation is given below.\n\n\\begin{align}\n\\langle \\chi_{\\rho_3}, \\chi_{\\text{\\bf 1}_{D_n}} \\rangle_{D_n} &= \\frac{1}{2 n} \\sum_{g\\in D_n} \\chi_{\\rho_3} \\left(g\\right) \n\\end{align}\n\nFollowing Remark ~\\ref{rem:d-n-subgroup-order}, $\\rho_3$ sends $n$ elements of $D_n$ to $\\begin{pmatrix}1\\end{pmatrix}$ and $2 n - n = n$ elements of $D_n$ to $\\begin{pmatrix}-1\\end{pmatrix}$. So,\n\n\\begin{align}\n\\langle \\chi_{\\rho_3}, \\chi_{\\text{\\bf 1}_{D_n}} \\rangle_{D_n} &= 0\n\\end{align}\n\nSo, the weak Fourier sampling will not be able to determine the labels of the sign representation which sends all elements in $\\langle x^2, y \\rangle$ to $\\begin{pmatrix}1\\end{pmatrix}$ and all elements outside $\\langle x \\rangle$ to $\\begin{pmatrix}-1\\end{pmatrix}$.\n\nWhen $i = 4$, $\\rho_4$ is the representation which sends all elements in $\\langle x^2, x y \\rangle$ to $\\begin{pmatrix}1\\end{pmatrix}$ and $x$ to $\\begin{pmatrix}-1\\end{pmatrix}$. The probability of measuring this representation is given below.\n\n\\begin{align}\n\\langle \\chi_{\\rho_4}, \\chi_{\\text{\\bf 1}_{D_n}} \\rangle_{D_n} &= \\frac{1}{2 n} \\sum_{g\\in D_n} \\chi_{\\rho_4} \\left(g\\right) \n\\end{align}\n\nFollowing Remark ~\\ref{rem:d-n-subgroup-order}, $\\rho_4$ sends $n$ elements of $D_n$ to $\\begin{pmatrix}1\\end{pmatrix}$ and $2 n - n = n$ elements of $D_n$ to $\\begin{pmatrix}-1\\end{pmatrix}$. So,\n\n\\begin{align}\n\\langle \\chi_{\\rho_4}, \\chi_{\\text{\\bf 1}_{D_n}} \\rangle_{D_n} &= 0\n\\end{align}\n\nSo, the weak Fourier sampling will not be able to determine the labels of the sign representation which sends all elements in $\\langle x^2, x y \\rangle$ to $\\begin{pmatrix}1\\end{pmatrix}$ and all elements outside $\\langle x \\rangle$ to $\\begin{pmatrix}-1\\end{pmatrix}$.\n\nThe case when $n$ is odd may be considered as a special case of when $n$ is even and show that only the trivial representation can be sampled with non-zero probability.\n\\end{proof}\n\n\n\nNow, the probability of measuring the labels of two dimensional irreducible representations will be computed in weak Fourier sampling. The discussion starts with the following lemma.\n\n\\begin{mylem}\n\\label{lem:d-n-2-d-rep-prob-zero}\nThe probability of measuring the labels of two dimensional irreducible representations of $D_n$ is always zero.\n\\end{mylem}\n\n\\begin{proof}[Proof of Lemma ~\\ref{lem:d-n-2-d-rep-prob-zero}]\nWhen $n$ is even, there are $\\frac{n-2}{2}$ such irreducible representations. Let the irreducible representations be denoted as $\\sigma_1, \\sigma_2, \\ldots, \\sigma_k, \\dots, \\sigma_{\\frac{n-2}{2}}$.\n\nThe probability of measuring $\\sigma_k$ is given below.\n\n\\begin{align}\n\\langle \\chi_{\\sigma_k}, \\chi_{\\text{\\bf 1}_{D_n}} \\rangle_{D_n} &= \\frac{1}{2 n} \\sum_{g\\in D_n} \\chi_{\\sigma_k} \\left(g\\right) \n\\end{align}\n\nFollowing Remark ~\\ref{rem:char-dn}, $\\sigma_k$ maps $n$ number of elements to the matrices for which the characters of the representations are zero. For the rest $n$ number  of the group elements, the character is $2 \\cos \\left(\\frac{2 \\pi k l}{n}\\right)$ where $0\\le l \\le n-1$. So,\n\n\\begin{align}\n\\langle \\chi_{\\sigma_k}, \\chi_{\\text{\\bf 1}_{D_n}} \\rangle_{D_n} &= \\frac{1}{2 n} \\sum^{n-1}_{l = 0} 2 \\cos \\left(\\frac{2 \\pi k l}{n}\\right)\n\\nonumber\\\\\n&= \\frac{1}{ n} \\sum^{n-1}_{l = 0}  \\cos \\left(\\frac{2 \\pi k l}{n}\\right)\n\\end{align}\n\nThe following formula for the sum of series of cosines when they are in arithmetic progression is worth mentioning as they have been proven in both \\cite{knapp2009sines} and ~\\cite{holdener2009math}.\n\n\\begin{align}\n\\sum_{l=1}^{n} \\cos (l\\theta)=\\frac{\\sin(n\\theta\/2)}{\\sin(\\theta\/2)}\\cos ((n+1)\\theta\/2),\\quad \\sin(\\theta\/2)\\neq0.\n\\end{align}\n\nWhen $\\theta=\\dfrac{2 \\pi k}{n}$,\n\n\\begin{align}\n\\sum^n_{l=1} \\cos \\left(\\frac{2 \\pi k l}{n}\\right) &=\\frac{\\sin(\\pi k)}{\\sin(\\pi k\/n)}\\cos ((n+1)\\pi k\/n)= 0\n\\end{align}\n\nIf the interval of $l$ is changed from $\\left[1, n\\right]$ to $\\left[0, n-1\\right]$ the sum still remains zero.\n\n\\begin{align}\n\\sum^{n-1}_{l=0} \\cos \\left(\\frac{2 \\pi k l}{n}\\right) &= \\cos (0)+0-\\cos(2\\pi k)=0.\n\\end{align}\n\nSo, the probability of measuring the labels of the two dimensional irreducible representations is:\n\n\\begin{align}\n\\langle \\chi_{\\sigma_k}, \\chi_{\\text{\\bf 1}_{D_n}} \\rangle_{D_n} &= \\frac{1}{ n} \\sum^{n-1}_{l = 0}  \\cos \\left(\\frac{2 \\pi k l}{n}\\right)\n\\nonumber\\\\\n&= 0\n\\end{align}\n\nThe case of $n$ being odd can be considered as a special case of $n$ being even and the same result can be proved.\n\\end{proof}\n\nSo, the weak Fourier sampling algorithm cannot determine the labels of any of the two dimensional irreducible representations. To summarize, weak Fourier sampling cannot determine any irreducible representation other than the trivial one. So, it cannot determine the automorphism group of the cycle graph which is a trivial problem in the classical paradigm. Lemmas ~\\ref{lem:d-n-1-d-rep-prob-zero}  and ~\\ref{lem:d-n-2-d-rep-prob-zero} may be consolidated into the following theorem.\n\n\\begin{mytheo}\n\\label{theo:cycle-graph-auto-weak-fails}\nWeak quantum Fourier sampling fails to solve the cycle graph automorphism problem.\n\\end{mytheo}\n\n\\begin{proof}\nFollows directly from the proofs of the Lemmas ~\\ref{lem:d-n-1-d-rep-prob-zero}  and ~\\ref{lem:d-n-2-d-rep-prob-zero}.\n\\end{proof}\n\nIt is observed that the success probability of strong Fourier sampling is a conditional probability which depends on the success probability of measuring the labels of representation. As the success probability of measuring the non-trivial representations of $D_n$ is zero, the success probability of measuring their individual matrix elements is also zero.\n\n\\begin{mycor}\n\\label{cor:cycle-graph-auto-strong-fails}\nStrong quantum Fourier sampling fails to solve the cycle graph automorphism problem.\n\\end{mycor}\n\nSo far, the discussion in this paper has been limited to the POVMs in the computational basis. One could ask if the results are comparable if one  were performing entangled measurements introduced in \\cite{hallgren2010limitations, moore2005tight}. It can be argued that quantum Fourier transform is guaranteed to fail also in the latter type of measurement. To perform an entangled measurement, it is assumed that one has already measured the labels of the irreducible representations with nonzero probability which is not possible for cycle graphs. Hence, the automorphism group of a cycle graph cannot be computed using the measurements in either computational or entangled bases. It is also natural to ask how the results relate to the results of \\cite{radhakrishnan2005power} on using random bases. As the bases are always being chosen from the complete set of computational bases, one can argue that strong random Fourier sampling does not change the probability of measuring the labels of irreducible representations from zero (where applicable) to a larger value. It can also be argued in a similar manner for both Kuperberg sieve \\cite{kuperberg2005subexponential, moore2010impossibility} and Pretty Good Measurement (PGM) as both algorithms are conditioned on weak sampling first. The Ettinger-H{\\o}yer-Knill theorem \\cite{ettinger2004quantum} that the quantum query complexity of the hidden subgroup problem is polynomial may also be mentioned here. The theorem cannot be applied here either as the assumption of the theorem is that the quantum algorithm will always output a subset of the hidden subgroup.\n\n\\section{Quantum Fourier transform for other graph automorphism problems}\n\\label{sec:qft-ga-plus}\nIn Section ~\\ref{sec:weak-cycle}, it has been shown that the  Fourier sampling fails to determine the automorphism group of the cycle graphs. It is natural to ask whether same is the case for the graphs which have the dihedral group as a subgroup in it's automorphism group. The question can be answered in the affirmative. Those cases can be instantiated using any of the following three approaches.\n\n\\subsection{Inductive approach}\nThis approach starts with a graph whose automorphism group is a dihedral group. Then, more complicated structures are inductively built on that graph. This is based of the following theorem reproduced from \\cite{dresselhaus2007group} and also mentioned as Theorem $10$ in Section $3.2$ in \\cite{serre2012linear}. To initiate the discussion,  an example can be provided, where a more complex graph is constructed from cycle graphs and show that quantum Fourier sampling is still guaranteed to fail to compute the automorphism group.\n\n\\begin{myexamp}[QFT for the automorphism group of $C_m \\sqcup C_n$]\n\\label{examp:c-m-c-n}\nThe graph $C_m \\sqcup C_n$ can be visualized as follows.\n\nIt is already known from \\cite{ganesan2012automorphism, jordan1869assemblages} that the automorphism group of $C_m \\sqcup C_n$ is $D_m \\times D_n$ where the dihedral groups are of order $2 m$ and $2 n$ respectively.\n\nFollowing the schemes introduced in Section ~\\ref{sec:weak-cycle},  the probabilities of measuring the labels irreducible representations of $D_m \\times D_n$ are computed.\n\nThe Theorem $10 (ii)$ in \\cite{serre2012linear} indicates that every irreducible representation of $D_m \\times D_n$ can be determined from the irreducible representations of $D_m$ and $D_n$.\n\nThe general expression for irreducible representations of dihedral groups is used as it is  given in Section ~\\ref{sec:weak-cycle}. Let the $i$-th irreducible $j$-dimensional representation of a dihedral group $D_n$ be $\\rho_{i,j, n}$ where $n$ is even. For example, for $D_m$, the irreducible representations are $\\rho_{1,1, m}$, $\\rho_{2,1, m}$, $\\rho_{3,1, m}$, $\\rho_{4,1, m}$, $\\rho_{1,2, m}$, $\\rho_{2,2, m}$, $\\rho_{3,2, m}$, and $\\rho_{4,2, m}$. The expressions of these representations can be derived from Section ~\\ref{sec:irrepdn}. It needs to be noted that in the two dimensional irreducible representations, there is an additional ordering parameter $k$. Hence, $k_m$ and $k_n$ will be used for the groups $D_m$ and $D_n$ respectively. So, for example, $\\begin{pmatrix}\ne^{\\frac{2 \\pi i k_{n}}{n}}&0\\\\\n0&e^{-\\frac{2 \\pi i k_{n}}{n}}\n\\end{pmatrix}$ denotes the $k_{n}$-th two dimensional irreducible representation for the element $x$. Another parameter $l$ is used to identify the elements $x^l$ in the group $D_n$ where $0 \\le l \\le n-1$. Let the parameter be $l_m$ for $D_m$ and $l_n$ for $D_n$.\n\nNow, the irreducible representations of $D_m \\times D_n$ are enumerated. The corresponding characters are also computed at the same time. There are sixteen one dimensional, thirty two two dimensional, and sixteen four dimensional irreducible representations. Table ~\\ref{table:irrep-cn-cm} summarizes the characters of the irreducible representations of $D_m \\times D_n$.  Curious readers may refer to Appendix ~\\ref{app:cmcn} for detailed derivations.\n\n\\begin{table}[H]\n\\centering\n\\begin{tabular}{ c| c   } \n \\hline\nIrreducible representation  & Character\\\\ \n \\hline\\hline\n  $\\rho_{7, 8, 15, 16, 23, 24, 31, 32, 39, 40, 47-64} $ &  $\\chi_{7, 8, 15, 16, 23, 24, 31, 32, 39, 40, 47-64} = 0$\\\\\n   $\\rho_{1}$ & $\\chi_{1} = 1$ \\\\\n $\\rho_{2-4, 9-12, 17-20, 25-28}$ & $\\chi_{2-4, 9-12, 17-20, 25-28} = \\pm 1$ \\\\\n $\\rho_5 $ &  $\\chi_5 =  2 \\cos \\left(\\frac{2 \\pi k_m}{m}\\right)$\\\\\n $\\rho_{6, 22} $ &  $\\chi_{6, 22} = 2 \\cos \\left(\\frac{2 \\pi k_m l_m}{m}\\right)$\\\\\n $\\rho_{13, 21, 29} $ & $\\chi_{13, 21, 29} = \\pm 2 \\cos \\left(\\frac{2 \\pi k_m}{m}\\right)$\\\\\n $\\rho_{14, 30} $ & $\\chi_{14, 30} = \\pm  2 \\cos \\left(\\frac{2 \\pi k_m l_m}{m}\\right)$\\\\\n $\\rho_{33} $ & $\\chi_{33} = 2 \\cos \\left(\\frac{2 \\pi k_n}{n}\\right)$\\\\\n $\\rho_{34-36}$ & $\\chi_{34-36} =\\pm 2 \\cos \\left(\\frac{2 \\pi k_n}{n}\\right)$\\\\\n $\\rho_{37} $ & $\\chi_{37} = 4 \\cos \\left(\\frac{2 \\pi  k_m}{m}\\right) \\cos \\left(\\frac{2 \\pi  k_n}{n}\\right)$\\\\\n $\\rho_{38} $ & $\\chi_{38} = 4 \\cos \\left(\\frac{2 \\pi  k_n}{n}\\right) \\cos \\left(\\frac{2 \\pi  k_m l_m}{m}\\right)$\\\\\n $\\rho_{41} $ & $\\chi_{41} = 2 \\cos \\left(\\frac{2 \\pi k_n l_n}{n}\\right)$\\\\\n $\\rho_{42-44} $ & $\\chi_{42-44} =\\pm 2 \\cos \\left(\\frac{2 \\pi k_n l_n}{n}\\right)$\\\\\n $\\rho_{45}$ & $\\chi_{45} = 4 \\cos \\left(\\frac{2 \\pi  k_m}{m}\\right) \\cos \\left(\\frac{2 \\pi  k_n l_n }{n}\\right)$\\\\\n $\\rho_{46} $ & $\\chi_{46} = 4 \\cos \\left(\\frac{2 \\pi  k_m l_m}{m}\\right) \\cos \\left(\\frac{2 \\pi  k_n l_n }{n}\\right)$\\\\\n\\hline\n\\end{tabular}\n\\caption{Irreducible representations of $\\text{Aut} \\left(C_m \\sqcup C_n\\right)$}\n\\label{table:irrep-cn-cm}\n\\end{table}\n\nIt can be readily seen that the quantum Fourier transform is guaranteed to fail to measure the labels of   $\\rho_{7, 8, 15, 16, 23, 24, 31, 32, 39, 40, 47-64}$. With little more algebraic steps, it can be shown to be true for few more irreducible representations.\n\\end{myexamp}\n\nFollowing Example ~\\ref{examp:c-m-c-n}, one can create arbitrarily large classes of graph automorphism problem by based on the known Theorem ~\\ref{lab:irrep-prod-grp} (not a result original to this paper) for which quantum Fourier transform will always fail. The proof of the Theorem ~\\ref{lab:irrep-prod-grp} is given as Theorem $10$ in \\cite{serre2012linear}\n\n\\begin{mytheo}\n\\label{lab:irrep-prod-grp}\nThe direct product of two irreducible representations of groups $H$ and $K$ yields an irreducible representation of the direct product group so that all irreducible representations of the direct product group can be generated from the irreducible representations of the original groups before they are joined.\n\\end{mytheo}\n\nThis paper provides Algorithm ~\\ref{algo:cycle-graph-infinite} which is a way to generate arbitrarily large class of graph automorphism problems for which quantum Fourier sampling is guaranteed to fail.\n\n\\begin{algorithm}[H]\n\\label{algo:arb-fail}\n\\caption{An algorithm to create arbitrarily large easy graph automorphism problem}\n     \\label{algo:cycle-graph-infinite}\n\\begin{algorithmic}[1]\n\\Procedure {DISJOINT-CYCLE-GRAPH}{$m, n$} \\Comment{$m$ is the number of nodes for the smallest cycle, $n$ is the number of cycles}\n\n\\State Create an $m$-cycle graph \\Comment{The first cycle}\n\n\\For{$i\\gets 1, n-1 $}\n\\State Increase $m$ by one\n\\State Create an $m$-cycle graph \\Comment{The next cycle}\n\\EndFor\n\\EndProcedure\n\\end{algorithmic}\n\\end{algorithm}\n\nBy the end of its execution, Algorithm ~\\ref{algo:cycle-graph-infinite} will generate a graph of $n$ cycles with a total of $m n$ nodes and the time complexity will be $O\\left(poly(m, n)\\right)$. One can be more creative about the Step 4 of Algorithm ~\\ref{algo:cycle-graph-infinite} to create other classes of graph automorphism problems for which quantum Fourier sampling is guaranteed to fail.\n\nAt this point following remark can be made.\n\n\\begin{myrem}\n\\label{rem:dn-prod-g-qft-fail}\nArbitrarily large classes of graph automorphism problems can be created for which quantum Fourier sampling is guaranteed to fail. Quantum Fourier sampling is guaranteed to fail to compute the automorphism group of a graph when the automorphism group is the product of a dihedral group and any finite group.\n\\end{myrem}\n\n\\subsection{Existential approach}\nIn the existential approach, a general class of graphs is chosen and it is proven that there is at least one graph in that class for which quantum hidden subgroup algorithm is guaranteed to fail to compute the automorphism group.\n\nThe discussion starts with the Frucht's theorem \\cite{frucht1939herstellung}.\n\n\\begin{mytheo}\nEvery abstract group is isomorphic to the automorphism group of some graph.\n\\end{mytheo}\n\nSo, any group which is a product of $D_n$ and a finite group $G$ is isomorphic to the automorphism group of some finite graph. It has also been shown in \\cite{babai1996automorphism} that  every finite group as the group of symmetries of a strongly regular graph. It indicates that there is a class of strongly regular graph whose automorphism group is isomorphic to $D_n \\times G$. According to the Theorem ~\\ref{lab:irrep-prod-grp} and Remark ~\\ref{rem:dn-prod-g-qft-fail}, one can argue that  quantum Fourier sampling should fail to construct the automorphism group of a subclass of strongly regular graphs.\n\nAnother example may be the  Cayley graph automorphism problem \\cite{xu1998automorphism}. It is well known that the automorphism group of the Cayley graph $\\text{Aut}\\left( C \\left(G, X\\right)\\right)$ of a group $G$ over a generating set $X$ contain an isomorphic copy of $G$ acting via left translations \\cite{jajcay2000structure}. In that case, the automorphism group of the Cayley graph of a dihedral group $D_n$  contains $D_n$ as a subgroup. So, following the result of the previous section, quantum Fourier sampling fails to compute the automorphism group of $\\text{Aut}\\left( C \\left(G, X\\right)\\right)$.\n\n\\subsection{Universal structures approach}\nA class $\\mathcal{C}$ of structures  is called {\\it universal} if every finite group is the automorphism group of a structure in $\\mathcal{C}$  \\cite{cameron2004automorphisms}. A series of works by Frucht, Sabidussi, Mendelsohn, Babai, Kantor, and others \\cite{cameron2004automorphisms} has shown the following classes of graphs to be universal - graphs of valency $k$ for any fixed $k > 2$ \\cite{frucht1949graphs}; bipartite graphs; strongly regular graphs \\cite{mendelsohn1978every}; Hamiltonian graphs \\cite{sabidussi1957graphs}; $k$-connected graphs \\cite{sabidussi1957graphs}, for $k > 0$; $k$-chromatic graphs, for $k > 1$; switching classes of graphs; lattices \\cite{birkhoff1946grupos}; projective planes (possibly infinite); and Steiner triple systems \\cite{mendelsohn1978groups}; and symmetric designs (BIBDs). It indicates that each of these classes has at least one graph which has its automorphism group isomorphic to $D_n \\times G$ where $G$ is any finite group. So, quantum Fourier sampling will fail to compute the automorphism group of each of these cases.\n\n\\section{Is hidden subgroup the ideal approach?}\nIt has been shown that there are instances of graph automorphism problem for which hidden subgroup algorithm can never be successful although they are trivial to solve on a classical computer. So, it can be argued that the space of the hidden subgroup representations of all graph automorphism problems cannot capture the structure of the space of all graph automorphism problem. As the graph isomorphism problem is believed to be at least as hard as the graph automorphism problem, it can also be added that the space of the hidden subgroup representations of all graph isomorphism problem cannot capture the structure of the space of all graph isomorphism problem. So, it would be appropriate to investigate alternative quantum algorithmic approach for these classes of problems.\n\n\\section{Conclusion}\nIt has been shown that, while solving the hidden subgroup representation of the graph isomorphism problem is equivalent to determining order $2$ subgroup of a symmetric group,  the hidden subgroup representation of the graph automorphism problem is equivalent to determining a hidden subgroup of higher order. This paper has identified a class of graph automorphism problem for which the quantum Fourier transform algorithm always fails. It also has shown how one can determine non-trivial classes of graphs for which the same algorithm always fails. With these negative results, one may be interested to ask whether the hidden subgroup representation is a practical representation of the graph isomorphism and automorphism problems in quantum regime.\n\n\\section*{Acknowledgement}\nOS thanks Dave Bacon, Aram Harrow, Robert Campbell, Marc Bogaerts, Andrew Childs,  Steven Gregory, Jef Laga, Dietrich Burde, Eric Wofsey, Alexander Hulpke, Michael Burr, Jyrki Lahtonen, Joshua Grochow and  Tobias Kildetoft for their helpful comments.\n\n\\newpage\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{sec:intro}\nMassive black holes (BHs) with masses >$10^{6}$$M_{\\odot}$\\ reside in most galaxies, including our own \\citep{Genzel:1996aa, Magorrian:1998aa, Schodel:2003aa,Ghez:1998aa, Ghez:2000aa, Ghez:2008aa}. Luminous quasars with $M_\\mathrm{BH}$\\ $\\sim10^{9}$$M_{\\odot}$\\ (\\citealt{Barth:2003ab}: $z = 6.4$, \\citealt{Willott:2003aa}: $z = 6.41$, \\citealt{Trakhtenbrot:2011aa}: $z\\sim4.8$) have been detected at $z\\sim5-7$ (\\citealt{Fan:2000aa, Fan:2001aa}: $z\\sim6$, \\citealt{Mortlock:2011aa}: $z = 7.085$). The BHs powering these quasars must therefore build up their mass in less than one billion years. Depending on the assumed seed formation model, almost constant Eddington accretion or even super-Eddington episodes are required to match these observations \\citep{Volonteri:2005aa, Volonteri:2014aa, Alexander:2014aa}.\nIn our current understanding, BHs grew out of $\\sim100-10^{5}$$M_{\\odot}$\\ seeds by accreting infalling matter or merging with a second BH \\citep{Rees:2007aa}.  \nTwo seed formation models are currently favored. One scenario predicts that the remnants of massive Population III (Pop III) stars constitute BH progenitors \\citep{Madau:2001aa, Haiman:2001aa, Bromm:2002aa, Alvarez:2009aa, Johnson:2012aa}. The second model is based upon the direct gravitational collapse of massive gas clouds \\citep{Loeb:1994aa, Bromm:2009aa, Volonteri:2010aa, Latif:2013ab}. Both models include uncertainties and predict markedly different BH growth histories. More exotic scenarios, including BH seed formation via stellar dynamical, rather than gas dynamical processes, have also been suggested ( see \\citealt{Volonteri:2010aa, Bromm:2011aa} and references therein). These scenarios primarily focus on reproducing the high-redshift quasar population. We must however also be able to explain the existence of less massive and luminous, but more abundant BHs that we find in galaxies such as the Milky Way \\citep{Treister:2011aa, Volonteri:2010aa}. A first step towards constraining seed formation models and determining if they are also valid for such 'normal' BHs, is measuring the BH luminosity function at high redshift.\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=0.49\\textwidth]{goods_scat_plot}\n\\end{centering}\n\\caption{\\label{fig:offset}Offset between GOODS\/ACS and \\textit{Chandra}\\ positions after the correction has been applied. Out of the 740 \\textit{Chandra}\\ X-ray sources 408 possess an optical counterpart. We determine a mean offset of (0.128$^{\\prime\\prime}$, -0.237$^{\\prime\\prime}$) for these 408 objects and correct for it by shifting the \\textit{Chandra}\\ positions. The grey triangle indicates the mean displacement before the correction. The green star illustrates the mean offset after our correction. The black points show the corrected object positions. The grey circle illustrates an offset of 0.5$^{\\prime\\prime}$.}\n\\end{figure}\n\nThe \\textit{Chandra}\\ 4-Ms catalog \\citep{Xue:2011aa} provides X-ray counts for the soft (0.5 keV - 2 keV), hard (2 keV - 8keV) and full (0.5 keV - 8 keV) band for the \\textit{Chandra}\\ Deep Field South (CDF-S). The on-axis flux limits lie at $9.1 \\times 10^{-18}$, $5.5 \\times 10^{-17}$ and $3.2 \\times 10^{-17} \\mathrm{erg}\\ \\mathrm{s}^{-1}\\ \\mathrm{cm}^{-2}$ for the soft, hard and full band, respectively. The CDF-S covers a $0.11$ $\\mathrm{deg}^2$ area. For our analysis we not only use the \\textit{Chandra}\\ 4-Ms data, but also require coverage by the CANDELS wide and deep surveys. The effective area of our field is hence 0.03 $\\mathrm{deg}^2$ \\footnote{README CANDELS GOODS-S Data Release v1.0 \\url{http:\/\/archive.stsci.edu\/pub\/hlsp\/candels\/goods-s\/gs-tot\/v1.0\/hlsp_candels_hst_acs-wfc3_gs-tot_readme_v1.0.pdf}}. \n\n\\cite{T13} used the \\textit{Chandra}\\ 4-Ms data in their search for high-redshift ($z > 6$) AGN. Using a sample of preselected $z = 6 - 8$ Lyman Break dropout and photometrically selected sources from the HUDF and CANDELS, \\cite{T13} showed that none of these sources are detected individually in the X-rays. Stacking the X-ray observations does not produce a significant detection either. \\cite{T13} suggested different processes that could account for the lack of X-ray counterparts to these high-redshift sources. \nThe sample could be contaminated by a large number of low-redshift interlopers. \nA low BH occupation fraction could explain the lack of X-ray counterparts. \nIt is possible that BH growth only occurs in dusty and\/or small galaxies which were not included in this analysis because they lie below the detection threshold. \nThe Treister et al. results can also be explained if large amounts of gas and dust obscure the X-ray emission of actively accreting BHs, \nas previously proposed by \\cite{Treister:2009aa} and \\cite{Fiore:2009aa}.  \nAdditionally, it is possible that accretion is not the dominant BH growth mode in the early universe. If BHs are primarily gaining mass by merging with other BHs, X-ray radiation might not probe BH activity. \n(See \\citealt{T13} for a more detailed discussion of these possible scenarios.)\nIn addition to the scenarios described above, short, super-Eddington growth episodes, as proposed by \\cite{Madau:2014aa} and \\cite{Volonteri:2014aa}, also present a possible solution. \nIn comparison to constant Eddington accretion, the amount of matter that is accreted during these super-Eddington growth phases is the same. However, \\cite{Madau:2014aa} showed that, if we allow super-Eddington accretion, a duty cycle of $20\\%$ is enough to grow a non-rotating $100$ $M_{\\odot}$\\ seed BH into a $10^9$ $M_{\\odot}$\\ object by $z\\sim7$. One could imagine that the seed BH grows via five $20\\ \\mathrm{Myr}$ long $\\dot{m}\/\\dot{m}_\\mathrm{Edd}=4$ growth modes, each followed by a $100\\ \\mathrm{Myr}$ phase of quiescence. For short, super-Eddington growth episodes, we would thus expect to find fewer BHs that are actively accreting at the same time. The \\cite{T13} sample could therefore not contain any BHs that are actively accreting at the time of observation. \n\n\\label{sec:AnalysisSteps}\n\\begin{figure*}\n\\begin{centering}\n\\input{MindMap2.tex}\n\\caption{\\label{fig:flowchart} Flowchart illustrating the analysis steps that we take to determine possible $z\\gtrsim5$ candidates. The number of objects that pass each step is given in brackets. After executing this analysis for all 740 Chandra sources, three $z\\gtrsim5$ candidates remain in our sample. Additional $K_S$-band and deep $Y$-band data does however show that they are most likely low-redshift sources.}\n\\end{centering}\n\\end{figure*}\n\nIn this work we carefully examine the \\textit{Chandra}\\ 4-Ms catalog for possible $z\\gtrsim5$ AGN. We combine the deep \\textit{Chandra}\\ observations with optical and infrared data from GOODS, CANDELS and \\textit{Spitzer}. We use the Lyman Break Technique and a photometric redshift code to estimate the redshift of our targets. We also use colour criteria, stacking and the X-ray Hardness Ratio. In contrast to \\citealt{T13}, we base this analysis on detected X-ray sources instead of trying to determine if a high-redshift object possesses a X-ray counterpart. We therefore also analyse X-ray sources that would not be classified as Lyman Break Galaxies because they are heavily obscured in the optical and the infrared. \n\nWe know that the CDF-S contains hundreds of well constrained $z\\gtrsim5$ Lyman Break Galaxies (see e.g. \\citealt{Stark:2009aa, Vanzella:2009aa, Wilkins:2010aa, Bouwens:2014aa, Duncan:2014aa}). These should all pass our manual inspection, stacking, colour criteria and photometric redshift measurement. To be considered as a high-redshift AGN candidate, they must however also be detected in the X-rays and pass our X-ray Hardness Ratio test.\n\n\\cite{Volonteri:2010ab} showed that the expected number density of high-redshift AGN depends on the assumed seed formation model. Both, a detection and a non-detection, of high-redshift AGN gives us a lower limit on this number density. Our search thus constrains possible seed formation scenarios and sheds light on BH growth modes.  \n\n\\cite{Vito:2013aa} searched for $z > 3$ AGN in the 4-Ms CDF-S. They mainly analysed the evolution of obscuration and AGN space density with redshift. In contrast to this work, \\cite{Vito:2013aa} based their analysis on already existing photometric and spectroscopic information on the \\textit{Chandra}\\ sources. We compare our results to \\cite{Vito:2013aa} in Section \\ref{sec:discussion}. \n\nThis paper is organized as follows. Section \\ref{sec:data} describes the data that is used in this work. Sections \\ref{sec:analysis} and \\ref{sec:combination} introduce our redshift tests and illustrate the results of their combination. We conclude with a discussion in Section \\ref{sec:discussion} and a summary in section \\ref{sec:summary}. Throughout this paper we assume a $\\Lambda$CDM cosmology with $h_0$ = 0.7, $\\Omega_\\mathrm{m}$ = 0.3 and $\\Omega_\\Lambda$ = 0.7. All magnitudes are given in the AB system \\citep{Oke:1983aa}.\n\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=0.49\\textwidth]{CDFs_goods_candels_field}\n\\end{centering}\n\\caption{\\label{fig:fields}Overview of the area covered by the HST\/ACS filters $B$ (red), $V$, $i$, $z$ (blue), the HST\/WFC3 bands $Y$ (green), $J$, $H$ (purple) and our 740 \\textit{Chandra}\\ 4-Ms sources. Grey points mark sources that are not of interest for this work because they are not covered by enough bands ($B$, $V$, $i$, $z$, $J$, $H$) to use the Lyman Break Technique. Black points indicate objects with enough filter coverage that were eliminated because the objects are clearly visible in all bands. According to the Lyman Break Technique this indicates $z < 4$. Yellow stars mark the positions of the 58 potential high-redshift AGN that we analyse more closely.}\n\\end{figure}\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=\\columnwidth]{photometry_measuredSens}\n\\caption{\\label{fig:comp_phot} Comparison between our own photometry and the flux values reported in the GOODS catalog  \\citep{Giavalisco:2004aa}. We show the AB magnitude values for the $z$-band. Note that the GOODS\/ACS catalog only contains flux values for an aperture with a 0.707$^{\\prime\\prime}$ and not a 0.75$^{\\prime\\prime}$ radius. Nonetheless, we compare these values to the brightness we measured within a 0.75$^{\\prime\\prime}$ radius aperture. We use a small 0.3$^{\\prime\\prime}$ radius aperture for two objects only. Since these two sources are not detected in the GOODS\/ACS catalog, they are not part of this comparison.}\n\\end{centering}\n\\end{figure}\n\n\\section{Data}\n\\label{sec:data}\n\nThe \\textit{Chandra}\\ 4-Ms source catalog by \\cite{Xue:2011aa} is the starting point of this work. It contains 740 sources and provides counts and observed frame fluxes in the soft (0.5 keV - 2 keV), hard (2 keV - 8 keV) and full (0.5 keV - 8 keV) band. All object IDs used in this work refer to the source numbers listed in the \\cite{Xue:2011aa} \\textit{Chandra}\\ 4-Ms catalog.\nWe make use of HST\/ACS data from the GOODS-south survey in the optical wavelength range. We use catalogs and images for filters F435W ($B$), F606W ($V$), F775W ($i$) and 850LP ($z$) from the second GOODS\/ACS data release (v2.0) \\citep{Giavalisco:2004aa}. \nWe use CANDELS WFC3\/IR data from the first data release (v1.0) for passbands F105W ($Y$), F125W ($J$) and F160W ($H$) \\citep{Grogin:2011aa,Koekemoer:2011aa}. \nTo determine which objects are red, dusty, low-redshift interlopers, we also include the 3.6 micron and 4.5 micron \\textit{Spitzer}\\ IRAC channels. We use SIMPLE  image data from the first data release (DR1, \\citealt{van-Dokkum:2005aa}) and the first version of the extended SIMPLE catalog by \\cite{Damen:2011aa}.  \n\nWhen comparing \\textit{Chandra}, GOODS\/ACS and CANDELS object positions, a clear offset in the \\textit{Chandra}\\ coordinates is apparent. We illustrate this inconsistency in Figure \\ref{fig:offset}. To correct for this discrepancy, we calculate  the mean displacement between the \\textit{Chandra}\\ and the GOODS\/ACS catalog. We determine a mean offset of $\\mathrm{RA}_{Chandra}-\\mathrm{RA}_\\mathrm{ACS}=0.128^{\\prime\\prime}$ and $\\mathrm{DEC}_{Chandra}-\\mathrm{DEC}_\\mathrm{ACS}=-0.237^{\\prime\\prime}$.  We adjust the GOODS\/ACS and CANDELS positions of each object by subtracting the mean displacement from the originally given catalog position. \n \n\\section{Analysis}\n\\label{sec:analysis}\nIn the following section we describe in detail the set of criteria we employed to the data in order to identify $z\\gtrsim5$ candidates. We first exclude objects with insufficient filter coverage, perform our own aperture photometry and determine the dropout band of each source by manual inspection. We  run a photometric redshift code, stack the GOODS\/ACS data and apply colour criteria. In addition, we use the X-ray data as a photometric redshift indicator. We combine all redshift tests in section \\ref{sec:combination}. Figure \\ref{fig:flowchart} illustrates and summarizes the complete analysis that is detailed in the following subsections. \n\n\\cite{Dahlen:2010aa}, \\cite{McLure:2011aa} and \\cite{Duncan:2014aa} showed that when selecting high-redshift objects, a selection based only on colour criteria is not as reliable as calculating photometric redshifts. Especially for faint, low signal-to-noise objects, errors and upper limits can lead to scattering out of the colour selection region. Similar to colour criteria, photometric redshifts strongly depend on the position of the Lyman Break. A photometric redshift code does however consider all filter information, including upper limits and errors. Furthermore, low-redshift interlopers can be identified by including the filters redward of the Lyman Break. \\cite{Dahlen:2010aa} illustrated the discrepancy between colour criteria and photometric redshifts for the GOODS-S field. Only 50$\\%$ of their photometrically selected $z\\sim4$ sources were also classified as $B$-dropouts according to colour criteria. We therefore primarily use a photometric redshift code. The colour criteria, our visual classification, the stacking of the GOODS\/ACS data and the X-ray Hardness Ratio provide additional redshift indications. \n\n\\subsection{Initial Sample Selection}\nOur initial sample consists  of the 740 objects given in the \\textit{Chandra}\\ 4-Ms source catalog. Figure \\ref{fig:fields} shows that not all \\textit{Chandra}\\ targets are covered by the GOODS\/ACS and CANDELS images. However, adjacent filter coverage is necessary for the application of the Lyman Break Technique. We therefore narrow the number of possible candidates down to 374 by removing sources that are not covered by $B$, $V$, $i$, $z$, $J$ and $H$. The $Y$-band area is small compared to the other filters. We therefore also include sources that are not covered by the $Y$-band provided that they are covered by all other GOODS\/ACS and CANDELS filters. The optical and infrared counterpart detection is primarily based upon the $H$-band image since this is the deepest band. The $H$-band images for objects 105 and 521 show significant artifacts, we therefore discard them. Source 366 is eliminated due to the object's position being at the edge of the GOODS\/ACS images. Hence, 371 possible candidates remain after this first visual preselection.  \n\n\\begin{figure*}\n\\begin{minipage}{\\textwidth}\n\\includegraphics[width=\\textwidth]{208_series_stamps}\n\\end{minipage}\n\\begin{minipage}{\\textwidth}\n$\\rightarrow \\textit{eliminated: lack of CANDELS coverage}$\n\\end{minipage}\n\n\\begin{minipage}{\\textwidth}\n\\includegraphics[width=\\textwidth]{141_series_stamps}\n\\end{minipage}\n\\begin{minipage}{\\textwidth}\n$\\rightarrow \\textit{eliminated: source is clearly visible by eye in the optical and the infrared, this indicates $z < 4$ according to}$\\\\\n$\\textit{the Lyman Break Technique}$\n\\end{minipage}\n\n\\begin{minipage}{\\textwidth}\n\\includegraphics[width=\\textwidth]{184_series_stamps}\n\\end{minipage}\n\\begin{minipage}{\\textwidth}\n$\\rightarrow  \\textit{kept in sample: by eye classified as a $B$-dropout, this indicates $z\\sim4$ according to the Lyman Break Technique}$\n\\end{minipage}\n\n\\caption{\\label{fig:visclass}Classification examples. We only kept source 184 in our sample. The images are 10$^{\\prime\\prime}$ x 10$^{\\prime\\prime}$ in size and were colour inverted. }\n\\end{figure*}\n\n\n\\begin{figure*}\n\n\\begin{minipage}{\\textwidth}\n\\includegraphics[width=\\textwidth]{190_series_stamps}\n\\end{minipage}\n\n\\begin{minipage}{\\textwidth}\n\\includegraphics[width=\\textwidth]{280_series_stamps}\n\\end{minipage}\n\n\\begin{minipage}{\\textwidth}\n\\includegraphics[width=\\textwidth]{333_series_stamps}\n\\end{minipage}\n\n\\begin{minipage}{\\textwidth}\n\\includegraphics[width=\\textwidth]{384_series_stamps}\n\\end{minipage}\n\n\\begin{minipage}{\\textwidth}\n\\includegraphics[width=\\textwidth]{643_series_stamps}\n\\end{minipage}\n\n\\begin{center}\n\\line(1,0){450}\n\\end{center}\n\n\\begin{minipage}{\\textwidth}\n\\includegraphics[width=\\textwidth]{104_series_stamps}\n\\end{minipage}\n\n\\begin{minipage}{\\textwidth}\n\\includegraphics[width=\\textwidth]{156_series_stamps}\n\\end{minipage}\n\n\\begin{minipage}{\\textwidth}\n\\includegraphics[width=\\textwidth]{276_series_stamps}\n\\end{minipage}\n\n\\caption{\\label{fig:mysteryobjects}Sources that can not be classified according to their dropout band. $Top$: Sources that are in the \\textit{Chandra}\\ 4-Ms catalog but do not show a clear counterpart in the optical and infrared, \\textit{'low significance objects'}. 333, 384 and 643 are close to a bright galaxy which is why we might not be able to detect the counterpart in the optical and infrared. Deeper observations would be needed to detect possible counterparts. $Bottom$: Sources with multiple possible counterparts. Out of these eight objects only 156, 276 and 333 are simultaneously detected in the hard and in the soft band. We are unable to gain redshift estimates for these eight objects and are thus unable to determine if they are low or high-redshift sources or if they might be spurious detections. The black circles are centered on the original \\textit{Chandra}\\ position and illustrate the positional uncertainty given in the \\protect\\cite{Xue:2011aa} catalog. All images were colour inverted and are 10$^{\\prime\\prime}$ x 10$^{\\prime\\prime}$ in size.} \n\\end{figure*}\n\n\\subsection{Aperture Photometry}\n\\label{sec:aperturephot}\nWe perform our own aperture photometry on the GOODS\/ACS and CANDELS images to gain flux values and to estimate parameters such as detection threshold and aperture size. We compare our results to the GOODS\/ACS catalog (Figure \\ref{fig:comp_phot}). \n\nTo perform aperture photometry we use Source-Extractor (SExtractor, \\citealt{Bertin:1996aa, 2002Terapix, Holwerda:2005aa}). We determine the counterpart position in the $H$ band in a first SExtractor run. We then run SExtractor on the remaining optical and infrared images to establish if the counterpart is present at the same location. The flux measurements are carried out within circular apertures with radii between 0.3$^{\\prime\\prime}$ and 1$^{\\prime\\prime}$. We alter the aperture size for faint sources and to prevent contamination through nearby objects. For sources with a signal-to-noise ratio $<1$ we use the $1\\sigma$ sensitivity limit of the corresponding filter as an upper limit. The SExtractor parameter values and aperture sizes are summarized in Table \\ref{tab:SE_para} and Table \\ref{tab:SE_ID}. For the 3.6 micron and 4.5 micron bands we rely on the flux values reported in the SIMPLE catalog \\citep{Damen:2011aa}. We make use of the flux values reported for a 1.5$^{\\prime\\prime}$ radius aperture. We show flux and error values for our main sample in Tables \\ref{tab:apphot1} and \\ref{tab:apphot2}.\n\n\n\\subsection{Lyman Break Technique and visual classification}\n\\label{sec:LBT}\nThe Lyman Break Technique \\citep{Steidel:1999fj, Giavalisco:2002aa, Dunlop:2013zr} employs the pronounced feature of the Lyman continuum discontinuity in the spectral energy distribution (SED) of young, star forming galaxies. We use the common terminology of referring to Lyman Break Galaxies as 'dropouts'. If a source is not detected in the $B$-band or any bluer passbands, but is visible in all redder filters, this indicates $z\\sim4$ and we refer to it as a '$B$-dropout'. $V$-dropouts, $i$-dropouts and $z$-dropouts correspond to redshifts of $\\sim5$, $\\sim6$ and $\\sim7$ respectively. \n\nWe classify the 371 possible candidates by eye according to their dropout band. If an object is clearly visible in all bands, this indicates $z < 4$. We exclude such sources from our sample. Figure \\ref{fig:visclass} illustrates the conditions sources have to fulfill to be included in the further analysis. \\\\\n\nEight sources are not classified according to their dropout band. These objects are shown in Figures \\ref{fig:mysteryobjects}. Figure \\ref{fig:counts} shows the hard and soft band counts for all eight sources in comparison to the entire sample. The hard, soft and full band counts are given in Table \\ref{tab:mysobscounts}.\\\\ \n\n\\begin{table*}\n\t\\begin{center}\n\t\t\\begin{tabular}{llllllllll}\n\t\t\\toprule\n\t\t\t{ID} & {Hard counts} & {$\\sigma_\\mathrm{Hard}$} & {$\\mathrm{SNR}_\\mathrm{Hard}$} & {Soft counts} & {$\\sigma_\\mathrm{Soft}$} & {$\\mathrm{SNR}_\\mathrm{Soft}$} & {Full counts} & {$\\sigma_\\mathrm{Full}$} & {$\\mathrm{SNR}_\\mathrm{Full}$}\\\\\n\n\t\t\t\\midrule\n\t\t\t$190$ & $27.55$ & $-1.00$ & $\\mathrm{-}$ & $17.29$ & $-1.00$ & $\\mathrm{-}$ & $21.99$ & $10.12$ & $2.17$\\\\\n\t\t\t$280$ & $15.13$ & $-1.00$ & $\\mathrm{-}$ & $9.36$ & $4.94$ & $1.89$ & $18.36$ & $-1.00$ & $\\mathrm{-}$\\\\\n\t\t\t$333$ & $51.15$ & $16.91$ & $3.02$ & $52.23$ & $11.59$ & $4.51$ & $103.21$ & $19.66$ & $5.25$\\\\\n\t\t\t$384$ & $11.37$ & $-1.00$ & $\\mathrm{-}$ & $7.25$ & $4.44$ & $1.63$ & $14.98$ & $-1.00$ & $\\mathrm{-}$\\\\\n\t\t\t$643$ & $34.34$ & $-1.00$ & $\\mathrm{-}$ & $27.16$ & $8.72$ & $3.11$ & $33.54$ & $13.21$ & $2.54$\\\\\n\t\t\t\\midrule\n\t\t\t$104$ & $49.95$ & $-1.00$ & $\\mathrm{-}$ & $38.57$ & $12.16$ & $3.17$ & $57.78$ & $-1.00$ & $\\mathrm{-}$\\\\\n\t\t\t$156$ & $65.01$ & $13.43$ & $4.84$ & $61.59$ & $10.59$ & $5.82$ & $126.33$ & $16.40$ & $7.70$\\\\\n\t\t\t$276$ & $22.17$ & $9.54$ & $2.32$ & $29.55$ & $7.95$ & $3.72$ & $51.61$ & $11.70$ & $4.41$\\\\\n\n\t\t\\bottomrule\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{\\label{tab:mysobscounts}Hard, soft and full band counts for sources that can not be classified according to their dropout band. We refer to the top 5 sources as \\textit{'low significance objects'}. For objects that are not detected we give an upper limit on the counts. We set $\\sigma$ to -1.00 and mark the signal-to-noise ratio with a dash. All values were directly extracted from the \\textit{Chandra}\\ 4-Ms catalog \\protect\\citep{Xue:2011aa}.}\n\\end{table*}\n\nFive (190, 280, 333, 384, 643) of these eight objects are especially interesting since they do not have a counterpart in the optical or the infrared. We refer to these sources as \\textit{'low significance objects'}. We show these five objects in the upper panel of Figure \\ref{fig:mysteryobjects}. For three (104, 156, 276) of these eight objects it is unclear which source represents the counterpart in the optical and infrared since multiple objects are visible in the GOODS\/ACS and CANDELS bands. These objects are shown in the lower panel of Figure \\ref{fig:mysteryobjects}. \n\nOnly three of the eight objects (333, 156, 276) are detected in the full, the hard and the soft band. 104 and 280 are detected in the soft band only. 384 and 643 are found in the soft and in the full band. 190 is detected in the full band only. Only one of the low significance objects (333) and one of the three sources with multiple counterparts in the optical and infrared (156) show a signal-to-noise ratio $\\geq5$ in at least one of the bands. Out of the 371 objects with enough filter coverage and intact images 140 (37.7\\%) have a signal-to-noise ratio $\\geq5$ in the soft, hard or full band. See Table \\ref{tab:SN} for the number of sources with signal-to-noise ratios $\\geq1$ and $\\geq5$ in the individual bands.\n\nThe objects for which we do not detect a counterpart in the optical and the infrared could be spurious detections. On the one hand, \\cite{Xue:2011aa} report that, for the entire catalog, the probability of a source not being real is < 0.004. The entire catalog does therefore contain up to three spurious sources per band. We do however only consider sources that are also covered by GOODS and CANDELS. The CANDELS wide and deep survey fields are $~0.03$ $\\mathrm{deg}^2$ in size and hence only make up $\\sim27\\%$ of the CDF-S ($0.11$ $\\mathrm{deg}^2$) area. According to this, we would expect to find $\\sim0.8$ spurious detections per band. So, we can not rule out the fact that we may have found one or more false detections. On the other hand, we find many sources of comparable X-ray brightness that do possess an optical and\/or infrared counterpart and that are detected in both bands (Figure \\ref{fig:counts}). 333, 384 and 643 are also close to a bright galaxy which might be why the counterpart remains undetected. This could indicate that at least some of the sources are real.  \n\nThe Lyman Break Technique is not applicable to the low significance objects and we are unable to measure a photometric redshift without a detection in the optical and the infrared. Since 333 is detected in the hard and the soft band, it is the only object for which we can apply our Hardness Ratio test (see Section \\ref{sec:HR}). With $HR=-0.01$ 333 could be a potential high-redshift AGN candidate. A negative Hardness Ratio alone does however not convince us of 333 indeed being at high-redshift. At this point we can thus not determine if our sources are real high-redshift AGN candidates, false detections or low-redshift objects that are optically faint. Hopefully, the forthcoming 7-Ms observations of the CDF-S (PI: William Brandt, Proposal ID: 15900132) will shed more light on our five low significance objects. We eliminate all eight sources from our sample. We stress that these targets could still be high-redshift AGN.   \n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=\\columnwidth]{hardsoftcounts}\n\\caption{\\label{fig:counts} Counts in the hard and in the soft band for the \\textit{Chandra}\\ 4-Ms sources. We highlight the eight objects that could not be classified according to the Lyman Break Technique. These eight sources might be real sources or spurious detections. The black points illustrate the 58 objects that are left in our sample. The grey points show the positions of the additional 502 \\textit{Chandra}\\ 4-Ms objects that are also classified as AGN. They were excluded because of insufficient filter coverage or quality or because they are clear low-redshift dropouts. \\protect\\cite{Xue:2011aa} categorize all of these eight objects and all 58 sample sources as AGN.}\n\\end{centering}\n\\end{figure}\n\nAfter discarding targets that are clearly visible in all bands and eliminating the eight objects that could not be classified according to the Lyman Break Technique, 58 $B$, $V$, $i$, $z$ and $Y$-dropouts remain in our main sample. For sources that we visually classify as $B$-dropouts the signal-to-noise in the $V$-band might be too low for a detection by SExtractor. By eliminating visually classified $B$-dropouts we could hence be missing objects that might be classified as $z\\sim5$ sources by other redshift tests. We therefore keep $B$-dropouts in our sample.  \n  \n\\subsection{Photometric Redshift measurements}\n\\label{sec:photoredcode}\nEven though the Lyman Break Technique provides a fast and easy way of identifying possible candidates, it is not without caveats. Dust in red, low-redshift galaxies can produce a sharp break in the SED that might be mistaken for the Lyman Break \\citep{Dunlop:2007aa, McLure:2010aa, Finkelstein:2012aa}. \nApplying the Lyman Break Technique therefore only produces a sample that may still contain low-redshift interlopers. \nYet, even the results of a photometric redshift code have to be treated carefully. While sources with a low photometric redshift are most likely indeed nearby objects, high photometric redshift results are less reliable. This is mainly due to template incompleteness and large photometric errors for faint high-redshift sources.   \n\nThe photometric redshifts we determine for our objects prove to be highly dependent on the filters used as input. We use the photometric redshift code EAZY \\citep{Brammer:2008aa}. We apply the default template set by \\cite{Brammer:2008aa} which is part of the EAZY distribution. Five of these six EAZY templates were created by using the \\cite{Blanton:2007aa} algorithm to reduce the template set by \\cite{Grazian:2006aa}. The sixth template describes a young starburst with a dust screen following the Calzetti law with $A_\\mathrm{v}$ = 2.75 \\citep{Calzetti:2000aa}.\nDue to large uncertainties in the luminosity function of high-redshift AGN, we do not include a luminosity prior when running EAZY.\nTo gain reliable redshift values we take the \\textit{Spitzer}\/IRAC images into account. We carefully compare the $H$ band CANDELS images to the \\textit{Spitzer}\\ 3.6 micron images to determine which of our candidates possesses a \\textit{Spitzer}\\ counterpart and for which sources the \\textit{Spitzer}\\ flux values can not be used due to source confusion. We include the \\textit{Spitzer}\\ images for 30 of our 58 objects. We show the $H$ band and \\textit{Spitzer}\\ stamps for these sources in the appendix.  We do not perform aperture photometry on the \\textit{Spitzer}\\ images, but simply extract the $1.5^{\\prime\\prime}$ aperture radius flux values from the \\textit{Spitzer}\\ catalog by \\cite{Damen:2011aa}.\n\nIf a source is not detected in a passband, we use the $1\\sigma$ sensitivity limit of this filter as an upper limit in the EAZY run. For the GOODS\/ACS bands $B$, $V$, $i$ and $z$ we determine the sensitivity limits by measuring the mean background flux. We measure the number of background counts for six of our objects. For each source, we determine the flux within apertures of varying size at five different positions. For the CANDELS bands $Y$, $J$ and $H$ we rely on the sensitivity limits reported in \\cite{Grogin:2011aa}. Table \\ref{tab:fluxlimits} shows the flux limits that we used in our analysis.\n\n\\begin{table}\n\t\\begin{center}\n\t\t\\begin{tabular}{lc}\n\t\t\t\\toprule\n\t\t\t{} & {$1\\sigma$ flux limit } \\\\\n\t\t\t{} & {in $\\mu$Jy \/ $\\mathrm{arcsec}^2$} \\\\ \n\t\t\t\\midrule\n\t\t\t$B$ & $4.636\\e{-2}$\\\\\n\t\t\t$V$ & $4.160\\e{-2}$\\\\\n\t\t\t$i$ & $8.255\\e{-2}$\\\\\n\t\t\t$z$ & $1.487\\e{-2}$\\\\ \n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{\\label{tab:fluxlimits}GOODS\/ACS flux limits. We determine these mean sensitivity limits by measuring the mean background flux for six different objects. For each object we determine the background counts within five apertures of varying size.}\n\\end{table}  \n\n\\begin{figure*}\n\\includegraphics[width=\\textwidth]{paper_SED_wS_nS.pdf}\n\\caption{\\label{fig:244} Best fit SED for 244 determined by running EAZY without the \\textit{Spitzer}\\ data ($top$) and with the \\textit{Spitzer}\\ data ($bottom$). If we only use the GOODS\/ACS and CANDELS filters as input for EAZY, the photometric redshift code will classify 402 as a $z\\sim6.54$ source. Even though the flux values are fit well by this SED, the shape of the SED seems unphysical for a $z\\sim6.54$ source. We would expect the continuum flux to be lower and bluer. A comparison to the $z\\sim7$ UV luminosity function by \\protect\\cite{Bouwens:2014aa} shows that 244 would indeed be bright if it was at  $z\\sim6.54$ ($M_{H}=-24.6$). The $\\chi^2$ distribution on the right shows multiple secondary low redshift solutions. SED shape and $\\chi^2$ distribution thus suggest that 244 might be a low-redshift object. In the bottom panel we illustrate the best fit SED determined by including the \\textit{Spitzer}\\ 3.6 and 4.5 micron IRAC channels. 244 is now exposed as a low-redshift source. Based upon $z\\sim1.9$ we reject 244 as a possible high-redshift candidate. The shown limits correspond to the 1$\\sigma$ sensitivity limits. }\n\\end{figure*}\n\nIncluding the \\textit{Spitzer}\\ data turns out to be crucial for our purposes. As an example we show the photometric redshift code results we get for source 244 in Figure \\ref{fig:244}. By running EAZY without the \\textit{Spitzer}\\ flux values we gain $z_\\mathrm{phot}$ = 6.54. Assuming that this photometric redshift value is correct, we determine an absolute magnitude of $M_{H}=-24.63$ for the $H$-band. \\cite{Bouwens:2014aa} however give $M_\\mathrm{UV}^{*}=-21.2$ for $z\\sim6$ Lyman Break Galaxies. 244 would therefore be very bright if it indeed was at $z\\sim6$. Including the 3.6 and 4.5 micron \\textit{Spitzer}\\ flux values in the photometric redshift code analysis results in  $z_\\mathrm{phot}$ = 1.9. Including the \\textit{Spitzer}\\ infrared data hence proves to be crucial in revealing low-redshift interlopers.  \n\n\\subsection{Stacking GOODS\/ACS data }\nWe combine the GOODS\/ACS images of each object into stacks. We generated three stacks per source: (1) combines $B$ and $V$, (2) $B$, $V$ and $i$ and (3) $B$, $V$, $i$, $z$. We examine these deeper stacks for detections by running SExtractor on them. We use a detection threshold of 1.5 $\\sigma$ (DETECT$\\_$TRESH = 1.5) and a minimum detection area of 15 pixels above the threshold (DETECT$\\_$MINAREA = 15). The remaining SExtractor parameter values are left at their default values. A detection in the first stack indicates a source that drops out before or in the $B$-band and therefore implies $z\\lesssim4$. Sources that are detected in (1), (2) and (3) are hence of no interest to us. We assume $z\\sim5$ if an object is detected in (2) and (3), but not in (1). Only being detected in (3) indicates $z\\sim6$. Finally, no detection in any stack signals $z\\gtrsim7$. Using this approach we find 35\\ $z\\lesssim4$, 5\\ $z\\sim5$, 3\\ $z\\sim6$ and 13\\ $z\\gtrsim7$ sources. The stacking analysis proves to be inconclusive for two objects (303, 651). Source 303 is detected in (1) and (3), but not in (2). 651 is only found in stack (2).  \n\n\n\\subsection{Colour criteria}\n\\label{sec:cc}\nWe obtain an additional redshift indication by applying colour criteria based upon population synthesis models \\citep{Guhathakurta:1990aa,Steidel:1992gf}. These colour criteria do not only depend on the position of the Lyman Break, but also use the overall SED shape. The criteria we apply here are based on \\cite{Vanzella:2009aa} who used 114 Lyman Break Galaxies from the GOODS field. The sample consisted of 51 $z\\sim4$, 31 $z\\sim5$ and 32 $z\\sim6$ Lyman Break Galaxies. These objects were first chosen by applying the Lyman Break Technique and then followed up spectroscopically.  \n\nObjects that fulfill the following condition are classified as $z\\sim4$ sources:\\\\\n\\begin{equation}\n(B-V)\\geq(V-z)\\,\\wedge\\,(B-V)\\geq1.1\\,\\wedge\\,(V-z)\\leq1.6\n\\end{equation}\n\n\\noindent Here $\\wedge$ and $\\vee$ represent the logical 'and' and 'or' operators respectively. $z\\sim5$ objects need to satisfy the constraints given here:\n\n\\begin{align}\n\\begin{split}\n[(V-i) >1.5+0.9\\times(i-z)]\\vee[(V-i)>2.0]\\wedge\\\\\n(V-i)\\geq1.2\\wedge (i-z)\\leq1.3\\wedge\\mathrm{(S\/N)}_{B}<2\n\\end{split}\n\\end{align}\n\n\\noindent For $z\\sim6$ galaxies these conditions apply:\n\n\\begin{equation}\n(i-z)>1.3\\wedge[(\\mathrm{S\/N)_{B}<2\\vee(\\mathrm{S\/N)}_{V}<2}]\n\\end{equation}\nTo classify the 58 possible candidates according to colour criteria, we use the magnitude values from our aperture photometry (see \\ref{sec:aperturephot}). If a source is not detected, we use the 1$\\sigma$ sensitivity limit as an upper limit. 49\\ of the 58 sources do not fulfill any colour criteria. Note that a source that simultaneously possesses upper limits in the $B$ and $V$ or $V$ and $z$ band is not included in the $z\\sim4$ diagram since its position cannot be determined. The same applies for the $z\\sim5$ and $z\\sim6$ diagrams. Furthermore, we only use colour criteria to determine $z\\sim4$, $z\\sim5$ and $z\\sim6$ dropouts. The 49\\ objects that can not be classified are hence not all necessarily at $z < 4$. Figure \\ref{fig:cc} shows the colour-colour diagrams that illustrate these criteria. Based on colour criteria we find 2\\ $z\\sim4$ (373, 444), 2\\ $z\\sim5$ (303, 321) and 5\\ $z\\sim6$ (226, 244, 296, 522, 589) objects.\\\\\n\n\\subsection{X-rays as a photometric redshift indicator}\n\\label{sec:HR}\n\n\\begin{figure*}\n\\begin{centering}\n\\begin{minipage}[t]{0.49\\textwidth}\n\\begin{centering}\n\\includegraphics[scale=0.31]{HR_model_z.pdf}\n\\end{centering}\n\\end{minipage}\n\\begin{minipage}[t]{0.49\\textwidth}\n\\begin{centering}\n\\includegraphics[scale=0.31]{HR_z_10.pdf}\n\\end{centering}\n\\end{minipage}\n\\caption{\\label{fig:HR}AGN X-ray spectrum simulation. Shown on the $left$ is our model that we generated by running Xspec \\citep{Arnaud:1996aa}. We assume a power law with a slope of 1.8 and a turnover due to photoelectric absorption (Xspec \\texttt{zphabs*zpow} model). This left plot illustrates the model SED for moderately high column densities (line colours) at several redshifts (line types). With increasing redshift the spectrum gets shifted to lower energies and the number of counts in the hard and in the soft band changes. $Right$: measured Hardness Ratio HR as a function of redshift ($HR=\\frac{H-S}{H+S}$, $H$ and $S$ representing the counts in the hard and in the soft band respectively). Based on our model, a HR > 0 indicates $z < 4.34$ for column densities up to $N_\\mathrm{H}$\\ = $10^{23.5} \\mathrm{cm}^{-2}$. In our analysis we thus dismiss objects with an HR > 0 as obscured AGN at low redshift.}\n\\end{centering}\n\\end{figure*}\n\n\nThe Hardness Ratio (HR), sometimes denoted as X-ray colour, represents an additional indicator for high-redshift AGN. The Hardness Ratio is defined as: \n\\begin{equation}\\label{eq:HR}\nHR=\\frac{H-S}{H+S}\n\\end{equation} \nHere H and S represent the observed frame hard (2 - 10 keV) and soft (0.5 - 2 keV) band counts respectively.\n\nTo zeroth-order, an AGN X-ray spectrum follows a powerlaw with an obscuration dependent turnover at lower energies. In the Compton-thin regime a higher column density $N_\\mathrm{H}$\\  means that, relative to the hard band, we will detect fewer counts in the soft band. In a galaxy's restframe the X-ray spectrum hence appears harder for higher obscuration. With increasing redshift the spectrum gets shifted to lower energies and the number of counts that we observe in the hard and in the soft band changes accordingly. We can therefore use the Hardness Ratio as an additional redshift indicator.\n\nA soft X-ray spectrum is then expected for Compton-thin ($N_\\mathrm{H}$\\ $< 10^{24}\\mathrm{cm}^{-2}$) objects. For sources in which $10^{24}\\lesssim$ $N_\\mathrm{H}$\\ $\\mathrm{H}\\lesssim 10^{25}\\mathrm{cm}^{-2}$, i.e. transmission-dominated Compton-thick AGN, the direct emission is still visible, although the E $<$ 10~keV radiation is completely obscured by photoelectric absorption, while the detected emission at higher energies is reduced by Compton scattering \\citep{Comastri:2010aa, Murphy:2009aa}. Hence, in these cases we expect to observe a hard X-ray spectrum even for sources at $z > 5$. When $N_\\mathrm{H}$\\ $ > 10^{25}\\mathrm{cm}^{-2}$, i.e. for reflection-dominated Compton-thick AGN, we only observe the small fraction of the initial emission which is reflected off the accretion disk or the obscuring material \\citep{Ajello:2009aa,2012MNRAS.422.1166B}. We can hence not make the same assumptions as for Compton-thin sources. Nevertheless, it is safe to assume that all $z\\gtrsim5$ sources that can be detected individually in the X-rays are Compton-thin objects. Compton-thick $z\\gtrsim5$ AGN would simply be too faint to be detected individually and are therefore not part of our sample.\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=0.49\\textwidth]{HR_hist}\n\\end{centering}\n\\caption{\\label{fig:HRhist}Number of objects per HR bin. This figure illustrates the distribution of our 54 objects in terms of HR. HR < 0  indicates that this source might be at $z > 4.34$. A positive HR suggests that this source might be at $z\\leq$ 4.34. Thus ten of our 54 objects might be at $z > 4.34$. Not shown are objects 496 and 578. These two sources are only detected in the full band and can therefore not be constrained by the Hardness Ratio. The numbers in each bar coincide with the object IDs in the corresponding HR bin.}\n\\end{figure}\n\nTo quantify the HR($z$) relation we use the X-ray spectral fitting tool Xspec to simulate X-ray spectra at different redshifts \\citep{Arnaud:1996aa}. We use a \\texttt{zphabs*zpow} model and assume a power law slope of 1.8 \\citep{Turner:1997aa, Tozzi:2006aa}. Figure \\ref{fig:HR} summarizes our results. The left panel illustrates our model for $N_\\mathrm{H}$\\ $ = 10^{22} \\mathrm{cm}^{-2}$, $10^{22.5} \\mathrm{cm}^{-2}$ and $10^{23.5} \\mathrm{cm}^{-2}$  (blue, yellow and green, respectively) at $z$ = 0.1, 3 and 6 (dot dashed, solid and dashed line, respectively). It is evident that the number of counts in the soft and hard band changes with redshift. After including the \\textit{Chandra}\\ Redistribution Matrix (RMF) and Auxiliary Response Files (ARF) for on-axis sources, we measure the spectral counts in the hard and in the soft band and determine the HR. The right panel of Figure \\ref{fig:HR} shows our results. According to our simulations HR > 0 signifies $z < 4.34$ for sources with $N_\\mathrm{H}$\\ up to $10^{23.5} \\mathrm{cm}^{-2}$. Allowing for a small amount of transmission, e.g. 1$\\%$ by using a \\texttt{zpcfabs*zpow} model does not change our results significantly (HR = 0 for $z$ = 4.3). In our analysis we hence discard objects with HR > 0. Figure \\ref{fig:HRhist}, which illustrates our results, shows that based solely on the HR, ten of our 58 candidates might be at $z > 4.34$. For 496 and 583 the HR can not be used to constrain $z$ since they are only detected in the full band. Table \\ref{tab:counts} summarizes the X-ray counts, signal-to-noise ratios and Hardness Ratios for each of our main sample sources.\n\nOur results are in good agreement with a similar analysis by \\cite{Wang:2004aa}. They also used Xspec \\citep{Arnaud:1996aa} to simulate the HR at different redshifts. Yet, \\cite{Wang:2004aa} chose a power law slope of 1.9 and used a \\texttt{wabs} model. Our analysis shows that $z > 5$ sources with $N_\\mathrm{H}$\\ up to $10^{23} \\mathrm{cm}^{-2}$ should have a HR of $\\sim-0.3$, \\cite{Wang:2004aa} find HR$\\sim-0.5$. For $N_\\mathrm{H}$\\ = $10^{23} \\mathrm{cm}^{-2}$ we determine HR = 0 at $z\\sim2$, \\cite{Wang:2004aa} find HR = 0 at $z\\sim1.5$.  \n\n\\newcommand\\low{\\textcolor{red}{$\\times$}}\n\\newcommand\\high{\\textcolor{green}{\\checkmark}}\n\\definecolor{green}{HTML}{3D9D93}\n\\definecolor{red}{HTML}{A75379}\n\n\\begin{table*}\n\t\\begin{center}\n\t\t\\begin{tabular}{llccccc}\n\t\t\\toprule\n\t\t\t{} & {ID} & {visual classification} & {colour criteria} & {Stacking} & {Hardness Ratio} & {photo-z}\\\\\n\t\t\t\\midrule\n\t\t\t{} & $121$ & \\low & 0 & \\low & \\low & \\low \\\\\n\t\t\t{} & $150$ & \\low & 0 & \\low & \\low & \\low \\\\\n\t\t\t{} & $173$ & \\low & 0 & \\low & \\high & \\low \\\\\n\t\t\t{} & $184$ & \\low & 0 & \\low & \\low & \\low \\\\\n\t\t\t{} & $189$ & \\low & 0 & \\low & \\low & \\low \\\\\n\t\t\t{} & $199$ & \\high & 0 & \\high & \\high & \\low \\\\\n\t\t\t{} & $211$ & \\low & 0 & \\low & \\low & \\low \\\\\n\t\t\t{} & $217$ & \\high & 0 & \\high & \\high & \\low \\\\\n\t\t\t{} & $221$ & \\low & 0 & \\low & \\low & \\low\\\\\n\t\t\t{} & $226$ & \\low & \\high & \\low & \\low & \\high \\\\\n\t\t\t{} & $242$ & \\low & 0 & \\low & \\low & \\low \\\\\n\t\t\t{} & $244$ & \\high & \\high & \\high & \\low & \\low \\\\\n\t\t\t{} & $258$ & \\high & 0 & \\high & \\low & \\low \\\\\n\t\t\t{} & $273$ & \\high & 0 & \\high & \\low & \\low \\\\\n\t\t\t{} & $296$ & \\high & \\high & \\high & \\low & \\low \\\\\n\t\t\t{} & $301$ & \\low & 0 & \\low & \\low & \\low \\\\\n\t\t\t{} & $302$ & \\low & 0 & \\low & \\low & \\low \\\\\n\t\t\t{} & $303$ & \\high & \\high & 0 & \\low & \\low \\\\\n\t\t\t{} & $306$ & \\high & 0 & \\high & \\low & \\low \\\\\n\t\t\t{} & $318$ & \\low & 0 & \\low & \\low & \\low \\\\\n\t\t\t{} & $321$ & \\low & \\high & \\low & \\low & \\low \\\\\n\t\t\t{} & $325$ & \\low & 0 & \\low & \\low & \\low \\\\\n\t\t\t{} & $328$ & \\low & 0 & \\low & \\low & \\low \\\\\n\t\t\t{} & $331$ & \\low & 0 & \\low & \\low & \\low\\\\\n\t\t\t{} & $348$ & \\high & 0 & \\high & \\low & \\low \\\\\n\t\t\t{} & $354$ & \\high & 0  & \\high & \\high & \\low \\\\\n\t\t\t{} & $371$ & \\high & 0  & \\high & \\low & \\high \\\\\n\t\t\t{} & $373$ & \\low & \\low  & \\low & \\low & \\low \\\\\n\t\t\t{} & $389$ & \\low & 0  & \\low & \\low & \\low \\\\\n\t\t\t{} & $392$ & \\high & 0  & \\high & \\low & \\low \\\\\n\t\t\t{} & $402$ & \\high & 0  & \\high & \\low & \\low \\\\\n\t\t\t{} & $403$ & \\low & 0 & \\low & \\low & \\low \\\\\n\t\t\t{} & $410$ & \\low & 0  & \\low & \\low & \\low \\\\\n\t\t\t{} & $428$ & \\low & 0  & \\low & \\low & \\low \\\\\n\t\t\t{} & $430$ & \\high & 0 & \\high & \\low & \\high \\\\\n \t\t\t{} & $444$ & \\low & \\low  & \\low & \\low & \\low \\\\\n\t\t\t{} & $455$ & \\low & 0  & \\low & \\low & \\low\\\\\n\t\t\t$\\rightarrow$ & $456$ & \\high & 0  & \\high & \\high & \\high \\\\\n\t\t\t{} & $460$ & \\high & 0  & \\high & \\high & \\low \\\\\n\t\t\t{} & $462$ & \\low & 0  & \\low & \\low & \\low \\\\\n\t\t\t{} & $466$ & \\low & 0  & \\low & \\low & \\low \\\\\n\t\t\t{} & $485$ & \\high & 0  & \\high & \\low & \\low \\\\\n\t\t\t{} & $496$ & \\low & 0  & \\low & 0 & \\low \\\\\n\t\t\t{} & $522$ & \\low & \\high  & \\high & \\high& \\low \\\\\n\t\t\t{} & $535$ & \\low & 0  & \\low & \\high & \\low \\\\\n\t\t\t{} & $539$ & \\low & 0  & \\low & \\low & \\low \\\\\n\t\t\t{} & $546$ & \\low & 0  & \\low & \\low & \\low\\\\\n\t\t\t{} & $556$ & \\low & 0  & \\low & \\low & \\low \\\\\n\t\t\t{} & $574$ & \\high & 0  & \\high & \\low & \\low \\\\\n\t\t\t$\\rightarrow$ & $578$ & \\high & 0  & \\high & 0 & \\high \\\\\n\t\t\t$\\rightarrow$ & $583$ & \\high & 0  & \\high & \\high & \\high \\\\\n\t\t\t{} & $589$ & \\low & \\high & \\low & \\low & \\low \\\\\n\t\t\t{} & $591$ & \\low & 0 & \\low & \\low & \\low \\\\\n\t\t\t{} & $620$ & \\low & 0  & \\low & \\high & \\low \\\\\n\t\t\t{} & $624$ & \\high & 0  & \\high & \\low & \\low \\\\\n\t\t\t{} & $625$ & \\low & 0  & \\low & \\low & \\low \\\\\n\t\t\t{} & $630$ &  \\high & 0  & \\low & \\low & \\low \\\\\n\t\t\t{} & $651$ & \\high & 0 & 0 & \\low & \\low \\\\\n\t\t\\bottomrule\n\t\t\\end{tabular}\n\t\\end{center}\n\\caption{\\label{tab:allresults} Overview showing all five redshift test results. Sources for which the test indicates $z < 5$ are marked with \\low. If the redshift test results in $z\\geq$ 5, we show a \\high. Objects that could not be classified via colour criteria, stacking or the Hardness Ratio are marked with 0. For the Hardness Ratio we can only distinguish between $z < 4.3$ (HR > 0) and $z\\geq$ 4.3 (HR $\\leq$ 0). Hence, for the Hardness Ratio \\high means $z\\geq$ 4.3. After combining all redshift tests we are left with three possible candidates (arrows). See Table \\ref{tab:overview} and Table \\ref{tab:counts} for a detailed overview of the results and the X-ray counts of each individual source.}\n\\end{table*}\n\\newpage\n\n\\section{Combining all redshift tests}\n\\label{sec:combination}\n\n\\begin{figure*}\n\\includegraphics[width=\\textwidth]{paper_HR_photz_chi2}\n\\caption{\\label{fig:HR_photz_chi2}After we combine our visual classifications with our $z_\\mathrm{phot}$, stacking and colour criteria results, we are left with seven possible candidates. We show their Hardness Ratio and photometric redshift values in the $left$ panel. The $right$ figure illustrates their $z_\\mathrm{phot}$ $\\chi^2$ distributions. By also eliminating objects with HR > 0, we are left with three final candidates (456, 578, 583). The left panel does not show sources 496 and 578. These two objects are only detected in the full band and can therefore not be constrained by the Hardness Ratio. While we find a low $z_\\mathrm{phot}$ for 496, 578 has $z_\\mathrm{phot}=6.77$ and is therefore one of our final high $z$ candidates. We show $1\\sigma$ error bars. For the X-ray counts and signal-to-noise ratios of each individual source see Table \\ref{tab:counts}.}\n\\end{figure*}\n\nWe now combine our stacking, colour criteria and photometric redshift code results. We exclude objects with  $z_\\mathrm{phot}<5$, $z_{\\mathrm{stacking}}<5$ and $z_{\\mathrm{colour}}<5$. Without the Hardness Ratio constraint six $z>5$ objects remain (Figure \\ref{fig:HR_photz_chi2}). Neither stacking, colour criteria nor our visual classification contradict $z>5$ for these six sources (226, 371, 456, 578, 583, 430). For 371, 456, 578, 583 and 430 the $1\\sigma$ error bars on $z_\\mathrm{phot}$ are asymmetric and reach below $z\\sim5$. The $\\chi^{2}$ distributions (Figure \\ref{fig:HR_photz_chi2}) also do not show clear global minima. For 430 and 578 the $\\chi^2$ distributions are flat and allow for a wide range of lower redshift solutions. The $z_\\mathrm{phot}$ solution for 456 is at the upper end of our allowed $z_\\mathrm{phot}$ range ($z_\\mathrm{phot}=$ 0 to 11). 371 shows additional minima at $z_\\mathrm{phot}\\sim1$. For 226 we determine an absolute magnitude of $M_H = -25.6$ assuming a photometric redshift of 5.43. 226 would therefore be extremely bright if it indeed was at $z \\sim 5$ (see section \\ref{sec:photoredcode}). There is thus only little evidence supporting the fact that these objects might be at high redshift. Only 226, 456 and 583 show global minima and are thus our most promising candidates.\n\n\\subsection{Our three final candidates}\n\\label{sec:final_three}\n\n\\begin{figure*}\n\\begin{minipage}{\\textwidth}\n\\includegraphics[width=\\textwidth]{456_series_stamps}\n\\end{minipage}\n\n\\begin{minipage}{\\textwidth}\n\\includegraphics[width=\\textwidth]{578_series_stamps}\n\\end{minipage}\n\n\\begin{minipage}{\\textwidth}\n\\includegraphics[width=\\textwidth]{583_series_stamps}\n\\end{minipage}\n\\caption{\\label{fig:candidates}Candidate $z > 5$ AGN that remain after combining all redshift tests. 456, 578 and 583 are the only sources that remain after we combine our stacking ($z_{\\mathrm{stacking}}\\sim7$ for all), colour criteria (all not classified), Hardness Ratio (456: $HR=-0.289$, 578: not classified, 583: $HR=-0.511$ ) and photometric redshift code results (456: $z_{\\mathrm{phot}}=10.913$, 578: $z_{\\mathrm{phot}}=6.766$, 583: $z_{\\mathrm{phot}}=9.364$). Visually we classify 456, 578 and 583 as $z$ dropouts. Due to source confusion, we do not use the \\textit{Spitzer}\\ 3.6 and 4.5 micron images when running the photometric redshift code for these objects. All images are colour inverted and are $10^{\\prime\\prime}$ x $10^{\\prime\\prime}$ in size.}\n\\end{figure*}\n\nWe now also take the Hardness Ratio information into account and exclude three of the six remaining sources based on a positive HR value. After combining all of our redshift tests we are hence left with three final high-redshift candidates (456, 578, 583, Figure \\ref{fig:candidates}). 456 and 583 have a negative Hardness Ratio whereas 578 cannot be constrained by HR since it is only detected in the full band.  \n\nFor 456 we find $z_\\mathrm{phot} = 10.91^{-0.92}_{-8.39}$ ($\\chi^2 \\sim 0$) and HR = -0.29. For 578 we determine $z_\\mathrm{phot} = 6.77^{+0.97}_{-5.10}$ ($\\chi^2 \\sim 0$). 583 has HR = -0.51 and $z_\\mathrm{phot}= 9.36^{+0.63}_{-4.82}$ ($\\chi^2 \\sim 0$). 456, 578 and 583 could not be classified according to our colour criteria. Nonetheless, our visual classification ($z$ dropouts) and our stacking analysis ($z_{\\mathrm{stacking}}\\sim7$ for all) indicate that these sources might be $z > 5$ AGN.   \n\n\\subsection{Expected number densities}\n\nWe note that 456, 578 and 583 seem bright for $z\\sim7$ and $z\\sim10$ sources. We use UV rest-frame galaxy luminosity functions by \\cite{Bouwens:2014aa} to quantify this statement. To justify our comparison to the galaxy, and not quasar luminosity function, we calculate $\\alpha_\\mathrm{ox}$ for our candidates. $\\alpha_\\mathrm{ox}$, the X-ray to optical-UV ratio, is defined as $\\alpha_\\mathrm{ox}=\\log\\big[{L_{2500\\AA}\/L_{2keV}}\\big]\/2.605$ (e.g. \\citealt{Tananbaum:1979aa, Wilkes:1994aa, Vignali:2003aa, Steffen:2006aa}). \\cite{Lusso:2010aa} analyse a sample of 545 X-ray selected Type 1 AGN from the XMM-COSMOS survey and find a mean $\\alpha_\\mathrm{ox}$ value of $\\sim1.37$ and a weak redshift dependence out to $z\\sim4$. We use the $H$-band and soft band flux values to calculate $\\alpha_\\mathrm{ox}$ for 456, 578 and 583. We find $\\alpha_{\\mathrm{ox}}=0.11$, $0.35$ and $-0.02$ for 456, 578 and 583, respectively. Assuming that $\\alpha_\\mathrm{ox}$ has no strong redshift dependence for $z>4$, this indicates that 456, 578 and 583 are not high-redshift quasars and justifies our comparison to the results by \\cite{Bouwens:2014aa}.     \n\nWe estimate the number of sources similar to 456, 578 and 583 that we expect to find in our field. Since we only analyse objects for which GOODS\/ACS and CANDELS data is available, we are only considering the area of the CANDELS deep and wide surveys (0.03 $\\mathrm{deg}^2$) and not the entire CDF-S (0.11 $\\mathrm{deg}^2$). Based on the surface density of $z\\sim7$ and $z\\sim10$ galaxies by \\cite{Bouwens:2014aa}, we expect to find $0.14\\pm0.14$ sources as bright as 456 at $z\\sim10$, $3.72\\pm0.80$ objects similar to 578 and less than $0.15$ $z\\sim10$ objects as bright as 583. We hence expect high-redshift sources as bright as 456, 578 and 583 to be rare. \n\n\\subsection{Deeper Y-band imaging for 583}\n\\label{sec:HUDF}\n\\begin{figure*}\n\\begin{minipage}{\\textwidth}\n\\includegraphics[width = \\textwidth]{paper_456}\n\\end{minipage}\n\n\\begin{minipage}{\\textwidth}\n\\includegraphics[width = \\textwidth]{paper_578}\n\\end{minipage}\n\n\\caption{\\label{fig:456_578}Photometric redshift code results for objects 456 and 578 before ($top\\ panels$) and after including the HUGS $K_S$-band ($bottom\\ panels $). After combining all of our redshift tests, we are left with three final candidates (for 583 see Figure \\ref{fig:583}). For these we include the Hawk-I UDS and GOODS Survey (HUGS) \\protect\\citep{Fontana:2014aa} $K_S$-band values when determining the photometric redshift. We show the best fit template SEDs ($left$), the $\\chi^2$ distributions ($middle$) and the corresponding images ($right$), before and after including the $K_S$-band. For 456 we redetermine the $J$-band upper limit. Our result is close to the value reported by \\protect\\cite{Grogin:2011aa} and does not change the $z_\\mathrm{phot}$ result significantly. For both, 456 and 578, a low redshift solution is allowed within the $2\\sigma$ error. These two final candidates are hence likely to be at low redshift. We use the $\\Delta\\chi^2$ method under the assumption of one free parameter ($z_\\mathrm{phot}$) to compute the $1, 2$ and $3\\sigma$ error bars that are shown in the middle panels. All images are colour inverted and are $5^{\\prime\\prime} \\times 5^{\\prime\\prime}$ in size.} \n\\end{figure*}\n\n\\begin{figure*}\n\\begin{minipage}{\\textwidth}\n\\includegraphics[width = \\textwidth]{paper_583}\n\\end{minipage}\n\\caption{\\label{fig:583} Photometric redshift code results for object 583 before ($top\\ panels$) and after including the HUDF $Y$-band ($middle\\ panels$) and the HUGS $K_S$-band ($bottom\\ panels$), analogue to Figure \\ref{fig:456_578}. After combining all of our redshift tests, we are left with three final candidates (for 456 and 578 see Figure \\ref{fig:456_578}). Fortunately, 583 is covered by the HUDF. We thus perform aperture photometry on the deep HST\/WFC3 HUDF $Y$-band and replace the CANDELS $Y$-band value with this flux ($middle\\ panels$). We also include the Hawk-I UDS and GOODS Survey (HUGS) \\protect\\citep{Fontana:2014aa} $K_S$-band value when determining the photometric redshift ($bottom\\ row$). With the HUDF $Y$-band and the HUGS $K_S$-band 583's photometric redshift value drops to $2.68$. It is hence most likely a low-redshift source. We show the best fit template SEDs ($left$), the $\\chi^2$ distributions ($middle$) and the corresponding images ($right$). We use the $\\Delta\\chi^2$ method under the assumption of one free parameter ($z_\\mathrm{phot}$) to compute the shown $1, 2$ and $3\\sigma$ error bars. All images are colour inverted and are $5^{\\prime\\prime} \\times 5^{\\prime\\prime}$ in size.}\n\\end{figure*}\n  \n583, one of our three final candidates, has a high photometric redshift ($z_\\mathrm{phot} = 9.36$) and shows a clear global minimum in the $\\chi^2$ distribution. In terms of the $\\chi^2$ distribution it is thus our most promising candidate. Fortunately, 583 is part of the $Hubble$ Ultra Deep Field (HUDF) \\citep{Beckwith:2006aa}. We are especially interested in deeper $Y$-band imaging since 583's high photometric redshift hinges on the upper limit in this band. We show our results in Figure \\ref{fig:583}. The top panels illustrate, that the upper limit in the $Y$-band, for which we used the official sensitivity limit given by \\cite{Grogin:2011aa}, is very low and thus forces a strong break in 583's SED. We hence combine all available HST\/WFC3 HUDF $Y$-band images (\\citealt{Koekemoer:2013aa}, HST Program ID 12498, PI: R. Ellis; \\citealt{Illingworth:2013aa}, HST Program ID 11563, PI: G. Illingworth, HST Program ID 12099, PI: A. Riess) using the HST DrizzlePac package. We measure the flux of 583 using simple aperture photometry and a $0.3^{\\prime\\prime}$ radius aperture (middle panels) and rerun EAZY. We use the $\\Delta\\chi^2$ method under the assumption of one free parameter ($z_\\mathrm{phot}$) to compute the $1, 2$ and $3\\sigma$ error bars. We still determine a high photometric redshift of $z_\\mathrm{phot}=7.41$, a low redshift solution at $z_\\mathrm{phot}\\sim2$ is however allowed within $1\\sigma$. 583 is hence unlikely to be at high redshift.  \n\n\\subsection{HUGS $K_S$-band data}\n\\label{sec:HUGS}\n\nTo gain more reliable photometric redshift values for all three of our final candidates, we now also take VLT\/Hawk-I $K_S$-band data into account. We take the $K_S$-band flux values for 456, 578 and 583 from the Hawk-I UDS and GOODS Survey (HUGS) catalog (v1.1) \\citep{Fontana:2014aa} and rerun EAZY. Our results are illustrated in Figures \\ref{fig:456_578} and \\ref{fig:583}. The photometric redshift value for 456 remains at $10.91^{-0.92}_{+8.39}$ ($\\chi^2 \\sim 0$). However, the $z_\\mathrm{phot}$ value seems highly dependent on the $J$-band upper limit. We thus redetermine the $J$-band sensitivity limit by measuring the background flux within 35 apertures which are scattered across the CANDELS field. We determine a flux limit close to the value reported by \\cite{Grogin:2011aa} (measured: $2.62 \\times 10^{-2} \\mu$Jy, reported: $1.97 \\times 10^{-2} \\mu$Jy for a $0.5^{\\prime\\prime}$ radius aperture). We rerun EAZY and find a value almost identical to the previous result ($z_\\mathrm{phot} = 10.91^{-1.14}_{-7.11}$, $\\chi^2 = 0.01$). 456 thus remains a source with a high photometric redshift. The $2\\sigma$ error bar, computed through the $\\Delta\\chi^2$ method, does however allow for a low-redshift solution at $z\\sim4$. For 578 the $K_S$-band causes the photometric redshift to increase from 6.77 to $7.49^{+0.54}_{-4.60}$ ($\\chi^2 = 0.08$). Nonetheless, similar to 456, 578's $2\\sigma$ error bar permits a $z\\sim3$ solution. For 583 we determine $z_\\mathrm{phot} = 2.68^{+5.61}_{+0.37}$ ($\\chi^2=0.21$) by including the HUDF deep $Y$-band and the HUGS $K_S$-band flux values. This $z_\\mathrm{phot}$ value matches what has previously been reported by \\cite{Szokoly:2004aa} for the galaxy next to 583 ( 53.1833, -27.7764). We thus suspect, that 583 might be part of a large clumpy galaxy at low redshift \\citep{Schawinski:2011aa}. With the X-ray emission being offset from what could be the main galaxy, this does remain a very interesting object. \n\nIn summary, the HUGS $K_S$-band and HUDF $Y$-band data causes the photometric redshifts of our three final candidates to either drop to low-redshift or to allow for a low-redshift solution within a $2\\sigma$ error bar. Considering these photometric redshift code results and how rare objects as bright as our candidates should be at high-redshift, we conclude that 456, 578 and 583 are unlikely to be at high redshift. They do however remain compelling candidates for follow-up observations. 456 remains a very interesting candidate since the $1\\sigma$ $z_\\mathrm{phot}$ error bar does not allow for lower redshift solutions. 578 could still be at high redshift and 583 is intriguing since the X-ray emission seems to be offset from the main galaxy. \n\n\\section{Discussion}\n\\label{sec:discussion}\n\n\\begin{figure*}\n\\begin{centering}\n\\includegraphics[width=\\textwidth]{four_red_all_bands}\n\\end{centering}\n\\caption{\\label{fig:BHmasses}Expected X-ray flux as a function of BH mass, Eddington ratio and redshift. We assume spherical accretion, $N_\\mathrm{H}$\\ $=10^{22}\\mathrm{cm}^{-2}$ and a bolometric correction of $k_\\mathrm{corr}=25$ to calculate the expected X-ray flux for BH masses ranging from $10^{6}\\ M_{\\odot}$ to $10^{9}\\ M_{\\odot}$, Eddington ratios between 0.01 and 1. and redshifts between $z=4$ to $z=10$. The dashed lines show the \\textit{Chandra}\\ 4-Ms flux limits in the soft, full and hard bands. The BH mass in a typical Lyman Break Galaxy at $z\\sim5$ is $\\sim10^{7}$$M_{\\odot}$. This figure illustrates that we are sensitive enough to detect such sources if we assume low obscuration and an Eddington ratio > 0.1. }\n\\end{figure*}\n\n\\subsection{Sensitivity and AGN number density}\nIn this section we show that our analysis should have been sensitive enough to detect an active BH in a typical high-redshift Lyman Break Galaxy and why we expected to find at least some high $z$ AGN. Furthermore, we determine an upper limit on the AGN number density. \n\nIn a first step we determine the BH masses and accretion rates we are sensitive to by calculating the expected X-ray flux as a function of BH mass, Eddington ratio  and redshift. We calculate the X-ray luminosity and translate it to observed flux. We use Eddington ratios between 0.01 and 1. and a redshift range from $z=4$ to $z=10$. For simplicity we assume a constant bolometric correction of $k_\\mathrm{corr}=25$ ($L_\\mathrm{bol}=k_\\mathrm{corr} L_\\mathrm{X}$, \\citealt{Vasudevan:2009aa}) and $N_\\mathrm{H}$\\ $=10^{22}\\mathrm{cm}^{-2}$. Figure \\ref{fig:BHmasses} illustrates our results. We also show the \\textit{Chandra}\\ 4-Ms flux limits which lie at $9.1\\times10^{-18}\\ \\mathrm{erg}\\ \\mathrm{s}^{-1}\\ \\mathrm{cm}^{-2}$ for the soft band and at $5.5\\times10^{-17}\\ \\mathrm{erg}\\ \\mathrm{s}^{-1}\\ \\mathrm{cm}^{-2}$ for the hard band \\citep{Xue:2011aa}. At $z\\sim5$ we are sensitive to luminosities as low as $\\sim10^{42}\\mathrm{erg}\\ \\mathrm{s}^{-1}$ in the soft band and $\\sim10^{43}\\mathrm{erg}\\ \\mathrm{s}^{-1}$ in the hard band.\n\nIn the GOODS field, the typical stellar mass of a Lyman Break Galaxy at $z\\sim5$ is $2.82\\times10^{10}$$M_{\\odot}$\\ \\citep{Lee:2012aa}. If we use the local BH-stellar mass relation ($M_{\\star, \\mathrm{total}}{\/}$ $M_\\mathrm{BH}$\\ $=562$, \\citealt{Jahnke:2009aa}) to determine the corresponding BH mass, we find $M_\\mathrm{BH}$\\ $=5\\times10^{7}$$M_{\\odot}$. Figure \\ref{fig:BHmasses} shows that, assuming low obscuration and Eddington ratios $>0.1$, we should have been capable of detecting such AGN at $z>5$.\n\nTo estimate the number of high $z$ AGN in the CDF-S we use the results by \\cite{Bouwens:2014aa}. They find 680 $z\\sim5$, 252 $z\\sim6$ and 113 $z\\sim7$ Lyman Break Galaxies in the CANDELS\/Deep and CANDELS\/Wide surveys for the GOODS-S field. \\cite{Nandra:2002aa} report an AGN fraction of $3\\%$ for $z\\sim3$ Lyman Break Galaxies in the Hubble Deep Field North. Assuming that this fraction does not evolve with redshift, we would expect the volume we looked at to contain $\\sim20$ AGN at $z\\sim5$, $\\sim8$ at $z\\sim6$ and $\\sim3$ at $z\\sim7$. Since we are sensitive enough to detect Compton-thin AGN in typical Lyman Break Galaxies, we would have expected to find at least some convincing high-redshift sources. Even if all of our three final candidates (456, 578, 583) would prove to be at high-redshift, the number of high-redshift AGN in our field would still be lower than expected. \n\nWe use the estimated number of high redshift AGN to determine an upper limit on the AGN number density. The combined CANDELS\/Wide and CANDELS\/Deep survey covers an area of 0.03 $\\mathrm{deg}^{2}$. At $z\\sim5$ the faint X-ray selected AGN number density is hence fewer than 655 AGN per $\\mathrm{deg}^{2}$. At $z\\sim6$ and $z\\sim7$ we would expect to find fewer than 262 and 98 AGN per $\\mathrm{deg}^{2}$, respectively. Here, we assumed the $z\\sim3$ Lyman Break Galaxy AGN fraction by \\cite{Nandra:2002aa}. This AGN fraction could however be especially low for high redshift Lyman Break Galaxies or generally evolve with redshift. We discuss this in more detail in Section \\ref{sec:explanations}.\n\n\\subsection{Comparison to existing work}\n\nWe compare our results to \\cite{Vito:2013aa}. They found three $z > 5$ AGN (139, 197, 485) in the CDF-S. Source 139 has a photometric redshift of $z_\\mathrm{phot}$ = 5.73 based on \\cite{Luo:2010aa}. Since 139 is not covered by $B$, $Y$, $J$ and $H$ it was not part of our analysis. Object 197 has $z_\\mathrm{phot}$ = 6.07 with a secondary solution at $z_\\mathrm{phot}$ = 4.39 \\citep{Luo:2010aa}. 197 is not in the GOODS field and was therefore immediately excluded by us. Source 485 has a photometric redshift of $z_\\mathrm{phot}$ = 7.62 with a secondary solution at $z_\\mathrm{phot}$ = 3.31 based on  \\cite{Luo:2010aa} and has $z_\\mathrm{phot}$ = 4.42 according to \\cite{Santini:2009aa}. Both, \\cite{Luo:2010aa} and \\cite{Santini:2009aa}, did not take CANDELS WFC3\/IR data into account when determining photometric redshifts. Instead they used the GOODS - MUSIC catalog \\citep{Grazian:2006aa} which contains the VLT\/ISAAC $J$, $H$, and $K$ bands. Taking CANDELS and \\textit{Spitzer}\\ data into account, we determined $z_\\mathrm{phot} = 2.83^{+0.54}_{-0.68}$ for object 485.\n\n\\cite{Vito:2013aa} also report spectroscopic and photometric redshifts for 31 additional AGN at lower redshifts. Seven of these 31 sources are also part of our sample. For three objects our photometric redshifts lie within $1\\sigma$ from the redshift reported by \\cite{Vito:2013aa} (331: $z_\\mathrm{phot,\\\/Vito} = 3.780$, $z_\\mathrm{phot} = 4.32^{+0.02}_{-3.63}$; 371: $z_\\mathrm{phot,\\\/Vito} = 3.10$, $z_\\mathrm{phot} = 5.56^{+0.04}_{-4.50}$; 546: $z_\\mathrm{spec,\\\/Vito} = 3.06$, $z_\\mathrm{phot} = 3.15^{+0.18}_{-0.25}$). For four sources our photometric redshifts do not lie within $1\\sigma$ (150: $z_\\mathrm{phot,\\\/Vito} = 3.34$, $z_\\mathrm{phot} = 3.63^{+0.14}_{-0.16}$; 403: $z_\\mathrm{spec,\\\/Vito} = 4.76$, $z_\\mathrm{phot} = 4.22^{+0.17}_{-0.12}$; 556: $z_\\mathrm{phot,\\\/Vito} = 3.53$, $z_\\mathrm{phot} = 2.68^{+0.17}_{-0.16}$; 651: $z_\\mathrm{phot,\\\/Vito} = 4.66$, $z_\\mathrm{phot} = 1.90^{+0.41}_{-0.34}$). The remaining 24 low-redshift AGN were not part of our sample because they were not covered by enough bands ($B$, $V$, $i$, $z$, $J$, $H$), because they were clearly visible in all bands and therefore discarded as $z<5$ sources or because their images were disturbed by artefacts.     \nExcept for source 485 and the four low-redshift AGN (150, 403, 556, 651) our findings do hence agree with the results by \\cite{Vito:2013aa}.\n\n\n\\cite{T13} searched for X-ray emission of $z = 6 - 8$ Lyman Break dropout and photometrically selected sources.  None of the $\\sim600$ $z\\sim6$, $\\sim150$ $z\\sim7$ or $\\sim80$ $z\\sim8$ sources could be detected individually in the X-rays. Stacking the X-ray data in redshift bins did not generate a significant detection either. In the stacks the $3\\sigma$ upper limits on the X-ray emission lay at $\\sim10^{41}\\mathrm{erg}\\ \\mathrm{s}^{-1}$ (soft) and $\\sim10^{42}\\mathrm{erg}\\ \\mathrm{s}^{-1}$ (hard) for the $z\\sim6$ and $z\\sim7$ bins. Assuming a bolometric correction of $k_\\mathrm{corr}=25$ \\citep{Vasudevan:2009aa} and an Eddington ratio of 0.1, these upper limits correspond to $\\sim10^{5}$ $M_{\\odot}$\\ (soft) and $\\sim10^{6}$ $M_{\\odot}$\\ (hard) in terms of BH mass. \nIn comparison to our work, \\cite{T13} based their search on a sample of optically selected sources, whereas we selected our objects in the X-rays. Nonetheless, the results are in agreement. \n\n\\subsection{$z\\gtrsim5$ AGN in the CDF-S}\n\nIn this work we searched for possible $z > 5$ AGN candidates in the CDF-S. In contrast to \\cite{T13} we started out with a sample of confirmed X-ray sources. We used visual classification, colour criteria, stacking, a photometric redshift code and the Hardness Ratio to obtain multiple redshift indications. After dismissing sources for which our redshift tests indicated $z < 5$, three final candidates with $z_\\mathrm{phot} \\sim 7$ (578), $z_\\mathrm{phot} \\sim 9$ (583) and $z_\\mathrm{phot} \\sim 10$ (456) remained in our sample. Our comparison to the galaxy luminosity function showed that $z\\sim7$, $z\\sim9$ and $z\\sim10$ objects as bright as 578, 583 and 456 are rare. By including the Hawk-I UDS and GOODS Survey (HUGS) $K_S$-band \\citep{Fontana:2014aa} and the HUDF $Y$-band data, the photometric redshifts for our three final candidates either dropped to low-redshift (583) or allowed for a low-redshift solution within $2\\sigma$ error bars (578, 583). Our three final candidates did hence not pass this extended redshift test. We conclude that considering our photometric redshift code results and the rarity of such high-redshift objects, 456, 578 and 583 are most likely low-redshift sources. We also found five low significance objects. These sources are detected in the X-rays, but they do not seem to possess a counterpart in the optical or infrared (including the \\textit{Spitzer}\/IRAC 3.6 and 4.5 micron channels and the VLT\/Hawk-I $K_S$-band). The currently available data does not allow us to constrain their redshifts or determine if these objects are spurious detections.  \n\nBased upon currently available GOODS\/ACS, CANDELS and \\textit{Spitzer}\\ data, the analysis did therefore yield three final $z > 5$ candidates and five low significance objects for this deep, but narrow field. Including HUGS and HUDF data did however show, that 456, 578 and 583 are likely to be at low redshift. Follow-up observations are necessary to gain more reliable redshifts for our three final candidates and to constrain the nature of our five low significance objects. \n\n\\subsection{Possible explanations}\n\\label{sec:explanations}\n\nBoth, the approach by \\cite{T13} and our approach, assumed that X-ray emission is a valid tracer for BH growth. If at high redshift BHs primarily grow through BH mergers instead of accretion, electromagnetic radiation might not be emitted during the growth process. X-ray emission might hence not be a indicator for active BHs.\n\nAt high redshift the number of actively accreting BHs could also be generally low. This could be caused by a low BH occupation fraction, a low AGN fraction or BH growth through short, super-Eddington episodes. We stress the difference between the BH occupation fraction and the AGN fraction since they describe different scenarios. \n\nIf the BH occupation fraction is low only very few haloes are seeded with BHs. Our sample could therefore be too small to not only contain a BH, but to contain a BH that is also actively accreting. \n\nEven if the BH occupation fraction is high, the AGN fraction could still be low. So, even if there are plenty of BHs in our field, only few of them could be active. For instance, BHs at high redshift could only grow in optically faint galaxies. In our analysis these faint galaxies could correspond to the low significance objects that do not seem to possess an optical or infrared counterpart (190, 280, 333, 384, 643, see Section \\ref{sec:LBT}). The currently available data does not allow us to investigate these sources further. We are however hopeful that the forthcoming 7-Ms survey for the CDF-S (PI: William Brandt, Proposal ID: 15900132) will show if these objects are real.  \n\nBHs could also grow through short, super-Eddington accretion phases \\citep{Madau:2014aa, Volonteri:2014aa}. \\cite{Madau:2014aa} illustrated that a duty cycle of $20\\%$ is enough to grow a  $100$ $M_{\\odot}$\\ non-rotating seed BH into a $\\sim10^9$ $M_{\\odot}$\\ BH by $z\\sim7$. This could, for instance, be realized through five $20\\ \\mathrm{Myr}$ long growth episodes with $\\dot{m}\/\\dot{m}_\\mathrm{Edd}=4$, each followed by a $100\\ \\mathrm{Myr}$ phase of quiescence. The \\cite{T13} and our sample, could thus not contain any BHs that are actively accreting at the time of observation. \n\nSimulations suggest that the BH occupation fraction should be high enough for our field to contain high-redshift BHs. \\cite{Menou:2001aa} ran Monte Carlo simulations of the merger history of dark matter haloes. They showed that to reproduce the local BH distribution, $>3\\%$ of the $M_\\mathrm{halo}\\gtrsim10^8$ $M_{\\odot}$\\ haloes should be seeded with BHs at $z\\sim5$.  \n\n \\cite{Bellovary:2011aa} ran SPH+$N$-body simulations in which only the local gas properties, such as density, temperature and metallicity, influence the BH formation and evolution. They showed that the BH occupation fraction is halo mass dependent. At $z\\sim5$ they found a BH occupation fraction of $\\sim50\\%$ for $10^8$ $M_{\\odot}$\\ $<M_\\mathrm{halo}<10^{9}$ $M_{\\odot}$\\ haloes and a fraction of $100\\%$ for $M_\\mathrm{halo}>10^{9}$ $M_{\\odot}$\\ haloes. \n\n\\cite{Alexander:2014aa} presented a sophisticated model in which a BH seed is being fed by dense cold gas flows while it is embedded in a nuclear star cluster. This can lead to supra-exponential accretion and could explain how a light ($\\sim10$ $M_{\\odot}$) Pop III remnant  BH seed could grow into a $\\gtrsim10^4$ $M_{\\odot}$\\ seed within $\\sim10^7$ years. Nevertheless, these $\\gtrsim10^4$ $M_{\\odot}$\\ seeds still need to grow into the massive $10^9$ $M_{\\odot}$\\ sources that we find at $z>6$. This most likely happens via Eddington limited accretion. The distribution of high-redshift quasars that we observe at $z>6$ can be reproduced if the supra-exponential accretion and the subsequent Eddington-limited growth work efficiently in at least $1-5\\%$ of the dark matter haloes.     \n\n\\cite{Stark:2009aa}, \\cite{Vanzella:2009aa}, \\cite{Wilkins:2010aa}, \\cite{Bouwens:2014aa}, \\cite{Duncan:2014aa}  and many more have shown that the GOODS-south field contains hundreds of $z\\gtrsim5$ Lyman Break Galaxies. These high-redshift sources should have passed our manual inspection, the colour criteria, the stacking and the photometric redshift measurement. So, according to the simulations and the number of high-redshift Lyman Break Galaxies in our field, we would expect our sample to include high-redshift BHs. The number of actively accreting BHs could however still be low. For instance, it would be possible that only the most massive haloes host AGN.  \n\nFinding one or more high-redshift AGN would have opened up the window to the early BH growth era. \\cite{Treister:2011aa} showed that the massive and luminous quasars we observe at $z>6$ are most likely not representative of the entire high-redshift BH population. Such quasars are rare and presumably only constitute the high mass end of the entire BH population \\citep{Volonteri:2010aa}. At $z\\sim6$ we expect to find only $\\sim2$ in a 1000 $\\mathrm{deg}^{2}$ field \\citep{Fan:2001aa, Fan:2000aa}. Furthermore, these objects only allow us limited insight into seed formation scenarios. For typical seed masses $\\lesssim10^{6}$$M_{\\odot}$, these objects have to undergo multiple Salpeter times \\citep{Salpeter:1964aa} to reach $M_\\mathrm{BH}\\sim10^9$ $M_{\\odot}$. By the time we observe them as quasars, all initial seed information will be lost. We are hence especially interested in the population of more abundant, less massive, less luminous AGN that will end up in galaxies similar to the Milky Way. If our analysis had yielded a convincing high-redshift AGN candidate, we would have been able to put first constraints on this more representative and revealing BH population.         \n\nTo constrain the explanations for our results and to gain further insights into BH formation and growth at high redshift, this analysis needs to be repeated for a larger sample. Especially constraining the BH occupation fraction and the short, super-Eddington growth scenario requires a larger field. The new field does not need to be deeper, but wider than the $0.11\\ \\mathrm{deg}^{2}$ CDF-S \\citep{Luo:2008aa}. The 2.8-Ms \\textit{Chandra}\\ COSMOS Legacy Survey, for which the data will be available soon, covers a $2.2\\ \\mathrm{deg}^{2}$\\ area \\citep{Civano:2014aa}. Not being as deep as the 4-Ms \\textit{Chandra}\\ data, the \\textit{Chandra}\\ COSMOS Legacy Survey will probe a slightly different parameter space (see \\citealt{Treister:2011aa} for an illustration of the high-redshift number density that is necessary for an individual detection). Nonetheless, it will provide data for a much wider field and will thus enable us to repeat this analysis for a larger sample. ATHENA, which is meant to be launched in 2028, will allow us to constrain the BH occupation fraction to new accuracy. With its X-ray Wide Field Imager, ATHENA is meant to detect over 400 $z = 6 -8$ and over 30 $z > 8$ X-ray selected active BHs \\citep{Nandra:2013aa,Aird:2013aa}. JWST data could help to investigate the nature of our five low significance objects. \n\n\n\\section{Summary}\n\\label{sec:summary}\n\nWe searched for $z\\gtrsim5$ AGN in the \\textit{Chandra}\\ Deep Field South (CDF-S). We used the \\textit{Chandra}\\ 4-Ms catalog and combined it with GOODS\/ACS, CANDELS\/WFC3 and \\textit{Spitzer}\/IRAC data. Our main sample contained 58 sources. We ran a photometric redshift code, stacked the GOODS\/ACS data, applied colour criteria and the Lyman Break Technique. Furthermore, we used the X-ray Hardness Ratio as a redshift indicator. After combining all redshift tests, three final $z\\gtrsim5$ AGN candidates remained. Redetermining their photometric redshifts with additional VLT\/Hawk-I HUGS $K_S$-band and HST\/WFC3 HUDF $Y$-band data showed, that they are most likely low-redshift sources. We also found five sources that are detected in the X-rays, but that do not seem to possess a counterpart in the optical or infrared (low significance objects). The currently available data did not allow us to determine if these objects are possible high-redshift AGN candidates, spurious detections or optically faint low-redshift sources. Assuming that our three final candidates are indeed low-redshift sources and that our five low significance objects are either spurious detections or also at low redshift, we concluded that the CDF-S does not contain any high-redshift AGN. We also showed that we should have been able to detect active BHs in typical $z\\sim5$ Lyman Break Galaxies and why we would have expected to find at least some high $z$ AGN. \nOur results could be explained by:    \n\\begin{itemize}\n\\item a low BH occupation fraction or a low AGN fraction. If at high redshift only very few haloes contain a BH or only very few BHs are actively accreting, our sample could be too small to contain an AGN.   \n\\item BH growth via short, super-Eddington growth modes. If BHs primarily grow through short accretion episodes, the number of actively accreting BHs in our sample might be zero.\n\\item BH growth in optically faint galaxies. Our five low significance objects could indicate that high-redshift AGN primarily grow in galaxies that we do not detect in the optical. We were however unable to constrain their redshifts and noted that these could be spurious detections.  \n\\item BH growth via BH-BH mergers. If at high $z$ BHs primarily grow through mergers instead of accretion, X-rays might not trace BH growth.  \n\\end{itemize}\n\n\\begin{table*}\n\\tiny\n\t\\begin{center}\n\t\t\\begin{tabular}{lllllllllllllll}\n\t\t\\toprule\n\t\t\t{ } & {ID} & {$\\mathrm{RA}$} & {$\\mathrm{DEC}$} & {$\\mathrm{RA}_{H}$} & {$\\mathrm{DEC}_{H}$} & {visual} & {stacking} & {colour} & {Hardness} & {$\\sigma_{\\mathrm{HR}}$} & {$z_\\mathrm{phot}$} & {$\\sigma_{\\mathrm{phot},{+}}$} & {$\\sigma_{\\mathrm{phot},{-}}$} &{$\\chi^2$}\\\\\n\t\t\t{ } & { } & {(\\textit{Chandra})} & {(\\textit{Chandra})} & { } & { } & { } & { } & {criteria} & {Ratio (HR) } & { } & { } & { }\\\\ \n\t\t\t\\midrule\n\t\t\t$*$ & $121$ & $53.0268$ & $-27.7653$ & $53.0267$ & $-27.7653$ & $4$ & $4$ & $0$ & $0.21^{l}$ & $-$ & $1.26$ & $0.64$ & $0.04$ & $14.31$\\\\\n\t\t\t$*$ & $150$ & $53.0400$ & $-27.7985$ & $53.0398$ & $-27.7985$ & $4$ & $4$ & $0$ & $0.25^{l}$ & $-$ & $3.63$ & $0.14$ & $0.16$ & $1.66$\\\\\n\t\t\t$ $ & $173$ & $53.0477$ & $-27.8351$ & $53.0477$ & $-27.8351$ & $4$ & $4$ & $0$ & $-0.37$ & $0.13$ & $1.28$ & $0.30$ & $0.03$ & $4.11$\\\\\n\t\t\t$*$ & $184$ & $53.0523$ & $-27.7748$ & $53.0522$ & $-27.7747$ & $4$ & $4$ & $0$ & $0.19$ & $0.28$ & $1.73$ & $0.05$ & $0.05$ & $2.19$\\\\\n\t\t\t$*$ & $189$ & $53.0546$ & $-27.7931$ & $53.0544$ & $-27.7931$ & $4$ & $4$ & $0$ & $0.31^{l}$ & $-$ & $1.57$ & $0.17$ & $0.07$ & $9.34$\\\\\n\t\t\t$*$ & $199$ & $53.0579$ & $-27.8336$ & $53.0579$ & $-27.8336$ & $6$ & $7$ & $0$ & $-0.09$ & $0.10$ & $2.65$ & $0.11$ & $0.08$ & $16.61$\\\\\n\t\t\t$ $ & $211$ & $53.0620$ & $-27.8511$ & $53.0625$ & $-27.8508$ & $4$ & $4$ & $0$ & $0.06$ & $0.06$ & $1.38$ & $0.29$ & $0.07$ & $11.58$\\\\\n\t\t\t$*$ & $217$ & $53.0639$ & $-27.8438$ & $53.0638$ & $-27.8433$ & $0$ & $7$ & $0$ & $-0.17$ & $0.12$ & $3.72$ & $0.40$ & $0.23$ & $38.86$\\\\\n\t\t\t$ $ & $221$ & $53.0657$ & $-27.8790$ & $53.0660$ & $-27.8787$ & $4$ & $4$ & $0$ & $0.12^{l}$ & $-$ & $2.05$ & $0.09$ & $0.06$ & $8.84$\\\\\n\t\t\t$ $ & $226$ & $53.0668$ & $-27.8166$ & $53.0668$ & $-27.8165$ & $4$ & $4$ & $6$ & $0.33^{u}$ & $-$ & $5.43$ & $0.08$ & $0.12$ & $0.64$\\\\\n\t\t\t$*$ & $242$ & $53.0716$ & $-27.7699$ & $53.0713$ & $-27.7696$ & $4$ & $4$ & $0$ & $0.16$ & $0.26$ & $1.60$ & $0.06$ & $0.04$ & $16.75$\\\\\n\t\t\t$*$ & $244$ & $53.0721$ & $-27.8190$ & $53.0717$ & $-27.8187$ & $6$ & $7$ & $6$ & $0.29^{u}$ & $-$ & $1.90$ & $4.32$ & $0.22$ & $2.15$\\\\\n\t\t\t$ $ & $258$ & $53.0766$ & $-27.8641$ & $53.0766$ & $-27.8644$ & $6$ & $6$ & $0$ & $0.09^{l}$ & $-$ & $4.22$ & $0.62$ & $3.26$ & $12.49$\\\\\n\t\t\t$*$ & $273$ & $53.0821$ & $-27.7673$ & $53.0826$ & $-27.7681$ & $6$ & $7$ & $0$ & $0.03^{u}$ & $-$ & $3.45$ & $0.20$ & $0.83$ & $1.01$\\\\\n\t\t\t$ $ & $296$ & $53.0907$ & $-27.7825$ & $53.0913$ & $-27.7820$ & $6$ & $7$ & $6$ & $0.56^{l}$ & $-$ & $1.99$ & $0.10$ & $0.09$ & $12.71$\\\\\n\t\t\t$ $ & $301$ & $53.0924$ & $-27.8033$ & $53.0918$ & $-27.8028$ & $4$ & $4$ & $0$ & $0.34$ & $0.10$ & $2.02$ & $0.30$ & $0.29$ & $6.78$\\\\\n\t\t\t$*$ & $302$ & $53.0924$ & $-27.8268$ & $53.0923$ & $-27.8260$ & $4$ & $4$ & $0$ & $0.04^{u}$ & $-$ & $1.05$ & $0.75$ & $0.05$ & $26.22$\\\\\n\t\t\t$ $ & $303$ & $53.0925$ & $-27.8771$ & $53.0921$ & $-27.8767$ & $5$ & $0$ & $5$ & $0.35^{l}$ & $-$ & $0.73$ & $3.57$ & $-0.07$ & $0.37$\\\\\n\t\t\t$ $ & $306$ & $53.0939$ & $-27.8258$ & $53.0944$ & $-27.8259$ & $7$ & $7$ & $0$ & $0.16$ & $0.36$ & $2.87$ & $0.41$ & $0.11$ & $12.30$\\\\\n\t\t\t$*$ & $318$ & $53.0965$ & $-27.7449$ & $53.0966$ & $-27.7447$ & $4$ & $4$ & $0$ & $0.38^{u}$ & $-$ & $1.38$ & $0.06$ & $0.05$ & $34.03$\\\\\n\t\t\t$*$ & $321$ & $53.0984$ & $-27.7671$ & $53.0983$ & $-27.7667$ & $4$ & $4$ & $5$ & $0.31^{l}$ & $-$ & $1.22$ & $0.05$ & $0.03$ & $18.91$\\\\\n\t\t\t$ $ & $325$ & $53.1000$ & $-27.8086$ & $53.0995$ & $-27.8085$ & $4$ & $4$ & $0$ & $0.46^{u}$ & $-$ & $2.83$ & $0.23$ & $0.55$ & $7.27$\\\\\n\t\t\t$ $ & $328$ & $53.1016$ & $-27.8217$ & $53.1012$ & $-27.8224$ & $4$ & $4$ & $0$ & $0.24^{u}$ & $-$ & $1.68$ & $0.71$ & $1.06$ & $6.38$\\\\\n\t\t\t$ $ & $331$ & $53.1027$ & $-27.8606$ & $53.1028$ & $-27.8610$ & $4$ & $4$ & $0$ & $0.21^{u}$ & $-$ & $4.32$ & $0.02$ & $3.63$ & $3.33$\\\\\n\t\t\t$*$ & $348$ & $53.1052$ & $-27.8752$ & $53.1058$ & $-27.8753$ & $6$ & $6$ & $0$ & $0.15^{u}$ & $-$ & $3.11$ & $0.12$ & $0.67$ & $3.85$\\\\\n\t\t\t$*$ & $354$ & $53.1076$ & $-27.8558$ & $53.1079$ & $-27.8558$ & $7$ & $7$ & $0$ & $-0.54$ & $0.15$ & $2.83$ & $3.58$ & $0.85$ & $0.38$\\\\\n\t\t\t$ $ & $371$ & $53.1116$ & $-27.7679$ & $53.1118$ & $-27.7680$ & $5$ & $5$ & $0$ & $0.15$ & $0.10$ & $5.56$ & $0.04$ & $4.50$ & $0.61$\\\\\n\t\t\t$ $ & $373$ & $53.1118$ & $-27.9096$ & $53.1113$ & $-27.9094$ & $4$ & $4$ & $4$ & $0.22^{u}$ & $-$ & $2.54$ & $0.15$ & $0.25$ & $12.99$\\\\\n\t\t\t$*$ & $389$ & $53.1193$ & $-27.7659$ & $53.1186$ & $-27.7658$ & $4$ & $4$ & $0$ & $0.16^{u}$ & $-$ & $2.02$ & $0.05$ & $0.48$ & $7.31$\\\\\n\t\t\t$*$ & $392$ & $53.1199$ & $-27.7432$ & $53.1198$ & $-27.7436$ & $7$ & $5$ & $0$ & $0.05^{u}$ & $-$ & $3.07$ & $3.10$ & $0.00$ & $1.31$\\\\\n\t\t\t$*$ & $402$ & $53.1219$ & $-27.7529$ & $53.1214$ & $-27.7531$ & $5$ & $5$ & $0$ & $0.27^{l}$ & $-$ & $2.37$ & $0.64$ & $0.21$ & $2.91$\\\\\n\t\t\t$ $ & $403$ & $53.1220$ & $-27.9388$ & $53.1224$ & $-27.9381$ & $4$ & $4$ & $0$ & $0.20^{l}$ & $-$ & $4.22$ & $0.17$ & $0.12$ & $6.97$\\\\\n\t\t\t$ $ & $410$ & $53.1241$ & $-27.8913$ & $53.1242$ & $-27.8917$ & $4$ & $4$ & $0$ & $0.54$ & $0.12$ & $3.49$ & $0.27$ & $0.27$ & $2.63$\\\\\n\t\t\t$*$ & $428$ & $53.1296$ & $-27.8278$ & $53.1295$ & $-27.8276$ & $4$ & $4$ & $0$ & $0.24^{u}$ & $-$ & $1.15$ & $0.07$ & $0.04$ & $17.71$\\\\\n\t\t\t$ $ & $430$ & $53.1305$ & $-27.7912$ & $53.1310$ & $-27.7911$ & $0$ & $7$ & $0$ & $0.29$ & $0.15$ & $6.54$ & $1.34$ & $5.07$ & $0.03$\\\\\t\t\t\n\t\t\t$*$ & $444$ & $53.1340$ & $-27.7811$ & $53.1340$ & $-27.7809$ & $4$ & $4$ & $4$ & $0.28$ & $0.07$ & $1.42$ & $0.10$ & $0.06$ & $41.98$\\\\\n\t\t\t$*$ & $455$ & $53.1378$ & $-27.8022$ & $53.1378$ & $-27.8021$ & $4$ & $4$ & $0$ & $0.21^{u}$ & $-$ & $1.26$ & $0.05$ & $0.03$ & $5.63$\\\\\n\t\t\t$ $ & $456$ & $53.1380$ & $-27.8683$ & $53.1381$ & $-27.8684$ & $7$ & $7$ & $0$ & $-0.29$ & $0.11$ & $10.91$ & $-0.92$ & $8.39$ & $0.00$\\\\\n\t\t\t$*$ & $460$ & $53.1393$ & $-27.8745$ & $53.1394$ & $-27.8746$ & $0$ & $5$ & $0$ & $-0.02$ & $0.27$ & $2.91$ & $2.68$ & $0.09$ & $0.73$\\\\\n\t\t\t$*$ & $462$ & $53.1403$ & $-27.7976$ & $53.1405$ & $-27.7973$ & $4$ & $4$ & $0$ & $0.06$ & $0.39$ & $1.63$ & $0.06$ & $0.04$ & $5.12$\\\\\n\t\t\t$ $ & $466$ & $53.1417$ & $-27.8167$ & $53.1416$ & $-27.8166$ & $4$ & $4$ & $0$ & $0.27$ & $0.13$ & $2.02$ & $0.20$ & $0.31$ & $5.73$\\\\\n\t\t\t$*$ & $485$ & $53.1466$ & $-27.8710$ & $53.1460$ & $-27.8711$ & $7$ & $7$ & $0$ & $0.45$ & $0.15$ & $2.83$ & $0.54$ & $0.68$ & $14.74$\\\\\n\t\t\t$ $ & $496$ & $53.1505$ & $-27.8890$ & $53.1507$ & $-27.8886$ & $4$ & $4$ & $0$ & $0.00$ & $-$ & $2.50$ & $0.31$ & $2.11$ & $1.44$\\\\\n\t\t\t$*$ & $522$ & $53.1585$ & $-27.7741$ & $53.1583$ & $-27.7738$ & $4$ & $5$ & $6$ & $-0.40$ & $0.04$ & $1.60$ & $0.21$ & $0.05$ & $4.06$\\\\\n\t\t\t$ $ & $535$ & $53.1627$ & $-27.7443$ & $53.1622$ & $-27.7442$ & $4$ & $4$ & $0$ & $-0.12$ & $0.06$ & $2.37$ & $0.30$ & $0.39$ & $3.44$\\\\\n\t\t\t$*$ & $539$ & $53.1632$ & $-27.8091$ & $53.1621$ & $-27.8097$ & $4$ & $4$ & $0$ & $0.53^{l}$ & $-$ & $0.97$ & $0.06$ & $0.08$ & $44.03$\\\\\n\t\t\t$ $ & $546$ & $53.1653$ & $-27.8142$ & $53.1648$ & $-27.8144$ & $4$ & $4$ & $0$ & $0.29$ & $0.03$ & $3.15$ & $0.18$ & $0.25$ & $8.09$\\\\\n\t\t\t$ $ & $556$ & $53.1701$ & $-27.9298$ & $53.1699$ & $-27.9304$ & $4$ & $4$ & $0$ & $0.40$ & $0.03$ & $2.68$ & $0.17$ & $0.16$ & $12.11$\\\\\n\t\t\t$*$ & $574$ & $53.1787$ & $-27.8027$ & $53.1782$ & $-27.8027$ & $6$ & $7$ & $0$ & $0.38$ & $0.25$ & $3.32$ & $0.42$ & $0.75$ & $2.72$\\\\\n\t\t\t$ $ & $578$ & $53.1806$ & $-27.7797$ & $53.1807$ & $-27.7796$ & $7$ & $7$ & $0$ & $0.00$ & $-$ & $6.77$ & $0.97$ & $5.10$ & $0.00$\\\\\n\t\t\t$ $ & $583$ & $53.1835$ & $-27.7766$ & $53.1834$ & $-27.7764$ & $7$ & $7$ & $0$ & $-0.51$ & $0.06$ & $9.36$ & $0.63$ & $4.82$ & $0.00$\\\\\n\t\t\t$ $ & $589$ & $53.1850$ & $-27.8198$ & $53.1851$ & $-27.8196$ & $4$ & $4$ & $6$ & $0.17^{u}$ & $-$ & $1.28$ & $0.26$ & $0.08$ & $5.77$\\\\\n\t\t\t$*$ & $591$ & $53.1852$ & $-27.7174$ & $53.1848$ & $-27.7173$ & $4$ & $4$ & $0$ & $0.29^{u}$ & $-$ & $1.35$ & $0.17$ & $0.21$ & $21.54$\\\\\n\t\t\t$ $ & $620$ & $53.1960$ & $-27.8927$ & $53.1957$ & $-27.8928$ & $4$ & $4$ & $0$ & $-0.02$ & $0.06$ & $2.57$ & $0.55$ & $0.45$ & $4.34$\\\\\n\t\t\t$ $ & $624$ & $53.1981$ & $-27.8323$ & $53.1979$ & $-27.8319$ & $5$ & $6$ & $0$ & $0.33^{l}$ & $-$ & $1.52$ & $2.23$ & $0.07$ & $0.56$\\\\\n\t\t\t$*$ & $625$ & $53.1989$ & $-27.8440$ & $53.1989$ & $-27.8439$ & $4$ & $4$ & $0$ & $0.71$ & $0.10$ & $1.68$ & $0.07$ & $0.04$ & $16.22$\\\\\n\t\t\t$*$ & $630$ & $53.2016$ & $-27.8443$ & $53.2027$ & $-27.8448$ & $7$ & $4$ & $0$ & $0.36^{u}$ & $-$ & $2.05$ & $5.04$ & $0.16$ & $0.37$\\\\\n\t\t\t$*$ & $651$ & $53.2153$ & $-27.8703$ & $53.2150$ & $-27.8695$ & $7$ & $0$ & $0$ & $0.40$ & $0.10$ & $1.90$ & $0.41$ & $0.34$ & $16.22$\\\\\n\t\t\\bottomrule\n\t\t\\end{tabular}\n\t\t\\end{center}\n\t\\caption{\\label{tab:overview}Overview over the redshift estimates that we gained for each object in the course of this analysis. All photometric redshifts were gained by using GOODS\/ACS, CANDELS and in some cases also \\textit{Spitzer}\\ data (marked with an asterisk). For our three final candidates (456, 578, 583) we also computed photometric redshifts using HUDF and HUGS data. Please see sections \\ref{sec:HUDF} and \\ref{sec:HUGS} for the corresponding values.  Note that the stacking procedure only gives a redshift indication. Hence, if '4' is given as the stacking result this corresponds to $z\\lesssim4$,  '7' indicates $z\\gtrsim7$. We mark sources for which we only gained upper or lower limits on the Hardness Ratio with '$u$' and '$l$'. '0' indicates that a source could not be classified by the corresponding redshift test.}\t\n\\end{table*}\n\\newpage\n\n\\section*{Acknowledgements}\nWe thank  Andreas Faisst for helpful discussions and Richard Ellis for suggesting the stacking analysis. We also thank the anonymous referee for helpful comments. AKW, KS and MK gratefully acknowledge support from Swiss National Science Foundation Grant PP00P2\\_138979\/1. Support for the work of ET was provided by the Center of Excellence in Astrophysics and Associated Technologies (PFB 06), by the FONDECYT regular grant 1120061 and by the CONICYT Anillo project ACT1101. This research has made use of NASA's ADS Service. This research made use of Astropy, a community-developed core Python package for Astronomy (Astropy Collaboration, 2013).\n\\newpage\n\n\\bibliographystyle{mn2e}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n    \n    \\para{Background}\n    The problem of simulating the time evolution of a quantum system is perhaps the most important application of quantum computers. Indeed, this was the reason Feynman proposed quantum computing~\\cite{Fey82}, and it remains an important practical application since a significant fraction of the world's supercomputing power is used to solve instances of this problem that arise in materials science, condensed matter physics, high energy physics, and chemistry~\\cite{NERSC16}.\n    \n    All known classical algorithms (i.e., algorithms that run on traditional non-quantum computers) for this problem run in exponential time. On the other hand, from the early days of quantum computing~\\cite{Fey82,Lloyd1996} it was known that quantum computers can solve this problem in polynomial time. More precisely, when formalized as a decision problem, the problem of simulating the time evolution of a quantum system is in the complexity class $\\mathsf{BQP}$, the class of problems solved by a quantum computer to bounded error in polynomial time. Furthermore, the problem is complete for $\\mathsf{BQP}$~\\cite{Feynman1985,Nagaj2009},\n    which means we do not expect there to be efficient classical algorithms for the problem, since that would imply $\\mathsf{BPP}=\\mathsf{BQP}$, which in turn would imply polynomial-time algorithms for problems such as integer factorization and discrete log~\\cite{Sho97}. \n    \n   \n    \n    \\para{Hamiltonian simulation problem} The Hamiltonian simulation problem is a standard formalization of the problem of simulating the time evolution\\footnote{This is sometimes referred to as ``real time evolution'', to distinguish it from ``imaginary time evolution'' which we will not talk about in this paper.} of a quantum system. In this problem, we assume the quantum system whose time evolution we wish to simulate consists of $n$ qubits and we want to simulate its time evolution for time $T$, in the sense that we are provided with the initial state $|\\psi(0)\\>$ and we want to compute the state of the system at time $T$, $|\\psi(T)\\>$. The goal of an efficient simulation is to solve the problem in time polynomial in $n$ and $T$.\n    \n    The state of a system of $n$ qubits can be described by a complex vector of dimension $2^n$ of unit norm. Since we are studying quantum algorithms for the problem, we are given the input as an $n$-qubit quantum state, and have to output an $n$-qubit quantum state. The relation between the output state at time $T$ and the initial state at time $0$ is given by the Schr\\\"odinger equation\n    \\begin{equation}\n    i\\frac{\\mathrm{d}}{\\mathrm{d}t}|\\psi(t)\\>=H(t)|\\psi(t)\\>,\n    \\end{equation}\n    where the Hamiltonian $H$, a $2^n \\times 2^n$ complex Hermitian matrix, has entries which may also be functions of time. The Hamiltonian captures the interaction between the constituents of the system and governs time dynamics. In the special case where the Hamiltonian is independent of time, the  Schr\\\"odinger equation can be solved to yield  $|\\psi(T)\\>=e^{-iHT}|\\psi(0)\\>$.\n    \n    More formally, the input to the Hamiltonian simulation problem consists of a Hamiltonian $H$ (or $H(t)$ in the time-dependent case), a time $T$, and an error parameter $\\epsilon$. The goal is to output a quantum circuit that approximates the unitary matrix that performs the time evolution above (e.g., for time-independent Hamiltonians, the quantum circuit should approximate the unitary $e^{-iHT}$). The notion of approximation used is the spectral norm distance between the ideal unitary and the one performed by the circuit. \n    The cost of a quantum circuit is measured by the number of gates used in the circuit, where the gates come from some fixed universal gate set. Note that it is important to describe how the Hamiltonian in the input is specified, since it is a matrix of size $2^n \\times 2^n$. This will be made precise when talking about specific classes of Hamiltonians that we would like to simulate.\n    \n    \\para{Geometrically local Hamiltonian simulation}\n    The most general class of Hamiltonians that is commonly studied in the literature is the class of sparse Hamiltonians~\\cite{AT03,Chi04,BAC+07,BerryEtAl2014,TS,BCK15,QSP,LC16, LC17USA}. A Hamiltonian on $n$ qubits is sparse if it has only $\\poly(n)$ nonzero entries in any row or column. For such Hamiltonians, we assume we have an efficient algorithm to compute the nonzero entries in each row or column, and the input Hamiltonian is specified by an oracle that can be queried for this information. In this model, recent quantum algorithms have achieved optimal complexity in terms of the queries made to the oracle~\\cite{QSP}.\n    \n    A very important special case of this type of Hamiltonian is a ``local Hamiltonian.'' Confusingly, this term is used to describe two different kinds of Hamiltonians in the literature. \n    We distinguish these two definitions of ``local'' by referring to them as ``non-geometrically local'' and ``geometrically local'' in this introduction. A non-geometrically local Hamiltonian is a Hamiltonian $H(t)$ that can be written as a sum of polynomially many terms $H_j(t)$, each of which acts nontrivially on only $k$ qubits at a time (i.e., the matrix acts as identity on all other qubits). A geometrically local Hamiltonian is similar, except that each term $H_j(t)$ must act on $k$ adjacent qubits. Since we refer to ``adjacent'' qubits, the geometry of how the qubits are laid out in space must be specified. In this paper we will deal with qubits laid out in a $D$-dimensional lattice in Euclidean space. I.e., qubits are located at points in $\\mathbb{Z}^D$ and are adjacent if they are close in Euclidean distance.\n    \n    Lattice Hamiltonians in $D$-dimensions (with $D\\leq 3$) already model all the physical systems we are interested in,\n    are of fundamental importance by the principle of locality.\n    From a practical perspective, this case captures a large fraction of all physical systems we are interested in.\\footnote{There are some physical situations where we do care about more general Hamiltonians. Even though the system we are given may be described by a geometrically local Hamiltonian, it is sometimes computationally advantageous to represent a given system with a non-geometrically local (or sparse) Hamiltonian.}\n    \n   \n    \n    \\para{Prior best algorithms}\n    To describe the known algorithms for this problem, we need to formally specify the problem. Although our results apply to very general time-dependent Hamiltonians, while comparing to previous work we assume the simpler case where the Hamiltonian is time independent.\n    \n    We assume our $n$ qubits are laid out in a $D$-dimensional lattice $\\Lambda$ in $\\mathbb{R}^D$, where $D=\\O(1)$, and every unit ball contains $\\O(1)$ qubits. We assume our Hamiltonian $H$ is given as a sum of terms $H = \\sum_{X\\subseteq\\Lambda} h_X$, where each $h_X$ only acts nontrivally on qubits in $X$ (and acts as identity on the qubits in $\\Lambda \\setminus X$), such that $h_X=0$ if $\\diam(X)>1$,\n    which enforces geometric locality. \n    (More formally, we rescale the metric\n    in such a way that $\\diam(X) > 1$ implies $h_X = 0$.)\n    We normalize the Hamiltonian by requiring $\\norm{h_X}\\leq 1$.\n    \n    We consider a quantum circuit simulating the time evolution due to such a Hamiltonian efficient if it uses $\\poly(n,T,1\/\\epsilon)$ gates. To get some intuition for what we should hope for, notice that in the real world, time evolution takes time $T$ and uses $n$ qubits. Regarding ``Nature'' as a quantum simulator, we might expect that there is a quantum circuit that uses $O(n)$ qubits, $O(T)$ circuit depth, and $O(nT)$ total gates to solve the problem. It is also reasonable to allow logarithmic overhead in the simulation since such overheads are common even when one classical system simulates the time evolution of another (e.g., when one kind of Turing machine simulates another).\n    \n    However, previous algorithms for this problem fall short of this expectation. The best Hamiltonian simulation algorithms for sparse Hamiltonians~\\cite{TS,QSP,LC16} have query complexity $\\O(nT \\polylog(nT\/\\epsilon))$, but the assumed oracle for the entries requires $O(n)$ gates to implement, yielding an algorithm that uses $\\O(n^2T \\polylog(nT\/\\epsilon))$ gates. This was also observed in a recent paper of Childs, Maslov, Nam, Ross, and Su~\\cite{Childs2017}, who noted that for $T=n$, all the sparse Hamiltonian simulation algorithms had gate cost proportional to $n^3$ (or worse). A standard application of high-order Lie-Trotter-Suzuki expansions~\\cite{Trotter1959,Suzuki1991,Lloyd1996,BAC+07} yields gate complexity $O(n^2T (nT\/\\epsilon)^{\\delta})$ for any fixed $\\delta > 0$. It has been argued~\\cite[Sec.~4.3]{JordanLeePreskill2014} that \n    this in fact yields an algorithm with gate complexity $O(nT (nT\/\\epsilon)^{\\delta})$ for any fixed $\\delta > 0$. We believe this analysis is correct, but perhaps some details need to be filled in to make the analysis rigorous. In any case, this algorithm still performs  worse than desired, and in particular does not have polylogarithmic dependence on $1\/\\epsilon$.\n    \n    \\subsection{Results} We exhibit a quantum algorithm that simulates the time evolution due to a time-dependent lattice Hamiltonian with a circuit that uses $\\O(nT \\polylog(nT\/\\epsilon))$ geometrically local $2$-qubit gates (i.e., the gates in our circuit also respect the geometry of the qubits), with depth  $\\O(T \\polylog(nT\/\\epsilon))$ using only $\\polylog(nT\/\\epsilon)$ ancilla qubits. We then also prove a matching lower bound, showing that no quantum algorithm can do better (up to logarithmic factors), even if we relax the output requirement significantly. We now describe our results more formally.\n    \n    \\para{Algorithmic results}\n    We consider a more general version of the problem with time-dependent Hamiltonians. In this case we will have $H(t) = \\sum_{X\\subseteq \\Lambda} h_X(t)$ with the locality and norm conditions as before. However, now the operators $h_X(t)$ are functions of time and we need to impose some reasonable constraints on the entries to obtain polynomial-time algorithms.\n    \n    First we need to be able to compute the entries of our Hamiltonian efficiently at a given time $t$.\n    We say that a function $\\alpha: [0,T] \\ni t \\mapsto \\alpha(t) \\in  {\\mathbb{R}}$ \n    is \\emph{efficiently computable}\n    if there is an algorithm that outputs $\\alpha(t)$ to precision $\\epsilon$ \n    for any given input $t$ specified to precision $\\epsilon$\n    in running time $\\polylog(T\/\\epsilon)$.\n    Note that any complex-valued analytic function \n    on a nonzero neighborhood of a closed real interval in the complex plane\n    is efficiently computable (see \\cref{app:chebyshev}).\n    We will assume that each entry in a local term $h_X(t)$ is efficiently computable.\n    \n    In addition to being able to compute the entries of the Hamiltonian, we require that the entries do not change wildly with time; otherwise, a sample of entries at discrete times may not predict the behavior of the entries at other times.\n    We say that a function $\\alpha$ on the interval $[0,T]$ ($T \\ge 1$) is \n    \\emph{piecewise slowly varying}\n    if there are $M = {{O}}(T)$ intervals $[t_{j-1},t_j]$ \n    with $0 = t_0 < t_1 < \\cdots < t_M = T$\n    such that $\\frac{\\mathrm{d}}{\\mathrm{d}t}\\alpha(t)$ exists and is bounded \n    by $1\/(t_j - t_{j-1})$ for $t \\in (t_{j-1},t_j)$.\n    In particular, a function is piecewise slowly varying if it is a sum of $O(T)$ pieces, each of which has derivative at most $O(1)$. \n    We will assume that each entry in a term $h_X(t)$ is piecewise slowly varying. \n    \n    We are now ready to state our main result, which is proved in \\Cref{sec:algorithm}\n    \n    \\begin{restatable}{theorem}{main}\n        Let $H(t) = \\sum_{X \\subseteq \\Lambda} h_X(t)$ be a time-dependent Hamiltonian\n        on a lattice $\\Lambda$ of $n$ qubits, embedded in the Euclidean metric space $\\mathbb R^D$.\n        Assume that every unit ball contains ${{O}}(1)$ qubits and\t$h_X = 0$ if $\\diam(X) > 1$.\n        Also assume that every local term $h_X(t)$ is efficiently computable (e.g., analytic), piecewise slowly varying on time domain $[0,T]$, and has  $\\norm{h_X(t)} \\le 1$ for any $X$ and $t$.\n        Then, there exists a quantum algorithm that can approximate the time evolution of $H$\n        for time $T$ to accuracy $\\epsilon$ using ${{O}}( Tn \\polylog(Tn\/\\epsilon))$ 2-qubit local gates,\n        and has depth ${{O}}(T \\polylog(Tn\/\\epsilon))$.\n        \\label{thm:main}\n    \\end{restatable}\n    \n    Our algorithm uses $O(1)$ ancillas per system qubit on which $H$ is defined.\n    The ancillas are interspersed with the system qubits,\n    and all the gates respect the locality of the lattice.\n    \n    \\para{Lower bounds}\n    We also prove a lower bound on the gate complexity of problem of simulating the time evolution of a time-dependent lattice Hamiltonian. This lower bound matches, up to logarithmic factors, the gate complexity of the algorithm presented in \\Cref{thm:main}. Note that unlike previous lower bounds on Hamiltonian simulation~\\cite{BAC+07,BerryEtAl2014,BCK15}, which prove lower bounds on query complexity, this is a lower bound on the number of gates required to approximately implement the time-evolution unitary. To our best knowledge, this is the first nontrivial lower bound on the gate complexity of the simulation problem. For concreteness, we focus on a 1-dimensional time-dependent local Hamiltonian in this section, although the lower bound extends to other constant dimensions with minor modifications. The lower bounds are proved in \\Cref{sec:lowerbound}.\n    \n    Before stating the result formally, let us precisely define the class of Hamiltonians for which we prove the lower bound.\n    We say a Hamiltonian $H(t)$ acting on $n$ qubits is a \n    ``piecewise constant 1D Hamiltonian''\n    if $H(t)=\\sum_{j=1}^{n-1}H_j(t)$, \n    where $H_j(t)$ is only supported on qubits $j$ and $j+1$ with $\\max_t \\norm{H_j(t)} = {{O}}(1)$,\n    and there is a time slicing $0 = t_0 < t_1 < \\cdots < t_M = T$\n    where $t_m - t_{m-1} \\le 1$ and $M = {{O}}(T)$ such that \n    $H(t)$ is time-independent within each time slice.\n    \n    For such Hamiltonians, the time evolution operator for time $T$ can be simulated with error at most $\\epsilon$ \n    using \\Cref{thm:main} with ${{O}}(Tn \\polylog(Tn\/\\epsilon))$ $2$-qubit local gates \n    (i.e., the 2-qubit gates only act on adjacent qubits). \n    In particular, for any constant error, the simulation only requires ${\\widetilde{\\calO}}(Tn)$ 2-qubit local gates. \n    We prove a matching lower bound, where the lower bound even holds against circuits that may use non-geometrically local (i.e., acting on non-adjacent qubits) $2$-qubit gates from a possibly infinite gate set and unlimited ancilla qubits. \n    \n    \\begin{restatable}{theorem}{lowerboundgeneral}\n        \\label{thm:lowerboundgeneral}\n        For any integers $n$ and $T \\leq 4^n$, \n        there exists a piecewise constant bounded 1D Hamil{-}tonian $H(t)$ on $n$ qubits,\n        such that any quantum circuit that approximates the time evolution due to $H(t)$ for time $T$ to constant error must use ${\\widetilde{\\Omega}}(Tn)$ $2$-qubit gates. \n        The quantum circuit may use unlimited ancilla qubits and the gates may be non-local and come from a possibly infinite gate set.\n    \\end{restatable}\n    \n    \n    Note that this lower bound only holds for $T\\leq 4^n$, because any unitary on $n$ qubits can be implemented with ${\\widetilde{\\calO}}(4^n)$ 2-qubit local gates~\\cite{BBC+95,Kni95}. \n    \n    We can also strengthen our lower bound to work in the situation where we are only interested in measuring a local observable at the end of the simulation. The simulation algorithm presented in \\Cref{thm:main} provides a strong guarantee: the output state is $\\epsilon$-close to the ideal output state in trace distance. Trace distance captures distinguishability with respect to arbitrary measurements, but for some applications it might be sufficient for the output state to be close to the ideal state with respect to local measurements only. We show that even in this limited measurement setting, it is not possible to speed up our algorithm in general. In fact, our lower bound works even if the only local measurement performed is a computational basis measurement on the first output qubit.\n    \n    \\begin{restatable}{theorem}{lowerboundlocal}\n        \\label{thm:lowerboundlocal}\n        For any integers $n$ and $T$ such that $1 \\le n\\leq T\\leq 2^n$, \n        there exists a piecewise constant bounded 1D Hamiltonian $H(t)$ on $n$ qubits, \n        such that any quantum circuit that approximates the time evolution due to $H(t)$ for time $T$ to constant error on any local observable must use ${\\widetilde{\\Omega}}(Tn)$ 2-qubit gates. \n        If $T\\leq n$, we have a lower bound of ${\\widetilde{\\Omega}}(T^2)$ gates. \n        (The quantum circuit may use unlimited ancilla qubits and the gates may be non-local and come from a possibly infinite gate set.)\n    \\end{restatable}\n    \n    Note that the fact that we get a weaker lower bound of ${\\widetilde{\\Omega}}(T^2)$ when $T\\leq n$ is not a limitation, but reflects the fact that small time evolutions are actually easier to simulate when the measurement is local. To see this, consider first simulating the time evolution using the algorithm in \\Cref{thm:main}. This yields a circuit with ${\\widetilde{\\calO}}(Tn)$ 2-qubit local gates. But if we only want the output of a local measurement after time $T$, qubits that are far away from the measured qubits cannot affect the output, since the circuit only consists of 2-qubit local gates. Hence we can simply remove all gates that are more than distance equal to the depth of the circuit, ${\\widetilde{\\calO}}(T)$, away from the measured qubits. We are then left with a circuit that uses ${\\widetilde{\\calO}}(T^2)$ gates, matching the lower bound in \\Cref{thm:lowerboundlocal}.\n    \n\\subsection{Techniques}\n    \n    \\para{Algorithm}\n    Our algorithm is based on a decomposition of the time evolution unitary using Lieb-Robinson bounds~\\cite{LiebRobinson1972,Hastings2004LSM,NachtergaeleSims2006,HastingsKoma2006,Hastings2010},\n    that was made explicit by Osborne~\\cite{Osborne2006} (see also Michalakis~\\cite[Sec.~III]{Michalakis2012}),\n    which when combined with recent advances in Hamiltonian simulation~\\cite{TS,QSP,LC16},\n    yields \\Cref{thm:main}. \n   \n   \n    \n    Lieb-Robinson bounds are theorems that informally state that information travels at a constant speed in geometrically local Hamiltonians. For intuition, consider a 1-dimensional lattice of qubits and a geometrically local Hamiltonian that is evolved for a short amount of time. If the time is too short, no information about the first qubit can be transmitted to the last qubit. Lieb-Robinson bounds make this intuition precise, and show that the qubit at position $n$ is only affected by the qubits and operators at position 1 after time $\\Omega(n)$. Note that if this were a small-depth unitary circuit of geometrically local $2$-qubit gates such a statement would follow using a ``lightcone'' argument. In other words, after one layer of geometrically local $2$-qubit gates, the influence of qubit 1 can only have spread to qubit 2. Similarly, after $k$ layers of $2$-qubit gates, the influence of qubit 1 can only have spread up to qubit $k$. The fact that this extends to geometrically local Hamiltonians is nontrivial, and is only approximately true. See \\Cref{lem:LR} for a formal statement of a Lieb-Robinson bound.\n    \n    We use these ideas to chop up the large unitary that performs time evolution for the full Hamiltonian $H$ into many smaller unitaries that perform time evolution for a small portion of the Hamiltonian. Quantitatively, we break Hamiltonian simulation for $H$ for time $O(1)$ into $O(n\/\\log(nT\/\\epsilon))$ pieces, each of which is a Hamiltonian simulation problem for a Hamiltonian on an instance of size $O(\\log(nT\/\\epsilon))$ to exponentially small error. At this point we can use any Hamiltonian simulation algorithm for the smaller piece as long as it has polynomial gate cost and has exponentially good dependence on $\\epsilon$. While Hamiltonian simulation algorithms based on product formulas do not have error dependence that is $\\polylog(1\/\\epsilon)$, recent Hamiltonian simulation algorithms, such as \\cite{BerryEtAl2014,TS,BCK15,QSP,LC16} have $\\polylog(1\/\\epsilon)$ scaling. Thus our result crucially uses the recent advances in Hamiltonian simulation with improved error scaling.\n    \n    \\para{Lower bound} As noted before, we lower bound the gate complexity (or total number of gates) required for Hamiltonian simulation, which is different from prior work which proved lower bounds on the query complexity of Hamiltonian simulation. As such, our techniques are completely different from those used in prior work. Informally, our lower bounds are based on a refined circuit-size hierarchy theorem for quantum circuits, although we are technically comparing two different resources in two different models, which are simulation time for Hamiltonians versus gate cost for circuits. \n    \n    As a simple motivating example, consider circuit-size hierarchy theorems for classical or quantum circuits more generally. Abstractly, a hierarchy theorem generally states that a computational model with $X$ amount of a resource (e.g., time, space, gates) can do more if given more of the same resource. For example, it can be shown that for every $G \\ll 2^n\/n$, there exists a Boolean function on $n$ bits that cannot be computed by a circuit of size $G$, but can be computed by a circuit of size $G+O(n)$. We show similar hierarchy theorems for quantum circuit size, except that we show that the circuit of larger size that computes the function actually comes from a weaker family of circuits. Informally, we are able to show that there are functions that can be computed by a larger circuit that uses only geometrically local $2$-qubit gates from a fixed universal gate set that cannot be a computed by a smaller circuit, even if we allow the smaller circuit access to unlimited ancilla bits and non-geometrically local $2$-qubit from an infinite gate set. We then leverage this asymmetric circuit size hierarchy theorem to show that there is a Hamiltonian whose evolution for time $T$ cannot be simulated by a circuit of size $\\ll nT$, by embedding the result of any quantum circuit with geometrically local $2$-qubit gates into a piecewise constant Hamiltonian with time proportional to the depth of the circuit.\n    \n    \n    \\section{Algorithm and analysis}\n    \\label{sec:algorithm}\n    \n    In this section we establish our main algorithmic result, restated below for convenience:\n    \n    \\begin{figure*}[t]\n        \\centering\n        \\includegraphics[width=0.85\\textwidth]{algorithm.pdf}\n        \\caption{Decomposition of time evolution operator for time $t = {{O}}(1)$.\n            The time is going upwards.\n            Each block $\\square$ represents the forward time evolution, $e^{-it H_\\square}$,\n            if the arrow is upward,\n            and backward time evolution, $e^{+it H_\\square}$, if the arrow is downward.\n            Here, $H_\\square$ is the sum of local terms in the Hamiltonian \n            supported within the block.\n            The overlap has size $\\ell$.\n            (a) shows a one-dimensional setting, \n            but a generalization to higher $D$ dimensions is readily achieved\n            by regarding each block as a $(D-1)$-dimensional hyperplane\n            so that the problem reduces to lower dimensions.\n            (b) shows a two-dimensional setting.\n            The approximation error from the depicted decomposition is $\\epsilon = {{O}}(e^{-\\mu \\ell} L^D \/ \\ell)$\n            where $L$ is the linear system size, $\\ell$ is the width of the overlap between blocks,\n            and $\\mu > 0$ is a constant that depends only on the locality of the Hamiltonian.\n            One can use any algorithm to further \n            decompose the resulting ``small'' unitaries on ${{O}}(\\log(L\/\\epsilon))$ qubits\n            into elementary gates.\n            To achieve gate count that is linear (up to logarithmic factors) in spacetime volume,\n            the algorithm for simulating the blocks needs to be polynomial in the block size and polylogarithmic in accuracy.\n        }\n        \\label{fig:algorithm}\n    \\end{figure*}\n    \n    \n    \n    \\main*\n    \n    The algorithm is depicted in \\Cref{fig:algorithm}.\n    Before showing why this algorithm works, we provide a high-level overview of the algorithm\n    and the structure of the proof.\n    Since a time evolution unitary $U(T;0)$ is equal to \n    $U(T=t_M; t_{M-1})U(t_{M-1},t_{M-2}) \\cdots U(t_2;t_1)U(t_1; t_0 = 0)$,\n    we will focus on a time evolution operator $U(t;0)$ where $t = {{O}}(1)$,\n    generated by a slowly varying bounded Hamiltonian.\n    The key idea, as shown in \\Cref{fig:algorithm}, is that the time-evolution operator, \n    $e^{-itH}$ due to the full Hamiltonian $\\sum_{X\\subseteq \\Lambda} h_X$ \n    can be approximately written as a product \n    \\begin{align}\n    e^{-itH} \\approx \n    \\left( e^{-itH_A} \\right) \\left(   e^{+itH_Y}  \\right) \\left(  e^{-itH_{Y\\cup B}} \\right).\n    \\end{align}\n    Here $A\\cup B = \\Lambda$ and \n    we think of $A$ and $B$ as large regions, but $Y$ as a small subset of $A$.\n    The error in the approximation is exponentially small as long as $Y$ is large enough.\n    This is formally proved in \\Cref{lem:decomposition}, which is supported by \n    \\Cref{lem:ode} and \\Cref{lem:LRB}. Applying this twice, using \n    \\begin{align}\n    e^{-itH_{Y\\cup B}} \n    \\approx \n    \\left( e^{-itH_B} \\right)\n    \\left( e^{+itH_Z}  \\right)\n    \\left( e^{-itH_{Y \\cup Z}} \\right)\n    \\end{align}\n    leads to a symmetric approximation\n    as depicted at the bottom left of \\Cref{fig:algorithm}.\n    This procedure can then be repeated\n    for the large operators supported on $A$ and $B$ to reduce the size of all the operators\n    involved, leading to the pattern in \\Cref{fig:algorithm} (a).\n    This reduces the problem of\n    implementing the time-evolution operator for $H$ into the problem of implementing smaller\n    time-evolution operators, which can be implemented using known quantum algorithms.\n    We now establish the lemmas needed to prove the result.\n    \n    \\begin{lemma}\n    Let $A_t$ and $B_t$ be continuous time-dependent Hermitian operators,\n    and let $U^A_t$ and $U^B_t$ with $U^A_0 = U^B_0 = {\\mathbf{1}}$ \n    be the corresponding time evolution unitaries.\n    Then the following hold:\n    \\begin{enumerate\n    \\item[(i)] $W_t=(U^B_t)^\\dagger U^A_t$ is the unique solution of \n    $i\\partial_t W_t = \\left((U^B_t)^\\dagger (A_t-B_t) U^B_t \\right) W_t$ \n    and $W_0 = {\\mathbf{1}}$.\n    \\item[(ii)] If $\\norm{A_s-B_s} \\le \\delta$ for all $s \\in [0,t]$,\n     then $\\norm{U^A_t - U^B_t} \\le t \\delta$.\n    \\end{enumerate}\n    \\label{lem:ode}\n    \\end{lemma}\n    \\begin{proof} \n    (i) Differentiate.\n    The solution to the ordinary differential equation is unique.\n    (ii) Apply Jensen's inequality for $\\norm{\\cdot}$ \n    (implied by the triangle inequality for $\\norm{\\cdot}$) \n    to the equation $W_t - W_0 = \\int_0^t {\\mathrm{d}} s \\partial_s W_s$.\n    Then, invoke (i) and the unitary invariance of $\\norm{\\cdot}$.\n    \\end{proof}\n\n    \\begin{lemma}[Lieb-Robinson \n    bound~\\cite{LiebRobinson1972,Hastings2004LSM,NachtergaeleSims2006,HastingsKoma2006}]\n    \\label{lem:LR}\\label{lem:LRB}\n    Let $H=\\sum_X h_X$ be a local Hamiltonian and \n    $O_X$ be any operator supported on $X$,\n    and put $\\ell = \\lfloor \\dist(X,\\Lambda\\setminus \\Omega) \\rfloor$.\n    Then\n    \\begin{align}\n    &\\norm{(U^{H}_t)^\\dagger O_X U^{H}_t\n        - (U^{H_{ \\Omega}}_t)^\\dagger O_X U^{H_{ \\Omega}}_t}\n    \\le \\nonumber \\\\\n    &|X|\\norm{O_X}\n    \\frac{(2\\zeta_0 |t|)^\\ell}{\\ell!},\n    \\label{eq:LRB}\n    \\end{align}\n    where $\\zeta_0 = \\max_{p \\in \\Lambda} \\sum_{Z \\ni p} |Z|\\norm{h_Z} = {{O}}(1)$.\n    In particular, there are constants $v_{LR} > 0$, called the Lieb-Robinson velocity,%\n    \\footnote{\n    Strictly speaking, the Lieb-Robinson velocity \n    is defined to be the infimum of any $v_{LR}$ such that \\cref{eq:exteriorLightcone} holds.\n    }\n    and $\\mu>0$, \n    such that for $\\ell \\geq v_{LR} |t|$, we have\n    \\begin{align}\n    &\\norm{(U^{H}_t)^\\dagger O_X U^{H}_t\n      - (U^{H_{ \\Omega}}_t)^\\dagger O_X U^{H_{ \\Omega}}_t} \\nonumber \\\\ \n      &\\le {{O}}(|X| \\norm{O_X} \\exp(- \\mu \\ell)).\n    \\label{eq:exteriorLightcone}\n    \\end{align}\n    \\end{lemma}\n    \\begin{proof}\n        See \\ref{app:pfLRB}.\n    \\end{proof}\n    We are considering strictly local interactions (as in \\Cref{thm:main}),\n    where $h_X = 0$ if $\\diam(X)>1$,\n    but similar results hold with milder locality conditions such as $\\norm{h_X} \\le e^{-\\diam(X)}$~\\cite{LiebRobinson1972,Hastings2004LSM,NachtergaeleSims2006,HastingsKoma2006,Hastings2010};\n    see the appendix for a detailed proof.\n    Below we will only use the result that \n    the error is at most ${{O}}(e^{ - \\mu \\ell})$ for some $\\mu > 0$ and fixed $t$.\n    For slower decaying interactions, the bound is weaker and\n    the overlap size $\\ell$ in \\Cref{fig:algorithm} will have to be larger.\n    \n    The Lieb-Robinson bound implies the following decomposition.\n    \\begin{lemma}\n    Let $H = \\sum_X h_X$ be a local Hamiltonian (as in \\Cref{thm:main}, or a more general definition\n    for which \\Cref{lem:LRB} still holds).\n    Then there are constants $v,\\mu >0$ such that\n    for any disjoint regions $A,B,C$, we have\n    \\begin{align}\n        &\\norm{U_t^{H_{A\\cup B}} (U_{t}^{H_{B}})^\\dagger U_t^{H_{B \\cup C}} \n        - U_t^{H_{A\\cup B \\cup C}}} \\le \\nonumber \\\\\n        &{{O}}( e^{vt - \\mu \\dist(A,C)} ) \\sum_{X: \\text{bd}(AB,C)}\\norm{h_X}\n    \\end{align}\n    where $X:\\text{bd}(AB,C)$ means that $X \\subseteq A \\cup B \\cup C$ \n    and $X \\not\\subseteq A \\cup B$  and $X \\not\\subseteq C$.\n    \\label{lem:decomposition}\n    \\end{lemma} \n    \\begin{proof}\n    We omit ``$\\cup$'' for the union of disjoint sets. \n    The following identity is trivial but important:\n    \\begin{align}\n        U^{H_{ABC}}_t =  \n        U^{H_{AB} + H_{C}}_t \\underbrace{(U^{H_{AB} + H_{C}}_t )^\\dagger  U^{H_{ABC}}_t}_{=W_t}.\n    \\end{align}\n    By \\Cref{lem:ode} (i), $W_t$ is generated by~\\cite{Osborne2006,Michalakis2012}\n    \\begin{align}\n    &(U^{H_{AB} + H_{C}}_t)^\\dagger \n    (\\underbrace{H_{ABC}-H_{AB}-H_C}_{H_\\text{bd}}) \n    U^{H_{AB} + H_{C}}_t \\nonumber \\\\\n    &\\quad = \n    (U^{H_{B} + H_{C}}_t)^\\dagger \n    H_\\text{bd} \n    U^{H_{B} + H_{C}}_t\n    +\n    \\underbrace{{{O}}(\\norm{H_\\text{bd}} e^{ v t -\\mu \\ell})}_{=\\delta} \\label{eq:truncation}\n    \\end{align}\n    where $\\ell$ \n    is the distance from the support of the boundary terms $H_\\text{bd}$ to $A$,\n    and the estimate of $\\delta$ is due to \\Cref{lem:LRB} \n    applied to local terms in $H_\\text{bd}$.\n    Since $H_\\text{bd}$ contains terms that cross between $AB$ and $C$,\n    the distance $\\ell$ is at least $\\dist(A,C)$ minus 2.\n\n    By \\Cref{lem:ode} (i) again,\n    the unitary generated by the first term of \\eqref{eq:truncation}\n    is $(U^{H_{B} + H_{C}}_t)^\\dagger U^{H_{BC}}_t$,\n    which can be thought of as the ``interaction picture'' time-evolution operator\n    of the Hamiltonian in \\eqref{eq:truncation}.\n    This is our simplification of the ``patching'' unitary,\n    which is $t \\delta$-close to $W_t$ by \\Cref{lem:ode} (ii).\n    \\end{proof}\n    \n    \\begin{proof}[Proof of \\protect{\\Cref{thm:main}}]\n    The circuit for simulating the Hamiltonian is described in \\Cref{fig:algorithm}.\n    The decomposition of time evolution unitary in \\Cref{fig:algorithm}\n    is a result of iterated application of \\Cref{lem:decomposition}.\n    For a one-dimensional chain, let $L$ be the length of the chain,\n    so there are ${{O}}(L)$ qubits.\n    Take a two contiguous blocks $Y$ and $Z$ of the chain that overlaps by length $\\ell \\ll L$.\n    Under periodic boundary conditions there are two components in the intersection,\n    and under  open boundary conditions,\n    there is one component in the intersection.\n    Applying \\Cref{lem:decomposition},\n    we decompose the full unitary into two blocks on $Y$ and $Z$, respectively,\n    and one or two blocks in the intersection.\n    Every block unitary in the decomposition is a time evolution operator\n    with respect to the sum of Hamitonian terms within the block,\n    and we can recursively apply the decomposition.\n    Making the final blocks as small as possible,\n    we end up with a layout of small unitaries as shown in \\Cref{fig:algorithm} (a).\n    The error from this decomposition is ${{O}}(\\delta L\/ \\ell)$,\n    which is exponentially small in $\\ell$ for $t = {{O}}(1)$.\n    \n    \n    Going to higher dimensions $D > 1$,\n    we first decompose the full time evolution \n    into unitaries on ${{O}}(L\/\\ell)$ hyperplanes (codimension 1).\n    This entails error ${{O}}(e^{-\\mu \\ell} L^D \/ \\ell)$ \n    since the boundary term has norm at most ${{O}}(L^{D-1})$.\n    For each hyperplane the decomposition into ${{O}}(L\/\\ell)$ blocks of codimension 2\n    gives error ${{O}}(e^{-\\mu \\ell} (\\ell L^{D-2}) (L\/\\ell))$.\n    Summing up all the hyperplanes, \n    we get ${{O}}(e^{-\\mu \\ell} L^{D}\/\\ell)$ for the second round of decomposition.\n    After $D$ rounds of the decomposition the total error is ${{O}}(e^{-\\mu \\ell} D L^{D}\/\\ell)$,\n    and we are left with ${{O}}((L\/\\ell)^D)$ blocks of unitaries for $t = {{O}}(1)$.\n    For longer times, apply the decomposition to each factor of \n    $U(T=t_M; t_{M-1})U(t_{M-1},t_{M-2}) \\cdots U(t_2;t_1)U(t_1; t_0 = 0)$\n    \n    It remains to implement the unitaries on $m = {{O}}(T L^D\/\\ell^D)$ blocks $\\square$ of ${{O}}(\\ell^D)$ qubits\n    where $\\ell = {{O}}(\\log (T L\/\\epsilon))$ to accuracy $\\epsilon \/ m$.\n    All blocks have the form $U^{H_{ \\square}}_t$, and can be implemented using any known\n    Hamiltonian simulation algorithm.\n    For a time-independent Hamiltonian, \n    if we use an algorithm that is polynomial in the spacetime volume\n    and polylogarithmic in the accuracy \n    such as those based on signal processing~\\cite{QSP,LC16} or \n    linear combination of unitaries~\\cite{BerryEtAl2014,TS,BCK15},\n    then the overall gate complexity is \n    ${{O}}(T L^D \\polylog(T L \/ \\epsilon)) = {{O}}(Tn \\polylog(Tn\/\\epsilon))$,\n    where the exponent in the $\\polylog$ factor depends on the choice of the algorithm.%\n    \\footnote{\n    If we use the quantum signal processing based algorithms~\\cite{QSP,LC16}\n    to implement the blocks of size ${{O}}(\\ell^D)$,\n    then we need ${{O}}(\\log \\ell)$ ancilla qubits for a block.\n    Thus, if we do not mind implementing them all in serial,\n    then it follows that the number of ancillas needed is \n    ${{O}}(\\log \\log (T n \/ \\epsilon))$,\n    which is much smaller than what would be needed if\n    the quantum signal processing algorithm was \n    directly used to simulate the full system.\n    }\n    For a slowly varying time-dependent Hamiltonian, we can use the \n    fractional queries algorithm~\\cite{BerryEtAl2014} or the Taylor series approach~\\cite{TS,Low2018,Kieferova2018} to\n    achieve the same gate complexity.\n    The Taylor series approach uses a subroutine\n    $\\ket t \\mapsto \\ket t \\left( \\sum_j \\abs{\\alpha_j(t)} \\right)^{-1\/2} \\sum_{j} \\sqrt{\\alpha_j(t)} \\ket j$,\n    where $\\alpha_j(t)$ is the real coefficient of Pauli operator $P_j$ in the Hamiltonian,\n    which must be efficiently evaluated.\n    \\end{proof}\n    For not too large system sizes $L$,\n    it may be reasonable to use a bruteforce method to decompose the block unitaries \n    into elementary gates~\\cite[Chap. 8]{KitaevBook}.\n    \n    \n    \n    \n\\section{Optimality}    \\label{sec:lowerbound}\n    \n    In this section we prove a lower bound on the gate complexity of problem of simulating the time evolution of a time-dependent local Hamiltonian. (Recall that throughout this paper we use \\emph{local} to mean geometrically local.) \n    \n\\subsection{Lower bound proofs}\n    \n    We now prove \\Cref{thm:lowerboundgeneral} and \\Cref{thm:lowerboundlocal}, starting with \\Cref{thm:lowerboundlocal}. This lower bound follows from the following three steps. First, in \\Cref{lem:circuittoHamiltonian}, we observe that for every depth-$T$ quantum circuit on $n$ qubits that uses  local $2$-qubit gates, there exists a Hamiltonian $H(t)$ of the above form such that time evolution due to $H(t)$ for time $T$ is equal to applying the quantum circuit. Then, in \\Cref{lem:countlower} we show that the number of distinct Boolean functions on $n$ bits computed by such quantum circuits is at least exponential in ${\\widetilde{\\Omega}}(Tn)$, where we say a quantum circuit has computed a Boolean function if its first output qubit is equal to the value of the Boolean function with high probability. Finally, in \\Cref{lem:countupper} we observe that the maximum number of Boolean functions that can be computed (to constant error) by the class of quantum circuits with $G$ arbitrary non-local $2$-qubit gates from any (possibly infinite) gate set is exponential in ${\\widetilde{\\calO}}(G\\log n)$. \n    Since we want this class of circuits to be able to simulate all piecewise constant bounded 1D Hamiltonians for time $T$, we must have $G = {\\widetilde{\\Omega}}(Tn)$.\n    \n    \\begin{lemma}\n        \\label{lem:circuittoHamiltonian}\n        Let $U$ be a depth-$T$ quantum circuit on $n$ qubits that uses local $2$-qubit gates from any gate set. \n        Then there exists a piecewise constant bounded 1D Hamiltonian $H(t)$ such that the time evolution due to $H(t)$ for time $T$ exactly equals $U$. \n    \\end{lemma}\n    \\begin{proof}\n        We first prove the claim for a depth-$1$ quantum circuit. This yields a Hamiltonian $H(t)$ that is defined for $t\\in[0,1]$, whose time evolution for unit time equals the given depth-$1$ circuit. Then we can apply the same argument to each layer of the circuit, obtaining Hamiltonians valid for times $t\\in[1,2]$, and so on, until $t\\in[T-1,T]$. This yields a Hamiltonian $H(t)$ defined for all time $t\\in[0,T]$ whose time evolution for time $T$ equals the given unitary. If the individual terms in a given time interval have bounded spectral norm, then so will the Hamiltonian defined for the full time duration. \n       \n        \n        For a depth $1$ circuit with local $2$-qubit gates, since the gates act on disjoint qubits we only need to solve the problem for one $2$-qubit unitary and sum the resulting Hamiltonians. Consider a unitary $U_j$ that acts on qubits $j$ and $j+1$. By choosing $H_j = i \\log U_j$, we can ensure that $e^{-iH_j} = U_j$ and $\\norm{H_j} = {{O}}(1)$. \n        \n        The overall Hamiltonian is now piecewise constant with $T$ time slices.\n    \\end{proof}\n    \n    Note that it also possible to use a similar construction to obtain a Hamiltonian that is continuous (instead of being piecewise constant) with a constant upper bound on the norm of the first derivative of the Hamiltonian.\n    \n    \\begin{lemma}\n        \\label{lem:countlower}\n        For any integers $n$ and $T$ such that $1 \\le n \\leq T \\leq 2^n$, the number of distinct Boolean functions $f:\\{0,1\\}^n\\to \\{0,1\\}$ that can be computed by depth-$T$ quantum circuits on $n$ qubits that use local $2$-qubit gates from a finite gate set is at least $2^{{\\widetilde{\\Omega}}(Tn)}$.\n    \\end{lemma}\n    \\begin{proof}\n        We first divide the $n$ qubits into groups of $k=\\log_2 T$ qubits, which is possible since $T \\leq 2^n$. \n        On these $k$ qubits, we will show that it is possible to compute $2^{{\\widetilde{\\Omega}}(T)}$ distinct Boolean functions \n        with a depth $T$ circuit that uses local $2$-qubit gates. \n        One way to do this is to consider all Boolean functions on $k'<k$ bits. \n        Any Boolean function $f_x$ that evaluates to $f_x(x)=1$ on exactly one input $x$ of size $k'$ \n        can be computed with a circuit of ${\\widetilde{\\calO}}(k')$ gates and ${\\widetilde{\\calO}}(k')$ depth \n        using only $2$-qubit gates and $1$ ancilla qubit in addition to one output qubit~\\cite[Corollary 7.4]{BBC+95}.\n        An arbitrary Boolean function $f:\\{0,1\\}^{k'} \\to \\{ 0,1\\}$ is a sum of such functions: \n        $f = \\sum_{x \\in f^{-1}(1)} f_x = \\bigoplus_{x \\in f^{-1}(1)} f_x$.\n        Implementing all $f_x$ for $x \\in f^{-1}(1)$ in serial using a common output qubit,\n        we obtain a circuit for the full function $f$.\n        Since $f^{-1}(1)$ consists at most $2^{k'}$ bit strings, \n        this will yield a circuit of size ${\\widetilde{\\calO}}(2^{k'})$ and depth ${\\widetilde{\\calO}}(2^{k'})$.\n        Making all the gates local using SWAP gates does not change these expressions by more than a log factor in the exponent. \n        By choosing $k' = k - \\Theta(\\log k)$,\n        we can compute all Boolean functions on $k'$ bits with depth at most $T$. \n        Since there are $2^{2^{k'}} = 2^{{\\widetilde{\\Omega}}(T)}$ distinct Boolean functions on $k'$ bits, \n        we have shown that circuits with depth $T$ using $k=\\log_2 T$ qubits can compute at least $2^{{\\widetilde{\\Omega}}(T)}$ distinct Boolean functions.\n        \n        We can compute $2^{{\\widetilde{\\Omega}}(T)}$ distinct Boolean functions on each of the $n\/k$ blocks of $k$ qubits to obtain $(2^{{\\widetilde{\\Omega}}(T)})^{n\/k} = 2^{{\\widetilde{\\Omega}}(Tn)}$ distinct Boolean functions with $n\/k$ outputs. I.e., we have computed a function $\\{0,1\\}^n \\to \\{0,1\\}^{n\/k}$. Since we want to obtain a single-output Boolean function, as the overall goal is to prove lower bounds against simulation algorithms correct on local measurements, we combine these Boolean functions into one. We do this by computing the parity of all the outputs of these $n\/k$ Boolean functions using CNOT gates. Computing the parity uses at most $n$ 2-qubit local gates and has depth $n$. The circuit now has depth $T + n \\leq 2T$ and by rescaling $T$ we can make this circuit have depth $T$, while retaining the lower bound of $2^{{\\widetilde{\\Omega}}(Tn)}$ distinct Boolean functions.\n        \n        Unfortunately, after taking the parity of these $n\/k$ functions, it is not true that the resulting functions are all distinct. For example, the parity of functions $f(x)$ and $g(y)$ is a new function $f(x) \\oplus g(y)$, which also happens to be the parity of the functions $\\neg f(x)$ and $\\neg g(y)$. To avoid this overcounting of functions, we do not use all possible functions on $k'$ bits in the argument above, but only all those functions that map the all-zeros input to 0. This only halves the total number of functions, which does not change the asymptotic expressions above. With this additional constraint, it is easy to see that if $f(x) \\oplus g(y) = f'(x) \\oplus g'(y)$, this implies that $f$ and $f'$ are the same, by fixing $y$ to be the all-zeros input, and similarly that $g$ and $g'$ are the same.\n    \\end{proof}\n    \n    \n    We say a quantum circuit $U$ computes a Boolean function $f:\\{0,1\\}^n\\to\\{0,1\\}$ with high probability if measuring the first output qubit of $U|x_1x_2\\cdots x_n0\\cdots 0\\rangle$ yields $f(x)$ with probability at least $2\/3$.\n    \\begin{lemma}\n        \\label{lem:countupper}\n        The number of Boolean functions $f:\\{0,1\\}^n\\to \\{0,1\\}$ that can be computed with high probability by quantum circuits with unlimited ancilla qubits using $G$ non-local $2$-qubit gates from any gate set is at most $2^{{\\widetilde{\\calO}}(G\\log n)}$.\n    \\end{lemma}\n    \\begin{proof}\n    First we note that if a circuit $U$ with $G$ arbitrary $2$-qubit gates from any gate set computes a Boolean function with probability at least $2\/3$, then there is another circuit with ${\\widetilde{\\calO}}(G)$ gates from a finite 2-qubit non-local gate set that computes the same function with probability at least $2\/3$. We do this by first boosting the success probability of the original circuit using an ancilla qubit to a constant larger than $2\/3$ and then invoking the Solovay--Kitaev theorem \\cite{NC00} to approximate each gate in this circuit to error ${{O}}(1\/G)$ with a circuit from a finite gate set of $2$-qubit gates. This increases the circuit size to ${\\widetilde{\\calO}}(G)$ gates. Since each gate has error ${{O}}(1\/G)$, the overall error is only a constant, and the new circuit approximates computes the Boolean function $f$ with high probability.\n        \n        We now have to show that the number of Boolean functions on $n$ bits computed by a circuit with ${\\widetilde{\\calO}}(G)$ non-local 2-qubit gates from a finite gate set is at most $2^{{\\widetilde{\\calO}}(G\\log n)}$. To do so, we simply show that the total number of distinct circuits with ${\\widetilde{\\calO}}(G)$ non-local 2-qubit gates from a finite gate set is at most $2^{{\\widetilde{\\calO}}(G \\log n)}$.\n        \n        First observe that a circuit with ${\\widetilde{\\calO}}(G)$ gates can only use ${\\widetilde{\\calO}}(G)$ ancilla qubits, since each $2$-qubit gate can interact with at most 2 new ancilla qubits. Furthermore, the depth of a circuit cannot be larger than the number of gates in the circuit. Let us now upper bound the total number of quantum circuits of this form. Each such circuit can be specified by listing the location of each gate and which gate it is from the finite gate set. Specifiying the latter only needs a constant number of bits since the gate set is finite, and the location can be specified using the gate's depth, and the labels of the two qubits it acts on. The depth only requires ${{O}}(\\log G)$ bits to specify, and since there are at most $n+{\\widetilde{\\calO}}(G)$ qubits, this only needs ${{O}}(\\log n + \\log G)$ bits to specify. In total, since there are ${\\widetilde{\\calO}}(G)$ gates, the entire circuit can be specified with ${\\widetilde{\\calO}}(G\\log n)$ bits. Finally, since any such circuit can be specified with ${\\widetilde{\\calO}}(G\\log n)$ bits, there can only be $2^{{\\widetilde{\\calO}}(G \\log n)}$ such circuits.\n    \\end{proof}\n    \n    \\begin{proof}[Proof of \\protect{\\Cref{thm:lowerboundlocal}}]\n        Suppose that any piecewise constant bounded 1D Hamiltonian on $n$ qubits \n        can be simulated for time $T$ using $G$ 2-qubit non-local gates from any (possibly infinite) gate set. \n        Then using \\Cref{lem:circuittoHamiltonian} and \\Cref{lem:countlower}, \n        we can compute at least $2^{{\\widetilde{\\Omega}}(Tn)}$ distinct Boolean functions using such Hamiltonians. \n        By assumption, each of these Boolean functions can be approximately computed by a circuit with $G$ gates. \n        Now invoking \\Cref{lem:countupper}, \n        we know that such circuits can compute at most $2^{{\\widetilde{\\calO}}(G\\log n)}$ $n$-bit Boolean functions. \n        Hence we must have $G\\log n = {\\widetilde{\\Omega}}(Tn)$, which yields $G = {\\widetilde{\\Omega}}(Tn)$.\n        \n        The proof for $T\\leq n$ follows in a black-box manner from the first part of the theorem statement by only using $T$ out of the $n$ available qubits. In this case the first part of the theorem applies and yields a lower bound of ${\\widetilde{\\Omega}}(T^2)$.\n    \\end{proof}\n    \n    We now prove \\Cref{thm:lowerboundgeneral}, which follows a similar outline. The first step is identical, and we can reuse \\Cref{lem:circuittoHamiltonian}. For the next step, instead of counting distinct Boolean functions, we count the total number of ``distinct'' unitaries. Unlike Boolean functions on $n$ bits, there are infinitely many  unitaries on $n$ qubits. Hence we count unitaries that are ``distinguishable.'' Formally, we say $U$ and $V$ are distinguishable if they are some constant, say $0.1$ for concreteness, apart in diamond norm. This is equivalent to the existence of a state $|\\psi\\rangle$ such that $U|\\psi\\rangle$ and $V|\\psi\\rangle$ have trace distance $0.1$. In \\Cref{lem:Ucountlower} we show that the number of distinguishable unitaries computed by quantum circuits on $n$ qubits with depth $T$ is exponential in ${\\widetilde{\\Omega}}(Tn)$. As before, we then show that the maximum number of distinguishable unitaries that can be computed (to constant error) by the class of quantum circuits with $G$ arbitrary non-local $2$-qubit gates from any (possibly infinite) gate set is exponential in ${\\widetilde{\\calO}}(G\\log n)$. \n    \n    \\begin{lemma}\n        \\label{lem:Ucountlower}\n        For any integer $n$ and integer $n\\leq T \\leq 4^n$, there exists a set of unitaries of size $2^{{\\widetilde{\\Omega}}(Tn)}$, such that every unitary in the set can be computed by a depth-$T$ quantum circuit on $n$ qubits that uses local $2$-qubit gates from a finite gate set, and the diamond norm distance between any $U \\neq V$ from this set is at least $0.1$. \n    \\end{lemma}\n    \\begin{proof}\n        We divide the $n$ qubits into groups of $k=\\log_4 T$ qubits, which is possible since $T \\leq 4^n$. On these $k$ qubits, we will \n        compute $2^{{\\widetilde{\\Omega}}(T)}$ distinguishable unitaries (i.e., unitaries that are at least distance $0.1$ apart from each other) with a depth $T$ circuit that uses local $2$-qubit gates. We can do this by considering a maximal set of unitaries on $k'$ qubits that is distinguishable. \n        More precisely, on $k'$ qubits there exist $2^{\\Omega(4^{k'})}$ unitaries \n        such that each pair of unitaries is at least distance $0.1$ apart; see e.g.~\\cite{Szarek1997}. \n        (This follows from the fact that in the group of $d \\times d$ unitaries with metric induced by operator norm, \n        a ball of radius $0.1$ has volume exponentially small in $d^2$.)\n        We know that any unitary on $k'$ qubits can be exactly written as a product of ${\\widetilde{\\calO}}(4^{k'})$ arbitrary 2-qubit gates~\\cite{BBC+95,Kni95}. Making these gates local and from a finite gate set only adds polynomial factors in $k'$. By choosing $k' = k - \\Theta(\\log k)$, we can compute this set of $2^\\Omega(4^{k'}) = 2^{{\\widetilde{\\Omega}}(T)}$ distinguishable unitaries with depth at most $T$. \n        \n        Since we can compute $2^{{\\widetilde{\\Omega}}(T)}$ pairwise-distant unitaries on each of the $n\/k$ blocks of $k$ qubits, we can compute $(2^{{\\widetilde{\\Omega}}(T)})^{n\/k} = 2^{{\\widetilde{\\Omega}}(Tn)}$ unitaries on all $n$ qubits by considering all possible combinations of unitaries on the different blocks. Finally, if $U$ and $V$ are distinguishable, then so are $U \\otimes X$ and $V \\otimes Y$, since the distinguisher can simply ignore the second register.\n    \\end{proof}\n    \n    \\begin{lemma}\n        \\label{lem:Ucountupper}\n        Let $S$ be a set of distinguishable unitaries (i.e., the diamond norm distance between any $U \\neq V$ from this set is at least $0.1$). Then if any unitary in $S$ can be computed by a quantum circuit with $G$ non-local $2$-qubit gates from any gate set, then $|S| = 2^{{\\widetilde{\\calO}}(G\\log n)}$.\n    \\end{lemma}\n    \\begin{proof}\n        This proof is essentially the same as that of \\Cref{lem:countupper}. We first observe that if a circuit over an arbitrary gate set computes a unitary $U$, we can approximate it to error less than $0.04$ using the Solovay--Kitaev theorem and increase the circuit size to ${\\widetilde{\\calO}}(G)$ gates. Importantly, since the unitaries are a distance $0.1$ apart, one circuit cannot simultaneously approximate two different unitaries to error $0.04$.\n        Then exactly the same counting argument as in \\Cref{lem:countupper} shows there can only be $2^{{\\widetilde{\\calO}}(G \\log n)}$ such circuits.\n    \\end{proof}\n    \n    \\begin{proof}[Proof of \\protect{\\Cref{thm:lowerboundgeneral}}]\n        Suppose that any piecewise constant bounded local Hamiltonian on $n$-qubits could be simulated for time $T$ using $G$ 2-qubit non-local gates from any (possibly infinite) gate set. Then using \\Cref{lem:circuittoHamiltonian} and \\Cref{lem:Ucountlower}, we can produce a set $S$ of distinguishable unitaries of size $2^{{\\widetilde{\\Omega}}(Tn)}$. By assumption, each of these unitaries can be approximately computed by a circuit with $G$ non-local 4-qubit gates. Now invoking \\Cref{lem:countupper}, we know that such circuits can approximate at most $2^{{\\widetilde{\\calO}}(G\\log n)}$ distinguishable unitaries on $n$ qubits. Hence we must have $G\\log n = {\\widetilde{\\Omega}}(Tn)$, which yields $G = {\\widetilde{\\Omega}}(Tn)$.\n    \\end{proof}\n    \n\\section{Discussion}\n    \n    We have only analyzed local Hamiltonians on (hyper)cubic lattices embedded in some Euclidean space,\n    but Lieb-Robinson bounds with exponential dependence on the separation distance\n    hold more generally.  However, on more general graphs, it may be more difficult to find an appropriate decomposition that gives a small error; this must be analyzed for each graph.\n    One advantage of the method here is that the accuracy improves for smaller Lieb-Robinson velocity.\n    This can occur if the terms in the Hamiltonian have a small commutator (see Appendix). \n    \n    The decomposition based on Lieb-Robinson bounds looks \n    very similar to higher order Lie-Trotter-Suzuki formulas.\n    The difference is in the fact that the overlap \n    is chosen to be larger and larger\n    (though very slowly) as the simulated spacetime volume increases.\n    If we want an algorithm that does not use any ancilla qubits,\n    similar to algorithms based on Lie-Trotter-Suzuki formulas,\n    then we can simulate the small blocks from Lieb-Robinson bounds\n    by high order Suzuki formulas~\\cite{Suzuki1991,BAC+07} \n    where the accuracy dependence is polynomial (power-law) of arbitrarily small exponent $a >0$.\n    This combination results in an algorithm of total gate complexity $O(Tn(Tn\/\\epsilon)^a)$,\n    similar to what is claimed to be achievable in Ref.~\\cite[Sec.~4.3]{JordanLeePreskill2014}.\n    \n    Application to fermions is straightforward but worth mentioning.\n    Since Hamiltonian terms always have fermion parity even,\n    Lieb-Robinson bounds hold without any change.\n    Given the block decomposition based on the Lieb-Robinson bound,\n    we should implement each small blocks in $\\polylog(Tn\/\\epsilon)$ gates.\n    The Jordan-Wigner transformation, a representation of Clifford algebra,\n    is a first method one may consider:\n    \\begin{align}\n    \\gamma_{2j-1} &\\mapsto \\sigma^z_1 \\otimes \\cdots \\otimes \\sigma^z_{j-1} \\otimes \\sigma^x_j,\\\\\n    \\gamma_{2j}   &\\mapsto \\sigma^z_1 \\otimes \\cdots \\otimes \\sigma^z_{j-1} \\otimes \\sigma^y_j,\n    \\end{align}\n    where $\\gamma_{2j-1},\\gamma_{2j}$ are Majorana (real) fermion operators,\n    and the right-hand side is a tensor product of Pauli matrices.\n    Often, the tensor factor of $\\sigma^z$ preceding $\\sigma^x$ or $\\sigma^y$ is called \n    a {\\it Jordan-Wigner string}.\n    In one spatial dimension, the above representation \n    where the ordering of $\\gamma$ is the same as the chain's direction\n    gives a local qubit Hamiltonian, since in any term Jordan-Wigner strings cancel.\n    The ordering of fermions is thus very important.\n    (Under periodic boundary conditions, at most one block may be nonlocal; \n    however, we can circumvent the problem \n    by regarding the periodic chain, a circle, \n    as a double line of finite length whose end points are glued:\n    $ \\big( [-1,+1] \\times \\{\\uparrow,\\downarrow\\} \\big) \/ \\{ (-1,\\uparrow) \\equiv (-1,\\downarrow), (+1,\\uparrow) \\equiv (+1,\\downarrow)\\} $.\n    This trick doubles the density of qubits in the system,\n    but is applicable in any dimensions for periodic boundary conditions.)\n    \n    In higher dimensions with fermions,\n    a naive ordering of fermion operators\n    turns most of the small blocks into nonlocal operators\n    under the Jordan-Wigner transformation.\n    However, fortunately, there is a way to get around this, at a modest cost,\n    by introducing auxiliary fermions and let them mediate the interaction of a target Hamiltonian~\\cite{Verstraete2005}.\n    The auxiliary fermions are ``frozen,'' during the entire simulation, \n    by an auxiliary Hamiltonian that commutes with the target Hamiltonian.\n    With a specific ordering for the fermions,\n    one can represent all the new interaction terms as local qubit operators.\n    The key is that if $c_j c_k$ is a fermion coupling whose Jordan-Wigner strings do not cancel,\n    we instead simulate $c_j c_k \\gamma_1 \\gamma_2$, where $\\gamma_{1,2}$ are auxiliary,\n    such that the Jordan-Wigner strings of $\\gamma_1,\\gamma_2$ cancel those of $c_j,c_k$, respectively.\n    The auxiliary $\\gamma$'s may be ``reused'' for other interaction terms \n    if the interaction term involves fermions that are close in a given ordering of fermions.\n    (Ref.~\\cite{Verstraete2005} explains the manipulation for quadratic terms\n    but it is straightforward that any higher order terms can be treated similarly.\n    They also manipulate the Hamiltonian for the auxiliary $\\gamma$ to make it local after Jordan-Wigner transformation,\n    but for our simulation purposes it suffices to initialize the corresponding auxiliary qubits.)\n    In this approach, if we insert ${{O}}(1)$ auxiliary fermion per fermion in the target system,\n    the gate and depth complexity is the same as if there were no fermions.\n    Note that we can make the density of auxiliary fermions arbitrarily small\n    by increasing the simulation complexity.\n    Divide the system with non-overlapping blocks of diameter $\\ell$, which is e.g. ${{O}}(\\log n)$,\n    that form a hypercubic lattice.\n    (These blocks have nothing to do with our decomposition by Lieb-Robinson bounds.)\n    Put ${{O}}(1)$ auxiliary fermions per block,\n    and order all the fermions lexicographically so that all the fermions in a block\n    be within a consecutive segment of length ${{O}}(\\ell^D)$ in the ordering.\n    Interaction terms within a block have Jordan-Wigner string of length at most ${{O}}(\\ell^D)$,\n    and so do the inter-block terms using the prescription of \\cite{Verstraete2005}.\n    The gate and depth complexity of this modified approach has $\\mathrm{poly}(\\ell)$ overhead.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nElectric tunnel junctions consist of two electrically conductive electrodes separated by an insulating tunnel barrier. In these tunnel junctions the barrier plays a crucial role for optimizing and engineering device properties. For example, by employing a barrier with a ferroelectric polarization it is possible to exploit the polarization state as a means to store information in the so called tunnel electroresistance states~\\cite{Garcia2014,Kim2013,Singh2015}, or utilizing a ferromagnetic barrier for spin filtering properties~\\cite{Moodera_2007}. In magnetic tunnel junctions (MTJs), consisting of two ferromagnetic electrodes, the tunnel magnetoresistance (TMR) can be further enhanced by utilizing wave function filtering of the tunnel barrier, as for example in Fe\/MgO\/Fe tunnel junctions with TMR ratios reaching up to $800\\;\\%$ at room temperature~\\cite{Zhang_Butler_2004,Parkin2004,Yuasa2004,Ikeda_TMR_2008}. However, for large TMR ratios not only the functional properties are relevant, but also a good lattice match between the electrode and barrier materials are necessary. This problem becomes apparent in MTJs based on Heusler electrodes with MgO barrier, where the lattice mismatch between electrodes and barrier reduces significantly the observed TMR~\\cite{ReviewHeusler_2012}.\n\nIn the following we investigate ferromagnetic La$_{0.67}$Sr$_{0.33}$MnO$_{3}$(LSMO) based MTJs utilizing a novel insulating barrier material: strontium tin oxide (SrSnO$_3$, SSO). SSO crystalizes in a cubic perovskite structure with a lattice constant $a=0.40254\\,\\mathrm{nm}$ (Ref.~\\onlinecite{SSO-crystal}). Up to now SSO has only been used as a transparent conductive layer, when doped with La, Ba,~\\cite{liu_structure_2012,kurre_studies_2011,wang_transparent_2010} as an insulation barrier with a high dielectric constant in single flux quantum circuits~\\cite{wakana_examination_2005}, and as a photoelectrochemical converter for the reduction of water~\\cite{bellal_visible_2009}. Our experiments show that it is possible to fabricate MTJs with an SSO barrier that exhibt large TMR ratios of up to $350\\:\\%$ at liquid He temperatures and large resistance-area products of up to $30\\;\\mathrm{M\\Omega \\mu m^2}$. We compare these results to reference MTJs with a SrTiO$_{3}$ (STO) barrier and compare the obtained results to already existing publications. We start with a short description of the experimental techniques used, then present the results of our experiments and conclude with a summary.\n\n\\section{Experiment}\n\nThe samples investigated here have been grown by pulsed laser deposition (PLD) from stoichiometric, ceramic LSMO, SSO and STO targets (99.99\\% purity) on (001)-oriented STO substrates. The pulsed laser deposition was carried out in an UHV-chamber with a base pressure of $2\\times10^{-7}\\;\\mathrm{Torr}$ using a KrF excimer laser ($248\\;\\mathrm{nm}$ wavelength, $2\\;\\mathrm{Hz}$ repetition rate) at a substrate temperature of $700^\\circ\\mathrm{C}$. For the LSMO, SSO, and STO deposition a laser fluence of $1.2\\;\\mathrm{J\/cm^2}$, $0.8\\;\\mathrm{J\/cm^2}$, and $1.2\\;\\mathrm{J\/cm^2}$ and an oxygen atmosphere $200\\;\\mathrm{mTorr}$, $200\\;\\mathrm{mTorr}$, and $100\\;\\mathrm{mTorr}$, respectively, has been used. After deposition the samples have been cooled down to room temperature in an oxygen atmosphere of $200\\;\\mathrm{mTorr}$\n\nTo determine the crystal phase and epitaxy of the resulting films, a standard 4 circle x-ray diffraction (XRD) setup (Phillips X'pert Pro) was used with a Cu $K\\alpha$ source. Transmission electron microscope (TEM) cross sectional samples were prepared by conventional mechanical polishing and Ar ion milling. The scanning TEM (STEM) imaging were carried out in an aberration-corrected Nion UltraSTEM 200 microscope operating at $200\\;\\mathrm{kV}$.\n\nFor the fabrication of the MTJs from the blanket films we utilized a three step photolithography process as detailed in Ref.~\\onlinecite{lu_large_1996}. In the first step, we defined the bottom contact mesa by an initial Ar ion milling etch followed by a chemical wet etch in diluted HCl (1:10) to avoid Ar-ion induced conductivity of the STO substrate.~\\cite{gross_situ_2011} After the etching process, a SiO$_2$ insulation layer was sputter deposited and the photoresist removed via lift-off. In the second step the tunnel junctions were formed by Ar ion milling and SiO$_2$ deposition. In the final step, top contacts were fabricated by Ru sputter deposition and lift-off.\n\nFor the electrical characterization the samples were mounted in the variable temperature insert of a Dynacool PPMS system at temperatures $5\\;\\mathrm{K} \\leq T \\leq 300\\;\\mathrm{K}$ and in magnetic fields $H$ applied in the film plane of up to $9\\;\\mathrm{T}$. A constant DC-bias voltage was applied to the MTJs, while recording the 4-pt DC-voltage drop $V_\\mathrm{4pt}$ using an Agilent HP 3478 voltmeter and the DC current flow $I$ using a Keithley 428 current amplifier and an Agilent HP 3458a voltmeter across the junction (See insert in Fig.~\\ref{figure:RT_LSMO_SSO}).\n\n\\section{Results and Discussion}\n\n\\begin{figure}[t,b]\n  \\includegraphics[width=85mm]{Fig1.pdf}\\\\\n  \\caption[XRD results for LSMO based Tunneljunctions]{(Color online) (a) Full range $2\\theta-\\omega$-scan for a LSMO(43)\/SSO(7.5)\/LSMO(17) trilayer on a STO (001) substrate. Only $(00l)$ reflections from STO, LSMO, and SSO are visible. (b) Enlarged scale of the $2\\theta-\\omega$-scan of (a) around the STO (002) reflection. (c) Rocking curve of the SSO(002) reflection exhibits a FWHM of $0.03^\\circ$ indicating a small mosaic spread.\n  \\label{figure:XRD_LSMO}}\n\\end{figure}\n\nWe first discuss the XRD data exemplarily obtained for a LSMO(43)\/SSO(7.5)\/LSMO(17) trilayer; here as throughout the text values in parenthesis are the corresponding film thicknesses in nm. The full range $2\\theta-\\omega$-scan in Fig.~\\ref{figure:XRD_LSMO}(a) exhibits no secondary phases. Only peaks that can be related to $(00l)$-reflections of the LSMO, SSO layers and the STO substrate are visible and even Laue oscillations around the SSO $(001)$-reflection are present. This indicates a good epitaxial (001)-oriented growth of SSO on LSMO. Moreover, in the high resolution $2\\theta-\\omega$-scan around the STO $(002)$-reflection the characteristic $K\\alpha_1$, $K\\alpha_2$ peak splitting of the LSMO peak indicates a high crystalline quality of the LSMO layers. A rocking curve of the LSMO $(002)$-reflection (not shown here) yields a full width at half maximum (FWHM) of the reflection peak of $0.02^\\circ$, which is close to the instrumental resolution of our setup. From the peak position of the SSO $(002)$-reflection we extract an out-of-plane lattice constant of $0.413\\;\\mathrm{nm}$, that is slightly larger then the bulk value of SSO, which we attribute to the compressive strain due to the growth on LSMO on a STO substrate (bulk lattice constant LSMO $a_\\mathrm{LSMO}=0.3894\\;\\mathrm{nm}$, bulk lattice constant STO $a_\\mathrm{STO}=0.3902\\;\\mathrm{nm}$, Ref.~\\onlinecite{Vailionis2011}). For the rocking curve of the SSO $(002)$-reflection shown in Fig.~\\ref{figure:XRD_LSMO}(c) we obtain a FWHM of $0.03^\\circ$, indicating a low mosaic spread of the SSO layer. We also employed $\\phi$-scans at the (101) and (202) reflections of substrate and deposited layers (data not shown here), which verified the in-plane epitaxial relation $[100]_\\mathrm{STO}\\parallel[100]_\\mathrm{SSO}\\parallel[100]_\\mathrm{LSMO}$. Taking all these findings together, our deposition parameters yield highly epitaxial LSMO\/SSO\/LSMO trilayers with excellent structural quality. Similar results have also been obtained with the very same growth conditions for our reference LSMO\/STO\/LSMO trilayers.\n\n\\begin{figure*}[b,t]\n  \\includegraphics[width=170mm]{Fig2.pdf}\\\\\n  \\caption[STEM images of SSO and STO trilayers]{(Color online)STEM Z-contrast images of $[110]_\\mathrm{pc}$-oriented (a) LSMO(43)\/SSO(7.5)\/LSMO(17) trilayer showing a dislocation in the SSO layer and (b) LSMO(50)\/STO(4)\/LSMO(30) trilayer. The colormaps below the Z-contrast images show the distribution of strain tensor components $\\epsilon_\\mathrm{xx}$ and $\\epsilon_\\mathrm{yy}$ in the respective images, where x and y are the in-plane and out-of-plane directions as shown by the arrows in (b). The scale bar in both the images is $2\\;\\mathrm{nm}$.\n  \\label{figure:TEM_LSMO}}\n\\end{figure*}\n\nWe now focus on the comparison of the structural quality of We now focus on the comparison of the structural quality of the two trilayers obtained from atomic resolution STEM. Figure~\\ref{figure:TEM_LSMO}(a) and (b) show high-angle annular dark field (HAADF) STEM images obtained for a LSMO(43)\/SSO(7.5)\/LSMO(17) and a LSMO(50)\/STO(4)\/LSMO(30) trilayer, respectively. Both the films are oriented along the $[110]_\\mathrm{pc}$ direction (pc denotes pseudocubic). As the contrast in a HAADF STEM image is roughly proportional to $Z^2$ (Ref.~\\onlinecite{Pennycook_2011}), with Z being the atomic number, the cation columns in both the trilayers can be clearly observed, while the lighter oxygen columns are invisible. Within the LSMO layers, the (La\/Sr)O layers appear brighter than the MnO$_2$ layers. Similarly in the STO barrier in Fig.~\\ref{figure:TEM_LSMO}(b), SrO layers appear brighter than TiO$_2$ layers. However, In the SSO barrier in Fig.~\\ref{figure:TEM_LSMO}(a), Sn (Z = 50) being heavier than Sr (Z = 38), the SnO$_2$ layers appear brighter than the SrO layers. While it is clear that the LSMO\/STO\/LSMO trilayer in Fig.~\\ref{figure:TEM_LSMO}(a) is free of any line defects and is of a high quality, the SSO layer and the LSMO\/SSO interfaces are defective with the presence of dislocations. We performed geometric phase analysis (GPA) of the two Z-contrast images to quantify the strain within the trilayers \\cite{Htch1998}, which are shown as colormaps below the respective images. For the LSMO\/SSO\/LSMO trilayer, GPA of $\\epsilon_\\mathrm{yy}$ component of the stress tensor shows presence of a very large dilation ($\\approx 9\\% $) in the out-of-plane direction in the SSO barrier, as expected due to the larger lattice constant of SSO. The $\\epsilon_\\mathrm{xx}$ component on the other hand shows large in-plane dilation and compression around the core of the dislocation, which is formed due to the lattice mismatch. In contrast, GPA of the STO barrier reveals only a small strain along the growth direction.\nFrom the TEM analysis we can conclude that the SSO barrier layer contains a higher density of defects, due to the larger lattice misfit between SSO and LSMO. This seems to be in contrast to our structural analysis via XRD, where we found excellent structural quality. This difference can be rationalized by the integral properties of XRD diffraction, while TEM reflects local properties of a sample. This highlights the importance of microstructural analysis via TEM imaging when analyzing the structural quality of samples.\n\n\\begin{figure}[b,t]\n  \\includegraphics[width=85mm]{Fig3.pdf}\\\\\n  \\caption[temperature dependence of resistance]{(Color online) Temperature dependence of the bottom contact resistance (black symbols) and of the MTJ resistance-area product (red line). The resistance of the bottom contact decreases with decreasing temperature. In contrast, the resistance-area product of the MTJ increases with decreasing $T$ (note the logarithmic scale), indicating that the resistance of the junction is dominated by the tunnel barrier and not the contact leads. The RA has been obtained under a bias voltage $V_\\mathrm{4pt}=100\\;\\mathrm{mV}$.\n  \\label{figure:RT_LSMO_SSO}}\n\\end{figure}\n\nIn the following we present our electrical transport data, focusing on the results obtained for a LSMO(43)\/SSO(2)\/LSMO(25) trilayer and a MTJ contact with a $2.5\\times2.5\\;\\mathrm{\\mu m^2}$ area. Note however, that similar results have been obtained for MTJs up to an area of $16\\;\\mathrm{\\mu m^2}$. A schematic drawing of the measurement setup is shown in the inset of Fig.~\\ref{figure:RT_LSMO_SSO}. We first discuss the temperature dependence of the resistance-area product (RA) of the MTJ obtained for a bias voltage $V_\\mathrm{4pt}=100\\;\\mathrm{mV}$ as depicted in Fig.~\\ref{figure:RT_LSMO_SSO}. Clearly, the RA increases with decreasing temperature $T$, while the LSMO bottom contact resistance measured simultaneously decreases with decreasing $T$. From this finding we conclude that the resistance of the MTJ is dominated by the tunnel barrier itself. In addition, the resistance of the junction is at least one order of magnitude larger than the bottom contact such that any magnetoresistance contributions from the bottom LSMO contact can be neglected in our measurements. Interestingly, the RA of MTJs based on a SSO barrier is rather large, reaching values of up to $30\\;\\mathrm{M\\Omega\\mu m^2}$ compared to values of up to $500\\;\\mathrm{k\\Omega\\mu m^2}$ obtained for the STO based reference junctions at $V_\\mathrm{4pt}=100\\;\\mathrm{mV}$. We attribute the increase to the larger bandgap of SSO ($4.1\\;\\mathrm{eV}$, Ref.~\\onlinecite{Zhang2006174}) as compared to STO ($3.3\\;\\mathrm{eV}$, Ref.~\\onlinecite{STO_bandgap_2001}), which directly leads to an increase in junction resistance. However, we would like to note that the high defect density in the SSO barrier might also play a crucial role for the RA, but would require further systematic studies.\n\n\\begin{figure}[b,t]\n  \\includegraphics[width=85mm]{Fig4.pdf}\\\\\n  \\caption[I-V characteristics at low T]{(Color online) (a) Bias dependence of the magnetoresistance at $T=5\\;\\mathrm{K}$ for a bias voltage of $0.2\\;\\mathrm{mV}$ (black), $22\\;\\mathrm{mV}$ (red), $42\\;\\mathrm{mV}$ (blue), and $100\\;\\mathrm{mV}$ (magenta) in a $2.5\\times2.5\\;\\mathrm{\\mu m^2}$ sized junction. The MR decreases with increasing bias voltage. Arrows in the figure indicate the sweep direction of the external magnetic field. (b) Current density vs applied bias voltage at $T=5\\;\\mathrm{K}$ for the very same sample as in (a) at two different external magnetic field values. For the electrode magnetization in the parallel state at $\\mu_0 H=-200\\mathrm{mT}$ (black symbols) a higher current flows as compared to the antiparallel arrangement at $\\mu_0 H=15\\mathrm{mT}$(red symbols). (c) Bias voltage dependence of the TMR extracted from the current-voltage characteristics of (b) (black line) and from the magnetic field sweeps at different $V_\\mathrm{4pt}$ of (a).\n  \\label{figure:IV_LSMO_SSO}}\n\\end{figure}\nAs a next step we look into the magnetic field-dependent magnetoresistance (MR) of the $2.5\\times2.5\\;\\mathrm{\\mu m^2}$ SSO junction at $T=5\\;\\mathrm{K}$. For this we measured $I(H)$ at fixed $V_\\mathrm{4pt}$ while sweeping the external applied magnetic field $H$ applied in the film plane along the bottom electrode strip from $\\mu_0 H=-200\\mathrm{mT}$ to $\\mu_0 H=200\\mathrm{mT}$ (upsweep) and back to $\\mu_0 H=-200\\mathrm{mT}$ (downsweep). From these measurements we calculated first the resistance $R(H)=V_\\mathrm{4pt}\/I(H)$ and then the magnetoresistance $MR(H)$ using\n\\begin{equation}\nMR(H)=\\frac{R(H)-R(-200\\;\\mathrm{mT})}{R(-200\\;\\mathrm{mT})}.\n\\label{equ:MR}\n\\end{equation}\nThe results of this procedure for different bias voltages $V_\\mathrm{4pt}$ are compiled in Fig.~\\ref{figure:IV_LSMO_SSO}(a). For all bias values used, we observe the typical pseudo spin-valve behaviour of a MTJ with two free magnetic layers: For large negative magnetic fields the magnetizations of the two LSMO electrodes are oriented parallel, which results in a low resistance state. When the external magnetic field is then increased to $\\mu_0 H=10\\;\\mathrm{mT}$ the bottom LSMO electrode changes its orientation, resulting in an (nearly) antiparallel alignment of the two magnetizations. We observe a high resistance state, yielding a positive MR. We verified that the first switching is indeed coming from the bottom electrode by an independent anisotropic magnetoresistance measurement of the bottom electrode. We find a nice agreement between the switching fields extracted from both independent measurements (data not shown here). By further increasing $H$ the magnetization of the top LSMO layer also changes its orientation resulting again in a parallel orientation of the two magnetizations and a low resistance state at $\\mu_0 H=670\\;\\mathrm{mT}$. A similar behaviour is observed when reversing the sweep direction only at negative fields. It is quite remarkable that bottom and top LSMO layer exhibit different switching fields. We attribute this to the difference in strain of the two layers induced by either the lattice mismatch between LSMO and SSO or the fabrication of the MTJs itself and the therefore resulting difference in lateral size for the top and bottom electrode. The difference in magnetic switching fields is further highlighted by the much steeper MR change occurring when the bottom electrode changes its orientation around $\\mu_0 H=10\\;\\mathrm{mT}$ and the more gradual change in MR when the top electrode starts to change into the parallel configuration for $30\\;\\mathrm{mT} \\leq \\mu_0 H \\leq 70\\;\\mathrm{mT}$. This may be explained by higher volume to surface ratio of the top LSMO electrode making it more susceptible to pinning defects, due to surface roughness, introduced by the MTJ fabrication process. In an independent series of experiments we measured the coercive fields of blanket LSMO films grown on (001)-oriented STO substrates with and without a $10\\;\\mathrm{nm}$ thick SSO layer between LSMO and substrate using a vibrating sample magnetometer (data not shown here). Within the experimental error we could not find any difference in the coercive fields for the different LSMO layers. These results suggest that the fabrication of the MTJs is an important factor for explaining the differences in switching fields for the top and bottom LSMO electrodes and further supports our assumption that pinning occurs due to the surface roughness introduced via the MTJ fabrication process.\n\nWe also studied the effect of the voltage bias on the observed $MR(H)$ for $V_\\mathrm{4pt}=0.2,22,42, 100\\;\\mathrm{mV}$ as shown in Fig.~\\ref{figure:IV_LSMO_SSO}(a) encoded in the graph as black, red, blue, and magenta symbols, respectively. The observed maximum MR increases with decreasing $V_\\mathrm{4pt}$. Such a behaviour should be expected if spin-flip scattering across the tunnel barrier and magnon scattering is enhanced for higher bias values.~\\cite{LeClair2005} In addition, the coercive field of the top electrode shifts to lower magnetic fields when increasing $V_\\mathrm{4pt}$. We attribute this to a slight temperature increase of the MTJ due to the higher current density flowing through it at larger voltage bias. Another possible explanation may be spin transfer torque due to the not perfectly antiparallel orientation of bottom and top electrode. However, to quantify this effect, further experiments have to be conducted, which go beyond the scope of this work.\n\nTo get a deeper insight into the bias dependence we also recorded I-V curves at a fixed applied magnetic field for parallel ($-200\\;\\mathrm{mT}$, black symbols) and antiparallel ($15\\;\\mathrm{mT}$) alignment of the ferromagnetic electrodes by changing $V_\\mathrm{4pt}$ and recording $I$ as shown in Fig.~\\ref{figure:IV_LSMO_SSO}(b). Comparing the two different I-V curves, it becomes apparent, that for antiparallel alignment the MTJ is in its high resistance state (lower current density for same bias voltage), while in the parallel state it is in its low resistance state (higher current density for same bias voltage). This finding is in agreement with the standard Julliere two spin current model and the difference in scattering rates due to different density of states at the fermi level for majority and minority spin carriers~\\cite{Julliere1975}. Moreover, we applied a Simmons fit~\\cite{Simmons1963} to the parallel magnetization state, which yields $3.2\\;\\mathrm{eV}$ barrier height and $2.6\\;\\mathrm{nm}$ barrier thickness for SSO. The barrier thickness agrees reasonably well with the $2\\;\\mathrm{nm}$ determined from X-ray reflectometry measurements. The determined barrier height is lower than the bulk bandgap of SSO ($4.1\\;\\mathrm{eV}$~\\cite{Zhang2006174}), which we attribute to the large concentration of defects in our SSO layer as observed in STEM images (see Fig.~\\ref{figure:TEM_LSMO}(a)). Moreover, the barrier height and bandgap of the material are not one and the same quantity: The barrier height will also depend on the position of the Fermi energy of the contact electrodes with respect to the bandgap, such that the barrier height should always be smaller than the band gap of the barrier material.\n\nGoing a step further we used the obtained I-V curves to directly calculate the bias dependence of the TMR (maximum of MR in Fig.~\\ref{figure:IV_LSMO_SSO}(a)) by determining the resistance for each bias voltage in the parallel ($R_\\mathrm{para}(V_\\mathrm{4pt})$ at $-200\\;\\mathrm{mT}$) and antiparallel ($R_\\mathrm{anti}(V_\\mathrm{4pt})$ at $15\\;\\mathrm{mT}$) alignment and using these values to calculate the TMR via\n\\begin{equation}\nTMR(V_\\mathrm{4pt})=\\frac{R_\\mathrm{anti}(V_\\mathrm{4pt})-R_\\mathrm{para}(V_\\mathrm{4pt})}{R_\\mathrm{para}(V_\\mathrm{4pt})}.\n\\label{equ:TMR}\n\\end{equation}\nThe obtained TMR bias dependence is shown in Fig.~\\ref{figure:IV_LSMO_SSO}(c) drawn as a black line. In addition, we included the MR obtained from the full $MR(H)$-loops [calculated also by Eq.~(\\ref{equ:TMR})] for the 4 different $V_\\mathrm{4pt}$ (Fig.~\\ref{figure:IV_LSMO_SSO}(a)) as black circles into Fig.~\\ref{figure:IV_LSMO_SSO}(c). We obtain a good agreement between the two differently determined TMR values for $V_\\mathrm{4pt}\\geq 42\\;\\mathrm{mV}$. For lower bias values both methods yield different results: While for the TMR determined from I-V curves we see a decrease of the TMR for low bias values, the TMR determined from $MR(H)$ loops monotonically increases with decreasing $V_\\mathrm{4pt}$. We attribute these discrepancies to small temperature fluctuations during the measurement of the I-V curves, which are most prominent at small bias values, leading to the observed bias dependence. Thus, we assume that we in reality observe an increase of the TMR with decreasing $V_\\mathrm{4pt}$. As already mentioned above, this behaviour is observed, when spin-flip and magnon scattering play a dominant role and influences on the TMR due to the band structure of the two ferromagnets can be neglected. A similar bias dependence has been observed in our STO based reference MTJs and also in other publications with STO as a tunnel barrier~\\cite{lu_large_1996}. The largest TMR value of $350\\;\\%$ in our junction for $T=5\\;\\mathrm{K}$ is obtained at $V_\\mathrm{4pt}= 0.2\\;\\mathrm{mV}$. Similar values have already been obtained in LSMO based MTJs with STO as a tunnel barrier, for example Garcia et al.~\\cite{garcia_temperature_2004} reported a maximum TMR of $540\\;\\%$ at $4.2\\;\\mathrm{K}$, while Sun et al.~\\cite{Sun1996} reported a maximum TMR of $\\approx200\\;\\%$ at $4.2\\;\\mathrm{K}$. For LSMO junctions with STO barriers the TMR could be enhanced by up to $1900\\;\\%$ at low temperatures by carefully optimizing growth, layer structures and patterning process of the MTJs~\\cite{Bowen2003,Werner2011}. This suggests that the TMR obtained in LSMO based MTJs with a SSO barrier could be further enhanced by further optimization. However, the TMR in these junctions is already comparable to values obtained with a STO barrier, despite the large density of defects present in the SSO barrier as determined by TEM analysis.\n\n\\begin{figure}[t,b]\n  \\includegraphics[width=85mm]{Fig5.pdf}\\\\\n  \\caption[TMR as a function of T]{(Color online) Temperature dependence of the TMR at fixed bias voltage $V_\\mathrm{4pt}= 100\\;\\mathrm{mV}$. (a) $MR(H)$ loops obtained at $T=5,10,25,50,75,100,150,200\\;\\mathrm{K}$, depicted with different coloured circles (see legend in the graph). (b) Temperature dependence of the TMR extracted from $MR(H)$. The TMR decreases with increasing $T$, for $T>200\\;\\mathrm{K}$ the TMR completely vanishes.\n  \\label{figure:TMRTemp_LSMO_SSO}}\n\\end{figure}\n\nWe investigated the temperature dependence of the TMR by measuring $I(H)$ loops at fixed $V_\\mathrm{4pt}= 100\\;\\mathrm{mV}$ and $T$. From these measurements we determined the $R(H)$ and $MR(H)$ via Eq.\\eqref{equ:MR}, the results of this procedure are depicted in Fig.~\\ref{figure:TMRTemp_LSMO_SSO}(a). The MR decreases with increasing temperature, for temperatures larger than $200\\;\\mathrm{K}$ no MR was visible within the noise limit of our setup. The vanishing of the MR at high temperatures can be attributed to a possible dead magnetic LSMO layer at the interface between LSMO and SSO, similar as already discussed in literature for LSMO\/STO interfaces.~\\cite{Sun1999,Bibes2001} Moreover, the switching field of the top LSMO electrode decreases from $\\mu_0H=60\\;\\mathrm{mT}$ at $T=5\\;\\mathrm{K}$ to $\\mu_0H=24\\;\\mathrm{mT}$ at $T=200\\;\\mathrm{K}$. In contrast, the switching field of the bottom electrode changes only slightly from $\\mu_0H=10\\;\\mathrm{mT}$ at $T=5\\;\\mathrm{K}$ to $\\mu_0H=6\\;\\mathrm{mT}$ at $T=200\\;\\mathrm{K}$. We attribute this different evolution of the switching fields with temperature, as already discussed above, to the difference in strain and contributions of magnetic pinning defects for the top and bottom LSMO layer. The contributions of both effects will be reduced when increasing the temperature, leading to a reduction of the switching field of the top LSMO electrode.\n\nFrom the $MR(H)$-loops at different $T$ we extracted the TMR as a function of temperature by taking the maximum value of each $MR(H)$ curve at each different $T$. The resulting TMR temperature dependence is depicted in Fig.~\\ref{figure:TMRTemp_LSMO_SSO}(b). Overall, the TMR decreases with increasing $T$ and vanishes for $T>200\\;\\mathrm{K}$. However, the TMR extracted for $T=10\\;\\mathrm{K}$ clearly deviates from the general monotonic decreasing trend and is significantly lower than for $T=25\\;\\mathrm{K}$. We attribute this difference to the fact that the $T=10\\;\\mathrm{K}$ data has been obtained in a second cooling cycle, which might have changed the magnetic domain configuration at the interface.~\\cite{Werner2011}\n\n\\section{Conclusions}\nIn summary, we have investigated for the first time the measurement of TMR in MTJs consisting of LSMO electrodes and SSO as the barrier. Our results suggest that SSO could be a promising barrier material for MTJs, with quite good insulating behaviour even when a high density of defects is present in the barrier. Our results show that the observed TMR in LSMO based MTJs with SSO as a barrier is comparable to the results obtained with STO as a barrier. However, further improvements in the SSO barrier properties, especially reducing the number of defects by lowering the lattice mismatch between the magnetic electrodes and the SSO barrier, are necessary to fully explore the potential of SSO as a barrier for MTJs. A very promising approach might be the usage of better lattice matched magnetic electrode materials, for example certain full Heusler compounds. For Co$_2$FeAl a very small lattice mismatch is obtained as $\\sqrt{2}a_\\mathrm{SSO}=0.5693\\;\\mathrm{nm}$ is extremely close to the bulk lattice constant of Co$_2$FeAl ($a_\\mathrm{Co_2FeAl}=0.569\\;\\mathrm{nm}$ Ref.~\\cite{Ebke2010}). We thus expect SSO to represent a very interesting material choice for further MTJ experiments.\n\\begin{acknowledgments}\nWe gratefully acknowledge financial support via NSF-ECCS Grant No.~1102263. Work at ORNL was supported by the U.S.~Department of Energy (DOE) Office of Science, Office of Basic Energy Sciences, Materials Science and Engineering Directorate.\n\\end{acknowledgments}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzgemg b/data_all_eng_slimpj/shuffled/split2/finalzzgemg
new file mode 100644
index 0000000000000000000000000000000000000000..c5fd48bae9054c69e14264932a6460ea592d9a54
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzgemg
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction and results}\n\\label{}\nLarge scale oceanic motions can be described - at first sight - by the linearized Saint-Venant equations for thin layers, with   Coriolis force, \ni.e. by the following two-dimensional system~ \\cite{P1}:\n\\begin{equation}\\label{syst}\n\\partial_tU+\\left(\\begin{array}{ccc}\n0&\\partial_{x_1}&\\partial_{x_2}\\\\\n\\partial_{x_1}&0&-B(x_2)\\\\\n\\partial_{x_2}&B(x_2)&0\\end{array}\\right)U=0,\\ \\mbox{where}\\  U=U(t,x_1,x_2)=\\left(\\begin{array}{c}\\rho\\\\u_1\\\\u_2\\end{array}\\right)\n\\end{equation}\n\n\n\\noindent where $B$ is the local vertical component of the Earth rotation vector (depending only on the latitude $x_2$),~$\\rho$ denotes \nthe fluctuation of water height and $u$  the horizontal velocity field. For the sake of simplicity, we assume that $(x_1,x_2) \\in {\\mathbb R}\\times\\mathbb T$, \nmeaning that we neglect the influence of the lateral boundaries.\n\nWaves associated to that linear system are usually classified in two ~\\\\families~: First, Poincar\\'e waves, which are fast dispersive \ngravity waves; Second, Rossby waves, due to the inhomogeneities of $B$,  which propagate much slower  \\cite{DU}, \\cite{GA}.\n\nIn this Note, we are interested in the propagation of Rossby waves. We show, in agreement with physical observations \\cite{P2}, that these waves \ncan be trapped in some regions, called ventilation zones, which are not influenced by external dynamics and sources such as continental \nrecirculation for instance.\n\n\n\n\n\nMore precisely, in the limit of large values of $B=b\/\\epsilon$, we   construct  a co-dimension $1$ submanifold~$\\Lambda$ of~$T^*({\\mathbb R}\\times\\mathbb T)$. \nThis set contains the $\\epsilon$-microlocalization\nregion inside which, essentially, an initial condition of (\\ref{syst})  remains trapped. \nLet us first recall that the $\\epsilon$-frequency set of a function $u$ \\cite{MA} is the (closed) subset of~$T^*({\\mathbb R}\\times\\mathbb T)$, complement of the set of points $(\\underline x,\\underline\\xi)$ such that there exists a $C^\\infty$ function $\\chi$, with $\\chi(\\underline x,\n\\underline \\xi)=1$, such that\n\\[\n\\left\\Vert\\int\\chi(\\frac{x+y}2,\\xi)e^{i\\frac{\\xi(x-y)}\\epsilon}u(y)dyd\\xi\\right\\Vert_{L^2}=O(\\epsilon^\\infty).\n\\]\n\n\\begin{theorem}\\label{main}\nLet us consider the system \n\\begin{equation}\\label{systempsilon}\n\\partial_tU+\\left(\\begin{array}{ccc}\n0&\\partial_{x_1}&\\partial_{x_2}\\\\\n\\partial_{x_1}&0&-\\frac{b(x_2)}\\epsilon\\\\\n\\partial_{x_2}&\\frac{b(x_2)}\\epsilon&0\\end{array}\\right)U=0\n\\ \\mbox{with an initial condition}\\ \n\\end{equation}\n\\begin{equation}\nU|_{t=0}=\\left(\\begin{array}{c}\n\\rho^0\\\\u_1^0\\\\u_2^0\\end{array}\\right)\n\\end{equation}\nwith $L^2({\\mathbb R}\\times \\mathbb T,dx_1dx_2)$ conditions (periodic in $x_2$). Let $\\Lambda=\\{F(\\xi_1,x_2,\\xi_2)=0\\}\\subset T^*{\\mathbb R}\\times \\mathbb T$, where $F$ is defined in Lemma\n\\ref{pm}.\n\nWe    suppose that the $\\epsilon$-frequency set of  $U|_{t=0}$ is contained in a compact set $\\mathcal C$ satisfying:\n\\begin{equation}\\label{cond}\n\\mathcal C\\cap\\{\\xi_1=0\\} =\\mathcal C\\cap\\{\\xi_2^2+b(x_2)^2=0\\}=\\emptyset.\n\\end{equation}\nLet us fix a compact set $\\Omega$ of ${\\mathbb R}\\times\\mathbb T$. Then there exists ($\\epsilon$-)pseudo-differential operators $P_\\rho^0, P_1^0, P_2^0$ of principal symbols\n$p_\\rho^0=ib(x_2)\\xi_1(\\xi_2^2+\\xi_1^2+b^2(x_2))^{-1}$, $p_1^0=-\\xi_1\\xi_2(\\xi_2^2+\\xi_1^2+b^2(x_2))^{-1}$, $p_2^0=\\xi_1^2(\\xi_2^2+\\xi_1^2+b^2(x_2))^{-1}$, such that:\n\n\n\\begin{enumerate}\n\\item if the $\\epsilon$-frequency set  of $P_\\rho^0\\rho^0+ P_1^0u_1^0+ P_2^0u_2^0$ does not intersect $\\Lambda\\cap T^*\\Omega$, then  $\\exists\nT>0$ such that:\n\\[\n||U(\\frac t{\\epsilon})||_{L^2(\\Omega)}=O(\\epsilon^\\infty)\\ \\mbox{for}\\ t\\geq  T .\n\\]\n\n\n\\item  if the $\\epsilon$-frequency set of $P_\\rho\\rho^0+ P_1u_1^0+ P_2u_2^0$ does  intersect $\\Lambda\\cap T^*\\Omega$, then, $\\forall  t\\geq 0 $,\n$\n||U(\\frac t{\\epsilon})||_{L^2(\\Omega)}\\neq O(\\epsilon^\\infty)\n$\n(in other words  the frequency set of $U(\\frac t{\\epsilon})$ intersects $T^*\\Omega$).\n\\end{enumerate}\n\nIn the case of a WKB initial condition the conclusion is more precise. Consider\n\\[\nU|_{t=0}=\\left(\\begin{array}{c}\nR^0(x)\\\\U_1^0(x)\\\\U_2^0(x)\\end{array}\\right)e^{i\\frac{S(x)}\\epsilon}.\n\\]\nSuppose than  the (Lagrangian) manifold \\\\$\\{(x,\\nabla S(x)),(p_\\rho^0 R^0+p_1^0U_1^0+p_2^0U_2^0)(x,\\nabla S(x)) \\neq 0\\}$\nintersects $\\Lambda\\cap T^*\\Omega$, then:\n\n\\begin{equation}\\label{wkb}\n||U(\\frac t{\\epsilon})||_{L^2(\\Omega)}\n\\sim C(t)+O(\\epsilon),\\ C(t)>0.\n\\end{equation}\n\n\\end{theorem}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Reduction to a scalar situation}\\label{reduction}\nPerforming first a Fourier analysis in $x_1$ (as the system does not contain explicitly this variable) and looking secondly at modes of the system (\\ref{systempsilon}), we can\nprove the following Proposition, heart of our results:\n\\begin{proposition}\\label{reduc}\nThere exist three pseudo-differential operators $T_\\pm,\\ T_0$, of leading symbols \\\\ $\\tau_{\\pm}= \\pm\\sqrt{\\xi_1^2+\\xi_2^2+b^2(x_2)},\\\n\\tau_0=\\epsilon\\frac{b'(x_2)\\xi_1}{\\xi_1^2+\\xi_2^2+b^2(x_2)}$, roots of (\\ref{the}), such that, if $u_{2,j}^0$ is microlocalized outside $\\xi_1^2-\\tau_j^2=0$, we have, $\\mbox{for}\\  j\\in\\{-,0,+\\}$:\n$$\\epsilon\\partial_tu_{2,j}=iT_ju_{2,j}   \\Longrightarrow \nU_j:=\\left(\\begin{array}{c}(i \\epsilon\\partial_{x_2}T_j+ \\epsilon\\partial_{x_1}b(x_2))(\\epsilon^2\\partial_{x_1}^2+T_j^2)^{-1}\\\\\n-(\\epsilon^2\\partial_{x_1}\\partial_{x_2}+ib(x_2)T_j)(\\epsilon^2\\partial_{x_1}^2+T_j^2)^{-1}\\\\\n\\mathbb Id\\end{array}\\right)u_{2,j}$$ $$\\mbox{ satisfies (\\ref{systempsilon}) up to } O(\\epsilon^\\infty).\n$$\n\\end{proposition}\nThe proof is a consequence of the result contained in the Appendix.\n\nThe following result shows that any initial condition of (\\ref{systempsilon}) can be decomposed on the three modes of the last section.\n\\begin{proposition}\n$\\forall \\rho,u_1, u_2,\\ \\exists  u_{2,j},\\ j\\in\\{+,0,-\\}$ such that:\n\\begin{equation}\\label{decomp}\n\\left(\\begin{array}{c}\\rho\\\\u_1\\\\u_2\\end{array}\\right)=\\sum_{j} \\left(\\begin{array}{c}(i \\epsilon\\partial_{x_2}T_j+ \\epsilon\\partial_{x_1}b(x_2))(\\epsilon^2\\partial_{x_1}^2+T_j^2)^{-1}\\\\\n-(\\epsilon^2\\partial_{x_1} \\partial_{x_2}+ib(x_2)T_j)(\\epsilon^2\\partial_{x_1}^2+T_j^2)^{-1}\\\\\n\\mathbb Id\\end{array}\\right)u_{2,j}+O(\\epsilon^\\infty)\n\\end{equation}\n\\[=:\\sum_{j} {\\mathbb Q}^j u_{2,j}+O(\\epsilon^\\infty) .\n\\]\n\\end{proposition}\nTo prove the Proposition one has just to invert the matrix:\n$(\n{\\mathbb Q}^- \\, {\\mathbb Q}^0 \\, {\\mathbb Q}^+ ).$\nSemiclassically it is enough to show that the matrix of the leading order symbol is invertible\n\\begin{equation}\\label{matrix}\\scriptsize\n\\left(\\begin{array}{ccc}\n{\\xi_2\\sqrt{\\xi_1^2+\\xi_2^2+b(x_2)^2} + i\\xi_1b(x_2)\\over \\xi_2^2+b(x_2)^2}&-{ i b(x_2)\\over \\xi_1}\n&{-\\xi_2\\sqrt{\\xi_1^2+\\xi_2^2+b(x_2)^2} + i\\xi_1b(x_2)\\over \\xi_2^2+b(x_2)^2}\\\\\n{\\xi_1\\xi_2+ib(x_2) \\sqrt{\\xi_1^2+\\xi_2^2+b(x_2)^2} \\over \\xi_2^2+b(x_2)^2}&-{\\xi_2 \\over \\xi_1}\n&{\\xi_1\\xi_2-ib(x_2) \\sqrt{\\xi_1^2+\\xi_2^2+b(x_2)^2} \\over \\xi_2^2+b(x_2)^2}\\\\1&1&1\n\\end{array}\\right).\n\\end{equation}\nA simple computation shows that the  jacobian\n$\nJ=\\frac{2(\\xi_1^2+\\xi_2^2+b^2(x_2))^{3\/2}}{(\\xi_2^2+b^2(x_2))|\\xi_1|}\\geq 2$.\nIn particular the inversion of the matrix $(\n{\\mathbb Q}^- \\, {\\mathbb Q}^0 \\, {\\mathbb Q}^+ )$ can be done symbolically at any order and gives the leading order symbols of the operators $\\mathbb\nP^j:=(P_\\rho^j,P_1^j,P_2^j)$\n such\nthat $u_{2,j}=P_\\rho^j\\rho+P_1^ju_1+P_2^ju_2,\\ j\\in\\{-,0,+\\}$. One gets~:\n$p_\\rho^0=ib(x_2)\\xi_1(\\xi_2^2+\\xi_1^2+b^2(x_2))^{-1},\\ p_1^0=-\\xi_1\\xi_2(\\xi_2^2+\\xi_1^2+b^2(x_2))^{-1},\\  p_2^0=\\xi_1^2(\\xi_2^2+\\xi_1^2+b^2(x_2))^{-1}$.\n\n\n\n\n\n\n\n\\section{Propagation under the Rossby Hamiltonian $\\tau_0$}\\label{lambda}\nThanks to  Proposition \\ref{reduc} it is enough, as far as  the Rossby mode is concerned, to look at propagation with respect to the Hamiltonian $\\tau_0\/\\epsilon$.\nThis Hamiltonian being  independent of $x_1$, $\\xi_1$ will be conserved. The flow is periodic in the variables $x_2,\\xi_2$ (one degree of freedom). \nTherefore, since \n$\\dot{x}_1=\\frac{b'(x_2)(\\xi_2^2-\\xi_1^2+b^2(x_2))}{(\\xi_2^2+\\xi_1^2+b^2(x_2))^2}\\mbox{ is periodic (the case of infinite and zero period is treated}$\n in \\cite{CGPS}),$\\ x_1(t)$ \nwill contain a part, linear in time except if\n$$F(\\xi_1,x_2(0),\\xi_2(0)):=\\int_0^{period}\\frac{b'(x_2(t))(\\xi_2(t)^2-\\xi_1^2+b^2(x_2(t)))}{(\\xi_2(t)^2+\\xi_1^2+b^2(x_2(t)))^2}dt=0.$$\n\\begin{lemma}\\label{pm}\nAs $b'(x_2)\\neq 0,\\ b'(x_2)F(\\xi_1,x_2,\\xi_2)>0\\mbox{ as } \\xi_1\\to\\pm\\infty,\\ <0\\mbox{ as }$\\\\$ \\xi_1\\to 0$ and is invariant under the flow of  $\\frac{\\tau_0}\\epsilon$.\n\nDefine \n$\nE(\\xi_1,x_2,\\xi_2)=\\frac{b'(x_2)\\xi_1}{\\xi_2^2+\\xi_1^2+b^2(x_2)}\\ \\ .\n$\n\nThen\n$\\displaystyle\n\\left\\vert F(\\xi_1,x_2,\\xi_2)\\right\\vert=\\left\\vert\\int_{x_-}^{x_+}\n\\frac{\\frac{b'(x)}{E(\\xi_1,x_2,\\xi_2)}-2\\xi_1}{\\sqrt{\\frac{b'(x)\\xi_1}{E(\\xi_1,x_2,\\xi_2)}-\\xi_1^2-b^2(x)}}dx\\right\\vert,\n$\nwhere\n~$]x_-,x_+[$\nis the largest interval of~${\\mathbb T}$ containing $x_2$ in which~$\\frac{b'(x)\\xi_1}{E(\\xi_1,x_2,\\xi_2)}-\\xi_1^2-b^2(x)>0$.\n\\end{lemma}\nWe define  \n$\\Lambda:=\\{(x_1,x_2;\\xi_1,\\xi_2)\/\\ F(\\xi_1,x_2,\\xi_2)=0\\}.$\n\n$\\mbox{ Thanks to Lemma \\ref{pm} } \\Lambda\\neq\\emptyset\\mbox{ and }dim\\Lambda=3$.\n\\begin{corollary}\nSuppose~$b'(x_2) \\neq 0$. Then $\\vert F(\\xi_1, x_2,\\xi_2)\\vert  \\geq\\frac{C(x_2,\\xi_2)}{\\xi_1}$ as $\\xi_1\\to 0$, with~$C(x_2,\\xi_2)>0$. \nThis implies that the trapping phenomenon will take place only with initial conditions oscillating enough in $x_1$.\n\\end{corollary}\n\n\\medskip\n\\begin{remark}: Since the Hamiltonian $E$ does not depend on $x_1$, we can  express it, for each value of $\\xi_1$, on the action variable \n$A$: $E(\\xi_1,x_2,\\xi_2)=H(A,\\xi_1)$. This allows to define the function\n  $A(\\xi_1,x_2,\\xi_2)$ by $E(\\xi_1,x_2,\\xi_2)=H(A(\\xi_1,x_2,\\xi_2),\\xi_1)$. One can easily show that:\n\\[F(\\xi_1,x_2,\\xi_2)=\\frac{\\partial_{\\xi_1}H(A,\\xi_1)}{\\partial_{A}H(A,\\xi_1)}|_{A=A(\\xi_1,x_2,\\xi_2)},\\]\n  and the following variational characterization of $\\Lambda$:\nLet us fix the energy to $E$ and let $\\Gamma(\\xi_1,E)$ be the energy shell $\\{E(\\xi_1,x_2,\\xi_2)=E\\}$. Then\n\n\\[\\Lambda=\\bigcup_{E,i}\\{(\\Gamma(\\xi_1^i,E),\\xi_1^i\\},\\]\\[\n\\mbox{ where $\\xi_1^i$ is such that the area inside $\\Gamma(\\xi_1^i,E)$ is extremal.}\\]\n\\end{remark}\n\n\n\n\n\n\n\n\n\\section{Dispersion of Poincar\\'e waves and proof of the Theorem}\nThe proof of the Theorem  involves, for the Rossby modes ($j=0$), the standard result of propagation of the frequency set. If\nthe initial  frequency set  is such that part of the Rossby mode is  trapped, in particular  if it intersects $\\{ b'(x_2)=0\\}$, this concludes the proof.\n\nIf not we have to prove some  dispersion for the Poincar\\'e modes ($j=\\pm$) for  times of the order $\\frac 1 \\epsilon$, for which the theorem of propagation of the frequency set is not enough. But, the system \nbeing integrable, \nwe can perform an expansion on (Bohr-Sommerfeld) eigenvalues of the Hamiltonians  $T_\\pm$  and a decomposition of~$U|_{t=0}$ on a compact set of\ncoherent states (thanks to the condition on the microlocalization of the initial datum) \\cite{PA}.\n\n\n\n\n\n\n\n\nSince  $\\tau_\\pm=\\pm\\sqrt{\\xi_2^2+b^2(x_2)+\\xi_1^2 }+O(\\epsilon)$ and we are microlocalized far away from $\\xi_2^2+b^2(x_2)+\\xi_1^2=0 $, \nwe can find  pseudo-differential operators $H_{2\\pm}$ of principal symbols \n$\\xi_2^2+b^2(x_2)$ such that $T_\\pm=\\pm\\sqrt{H_{2\\pm}+\\xi_1^2 }$. The Bohr-Sommerfeld\nquantization condition (with subsymbol) gives that the eigenvalues of $H_{2\\pm}$ are of the form:\n\\[\n\\lambda_\\pm^k=\\lambda_\\pm\\left((k+\\frac 1 2 )\\epsilon\\right)+\\epsilon\\mu^{k}_\\pm(\\xi_1)+O(\\epsilon^2),\n\\]\nwhere $\\lambda_\\pm$ is the energy $\\xi_2^2+b^2(x_2)$ defined on action variable, and $\\epsilon\\mu^{k}_\\pm(\\xi_1)\\in C^\\infty$ is the correction due to the subsymbol.\nPropagating at time $t= s \/\\epsilon$ a function, product of a coherent state at $(q,p)$ (in $x_1$) and  \nan eigenfunction of $T_\\pm$ (in $x_2 $),  gives rise to expressions of the type:\n\\[\n\\int \\exp{i\\frac{\\epsilon(x_1-q)\\xi_1\\pm(\\lambda_\\pm^k+\\xi^2_1)^{\\frac 1 2}s+i\\epsilon(\\xi_1-p)^2}{\\epsilon}}d\\xi_1.\n\\]\nThe stationary phase lemma then  gives that this integral is $O(\\epsilon^\\infty)$ except if there exists a stationary point,  given by the conditions:\n\\[\n\\xi_1=p\\ \\mbox{and}\\  \\epsilon(x_1-q)\\pm\\frac{(2\\xi_1 +\\epsilon\\partial_{\\xi_1}\\mu^{k}_\\pm)}{2\\sqrt{\\lambda_\\pm^k+\\xi^2_1}}s=0 . \\]\nThe second condition gives:\n$\n2\\sqrt{\\lambda_\\pm^k+\\xi^2_1}(x_1-q)=\\mp\\left(\\frac{2p}\\epsilon+\\partial_{\\xi_1}\\mu^{k}_\\pm\\right)s.\n$\nTherefore, since $p\\neq 0$ and the $\\lambda^k$\\ 's are bounded by the above condition (\\ref{cond}) on $\\mathcal C$, there is no critical point for $x_1$ in a compact set.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intr}\n Anomaly detection, also known as outlier detection, is the process of discovering patterns in a given data set that do not conform to expected behavior \\citep{Akyildiz}. Anomaly detection is to find the events that happen relatively infrequently, which has been extensively used in a wide variety of applications, including fraud detection for credit cards, insurance, health care, intrusion detection for cyber-security, fault detection in safety critical systems, and many others \\citep{Akyildiz}. Smart meter analytics attracts the growing research effort \\eg, \\citep{sdewes2015,icde2015,energy2016,edbt2015}, due to the wide installation of smart meters. Anomaly detection can be applied for analyze live smart meter data, which aims to help energy consumers identify unusual behaviors, e.g., forgetting to turn off stoves after cooking; and to help utilities detect extraordinary events, e.g., energy leakage and theft. Since abnormal consumption may also result from user activities, such as using inefficient appliances, or over-lighting and working overtime in office buildings, anomalous feedback can be used to warn energy consumers to minimize energy usage and to help them identify inefficient appliances or over-lighting. Furthermore, anomaly detection can help utilities to establish the baseline for providing more accurate demand-response programs to their customers \\citep{zhang2011}. Abnormal energy consumption detection is related to finding patterns in data, and the statistical and data mining techniques are used to detect the patterns, e.g., \\citep{zhang2011,nadai2015,liu2011,Chou2014}, which can perform close to or better than domain experts. \n \nToday, smart meters are widely used worldwide. Smart meters are the digital devices that can record energy consumption at the interval of an hour or fewer \\citep{Depuru2011}. Smart meters record detailed consumption readings in real-time or near real-time, which provide the opportunity to monitor timely unusual events or consumption behaviors. However, the enabling anomaly detection for smart meters typically uses data mining technologies, which require large amounts of training data sets, as well as significantly complex systems. In typical applications of data mining to anomaly detection, the detection models are produced off-line because the learning algorithms have to process tremendous amounts of data \\citep{lee2001}. The generated models are naturally used for offline anomaly detection, i.e., analyzing consumption data after being loaded into an energy management system (EMS). Effective anomaly detection, however, should happen in real-time in order to minimize the compromises to the use of energy. The efficiency of updating the detection model and the accuracy of the detection are the important consideration for constructing such a real-time anomaly detection system. \n \n \nIn this paper, we propose a statistical anomaly detection method based on the consumption patterns of smart grid data. For residential electricity consumption, the daily consumption patterns of a customer usually show quite similar. The proposed algorithm can detect the unusual energy usage from one's history consumption patterns, e.g., the abnormal high usage than the expected (see Fig.~\\ref{fig:anomalycase}). \n\\begin{figure}[htp] \n\\centering \n\\includegraphics[width=0.6\\textwidth]{anomalycase} \n\\caption{Daily pattern of a typical household and anomaly consumption} \n\\label{fig:anomalycase} \n\\end{figure} \n\nTo detect anomalies in time and obtain a better accuracy, we make use of the so-called {\\em Lambda} architecture \\citep{Marz2013}, that can detect anomalies in near real-time, and can efficiently update detection models regularly according to a user-specified time interval. A lambda architecture enables real-time updates through a three-layer structure, including speed layer (or real-time layer), batch layer and serving layer. It is a generic system architecture for obtaining near real-time capability, and its three layers use different technologies in processing data. It is well-suited for constructing an anomaly detection system that requires real-time anomaly detection and efficient model refreshment (we will detail it in the next section). To support big data capability, we choose the Spark Streaming as the speed layer technology for detecting anomalies on a large amount of data streams, Hive as the batch layer technology for computing anomaly detection models, and PostgreSQL as the serving layer for saving the models and detected anomalies; and sending feedbacks to customers. The proposed system can be integrated with smart meters for detecting anomalous energy consumption online. To summarize, we make the following contributions: 1) we propose the statistical-based anomaly detection algorithm based on customers' history consumption patterns; 2) we propose making use of the lambda architecture for the efficiency of the model updating and real-time anomaly detection; 3) we implement the system with a lambda architecture using hybrid technologies; 4) we evaluate our system in a cluster environment using realistic data sets, and show the efficiency and effectiveness of using the lambda architecture in a real-time anomaly detection system. \n \n \nThe rest of this paper is organized as follows. Section 2 discusses the anomaly detection algorithm used in the paper. Section 3 describes the implementation of the lambda detection system. Section 4 evaluates the system. Section 5 surveys the related works. Section 6 concludes the paper and provides the direction for the future works.   \n\\section{Preminaries}\n\n\\subsection{Anomaly Detection Model}\nThe used anomaly detection model is a combination of a short-term energy consumption prediction algorithm, called {\\em periodic auto-regression with eXogenous variables (PARX)} \\citep{omid}, and {\\em Gaussian statistical distribution}. We now first describe the PARX algorithm, which will be used for the prediction based on history consumption patterns. Generally speaking, residential electricity consumption is highly correlated to the weather temperature. In winter, electricity consumption increases since the temperature decrease because of the heating needs. Similarly, in summer, electricity consumption increases when the temperature is higher because of cooling loads. A similar daily consumption pattern may appear repeatedly for a customer, e.g., due to the living habit of the customer. For example, if a customer usually gets up at 7 o'clock, then the consumption pattern will have the morning peak between 7 and 8 o'clock; In the evening, if the customer gets home at 5 o'clock after work, the consumption pattern typically will have the evening peak between 17 and 20 o'clock, due to cooking and washing. \n \n \nThe PARX model, thus, uses a daily period, taking 24 hours of the day as the seasons, \\ie, $t=0...23$,  and uses the previous $p$ days' consumptions at the time $t$ for auto-regression. The PARX model at the $s$-th season and at the $n$-th period is formulated as    \n\n\\vspace{-10pt}\n\\begin{equation}\n\tY_{s,n} = \\sum_{i=1}^{p} \\alpha_{s,i} Y_{s,n-i}  + \\beta_{s,1} XT1 + \\beta_{s,2} XT2 + \\beta_{s,3} XT3 \t+ \\epsilon_s, \\enspace s \\in t\n\\label{eq:parx}\n\\end{equation}\nwhere $Y$ is the data point in the consumption time-series; $p$ is the number of order in the auto-regression; $XT1, XT2$ and $XT3$ are the exogenous variables accounting for the weather temperature, defined in the equations of $(2) - (4)$; $\\alpha$ and $\\beta$ are the coefficients; and $\\epsilon$ is the value of the white noise.   \n{\\small\n\\begin{equation}\nXT1 = \\begin{cases}\nT-20 & \\text{if } T>20 \\\\ \n0 & \\text{otherwise}  \n\\end{cases}\n\\qquad\n\tXT2 = \\begin{cases}\n16-T & \\text{if } T<16 \\\\ \n0 & \\text{otherwise}  \n\\end{cases} \n\\qquad\n\tXT3 = \\begin{cases}\n5-T & \\text{if } T<5 \\\\ \n0 & \\text{otherwise}  \n\\end{cases}\n\\end{equation}\n}\n\nThe variables represent the cooling (temperature above 20 degrees), heating \n(temperature below 16 degrees), and overheating (temperature below 5 degrees), respectively. \nThe anomaly detection algorithm uses unique variate Gaussian distribution described in the following. \nGiven the training data set, $X=\\{x_1, x_2, ..., x_n\\}$ whose data points obey the normal distribution with the mean $\\mu$ and the variance $\\delta^2$, the detection function is defined as \n\\begin{equation} \np(x; \\mu, \\delta)=\\frac{1}{\\delta \\sqrt{2\\pi}}e^{-\\frac{(x-\\mu)^2}{2\\delta^2}} \n\\end{equation} \nwhere $\\mu=\\frac{1}{n} \\sum^n_{i=1} x_i$ and $\\delta^2=\\frac{1}{n}\\sum_{i=1}^{n}(x_i-\\mu)^2$. For a new data point, $x$, this function computes its probability density. If the probability is less than a user-defined threshold, \\ie, $p(x)<\\epsilon$, it is classified as an anomaly, otherwise, it is a normal data point. In our model training process, we compute the L1 distance between the actual and predicted consumptions, \\ie, $||Y_t - \\hat{Y}_t||$, where $Y_t$ is the actual hourly consumption at the time $t$, and $\\hat{Y}_t$ is the predicted hourly consumption at the time $t$. The predicted hourly consumption, $\\hat{Y}_t$, is computed using the PARX model in Equation~\\ref{eq:parx}. We find that the L1 distances obey to a log-normal distribution (see Section~\\ref{sec:accuracy}). Therefore, the $x$ in the normal distribution will be the log value of the distance, \\ie, $ln||Y_t - \\hat{Y}_t||$.   \n\n\n\\subsection{Lambda Architecture}\nWe now introduce the Lambda architecture that will be used in our anomaly detection system. As mentioned in Section~\\ref{sec:intr}, the lambda architecture consists of three layers, including speed layer, batch layer and serving layer, illustrated in Fig.~\\ref{fig:lambdaarch}. The speed layer directly ingests data streams from data sources, processes them, and continuously updates the results into the real-time views in the database in the serving layer. The speed layer does not keep any history records, and typically uses main memory based technologies to analyze the incoming data. In contrast, the batch layer runs iteratively and starts from the beginning of the data set once a batch job has finished. When a batch job starts, all the available data in the batch layer storage will be processed. Therefore, the data arriving after the job starts will not be processed until the next job. Since all the data are analyzed in each iteration, each of the new result views will replace its predecessor. As the batch layer does not rely on incremental processing, it is robust to any system failures, which the batch job simply processes all the available data sets in each iteration. The speed and batch usually use different technologies because of their distinct requirements regarding read and write operations. Any query against the data is answered through the serving layer, \\ie, the query processor queries both the views from the speed and the batch layers, and merges them. \n\\begin{figure*}[htp] \n\\vspace{-10pt}\n\\centering \n\\includegraphics[width=0.9\\textwidth]{lambdaarch} \n\\caption{Lambda architecture} \n\\label{fig:lambdaarch} \n\\vspace{-15pt}\n\\end{figure*} \n\nThe lambda architecture itself is only a paradigm. The technologies with which the different layers are implemented are independent of the general idea. The speed layer only deals with new data and compensates for the high latency updates of the batch layer. It can typically leverage stream processing systems, such as Storm, S4, and Spark Streaming, etc. The batch layer needs to be horizontally scalable and supporting random reads, where the technologies like Hadoop with Cascading, Scalding, Pig, and Hive, are suitable. The serving layer requires a system with the ability to perform fast random reads and writes. The system can be a high-performance RDBMS (e.g., PostgreSQL), an in-memory data store (\\eg, Redis, or Memcache), or a high scalable NoSQL system (\\eg, HBase, Cassandra, ElephantDB, MongoDB, or DynamoDB).     \n\n\n\\section{\\uppercase{Implementation}}\n\n\\subsection{System Overview}\nWe now describe the implementation of the anomaly detection system. We choose Spark Streaming, Spark, and PostgreSQL as the speed layer, batch layer and serving layer technology, respectively (see Fig.~\\ref{fig:systemoverview}). The system employs Spark to compute the models for anomaly detection, which reads the data from the Hadoop distributed file system (HDFS) in the batch layer. The batch job runs at a regular time interval, computes and updates the detection models to the table in PostgreSQL database. Spark Streaming is used for processing real-time data streams, e.g., it directly gets data from smart meters, and detects abnormal consumption by the detection algorithm. The detection algorithm always uses the latest models getting from the PostgreSQL database. Spark Stream writes the detected anomalies back to the PostgreSQL database, which will be used for the notification of customers.   \n\\begin{figure}[htp]\n\\vspace{-15pt}\n\\centering\n\\includegraphics[width=0.8\\textwidth]{systemoverview}\n\\vspace{-5pt}\n\\caption{The anomaly detection system}\n\\label{fig:systemoverview}\n\\vspace{-20pt}\n\\end{figure}\n\n\n\\subsection{Training Anomaly Detection Models}\n\nWe employ Spark to train the detection model by running regular batch jobs. All the consumption data from smart meters are written to the append-only HDFS. In each iteration of the batch jobs, Spark uses all the available data in HDFS to compute the detection model. The use of Spark and HDFS supports the computation of the model based on scalable data sets, and since they both are the distributed computing technology, the computation can be finished within a certain time limit, which means that the detection algorithm can use the latest data model for the anomaly detection. Fig.~\\ref{fig:training} illustrates the training process of generating PARX model and Gaussian model using energy consumption and weather temperature time series at the season from 0 to 23. That is, for each season, e.g., $s=0$, we create a new time series with the hourly reading at 0 o'clock of all days, then use the Equation~\\ref{eq:parx} to compute the PARX model (or parameters), and compute the Gaussian model, \\ie, $N(\\mu, \\delta^2)$. Therefore, there are 24 PARX and 24 Gaussian models in total for the hours of the day.   \n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=0.7\\textwidth]{training}\n\\caption{Process of training detection models}\n\\label{fig:training}\n\\end{figure}\n\nAlgorithm~\\ref{alg:training} gives more details about the implementation. This algorithm computes the anomaly detection models with the given training time series collection $\\mathcal{TS}$, weather temperature time series $ts'$, and auto-regression order $p$. Each time series in $\\mathcal{TS}$ represents the hourly energy consumption of a customer. To compute the detection models for each season $s$, we first need to create a new consumption time series and  a new temperature time series (see line 7), then use the two new time series to compute the PARX model (see line 8). According to our analysis in Section~\\ref{sec:accuracy}, the $L1$ distances between predict consumption and actual consumption at season $s$ for all days observes to a log-normal distribution. Therefore, we compute Gaussian statistical model based on the $L1$ distance log values (see line 12-18). The total number of PARX models for all the time series is $||\\mathcal{TS}|| \\times 24$, which is same as the number of the Gaussian models. In the end, all the models are updated to the PostgreSQL database  that will be used for the online anomaly detection in the speed layer. \n\\begin{algorithm}\n\\caption{Training of anomaly detection models}\n{\\scriptsize\n\\begin{algorithmic}[1]\n\\Function {Train}{TimeSeriesCollection $\\mathcal{TS}$, TemperatureTimeSeries $ts'$ Order $p$}\n    \\State $\\mathcal{M} \\gets \\{ \\}$ \\Comment{Initialize the collection of PARX parameters}\n    \\State $\\mathcal{N} \\gets \\{ \\}$ \\Comment{Initialize the collection of the statical model parameters}\n    \\ForAll{$ ts \\in \\mathcal{TS}$ }\n        \\State $id \\gets$ Get the unique identity of $ts$\n        \\ForAll{$ s \\in 0...23$ }\n        \t\\State $ts^c, ts^t \\gets$ Construct a new consumption time series using $ts$, and a new temperature time $ts^t$ using $ts'$ at season $s$\n\t\t    \\State $\\alpha_{1},...,\\alpha_{p}, \\beta_{1},\\beta_{2},\\beta_{3} \\gets$ Compute PARX model using $ts^c$ and $ts^t$\n\t\t    \\State Insert $(id, s, \\alpha_{1},...,\\alpha_{p}, \\beta_{1},\\beta_{2},\\beta_{3})$ into $\\mathcal{M}$\n\t\t     \\State $\\mathcal{L} \\gets \\{ \\}$\n\t\t     \\State $\\mathcal{D} \\gets$ Get the days of $ts$\n\t\t    \\ForAll{$ d \\in \\mathcal{D}$ }\n\t\t        \\State $\\hat{v} \\gets$ Compute the predict reading of the season $s$ using PARX\n\t\t        \\State $v \\gets$ Get the actual hourly reading from $ts$ of the day $d$\n\t\t        \\State $l \\gets$ Compute the ln value of $L_1$ distance of the day $d$, $ln(||\\hat{v} - v||)$\n\t\t        \\State Add $l$ into  $\\mathcal{L}$\n\t\t    \\EndFor\n\t\t    \\State $\\mu, \\beta \\gets$ Compute the mean and standard deviation using the normal distribution statistical model on  $\\mathcal{L}$\n\t\t    \\State Insert $(id, s, \\mu, \\delta )$ into $N$\n\t\t    \\EndFor\n     \\EndFor\t\n    \\Return $\\mathcal{M}, \\mathcal{N}$\n \\EndFunction\n\\end{algorithmic}\n}\n\\label{alg:training}\n\\end{algorithm}\n\nThe implementation is a Spark program. The consumption time series, as well as temperature time series, are read into the distributed main memory as {\\em resilient distributed datasets (RDDs)}, which are fault-tolerant, immutable and partitioned parallel data structures that can be operated in parallel, e.g., by using the operators, including map, reduce, groupByKey, filter, collect, etc \\citep{Zaharia}. To generate the new time series, we use the {\\em groupByKey} operator to aggregate the consumption series by the composite key of meter ID and season (or hours); while use only the season as the key to the temperature time series. Then, we merge the generated time series by the \\texttt{join} operator on the key of the season. The PARX, in fact, can be regarded as a multi-linear regression model, which simply takes the auto-regressors and the exogenous variables as the independent variables. We, then, apply the multiple linear regression function from the Spark machine learning the library, MLib \\citep{Meng}, to compute the coefficients. For all of these operations, the transformation functions are directly applied on RDDs for doing the data processing.    \n\n\\subsection{Real-time Anomaly Detection}\nThe real-time anomaly detection is carried out in the speed layer. Algorithm~\\ref{alg:realtimedetection} describes the anomaly detection process, which is self-explanatory. First, the detection algorithm reads meter readings from all the incoming data streams, and reads the weather temperature and detection models from the PostgreSQL database each hour. For each data stream, the algorithm predicts the reading using the PARX algorithm, with the pre-computed parameters, the previous $p$ day's readings at the current hour, \\ie, the season $s$, and weather temperature (see line 4--7). Then, the algorithm calculates the log value of the $L1$ distance between the predict and the actual readings, then uses it compute the probability using the Gaussian model (see line 8-10). In the end, the algorithm decides whether the current reading is an anomaly or not based on the computed probability value, \\ie, if its value is below the user-defined threshold, $\\epsilon$. If the current reading is classified as an anomaly, it will be written into the database for the customer notification (see line 11--12).\n\\begin{algorithm}[htp]\n\\caption{Real-time anomaly detection}\n{\\scriptsize\n\\begin{algorithmic}[1]\n\\Function {Detect}{CurrentReadingCollection $\\mathcal{V}$, Temperature $t$, PredictModel $\\mathcal{M}$, StaticalModel $\\mathcal{N}$, Threshold $\\epsilon$ }\n    \\State $\\mathcal{R} \\gets \\{ \\}$ \\Comment{Initialize the detection results}\n    \\ForAll{$ v \\in \\mathcal{V}$ }\n        \\State $id \\gets$ Get the unique identity of $v$\n        \\State $s \\gets$ Get the season of $v$\n        \\State $\\alpha_{1},...,\\alpha_{p}, \\beta_{1},\\beta_{2},\\beta_{3} \\gets$ Get the parameters from  $\\mathcal{M}$ by $id$\n        \\State $\\hat{v} \\gets$ Compute the predict reading at $s$ using PARX with the parameters, the $p$ days' readings at $s$, and temperature $t$\n        \\State $x \\gets ln||\\hat{v} - v||$ \\Comment{Compute the ln value of $L_1$ distance at the season $s$}\n        \\State $\\mu, \\delta \\gets$ Get the statical model parameters from $\\mathcal{N}$ by $id$ and $s$\n        \\State $p \\gets$ Compute the probability using the normal distribution function, $\\frac{1}{\\delta \\sqrt{2\\pi}}e^{-\\frac{(x-\\mu)^2}{2\\delta^2}}$\n        \\If {$p<\\epsilon$}\n         \\State  Add $(id, s, p, v, \\hat{v})$ into $\\mathcal{R}$\n        \\EndIf\n     \\EndFor\t\n    \\Return $\\mathcal{R}$\n \\EndFunction\n\\end{algorithmic}\n}\n\\label{alg:realtimedetection}\n\\end{algorithm}\n\nWe implement the algorithm to process the real-time data on Spark Streaming. Spark Streaming allows for continuous processing via short interval batches, and its basic data abstraction is called {\\em discretized streams (D-Streams)}, a continuous stream of data \\citep{sparkstreaming}. The data are received in each interval batch, {\\em hourly} in our case, and operations will run upon the data for doing transformations, such as filter unnecessary attribute values, extracting the hour from the timestamp, etc. (see Fig.~\\ref{fig:rddslidewin}). When using PARX for the prediction, we fetch the previous $p$ days' readings of the current hour for auto-regression. For example, in Fig.~\\ref{fig:rddslidewin} we set the order, $p=3$, therefore, the window size is set to 72 hours (\\ie, 3 days)  to keep the past three days' readings at the particular hour within the same window (e.g., the RDDs colored by green). This is done by using the window function, \\texttt{reduceByKeyAndWindow(func, windowLength, slideInterval)}, to aggregate the data with specified key, window length and slide interval (\\eg, meter ID and season as the composite key in this case,  $windowLength=72$ hours and $slideInterval=1$ hour). In the underlying, Spark uses  the data {\\em checkpointing} mechanism to keep the past RDDs in HDFS. At the beginning of each interval batch, the data models are read from the PostgreSQL database in the serving layer, and broadcast to all DStreams. Therefore, the detection program can always use the latest detection models to detect anomalies.   \n\\begin{figure\n\\centering\n\\includegraphics[width=0.85\\textwidth]{rddslidewin}\n\\vspace{-5pt}\n\\caption{Slide windows in the real-time anomaly detection}\n\\label{fig:rddslidewin}\n\\end{figure}\n\n\n\n\n\\section{\\uppercase{Evaluation}}\n\\subsection{Experimental settings}\nIn this section, we will evaluate the effectiveness and the scalability of our anomaly detection system. We conduct the experiments in a cluster with 17 servers.  Five servers\nare used for running the speed layer, while twelve servers are used for the batch layer.\nWe also exploit one of the servers in the speed layer as the serving layer for managing\nthe detection models and sending anomaly detection messages. All the servers have the identical settings, configured with an Intel(R) Core(TM) i7-4770 processor (3.40GHz, 4 Cores, hyper-threading is enabled, two hyper-threads per core), 16GB RAM, and a Seagate Hard driver (1TB, 6 GB\/s, 32 MB Cache and 7200 RPM), running Ubuntu 12.04 LTS with 64bit Linux 3.11.0 kernel. The serving layer uses PostgreSQL 9.4 database with the settings ``shared buffers=4096MB, temp buffers=512MB, work mem=1024MB, checkpoint segments=64'' and default values for the other configuration parameters.\n\nWe have a real-world residential electricity consumption data set (27,300 time series), which will be used for evaluating the accuracy of the anomaly detection. The data are the 2-year time series of hourly resolution. To evaluate the scalability, we use the synthetic data set generated by our data generator with the real-world data as the seed. The size of data tested in the cluster environment is scaled up to one terabyte, corresponding to over twenty million time series. \n\n\\subsection{Anomaly Detection Accuracy}\n\\label{sec:accuracy}\nWe start by evaluating the accuracy of our anomaly detection system using a randomly-selected time series from the real-world data set.\n\nTo provide a basis for comparison, we perform the anomaly detection using a standard boxplot analysis as well. Boxplot is a quick graphic approach for examining data sets, and has been used for decades. A boxplot uses five parameters to describe a numeric data set, including lower fence, lower quartile, media, upper quartile and upper fence (see Fig.~\\ref{fig:boxplot}). According to Fig.~\\ref{fig:boxplot}, a boxplot is constructed by drawing a rectangle between the upper and lower quartiles with a solid line indicating the median. The length of the box is called interquartile range, $IQR$. The sample data points lying outside the fences, $1.5*IQR$, are classified as the outliers, which has been indicated to be acceptable for most situations \\citep{frigge1989}.\n\nTo align boxplot with our detection method, we test the anomalies based on the 24 seasons. There are 17,520 data points in total in the selected time series. Fig.~\\ref{fig:anomalyboxplot} shows the boxplot result where the blue points located on the top of the upper fence represent the anomalies, a total of 1,260 data points. Since the boxplot approach is merely able to detect energy consumption lying unusually far from the main body of the data,  it is difficult to determine which ones are the true anomalies, and to identify the potential reasons for these anomalies because there are too many false positives.\n\n\n\\begin{figure}[htp]\n\t\\vspace{-10pt}\n\\begin{minipage}[b]{0.4\\textwidth}\n\t\\centering\n\t\\includegraphics[width=1.35\\textwidth]{boxplot}\n\t\\vspace{-18pt}\n\t\\caption{Box plot}\n\t\\label{fig:boxplot}\n\\end{minipage}\n\\begin{minipage}[b]{0.75\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.56\\textwidth]{anomalyboxplot}\n\t\t\\vspace{-10pt}\n\t\t\\caption{Anomaly detection using box plot}\n\t\t\\label{fig:anomalyboxplot}\n\t   \\end{minipage}\n\\vspace{-20pt}\n\\end{figure}\n\n\\begin{figure}[t]\n\\vspace{-5pt}\n  \\begin{minipage}[b]{0.33\\textwidth}\n    \\centering\n\t\\includegraphics[width=1.02\\textwidth]{l1distancedist}\n\t\\vspace{-20pt}\n\t\\caption{Log-normal distributions across the L1 distances }\n\t\\label{fig:l1distancedist}\n  \\end{minipage}\n  \\begin{minipage}[b]{0.33\\textwidth}\n    \\centering\n\t\\includegraphics[width=0.9\\textwidth]{detectedanomalies}\n\t\\vspace{-10pt}\n\t\\caption{Anomaly detection using PARX and statistical method}\n\t\\label{fig:detectedanomalies}\n  \\end{minipage}\n   \\begin{minipage}[b]{0.33\\textwidth}\n     \\centering\n\t\\includegraphics[width=0.9\\textwidth]{detectmodelupdate}\n\t\\vspace{-10pt}\n\t\\caption{Impact of detection model update frequency}\n\t\\label{fig:detectmodelupdate}\n  \\end{minipage}\n  \\vspace{-10pt}\n\\end{figure}\n\nWe now use the proposed detection algorithm to analyze the same time series. Fig.~\\ref{fig:l1distancedist} depicts the distribution of the $L1$ distances of a season by the histogram. As shown, the distribution has the shape of a log-normal distribution. We have checked the $L1$ distance distributions for all the 24 seasons, and found that they all have a  similar shape. This is the reason that we choose log-normal distribution in our statistical-based anomaly detection. Besides, we test the anomalies by treating all the days the same, and different, \\ie, discriminating the days into workdays, weekend \\& holidays.  Moreover, we increase the threshold value, $\\epsilon$, from 0.05 to 0.15, and do the test. The results in Fig.~\\ref{fig:detectedanomalies} demonstrate that the detection identifies more anomalies for treating all the days the same than different. The reason is that during weekends and holidays, people tend to stay at home more time, thus use more energy. The consumptions are more likely higher than in the weekdays. For the threshold parameter, its value is for classifying a usual or unusual reading. According to the results, if the value increases, the number of detected anomalies changes significantly. For the real-world deployment of this system, the threshold value can be set by the residents to decide when to receive anomaly alerting messages.\n\n\nWe now evaluate the impact of the model update frequency on the detection accuracy. We use half-year's time series as the initial data set to train the detection models. We design the following three scenarios for model updating: 1) update per day; 2) update per 10 days behind the detection; and 3) without an update. We measure the detected anomalies for the three scenarios by treating all days the same. According to the results shown in Fig.~\\ref{fig:detectmodelupdate}, the frequent updates of the models help to decrease the detected anomalies. It is due to the improvement of the prediction accuracy of the PARX model. Thus, less large $L1$ distances are identified as the anomalies. However, although the update frequency does help to determine the real anomalies, the results do not show a big difference if the models are updated within a certain short-time interval, \\eg, the results of the scenario 1) and 2).\n\nIn the end, we compare our approach with the boxplot, and the result shows that the number of the anomalies reported by the PARX prediction and statistical method can be decreased notably. This increases the chance to determine accurately real anomalies for an energy consumption time series. \n\n\n\n\n\\subsection{System Scalability}\nWe now evaluate the scalability of our anomaly detection system. As our system can scale-out and to efficiently cope with large amounts of data, we vary both the number of executors and the volume of the input data in the following.\n\n{\\bf Scale-out experiment.} Parallel processing is a key feature of the proposed system. To evaluate the scalability of our implementation, we conduct an experiment with a varying number of the executors in Spark. In this experiment, we use a fix-sized synthetic data set with eight million time series of a one-year length (275GB), which were generated by our data generator seeded with the real-world data set. Since we are interested in the real-time and batch capability of our system, we use the full sets of the nodes to test  batch model training and real-time anomaly detection separately. We first test the batch capability by running the training program with the number of executors increased from 8 to 256. We repeat each test for ten times, and record their execution times. The results are depicted by the boxplot shown in Fig.~\\ref{fig:scaleoutbatch}. According to the results, the execution time and the time variance decrease when more executors are added. But, when the number reaches 64, the increasing parallelism does not speed up the batch processing further, which is due to the overhead of the Spark master when managing a large number of executors. We conduct the real-time anomaly detection on Spark Streaming, and likewise, we scale the number of executors from 8 to 256. Since we only study the real-time detection scalability, we use the detection models without updating. Fig.~\\ref{fig:scaleoutstream} shows the results, which indicate that the variance of execution time is larger than the batch model training. It might be due to the variability of real-time batch executions on Spark Streaming when doing the anomaly detection for each hour.\n\n\n\\begin{figure*}[t]\n  \\begin{minipage}[b]{0.33\\textwidth}\n  \\centering\n\t\\includegraphics[width=0.95\\textwidth]{scaleoutbatch}\n\t\\caption{Batch model training}\n\t\\label{fig:scaleoutbatch}\n  \\end{minipage}\n  \\begin{minipage}[b]{0.33\\textwidth}\n    \\centering\n\t\\includegraphics[width=0.95\\textwidth]{scaleoutstream}\n\t\\caption{Real-time detection}\n\t\\label{fig:scaleoutstream}\n\t\\end{minipage}\n  \\begin{minipage}[b]{0.33\\textwidth}\n    \\centering\n\t\\includegraphics[width=1\\textwidth]{sizeup}\n\t\\caption{Size-up experiment}\n\t\\label{fig:sizeup}\n\t\\end{minipage}\n\\end{figure*}\n\n{\\bf Increased data load experiment.}  To evaluate the scalability of the algorithms over large volumes of data, we compare different workloads. According to the above experiments, the optimal number of parallel executors for model training and anomaly detection are 64 and 128, respectively. We use the optimal executor number (the memory of executor is configured to 4GB) in our experiments, but vary the number of time series from 8 to 24 million (the size is from 275GB to 825GB). The processing times of varying data workloads are displayed in Fig.~\\ref{fig:sizeup}. We observe that both of the training and detection processes can scale near linearly with the quantity of the time series. The time of detection, in this case, is the total execution time of handling all the time series of a one-year length, \\eg, it takes less than two minutes to finish eight million time series with the optimized settings. The average time of each real-time batch only takes a few seconds (recall that a batch in Spark Streaming processes the data of each hour). In the real-world deployment, the detection program can be set to run every hour to inspect hourly smart meter readings. According to the results, the anomaly detection system has a very good scalability which can meet the fast-growth of smart meter data.\n\nUnlike the anomaly detection, the training process uses the full set of the data to generate new models each time.  Anomaly detection, instead, is performed for each hour where Spark Streaming runs periodical batch (or impulse) to process the data, which needs more time in overall. The training and detection programs can be deployed either in different clusters or the same cluster. If deployed in the same cluster, it is necessary to allocate computing resources in a reasonable way. For example, since the batch job takes a much longer time, it can be scheduled to run immediately after the anomaly detection job. A scheduling system is, thus, necessary, and this will be for our future work.\n\n\n\\section{\\uppercase{Related Works}}\n{\\bf Anomaly energy consumption detection.} \nAnomaly detection is an important aspect in energy consumption time series management. Chandola et al.  present a survey of different anomaly detection techniques in various application domains including energy \\citep{Chandola2009}. Statistical and data mining are the commonly used techniques for discovering abnormal consumption behaviors \\citep{Janetzko}. Statistical methods are based on modeling data using distributions, and see if the data under test observes to the distributions. Accordingly, the approaches presented in this paper combine PARX and log-normal distribution function to detect anomalies in energy consumption time series. Jakkula and Cook use statistics and clustering to identify outliers in power datasets collected from smart environments \\citep{Jakkula2010}, but they have not considered the impact of the exogenous variables, \\eg, weather temperature,  on the electricity consumption. Linear regression can extract time series features when the dependent variables are well-defined \\citep{Magld2012}. The early experience of identifying outliers in linear regression is through setting a threshold limit, but this yields many false positives for large data sets \\citep{lee1997}. Adnan et al.  combine linear regression with clustering techniques for getting better results \\citep{Adnan2003}. Zhang et al.  \\citep{zhang2011} further  use piecewise linear regression to fit the relation between energy consumption and weather temperature. The results obtained are more favorable than entropy and clustering methods. But, their approach does not take the changes of consumption pattern into account. Brown et al.  use K-nearest neighborhood (KNN) in fast kernel regression to predict electricity consumption \\citep{brown2011}, which requires large datasets. The resulting models are static, thus it is not preferable for online anomaly detection and the situation when consumption pattern is changed. Nadai et al.  combine ARIMA and adaptive artificial neural network (ANN) to detect anomaly consumption \\citep{nadai2015} using a relatively small data set that is from a few buildings. In comparison, we propose the prediction and statistical anomaly models and combine with the lambda architecture for supporting regular model refreshment, and real-time anomaly detections. Besides, the proposed approach can handle scalable data sets, and consumption pattern changing owing to using the PARX model. \n\n\n{\\bf Batch and stream processing on big data.} \nBatch and realtime\/stream processings have attracted much research effort in recent year, with the popularity of Internet of Things (IoT). Liu et al.  make a survey of the existing stream processing systems, and discuss the potential technologies used for lambda architecture \\citep{Liu2014}. The tools \\citep{dawak2011,tldks2011,pvldb2011,ideas2014} Cheng et al.  propose a smart city data platform that supports both batch and real-time data processing \\citep{Cheng}, and they suggest that anomaly detection should be implemented as the chief component of any platform for processing sensor data. Different to proposing the generic lambda architecture  \\citep{Marz2013}, Preuveneers et al.  \\citep{Preuveneers} and Gao \\citep{Gao} present the big data architectures for processing domain-specific big data, including health care, context-aware user authentication and social media.  Schneider et al.  study batch data and streaming data anomaly detection, respectively \\citep{Schneider2016}. The used detection model, however, is static, and the use case is different to ours which employs the batch job to update the models iteratively while use the real-time job to detect the anomalies online in data streams.  \n\n\\begin{comment}\nBatch and stream processing on big data The data explosion of the Internet of Things (IoT) is often linked with the Big Data paradigm. MapReduce [10] \u2013 and its Hadoop [48] implementation \u2013 is a software framework and programming model that allows\ndevelopers to write programs that process massive amounts of unstructured data in parallel across a distributed cluster of computers. The shortcomings and drawbacks of batch-oriented data processing have been widely recognized as many applications are in need of real-time [5,44] and in-stream processing capabilities [9]. This concept got a lot of traction with\nvarious distributed event stream processing (ESP) engines emerging. Yahoo's S4 [29] and Twitter's Storm project [26,47] were among the first to attract a lot of attention. Also Google acknowledged the limitations of MapReduce, with its MillWheel [2] framework and programming model dedicated to fault-tolerant stream processing at Internet scale. Spark [49] is another\nstate-of-practice software solution for large-scale data processing. It runs up to 100 times faster in memory than Hadoop MapReduce and supports scalable fault-tolerant streaming applications. Spark Streaming [50] builds upon the Spark's foundations to build big data application that act on data in real time. Apache Samza2 \u2013 originally developed at LinkedIn \u2013 is\nanother popular open source frameworks for scalable stream processing.\n\\end{comment}\n\n{\\bf The use of lambda architecture.}\nLambda architecture has attracted a growing interest due to its mix capabilities to process both real-time and batch data. Sequeira et al.  use lambda architecture in an industrial EMS solution with cloud computing capabilities \\citep{Sequeira2014}. Kro\\ss et al.  develop on-demand stream processing within the lambda architecture to optimize the usage of computing resources in clusters \\citep{Brunnert}. Martnez-Prieto et al.  adapts the lambda architecture in semantic data processing \\citep{Martnez2015}; Liu et al.  applies it to smart grid complex event processing (CEP) \\citep{Liu2015}; Villari et al.  proposes AllJoyn Lambda, the platform for managing embedded devices of smart homes \\citep{Villari}; and Hasani et al.  use it for real-time big data analytics \\citep{Hasani}. Besides, the works \\citep{Casado2015,Liu2014} both give an extensive review of the technologies for the lambda architecture. There are various use cases of the lambda architecture. In contrast, we focus on a particular use case, \\ie, energy consumption anomaly detection as has been mentioned in future extensions of \\citep{Kiran} work. More specifically, we use it in the model update and the real-time anomaly detection using the models, which is significant to the large deployment of smart meters and sensors of IoT today.\n\n\n\\section{\\uppercase{Conclusions and Future Work}}\n\\label{sec:conclusion}\nAnalyzing and detecting anomalies is an important task for live energy consumption data while the improvement of detection accuracy and scalability is quite challenging. In this paper, we applied the novel lambda architecture technique to an anomaly detection system, which supports iterative batch updates of the detection models, and real-time anomaly detection on scalable data streams. We have proposed the detection algorithm for finding the anomalies based on one's history consumption pattern via the supervised learning and statistical algorithm. Furthermore, the system supports alerting service for customers by setting a personalized threshold value for conspicuous energy consumption. We have evaluated the accuracy of the anomaly detection algorithm using a real-world data set, and the scalability using a large synthetic data set. The results have validated the effectiveness and the efficiency of the proposed system with the lambda architecture. \n\n\nFor the future work, we will implement a scheduling system that can coordinate the running of the batch and real-time jobs within the same cluster. We intend to explore the ways to detect a greater range of anomalies, such as missing values, negative energy consumption and device errors. Besides, we plan to support additional types of data, such as gas, heating and water data, and to implement the corresponding detection algorithm.   \n\n \n\n\\section*{Acknowledgements}\n\\noindent This work is part of the CITIES (NO. 1035-00027B) research project funded by Innovation Found Denmark.\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nStatistical topic models are generative unsupervised models that describe the content of documents in large textual collections. Prior research has investigated the application of topic models such as Latent Dirichlet Allocation (LDA) \\cite{blei2003latent} in a variety of domains ranging from image analysis to political science. \nMost of the work on topic models assumes exchangeability between words and treats documents in a bag-of-words fashion. As a result, the words' grouping in coherent text segments, such as sentences or phrases, is lost. \n\nHowever, the inner structure of documents is generally useful, when identifying topics. For instance, one would expect that in each sentence, after standard pre-processing steps such as stop-word removal, only a very limited number of latent topics would appear. Thus, we argue that coherent text segments should pose ``constraints'' on the amount of topics that appear inside those segments.  \n\nIn this paper, we propose \\textit{sentenceLDA} (\\textit{senLDA}), whose purpose is to incorporate part of the text structure in the topic model. Motivated by the argument that coherent text spans should be produced by only a handful of topics, we propose to modify the generative process of LDA. Hence, we argue that the latent topics of short text spans should be consistent across the units of those spans. In our approach, such text spans can vary from paragraphs to sentences and phrases depending on the task's purpose. Also, note that in the extreme case where words are the coherent text segments, the standard LDA model becomes a special case of \\textit{senLDA}. \n\nIn the remainder of the paper we present the \\textit{senLDA }  and we derive its collapsed Gibbs sampler in Section 2, we illustrate its advantages by comparing it with  LDA on intrinsic (\\textit{in vitro}) and extrinsic (\\textit{ex vivo}) evaluation experiments using collections of Wikipedia and PubMed articles in Section 3, and we conclude in Section 4.\n\n\\section{The proposed model}\nA statistical topic model represents the words in a collection of $D$ documents\nas mixtures of $K$ ``topics'', which\nare multinomials over a vocabulary of size $V$. In the case of LDA, for each document\n$d_i$ a multinomial over topics is sampled from a Dirichlet prior with parameters $\\boldsymbol{\\alpha}$. \nThe probability $p(w|z=k)$ of a term $w$, given the topic $k$, is represented by\n$\\phi_{k,t}$. We refer to the complete $K\\times V$ matrix of word-topic \nprobabilities as $\\Phi$. \nThe multinomial parameters $\\phi_k$\nare again drawn from a Dirichlet prior parametrized by $\\boldsymbol{\\beta}$. Each observed term $w$ in the collection is drawn from a multinomial for the topic\nrepresented by a discrete hidden indicator variable $z_i$. For simplicity in the mathematical development and notation, we assume symmetric Dirichlet priors but the extension to the asymmetric case is straightforward. Hence, the values of $\\alpha$ and $\\beta$ are model hyper-parameters.\n \n\\input{senLDAfig.tex}\n\nWe extend LDA by adding an extra plate denoting the coherent text segments of a document. In the rest, without loss of generality we use sentences as coherent segments. A finer level of granularity can be achieved though, by analysing the structure of sentences and using phrases as such segments. The graphical representation of the \\textit{senLDA } model is shown in Figure \\ref{fig:senlda} and the generative process of a document collection using \\textit{senLDA } is described in Algorithm \\ref{algo:senlda}. \nFor inference, we use a collapsed Gibbs sampling method \\cite{griffiths2004finding}. \nWe now derive the Gibbs sampler equations by estimating the hidden topic variables. \n\n\nIn \\textit{senLDA } the joint distribution can be factored:\n\\begin{equation}\np(w,z|\\alpha,\\beta) = p(w |z,\\beta)p(z|\\alpha)\n\\label{eq:factors}\n\\end{equation}\nbecause the first term is independent of $\\alpha$ and the second from $\\beta$. \nAfter standard manipulations as in the paradigm of \\cite{heinrich2005parameter} one arrives at: \n\\begin{equation}\np(\\vec{z}, \\vec{w}|\\alpha,\\beta) = \\prod_{z=1}^{K}\\frac{\\Delta(\\vec{n}_z+\\beta)}{\\Delta(\\beta)} \\prod_{m=1}^D \\frac{\\Delta(\\vec{n}_m+\\alpha)}{\\Delta(\\alpha)}\\label{eq:joint}\n\\end{equation}\n\\noindent where $\\Delta(\\vec{x})=Beta(x_1,\\ldots,x_m)=\\frac{\\prod_{k=1}^{dim \\vec{x}} \\Gamma(x_k)}{\\Gamma(\\sum_{k=1}^{dim \\vec{x}} x_k)}$ is a multidimensional extension of the beta function used for notation convenience, and $\\vec{n}_m$, $\\vec{n}_z$ refer to the occurrences of topics with documents and topics with terms respectively. To calculate the full conditional\nwe take into account the structure of the document $d$ and the fact that  $\\vec{w_d}=\\{\\vec{w}_{d\\neg{s}}, \\vec{w}_{\\neg{s}}\\}$, $\\vec{z}=\\{\\vec{z}_{d\\neg{s}}, \\vec{z}_{\\neg{s}}\\}$. The subscript $s$ in $\\vec{w}_s,\\vec{z}_s$ denotes the words and the topic respectively of sentence $s$. For the full conditional  of topic $k$ we have:\n\\begin{equation}\n\\begin{split}\np(z_s=k|\\vec{z}_{\\neg{s}}, \\vec{w}) = \\frac{p(\\vec{w},\\vec{z})}{p(\\vec{w}, \\vec{z}_{\\neg{s}})} = \\frac{p(\\vec{w}|\\vec{z})}{p(\\vec{w}_{\\neg{s}}|\\vec{z}_{\\neg{s}}) p(w_s)}\\frac{p(\\vec{z})}{p(\\vec{z}_{\\neg{s}})}=\\\\\n= \\frac{p(\\vec{w},\\vec{z})}{p(\\vec{w}_{\\neg{s}}, \\vec{z}_{\\neg{s}})} \\propto\n \\frac{\\Delta(\\vec{n}_z+\\beta)}{\\Delta(\\vec{n}_{z,\\neg{s}}+\\beta)} \\frac{\\Delta(\\vec{n}_m+\\alpha)}{\\Delta(\\vec{n}_{m,\\neg{s}}+\\alpha)} \n\n\\end{split}\n\\label{eq:fullConditional}\n\\end{equation}\n\n\\noindent For the first term of equation Eq. \\eqref{eq:fullConditional} we have:\n\\begin{multline}\n  \\frac{\\Delta(\\vec{n}_z+\\beta)}{\\Delta(\\vec{n}_{z,\\neg{s}}+\\beta)} =\n  \\frac{ \\frac{\\prod_{w\\in s} \\Gamma(\\vec{n}_z+\\beta)}{\\Gamma(\\sum_{w\\in s}(\\vec{n}_z+\\beta))}}\n  {\\frac{\\prod_{w\\in s} \\Gamma(\\vec{n}_{z,\\neg{s}}+\\beta)}{\\Gamma(\\sum_{w\\in s}(\\vec{n}_{z,\\neg{s}}+\\beta))}}= \\\\ =\\prod_{w\\in s} (\\frac{ \\Gamma(\\vec{n}_z+\\beta)}{\\Gamma(\\vec{n}_{z,\\neg{s}}+\\beta)}) \\frac{\\Gamma(\\sum_{w\\in s}(n_{z,\\neg{s}}+\\beta))}{\\Gamma(\\sum_{w\\in s}(n_z+\\beta))} = \\\\\n  =   \\underbrace{\\frac{\\overbrace{\\prod_{w\\in s} (n_{k,\\neg{s}}^{(w)}+\\beta)\\cdots (n_{k,\\neg{s}}^{(w)}+\\beta+(n_{k,s}^{(w)}-1))  }^{\\mbox{A}}}{(\\sum_{w\\in V} (n_{k,\\neg{s}}^{(w)}+\\beta))\\cdots(\\sum_{w\\in V} n_{k,\\neg{s}}^{(w)}+\\beta+(N_{k,s}^{(w)}-1))} }_{\\mbox{B}}\n\\label{eq:inferenceDetails}\n\\end{multline}\nHere, for the generation of A and B  we used the recursive property of the $\\Gamma$ function: $\\Gamma(x+m)= (x+m-1)(x+m-2)\\cdots(x+1)x\\Gamma(x)$; $w$ is a term that can occur many times in a sentence and $n_{k,s}^{(w)}$ denotes $w$'s frequency in sentence $s$ given that the sentence $s$ belongs to topic $k$; $N_{k,s}^{(w)}$ denotes how many words of sentence $s$ belong to topic $t$. \n\n\\RestyleAlgo{boxruled}\n\\begin{algorithm}[t]\\small\n\\DontPrintSemicolon\n \\For{ \\upshape document $d \\in[1,\\ldots,D]$}{\n  sample mixture of topics $\\theta_m \\sim$ Dirichlet(a)\\;\n  sample sentence number $S_d \\sim  Poisson (\\xi)$  \\;\n   \/\/Sentence plate\\;\n  \\For{ \\upshape sentence $s\\in[1,S_d]$}{\n  sample number of words  $W_{s} \\sim  Poisson (\\xi_d)$ \\;\n  sample topic $z_{d,s} \\sim Multinomial(\\theta_m)$\\;\n  \/\/Word plate in each language\\;\n  \\For{ \\upshape words $w\\in [1, W_{d,s}]$ in sentence $s$ }{\n  sample term for $w \\sim Multinomial(\\phi_{z_{d,s}}) $\\;\n  }\n  } \n }\n\\caption{Text collection generation with \\textit{senLDA}}\\label{algo:senlda}\n\\end{algorithm}\n\nThe development of the second factor in the final step of Eq. \\eqref{eq:fullConditional} is similar to the LDA calculations with the difference that the counts of topics per document are calculated given the allocation of sentences to topics and not the allocation of words to topics. This yields:\n\\begin{multline}\n p(z_s=k|\\vec{z}_{\\neg{s}}, \\vec{w}) = (n_{m,\\neg{s}}^{(k)} + \\alpha) \\times \\\\ \\times \\frac{\\prod_{w\\in s} (n_{k,\\neg{s}}^{(w)}+\\beta)\\cdots (n_{k,\\neg{s}}^{(w)}+\\beta+(n_{k,s}^{(w)}-1))}{(\\sum_{w\\in V} (n_{k,\\neg{s}}^{(w)}+\\beta))\\cdots(\\sum_{w\\in V} n_{k,\\neg{s}}^{(w)}+\\beta+(N_{k,s}^{(w)}-1))}  \n\\label{eq:finalMonolingual}\n \\end{multline}\nwhere $n_{m,\\neg{s}}^{(w)}$ denotes the number of times that topic $k$ has been observed with a sentence from document $d$, excluding the sentence currently sampled. \nNote that Eq. \\eqref{eq:finalMonolingual} reduces to the standard  LDA collapsed Gibbs sampling inference equations if the coherent text spans are reduced to words.\n\n\nThe idea of integrating the sentence limits in the LDA model has been previously investigated. For instance, in \\cite{wang2009multi} in the context of summarization the authors combine the unigram language model with topic models over sentences so that the latent topics are represented by sentences instead of terms. In \\cite{chen2010adaptation} the notion of \\textit{sentence topics} is introduced and they are sampled from separate topic distributions and co-exist with the word topics. Also, Boyd et al. \\cite{boyd2009syntactic} propose an adaptation of topic models to the text structure obtained by the parsing tree of a document. Our method resembles these works in that it integrates the notion of sentences to extend LDA. In our case though, we directly extend LDA maintaining the association of words to topics, we retain its simplicity without adding extra hyper-parameters thus allowing a fast, gibbs sampling inference, and we do not require any language-dependent tools such as parsers.\n\n\\section{Empirical results}\n\n\nWe conduct experiments to verify the applicability and evaluate the performance of \\textit{senLDA }  compared to LDA. The process is divided into two steps: (i) the training phase, where the topic models are trained to learn the their parameters, and (ii) the inference phase that is for new, unseen documents their topic distributions are estimated. We use the Gibbs sampling inference approach given by Eq. \\eqref{eq:finalMonolingual}. The hyper-parameters $\\alpha$ and $\\beta$ are set to $\\frac{1}{K}$, with $K$ being the number of topics. Table \\ref{table:trainData} shows the  datasets we used. They come from the publicly available collections of Wikipedia \\cite{partalas15lshtc} and PubMed \\cite{tsatsaronis2015overview}. The first four datasets (WikiTrain* and PubMedTrain*) were used for learning the topic model parameters; they differ in their respective size. Also, the vocabulary of the PubMed datasets is significantly larger due to the medical terms that appear. During preprocessing we only applied lower-casing, stop-word removal and lemmatization using the WordNet Lemmatizer.\\footnote{The code and the data are publicly available  at \\url{https:\/\/github.com\/balikasg\/topicModelling\/}}  The rest of the document collections of Table \\ref{table:trainData} are used for classification purposes and are discussed later in the section. \n\n\\begin{table}[t]\n\\small\\centering\n \\begin{tabular}{lcccc}\n  \\toprule\n  &Documents & $|V|$ & Classes & Timing (sec) \\\\\\midrule\n  WikiTrain1 & 10,000 & 46,051 & - &182|271 \\\\\n  WikiTrain2 & 30,000 & 65,820 & - &332|434 \\\\\n  PubMedTrain1 &10,000& 55,115 & - &304|433 \\\\\n  PubMedTrain2 & 60,000& 150,440 & - & 1830|2799 \\\\\n  Wiki37& 2,459 & 23,559& 37 & -\\\\\n  Wiki46& 3,657 & 27,914& 46 & -\\\\\n  PubMed25 & 7,248 &40,173 & 25 & -\\\\\n  PubMed50 & 9,035 &47,199 & 50 & -\\\\\n\\bottomrul\n \\end{tabular}\n\\caption{Description of the data used after pre-processing. ``Timing'' refers to the 25 first training iterations with the left (resp. right) values corresponding to \\textit{senLDA } (resp. LDA).}\\label{table:trainData}\n\\end{table}\n\n\\noindent\\textbf{Intrinsic evaluation} Topic model evaluation has been the subject of intense research. For intrinsic evaluation we report here perplexity \\cite{azzopardi2003investigating}, which is probably the dominant measure for topic models evaluation in the bibliography. The perplexity of $d$ held out documents given the model parameters $\\vec{\\vartheta}$ is defined as the reciprocal geometric mean of the token likelihoods of those data, given the parameters of the model:\n\\begin{equation}\n p(w_{\\text{heldOut}}) = \\exp-\\frac{\\sum_{i=1}^d \\sum_{j=1}^{w_i} \\log p(w_{i,j}|\\vec{\\vartheta})}{\\sum_{i=1}^d \\sum_{j=1}^{w_i} 1}  \n\\end{equation}\nNote that \\textit{senLDA }  samples per sentence and thus results in less flexibility at the word level where perplexity is calculated. Even though, the comparison between \\textit{senLDA } and LDA, at word level using perplexity, gives insights in the relative merits of the the proposed model. \n\\begin{figure}[t]\n\\centering\n \\includegraphics[width=0.47\\textwidth]{perplexity_both.pdf}\n \\caption{The ratio of perplexities of \\textit{senLDA } and LDA calculated on Wiki37 and PubMed25.  }\\label{fig:perplexity}\n\\end{figure}\n\nFigure \\ref{fig:perplexity} depicts the ratio of the perplexity values between \\textit{senLDA } and LDA. We set $K=125$ after grid searching $K\\in \\{25, 75, 125, 175\\}$ for perplexity with 5-fold cross-validation on the training data. Values higher (resp. lower) than one signify that \\textit{senLDA } achieves lower (resp. higher) perplexity than LDA. The figure demonstrates that in the first iterations before convergence of both models, \\textit{senLDA } performs better. What is more, \\textit{senLDA } converges after only around 30 iterations, whereas LDA converges after 160 iterations on Wikipedia and 200 iterations on the PubMed  datasets respectively. We define convergence as the situation where the model's perplexity does not any more decrease over training iterations. The shaded area in the figure highlights the period while \\textit{senLDA } performs better. It is to be noted, that although competitive, \\textit{senLDA }  does not outperform  LDA given unlimited time resources. However, that was expected since for \\textit{senLDA } the training instances are sentences, thus the model's flexibility is restricted when evaluated against a word-based \nmeasure. \n\n\\begin{figure*}[ht]\n  \\centering\n \\includegraphics[width=0.90\\textwidth]{classification_30k.pdf}\n \\caption{Classification performance on PubMed and Wikipedia text collections using $F_1$ measure.}\\label{fig:classfication}\n\\end{figure*}\n\nAn important difference between the models however, lies in the way they converge. From Figure \\ref{fig:perplexity} it is clear that \\textit{senLDA } converges faster. We highlight this by providing exact timings for the first 25 iterations of the models (column ``Timing'' of Table \\ref{table:trainData}) on a machine using an Intel Xeon CPU E5-2643 v3 at 3.40GHz. For both models we use our own Python implementations with the same speed optimisations.  Using ``WikiTrain2'' and 125 topics, for 25 iterations the \\textit{senLDA } needs 332 secs, whereas LDA needs 434 sec., an improvement of 30\\%. Furthermore, comparing the convergence, \\textit{senLDA } needs 332 secs (25 iterations) whereas LDA needs more than 2770 secs (more than 160 iterations) making \\textit{senLDA } more than 8 times faster. Similarly for the ``PubMedTrain2'' dataset which is more complex due to its larger vocabulary size, \\textit{senLDA } converges around 12 times (an order of magnitude) faster.\nNote that \\textit{senLDA}'s fast convergence is a strong advantage and can be highly appreciated in different application scenarios where unlimited time resources are not available.  \n\n\n\\noindent\\textbf{Extrinsic evaluation} Previous studies have shown that perplexity does not always agree with human evaluations of topic  models \\cite{azzopardi2003investigating} and it is recommended to evaluate topic models on real tasks. To better support  our development for \\textit{senLDA } applicability we also evaluate it using text classification as the evaluation task. \n For text classification, each document is represented by its topic distribution, which is the vectorial input to Support Vector Machines (SVMs). The classification collections are split on train\/test (75\\%\/25\\%) parts. The SVM regularization hyper-parameter $\\lambda$ is selected from $\\lambda\\in[10^{-4},\\ldots,10^4]$ using 5-fold cross-validation on the training part of the classification data.\nThe PubMed testsets are multilabel, that is each instance is associated with several classes, 1.4 in average in the sets of Table \\ref{table:trainData}. For the multilabel problem with the SVMs we used a binary relevance approach.    \nTo assess the classification performance, we report the $F_1$ evaluation measure, which is the harmonic mean of precision and recall.\n\n\n\nThe classification performance on $F_1$ measure for the different classification datasets is shown in Figure \\ref{fig:classfication}. \nFirst note that in the majority of the classification scenarios, \\textit{senLDA }  outperforms LDA. In most cases, the performance difference increases when the larger train sets (``WikiTrain2'' and ``PubMedTrain2'') are used. For instance, in the second line of figures with the PubMed classification experiments, increasing the topic models' training data benefits both LDA and \\textit{senLDA }, but \\textit{senLDA } still performs better. \nMore importantly though and in consistence with the perplexity experiments, the advantage of \\textit{senLDA } remains: the faster \\textit{senLDA } convergence benefits the classification performance. The \\textit{senLDA } curves are steeper in the first training iterations and stabilize after roughly 30 iterations when the model converges. We believe that assigning the latent topics to coherent groups of words such as sentences results in document representations of finer level. In this sense,  spans larger than single words can capture and express the document's content more efficiently for discriminative tasks like classification. \n\n\nTo investigate the correlation of topic model representations learned on different levels of text, we report the classification  performance using as document representations the concatenation of a document's topic distributions output by LDA and \\textit{senLDA }. For instance, the concatenated vectorial representation of a document when $K=125$ for each model is a vector of 250 dimensions. The resulting concatenated representations are denoted by ``senLDA+'' in Figure \\ref{fig:classfication}. As it can be seen, ``senLDA+'' performs better compared to both LDA and \\textit{senLDA }. Its performance combines the advantages of both models: during the first iterations it is as steep as the \\textit{senLDA } representations and in the later iterations benefits by the LDA convergence to outperform the simple \\textit{senLDA } representation. Hence,  the concatenation of the two distributions creates a richer representation where the two models contribute complementary information that achieves the best classification performance. Achieving the optimal performance using those representations suggests that the relaxation of the independence assumptions between the text structural units can be beneficial; this is also among the contributions of this work.\n\n\n\\section{Conclusion}\nWe proposed \\textit{senLDA}, an extension of LDA where topics are sampled per coherent text spans. This resulted in very fast convergence and good classification and perplexity performance. LDA and \\textit{senLDA } differ in that the second assumes a very strong dependence of the latent topics between the words of sentences, whereas the first assumes independence between the words of documents in general. In our future research, our goal is to investigate this dependence and further adapt the sampling process of topic models to cope with the rich text structure.\n\n\n\n\\section{Acknowledgements} \nThis work is partially supported by the CIFRE N 28\/2015.\n\n\\bibliographystyle{abbrv}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction.}\\ \n\n\\vspace{-1em}\n\n\n\nLet $G$ be a complex connected semisimple algebraic group (e.g.\\ $\\mathrm{SL}_n$ or $\\mathrm{SO}_n$), $B$ a Borel subgroup of $G$, and $U$ the unipotent radical of $B$. For example, in the case $G=\\mathrm{SL}_n$, we can take $U$ to be the collection of all upper triangular matrices with all diagonal entries equal to $1$.\n\nThe homogeneous space $G\/U$ is called the ``basic affine space\". While $G\/B$ is projective, the basic affine space $G\/U$ is a quasi-affine variety. It turns out that many interesting problems in representation theory are related to the basic affine space. In particular, the algebra $\\mathcal{D}(G\/U)$ of algebraic differential operators on $G\/U$ is well-studied, for example in  \\cite{Gelfand}\\cite{GK}\\cite{LS}\\cite{BBP}.\n\nIn this paper, we study the cotangent bundle $T^*(G\/U)$, of which the coordinate ring $\\mathbb{C}[T^*(G\/U)]$ is the quasi-classical counterpart of $\\mathcal{D}(G\/U)$. From a result by Ginzburg and Riche \\cite{GR}, the coordinate ring $\\mathbb{C}[T^*(G\/U)]$ is finitely generated, and the affine closure of the basic affine space is defined as $$\\overline{T^*(G\/U)}\\coloneqq \\Spec \\mathbb{C}[T^*(G\/U)].$$\n\nFollowing Beauville \\cite{B}, a normal variety $X$ has\nsymplectic singularities if its regular locus carries a symplectic $2$-form whose pull-back\nalong any resolution $\\widetilde{X}\\rightarrow X$ extends to a holomorphic $2$-form on $\\widetilde{X}$.\nThere are many interesting and important examples of symplectic singularities. These include finite quotient singularities \\cite{B}, normal closures of nilpotent coadjoint orbits \\cite{Pan}, Nakajima quiver varieties \\cite{BS}, and more recently Coulomb branches associated to simple quivers \\cite{Wee}.\n\nSymplectic singularities play an important role in representation theory, and they have a lot of interesting properties. For instance, in the conical case (affine symplectic singularities with a good $\\mathbb{C}^*$-action), there is a universal family of deformations \\cite{Nam}\\cite{Nam2}\\cite{KV}\nand filtered quantizations \\cite{BK}\\cite{BLPW1}\\cite{Los}. Thanks to these two properties, one can generalize results in Lie theory to conical symplectic singularities. For example, there is a category $\\mathscr{O}$ for a conical symplectic resolution \\cite{BLPW2}. And in this general setting, the classical BGG category $\\mathscr{O}$ \\cite{BGG} can be viewed as the category $\\mathscr{O}$ associated with the Springer resolution \\cite{Springer}. Hence A. Okounkov once said, ``Symplectic singularities are the Lie algebras of the 21st century\".\n\n\nIn \\cite{GK}, Ginzburg and Kazhdan conjectured that the affine closure of the basic affine space also has symplectic singularities.\nIn the case $G=\\mathrm{SL}_n$, we have the following result proved in section \\ref{dq}.\n\\begin{thm}\\label{main}\n    The affine closure $\\overline{T^*(\\mathrm{SL}_n\/U)}$ has symplectic singularities.\n\\end{thm}\n\nIn the Lie algebra $\\mathfrak{so}_8$, there is a unique nilpotent adjoint orbit $\\mathcal{O}_{\\textrm{min}}\\subset\\mathfrak{so}_8$ of minimal (positive) dimension $10$. The closure of the minimal orbit is \\mbox{$\\overline{\\mathcal{O}}_\\textrm{min}=\\mathcal{O}_{\\textrm{min}}\\cup\\{0\\}$}. In section \\ref{kostant}, we discuss Kostant's theorem on the highest weight variety and its application to the minimal nilpotent orbits of type $D$.\nIn section \\ref{sl3}, we show that there is an isomorphism of affine varieties\n    \\begin{equation}\\label{1.3}\n        \\overline{T^*(\\mathrm{SL}_3\/U)}\\rightarrow\\overline{\\mathcal{O}}_\\textrm{min}.\n    \\end{equation}\nIn section \\ref{symplectic}, we show that this map (\\ref{1.3}), when restricted on smooth points, is a symplectic isomorphism.\n\nIn \\cite{GK}, Ginzburg and Kazhdan constructed an action on $\\overline{T^*(G\/U)}$ by the Weyl group $W$ of $G$, called the ``Gelfand--Graev action\". So in the case $G=\\mathrm{SL}_3$ we have an $S_3$-action on $\\overline{T^*(\\SL_3\/U)}$. On the other hand, the Lie algebra $\\mathfrak{so}_8$ has an $S_3$-symmetry called the triality action \\cite{Cartan}\\cite{MW}, and the restriction of the triality action gives an $S_3$-action on $\\overline{\\mathcal{O}}_\\textrm{min}$. In section \\ref{triality}, we give a new interpretation of this triality action. In section \\ref{ggaction}, we show that the isomorphism (\\ref{1.3}) is $S_3$-equivariant. So we have proved\n\\begin{thm}\nThere is an $S_3$-equivariant isomorphism\n    \\begin{equation}\\label{2.7}\n        \\overline{T^*(\\mathrm{SL}_3\/U)}\\rightarrow\\overline{\\mathcal{O}}_\\textrm{min}\n    \\end{equation} of affine Poisson varieties.\n\\end{thm}\n\n\n\n\n\\vspace{2em}\n\\textbf{Acknowledgments.}\\ The author is very grateful to his advisor, Victor Ginzburg, for all the encouragement, suggestions, and discussions. He would like to thank David Kazhdan, for the helpful correspondence containing statements related to the affine closure $\\overline{T^*(\\SL_3\/U)}$, the minimal nilpotent orbit $\\mathcal{O}_{\\textrm{min}}$ in $\\mathfrak{so}_8$ and the triality action. He would also like to thank Sam Evens, Yu Li, and Minh-Tam Trinh for their useful suggestions and comments.\n\n\\vspace{3em}\n\\section{The affine closure $\\overline{T^*(SL_n\/U)}$ has symplectic singularities.}\\label{dq}\n\nA quiver $Q$ is a finite directed graph consisting of \na vertex set $I$ and an edge set $E$.\nWrite $Q^\\textrm{op}=(I,\\overline{E})$\nfor the opposite quiver obtained from $Q$ by reversing the orientation of edges.\nThe\ndouble quiver of $Q$ is defined by $\\overline Q=(I,E\\sqcup \\overline{E})$. \nA representation of $Q$ assigns a vector space $V(i)$ to each vertex $i\\in I$ and a linear map $V(e):V(i)\\rightarrow V(j)$ to each arrow $e\\in E$ whose source and target are $i$ and $j$ respectively.\n\nLet $Q$ be the following Dynkin quiver of type $\\mathrm{A}_{n}$.\n$$\n    \\begin{tikzcd}\n\\bullet \\arrow[r]\n\t& \\bullet\n\t\\arrow[r] \n\t& \\cdots\n\t\\arrow[r] \n\t& \\bullet\n\t\\arrow[r] \n\t& \\bullet\n\\end{tikzcd}\n$$\nLet $V=\\bigoplus_{k=1}^{n-1}\\Hom(\\mathbb{C}^k,\\mathbb{C}^{k+1})$, so each element of $V$ defines a representation of the quiver $Q$.\nThe cotangent space $T^*V$ is identified with $$\\bigoplus_{k=1}^{n-1}\\Hom(\\mathbb{C}^k,\\mathbb{C}^{k+1})\\oplus\\Hom(\\mathbb{C}^{k+1},\\mathbb{C}^{k})$$ via the  trace pairing\n$\n    \\sum_{k=1}^{n-1}\\mathrm{Tr} (\\beta_k\\circ\\alpha_k),\n$ for $\\alpha_k\\in\\Hom(\\mathbb{C}^k,\\mathbb{C}^{k+1})$ and $ \\beta_k\\in\\Hom(\\mathbb{C}^{k+1},\\mathbb{C}^{k})$.\nSo each element of $T^*V$ gives a representation of the double quiver $\\overline{Q}$  \n\\begin{equation}\\label{double quiver}\n    \\begin{tikzcd}\n {\\mathbb{C}} \\arrow[r,shift left=.7ex,\"\\alpha_1\"]\n\t& {\\mathbb{C}^2} \\arrow[l,shift left=.7ex,\"\\beta_1\"] \n\t\\arrow[r,shift left=.7ex,\"\\alpha_2\"] \n\t& {\\cdots} \\arrow[l,shift left=.7ex,\"\\beta_2\"] \n\t\\arrow[r,shift left=.7ex,\"\\alpha_{n-2}\"] \n\t& {\\mathbb{C}^{n-1}} \\arrow[l,shift left=.7ex,\"\\beta_{n-2}\"] \n\t\\arrow[r,shift left=.7ex,\"\\alpha_{n-1}\"] \n\t& \\mathbb{C}^n \\arrow[l,shift left=.7ex,\"\\beta_{n-1}\"] \n\\end{tikzcd}.\n\\end{equation}\nThis construction was applied in \\cite{KP} to show the normality of the nilpotent cone of $\\mathfrak{gl}_n$. Similar constructions can also be found in \\cite{KS}\\cite{CB1}\\cite{CB2}.\n\n\nThere is an action of $H\\coloneqq\\prod_{k=2}^{n-1}\\mathrm{SL}_i(\\mathbb{C})$ on $V$ by $$(g_2,g_3,\\cdots,g_{n-1}).(\\alpha_1,\\alpha_2,\\cdots,\\alpha_{n-1})=(g_{2}\\circ\\alpha_1,g_{3}\\circ\\alpha_2\\circ g_{2}^{-1},\\cdots,g_{n}\\circ\\alpha_k\\circ g_{n-1}^{-1}).$$ This $H$-action on $V$\ninduces a Hamiltonian $H$-action on $T^*V$, of which the moment map is\n$$\n    \\mu_H(\\alpha,\\beta)(X)=\\sum_{k=2}^{n-1}\\mathrm{Tr}\\left((\\alpha_{k-1}\\beta_{k-1}-\\beta_k\\alpha_k)X_k\\right),\n$$\nwhere $(\\alpha,\\beta)=\\bigoplus_{k=1}^{n-1}(\\alpha_k,\\beta_k)\\in T^*V$, and $X=(X_2,X_3,\\cdots,X_{n-1})\\in\\mathfrak{sl}_2\\times\\mathfrak{sl}_3\\times\\cdots\\times\\mathfrak{sl}_{n-1}$.\n\n\n\\begin{defn}\n    We denote the zero fiber of the moment map by\n    $$\n        N\\coloneqq\\mu_H^{-1}(0)\\subset T^*V\n    $$\n    So $N$ is the subvariety consists of all the $(\\alpha,\\beta)$ in  $T^*V$ such that for all $k\\in\\{1,2,3,\\cdots,n-1\\}$\n\\begin{equation}\\label{N}\n    \t\\beta_k\\alpha_k-\\alpha_{k-1}\\beta_{k-1}=\\lambda_k\\,\\textrm{Id}_{\\mathbb{C}^k},\n\\end{equation}\nfor some $\\lambda_k\\in\\mathbb{C}.$\n\\end{defn}\n\n\n\\begin{notn}\nThroughout this paper, we use the expression $(\\alpha,\n\\beta)$ to denote an element $$\\bigoplus_{k=1}^{n-1}(\\alpha_k,\\beta_k)\\in\\bigoplus_{k=1}^{n-1}\\Hom(\\mathbb{C}^k,\\mathbb{C}^{k+1})\\oplus\\Hom(\\mathbb{C}^{k+1},\\mathbb{C}^{k})=T^*V.$$\n\\end{notn}\n\nWe recall the following theorem.\n\\begin{thm}[\\textrm{Theorem 7.18} in \\cite{DKS}]\nThe affine closure $\\overline{T^*(\\mathrm{SL}_n\/U)}$ is isomorphic to the categorical quotient $N\\sslash H$ as affine varieties.\n\\end{thm}\n\n\n\\begin{lem}\n\tLet $0=m_0\\leq m_1\\leq m_2\\leq\\cdots\\leq m_{n-1}\\leq m_n=n$.\n\tSuppose $m_k\\leq k$ for all $k$. Then\n\t$$\n\t\\sum_{k=1}^{n-1}m_k(m_{k+1}-m_k)\\leq\\sum_{k=1}^{n-1}k,\n\t$$\n\tand the equality holds if and only if $m_k=k$ for all $k$.\n\\end{lem}\n\\begin{proof}\n\tSince $m_0=0$, we show that \n\t\\begin{align*}\n\t\t\\sum_{k=0}^{n-1}m_k(m_{k+1}-m_k)\n\t\t=\\sum_{k=0}^{n-1}\\sum_{j=m_k+1}^{m_{k+1}}m_k\n\t\t\\leq\\sum_{k=0}^{n-1}\\sum_{j=m_k+1}^{m_{k+1}}(j-1)\n\t\t=\\sum_{k=1}^{n-1}k.\n\t\\end{align*}\n\tIn fact, the LHS is the dimension of some (partial) flag variety of $\\mathrm{GL}_n$, so it obtains maximum if and only if in the case of complete flag variety, which has dimension RHS.\n\\end{proof}\n\n\\pagebreak\n\\begin{lem}\\label{codim}\nThe singular locus of $\\overline{T^*(\\SL_n\/U)}$ has codimension at least $4$.\n\\end{lem}\n\\begin{proof}\n\tFor each $\\underline{m}=(m_1,m_2,m_3,\\cdots,m_{n-1})$ satisfying the condition in the previous lemma, we define $H(\\underline{m})\\coloneqq\\prod_{k=1}^{n-1}\\mathrm{SL}_{m_k}(\\mathbb{C})$,\n\t$\\tilde{H}(\\underline{m})\\coloneqq\\prod_{k=1}^{n-1}\\mathrm{GL}_{m_k}(\\mathbb{C})$, $V(\\underline{m})=\\bigoplus_{k=1}^{n-1}\\Hom(\\mathbb{C}^{m_k},\\mathbb{C}^{m_{k+1}})$.\n\tThen by Theorem 6.13 in \\cite{DKS} (page 30), the affine variety $X$ can be written as disjoint union \n\t   $$\n\t           X=\\bigsqcup_{S,\\delta}Q_{(S,\\delta)}\n\t   $$\n\twhere each $Q_{(S,\\delta)}$ is a smooth hyperk\\\"ahler manifold that is a locally closed subset of $X$.\n\tLet $x$ be a singular point of $\\overline{T^*(\\SL_n\/U)}$. Then $x$ lies in some strata $Q_{(S,\\delta)}$ with the same dimension as the Hamiltonian reduction of $T^*V(\\underline{m})$ by a subgroup $H_S\\leq\\tilde{H}(\\underline{m})$, where $\\underline{m}\\neq(1,2,3,\\cdots,n-1)$ and $H_S$ is defined after (6.1) in \\cite{DKS} (page 26), from which we also know $H(\\underline{m})$ is a proper subgroup of $H_S$.\n\tSo \n\t\\begin{align*}\n\t\t\\dim Q_{(S,\\delta)}& =\\dim(T^*V(\\underline{m})\/\\!\/\\!\/ H_S)\\\\\n\t\t& = 2(\\dim V(\\underline{m})-\\dim H_S)\\\\\n\t\t& \\leq 2(\\dim V(\\underline{m})-\\dim H(\\underline{m})-1)\\\\\n\t\t& =2\\left(\\sum_{k=1}^{n-1}m_km_{k+1}-(m^2_k-1)\\right)-2\\\\\n\t\t& =2\\left(n-2+\\sum_{k=1}^{n-1}m_k(m_{k+1}-m_k)\\right)\n\t\\end{align*}\n\tSince $\\underline{m}\\neq(1,2,3,\\cdots,n-1)$, by the previous lemma, we know $\\sum_{k=1}^{n-1}m_k(m_{k+1}-m_k)\\leq\\left(\\sum_{k=1}^{n-1}k\\right)-1$.\n\tThus\n\t$$\n\t\\dim Q_{(S,\\delta)}\\leq2\\left((n-3)+\\sum_{k=1}^{n-1}k\\right)\n\t=2\\left(-1+\\sum_{k=1}^{n}k\\right)-4=\\dim(\\overline{T^*(\\SL_n\/U)})-4.\n\t$$\n\\end{proof}\n\n\\begin{defn}\\label{Nsurj}\n    We define the surjective part (resp.\\ principal part) of $N$ as follows.\n    \\begin{align*}\n        N_\\textrm{surj}& =\\{(\\alpha,\\beta)\\in N\\,|\\, \\textrm{all the $\\beta_k$ are surjective}\\},\\\\\n        N_\\textrm{pr}& =\\{(\\alpha,\\beta)\\in N\\,|\\, \\textrm{the $H$-orbit of $(\\alpha,\\beta)$ is closed and the stabilizer }H_{(\\alpha,\\beta)}=1\\}.\n    \\end{align*}\n    Then we have\n    $\n        N_\\textrm{surj}\\subset N_\\textrm{pr}.\n    $\n\\end{defn}\n\n\\begin{prop}\\label{symp}\n\tThe smooth locus of $\\overline{T^*(\\SL_n\/U)}$ admits a holomorphic symplectic form.\n\\end{prop}\n\\begin{proof}\n\tLet $X=\\overline{T^*(\\SL_n\/U)}$. Let $\\pi:N\\rightarrow N\\sslash H=X$ be the categorical quotient map. Let $X_\\textrm{sm}$ be the smooth locus of $X$.\n\tBy proposition 5.7 in \\cite{Dre}, we have $\\pi(N_\\textrm{pr})\\subset X_\\textrm{sm} \\subset X$. From the proof of \\textrm{Lemma 7.17} in \\cite{DKS} (page 38), we know $N\\setminus N_\\textrm{surj}$ has codimension at least 2, so $$\\mathrm{codim}(N\\setminus N_\\textrm{pr})\\geq2.$$ By the upper semicontinuity of the dimensions of the fibers of $\\pi$ we have $$\\mathrm{codim}(X_\\textrm{sm}\\setminus \\pi(N_\\textrm{pr}))\\geq2.$$  From the Marsden-Weinstein theorem, there is a holomorphic symplectic form $\\omega_0$ on $\\pi(N_\\textrm{pr})$. Then by Hartog's lemma, we can extend $\\omega_0$ to a closed holomorphic 2-form $\\omega$ on $X_\\textrm{sm}$. By taking the top wedge of $\\omega$, we have its points of degeneracy has codimension $1$, so $\\omega$ has to be non-degenerated on $X_\\textrm{sm}$, hence symplectic. \n\\end{proof}\n\n\\pagebreak\n\\begin{thm}\\label{main2}\n\tThe affine closure $\\overline{T^*(\\mathrm{SL}_n\/U)}$ has symplectic singularities.\n\\end{thm}\n\\begin{proof}\n   Applying Theorem 18.4 in \\cite{gros}, we know that $X\\coloneqq\\overline{T^*(\\SL_n\/U)}$ is a normal variety. Let $\\omega$ be the holomorphic symplectic form on the smooth locus of $X$ as in Proposition \\ref{symp}. Let $$\\nu:\\widetilde{X}\\rightarrow X$$ be any resolution of singularity. By Lemma \\ref{codim} and the main theorem in \\cite{Flenner}, the pull back map $$\\nu^*:\\Omega^2_\\textrm{hol}(X)\\rightarrow\\Omega^2_\\textrm{hol}(\\widetilde{X})$$ is a bijection. So $\\nu^*\\omega$ extends holomorphically everywhere on $\\widetilde{X}$.\n\\end{proof}\n\\begin{rem}\n    Let $P$ be any parabolic subgroup of $\\mathrm{SL}_n$.\n    Let $[P,P]$ denote the commutator subgroup of $P$.\n    Then I believe one can use similar method to prove that the affinization $\\overline{T^*(\\mathrm{SL}_n\/[P,P])}$ also has symplectic singularities, which generalizes Theorem \\ref{main2}.\n\\end{rem}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\vspace{3em}\n\\section{The case of $n=3$ and minimal nilpotent orbits.}\n\n\\subsection{The minimal nilpotent orbit of $\\mathfrak{so}_{2m}$}\\label{kostant}\nLet $\\mathcal{O}_{\\textrm{min}}$ be the minimal nilpotent orbit of a semisimple Lie algebra $\\mathfrak{g}$.\nLet $\\mathfrak{g} = \\mathfrak{n} \\oplus \\mathfrak{h} \\oplus \\mathfrak{n}^{-}$ be a triangular decomposition of $\\mathfrak{g},$ such that $\\mathfrak{n}=\\mathrm{Lie}(U)$.\nWrite $\\Delta = \\Delta^{+} \\sqcup \\Delta^{-}$\nfor the root system of $(\\mathfrak{g}, \\mathfrak{h})$ so that the roots in $\\Delta^+$ (resp.\\ in $\\Delta^-$) correspond to $\\mathfrak{n}$ (resp.\\ $\\mathfrak{n}^-$).\nLet $\\psi\\in\\Delta^{+}$ be the highest root of $\\mathfrak{g}$. Let $v_\\psi$ be a nonzero root vector corresponding to $\\psi$.\n\\begin{prop}[\\textrm{Theorem 4.3.3} in \\cite{CM}] The minimal nilpotent orbit $\\mathcal{O}_{\\textrm{min}}$ is the adjoint orbit of $v_\\psi$.\\end{prop}\n\nLet $\\mathcal{I}\\subset\\mathbb{C}[\\mathfrak{g}]$ be the ideal of polynomials which vanish identically on $\\mathcal{O}_{\\textrm{min}}$.\nFor each dominant weight $\\lambda\\in\\mathfrak{h}^*$, define $V_\\lambda$ as an irreducible $\\mathfrak{g}$-representation with highest weight $\\lambda$. One has the following result\n\\begin{thm}[Kostant's Theorem (\\textrm{Theorem III.2.1} in \\cite{Garfinkle})]\nDecompose the $\\mathfrak{g}$-representation\n$$\\mathrm{Sym}^2(\\mathfrak{g})^*=V_{2\\psi}^*\\oplus C.$$\nThe ideal $\\mathcal{I}$ is generated by $C$ in $\\mathrm{Sym}(\\mathfrak{g})^*=\\mathbb{C}[\\mathfrak{g}]$.\n\\end{thm}\n\n\n\nLet $m\\geq4$. We consider the case $\\mathfrak{g}=\\mathfrak{so}_{2m}$.\nLet $e_1,e_2,\\cdots,e_{2m}$ be the natural basis of $\\mathbb{C}^{2m}$.\nDefine an Euclidean inner product $(\\ ,\\ )$ on $\\mathbb{C}^{2m}$ by\n    \\begin{equation}\\label{inner}\n        (e_i,e_j)=\\delta_{i,2m+1-j}.\n    \\end{equation}\nFor $v_1\\wedge v_2\\in\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}$, we define\n\\begin{align}\\label{varphi}\n\t\\qquad \\varphi_{v_1\\wedge v_2}:\\ \\mathbb{C}^{2m}\\ &\\longrightarrow \\ \\mathbb{C}^{2m}\\\\\n\tu\\ \\ &\\longmapsto\\ \\, \\left(v_1,u\\right)v_2-\n\t\\left(v_2,u\\right)v_1.\\nonumber\n\\end{align}\nExtend by linearity, we get the following isomorphism of $\\mathfrak{so}_{2m}$-representations:\n\\begin{equation}\\label{Lso}\n    \\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}=\\mathfrak{so}_{2m}. \n\\end{equation}\n\n\n\\begin{defns}\nWe say a subspace $W\\subset \\mathbb{C}^{2m}$ is isotropic if $(W,W)=0$. An element $f\\in\\Hom(\\mathbb{C}^2,\\mathbb{C}^{2m})$ is isotropic if its image is isotropic.\nAn element $\\alpha\\in\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}$ is isotropic if for all $v_1^*, v_2^*\\in (\\mathbb{C}^{2m})^*$,\n\t$$\n\t    (\\iota_{v_1^*}\\alpha, \\iota_{v_2^*}\\alpha)=0.\n\t$$\nWe say $\\alpha\\in\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}$ is decomposable if there exist some $v_1,v_2\\in\\mathbb{C}^{2m}$ such that $\\alpha=v_1\\wedge v_2$. \n\\end{defns}\n\\begin{rem}\n    By a theorem of Pl\\\"ucker we know that $\\alpha\\in\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}$ is decomposable if and only if\n$$\\alpha\\wedge\\alpha=0\\in\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^4\\mathbb{C}^{2m}.$$\n\\end{rem}\n\\begin{lem}\\label{iso}\n\tLet $v_1\\wedge v_2\\in\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}$ \n\tbe a nonzero decomposable element. \n\tThen \n\t$$\n\t\\mathrm{span} (v_1,v_2) \\textrm{ is isotropic }\n\t\\iff v_1\\wedge v_2 \\textrm{ is isotropic.}\n\t$$\n\\end{lem}\n\\begin{proof}\n\t($\\Longrightarrow$)\\ Clear.\\\\\n\t($\\Longleftarrow$)\\ It suffices to show that \n\t$$\n\t(v_1,v_1)=(v_1,v_2)=(v_2,v_2)=0.\n\t$$\n\tSince $v_1\\wedge v_2\\neq0$, we have $v_1$ and $v_2$ are linearly independent. So there exist $v_1^*,v_2^*\\in(\\mathbb{C}^{2m})^*$ such that\n\t$$\n\tv_1^*(v_1)=v_2^*(v_2)=1,\\textrm{ and } \n\tv_1^*(v_2)=v_2^*(v_1)=0.\n\t$$\n\tThen\n\t\\vspace{-1em}\n\t$$\n\tv_1=-\\iota_{v_2^*}(v_1\\wedge v_2)\\textrm{, }\n\tv_2=\\iota_{v_1^*}(v_1\\wedge v_2),\n\t$$\n\tand our conclusion follows from the definition.\n\\end{proof}\n\nLet $\\overline{\\mathcal{O}}_\\textrm{min}$ be the  closure of the minimal nilpotent orbit $\\mathcal{O}_{\\textrm{min}}\\subset\\mathfrak{so}_{2m}$.\n\\begin{prop}\\label{ominiso} Under the identification (\\ref{Lso}),\n\t\\begin{equation}\\label{Omin}\n\t    \\overline{\\mathcal{O}}_\\textrm{min}=\\{\\alpha\\in\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}\\,|\\, \\alpha\\textrm{ is decomposable and isotropic}\\}\n\t\\end{equation}\n\\end{prop}\n\\begin{proof}\n\tSince $\\overline{\\mathcal{O}}_\\textrm{min}=\\mathcal{O}_{\\textrm{min}}\\cup\\{0\\}$, it suffices to show that\n\t$$\n\t\\mathcal{O}_{\\textrm{min}}=\\{\\alpha\\in\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}\\,|\\, \\alpha\\textrm{ is a nonzero isotropic decomposable element}\\}.\n\t$$\n\tSince both decomposable and isotropic properties are invariant under the $\\mathrm{SO}_{2m}$ action, and $e_1\\wedge e_2$ is isotropic decomposable, so all elements in the minimal orbit are isotropic decomposable. But $\\mathrm{SO}_{2m}$ acts transitively on the isotropic planes in $\\mathbb{C}^{2m}$, and there is a scaling  $\\mathbb{C}^*$-action on $\\mathcal{O}_{\\textrm{min}}$, so $\\mathrm{SO}_{2m}$ acts transitively on the nonzero isotropic decomposable elements.\n\\end{proof}\n\n\n\\begin{defns}\n    Define the following two maps:\n\t\\begin{align*}\n\t\t\\Phi:(\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^4\\mathbb{C}^{2m})^* & \\longrightarrow\\ \\mathbb{C}[\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}]\\\\\n\t\t\\varphi\\quad\\ \\ & \\longmapsto\\  [\\alpha\\mapsto\\varphi(\\alpha\\wedge\\alpha)],\\\\[0.5em]\n        \\Psi:\\mathrm{Sym}^2(\\mathbb{C}^{2m})^* & \\longrightarrow\\mathbb{C}[\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}]\\\\\n\t    v_1^*\\cdot v_2^* & \\longmapsto[\\alpha\\mapsto(\\iota_{v_1^*}\\alpha,\\iota_{v_2^*}\\alpha)].\n\t\\end{align*}\n\tLet $V_\\textrm{dec}$ (resp.\\ $V_\\textrm{iso}$) denote the image of $\\Phi$ (resp.\\ $\\Psi$).\n\tNote that $V_\\textrm{dec}$ (resp.\\ $V_\\textrm{iso}$) generates the ideal $\\mathcal{I}_\\textrm{dec}$ (resp.\\ $\\mathcal{I}_\\textrm{iso}$) of polynomial functions on $\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}$ vanishing on decomposable (resp.\\ isotropic) elements of $\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}$.\n\\end{defns}\n\n\n\\begin{prop}\nBoth $\\Phi$ and $\\Psi$ are injective, and\n\t$$\n\t    \\mathrm{Sym}^2(\\mathfrak{so}_{2m})^*=V^*_{2\\psi}\\oplus V_\\textrm{dec}\\oplus V_\\textrm{iso}\n\t$$\nas $\\mathfrak{so}_{2m}$-representions.\n\\end{prop}\n\\begin{proof}\nFirst we show the following elements are linearly independent in $\\mathbb{C}[\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}]$:\n$$\n    \\{\\Phi(e_i^*\\wedge e_j^*\\wedge e_k^*\\wedge e_l^*)\\suchthat1\\leq i<j<k<l\\leq 2m\\}\\cup\\{\\Psi(e_r^*\\cdot e_s^*)\\suchthat1\\leq r\\leq s\\leq 2m\\}.\n$$\nSuppose there exist some $A_{ijkl},B_{rs}\\in\\mathbb{C}$ such that\n\\begin{equation}\\label{sum}\n    \\sum_{1\\leq i<j<k<l\\leq 2m}\\!\\!A_{ijkl}\\ \\Phi(e_i^*\\wedge e_j^*\\wedge e_k^*\\wedge e_l^*)\\ +\\sum_{1\\leq r\\leq s\\leq 2m}\\!\\!B_{rs}\\,\\Psi(e_r^*\\cdot e_s^*)=0.\n\\end{equation}\nLet $1\\leq t\\leq 2m$. Evaluating the LHS of (\\ref{sum}) at $e_{t}\\wedge e_{2m+1-t}$, we have $B_{t,2m+1-t}=0$. So\n\\begin{equation}\\label{sum1}\n    \\sum_{1\\leq i<j<k<l\\leq 2m}\\!\\!A_{ijkl}\\ \\Phi(e_i^*\\wedge e_j^*\\wedge e_k^*\\wedge e_l^*)\\ +\\sum_{s\\neq 2m+1-r}\\!\\!B_{rs}\\,\\Psi(e_r^*\\cdot e_s^*)=0.\n\\end{equation}\nLet $1\\leq r\\leq s\\leq 2m$ with $s\\neq 2m+1-r$.\nEvaluating the LHS of (\\ref{sum1}) at $e_{r}\\wedge (e_s+e_{2m+1-r})$, we have $B_{rs}=0$. \nSo\n\\begin{equation}\\label{sum2}\n    \\sum_{1\\leq i<j<k<l\\leq 2m}\\!\\!A_{ijkl}\\ \\Phi(e_i^*\\wedge e_j^*\\wedge e_k^*\\wedge e_l^*)=0.\n\\end{equation}\nLet $1\\leq i<j<k<l\\leq 2m$, evaluating the LHS of (\\ref{sum2}) at $(e_i\\wedge e_j+e_k\\wedge e_l)$, we have $A_{ijkl}=0$.\nSo both $\\Phi$ and $\\Psi$ are injective, and \n$$\n    V_\\textrm{dec}\\cap V_\\textrm{iso}=\\{0\\}.\n$$\nSince the highest weight vector ${(e_{1}^*}\\wedge {e_{2}^*})^2\\in \\mathrm{Sym}^{2}(\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^{2}(\\mathbb{C}^{2m})^{*})$ does not belong to the $\\mathfrak{so}_8$-representation $V_\\textrm{dec}\\oplus V_\\textrm{iso}$. By the irrducibility of $V_{2\\psi}^*$ we have\n$$\n    V_{2\\psi}^*\\cap(V_\\textrm{dec}\\oplus V_\\textrm{iso})=\\{0\\}.\n$$\nBy the injectivity of $\\Phi$ and $\\Psi$ we have\n$$\n        \\dim V_\\textrm{dec} =\\binom{2m}{4},\\  \n        \\dim V_\\textrm{iso} =\\binom{2m+1}{2}.\n$$\nBy Weyl's dimension formula\n    $$\n        \\dim V_{2\\psi}^*=\\frac{m}{3}(4m^3-7m-3).\n    $$\nSo, we find\n    \\begin{align*}\n    \\dim V_{2\\psi}^*+\\dim V_\\textrm{dec}+\\dim V_\\textrm{iso}&=\\frac{m}{3}(4m^3-7m-3)+\\binom{2m}{4}+\\binom{2m+1}{2}\\\\\n        &=2 m^4 - 2 m^3 + \\frac{3 m^2}{2} - \\frac{m}{2}\n        =\\binom{\\binom{2m}{2}+1}{2}\\\\\n        &=\\dim\\mathrm{Sym}^2(\\mathfrak{so}(2m)).\n    \\end{align*}\nTherefore, we obtain\n\t$$\n\t    \\mathrm{Sym}^2(\\mathfrak{so}_{2m})=V_{2\\psi}^*\\oplus V_\\textrm{dec}\\oplus V_\\textrm{iso}.\n\t$$\n\\end{proof}\n\nThe following Hamiltonian reduction construction of the closure of the minimal nilpotent orbit in $\\mathfrak{so}_{2m}$ is known. We repeat this construction to set up the notations.\nFix the natural basis $e_1,e_2\\in\\mathbb{C}^2$, and define a symplectic bilinear form $\\omega_1$ on $\\Hom(\\mathbb{C}^2,\\mathbb{C}^{2m})$ by\n\\begin{equation}\\label{o1}\n    \\omega_1(f_1,f_2)=(f_1(e_1),f_2(e_2))-(f_1(e_2),f_2(e_1)).\n\\end{equation}\n\nThe action of $\\mathrm{SL}_2$ on  $\\Hom(\\mathbb{C}^2,\\mathbb{C}^{2m})$ by precomposing is Hamiltonian with moment map\n\\begin{align*}\n    & \\mu_{\\mathrm{SL}_2}:\\Hom(\\mathbb{C}^2,\\mathbb{C}^{2m}) \\longrightarrow \\mathfrak{sl}_2^*\\cong\\mathfrak{sl}_2,\\\\\n    & \\mu_{\\mathrm{SL}_2}(f) =\n    \\renewcommand*{\\arraystretch}{1.5}\n    \\begin{pmatrix*}\n    (f(e_1),f(e_2)) && \\hspace{-1em}(f(e_1),f(e_1))\\\\\n    (f(e_2),f(e_2)) && \\hspace{-1em}-(f(e_1),f(e_2))\n    \\end{pmatrix*}.\n\\end{align*}\n\n\nSo the zero fiber of the $\\mathrm{SL}_2$-moment map on \\mbox{$\\Hom(\\mathbb{C}^2,\\mathbb{C}^{2m})$ is}\n    \\begin{equation}\n        N_1\\coloneqq\\mu_{\\mathrm{SL}_2}^{-1}(0)=\\{f\\in\\Hom(\\mathbb{C}^2,\\mathbb{C}^{2m})\\,|\\, f \\textrm{ is isotropic with respect to }(\\ ,\\ )\\}.\n    \\end{equation}\n\n\\pagebreak\nFor the reader's convenience we include the proof of the following result:\n\\begin{prop}\\label{ominhamil} The Hamiltonian reduction $N_1\\sslash\\mathrm{SL}_2$ is isomorphic to the closure $\\overline{\\mathcal{O}}_\\textrm{min}$ of the minimal orbit in $\\mathfrak{so}_{2m}$ as affine varieties.\n\\end{prop}\n\\begin{proof}\nRecall from Proposition \\ref{ominiso}, $$\\overline{\\mathcal{O}}_\\textrm{min}=\\{\\alpha\\in\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}\\,|\\, \\alpha\\textrm{ is decomposable and isotropic}\\}.$$\nDenote\n\t\\vspace{-1em}\n\t\\begin{align*}\n\t\t& \\left(\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}\\right)_\\textrm{dec}\\coloneqq\\{\\alpha\\in\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}\\,|\\, \\alpha\\mathrm{\\ is\\ decomposable}\\},\\\\\n\t\t& \\left(\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}\\right)_{\\textrm{dec},\\,\\textrm{iso}}\\coloneqq\\{\\alpha\\in\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}\\,|\\, \\alpha\\mathrm{\\ is\\ decomposable\\ and\\ isotropic}\\}.\n\t\\end{align*}\n\t\n\t\n\tBy the fundamental theorem of invariant theory for $\\mathrm{SL}_2$ (\\textrm{Theorem 2} in page 387 of \\cite{Procesi}), we have $$\\mathbb{C}[(\\mathbb{C}^2)^*\\otimes\\mathbb{C}^{2m}]^{\\mathrm{SL}_2}=\\mathbb{C}\\left[\\left(\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}\\right)_\\textrm{dec}\\right].$$ \n\tSince $\\mathrm{SL}_2$ is reductive and $N_1\\subset\\Hom(\\mathbb{C}^2,\\mathbb{C}^{2m})=(\\mathbb{C}^2)^*\\otimes\\mathbb{C}^{2m}$ is an $\\mathrm{SL}_2$-invariant sub-variety, the restriction from $(\\mathbb{C}^2)^*\\otimes\\mathbb{C}^{2m}$ to $N_1$ gives a surjective map\n\t\\begin{align}\\label{surj}\n\t\t\\mathbb{C}\\left[\\left(\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}\\right)_\\textrm{dec}\\right]\\quad& \\longrightarrow\\qquad\\qquad\\mathbb{C}[N_1]^{\\mathrm{SL}_2}\\\\\n\t\tf\\qquad \\qquad& \\longmapsto \\quad\\left(e_1^*\\otimes v_1+e_2^*\\otimes v_2\\mapsto f(v_1\\wedge v_2)\\right)\\nonumber\n\t\\end{align}\nBy Lemma \\ref{iso} we have the kernel of (\\ref{surj}) is precisely the ideal $\\mathcal{I}_\\textrm{iso}$ of functions on $\\left(\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}\\right)_\\textrm{dec}$ vanishing identically on the isotropic elements. Hence\n$$\n    \\mathbb{C}[N_1]^{\\mathrm{SL}_2}=\\mathbb{C}\\left[\\left(\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}\\right)_\\textrm{dec}\\right]\/\\mathcal{I}_\\textrm{iso}=\\mathbb{C}\\left[\\left(\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}\\right)_\\textrm{dec,\\,iso}\\right].\n$$\n\\end{proof}\t\n\n\n\\vspace{3em}\n\\subsection{The Lie algebra $\\mathfrak{so}_8$ as an $\\mathfrak{sl}_3$-representation}\nFix the following set of simple roots for the root system of type $D_4$.\n\\begin{align*}\n\t\\alpha_1& =(1,-1,0,0)\\\\\n\t\\alpha_2& =(0,1,-1,0)\\\\\n\t\\alpha_3& =(0,0,1,-1)\\\\\n\t\\alpha_4& =(0,0,1,1)\n\\end{align*}\nSo that the Dynkin Diagram $D_4$ is labeled as follows.\n    $$\n     \\dynkin[text style\/.style={scale=1},label,label macro\/.code={\\alpha_{\\drlap{#1}}},scale=1.2,edge\nlength=0.9cm]D4\n    $$\nChoose a Chevalley basis of $\\mathfrak{so}_8$ as in the \\hyperref[appendix]{Appendix}  $$\\{X_\\alpha\\}_{\\alpha\\in\\Delta^+},\\{Y_{-\\alpha}\\}_{\\alpha\\in\\Delta^+},\\{H_{\\alpha_i}\\}_{i=1,2,3,4}.$$\nChoose simple roots $\\alpha=(1,-1,0),\\ \\beta=(0,1,-1)$ of $\\mathfrak{sl}_3$. There exists a unique embedding of Lie algebras\n$$\n    \\varphi_1:\\mathfrak{sl}_3\\longhookrightarrow\\mathfrak{so}_8.\n$$\nwhich restricts to an embedding $\\varphi_1|_{\\mathfrak{h}_{\\mathfrak{sl}_3}}:\\mathfrak{h}_{\\mathfrak{sl}_3}\\hookrightarrow\\mathfrak{h}_{\\mathfrak{so}_8}$ such that\n$$\n\\varphi_1^*(\\alpha_2)=\\alpha,\\textrm{ and } \\varphi_1^*(\\alpha_1+\\alpha_2+\\alpha_3+\\alpha_4)=\\beta.\n$$\nSo\n$\\mathfrak{so}_8$ becomes an $\\mathfrak{sl}_3$-representation with respect to the adjoint action. \n\n\\begin{defn}\\label{F}\n    Define an isomorphism $\\eta:\\mathbb{C}^3\\oplus\\mathbb{C}\\oplus\\mathbb{C}^*\\oplus(\\mathbb{C}^3)^*\\longrightarrow\\mathbb{C}^8$ by the formula\n    \\begin{equation}\\label{eta}\n          \\eta(z_1,z_2,z_3,a,b,w_1,w_2,w_3)=z_1e_2+z_2e_3-z_3e_8+ae_4\n        +be_5-w_3e_1+w_2e_6+w_1e_7\n    \\end{equation}\n\\end{defn}\n\n\\begin{prop}\n    The map $\\eta$ induces an isomorphism of $\\mathfrak{sl}_3$-representations\n    $$\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\eta:\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2(\\mathbb{C}^3\\oplus\\mathbb{C}\\oplus\\mathbb{C}^*\\oplus(\\mathbb{C}^3)^*)\\longrightarrow\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2(\\mathbb{C}^8)=\\mathfrak{so}_8.$$  \n\\end{prop}\n\\begin{proof}\n    First, let $V_0\\coloneqq\\mathbb{C}^3\\oplus\\mathbb{C}\\oplus\\mathbb{C}^*\\oplus(\\mathbb{C}^3)^*$, and $<\\,,\\,>$ denote the inner product on $V_0$ given by the natural pairing in between $\\mathbb{C}^3\\oplus\\mathbb{C}$ and $\\mathbb{C}^*\\oplus(\\mathbb{C}^3)^*$. Then\n    $$\n        \\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2(\\mathbb{C}^3\\oplus\\mathbb{C}\\oplus\\mathbb{C}^*\\oplus(\\mathbb{C}^3)^*)=\\mathfrak{so}(V_0,<\\,,\\,>).\n    $$\n    Since $\\eta$ is an isometry between $(V_0,<\\,,\\,>)$ and $(\\mathbb{C}^8,(\\,,\\,))$, the map $\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\eta$ is a Lie algebra isomorphism between $\\mathfrak{so}(V_0,<\\,,\\,>)$ and $\\mathfrak{so}_8$.\n    We identify $$\\mathfrak{sl}_3=\\{A\\in\\mathbb{C}^3\\otimes(\\mathbb{C}^3)^*\\,|\\, \\mathrm{tr}(A)=0\\}\\subset\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2(\\mathbb{C}^3\\oplus\\mathbb{C}\\oplus\\mathbb{C}^*\\oplus(\\mathbb{C}^3)^*).$$\n    Then $\\mathfrak{sl}_3$ becomes an Lie subalgebra of $\\mathfrak{so}(V_0,<\\,,\\,>)$, and the $\\mathfrak{sl}_3$-representation structure on $\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2(\\mathbb{C}^3\\oplus\\mathbb{C}\\oplus\\mathbb{C}^*\\oplus(\\mathbb{C}^3)^*)$ is given by the adjoint action.\n    By an explicit calculation, we have the restriction of $\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\eta$ to $\\mathfrak{sl}_3$ equals to the embedding $\\varphi_1$. For example, for positive root vectors,\n    \\begin{align*}\n        \\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\eta\\,(e_1\\wedge e^*_2)& =e_2\\wedge e_6= X_{\\alpha_2},\\\\\n        \\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\eta\\,(e_2\\wedge e^*_3)& =e_3\\wedge (-e_1)= X_{\\alpha_1+\\alpha_2+\\alpha_3+\\alpha_4},\\\\\n        \\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\eta\\,(e_1\\wedge e^*_3)& =e_2\\wedge (-e_1)= X_{\\alpha_1+2\\alpha_2+\\alpha_3+\\alpha_4}.\n    \\end{align*}\n    Since the $\\mathfrak{sl}_3$-representation structure on $\\mathfrak{so}_8$ is given by the Lie algebra embedding $\\varphi_1$, our statement follows. \n\\end{proof}\n\nDecompose $\\mathfrak{so}_8=\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2(\\mathbb{C}^3\\oplus\\mathbb{C}\\oplus\\mathbb{C}^*\\oplus(\\mathbb{C}^3)^*)$ into irreducible $\\mathfrak{sl}_3$-representations, \n\\begin{equation}\\label{decomp}\n    \\mathfrak{so}_8=\\mathfrak{sl}_3\\oplus\\mathbb{C}_{\\textrm{trace}}\\oplus\\mathbb{C}\\oplus\\mathbb{C}^3\\oplus\\mathbb{C}^3\\oplus\\mathbb{C}^3\\oplus(\\mathbb{C}^3)^*\\oplus(\\mathbb{C}^3)^*\\oplus(\\mathbb{C}^3)^*.\n\\end{equation}\n\n\n\\begin{defns}\\label{decomp2}\nLet $\\mathfrak{h}=\\{(c_1,c_2,c_3)\\in\\mathbb{C}^3\\,|\\,c_1+c_2+c_3=0\\}$. Define $\\varphi_2:\\mathfrak{h}\\rightarrow\\mathfrak{so}_8$ by \n\\begin{align*}\n\t\\varphi_2(c_1,c_2,c_3)\\coloneqq\\ & c_1H_{\\alpha_1}+c_2H_{\\alpha_2}+c_3H_{\\alpha_3}\\\\\n\t=\\ & c_1(H_1-H_2)+c_2(H_3-H_4)+c_3(H_3+H_4)\\\\\n\t=\\ & -c_1(H_2+H_3-H_1)+(c_3-c_2)H_4\\in\\mathbb{C}_{\\textrm{trace}}\\oplus\\mathbb{C},\n\\end{align*}\nwhere $H_i\\coloneqq e_i\\wedge e_{9-i}$ for each $i$.\nSo the image of $\\varphi_2$ is precisely the sub-representation $\\mathbb{C}_{\\textrm{trace}}\\oplus\\mathbb{C}\\subset\\mathfrak{so}_8.$\n\n\n\nLet $V_1,V_2,V_3$ (resp.\\ $V^*_1,V^*_2,V^*_3$) be the three copies of $\\mathbb{C}^3$ (resp.\\ $(\\mathbb{C}^3)^*$) in the decomposition (\\ref{decomp}) whose highest weight vectors are root vectors for $\\mathfrak{so}_8$ corresponding to the roots\n$\\alpha_1+\\alpha_2, \\alpha_2+\\alpha_3,\\alpha_2+\\alpha_4$ (resp.\\ $\\alpha_2+\\alpha_3+\\alpha_4, \\alpha_1+\\alpha_2+\\alpha_4, \\alpha_1+\\alpha_2+\\alpha_3$). Identify\n $V_1,V_2,V_3$ (\\textrm{resp.\\ }$V^*_1,V^*_2,V^*_3$) via the unique $\\mathfrak{sl}_3$-equivariant linear isomorphisms which map the above choice of highest weight vectors to each other.\n Explicitly,\n\\begin{align*}\n& V_1=\\mathbb{C}\\langle -X_{\\alpha_1+\\alpha_2},X_{\\alpha_1},Y_{-\\alpha_2-\\alpha_3-\\alpha_4}\\rangle=\\mathbb{C}^3,\\\\\n& V_2=\\mathbb{C}\\langle X_{\\alpha_2+\\alpha_3},X_{\\alpha_3},Y_{-\\alpha_1-\\alpha_2-\\alpha_4}\\rangle=\\mathbb{C}^3,\\\\\n& V_3=\\mathbb{C}\\langle X_{\\alpha_2+\\alpha_4},X_{\\alpha_4},Y_{-\\alpha_1-\\alpha_2-\\alpha_3}\\rangle=\\mathbb{C}^3,\\\\\n& V_1^*=\\mathbb{C}\\langle X_{\\alpha_2+\\alpha_3+\\alpha_4},Y_{-\\alpha_1},-Y_{-\\alpha_1-\\alpha_2}\\rangle=(\\mathbb{C}^3)^*,\\\\\n& V_2^*=\\mathbb{C}\\langle X_{\\alpha_1+\\alpha_2+\\alpha_4},Y_{-\\alpha_3},Y_{-\\alpha_2-\\alpha_3}\\rangle=(\\mathbb{C}^3)^*,\\\\\n& V_3^*=\\mathbb{C}\\langle X_{\\alpha_1+\\alpha_2+\\alpha_3},Y_{-\\alpha_4},Y_{-\\alpha_2-\\alpha_4}\\rangle=(\\mathbb{C}^3)^*.\n\\end{align*}\nSo we have an identification\n\\begin{equation}\\label{phi}\n       \\varphi: \\mathfrak{sl}_3\\oplus\\mathfrak{h}\\oplus V_1\\oplus V_2\\oplus V_3\\oplus V^*_1\\oplus V^*_2\\oplus V^*_3\\longrightarrow\\mathfrak{so}_8\n\\end{equation}\n\\end{defns}\n\n\n\\begin{rem}\nFix the standard basis $e_1,e_2,e_3$ of $\\mathbb{C}^3$, and identify\n$$\n\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^3\\mathbb{C}^3=\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^3(\\mathbb{C}^3)^*=\\mathbb{C},\\ \\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^3=(\\mathbb{C}^3)^*,\\ \\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2(\\mathbb{C}^3)^*=\\mathbb{C}^3.\n$$\nFor any given $A\\in\\End(\\mathbb{C}^3)$ and $v\\in\\mathbb{C}^3$ we define $A\\wedge v\\in\\mathrm{Sym}^2(\\mathbb{C}^3)^*$ by\n$$\n    (A\\wedge v)(w_1,w_2)=(Aw_1)\\wedge v\\wedge w_2+(Aw_2)\\wedge v\\wedge w_1.\n$$\nSimilarly we can define $A\\wedge v^*\\in\\mathrm{Sym}^2\\mathbb{C}^3$ for $v^*\\in(\\mathbb{C}^3)^*$.\n\nLet $(M,{c},{u},{u}^*)$ be an element in $\\mathfrak{sl}_3\\oplus\\mathfrak{h}\\oplus(\\mathbb{C}^3)^3\\oplus((\\mathbb{C}^3)^*)^3$. Then \n$\\varphi(M,{c},{u},{u}^*)\\in\\overline{\\mathcal{O}}_\\textrm{min}$ if and only if:\n\\begin{align*}\n\t& u_i^*(u_j)=0\\textrm{ for all distinct $i,j\\in\\{1,2,3\\}$},\\\\\n\t& u^*_1(u_1)=(c_1-c_2)(c_1-c_3),\\textrm{ and its cyclic permutations},\\\\\n\t& u_1\\wedge u_2 = (c_1-c_2)u_3^*,\\textrm{ and its cyclic permutations},\\\\\n\t& u_1^*\\wedge u_2^* = (c_1-c_2)u_3,\\textrm{ and its cyclic permutations},\\\\\n\t& (M-c_3\\,\\mathrm{Id}_{\\mathbb{C}^3})\\wedge u_3+u_1^*\\cdot u_2^*=0, \\textrm{ and its cyclic permutations},\\\\\n\t& (M-c_3\\,\\mathrm{Id}_{\\mathbb{C}^3})\\wedge u_3^*+u_1\\cdot u_2=0, \\textrm{ and its cyclic permutations},\\\\\n\t& (M-c_3\\,\\mathrm{Id}_{\\mathbb{C}^3})^2+u_1\\otimes u_1^*+u_2\\otimes u_2^*-u_3\\otimes u_3^*+u_3^*(u_3)\\,\\mathrm{Id}_{\\mathbb{C}^3}=0, \\textrm{ and its cyclic permutations}.\n\\end{align*}\n\\end{rem}\n\n\n\\vspace{3em}\n\\subsection{Isomorphism between $\\overline{T^*(\\mathrm{SL}_3\/U)}$ and $\\overline{\\mathcal{O}}_\\textrm{min}\\subset\\mathfrak{so}_8$}\\label{sl3}\n\nIn the special case $n=3$, the construction of $N$ defined in section 2 consist of elements in\n\\begin{equation*}\n    \\begin{tikzcd}\n\t\\mathbb{C} \\arrow[r,shift left=.7ex,\"\\alpha_1\"]   \n\t& {\\mathbb{C}^2} \\arrow[l,shift left=.7ex,\"\\beta_1\"] \n\t\\arrow[r,shift left=.7ex,\"\\alpha_2\"] \n\t& \\mathbb{C}^3 \\arrow[l,shift left=.7ex,\"\\beta_2\"] \n\\end{tikzcd}\n\\vspace{-1em}\n\\end{equation*}\nsuch that\n$$(\\beta_2 \\alpha_2-\\alpha_1 \\beta_1)=\\lambda_2\\,\\mathrm{Id}_{\\mathbb{C}^2}\\text{ for some }\\lambda_2\\in\\mathbb{C}$$\nWe denote the scalar $\\beta_1\\alpha_1$ by $\\lambda_1$.\nBy \\cite{DKS} again, we have the affine closure $\\overline{T^*(\\mathrm{SL}_3\/U)}$ is identified with $N\\sslash \\mathrm{SL}_2$ as affine varieties.\nIn the special case $m=4$, Proposition \\ref{ominhamil} says that $\\overline{\\mathcal{O}}_\\textrm{min}=N_1\\sslash\\mathrm{SL}_2$ as affine varieties, where\n$$\n    N_1=\\{f\\in\\Hom(\\mathbb{C}^2,\\mathbb{C}^{8})\\,|\\, f \\textrm{ is isotropic with respect to }(\\ ,\\ )\\}.\n$$\n\n\n\tWe identify $\\mathbb{C}^8=(\\mathbb{C}^3\\oplus\\mathbb{C})\\oplus(\\mathbb{C}^*\\oplus(\\mathbb{C}^3)^*)=\\mathbb{C}^4\\oplus(\\mathbb{C}^4)^*$ via $\\eta$ in (\\ref{eta}). Then $\\eta$ is an isometry with respect to the inner product on $\\mathbb{C}^4\\oplus(\\mathbb{C}^4)^*$ given by the natural pairing of $\\mathbb{C}^4$ and $(\\mathbb{C}^4)^*$ and the inner product $(\\,,\\,)$ on $\\mathbb{C}^8$ defined by taking $m=4$ in (\\ref{inner}).\n\\begin{defn}\n    We define the map\n\\begin{align*}\n\tF:\n\tT^*V& \\longrightarrow \\Hom(\\mathbb{C}^2,\\mathbb{C}^8)=\\Hom(\\mathbb{C}^2,\\mathbb{C}^4\\oplus(\\mathbb{C}^4)^*)\\\\\n\t(\\alpha,\\beta) &\\longmapsto\\left(v\\mapsto (\\alpha_2\\oplus-\\beta_1)(v)\\oplus\\frac{(\\beta_2\\oplus\\alpha_1)\\wedge v}{e_1\\wedge e_2}\\right)\n\\end{align*}\n\\end{defn}\n\n\\begin{prop}\\label{Fthm}\n\tThe map $F$ is a $\\mathrm{SL}_2$-equivariant symplectic isomorphism between $T^*V$ and $(\\Hom(\\mathbb{C}^2,\\mathbb{C}^8),\\omega_1)$. \n\\end{prop}\n\\begin{proof}\n\tConsider the following $\\mathrm{SL}_2$-equivariant symplectic isomorphism.\n\tFirst, we have a symplectic isomorphism $$F_0:T^*V\\longrightarrow T^*(\\Hom(\\mathbb{C}^2,\\mathbb{C}^4)$$ which maps $(\\alpha,\\beta)$ to the following element in $\\Hom(\\mathbb{C}^2,\\mathbb{C}^4)\\oplus\\Hom(\\mathbb{C}^4,\\mathbb{C}^2)$:\n\t$$\n\t\\begin{tikzcd}[row sep=large, column sep = large]\n\t\t\\mathbb{C}^2 \\arrow[r,shift left=.7ex,\"\\alpha_2\\oplus-\\beta_1\"]   \n\t\t& {(\\mathbb{C}^3\\oplus\\mathbb{C})=\\mathbb{C}^4} \\arrow[l,shift left=.7ex,\"\\beta_2\\oplus\\alpha_1\"] \n\t\\end{tikzcd}.\n\t$$\n\t\n\tNext, with respect to the basis $e_1,e_2\\in\\mathbb{C}^2$ we decompose\n\t\\begin{align*}\n\t\t& T^*(\\Hom(\\mathbb{C}^2,\\mathbb{C}^4))=T^*(\\mathbb{C}^4\\oplus\\mathbb{C}^4)=(\\mathbb{C}^4\\oplus\\mathbb{C}^4)\\oplus((\\mathbb{C}^4)^*\\oplus(\\mathbb{C}^4)^*).\\\\\n\t\t& \\Hom(\\mathbb{C}^2,\\mathbb{C}^4\\oplus(\\mathbb{C}^4)^*)=(\\mathbb{C}^4\\oplus(\\mathbb{C}^4)^*)\\oplus(\\mathbb{C}^4\\oplus(\\mathbb{C}^4)^*).\n\t\\end{align*}\n\tWe define the map\n\t\\begin{align*}\n\t\tF_1:\\qquad T^*(\\Hom(\\mathbb{C}^2,\\mathbb{C}^4))\\ &\\longrightarrow\\ \\Hom(\\mathbb{C}^2,\\mathbb{C}^4\\oplus(\\mathbb{C}^4)^*)\\\\\n\t\t(v_1\\oplus v_2)\\oplus(v_1^*\\oplus v_2^*)\\ &\\longmapsto\\ (v_1\\oplus (-v_2^*))\\oplus(v_2\\oplus v_1^*).\n\t\\end{align*}\n\tAnd we check $F_1$ is a symplectic isomorphism:\n\t\\begin{align*}\n\t\t&\\ \\ \\ F_1^*\\omega_1\\big((v_1\\oplus v_2)\\oplus(v_1^*\\oplus v_2^*),(w_1\\oplus w_2)\\oplus(w_1^*\\oplus w_2^*)\\big)\\\\\n\t\t& = \\omega_1\\big((v_1\\oplus (-v_2^*))\\oplus(v_2\\oplus v_1^*),(w_1\\oplus (-w_2^*))\\oplus(w_2\\oplus w_1^*)\\big)\\\\\n\t\t& =\\big(\\!-\\!v_2^*(w_2)+w_1^*(v_1)\\big)-\\big(v_1^*(w_1)-w_2^*(v_2)\\big)\\\\\n\t\t& =(w_1^*\\oplus w_2^*)(v_1\\oplus v_2)-(v_1^*\\oplus v_2^*)(w_1\\oplus w_2),\n\t\\end{align*}\n\twhich is the natural symplectic form on $T^*(\\Hom(\\mathbb{C}^2,\\mathbb{C}^4))$.\n\tNotice that all three spaces $$T^*V,\\quad T^*(\\Hom(\\mathbb{C}^2,\\mathbb{C}^4)),\\quad \\Hom(\\mathbb{C}^2,\\mathbb{C}^4\\oplus(\\mathbb{C}^4)^*)$$ have natural Hamiltonian $\\mathrm{SL}_2$-actions. Since both $F_0$ and $F_1$ are $\\mathrm{SL}_2$-equivariant, $F$ is also an $\\mathrm{SL}_2$ equivariant symplectic isomorphism as\n\t$$\n\t\tF=F_1\\circ F_0.\n\t$$\n\\end{proof}\n\\begin{thm}\\label{TO}\n\tThe map $F$ induces an isomorphism between $\\overline{T^*(\\mathrm{SL}_3\/U)}$ and $\\overline{\\mathcal{O}}_\\textrm{min}$ as affine varieties.\n\\end{thm}\n\\begin{proof}\nThis is a direct corollary of Propositions \\ref{Fthm} and \\ref{ominhamil}.\n\\end{proof}\n\\begin{rem}\n    Let $P$ be the parabolic subgroup of $\\mathrm{SL}_n$ corresponding to the standard partial flag \n$$\\mathbb{C}^1 \\subset \\mathbb{C}^2 \\subset \\mathbb{C}^n.$$\nLet $[P,P]$ denote the commutator subgroup of $P$.\nThen I expect that one can use similar method to prove that the affinization $\\overline{T^*(\\mathrm{SL}_n\/[P,P])}$ is isomorphic to the closure of the minimal orbit in $so(2n+2)$.\n\\end{rem}\n\n\n\n\n\n\n\\vspace{3em}\n\\subsection{Symplectic form on the smooth locus}\\label{symplectic}\nLet $\\mathcal{O}_{\\textrm{nilp}}$ be a nilpotent adjoint orbit in $\\mathfrak{g}$.\nUsing Killing form $\\kappa$ we identify $\\mathcal{O}_{\\textrm{nilp}}$ with a nilpotent coadjoint orbit in $\\mathfrak{g}^*$. For any $Y\\in\\mathfrak{g}$, we denote $\\xi_Y$ to be the vector field on $\\mathcal{O}_{\\textrm{nilp}}$ generated by the infinitesimal action by $Y$.\nAnd the Kirillov-Kostant-Souriau symplectic strucutre $\\omega^\\textrm{KKS}$ on $\\mathcal{O}_{\\textrm{nilp}}$ is given by the formula:\n$$\n    \\omega^\\textrm{KKS}_{X}(\\xi_Y,\\xi_Z)=\\kappa(X,[Y,Z]).\n$$\nSince $\\mathcal{O}_{\\textrm{nilp}}$ is a nilpotent orbit, there is a $\\mathbb{C}^*$-action on it such that $\\omega^\\textrm{KKS}=\\mathcal{L}_{\\textsl{Eu}}\\omega^\\textrm{KKS},$ where $\\textsl{Eu}$ is the Euler vector field on $\\mathcal{O}_{\\textrm{nilp}}$. Define $$\\lambda^\\textrm{KKS}\\coloneqq\\iota_{\\textsl{Eu}}\\omega^\\textrm{KKS}.$$\nSo we have $$d\\lambda^\\textrm{KKS}=d\\iota_{\\textsl{Eu}}\\omega^\\textrm{KKS}=\\mathcal{L}_{\\textsl{Eu}}\\omega^\\textrm{KKS}=\\omega^\\textrm{KKS}.$$\nLet us recall the following known result.\n\\begin{prop}\n    Let $X\\in\\mathcal{O}_{\\textrm{nilp}}$ and $Y\\in\\mathfrak{g}$.  Then\n    $$\\lambda^\\textrm{KKS}_X(\\xi_Y)=\\kappa(X,Y).$$\n\\end{prop}\n\\begin{proof}\nSince $X$ is nilpotent, we have $Y\\in\\mathfrak{g}_X$ if and only if $\\kappa(X,Y)=0$. So the RHS of the formula only depends on $\\xi_Y$ (i.e. independent of the choice of $Y$). Let $E\\in\\mathcal{O}_{\\textrm{nilp}}$. Fix some $\\mathfrak{sl}_2$-triple $(E,H,F)$ in $\\mathfrak{g}$. Then we have\n$$\n    (\\xi_H)_E=2\\,\\textsl{Eu}_{E}\\in T_E\\mathcal{O}_{\\textrm{nilp}}.\n$$\nSince $\\lambda^\\textrm{KKS}$ is invariant under the $G$-action, it suffices to prove the statement for $X=E$. Let $Y\\in\\mathfrak{g}$. We compute\n$$\n    \\lambda^\\textrm{KKS}_E(\\xi_Y)=\\iota_{\\textsl{Eu}}\\omega_E^\\textrm{KKS}(\\xi_Y)=\\frac{1}{2}\\omega_E^\\textrm{KKS}(\\xi_Y,\\xi_H)=\\frac{1}{2}\\kappa(E,[Y,H])=\\frac{1}{2}\\kappa([H,E],Y)=\\kappa(E,Y).\n$$\n\\end{proof}\n\n\\begin{thm}\n\tThe symplectic form on $\\overline{T^*(\\mathrm{SL}_3\/U)}_\\textrm{sm}$ given in Proposition \\ref{symp} coincides (up to a scalar multiple) with $\\omega^\\textrm{KKS}$ on $\\mathcal{O}_{\\textrm{min}}\\subset\\mathfrak{so}_8$.\n\\end{thm}\n\\begin{proof}\n\tLet $v_1\\wedge v_2\\in \\mathcal{O}_{\\textrm{min}}\\subset\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2 {\\mathbb{C}^8}$.\n\tRecall the element $\\varphi_{v_1\\wedge v_2}\\in \\mathfrak{so}_8$ is defined in (\\ref{varphi}) by the formula\n\t$$\n\t\\varphi_{v_1\\wedge v_2}(u)=(v_1,u)v_2-(v_2,u)v_1.\n\t$$\n\tObserve that the tangent space $T_{v_1\\wedge v_2}\\mathcal{O}_{\\textrm{min}}$ is spanned by vectors given by infinitesimal actions of some $w_1\\wedge w_2\\in\\mathfrak{so}_8$. Let \n\t$$\n\t\\xi_{w_1\\wedge w_2}=[w_1\\wedge w_2,v_1\\wedge v_2]\n\t$$\n\tbe such a tangent vector.\n\tTo calculate $\\xi_{w_1\\wedge w_2}$ we first we compute\n\t\\begin{align*}\n\t\t(\\varphi_{v_1\\wedge v_2}\\circ \\varphi_{w_1\\wedge w_2})(u)\\ \\ =\\ \\ \\ &[(w_2,u)(v_2,w_1)-\\,(w_1,u)(v_2,w_2)] v_1\\\\\n\t\t+&[(w_1,u)(v_1,w_2)-\\,(w_2,u)(v_1,w_1)] v_2,\\\\\n\t\t(\\varphi_{w_1\\wedge w_2}\\circ \\varphi_{v_1\\wedge v_2})(u)\\ \\ =\\ \\ \\ &[(v_2,u)(w_2,v_1)-\\,(v_1,u)(w_2,v_2)] w_1\\\\\n\t\t+&[(v_1,u)(w_1,v_2)-\\,(v_2,u)(w_1,v_1)] w_2.\n\t\\end{align*}\n\tThen we compute the Lie bracket $[\\varphi_{v_1\\wedge v_2}, \\varphi_{w_1\\wedge w_2}]$ and get\n\t\\begin{align*}\n\t\t\\xi_{w_1\\wedge w_2}& =(v_2,w_2)\\, w_1\\wedge v_1-(v_1,w_2)\\, w_1\\wedge v_2-(v_2,w_1)\\, w_2\\wedge v_1+(v_1,w_1)\\, w_2\\wedge v_2\\\\\n\t\t& =\\varphi_{w_1\\wedge w_2}(v_1)\\wedge v_2+ v_1\\wedge \\varphi_{w_1\\wedge w_2}(v_2).\n\t\\end{align*}\n\tThe value of the one form $\\lambda^\\textrm{KKS}$ on $\\mathcal{O}_{\\textrm{min}}$ at $v_1\\wedge v_2$ is\n\t\\begin{align*}\n\t\t\\lambda^\\textrm{KKS}_{v_1\\wedge v_2}(\\xi_{w_1\\wedge w_2})=\\mathrm{Tr}(\\varphi_{v_1\\wedge v_2}\\circ \\varphi_{w_1\\wedge w_2})=2((v_2,w_1)(v_1,w_2)-(v_1,w_1)(v_2,w_2)).\n\t\\end{align*}\n\tRecall \\textrm{Proposition\\ \\ref{Fthm}}, the symplectic form on $\\overline{T^*(\\mathrm{SL}_3\/U)}_\\textrm{sm}$ given as in \\textrm{Proposition \\ref{symp}} coincides with the pull-back of the symplectic form on $(N_1\\sslash \\mathrm{SL}_2)_\\textrm{sm}$ by the induced isomorphism $$F:\\left(\\overline{T^*(\\SL_3\/U)}\\right)_\\textrm{sm}\\rightarrow \\mathcal{O}_{\\textrm{min}}.$$\n\tSo it suffices to show that the symplectic form on $(N_1\\sslash \\mathrm{SL}_2)_\\textrm{sm}$ coming from the Hamiltonian reduction coincides (up to a scalar multiple) with $\\omega^\\textrm{KKS}$. It suffices to check for one forms.\n\tWe know on $\\Hom(\\mathbb{C}^2,\\mathbb{C}^8)=\\mathbb{C}^8\\oplus\\mathbb{C}^8$, the one form\n\t$$\\lambda'_{v_1\\oplus v_2}(x_1\\oplus x_2)=(v_1,x_2)-(v_2,x_1)$$\n\tis $\\mathrm{SL}_2$-invariant. \n\tRecall from (\\ref{o1}), we have $\\omega_1=(1\/2)d\\lambda'$.\n\tLift the tangent vector $\\xi_{w_1\\wedge w_2}$ to $(x_1\\oplus x_2)\\in T_{v_1\\oplus v_2}(\\mathbb{C}^8\\oplus\\mathbb{C}^8)$, where\n\t\\begin{align*}\n\t\tx_1& =\\varphi_{w_1\\wedge w_2}(v_1)=(w_1,v_1)w_2-(w_2,v_1)w_1,\\\\\n\t\tx_2& =\\varphi_{w_1\\wedge w_2}(v_2)=(w_1,v_2)w_2-(w_2,v_2)w_1.\n\t\\end{align*}\n\tSo we have $$\\lambda'_{v_1\\oplus v_2}(x_1\\oplus x_2)=2((w_1,v_2)(v_1,w_2)-(w_2,v_2)(v_1,w_1))=\\lambda^\\textrm{KKS}_{v_1\\wedge v_2}(\\xi_{w_1\\wedge w_2}).$$\n\\end{proof}\n\n\n\n\n\n\n\n\\vspace{3em}\n\\section{The Gelfand-Graev action on $\\overline{T^*(\\mathrm{SL}_3\/U)}$.}\n\n\n\n\\subsection{The triality action on $\\mathfrak{so}_8$}\\label{triality}\n\nThe triality action was first discovered by E.\\ Cartan in his 1925 paper \\cite{Cartan} in which he constructed a $S_3$-action on $\\mathfrak{so}_8$ preserving the Lie bracket lifting the $S_3$-symmetry of the Dynkin Diagram $D_4$:\n    $$\n     \\dynkin[text style\/.style={scale=1},label,label macro\/.code={\\alpha_{\\drlap{#1}}},scale=1.2,edge\nlength=0.9cm]D4\n    $$\nThe action of an order three element of the triality action is constructed from the following matrix\n$$\n\\begingroup\n\\renewcommand*{\\arraystretch}{1.5}\n\\begin{pmatrix*}[r]\n    -\\frac{1}{2} && -\\frac{1}{2} && -\\frac{1}{2} && -\\frac{1}{2}\\\\\n    \\frac{1}{2} && \\frac{1}{2} && -\\frac{1}{2} && -\\frac{1}{2}\\\\\n    \\frac{1}{2} && -\\frac{1}{2} && \\frac{1}{2} && -\\frac{1}{2}\\\\\n    \\frac{1}{2} && -\\frac{1}{2} && -\\frac{1}{2} && \\frac{1}{2}\n\\end{pmatrix*}.\n\\endgroup\n$$\n\n\nGiven the Chevalley basis of $\\mathfrak{so}_8$ as in the \\hyperref[appendix]{Appendix}, each automorphism of the Dynkin diagram does lift uniquely to a Lie algebra automorphism (c.f. Theorem 2.108 in \\cite{Knapp}, and Lemma 2.6 in \\cite{EL}). But a priori it is not clear that the these lifts gives an $S_3\\hookrightarrow\\mathrm{Aut}(\\mathfrak{so}_8)$. For example, not every lift of the cyclic permutation of the simple roots $\\alpha_1,\\alpha_3,\\alpha_4$ has order three in $\\mathrm{Aut}(\\mathfrak{so}_8)$.\nIn the work of \\cite{MW}, a lifted triality action on the Lie algebra $\\mathfrak{so}(4,4)$ is constructed, but the authors proved their result by an explicit calculation in coordinates by computer.\n\nIn this section, a new interpretation of the triality action on $\\mathfrak{so}_8=\\mathfrak{so}(4,4)_\\mathbb{C}$ is given by applying the decomposation of $\\mathfrak{so}_8$ into irreducible $\\mathfrak{sl}_3$-representations as in (\\ref{decomp}).\nRecall the notations and definitions in Definition \\ref{decomp2}, we have an isomorphism (\\ref{phi}) of $\\mathfrak{sl}_3$-representations\n$$\n       \\varphi: \\mathfrak{sl}_3\\oplus\\mathfrak{h}\\oplus V_1\\oplus V_2\\oplus V_3\\oplus V^*_1\\oplus V^*_2\\oplus V^*_3\\longrightarrow\\mathfrak{so}_8.\n$$\n\n\\begin{defn}\n\nDefine an $S_3$-action on $\\mathfrak{sl}_3\\oplus\\mathfrak{h}\\oplus V_1\\oplus V_2\\oplus V_3\\oplus V^*_1\\oplus V^*_2\\oplus V^*_3$ by fixing $\\mathfrak{sl}_3$, acting on $\\mathfrak{h}$ as the Weyl group $S_3$-action, and permuting subscripts of $V_i$ and $V^*_i$. Then under the identification $\\varphi$, we have an $S_3$-action $\\mathsf{act}$ on $\\mathfrak{so}_8$.\n\\end{defn}\n\n\\begin{thm}\n    The $S_3$-action $\\mathsf{act}$ gives an embedding $S_3\\hookrightarrow \\mathrm{Aut}(\\mathfrak{so}_8)$ which coincides with the triality action.\n\\end{thm}\n\n\\begin{proof}\n\t\n\tLet $(M,{c},{u},{u}^*)$ be an element in $\\mathfrak{sl}_3\\oplus\\mathfrak{h}\\oplus(\\mathbb{C}^3)^3\\oplus((\\mathbb{C}^3)^*)^3$. Express each components in terms of the Chevalley basis, we see the $S_3$-action $\\mathsf{act}$ does permutes the root vectors accordingly (i.e. fixing $\\alpha_2$, and permutes $\\alpha_1,\\alpha_3,\\alpha_4$). So it suffices to show that this action preserves the Lie bracket of $\\mathfrak{so}_8$.\n\t\nThere is a linear action of $(M,{c},{u},{u}^*)$ on elements $(v,a,b,v^*)\\in\\mathbb{C}^3\\oplus\\mathbb{C}\\oplus\\mathbb{C}^*\\oplus(\\mathbb{C}^3)^*$ coming from the linear action of $\\mathfrak{so}_8$ on $\\mathbb{C}^8$ given by\n\t$$\n\t(M,c,u,u^*).\\begin{pmatrix}\n\t\tv\\\\\n\t\ta\\\\\n\t\tb\\\\\n\t\t\\ v^*\n\t\\end{pmatrix}=\n\t\\begin{pmatrix}\n\t\tM v+c_1v+au_2+bu_3+v^*\\wedge u_1^*\\\\\n\t\tu_2^*(v)+(c_3-c_2)a-v^*(u_3)\\\\\n\t\tu_3^*(v)-(c_3-c_2)b-v^*(u_2)\\\\\n\t\t-v^* M-c_1v^*-au_3^*-bu_2^*+v\\wedge u_1\n\t\\end{pmatrix}.\n\t$$\n\t\n\tWe calculate the Lie bracket structure on $\\mathfrak{sl}_3\\oplus\\mathfrak{h}\\oplus(\\mathbb{C}^3)^3\\oplus((\\mathbb{C}^3)^*)^3$.\n\tLet $A=(M_A,{c}_A,{u}_A,{u}^*_A)$ and $B=(M_B,c_B,u_B,u^*_B)$ be two elements in $\\mathfrak{sl}_3\\oplus\\mathfrak{h}\\oplus(\\mathbb{C}^3)^3\\oplus((\\mathbb{C}^*)^3)^3$. Then the Lie bracket can be written as $[A,B]=(M_C,c_C,u_C,u^*_C)$ where\n\t\\renewcommand{\\arraystretch}{1.5}\n\t\\begin{align*}\n\t\t& M_C=[M_A,M_B]+\\sum_{i=1}^{3}(u_{A,i}\\otimes u^*_{B,i}-u_{B,i}\\otimes u^*_{A,i})+\\frac{1}{3}\\sum_{i=1}^{3}(u^*_{A,i}(u_{B,i})-u^*_{B,i}(u_{A,i}))\\,\\mathrm{Id}_{\\mathbb{C}^3}\\\\\n\t\t& c_C=\n\t\t\\begin{pmatrix}\n\t\t\t-2\/3 && 1\/3 && 1\/3\\\\\n\t\t\t1\/3 && -2\/3 && 1\/3\\\\\n\t\t\t1\/3 && 1\/3 && -2\/3\n\t\t\\end{pmatrix}\n\t\t\\begin{pmatrix}\n\t\t\tu^*_{A,1}(u_{B,1})-u^*_{B,1}(u_{A,1}) \\\\\n\t\t\tu^*_{A,2}(u_{B,2})-u^*_{B,2}(u_{A,2}) \\\\\n\t\t\tu^*_{A,3}(u_{B,3})-u^*_{B,3}(u_{A,3})\n\t\t\\end{pmatrix}\\\\\n\t\t& u_C=\t\t\\begin{pmatrix}\n\t\t\tM_A\\,u_{A,1}-M_B\\,u_{B,1}+u^*_{A,2}\\wedge u^*_{B,3}-u^*_{B,2}\\wedge u^*_{A,3}+2c_{A,1}\\,u_{B,1}-2c_{B,1}\\,u_{A,1}\\\\\n\t\t\tM_A\\,u_{A,2}-M_B\\,u_{B,2}+u^*_{A,3}\\wedge u^*_{B,1}-u^*_{B,3}\\wedge u^*_{A,1}+2c_{A,2}\\,u_{B,2}-2c_{B,2}\\,u_{A,2}\\\\\n\t\t\tM_A\\,u_{A,3}-M_B\\,u_{B,3}+u^*_{A,1}\\wedge u^*_{B,2}-u^*_{B,1}\\wedge u^*_{A,2}+2c_{A,3}\\,u_{B,3}-2c_{B,3}\\,u_{A,3}\\\\\n\t\t\\end{pmatrix}\\\\\n\t\t& u^*_C=\t\t-\\begin{pmatrix}\n\t\t\tu^*_{A,1}M_A-u^*_{B,1}M_A+u_{A,2}\\wedge u_{B,3}-u_{B,2}\\wedge u_{A,3}+2c_{A,1}\\,u^*_{B,1}-2c_{B,1}\\,u^*_{A,1}\\\\\n\t\t\tu^*_{A,2}M_A-u^*_{B,2}M_A+u_{A,3}\\wedge u_{B,1}-u_{B,3}\\wedge u_{A,1}+2c_{A,2}\\,u^*_{B,2}-2c_{B,2}\\,u^*_{A,2}\\\\\n\t\t\tu^*_{A,3}M_A-u^*_{B,3}M_A+u_{A,1}\\wedge u_{B,2}-u_{B,1}\\wedge u_{A,2}+2c_{A,3}\\,u^*_{B,3}-2c_{B,3}\\,u^*_{A,3}\\\\\n\t\t\\end{pmatrix}\n\t\\end{align*}\n\tThen we see that the Lie bracket is manifestly equivariant under the lifted triality action. \n\\end{proof}\n\n\\begin{rem}\n\tThe Killing form can also be written in this fashion.\n\t$$\n\t\\frac{1}{2}\\,\\mathrm{Tr}(A B)=\\mathrm{Tr}(M_A M_B)+\\sum_{i=1}^{3}(u^*_{A,i}(u_{B,i})+u^*_{B,i}(u_{A,i}))+\\sum_{i=1}^{3}2\\,c_{A,i}\\,c_{B,i}\n\t$$\n\t\n\tThen the Cartan three form can be expressed as\n\t\\begin{align*}\n\t\t\\frac{1}{2}\\,\\mathrm{Tr}(A\\,[B,C])& =\\mathrm{Tr}(M_A\\,[M_B,M_C])+\\sum_{\\sigma\\in S_3}\\left(u^*_{A,\\sigma(1)}\\wedge u^*_{B,\\sigma(2)}\\wedge u^*_{C,\\sigma(3)}-u_{A,\\sigma(1)}\\wedge u_{B,\\sigma(2)}\\wedge u_{C,\\sigma(3)}\\right)\\\\\n\t\t&\\quad +\\sum_{\\textrm{Cyclic } A,B,C}\\left(2c_A\\cdot\\left(u^*_C(u_B)-u^*_B(u_C)\\right)+\\sum_{i=1}^{3}\\,\\left(u^*_{C,i}M_Au_{B,i}-u^*_{B,i}M_Au_{C,i}\\right)\\right).\n\t\\end{align*}\n\\end{rem}\n\n\n\n\n\n\\vspace{3em} \n\\subsection{The Gelfand-Graev action on $\\overline{T^*(\\SL_3\/U)}$}\\label{ggaction}\n\nRecall that $N$ is defined as elements in the following double quiver representation.\n\\[\n\\begin{tikzcd}\n\t0 \\arrow[r,shift left=.7ex,\"\\alpha_0\"]   \n\t& {\\mathbb{C}} \\arrow[r,shift left=.7ex,\"\\alpha_1\"]\n\t\\arrow[l,shift left=.7ex,\"\\beta_0\"] \n\t& {\\mathbb{C}^2} \\arrow[l,shift left=.7ex,\"\\beta_1\"] \n\t\\arrow[r,shift left=.7ex,\"\\alpha_2\"] \n\t& {\\cdots} \\arrow[l,shift left=.7ex,\"\\beta_2\"] \n\t\\arrow[r,shift left=.7ex,\"\\alpha_{n-2}\"] \n\t& {\\mathbb{C}^{n-1}} \\arrow[l,shift left=.7ex,\"\\beta_{n-2}\"] \n\t\\arrow[r,shift left=.7ex,\"\\alpha_{n-1}\"] \n\t& \\mathbb{C}^n \\arrow[l,shift left=.7ex,\"\\beta_{n-1}\"]\n\\end{tikzcd},\n\\]\nsuch that $\\forall k\\in\\{1,2,3,\\cdots,n-1\\}$\n\\begin{align*}\n\t\\beta_k\\alpha_k-\\alpha_{k-1}\\beta_{k-1}=\\lambda_k\\,\\textrm{Id}_{\\mathbb{C}^k}\n\\end{align*}\nfor some $\\lambda_k\\in\\mathbb{C}.$\n\n\n\\begin{defn}[Following \\cite{Wang} with slightly different convention]\\ \n\t\t\n\tLet $B=(\\alpha,\\beta)\\in N$.\n\tLet $k\\in\\{1,2,3,\\cdots,n-1\\}$. Define\n\t\\begin{align*}\n\t\t& \\mathrm{out}_k(B) \\coloneqq\\alpha_k\\oplus\\beta_{k-1}\\in\\Hom(\\mathbb{C}^k,\\mathbb{C}^{k+1}\\oplus\\mathbb{C}^{k-1}),\\\\\n\t\t& \\mathrm{in}_k(B) \\coloneqq\\beta_k\\oplus(-\\alpha_{k-1})\\in\\Hom(\\mathbb{C}^{k+1}\\oplus\\mathbb{C}^{k-1},\\mathbb{C}^k).\n\t\\end{align*}\nThen\n$$\n\t\\mathrm{in}_k(B)\\,\\mathrm{out}_k(B)=\\lambda_k\\,\\textrm{Id}_{\\mathbb{C}^k}.\n$$\n\tWe also define $Z_k$ to be a subvariety of $N\\times N$ consisting of pairs $$(B,B')=((\\alpha,\\beta),(\\alpha',\\beta')),$$ such that\\\\\n(1)\\quad $\\forall j\\notin\\{k-1,k\\},\\ \\alpha_j=\\alpha'_j,\\textrm{ and }\\beta_j=\\beta'_j.$\\\\\n(2)\\quad The following short sequence is exact\n$$\n\\begin{tikzcd}[column sep=large]\n\t0\\arrow[r]& \\mathbb{C}^k\\arrow[r,\"\\mathrm{out}_k(B')\"]& \\mathbb{C}^{k+1}\\oplus\\mathbb{C}^{k-1}\\arrow[r,\"\\mathrm{in}_k(B)\"]& \\mathbb{C}^k\\arrow[r] & 0\n\\end{tikzcd}\n$$\nMoreover, for each $j$ we fix some volume form $\\mathsf{vol}_j\\in\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^j\\mathbb{C}^j$, and require\n$$\n\t\\mathrm{out}_k(B')(\\mathsf{vol}_k)\\wedge\\mathrm{in}_k(B)^{-1}(\\mathsf{vol}_k)=\\mathsf{vol}_{k-1}\\wedge \\mathsf{vol}_{k+1}\\in\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^{2k}(\\mathbb{C}^{k+1}\\oplus\\mathbb{C}^{k-1}).\n$$\n(Since the short sequence is exact, we may choose any representative of $\\mathrm{in}_k(B)^{-1}(\\mathsf{vol}_k)$.)\\\\\n(3)\n$$\n\t\\mathrm{out}_k(B')\\,\\mathrm{in}_k(B')=\\mathrm{out}_k(B)\\,\\mathrm{in}_k(B)-\\lambda_k\\,\\textrm{Id}_{\\mathbb{C}^{k+1}\\oplus\\mathbb{C}^{k-1}}.\n$$\n\\end{defn}\n\n\\begin{defn}\n\t  As in Proposition-Definition 3.6.2 in \\cite{Wang}, for any simple reflection $s_k\\in W=S_n$, one constructs an automorphism $S_k$ on $N\\sslash H$, such that for all $(B,B')\\in Z_k$,\n\t  $$\n\t  \t\tS_k(\\pi(B))=\\pi(B').\n\t  $$\n\t  (Remark: for a given $B$, with $\\mathrm{in}(B)$ surjective, the choice of $B'$ such that $(B,B')\\in Z_k$ is unique up to an $\\mathrm{SL}_k$ action, see Lemma 3.5.3 in \\cite{Wang}, hence $\\pi(B')$ is well defined.\n\t  This property also uniquely determines $S_k$. Then in that paper, the author verified all the Coxeter relations, hence constructed a Weyl group action on $N\\sslash H$.)\n\\end{defn}\n\n\\begin{thm}[Theorem 4.7.1 in \\cite{Wang}]\n\tThe Weyl group action on $N\\sslash H$ we described above coincides with the Gelfand-Graev action on $\\overline{T^*(\\SL_3\/U)}$ constructed in \\cite{GK}, Theorem 5.2.7.\n\\end{thm}\n\n\\begin{prop}\n\tIn the case of $n=3$, and under the identification $$\\overline{T^*(\\SL_3\/U)}=\\overline{\\mathcal{O}}_\\textrm{min}\\subset\\mathfrak{so}_8,$$ the Gelfand-Graev action coincides with the triality $S_3$-action $\\mathsf{act}$ on $\\mathfrak{so}_8$ restricted on $\\overline{\\mathcal{O}}_\\textrm{min}$.\n\\end{prop}\n\n\\begin{proof}\n\tSince Lie algebra automorphism preserves the minimal orbit, so the triality action can be restricted to an $S_3$ action on $\\overline{\\mathcal{O}}_\\textrm{min}$. It suffices to check for two of the simple transpositions $s_1=(23),s_2=(13)\\in S_3$, the triality action satifies the condition that for each $k\\in\\{1,2\\}$, and $(B,B')\\in Z_k$, we have\n\t$$\n\t\tS_k(\\pi(B))=\\pi(B').\n\t$$\n\tSince the Gelfand-Graev action is uniquely determined by its restriction on the regular semisimple open part, we take $(M,{c},{u},{u}^*)$ to be an element in $\\mathcal{O}_{\\textrm{min}}\\subset\\mathfrak{sl}_3\\oplus\\mathfrak{h}\\oplus(\\mathbb{C}^3)^3\\oplus((\\mathbb{C}^3)^*)^3$ with distinct $c_1, c_2, c_3$.\n\tThen the corresponding isotropic decomposable element in $\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2 {\\left(\\mathbb{C}^3\\oplus\\mathbb{C}\\oplus\\mathbb{C}^*\\oplus(\\mathbb{C}^3)^*\\right)}$ is\n\t$$\n\t\n\t\t\\left(u_3\\oplus 0 \\oplus -(c_3-c_2)\\oplus -u_2^*\\right)\\wedge\\left(\\frac{u_2}{c_3-c_2}\\oplus 1 \\oplus 0 \\oplus -\\frac{u_3^*}{c_3-c_2}\\right).\n\t$$\n\tSo with this choice of representative, we have a lift $B=(\\alpha,\\beta)\\in N$\n$$\n\t\\alpha_1=\n\t\\begin{pmatrix}\n\t\t0\\\\\n\t\tc_3-c_2\n\t\\end{pmatrix},\\quad\n\t\\beta_1=\n\t\\begin{pmatrix}\n\t\t0 & 1\n\t\\end{pmatrix},\\quad\n\t\\alpha_2=\\hspace{-1.5em}\n\t\\vcenter{\\begin{tikzpicture}\\hspace{-1.5em}\n\t\t\t\\matrix (m)[matrix of math nodes, nodes in empty cells, left delimiter={(},right delimiter={)},minimum width=3.2em,minimum height=2.2em)]\n\t\t\t{\n\t\t\t\t\\   & \\    \\\\\n\t\t\t\tu_3 & \\!\\!\\!\\displaystyle\\frac{u_2}{c_3\\!-\\!c_2} \\!\\!\\!  \\\\\n\t\t\t\t\\   & \\     \\\\\n\t\t\t} ;\n\t\t\t\\draw (m-1-1.north west) rectangle (m-3-2.south east);\n\t\t\t\\draw (m-1-1.north east) -- (m-3-1.south east);\t\n\t\\end{tikzpicture}}\\hspace{-36em},\\quad\n\t\\beta_2=\n\\hspace{-1.5em}\n\\vcenter{\\begin{tikzpicture}\n\t\t\\matrix (m)[matrix of math nodes, nodes in empty cells, left delimiter={(},right delimiter={)},minimum width=3.6em,minimum height=2.75em)]\n\t\t{\n\t\t\t\\quad\\ \\displaystyle\\ \\ -\\frac{u_3^*}{c_3-c_2} \\qquad \\\\\n\t\t\t\\qquad\\ \\ \\ \\ u_2^* \\qquad\\ \\ \\ \\\\\n\t\t} ;\n\t\t\\draw (m-2-1.south west) rectangle (m-1-1.north east);\n\t\t\\draw (m-2-1.north west) -- (m-1-1.south east);\t\n\\end{tikzpicture}}\n\\hspace{-36em}\\qquad.\n$$\n\nFirst, for $k=1$, we have\n\t$$\n\t\\mathrm{out}_1(B)=\n\t\t\\begin{pmatrix}\n\t\t\t0\\\\\n\t\t\tc_3-c_2\n\t\t\\end{pmatrix},\\quad\n\t\\mathrm{in}_1(B)=\n\t\t\\begin{pmatrix}\n\t\t\t0 & 1\n\t\t\\end{pmatrix},\\quad\n\t\\lambda_1=c_3-c_2.\n\t$$\n\tApply the action by $s_1=(23)$, and rewrite the bivector so that the first defining property of $Z_1$ holds.\n\t\\begin{align*}\n\t\t&\\ \\ \\ \\left(u_2\\oplus 0 \\oplus -(c_2-c_3)\\oplus -u_3^*\\right)\\wedge\\left(\\frac{u_3}{c_2-c_3}\\oplus 1 \\oplus 0 \\oplus -\\frac{u_2^*}{c_2-c_3}\\right)\\\\\n\t\t& = \\left(u_3\\oplus (c_2-c_3) \\oplus 0\\oplus -u_2^*\\right)\\wedge\\left(\\frac{u_2}{c_3-c_2}\\oplus 0 \\oplus 1 \\oplus -\\frac{u_3^*}{c_1-c_2}\\right).\n\t\\end{align*}\nSo we have\n$$\n\t\\quad\\ \\mathrm{out}_1(B')=\n\t\\begin{pmatrix}\n\t\t1\\\\\n\t\t0\n\t\\end{pmatrix},\\quad\n\t\\mathrm{in}_1(B')=\\begin{pmatrix}\n\t\t\\,c_2\\!-\\!c_3 & 0\\,\n\t\\end{pmatrix}.\n$$\nThen $(B,B')\\in Z_1$. Hence the triality action of $s_1$ is the same as the Gelfand-Graev action by $S_1$.\n\nNext, for $k=2$, we have\n$$\n\\mathrm{out}_2(B)=\\hspace{-1.5em}\n\\vcenter{\\begin{tikzpicture}\n\t\t\\matrix (m)[matrix of math nodes, nodes in empty cells, left delimiter={(},right delimiter={)},minimum width=3.2em,minimum height=2em)]\n\t\t{\n\t\t\t\\   & \\    \\\\\n\t\t\tu_3 & \\!\\!\\!\\displaystyle\\frac{u_2}{c_3\\!-\\!c_2} \\!\\!\\!  \\\\\n\t\t\t\\   & \\     \\\\\n\t\t\t0 & \\!\\!\\!1  \\!\\!\\! \\\\\n\t\t} ;\n\t\t\\draw (m-1-1.north west) rectangle (m-4-2.south east);\n\t\t\\draw (m-4-1.north west) -- (m-4-2.north east);\n\t\t\\draw (m-1-1.north east) -- (m-4-1.south east);\t\n\\end{tikzpicture}}\t\n\\hspace{-36em},\\quad\\mathrm{in}_2(B)=\\hspace{-1.5em}\n\\vcenter{\\begin{tikzpicture}\n\t\t\\matrix (m)[matrix of math nodes, nodes in empty cells, left delimiter={(},right delimiter={)},minimum width=3.6em,minimum height=2.75em)]\n\t\t{\n\t\t\t\\quad\\ \\  \\displaystyle-\\frac{u_3^*}{c_3-c_2} \\qquad  & 0\\\\\n\t\t\t\\qquad\\qquad u_2^* \\ \\ \\ \\qquad\\ \\ \\ & c_2\\!-\\!c_3\\\\\n\t\t} ;\n\t\t\\draw (m-2-1.south west) rectangle (m-1-2.north east);\n\t\t\\draw (m-1-2.north west) -- (m-2-2.south west);\t\n\t\t\\draw (m-2-1.north west) -- (m-2-2.north east);\t\n\\end{tikzpicture}}\n\\hspace{-29.5em},\\quad\\lambda_2=c_1-c_3\\,.\n$$\nApplying the triality action by element $s_2=(13)\\in S_3$, we get $B''$ such that\n$$\n\\mathrm{out}_2(B'')=\\hspace{-1.5em}\n\\vcenter{\\begin{tikzpicture}\n\t\t\\matrix (m)[matrix of math nodes, nodes in empty cells, left delimiter={(},right delimiter={)},minimum width=3.2em,minimum height=2em)]\n\t\t{\n\t\t\t\\   & \\    \\\\\n\t\t\tu_1 & \\!\\!\\!\\displaystyle\\frac{u_2}{c_1\\!-\\!c_2} \\!\\!\\!  \\\\\n\t\t\t\\   & \\     \\\\\n\t\t\t0 & \\!\\!\\!1  \\!\\!\\! \\\\\n\t\t} ;\n\t\t\\draw (m-1-1.north west) rectangle (m-4-2.south east);\n\t\t\\draw (m-4-1.north west) -- (m-4-2.north east);\n\t\t\\draw (m-1-1.north east) -- (m-4-1.south east);\t\n\\end{tikzpicture}}\t\n\\hspace{-36 em},\\qquad\\mathrm{in}_2(B'')=\\hspace{-1.5em}\n\\vcenter{\\begin{tikzpicture}\n\t\t\\matrix (m)[matrix of math nodes, nodes in empty cells, left delimiter={(},right delimiter={)},minimum width=3.6em,minimum height=2.75em)]\n\t\t{\n\t\t\t\\quad\\ \\  \\displaystyle-\\frac{u_1^*}{c_1-c_2} \\qquad  & 0\\\\\n\t\t\t\\qquad\\qquad u_2^* \\ \\ \\ \\qquad\\ \\ \\ & c_2\\!-\\!c_1\\\\\n\t\t} ;\n\t\t\\draw (m-2-1.south west) rectangle (m-1-2.north east);\n\t\t\\draw (m-1-2.north west) -- (m-2-2.south west);\t\n\t\t\\draw (m-2-1.north west) -- (m-2-2.north east);\t\n\\end{tikzpicture}}\n\\hspace{-29.5em}.\n$$\nThen we can check\n$$\n\\mathrm{out}_2(B)\\,\\mathrm{in}_2(B) =\\hspace{-1.5em}\n\\vcenter{\\begin{tikzpicture}\n\t\t\\matrix (m)[matrix of math nodes, nodes in empty cells, left delimiter={(},right delimiter={)},minimum width=3.3em,minimum height=2.5em)]\n\t\t{\n\t\t\t\\ \\hspace{8em}\\ & \\    \\\\\n\t\t\t\\displaystyle\\frac{u_2\\wedge u^*_2-u_3\\wedge u_3^*}{c_3-c_2} & u_2   \\\\\n\t\t\t\\ \\hspace{8em}\\ & \\     \\\\\n\t\t\tu^*_2 & c_2\\!-\\!c_3 \\\\\n\t\t} ;\n\t\t\\draw (m-1-1.north west) rectangle (m-4-2.south east);\n\t\t\\draw (m-3-1.south west) -- (m-3-2.south east);\n\t\t\\draw (m-1-2.north west) -- (m-4-2.south west);\t\n\\end{tikzpicture}}\n\\hspace{-30em}=\\hspace{-1.2em}\\vcenter{\\begin{tikzpicture}\n\t\t\\matrix (m)[matrix of math nodes, nodes in empty cells, left delimiter={(},right delimiter={)},minimum width=3.3em,minimum height=2.5em)]\n\t\t{\n\t\t\t\\ \\hspace{7em}\\ & \\    \\\\\n\t\t\t-M+c_1\\mathrm{Id}_{\\mathbb{C}^3} & u_2   \\\\\n\t\t\t\\ \\hspace{7em}\\ & \\     \\\\\n\t\t\tu^*_2 & c_2\\!-\\!c_3 \\\\\n\t\t} ;\n\t\t\\draw (m-1-1.north west) rectangle (m-4-2.south east);\n\t\t\\draw (m-3-1.south west) -- (m-3-2.south east);\n\t\t\\draw (m-1-2.north west) -- (m-4-2.south west);\t\n\\end{tikzpicture}}\\hspace{-30.5em},\n$$\n\n$$\n\\mathrm{out}_2(B'')\\,\\mathrm{in}_2(B'') =\\hspace{-1.5em}\n\\vcenter{\\begin{tikzpicture}\n\t\t\\matrix (m)[matrix of math nodes, nodes in empty cells, left delimiter={(},right delimiter={)},minimum width=3.3em,minimum height=2.5em)]\n\t\t{\n\t\t\t\\ \\hspace{8em}\\ & \\    \\\\\n\t\t\t\\displaystyle\\frac{u_2\\wedge u^*_2-u_1\\wedge u_1^*}{c_1-c_2} & u_2   \\\\\n\t\t\t\\ \\hspace{8em}\\ & \\     \\\\\n\t\t\tu^*_2 & c_2\\!-\\!c_1 \\\\\n\t\t} ;\n\t\t\\draw (m-1-1.north west) rectangle (m-4-2.south east);\n\t\t\\draw (m-3-1.south west) -- (m-3-2.south east);\n\t\t\\draw (m-1-2.north west) -- (m-4-2.south west);\t\n\\end{tikzpicture}}\n\\hspace{-30em}=\\hspace{-1.2em}\\vcenter{\\begin{tikzpicture}\n\t\t\\matrix (m)[matrix of math nodes, nodes in empty cells, left delimiter={(},right delimiter={)},minimum width=3.3em,minimum height=2.5em)]\n\t\t{\n\t\t\t\\ \\hspace{7em}\\ & \\    \\\\\n\t\t\t-M+c_3\\mathrm{Id}_{\\mathbb{C}^3} & u_2   \\\\\n\t\t\t\\ \\hspace{7em}\\ & \\     \\\\\n\t\t\tu^*_2 & c_2\\!-\\!c_1 \\\\\n\t\t} ;\n\t\t\\draw (m-1-1.north west) rectangle (m-4-2.south east);\n\t\t\\draw (m-3-1.south west) -- (m-3-2.south east);\n\t\t\\draw (m-1-2.north west) -- (m-4-2.south west);\t\n\\end{tikzpicture}}\\hspace{-30.5em}.\n$$\n\n\nSo \n$$\n\\mathrm{out}(B'')\\,\\mathrm{in}(B'')=\\mathrm{out}(B)\\,\\mathrm{in}(B)-\\lambda_2\\,\\mathrm{Id}_{\\mathbb{C}^4}\n$$\nSo $(B,B'')\\in Z_2$.\nThis shows that the Gelfand-Graev action coincides with the triality action.\n\\end{proof}\n\n\n\n\n\\vspace{3em}\n\\section{A $\\mathbb{C}^*$-action on $\\overline{T^*(\\mathrm{SL}_n\/U)}$.}\\label{TDS}\n\nLet $\\mathfrak{b}$ (resp.\\ $\\mathfrak{n}$) be the Lie algebra of $B$ (resp.\\ $U$). Then $\\mathfrak{b}$ is the orthogonal complement of $\\mathfrak{n}$ with respect to the Killing Form, and the cotangent bundle $T^*(\\mathrm{SL}_n\/U)$ can be identified with\n$$\n\\mathrm{SL}_n\\times_U \\mathfrak{b}.\n$$\n\\begin{defn}[Definition 4.3 in \\cite{Wang}]\\label{Xi}\nLet $(\\alpha,\\beta)\\in N_\\textrm{surj}$ as in Definition \\ref{Nsurj}.\nFor each $k$, we choose some matrices of special forms,\n$\n    \\beta^0_k=(\\textbf{0}_{k,1}\\,|\\, \\mathrm{Id}_{\\mathbb{C}^k} )\\in\\Hom(\\mathbb{C}^{k+1},\\mathbb{C}^k).\n$\nChoose some\n$$\n    (g_1,g_2,g_3,\\cdots,g_{n-1},g_n)\\in\\mathrm{SL}_1\\times\\mathrm{SL}_2\\times\\cdots\\times\\mathrm{SL}_{n-1}\\times\\mathrm{SL}_n,\n$$\nsuch that for each $k\\in\\{1,2,3,4,\\cdots,n-1\\}$,\n$$\n    g_{k}\\beta'_kg_{k+1}^{-1}=\\beta^0_k.\n$$\nLet $\\Gamma:\\mathfrak{gl}_n\\rightarrow\\mathfrak{sl}_n$ denote the projection $\\Gamma(X)=X-(1\/n)\\mathrm{Tr}(X)\\mathrm{Id}_{\\mathbb{C}^n}$.\nThen as in equation (4.5) in \\cite{Wang}, we define the following map:\n\\begin{align}\n    \\Xi:N_\\textrm{surj}\/H & \\longrightarrow\\mathrm{SL}_n\\times_U \\mathfrak{b}\\\\\n    [(\\alpha,\\beta)]& \\longmapsto [g_n^{-1},\\Gamma(g_n\\alpha_{n-1}\\beta_{n-1}g_n^{-1})]\\nonumber\n\\end{align}\nThen the map $\\Xi$ is well defined and extends to an isomorphism\n$$\n    \\Xi:N\\sslash H\\rightarrow\\overline{T^*(\\SL_n\/U)}.\n$$\n\\end{defn}\n\nFor each $k\\geq2$, fix the following principal $\\mathfrak{sl}_2$-triple in $\\mathfrak{sl}_k$\n$$\n        e_k={\\begin{pmatrix}\n        0 & 1 \\\\\n        \\ & 0 & \\!1 \\\\\n        \\ & \\ & \\!\\ddots & 1 \\\\\n        \\ & \\ & \\ & 0 & 1 \\\\  \n        \\ & \\ & \\ & \\ & 0 \\\\ \n    \\end{pmatrix}},\\ \\ \n   h_k={\\begin{pmatrix}\n        {k\\!-\\!1} & \\hspace{-1em}0 \\\\\n         \\hspace{-.5em}0 & \\hspace{-1em}{k\\!-\\!3}\\\\\n        \\ & \\ & \\hspace{-1em}\\ddots\\\\\n        \\ & \\ & \\ & \\hspace{-2em}{-(k\\!-\\!3)} & \\hspace{-1em}0 \\\\  \n        \\ & \\ & \\ & \\hspace{-2em}0 & \\hspace{-2em}{-(k\\!-\\!1)} \\\\ \n    \\end{pmatrix}},\\ \\ \n  f_k={\\begin{pmatrix}\n        0 &\\\\\n        1(k\\!-\\!1) & \\hspace{-1.5em}0 \\\\\n        \\ & \\hspace{-2em}2(k\\!-\\!2) \\\\\n        \\ & \\ &\\hspace{-2.5em}\\ddots & \\hspace{-1em}0 & \\\\  \n        \\ & \\ & \\ & \\hspace{-2.5em}{(k\\!-\\!1)1} & 0 \\\\ \n    \\end{pmatrix}}.\n$$\n\n\\begin{defn}\n    Let $\\gamma:\\mathbb{C}^*\\rightarrow \\mathrm{SL}_n$ be the following one-parameter subgroup generated by the element $h_n$ of our fixed\n$\\mathfrak{sl}_2$-triple,\n$$\n    \\gamma(z)\\coloneqq z^{h_n}={\\begin{pmatrix}\n        z^{n\\!-\\!1} & 0 \\\\\n         0 & z^{n\\!-\\!3}\\\\\n        \\ & \\ & \\ddots\\\\\n        \\ & \\ & \\ & z^{-(n\\!-\\!3)} & 0 \\\\  \n        \\ & \\ & \\ & 0 & z^{-(n\\!-\\!1)}\n    \\end{pmatrix}}.\n$$\nLet $z\\in\\mathbb{C}^*, [g,X]\\in \\mathrm{SL}_n\\times_U\\mathfrak{b}$. Define a $\\mathbb{C}^*$-action on $T^*(\\mathrm{SL}_n\/U)$ by\n\\begin{equation}\\label{Cstar}\n        z.[g,X]=[g.\\gamma(z),z^2\\, \\mathrm{Ad}_{\\gamma(z^{-1})}X].\n\\end{equation}\n\\end{defn}\n\n\\begin{prop}\n    In terms of the construction of $\\overline{T^*(\\SL_n\/U)}$ in section \\ref{dq}, this $\\mathbb{C}^*$-action (\\ref{Cstar}) is given by \n    \\begin{equation}\n        z.\\bigoplus_{k}(\\alpha_k,\\beta_k)\\coloneqq\\bigoplus_{k}(z\\alpha_k,z\\beta_k)\n    \\end{equation}\n\\end{prop}\n\\begin{proof}\nLet $[g,X]\\in \\mathrm{SL}_n\\times_U\\mathfrak{b}$. Take a representative\n$$\n    (g,X)\\in\\mathrm{SL}_n\\times\\mathfrak{b}.\n$$\nWe are going to construct an $(\\alpha,\\beta)\\coloneqq\\bigoplus_{k=1}^{n-1}(\\alpha_k,\\beta_k)\\in N$ such that\n$$\\Xi(\\alpha,\\beta)=[g,X].$$ \nLet $\\iota:\\mathbb{C}^{n-1}\\hookrightarrow\\mathbb{C}^n$ be the embedding of $\\mathbb{C}^{n-1}$ into the last $(n-1)$ coordinates of $\\mathbb{C}^n$, i.e.\n$$\n    \\iota(z_1,z_2,\\cdots,z_{n-1})=(0,z_1,z_2,\\cdots,z_{n-1})\n$$\nDefine\n$$\n    \\alpha_{n-1}=g\\circ(X-x_{11}\\mathrm{Id}_{\\mathbb{C}^n})\\circ\\iota,\n$$\nwhere $x_{11}$ is the first row and first column entry of the matrix $X$.\nDefine\n$$\n    \\beta_{n-1}=\\beta^0_{n-1}\\circ g^{-1},\n$$\nand for each $k\\in\\{1,2,3,\\cdots,n-2\\}$, define\n$$\n    \\beta_k=\\beta^0_k.\n$$\nAnd all the linear maps $\\alpha_1,\\alpha_2,\\cdots,\\alpha_{n-2}$ are uniquely determined by $\\alpha_{n-1},\\beta_1,\\beta_2,\\cdots,\\beta_{n-1}$ and the condition (\\ref{N}). We check\n\\begin{align*}\n    \\Xi(\\alpha,\\beta)& =[g,\\Gamma(g^{-1}\\alpha_{n-1}\\beta_{n-1}g)]\\\\\n    & =[g,\\Gamma[g^{-1}g(X-x_{11}\\mathrm{Id}_{\\mathbb{C}^n})\\circ\\iota\\circ \\beta^0_{n-1}g^{-1}g]\\\\\n        & =[g,\\Gamma((X-x_{11}\\mathrm{Id}_{\\mathbb{C}^n})\\circ\\mathrm{diag}(0,1,1,\\cdots,1))]\\\\\n        & =[g,\\Gamma(X-x_{11}\\mathrm{Id}_{\\mathbb{C}^n})]=[g,X].\n\\end{align*}\n\nLet $z\\in\\mathbb{C}^*$, and define\n$$\n    (\\alpha',\\beta')=\\bigoplus_{k}(\\alpha'_k,\\beta'_k)\\coloneqq\\bigoplus_{k}(z\\alpha_k,z\\beta_k).\n$$\nTake\n$$\n    (g_1,g_2,g_3,\\cdots,g_{n-1},g_n)\\coloneqq(1,z^{-h_2},z^{-h_3},\\cdots,z^{-h_{n-1}},\\gamma(z)^{-1}g^{-1})\\in\\mathrm{SL}_1\\times\\mathrm{SL}_2\\times\\cdots\\times\\mathrm{SL}_{n-1}\\times\\mathrm{SL}_n.\n$$\nThen for each $k\\in\\{1,2,3,4,\\cdots,n-1\\}$\n$$\n    g_{k}\\beta'_kg_{k+1}^{-1}=\\beta^0_k.\n$$\nSo by Definition \\ref{Xi} again,\n$$\n        \\Xi(\\alpha',\\beta')=[g_n^{-1},\\Gamma(g_n\\alpha'_{n-1}\\beta'_{n-1}g_n^{-1})]=[g_n^{-1},z^2\\mathrm{Ad}_{\\gamma(z^{-1})}\\Gamma(g^{-1}\\alpha_{n-1}\\beta_{n-1}g)]=[g.\\gamma(z),z^2\\,\\mathrm{Ad}_{\\gamma(z^{-1})}X]\n$$\n\n\\end{proof}\nIn the special case of $n=3$, applying the identification $\\overline{T^*(\\SL_3\/U)}=\\overline{\\mathcal{O}}_\\textrm{min}$ in \\textrm{Corollary \\ref{TO}}, we have\n\\begin{cor}\n    The above $\\mathbb{C}^*$-action (\\ref{Cstar}) on $\\overline{T^*(\\SL_3\/U)}$ corresponds to the action of $z^2$ for the natural \\mbox{$\\mathbb{C}^*$-action on $\\overline{\\mathcal{O}}_\\textrm{min}$.}\n\\end{cor}\n\n\n\n\\section{Appendix: a Chevalley basis of $\\mathfrak{so}_8={\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2(\\mathbb{C}^8)}$.}\\label{appendix}\n\n\n\\begin{align*}\n\t& X_{\\alpha_1}=e_1\\wedge e_7 \n\t&& Y_{-\\alpha_1}=e_2\\wedge e_8\\\\\n\t& X_{\\alpha_2}=e_2\\wedge e_6\n\t&& Y_{-\\alpha_2}=e_3\\wedge e_7\\\\\n\t& X_{\\alpha_3}=e_3\\wedge e_5\n\t&& Y_{-\\alpha_3}=e_4\\wedge e_6\\\\\n\t& X_{\\alpha_4}=e_3\\wedge e_4\n\t&& Y_{-\\alpha_4}=e_5\\wedge e_6\\\\\n\t& X_{\\alpha_1+\\alpha_2}=e_1\\wedge e_6 \n\t&& Y_{-\\alpha_1-\\alpha_2}=e_3\\wedge e_8\\\\\n\t& X_{\\alpha_2+\\alpha_3}=e_2\\wedge e_5 \n\t&& Y_{-\\alpha_2-\\alpha_3}=e_4\\wedge e_7\\\\\n\t& X_{\\alpha_2+\\alpha_4}=e_2\\wedge e_4\n\t&& Y_{-\\alpha_2-\\alpha_4}=e_5\\wedge e_7\\\\\n\t& X_{\\alpha_2+\\alpha_3+\\alpha_4}=e_2\\wedge e_3\\hspace{-3em}\n\t&& Y_{-\\alpha_2-\\alpha_3-\\alpha_4}=e_6\\wedge e_7\\\\\n\t& X_{\\alpha_1+\\alpha_2+\\alpha_4}=e_1\\wedge e_4\\hspace{-3em} \n\t&& Y_{-\\alpha_1-\\alpha_2-\\alpha_4}=e_5\\wedge e_8\\\\\n\t& X_{\\alpha_1+\\alpha_2+\\alpha_3}=e_1\\wedge e_5\\hspace{-3em} \n\t&& Y_{-\\alpha_1-\\alpha_2-\\alpha_3}=e_4\\wedge e_8\\\\\n\t& X_{\\alpha_1+\\alpha_2+\\alpha_3+\\alpha_4}=e_1\\wedge e_3\\hspace{-3em}\n\t&& Y_{-\\alpha_1-\\alpha_2-\\alpha_3-\\alpha_4}=e_6\\wedge e_8\\\\\n\t& X_{\\alpha_1+2\\alpha_2+\\alpha_3+\\alpha_4}=e_1\\wedge e_2\\hspace{-3em}\n\t&& Y_{-\\alpha_1-2\\alpha_2-\\alpha_3-\\alpha_4}=e_7\\wedge e_8\\\\\n\t& H_{\\alpha_1}=e_1\\wedge e_8-e_2\\wedge e_7\\hspace{-3em}\n\t&& H_{\\alpha_3}=e_3\\wedge e_6-e_4\\wedge e_5\\\\\n    &  H_{\\alpha_2}=e_2\\wedge e_7-e_3\\wedge e_6\\hspace{-3em}\n    && H_{\\alpha_4}=e_3\\wedge e_6+e_4\\wedge e_5\n\\end{align*}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\vspace{2em}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nA covering of a topological space $X$ is said to be \\emph{categorical} if every set in the covering is open and contractible in $X$.\nThat is, the inclusion map of each set into $X$ is nullhomotopic.\nThe \\emph{Lusternik-Schnirelmann category} (or simply \\emph{category}) $\\cat(X)$ of $X$ is the smallest integer $k$\nsuch that $X$ admits a categorical covering by $k+1$ open sets $\\{U_0,\\ldots,U_k\\}$. \n\nLusternik-Schnirelmann category and related invariants have been computed for polyhedral products of the form $(X,\\ast)^K$ \nfor certain nice spaces $X$ in~\\cite{MR2457428,2015arXiv150107474G}, \nthough the methods there do not extend to moment-angle complexes $\\mathcal Z_K=(D^2,\\ensuremath{\\partial} D^2)^K$ for several reasons.\nFor example, the contractibility of the disk $D^2$ does not give nice lower bounds in terms of the dimension of $K$. \nIndeed, there is a fairly large literature that is focused on determining those $K$ for which the moment-angle complex $\\mathcal Z_K$ has category $1$ \n(c.f.~\\cite{2014arXiv1409.4462B,2015arXiv151005283B,MR3461047,MR3084441,MR2321037,2014arXiv1412.4866I,2013arXiv1306.6221I,BerglundRational}).\n\nWe will mostly be interested in moment-angle complexes $\\mathcal Z_K$ over triangulated spheres $K$.\nThese are known as \\emph{moment-angle manifolds} since it is here that $\\mathcal Z_K$ takes the form of a topological manifold. \nMoment-angle complexes of this form have generated a lot of interest due to their connections to quasitoric manifolds in toric topology \nand intersection of quadrics in complex geometry, amongst other things.\nTheir topology and cohomology is, however, very intricate, with many questions remaining open even for low dimensional $K$ \n(see for example~\\cite{MR3363157,MR2330154,MR3073929,MR1104531,MR531977,MR2285318,2015arXiv150904302L,2015arXiv151007778L}). \nNone-the-less, we will characterize those triangulations of $d$-spheres where $d\\leq 2$ for which $\\mathcal Z_K$ has a given category,\nas well as certain higher dimensional spheres built up via join, connected sum, and vertex doubling operations.\n\nOur motivation for doing this is a combinatorial and algebraic characterization of Golod complexes $K$ and $co$-$H$-space \n(category $\\leq 1$) moment-angle complexes $\\mathcal Z_K$ given in~\\cite{MR3461047} in the case of flag complexes $K$.\nThe authors there showed that both of these concepts are equivalent, and moreover, that they both coincide with chordality \nof the $1$-skeleton of $K$ and the triviality of the multiplication on $\\mathrm{Tor}^+_{R[v_1,\\ldots,v_n]}(R[K],R)$\nfor $R=\\mb Z$ or $R$ any field, where $R[K]$ is the Stanley-Reisner ring of $K$ over $R$.\nAn interesting algebraic consequence of this from the perspective of commutative algebra was that triviality of the multiplication on \n$\\mathrm{Tor}^+_{R[v_1,\\ldots,v_n]}(R[K],R)$ implies that all higher Massey products are also trivial, at the very least when $K$ is flag.\n\\footnote{A theorem in~\\cite{MR2344344} claims this is true for all $K$, but a recent paper~\\cite{2015arXiv151104883K} provides a simple counter-example.}  \nThis depended on the general fact that the cohomology ring of a space of category $\\leq 1$ has trivial multiplication and Massey products vanish~\\cite{MR0231375,MR0356037}.\nIt is natural to ask what the corresponding statement is for spaces with larger category, \nmore so, if the characterization for Golod flag complexes in~\\cite{MR3461047} can be generalized in this sense. \nAn answer to the first question was given by Rudyak in~\\cite{MR1644063}, which inspires the following.\n\\begin{definition}\nA simplicial complex $K$ on vertex set $[n]$ is $m$-\\emph{Golod} over $R$ if\n\\begin{itemize}\n\\item[(1)] $\\Nill(\\mathrm{Tor}_{R[v_1,\\ldots,v_n]}(R[K],R))\\leq m+1$;\n\\item[(2)] Massey products $\\vbr{v_1,\\ldots,v_k}$ vanish in $\\mathrm{Tor}^+_{R[v_1,\\ldots,v_n]}(R[K],R)$ \nwhenever $v_i=a_1\\cdots a_{m_i}$ and $v_j=b_1\\cdots b_{m_j}$, and $m_i+m_j>m$ for some odd $i$ and even $j$ and\n$a_s,b_t\\in\\mathrm{Tor}^+_{R[v_1,\\ldots,v_n]}(R[K],R)$.\n\\end{itemize}\n\\end{definition}\n\n\\begin{proposition}\nIf $\\cat(\\mathcal Z_K)\\leq m$, then $K$ is $m$-Golod.~$\\hfill\\square$ \n\\end{proposition}\n\nHere the \\emph{nilpotency} $\\Nill A$ of a graded algebra $A$ is the smallest integer $k$ such that all length $k$ products in the \npositive degree part $A^+$ vanish. \nNotice that $K$ is $(m+1)$-Golod whenever it is $m$-Golod, and $1$-Golodness of $K$ coincides with the classical notion of Golodness~\\cite{MR0138667},\nnamely, that all products and (higher) Massey products are trivial in $\\mathrm{Tor}^+_{R[v_1,\\ldots,v_n]}(R[K],R)$. \nAll of this can be restated equivalently in terms of the cohomology of $\\mathcal Z_K$ due to an isomorphism of graded commutative algebras\n$H^*(\\mathcal Z_K;R)\\cong\\mathrm{Tor}_{R[v_1,\\ldots,v_n]}(R[K],R)$ when $R$ is a field or $\\ensuremath{\\mathbb{Z}}$~\\cite{MR2117435}.\nWe will consider (co)homology with integer coefficients. \n\n\\begin{theorem}\n\\label{TM0}\nIf $K$ is $k$-neighbourly, $\\cat(\\mathcal Z_K)\\leq \\frac{1+\\dim K}{k}$ and $K$ is $\\paren{\\frac{1+\\dim K}{k}}$-Golod.~$\\hfill\\square$\n\\end{theorem}\n\n\n\\begin{theorem}\n\\label{TM1}\nIf $K$ is any $d$-sphere for $d\\leq 2$, or (under a few conditions) built up as a connected sum of joins of such spheres,\nthen the following are equivalent: (a) $\\cat(\\mathcal Z_K)\\leq k$; (b) $K$ is $k$-Golod; \n(c) length $k+1$ cup products of positive degree elements in $H^*(\\mathcal Z_K)$ vanish;\n(d) there does not exist a spherical filtration of full subcomplexes of $K$ of length more than $k$. Moreover, $k\\leq d+1$.~$\\hfill\\square$\n\\end{theorem}\n\nFor instance, $k$ is precisely $3$ when $K$ is the boundary of the dual of a \\emph{fullerene} $P$. \n\n\\begin{theorem}\nFor fullerences $P$, $\\cat(\\z{\\ensuremath{\\partial} P^\\ast})=3$ and $\\ensuremath{\\partial} P^\\ast$ is $3$-Golod.\nIn particular, all Massey products consisting of decomposable elements in $H^+(\\z{\\ensuremath{\\partial} P^\\ast})$ must vanish.~$\\hfill\\square$\n\\end{theorem}\n\nApplying \\emph{vertex doubling operations}, the range of spheres in Theorem~\\ref{TM1} can be extended using the following.\n\n\n\\begin{theorem}\n\\label{TM2}\nIf $K(J)$ is the simplicial wedge of $K$ for some integer sequence $J=(j_1,\\ldots,j_n)$, then $\\cat(\\z{K(J)})\\leq \\cat(\\mathcal Z_K)$.~$\\hfill\\square$\n\\end{theorem}\n\nMore general bounds for $\\cat(\\mathcal Z_K)$ will be given along the way,\nfor example, when $K$ is formed by gluing simplicial complexes along common full subcomplexes.  \nWe remark that many of the results in this paper extend to polyhedral products of the form $(Cone(X),X)^K$ in place of $\\mathcal Z_K=(D^2,S^1)^K$.\n\n\n\\section{Preliminary}\n\nRecall the following concepts from~\\cite{MR516214,MR1990857,MR0004108,MR0229240}. \nThe \\emph{geometric category} $\\gcat(X)$ of a space $X$ is the smallest integer $k$ such that $X$ admits\na categorical covering $\\{U_0,\\ldots,U_k\\}$ of $X$ with each $U_i$ contractible (in itself),\nand the \\emph{category} $\\cat(f)$ of a map $f\\colon\\seqm{X}{}{Y}$ is the smallest $k$ such that $X$ admits\nan open covering $\\{V_0,\\ldots,V_k\\}$ such that $f$ restricts to a nullhomotopic map on each $V_i$.\nIt is easy to see that $\\cat(X)=\\cat(\\ensuremath{\\mathbbm{1}}\\colon\\seqm{X}{}{X})$, $\\cat(f)\\leq\\min\\{\\cat(X),\\cat(Y)\\}$, $\\cat(h\\circ h')\\leq \\cat(h')$.\nFor path-connected paracompact spaces,\n$$\n\\cat(f\\times g)\\leq \\cat(f)+\\cat(g),\n$$ \nwhich follows from the also well-known fact that $\\cat(X\\times Y)\\leq \\cat(X)+\\cat(Y)$ together with the preceding inequalities.\nUnlike $\\cat()$, $\\gcat()$ is not a homotopy invariant,\nthough one can obtain a homotopy invariant from $\\gcat()$ by defining the \\emph{strong category}\n$$\n\\Cat(X)=\\min\\cset{\\gcat(Y)}{Y\\simeq X}.\n$$\nIn fact, the strong category satisfies $\\Cat(X)-1\\leq \\cat(X)\\leq\\Cat(X)\\leq \\gcat(X)$.\nWe shall let $\\cuplen(X)=\\Nill H^*(X)-1$ denote the length of the longest non-zero cup product of positive degree elements in $H^*(X)$. \nThe main use of this is the classical lower bound \n$$\n\\cuplen(X)\\leq \\cat(X). \n$$\n\n\n\n\\subsection{Some General Bounds}\n\nWe begin by giving upper bounds for the Lusternik-Schnirelmann category over some general spaces.\n\n\n\\begin{lemma}\n\\label{LOpen}\nLet $A$ be a subcomplex of $X$ and $S$ an open subset of $A$. \nThen $S$ is a deformation retract of an open subset $U$ of $X$ such that $U\\cap A=S$.\n\\end{lemma}\n\n\\begin{proof}\n\nLet $\\mc I_j$ be an index set for the $j$-cells $e_\\alpha^j$ of $X-A$, $\\Phi_\\alpha\\colon\\seqm{D^j}{}{X}$ its characteristic map,\nand $\\phi_\\alpha\\colon\\ensuremath{\\partial} D^j\\incladdr{}{D^j}\\mapaddr{\\Phi_\\alpha}{X}$ its attaching map. \nGiven a subset $B\\subseteq X$, let $V_{\\alpha,B}$ be the image of \n$\\phi_\\alpha^{-1}(B)\\times[0,\\frac{1}{2})\\subseteq D^j\\cong (\\ensuremath{\\partial} D^j\\times [0,1])\/(\\ensuremath{\\partial} D^j\\times\\{1\\})$ under $\\Phi_\\alpha$.\nNotice $V_{\\alpha,B}$ deformation retracts onto a subspace of $\\phi_\\alpha(\\ensuremath{\\partial} D^j)\\cap B$,\nand if $B\\cap e_\\alpha^j=\\emptyset$, $B\\cup V_{\\alpha,B}$ deformation retracts onto $B$.\n\nConstruct $R_{i+1}\\subseteq X$ such that $R_i\\subseteq R_{i+1}$, $R_i$ is a deformation retract of $R_{i+1}$, \nand $R_i\\cap X^{\\vbr{i}}$ is open in the $i$-skeleton $X^{\\vbr{i}}$, \nby letting $R_0=S$ and $R_{i+1}=R_i\\cup \\bigcup_{\\alpha\\in\\mc I_{i+1}} V_{\\alpha,S}$.\nThen $U=\\bigcup_{i\\geq 0} R_i$ is open in $X$, deformation retracts onto $S$, and $U\\cap A=S$. \n\n\\end{proof}\n\n\n\n\\begin{lemma}\n\\label{LFiltration}\n\nGiven a filtration $X_0\\subseteq\\cdots\\subseteq X_m=X$ of subcomplexes of a $CW$-complex $X$,\nsuppose $X_{i+1}-X_i$ is contractible in $X$ for each $i$. Then $\\cat(X)\\leq \\cat(\\incl{X_0}{}{X})+m\\leq \\cat(X_0)+m$.\n\n\\end{lemma}\n\n\\begin{proof}\n\nLet $k=\\cat(\\incl{X_0}{}{X})$, and $\\{U_0,\\ldots, U_k\\}$ be a categorical cover of the inclusion \\incl{X_0}{}{X}.\nNote $V_i=X_{i+1}-X_i$ is open in $X_{i+1}$ since the subcomplex $X_i$ is closed in $X_{i+1}$. \nThen iterating Lemma~\\ref{LOpen}, we have open subsets $\\bar U_i$ and $\\bar V_i$ that deformation retract onto\neach $U_i$ and $V_i$ respectively. Since the $U_i$'s and $V_i$'s cover $X$ and are contractible in $X$,\nso do the $\\bar U_i$'s and $\\bar V_i$'s, thus they form a categorical cover of $X$.\n\n\\end{proof}\n\n\n\nFor any spaces $X$ and $Y$, and a fixed basepoint $\\ast\\in X$, \nwe let $X\\rtimes Y=(X\\times Y)\/(\\ast\\times Y)$ denote the \\emph{right half-smash} of $X$ and $Y$,\nand $Y\\ltimes X=(Y\\times X)\/(Y\\times\\ast)$ the \\emph{left half-smash}.\n\n\\begin{lemma}\n\\label{LHalfSmash}\nIf $X$ and $Y$ are $CW$-complexes and $X$ is path-connected, then $\\cat(X\\rtimes Y)=\\cat(X)$. \n\\end{lemma}\n\n\\begin{proof}\n\nLet  $\\tilde X$ be given by attaching the interval $[0,1]$ to $X$ \nby identifying $0\\in [0,1]$ with the basepoint $\\ast\\in X$, and fix $1\\in\\tilde X$ to be the basepoint.\nGiven $k=\\cat(\\tilde X)$ and $\\{U_0,\\ldots,U_k\\}$ a categorical cover of $\\tilde X$,\ntake the open cover $\\{U_0\\rtimes Y,\\ldots, U_k\\rtimes Y\\}$ of $\\tilde X\\rtimes Y=(\\tilde X\\times Y)\/(1\\times Y)$\n(here $U_i\\rtimes Y=U_i\\times Y$ if $1\\nin U_i$).\nNotice that the contractions of each $U_i$ in $\\tilde X$ can be taken so that $1$ remains fixed if $1\\in U_i$.\nIf $U_i$ contracts to a point $b_i$ in $\\tilde X$, $U_i\\rtimes Y$ deforms onto $\\{b_i\\}\\rtimes Y$ in $\\tilde X\\rtimes Y$,\nwhich in turn contracts to the basepoint in $\\tilde X\\rtimes Y$ by homotoping the coordinate $b_i$ to $1$.\nTherefore $\\cat(\\tilde X\\rtimes Y)\\leq k$, and we have $\\cat(X\\rtimes Y)\\leq k$ since $X\\simeq\\tilde X$ \nand $X\\rtimes Y\\simeq \\tilde X\\rtimes Y$. Moreover, $\\cat(X\\rtimes Y)\\geq k$ since $X$ is a retract of $X\\rtimes Y$.\n\n\\end{proof}\n\n\nLet $\\mc S$ be $m$ copies of the interval $[0,1]$ glued together at the endpoints $1$ in some order.\nGiven a collection of maps \\seqm{X}{f_i}{Y_i} for $i=1,\\ldots,m$, the \\emph{homotopy pushout} $P$ of the maps $f_i$ \nis the $m$-fold mapping cylinder \n$$\nP=\\paren{Y_1\\coprod\\cdots\\coprod Y_m\\coprod (X\\times\\mc S)}\/\\sim\n$$ \nunder the identification $(x,t)\\sim f_i(x)$ whenever $t$ is in the $i^{th}$ copy of $[0,1]$ in $\\mc S$ and $t=0$.  \n\n\n\n\n\\begin{lemma}\n\\label{LGT}\nFix $m\\geq 2$. For $i=1,\\ldots, m$, \nlet $A_i$ and $C_i$ be basepointed $CW$-complexes, $B_i=\\prod_{j\\neq i} A_j$, and $E$ be a contractible space.\nSuppose \\seqm{A_i\\times E}{f_i}{C_i} are nullhomotopic maps,\nand $P$ is the homotopy pushout of the maps \n\\seqm{A_i\\times E\\times B_i}{f_i\\times\\ensuremath{\\mathbbm{1}}_{B_i}}{C_i\\times B_i} for $i=1,\\ldots, m$.\nThen \n$$\n\\cat(P)\\leq \\max\\{1,\\cat(C_1),\\ldots,\\cat(C_m)\\}. \n$$\n\\end{lemma}\n\n\n\\begin{proof}\n\nWe proceed by induction on $m$. Start with $m=2$. By Lemma~$7.1$ in~\\cite{MR3084441}, there is a splitting \n$P\\simeq (\\Sigma A_1\\wedge A_2)\\vee (C_1\\rtimes A_2)\\vee(C_2\\rtimes A_1)$. Thus using Lemma~\\ref{LHalfSmash}, \n$$\n\\cat(P)=\\max\\{\\cat(\\Sigma A_1\\wedge A_2),\\cat(C_1\\rtimes A_2),\\cat(C_2\\rtimes A_1)\\}=\\max\\{1,\\cat(C_1),\\cat(C_2)\\}.\n$$\nThe statement holds when $m=2$.\n\n\nTake $\\mc B_0=\\ast$, $\\mc B_\\ell=\\prod_{j\\leq \\ell} A_j$, $B'_i=\\prod_{j\\neq i,j<m} A_j$, and $B_i$ as basepointed subspaces of $\\mc B=\\prod_j A_j$.\nLet $P'$ be the homotopy pushout of $f_i\\times\\ensuremath{\\mathbbm{1}}_{B'_i}$ for $i=1,\\ldots, m-1$ (these are all maps from $E\\times B_m=E\\times \\mc B_{m-1}$).\nSuppose the lemma holds whenever $m<m'$ for some $m'>2$. Let $m=m'$. Then $\\cat(P')\\leq \\max\\{1,\\cat(C_1),\\ldots,\\cat(C_{m-1})\\}$. \nNotice that $P$ is the homotopy pushout of $f_m\\times\\ensuremath{\\mathbbm{1}}_{B_m}$ and the inclusion \\seqm{A_m\\times E\\times B_m}{\\ensuremath{\\mathbbm{1}}_{A_m}\\times g}{A_m\\times P'},\nwhere $g$ is the inclusion $W_{m-1}\\subset P'$, and $W_\\ell=E\\times \\mc B_\\ell\\times\\{1\\}$. \nWe can deform $W_\\ell$ into $W_{\\ell-1}$ in $P'$ as follows. First deform $W_\\ell$ onto $f_\\ell(A_\\ell\\times E)\\times \\mc B_{\\ell-1}$ by moving it\ndown the mapping cylinder $M=((E\\times B_m\\times [0,1])\\coprod (C_\\ell\\times B_\\ell))\/\\sim\\,$ of $P'$ and onto the base $C_\\ell\\times B_\\ell$,\nthen deform it onto $\\ast\\times \\mc B_{\\ell-1}$ in $C_\\ell\\times B_\\ell$ using the nullhomotopy of $f_\\ell$. Finally, \nmove $\\mc B_{\\ell-1}$ back up towards the top of the mapping cylinder $M$ and into $W_{\\ell-1}$. Composing these deformations for $\\ell=m-1,\\ldots,1$\ngives a contraction in $P'$ of $W_{m-1}$ to a point. Thus, $g$ is nullhomotopic, as is $f_m$. \nSince the lemma holds for the base case $m=2$, $\\cat(P)=\\max\\{1,\\cat(P'),\\cat(C_m)\\}\\leq \\max\\{1,\\cat(C_1),\\ldots,\\cat(C_m)\\}$.\n\n\\end{proof}\n\n\n\\begin{lemma}\n\\label{LGluing}\nFix $m\\geq 2$, and for $i=1,\\ldots, m$, let $A_i$, $C_i$, $E$ be basepointed $CW$-complexes, $E$ is path-connected, and let $B_i=\\prod_{j\\neq i} A_j$.\nSuppose \\seqm{A_i\\times E}{f_i}{C_i} are maps such that the restriction $(f_i)_{|A_i\\times\\ast}$ of $f_i$ to $A_i\\times\\ast$ is nullhomotopic,\nand $P$ is the homotopy pushout of the maps \\seqm{A_i\\times E\\times B_i}{f_i\\times\\ensuremath{\\mathbbm{1}}_{B_i}}{C_i\\times B_i} for $i=1,\\ldots, m$.\nThen \n$$\n\\cat(P)\\leq \\max\\{1,\\cat(C_1),\\ldots,\\cat(C_m)\\}+\\Cat(E). \n$$\n\nMoreover, if each $C_i\\times B_i$ is a subcomplex of some $CW$-complex $X_i$ such that \n$X_i-C_i\\times B_i$ is contractible in $X_i$, \nand $P'$ is the homotopy pushout of the maps $A_i\\times E\\times B_i\\mapaddr{f_i\\times\\ensuremath{\\mathbbm{1}}_{B_i}}{C_i\\times B_i}\\incladdr{}{X_i}$\nfor $i=1,\\ldots, m$, then also\n$$\n\\cat(P')\\leq \\max\\{1,\\cat(C_1),\\ldots,\\cat(C_m)\\}+\\Cat(E).\n$$ \n\\end{lemma}\n\n\n\\begin{proof}[Proof (part $1$)]\nLet $\\mc B=A_1\\times\\cdots\\times A_m$, and $\\mc D=\\coprod_{i=1,\\ldots,m}(C_i\\times B_i)$,\nand let $\\mc S_t$ for $t<1$ be $m$ copies of the interval $[t,1]$ glued together at the endpoints $1$, \nand $\\mc S'_t$ be its interior, namely, $m$ copies of $(t,1]$ glued at $1$.\n\nLet $k=\\Cat(E)$ and take $E'\\simeq E$ to be such that $k=\\gcat(E')$.\nThen $P$ is homotopy equivalent to the homotopy pushout $Q$ of the maps\n\\seqm{A_i\\times E'\\times B_i}{f'_i\\times\\ensuremath{\\mathbbm{1}}_{B_i}}{C_i\\times B_i} for $i=1,\\ldots, m$, \nwhere $\\seqm{A_i\\times E'}{f'_i}{C_i}$ is the composite of $f_i$ with the homotopy equivalence\n$\\seqm{A_i\\times E'}{\\ensuremath{\\mathbbm{1}}_{A_i}\\times\\simeq}{A_i\\times E}$.\nSince $E$ is path-connected and $\\gcat()$ is unaffected by attaching an interval $[0,1]$ to a space,\nwe may assume that the homotopy equivalence \\seqm{E'}{\\simeq}{E} is basepointed for some $\\ast\\in E'$. \n\nLet $U_0,\\ldots,U_k$ be an open cover of $E'$ with each $U_i$ a contractible subspace.\nTake $Q_j$ to be the homotopy pushout of \n\\seqm{A_i\\times U_j\\times B_i}{g_{i,j}\\times\\ensuremath{\\mathbbm{1}}_{B_i}}{C_i\\times B_i} for $i=1,\\ldots, m$, \nwhere $g_{i,j}$ is the restriction of $f'_i$ to $A_i\\times U_j$, and let\n$V_j=Q_j-\\mc D\\,\\cong\\,U_j\\times\\mc B\\times \\mc S'_0$. \nSince $g_{i,j}\\times\\ensuremath{\\mathbbm{1}}_{B_i}$ restricts $f'_i\\times\\ensuremath{\\mathbbm{1}}_{B_i}$, \n$Q_j$ is a subspace of $Q$ and $V_j$ is open in $Q$.\nMoreover, we may contract $V_j$ in $Q$ to a point as follows.\nLet $\\mc B_0=\\ast$ and $\\mc B_\\ell=\\prod_{i\\leq\\ell} A_i\\,\\subseteq\\,\\mc B$,\nand take the subspace $W_\\ell=\\ast\\times\\mc B_\\ell\\times\\{1\\}$ of $E'\\times\\mc B\\times\\mc S'_0\\subset Q$.\nWe can deform $W_\\ell$ into $W_{\\ell-1}$ in $Q$, first by deforming $W_\\ell$ onto \n$f'_\\ell(A_\\ell\\times\\ast)\\times\\mc B_{\\ell-1}$ by moving it down the mapping cylinder \n$M=((E'\\times \\mc B\\times [0,1])\\coprod (C_\\ell\\times B_\\ell))\/\\sim\\,$ of $Q$ and onto $C_\\ell\\times B_\\ell$,\nthen deforming it onto $\\ast\\times \\mc B_{\\ell-1}$ in $C_\\ell\\times B_\\ell$ \nusing the nullhomotopy of $(f'_\\ell)_{|A_\\ell\\times\\ast}$, and finally, \nmoving $\\mc B_{\\ell-1}$ back up towards the top of the mapping cylinder $M$ and into $E'\\times \\mc B\\times\\{1\\}$.\nComposing these deformations of $W_\\ell$ into $W_{\\ell-1}$ in $Q$ for $\\ell=m,m-1,\\ldots,1$,\nand deforming $V_j$ onto $W_m$ using contractibility of $U_j$ and $\\mc S'_0$ (onto $1$),\ngives our contraction of $V_j$ in $Q$ to a point.\n\nAssume $\\ast\\in U_0$. \nSince $U_0$ is contractible and $(g_{i,0})_{|A_i\\times\\ast}=(f'_i)_{|A_i\\times\\ast}$ is nullhomotopic,\n$g_{i,0}$ is also nullhomotopic. Lemma~\\ref{LGT} then applies to $Q_0$, namely, we have \n$$\n\\cat(Q_0)\\leq \\max\\{1,\\cat(C_1),\\ldots,\\cat(C_m)\\}.\n$$\nLet $\\mc R=\\mc S'_0-\\mc S_{\\frac{1}{2}}\\,\\cong\\,\\coprod_{i=1,\\ldots,m}(0,\\frac{1}{2})$\nand $\\bar{\\mc R}=\\mc S_0-\\mc S_{\\frac{1}{2}}\\,\\cong\\,\\coprod_{i=1,\\ldots,m}[0,\\frac{1}{2})$,\nand consider the open subspace $Q'_0 = Q_0\\cup(E'\\times \\mc B\\times\\mc R)$ of $Q$.\nNotice $Q'_0$ deformation retracts in the weak sense onto $Q_0$ by deformation retracting the subspace of $Q_0$\n$$\n\\paren{(E'\\times \\mc B\\times\\bar{\\mc R})\\,\\coprod\\,\\mc D\\,}\/\\sim\n$$ \nonto $\\mc D$, this being done by contracting each copy of $[0,\\frac{1}{2})$ in the factor $\\bar{\\mc R}$ to $0$, \nat the same time expanding $(U_0\\times\\mc B)\\times \\mc S_{\\frac{1}{2}}$ in $Q'_0$ by expanding each copy of $[\\frac{1}{2},1]$ \nin the factor $\\mc S_{\\frac{1}{2}}$ outwards to $[0,1]$. Then $\\cat(Q'_0)=\\cat(Q_0)$.\nSo take $k'=\\max\\{1,\\cat(C_1),\\ldots,\\cat(C_m)\\}$ and $\\{U'_0,\\ldots,U'_{k'}\\}$ to be a categorical cover for $Q'_0$.\nNotice that $U'_i$ is open in $Q$ since $Q'_0$ is, and $Q=\\bigcup_{j=0}^n Q_j = Q'_0\\cup \\bigcup_{j=1}^n V_j$.\nAs each $V_j$ is open and contractible in $Q$, \nthen $\\{U'_0,\\ldots,U'_{k'},V_1,\\ldots,V_k\\}$ is a categorical cover of $Q$. \nTherefore $\\cat(P)=\\cat(Q)\\leq k'+k$.\n\n\\end{proof}\n\n\n \n\\begin{proof}[Proof (part $2$)]\n\nSince $C_i\\times B_i$ is a subcomplex of $X_i$, $P$ is a subspace of $P'$ with \n$$\nP'-P=\\cdunion{i=1,\\ldots,m}{}{(X_i-C_i\\times B_i)},\n$$ \nso $P'-P$ is open and contractible in $P'$. Notice each $V_j$ is an open (and contractible) subset of $P'$, \nwhile $S_j=U'_j\\cap(\\coprod_{i=1,\\ldots,m} X_i)$ is an open subset of $\\mc D$. \nBy Lemma~\\ref{LOpen}, there exists an open subset $R_j$ of $\\coprod_{i=1,\\ldots,m} X_i$ that deformation retracts onto $S_j$\nsuch that $R_j\\cap \\mc D=S_j$.\nThen $R'_j=R_j\\coprod (U'_j-S_j)$ is an open subset of $P'$ that deformation retracts onto $U'_j$, \nthus is contractible in $P'$. Since $P'-P$ and $V_j$ are both open in $P'$, and $(P'-P)\\cap V_j=\\emptyset$,\nthen the subspace $(P'-P)\\coprod V_j$ is contractible in $P'$. \nWe can therefore take $\\{R'_0,\\ldots,R'_{k'},(V_1\\coprod(P'-P)),V_2,\\ldots,V_k\\}$ as a categorical cover for $P'$,\nso $\\cat(P')\\leq k'+k$.\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Moment-Angle Complexes}\n\n\nGiven a simplicial complex $K$ on vertex set $[n]$ and \na sequence of of pairs of spaces \n$$\n\\mc S=((X_1,A_1),\\ldots,(X_n,A_n)), \n$$ \n$A_i\\subseteq X_i$, the \\emph{polyhedral product} $\\mc S^K$ is the subspace of $X^{\\times n}$ defined by\n$$\n\\mc S^K=\\bigcup_{\\sigma\\in K}Y^\\sigma_1\\times\\cdots\\times Y^\\sigma_n,\n$$ \nwhere $Y^\\sigma_i=X_i$ if $i\\in\\sigma$, or $Y^\\sigma_i=A_i$ if $i\\nin\\sigma$.\nIf the pairs $(X_i,A_i)$ are all equal to the same pair $(X,A)$, we usually write $\\mc S^K$ as $(X,A)^K$.\nThe \\emph{moment-angle complex} $\\mathcal Z_K$ is defined as the polyhedral product $(D^2,\\ensuremath{\\partial} D^2)^K$,\nand the \\emph{real moment-angle complex} $\\mathbb R\\mathcal Z_K$ is the polyhedral product $(D^1,\\ensuremath{\\partial} D^1)^K$.\n\nThe \\emph{join} of two simplicial complexes $K$ and $L$ is the simplicial complex \n$K\\ast L=\\cset{\\sigma\\sqcup\\tau}{\\sigma\\in K,\\,\\tau\\in L}$,\nand one has $|K\\ast L|\\cong |K|\\ast|L|\\simeq\\Sigma|K|\\wedge|L|$ and $\\z{K\\ast L}\\cong\\mathcal Z_K\\times\\mathcal Z_L$.\nIf $I\\subseteq [n]$, $K_I=\\cset{\\sigma\\in K}{\\sigma\\subseteq I}$ denotes the \\emph{full subcomplex} of $K$ on vertex set $I$,\nin which case $\\z{K_I}$ is a retract of $\\mathcal Z_K$.\nNotice that if $K_I$ and $L_J$ are full subcomplexes of $K$ and $L$, \nthen $K_I\\ast L_J$ is the full subcomplex $(K\\ast L)_{I\\sqcup J}$ of $K\\ast L$.\nAs a convention, we let $\\z{\\emptyset}=\\ast$ when $\\emptyset$ is on empty vertex set. \n\n\n\nWe let $S^0$ denote both the $0$-sphere and the simplicial complex $\\ensuremath{\\partial}\\Delta^1$ consisting of only two vertices. \nGenerally, we assume our simplicial complexes (except $\\emptyset$) are non-empty and have no ghost vertices,\nunless stated otherwise. Under this assumption, it is not difficult to see that the following holds.\n\n\n\n\n\n\\subsection{The Hochster Theorem}\n\\label{SHochster}\n\nWhen $R$ is a field or $\\mb Z$, it was shown in~\\cite{MR1897064,MR2255969,MR2117435,MR0441987} that there are isomorphisms \nof graded commutative algebras\n\\begin{equation}\n\\label{EHochster}\nH^*(\\mathcal Z_K;R)\\cong \\mathrm{Tor}_{R[v_1,\\ldots,v_n]}(R[K],R) \\cong \\cplus{I\\subseteq[n]}{}{\\tilde H^*(\\Sigma^{|I|+1}|K_I|;R)}.\n\\end{equation}\nThe isomorphism on the left is induced by a quasi-isomorphism of DGAs between the Koszul complex of the Stanley-Reisner ring $R[K]$\nand the cellular cochain complex of $\\mathcal Z_K$ with coefficients in $R$. \nThe multiplication on the right is given by maps \\seqm{H^*(K_I)\\otimes H^*(K_J)}{}{H^{*+1}(K_{I\\cup J})} \nthat are zero when $I\\cap J\\neq\\emptyset$, otherwise they are induced by maps\n$\\iota_{I,J}\\ensuremath{\\,\\colon\\,}\\seqm{|K_{I\\cup J}|}{}{|K_I\\ast K_J|\\cong |K_I|\\ast |K_J|\\simeq\\Sigma |K_I|\\wedge |K_J|}$\ngeometrically realizing the canonical inclusions \\incl{K_{I\\cup J}}{}{K_I\\ast K_J}.\nOne can iterate so that any length $\\ell$ product\n\\seqm{\\bigotimes^\\ell_{i=1} H^*(K_{I_i})}{}{H^{*+\\ell-1}(K_{I_1\\cup\\cdots\\cup I_\\ell})} is induced by the inclusion \n$$\n\\iota_{I_1,\\ldots,I_\\ell}\\ensuremath{\\,\\colon\\,}\\incl{|K_{I_1\\cup\\cdots\\cup I_\\ell}|}{}{|K_{I_1}\\ast\\cdots\\ast K_{I_\\ell}|}\n$$ \nwhere the $I_i$'s are mutually disjoint.\n\n\n\\subsection{A Necessary Condition}\n\nSuppose $\\cat(\\mathcal Z_K)\\leq \\ell-1$,\nso cup products of length $l$ vanish in $H^+(\\mathcal Z_K)$.\nThen in light of the Hochster theorem, the inclusions $\\iota_{I_1,\\ldots,I_\\ell}$ must induce trivial maps on cohomology. \nIn fact, their suspensions must be nullhomotopic by the following argument. \n\nLet $\\widehat{\\mathcal Z}_K=\\mathcal Z_K\/\\cset{(x_1,\\ldots,x_n)\\in \\mathcal Z_K}{\\mbox{at least one }x_i=\\ast}$.\nFix $m=|I_1\\cup\\cdots\\cup I_\\ell|$, $Y=\\z{K_{I_1\\cup\\cdots\\cup I_\\ell}}$, and $\\hat Y=\\hz{K_{I_1\\cup\\cdots\\cup I_\\ell}}$. \nSince $Y$ is a retract of $\\mathcal Z_K$, $\\cat(Y)\\leq\\ell-1$.\nRecall from~\\cite{MR516214} that a path-connected basepointed $CW$-complex such as $Y$ satisfies $\\cat(Y)\\leq\\ell-1$ if and only if \nthere is a map \\seqm{Y}{\\psi}{FW_\\ell(Y)} such that the diagonal map \\seqm{Y}{\\vartriangle}{Y^{\\times\\ell}} \nfactors up to homotopy as \\seqmm{Y}{\\psi}{FW_\\ell(Y)}{include}{Y^{\\times\\ell}}.\nHere $FW_\\ell(Y)=\\cset{(y_1,\\ldots,y_\\ell)\\in Y^{\\times \\ell}}{\\mbox{at least one }y_i=\\ast}$ is the fat wedge.  \nThis implies the reduced diagonal map\n$\\bar\\vartriangle\\ensuremath{\\,\\colon\\,}\\seqmmm{Y}{\\vartriangle}{Y^{\\times\\ell}}{}{Y^{\\times\\ell}\/FW_\\ell(Y)}{\\cong}{Y^{\\wedge\\ell}}$\nis nullhomotopic. Then so is\n$\\zeta\\ensuremath{\\,\\colon\\,}\\seqmmm{Y}{\\bar\\vartriangle}{Y^{\\wedge\\ell}}{}{\\bigwedge_j\\z{K_{I_j}}}{}{\\bigwedge_j\\hz{K_{I_j}}}$,\nwhere the second last map is the smash of the coordinate-wise projection maps onto each $\\z{K_{I_j}}$, \nand the last map is the smash of quotient maps. \nThis last nullhomotopic map $\\zeta$ coincides with $\\seqmm{Y}{q}{\\hat Y}{\\hat\\iota}{\\bigwedge_j\\hz{K_{I_j}}}$,\nwhere $q$ is the quotient map and $\\hat\\iota$ is the inclusion given simply by rearranging coordinates. \nMoreover, \n$\\hat\\iota$ is homeomorphic to $\\Sigma^{m+1}\\iota_{I_1,\\ldots,I_\\ell}$ and $\\Sigma q$ has a right homotopy inverse\n(c.f.~\\cite{MR2673742}, and also the proof of Proposition~$2.5$ and pg. $23$ in~\\cite{2014arXiv1409.4462B}).\nIt follows that $\\Sigma^{m+1}\\iota_{I_1,\\ldots,I_\\ell}$ is nullhomotopic.\n\n\n\n\\subsection{Skeleta and Suspension on Coordinates}\n\n\nLet $\\sk{K}{i}$ denote the $i$-skeleton of $K$, and $\\sk{K}{-1}=\\emptyset$ on vertex set $[n]$. \nAn inclusion of simplicial complexes \\incl{L}{}{K} induces a canonical inclusion of $CW$-complexes \\incl{\\mathcal Z_L}{}{\\mathcal Z_K}. \nThis gives $\\zsk{K}{i}$ and $\\zsk{K}{-1}=(\\ensuremath{\\partial} D^2)^{\\times n}=(S^1)^{\\times n}$ as subcomplexes of $\\mathcal Z_K$.\n\n\n\\begin{lemma}[Corollary~$3.3$ in~\\cite{MR3084441}]\n\\label{LGT2}\nIf $K$ is on vertex set $[n]$ with no ghost vertices, then $\\zsk{K}{-1}=(\\ensuremath{\\partial} D^2)^{\\times n}$ is contractible in $\\mathcal Z_K$.~$\\hfill\\square$\n\\end{lemma} \n\n\\begin{lemma}\n\\label{LSkeleta}\nIf $0\\leq l\\leq \\dim K$, then $\\zsk{K}{\\ell}-\\zsk{K}{\\ell-1}$ is contractible in $\\mathcal Z_K$.\n\\end{lemma}\n\n\\begin{proof}\nWe have a decomposition\n$$\n\\zsk{K}{\\ell}-\\zsk{K}{\\ell-1}=\\cdunion{\\sigma\\in K,\\,|\\sigma|=\\ell+1}{}{\\tilde Y_1^\\sigma\\times\\cdots\\times \\tilde Y_n^\\sigma}\n$$\nwhere $\\tilde Y_i^{\\sigma}=D^2-\\ensuremath{\\partial} D^2$ if $i\\in\\sigma$ and $\\tilde Y_i^{\\sigma}=\\ensuremath{\\partial} D^2$ if $i\\nin\\sigma$. \nThis being a disjoint union of open subspaces of $\\zsk{K}{l}$, \neach of which can be deformed into $\\zsk{K}{-1}$ in $\\mathcal Z_K$\n(by contracting $\\tilde Y_i^{\\sigma}$ to a point in $\\ensuremath{\\partial} D^2$ whenever $i\\in\\sigma$). \nThus $\\zsk{K}{\\ell}-\\zsk{K}{\\ell-1}$ can also be deformed into $\\zsk{K}{-1}$.\nThen $\\zsk{K}{\\ell}-\\zsk{K}{\\ell-1}$ is contractible in $\\mathcal Z_K$ by Lemma~\\ref{LGT2}.\n\n\\end{proof}\n\n\\begin{lemma}\n\\label{LSkeleta2}\nIf $-1\\leq j\\leq \\dim K$, then \n$$\n\\cat(\\mathcal Z_K)\\leq \\cat(\\incl{\\zsk{K}{j}}{}{\\mathcal Z_K})+\\dim K-j.\n$$\nIn particular, $\\cat(\\mathcal Z_K)\\leq \\dim K+1$.\n\\end{lemma}\n\n\\begin{remark}\nLemma~\\ref{LSkeleta} and~\\ref{LSkeleta2} can be generalized to any filtration \n$L_j\\subseteq \\cdots\\subseteq L_k=K$ satisfying $\\ensuremath{\\partial}\\sigma\\subseteq L_i$ whenever $\\sigma\\in L_{i+1}$\n(in place of the skeletal filtration).\n\\end{remark}\n\n\n\\begin{proof}\nThe skeletal filtration $\\sk{K}{j}\\subseteq\\cdots\\subseteq\\sk{K}{\\dim K}=K$\ninduces a filtration of subcomplexes $\\zsk{K}{j}\\subseteq\\cdots\\subseteq\\mathcal Z_K$, \nand  for $0\\leq j\\leq \\dim K$, $\\zsk{K}{j}-\\zsk{K}{j-1}$ is contractible in $\\mathcal Z_K$ by Lemma~\\ref{LSkeleta}.\nThe result then follows using Lemma~\\ref{LFiltration}. In particular, when $j=-1$, \nwe get $\\cat(\\mathcal Z_K)\\leq \\dim K+1$ since $\\cat(\\incl{\\zsk{K}{-1}}{}{\\mathcal Z_K})=0$ by Lemma~\\ref{LGT2}.\n\\end{proof}\n\n\n\n\n\\begin{proposition}\n\\label{PCoordSusp}\nLet $\\mc S=((X_1,A_1),\\ldots,(X_n,A_n))$ \nand $\\mc T=((\\Sigma^{m_1}X_1,\\Sigma^{m_1} A_1),\\ldots,(\\Sigma^{m_n} X_n,\\Sigma^{m_n} A_n))$\nbe sequences of pairs of spaces for some integers $m_i$ and connected basepointed $X_i$. \nThen for any $K$ (with no ghost vertices),\n$$\n\\cat(\\mc T^K)\\leq \\cat(\\mc S^K).\n$$\n\\end{proposition}\n\n\\begin{proof}\nLet $K$ be on vertex set $[n]$, $k=\\cat(\\mc S^K)$, and take a categorical cover $\\{U_0,\\ldots,U_k\\}$ of $\\mc S^K$.\nFor any open subset $V$ of $\\mc S^K$, define the following open subset $V^1$ of $\\mc T^K$\n$$\nV^1 = \\cset{((t_1,x_1),\\ldots,(t_n,x_n))\\in \\prod_{i=1}^n \\Sigma^{m_i}X_i}{(x_1,\\ldots,x_n)\\in V,\\,t_i\\in D^{m_i}}.\n$$\nIn particular, $\\mc T^K=(\\mc S^K)^1$. Then $\\{U^1_0,\\ldots,U^1_k\\}$ is an open cover of $\\mc T^K$. \nSince $K$ has no ghost vertices, $A_i\\subseteq X_i$, and each $X_i$ is path-connected, then $\\mc S^K$ is path-connected. \nSince $\\Sigma^{m_i}X_i$ is the reduced suspension of the basepointed space $X_i$, \nwe have identifications $(t,\\ast)\\sim\\ast\\in\\Sigma^{m_i}X_i$. \nThen we can define a contraction of $U^1_i$ in $\\mc T^K$ by contracting $U_i$ in $\\mc S^K$ to a point $p$ \nand homotoping $p$ to the basepoint $(\\ast,\\ldots,\\ast)\\in\\mc S^K$.\nTherefore, $\\{U^1_0,\\ldots,U^1_k\\}$ is a categorical cover of $\\mathcal Z_K$. \n\n\\end{proof}\n\n\n\nNotice that the $(i+1)$-skeleton $\\sk{(\\mathbb R\\mathcal Z_K)}{i+1}$ of $\\mathbb R\\mathcal Z_K$ is equal to $\\rzsk{K}{i}$\n(this is not true for the complex moment-angle complex $\\mathcal Z_K$).\n\n\n\\begin{corollary}\n\\label{CRealMAC}\nFor any $K$ (with no ghost vertices),\n\\begin{equation}\n\\label{mac}\n\\cat(\\mathcal Z_K)\\leq \\cat(\\mathbb R\\mathcal Z_K)\n\\end{equation}\nand if $\\mathbb R\\mathcal Z_K$ is not contractible and $i\\geq 0$, then\n\\begin{equation}\n\\label{RmacCmac}\n\\cat(\\zsk{K}{i})\\leq \\cat(\\rzsk{K}{i})\\leq \\cat(\\mathbb R\\mathcal Z_K).\n\\end{equation}\n\\end{corollary}\n\n\\begin{proof}\nInequality~\\eqref{mac} and the first inequality in~\\eqref{RmacCmac} follow from Proposition~\\ref{PCoordSusp}.  \nBy the main corollary of Theorem~$1$ in~\\cite{MR1933583},\nthe $i$-skeleton $\\sk{X}{i}$ of any connected non-contractible $CW$-complex $X$\nsatisfies that $\\cat(\\sk{X}{i})\\leq\\cat(X)$. \nSince $\\sk{(\\mathbb R\\mathcal Z_K)}{i+1}=\\rzsk{K}{i}$ holds for real moment-angle complexes, the last inequality follows.\n\n\\end{proof}\n\n\nIt is plausible that the second bound can be strengthened to $\\cat(\\zsk{K}{i})\\leq \\cat(\\mathcal Z_K)$.\nIn any case, even if it is true, we will sometimes need a sharper bound.\n\nLet $X$ and $Y$ be path-connected paracompact spaces, and $\\mc U=\\{U_0,\\ldots,U_k\\}$ and $\\mc V=\\{V_0,\\ldots,V_{\\ell}\\}$ \nbe categorical covers of $X$ and $Y$, respectively. We recall James' construction of a categorical cover $\\mc W=\\{W_0,\\ldots,W_{k+\\ell}\\}$\nof $X\\times Y$ from the covers $\\mc U$ and $\\mc V$ (see~\\cite{MR516214}, page $333$). \n\nLet $\\{\\pi_j\\}_{j\\in\\{0,\\dots,k\\}}$ be a partition of unity subordinate to the cover $\\mc U$. \nFor any subset $S\\subseteq \\{0,\\dots,k\\}$, define \n$$\nW_{\\mc U}(S)=\\cset{x\\in X}{\\pi_j(x)>\\pi_i(x)\\mbox{ for any }j\\in S\\mbox{ and }i\\nin S},\n$$ \nand for any point $p\\in X$, let \n$$\nS_{\\mc U}(p)=\\cset{j\\in \\{0,\\dots,k\\}}{\\pi_j(p)>0}\n$$\n(since the context is clear, let $W(S)=W_{\\mc U}(S)$ and $S(p)=S_{\\mc U}(p)$).\nThen $W(S)$ is an open subset of $X$ and $X=\\bigcup_{S\\subseteq\\{0,\\dots,k\\}}W(S)$ \n(given $x\\in X$, $x\\in W(S)$ where $S=\\cset{i}{\\,\\pi_i(x)=\\max\\{\\pi_1(x),\\ldots,\\pi_k(x)\\}\\,}$).\nMoreover, $W(S')\\cap W(S)=\\emptyset$ when $S\\nsubseteq S'$ and $S'\\nsubseteq S$ \n(in particular, when $|S|=|S'|$ and $S\\neq S'$), and $W(S)\\subseteq U_j$\nwhenever $j\\in S$. \nTherefore $W(S)$ is contractible in $X$. Then so is the disjoint union of open sets\n\\begin{equation}\n\\label{EUi}\nU'_i=\\cdunionmulti{S=S(p)\\mbox{\\tiny{ for some} }p\\in X}{|S|=i+1}{}{W(S)}.\n\\end{equation}\nSince $W(S)=\\emptyset$ when $S\\neq S(p)$ for every $p\\in X$,\nthe set $\\{U'_0,\\ldots,U'_k\\}$ forms a categorical cover of $X$.\nWe obtain a categorical cover $\\{V'_0,\\ldots,V'_\\ell\\}$ of $Y$ from $\\mc V$ by an analogous construction.\n\n\nNow let $\\bar U_i=U'_{k-i}\\cup\\cdots\\cup U'_k$ and $\\bar V_j=V'_{\\ell-j}\\cup\\cdots\\cup V'_\\ell$,\nand for $-1\\leq s\\leq k+\\ell$, let $C_{-1}=\\emptyset$ and\n$$\nC_s=\\cunionmulti{i+j=s}{i\\leq k,\\,j\\leq\\ell}{}{\\bar U_i\\times\\bar V_j}\n$$\nTake $W_s=C_s-C_{s-1}$. Notice that\n\\begin{equation}\n\\label{EWs}\nW_s=\\cdunionmulti{i+j=s}{i\\leq k,\\,j\\leq\\ell}{}{U'_i\\times V'_j}.\n\\end{equation}\nThis defines a categorical cover $\\mc W$ of $X\\times Y$.\n\nGiven subcomplexes $B\\subseteq Y$ and $A\\subseteq X$,\nconsider the polyhedral product \n$$\n\\mc X^{S^0}=X\\times B \\cup_{A\\times B} A\\times Y\n$$\nover the sequence $\\mc X=((X,A),(Y,B))$.\n\n\n\\begin{lemma}\n\\label{LPoly}\nIf $X-A$ is contractible in $X$ and $Y-B$ is contractible in $Y$, then\n$$\n\\cat(\\mc X^{S^0})\\leq \\cat(A)+\\cat(B)+1.\n$$\n\\end{lemma} \n\n\\begin{proof}\nSuppose we have categorical covers $\\{R_1,\\ldots,R_k\\}$ and $\\{S_1,\\ldots,S_\\ell\\}$\nof $A$, and $B$. By Lemma~\\ref{LOpen}, we have open subsets $U_i\\subseteq X$ and $V_i\\subseteq Y$\nsuch that $U_i$ and $V_i$ deformation retract onto $R_i$ and $S_i$ respectively, \nand $U_i\\cap A=R_i$ and $V_i\\cap B=S_i$ for $i\\geq 1$. Then taking $U_0=X-A$ and $V_0=Y-B$, \n$\\mc U=\\{U_0,\\ldots,U_k\\}$ and $\\mc V=\\{V_0,\\ldots,V_\\ell\\}$ are categorical covers of $X$ and $Y$.\n\nNotice that\n$$\nX\\times Y-U_0\\times V_0=\\mc X^{S^0},\n$$\nand since $R_i=U_i\\cap A=U_i-U_0$ and $S_j=V_j\\cap B=V_j-V_0$ for $i,j\\geq 1$, \n\\begin{equation}\n\\label{EDiff}\nD_{i,j}\\,:=\\,U_i\\times V_j-U_0\\times V_0\\,=\\,(R_i\\times V_j)\\cup_{R_i\\times S_j}(U_i\\times S_j). \n\\end{equation}\nNotice that $D_{i,j}$ is contractible in $\\mc X^{S^0}$ by deformation retracting the factor\n$U_i$ onto $R_i$ and $V_j$ onto $S_j$,\nthen contracting $R_i\\times S_j$ in $A\\times B$.\n\n\nTake the categorical cover $\\mc W=\\{W_0,\\ldots,W_{k+\\ell}\\}$ of $X\\times Y$ constructed from \n$\\mc U$ and $\\mc V$ as above. By~\\eqref{EWs},\n\\begin{align*}\nW_s-U_0\\times V_0 &=\\cdunionmulti{i+j=s}{i\\leq k,\\,j\\leq\\ell}{}{(U'_i\\times V'_j-U_0\\times V_0)},\n\\end{align*}\nand by~\\eqref{EUi},\n\\begin{align*}\nU'_i\\times V'_j-U_0\\times V_0=\\cdunion\n{\\indexsize{\\begin{matrix}\n\\scriptstyle S=S_{\\mc U}(p)\\mbox{\\tiny{ for some} }p\\in X \\cr \n\\scriptstyle T=S_{\\mc V}(q)\\mbox{\\tiny{ for some} }q\\in Y \\cr \n\\scriptstyle |S|=i+1,\\,|T|=j+1\n\\end{matrix}}}\n{}{(W_{\\mc U}(S)\\times W_{\\mc V}(T)-U_0\\times V_0)}.\n\\end{align*}\nThese are disjoint unions of open subsets of $\\mc X^{S^0}$.\nSince $W_{\\mc U}(S)$ is contained in some $U_{i'}$ and\n$W_{\\mc V}(T)$ is contained in some $V_{j'}$, it follows that\n$(W_{\\mc U}(S)\\times W_{\\mc V}(T)-U_0\\times V_0)$ is contained in $D_{i',j'}$, \nso it is contractible in $\\mc X^{S^0}$. \nTherefore, \nso are the disjoint unions $U'_i\\times V'_j-U_0\\times V_0$ and $W_s-U_0\\times V_0$. \nMoreover, since $W_{k+\\ell}=U'_k\\times V'_\\ell$,\nand $U'_k=W_{\\mc U}(\\{0,\\ldots,k\\})$ and $V'_\\ell=W_{\\mc V}(\\{0,\\ldots,\\ell\\})$ \nare contained in $U_{i'}$ and $V_{j'}$ respectively for each $i'\\in\\{0,\\ldots,k\\}$ and $j'\\in\\{0,\\ldots,\\ell\\}$,\n$W_{k+\\ell}-U_0\\times V_0=\\emptyset$. Then\n$$\n\\{(W_0-U_0\\times V_0),\\ldots,(W_{k+\\ell-1}-U_0\\times V_0)\\}\n$$\nis a categorical cover of $\\mc X^{S^0}$.\n\n\\end{proof}\n\n\n\n\n\\begin{corollary}\n\\label{CJoinSkeleton}\nLet $K$ and $L$ be simplicial complexes with $d=\\dim K$ and $d'=\\dim L$ (so $\\dim K\\ast L=d+d'+1$). Then\n$$\n\\cat(\\zsk{(K\\ast L)}{d+d'})\\leq \\cat(\\zsk{K}{d-1})+\\cat(\\zsk{L}{d'-1})+1.\n$$\n\\end{corollary}\n\n\\begin{proof}\nNotice $\\sk{(K\\ast L)}{d+d'}=(K\\ast\\sk{L}{d'-1})\\cup_{(\\sk{K}{d-1}\\ast\\sk{L}{d'-1})} (\\sk{K}{d-1}\\ast L)$, \n$\\z{K\\ast L}=\\mathcal Z_K\\times\\mathcal Z_L$, so\n\\begin{align*}\n\\zsk{(K\\ast L)}{d+d'} &= (\\z{K\\ast\\sk{L}{d'-1}}) \\cup_{\\z{(\\sk{K}{d-1}\\ast\\sk{L}{d'-1})}} (\\z{\\sk{K}{d-1}\\ast L})\\\\\n&= (\\mathcal Z_K\\times\\zsk{L}{d'-1})\\cup_{\\zsk{K}{d-1}\\times\\zsk{L}{d'-1}} (\\zsk{K}{d-1}\\times\\mathcal Z_L),\n\\end{align*}\nand $\\mathcal Z_K-\\zsk{K}{d-1}$ and $\\mathcal Z_L-\\zsk{L}{d'-1}$ are contractible in $\\mathcal Z_K$ and $\\mathcal Z_L$ by Lemma~\\ref{LSkeleta}.\nThe result follows by Lemma~\\ref{LPoly}.\n\n\\end{proof}\n\n\n\n\n\n\\subsection{Missing Face Complexes}\n\nTake $K$ on vertex set $[n]$.\nWe fix the basepoint in the unreduced suspension $\\Sigma|K|=(|K|\\times [0,1]) \/\\sim$ \nto be the tip of the double cone corresponding to $1$ under the indentifications $(x,0)\\sim 0$ and $(x,1)\\sim 1$.\nLet $MF(K)=\\cset{\\sigma\\subseteq[n]}{\\sigma\\nin K,\\,\\ensuremath{\\partial}\\sigma\\subseteq K}$ be the collection of (minimal) missing faces of $K$.\nWe will need a somewhat more flexible alternative to the \\emph{directed missing face complexes} defined in~\\cite{2010arXiv1011.2133G}.\n\n\\begin{definition}\n$K$ on vertex set $[n]$ is called a \\emph{homology missing face complex} (or \\emph{HMF-complex}) if for each non-empty $I\\subseteq [n]$, \n$K_I$ is a simplex or there exists a subcollection $\\mc C_I\\subseteq MF(K_I)$ such that the wedge sum of suspended inclusions\n$$\n\\gamma_I\\ensuremath{\\,\\colon\\,}\\seqm{\\cvee{\\sigma\\in \\mc C_I}{}{\\Sigma|\\ensuremath{\\partial}\\sigma|}}{}{\\Sigma|K_I|}\n$$\ninduces an isomorphism on homology\n(therefore it is a homotopy equivalence since it is a map between suspensions).\n\\end{definition}\n\n\\begin{bigremark}\nGiven $H_*(K_I)$ is torsion-free, since each $\\Sigma|\\ensuremath{\\partial}\\sigma|$ is a sphere, \none needs only to find $\\gamma_I$ that induces surjection on homology in order for $K$ to be an $HMF$-complex.\n\\end{bigremark}\n\n\\begin{proposition}\n\\label{PExtractible}\nIf $K$ is an $HMF$-complex, then $\\mathcal Z_K$ is homotopy equivalent to a wedge of spheres or is contractible.\nTherefore $\\cat(\\mathcal Z_K)\\leq 1$ and $\\Cat(\\mathcal Z_K)\\leq 1$.\n\\end{proposition}\n\n\\begin{proof}\nFor each $I\\subseteq [n]$, either $K_I$ is a simplex, boundary of a simplex, or else for each $\\sigma\\in\\mc C_I$, \nwe can pick an $i_\\sigma\\in I$ such that $\\ensuremath{\\partial}\\sigma\\subseteq K_{I-\\{i_\\sigma\\}}$, \nso each inclusion \\seqm{|\\ensuremath{\\partial}\\sigma|}{}{|K_I|} factors through inclusions \n\\seqmm{|\\ensuremath{\\partial}\\sigma|}{}{|K_{I-\\{i_\\sigma\\}}|}{}{|K_I|}. Take the composite\n$$\nf\\ensuremath{\\,\\colon\\,}\\seqmmm{\\Sigma|K_I|}{\\gamma^{-1}_I}{\\cvee{\\sigma\\in\\mc C_I}{}{\\Sigma|\\ensuremath{\\partial}\\sigma|}}{}\n{\\cvee{i\\in I}{}{\\Sigma|K_{I-\\{i_\\sigma\\}}|}}{}{\\Sigma|K_I|}\n$$\nwhere $\\gamma^{-1}_I$ is a homotopy inverse of $\\gamma_I$,\nthe second last map includes the summand $\\Sigma|\\ensuremath{\\partial}\\sigma|$ into the summand $\\Sigma|K_{I-\\{i_\\sigma\\}}|$,\nand the last map is the standard inclusion on each summand. \nSince the composite of the last two maps is $\\gamma_I$, $f$ is a homotopy equivalence.\nThen $K$ is an \\emph{extractible complex} as defined in~\\cite{2013arXiv1306.6221I}. \nTherefore $\\mathcal Z_K$ is homotopy equivalent to a wedge of spheres or contractible by Corollary~$3.3$ therein.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\subsection{Gluings and Connected Sums}\n\nIf $L$ and $K$ are simplicial complexes and $C$ is a full subcomplex common to both $L$ and $K$, then we obtain a new simplicial complex\n$L\\cup_C K$ by gluing $L$ and $K$ along $C$. One can always glue along simplices since they are always full subcomplexes.\nWhen $C=\\emptyset$, $L\\cup_C K$ is just the disjoint union $L\\sqcup K$. \n\nGiven $\\sigma\\in K$, define the \\emph{deletion} of the face $\\sigma$ from $K$ to be the simplicial complex\ngiven by \n$$\nK\\backslash\\sigma=\\cset{\\tau\\in K}{\\sigma\\not\\subseteq\\tau}.\n$$\nIf $\\sigma$ is a common face of $L$ and $K$, define the \\emph{connected sum} $L\\#_\\sigma K$ to be the simplicial complex \n$(L\\backslash\\sigma)\\cup_{\\ensuremath{\\partial}\\sigma}(K\\backslash\\sigma)$.\nIn other words, $L\\#_\\sigma K$ is obtained by deleting $\\sigma$ from $L$ and $K$ and gluing along the boundary $\\ensuremath{\\partial}\\sigma$.\nAs a convention, we let $\\z{\\emptyset}=\\ast$ when $\\emptyset$ is on empty vertex set. \n\n\n\n\n\n\\begin{proposition}\n\\label{PGluing2}\nIf $C$ is a (possibly empty) full subcomplex common to $K_1,\\ldots,K_m$, then \n$$\n\\cat(\\z{K_1\\cup_C\\cdots\\cup_C K_m})\\leq \\max\\{1,\\cat(\\z{K_1}),\\ldots,\\cat(\\z{K_m})\\}+\\Cat(\\z{C}).\n$$\nMoreover, if each $K_i$ is the $d_i-1$ skeleton of some $d_i$ dimensional simplicial complex $\\bar K_i$, \nand $C$ is also a full subcomplex of each $\\bar K_i$, then \n$$\n\\cat(\\z{\\bar K_1\\cup_C\\cdots\\cup_C \\bar K_m})\\leq \\max\\{1,\\cat(\\z{K_1}),\\ldots,\\cat(\\z{K_m})\\}+\\Cat(\\z{C}).\n$$\n\\end{proposition}\n\n\n\\begin{proof\n\nLet $K_i$ be on vertex set $[n_i]$, and $C$ has $\\ell$ vertices. \nIf $C$ is on vertex set $[n_i]$, possibly with ghost vertices, \nthe inclusion \\incl{C}{}{K_i} induces a coordinate-wise inclusion \\incl{(\\ensuremath{\\partial} D^2)^{\\times n_i-\\ell}\\times\\z{C}}{f_i}{\\z{K_i}}.\nBy Lemma~\\ref{LGT2}, $f_i$ is nullhomotopic when restricted to $(\\ensuremath{\\partial} D^2)^{\\times n_i-\\ell}\\times\\ast$.\nLet $N_i=\\Sigma_{j\\neq i}n_j$. Note $\\z{K_1\\cup_C\\cdots\\cup_C K_m}$ is the pushout of\n\\incl{(\\ensuremath{\\partial} D^2)^{\\times n_i-\\ell}\\times\\z{C}\\times(\\ensuremath{\\partial} D^2)^{\\times N_i-\\ell}}{f_i\\times\\ensuremath{\\mathbbm{1}}}{\\z{K_i}\\times(\\ensuremath{\\partial} D^2)^{\\times N_i-\\ell}}\nfor $i=1,\\ldots,m$. \nSince each of these maps are inclusions of subcomplexes, $\\z{K_1\\cup_C\\cdots\\cup_C K_m}$ is homotopy equivalent to the homotopy pushout $P$ \nof these maps. The first inequality therefore follows from the first part of Lemma~\\ref{LGluing}.\n\nBy Lemma~\\ref{LSkeleta}, $\\z{\\bar K_i}-\\z{K_i}$ is contractible in $\\z{\\bar K_i}$,  \nso the second equality follows from the second part of Lemma~\\ref{LGluing}. \n\n\\end{proof}\n\n\\begin{example}\n\\label{EGluing}\nIn particular, when $C$ is a simplex \n$\\cat(\\z{L\\cup_C K})\\leq \\max\\{1,\\cat(\\mathcal Z_L),\\cat(\\mathcal Z_K)\\}$ and\n$\\cat(\\z{\\bar L\\cup_C \\bar K})\\leq \\max\\{1,\\cat(\\mathcal Z_L),\\cat(\\mathcal Z_K)\\}$ \nsince $\\z{C}$ is contractible.\nThese also hold when $C$ is the empty simplex and $\\z{C}=\\ast$\n(in which case $\\bar L\\cup_C \\bar K=\\bar L\\sqcup \\bar K$ and $L\\cup_C K=L\\sqcup K$).\nWhen $C$ is the boundary of a simplex,   \n$\\cat(\\z{L\\cup_C K})\\leq \\max\\{1,\\cat(\\mathcal Z_L),\\cat(\\mathcal Z_K)\\}+1$\nsince $\\z{C}$ here is a sphere.\n\\end{example}\n\nThe bound in Proposition~\\ref{PGluing2} is not always optimal, sometimes far from it. \nIf $K$ and $\\Delta^{n-1}$ are on vertex set $[n]$ and $L$ is formed by gluing $\\Delta^{n-1}$ and $\\{n+1\\}\\ast K$ along $K$,\nthen $\\mathcal Z_L$ is a $co$-$H$-space by~\\cite{2013arXiv1306.6221I} so $\\cat(\\mathcal Z_L)=1$ \n(in fact, it is not difficult to directly show that $\\mathcal Z_L\\simeq\\Sigma^2\\mathcal Z_K$). \nOn the other hand, \nProposition~\\ref{PGluing2} gives $\\cat(\\mathcal Z_L)\\leq \\max\\{1,\\cat(\\mathcal Z_K)\\}+\\Cat(\\mathcal Z_K)$ since $\\z{\\{n+1\\}\\ast K}\\cong D^2\\times\\mathcal Z_K\\simeq\\mathcal Z_K$ and \n$\\z{\\Delta^n}$ is contractible.\n\n\n\n\\begin{corollary}\n\\label{CConnectedSum}\nSuppose\n$$\nK=L_1\\#_{\\sigma_1} L_2\\#_{\\sigma_2}\\cdots\\#_{\\sigma_{k-1}} L_k  \n$$\nwhere $\\dim L_i=d$, $\\sigma_i$ is a $d$-face common to $L_i$ and $L_{i+1}$, \nand $\\sigma_i\\cap\\sigma_j=\\emptyset$ when $i\\neq j$. Then\n$$\n\\cat(\\mathcal Z_K) \\leq \\max\\{1,\\cat(\\zsk{L_1}{d-1}),\\ldots,\\cat(\\zsk{L_k}{d-1})\\}+1.\n$$\n\\end{corollary}\n\n\\begin{proof}\nTake the disjoint unions \n$$\nC=\\ensuremath{\\partial}\\sigma_1\\sqcup\\cdots\\sqcup\\ensuremath{\\partial}\\sigma_{k-1},\n$$ \n$$\nK_1=\\csqdunion{1\\leq 2i+1\\leq k-1}{}{L_{2i+1}}\n$$\n$$\nK_2=\\csqdunion{2\\leq 2i\\leq k-1}{}{L_{2i}},\n$$\nand take the iterated face deletions $K'_1=K_1\\backslash(\\sigma_1\\sqcup\\cdots\\sqcup\\sigma_{k-1})$ and\n$K'_2=K_2\\backslash(\\sigma_1\\sqcup\\cdots\\sqcup\\sigma_{k-1})$. \nThen $C$ is a full subcomplex common to both $K'_1$ and $K'_2$, \nand to both $\\sk{K'_1}{d-1}$ and $\\sk{K'_2}{d-1}$.\nMoreover, $\\sk{K'_1}{d-1}=\\sk{K_1}{d-1}$ and $\\sk{K'_2}{d-1}=\\sk{K_2}{d-1}$,\nand $K=K'_1\\cup_C K'_2$, so by the second part of Proposition~\\ref{PGluing2},\n$$\n\\cat(\\z{K})\\leq \\max\\{1,\\cat(\\zsk{K_1}{d-1}),\\cat(\\zsk{K_2}{d-1})\\}+\\Cat(\\z{C})\n$$\nIt is clear that $C$ is an $HMF$-complex, so $\\z{C}$ is homotopy equivalent to a wedge of spheres \nand $\\Cat(\\z{C})=1$ (alternatively, this follows from Theorem~$10.1$ in~\\cite{MR2321037}).\nMoreover, we can think of $\\zsk{K_1}{d-1}$ as being built up iteratively by gluing \n$\\bigsqcup_{1\\leq 1\\leq j}{}{\\zsk{L_{2i+1}}{d-1}}$ and $\\zsk{L_{2j+3}}{d-1}$ along the empty simplex,\nso iterating the first inequality in Example~\\ref{EGluing},\n$$\n\\cat(\\zsk{K_1}{d-1})\\leq \\max\\{\\,1,\\,\\cat(\\zsk{L_1}{d-1}),\\,\\cat(\\zsk{L_3}{d-1}),\\,\\ldots\\,\\}.\n$$ \nLikewise,\n$\n\\cat(\\zsk{K_2}{d-1})\\leq \\max\\{\\,1,\\,\\cat(\\zsk{L_2}{d-1}),\\,\\cat(\\zsk{L_4}{d-1}),\\,\\ldots\\,\\}.\n$ \nThe inequality in the lemma follows.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Triangulated Spheres}\n\n\n\n\nLet $\\mc C_0=\\{S^0\\}$, $\\mc C_1$, and $\\mc C_2$ consist of all triangulated $0$,$1$, and $2$-spheres, \nand for $d\\geq 3$, let $\\mc C_d$ be the class of triangulated $d$-spheres defined by $K\\in \\mc C_d$ if  \n\\begin{itemize}\n\\item[(1)] $K=L_1\\ast\\cdots\\ast L_k$ for some $L_i\\in\\mc C_{d_i}$, $d_i\\leq 2$, and $d_1+\\cdots+d_k=d-k+1$;  \n\\item[(2)] $K=K_1\\#_{\\sigma_1}\\cdots\\#_{\\sigma_{\\ell-1}}K_\\ell$ where $\\sigma_i$ is a $d$-face common to $K_i$ and $K_{i+1}$ \nwith $\\sigma_i\\cap\\sigma_j=\\emptyset$ when $i\\neq j$, and each $K_i=L_{1,i}\\ast\\cdots\\ast L_{k_i,i}$ \nis of the form (1) such that each $L_{j,i}$ is not the boundary of a simplex.\n\\end{itemize}\n\nThe join $L\\ast L'$ is the simplicial complex $\\cset{\\sigma\\sqcup\\sigma'}{\\sigma\\in L,\\,\\sigma'\\in L'}$, \nand the connected sum $K\\#_{\\sigma} K'$ is given topologically by gluing triangulations $K$ and $K'$ of $S^d$ \nalong a common $d$-face $\\sigma$, and deleting its interior.\n\n\n\n\\begin{bigremark}\nIf $L$ and $L'$ are boundaries $\\ensuremath{\\partial} P^*$ and $\\ensuremath{\\partial} P'^*$ of the duals of  simple polytopes $P$ and $P'$,\nthen $L\\ast L'$ is the boundary of $(P\\times P')^*$, while $L\\#_{\\sigma} L'$ is the boundary of dual $Q^*$, \nwhere $Q$ is obtained by taking the vertex cut at the vertices of $P$ and $P'$ that are dual to $\\sigma$, \ngluing along the new hyperplane and removing it after gluing. \n\\end{bigremark}\n\n\n\nOur goal in this section will be to show the following.\n\n\\begin{theorem}\n\\label{TMain}\nIf $K$ on vertex set $[n]=\\{1,\\ldots,n\\}$ is any triangulated $d$-sphere for $d=0,1,2$, \nor $K\\in \\mc C_d$ for $d\\geq 3$, then the following are equivalent.\n\\begin{itemize}\n\\item[(1)] $K$ is $m$-Golod over $\\ensuremath{\\mathbb{Z}}$;\n\\item[(2)] $\\Nill(\\mathrm{Tor}_{\\ensuremath{\\mathbb{Z}}[v_1,\\ldots,v_n]}(\\ensuremath{\\mathbb{Z}}[K],\\ensuremath{\\mathbb{Z}}))\\leq m+1$ (equivalently $\\cuplen(\\mathcal Z_K)\\leq m$);\n\\item[(3)] for any filtration of full subcomplexes \n$$\n\\ensuremath{\\partial}\\Delta^{d+2-\\ell}=K_{I_\\ell}\\subsetneq K_{I_{\\ell-1}}\\subsetneq\\cdots\\subsetneq K_{I_1}=K\n$$ \nsuch that $|K_{I_i}|\\cong S^{d+1-i}$, we have $\\ell\\leq m$; \n\\item[(4)] $\\cat(\\mathcal Z_K)\\leq m$.\n\\end{itemize}\nMoreover, $1\\leq m\\leq d+1$; that is, $K$ satisfies any of the above for some $m$ which cannot be greater than $d+1$.~$\\hfill\\square$ \n\\end{theorem}\n\n\n\n\n\n\n\\subsection{Well-Behaved Triangulations}\n\n\\begin{definition}\n\\label{DWell}\nWe say a triangulation $K$ of a $d$-sphere $S^d$ on vertex set $[n]$ is \\emph{well-behaved} if \nfor all $I\\subseteq[n]$ and any $k<d$:\n\\begin{itemize}\n\\item[(1)] if $|K_I|\\cong S^k$ and $K_I\\subseteq K_J$ for some $J\\subseteq[n]$ such that $|K_J|\\cong S^{k+1}$, \nthen $|K_{J-I}|\\simeq S^0$;\n\\item[(2)] if $H_*(K_I)\\cong H_*(S^k)$, then there is $I'\\subseteq[n]$ such that $K_{I'}\\subseteq K_I$ and $|K_{I'}|\\cong S^k$;\n\\item[(3)] if $I$ is a face in $K$, then $|K_{[n]-I}|\\simeq\\ast$;\n\\item[(4)] $H_*(K_I)$ is torsion-free.\n\\end{itemize}\n\\end{definition}\n\nAs we will see, conditions $(2)$-$(4)$ are there to keep the well-behaved condition invariant under join and connected sum operations.\n\n\n\\begin{lemma}\n\\label{Ld012Well}\nTriangulations $K$ of $0$, $1$, and $2$-spheres are well-behaved.\n\\end{lemma}\n\\begin{proof}\nThis is trivial for $d=0,1$. When $d=2$, Conditions~$(1)$ and~$(3)$ are also clear.\nSince $|K|$ is a sphere and $|K_I|$ has the homotopy type of a dimension $\\leq 1$ $CW$-complex when \n$I\\subsetneq [n]$, $H_*(K_I)$ is torsion-free. To see that Condition~$(2)$ holds, \nnotice that if $H_*(K_I)\\cong H_*(S^0)$ then we can pick $K_{I'}=S^0\\subseteq K_I$ ($|I'|=2$).\nIf $H_*(K_I)\\cong H_*(S^1)$, we obtain a $1$-dimensional full subcomplex $K_{I''}\\subseteq K_I$\nsuch that $H_*(K_{I''})\\cong H_*(S^1)$ via a sequence of elementary collapses of $2$-faces in $K_I$, \nand then we can choose $K_{I'}\\subseteq K_{I''}$ such that $|K_{I'}|\\cong S^1$.   \n\\end{proof}\n\n\nIf $K$ and $L$ are triangulations of the $d$-sphere $S^d$, then so is $K\\#_\\sigma L$.\nIf $K$ and $L$  are triangulations of $S^d$ and $S^{d'}$, \nthen $K\\ast L$ is a triangulation of $S^{d+d'+1}$ since $|K\\ast L|\\cong |K|\\ast|L|$.\n\n\n\n\n\n\n\\begin{lemma}\n\\label{LCSumWell}\nSuppose $K$ and $L$ are both triangulations of $S^d$, and $\\sigma$ is a $d$-face common to $K$ and $L$.\nIf $K$ and $L$ are well-behaved, then so is $K\\#_\\sigma L$.\n\\end{lemma}\n\n\\begin{lemma}\n\\label{LJoinWell}\nIf $K$ and $L$ are well-behaved triangulations of $S^d$ and $S^{d'}$, \nthen $K\\ast L$ is a well-behaved triangulation of $S^{d+d'+1}$.\n\\end{lemma}\n\nWe will need three lemmas before we can prove these.\n\n\n\n\n\n\n\n\\begin{lemma}\n\\label{LUseful}\nSuppose $K$ and $L$ are well-behaved triangulations on vertex sets $\\mc V$ and $\\mc V'$,\nand let $N$ be a full subcomplex of $K\\ast L$. \n\nIf $H_*(N)\\cong H_*(S^k)$ for some $k$, then for some $I\\subseteq\\mc V$ and $J\\subseteq\\mc V'$, \nwe have $N=K_I\\ast L_J$ such that $H_*(K_I)\\cong H_*(S^i)$ and $H_*(L_J)\\cong H_*(S^j)$.\nMoreover, if $|N|\\cong S^k$, then $|K_I|\\cong S^i$ and $|L_J|\\cong S^j$.\n\\end{lemma}\n\n\n\\begin{proof}\n\nSince $N$ is a full subcomplex of $K\\ast L$,\n$N=K_I\\ast L_J$ where $I\\sqcup J$ is the vertex set of $N$. \nSince $K$ and $L$ are well-behaved, \n$H_*(K_I)$ and $H_*(L_J)$ are torsion-free, and since \n\\begin{equation}\n\\label{ETorsion}\n\\tilde H_{*+1}(N)\\cong \\tilde H_{*+1}(|K_I|\\ast|L_J|)\\cong \n\\tilde H_*(|K_I|\\wedge|L_J|)\\cong \\tilde H_*(K_I)\\otimes\\tilde H_*(L_J),\n\\end{equation}\nthen $H_*(N)$ is torsion-free.\n\nIf $I=\\emptyset$ then $N=K_\\emptyset\\ast L_J=\\emptyset\\ast L_J=L_J$, and the lemma follows.\nSimilarly when $J=\\emptyset$. Assume $I\\neq\\emptyset$ and $J\\neq\\emptyset$. \nSuppose $H_*(N)\\cong H_*(S^k)$. Then by~\\eqref{ETorsion}, \nwe must have $H_*(K_I)\\cong H_*(S^i)$ and $H_*(L_J)\\cong H_*(S^j)$ for some $i,j$ such that $i+j+1=k$.\n\nNow suppose, moreover, that we have $|N|\\cong S^k$. Suppose $|K_I|\\not\\cong S^i$. \nSince $K$ is well-behaved and $H_*(K_I)\\cong H_*(S^i)$ , there exists a full subcomplex $K_{I'}\\subseteq K_I$\nsuch that $|K_{I'}|\\cong S^i$, and $K_{I'}\\neq K_I$ since $|K_I|\\not\\cong S^i$.\nThen $H_k(K_{I'}\\ast L_J)\\cong H_k(K_I\\ast L_J)=H_k(N)\\cong\\ensuremath{\\mathbb{Z}}$, \nso under the inlusion $K_{I'}\\ast L_J\\subseteq K_I\\ast L_J=N$, \nthere exists a cycle $\\omega$ in the simplicial chain subgroup $C_k(K_{I'}\\ast L_J)\\subseteq C_k(N)$ that represents  \na generator $\\ensuremath{\\mathbb{Z}}\\subseteq H_k(K_{I'}\\ast L_J)$. But since $\\dim (K_{I'}\\ast L_J)=\\dim N=k$, $K_{I'}\\ast L_J\\neq K_I\\ast L_J$, \nand the $k$-faces of $N$ generate $N$ (since $|N|$ is a sphere), $K_{I'}\\ast L_J$ does not contain all $k$-faces of $N$, \nso $\\omega$ must be a cycle in $C_{k}(N)$ that does not contain all $k$-faces of $N$. \nBut this is impossible since $|N|\\cong S^k$, so any $k$-dimensional cycle must contain all the $k$-faces, \nso we must have $|K_I|\\cong S^i$. By a similar argument, $|L_J|\\cong S^j$.\n\n\n\\end{proof}\n\n\n\n\n\\begin{lemma}\n\\label{LTorsion1}\nLet $K$ be a triangulation of a $d$-sphere $S^d$ on vertex set $[n]$. \nIf $H_*(K_I)$ is torsion-free for some $I\\subseteq[n]$, $I\\neq [n]$, \nthen so is $H_*(K_I\\backslash\\sigma)$ for any $d$-face $\\sigma\\in K_I$,\nand the inclusion \\incl{|\\ensuremath{\\partial}\\sigma|}{}{|K_I\\backslash\\sigma|} \ninduces an injection on homology groups with torsion-free cokernel.\n\\end{lemma}\n\n\\begin{proof}\nSince $|K_I|\\not\\cong|K|\\cong S^d$, \nany cycle in the simplicial chain group $C_d(K)$ consists of all the $d$-faces of $K$,\nand there are no cycles in $C_d(K_I)$ and $C_d(K_I\\backslash\\sigma)$, \nso $H_i(K_I)\\cong H_i(K_I\\backslash\\sigma)=0$ for $i\\geq d$.\nBy the homology long exact sequence for the cofibration sequence \n$|K_I\\backslash\\sigma|\\incladdr{}{|K_I|}\\mapaddr{}{|K_I|\/|K_I\\backslash\\sigma|\\cong S^d}$,\nwe have $H_i(K_I\\backslash\\sigma)\\cong H_i(K_I)$ for $i<d-1$, and a short exact sequence\n$$\n\\seqmmmm{0}{}{H_d(S^d)}{}{H_{d-1}(K_I\\backslash\\sigma)}{}{H_{d-1}(K_I)}{}{0}.\n$$\nTherefore $H_{d-1}(K_I\\backslash\\sigma)$ is torsion-free since $H_{d-1}(K_I)$ and $H_d(S^d)\\cong\\mb Z$ are,\nand the second map in the sequence has torsion-free cokernel.\nSince $|K_I|$ is the homotopy cofiber of \\incl{S^{d-1}\\cong|\\ensuremath{\\partial}\\sigma|}{}{|K_I\\backslash\\sigma|},\nthe second map is induced by this inclusion in degree $d-1$, \nso it induces an injection with torsion-free cokernel on each homology group.\n\\end{proof}\n\n\n\n\\begin{lemma}\n\\label{LTorsion2}\nSuppose $K$ and $L$ are triangulations of $S^d$ on vertex sets $\\mc V$ and $\\mc V'$,\n$\\sigma$ is a $d$-face common to $K$ and $L$, and take $K\\#_{\\sigma} L$ on vertex set $\\mc W$. \n\nIf $H_*(K_I)$ and $H_*(L_J)$ are torsion-free for each $I\\subseteq\\mc V$ and $J\\subseteq\\mc V'$, \nthen $H_*((K\\#_{\\sigma} L)_{\\mc I})$ is torsion-free for each subset $\\mc I\\subseteq\\mc W$.\n\\end{lemma}\n\n\\begin{proof}\nLet $Z=K\\#_{\\sigma} L$.\nGiven $\\mc I\\subseteq\\mc W$, and $I=\\mc V\\cap\\mc I$ and $J=\\mc V'\\cap\\mc I$, \nwe have $|Z_{\\mc I}|$ is homotopy equivalent to: \n(a) $|K_I|\\vee |L_J|$ when $\\mc I\\cap\\sigma\\neq\\emptyset$ and $\\sigma\\not\\subseteq\\mc I$;\n(b) $|K_I|\\coprod |L_J|$ when $\\mc I\\cap\\sigma=\\emptyset$;\n(c) otherwise it is equal to $|K_I|\\#_{\\sigma}|L_J|$ when $\\sigma\\subseteq\\mc I$. \nTherefore $H_*(Z_{\\mc I})$ is torsion free in the first two cases, or the last case when $\\mc I=\\mc W$. \nTo see that it is torsion-free in the last case when $\\mc I\\neq\\mc W$, \nnotice that one of $I\\neq\\mc V$ or $J\\neq\\mc V'$. If $I=\\mc V$ or $J=\\mc V'$, \nthen $|Z_{\\mc I}|\\cong |L_J|$ or $|Z_{\\mc I}|\\cong |K_I|$ respectively, and we are done. \nAssume $I\\neq\\mc V$ and $J\\neq\\mc V'$. \nNotice that $|K_I\\backslash\\sigma|\\cup|L_J\\backslash\\sigma|=|Z_{\\mc I}|$, \n$|K_I\\backslash\\sigma|\\cap|L_J\\backslash\\sigma|=|\\ensuremath{\\partial}\\sigma|$, and by Lemma~\\ref{LTorsion1},\n$H_*(K_I\\backslash\\sigma)$ and $H_*(L_J\\backslash\\sigma)$ are torsion-free,\nwith the inclusions \\incl{|\\ensuremath{\\partial}\\sigma|}{}{|K_I\\backslash\\sigma|} and \\incl{|\\ensuremath{\\partial}\\sigma|}{}{|L_J\\backslash\\sigma|}  \ninducing injections on homology groups with torsion-free cokernel. \nThen it follows by the Mayer-Veitoris sequence for $(|Z_{\\mc I}|,|K_I\\backslash\\sigma|,|L_J\\backslash\\sigma|)$\nthat $H_*(Z_{\\mc I})$ is torsion free. \n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Lemma~\\ref{LCSumWell}]\nLet $Z=K\\#_\\sigma L$ on vertex set $\\mc W$, and $K$ and $L$ be on vertex sets $\\mc V$ and $\\mc V'$.\nBy Lemma~\\ref{LTorsion2}, any full subcomplex $Z_{\\mc I}$ has torsion-free homology,\nso condition~$(4)$ in Definition~\\ref{DWell} holds for $Z_{\\mc I}$.\n\nSince $Z$ is $K\\backslash\\sigma$ and $L\\backslash\\sigma$ glued along the boundary of the $d$-simplex $\\ensuremath{\\partial}\\sigma$, \n$\\mc I$ is a face in $K$ or $L$ whenever it is a face in $Z$.\nThen $K$ and $L$ being well-behaved, \nwe have $|Z_{\\mc W-\\mc I}|\\simeq |K_{\\mc V-\\mc I}|\\vee|L_{\\mc V'-\\mc I}|\\simeq\\ast$ when $\\mc I\\cap\\sigma\\neq\\emptyset$. \nSince $|L\\backslash\\sigma|\\cong |K\\backslash\\sigma|\\cong D^d$,\nthen $|Z_{\\mc W-\\mc I}|\\cong |K_{\\mc V-\\mc I}|\\simeq\\ast$ or $|Z_{\\mc W-\\mc I}|\\cong |L_{\\mc V'-\\mc I}|\\simeq\\ast$\nwhen $\\mc I\\cap\\sigma=\\emptyset$. Condition~$(3)$ holds as well.\n\nIf $|Z_{\\mc I}|\\cong S^k$ for some $k<d$, \nthen $Z_{\\mc I}$ is either a full subcomplex of one of $K$ or $L$, or else $Z_{\\mc I}=\\ensuremath{\\partial}\\sigma$\n(otherwise $Z_{\\mc I}$ would have a $(k-1)$-face contained in three $k$-faces)\nso the only case to check is the last one $Z_{\\mc I}=\\ensuremath{\\partial}\\sigma$. \nHere, condition~$(2)$ in Definition~\\ref{DWell} clearly holds. \nSince $K$ and $L$ are well-behaved, $|K_{\\mc V-\\sigma}|\\simeq|L_{\\mc V'-\\sigma}|\\simeq\\ast$,\nso it follows that $|Z_{\\mc W-\\sigma}|\\simeq S^0$, and condition~$(1)$ holds as well.\n\n\\end{proof}\n\n\n\n\n\\begin{proof}[Proof of Lemma~\\ref{LJoinWell}]\n\nLet $K$ and $L$ be on vertex sets $\\mc V$ and $\\mc V'$, $Z=K\\ast L$ be on vertex set $\\mc W=\\mc V\\sqcup\\mc V'$,\n and let $\\mc I\\subseteq \\mc W$. \nThen $Z_{\\mc I}=K_I\\ast L_J$ where $I\\sqcup J=\\mc I$, and since $H_*(K_I)$ and $H_*(L_J)$ are torsion-free,\n$\\tilde H_{*+1}(Z_{\\mc I})\\cong \\tilde H_*(K_I)\\otimes\\tilde H_*(L_J)$ is torsion-free.    \n\nIf $\\mc I$ is a face in $Z$, then $I$ and $J$ are faces in $K$ and $L$,\nso we have $|K_{\\mc V-I}|\\simeq\\ast$ and $|L_{\\mc V'-J}|\\simeq\\ast$ since $K$ and $L$ are well-behaved.\nTherefore $|Z_{\\mc W-\\mc I}|=|K_{\\mc V-I}\\ast L_{\\mc V'-J}|\\simeq\\ast$,\nand condition~$(3)$ of Definition~\\ref{DWell} holds.\n\nSuppose $H_*(Z_{\\mc I})\\cong H_*(S^k)$. \nSuppose $I\\neq\\emptyset$ and $J\\neq\\emptyset$. \nBy Lemma~\\ref{LUseful}, $H_*(K_I)\\cong H_*(S^i)$ and $H_*(L_J)\\cong H_*(S^j)$ for some $i$,$j$ such that $i+j+1=k$.\nSince $K$ and $L$ and well-behaved, this implies there are $K_{I'}\\subseteq K_I$ and $L_{J'}\\subseteq L_J$ \nsuch that $|K_I|\\cong S^i$ are $|L_J|\\cong S^j$. \nThen \n$Z_{I'\\sqcup J'}=K_{I'}\\ast L_{J'}\\subseteq K_I\\ast K_J=Z_{\\mc I}$\nand $|Z_{I'\\sqcup J'}|\\cong S^{i+j+1}$.\nThus condition~$(2)$ holds when $I\\neq\\emptyset$ and $J\\neq\\emptyset$.  \n\nNow consider that case where $I=\\emptyset$ or $J=\\emptyset$. \nWithout loss of generality, suppose $I=\\emptyset$.\nWe have $Z_{\\mc I}=K_\\emptyset\\ast L_J=\\emptyset\\ast L_J=L_J$,\nso if $H_*(Z_{\\mc I})\\cong H_*(S^k)$, then $H_*(L_J)\\cong S^k$, and we have $L_{J'}\\subseteq L_J=Z_{\\mc I}$\nsuch that $|L_{J'}|\\cong S^k$ since $L$ is well-behaved.\nThus condition~$(2)$ holds when $I=\\emptyset$ or $J=\\emptyset$.  \n\n\n\nLet $S^{-1}=\\emptyset$.\nSuppose $|Z_{\\mc I}|\\cong S^k$ and $Z_{\\mc I}\\subseteq Z_{\\mc J}$ such that $|Z_{\\mc J}|\\cong S^{k+1}$.\nThen $Z_{\\mc J}=K_{I'}\\ast L_{J'}$ where $I'\\sqcup J'=\\mc J\\subsetneq\\mc I=I\\sqcup J$, \nso $K_I\\subseteq K_{I'}$ and $L_J\\subseteq L_{J'}$,\nand by Lemma~\\ref{LUseful}, we must have $|K_I|\\cong S^i$, $|L_J|\\cong S^j$, $|K_{I'}|\\cong S^{i'}$, $|L_{J'}|\\cong S^{j'}$, \nwhere $i+j+1=k$ and $i'+j'+1=k+1$. Since $K_I\\subseteq K_{I'}$ and $L_J\\subseteq L_{J'}$, then $i\\leq i'$ and $j\\leq j'$,\nso either (a) $i'=i$ and $j'=j+1$; or (b) $i'=i+1$ and $j'=j$. Without any loss of generality, assume (a) holds. \nThen $K_I=K_{I'}$ (i.e. $I=I'$), $K_{I-I'}=\\emptyset$, and\n$$\n|Z_{\\mc J-\\mc I}|=|K_{I'-I}\\ast L_{J'-J}|=|L_{J'-J}|\\simeq S^0,\n$$\nwith the last homotopy equivalence since $L$ is well-behaved. \nIt follows that condition~$(1)$ of Definition~\\ref{DWell} holds. \n\n\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Spherical Filtrations and Filtration Length}\n\n\n\n\n\n\\begin{definition}\nGiven a triangulation of a $d$-sphere $K$ on vertex set $[n]$, \nsuppose we have a filtration of full subcomplexes\n$$\n\\ensuremath{\\partial}\\Delta^{d+2-\\ell}=K_{I_\\ell}\\subsetneq\\cdots\\subsetneq K_{I_1}=K\n$$ \nsuch that $K_{I_i}$ is a triangulation of a $(d+1-i)$-sphere.\nThen we say that this is a \\emph{spherical filtration} of $K$ of length $\\ell$. \n\\end{definition}\n\n\\begin{remark}\nImplicitly, $I_\\ell\\subsetneq\\cdots\\subsetneq I_1=[n]$ and $|I_\\ell|=d+3-\\ell$. \n\\end{remark}\n\n\\begin{definition}\nFor any triangulated sphere $K$, \ndefine the \\emph{filtration length} $\\filt(K)$ to be the largest integer $\\ell$ \nsuch that $K$ admits a spherical filtration of length $\\ell$.\n\\end{definition}\n\n\nFor example, $\\filt(\\ensuremath{\\partial}\\Delta^{d+1})=1$. \nGenerally, there are the following bounds with respect to joins, connected sums, and cup product length. \n\n\n\n\\begin{lemma}\n\\label{LFiltCSum}\nIf $K$ and $L$ are both triangulations of $S^d$, and $\\sigma$ is a $d$-face common to $K$ and $L$, \nthen \n$$\n\\filt(K\\#_\\sigma L)\\geq \\max\\{2,\\filt(K),\\filt(L)\\}.\n$$\n\\end{lemma}\n\n\\begin{proof}\nAs in the proof of Lemma~\\ref{LCSumWell},\na full subcomplex $Z_{\\mc I}$ of $K\\#_\\sigma L$ satisfying $|Z_{\\mc I}|\\cong S^k$ for some $k<d$ must either be a full subcomplex\nof exactly one of $K$ or $L$, or else $Z_{\\mc I}=\\ensuremath{\\partial}\\sigma$. \nMoreover, $K\\#_\\sigma L$ always has the length $2$ spherical filtration $\\ensuremath{\\partial}\\sigma\\subsetneq K\\#_\\sigma L$.\nThe lemma follows immediately. \n\\end{proof}\n\n\n\n\n\n\n\n\\begin{lemma}\n\\label{LFiltJoin}\nIf $K$ and $L$ are any triangulated spheres, then \n$$\n\\filt(K\\ast L)\\geq \\filt(K)+\\filt(L).\n$$\n\\end{lemma}\n\n\n\n\\begin{proof}\n\nLet $K$ and $L$ be on vertex sets $[n]$ and $[m]$.\nLet $d=\\dim K$, $d'=\\dim L'$, $\\ell=\\filt(K)$, $\\ell'=\\filt(L)$, and take \n$\\ensuremath{\\partial}\\Delta^{d+2-\\ell}=K_{I_\\ell}\\subsetneq\\cdots\\subsetneq K_{I_1}=K$ and\n$\\ensuremath{\\partial}\\Delta^{d'+2-\\ell'}=L_{J_{\\ell'}}\\subsetneq\\cdots\\subsetneq L_{J_1}=L$ \nto be spherical filtrations of $K$ and $L$.\n\nSince $|K_{I_i}\\ast L_{J_j}|\\cong |K_{I_i}|\\ast|L_{J_j}|\\cong S^{d+1-i}\\ast S^{d'+1-j}\\cong S^{d+d'-i-j+3}$,\nand  $|K_{I_i}\\ast L_{J_j}|$ is a full subcomplex of $|K_{I_{i'}}\\ast L_{J_{j'}}|$ when $i\\leq i'$ and $j\\leq j'$,\nwe have a length $\\ell+\\ell'$ spherical filtration of $K\\ast L$\n$$\n\\ensuremath{\\partial}\\Delta^{d+2-\\ell}\n\\subsetneq(K_{I_\\ell}\\ast L_{J_{\\ell'}})\\subsetneq\\cdots\n\\subsetneq (K_{I_\\ell}\\ast L_{J_2})\n\\subsetneq (K_{I_\\ell}\\ast L_{J_1})\n\\subsetneq (K_{I_{\\ell-1}}\\ast L_{J_1})\n\\subsetneq\\cdots\\subsetneq (K_{I_1}\\ast L_{J_1})=K\\ast L, \n$$\nwhere $\\ensuremath{\\partial}\\Delta^{d+2-\\ell}$ is the full subcomplex of \n$K_{I_\\ell}\\ast L_{J_{\\ell'}}=\\ensuremath{\\partial}\\Delta^{d+2-\\ell}\\ast\\ensuremath{\\partial}\\Delta^{d'+2-\\ell'}$\non the vertices of $\\ensuremath{\\partial}\\Delta^{d+2-\\ell}$. Therefore $\\filt(K\\ast L)\\geq\\ell+\\ell'$. \n\n\\end{proof}\n\n\n\n\\vspace{5mm}\n\n\n\n\n\n\n\n\\begin{lemma}\n\\label{LFiltCup}\nIf $K$ is a well-behaved triangulation of $S^d$, then $\\filt(K)\\leq\\cuplen(\\mathcal Z_K)$.\n\\end{lemma}\n\n\\begin{proof}\n\nLet $\\ell=\\filt(K)$ and $\\ensuremath{\\partial}\\Delta^{d+1-\\ell}=K_{I_\\ell}\\subsetneq\\cdots\\subsetneq K_{I_1}=K$ be a spherical filtration\nof length $\\ell$. \n\nLet $J_{i+1}=I_i-I_{i+1}$ for $i<\\ell$. Since $K$ is well-behaved, $|K_{J_{i+1}}|\\simeq S^0$,\nso $|K_{J_{i+1}}|\\cong D_i\\coprod D'_i$ for some contractible subcomplexes $D_i$ and $D'_i$ of $|K_{I_i}|$.\nTake the inclusion induced by \\incl{K_{I_i}}{}{K_{I_{i+1}}\\ast K_{J_{i+1}}}\n$$\n\\iota_i\\ensuremath{\\,\\colon\\,}\\incl{|K_{I_i}|}{}{|K_{I_{i+1}}\\ast K_{J_{i+1}}|\\cong |K_{I_{i+1}}|\\ast |K_{J_{i+1}}|},\n$$ \nand let $h_i\\colon\\seqm{|K_{I_{i+1}}|\\ast |K_{J_{i+1}}|}{}{|K_{I_{i+1}}|\\ast S^0}$\nbe the join of the identity map on the left factor with the map collapsing $D_i$ and $D'_i$ to $-1$ and $1$ \nin $S^0=\\{-1,1\\}$ on the right factor. Then $h_i$ is a homotopy equivalence since $D_i$ and $D'_i$ are contractible. \nNotice $h_i\\circ\\iota_i$ is the quotient map that collapses $D_i$ to a point and $D'_i$ to another point.\nSince $D_i$ and $D'_i$ are disjoint contractible subcomplexes of $|K_{I_i}|$, \n$h_i\\circ\\iota_i$ is a homotopy equivalence. Therefore, $\\iota_i$ is a homotopy equivalence.\n\nTake the composite of inclusions\n\\begin{equation}\n\\label{EComposite}\n|K|=|K_1|\\incladdr{\\iota_1}{|K_{I_2}\\ast K_{J_2}|}\\incladdr{\\iota'_2}{|K_{I_3}\\ast K_{J_3}\\ast K_{J_2}|}\\incladdr{\\iota'_3}{\\cdots}\n\\incladdr{\\iota'_{\\ell-1}}{|K_{I_\\ell}\\ast K_{J_\\ell}\\ast\\cdots\\ast K_{J_2}|}.\n\\end{equation}\nThe $i^{th}$ map ($i\\geq 2$) in this composite  \n$$\n\\iota'_i\\ensuremath{\\,\\colon\\,}\n\\incl{|K_{I_i}\\ast K_{J_i}\\ast\\cdots\\ast K_{J_2}|}{}{|K_{I_{i+1}}\\ast K_{J_{i+1}}\\ast K_{J_i}\\ast\\cdots\\ast K_{J_2}|}\n$$\nis the join of the homotopy equivalence $\\iota_i$ and the identity \\seqm{|K_{J_i}\\ast\\cdots\\ast K_{J_2}|}{\\ensuremath{\\mathbbm{1}}_i}{|K_{J_i}\\ast\\cdots\\ast K_{J_2}|}, \nand since each $|K_{J_j}|\\simeq S^0$, then $\\ensuremath{\\mathbbm{1}}_i$ is homotopy equivalent to the identity \\seqm{S^{i-2}}{\\ensuremath{\\mathbbm{1}}}{S^{i-2}}.\nEach $\\iota'_i$ (and $\\iota_1$) is therefore a homotopy equivalence, and then so is composite~\\eqref{EComposite}\n(in fact, since $|K_{I_i}|\\cong S^{d+1-i}$, $\\iota_i$ and~\\eqref{EComposite} are homotopy equivalent to homeomorphisms \n\\seqm{S^{d+1-i}}{\\cong}{S^{d+1-i}} and \\seqm{S^d}{\\cong}{S^d}). \nTherefore~\\eqref{EComposite} induces a non-trivial map on cohomology,\nand so the Hochster theorem implies there is a non-trivial length $\\ell$ cup product in $H^+(\\mathcal Z_K)$.\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Triangulated $d$-spheres for $d=0,1,2$}\nThe only triangulated $0$-sphere is $S^0$, and the only triangulated $1$-sphere with $n\\geq 3$ vertices is the $n$-gon. \nWe will say that $C$ is a \\emph{chordless cycle} in $K$ if $C$ is a full subcomplex $K_I$ of $K$, \nand $C$ is an $m$-gon for some $m\\geq 3$. \nWhen $K$ is a graph, this is the same as $C$ being an induced cycle of $G$.\n$K$ is said to be \\emph{chordal} if it contains no chordless cycles with $4$ or more vertices.\n\n\n\\begin{lemma} \n\\label{LCup1}\nIf $K$ is a triangulation of $S^1$ on vertex set $[n]$, then\n\\begin{itemize}\n\\item[(i)] $\\filt(K)\\geq 2$ whenever $K$ has at least $4$ vertices.\n\\end{itemize}\nIf $K$ is a triangulation of $S^2$, then \n\\begin{itemize}\n\\item[(ii)] $\\filt(K)\\geq 2$ whenever $K$ has a chordless cycle;\n\\item[(iii)] $\\filt(K)\\geq 3$ whenever $K$ has a chordless cycle with at least $4$ vertices.\n\\end{itemize}\n\\end{lemma}\n\n\n\\begin{proof}\n\nLet $|K|\\cong S^1$. If $K$ has at least $n\\geq 4$ vertices, \nthen $K$ being an $n$-gon means we can take $I'\\subset [n]$, $|I'|=2$, \nsuch that $|K_{I'}|=S^0$. Then $S^0\\subsetneq K$ is a length $2$ spherical filtration of $K$. \n\nLet $|K|\\cong S^2$, and $C=K_I$ be a chordless cycle for some $I\\subset [n]$. \nWe have $|K_I|\\cong S^1$ and $K_I=\\ensuremath{\\partial}\\Delta^2$ when $K_I$ has $3$ vertices, \nin which case $K_I\\subsetneq K$ is a length $2$ spherical filtration.\nOtherwise, when $K_I$ has at least $4$ vertices, $\\filt(K)\\geq 3$ since \nthere is a spherical filtration $S^0\\subsetneq K_I\\subsetneq K$ by part~$(i)$.\n\\end{proof}\n\n\n\n\\begin{proposition} \n\\label{Pd012}\nLet $K$ be a triangulated $d$-sphere, $d=0,1,2$. Then \n$$\n1\\leq\\filt(K)=\\cuplen(\\mathcal Z_K)=\\cat(\\mathcal Z_K)\\leq d+1.\n$$\nIn particular, letting $m=\\cat(\\mathcal Z_K)$, when $d=1$ we have $m=1$ iff $K=\\ensuremath{\\partial}\\Delta^2$, and $m=2$ iff $K$ at least $4$-vertices;\nand when $d=2$ we have $m=1$ iff $K=\\ensuremath{\\partial}\\Delta^3$, $m=2$ iff $K$ has a chordless cycle but none with more than $3$ vertices,\nand $m=3$ iff $K$ has a chordless cycle with at least $4$ vertices.\n\n\\end{proposition}\n\n\\begin{proof} \n\n\nThe case $d=0$ is immediate since $S^0$ is the only triangulated $0$-sphere, and $\\z{S^0}\\cong S^3$. \nSince $K$ is well-behaved by Lemma~\\ref{Ld012Well},\nwe have by Lemma~\\ref{LSkeleta2} and Lemma~\\ref{LFiltCup}\n$$\n1\\leq\\filt(K)\\leq\\cuplen(\\mathcal Z_K)\\leq\\cat(\\mathcal Z_K)\\leq d+1,\n$$ \nso it remains to show that $\\cat(\\mathcal Z_K)\\leq\\filt(K)$. \n\nFix $d=1$. Using Lemma~\\ref{LCup1}, if $K$ has at least $4$ vertices, \nthen $\\filt(K)\\geq 2$, so $\\cat(\\mathcal Z_K)\\leq\\filt(K)$ since $\\cat(\\mathcal Z_K)\\leq d+1=2$. \nOtherwise, if $K$ has $3$-vertices, then $K=\\ensuremath{\\partial}\\Delta^2$ and $\\mathcal Z_K\\cong S^5$, so $\\filt(K)=\\cat(\\mathcal Z_K)=1$. \n\nNow fix $d=2$. \nBy Lemma~\\ref{LCup1}, $\\filt(K)\\geq 3$ whenever $K$ has chordless cycles with at least $4$ vertices, \nso $\\cat(\\mathcal Z_K)\\leq\\filt(K)$ since $\\cat(\\mathcal Z_K)\\leq d+1=3$. \nOn the other hand, suppose $K$ has chordless cycles, but none with more than $3$ vertices. \nThen $\\filt(K)=2$ by Lemma~\\ref{LCup1}, and the $1$-skeleton $\\sk{K}{1}$ is a chordal graph. \nThe chordal property is closed under vertex deletion (taking full subcomplexes).\nMoreover, recall from~\\cite{MR0186421} that chordal graphs have a \\emph{total elimination ordering}, that is,\nthey can be built up one vertex $v$ at a time in some order such that at each step the neighbours of $v$ form a clique. \nInducting on this ordering, one sees that chordal graphs are $HMF$-complexes,\ntherefore $\\cat(\\zsk{K}{1})\\leq 1$ by Proposition~\\ref{PExtractible} (this also follows the main result in~\\cite{2015arXiv150608970I}).\nUsing Lemma~\\ref{LSkeleta2}, we have $\\cat(\\mathcal Z_K)\\leq \\cat(\\zsk{K}{1})+1\\leq 2=\\filt(K)$.\nOtherwise, $K=\\ensuremath{\\partial}\\Delta^3$ when $K$ has no chordless cycles at all, so we have $\\mathcal Z_K\\cong S^7$ and $\\filt(K)=\\cat(\\mathcal Z_K)=1$. \n\n\\end{proof}\n\n\n\n\\begin{corollary}\n\\label{Cd012}\nLet $K$ be a triangulated $d$-sphere, $d=1,2$, then \n$$\n\\cat(\\zsk{K}{d-1})\\leq\\max\\{1,\\cat(\\mathcal Z_K)-1\\}=\\max\\{1,\\filt(K)-1\\}.\n$$\n\\end{corollary}\n\n\\begin{proof}\nThe last equality $\\filt(K)=\\cat(\\mathcal Z_K)$ is from the previous proposition. Let $m=\\cat(\\mathcal Z_K)$. \nUsing Proposition~\\ref{Pd012}, the statement simplifies to $\\cat(\\zsk{K}{0})\\leq 1$ when $d=1$, or $d=2$ and $m=1$.\nThis is true since $\\zsk{K}{0}$ has the homotopy type of a wedge of spheres by~\\cite{MR2138475}, \nor by Proposition~\\ref{PExtractible} (as $\\sk{K}{0}$ is a collection of disjoint points).\nFix $d=2$. As in the proof of Propostion~\\ref{Pd012}, $\\cat(\\zsk{K}{1})\\leq 1<m$ when $m=2$, since $\\filt(K)=2$. \nThis last inequality also holds when $m=3$ since $\\cat(\\zsk{K}{1})\\leq \\dim\\sk{K}{1}+1=2$ is always true by Lemma~\\ref{LSkeleta2}. \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Triangulated $d$-spheres for $d\\geq 3$}\n\n\n\n\n\\begin{proposition}\n\\label{Pdinfinity}\nSuppose $K\\in\\mc C_d$, $d\\geq 0$. Then \n$$\n1\\leq\\filt(K)=\\cuplen(\\mathcal Z_K)=\\cat(\\mathcal Z_K)\\leq d+1.\n$$\n\\end{proposition}\n\n\n\n\\begin{proof}\nTriangulations of $0,1,2$-spheres are well-behaved by Lemma~\\ref{Ld012Well}, \nso iterating Lemmas~\\ref{LJoinWell} and~\\ref{LCSumWell}, any $K\\in\\mc C_d$ is well-behaved.\nThen by Lemmas~\\ref{LSkeleta2} and~\\ref{LFiltCup},\n$$\n1\\leq\\filt(K)\\leq\\cuplen(\\mathcal Z_K)\\leq\\cat(\\mathcal Z_K)\\leq d+1.\n$$ \nIt remains to show that $\\cat(\\mathcal Z_K)\\leq\\filt(K)$. The $d=0,1,2$ case is Proposition~\\ref{Pd012}.\n\nSuppose $K=L_1\\ast\\cdots\\ast L_k\\in \\mc C_d$ for some $L_i\\in\\mc C_{d_i}$, $d_i\\leq 2$, and $d_1+\\cdots+d_k=d-k-1$.\nThen $\\mathcal Z_K=\\z{L_1}\\times\\cdots\\times\\z{L_k}$, and so using Proposition~\\ref{Pd012} and iterating Lemma~\\ref{LFiltJoin},\n$$\n\\cat(\\mathcal Z_K)\\leq \\csum{i=1,\\ldots,k}{}{\\cat(\\z{L_i})} = \\csum{i=1,\\ldots,k}{}{\\filt(L_i)} \\leq \\filt(L_1\\ast\\cdots\\ast L_k)=\\filt(K).\n$$ \nMoreover, iterating Corollary~\\ref{CJoinSkeleton}, and using Corollary~\\ref{Cd012}, when each $L_i\\neq S^0$\n$$\n\\cat(\\zsk{K}{d-1})\\leq \\csum{i=1,\\ldots,k}{}{\\cat(\\zsk{L_i}{d_i-1})}+k-1\\leq\\csum{i=1,\\ldots,k}{}{\\max\\{1,\\filt(L_i)-1\\}}+k-1,\n$$\nand when each $L_i\\neq\\ensuremath{\\partial}\\Delta^{d_i+1}$, we have $\\filt(L_i)>1$, therefore \n\\begin{equation}\n\\label{ESk}\n\\cat(\\zsk{K}{d-1})\\leq \\paren{\\csum{i=1,\\ldots,k}{}{\\filt(L_i)}}-1\\leq \\filt(L_1\\ast\\cdots\\ast L_k)-1=\\filt(K)-1,\n\\end{equation}\nthe second inequality by iterating Lemma~\\ref{LFiltJoin}.\n\n\nSuppose $K=K_1\\#_{\\sigma_1}\\cdots\\#_{\\sigma_{\\ell-1}}K_\\ell$ where each $K_i=L_{1,i}\\ast\\cdots\\ast L_{k_i,i}$ \nis a join of the above form such that each $L_{j,i}$ is not the boundary of a simplex,\nand $\\sigma_i$ is a $d$-face common to $K_i$ and $K_{i+1}$ with $\\sigma_i\\cap\\sigma_j=\\emptyset$ when $i\\neq j$.\nBy Corollary~\\ref{CConnectedSum} and inequality~\\eqref{ESk}, we have \n\\begin{align*}\n\\cat(\\mathcal Z_K) & \\leq \\max\\{1,\\cat(\\zsk{K_1}{d-1}),\\ldots,\\cat(\\zsk{K_\\ell}{d-1})\\}+1\\\\\n& \\leq \\max\\{1,\\filt(K_1)-1,\\ldots,\\filt(K_\\ell)-1\\}+1\\\\\n& = \\max\\{2,\\filt(K_1),\\ldots,\\filt(K_\\ell)\\}\\\\\n& = \\filt(K_1\\#_{\\sigma_1}\\cdots\\#_{\\sigma_{\\ell-1}}K_\\ell)\\\\\n& = \\filt(K),\n\\end{align*}\nwhere the second last inequality follows from iterating Lemma~\\ref{LFiltCSum}.\n\n\\end{proof}\n\n\n\\subsection{Proof of Theorem~\\ref{TMain}}\n\nThe following is an immediate consequence of Theorem~$4.4$ in~\\cite{MR1644063} and the fact that \n\\emph{category weight} $cwgt()$ as defined there is bounded below by $1$, \nand linearly below with respect to cup products.\n\n\n\\begin{theorem}[Rudyak~\\cite{MR1644063}]\nIf $\\cat(X)\\leq m$, then\n\\begin{itemize}\n\\item[(1)] $\\cuplen(X)\\leq m$;\n\\item[(2)] Massey products $\\vbr{v_1,\\ldots,v_k}$ vanish in $H^*(X)$ \nwhenever $v_i=a_1\\cdots a_{m_i}$ and $v_j=b_1\\cdots b_{m_j}$, and $m_i+m_j>m$, for some odd $i$ and even $j$ \nand $a_s,b_t\\in H^+(X)$.~$\\hfill\\square$\n\\end{itemize}\n\\end{theorem}\n\nBy our remarks in Section~\\ref{SHochster}, and by definition $\\cuplen(X)=\\Nill H^*(X)-1$, we obtain\n\n\\begin{proposition}\n\\label{PRudyak}\nIf $\\cat(\\mathcal Z_K)\\leq m$, then $K$ is $m$-Golod.~$\\hfill\\square$ \n\\end{proposition}\n\n\nNow Theorem~\\ref{TMain} follows from Propositions~\\ref{Pd012} and~\\ref{Pdinfinity}, \nand the fact that $\\cuplen(\\mathcal Z_K)\\leq m$ when $K$ is $m$-Golod (by definition).\n\n\n\n\n\n\n\n\n\n\n\\section{Further Applications}\n\n\n\n\n\n\\subsection{Fullerenes}\nA \\emph{fullerene} $P$ is a simple $3$-polytope all of whose $2$-faces are pentagons and hexagons. \nThese are mathematical idealisations of physical fullerenes - \nspherical molecules of carbon such that each carbon atom belongs to three carbon rings, and each carbon ring is either a pentagon or hexagon. \n\n\nThe authors in~\\cite{2015arXiv151103624F} have shown that the cohomology ring of moment-angle complexes is a complete combinatorial invariant of fullerenes,\nwhile Buchstaber and Erokhovets~\\cite{2015arXiv151002948B,fullerene} show that the finer details of their cohomology encode many interesting properties of fullerenes.\nFor example, if $P^\\ast$ is the dual of $P$, then the bigraded Betti numbers of $\\z{\\ensuremath{\\partial} P^\\ast}$ count the number $k$-\\emph{belts} in $P$. \nHere, a $k$-belt of a simple polytope such as $P$ is a sequence of $2$-faces $(F_1,\\ldots,F_k)$ such that\n$F_k\\cap F_1\\neq\\emptyset$, $F_i\\cap F_{i+1}\\neq\\emptyset$ for $1\\leq i\\leq k-1$, and all other intersections are empty.\nNotice that the $k$-belts of $P$ correspond to full subcomplexes of $\\ensuremath{\\partial} P^\\ast$ that are $n$-gons.  \nBut since fullerenes can have no $3$-belts~\\cite{2015arXiv151002948B,fullerene}, $\\ensuremath{\\partial} P^\\ast$ must only have $n$-gons as full subcomplexes for $n\\geq 4$.\nMoreover, since $\\ensuremath{\\partial} P^\\ast$ is a triangulated $2$-sphere that is not a boundary of the $2$-simplex, it must have at least one such $n$-gon as a full subcomplex. \nThus, $\\filt(\\ensuremath{\\partial} P^\\ast)=3$, so by Theorem~\\ref{TMain}, \n\\begin{proposition}\nFor fullerences $P$, $\\cat(\\z{\\ensuremath{\\partial} P^\\ast})=3$ and $\\ensuremath{\\partial} P^\\ast$ is $3$-Golod.\nIn particular, all Massey products consisting of decomposable elements in $H^+(\\z{\\ensuremath{\\partial} P^\\ast})$ must vanish.~$\\hfill\\square$\n\\end{proposition}\n\n\n\n\n\n\n\\subsection{Neighbourly Complexes}\n\nFor any finite simply-connected $CW$-complex $X$, let \n$$\n\\hd(X)=\\max\\br{\\,\\max\\cset{i}{\\tilde H^i(X)\\otimes\\mb Q\\neq 0},\\,\\max\\cset{i}{Torsion(\\tilde H^{i-1}(X))\\neq 0}\\,}\n$$ \nand \n$$\n\\hc(X)=\\min\\cset{i}{\\tilde H^{i+1}(X)\\neq 0}.\n$$\nThese coincide with the dimension and connectivity of $X$ up to homotopy equivalence.\nIt is well known (c.f.~\\cite{MR516214}) that $X$ satisfies \n\\begin{equation}\n\\label{EOld}\n\\cat(X)\\leq \\frac{\\hd(X)}{\\hc(X)+1}.\n\\end{equation} \n\nA version of the Hochster formula also holds for real moment-angle complexes, namely,\n\\begin{equation}\nH^*(\\mathbb R\\mathcal Z_K)\\cong\\bigoplus_{I\\subseteq[n]}\\tilde H^*(\\Sigma|K_I|). \n\\end{equation} \nThus, \n$$\n\\hd(\\mathbb R\\mathcal Z_K)=1+\\max\\cset{\\,\\hd(|K_I|)}{I\\subseteq[n]\\,}\\,\\leq\\, 1+\\dim K\n$$\nand\n$$\n\\hc(\\mathbb R\\mathcal Z_K)=1+\\min\\cset{\\,\\hc(|K_I|)}{I\\subseteq[n]\\,},\n$$\nand using the inequality $\\cat(\\mathcal Z_K)\\leq\\cat(\\mathbb R\\mathcal Z_K)$ from Corollary~\\ref{CRealMAC}, \n\\begin{proposition}\n\\label{PImproved}\nIt holds that \n$$\\cat(\\mathcal Z_K)\\leq \\frac{\\hd(\\mathbb R\\mathcal Z_K)}{\\hc(\\mathbb R\\mathcal Z_K)}.$$ \n\\end{proposition} \nComparing the Hochster formula for $H^*(\\mathbb R\\mathcal Z_K)$ to the Hochster formula for $H^*(\\mathcal Z_K)$ in Section~\\ref{SHochster}, \none sees that the inequality $\\frac{\\hd(\\mathbb R\\mathcal Z_K)}{\\hc(\\mathbb R\\mathcal Z_K)}\\leq \\frac{\\hd(\\mathcal Z_K)}{\\hc(\\mathcal Z_K)}$ usually holds, \nwith the disparity between these two often being very large. In such case, the bound in Proposition~\\ref{PImproved}\nis an improvement over what one would get by applying~\\eqref{EOld} directly to $X=\\mathcal Z_K$.\n\nConsider, for instance, the case of $k$-neighbourly complexes. \nA simplicial complex $K$ on vertex set $[n]$ is said to be $k$-\\emph{neighbourly} if every subset of $k$ or less vertices in $[n]$ is a face of $K$.\nIn this case $H_i(K_I)=0$ for $i\\leq k-2$ and each $I\\subseteq[n]$, so $\\hc(\\mathbb R\\mathcal Z_K)\\geq k-1$. Therefore\nwe have the following result.\n\n\\begin{theorem}\nIf $K$ is $k$-neighbourly, $\\cat(\\mathcal Z_K)\\leq \\frac{1+\\dim K}{k}$. In particular, $K$ is $\\paren{\\frac{1+\\dim K}{k}}$-Golod.~$\\hfill\\square$\n\\end{theorem}\n\n\\begin{example}\nSuppose $K$ is $\\ceil{\\frac{n}{2}}$-neighbourly. \nIf $K\\neq\\Delta^{n-1}$, then $\\dim K\\leq n-2$.\nThus, $\\cat(\\mathcal Z_K)\\leq 1$ ($\\mathcal Z_K$ is a $co$-$H$-space) and $K$ is $1$-Golod.\n\\end{example}\n\n\n\n\n\n\n\n\n\n\\subsection{Simplicial Wedges}\n\n\nWe recall the \\emph{simplicial wedge} construction defined in~\\cite{MR593648,MR3426378}. \nLet $K$ be a simplicial complex on vertex set $\\{v_1,...,v_n\\}$, and for any face $\\sigma\\in K$, \ndefine the \\emph{link} of $\\sigma$ in $K$ the subcomplex of $K$ given by\n$$\n\\link_K(\\sigma)=\\cset{\\tau\\in K}{\\tau\\cap\\sigma=\\emptyset,\\,\\tau\\cup\\sigma\\in K}.\n$$\nBy \\emph{doubling} a vertex $v_i$ in $K$, we obtain a new simplicial complex $K(v_i)$ on vertex set \n$$\n\\{v_1,\\ldots,v_{i-1},v_{i1},v_{i2},v_{i+1},\\ldots,v_n\\}\n$$ \ndefined by\n$$\nK(v_i) = (v_{i1},v_{i2})\\ast \\link_K(v_i)\\cup_{\\{v_{i1},v_{i2}\\}\\ast\\link_K(v_i)} \\{v_{i1},v_{i2}\\}\\ast K\\backslash\\{v_i\\},\n$$\nwhere $(v_{i1},v_{i2})$ is the $1$-simplex with vertices $\\{v_{i1},v_{i2}\\}$.\nOne can of course iterate this construction by reapplying the doubling operation to successive complexes,\nand the order of vertices on which this is done is irrelevant. \nTo this end, take any sequence $J=(j_1,\\ldots,j_n)$ of non-negative integers, \nlet $u_j$ be the $j^{th}$ vertex in the sequence $v_1,v_{12},\\ldots,v_{1j_1},v_2,\\ldots,v_n,v_{n2},\\ldots,v_{nj_n}$\nand $N=j_1+\\cdots+j_n$, and define \n$$\nK(J)=K_N\n$$ \nwhere $K_{j+1}=K_j(u_{j+1})$ and $K_0=K$.\nIn algebraic terms, the Stanley-Reisner ideal of $K(J)$ is obtained from the Stanley-Reisner ideal of $K$ by replacing each vertex $v_i$ \nby $v_{i1},v_{i2},\\ldots,v_{ij_i}$ in each monomial. \nThis construction arises in combinatorics (see~\\cite{MR593648}) and has the important property that if $K$ is the boundary of the dual of $d$-polytope, \nthen $K(J)$ is the boundary of the dual of a $(d+N)$-polytope.\n\n\n\n\n\\begin{theorem}\n\\label{TSW} For any $J$, $\\cat(\\z{K(J)})\\leq \\cat(\\mathcal Z_K)$.\n\\end{theorem}\n\n\n\\begin{proof}\nLet $(D^J,S^J)$ be the sequence of pairs $((D^{2j_1+2},S^{2j_1+1}),\\ldots,(D^{2j_n+2},S^{2j_n+1}))$.\nBy~\\cite{MR3426378,MR3084441}, there is a homeomorphism\n$$\n\\z{K(J)}\\,=\\,(D^2,S^1)^{K(J)}\\,\\cong\\,(D^J,S^J)^K,\n$$\nand by Proposition~\\ref{PCoordSusp}, $\\cat((D^J,S^J)^K)\\leq \\cat((D^2,S^1)^K)=\\cat(\\mathcal Z_K)$.\n\\end{proof}\n\n\nThis result becomes algebraically useful when a good bound on $\\cat(\\mathcal Z_K)$ is known.\nFor instance, there are many examples of complexes $K$ for which $\\cat(\\mathcal Z_K)=1$, \nduals of sequential Cohen Macaulay and shellable complexes, and chordal flag complexes to name a few~\\cite{MR3461047,2013arXiv1306.6221I}.\nIn each of these examples $\\cat(\\z{K(J)})\\leq 1$, so $K(J)$ is Golod. \nGenerally, $K(J)$ is at least $(1+\\dim K)$-Golod since $\\cat(\\mathcal Z_K)\\leq 1+\\dim K$, \neven though $\\dim K(J)-\\dim K$ can be arbitrarily large.\n\nNotice $K(J)$ is a triangulation of a $(d+N)$-sphere whenever $K$ is a triangulation of a $d$-sphere.\nCombining Theorem~\\ref{TSW} and Proposition~\\ref{PRudyak},\nthe range of spheres for which Theorem~\\ref{TMain} holds generalizes as follows. \n\n\n\\begin{corollary}\nLet $K$ be be any triangulated $d$-sphere for $d=0,1,2$, or $K\\in \\mc C_d$ when $d\\geq 3$,\nand let $m=\\filt(K)$ (equivalently $m=\\cuplen(\\mathcal Z_K))$. Then $\\cat(\\z{K(J)})\\leq m$ and $K(J)$ is $m$-Golod.~$\\hfill\\square$\n\\end{corollary}\n\n\n\n\n\n\n\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzgzpm b/data_all_eng_slimpj/shuffled/split2/finalzzgzpm
new file mode 100644
index 0000000000000000000000000000000000000000..7ae202c1e04c57c277af3b4ef15d1a8b0440500d
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzgzpm
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\nIn this paper, we consider a stochastic minimization problem. Let $\\Xi$ be a statistical sample space with probability\n distribution $\\Pi$ (we omit the underlying $\\sigma$-algebra). Let $X\\subseteq\\mathbb{R}^n$ be a closed convex set, which represents the parameter space. $F(\\cdot;\\xi):X\\rightarrow \\mathbb{R}$ is a closed convex function associated with $\\xi\\in\\Xi$. We aim to solve the following problem:\n\\begin{align}\\label{minex}\n    \\textrm{minimize}_{x\\in X\\subseteq \\mathbb{R}^n} ~~\\mathbb{E}_{\\xi}\\big(F(x;\\xi)\\big)=\\int_{\\Pi} F(x,\\xi) d\\Pi(\\xi).\n\\end{align}\n\\\\\nA common method to minimize \\eqref{minex} is Stochastic Gradient Descent (SGD) \\cite{robbins1951stochastic}:\n\\begin{align}\\label{sgd}\n    x^{k+1}=\\textbf{Proj}_{X}\\big(x^k-\\gamma_k\\partial F(x^k;\\xi^k)\\big),\\quad \\xi^k\\stackrel{\\text{i.i.d}}{\\sim} \\Pi.\n\\end{align}\n\nHowever, for some problems and distributions, direct sampling from $\\Pi$ is expensive or impossible, and it is possible that $\\Xi$ is not known. In these cases, it can be much cheaper to sample by following a Markov chain that has the desired distribution $\\Pi$ as its equilibrium distribution.\n\nTo be concrete, imagine solving problem \\eqref{minex} with a discrete space $\\Xi:=\\{x\\in \\{0,1\\}^n\\mid\\langle a,x\\rangle\\leq b\\}$, where $a\\in\\mathbb{R}^n$ and $b\\in \\mathbb{R}$, and the uniform distribution $\\Pi$ over $\\Xi$.\nA straightforward way to obtain a uniform sample is iteratively randomly sampling $x\\in\\{0,1\\}^n$ until the constraint $\\langle a,x\\rangle\\leq b$ is satisfied. Even if the feasible set is small, it may take up to $O(2^n)$ iterations to get a feasible sample.\nInstead, one can sample a trajectory of a Markov chain described in \\cite{jerrum1996markov}; to obtain a sample $\\varepsilon$-close to the distribution $\\Pi$, one only needs $\\log(\\frac{\\sqrt{\\mid \\Xi\\mid}}{\\varepsilon})\\exp(O(\\sqrt{n}(\\log n)^{\\frac{5}{2}}))$ samples \\cite{dyer1993mildly}, where  $|\\Xi|$ is the cardinality  of $\\Xi$. This presents a signifant saving in sampling cost.\n\nMarkov chains also naturally arise in some applications. Common examples are systems that evolve according to Markov chains, for example, linear dynamic systems with random transitions or errors. Another example is a distributed system in which every node locally stores a subset of training samples; to train a model using these samples, we can let a token that holds all the model parameters traverse the nodes following a random walk, so the samples are accessed according to a Markov chain.\n\n\nSuppose that the Markov chain has a stationary distribution $\\Pi$ and a finite mixing time $T$, which is how long a random trajectory needs to be until its current state has a distribution that roughly matches $\\Pi$. A larger $T$ means a closer match. Then, in order to run one iteration of \\eqref{sgd}, we can generate a trajectory of samples $\\xi^1,\\xi^2,\\xi^3,\\ldots,\\xi^T$  and only take the last sample $\\xi:=\\xi^T$.\nTo run another iteration of \\eqref{sgd}, we repeat this process, i.e., sample a new trajectory $\\xi^1,\\xi^2,\\xi^3,\\ldots,\\xi^T$ and take $\\xi:=\\xi^T$.\n\nClearly, sampling a long trajectory just to use the last sample wastes a lot of samples, especially when $T$ is large. But, this may seem necessary because $\\xi^t$, for all small $t$, have large biases. After all, it can take a long time for the random trajectory to explore all of the space,  and it will often double back and visit states that it previously visited. Furthermore, it is also difficult to choose an appropriate $T$. A small $T$ will cause large bias in $\\xi^T$, which slows the SGD convergence and reduces its final accuracy. A large $T$, on the other hand, is wasteful especially when $x^k$ is still far from convergence and some bias does not prevent \\eqref{sgd} to make good progress. Therefore, $T$ should increase adaptively as $k$ increases --- this makes the choice of $T$ even more difficult.\n\n\nSo, why waste samples, why worry about $T$, and why not just apply every sample immediately in stochastic gradient descent?\nThis approach has appeared in \\cite{johansson2007simple,johansson2009randomized}, which we call the Markov Chain Gradient Descent (MCGD) algorithm for problem \\eqref{minex}:\n\\begin{align}\\label{online-alg1}\n    x^{k+1}=\\textbf{Proj}_{X}\\big(x^k-\\gamma_k \\hat{\\nabla} F(x^k;\\xi^k)\\big),\n\\end{align}\nwhere $\\xi^0,\\xi^1,\\ldots$ are samples on a Markov chain trajectory and $\\hat{\\nabla} F(x^k;\\xi^k)\\in \\partial F(x^k;\\xi^k)$ is a subgradient.\n\nLet us examine some special cases.\nSuppose the distribution $\\Pi$ is supported on a set of $M$ points, $y^1,\\dots,y^M$. Then, by letting $f_i(x) := M\\cdot\\mathrm{Prob}(\\xi=y^i)\\cdot F(x,y^i)$, problem \\eqref{minex} reduces to the finite-sum problem:\n\\begin{equation}\\label{model}\n    \\textrm{minimize}_{x\\in X\\subseteq \\mathbb{R}^d}f(x)\\equiv\\frac{1}{M} \\sum_{i=1}^M f_i(x).\n\\end{equation}\nBy the definition of $f_i$, each state $i$ has the uniform probability $1\/M$.\nAt each iteration $k$ of MCGD, we have\n\\begin{equation}\\label{scheme-sgd}\n    x^{k+1}=\\textbf{Proj}_{X}\\big(x^k-\\gamma_k\\hat{\\nabla} f_{j_k}(x^k)\\big),\n\\end{equation}\nwhere $(j_k)_{k\\geq 0}$ is a trajectory of a Markov chain on $\\{1,2,\\dots, M\\}$ that has a uniform stationary distribution. Here, $(\\xi^k)_{k\\geq 0}\\subseteq\\Pi$ and $(j_k)_{k\\geq 0}\\subseteq [M]$ are two different, but related Markov chains.\nStarting from a deterministic and arbitrary initialization $x^0$, the iteration is illustrated by the following diagram:\n\\begin{equation}\\label{tutu-sgd1}\n    \\CD\n  @.       j_0 @>  >>j_1 @> >> j_2  @> >> \\ldots\\\\\n  @.       @V  VV @V  VV @V    VV @.   \\\\\n  x^0 @> >> x^1 @> >> x^2 @> >> x^3 @>>>\\ldots\n    \\endCD\n\\end{equation}\nIn the diagram, given each $j_k$, the next state $j_{k+1}$ is statistically independent of $j_{k-1},\\dots,j_0$; given $j_k$ and $x^k$, the next iterate $x^{k+1}$ is statistically independent of $j_{k-1},\\dots,j_0$ and $x^{k-1},\\dots,x^0$.\n\nAnother application of MCGD involves a network: consider a strongly connected graph $\\mathcal{G} = (\\mathcal{V}, \\mathcal{E})$ with the set of vertices $\\mathcal{V}=\\{1,2,\\ldots,M\\}$ and set of edges $\\mathcal{E}\\subseteq \\mathcal{V}\\times \\mathcal{V}$.\nEach node $j\\in \\{1,2,\\ldots,M\\}$ possess some data and can compute $\\nabla f_j(\\cdot)$.  To run MCGD,\nwe employ a token that carries the variable $x$, walking randomly over the network. When it reaches a node $j$, node $j$ reads $x$ form the token and computes $\\nabla f_j(\\cdot)$ to update $x$ according to \\eqref{scheme-sgd}. Then, the token walks away to a random neighbor of node $j$.\n\n\\subsection{Numerical tests}\nWe present two kinds of numerical results. The first one is to show that MCGD uses fewer samples to train both a convex model and a nonconvex model. The second one demonstrates the advantage of the faster mixing of a non-reversible Markov chain. Our results on nonconvex objective and  non-reversible chains are new.\\\\\n\\\\\n\\textit{1. Comparision with SGD}\\\\\nLet us compare:\n\\begin{enumerate}\n    \\item MCGD \\eqref{online-alg1}, where $j_k$ is taken from one  trajectory of the Markov chain;\n    \\item SGD$T$, for $T=1,2,4,8,16,32$, where each $j_k$ is the $T$th sample of a fresh, independent  trajectory. All trajectories are generated by starting from the same state $0$.\n\\end{enumerate}\nTo compute $T$ gradients, SGD$T$ uses $T$ times as many samples as MCGD. We did not try to adapt $T$ as $k$ increases because there lacks a theoretical guidance.\n\nIn the first test, we recover a vector $u$ from an auto regressive process, which closely resembles the first experiment in \\cite{duchi2012ergodic}. Set matrix A as a subdiagonal matrix with random entries $A_{i,i-1}\\overset{\\text{i.i.d}}{\\sim} \\mathcal{U}[0.8,0.99]$. Randomly sample a vector $u\\in\\mathbb{R}^d$, $d=50$, with the unit 2-norm. Our data $(\\xi_t^1,\\xi_t^2)_{t=1}^{\\infty}$ are generated according to the following auto regressive process:\n\\begin{align*}\n    &\\xi^1_t = A\\xi^1_{t-1}+e_1W_t,~ W_t\\stackrel{\\text{i.i.d}}{\\sim} N(0,1)\\\\\n    &\\bar{\\xi}^2_t = \\left\\{\\begin{array}{ll}\n        1, &  \\text{if } \\langle u, \\xi^1_t\\rangle>0,\\\\\n        0, &  \\text{otherwise};\n    \\end{array}\\right.\\\\\n    &\\xi^2_t = \\left\\{\\begin{array}{ll}\n    \\bar{\\xi}^2_t, &\\text{ with probability 0.8,}\\\\\n    1-\\bar{\\xi}^2_t, &\\text{  with probability 0.2.}\n    \\end{array}\\right.\n\\end{align*}\n\nClearly, $(\\xi_t^1,\\xi_t^2)_{t=1}^{\\infty}$ forms a Markov chain. Let $\\Pi$ denote the stationary distribution of this Markov chain. We recover $u$ as the solution to the following problem:\n\\begin{align*}\n   \\textrm{minimize}_{x} ~~\\mathbb{E}_{(\\xi^1,\\xi^2)\\sim\\Pi}\\ell(x;\\xi^1,\\xi^2).\n\\end{align*}\n\nWe consider both convex and nonconvex loss functions, which were not done before in the literature. The convex one is the logistic loss\n\\begin{align*}\n\\ell(x;\\xi^1,\\xi^2)=-\\xi^2\\log(\\sigma(\\langle x, \\xi^1 \\rangle))-(1-\\xi^2)\\log(1-\\sigma(\\langle x,\\xi^1\\rangle)),\n\\end{align*}\n where $\\sigma(t)=\\frac{1}{1+\\exp(-t)}$. And the nonconvex one is taken as\n\\begin{align*}\n\\ell(x;\\xi^1,\\xi^2)=\\frac{1}{2}(\\sigma(\\langle x,\\xi^1 \\rangle)-\\xi^2)^2\n\\end{align*}\nfrom \\cite{mei2016landscape}. We choose $\\gamma_k = \\frac{1}{k^q}$ as our stepsize, where $q = 0.501$. This choice is consistently with our theory below.\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.245\\textwidth]{convexcase.pdf}\\includegraphics[width=0.245\\textwidth]{convexcase1.pdf}\n    \\includegraphics[width=0.245\\textwidth]{nonconvexcase.pdf}\\includegraphics[width=0.245\\textwidth]{nonconvexcase1.pdf}\n    \\caption{Comparisons of MCGD and SGD$T$ for $T=1,2,4,8,16,32$. $\\overline{x^k}$ is the average of $x^1,\\ldots,x^k$.}\n    \\label{fig:my_label}\n\\end{figure}\n\nOur results in Figure 1 are surprisingly positive on MCGD,   more so to our expectation. As we had expected, MCGD used significantly fewer total samples than SGD on every $T$. But, it is surprising that MCGD did not need even more gradient evaluations. Randomly generated data must have helped homogenize the samples over the different states, making it less important for a trajectory to converge. It is important to note that SGD1 and SGD2, as well as SGD4, in the nonconvex case, stagnate at noticeably lower accuracies because their $T$ values are too small for convergence. \\\\\n\\\\\n\\textit{2. Comparison of reversible and non-reversible Markov chains}\\\\\nWe also compare the convergence of MCGD when working with reversible and non-reversible Markov chains (the definition  of reversibility is   given in next section). As mentioned in \\cite{turitsyn2011irreversible}, transforming a reversible Markov chain into non-reversible Markov chain can significantly accelerate the mixing process. This technique also helps to accelerate the convergence of MCGD.\n\nIn our experiment, we first construct an undirected connected graph with $n=20$ nodes with edges randomly generated. Let $G$ denote the adjacency matrix of the graph, that is,\n\\begin{align*}\n    G_{i,j}=\\left\\{\\begin{array}{ll}\n    1, &\\text{if $i, j$ are connected;}\\\\\n    0, &\\text{otherwise.}\n    \\end{array}\\right.\n\\end{align*}\nLet $d_{\\max}$ be the maximum number of outgoing edges of a node.\nSelect $d=10$ and compute $\\beta^*\\sim \\mathcal{N}(0,I_d)$.\nThe transition probability of the reversible Markov chain is then defined by, known as    Metropolis-Hastings markov chain,\n\\begin{align*}\n    P_{i,j} = \\left\\{\\begin{array}{ll}\n        \\frac{1}{d_{\\max}}, &  \\text{if $j\\neq i$, $G_{i,j}=1$;}\\\\\n        1-\\frac{\\sum_{j\\neq i}G_{i,j}}{d_{\\max}}, & \\text{if $j=i$;}\\\\\n        0, & \\text{otherwise.}\n    \\end{array}\\right.\n\\end{align*}\nObviously, $P$ is symmetric and the stationary distribution is uniform.\nThe non-reversible Markov chain is constructed by adding cycles. The edges of these cycles are directed and let $V$ denote the adjacency matrix of these cycles. If $V_{i,j}=1$, then $V_{j,i}=0$. Let $w_0>0$ be the weight of flows along these cycles. Then we construct the transition probability of the non-reversible Markov chain as follows,\n\\begin{align*}\nQ_{i,j} = \\frac{W_{i,j}}{\\sum_l W_{i,l}},\n\\end{align*}\nwhere $W = d_{\\max}P+w_0 V$. See [14] for an explanation why this change makes the chain mix faster.\n\n\n In our experiment, we add 5 cycles of length 4, with edges existing in $G$. $w_0$ is set to be $\\frac{d_{\\max}}{2}$.  We test MCGD on a least square problem. First, we select $\\beta^*\\sim\\mathcal{N}(0, I_d)$; and  then for each node $i$, we generate $x_i\\sim\\mathcal{N}(0, I_d)$, and $y_i = x_i^T\\beta^*$. The objective function is defined as,\n\\begin{align*}\n    f(\\beta)=\\frac{1}{2}\\sum\\limits_{i=1}^{n}(x_i^T\\beta-y_i)^2.\n\\end{align*}\nThe convergence results   are depicted in Figure \\ref{fig:irreversible}.\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{non-reversible.pdf}\\\\\n    \\caption{Comparison of reversible and irreversible Markov chains. The second largest eigenvalues of reversible and non-reversible Markov chains are 0.75 and 0.66 respectively. }\n    \\label{fig:irreversible}\n\\end{figure}\n\n\n\n\n\n\n\\subsection{Known approaches and results}\nIt is more difficult to analyze MCGD due to its biased samples. To see this, let $p_{k,j}$ be the probability to select $\\nabla f_j$ in the $k$th iteration.\nSGD's uniform probability selection ($p_{k,j}\\equiv\\frac{1}{M}$) yields an unbiased gradient estimate\n\\begin{align}\\label{unba}\n    \\mathbb{E}_{j_k}(\\nabla f_{j_k}(x^k))=C\\nabla f(x^k)\n\\end{align}\nfor some $C>0$.\nHowever, in MCGD, it is possible to have $p_{k,j}=0$ for some $k,j$. Consider a ``random wal''. The probability $p_{j_k,j}$ is determined by the current state $j_k$, and we have $p_{j_k,i}>0$ only for $i\\in \\mathcal{N}(j_k)$ and $p_{j_k,i}=0$ for $i\\notin \\mathcal{N}(j_k)$, where $\\mathcal{N}(j_k)$ denotes the neighborhood of $j_k$. Therefore, we no longer have \\eqref{unba}.\n\nAll analyses of MCGD must deal with the biased expectation. Papers \\cite{johansson2009randomized,johansson2007simple} investigate the conditional expectation $\\mathbb{E}_{j_{k+\\tau}\\mid j_k}(\\nabla f_{j_{k+\\tau}}(x^{k}))$. For a sufficiently large $\\tau\\in\\mathbb{Z}^+$, it is sufficiently close to $\\frac{1}{M}\\nabla f(x^{k})$  (but still different).\nIn \\cite{johansson2009randomized,johansson2007simple}, the authors proved that, to achieve an $\\epsilon$ error, MCGD with stepsize $O(\\epsilon)$ can return a solution in $O(\\frac{1}{\\epsilon^2})$ iteration. Their error bound is given in the ergodic sense and using $\\mathrm{liminf}$. The authors of \\cite{ram2009incremental}\nproved a  $\\liminf f(x^k)$ and $\\mathbb{E} \\mathrm{dist}^2(x^k,X^*)$  have almost sure convergence under diminishing stepsizes  $\\gamma_k=\\frac{1}{k^q}$, $\\frac{2}{3}< q\\leq 1$. Although the authors did not compute any rates, we computed that their stepsizes will lead to\na solution with $\\epsilon$ error in $O(\\frac{1}{\\epsilon^{\\frac{1}{1-q}}})$ iterations, for $\\frac{2}{3}<q<1$, and $O(e^\\frac{1}{\\epsilon})$ for $q=1$. In \\cite{duchi2012ergodic}, the authors improved the stepsizes to $\\gamma_k=\\frac{1}{\\sqrt{k}}$ and showed ergodic convergence; in other words, to achieve $\\epsilon$ error, it is enough to run MCGD for $O(\\frac{1}{\\epsilon^2})$ iterations. There is no non-ergodic result regarding the convergence of $f(x^k)$.\nIt is worth mentioning that \\cite{ram2009incremental,duchi2012ergodic} use time non-homogeneous Markov chains, where the transition probability can change over the iterations as long as there is still a finite mixing time. In \\cite{duchi2012ergodic}, MCGD is generalized from gradient descent to mirror descent.\nIn all these works, the Markov chain is required to be reversible, and all functions $f_i$, $i\\in[M]$, are assumed to be convex. However, non-reversible chains can have substantially faster convergence and thus more numerically efficient.\n\n\n\n\\subsection{Our approaches and results}\nIn this paper, we improve the analyses of MCGD to non-reversible finite-state Markov chains and to nonconvex functions. The former allows us to have faster mixing, and the latter frequently appears in applications. Our convergence result is given in the non-ergodic sense though the rate results are still given the ergodic sense. It is important to mention that, in our analysis, the mixing time of the underlying Markov chain is not tied to a fixed mixing level but can vary to different levels. This is essential because MCGD needs time to reduce its objective error from its current value to a lower one, and this time becomes longer when the current value is lower since a more accurate Markov chain convergence and thus a longer mixing time are required.\nWhen $f_1,f_2,\\ldots,f_M$ are all convex, we allow them to be non-differentiable and MCGD to use subgradients, provided that $X$ is bounded. When any of them is nonconvex, we assume $X$ is the full space and $f_1,f_2,\\ldots,f_M$ are differentiable with bounded gradients. The bounded-gradient assumption is due to a technical difficulty associated with nonconvexity.\n\n\nSpecifically, in the convex setting, we prove $\\lim_k\\mathbb{E} f(x^k)=f^*$ (minimum of $f$ over $X$) for both exact and inexact MCGD with stepsizes $\\gamma_k=\\frac{1}{k^q}$, $\\frac{1}{2}<q<1$.\nThe convergence rates of MCGD with exact and inexact subgradient computations are presented.\nThe first analysis of nonconvex MCGD is also presented with its convergence given in the expectation of $\\|\\nabla f(x^k)\\|$.\nThese results hold for non-reversible finite-state Markov chains and can be extended to time non-homogeneous Markov chain under extra assumptions \\cite[Assumptions 4 and 5]{ram2009incremental} and \\cite[Assumption C]{duchi2012ergodic}, which essentially ensure finite mixing.\n\nOur results for finite-state Markov chains are first presented in Sections 3 and 4. They are extended to continuous-state reversible Markov chains in Section 5.\n\nSome novel results are are developed based on new techniques and  approaches developed in this paper. To get the stronger results in general cases, we used the varying mixing time rather than fixed ones. Several technical lemmas (Lemmas 2,3,4,5) are given in next section, in which, Lemma 2 plays a core role in our analyses.\n\nWe list the possible extensions of MCGD that are not discussed in this paper. The first one is the accelerated versions including the Nesterov's acceleration and variance reduction schemes. The second one is the design and optimization of Markov chains to improve the convergence of MCGD.\n\n\n\n\n\\section{Preliminaries}\n\\subsection{Markov chain}\nWe recall some definitions, properties, and existing results about the\nMarkov chain. Although we use\nthe finite-state time-homogeneous Markov chain,\nresults can be extended to more general chains under similar extra assumptions in \\cite[Assumptions 4, 5]{ram2009incremental} and \\cite[Assumption C]{duchi2012ergodic}.\n\n\\begin{definition}[finite-state time-homogeneous Markov chain]\nLet $P$ be an $M\\times M$-matrix with real-valued elements.\nA stochastic process $X_1,X_2,...$ in a finite state space $[M]:=\\{1,2,\\ldots,M\\}$ is called a time-homogeneous Markov chain with transition matrix $P$ if, for $k\\in \\mathbb{N}$, $i,j\\in [M]$, and  $i_0,i_1,\\ldots,i_{k-1}\\in [M]$, we have\n\\begin{equation}\n    \\mathbb{P}(X_{k+1}=j\\mid X_0=i_0,X_1=i_1,\\ldots,X_k=i)=\\mathbb{P}(X_{k+1}=j\\mid X_k=i)=P_{i,j}.\n\\end{equation}\n\\end{definition}\nLet the probability distribution of $X_k$ be denoted as the non-negative row vector $\\pi^k=(\\pi^k_1,\\pi^k_2,\\ldots,\\pi^k_M)$, that is, $\\mathbb{P}(X_k=j)=\\pi_j^k$. $\\pi$ satisfies  $\\sum_{i=1}^M \\pi^k_i=1.$\nWhen the Markov chain is time-homogeneous, we have $\\pi^k=\\pi^{k-1} P$ and\n\\begin{equation}\n    \\pi^k=\\pi^{k-1} P=\\cdots=\\pi^0 P^{k},\n\\end{equation}\nfor $k\\in \\mathbb{N}$, where $P^{k}$ denotes the $k$th power of $P$.\nA Markov chain is irreducible if, for any $i,j\\in [M]$, there exists $k$ such that $(P^k)_{i,j}>0$.\nState $i\\in[M]$ is said to have a period $d$ if $P^k_{i,i} = 0$ whenever $k$ is \\emph{not} a\nmultiple of $d$ and $d$ is the greatest integer with\nthis property. If $d=1$, then we say state $i$ is aperiodic. If  every state is aperiodic, the Markov chain is said to be aperiodic.\n\n\nAny time-homogeneous, irreducible, and aperiodic\nMarkov chain has a stationary distribution $\\pi^*=\\lim_k \\pi^k\n=[\\pi^*_1,\\pi^*_2,\\ldots,\\pi^*_M]$ with $\\sum_{i=1}^M \\pi^*_i=1$ and $\\min_i\\{\\pi^*_i\\}>0$, and $\\pi^*= \\pi^* P$. It also holds that\n\\begin{equation}\\label{convermatrix}\n    \\lim_{k}P^{k}=[(\\pi^*);(\\pi^*);\\ldots;(\\pi^*)]=:\\Pi^*\\in \\mathbb{R}^{M\\times M}.\n\\end{equation}\nThe largest eigenvalue of $P$ is 1, and the corresponding left eigenvector is $\\pi^*$.\n\n\\begin{assumption}\\label{ass:mc}{The Markov chain $(X_k)_{k \\ge 0}$ is time-homogeneous, irreducible, and aperiodic.  It has a transition matrix $P$ and has stationary distribution $\\pi^*$.}\n\\end{assumption}\n\n\n\\subsection{Mixing time}\\label{mixt}\nMixing time is how long a Markov chain evolves until its current state has a distribution very close to its stationary distribution. The literature has a thorough investigation of various kinds of mixing times, with the majority for\n reversible Markov chains (that is,  $\\pi_i P_{i,j} = \\pi_j P_{j,i}$). Mixing times of non-reversible Markov chains are discussed in \\cite{fill1991eigenvalue}. In this part, we consider a new type of mixing time of non-reversible Markov chain. The proofs are based on basic matrix analysis. Our mixing time gives us a direct relationship between $k$ and the deviation of the distribution of the current state from the stationary distribution.\n\nTo start a lemma, we review some basic notions in linear algebra. Let $\\mathbb{C}$ be the $n$-dimensional complex field.\nThe modulus of a complex number $a\\in\\mathbb{C}$ is given as $|a|$.\nFor a vector $x\\in \\mathbb{C}^{n}$, the $\\ell_{\\infty}$ and $\\ell_2$ norms are defined as $\\|x\\|_{\\infty}:=\\max_{i}|x_i|$, $\\|x\\|_{2}:=\\sqrt{ \\sum_{i=1}^n|x_i|^2}$.\nFor a matrix $A=\\begin{bmatrix}a_{i,j}\\end{bmatrix}\\in \\mathbb{C}^{m\\times n}$, its $\\infty$-induced and Frobenius norms are $\\|A\\|_{\\infty}:=\\max_{i,j}|a_{i,j}|$,  $\\|A\\|_{F}:=\\sqrt{\\sum_{i,j=1}^n|a_{i,j}|^2}$, respectively.\n\nWe know $P^k\\rightarrow\\Pi^*$, as $k\\to\\infty$. The following lemma presents a deviation bound for finite $k$.\n\\begin{lemma}\\label{le1}\nLet Assumption \\ref{ass:mc} hold and let $\\lambda_i(P)\\in \\mathbb{C}$ be  the $i$th largest eigenvalue of $P$, and\n$$\\lambda(P):=\\frac{\\max\\{|\\lambda_2(P)|,|\\lambda_M(P)|\\}+1}{2}\\in [0,1).$$\nThen, we can bound the largest entry-wise absolute value of the deviation matrix $\\delta^k :=\\Pi^*-P^{k}\\in \\mathbb{R}^{M\\times M}$ as\n\\begin{equation}\\label{core1}\n    \\|\\delta^k\\|_{\\infty}\\leq C_P\\cdot\\lambda^k(P)\n\\end{equation}\nfor $k\\geq K_P$,  where $C_P$ is a constant that also depends on the Jordan canonical form of $P$ and  $K_P$ is a constant that depends on $\\lambda(P)$ and $\\lambda_2(P)$. Their formulas are given in \\eqref{cpformulate} and \\eqref{kpformulate} in the Supplementary Material.\n\n\\end{lemma}\n\\begin{remark}\nIf $P$ is symmetric, then all $\\lambda_{i}(P)$'s are all real and nonnegative, $K_P=0$, and\n    $C_P\\leq M^{\\frac{3}{2}}$.\nFurthermore,  \\eqref{2eq2} can be improved by directly using $\\lambda_2^k(P)$ for the right side as\n$$    \\|\\delta^k\\|_{\\infty}\\leq \\|\\delta^k\\|_{F}\\leq M^{\\frac{3}{2}}\\cdot\\lambda_2^k(P),~~k\\geq 0.$$\n\\end{remark}\n\n\n\n\n\n\n\n\\section{Convergence analysis for convex minimization}\nThis part considers the convergence of  MCGD in the convex cases, i.e.,  $f_1, f_2,\\ldots, f_M$ and $X$ are all convex. We investigate the convergence of scheme \\eqref{scheme-sgd}.\nWe prove non-ergodic convergence of the expected objective value sequence under diminishing non-summable stepsizes, where the stepsizes are required to be ``almost\" square summable. Therefore,   the convergence requirements are almost equal to SGD.\nThis section uses the following assumption.\n\\begin{assumption}  The set $X$ is assumed to be convex and  compact.\n\\end{assumption}\n\n\n\nNow, we present the convergence results for MCGD in the convex (but not necessarily differentiable) case. Let $f^*$ be the minimum value of $f$ over $X$.\n\\begin{theorem}\\label{th-sgd}\nLet Assumptions 1 and 2 hold and  $(x^k)_{k\\geq 0}$ be generated by scheme (\\ref{scheme-sgd}). Assume that $f_i$, $i\\in[M]$, are convex functions, and the stepsizes satisfy\n\\begin{align}\\label{stepsize-1}\n    \\sum_k\\gamma_k=+\\infty,\\quad\\sum_k\\ln k\\cdot\\gamma_k^2<+\\infty.\n\\end{align}\nThen, we have\n\\begin{equation}\\label{th-sgd-r1}\n    \\lim_{k}  \\mathbb{E} f(x^k)=f^*.\n\\end{equation}\nDefine\n$$\\psi(P):=\\max\\{1,\\frac{1}{\\ln(1\/\\lambda(P))}\\}.$$\nWe have:\n\\begin{equation}\\label{th-sgd-r2}\n     \\mathbb{E}(f(\\overline{x^k})- f^*)=O\\Big(\\frac{\\psi(P)}{\\sum_{i=1}^k\\gamma_i}\\Big),\n\\end{equation}\nwhere $\\overline{x^k}:=\\frac{\\sum_{i=1}^k\\gamma_i x^i}{\\sum_{i=1}^k\\gamma_i}$.\nTherefore, if we select the stepsize $\\gamma_k=O(\\frac{1}{k^{q}})$ as $\\frac{1}{2}<q<1$, we get the rate  $\\mathbb{E}(f(\\overline{x^k})- f^*)=O(\\frac{\\psi(P)}{k^{1-q}})$.\n\nFurthermore, consider the inexact version of  MCGD:\n\\begin{equation}\\label{scheme-sgd-in}\n    x^{k+1}=\\emph{\\textbf{Proj}}_{X}\\big(x^k-\\gamma_k(\\hat{\\nabla} f_{j_k}(x^k)+e^k)\\big),\n\\end{equation}\nwhere the noise sequence $(e^k)_{k\\geq 0}$ is arbitrary but obeys\n\\begin{align}\\label{noise-1}\n    \\sum_{k=2}^{+\\infty}\\frac{\\|e^k\\|^2}{\\ln k}<+\\infty.\n\\end{align}\nThen, for iteration \\eqref{scheme-sgd-in}, results \\eqref{th-sgd-r1} and \\eqref{th-sgd-r2} still hold; furthermore,\nif $\\|e^k\\|=O(\\frac{1}{k^{p}})$ with $p>\\frac{1}{2}$ and $\\gamma_k=O(\\frac{1}{k^{q}})$ as $\\frac{1}{2}<q<1$, the rate  $\\mathbb{E}(f(\\overline{x^k})- f^*)=O(\\frac{\\psi(P)}{k^{1-q}})$ also holds.\n\\end{theorem}\n\n\nThe stepsizes requirement \\eqref{stepsize-1} is nearly identical to the one of SGD and subgradient algorithms.\nIn the theorem above, we use the stepsize setting $\\gamma_k=O(\\frac{1}{k^{q}})$ as $\\frac{1}{2}<q<1$. This kind of stepsize requirements  also works for  SGD and subgradient algorithms.\nThe convergence rate of MCGD is  $O(\\frac{1}{\\sum_{i=1}^k\\gamma_i})=O(\\frac{1}{k^{1-q}})$, which is also as the same as SGD and subgradient algorithms for $\\gamma_k=O(\\frac{1}{k^{q}})$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Convergence analysis for nonconvex minimization}\nThis section considers the convergence of  MCGD when one or more of $f_i$ is nonconvex. In this case, we assume  $f_i$, $i=1,2,\\ldots,M$, are differentiable and  $\\nabla f_i$ is Lipschitz with $L$\\footnote{This is for the convenience of the presentation in the proofs. If each $f_i$ has a $L_i$, it is possible to improve our results slights. But, we simply set $L:=\\max_{i}\\{L_i\\}$}. We also set $X$ as the full space. We study the following scheme\n\\begin{equation}\\label{scheme-sgd2}\n    x^{k+1}=x^k-\\gamma_k\\nabla f_{j_k}(x^k).\n\\end{equation}\nWe  prove non-ergodic convergence of the expected gradient norm of $f$ under diminishing non-summable stepsizes. The stepsize requirements in this section are slightly stronger than those in the convex case with an extra $\\ln k$ factor. In this part, we use the following assumption.\n %\n\n \\begin{assumption}\\label{ass-bound-2}\n  The gradients of $f_i$ are assumed to be bounded, i.e., there exists $D>0$ such that\n\\begin{equation}\\label{boundx}\n    \\|\\nabla f_i(x)\\|\\leq D,\\quad i\\in[M].\n\\end{equation}\n \\end{assumption}\nWe use this new assumption because $X$ is now the full space, and we have to directly bound the size of $\\|\\nabla f_i(x)\\|$. In the nonconvex case, we cannot obtain objective value convergence, and we only bound the gradients.\nNow, we are prepared to present our convergence results of nonconvex MCGD.\n\n\\begin{theorem}\\label{th-sgd2}\nLet   Assumptions 1 and 3 hold and $(x^k)_{k\\geq 0}$ be generated by scheme (\\ref{scheme-sgd2}). Also, assume $f_i$ is differentiable and $\\nabla f_i$ is $L$-Lipschitz, and the stepsizes satisfy\n\\begin{align}\\label{stepsize-2}\n    \\sum_k\\gamma_k=+\\infty,\\sum_k\\ln^2 k\\cdot\\gamma_k^2<+\\infty.\n\\end{align}\nThen, we have\n\\begin{equation}\\label{th-sgd2-r1}\n    \\lim_{k} \\mathbb{E}\\|\\nabla f(x^k)\\|=0.\n\\end{equation}\nand\n\\begin{equation}\\label{th-sgd2-r2}\n     \\mathbb{E}\\big(\\min_{1\\leq i\\leq k}\\{\\|\\nabla f(x^i)\\|^2\\}\\big)=O\\Big(\\frac{\\psi(P)}{\\sum_{i=1}^k\\gamma_i}\\Big),\n\\end{equation}\nwhere $\\psi(P)$ is given in Lemma \\ref{le1}.\nIf we select the stepsize as $\\gamma_k=O(\\frac{1}{k^{q}})$, $\\frac{1}{2}<q<1$, then we get the rate $\\mathbb{E}\\big(\\min_{1\\leq i\\leq k}\\{\\|\\nabla f(x^i)\\|^2\\}\\big)=O(\\frac{\\psi(P)}{k^{1-q}})$.\n\nFurthermore, let $(e^k)_{k\\geq 0}$ be a sequence of noise and consider the inexact nonconvex MCGD iteration:\n\\begin{equation}\\label{scheme-sgd2-in}\n    x^{k+1}=x^k-\\gamma_k\\big(\\nabla f_{j_k}(x^k)+e^k\\big).\n\\end{equation}\nIf the noise sequence obeys\n\\begin{align}\\label{noise-2}\n    \\sum_{k=1}^{+\\infty}\\gamma_k\\cdot\\|e^k\\|<+\\infty,\n\\end{align}\nthen the convergence results \\eqref{th-sgd2-r1} and \\eqref{th-sgd2-r2} still hold for inexact nonconvex MCGD. In addition, if we set $\\gamma_k=O(\\frac{1}{k^{q}})$ as $\\frac{1}{2}<q<1$ and the noise satisfy  $\\|e^k\\|=O(\\frac{1}{k^{p}})$ for  $p+q>1$, then \\eqref{th-sgd2-r1} still holds and  $\\mathbb{E}\\big(\\min_{1\\leq i\\leq k}\\{\\|\\nabla f(x^i)\\|^2\\}\\big)=O(\\frac{\\psi(P)}{k^{1-q}})$.\n\\end{theorem}\nThis proof of Theorem \\ref{th-sgd2} is different from previous one. In particular, we cannot expect some sort of convergence to $f(x^*)$, where $x^*\\in\\mathop{\\mathrm{argmin}} f$ due to nonconvexity. To this end, we use the  Lipschitz continuity of $\\nabla f_i$ ($i\\in[M]$) to derive the ``descent\". Here, the ``$O$\" contains a polynomial compisition of constants $D$ and $L$.\n\nCompared with MCGD in the convex case, the stepsize requirements of nonconvex MCGD become a tad higher; in summable part, we need $\\sum_k\\ln^2 k\\cdot\\gamma_k^2<+\\infty$ rather than $\\sum_k\\ln k\\cdot\\gamma_k^2<+\\infty$. Nevertheless, we can still use  $\\gamma_k=O(\\frac{1}{k^{q}})$ for $\\frac{1}{2}<q<1$.\n\n\n\n\n\n\n\n\n\\section{Convergence analysis for continuous state space}\nWhen the state space $\\Xi$ is a continuum, there are infinitely many possible states.\nIn this case, we consider an infinite-state Markov chain that is time-homogeneous and reversible. Using the results in \\cite[Theorem 4.9]{montenegro2006mathematical}, the mixing time of this kind of Markov chain still has geometric decrease like \\eqref{core1}. Since Lemma \\ref{le1} is based on a linear algebra analysis, it no longer applies to the continuous case. Nevertheless, previous results still hold with nearly unchanged proofs under the following assumption:\n\\begin{assumption}\\label{Assup3}\nFor any $\\xi\\in\\Xi$, $|F(x;\\xi)-F(y;\\xi)|\\leq L\\|x-y\\|$, $\\sup_{x\\in X,\\xi\\in\\Xi}\\{\\|\\hat{\\nabla} F(x;\\xi)\\|\\}\\leq D$, $\\mathbb{E}_{\\xi}\\hat{\\nabla} F(x;\\xi)\\in \\partial \\mathbb{E}_{\\xi} F(x;\\xi)$, and $\\sup_{x,y\\in X,\\xi\\in\\Xi}|F(x;\\xi)-F(y;\\xi)|\\leq H$.\n\\end{assumption}\nWe consider the general scheme\n\\begin{align}\\label{online-alg-1i}\n    x^{k+1}=\\textbf{Proj}_{X}\\big(x^k-\\gamma_k (\\hat{\\nabla} F(x^k;\\xi^k)+e^k)\\big),\n\\end{align}\nwhere $\\xi^k$ are samples on a Markov chain trajectory.\nIf $e^k\\equiv \\textbf{0}$, the scheme then reduces to \\eqref{online-alg1}.\n\\begin{corollary}\\label{online-1}\nAssume $F(\\cdot;\\xi)$ is convex for each $\\xi\\in\\Xi$. Let the stepsizes satisfy \\eqref{stepsize-1} and $(x^k)_{k\\geq 0}$ be generated by Algorithm \\eqref{online-alg-1i}, and $(e^k)_{k\\geq 0}$ satisfy \\eqref{noise-1}. Let $F^*:=\\min_{x\\in X} \\mathbb{E}_{\\xi}(F(x;\\xi))$. If Assumption \\ref{Assup3} holds and the Markov chain is time-homogeneous, irreducible, aperiodic, and reversible,\nthen we have\n\\small\n\\begin{equation*}\n    \\lim_{k}  \\mathbb{E}\\big(\\mathbb{E}_{\\xi}(F(x^k;\\xi))-F^*\\big)=0,\\quad \\mathbb{E}(\\mathbb{E}_{\\xi}(F(\\overline{x^k};\\xi))-F^*)=O\\Big(\\frac{\\max\\{1,\\frac{1}{\\ln(1\/\\lambda)}\\}}{\\sum_{i=1}^k\\gamma_i}\\Big),\n\\end{equation*}\nwhere $0<\\lambda<1$ is the geometric rate of the mixing time of the Markov chain (which corresponds to $\\lambda(P)$ in the finite-state case).\n\\end{corollary}\n\nNext, we present our result for a possibly nonconvex objective function $F(\\cdot;\\xi)$ under the following assumption.\n\\begin{assumption}\\label{Assup4}\nFor any $\\xi\\in\\Xi$, $F(x;\\xi)$ is differentiable,   and $\\| \\nabla F(x;\\xi)-\\nabla F(y;\\xi)\\|\\leq L\\|x-y\\|$. In addition, $\\sup_{x\\in X,\\xi\\in\\Xi}\\{\\|\\nabla F(x;\\xi)\\|\\}<+\\infty$, $X$ is the full space,  and $\\mathbb{E}_{\\xi}\\nabla F(x;\\xi)=\\nabla \\mathbb{E}_{\\xi} F(x;\\xi)$.\n\\end{assumption}\nSince $F(x,\\xi)$ is differentiable  and $X$ is the full space, the iteration reduces to\n\\begin{align}\\label{online-alg2}\n    x^{k+1}=x^k-\\gamma_k (\\nabla F(x^k;\\xi^k)+e^k).\n\\end{align}\n\n\\begin{corollary}\\label{online-2}\nLet the stepsizes satisfy \\eqref{stepsize-2}, $(x^k)_{k\\geq 0}$ be generated by Algorithm \\eqref{online-alg2}, the noises obey \\eqref{noise-2}, and Assumption \\ref{Assup4} hold. Assume the Markov chain is time-homogeneous, irreducible, and aperiodic and reversible.\nThen, we have\n\\begin{equation}\\label{online-2-r1}\n    \\lim_{k}  \\mathbb{E}\\|\\nabla \\mathbb{E}_{\\xi}( F(x^k;\\xi))\\|=0,\\quad      \\mathbb{E}(\\min_{1\\leq i\\leq k}\\{\\|\\nabla \\mathbb{E}_{\\xi}( F(x^i;\\xi))\\|^2\\})=O\\Big(\\frac{\\max\\{1,\\frac{1}{\\ln(1\/\\lambda)}\\}}{\\sum_{i=1}^k\\gamma_i}\\Big),\n\\end{equation}\nwhere $0<\\lambda<1$ is geometric rate for the mixing time of the Markov chain.\n\\end{corollary}\n\n\n\n\n\n\n\n\\section{Conclusion}\nIn this paper, we have analyzed the stochastic gradient descent method where the samples are taken on a trajectory of Markov chain. One of our main contributions is non-ergodic convergence analysis for convex MCGD, which uses a novel line of analysis. The result is then extended to the inexact gradients. This analysis lets us establish convergence for non-reversible finite-state Markov chains and for nonconvex minimization problems. Our results are useful in the cases where it is impossible or expensive to directly take samples from a distribution, or the distribution is not even known, but sampling via a Markov chain is possible. Our results also apply to decentralized learning over a network, where we can employ a random walker to traverse the network and minimizer the objective that is defined over the samples that are held at the nodes in a distribute fashion.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nRecently, the entanglement has been recognized as an important\nresource of some quantum mechanical phenomena, such as quantum\nteleportation, quantum cryptography, quantum computation, and\nviolation of Bell's inequality \\cite{Nielson,Bennett,Murao}. Many\ninvestigations show that entanglement exists naturally in the spin\nchain when the temperature of system is at zero. A useful many-body\nentanglement measure of a pure state is the von Neumann entanglement\nentropy \\cite{Bennett1}, which can also quantify quantum phase\ntransition \\cite{Sachdev,Gu1,Anfossi,Lambert,Larsson,Molina,Legeza}.\nThe bipartite entanglement in systems of atomic Bose-Einstein\ncondensate was studied \\cite{Milburn,Tonel}. The entanglement\nentropy in the antiferromagnetic Heisenberg $XX$ chain and Ising\nmodel was investigated \\cite{Vidal}. In isotropic antiferromagnetic\nHeisenberg model, the universal form of the entropy is predicted by\n\\cite{Latorre}\n\\begin{equation}\n\\label{eq1}S_L=\\frac{c}{3}log_2L+k,\n\\end{equation}\nwhere $c$ is the central charge and $k$ is a non-universal constant.\nFor a spin chain of open boundary condition, the analogous formula\n$c\/3$ should be replaced by $c\/6$ for a part of length $L$ in an\ninfinite one-dimension system \\cite{Holzhey,Calabrese}. The\nanalogous formula is dependent on the boundary conditions of the\nblock and the rest of the chain. If the block has two boundaries\nwith the rest of the chain then the factor is $c\/3$, while if the\nblock has just one boundary as in the case of a block consisting of\nthe first adjacent spins of a semi-infinite chain, then the factor\nis $c\/6$. Recently, it is shown that a feeble central bound defect\n\\cite{Zhao} or single impurity in the boundary \\cite{Es1} has strong\ninfluence on the entropy, though the entropy measures the mutual\ncoupling of the two parts of a system in wave function. A weak\ntransverse boundary magnetic field impurity \\cite{Zhou} and domain\nwalls \\cite{Ren} generated by antiparallel magnetic field have\ndifferent effects on the entanglement entropy. Moreover, impurities\nshow their strong influences on spin correlation function\n\\cite{Lou,Fath}. It would be interesting to investigate the effects\nof boundary impurities on the entanglement entropy in an opened\nantiferromagnetic Heisenberg spin chain.\n\nIn this paper, the entanglement entropy of a spin$-1\/2$\nantiferromagnetic Heisenberg chain with boundary impurities located\nat two ends is investigated. In scetion II, the Hamiltonian of an\nantiferromagnetic Heisenberg spin$-1\/2$ chain is presented. By using\nthe method of density-matrix renormalization-group (DMRG)\n\\cite{white,U}, the entropy of ground state is calculated and the\neffects of the impurities are analyzed in section III. A discussion\nconcludes the paper.\n\n\\section{Hamiltonian of Heisenberg Spin Chain}\n\nThe Hamiltonian of a spin$-1\/2$ Heisenberg opened chain with\nboundary impurities at two ends can be written as\n\\begin{equation}\n\\label{eq2} H\n=\\sum_{i=2}^{N-2}J\\vec{S}_i\\vec{S}_{i+1}+\\alpha(\\vec{S}_1\\vec{S}_2+\\vec{S}_{N-1}\\vec{S}_{N}).\n\\end{equation}\nWhere the coupling exchange $J>0$ corresponds to the\nantiferromagnetic case, $\\vec{S}_j$ are spin operators, $N$ is the\nlength of the spin chain. The coupling exchange $\\alpha$ is impurity\ninteraction. For simplicity, $J=1$ is assumed in this paper.\n\nThe entropy is used as a measure of the bipartite entanglement. If\n$|Gs\\rangle$ is the ground state of a chain of N qubits, a reduced\ndensity matrix of $L$ contiguous qubits can be written as\n\\begin{equation}\n\\label{eq3}\\rho_L=Tr_{N-L}|Gs\\rangle\\langle Gs|.\n\\end{equation}\nThe bipartite entanglement between the right-hand $L$ contiguous\nqubits and the rest of the system can be measured by the entropy\n\\begin{equation}\n\\label{eq4}S_L=-Tr(\\rho_L\\log_2\\rho_L).\n\\end{equation}\nOne of the properties of the entropy of a block of the system can be\ngiven by\n\\begin{equation}\n\\label{eq5}S_L=S_{N-L},\n\\end{equation}\nsince the spectrum of the reduced density matrix $\\rho_L$ is the\nsame as that of $\\rho_{N-L}$.\n\n\\section{Entanglement of Entropy with Impurities}\n\nIn order to calculate the entropy accurately using the method of\nDMRG, the length of the spin chain needs to be relatively long. The\nlength of the spin chain is chosen to be $N=256$. The total number\nof the density matrix eigenstates held in the system block is\n$m=128$ in the basis truncation procedure.\n\nTo check the accuracy of the results from the method of DMRG, the\nopen boundary condition without impurities of $\\alpha=1$ is\nconsidered. The corresponding results of finite spin chain, which is\npredicted by conformal field theory(CFT), can be considered as a\nbenchmark. It can be written as\n\\begin{equation}\n\\label{eq6}S_L=\\frac{c}{6}\\log_2[\\frac{N}{\\pi}\\sin(\\frac{\\pi}{N}L)]+A,\n\\end{equation}\nwhere $c$ is the central charge and $A$ is a non-universal constant\n\\cite{Chiara,Nicolas}. There are large oscillations between even and\nodd $L$-value entropy. To avoid these relatively large oscillations,\nthe even-value entropy is chosen. The entropy $S_L$ between\ncontiguous $L$ qubits and the remain $N-L$ qubits is plotted as a\nfunction of the subsystem L in Fig. 1 when $N=160, 200$ and $256$.\nFor $L<8$, the results of DMRG are slightly lower than that of CFT.\nFor $L>8$, almost perfect agreement between the two results is\nobtained. For different values of $N$ with large $L$, $S_L$ is small\nfor small values of $N$. For small $L$, there is almost no\ndifference between different values of $N$. It seems that $S_L$\napproaching a constant for very large $L$ is mainly due to finite\nsize effect. The entropy $S_L$ is also plotted as a function of\n$\\log_2[\\frac{N}{\\pi}\\sin(\\frac{\\pi}{N}L)]$ in the inset of Fig. 1.\nIt is shown that the entropy appears as a straight line whose slope\nis very close to $c\/6$.\n\nThe entanglement entropy $S_{L}$ is plotted as a function of the\nsubsystem length $L$ for different values of the impurity\ninteraction $\\alpha$ in Fig. 2(a). It is seen that the entropy $S_L$\nincreases with the subsystem length $L$ and then approaches a\nconstant when $L$ is very large for $\\alpha=0.1, 2.0$. When\n$\\alpha=0.3, 0.5$, the entropy $S_L$ decreases slightly, then\nincreases and approaches a constant for very large $L$. The minimal\nvalue is $1.53$ at $L=6$ for $\\alpha=0.3$ and $1.16$ at $L=4$ for\n$\\alpha=0.5$. When $\\alpha=0.1$, $S_L$ approaches the value about\n$2.4$ for very large $L$. While for $\\alpha=0.3, 0.5, 2.0$, $S_L$\napproaches the value about $1.6$ for very large $L$. The influence\nof the impurity at two ends of the Heisenberg spin$-1\/2$ chain\ndepends on the value of the impurity interaction $\\alpha$. For\n$\\alpha=\\alpha_0=0.235$, the strength alternation of even bond and\nodd bond in the center of the spin chain is minimized to almost\nclose to zero \\cite{white}. For $\\alpha<\\alpha_0$, the even\nsublattice is favored. This induces larger value of $S_L$. While for\n$\\alpha>\\alpha_0$, the odd sublattice is favored. This induces\nsmaller value of $S_L$. If the value of $\\alpha$ is less than\n$\\alpha_0=0.235$, the effects of the impurity on the entanglement\nentropy $S_L$ is stronger. Therefore, the entanglement entropy $S_L$\nof $\\alpha=0.1$ is much larger than the $S_L$ of $\\alpha=0.3, 0.5$\nand $2.0$ \\cite{Zhao,Es1,white}. The entropy $S_L$ is also plotted\nas a function of $\\log_2[\\frac{N}{\\pi}\\sin(\\frac{\\pi}{N}L)]$ in the\ninset of Fig. 2(a). It is seen that $S_L$ is almost a straight as a\nfunction of $\\log_2[\\frac{N}{\\pi}\\sin(\\frac{\\pi}{N}L)]$ for very\nlarge $L$.\n\nIf $S_{L\\alpha}$ is the entropy with impurities at two ends and\n$S_{L0}$ is the entropy without impurities, the difference of the\nentropy $\\Delta S_L$ can be defined as\n\\begin{equation}\n\\label{eq7}\\Delta S_L=S_{L\\alpha}-S_{L0}.\n\\end{equation}\nThe entropy difference $\\Delta S_{L}$ may also be called \"impurity\nentanglement entropy\" that is induced by adding impurities at two\nends of the spin chain \\cite{Es1}. The entropy difference $\\Delta\nS_{L}$ is plotted as a function of the subsystem length $L$ for\ndifferent values of the impurity interaction $\\alpha$ in Fig. 2(b).\nIt is shown that $\\Delta S_L$ decreases and then approaches a\nconstant when the subsystem $L$ increases for $\\alpha=0.1, 0.3,\n0.5$. The value of $\\Delta S_L$ increases and then approaches a\nconstant with increase of the subsystem length $L$ when\n$\\alpha=2.0$. It seems that the effect of impurity decreases with\nthe increase of the subsystem $L$ when $\\alpha<\\alpha_0=1.0$.\nHowever, the effect of the impurity increases when\n$\\alpha>\\alpha_0=1.0$. The entropy difference $\\Delta S_{L}$ is also\nplotted as a function of $\\log_2[\\frac{N}{\\pi}\\sin(\\frac{\\pi}{N}L)]$\nin the inset of Fig. 2(b). It is seen that $\\Delta S_L$ is almost a\nstraight as a function of\n$\\log_2[\\frac{N}{\\pi}\\sin(\\frac{\\pi}{N}L)]$ for very large $L$.\nSimilar to that shown in Fig. 2(a), the entropy difference $\\Delta\nS_{L}$ of $\\alpha=0.1$ is much larger than $\\Delta S_L$ of\n$\\alpha=0.3, 0.5$ and $2.0$. This is mainly due to the fact that\nsmall value of $\\alpha<\\alpha_0=0.235$ can induce stronger effect of\nimpurity on $\\Delta S_{L}$ since the even sublattice is favored\n\\cite{Zhao,Es1,white}.\n\nFrom Fig. 2, it is seen that both values of $S_L$ and $\\Delta S_L$\nare decreases when $\\alpha$ increases especially for small number of\n$L$. There are large differences of $S_L$ and $\\Delta S_L$ for\ndifferent impurity interactions $\\alpha$ when $L$ is small. If $L$\nis quite large, $S_L$ and $\\Delta S_L$ approach to constants. If\n$\\alpha=0.3, 0.5$ and $2.0$, $S_L$ approaches to about $1.6$ while\n$\\Delta S_L$ approaches to about zero for quite large $L$. It seems\nthat the effect of the impurity at two ends is very small for large\n$L$. If $\\alpha=0.1$, both $S_L$ and $\\Delta S_L$ are quite large.\nIt seems that the small value of impurity interaction\n$\\alpha<\\alpha_0=0.235$ can induce strong effect on $S_L$ and\n$\\Delta S_L$.\n\nThe central charge $c$ in Eq. (6) plays an important role in the\nmeasurement of the entanglement entropy. The central charge $c$ can\nbe calculated numerically by \\cite{Calabrese,Es1}\n\\begin{equation}\n\\label{eq8}c(L)=6[\\frac{S_{L+2}-S_{L-2}}{T(L+2)-T(L-2)}],\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{eq9}T(L)=\\log_2[\\frac{N}{\\pi}\\sin(\\frac{\\pi}{N}L)].\n\\end{equation}\nThe central charge labeled by $c(L)$ is plotted in Fig. 3(a) as a\nfunction of the subsystem length $L$ for different values of the\nimpurity interaction $\\alpha$. When $\\alpha=0.1$, the central charge\n$c(L)$ increases to a peak and then decreases slowly with the\nincrease of the subsystem length $L$. The central charge $c(L)$\ndecreases and then approaches a constant with the increase of the\nsubsystem $L$ when $\\alpha=2.0$. When $\\alpha=0.3, 0.5$, the central\ncharge $c(L)$ increases and almost approaches a constant with the\nincrease of the subsystem length $L$. The central charges of\n$c(L=6)$ and $c(L=4)$ are negative when $\\alpha=0.3$ and $0.5$\nrespectively. This corresponds to the minimum values of $S_L$ shown\nin Fig. 2(a). For $\\alpha>0.235$, the central charge $c(L)$\nincreases with increasing value of $\\alpha$. For $\\alpha=0.1<0.235$,\nc(L) of $\\alpha=0.1$ is larger than that of $\\alpha=0.3$, it\ndecreases and finally approaches to that of $\\alpha=0.3$ when $L$ is\nvery large. The value of $c(L)$ of $\\alpha=0.1$ is larger than that\nof $\\alpha=0.5$ if $L<20$. If $L>20$, $c(L)$ of $\\alpha=0.1$ is\nsmaller than that of $\\alpha=0.5$. This is mainly due to the\nstronger singlet bonds on the even numbered links of the chain for\n$\\alpha<0.235$ \\cite{white}. Since the central charge may clarify\nthe behavior of the entropy for large values of subsystem, the\ncentral charge $c(L=80)$ is plotted as a function of impurity\ninteraction $\\alpha$ in Fig. 2(b) when $1\\ll L(=80)< N\/2(=128)$. It\nis seen that the central charge c reaches a minimum value when\nimpurity interaction $\\alpha=0.235$. When impurity interaction\n$\\alpha<0.235$, the central charge $c$ decreases with impurity\ninteraction $\\alpha$ increases. The central charge $c$ increases\nwith impurity interaction $\\alpha$ increases when $\\alpha>0.235$. It\napproaches to $1.0$ when impurity interaction $\\alpha$ is close to\n$2.0$.\n\nIf the impurities are qutrit with spin-$1$ operators $\\vec{S'}$, the\neffects of qutrit-impurities on the entropy can also be\ninvestigated. The entanglement entropy $S_L$ and the difference of\nthe entropy $\\Delta S_{L}$ are plotted in Figs. 4(a) and 4(b)\nrespectively as a function of the subsystem length $L$ for impurity\ninteraction $\\alpha$ and different impurities. The entropy $S_L$ and\nthe difference of the entropy $\\Delta S_{L}$ are also plotted as a\nfunction of $\\log_2[\\frac{N}{\\pi}\\sin(\\frac{\\pi}{N}L)]$ in the\ninsets of Figs. 4(a) and 4(b). Similar to that shown in Fig. 2, both\n$S_L$ and $\\Delta S_L$ of spin-$1$ decreases with the increase of\n$\\alpha$. The entropy $S_L$ increases and then almost approaches a\nconstant when the the subsystem length $L$ increases. The entropy of\nimpurity with spin-$1$ is much larger than that with spin-$1\/2$. For\nthe impurity of spin-$1\/2$, the entropies of $\\alpha=0.5$ and\n$\\alpha=2.0$ are almost not distinguishable when the subsystem\nlength $L$ is very large. While for the impurity of spin-$1$, the\ndifference of the entropies of $\\alpha=0.5$ and $\\alpha=2.0$ are\nquite large and approach a constant when $L$ is very large. It is\nclear that the effects of qutrit impurity on the entanglement\nentropy are much stronger than that of qubit impurity. It seems that\nit is more easily to control the entropy of the system using qutrit\nimpurity.\n\n\\section{Discussion}\n\nIt is clear that the impurity interaction and the impurity spin have\na strong influence on the entanglement of the two subsystems\n\\cite{Es1,Wang1,E02}. For pairwise entanglement between the impurity\nspin and the spin chain, the two boundary spins will have a strong\ntendency to form a singlet pair when the impurity interaction is\nlarge. This will reduce the entanglement between the boundary of the\ntwo spin subsystems and the rest of the system. The value of\nentanglement entropy is mainly determined by the density-matrix\nspectra, extremely by the few largest eigenvalues of the reduced\ndensity matrix \\cite{U,Zhou,Zhao}. For qubit impurities, impurities\ncan affect the entropy between two subsystems by changing the\ndistribution of the reduced density-matrix spectra. If the\nimpurities are qutrits with the same impurity interaction, not only\nthe distribution of the reduced density-matrix spectra is changed,\nbut also the degree of the freedom of density-matrix spectra of the\nsubsystem is enlarged in the Hilbert space. This is similar to the\nresult of the entropy with the increase of subsystems\n\\cite{Latorre,Zhou,Ren}.\n\nThe effects of boundary impurities on the bipartite entanglement in\nan antiferromagnetic Heisenberg open spin chain are discussed. Using\nthe method of density-matrix renormalization-group, entanglement\nentropy is calculated for the even number subsystem. The\nentanglement entropy decreases with the increase of the impurity\ninteraction while it increases with the increase of the subsystem\nlength. When the system length is very large, the entanglement\nentropy approaches a constant due to finite size effect. The\ninfluences of boundary impurities with qutrit of spin-$1$ are much\nstronger than that of qubit of spin-$1\/2$. With the same impurity\ninteraction, qutrit impurities can increase the entanglement. All\nthe results are dependent with the selection of even subsystem. This\nis shown that the entropy of the system with qutrit impurity can be\nmore easily controlled.\n\n\\vskip 0.4 cm {\\bf Acknowledgements}\n\nIt is a pleasure to thank Yinsheng Ling and Jianxing Fang for their\nmany helpful discussions. The financial supports from the National\nNatural Science Foundation of China (Grant No. 10774108) and the\nCreative Project for Doctors of Jiangsu Province of China are\ngratefully acknowledged.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\nAxions are the most natural solution to the strong CP problem \\cite{Peccei:1977hh,Weinberg:1977ma,Wilczek:1977pj}.\nThere are significant phenomenological constraints on axions,  such that only so-called ``invisible axions\" remain\nas viable candidates (for a review, see \\cite{Kim:2008hd}). These are known as the KSVZ axion \\cite{Kim:1979if,Shifman:1979if} or\nDFSZ axion \\cite{Dine:1981rt,Zhitnitsky:1980tq}.  The supersymmetric version of invisible axions was given in \\cite{Nilles:1981py,Kim:1983dt}.\nAxions, with a decay constant $f_a \\sim 10^{12}$ GeV can make up the dominant form of dark matter in the universe.  They form a smooth coherent\nbackground state.   Laboratory experiments such as ADMX \\cite{Rosenberg:2010zz,Carosi:2012zz} and light shining through walls \\cite{Redondo:2010dp,Betz:2013dza} have been designed to look for them.  These are based\non the fact that an axion in a strong magnetic field produces light.   In a recent paper, Graham and Rajendran \\cite{Graham:2013gfa}, have used the generic phenomenon that axions are responsible for a time dependent electric dipole moment of nuclei (see, for example, \\cite{Pospelov:1999ha}).  They propose searching for this tiny effect.\n\nOn another line of reasoning,  it has been known that a self-gravitating Bose gas can form stable clumps of matter \\cite{Kaup:1968zz,Ruffini:1969qy,Breit:1983nr,Tkachev:1991ka,Wang:2001wq}.  In the case of axions these clumps are termed ``axion stars\"  \\cite{Barranco:2010ib,Eby:2014fya,Chavanis:2011zm,Guth:2014hsa}.  These objects are very dilute with $M_* \\sim 10^{-14} M_\\odot$ and radius $R_a \\sim 10^{-4} R_\\odot$.  Recently the possibility of dense axion stars have been discussed \\cite{Braaten:2015eeu}.  Note, if primordial density perturbations in the axion field leads to most of the axion background contained in axion stars then it will be very difficult to find axions in laboratory experiments.\n\nIn the case of axion stars, other methods for discovery are necessary.  One possibility was suggested by Iwazaki \\cite{Iwazaki:2014wta}.  He uses   the electromagnetic coupling of axions, given by ${\\cal L} \\supset  \\frac{\\alpha}{f_a \\pi} a(\\vec{x},t) \\vec{E} \\cdot \\vec{B}$  where $\\vec{E}, \\ \\vec{B}$ are the electric and magnetic fields.   Neutron stars have strong magnetic fields and free electrons moving in the thin atmosphere on the surface. When an axion star enters the atmosphere of a neutron star, a time dependent electric field is produced by the axion star in the background magnetic field of the neutron star.  Then an immense amount of energy is released due to the time dependent coherent dipole oscillations\nof the electrons.   The energy released is in the radio frequency (note $m_a c^2 = 10^{-5}$ eV corresponds to a frequency $\\frac{m_a c^2}{2 \\pi} = 2.5 $  GHz).  This suggests that the encounter of axion stars with neutron stars might be the source for Fast Radio Bursts [FRBs]  \\cite{Lorimer:2007qn,Keane:2009cz,Thornton:2013iua,Spitler:2014fla}.  For a review on FRBs, see \\cite{Katz:2016dti}.  There are only 17 known FRBs in the list.  Recently a repeating FRB has been discovered \\cite{Spitler:2016dmz}.  This may make the chance encounter of an axion star with a neutron star an unlikely source for this phenomena,  but it does not exclude this possibility.\n\nIn this paper we consider the effect of the time dependent electric dipole moment for neutrons in the upper core of the neutron star.\nThese also radiate in the presence of an axion star and we argue that a significant portion of this radiation is coherent.   As a consequence, much more energy is available which can be released as FRBs.\n\n \\section*{Axion stars}\nLet us first note some of the parameters for, so-called, dilute axion stars.  They have a minimum radius, $R_a\/R_\\odot \\sim 3 \\times 10^{-4}$, with a maximum mass,  $M_a\/M_\\odot \\sim 6 \\times 10^{-14 \\mp 4}$ for an axion mass, $m_a c^2 \\sim 10^{-4 \\pm 2}$ eV \\cite{Chavanis:2011zm}.\\footnote{$M_\\odot \\approx 2 \\times 10^{30} Kg = 1.2 \\times 10^{57} \\ m_N$ and $R_\\odot \\approx 7 \\times 10^8 m$.}\nFor later use we note that in a dilute axion star the central value of the axion field, $a_0$, for an axion star with maximum mass is given by $a_0\/f_a \\sim 10^{-7 \\mp 2}$.  In general, we have $a_0\/f_a \\sim 10^{-7 \\mp 2} (M_a\/M_a^{max})^4$  where $R_a\/R_\\odot \\sim 3 \\times 10^{-4}  (M_a^{max}\/M_a)$.\n\nFor a dense axion star, on the other hand, the minimum mass is approximately equal to $10^{-20 \\mp 6} \\ M_\\odot$ with a minimum radius of order $R_a\/R_\\odot \\sim 10^{-11 \\mp 2}$.  It has a maximum mass of order $M_a\/M_\\odot \\sim 1$ and maximum radius, $R_a\/R_\\odot \\sim 10^{-5}$ \\cite{Braaten:2015eeu}.   In this case we have $a_0\/f_a \\sim 1$.\n\nThe flux, $F_a$, of axion stars in the galaxy is roughly given in terms of the energy density of dark matter in the galaxy, $\\rho_{DM} \\sim 0.3 {\\rm GeV\/cm^3}$, times the virial velocity, $v_{vir}\\sim 300 {\\rm km\/s}$, of objects in the galaxy.  Thus $F_a \\sim \\left( \\rho_{DM}\/M_a \\right) v_{vir}$. For the calculations in this letter we take $m_a c^2 = 10^{-5}$ eV, corresponding to $f_a \\sim 10^{12}$ GeV, which is in the regime required for a natural abundance of axion cold dark matter \\cite{Kim:2008hd}.  This gives a dilute axion star with maximum mass, $M_a\/M_\\odot = 6 \\times 10^{-12}$  or $M_a = 6.6 \\times 10^{45}$ GeV\/c$^2$.  Note, the more massive the axion star, the smaller the flux of axion stars.  In addition, the denser the axion star,  the smaller is its radius and thus the collision rate for axion - neutron star collisions is suppressed.   For both these reasons, we only consider dilute axion stars in the following.  In order to evaluate the rate of axion-neutron star collisions we need to know the cross-section for this process.  This cross-section is given by the area, $A = \\pi b^2$,  where $b$ is the impact parameter for the collision.   Due to the classical counterpart of the Sommerfeld enhancement we have $b^2 = (R_{ns} + R_a)^2  \\left[1 + \\frac{2 G_N M_{ns}}{(R_{ns} + R_a) v_{vir}^2} \\right]$, where $R_{ns} = 10$ km is the typical radius of a neutron star, $M_{ns} = 1.4 M_\\odot$ is its mass and $G_N$ is Newton's constant.  The enhancement factor $\\frac{2 G_N M_{ns}}{(R_{ns} + R_a) v_{vir}^2} \\sim 20$ and we obtain $A = 3.2 \\times 10^6  \\left( \\frac{R_{ns} + R_a}{220 {\\rm km}} \\right) $ km$^2$.  Assuming of order $10^9$ neutron stars in a galaxy we obtain an event rate per galaxy, \\begin{equation} R_{a-ns} \\sim 10^{-6} (6 \\times 10^{-12} M_\\odot\/M_a)\\left( \\frac{R_{ns} + R_a}{220 {\\rm km}} \\right) {\\rm y}^{-1} . \\end{equation} Note, the radius of a dilute axion star increases as its mass decreases. Thus an axion star with mass, $M_a\/M_\\odot \\sim 2 \\times 10^{-13}$ and $R_a\/R_\\odot \\sim 9 \\times 10^{-3}$ would produce an event rate per galaxy of order, $R_{a-ns} \\sim 10^{-3} y^{-1}$.  Now, following Iwazaki \\cite{Iwazaki:2014wta}, we note that this is the observed rate for fast radio bursts [FRBs] \\cite{Thornton:2013iua}.\n\n\\section*{Fast Radio Bursts and Axion stars}\n\n  FRBs have been observed at a frequency, $\\nu_{FRB} \\sim 1.4$ GHz.  This happens to be about the axion oscillation frequencey, $\\nu_a = \\frac{m_a c^2}{2 \\pi} = 2.5 \\left( \\frac{m_a c^2}{10^{-5} {\\rm eV}} \\right)$ GHz. Moreover, the observed frequency will be red shifted.   Iwazaki discussed the FRBs produced when the axion star interacts with the strong magnetic field in the thin atmosphere of the neutron star.  It causes coherent dipole oscillations of the electrons which then radiate in the frequency set by the axion star over a time scale of order a milli-second or less.   In the following we describe an additional effect and perhaps the dominant effect.  FRBs are most likely due to events at redshifts of 0.5 to 1.  They are associated with a huge release of energy of order $10^{38} - 10^{40}$ ergs or $6.25 \\times (10^{40} - 10^{42})$ GeV corresponding to a rate of energy production, $P$, exceeding $6.25 \\times (10^{43} - 10^{45})$ GeV\/s.\n\n\\section*{Neutron stars}\n\nThe outer core of neutron stars contains mostly neutrons with 3 to 5\\% free protons and electrons \\cite{Potekhin:2011xe}.  The neutron star cools to a temperature, $T \\sim 10^6 {^\\circ}K$ with a strong magnetic field, $B \\geq 10^{12}$ G.  When the axion star enters the outer core of the neutron star it induces a time dependent electric dipole moment in the neutrons, $d(t) = d_0 \\sin(\\omega t)$ with $\\omega = m_a c^2\/\\hbar$.   We have $d_0 = 2.4 \\times 10^{-16} \\left(\\frac{a_0}{f_a}\\right)$ e cm \\cite{Pospelov:1999ha,Graham:2013gfa} and we take $a_0\/f_a =  10^{-10}$.  This time dependent electric dipole moment radiates radio waves with frequency $\\nu_a$. The total power radiated is given by \\begin{equation} P = \\frac{(m_a c^2)^4 \\alpha}{3 \\pi \\hbar} \\left(\\frac{d_0 N_n \\rho}{e \\hbar c}\\right)^2  \\end{equation} where $\\alpha = 1\/137$ is the fine structure constant, $N_n$ is the number of neutrons affected by the axion star and $\\rho$ is the net polarization of the neutron spins given by $\\rho = \\frac{N_n \\uparrow - N_n \\downarrow}{N_n \\uparrow + N_n \\downarrow}$.\\footnote{We expect the polarized neutrons to radiate coherently \\cite{Dicke:1954zz}.}  The neutrons are polarized by the strong magnetic field.  The magnetic field inside a neutron star has been modeled \\cite{Braithwaite:2005xi,Ciolfi:2014jha}.   It includes both poloidal and toroidal components.   Neutrons have a magnetic dipole moment, $|\\mu_n| \\approx 6 \\times 10^{-14}$ MeV\/T.  Thus $2 |\\mu_n| B = 1.2 \\times 10^{-2} (B\/10^{15} {\\rm G})$ MeV  and the Boltzmann factor determining the ratio of spin up to spin down neutrons, $e^x$ is given by $ x \\equiv \\frac{2 |\\mu_n| B}{k T} = 1.4 \\ (B\/10^{15} {\\rm G})\/(T\/ 10^8 {^\\circ} {\\rm K})$  with $k T = 8.6 \\ (T\/ 10^8 {^\\circ}{\\rm K})$ keV.\\footnote{The neutrons in the neutron star are believed to be in a $^3P_2$ superfluid state \\cite{Schwenk:2003bc,Page:2010aw}. Thus the neutron spins are alligned in the condensate.  In addition we have assumed a large magnetic field, associated with that of neutron stars, known as magnetars.}   The net polarization is then $\\rho = \\tanh(x\/2) \\approx 0.6$.  We then find the power radiated given by\n\\begin{eqnarray}  P = & 1.8 \\times 10^{55} \\ \\left( \\frac{m_a c^2}{10^{-5} \\ {\\rm eV}} \\right)^4 \\left( \\frac{N_n}{1.7 \\times 10^{57}} \\ \\frac{\\rho}{0.6} \\right)^2  \\ {\\rm GeV}\/s & \\\\\n= & 2.9 \\times 10^{52} \\ \\left( \\frac{m_a c^2}{10^{-5} \\ {\\rm eV}} \\right)^4 \\left( \\frac{N_n}{1.7 \\times 10^{57}} \\ \\frac{\\rho}{0.6} \\right)^2 \\ {\\rm erg}\/s. & \\nonumber \\end{eqnarray}  Since we have assumed that the total mass of the axion star is $M_a\/M_\\odot \\sim 2 \\times 10^{-13}$ or $M_a \\sim 2.2 \\times 10^{44}$ GeV\/c$^2$, the star loses all its mass to radiation in a small fraction of a second.   This pulse of energy would typically last much less than a milli-second. Note, however, that the rate depends sensitively on the axion mass, the net polarization, $\\rho$, of the neutrons in the neutron star core (which depends on the magnitude of the magnetic field and the temperature in the neutron star), the fraction of the neutron star covered by the axion star, i.e. $f = \\frac{N_n}{1.7 \\times 10^{57}}$\nand the timescale for the traversal of the axion star through the neutron star.  The last two effects depend crucially on the the amount of stretching of the axion star due to tidal forces and the velocity of the axion star.  We will comment on the issue of tidal forces in the next section.\n\nOf course, the next question is does this radiation escape from the neutron star.  This is a very difficult question which would require serious numerical simulations.  But in the absence of this analysis, let me make some simplifying assumptions.   There are two properties of the interior neutron star which can dramatically affect the result.  First there is a huge magnetic field which is most likely a combination of poloidal and toroidal. Secondly, about 1\\% of the outer core of the neutron star is a plasma of ionized protons and electrons.  It is believed that the protons in the outer core of a $10^{12}$ G neutron star are in a superconducting state.  As a result, the magnetic fields either form a lattice of Abrikosov flux tubes for a type II superconductor or an intermediate state of normal and superconducting matter for a type I superconductor \\cite{Huebener:1974ui,PhysRevB.57.3058}.  In either case,  electromagnetic waves would be severely constrained.   However, if the interior magnetic field is large enough, $B > 2 \\times 10^{16}$ G, then it is believed that the core is not superconducting \\cite{Sinha:2015bva,Sinha2015}.  Moreover, for smaller magnetic fields, $ 10^{15} \\ {\\rm G} \\leq B \\leq 2 \\times 10^{16} \\ {\\rm G}$,  the core may be only partially superconducting \\cite{Sinha:2015bva,Sinha2015}.  Hence radiation may then propagate in this non-superconducting core. Neutron stars with surface magnetic fields of order $10^{15}$ G are known as ``magnetars.\"   It is believed that magnetars may be just as numerous as neutron stars with magnetic fields in the $10^{12}$ G range \\cite{Duncan:1992hi,Thompson:1993hn,Thompson:1995gw,Duncan:2003,Turolla:2015mwa}.\n\nThe magnetic field might make it easier for the radiation to escape.  It can be shown that\nthe optical opacity of a medium is severely suppressed in the presence of a strong magnetic field \\cite{Canuto:1971cd}.  In the\nfully ionized atmosphere of a neutron star the opacity has been evaluated in Ref. \\cite{Potekhin:2002nk}.  The absorption cross-section for transverse electromagnetic waves with $\\alpha = \\pm 1$ and frequency $\\omega$ is given by (Eqn. 51, Ref. \\cite{Potekhin:2002nk})\n\\begin{equation} \\sigma_\\alpha^{ff} \\approx \\frac{\\omega^2}{(\\omega + \\alpha \\omega_{ce})^2 (\\omega - \\alpha \\omega_{cp})^2 + \\omega^2 \\tilde \\nu_\\alpha^2} \\frac{4 \\pi e^2 \\nu_\\alpha^{ff}}{m_e c} \\end{equation}  where $\\omega_{ce}, \\ \\omega_{cp}$ are cyclotron frequencies for electrons and protons, respectively.   For $\\omega \\ll \\omega_{ce}, \\ \\omega_{cp}$ the absorption cross-sections vanishes $\\propto \\omega^2$.   In Fig. 7, Ref. \\cite{Potekhin:2002nk}, the opacity was calculated for a plasma with\ndensity, $\\rho = 500$ g\/cm$^3$, temperature, $T = 5 \\times 10^6 {^\\circ}$ K and magnetic field, $B = 5 \\times 10^{14}$ G.  These results will change with a stronger magnetic field.   At magnetic fields of order $10^{16}$ G, the proton cyclotron frequency is two orders of magnitude larger than the 3.15 keV of Fig. 7, Ref. \\cite{Potekhin:2002nk}.  Moreover, we are interested in radio frequencies of order $10^{-5}$ eV, which are much lower.\n\n\nAssuming we can scale the opacity of Fig. 7, Ref. \\cite{Potekhin:2002nk} (defined for $T \\sim 5\\times 10^6 {^\\circ}$K and $B \\sim 5 \\times 10^{14}$ G) from $\\kappa_0 \\sim 10^{-4.5} cm^2\/g$ at $\\hbar \\omega_0 = 10^3$ eV to lower frequencies with $\\kappa(\\hbar \\omega_a) = (\\frac{\\hbar \\omega_a}{\\hbar \\omega_0})^2 \\ \\kappa_0$  (where $\\hbar \\omega_a = m_a c^2 = 10^{-5}$ eV), we find an extinction coefficient $\\rho \\kappa = \\tau^{-1}$ with $\\tau \\sim 32 \\ (\\frac{\\rho}{10^{14} \\ g\/cm^3})^{-1} \\ km$  for a core density, $\\rho \\sim 10^{14}$ g\/cm$^3$.\\footnote{For a stronger magnetic field the frequency $\\hbar \\omega_0$ would increase. Also note, the core temperature is actually closer to $T \\sim 10^8 {^\\circ}$K.}   Clearly most of the energy emitted by that fraction of the axion star which covers the outer core, located at approximately 1 km below the surface of the neutron star, makes it out of the neutron star.  The rest of the energy goes into heating the plasma.\nFinally, there is necessarily a back reaction on the axion star, as its mass is converted into radiation.\n\n\\section*{Tidal distortion of the axion star}\n\nOnce the axion star approaches a distance $R_{tidal}$ from the neutron star given by \\begin{equation} R_{tidal} \\approx (\\frac{M_{ns}}{M_a})^{1\/3} \\ R_a, \\end{equation} i.e, when the gravitational force at the surface of the axion star is balanced by the tidal forces due to the neutron star, then the axion star stretches in the direction of the neutron star. If it approaches the neutron star with zero impact parameter, $b$, the radio burst may be prolonged \\cite{Pshirkov:2016bjr}, of order a second.  However, if it grazes the neutron star at an impact parameter, $b \\gg R_{ns}$, then only a fraction of the axion star will actually cross the neutron star.  Then the crossing timescale can be of order milli-seconds.\n\n\n\\section*{Conclusion}\n\nIn this paper we discuss the possibility that FRBs are caused by the collisions of dilute axion stars with\nneutron stars of the type known as magnetars.   Unlike the previous analysis of Iwazaki \\cite{Iwazaki:2014wta} which focused on axion stars producing electric dipole radiation from the thin atmosphere of a neutron star, the proposed mechanism can be a much larger volume effect having to do with the\ninduced time dependent electric dipole moment of the neutrons interior to the star.   Further calculations are needed to evaluate the viability of the proposed mechanism.\n\n\\section*{Acknowledgments}\n\nI acknowledge illuminating conversations with E. Braaten, C. Hirata, R. Furnstahl, T. Thompson and  H. Zhang.\nI also received partial support for this work from DOE grant, DE-SC0011726.\n\n\n\n\n\\clearpage\n\\newpage\n\n\\bibliographystyle{utphys}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\nSoftly broken supersymmetry (SUSY) is arguably our best candidate for an explanation of what cuts off the quadratic divergent quantum corrections to the squared Higgs mass in the standard model (SM).  Because of these corrections, the SM alone, with a cutoff much higher than the weak scale, suffers from a fine-tuning problem.  The mass scale of superpartners effectively represent the cutoff and above that scale the tuning is no longer necessary.  Thus, to completely remove tuning from the Higgs, the natural scale for superpartner masses is on the order of the weak scale.\n\n\n\nOf course neither the Higgs superpartners have been discovered.  The bound on the SM Higgs mass is 114.4 GeV \\cite{LEPhiggs}, and this bound, to a good approximation, applies to the lightest Higgs in most of the parameter space of the so-called 'mSUGRA' version of the minimal supersymmetric standard model (MSSM) \\cite{lepSUSYhiggs}.  These bounds are strong enough to require fine tuning within the mSUGRA model, the most restrictive being the Higgs mass bound - the physical Higgs mass in mSUGRA lies naturally below the $Z$ mass, and increases above the $Z$ with contributions only logarithmically sensitive to the superpartner mass scale.  Superpartner masses (mainly the stop mass) need to be pushed beyond the electroweak scale in order to satisfy the bound.\n\n\nGoing beyond mSUGRA, Higgs and superpartner mass bounds can change - and sometimes radically.  Here we study a version of the MSSM with R-parity violation.  We also consider regions where the standard gaugino mass relations do not hold.  The main phenomenological feature of violating R-parity is that the lightest superpartner is no longer stable and that 'missing energy' is either strongly reduced or eliminated in events with superpartners produced.  This single change, allowing the lightest supersymmetric particle (LSP) to decay, reduces bounds on nearly every superpartner (save the chargino) to below 100 GeV, and sometimes well below.  This keeps open the possibility of the Higgs boson decaying into superpartners.\n\n\n\nIn this paper, we explore Higgs decays into a pair of LSPs which result in multi-body final states.  If the Higgs decays in a non-SM way, it affects the lower bound on the Higgs mass as the standard LEP searches are less efficient or simply do not apply.  Based on a wide range of searches, we estimate that many of these new decays reduce the lower bound on the Higgs mass to 105 GeV or less.  While this seems like a small change, the exponential sensitivity of the superpartner ({\\it e.g.}, stop) masses to the needed Higgs mass correction makes this reduction relevant for re-opening SUSY parameter space. The key is beating the standard decays of $h \\rightarrow b\\overline{b}$ by a large amount - and this happens in broad regions of parameter space the Higgs sector is in the decoupling limit.\n\n\nThis paper is organized as follows, section 2 reviews R parity violating operators and their bounds. Section 3 discusses constraints, signatures and parameter space for neutral gaugino LSPs.  Section 4 discusses constraints, signatures and parameter space for third generation scalar LSPs, and section 5 concludes.  In the appendix, we discuss the fit to $e^+ e^-$ hadronic cross section data when a new light charged scalar is in the spectrum.\n\n\n\n\\section{R parity violation}\n\nIn the MSSM generally one imposes a symmetry known as R parity whereby fields and superfields are given opposite parity.  Only operators with positive parity are allowed to appear.  The LSP is stable; if one starts with N superparticles, one must end with N mod 2 due to their negative parity. However we are free to introduce parity violating operators into the superpotential:\n\\begin{eqnarray}\n W & \\supset &\n \\mu_i L_i {\\bar H} + \\lambda_{ijk} L_i L_j E^c_k + \\lambda'_{ijk} L_i Q_j D^c_k \\nonumber \\\\ & + & \\lambda''_{ijk} U^c_i D^c_j D^c_k ,\n \\label{eq:rpv}\n \\end{eqnarray}\nwhere $L$, $E^c$, $\\bar{H}$, $Q$, $U^c$, and $D^c$ are lepton doublet, lepton singlet, up-type Higgs, quark doublet, up-type quark singlet, and down-type quark singlet superfields respectively, and the $ijk$ are flavor indices.  The first operators violate lepton number conservation, while the third violates baryon number conservation.  An acceptable theory cannot have both LNV and BNV couplings of non-negligible size as it would cause rapid proton decay.\n\nFor a general flavor of RPV bounds see \\cite{Allanach:1999ic,Bhattacharyya:1997vv}.  In general LLE operators are of order a few times $10^{-2}$, while LQD and UDD operators are bounded at $10^{-1}$ though a few couplings have much harsher bounds and some are order 1.  LNV bounds come from a great variety of sources.  Semi-leptonic meson decays put bounds on a many products of couplings \\cite{Dreiner:2006gu} most of which fall in the range $10^{-2}-10^{-4}$.  The most stringent bound for LLE operators is on the $\\lambda_{133}$ operator and comes from contributions to the electron neutrino' majorana mass and is less than $10^{-3}$ for 100 GeV superpartner masses.  Other LLE bounds come from measurements of $R_{\\tau}$ and $R_{\\tau \\mu}$.\nThe LQD constraints come from a variety of sources.  Neutrinoless double beta decay\nputs a strict bound on the $\\lambda_{111}^{'}$ operator of $\\sim 10^{-4} \\times (m_{\\chi}\/100GeV)^{1\/2}(m_{\\tilde{e}}\/100GeV)^2$.  Charged current universality, $D_s$ and $D$ meson decays and the measurement of $R_{\\tau}$ also place bounds and in general operators involving third generation couplings $\\lambda_{331}^{'}, \\lambda_{332}^{'},\\lambda_{333}^{'}$ and $\\lambda_{232}^{'}$ are relatively unconstrained.\nThe strictest BNV bounds come from  double nucleon decay and apply to first generation couplings $\\lambda_{112}^{''} \\sim 10^{-7}$.  Neutron anti neutron oscillation limits $\\lambda_{113}^{''}$ to $\\sim 10^{-4}$.  Other couplings are less constrained and bounds vary from $10^{-1}$ to 1.  A generally safe statement is that with the exception of a few pathological couplings cited in the canon, nearly all bounds allow for reasonably prompt decay of a superpartner into SM particles.  Avoiding stringent constraints on some of the first generation RPV operators may be remedied easily by requiring a Yukawa-hierarchical coupling scheme where third generation couplings are dominant; this scheme will be of particular value when considering LNV decays of gaugino LSPs.  Here we favor a Yukawa hierarchical scheme and consider one only one RPV coupling at a time to be turned on.  We now go on to list possible LSPs which both may be lighter than half the Higgs mass, and then may decay via RPV.\n\n\n\\section{Gaugino LSPs}\n\\subsection{Topology and General Constraints}\nThe Higgs may decay into a pair of neutral gaugino LSPs.  The RPV decay then proceeds as each gaugino decays to a fermion and virtual sfermion which itself decays via an RPV vertex to two SM particles.  It would be simple here to think of gauginos decaying through an effective four fermi interaction to three fermions $\\chi_0 g_i\\lambda_{jk}\/M_{\\overline{f}}$.  The topology of these decays is Higgs to 6 SM fermions.  In the case of BNV we would have Higgs to six jets.  For LNV, Higgs to 4 jets plus 2 leptons, and LLE to 6 leptons.\n\nThere are several general constraints on this scenario applicable to all gaugino LSPs.\nThe first is the mass requirement that gauginos be less than half of the Higgs mass.  What usually stands in the way of gauginos this light are assumptions about mass relations in standard SUSY mass schemes.  Gauge mediation, anomaly mediation, and MSUGRA relate the gauginos by a single mass parameter.  There is a tight experimental lower bound on the chargino mass of 102.7 GeV, which then constrains the mass of the lightest neutralino through standard gaugino mass relations.  In addition, most standard mass schemes also predict a gluino much heavier than the other gauginos.  For example, MSUGRA predicts the M2:M3 ratio to be about 3:1 thus a chargino passing the lower mass bound implies a heavy gluino.  By rejecting the standard gaugino mass relationships we open up large regions of SUSY parameter space.\n\nWe must also address lower mass bounds on sparticles from existing RPV searches.  DELPHI and L3 have put lower bounds on the neutralino as the LSP decaying through RPV \\cite{Abdallah:2003xc} \\cite{Achard:2001ek} \\cite{Acciarri:2000nf}.  L3's mass limits for $\\chi_0$ decaying via LQD is 32.6GeV, while it's limits on decay via LLE and UDD are 40.2GeV and 39.9GeV respectively.  DELPHI has put lower mass limits on decay via LLE and UDD as 39.5GeV and 38GeV.  At DELPHI, a simple counting experiment was done assuming gaugino mass unification. We will examine the LLE decay count, since it was the lowest; if we avoid the LLE constraint we avoid them all.  In the LLE channel DELPHI found only 1.5 events above the background at 95 percent confidence.  Recalling that $N= \\epsilon L \\sigma $ where $\\epsilon$ is the efficiency, we see that for $\\epsilon$ between .11 and .38 and an integrated luminosity of 437.8$pb^{-1}$, the production limit is $\\sim$ .03-.01 pb.  If we assume no contribution from charginos, a selectron mass of between 380 and 500 GeV will sufficiently suppress this process for a 30 GeV neutralino.  There were additional searches at ALEPH that placed lower bounds of 29 GeV and 23 GeV on LQD and LLE decays of neutralinos respectively \\cite{Barate:1997ra}\n\\cite{Barate:1998gy}.  These analyses also relied on MSUGRA.  For example, they put an upper bound on neutralino production from LLE decay which is at least .5pb.  Without MSUGRA, we may easily beat this production bound by picking the selectron mass.\n\nIn the case of gluino as the LSP no direct decay bounds are quoted.  An indirect search exists  which looks for direct decay of a neutralino produced in the decay of a squark, and assuming gaugino mass unification \\cite{Abazov:2001nt}.  This search concerns the coupling $\\lambda_{2jk}^{'}$ only and looks for muons in the final sate. For reason which will be explained this does not fall into our scenarios.  Interestingly, a lower mass limit on the gluino is set at 6.3 GeV \\cite{Janot:2003cr}. This is obtained from the contribution of hadronic decay width of the Z when $e^+ e^- \\rightarrow q\\overline{q}$ and a quark radiates a gluon which decays to a gluino pair.\n\nThe decay of Higgs to neutralinos must also beat the standard Higgs decays if current searches are not to have already ruled out the Higgs.  The largest standard decay rate is Higgs$\\rightarrow b\\overline{b}$.  The bottom Yukawa coupling is not particularly large and we shall show that it does not require special fine tuning in parameter space in order for Higgs decays to gauginos to beat the bottom decay by a factor of 5 or so.\n\n\n\nThe next constraints heavily involve the couplings to Z bosons.  Decays of light neutral gauginos may not substantially  change the Z width, nor may they have a large effect on the predictions for total e+ e- to hadrons cross section.  These issues may be solved by decoupling the lightest neutralino sufficiently from the Z, and making sure loop induced couplings from of the Z to the gluino are not too great.\n\nThere is a constraint from supersymmetric contributions to $b \\rightarrow s\\gamma$ which come mainly from the charged Higgs and the chargino (diagrams involving the gluino also make contributions but are suppressed by insertions of the CKM matrix).  Diagrams with the $ \\chi ^{+}$ and $H^+$ have a stop in the loop.   Avoiding large contributions to this process  is accomplished in two ways: if the stop and the charged Higgs are heavy and all diagrams are suppressed, or if the diagrams cancel.  Often there is some tension between a heavy stop, which could tune the Higgs sector, and small $b- s \\gamma$.  The current experimental bound is $3.55 +- .24^{+.09}_{-.1} +- .03 x 10^{-4}$\\cite{Barberio:2007cr}.  The current Standard Model theoretical prediction made at NNLO is now significantly below the measurement at $3.15 +- .23 x 10^{-4}$\\cite{Misiak:2006zs}, so there is some small space for supersymmetric contributions.\n\nThere are generational constraints on LNV decays which have charged leptons in the final state.  Tight constraints from Tevatron exist on like sign dilepton signals \\cite{Abulencia:2007rd}.  This search found a slight excess of like sign dilepton events, 20 events at 95 percent confidence, at an integrated luminosity of 1$fb^{-1}$.  The efficiencies for seeing new physics that produced a WZ were assumed to be around 8 percent.  In SUSY optimized scenarios the observed excess was only 8 events and the expected efficiency was slightly lower but this scenario required a large transverse momentum imbalance which presents difficulties for a symmetric decay.  If we consider that the standard Higgs production cross section at Tevatron is over $10^3$ fb, we see that if the generational LLE couplings were the same we would have expected at least a 40 event excess in like sign dilepton events; as two neutralinos decaying through a virtual charged slepton will produce like sign dileptons half the time.  Taus were not covered in the dilepton search since they looked for well isolated lepton pairs that could be tracked back to a single vertex.   This tells us that the LNV operator must favor third generation processes in order to avoid constraints.\n\nFinally, since these decays involve virtual sfermions and small couplings they may not be prompt.  The general decay length has the form\n\\begin{equation}\nL \\sim \\frac{Nm_{\\tilde{m}}^4}{c^2 \\lambda^2 m_{\\chi}^5}  \\beta \\gamma\n\\end{equation}\nwhere $\\lambda$ is the RPV coupling, $m_{\\tilde{m}}$ is the sfermion mass and N is a factor specific to the decay.  If the decay length is long enough, the decay will have two displaced vertices that are separated from the primary decay vertex.  However if the decay length is too long, the decay will occur outside of the detector and will thus be ruled out by existing missing energy searches.  In this way we may put bounds on the RPV coupling and sfermion masses.  We go on to do specific analyses of gaugino decays.\n\n\\subsection{Neutralino LSP}\n\nThe possibility of a light neutralino LSP decaying via BNV was discussed in detail in ref \\cite{Carpenter:2006hs}.  Here we discuss the possibility of a light neutralino decaying via LNV as well.\n\nThe LLE signal would look like\nHiggs to 4 leptons plus missing energy, where the neutralino decays through a virtual lepton.\nFor the LQD channel the neutralino may decay either through a virtual slepton or squark.  The signal would be\nHiggs to missing energy plus 4 jets, 2 leptons plus four jets, or 4 jets one lepton plus missing energy. All of the signals\nfor LNV decay of neutralinos may have multiple b squarks.\n\nBNV decays of the Higgs all have a 6 jet topology, however, the flavor antisymmetry of BNV operators does not allow two down type squarks of\nthe same flavor to be produced by the same vertex.\n\n\\begin{figure}[t]\n\\centerline{\\includegraphics{binoparameterspace.eps}}\n\\caption{Plot of mu term vs tanb for a Higgs to neutralino cross section which beats Higgs to bbar by a\nfactor of 5 and satisfies Z width, b to s gamma and chargino mass constraints}\n\\label{fig:binos}\n\\end{figure}\n\n\\begin{figure}[t]\n\\centerline{\\includegraphics{binovtanb.eps}}\n\\caption{Plot of bino mass vs. $tan\\beta$}\n\\label{fig:binovtanb}\n\\end{figure}\n\n\n  Since the decay happens through a virtual heavy squarks or sleptons, we may calculate the decay length.  Here we calculate the decay length for neutralinos decaying via the BNV operator.\n\\begin{eqnarray}\nL & \\simeq & \\frac{384\\pi^2 \\cos^2\\theta_w}{\\alpha \\left| U_{21}\\right|^2 \\lambda^2}\\frac{m_{\\tilde{m}}^4}{m_\\chi^5} ( \\beta\\gamma)\\\\\\nonumber\n& \\sim & \\frac{3 {\\mu}m}{\\left| U_{21}\\right|^2} \\left(\\frac{10^{-2}}{\\lambda}\\right)^2 \\left(\\frac{m_{\\tilde m}}{100\\; {\\rm GeV}}\\right)^4 \\left(\\frac{30\\; {\\rm GeV}}{m_\\chi}\\right)^5 \\frac{p_\\chi}{m_\\chi},\n\\end{eqnarray}\nwhere $|U_{21}|$ is an element of the mixing matrix  and $p_\\chi$ is the neutralino's momentum.  The\nneutralino may decay via BNV or either LNV operator.  For large RPV couplings and small offshell masses this decay is prompt.  However if $(\\lambda^2\/ m_{\\chi_0})^4$ is too small the decay will not happen inside the detector and this process will be ruled out by searches for invisible decays of the Higgs.  There is an intermediate range where the decay happens inside the detector but there is a secondary displaced vertex.  In this case, the decay products of each $\\chi_0$ will track back to these vertices, which are separate from the  initial vertex where the $\\chi_0$'s are made.  To illustrate the point we have constructed a plot assuming the BNV decay of the neutralinos which shows us the RPV coupling vs. offshell squark mass for different displacements of the secondary vertex.  Though it is easy to be in a region of this parameter space with a prompt decay, there is also quite a bit of space that allows for a displaced vertex of a few microns.\n\n\\begin{figure}[t]\n\\centerline{\\includegraphics{squarks.eps}}\n\\caption{For neutralino masses of 25 and 35 GeV, squark mass vs RPV coupling for displaced vertex of 4 microns, 4cm, and 1m }\n\\label{fig:squarks}\n\\end{figure}\n\nSome attention should be paid to the specific issue of like sign dilepton constraints in LNV scenarios.  As mentioned earlier, like sign dilepton measurements force us to consider only decays with taus leptons in the final state.  For decays proceeding through the LQD operator we may easily choose to turn on RPV couplings such that only a tau will appear in the final state.  The LLE operator requires each neutralino decay to two charged leptons and a neutrino.  Since the LLE operator is antisymmetric in left handed fields, and we require the charges leptons to be taus, the neutrino may not be a tau neutrino.  For example the decay may proceed via the coupling 323 to two taus and a muon neutrino.  In this case the coupling 233 has an equal magnitude and the decay products are just as likely to contain a tau a muon and a tau neutrino.  In this case $frac{1}{8}$ of the events will contain like sign (non-tau) dileptons, and if we a standard Higgs production cross section and an efficiency of 8 percent at 1 $fb^{-1}$ we expect to see 10 events over background at 95 percent confidence.  This is within the inclusive excess for the Tevatron search.\n\nWe now mention constraints and discuss the parameter space of these decays.\n\n\nIn the case of UDD and LQD operators, there is a\ncontribution of the Z width to hadrons and the LLE operator contributes to the total Z width. Since our lightest neutralino is mostly Bino, we may sufficiently decouple from the Z to suppress large contributions to the width.  All scans show points within $1\\sigma$ of the measured value.  We require that the decay of Higgs to neutralinos beat Higgs $\\rightarrow b\\overline{b}$ by a factor of 5.  In addition we have used to decoupling of the M1 parameter from M2 to ensure that we may satisfy the chargino lower bound of 102.7 GeV.  We have also plotted points such that the value of $b\\rightarrow s \\gamma$ does not exceed $2\\sigma$ of the measured value while scanning over stop mass parameters and charged Higgs masses.  We have plotted points scanning for a Higgs mass of 91 GeV, however small regions of parameter space exist for Higgs masses as low as 87 GeV.\n\\subsection{Gluino LSP}\n\n\\begin{figure}[t]\n\\centerline{\\includegraphics{gluinoparameterspace.eps}}\n\\caption{Plot of mu term vs tanb for a Higgs to gluino cross section which beats Higgs to bbar by a\nfactor of 5 and satisfies Z width, b to s gamma and chargino mass constraints}\n\\label{fig:gluinos}\n\\end{figure}\n\nThe second possibility for gaugino LSPs are gluinos.  Light gluinos may easily dominate the Higgs decay width as shown in\n \\cite{Djouadi:1994js}.  Again, gluinos less than half of the Higgs mass are maximally incompatible with the standard gaugino mass ratio predictions.  However light gluinos do have a model\nbuilding benefit, light gluinos do not contribute substantially to squark masses in loops.  When gluinos have mass, there is a two loop gauge mediation-like contribution to squark masses which\n goes like $\\frac{\\alpha_3}{\\pi} M_32 log(\\frac{\\Lambda}{m})$.  Here $\\Lambda$ is the cutoff of the theory, and M3 is the gluino mass parameter.  If the cutoff of the theory is high the squarks get a large mass contribution  and the Higgs sector may become tuned as noted in \\cite{Schuster:2005py}.\n\n\nThe Higgs decays to gluinos through a triangle diagram involving top squarks and top quark.  The diagram has a loop factor and is cut off by the largest scale in the loop.  A similar diagram generates Higgs to gluons except only tops or bottoms appear in the loop.  In this case however, the coupling of Higgs to tops is only the Yukawa coupling which is not enhanced by mixing.  The stop's left right eigenstates are not their mass eigenstates.  If the left right eigenstates have a large mixing, one stop will be much lighter than the other.  In addition the Higgs coupling to $t_{1}t_{1}$  is proportional to the off-diagonal element of the mass matrix $\\sim m_{t1}^2-m_{t2}^2$.  As the coupling becomes large, this decay may easily beat Higgs $\\rightarrow b\\overline{b}$. See figure \\ref{fig:gluinos} for the stop sector parameters needed to beat the bbar cross section by a factor of 5, where we have plotted parameter space in the limit of light gluinos $\\sim 20 GeV$ or less and a 100 GeV Higgs.\n\n\nThe gluinos are then free to decay via an LNV or BNV operator. The topology of this decay is similar to that of the neutralino, the decay will proceed trough a virtual squark, and result in six fermions in the final state.  In the case of BNV, the gluino will decay to a quark\nand an offshell squark, which will then decay to two squarks; thus the signal is Higgs to six jets.  If the gluino were to decay via an LNV vertex, it would be via the LQD operator.\n In this case the gluino would decay to a quark and an offshell squark.  The squark then decays to a quark and lepton.  The signal would then be Higgs to 4 jets plus missing energy or Higgs to 4 jets plus 2 charged leptons.  Because there is no flavor antisymmetry in the LNV operator, these signals may contain multiple quarks of heavy flavor.  In particular, if the final state leptons are neutrinos, we may have the signals 4b plus missing energy, 4c plus missing energy or 2b and 2 c plus missing energy(since each neutralino is free to decay to bottom or charm pairs).   Perhaps the most striking signal would contain 2b, 2c and 2$\\tau$.  There may also be signals in which one gluino decays with a neutrino in the final state, while one decays with a tau.  In this case the decay products may contain an odd number of a heavy quark such as 3b, c, tau,and missing energy or 3c, b, tau and missing energy.\n\n  One might worry that a similar process also contributes to Z to gluinos which would negatively effect the measured $e^+ e^- \\rightarrow$ hadrons, and the total hadronic width of the Z.  However the coupling of stops to the Z is only a gauge coupling which can't compete with the coupling to the Higgs.  Further, the coupling of the stops or sbottoms (but not both at the same time) to the Z may be tuned away while remaining large for the Higgs.   The $b \\rightarrow s\\gamma$ measurement mostly constrains allowed values of $tan\\beta$, as part of the $b \\rightarrow s\\gamma$ amplitude goes like $1\/cos\\beta$.  Our plot shows parameter space for which the $b\\rightarrow s \\gamma$ prediction is within 2$\\sigma$ of the measured value.\n  Like the neutralino, since the decay of the LSP proceeds through an offshell sparticle, there exists the possibility of a secondary displaced vertex.  For sufficiently large RPV coupling and small mass of the offshell sparticle, the decay will occur inside the detector.  In addition, we note that the decay length in this scenario will be more likely to be shorter than that of neutralino LSPs, as it goes like $\\frac{1}{\\alpha}$.  Finally, as is the case with neutralinos, like sign dilepton constraints require that the LNV operator must have a structure that favors third generation couplings.\n\n\n\n\\section {Scalars}\n\\subsection{Topology and General Constraints}\nThe Higgs may decay to a pair of scalars, which then each decay directly through an R parity violating operator.  These decays of the Higgs are usually prompt, and have a 4 particle final state topology.\nThese scenarios are constrained by the chargino lower mass bound and $b\\rightarrow s\\gamma$ as were the gaugino LSP scenario.  Since the scalar LSP pair must have opposite charges, like sign dileptons do not constrain this scenario.  Again we must insure that these decays can beat Higgs $\\rightarrow b\\overline{b}$ by a suitable factor.  However, as we explain, mass bounds on the scalar LSPs themselves are of the biggest concern.\n\nSeveral experiments have put lower mass bounds or production cross section limits on scalars decaying through RPV.  The least stringent bounds for $\\tilde{\\tau}$ decay are from L3 and OPAL respectively and are $m_{\\tilde{\\tau}}> 61$ for LLE and $m_{\\tilde{\\tau}}> 74$ for LQD couplings.   For sbottoms the least stringent limits quoted by L3 for UDD decays and is $m_{\\tilde{b}}> 55$.  These limits would seem to exclude sbottom and stau as LSPs in our scenario.  However none of these searches looked for RPV decays of these particles in mass windows much below half of the Z mass. Table 1 compiles a list of the lowest masses used in direct scalar searches  \\cite{Abdallah:2003xc},\\cite{Achard:2001ek}(table 3),\\cite{Abbiendi:2003rn},\\cite{Heister:2002jc}.  It was assumed that particles less than half of the Z mass could be ruled out with Z width measurements and with contributions to $e^{+}e^{-}\\rightarrow hadrons$.\n\nAs we shall elaborate, third generation scalars may have substantial mixing in their mass matrices.  By varying the scalar mixing angle, we may change their coupling to the Z.  By choosing a small coupling to the Z, we may suppress contributions to $e^{+}e^{-}\\rightarrow hadrons$ and the Z width for scalars with masses below half of the Higgs mass.  In this way we may avoid RPV lower mass bounds for third generation scalars.\n\nFor scalars, the mass eigenstates and left right eigenstates are not the same.  We may write,\n\\begin{equation}\ns1= Sin\\theta s_L + Cos\\theta s_R\n\\end{equation}\n\\begin{equation}\ns2= Cos\\theta s_L-  Sin\\theta s_R\n\\end{equation}\nwhere $\\theta$ is the $s_1$ $s_2$ mixing angle.  The mass matrix for third generation scalars is given by,\n\n\n\n\\centerline{$\\left(\n\\begin{array}{cc}\nM_{sL}^2 + ms^2 + D_{\\tilde s_L} & m_s A \\\\\n m_s A & M_{sR}^2 + ms^2 + D_{\\tilde s_R}\\\\\n\\end{array}\n\\right)$}\n\nwhere $A=A_{su}-\\mu cot\\beta$ for up-type, $A=A_{sb}-\\mu tan\\beta$ for down-type.  The mixing angle is\n\n\\begin{equation}\nsin2\\theta =2 m_b A\/(m_{s1}^2-m_{s2}^2)\n\\end{equation}\n\nFor third generation sparticles it is possible to get large off-diagonal terms and we may have one mass eigenstate which is  much lighter than the other.  In general, it is not too hard to achieve a light mass eigenstate less than half of the Higgs mass.  In addition the couplings of the light mass eigenstate to the Z are controlled by the mixing angle, \\begin{equation}\ng_{Zs_2 s_2}=g(I_3 sin_w^2-Q_2 sin^2\\theta)\n\\end{equation}\nwhich may be adjusted to vanish.  Due to differing charges, the angle at which the scalar completely decouples will be different for stops, sbottoms and staus.  It is worth noting that though third generation scalars can be decoupled from the Z, nothing will decouple them from photons. Thus while we can suppress the contribution of scalars to $e^+ e^- \\rightarrow hadrons$, we cannot eliminate it.  Henceforward we will concentrate on Higgs decaying through stau and sbottom LSPs.\n\n\n\\begin{table}\n\\label{tab:points}\n\\begin{center}\n\\begin{tabular}{c|c|c|c}\nEXP &  LLE & LQD & UDD\\\\\n\\hline\n\nDELPHI & $\\tau >$ 45 & - & b $>$ 45\\\\\nOPAL & $\\tau > $ 45 & $\\tau >$ 45   & b $>$ 45 \\\\\nL3 & $\\tau >$ 70 & - & b $>$ 30 \\\\\nALEPH & $\\tau >$ 45 & $\\tau >$40, b$>$ 30 & b $>$ 45 \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\caption{LEP2 experiment searches for scalars decaying directly through RPV}\n\\end{center}\n\\end{table}\n\n\\subsection{Sbottom LSP}\n\n\n\nThe bottom squark is a good candidate for a light LSP.  Its decay may proceed either through LNV or BNV operator.\nIn the case of BNV the decay proceeds as Higgs to light sbottoms with each light sbottom decaying to an up\n and down quark directly through the RPV coupling.  Because of the down type flavor antisymmetry no bottoms appear\nin the final state.  The signal is thus Higgs to 4 jets, at most 2 of heavy flavor(charm).  LNV decays proceed through the LQD operator, with the signal two quarks and two leptons.  In the case of the LQD operator there is no flavor antisymmetry, thus there may be a b and a $\\overline{b}$ in the final state.  If the final state leptons are neutrinos, the most visible decay will be Higgs to 2b plus missing energy.   The most visible signal in this case would be $h \\rightarrow 2c + 2\\tau$.\n\n\n\\begin{figure}[t]\n\\centerline{\\includegraphics{sbottombsg2.eps}}\n\\caption{Plot of mu term vs tanb for a Higgs to sbottom cross section for sbottoms between 7.5 and 30 GeV which beats Higgs to bbar by a\nfactor of 5 and satisfies Z width, b to s gamma and chargino mass constraints}\n\\label{fig:sbs}\n\\end{figure}\n\n\n\nWe will now discuss constraints and the parameter space.  Long lived sbottoms are ruled\nout under 92 GeV.  However, as long as the RPV couplings are large enough and the sbottom is sufficiently more massive than its decay products, the decay should be prompt.\nWe see that by adjusting the mixing angle of the sbottoms, we decrease the coupling to Z.  Following the formula quoted in the beginning of this section, we see that this coupling is turned off when $sin\\theta \\sim .39$.  A lower bound on sbottom masses\nhas been set at 7.5GeV by measuring contributions to the over\nall cross sections of $e^+e^- \\rightarrow hadrons$ while turning off the sbottom coupling to Z's \\cite{Janot:2004cy}.   As quoted in table 1, the L3 experiment put an effective upper bound of 30 GeV on sbottom squarks decaying through RPV.  Previous bounds had been set on the mass splitting of very light sbottoms and the lightest stop \\cite{Carena:2000ka}.  These took into account a large stop loop contribution to the Higgs mass and limits on the $\\rho$ parameter and required a light stop lighter than 300 GeV.  In our scenario however the bound is even more relaxed since we do not require a large stop loop contribution to the Higgs, and our stop sbottom splitting is not quite as extreme.\n\nThe decay rate of Higgs to sbottoms must beat that of bottoms by a factor of a few.  The ratio of decays rates is given by \\cite{Berger:2002vs} as\n\\begin{equation}\n \\Gamma_{\\tilde{b}}\/ \\Gamma_b= \\mu tan\\beta^2 \/ 2 m_h^2 sin2\\theta^2 (1-4m_{b}^2\/m_h^2)^{1\/2}\n\\end{equation}\n which gets large for large values of $\\mu$ and tan$\\beta$.  In this scenario there are 5 free parameters; the A term, $\\mu$, tan$\\beta$, and the soft masses.  We have plotted $\\mu$ vs tan$\\beta$ in parameter space for this window of allowed sbottom mixing angles.  We see in the plot of that lower values of $\\mu$ and tan$\\beta$ are ruled out by the upper mass limit on the sbottom.  If the product $\\mu$tan$\\beta$ is too small, the off diagonal elements of the mass matrix will not be big enough to produce a light mass eigenstate.  We therefore expect to find parameter space at larger values of $tan\\beta$ than we did for neutralino LSPs.  The measurements of $e^{+}e^{-} \\rightarrow hadrons$ constrain allowed values of $\\theta$.  Janot tells us the largest allowed mass range for sbottoms occurs between mixing angles $.3 < sin\\theta < .45$ \\cite{Janot:2004cy}(see fig 8).  For A terms and $\\mu$ terms of a few hundred GeV we see that we may fall into this range of mixing parameter if A and $\\mu$ cancel to within 20 or 30 percent of their value.  The constraint from b to s gamma follows as before and we scan over stop sector parameters and obey the chargino constraint. However neither of these constraints is as restrictive as for the case of light neutralinos since we are free to make the gaugino sector heavier in general.\n\n\\subsection{Stau LSP}\n\n\n\nThe decay of the Higgs may proceed through a stau anti-stau LSP pair.  The direct RPV decays of the tau happen through LNV operators only.\n\nFor LQD decays of the stau, the final state will have 4 quarks, 2 up type and 2 down type.  Because there is no flavor antisymmetry, the final state may consist of all quarks of heavy flavor, $h \\rightarrow b\\overline{b}c\\overline{c}$.\n\n\\begin{figure}[h]\n\\centerline{\\includegraphics{stausigmahadrons.eps}}\n\\caption{Plot of stau mixing vs hadronic cross section at the Z pole.  The lower region is ruled out by PEP PETRA and TRISTAN measurements, the regions to the left and right by Z pole data from LEP 1.}\n\\label{fig:hadrons}\n\\end{figure}\n\n\n\n\n\\begin{figure}[h]\n\\centerline{\\includegraphics{staumassangle.eps}}\n\\caption{Plot of allowed stau mass vs. stau mixing angle}\n\\label{fig:massangle}\n\\end{figure}\n\n\n\n\n For decays with quarks in the final state, the $\\tilde{\\tau}$ masses below half the Z mass are constrained by the measurement of $e^{+} e^{-} \\rightarrow hadrons $.  Following the method of Janot for light sbottoms, we may go back to LEP2, PEP, PETRA, and TRISTAN measurements of the total $e^{+} e^{-} \\rightarrow hadrons$ cross sections to set lower mass bounds on the staus \\cite{Janot:2004cy}.  Ref \\cite{Janot:2004cy} contains a compilation of the total $\\sigma_{had}$ measurements for various experimental energies.  We first calculate the absolute minimum allowed $\\tilde{\\tau}$ mass by assuming complete decoupling from the Z and calculating the resulting contribution to $e^{+}e^{-} \\rightarrow hadrons$ for production from photons alone for different experimental energies.  We then perform a $\\chi^2$ fit for the hadronic cross section contribution.  We find that at 95 percent confidence the lower limit on stau masses is 11 GeV with a best fit value of 57 GeV.  A plot of the LEP and low energy cross section measurements and cross section predictions for light staus appears in the appendix.  There seems to be significant parameter space for light stuas.  We must note that Janot performed a more sophisticated analysis of the low energy data which may further constrain the lighter stau masses.  However for stau mass heavier than 28 GeV the lowest energy data does not constrain our scenario.  We will therefore conservatively consider the viable allowed stau mass window to be above 28 GeV.  \n\n\nIn addition we may analyze the Z pole data from LEP 1.  As quoted by Janot, the Z pole measurements limit new contributions to the hadronic cross section to 56pb at 95 percent confidence.  Taking these measurements into account, we may exclude stau masses for different couplings of staus to the Z.  When the $\\tilde{\\tau}$ has small coupling to the Z, we find a mass window  between 28 and 45 GeV.  Following the formula in section 4, we see that the Z coupling to $\\tilde{\\tau}$ is turned off when $sin \\theta=.67$, and allowed mixing angles range from  $.6< sin\\theta < .7$ in the case of light staus.  We have plotted the total contribution from light stau production at the Z pole vs mixing angle, as well as allowed stau mixing angles vs stau mass taking into account both Z pole and PEP,PETRA and TRISTAN data.\n\n\\begin{figure}[h]\n\\centerline{\\includegraphics{stauparameterspace.eps}}\n\\caption{Plot of mu term vs tanb for a Higgs to staus cross section, with a Higgs mass of 105 GeV, which beats Higgs to bbar by a\nfactor of 5 and satisfies Z width, b to s gamma and chargino mass, and hadronic cross section constraints}\n\\label{fig:stauspace}\n\\end{figure}\n\n\n\n In principle the parameter space for this decay looks much like the parameter space for sbottoms.  However, the constraint on the allowed mixing angle is tighter since a large contribution of $e^+e^- \\rightarrow hadron$ comes directly from the photon coupling to tau with no coupling to the Z at all.  Again getting a light stau requires large contributions to the off-diagonal mass matrix elements, in this case given by $m_{\\tau}(A-\\mu tan\\beta)$.  Since $m_{\\tau}$ is small, a large product $\\mu$ $tan\\beta$ is required for large mixing, even larger than what was required for light sbottoms.  We keep in mind that we do not want excessively high values of $\\mu$, since this would again introduce tuning into the Higgs potential.  In addition tan$\\beta $ must not become too large or the bottom Yukawa coupling will become nonperturbative.  We also need to choose $\\mu$ and $tan\\beta$ such that we may obey the lower mass bound on the lightest chargino.  In this case, the variables effecting the stau sector that are at play in the $b \\rightarrow s\\gamma$ process are only $tan\\beta$ and $\\mu$, the stau and stop sectors are fairly uncoupled and $b \\rightarrow s\\gamma$ is not a strong constraint in the light stau scenario.  We have plotted parameter space for all constraints.\n\n\n\n\nThe LLE decay of the stau is an interesting subject.  If allowed, each stau would decay into a charged lepton plus a neutrino, and the signal would be opposite sign dileptons plus missing energy.  The most striking signal here would be $h \\rightarrow 2\\tau + \\not{E}$.  A search with similar signature, gauge mediated decays of stau to tau plus gravitino \\cite{Abreu:2000nm}, highly constrains this scenario.  This search rules out the tau plus missing energy signal for stau mass larger than 2GeV.  It should be noted that this search found a small window for $m_{\\tau} < m_{\\tilde{\\tau}} < 2 GeV$.  One might guess that in the case where both staus decay hadronically, contributions to $e^{+}e^{-} \\rightarrow hadrons$ rule out this possibility.  However, we  see that the PEP, PETRA, and TRISTAN data placed cuts specifically to rule exclude the background from $\\tau^{+}\\tau^{-}$ and it is likely these decays may have been missed.  See for example \\cite{Von:1990}).  Thus there may be a signal for Higgs to 2 stau plus missing energy in the improbable event that there exist super-light staus.  In this case, since there is a only a small window between the tau and the stau, if the RPV coupling is small, the stau may live for some time before it decays.  In this case, in addition to a 2$\\tau$ plus missing energy decay, they may also be a secondary displaced vertex.  For example for LLE coupling of the size a few times $10^{-4}$ a 1.8 GeV stau may live for 100 microns.  Stau pair decays with one or two light leptons may also occur.\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusions}\nIn the R parity violating MSSM without standard gaugino mass relationships, we find a menagerie of new decays for a light Higgs.  We present the list of possible LSPs and topologies in the table found above.\n\n\n\nOne may wonder how all of these signals pass current bounds.  There are several current searches that one must consider and we will first mention general of these.  DELPHI did an analysis for Higgs goes to anything which could put a lower mass bound on the Higgs of 82 GeV \\cite{Abbiendi:2002qp}.  There has been a Higgs to missing energy search which put a lower bound on the Higgs mass of 114 GeV \\cite{Schael:2006cr}.  This is a concern\nfor us only when the Higgs decay is not prompt as is the case for gaugino LSPs.  It is easy to avoid falling into this search if RPV couplings are large or if squark or slepton masses are of reasonable size.  In fact, in the case of squarks, we would not want masses to go much beyond 1TeV, for then we encounter the same tuning problem in the Higgs potential that we wished to avoid.\n\n\nWe shall now consider searches which apply to decays with all hadronic final states.  First there is the 2 jet flavorless search which puts a lower bound on the Higgs of 113 GeV \\cite{:2001yb}.  In order for this search to be sensitive to our scenario, we would have to force our 4 and 6 jet final states into two jets.  Such forcing is used in a set of searches where members of the Higgs multiplet undergoes cascade decay \\cite{Abbiendi:2004ww} \\cite{Abdallah:2004wy}.  Thus we worry about decays $e^+e^- \\rightarrow H_2Z \\rightarrow H_1H_1Z$, where $H_1$ decays to bottoms or taus.  In the all hadronic case, the final state is 4b.  The event is forced into two 2b jets and the efficiency is calculated of a 4b even being picked up by a 2b search.  Using the DELPHI search table we can guess our efficiency of being picked up by the 2b search.  This should give us a good idea about matching our 4 jet signals to 2 jets, for 6 jets the efficiency would be even worse.  As an example, notice that to rule out the Higgs at 80 GeV, the 4b search needs about 5.5 times the number of events as the 2b.  Even assuming both of our b's were tagged, our scenario would only be picked up 18 percent of the time.  The flavorless search is the same as the 2b search without the b tags\\cite{:2001yb}.  LEP performed a search in which HZ was produced and Higgs decayed to $WW^{*}$ \\cite{Schael:2006ra}.  An analysis was done for the final state $Z \\rightarrow \\nu\\nu, H \\rightarrow q\\overline{q}q\\overline{q} $, which is relevant to our light stau scenario.  This search places a lower mass bound on the Higgs of 105 GeV.  In addition the $WW^{*} $ search considered a final state in which $e^+e^- \\rightarrow  HZ\\rightarrow 6q$.  In this case however, cuts are applied which reconstruct the masses of the final state Z and Ws, so this search is not likely to be sensitive to our 4 or 6 jet signals.\n\nWe now turn to searches with quarks and charged leptons in the final states.  The cascade decays mentioned above fall into this category produce final states with 4$\\tau$ and 2b+2$\\tau$.  We can compare these final states to all of ours and see that these searches are not directly sensitive to any of our final states.  Of our final states without missing energy these signals come closest are the six body decay 2b+2c+2$\\tau$ and the four body decay, 2c+2$\\tau$.  Reconstruction of the cascade decay requires multiple b tags.  It is unlikely that the 2 charms will receive a large b-likeness parameter.  In addition to the b tag problem, our 2b+2c+2$\\tau$ decay must be force four jets into two loosing efficiency.  We do not expect that these searches will constrain our scenario.  Cascade decays with even more final state b's and $\\tau$s occur through the process $e^+e^- \\rightarrow  H_2H_2\\rightarrow H_1H_1H_1$.  Again, none of these final states match directly to our signals.  Events with 4 or 6 b's require even more b tags, which are unlikely to match up to our 6 particle final states that have at most 2 b's and none of our final states contain more than two taus.\n\nFinally there are decays with missing energy in the final state.  A compendium of some searches that constrain Higgs signals with missing energy is found in \\cite{Chang:2007de}.  In particular this paper quotes upper bounds on the Higgs production cross section for Higgs decays with final states and $2q + \\not E$.  These limits do not come from a Higgs search but from the LEP2 squark searches where sparticles are pair produced and decay to quarks and neutralinos.  This signal is identical to our scenario with quarks and missing energy in the final state.  In this case a lower bound may be set on the Higgs mass of 103 GeV if the final state quarks are light and 111 GeV if both of the final state quarks are b quarks.  These bounds do not apply in cases where the final state quarks are not the same flavor.  One might consider the cascade decays $e^+e^- \\rightarrow H_2Z \\rightarrow H_1H_1Z\\rightarrow 4b\/4\\tau$ $Z$.  If in this case the Z decays invisibly the overall signal would be identical to our $H \\rightarrow 4b\/4\\tau + 2\\nu$ final state.  The 4b search involved cuts which considered parameters such as the missing mass and the $ln(\\chi_{mZ})$ parameter where the missing mass is forced to the Z mass.  These cuts are bound to exclude data in our scenario.  Further we can see from the exclusion plot in fig 12 that if the efficiency of observing out scenario is 60 percent, all of parameter space is open up down to the 82 GeV Higgs mass bound.  The $4\\tau$ search constraints on the decay topology such as restrictions on the angle between charged particle pairs, in addition to cuts on the missing momentum which make it insensitive to our decay.  However the in the $WW^{*}$ search an analysis was done for the channel $e^{+}e^{-} \\rightarrow HZ \\rightarrow WW^{*}Z$ where the Ws decay hadronically and the Z invisibly.  Here we cannot avoid constraints and a lower limit of 105 GeV may be placed on the Higgs mass.  Finally, the $WW^{*}$ search analyzed a Higgs channel with charged light leptons and missing energy in the final state.  This signal is identical to out final state where scalar LSPs decay through the LLE operator.  Even in the case that we consider an LLE operator like 313, where the final state leptons may be only taus, the antisymmetry of the RPV operator forces us the coupling 133 to be the same size. Thus we get light leptons a quarter of the time and this search will also constrain the 2 tau plus missing energy channel.  The search places an upper bound on the Higgs production cross section of .044pb and in our case, this translates to a Higgs mass lower bound of around 104 GeVfor final states with taus or light leptons plus missing energy.  We may lower the bound by turning on more than one RPV coupling at once and decreasing the likelihood that the decay products are symmetric.  For example, turning on two coupling at once which are equal in size decreases the Higgs mass bound to 95 GeV.\n\n\nOther asymmetric decays are also listed in table 2 which involve 2 RPV coupling being turned on at once.  These scenarios are bizarre enough to be mostly unconstrained.  The final state b+c+$\\tau +\\nu$ is mentioned in the $WW^{*}$ search, however the combinatorics of turning on two RPV operators at once allow us to avoid the Higgs mass bound of 95 GeV set by this scenario.  Stranger 6 body decays listed in our table have are not constrained by any relevant search.\n\n\\begin{table}\n\\label{tab:points}\n\\begin{center}\n\\begin{tabular}{c|c|c|c}\n LSP &  LLE & LQD & UDD\\\\\n\\hline\n\n$\\chi_0$ & 4$\\tau$+$2\\nu$  & 4b\/4c+2$\\nu$, 2b+2c+2$\\nu$, 2b+2c+2$\\tau$,3b+c+$\\tau$+$\\nu$, b+3c+$\\tau$+$\\nu$ & 2b+2c+2q\\\\\ng & - & 4b\/4c+2$\\nu$, 2b+2c+2$\\nu$, 2b+2c+2$\\tau$,3b+c+$\\tau$+$\\nu$, b+3c+$\\tau$+$\\nu$   & 2b+2c+2q \\\\\nb & - & 2b+2$\\nu$, 2c+2$\\tau$, b+c+$\\nu$+$\\tau$ & 2c+2q  \\\\\n$\\tau$& 2$\\tau$+2$\\nu$ & 2b+2c & - \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\caption{Higgs decay signals for all possible LSPs and RPV operators}\n\\end{center}\n\\end{table}\n\n\\begin{table}\n\\label{tab:points}\n\\begin{center}\n\\begin{tabular}{c|c|c|c}\n LSP &  Signature & Mass Bound & Search\\\\\n\\hline\n\n$\\chi_0$ & $4q+2\\nu$ & 105 GeV & $WW^{*}$ with invisible Z decay\\\\\n$\\tilde{g}$ & $4q+2\\nu$ & 105 GeV &  $WW^{*}$ with invisible Z decay \\\\\n$\\tilde{b}$ & $2q+2\\nu$ & 103 GeV & SUSY squark search\\\\\n- & $2b+2\\nu$ & 111 GeV & SUSY squark search\\\\\n- & $4q$ & 105 GeV & $WW^{*}$ with invisible Z decay\\\\\n$\\tilde{\\tau}$ & $\\tau\\overline{\\tau}+2\\nu$ & 104 GeV & $WW^{*}$ \\\\\n- & $l\\overline{l}+2\\nu$ & 104 GeV & $WW^{*}$ \\\\\n- & 4q & 105 GeV &$WW^{*}$ with invisible Z decay \\\\\n\n\\hline\n\\hline\n\\end{tabular}\n\\caption{Higgs Mass Lower Bounds for Various Channels. For decays not listed current searches do not severely constrain the Higgs mass.}\n\\end{center}\n\\end{table}\n\n\nOverall we have proposed over a dozen distinct channels for Higgs discovery with multiple heavy flavor particles in their final states.  Some of these decays would be very hard to detect, for example those decays in which the final state is 4 or 6 jets.  However, many of these decays have interesting missing energy signatures, some of which are quite bizarre -  for example the b+3c+$\\tau$ +$\\nu$ decay of the gauginos.  Those decays that proceed through a gaugino LSP have the added bonus of possible secondary displaced vertices.  We may imagine modifying existing searches to look in some of these channels, for example modifying existing 2 or 4 b and $\\tau$ searches to be sensitive to missing energy.  In addition we might hope to detect events which contain bottoms at LHCb as has been recently proposed \\cite{Kaplan:2007ap}.\n\n\n\\section{Appendix}\n\nBelow we have plotted hadronic cross section vs center of mass energies.  The points are the measurement of the hadronic crossection minus the SM theory prediction with two sigma error bars.  The curves are contributions to the hadronic cross section from the decay of low energy staus of various masses.\n\n\\begin{figure}[h]\n\\centerline{\\includegraphics{staucross.eps}}\n\\caption{Plot of Energy vs hadronic cross section for low energy data and stau contributions.  from the top down the curves are for staues of mass 10, 15, 22, 28, and 45 GeV }\n\\label{fig:stlength}\n\\end{figure}\n\n\n\n{\\bf Acknowledgments}\n\nThis work was supported in part by funding from the US Department of Energy.  We would like to thank Tom Banks and Jason Neilsen for enlightening discussion about signals and constraints.\n\n\\bibliographystyle{apsrev}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe Drell-Yan process in hadron-hadron collisions has for many\nyears yielded important information on the quark distributions of\nhadrons \\cite{DRELL}. In proton-nucleon collisions, for example, the sea quark\ndistribution at medium and large $x$  -- which is overwhelmed in deep\ninelastic scattering by the valence quarks --  can be directly measured.\nIt was pointed out several years ago \\cite{KUNSZT} that the Drell-Yan\nprocess at HERA could also be of interest. In particular, since the\nphotoproduction of lepton pairs arises mainly from  $q\\bar{q}$ annihilation,\nthe quark distributions in the photon can be studied.  For example,\nconsider the cross section differential in both\nlepton-pair mass $M$ and rapidity $y$, with the proton direction\ndefining $y>0$. In the centre-of mass frame of\nthe $\\gamma p$ collision   in leading order, the momentum fractions of\nthe annihilating quarks are given simply\nby $x_{\\gamma} = \\sqrt{\\tau} \\exp(-y)$, $x_{p} = \\sqrt{\\tau} \\exp(y)$, where\n$\\tau = M^2\/s_{\\gamma p}$.\nAt large negative rapidities, therefore, the quark distribution in the photon\nis probed at large $x$, while that in the proton is probed at small $x$.\nNow at HERA, the fact that the photon-proton system has a large\npositive boost in the\nlab frame means that $y_{\\rm lab} < 0$ for the lepton pairs already implies\na very large rapidity in the photon-proton centre-of-mass frame. In fact, as we\nshall demonstrate below, lepton pairs with $y_{\\rm lab} \\sim -1$ probe\n$x_\\gamma \\sim 1$ and $x_p \\sim 10^{-4} - 10^{-3}$. In addition, such lepton\npairs should be relatively easy to identify, being produced in the\n\\lq backward' part of the\ndetector which is relatively free from hadronic activity.\n\nThe importance of such a measurement is  that it allows\nan independent  probe of the small $x$ structure of the proton. Recent deep\ninelastic structure function measurements from the H1 \\cite{H1} and\nZEUS \\cite{ZEUS} collaborations at HERA indicate\nthat $F_2(x,Q^2)$ rises\nsteeply at small $x$ and modest $Q^2$, in line with predictions\nbased on the \\lq\\lq perturbative pomeron\" of QCD \\cite{SMALLX}.\nIt is vitally important to obtain additional evidence of this effect,\nand the Drell-Yan cross section at negative $y_{\\rm lab}$ provides just such\na measurement. We note that the ${\\gamma p}$ Drell-Yan cross section has already\nbeen advocated as a probe of the small $x$ structure of the {\\it photon}\n\\cite{KUNSZT,GLUCKA,GLUCKB},\nbut this corresponds to large, positive $y_{\\rm lab}$ values. In practice, the\nhadronic activity in this region presumably makes a clean measurement\nrather difficult.\n\nIn this paper we  study the ${\\gamma p}$ Drell-Yan cross section for $y_{\\rm lab} < 0$.\nTo test the discriminating power in the  $x_p \\ll 1$ region, we use the recent\nMRS-D$_0'$ and MRS-D$_-'$ parton distributions \\cite{MRS92P}.\nThe former have \\lq\\lq conventional\" Regge small $x$ behaviour,\n$xq, xg \\sim x^0$ as $x \\to 0$, while the latter have more singular\n$xq, xg \\sim x^{-1\/2}$ behaviour as expected  from perturbative\npomeron arguments \\cite{MRS92P}.\n Note that the recent $F_2$ data from H1 \\cite{H1} and ZEUS\n\\cite{ZEUS} at HERA strongly favour the D$_-'$ distributions.\nA crucial part of our argument is that for $y_{\\rm lab} < 0$\n the parton distributions of the photon are\n being probed in a $x$ region where they are\n already constrained by $F_2^\\gamma$ data. The difficulty here\nstems from the  fact that in practice the $F_2^\\gamma$ data in the relevant\n$x,Q^2$ regions are not very precise, and this introduces some uncertainty\ninto the  Drell-Yan predictions which in turn weakens the sensitivity\nto the proton structure. To try to quantify this, we use a variety\nof recent photon parton-distribution parametrizations.\n\nThe paper is organised as follows. In the next section we describe the\ntheoretical formalism\nfor computing the Drell-Yan cross section at next-to-leading\norder in photon-proton collisions.\nNumerical calculations then quantify the overall event rates and\nthe differences obtained from using different sets of\nparton distributions.\nIn Section 3 we perform a Monte Carlo event simulation to study in detail\nthe $x$ values which are being probed, and also to investigate the\nenergy and angular distributions of the leptons in the lab.\nSection 4 contains our conclusions.\n\n\n\\section{The Drell-Yan cross section at next-to-leading order}\n\n\\subsection{Expression for the inclusive cross section}\nIn this study we  use both next-to-leading order matrix elements and\nparton distributions, in the $\\overline{\\rm MS}$ scheme. We  calculate\n the inclusive cross sections of dilepton production at HERA:\n\\begin{eqnarray}\n\\gamma p & \\rightarrow & l^+l^-+X\\\\\ne p & \\rightarrow & l^+l^-+X      \\; ,\n\\end{eqnarray}\nusing  the expression  from \\cite{GLUCKA}:\n\\begin{eqnarray}\n\\lefteqn{\n{d\\sigma\\over{dM^2}} = {4\\pi\\alpha^2\\over{3M^2}}\\int_\\tau^1{dx_p\\over{x_p}}\n\\int_{\\tau\/x_p}^1{dx_\\gamma\\over{x_\\gamma}}\\sum_{q=u,d,s,c}e_q^2\n         }\n\\nonumber\\\\\n&&\n\\times \\Biggl\\{\n\\left[q_p(x_p,Q^2)\\bar q_\\gamma (x_\\gamma,Q^2)+\\bar q_p(x_p,Q^2)\nq_\\gamma(x_\\gamma,Q^2)\\right] \\left[\\delta(1-{\\tau\\over{x_px_\\gamma}})\n+{\\alpha_s(\\mu^2)\\over{2\\pi}}f_{q\\bar{q}} \\left({\\tau\\over{x_px_\\gamma}}\\right)\n\\right]\n \\nonumber\\\\\n&&\n +{\\alpha_s(\\mu^2)\\over{2\\pi}}\\;   f_{gq}\\left({\\tau\\over{x_px_\\gamma}}\\right)\n\\nonumber\\\\\n&&\n \\times\\left[g_p(x_p,Q^2)\n\\left(q_\\gamma(x_\\gamma,Q^2)+\\bar q_\\gamma (x_\\gamma,Q^2)\\right)\n+\\left(q_p(x_p,Q^2)+\\bar q_p(x_p,Q^2)\\right)g_\\gamma(x_\\gamma,Q^2)\\right]\n\\nonumber \\\\\n&&\n +{6e_q^2\\alpha\\over{2\\pi}}\\; f_{gq}\\left({\\tau\\over{x_px_\\gamma}}\\right)\n\\left(q_p(x_p,Q^2)+\\bar q_p(x_p,Q^2)\\right)\\delta(1-x_\\gamma)\n\\Biggr\\}\n\\end{eqnarray}\nwhere $\\tau=M^2\/s$ and the higher-order terms $f_{q\\bar{q}}$ and $f_{gq}$ are given\nin Eqn.(5) in \\cite{GLUCKA}. In this expression, the first term contains\nthe dominating LO $q\\bar{q}$ initial state and the associated\nreal  and virtual corrections.\n The second term contains the $g_p+q_\\gamma$ and $q_p+g_\\gamma$\ninitial states and finally the third term is the Compton term given by\n$q_p+\\gamma\\rightarrow \\gamma*+q$. Throughout this calculation we set the\nfactorization and renormalization scales at the invariant mass of the dilepton\npair; that is, $\\mu^2=Q^2=M^2$.\n\nFor the parton distributions in the photon  which appear\nin the above expression, we take  the GRV(NLO)\nnext-to-leading order set  \\cite{GRV}  as  standard\nand probe the sensitivity of the\ncross sections to the MRS-D$_0'$ and MRS-D$_-'$ parton distributions.\nWe also study the sensitivity of the cross sections to different photon\nparametrizations, to attempt to quantify the uncertainty in extracting\ninformation on the small-$x$ proton distributions from this source.\nTo obtain the relevant cross sections for the $ep$ initial state we\nconvolute the cross-section expression with the simplest form of the\nequivalent photon approximation \\cite{WWA}, in which we have set the\nenergy scale to be $\\hat s=x_px_es$.\n\n\\subsection{Event rates and sensitivity to photon and proton\nparton distributions}\nThe cross-section calculations are performed for the energy parameters of\nHERA. For the $\\gamma p$ initial state we take $\\sqrt s_{\\gamma p}\n=200$ GeV, and for the $ep$ initial state we assume $E_e=30$ GeV and $E_p=820$\nGeV.\nIn Figure 1 we present the results of the calculation of the cross section\n$d\\sigma\/dM$ as a function of $M$, where $M$ is the invariant mass of the\ndilepton pair. The $\\gamma p$ cross section is presented in Figure 1(a) for\nboth the MRS-D$_0'$ and MRS-D$_-'$ parton distributions. The MRS-D$_-'$ cross\nsection is larger, as expected,\n but the difference between the two decreases substantially\nas $M$ increases, since at small $M$ the sensitivity to small-$x$ in the proton\nis large and the differences between the two sets of proton distribution\nfunctions is accentuated. At $M=2$ GeV\/c$^2$ the cross section is approximately\n76~pb\/GeV for the MRS-D$_0'$ case and 155~pb\/GeV\n for the MRS-D$_-'$ case, and\nat $M=4$ GeV\/c$^2$ the cross sections have fallen to  10~pb\/GeV and\n14~pb\/GeV, respectively. The\nnext-to-leading contribution to this cross section is approximately 1.5 to 2\ntimes that of the leading order one. In Figure 1(b) the corresponding\ncross section for the $ep$ initial state is shown. Evidently\nthese cross sections are approximately one order of magnitude smaller\nthan those for $\\gamma p$. The reason  is that\nat these energies  ${\\cal L}_{\\gamma p}\n\\simeq 0.1\\! {\\cal L}_{ep}$. The sensitivity of the $ep$ cross sections to the\ntwo sets of parametrizations is somewhat smaller than in the $\\gamma p$ case.\n\nTo emphasize the sensitivity of the Drell-Yan cross sections to the small-$x$\nbehaviour of the proton distribution functions we next calculate the rapidity\ndistribution of the dilepton pairs in the $\\gamma p$ case. The\nkinematics are such that $x_p=\\sqrt\\tau {\\rm e}^{y_{{\\gamma p}}}$ and\n$x_\\gamma=\\sqrt\\tau {\\rm e}^{-y_{{\\gamma p}}}$,\nwhich implies that $x_p\\rightarrow 0$ in the backward direction, which is the\nregion\nwhere differences between the MRS-D$_0'$ and MRS-D$_-'$ should be\nobserved.\nIn Figure 2 we present the plots for the cross section $d\\sigma\/dMdy$ as a\nfunction of $y$ for $M=4,6$ GeV\/c$^2$. The $M=4$ GeV\/c$^2$ case is shown\nin Figure 2(a) and we see that at large negative $ y$  ($\\equivy_{\\rm lab}$),\n the MRS-D$_-'$\ncross section is larger than the MRS-D$_0'$ one by a factor larger than 2.\nAt this invariant mass the proton is probed down to $x_p\\simeq 4\\times10^{-4}$\n(see below).\nAt smaller $M$ the difference is considerably larger; for instance at $M=2$\nGeV\/c$^2$ there is a factor of 4 between the two distributions at large\nnegative rapidities. However, at such low masses data from\nhadron-hadron collisions suggest that  the background to the dilepton\ncross section arising from\n$J\/\\psi$ production may be substantial \\cite{DRELL}. In what follows,\ntherefore, we impose a lower limit of $M > 4$ GeV\/c$^2$ on the dilepton\nmass.\nIn Figure 2(b) the same distribution is plotted for\n$M=6$ GeV\/c$^2$ and as expected, the effect washes out, due to the fact\nthat a larger invariant mass forces larger $x_p$ values; at large\nnegative rapidities the cross section ratio,\nMRS-D$_-'$\/MRS-D$_0'$ reduces to approximately 1.5. In the positive rapidity\nregion the ratio is very nearly 1 in both Figures 2(a),(b).\n\nThe usefulness of this process to probe the small-$x$ behaviour of the proton\nstructure function depends on the\ninsensitivity of the cross sections to the large-$x$ behaviour of the\nphoton structure function. To investigate this, we repeat the\ncalculations using several other sets of photon distributions,\nkeeping the proton distributions (MRS-D$_-'$) fixed.\nFigure 3 shows  the rapidity\ndistribution, $d\\sigma\/dMdy$ for $M=4$ GeV\/c$^2$ as a function of $ y$\n for various\nchoices of the photon distributions: (i) the reference GRV(NLO) set\n\\cite{GRV} (solid line), (ii) the GS(NLO) set \\cite{GS} (dashed line)\nand the LAC1(LO) set \\cite{LAC} (dotted line).\nIn the interesting region of phase space (large negative\nrapidities) we see that  the different distributions give roughly the\nsame shape in rapidity but a spread in normalisation of order $10\\%$.\nThis can be understood in terms of the spread in predictions of these\nsets for the photon structure function $F_2^\\gamma$. Figure 4 shows\n data on $F_2^\\gamma$ together with the predictions of the same three\nsets as in Figure 3. We have chosen data in a similar $Q^2$ range\nas the $M^2$ values relevant to the Drell-Yan cross section at\nHERA. We see immediately that the spread in the predictions\nsimply reflects the uncertainty in the structure function\nmeasurements. More precise measurements of $F_2^\\gamma$ would of course\npin down the Drell-Yan cross section more accurately. In the meantime\nwe should regard the $O(10\\%)$ spread in normalisation as\na systematic uncertainty in the extraction of the {\\it proton}\ndistributions from the Drell-Yan data.\n\n\\section{Event simulation}\nIn the previous section we have seen how the $\\gamma p$ Drell-Yan\ncross section for $y_{\\rm lab} < 0$ and $ M > 4$ GeV\/c$^2$ offers a good\npossibility of obtaining information on the quark distribution\nin the proton at small $x$. An important question is whether the bulk\nof these events actually fall within the acceptance of the HERA\ndetectors. To study this, we perform a Monte Carlo simulation using the\nPYTHIA5.6 \\cite{PYTHIA} MC generator for ${\\gamma p} \\to l^+l^- + X$\nat $\\sqrt{s_{{\\gamma p}}} = 200$ GeV.\nThis simulation also allows us to investigate which  $x_p$, $x_\\gamma$\nvalues give the dominant contribution to the cross section.\n\nFigure 5 show the distributions in (a) $x_{\\gamma}$ and (b) $x_p$\nfor $M > 4$ GeV\/c$^2$. The two histograms correspond to $y_{\\rm lab} < 0$\n(solid lines) and $y_{\\rm lab} > 0$ (dashed lines). The former is dominated\nby $x_p$ values in the $10^{-4} - 10^{-3}$ region, as expected.\nSelecting only those events with $M > 4$ GeV\/c$^2$, and\ndefining $\\vartheta_l$ and $E_l$ to be the lepton polar angle and energy\nin the lab frame respectively, Figure 6 shows the scatter plots for\n(a) $x_\\gamma$ {\\it vs.} $\\vartheta_l$,\n(b) $x_p$ {\\it vs.} $\\vartheta_l$,\n(c) $E_l$ {\\it vs.} $\\vartheta_l$,\nand (d) $M$ {\\it vs.} $\\vartheta_l$.\nWe see immediately that most of the events have leptons which\nare produced with reasonable  energy and at sizeable angles to the\nbeam direction $\\vartheta = 0^{\\rm o}$, $180^{\\rm o}$. Given that\nwe expect very little additional  hadronic activity in these regions\nof phase space, the acceptance for both muon and electron\npairs should be quite high.\n\n\\section{Conclusions}\n\nThe structure of the proton at small $x$ is an important new area\nof interest, and  the first $F_2$ data from HERA provide tentative\nevidence for the \\lq\\lq perturbative pomeron\". The Drell-Yan cross\nsection in the backward rapidity region should in the very near future\n give a confirmation of this small-$x$\n behaviour from a completely different process.\nThe decrease in hadronic activity as $y_{\\rm lab}$\nbecomes more negative and the effect of the boost on the lepton lab angle\nshould make  this a particularly clean measurement.\nOur study has concentrated  on ${\\gamma p}$ collisions at $\\sqrt{s_{{\\gamma p}}} =\n200$ GeV, tacitly assuming that the final-state electron can be tagged\nwith reasonable efficiency at HERA. However, as is clear from Figure 1(b),\nthe effects we describe should also be observable in the $ep$ cross section.\n\nObservation  of the dramatic differences induced by \\lq\\lq standard\"\nand \\lq\\lq singular\"  small-$x$ behaviour  in the proton in the $y_{\\rm lab} < 0$\nDrell-Yan cross section relies\non  knowledge of the large-$x$ photon  structure.\n We have shown that\nthe uncertainty in the photon  at large $x$ is of the order of $10$\\%,\nwhich for $M=4 - 6$ GeV is small in comparison to the D$_0'$\/D$_-'$ effect.\nBut we should\nstress again that once these $F_2^p$ become well-measured at HERA, the\nDrell-Yan\ncross section will provide a complementary measurement of the\nlarge-$x$ behaviour of the photon.\n\n\\bigskip\n\n\\vspace{1cm}\n\\noindent {\\Large\\bf Acknowledgments}\n\n\\bigskip\nKCh would like to thank the Physics Department and Grey College\nat the  University  of Durham for their warm hospitality.\nThe work of KCh was supported   by a EC \\lq\\lq Go--West\" Scholarship.\nACB would like to thank the Physics Department at the University\nof Durham for its kind hospitality when this project was conceived,\nand the Foundation for Research Development for partial funding.\nWe are grateful to Lionel Gordon for discussions concerning the\nGS parametrizations.\n\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzhqdz b/data_all_eng_slimpj/shuffled/split2/finalzzhqdz
new file mode 100644
index 0000000000000000000000000000000000000000..192777e8f0a599cc68cca93df920e74e25894338
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzhqdz
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nThe motivic nearby fiber and the motivic vanishing cycles were\nintroduced by J.~Denef and F.~Loeser (see\n\\cite{denef-loeaser-motivic-igusa-zeta, denef-loeser-motivic,\n  denef-loeser-arc-spaces, looijenga-motivic-measures}). Let $V\n\\colon X \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}$ \nbe a morphism of ${\\mathsf{k}}$-varieties where ${\\mathsf{k}}$ is an algebraically\nclosed field of characteristic zero and $X$ is smooth over ${\\mathsf{k}}$\nand connected.\nThe motivic nearby fiber $\\psi_{V,a}$ and the motivic vanishing\ncycles $\\phi_{V,a}$ of $V$\nat a point $a \\in {\\mathsf{k}}={\\mathbb{A}}^1_{\\mathsf{k}}({\\mathsf{k}})$ are elements of\na localization $\\mathcal{M}_{|X_a|}^{\\hat{\\upmu}}$ of the\nequivariant Grothendieck ring $K_0({\\op{Var}}_{|X_a|}^{\\hat{\\upmu}})$ \nof varieties over the reduced fiber $|X_a|$ of $V$ over $a$.\nWe refer the reader to the main body of this article for precise\ndefinitions. We will often view $\\psi_{V,a}$ and $\\phi_{V,a}$ as\nelements of $\\mathcal{M}_{{\\mathsf{k}}}^{\\hat{\\upmu}}$ in this introduction.\n\nThe motivic nearby fiber is additive on the Grothendieck group\n$K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}})$ of varieties over ${\\mathbb{A}}^1_{\\mathsf{k}}$, as shown\nby F.~Bittner \n\\cite{bittner-motivic-nearby} \nand by G.~Guibert, F.~Loeser and M.~Merle \n\\cite[Thm.~3.9]{guibert-loeser-merle-convolution}.\nNamely, for any $a \\in {\\mathsf{k}}$,\nthere is a map\n\\begin{equation*}\n  K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}) \\rightarrow \\mathcal{M}_{\\mathsf{k}}^{\\hat{\\upmu}}  \n\\end{equation*}\nof $K_0({\\op{Var}}_{\\mathsf{k}})$-modules which maps the class of a proper\nmorphism $V \\colon X \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}$ with $X$ as above to \nthe motivic nearby fiber $\\psi_{V,a}$.\n  \nThe motivic Thom-Sebastiani theorem\n\\cite{guibert-loeser-merle-convolution} \nis a\nlocal multiplicativity result for motivic vanishing cycles. Given\nanother morphism $W \\colon Y \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}$ as above define\n$V \\circledast W \\colon X \\times Y \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}$ by $(V\n\\circledast W)(x,y)=V(x)+W(y)$. Then the motivic\nThom-Sebastiani Theorem states that a certain convolution of the\nmotivic vanishing cycles $\\phi_{V,a}$ and $\\phi_{W,b}$ determines\nsome part of the motivic vanishing cycles\n$\\phi_{V \\circledast W,a+b}$ (see Theorem\n\\ref{t:thom-sebastiani-GLM-motivic-vanishing-cycles}).\n\nOur main result states that after small adjustments - the\nmotivic vanishing cycles $\\phi_{V,a}$ we use differ by a factor\n$(-1)^{\\dim X}$ from the usual motivic vanishing cycles (see\nRemark~\\ref{rem:motivic-fibers-signs}) - the motivic vanishing\ncycles are \nboth additive and multiplicative.\n\n\\begin{theorem}\n  [{see Theorem~\\ref{t:total-motivic-vanishing-cycles-ring-homo}}]\n  \\label{t:main-intro}\n  There is a morphism\n  \\begin{equation}\n    \\label{eq:60}\n    (K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}), \\star) \\rightarrow\n    (\\mathcal{M}_{\\mathsf{k}}^{\\hat{\\upmu}},*) \n  \\end{equation}\n  of $K_0({\\op{Var}}_{\\mathsf{k}})$-algebras - called \\textbf{motivic vanishing\n    cycles measure} -\n  which is uniquely\n  determined by the following \n  property: it maps the class of each proper\n  morphism $V \\colon  X \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}$ from a smooth and connected\n  ${\\mathsf{k}}$-variety $X$ to the sum \n  $\\sum_{a \\in {\\mathsf{k}}} \\phi_{V,a}$ of its motivic vanishing cycles.\n\\end{theorem}\n\nThe motivic vanishing cycles measure is a motivic\nmeasure in the sense that it is a ring morphism from\nsome Grothendieck ring of\nvarieties to another ring.\nThe multiplication $*$ on the target of our measure is a convolution\nproduct whose definition is due to Looijenga and involves Fermat\nvarieties. The \nmultiplication $\\star$ on the source is given by\n$[X \\xra{V} {\\mathbb{A}}^1_{\\mathsf{k}}] \\star [Y \\xra{W} {\\mathbb{A}}^1_{\\mathsf{k}}] = \n[X \\times Y \\xra{V \\circledast W} {\\mathbb{A}}^1_{\\mathsf{k}}]$.\nApart from the additivity and local multiplicativity results\nmentioned \nabove, the main ingredient in the proof of\nTheorem~\\ref{t:main-intro} is a compactification construction\ndescribed in \n\\cite{valery-olaf-matfak-motmeas}. \nIn fact, \nwe prove a slightly stronger statement\nin Theorem~\\ref{t:total-motivic-vanishing-cycles-ring-homo}: the\nmotivic \nvanishing cycles \nmeasure \\eqref{eq:60}\ncomes from a\nmorphism $(K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}), \\star) \\rightarrow\n(\\tilde{\\mathcal{M}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\hat{\\upmu}},\\star)$ \nof $K_0({\\op{Var}}_{\\mathsf{k}})$-algebras. \nLet us mention that our sign adjustments are already\nnecessary for additivity (see\nRemark~\\ref{rem:justify-sign-vanishing}).\n\nIn the last part of this article we compare the motivic vanishing\ncycles measure \nwith a motivic measure of a completely different categorical\nnature (in case \n${\\mathsf{k}}=\\mathbb{C}$). Mapping a projective morphism $W \\colon X \\rightarrow {\\mathbb{A}}^1_\\mathbb{C}$\nfrom a smooth complex variety $X$ to its category of matrix\nfactorizations gives rise to\na ``matrix factorization'' motivic\nmeasure\n\\begin{equation*}\n  \\mu \\colon \n  (K_0({\\op{Var}}_{{\\mathbb{A}}^1_\\mathbb{C}}), \\star)  \n \n  \\rightarrow K_0(\\op{sat}_\\mathbb{C}^{{\\mathbb{Z}}_2})  \n\\end{equation*}\nas we explained in \n\\cite{valery-olaf-matfak-semi-orth-decomp,\n  valery-olaf-matfak-motmeas}. The target of this ring morphism\nis the Grothendieck\nring of saturated differential $\\mathbb{Z}_2$-graded categories.\nHere is our comparison result.\n\n\\begin{theorem} \n  [{see Theorem~\\ref{t:comparison}}]\n  \\label{t:comparison-intro}\n  We have the following commutative diagram of ring homomorphisms\n  \\begin{equation*}\n    \\xymatrix{\n      {(K_0({\\op{Var}}_{{\\mathbb{A}}^1_\\mathbb{C}}), \\star)} \n      \\ar[r]^-{\\mu} \\ar[d]\n      & {K_0(\\op{sat}_{\\mathbb{C}}^{{\\mathbb{Z}}_2})} \\ar[d]^-{\\chi_{{\\operatorname{HP}}}} \\\\\n      {(\\mathcal{M}_{\\mathbb{C}}^{\\hat{\\upmu}},*)} \\ar[r]^-{\\chi_{\\op{c}}} \n      & {\\mathbb{Z}}\n    }\n  \\end{equation*}\n  where the left vertical arrow is \n  the motivic vanishing cycles measure \\eqref{eq:60}\n  from Theorem~\\ref{t:main-intro},\n  the lower horizontal arrow is induced by forgetting the group\n  action and taking the Euler\n  characteristic with compact support, and the \n  right vertical arrow is induced by taking the Euler\n  characteristic of periodic cyclic homology.\n\\end{theorem}\n\nThe main ingredients in the proof of this theorem are the\ncomparison between the periodic cyclic homology of the dg\ncategory of matrix factorizations of a given potential $V$ with\nthe vanishing cohomology of $V$ due to A.~Efimov\n\\cite{efimov-cyclic-homology-matrix-factorizations}, and the\ncomparison between the motivic and geometric vanishing cycles due\nto G.~Guibert, F.~Loeser and M.~Merle\n\\cite{guibert-loeser-merle-convolution}.\n \n\\subsection{Structure of the article}\n\\label{sec:structure-article}\n\n\\begin{itemize}\n\\item[\\S\\ref{sec:grothendieck-rings}]\n  We remind the reader of various (equivariant) Grothendieck\n  abelian groups of \n  varieties and multiplications (or ``convolutions'') on them.\n  We recall Looijenga's convolution\n  product $*$ in section~\\ref{sec:convolution} and include a direct\n  proof of associativity (see\n  Proposition~\\ref{p:convolution-comm-ass-unit}); \n  this reproves results of \n  \\cite[5.1-5.5]{guibert-loeser-merle-convolution}.\n  We also define a variant of Looijenga's convolution product for\n  varieties over ${\\mathbb{A}}^1_{\\mathsf{k}}$ in section~\\ref{sec:conv-vari-over}.\n\\item[\\S\\ref{sec:motiv-vanish-fiber}]\n  We recall the definition of the motivic nearby\n  fiber $\\psi_{V,a}$ and the motivic vanishing\n  cycles $\\phi_{V,a}$ and show that $\\phi_{V,a}$ lies in \n  $\\mathcal{M}^{\\hat{\\upmu}}_{|{\\op{Sing}}(V) \\cap X_a|}$ (see\n  Proposition~\\ref{p:motivic-vanishing-cycles-over-singular-part}). \n  We also show an invariance property of $\\phi_{V,a}$ \n  in Corollary~\\ref{c:mot-van-fiber-compactification}.\n\\item[\\S\\ref{sec:thom-sebastiani}]\n  We state the motivic Thom-Sebastiani Theorem\n  \\cite[Thm.~5.18]{guibert-loeser-merle-convolution}\n  as\n  Theorem~\\ref{t:thom-sebastiani-GLM-motivic-vanishing-cycles} and\n  give some corollaries. In particular, we globalize the\n  Thom-Sebastiani Theorem to \n  Corollary~\\ref{c:global-thom-sebastiani}.\n\\item[\\S\\ref{sec:motiv-vanish-fiber-1}]\n  A corollary of\n  \\cite[Thm.~3.9]{guibert-loeser-merle-convolution} is given as\n  Theorem~\\ref{t:GLM-measure}. We obtain additivity of the\n  motivic vanishing cycles in \n  Theorem~\\ref{t:total-vanishing-cycles-group-homo}.\n  Then we deduce our main\n  Theorem~\\ref{t:total-motivic-vanishing-cycles-ring-homo} \n  using the previous Thom-Sebastiani results\n  and\n  the compactification result stated as\n  Proposition~\\ref{p:compactification}.\n\\item [\\S\\ref{sec:comp-with-matr}]\n  We remind the reader of the categorical motivic measure\n  in \\cite{bondal-larsen-lunts-grothendieck-ring}\n  and its relation to the matrix factorization measure.\n  Then we prove Theorem~\\ref{t:comparison}. We finish by giving\n  two examples and by drawing a diagram relating the motivic\n  measures considered in this article.\n\\end{itemize}\n\n\\subsection{Acknowledgements}\n\\label{sec:acknowledgements}\n\nWe thank Daniel Bergh, Alexander Efimov, Annabelle Hartmann,\nFran\\-\\c{c}ois Loeser, and \nAnatoly Preygel for useful discussions, and the referee for\ncareful reading.\n\nThe first author was supported by \nNSA grant 43.294.03.\nThe second author was supported by a postdoctoral\nfellowship of the DFG\nand by \nSFB\/TR 45 of the DFG.\n\n\\subsection{Conventions}\n\\label{sec:conventions}\n\nWe fix an algebraically closed field ${\\mathsf{k}}$ of characteristic\nzero.\nBy a ${\\mathsf{k}}$-variety we mean a separated reduced scheme of finite\ntype over ${\\mathsf{k}}$.\nA morphism of ${\\mathsf{k}}$-varieties is a morphism of ${\\mathsf{k}}$-schemes.\nLet ${\\op{Var}}_{\\mathsf{k}}$ be the category of\n${\\mathsf{k}}$-varieties.\nWe write $\\times$ instead of $\\times_{\\operatorname{Spec} {\\mathsf{k}}}$. By our\nassumptions on ${\\mathsf{k}}$, the product of two ${\\mathsf{k}}$-varieties is again\nreduced and hence a ${\\mathsf{k}}$-variety.\nIf $X$ is a scheme we denote by $|X|$ the\ncorresponding reduced closed subscheme. \n\n\\section{Grothendieck rings of varieties}\n\\label{sec:grothendieck-rings}\n\n\\subsection{Grothendieck rings of varieties over a base variety}\n\\label{sec:groth-rings-vari}\n\n\nFix a ${\\mathsf{k}}$-variety $S$. By an $S$-variety we mean a morphism\n$X \\rightarrow S$ of ${\\mathsf{k}}$-varieties. Let ${\\op{Var}}_S$ be the category of\n$S$-varieties.\nThe Grothendieck group $K_0({\\op{Var}}_S)$ of $S$-varieties is the\nquotient of the free abelian group on \nisomorphism \nclasses $\\langle X \\rightarrow S\\rangle$ of $S$-varieties $X \\rightarrow S$ by the subgroup generated by the scissor\nexpressions $\\langle X \\rightarrow S \\rangle - \\langle (X - Y) \\rightarrow S\n\\rangle - \\langle Y \\rightarrow S\\rangle$ where $Y \\subset X$ is a\nclosed reduced subscheme.\nAny $S$-variety $X \\rightarrow S$  \ndefines an element $[X \\rightarrow S]$ of $K_0({\\op{Var}}_S)$. \n\nGiven $S$-varieties \n$X \\rightarrow S$ and $Y \\rightarrow S$, the composition\n$|X \\times_S Y| \\rightarrow X \\times_S Y \\rightarrow S$ is an $S$-variety; this\noperation turns $K_0({\\op{Var}}_S)$ into a commutative associative ring\nwith identity element\n$[S \\xra{\\operatorname{id}} S]$ (use \\cite[Prop.~4.34]{goertz-wedhorn-AGI} for\nassociativity). \n\nLet $\\mathcal{M}_S:= K_0({\\op{Var}}_S)[\\mathbb{L}_S^{-1}]$ be the ring obtained\nfrom  \n$K_0({\\op{Var}}_S)$ by inverting $\\mathbb{L}_S=[{\\mathbb{A}}^1_S \\rightarrow S]$.\n\nWe usually write $K_0({\\op{Var}}_{\\mathsf{k}})$ instead of $K_0({\\op{Var}}_{\\operatorname{Spec}\n  {\\mathsf{k}}})$, $\\mathbb{L}=\\mathbb{L}_{\\mathsf{k}}$ instead of $\\mathbb{L}_{\\operatorname{Spec} {\\mathsf{k}}}$, and $\\mathcal{M}_{\\mathsf{k}}$ instead of $\\mathcal{M}_{\\operatorname{Spec} {\\mathsf{k}}}$.\n\n\\begin{remark}\n  \\label{rem:compare-Grothendieck-rings}\n  Note that the Grothendieck ring $K_0({\\op{Var}}_S)$ defined here is\n  canonically isomorphic to the Grothendieck ring defined in\n  \\cite[3.1]{nicaise-sebag-grothendieck-ring-of-varieties}, by\n  \\cite[3.2.2]{nicaise-sebag-grothendieck-ring-of-varieties}.\n\\end{remark}\n\n\\subsubsection{Pullback}\n\\label{sec:pullback}\n\nLet $f \\colon T \\rightarrow S$ be a morphism of ${\\mathsf{k}}$-varieties.  Then\nthe functor ${\\op{Var}}_S \\rightarrow {\\op{Var}}_T$,\n$(X \\rightarrow S) \\mapsto (|T \\times_S X| \\rightarrow T \\times_S X \\rightarrow T)$,\ninduces a morphism\n\\begin{equation}\n  \\label{eq:1}\n  f^* \\colon K_0({\\op{Var}}_S) \\rightarrow K_0({\\op{Var}}_T)\n\\end{equation}\nof commutative unital rings\nwhich satisfies $f^*(\\mathbb{L}_S)=\\mathbb{L}_T$\nand hence \ninduces a morphism\n\\begin{equation}\n  \\label{eq:2}\n  f^* \\colon \\mathcal{M}_S \\rightarrow \\mathcal{M}_T\n\\end{equation}\nof rings. \nIf $g \\colon U \\rightarrow T$ is another morphism of\n${\\mathsf{k}}$-varieties, we have $g^* f^* =(fg)^*$, by \n\\cite[Prop.~4.34]{goertz-wedhorn-AGI}.\n\nIn particular, $K_0({\\op{Var}}_S)$ (resp.\\ $\\mathcal{M}_S$) becomes a\n$K_0({\\op{Var}}_{\\mathsf{k}})$-algebra (resp.\\ $\\mathcal{M}_{\\mathsf{k}}$-algebra), and \n\\eqref{eq:1} (resp.\\ \\eqref{eq:2}) is a morphism of\n$K_0({\\op{Var}}_{\\mathsf{k}})$-algebras (resp of $\\mathcal{M}_{\\mathsf{k}}$-algebras).\nNote that the obvious map defines a canonical\nisomorphism  \n\\begin{equation*}\n  \\mathcal{M}_{\\mathsf{k}} \\otimes_{K_0({\\op{Var}}_{\\mathsf{k}})} K_0({\\op{Var}}_S) \\xra{\\sim}\n  \\mathcal{M}_S \n\\end{equation*}\nof $\\mathcal{M}_{\\mathsf{k}}$-algebras.\n\n\\subsubsection{Pushforward}\n\\label{sec:push-forward}\n\nLet $f \\colon T \\rightarrow S$ be a morphism of ${\\mathsf{k}}$-varieties.\nThe functor ${\\op{Var}}_T \\rightarrow {\\op{Var}}_S$, $(Y \\xra{y} T) \\mapsto (Y\n\\xra{fy} S)$, induces a morphism\n\\begin{equation*}\n \n  f_! \\colon K_0({\\op{Var}}_T) \\rightarrow K_0({\\op{Var}}_S)\n\\end{equation*}\nof $K_0({\\op{Var}}_{\\mathsf{k}})$-modules. \nTensoring with $\\mathcal{M}_{\\mathsf{k}}$ yields a morphism\n\\begin{equation*}\n \n  f_! \\colon \\mathcal{M}_T \\rightarrow \\mathcal{M}_S\n\\end{equation*}\nof $\\mathcal{M}_{\\mathsf{k}}$-modules which sends\n$[Y \\xra{y} T] \\cdot \\mathbb{L}_T^{-i}$ to\n$[Y \\xra{fy} S] \\cdot \\mathbb{L}_S^{-i}$.\n\n\\begin{remark}\n  \\label{rem:compare-Grothendieck-rings-and-pullback}\n  The canonical isomorphisms from \n  Remark~\\ref{rem:compare-Grothendieck-rings}\n  are compatible with pullback and pushforward, \n  by \\cite[Prop.~4.34]{goertz-wedhorn-AGI}.\n\\end{remark}\n\n\\subsection{Grothendieck rings of equivariant varieties over a\n  base variety}\n\\label{sec:groth-rings-equiv}\n\nFor $n \\in \\mathbb{N}_{>0}$ let $\\upmu_n=\\operatorname{Spec}({\\mathsf{k}}[x]\/(x^n-1))$ be the\ngroup ${\\mathsf{k}}$-variety of $n$-th roots of unity. Note that actions\nof $\\upmu_n$ on a ${\\mathsf{k}}$-variety $X$ correspond bijectively to group\nmorphisms $\\upmu_n({\\mathsf{k}}) \\rightarrow \\operatorname{Aut}_{{\\op{Var}}_{\\mathsf{k}}}(X)$. \n\nFix a ${\\mathsf{k}}$-variety $S$ and let $n \\in \\mathbb{N}_{>0}$. Recall that a\ngood $\\upmu_n$-action on a ${\\mathsf{k}}$-variety is a $\\upmu_n$-action\nsuch that each $\\upmu_n({\\mathsf{k}})$-orbit is contained in an affine\nopen subset of $X$. An $S$-variety with a good $\\upmu_n$-action is\nan $S$-variety $p \\colon X \\rightarrow S$ together with a good\n$\\upmu_n$-action on $X$. So $p$ is $\\upmu_n$-equivariant if we equip\n$S$ with the trivial $\\upmu_n$-action. We obtain the category\n${\\op{Var}}_S^{\\upmu_n}$ of $S$-varieties with good $\\upmu_n$-action.\n \nThe definition of the Grothendieck ring $K_0({\\op{Var}}_S^{\\upmu_n})$ of\n$S$-varieties with good $\\upmu_n$-action is evident from\n\\cite[2.2-2.5]{guibert-loeser-merle-convolution}; apart from the\nusual scissor relations there is another family of relations,\ncf.\\ \\cite[(2.2.1)]{guibert-loeser-merle-convolution}.  Any\n$S$-variety $X \\rightarrow S$ with good $\\upmu_n$-action gives rise to an\nelement $[X \\rightarrow S]=[X]$ of $K_0({\\op{Var}}_S^{\\upmu_n})$. The\nproduct of $[X \\rightarrow S]$ and $[Y \\rightarrow S]$ is the element obtained\nfrom $|X \\times_S Y| \\rightarrow S$ with the obvious diagonal\n$\\upmu_n$-action.  Define\n$\\mathbb{L}_S=\\mathbb{L}_{S, \\upmu_n}=[{\\mathbb{A}}^1_S \\rightarrow S] \\in K_0({\\op{Var}}_S^{\\upmu_n})$\nwhere $\\upmu_n$ acts trivially on ${\\mathbb{A}}^1_S$. Let\n$\\mathcal{M}_S^{\\upmu_n}:= K_0({\\op{Var}}_S^{\\upmu_n})[\\mathbb{L}_S^{-1}]$.\n\n\nWe write \n$K_0({\\op{Var}}_{\\mathsf{k}}^{\\upmu_n})$ and $\\mathcal{M}_{\\mathsf{k}}^{\\upmu_n}$ instead of \n$K_0({\\op{Var}}_{\\operatorname{Spec} {\\mathsf{k}}}^{\\upmu_n})$ and $\\mathcal{M}_{\\operatorname{Spec} {\\mathsf{k}}}^{\\upmu_n}$.\n\nIf $f \\colon T \\rightarrow S$ is a morphism of ${\\mathsf{k}}$-varieties we obtain\nas above a pullback morphism\n$f^* \\colon K_0({\\op{Var}}_S^{\\upmu_n}) \\rightarrow K_0({\\op{Var}}_T^{\\upmu_n})$\nof $K_0({\\op{Var}}_{\\mathsf{k}}^{\\upmu_n})$-algebras\nsatisfying $f^*(\\mathbb{L}_S)=\\mathbb{L}_T$ and an induced\npullback morphism\n$f^* \\colon \\mathcal{M}_S^{\\upmu_n} \\rightarrow \\mathcal{M}_T^{\\upmu_n}$\nof $\\mathcal{M}_{\\mathsf{k}}^{\\upmu_n}$-algebras.\nWe also have a pushforward morphism\n$f_! \\colon K_0({\\op{Var}}_T^{\\upmu_n}) \\rightarrow K_0({\\op{Var}}_S^{\\upmu_n})$\nof $K_0({\\op{Var}}_{\\mathsf{k}}^{\\upmu_n})$-modules, and a pushforward morphism\n$f_! \\colon \\mathcal{M}_T^{\\upmu_n} \\rightarrow \\mathcal{M}_S^{\\upmu_n}$\nof $\\mathcal{M}_{\\mathsf{k}}^{\\upmu_n}$-modules. \nFor $n=1$ we recover the notions from \n\\ref{sec:groth-rings-vari}.\n\nWhenever $n'$ is a multiple of $n$ there is a morphism\n$\\upmu_{n'} \\rightarrow \\upmu_n$, $\\lambda\n\\mapsto \\lambda^{n'\/n}$, of ${\\mathsf{k}}$-group varieties inducing\nmorphisms \n\\begin{align}\n  \\label{eq:28}\n  K_0({\\op{Var}}_S^{\\upmu_n}) & \\rightarrow K_0({\\op{Var}}_S^{\\upmu_{n'}}),\\\\\n  \\label{eq:29}\n  \\mathcal{M}_S^{\\upmu_n} & \\rightarrow \\mathcal{M}_S^{\\upmu_{n'}},\n\\end{align}\nof rings.\nThese \nmorphism are compatible with pullback and pushforward morphisms.\n\nIn particular, $K_0({\\op{Var}}^{\\upmu_n}_S)$ (resp.\\\n$\\mathcal{M}^{\\upmu_n}_S$) becomes a $K_0({\\op{Var}}_{\\mathsf{k}})$-algebra\n(resp.\\ $\\mathcal{M}_{\\mathsf{k}}$-algebra)\nand the morphisms \n\\eqref{eq:28}\nand \\eqref{eq:29} are morphisms of algebras. We have a canonical\nisomorphism \n\\begin{equation}\n  \\label{eq:15}\n  \\mathcal{M}_{\\mathsf{k}} \\otimes_{K_0({\\op{Var}}_{\\mathsf{k}})} K_0({\\op{Var}}^{\\upmu_n}_S)\n  \\xra{\\sim} \\mathcal{M}_S^{\\upmu_n}  \n\\end{equation}\nof $\\mathcal{M}_{\\mathsf{k}}$-algebras\ngiven by $\\tfrac{r}{(\\mathbb{L}_{\\mathsf{k}})^n}\n\\otimes a \\mapsto \\tfrac{ra}{(\\mathbb{L}_S)^n}$. \n\nLet $\\hat{\\upmu}$ be the (inverse) limit of the\n$(\\upmu_n({\\mathsf{k}}))_{n \\in \\mathbb{N}_{>0}}$ with respect to the morphisms\n$\\upmu_{n'}({\\mathsf{k}}) \\rightarrow \\upmu_n({\\mathsf{k}})$,\n$\\lambda \\mapsto \\lambda^{n'\/n}$, whenever $n'$ is a multiple of\n$n$.\n\nAn $S$-variety with good $\\hat{\\upmu}$-action is by definition an\n$S$-variety $X \\rightarrow S$ together with a group morphism $\\hat{\\upmu} \\rightarrow\n\\operatorname{Aut}_{{\\op{Var}}_{\\mathsf{k}}}(X)$ that comes from a good $\\upmu_n$-action on $X$,\nfor some $n \\in \\mathbb{N}_{>0}$.\nAs in \\cite[2.2]{guibert-loeser-merle-convolution} we obtain \nthe category ${\\op{Var}}_S^{\\hat{\\upmu}}$ of $S$-varieties with good\n$\\hat{\\upmu}$-action.\nWe define $K_0({\\op{Var}}_S^{\\hat{\\upmu}})$ and\n$\\mathcal{M}_S^{\\hat{\\upmu}}$ in the obvious way so that we have\n\\begin{align*}\n \n  K_0({\\op{Var}}_S^{\\hat{\\upmu}}) & = \\operatorname{colim}_n K_0({\\op{Var}}_S^{\\upmu_n}),\\\\\n  \\mathcal{M}_S^{\\hat{\\upmu}} & = \\operatorname{colim}_n \\mathcal{M}_S^{\\upmu_n}.\n\\end{align*}\nThe Grothendieck ring \n$K_0({\\op{Var}}_S^{\\hat{\\upmu}})$\nis an $K_0({\\op{Var}}_{\\mathsf{k}})$-algebra (even a \n$K_0({\\op{Var}}_{\\mathsf{k}}^{\\hat{\\upmu}})$-algebra), and  \n$\\mathcal{M}_S^{\\hat{\\upmu}}$ is a $\\mathcal{M}_{\\mathsf{k}}$-algebra\n(even a \n$\\mathcal{M}_{\\mathsf{k}}^{\\hat{\\upmu}}$-algebra). We have \n\\begin{equation*}\n \n  \\mathcal{M}_{\\mathsf{k}} \\otimes_{K_0({\\op{Var}}_{\\mathsf{k}})} K_0({\\op{Var}}^{\\hat{\\upmu}}_S)\n  \\cong \\mathcal{M}_S^{\\hat{\\upmu}}  \n\\end{equation*}\ncanonically as rings.\nIf $f \\colon T \\rightarrow S$ is a morphism of ${\\mathsf{k}}$-varieties, we obtain\na pullback morphism\n$f^* \\colon K_0({\\op{Var}}_S^{\\hat{\\upmu}}) \\rightarrow  K_0({\\op{Var}}_T^{\\hat{\\upmu}})$\nof $K_0({\\op{Var}}_{\\mathsf{k}})$-algebras and a pushforward morphism\n$f_! \\colon\n K_0({\\op{Var}}_T^{\\hat{\\upmu}})\n\\rightarrow\n K_0({\\op{Var}}_S^{\\hat{\\upmu}})$\nof $K_0({\\op{Var}}_{\\mathsf{k}})$-modules. The base changes of these morphisms along\nthe ring morphism $K_0({\\op{Var}}_{\\mathsf{k}}) \\rightarrow \\mathcal{M}_{\\mathsf{k}}$ are denoted\nby the same symbols.\n\nInstead of working with $\\upmu_n$ we could work more generally with\n$\\upmu_{n_1} \\times \\dots \\upmu_{n_r}$ (for $r \\in \\mathbb{N}$ and $n_1,\n\\dots, n_r \\in \n\\mathbb{N}_{>0}$), and instead of $\\hat{\\upmu}$ we could work\nwith $\\hat{\\upmu}^r$ (for $r \\in \\mathbb{N}$). We extend our notation\naccordingly. \n\n\\begin{remark}\n  \\label{rem:dictionary}\n  There is an alternative description of \n  $K_0({\\op{Var}}_S^{\\hat{\\upmu}^r})$ and\n  $\\mathcal{M}_S^{\\hat{\\upmu}^r}$,\n  see \n \n  the dictionary in\n  \\cite[2.3-2.6]{guibert-loeser-merle-convolution}. \n  When referring to results of\n  \\cite{guibert-loeser-merle-convolution} we will usually\n  translate \n  them using this dictionary.\n\\end{remark}\n\n\\begin{lemma}\n  \\label{l:open-closed-decomposition}\n  Let $S$ be a ${\\mathsf{k}}$-variety and $F \\subset S$ a closed reduced\n  subscheme with open complement $U$.\n  Let $i \\colon F \\rightarrow S$ and $j \\colon U \\rightarrow S$ denote the\n  inclusions. Then \n  \\begin{align*}\n    (j^*, i^*) \\colon K_0({\\op{Var}}_S^{\\hat{\\upmu}}) \n    & \\xra{\\sim}\n    K_0({\\op{Var}}_U^{\\hat{\\upmu}})\n    \\times\n    K_0({\\op{Var}}_F^{\\hat{\\upmu}}),\\\\\n    A & \\mapsto (j^*(A), i^*(A)),\n  \\end{align*}\n  is an isomorphism of $K_0({\\op{Var}}_{\\mathsf{k}})$-algebras, with\n  inverse given by $(B,C) \\mapsto j_!(B) + i_!(C)$.\n  Similarly, \n  \\begin{equation*}\n   \n    (j^*, i^*) \\colon \\mathcal{M}_S^{\\hat{\\upmu}} \\xra{\\sim}\n    \\mathcal{M}_U^{\\hat{\\upmu}}\n    \\times \\mathcal{M}_F^{\\hat{\\upmu}}\n  \\end{equation*}\n  is an isomorphism of $\\mathcal{M}_{\\mathsf{k}}$-algebras.\n\\end{lemma}\n\n\\begin{proof}\n  This is obvious from the definitions.\n\\end{proof}\n\n\\begin{remark}\n  \\label{rem:standard-multiplication}\n  Recall that $K_0({\\op{Var}}_S)$,\n  $K_0({\\op{Var}}_S^{\\upmu_n})$ and\n  $K_0({\\op{Var}}_S^{\\hat{\\upmu}})$ are $K_0({\\op{Var}}_{\\mathsf{k}})$-algebras\n  whose multiplications are\n  induced from the fiber product over\n  $S$. In the rest of this article mainly the \n  underlying $K_0({\\op{Var}}_{\\mathsf{k}})$-module structure on \n  $K_0({\\op{Var}}_S)$,\n  $K_0({\\op{Var}}_S^{\\upmu_n})$ and\n  $K_0({\\op{Var}}_S^{\\hat{\\upmu}})$ will be important. \n  Given $(T \\rightarrow \\operatorname{Spec} {\\mathsf{k}})$ in ${\\op{Var}}_{\\mathsf{k}}$ and $(Z \\rightarrow S)$ in ${\\op{Var}}_S$\n  or ${\\op{Var}}_S^{\\upmu_n}$ or\n  ${\\op{Var}}_S^{\\hat{\\upmu}}$ it is given by\n  \\begin{equation*}\n   \n    [T \\rightarrow \\operatorname{Spec} {\\mathsf{k}}].[Z \\rightarrow S] = [T \\times Z \\rightarrow S].\n  \\end{equation*}\n  In fact, we will introduce other multiplications on \n  the \n  $K_0({\\op{Var}}_{\\mathsf{k}})$-modules\n \n  $K_0({\\op{Var}}_S^{\\upmu_n})$ and\n  $K_0({\\op{Var}}_S^{\\hat{\\upmu}})$\n  turning them into \n  $K_0({\\op{Var}}_{\\mathsf{k}})$-algebras.\n\\end{remark}\n\n\\subsection{Convolution}\n\\label{sec:convolution}\n\nAfter some preparations we define the convolution product $*$ on\n$K_0({\\op{Var}}_S^{\\upmu_n})$ (Definition~\\ref{d:convolution})\nand show that it turns\n$K_0({\\op{Var}}_S^{\\upmu_n})$ into a \n$K_0({\\op{Var}}_{\\mathsf{k}})$-algebra\n(Proposition~\\ref{p:convolution-comm-ass-unit}). \nThis is not a new result: see \n\\cite[5.1-5.5]{guibert-loeser-merle-convolution} and use the\ndictionary from Remark~\\ref{rem:dictionary}. \nNevertheless we liked the exercise of showing associativity \nwithout using this dictionary.\n\n\n\nLet $S$ be a ${\\mathsf{k}}$-variety and $n \\in \\mathbb{N}_{>0}$.  Let\n$p \\colon Z \\rightarrow S$ be an object of ${\\op{Var}}^{\\upmu_n \\times \\upmu_n}_S$.\nWe assume that $\\upmu_n \\times \\upmu_n$ acts on $Z$ from the right.\nThe group $\\upmu_n \\times \\upmu_n$ acts on the ${\\mathsf{k}}$-variety\n$Z \\times {\\DG_\\op{m}} \\times {\\DG_\\op{m}}$ via\n$(z,x,y).(s,t):=(z.(s,t), s^{-1} x, t^{-1} y)$. The quotient with\nrespect to this action is the balanced\nproduct \n$Z \\times^{\\upmu_n \\times \\upmu_n} {\\DG_\\op{m}} \\times {\\DG_\\op{m}}$ which is again\na ${\\mathsf{k}}$-variety (use \\cite[Exp.~V.1]{SGA-1}).\nWe equip it with the\ndiagonal $\\upmu_n$-action given by \n$[z,x,y].t= [z, tx, ty]=[z.(t,t), x, y]$.\nWith the obvious morphism to $S$ induced by $p$ it is \nan object of\n${\\op{Var}}_S^{\\upmu_n}$.\nSimilarly, starting from the two closed $S$-subvarieties\nof $Z \\times {\\DG_\\op{m}} \\times {\\DG_\\op{m}}$ defined by the equations $x^n+y^n=1$ and\n$x^n+y^n=0$, we obtain the two objects\n$(Z \\times^{\\upmu_n \\times \\upmu_n} {\\DG_\\op{m}} \\times {\\DG_\\op{m}})|_{x^n+y^n=1}$\nand\n$(Z \\times^{\\upmu_n \\times \\upmu_n} {\\DG_\\op{m}} \\times {\\DG_\\op{m}})|_{x^n+y^n=0}$\nof ${\\op{Var}}_S^{\\upmu_n}$.\n\nGiven $(Z \\xra{p} S) \\in {\\op{Var}}_S^{\\upmu_n \\times \\upmu_n}$ as above define\n\\begin{align}\n  \\label{eq:psi-two-mu-s}\n  \\Psi(Z \\xra{p} S)\n  := &\n  -[(Z \\times^{\\upmu_n \\times \\upmu_n} \\underset{x}{{\\DG_\\op{m}}} \\times\n  \\underset{y}{{\\DG_\\op{m}}})|_{x^n+y^n=1} \\xra{[z,x,y] \\mapsto p(z)}\n  S]\\\\\n  \\notag\n  & +[(Z \\times^{\\upmu_n \\times \\upmu_n} \\underset{x}{{\\DG_\\op{m}}} \\times\n  \\underset{y}{{\\DG_\\op{m}}})|_{x^n+y^n=0}\n  \\xra{([z,x,y]) \\mapsto p(z)} S] \n  \\in K_0({\\op{Var}}_S^{\\upmu_n}).\n\\end{align}\nHere the symbols $x$ and $y$ below ${\\DG_\\op{m}} \\times {\\DG_\\op{m}}$ indicate\nthat $(x,y)$ forms a system \nof coordinates \non ${\\DG_\\op{m}} \\times {\\DG_\\op{m}}$.\nSimilar notation will be\nused below without further explanations.\n\n\\begin{example}\n  \\label{ex:trivial-action}\n  Let $p \\colon Z \\rightarrow S$ be as above and assume that\n  $\\upmu_n \\times \\upmu_n$ acts trivially on $Z$. Then\n  $Z \\times^{\\upmu_n \\times \\upmu_n} {\\DG_\\op{m}} \\times {\\DG_\\op{m}} \\xra{\\sim} Z\n  \\times {\\DG_\\op{m}} \\times {\\DG_\\op{m}}$,\n  $[z,x,y] \\mapsto (z,x^n, y^n)$, is an isomorphism which is\n  $\\upmu_n$-equivariant if we equip \n  $Z \\times {\\DG_\\op{m}} \\times {\\DG_\\op{m}}$\n  with the trivial $\\upmu_n$-action. This implies\n  $\\Psi(Z \\xra{p} S) = [Z \\xra{p} S]$ in $K_0({\\op{Var}}_S^{\\upmu_n})$\n  where $Z$ is considered as a $\\upmu_n$-variety over $S$ with\n  trivial action. In particular, we obtain\n  $\\Psi(S \\xra{\\operatorname{id}} S) = [S \\xra{\\operatorname{id}} S]$.\n \n \n\\end{example}\n\n\\begin{example}\n  \\label{ex:p-product-over-k-trivial-on-one-factor}\n  Assume that $p=p_1 \\times p_2 \\colon Z=Z_1 \\times Z_2 \\rightarrow S=S_1\n  \\times S_2$ where $S_1$ and $S_2$ are ${\\mathsf{k}}$-varieties and \n  $p_i \\colon Z_i \\rightarrow S_i$\n  is an object of ${\\op{Var}}_{S_i}^{\\upmu_n}$, for $i=1,2$.\n  Moreover assume that the action of $\\upmu_n$ on\n  $Z_2$ is trivial. Then \n  $Z_1 \\times Z_2 \\times^{\\upmu_n \\times \\upmu_n} {\\DG_\\op{m}} \\times\n  {\\DG_\\op{m}} \\xra{\\sim} (Z_1 \\times^{\\upmu_n} {\\DG_\\op{m}}) \\times (Z_2 \\times {\\DG_\\op{m}})$,\n  $[z_1,z_2,x,y] \\mapsto ([z_1,x], z_2, y^n)$, is an isomorphism\n  over $S$, and we can simplify \n  \\eqref{eq:psi-two-mu-s} to\n  \\begin{align*}\n   \n    \\Psi(Z_1 \\times Z_2 \\xra{p_1 \\times p_2} S_1 \\times S_2)\n    = \n    &\n      -[(Z_1 \\times^{\\upmu_n} \\underset{x}{{\\DG_\\op{m}}})|_{x^n\\not=1} \\times Z_2\n      \\rightarrow S_1 \\times S_2]\\\\\n    \\notag\n    &\n      +[(Z_1 \\times^{\\upmu_n} \\underset{x}{{\\DG_\\op{m}}}) \\times Z_2\n      \\rightarrow S_1 \\times S_2]\\\\\n    \\notag\n    =\n    & [(Z_1 \\times^{\\upmu_n} \\upmu_n) \\times Z_2\n      \\rightarrow S_1 \\times S_2]\\\\\n    \\notag\n    =\n    & [Z_1 \\times Z_2\n      \\rightarrow S_1 \\times S_2].\n  \\end{align*}\n  This example will be useful later on.\n\\end{example}\n\nIn fact, $\\Psi$ induces a morphism\n\\begin{equation}\n  \\label{eq:10}\n  \\Psi \\colon  K_0({\\op{Var}}_S^{\\upmu_n \\times \\upmu_n}) \\rightarrow K_0({\\op{Var}}_S^{\\upmu_n})\n\\end{equation}\nof $K_0({\\op{Var}}_{\\mathsf{k}})$-modules.\n\nOur next aim is to prove Proposition~\\ref{p:Psi-associative}\nwhich will later on imply associativity of the convolution\nproduct.\n\nLet $p \\colon Z \\rightarrow S$ be an object of ${\\op{Var}}^{\\upmu_n \\times \\upmu_n\n  \\times \\upmu_n}_S$. Similarly as above we \ndefine\n\\begin{multline}\n  \\label{eq:psi-123}\n  \\Psi_{123}(Z \\xra{p} S) := \n  -[(Z \\times^{\\upmu_n \\times \\upmu_n \\times\n    \\upmu_n} \\underset{x_1}{{\\DG_\\op{m}}} \\times \\underset{x_2}{{\\DG_\\op{m}}}\n    \\times \\underset{x_3}{{\\DG_\\op{m}}})|_{x_1^n+x_2^n+x_3^n=1}\n    \\xra{[z,x_1,x_2,x_3] \\mapsto p(z)}\n    S]\\\\\n \n    +[(Z\n    \\times^{\\upmu_n \\times \\upmu_n \\times\n    \\upmu_n} \\underset{x_1}{{\\DG_\\op{m}}} \\times \\underset{x_2}{{\\DG_\\op{m}}}\n    \\times \\underset{x_3}{{\\DG_\\op{m}}})|_{x_1^n+x_2^n+x_3^n=0}\n    \\xra{([z,x_1,x_2,x_3]) \\mapsto p(z)} S] \\in K_0({\\op{Var}}_S^{\\upmu_n})\n\\end{multline}\nwhere the closed subvarieties of $Z \\times^{\\upmu_n \\times \\upmu_n \\times \\upmu_n}\n\\underset{x_1}{{\\DG_\\op{m}}} \\times \\underset{x_2}{{\\DG_\\op{m}}} \\times\n\\underset{x_3}{{\\DG_\\op{m}}}$ are equipped with the $\\upmu_n$-action\n$[z,x_1,x_2,x_3].t=[z,tx_1, tx_2, tx_3]=[z.(t,t,t), x_1, x_2,\nx_3]$.\nAgain we obtain a morphism\n\\begin{equation*}\n \n  \\Psi_{123} \\colon  K_0({\\op{Var}}_S^{\\upmu_n \\times \\upmu_n \\times \\upmu_n}) \\rightarrow\n  K_0({\\op{Var}}_S^{\\upmu_n}) \n\\end{equation*}\nof $K_0({\\op{Var}}_k)$-modules.\n\nSimilarly we associate to \n$(p \\colon Z \\rightarrow S) \\in {\\op{Var}}^{\\upmu_n \\times \\upmu_n\n  \\times \\upmu_n}_S$ the element\n\\begin{align}\n  \\label{eq:psi-13}\n  \\Psi_{13}&(Z) := \n  -[(Z \\times^{\\upmu_n \\times \\{1\\} \\times\n    \\upmu_n} \\underset{x_1}{{\\DG_\\op{m}}} \\times \\{1\\}\n    \\times \\underset{x_3}{{\\DG_\\op{m}}})|_{x_1^n+x_3^n=1}\n    \\xra{[z,x_1,1,x_3] \\mapsto p(z)}\n    S]\n  \\\\\n  \\notag \n  & +[(Z\n    \\times^{\\upmu_n \\times \\{1\\} \\times\n    \\upmu_n} \\underset{x_1}{{\\DG_\\op{m}}} \\times \\{1\\}\n    \\times \\underset{x_3}{{\\DG_\\op{m}}})|_{x_1^n+x_3^n=0}\n    \\xra{([z,x_1,1,x_3]) \\mapsto p(z)} S] \n    \\in K_0({\\op{Var}}_S^{\\upmu_n \\times \\upmu_n}).\n\\end{align}\nHere the $\\upmu_n \\times \\upmu_n$-action is\ngiven by the two commuting $\\upmu_n$-actions\n$[z,x_1,1,x_3].s=[z,sx_1, 1, sx_3]=[z.(s,1,s), x_1, 1, x_3]$\nand\n$[z,x_1,1,x_3].t=[z.(1,t,1),x_1, 1, x_3]$, i.\\,e.\\ we have\n$[z,x_1,1,x_3].(s,t) =[z.(s,t,s), x_1, 1, x_3]$.\nAs above we obtain a morphism\n\\begin{equation*}\n  \\Psi_{13} \\colon  K_0({\\op{Var}}_S^{\\upmu_n \\times \\upmu_n \\times \\upmu_n}) \\rightarrow\n  K_0({\\op{Var}}_S^{\\upmu_n \\times \\upmu_n}) \n\\end{equation*}\nof $K_0({\\op{Var}}_k)$-modules.\nSimilarly we define $\\Psi_{12}$ and $\\Psi_{23}$.\n\n\\begin{remark}\n  \\label{rem:Psi-compatible-with-pushforward-pullback}\n  If $f \\colon  S \\rightarrow S'$ is a morphism of ${\\mathsf{k}}$-varieties, all maps\n  $\\Psi$, \n  $\\Psi_{123}$, \n  $\\Psi_{12}$,\n  $\\Psi_{13}$,\n  $\\Psi_{23}$\n  are compatible with $f_!$ and $f^*,$ for example\n  $\\Psi(f_!(Z))=f_!(\\Psi(Z))$ for $Z \\in\n  K_0({\\op{Var}}_{S}^{\\upmu_n \\times \\upmu_n})$\n  and  \n  $\\Psi(f^*(Z))=f^*(\\Psi(Z))$ for $Z \\in\n  K_0({\\op{Var}}_{S'}^{\\upmu_n \\times \\upmu_n})$.\n  For $f_!$ this is obvious. For $f^*$ one uses the fact that\n  $Z \\times {\\DG_\\op{m}} \\times {\\DG_\\op{m}} \\rightarrow Z \\times^{\\upmu_n \\times\n    \\upmu_n} {\\DG_\\op{m}} \\times {\\DG_\\op{m}}$ is a $(\\upmu_n \\times\n  \\upmu_n)$-torsor and hence its pullback under the base change\n  morphism $f$ is again such a torsor.\n\\end{remark}\n\n\\begin{proposition}\n \n  [{\\cite[Prop.~5.5]{guibert-loeser-merle-convolution}}]\n  \\label{p:Psi-associative}\n  We have\n  \\begin{equation*}\n    \\Psi_{123}=\\Psi \\circ \\Psi_{13}\n    =\\Psi \\circ \\Psi_{12}\n    =\\Psi \\circ \\Psi_{23}\n  \\end{equation*}\n  as morphisms \n  $K_0({\\op{Var}}_S^{\\upmu_n \\times \\upmu_n \\times \\upmu_n}) \\rightarrow\n  K_0({\\op{Var}}_S^{\\upmu_n})$\n  of $K_0({\\op{Var}}_{\\mathsf{k}})$-modules.\n\\end{proposition}\n\n\\begin{proof}\n  Let $p \\colon Z \\rightarrow S$ be an object of\n  ${\\op{Var}}^{\\upmu_n \\times \\upmu_n \\times \\upmu_n}_S$. It is enough to\n  show that\n  $\\Psi_{123}(Z) =\\Psi(\\Psi_{13}(Z)) =\\Psi(\\Psi_{12}(Z))\n  =\\Psi(\\Psi_{23}(Z))$\n  in $K_0({\\op{Var}}_S^{\\upmu_n})$. We only prove\n  $\\Psi_{123}(Z) =\\Psi(\\Psi_{13}(Z))$ and leave the remaining\n  cases to the reader.\n  \n  From \n  \\eqref{eq:psi-two-mu-s}\n  and \n  \\eqref{eq:psi-13}\n  we obtain\n  \\begin{align}\n    \\label{eq:psi-psi13}\n    \\Psi(\\Psi_{13}(Z)) = \n    & -\\Psi([(Z \\times^{\\upmu_n \\times \\{1\\}\n      \\times \\upmu_n} \\underset{x_1}{{\\DG_\\op{m}}} \\times \\{1\\} \\times\n    \\underset{x_3}{{\\DG_\\op{m}}})|_{x_1^n+x_3^n=1} \\rightarrow\n    S])\\\\\n    \\notag \n    & +\\Psi([(Z \\times^{\\upmu_n \\times \\{1\\} \\times \\upmu_n}\n      \\underset{x_1}{{\\DG_\\op{m}}} \\times \\{1\\} \\times\n      \\underset{x_3}{{\\DG_\\op{m}}})|_{x_1^n+x_3^n=0} \\rightarrow S])\\\\\n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n    \\notag = &\n    \\sum_{\\delta, \\epsilon \\in \\{0,1\\}}\n    (-1)^{\\delta+\\epsilon}\n    [D_{\\delta, \\epsilon} \\rightarrow S]\n  \\end{align}\n  where \n  \\begin{align*}\n   \n    D_{\\delta, \\epsilon} & :=\n    ((Z \\times^{\\upmu_n \\times \\{1\\} \\times \\upmu_n}\n    \\underset{x_1}{{\\DG_\\op{m}}} \\times \\{1\\} \\times\n    \\underset{x_3}{{\\DG_\\op{m}}})|_{x_1^n+x_3^n=\\delta} \\times^{\\upmu_n \\times\n      \\upmu_n} \\underset{y_1}{{\\DG_\\op{m}}} \\times\n    \\underset{y_2}{{\\DG_\\op{m}}})|_{y_1^n+y_2^n=\\epsilon}\\\\\n    \\notag\n    & =\n    (Z \\times^{\\upmu_n \\times \\upmu_n}\n    \\underset{x_1}{{\\DG_\\op{m}}} \\times\n    \\underset{x_3}{{\\DG_\\op{m}}} \\times^{\\upmu_n \\times\n      \\upmu_n} \\underset{y_1}{{\\DG_\\op{m}}} \\times\n    \\underset{y_2}{{\\DG_\\op{m}}})|_{x_1^n+x_3^n=\\delta, \\atop\n      y_1^n+y_2^n=\\epsilon}\\\\\n    \\notag\n    & =\n    \\frac{(Z \\times\n      \\underset{x_1}{{\\DG_\\op{m}}} \\times\n      \\underset{x_3}{{\\DG_\\op{m}}} \\times \\underset{y_1}{{\\DG_\\op{m}}} \\times\n      \\underset{y_2}{{\\DG_\\op{m}}})|_{x_1^n+x_3^n=\\delta, \\atop\n        y_1^n+y_2^n=\\epsilon}}\n    {\\upmu_n \\times \\upmu_n \\times \\upmu_n \\times \\upmu_n}.\n  \\end{align*}\n  Here, by the definitions of the quotients in\n  \\eqref{eq:psi-two-mu-s}\n  and \\eqref{eq:psi-13},\n  the quotient is formed with respect to the\n  $(\\upmu_n)^{\\times 4}$-action \n  \\begin{equation*}\n   \n    (z,x_1, x_3, y_1, y_2).(s,t,u,v)=\n    (z.(su,v,tu),s^{-1}x_1, t^{-1}x_3, u^{-1}y_1, v^{-1}y_2).    \n  \\end{equation*}\n  and $D_{\\delta,\\epsilon}$ is a $\\upmu_n$-variety with action\n  \\begin{equation*}\n   \n    [z,x_1,x_3,y_1,y_2].m=[z,x_1,x_3,my_1,my_2]=\n    [z.(m,m,m),x_1,x_3,y_1,y_2].\n  \\end{equation*}\n\n  The coordinate changes $a_1=x_1y_1$, $a_2=x_3y_1$, $b=y_1$,\n  $a_3=y_2$ in $({\\DG_\\op{m}})^{\\times 4}$ and $s'=su$,\n  $t'=tu$, $u=u$, $v=v$ in $(\\upmu_n)^{\\times 4}$ show that\n  \\begin{equation*}\n   \n    D_{\\delta,\\epsilon}\n    \\cong\n    \\frac{(Z \\times\n      \\underset{a_1}{{\\DG_\\op{m}}} \\times\n      \\underset{a_2}{{\\DG_\\op{m}}} \\times \\underset{b}{{\\DG_\\op{m}}} \\times\n      \\underset{a_3}{{\\DG_\\op{m}}})|_{a_1^n+a_2^n=\\delta b^n, \\atop\n        b^n+a_3^n=\\epsilon}}\n    {\\upmu_n \\times \\upmu_n \\times \\upmu_n \\times \\upmu_n}.\n  \\end{equation*}\n  where the quotient is formed with respect to the\n  $(\\upmu_n)^{\\times 4}$-action \n  \\begin{equation*}\n   \n    (z,a_1, a_2, b, a_3).(s',t',u,v)=\n    (z.(s',v,t'),s'^{-1}a_1, t'^{-1}a_2, u^{-1}b, v^{-1}a_3)    \n  \\end{equation*}\n  and the $\\upmu_n$-action on this quotient is given by \n  \\begin{equation*}\n    [z,a_1,a_2,b,a_3].m=[z,ma_1,ma_2,mb,ma_3]=\n    [z.(m,m,m),a_1,a_2,b,a_3].\n  \\end{equation*}\n  The quotient of \n  $\\underset{a_1}{{\\DG_\\op{m}}} \\times\n  \\underset{a_2}{{\\DG_\\op{m}}} \\times \n  \\underset{b}{{\\DG_\\op{m}}} \\times \n  \\underset{a_3}{{\\DG_\\op{m}}}|_{a_1^n+a_2^n=\\delta b^n, \\atop\n        b^n+a_3^n=\\epsilon}$\n  under the obvious action of $\\{1\\} \\times \\{1\\} \\times \\upmu_n\n  \\times \\{1\\}$ on the factor ${\\DG_\\op{m}}$ with coordinate $b$ is \n  clearly isomorphic to \n  \\begin{equation*}\n   \n    Q_{\\delta, \\epsilon}:=(\\underset{a_1}{{\\DG_\\op{m}}} \\times\n    \\underset{a_2}{{\\DG_\\op{m}}} \\times \n    \\underset{a_3}{{\\DG_\\op{m}}})|_{a_1^n+a_2^n=\\delta (\\epsilon -a_3^n),\n      \\atop \n      a_3^n\\not=\\epsilon}\n  \\end{equation*}\n  So we obtain\n  \\begin{equation*}\n   \n    D_{\\delta, \\epsilon} \\cong Z \\times^{\\upmu_n \\times \\upmu_n \\times\n      \\upmu_n} Q_{\\delta,\\epsilon} \n  \\end{equation*}\n  where the quotient is formed with respect\n  to the $(\\upmu_n)^{\\times 3}$-action\n  \\begin{equation*}\n   \n    (z,a_1, a_2, a_3).(s',t',v)=\n    (z.(s',v,t'),s'^{-1}a_1, t'^{-1}a_2, v^{-1}a_3)    \n  \\end{equation*}\n  and the $\\upmu_n$-action on this quotient is given by \n  \\begin{equation*}\n    [z,a_1,a_2,a_3].m=\n    [z.(m,m,m),a_1,a_2,a_3]=\n    [z,ma_1,ma_2,ma_3].\n  \\end{equation*}\n  Continuing the computation\n  \\eqref{eq:psi-psi13} we obtain\n  \\begin{align*}\n   \n    \\Psi(\\Psi_{13}(Z)) \n   \n   \n   \n   \n   \n    = &\n    + [Z \\times^{\\upmu_n \\times \\upmu_n\n      \\times \\upmu_n}\n    (\\underset{a_1}{{\\DG_\\op{m}}} \\times\n    \\underset{a_2}{{\\DG_\\op{m}}} \\times\n    \\underset{a_3}{{\\DG_\\op{m}}})|_{a_1^n+a_2^n+a_3^n=1\n      \\atop 1 \\not= a_3^n}\n    \\rightarrow S]\\\\\n    \\notag\n    & -[Z \\times^{\\upmu_n \\times \\upmu_n\n      \\times \\upmu_n}\n    (\\underset{a_1}{{\\DG_\\op{m}}} \\times\n      \\underset{a_2}{{\\DG_\\op{m}}} \\times\n      \\underset{a_3}{{\\DG_\\op{m}}})|_{a_1^n+a_2^n+a_3^n=0}\n    \\rightarrow S]\\\\\n    \\notag\n    & -[Z \\times^{\\upmu_n \\times \\upmu_n\n      \\times \\upmu_n}\n    (\\underset{a_1}{{\\DG_\\op{m}}} \\times\n    \\underset{a_2}{{\\DG_\\op{m}}} \\times \\underset{a_3}{{\\DG_\\op{m}}})|_{a_1^n+a_2^n=0\n      \\atop 1 \\not= a_3^n}\n    \\rightarrow S]\\\\\n    \\notag\n    & + [Z \\times^{\\upmu_n \\times \\upmu_n\n      \\times \\upmu_n}\n    (\\underset{a_1}{{\\DG_\\op{m}}} \\times\n    \\underset{a_2}{{\\DG_\\op{m}}} \\times \\underset{a_3}{{\\DG_\\op{m}}})|_{a_1^n+a_2^n=0}\n    \\rightarrow S]\n  \\end{align*}\n  The last two summands simplify to \n  \\begin{equation*}\n    +[Z \\times^{\\upmu_n \\times \\upmu_n\n      \\times \\upmu_n}\n    (\\underset{a_1}{{\\DG_\\op{m}}} \\times\n    \\underset{a_2}{{\\DG_\\op{m}}} \\times\n    \\underset{a_3}{{\\DG_\\op{m}}})|_{a_1^n+a_2^n=0 \n      \\atop 1=a_3^n} \n    \\rightarrow S].\n  \\end{equation*}\n  The two conditions\n  $a_1^n+a_2^n=0$ and $1=a_3^n$ are equivalent to\n  the two conditions\n  $a_1^n+a_2^n+a_3^n=1$ and $1=a_3^n$.\n  Hence we can further simplify and obtain\n  \\begin{align*}\n    \\Psi(\\Psi_{13}(Z)) = &\n    + [Z \\times^{\\upmu_n \\times \\upmu_n\n      \\times \\upmu_n}\n    (\\underset{a_1}{{\\DG_\\op{m}}} \\times\n    \\underset{a_2}{{\\DG_\\op{m}}} \\times\n    \\underset{a_3}{{\\DG_\\op{m}}})|_{a_1^n+a_2^n+a_3^n=1}\n    \\rightarrow S]\\\\\n    & -[Z \\times^{\\upmu_n \\times \\upmu_n\n      \\times \\upmu_n}\n    (\\underset{a_1}{{\\DG_\\op{m}}} \\times\n      \\underset{a_2}{{\\DG_\\op{m}}} \\times\n      \\underset{a_3}{{\\DG_\\op{m}}})|_{a_1^n+a_2^n+a_3^n=0}\n    \\rightarrow S]\\\\\n    = &\n    \\Psi_{123}(Z).\n  \\end{align*}\n  where the last equality holds by definition\n  \\eqref{eq:psi-123}.\n\\end{proof}\n\n\\begin{definition}\n  [{Convolution product}]\n  \\label{d:convolution}\n \n  The convolution product $*$ on $K_0({\\op{Var}}_S^{\\upmu_n})$ is\n  defined as the $K_0({\\op{Var}}_{\\mathsf{k}})$-linear composition\n  \\begin{equation}\n    \\label{eq:25}\n    * \\colon K_0({\\op{Var}}_S^{\\upmu_n}) \\otimes_{K_0({\\op{Var}}_{\\mathsf{k}})}\n    K_0({\\op{Var}}_S^{\\upmu_n})  \n    \\xra{\\times_S} K_0({\\op{Var}}_S^{\\upmu_n \\times \\upmu_n})\n    \\xra{\\Psi}  K_0({\\op{Var}}_S^{\\upmu_n})\n  \\end{equation}\n  where the first map $\\times_S$ is \n  the $K_0({\\op{Var}}_{\\mathsf{k}})$-linear map\n  induced by mapping a pair\n  $(A,B)$ of \n  $S$-varieties with good $\\upmu_n$-action to the class of the\n  $S$-variety $|A \\times_S B|$ with good $(\\upmu_n \\times\n  \\upmu_n)$-action. \n\\end{definition}\n\nMore explicitly, if $A \\rightarrow S$ and $B \\rightarrow S$ are\n$S$-varieties with good $\\upmu_n$-action, then\n\\begin{align}\n  \\label{eq:convolution-explicit}\n  [A \\rightarrow S] * [B \\rightarrow S] = \n  & -[(|A \\times_S B| \\times^{\\upmu_n \\times \\upmu_n}\n    \\underset{x}{{\\DG_\\op{m}}} \\times \n    \\underset{y}{{\\DG_\\op{m}}})|_{x^n+y^n=1} \\rightarrow S]\\\\\n  \\notag\n  & +[(|A \\times_S B| \\times^{\\upmu_n \\times \\upmu_n}\n    \\underset{x}{{\\DG_\\op{m}}} \\times \n    \\underset{y}{{\\DG_\\op{m}}})|_{x^n+y^n=0}\n    \\rightarrow S]\n  \\\\\n  \\notag\n  =\n  & -[|(A \\times_S B \\times^{\\upmu_n \\times \\upmu_n}\n    \\underset{x}{{\\DG_\\op{m}}} \\times \n    \\underset{y}{{\\DG_\\op{m}}})|_{x^n+y^n=1}| \\rightarrow S]\n  \\\\\n  \\notag\n  & +[|(A \\times_S B \\times^{\\upmu_n \\times \\upmu_n}\n    \\underset{x}{{\\DG_\\op{m}}} \\times \n    \\underset{y}{{\\DG_\\op{m}}})|_{x^n+y^n=0}|\n    \\rightarrow S]. \n\\end{align}\nThe second equality comes from the fact that taking the reduced\nsubscheme structure commutes with fiber \nproducts \n(\\cite[Prop.~4.34]{goertz-wedhorn-AGI})\nand with quotients under the action of a finite group.\n\n\\begin{remark}\n  \\label{rem:action-on-B-trivial}\n  Let $A \\rightarrow S$ and $B \\rightarrow S$ be\n  $S$-varieties with good $\\upmu_n$-action, and assume that the\n  $\\upmu_n$-action on $B$ is trivial.\n \n \n  Similar as in\n  Example~\\ref{ex:p-product-over-k-trivial-on-one-factor}\n  we deduce from \\eqref{eq:convolution-explicit} that\n  \\begin{multline*}\n   \n    [A] * [B]\n    = \n    -[|((A \\times^{\\upmu_n}\n    \\underset{x}{{\\DG_\\op{m}}}) \\times_S \n    (B \\times \\underset{y'}{{\\DG_\\op{m}}}))|_{x^n+y'=1}|]\n    +[|((A \\times^{\\upmu_n}\n    \\underset{x}{{\\DG_\\op{m}}}) \\times_S \n    (B \\times \\underset{y'}{{\\DG_\\op{m}}}))|_{x^n+y'=0}|]\n    \\\\\n    = \n    -[|((A \\times^{\\upmu_n}\n    \\underset{x}{{\\DG_\\op{m}}}) \\times_S B)|_{x^n\\not=1}|]\n    +[|(A \\times^{\\upmu_n} {\\DG_\\op{m}}) \\times_S B|]\n    \\\\\n    = [|(A \\times^{\\upmu_n} \\upmu_n) \\times_S B|]\n    = [|A \\times_S B|]= [A] [B].\n  \\end{multline*}\n\\end{remark}\n\n\\begin{proposition}\n  [{\\cite[Prop.~5.2]{guibert-loeser-merle-convolution}}]\n  \\label{p:convolution-comm-ass-unit}\n  Let $S$ be a ${\\mathsf{k}}$-variety and $n \\geq 1$.\n  The convolution product $*$ turns $K_0({\\op{Var}}_S^{\\upmu_n})$ into\n  an associative commutative unital $K_0({\\op{Var}}_{\\mathsf{k}})$-algebra. The\n  identity element is  \n  the class of $(\\operatorname{id}_S \\colon S \\rightarrow S)$ where $\\upmu_n$ acts\n  trivially on $S$.  \n  We denote this ring as $(K_0({\\op{Var}}_S^{\\upmu_n}),*)$.\n\\end{proposition}\n\n\\begin{proof}\n  Clearly, the convolution product is commutative.\n  Remark~\\ref{rem:action-on-B-trivial}\n  shows that $[\\operatorname{id}_S \\colon S \\rightarrow S]$ is the identity with respect to\n  the convolution \n  product.  \n  Associativity follows from \n  Proposition~\\ref{p:Psi-associative}:\n  \\begin{multline*}\n    ([A] * [B]) * [C] \n    = \\Psi(\\Psi_{12}([|A \\times_S B \\times_S C|]) \n    = \\Psi(\\Psi_{23}([|A \\times_S B \\times_S C|]) \n    = [A] * ([B] * [C]).\n  \\end{multline*}\n  Here we again use that passing to the reduced subscheme structure\n  commutes with fiber products and taking quotients under the\n  action of a finite group.\n\\end{proof}\n\n\\begin{remark}\n  \\label{rem:convolution-n=1}\n  For $n=1$ the convolution product $*$ on $K_0({\\op{Var}}_S^{\\upmu_1})$\n  coincides with the product on\n  $K_0({\\op{Var}}_S)=K_0({\\op{Var}}_S^{\\upmu_1})$,\n  so $K_0({\\op{Var}}_S)=(K_0({\\op{Var}}_S^{\\upmu_1}),*)$ as $K_0({\\op{Var}}_{\\mathsf{k}})$-algebras.\n  This follows immediately from \n  Remark~\\ref{rem:action-on-B-trivial}.\n\\end{remark}\n\nLet $(Z \\rightarrow S) \\in {\\op{Var}}_S^{\\upmu_n \\times \\upmu_n}$ and assume\nthat $n'=dn$ is a multiple of $n$. Then the morphism\n$Z \\times {\\DG_\\op{m}} \\times {\\DG_\\op{m}} \\rightarrow Z \\times {\\DG_\\op{m}} \\times {\\DG_\\op{m}}$,\n$(z,x,y) \\mapsto (z,x^d,y^d)$ defines an isomorphism\n\\begin{equation}\n  \\label{eq:22}\n  Z \\times^{\\upmu_{n'} \\times \\upmu_{n'}} {\\DG_\\op{m}} \\times {\\DG_\\op{m}} \\xra{\\sim} \n  Z \\times^{\\upmu_n \\times \\upmu_n} {\\DG_\\op{m}} \\times {\\DG_\\op{m}}\n\\end{equation}\nin ${\\op{Var}}_S^{\\upmu_{n'}}$.\nThis implies that $\\Psi$ is compatible with the morphisms\n$K_0({\\op{Var}}_S^{\\upmu_n \\times \\upmu_n}) \\rightarrow K_0({\\op{Var}}_S^{\\upmu_{n'}\n  \\times \\upmu_{n'}})$\nand \n$K_0({\\op{Var}}_S^{\\upmu_n}) \\rightarrow K_0({\\op{Var}}_S^{\\upmu_{n'}})$,\ncf.\\ \\eqref{eq:28}, and so is the \nfirst map in \\eqref{eq:25}.\nWe deduce that the obvious morphism\n\\begin{equation*}\n \n  (K_0({\\op{Var}}_S^{\\upmu_n}),*) \\rightarrow (K_0({\\op{Var}}_S^{\\upmu_{n'}}),*)\n\\end{equation*}\nis a map of $K_0({\\op{Var}}_{\\mathsf{k}})$-algebras.\nHence convolution turns $K_0({\\op{Var}}_S^{\\hat{\\upmu}})$ \ninto an associative commutative unital \n$K_0({\\op{Var}}_{\\mathsf{k}})$-algebra; we denote this algebra by \n$(K_0({\\op{Var}}_S^{\\hat{\\upmu}}),*)$. \n \nIf $f \\colon T \\rightarrow S$ is a morphism of ${\\mathsf{k}}$-varieties, \nthe \npullback maps \n$f^* \\colon (K_0({\\op{Var}}_S^{\\upmu_n}),*) \\rightarrow\n(K_0({\\op{Var}}_T^{\\upmu_n}),*)$\nand\n$f^* \\colon (K_0({\\op{Var}}_S^{\\hat{\\upmu}}),*) \\rightarrow\n(K_0({\\op{Var}}_T^{\\hat{\\upmu}}),*)$\nare maps of $K_0({\\op{Var}}_{\\mathsf{k}})$-algebras (use\nRemark~\\ref{rem:Psi-compatible-with-pushforward-pullback}\nand that the first map in \\eqref{eq:25} is compatible\nwith pullbacks). \n\n\nWe also want to define a convolution product on\n$\\mathcal{M}_S^{\\upmu_n}$ and $\\mathcal{M}_S^{\\hat{\\upmu}}$.\n\nConsider \nthe localization of  \n$(K_0({\\op{Var}}_S^{\\upmu_n}),*)$ at the multiplicative set $\\{1,\n\\mathbb{L}_S, \\mathbb{L}_S * \\mathbb{L}_S, \\dots\\}$. The $n$-fold convolution\nproduct of $\\mathbb{L}_S=[{\\mathbb{A}}^1_S]$ with itself is $[{\\mathbb{A}}^n_S]$ \nand we have \n$[A]*[{\\mathbb{A}}^n_S]=[A][{\\mathbb{A}}^n_S]$ for $[A] \\in K_0({\\op{Var}}_S^{\\upmu_n})$,\nby\nRemark~\\ref{rem:action-on-B-trivial}. Hence the underlying\nabelian group of\nthis localization is canonically identified with the underlying\nabelian group of $\\mathcal{M}_S^{\\upmu_n}$. We can therefore\ndenote the above localization by $(\\mathcal{M}_S^{\\upmu_n},*)$.\n\nBecause the structure morphism $K_0({\\op{Var}}_{\\mathsf{k}}) \\rightarrow\n(K_0({\\op{Var}}_S^{\\upmu_n}),*)$ sends $\\mathbb{L}_{\\mathsf{k}}$ to $\\mathbb{L}_S$ we obtain\na canonical isomorphism\n\\begin{equation}\n  \\label{eq:26}\n  \\mathcal{M}_{\\mathsf{k}} \\otimes_{K_0({\\op{Var}}_{\\mathsf{k}})} (K_0({\\op{Var}}_S^{\\upmu_n}),*)\n  \\xra{\\sim}\n  (\\mathcal{M}_S^{\\upmu_n},*)\n\\end{equation}\nof $\\mathcal{M}_{\\mathsf{k}}$-algebras which we will often treat as an\nequality in the following. Its underlying morphism of \n$\\mathcal{M}_{\\mathsf{k}}$-modules coincides with \\eqref{eq:15}.\n\nSimilarly, we define the convolution product $*$ on\n$\\mathcal{M}_S^{\\hat{\\upmu}}$ and obtain the $\\mathcal{M}_{\\mathsf{k}}$-algebra\n$\\mathcal{M}_{\\mathsf{k}} \\otimes_{K_0({\\op{Var}}_{\\mathsf{k}})} (K_0({\\op{Var}}_S^{\\hat{\\upmu}}),*)=\n(\\mathcal{M}_S^{\\hat{\\upmu}},*)$. \nThe map $\\Psi$ from \\eqref{eq:10} gives in the obvious way rise to a morphism\n\\begin{equation*}\n \n  \\Psi \n  \\colon  \n  \\mathcal{M}_S^{\\hat{\\upmu} \\times \\hat{\\upmu}} \n  \\rightarrow \n  \\mathcal{M}_S^{\\hat{\\upmu}}\n\\end{equation*}\nof $\\mathcal{M}_{\\mathsf{k}}$-modules; \nthe \nconvolution product $*$ on\n$\\mathcal{M}_S^{\\hat{\\upmu}}$ is then given as the composition\n\\begin{equation*}\n  * \\colon \\mathcal{M}_S^{\\hat{\\upmu}} \\otimes_{\\mathcal{M}_{\\mathsf{k}}}\n  \\mathcal{M}_S^{\\hat{\\upmu}}\n  \\xra{\\times_S} \\mathcal{M}_S^{\\hat{\\upmu} \\times \\hat{\\upmu}}\n  \\xra{\\Psi} \\mathcal{M}_S^{\\hat{\\upmu}}.\n\\end{equation*}\n\nGiven $f \\colon T \\rightarrow S$ as above, we obtain pullback maps\n$f^* \\colon (\\mathcal{M}_S^{\\upmu_n},*) \\rightarrow (\\mathcal{M}_T^{\\upmu_n},*)$\nand\n$f^* \\colon (\\mathcal{M}_S^{\\hat{\\upmu}},*) \\rightarrow\n(\\mathcal{M}_T^{\\hat{\\upmu}},*)$ of\n$\\mathcal{M}_{\\mathsf{k}}$-algebras. \nUnder the isomorphisms \\eqref{eq:26} they are just obtained by\nscalar extension along $K_0({\\op{Var}}_{\\mathsf{k}}) \\rightarrow \\mathcal{M}_{\\mathsf{k}}$ from\nthe previous pullback maps.\n\n\\subsection{Convolution of varieties over\n  \\texorpdfstring{${\\mathbb{A}}^1_{\\mathsf{k}}$}{A1}}\n\\label{sec:conv-vari-over}\n\nWe now use that ${\\mathbb{A}}^1_{\\mathsf{k}}$ is a commutative \ngroup\n${\\mathsf{k}}$-variety.\nLet $\\operatorname{add} \\colon  {\\mathbb{A}}^1_{\\mathsf{k}} \\times {\\mathbb{A}}^1_{\\mathsf{k}} \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}$, $(x,y)\n\\mapsto x+y$, be the \naddition morphism. Let $n \\geq 1$.\n\n\\begin{definition}\n  [{Convolution over ${\\mathbb{A}}^1_{\\mathsf{k}}$}]\n  \\label{d:convolution-over-A1}\n \n  The convolution product $\\star$ on $K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\upmu_n})$ is\n  defined as the $K_0({\\op{Var}}_{\\mathsf{k}})$-linear composition\n  \\begin{equation}\n    \\label{eq:convolution-A1}\n    \\star \\colon K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\upmu_n}) \\otimes_{K_0({\\op{Var}}_{\\mathsf{k}})}\n    K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\upmu_n})  \n    \\xra{\\times}\n    K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}} \\times {\\mathbb{A}}^1_{\\mathsf{k}}}^{\\upmu_n \\times \\upmu_n})\n    \\xra{\\operatorname{add}_!}\n    K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\upmu_n \\times \\upmu_n})\n    \\xra{\\Psi}  K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\upmu_n})\n  \\end{equation}\n  where the first map $\\times$ is \n  the $K_0({\\op{Var}}_{\\mathsf{k}})$-linear map\n  induced by mapping a pair\n  $(A,B)$ of \n  ${\\mathbb{A}}^1_{\\mathsf{k}}$-varieties with good $\\upmu_n$-action to the\n  $S$-variety $A \\times B$ with good $(\\upmu_n \\times\n  \\upmu_n)$-action. \n\\end{definition}\n\nBy Remark~\\ref{rem:Psi-compatible-with-pushforward-pullback} we\nhave\n\\begin{equation*}\n  A \\star B\n  =\\Psi(\\operatorname{add}_!(A \\times B)) \n  =\\operatorname{add}_!(\\Psi(A \\times B))\n\\end{equation*}\nfor $A$, $B \\in \nK_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\upmu_n})$.\n\n\n\\begin{remark}\n  \\label{rem:action-on-B-trivial-over-A1}\n  Let $A \\xra{\\alpha} {\\mathbb{A}}^1_{\\mathsf{k}}$ and $B \\xra{\\beta} {\\mathbb{A}}^1_{\\mathsf{k}}$ be\n  ${\\mathbb{A}}^1_{\\mathsf{k}}$-varieties with good $\\upmu_n$-action, and assume\n  that the \n  $\\upmu_n$-action on $B$ is trivial.\n  Then \n  Example~\\ref{ex:p-product-over-k-trivial-on-one-factor}\n \n \n  implies that\n  \\begin{equation*}\n   \n    [A \\xra{\\alpha} {\\mathbb{A}}^1_{\\mathsf{k}}] \\star [B \\xra{\\beta} {\\mathbb{A}}^1_{\\mathsf{k}}] =\n    [A \\times B \\xra{\\alpha \\circledast \\beta} {\\mathbb{A}}^1_{\\mathsf{k}}]\n  \\end{equation*}\n  where $(\\alpha \\circledast \\beta)(a,b)=\\alpha(a)+\\beta(b)$;\n  the $\\upmu_n$-action on $A \\times B$ is the obvious diagonal\n  action $(a,b).t=(a.t,b.t)=(a.t,b)$.\n\\end{remark}\n\n\\begin{remark}\n  \\label{rem:no-group-action}\n  In the case $n=1$ the convolution product $\\star$ on\n  $K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\upmu_1})= K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}})$\n  satisfies\n  \\begin{equation*}\n   \n    [A \\xra{\\alpha} {\\mathbb{A}}^1_{\\mathsf{k}}] \\star [B \\xra{\\beta} {\\mathbb{A}}^1_{\\mathsf{k}}] =\n    [A \\times B \\xra{\\alpha \\circledast \\beta} {\\mathbb{A}}^1_{\\mathsf{k}}]\n  \\end{equation*}\n  where $A \\xra{\\alpha} {\\mathbb{A}}^1_{\\mathsf{k}}$ and $B \\xra{\\beta} {\\mathbb{A}}^1_{\\mathsf{k}}$ \n  are \n  ${\\mathbb{A}}^1_{\\mathsf{k}}$-varieties.\n  This is a special case of\n  Remark~\\ref{rem:action-on-B-trivial-over-A1}.   \n\\end{remark}\n\n\\begin{proposition}\n  \\label{p:A1-convolution-comm-ass-unit}\n  The convolution product $\\star$ \n  turns \n  $K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\upmu_n})$\n \n  into an associative commutative $K_0({\\op{Var}}_{\\mathsf{k}})$-algebra with\n  identity element \n  $[\\operatorname{Spec} k \\xra{0} {\\mathbb{A}}^1_{\\mathsf{k}}]$.\n \n  We denote this ring by $(K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\upmu_n}), \\star)$.\n\\end{proposition}\n\n\\begin{proof}\n  Commutativity follows from commutativity of ${\\mathbb{A}}^1_{\\mathsf{k}}$.\n  That \n  $[\\operatorname{Spec} k \\xra{0} {\\mathbb{A}}^1_{\\mathsf{k}}]$ is the identity element with respect\n  to $\\star$ follows from\n  Remark~\\ref{rem:action-on-B-trivial-over-A1}.\n  Denote the morphism $({\\mathbb{A}}^1_{\\mathsf{k}})^{\\times 3} \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}$,\n  $(x,y,z) \n  \\mapsto x+y+z$ by $\\op{addd}$.\n  Remark~\\ref{rem:Psi-compatible-with-pushforward-pullback} and\n  Proposition~\\ref{p:Psi-associative}\n  show that\n  \\begin{align*}\n    (A \\star B) \\star C \n    & = \n    \\operatorname{add}_!(\\Psi(\\operatorname{add}_!(\\Psi(A \\times B)) \\times C))\\\\\n    & = \n    \\operatorname{add}_!((\\operatorname{add} \\times \\operatorname{id})_!(\\Psi(\\Psi(A \\times B) \\times C)))\\\\\n    & = \\op{addd}_!\n      (\\Psi(\\Psi_{12}(A \\times B \\times C)))\\\\\n    & = \n    \\op{addd}_!(\\Psi_{123}(A \\times B \\times C))\n  \\end{align*}\n  A similar computation shows that the last term equals $A\n  \\star \n  (B \\star C)$. This proves associativity.\n\\end{proof}\n\nMapping a ${\\mathsf{k}}$-variety $(A \\rightarrow \\operatorname{Spec} {\\mathsf{k}})$ to $(A \\xra{0}\n{\\mathbb{A}}^1_{\\mathsf{k}})$ induces a morphism of unital rings\n\\begin{equation}\n  \\label{eq:46}\n  K_0({\\op{Var}}_{\\mathsf{k}}) \\rightarrow (K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\upmu_n}), \\star)\n\\end{equation}\nas follows immediately from\nRemark~\\ref{rem:action-on-B-trivial-over-A1}. \nThis map is the structure map of the  $K_0({\\op{Var}}_{\\mathsf{k}})$-algebra\n$(K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\upmu_n}), \\star)$.\nDenote the image of $\\mathbb{L}_{\\mathsf{k}}$ under this map by\n\\begin{equation}\n  \\label{eq:def-L-A1-0}\n  \\mathbb{L}_{({\\mathbb{A}}^1_{\\mathsf{k}},0)}:=[{\\mathbb{A}}^1_{\\mathsf{k}} \\xra{0} {\\mathbb{A}}^1_{\\mathsf{k}}] \\in\n  K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\upmu_n}). \n\\end{equation}\n\nLet us denote the localization of \n$(K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\upmu_n}), \\star)$ with respect to the\nmultiplicative set $\\{1, \\mathbb{L}_{({\\mathbb{A}}^1_{\\mathsf{k}},0)}, \\mathbb{L}_{({\\mathbb{A}}^1_{\\mathsf{k}},0)}\n\\star \\mathbb{L}_{({\\mathbb{A}}^1_{\\mathsf{k}},0)}, \\dots\\}$ by\n$(\\tilde{\\mathcal{M}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\upmu_n}, \\star)$. Then there is a canonical isomorphism\n\\begin{equation*}\n \n  \\mathcal{M}_{\\mathsf{k}} \\otimes_{K_0({\\op{Var}}_{\\mathsf{k}})}\n  (K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\upmu_n}), \\star) \n  \\xra{\\sim}\n  (\\tilde{\\mathcal{M}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\upmu_n}, \\star)\n\\end{equation*}\nof $\\mathcal{M}_{\\mathsf{k}}$-algebras given by $\\tfrac{r}{(\\mathbb{L}_{\\mathsf{k}})^n}\n\\otimes a \\mapsto \\tfrac{ra}{(\\mathbb{L}_{({\\mathbb{A}}^1_{\\mathsf{k}},0)})^n}$. \nIf we compare with \nthe isomorphism~\\eqref{eq:15} we see that\n$\\tfrac{b}{(\\mathbb{L}_{{\\mathbb{A}}^1_{\\mathsf{k}}})^n} \\mapsto\n\\tfrac{b}{(\\mathbb{L}_{({\\mathbb{A}}^1_{\\mathsf{k}},0)})^n}$\ndefines an isomorphism \n\\begin{equation}\n  \\label{eq:71}\n  \\mathcal{M}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\upmu_n}\n  \\xra{\\sim}\n  \\tilde{\\mathcal{M}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\upmu_n}\n\\end{equation}\nof $\\mathcal{M}_{\\mathsf{k}}$-modules.\n\nSimilarly as above (cf.\\ the reasoning around \\eqref{eq:22}),\nthe various $K_0({\\op{Var}}_{\\mathsf{k}})$-algebras\n$(K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\upmu_n}), \\star)$ for $n \\geq 1$ are\ncompatible. Hence we obtain the \n$K_0({\\op{Var}}_{\\mathsf{k}})$-algebras\n$(K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\hat{\\upmu}}), \\star)$\nand\n\\begin{equation*}\n \n  \\mathcal{M}_{\\mathsf{k}} \\otimes_{K_0({\\op{Var}}_{\\mathsf{k}})}\n  (K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\hat{\\upmu}}), \\star) \n  \\xra{\\sim}\n  (\\tilde{\\mathcal{M}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\hat{\\upmu}}, \\star)\n\\end{equation*}\nand an isomorphism\n\\begin{equation}\n  \\label{eq:62}\n  \\mathcal{M}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\hat{\\upmu}}\n  \\xra{\\sim}\n  \\tilde{\\mathcal{M}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\hat{\\upmu}}\n\\end{equation}\nof $\\mathcal{M}_{\\mathsf{k}}$-modules.\n\n\\begin{lemma}\n  \\label{l:convolution-products-compatible}\n  Let $\\epsilon \\colon  {\\mathbb{A}}^1_{\\mathsf{k}} \\rightarrow \\operatorname{Spec} {\\mathsf{k}}$ be the structure\n  morphism. \n  Then mapping an object $(A \\xra{\\alpha} {\\mathbb{A}}^1_{\\mathsf{k}}) \\in\n  {\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\upmu_n}$ to $(A \\xra{\\epsilon \\alpha} \\operatorname{Spec} {\\mathsf{k}})\n  \\in {\\op{Var}}^{\\upmu_n}_{\\mathsf{k}}$ induces a morphism\n  \\begin{equation}\n    \\label{epsilon!-star-*}\n    \\epsilon_! \\colon  (\\tilde{\\mathcal{M}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\hat{\\upmu}}, \\star)\n    \\rightarrow (\\mathcal{M}_k^{\\hat{\\upmu}}, *)\n  \\end{equation}\n \n  of $\\mathcal{M}_{\\mathsf{k}}$-algebras.\n\\end{lemma}\n\n\\begin{proof}\n  Certainly we have a morphism\n  \\begin{equation}\n    \\label{eq:33}\n    \\epsilon_! \\colon  K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\upmu_n})\n    \\rightarrow K_0({\\op{Var}}_k^{\\upmu_n})  \n  \\end{equation}\n  of $K_0({\\op{Var}}_{\\mathsf{k}})$-modules.\n  If $Z$ is a ${\\mathsf{k}}$-variety we denote its structure morphism\n  $Z \\rightarrow \\operatorname{Spec} {\\mathsf{k}}$ by $\\epsilon^Z$.\n  Let $A, B \\in  K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\upmu_n})$.\n  Since $\\epsilon_!(A)$ and $\\epsilon_!(B)$ are in\n  $K_0({\\op{Var}}_k^{\\upmu_n})$,  \n  $\\epsilon_!(A) * \\epsilon_!(B)$ is defined using the fiber\n  product over ${\\mathsf{k}}$.\n  Using Remark~\\ref{rem:Psi-compatible-with-pushforward-pullback}\n  we obtain\n  \\begin{multline*}\n     \\epsilon_!(A \\star B) \n     = \\epsilon_!(\\operatorname{add}_!(\\Psi(A \\times B)))\n     = \\epsilon^{{\\mathbb{A}}^1_{\\mathsf{k}} \\times {\\mathbb{A}}^1_{\\mathsf{k}}}_!(\\Psi(A \\times B))\n     = \\Psi(\\epsilon^{{\\mathbb{A}}^1_{\\mathsf{k}} \\times {\\mathbb{A}}^1_{\\mathsf{k}}}_!(A \\times B))\\\\\n     = \\Psi(\\epsilon_!(A) \\times \\epsilon_!(B))\n     = \\epsilon_!(A) * \\epsilon_!(B).\n  \\end{multline*}\n  Clearly, \\eqref{eq:33} maps $[\\operatorname{Spec} {\\mathsf{k}} \\xra{0} {\\mathbb{A}}^1_{\\mathsf{k}}]$ to\n  $[\\operatorname{Spec} {\\mathsf{k}} \\rightarrow \\operatorname{Spec} {\\mathsf{k}}]$. Therefore it is a morphism of\n  $K_0({\\op{Var}}_{\\mathsf{k}})$-algebras\n  $\\epsilon_! \\colon  (K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\upmu_n}), \\star)\n  \\rightarrow (K_0({\\op{Var}}_k^{\\upmu_n}),*)$.  \n  We can pass to $\\hat{\\upmu}$. Then\n  base change along $K_0({\\op{Var}}_{\\mathsf{k}}) \\rightarrow\n  \\mathcal{M}_{\\mathsf{k}}$ (or noting that\n  $\\mathbb{L}_{({\\mathbb{A}}^1_{\\mathsf{k}},0)}$ goes to $\\mathbb{L}_{\\operatorname{Spec} {\\mathsf{k}}}$) yields a\n  morphism\n  $\\epsilon_! \\colon (\\tilde{\\mathcal{M}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\upmu_n},\n  \\star) \\rightarrow (\\mathcal{M}_k^{\\upmu_n}, *)$\n  of $\\mathcal{M}_{\\mathsf{k}}$-algebras.  The lemma follows.\n\\end{proof}\n\n\\section{Motivic vanishing cycles}\n\\label{sec:motiv-vanish-fiber}\n\nLet $X$ be a smooth connected (nonempty) ${\\mathsf{k}}$-variety and let \n$V \\colon  X \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}$ be a morphism. \nGiven $a \\in {\\mathsf{k}} = {\\mathbb{A}}^1_{\\mathsf{k}}({\\mathsf{k}})$ we\ndenote by $X_a$ the scheme theoretic fiber of $V$ over $a$.\n\nWe quickly review the definition of the motivic vanishing\ncycles. For details we refer to  \n\\cite[Sect.~3]{guibert-loeser-merle-convolution};\nnote however that we use slightly different signs, see Remark \n\\ref{rem:motivic-fibers-signs} below. \nFollowing Denef and Loeser, the \\define{motivic zeta function of\n  $V$ at $a$} is a certain power series \n\\begin{equation*}\n  Z_{V,a}(T) \\in \\mathcal{M}^{\\hat{\\upmu}}_{|X_a|}[[T]]\n\\end{equation*}\nwhose coefficients are defined using arc spaces, see\n\\cite[(3.2.2)]{guibert-loeser-merle-convolution}.\nIt is possible to evaluate $Z_{V,a}$ at $T=\\infty$. \nThis is clear if $V$ is constant because then $Z_{V,a}=0$. \nIf $V$ is not \nequal to $a$\nthere is a formula expressing $Z_{V,a}$ in terms of an embedded\nresolution of $|X_a| \\subset X$\nwhich makes it evident that the\nevaluation at $T=\\infty$ exists.\n \nThe \\define{motivic nearby fiber\n  $\\psi_{V,a}$ of $V$ at $a$} is defined to be\nthe negative of this value at \ninfinity, i.\\,e.\\\n\\begin{equation*}\n  \\psi_{V,a} := -Z_{V,a}(\\infty) \\in\n  \\mathcal{M}^{\\hat{\\upmu}}_{|X_a|}. \n\\end{equation*}\nSee \\eqref{eq:formula-mot-nearby-fiber} below for a formula for\n$\\psi_{V,a}$ in terms of an embedded resolution.\nThe \\define{motivic vanishing cycles of $V$ at $a$} are defined by\n\\begin{equation}\n  \\label{eq:defi-mot-van-fiber}\n  \\phi_{V,a}:= [|X_a| \\xra{\\operatorname{id}} |X_a|] \n  -\\psi_{V,a}\n  \\in\n  \\mathcal{M}^{\\hat{\\upmu}}_{|X_a|}.  \n\\end{equation}\nHere $|X_a|$ is endowed with the trivial\n$\\hat{\\upmu}$-action.\n\n\\begin{remark}\n  \\label{rem:constant-potential}\n  If $V$ is constant we have $\\psi_{V,a}=0$. If $V$ is\n  constant \n  $\\not=a$ we have $\\phi_{V,a}=0$. If $V$ is constant $=a$ we\n  have $X=|X_a|$ and $\\phi_{V,a}=[X \\xra{\\operatorname{id}} X]$.\n\\end{remark}\n\n\\begin{remark}\n  \\label{rem:motivic-fibers-signs} \n  Denef and Loeser choose different signs in the definition of\n  the motivic vanishing cycles. In \n  \\cite{guibert-loeser-merle-convolution}, the motivic nearby\n  fiber (resp.\\ motivic vanishing cycles) \n  of $V$ at $0$ is denoted $\\mathcal{S}_V$ (resp.\\\n  $\\mathcal{S}_V^\\phi$). They are related to our definitions by\n  \\begin{align*}\n   \n    \\psi_{V,a} & = \\mathcal{S}_{V-a},\\\\\n   \n    \\phi_{V,a} & = (-1)^{\\dim X} \\mathcal{S}_{V-a}^\\phi.\n  \\end{align*}\n  Our sign choice for the motivic vanishing cycles is justified in \n  Remark~\\ref{rem:justify-sign-vanishing}.\n\\end{remark}\n\nLet ${\\op{Sing}}(V) \\subset X$ be the closed subscheme defined by the \nvanishing of the section $dV \\in \\Gamma(X, \\Omega^1_{X\/{\\mathsf{k}}})$\nof the cotangent bundle.\nThe closed points of ${\\op{Sing}}(V)$ are the critical\npoints of $V$. Let ${\\op{Crit}}(V) =V({\\op{Sing}}(V)({\\mathsf{k}})) \\subset\n{\\mathbb{A}}^1({\\mathsf{k}})={\\mathsf{k}}$ \nbe the set of critical values of $W$; it is finite by generic\nsmoothness on the target. Trivially we have ${\\op{Sing}}(V) \\cap\nX_a=\\emptyset$ if $a$ is not a critical value.\n\nIf $Z$ is a scheme locally of finite type over ${\\mathsf{k}}$ we denote\nits open\nsubscheme\nconsisting of\nregular points by $Z^{\\op{reg}}$. The closed subset $Z^{\\op{sing}} \\subset Z$ of\nsingular points has a unique structure of a reduced closed\nsubscheme of $Z,$ denoted by $|Z^{\\op{sing}}|$.\n\n\\begin{remark}\n  \\label{rem:SingV-cap-Xa-versus-Xa-sing}\n  If $V=a$ then ${\\op{Sing}}(V) \\cap X_a=X$ and\n  $(X_a)^{\\op{sing}}=\\emptyset$. \n  Otherwise\n \n  the singular\n  points of $X_a$ are precisely the elements of the\n  scheme-theoretic intersection ${\\op{Sing}}(V) \\cap X_a,$ i.\\,e.\\ we\n  have the equality\n  \\begin{equation}\n    \\label{eq:reduced-SingV-cap-Xa-equals-reduced-Xa-sing}\n    |{\\op{Sing}}(V) \\cap X_a| = |(X_a)^{\\op{sing}}|  \n  \\end{equation}\n  of ${\\mathsf{k}}$-varieties. This is\n  trivial if $V$ is constant $\\not=a,$ and otherwise it follows\n  by considering Jacobian matrices.\n \n\\end{remark}\n\nLet us prove that the motivic vanishing cycles $\\phi_{V,a}$ live \nover $|{\\op{Sing}}(V) \\cap X_a|$.\n\n\\begin{proposition}\n  \\label{p:motivic-vanishing-cycles-over-singular-part}\n  We have $\\phi_{V,a} \\in\n  \\mathcal{M}^{\\hat{\\upmu}}_{|{\\op{Sing}}(V) \\cap X_a|}$ canonically.\n\\end{proposition}\n\nTherefore we will often view the motivic vanishing cycles\n$\\phi_{V,a}$ as an \nelement of $\\mathcal{M}^{\\hat{\\upmu}}_{|({\\op{Sing}}(V) \\cap X_a|}$ in\nthe following. \n\n\\begin{proof}\n  If $V$ is constant this follows directly from\n  Remarks~\\ref{rem:constant-potential}\n  and \\ref{rem:SingV-cap-Xa-versus-Xa-sing}\n \n  So let us assume that $V$ is not constant.  \n  As in \\cite[3.3]{denef-loeser-arc-spaces}\n  let\n  $h \\colon R \\rightarrow X$ be an embedded resolution of $|X_a|$\n  so that the ideal sheaf of $h^{-1}(|X_a|)$ is the ideal sheaf of\n  a simple normal crossing divisor\n  (cf.\\ \\cite[Thm.~3.26]{kollar-singularities} or\n  \\cite[Thm.~2.2]{villamayor-on-constructive-desingularization} \n  for existence).\n  Let $E=h^{-1}(X_a)$ be the divisor\n  on $R$ defined by $V \\circ h$.\n  Let\n  $(E_i)_{i \\in \\op{Irr}(|E|)}$ \n  denote the irreducible components of $|E|$.\n  Then $E=\\sum_{i \\in \\op{Irr}(|E|)} m_i E_i$ for unique\n  $m_i \\in \\mathbb{N}_+$. Let $I \\subset \\op{Irr}(|E|)$ be given.  Define\n  $E_I:= \\cap_{i \\in I} E_i$ and\n  $E_I^\\circ:= E_I \\setminus \\bigcup_{j \\in \\op{Irr}(|E|) \\setminus I}\n  E_j$.\n  Let $m_I$ be the greatest common divisor of the $m_i$ for\n  $i \\in I$. Then Denef and Loeser define an unramified Galois\n  cover $\\tildew{E}^\\circ_I \\rightarrow E^\\circ_I$ with Galois group\n  $\\upmu_{m_I}$. They establish the formula\n  \\begin{equation}\n    \\label{eq:formula-mot-nearby-fiber}\n    \\psi_{V,a}= \\mathcal{S}_{V-a} = \\sum_{\\emptyset \\not=I \\subset \\op{Irr}(|E|)}\n    (1-\\mathbb{L})^{|I|-1}[\\tildew{E}^\\circ_I \\rightarrow |X_a|],\n  \\end{equation}\n  see \\cite[Sect. 3.3 and Def.~3.5.3]{denef-loeser-arc-spaces}.\n\n \n  Note that $h$ induces an\n  isomorphism $h^{-1}(U) \\xra{\\sim} U$ where\n  $U:= X - |X_a|^{\\op{sing}}$, by \n  part (ii) of\n  \\cite[Thm.~2.2]{villamayor-on-constructive-desingularization}.\n  We can also deduce this from principalization \n  \\cite[Thm.~3.26]{kollar-singularities} as follows.\n  Since $|X_a|$ has codimension one and $h$ is a\n  composition of blow-ups in smooth centers of codimension two\n  and higher,  \n  $h$ is an isomorphism over an open neighborhood of some regular\n  point \n  of $|X_a|$ if $|X_a|$ is non-empty. \n  Since principalization \n  is \n  functorial with respect to \\'etale morphisms, $h$ is an\n  isomorphism over all regular points of $|X_a|$. \n\n  We obviously have open embeddings $(X_a)^{\\op{reg}} = |(X_a)^{\\op{reg}}| \\subset\n  |X_a|^{\\op{reg}} \\subset |X_a|$ and hence a closed embedding\n  $||X_a|^{\\op{sing}}| \\subset |(X_a)^{\\op{sing}}|$.  \n  Let $U':= X - (X_a)^{\\op{sing}} \\subset U$, so $h^{-1}(U') \\xra{\\sim} U'$\n  is an isomorphism. Over $X_a \\cap U' =(X_a)^{\\op{reg}}$ we obtain the\n  isomorphism \n  \\begin{equation}\n    \\label{eq:36}\n    h \\colon E \\cap h^{-1}(U') \\xra{\\sim} (X_a)^{\\op{reg}},\n  \\end{equation}\n  so $E \\cap h^{-1}(U')$ is regular.\n\n  If $|I|\\geq 2$, then every element $e \\in E_I$ lies in $|E|^{\\op{sing}}\n  \\subset E^{\\op{sing}}$, so $e \\not\\in E \\cap h^{-1}(U')$ and hence\n  $h(e) \\in (X_a)^{\\op{sing}}$. Therefore  \n  $\\tildew{E}^\\circ_I \\rightarrow |X_a|$ factors as \n  $\\tildew{E}^\\circ_I \\rightarrow |(X_a)^{\\op{sing}}| \\subset |X_a|$.\n\n  If $r \\colon (X_a)^{\\op{reg}} = |(X_a)^{\\op{reg}}| \\rightarrow |X_a|$ is the\n  inclusion we hence obtain\n  \\begin{equation*}\n   \n    r^*(\\psi_{V,a}) = \\sum_{i \\in \\op{Irr}(|E|)}\n    [\\tildew{E}^\\circ_i|_{E_i^\\circ \\cap h^{-1}(U')} \\rightarrow (X_a)^{\\op{reg}}].\n  \\end{equation*}\n  If $E_i^\\circ \\cap h^{-1}(U')$ is nonempty then $m_i=1$ because\n  $E \\cap h^{-1}(U')$ is reduced, so $\\tildew{E}^\\circ_i \\rightarrow\n  E^\\circ_i$ is an isomorphism. Moreover, $E \\cap h^{-1}(U')$ is\n  the disjoint union of the \n  $E_i^\\circ \\cap h^{-1}(U')$, for $i \\in \\op{Irr}(|E|)$.\n  These facts and the isomorphism \\eqref{eq:36} imply that\n  \\begin{equation*}\n   \n    r^*(\\psi_{V,a}) \n    = \n    [(X_a)^{\\op{reg}} \\xra{\\operatorname{id}} (X_a)^{\\op{reg}}].\n  \\end{equation*}\n  Hence $r^*(\\phi_{V,a})=0$ by definition\n  \\eqref{eq:defi-mot-van-fiber}.\n  The decomposition\n  $(X_a)^{\\op{reg}} \\subset X_a \\supset |(X_a)^{\\op{sing}}|$ of $X_a$ into an\n  open and a closed reduced subscheme gives rise to a similar\n  decomposition\n  $(X_a)^{\\op{reg}}=|(X_a)^{\\op{reg}}| \\subset |X_a| \\supset |(X_a)^{\\op{sing}}|$\n  of $|X_a|$. \n  Hence Lemma~\\ref{l:open-closed-decomposition} shows that\n  $\\phi_{V,a} \\in \\mathcal{M}_{|(X_a)^{\\op{sing}}|}^{\\hat{\\upmu}}$.\n  Since $V$ is not constant, \n  \\eqref{eq:reduced-SingV-cap-Xa-equals-reduced-Xa-sing} holds\n  true. \n\\end{proof}\n\n\\begin{corollary}\n  \\label{c:mot-van-fiber-zero-if-Xa-smooth}\n  If $V$ is not constant and $X_a$ is smooth\n  then $\\phi_{V,a}=0$. \n\\end{corollary}\n\n\\begin{proof}\n  In this case we have \n  $|{\\op{Sing}}(V) \\cap X_a| = |(X_a)^{\\op{sing}}|= \\emptyset$\n  by \\eqref{eq:reduced-SingV-cap-Xa-equals-reduced-Xa-sing}.\n  More directly, we can take $h=\\operatorname{id}$ as an embedded resolution\n  in the above proof and obtain \n  $\\psi_{V,a}= [|X_a| \\xra{\\operatorname{id}} |X_a|]$ from\n  \\eqref{eq:formula-mot-nearby-fiber} and\n  hence $\\phi_{V,a}=0$. \n\\end{proof}\n\n\\begin{corollary}\n  \\label{c:mot-van-fiber-compactification}\n  Assume that $X$ is a dense open subset of a smooth\n  ${\\mathsf{k}}$-variety $X'$ and that $V \\colon X \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}$ extends\n  to a morphism $V' \\colon X' \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}$ such that all\n  critical points of $V'$ are contained in $X$, i.\\,e.\\\n  ${\\op{Sing}}(V')={\\op{Sing}}(V') \\cap X={\\op{Sing}}(V)$. \n  Then $\\phi_{V,a}=\\phi_{V',a}$.\n\\end{corollary}\n\n\\begin{proof}\n  If $V$ is constant then $V'$ is constant and\n  $X={\\op{Sing}}(V)={\\op{Sing}}(V')=X'$, so the claim is trivial.\n\n  Assume that $V$ is not constant. Then we can assume that the \n  embedded resolution $h \\colon R \\rightarrow X$ of $|X_a|$ from the\n  proof of  \n  Proposition~\\ref{p:motivic-vanishing-cycles-over-singular-part}\n  is the restriction to $X$ of a similar embedded resolution \n  $h' \\colon R' \\rightarrow X'$ \n  of $|X'_a|$.\n  Let $s \\colon |(X_a)^{\\op{sing}}| \\rightarrow |X_a|$ \n  and $s' \\colon |(X'_a)^{\\op{sing}}| \\rightarrow |X'_a|$ denote the closed\n  embeddings. \n  Then $\\phi_{V,a}=s_! s^*(\\phi_{V,a})$ \n  by (the proof of)\n  Proposition~\\ref{p:motivic-vanishing-cycles-over-singular-part}\n  and\n  \\begin{equation*}\n   \n    s^*\\phi_{V,a} = \n    [|(X_a)^{\\op{sing}}| \\rightarrow |(X_a)^{\\op{sing}}|]\n    -s^*(\\psi_{V,a})\n  \\end{equation*}\n  by definition \n  \\eqref{eq:defi-mot-van-fiber}.\n  Similarly, we have $\\phi_{V',a}=s'_! s'^*(\\phi_{V',a})$ and\n  \\begin{equation*}\n   \n    s'^*\\phi_{V',a}= \n    [|(X'_a)^{\\op{sing}}| \\rightarrow |(X'_a)^{\\op{sing}}|]\n    -s'^*(\\psi_{V',a}).\n  \\end{equation*}\n  By assumption and Remark~\\ref{rem:SingV-cap-Xa-versus-Xa-sing}\n  we have $|(X_a)^{\\op{sing}}|=|{\\op{Sing}}(V) \\cap X_a|=|{\\op{Sing}}(V') \\cap\n  X'_a|=|(X'_a)^{\\op{sing}}|$. Therefore it is enough to show that\n  $s^*(\\psi_{V,a})=s'^*(\\psi_{V',a})$. But both\n  expressions have \n  explicit formulas obtained from\n  equation \\eqref{eq:formula-mot-nearby-fiber}; these expressions\n  coincide since the Galois coverings \n  $\\tildew{E}^\\circ_I \\rightarrow E^\\circ_I$ and \n  $\\tildew{E}'^\\circ_{I'} \\rightarrow E'^\\circ_{I'}$ \n  constructed for \n  $h \\colon R \\rightarrow X$ \n  and $h' \\colon R' \\rightarrow X'$ are compatible and give rise to\n  isomorphic varieties \n  with $\\hat{\\upmu}$-action\n  over  \n  $|(X_a)^{\\op{sing}}|=|(X'_a)^{\\op{sing}}|$.\n\\end{proof}\n\n\\section{Motivic Thom-Sebastiani theorem}\n\\label{sec:thom-sebastiani}\n\nLet $V \\colon  X \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}$ be as in the previous\nsection~\\ref{sec:motiv-vanish-fiber}.  \nLet $Y$ be a smooth connected (nonempty) ${\\mathsf{k}}$-variety with a \nmorphism $W \\colon Y \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}$. \nWe define $V \\circledast W$ to be the composition\n\\begin{equation*}\n \n  V \\circledast W \\colon X \\times Y \\xra{V \\times W} {\\mathbb{A}}^1_{\\mathsf{k}} \\times\n  {\\mathbb{A}}^1_{\\mathsf{k}} \\xra{+} \n  {\\mathbb{A}}^1_{\\mathsf{k}}.  \n\\end{equation*}\n\n\\begin{theorem}\n  [{Motivic Thom-Sebastiani Theorem,\n  \\cite[Thm.~5.18]{guibert-loeser-merle-convolution}}] \n  \\label{t:thom-sebastiani-GLM-motivic-vanishing-cycles}\n  Consider morphisms $V \\colon X \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}$ and $W \\colon Y \\rightarrow\n  {\\mathbb{A}}^1_{\\mathsf{k}}$ \n  as above and let $a, b \\in {\\mathsf{k}}$. Let $i_{a,b}$ be the\n  inclusion $|X_a| \\times \n  |Y_b| \\rightarrow |(X \\times Y)_{a+b}|$. \n  Then\n  \\begin{equation}\n    \\label{eq:thom-sebastiani-GLM-motivic-vanishing-cycles}\n    i_{a,b}^*(\\phi_{V \\circledast W,a+b})\n    =\n    \\Psi(\\phi_{V,a} \\times \\phi_{W,b}) \n  \\end{equation}\n  in $\\mathcal{M}^{\\hat{\\upmu}}_{|X_a| \\times |Y_b|}$ where \n  $\\phi_{V,a} \\times \\phi_{W,b}$ is the obvious element of\n  $\\mathcal{M}^{\\hat{\\upmu}\n  \\times \\hat{\\upmu}}_{|X_a| \\times |Y_b|}$.\n\\end{theorem}\n\n\\begin{proof}\n  Using Remark~\\ref{rem:motivic-fibers-signs}\n  this is precisely\n  \\cite[Thm.~5.18]{guibert-loeser-merle-convolution}.  \n\n  We would like to emphasize that\n  \\eqref{eq:thom-sebastiani-GLM-motivic-vanishing-cycles} also\n  holds if $V$ or $W$ is constant. \n  Assume first that both $V$ and $W$ are constant; if $V=a$ and\n  $W=b$ then both sides of\n  \\eqref{eq:thom-sebastiani-GLM-motivic-vanishing-cycles} are\n  equal to $[X \\times Y \\xra{\\operatorname{id}} X \\times Y]$ (use\n  Remark~\\ref{rem:constant-potential} and \n  Example~\\ref{ex:p-product-over-k-trivial-on-one-factor}); otherwise\n  both sides are zero.\n\n  Now assume that $V$ is not constant but $W$ is. If $W\n  \\not= b$ then again both sides of \n  \\eqref{eq:thom-sebastiani-GLM-motivic-vanishing-cycles} are\n  zero. If $W=b$ choose an embedded resolution of $|X_a| \\subset\n  X$ as in the\n  proof of \n  Proposition~\\ref{p:motivic-vanishing-cycles-over-singular-part}\n  and obtain an explicit expression for $\\phi_{V,a}$. If we take\n  the product of \n  this embedded resolution with $Y=Y_b$ we obtain an embedded\n  resolution of $|(X \\times Y)_{a+b}|=|X_a| \\times Y \\subset X\n  \\times Y$ and an explicit expression for $\\phi_{V \\circledast W,a+b}$.\n  Now use again Remark~\\ref{rem:constant-potential} and \n  Example~\\ref{ex:p-product-over-k-trivial-on-one-factor}.\n\\end{proof}\n\nWe want to globalize this theorem.\nSince the set ${\\op{Crit}}(V) \\subset {\\mathsf{k}}$ of\ncritical values of $V$ is finite, we have\n\\begin{equation}\n  \\label{eq:63}\n  |{\\op{Sing}}(V)| = \\coprod_{a \\in {\\op{Crit}}(V)} |{\\op{Sing}}(V) \\cap X_a|.\n\\end{equation}\nProposition~\\ref{p:motivic-vanishing-cycles-over-singular-part}\nshows that we can view \n$\\phi_{V,a}$ as an element of\n$\\mathcal{M}^{\\hat{\\upmu}}_{|{\\op{Sing}}(V)|}$. Define\n\\begin{equation*}\n \n  \\tildew{\\phi}_V:= \\sum_{a \\in {\\op{Crit}}(V)} \\phi_{V,a}\n  \\in \\mathcal{M}^{\\hat{\\upmu}}_{|{\\op{Sing}}(V)|}. \n\\end{equation*}\nOf course we could have equivalently taken the sum over all $a\n\\in {\\mathsf{k}}$,\nby Corollary~\\ref{c:mot-van-fiber-zero-if-Xa-smooth}\nand Remark~\\ref{rem:constant-potential}.\n\n\nWe obviously have\n\\begin{equation}\n  \\label{eq:Sing-V*W}\n  {\\op{Sing}}(V \\circledast W) =   {\\op{Sing}}(V) \\times {\\op{Sing}}(W)\n\\end{equation}\nand hence\n${\\op{Crit}}(W*V)={\\op{Crit}}(W)+{\\op{Crit}}(V):=\\{a+b\\mid a \\in\n{\\op{Crit}}(W), b \\in {\\op{Crit}}(V)\\}$. \n\n\\begin{corollary}\n  [{Global motivic Thom-Sebastiani}]\n  \\label{c:global-thom-sebastiani}\n  Let $V \\colon X \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}$ and $W \\colon Y \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}$\n  be as above. Then\n  \\begin{equation}\n    \\label{eq:global-thom-sebastiani}\n    \\tildew{\\phi}_{V \\circledast W}\n    =\n    \\Psi(\\tildew{\\phi}_V \\times \\tildew{\\phi}_W) \n  \\end{equation}\n  in $\\mathcal{M}^{\\hat{\\upmu}}_{|{\\op{Sing}}(V)| \\times |{\\op{Sing}}(W)|}$ where \n  $\\tildew{\\phi}_V \\times \\tildew{\\phi}_W$ is the obvious element of\n  $\\mathcal{M}^{\\hat{\\upmu}\n  \\times \\hat{\\upmu}}_{|{\\op{Sing}}(V)| \\times |{\\op{Sing}}(W)|}$.\n\\end{corollary}\n\n\\begin{proof}\n  Let $s_a \\colon |{\\op{Sing}}(V) \\cap X_a| \\rightarrow |{\\op{Sing}}(V)|$ be\n  the closed embedding. Then obviously\n  $s_a^*(\\tildew{\\phi})=\\phi_{V,a}$. \n  Define $s'_b$ and $s''_c$ similarly for $W$ and $V \\circledast W$.\n  From \\eqref{eq:63} and \\eqref{eq:Sing-V*W} we see that\n  $|{\\op{Sing}}(V \\circledast W)|$ is the disjoint finite union of its\n  closed  \n  subvarieties $|{\\op{Sing}}(V) \\cap X_a| \\times |{\\op{Sing}}(W) \\cap Y_b|$ where\n  $(a,b) \\in {\\op{Crit}}(V) \\times {\\op{Crit}}(W)$.\n  By Lemma~\\ref{l:open-closed-decomposition} it is therefore\n  enough to show that both sides of\n  \\eqref{eq:global-thom-sebastiani}\n  coincide when restricted to each of these subvarieties.\n  Consider the following commutative diagram\n  \\begin{equation*}\n   \n    \\xymatrix{\n      {|{\\op{Sing}}(V) \\cap X_a| \\times |{\\op{Sing}}(W) \\cap Y_b|}\n      \\ar[d]^-{s_a \\times s'_b} \n      \\ar[r]^-{\\iota_{a,b}}\n      &\n      {|{\\op{Sing}}(V \\circledast W) \\cap (X \\times\n        Y)_{a+b}|}\n      \\ar[d]^-{s''_{a+b}}\\\\\n      {|{\\op{Sing}}(V)| \\times |{\\op{Sing}}(W)|}\n      \\ar@{}[r]|-{=}\n      & \n      {|{\\op{Sing}}(V \\circledast W)|.}\n    }\n  \\end{equation*}\n  If we apply $(s_a \\times s'_b)^*$ to both sides of\n  \\eqref{eq:global-thom-sebastiani}\n  we obtain on the left\n  \\begin{equation*}\n   \n   \n   \n    \\iota_{a,b}^*((s''_{a+b})^*(\\tildew{\\phi}_{V \\circledast W}))\n    = \\iota_{a,b}^*(\\phi_{V \\circledast W,a+b})\n  \\end{equation*}\n  and on the right\n  \\begin{equation*}\n   \n   \n   \n    \\Psi((s_a \\times s'_b)^*(\\tildew{\\phi}_V \\times\n    \\tildew{\\phi}_W)) \n    =\n    \\Psi(s_a^*(\\tildew{\\phi}_V) \\times (s'_b)^*(\\tildew{\\phi}_W)) \n    =\n    \\Psi(\\phi_{V,a} \\times \\phi_{W,b})\n  \\end{equation*}\n  where we use\n  Remark~\\ref{rem:Psi-compatible-with-pushforward-pullback}. \n  But \n  $\\iota_{a,b}^*(\\phi_{V \\circledast W,a+b})\n  = \\Psi(\\phi_{V,a} \\times \\phi_{W,b})$\n  is just a reformulation of\n  Theorem~\\ref{t:thom-sebastiani-GLM-motivic-vanishing-cycles}\n  using \n  Proposition~\\ref{p:motivic-vanishing-cycles-over-singular-part}\n  and Remark~\\ref{rem:Psi-compatible-with-pushforward-pullback}. \n\\end{proof}\n\n\n\\begin{definition}\n  \\label{d:total-motivic-vanishing-cycles}\n  For $V \\colon  X \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}$ as above we define \n \n  \\begin{equation}\n    \\label{eq:phi-V-DA1}\n    (\\phi_V)_{{\\mathbb{A}}^1_{\\mathsf{k}}} := V_!(\\tildew{\\phi}_V)=\\sum_{a \\in {\\mathsf{k}}}\n    V_!(\\phi_{V,a}) \n    \\in \\tilde{\\mathcal{M}}^{\\hat{\\upmu}}_{{\\mathbb{A}}^1_{\\mathsf{k}}} \n  \\end{equation}\n  where we use the isomorphism \\eqref{eq:62}\n  in order to change the target of\n  $V_! \\colon \\mathcal{M}_{|{\\op{Sing}}(V)|}^{\\hat{\\upmu}} \\rightarrow \n  \\mathcal{M}^{\\hat{\\upmu}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}$ to\n  $\\tilde{\\mathcal{M}}^{\\hat{\\upmu}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}$.\n\\end{definition}\n\nRecall the convolution product\n\\eqref{eq:convolution-A1}. \n\n\\begin{corollary}\n  \\label{c:thom-sebastiani-motivic-vanishing-cycles}\n  Let $V \\colon X \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}$ and $W \\colon Y \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}$\n  be as above. \n  Then\n  \\begin{equation*}\n   \n    (\\phi_{V \\circledast W})_{{\\mathbb{A}}^1_{\\mathsf{k}}}\n    =\n    (\\phi_{V})_{{\\mathbb{A}}^1_{\\mathsf{k}}}\n    \\star\n    (\\phi_{W})_{{\\mathbb{A}}^1_{\\mathsf{k}}}\n  \\end{equation*}\n  in\n  $(\\tilde{\\mathcal{M}}^{\\hat{\\upmu}}_{{\\mathbb{A}}^1_{\\mathsf{k}}},\\star)$.\n\\end{corollary}\n\n\\begin{proof}\n  Just apply $(V \\circledast W)_!=\\operatorname{add}_!(V \\times W)_!$ to \n  \\eqref{eq:global-thom-sebastiani} and note that\n  \\begin{multline*}\n   \n    \\operatorname{add}_!((V \\times W)_!(\\Psi(\\tildew{\\phi}_V \\times\n    \\tildew{\\phi}_W)))\n    =\\operatorname{add}_!(\\Psi((V \\times W)_!(\\tildew{\\phi}_V \\times\n    \\tildew{\\phi}_W)))\\\\\n    =\\operatorname{add}_!(\\Psi(V_!(\\tildew{\\phi}_V) \\times\n    W_!(\\tildew{\\phi}_W)))\n    =\n    (\\phi_{V})_{{\\mathbb{A}}^1_{\\mathsf{k}}}\n    \\star\n    (\\phi_{W})_{{\\mathbb{A}}^1_{\\mathsf{k}}}.\n  \\end{multline*}\n  using\n  Remark~\\ref{rem:Psi-compatible-with-pushforward-pullback}. \n\\end{proof}\n\n\nDefine\n$\\ul{\\phi_V}$ as the image of\n$(\\phi_V)_{{\\mathbb{A}}^1_{\\mathsf{k}}}$ under the morphism $\\epsilon_!$ \nof rings from Lemma~\\ref{l:convolution-products-compatible}, i.\\,e.\\\n\\begin{equation}\n  \\label{eq:phi-V-k}\n  \\ul{\\phi_V} := \n  \\epsilon_!((\\phi_V)_{{\\mathbb{A}}^1_{\\mathsf{k}}})\n  =\n  \\sum_{a \\in {\\mathsf{k}}} (\\epsilon_a)_!(\\phi_{V,a})\n  \\in \\mathcal{M}^{\\hat{\\upmu}}_{{\\mathsf{k}}} \n\\end{equation}\nwhere $\\epsilon_a \\colon |{\\op{Sing}}(V) \\cap X_a| \\rightarrow \\operatorname{Spec} {\\mathsf{k}}$\ndenotes the structure morphism.\n\n\\begin{corollary}\n  \\label{c:thom-sebastiani-motivic-vanishing-cycles-over-k}\n  We have\n  $\\ul{\\phi_{V \\circledast W}}\n  =\n  \\ul{\\phi_{V}}\n  *\n  \\ul{\\phi_{W}}$\n  in\n  $\\mathcal{M}^{\\hat{\\upmu}}_{{\\mathsf{k}}}$.\n\\end{corollary}\n\n\\begin{proof}\n  This is obvious from\n  Lemma~\\ref{l:convolution-products-compatible} \n  and Corollary~\\ref{c:thom-sebastiani-motivic-vanishing-cycles}.\n\\end{proof}\n\n\\section{Motivic vanishing cycles measure}\n\\label{sec:motiv-vanish-fiber-1}\n\n\\subsection{Some reminders}\n\\label{sec:some-reminders}\n\nWe recall some facts from\n\\cite[3.7-3.9]{guibert-loeser-merle-convolution}.\nLet $X$ be a smooth connected ${\\mathsf{k}}$-variety and $V\n\\colon X \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}$ a morphism. Let $U \\subset X$ be a dense\nopen subvariety. Then Guibert, Loeser and Merle\ndefine \nin \\cite[Prop.~3.8]{guibert-loeser-merle-convolution}\nan element \n\\begin{equation*}\n \n  \\mathcal{S}_{V, U, X} \\in\n  \\mathcal{M}_{|X_0|}^{\\hat{\\upmu}}  \n\\end{equation*}\n(they denote this element just\nby $\\mathcal{S}_{V,U}$).\n\n\\begin{remark}\n  The remarks at the end of \n  \\cite[3.8]{guibert-loeser-merle-convolution}\n  (and Remarks~\\ref{rem:motivic-fibers-signs} and\n  \\ref{rem:constant-potential})\n  imply:\n  if $U=X$ we have\n  \\begin{equation}\n    \\label{eq:43} \n    \\mathcal{S}_{V, X, X}=\n    \\mathcal{S}_V=\\psi_{V,0};\n  \\end{equation}\n  if $V=0$ we have\n  \\begin{equation*}\n   \n     \\mathcal{S}_{V, U, X}=0=\\psi_{V,0}=\\mathcal{S}_{V}\n  \\end{equation*}\n\\end{remark}\n\n\\begin{theorem}\n  [{\\cite[Thm.~3.9]{guibert-loeser-merle-convolution}}]  \n  \\label{t:GLM-measure}\n  Let $\\alpha \\colon A \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}$ be an ${\\mathbb{A}}^1_{\\mathsf{k}}$-variety. \n  Then there exists a unique $\\mathcal{M}_{\\mathsf{k}}$-linear map\n  \\begin{equation*}\n   \n    \\mathcal{S}_\\alpha^{\\mathcal{M}_A} \\colon \\mathcal{M}_A \\rightarrow\n    \\mathcal{M}^{\\hat{\\upmu}}_{|A_0|} \n  \\end{equation*}\n  such that for every proper morphism $V' \\colon Z \\rightarrow A$\n  where $Z$ \n  is a smooth and connected ${\\mathsf{k}}$-variety, and every dense open\n  subvariety $U$ of $Z$ \n  we have\n  \\begin{equation*}\n   \n    \\mathcal{S}_\\alpha^{\\mathcal{M}_A}([U \\rightarrow A]) =\n    V'_!(\\mathcal{S}_{\\alpha \\circ V', U, X}).\n  \\end{equation*}\n\\end{theorem}\n\nNote that given any morphism $V \\colon U \\rightarrow A$ where $U$ is a smooth\nconnected ${\\mathsf{k}}$-variety, there is a smooth connected\n${\\mathsf{k}}$-variety $Z$ containing $U$ as a dense open subscheme and a\nproper morphism $V' \\colon Z \\rightarrow A$ extending $V$ (use Nagata\ncompactification and resolve the singularities).\n\nIn particular, if $V$ is a proper morphism, then by definition\nof $\\mathcal{S}_\\alpha^{\\mathcal{M}_A}$ and using\n\\eqref{eq:43} we have\n\\begin{equation}\n  \\label{eq:42}\n  \\mathcal{S}_\\alpha^{\\mathcal{M}_A}([U \\xra{V} A]) \n  = V_!(\\mathcal{S}_{\\alpha \\circ V, U, U})\n  = V_!(\\mathcal{S}_{\\alpha \\circ V})\n  = V_!(\\psi_{\\alpha \\circ V,0}).\n\\end{equation}\nIn particular, if $A$ is smooth and connected we obtain \n$\\mathcal{S}_\\alpha^{\\mathcal{M}_A}([A \\xra{\\operatorname{id}} A]) \n= \\mathcal{S}_{\\alpha}=\\psi_{\\alpha,0}$ which justifies the notation\n$\\mathcal{S}_\\alpha^{\\mathcal{M}_A}$.\n\nWe will apply Theorem~\\ref{t:GLM-measure} only in the case that\n$\\alpha$ is a translation ${\\mathbb{A}}^1_{\\mathsf{k}} \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}$, $x \\mapsto\nx-a$, for some $a \\in {\\mathsf{k}}$.\n\n\\subsection{Additivity of the motivic vanishing cycles}\n\\label{sec:addit-vanish-cycl}\n\nTheorem~\\ref{t:GLM-measure} has the following consequence.\n\n\n\\begin{theorem}\n  \\label{t:total-vanishing-cycles-group-homo}\n  There exists a unique $\\mathcal{M}_{\\mathsf{k}}$-linear map\n  \\begin{equation*}\n   \n    \\Phi' \\colon  \\mathcal{M}_{{\\mathbb{A}}^1_{\\mathsf{k}}} \\rightarrow\n    \\tilde{\\mathcal{M}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\hat{\\upmu}} \n  \\end{equation*}\n  such that \n  \\begin{equation*}\n   \n    \\Phi'([X \\xra{V} {\\mathbb{A}}^1_{\\mathsf{k}}]) = (\\phi_{V})_{{\\mathbb{A}}^1_{\\mathsf{k}}}\n  \\end{equation*}\n  for all proper morphisms $V \\colon  X \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}$ of\n  ${\\mathsf{k}}$-varieties \n  where $X$ is smooth over ${\\mathsf{k}}$ and connected\n  (for the definition of $(\\phi_{V})_{{\\mathbb{A}}^1_{\\mathsf{k}}}$ see\n  Definition~\\ref{d:total-motivic-vanishing-cycles}). \n\\end{theorem}\n\n\\begin{proof}\n  Uniqueness is clear \n  since $K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}})$ is generated by the classes of\n  proper morphisms $V \\colon  X \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}$ of ${\\mathsf{k}}$-varieties\n  with $X$ \n  connected and smooth over ${\\mathsf{k}}$ \n  (and relations given by the blowing-up relations), see\n  \\cite[Thm.~5.1]{bittner-euler-characteristic}.\n\n  Let $a \\in {\\mathsf{k}}$ and let\n  $\\gamma_a \\colon |X_a| \\rightarrow \\{a\\}=\\operatorname{Spec} {\\mathsf{k}}$ be\n  the obvious morphism. Apply Theorem~\\ref{t:GLM-measure} to the\n  morphism $\\alpha \\colon {\\mathbb{A}}^1_{\\mathsf{k}} \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}$,\n  $x \\mapsto x-a$. We obtain an $\\mathcal{M}_{\\mathsf{k}}$-linear map\n  $\\mathcal{M}_{{\\mathbb{A}}^1_{\\mathsf{k}}} \\rightarrow \\mathcal{M}_{\\{a\\}}^{\\hat{\\upmu}}$\n  that maps $[V \\colon X \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}]$ to\n  $-(\\gamma_a)_!(\\psi_{V,a})$\n  (use \\eqref{eq:42}; we add a global minus sign)\n  whenever $V \\colon X \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}$\n  is proper with $X$ connected and smooth over ${\\mathsf{k}}$.  \n\n  Obviously there\n  is a unique \n  $\\mathcal{M}_{\\mathsf{k}}$-linear map\n  $\\mathcal{M}_{{\\mathbb{A}}^1_{\\mathsf{k}}} \\rightarrow\n  \\mathcal{M}_{\\{a\\}}^{\\hat{\\upmu}}$\n  mapping $[V \\colon  X \\rightarrow\n  {\\mathbb{A}}^1_{\\mathsf{k}}]$ to \n  $[|X_a| \\rightarrow \\{a\\}]=(\\gamma_a)_!([|X_a| \\rightarrow |X_a|)$\n  for any morphism $V \\colon  X \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}$ of \n  ${\\mathsf{k}}$-varieties.\n  \n  Let $\\Phi'_a \\colon  \\mathcal{M}_{{\\mathbb{A}}^1_{\\mathsf{k}}} \\rightarrow\n  \\mathcal{M}_{\\{a\\}}^{\\hat{\\upmu}}$ be the sum of these two\n  maps. If\n  $V \\colon X \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}$\n  is proper with $X$ connected and smooth over ${\\mathsf{k}}$ we have \n  $\\Phi'_a([V \\colon X \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}]) =(\\gamma_a)_!(\\phi_{V,a})$\n  by the  \n  definition of the motivic vanishing cycles\n  \\eqref{eq:defi-mot-van-fiber}. \n\n  For any $a \\in {\\mathsf{k}},$ let $i_a \\colon \\{a\\} \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}$ be the\n  inclusion. Observe that \n  \\begin{equation*}\n   \n    \\sum_{a \\in {\\mathsf{k}}} (i_a)_!(\\Phi'_a) \\colon \n    \\mathcal{M}_{{\\mathbb{A}}^1_{\\mathsf{k}}} \\rightarrow \\mathcal{M}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\hat{\\upmu}}    \n  \\end{equation*}\n  is\n  well defined since for any given $m \\in \\mathcal{M}_{{\\mathbb{A}}^1_{\\mathsf{k}}}$\n  only finitely many $\\Phi'_a(m)$ are nonzero. \n  The composition of this morphism with the isomorphism \\eqref{eq:62}\n  has the required properties. This proves existence.\n\\end{proof}\n\n\\begin{remark}\n  \\label{rem:Phi'-on-unity-and-A1-0-A1}\n  If $Z$ is a smooth ${\\mathsf{k}}$-variety we have $\\Phi'([Z \\xra{0}\n  {\\mathbb{A}}^1_{\\mathsf{k}}])= [Z \\xra{0} {\\mathbb{A}}^1_{\\mathsf{k}}]$. \n  If $Z$ is proper over ${\\mathsf{k}}$ this follows from \n  Remark~\\ref{rem:constant-potential}. Otherwise we can\n  compactify $Z$ to a smooth proper ${\\mathsf{k}}$-variety $\\ol{Z}$ such\n  that $\\ol{Z}-Z$ is a simple normal crossing divisor and then\n  express the class of $Z$ in terms of $\\ol{Z}$ and the various\n  smooth intersections of the involved smooth prime divisors.\n  \n  In particular we have\n  $\\Phi'([\\operatorname{Spec} {\\mathsf{k}} \\xra{0} {\\mathbb{A}}^1_{\\mathsf{k}}])=[\\operatorname{Spec} {\\mathsf{k}} \\xra{0}\n  {\\mathbb{A}}^1_{\\mathsf{k}}]$\n  and \n  $\\Phi'([{\\mathbb{A}}^1_{\\mathsf{k}} \\xra{0} {\\mathbb{A}}^1_{\\mathsf{k}}])=[{\\mathbb{A}}^1_{\\mathsf{k}} \\xra{0}\n  {\\mathbb{A}}^1_{\\mathsf{k}}]$\n\\end{remark}\n\n\\begin{remark}\n  \\label{rem:justify-sign-vanishing}\n  We \n  keep our promise from\n  Remark~\\ref{rem:motivic-fibers-signs}\n  to justify our sign choice.\n  We do this\n  by showing that\n  Theorem~\\ref{t:total-vanishing-cycles-group-homo} \n  does not hold if the right hand side of\n  \\eqref{eq:phi-V-DA1} \n  is replaced by\n  $\\sum_{a \\in {\\mathsf{k}}} V_!(\\mathcal{S}^\\phi_{V-a})$.\n  Assume that there is morphism $\\Xi \\colon\n  \\mathcal{M}_{{\\mathbb{A}}^1_{\\mathsf{k}}} \\rightarrow\n  \\mathcal{M}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\hat{\\upmu}}\n  \\cong\n  \\tilde{\\mathcal{M}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\hat{\\upmu}}$\n  of abelian groups\n  such that \n  $\\Xi([X \\xra{V} {\\mathbb{A}}^1_{\\mathsf{k}}]) = \n  \\sum_{a \\in {\\mathsf{k}}} V_!(\\mathcal{S}^\\phi_{V-a})$\n  for all proper morphisms $V \\colon  X \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}$ of\n  ${\\mathsf{k}}$-varieties \n  where $X$ is smooth over ${\\mathsf{k}}$ and connected.\n  Remark~\\ref{rem:constant-potential} implies that\n  $\\Xi([Z \\xra{0}\n  {\\mathbb{A}}^1_{\\mathsf{k}}])= (-1)^{\\dim Z}[Z \\xra{0} {\\mathbb{A}}^1_{\\mathsf{k}}]$ for all smooth proper\n  connected ${\\mathsf{k}}$-varieties $Z$.\n\n  Let $X$ be a smooth proper connected 2-dimensional\n  ${\\mathsf{k}}$-variety \n  and $\\tildew{X}$ its blowup in a \n  closed point $Y =\\{x\\} \\subset X$. Let $E$ be the exceptional\n  divisor. We view $X, \\tildew{X}, Y, E$ as ${\\mathbb{A}}^1_{\\mathsf{k}}$-varieties\n  via the zero morphism to ${\\mathbb{A}}^1_{\\mathsf{k}}$.\n  In $K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}})$ we obviously have $[X]-[Y]=\n  [\\tildew{X}] -[E]$. So if we apply $\\Xi$ we obtain\n  $[X]-[Y]=[\\tildew{X}] +[E]$ since $E$ has odd dimension. \n  We obtain $2[E]=0$ in \n  $\\mathcal{M}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\hat{\\upmu}}$.\n  Let us explain why this is a contradiction.\n  Note that $E \\cong\n  {\\mathbb{P}}^1_{\\mathsf{k}}$. Pulling back via the inclusion $\\operatorname{Spec} {\\mathsf{k}} \\xra{0}\n  {\\mathbb{A}}^1_{\\mathsf{k}}$ and forgetting the group action shows that\n  $2[{\\mathbb{P}}^1_{\\mathsf{k}}]=0$ in $\\mathcal{M}_{\\mathsf{k}}$. \n  Taking the topological Euler characteristic with compact\n  support (see \\cite[Example\n  4.3]{nicaise-sebag-grothendieck-ring-of-varieties}) yields the\n  contradiction $4=0$ in $\\mathbb{Z}$.\n\\end{remark}\n\n\\begin{remark}\n  \\label{rem:Phi'-on-smooth-proper-morphisms}\n  If $V \\colon X \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}$ is a smooth and proper morphism\n  then \n  $\\Phi'([X \\xra{V} {\\mathbb{A}}^1_{\\mathsf{k}}])=0$. This follows from\n  Corollary~\\ref{c:mot-van-fiber-zero-if-Xa-smooth};\n  note that\n  $X$ is smooth over ${\\mathsf{k}}$ and $V$ is not constant (if $X$ is\n  nonempty). \n\\end{remark}\n\n\\begin{remark}\n  \\label{rem:Phi'-of-L-A1}\n  We claim that $\\Phi'(\\mathbb{L}_{{\\mathbb{A}}^1_{\\mathsf{k}}})=0$.\n  Indeed, we have\n  \\begin{equation*}\n   \n    \\mathbb{L}_{{\\mathbb{A}}^1_{\\mathsf{k}}}=\n    [{\\mathbb{A}}^1_{{\\mathbb{A}}^1_{\\mathsf{k}}}]=[{\\mathbb{A}}^1_{\\mathsf{k}} \\times {\\mathbb{A}}^1_{\\mathsf{k}} \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}]\n    =\n    [{\\mathbb{A}}^1_{\\mathsf{k}} \\times {\\mathbb{P}}^1_{\\mathsf{k}} \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}]\n    -\n    [{\\mathbb{A}}^1_{\\mathsf{k}} \\times \\operatorname{Spec} {\\mathsf{k}} \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}]\n  \\end{equation*}\n  in $K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}})$. Now apply \n  Remark~\\ref{rem:Phi'-on-smooth-proper-morphisms}.\n  In fact, this argument \n  together with the compactification argument from\n  Remark~\\ref{rem:Phi'-on-unity-and-A1-0-A1}\n  shows: if $Z$ is any smooth\n  ${\\mathsf{k}}$-variety, then $\\Phi'$ maps the class of the projection\n  ${\\mathbb{A}}^1_{\\mathsf{k}} \\times Z \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}$ to zero.\n\\end{remark}\n\n\\begin{remark}\n  \\label{rem:Phi'-cannot-be-ring-morphism}\n  Remark~\\ref{rem:Phi'-of-L-A1} shows that\n  the morphism $\\Phi'$ from\n  Theorem~\\ref{t:total-vanishing-cycles-group-homo}\n  is not a morphism of rings if we consider the usual \n  multiplication on\n  $\\mathcal{M}_{{\\mathbb{A}}^1_{\\mathsf{k}}}$: it\n  maps the invertible element \n  $\\mathbb{L}_{{\\mathbb{A}}^1_{\\mathsf{k}}}$ to zero and hence would be the zero morphism\n  (which it is not, by\n  Remark~\\ref{rem:Phi'-on-unity-and-A1-0-A1}).\n  Therefore it seems presently more appropriate to restrict\n  $\\Phi'$ to  \n  $K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}})$. \n  See however Remark~\\ref{rem:A1-0-A1-sent-to-invertible} below.\n\\end{remark}\n\n\\subsection{The motivic vanishing cycles measure}\n\\label{sec:motiv-vanish-cycl}\n\nWe define $\\Phi$ to be the $K_0({\\op{Var}}_{\\mathsf{k}})$-linear composition\n\\begin{equation}\n  \\label{eq:52}\n  \\Phi \\colon K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}) \\rightarrow \n  \\mathcal{M}_{{\\mathbb{A}}^1_{\\mathsf{k}}} \\xra{\\Phi'}\n  \\tilde{\\mathcal{M}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\hat{\\upmu}} \n\\end{equation}\nwhere the second map is the morphism $\\Phi'$ from\nTheorem~\\ref{t:total-vanishing-cycles-group-homo}.\n\nNow we can state our main theorem which says that $\\Phi$ is a\nring morphism if we equip source \n$K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}})=  K_0({\\op{Var}}^{\\upmu_1}_{{\\mathbb{A}}^1_{\\mathsf{k}}})$ and\ntarget $\\tilde{\\mathcal{M}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\hat{\\upmu}}$ \nwith the convolution product $\\star$ \nfrom\nsection~\\ref{sec:conv-vari-over}, see in particular\nRemark~\\ref{rem:no-group-action} and Proposition~\\ref{p:A1-convolution-comm-ass-unit}.\n\n\\begin{theorem}\n  \\label{t:total-motivic-vanishing-cycles-ring-homo}\n  The map \\eqref{eq:52}\n  from Theorem~\\ref{t:total-vanishing-cycles-group-homo}\n  is a morphism \n  \\begin{equation*}\n    \\Phi \\colon  (K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}), \\star) \\rightarrow\n    (\\tilde{\\mathcal{M}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\hat{\\upmu}},\\star) \n  \\end{equation*}\n  of $K_0({\\op{Var}}_{\\mathsf{k}})$-algebras.\n  By composing with\n  \\eqref{epsilon!-star-*}\n  we obtain a morphism\n  \\begin{equation}\n    \\label{eq:epsilon-Phi}\n    \\epsilon_! \\circ \\Phi \\colon  (K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}), \\star) \\rightarrow\n    (\\mathcal{M}_{{\\mathsf{k}}}^{\\hat{\\upmu}},*) \n  \\end{equation}\n  of $K_0({\\op{Var}}_{\\mathsf{k}})$-algebras. We call these two morphisms\n  \\textbf{motivic vanishing cycles measures}.\n\\end{theorem}\n\n\\begin{proof}\n  The second claim is obvious from\n  Lemma~\\ref{l:convolution-products-compatible}, so let us prove\n  the first claim.\n  Remark~\\ref{rem:Phi'-on-unity-and-A1-0-A1} shows that $\\Phi$ maps\n  the identity element to the identity element.\n  Remark~\\ref{rem:Phi'-on-unity-and-A1-0-A1} shows that $\\Phi$ is\n  compatible with the algebra structure maps, cf.\\ \\eqref{eq:46}.\n\n  We use that $K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}})$ is generated by the classes of\n  projective morphisms $V \\colon  X \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}$ with $X$\n  a connected quasi-projective ${\\mathsf{k}}$-variety that is smooth over\n  ${\\mathsf{k}}$ \n  (and relations given by the blowing-up relations), see\n  \\cite[Thm.~5.1]{bittner-euler-characteristic}.\n\n  So let $X$ and $Y$ be connected quasi-projective ${\\mathsf{k}}$-varieties that\n  are smooth over ${\\mathsf{k}}$ and let $V \\colon X \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}$ and $W \\colon Y \\rightarrow\n  {\\mathbb{A}}^1_{\\mathsf{k}}$ be projective morphisms.\n  Then we know by\n  Theorem~\\ref{t:total-vanishing-cycles-group-homo}\n  and\n  Corollary\n  \\ref{c:thom-sebastiani-motivic-vanishing-cycles}\n  that\n  \\begin{equation*}\n    \\Phi([X \\xra{V} {\\mathbb{A}}^1_{\\mathsf{k}}]) \\star\n    \\Phi([Y \\xra{W} {\\mathbb{A}}^1_{\\mathsf{k}}])\n    =\n    (\\phi_{V})_{{\\mathbb{A}}^1_{\\mathsf{k}}}\n    \\star\n    (\\phi_{W})_{{\\mathbb{A}}^1_{\\mathsf{k}}}\n    = (\\phi_{V \\circledast W})_{{\\mathbb{A}}^1_{\\mathsf{k}}}.\n  \\end{equation*}\n  Our aim is to show that\n  \\begin{equation*}\n   \n    (\\phi_{V \\circledast W})_{{\\mathbb{A}}^1_{\\mathsf{k}}}=  \n    \\Phi([X \\times Y \\xra{V \\circledast W} {\\mathbb{A}}^1_{\\mathsf{k}}]).\n  \\end{equation*}\n  This is not obvious since $V \\circledast W \\colon  X \\times Y \\rightarrow\n  {\\mathbb{A}}^1_{\\mathsf{k}}$ is not \n  proper in general. \n\n  We apply Proposition~\\ref{p:compactification} below and\n  use notation from there.\n  We obtain the equality\n  \\begin{multline*}\n    [X\\times Y \\xra{V \\circledast W} {\\mathbb{A}}^1_{\\mathsf{k}}]\n    =\n    [Z \\xra{h} {\\mathbb{A}}^1_{\\mathsf{k}}]\n    -\\sum_{i}[D_i\\xra{h_i} {\\mathbb{A}}^1_{\\mathsf{k}}]\n    +\\sum_{i<j}[D_{ij} \\xra{h_{ij}} {\\mathbb{A}}^1_{\\mathsf{k}}]\\\\\n    - \\dots + (-1)^s[D_{12 \\dots s} \\xra{h_{12 \\dots s}} {\\mathbb{A}}^1_{\\mathsf{k}}]\n  \\end{multline*}\n  in $K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}})$.\n  On the right-hand side, $Z$ and all $D_{i_1 \\dots i_p}$ \n  are smooth quasi-projective ${\\mathsf{k}}$-varieties, $h$ is a projective\n  morphism, and all \n  $h_{i_1 \\dots i_p}$ are projective and smooth morphisms, \n  by part\n  \\ref{enum:compactification-intersections-proper-and-smooth} \n  of\n  Proposition~\\ref{p:compactification}.\n  Hence we can compute $\\Phi([X\\times Y \\xra{V \\circledast W}\n  {\\mathbb{A}}^1_{\\mathsf{k}}]$ now. \n  Remark~\\ref{rem:Phi'-on-smooth-proper-morphisms}\n  shows that\n  $\\Phi$ vanishes\n  on all\n  $[D_{i_1 \\dots i_p}\\xra{h_{i_1 \\dots i_p}} {\\mathbb{A}}^1_{\\mathsf{k}}]$. We\n  obtain\n  \\begin{equation*}\n    \\Phi([X\\times Y \\xra{V \\circledast W} {\\mathbb{A}}^1_{\\mathsf{k}}])=\\Phi([Z \\xra{h}\n    {\\mathbb{A}}^1_{\\mathsf{k}}]) = (\\phi_{h})_{{\\mathbb{A}}^1_{\\mathsf{k}}}\n  \\end{equation*}\n  and are left to show that\n  \\begin{equation*}\n    (\\phi_{h})_{{\\mathbb{A}}^1_{\\mathsf{k}}} = (\\phi_{V \\circledast W})_{{\\mathbb{A}}^1_{\\mathsf{k}}}.\n  \\end{equation*}\n  But this holds true by \n  Corollary~\\ref{c:mot-van-fiber-compactification}\n  which we can apply by\n  parts\n  \\ref{enum:compactification-diagram},\n  \\ref{enum:compactification-critical-points},\n  \\ref{enum:compactification-boundary-snc}\n  of Proposition~\\ref{p:compactification}.\n\\end{proof}\n\n\\begin{remark}\n  \\label{rem:evaluate-on-product}\n  If $X$ and $Y$ are smooth connected ${\\mathsf{k}}$-varieties and $V\n  \\colon X \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}$ and $W \\colon Y \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}$ are\n  proper morphisms then\n  Theorems~\\ref{t:total-motivic-vanishing-cycles-ring-homo}\n  and \\ref{t:thom-sebastiani-GLM-motivic-vanishing-cycles}\n  show that\n  \\begin{equation*}\n   \n    \\Phi([X \\times Y \\xra{V \\circledast W} {\\mathbb{A}}^1_{\\mathsf{k}}]) =\n    \\Phi([X \\xra{V} {\\mathbb{A}}^1_{\\mathsf{k}}]) \\ast \\Phi([Y \\xra{W} {\\mathbb{A}}^1_{\\mathsf{k}}]) \n    =\n    (\\phi_{V})_{{\\mathbb{A}}^1_{\\mathsf{k}}}\n    \\ast \n    (\\phi_{V})_{{\\mathbb{A}}^1_{\\mathsf{k}}}\n    =\n    (\\phi_{V \\circledast W})_{{\\mathbb{A}}^1_{\\mathsf{k}}}.\n  \\end{equation*}\n  So even though $V \\circledast W$ might not be proper, the\n  motivic vanishing cycles measure $\\Phi$ sends it to \n  $(\\phi_{V \\circledast W})_{{\\mathbb{A}}^1_{\\mathsf{k}}}$. \n \n \n \n\\end{remark}\n\n\\begin{remark}\n  \\label{rem:A1-0-A1-sent-to-invertible}\n  Recall the element\n  $\\mathbb{L}_{({\\mathbb{A}}^1_{\\mathsf{k}},0)}=[{\\mathbb{A}}^1_{\\mathsf{k}} \\xra{0} {\\mathbb{A}}^1_{\\mathsf{k}}] \\in\n  K_0({\\op{Var}}^{\\upmu_n}_{{\\mathbb{A}}^1_{\\mathsf{k}}})$\n  defined in \\eqref{eq:def-L-A1-0}. For $n=1$ it is an element of\n  $K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}})$ and we have\n  $(K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}), \\star)[(\\mathbb{L}_{({\\mathbb{A}}^1_{\\mathsf{k}},0)})^{-1}]=\n  (\\tilde{\\mathcal{M}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}, \\star)$.\n  Remark~\\ref{rem:Phi'-on-unity-and-A1-0-A1}  \n  shows that $\\Phi(\\mathbb{L}_{({\\mathbb{A}}^1_{\\mathsf{k}},0)})= \\mathbb{L}_{({\\mathbb{A}}^1_{\\mathsf{k}},0)}$\n  where we view $\\mathbb{L}_{({\\mathbb{A}}^1_{\\mathsf{k}},0)}$ in the obvious way as an element of\n  $K_0({\\op{Var}}^{\\hat{\\upmu}}_{{\\mathbb{A}}^1_{\\mathsf{k}}})$. \n  Therefore, $\\Phi$ factors as the composition\n  \\begin{equation*}\n   \n    \\Phi \\colon  (K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}), \\star) \n    \\rightarrow \n    (\\tilde{\\mathcal{M}}_{{\\mathbb{A}}^1_{\\mathsf{k}}},\\star) \n    \\rightarrow\n    (\\tilde{\\mathcal{M}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}^{\\hat{\\upmu}},\\star). \n  \\end{equation*}\n  The second map is a morphism of $\\mathcal{M}_{\\mathsf{k}}$-algebras.\n  It is, up to the isomorphism\n  $\\mathcal{M}_{{\\mathbb{A}}^1_{\\mathsf{k}}}\n  \\xra{\\sim}\n  \\tilde{\\mathcal{M}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}$\n  from \n  \\eqref{eq:71} (for $n=1$), the morphism\n  $\\Phi'$ from \n  Theorem~\\ref{t:total-vanishing-cycles-group-homo}.\n  This makes up for\n  Remark~\\ref{rem:Phi'-cannot-be-ring-morphism}.\n  As observed in Remarks~\\ref{rem:Phi'-on-smooth-proper-morphisms}\n  and~\\ref{rem:Phi'-of-L-A1}, \n  $\\Phi$ vanishes on many other elements, for example on\n  $[{\\mathbb{A}}^1_{\\mathsf{k}} \\xra{\\operatorname{id}} {\\mathbb{A}}^1_{\\mathsf{k}}]$ or on\n  $\\mathbb{L}_{{\\mathbb{A}}^1_{\\mathsf{k}}}$.\n\\end{remark}\n\n\\subsection{Compactification}\n\\label{sec:compactification}\n\nFor the convenience of the reader we recall our compactification\nresult from \\cite{valery-olaf-matfak-motmeas}.\n\n\\begin{proposition}\n  [{\\cite[Prop.~6.1]{valery-olaf-matfak-motmeas}}]\n  \\label{p:compactification}\n  Let ${\\mathsf{k}}$ be an algebraically closed field of characteristic zero.\n  Let $X$ and $Y$ be smooth \n  ${\\mathsf{k}}$-varieties and let\n  $V \\colon  X \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}$ and $W \\colon Y\\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}$ be \n  projective\n  morphisms (hence $X$ and $Y$ are\n  quasi-projective ${\\mathsf{k}}$-varieties).\n  Consider the convolution\n  \\begin{equation*}\n    V \\circledast W \\colon X\\times Y\\xra{V \\times W} {\\mathbb{A}}^1_{\\mathsf{k}}\\times\n    {\\mathbb{A}}^1_{\\mathsf{k}}\\xra{+} {\\mathbb{A}}^1_{\\mathsf{k}}. \n  \\end{equation*}\n  Then there exists a smooth quasi-projective ${\\mathsf{k}}$-variety $Z$ with\n  an open embedding $X\\times Y \\hookrightarrow Z$ and a\n  projective morphism $h \\colon Z\\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}$ such that the\n  following conditions are satisfied.\n  \\begin{enumerate}[label=(\\roman*)]\n  \\item\n    \\label{enum:compactification-diagram}\n    The diagram\n    \\begin{equation*}\n      \\xymatrix{\n        {X\\times Y} \n        \\ar@{^{(}->}[r]\n        \\ar[d]^-{V \\circledast W} & {Z} \\ar[d]^-h\\\\\n        {{\\mathbb{A}}^1_{\\mathsf{k}}} \\ar@{}[r]|-{=} & {{\\mathbb{A}}^1_{\\mathsf{k}}}\n      }\n    \\end{equation*}\n    commutes.\n  \\item\n    \\label{enum:compactification-critical-points}\n    All critical points of $h$ are contained\n    in $X\\times Y,$ i.\\,e.\\ ${\\op{Sing}}(V \\circledast W)={\\op{Sing}}(h)$.\n  \\item\n    \\label{enum:compactification-boundary-snc}\n    We have $Z \\setminus X \\times Y =  \\bigcup_{i=1}^s D_i$ where\n    the\n    $D_i$ are pairwise distinct smooth\n    prime divisors.\n    More precisely, $Z \\setminus X \\times Y$ is the support\n    of a simple normal crossing divisor.\n  \\item\n    \\label{enum:compactification-intersections-proper-and-smooth}\n    For every $p$-tuple $(i_1, \\dots, i_p)$ of indices\n    (with $p \\geq 1$) the morphism\n    \\begin{equation*}\n      h_{i_1 \\dots i_p} \\colon  D_{i_1 \\dots i_p}:=D_{i_1} \\cap \\dots \\cap\n      D_{i_p} \\rightarrow {\\mathbb{A}}^1_{\\mathsf{k}}   \n    \\end{equation*}\n    induced by $h$ is projective and smooth. In\n    particular, all\n    $D_{i_1 \\dots i_p}$ are smooth quasi-projective ${\\mathsf{k}}$-varieties.\n  \\end{enumerate}\n\\end{proposition}\n\n\\section{Comparison with the matrix factorization motivic\n  measure} \n\\label{sec:comp-with-matr}\n\n\nWe would like to place\nTheorem~\\ref{t:total-motivic-vanishing-cycles-ring-homo} \nin a certain context and compare the motivic measure $\\Phi$ \nor rather $\\epsilon_! \\circ \\Phi$ with\nanother motivic measure of a different nature. \n\n\\subsection{Categorical motivic measures}\n\\label{sec:categorical-measures}\n\nFirst let us recall the {\\it categorical} measure\n\\begin{equation*}\n  \\nu \\colon K_0({\\op{Var}}_{\\mathsf{k}})\\rightarrow K_0(\\op{sat}^\\mathbb{Z}_{\\mathsf{k}})  \n\\end{equation*}\nconstructed in \\cite{bondal-larsen-lunts-grothendieck-ring}.\nHere $K_0(\\op{sat}^\\mathbb{Z}_{\\mathsf{k}})$ is the free abelian group generated by\nquasi-equivalence classes of saturated differential\n$\\mathbb{Z}$-graded \n${\\mathsf{k}}$-categories with relations coming from semiorthogonal\ndecompositions into admissible subcategories on the level of\nhomotopy categories. The map $\\nu $ sends the class $[X]$ of a\nsmooth projective ${\\mathsf{k}}$-variety $X$ to the class\n$[{\\operatorname{D}}^{\\op{b}}({\\op{Coh}}(X))]$ \nof (a suitable\ndifferential $\\mathbb{Z}$-graded ${\\mathsf{k}}$-enhancement of) its derived\ncategory\n${\\operatorname{D}}^{\\op{b}}({\\op{Coh}}(X))$. The\ntensor product of differential $\\mathbb{Z}$-graded ${\\mathsf{k}}$-categories induces a ring structure on\n$K_0(\\op{sat}^\\mathbb{Z}_{\\mathsf{k}})$ and $\\nu $ is a ring homomorphism.  In recent\npapers\n\\cite{valery-olaf-matfak-semi-orth-decomp,\n  valery-olaf-matfak-motmeas} we have constructed a motivic\nmeasure\n\\begin{equation*}\n  \\mu \\colon \n  (K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}), \\star)  \n \n  \\rightarrow K_0(\\op{sat}_{\\mathsf{k}}^{{\\mathbb{Z}}_2})  \n\\end{equation*}\nwhich is a relative analogue of the measure $\\nu $. Here\n$K_0(\\op{sat}_{\\mathsf{k}}^{{\\mathbb{Z}}_2})$ is defined in exactly the same way as\n$K_0(\\op{sat}^\\mathbb{Z}_{\\mathsf{k}})$ except that this time we consider saturated\ndifferential ${{\\mathbb{Z}}_2}$-graded ${\\mathsf{k}}$-categories. If\n$[X \\xra{W} {{\\mathbb{A}}}^1_{\\mathsf{k}}]\\in K_0({\\op{Var}}_{{{\\mathbb{A}}}^1_{\\mathsf{k}}})$ where $X$ is\nsmooth over ${\\mathsf{k}}$ and $W$ is proper, then $\\mu([X\\xra{W}{{\\mathbb{A}}}^1_{\\mathsf{k}}])$\nis defined as the class \n$[{\\mathbf{MF}}(W)]$\nof (a suitable differential $\\mathbb{Z}_2$-graded ${\\mathsf{k}}$-enhancement of)\nthe category\n\\begin{equation*}\n  {\\mathbf{MF}}(W):=\\prod_{a\\in k}{\\mathbf{MF}}(X,W-a)^\\natural  \n\\end{equation*}\nHere ${\\mathbf{MF}}(X,W-a)^\\natural$ is the Karoubi envelope of the\ncategory ${\\mathbf{MF}}(X,W-a)$ of matrix factorizations of the potential\n$W-a$.\n\nThe measures $\\nu$ and $\\mu$ are related by the commutative\ndiagram of ring homomorphisms \n(as announced in the introduction of \\cite{valery-olaf-matfak-semi-orth-decomp})\n\\begin{equation}\n  \\label{eq:folding-diagram}\n  \\xymatrix{\n    {K_0({\\op{Var}}_{\\mathsf{k}})} \\ar[r]^-{\\nu} \\ar[d] \n    \\ar[d]\n    & {K_0(\\op{sat}_{\\mathsf{k}}^{\\mathbb{Z}})} \\ar[d]\\\\\n    {(K_0({\\op{Var}}_{{\\mathbb{A}}^1_{\\mathsf{k}}}), \\star)} \\ar[r]^-{\\mu} \n    & {K_0(\\op{sat}_{\\mathsf{k}}^{\\mathbb{Z}_2})}\n  }\n\\end{equation}\nwhere $K_0({\\op{Var}}_{\\mathsf{k}}) \\rightarrow  K_0({\\op{Var}}_{{{\\mathbb{A}}}^1_{\\mathsf{k}}})$ is the ring\nhomomorphism \\eqref{eq:46} (for $n=1$)\nand\n$K_0(\\op{sat}^\\mathbb{Z}_{\\mathsf{k}})  \\rightarrow  K_0(\\op{sat}_{\\mathsf{k}}^{{\\mathbb{Z}}_2})$ is induced from\nthe {\\it folding} \n(see \\cite{olaf-folding-derived-categories-in-prep})\nwhich assigns to a differential ${\\mathbb{Z}}$-graded ${\\mathsf{k}}$-category the\ncorresponding differential ${{\\mathbb{Z}}}_2$-graded ${\\mathsf{k}}$-category. \n\n\\subsection{Comparing vanishing cycles and matrix factorization\n  measures}\n\\label{sec:comp-vanish-cycl}\n\n\n\nTo each saturated differential ${{\\mathbb{Z}}}_2$-graded ${\\mathsf{k}}$-category $A$\none assigns the finite dimensional ${{\\mathbb{Z}}}_2$-graded vector\nspace\n\\begin{equation*}\n  {\\operatorname{HP}}(A)={\\operatorname{HP}}_0(A)\\oplus {\\operatorname{HP}}_1(A)  \n\\end{equation*}\nover the Laurent power series field ${\\mathsf{k}}((u))$ - the periodic\ncyclic homology of $A$ (see \\cite{keller-invariance}).\n\nPut\n$\\chi_{{\\operatorname{HP}}}(A):=\\dim_{{\\mathsf{k}}((u))}{\\operatorname{HP}}_0(A)-\\dim_{{\\mathsf{k}}((u))}{\\operatorname{HP}}_1(A)$. Since\n${\\operatorname{HP}}$ is additive on semiorthogonal decompositions of\ntriangulated categories (see\n\\cite{keller-cyclic-homology-exact-cat}) \nthis\nassignment descends to a group homomorphism\n\\begin{equation*}\n  \\chi_{{\\operatorname{HP}}}\\colon K_0(\\op{sat}_{k}^{{\\mathbb{Z}}_2})\\rightarrow {\\mathbb{Z}}  \n\\end{equation*}\nBecause of the K\\\"unneth property for ${\\operatorname{HP}}$ \n(see \\cite{shklyarov-hodge-theoretic-property-kuenneth} and references therein) the map $\\chi_{{\\operatorname{HP}}}$ is in fact a ring homomorphism.\n\nOn the other hand, if ${\\mathsf{k}}={\\mathbb{C}}$ we have the usual ring\nhomomorphism (see \\cite{looijenga-motivic-measures})\n\\begin{equation}\n  \\label{eq:chi_c}\n  \\chi_{\\op{c}}:=\n  \\sum (-1)^i\\dim {\\operatorname{H}}_{\\op{c}}^i \\ \\colon \\mathcal{M}_{\\mathbb{C}}\\rightarrow {\\mathbb{Z}}\n\\end{equation}\nNotice that $\\chi_{\\op{c}}({\\mathbb{L}})=1$, hence $\\chi_{\\op{c}}$ is indeed well-defined as a homomorphism from\n$\\mathcal{M}_{\\mathbb{C}}$.\n\nForgetting the action of $\\hat{\\upmu}$ obviously defines a \nmap\n\\begin{equation}\n  \\label{eq:14}\n  \\mathcal{M}_\\mathbb{C}^{\\hat{\\upmu}} \\rightarrow \\mathcal{M}_\\mathbb{C}\n\\end{equation}\nof $\\mathcal{M}_\\mathbb{C}$-modules.\nClearly, this map is a ring\nhomomorphism if we equip its source with the usual\nmultiplication.\nHowever, this is not true if we equip its source with the \nconvolution product $*$ as we will explain in\nLemma~\\ref{l:forgetful-convolution} below.\nNevertheless we have the following result.\n\n\\begin{proposition}\n  \\label{p:chi_c-covolution}\n  The composition of $\\chi_{\\op{c}}$ (see \\eqref{eq:chi_c}) with the map ``forget the\n  \\mbox{$\\hat{\\upmu}$-action}''\n  \\eqref{eq:14}\n  defines a ring homomorphism\n  \\begin{equation}\n    \\label{eq:72}\n    \\chi_{\\op{c}} \\colon (\\mathcal{M}_\\mathbb{C}^{\\hat{\\upmu}},*) \\rightarrow \\mathbb{Z}\n  \\end{equation}\n  which we denote again by $\\chi_{\\op{c}}$.\n\\end{proposition}\n\n\\begin{proof}\n  Let $A$ and $B$ be complex varieties\n  with\n  a good $\\upmu_n$-action for some $n \\geq 1$. We need to show\n  that $A \\times B$ and \n  \\begin{equation*}\n   \n    [A]*[B] = \n    [((A \\times^{\\upmu_n} \\underset{x}{{\\DG_\\op{m}}})\n    \\times \n    (B \\times^{\\upmu_n} \\underset{y}{{\\DG_\\op{m}}}))|_{x^n+y^n=0}]\n    -[((A \\times^{\\upmu_n} \\underset{x}{{\\DG_\\op{m}}})\n    \\times \n    (B \\times^{\\upmu_n} \\underset{y}{{\\DG_\\op{m}}}))|_{x^n+y^n=1}]\n  \\end{equation*}\n  (see\n  \\eqref{eq:convolution-explicit})\n \n  have the same Euler characteristic\n  with compact support.\n  Since ${\\DG_\\op{m}} \\rightarrow {\\DG_\\op{m}}$, $x \\mapsto x^n$, is a $\\upmu_n$-torsor\n  (in the \\'etale topology) we have a pullback square\n  \\begin{equation*}\n    \\xymatrix{\n      {A \\times {\\DG_\\op{m}}}  \n      \\ar[rr]^-{(a,x) \\mapsto x}\n      \\ar[d]\n      &&\n      {{\\DG_\\op{m}}} \n      \\ar[d]^-{x \\mapsto x^n}\n      \\\\\n      {A \\times^{\\upmu_n} {\\DG_\\op{m}}}  \n      \\ar[rr]^-{[a,x] \\mapsto x^n}\n      &&\n      {{\\DG_\\op{m}}.}\n    }\n  \\end{equation*}\n  Its lower horizontal morphism is an \\'etale-locally\n  trivial fibration with fiber $A$. Therefore it is a locally\n  trivial fibration if we pass to the analytic topologies.\n  In this way we obtain a locally trivial fibration\n  \\begin{equation*}\n    f \\colon (A^{\\op{an}} \\times^{\\upmu_n(\\mathbb{C})} \\mathbb{C}^\\times)\n    \\times \n    (B^{\\op{an}} \\times^{\\upmu_n(\\mathbb{C})} \\mathbb{C}^\\times)\n    \\xra{([a,x],[b,y]) \\mapsto (x^n,y^n)} \\mathbb{C}^\\times \\times \\mathbb{C}^\\times\n  \\end{equation*}\n  with fiber $A^{\\op{an}} \\times B^{\\op{an}}$. \n  Consider the subsets $N:=\\{x'+y'=0\\} \\cong \\mathbb{C}^\\times$ and \n  $E:=\\{x'+y'=1\\} \\cong \\mathbb{C}^\\times -\\{1\\}$ of the base\n  of this\n  fibration where\n  $x'$ and $y'$ are the obvious coordinates.\n  Then\n  \\begin{multline*}\n    \\chi_{\\op{c}}([A]*[B]) =\n    \\chi_{\\op{c}}(f^{-1}(N))-\\chi_{\\op{c}}(f^{-1}(E))\\\\\n    =\n    \\chi_{\\op{c}}(A^{\\op{an}} \\times B^{\\op{an}})\n    (\\chi_{\\op{c}}(N) - \\chi_{\\op{c}}(E))\n    =\\chi_{\\op{c}}(A^{\\op{an}} \\times B^{\\op{an}}).\n  \\end{multline*}\n  This proves what we need.\n\\end{proof}\n\nAlthough not strictly needed for our purposes we would like to\ninclude the following result (which is also true for ${\\mathsf{k}}$\ninstead of $\\mathbb{C}$).\n\n\\begin{lemma}\n  \\label{l:forgetful-convolution}\n  The map ``forget the \\mbox{$\\hat{\\upmu}$-action}''\n  $f \\colon \\mathcal{M}_\\mathbb{C}^{\\hat{\\upmu}} \\rightarrow \\mathcal{M}_\\mathbb{C}$\n  (see\n  \\eqref{eq:14}) does not define a\n  ring homomorphisms \n  $(\\mathcal{M}_\\mathbb{C}^{\\hat{\\upmu}},*) \\rightarrow \\mathcal{M}_\\mathbb{C}$.\n \n \n \n \n \n \n \n  %\n \n\\end{lemma}\n\n\\begin{proof}\n  Let $M=\\upmu_2 \\in {\\op{Var}}^{\\upmu_2}_\\mathbb{C}$ with obvious action of\n  $\\upmu_2$.\n  We claim that $f([M] * [M]) \\not= f([M])f([M])$.\n\n  We clearly have $f([M])f([M])=4 [\\operatorname{Spec} \\mathbb{C}]=4$.\n  On the other hand multiplication defines an isomorphism\n  $M \\times^{\\upmu_2} {\\DG_\\op{m}} \\xra{\\sim} {\\DG_\\op{m}}$ and therefore\n  \\eqref{eq:convolution-explicit} yields\n  \\begin{equation*}\n    [M] * [M] = \n    [(\\underset{x}{{\\DG_\\op{m}}} \\times \n      \\underset{y}{{\\DG_\\op{m}}})|_{x^2+y^2=0}]\n      -[(\\underset{x}{{\\DG_\\op{m}}} \\times \n      \\underset{y}{{\\DG_\\op{m}}})|_{x^2+y^2=1}]\n  \\end{equation*}\n  The $\\upmu_2$-action on ${\\DG_\\op{m}} \\times {\\DG_\\op{m}}$ is the diagonal action.\n  Instead of using the coordinates $(x,y)$ on ${\\mathbb{A}}^2_\\mathbb{C}$ let us\n  use the coordinates $(a,b)$ where $a=x+iy$ and $b=x-iy$.\n \n  Then $x^2+y^2=ab$ and the conditions $x \\not= 0$ and $y\\not=0$\n  are equivalent to $a+b \\not=0$ and $a-b \\not=0$. \n  Hence\n  \\begin{equation*}\n    [M] * [M] \n    = \n    [\\underset{(a,b)}{{\\mathbb{A}}^2_\\mathbb{C}}|_{ab=0,\\; a \\not=\\pm b}]\n    -[\\underset{(a,b)}{{\\mathbb{A}}^2_\\mathbb{C}}|_{ab=1,\\; a \\not=\\pm b}]\n  \\end{equation*}\n  The $\\upmu_2$-action on ${\\mathbb{A}}^2_\\mathbb{C}$ is again the diagonal\n  action.\n  The first summand is the coordinate cross without the origin\n  and equal to $2[{\\DG_\\op{m}}]$ with obvious $\\upmu_2$-action. \n  To treat the second summand note that the map $({\\DG_\\op{m}} - \\upmu_4)\n  \\rightarrow  \n  {{\\mathbb{A}}^2_\\mathbb{C}}|_{ab=1, a \\not=\\pm b}$, $a \\mapsto (a, a^{-1})$\n  defines a $\\upmu_2$-equivariant isomorphism.\n  Hence\n  \\begin{equation}\n    \\label{eq:mu2-convolution-squared}\n    [M] * [M] = 2[{\\DG_\\op{m}}]-[{\\DG_\\op{m}}]+[\\upmu_4]=[{\\DG_\\op{m}}]+2[\\upmu_2]\n  \\end{equation}\n  and $f([M] * [M])=[{\\DG_\\op{m}}]+4$.\n\n  But the element $[{\\DG_\\op{m}}]=f([M] * [M])-f([M])f([M])$ is certainly\n  not zero in $\\mathcal{M}_\\mathbb{C}$: taking the  \n  Hodge-Deligne polynomial defines a ring homomorphism\n  $\\mathcal{M}_\\mathbb{C} \\rightarrow \\mathbb{Z}[u,v,u^{-1},v^{-1}]$ which sends\n  $[{\\DG_\\op{m}}]$ to $uv-1$, cf.\\ \\cite[Example\n  4.6]{nicaise-sebag-grothendieck-ring-of-varieties}. \n\\end{proof}\n\n\\begin{theorem} \n  \\label{t:comparison}\n  We have the following commutative diagram of ring homomorphisms\n  \\begin{equation*}\n    \\xymatrix{\n      {(K_0({\\op{Var}}_{{\\mathbb{A}}^1_\\mathbb{C}}), \\star)} \n      \\ar[r]^-{\\mu} \\ar[d]_-{\\epsilon_!\\circ \\Phi}\n      & {K_0(\\op{sat}_{\\mathbb{C}}^{{\\mathbb{Z}}_2})} \\ar[d]^-{\\chi_{{\\operatorname{HP}}}} \\\\\n      {(\\mathcal{M}_{\\mathbb{C}}^{\\hat{\\upmu}},*)} \\ar[r]^-{\\chi_{\\op{c}}} \n      & {\\mathbb{Z}}\n    }\n  \\end{equation*}\n  where the left vertical arrow is \n  the map \n  \\eqref{eq:epsilon-Phi}\n  from Theorem~\\ref{t:total-motivic-vanishing-cycles-ring-homo}\n  and the lower horizontal map is \n  the ring homomorphism \\eqref{eq:72}.\n\\end{theorem}\n\n\\begin{proof} \n  The abelian group $K_0({\\op{Var}}_{{{\\mathbb{A}}}^1_{\\mathbb{C}}})$ is\n  generated by classes \n  $[X\\xra{W} {{\\mathbb{A}}}^1_{\\mathbb{C}}]$ where $X$ is smooth over ${\\mathbb{C}}$ and\n  the map $W$ is projective\n  (see \\cite{bittner-euler-characteristic}).\n  So it suffices to prove commutativity on such generators.\n\n  Fix a projective map $W \\colon X\\rightarrow {{\\mathbb{A}}}^1_{\\mathbb{C}}$ of a smooth\n  ${\\mathbb{C}}$-variety $X$. Then by definition\n  \\begin{equation*}\n    \\mu (W)=\\sum_{a\\in {\\mathbb{C}}}[{\\mathbf{MF}}(X,W-a)^\\natural ]\n    \\in\n    K_0(\\op{sat}_{{\\mathbb{C}}}^{{\\mathbb{Z}}_2})   \n  \\end{equation*}\n  and\n  \\begin{equation*}\n    \\epsilon_!\\circ \\Phi (W)\n    =\\sum_{a \\in {\\mathbb{C}}}(\\epsilon_a)_!\\phi_{W,a}\\in \\mathcal{M}_{{\\mathbb{C}}}   \n  \\end{equation*}\n  with notation as in \\eqref{eq:phi-V-k}.\n  So it suffices to prove that\n  \\begin{equation*} \n   \n    \\chi_{{\\operatorname{HP}}}({\\mathbf{MF}}(X,W-a)^\\natural)=\\chi_{\\op{c}}((\\epsilon_a)_!\\phi_{W,a})\n  \\end{equation*}\n  for any given $a\\in {\\mathbb{C}}$. We may and will assume that $a=0$.\n\n  Let $X^{{\\op{an}}}$ denote the space $X$ with the analytic\n  topology. Recall the classical functors of nearby and vanishing\n  cycles\n  \\begin{equation*}\n    \\psi^{{\\op{geom}}}_W,\\phi^{{\\op{geom}}}_W\\colon {\\operatorname{D}}^{\\op{b}}_{\\op{c}}(X^{{\\op{an}}})\\to\n    {\\operatorname{D}}^{\\op{b}}_{\\op{c}}(X_0^{{\\op{an}}}) \n  \\end{equation*}\n  between the corresponding derived categories of constructible\n  sheaves with complex coefficients.\n  For $F\\in {\\operatorname{D}}^{\\op{b}}_{\\op{c}}(X^{{\\op{an}}})$ we have a distinguished triangle\n  \\begin{equation}\n    \\label{eq:extr}\n    F\\vert_{X_0^{{\\op{an}}}}\\rightarrow \\psi^{{\\op{geom}}}_WF\\rightarrow \\phi^{{\\op{geom}}}_WF\\to\n    F\\vert_{X_0^{{\\op{an}}}}[1] \n  \\end{equation}\n  in  ${\\operatorname{D}}^{\\op{b}}_{\\op{c}}(X_0^{{\\op{an}}})$ (see\n  \\cite[Exp.~XIII]{SGA7-II}).\n \n \n\n  In particular for the constant sheaf ${\\mathbb{C}}_{X^{{\\op{an}}}}$ we have\n  the complex $\\phi^{{\\op{geom}}}_W{\\mathbb{C}}_{X^{{\\op{an}}}}$ of sheaves on\n  $X_0^{{\\op{an}}}$. Consider its hypercohomology with compact\n  supports\n  ${\\operatorname{H}}_{\\op{c}}^\\bullet (X_0^{{\\op{an}}}, \\phi^{{\\op{geom}}}_W{\\mathbb{C}}_{X^{{\\op{an}}}})$\n  and its Euler characteristic\n  $\\sum_i(-1)^i\\dim {\\operatorname{H}}^i_{\\op{c}}(X_0^{{\\op{an}}}, \\phi^{{\\op{geom}}}_W{\\mathbb{C}}_{X^{{\\op{an}}}})$.\n  (Note that in our case we may as well consider the\n  hypercohomology ${\\operatorname{H}}^\\bullet$ instead of ${\\operatorname{H}}^\\bullet_{\\op{c}}$,\n  since $X_0^{\\op{an}}$ is compact.) It follows from\n  \\cite[Thm.~1.1]{efimov-cyclic-homology-matrix-factorizations}\n  that \n  \\begin{equation*}\n   \n    \\chi_{{\\operatorname{HP}}}({\\mathbf{MF}}(X,W))\n    =-\\sum_i(-1)^i\\dim {\\operatorname{H}}^i_{\\op{c}}(X_0^{{\\op{an}}},\\phi^{{\\op{geom}}}_W{\\mathbb{C}}_{X^{{\\op{an}}}}). \n  \\end{equation*}\n  By the localization theorem in cyclic homology it follows that the\n  Karoubi closure ${\\mathbf{MF}}(X,W)^\\natural$ has the same cyclic\n  homology as ${\\mathbf{MF}}(X,W)$, i.\\,e.\\\n  $\\chi_{{\\operatorname{HP}}}({\\mathbf{MF}}(X,W))=\\chi_{{\\operatorname{HP}}}({\\mathbf{MF}}(X,W)^\\natural)$. \n \n \n \n \n \n \n \n  Hence it remains to prove the equality\n  \\begin{equation}\n    \\label{eq:val4} \n    \\chi_{\\op{c}}((\\epsilon_0)_!\\phi_{W,0})=-\\sum_i(-1)^i\\dim\n    {\\operatorname{H}}^i_{\\op{c}}(X_0^{{\\op{an}}}, \\phi^{{\\op{geom}}}_W{\\mathbb{C}}_{X^{{\\op{an}}}}).\n  \\end{equation}\n\n  \\begin{lemma}\n    \\label{l:SH}\n    \\begin{enumerate}\n    \\item \n      \\label{enum:SH-exist}\n      For every variety $Y$ there\n      exists a unique \n      group homomorphism\n      \\begin{equation*}\n        {\\operatorname{SH}}_Y\\colon K_0({\\op{Var}}_Y)\\rightarrow K_0({\\operatorname{D}}_{\\op{c}}^{\\op{b}}(Y^{{\\op{an}}}))  \n      \\end{equation*}\n      such that ${\\operatorname{SH}}_Y([Z\\xra{f}Y])=[{{\\mathbf{R}}}f_!{\\mathbb{C}}_{Z^{{\\op{an}}}}]$.\n    \\item \n      \\label{enum:SH-push}\n      Given a morphism of varieties $g\\colon Y\\rightarrow T$ the diagram\n      \\begin{equation*}\n        \\xymatrix{\n          {K_0({\\op{Var}}_Y)} \\ar[r]^-{{\\operatorname{SH}}_Y} \\ar[d]_-{g_!}\n          & {K_0({\\operatorname{D}}_{\\op{c}}^{\\op{b}}(Y^{{\\op{an}}}))} \\ar[d]^-{K_0({{\\mathbf{R}}}g_!)}\\\\\n          {K_0({\\op{Var}}_T)} \\ar[r]^-{{\\operatorname{SH}}_T} \n          & {K_0({\\operatorname{D}}_{\\op{c}}^{\\op{b}}(T^{{\\op{an}}}))}\n        }\n      \\end{equation*}\n      commutes.\n    \\item\n      \\label{enum:SH-C}\n      If $Y=\\operatorname{Spec} {\\mathbb{C}}$, then $K_0({\\operatorname{D}}^{\\op{b}}_{\\op{c}}((\\operatorname{Spec}\n      {\\mathbb{C}})^{\\op{an}}))={\\mathbb{Z}}$ (by taking the alternating sum of the\n      cohomologies) and ${\\operatorname{SH}}_{\\operatorname{Spec} {\\mathbb{C}}}([Z\\rightarrow \\operatorname{Spec}\n      {\\mathbb{C}}])=\\chi_{\\op{c}}([Z])$. \n    \\end{enumerate}\n  \\end{lemma}\n\n  \\begin{proof} \n    \\ref{enum:SH-exist}\n    For a variety $S$ and an open embedding $j\\colon\n    U\\hookrightarrow S$ with \n    complementary closed embedding $i\\colon\n    Z=S-U\\hookrightarrow S$ recall the short exact sequence \n    of sheaves\n    \\begin{equation*}\n     \n      0\\rightarrow j_!{\\mathbb{C}}_{U^{{\\op{an}}}}\\rightarrow {\\mathbb{C}}_{S^{{\\op{an}}}}\\rightarrow i_!{\\mathbb{C}}_{Z^{{\\op{an}}}}\\rightarrow 0.\n    \\end{equation*}\n    This implies that the map\n    ${\\operatorname{SH}}_Y([Z\\xra{f}Y])=[{{\\mathbf{R}}}f_!{\\mathbb{C}}_{Z^{{\\op{an}}}}]$ indeed\n    descends to a homomorphism ${\\operatorname{SH}}_Y\\colon K_0({\\op{Var}}_Y)\\to\n    K_0({\\operatorname{D}}_{\\op{c}}^{\\op{b}}(Y^{{\\op{an}}}))$. Uniqueness is obvious. \n\n    \\ref{enum:SH-push}\n    Given a morphism $f\\colon Z\\rightarrow Y$ we have by definition\n    \\begin{equation*}\n      K_0({{\\mathbf{R}}}g_!)\\cdot {\\operatorname{SH}}_Y([Z\\xra{f}Y])=\n      K_0({{\\mathbf{R}}}g_!)[{{\\mathbf{R}}}f_!{\\mathbb{C}}_{Z^{{\\op{an}}}}]=[{{\\mathbf{R}}}(gf)_!{\\mathbb{C}}_{Z^{{\\op{an}}}}]  \n    \\end{equation*}\n    and\n    \\begin{equation*}\n      {\\operatorname{SH}}_T \\cdot g_!([Z\\xra{f}Y])\n      ={\\operatorname{SH}}_T([Z\\xra{gf}T]=[{{\\mathbf{R}}}(gf)_!{\\mathbb{C}}_{Z^{{\\op{an}}}}]  \n    \\end{equation*}\n\n    \\ref{enum:SH-C}\n    This is clear.\n  \\end{proof}\n  \n  Now \\cite[Prop.~3.17]{guibert-loeser-merle-convolution} implies\n  the following equality in \n  $K_0({\\operatorname{D}}^{\\op{b}}_{\\op{c}}(X^{{\\op{an}}}_0))$: \n  \\begin{equation*}\n   \n    {\\operatorname{SH}}_{X_0}(\\psi_{W,0})=[\\psi_W^{{\\op{geom}}}({\\mathbb{C}}_{X^{{\\op{an}}}})].\n  \\end{equation*}\n  Applying part \\ref{enum:SH-push}\n  of Lemma~\\ref{l:SH}\n  to the map $\\epsilon_0\\colon\n  X_0\\rightarrow {\\operatorname{Spec} {\\mathbb{C}}}$ and\n  using part \\ref{enum:SH-C}\n  we conclude that\n  \\begin{equation*}\n   \n    \\chi_{\\op{c}}((\\epsilon_0)_!\\psi_{W,0})=\\sum_i(-1)^i\\dim\n    {\\operatorname{H}}^i_{\\op{c}}(X_0^{{\\op{an}}},\\psi_W^{{\\op{geom}}}({\\mathbb{C}}_{X^{{\\op{an}}}})). \n  \\end{equation*}\n  Notice that on one hand by definition of $\\phi_{W,0}$ we have\n  \\begin{equation*}\n    \\chi_{\\op{c}}((\\epsilon_0)_!\\phi_{W,0})\n    =\\chi_{\\op{c}}((\\epsilon_0)_![X_0\\xra{\\operatorname{id}}\n    X_0])-\\chi_{\\op{c}}((\\epsilon_0)_!\\psi_{W,0})    \n  \\end{equation*}\n  and on the other hand by the distinguished triangle\n  \\eqref{eq:extr} we have \n  \\begin{multline*}\n    \\sum_i(-1)^i\\dim {\\operatorname{H}}^i_{\\op{c}}(X_0^{{\\op{an}}},\\phi_W^{{\\op{geom}}}({\\mathbb{C}}_{X^{{\\op{an}}}}))=\n    \\sum_i(-1)^i\\dim\n    {\\operatorname{H}}^i_{\\op{c}}(X_0^{{\\op{an}}},\\psi_W^{{\\op{geom}}}({\\mathbb{C}}_{X^{{\\op{an}}}}))\n    \\\\\n    -\\sum_i(-1)^i\\dim {\\operatorname{H}}^i_{\\op{c}}(X_0^{{\\op{an}}},{\\mathbb{C}}_{X_0^{{\\op{an}}}})  \n  \\end{multline*}\n  It remains to notice that\n  \\begin{equation*}\n    \\chi_{\\op{c}}((\\epsilon_0)_![X_0\\xra{\\operatorname{id}} X_0])=\\sum_i(-1)^i\\dim\n    {\\operatorname{H}}^i_{\\op{c}}(X_0^{{\\op{an}}},{\\mathbb{C}}_{X_0^{{\\op{an}}}})   \n  \\end{equation*}\n  This proves equality \\eqref{eq:val4} and finishes the proof of the\n  theorem. \n\\end{proof}\n\nWe give two simple examples in which the equality \\eqref{eq:val4} can\nbe verified directly. \n\n\\begin{example} \n  \\label{expl:a^n}\n  Let $X={{\\mathbb{A}}}^1_\\mathbb{C}$ and $W(a)=a^n$ for some $n\\geq 1$.\n  Then $\\phi^{{\\op{geom}}}_W{\\mathbb{C}}_{X^{{\\op{an}}}}={\\mathbb{C}}^{\\oplus n-1}_{(0)}$. \n  Hence the right-hand side of \n  equation \\eqref{eq:val4} is equal to $-(n-1)$.\n\n  On the other hand, in the notation of \n  the proof of\n  Proposition~\\ref{p:motivic-vanishing-cycles-over-singular-part}\n  (with the identity as embedded resolution) \n  the divisor $E$ is $n\\cdot (0)$ and\n  hence its $\\upmu_n$-Galois covering $\\tilde{E}$ is\n  isomorphic to $\\upmu_n$. \n  From \\eqref{eq:formula-mot-nearby-fiber}\n  we obtain\n  \\begin{equation*}\n    \\phi_{W,0}=[|X_0|]-\\psi_{W,0}=[(0)]-\\upmu_n.  \n  \\end{equation*}\n  Thus $\\chi_{\\op{c}}((\\epsilon_0)_!\\phi_{W,0})$ is also equal to $-(n-1)$.\n\\end{example}\n\n\\begin{example} \n  \\label{expl:ab}\n  Let $X={{\\mathbb{A}}}^2_\\mathbb{C}$ and $W\\colon X\\rightarrow {{\\mathbb{A}}}^1_\\mathbb{C}$,\n  $W(a,b)=ab$. (This is not proper, but should make no \n  difference since the complex $\\phi^{{\\op{geom}}}_W{\\mathbb{C}}_{X^{{\\op{an}}}}$\n  has compact support.) \n  Then $\\phi^{{\\op{geom}}}_W{\\mathbb{C}}_{X^{{\\op{an}}}}={\\mathbb{C}}_{(0,0)}[-1]$. Hence\n  the right-hand side of \\eqref{eq:val4} is equal to 1. \n\n  On the other hand, in the notation of \n  the proof of\n  Proposition~\\ref{p:motivic-vanishing-cycles-over-singular-part}\n  the divisor $E$ is\n  the coordinate cross (with components of multiplicity one) \n  and so  \\eqref{eq:formula-mot-nearby-fiber}\n  yields\n  \\begin{equation*}\n    \\phi_{W,0}=[X_0]-\\psi_{W,0}=({\\DG_\\op{m}} +{\\DG_\\op{m}}+pt)-({\\DG_\\op{m}}+{\\DG_\\op{m}}-{\\DG_\\op{m}})={\\mathbb{L}}  \n  \\end{equation*}\n  Hence $\\chi_{\\op{c}}((\\epsilon_0)_!\\phi_{W,0})=1$.\n\n  Here is another way to compute this example.\n  Using coordinates $(s,t)$ on ${\\mathbb{A}}^2_\\mathbb{C}$ so that $a=s+it$ and\n  $b=s-it$ we have $W(a,b)=ab=s^2+t^2= s^2 \\circledast t^2$. \n  Example~\\ref{expl:a^n} shows that\n  $\\phi_{s^2,0}=[(0)]-\\upmu_2$ and\n  $\\chi_{\\op{c}}((\\epsilon_0)_!\\phi_{s^2,0})=\n  -1$.\n  We have $\\epsilon_!\\Phi(s^2)=(\\epsilon_0)_!\\phi_{s^2,0}$\n  and\n  \\begin{equation*}\n    \\chi_{\\op{c}}((\\epsilon_0)_!\\phi_{W,0})=\n    \\chi_{\\op{c}}(\\epsilon_!\\Phi(ab))=\n    \\chi_{\\op{c}}(\\epsilon_!\\Phi(s^2))\n    \\chi_{\\op{c}}(\\epsilon_!\\Phi(t^2))\n    =(-1)^2=1\n  \\end{equation*}\n  using multiplicativity of our motivic measures.\n  We can also use the motivic Thom-Sebastiani \n  Theorem~\\ref{t:thom-sebastiani-GLM-motivic-vanishing-cycles}\n  and compute \n  (use Remark~\\ref{rem:action-on-B-trivial}\n  and (the computation leading to) equation \n  \\eqref{eq:mu2-convolution-squared})\n  \\begin{multline*}\n    \\phi_{W,0}=\n    \\Psi(\\phi_{s^2,0} \\times \\phi_{t^2,0})=\n    \\Psi([(0)] \\times [(0)])\n    -\n    2 \\Psi([(0)] \\times [\\upmu_2])\n    + \\Psi([\\upmu_2] \\times [\\upmu_2])\\\\\n    = [(0)]-2 [\\upmu_2]+([{\\DG_\\op{m}}] +2[\\upmu_2])) =\\mathbb{L}.\n  \\end{multline*}\n  Here the $\\mu_2$-action on ${\\mathbb{G}}_m$ is a priori the obvious one\n  but can then also be assumed to be trivial by the defining\n  relations \n  of the equivariant Grothendieck group.\n\\end{example}\n\n\\subsection{Summarizing diagram}\n\\label{sec:summarizing-diagram}\n\nWe collect the motivic measures considered in this paper in the following\ncommutative diagram (in case ${\\mathsf{k}}={\\mathbb{C}}$; see\n\\eqref{eq:folding-diagram} and Theorem~\\ref{t:comparison}).\n\\begin{equation*}\n  \\xymatrix{\n    {K_0({\\op{Var}}_\\mathbb{C})} \\ar[r]^-{\\nu} \\ar[d]\n    & {K_0(\\op{sat}_\\mathbb{C}^{\\mathbb{Z}})} \\ar[d]\\\\\n    {(K_0({\\op{Var}}_{{\\mathbb{A}}^1_\\mathbb{C}}), \\star)} \\ar[r]^-{\\mu} \\ar[d]_{\\epsilon_!\n      \\circ \\Phi}\n    & {K_0(\\op{sat}_\\mathbb{C}^{\\mathbb{Z}_2})} \\ar[d]^-{\\chi_{\\operatorname{HP}}}\n    \\\\\n    {(\\mathcal{M}^{\\hat{\\upmu}}_\\mathbb{C}, *)} \n    \\ar[r]^-{\\chi_{\\op{c}}}\n    & {\\mathbb{Z}}\n  }\n\\end{equation*}\nThe upper left vertical arrow and the vertical composition on the\nleft are the algebra structure maps. The composition from the top\nleft corner to the bottom right corner is induced by mapping a\ncomplex variety to its Euler characteristic with compact support.\n\n\n\\def$'$} \\def\\cprime{$'${$'$} \\def$'$} \\def\\cprime{$'${$'$} \\def$'$} \\def\\cprime{$'${$'$} \\def$'$} \\def\\cprime{$'${$'$}\n  \\def\\leavevmode\\lower.6ex\\hbox to 0pt{\\hskip-.23ex \\accent\"16\\hss}D{\\leavevmode\\lower.6ex\\hbox to 0pt{\\hskip-.23ex \\accent\"16\\hss}D}\n  \\def$'$} \\def\\cprime{$'${$'$} \\def$'$} \\def\\cprime{$'${$'$}\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]{%\n  \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nRecent experimental efforts have been devoted to measurements of the weak mixing angle in electron-proton scattering \\cite{Androic,Becker}. Precise measurements at low energy are of high interest due to the sensitivity of the running of the weak mixing angle in extensions of the Standard Model \\cite{Davoudiasl}. In order to control the effects of proton structure in the scattering experiments, knowledge of nucleon form factors is required. Among others, the strange form factors $G_E^s(Q^2)$, $G_M^s(Q^2)$ and $G_A^s(Q^2)$ need to be known precisely, which is a challenging task in experiments \\cite{Pate}. Here, we extract these form factors from a first-principles Lattice QCD study, based on our previous work \\cite{Djukanovic}.  Furthermore, we calculate the contributions of the $u$-, $d$- and $s$-quarks to the axial charge. These correspond to the contributions of the intrinsic spin of the respective quarks to the total spin of the nucleon \\cite{Ji}. Following the same procedure, we also obtain the $u$-, $d$- and $s$-contributions to the tensor charge of the nucleon. \n\n\\section{Extracting Form Factors from Lattice QCD}\nTo extract nucleon form factors, we consider the following ratio of nucleon three- and two-point functions \\cite{Brandt}\n\\begin{align}\n\\begin{split}\nR_{J_\\mu}(\\vec{q};\\vec{p}^\\prime;\\Gamma_\\nu) &= \\frac{C_{3,J_\\mu}^N(\\vec{q},z_0;\\vec{p}^\\prime,y_0;\\Gamma_\\nu)}{C_2^N(\\vec{p}^\\prime,y_0)}\\sqrt{\\frac{C_2^N(\\vec{p}^\\prime,y_0)C_2^N(\\vec{p}^\\prime,z_0)C_2^N(\\vec{p}^\\prime\\text{-}\\vec{q},y_0\\text{-}z_0)}{C_2^N(\\vec{p}^\\prime\\text{-}\\vec{q},y_0)C_2^N(\\vec{p}^\\prime\\text{-}\\vec{q},z_0)C_2^N(\\vec{p}^\\prime,y_0\\text{-}z_0)}}\\\\[0.25cm]\t\n\t&\\stackrel{\\text{s.d.}}{=} M_{\\nu\\mu}^1(\\vec{q},\\vec{p}^\\prime) G_1(Q^2) + M_{\\nu\\mu}^2(\\vec{q},\\vec{p}^\\prime) G_2(Q^2)\\ ,\n\\end{split}\n\\label{eq:ratio}\n\\end{align}\nwhere the second line corresponds to the spectral decomposition at large Euclidean times. At each value of $Q^2$, the ratios belonging to different $\\mu$, $\\nu$ and momenta $\\vec{q}$, $\\vec{p}^\\prime$ can be grouped into a vector $\\bm{R}$. Ordering the kinematic factors $M^1$ and $M^2$ accordingly into a matrix $M$, a (generally overdetermined) system of equations can be defined\n\\begin{equation}\nM\\ \\bm{G} = \\bm{R}\\ \\ ;\\ \\ M = \\left( \\begin{array}{c}\nM^1_1\\\\\n\\vdots\\\\\nM^1_N\\\\\n\\end{array}\\ \\begin{array}{c}\nM^2_1\\\\\n\\vdots\\\\\nM^2_N\\\\\n\\end{array} \\right)\\ \\ ,\\ \\ \\bm{G} = \n\\left(\n\\begin{array}{c}\nG_1\\\\\nG_2\\\\\n\\end{array}\n\\right),\\ \\ \\bm{R} = \n\\left(\n\\begin{array}{c}\nR_1\\\\\n\\vdots\\\\\nR_N\\\\\n\\end{array}\n\\right)\\ .\n\\label{eq:system}\n\\end{equation}\n\nIn the case of the disconnected contributions, we reduce the size of the system by first averaging equivalent three-point functions, where the momenta at the source and the sink are related by spatial symmetry \\cite{Syritsyn}, followed by a construction of the ratios from the resulting averages. Concerning the connected contributions, we perform an average of equivalent ratios. In both cases we drop non-contributing ratios, for which both kinematic factors $M^1$ and $M^2$ are zero. Furthermore, we average the two-point functions over equivalent momentum classes. The system of equations can be solved for the form factors $G_1$ and $G_2$ by applying a two-parameter fit corresponding to minimizing the $\\chi^2$-function given by $\\chi^2 = (\\bm{R}-M\\bm{G})^T\\ C^{-1}\\ (\\bm{R}-M\\bm{G})$, where $C$ denotes the covariance matrix. Identifying $J_\\mu$ with the vector current $V_\\mu$ in \\Eq{eq:ratio} yields the electromagnetic form factors $G_E$ and $G_M$, whereas the axial current $A_\\mu$ leads to the axial vector form factors $G_A$ and $G_P$. We obtain the tensor charge $g_T$ at $Q^2=0$ from the tensor current $T_{\\mu\\nu}$, where only one independent ratio contributes.\n\n\\section{Setup}\nThis work makes use of the CLS $N_f=2+1$ $\\mathcal{O}(a)$-improved Wilson fermion ensembles \\cite{Bruno} listed in Tab.~\\ref{tab:ensembles}. The tree-level improved L\\\"uscher-Weisz gauge action is employed in the gauge sector. These ensembles obey open boundary conditions in time in order to prevent topological freezing \\cite{Luscher}. The physical quark masses are approached along a trajectory where the trace of the quark mass matrix is kept (approximately) constant, $\\mathrm{tr}\\ M_q = \\text{const}$. To set the scale, we rely on the gradient flow time $t_0$ determined in Ref. \\cite{Bruno2}.\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{l|ccccccccc}\n\t&$\\beta$\t&$a$ [fm] &$N_s^3\\times N_t$\t&$m_\\pi$[MeV]\t&$m_K$[MeV]\t&$N_{\\text{cfg}}^{\\text{dis}}$\t&$N_{\\text{cfg}}^{\\text{con}}$\\\\\n\\hline\nH102\t&3.40\t&0.08636\t&$32^3\\times 96$\t&352\t&438\t&997\t&997\\\\\nH105\t&3.40\t&0.08636 &$32^3\\times 96$\t&278\t&460\t&1019\t&1019\\\\\nC101\t&3.40\t&0.08636\t&$48^3\\times 96$\t&223\t&472\t&1000\t&2000\\\\\n\\hline\nN401\t&3.46\t&0.07634\t&$48^3\\times 128$\t&289\t&462\t&701\t&701\\\\\n\\hline\nN203\t&3.55 &0.06426\t&$48^3\\times 128$ &345\t&441\t&768\t&1536\\\\\nN200\t&3.55 &0.06426\t&$48^3\\times 128$\t&283\t&463\t&852\t&852\\\\\nD200\t&3.55 &0.06426\t&$64^3\\times 128$\t&200\t&480\t&234\t&936\\\\\n\\hline\nN302\t&3.70\t&0.04981\t&$48^3\\times 128$\t&354\t&458\t&1177\t&1177\\\\\n\\end{tabular}\n\\end{center}\n\\caption{The ensembles used for this work. The last two columns give the number of configurations for the disconnected and connected diagrams, respectively.}\n\\label{tab:ensembles}\n\\end{table}\n\nFor the disconnected three-point function it is straightforward to check that it factorizes into separate traces of the quark loop and the two-point function\n\\begin{equation}\nC_{3,J_\\mu}^{N,l\/s}(\\bm{q},z_0;\\bm{p}^\\prime,y_0;\\Gamma_\\nu) = \\left\\langle \\mathcal{L}_{J_\\mu}^{l\/s}(\\bm{q},z_0)\\cdot \\mathcal{C}_2^N(\\bm{p}^\\prime,y_0;\\Gamma_\\nu) \\right\\rangle_G\\ .\n\\end{equation}\n\nThe quark loops have been calculated with an improved stochastical estimator provided by hierarchical probing \\cite{Stathopoulos}. Here the sequence of noise vectors is replaced by a sequence of Hadamard vectors, which are element-wise multiplied by one noise vector. In total, we considered two noise vectors, each multiplied with a sequence of 512 Hadamard vectors on every configuration and for each quark flavor. For the two-point function we used the standard nucleon interpolator \n\\begin{equation}\nN_{\\alpha}(x) = \\epsilon_{abc}\\left( u_\\beta^a(x)\\ \\left(C\\gamma_5\\right)_{\\beta\\gamma}\\ d_\\gamma^b(x) \\right)\\ u_\\alpha^c(x)\\ .\n\\end{equation}\n\nTo increase the overlap with the ground state, Wuppertal smearing \\cite{Gusken} was used at the source and the sink. The statistical precision of the two-point function has been improved by employing the truncated-solver method \\cite{Bali,Shintani}. For each ensemble we performed 32 low-precision solves on seven timeslices equally distributed around the center of the lattice and separated by seven timeslices without sources. On each of these timeslices one low-precision solve was considered for the bias correction, except for ensemble H105, where we considered four low-precision solves on each timeslice. For each current we considered all components combined with four choices of the projector $\\Gamma$: $\\Gamma_0 = \\frac{1}{2}(1+\\gamma_0)\\ ,\\ \\Gamma_k = \\frac{1}{2}(1+\\gamma_0)\\ \\mathrm{i}\\gamma_5\\gamma_k\\ ,\\ k\\in\\{1,2,3\\}$. Details on the two- and three-point functions for the connected contributions at $Q^2=0$ can be found in Ref. \\cite{Harris}. The same details also carry over to $Q^2\\neq 0$. Here, we did not make use of the zeroth component of the axial vector current, due to its large excited-state contamination. Furthermore, the electromagnetic form factors can be obtained directly, as shown in Ref. \\cite{Capitani}, without fitting the system in \\Eq{eq:system}. Note that for the connected three-point function the nucleon at the sink is always at rest, whereas in the disconnected contributions we consider up to two units of the squared lattice integer momentum. For the local axial vector and the conserved vector current we implemented the improved versions\n\\begin{equation}\nA_\\mu(\\vec{z},z_0)^{\\text{Imp.}}=A_\\mu(\\vec{z},z_0)+ac_A\\ \\partial_\\mu P(\\vec{z},z_0)\\ ,\\ V_\\mu(\\vec{z},z_0)^{\\text{Imp.}}=V_\\mu(\\vec{z},z_0)+ac_V\\ \\partial_\\nu T_{\\nu\\mu}(\\vec{z},z_0)\\ ,\n\\end{equation}\nwhere the improvement coefficients have been determined non-perturbatively in Ref. \\cite{Bulava,Gerardin}. For the tensor charge we used the local tensor current.\n\n\\section{Renormalization}\nAs starting point for the renormalization procedure we consider the flavor-diagonal operators \n\\begin{equation}\nO^a_\\Gamma(x) = \\bar{\\psi}(x) \\Gamma \\lambda^a \\psi(x)\\ \\ ,\\ \\ \\psi = (u,d,s)^T\\ \\ ,\\ \\ a\\in\\{3,8,0\\}\\ ,\n\\end{equation}\n\\begin{equation}\n\\lambda^3 = \\dfrac{1}{\\sqrt{2}}\\text{diag}\\left(1,-1,0\\right)\\ ,\\ \\lambda^8 = \\dfrac{1}{\\sqrt{6}}\\text{diag}\\left(1,1,-2\\right)\\ ,\\ \\lambda^0 = \\dfrac{1}{\\sqrt{3}}\\text{diag}\\left(1,1,1\\right)\\ .\n\\end{equation}\n\nWe make use of $N_f=3$ $\\mathcal{O}(a)$-improved Wilson fermion ensembles, and hence, the renormalization matrix of the flavor-diagonal operators is given by\n\\begin{equation}\nZ_\\Gamma = \\text{diag}\\left(Z_\\Gamma^{33},Z_\\Gamma^{88},Z_\\Gamma^{00}\\right)\\ \\ ,\\ \\ Z_\\Gamma^{33} = Z_\\Gamma^{88}\\ \\ .\n\\end{equation}\n\nAll details on our renormalization procedure for the $Z_\\Gamma^{33}$ can be found in Ref. \\cite{Harris}. We follow the same procedure for $Z_\\Gamma^{00}$ which, however, necessitates the calculation of additional diagrams. The required quark loops have been estimated with hierarchical probing, using 512 Hadamard vectors on each configuration. Furthermore, in the case of the singlet axial operator, an anomalous dimension arises (given in Ref. \\cite{Larin}) and the conversion factors $Z^{\\overline{\\text{MS}}}_{\\text{RI}^\\prime\\text{-MOM}}$ for the singlet vector and axial vector operators are only known to one loop. Finally, we apply a basis transformation to the physical basis given by\n\\begin{equation}\n\\left(\\begin{array}{c}\nO_\\Gamma^{u-d}(x)_R\\\\\nO_\\Gamma^{u+d}(x)_R\\\\\nO_\\Gamma^{s}(x)_R\\\\\n\\end{array}  \\right) = \\left(\\begin{array}{ccc} Z_\\Gamma^{u-d,u-d} &0 &0\\\\ 0 &Z_\\Gamma^{u+d,u+d} &Z_\\Gamma^{u+d,s}\\\\ 0 &Z_\\Gamma^{s,u+d} &Z_\\Gamma^{s,s} \\end{array}\\right)\\ \\left(\\begin{array}{c}\nO_\\Gamma^{u-d}(x)\\\\\nO_\\Gamma^{u+d}(x)\\\\\nO_\\Gamma^{s}(x)\\\\\n\\end{array}  \\right)\\ .\n\\end{equation}\n\nThe final renormalization constants are taken in the $\\overline{\\text{MS}}$-scheme at $\\mu=2\\,\\text{GeV}$. Note that the renormalization constants at $\\beta=3.7$ have been obtained from a linear extrapolation. To account for the systematic uncertainty introduced through the extrapolation, the error of the renormalization constants at this $\\beta$ has been inflated by a factor of ten. The reliability of the extrapolation was checked for the isovector axial charge $g_A$ in Ref. \\cite{Harris}, where consistent results from our renormalization procedure compared to renormalizing through the Schr\\\"odinger functional were found.\n\n\n\\section{Results}\nIn \\Fig{fig:svecff}, we show the resulting strange electromagnetic form factors for three lattice spacings at constant kaon mass of $m_K\\approx 460\\,\\text{MeV}$. Excited-state contamination has been handled with the summation method with $y_0\\in[0.5,1.3]\\,\\text{fm}$. The bands correspond to $z$-expansion fits \\cite{Hill} to fifth order using the two-kaon production threshold for the conformal map, where Gaussian priors of the form $\\tilde{a}_k = 0 \\pm 5\\max\\{|a_0|,|a_1|\\}\\ \\forall\\ k>1$ have been employed to stabilize the fits. \n\n\\begin{figure}\n\\hspace{1cm}\n\\includegraphics[scale=0.39]{plots\/zexp_fit_GEs.pdf}\n\\hspace{1cm}\n\\includegraphics[scale=0.39]{plots\/zexp_fit_GMs.pdf}\n\\caption{The $Q^2$-dependence of the strange electromagnetic form factors at a kaon mass of $m_K\\approx 460\\,\\text{MeV}$.}\n\\label{fig:svecff}\n\\end{figure}\n\nFrom the $z$-expansion fits we can extract the strange magnetic moment $\\mu^s=G^s_M(0)$ and the strange electromagnetic charge radii $(r^2)_{E\/M}^s$, shown in \\Fig{fig:extsEMff}. Note that for our main result we impose a cut in $Q^2$ at $0.5\\,\\text{GeV}^2$. We extrapolate to the physical kaon mass by performing linear fits of the form \n\\begin{equation}\n(r^2_E)^s(m_K) = a_0 + a_1 \\log(m_K)\\ ,\\ \\mu^s(m_K) = a_2 + a_3 m_K\\ ,\\ (r^2_M)^s(m_K) = a_4 + a_5\/m_K\\ ,\n\\end{equation}\nderived from leading-order SU(3) HBChPT \\cite{Hemmert}, where we add the higher-order $a_4$-term as our data does not show the leading-order divergence expected by HBChPT. \n\n\\begin{figure}\n\\includegraphics[scale=0.325]{plots\/elec_charge_radius_phys.pdf}\n\\includegraphics[scale=0.325]{plots\/magn_mom_phys.pdf}\n\\includegraphics[scale=0.325]{plots\/magn_charge_radius_phys.pdf}\n\\caption{The kaon mass dependence of the strange magnetic moment and strange charge radii. The orange band is a linear extrapolation to the physical kaon mass (black data point).}\n\\label{fig:extsEMff}\n\\end{figure}\n\nIn order to estimate systematic errors we consider four variations: the inclusion of $\\mathcal{O}(a^2)$ lattice artifacts, doubling the prior width in the $z$-expansion fits, handling excited states with the plateau method at $\\sim 1\\,\\text{fm}$, and removing the cut in $Q^2$. The respective errors are taken from the difference to the main result and then added in quadrature leading to our final result\n\\begin{equation}\n(r^2_E)^s = -0.0048(11)(23)\\,\\text{fm}^2\\ ,\\ \\mu^s = -0.020(4)(11)\\ ,\\ (r^2_M)^s = -0.011(5)(12)\\,\\text{fm}^2\\ .\n\\end{equation}\n\nWe follow the same procedure for the strange axial vector form factor, shown in \\Fig{fig:saxvecffagts}. Also shown is a linear extrapolation in $m_\\pi^2$ to the physical point for the strange contribution to the axial charge. The same sources of systematics are considered in this case with the adjustment to $\\mathcal{O}(a)$ lattice artifacts, as the renormalization constants are not $\\mathcal{O}(a)$-improved. Our final result is given by $g_A^s = -0.044(4)(5)$. Furthermore, in \\Fig{fig:saxvecffagts}, we show the analogous linear extrapolation of the strange tensor charge, again obtained from the summation method. In the estimation of the systematic error, we consider the plateau method at $\\sim 1\\,\\text{fm}$ and the inclusion of $\\mathcal{O}(a)$ lattice artifacts in the extrapolation. The final result is $g_T^s = -0.0026(73)(424)$.\n\\begin{figure}\n\\includegraphics[scale=0.325]{plots\/zexp_fit_GAs.pdf}\n\\includegraphics[scale=0.325]{plots\/gas_phys.pdf}\n\\includegraphics[scale=0.325]{plots\/gts_phys.pdf}\n\\caption{The $Q^2$-dependence of the strange axial vector form factor for three lattice spacings at constant kaon mass of $m_K\\approx 460\\,\\text{MeV}$ (left), and the extrapolation of the strange axial charge (middle) and strange tensor charge (right) to the physical point (black data point).}\n\\label{fig:saxvecffagts}\n\\end{figure}\n\nWe make use of the following decomposition of the $u$- and $d$-charges\n\\begin{equation}\ng^u_{A\/T} = \\frac{1}{2}\\left( g_{A\/T}^{u+d-2s} + 2 g_{A\/T}^{s} + g_{A\/T}^{u-d} \\right)\\ ,\\ g^d_{A\/T} = \\frac{1}{2}\\left( g_{A\/T}^{u+d-2s} + 2 g_{A\/T}^{s} - g_{A\/T}^{u-d} \\right)\\ .\n\\label{eq:qcdec}\n\\end{equation}\n\nThe strange contributions have been shown above and our results for the isovector charges can be found in Ref. \\cite{Harris}. It is favorable to consider the $(u+d-2s)$-contribution, as it renormalizes like the isovector charges, so that no mixing occurs, and the disconnected contributions can be combined to $(l-s)$, which leads to a noise reduction. To handle the excited state contamination, we perform two-state fits simultaneously to all source-sink separations $y_0$ of the form\n\\begin{equation*}\ng_{A\/T}(z_0,y_0) = g_{A\/T} + A_{A\/T} \\left( e^{-\\Delta z_0} + e^{-\\Delta (y_0 - z_0)} \\right) + B_{A\/T} e^{-\\Delta y_0}\\ ,\n\\end{equation*}\nwhere the energy gap $\\Delta$ to the first excited state has been determined in a simultaneous two-state fit to all six isovector charges, as explained in detail in Ref. \\cite{Harris}. We show an example of this fit in \\Fig{fig:qcat}, where we also compare to the summation method. We then perform linear extrapolations in $m_\\pi^2$ to the physical point, also shown in \\Fig{fig:qcat}.\n\\begin{figure}\n\\includegraphics[scale=0.325]{plots\/twostatefit_N302.pdf}\n\\includegraphics[scale=0.325]{plots\/gaUpDm2s_phys.pdf}\n\\includegraphics[scale=0.325]{plots\/gtUpDm2s_phys.pdf}\n\\caption{Two-state fit to $g_A^{u+d-2s}$ at the smallest $y_0$ on ensemble N302 (left), where also the summation method result is shown (blue data point), and extrapolation of $g_A^{u+d-2s}$ (middle) and $g_T^{u+d-2s}$ (right) to the physical point (black data point).}\n\\label{fig:qcat}\n\\end{figure}\n\nTo assign a systematic error, we consider the difference to the extrapolation of the summation method results and an extrapolation with the inclusion of $\\mathcal{O}(a)$ lattice artifacts. The final results are given by\n\\begin{equation}\ng_A^{u} = 0.84(3)(4)\\ ,\\ g_A^{d} = -0.40(3)(4)\\ ,\\ g_T^{u} = 0.77(4)(6)\\ ,\\ g_T^{d} = -0.19(4)(6)\\ .\n\\end{equation}\n\n\\paragraph{Acknowledgements:} This research is supported by the DFG through the SFB 1044, and under grant HI 2048\/1-1. Calculations for this project were partly performed on the HPC clusters \"{}Clover\"{} and \"{}HIMster II\"{} at the Helmholtz-Institut Mainz and \"{}Mogon II\"{} at JGU Mainz. Additional computer time has been allocated through projects HMZ21 and HMZ36 on the BlueGene supercomputer system \"{}JUQUEEN\"{} at NIC, J\\\"ulich. Our programmes use the QDP++ library \\cite{QDPpp} and deflated SAP+GCR solver from the openQCD package \\cite{openQCD}, while the contractions have been explicitly checked using \\cite{QCT}. We are grateful to our colleagues in the CLS initiative for sharing ensembles. The ensembles for the calculation of the renormalization constants have been generated in a joint effort with the RQCD collaboration.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\emph{Address Parsing} is the task of decomposing an address into the different components it is made of. This task is an essential part of many applications, such as geocoding and record linkage. Indeed, to find a particular location based on textual data, it is quite useful to detect the different parts of an address to make an informed decision. Similarly, comparing two addresses to decide whether two or more database entries refer to the same entity can prove to be quite difficult and prone to errors if based on methods such as edit distance algorithms given the various address writing standards.\n\nThere have been many efforts to solve the address parsing problem. From rule-based techniques \\cite{rule-based} to probabilistic approaches and neural network models \\cite{8615844}, a lot of progress has been made in reaching an accurate segmentation of addresses. These previous pieces of work did a remarkable job at finding solutions for the challenges related to the address parsing task. However, most of these approaches either do not take into account parsing addresses from different countries or do so but at the cost of a considerable amount of meta-data and elaborate data pre-processing pipelines \\cite{rnn-parsing, hmm-parsing, crf-parsing, feedforward-parsing}. \n\nOur work comes with three contributions. First, we propose an approach for multinational address parsing using a Recurrent Neural Network (RNN) architecture by building on, and improving, our previous work on the subject \\cite{9357170}. Secondly, we evaluate the degree to which the improved models trained on countries' addresses data can perform well at parsing addresses from other countries. Finally, we evaluate the performance of our models on incomplete addresses and propose a method to improve their accuracy.\nThe article outline is as follows: we present the related work in \\autoref{sec:relatedwork}, our proposed architectures and approaches in \\autoref{sec:architectures}, the datasets that we used for our experiments in \\autoref{sec:data}, our experimental settings in \\autoref{sec:exp}, our results on complete addresses in \\autoref{sec:completeresults} and on addresses with missing values in \\autoref{sec:missingresults}, and finally we conclude our article with an overview of our work and future work in \\autoref{sec:conclusion}.\n\n\\section{Related Work}\n\\label{sec:relatedwork}\n\nSince address parsing is a sequence tagging task, it has been tackled using probabilistic methods mainly based on Hidden Markov Models (HMM) and Conditional Random Fields (CRF) \\cite{hmm-parsing, crf-parsing, 8615844}. For instance, \\cite{hmm-parsing} proposed a large scale HMM-based parsing technique capable of segmenting a large number of addresses whilst being robust to possible irregularities in the input data. In addition, \\cite{crf-parsing} implemented a discriminative model using a linear-chain CRF coupled with a learned Stochastic Regular Grammar (SRG). This approach allowed the authors to better address the complexity of the features while capturing higher-level dependencies by applying the SRG on the CRF outputs as a score function, thus taking into account the possible lack of features for a particular token in a lexicon-based model. These probabilistic methods usually rely on structured data as well as some sort of prior knowledge of this data for feature extraction or in order to implement algorithms such as Viterbi \\cite{viterbi}, especially in the case of generative methods. \n\nIn recent years, new methods \\cite{rnn-parsing, 8615844} utilizing the power of neural networks have been proposed as solutions for the address parsing problem. Using a single hidden layer feed-forward model, \\cite{feedforward-parsing} achieved good performance. However, their approach relied on a pre-processing and post-processing pipeline to deal with the different structures of address writing and the possible prediction errors. For instance, the input data is normalized to reduce noise and standardize the many variations that can refer to the same word, such as \\emph{road} and \\emph{rd}. In addition, the model's predictions are put through a rule-based validation step to make sure that they fit known patterns. In contrast, \\cite{rnn-parsing} proposed a deep learning approach based on the use of RNN and did not use any pre or post-processing. Their experiments focused on comparing the performance of both unidirectional and bidirectional vanilla RNN and Long-Short Term Memory Models (LSTM) \\cite{LSTM}, as well as a Seq2Seq. The models achieved high accuracy on test sets with the Seq2Seq leading the scoreboard on most of them with no particular pre-processing needed during the inference process.\n\nDespite reaching notable performances, the aforementioned approaches are limited to parsing addresses from a single country and would need to be adjusted to support a multinational scope of address parsing. To tackle this problem, Libpostal\\footnote{\\href{https:\/\/github.com\/openvenues\/libpostal}{\\url{https:\/\/github.com\/openvenues\/libpostal}}}, a library for international address parsing, has been proposed. This library uses a CRF-based model trained with an averaged Perceptron for scalability. The model was trained on data from each country in the world and was able to achieve a $99.45~\\%$ full parse accuracy\\footnote{The accuracy was computed considering the entire sequence and was not focused on individual tokens.}, thus defining a new state-of-the-art\\footnote{For a comparison of some of our models with Libpostal, visit our previous article \\cite{9357170}.}. However, this requires putting addresses through a heavy pre-processing pipeline before feeding them to the prediction model. \n\nTaking into account that some countries' addresses share the same language and format (i.e: the order of the address elements), \\cite{9357170} have proposed to leverage subword embeddings with a Seq2Seq architecture to achieve multinational address parsing. They achieved a $\\mathbf{99~\\%}$ parse accuracy on 20 countries. They also proposed to explore whether their trained model had learned transferable representation in a zero-shot evaluation procedure. Finally, they tested the parsing accuracy on 41 countries not seen during training and achieved at least $\\mathbf{80~\\%}$ parse accuracy for most of these countries.\n\nThis work aims to enhance the performance of a single model capable of parsing addresses from multiple countries, to explore the possibility of zero-shot transfer from some countries' addresses to others' and to explore (plus improve) the performance on incomplete addresses. In this respect, attention mechanisms have become ubiquitous when it comes to improving the performance of natural language processing models. These mechanisms come in different formats \\cite{attn_nlp} and have been applied to various tasks such as Machine Translation \\cite{luong-etal-2015-effective} Named Entity Recognition \\cite{attn_ner, attn_ner_2} and Sentiment Analysis \\cite{wang-etal-2016-attention}. Moreover, the Transformer architecture \\cite{NIPS2017_3f5ee243} which is based on self-attention has become the state-of-the-art in an overwhelming majority of natural language processing tasks. Zero-shot transfer has also been applied in similar settings to ours, particularly in cross lingual transfer. For instance, \\cite{zero-shot-bert} propose a linear transformation which projects the contextual representations obtained by BERT \\cite{devlin-etal-2019-bert} for certain target languages into the space of a source language, on which the model was trained to achieve the task of dependency parsing. This strategy managed to improve the model's performance in zero-shot cross lingual dependency parsing. As for \\cite{DBLP:conf\/aaai\/JiZDZCL20}, they have used cross-lingual pre-training of a Transformer encoder in order to achieve zero-shot machine translation for source-target language pairs with scarce parallel training data. Another technique that has proven to be effective in the context of zero-shot transfer is domain adaptation. In this regard, \\cite{peng2018zeroshot} and \\cite{9447890} approach the problem of zero-shot domain adaptation, in which target domain data for a particular task is completely absent, by training a model on an irrelevant task for which data from both source and target domains is available, while also utilizing data from the source domain to learn the objective task. We leverage both attention mechanisms and domain adaptation in an attempt to build a model capable of performing well on the task of address parsing for training countries and transferring the obtained knowledge to other countries without further training.\n\n\n\\section{Architectures}\n\\label{sec:architectures}\nOur models' architectures are based on previous work conducted in \\cite{9357170}. They are composed of an embedding model and a tagging model. The base architectures are improved using an attention mechanism as well as domain adaptation. These improvements are discussed in Subsections \\ref{subsec:attention} and \\ref{subsec:adann} respectively. Subsection~\\ref{subsec:base} introduces the base architecture.\n\n\\subsection{Base Architecture}\n\\label{subsec:base}\n\n\\subsubsection{Embedding Model}\n\\label{subsec:embedding}\nAn embedding model is essential in deep learning architectures solving natural language processing problems since it converts textual data into vector representations which are then used as inputs to the downstream models. Since we are dealing with a multilingual dataset, the used embedding model must be able to handle multiple languages.\n\nFirstly, we use a fixed pre-trained monolingual fastText model \\cite{fasttext} (pre-trained on the French language). The French embeddings were chosen since the French language shares Latin roots with many languages in our test set. It is also due to the considerable size of the corpus on which these embeddings were trained. Despite its monolingual characteristic, this embedding model is able to generate vector representations for words outside its training vocabulary by utilizing subword information in the form of character n-grams. We refer to this embedding model as FastText.\n\nSecondly, we use an encoding of words using multilingual subword embeddings provided by MultiBPEmb \\cite{bpemb}. This model relies on subword units to produce vector representations, therefore producing one or more vectors for each word. Consequently, we merge the obtained embeddings into a single word embedding using a Bidirectional LSTM (Bi-LSTM) with a hidden state dimension of 300. We build the word embeddings by running the concatenated forward and backward hidden states corresponding to the last time step for each word decomposition through a fully connected layer of which the number of neurons is equal to the dimension of the hidden states. This approach produces 300-dimensional word embeddings. We refer to these embeddings as BPEmb.\n\n\\subsubsection{Tagging Model}\n\\label{subsec:tagging}\n\nThe used downstream tagging model is a Seq2Seq model consisting of a one-layer unidirectional LSTM encoder and a one-layer unidirectional LSTM decoder followed by a fully-connected linear layer with a softmax activation. Both the encoder's and decoder's hidden states are of dimension $1024$. The embedded address sequence is fed to the encoder that produces hidden states, the last of which is used as a context vector to initialize the decoder's hidden states. The decoder is then given a Beginning Of Sequence (BOS) token as input and, at each time step, the prediction from the last step is used as input. To better adapt the model to the task of sequence labeling and to facilitate the convergence process, we only require the decoder to produce a sequence with the same length as the input address. The decoder's outputs are forwarded to the linear layer of which the number of neurons is equal to the tag space dimensionality. The softmax activation function computes probabilities over the linear layer's outputs to predict the most likely token at each time step.\n\n\\subsection{Attention Architecture}\n\\label{subsec:attention}\nAttention mechanisms are neural network components that can produce a distribution describing the interdependence between a model's inputs and outputs (general attention) or amongst model inputs themselves (self-attention). These mechanisms are common in natural language processing encoder-decoder architectures such as neural machine translation models \\cite{nmt} since they have shown to improve models' performance and help address some of the issues recurrent neural networks suffer from when it comes to dealing with long sequences.\n\nAttention mechanisms are also exploited for the interpretability of neural networks, where they are considered to provide insights about the impact of some neural network's inputs on its predictions. This use of attention mechanisms has been contested in \\cite{jain-wallace-2019-attention} because of a lack of consistency with feature importance measures, among other things. However, other work has suggested that attention mechanisms provide a certain degree of interpretability depending on the task at hand \\cite{wiegreffe2019attention} \\cite{vashishth2019attention}. In this work, we focus on the performance enhancement of address tagging models using attention.\n\n\\subsubsection{Attention models}\nThe implementation of our attention models was inspired by \\cite{nmt}. The models' architecture remains similar to that of our base models, with some alterations to the decoding process. Indeed, instead of feeding the last predicted tag as an input to the decoder at the current time step $i$, we compute an input using the encoder's outputs $\\vec{O}$, the last decoder hidden state $h_{i-1}$, and the last predicted tag's representation $\\vec{t}$. \n\nWe start by computing attention weights as follows:\n\n\\begin{center}\n$ \\displaystyle \\alpha_{i, j} = \\frac{\\exp({a_{i, j})}}{\\sum_{k} \\exp({a_{i, k}})}$\n\\end{center}\n\nwhere\n\n\\begin{center}\n$ a_{i, j} = p \\times tanh(W_{h} h_{i-1} + W_{o} O_{j})$\n\\end{center}\n\nand $W_{h}$, $W_{o}$ and $p$ are learnable parameters.\n\nNext, we compute a context vector by weighting the encoder's outputs with the obtained attention weights:\n\n\\begin{center}\n    $\\vec{c_{i}} = \\sum_{k} \\alpha_{i, k} O_{k}$\n\\end{center}\n\nFinally, we obtain the decoder's input by concatenating the last prediction $\\vec{t}$ with the context vector $\\vec{c_{i}}$.\nNote that the first decoder's input is computed using the last encoder's hidden state.\n\nWe use this approach with the two aforementioned embedding methods and name the obtained models \\fasttextatt{} and \\textbf{BPEmbAtt}{} respectively.\n\n\\subsection{Domain Adaptation}\n\\label{subsec:adann}\nDomain adaptation is a branch of transfer learning which aims at applying a model trained on data from a source domain to data from a target domain which somewhat differs from one another but still retains a certain degree of similarity. More specifically, it is a technique used when the input and output belong to the same space, but the probability distribution which associates them changes as we move from one domain to the other \\cite{da-tl}. Our objective is to generalize the performance of address parsing models to countries of which no data is used at training time. In order to extend our models to cope with this domain adaptation problem, we enhance our base approach using domain adversarial neural networks. \n\n\\subsubsection{Domain Adversarial Neural Networks}\nDomain adversarial neural networks \\cite{dann} achieve domain adaptation by appending a second parallel output layer to a neural network classifier, the purpose of which is to predict the domain of the network's input. Since the target domain labels are not available during training, two losses are computed when the input belongs to the source domain (i.e. the labels classification loss and the domain classification loss), whilst only the domain classification loss is computed when the input belongs to the target domain. Moreover, the gradient associated with the domain classification layer is reversed during backpropagation. This aims to hinder the neural network's ability to differentiate between the source and target domains while still learning to perform well on classifying data from the source domain.\n\n\\subsubsection{Implementation}\nFirst of all, we modified our neural network architecture by adding a domain discriminator in the form of a fully connected layer with two output neurons which takes the context vector produced by the Seq2Seq encoder as input. This layer is preceded by a gradient reversal layer that reverses the computed gradient sign during backpropagation. The domain discriminator is followed by a Softmax activation function, and its loss is computed using a Cross-Entropy loss function. Secondly, we used the ADANN \\cite{adann} training algorithm to train our model since it is designed to enable multi-domain adversarial training by considering, during the forward pass of each batch, one domain as a source domain and another random domain as a target domain, and so on for each of the available domains; our domains being the countries for which training data is available. We hope, by using this approach, to construct models that can learn to parse addresses from countries with different address formats without making a significant distinction between them. Therefore these models would perform better on parsing addresses from countries not seen during their training.\n\nWe use this approach with the two aforementioned embedding methods and name the obtained models \\textbf{fastTextADANN}{} and \\textbf{BPEmbADANN}{} respectively.\n\n\\section{Data}\n\\label{sec:data}\n\\subsection{Complete Address Dataset}\n\nWe use the same dataset as per \\cite{9357170}\\footnote{\\href{https:\/\/github.com\/GRAAL-Research\/deepparse-address-data}{\\url{https:\/\/github.com\/GRAAL-Research\/deepparse-address-data}}}, Deepparse dataset.\nWhich is built on the open-source dataset on which Libpostal's models were trained.\nLibpostal dataset is based on three open source addresses datasets: OpenStreetMap, Yahoo GeoPlanet and OpenAddresses. \nIt contains over 1 billion addresses from nearly 241 countries \\cite{libpostal}.\nUsing this dataset, \\cite{9357170} have selected 61 countries based on twofold: availability of at least 500 addresses, availability of the addresses components type.\nThe address data of those 61 countries are in their official language.\nFor example, Korean addresses are in Korean while Canada's addresses are in the two official languages, French and English. \nAmong the 61 selected countries, twenty countries are used for multinational training, and forty-one are used for zero-shot transfer evaluation. \nThe dataset uses the following eight tags: StreetNumber, StreetName, Unit, Municipality, Province, PostalCode, Orientation, and GeneralDelivery.\n\n\\subsection{Incomplete Address Dataset}\n\\label{subsec:iad}\nWe introduce a second dataset based on Deepparse dataset. It is similar to the Complete address dataset. \nIt is composed of the same twenty countries used for multinational training but with a sample size of \\num{50000} for training and \\num{25000} for holdout evaluation. \nThe dataset consists only of addresses where each one is missing at least one of the following four tags: StreetName, PostalCode, Municipality and Province. We consider an address incomplete if it is not composed of at least all of the four tags. \nFor example, the sequence of tags for the address ``221 B Baker Street\" is \\{StreetNumber, Unit, StreetName, StreetName\\}, and it is incomplete since the PostalCode and the Municipality tags are missing. We will refer to this dataset as the ``incomplete address\" dataset.\n\n\\section{Experiments}\n\\label{sec:exp}\nFor our experiments, as per \\cite{9357170} we trained each of our four models (\\fasttextatt{}, \\textbf{fastTextADANN}{}, \\textbf{BPEmbAtt}{} and \\textbf{BPEmbADANN}{}) five times\\footnote{Using each of the following seeds $\\{5, 10, 15, 20, 25\\}$. When a model did not converge (a high loss value on train and validation), we retrain the model using a different seed (30).} for 200 epochs with a batch size of \\num{512} for the base approach and the attention models and \\num{256} for the ADANN one. An early stopping with a patience of fifteen epochs was also applied during training. We initialize the learning rate at 0.1 and use learning rate scheduling to lower it by a factor of 0.1 after ten epochs without loss reduction. Our loss function of choice is the Cross-Entropy loss due to its suitability for the softmax function. The optimization is done through Stochastic Gradient Descent.\n\nAs per \\cite{9357170}, we use teacher forcing \\cite{6795228} to speed up the convergence. The architecture and the training of the models were implemented using Pytorch \\cite{pytorch}, and Poutyne \\cite{poutyne}.\n\n\\subsection{Evaluation Procedure}\nWe train our four models on our multinational dataset, the difference between the models being (1) the word embedding method employed (fastText and BPEmb) and (2) the use of attention mechanism or domain adaption learning. Each model has been trained five times, and we report the models' mean accuracy and standard deviation on the per-country zero-shot data. The accuracy for each sequence is computed as the proportion of the tags predicted correctly by the model. As such, predicting all the tags for a sequence correctly yields a perfect accuracy. More precisely, errors in tag predictions have an impact on the accuracy for a given sequence. However, the accuracy will not be null unless all the predicted tags for the sequence are incorrect. These results will be discussed in \\autoref{sec:completeresults}.\n\n\\subsection{Incomplete Address Evaluation Procedure}\nSince addresses do not always include all the components, we also evaluate our four models on the incomplete addresses dataset introduced in the Subsection~\\ref{subsec:iad}. We hypothesize that an incomplete address can confuse our models since we use a Seq2Seq architecture, and the compressed representation of an incomplete address will not be the same as the same complete one. For example, the address ``221 B Baker Street London NW1 6XE\" is complete and is a typical way to write an address. But, many addresses are not always in such a form. Such as the address ``221 B Baker Street\", which is the same as the previous one but without the city and the postal code. That difference can be more challenging for our models since postal code is usually a good way to tell the difference between the pattern. We will also evaluate the performance of two new models, \\textbf{fastTextADANNNoisy}{} and \\textbf{BPEmbADANNNoisy}{}), trained on the complete and incomplete addresses dataset to investigate if the addition of incomplete address help improves performance on that type of data. These results will be discussed in \\autoref{sec:missingresults}.\n\n\\section{Complete addresses Results}\n\\label{sec:completeresults}\nIn this section, we present and discuss the results of all our four trained models plus the two models in \\cite{9357170} (\\fasttext{} and \\textbf{BPEmb}{}). We first evaluated them on the holdout addresses dataset and the zero-shot addresses dataset.\n\n\\subsection{Multinational Evaluation}\n\\label{subsec:multieval}\n\\begin{table*}\n    \\centering\n    \\caption{Mean accuracy and standard deviation obtained with multinational models on the holdout dataset for training countries. \\textbf{bold} values indicate the best accuracy for each country.}\n    \\label{table:multinational-holdout}\n   \\resizebox{0.85\\textwidth}{!}{%\n    \\begin{tabular}{lrrrrrr}\n        \\toprule\n        Country & FastText & BPEmb & FastTextAtt & BPEmbAtt & FastTextADANN & BPEmbADANN\\\\\n        \\midrule\n        United States & $99.61 \\pm 0.09$ & $99.67 \\pm 0.09$ & $\\mathbf{99.73 \\pm 0.02}$ & $99.65 \\pm 0.23$ & $99.70 \\pm 0.03$ & $99.68 \\pm 0.16$\\\\\n        Brazil & $99.40 \\pm 0.10$ & $99.42 \\pm 0.15$ & $\\mathbf{99.58 \\pm 0.04}$ & $99.42 \\pm 0.39$ & $99.53 \\pm 0.04$ & $99.42 \\pm 0.34$\\\\\n        South Korea & $99.96 \\pm 0.01$ & $\\mathbf{100.00 \\pm 0.00}$ & $99.98 \\pm 0.01$ & $\\mathbf{100.00 \\pm 0.00}$ & $99.98 \\pm 0.01$ & $\\mathbf{100.00 \\pm 0.00}$\\\\\n        Australia & $99.68 \\pm 0.05$ & $\\mathbf{99.80 \\pm 0.05}$ & $99.77 \\pm 0.03$ & $99.78 \\pm 0.13$ & $99.76 \\pm 0.03$ & $99.77 \\pm 0.17$\\\\\n        Mexico & $99.60 \\pm 0.06$ & $99.68 \\pm 0.06$ & $\\mathbf{99.71 \\pm 0.03}$ & $99.70 \\pm 0.14$ & $99.68 \\pm 0.02$ & $99.69 \\pm 0.12$\\\\\n        Germany & $99.77 \\pm 0.04$ & $99.89 \\pm 0.03$ & $99.85 \\pm 0.02$ & $99.90 \\pm 0.08$ & $99.84 \\pm 0.01$ & $\\mathbf{99.91 \\pm 0.05}$\\\\\n        Spain & $99.75 \\pm 0.05$ & $99.85 \\pm 0.04$ & $99.83 \\pm 0.02$ & $\\mathbf{99.86 \\pm 0.09}$ & $99.80 \\pm 0.02$ & $99.83 \\pm 0.11$\\\\\n        Netherlands & $99.61 \\pm 0.07$ & $99.88 \\pm 0.03$ & $99.75 \\pm 0.03$ & $99.90 \\pm 0.07$ & $99.72 \\pm 0.02$ & $\\mathbf{99.91 \\pm 0.05}$\\\\\n        Canada & $99.79 \\pm 0.05$ & $\\mathbf{99.87 \\pm 0.04}$ & $99.87 \\pm 0.02$ & $99.87 \\pm 0.10$ & $99.85 \\pm 0.01$ & $99.86 \\pm 0.10$\\\\\n        Switzerland & $99.53 \\pm 0.09$ & $99.75 \\pm 0.08$ & $99.62 \\pm 0.05$ & $99.77 \\pm 0.14$ & $99.59 \\pm 0.05$ & $\\mathbf{99.82 \\pm 0.12}$\\\\\n        Poland & $99.69 \\pm 0.07$ & $99.89 \\pm 0.04$ & $99.80 \\pm 0.02$ & $99.90 \\pm 0.07$ & $99.78 \\pm 0.02$ & $\\mathbf{99.92 \\pm 0.04}$\\\\\n        Norway & $99.46 \\pm 0.06$ & $98.41 \\pm 0.63$ & $99.44 \\pm 0.11$ & $98.20 \\pm 1.13$ & $\\mathbf{99.53 \\pm 0.04}$ & $97.95 \\pm 0.44$\\\\\n        Austria & $99.28 \\pm 0.03$ & $98.98 \\pm 0.22$ & $\\mathbf{99.38 \\pm 0.06}$ & $98.96 \\pm 0.37$ & $99.30 \\pm 0.07$ & $99.34 \\pm 0.32$\\\\\n        Finland & $99.77 \\pm 0.03$ & $\\mathbf{99.87 \\pm 0.01}$ & $99.83 \\pm 0.02$ & $99.86 \\pm 0.01$ & $99.82 \\pm 0.01$ & $99.84 \\pm 0.01$\\\\\n        Denmark & $99.71 \\pm 0.07$ & $99.90 \\pm 0.03$ & $99.82 \\pm 0.03$ & $99.91 \\pm 0.06$ & $99.80 \\pm 0.02$ & $\\mathbf{99.93 \\pm 0.05}$\\\\\n        Czechia & $99.57 \\pm 0.09$ & $99.89 \\pm 0.04$ & $99.73 \\pm 0.02$ & $99.89 \\pm 0.10$ & $99.70 \\pm 0.02$ & $\\mathbf{99.90 \\pm 0.06}$\\\\\n        Italy & $99.73 \\pm 0.05$ & $99.81 \\pm 0.05$ & $99.83 \\pm 0.02$ & $\\mathbf{99.83 \\pm 0.11}$ & $99.80 \\pm 0.02$ & $99.82 \\pm 0.11$\\\\\n        France & $99.66 \\pm 0.08$ & $99.69 \\pm 0.11$ & $\\mathbf{99.79 \\pm 0.04}$ & $99.69 \\pm 0.22$ & $99.77 \\pm 0.03$ & $99.70 \\pm 0.17$\\\\\n        UK & $99.61 \\pm 0.10$ & $99.74 \\pm 0.08$ & $\\mathbf{99.77 \\pm 0.05}$ & $99.72 \\pm 0.20$ & $99.73 \\pm 0.03$ & $99.72 \\pm 0.20$\\\\\n        Russia & $99.03 \\pm 0.24$ & $\\mathbf{99.67 \\pm 0.11}$ & $99.40 \\pm 0.13$ & $99.54 \\pm 0.39$ & $99.23 \\pm 0.12$ & $99.59 \\pm 0.31$\\\\\n        \\midrule\n        Mean & $99.61 \\pm 0.20$ & $99.68 \\pm 0.36$ & $\\mathbf{99.72 \\pm 0.16}$ & $99.67 \\pm 0.40$ & $99.70 \\pm 0.18$ & $99.68 \\pm 0.43$\\\\\n        \\bottomrule\n    \\end{tabular}%\n    }\n\\end{table*}\n\n\\autoref{table:multinational-holdout} presents all the models' mean accuracy and standard deviation on the holdout dataset for training countries. First, we find that South Korea is the only country where a perfect accuracy was achieved when using byte-pairs embeddings (BPemb) or almost all the time (four seeds out of five) when using fastText embeddings. Since South Korea is the only country using a different pattern in the training set where the province and municipality occur before the street name, it seems that our models might have memorized this particular pattern. To validate this intuition, we randomly reordered \\num{6000} South Korean addresses to follow either the first (red) or the second (brown) address pattern (equally divided between the two). We observe, after this reordering, that the mean accuracy drops to $28.04~\\%$ considering that using a random tags annotation, we get a $12.29~\\%$ accuracy.\n\nIt is also interesting to notice that the models' accuracies are good when using fastText monolingual word embeddings, especially on South Korean addresses despite the entirely different alphabet. These results illustrate that our model, regardless of the embeddings model, learned the representation of an address sequence even if the words' representations are not native to the language (French vs Korean). \n\nFinally, all our models achieve state-of-the-art performance on our holdout dataset while using less data than previous approaches (e.g. Libpostal) and neither pre nor post-processing. However, at this point, it is difficult to conclude which of our models is the leading one. In the following subsection, we investigate the zero-shot performance of our models on countries not seen during training.\n\n\\subsection{Zero-Shot Evaluation}\nSince training a deep learning model to parse addresses from every country in the world would require a significant amount of data and resources, our ongoing work aims at achieving domain adaptation to be able to train on a reasonable amount of data and generalize to data from different sources. We begin by exploring how well our architecture can generalize in a zero-shot manner. To this end, we test each of our four models on address data from countries not seen during the training. The results are reported in \\autoref{table:zero-shot-results}.\n\n\\subsubsection{Attention Models}\nIn an attempt to improve our models' zero-shot performance, the base architecture was augmented with an attention mechanism as described in Subsection~\\ref{subsec:attention}. The results of those two models (\\fasttextatt{} and \\textbf{BPEmbAtt}{}) are compared to their base approach equivalents respectively in \\autoref{table:zero-shot-results}. For the model \\fasttextatt{} (left section of the table), we observe that attention mechanisms improve the performance for 20 out of 41 countries by more than $1~\\%$. Where 10 out of 41 improves with more than $2~\\%$. Also, 10 out of 41 countries' accuracies were increased by less than $1~\\%$. However, for the other countries (11 out of 41), we observe that the accuracy is less than $0.5~\\%$ poorer than the base approach for most of them. The attention mechanism improved two countries' accuracies above the $90~\\%$ accuracy (Belgium and Lithuania) and one above the $80~\\%$ (Philippines). \\textbf{FastTextAtt} achieves good results for $83~\\%$ of the countries (34), almost $52~\\%$ of which (21) are near state-of-the-art performance. \n\nIn contrast, the results for the \\textbf{BPEmbAtt}{} model (right section of the table) are not as good. We observe that results were be increased by at least more than $1~\\%$ for 14 out of 41 countries. While 11 out of 41 saw improvement by more than $2~\\%$. Also, 9 out of 41 countries improved by less than $1~\\%$. However, for the other countries (20), we observe that the accuracy is less than $0.5~\\%$ poorer than the base approach for half of them. For the other half, results are between $1~\\%$ and $2~\\%$ lower, thus lowering performance for two countries (India and Bangladesh) below $80~\\%$ and one (Malaysia) below $90~\\%$. Hence, the use of attention mechanism with byte-pair multilingual embeddings lowers performance overall, especially since only one country (Estonia) yields results above the $80~\\%$ compared to the base approach. \\textbf{BPEmbAtt}{} achieves good results for $78~\\%$ of the countries (32), almost $50~\\%$ of which (20) is near state-of-the-art performance. \n\nFinally, we observe that in some cases, the use of an attention mechanism can substantially improve performance, such as Greece where the increase is $4~\\%$ for \\fasttextatt{} and near $16~\\%$ for \\textbf{BPEmbAtt}{}. We also observe a smaller variance for both the models using attention mechanisms, meaning those models are more stable during training and converge to a better optimum. Overall, these results show that our attention mechanisms architecture can generally yield better accuracies during training.\n\n\\begin{table*}\n    \\centering\n    \\caption{Mean accuracy (and standard deviation) per country for zero-shot transfer models - base approach versus attention models. \\textbf{bold} values indicate the best accuracy for each embeddings type.}\n    \\label{table:zero-shot-results}\n    \\resizebox{\\textwidth}{!}{%\n    \\begin{tabular}{lrr|rrlrr|rr}\n    \\toprule\n    Country & FastText & FastTextAtt & BPEmb & BPEmbAtt & Country & FastText & FastTextAtt & BPEmb & BPEmbAtt\\\\\n    \\midrule\n    Belgium & $88.14 \\pm 1.04$ & $\\mathbf{90.01 \\pm 1.59}$ & $87.29 \\pm 1.40$ & $\\mathbf{89.35 \\pm 0.48}$ & Faroe Islands & $\\mathbf{74.14 \\pm 1.83}$ & $74.00 \\pm 1.00$ & $85.50 \\pm 0.11$ & $\\mathbf{85.89 \\pm 1.08}$\\\\\n    Sweden & $81.59 \\pm 4.53$ & $\\mathbf{85.73 \\pm 2.30}$ & $\\mathbf{90.76 \\pm 3.03}$ & $90.64 \\pm 5.07$ & R\u00e9union & $96.80 \\pm 0.45$ & $\\mathbf{96.86 \\pm 1.27}$ & $93.67 \\pm 0.26$ & $\\mathbf{94.19 \\pm 0.27}$\\\\\n    Argentina & $86.26 \\pm 0.47$ & $\\mathbf{87.76 \\pm 0.71}$ & $\\mathbf{88.04 \\pm 0.83}$ & $87.45 \\pm 2.57$ & Moldova & $90.18 \\pm 0.79$ & $\\mathbf{91.25 \\pm 0.84}$ & $86.89 \\pm 3.01$ & $\\mathbf{89.99 \\pm 2.22}$\\\\\n    India & $69.09 \\pm 1.74$ & $\\mathbf{71.94 \\pm 2.06}$ & $\\mathbf{80.04 \\pm 3.24}$ & $79.73 \\pm 3.65$ & Indonesia & $64.31 \\pm 0.84$ & $\\mathbf{65.98 \\pm 0.97}$ & $70.28 \\pm 1.64$ & $\\mathbf{72.05 \\pm 2.81}$\\\\\n    Romania & $94.49 \\pm 1.52$ & $\\mathbf{96.01 \\pm 0.73}$ & $91.65 \\pm 1.21$ & $\\mathbf{93.01 \\pm 0.84}$ & Bermuda & $92.31 \\pm 0.60$ & $\\mathbf{92.82 \\pm 0.68}$ & $93.70 \\pm 0.35$ & $\\mathbf{93.82 \\pm 0.08}$\\\\\n    Slovakia & $82.10 \\pm 0.98$ & $\\mathbf{84.16 \\pm 2.60}$ & $90.31 \\pm 3.88$ & $\\mathbf{92.60 \\pm 4.64}$ & Malaysia & $\\mathbf{78.93 \\pm 3.78}$ & $75.79 \\pm 2.77$ & $\\mathbf{94.16 \\pm 0.49}$ & $89.06 \\pm 5.22$\\\\\n    Hungary & $\\mathbf{48.92 \\pm 3.59}$ & $48.48 \\pm 4.55$ & $\\mathbf{25.51 \\pm 2.60}$ & $24.00 \\pm 1.09$ & South Africa & $\\mathbf{95.31 \\pm 1.68}$ & $94.82 \\pm 2.79$ & $\\mathbf{96.87 \\pm 0.96}$ & $96.83 \\pm 2.11$\\\\\n    Japan & $41.41 \\pm 3.21$ & $\\mathbf{42.63 \\pm 4.13}$ & $35.33 \\pm 1.28$ & $\\mathbf{42.81 \\pm 5.73}$ & Latvia & $\\mathbf{93.66 \\pm 0.64}$ & $93.04 \\pm 0.85$ & $\\mathbf{74.78 \\pm 4.33}$ & $72.14 \\pm 2.00$\\\\\n    Iceland & $\\mathbf{96.55 \\pm 1.20}$ & $96.51 \\pm 0.50$ & $\\mathbf{97.38 \\pm 1.18}$ & $96.77 \\pm 0.43$ & Kazakhstan & $86.33 \\pm 3.06$ & $\\mathbf{89.52 \\pm 4.82}$ & $\\mathbf{94.12 \\pm 1.94}$ & $92.60 \\pm 5.40$\\\\\n    Venezuela & $94.87 \\pm 0.53$ & $\\mathbf{95.37 \\pm 0.45}$ & $93.05 \\pm 2.02$ & $\\mathbf{93.79 \\pm 3.07}$ & New Caledonia & $99.48 \\pm 0.15$ & $\\mathbf{99.52 \\pm 0.08}$ & $99.25 \\pm 0.19$ & $\\mathbf{99.38 \\pm 0.34}$\\\\\n    Philippines & $77.76 \\pm 3.97$ & $\\mathbf{82.87 \\pm 5.17}$ & $\\mathbf{81.95 \\pm 8.07}$ & $81.30 \\pm 3.50$ & Estonia & $87.08 \\pm 1.89$ & $\\mathbf{89.29 \\pm 2.49}$ & $77.30 \\pm 1.22$ & $\\mathbf{80.24 \\pm 5.49}$\\\\\n    Slovenia & $95.37 \\pm 0.23$ & $\\mathbf{95.74 \\pm 0.44}$ & $\\mathbf{97.47 \\pm 0.45}$ & $97.04 \\pm 0.38$ & Singapore & $\\mathbf{86.42 \\pm 2.36}$ & $86.40 \\pm 2.59$ & $86.87 \\pm 2.01$ & $\\mathbf{87.53 \\pm 1.30}$\\\\\n    Ukraine & $92.99 \\pm 0.70$ & $\\mathbf{93.45 \\pm 0.76}$ & $92.60 \\pm 1.84$ & $\\mathbf{93.52 \\pm 1.81}$ & Bangladesh & $\\mathbf{78.61 \\pm 0.43}$ & $76.48 \\pm 1.06$ & $\\mathbf{82.45 \\pm 2.54}$ & $79.01 \\pm 2.18$\\\\\n    Belarus & $91.08 \\pm 3.08$ & $\\mathbf{94.43 \\pm 3.59}$ & $\\mathbf{96.40 \\pm 1.76}$ & $94.44 \\pm 4.19$ & Paraguay & $96.01 \\pm 1.23$ & $\\mathbf{96.68 \\pm 0.32}$ & $\\mathbf{97.20 \\pm 0.35}$ & $96.25 \\pm 1.17$\\\\\n    Serbia & $95.31 \\pm 0.48$ & $\\mathbf{95.68 \\pm 0.33}$ & $92.62 \\pm 3.83$ & $\\mathbf{93.09 \\pm 1.81}$ & Cyprus & $\\mathbf{97.67 \\pm 0.34}$ & $97.57 \\pm 0.31$ & $94.31 \\pm 7.21$ & $\\mathbf{98.32 \\pm 0.37}$\\\\\n    Croatia & $94.59 \\pm 2.21$ & $\\mathbf{96.40 \\pm 0.57}$ & $88.04 \\pm 4.68$ & $\\mathbf{90.66 \\pm 3.76}$ & Bosnia & $84.04 \\pm 1.47$ & $\\mathbf{87.42 \\pm 1.95}$ & $84.46 \\pm 5.76$ & $\\mathbf{88.61 \\pm 5.06}$\\\\\n    Greece & $81.98 \\pm 0.60$ & $\\mathbf{85.00 \\pm 1.61}$ & $40.97 \\pm 14.89$ & $\\mathbf{56.01 \\pm 10.98}$ & Ireland & $87.44 \\pm 0.69$ & $\\mathbf{87.69 \\pm 0.95}$ & $86.49 \\pm 1.31$ & $\\mathbf{87.56 \\pm 3.01}$\\\\\n    New Zealand & $94.27 \\pm 1.50$ & $\\mathbf{95.91 \\pm 1.41}$ & $\\mathbf{99.44 \\pm 0.29}$ & $98.21 \\pm 1.37$ & Algeria & $85.37 \\pm 2.05$ & $\\mathbf{86.03 \\pm 1.55}$ & $84.65 \\pm 4.47$ & $\\mathbf{85.08 \\pm 2.50}$\\\\\n    Portugal & $93.65 \\pm 0.46$ & $\\mathbf{94.54 \\pm 0.67}$ & $92.68 \\pm 1.46$ & $\\mathbf{93.33 \\pm 0.59}$ & Colombia & $87.81 \\pm 0.92$ & $\\mathbf{89.09 \\pm 0.64}$ & $\\mathbf{89.51 \\pm 0.88}$ & $88.32 \\pm 2.55$\\\\\n    Bulgaria & $\\mathbf{91.03 \\pm 2.07}$ & $90.87 \\pm 2.63$ & $\\mathbf{93.47 \\pm 3.07}$ & $92.97 \\pm 3.66$ & Uzbekistan & $86.76 \\pm 1.13$ & $\\mathbf{87.36 \\pm 0.74}$ & $75.18 \\pm 1.92$ & $\\mathbf{77.52 \\pm 2.62}$\\\\\n    Lithuania & $87.67 \\pm 3.05$ & $\\mathbf{90.88 \\pm 1.73}$ & $\\mathbf{76.41 \\pm 1.66}$ & $76.16 \\pm 1.54$ &  && & &\\\\\n    \\bottomrule\n    \\end{tabular}%\n    }\n\\end{table*}\n\n\\subsubsection{ADANN}\nIn a second attempt to improve our models' zero-shot performance, the base architecture was augmented with a domain adaptation approach as described in Subsection~\\ref{subsec:adann}. The results of those two models (\\textbf{fastTextADANN}{} and \\textbf{BPEmbADANN}{}) are compared to their attention approach equivalents in \\autoref{table:zero-shot-results}. For both the \\textbf{fastTextADANN}{} and \\textbf{BPEmbADANN}{} models (left and right section of the table respectively), we observe similar results. \n\nFirst, the ADANN algorithm was able to improve the performance for a minority (4 and 8 respectively for \\textbf{fastTextADANN}{} and \\textbf{BPEmbADANN}{} respectively) of the countries by more than $1~\\%$. A few (2 or 3) out of the 41 improve by more than $2~\\%$ for both models. Also, near a fourth of the 41 countries' accuracies increased by less than $1~\\%$. For the other countries (the majority), we observe that for half of them, the difference is less than $1~\\%$ poorer than the attention approaches, and the other half is a couple of percent poorer. Sometimes the difference can be as much as $5~\\%$ (e.g. Sweden for \\textbf{fastTextADANN}{} or Estonia for \\textbf{BPEmbADANN}{}). Also, for both models, two countries drop below $80~\\%$ (Lithuania and Moldova for \\textbf{fastTextADANN}{} and Philippines and Bosnia for \\textbf{BPEmbADANN}{}) and for \\textbf{BPEmbADANN}{} Sweden accuracy's drops below $90~\\%$. Thus, the use of domain adaptation technique during training lowers the performance overall, especially since only one country for each model (Malaysia and India respectively) yields results above $80~\\%$ compared to the base approach and one above $90~\\%$ (Kazakhstan and Malaysia respectively). Although, overall, the two models achieve good results for nearly $80~\\%$ of the countries, almost $50~\\%$ of which are near state-of-the-art performance. However, these results are less good than models using attention mechanisms in many cases.\n\nSecond, we observe that our model does not seem to have fit the training data as much as possible, as shown in \\autoref{table:multinational-holdout-attention}. This table presents the multinational models' mean accuracy (and a standard deviation) on the holdout dataset for \\textbf{fastTextADANN}{}. Nevertheless, we are surprised by our results since the ADANN algorithm is a transfer learning technique. An advantage of ADANN is that the network's weights have strong incentives to be subject-agnostic, meaning that the learned representation extracted from the network can be thought of as general features for the prediction layer \\cite{adann}. We argue that it is more challenging to train models using the ADANN algorithm since the time needed for one epoch is nearly 5 hours. Meaning that the expected time to train for the 200 epochs is near 41 days per model (and we train five seeds) compared to a couple of days for the attention models. So we did not have much opportunity to fine-tune our model. We also hypothesize that the domain choice (i.e. the country) might be too granular since many countries have similar patterns or similar language, making the task more difficult. An idea of improvement could be to use more definitions for the domain, such as the language and the address pattern type. For example, we could use a categorical variable representing the address pattern number and a categorical variable for the language. That way, we could help guide the training into a better understanding of the context of an address which is not necessary the country but rather the language and the pattern. \n\nFinally, on average, performance is similar between the attention and ADANN approaches, but FastText models perform slightly better. They are, on average, $2~\\%$ better than the BPEmb approaches. For example, \\textbf{fastTextADANN}{} yields in average $86.45 \\pm 12.13~\\%$ accuracy across the zero-shot countries and \\textbf{BPEmbADANN}{} yields a $85.32 \\pm 15.31~\\%$. Again, both BPEmb models have higher variance than FastText one. These results show that BPEmb models' performance tends to be more variable and more sensitive to changes in seeds.\n\n\\begin{table*}\n    \\centering\n    \\caption{Mean accuracy (and standard deviation) per country for zero-shot transfer models - attention models versus ADANN models.}\n    \\label{table:zero-shot-results-adann}\n    \\resizebox{\\textwidth}{!}{%\n    \\begin{tabular}{lrr|rrlrr|rr}\n    \\toprule\n    Country & FastTextAtt & FastTextADANN & BPEmbAtt & BPEmbADANN & Country & FastTextAtt & FastTextADANN & BPEmbAtt & BPEmbADANN\\\\\n    \\midrule\n    Belgium & $90.01 \\pm 1.59$ & $\\mathbf{90.90 \\pm 3.76}$ & $\\mathbf{89.35 \\pm 0.48}$ & $88.74 \\pm 0.59$ & Faroe Islands & $\\mathbf{74.00 \\pm 1.00}$ & $69.76 \\pm 2.08$ & $\\mathbf{85.89 \\pm 1.08}$ & $85.34 \\pm 0.20$\\\\\n    Sweden & $\\mathbf{85.73 \\pm 2.30}$ & $80.99 \\pm 3.78$ & $\\mathbf{90.64 \\pm 5.07}$ & $89.65 \\pm 1.08$ & R\u00e9union & $\\mathbf{96.86 \\pm 1.27}$ & $96.48 \\pm 0.96$ & $\\mathbf{94.19 \\pm 0.27}$ & $93.99 \\pm 0.59$\\\\\n    Argentina & $\\mathbf{87.76 \\pm 0.71}$ & $87.72 \\pm 1.33$ & $\\mathbf{87.45 \\pm 2.57}$ & $87.13 \\pm 2.65$ & Moldova & $\\mathbf{91.25 \\pm 0.84}$ & $89.87 \\pm 0.56$ & $\\mathbf{89.99 \\pm 2.22}$ & $88.67 \\pm 1.66$\\\\\n    India & $\\mathbf{71.94 \\pm 2.06}$ & $70.30 \\pm 1.22$ & $79.73 \\pm 3.65$ & $\\mathbf{81.12 \\pm 2.97}$ & Indonesia & $\\mathbf{65.98 \\pm 0.97}$ & $65.42 \\pm 0.75$ & $\\mathbf{72.05 \\pm 2.81}$ & $71.71 \\pm 2.79$\\\\\n    Romania & $96.01 \\pm 0.73$ & $\\mathbf{96.22 \\pm 1.11}$ & $\\mathbf{93.01 \\pm 0.84}$ & $92.58 \\pm 1.53$ & Bermuda & $\\mathbf{92.82 \\pm 0.68}$ & $92.02 \\pm 0.52$ & $93.82 \\pm 0.08$ & $\\mathbf{95.02 \\pm 0.58}$\\\\\n    Slovakia & $\\mathbf{84.16 \\pm 2.60}$ & $81.93 \\pm 1.80$ & $\\mathbf{92.60 \\pm 4.64}$ & $91.14 \\pm 3.81$ & Malaysia & $75.79 \\pm 2.77$ & $\\mathbf{83.48 \\pm 3.42}$ & $89.06 \\pm 5.22$ & $\\mathbf{90.52 \\pm 4.27}$\\\\\n    Hungary & $\\mathbf{48.48 \\pm 4.55}$ & $46.42 \\pm 5.62$ & $\\mathbf{24.00 \\pm 1.09}$ & $23.54 \\pm 1.69$ & South Africa & $\\mathbf{94.82 \\pm 2.79}$ & $93.09 \\pm 2.35$ & $\\mathbf{96.83 \\pm 2.11}$ & $96.02 \\pm 1.55$\\\\\n    Japan & $42.63 \\pm 4.13$ & $\\mathbf{43.88 \\pm 1.26}$ & $\\mathbf{42.81 \\pm 5.73}$ & $38.64 \\pm 5.26$ & Latvia & $\\mathbf{93.04 \\pm 0.85}$ & $90.64 \\pm 1.26$ & $\\mathbf{72.14 \\pm 2.00}$ & $71.92 \\pm 2.60$\\\\\n    Iceland & $\\mathbf{96.51 \\pm 0.50}$ & $96.18 \\pm 0.35$ & $96.77 \\pm 0.43$ & $\\mathbf{97.15 \\pm 0.64}$ & Kazakhstan & $89.52 \\pm 4.82$ & $\\mathbf{92.00 \\pm 1.36}$ & $92.60 \\pm 5.40$ & $\\mathbf{96.49 \\pm 1.94}$\\\\\n    Venezuela & $\\mathbf{95.37 \\pm 0.45}$ & $95.02 \\pm 0.76$ & $93.79 \\pm 3.07$ & $\\mathbf{94.79 \\pm 0.57}$ & New Caledonia & $\\mathbf{99.52 \\pm 0.08}$ & $99.52 \\pm 0.11$ & $99.38 \\pm 0.34$ & $\\mathbf{99.40 \\pm 0.26}$\\\\\n    Philippines & $82.87 \\pm 5.17$ & $\\mathbf{83.81 \\pm 4.95}$ & $\\mathbf{81.30 \\pm 3.50}$ & $74.94 \\pm 5.57$ & Estonia & $\\mathbf{89.29 \\pm 2.49}$ & $86.72 \\pm 4.00$ & $80.24 \\pm 5.49$ & $\\mathbf{85.84 \\pm 4.87}$\\\\\n    Slovenia & $95.74 \\pm 0.44$ & $\\mathbf{95.75 \\pm 0.53}$ & $\\mathbf{97.04 \\pm 0.38}$ & $96.77 \\pm 0.57$ & Singapore & $\\mathbf{86.40 \\pm 2.59}$ & $84.96 \\pm 2.84$ & $\\mathbf{87.53 \\pm 1.30}$ & $84.70 \\pm 1.86$\\\\\n    Ukraine & $93.45 \\pm 0.76$ & $\\mathbf{93.70 \\pm 0.28}$ & $93.52 \\pm 1.81$ & $\\mathbf{94.17 \\pm 2.30}$ & Bangladesh & $76.48 \\pm 1.06$ & $\\mathbf{76.80 \\pm 1.21}$ & $\\mathbf{79.01 \\pm 2.18}$ & $78.68 \\pm 3.96$\\\\\n    Belarus & $94.43 \\pm 3.59$ & $\\mathbf{96.05 \\pm 0.57}$ & $94.44 \\pm 4.19$ & $\\mathbf{98.15 \\pm 0.97}$ & Paraguay & $96.68 \\pm 0.32$ & $\\mathbf{97.00 \\pm 1.08}$ & $96.25 \\pm 1.17$ & $\\mathbf{96.28 \\pm 0.96}$\\\\\n    Serbia & $95.68 \\pm 0.33$ & $\\mathbf{95.75 \\pm 0.25}$ & $93.09 \\pm 1.81$ & $\\mathbf{93.45 \\pm 1.44}$ & Cyprus & $\\mathbf{97.57 \\pm 0.31}$ & $97.39 \\pm 0.38$ & $\\mathbf{98.32 \\pm 0.37}$ & $97.53 \\pm 1.49$\\\\\n    Croatia & $\\mathbf{96.40 \\pm 0.57}$ & $94.98 \\pm 2.01$ & $90.66 \\pm 3.76$ & $\\mathbf{91.37 \\pm 4.39}$ & Bosnia & $\\mathbf{87.42 \\pm 1.95}$ & $84.95 \\pm 3.47$ & $\\mathbf{88.61 \\pm 5.06}$ & $79.11 \\pm 3.07$\\\\\n    Greece & $\\mathbf{85.00 \\pm 1.61}$ & $83.18 \\pm 3.31$ & $56.01 \\pm 10.98$ & $\\mathbf{57.47 \\pm 7.75}$ & Ireland & $\\mathbf{87.69 \\pm 0.95}$ & $87.44 \\pm 0.20$ & $87.56 \\pm 3.01$ & $\\mathbf{88.42 \\pm 0.97}$\\\\\n    New Zealand & $\\mathbf{95.91 \\pm 1.41}$ & $93.60 \\pm 2.42$ & $98.21 \\pm 1.37$ & $\\mathbf{99.13 \\pm 0.37}$ & Algeria & $\\mathbf{86.03 \\pm 1.55}$ & $83.07 \\pm 3.85$ & $\\mathbf{85.08 \\pm 2.50}$ & $83.64 \\pm 2.60$\\\\\n    Portugal & $\\mathbf{94.54 \\pm 0.67}$ & $94.53 \\pm 0.41$ & $\\mathbf{93.33 \\pm 0.59}$ & $91.07 \\pm 1.26$ & Colombia & $\\mathbf{89.09 \\pm 0.64}$ & $87.76 \\pm 1.30$ & $88.32 \\pm 2.55$ & $\\mathbf{88.99 \\pm 1.83}$\\\\\n    Bulgaria & $\\mathbf{90.87 \\pm 2.63}$ & $90.40 \\pm 1.10$ & $92.97 \\pm 3.66$ & $\\mathbf{93.93 \\pm 2.59}$ & Uzbekistan & $\\mathbf{87.36 \\pm 0.74}$ & $86.23 \\pm 2.18$ & $\\mathbf{77.52 \\pm 2.62}$ & $73.83 \\pm 2.92$\\\\\n    Lithuania & $\\mathbf{90.88 \\pm 1.73}$ & $88.58 \\pm 2.35$ & $76.16 \\pm 1.54$ & $\\mathbf{77.26 \\pm 2.54}$ & &&&\\\\\n    \\bottomrule\n    \\end{tabular}%\n    }\n\\end{table*}\n\n\\begin{table}\n    \\centering\n    \\caption{Mean accuracy (and standard deviation) for multinational models on holdout dataset for training countries - \\fasttextatt{} versus \\textbf{fastTextADANN}{}.}\n    \\label{table:multinational-holdout-attention}\n    \\resizebox{0.49\\textwidth}{!}{%\n    \\begin{tabular}{lrrlrr}\n    \\toprule\n    Country & FastTextAtt & FastTextADANN & Country & FastTextAtt & FastTextADANN\\\\\n    \\midrule\n    United States & $\\mathbf{99.73 \\pm 0.02}$ & $99.70 \\pm 0.03$ & Poland & $\\mathbf{99.80 \\pm 0.02}$ & $99.78 \\pm 0.02$\\\\\n    Brazil & $\\mathbf{99.58 \\pm 0.04}$ & $99.53 \\pm 0.04$ & Norway & $99.44 \\pm 0.11$ & $\\mathbf{99.53 \\pm 0.04}$\\\\\n    South Korea & $\\mathbf{99.98 \\pm 0.01}$ & $\\mathbf{99.98 \\pm 0.01}$ & Austria & $\\mathbf{99.38 \\pm 0.06}$ & $99.30 \\pm 0.07$\\\\\n    Australia & $\\mathbf{99.77 \\pm 0.03}$ & $99.76 \\pm 0.03$ & Finland & $\\mathbf{99.83 \\pm 0.02}$ & $99.82 \\pm 0.01$\\\\\n    Mexico & $\\mathbf{99.71 \\pm 0.03}$ & $99.68 \\pm 0.02$ & Denmark & $\\mathbf{99.82 \\pm 0.03}$ & $99.80 \\pm 0.02$\\\\\n    Germany & $\\mathbf{99.85 \\pm 0.02}$ & $99.84 \\pm 0.01$ & Czechia & $\\mathbf{99.73 \\pm 0.02}$ & $99.70 \\pm 0.02$\\\\\n    Spain & $\\mathbf{99.83 \\pm 0.02}$ & $99.80 \\pm 0.02$ & Italy & $\\mathbf{99.83 \\pm 0.02}$ & $99.80 \\pm 0.02$\\\\\n    Netherlands & $\\mathbf{99.75 \\pm 0.03}$ & $99.72 \\pm 0.02$ & France & $\\mathbf{99.79 \\pm 0.04}$ & $99.77 \\pm 0.03$\\\\\n    Canada & $\\mathbf{99.87 \\pm 0.02}$ & $99.85 \\pm 0.01$ & UK & $\\mathbf{99.77 \\pm 0.05}$ & $99.73 \\pm 0.03$\\\\\n    Switzerland & $\\mathbf{99.62 \\pm 0.05}$ & $99.59 \\pm 0.05$ & Russia & $\\mathbf{99.40 \\pm 0.13}$ & $99.23 \\pm 0.12$\\\\\n    \\bottomrule\n    \\end{tabular}%\n    }\n\\end{table}\n\n\\section{Missing Values Handling}\n\\label{sec:missingresults}\nIn this section, we aim to evaluate and improve the results of our four best models, \\fasttextatt{}, \\textbf{BPEmbAtt}{}, \\textbf{fastTextADANN}{} and \\textbf{BPEmbADANN}{} on the incomplete address data. As presented in Subsection~\\ref{subsec:iad}, we introduce an incomplete address dataset where the addresses do not always include all the components. \\autoref{table:incomplete-holdout} presents the results of the four models evaluated on the incomplete holdout dataset without any prior training on incomplete addresses. Since we observe that performances for all of the training countries are lower by $20~\\%$ to $40~\\%$ than previous scores, we choose to only evaluate our models on the countries seen during training (holdout). The lower accuracy is that of South Korea for both the models (nearly over the $45~\\%$ accuracy)(will be discussed in more detail later). We also observe that the ADANN approach yields better results (12 out of 20) and is better by less than $1~\\%$ on average (last row). Finally, we observe that the BPEmb approach still has higher variance than the FastText one since we have observed some seeds converging to suboptimal loss. Nevertheless, we observe that despite poorer performance on zero-shot evaluation, the ADANN approaches yield better results on incomplete addresses than attention approaches. We hypothesize that ADANN models have not overfitted the representation of an address structure seen in the training and more on the general features of address structures and domain type (e.g. the country and language). Also, knowing the domain, and indirectly the address structure and language, makes it easier to parse an address when incomplete.\n\n\\begin{table*}\n    \\centering\n    \\caption{Mean accuracy (and standard deviation) for multinational models without any prior training on incomplete dataset.}\n    \\label{table:incomplete-holdout}\n        \\resizebox{0.6\\textwidth}{!}{%\n    \\begin{tabular}{lrrrr}\n    \\toprule\n    Country & FastTextAtt & FastTextADANN & BPEmbAtt & BPEmbADANN\\\\\n    \\midrule\n    United States & $\\mathbf{68.59 \\pm 2.61}$ & $65.57 \\pm 0.80$ & $66.14 \\pm 3.88$ & $\\mathbf{67.94 \\pm 4.94}$\\\\\n    Brazil & $57.00 \\pm 2.41$ & $\\mathbf{57.39 \\pm 3.63}$ & $\\mathbf{55.01 \\pm 3.90}$ & $51.10 \\pm 4.08$\\\\\n    South Korea & $45.91 \\pm 4.22$ & $\\mathbf{46.28 \\pm 2.29}$ & $\\mathbf{49.84 \\pm 8.17}$ & $35.85 \\pm 5.32$\\\\\n    Australia & $75.90 \\pm 1.28$ & $\\mathbf{76.02 \\pm 0.36}$ & $\\mathbf{77.18 \\pm 0.80}$ & $76.49 \\pm 1.56$\\\\\n    Mexico & $\\mathbf{61.52 \\pm 2.42}$ & $59.88 \\pm 2.83$ & $\\mathbf{61.25 \\pm 1.42}$ & $58.93 \\pm 6.36$\\\\\n    Germany & $48.39 \\pm 1.40$ & $\\mathbf{50.32 \\pm 6.96}$ & $\\mathbf{58.98 \\pm 0.86}$ & $58.62 \\pm 3.31$\\\\\n    Spain & $70.74 \\pm 1.26$ & $\\mathbf{72.39 \\pm 1.13}$ & $79.51 \\pm 0.49$ & $\\mathbf{81.05 \\pm 1.17}$\\\\\n    Netherlands & $59.72 \\pm 7.54$ & $\\mathbf{63.74 \\pm 3.47}$ & $75.34 \\pm 1.32$ & $\\mathbf{77.39 \\pm 3.52}$\\\\\n    Canada & $\\mathbf{72.77 \\pm 0.76}$ & $71.04 \\pm 1.98$ & $73.00 \\pm 2.51$ & $\\mathbf{81.29 \\pm 3.22}$\\\\\n    Switzerland & $\\mathbf{64.13 \\pm 6.21}$ & $62.66 \\pm 6.31$ & $\\mathbf{73.08 \\pm 0.89}$ & $72.55 \\pm 3.22$\\\\\n    Poland & $43.99 \\pm 3.35$ & $\\mathbf{47.04 \\pm 4.06}$ & $48.50 \\pm 1.08$ & $\\mathbf{54.47 \\pm 4.73}$\\\\\n    Norway & $\\mathbf{64.69 \\pm 9.77}$ & $62.67 \\pm 8.64$ & $76.43 \\pm 1.62$ & $\\mathbf{78.28 \\pm 4.04}$\\\\\n    Austria & $\\mathbf{69.33 \\pm 6.67}$ & $69.21 \\pm 5.21$ & $77.69 \\pm 1.08$ & $\\mathbf{78.56 \\pm 2.39}$\\\\\n    Finland & $57.94 \\pm 8.91$ & $\\mathbf{60.83 \\pm 6.98}$ & $74.65 \\pm 1.36$ & $\\mathbf{76.31 \\pm 3.46}$\\\\\n    Denmark & $\\mathbf{56.94 \\pm 3.24}$ & $55.54 \\pm 5.48$ & $\\mathbf{65.32 \\pm 2.40}$ & $63.73 \\pm 3.04$\\\\\n    Czechia & $61.16 \\pm 3.21$ & $\\mathbf{64.40 \\pm 2.91}$ & $72.58 \\pm 1.72$ & $\\mathbf{74.79 \\pm 3.53}$\\\\\n    Italy & $66.42 \\pm 2.49$ & $\\mathbf{71.16 \\pm 1.67}$ & $73.89 \\pm 0.77$ & $\\mathbf{76.33 \\pm 1.46}$\\\\\n    France & $70.15 \\pm 0.88$ & $\\mathbf{73.14 \\pm 2.38}$ & $71.91 \\pm 3.46$ & $\\mathbf{77.45 \\pm 1.08}$\\\\\n    UK & $53.63 \\pm 1.28$ & $\\mathbf{54.58 \\pm 1.24}$ & $51.27 \\pm 1.96$ & $\\mathbf{53.62 \\pm 3.30}$\\\\\n    Russia & $\\mathbf{58.41 \\pm 1.12}$ & $57.94 \\pm 0.99$ & $58.80 \\pm 5.21$ & $\\mathbf{61.91 \\pm 2.86}$\\\\\\midrule\n    Mean  &       $61.37 \\pm 8.65$&         $\\mathbf{62.09 \\pm 8.41}$ &    $67.02 \\pm 9.91$ &      $\\mathbf{67.83 \\pm 12.15}$ \\\\\n    \\bottomrule\n    \\end{tabular}%\n    }\n\\end{table*}\n\nTo improve our models' incomplete addresses performance, we have trained two new models using the complete and incomplete addresses datasets. This merged dataset consists of \\num{150000} addresses per country using the same settings as described in \\autoref{sec:exp}. We choose to only train the two best models on incomplete addresses, \\textbf{fastTextADANN}{} and \\textbf{BPEmbADANN}{}. We refer to these new models by \\textbf{fastTextADANNNoisy}{} and \\textbf{BPEmbADANNNoisy}{} where the difference between the two is the embeddings. \\autoref{table:incomplete-holdout-train-clean-noisy} present the mean accuracy and a standard deviation of our two models tested on the incomplete dataset. \n\nFirst, we observe that using incomplete addresses during training substantially improves the accuracy for all the countries. We also observe that the \\textbf{fastTextADANNNoisy}{} is the leading model across the board with the best accuracy on all the twenty countries, where results are nearly always more than $98~\\%$ in accuracy. In contrast, we observe that, again, despite using an embeddings layer to learn the representation of the byte-pairs embeddings (BPEmb), \\textbf{BPEmbADANNNoisy}{} shows poor results compare to \\textbf{fastTextADANNNoisy}{}. Results are, on average, nearly $6~\\%$ worse and have a higher variance of more than the double, with results as low as $84~\\%$ (which is lower than some results observe in zero-shot evaluation). Again, this shows that the trained embeddings layer is overfitting or that the byte-pair embeddings are not well suited for address (such as the embeddings of postal code). This also could mean that the French fastText embeddings approach to construct out-of-vocabulary embeddings are more generalized than the one that we retrain using multi-lingual embeddings.\n\nSecond, it is interesting to note that even with a relatively large number of incomplete addresses in the training dataset (\\num{50000}), we did not achieve scores as good as seen with the complete dataset (\\autoref{table:multinational-holdout}). Results are in average of $98~\\%$ and near $93~\\%$ for \\textbf{fastTextADANN}{} and \\textbf{BPEmbADANN}{} respectively. Also, accuracies on the complete addresses holdout dataset are nearly as good as those presented in \\autoref{table:multinational-holdout}. We observe similar results for most of the training countries with a difference of $0.5~\\%$ lower.\n\nFinally, we can see that the worse results for both models are for South Korea. These results contrast with the nearly perfect score observed for all the models in \\autoref{table:multinational-holdout}. This highlights our hypothesis (Subsection~\\ref{subsec:multieval}) that our model has memorized the particular pattern of South Korea during training for complete addresses. However, since we also have trained using incomplete addresses, some incomplete addresses are now not so different from the other address patterns, confusing our models. For example, if we remove the tags Province and Municipality of an address, it can be in 4 of the five patterns making more harder for our models. This shows that our models have overfitted in that case, and adding noise in the training data helps reduce that overfitting but lowers the accuracy. We also observe that using a domain adversarial technique substantially improves performance for that specific case where we observe the best improvement with the accuracy nearly doubling.\n\n\\begin{table*}\n    \\centering\n    \\caption{Mean accuracy (and standard deviation) for multinational models trained on complete and incomplete datasets and evaluated on the incomplete dataset for training countries. \\textbf{bold} values indicate the best accuracy for each country.}\n    \\label{table:incomplete-holdout-train-clean-noisy}\n    \\resizebox{0.65\\textwidth}{!}{%\n    \\begin{tabular}{lrrrr}\n    \\toprule\n    Country & FastTextADANN & FastTextADANNNoisy & BPEmbADANN & BPEmbADANNNoisy\\\\\n    \\midrule\n    United States & $65.57 \\pm 0.80$ & $\\mathbf{97.88 \\pm 2.04}$ & $67.94 \\pm 4.94$ & $92.12 \\pm 3.60$\\\\\n    Brazil & $57.39 \\pm 3.63$ & $\\mathbf{98.39 \\pm 1.90}$ & $51.10 \\pm 4.08$ & $89.57 \\pm 6.46$\\\\\n    South Korea & $46.28 \\pm 2.29$ & $\\mathbf{92.25 \\pm 3.72}$ & $35.85 \\pm 5.32$ & $84.58 \\pm 9.86$\\\\\n    Australia & $76.02 \\pm 0.36$ & $\\mathbf{98.86 \\pm 1.46}$ & $76.49 \\pm 1.56$ & $93.32 \\pm 5.75$\\\\\n    Mexico & $59.88 \\pm 2.83$ & $\\mathbf{97.56 \\pm 1.63}$ & $58.93 \\pm 6.36$ & $88.79 \\pm 5.12$\\\\\n    Germany & $50.32 \\pm 6.96$ & $\\mathbf{99.20 \\pm 0.74}$ & $58.62 \\pm 3.31$ & $96.79 \\pm 2.05$\\\\\n    Spain & $72.39 \\pm 1.13$ & $\\mathbf{98.24 \\pm 1.82}$ & $81.05 \\pm 1.17$ & $90.23 \\pm 7.02$\\\\\n    Netherlands & $63.74 \\pm 3.47$ & $\\mathbf{98.65 \\pm 0.98}$ & $77.39 \\pm 3.52$ & $97.21 \\pm 1.77$\\\\\n    Canada & $71.04 \\pm 1.98$ & $\\mathbf{97.88 \\pm 2.65}$ & $81.29 \\pm 3.22$ & $91.93 \\pm 3.74$\\\\\n    Switzerland & $62.66 \\pm 6.31$ & $\\mathbf{99.03 \\pm 0.78}$ & $72.55 \\pm 3.22$ & $97.12 \\pm 1.52$\\\\\n    Poland & $47.04 \\pm 4.06$ & $\\mathbf{99.28 \\pm 0.52}$ & $54.47 \\pm 4.73$ & $97.89 \\pm 1.80$\\\\\n    Norway & $62.67 \\pm 8.64$ & $\\mathbf{99.35 \\pm 0.54}$ & $78.28 \\pm 4.04$ & $98.25 \\pm 1.06$\\\\\n    Austria & $69.21 \\pm 5.21$ & $\\mathbf{99.10 \\pm 1.16}$ & $78.56 \\pm 2.39$ & $94.74 \\pm 2.88$\\\\\n    Finland & $60.83 \\pm 6.98$ & $\\mathbf{98.97 \\pm 0.64}$ & $76.31 \\pm 3.46$ & $98.75 \\pm 0.57$\\\\\n    Denmark & $55.54 \\pm 5.48$ & $\\mathbf{97.38 \\pm 2.56}$ & $63.73 \\pm 3.04$ & $92.15 \\pm 3.77$\\\\\n    Czechia & $64.40 \\pm 2.91$ & $\\mathbf{98.14 \\pm 1.74}$ & $74.79 \\pm 3.53$ & $93.75 \\pm 3.79$\\\\\n    Italy & $71.16 \\pm 1.67$ & $\\mathbf{98.82 \\pm 1.15}$ & $76.33 \\pm 1.46$ & $94.49 \\pm 3.72$\\\\\n    France & $73.14 \\pm 2.38$ & $\\mathbf{98.98 \\pm 1.32}$ & $77.45 \\pm 1.08$ & $91.10 \\pm 6.85$\\\\\n    UK & $54.58 \\pm 1.24$ & $\\mathbf{96.01 \\pm 4.75}$ & $53.62 \\pm 3.30$ & $84.95 \\pm 7.63$\\\\\n    Russia & $57.94 \\pm 0.99$ & $\\mathbf{96.05 \\pm 4.05}$ & $61.91 \\pm 2.86$ & $86.71 \\pm 5.71$\\\\\\midrule\n    Mean             &         $62.09 \\pm 8.41$ &              $\\mathbf{98.00 \\pm 1.62}$ &      $67.83 \\pm 12.15$ &           $92.72 \\pm 4.22$ \\\\\n    \\bottomrule\n    \\end{tabular}%\n    }\n\\end{table*}\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nWe estimate that we have reached our first objective, which was to build a single model capable of learning to parse addresses of different formats and languages using a multinational dataset and subword embeddings. Indeed, all our approaches achieve accuracies around $99~\\%$ on all the twenty countries used for training. Our experiments with zero-shot transfer learning also yielded interesting results. First, our baseline approaches obtain good results achieving near $50~\\%$ of state-of-the-art performance. Second, using an attention mechanism helps to improve our results and could also provide insights about the address elements on which the model focuses to make a tag prediction. However, this analysis is left as future work. Third, our experiments indicate that using a domain adversarial training algorithm does not necessarily improve our results on countries not seen during training. However, they bring a significant contribution on incomplete addresses. Finally, we tested some of our models on incomplete addresses to evaluate their performance on such data. We also improved performance by using some incomplete addresses during training improved performance. These results provide insights into the direction that our future work should take. It would be interesting to explore how other subword embeddings techniques, such as character-based ones \\cite{kim2015characteraware}, would perform on the multinational address parsing task. Additional qualitative analysis of the results would also be required to investigate the models' typical errors further. \n\n\\section*{Acknowledgment}\nThis research was supported by the Natural Sciences and Engineering Research Council of Canada (IRCPJ 529529-17) and a Canadian insurance company. We wish to thank the reviewers for their comments regarding our work and methodology.\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n{\\bf Goal}\nElliptical galaxies are presumed to be the product of galaxy merging, where different mass ratios\nresult in different classes of elliptical galaxies (disky, boxy\\dots) \\cite{naa99}. This\nseems consistent with the hierarchical scenario of galaxy formation\n\\cite{pre74}, although it is clear that more standard collapse processes and secular \nevolution contribute to the shaping of galaxies.\nThere are many different and often complementary approaches to constrain these scenarios. \nThe one we adopted is to study the structures of nearby elliptical galaxies in detail, \nin order to probe the signatures of their evolution and to be able to trace their formation\nhistory. This fossil search requires data both on the chemical and\ndynamical status of the stellar components, and if present of the gaseous\nsystem, both being provided by spectroscopy. \\\\\n\n\\hspace{-.5cm}{\\bf 3D spectroscopy} Spectroscopy can indeed deliver information on the stars'\ndynamics and chemistry via the analysis of their absorption lines. Two-dimensional spatial\ncoverage provided by integral-field spectroscopy is critical to homogeneously sample\nan extended target, while more traditional long-slit spectroscopy generally restricts our view \nto a few a priori fixed axes. \nA qualitative assessment of the obtained maps is clearly not sufficient to fully\naddress the issues mentioned above. We need to go further by\nmodelling the observed galaxies using constraints retrieved from 3D spectroscopy, such as\nluminosity, stellar velocity distribution and stellar populations. As emphasized by  \\cite{bin05},\n3D spectroscopy seems to be essential to properly constrain the dynamics of\nthe galaxy under scrutiny.\\\\\n\n\\hspace{-.5cm}{\\bf Existing models}\nOne of the key goal of galaxy modelling is to retrieve the full distribution function (DF), \n{\\it i.e.} the density of stars in phase-space.\nThere are already several known methods to address this problem,\neach of them having both advantages and drawbacks. We can distinguish different\nclasses of techniques~:\n\\begin{itemize}\n\\item DF-based method, where models are often restricted to simple geometries.\n\\item Moment-based method, which consists in solving Boltzmann and\nPoisson equations via the use of a closed system of relations (Jeans equations). The main issue here is that the final DF may not be positive everywhere.\n\\item Orbit-based method, or Schwarzschild modelling \\cite{sch79}. Libraries\n of orbits are built within a fixed potential, and each orbit is weighted\n as to reproduce the observed galaxy (photometry and kinematics).\n\\item Particle-based method, where the particles represent groups of stars.\n One has to guess the right initial conditions which will evolve in a configuration\n resembling the chosen galaxy.\n\\end{itemize}\n\\section{Method}\nThe method we wish to present here is an hybrid scheme between the Schwarzschild and N-body techniques.\nIt has been first proposed and developed by \\cite{sye96} and consists in an N-body realization where\nthe weight of each particle is gradually changed in order to fit the observables.\\\\\n\n\\hspace{-.5cm}{\\bf Previous applications}\nThis method has been tested by \\cite{sye96} on fake galaxy models, adjusting\nthe photometry alone. It has been recently applied on the Milky Way by \\cite{bis04}, \nfitting the photometry of a frozen snapshot extracted from a full N-body\nsimulation\\footnote{An extended independent implementation of this algorithm has been recently presented by de Lorenzi et al. 2007: see their paper for details}.\\\\\n\n\\hspace{-.5cm}{\\bf Algorithm}\nThe algorithm we use is illustrated in Fig.~\\ref{jou:fig1}.\nParticles start from (N-body) initial conditions and are integrated along their orbits. \nObservables are used as reference input to the modelling, particles being projected \nas to mimic these observables. A weight prescription is derived from the\ndifference between the model and the data, \nthis prescription being then applied on each particle accordingly. \nThis procedure (integration, projection, comparison, weight changing) is\nrepeated until the weights converge.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.6\\textwidth]{jourdeuilF1.eps}\n\\caption{Algorithm of the Syer \\& Tremaine method.}\n\\label{jou:fig1}\n\\end{figure}\n\\\\\n\n\\hspace{-.5cm}{\\bf Initial conditions}\nThe initial conditions have to properly sample the galaxy, {\\it ie} in position\/velocity space or using integrals of motions. This is not an easy task when we do not {\\it a priori} know the internal structure of the galaxy we wish to model.\nIf we want to model real galaxies, we have to find\nappropriate initial conditions, without a priori knowing its DF. We thus first\nbuild an initial mass distribution, by using the known photometry and assuming a\nconstant mass-to-light ratio. This is achieved here via the Multi Gaussian Expansion (MGE) \nproposed by \\cite{mon92}, which allows us to decompose the mass distribution \ninto a sum of tri-dimensional Gaussians, and to derive the corresponding gravitational\npotential analytically \\cite{ems94}. Jeans equations are then used to obtain a first guess of\nthe dynamics and set up initial conditions.\\\\\n\n\\hspace{-.5cm}{\\bf Integration}\nIn a first version of the code, we keep the potential steady. The code has been\ndeveloped in a very modular and flexible fashion, such as to allow the easy\nimplementation of a self-consistency module~: this will be described in \na forthcoming paper. As we need to evaluate the observables of all particles at the same time, \nwe chose a second-order synchronized leapfrog scheme, where positions and velocities can be evaluated\nsimultaneously. The leapfrog technique is perhaps not the best choice in terms of accuracy, but \nwas found sufficient for this first development step. Other schemes (e.g. Runge-Kutta) can be easily\nimplemented.\nAdaptive time-steps have been included, in order to optimally sample all \nthe orbits. An independent integration of each particle would result in a very\ninefficient algorithm. We thus chose to follow the algorithm described in \nthe N-body code GADGET \\cite{spr01}, in which particles advance in bunches \nso that they are always distributed in a tight time range around the current time.\\\\\n\n\\hspace{-.5cm}{\\bf Observables}\nOur code has been designed around two new main items.\nFirst, we do not restrict ourselves to fit the photometry~: the code also allows the fitting of the kinematics, via the use of Line of Sight Velocity\nDistributions (LOSVDs). Second, the code has been developed as to permit the use of \n3D spectroscopic data, including spatial adaptive binning (Voronoi bins, see \\cite{cap03}). We have thus used the data obtained with the SAURON spectrograph (WHT,\nCanary Islands) for its survey of early-type galaxies \\cite{dez02,ems04}.\\\\\n\n\\hspace{-.5cm}{\\bf Prescription}\nThe heart of the code is the prescription developed by \\cite{sye96}, which\nrules the weight changing of each particle so that the cumulated\nobservables reproduce the observations~:\n\\begin{equation}\n  \\frac{dw_i(t)}{dt} = -\\epsilon w_i(t)\\sum_{j=1}^J\\frac{K_{ij}}{Z_j}\\Delta_j\n\\label{jou:eq1}\n\\end{equation}\nwhere $i$ describes the particles, $j$ the observables, $w$ the weights,\n$K_{ij}$ the contribution of the particle $i$ to the observable $j$, and $\\Delta_j\n\\equiv y_j(t)\/Y_j -1$ the contribution of all the particles to this same observable. $\\epsilon$ is the strength of the weight changing, $w_i$\navoids the weights to be negative when $w_i$ approaches $0$. \nThe input parameters are the following~: \n\\begin{itemize} \\item $N$ is the number of particles,\n\\item $d_{time}$ is the time during which the observables are averaged,\n\\item $\\epsilon$ is the strength parameter mentioned in Eq.~\\ref{jou:eq1},\n\\item $\\alpha$ is the smoothing parameter of the observables, used in the equation~:\n\\begin{equation}\n      \\tilde\\Delta_j(t) = \\alpha \\int_0^\\infty\\Delta_j(t-\\tau)e^{-\\alpha \\tau}d\\tau\n    \\end{equation}\nso that $\\tilde\\Delta_j(t)$ replaces $\\Delta_j(t)$ in Eq.~\\ref{jou:eq1}.\n\\end{itemize}\nThe choice of these parameters is crucial, each galaxy requiring an\nappropriate set.\n\\section{Results}\n\nA preliminary model has been obtained on the nearby disky galaxy\nNGC~3377. In Fig.~\\ref{jou:fig2} we illustrate the fit of the photometry and LOSVDs \nobtained with the SAURON instrument on this galaxy, for a given mass\n(MGE) distribution. The maps are globally well reproduced ($I$, $\\sigma$, $h_3$).\nSome remaining discrepancy in the velocity field is probably due to an observed asymmetry in\nthe kinematics of this galaxy (tilt between the photometric and kinematic\nminor-axis). The relatively high $h_4$ observed level is not reached by the\nmodel~: this may be partly related to a template mismatch effect which\nmostly affects the even moments of the LOSVD.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.7\\textwidth]{jourdeuilF2.eps}\n\\caption{$I, V, \\sigma, h_3, h_4$ from the observations (top) and the model (bottom) of NGC~3377}\n\\label{jou:fig2}\n\\end{figure}\n\\section{Conclusion and perspectives}\nAlthough the results shown here are very preliminary,\nthey illustrate the possibility of fitting the kinematics,\nwhich is an important step in the development of such a code. \nIt has also been designed to adjust complex data such as the\none provided by 3D spectrographs. A few improvements have been or are now being implemented~:\n\\begin{itemize}\n\\item Self-consistency~: it is technically easy to replace the current integration \nmodule with a self-consistent scheme. The full implications of a continuous weight changing\nremains however unclear and have to be examined.\n\\item Addition of a central black hole~: this would allow the use of higher\nspatial resolution data of nearby galaxies.\n\\item Stellar populations~: this new class of observables would allow us\nto link the dynamics and chemistry of the stellar component.\n\\end{itemize}\n \\bibliographystyle{abbrv}\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn 1965 Buchberger invented Gr\\\"obner basis theory, techniques enabling the \ncomputation of ideals in commutative polynomial rings over fields.\nImplementations of Buchberger's algorithm are now provided by all major\ncomputer algebra systems, a good cross-section of the ways in which the \ntheory has developed may be found in \\cite{RISC}. \nMora generalised Gr\\\"obner basis theory to noncommutative polynomial rings\n(algebras) \\cite{FMora}. \nIntroductions to these procedures may be found in \\cite{Ufn,TMora2}.\nThis paper presents an extension of the noncommutative Gr\\\"obner\nbasis procedures for polynomials to what we call {\\em tagged polynomials}.\nThe intention is to describe methods of computation that may be applied\nto the problem of computing right (or left) ideals in finitely presented\n$K$-algebras.\n\\\\\n\nThe data defining the problem consists of the field $K$, a set of\nnoncommuting\nvariables $X$, a set of generators $P \\subseteq K[X^\\dagger]$ for a two-\nsided\nideal $\\langle P \\rangle$, defining an algebra \n$A=K[X^\\dagger]\/ \\langle P \\rangle$ and a set of generators $Q'\n\\subseteq A$\nfor a right ideal $\\langle Q' \\rangle^r$.\nWe expect elements of $A$ to be given in terms of $K[X^\\dagger]$, so\n$Q'$ is\nspecified by a set $Q \\subseteq K[X^\\dagger]$.\nThe problem we address is that of computing the right $A$-ideal\ngenerated by $Q'$, written $\\langle Q' \\rangle^r$.\n\\\\\n\nOur solution lies in using the free right $K$-module $K[\\dashv \\!\nX^\\dagger]$. \nHere  $\\dashv$ is just a symbol or tag and \n$K[\\dashv\\!X^\\dagger]$ is bijective with $K[X^\\dagger]$.\nWe call elements of $K[\\dashv\\!X^\\dagger]$ \\emph{tagged polynomials} \n(Definition 3.1) and\nwrite them $k_1\\dashv\\!m_1+ \\cdots + k_n\\dashv\\!m_n$ where \n$k_1,\\ldots,k_t \\in K$ and $m_1,\\ldots,m_t \\in X^\\dagger$. \nOrdinary polynomials $F_P$ defining the two-sided ideal\n$\\langle P \\rangle$ which defines $A$ are combined with\ntagged polynomials $F_T$ defining the one-sided ideal. \nThe mixed set of polynomials $F:=(F_T,F_P)$ determines a\nreduction relation $\\to_F$ (Definition 3.2) on the tagged polynomials\n$K[\\dashv \\! X^\\dagger]$.\n\\\\\n\n\nThe value of this combination and use of tagging is in computation, as\nwill be shown in \nthe main result (Algorithm 4.9), which describes a variant of the Buchberger\nalgorithm. The initial mixed set of polynomials $F$ is appended with tagged \nand non-tagged polynomials until the relation $\\to_F$ is complete on\n$K[\\dashv\\!X^\\dagger]$. When the procedure terminates the usual\nnormal form arguments apply and reduction modulo $F$ can be used to\nsolve \nthe membership problem for the right ideal $\\langle Q' \\rangle^r$\nof the finitely presented algebra $A$.\n\\\\\n \nPrevious work \\cite{Birgit,MRC} attempt the computation of one-sided ideals\nby using different definitions of \npurely one-sided reduction relation in particular algebras \n(e.g. $\\mathbb{Q}[M]$ for a monoid $M$ presented by a complete rewrite system). \nThe main problem encountered is that of computing in a non-free algebra,\nwe avoid this and base all the computations specifying the algebra at\nthe same level (in a particular free right module) \nas those for the ideal and compute the two simultaneously. \nIn other words, the methods we describe \nprovide for local computations, concerning single ideals \n$\\langle Q' \\rangle^r$ without the requirement to compute \nthe global structure of the algebra $A$ or face the\ndifficulties of calculations with elements of $A$.\nThis idea follows the philosophy that \ncomputations take place in free objects.\n\n \n\\section{Algebra Presentations and One-sided Ideals}\n\nIf $X$ is a set, then \n$X^\\dagger$ is the {\\em free semigroup} \nof all strings of elements of $X$, and\n$X^*$ is the {\\em free monoid} of all strings \ntogether with the empty string, which acts as the identity $i\\!d$ for\n$X^*$.\nA {\\em semigroup presentation} is a pair $sgp \\langle X | R \\rangle$\nwhere\n$X$ is a set and $R \\subseteq X^\\dagger \\times X^\\dagger$. \nIt {\\em presents a semigroup} $S$ if $X$ is a set of generators of $S$\nand the natural morphism $\\theta:X^\\dagger \\to S$ induces an isomorphism\nfrom $X^\\dagger\\!\/=_R$ to $S$, where $=_R$ is the\ncongruence generated on $X^\\dagger$ by $R$.\nSimilarly, a {\\em monoid presentation} is a pair $mon \\langle X | R \\rangle$\nwhere $X$ is a set and $R \\subseteq X^* \\times X^*$. \nIt {\\em presents a monoid} $M$ if $X$ is a set of generators of $M$\nand the natural morphism $\\theta:X^* \\to M$ induces an isomorphism\nfrom $X^*\\!\/=_R$ to $M$, where $=_R$ is the\ncongruence generated on $X^*$ by $R$.\n\\\\\n\nLet $K$ be a field.\nThe { \\em free $K$-algebra} $K[S]$ on a semigroup $S$ \nconsists of all sums of $K$-multiples\nof elements of $S$ with the operations of \naddition and multiplication defined\nin the obvious way. In particular the elements of\n$K[X^\\dagger]$ are called {\\em polynomials} and written\n$k_1m_1+\\cdots+k_nm_n$ where $k_1,\\ldots,k_n \\in K$ and \n$m_1,\\ldots,m_n \\in X^\\dagger$.\\\\\n\n\nIf $P$ is a subset of an algebra $Z$ then the \n{\\em two-sided ideal} generated by $P$ in \n$Z$ is denoted $\\langle P \\rangle$. In the case $Z=K[X^\\dagger]$ this\nconsists of all sums of multiples of\nelements of $P$, i.e.\n$$\\langle P \\rangle := \\{ k_1u_1p_1v_1+ \\cdots + k_nu_np_nv_n \n               \\ | \\ p_1, \\ldots,p_n \\in P, k_1,\\ldots,k_n \\in K, \n               u_1,v_1,\\ldots,u_n,v_n \\in X^*\\}.\n$$\nGiven an ideal in an algebra the {\\em membership problem} is that of\ndetermining,\nfor a given element of the algebra, whether it is an element of the\nideal.\n\\\\\n\nA {\\em congruence} on an algebra $Z$ is an equivalence relation $\\sim$\non its elements such that if $p \\sim q$ then $p+u \\sim q+u$ and \n$upv \\sim uqv$ for all $u,v \\in Z$.\nGiven an algebra $Z$ and an ideal $\\langle P \\rangle$ \nideal membership defines a congruence on the algebra by\n$$\np \\sim q \\Leftrightarrow p-q \\in \\langle P \\rangle.\n$$\nThe {\\em quotient algebra} $Z\/\\langle P \\rangle$ is the algebra of\ncongruence \nclasses of $Z$ under $\\langle P \\rangle$.\nA $K$-\\emph{algebra presentation} is a pair $alg\\langle X | P \\rangle$, where\n$P \\subseteq K[X^\\dagger]$.\nA $K$-algebra $A$ is presented by $alg\\langle X|P \\rangle$ if \n$X$ is a set of generators of $A$ and the natural morphism\n$K[X^\\dagger] \\to A$ induces an isomorphism $K[X^\\dagger]\/\\langle P\n\\rangle \\to A$.\\\\\n\n\nNoncommutative Gr\\\"obner basis theory (as described in \\cite{TMora, TMora2})\nuses the notion of an ordering on $X^\\dagger$, thereby allowing the \nconcepts of {\\em leading monomial}, \n{\\em leading term} and {\\em leading coefficient} \n on the polynomials of $K[X^\\dagger]$. \nGiven any subset $P$ of $K[X^\\dagger]$ \nan ordering determines a Noetherian \nreduction relation $\\to_P$ on the\nelements of $K[X^\\dagger]$. \nThe reflexive, symmetric, transitive closure of this\nrelation is a congruence relation \ncoinciding with the ideal membership of $\\langle P \\rangle$.\n\\\\\n\n\nLet $A$ be the $K$-algebra presented by\n$alg\\langle X | P \\rangle$ and let $Q' \\subseteq A$.\nWe wish to consider  the right ideal $\\langle Q'\\rangle^r $\ngenerated in $A$ by $Q'$,\ni.e.\n$$\n\\langle Q'\\rangle^r :=\n  \\{q_1'a_1+ \\cdots +q_n'a_n \\ | \\ a_1, \\ldots, a_n \\in A,\n                                           q_1', \\ldots, q_n' \\in Q'\\}.\n$$\n\nA {\\em right congruence} on an algebra $A$ is an equivalence relation\n$\\stackrel{r}{\\sim}$ such that for all $a,b,y \\in A$\n$$a \\stackrel{r}{\\sim} b  \\Rightarrow \n       a+y \\stackrel{r}{\\sim} b+y  \\text{ and } \n       ay \\stackrel{r}{\\sim} by.$$\n\n\nMembership of a right ideal $\\langle Q' \\rangle^r$ defines \na right congruence on $A$,  \nby $a \\stackrel{r}{\\sim}_{Q'} b \\Leftrightarrow a-b \\in \\langle Q'\n\\rangle^r$.\nThe quotient $A\/\\langle Q' \\rangle^r$ is \nthe set of all the right congruence classes of $A$ under \n$\\stackrel{r}{\\sim}_{Q'}$ \nwhere classes are denoted $[a]_{Q'}$ for $a \\in A$.\nNote that for $a, b \\in A$,\n$[a+b]_{Q'} = [a]_{Q'} + [b]_{Q'}$ \nand\n$[a]_{Q'}[b]_{Q'} = [a]_{Q'}$.\\\\\n\n\n {\\em Buchberger's algorithm} \nis a critical pair completion procedure. \nThe algorithm begins with a set of polynomials \n$P$ of a free algebra. Set $F:=P$ and a search for \noverlapping leading terms \nwill find all critical terms of the \nreduction relation $\\to_F$. \nThis enables a test for local confluence. \nOverlaps which cannot be resolved result in \n{\\em S-polynomials} all of which are \nadded to $F$ at each stage\n(though some elimination is possible, \nfor efficiency). The algorithm\nterminates when all the overlaps of \n$F$ can be resolved, i.e.\n$\\to_F$ is complete \n(Noetherian and confluent), when\nthis occurs $F$ is said to be a \n{\\em Gr\\\"obner basis} for \nthe ideal $\\langle P \\rangle$.\nObtaining a Gr\\\"obner basis $F$ \nallows, in particular, the \nsolution of the  membership problem \nby using $\\to_F$ as a normal\nform function on $K[X^\\dagger]$.\nThus, if $F$ is a Gr\\\"obner basis for \nthe ideal $\\langle P \\rangle$ on $K[X^\\dagger]$ \nand $p,q \\in K[X^\\dagger]$, then\n$$\n  p \\sim q \\Leftrightarrow \n  p-q \\in \\langle P \\rangle \\Leftrightarrow\np \\stackrel{*}{\\to}_{F} u \\text{ and }\nq \\stackrel{*}{\\to}_{F} u \n  \\text{ for some } u \\in K[X^\\dagger].\n$$\n\n\nThe Noetherian property ensures that \nthe process of reduction terminates with an \nirreducible element; confluence ensures that \nany two elements of the same class\nreduce to the same form. In practice: reduce \neach polynomial as far as \npossible using $\\to_{F}$, the original \npolynomials are equivalent only \nif their irreducible forms are equal.\\\\\n\n\nIn the next sections we show how to apply Buchberger's algorithm to obtain --\nwhen possible -- a Gr\\\"obner basis of (two types of) polynomials, which will\nenable the use of normal forms arguments.\n\n\n\\section{One-sided Noncommutative Gr\\\"obner Basis Procedures}\n\nGiven a finitely presented $K$-algebra $A$ and a subset $Q'$ of $A$ we\nwish to compute the right ideal $\\langle Q' \\rangle^r$. \nThe meaning of `computing the ideal' in this context is that of solving  \nthe ideal membership problem for $\\langle Q' \\rangle^r$ in $A$.\nThe $K$-algebra $A$ is presented by \n$alg\\langle X | P \\rangle$ and to obtain normal forms for $A$ we would\ntherefore apply Gr\\\"obner basis procedures\nto $P$ in the free algebra $K[X^\\dagger]$.\nSince we are interested in a one-sided ideal we introduce the \ntagging notation which will allow the combination of $P$ and $Q$. \n\n\n\\begin{Def}[Tagged polynomials]\nLet $K$ be a field, let $X$ be a set and let $\\dashv$ be a symbol.\nThen $\\dashv\\!X^\\dagger$ is the set of \\textbf{tagged terms}\n$\\dashv\\!m$\nwhere $m \\in X^\\dagger$ and $K[\\dashv \\! X^\\dagger]$ is the free right \n$K[X^\\dagger]$-module of \\textbf{tagged polynomials},\ni.e. elements $k_1 t_1 + \\cdots + k_n t_n$ for\n$k_1, \\ldots, k_n \\in K$, $t_1, \\ldots, t_n \\in \\dashv\\! X^\\dagger$.\n\\end{Def}\n\nLet $>$ be a {\\em semigroup ordering} on $X^\\dagger$, i.e.\n$>$ is an irreflexive, antisymmetric, transitive relation on $X^\\dagger$\nsuch\nthat if $m_1>m_2$ then $um_1v>um_2v$ for all $u,v \\in X^*$.\nFurther we require the {\\em well-ordering} property, that there is no\ninfinite sequence $m_1>m_2>m_3> \\cdots$.\\\\\n\nLet $p=k_1m_1 + \\cdots + k_nm_n \\in K[X^\\dagger]$. \nThe $k_im_i$ are referred to as the {\\em monomials} of the polynomial, where\n$m_i$ is the {\\em term} and $k_i$ the {\\em coefficient}.\nAssuming the well-ordering on $X^\\dagger$, \nthe leading monomial $\\mathtt{LM}(p)$ is defined to be the monomial with\nthe largest term. The leading term $\\mathtt{LT}(p)$ and\nleading\ncoefficient $\\mathtt{LC}(p)$ are the coefficient and term of\nthis monomial.\\\\ \n\nTo simplify the definitions throughout this paper \nwe will assume all polynomials to be {\\em monic},\ni.e. their leading coefficients are all 1.\nThere is no loss in doing this: $K$ is a field so \nthe polynomials $F$ may always be divided by their leading \ncoefficients and still generate the same ideal.\\\\\n\n\nThe well-ordering on $X^\\dagger$ induces a well-ordering on\n$\\dashv\\!X^\\dagger$\ndefined by $\\dashv\\!m_1>\\dashv\\!m_2 \\Leftrightarrow m_1>m_2$.\nThis gives corresponding notions of leading monomial, leading term \nand leading coefficient for the tagged polynomials. \nIn detail: if $p=k_1m_1+ \\cdots + k_nm_n$ where $k_1,\\ldots,k_n \\in K$ and\n$m_1,\\ldots,m_n\\in X^\\dagger$ is a polynomial with leading term\n$\\mathtt{LT}(p)=m_i$ then the tagged polynomial\n$\\dashv\\!p:=k_1\\dashv\\!m_1 + \\cdots + k_n \\dashv\\!m_n$ has a tagged leading \nterm $\\mathtt{LT}(\\dashv\\!p)=\\dashv\\!m_i$. \n\\\\\n\nWe will now introduce the definition of a reduction relation \non $K[\\dashv\\!X^\\dagger]$, defined by a mixed set of polynomials\n$F=(F_T,F_P)$ where $F_T$ is a set of tagged polynomials, elements of\nthe module, and $F_P$ is a set of polynomials, \nelements of the algebra acting on the right of the module.\nThe reduction relation $\\to_F$ combines the two relations so\nthat they are defined on the free right module of tagged polynomials. \n\n\n\n\n\\begin{Def}[Reduction of tagged polynomials]\nLet $F:=(F_T,F_P)$ where \n$F_T \\subseteq K[\\dashv\\! X^\\dagger]$ and \n$F_P \\subseteq K[X^\\dagger]$. \nDefine the reduction relation $\\to_F$ on\ntagged polynomials $f \\in K[\\dashv \\! X^\\dagger]$ by\n$$\nf \\to_{F}  f - k u (f_i) v\n$$\nif $u \\mathtt{LT}(f_i) v$ occurs in $f$ with coefficient $k \\in K$\nfor some $u \\in \\dashv\\!X^* \\cup \\{i\\!d\\}$, $v \\in X^*$, $f_i \\in F$.  \n\\end{Def}\n\nA one-step reduction like that of the definition may also be\nwritten $f \\to_{f_i}  f - k u (f_i) v$. \nThis relation may be understood to be a rewrite\nsystem on the polynomials (similarly to observations made in \\cite{Birgit}\non Mora's definitions of reduction \\cite{FMora}). When a multiple of the \nleading term of $f_i$ for $f_i \\in F$ occurs in the polynomial that is \nto be reduced, the rest of $f_i$ is substituted for the leading monomial \nof $f_i$. \n\n\nRegarding $F$ as a rewrite system with two types of rules that may be\napplied to monomials of polynomials, we could say that\nthe non-tagged polynomials can be applied anywhere in a term, \nbut the tagged ones apply only at the tagged side of a term.\n\n\\begin{example}[Reduction]\n\\emph{For example let $F_T:=\\{f_1,f_2\\}$ where \n$f_1:=\\dashv\\!xyx+\\dashv\\!yx+2\\dashv\\!y$,\n$f_2:=\\dashv\\!yx^2+\\dashv\\!x^2$ and\n$F_P:=\\{f_3,f_4\\}$ where $f_3:=x^2y-3yx$, $f_4:=yx^3-2xy$. \nThen the tagged polynomial \n$f:=8\\dashv\\!xyx^2y^3+5\\dashv\\!y$ cannot be reduced by $f_2$ or $f_4$ \nbut can be reduced by $f_1$ to \n$f-8f_1xy^3=5\\dashv\\!y-8\\dashv\\!yx^2y^3-16\\dashv\\!yxy^3$ \nor by $f_3$ to\n$f-8\\dashv\\!xyf_3y^2=5\\dashv\\!y+24\\dashv\\!xy^2xy^2$.}\n\\end{example}\n\n\nThese results allow the combination of two-sided and one-sided\ncongruences,\nby the use of tagged polynomials. A mixed set of polynomials $F$\ndefines a reduction relation on the module of all tagged polynomials.\nThe reflexive, symmetric, transitive closure of $\\to_F$ will be \ndenoted $\\stackrel{*}{\\leftrightarrow}_F$. \nThe class of $f \\in K[\\dashv \\! X^\\dagger]$ under the equivalence\nrelation \n$\\stackrel{*}{\\leftrightarrow}_F$ will be denoted $[f]_F$.\n\n\n\n\n\\begin{thm}\nLet $A$ be a $K$-algebra finitely presented by $alg\\langle X | P\n\\rangle$ with quotient morphism $\\theta$.\nLet $Q \\subseteq K[X^\\dagger]$ and define $Q':=\\theta Q$.\nDefine $F:=(\\dashv\\!Q, P)$ where $\\dashv\\!Q:=\\{\\dashv\\!q:q\\in Q\\}$.\nThen there is a bijection of sets\n$$\n\\frac{ K[\\dashv \\! X^\\dagger] }{ \\stackrel{*}{\\leftrightarrow}_F }\n\\cong  \\frac{A}{ \\langle Q' \\rangle^r }$$\n\\end{thm}\n\n\\begin{proof}\nThe quotient\nmorphism $\\theta:K[X^\\dagger] \\to A$, \ndefines a surjection\n$\\theta^\\dashv:K[\\dashv \\! X^\\dagger] \\to A$. \nThen $\\theta^\\dashv(\\dashv \\! Q)=Q'$.\\\\\n\nDefine $\\phi:K[X^\\dagger]\/\\!\\stackrel{*}{\\leftrightarrow}_F \\; \n            \\to A\/\\langle Q' \\rangle^r$ by \n\\, $\\phi([f]_F) := [\\theta^\\dashv(f)]_{Q'}.$\n\nTo prove that $\\phi$ is well-defined we show that it preserves the right \ncongruence $\\stackrel{*}{\\leftrightarrow}_F$.\nWe assume all polynomials of $F$ are monic.\nLet $f \\in K[\\dashv\\!X^\\dagger]$\nand $f_i \\in F$ and suppose that $f \\to_F f - kuf_iv$ for some\n$k \\in K$, $u \\in \\dashv\\!X^* \\cup \\{i\\!d\\}$, $v \\in X^*$.\nBy definition \n$\\phi([ f-kuf_iv ]_F) = [ \\theta^\\dashv(f) - \\theta^\\dashv(kuf_iv)\n]_{Q'}$.\n\\\\\n\nNow either\\\\ \n(i) $f_i \\in P \\subseteq K[X^\\dagger]$ and  \n      $\\theta^\\dashv(kuf_iv)=0$, since $kuf_iv \\in \\langle P \\rangle$,\\\\\nor else \\\\\n(ii) $f_i \\in \\dashv\\! Q \\subseteq K[\\dashv\\!X^\\dagger]$ and \n      $u=i\\!d$, so $\\theta^\\dashv(kuf_iv)=k\\theta^\\dashv (f_iv)$.\\\\\nIn either case $\\theta^\\dashv(kuf_iv) \\in \\langle Q' \\rangle^r$, so\n$\\phi(  [f ]_F ) = \\phi( [ f-kuf_iv ]_F )$, \ni.e. $\\phi$ preserves the relation $\\to_F$. \n\nFurthermore if $[f]_F=[g]_F$ for some $f,g\\in K[\\dashv\\!X^\\dagger]$, then\nfor all $v \\in X^*$,\n\\begin{align*} \n\\phi([fv]_F) &= [\\theta^\\dashv(fv)]_{Q'}\\\\\n             &= [\\theta^\\dashv(f)]_{Q'} \\theta(v) \\\\\n             &= [\\theta^\\dashv(g)]_{Q'} \\theta(v) \\\\\n             &= [\\theta^\\dashv(gv)]_{Q'} \\\\\n             &= \\phi([gv]_F).\n\\end{align*}\nTherefore $\\phi$ preserves the right congruence \n$\\stackrel{*}{\\leftrightarrow}_F$.\\\\\n\n\nWe now prove that $\\phi$ is surjective.\nLet $a \\in A$. \nThen there exists $f \\in K[\\dashv \\! X^\\dagger]$ such that \n$\\theta^\\dashv(f)=a$, because $\\theta^\\dashv$ is a surjection. \nThus for all $[a]_{Q'} \\in A\/\\langle Q' \\rangle^r$\nthere exists $[f]_F \n\\in K[\\dashv \\! X^\\dagger]\/\\stackrel{*}{\\leftrightarrow}_F$\nsuch that $\\phi([f]_F)=[\\theta^\\dashv f]_{Q'}=[a]_{Q'}$.\\\\\n\n\nFinally, we prove that $\\phi$ is injective.\nLet $f,g \\in K[\\dashv \\! X^\\dagger]$ such that \n$\\phi[f]_F = \\phi[g]_F$.\nThen $[\\theta^\\dashv(f)]_{Q'} = [\\theta^\\dashv(g)]_{Q'}$. \nTherefore there exist $q_1',\\ldots,q_n' \\in Q'$ and $k_1,\\ldots,k_n \\in\nK$, \n$a_1,\\ldots,a_n \\in A$, such that\n$$\\theta^\\dashv(f)-\\theta^\\dashv(g)=k_1q_1'a_1+\\cdots + k_nq_n'a_n.$$\nFor $i=1,\\ldots,n$ there exists $q_i \\in Q$, $y_i \\in X^*$ such that\n$\\theta^\\dashv(q_iy_i)=q_i'a_i$.\nHence \n$$\\theta^\\dashv(f)-\\theta^\\dashv(g)=k_1q_1y_1+ \\cdots + k_nq_ny_n.$$\nNow $\\theta^\\dashv$ preserves $+$ and therefore\n$\\theta^\\dashv( f - g - k_1q_1y_1 - \\cdots - k_nq_ny_n ) = 0$.\nBy the definition of $\\theta$, and so $\\theta^\\dashv$, therefore\n$$f-g-k_1q_1y_1 - \\cdots - k_nq_ny_n = l_1u_1p_1v_1 + \\cdots +\nl_mu_mp_mv_m$$\nfor some $p_1,\\ldots,p_m \\in P$, \n$l_1,\\ldots,l_m\\in K$ and $u_1,\\ldots,u_m \\in \\dashv\\!X^* \\cup \\{i\\!d\\}$, \n$v_1,\\ldots,v_m \\in X^*$.\nTherefore $f \\stackrel{*}{\\leftrightarrow}_F g$, from the definition.\nTherefore $\\phi$ is a well-defined bijection of sets.\n\\end{proof}\n\n\\begin{cor}\nLet $S$ be a semigroup with presentation $sgp\\langle X | R\\rangle$.\nLet $P:=\\{l-r:(l,r)\\in R\\}$, $Q \\subseteq K[X^\\dagger]$.\nDefine $F:=(P,\\dashv\\!Q)$. Then there is a bijection of sets\n$$\\frac{K[S]}{\\langle Q' \\rangle^r} \\cong \n  \\frac{K[\\dashv\\!X^\\dagger]}{\\stackrel{*}{\\leftrightarrow}_F}$$\n\\end{cor}\n\nHere it is appropriate to observe the link to rewrite systems which is used\nin the proof of this corollary, in particular, $alg\\langle X|P\\rangle$ is\na presentation of $K[S]$ \\cite{TMora,Birgit,paper3}. \nThis corollary (also see the next result) provides an alternative\napproach to that of Reinert and Zecker \\cite{MRC} for attempting the computation of ideals in \n$\\mathbb{Q}[M]$, where $M$ is a monoid. \nOur computations are based in $\\mathbb{Q}[X^*]$ where $X$ is a set of \ngenerators for $M$, the computations of Reinert and Zecker are made within\n$\\mathbb{Q}[M]$, (also using a presentation of $M$).\n\n\n\\begin{thm}\n Let $X$ be a set of generators for the terms of a $K$-algebra $A$ and let\n$P \\subseteq K[X^*]$ such that the natural morphism $\\theta:K[X^*] \\to A$\ninduces an isomorphism $K[X^*]\/\\langle P \\rangle \\to A$.\nLet $Q \\subseteq K[X^*]$ and define $Q':=\\theta Q$.\nDefine $F:=(\\dashv\\!Q, P)$ where $\\dashv\\!Q:=\\{\\dashv\\!q:q\\in Q\\}$.\nThen there is a bijection of sets\n$$\n\\frac{ K[\\dashv \\! X^*] }{ \\stackrel{*}{\\leftrightarrow}_F }\n\\cong  \\frac{A}{ \\langle Q' \\rangle^r }$$\n\\end{thm}\n\\begin{proof}\nDefine \n$\\phi:K[\\dashv\\!X^*]\/\\stackrel{*}{\\leftrightarrow}_F \\; \\to A\/ \\langle Q' \\rangle^r$ \nby\n$\\phi([f]_F):=[\\theta^\\dashv(f)]_{Q'}$.\nThe verification that $\\phi$ is a well-defined bijection on the congruence \nclasses is similar to that detailed in the proof of Theorem 3.4.\n\\end{proof}\n\n\n\n\n\n\\section{The Noncommutative Buchberger Algorithm for One-sided Ideals}\n\nRecall that $F=(F_T,F_P)$ is a mixed set of polynomials \n$F_T \\subseteq K[\\dashv\\!X^\\dagger]$ and\n$F_P \\subseteq K[X^\\dagger]$.\nThe definition of reduction of a tagged polynomial\n$f$ requires that a tagged term $\\dashv \\! m$ of $f$ is some\nmultiple\nof a leading term from the polynomials $f_i$ of $F$. \nThis definition of reduction will \nallow the application of the standard noncommutative\nBuchberger algorithm to $F$ to attempt to complete \n$\\to_F$.\n\n\\begin{Def}[Gr\\\"obner basis of mixed polynomials]\nA set $F=(F_T,F_P)$ where $F_T \\subseteq K[\\dashv\\!X^\\dagger]$ and\n$F_P \\subset K[X^\\dagger]$ is a \\textbf{Gr\\\"obner basis} on \n$K[\\dashv\\!X^\\dagger]$ with respect to $>$ if $\\to_F$ is complete.\n\\end{Def} \n\n\n\\begin{lem}[Noetherian property]\nLet $F=(F_T,F_P)$ where $F_T \\subseteq K[\\dashv \\! X^\\dagger]$ and \n$F_P \\subseteq K[X^\\dagger]$. \nLet $>$ be a semigroup well-ordering on $X^\\dagger$.\nThen the reduction relation $\\to_{F}$ is Noetherian on $K[\\dashv \\!\nX^\\dagger]$.\n\\end{lem}\n\n\\begin{proof}\nAccording to the definition, the process of reduction replaces one monomial\nwith monomials which are smaller with respect to $>$ \n(since $>$ is a term order on $X^\\dagger$).\nThe existence of an infinite sequence of reductions\n$f_1 \\to_{F} f_2 \\to_{F} \\cdots$ of polynomials \n$f_1,f_2, \\ldots \\in K[\\dashv \\! X^\\dagger]$ would imply the existence\nof\nan infinite sequence $m_1>m_2> \\cdots$ of terms \n$m_1,m_2, \\ldots \\in X^\\dagger$.\nTherefore $\\to_{F}$ is Noetherian.\n\\end{proof}\n\n\\begin{Def}[Matches and S-polynomials of tagged and non-tagged\npolynomials]\n\\mbox{ }\\\\\nLet $F=(F_T,F_P)$ where \n$F_T \\subseteq K[\\dashv\\!X^\\dagger]$ and \n$F_P \\subseteq K[X^\\dagger]$.\nA pair of polynomials $f_1,f_2 \\in F$ \nhas a \\textbf{match} if their\nleading terms $m_1,m_2$ coincide. If a pair of polynomials have a match then\nan \\textbf{S-polynomial} is defined. There are five possible cases:\n\\begin{center}\n\\begin{tabular}{|l|ll|l|l|}\n\\hline\n      & & match & S-polynomial & \\\\\n\\hline\nboth $f_1$ and $f_2$ in $F_T$  &    (i)    & $m_1 v = m_2$   & $f_1v-f_2$  & where $v\\in X^*$ \\\\\n\\hline\n$f_1$ in $F_T$ and $f_2$ in $F_P$ & (ii)   & $m_1 v = u m_2$ & $f_1v-uf_2$ &  \\\\ \n& (iii)                                    & $m_1 = u m_2 v$ & $f_1-uf_2v$ & where $u \\in \\dashv\\!X^*\\cup\\{i\\!d\\}, v \\in X^*$ \\\\\n\\hline\nboth $f_1$ and $f_2$ in $F_P$ & (iv)       & $u m_1 = m_2 v$ & $f_1v-uf_2$ & \\\\\n&  (v)                                     & $m_1 = u m_2 v$ & $f_1-uf_2v$ & where $u,v \\in X^*$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\nA match is said to \\textbf{resolve} if the resulting S-polynomial can be\nreduced to zero by $F$.\n\\end{Def}\n\n\n\\begin{rem}\n{\\em\nIf a match of any of the types above occurs between $f_1$ and $f_2$\nthen the match may be represented:\n$u_1 m_1 v_1 =u_2 m_2 v_2$, where $u_1,u_2,v_1,v_2 \\in \\dashv\\!X^* \\cup X^*$.\nA match of $f_1$ and $f_2$ may occur when either, neither, or both of\n$f_1$ and $f_2$ are tagged.\nHowever, if one or both has a tag, the tag forms part of the match and\nthe resulting S-polynomial will be tagged.}\n\\end{rem}\n\nThe following lemma is proved in the same way as in the standard \ncommutative non-tagged situation as described in \\cite{TAT}.\n\n\\begin{lem}\n\\label{zero}\nLet $F=(F_T,F_P)$ where $F_T \\subseteq K[\\dashv \\! X^\\dagger]$\nand $F_P \\subseteq K[X^\\dagger]$.\nLet $g_1, g_2 \\in K[\\dashv\\!X^\\dagger]$ where $g_1-g_2 \\to_{F}^* 0$. \nThen there exists a tagged polynomial $h \\in K[\\dashv\\!X^\\dagger]$ such\nthat \n$g_1 \\stackrel{*}{\\to}_{F} h$ and\n$g_2 \\stackrel{*}{\\to}_{F} h$.\n\\end{lem}\n\n\\begin{proof} \nThe {\\em length} of a reduction sequence is \ndefined to be the number of one-step reductions of which it is made up.\nThis proof is by induction on the length of the reduction sequence\n$g_1-g_2 \\stackrel{*}{\\to}_{F} 0$.\\\\\n\nFor the basis of induction \nsuppose the length of the reduction sequence is zero.\nThen $g_1-g_2=0$ so $g_1=g_2$.\\\\\n\nFor the induction step, assume that if $g_1'-g_2' \\stackrel{*}{\\to}_{F}\n0$\nis a reduction sequence of length $n$ then there exists \n$h \\in K[\\dashv\\!X^\\dagger]$ such that \n$g_1' \\stackrel{*}{\\to}_{F} h$ and\n$g_2' \\stackrel{*}{\\to}_{F} h$.\\\\\n\nSuppose $g_1-g_2 \\to_{f_i} g \\stackrel{*}{\\to}_{F} 0$ where\n$g \\stackrel{*}{\\to}_{F} 0$ is a reduction sequence of length $n$.\n\\\\\n\nLet $t \\in {\\dashv\\!X^\\dagger}$ be the tagged term in $g_1-g_2$ to\nwhich the reduction by $f_i$ is applied.\nLet $u \\in \\dashv\\!X^* \\cup \\{i\\!d\\}$, $v \\in X^*$ such that \n$t=u \\mathtt{LT}(f_i) v$, and let $k_1,k_2$ be the coefficients\nof $t$ in $g_1,g_2$ respectively.\nNow $k_1-k_2 \\not= 0$ since it is the coefficient of $t$ in $g_1-g_2$.\\\\\n\nDepending on whether $k_1$ and $k_2$ are zero or not we have the\nfollowing\nzero- or one-step reductions:\n$$g_1 \\stackrel{=}{\\to}_{f_i} g_1-k_1uf_iv, \\quad\ng_2 \\stackrel{=}{\\to}_{f_i} g_2-k_2uf_iv.$$\n\nSince $g=(g_1-k_1uf_iv)-(g_2-k_2uf_iv)$ and $g \\stackrel{*}{\\to} 0$ in\n$n$ \nsteps, by the induction hypothesis there exists $h \\in\nK[\\dashv\\!X^\\dagger]$\nsuch that $g_1 - k_1uf_iv \\stackrel{*}{\\to}_{F} h$ and \n$g_2 - k_2uf_iv \\stackrel{*}{\\to}_{F} h$. Hence $g_1 \\stackrel{*}{\\to}_F\nh$\nand $g_2 \\stackrel{*}{\\to}_F h$.\n\\end{proof}\n\n\\begin{thm}[Test for confluence]\nThe reduction relation $\\to_{F}$ generated by $F$ \nis complete on \n$K[\\dashv \\! X^\\dagger]$ if and only\nif all matches of $F$ resolve.\n\\end{thm}\n\n\\begin{proof}\nIn Lemma 4.2 we proved that $\\to_{F}$ is Noetherian and therefore,\nby Newman's Lemma for reduction relations on sets, we need only to prove\nthat\n$\\to_{F}$ is locally confluent.\\\\\n  \n  Let $f,g_1,g_2 \\in K[\\dashv\\!X^\\dagger]$ such that \n$f \\to_{F} g_1$ and \n$f \\to_{F} g_2$.\n  Then \n$g_1=f-k_1u_1f_1v_1$ and\n$g_2=f-k_2u_2f_2v_2$ for some $f_1,f_2 \\in F$, $k_1,k_2 \\in K$,\n$u_1,u_2,v_1,v_2 \\in \\dashv\\!X^* \\cup X^*$.\n  Let $m_1:=\\mathtt{LT}(f_1)$ and $m_2:=\\mathtt{LT}(f_2)$.\\\\\n  \n  If the reductions do not overlap on $f$, i.e. \n$u_1m_1v_1 \\not= u_2m_2v_2$ then it is immediate that \n$g_1 \\to_{F} h$ and \n$g_2 \\to_{F} h$ where \n$h = f - k_1u_1f_1v_1 - k_2u_2f_2v_2$.\\\\\n\nOtherwise $u_1m_1v_1=u_2m_2v_2$. \nIn this case $m_1$ and $m_2$ may or may not coincide.\\\\\n\nIf they do not coincide, i.e. if there exists $w \\in X^*$ such that\n$u_1m_1v_1=u_1m_1wm_2v_2$ or $u_1m_1v_1=u_2m_2wm_1v_1$ then again\n$g_1 \\stackrel{*}{\\to}_{F} h$ and \n$g_2 \\stackrel{*}{\\to}_{F} h$ where \n$h = f - u_1f_1wm_2v_2 - u_1m_1wf_2v_2$ or \n$h = f - u_2m_2wf_1v_1 - u_2f_2wm_1v_1$ respectively.\\\\\n\nIf the leading terms $m_1$ and $m_2$ do coincide then \n$u_1m_1v_1=u_2m_2v_2$ represents a multiple of a match between $f_1$ and\n$f_2$,\ni.e. there exist $u_1',u_2',v_1',v_2',w,z \\in \\dashv\\!X^* \\cup X^*$, \nsuch that $u_1=wu_1'$, $v_1=v_1'z$, $u_2=wu_2'$, $v_2=v_2'z$ and\n$u_1'm_1v_1'=u_2'm_2v_2'$ represents a match between $f_1$ and $f_2$.\nIn this case $u_1'f_1v_1'-u_2'f_2v_2' \\stackrel{*}{\\to}_{F} 0$ by\nassumption, and therefore \n$wu_1'f_1v_1'z-wu_2'f_2v_2'z \n  = u_1m_1v_1-u_2m_2v_2 \\stackrel{*}{\\to}_{F} 0$. By Lemma 4.5 this\n  implies that there exists $h \\in K[\\dashv \\! X^\\dagger]$ such that\n$g_1 \\stackrel{*}{\\to}_{F} h$ and \n$g_2 \\stackrel{*}{\\to}_{F} h$.\\\\\n \n\nThe converse of the above is easily checked. \nSuppose that $\\to_{F}$ is confluent. Then any\nS-polynomial arising from a match between polynomials is the result of\nreducing one term in two different ways, i.e. \n$f \\to_{F} g_1$ and\n$f \\to_{F} g_2$ for some $f,g_1,g_2\\in K[X^\\dagger]$. \nThe S-polynomial is\nequal to\n$g_1- g_2$. The relation $\\to_{F}$ is locally confluent and so\nthere\nexists $h \\in K[\\dashv \\! X^\\dagger]$ such that \n$g_1 \\to_{F} h$ and \n$g_2 \\to_{F} h$. Therefore\n$g_1 - g_2 \\stackrel{*}{\\to} h - h = 0$ as required.\n\\end{proof}\n\n\nWe may now apply the noncommutative version of Buchberger's algorithm \n(as described in \\cite{TMora2}) to \nattempt to complete a mixed set of tagged and non-tagged polynomials.\nTo verify steps 4 and 5 of the algorithm we observe the following two \ntechnical lemmas.\n\n\\begin{lem}[Addition of S-polynomials]\nLet $F=(F_T,F_P)$. If $f$ is an S-polynomial resulting from a match of $F$, \nthen the congruences $\\stackrel{*}{\\leftrightarrow}_F$ and\n$\\stackrel{*}{\\leftrightarrow}_{F\\cup\\{f\\}}$ coincide.\n\\end{lem}\n\\begin{proof}\nThe result is proved by showing that, in each of the five cases, an\nS-polynomial $f$ resulting from a match of polynomials $f_1,f_2 \\in F$ \ncan be written in the form $u_1f_1v_1-u_1f_2v_2$ and therefore \n$f \\stackrel{*}{\\leftrightarrow}_F 0$.\n\\end{proof}\n\n\\begin{lem}[Elimination of redundancies]\nLet $F=(F_T,F_P)$. If $f\\in F$ is such that $f \\to_{F \\setminus\\{f\\}} 0$ then \nthe relations $\\stackrel{*}{\\to}_F$ and\n$\\stackrel{*}{\\to}_{F\\setminus\\{f\\}}$ coincide.\n\\end{lem}\n\\begin{proof}\nThe result is immediate, since for all $g \\to_{\\{f\\}} h$ then \n$g = h+ kufv \\stackrel{*}{\\to}_{F\\setminus\\{f\\}} h$ where $k \\in K$, \n$ufv \\in \\dashv\\!X^\\dagger$.\n\\end{proof} \n\n\\begin{alg}[Noncommutative Buchberger Algorithm with tags]\n\nGiven a set of tagged and non-tagged polynomials the algorithm \nattempts to complete the set with respect to a given ordering\nso that the reduction relation generated is complete.\n\\begin{enumerate}\n\\item\n(Input:) A mixed set of tagged and non-tagged polynomials \n$F=(F_T,F_P)$ where $F_T \\subseteq K[\\dashv\\!X^\\dagger]$ and \n$F_P \\subseteq K[X^\\dagger]$.\n\\item\n(Initialise:)\nPut $\\mathtt{OLD} := F$ and $\\mathtt{SPOL}:=\\emptyset$. \n\\item\n(Search for matches:)\nIf the leading monomials of any of the polynomials overlap then calculate\nthe\nresulting S-polynomial and attempt to reduce it using $\\to_F$.\nRecord all non-zero reduced S-polynomials in the list $\\mathtt{SPOL}$.\n\\item\n(Add unresolved S-polynomials:)\nWhen all matches have been considered define $\\mathtt{NEW:=OLD \\cup\nSPOL}$.\n\\item\n(Eliminate redundancies:)\nPass through $\\mathtt{NEW}$ removing each polynomial in turn and\nreducing it with respect to the other polynomials in $\\mathtt{NEW}$. \nIf a polynomial reduces to zero, delete it from\n$\\mathtt{NEW}$. Otherwise replace each with its reduced form.\n\\item\n(Loop:)\nWhilst $\\mathtt{OLD} \\not= \\mathtt{NEW}$ set $\\mathtt{OLD:=NEW}$,\n$\\mathtt{SPOL}:=\\emptyset$ and return to step 3.\n\\item\n(Output:)\nA set $F:=\\mathtt{NEW}$ of polynomials such that $\\to_F$\nis a complete reduction relation on $K[\\dashv\\!X^\\dagger]$.\n\\end{enumerate}\n\\end{alg}\n\n\\begin{rem}[Left Ideals]\n\\emph{Placing tags to the right of polynomials rather than the left, \ni.e. working in $K[X^\\dagger\\vdash]$ or $K[X^*\\vdash]$, by similar arguments\nwe can compute left ideals. The tags act as a block to multiplication: \nto calculate left ideals, one blocks the right multiplication with a tag on \nthe right; to calculate right ideals, one blocks the left multiplication \nwith a tag on the left. It is natural that two-sided ideals have no tags, \nsince both multiplications are defined.}\n\\end{rem}\n\n\n\\begin{rem}[Implementation]\n\\emph{The use of the free monoid $(X \\cup \\{ \\dashv \\})^*$ is possible\nin Definition 3.2, i.e. $f \\to_F f-kuf_iv$ if \n$u\\mathtt{LT}(f_i)v$ occurs in $f$ with coefficient $k$\n$u,v \\in (X \\cup \\{ \\dashv \\})^*$.  \nIf a match of any of the five types described above occurs between \n$f_1$ and $f_2$ then there exist\n$u_1,u_2,v_1,v_2 \\in (X \\cup \\{ \\dashv \\})^*$ such that\n$u_1 m_1 v_1 =u_2 m_2 v_2$ (the converse is not true).\nUnmeaningful monomials \nsuch as $\\dashv\\!xx\\!\\dashv\\dashv\\!x$ do not arise as a result of\nany procedure we describe, including Algorithm 4.9.\nTherefore there is no problem with the\ncomputations taking place inside the free $K$-algebra \n$K[(X \\cup\\{\\dashv\\})^*]$.\nThis is useful computationally as it allows us to use a standard \nnoncommutative Gr\\\"obner basis program as an implementation of the procedures.\nIn other words, this widens the scope of a noncommutative Gr\\\"obner basis \nprogram without modifying it: the program can now attempt to compute bases \nfor one-sided ideals in finitely presented algebras.}\n\\end{rem}\n\n\n\\section{Application to Green's Relations}\n\nThe standard way of expressing the structure of an (abstract) semigroup\nis in terms of Green's relations. The relations enable the expression\nof the local structure of the semigroup in terms of groups with certain\nactions \non them. Eggbox diagrams depict the partitions of a semigroup into their\n$L$-classes $R$-classes, $D$-classes and $H$-classes as defined by\nGreen's\nrelations. \nWe can sometimes determine the classes by using Gr\\\"obner bases applied \ndirectly to the presentation. The\nexamples show that there is also the possibility of dealing with\ninfinite\nsemigroups having infinitely many $H$-classes, $L$-classes or\n$R$-classes.\nFirst we recall some definitions \\cite{Howie}.\\\\\n\nA nonempty subset $A$ of a\nsemigroup $S$ is a {\\em right ideal} of $S$ if $AS \\subseteq A$, where\n$AS:=\\{as:a \\in A, s \\in S\\}$.\nIt is a {\\em left ideal} of $S$ if $SA \\subseteq A$.\nIf $x$ is an element of $S$ then the smallest right ideal of $S$\ncontaining $x$\nis $xS \\cup \\{x\\}$, we denote this $\\langle x\\rangle^r $ as it is called\nthe\n{\\em right ideal generated by $x$}. Similarly the \n{\\em left ideal generated by $x$} is $Sx \\cup \\{x\\}$ and is denoted \n$\\langle x \\rangle^l$.\\\\\n\n\\textbf{Green's Relations}\\\\\nLet $S$ be a semigroup and let $s$ and $t$ be elements of $S$.\nWe say that $s$ and $t$ are {\\em L-related} if the left ideal\ngenerated\nby $s$ in $S$ is equal to the left ideal generated by $t$:\n$$s \\sim_L t \\Leftrightarrow \\langle s\\rangle^l =\\langle t\\rangle^l .$$\nSimilarly they are {\\em R-related} if the right ideals are the same:\n$$s \\sim_R t \\Leftrightarrow \\langle s\\rangle^r =\\langle t\\rangle^r.$$\n\nThe $L$-relation is a right congruence on $S$ and the $R$-relation is a\nleft\ncongruence on $S$. (The right action of $S$ on itself is preserved by\nthe\nmapping to the\n$L$-classes - so $[x^y]_{\\sim_L}=[xy]_{\\sim_L}={[x]^y}_{\\sim_L}$,\nsimilarly for\nthe left action and $R$-classes.)\nThe elements $s$ and $t$ are said to be {\\em H-related} if they are\n\\emph{both} $L$-related \\emph{and} $R$-related, and are\n{\\em D-related} if they are \\emph{either} $L$-related \\emph{or}\n$R$-related.\\\\\n\nTo determine whether $s$ and $t$ are $R$ (or $L$)-related\nwe can compute the appropriate Gr\\\"obner bases and compare them.\nFirst let $K$ be (any) field.\nLet $S$ have presentation $sgp\\langle X|Rel\\rangle$\nLet $P$ be a Gr\\\"obner basis for $K[S]$ (so $K[X^\\dagger]\/\\! =_P\\cong\nK[S]$).\nWe would add\nthe polynomial $\\dashv \\! s$ to the Gr\\\"obner basis system for $K[S]$\nand\ncompute\nthe Gr\\\"obner basis, and see whether this was equivalent to the basis\nobtained\nfor $\\dashv \\! t$.\n\n\n\\section{Examples}\n\nThroughout the examples we will use the field $\\mathbb{Q}$ and the \nstandard length-lexicographical ordering $>$.\n\n\\begin{example}\n\\emph{The first example is a two element semigroup with presentation}\n$sgp \\langle x |x^3=x^2 \\rangle.$\\\\\n\n\\emph{The Gr\\\"obner basis for the right ideal $\\langle x\\rangle^r$ is\n$\\{\\dashv\\!x, x^3-x^2\\}$ and the Gr\\\"obner basis for $\\langle x^2 \\rangle^r$\nis $\\{\\dashv\\!x^2, x^3-x^2\\}$. The Gr\\\"obner bases are different and therefore \n$x$ and $x^2$ are not $R$-related.\nSimilarly, the Gr\\\"obner basis for the left ideal $\\langle x\\rangle^l$  is\n$\\{x\\!\\vdash, x^3-x^2\\}$ and the Gr\\\"obner basis for $\\langle x^2\\rangle^l$ is\n$\\{x^2\\!\\vdash, x^3-x^2\\}$ so the elements are not $L$-related. Therefore\nthis semigroup has two $H$-classes.} \n\\end{example}\n\n\n\\begin{example}\n\\emph{The following example is for the \nfinite monoid $Sym(2)$ with semigroup \npresentation}\n\n\\vspace{-.4cm}\n$$\nsgp\\langle e,s| e^2=e, s^3=s, s^2e=e, es^2=e, sese=ese, eses=ese\\rangle.\n$$ \n\\vspace{-.5cm}\n\n\\emph{The Gr\\\"obner basis equivalent to the rewrite system is}\n\n\\vspace{-.4cm}\n$$\nF:=\\{e^2-e, \\ s^3-s, \\ s^2e-e, \\ es^2-e, \\ eses-ese, \\ sese-ese\\}.\n$$\n\\vspace{-.5cm}\n\n\\emph{The elements are $\\{ e, s, es, se, s^2, ese, ses\\}$, \nwhere $s^2$ is the identity element.\nWe calculate Gr\\\"obner bases for \nthe right and left ideals for each of\nthe elements. The results are displayed \nin the table below. In detail, a Gr\\\"obner\nbasis for $\\langle ses \\rangle^r$ in $K[S]$ in \n$K[\\dashv\\! X^\\dagger]$ is calculated by \nadding $\\dashv\\! ses$ to the set of polynomials \n$F$.\nA match $s$ occurs on $\\dashv\\! sesse$ \nbetween $sse-e$ and $\\dashv\\! ses$. \nThis results in the S-polynomial\n$\\dashv \\! se(e)-(0)se$ \nwhich reduces to $\\dashv \\! se$. \nAnother match of $es$ occurs on $\\dashv \\! seses$ \nbetween $eses-ese$ and $\\dashv \\! ses$. \nThis results in the \nS-polynomial $\\dashv \\! s(ese)-(0)es$ \nwhich reduces to $\\dashv \\! ese$.\nAll further matches result in \nS-polynomials which reduce to zero. \nThe polynomials we add to $F$ \nto obtain a Gr\\\"obner basis are \n$\\{ \\dashv \\! se, \\dashv \\! ese \\}$ \n(note that $\\dashv \\! ses$ is a \nmultiple of $\\dashv \\! se$ so it is \nnot required in the Gr\\\"obner basis).\nThe table lists the polynomials which, \ntogether with $F$, will give the \nGr\\\"obner bases for the right and left \nideals generated by single elements.}\n\n\\begin{center}\n\\begin{tabular}{l|l|l}\nelement & right ideal & left ideal\\\\\n\\hline\n\n$e$    & $\\dashv\\! e                $    & $e \\!\\vdash\n$\\\\\n$s$    & $\\dashv\\! e,  \\dashv\\! s   $    & $e \\!\\vdash,   s \\!\\vdash\n$\\\\\n$es$   & $\\dashv\\! e                $    & $es \\!\\vdash , ese \\!\\vdash\n$\\\\\n$se$   & $\\dashv\\! se, \\dashv\\! ese $    & $e \\!\\vdash\n$\\\\\n$ss$   & $\\dashv\\! e,  \\dashv\\! s   $    & $e \\!\\vdash,   s \\!\\vdash\n$\\\\\n$ese$  & $\\dashv\\! ese              $    & $ese \\!\\vdash\n$\\\\\n$ses$  & $\\dashv\\! se, \\dashv\\! ese $    & $es \\!\\vdash , ese \\!\\vdash\n$\n\\end{tabular}\n\\end{center}\n\n\\emph{Two elements whose right ideals are generated by the same\nGr\\\"obner basis have \nthe same right ideal (similarly left), and so\nit is immediately deducible that\\\\\nthe $R$-classes are $\\{s,s^2\\},\\{e,es\\},\\{se,ses\\}$ and $\\{ese\\}$, \nthe $L$-classes are $\\{s,s^2\\},\\{e,se\\},\\{es,ses\\}$ and $\\{ese\\}$,\nthe $H$-classes are $\\{s,s^2\\},\\{e\\},\\{se\\},\\{es\\},\\{ses\\}$ and\n$\\{ese\\}$ and\nthe $D$-classes are $\\{s,s^2\\},\\{e,es,se,ses\\}$ and $\\{ese\\}$.\\\\\nThe eggbox diagram is as follows\nwhere $L$-classes are columns, $R$-classes are rows, \n$D$-classes are diagonal boxes\nand $H$-classes are the small boxes:}\n\\\\\n\n\\setlength{\\unitlength}{0.7cm}\n\\begin{picture}(10,4)\n\\put(9,3){\\line(0,0){1}}\n\\put(10,1){\\line(0,0){3}}\n\\put(11,1){\\line(0,0){2}}\n\\put(12,0){\\line(0,0){3}}\n\\put(13,0){\\line(0,0){1}}\n\\put(9,4){\\line(1,0){1}}\n\\put(9,3){\\line(1,0){3}}\n\\put(10,2){\\line(1,0){2}}\n\\put(10,1){\\line(1,0){3}}\n\\put(12,0){\\line(1,0){1}}\n\\put(9.1,3.4){$s,\\!s^2$}\n\\put(10.3,1.4){$se$}\n\\put(10.4,2.4){$e$}\n\\put(11.2,1.4){$ses$}\n\\put(11.3,2.4){$es$}\n\\put(12.2,0.4){$ese$}\n\\end{picture}\n\n\n\\emph{This example could have been calculated by enumerating the\nelements of each of the fourteen ideals -- a time consuming procedure which\ncalculates details which we do not require.}\n\\end{example}\n\n\n\\begin{example}\n\\emph{The next example is the Bicyclic monoid which is infinite.\nWe use the semigroup presentation} \\;\n$sgp \\langle p,q, i | p i=p, q i= q, i p=p, iq=q, pq=i \\rangle.$\\\\\n\n\n\\emph{The equivalent Gr\\\"obner basis, defined on $K[\\{p,q,i\\}^\\dagger]$, is \n$\\{p i-p, q i- q, i p-p, iq-q, pq-i\\}$.\nWe begin the table as before:}\n\n\\begin{center}\n\\begin{tabular}{l|l|l}\nelement & right ideal & left ideal\\\\\n\\hline\n$i\\!d$    & $\\dashv\\! i$. & $i\\! \\vdash $.\\\\\n$p$      & $\\dashv\\! i$. & $p\\! \\vdash $.\\\\\n$q$      & $\\dashv\\! q$.   & $q\\! \\vdash $.\\\\\n$p^2$    & $\\dashv\\! i$. & $p^2\\! \\vdash $.\\\\\n$qp$     & $\\dashv\\! q$.   & $p\\! \\vdash $.\\\\\n$q^2$    & $\\dashv\\! q^2$. & $i\\! \\vdash $.\\\\\n$\\cdots$ & $\\cdots$        & $\\cdots$\\\\\n$q^np^m$ & $\\dashv\\! q^n$. & $p^m\\! \\vdash $.\\\\\n\\end{tabular}\n\\end{center}\n\\emph{It can be seen that there are infinitely many $L$-classes \nand infinitely many $R$-classes.\nRepresentatives for the $L$-classes are the elements of $\\{q\\}^*$\nbecause \n$q^np^m\\! \\vdash  \\to q^n\\! \\vdash $ (using the S-polynomial resulting\nfrom\n$p^n(q^np^m\\! \\vdash )\\to p^n\\! \\vdash $ with  $(p^nq^n)p^m\\! \\vdash\n\\to p^m\\! \\vdash $).\nSimilarly the elements of $\\{p\\}^*$ are representatives for the\n$R$-classes.\nAll elements are $D$-related and none of them are $H$-related. So the\neggbox\ndiagram would be an infinitely large box of cells with one element in\neach\ncell. This means that the monoid is} bisimple.\n\\end{example}\n\n\n\\begin{example}\n\\emph{Now consider the Polycyclic monoid $P_n$ which has monoid\npresentation}\n\n\\vspace{-.4cm}\n$$\nmon\\langle x_1,\\ldots,x_n,y_1,\\ldots,y_n,o, i\\!d\n\\; | \\; ox_i \\! = \\! x_io \\! = \\! oy_i \\! = \\! y_io \\! = \\! o, x_iy_i \\!\n= \\! i\\!d,\nx_iy_j \\! = \\! o  \\text{\\emph{ for }} i,j \\! = \\! 1,\\ldots,{n\\!-\\!1},\ni\\not \\! = \\! j\\rangle\n$$\n\\vspace{-.4cm}\n\n\\emph{and therefore the Gr\\\"obner basis for $K[P_n]$, where $K$ is a\nfield, is}\n\n\\vspace{-.2cm}\n$$\n\\{x_iy_i-i\\!d, x_iy_j-0 \\text{ for } i,j=1,\\ldots,{n-1},\\, i \\not= j \\}.\n$$\n\\vspace{-.2cm}\n\n\\emph{Green's relations for the polycyclic monoids are naturally similar to\nthose for the Bicyclic monoid. The $L$-classes are represented by\nsequences of\n$y_i$'s and the\n$R$-classes are represented by sequences of $x_i$'s. To verify this, let\n$X=x_{i_1}\\cdots x_{i_k}$ be a\ngeneral word in the $x_i$'s, and let $Y$ be $y_{j_1}\\cdots y_{j_l}$ a\ngeneral\nword in the $y_j$'s.\nThen we can show that $YX \\sim_L X$.\nTo do this consider $\\langle YX\\! \\vdash  \\rangle$ and $\\langle X\\!\n\\vdash  \\rangle$.\n To find a Gr\\\"obner basis for $\\langle YX\\! \\vdash  \\rangle$ consider\nthe match\n $x_{j_l} \\cdots x_{j_1}y_{j_1} \\cdots y_{j_l}x_{i_1} \\cdots\nx_{i_k}\\! \\vdash $.\nThis results in the S-polynomial\n $(i\\!d)x_{i_1} \\cdots x_{i_k}\\! \\vdash  - x_{j_l} \\cdots x_{j_1}(0)$\nwhich simplifies to $x_{i_1} \\cdots x_{i_k}\\! \\vdash  = X\\! \\vdash $.\nThis is\na\nGr\\\"obner basis for $\\langle YX\\! \\vdash  \\rangle$, and so $\\langle YX\\!\n\\vdash  \\rangle\n=\\langle\nX\\! \\vdash  \\rangle$. Similarly $\\langle \\dashv \\! YX \\rangle= \\langle\n\\dashv \\! Y \\rangle$ so\n$YX\n\\sim_R X$ for any $Y = y_{j_1}\\cdots y_{j_l}$, $X = x_{i_1}\\cdots\nx_{i_k}$.}\n\\\\\n\n\\emph{The eggbox diagram is drawn below. As before the $L$ classes are the\ncolumns\nand the $R$-classes the rows, $H$-classes are the cells, and there is\njust one\n$D$-class other than the one containing the zero. This proves that the\npolycyclic monoids are bisimple.\nThe diagram is more conventional than the\nprevious one, as classes are listed but not individual elements, instead\nthe\nnumber of elements in each cell is indicated.}\n\n\n\\setlength{\\unitlength}{0.8cm}\n\\begin{picture}(16,12)\n\\put(5,10){\\line(0,0){1}}\n\\put(6,1){\\line(0,0){1}}\n\\put(6,4){\\line(0,0){2}}\n\\put(6,7){\\line(0,0){4}\n\\put(7,1){\\line(0,0){1}}\n\\put(7,4){\\line(0,0){2}}\n\\put(7,7){\\line(0,0){3}}\n\\put(8,1){\\line(0,0){1}}\n\\put(8,4){\\line(0,0){2}}\n\\put(8,7){\\line(0,0){3}}\n\\put(9,1){\\line(0,0){1}}\n\\put(9,4){\\line(0,0){2}}\n\\put(9,7){\\line(0,0){3}}\n\\put(10,1){\\line(0,0){1}}\n\\put(10,4){\\line(0,0){2}}\n\\put(10,7){\\line(0,0){3}}\n\\put(11,1){\\line(0,0){1}}\n\\put(11,4){\\line(0,0){2}}\n\\put(11,7){\\line(0,0){3}}\n\\put(12,1){\\line(0,0){1}}\n\\put(12,4){\\line(0,0){2}}\n\\put(12,7){\\line(0,0){3}}\n\\put(14,1){\\line(0,0){1}}\n\\put(14,4){\\line(0,0){2}}\n\\put(14,7){\\line(0,0){3}}\n\\put(15,1){\\line(0,0){1}}\n\\put(15,4){\\line(0,0){2}}\n\\put(15,7){\\line(0,0){3}}\n\\put(5,10){\\line(1,0){4}\n\\put(5,11){\\line(1,0){1}}\n\\put(6,9){\\line(1,0){3}}\n\\put(6,8){\\line(1,0){3}}\n\\put(6,7){\\line(1,0){3}}\n\\put(6,6){\\line(1,0){3}}\n\\put(6,5){\\line(1,0){3}}\n\\put(6,4){\\line(1,0){3}}\n\\put(6,2){\\line(1,0){3}}\n\\put(6,1){\\line(1,0){3}}\n\\put(10,9){\\line(1,0){2}}\n\\put(10,8){\\line(1,0){2}}\n\\put(10,7){\\line(1,0){2}}\n\\put(10,6){\\line(1,0){2}}\n\\put(10,5){\\line(1,0){2}}\n\\put(10,4){\\line(1,0){2}}\n\\put(10,2){\\line(1,0){2}}\n\\put(10,1){\\line(1,0){2}}\n\\put(10,10){\\line(1,0){2}}\n\\put(14,9){\\line(1,0){1}}\n\\put(14,8){\\line(1,0){1}}\n\\put(14,7){\\line(1,0){1}}\n\\put(14,6){\\line(1,0){1}}\n\\put(14,5){\\line(1,0){1}}\n\\put(14,4){\\line(1,0){1}}\n\\put(14,2){\\line(1,0){1}}\n\\put(14,1){\\line(1,0){1}}\n\\put(14,10){\\line(1,0){1}}\n\\put(4.2,1.4){$[X]$}\n\\put(4.2,4.4){$[x_1x_2]$}\n\\put(4.2,5.4){$[{x_1}^2]$}\n\\put(4.2,7.4){$[x_2]$}\n\\put(4.2,8.4){$[x_1]$}\n\\put(4.2,9.4){$[i\\!d]$}   \\put(4.2,10.4){$[0]$} \\put(5.2,11.4){$[0]$}\n\\put(6.2,11.4){$[i\\!d]$}\n\\put(7.2,11.4){$[y_1]$}\n\\put(8.2,11.4){$[y_2]$}\n\\put(10.2,11.4){$[{y_1}^2]$}\n\\put(11.2,11.4){$[y_1y_2]$}\n\\put(14.2,11.4){$[Y]$}\n\\put(6,0.3){\\line(0,0){0.4}}\n\\put(7,0.3){\\line(0,0){0.4}}\n\\put(8,0.3){\\line(0,0){0.4}}\n\\put(9,0.3){\\line(0,0){0.4}}\n\\put(10,0.3){\\line(0,0){0.4}}\n\\put(11,0.3){\\line(0,0){0.4}}\n\\put(12,0.3){\\line(0,0){0.4}}\n\\put(14,0.3){\\line(0,0){0.4}}\n\\put(15,0.3){\\line(0,0){0.4}}\n\\put(6,2.3){\\line(0,0){0.4}}\n\\put(7,2.3){\\line(0,0){0.4}}\n\\put(8,2.3){\\line(0,0){0.4}}\n\\put(9,2.3){\\line(0,0){0.4}}\n\\put(10,2.3){\\line(0,0){0.4}}\n\\put(11,2.3){\\line(0,0){0.4}}\n\\put(12,2.3){\\line(0,0){0.4}}\n\\put(14,2.3){\\line(0,0){0.4}}\n\\put(15,2.3){\\line(0,0){0.4}}\n\\put(6,3.3){\\line(0,0){0.4}}\n\\put(7,3.3){\\line(0,0){0.4}}\n\\put(8,3.3){\\line(0,0){0.4}}\n\\put(9,3.3){\\line(0,0){0.4}}\n\\put(10,3.3){\\line(0,0){0.4}}\n\\put(11,3.3){\\line(0,0){0.4}}\n\\put(12,3.3){\\line(0,0){0.4}\n\\put(14,3.3){\\line(0,0){0.4}}\n\\put(15,3.3){\\line(0,0){0.4}}\n\\put(6,6.3){\\line(0,0){0.4}}\n\\put(7,6.3){\\line(0,0){0.4}}\n\\put(8,6.3){\\line(0,0){0.4}}\n\\put(9,6.3){\\line(0,0){0.4}}\n\\put(10,6.3){\\line(0,0){0.4}}\n\\put(11,6.3){\\line(0,0){0.4}}\n\\put(12,6.3){\\line(0,0){0.4}}\n\\put(14,6.3){\\line(0,0){0.4}}\n\\put(15,6.3){\\line(0,0){0.4}}\n\\put(9.3,1){\\line(1,0){0.4}}\n\\put(9.3,2){\\line(1,0){0.4}}\n\\put(9.3,4){\\line(1,0){0.4}}\n\\put(9.3,5){\\line(1,0){0.4}}\n\\put(9.3,6){\\line(1,0){0.4}}\n\\put(9.3,7){\\line(1,0){0.4}}\n\\put(9.3,8){\\line(1,0){0.4}}\n\\put(9.3,9){\\line(1,0){0.4}}\n\\put(9.3,10){\\line(1,0){0.4}}\n\\put(12.3,1){\\line(1,0){0.4}}\n\\put(12.3,2){\\line(1,0){0.4}}\n\\put(12.3,4){\\line(1,0){0.4}}\n\\put(12.3,5){\\line(1,0){0.4}}\n\\put(12.3,6){\\line(1,0){0.4}}\n\\put(12.3,7){\\line(1,0){0.4}}\n\\put(12.3,8){\\line(1,0){0.4}}\n\\put(12.3,9){\\line(1,0){0.4}}\n\\put(12.3,10){\\line(1,0){0.4}}\n\\put(13.3,1){\\line(1,0){0.4}}\n\\put(13.3,2){\\line(1,0){0.4}}\n\\put(13.3,4){\\line(1,0){0.4}}\n\\put(13.3,5){\\line(1,0){0.4}}\n\\put(13.3,6){\\line(1,0){0.4}}\n\\put(13.3,7){\\line(1,0){0.4}}\n\\put(13.3,8){\\line(1,0){0.4}}\n\\put(13.3,9){\\line(1,0){0.4}}\n\\put(13.3,10){\\line(1,0){0.4}}\n\\put(15.3,1){\\line(1,0){0.4}}\n\\put(15.3,2){\\line(1,0){0.4}}\n\\put(15.3,4){\\line(1,0){0.4}}\n\\put(15.3,5){\\line(1,0){0.4}}\n\\put(15.3,6){\\line(1,0){0.4}}\n\\put(15.3,7){\\line(1,0){0.4}}\n\\put(15.3,8){\\line(1,0){0.4}}\n\\put(15.3,9){\\line(1,0){0.4}}\n\\put(15.3,10){\\line(1,0){0.4}}   \\put(5.4,10.4){1}\n\\put(6.4,9.4){1} \\put(6.4,8.4){1} \\put(6.4,7.4){1}\n\\put(6.4,5.4){1} \\put(6.4,4.4){1} \\put(6.4,1.4){1}\n\\put(7.4,9.4){1} \\put(7.4,8.4){1} \\put(7.4,7.4){1}\n\\put(7.4,5.4){1} \\put(7.4,4.4){1} \\put(7.4,1.4){1}\n\\put(8.4,9.4){1} \\put(8.4,8.4){1} \\put(8.4,7.4){1}\n\\put(8.4,5.4){1} \\put(8.4,4.4){1} \\put(8.4,1.4){1}\n\\put(10.4,9.4){1} \\put(10.4,8.4){1} \\put(10.4,7.4){1}\n\\put(10.4,5.4){1} \\put(10.4,4.4){1} \\put(10.4,1.4){1}\n\\put(11.4,9.4){1} \\put(11.4,8.4){1} \\put(11.4,7.4){1}\n\\put(11.4,5.4){1} \\put(11.4,4.4){1} \\put(11.4,1.4){1}\n\\put(14.4,9.4){1} \\put(14.4,8.4){1} \\put(14.4,7.4){1}\n\\put(14.4,5.4){1} \\put(14.4,4.4){1} \\put(14.4,1.4){1}\n\\end{picture}\n\\end{example}\n\n\nThese examples illustrate the fact that Buchberger's algorithm can be used to\ncompute Green's relations for (infinite) semigroups which have finite\ncomplete presentations. \nPrevious methods for calculating minimal ideals from presentations of\nsemigroups were variations on the classical Todd-Coxeter enumeration\nprocedure \\cite{Campbell95}.\nThis is an alternative computational approach \nto that given in \\cite{Linton,Linton2} which uses the transformation \nrepresentation of a semigroup rather than a presentation.\nAs with \\cite{Linton2} the methods described in this paper provide for local computations, \nconcerning a single $R$-class, without computing the whole semigroup.\nThe one-sided Gr\\\"obner basis methods have\nlimitations in that a complete rewrite system with respect to the chosen\norder might not be found, but they do give the possibility of calculating the\nstructure of infinite semigroups and do not require the determination of a\ntransformation representation for those semigroups which arise \nnaturally as presentations.\\\\\n\n\nThe calculations of the examples were achieved using a $\\mathsf{GAP3}$ \nimplementation of the Gr\\\"obner basis procedures  for polynomials in \nnoncommutative variables over $\\mathbb{Q}$ as described in \\cite{TMora}. \nFurther details of this program can be found in \\cite{Annethesis} or e-mail\nthe author.\nOther implementations (e.g. OPAL, Bergman) are more powerful: the key point \nof this paper is to point out that such programs can be used \nfor a wider range of problems than has previously been recorded.\\\\\n\n\n\n{\\small\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzivkm b/data_all_eng_slimpj/shuffled/split2/finalzzivkm
new file mode 100644
index 0000000000000000000000000000000000000000..0610177b01bc99c48519c0f18ae268f611b15d32
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzivkm
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\nA key idea in noncommutative geometry is to access the study of topological dynamical systems (for example a discrete group acting on a compact space by homeomorphisms) through a naturally associated noncommutative $\\mathrm{C}^*$-algebra. Many of these constructions can be conveniently studied simultaneously in the framework of topological groupoids.\nIndeed, large classes of $\\mathrm{C}^*$-algebras have been shown to admit groupoid models since Renault's seminal work on groupoid $\\mathrm{C}^*$-algebras in \\cite{Renault1980}. The range of examples covers crossed products associated to transformation groups, (higher rank) graph-$\\mathrm{C}^*$-algebras \\cite{KumjianPask00}, the uniform Roe algebra \\cite{STY02}, and the $\\mathrm{C}^*$-algebras associated to various classes of semigroups \\cite{Paterson1999,MR3077884} to name but a few prominent ones.\nA slight alteration of the construction takes certain cohomological objects called twists associated with a groupoid $G$ into account. A twist over $G$ is an extension of $G$ by the trivial circle bundle $G^{(0)}\\times\\TT$.\n\nTwisted groupoid $\\mathrm{C}^*$-algebras are useful in the study of untwisted algebras \\cite{MR3162248}, but the addition of a twist as input data has much further reaching consequences: Indeed, building on the theory of Feldman-Moore in the von Neumann setting \\cite{MR0578656,MR0578730}, Kumjian and Renault showed \\cite{MR854149,MR2460017} that whenever a $\\mathrm{C}^*$-algebra $A$ admits a Cartan subalgebra, then $A$ is in fact isomorphic to a twisted groupoid $\\mathrm{C}^*$-algebra. In other words, (twisted) groupoid $\\mathrm{C}^*$-algebras appear intrinsically in $\\mathrm{C}^*$-algebra theory.\n\n\nThis article is concerned with the $\\K$-theory of twisted group\\-oid $\\mathrm{C}^*$-algebras. The main result establishes that the $\\mathrm{K}$-theory of a twisted groupoid $\\mathrm{C}^*$-algebra is invariant with respect to homotopies of twists for a large class of groupoids and reads as follows:\n\\begin{thmx}\\label{MainTheoremIntro}(see Theorem \\ref{Thm:Evalutation Induces Isomorphism on K-theory})\n\tLet $G$ be a second countable ample groupoid, which satisfies the Baum-Connes conjecture with coefficients and let $\\Sigma$ be a  homotopy of twists for $G$. Then the canonical map $q_t:C_r^*(G\\times [0,1];\\Sigma)\\rightarrow C_r^*(G,\\Sigma_t)$ given by evaluation induces an isomorphism $$(q_{t})_*:\\K_*(C_r^*(G\\times[0,1];\\Sigma))\\rightarrow \\K_*(C_r^*(G,\\Sigma_t)).$$\n\\end{thmx}\n\nSome remarks on Theorem \\ref{MainTheoremIntro} are in order: \nFirst, the result is non-trivial since a homotopy of twists does not result in a homotopy-equivalence of the twisted groupoid $\\mathrm{C}^*$-algebras. A basic example is provided by the irrational rotation algebra $\\mathcal{A}_\\theta$, which can be realized as a twisted group algebra $\\mathcal{A}_\\theta=C^*(\\ZZ^2,\\omega_\\theta)$ with respect to the $2$-cocycle $\\omega_\\theta((k,l),(n,m))=e^{2\\pi i \\theta ln}$. This cocycle is homotopic to the constant cocycle but $\\mathcal{A}_\\theta$ is not homotopy-equivalent to $C(\\TT^2)$ since this would require the existence of a $\\ast$-homomorphism $\\mathcal{A}_\\theta\\rightarrow C(\\TT^2)$ of which there are none since $\\mathcal{A}_\\theta$ is a simple, non-trivial noncommutative algebra.\nOur second remark concerns the assumptions of Theorem \\ref{MainTheoremIntro}:\nAlthough the Baum-Connes conjecture with coefficients is known to be false in general \\cite{MR1911663}, it has been confirmed for large classes of groupoids, including all amenable groupoids \\cite{Tu98}. Thus, our result applies to large classes of examples.\n\nMany concrete twists arising in examples of interest are actually homotopic to the trivial twist.\nIn these cases, our result says that the $\\K$-theory of the twisted groupoid $\\mathrm{C}^*$-algebra does not depend on the twist at all.\nExamples include the already mentioned irrational rotation algebras $\\mathcal{A}_\\theta$ and their higher-dimensional analogues and the Heegaard-type quantum spheres \\cite[Example~3.5]{MR3011251}.\n\n\n\nThe motivation for this work is twofold: There is a strong interest in understanding the $\\mathrm{K}$-theory of twisted groupoid $\\mathrm{C}^*$-algebras coming from the classification program for simple, unital, nuclear $\\mathrm{C}^*$-algebras on the one hand and coming from the physics of quasicrystals on the other.\n\nBy the work of numerous researchers (see \\cite{MR3583354} and the references therein), there is now a complete classification of separable, simple, unital $\\mathrm{C}^*$-algebras of finite nuclear dimension satisfying Rosenberg and Schochet's universal coefficient theorem (UCT).\nThe classifying invariant is the so called Elliott-invariant, whose main ingredient is operator $\\K$-theory.\nCombining the astonishing results from \\cite{1802.01190,MR2329002}, every $\\mathrm{C}^*$-algebra falling within the scope of classification admits a twisted groupoid model. Even better, the nuclearity assumption in the classification theorems implies that the underlying groupoid is amenable and hence satisfies the Baum-Connes conjecture.\nThis is closely linked to the question whether all nuclear $\\mathrm{C}^*$-algebras satisfy the UCT and to the existence of Cartan subalgebras \\cite{MR3672919}.\n\nIn another direction, twisted groupoid $\\mathrm{C}^*$-algebras and their $\\K$-theo\\-ry play an important role in physics: In the article \\cite{MR2119241} and the references therein a link to classification of $D$-brane charges in string theory is explained.\nMore directly related to our work is another important application studied in \\cite{MR3438328}: A twisted version of Bellissard's Gap Labelling Conjecture \\cite{MR1798994}.\nThe classical (untwisted) version of the conjecture is a statement about the possible gaps in the spectrum of certain Schr\u00f6dinger operators which arises in solid state physics. Mathematically, the conjecture is about a description of the image of a canonical trace map on the $\\K_0$-group of a certain groupoid $\\mathrm{C}^*$-algebra. A twist over that groupoid enters the picture once one allows for the presence of a non-zero magnetic field.\n\nIn conclusion, the $\\K$-theory of twisted groupoid $\\mathrm{C}^*$-algebras plays a major role in the classification program for simple $\\mathrm{C}^*$-algebras as well as in the applications from physics. This provides a strong motivation to study methods that enable the computation of $\\K$-theory for twisted groupoid $\\mathrm{C}^*$-algebras.\nThis question has been addressed before in the case of groups by Echterhoff, L\u00fcck, Phillips and Walters in \\cite{MR2608195} and Gillaspy treated the cases of transformation groups, higher-rank graphs and group bundles in a series of articles \\cite{MR3407179,MR3346133,MR3563179}. Using the machinery developed by the author in \\cite{1806.00391} this article presents a unified approach and considerable extension of the above mentioned results.\n\nTo make the result accessible for readers with different backgrounds we also interpret Theorem \\ref{MainTheoremIntro} in the setting of twisted actions of inverse semigroups in the sense of \\cite{MR2821242,MR1620499}.\nFinally, we apply our results to semidirect product groupoids arising from (partial) actions of subsemigroups of groups. These groupoids can be viewed as a generalisation of the transformation groupoids associated to ordinary group actions. In particular, we can recover and generalize the main results of \\cite{MR3407179}.\n\nThe article is organized as follows: \nAfter recalling some basic facts about \u00e9tale groupoids, groupoid dynamical systems and the associated reduced crossed products, we turn to the study of twists. We show in Proposition \\ref{Prop:LocalSections}, that every twist over an \u00e9tale groupoid locally admits continuous cross sections and conclude that every twist over an ample groupoid admits a global continuous cross section.\nIn section \\ref{Section:MainSection} we prove our main result. The idea is to use the Going-Down principle to reduce to the case of compact groupoids, where the result can be proven directly.\nIn the two final sections we look at some examples: First, we interpret our main result in terms of twisted actions of inverse semigroups, and second, we study certain semidirect-product groupoids arising from actions of subsemigroups of groups.\n\n\\section{Preliminaries}\n\\subsection{\u00c9tale and ample groupoids}\nRecall, that a \\textit{groupoid} is a set $G$ together with a subset $G^{(2)}\\subseteq G\\times G$, called the set of \\textit{composable pairs,} a \\textit{product} map $(g,h)\\mapsto gh$ from $G^{(2)}$ to $G$ and an \\textit{inverse} map $g\\mapsto g^{-1}$ from $G$ onto $G$, such that the following hold:\n\t\\begin{enumerate}\n\t\t\\item The product is associative: If $(g_1,g_2),(g_2,g_3)\\in G^{(2)}$ for some $g_1,g_2,g_3\\in G$, then we also have $(g_1g_2,g_3),(g_1,g_2g_3)\\in G^{(2)}$ and \n\t\t\\begin{equation*}\n\t\t(g_1g_2)g_3=g_1(g_2g_3).\n\t\t\\end{equation*}\n\t\t\\item The inverse map is involutive, i.e. $(g^{-1})^{-1}=g$ for all $g\\in G$.\n\t\t\\item $(g,g^{-1})\\in G^{(2)}$ for all $g\\in G$ and if $(g,h)\\in G^{(2)}$, then \n\t\t\\begin{equation*}\n\t\tg^{-1}(gh)=h \\ \\textup{ and }\\ (gh)h^{-1}=g.\n\t\t\\end{equation*}\n\t\\end{enumerate}\nThe fact that multiplication is partially defined implies that multiple elements may act as (partial) units:\n\tThe set $G^{(0)}:=\\lbrace g\\in G\\mid g=g^{-1}=g^2\\rbrace$ is called the \\textit{set of units} in $G$. There are canonical maps $d:G\\rightarrow G^{(0)}$ given by $d(g)=g^{-1}g$ and $r:G\\rightarrow G^{(0)}$ given by $r(g)=gg^{-1}$, called the \\textit{domain} and \\textit{range} \\textit{map} respectively.\n\nFor subsets $A,B\\in G^{(0)}$ we will write $G_A:=d^{-1}(A)$, $G^B:=r^{-1}(B)$ and $G_A^B:=G_A\\cap G^B$. If $A$ (and\/or $B$) consists just of a single unit $u\\in G^{(0)}$ we will omit the braces (e.g.: we will write $G^u:=r^{-1}(\\lbrace u\\rbrace)$).\n\nIn this article we will be concerned with topological groupoids:\nWe say that $G$ is a locally compact Hausdorff groupoid, if $G$ is a groupoid, which is equipped with a locally compact Hausdorff topology, such that the multiplication and inversion map are continuous. The fact that $G$ is Hausdorff ensures that the unit space $G^{(0)}$ is closed in $G$.\n\nA locally compact groupoid is called \\textit{\u00e9tale}, if $d:G\\rightarrow G$ is a local homeomorphism, i.e. every point $g\\in G$ has an open neighbourhood $U\\subseteq G$, such that $d(U)$ is open in $G$ and $d_{\\mid U}:U\\rightarrow d(U)$ is a homeomorphism.\nIt follows easily from the definition that for an \u00e9tale groupoid $G$\nthe unit space $G^{(0)}$ is open in $G$ and for each $u\\in G^{(0)}$ the sets $G^u$ and $G_u$ are discrete (in the subspace topology).\n\nThe results of the present article apply to a large subclass of the \u00e9tale groupoids:\nAn \u00e9tale groupoid $G$ is called \\textit{ample}, if its space of units $G^{(0)}$ is totally disconnected.\nThere is a wealth of groupoids studied in the literature that belong to this class, e.g groupoids associated to aperiodic tilings and quasicrystals (see \\cite{MR2658985}), groupoids associated to directed graphs (see \\cite{MR1432596}) and higher-rank graphs (see \\cite{KumjianPask00}), groupoids associated to inverse semigroups (see \\cite{Paterson1999}), and the coarse groupoid studied in large scale geometry (see \\cite{STY02}).\n\n\\subsection{Groupoid crossed products}\nIn this short paragraph we remind the reader of the definitions of groupoid dynamical systems and the associated reduced crossed products.\nLet $G$ be an \u00e9tale groupoid.\nA \\textit{groupoid dynamical system} $(A,G,\\alpha)$ consists of a $C_0(G^{(0)})$-algebra $A$ and a family $(\\alpha_g)_{g\\in G}$ of $*$-isomorphisms $\\alpha_g:A_{d(g)}\\rightarrow A_{r(g)}$ such that $\\alpha_{gh}=\\alpha_g\\circ \\alpha_h$ for all $(g,h)\\in G^{(2)}$ and such that $g\\cdot a:=\\alpha_g(a)$ defines a continuous action of $G$ on the upper-semicontinuous bundle $\\mathcal{A}$ over $G^{(0)}$ associated with $A$.\n\nAssociated with this data is a $\\mathrm{C}^*$-algebra called the reduced crossed product. We will briefly run through the constructions roughly following \\cite{MR1900993}.\nGiven a groupoid dynamical system $(A,G,\\alpha)$ as above, consider the complex vector space $\\Gamma_c(G,r^*\\mathcal{A})$. It carries a canonical $*$-algebra structure with respect to the following operations:\n$$(f_1\\ast f_2)(g)=\\sum\\limits_{h\\in G^{r(g)}} f_1(h)\\alpha_h(f_2(h^{-1}g))$$\nand\n$$f^*(g)=\\alpha_g(f(g^{-1})^*).$$\nSee for example \\cite[Proposition~4.4]{MR2547343} for a proof of this fact.\nFor $u\\in G^{(0)}$ consider the Hilbert $A_u$-module $\\ell^2(G^u,A_u)$. It is the completion of the space of finitely supported $A_u$-valued functions on $G^u$, with respect to the inner product \n$$\\lk \\xi,\\eta\\rk=\\sum\\limits_{h\\in G^u}\\xi(h)^*\\eta(h).$$\nWe can then define a $*$-representation $\\pi_u:\\Gamma_c(G,r^*\\mathcal{A})\\rightarrow \\mathcal{L}(\\ell^2(G^u,A_u))$ by\n$$\\pi_u(f)\\xi(g)=\\sum\\limits_{h\\in G^u}\\alpha_g(f(g^{-1}h))\\xi(h).$$\nUsing this family of representations, we can define a $\\mathrm{C}^*$-norm on the convolution algebra $\\Gamma_c(G,r^*\\mathcal{A})$ by\n$$\\norm{f}_r:=\\sup\\limits_{u\\in G^{(0)}}\\norm{\\pi_u(f)}.$$\nThe reduced crossed product $A\\rtimes_r G$ is defined to be the completion of $\\Gamma_c(G,r^*\\mathcal{A})$ with respect to $\\norm{\\cdot}_r$.\n\n\\section{Twists over ample groupoids}\nLet us recall the definition of a twist over a groupoid:\n\t\\begin{defi}\n\t\tLet $G$ be a topological groupoid. A \\textit{twist} $\\Sigma$ over $G$ is a central groupoid extension\n\t\t$$G^{(0)}\\times\\TT\\stackrel{i}{\\longrightarrow} \\Sigma\\stackrel{j}{\\longrightarrow} G,$$\n\t\tby which we mean:\n\t\t\\begin{enumerate}\n\t\t\t\\item The map $i$ is a homeomorphism onto $j^{-1}(G^{(0)})\\subseteq \\Sigma$,\n\t\t\t\\item the map $j$ is a continuous and open surjection, and\n\t\t\t\\item the extension is central meaning that $i(r(\\sigma),z)\\sigma=\\sigma i(d(\\sigma),z)$ for all $\\sigma\\in \\Sigma$ and $z\\in\\TT$.\n\t\t\\end{enumerate}\n\t\tWe say that $\\Sigma$ is a \\textit{continuous twist} over $G$, if $j$ admits a continuous cross section.\n\t\\end{defi}\n\tNote, that we will canonically identify $\\Sigma^{(0)}$ with $G^{(0)}$ and for all $u\\in G^{(0)}$ we have $j(i(u,z))=u$. Moreover, $\\Sigma$ admits a canonical left action of $\\TT$ given by $z\\cdot \\sigma:=i(r(\\sigma),z)\\sigma$. Hence a twist can also be viewed as a principal $\\TT$-bundle over $G$.\n\t\n\t\\begin{bem}\\label{Rem:Cocycles}\n\t\tTwists over groupoids are closely related to $2$-cocycles on them.\n\t\tRecall, that a $2$-cocycle for $G$ is a map $\\omega:G^{(2)}\\rightarrow \\TT$, such that $$\\omega(g_1,g_2)\\omega(g_1g_2,g_3)=\\omega(g_1,g_2g_3)\\omega(g_2,g_3)$$\n\t\tfor all $g_1,g_2,g_3\\in G$ with $(g_1,g_2),(g_2,g_3)\\in G^{(2)}$. It is called \\textit{normalized} if in addition one has\n\t\t$$\\omega(g,d(g))=1=\\omega(r(g),g)$$\n\t\tfor all $g\\in G$.\n\t\t\n\t\tGiven such a $2$-cocycle $\\omega$ on $G$ we can define a groupoid structure on $\\Sigma_\\omega:=G\\times  \\TT$ as follows: Two elements $(g_1,s_1),(g_2,s_2)\\in \\Sigma_\\omega$ are composable if $(g_1,g_2)\\in G^{(2)}$ and their product is defined as\n\t\t$$(g_1,s_1)(g_2,s_2):=(g_1g_2,s_1s_2\\omega(g_1,g_2)).$$\n\t\tThe inverse of $(g,s)\\in \\Sigma_\\omega$ is given by\n\t\t$$(g,s)^{-1}:=(g^{-1},\\overline{s\\omega(g^{-1},g)}).$$\n\t\tIf $\\omega$ is continuous, it is not hard to check that $\\Sigma_\\omega$ is a locally compact Hausdorff groupoid in the product topology. Thus, we obtain a central extension of groupoids\n\t\t$$ G^{(0)}\\times\\TT \\stackrel{i}{\\longrightarrow} \\Sigma_\\omega\\stackrel{j}{\\longrightarrow} G,$$\n\t\twhere the first map is the canonical inclusion and the second map is the projection onto the first factor. Note, that $j$ has a canonical continuous cross section $s$ given by $s(g)=(g,1)$.\n\t\t\n\t\tConversely, starting with a twist $\\Sigma$ over $G$ admitting a continuous section $s:G\\rightarrow \\Sigma$, we note that $j(s(gh)^{-1}s(g)s(h))\\in G^{(0)}$. Hence, by exactness we get $s(gh)^{-1}s(g)s(h)\\in i(G^{(0)}\\times \\TT)$. Since $i$ is a homeomorphism onto its image, we obtain a continuous map $\\omega:G^{(2)}\\rightarrow\\TT$ by letting $\\omega(g,h)=i^{-1}(s(gh)^{-1}s(g)s(h))$. It is then routine to check that $\\omega$ satisfies the cocycle identity. It is normalized, provided that the section $s$ restricts to the identity on $G^{(0)}$.\n\t\tIf $G$ is an \u00e9tale Hausdorff groupoid, this can always be arranged: If $s:G\\rightarrow \\Sigma$ is any continuous section, then define a new section $s':G\\rightarrow \\Sigma$ by $s'(g)=s(g)$ for all $g\\in G\\setminus G^{(0)}$ and $s'(u)=u$ for all $u\\in G^{(0)}$. Then $s'$ is still continuous since $G^{(0)}$ is clopen in $G$.\n\t\t\n\t\tWe remark, that it is a well-known fact that twists do not generally admit a continuous cross section and thus are more general than $2$-cocycles (see \\cite{MR1174207}).\n\t\\end{bem}\n\t\n\t\n\tAs a first goal we wish to show that twists over ample groupoids are automatically continuous. To this end we need to introduce some technology developed in \\cite{VanErp}.\n\tLet $X$ be a locally compact Hausdorff space and $E$ be a Hilbert $C_0(X)$-module. Write $\\mathcal{E}$ for the associated bundle of Hilbert spaces $\\mathcal{E}=\\coprod_{x\\in X} E_x$, which can be equipped with a topology turning it into a continuous Hilbert bundle over $X$ such that $E$ is isomorphic to the continuous sections of the bundle $\\mathcal{E}\\rightarrow X$ vanishing at infinity denoted by $\\Gamma_0(X,\\mathcal{E})$.\n\tAssociated with such a module is the groupoid\n\t$$Iso(E):=\\lbrace (x,V,y)\\mid x,y\\in X\\textit{ and }V:E_y\\rightarrow E_x\\textit{ is a unitary}\\rbrace,$$\n\tequipped with the obvious structure:\n\t$$(x,V,y)(y,W,z):=(x,VW,z)\\textit{ and }(x,V,y)^{-1}:=(y,V^*,x).$$\n\tIt was shown in \\cite[Proposition~3.3]{VanErp} that $Iso(E)$ can be equipped with a canonical Hausdorff topology making it into a topological group\\-oid.\n\t\n\tIn a similar spirit one can define the automorphism groupoid of an upper semicontinuous $\\mathrm{C}^*$-bundle $p:\\mathcal{A}\\rightarrow X$ over $X$ by\n\t$$Aut(\\mathcal{A}):=\\lbrace (x,\\alpha,y)\\mid x,y\\in X,\\ \\alpha:A_y\\rightarrow A_x \\textit{ is a }\\ast-isomorphism\\rbrace.$$\n\tAgain, $Aut(\\mathcal{A})$ can be equipped with a Hausdorff topology making it into a topological groupoid by \\cite[Proposition~3.1]{VanErp}.\n\t\n\tThe two constructions presented above are closely related. If $E$ is a Hilbert $C_0(X)$-module as above, then we can consider the $\\mathrm{C}^*$-algebra $K(E)$ of compact operators on $E$. This algebra is a $C_0(X)$-algebra in a canonical way with fibres $K(E)_x=K(E_x)$. Consequently, we can form the associated upper semicontinuous $\\mathrm{C}^*$-bundle denoted by $\\mathcal{K}\\rightarrow X$, such that $\\Gamma_0(X,\\mathcal{K})\\cong K(E)$ as $C_0(X)$-algebras.\n\tEvery unitary operator $V:E_y\\rightarrow E_x$ clearly defines a $\\ast$-isomorphism $\\mathrm{Ad}\\ V:K(E_y)\\rightarrow K(E_x)$ given by $(\\mathrm{Ad}\\ V)(T)=VTV^*$. Hence we obtain a canonical groupoid homomorphism $\\mathrm{Ad}:Iso(E)\\rightarrow Aut(\\mathcal{K})$. It was shown in \\cite[Proposition~3.4]{VanErp} that $\\mathrm{Ad}$ fits into a short exact sequence of topological groupoids\n\t$$X\\times \\TT\\stackrel{i}{\\rightarrow} Iso(E)\\stackrel{\\mathrm{Ad}}{\\rightarrow} Aut(\\mathcal{K}),$$\n\tsuch that $\\mathrm{Ad}$ is a continuous and open surjection, and $i$ is a homeomorphism onto the kernel of $\\mathrm{Ad}$.\n\t Moreover, the map $\\mathrm{Ad}$ admits local continuous sections, in the sense that for every $(x,\\alpha,y)\\in Aut(\\mathcal{K})$ there exists a neighbourhood $N$ of $(x,\\alpha,y)\\in Aut(\\mathcal{K})$ and a continuous map $\\beta:N\\rightarrow Iso(E)$ such that $\\mathrm{Ad} \\circ\\beta=\\id_N$.\n\t\n\tWe want to apply this machinery to twists over groupoids. Although much of what follows also works for more general groupoids we restrict ourselves to the class of \u00e9tale groupoids at this point to avoid unnecessary technicalities. So let $\\Sigma$ be a twist over an \u00e9tale groupoid $G$. \n\tConsider the complex vector space\n\t$$C_c(G;\\Sigma):=\\lbrace f\\in C_c(\\Sigma)\\mid f(z\\sigma)=zf(\\sigma)\\rbrace.$$ \n\tThen we obtain a Hilbert $C_0(G^{(0)})$-module $E$ by separation and completion of $C_c(G;\\Sigma)$ with respect to the inner product\n\t\\begin{equation}\\label{Eq:TwistedInnerProduct}\n\t\\lk f_1,f_2\\rk_{C_0(G^{(0)})}(u)=\\sum\\limits_{j(\\sigma)\\in G_u} \\overline{f_1(\\sigma)}f_2(\\sigma).\n\t\\end{equation}\n\t\tEach of the fibres $E_u$ is a Hilbert space and the following lemma gives a convenient description of those.\n\t\t\\begin{lemma}\\label{Lem:ContinuousTwist}\n\t\t\tIf $G^{(0)}\\times \\TT\\rightarrow\\Sigma\\stackrel{j}{\\rightarrow}G$ is a twist over $G$ then the fibre $E_u$ over $u\\in G^{(0)}$ can be identified with the Hilbert space obtained by completion of $E_0(u)=\\lbrace f\\in C_c(\\Sigma_u)\\mid f(z\\sigma)=zf(\\sigma)\\rbrace$ with respect to the inner product $\\lk f_1,f_2\\rk=\\sum_{j(\\sigma)\\in G_u} \\overline{f_1(\\sigma)}f_2(\\sigma).$\n\t\t\\end{lemma}\n\t\t\\begin{proof}\n\t\t\tOne easily sees that the restriction map $C_c(G;\\Sigma)\\rightarrow E_0(u)$ factors through an isometric linear map $E_u\\rightarrow \\overline{E_0(u)}$. The only issue is the surjectivity of the restriction map.\n\t\t\tTo this end let $f_0\\in C_c(\\Sigma)$ be an extension of $f$.\n\t\t\tThen $f_1:\\Sigma\\rightarrow \\TT$ defined by\n\t\t\t$$f_1(\\sigma):=\\int_{\\TT}\\overline{z}f_0(z\\sigma)dz,$$\n\t\t\tis an element of $E_0$, that extends $f$. \n\t\t\t\n\t\t\n\t\t\\end{proof}\n\t\tLet $\\rho:\\Sigma\\rightarrow Iso(E)$ be the representation given by $(\\rho(\\sigma)\\xi)(\\tau)=\\xi(\\tau\\sigma)$. Then $\\rho(z\\sigma)=z\\rho(\\sigma)$ for all $z\\in \\TT$ and $\\sigma\\in \\Sigma$. Consequently, denoting the upper semicontinuous $\\mathrm{C}^*$-bundle associated to the $C_0(G^{(0)})$-algebra $K(E)$ by $\\mathcal{K}$ again, we obtain a well-defined continuous groupoid homomorphism $\\alpha:G\\rightarrow Aut(\\mathcal{K})$ by $\\alpha_{j(\\sigma)}=\\mathrm{Ad}\\ \\rho(\\sigma)$.\n\t\tThis homomorphism fits into the following commutative diagram:\n\t\t\\begin{center}\n\t\t\t\\begin{tikzpicture}[description\/.style={fill=white,inner sep=2pt}]\n\t\t\t\\matrix (m) [matrix of math nodes, row sep=3em,\n\t\t\tcolumn sep=2.5em, text height=1.5ex, text depth=0.25ex]\n\t\t\t{ G^{(0)}\\times\\TT & \\Sigma & G \\\\\n\t\t\t\tG^{(0)}\\times \\TT & Iso(E) & Aut(\\mathcal{K}) \\\\\n\t\t\t};\n\t\t\t\\path[->,font=\\scriptsize]\n\t\t\t(m-1-1) edge node[auto] {$  $} (m-1-2)\n\t\t\t\n\t\t\t(m-2-1) edge node[auto] {$  $} (m-2-2)\n\t\t\t\n\t\t\t(m-1-1) edge node[auto] {$ \\id $} (m-2-1)\n\t\t\t(m-1-2) edge node[auto] {$ \\rho $} (m-2-2)\n\t\t\t(m-1-2) edge node[auto] {$ j $} (m-1-3)\n\t\t\t\t\t\t(m-2-2) edge node[auto] {$ \\mathrm{Ad} $} (m-2-3)\n\t\t\t\t\t\t(m-1-3) edge node[auto] {$ \\alpha $} (m-2-3)\n\t\t\t;\n\t\t\t\\end{tikzpicture}\n\t\t\\end{center}\t\n\t\n\tThe following result seems to be folklore but we could not locate it in the literature, so we provide a proof.\n\t\\begin{prop}\\label{Prop:LocalSections}\n\t\tLet $G^{(0)}\\times\\TT\\stackrel{i}{\\longrightarrow} \\Sigma\\stackrel{j}{\\longrightarrow} G$ be a twist over an \u00e9tale groupoid $G$. Then $j$ admits local cross sections in the sense that for every $g\\in G$ there exists a neighbourhood $U\\subseteq G$ of $g$ and a continuous map $s:U\\rightarrow \\Sigma$ such that $j(s(h))=h$ for all $h\\in U$.\n\t\\end{prop}\n\t\\begin{proof}\n\t\tLet $E$ be the Hilbert $C_0(G^{(0)})$-module associated with $\\Sigma$ and $Iso(E)$ and $Aut(\\mathcal{K})$ as above. Since the canonical map $\\mathrm{Ad}:Iso(E)\\rightarrow Aut(\\mathcal{K})$ has local cross sections, so does every pullback along this map. Hence it suffices to show that $\\Sigma$ is isomorphic to the pullback groupoid $$\\Sigma'=\\lbrace (V,g)\\mid g\\in G, V:E_{d(g)}\\rightarrow E_{r(g)}\\textit{ unitary with } \\mathrm{Ad}\\ V=\\alpha_g\\rbrace.$$ \n\t\t\n\t\tFirst of all, there exists a canonical groupoid homomorphism $\\varphi:\\Sigma\\rightarrow \\Sigma'$ given by $\\varphi(\\sigma):=(\\rho(\\sigma),j(\\sigma))$.\n\t\tFor the injectivity assume that $\\varphi(\\sigma)=\\varphi(\\sigma')$. Then $j(\\sigma)=j(\\sigma')$, so using exactness we get that there exists a $z\\in\\TT$ such that $\\sigma'=z\\sigma$. From the definition of $\\rho$ we get that for any $f\\in C_c(G;\\Sigma)$ and any $\\tau\\in \\Sigma$ such that $d(\\tau)=r(\\sigma)=r(\\sigma')$ we have $f(\\tau\\sigma)=f(\\tau\\sigma')$. In particular, we get $f(\\sigma)=f(\\sigma')=zf(\\sigma)$ for all $f\\in C_c(G;\\Sigma)$. We conclude that $z=1$ and hence $\\sigma=\\sigma'$ as desired.\n\t\t\n\t\tFor the surjectivity let $(V,g)\\in \\Sigma'$. First choose $\\sigma_0\\in \\Sigma$ with $j(\\sigma_0)=g$. Then $\\mathrm{Ad}\\ \\rho(\\sigma_0)=\\alpha_{j(\\sigma_0)}=\\alpha_g=\\mathrm{Ad}\\ V$ and hence $\\rho(\\sigma_0)=z V$ for some $z\\in\\TT$. Now put $\\sigma:=\\overline{z}\\sigma_0$. Then $j(\\sigma)=g$ and $\\rho(\\sigma)=\\overline{z}\\rho(\\sigma_0)=V$ and hence $\\varphi(\\sigma)=(V,g)$ as desired.\n\t\t\n\t\tSince $\\varphi$ is clearly continuous it remains to prove that it is also open. To see this we will employ \\cite[Proposition~1.15]{Williams}. So let $(V_\\lambda,g_\\lambda)_\\lambda$ be a net converging to $\\varphi(\\sigma)=(\\rho(\\sigma),j(\\sigma))$ in $\\Sigma'$. In particular we get $g_\\lambda\\rightarrow j(\\sigma)$ in $G$. Since $j$ is an open surjection we can pass to a subnet and relabel to assume that there exists a net $(\\sigma_\\lambda)$ in $\\Sigma$ such that $j(\\sigma_\\lambda)=g_\\lambda$ and $\\sigma_\\lambda\\rightarrow \\sigma$.\n\t\tMoreover, $\\mathrm{Ad}\\ \\rho(\\sigma_\\lambda)=\\alpha_{j(\\sigma_\\lambda)}=\\mathrm{Ad}\\ V_\\lambda$ and consequently $\\rho(\\sigma_\\lambda)=z_\\lambda V_\\lambda$ for a suitable net $(z_\\lambda)_\\lambda$ in $\\TT$. Now since $\\TT$ is compact, we can pass to yet another subnet (and relabel) to assume that the net $(z_\\lambda)_\\lambda$ converges to some $z\\in \\TT$. Since $\\rho(\\sigma)=\\lim_\\lambda \\rho(\\sigma_\\lambda)=\\lim_\\lambda z_\\lambda V_\\lambda=z\\rho(\\sigma)$, we get $z=1$.\n\t\tTo sum up we found a (sub)net of $(V_\\lambda,g_\\lambda)$ and a net $(\\overline{z_\\lambda}\\sigma_\\lambda)_\\lambda$ in $\\Sigma$ converging to $\\sigma$ such that $\\varphi(\\overline{z}_\\lambda\\sigma_\\lambda)=(\\rho(\\overline{z}_\\lambda\\sigma_\\lambda),j(\\sigma_\\lambda))=(V_\\lambda,g_\\lambda)$.\n\t\\end{proof}\n\t\n\t\\begin{kor}\\label{Cor:TwistsOnAmpleGroupoidsAreContinuous}\n\tEvery twist $\\Sigma$ over a $\\sigma$-compact ample groupoid $G$ is continuous.\n\t\\end{kor}\n\t\\begin{proof}\n\t\tBy the previous proposition the map $j:\\Sigma\\rightarrow G$ admits local cross sections. Combining this with the assumptions that $G$ is $\\sigma$-compact and totally disconnected, we can find a countable covering of $G=\\bigcup U_n$ by compact open subsets which are the domains of local sections of $j$. Using a standard inclusion-exclusion argument we can recursively define a partition $G=\\bigsqcup_{n\\in\\NN} V_n$ into compact open subsets $V_n$, which are the domains of local sections $s_n:V_n\\rightarrow \\Sigma$ of $j$. Then we can piece all of these sections together in the obvious way to obtain a global continuous section $s:G\\rightarrow \\Sigma$ of $j$ as desired.\n\t\\end{proof}\n\t\n\tFollowing \\cite{MR1174207}, we associate a $\\mathrm{C}^*$-algebra to a twist $\\Sigma$ over an \u00e9tale groupoid $G$ as follows:\n\tConsider the complex vector space\n\t$$C_c(G;\\Sigma):=\\lbrace f\\in C_c(\\Sigma)\\mid f(z\\sigma)=zf(\\sigma)\\rbrace.$$ \n\tThen $C_c(G;\\Sigma)$ becomes a $\\ast$-algebra with repect to the operations\n\t$$f_1\\ast f_2(\\sigma)=\\sum\\limits_{j(\\tau)\\in G^{r(\\sigma)}} f_1(\\tau)f_2(\\tau^{-1}\\sigma)\\text{ and }f^*(\\sigma)=\\overline{f(\\sigma^{-1})}.$$\n\tObserve that the sum makes sense, since the expression $f_1(\\tau)f_2(\\tau^{-1}\\sigma)$ only depends on $j(\\tau)\\in G$. \n\tFor each $u\\in G^{(0)}$ let $E_u$ be the fibre of the Hilbert $C_0(G^{(0)})$-module $E$ associated with $\\Sigma$ as above.\n\tThen, for $f\\in C_c(G;\\Sigma)$ we can define an operator $\\pi_u(f)$ on $E_u$ by \t$\\pi_u(f)\\xi)=f\\ast\\xi$. The operator $\\pi_u(f)$ is bounded and we define $C_r^*(G;\\Sigma)$ to be the completion of $C_c(G;\\Sigma)$ with respect to the norm\n\t$$\\norm{f}_r:=\\sup\\limits_{u\\in G^{(0)}} \\norm{\\pi_u(f)}.$$\n\t\n\t\n\t\n\t\\begin{bem}\n\t\tIn the literature one often finds a direct construction of the twisted groupoid $\\mathrm{C}^*$-algebra associated to a continuous $2$-cocycle $\\omega$, that does not pass through the canonical extension $\\Sigma_\\omega$ explained above. It is defined as a completion of the convolution algebra $C_c(G)$ with product and involution given by\n\t\t$$f_1\\ast_\\omega f_2(g)=\\sum\\limits_{h\\in G^{r(g)}} f_1(h)f_2(h^{-1}g)\\omega(h,h^{-1}g)\\text{ and }$$\n\t\t$$f^*(g)=\\overline{f(g^{-1})\\omega(g,g^{-1})},$$\n\t\tand we will denote it by $C_r^*(G,\\omega)$.\n\t\tNote, that both constructions yield the same $\\mathrm{C}^*$-algebras, since there is a canonical isomorphism $\\Phi:C_r^*(G,\\Sigma_\\omega)\\rightarrow C_r^*(G,\\omega)$, given by $\\Phi(f)(g)=f(g,1)$. One can easily define an inverse map $\\Psi:C_r^*(G,\\omega)\\rightarrow C_r^*(G,\\Sigma_\\omega)$ by $\\Psi(f)(g,z)=zf(g)$.\n\t\\end{bem}\n\t\n\t\n\tWe shall need the following version of the Packer-Raeburn stabilisation trick in the groupoid framework.\n\t\\begin{prop}\\label{Prop:PackerRaeburn}\\cite[Proposition~5.1]{VanErp}\n\t\tLet $\\Sigma$ be a twist over an \u00e9tale groupoid $G$. If $E$ is the associated Hilbert $C_0(G^{(0)})$-module and $\\alpha:G\\rightarrow Aut(\\mathcal{K})$ is the homomorphism constructed above, then $\\alpha$ defines a groupoid dynamical system $(K(E),G,\\alpha)$ such that $K(E)\\rtimes_{\\alpha,r} G$ is Morita equivalent to $C_r^*(G;\\Sigma)$.\n\t\\end{prop}\n\tFor later reference, we also need to briefly recall the construction of the bimodule that provides the Morita equivalence:\t\n\tLet $A_0$ be the dense subalgebra $\\Gamma_c(G,r^*\\mathcal{K})\\subseteq K(E)\\rtimes_{r,\\alpha} G$.\n\tFollowing \\cite[Theorem~6.4]{MR2446021} together with the formulas given in the proof of \\cite[Proposition~5.1]{VanErp} one then defines a pre-Hilbert bimodule-structure on $X_0:=\\Gamma_c(G,d^*\\mathcal{E})$ as follows: For $\\xi,\\eta\\in X_0$ and $f\\in A_0$ define\n\t$$(f\\xi)(g)=\\sum\\limits_{h\\in G^{r(g)}} \\alpha_{g^{-1}h}(f(h^{-1}))\\xi(h^{-1}g),$$\n\t\n\t$${}_{A_0}\\lk \\xi,\\eta\\rk(g)=\\sum\\limits_{h\\in G^{s(g)}} \\alpha_{gh}({}_{K(E_x)}\\lk\\xi(h),\\eta(gh)\\rk).$$\n\tNote that in \\cite{VanErp}, the authors construct the crossed product by completing $\\Gamma_c(G,s^*\\mathcal{K})$. Thus, in order to obtain the formulas above we need to pass through the canonical isomorphism, sending $f\\in A_0$ to the function $\\check{f}\\in \\Gamma_c(G,s^*\\mathcal{K})$, given by $\\check{f}(g):=f(g^{-1})$.\n\t\n\tFor $\\xi,\\eta\\in X_0$ and $f\\in C_c(G;\\Sigma)$ define\n\t$$(\\xi f)(g)=\\sum\\limits_{j(\\sigma)\\in G^{d(g)}}f(\\sigma^{-1})\\rho(\\sigma)\\xi(gj(\\sigma))$$\n\tand a $C_c(G;\\Sigma)$-valued inner product by\n\t$$\\lk \\xi,\\eta\\rk (\\tau)=\\sum\\limits_{j(\\sigma)\\in G^{d(\\tau)}}\\lk \\rho(\\sigma^{-1}\\tau^{-1})\\xi(j(\\sigma^{-1}\\tau^{-1})),\\rho(\\sigma^{-1})\\eta(j(\\sigma^{-1}))\\rk (d(\\sigma))$$\n\tThe completion $X$ of $X_0$ then implements a Morita equivalence between $K(E)\\rtimes_r G$ and $C_r^*(G;\\Sigma)$. With this description at hand we can prove the following technical little lemma, which will turn out useful later:\n\t\\begin{lemma}\\label{Lemma:PR-Inductive Limit}\n\tIf $(\\xi_i)_i$ is a net in $X_0$ converging to $\\xi\\in X_0$ in the inductive limit topology, then $\\norm{\\xi_i-\\xi}\\rightarrow 0$.\n\t\\end{lemma}\n\t\\begin{proof}\n\tLet $\\eta_i:=\\xi-\\xi_i\\in X_0$. We will show, that ${}_{A_0}\\lk \\eta_i,\\eta_i\\rk$ converges to zero in the inductive limit topology. Then it will also converge to zero in the reduced norm and hence $\\norm{\\xi-\\xi_i}^2=\\norm{{}_{A_0}\\lk \\eta_i,\\eta_i\\rk}\\rightarrow0$ as desired.\n\tBy assumption, there exists a compact subset $K\\subseteq G$ such that $supp(\\eta_i)\\subseteq K$ for all $i$. Since the action of $G$ on itself by multiplication is always proper, the set $C:=\\lbrace g\\in G\\mid g^{-1}K\\cap K\\neq\\emptyset\\rbrace$ is also compact. Now if $0\\neq {}_{A_0}\\lk \\eta_i,\\eta_i\\rk(g)=\\sum_{h\\in G^{d(g)}}\\alpha_{gh}(\\lk \\eta_i(h),\\eta_i(gh)\\rk)$, there exists some $h\\in G^{d(g)}$ such that $\\lk \\eta_i(h),\\eta_i(gh)\\rk\\neq 0$.\n\tBut then necessarily $h\\in g^{-1}K\\cap K$, which implies $g\\in C$. Thus $supp({}_{A_0}\\lk \\eta_i,\\eta_i\\rk)\\subseteq C$ for all $i$.\n\tNow let $\\varepsilon>0$ be given. Choose $M>0$ such that $\\sup_{u\\in G^{(0)}}\\betrag{K^u}\\leq M$. Then we have $\\sup_{g\\in G}\\norm{\\lk \\eta_i(g),\\eta_i(g)\\rk}<\\frac{\\sqrt{\\varepsilon}}{M}$ for $i$ large enough.\n\tFor $i$ large enough we can then compute \n\t\\begin{align*}\n\t\\norm{\\lk \\eta_i,\\eta_i\\rk(g)}&\\leq \\sum\\limits_{h\\in G^{d(g)}}\\norm{\\lk \\eta_i(h),\\eta_i(gh)}\\\\\n\t&\\leq \\sum\\limits_{h\\in G^{d(g)}}\\norm{\\lk \\eta_i(h),\\eta_i(h)}\\norm{\\lk \\eta_i(gh),\\eta_i(gh)}<\\varepsilon,\n\t\\end{align*}\n\twhich finishes the proof.\n\t\\end{proof}\n\t\\begin{bem}\\label{Rem:PackerRaeburnRestrict}\n\t\tLet $\\Sigma$ be a twist over the \u00e9tale groupoid $G$ and $H\\subseteq G$ a compact open subgroupoid. Then $\\Sigma':=j^{-1}(H)$ is easily seen to be a twist over $H$.\n\t\tLet $E$ and $\\alpha$ be as above. Then we can restrict the action $\\alpha$ to an action of $H$ on $K(E)_{\\mid H}$. We claim that the resulting crossed product $K(E)_{\\mid H}\\rtimes_r H$ is then Morita equivalent to $C_r^*(H;\\Sigma')$. The proof is basically the same as in \\cite[Proposition~5.1]{VanErp}, we just restrict all the appearing bundles to the subgroupoid $H$ and use the fact that $H$ is an $(H,H)$-equivalence.\n\t\\end{bem}\n\t\\section{Homotopies of twists}\\label{Section:MainSection}\n\tOur goal is to prove that the $\\K$-theory of $C_r^*(G;\\Sigma)$ only depends on the homotopy class of $\\Sigma$. We will start by formalizing what we mean by a homotopy:\n\tGiven a locally compact Hausdorff groupoid $G$, consider the trivial bundle of groupoids $G\\times[0,1]$ with the product topology. This bundle is itself a locally compact groupoid, where $(g,s)$ and $(h,t)$ are composable if $g$ and $h$ are composable in $G$ and $s=t$. In this case we define their product by $(g,s)(h,s):=(gh,s)$ and an inverse by $(g,s)^{-1}:=(g^{-1},s)$. Consequently, the unit space is given by $G^{(0)}\\times [0,1]$.\n\t\n\t\\begin{defi}\n\t\tA (continuous) twist $\\Sigma$ over $G\\times[0,1]$ is called a (continuous) \\textit{homotopy of twists} for $G$.\n\t\\end{defi}\n\tIf $\\Sigma$ is a homotopy of twists over $G$ then $\\Sigma$ is a continuous field of groupoids over $[0,1]$ in the sense of \\cite[Definition~8.9]{Delaroche} since for all $\\sigma\\in\\Sigma$ we have $pr_{[0,1]}(d(\\sigma))=pr_{[0,1]}(r(\\sigma))$.\n\t In particular, for each $t\\in[0,1]$ we obtain a twist $\\Sigma_t$ over $G$ by letting $\\Sigma_t:=(pr_{[0,1]}\\circ r)^{-1}(\\Sigma)=\\Sigma_{\\mid G^{(0)}\\times \\lbrace t\\rbrace}$.\n\t \n\tIf $G$ is ample, then each $\\Sigma_t$ is a continuous twist by Corollary \\ref{Cor:TwistsOnAmpleGroupoidsAreContinuous}, so one may wonder if $\\Sigma$ is always continuous as a twist over $G\\times [0,1]$. Although $G\\times [0,1]$ is no longer ample, this can be arranged by a standard patching argument:\n\t\n\t\\begin{prop}\\label{Prop:HomotopiesAreContinuous}\n\t\tLet $G$ be a $\\sigma$-compact ample groupoid and $\\Sigma$ be a homotopy of twists for $G$. Then $\\Sigma$ is automatically a continuous homotopy of twists, in the sense that there exists a continuous section $s:G\\times [0,1]\\rightarrow \\Sigma$.\n\t\\end{prop}\n\t\\begin{proof}\n\t\tWe start to prove a preliminary claim:\n\t\tSuppose $V\\subseteq G$ is an open subset and $0\\leq a<b<c\\leq 1$ are such that there exist continuous sections $s_1:V\\times [a,b]\\rightarrow \\Sigma$ and $s_2:V\\times [b,c]\\rightarrow \\Sigma$. Then we claim that there exists a continuous section $s:V\\times [a,c]\\rightarrow \\Sigma$.\n\t\tIndeed, since $s_1$ and $s_2$ are continuous sections, we can define a continuous map $f:V\\rightarrow \\TT$ by $f(g)=s_1(g,b)s_2(g,b)^{-1}$. Then $\\tilde{s}_2(g,t):=f(g)s_2(g,t)$ is still a continuous section, but it has the virtue that $\\tilde{s}_2(g,b)=s_1(g,b)$ for all $g\\in V$. Hence we can piece $s_1$ and $\\tilde{s}_2$ together to obtain the desired function $s:V\\times [a,c]\\rightarrow \\Sigma$.\n\t\t\n\t\tWe now return to the proof of the proposition: Fix $g\\in G$ for the moment. For each $t\\in[0,1]$ we can apply Proposition \\ref{Prop:LocalSections} to find a compact open neighbourhood $V_t$ of $g$ and some open interval $U_t\\subseteq [0,1]$ and a continuous section $s_t:V_t\\times U_t\\rightarrow \\Sigma$. Using compactness of $[0,1]$ and intersecting the resulting finite number of neighbourhoods among the $V_t$ we find a finite sequence of numbers $0=t_0<t_1<\\ldots <t_n=1$, a compact open neighbourhood $V$ of $g$, and finitely many continuous sections $s_i:V\\times [t_{i-1},t_i]\\rightarrow \\Sigma$. Now we successively apply the first paragraph of this proof to obtain a continuous section $s_g:V\\times [0,1]\\rightarrow \\Sigma$.\n\t\tFinally, we use $\\sigma$-compactness to proceed as in the proof of Corollary \\ref{Cor:TwistsOnAmpleGroupoidsAreContinuous} and piece these sections together to a globally continuous cross section $s:G\\times [0,1]\\rightarrow \\Sigma$.\n\t\\end{proof}\n\t\n\tFor every $t\\in[0,1]$ we obtain a canonical $\\ast$-homo\\-morphism \n\t$$q_t:C_r^*(G\\times[0,1];\\Sigma)\\rightarrow C_r^*(G;\\Sigma_t),$$\n\twhich for $f\\in C_c(G\\times[0,1];\\Sigma)$ is\n\tgiven by $q_t(f)=f_{\\mid \\Sigma_t}$. An argument very similar to the proof of Lemma \\ref{Lem:ContinuousTwist} shows, that $q_t$ is surjective.\n\tThe main goal of this article is to prove the following result:\n\t\\begin{satz}\\label{Thm:Evalutation Induces Isomorphism on K-theory}\n\t\tLet $G$ be an ample groupoid, which satisfies the Baum-Connes conjecture with coefficients and let $\\Sigma$ be a homotopy of twists for $G$ and $t\\in[0,1]$. Then $$(q_{t})_*:\\K_*(C_r^*(G\\times[0,1];\\Sigma))\\rightarrow \\K_*(C_r^*(G,\\Sigma_t))$$\n\t\tis an isomorphism.\n\t\\end{satz}\n\t\n\tThe following result settles the case of continuous homotopies of twists over compact groupoids.\n\t\\begin{prop}\\cite[Proposition~3.1]{MR3346133}\\label{Prop:CompactCase} If $\\Sigma$ is a continuous homotopy of twists on a compact Hausdorff groupoid $G$, then the canonical $\\ast$-homomorphism $q_t:C_r^*(G\\times[0,1];\\Sigma)\\rightarrow C_r^*(G;\\Sigma_t)$ is a homotopy equivalence.\n\t\\end{prop}\n\n\tThe rough idea in proving Theorem \\ref{Thm:Evalutation Induces Isomorphism on K-theory} is to use the Going-Down principle to reduce the question to the case of compact groupoids and then appeal to Proposition \\ref{Prop:CompactCase} above.\n\tBefore proceeding to the precise form of the Going-Down principle we are going to use, the reader may wish to recall the definitions of Le Gall's $\\KK^G$-theory \\cite{LeGall99}, the topological $\\K$-theory $\\K_*^{\\mathrm{top}}(G;A)$ of a groupoid with coefficients in a $G$-algebra $A$, and the Baum-Connes assembly map $\\mu_A:\\K_*^{\\mathrm{top}}(G;A)\\rightarrow \\K_*(A\\rtimes_r G)$ \\cite{MR1798599}. See also \\cite{Bonicke,1806.00391} for a detailed overview.\n\t\\begin{satz}\\cite[Theorem~7.10]{1806.00391}\n\t\tLet $G$ be an ample, second countable, locally compact Hausdorff groupoid and let $A$ and $B$ be separable $G$-algebras. Suppose there is an element $x\\in \\KK^G(A,B)$ such that \n\t\t\\[ \\KK^H(C(H^{(0)}),A_{\\mid H})\\stackrel{\\cdot\\otimes res_H^G(x)}{\\rightarrow}\\KK^H(C(H^{(0)}), B_{\\mid H})\\]\n\t\tis an isomorphism for all compact open subgroupoids $H\\subseteq G$. Then the Kasparov-product with $x$ induces an isomorphism\n\t\t$$ \\cdot\\otimes x:\\K_*^{\\mathrm{top}}(G;A)\\rightarrow \\K_*^{\\mathrm{top}}(G;B).$$\n\t\\end{satz}\n\t\n\tFrom now on fix a homotopy of twists $\\Sigma$ over an \u00e9tale Hausdorff groupoid $G$.\n\tConsider the canonical Hilbert $C_0(G^{(0)}\\times [0,1])$-module $E$, defined as the completion of $C_c(G\\times[0,1];\\Sigma)$ with respect to the inner product defined in equation \\ref{Eq:TwistedInnerProduct}.\n\tNow by Proposition \\ref{Prop:PackerRaeburn} and the discussion thereafter we obtain an action $\\alpha$ of $G\\times[0,1]$ on $K(E)$. Observe, that there is a canonical action of $G$ on $G^{(0)}\\times[0,1]$ given by $g\\cdot (d(g),t)=(r(g),t)$, such that $G\\times[0,1]\\cong G\\ltimes(G^{(0)}\\times [0,1])$. Taking this point of view we can use the pushforward construction (see for example \\cite[Proposition~3.10]{1806.00391}) to obtain an action $\\beta$ of $G$ on $K(E)$. One has the following:\n\t\\begin{prop}\\cite[Theorem~3.8]{1703.05190}\n\tThe canonical map\n\t$\\Phi:\\Gamma_c(G\\times[0,1],r^*\\mathcal{K})\\rightarrow \\Gamma_c(G,r^*\\mathcal{K})$ given by\n\t$\\Phi(f)(g)(t)=f(g,t)$\n\tis a $\\ast$-homo\\-morphism and extends to a $\\ast$-isomorphism $$\\Phi:K(E)\\rtimes_{\\alpha,r} (G\\times[0,1])\\rightarrow K(E)\\rtimes_{\\beta,r} G.$$\n\t\\end{prop}\n\tOn the other hand for each $t\\in[0,1]$ we can apply (the proof of) Proposition \\ref{Prop:PackerRaeburn} to the twist $\\Sigma_t$ over $G$, in order to obtain a Hilbert $C_0(G^{(0)})$-module $E_t$ and an action $\\alpha_t$ of $G$ on $K(E_t)$. Let us record the following easy observations concerning the relationship between $E$ and $E_t$:\n\t\\begin{lemma}\n\t\tThe restriction map $C_c(G\\times[0,1];\\Sigma)\\rightarrow C_c(G;\\Sigma_t)$, $f\\mapsto f_{\\mid \\Sigma_t}$ extends to a surjective bounded linear map $p_t:E\\rightarrow E_t$.\n\t\\end{lemma}\n\t\\begin{proof}\n\tIt is routine to check, that the restriction map is bounded and linear. Using an argument similar to the proof of Lemma \\ref{Lem:ContinuousTwist} one sees that the restriction map $C_c(G\\times[0,1];\\Sigma)\\rightarrow C_c(G;\\Sigma_t)$ is surjective. This is not quite enough to conclude that $p_t$ is surjective. However, if $i_t:G^{(0)}\\rightarrow G^{(0)}\\times[0,1]$ denotes the inclusion at $t\\in[0,1]$, then $p_t$ factors through an isometric linear map $i_t^*E\\rightarrow E_t$. Since this map is isometric, it is enough to know that the dense subset $C_c(G;\\Sigma_t)$ is contained in the image to conclude surjectivity. Using, that the canonical map $E\\cong E\\otimes_{C_0(G^{(0)}\\times[0,1])}C_0(G^{(0)}\\times[0,1])\\rightarrow i_t^*E$ is surjective, the result follows.\n\t\\end{proof}\n\tWe can use an argument similar to the proof of Lemma \\ref{Lem:ContinuousTwist} again, to show that for $u\\in G^{(0)}$ we can canonically identify the Hilbert spaces $E_{(u,t)}$ and $(E_t)_u$ and hence also $K(E)_{(u,t)}$ with $K(E_t)_u$.\n\tLet $X$ and $X_t$ be the equivalence bimodules obtained from applying Proposition \\ref{Prop:PackerRaeburn} to the twists $\\Sigma$ and $\\Sigma_t$, respectively. We have the following:\n\t\\begin{prop}\\label{Prop:Bimodules}\n\t\tThe canonical restriction map $\\Gamma_c(G\\times[0,1],d^*\\mathcal{E})\\rightarrow \\Gamma_c(G,d^*\\mathcal{E}_t)$, $\\xi\\mapsto \\xi_{\\mid G\\times\\lbrace t\\rbrace}$ extends to a bounded linear map $\\Psi_t:X\\rightarrow X_t$ and factors through an isomorphism\n\t\t$$\\Theta_t:q_t^*(X)\\rightarrow X_t$$\n\t\tof Hilbert $C_r^*(G;\\Sigma_t)$-modules.\n\t\\end{prop}\n\t\\begin{proof}\n\t\tFrom the definition of the respective inner products it is obvious that $\\lk \\Psi_t(\\xi),\\Psi_t(\\eta)\\rk=q_t(\\lk \\xi, \\eta\\rk)$ for all $\\xi,\\eta\\in \\Gamma_c(G\\times[0,1],d^*\\mathcal{E})$. It follows that $\\Psi_t$ is bounded and hence extends to all of $X$. Define $\\Theta_t:q_t^*X=X\\otimes_{q_t}C_r^*(G;\\Sigma_t)\\rightarrow X_t$ on elementary tensors by $\\Theta_t(\\xi\\otimes a)=\\Psi_t(\\xi)a$. Then $\\Theta_t$ extends to an isometric map on all of $q_t^*X$, since\n\t\tfor $\\xi,\\eta\\in X$ and $a,b\\in C_r^*(G;\\Sigma_t)$ we can compute\n\t\t\\begin{align*}\n\t\t\t\\lk \\xi\\otimes a,\\eta\\otimes b\\rk & = (q_t(\\lk \\eta,\\xi\\rk)a)^*b\\\\\n\t\t\t& = (\\lk \\Psi_t(\\eta),\\Psi_t(\\xi)\\rk a)^*b\\\\\n\t\t\t& = \\lk \\Psi_t(\\xi)a, \\Psi_t(\\eta) b\\rk\\\\\n\t\t\t& = \\lk \\Theta(\\xi\\otimes a),\\Theta(\\eta\\otimes b)\\rk.\n\t\t\\end{align*}\n\t\tFinally, to see that $\\Theta_t$ is surjective, it is enough to show that it has dense image. First, let $\\xi\\in \\Gamma_c(G,d^*\\mathcal{E}_t)$ be of the form $\\xi=\\varphi\\otimes e$ with $\\varphi\\in C_c(G)$ and $e\\in E_t$, i.e. $\\xi(g)=\\varphi(g)e(d(g))$. Since $p_t:E\\rightarrow E_t$ is surjective, we can find an element $e'\\in E$ such that $p_t(e')=e$. Also, pick any map $\\varphi'\\in C_c(G\\times[0,1])$, such that $\\varphi'(g,t)=\\varphi(g)$. Then $\\varphi'\\otimes e'\\in X$ such that $\\Psi_t(\\varphi'\\otimes e')=\\xi$. Now if $\\xi\\in \\Gamma_c(G,d^*\\mathcal{E}_t)$ is arbitrary we can approximate it in the inductive limit topology by finite sums of elements of the form $\\varphi\\otimes e$ as above.\n\t\tAn application of Lemma \\ref{Lemma:PR-Inductive Limit} completes the proof.\n\t\\end{proof}\n\tLet $x\\in \\KK(K(E)\\rtimes_r G,C_r^*(G\\times[0,1];\\Sigma))$ and $x_t\\in \\KK(K(E_t)\\rtimes_r G,C_r^*(G;\\Sigma_t))$ be the canonical KK-equivalences associated to the equivalence bimodules $X$ and $X_t$ respectively. \n\t\\begin{lemma}\\label{Lem:Evaluation1}\n\tFor each $t\\in[0,1]$ restriction of functions induces a $G$-equi\\-variant $\\ast$-homo\\-morphism $\\Phi_t:K(E)\\rightarrow K(E_t)$, such that the following diagram commutes:\\begin{center}\n\t\t\t\\begin{tikzpicture}[description\/.style={fill=white,inner sep=2pt}]\n\t\t\t\\matrix (m) [matrix of math nodes, row sep=3em,\n\t\t\tcolumn sep=2.5em, text height=1.5ex, text depth=0.25ex]\n\t\t\t{ \\K_*(K(E)\\rtimes_r G) & \\K_*(K(E_t)\\rtimes_{\\alpha_t,r}G) \\\\\n\t\t\t\t\\K_*(C_r^*(G\\times[0,1];\\Sigma)) & \\K_*(C_r^*(G;\\Sigma_t)) \\\\\n\t\t\t};\n\t\t\t\\path[->,font=\\scriptsize]\n\t\t\t(m-1-1) edge node[auto] {$ (\\Phi_t\\rtimes G)_* $} (m-1-2)\n\t\t\t\n\t\t\t(m-2-1) edge node[auto] {$ (q_t)_* $} (m-2-2)\n\t\t\t\n\t\t\t(m-1-1) edge node[auto] {$ \\cdot\\otimes x $} (m-2-1)\n\t\t\t(m-1-2) edge node[auto] {$ \\cdot\\otimes x_t $} (m-2-2)\n\t\t\t\n\t\t\t;\n\t\t\t\\end{tikzpicture}\n\t\t\\end{center}\t\n\t\\end{lemma}\n\t\n\t\\begin{proof}\n\tRecall that $K(E)$ is a $C_0(G^{(0)}\\times[0,1])$-algebra. Let $\\mathcal{K}(E)$ denote the associated bundle. Similarly $K(E_t)$ is a $C_0(G^{(0)})$-algebra with associate bundle $\\mathcal{K}(E_t)$. For $f\\in \\Gamma_0(G^{(0)}\\times[0,1],\\mathcal{K}(E))=K(E)$ and $u\\in G^{(0)}$ define\n\t$$\\Phi_t(f)(u):=f(u,t).$$\n\tThen $\\Phi_t(f)\\in \\Gamma_0(G^{(0)},\\mathcal{K}(E_t))\\cong K(E_t)$ and it is straightforward to verify, that $\\Phi_t$ is a $G$-equivariant $\\ast$-homomorphism.\n\tTo see commutativity of the diagram, it is enough to check that $[\\Phi_t\\rtimes G]\\otimes x_t=x\\otimes [q_t]$ in $\\KK(K(E)\\rtimes_r G,C_r^*(G;\\Sigma_t))$. Since all the elements involved can be represented by Kasparov-triples, where the operator is zero, these products are easy to describe: The element on the left handside can be represented by the triple $(X_t,\\Phi_t\\rtimes G,0)$, while the right handside is given by the class of $(X\\otimes_{q_t}C_r^*(G;\\Sigma_t),\\psi\\otimes 1,0)$, where $\\psi$ is the left action of $K(E)\\rtimes_r G$ on $X$.\n\tFrom Proposition \\ref{Prop:Bimodules} we have an isomorphism of  right Hilbert $C_r^*(G;\\Sigma_t)$-bimodules\n\t$\\Theta:X\\otimes_{q_t}C_r^*(G;\\Sigma_t)\\rightarrow X_t$ given on elementary tensors by $\\xi\\otimes a\\mapsto \\xi_{\\mid G\\times\\lbrace t\\rbrace}a$.\n\tThus, to complete the proof, we observe that $\\Theta$ intertwines the left actions of $K(E)\\rtimes_r G$.\n\t\\end{proof}\n\t\n\t\\begin{proof}[Proof of Theorem \\ref{Thm:Evalutation Induces Isomorphism on K-theory}]\n\tFix $t\\in[0,1]$. In light of Lemma \\ref{Lem:Evaluation1} and the fact that $G$ satisfies the Baum-Connes conjecture with coefficients (and the naturality of the Baum-Connes assembly map),\n\tit is enough to show that $\\Phi_t:K(E)\\rightarrow K(E_t)$ induces an isomorphism\n\t$$(\\Phi_t)_*:\\K_*^{\\mathrm{top}}(G;K(E))\\rightarrow \\K_*^{\\mathrm{top}}(G;K(E_t)).$$\n\tHence we are in the position to apply \\cite[Theorem~7.10]{1806.00391} to deduce that it is enough to show that\n\t$$(\\Phi_t\\rtimes H)_*:\\K_*(K(E)_{\\mid H}\\rtimes H)\\rightarrow \\K_*(E_t)_{\\mid H}\\rtimes H)$$\n\tis an isomorphism for all compact open subgroupoids $H\\subseteq G$.\n\tUsing Remark \\ref{Rem:PackerRaeburnRestrict} and the same arguments as in Lemma \\ref{Lem:Evaluation1} for $H$, we conclude that it is enough to prove that \n\t$$(q_t)_*:\\K_*(C_r^*(H\\times[0,1];\\Sigma))\\rightarrow \\K_*(C_r^*(H;\\Sigma_t))$$\n\tis an isomorphism.\n\tSince $G$ is ample, $\\Sigma$ is a continuous homotopy of twists by Proposition \\ref{Prop:HomotopiesAreContinuous}. Clearly, the restriction of $\\Sigma$ to $H$ has the same property. Consequently, we may apply Proposition \\ref{Prop:CompactCase} to finish the proof.\n\t\\end{proof}\n\n\t\t\n\t\n\n\n\n\n\n\n\n\n\t\n\n\n\t\n\n\n\n\t\n\t\\section{Twisted actions of inverse semigroups}\n\tIn this section we phrase our results in terms of twisted actions of inverse semigroups. These have been studied by several authors and we will recall the definition in the setting we want to work in for the readers convenience. In what follows we will write $E$ for the lattice of idempotents in an inverse semigroup $S$.\n\tFor convenience, we will also assume that there is an element $0\\in S$ such that $0s=s0=0$ for all $s\\in S$. If a semigroup does not have a zero element, one can adjoin a zero element and extend the multiplication in the obvious way.\n\t\\begin{defi}\\label{Def:TwistedAction}\n\tA \\textit{twisted action} $\\theta$ of an inverse semigroup $S$ on a locally compact Hausdorff space $X$ is a triple $$\\theta=(\\lbrace D_s\\rbrace_{s\\in S},\\lbrace \\theta_s\\rbrace_{s\\in S},\\lbrace \\omega_{s,t}\\rbrace_{s,t\\in S}),$$ consisting of a family of open subsets $D_s$ of $X$ whose union covers $X$, a family of homeomorphisms $\\theta_s:D_{s}\\rightarrow D_{s^*}$, and a family of continuous maps $\\omega(s,t):D_{st}\\rightarrow \\TT$, such that the following hold:\n\t\\begin{enumerate}\n\t\\item The pair $(\\lbrace D_s\\rbrace_{s\\in S},\\lbrace \\theta_s\\rbrace_{s\\in S})$ defines an action of $S$ on $X$, i.e. $\\theta_r\\circ \\theta_s=\\theta_{rs}$;\n\t\\item $\\omega(s,t)(\\theta_r^{-1}(x))\\omega(r,st)(x)=\\omega(r,s)(x)\\omega(rs,t)(x)$, whenever $x\\in D_r\\cap D_{st}$; and\n\t\\item $\\omega(s,e)=1_{se}$ and $\\omega(e,s)=1_{es}$ for all $s\\in S$ and $e\\in E$.\n\t\\end{enumerate}\n\t\\end{defi}\n\n\tIt is is routine to check that a twisted action of $S$ on $X$ as defined above induces a twisted action of $S$ on $C_0(X)$ in the sense of  \\cite{MR1620499} and \\cite{MR2821242}.\n\tFollowing \\cite[Theorem~7.2]{MR2821242}, for any such twisted action we can construct an \u00e9tale groupoid $G$ with unit space $X$ and a twist $\\Sigma$ over $G$ such that the twisted groupoid $\\mathrm{C}^*$-algebra $C_r^*(G;\\Sigma)$ is isomorphic to the twisted crossed product $C_0(X)\\rtimes_{\\theta,\\omega}^r S$.\n\tIt follows from \\cite[Proposition~7.4]{MR2821242} that twisted actions in the sense above yield groupoids with continuous twists since we insist on condition $(3)$ above. One should note that Buss and Exel study a more general version of twisted actions. But since our main result applies to ample groupoids (and hence to continuous twists), we can stick to the definition given above without loosing any generality. More precisely, we have the following interpretation of Corollary \\ref{Cor:TwistsOnAmpleGroupoidsAreContinuous} in the setting of inverse semigroups:\n\t\\begin{kor} Let $S$ be an inverse semigroup and $X$ be a totally disconnected locally compact Hausdorff space.\n\tThen every twisted action $\\theta$ of $S$ on $X$ in the sense of \\cite[Definition~4.1]{MR2821242} is a twisted action in the sense of Definition \\ref{Def:TwistedAction}.\n\t\\end{kor}\n\tAnother obstruction to apply our main results is the requirement that the groupoid in question must be Hausdorff. The groupoid associated to a twisted action of an inverse semigroup is constructed as a certain groupoid of germs. The question when such a groupoid is Hausdorff is a subtle one. It has been studied by Steinberg and was refined by Exel and Pardo. In order to state their result recall, that there is a canonical order relation on an inverse semigroup $S$ given by \n\t$$s\\leq t\\Leftrightarrow s=ts^*s \\text{ for all } s,t\\in S.$$\n\t\\begin{satz}\\cite[Theorem~3.15]{MR3448414}\n\t\tLet $(\\lbrace D_s\\rbrace_{s\\in S},\\lbrace \\theta_s\\rbrace_{s\\in S})$ be an action of the inverse semigroup $S$ on a locally compact Hausdorff space $X$ and let $G(\\theta)$ denote the associated groupoid of germs. Then $G(\\theta)$ is Hausdorff if and only if for every $s\\in S$ the union $\\bigcup_{\\lbrace e\\in E:e\\leq s\\rbrace} D_e$ is closed relative to $D_{s^*s}$.\t\t\n\t\\end{satz}\n\tIn particular, if $S$ is a \\textit{weak semilattice} then the groupoid of germs $G(\\theta)$ is Hausdorff for any action $\\theta$ of $S$ on a locally compact Hausdorff space $X$ such that $D_e$ is clopen for every $e\\in E$ (see \\cite[Theorem~5.17]{MR2565546}).\n\t\n\tThe previous discussion applies in particular to twisted versions of the canonical action of $S$ on the spectrum of its idempotents:\n\tRecall, that a \\textit{character} on the idempotent semilattice $E$ of $S$ is a non-zero map $\\chi:E\\rightarrow \\lbrace 0,1\\rbrace$, such that $\\chi(ef)=\\chi(e)\\chi(f)$ and $\\chi(0)=0$. The set of all characters\n\t$$\\widehat{E}=\\lbrace \\chi:E\\rightarrow \\lbrace 0,1\\rbrace\\mid \\chi\\textit{ is a character}\\rbrace$$\n\tis called the spectrum of $E$. Equipped with the topology inherited from the product topology on $\\lbrace 0,1\\rbrace^E$, the spectrum $\\widehat{E}$ is a locally compact, totally disconnected Hausdorff space.\n\tThere is a canonical action of $S$ on $\\widehat{E}$ given as follows:\n\tFor $s\\in S$ let $D_s:=\\lbrace \\chi\\in \\widehat{E}\\mid \\chi(s^*s)=1\\rbrace$ and $\\theta_s:D_s\\rightarrow D_{s^*}$ be given by $\\theta_s(\\chi)(e):=\\chi(s^*es)$. Then each $D_s$ is clopen in $\\widehat{E}$ and $\\theta$ is an action of $S$ on $\\widehat{E}$.\n\tThe groupoid of germs of this action is usually called the \\textit{universal groupoid} of $S$ and denoted by $G(S)$ (see \\cite{Paterson1999,zbMATH05553518} for more details on this construction).\n\t\n\tBy \\cite{MR2821242} again, twisted actions of $S$ on $\\widehat{E}$ are in one-to-one correspondence with continuous twists over the universal groupoid $G(S)$ of $S$. A twisted action of $S$ on $\\widehat{E}$ whose underlying action is the canonical one described above (i.e. the data mainly consists of a family of maps $\\omega_{s,t}:D_{st}\\rightarrow \\TT$ satisfying the appropriate conditions from Definition \\ref{Def:TwistedAction}) will also be called a \\textit{twist} over $S$ or a \\textit{$2$-cocycle} on $S$.\n\t\n\t\n\tIn light of the above discussion there is an obvious notion of homotopy of twisted actions:\n\t\\begin{defi}\n\tA \\textit{homotopy of twisted actions} of $S$ on $X$ is a triple $$(\\lbrace D_s\\rbrace_{s\\in S},\\lbrace \\theta_s\\rbrace_{s\\in S},\\lbrace \\omega_{r,s}\\rbrace_{r,s\\in S})$$ consisting of a family of open subsets $D_s$ of $X$ whose union covers $X$, a family of homeomorphisms $\\theta_s:D_{s}\\rightarrow D_{s^*}$, and a family of continuous maps $\\omega(r,s):D_{rs}\\times [0,1]\\rightarrow \\mathbb{T}$, such that\n\t$(\\lbrace D_s\\times [0,1]\\rbrace_{s\\in S},\\lbrace\\theta_s\\times\\id_{[0,1]}\\rbrace_{s\\in S},\\lbrace\\omega(r,s)\\rbrace_{r,s\\in S})$ is a twisted action of $S$ on $X\\times [0,1]$.\n\t\\end{defi}\n\tNote that a homotopy defines a family of twisted actions of $S$ on $X$, indexed by the unit interval $[0,1]$.\n\t\n\tInvoking \\cite[Theorem~7.2]{MR2821242} again, a homotopy of twisted actions provides us with an \u00e9tale groupoid $\\mathcal{G}$ with $\\mathcal{G}^{(0)}=X\\times[0,1]$ and a twist $\\Sigma$ over $\\mathcal{G}$ such that $C_r^*(\\mathcal{G};\\Sigma)\\cong C_0(X\\times[0,1])\\rtimes_{\\theta,\\omega}^r S$.\n\tSince in our definition of homotopy the action of $S$ on $X$ does not depend on $t\\in [0,1]$, the groupoid $\\mathcal{G}$ is just the trivial bundle of groupoids $\\mathcal{G}=G\\times [0,1]$, where $G$ is the groupoid of germs associated to the action $(\\lbrace D_s\\rbrace_{s\\in S},\\lbrace \\theta_s\\rbrace_{s\\in S})$ of $S$ on $X$.\n\tHence the following result is a direct consequence of Theorem \\ref{Thm:Evalutation Induces Isomorphism on K-theory} and \\cite{Tu98}:\n\t\\begin{satz}\n\tLet $(\\lbrace D_s\\rbrace_{s\\in S},\\lbrace \\theta_s\\rbrace_{s\\in S},\\lbrace \\omega_{s,t}\\rbrace_{s,t\\in S})$ be a homotopy of twisted actions of $S$ on a locally compact totally disconnected Hausdorff space $X$. Suppose further, that\n\t\\begin{enumerate}\n\t\t\\item for every $s\\in S$ the union $\\bigcup_{\\lbrace e\\in E:e\\leq s\\rbrace} D_e$ is closed relative to $D_{s^*s}$, and\n\t\t\\item the action $\\theta$ of $S$ on $X$ is amenable in the sense of \\cite[Definition~3.2]{MR3614034}.\n\t\\end{enumerate}  Then the evaluation homomorphism $$q_t:C_0(X\\times[0,1])\\rtimes_{\\theta\\times \\id,\\omega}^r S\\rightarrow C_0(X)\\rtimes_{\\theta,\\omega_t}^r S$$\n\tinduces an isomorphism in $\\K$-theory.\n\t\\end{satz}\n\tSpecialising to the canonical action of $S$ on $\\widehat{E}$ again we get the following:\n\tFor a homotopy of $2$-cocycles $\\omega$ on $S$ and $t\\in[0,1]$ let $\\omega^t$ be the $2$-cocycle on $S$ given by $\\omega^t_{r,s}(\\chi):=\\omega_{r,s}(\\chi,t)$.\n\tGiven a $2$-cocycle $\\omega=(\\omega_{s,t})_{s,t\\in S}$ on an inverse semigroup $S$ let $C_r^*(S,\\omega):=C_0(\\widehat{E})\\rtimes_{\\theta,\\omega}^r S$.\n\tThen we have the following immediate corollary:\n\t\\begin{kor}\n\t\tLet $S$ be an inverse semigroup. Suppose that $S$ is a weak semilattice and that the canonical action of $S$ on $\\widehat{E}$ is amenable.\n\t\tIf $\\omega$ is a homotopy of $2$-cocycles on $S$, then\n\t\t$$\\K_*(C_r^*(S,\\omega^0))\\cong \\K_*(C_r^*(S,\\omega^1)).$$\n\t\\end{kor}\n\t \n\t \n\t \n\t \\section{Applications}\n\t \\subsection{Generalized Renault-Deaconu groupoids}\n\t Our results apply nicely to actions of certain semigroups studied in the literature. Recall the following definition from \\cite{MR3592511}:\n\t \\begin{defi} Let $P$ be a subsemigroup of a discrete group $\\Gamma$ containing the identity element $e$ of $\\Gamma$ and let $X$ be a locally compact Hausdorff space. Then a (continuous) \\textit{left} action of $P$ consists of a family of local homeomorphisms $(T_p)_{p\\in P}$ with $T_p:dom(p)\\rightarrow ran(p)$ such that \n\t \t\\begin{enumerate}\n\t \t\t\\item $dom(p),ran(p)$ are open subsets of $X$ for all $p\\in P$,\n\t \t\t\\item $dom(e)=ran(e)=X$ and $T_e=\\id_X$, and\n\t \t\t\\item for all $p,q\\in P$ we have $x\\in dom(pq)$ if and only if $x\\in dom(q)$ and $T_q(x)\\in dom(p)$ and in that case $T_{pq}(x)=(T_p\\circ T_q)(x)$.\n\t \t\\end{enumerate}\n\t Moreover, such an action is called \\textit{directed}, provided that for every pair of elements $p,q\\in P$ with $dom(p)\\cap dom(q)\\neq \\emptyset$ there exists an upper bound $r\\in P$ of $p$ and $q$ such that $dom(p)\\cap dom(q)\\subseteq dom(r)$.\n\t \\end{defi}\n\t \n\t For example, every action of an Ore semigroup is directed, and the action of a quasi-latticed ordered semigroup on its spectrum is directed.\n\t \n\t Given a directed action $T$ of $P$ on $X$, where $P$ is a subsemigroup of the discrete group $\\Gamma$ one can define the \\textit{semidirect product groupoid} (or generalized Renault-Deaconu groupoid) $G(X,P,T)$ of the action as the set\n\t $$\\lbrace (x,\\gamma,y)\\in X\\times\\Gamma\\times X\\mid \\exists p,q\\in P: \\gamma=p^{-1}q\\textit{ and }T_p(x)=T_q(y)\\rbrace,$$\n\t and equip it with the groupoid structure it inherits as a subset of $X\\times\\Gamma\\times X$, equipped with the obvious groupoid structure over $X$.\n\t It can be shown that the sets\n\t $$Z(U,p,q,V)=\\lbrace (x,p^{-1}q,y)\\in G(X,P,T)\\mid x\\in U, y\\in V\\rbrace,$$\n\twith $U,V$ ranging over the open subsets of $X$ and $p,q\\in P$ form a basis for a locally compact Hausdorff topology on $G(X,P,T)$ giving it the structure of an \u00e9tale groupoid over $X$, such that the canonical homomorphism $c:G(X,P,T)\\rightarrow \\Gamma$ given by $c(x,\\gamma,y)=\\gamma$ is continuous (see \\cite[Proposition~5.12]{MR3592511} for the details). In particular, if $X$ happens to be totally disconnected, then $G(X,P,T)$ is an ample groupoid.\n\tMoreover, if the ambient group $\\Gamma$ is amenable then the groupoid $G(X,P,T)$ is topologically amenable by \\cite[Theorem~5.13]{MR3592511}.\n\tHence we can apply our main result to twists over such groupoids.\n\t\n\tThe following lemma illustrates how twists on such a semidirect product groupoid can arise in practice. We omit its routine proof.\n\t\\begin{lemma}\n\t\tLet $(X,P,T)$ be a directed action of $P\\subseteq \\Gamma$ on a locally compact Hausdorff space $X$.\n\t\tIf $\\omega:\\Gamma^2\\rightarrow \\TT$ is a (normalized) $2$-cocycle for $\\Gamma$, then $$\\tilde{\\omega}:G(X,P,T)^{(2)}\\rightarrow \\TT; \\ \\ \\ \\tilde{\\omega}((x,\\gamma,y),(y,\\gamma',z)):=\\omega(\\gamma,\\gamma')$$ defines a continuous (normalized) $2$-cocycle on $G(X,P,T)$.\n\t\tMoreover, if $\\omega_1,\\omega_2:\\Gamma ^2\\rightarrow \\TT$ are cohomologous then so are $\\tilde{\\omega_1},\\tilde{\\omega_2}$.\n\t\\end{lemma}\n\tSimilarly, homotopies of $2$-cocycles on $\\Gamma$ induce homotopies of continuous $2$-cocycles on $G(X,P,T)$.\n\t\n\t\\begin{ex}\n\t\t\tConsider the case $(P,\\Gamma)=(\\NN^k,\\ZZ^k)$. It is a well-known fact that every class in $H^2(\\ZZ^k,\\TT)$ has a representative of the form $\\omega_\\Theta(n,m)=e^{i\\langle \\Theta n,m\\rangle}$ for a skew symmetric real $k\\times k$-matrix $\\Theta$. Each of these cocycles is clearly homotopic to the constant $2$-cocycle $\\omega_1$.\n\t\t\tIt follows that the associated twists $\\Sigma_{\\Theta}$ (via the previous Lemma and Remark \\ref{Rem:Cocycles}) over $G(X,\\NN^k,T)$ are all homotopic to the trivial twist and hence\n\t\t\t$\\K_*(C_r^*(G(X;\\NN^k,T);\\Sigma_{\\Theta}))\\cong \\K_*(C_r^*(G(X,\\NN^k,T)))$.\n\t\t\\end{ex}\n\t\t\n\t\n\tWe remark that certainly not all twists on $G(X,P,T)$ arise in this way, unless $X$ is zero-dimensional. To see this consider the ordinary Renault-Deaconu groupoid $G(X,\\NN,\\sigma)$ associated to a local homeomorphism $\\sigma:X\\rightarrow X$. In this case the ambient group is $\\ZZ$ and it is well-known, that every $2$-cocycle on $\\ZZ$ is a coboundary, and hence the cohomology group $H^2(\\ZZ,\\TT)$ is trivial. On the other hand it was observed in \\cite{zbMATH01752404} that the group of (isomorphism classes of) twists $Tw(G(X,\\NN,\\sigma))$ is isomorphic to the sheaf cohomology group $H^1(X,\\mathcal{S})$ of $X$, where $\\mathcal{S}$ is the sheaf of germs of continuous $\\TT$-valued functions on $X$.\n\tIf $X$ is zero-dimensional, then $H^1(X,\\mathcal{S})$ vanishes and hence ordinary Renault-Deaconu groupoids (i.e. the case where $P=\\NN)$ over totally disconnected spaces do not admit any non-trivial twists.\n\t\n\tThis begs the following question: If $(X,P,T)$ is a directed action of $P\\subseteq \\Gamma$ on a locally compact totally disconnected Hausdorff space $X$, is every twist on $G(X,P,T)$ induced by a $2$-cocycle on $\\Gamma$?\n\n\t\\subsection{$P$-graphs}\n\tInteresting examples of the semidirect product group\\-oids studied above arise from higher-rank graphs or more general $P$-graphs as introduced in \\cite{MR3592511,MR3871826}. Let $P$ be a subsemigroup of a countable discrete group $\\Gamma$. According to \\cite[Definition~6.1]{MR3592511} a (discrete) $P$-graph $\\Lambda$ is a small category $\\Lambda$ endowed with a \\textit{degree} functor $d:\\Lambda\\rightarrow P$ satisfying the unique factorisation property.\n\t\n\tAssociated with a $P$-graph $\\Lambda$ is a canonical action $T_\\Lambda$ of $P$ on the infinite path space $X_\\Lambda$, which in turn gives rise to the ample groupoid $G_\\Lambda:=G(X_\\Lambda,P,T_\\Lambda)$.\n\t\n\tLet us specialise to the case where $P= \\NN^k\\times F$ for some countable abelian group $F$.\n\tBy \\cite[Proposition~5.7, Corollary~5.10]{MR3871826} every $2$-cocycle $c$ on $\\Lambda$ gives rise to a continuous $2$-cocycle $\\omega_c$ and hence a twist $\\Sigma_c$ on $G_\\Lambda$ such that\n\t$C^*(\\Lambda,c)$ is canonically isomorphic to $C^*(G_\\Lambda;\\Sigma_c)$.\n\t\n\tA homotopy of $2$-cocycles for a $P$-graph $\\Lambda$ is a family $(c_t)_{t\\in [0,1]}$ of $2$-cocycles, such that for each pair $(\\lambda,\\mu)\\in \\Lambda^{\\ast 2}$ the map $t\\mapsto c_t(\\lambda,\\mu)\\in\\TT$ is continuous. Following the line of argument in \\cite[Proposition~3.8]{MR3407179} every homotopy of $2$-cocycles on $\\Lambda$ gives rise to a continuous homotopy of twists over $G_\\Lambda\\times [0,1]$.\n\t\\begin{kor}\n\t\tLet $P= \\NN^k\\times F$ for some countable abelian group $F$. Let $\\Lambda$ be a row-finite $P$-graph with no sources and let $(c_t)_{[t\\in[0,1]}$ be a homotopy of $2$-cocycles on $\\Lambda$. Then $$\\K_*(C^*(\\Lambda,c_0))\\cong \\K_*(C^*(\\Lambda,c_1)).$$\n\t\\end{kor}\n\t\\begin{proof}\n\t\tSince $\\ZZ^k\\times F$ is amenable, so is $G_\\Lambda$ by \\cite[Corollary~6.19]{MR3592511}. Thus, Theorem \\ref{Thm:Evalutation Induces Isomorphism on K-theory} implies that \n\t\t$$\\K_*(C^*(\\Lambda,c_0))\\cong\\K_*(C_r^*(G_\\Lambda;\\Sigma_{c_0}))\\cong \\K_*(C_r^*(G_\\Lambda;\\Sigma_{c_1}))\\cong \\K_*(C^*(\\Lambda,c_1)).$$\n\t\\end{proof}\n\tWe expect this result to hold in much greater generality. In fact we believe it is true for all $(r,d)$-proper $P$-graphs $\\Lambda$ (in the sense of \\cite{MR3592511}), where $P$ is a quasi-lattice ordered subsemigroup of a countable discrete amenable group $\\Gamma$. This would require a generalisation of several results in \\cite{MR3871826} to more general $P$-graphs, which we did not seriously attempt.\n\t\n\n\n\\ \\newline\n{\\bf Acknowledgments}. The content of this paper is based on some of the findings from \nthe author's doctoral thesis. He would like to thank his supervisor Siegfried Echterhoff for his support and advice and Stuart White for commenting on an early draft of this manuscript. He is further indebted to the anonymous referee for useful comments and remarks.\n\n\\bibliographystyle{plain}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\noindent Black holes, coined by Wheeler \\cite{wh67} were generally considered as rather esoteric objects of purely theoretical interest and little physical relevance. It is well known that  a black hole is thermodynamical system, there exist fundamental connections between the thermodynamics and general relativity since the  discovery of black hole entropy by Bekenstein \\cite{JD,JD1,JD2,JD3,JD4}  and Hawking \\cite{SWH}. During the past years the development of Maldasena conjucture (AdS\/CFT correspondence) \\cite{jm}, has attracted significant attention to AdS black holes.  The properties of  black hole physics have changed dramatically and subsequent studies found that the AdS black hole behave like the Van der Walls fluid. \n\nThe first regular black hole was proposed by Bardeen \\cite{Regular:1968} which means there is no singularity. Later on an exact solution of  Bardeen black hole was given by Ayon-Beato and Garcia \\cite{AGB,AGB1,AGB2} wherein general relativity coupled to nonlinear electrodynamics. Thereafter, more intense emphasis was given by researchers investing the regular black holes and mor regular black holes were discovered recently \\cite{hc,lbev,Balart:2014cga,Xiang,Bronnikov:2000vy,Zaslavskii:2009kp,Lemos:2011dq,Ansoldi:2008jw,Ghosh:2014pba,dvs99}. But most of these solutions are based on Bardeen's Model \\cite{Regular:1968}. In later years the generalized solutions of Bardeen black hole model were developed which includes Bardeen de Sitter black hole  \\cite{singh,fr1}, rotating  Bardeen solution \\cite{Bambi},  noncommutative Bardeen solution \\cite{sharif}, higher dimensional black holes  \\cite{sabir} and EGB black holes \\cite{dvs19,kumar19,Kumar:2020bqf,Singh:2020xju,Kumar:2020uyz} etc. However,  many singular black hole solutions with massive gravity  \\cite{Cai:2014znn,Babichev:2015xha,EslamPanah:2019fci,EslamPanah:2018rob} are already present in the literature, but they  are not regular black hole solutions. Therefore, the aim of this work to get $4\\-D$  spherically symmetric  Bardeen-like black hole solution in massive gravity  in AdS spacetimes viz., Bardeen AdS massive metric.  It is  shown that the Bardeen AdS massive metric is an exact black  hole solution  in AdS spacetime thereby generalizing the  Bardeen black hole solution \\cite{Tzikas:2018cvs} which is encompassed as a special case. We analyze  their thermodynamical properties  and also perform a  phase structure  analysis of the Bardeen AdS  massive black holes.\n\nIn the extended phase space cosmological constant ($\\Lambda$) is identified as thermodynamical  pressure ($P=-\\Lambda\/8\\pi$) and its conjugate quantity is  thermodynamical volume \\cite{Cvetic:2010jb, dolan13}.  Hawking and Page \\cite {hp} have first studied the phase transition between the AdS black hole and thermal AdS space  and also confinement\/deconfinement phase of gauge field  was studied by Witten \\cite{witten}. The phase transition behaviour of charged AdS black hole and  the Van der Walls liquid-gas system were studied by Chamblin \\cite{Chamblin:1999tk,Chamblin:1999hg}. In fact, phase transition plays an important role to investigate the thermodynamical properties of the objects at the critical point.  This consideration  has been investigated for different type   of black holes  \\cite{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24}.\n\nThe paper is organized as follows. In Section II we find the exact solution of Bardeen black hole in the presence of massive gravity and studied its physical and thermodynamical properties.  Section III,  gives the investigation of  phase structure for  Bardeen AdS  black holes in massive gravity. The thermodynamical stability and phase diagrams are studied in Sec IV.   Finally, we discuss our result and conclusions in Section V. (Here we use the units\n$G_4 =  k_B = c = 1$).\n\\section{\\label{sec:level3}Bardeen Black Hole solution in Massive gravity}\nThe Einstein- Hilbert action coupled to nonlinear electrodynamics in the presence of massive gravity  with negative cosmological constant \\cite{cao} is given by\n\\begin{eqnarray}\nS &=&\\frac{1}{2 }\\int d^{4}x\\sqrt{-g}\\Big[R\n-2\\Lambda +m^{2}\\sum_{i}^{2}c_{i}\\,\\mathcal{U}_{i}(g,f)- \\frac{1}{4\\pi}\\mathcal{ L}(F)\\Big],\n\\label{action}\n\\end{eqnarray}\nwhere $R$ is scalar curvature, $\\Lambda$ is a cosmological constant,  $m$ is a parameter of massive gravity,  $c_i$ are constants, $f$ is symmetric tensor and $\\mathcal{U}_{i}$ are polynomials of eigenvalues of the ($4\\times 4$) matrix $\\mathcal{K}_{\\nu }^{\\mu }=\\sqrt{g^{\\mu \\alpha }f_{\\alpha \\nu }}$ which can be written as   \n\\begin{eqnarray}\n\\mathcal{U}_{1} &=&\\left[ \\mathcal{K}\\right] ,\\nonumber \\\\\n\\mathcal{U}_{2} &=&\\left[ \\mathcal{K}\\right] ^{2}-\\left[ \\mathcal{K}^{2}\n\\right] ,\\nonumber\\\\\n\\mathcal{U}_{3} &=&\\left[ \\mathcal{K}\\right] ^{3}-3\\left[ \\mathcal{K}\\right] \n\\left[ \\mathcal{K}^{2}\\right] +2\\left[ \\mathcal{K}^{3}\\right] ,\\nonumber \\\\\n\\mathcal{U}_{4} &=&\\left[ \\mathcal{K}\\right] ^{4}-6\\left[ \\mathcal{K}^{2}\n\\right] \\left[ \\mathcal{K}\\right] ^{2}+8\\left[ \\mathcal{K}^{3}\\right] \\left[\n\\mathcal{K}\\right] +3\\left[ \\mathcal{K}^{2}\\right] ^{2}-6\\left[ \\mathcal{K}\n^{4}\\right],\n\\end{eqnarray}\nwith $\\left[ \\mathcal{K}\\right]=\\left[ \\mathcal{K}_{\\mu }^{\\mu }\\right]$ and $\\mathcal{ L}(F)$ is the function of $F\\equiv F_{\\mu \\nu } F^{\\mu \\nu}$   be  the electromagnetic field  tensor  which is  the generalization of Maxwell field and it reduces to Maxwell field in the weak field limit. We consider Lagrangian density of  the non-linear electromagnetic field as \\cite{AGB2}\n\\begin{eqnarray}\n{\\cal{L}}(F)=\\frac{1}{2se^2}\\left(\\frac{\\sqrt{2e^2F}}{1+\\sqrt{2e^2F}}\\right)^{\\frac{5}{2}},\n\\label{matter}\n\\end{eqnarray}\nwhere $\\mathcal{ L}(F)$ is the Lagrangian density of Bardeen source \\cite{AGB2}  and parameters  $M$ and  $e$ connected with $s$ by relation $s=e\/2M$. We would like to consider the spherically-symmetric  static metric of the spacetime, given by the following  line element\n\\begin{equation}\nds^2=-f(r)dt^2 +\\frac{1}{f(r)}dr^2+r^2d\\Omega^2, \\qquad\\text{with}\\qquad f(r)=1-\\frac{2m(r)}{r}.\n\\label{m2}\n\\end{equation}\nWe consider the following metric ansatz \\cite{cao} for the reference metric as\n\\begin{equation}\nf_{\\mu \\nu }=diag(0,0,c^{2}h_{ij}),\n  \\label{f11}\n\\end{equation}\nwhere $c$ is  the positive constant. Using the metric ansatz (\\ref{f11}), one can easily write $\\mathcal {U}_i$ as\n\\begin{eqnarray}\n&&\\mathcal {U}_1=\\frac{2}{r},\\qquad \\mathcal {U}_2=\\frac{2}{r^2} ,\\qquad \\mathcal {U}_3=0 \\quad\\text{and}\\quad \\mathcal {U}_4=0.\n\\end{eqnarray}\nUsing the action (\\ref{action}) and variation of this action with respect to the metric tensor ($g_{\\mu \\nu }$) and the electromagnetic  potential ($A_{\\mu}$), respectively, leads to\n\\begin{eqnarray}\nG_{\\mu \\nu }&+&\\Lambda g_{\\mu \\nu }+m^{2}\\chi _{\\mu \\nu }=T_{\\mu\\nu}  \\equiv2\\left[\\frac{\\partial {\\cal{L}}(F)}{\\partial F}F_{\\mu\\rho}F^{\\rho}_{\\nu}-g_{\\mu\\nu}{\\cal{L}}(F)\\right],\\nonumber\\\\\n&&\\nabla_{\\mu}\\left(\\frac{\\partial {\\cal{L(F)}}}{\\partial F}F^{\\mu \\nu}\\right)=0,\\quad \\text{and}\\quad \\nabla_{\\mu}(* F^{\\mu\\nu})=0,\n\\label{Field equation}\n\\end{eqnarray}\n where $T_{\\mu\\nu}$ is the energy momentum tensor, $G_{\\mu\\nu}$   and $\\chi_{\\mu\\nu}$ are  the Einstein tensor and massive gravity tensor  respectively \\cite{cao} are given as\n\\begin{eqnarray}\nG_{\\mu\\nu}&=&R_{\\mu\\nu}-\\frac{1}{2}g_{\\mu\\nu}R,\\\\\n\\chi _{\\mu \\nu }& =&-\\frac{c_{1}}{2}\\left( \\mathcal{U}_{1}g_{\\mu \\nu }-\n\\mathcal{K}_{\\mu \\nu }\\right) -\\frac{c_{2}}{2}\\left( \\mathcal{U}_{2}g_{\\mu\n\\nu }-2\\mathcal{U}_{1}\\mathcal{K}_{\\mu \\nu }+2\\mathcal{K}_{\\mu \\nu\n}^{2}\\right). \n\\end{eqnarray}\nThe non-vanishing components of $F_{ab}$ are  $F_{\\theta\\phi}=g(r)\\sin\\theta$ and with potential $A_{\\phi}=-g(r)\\cos\\theta$, reads \\cite{AGB2}. The non-zero components of energy momentum tensor is  given by,\n\\begin{eqnarray}\n&&T^t_t=T^r_r=\\frac{8Me^{2}}{(r^2+e^2)^{\\frac{5}{2}}}\\\\\n&&T^{\\theta}_{\\theta}= T^{\\phi}_{\\phi}=\\frac{8M e^2(r^2-4)}{(r^2+e^2)^{\\frac{5}{2}}}.\n\\end{eqnarray}\nUsing the Eq. \\ref{m2}, the $(r,r)$ components of the Eq. \\ref{Field equation} are\n\\begin{eqnarray}\n&&m'(r)+\\frac{3 r^2}{2l^2}+\\frac{m^2}{2}\\left(\\frac{cc_1 r^2}{2}+c^2c_2\\right)=\\frac{2Me^{2}}{{(r^2+e^2)^{\\frac{5}{2}}}},\n\\label{rreq1}\n\\end{eqnarray}\nwhere the prime is the first derivatives with respect to $r$. We can obtain\nthe metric function $f(r)$ using the Eq. (\\ref{rreq1}). Integrating the Eq. (\\ref{rreq1}) in the limit $r\\to \\infty$, \n\\begin{equation}\nm(r)+\\frac{ r^3}{2l^2}+\\frac{m^2}{2}\\left(\\frac{cc_1 r^2}{2}+c^2c_2\\right)=M\n\\end{equation} \n by substituting $m(r)$ in $f(r)$ Eq. (\\ref{m2}), then the solution becomes\n\\begin{eqnarray}\nf(r)=1-\\frac{2M r^2}{(r^{2}+e^{2})^{\\frac{3}{2}}}+\\frac{ r^2}{l^2}+m^2\\left (c^2c_2+\\frac{cc_1 r}{2}\\right) ,\n\\label{sol2}\n\\end{eqnarray}\nThis solutions describe the four dimensional  $AdS$ Bardeen  black hole in the presence of massive gravity and it is characterized by the mass $M$, cosmological constant $\\Lambda=-3\/l^2$,  magnetic charge $e$, and mass parameter $m$.   In the absence of magnetic charge $e$ this black hole solution (\\ref{sol2}) interpolated with $4D$ AdS massive black hole \\cite{Babichev:2015xha},\n\\begin{eqnarray}\nf(r)=1-\\frac{2M}{r}+\\frac{ r^2}{l^2}+m^2\\left (c^2c_2+\\frac{cc_1 r}{2}\\right) ,\n\\label{sol3}\n\\end{eqnarray}\n for $m=0$, it reduces to AdS Bardeen black hole as \\cite{Tzikas:2018cvs}\n\\begin{eqnarray}\nf(r)=1-\\frac{2M r^2}{(r^{2}+e^{2})^{\\frac{3}{2}}}+\\frac{ r^2}{l^2},\n\\label{sol4}\n\\end{eqnarray}\n and for   $e=0,m=0$ it gives to AdS Schwazshild black hole.  It would be more convenient to study the horizon structure. The horizon of the black hole  can be achieved when $f(r_+)=0$. \n\\begin{eqnarray}\n 1-\\frac{2GM r^2}{(r^{2}+e^{2})^{\\frac{3}{2}}}+\\frac{ r^2}{l^2}+m^2 \\left(c^2c_2+\\frac{cc_1 r}{2}\\right)=0\n\\end{eqnarray}\n This equation can not be solved analytically.  The numerical analysis of the $f(r_+)=0$ on the varying the massive gravity parameters is depicted in the Fig. (\\ref{fig:1}). The numerical analysis of $f (r_+ ) = 0$  reveals that it is possible to find non-vanishing\nvalue of magnetic monopole charge ($e$),  cosmological  constant ($\\Lambda$) and massive parameter ($m$)  for which metric function $f (r )$ is minimum, i.e, $f (r_+ ) = 0$ this will give four roots $r_+, r_-, r_c$ and $r_m$  which  correspond to the Cauchy horizon, event horizon, cosmological horizon and fourth horizon due to mass parameter ($m$) of the black holes respectively. The size of black hole increase with decrease the massive parameter ($m$) with constant magnetic charge ($e$). \n\\begin{figure*}[ht]\n\\begin{tabular}{c c c c}\n\\includegraphics[width=.55\\linewidth]{f1e.eps}\n\\includegraphics[width=.55\\linewidth]{f1m.eps}\n\\end{tabular}\n\\caption{The plot of $f(r)$ vs $r$ for different values of magnetic charge  $e$ and mass parameter $m$  with fixed value of $c=1,c_1=-1, c_2=1,l=20$ and $m=1.12$ (left) and magnetic charge $e=0.40$ (right).}\n\\label{fig:1}\n\\end{figure*}\n\n\\noindent The horizon limit of the Bardeen AdS massive black hole are: \\\\\n1) The size of horizon increase with decrease  the magnetic charge and  the magnetic charge does not affect  the cosmological and massive horizon.\\\\\n2)  The cosmological and massive horizon increases with the mass parameter $m$ and no cosmological and massive horizon for $m<1.125$.\\\\\n3)  The Cauchy and event horizon independent of mass parameter $m$ and the cosmological and massive horizon independent of magnetic charge $e$.\\\\\n\nNow, let us study the nature of singularity structure at $r=0$. It becomes useful to consider the curvature invariants of the spacetime such as the   Ricci square ($R_{ab}R^{ab}$), and Kretshmann scalars ($R_{abcd}R^{abcd}$)\n\\begin{eqnarray}\n&&\\lim_{r\\to 0}R_{ab}R^{ab}=\\frac{36}{l^2}-\\frac{144M}{e^3}\\left(\\frac{1}{l^2}-\\frac{M}{e^3}+\\frac{ m^2cc_1}{4} \\right)+\\frac{42m^2c^2}{e^2}\\left(\\frac{m^2c^2c_2^2}{e^2}+\\frac{2c_2}{l^2}+\\frac{5c_1^2}{12}+\\frac{c^4c_2^2}{e^2} \\right),\\nonumber\\\\\n&&\\lim_{r\\to 0}R_{abcd}R^{abcd}=\\frac{12}{l^4}-\\frac{48M}{e^3}\\left(\\frac{1}{l^2}+\\frac{M}{e^3}+\\frac{5 m^2 c_2c^2}{6e^2}\\right)+\\frac{7m^4 c^2}{e^2}\\left(c_1^2\n+\\frac{6 c^2c_2^2}{e^2}+\\frac{2e^2c^2c_2^2}{7}+\\frac{4c_2}{m^2l^2}\\right).\\nonumber\\\\\n\\label{RR}\n\\end{eqnarray}\nThese curvature invariants show that the   solution (\\ref{sol2}) is regular everywhere including origin ($r=0$). The singularity of the solution is removed due to the presence of non-linear source (\\ref{matter}). \n\n\\section{\\label{sec:level5} Thermodynamics}\nIn this section, we explore the thermodynamics of the Bardeen-AdS  massive black hole solutions (\\ref{sol2}). The thermodynamical quantities temperature ($T_+$), entropy ($S_+$) and heat capacity ($C_+$) associated with the black hole solutions. The mass of the Bardeen-massive-AdS black hole  is \n\\begin{eqnarray}\nM_+=\\frac{(r_+^{2}+e^{2})^{\\frac{3}{2}}}{2r_+^2}\\left[1+\\frac{r_+^2 }{l^2}+{m^2}\\left(c^2c_2+\\frac{\ncc_1r_+}{2}\\right)\\right]\n\\label{mass}\n\\end{eqnarray}\nand the mass of   AdS Bardeen  black hole \\cite{Tzikas:2018cvs} for $m \\to 0$, one recovers mass of the AdS Schwarzschild black hole from (\\ref{mass}), when  $m =0, e=0$. The temperature of the black hole  associated with it, known as Hawking temperature. Hawking temperature of the black hole can be defined by \n\\begin{eqnarray}\nT=\\frac{\\kappa}{2\\pi}=\\frac{\\sqrt{-\\frac{1}{2}\\nabla_\\mu\\xi_\\nu \\nabla^\\mu \\xi^\\nu}}{2\\pi}=\\frac{1}{4\\pi}\\frac{\\partial{\\sqrt{-g^{rr} g_{tt}}}}{\\partial r}\\mid_{r=r_+},\n\\end{eqnarray}\nwhere $\\kappa$ is surface gravity, $\\xi^\\mu= \\partial^\\mu_t$ is a Killing vector for static spherically symmetric case. Now, on inserting the temperature $T_+$ associated with the Bardeen massive black hole can  be calculated as\n \\begin{eqnarray}\nT_+ =\\frac{1}{4\\pi r_+( e^2+r_+^{2})}\\left[{r_+^2- e^2+\\frac{m^2}{2}(c_2c^2r_+(2r^2_+-e^2)+2r_+c c_1(r_+^2-2e^2))}+\\frac{3r_+^2}{l^2}\\right],\n\\label{temp1}\n\\end{eqnarray}\nand the  AdS Bardeen  black hole \\cite{Tzikas:2018cvs}, when $m \\to 0$, one recovers from (\\ref{temp1}) the temperature of   Bardeen black hole \\cite{dvs19}\n\\begin{equation}\nT_+= \\frac{1}{4\\pi r_+}\\left(\\frac{r_+^2-e^2}{r_+^2+e^2}\\right).\n\\label{eqT1}\n\\end{equation}\nIn turns the temperature reduces to $T_+=1\/4\\pi r_+$ of the  Schwarzschild  black holes, can be obtained by taking   $e=0$ in (\\ref{eqT1}). The Fig. \\ref{fig:1} displays the behaviour of temperature for various values of $e$ and $m$, which  shows that the temperature grows to a maximum $T_{max}$ then it drop to minimum value and increase again. The maximum temperature $T_{max}$ depends on charge $e$ and is shown in the Table \\ref{tab:temp}.  Thus a remarkable result is that, unlike the  Schwarzschild black hole, for the  radius corresponding to the  temperature of Bardeen AdS massive black hole  increases with $e$ and decreses with the mass parameter $m$(See in the Fig. \\ref{fig1}).  A maximum of  Hawking temperature occurs at the critical radius shown in Table \\ref{tab:temp}. It turns out that the maximum temperature decreases with increase in the values of $e$ and $m$ which diverges when the horizon radius shrinks to zero.\n\\begin{figure*}[ht]\n\\begin{tabular}{c c c c}\n\\includegraphics[width=.55\\linewidth]{t1.eps}\n\\includegraphics[width=.55\\linewidth]{t2.eps}\\\\\n\\end{tabular}\n\\caption{The plot of temperature vs horizon radius $r_+$ with different value of magnetic charge $e$ (Left) and different value of mass parameter $m$ (Right) with fixed value of $c=1,c_1=-1, c_2=1$ and $l=20$. }\n\\label{fig:1}\n\\end{figure*}\n\n\\begin{center}\n\t\\begin{table}[h]\n\t\t\\begin{center}\n\t\t\t\\begin{tabular}{l|l r l r l| r l r r r}\n\t\t\t\t\\hline\n\t\t\t\t\\hline\n\t\t\t\t\\multicolumn{1}{c}{ }&\\multicolumn{1}{c}{ }&\\multicolumn{1}{c}{ }&\\multicolumn{1}{c}{m= fixed  }&\\multicolumn{1}{c}{ \\,\\,\\,\\,\\,\\, }&\\multicolumn{1}{c|}{ }&\\multicolumn{1}{c}{  }&\\multicolumn{1}{c}{ }&\\multicolumn{1}{c}{e=fixed}\\,\\,\\,\\,\\,\\,\\\\\n\t\t\t\t\\hline\n\t\t\t\t\\multicolumn{1}{c|}{ } &\\multicolumn{1}{c}{e= 0.55 } &\\multicolumn{1}{c}{ 0.57 } & \\multicolumn{1}{c}{ 0.592 }& \\multicolumn{1}{c}{0.63}& \\multicolumn{1}{c|}{} &\\multicolumn{1}{c}{m= 0.25}&\\multicolumn{1}{c}{0.50} &\\multicolumn{1}{c}{ 0.75}   & \\multicolumn{1}{c}{1.0}& \\multicolumn{1}{c}{} \\\\\n\t\t\t\t\\hline\n\t\t\t\t$r_c^T$ & 1.485& 1.531 &1.1578 &1.647 &&1.12\\,\\,\\,&1.11&1.05&1.03&\n\t\t\t\t\\\\\n\t\t\t\t$T_+^{Max}$&0.0296& 0.0281 &0.0266& 0.0243 &&0.0255& 0.0248&0.0235&0.0228&\\\\\n\t\t\t\t\\hline\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}\n\t\t\\end{center}\n\t\t\\caption{Maximum Hawking temperature ($T_+^{max}$) at critical radius ($r_c^{T}$) for the $4D$ AdS Bardeen massive black hole.}\n\t\t\\label{tab:temp}\n\t\\end{table}\n\\end{center}\n\\noindent One can easily obtain the following formula by using the first\nlaw of thermodynamics $dM_+=T_+dS_++\\phi de$ for the entropy of black hole\n\\begin{eqnarray}\\label{sf}\nS_+=\\int T_+^{-1} dM_+=\\int T_+^{-1}\\frac{d M_+}{d r_+}dr_+.\n\\end{eqnarray}\nNow, substituting the value of $M_+$ and $T_+$  form Eqs. (\\ref{mass}), (\\ref{temp1}) into Eq. (\\ref{sf}), we obtained the entropy of Bardeen AdS -massive black hole as\n\\begin{eqnarray}\nS_+&&=\\pi\\Big[(r_+^2-2e^2)(r_+^2+e^2)^{1\/2}+3e^2r_+\\text{log}[r_++(r_+^2+e^2)^{1\/2}]\\Big].\n\\label{entropy}\n\\end{eqnarray}\nThus, the area law is no longer valid for regular black holes.   We can also be derived the temperature according to the entropy using the first law of thermodynamics.\n\nThe  thermodynamic or local stability of a black hole is performed by studying the behaviour of  its heat capacity ($C_+$). The  positive ($C_+>0$) heat capacity indicates that the  black hole is stable; when it is negative ($C_+<0$), the black hole is said to be unstable \\cite{dvs19,kumar19,sabir}.    The heat capacity of the black hole is defined as\n\\begin{eqnarray}\\label{SH}\nC_+&=&\\frac{\\partial{M_+}}{\\partial{T_+}}=\\left(\\frac{\\partial{M_+}}{\\partial{r_+}}\\right)\\left(\\frac{\\partial{r_+}}{\\partial{T_+}}\\right).\n\\end{eqnarray}\n The heat capacity of Bardeen -AdS-massive  black hole, by using Eqs.  (\\ref{mass}), (\\ref{temp1}), and (\\ref{SH}),  we obtained\n\\begin{widetext}\n\\begin{eqnarray}\nC_+&=&\\frac{\\pi (e^2+r_+)^{5\/2}(6 r_+^4+2l^2(2r_+^2-e^2)+m^2l^2(2c^2c_2r_+^2+2cc_1r_+^3-4c^2c_2e^2-cc_1e^2)}{r_+(2e^4l^2+7e^2l^2r_+^2+9e^2r_+^4-l^2r_+^4+3r_+^6)+m^2l^2(2c^2c_2e^4+7c^c_2e^2r_+^2+cc_1e^2r_+^3-c^2c_2r_+^4)},\\nonumber\\\\\n\\label{SH1}\n\\end{eqnarray}\n\\end{widetext}\n\n\\begin{figure}[ht]\n\\begin{tabular}{c c c c}\n\\includegraphics[width=.5\\linewidth]{c1.eps}\n\\includegraphics[width=.5\\linewidth]{c2.eps}\\\\\n\\includegraphics[width=.5\\linewidth]{c3.eps}\n\\includegraphics[width=.5\\linewidth]{c4.eps}\n\\end{tabular}\n\\caption{The plot of specific heat vs horizon radius  with fixed value of $c=1,c_1=-1, c_2=1, e=0.40$ and $l=20$. }\n\\label{fig:1}\n\\end{figure}\n\nIt can be seen clearly, that the heat capacity depends on the charge ${e}$,  mass parameter $m$, and  cosmological constant $\\Lambda$.   The heat capacity is plotted  in Fig. \\ref{fig:1} for different values of $e$. The  heat capacity is positive (negative) $r_+<r_C$ $ (r_+>r_C) $ suggesting  thermodynamic stability (instability) of smaller (larger) regular black holes. The heat capacity is discontinuous  at $r_+=r_C$ which means the second order phase transition occurs  \\cite{hp,dvs99}. Interestingly, the discontinuity of the heat capacity occurs at $r_+=1.439$, point at which point  the Hawking temperature has the  maximum value $T_+=0.0296$ for  $e=0.55$. Hence the phase transition occurs from the lower to higher mass black holes corresponding from positive to negative heat capacity of the black hole.  In the absence of  mass parameter $m$, it reduce to the expression for heat capacity of  Bardeen  black hole  \\cite{dvs19}.    The  heat capacity for Schwarzschild  black hole $C_{+} =-2\\pi r^{3}_+.$  in the limit $m=0$ .\n\n\n\\section{\\label{sec:level6} Thermodynamic Stability and Phase Diagrams}\n\\noindent Now we study the phase transition of the Bardeen AdS massive black hole by using the P-V criticality and phase diagrams.  The thermodynamical relation between the pressure and cosmological constant given as\n\\begin{equation}\nP=-\\frac{\\Lambda}{8\\pi}=\\frac{3}{8\\pi l^2}.\n\\label{lpre}\n\\end{equation}\n  The mass of  the black hole interpretes  enthalpy of thermodynamical system. The equation of state can by obtained by using the Eq. (\\ref{temp1})  and  Eq. (\\ref{lpre}) as\n\\begin{eqnarray}\nP_+=\\frac{1}{2r_+ T_+}\\left(1+\\frac{e^2}{r_+^2}\\right)-\\frac{1}{4\\pi r_+^2}\\left(1-\\frac{e^2}{2r_+^2}\\right)+\\frac{m^2}{16\\pi r_+^4}\\left(cc_1r_+(e^2-2r_+^2)+2c^2c_2(r_+^2-2e^2)\\right),\\nonumber\\\\\n\\label{pv4}\n\\end{eqnarray}\nand the corresponding thermodynamics volume is\n\\begin{eqnarray}\n V=\\left(\\frac{\\partial H_+}{\\partial P_+}\\right)_{S_+,e}= \\frac{4\\pi}{3}{\\left(r_+^{2}+e^{2}\\right)^{\\frac{1}{2}}}\n\\label{pv3}\n\\end{eqnarray}\nAt the  inflection point, we can determine the critical temperature and critical  pressure b following equations \n\\begin{equation}\n\\frac{\\partial P_+}{\\partial r_+}=0,\\qquad \\qquad \\frac{\\partial ^2P_+}{\\partial r_+^2}=0.\n\\label{pv5}\n\\end{equation}\nSubstitution Eq.  (\\ref{pv4}) into Eq.  (\\ref{pv5}), we find the critical points and  the  horizon radius satisfy the following equation\n\\begin{eqnarray}\n\\frac{24e^2+30e^2r_+^2-2r_+^4+m^2[c_2c^2(24e^4+30r_+^2e^2-2r_+^4)+9cc_1e^2r_+^3]}{128\\pi^3r_+^{8}(e^2+r_+^2)(3e^2+r_+^2)}.\n\\label{pv6}\n\\end{eqnarray}\nThe  Eq. (\\ref{pv6}) can not be solved analytically, so we can calculate the critical radius $r_+$, critical pressure $P_+$ and temperature  $T_+$  numerically and the numerical results are presented in  Table \\ref{tr10} and  Table \\ref{tr12} for different value of mass parameter $m$ and magnetic charge $e$.  We can see that the  critical  radius $r_+$ increases with magnetic charge $e$   and decrease with the mass parameter $m$. The universal ratio $P_cr_c\/T_c$ are increasing function of the magnetic charge $e$ and  massive parameter $m$. \n\\begin{table}[ht]\n \\begin{center}\n \\begin{tabular}{ l | l   | l   | l   | l   }\n\\hline\n            \\hline\n  \\multicolumn{1}{c|}{ $m$} &\\multicolumn{1}{c}{$r_C$}  &\\multicolumn{1}{|c|}{$T_C$}  &\\multicolumn{1}{c|}{$P_C$} &\\multicolumn{1}{c}{$\\frac{P_C\\,r_C}{T_C}$}\\\\\n            \\hline\n            \\,\\,\\,\\,\\,0 ~~  &~~0.3970~~  & ~~0.2513~~ & ~~0.1161~ & ~~0.1821~~ \\\\    \n            %\n            \\,\\,\\,\\,\\,1~~ &~~0.3864~~ & ~~0.4425~~ & ~~0.2489~~ &  ~~0.2173~~      \\\\\n            %\n            \\,\\,\\,\\,\\,2~~ &~~0.3803~~  & ~~1.0174~~ & ~~0.6488~~ &  ~~0.2425~~    \\\\\n            %\n            \\,\\,\\,\\,\\,3~~ &~~0.3782~~ & ~~1.9758~~ & ~~1.3158~~ &    ~~0.2518~~  \\\\\n            %\n            \\,\\,\\,\\,\\,4~~ &~~0.3774~~  & ~~3.3178~~ & ~~2.2496~~ &    ~~0.2558~~   \\\\\n            %\n            \\,\\,\\,\\,\\,5~~ &~~0.3770~~  & ~~5.0432~~ & ~~3.4504~~ &    ~~0.2579~~    \\\\\n            \\hline \n\\hline\n        \\end{tabular}\n        \\caption{The table for critical temperature $T_C$, critical pressure $P_C$ and $P_C\\,r_C\/T_C$ corresponding different value of mass $m$ with fixed value of $c=1,c_1=-1, c_2=1,e=0.1$ .}\n\\label{tr10}\n    \\end{center}\n\\end{table}\n\\begin{table}\n \\begin{center}\n \\begin{tabular}{ l | l   | l   | l   |  l   }\n\\hline\n            \\hline\n  \\multicolumn{1}{c|}{ $e$} &\\multicolumn{1}{c|}{$r_C$}  &\\multicolumn{1}{c|}{$T_C$}  &\\multicolumn{1}{c|}{$P_C$} &\\multicolumn{1}{|c}{$P_C\\,r_C\/T_C$}\\\\\n            \\hline\n            \\,\\,\\,\\,\\,0.1 ~~  &~~0.3864~~  & ~~0.4425~~ & ~~0.2489~~ & ~~0.1821~~ \\\\            \n            %\n            \\,\\,\\,\\,\\,0.2~~ &~~0.7524~~ & ~~0.1917~~ & ~~0.0667~~ &  ~~0.2617~~      \\\\\n            %\n            \\,\\,\\,\\,\\,0.3~~ &~~1.099~~  & ~~0.1083~~ & ~~0.0317~~ &  ~~0.3216~~    \\\\\n            %\n            \\,\\,\\,\\,\\,0.4~~ &~~1.427~~ & ~~0.06697~~ & ~~0.0191~~ &    ~~0.4069~  \\\\\n            %\n            \\,\\,\\,\\,\\,0.5~~ &~~1.738~~  & ~~0.04231~~ & ~~0.0131~~ &    ~~0.5381~~   \\\\\n            %\n            \\hline \n\\hline\n        \\end{tabular}\n        \\caption{The table for critical temperature $T_C$, critical pressure $P_c$ and $P_C\\,r_C\/T_c$ corresponding to the different value of  magnetic charge $e$ with fixed value of $c=1,c_1=-1, c_2=1, m=1$ .}\n\\label{tr12}\n    \\end{center}\n\\end{table}\nIn order to obtained the phase transition of the black holes an analog with the vander Walls phase transition, we can identify the free energy of the black hole. to see the effect of magnetic charge $e$ and mass parameter $m$ on the phase structure of the system, we fix the magnetic charge $e$ first and vary the mass parameter $m$. The   critical pressure and  critical temperature  increase  with critical radius (see Fig. \\ref{tr10}) and the opposite nature show when we fix the mass parameter $m$ and vary of  magnetic charge $e$.\n\n In $ T_+-r_+$  plots  we can  see that the three  black hole (small, intermediate and large )  for the $P< P_c$, $P=P_c$ and $P> Pc$ for  Bardeen AdS   massive black hole with  the same magnetic charge $e$ and the mass parameter $m$ for a certain range of temperature. The small and large are stable but intermediate black hole is unstable, since the heat capacity is negative (see  Fig. \\ref{fig:1}). When $ T_+<T_{\\star} $ the small black hole and $ T_+>T_{\\star} $  corresponding to large black hole due to small free energy.  We can transit one phase to other phase at critical temperature due to same free energy.  In    Fig. \\ref{fig1} (middle)  isotherms represents the first order $T_+<T_c$ and second order  phase transition $T_+=T_{\\star}$, which is obtained from the free energy diagram (see Fig. \\ref{fig1} (lower)  and  the corresponding temperature is $T_{\\star}=0.335$.  In $G_+ - T_+$  plots the appearance of characteristic swallow tail in   show that the obtained values are critical ones in which the phase transition take place .  In  Fig. \\ref{fig1} (lower) , we can see that swallow tail shape $P<P_c$,  for the first order phase transition and $P=P_c=0.0249$ for the second order phase transition.\n\\begin{figure}[ht]\n\\begin{tabular}{c c c c}\n\\includegraphics[width=.5\\linewidth]{p1.eps}\n\\includegraphics[width=.5\\linewidth]{p3.eps}\\\\\n\\includegraphics[width=.5\\linewidth]{t11.eps}\n\\includegraphics[width=.5\\linewidth]{t13.eps}\\\\\n\\includegraphics[width=.49\\linewidth]{g1.eps}\n\\includegraphics[width=.49\\linewidth]{g3.eps}\n\\end{tabular}\n\\caption{The plots of pressure and temperature vs horizon radius  and Gibbs free energy ($G_+$) vs temperature ($T_+$)  for $e=0.1$ (left) and $e=0.5$ (right) with fixed values of $c=1,c_1=-0.75,c_2=0.75$ and $m=1$ .}\n\\label{fig1}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\begin{tabular}{c c c c}\n\\includegraphics[width=.5\\linewidth]{p2.eps}\n\\includegraphics[width=.5\\linewidth]{p4.eps}\\\\\n\\includegraphics[width=.5\\linewidth]{t12.eps}\n\\includegraphics[width=.5\\linewidth]{t14.eps}\\\\\n\\includegraphics[width=.5\\linewidth]{g2.eps}\n\\includegraphics[width=.5\\linewidth]{g4.eps}\n\\end{tabular}\n\\caption{The plots of pressure and temperature vs horizon radius  and Gibbs free energy ($G_+$) vs temperature ($T_+$) for $e=0.1$ (left) and $e=0.5$ (right) with fixed values of $c=1,c_1=-0.75,c_2=0.75$ and $m=1$ . }\n\\label{fig2}\n\\end{figure}\n\nFig. 5 shows   the effects of variation of mass parameter $m$ and magnetic charge $e$ on critical values.  The isobars are increase function of mass parameter $m$ and  decreasing with the magnetic charge $e$  respectively. The gap between the two sub-critical bar are the increases      \n mass parameter $m$ as well as  magnetic charge $e$. The sub-critical isobar is the region, where the phase transition take place. The only difference is that Gibbs free energy of two phases are increasing function of massive parameter $m$ and decreasing function of magnetic charge $e$.  Interestingly, the universal relation $P_c\\,r_c\/T_c$, shows that the the effects of variation of mass parameter $m$ and magnetic charge $e$ are decreasing function massive parameter whereas critical temperature   and pressure are increasing function of it. \n\n\n\\section{\\label{sec:level6} Conclusions}\nIn this paper, we have presented exact  solution of Bardeen AdS  black holes in massive gravity with a negative cosmological constant, thereby generalizing bardeen black holes which are included as a special case ($e=0$, $l^2 \\to \\infty$). The AdS Bardeen black holes are characterized by analyzing horizons, which at most could be four, viz. inner Cauchy , event, cosmological  and massive horizons. We have analysed the thermodynamical properties and phase structure of  Bardeen AdS massive  black holes.  Despite complicated solutions,  exact expression for the thermodynamical quantities like the black hole mass, Hawking temperature, entropy and free energy at event horizon $r_+$ are obtained. The thermodynamical stability of the black holes is also analysed by studying the  heat capacity. The entropy (\\ref{entropy}) of the black holes is modified due to the magnetic charge $e$ and the mass parameter $m$ resulting area law $S=A\/4$ no longer valid. The phase transition is detectable by the divergence of the heat capacity ($C_+$) at a critical radius $r_c$ (changes with $e, m$ ), such that the black hole is stable  with positive heat capacity ($C_+>0$), and unstable with negative heat capacity ($C_+<0$).\n\nThe phase transition of  Bardeen AdS massive black holes have also been studied in the extended phase by considering the cosmological constant as thermodynamic pressure.  It is seen  that the thermodynamic volume is independent of  mass parameter $m$ and magnetic charge $e$. The phase diagram shows that the obtained critical values of pressure and temperature are one in which phase transition take place.   Interestingly, it is pointed out that the nature of  massive parameter $m$ and magnetic charge $e$ are opposite to each other. The critical temperature and pressure  were  found highly sensitive to the variation parameters ( m and e).  \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n The defining equation \n\\begin{align}\\label{eq-ast}\n\\dot \\omega_t=-F'(\\omega_t),\n\\end{align}\nof a gradient flow is \\emph{kinetic} in essence. This article explores some \\emph{dynamical} (rather than kinetic) features of gradient flows  (i) by embedding equation \\eqref{eq-ast} into the family of slowed down gradient flow equations: $\\dot \\omega ^{  \\varepsilon}_t=- \\varepsilon F'(\\omega ^{ \\varepsilon}_t),$ where $\\varepsilon>0$, and (ii) by considering   the \\emph{accelerations} $\\ddot \\omega ^{ \\varepsilon}_t$. We shall focus on Wasserstein gradient flows, i.e.\\ gradient flows with respect to the Wasserstein metric on a space of probability measures. Our approach in this article is mainly heuristic, using Otto calculus.\n\nOtto calculus is a powerful tool to understand the geometry of the Wasserstein space on a Riemannian manifold $N$. It offers a heuristic for considering the space $ \\mathcal{P}_2(N)$ of\n probability measures with finite second moments on the manifold, see \\eqref{eq-05},  as an infinite dimensional Riemannian manifold, allowing one to address natural conjectures.  Roughly speaking,  Otto calculus is twofold. Firstly, it leads to the definition of the  Wasserstein metric on $ \\mathcal{P}_2(N)$ whose corresponding squared \n Wasserstein  distance  between two probability measures $\\mu$ and $\\nu$  is given by the Benamou-Brenier formula\n \\begin{align*}\n W_2^2(\\mu,\\nu)=\\inf _{ (( \\mu_s),(v_s))} \\int _{ [0,1]\\times N}|v_s(x)|_{x}^2\\, \\mu_s(dx)\\,ds\n \\end{align*}\nwhere the   infimum is taken over all   $(\\mu_s, v_s) _{ 0\\le s\\le 1}$ such that $(\\mu_s)$ is a trajectory in $ \\mathcal{P}_2(N)$, starting from $\\mu$ and arriving at $\\nu$ and $(v_s)$ is its velocity field, meaning that the transport equation $ \\partial_s \\mu+ \\mathrm{div}\\, (\\mu v)=0$ is satisfied. \n\\\\\nDenoting the squared length  of the velocity $\\dot\\mu=v$ by \n\\begin{align}\\label{eq-96}\n|\\dot\\mu_s|_{\\mu_s}^2:=\\inf\\left\\{\\int_N |v|^2\\, d\\mu_s ;v:  \\partial_s \\mu+ \\mathrm{div}\\, (\\mu_s v)=0\\right\\},\n\\end{align} \nwe obtain the Riemannian distance like formula\n \\begin{align}\\label{eq-01}\n W_2^2(\\mu,\\nu)\n \t=\\inf _{ ( \\mu_s)} \\int_0^1|\\dot\\mu_s|_{\\mu_s}^2\\, ds.\n \\end{align}\nThis provides us with natural definitions on $ \\mathcal{P}_2(N)$ of geodesics, gradients,  Hessians and so on. We call  the squared distance $W_2^2$ the \\emph{Wasserstein cost}.\n\n \nSecondly, it appears that several PDEs whose solutions $(\\mu_t)_{t\\ge 0}$ are flows of probability  measures, are gradient flows  with respect to the Wasserstein metric, of some function $\\cF$:\n\\begin{equation}\n\\label{eq-90}\n\\dot\\mu_t=- \\ {\\rm grad}_{\\mu_t}\\cF,\n\\end{equation}\nwhere  the velocity $\\dot\\mu_t$  and  the gradient of $\\cF$ are understood with respect to the Wasserstein metric. \nFor instance it is well known since \\cite{jko1998} that  the heat equation is the  gradient flow  with respect to the Wasserstein metric of the usual entropy\n\\begin{equation}\\label{eq-02}\n {\\mathcal En}(\\mu):=\\int_N \\mu\\log \\mu \\,d\\!\\vol  .\n\\end{equation}\nWe introduce a cost function which is a perturbed version of the  Wasserstein cost $W_2^2$. It is defined for any regular function $\\cF$ on the set of probability measures, any   $\\ep\\ge 0$ and any  probability measures $\\mu,\\nu$ on the manifold, by \n$$\n \\mathcal{A} ^{ \\ep} _{ \\mathcal{F}}(\\mu,\\nu)\n\t:=\\inf _{ (\\mu_s)} \\int_0^1\\Big(\\frac{1}{2}|\\dot\\mu_s|_{\\mu_s}^2+\\frac{\\ep^2}{2}|{\\rm grad}_{\\mu_s} \\cF|^2_{\\mu_s}\\Big)ds,\n$$ \nwhere as in \\eqref{eq-01} the infimum runs through all paths   $(\\mu_s) _{ 0\\le s\\le 1}$ in $ \\mathcal{P}_2(N)$ from $\\mu$ to $\\nu$. \nRemark that this family of cost functions embeds the Wasserstein cost \n$W_2^2=\\cA_\\cF ^{ \\ep=0}$ as a specific limiting case, see \\eqref{eq-01}.\nThis paper investigates  basic properties of $ \\mathcal{A} ^{ \\ep} _{ \\mathcal{F}}$ and of its minimizers which are called $ \\ep\\mathcal{F}$-interpolations. It also  provides heuristic results which extend to  $ \\mathcal{A} ^{ \\ep} _{ \\mathcal{F}}$\n several known theorems about the optimal transport cost $W^2_2$ and the convexity properties of $ \\mathcal{F}.$ \nThe main motivation for introducing $ \\mathcal{A} ^{ \\ep} _{ \\mathcal{F}}$ is that the gradient flow solving\n\\begin{equation}\\label{eq-90b}\n\\dot\\mu_t=- \\ep\\ {\\rm grad}_{\\mu_t}\\cF,\n\\end{equation}\nis naturally associated to the Lagrangian $|\\dot\\mu |_{\\mu}^2\/2+{\\ep^2}|{\\rm grad}_{\\mu} \\cF|^2_{\\mu}\/2.$ Indeed, any solution of \\eqref{eq-90b} and any $ \\ep \\mathcal{F}$-interpolation  satisfies the same Newton equation\n$$\n\\ddot\\mu_s=\\frac{\\ep^2}{2}{\\rm grad}_{\\mu_s}|{\\rm grad}_{\\mu_s} \\cF|^2_{\\mu_s}\n$$\nwhere $\\ddot\\mu$ denotes the acceleration  with respect to the Wasserstein metric and a Wasserstein version of the Levi-Civita connection.\nWhen $ \\ep=1$ this is the equation of motion of the gradient flow \\eqref{eq-90}, while when $ \\ep=0,$ this is the equation of the  free motion in the Wasserstein space characterizing McCann's displacement interpolations.\n\n\n\n\n\n\n Let us quote some of our results.\n\\begin{itemize}\n\\item  We denote the solution of \\eqref{eq-90} with initial state $ \\mu_0$ by the semigroup notation: $\\mu_t=\\mathsf{S} ^{ \\mathcal{F}}_t(\\mu_0)$.  The cost $ \\mathcal{A} ^{ \\ep} _{ \\mathcal{F}}$ satisfies the same  contraction inequalities along the gradient flow $(\\mathsf{S} ^{ \\mathcal{F}}_t)$ as the one satisfied by the Wasserstein cost. The simplest contraction result states that, under a nonnegative Ricci curvature condition,   \n$$\n \\mathcal{A} ^{ \\ep} _{ \\mathcal{F}}(\\mathsf{S} ^{ \\mathcal{F}}_t\\mu,\\mathsf{S} ^{ \\mathcal{F}}_t\\nu)\\le   \\mathcal{A} ^{ \\ep} _{ \\mathcal{F}}(\\nu,\\mu), \n$$\nfor any $t\\ge 0$ and any probability measures $\\mu,\\nu$. This extends the well-known result by von Renesse and Sturm  for the Wasserstein cost \\cite{renesse-sturm2005}.\n\\item Newton's equation satisfied by the $ \\ep\\mathcal{F}$-interpolations allows us to prove convexity properties of $\\cF$ along $ \\ep\\cF$-interpolations.  For instance, when the Ricci curvature is nonnegative,   $\\cF$ is convex along the $ \\ep\\cF$-interpolations. This generalizes  McCann's result about the Wasserstein cost \\cite{mccann1995}.\n\\end{itemize}\nIt is remarkable that, similarly to the Wasserstein cost,  $ \\mathcal{A} ^{ \\ep} _{ \\mathcal{F}}$ behaves pretty well  in presence of the Bakry-\\'Emery curvature-dimension  condition, see Section \\ref{sec-ineq}.\n\n\\medskip\n\\noindent \\emph{Schr\u00f6dinger problem.}\\ \nA particular and fundamental case is when $\\cF$ is the standard entropy ${\\mathcal En}$, see \\eqref{eq-02}. The associated cost $ \\mathcal{A} ^{ \\ep} _{\\entro}$ is related to the Schr\u00f6dinger  problem by a  Benamou-Brenier formula, see Section \\ref{sec-schrodinger}. It is identical up to an additive constant to an entropy minimization problem on the path space. The Newton equation satisfied by the $ \\ep{\\mathcal En}$-interpolations is a recent result by  Conforti \\cite{conforti2017}. It is a keystone \non this field. \n\\\\\nUnlike the remainder of this article, our results about the Schr\u00f6dinger problem are rigorous. In particular we prove that the entropic cost satisfies a general contraction inequality for the heat\nsemigroup $(\\ss_t^{\\mathcal En})_{t\\ge 0}$ under the assumption that the Ricci curvature is bounded from below by $\\rho$ in an $n$-dimensional Riemannian manifold: For any $t\\ge 0$ and any  \n probability measures $\\mu,\\nu$, \n$$\n\\mathcal A_{{\\mathcal En}}^\\varepsilon(\\ss_t^{\\mathcal En}\\mu,\\ss_t^{\\mathcal En}\\nu)\n\t\\le  e^{-2\\rho t}\\mathcal A_{{\\mathcal En}}^\\varepsilon(\\mu,\\nu)\n\t-\\frac{1}{n}\\int_0^te^{-2\\rho(t-u)}({\\mathcal En}(\\ss_u^{\\mathcal En}\\mu)-{\\mathcal En}(\\ss_u^{\\mathcal En}\\nu))^2du.\n$$\n\\medskip\n\nThe paper is organized as follows. In  next Section \\ref{sec-1} we treat the finite dimensional case where the state space is $\\R^n$ equipped with the Euclidean metric. In this case, we are able to do explicitly all the computations using classical differential calculus. \nIn Section~\\ref{sec-newton} we treat the infinite \ndimensional case using the Otto calculus:  we start recalling in a simple way the Otto calculus, then a heuristic derivation of  the Newton equation is presented. Convexity properties are explored at Section~\\ref{sec-ineq}. Finally in Section~\\ref{sec-schrodinger},  the special case of the Schr\u00f6dinger problem is investigated. \n\n\n\\bigskip\n\nWe have to mention again that, except for  Section~\\ref{sec-schrodinger}, all the results  in this paper are heuristic, even if we believe that there is a way to prove them rigorously.  Heuristic results are denoted with quotation marks.  Some parts of the paper are related to other mathematical domains such as Euler equations and mean field games ; we tried to extract references to known results from the large related literature. In order to propose  a comprehensive document, we do not provide the detailed proofs in the finite dimensional case at Section~\\ref{sec-1}. \n\n\n\n\\section{Warm up in $\\R^n$}\n\\label{sec-1}\n\n\n\nIn  this section, $F:\\R^n\\to\\mathbf{R}$ is a smooth function ($\\mathcal C^\\infty$) with its first derivative (gradient) and second derivative (Hessian) denoted by $F'$ and $F''$. All the  results about gradient flows which are stated below are well known, see for instance~\\cite{daneri-savare2008}.\n\n\n\\subsection{Gradient flows in $\\R^n$}\n\n The equation of a gradient flow: $[0, \\infty)\\ni t\\mapsto \\omega_t\\in\\R^n$  is\n\\begin{equation}\\label{eq-04}\n\\dot \\omega _t= -  F'(\\omega _t),\\quad t\\ge  0,\n\\end{equation}\nwhere  $\\dot \\omega_t$ is the time derivative at time $t$ of the path $ \\omega$. This evolution equation   makes sense in a  Riemannian manifold if it is replaced by \n$\n\\dot \\omega _t=-{\\rm grad}_{\\omega _t} F.\n$\n\\begin{rem}\n\\label{rem-1}\nWhen $F$ is $ \\rho$-convex for some $ \\rho\\in\\R,$ that is\n\\begin{equation}\\label{eq-11}\nF''\\ge  \\rho \\mathrm{Id}\n\\end{equation}\nin the sense of quadratic forms,  there exists a unique  solution of \\eqref{eq-04} for any initial state. \n\\end{rem}\n\n\n\\begin{defi}[Semigroup]\nFor any $x\\in\\R^n$, we denote \n$$\n\\ss_t(x):= \\omega_t^x,\\qquad t\\ge 0, x\\in\\R^n,\n$$ \nwhere  $(\\omega_t^x)_{t\\ge  0}$ is the solution of~\\eqref{eq-04} starting form $x$.  \\\\\nFrom Remark~\\ref{rem-1}, it follows that $(\\ss_t)_{t\\ge 0}$ defines a semigroup, i.e.:\n\t$\\ss _{ s+t}(x)=\\ss_t\\big(\\ss_s(x)\\big)$  and $\\ss_0(x)=x$, for all $s,t\\ge  0,$ and all $x\\in\\R^n.$ This semigroup is called the gradient flow of $F$ (with respect to the Euclidean metric).\n\\end{defi}\n\nFor any $t$, $x\\mapsto \\ss_t(x)$ is continuously differentiable.\n\n\n\n\\subsubsection*{Equilibrium state}\nAny critical point $\\hat x$ of $F$ is an equilibrium of \\eqref{eq-04} since $F'(\\hat x)=0$ implies that $\\dot \\omega _t=0$ for all $t$ as soon as $\\omega _0=\\hat x.$  If $F''(\\hat x)>0$ (in the sense of quadratic forms), then $\\hat x$ is a stable equilibrium, while when $F''(\\hat x)<0,$ it is unstable. See Figure \\ref{fig-01} for an illustration in dimension one. In the multidimensional case $F''(\\hat x)$ may admit both stable and unstable directions.\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=12cm]{GF-a.pdf}\n\\caption{Graphical representations of $F$ and $-F'$}\\label{fig-01}\n\\end{center}\n\\end{figure}\n\n\\subsubsection*{Two metric formulations of gradient flows}\n\nThe evolution equation \\eqref{eq-04} only makes sense in presence  of a  differential geometric structure. It is worth rephrasing \\eqref{eq-04} in terms allowing a natural extension to a metric setting. The main idea is to express everything with the scalar quantities $|\\dot \\omega _t|$ and $|F'(\\omega _t)|$ which admit metric analogues. \n\n\\begin{prop}\nThe gradient flow equation \\eqref{eq-04} is equivalent to\n\\begin{equation*\n\\frac{1}{2}\n \\int_s^t \\Big(|\\dot \\omega _r|^2+  |F'(\\omega _r)|^2\\Big)\\, dr\n\t\\le  F(\\omega _s)-F(\\omega _t),\n\t\\quad \\forall \\;0\\le  s\\le  t.\n\\end{equation*}\nIn such case, the curve $\\omega $ is said to be of \\emph{maximal slope} with respect to $F$.\n\\end{prop}\n\n\n\n\\begin{theo}[Evolution variational inequality]\nLet $F$ be a twice differentiable function on $\\R^n.$\n\\begin{enumerate}[(a)]\n\\item\nThe  function $F$ is $\\rho$-convex if and only if \n\\begin{equation}\\label{eq-10a}\n\\frac{d}{dt} \\frac{1}{2} |y-\\ss _t(x)|^2+ \\frac{\\rho}{2}|y-\\ss _t(x)|^2\\le F(y)-F(\\ss _t(x)),\\quad \\forall t\\ge  0, \\forall x,y.\n\\end{equation}\n\n\\item\nIf $ \\omega$ is a $ \\mathcal{C}^1$ path satisfying \n\\begin{equation}\\label{eq-10}\n\\frac{d}{dt} \\frac{1}{2} |y- \\omega _t|^2+ \\frac{\\rho}{2}|y- \\omega _t|^2\\le F(y)-F(\\omega _t),\n\\quad\\forall t\\ge  0, \\forall y,\n\\end{equation}\n then it solves \\eqref{eq-04}.\n\n\\item\nThe function $F$ is $\\rho$-convex if and only if there exists a semigroup $ (\\mathsf{T}_t)$ such that for any $x$, $ \\mathsf{T}_t(x)$ is $t$-differentiable and\n\\begin{equation}\\label{eq-10b}\n\\frac{d}{dt}_{ |t=0^+}  \\frac{1}{2}|y- \\mathsf{T} _t(x)|^2 + \\frac{\\rho}{2}|y- x|^2\\le F(y)-F(x),\\quad  \\forall x,y.\n\\end{equation}\nIn this case, $ \\mathsf{T}=\\ss$.\n\\end{enumerate}\n\\end{theo}\n\n\nThe evolution variational inequality~\\eqref{eq-10} (EVI in short) is a key inequality for extending the notion of gradient flow to  general  spaces. For instance, it leads to the definition of a gradient flow in a geodesic space, see \\cite[Definition~23.7]{villani2009}. This is a reason why many research papers  focus on  EVI.\n\n\n\n\n\\subsection{The cost $A_F^\\ep$ and the $\\ep F$-interpolations in $\\R^n$}\n\nIn this section we see that gradient flows are special solutions of some Hamilton evolution equations and  the related action minimizing problem is considered. \n\n\n\n\\subsubsection*{Free energy and Fisher information}\nTo draw an analogy with the infinite dimensional setting to be explored later on, where $\\R^n$ will be replaced by the state space $ \\mathcal{P}_2(N)$   consisting of all probability measures with a finite second moment on some configuration space described by  a Riemannian manifold $N$, $\\R^n$ should be interpreted as the state space (not to be confused with the configuration space) and the function $F$  as the ``free energy'' of the system.\nWith this analogy in mind, we define the ``Fisher information''  $I$ by\n\\begin{equation*}\nI:=|F'|^2, \n\\end{equation*}\nwhich appears as minus  the  free energy production along the gradient flow \\eqref{eq-04} since for any $t\\ge  0,$\n\\begin{equation}\\label{eq-06}\n\\frac{d}{dt} F(\\omega _t)= F'(\\omega _t)\\cdot \\dot \\omega _t=-|F'(\\omega _t)|^2=-I(\\omega _t).\n\\end{equation}\nAs $I\\ge  0,$ this implies that $t\\mapsto F(\\omega _t)$  decreases as time passes. In mathematical terms: $F$ is a Lyapunov function of the system, while with a statistical point of view, this property is an avatar of the second principle of thermodynamics.  \n\n\n\\subsubsection*{Gradient flows as solutions of a Newton equation}\n\nLet us introduce a parameter $ \\ep\\ge 0.$\nFor any $x\\in\\R^n$, the path\n\\begin{equation*}\n\\omega_t ^{  \\ep}:=\\ss_{\\ep t}(x)\n\\end{equation*} satisfies  the evolution equation\n\\begin{equation*}\n\\dot \\omega _t ^{  \\ep}=- { \\ep}F'( \\omega _t ^{ \\ep}),\n\\quad t\\ge  0.\n\\end{equation*}\nWhen $ \\ep$ is small, this corresponds to a \\emph{slowing down} of \\eqref{eq-04}. The acceleration of its solution is given by\n$\n\\ddot \\omega ^{ \\ep} _t=- \\ep\\,F''(\\omega ^{ \\ep} _t)\\cdot \\dot \\omega ^{ \\ep} _t= \\ep^2\\,F'' F'(\\omega ^{ \\ep} _t).\n$\nHence $ \\omega ^{ \\ep}$ satisfies\n the Newton equation\n\\begin{equation}\\label{eq-17b}\n\\ddot \\omega ^{ \\ep} =- U ^{ \\ep\\prime}(\\omega ^{ \\ep} )\n\\end{equation} \nwith the scalar potential \n\\begin{equation*}\nU^ \\ep:=- \\frac{ \\ep^2}{2}I=-\\frac{ \\ep^2}{2}|F'|^2.\n\\end{equation*}\nA graphical representation of $U ^{ \\ep=1}=-|F'|^2\/2$ corresponding to the free energy of Figure~\\ref{fig-01} is given at Figure~\\ref{fig-02}. \n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=6cm]{GF-b.pdf}\n\\caption{Graphical representation of $U=-|F'|^2\/2$}\\label{fig-02}\n\\end{center}\n\\end{figure}\n\n\n\\subsubsection*{$ \\ep F$-interpolations}\n\nIt is tempting to investigate the dynamical properties of the trajectories solving the corresponding Hamilton minimization problem. It is associated with the Lagrangian\n\\begin{equation}\\label{eq-03}\nL^{ \\ep}(q,v)=\\frac{|v|^2}{2}-  U ^{ \\ep}(q)=\\frac{|v|^2}{2}+ \\frac{\\ep^2}{2} |F'(q)|^2.\n\\end{equation}\nThe associated Hamilton minimization principle is expressed below at \\eqref{eq-def-A-1}.\n\n\\begin{defi}[$ \\ep F$-cost, $ \\ep F$-interpolations]\n\\label{def-inter-F}\nFor any $x,y\\in\\R^n$ and $\\ep\\ge 0$, we define \n\\begin{equation}\n\\label{eq-def-A-1}\nA_F^\\ep(x,y)=\\inf _{ \\omega\\in\\OO ^{ xy}}\\int_0^1 \\Big(\\frac12|\\dot\\omega _s|^2+ \\frac{ \\ep^2}{2}|F'(\\omega_s)|^2\\Big)\\,ds,\n\\end{equation}\nwhere the infimum runs through all the subset $\\OO ^{ xy}$ of all paths   starting from $x$ and arriving at $y$. \nWe call $A ^ \\ep_F(x,y)$   the $ \\ep F$-cost between $x$ and $y$, and any minimizer $\\omega ^{ \\ep, xy}$ of   \\eqref{eq-def-A-1} is called an $ \\ep F$-interpolation between $x$ and $y$. \n\\end{defi}\n\n\n\\begin{itemize}\n\\item\nRemark that $A_ F^ \\ep=A _{ \\ep F}^1.$ Therefore, the above definitions of the $ \\ep F$-cost and  $ \\ep F$-interpolation could also be called  $( \\ep, F)$-cost and $( \\ep, F)$-interpolation.\n\n\\item\nFor any $x,y\\in\\R^n$, $A_F^\\ep(x,y)\\ge 0$ but unless $ \\ep=0,$ it is not a squared distance on $\\R^n$. For instance whenever $F'(x)\\neq 0$, we see that  $A_F^\\ep(x,x)>0$. \n\\end{itemize}\n\nA key remark about this article is the following\n\\begin{rem}[Gradient flows are $ \\ep F$-interpolations]\n\\label{rem-uniqueness}\nSuppose that for any $\\varepsilon>0$ and all $x,y$ there exists a unique $ \\ep F$-interpolation  $ \\omega ^{ \\ep,xy}$. Then, for any $x$, the $ \\ep F$-interpolation between $x$ and $y=\\ss _{ \\ep} x$  is\n$$\n \\omega_s^{ \\ep,x\\ss _{ \\ep} x}=\\ss_{\\varepsilon s}x, \\quad 0\\le s\\le1.\n$$\nIn other words, the $ \\ep F$-interpolation matches with the gradient flow when the endpoint is well appropriate. The reason is clear from of the uniqueness of the minimizer and the fact that the map $s\\mapsto \\ss_{\\varepsilon s}x$ satisfies  Newton's equation~\\eqref{eq-17b}. \n\\end{rem} \nThe Hamiltonian corresponding to the Lagrangian \\eqref{eq-03} is\n\\begin{align*}\nH ^{ \\ep}(q,p)=\\frac{|p|^2}{2}+  U ^{ \\ep}(q)=\\frac{|p|^2}{2}- \\frac{\\ep^2}{2} |F'(q)|^2,\n\\end{align*}\nand the equation of motion of any minimizer of \\eqref{eq-def-A-1}, i.e.\\ any $ \\ep F$-interpolation $ \\omega$, is given by the Hamilton system of equations:\n$ \\displaystyle{\n\\left\\{ \\begin{array}{lcl}\n\\dot \\omega_s&=& p_s,\\\\\n\\dot p_s&=&-U ^{ \\ep\\prime}( \\omega_s),\n\\end{array}\\right.\n}\\  0\\le s\\le 1.$\n\n\n\n\n\n\\begin{prop}[Properties of the $ \\ep F$-interpolations]\n\\label{res-01}\\ \n\\begin{enumerate}[(a)]\n\\item\nFor any $x,y\\in\\R^n$ and any $ \\ep\\ge 0,$ the minimization problem~\\eqref{eq-def-A-1} admits at least a solution $ \\omega ^{ \\ep,xy}.$ \n\\item\nFurthermore, $\\lime  \\omega ^{ \\ep,xy}= \\omega ^{ xy}$ pointwise, where $ \\omega ^{ xy}$ is the constant speed geodesic from $x$ to $y$.\n\\item Any $ \\ep F$-interpolation  is $\\cC^2$ and satisfies the Newton equation\n\\begin{equation}\\label{eq-17}\n\\ddot \\omega _s ^{ \\ep,xy}= \\frac{\\ep^2}{2} I'( \\omega _s ^{ \\ep,xy})\n= \\ep^2 F''F'( \\omega ^ {\\ep,xy}_s),\n\\end{equation}\nwhich is also \\eqref{eq-17b}.\n\\item\nAlong any $ \\ep F$-interpolation $ \\omega$,  the Hamiltonian is conserved as a function of time:\n\\begin{align*}\nH( \\omega_s,\\dot \\omega_s)\n\t=\\frac{|\\dot \\omega_s|^2}{2}- \\frac{\\ep^2}{2} |F'( \\omega_s)|^2\n\t=H( \\omega_0,\\dot \\omega_0),\n\\quad \\forall 0\\le s\\le 1.\n\\end{align*}\n\\end{enumerate}\n\\end{prop}\n\n\\begin{itemize}\n\\item\n As already noticed at Remark \\ref{rem-uniqueness}, the path $(\\ss_{\\ep s}(x))_{0\\le s\\le 1}$ is an $\\ep F$-interpolation between $x$ and $\\ss_\\ep(x)$ satisfying Newton's  equation \\eqref{eq-17} and one immediately sees with the very definition \\eqref{eq-04} of its evolution   that the  conserved value of the Hamiltonian along it is zero.\n\\item\nWhen $ \\ep=0,$ the $\\ep F$-interpolation is the standard constant speed geodesic between $x$ and $y$. \nFor any small $ \\ep>0$, one can think of the $\\ep F$-interpolation $ \\omega ^{ \\ep,xy}$ as a small perturbation of the geodesic.\n\\end{itemize}\n\n \\bigskip\\noindent \\textbf{Dual formulation of the cost $A_F^\\ep$.}\\ \n\\medskip\n\nLet  $h:\\R^n\\mapsto \\R$ be a Lipschitz function and let $\\ep>0$. We define the Hamilton-Jacobi semigroup for any $t\\geq0$ and $y\\in\\R^n$ by, \n\\begin{equation}\n\\label{eq-93}\nQ_t^{\\ep F} h(y)= \\inf _{ \\omega: \\omega_t=y} \\left\\{ h(\\omega_0)+\\int_0^t \\frac{|\\dot \\omega_s|^2}{2} +\\frac{\\ep^2}{2}| F'(\\omega_s)|^2 ds\\right\\},\n\\end{equation}\nwhere the infimum is running over all $\\cC^1$ path $\\omega$ such that $\\omega_t=y$.  \nThe function $U:(t,y)\\mapsto Q_t^{\\ep F} h(y)$, satisfies, in the sense of viscosity solutions,  the Hamilton-Jacobi equation \n\\begin{equation}\n\\label{eq-933}\n\\left\\{\n\\begin{array}{l}\n\\displaystyle\\partial_t U(t,y)+\\frac{1}{2}|U'(t,y)|^2=\\frac{\\ep^2}{2}|F'(y)|^2;\\\\\nU(0,\\cdot)=h(\\cdot).\n\\end{array}\n\\right.\n\\end{equation}\nMinimizers of~\\eqref{eq-93} are  solutions of the system, \n\\begin{equation}\n\\label{eq-94}\n\\left\\{\n\\begin{array}{ll}\n\\ddot \\omega_s=\\big[\\frac{\\ep^2}{2}  | F'(\\omega_s)|^2\\big]', \\quad 0\\leq s\\leq t,\\\\ \\\\\n\\displaystyle\\dot \\omega_0= h'(\\omega_0),\\quad \\omega_t=y.\n\\end{array}\\right.\n\\end{equation}\n\n\n\\begin{prop}[Dual formulation of $A_F^\\ep$]\\label{prop-dual}\nFor any $x,y\\in\\R^n$, \n$$\nA_F^\\ep (x,y)=\\sup_h\\{ Q_1 ^{ \\epsilon F} h(y)-h(x)\\}=\\sup_h\\{ Q_1 ^{ \\epsilon F} h(x)-h(y)\\}, \n$$\n where the supremum  runs through all regular enough functions $h$.  \n\\end{prop}\n\n\n\\begin{fproof}\nFor any smooth  path $(\\omega_t)_{0\\leq t\\leq 1}$ between $\\omega_0=x$ and $\\omega_1=y$, from the definition of $Q_1^{\\ep F}$, \n\\begin{equation*}\nQ_1^{\\ep F} h(y)-h(x)  \\leq \\int_0^1  \\left( \\frac{|\\dot \\omega_t|^2}{2} +\\frac{\\ep^2}{2}| F'(\\omega_t)|^2 \\right) dt,\n\\end{equation*}\nthen\n$$\n\\sup_{h\\,\\, \\text{linear}} \\left\\{ Q_1^\\ep h(y)-h(x) \\right\\} \\leq A_F^\\ep(x,y).\n$$ \nNow, let $\\gamma$ be a minimizer of \\eqref{eq-93} with $t=1$ and choose $h(z)=\\dot \\gamma_0\\cdot z$.  By \\eqref{eq-94}, we know that $ \\gamma$ satisfies  Newton equation. Hence it is  an $ \\epsilon F$-interpolation, that is a minimizer of the action between $ \\gamma_0=x$ at time $t=0$ and $ \\gamma_1=y$ at time $t=1.$ In other words\n$$\nQ_1^{\\ep F} h(y)-h(x)= \\int_0^1 \\left(\\frac{|\\dot \\gamma_t|^2}{2}+\\frac{\\ep^2}{2}| F'( \\gamma_t)|^2 \\right) dt= A_F^\\ep(x,y).\n$$\nThis proves the first equality: $A_F^\\ep(x,y)=\\sup\\{ Q_1^\\ep h(y)-h(x)\\}.$\n\\\\\nAs the action appearing at formula \\eqref{eq-def-A-1} is invariant with respect to time reversal, we see that $A_F^\\ep(x,y)=A_F^\\ep(y,x)$. It follows immediately that $A_F^\\ep(x,y)=\\sup\\{ Q_1^\\ep h(x)-h(y)\\}.$\n\\end{fproof}\n\n\n\n\\bigskip\\noindent \\textbf{Alternate formulations of the cost $A_F^\\ep$.}\\ \nSince for any $x,y\\in\\R^n$ and any path $ \\omega$ from  $x$ to $y$, we have\n\\begin{multline*}\n\\int_0^1 \\Big[|\\dot{\\omega}_s|^2+\\ep^2|F'(\\omega_s)|^2\\Big]ds\n\t=\\int_0^1 \\Big[|\\dot{\\omega}_s+\\ep F'(\\omega_s)|^2-2\\ep F'(\\omega_s)\\cdot \\dot \\omega_s\\Big]ds\\\\\n\t=\\int_0^1 |\\dot{\\omega}_s+\\ep F'(\\omega_s)|^2ds-2\\ep(F(y)-F(x)),\n\\end{multline*}\nwe obtain the forward expression of the cost\n\\begin{equation*\nA_F^\\ep(x,y)=\\frac{1}{2}\\inf _{ \\omega\\in\\OO ^{xy}}\\BRA{\\int_0^1 |\\dot{\\omega}_s+\\ep F'(\\omega_s)|^2ds}-\\ep(F(y)-F(x)).\n\\end{equation*}\n With the same way of reasoning, its backward formulation is\n\\begin{align}\\label{eq-backward}\nA_F^\\ep(x,y)=\\frac{1}{2}\\inf _{ \\omega\\in\\OO ^{xy}}\\BRA{\\int_0^1 |\\dot{\\omega}_s-\\ep F'(\\omega_s)|^2ds}+\\ep(F(y)-F(x)).\n\\end{align}\nIt follows that  its symmetric expression is\n$$\nA_F^\\ep(x,y)=\\frac{1}{4}\\inf _{ \\omega\\in\\OO ^{xy}}\\BRA{\\int_0^1 |\\dot{\\omega}_s+\\ep F'(\\omega_s)|^2ds}+\\frac{1}{4}\\inf _{ \\omega\\in\\OO ^{xy}}\\BRA{\\int_0^1 |\\dot{\\omega}_s-\\ep F'(\\omega_s)|^2ds}.\n$$\n\n\n\\bigskip\\noindent \\textbf{Example.}\\ \nLet us treat the simple case corresponding to $F(x)=|x|^2\/2$, $x\\in\\R^n,$ where computations are simple and explicit. \n\\begin{itemize}\n\\item For any $t\\ge 0$ and $x\\in\\R^n$, $\\ss_t(x)=e^{-t}x$.\n\\item For any $x,y\\in\\R^n$, the $\\ep F$-interpolation between $x$ and $y$  is given by\n$$\n\\omega ^{ \\ep}_s=\\ss_{\\ep s}(\\alpha)+\\ss_{\\ep (1-s)}(\\beta),\n\\quad 0\\le s\\le 1,\n$$\nwhere \\quad\n$ \\displaystyle{\n\\alpha=\\frac{x-ye^{-\\ep}}{1-e^{-2\\ep}},\\quad \\beta=\\frac{y-xe^{-\\ep}}{1-e^{-2\\ep}}.\n}$\n\\item\nFor any $x\\in\\R^n$, \\quad\n$ \\displaystyle{\nA_F^\\ep(x,x)=\\frac{\\ep}{2}\\frac{1-e^{-\\ep}}{1+e^{-\\ep}}|x|^2.}\n$\n\\item  Moreover, the corresponding Hamilton-Jacobi equation~\\eqref{eq-933} takes the form \n$$\n\\left\\{ \n\\begin{array}{ll}\n\\partial_t U(t,y)+\\frac{1}{2} |U'(t,y)|^2 = {\\ep^2}|y|^2\/2, \\quad t>0\\\\\nU(0,x)=h(x),\n\\end{array}\n\\right.\n$$\n and has an explicit solution, \n$$\nQ_t^{\\ep F} h(y)= {Q^0_{1-e^{-2\\ep t}}} f(e^{-\\ep t}y) \/(2\\ep)+{\\ep}|y|^2\/2\n$$\nwith $f(x)=h(x)-\\ep |x|^2\/2.$ \n\\end{itemize}\n\n\n\\subsection{Convexity properties  of the cost $A_F^\\ep$}\n\n\n\n\\begin{defi}[$(\\rho, \\mathsf{n})$-convexity]\n\\label{def-23}\nLet $\\rho\\in\\R$ and $ \\mathsf{n}\\in(0,\\infty]$. We say that the twice   differentiable   function $F$ on $\\R^n$ is  $(\\rho, \\mathsf{n})$-convex  if  \n\\begin{equation}\n\\label{eq-85}\nF''\\ge  \\rho \\,{\\rm{Id}}+F'\\otimes F'\/ \\mathsf{n}.\n\\end{equation}\n\\end{defi}\n\n\\bigskip\\noindent \\textbf{Example.}\\ \nLet us give some examples  where $n=1$, $ \\mathsf{n}>0$ and \\eqref{eq-85} is an equality.\n\\begin{itemize}\n\\item The map $x\\mapsto - \\mathsf{n}\\log x$ is $(0, \\mathsf{n})$-convex on $(0,\\infty)$. \n\\item When  $\\rho>0$, the map $x\\mapsto - \\mathsf{n}\\log \\cos(x\\sqrt{\\rho\/ \\mathsf{n}})$ is $(\\rho, \\mathsf{n})$-convex on the interval $(-\\pi\/2\\sqrt{ \\mathsf{n}\/\\rho},\\pi\/2\\sqrt{ \\mathsf{n}\/\\rho})$.\n\\item When  $\\rho<0$, the map $x\\mapsto - \\mathsf{n}\\log \\sinh(x\\sqrt{-\\rho\/ \\mathsf{n}})$ is $(\\rho, \\mathsf{n})$-convex on the interval $(0,\\infty)$.\n\\end{itemize}\n\n\n\n\\subsubsection*{Contraction inequality under a convexity assumption}\n\n\n\\begin{prop}[Contraction of the gradient flow]\n\\label{prop-contraction}\nLet us assume that $F$ is $(\\rho, \\mathsf{n})$-convex. Then, for all $t\\ge  0$, $\\ep\\ge  0$ and $x,y\\in\\ZZ$,\n\\begin{equation}\n\\label{eq-87}\nA_F^\\ep (\\ss_t(x),\\ss_t(y))\n\t\\le e ^{ -\\rho t} A_F^\\ep (x,y)\n\t\t- \\frac{1}{ \\mathsf{n}}\\int_0^t e ^{ -2\\rho(t-u)}[F(\\ss_u(x))-F(\\ss_u(y))]^2\\,du.\n\\end{equation}\n\\end{prop}\n\n\\begin{eproof}\nLet $ (\\omega_s)_{0\\le s\\le 1}$ be any smooth path between $x$ and $y$. For all $t\\ge  0,$ the composed path $[\\ss_t( \\omega_s)]_{0\\le s\\le 1}$ is a path between $\\ss_t(x)$ and $\\ss_t(y)$ and it follows from the definition of $A_F^\\ep $ that the first inequality in the subsequent chain, where $L ^{ \\ep}$ is defined at \\eqref{eq-03}, is satisfied:\n$$\n\\begin{array}{ll}\nA_F^\\ep  (\\ss_t(x),\\ss_t(y))\n\t&\\displaystyle\\le \\int_0^1 L^{ \\ep}(\\ss_t( \\omega_s), \\partial_s \\ss_t( \\omega_s)) \\,ds\\\\\n\t&\\displaystyle\\le e ^{ -2\\rho t} \\int_0^1 L^{ \\ep} ( \\omega_s,\\dot \\omega_s)\\,ds\n\t\t-\\frac{1}{ \\mathsf{n}} \\int _{ 0}^t \\int _{ 0}^1 e ^{ -2\\rho(t-u)}[F'(\\ss_u( \\omega_s))\\cdot \\ss_u'( \\omega_s)\\dot \\omega_s]^2\\,dsdu\\\\\n\t&\\displaystyle\\le e ^{ -2\\rho t} \\int_0^1 L^{ \\ep} ( \\omega_s,\\dot \\omega_s)\\,ds\n\t\t-\\frac{1}{ \\mathsf{n}} \\int _{ 0}^t e ^{ -2\\rho(t-u)}[F(\\ss_u( y))-F(\\ss_u( x))]^2\\,du.\n\\end{array}\n$$\nThe second inequality is a consequence of Lemma \\ref{res-03} below, and the last inequality is implied by Jensen's inequality. This concludes the proof of the proposition. \n\\end{eproof}\n\n\n\\begin{lem}\\label{res-03}\nLet $\\rho\\in\\R$ and $ \\mathsf{n}\\in (0,\\infty)$, the following assertions are equivalent. \n\\begin{enumerate}[(i)]\n\\item The function $F$ is $(\\rho, \\mathsf{n})$-convex. \n\\item For any differentiable path $ (\\omega_s)_{0\\le s\\le 1}$, any $\\ep\\ge 0$ and any $ t\\ge  0$,  $0\\le s\\le1,$  we have\n\\begin{equation}\n\\label{eq-86}\nL^{ \\ep}(\\ss_t( \\omega_s), \\partial_s \\ss_t( \\omega_s))\n\t\\le e ^{ -2\\rho t} L^{ \\ep}(\\omega_s, \\dot \\omega_s)\n\t- \\frac{1}{ \\mathsf{n}}\\int_0^t\n e ^{ -2\\rho(t-u)}[F'(\\ss_u( \\omega_s))\\cdot\\ss_u'( \\omega_s)\\dot \\omega_s]^2\\,du.\n\\end{equation}\n\\end{enumerate}\n\\end{lem}\n\n\\begin{eproof}\nLet us prove that  {\\it (i)} implies {\\it (ii)}. We have $ \\partial_s \\ss_t( \\omega_s)=\\ss_t'( \\omega_s)\\dot \\omega_s$, and for simplicity, we denote $x= \\omega_s$ and $ v=\\dot \\omega_s.$ We shall use: \n\t$ \\partial_t \\ss_t(x)=-F'(\\ss_t(x)),$\n\t$ \\partial_t\\ss_t'(x)=-F''(\\ss_t(x))\\ss_t'(x)$ and\n\t$ \\partial_tF'(\\ss_t(x))=-F''(\\ss_t(x))F'(\\ss_t(x)).$\n Let us set\n $$ \n \\Lambda(t):=L^{ \\ep}(\\ss_t( \\omega_s), \\partial_s \\ss_t( \\omega_s))\n\t\t=\\frac{1}{2}|\\ss_t'(x)v|^2+ \\frac{\\ep^2}{2}|F'(\\ss_t(x))|^2.\n $$ \n Its  derivative is \n\\begin{multline*}\n\\begin{split}\n\\Lambda'(t)\n\t&=-\\ss_t'(x)v\\cdot F''(\\ss_t(x))\\ss_t'(x) v\n\t- \\ep^2 F'(\\ss_t(x))\\cdot F''(\\ss_t(x))F'(\\ss_t(x))\\\\\n\t&\\overset{(i)}{\\le} -2\\rho \\Lambda(t)-\\frac{1}{ \\mathsf{n}}[F'(\\ss_t(x))\\ss_t'(x)v]^2\n\t\t- \\frac{\\ep^2}{ \\mathsf{n}} |F'(\\ss_t(x))|^4\\\\\n\t&\\le-2\\rho \\Lambda(t)-\\frac{1}{ \\mathsf{n}}[F'(\\ss_t(x))\\ss_t'(x)v]^2,\n\\end{split}\n\\end{multline*}\nwhich implies~\\eqref{eq-86}.\n\\\\\n Let us now assume {\\it (ii)} and show that it implies  {\\it (i)}. Since~\\eqref{eq-86} is an equality at time $t=0,$ fixing $s=0$,  $\\omega_0=x$ and $\\dot\\omega_0=v$, the first order Taylor expansion of~\\eqref{eq-86} in $t$  around $t=0$ implies that for any $v,x\\in\\R^n,$\n\\begin{equation}\n\\label{eq-92}\nv\\cdot (F''(x)-\\rho \\,{\\rm{Id}})v-\\frac{1}{ \\mathsf{n}}(F'(x)\\cdot v)^2\n\t+\\ep^2F'(x)\\cdot (F''(x)- \\rho\\,{\\rm{Id}}) F'(x)\\ge  0.\n\\end{equation}\nSuppose ad absurdum that {\\it (i)} is false. Then  \\eqref{eq-85} fails and there exist $x_o,v_o\\in\\R^n$ such that \n$$\nv_o\\cdot (F''(x_o)-\\rho \\,{\\rm{Id}})v_o-\\frac{1}{ \\mathsf{n}}(F'(x)\\cdot v_o)^2<0. \n$$\nBut taking $x=x_o$, $v=\\lambda v_o$ in~\\eqref{eq-92} and  sending  $\\lambda$  to infinity leads to a contradiction. \n\\end{eproof}\n\nActually we believe that the contraction inequality~\\eqref{eq-87} is equivalent to the $(\\rho, \\mathsf{n})$-convexity of the function $F$ as in the case when $\\ep=0$. \n\n\n\\subsubsection*{Convexity properties along $\\ep F$-interpolations}\nLet us introduce the notation\n$$\n\\theta_a(s):= \\frac{1- e ^{ -2as}}{1- e ^{ -2a}}.\n$$\nNote that $\\lim _{ a\\to 0} \\theta_a(s)=s.$\n\n\\begin{prop}[Convexity under the $(\\rho,\\infty)$-condition]\n\\label{prop-86}\nLet  $ F$ be a $(\\rho,\\infty)$-convex function with $\\rho\\in\\R$. Then  any $ \\ep F$-interpolation $ \\omega$ satisfies \n\\begin{equation}\\label{eq-conforti-fini}\n\\begin{split}\nF(\\omega_s)\\le &\\ \\theta _{ \\rho \\ep}(1-s) F(\\omega_0)+\n\\theta _{ \\rho \\ep}(s)F(\\omega_1)\\\\\n &- \\frac{1- e ^{ -2 \\rho \\ep}}{2 \\ep} \\theta _{ \\rho \\ep}(s) \\theta _{ \\rho \\ep}(1-s)\n [A_{ F}^\\varepsilon(\\omega_0,\\omega_1)+ \\ep F(\\omega_0)+ \\ep\t F(\\omega_1)],\n \\quad \\forall \\;0\\le s\\le 1.\n\\end{split}\n\\end{equation}\n\\end{prop}\n\n\n\n\n\n\\begin{eproof}\nWe start following  the smart proof of Conforti~\\cite[Thm~1.4]{conforti2017}. \nLet $(\\omega_s)_{0\\le s\\le 1}$ be an $\\ep F$-interpolation, and let $\\overleftarrow{h} $ and $\\overrightarrow{h} $ be two functions on $[0,1]$ such that for any $s\\in[0,1]$,  \n$$\nF(\\omega_s)=\\overrightarrow{h} (s)-\\overleftarrow{h} (s)\n$$\nand \n$$\n\\overrightarrow{h}'(s)=\\frac{1}{4\\varepsilon}|\\dot{\\omega}_s+\\varepsilon F'({\\omega_s}) |^2,\n\\qquad \\overleftarrow{h}'(s)=\\frac{1}{4\\varepsilon}|\\dot{\\omega}_s-\\varepsilon F'({\\omega_s})|^2.\n$$\nThis is possible since \n$ \\displaystyle{\n\\frac{d}{ds} F(\\omega_s)=F'(\\omega_s)\\cdot \\dot{\\omega}_s=\\overrightarrow{h}'(s)-\\overleftarrow{h}'(s).\n}$\n\\\\\nThen, for any $s\\in[0,1]$,  using the Newton equation~\\eqref{eq-17} satisfied by $ \\omega$ and the $(\\rho,\\infty)$-convexity~\\eqref{eq-85} of $ F$,  we obtain\n$$\n\\overrightarrow{h}''(s)=\\frac{1}{2}F''(\\varepsilon F'+\\dot{\\omega}_s,\\varepsilon F'+\\dot{\\omega}_s)\n\\ge  2\\rho\\varepsilon \\overrightarrow{h}'(s).\n$$\nSimilarly we have  $\\overleftarrow{h}''(s)\\le - 2\\rho\\varepsilon \\overleftarrow{h}'(s)$. We know by \\cite[Lem.~4.1]{conforti2017} that these inequalities imply\n\\begin{align*}\n\\overrightarrow{h} (s)\n\t\\le \\overrightarrow{h}(1)- \\theta _{ \\rho \\ep}(1-s)[\\overrightarrow{h}(1)-\\overrightarrow{h}(0)],\n\\qquad\n\\overleftarrow{h} (s)\n\t\\ge  \\overleftarrow{h}(0)+ \\theta _{ \\rho \\ep}(s)[\\overleftarrow{h}(1)-\\overleftarrow{h}(0)].\n\\end{align*}\nArranging the terms in $ F( \\omega_s)=\\overrightarrow{h} (s)-\\overleftarrow{h} (s)$, we see that\n\\begin{equation*}\nF(\\omega_s)\n\t\\le \\theta _{ \\rho \\ep}(1-s)  F(\\omega_0)+\n\\theta _{ \\rho \\ep}(s)F(\\omega_1)\n- (1- e ^{ -2 \\rho \\ep}) \\theta _{ \\rho \\ep}(s) \\theta _{ \\rho \\ep}(1-s)[\\overrightarrow{h} (1)-\\overleftarrow{h} (0)].\n\\end{equation*}\nNow the proof differs from Conforti's one. By the  definitions of $\\overrightarrow{h}$ and $\\overleftarrow{h},$ and using the backward formulation \\eqref{eq-backward} of the cost $A_F^\\ep$, we obtain \n$$\n2 \\ep[\\overrightarrow{h} (1)-\\overleftarrow{h} (0)]=\\int_0^1\\frac{1}{2}|\\dot{\\omega}_s-\\varepsilon F'(\\omega_s)|^2ds+2 \\ep F(\\omega_1)\n=  A_{ F}^\\varepsilon(x,y)+ \\ep F(\\omega_0)+ \\ep F(\\omega_1),\n$$\nwhich gives us the desired inequality~\\eqref{eq-conforti-fini}.\n\\end{eproof}\n\nProposition \\ref{prop-86} implies a Talagrand-type inequality, i.e.\\ a  comparison between a cost function and an entropy, see \\cite{conforti2017}.    \n\n\\begin{cor}[Talagrand-type inequality for the cost $A_{ F}^\\ep$] \\label{res-a01}\nAssume that $F$ is  $(\\rho,\\infty)$-convex with $\\rho>0$ and that it is normalized by $\\inf F=0.$ Then for any $ \\ep>0$ and  $x,y\\in \\R^n$, \n$$\nA_{ F}^\\ep(x,y)\\le  \\frac{\\ep(1+e^{-\\rho\\ep})}{1-e^{-\\rho\\ep}}\\PAR{ F(x)+F(y)}.\n$$ \nIn particular, if $\\inf F=F(x^*)=0$ at $x^*,$ then for any $x\\in\\R^n,$\n\\begin{align}\\label{eq-a01}\nA_{ F}^\\ep(x^*,x)\\le  \\frac{\\ep(1+e^{-\\rho\\ep})}{1-e^{-\\rho\\ep}}F(x).\n\\end{align}\n\\end{cor}\n\n\\begin{eproof}\nIt is a direct consequence of \\eqref{eq-conforti-fini} at time $s=1\/2$ with $F( \\omega _{ 1\/2})\\ge 0.$\n\\end{eproof}\nLetting $ \\ep$ tend to zero in \\eqref{eq-a01}, we see that: $F(x)\\ge \\rho |x-x^*|^2\/4.$ We  are a factor 2 below the optimal result.\n\n\nCosta's lemma states the concavity of the exponential entropy along the heat semigroup.  Here is an analogous result. \n\n\\begin{prop}[Convexity under the $(0, \\mathsf{n})$-condition]\n\\label{prop-85}\nLet us  assume that $ F$ is $(0, \\mathsf{n})$-convex. Then, for any $ \\ep\\ge 0$ and any $\\ep F$-interpolation $ \\omega$, the function \n$$\n[0,1]\\ni s\\mapsto e^{- F( \\omega_s)\/ \\mathsf{n}}\n$$ \nis concave.\n\\end{prop}\n\n\n\\begin{eproof}\nDifferentiating twice  $ \\Lambda(s):=\\exp\\{-F( \\omega_s)\/ \\mathsf{n} \\}$, we obtain:\n\t$ \\Lambda'(s)=-  \\Lambda(s) F'( \\omega)\\cdot \\dot\\omega_s\/ \\mathsf{n} ,$ and \n\\begin{multline*}\n\\begin{split}\n \\Lambda''(s)&=-\\frac{1}{ \\mathsf{n}} \\Lambda'(s) F'( \\omega_s)\\cdot\\dot \\omega_s\n\t\t-\\frac{1}{ \\mathsf{n}} \\Lambda(s)F''( \\omega_s)(\\dot \\omega_s, \\dot \\omega_s)\n\t\t-\\frac{1}{ \\mathsf{n}} \\Lambda(s)F'( \\omega_s)\\cdot \\ddot \\omega_s\\\\\n\t\t&=- \\frac{1}{ \\mathsf{n}}\\Lambda(s) \\Big[-\\frac{1}{ \\mathsf{n}}|F'( \\omega_s)\\dot \\omega_s|^2\n\t\t\t+\\underbrace{F''( \\omega_s)(\\dot \\omega_s, \\dot \\omega_s)} _{ \\ge  \\frac{1}{ \\mathsf{n}}(F'( \\omega_s)\\cdot \\dot \\omega_s)^2}\n\t\t\t+\\underbrace{ \\ep^2 F''( \\omega_s)(F'( \\omega_s),F'( \\omega_s))} _{ \\ge   \\frac{\\ep^2}{ \\mathsf{n}}|F'( \\omega_s)|^4}\\Big]\\\\\n\t\t\t&\\le 0,\n\\end{split}\n\\end{multline*}\nwhich is the desired result. We used \\eqref{eq-17} at second equality and the $(0, \\mathsf{n})$-convexity of $F$ at last inequality.\n\\end{eproof}\n\nAs a direct consequence of Propositions~\\ref{prop-86} and~\\ref{prop-85}, we obtain the following   \n\\begin{cor}\n\\label{cor-15}\nLet  $ \\omega$ be an $\\ep F$-interpolation with $\\ep>0$.\n\\begin{enumerate}[(a)]\n\\item If  $F$ is $(\\rho,\\infty)$-convex with $\\rho\\in\\R$ then \n\\begin{equation*}\n-\\frac{d^+}{ds}F(\\omega_s)\\Big|_{s=1}\n\t+\\rho A_{ F}^\\varepsilon(\\omega_0,\\omega_1)\n\t\\le  \\frac{\\rho\\varepsilon(1+e ^{ -2 \\rho \\ep})}{1-e^{-2\\rho\\varepsilon}} [F(\\omega_0)-F(\\omega_1)].\n \\end{equation*}\n\\item If  $ F$ is $(0, \\mathsf{n})$-convex  with $ \\mathsf{n}>0$ then \n\\begin{equation}\n\\label{eq-cor-rip3}\n-\\frac{d^+}{ds} F( \\omega_s)\\Big|_{s=1}\\le   \\mathsf{n}[1-e^{-( F(\\omega_0)- F(\\omega_1))\/ \\mathsf{n} }].\n\\end{equation}\n\\end{enumerate}\n\\end{cor}\n\nIn order to prepare the proof of the analogue of EVI at Proposition \\ref{prop-88}, we need the next  result.\n\n\\begin{prop}[Derivative formula]\n\\label{prop-87}\nFor any $x,y\\in\\R^n$,\n$$\n\\frac{d}{dt}^+\\Big|_{t=0}A_F^\\ep(\\ss_t(x),y)\\le -\\frac{d}{ds}\\Big|_{s=1}F(\\omega_s^{yx}),\n$$\nwhere $\\omega^{yx}$ is any $\\ep F$-interpolation from $y$ to $x$. \n\\end{prop}\n\n\\begin{eproof}\nLet $\\omega$ be an $\\ep F$-interpolation  from $y$ to $x$, (we drop the superscript $yx$ for simplicity). Then for any $t\\ge 0,$ $(\\eta_{s,t})_{0\\le s\\le 1}=(\\ss_{st}(\\omega_s ))_{0\\le s\\le 1}$ is a path from $y$ to $\\ss_t(x)$ and by definition of $A_F^\\ep$,  we have\n$$\nA_F^\\ep(\\ss_t(x),y)\\le  \\int_0^1\\left[\\frac12|\\partial_s\\eta_{s,t}|^2+\\frac{\\ep^2}{2}|F'(\\eta_{s,t})|^2\\right]ds.\n$$\nSince at $t=0$ this is an equality,   we see that\n$$\n\\frac{d}{dt}^+\\Big|_{t=0}A_F^\\ep(\\ss_t(x),y)\n\t\\le  \\int_0^1\\Big[-F'(\\omega_s )\\cdot\\dot\\omega_s -s F''(\\omega_s )(\\dot\\omega_s,\\dot\\omega_s) -s\\ep^2 F''(\\omega_s )(F'(\\omega_s ),F'(\\omega_s ))\\Big]\\,ds.\n$$\nDifferentiating $H(s):=F(\\omega_s )$, we obtain \n$H'(s)=F'( \\omega_s)\\cdot \\dot\\omega_s$ and \\\\\n\t$H''(s)=F''( \\omega_s)(\\dot \\omega_s, \\dot \\omega_s)+F'( \\omega_s)\\cdot \\ddot \\omega_s.$ With the Newton equation \\eqref{eq-17}: $\\ddot \\omega_s= \\ep^2 F''F'( \\omega_s),$ we arrive at\n$$\n\\frac{d}{dt}^+\\Big|_{t=0}A_F^\\ep(\\ss_t(x),y)\n\t\\le  \\int_0^1[-H'(s)-sH''(s)]ds=-H'(1),\n$$\nwhich is   the announced result. \n\\end{eproof}\n\nThis specific method of applying the gradient flow $\\ss _{ st}( \\omega_s)$ to a path $ \\omega$ has been successfully used in \\cite{daneri-savare2008}. \n\n\nOne  derives immediately from Corollary~\\ref{cor-15} and Proposition~\\ref{prop-87} the following \n\n\\begin{prop}[EVI  under $(\\rho,\\infty)$ or $(0, \\mathsf{n})$-convexity]\n\\label{prop-88}\n\\ \n\\begin{enumerate}[(a)]\n\\item Assume that $ F$ is $(\\rho,\\infty)$-convex. Then, for any $x,y\\in\\R^n$,\n\\begin{equation}\n\\label{eq-evi-fini}\n\\frac{d}{dt}^+\\Big|_{t=0}A_{ F}^\\varepsilon(\\ss_t(x),y)\n\t+ \\rho A_{ F}^\\varepsilon(x,y)\n\t\\le  \\frac{\\rho\\varepsilon(1+e^{-2\\rho\\varepsilon})}{1-e^{-2\\rho\\varepsilon}} [F(y)-F(x)].\n\\end{equation}\n\n\\item Assume that $ F$ is $(0, \\mathsf{n})$-convex. Then, for any $x,y\\in\\R^n$,\n$$\n\\frac{d}{dt}^+\\Big|_{t=0} A_{ F}^\\varepsilon(\\ss_t(x),y)\\le  n[1-e^{-(F(y)- F(x))\/ \\mathsf{n} }].\n$$\n\\end{enumerate}     \n\\end{prop}\nThese inequalities should be compared with the EVI formulation \\eqref{eq-10b} which holds in the case where $ \\ep =0$ and $ \\mathsf{n}= \\infty.$ Note that one recovers  \\eqref{eq-10b}  by letting $ \\ep$ tend to zero in \\eqref{eq-evi-fini}, or by sending $ \\mathsf{n}$ to infinity in the last inequality.\n\n\n\\section{Newton equation in the Wasserstein space} \\label{sec-newton}\n\nThis section is dedicated to the infinite dimension case where the states live in  the Wasserstein space    on a Riemannian manifold.\nThis is done by relying on the results of  previous section in $\\R^n$ as an analogical  guideline, and by using  Otto's heuristic.  Only  non-rigorous statements  and ``proofs'' are presented in this section.  For the sake of completeness, we start recalling Otto calculus which was introduced in the seminal paper~\\cite{otto2001}. This heuristic theory is known for long and  well explained at many places \\cite{villani2009,gigli2012,ambrosio-gigli2013}.\nOur contribution is the introduction of the Newton equation, a key point of our computations. \n\n\\subsection{The setting}\n\\label{sec-setting}\n\nThe configuration space is a Riemannian manifold  $(N,\\gg)$  equipped with a Riemannian measure  $ \\mathrm{vol}$ and the state space to be considered later on is the Wasserstein space \n\\begin{align}\\label{eq-05}\nM:= \\mathcal{P}_2(N)\n\\end{align}\nof all probability measures $ \\mu$ on $N$ such that $\\int_N d^2(x_o,\\cdot) \\,d\\mu < \\infty$, where $d$ stands for the Riemannian distance.  \n\n\\subsubsection*{Carr\u00e9 du champ}\n\nThe gradient in $(N,\\gg)$ is denoted by $\\nabla$ and the  divergence by  $\\nabla \\cdot\\ $. Functions and vector fields are assumed to be smooth enough for allowing  all the computations.  In particular, for any vector field $\\vec A$ and function $f$, the following  integration by parts formula holds   \n$$\n\\int f\\,\\nabla \\!\\cdot\\! \\vec A \\,d\\!\\vol  = -\\int \\nabla f\\cdot \\vec A \\,d\\!\\vol  , \n$$\nusing the inner product on $(N,\\gg)$.\n The  carr\u00e9 du champ $ \\Gamma$ in $N$ is defined for any functions $f,g$ on $N$ by\n$$ \n\\Gamma(f,g): x\\mapsto \\nabla f(x)\\cdot \\nabla g(x),\\quad x\\in N.\n$$\nAs usual we write $ \\Gamma(f,g)=\\nabla f\\cdot\\nabla g,$ and state $\\Gamma(f)=\\nabla f\\cdot\\nabla f=|\\nabla f|^2$.  Denoting $\\Delta_\\gg= \\Delta=\\nabla\\cdot\\nabla$ (we  drop the subscript $\\gg$ for simplicity),   the  Laplace-Beltrami operator on $N$, we have \n$$\n\\int \\Gamma(f,g)\\,d\\!\\vol  =-\\int f\\Delta g\\,d\\!\\vol  .\n$$\n\n\n\n\\subsubsection*{Iteration of $ \\Gamma$, hidden connection}\n\n\nThe iterated carr\u00e9 du champ operator in $(N,\\gg)$ introduced by Bakry and \\'Emery in~\\cite{bakry-emery1985} (see also~\\cite{bgl-book})   is defined for any $f$ by\n\\begin{equation}\n\\label{eq-def-gamma2}\n\\Gamma_2(f)=\\frac{1}{2}\\Delta\\Gamma(f)-\\Gamma(f,\\Delta f).\n\\end{equation}\nIt happens to be the left-hand side of the Bochner-Lichnerowicz identity\n\\begin{align}\\label{eq-13}\n\\Gamma_2(f)=||\\nabla^2f||^2_{\\mathrm{HS}}+{\\rm Ric}_\\gg(\\nabla f,\\nabla f),\n\\end{align}\nwhere $||\\nabla^2f||^2_{\\mathrm{HS}}$ is the Hilbert-Schmidt norm of the Hessian $\\nabla^2f$ of $f$ and ${\\rm Ric}_\\gg$ is the Ricci tensor of $(N,\\gg)$.    \\\\\nFor any functions $f,g,h:N\\mapsto \\R$, the evaluation of the Hessian of $f$ applied to two gradients $\\nabla g,\\nabla h$ only depends on the carr\u00e9 du champ $ \\Gamma:$ \n\\begin{equation}\n\\label{eq-hessian1}\n\\nabla^2 f(\\nabla g,\\nabla h)=\\frac{1}{2}\\big[\\Gamma(\\Gamma( f, g), h)+\\Gamma(\\Gamma( f, h), g)-\\Gamma(\\Gamma( g, h), f)\\big]\n\\end{equation}\nand of course $2\\nabla^2 f(\\nabla f,\\nabla f)=\\Gamma(\\Gamma( f), f)$. \n\\\\\nThe Levi-Civita connection is implicitly invoked in the Bochner-Lichnerowicz identity, in particular for computing the Hessian of $f$, and it is hidden   in \\eqref{eq-hessian1}. This  is commented on at~\\cite[p.158]{bgl-book}.\n\n\n\n\\subsection{Some PDEs are Wasserstein gradient flows} \n\\label{sec-gradient-flow}\n\n\nWe present basic notions which are useful to visualize heuristically the Wasserstein space $M= \\mathcal{P}_2(N)$ as an infinite Riemannian manifold.\n\nThe same notation is used for the probability measure $ \\mu$ and its density $ d \\mu\/d\\! \\mathrm{vol}$.\n \n\n\\subsubsection*{Velocity, tangent space and Wasserstein metric}\n\nIf $(\\mu_t)$ is a path in $M$, then $\\partial_t\\mu_t$ satisfies $\\int_N\\partial_t\\mu_t\\,d\\!\\vol =0$. In other words the tangent space at some $\\mu\\in M$ is a subset of functions $f$ satisfying $\\int_N f\\,d\\!\\vol  =0$. \n\nLet us explore  another representation of the tangent space. Again, let $(\\mu_t)$ be a path in $M$. Then for each $t$, there exists a unique (up to some identification) map $\\Phi_t:N\\mapsto \\R$ such that the continuity equation\n\\begin{align}\\label{eq-a02}\n\\partial_t\\mu_t=-\\nabla \\cdot(\\mu_t \\nabla \\Phi_t)\n\\end{align}\nholds. As a hint for the proof of this statement,  denoting $\\aa_\\mu(\\nabla f):=-\\nabla\\cdot(\\mu\\nabla f)$, we see that $\\aa_\\mu(\\nabla f)= v$ is an elliptic equation in $f$ and \n$\n\\nabla \\Phi_t=\\aa^{-1} _{ \\mu_t}(\\partial_t \\mu_t)\n$\nexists because   $v= \\partial_t \\mu_t$ satisfies \n  $\\int \\partial_t \\mu_t \\,d\\!\\vol  =0$. This allows one to identify the velocity $ \\partial_t \\mu_t$ with the gradient vector field $\\nabla \\Phi_t,$ via the mapping \n  \\begin{align}\\label{eq-a03}\n\\dot \\mu_t:=\\nabla \\Phi_t=\\aa^{-1} _{ \\mu_t}(\\partial_t \\mu_t).\n\\end{align}\nWithin Otto's heuristics,  the tangent space of $M$ at $\\mu$ is represented by the set of gradients\n$$\n\\mathrm{T}_\\mu M=\\{\\nabla \\Phi,\\ \\Phi: N\\mapsto\\R\\}.\n$$ \nTo be rigorous it is necessary  to define  $\\mathrm{T}_\\mu M$ as the closure in $L^2(N,\\mu)$ of  $\\{\\nabla \\Phi,\\,\\, \\Phi\\in\\cC_c^\\infty(N)\\}.$ \n\\emph{\nWe call $\\dot\\mu_t\\in \\mathrm{T}_\\mu M$ the Wasserstein velocity to distinguish it from the standard velocity $ \\partial_t\\mu_t.$ }\n\nThe continuity equation \\eqref{eq-a02} is a keystone of the theory. It is valid for instance when $N$ is a smooth compact Riemannian manifold and \nthe density $\\mu(t,x)$ is smooth and positive. This extends to more general settings, as  explained in~\\cite[Ch.\\,8]{ambrosio-gigli2008}. \n    \n\\begin{defi}[Wasserstein metric] The inner product on  $\\mathrm{T}_\\mu M$ is defined for any $\\nabla\\Phi, \\nabla\\Psi$ by\n$$\n\\langle \\nabla \\Phi,\\nabla \\Psi \\rangle_\\mu=\\int \\nabla \\Phi\\cdot \\nabla \\Psi \\,d\\mu=\\int \\Gamma(\\Phi,\\Psi)\\,d\\mu.\n$$\n Hence the speed of a path $(\\mu_t)$ in $M$ is:\n$\n|\\dot{\\mu}_t|_{\\mu_t}=\\PAR{\\int  | \\aa^{-1}_{\\mu_t}(\\partial_t\\mu_t)|^2d \\mu_t}^{1\/2}.\n$\n\\end{defi}\nComparing with \\eqref{eq-96}, we see that minimizing velocity fields satisfying the continuity equation \\eqref{eq-a02} are gradient fields.\n\n\n\nAs a definition, the Wasserstein cost for transporting $\\mu$ onto $\\nu$ is\n\\begin{align*}\nW_2^2( \\mu, \\nu):=\\inf _{ \\pi}\\int _{ N\\times N} d^2(x,y)\\,d\\pi,\n\\end{align*}\nwhere $d$ is the Riemannian distance on $N$ and $\\pi$ runs through the set of all couplings between $\\mu$ and $\\nu.$\nThe Benamou-Brenier theorem \\cite{benamou-brenier2000} states that\n\\begin{align}\\label{eq-a06}\n\\inf _{ (\\xi,v)} \\int _{ [0,1]\\times N} |v_t|^2\\,d\\xi_tdt=W_2^2( \\mu, \\nu),\n\\end{align}\nwhere the infimum runs through the set of all $(\\xi,v)$ such that for all $t$, $\\xi_t\\in \\mathcal{P}_2(N)$ and $v_t\\in\\mathbf{R}^n$ satisfying the continuity equation \n\t$ \\partial_t\\xi+\\nabla\\!\\cdot\\!( \\xi v)=0$ and the endpoint constraint $\\xi_0=\\mu, \\xi_1=\\nu$. This means that the Riemannian distance associated to the Wasserstein metric is  $W_2$, recall \\eqref{eq-01}. \nThe proof of the analogous result in a metric space can be found in~\\cite{ambrosio-gigli2008}. \n\n\n\n\\subsubsection*{Other metrics}\n\nThe standard velocity $ \\partial_t\\mu_t$ lives in the space \n\\[\n\\mathcal{T}_\\mu:=\\{\\partial\\nu:\\ \\int \\partial\\nu\\,d\\!\\vol =0\\}\n\\]\nequipped with the inner product of $L^2( \\mathrm{vol}),$ while the Wasserstein velocity $\\dot \\mu_t$ lives in $\\mathrm{T}_\\mu M$ equipped with the above Wasserstein inner product. Both of them represents the same object, they are linked by \\eqref{eq-a03}, but they give rise to distinct gradient flows.\n\n\n Although the present article focuses on the Wasserstein metric, let us give some examples of alternate metrics based on the tangent space  $ \\mathcal{T}_\\mu$. \n\\begin{enumerate}[1)]\n\\item The first example is  described  in~\\cite[Sec.~9.6]{evans1998}. For any $\\mu\\in M$ and two perturbations $\\partial \\nu,\\partial \\nu'$ around $\\mu$,  the inner product is the standard scalar product in $L^2( \\mathrm{vol} )$:\n\\begin{equation}\n\\label{eq-def-usualsp2}\n\\langle \\partial\\nu,\\partial\\nu'\\rangle_{\\mu,1}=\\int\\partial\\nu \\,\\partial\\nu'\\,d\\!\\vol  .\n\\end{equation}\n\n\\item\n\nThe second example is rather similar. For any $\\mu\\in M$ and $\\partial \\nu,\\partial\\nu'\\in  \\mathcal{T}_\\mu$,  \n\\begin{equation}\n\\label{eq-def-usualsp}\n\\langle \\partial \\nu,\\partial\\nu'\\rangle_{\\mu,2}=\\int \\nabla \\Delta^{-1}(\\partial\\nu)\\cdot\\nabla \\Delta^{-1}(\\partial\\nu') \\,d\\!\\vol  =-\\int \\Delta^{-1}(\\partial\\nu) \\partial\\nu' \\,d\\!\\vol .\n\\end{equation}\nThis metric is  explained in~\\cite[Sec.\\,1.2]{otto2001}.\n\nBoth inner products  $\\langle \\cdot,\\cdot\\rangle_{\\mu,1}$ and $\\langle \\cdot,\\cdot\\rangle_{\\mu,2}$ do not depend on $\\mu$ and one can show that  the geodesic $(\\mu_s)_{0\\le s\\le 1}$ between $\\mu$ and $\\nu$ satisfies: $\\partial^2_{ss}\\mu_s=0$. It follows that $( \\mu_s)$ is an affine interpolation, meaning that the mass is not transported but teleported. The geometric content of these metrics is poorer than the Wasserstein metric one. \n\n\\item  For any $\\mu\\in M$ and two perturbations $\\partial\\nu,\\partial\\nu'\\in \\mathcal{T}_\\mu$ around $\\mu$, \n\\begin{equation}\n\\label{eq-def-mtd}\n\\langle \\partial\\nu,\\partial\\nu'\\rangle_{\\mu,3}=\\int \\frac{\\nabla \\Delta^{-1}(\\partial\\nu)\\cdot\\nabla \\Delta^{-1}(\\partial\\nu')}{\\mu} \\,d\\!\\vol  .\n\\end{equation}\nThis  is  the Markov transportation metric which is defined and used in~\\cite{bolley-gentil2014}.\n\\end{enumerate}\nThe heat equation \n\\begin{equation}\\label{eq-heat-eq}\n\\partial_t u=\\Delta u\n\\end{equation}\nis the gradient flow of $\\cF_1(\\mu)=\\int |\\nabla \\mu|^2\\,d\\!\\vol  $ with respect to the metric $\\langle \\cdot,\\cdot\\rangle_{\\mu,1}$ defined at \\eqref{eq-def-usualsp2} and also of $\\mathcal F_2(\\mu)=\\int \\mu^2\/2\\,\\,d\\!\\vol  $ with respect to the metric $\\langle \\cdot,\\cdot\\rangle_{\\mu,2}$ defined at \\eqref{eq-def-usualsp}, cf.~\\cite{evans1998}.\nThe heat equation\n is also  the gradient flow of the standard   entropy defined below at \\eqref{eq-a07},\n with respect to the Markov transportation metric defined in~\\eqref{eq-def-mtd}. We shall see in a moment at \\eqref{eq-12} that this equation is also the gradient flow of the same entropy with respect to the Wasserstein metric. \n\n\n\n\\subsubsection*{Wasserstein gradient flow}\n\nLet us go back to the Wasserstein metric and define the gradient  with respect to this metric of a function  $\\mathcal F : M\\mapsto\\R$. If $ \\mathcal{F}$ is differentiable at $ \\mu$, there exists a gradient field $\\nabla \\Phi$ such that for any path $(\\mu_t)$ satisfying $\\dot{\\mu}_\nt=\\nabla {\\Psi_t}$ and $ \\mu _{ t_o}= \\mu$, we have   \n$$\n\\frac{d}{dt}\\mathcal F(\\mu_t) _{ \\big|t=t_o}=\\langle \\nabla \\Phi,\\nabla {\\Psi _{ t_o}} \\rangle_{\\mu}.\n$$\nAs a definition, the gradient of $\\mathcal F$ at  $\\mu$ is\n$$\n{\\rm grad}_{\\mu} \\mathcal F:=\\nabla \\Phi\\in \\mathrm{T}_{\\mu}M.\n$$\nSimilarly to the finite dimensional setting, one can define a  gradient flow in $M$ with respect to the Wasserstein metric. \n\\begin{defi}[Wasserstein gradient flow]\nA path $(\\mu_t)_{t\\ge 0}$ in $M$ is a Wasserstein gradient flow of a function $\\mathcal F$  if for any $t\\ge 0$, \n$$\n\\dot{\\mu}_t=-{\\rm grad}_{\\mu_t}  \\mathcal F. \n$$\nWe denote by \n$\n\\mu_t=\\mathsf{S} ^{ \\mathcal{F}}_t\\mu, \n$\nthe solution of the above gradient flow equation starting from $\\mu\\in M$. \n\\end{defi}\n\n\n\\subsubsection*{Examples of Wasserstein gradient flows}\n\nConsider the  following interesting type of functions\n\\begin{align}\n\\label{eq-def-F}\n\\mathcal F(\\mu)&=\\int f(\\mu)\\,d\\!\\vol  , \n\\end{align}\nwhere  $f:\\R\\mapsto \\R$ and  the Wasserstein gradient flow equation\n\\begin{align}\\label{eq-a11}\n\\dot\\mu_t=-{\\rm grad} _{ \\mu_t} \\mathcal{F}.\n\\end{align}\nWith a path $(\\mu_t)$ satisfying  $\\dot{\\mu}_\nt=\\nabla {\\Psi_t}$, we obtain\n\\begin{align*}\n\\frac{d}{dt}\\mathcal F(\\mu_t)\n\t&=\\int f'(\\mu_t) \\partial_t \\mu_t\\,d\\!\\vol \n\t=-\\int f'(\\mu_t)\\nabla \\!\\cdot\\!(\\mu_t \\nabla \\Psi_t)\\,d\\!\\vol  \\\\\n\t&=\\int  \\nabla[ f'(\\mu_t)] \\cdot \\nabla \\Psi_t\\, d\\mu_t \n\t=\\langle \\nabla [f'(\\mu_t)],\\nabla\\Psi_t \\rangle_{\\mu_t}. \n\\end{align*}\nTherefore \n\\begin{equation}\\label{eq-def-delta}\n{\\rm grad}_\\mu \\mathcal F=\\nabla [f'(\\mu)]\\in \\mathrm{T}_{\\mu}M.\n\\end{equation}\nWe  mention three standard  examples of PDEs whose solutions are Wasserstein gradient flows.  \n\\begin{enumerate}[(1)]\n\\item \n\\label{ex-1}\nThe standard  entropy, that is the relative entropy with respect to the volume measure,  is  defined for any $\\mu\\in M$ by \n\\begin{align}\\label{eq-a07}\n {\\mathcal En}(\\mu):=\\int \\mu\\log \\mu \\,d\\!\\vol  .\n\\end{align}\nLet us show that \\emph{the Wasserstein gradient flow of the entropy is the heat equation.}\n\\\\\nThe function $ \\mathcal{F}={\\mathcal En}$ corresponds to $f(a)=a\\log a$ and its Wasserstein gradient \\eqref{eq-def-delta} is\n\\begin{align}\\label{eq-a08}\n{\\rm grad}_\\mu{\\mathcal En}=\\nabla\\log\\mu.\n\\end{align}\nWith \\eqref{eq-a02}, we see that the gradient flow equation \n\\eqref{eq-a11} writes as \n\\begin{align}\\label{eq-12}\n\\partial_t\\mu_t=\\nabla\\!\\cdot\\!(\\mu_t \\nabla\\log\\mu_t)= \\Delta\\mu_t,\n\\end{align}\nwhich is the heat equation \\eqref{eq-heat-eq}.\nThis  was discovered in the seminal paper~\\cite{jko1998}.  \n\\\\\n This example can be generalized by considering the Fokker-Planck equation with a potential $V$   \n\\begin{equation}\n\\label{eq-def-fp}\n\\partial_t \\mu_t=\\Delta \\mu_t+\\nabla\\cdot(\\mu_t \\nabla V)=\\nabla\\cdot (\\mu_t\\nabla (\\log\\mu_t+V))\n\\end{equation}\nwhose solution is the Wasserstein gradient flow of \n$$\n\\mathcal F(\\mu)={\\mathcal En}(\\mu)+\\int Vd\\mu.\n$$\n\n\n\\item\nThe R\u00e9nyi entropy of order $p>0$ with $p\\neq 1,$ is \n\\begin{align}\\label{eq-a04}\n\\mathcal R_p(\\mu):=\\frac{1}{p-1}\\int \\mu^{p} \\,d\\!\\vol .\n\\end{align} \nLet us show that \\emph{the Wasserstein gradient flow equation \\eqref{eq-a11} of the R\u00e9nyi entropy is the porous media equation (or fast diffusion equation)}\n$$\n\\partial_t \\mu_t=\\Delta \\mu_t^p.\n$$\nThe function $ \\mathcal{F}= \\mathcal{R}_p$ corresponds to $f(a)=a^p\/(p-1)$ and its Wasserstein gradient \\eqref{eq-def-delta} is\n\\begin{align}\\label{eq-a09}\n{\\rm grad}_\\mu \\mathcal{R}_p= \\frac{p}{p-1}\\nabla\\mu ^{ p-1}.\n\\end{align}\nWith \\eqref{eq-a02}, we see that the gradient flow equation \nwrites as \n$\n\\partial_t\\mu_t=\\nabla\\!\\cdot\\! \\left(\\mu_t \\frac{p}{p-1}\\nabla \\mu_t^{p-1}\\right)\n\t= \\Delta\\mu_t^p.$\n This was proved rigorously  in~\\cite{otto2001}.\n \n\\item The granular media equation in $\\R^n$ is \n\\begin{equation}\n\\label{eq-def-granular}\n\\partial_t \\mu_t=\\nabla\\cdot(\\mu_t\\nabla(\\log \\mu_t+V+ W*\\mu_t)), \n\\end{equation}\nwhere $V:\\R^n\\mapsto \\R$ is a confinement potential and $W:\\R^n\\mapsto \\R$ is an interaction potential.  We assume that $W(-x)=W(x)$ and $W*\\mu_t(x)=\\int W(x-y)\\mu_t(x)\\,d\\!\\vol  $ is the usual convolution in $\\R^n$. We directly derive from this equation\n$$\n\\dot{\\mu}_t =-\\nabla(\\log \\mu_t+ V+W*\\mu_t). \n$$\nLet us introduce \n$$\n\\mathcal F(\\mu)=\\int \\Big(\\log \\mu+ V+\\frac{1}{2}W*\\mu\\Big)d\\mu.\n$$\nBy the same computation, we obtain\n$$\n{\\rm grad}_{\\mu} \\mathcal F=\\nabla(\\log \\mu+ V+ W*\\mu).\n$$\nSo the granular media equation~\\eqref{eq-def-granular} is the  Wasserstein gradient flow equation of the above functional $\\mathcal F$. This result is derived and used in~\\cite{carrillo-mccann2003}.\n\\end{enumerate}\n\n\n\n\n\\subsection{Second order derivative in $M$}\n\\label{sec-second-derivative}\n\n\n\\subsubsection*{Covariant derivative of a gradient field}\n\nLet us show that the covariant derivative $ D_t\\nabla\\Phi_t$  of the vector field $\\nabla \\Phi_t$ in the tangent space $\\mathrm{T}_{\\mu_t} M$, along a path  $(\\mu_t)_{t\\ge 0}$  with velocity $\\dot{\\mu}_t =\\nabla \\Theta _t$, is\n\\begin{equation}\n\\label{eq-covariant-meca}\n D_t\\nabla\\Phi_t = \\mathsf{Proj} _{ \\mu_t}\\PAR{\\nabla\\partial_t\\Phi_t+ \\nabla^2\\Phi_t \\nabla \\Theta_t},\n\\end{equation}\nwhere $ \\mathsf{Proj} _{ \\mu_t} $ is the orthogonal projection on the tangent space $\\mathrm{T}_{\\mu_t} M$. To see this, we compute the time-derivative of \n$\n\\langle \\nabla \\Phi_t,\\nabla\\Psi_t\\rangle_{\\mu_t}=\\int\\Gamma(\\Phi_t,\\Psi_t) \\,d\\mu_t , \n$\nwhere   $\\nabla\\Phi_t$ and $\\nabla\\Psi_t$  belong to $ \\mathrm{T}_{\\mu_t}M$:\n$$\n\\frac{d}{dt}\\langle \\nabla \\Phi_t,\\nabla\\Psi_t\\rangle_{\\mu_t}\n\t=\\int [\\Gamma(\\partial_t \\Phi_t,\\Psi_t) +\\Gamma(\\Phi_t,\\partial_t\\Psi_t)+ \\Gamma(\\Gamma(\\Phi_t,\\Psi_t),\\Theta _t)]\\,d\\mu_t .\n$$\nBut from~\\eqref{eq-hessian1},  \n$$\n\\Gamma(\\Gamma(\\Phi_t,\\Psi_t),\\Theta _t)=\\nabla^2\\Psi_t (\\nabla \\Phi_t,  \\nabla \\Theta _t)+ \\nabla^2\\Phi_t (\\nabla \\Psi_t,  \\nabla \\Theta _t).\n$$\nHence \n$$\n\\frac{d}{dt}\\langle \\nabla \\Phi_t,\\nabla\\Psi_t\\rangle_{\\mu_t}=\\int\\PAR{\\nabla\\partial_t\\Phi_t+ \\nabla^2\\Phi_t \\nabla \\Theta _t}\\cdot\\nabla \\Psi_t\\,d\\mu_t +\\int\\PAR{\\nabla\\partial_t\\Psi_t+ \\nabla^2\\Psi_t \\nabla \\Theta _t}\\cdot \\nabla\\Phi_t\\,d\\mu_t .\n$$\nAs the covariant derivative  must  obey  the chain rule: \n\\begin{equation}\n\\label{eq-chain-rule}\n\\frac{d}{dt}\\langle \\nabla \\Phi_t,\\nabla\\Psi_t\\rangle_{\\mu_t}\n\t=\\langle  D_t\\nabla\\Phi_t ,\\nabla\\Psi_t\\rangle_{\\mu_t}+\\langle {\\nabla \\Phi_t}, D_t\\nabla\\Psi_t \\rangle_{\\mu_t},\n\\end{equation}\nwe have shown   \\eqref{eq-covariant-meca}. Introducing the convective derivative\n$$\n \\mathsf{D} _{ \\dot\\mu_t}:=\\partial_t+\\dot\\mu_t\\!\\cdot\\!\\nabla\n$$\n used in fluid mechanics,  \\eqref{eq-covariant-meca} writes as\n\\begin{align}\\label{eq-covariant-meca2}\n  D_t\\nabla\\Phi_t=  \\mathsf{Proj} _{ \\mu_t}  \\mathsf{D} _{ \\dot\\mu_t}\\nabla\\Phi_t.\n \\end{align}\n\n\n\\subsubsection*{Acceleration}\n\nThe acceleration $\\ddot \\mu_t= D_t\\dot\\mu_t$ of the path $(\\mu)$ is the covariant derivative of $\\dot\\mu_t=\\nabla \\Phi_t$.  It is given by \\eqref{eq-covariant-meca} and \\eqref{eq-covariant-meca2} with $\\nabla\\Phi=\\nabla \\Theta :$\n\\begin{equation}\n\\label{eq-def-secdon-derivative}\n\\ddot{\\mu}_t=\\nabla\\PAR{\\partial_t\\Phi_t+ \\frac{1}{2}\\Gamma(\\Phi_t)}\n\t= \\mathsf{D} _{ \\dot\\mu_t}\\dot\\mu_t\\in \\mathrm{T}_{\\mu_t}M.\n\\end{equation}\n It follows from \\eqref{eq-def-secdon-derivative} that any geodesic $(\\mu_t)$ in the Wasserstein space satisfies \n\\begin{equation}\n\\label{eq-mccann}\n\\left\\{\n\\begin{array}{l}\n\\displaystyle\\partial_t \\mu_t=-\\nabla\\cdot\\PAR{\\mu_t\\nabla \\Phi_t}\\\\\n\\displaystyle\\partial_t\\Phi_t+ \\frac{1}{2}\\Gamma(\\Phi_t)=0,\n\\end{array}\n\\right.\n\\end{equation}\nsee for instance \\cite{villani2009}. Such a geodesic is sometimes called a displacement interpolation or a McCann geodesic.\n\n\n\\subsubsection*{Hessian}\n\nLet $(\\mu_t)_{t\\ge 0}$ be a Wasserstein geodesic and $\\mathcal F$ a function on $M$. With \\eqref{eq-covariant-meca2} and $\\ddot\\mu_t=0,$ we see that \nthe Wasserstein Hessian of $\\mathcal F$ at $\\mu_t\\in M$, applied to $\\dot{\\mu}_t$ is \n\\begin{align*}\n\\Hess _{ \\mu_t}\\mathcal F(\\dot{\\mu}_t,\\dot{\\mu}_t)\n\t=\\frac{d^2}{dt^2}\\mathcal F(\\mu_t)\n\t=\\frac{d}{dt}\\langle {\\rm grad}_{\\mu_t} \\mathcal F,\\dot{\\mu}_t\\rangle_{\\mu_t}\n\t=\\langle D_t {\\rm grad}_{\\mu_t} \\mathcal F,\\dot{\\mu}_t\\rangle_{\\mu_t}\n=\\langle \\mathsf{D} _{ \\dot\\mu_t}{\\rm grad}_{\\mu_t} \\mathcal F,\\dot{\\mu}_t\\rangle_{\\mu_t}.\n\\end{align*}\nActually, we need some additional information about $\\mathcal F$ and  its gradient to give a more explicit expression of the Hessian of $\\mathcal F$.\n Let us have a look at the important case where  $$\\mathcal F(\\mu)=\\int f(\\mu)\\,d\\!\\vol  .$$\n In the subsequent lines,  $(\\mu_t)$ is a generic path with velocity $\\dot\\mu=\\nabla \\Phi$.\n\\\\\nSince,\n$\n D_t {\\rm grad}_{\\mu_t} \\mathcal F\n\t= -\\nabla\\!\\cdot\\!(\\mu\\nabla \\Phi_t))\\nabla (f''(\\mu_t))+ \\mathsf{Proj} _{ \\mu_t}\\nabla^2(f'(\\mu_t))\\nabla \\Phi_t,\n$\nwe see that\n$$\n\\Hess_\\mu\\mathcal F(\\dot{\\mu},\\dot{\\mu})\n\t=\\int \\PAR{-\\Gamma\\Big[f''(\\mu)\\nabla \\!\\cdot\\!(\\mu\\nabla \\Phi),\\Phi\\Big]+\\Gamma\\Big[f'(\\mu),\\Gamma(f'(\\mu),\\Phi)\\Big]-\\frac{1}{2}\\Gamma\\Big[f'(\\mu),\\Gamma(\\Phi)\\Big]}\\,d\\mu,\n$$\nwhere we  used\n\\begin{equation*}\n \\nabla^2 [f'(\\mu)](\\nabla \\Phi,\\nabla\\Phi)\n \t= \\Gamma\\Big( \\Gamma(f'(\\mu),\\Phi),\\Phi\\Big)\n\t\t- \\frac{1}{2} \\Gamma\\Big( \\Gamma(\\Phi),f'(\\mu)\\Big).\n\\end{equation*}\nIntegrating by parts, we obtain\n$$\n\\Hess_\\mu\\mathcal F(\\dot{\\mu},\\dot{\\mu})=\\int \\Big(\\SBRA{\\mu f''(\\mu)-\\mu f'(\\mu)+f(\\mu)}(\\Delta \\Phi)^2+\\SBRA{\\mu f'(\\mu)-f(\\mu)}\\Gamma_2(\\Phi)\\Big)\\,d\\!\\vol  .\n$$\nThese computations appear in~\\cite[p.\\,9]{otto-westickenberg2006}, see also~\\cite[p.\\,425]{villani2009}. \nThe Hessian of the entropy is\n\\begin{equation}\n\\label{eq-gamma2-3}\n\\Hess_\\mu {\\mathcal En}(\\dot{\\mu},\\dot{\\mu})=\\int\\Gamma_2(\\Phi)\\,d\\mu.\n\\end{equation}\nIn the case of a gradient flow where $\\nabla\\Phi=-{\\rm grad}_{\\mu} {\\mathcal En}=-\\nabla\\log \\mu$, we obtain\n\\begin{equation}\n\\label{eq-belle-formule}\n\\Hess_\\mu {\\mathcal En}({\\rm grad}_\\mu {\\mathcal En},{\\rm grad}_\\mu {\\mathcal En})=\\int \\Gamma_2(\\log\\mu)\\,d\\mu.\n\\end{equation}\nThe  Hessian of the R\u00e9nyi entropy \\eqref{eq-a04}, corresponding to \n$f(a)=a^p\/(p-1)$, is\n\\begin{equation}\n\\label{eq-gemma2-re}\n\\Hess_\\mu\\mathcal R_p(\\dot{\\mu},\\dot{\\mu})=\\int \\Big((p-1)(\\Delta \\Phi)^2+\\Gamma_2(\\Phi)\\Big)\\mu^{p-1}\\,d\\mu.\n\\end{equation}\n\n\n\n\\subsubsection*{Schwarz theorem}\n\n\nNext  result is a Schwarz theorem for a path in the Wasserstein space depending both on the time parameters $s$ and $t$. For the special purpose of next statement, we introduce the notation $$  \\mathsf{d}_t \\mu_t:=\\dot\\mu_t$$ for the expression \\eqref{eq-a03} of the dot derivative.\n\n\\begin{hlemma}[Commutation of the dot derivatives]\n\\label{fact-commu}\nFor any map $(\\mu_{s,t})_{s,t\\ge 0}$  from $[0,\\infty)^2$ to $M$,  \n\\begin{equation}\n\\label{eq-commu}\n  \\mathsf{d}_s  \\mathsf{d}_t {\\mu}_{s,t}=  \\mathsf{d}_t  \\mathsf{d}_s{\\mu}_{s,t}\\,\\in \\mathrm{T}_{{\\mu}_{s,t}}M.\n\\end{equation}\n\\end{hlemma}\n\nThis will be used later during the heuristic proof of Proposition \\ref{fact-newton}.\n\n\\begin{fproof}\nThere exist two functions $\\Phi_{s,t},\\Psi_{s,t}$ such that $  \\mathsf{d}_s \\mu _{ s,t}=\\nabla \\Phi _{ s,t}$ and $  \\mathsf{d}_t \\mu _{ s,t}=\\nabla \\Psi _{ s,t}$.  By \\eqref{eq-a03}, this means \n$$\n\\partial_s\\mu_{s,t}=-\\nabla\\cdot(\\mu_{s,t}\\nabla\\Phi_{s,t})\\quad{\\rm and}\\quad\\partial_t\\mu_{s,t}=-\\nabla\\cdot(\\mu_{s,t}\\nabla\\Psi_{s,t}).\n$$\nFrom~\\eqref{eq-covariant-meca}, \n\\begin{align*}\n \\mathsf{d}_s  \\mathsf{d}_t \\mu _{ s,t}\n\t&= D_s\\nabla \\Psi_{s,t}\n\t= \\mathsf{P}_{\\mu_{s,t}}\\PAR{\\nabla\\partial_s\\Psi_{s,t}+\\nabla^2\\Psi_{s,t}\\nabla\\Phi_{s,t}}\\\\\n\t \\mathsf{d}_t  \\mathsf{d}_s \\mu _{ s,t}\n\t&= D_t \\nabla \\Phi_{s,t}= \\mathsf{P}_{\\mu_{s,t}}\\PAR{\\nabla\\partial_t\\Phi_{s,t}+\\nabla^2\\Phi_{s,t}\\nabla\\Psi_{s,t}}.\n\\end{align*}\nOn the other hand, by the standard Schwarz theorem: $ \\partial_s \\partial_t\\mu_{s,t}= \\partial_t \\partial_s\\mu_{s,t}=:\\partial^2_{st}\\mu_{s,t},$ \n\\begin{align}\n\\label{eq-comm}\n\\begin{split}\n\\partial^2_{st}\\mu_{s,t}\n\t&=-\\nabla\\cdot(-\\nabla\\cdot(\\mu_{s,t}\\nabla\\Psi_{s,t})\\nabla\\Phi_{s,t}+\\mu_{s,t}\\nabla\\partial_t\\Phi_{s,t})\\\\\n\t&=-\\nabla\\cdot(-\\nabla\\cdot(\\mu_{s,t}\\nabla\\Phi_{s,t})\\nabla\\Psi_{s,t}+\\mu_{s,t}\\nabla\\partial_s\\Psi_{s,t}).\n\\end{split}\n\\end{align}\nIt is enough to prove that for any $\\nabla\\raisebox{0.35ex}{\\( \\chi \\)}\\in \\mathrm{T}_{\\mu_{s,t}}M$,\n\\begin{align}\\label{eq-a05}\n\\langle  \\mathsf{d}_s  \\mathsf{d}_t{\\mu}_{s,t}-  \\mathsf{d}_t  \\mathsf{d}_s{\\mu}_{s,t},\\nabla\\raisebox{0.35ex}{\\( \\chi \\)}\\rangle_{{\\mu}_{s,t}}=0.\n\\end{align}\nDropping the subscript $s,t$  and using \\eqref{eq-comm},  \n\\begin{multline*}\n\\langle( \\mathsf{d}_s  \\mathsf{d}_t{\\mu}-  \\mathsf{d}_t  \\mathsf{d}_s{\\mu},\\nabla\\raisebox{0.35ex}{\\( \\chi \\)}\\rangle_{{\\mu}}\n\t=\\int \\mu\\,\\big[\\nabla\\partial_s\\Psi+\\nabla^2\\Psi\\nabla\\Phi-\\nabla\\partial_t\\Phi-\\nabla^2\\Phi\\nabla\\Psi\\big]\\cdot\\nabla\\raisebox{0.35ex}{\\( \\chi \\)} \\,d\\!\\vol  \\\\\n\t=\\int \\Big[\\nabla^2\\Psi(\\nabla \\Phi,\\nabla \\raisebox{0.35ex}{\\( \\chi \\)})-\\nabla^2\\Phi(\\nabla \\Psi,\\nabla \\raisebox{0.35ex}{\\( \\chi \\)})-\\Delta\\Psi\\Gamma(\\Phi,\\raisebox{0.35ex}{\\( \\chi \\)})-\\Gamma(\\log\\mu,\\Psi)\\Gamma(\\Phi,\\raisebox{0.35ex}{\\( \\chi \\)})\\\\\n+\\Delta\\Phi\\Gamma(\\Psi,\\raisebox{0.35ex}{\\( \\chi \\)})+\\Gamma(\\log\\mu,\\Phi)\\Gamma(\\Psi,\\raisebox{0.35ex}{\\( \\chi \\)})\\Big]\\,d\\mu.\n\\end{multline*}\nFrom~\\eqref{eq-hessian1} we see that\n$$\n\\int \\Big[\\nabla^2\\Psi(\\nabla \\Phi,\\nabla \\raisebox{0.35ex}{\\( \\chi \\)})-\\nabla^2\\Phi(\\nabla \\Psi,\\nabla \\raisebox{0.35ex}{\\( \\chi \\)})\\Big]\\,d\\mu=\\int \\Big[\\Gamma(\\Phi,\\Gamma(\\Psi,\\raisebox{0.35ex}{\\( \\chi \\)}))-\\Gamma(\\Psi,\\Gamma(\\Phi,\\raisebox{0.35ex}{\\( \\chi \\)}))\\Big]\\,d\\mu.\n$$\nIntegrating by parts, \n$$\n\\int \\Delta\\Phi\\Gamma(\\Psi,\\raisebox{0.35ex}{\\( \\chi \\)})\\,d\\mu=-\\int \\Gamma(\\Phi,\\mu\\Gamma(\\Psi,\\raisebox{0.35ex}{\\( \\chi \\)}))\\,d\\!\\vol  =-\\int \\Big[\\Gamma(\\Phi,\\Gamma(\\Psi,\\raisebox{0.35ex}{\\( \\chi \\)}))+\\Gamma(\\Psi,\\raisebox{0.35ex}{\\( \\chi \\)})\\Gamma(\\Phi,\\log \\mu)\\Big]\\,d\\mu\n$$\nand something similar for $\\int \\Delta\\Psi\\Gamma(\\Phi,\\raisebox{0.35ex}{\\( \\chi \\)})\\,d\\mu$. Adding all these quantities, we obtain \\eqref{eq-a05}.\n\\end{fproof}\n\n\n\n\n\n\\subsection{$ \\ep\\mathcal F$-interpolations solve Newton equation}\n\\label{sec-newton-inf}\n\nIn this section we investigate the analogue in the Wasserstein space of Proposition \\ref{res-01}.\n\\\\\n   The carr\u00e9 du champ operator on $M$ is defined  for any functions $\\mathcal F, \\mathcal G: M\\mapsto \\R$,    by\n\\begin{equation}\n\\label{eq-def-grosgamma}\n\\boldsymbol{ \\Gamma}(\\mathcal F,\\mathcal G)(\\mu)\n\t:=\\langle{\\rm grad}_\\mu \\mathcal F, {\\rm grad}_\\mu \\mathcal G\\rangle_\\mu, \\quad \\mu\\in M,\n\\end{equation}\nwith  $\\boldsymbol{ \\Gamma}(\\mathcal F,\\mathcal F)(\\mu)=\\boldsymbol{ \\Gamma}(\\mathcal F)(\\mu)=|{\\rm grad}_\\mu\\mathcal F|^2_\\mu\\geq0$. \\\\\nFor instance, if $\\mathcal F(\\mu)=\\int f(\\mu)dx$ and $\\mathcal G(\\mu)=\\int g(\\mu)dx$ for some real functions $f$ and $g$, then \n$$\n\\boldsymbol{ \\Gamma}(\\mathcal F,\\mathcal G)(\\mu)=\\int \\nabla f'(\\mu)\\cdot\\nabla g'(\\mu)d\\mu=\\int \\Gamma(f'(\\mu),g'(\\mu))d\\mu=\\int f''(\\mu)g''(\\mu)\\Gamma(\\mu)d\\mu,\n$$\nwhere we have used the chain rule in $(N,g)$.   \n\nIn analogy with the Definition \\ref{def-inter-F} of the $ \\ep F$-cost  and $ \\ep F$-interpolations, we introduce the following \n\n\\begin{hdefi}[$ \\ep{\\mathcal F}$-cost, $ \\ep \\mathcal{F}$-interpolations]\nLet $\\mathcal F$ be a (regular enough) function on $M$. For any $ \\ep>0$ and $\\mu,\\nu\\in M$, we define the $ \\ep\\cF$-cost between $\\mu$ and $\\nu$ by\n\\begin{equation}\n\\label{eq-def-A}\n\\mathcal A^\\varepsilon_{\\mathcal F}(\\mu,\\nu)\n\t=\\inf _{ ( \\mu)}\\int_0^1\\PAR{\\frac{1}{2}|\\dot{\\mu}_s|^2_{\\mu_s}\n\t+ \\frac{\\varepsilon^2}{2}\\boldsymbol{ \\Gamma}(\\mathcal F)(\\mu_s)}ds,\n\\end{equation}\nwhere $|\\dot{\\mu}_s|^2_{\\mu_s}=\\langle \\dot{\\mu}_s,\\dot{\\mu}_s\\rangle_{\\mu_s}$, \nand the infimum runs over all  paths $(\\mu_s)_{0\\le s\\le 1}$ in $M$ such that $\\mu_0=\\mu$ and $\\mu_1=\\nu$. \n\\\\\nMinimizers are called $ \\ep\\mathcal F${\\it-interpolations} and are denoted by $(\\mu_s^{ \\ep,\\mu\\nu})_{0\\le s\\le 1}$ (for simplicity we sometimes omit $ \\ep,$ $\\mu$ and $\\nu$).\n\\end{hdefi}\n\\begin{itemize}\n\\item\nWhen $\\varepsilon=0$, we recover the   formula \\eqref{eq-01}.   \n\\item\nSince   $\\mathcal A^\\varepsilon_{\\mathcal F}\\ge \\frac{1}{2}W_2^2$, $\\mathcal A^\\varepsilon_{\\mathcal F}$ appears as an approximation from above of the squared  Wasserstein distance. \n\\end{itemize}\nThe $ \\ep\\mathcal F$-cost is the action  associated with the Lagrangian\n\\begin{align*}\n \\mathcal{L} ^{ \\ep}(\\mu,\\dot\\mu)\n \t=|\\dot{\\mu}|^2_{\\mu}\/2+\\varepsilon^2\\boldsymbol{ \\Gamma}(\\mathcal F)( \\mu)\/2\n\\end{align*}\ncorresponding to the scalar potential\n$$\\mathcal U ^{ \\ep}:=- \\frac{ \\ep^2}{2}\\boldsymbol{ \\Gamma}(\\mathcal F).$$\n\n\n\\begin{hprop}[Interpolations solve Newton equation]\n\\label{fact-newton}\nAny $ \\ep \\mathcal{F}$-interpolation $ \\mu$ satisfies the Newton equation\n\\begin{equation}\n\\label{eq-def-newton-3}\n\\ddot{\\mu}=\\frac{\\varepsilon^2}{2}{\\rm grad}_{\\mu}\\boldsymbol{ \\Gamma}(\\mathcal F).\n\\end{equation}\n\\end{hprop}\n\\begin{fproof}\nLet $ \\mu$ be an $ \\ep\\cF$-interpolation and take any    perturbation $(\\mu_{s,t})_{s\\in[0,1],t\\in\\mathbf{R}}$ of $ \\mu$ verifying:   $\\mu_{s,0}=\\mu_s$ for any $s\\in[0,1]$,   $\\mu_{0,t}=\\mu_0$, $\\mu_{1,t}=\\mu_1$ for any $t$, and $\n\\stackrel{\\odot}{\\mu}_{0,0}=\\stackrel{\\odot}{\\mu}_{1,0}=0,\n$\nwhere the Wasserstein $t$-velocity of $\\mu _{ s,t}$ is denoted by $ \\stackrel{\\odot}{\\mu}_{s,t}$, while its   $s$-velocity  is denoted as usual by $\\dot \\mu _{ s,t}.$\n  Let us differentiate the action\n$\n\\Lambda(t):=\\int_0^1\\PAR{|\\dot{\\mu}_{s,t}|^2_{\\mu_{s,t}}\/2-\\varepsilon^2\\mathcal U(\\mu_{s,t})}ds,\\ t\\in\\mathbf{R},\n$\nof the $t$-perturbation:\n$$\n\\Lambda'(t)=\\int_0^1\\PAR{\\langle\\dot{\\mu}_{s,t},\\stackrel{\\cdot\\odot}{\\mu}_{s,t}\\rangle_{\\mu_{s,t}}-\\varepsilon^2\\langle{\\rm grad}_{\\mu_{s,t}}\\mathcal U,\\stackrel{\\odot}{\\mu}_{s,t}\\rangle_{\\mu_{s,t}}}ds.\n$$ \nLet us integrate by parts $\\int_0^1\\langle\\dot{\\mu}_{s,t},\\stackrel{\\cdot\\odot}{\\mu}_{s,t}\\rangle_{\\mu_{s,t}}\\,ds.$\nBy the chain rule~\\eqref{eq-chain-rule}\n$$\n\\frac{d}{ds}\\langle\\dot{\\mu}_{s,t},\\stackrel{\\odot}{\\mu}_{s,t}\\rangle_{\\mu_{s,t}}=\\langle\\ddot{\\mu}_{s,t},\\stackrel{\\odot}{\\mu}_{s,t}\\rangle_{\\mu_{s,t}}+\\langle\\dot{\\mu}_{s,t},\\stackrel{\\cdot\\odot}{\\mu}_{s,t}\\rangle_{\\mu_{s,t}},\n$$\nand  by Lemma~\\ref{fact-commu}\n$$\n\\int_0^1{\\langle\\dot{\\mu}_{s,t},\\stackrel{\\cdot\\odot}{\\mu}_{s,t}\\rangle_{\\mu_{s,t}}}ds=-\\int_0^1\\langle\\ddot{\\mu}_{s,t},\\stackrel{\\odot}{\\mu}_{s,t}\\rangle_{\\mu_{s,t}}ds+\\langle\\dot{\\mu}_{1,t},\\stackrel{\\odot}{\\mu}_{1,t}\\rangle_{\\mu_{1,t}}-\\langle\\dot{\\mu}_{0,t},\\stackrel{\\odot}{\\mu}_{0,t}\\rangle_{\\mu_{0,t}}.\n$$\nTaking into account the boundary   conditions $\n\\stackrel{\\odot}{\\mu}_{0,0}=\\stackrel{\\odot}{\\mu}_{1,0}=0,\n$ we obtain at $t=0$\n$$\n\\Lambda'(0)=-\\int_0^1{\\big\\langle\\PAR{\\ddot{\\mu}_{s}+\\varepsilon^2{\\rm grad}_{\\mu_{s}}\\mathcal U},\\stackrel{\\odot}{\\mu}_{s,0}\\big\\rangle_{\\mu_{s}}}ds. \n$$ \nSince $(\\mu_s)_{0\\le s\\le 1}$ is a minimizer, $\\Lambda'(0)\\ge 0$. As $(\\stackrel{\\odot}{\\mu}_{s,0})$ can be chosen arbitrarily on any interval $s\\in [ \\delta, 1- \\delta]$ with $ \\delta>0,$  this shows that  $\\Lambda'(0)=0$, showing that the Newton equation~\\eqref{eq-def-newton-3} holds.\n\\end{fproof}\nFrom Newton's equation \\eqref{eq-def-newton-3} and the expression \\eqref{eq-def-secdon-derivative} of the acceleration $\\ddot \\mu$, one  deduces the subsequent result.\n\n\n\\begin{hcor}[Equations of motion of the  interpolations]\nThe $ \\ep\\mathcal F$-interpolation between $\\mu$ and $\\nu$ satisfies the system of equations\n\\begin{equation}\n\\label{eq-sys-int}\n\\left\\{\n\\begin{array}{l}\n\\displaystyle\\partial_s\\mu_s=-\\nabla\\cdot (\\mu_s\\nabla\\Phi_s),\\\\\n\\displaystyle\\nabla\\partial_s\\Phi_s+\\nabla\\Gamma(\\Phi_s)\/2\n\t=\\frac{\\varepsilon^2}{2}{\\rm grad}_{\\mu_s}\\boldsymbol{ \\Gamma}(\\mathcal F),\\\\\n\\displaystyle \\mu_0=\\mu,\\quad\\mu_1=\\nu.\n\\end{array}\n\\right.\n\\end{equation}\n\\end{hcor}\nWhen $\\varepsilon=0$, one recovers the system \\eqref{eq-mccann} satisfied by the McCann geodesics.    We also believe that \\eqref{eq-sys-int} implies regularity properties of the $ \\ep\\mathcal F$-interpolation. \n\n In $\\R^n$, the system~\\eqref{eq-sys-int} appears as the Euler equation where the initial and the final configuration are prescribed, \n$$\n\\left\\{\n\\begin{array}{l}\n\\displaystyle\\partial_s\\mu_s=-\\nabla\\cdot (\\mu_s\\nabla\\Phi_s),\\\\\n \\displaystyle{\\partial_s(\\mu_s\\nabla\\Phi_s)+\\nabla\\cdot(\\mu_s\\nabla\\Phi_s\\otimes\\nabla\\Phi_s)= \\frac{\\ep^2}{2}\\mu_s\\,{\\rm grad}_{\\mu_s}\\boldsymbol{ \\Gamma}(\\mathcal F)}, \\\\\n\\displaystyle \\mu_0=\\mu,\\quad\\mu_1=\\nu,\n\\end{array}\n\\right.\n$$\nwhere $\\nabla\\cdot$ is the divergence  applied to each column. This system is a particular case of~\\cite[Eq.\\,(17)]{gangbo-nguyen2008} where the solution   also  minimizes some action. \n\n\n\n\\medskip\\noindent \\textbf{Examples.}\nLet us treat two examples. \n\\begin{enumerate}[(1)]\n\\item In the case where $\\mathcal F={\\mathcal En}$ is the usual entropy,  the system~\\eqref{eq-sys-int} becomes \n\\begin{equation}\n\\label{eq-sys-entro}\n\\left\\{\n\\begin{array}{l}\n\\displaystyle\\partial_s\\mu_s=-\\nabla\\cdot (\\mu_s\\nabla\\Phi_s)\\\\\n\\displaystyle\\partial_s\\Phi_s+\\Gamma(\\Phi_s)\/2\n\t=\\varepsilon^2\n[\\Gamma(\\log\\mu_s)\/2-\\Delta\\mu_s\/\\mu_s]\\\\\n\\displaystyle \\mu_0=\\mu,\\quad\\mu_1=\\nu,\n\\end{array}\n\\right.\n\\end{equation}\nThis fundamental example is  related to the Schr\u00f6dinger problem. It will be developped carefully at Section~\\ref{sec-schrodinger}. \n\n\\item In the case where  $\\mathcal F= \\mathcal{R}_p$ is the R\u00e9nyi entropy \\eqref{eq-a04},  the system~\\eqref{eq-sys-int} becomes \n\\begin{equation*}\n\\left\\{\n\\begin{array}{l}\n\\displaystyle\\partial_s\\mu_s=-\\nabla\\cdot (\\mu_s\\nabla\\Phi_s);\\\\\n\\displaystyle\\partial_s\\Phi_s+\\frac{1}{2}\\Gamma(\\Phi_s)\n\t=\\frac{\\varepsilon^2 p^2}{2}\\PAR{(3-2p)\\mu_s^{2p-4}\\Gamma(\\mu_s)-2\\mu_s^{2p-3}\\Delta\\mu_s};\\\\\n\\displaystyle \\mu_0=\\mu,\\quad\\mu_1=\\nu.\n\\end{array}\n\\right.\n\\end{equation*}\nWith $p=1$, we are are back to the entropy ${\\mathcal En}$. When $p\\neq1$, we don't know how to solve this system. \n\\end{enumerate}\nIn the case where $\\mathcal F(\\mu)=\\int f(\\mu)\\,\\,d\\!\\vol ,$ we have  \n$$\n{\\rm grad}_{\\mu} \\mathcal F=\\nabla [f'(\\mu)]=f''(\\mu)\\nabla \\mu,\n$$ \nand\n$$\n\\boldsymbol{ \\Gamma}(\\mathcal F)(\\mu)=\\int f''(\\mu)^2\\,\\Gamma(\\mu)\\,d\\mu.\n$$\nMoreover, after some computations, we obtain \n$$\n{\\rm grad}_\\mu \\boldsymbol{ \\Gamma}(\\mathcal F)=-\\nabla\\big[2f''(\\mu)^2\\mu\\Delta\\mu+(2f'''(\\mu)f''(\\mu)\\mu+f''(\\mu)^2)\\Gamma(\\mu)\\big].\n$$\n\n \\bigskip\\noindent \\textbf{Dual formulation of the cost $\\mathcal A_\\mathcal F^\\ep$.}\\ \n\\medskip\n\nLet $\\mathcal H$ be a functional on $M$ and let $\\ep>0$. We define the Hamilton-Jacobi semigroup for any $t\\geq0$ and $\\nu \\in M$ by, ,\n\\begin{equation}\n\\label{eq-93-inf}\n\\mathcal Q_t^{\\ep \\mathcal F} \\mathcal{H}(\\nu)= \\inf \\left\\{ \\mathcal{H}(\\mu_0)+\\int_0^t \\frac{|\\dot \\mu_s|^2}{2} +\\frac{\\ep^2}{2}\\boldsymbol{ \\Gamma}(\\mathcal F)(\\mu_s) ds\\right\\},\n\\end{equation}\nwhere the infimum is running over all path $(\\mu_s)_{0\\leq s\\leq t}$ such that  $\\mu_t=\\nu$.  \nThe function $\\mathcal U:(t,\\nu) \\mapsto \\mathcal Q_t^{\\ep \\mathcal F} \\mathcal H (\\nu)$, satisfies  the Hamilton-Jacobi equation \n\\begin{equation}\n\\label{eq-94-inf}\n\\left\\{\n\\begin{array}{l}\n\\displaystyle\\partial_t \\mathcal U(t,\\nu)+\\frac{1}{2}\\boldsymbol{ \\Gamma} (\\mathcal U) (t,\\nu)=\\frac{\\ep^2}{2}\\boldsymbol{ \\Gamma}(\\mathcal F)(\\nu);\\\\\n\\mathcal U(0,\\cdot)=\\mathcal H(\\cdot).\n\\end{array}\n\\right.\n\\end{equation}\nEven if the definition of the function $\\mathcal Q_t^{\\ep \\mathcal F} \\mathcal{H}$ is formal, following the work initiated by Gangbo-Nguyen-Tudorascu~\\cite{gangbo-nguyen2008} (see also~\\cite{ambrosio-feng2014,hynd-kim2015} and also \\cite{gangbo2015} for their applications to the mean field games), the functional $\\mathcal U$ satisfies the Hamilton-Jacobi equation in the Wasserstein space in the sense of viscosity solutions. \n\\\\\nMinimizers of~\\eqref{eq-93-inf} are  solutions of the system, \n\\begin{equation}\n\\label{eq-95-inf}\n\\left\\{\n\\begin{array}{ll}\n\\displaystyle\\ddot \\mu_s=\\frac{\\ep^2}{2} {\\rm grad}_{\\mu_t} \\boldsymbol{ \\Gamma} (\\mathcal F), \\quad 0\\leq s\\leq t ;\\\\\n\\displaystyle\\dot \\mu_0={\\rm grad}_{\\mu_0} \\mathcal H, \\quad \\mu_t=\\nu .\\\\\n\\end{array}\\right.\n\\end{equation}\n\n\n\\begin{hprop}[Dual formulation of $\\mathcal A_\\mathcal F^\\ep$]\n\\label{prop-55}\nFor any $\\mu,\\nu \\in M$, \n$$\n\\mathcal A_\\mathcal F^\\ep (\\mu,\\nu)=\\sup _{ \\mathcal{H}}\\{ \\mathcal Q_1^{\\ep \\mathcal F} \\mathcal H(\\mu)-\\mathcal H(\\nu)\\}, \n$$\n where the supremum  runs through all functionals  $\\mathcal H$ on $M$.  \n\\end{hprop}\n\n\\begin{fproof}\nIt is the analogue of Proposition~\\ref{prop-dual}'s proof.\n\\end{fproof}\n\n\n\n\\begin{hprop}[Conserved quantity]\nLet $(\\mu_s)_{0\\le s\\le 1}$ be an $ \\ep\\mathcal F$-interpolation, then the map\n$$\ns\\mapsto |\\dot{\\mu}_s|^2_{\\mu_s}- \\ep ^2 \\boldsymbol{ \\Gamma} \\mathcal{F}(\\mu_s)\n$$\nis constant on $[0,1]$. \n\\end{hprop}\nThe result is obvious from the Newton equation~\\eqref{eq-def-newton-3} satisfied by the $ \\ep\\mathcal F$-interpolations $(\\mu_s)_{0\\le s\\le 1}$. \n\n\n\n\n\\section{Inequalities based upon $\\mathcal A_\\mathcal F^\\ep$} \\label{sec-ineq}\n\nWe go on  using  Section~\\ref{sec-1} as a guideline.  To this aim we need to introduce the curvature-dimension property in the Wasserstein space. \n\n\\subsection{$(\\rho, \\mathsf{n})$-convexity in the Wasserstein space}\n\n\n\\begin{defi}[$(\\rho, \\mathsf{n})$-convexity]\n\\label{def-cd-inf}\nA function $\\mathcal F$  on $M$  is $(\\rho, \\mathsf{n})$-convex, where $\\rho\\in\\R$ and $ \\mathsf{n}\\in(0,\\infty],$ if \n\\begin{equation}\n\\label{eq-def-cd-inf}\n\\Hess_\\mu \\mathcal F(\\dot{\\mu},\\dot{\\mu})\\ge  \\rho|\\dot{\\mu}|^2_\\mu+\\frac{1}{ \\mathsf{n}}\\langle {\\rm grad}_\\mu \\mathcal F,\\dot{\\mu}\\rangle_\\mu^2,\n\\end{equation}\nfor any $\\mu\\in M$ and $\\dot{\\mu}\\in T _{ \\mu}M$.\n\\end{defi}\nThis   analogue of Definition \\ref{def-23}  was introduced in~\\cite[Sec.\\,2.1]{erbar-kuwada2015}.\n\n\\begin{ex}\nLet us look at the examples of the standard and R\u00e9nyi entropies on the $n$-dimensional manifold  $(N,\\gg)$.\n\\begin{enumerate}[(1)]\n\\item  Suppose that  ${\\rm Ric}_\\gg\\ge \\rho\\, {\\rm Id}$ for some real $ \\rho$. Applying this inequality and   the Cauchy-Schwarz inequality: $||\\nabla^2\\Phi||^2_{\\mathrm{HS}}\\ge  (\\Delta \\Phi)^2\/n$,  to the expression \\eqref{eq-13} of $ \\Gamma_2,$ we see that the Bakry-\\'Emery curvature-dimension condition holds with $ \\mathsf{n}=n$, i.e.: for any function $f$,\n$$\n\\Gamma_2(f)\\ge  \\rho\\Gamma(f)+\\frac{1}{n}(\\Delta f)^2. \n$$\nBy \\eqref{eq-gamma2-3},  for any $\\dot{\\mu}=\\nabla \\Phi\\in \\mathrm{T}_\\mu M$, \n\\begin{multline*}\n\\Hess_\\mu {\\mathcal En}(\\dot{\\mu},\\dot{\\mu})=\\int \\Gamma_2(\\Phi)\\,d\\mu\\ge  \\rho\\int \\Gamma(\\Phi)\\,d\\mu+\\frac{1}{n}\\int (\\Delta \\Phi)^2\\,d\\mu\\\\\n\\ge  \n\\rho|\\dot{\\mu}|^2_\\mu+\\frac{1}{n}\\PAR{\\int \\Delta \\Phi \\,d\\mu}^2=\\rho|\\dot{\\mu}|^2_\\mu+\\frac{1}{n}\\langle{\\rm grad}_\\mu{\\mathcal En},\\dot{\\mu}\\rangle_\\mu^2,\n\\end{multline*}\nwhere we used  ${\\rm grad}_\\mu{\\mathcal En}=\\nabla\\log\\mu,$ see \\eqref{eq-a08}.\n\\\\\nIn other words, ${\\mathcal En}$ is $(\\rho,n)$-convex. \n\n\\item The R\u00e9nyi entropy $ \\mathcal{R}_p$, defined at \\eqref{eq-a04}, is convex whenever ${\\rm Ric}_\\gg\\ge  0$ and $p\\ge  1-1\/n ,$ $p\\neq1$. Indeed, it follows from~\\eqref{eq-gemma2-re} that\n$$\n\\Hess_\\mu \\mathcal R_p(\\dot{\\mu},\\dot{\\mu})\\ge (p-1+1\/n )\\int(\\Delta \\Phi)^2\\mu^{p-1}\\,d\\mu\\ge 0,\n$$\nwhere $\\dot{\\mu}=\\nabla \\Phi$.\n\\end{enumerate}\n\\end{ex}\n\n\n\n\\subsection{Contraction inequality}\n\n\nNext proposition is the infinite dimensional analogue of Proposition~\\ref{prop-contraction}.\n\\begin{hprop}[Contraction under $(\\rho, \\mathsf{n})$-convexity]\n\\label{prop-contration-inf}\nSuppose that $\\mathcal F$ is $(\\rho, \\mathsf{n})$-convex. Then for any $\\mu,\\nu\\in M$ and $t\\ge 0$, \n\\begin{equation}\n\\label{eq-con-lambda-n}\n\\mathcal A_{\\mathcal F}^\\varepsilon(\\mathsf{S} ^{ \\mathcal{F}}_t \\mu,\\mathsf{S} ^{ \\mathcal{F}}_t\\nu)\\le  e^{-2\\rho t}\\mathcal A_{\\mathcal F}^\\varepsilon(\\mu,\\nu)-\\frac{1}{ \\mathsf{n}}\\int_0^te^{-2\\rho(t-u)}(\\cF(\\mathsf{S} ^{ \\mathcal{F}}_u\\mu)-\\cF(\\mathsf{S} ^{ \\mathcal{F}}_u\\nu))^2du.\n\\end{equation}\n\\end{hprop}\n\n\n\\begin{fproof}\nThe proof follows the line of Proposition \\ref{prop-contraction}. The Wasserstein setting analogue of the implication $(i)\\Rightarrow (ii)$ of Lemma \\ref{res-03} which is used in Proposition \\ref{prop-contraction}'s proof, still holds true. Its relies on the commutation of the dot derivatives stated at ``Lemma'' \\ref{fact-commu}.\n\\end{fproof}\n\nThis result is proved and stated rigorously at Theorem~\\ref{theo-contration-inf-ri} in the context of a compact and smooth Riemannian manifold. It gives a better contraction inequality than \\cite[Thm.\\,6.1]{gentil-leonard2017}. \n\n\nIn the case of $\\varepsilon=0$, we recover the dimensional contraction proved in~\\cite{bolley-gentil2014,gentil2015,bolley-gentil2018}. See also~\\cite{bolley-gentil-guillin2018}. When $\\varepsilon>0$ and $\\rho=0$, an improved contraction inequality is also given in~\\cite{ripani2017} for the usual entropy. The equivalence between contraction inequality for the Wasserstein distance ($\\ep=0$) with $ \\mathsf{n}=\\infty$ and a lower bound on the Ricci curvature was first proved  in~\\cite{renesse-sturm2005}. See also~\\cite{kuwada2015,bakry-gentil2015} for a dimensional contraction with two different times. \n\n\n \n\n\\subsection{Convexity properties along $ \\ep\\cF$-interpolations}\n\nTwo kinds of convexity properties can be explored for the cost $\\mathcal A_\\mathcal F^\\ep$. When $\\mathcal F={\\mathcal En}$ is the usual entropy,  the first convexity property  has been introduced by Conforti under a $(\\rho,\\infty)$-convexity assumption and the second one by the third author under a $(0, \\mathsf{n})$-convexity assumption.  \n\n\nNext result  extends in our context   Conforti's convexity inequality  \\cite[Thm~1.4]{conforti2017}. \nLet us recall our notation\n$ \\displaystyle{\n\\theta_a(s):= \\frac{1- e ^{ -2as}}{1- e ^{ -2a}}.\n}$\nNote that $\\lim _{ a\\to 0} \\theta_a(s)=s.$ \n\n\n\\begin{hprop}[Convexity under the $(\\rho,\\infty)$-condition]\n\\label{prop-conforti}\nLet  $ \\mathcal{F}$ be a $(\\rho,\\infty)$-convex function with $\\rho\\in\\R$. Then  any $ \\ep \\mathcal{F}$-interpolation $(\\mu_s)_{0\\le s\\le 1}$ satisfies \n\\begin{equation}\\label{eq-conforti}\n\\begin{split}\n \\mathcal{F}(\\mu_s)\\le \\theta _{ \\rho \\ep}(1-s) \\mathcal{F}(\\mu_0)&+\n\\theta _{ \\rho \\ep}(s)\\mathcal{F}(\\mu_1)\\\\\n &- \\frac{1- e ^{ -2 \\rho \\ep}}{2 \\ep} \\theta _{ \\rho \\ep}(s) \\theta _{ \\rho \\ep}(1-s)\n [ \\mathcal{A}_{ \\mathcal{F}}^\\varepsilon(\\mu_0,\\mu_1)+ \\ep \\mathcal{F}(\\mu_0)+ \\ep\t \\mathcal{F}(\\mu_1)].\n\\end{split}\n\\end{equation}\n\\end{hprop}\n\n\\begin{fproof}\nExact analogue of Proposition \\ref{prop-86}'s proof.\n\\end{fproof}\n\nWhen $\\rho=0$ the inequality~\\eqref{eq-conforti} simply implies that the map $s\\mapsto \\mathcal F(\\mu_s)$ is convex: a result obtained by the second author for the usual entropy in~\\cite{leonard2017}. When $\\varepsilon=0$, we recover the usual convexity of $\\mathcal F$ along McCann interpolations: the starting point of the Lott-Sturm-Villani theory~\\cite{sturm2006,lott-villani2009}, \n$$\n\\mathcal F(\\mu_s)\\le  (1-s)\\mathcal F(\\mu_0)+\ns\\mathcal F(\\mu_1)\n -2\\rho s(1-s)W_2^2(\\mu_0,\\mu_1),\n \\quad \\forall 0\\le s\\le 1.\n$$\nFrom the above inequality, when $\\rho>0$, we can deduce a Talagrand inequality relating the Wasserstein distance with the entropy. Indeed, taking $s=1\/2$, one obtains \n$$\nW_2^2(\\mu_0,\\mu_1)\\le  \\frac{1}{\\rho}\\PAR{\\mathcal F(\\mu_0)+\\mathcal F(\\mu_1)}.\n$$\nThe same property holds for the cost $\\mathcal A_{\\mathcal F}^\\ep$ as proposed in~\\cite[Cor.\\,1.2]{conforti2017}. \n\n\n\\begin{hcor}[Talagrand  inequality for the cost $\\mathcal A_{\\mathcal F}^\\ep$]\nAssume that $ \\mathcal{F}$ is  $(\\rho,\\infty)$-convex with $\\rho>0$ and that it is normalized by $\\inf  \\mathcal{F}=0.$ Then for any $ \\ep>0$ and  $\\mu,\\nu\\in M$, \n$$\n\\mathcal A_{\\mathcal F}^\\ep(\\mu,\\nu)\\le  \\frac{\\ep(1+e^{-\\rho\\ep})}{1-e^{-\\rho\\ep}}\\PAR{ \\mathcal{F}(\\mu)+ \\mathcal{F}(\\nu)}.\n$$ \nIn particular, if $m\\in M$ minimizes $ \\mathcal{F}:$ $\\inf \\mathcal{F}= \\mathcal{F}( m)=0$, then for any $\\mu,$\n\\begin{align*}\n\\mathcal A_{\\mathcal F}^\\ep(\\mu,m)\\le  \\frac{\\ep(1+e^{-\\rho\\ep})}{1-e^{-\\rho\\ep}}\\mathcal{F}(\\mu).\n\\end{align*}\n\\end{hcor}\nThis is the exact analogue  of Proposition \\ref{res-a01}.\n\n\nNext result is a generalization of a result proved for the usual entropy by the third author~\\cite{ripani2017}.\n\n\\begin{hprop}[Convexity under the $(0, \\mathsf{n})$-condition]\n\\label{prop-rip-inf}\nSuppose that $\\mathcal F$ is $(0, \\mathsf{n})$-convex  with $ \\mathsf{n}>0$. Then, for any $\\mathcal F$-interpolation $(\\mu_s)_{0\\le s\\le 1}$, the map  \n\\begin{equation}\n\\label{eq-rip-inf}\n[0,1]\\ni s\\mapsto \\exp\\PAR{-\\mathcal F (\\mu_s)\/ \\mathsf{n} },\n\\end{equation}\nis concave.\n\\end{hprop}\n\\begin{fproof}\nAnalogous to the proof of Proposition \\ref{prop-85}.\n\\end{fproof}\n\n\n\n\n\n\\begin{hprop}[EVI inequality under $(\\rho,\\infty)$ or $(0, \\mathsf{n})$-convexity]\n\\label{prop-derivative-evi-inf}\n\\ \n\\begin{enumerate}[(a)]\n\\item Assume that $ \\mathcal{F}$ is $(\\rho,\\infty)$-convex. Then, for any $\\mu,\\nu\\in M$,\n\\begin{equation}\n\\label{eq-evi-inf}\n\\frac{d}{dt}^+\\Big|_{t=0} \\mathcal{A}_{ \\mathcal{F}}^\\varepsilon(\\mathsf{S} ^{ \\mathcal{F}}_t\\mu,\\nu)\n\t+ \\rho \\mathcal{A}_{ \\mathcal{F}}^\\varepsilon(\\mu,\\nu)\n\t\\le  \\frac{\\rho\\varepsilon(1+e^{-2\\rho\\varepsilon})}{1-e^{-2\\rho\\varepsilon}} [ \\mathcal{F}(\\nu)- \\mathcal{F}(\\mu)].\n\\end{equation}\n\n\\item Assume that $ \\mathcal{F}$ is $(0, \\mathsf{n})$-convex. Then, for any $\\mu,\\nu\\in M$,\n$$\n\\frac{d}{dt}^+\\Big|_{t=0} A_{ \\mathcal{F}}^\\varepsilon(\\mathsf{S} ^{ \\mathcal{F}}_t\\mu,\\nu)\\le  n[1-e^{-(\\mathcal{F}(\\nu)- \\mathcal{F}(\\mu))\/ \\mathsf{n} }].\n$$\n\\end{enumerate}     \n\\end{hprop}\n\n\\begin{fproof}\nIt follows exactly the line of proof of Proposition \\ref{prop-88}. The analogues of the preliminary results Corollary \\ref{cor-15} and Proposition \\ref{prop-87} are derived by means of the commutation of the dot derivatives obtained at ``Lemma\" \\ref{fact-commu}.\n\\end{fproof}\n\n\nWhen $\\ep=0$, we recover from~\\eqref{eq-evi-inf}, the EVI inequality under the $(\\rho,\\infty)$-convexity of $\\mathcal F$,\n$$\n\\frac{d}{dt}^+\\Big|_{t=0}\\frac{1}{2}W_2^2(\\mathsf{S} ^{ \\mathcal{F}}_t \\mu,\\nu)\n\t+\\frac{\\rho}{2}W_2^2(\\mu,\\nu)\n\t\\le \\mathcal F(\\nu)-\\mathcal F(\\mu).\n$$\nThis inequality is commented on for instance in~\\cite{ambrosio-gigli2008}. \n\\\\Under the $(0, \\mathsf{n})$-convexity of $\\mathcal F$, the inequality has the same form \n$$\n\\frac{d}{dt}^+\\Big|_{t=0}\\frac{1}{2}W_2^2(\\mathsf{S} ^{ \\mathcal{F}}_t \\mu,\\nu)\\le  n\\PAR{1-e^{-\\frac{1}{ \\mathsf{n}}(\\mathcal F(\\nu)-\\mathcal F(\\mu))}}.\n$$\nIt was proved in~\\cite{erbar-kuwada2015}.\n\n\n\n\n\\section{Link with the Schr\u00f6dinger problem}\n\\label{sec-schrodinger}\n\n\\subsection{Entropic cost} \n\nThis section is dedicated to the specific setting where \n\\begin{align*}\n \\mathcal{F}( \\mu)={\\mathcal En}(\\mu)=\\int \\mu\\log\\mu \\,d\\!\\vol  .\n\\end{align*}\n  We depart from the heuristic style of previous sections and provide rigorous results. This requires  some regularity hypotheses: $(N,\\gg)$ is assumed to a compact, connected and smooth Riemannian manifold. \n\\\\\nThe Riemannian measure $ \\mathrm{vol} $ is normalized  to be  a probability measure.  Since all the measures considered in this section are smooth and absolutely continuous with respect to  $ \\mathrm{vol} $, we identify  measures and  densities. \nThe heat semigroup is denoted by $( \\mathsf{P}_t) _{ t\\ge 0}$ and defined  for any smooth function $f$ by $ \\mathsf{P}_tf(x)=u(t,x)$ where $u$ solves \n$\n\\left\\{ \\begin{array}{l}\n\\partial_t u=\\Delta u,\\\\\nu(0,\\cdot)=f,\n\\end{array}\\right.\n\\ t\\ge0.\n$\n\\\\\n We already saw that the heat equation is the  Wasserstein gradient flow of ${\\mathcal En}$, see \\eqref{eq-12}.  We also know   that\n\\begin{align*}\n{\\rm grad}_{\\mu} {\\mathcal En}=\\nabla{\\log\\mu},\\qquad\n\\boldsymbol{ \\Gamma}({\\mathcal En})(\\mu)=\\int\\big|\\nabla\\log {\\mu}\\big|^2\\,d\\mu,\n\\end{align*}\nand we expect that any $ \\ep{\\mathcal En}$-interpolation $( \\mu)$ solves the Newton equation \\eqref{eq-def-newton-3}:\n\\begin{align}\\label{eq-16}\n\\ddot{\\mu}=\\frac{\\varepsilon^2}{2}{\\rm grad}_{\\mu}\\boldsymbol{ \\Gamma}({\\mathcal En}).\n\\end{align}\n The gradient flow of the usual entropy is denoted by $\\mathsf{S} ^{ \\entro}_t$ and for any $\\mu\\in\\mathcal P(M)$, \n$$\n\\mathsf{S} ^{ \\entro}_t(\\mu)= \\mathsf{P}_t\\PAR{\\frac{d\\mu}{d{\\rm vol}}}{\\rm vol},\n$$\nwhere $ \\mathsf{P}_t=\\exp(t \\Delta)$ is the usual heat semigroup.\n\nThe rigorous definition of the cost $ \\mathcal{A} ^{ \\ep} _{\\entro}$ on the space $(N,\\gg)$ is the following.\n\\begin{defi}[Entropic cost]\nFor any   positive and smooth densities $\\mu,\\nu\\in M$,  \n$$\n\\mathcal A^\\varepsilon_{{\\mathcal En}}(\\mu,\\nu)\n\t=\\inf _{ ( \\mu_t,\\nabla\\Phi_t) _{ 0\\le t\\le 1}}\\BRA{\\int_0^1\\PAR{\\frac{1}{2}|\\nabla\\Phi_t|^2_{\\mu_t}+\\frac{\\varepsilon^2}{2}\\int\\big|\\nabla\\log{\\mu_t}\\big|^2\\,d\\mu_t }dt},\n$$\nwhere the infimum runs through all smooth and positive paths $(\\mu_t)_{ 0\\le t\\le 1}$ and smooth gradient vector fields $(\\nabla\\Phi_t)_{ 0\\le t\\le 1}$ such that \n\\begin{align}\\label{eq-a10}\n\\left\\{\n\\begin{array}{l}\n\\partial_t\\mu_t=-\\nabla\\cdot(\\mu_t\\nabla \\Phi_t),\\\\\n\\mu_0=\\mu,\\quad \n\\mu_1=\\nu.\n\\end{array}\n\\right.\n\\end{align} \n\n\\end{defi}\nAnother equivalent formulation is\n\\begin{align}\\label{eq-14}\n\\mathcal A^\\varepsilon_{{\\mathcal En}}(\\mu,\\nu)\n\t=\\inf _{ ( \\mu_t)}\\BRA{\\int_0^1\\PAR{\\frac{1}{2}|\\dot{\\mu}_t|^2_{\\mu_t}+\\frac{\\varepsilon^2}{2}\\int\\big|\\nabla\\log{\\mu_t}\\big|^2\\,d\\mu_t }dt},\n\\end{align}\nwhere the infimum is taken over all smooth $( \\mu_t)_{ 0\\le t\\le 1}$  such that $ \\mu_0=\\mu$ and $ \\mu_1=\\nu.$\n\n\n\\subsection{Schr\u00f6dinger problem} \n\nLet us recall the Schr\u00f6dinger problem as explained for instance in the survey~\\cite{leonard2014}. \nFor any two probability measures $ \\mathsf{q}, \\mathsf{r}$ on some measure space, the  {\\it relative entropy} of $ \\mathsf{q}$ with respect to $ \\mathsf{r}$ is  \n$$\nH( \\mathsf{q}| \\mathsf{r})=\n\\left\\{\n\\begin{array}{ll}\n\\displaystyle\\int \\log\\frac{d \\mathsf{q}}{d \\mathsf{r}}\\, d \\mathsf{q},\\quad& {\\rm if}\\ \\mathsf{q}\\ll \\mathsf{r};\\\\\n+\\infty,&{\\rm otherwise}.\n\\end{array}\n\\right.\n$$\nThe usual entropy is   ${\\mathcal En}(\\mu)=H(\\mu| \\mathrm{vol} ), $ $ \\mu\\in \\mathcal{P}(N)$.\nThe set of all probability measures on the path space $\\Omega=\\cC([0,1],N)$ is denoted by $\\mathcal P(\\Omega).$ For any $\\ep>0$,  $$R^\\ep\\in \\mathcal P(\\Omega)$$  is the law of the reversible Brownian motion with generator \n${\\ep}\\Delta $\nand initial measure $ \\mathrm{vol} $. Note that this motion has the same law as  $ \\sqrt{2 \\ep}$ times a standard reversible Brownian motion.  The joint law of the initial and final positions of this  process is the Gaussian measure \n$$\nR_{01}^\\ep(dxdy)=(4\\pi \\ep) ^{ -n\/2}\\exp(-d^2(x,y)\/4 \\ep)\\, \\mathrm{vol}(dx)  \\mathrm{vol}(dy)  \\in \\mathcal P(N\\times  N)\n$$ \nwhere $d$ is the Riemannian distance.\nFor any couple of measures $\\mu,\\nu\\in M=\\mathcal P(N)$ such that  $H(\\mu| \\mathrm{vol} )={\\mathcal En}(\\mu)<+\\infty$ and $H(\\nu| \\mathrm{vol} )={\\mathcal En}(\\nu)<+\\infty$, the  Schr\u00f6dinger problem is\n\\begin{equation}\\label{eq-15}\n \\textrm{minimize}\\quad \\ep H(Q|R^\\ep) \\quad   \\textrm{subject to }\\  Q \\in \\mathcal P(\\Omega): Q_{0}=\\mu, Q_{1}=\\nu,\n\\end{equation}\nwhere for any $0\\le s\\le 1,$ $Q_s=(X_s)\\pf Q\\in \\mathcal{P}(N)$ denotes the $s$-marginal of $Q$, i.e.\\ the law of the position $X_s$ at time $s$ of the canonical process with law $Q.$  Its value \n\\begin{align*}\n{\\rm Sch}^\\ep(\\mu,\\nu) := \\inf \\eqref{eq-15}\n\\end{align*}\nis called the Schr\u00f6dinger cost, its solution $Q$ the Schr\u00f6dinger bridge and the time-marginal flow $(Q_s) _{ 0\\le s\\le 1}$  the entropic interpolation, between $ \\mu$ and $\\nu.$ The Schr\u00f6dinger problem admits an equivalent static formulation:\n\\begin{equation*}\n{\\rm Sch}^\\ep(\\mu,\\nu) = \\inf \\{\\ep H(\\pi|R_{01}^\\ep) ; \\  \\pi \\in \\mathcal P(N\\times N): \\pi_{0}=\\mu, \\pi_{1}=\\nu\\},\n\\end{equation*}\nwhere $\\pi_0,\\pi_1\\in \\mathcal{P}(N)$ are the marginals of $\\pi.$\nIt was proved by Beurling \\cite{Beu60} and F\u00f6llmer \\cite{follmer1988}, see also~\\cite{leonard2014},  that for any probability measures $\\mu,\\nu\\in\\mathcal P(N)$ with finite entropy and finite second order moments, the minimum of \\eqref{eq-15} is achieved by a unique probability measure, whose shape is characterized by the product form\n\\begin{equation}\n\\label{eq-opq}\nP=f(X_0)g(X_1) R^\\ep\\in\\mathcal P(\\Omega),\n\\end{equation}\nwhere $f$ and $g$ are nonnegative functions (the $\\ep$-dependence is omitted on the functions $f$ and $g$). The logarithm of these functions play the same role as the Kantorovich potentials in the optimal transportation theory. \n\\\\\nThe time-marginals of the minimizer $P$ define a  flow of probability measures $(\\mu_s)_{0\\le s\\le 1}$ called the \\emph{entropic $ \\ep$-interpolation} between $\\mu$ and $\\nu$. It follows from the Markov property of the Brownian motion and \\eqref{eq-opq} that  $(\\mu_s)_{0\\le s\\le 1}$ takes the particular form \n\\begin{equation}\\label{eq-entropic-int}\n\\mu_s= \\mathsf{P}_{\\ep s}f\\,  \\mathsf{P}_{\\ep(1-s)}g \\, \\mathrm{vol} ,\n\\end{equation}\nwhere  in the present context $ \\mathsf{P}_t=\\exp(t \\Delta)$ is the heat  semigroup.\n\\\\\nIt is proved in~\\cite{mikami2004}, see also \\cite{leonard2012} for a more general result, that\n$$\n\\lim_{\\ep\\rightarrow 0}{\\rm Sch}^\\ep(\\mu,\\nu)=W_2^2(\\mu,\\nu)\/4.\n$$\nNext lemma establishes a connection  between the  Schr\u00f6dinger problem and  previous sections.  \n\n\\begin{lem}[Entropic interpolation  solves \\eqref{eq-sys-entro}]\n\\label{lem-systeme}\nLet $(\\mu_s)_{0\\le s\\le 1}$ be a path of probability measures on $N$ such that \n$$\n\\mu_s= \\mathsf{P}_{\\ep s}f  \\mathsf{P}_{\\ep(1-s)}g\\  \\mathrm{vol} .\n$$ \nwhere  $f$ and $g$ are smooth and positive functions on $N$. Then $(\\mu_s)_{0\\le s\\le 1}$ solves \\eqref{eq-sys-entro}, i.e.:\n\\begin{equation}\n\\label{eq-sys-entro2}\n\\left\\{\n\\begin{array}{l}\n\\displaystyle\\partial_s\\mu_s=-\\nabla\\cdot (\\mu_s\\nabla\\Phi_s)\\\\\n\\displaystyle\\partial_s\\Phi_s+\\Gamma(\\Phi_s)\/2\n\t={\\varepsilon^2}\\PAR{\\Gamma(\\log\\mu_s)\/2-{\\Delta\\mu_s}\/{\\mu_s}},\n\\end{array}\n\\right.\n\\end{equation}\nwith $\\Phi_s= \\ep\\log  \\mathsf{P}_{\\ep(1-s)}g- \\ep\\log  \\mathsf{P}_{\\ep s}f$.\n\\end{lem}\nThis system has an explicit solution  given by $\\mu_s=\\exp\\big(({\\phi_s+\\psi_s})\/{\\ep}\\big)$ where $  \\varphi_s:= \\ep\\log  \\mathsf{P}_{\\ep s}f$ and $ \\psi_s:=  \\ep\\log \\mathsf{P}_{\\ep(1-s)}g.$    \nThis gives  $\\Phi_s=\\psi_s-\\phi_s$ and $\\phi_s$, $\\psi_s$ satisfy the Hamilton-Jacobi-Bellman \nequations\n\\begin{equation}\n\\label{eq-inter-2}\n\\partial_s \\phi_s=\\ep\\Delta \\phi_s+\\Gamma(\\phi_s),\\qquad \\partial_s \\psi_s=-\\ep\\Delta \\psi_s-\\Gamma(\\psi_s).\n\\end{equation}\nA heuristic presentation of this computation was proposed at page \\pageref{eq-sys-entro}. \n\n\nIn $\\R^n$, the system~\\eqref{eq-sys-entro2} appears as the pressureless Euler equation:\n$$\n\\left\\{\n\\begin{array}{l}\n\\displaystyle\\partial_s\\mu_s=-\\nabla\\cdot (\\mu_s\\nabla\\Phi_s),\\\\\n\\displaystyle{\n\\partial_s(\\mu_s\\nabla\\Phi_s)+\\nabla\\cdot(\\mu_s\\nabla\\Phi_s\\otimes\\nabla\\Phi_s)=-\\frac{\\ep^2}{2}\\mu_s\\,\\nabla\\PAR{\\frac{\\Delta\\sqrt{\\mu_s}}{\\sqrt{\\mu_s}}},} \\\\\n\\displaystyle \\mu_0=\\mu,\\quad\\mu_1=\\nu.\n\\end{array}\n\\right.\n$$\nThis system is a particular case of~\\cite[Eq.\\,(1.1)]{feng-nguyen2012}.  Let us emphasize that this specific system admits the entropic interpolation as an explicit smooth solution; this was   unnoticed in~\\cite{feng-nguyen2012}.\n\n\n\n\\subsubsection*{Newton equation}\nRecall that the acceleration is given at \\eqref{eq-def-secdon-derivative} by\n$\n\\ddot{\\mu}_t=\\nabla\\PAR{\\partial_t\\Phi_t+ \\frac{1}{2}\\Gamma(\\Phi_t)}\n\t\\in \\mathrm{T}_{\\mu_t}M.\n$\nTogether with \\eqref{eq-sys-entro2}, this leads us to the Newton equation \\eqref{eq-16}: \n\\begin{align}\\label{eq-16b}\n\\ddot \\mu=  \\ep^2\\nabla ( \\Gamma(\\log \\mu)\/2- \\Delta \\mu\/ \\mu),\n\\end{align}\nwhich was recently derived by Conforti in \n\\cite{conforti2017}.\n\nAfter the seminal works of Schr\u00f6dinger in 1931 \\cite{Sch31,Sch32}, different aspects of entropic interpolations were investigated  by Bernstein  \\cite{bernstein1932}  and almost thirty years later by  Beurling \\cite{Beu60}.  During the 70's, Jamison rediscovered this theory \\cite{Jam75} and  this also happened independently during the 80's to Zambrini  \\cite{Zam86} who initiated the theory of Euclidean quantum theory  \\cite{CZ91,CZ08}. The works by Jamison and Zambrini opened the way to the study of second order stochastic differential equations in order to derive Newton equations for the Schr\u00f6dinger bridges in terms of random paths, see for instance the papers by Krener, Thieullen and Zambrini \\cite{Th93,Kre97,TZ97a}.\n\nThe equation \\eqref{eq-16b} in the Wasserstein space has a different nature than the above mentioned Newton equations in the path space. It   sheds  a new light on the dynamics of the entropic interpolations.\n\n\n\\subsection{Benamou-Brenier and Kantorovich formulas} \n\nWe show that the Schr\u00f6dinger  cost ${\\rm Sch}^\\ep$ is equal, up to an additive constant, to the entropic cost $\\mathcal A^\\varepsilon_{{\\mathcal En}}$, using the Benamou-Brenier-Schr\u00f6dinger formula.  This was proved in~\\cite{chengeorgiou2016} and in~\\cite[Cor.\\,5.3]{gentil-leonard2017} in $\\R^n$ with a Kolmogorov generator and in a more general case, like  a $ \\mathrm{RCD}^*(K,N)$ space, in~\\cite[Th.5.4.3]{tamanini2017}, for instance in the compact Riemannian manifold $(N,\\gg)$. \n\n\n\\begin{theo}[Benamou-Brenier-Schr\u00f6dinger formula] For any couple of positive and smooth probability measures $\\mu,\\nu$ on $N$, \n\\begin{equation}\n\\label{eq-bbsch} \n{\\rm Sch}^\\ep(\\mu,\\nu)\n\t=\\mathcal A^\\varepsilon_{{\\mathcal En}}(\\mu,\\nu)\n\t+ \\ep[{\\mathcal En}(\\mu)+{\\mathcal En}(\\nu)]\/2.\n\\end{equation}\nSee \\cite{chengeorgiou2016,gentil-leonard2017,tamanini2017,gigli-tamanini2018}.\n\\end{theo}\n\n\n\\begin{eproof}\nLet us recall, in our context, the proof proposed in~\\cite[Thm.\\,5.4.3]{tamanini2017}.   Let $(\\mu_s)_{0\\le s\\le 1}$ be a path  from  $\\mu$ to $\\nu$ with velocity $\\nabla \\Psi_s$, and let $(\\nu_s)_{0\\le s\\le 1}$ be the entropic interpolation between $\\mu$ and $\\nu$ that is the path given by \\eqref{eq-entropic-int} for some functions $f$ and $g$. If $(\\nabla\\Phi_s)_{0\\le s\\le 1}$ denotes its velocity, then from Lemma~\\ref{lem-systeme}, the couple $(\\nu_s,\\nabla\\Phi_s)_{0\\le s\\le 1}$ satisfies \\eqref{eq-sys-entro2}. We have\n$$\n\\frac{d}{ds}\\int \\Phi_s\\,d\\mu_s\n\t=\\int\\partial_s\\Phi_s\\,d\\mu_s+\\int\\Phi_s \\partial_s\\mu_s\\,\\,d\\!\\vol \n\t=\\int\\partial_s\\Phi_s\\,d\\mu_s+\\int\\Gamma(\\Phi_s,\\Psi_s)\\,d\\mu_s,\n$$\nand from~\\eqref{eq-sys-entro2} and an integration by parts,  \n\\begin{multline*}\n\\frac{d}{ds}\\int \\Phi_s\\,d\\mu_s=\\int\\SBRA{-\\frac{1}{2}\\Gamma(\\Phi_s)+\\frac{\\ep^2}{2}\\PAR{\\Gamma(\\log\\nu_s)-2\\frac{\\Delta\\nu_s}{\\nu_s}}+\\Gamma(\\Phi_s,\\Psi_s)}\\,d\\mu_s\\\\\n\t=\\int\\SBRA{-\\frac{1}{2}\\Gamma(\\Phi_s)+\\Gamma(\\Phi_s,\\Psi_s)+\\frac{\\ep^2}{2}\\Big(2\\Gamma(\\log\\mu_s,\\log\\nu_s)-\\Gamma(\\log\\nu_s)\\Big)}\\,d\\mu_s.\n\\end{multline*}\nBy \\cite[Prop.~4.1.5]{tamanini2017}, the functions $f$ and $g$ are positive and smooth. Applying  Cauchy-Schwarz inequality  to  $\\Gamma$, we obtain \n$$\n\\frac{d}{ds}\\int \\Phi_s\\,d\\mu_s\\le \\int \\SBRA{\\frac{1}{2}\\Gamma(\\Psi_s)+\\frac{\\ep^2}{2}\\Gamma(\\log\\mu_s)}\\,d\\mu_s,\n$$\nwith equality if $\\Psi_s=\\Phi_s$ (which implies that $\\mu_s=\\nu_s$). Hence, \n\\begin{equation}\\label{eq-equality}\n\\int \\Phi_1d\\nu-\\int \\Phi_0\\,d\\mu\\le \\int_0^1\\int \\SBRA{\\frac{1}{2}\\Gamma(\\Psi_s)+\\frac{\\ep^2}{2}\\Gamma(\\log\\mu_s)}\\,d\\mu_sds,\n\\end{equation}\nand taking the minimum over all paths $(\\mu_s)_{0\\le s\\le 1}$, we have obtained\n$$\n\\int \\Phi_1d\\nu-\\int \\Phi_0\\,d\\mu\\le \\mathcal A^\\varepsilon_{{\\mathcal En}}(\\mu,\\nu),\n$$\nBut when $\\mu_s=\\nu_s$, \\eqref{eq-equality} is an equality. This implies that\n$$\n\\int \\Phi_1d\\nu-\\int \\Phi_0\\,d\\mu=\\mathcal A^\\varepsilon_{{\\mathcal En}}(\\mu,\\nu).\n$$\nSince $(\\nu_s)$ is  optimal, we know by \\eqref{eq-entropic-int} that  $\\nu_s= \\mathsf{P}_{\\ep s}f \\mathsf{P}_{\\ep(1-s)}g$ and $\\Phi_s=\\ep\\log  \\mathsf{P}_{\\ep(1-s)}g-\\ep\\log  \\mathsf{P}_{\\ep s}f$. Hence, \n\\begin{multline*}\n\\int \\Phi_1d\\nu-\\int \\Phi_0\\,d\\mu=2\\ep\\PAR{\\int f \\mathsf{P}_{\\ep }g\\log f\\,d\\!\\vol  +\\int g \\mathsf{P}_{\\ep }f\\log g\\,d\\!\\vol  }-\\ep({\\mathcal En}(\\mu)+{\\mathcal En}(\\nu))\\\\\n=2\\ep H(P|R_{01}^\\ep)-\\ep({\\mathcal En}(\\mu)+{\\mathcal En}(\\nu)),\n\\end{multline*}\nwhere $P$ is given in~\\eqref{eq-opq}. Then \n$$\n\\int \\Phi_1d\\nu-\\int \\Phi_0\\,d\\mu=2\\rm Sch^{\\varepsilon} (\\mu,\\nu)-{\\varepsilon}\\PAR{{\\mathcal En}(\\mu)+{\\mathcal En}(\\nu)},\n$$\nand the Benamou-Brenier-Schr\u00f6dinger formula~\\eqref{eq-bbsch} is proved. \n\\end{eproof}\n\nThis result tells us that the action minimization problem \\eqref{eq-14} and the Schr\u00f6dinger problem \\eqref{eq-15} are equivalent. In particular, the minimizers coincide and satisfy  Newton's equation~\\eqref{eq-16}.  This was  proved rigorously  in~\\cite[Th.\\,1.3]{conforti2017} in the same setting.  A similar formal computation is done in~\\cite{renesse2012} for the Schr\u00f6dinger equation (not the problem).  \n\nWe propose now a new proof of the dual formulation of the Schr\u00f6dinger problem. The proof is given in~\\cite{mikami-thieullen2006} by using stochastic control and in~\\cite[Thm~4.1]{gentil-leonard2017} in a more genera context. The one proposed here is simpler.  Let us note that independently and at the same moment, the result was also proved in a $ \\mathrm{RCD}^*(\\rho, \\mathsf{n})$ space in~\\cite{gigli-tamanini2018}.\n\n\n\\begin{theo}[Kantorovich-Schr\u00f6dinger dual formulation]\n\\label{theo-kanto}\nFor any couple of positive and smooth probability measures $\\mu,\\nu$ on $N$, \n\\begin{equation}\n\\label{eq-kanto} \n\\varepsilon\\sup_{h\\in\\cC(N)}\\left\\{\\int\\! \\log h\\, d\\nu-\\int\\! \\log  \\mathsf{P}_\\ep h\\,d\\mu\\right\\}=\\frac{1}{2} \\mathcal{A} ^{ \\ep} _{\\entro}(\\mu,\\nu)+\\frac{\\varepsilon}{2}({\\mathcal En}(\\nu)-{\\mathcal En}(\\mu)).\n\\end{equation}\nSee \\cite{mikami-thieullen2006,gentil-leonard2017,gigli-tamanini2018}.\n\\end{theo}\n\n\n\\begin{eproof}  \nFor any   $\\ep{\\mathcal En}$-interpolation $(\\nu_s)_{0\\le s\\le 1}$ between $\\mu$ and $\\nu$ with velocity  $(\\nabla\\Phi_s)_{0\\le s\\le 1}$ and any smooth and positive  function $h$ on $N$,\n\\begin{align*}\n&\\int \\log h\\,d\\nu-\\int \\log  \\mathsf{P}_\\ep h\\,d\\mu\\\\\n\t= &\\int_0^1\\frac{d}{ds}\\int \\log  \\mathsf{P}_{\\ep(1-s)}h\\,d\\nu_s \\,ds\\\\\n\t= &\\int_0^1\\int \\PAR{-\\ep\\frac{\\Delta  \\mathsf{P}_{\\ep(1-s)}h}{ \\mathsf{P}_{\\ep(1-s)}h}+\\Gamma(\\log  \\mathsf{P}_{\\ep(1-s)}h,\\Phi_s)}d\\nu_s\\,ds\\\\\n\t= &\\int_0^1\\PAR{\\int \\ep\\Gamma\\Big(\\frac{\\nu_s}{ \\mathsf{P}_{\\ep(1-s)}h},  \\mathsf{P}_{\\ep(1-s)}h\\Big)\\,d\\!\\vol  +\\int\\Gamma(\\log  \\mathsf{P}_{\\ep(1-s)}h,\\Phi_s)d\\nu_s}ds\\\\\n\t= &\\int_0^1\\int\\PAR{ \\Gamma(\\ep\\log \\nu_s+\\Phi_s,\\log  \\mathsf{P}_{\\ep(1-s)}h)-\\ep \\Gamma(\\log  \\mathsf{P}_{\\ep(1-s)}h)}d\\nu_s\\,ds.\n\\end{align*}\nNow, because the $\\ep{\\mathcal En}$-interpolation writes as $\\nu_s= \\mathsf{P}_{\\ep s}f \\mathsf{P}_{\\ep(1-s)}g\\, \\mathrm{vol} $ for some positive smooth and bounded functions $f$ and $g$ (see  \\cite{tamanini2017} again), and\n$\\Phi_s=\\ep\\log  \\mathsf{P}_{\\ep(1-s)}g-\\ep\\log  \\mathsf{P}_{\\ep s}f$, we have:  $\\ep\\log \\nu_s+\\Phi_s=2\\ep\\log  \\mathsf{P}_{\\ep(1-s)}g $.\nThis leads to the following inequality\n\\begin{multline*}\n\\int\\! \\log hd\\nu-\\int\\! \\log  \\mathsf{P}_\\ep h\\,d\\mu=\n\\ep\\int_0^1\\!\\!\\!\\int\\!\\!\\!\\PAR{ 2\\Gamma(\\log  \\mathsf{P}_{\\ep(1-s)}g,\\log  \\mathsf{P}_{\\ep(1-s)}h)-\\Gamma(\\log  \\mathsf{P}_{\\ep(1-s)}h)}d\\nu_s\\,ds\\\\\n\\le \\ep\\int_0^1\\!\\int\\! \\Gamma(\\log  \\mathsf{P}_{\\ep(1-s)}g)d\\nu_s\\,ds=\\int\\! \\log gd\\nu-\\int\\! \\log  \\mathsf{P}_\\ep g\\,d\\mu,\n\\end{multline*}\nwith equality when $h=g$. In other words, \n$$\n\\sup_{h\\in\\cC(N)}\\left\\{\\int\\! \\log hd\\nu-\\int\\! \\log  \\mathsf{P}_\\ep h\\,d\\mu\\right\\}=\\int\\! \\log gd\\nu-\\int\\! \\log  \\mathsf{P}_\\ep g\\,d\\mu.\n$$\nMoreover, \n\\begin{multline*}\n\\int\\! \\log gd\\nu-\\int\\! \\log  \\mathsf{P}_\\ep g\\,d\\mu=\\int\\! (\\log g)g \\mathsf{P}_{\\ep}f\\,d\\!\\vol  -\\int\\! (\\log  \\mathsf{P}_\\ep g) f \\mathsf{P}_\\ep g\\,d\\!\\vol  \\\\\n=\\int\\! (\\log g)g \\mathsf{P}_{\\ep}f\\,d\\!\\vol  +\\int\\! (\\log f) f \\mathsf{P}_\\ep g\\,d\\!\\vol  -\\int\\! (\\log f  \\mathsf{P}_\\ep g) f \\mathsf{P}_\\ep g\\,d\\!\\vol  =H(P|R^\\ep)-{\\mathcal En}(\\mu),\n\\end{multline*}\nwhere $P$ is the optimal probability measure given in~\\eqref{eq-opq}. Taking equation \\eqref{eq-bbsch} into account,  this concludes the proof. \n\\end{eproof}\n\n\nLet us see how the dual formulation proposed at Proposition~\\ref{prop-55} is in fact the same as the one in~\\eqref{eq-kanto}. Let us define for any $t\\geq0$ and $\\mu\\in M$, \n\\begin{equation}\n\\label{eq-216}\n\\mathcal U(t,\\mu):=\\int Q_t^\\ep fd\\mu+\\ep{\\mathcal En}(\\mu), \n\\end{equation}\nwhere $Q_t^\\ep f=-2\\ep\\log  \\mathsf{P}_{\\ep t} {\\exp(-{f}\/{2\\ep})}$ is the solution of the Hamilton-Jacobi-Bellman equation in $N$, \n$$\n\\partial_t u+\\frac{1}{2}\\Gamma(u)=\\ep\\Delta u.\n$$\nThen, from a direct computation, the map $\\mathcal U$ solves the Hamilton-Jacobi equation in the Wasserstein space:\n\\begin{equation}\n\\label{eq-215}\n\\partial_t \\mathcal U(t,\\mu)+\\frac{1}{2}\\boldsymbol{ \\Gamma} (\\mathcal U) (t,\\nu)=\\frac{\\ep^2}{2}\\boldsymbol{ \\Gamma}({\\mathcal En})(\\mu),\n\\end{equation}\nstarting from the initial condition $\\mu\\mapsto \\int fd\\mu+\\ep{\\mathcal En}(\\mu).$\n\nBy Proposition~\\ref{prop-55}, we have \n$$\n\\sup_{f}\\BRA{\\int Q_t^\\ep fd\\mu+\\ep{\\mathcal En}(\\mu)-\\int fd\\nu-\\ep{\\mathcal En}(\\nu)}=\\cA_{\\mathcal En}^\\ep(\\mu,\\nu).\n$$\nOn the other hand from the definition of $Q_t^\\ep f$, \n\\begin{multline*}\n\\sup_{f}\\BRA{\\int Q_t^\\ep f\\,d\\mu+\\ep{\\mathcal En}(\\mu)-\\int f\\,d\\nu-\\ep{\\mathcal En}(\\nu)}=\\\\\n\\ep\\sup_{h}\\BRA{\\int \\log h\\,d\\nu-\\log P_{\\ep}h\\,d\\mu }+\\frac{\\ep}{2}\\PAR{{\\mathcal En}(\\mu)-{\\mathcal En}(\\nu)},\n\\end{multline*}\nhence we recover formally the identity~\\eqref{eq-kanto}. It is interesting to notice that the Hamilton-Jacobi equation~\\eqref{eq-215} has a ``smooth\" solution given by~\\eqref{eq-216}. Let us note that the dual formulation of the Schr\\\"odinger problem has also been computed recently as a particular case  in~\\cite{bowles-ghoussoub2018}.\n\n\n\n\\subsection{Contraction inequality} \n\nWe are now able to give a new and  rigorous result. \n\\begin{theo}[Contraction under $(\\rho,n)$-convexity]\n\\label{theo-contration-inf-ri}\nSuppose that $(N,\\gg)$ is a smooth, connected $n$-dimensional Riemannian manifold with $Ric_g\\ge  \\rho$ for some $\\rho\\in\\R$. Then for any smooth and positive probability measures $\\mu$, $\\nu$,  \n\\begin{equation}\n\\label{eq-con-lambda-ri}\n\\mathcal A_{{\\mathcal En}}^\\varepsilon(\\mathsf{S} ^{ \\entro}_t\\mu,\\mathsf{S} ^{ \\entro}_t\\nu)\\le  e^{-2\\rho t}\\mathcal A_{{\\mathcal En}}^\\varepsilon(\\mu,\\nu)-\\frac{1}{n}\\int_0^te^{-2\\rho(t-u)}({\\mathcal En}(\\mathsf{S} ^{ \\entro}_u\\mu)-{\\mathcal En}(\\mathsf{S} ^{ \\entro}_u\\nu))^2du,\n\\end{equation}\nfor any $t\\ge 0$. \n\\end{theo}\n\\begin{eproof}\nLet $(\\mu_s)_{0\\le s\\le 1}$ be any smooth and positive path between $\\mu$ and $\\nu$ satisfying \n$$\n\\partial_s\\mu_s=-\\nabla\\cdot(\\mu_s\\nabla\\Phi_s), \n$$ \nwith a smooth velocity  $\\nabla\\Phi_s$. \nLet us denote for any $t\\ge 0$, $\\mu_{s,t}:=\\mathsf{S} ^{ \\entro}_t(\\mu_s)$. It is a path between $\\mathsf{S} ^{ \\entro}_t(\\mu)$ and $\\mathsf{S} ^{ \\entro}_t(\\nu)$. From the continuity equation, \n$$\n\\partial_s\\mathsf{S} ^{ \\entro}_t(\\mu_s)=-\\mathsf{S} ^{ \\entro}_t(\\nabla\\cdot(\\mu_s\\nabla\\Phi_s))=-\\nabla\\cdot \\PAR{\\mathsf{S} ^{ \\entro}_t(\\mu_s)\\frac{\\ensuremath{\\mathbf{R}}_t(\\mu_s\\nabla\\Phi_s)}{\\mathsf{S} ^{ \\entro}_t(\\mu_s)}}, \n$$\nwhere $(\\ensuremath{\\mathbf{R}}_t)_{t\\ge 0}$ is the Hodge-de Rham semigroup on forms. The definition and properties of this semigroup can be found in this context in~\\cite{gentil2015}. Of course the $s$-velocity  $\\dot{\\mu}_{s,t}$ of $\\mu_{s,t}$ is not equal to  \n${\\ensuremath{\\mathbf{R}}_t(\\mu_s\\nabla\\Phi_s)}\/{\\mathsf{S} ^{ \\entro}_t(\\mu_s)}$ since the latter is not a gradient. But  some inequality remains valid:\n$$\n|\\dot{\\mu}_{s,t}|_{{\\mu}_{s,t}}^2\\le  \\int \\left|\\frac{\\ensuremath{\\mathbf{R}}_t(\\mu_s\\nabla\\Phi_s)}{\\mathsf{S} ^{ \\entro}_t(\\mu_s)}\\right|^2 \\mathsf{S} ^{ \\entro}_t(\\mu_s)\\,d\\!\\vol  .\n$$\nHence \n\\begin{equation}\n\\label{eq-longue}\n\\mathcal A_{{\\mathcal En}}^\\varepsilon( \\mathsf{P}_tf,  \\mathsf{P}_tg)\\le \\int_0^1\\int\\SBRA{ \\frac12\\left|\\frac{\\ensuremath{\\mathbf{R}}_t(\\mu_s\\nabla\\Phi_s)}{ \\mathsf{P}_t(\\mu_s)}\\right|^2+\\frac{\\varepsilon^2}{2}\\big|\\nabla\\log{ \\mathsf{P}_t(\\mu_s)}\\big|^2}  \\mathsf{P}_t(\\mu_s)\\,d\\!\\vol  ds.\n\\end{equation}\nNow we follow \\cite[Th~3.8]{gentil2015}. Let $t$ be fixed and for any $r\\in[0,t]$,  \n$$\n\\Lambda(r):= \\mathsf{P}_r\\PAR{\\frac{|\\ensuremath{\\mathbf{R}}_{t-r}(\\omega)|^2}{ \\mathsf{P}_{t-r}(g)}}+\\varepsilon^2 \\mathsf{P}_r\\PAR{\\Gamma(\\log  \\mathsf{P}_{t-r}g) \\mathsf{P}_{t-r}g},\n$$\nwhere $\\omega$ is some given 1-form  and $g$ is some given smooth and positive function on $N$. First, from the usual computations on Markov semigroups by using~\\cite[Th~5.5.3]{bgl-book} and the Ricci bound on $(N,\\gg)$, \n$$\n\\frac{\\partial}{\\partial_r} \\mathsf{P}_r\\PAR{\\Gamma(\\log  \\mathsf{P}_{t-r}g) \\mathsf{P}_{t-r}g}=2 \\mathsf{P}_r\\PAR{\\Gamma_2(\\log  \\mathsf{P}_{t-r}g) \\mathsf{P}_{t-r}g}\\ge  2\\rho  \\mathsf{P}_r\\PAR{\\Gamma(\\log  \\mathsf{P}_{t-r}g) \\mathsf{P}_{t-r}g}.\n$$\nAs in the proof of~\\cite[Th~3.8]{gentil2015} we obtain \n\\begin{multline*}\n\\Lambda'(r)\\ge  2\\rho  \\mathsf{P}_r\\PAR{\\frac{|\\ensuremath{\\mathbf{R}}_{t-r}(\\omega)|^2}{ \\mathsf{P}_{t-r}(g)}}+2\\rho \\ep^2  \\mathsf{P}_r\\PAR{\\Gamma(\\log  \\mathsf{P}_{t-r}g) \\mathsf{P}_{t-r}g}\\\\\n+\\frac{2}{n}\\frac{\\SBRA{ \\mathsf{P}_t(\\nabla\\cdot\\omega)- \\mathsf{P}_r(\\nabla(\\log  \\mathsf{P}_{t-r}g)\\cdot \\ensuremath{\\mathbf{R}}_{t-r}\\omega)}^2}{ \\mathsf{P}_rg}\\\\\n=2\\rho\\Lambda(r)+\\frac{2}{n}\\frac{\\SBRA{ \\mathsf{P}_t(\\nabla\\cdot\\omega)- \\mathsf{P}_r(\\nabla(\\log  \\mathsf{P}_{t-r}g)\\cdot \\ensuremath{\\mathbf{R}}_{t-r}\\omega)}^2}{ \\mathsf{P}_rg}.\n\\end{multline*}\nIntegrating over  $[0,t]$, \n$$\n\\Lambda(0)\\le  e^{-2\\rho t}\\Lambda(t)-\\frac{2}{n}\\int_0^te^{-2\\rho r}\\frac{\\SBRA{ \\mathsf{P}_t(\\nabla\\cdot \\omega)- \\mathsf{P}_r(\\nabla(\\log  \\mathsf{P}_{t-r}g)\\cdot \\ensuremath{\\mathbf{R}}_{t-r}\\omega)}^2}{ \\mathsf{P}_rg}dr. \n$$\nLet us choose $\\omega=\\mu_s\\nabla\\Phi_s$ and $g=\\mu_s$. Integrate this inequality with respect to the probability measure $\\,d\\!\\vol  $, by~\\eqref{eq-longue}\n\\begin{multline*}\n\\mathcal A_{{\\mathcal En}}^\\varepsilon( \\mathsf{P}_tf,  \\mathsf{P}_tg)\\le  e^{-2\\rho t}\\frac{1}{2}\\int_0^1\\int\\SBRA{|\\nabla\\Phi_s|^2\\,d\\mu_s+{\\ep^2}{\\Gamma(\\log \\mu_s)}}\\,d\\mu_sds\\\\\n-\\frac{1}{n}\\int_0^t\\int_0^1\\int e^{-2\\rho r}\\frac{\\SBRA{ \\mathsf{P}_t(\\nabla\\cdot(\\mu_s\\nabla\\Phi_s))- \\mathsf{P}_r(\\nabla(\\log  \\mathsf{P}_{t-r}\\mu_s)\\cdot \\ensuremath{\\mathbf{R}}_{t-r}(\\mu_s\\nabla\\Phi_s))}^2}{ \\mathsf{P}_r\\mu_s}\\,d\\!\\vol  dsdr.\n\\end{multline*}\nApplying twice  the Cauchy-Schwarz inequality, \n\\begin{multline*}\n\\int_0^t\\int_0^1\\int e^{-2\\rho r} \\frac{\\SBRA{ \\mathsf{P}_t(\\nabla\\cdot(\\mu_s\\nabla\\Phi_s))- \\mathsf{P}_r(\\nabla(\\log  \\mathsf{P}_{t-r}\\mu_s)\\cdot \\ensuremath{\\mathbf{R}}_{t-r}(\\mu_s\\nabla\\Phi_s))}^2}{ \\mathsf{P}_r\\mu_s}\\,d\\!\\vol  dsdr\\ge \\\\ \n\\int_0^te^{-2\\rho (t-r)}\\PAR{\\int_0^1\\int \\nabla(\\log  \\mathsf{P}_{r}\\mu_s)\\cdot \\ensuremath{\\mathbf{R}}_{r}(\\mu_s\\nabla\\Phi_s)\\,d\\!\\vol  ds}^2dr.\n\\end{multline*}\nNow, since\n$$\n\\int \\nabla(\\log  \\mathsf{P}_{r}\\mu_s)\\cdot \\ensuremath{\\mathbf{R}}_{r}(\\mu_s\\nabla\\Phi_s)\\,d\\!\\vol  =-\\int \\log ( \\mathsf{P}_{r}\\mu_s) \\mathsf{P}_{r}(\\mu_s\\nabla\\Phi_s)\\,d\\!\\vol  =\\frac{d}{ds}{\\mathcal En}( \\mathsf{P}_r\\mu_s), \n$$\nwe have\n$$\n\\int_0^1\\int \\nabla(\\log  \\mathsf{P}_{r}\\mu_s)\\cdot \\ensuremath{\\mathbf{R}}_{r}(\\mu_s\\nabla\\Phi_s)\\,d\\!\\vol  ds={\\mathcal En}( \\mathsf{P}_r\\nu)-{\\mathcal En}( \\mathsf{P}_r\\mu).\n$$\nTaking the infimum over all smooth and positive paths $(\\mu_s)$ leads to the announced inequality.  \n\\end{eproof}\n\n\n\\begin{rem}\nThis new result is interesting since the first  one, proved in~\\cite{gentil-leonard2017}, was not satisfactory at the light of this one. Indeed, we don't need to use a distortion of the time to obtain a contraction inequality. The Benamou-Brenier-Schr\u00f6dinger formulation, namely the definition of the cost $ \\mathcal{A} ^{ \\ep} _{ \\mathcal{F}}$, appears  as a efficient  method to deal with the Schr\u00f6dinger problem and many others inequalities. \n\nOf course, this contraction inequality can be generalized to a more general operator $\\Delta_g+V$ where $V$ is a vector field in $(N,\\gg)$. \n\\end{rem}\n\n\\subsection{Conclusion}\n\\begin{itemize}\n\\item  The general cost $ \\mathcal{A} ^{ \\ep} _{ \\mathcal{F}}$ shares the same curvature properties than the Wasserstein distance, which is quite surprising for the Schr\u00f6dinger problem as a minimisation problem of the relative entropy. \n\n\\item  We believe that the $\\cF$-interpolations are smooth as in the case of the Schr\u00f6dinger problem. This gives a way used in~\\cite{gigli-tamanini} to reach the second order differentiation formula  on the Wasserstein space. \n\\end{itemize}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{SEC:intro}\n\nWe consider the following Cauchy problem for the quadratic nonlinear Schr\\\"odinger equation (NLS) on the two-dimensional torus $\\mathbb{T}^2 = (\\mathbb{R} \/ \\mathbb{Z})^2$:\n\\begin{equation}\n\\begin{cases}\ni \\partial_t u + \\Delta u = |u|^2 \\\\\nu|_{t = 0} = u_0.\n\\end{cases}\n\\label{qNLS}\n\\end{equation}\n\n\n\\noindent\nOur main goal is to prove local well-posedness of \\eqref{qNLS} in the low regularity setting.\nOver the  last forty years, the Cauchy problems of NLS with various types of nonlinearities \nhave been studied extensively.\nThe following NLS with a gauge-invariant nonlinearity:\n\\begin{equation}\ni \\partial_t u + \\Delta u =  \\pm |u|^{p-1} u \n\\label{NLS}\n\\end{equation}\n\n\\noindent\nis the most well-known and well-studied example.\nIn the case of the $d$-dimensional torus $\\mathbb{T}^d = (\\mathbb{R}\/\\mathbb{Z})^d$, Bourgain  \\cite{B93} \nintroduced the Fourier restriction norm method (via the $X^{s, b}$-spaces)\nand proved local well-posedness of \\eqref{NLS}\nin  the low regularity setting.\nIn particular, when $d = 2$, he proved the following \n$L^4$-Strichartz estimate (with a derivative loss):\n\\begin{align}\n\\| e^{it \\Delta}  f\\|_{L^4([0, 1]; L^4(\\mathbb{T}^2))} \\lesssim \\|f\\|_{H^s}\n\\label{Str1}\n\\end{align}\n\n\\noindent\nfor any $s> 0$, which allowed him to prove\nlocal well-posedness of the cubic NLS, \\eqref{NLS} with $p = 3$, \nin $H^s(\\mathbb{T}^2)$, for any $s> 0$. We point out that the estimate \\eqref{Str1}\nfails when $s = 0$ and \nthe well-posedness issue of the cubic NLS\nin the critical space $L^2(\\mathbb{T}^2)$ remains\na challenging open problem; see \\cite{B93, TTz, Kishi}.\n\nLet us now turn to the quadratic NLS \\eqref{qNLS}.\nBy interpolating \\eqref{Str1} with the trivial $L^2$-bound, \nwe obtain the $L^3$-Strichartz estimate with a derivative loss.\nThen by proceeding as in~\\cite{B93}, \nwe immediately obtain local well-posedness\nof \\eqref{qNLS} in $H^s(\\mathbb{T}^2)$ for any $s > 0$.\nDue to the derivative loss in the Strichartz estimate, \nhowever, its well-posedness in $L^2(\\mathbb{T}^2)$ has remained open \nfor the last thirty years.\nIn this paper, we prove that \\eqref{qNLS} is\nindeed locally well-posed in $L^2(\\mathbb{T}^2)$.\n\n\n\n\\begin{theorem}\\label{THM:LWP}\nThe quadratic NLS \\eqref{qNLS} is locally well-posed in $L^2(\\mathbb{T}^2)$. More precisely, given any $u_0 \\in L^2(\\mathbb{T}^2)$, there exists $T = T(\\| u_0 \\|_{L^2}) > 0$ and a unique solution $u\n\\in C([-T, T]; L^2(\\mathbb{T}^2))$ to \\eqref{qNLS} with $u|_{t=0} = u_0$.\n\\end{theorem}\n\n\nOur proof is based on the Fourier restriction norm method and, as such, \n the uniqueness holds only in (the local-in-time version of) the relevant $X^{s,b}$-space.\n\nA few remarks are in order.\nIn \\cite{Kishi3}, Kishimoto proved ill-posedness of \\eqref{qNLS} in $H^s(\\mathbb{T}^2)$ for $s < 0$\nand thus\nTheorem \\ref{THM:LWP} is sharp.  See Remark \\ref{REM:1} below.\nTheorem \\ref{THM:LWP} is also sharp in the sense that \nlocal-in-time solutions constructed in Theorem \\ref{THM:LWP}\ncan not be in general extended globally in time.\nIndeed,  the second author \\cite{O1} proved a finite-time blowup result\nfor \\eqref{qNLS}. See also~\\cite{FO}.\nThis argument can be easily extended to $L^2(\\mathbb{T}^2)$, yielding the following proposition.\n\n\\begin{proposition}\nLet $s \\geq 0$ and $u_0 \\in H^s(\\mathbb{T}^2)$. If the initial data $u_0$ satisfies\n\\[ \\Im \\int_{\\mathbb{T}^2} u_0 dx < 0 \\quad \\text{or} \\quad \\Re \\int_{\\mathbb{T}^2} u_0 dx \\neq 0,  \\]\nthen the forward maximal existence time $T^*$ of the solution $u$ to \\eqref{qNLS} \nwith $u|_{t = 0} = u_0$ is finite\nand we have  $\\liminf_{t \\nearrow T^*} \\| u (t) \\|_{H^s} = \\infty$.\n\\end{proposition}\n\n\nSee \\cite{FO} on the lifespan of solutions to \\eqref{qNLS}.\nWe point out that, in the one-dimensional case, Fujiwara and Georgiev \\cite{FG}\nrecently proved the criterion for global existence.\nConsider the quadratic NLS \\eqref{qNLS} on the one-dimensional torus $\\mathbb{T}$.\nThen, there exists a global $L^2$-solution $u$ to \\eqref{qNLS} on $\\mathbb{T}$\nif and only if $\\Re u_0 = 0$ and $\\Im u_0 = \\mu$\nfor some $\\mu \\geq 0$.  In particular, any global $L^2$-solution\nis necessarily constant in space. It would be of interest to investigate this issue\nin the two-dimensional case.\n\n\n\nAs mentioned above, our proof of Theorem \\ref{THM:LWP} is based\non the Fourier restriction norm method.\nMore precisely, Theorem \\ref{THM:LWP} follows\nfrom a standard contraction argument, once we proof the following bilinear estimate.\n\n\\begin{proposition}\\label{PROP:bilin}\nLet $0 < T \\leq 1$. Let $\\delta_1 > \\delta_2 > 0$ be sufficiently small.\nThen,  for $s\\ge 0 $, we have\n\\begin{align} \\| u \\overline{v} \\|_{X_T^{s,-\\frac 12 + \\delta_1}} \\lesssim \\| u \\|_{X_T^{s, \\frac 12 + \\delta_2}} \\| v \\|_{X_T^{s, \\frac 12 + \\delta_2}}. \n\\label{bilin1}\n\\end{align}\n\\end{proposition}\n\nHere, $X^{s, b}_T$ denotes the local-in-time version of the $X^{s, b}$-space.\nSee Section \\ref{SEC:2} for the definition.\nAs mentioned above, the $L^3$-Strichartz comes with a derivative loss\nand hence can not be used directly to prove \\eqref{bilin1}.\nWe instead separate the proof into two cases:\n(i)~non-resonant interaction and (ii) nearly resonant interaction.\nIn the non-resonant case, thanks to the gain of derivative via\nmultilinear dispersion, \nwe can make up for the loss of the derivative in the $L^3$-Strichartz estimate.\nIn the nearly resonant case, we notice that the angle\nbetween the second incoming wave and the outgoing wave\nis almost perpendicular.\nThis angular restriction allows us to prove the estimate without any derivative loss.\nSee also \\cite{Tao4, CKSTT08}.\nSee Section~\\ref{SEC:bilin} for details.\n\n\nWe conclude this introduction by several remarks.\n\n\n\\begin{remark}\\label{REM:1} \\rm\n(i) Unlike \\cite{B93}, \nour argument does not rely on intricate number theoretic properties\nand thus Theorem \\ref{THM:LWP} also holds on a general torus\n$\\mathbb{T}^2_{\\pmb{\\alpha}} = (\\mathbb{R}\/\\alpha_1 \\mathbb{Z}) \\times (\\mathbb{R}\/\\alpha_2 \\mathbb{Z}) $\nfor any ratio $\\pmb{\\alpha} = (\\alpha_1, \\alpha_2)$ with  $\\alpha_1, \\alpha_2 > 0$.\n   This is essentially due to the fact that\n   the key bilinear Strichartz estimate\n   and the counting lemma (see Lemmas \\ref{LEM:bilin} and \\ref{LEM:count} below) hold on a general flat torus (see Lemmas 2.5 and 2.9 in~\\cite{Kish13}) and the $L^4$-Strichartz estimate in Lemma \\ref{LEM:L4} also holds on  a general torus with an $\\varepsilon$ derivative loss (see Theorem 2.4 in \\cite{BD15}).\nWe also mention\n\\cite{B07, CW, GOW, BD15, KV, DGG}\nfor further discussions on the Strichartz estimates\nand well-posedness of NLS on irrational tori.\n\n\n\n\n\\smallskip\n\n\\noindent\n(ii) The bilinear estimate \\eqref{bilin1} also holds\nfor $u v$ and $\\overline{u} \\overline{v}$ in place of $u \\overline{v}$ on the left-hand side.\nIndeed,  for $\\overline{u} \\overline{v}$, Gr\\\"unrock \\cite{Gr}\nproved the corresponding bilinear estimate for $s > - \\frac 12$.\nAs for $uv$, a slight modification of the proof of Proposition \\ref{PROP:bilin}\nyields the corresponding bilinear estimate for $s \\ge 0$.\nSee Remark \\ref{REM:bilin}.\nThese bilinear estimates yield local well-posedness\nin the corresponding ranges.\n\nThe quadratic NLS on $\\mathbb{T}^2$ is critical\nin $H^{-1}(\\mathbb{T}^2)$ with respect to the scaling symmetry.\nNonetheless, Kishimoto \\cite{Kishi3} proved ill-posedness of the \nquadratic NLS on $\\mathbb{T}^2$ in $H^s(\\mathbb{T}^2)$\nfor (a) $s < 0 $ with the nonlinearity $\\mathcal{N}(u) =  |u|^2$\nand  \n(b) $s \\le -1$ with $\\mathcal{N}(u) = u^2$ or $\\overline{u}^2$.\nAs mentioned above, the well-posedness theory of \nthe quadratic NLS~\\eqref{qNLS} with the nonlinearity $|u|^2$\nis now complete.\nOn the other hand, as for the nonlinearity $u^2$ or $\\overline u^2$, \nthere is still a gap between the well-posedness and ill-posedness regularities.\n\n\n\\smallskip\n\n\n\\noindent\n(iii) \nIt is conjectured in \\cite{B93} that, on $\\mathbb{T}^2$, the $L^p$-Strichartz estimate holds\nwithout any derivative loss as long as $  p < 4$.\nAt this point, the $L^p$-Strichartz estimate on $\\mathbb{T}^2$\nfor $2< p < 4$ is known to hold with a slight loss of derivative\nand  this conjecture remains open.\nBy considering a multilinear version of the Strichartz estimate, however, \nwe obtain \nthe following tri-linear version of the $L^3$-Strichartz estimate without any derivative loss:\n\\begin{align}\n\\bigg|\\int_0^1 \\int_{\\mathbb{T}^2} \\prod_{j = 1}^3 \\big(e^{it \\Delta} \\phi_j\\big)^*\ndx dt \\bigg|\n\\lesssim \\prod_{j = 1}^3 \\|\\phi_j \\|_{L^2},\n\\label{tri1}\n\\end{align}\n\n\\noindent\nwhere $u^*$ denotes $ u$ or $\\overline u$.\nThe estimate \\eqref{tri1} follows easily from \nthe bilinear estimate \\eqref{bilin1} (also for $uv$ and $\\overline u \\overline v$)\nand the duality.\nWe point out that it is crucial that \nwe have a product structure on the left-hand side of \\eqref{tri1}.\n\n\n\\smallskip\n\n\\noindent\n(iv) Lastly, let us consider the following quadratic NLS on $\\mathbb{T}^2$\nwith a gauge-invariant nonlinearity:\n\\begin{equation}\ni \\partial_t u + \\Delta u = \\pm |u| u .\n\\label{NLS9}\n\\end{equation}\n\n\\noindent\nThis equation is of particular interest in view of the mass and energy conservations.\nIn particular, local well-posedness in $L^2(\\mathbb{T}^2)$\ntogether with the mass conservation would imply\nglobal well-posedness in the same space.\n\n\nWhen $s> 0$, local well-posedness of \\eqref{NLS9} in $H^s(\\mathbb{T}^2)$\neasily \nfollows from the $L^4$-Strichartz estimate~\\eqref{Str1}\n(more precisely, Lemma \\ref{LEM:L4} below).\nWhen $s = 0$, however, the non-algebraic nature of the nonlinearity\nmakes the local well-posedness problem of \\eqref{NLS9} in $L^2(\\mathbb{T}^2)$\nrather challenging.\nFor example, the tri-linear estimate \\eqref{tri1}\nis not useful to study \\eqref{NLS9}\nand,  moreover, multilinear analysis via the Fourier restriction\nnorm method (such as that presented in this paper)\nis not applicable due to the presence of \n$|u|$.\nWhile there are well-posedness results\n\\cite{OOP, Lee}\non the periodic  NLS with non-algebraic gauge-invariant nonlinearities, \none would need a much more intricate argument\nto prove local well-posedness of \\eqref{NLS9} in $L^2(\\mathbb{T}^2)$.\n\n\n\\end{remark}\n\n\n\n\n\n\n\n\n\n\n\\section{Notations and preliminary lemmas}\n\\label{SEC:2}\n\n\n\n\\subsection{Notations}\nFor a spacetime function $u$ defined on $\\mathbb{R} \\times \\mathbb{T}^2$, we write $\\mathcal{F}_{t,x} u$ or $\\widehat u$ to denote the spacetime Fourier transform of $u$.\nIn the following, we drop the inessential factor of $2\\pi$.\nWe  also set $\\jb{\\,\\cdot\\,} = (1 + |\\cdot|^2)^\\frac{1}{2}$.\n\nGiven a dyadic number $N  \\ge 1$, we let $P_N$ be the spatial  frequency projector onto the\n frequencies \n\\[ \\mathfrak{P}_N := \\big\\{ (\\tau, n)\\in \\mathbb{R}\\times \\mathbb{Z}^2 : \\tfrac{N}{2} < |n| \\leq N \\big\\}.\\] \nAlso, given a dyadic number $L \\geq 1$, we define $Q_L$ to be the \nmodulation projector onto the space-time frequencies \n\\[\\mathfrak{S}_L :=\\big \\{ (\\tau, n)\\in \\mathbb{R}\\times\\mathbb{Z}^2: \\tfrac{L}{2} < \\big| \\tau + |n|^2 \\big| \\leq L \\big\\}.\\]\n For brevity, we also set  $P_{N,L} = P_N Q_L$. \n\n\nIn what follows, $N$ and $L$ (possibly with subscripts) always denote dyadic numbers $\\geq 1$. We write $\\overline{N}_{ij\\cdots} := \\max(N_i, N_j, \\dots )$ and $\\underline{N}_{ij\\cdots} := \\min ( N_i, N_j, \\dots )$. We also write $N_{\\text{max}} := \\overline{N}_{012}$, $N_{\\text{min}} := \\underline{N}_{012}$, $L_{\\text{max}} := \\overline{L}_{012}$, $L_{\\text{min}} := \\underline{L}_{012}$, and $L_{\\text{med}} := L_0 L_1 L_2 \/ L_{\\text{min}} L_{\\text{max}}$.\n\nWe use $A \\lesssim B$ to denote $A \\leq CB$ for some constant $C > 0$, and we write $A \\sim B$ to denote $A \\lesssim B$ and $B \\lesssim A$. We also write $A \\ll B$ if $A \\leq cB$ for some sufficiently small $c > 0$.\nWe may use subscripts to denote dependence on external parameters; for example,\n $A\\lesssim_{p, q} B$ means $A\\le C(p, q) B$,\n where the constant $C(p, q)$ depends on parameters $p$ and $q$.\n In addition, we use $a+$ (and $a-$, respectively) to denote $a+\\varepsilon$ (and $a-\\varepsilon$, respectively)\nfor arbitrarily small $\\varepsilon > 0$.\n\n\n\\subsection{Preliminary lemmas}\nIn this subsection, we recall some useful lemmas. \nWe first recall the following bilinear  Strichartz estimate. For the proof, see Lemma 2.5 in~\\cite{Kish13}.\n\\begin{lemma}\\label{LEM:bilin}\nLet $N_j, L_j \\geq 1$,  $j = 0,1,2$,  be dyadic numbers. Suppose that $u_1, u_2 \\in L^2(\\mathbb{R} \\times \\mathbb{T}^2)$ satisfy\n\\[ \\textup{supp } \\widehat{u_1} \\subset \\mathfrak{P}_{N_1} \\cap \\mathfrak{S}_{L_1}\n\\quad\\text{and}\\quad \\textup{supp } \\widehat{u_2} \\subset \\mathfrak{P}_{N_2} \\cap \\mathfrak{S}_{L_2}.  \\]\nThen, we have\n\\[ \\|  \\mathcal{F}_{t, x}(u_1 \\overline{u_2}) \\|_{L_\\tau^2 \\ell_n^2(\\mathfrak{P}_{N_0})} \n\\lesssim \\underline{L}_{12}^{\\frac 12} \\bigg( \\frac{\\overline{L}_{12}}{N_0} + 1 \\bigg)^{\\frac 12} N_{\\textup{min}}^{\\frac 12} \\| u_1 \\|_{L_{t,x}^2} \\| u_2 \\|_{L_{t,x}^2}. \\]\n\\end{lemma}\n\nWe also recall the following counting lemma. For the proof, see Lemma 2.9\\,(ii) in \\cite{Kish13}, which was stated in general dimension $d \\geq 2$ but we only need the $d = 2$ case.\n\n\\begin{lemma}\\label{LEM:count}\nLet $N \\gg 1$, $N^{-1} \\leq \\mu, \\nu \\ll N$, $M \\geq 0$, and set\n\\[ D := \\{ \\xi = (\\xi_1, \\xi_2) \\in \\mathbb{R}^2: N \\leq |\\xi| \\leq N + \\mu, M \\leq \\xi_1 \\leq M + \\nu \\}. \\]\nLet $\\mathcal{R}$ be an arbitrary rotation operator on $\\mathbb{R}^2$. \nMoreover, with $e_1 = ( 1,0) \\in \\mathbb{R}^2$, set \n\\[ K := \\Big\\{ \\xi \\in \\mathbb{R}^2: \\frac{\\alpha}{2} \\leq \\angle(\\xi, e_1) \\leq 2\\alpha \\Big\\},\\]\n\n\n\\noindent\nwhere $\\angle(\\xi, e_1) $ denotes the angle between $\\xi$ and $e_1$.\nSuppose that \n\\[ \n\\bigg( \\frac{\\mu + \\min\\{\\nu, 1\\}}{N} \\bigg)^{\\frac 12 } \\ll \\alpha \\leq \\frac{\\pi}{4}. \\]\nThen, we have\n\\[ |\\mathbb{Z}^2 \\cap \\mathcal{R}(D \\cap K)| \\lesssim \\max\\{ \\nu, 1 \\} \\big(\\alpha^{-1}(\\mu + \\min\\{ \\nu, 1 \\}) + 1\\big). \\]\n\\end{lemma}\n\n\n\\subsection{Fourier restriction norm method}\nIn studying well-posedness of NLS on $\\mathbb{T}^d$, Bourgain \\cite{B93}\nintroduced the Fourier restriction norm method (see also \\cite{KM93}), \nutilizing the following $X^{s, b}$-spaces.\n Given $s, b \\in \\mathbb{R}$, \n we define the space $X^{s,b}(\\mathbb{R} \\times \\mathbb{T}^2)$ \n to be  the completion of functions that are $C^\\infty$ in space and Schwartz in time with respect to the following norm:\n\\begin{align}\n \\| u \\|_{X^{s,b}(\\mathbb{R} \\times \\mathbb{T}^2)} := \\big\\| \\jb{n}^s \\jb{\\tau + |n|^2}^b \\widehat u(\\tau, n) \\big\\|_{ L_\\tau^2 \\ell_n^2(\\mathbb{R} \\times \\mathbb{Z}^2) }. \n\\label{Xsb}\n\\end{align}\n\n\\noindent\nFor $T > 0$, we define the space $X_T^{s,b}$ to be the restriction of \nthe $X^{s, b}$-space onto the time interval \n$[-T, T]$ via the norm:\n\\begin{equation}\n\\| u \\|_{X_T^{s,b}} := \\inf \\big\\{ \\| v \\|_{X^{s,b}}: v|_{[-T, T]} = u \\big\\}.\n\\label{XT}\n\\end{equation}\n\n\n\\noindent\nNote that $X^{s, b}_T$ is complete.\nGiven any $s \\in \\mathbb{R}$ and $b > \\frac 12$, we have $X_T^{s,b} \\subset C([-T, T]; H^s(\\mathbb{T}^2))$.\n\nWe now recall the following linear estimates. See \\cite{B93, GTV, Tao}.\n\n\\begin{lemma}\\label{LEM:Xlin}\nLet $s \\in \\mathbb{R}$ and $0 < T \\leq 1$.\n\n\\smallskip \\noindent\n\\textup{(i)} For any $b \\in \\mathbb{R}$, we have\n\\[ \\| e^{it\\Delta} \\phi \\|_{X_T^{s,b}} \\lesssim_b \\| \\phi \\|_{H^s}. \\]\n\n\\smallskip \\noindent\n\\textup{(ii)} Let $-\\frac 12 < b' \\leq 0 \\leq b \\leq b'+1$. Then, we have\n\\[ \\bigg\\| \\int_0^t e^{i(t-t')\\Delta} F(t') dt' \\bigg\\|_{X_T^{s,b}} \\lesssim_{b,b'} T^{1-b+b'} \\| F \\|_{X_T^{s,b'}}. \\]\n\\end{lemma}\n\nLastly,  we record the following $L^4$-Strichartz estimate on $\\mathbb{T}^2$. \n\n\\begin{lemma}\\label{LEM:L4}\nLet $s > 0$ and $b > \\frac 12$. Then, we have\n\\begin{align} \\| u \\|_{L_{t,x}^4([0,1] \\times \\mathbb{T}^2)} \\lesssim_{s, b} \\| u \\|_{X^{s, b}}. \n\\label{L42}\n\\end{align}\n\\end{lemma}\n\nFor the proof, see \\cite{B93, B95}.  We point out that the $L^4$-Strichartz estimate \n\\eqref{L42} does not hold when $s=0$.  See \\cite{TTz}.\nAs mentioned in the introduction, the $L^4$-Strichartz estimate \\eqref{Str1}\nwith an $\\varepsilon$-loss of derivative\nholds on a general flat torus\n$\\mathbb{T}^2_{\\pmb{\\alpha}} = (\\mathbb{R}\/\\alpha_1 \\mathbb{Z}) \\times (\\mathbb{R}\/\\alpha_2 \\mathbb{Z}) $\nfor any ratio $\\pmb{\\alpha} = (\\alpha_1, \\alpha_2)$ with  $\\alpha_1, \\alpha_2 > 0$;\nsee  Theorem~2.4 in \\cite{BD15}.\nThen, the estimate \\eqref{L42}\non a general flat torus $\\mathbb{T}^2_{\\pmb{\\alpha}}$ follows from a standard transference principle (see Lemma 2.9 in \\cite{Tao}).\n\n\n\\section{Proof of Proposition \\ref{PROP:bilin}}\n\\label{SEC:bilin}\n\nIn this section, we  present the proof of  Proposition~\\ref{PROP:bilin}.\nBy the triangle inequality $\\jb{n}^s \\lesssim \\jb{n_1}^s \\jb{n_2}^s$\nunder $n = n_1 + n_2$ for $s \\ge 0$, \n our goal is to prove  the following bilinear estimate (with $s = 0$):\n\\begin{align}\n \\| u \\overline{v} \\|_{X_T^{0,-\\frac 12 + \\delta_1}} \\lesssim \\| u \\|_{X_T^{0, \\frac 12 + \\delta_2}} \n \\| v \\|_{X_T^{0, \\frac 12 + \\delta_2}}, \n\\label{bilin1a}\n\\end{align}\n\n\n\\noindent\nwhere $0 < T \\leq 1$ and $\\delta_1, \\delta_2 > 0$ are sufficiently small, \nsatisfying  $\\delta_2 < \\delta_1 < 2\\delta_2$. \nIn view of the definitions \\eqref{Xsb} and \\eqref{XT} of the $X^{s, b}$-spaces, \nthe bilinear estimate \\eqref{bilin1a} follows\nonce we prove\n\\begin{equation}\n\\begin{split}\n\\bigg\\| & \\sum_{\\substack{n_1\\in \\mathbb{Z}^2\\\\ n = n_1 - n_2}} \n\\intt_{\\tau = \\tau_1 - \\tau_2} \\frac{\\widehat u(\\tau_1, n_1) \\overline{\\widehat v(\\tau_2, n_2)}}{\\jb{\\tau_1 + |n_1|^2}^{\\frac 12 + \\delta_2} \\jb{\\tau_2 + |n_2|^2}^{\\frac 12 + \\delta_2} \\jb{\\tau + |n|^2}^{\\frac 12 - \\delta_1}} d\\tau_1 \\bigg\\|_{L_\\tau^2 \\ell_n^2} \\\\\n& \\lesssim \\| u \\|_{L_{t,x}^2} \\| v \\|_{L_{t,x}^2}.\n\\end{split}\n\\label{bilin2}\n\\end{equation}\n\n\\noindent\nBy duality, the estimate \\eqref{bilin2} \nfollows once we prove the following estimate:\\\n\\begin{equation}\n\\begin{split}\n\\bigg| &\\sum_{\\substack{n, n_1\\in \\mathbb{Z}^2\\\\ n = n_1 - n_2}} \n\\iintt_{\\tau = \\tau_1 - \\tau_2} \\frac{\\widehat u (\\tau_1, n_1) \\overline{\\widehat v (\\tau_2, n_2) }\\overline{\\widehat w (\\tau, n)}}{\\jb{\\tau_1 + |n_1|^2}^{\\frac 12 + \\delta_2} \\jb{\\tau_2 + |n_2|^2}^{\\frac 12 + \\delta_2} \\jb{\\tau + |n|^2}^{\\frac 12 - \\delta_1}}\nd\\tau_1 d\\tau \\bigg| \\\\\n& \\lesssim \\| u \\|_{L_{t,x}^2} \\| v \\|_{L_{t,x}^2} \\| w \\|_{L_{t,x}^2}.\n\\end{split}\n\\label{bilin3}\n\\end{equation}\n\n\\noindent\nWe first note that if  $n = 0$, $n_1 = 0$, or $n_2 = 0$, \nthen the estimate \\eqref{bilin3} follows easily from  the Cauchy-Schwarz inequality,\nprovided that $\\delta_1 > 0$ is sufficiently small.\n Hence,  for the remaining of this section, we assume that $n \\neq 0$, $n_1 \\neq 0$, and $n_2 \\neq 0$.\n\nBefore proceeding further, we recall\nthe following key algebraic relation:\n\\begin{equation}\n(\\tau + |n|^2) - (\\tau_1 + |n_1|^2) + (\\tau_2 + |n_2|^2) = -2 n \\cdot n_2, \n\\label{phase}\n\\end{equation}\n\n\\noindent\nwhere  $n = n_1 - n_2$ and $\\tau = \\tau_1 - \\tau_2$.\nThe main difficulty in proving \\eqref{bilin3} (namely, the bilinear estimate \\eqref{bilin1} with $s = 0$)\ncomes from the failure of \nthe $L^4$-Strichartz estimate (without a derivative loss).\nIn order to overcome this difficulty, we separately estimate the contributions coming\nfrom \n(i) non-resonant case: $|n \\cdot n_2| \\gtrsim |n|^\\varepsilon |n_2|^\\varepsilon$\nand (ii) nearly resonant case: $|n \\cdot n_2| \\ll |n|^\\varepsilon |n_2|^\\varepsilon$\nfor some small $ \\varepsilon > 0$.\nIn the non-resonant case, \nwe can use the multilinear dispersion\nto make up for the derivative loss in the $L^4$-Strichartz estimate (Lemma~\\ref{LEM:L4}).\nIn the nearly resonant case, by noting that the frequencies $n$ and $n_2$\nare almost perpendicular, \nwe make use of this angular restriction to prove\na multilinear estimate without a derivative loss\nin the spirit of \\cite{Tao4, CKSTT08}.\n\n\n\n\n\n\n\\subsection{Non-resonant interaction}\nIn this subsection, we consider the non-resonant case:\n\\begin{align}\n|n \\cdot n_2| \\gtrsim |n|^\\varepsilon |n_2|^\\varepsilon\n\\label{nr1}\n\\end{align}\n\n\\noindent\n for some small $\\varepsilon > 0$ sufficiently small.\nFrom \\eqref{phase} and \\eqref{nr1}, we have\n\\begin{align}\n\\begin{split}\n\\MAX :\\! & =  \\max \\big( \\jb{\\tau + |n|^2}, \\jb{\\tau_1 + |n_1|^2}, \\jb{\\tau_2 + |n_2|^2} \\big)\\\\\n& \\gtrsim \\jb{n \\cdot n_2}\n\\gtrsim \\jb{n}^\\varepsilon \\jb{n_2}^\\varepsilon.\n\\end{split}\n\\label{nr2}\n\\end{align}\nWe then consider the following three cases.\nIn the following, we set   $0 < \\delta_1 < \\frac 12 $.\n\n\n\\medskip \\noindent\n$\\bullet$ \\textbf{Case 1:} $\\text{MAX} = \\jb{\\tau + |n|^2}$. \n\nIn this case, we directly prove \\eqref{bilin2}.\nThen, with  $\\varepsilon_1 = \\frac \\varepsilon 2(\\frac 12 - \\delta_1)> 0$, \nit follows from \\eqref{nr2} that \n\\begin{equation}\n\\jb{n_1}^{\\varepsilon_1} \\jb{n_2}^{\\varepsilon_1} = \\jb{n+n_2}^{\\varepsilon_1} \\jb{n_2}^{\\varepsilon_1} \\lesssim \\jb{n}^{2\\varepsilon_1} \\jb{n_2}^{2\\varepsilon_1} \\lesssim \\jb{\\tau + |n|^2}^{\\frac 12 - \\delta_1}.\n\\label{tn0}\n\\end{equation}\n\n\\noindent\nLet $\\widehat U(\\tau_1, n_1) = \\jb{n_1}^{-\\varepsilon_1} \\jb{\\tau_1 + |n_1|^2}^{-\\frac 12 - \\delta_2} \\widehat u(\\tau_1, n_1)$\nand\n$\\widehat V(\\tau_2, n_2) = \\jb{n_2}^{-\\varepsilon_1} \\jb{\\tau_2 + |n_2|^2}^{-\\frac 12 - \\delta_2} \\widehat v(\\tau_2, n_2)$.\nThen, by \\eqref{tn0}, Plancherel's theorem, H\\\"older's inequality, and the $L^4$-Strichartz estimate (Lemma \\ref{LEM:L4}), we have\n\\begin{align*}\n\\text{LHS of \\eqref{bilin2}} \n&\\lesssim \\| U \\overline V\\|_{L^2_{t, x}}\n\\leq \\| U \\|_{L^4_{t, x}}\\|  V\\|_{L^4_{t, x}}\n\\lesssim \\|U\\|_{X^{\\varepsilon_1, \\frac 12 + \\delta_2}}\n\\|V\\|_{X^{\\varepsilon_1, \\frac 12 + \\delta_2}}\\\\\n&= \\| u \\|_{L_{t,x}^2} \\| v \\|_{L_{t,x}^2}\n\\end{align*}\n\n\\noindent\nfor any $\\delta_2 > 0$.\n\n\\medskip \\noindent\n$\\bullet$ \\textbf{Case 2:} $\\text{MAX} = \\jb{\\tau_1 + |n_1|^2}$. \n\nIn this case, we prove \\eqref{bilin3}. \nWith  $\\varepsilon_2 = \\varepsilon(\\frac 12 - \\delta_1)> 0$, \nit follows from \\eqref{nr2} that \n\\begin{equation}\n\\jb{n}^{\\varepsilon_2} \\jb{n_2}^{\\varepsilon_2} \\lesssim \\jb{\\tau_1 + |n_1|^2}^{\\frac 12 - \\delta_1}.\n\\label{tn1}\n\\end{equation}\n\n\\noindent\nThen, letting\n$\\widehat V(\\tau_2, n_2) = \\jb{n_2}^{-\\varepsilon_2} \\jb{\\tau_2 + |n_2|^2}^{-\\frac 12 - \\delta_2} \\widehat v(\\tau_2, n_2)$\nand \n$\\widehat W(\\tau, n) = \\jb{n}^{-\\varepsilon_2} \\jb{\\tau + |n|^2}^{-\\frac 12 - \\delta_2} \\widehat w(\\tau, n)$, \nit follows from \n the Cauchy-Schwarz inequality in $\\tau_1$ and $n_1$, \\eqref{tn1}, Plancherel's theorem, H\\\"older's inequality, and the $L^4$-Strichartz estimate (Lemma \\ref{LEM:L4}) that \n\\begin{align*}\n \\text{LHS of \\eqref{bilin3}} \n&\\lesssim \\|u\\|_{L^2_{t, x}} \n\\| V \\|_{L^4_{t, x}}\\|W\\|_{L^4_{t, x}}\\\\\n&\\lesssim \\|u\\|_{L^2_{t, x}} \n\\| V \\|_{X^{\\varepsilon_2, \\frac 12 +\\delta_2}} \\|W\\|_{X^{\\varepsilon_2, \\frac 12 +\\delta_2}} \\\\\n&= \\| u \\|_{L_{t,x}^2} \\| v \\|_{L_{t,x}^2} \\| w \\|_{L_{t,x}^2}\n\\end{align*}\n\n\\noindent\nfor any $\\delta_2 > 0$.\n\n\\medskip\n\\noindent\n$\\bullet$ \\textbf{Case 3:} $\\text{MAX} = \\jb{\\tau_2 + |n_2|^2}$.\n\nThis case follows from proceeding as in Case 2\nand thus we omit details.\n\n\n\n\n\\subsection{Resonant interaction}\n\\label{SUBSEC:res}\nWe now consider the resonant case $|n \\cdot n_2| \\ll |n|^\\varepsilon |n_2|^\\varepsilon$.\nWith $\\theta = \\theta (n, n_2) = \\frac{1}{|n|^{1-\\varepsilon} |n_2|^{1-\\varepsilon}}$, we rewrite this condition as\n\\begin{align*}\n|n \\cdot n_2| \\ll \\theta |n| |n_2|.\n\\end{align*}\n\n\\noindent\nIn the following, we divide the argument  into three subcases, depending on the sizes\nof $n$ and~$n_2$.\n\n\\medskip \\noindent\n$\\bullet$ \\textbf{Case 1:} $|n|^{\\frac 12} \\lesssim |n_2| \\lesssim |n|^2$.\n\nIn this case, we follow the idea from the proof of Proposition 10.1 in \\cite{Tao4} (see also \\cite{CKSTT08}). \n\n\n\\smallskip \\noindent\n$\\circ$ \\textbf{Subcase 1.a:} $|n| \\sim |n_2|$.\n\nBy applying the dyadic decompositions to the frequencies $n$ and $n_2$, we have\n\\begin{equation}\n\\begin{split}\n\\text{LHS of } \\eqref{bilin3} \n&\\lesssim \\sum_{\\substack{N \\sim N_2 \\geq 1\\\\\\text{dyadic}} }\n\\bigg| \\sum_{\\substack{n, n_1\\in \\mathbb{Z}^2\\\\ n = n_1 - n_2}} \\iintt_{\\tau = \\tau_1 - \\tau_2} \n\\mathbf 1_{|\\cos \\angle(n, n_2)| \\ll \\theta} \\\\\n&\\quad \\times \\frac{\\widehat u (\\tau_1, n_1) \\overline{\\widehat{P_{N_2}v} (\\tau_2, n_2)} \\overline{\\widehat{P_N w} (\\tau, n)}}{\\jb{\\tau_1 + |n_1|^2}^{\\frac 12 + \\delta_2} \\jb{\\tau_2 + |n_2|^2}^{\\frac 12 + \\delta_2} \\jb{\\tau + |n|^2}^{\\frac 12 - \\delta_1}} d\\tau_1 d\\tau \\bigg|.\n\\end{split}\n\\label{bilin_a1}\n\\end{equation}\n\n\\noindent\nNote that when $|n| \\sim |n_2| \\sim N$, we have $\\theta = \\frac{1}{|n|^{1 - \\varepsilon} |n_2|^{1 - \\varepsilon}} \\sim N^{-2 + 2 \\varepsilon}$. We now restrict $n$ and $n_2$ to the following angular sectors:\n\\begin{align}\n\\begin{split}\nA_\\ell : \\!& = \\big\\{ n \\in \\mathbb{Z}^2: |n| \\sim N, \\,  \\arg(n) = \\ell N^{-2 + 2 \\varepsilon} + O(N^{-2 + 2 \\varepsilon}) \\big\\}, \\\\\nA_{\\ell_2} :\\!&= \\big\\{ n_2 \\in \\mathbb{Z}^2: |n_2| \\sim N_2, \\, \\arg(n_2) = \\ell_2 N^{-2 + 2 \\varepsilon} + O(N^{-2 + 2 \\varepsilon}) \\big\\},\n\\end{split}\n\\label{A0}\n\\end{align}\n\n\\noindent\nwhere $\\ell, \\ell_2 \\in I_N: = [1, 2\\pi N^{2 - \\varepsilon}] \\cap \\mathbb{Z}$.\nSince $|\\cos \\angle (n, n_2)| \\ll \\theta \\sim N^{-2 + 2\\varepsilon}$, we have $|\\arg(n) - \\arg(n_2)| = \\frac{\\pi}{2} + O(N^{-2 + 2\\varepsilon})$ or $|\\arg(n) - \\arg(n_2)| = \\frac{3\\pi}{2} + O(N^{-2 + 2\\varepsilon})$. This means that for each fixed $\\ell$, there exists a set  $L_2(\\l)$ of size $O(1)$ such that \n$\\mathbf 1_{A_\\ell} (n)\\cdot \\mathbf 1_{A_{\\ell_2}}(n_2)= 0$\nunless $\\l_2 \\in L_2(\\l)$.\n\n\nContinuing with the right-hand side of \\eqref{bilin_a1}, \nwe insert the angular restrictions \\eqref{A0}\nand set \n\\[ \\widehat v_{N_2, \\l_2}(\\tau_2, n_2) =  \\mathbf 1_{A_{\\ell_2}}(n_2)  \\cdot \\widehat{P_{N_2}v} (\\tau_2, n_2)\n\\quad \\text{and}\\quad\n\\widehat w_{N, \\l} (\\tau, n) = \\mathbf 1_{A_\\ell}(n) \\cdot \\widehat{P_N w} (\\tau, n). \\]\n\n\\noindent\n Then, by \napplying  the Cauchy-Schwarz inequality in $\\tau_1$ and $n_1$, \nthen  in $\\tau$ and $n$, and H\\\"older's inequality in $\\tau_1$ and $n_1$, \nwe obtain\n\\begin{equation}\n\\begin{split}\n\\text{LHS of } \\eqref{bilin3}\n &\\lesssim \\sum_{\\substack{N \\sim N_2 \\ge 1\\\\\\text{dyadic}}} \n \\sum_{\\ell  \\in I_N} \n \\sum_{ \\ell_2 \\in L_2(\\ell)} \n \\bigg| \\sum_{\\substack{n, n_1\\in \\mathbb{Z}^2\\\\ n = n_1 - n_2}} \n \\iintt_{\\tau = \\tau_1 - \\tau_2}  \\mathbf 1_{|\\cos \\angle (n, n_2)| \\ll N^{-2 + 2\\varepsilon}} \\\\\n&\\quad \\times \\frac{ \\widehat u (\\tau_1, n_1) \\overline{\\widehat v_{N_2, \\l_2} (\\tau_2, n_2) }\n  \\overline{\\widehat  w_{N, \\l} (\\tau, n) }}\n  {\\jb{\\tau_1 + |n_1|^2}^{\\frac 12 + \\delta_2} \\jb{\\tau_2 + |n_2|^2}^{\\frac 12 + \\delta_2} \\jb{\\tau + |n|^2}^{\\frac 12 - \\delta_1}} d\\tau_1 d\\tau \\bigg| \\\\\n&\\leq  \\sum_{\\substack{N \\sim N_2 \\ge 1\\\\\\text{dyadic}}} \n \\sum_{\\ell  \\in I_N} \n \\sum_{ \\ell_2 \\in L_2(\\ell)} \n  \\| \\widehat u \\|_{L_{\\tau_1}^2 \\ell_{n_1}^2} \\bigg\\| \\sum_{n\\in \\mathbb{Z}^2} \\int \\mathbf 1_{|\\cos \\angle (n, n_1 - n)| \\ll N^{-2 + 2\\varepsilon}} \\\\\n&\\quad \\times \\frac{ \\widehat v_{N_2, \\l_2} (\\tau_1 - \\tau , n_1- n) \n  \\widehat  w_{N, \\l} (\\tau, n) }{ \\jb{\\tau_1 - \\tau + |n_1 - n|^2}^{\\frac 12 + \\delta_2} } d\\tau \\bigg\\|_{L_{\\tau_1}^2 \\ell_{n_1}^2} \\\\\n&\\leq \\| u \\|_{L_{t, x}^2} \\sum_{\\substack{N \\sim N_2 \\ge 1\\\\\\text{dyadic}}} \n \\sum_{\\ell  \\in I_N} \n \\sum_{ \\ell_2 \\in L_2(\\ell)} \n  \\| \\widehat v_{N_2, \\l_2} \\|_{L_{\\tau_2}^2 \\ell_{n_2}^2} \\| \\widehat w_{N, \\l}\\|_{L_\\tau^2 \\ell_n^2} \\\\\n&\\quad \\times \\sup_{\\tau_1, n_1} \\bigg( \\sum_n \\int \\frac{ \\mathbf 1_{|\\cos \\angle (n, n_1 - n)| \\ll N^{-2 + 2\\varepsilon}} \\mathbf 1_{A_{\\ell_2}}(n_1 - n)  \\cdot \\mathbf 1_{A_\\ell}(n)}{\\jb{\\tau_1 - \\tau + |n_1 - n|^2}^{1 + 2 \\delta_2}} d\\tau \\bigg)^{\\frac 12} \\\\\n&\\lesssim \\| u \\|_{L_{t, x}^2 }  \\sum_{\\substack{N \\sim N_2 \\ge 1\\\\\\text{dyadic}}} \n\\sum_{\\ell  \\in I_N} \n \\sum_{ \\ell_2 \\in L_2(\\ell)} \n  \\| \\widehat v_{N_2, \\l_2} \\|_{L_{\\tau_2}^2 \\ell_{n_2}^2} \\| \\widehat w_{N, \\l}\\|_{L_\\tau^2 \\ell_n^2}\\\\\n& \\quad \\times \\sup_{n_1, \\l, \\l_2} |\\mathcal{A}_{N, \\l, \\l_2}(n_1)|^\\frac 12 , \n\\end{split}\n\\label{bilin_a2}\n\\end{equation}\n\n\\noindent\nwhere $\\mathcal{A}_{N, \\l, \\l_2}(n_1)$ is defined by \n\\[  \\mathcal{A}_{N, \\l, \\l_2}(n_1) = \\big\\{ n \\in \\mathbb{Z}^2: |\\cos \\angle (n, n_1 - n)| \\ll N^{-2 + 2\\varepsilon}, \\, n \\in A_\\ell,  \\, n_1 - n \\in A_{\\ell_2}\\big \\}.\\]\n\nIn view of \\eqref{A0}, \nfor  fixed $n_1 \\in \\mathbb{Z}^2$ and fixed $\\ell, \\ell_2\\in I_N$, \nwe note that any $n\\in \\mathcal{A}_{N, \\l. \\l_2}$\nis contained in  a rectangle with sides of length $\\sim$\n\\[ |n| N^{-2 + 2\\varepsilon} \\sim N^{-1 + 2\\varepsilon} \\quad \\text{and} \\quad |n_1 - n| N^{-2 + 2\\varepsilon} \\sim N^{-1 + 2\\varepsilon}. \\] \n\n\n\\noindent\nHence, for  $0 < \\varepsilon < \\frac 12$,\nwe have \n\\begin{align}\n\\sup_{n_1, \\l,  \\l_2} |\\mathcal{A}_{N, \\l, \\l_2}(n_1)|\\lesssim 1.\n\\label{A2}\n\\end{align}\n\n\\noindent\nTherefore, \nby estimating  \\eqref{bilin_a2} \nwith \\eqref{A2}, \n the Cauchy-Schwarz inequality in $\\ell$, and then in $N \\sim N_2$ we obtain the desired inequality \\eqref{bilin3}.\n\n\n\n\n\\smallskip \\noindent\n$\\circ$ \\textbf{Subcase 1.b:} $|n_2| \\ll |n| \\lesssim |n_2|^2$.\n\n Since $|n| \\gg |n_2|$ and $n_1 = n + n_2$, we have $|n_1| \\sim |n|$. \nNote that when  $|n| \\sim N \\lesssim |n_2|^2$, we have $\\theta = \\frac{1}{|n|^{1 - \\varepsilon} |n_2|^{1 - \\varepsilon}} \\lesssim N^{-\\frac 32 + \\frac 32 \\varepsilon}$. We proceed as in Subcase 1.a by restricting $n$ and $n_2$ to the following angular sectors:\n\\begin{align}\n\\begin{split}\nB_\\l : \\! & = \\big\\{ n \\in \\mathbb{Z}^2: |n| \\sim N, \\arg(n) = \\ell N^{-\\frac 32 + \\frac 32 \\varepsilon} + O(N^{-\\frac 32 + \\frac 32 \\varepsilon}) \\big\\}, \\\\\nB_{\\l_2} :\\! & = \\big\\{ n_2 \\in \\mathbb{Z}^2: N^{\\frac 12} \\lesssim |n_2| \\ll N, \\arg(n_2) = \\ell_2 N^{-\\frac 32 + \\frac 32 \\varepsilon} + O(N^{-\\frac 32 + \\frac 32 \\varepsilon}) \\big\\},\n\\end{split}\n\\label{B0}\n\\end{align}\n\n\\noindent\nwhere $\\ell, \\ell_2 \\in I_N': = [1, 2\\pi N^{\\frac 32 - \\frac 32\\varepsilon}] \\cap \\mathbb{Z}$.\nUnder $|\\cos \\angle (n, n_2)| \\ll N^{-\\frac 32 + \\frac 32 \\varepsilon}$, we have $|\\arg(n) - \\arg(n_2)| = \\frac{\\pi}{2} + O(N^{-\\frac 32 + \\frac 32 \\varepsilon})$ or $|\\arg(n) - \\arg(n_2)| = \\frac{3\\pi}{2} + O(N^{-\\frac 32 + \\frac 32 \\varepsilon})$. \nHence, \n for each fixed $\\ell$, there exists a set  $L_2'(\\l)$ of size $O(1)$ such that \n$\\mathbf 1_{B_\\ell} (n)\\cdot \\mathbf 1_{B_{\\ell_2}}(n_2)= 0$\nunless $\\l_2 \\in L_2'(\\l)$.\nWith a slight abuse of notation, set \n\\[ \\widehat v_{N, \\l_2}(\\tau_2, n_2) =  \\mathbf 1_{B_{\\ell_2}}(n_2)  \\cdot \\widehat{v} (\\tau_2, n_2)\n\\quad \\text{and}\\quad\n\\widehat w_{N, \\l} (\\tau, n) = \\mathbf 1_{B_\\ell}(n) \\cdot \\widehat{P_N w} (\\tau, n). \\]\n\n\\noindent\n Then, by proceeding as in \\eqref{bilin_a2}, i.e.~by\napplying  the Cauchy-Schwarz inequality in $\\tau_1$ and $n_1$, \nthen  in $\\tau$ and $n$, and H\\\"older's inequality in $\\tau_1$ and $n_1$, \nwe obtain\n\\begin{align*}\n\\text{LHS of } \\eqref{bilin3}\n &\\lesssim \\sum_{\\substack{N \\sim N_1\\ge 1\\\\\\text{dyadic}}}\n   \\sum_{\\ell  \\in I_N'} \n \\sum_{ \\ell_2 \\in L_2'(\\ell)} \n   \\bigg| \\sum_{\\substack{n, n_1\\in \\mathbb{Z}^2\\\\ n = n_1 - n_2}} \\iintt_{\\tau = \\tau_1 - \\tau_2} \\mathbf 1_{|\\cos \\angle (n, n_2)| \\ll N^{-\\frac 32 + \\frac 32 \\varepsilon}}  \\\\\n&\\quad \\times \\frac{\\widehat{P_{N_1} u} (\\tau_1, n_1) \\overline{\\widehat v_{N, \\l_2} (\\tau_2, n_2)}\n \\overline{\\widehat w_{N, \\l} (\\tau, n)}}{\\jb{\\tau_1 + |n_1|^2}^{\\frac 12 + \\delta_2} \\jb{\\tau_2 + |n_2|^2}^{\\frac 12 + \\delta_2} \\jb{\\tau + |n|^2}^{\\frac 12 - \\delta_1}} d\\tau_1 d\\tau \\bigg| \\\\\n&\\leq  \\sum_{\\substack{N \\sim N_1  \\ge 1\\\\\\text{dyadic}}}\n   \\sum_{\\ell  \\in I_N'} \n \\sum_{ \\ell_2 \\in L_2'(\\ell)} \n\\big\\| \\widehat{P_{N_1} u} \\big\\|_{L_{\\tau_1}^2 \\ell_{n_1}^2} \\bigg\\| \\sum_{n \\in  \\mathbb{Z}^2} \\int \\mathbf 1_{|\\cos \\angle (n, n_1 - n)| \\ll N^{-\\frac 32 + \\frac 32 \\varepsilon}}  \\\\\n&\\quad \\times \\frac{\\widehat v_{N, \\l_2}  (\\tau_1 - \\tau, n_1 - n)\\widehat w_{N, \\l}(\\tau, n) }{\\jb{\\tau_1 - \\tau + |n_1 - n|^2}^{\\frac 12 + \\delta_2}} d\\tau \\bigg\\|_{L_{\\tau_1}^2 \\ell_{n_1}^2} \\\\\n&\\leq  \\sum_{\\substack{N \\sim N_1 \\ge 1\\\\\\text{dyadic}}}\n    \\sum_{\\ell  \\in I_N'} \n \\sum_{ \\ell_2 \\in L_2'(\\ell)} \n  \\| \\widehat{P_{N_1} u} \\|_{L_{\\tau_1}^2 \\ell_{n_1}^2} \\| \\widehat v \\_{N, \\l_2}  \\|_{L_{\\tau_2}^2 \\ell_{n_2}^2} \n  \\| w_{N, \\l} \\|_{L_\\tau^2 \\ell_n^2} \\\\\n&\\quad \\times \\sup_{\\tau_1, n_1} \\bigg( \\sum_n \\int \\frac{\\mathbf 1_{|\\cos \\angle (n, n_1 - n)| \\ll N^{-\\frac 32 + \\frac 32 \\varepsilon}} \\mathbf 1_{B_{\\ell_2}} (n_1 - n)\\cdot  \\mathbf 1_{B_\\ell} (n)}{\\jb{\\tau_1 - \\tau + |n_1 - n|^2}^{1 + 2 \\delta_2}} d\\tau \\bigg)^{1\/2} \\\\\n&\\lesssim  \\sum_{\\substack{N \\sim N_1 \\ge 1\\\\\\text{dyadic}}}\n    \\sum_{\\ell  \\in I_N'} \n \\sum_{ \\ell_2 \\in L_2'(\\ell)} \n  \\| \\widehat{P_{N_1} u} \\|_{L_{\\tau_1}^2 \\ell_{n_1}^2} \\| \\widehat v_{N, \\l_2} \\|_{L_{\\tau_2}^2 \\ell_{n_2}^2} \n  \\| \\widehat w_{N, \\l_2} \\|_{L_\\tau^2 \\ell_n^2} \\\\\n& \\quad \\times \\sup_{n_1} |\\mathcal{B}_{N, \\l, \\l_2}(n_1)|^\\frac 12, \n\\end{align*}\n\n\n\\noindent\nwhere $\\mathcal{B}_{N, \\l, \\l_2}(n_1)$ is defined by \n\\[ \\mathcal{B}_{N, \\l, \\l_2}(n_1)\n = \\big \\{ n \\in \\mathbb{Z}^2: |\\cos \\angle (n, n_1 - n)| \\ll N^{-\\frac 32 + \\frac 32 \\varepsilon},, n \\in B_\\ell,,  n_1 - n \\in B_{\\ell_2} \\big\\}. \\]\n\n\\noindent\nThen, the desired bound \\eqref{bilin3} follows from \n the Cauchy-Schwarz inequality in $\\ell$ and then in  the Cauchy-Schwarz inequality in $N \\sim N_1$,\n once we prove \n\\begin{align}\n\\sup_{n_1, \\l,  \\l_2} |\\mathcal{B}_{N, \\l, \\l_2}(n_1)|\\lesssim 1.\n\\label{B2}\n\\end{align}\n\n\n\nIn view of \\eqref{B0}, \nfor  fixed $n_1 \\in \\mathbb{Z}^2$ and fixed $\\ell, \\ell_2\\in I_N'$, \nwe note that any $n\\in \\mathcal{B}_{N, \\l. \\l_2}$\nis contained in  a rectangle with sides of length $\\sim$\n\\[ |n| N^{-\\frac 32 + \\frac 32 \\varepsilon} \\sim N^{-\\frac 12 + \\frac 32 \\varepsilon} \\quad \\text{and} \\quad |n_1 - n| N^{-\\frac 32 + \\frac 32 \\varepsilon} \\ll N^{-\\frac 12 + \\frac 32 \\varepsilon}. \\] \n\n\\noindent\nTherefore, as long as $0 < \\varepsilon < \\frac 13$, there are at most $O(1)$ many \n$n'$s in $ \\mathcal{B}_{N, \\l. \\l_2}$, yielding \\eqref{B2}.\n\n\n\\smallskip \\noindent\n$\\circ$ \\textbf{Subcase 1.c:} $|n| \\ll |n_2|\\lesssim |n|^2$.\n\\\\ \\indent\nThis subcase follows from Subcase 1.b by switching the roles of $(\\tau, n)$ and $(\\tau_2, n_2)$\nand thus we omit details.\n\n\n\n\n\n\n\n\n\\medskip \\noindent\n$\\bullet$ \\textbf{Case 2:} $|n|^2 \\ll |n_2|$.\n\nWe divide the argument into two subcases, depending on the sizes\nof  $N_2$ and the largest modulation $L_{\\textup{max}} = \\max (L_0, L_1, L_2)$. \n\n\\medskip \\noindent\n$\\circ$ \\textbf{Subcase 2.a:} $L_{\\textup{max}} \\gtrsim N_2$ (high-modulation case).\n\nIn this subcase, recalling the notations in Section \\ref{SEC:2}, we dyadically decompose\n the spatial frequencies and modulations of  $u$, $v$, and $w$ so that\n  $\\textup{supp } \\widehat {P_{N_1, L_1} u} \\subset \\mathfrak{P}_{N_1} \\cap \\mathfrak{S}_{L_1}$, \n   $\\textup{supp } \\widehat {P_{N_2, L_2} v } \\subset \\mathfrak{P}_{N_2} \\cap \\mathfrak{S}_{L_2}$, and $\\textup{supp } \\widehat{P_{N_0, L_0} w} \\subset \\mathfrak{P}_{N_0} \\cap \\mathfrak{S}_{L_0}$ for dyadic $N_j, L_j \\geq 1$, $j = 0,1,2$. Note that we have $N_0^2 \\ll N_2 \\sim N_1$. Our main goal is to show the following lemma.\n\n\n\\begin{lemma}\\label{LEM:high}\nLet $N_j, L_j \\geq 1$, $j = 0,1,2$, be dyadic numbers with $N_0^2 \\ll N_2 \\sim N_1$ and $L_{\\textup{max}} \\gtrsim N_2$. Suppose that $f, g, h \\in L^2(\\mathbb{R} \\times \\mathbb{Z}^2)$ are real-valued nonnegative functions such that \n\\begin{align}\n \\textup{supp } f \\subset \\mathfrak{P}_{N_1} \\cap \\mathfrak{S}_{L_1}, \\quad \\textup{supp } g \\subset \\mathfrak{P}_{N_2} \\cap \\mathfrak{S}_{L_2}, \\quad \\text{and} \\quad  \\textup{supp } h \\subset \\mathfrak{P}_{N_0} \\cap \\mathfrak{S}_{L_0}.  \n\\label{high1}\n \\end{align}\nThen, we have\n\\begin{equation}\n\\begin{split}\n\\bigg| \\sum_{\\substack{n, n_1 \\in \\mathbb{Z}^2\\\\ n = n_1 - n_2}} &\\iintt_{\\tau = \\tau_1 - \\tau_2} f(\\tau_1, n_1) g(\\tau_2, n_2) h(\\tau, n) d\\tau d\\tau_1 \\bigg| \\\\\n&\\lesssim L_1^{\\frac 12 +} L_2^{\\frac 12 +} L_0^{\\frac 14 +} N_2^{0-} \\| f \\|_{L_{\\tau_1}^2 \\ell_{n_1}^2} \\| g \\|_{L_{\\tau_2}^2 \\ell_{n_2}^2} \\| h \\|_{L_\\tau^2 \\ell_n^2}.\n\\end{split}\n\\label{high}\n\\end{equation}\n\\end{lemma}\n\nWe first present the proof of \\eqref{bilin3} in this case by assuming Lemma \\ref{LEM:high}.\nUnder $N_0^2 \\ll N_2 \\sim N_1$ and $L_{\\textup{max}} \\gtrsim N_2$, \nby  applying Lemma \\ref{LEM:high}, we have \n\\begin{align*}\n\\text{LHS of \\eqref{bilin3}} &\\lesssim \\sum_{\\substack{ N_1, N_2, N_0\\ge 1\\\\\\text{dyadic}}}\n \\sum_{\\substack{L_1, L_2, L_0 \\ge 1\\\\\\text{dyadic}}} L_1^{-\\frac 12 - \\delta_2} L_2^{-\\frac 12 - \\delta_2} L_0^{-\\frac 12 + \\delta_1} \\\\\n&\\qquad \\times L_1^{\\frac 12 +} L_2^{\\frac 12 +} L_0^{\\frac 14 +} N_2^{0-} \\| P_{N_1, L_1} u \\|_{L_{t,x}^2} \\| P_{N_2, L_2} v \\|_{L_{t,x}^2} \\| P_{N_0, L_0} w \\|_{L_{t,x}^2} \\\\\n&\\leq \\sum_{\\substack{N_1, N_2, N_0\\ge 1\\\\\\text{dyadic}}} \\sum_{\\substack{ L_1, L_2, L_0\\ge 1\\\\\\text{dyadic}}} L_1^{0-} L_2^{0-} L_0^{0-} N_2^{0-} \\| u \\|_{L_{t,x}^2} \\| v \\|_{L_{t,x}^2} \\| w \\|_{L_{t,x}^2} \\\\\n&\\lesssim \\| u \\|_{L_{t,x}^2} \\| v \\|_{L_{t,x}^2} \\| w \\|_{L_{t,x}^2},\n\\end{align*}\nas desired.\n\nWe now present the proof of Lemma \\ref{LEM:high}.\n\n\\begin{proof}[Proof of Lemma \\ref{LEM:high}]\nBy the Cauchy-Schwarz inequality in $\\tau$ and $n$, Lemma \\ref{LEM:bilin}, and the condition $L_{\\textup{max}} \\gtrsim N_2 \\gg N_0^2$, we have\n\\begin{align*}\n\\text{LHS of \\eqref{high}} &\\lesssim \\| h \\|_{L_\\tau^2 \\ell_n^2} \\underline{L}_{12}^{\\frac 12} \n\\bigg( \\frac{\\overline{L}_{12}}{N_0} + 1 \\bigg)^{\\frac 12} N_0^{\\frac 12} \\| f \\|_{L_{\\tau_1}^2 \\ell_{n_1}^2} \\| g \\|_{L_{\\tau_2}^2 \\ell_{n_2}^2} \\\\\n&\\lesssim \\big( L_1^{\\frac 12} L_2^{\\frac 12} + \\underline{L}_{12}^{\\frac 12} N_2^{\\frac 14} \\big) \\| f \\|_{L_{\\tau_1}^2 \\ell_{n_1}^2} \\| g \\|_{L_{\\tau_2}^2 \\ell_{n_2}^2} \\| h \\|_{L_\\tau^2 \\ell_n^2} \\\\\n&\\lesssim L_1^{\\frac 12 +} L_2^{\\frac 12 +} L_0^{\\frac 14 +} N_2^{0-} \\| f \\|_{L_{\\tau_1}^2 \\ell_{n_1}^2} \\| g \\|_{L_{\\tau_2}^2 \\ell_{n_2}^2} \\| h \\|_{L_\\tau^2 \\ell_n^2}, \n\\end{align*}\n\n\\noindent\nyielding \\eqref{high}.\n\\end{proof}\n\n\\medskip \\noindent\n$\\circ$  \\textbf{Subcase 2.b:} $L_{\\textup{max}} \\ll N_2$ (low-modulation case).\n\nOur goal in this subcase is to show the following lemma.\n\\begin{lemma}\\label{LEM:low}\nLet $N_1, N_2, L_0, L_1, L_2 \\geq 1$ be dyadic numbers with $N_1 \\sim N_2$ and $L_{\\textup{max}} \\ll N_2$. Suppose that $f, g, h \\in L^2(\\mathbb{R} \\times \\mathbb{Z}^2)$ are real-valued nonnegative functions such that\n\\begin{align*}\n&\\textup{supp } f \\subset \\mathfrak{P}_{N_1} \\cap \\mathfrak{S}_{L_1}, \\quad \\textup{supp } g \\subset \\mathfrak{P}_{N_2} \\cap \\mathfrak{S}_{L_2}, \n\\quad \\text{and}\\\\\n&\\textup{supp } h \\subset \\{n \\in \\mathbb{Z}^2: 1 \\leq |n|^2 \\ll N_2 \\} \\cap \\mathfrak{S}_{L_0}.\n\\end{align*}\nThen, we have \n\\begin{equation}\n\\begin{split}\n\\bigg| \\sum_{\\substack{n, n_1\\in \\mathbb{Z}^2\\\\ n = n_1 - n_2}} &\\iintt_{\\tau = \\tau_1 - \\tau_2} f(\\tau_1, n_1) g(\\tau_2, n_2) h(\\tau, n) \\cdot \\mathbf 1_{|\\cos \\angle(n, n_2)| \\ll \\theta} \\, d\\tau d\\tau_1 \\bigg| \\\\\n&\\lesssim L_{\\textup{med}}^{\\frac 38} L_{\\textup{max}}^{\\frac 38} \\| f \\|_{L_{\\tau_1}^2 \\ell_{n_1}^2} \\| g \\|_{L_{\\tau_2}^2 \\ell_{n_2}^2} \\| h \\|_{L_\\tau^2 \\ell_n^2},\n\\end{split}\n\\label{low}\n\\end{equation}\nwhere $\\theta = \\frac{1}{|n|^{1-\\varepsilon} |n_2|^{1-\\varepsilon}} \\ll 1$.\n\\end{lemma}\n\nWe first assume Lemma \\ref{LEM:low} and prove \\eqref{bilin3}. \nRecall that  $|n|^2 \\ll |n_2|\\sim |n_1|$ in this case. \nThen,  by Lemma \\ref{LEM:low} (with $\\theta$ as above) and the Cauchy-Schwarz inequality in $N_1 \\sim N_2$, we have\n\\begin{align*}\n& \\text{LHS of } \\eqref{bilin3}\\\\ \n&\\hphantom{X} \\lesssim \\sum_{\\substack{N_1 \\sim N_2\\ge 1\\\\\\text{dyadic}}} \n\\sum_{\\substack{ L_1, L_2, L_0\\ge 1\\\\\\text{dyadic}}} L_1^{-\\frac 12 - \\delta_2} L_2^{-\\frac 12 - \\delta_2} L_0^{-\\frac 12 + \\delta_1} \n\\bigg| \\sum_{\\substack{n, n_1\\in \\mathbb{Z}^2\\\\ n = n_1 - n_2}} \\iintt_{\\tau = \\tau_1 - \\tau_2}\n \\mathbf 1_{|\\cos \\angle (n, n_2)| \\ll \\theta} \\\\\n&\\hphantom{XXX}\n\\times \\mathbf 1_{|n|^2 \\ll N_2} \\widehat{P_{N_1, L_1} u} (\\tau_1, n_1) \\overline{\\widehat{P_{N_2, L_2} v}} (\\tau_2, n_2) \\overline{\\widehat{Q_{L_0} w}} (\\tau, n) d\\tau_1 d\\tau \\bigg| \\\\\n&\\hphantom{X}\n\\lesssim  \\sum_{\\substack{N_1 \\sim N_2\\ge 1\\\\\\text{dyadic}}} \n\\sum_{\\substack{ L_1, L_, L_0\\ge 1\\\\\\text{dyadic}}}\n L_1^{0-} L_2^{0-} L_0^{0-} \\big\\| \\widehat{P_{N_1, L_1} u} \\big\\|_{L_{\\tau_1}^2 \\ell_{n_1}^2} \\big\\| \\widehat{P_{N_2, L_2} v} \\big\\|_{L_{\\tau_2}^2 \\ell_{n_2}^2} \\big\\| \\widehat{Q_{L_0} w} \\big\\|_{L_\\tau^2 \\ell_n^2} \\\\\n&\\hphantom{X}\n\\lesssim \\| \\widehat u \\|_{L_{\\tau_1}^2 \\ell_{n_1}^2} \\| \\widehat v \\|_{L_{\\tau_2}^2 \\ell_{n_2}^2} \\| \\widehat w \\|_{L_\\tau^2 \\ell_n^2},\n\\end{align*}\n\n\\noindent\nas desired.\n\n\n\nWe now present the proof of Lemma \\ref{LEM:low}, \nwhere we use the idea  from the proofs of  of Propositions 3.6 and 3.9 in \\cite{Kish13}.\n\n\n\\begin{proof}[Proof of Lemma \\ref{LEM:low}]\n\nWe first consider the case when $|n|\\lesssim L_{\\text{max}}$.\n In this case, we decompose the spatial frequencies of $h$ on the left-hand side  of \\eqref{low} into dyadic blocks $\\{|n| \\sim N_0\\}$, where the dyadic number  $N_0 \\ge 1$ satisfies  $N_0^2 \\ll N_2$.\n  For  fixed $N_0$, we decompose the spatial frequencies of $f$ and $g$ into  balls of radius $\\sim N_0$,\n  indexed by $J_1 \\in \\mathcal{J}_1$ and $J_2 \\in \\mathcal{J}_2$, respectively.\n  With a slight abuse of notation, \n  we also use $J_1$ and $J_2$ to denote the balls themselves.\n  Note that for each fixed $J_1 \\in \\mathcal{J}_1$, the product $\\mathbf 1_{J_1} (\\tau_1, n_1)\\cdot  \\mathbf 1_{J_2} (\\tau_1 - \\tau, n_1 - n)$ is nonzero for at most $O(1)$ many $J_2 \\in \\mathcal{J}_2$, and\n  we denote the set of these indices $J_2$'s as $\\mathcal{J}_2 (J_1)$.\n\nNoting that \n\\[\n\\big| |n_1|^2 - |n_2|^2 \\big| = \\big| |n|^2 -  (\\tau + |n|^2) + (\\tau_1 + |n_1|^2) - (\\tau_2 + |n_2|^2) \\big| \\lesssim N_0^2 + L_{\\textup{max}},\n\\]\n\n\\noindent\nwe have \n\\begin{align}\n\\big| |n_1| - |n_2| \\big| \\lesssim N_2^{-1}  (N_0^2 + L_{\\textup{max}})  \\ll 1.\n\\label{C1}\n\\end{align}\n \n \n \\noindent\n Given $j \\in \\mathbb{N}$, let $A_j = \\{ ( \\tau_1, n_1) : j < |n_1| \\leq j+1 \\}$\n and  $B_j = \\{ ( \\tau_2, n_2) : j  < |n_2| \\leq j + 1 \\}$.\n In view of \\eqref{C1}, we see that  \n  the product $\\mathbf 1_{A_{j_1}} (\\tau_1, n_1) \\cdot \\mathbf 1_{B_{j_2}} (\\tau_2, n_2)$\n   is nonzero if and only if $|j_1 - j_2| \\le 1$.\nHence, by letting $f_{j_1,  J_1}(\\tau_1, n_1) = \\mathbf 1_{A_{j_1} \\cap J_1} (\\tau_1, n_1)\\cdot f(\\tau_1, n_1)$\nand $g_{j_2, J_2}(\\tau_2, n_2) = \n \\mathbf 1_{B_{j_2} \\cap J_2} (\\tau_2, n_2)\\cdot g(\\tau_2, n_2)$, \n we can estimate  \n the left-hand side of \\eqref{low}  by \n\\begin{align*}\n\\sum_{\\substack{N_0\\ge 1\\\\ \\text{dyadic}}} \\sum_{\\substack{J_1 \\in \\mathcal{J}_1 \\\\ J_2 \\in \\mathcal{J}_2 (J_1)}} \\sum_{\\substack{j_1, j_2\\in \\mathbb{N}\\\\ |j_1 - j_2| \\le 1}} \n\\bigg| \n\\sum_{\\substack{n, n_1 \\in \\mathbb{Z}^2\\\\ n = n_1 - n_2\\\\|n| \\sim N_0}} &  \\iintt_{\\tau = \\tau_1 - \\tau_2}  f_{j_1, J_1} (\\tau_1, n_1)\ng_{j_2, J_2} (\\tau_2, n_2) \n\\\\\n&\\times \nh(\\tau, n) \\cdot \\mathbf 1_{S_{\\tau, n, j_1, j_2, J_1, J_2}}(\\tau_1, n_1) d\\tau d\\tau_1 \\bigg|,\n\\end{align*}\n\n\\noindent\nwhere the set $S_{\\tau, n, j_1, j_2, J_1, J_2}$ is defined by\n\\begin{align}\n\\begin{split}\nS_{\\tau, n, j_1, j_2, J_1, J_2} := \\big\\{ &(\\tau_1, n_1) \\in \\mathbb{R} \\times \\mathbb{Z}^2: \n(\\tau_1, n_1) \\in   \\mathfrak{P}_{N_1} \\cap \\mathfrak{S}_{L_1} \\cap A_{j_1},  \\, n_1 \\in J_1 \\\\\n&(\\tau_1 - \\tau, n_1 - n) \\in  \\mathfrak{P}_{N_2} \\cap \\mathfrak{S}_{L_2}\\cap B_{j_2}, \\, \nn_1 - n \\in J_2, \\\\\n&  |\\cos \\angle (n, n_1 - n)| \\ll \\theta \\big\\}.\n\\end{split}\n\\label{C1a}\n\\end{align}\n\n\nBy the Cauchy-Schwarz inequality in $\\tau_1$ and $n_1$, H\\\"older's inequality in $\\tau$ and $n$, and H\\\"older's inequality in $j$, we then have\n\\begin{align*}\n&\\text{LHS of }\\eqref{low} \\\\\n&\\leq \\sum_{\\substack{N_0\\ge 1\\\\ \\text{dyadic}}} \\sum_{\\substack{J_1 \\in \\mathcal{J}_1\\\\ J_2 \\in \\mathcal{J}_2 (J_1)}} \\sum_{\\substack{j_1, j_2\\in \\mathbb{N}\\\\ |j_1 - j_2| \\le 1}} \n \\sum_{n: |n| \\sim N_0} \\int \n\\| \\mathbf 1_{S_{\\tau, n, j_1, j_2, J_1, J_2}}\\|_{L^2_{\\tau_1} \\l^2_{n_1}}\\\\ \n&\\qquad \\times \n\\| f_{j_1, J_1} (\\tau_1, n_1) g_{j_2, J_2} (\\tau_1 - \\tau, n_1 - n)\\|_{L^2_{\\tau_1} \\l^2_{n_1}}\n |h(\\tau, n)| d\\tau \\\\\n&\\leq \\sum_{\\substack{N_0\\ge 1\\\\ \\text{dyadic}}} \\sum_{\\substack{J_1 \\in \\mathcal{J}_1 \\\\ J_2 \\in \\mathcal{J}_2 (J_1)}} \\sum_{\\substack{j_1, j_2\\in \\mathbb{N}\\\\ |j_1 - j_2| \\le 1}} \n \\underline{L}_{12}^{\\frac 12}  \\| f _{j_1, J_1}  \\|_{L_{\\tau_1}^2 \\l_{n_1}^2}\n  \\| g_{j_2, J_2} \\|_{L_{\\tau_2}^2 \\l_{n_2}^2} \\| h \\|_{L_\\tau^2 \\l_n^2} \\\\\n&\\qquad \\times \\sup_{\\substack{\\tau, \\tau_1, n\\\\ |n| \\sim N_0}} \\big|\\widetilde S_{\\tau, \\tau_1, n, j_1, j_2, J_1, J_2}\\big|^{\\frac 12} \\\\\n&\\lesssim \n\\sum_{\\substack{N_0\\ge 1\\\\ \\text{dyadic}}} \\sum_{\\substack{J_1 \\in \\mathcal{J}_1 \\\\ J_2 \\in \\mathcal{J}_2 (J_1)}} \\underline{L}_{12}^{\\frac 12} \\| \\mathbf 1_{J_1}\\cdot f  \\|_{L_{\\tau_1}^2 \\l_{n_1}^2} \\|  \\mathbf 1_{J_2} \\cdot g\\|_{L_{\\tau_2}^2 \\l_{n_2}^2} \\| h \\|_{L_\\tau^2 \\l_n^2} \\\\\n&\\qquad \\times \\sup_{\\substack{\\tau, \\tau_1, n, j_1, j_2\\\\ |n| \\sim N_0}}  \\big|\\widetilde S_{\\tau, \\tau_1, n, j_1, j_2, J_1, J_2}\\big|^{\\frac 12} , \n\\end{align*}\n\n\n\\noindent\nwhere the set $\\widetilde S_{\\tau, \\tau_1, n, j_1, j_2,  J_1, J_2}$ is defined by\n\\begin{align}\n\\widetilde S_{\\tau, \\tau_1, n, j_1, j_2,  J_1, J_2} := &\\big\\{ n_1 \\in \\mathbb{Z}^2: (\\tau_1, n_1) \\in \nS_{\\tau, n, j_1, j_2, J_1, J_2} \\big\\}.\n\\label{C1b}\n\\end{align}\n\nWe claim that the following bound holds:\n\\begin{align}\n\\sup_{\\substack{J_1 \\in \\mathcal{J}_1\\\\J_2 \\in \\mathcal{J}_2(J_1)}}\n \\sup_{\\tau, \\tau_1, n, j: |n| \\sim N_0} \\big| \\widetilde  S_{\\tau, \\tau_1, n, j_1, j_2, J_1, J_2}\\big| \\lesssim \\min\\bigg( N_0,  \\frac{L_{\\textup{max}}}{N_0} \\bigg). \n \\label{C2}\n\\end{align}\n\n\\noindent\nFor now, let us assume \\eqref{C2}.\nNote that the right-hand side of \\eqref{C2} is bounded by $L_{\\max}^\\frac{1}{2}$.\nThen, by applying  the Cauchy-Schwarz inequality in $J_1 \\in \\mathcal{J}_1$ and\nsumming over dyadic $N_0 \\geq 1$ together with \\eqref{C2}, \nwe  obtain the desired bound \\eqref{low}.\n \n\n\n\nIt remains to prove \\eqref{C2}.\nFor $n_1 \\in \\widetilde  S_{\\tau, \\tau_1, n, j_1, j_2, J_1, J_2}$, \n  it follows from \\eqref{C1b} with \\eqref{C1a} that \n\\[ \\tau_1 \\in \\big( - |n_1|^2 + [-C L_1, C L_1] \\big) \\cap \\big( \\tau - |n_1 - n|^2 + [-C L_2, C L_2] \\big), \\]\nfor some constant $C > 0$. Since this intersection of  the two sets is nonempty,\nwe must have $\\big| -|n_1|^2 - (\\tau - |n_1 - n|^2) \\big| = O(\\overline{L}_{12})$, \nwhich in turn implies \n\\begin{align}\n n_1 \\cdot \\frac{n}{|n|} = -\\frac{\\tau}{2|n|} + \\frac{|n|}{2} + O\\bigg( \\frac{\\overline{L}_{12}}{N_0} \\bigg). \n\\label{C3}\n \\end{align}\n\n\\noindent\nOn the other hand,   \nfor $n_1 \\in \\widetilde  S_{\\tau, \\tau_1, n, j_1, j_2, J_1, J_2}$, \nwe also have\n$n_1 \\in J_1$ where $J_1$ is a ball of radius $\\sim N_0$.\nHence, together with \\eqref{C3}, we see that  the component of $n_1$ which is parallel to $n$ is restricted to an interval of length $\\sim \\min(N_0, L_{\\text{max}} \/ N_0)$. Furthermore, we have\n\\begin{align*}\n\\begin{split}\n |\\cos \\angle (n_1, n)| & = \\frac{|n_1 \\cdot n|}{|n_1| | n|} \n \\leq \\frac{|n|}{|n_1|}  + \\frac{|(n_1 - n)\\cdot n|}{|n_1||n|}\\\\\n &  \\leq \\frac{|n|}{|n_1|} + |\\cos \\angle (n_1 - n, n)| \\frac{|n_1 - n|}{|n_1|} \\ll \\frac{|n|}{|n_1|} + \\frac{|n_1 - n|^\\varepsilon}{|n|^{1-\\varepsilon} |n_1|} \\ll 1, \n \\end{split}\n\\end{align*}\n\n\\noindent\nsince $|n_1|\\sim |n_1 - n|\\gg |n|^2$.\nHence, by Lemma \\ref{LEM:count} with $\\mu  =  1$, \n $\\nu \\sim  \\min(N_0, L_{\\textup{max}} \/ N_0)$, and $\\alpha = \\frac \\pi 4$, we obtain\n \\eqref{C2}.\n \n \n \nThe case  $L_{\\text{max}} \\ll |n|$ is similar and much simpler. We apply the same steps as in the previous case ($L_{\\max}\\gtrsim |n|$), except that we do not need to decompose the spatial frequencies of $n$ into dyadic piece or  to localize $n_1$ and $n_2$ on balls of smaller radii.\nIn this case,  we  apply Lemma \\ref{LEM:count} with $\\nu \\sim 1$ to obtain the desired bound.\n\\end{proof}\n\n\n\\medskip \\noindent\n$\\bullet$ \\textbf{Case 3:} $|n_2|^2 \\ll |n|$.\n\nThis case is similar to Case 2, so we will be brief here. For the low-modulation case (i.e.~$L_{\\text{max}} \\ll N_0$), the same argument works by switching the roles of $(\\tau, n)$ and $(\\tau_2, n_2)$. \n\nFor the high-modulation case (i.e.~$L_{\\text{max}} \\gtrsim N_0$), we need to ensure that on the \nright-hand side of \\eqref{high}  in Lemma \\ref{LEM:high}, the power of $L_0$ is less than $\\frac 12$ (so that simply switching the roles of $(\\tau, n)$ and $(\\tau_2, n_2)$ does not work). With $f$, $g$, and $h$ as in the statement of Lemma~\\ref{LEM:high}, when $N_2^2 \\ll N_0 \\sim N_1$ and $L_{\\text{max}} \\gtrsim N_0$, we use the Cauchy-Schwarz inequality in $\\tau$ and $n$ and apply Lemma \\ref{LEM:bilin} to obtain\n\\begin{align*}\n\\text{LHS of } \\eqref{high} &\\lesssim \\| h \\|_{L_\\tau^2 \\ell_n^2}\n \\underline{L}_{12}^{\\frac 12} \\bigg( \\frac{\\overline{L}_{12}}{N_0} + 1 \\bigg)^{\\frac 12 } N_2^{\\frac 12} \\| f \\|_{L_{\\tau_1}^2 \\ell_{n_1}^2} \\| g \\|_{L_{\\tau_2}^2 \\ell_{n_2}^2} \\\\\n&\\lesssim \\big( L_1^{\\frac 12} L_2^{\\frac 12} + \\underline{L}_{12}^{\\frac 12} N_0^{\\frac 14} \\big) \\| f \\|_{L_{\\tau_1}^2 \\ell_{n_1}^2} \\| g \\|_{L_{\\tau_2}^2 \\ell_{n_2}^2} \\| h \\|_{L_\\tau^2 \\ell_n^2} \\\\\n&\\lesssim \\big( L_1^{\\frac 12} L_2^{\\frac 12} L_{\\text{max}}^{0+} + \\underline{L}_{12}^{\\frac 12} L_{\\text{max}}^{\\frac 14 +} \\big) N_0^{0-} \\| f \\|_{L_{\\tau_1}^2 \\ell_{n_1}^2} \\| g \\|_{L_{\\tau_2}^2 \\ell_{n_2}^2} \\| h \\|_{L_\\tau^2 \\ell_n^2},\n\\end{align*}\n\n\\noindent\nwhich suffices for our purpose.\n\n\n\n\\begin{remark}\\label{REM:bilin}\n\\rm\nThe bilinear estimate \\eqref{bilin1} in Proposition \\ref{PROP:bilin}\nalso holds if we replace $u \\overline{v}$ by $uv$.\nIn fact, \na slight modification of the argument allows us to show \n\\begin{equation}\n\\| u v \\|_{X_T^{0,-\\frac 12 + \\delta_1}} \\lesssim \\| u \\|_{X_T^{0, \\frac 12 + \\delta_2}} \\| v \\|_{X_T^{0, \\frac 12 + \\delta_2}},\n\\label{bilin_uv}\n\\end{equation}\n\n\\noindent\nwhere $0 < T \\leq 1$ and $\\delta_1 >  \\delta_2 > 0$ are sufficiently small. \nNote that, in this case, by denoting $n_1$, $n_2$, and $n$ as the frequency of $u$, $v$, and \nthe duality term $w$, respectively.  we have $n = n_1 + n_2$ and $\\tau = \\tau_1 + \\tau_2$ so that \n\\[ (\\tau + |n|^2) - (\\tau_1 + |n_1|^2) - (\\tau_2 + |n_2|^2) = 2n_1 \\cdot n_2. \\]\n\n\\noindent\nHence, we need to perform case-by-case analysis, depending on the interaction between $n_1$ and $n_2$. \n\n\nThe proof of \\eqref{bilin_uv} follows essentially as in the proof of \\eqref{bilin1} except for Lemma \\ref{LEM:high}. With $f$, $g$, and $h$ as in the statement of Lemma \\ref{LEM:high}, when $N_1^2 \\ll N_2 \\sim N_0$ and $L_{\\text{max}} \\gtrsim N_2$,  the Cauchy-Schwarz inequality in $\\tau_2$ and $n_2$ and\n Lemma \\ref{LEM:bilin} yield \n\\begin{equation}\n\\begin{split}\n\\bigg| \\sum_{\\substack{n, n_1\\in \\mathbb{Z}^2\\\\ n = n_1 + n_2}} &\n\\iintt_{\\tau = \\tau_1 + \\tau_2} f(\\tau_1, n_1) g(\\tau_2, n_2) h(\\tau, n) d\\tau d\\tau_1 \\bigg| \\\\\n&\\lesssim \\big( L_0^{\\frac 12} L_1^{\\frac 12} N_2^{-\\frac 14} + \\underline{L}_{01}^{\\frac 12} N_2^{\\frac 14} \\big) \\| f \\|_{L_{\\tau_1}^2 \\ell_{n_1}^2} \\| g \\|_{L_{\\tau_2}^2 \\ell_{n_2}^2} \\| h \\|_{L_\\tau^2 \\ell_n^2}.\n\\end{split}\n\\label{uv1}\n\\end{equation}\n\n\\noindent\nIn order to repeat the argument in Subcase 2.a in the proof of Proposition \\ref{PROP:bilin}, \nwe need the power of $L_0$ to be less than $\\frac 12$, especially  when $L_{\\text{max}} = L_0$. \nNamely, the bound \\eqref{uv1} can not be used directly as it is.\n\n\nBy H\\\"older's inequality in $\\tau$ and $n$, Young's inequality, and H\\\"older's inequality\nwith \\eqref{high1}, we obtain\n\\begin{equation}\n\\begin{split}\n\\bigg| \\sum_{\\substack{n, n_1\\in \\mathbb{Z}^2\\\\ n = n_1 + n_2}} &\n\\iintt_{\\tau = \\tau_1 + \\tau_2} f(\\tau_1, n_1) g(\\tau_2, n_2) h(\\tau, n) d\\tau d\\tau_1 \\bigg| \\\\\n&\\lesssim \\| f \\|_{\\ell_{n_1}^2 L_{\\tau_1}^{\\frac 8 7}} \\| g \\|_{\\ell_{n_2}^1 L_{\\tau_2}^2} \\| h \\|_{\\ell_n^2 L_{\\tau}^{\\frac 85}} \\\\\n&\\lesssim L_0^{\\frac 18} L_1^{\\frac 38} N_2 \\| f \\|_{L_{\\tau_1}^2 \\ell_{n_1}^2} \\| g \\|_{L_{\\tau_2}^2 \\ell_{n_2}^2} \\| h \\|_{L_\\tau^2 \\ell_n^2}.\n\\end{split}\n\\label{uv2}\n\\end{equation}\n\n\\noindent\nThen, by interpolating \\eqref{uv1} and \\eqref{uv2}, we  obtain the desired inequality. A similar inequality also holds  when $N_2^2 \\ll N_1 \\sim N_0$ and $L_{\\text{max}} \\gtrsim N_1$ by switching the roles of $(\\tau_1, n_1)$ and $(\\tau_2, n_2)$.\n\n\n\\end{remark}\n\n\n\n\\section{Local well-posedness of the quadratic NLS}\n\\label{SEC:LWP}\n\nIn this section, we present the proof of Theorem \\ref{THM:LWP}, i.e. local well-posedness of the quadratic NLS \\eqref{qNLS} in $L^2(\\mathbb{T}^2)$.\nBy writing \\eqref{qNLS} in the following Duhamel formulation, we have \n\\begin{equation}\nu(t) = \\Gamma(u) := e^{it\\Delta} u_0 - i \\int_0^t e^{i(t-t')\\Delta} |u|^2(t') dt'.\n\\label{Duh}\n\\end{equation}\n\n\\noindent\nLet $\\varepsilon > 0$ be sufficiently small. Then, by \\eqref{Duh}, Lemma \\ref{LEM:Xlin}, and Proposition \\ref{PROP:bilin}, we have\n\\begin{align*}\n\\| \\Gamma(u) \\|_{X_T^{0, \\frac 12 + \\varepsilon}} &\\leq \\| e^{it\\Delta} u_0 \\|_{X_T^{0, \\frac 12 + \\varepsilon}} + \\bigg\\| \\int_0^t e^{i(t-t')\\Delta} u(t') \\overline{u}(t') dt' \\bigg\\|_{X_T^{0, \\frac 12 + \\varepsilon}} \\\\\n&\\lesssim \\| u_0 \\|_{L^2} + T^{\\varepsilon }\\| u \\overline{u} \\|_{X^{0, -\\frac 12 + 2 \\varepsilon}} \\\\\n&\\lesssim \\| u_0 \\|_{L^2} + T^{\\varepsilon } \\| u \\|_{X_T^{0, \\frac 12 + \\varepsilon}}^2.\n\\end{align*}\n\n\\noindent\n Similarly, we  obtain the following difference estimate:\n\\begin{align*}\n\\| \\Gamma(u) - \\Gamma(v) \\|_{X_T^{0, \\frac 12 + \\varepsilon}}\n\\lesssim T^{ \\varepsilon}\\Big( \\| u \\|_{X_T^{0, \\frac 12 + \\varepsilon}} + \\| v \\|_{X_T^{0, \\frac 12 + \\varepsilon}} \\Big)\n \\| u-v \\|_{X_T^{0, \\frac 12 + \\varepsilon}} .\n\\end{align*}\n\n\\noindent\nTherefore,  by choosing $T = T(\\| u_0 \\|_{L^2}) > 0$\nsufficiently small, we conclude  that $\\Gamma$ is a contraction on the ball $B_R \\subset X^{0, \\frac 12 + \\varepsilon}_T$ of radius $R \\sim \\| u_0 \\|_{L^2}$.\nThis proves  Theorem~\\ref{THM:LWP}.\n\n\n\n\n\n\n\n\\begin{ackno}\\rm\nR.L.~and T.O.~were supported by the European Research Council (grant no.~864138 ``SingStochDispDyn'').\n They are also grateful to the anonymous referee for the helpful comments.\n\n\\end{ackno}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec1}\\label{chp3intro}\nA common technique for eliciting consumption in studies of substance\nabuse is the time-line follow-back (TLFB) method, in which one asks\nsubjects to report daily consumption retrospectively over the preceding\nweek, month or other designated period. In smoking cessation research,\nfor example, TLFB is one important method for measuring cigarette\nconsumption and defining periods of quit and lapse.\n\nAlthough TLFB is a practical approach to quantifying average smoking\nbehavior [\\citet{Brown}], TLFB data can harbor substantial\nerrors as measures of daily consumption [\\citet{Klesges}].\nTLFB questionnaires request exact daily cigarette counts, which smokers\nare unlikely to remember, particularly after several days have passed.\nMoreover, some smokers may understate consumption to avoid the social\nstigma attached to excessive smoking or an inability to quit\n[\\citet{Boyd}]. Thus, smoking cessation studies typically require\nvalidation of TLFB reports of zero consumption by biochemical\nmeasurement of exhaled carbon monoxide or nicotine metabolites from\nsaliva or blood.\n\nA second concern is that histograms of TLFB-derived daily cigarette\ncounts commonly exhibit spikes at multiples of 20, 10 or even 5\ncigarettes. This phenomenon, known as ``digit preference'' or\n``heaping,'' is thought to reflect a tendency to report consumption in\nterms of packs (each pack in the US contains 20 cigarettes) or half or\nquarter packs. The heaps presumably arise because many smokers do not\nremember precisely how many cigarettes they smoked and therefore report\ntheir count rounded off to a nearby convenient number. It has also been\nhypothesized that some smokers consume exactly an integral number of\npacks per day as a self-rationing strategy [\\citet{Farrell}], but\nevidence so far suggests that such behavior, if it exists, causes only\na small fraction of the observed heaping [\\citet{WangHeitjan}].\nIndeed, \\citet{Klesges} observed that the distribution of\nbiochemical residues of smoking is smooth, suggesting that heaping is a\nphenomenon of reporting rather than consumption.\n\nRecall bias and heaping bias in self-reported longitudinal cigarette\ncounts potentially affect estimates of both means and treatment\neffects. Moreover, heaping may lead to underestimation of\nwithin-subject variability, thanks to smokers who regularly report one\npack rather than a precise count that varies around some mean in the\nvicinity of 20. If a large enough fraction of subjects in a study are\nof this kind, estimates of both within-subject and between-subject\nvariability can be distorted.\n\nAlthough there has been substantial research on statistical modeling of\nheaping and digit preference in a range of disciplines\n[Heitjan and Rubin (\\citeyear{HeitjanRubin,HR2}),\n\\citet{RidoutMorgan}, \\citet{Pickering},\n\\citet{Klerman}, \\citet{Torelli}, \\citet{Dellaportas},\n\\citet{Roberts}, \\citet{Wright} and \\citet{Wolff}], the\nonly such application in smoking cessation research is that of\n\\citet{WangHeitjan}, who described a latent-variable rounding\nmodel for heaped univariate TLFB cigarette count data. They postulated\nthat the reported cigarette count is a function of the unobserved true\ncount and a latent heaping behavior variable. The latter can take one\nof four values, representing exact reporting, rounding to the nearest\n5, rounding to the nearest 10, and rounding to the nearest 20. Except\nfor ``exact'' reporters (i.e., those who report counts not divisible by\n5), one obtains at best partial information on the true count and the\nheaping behavior. They analyzed univariate count data from a smoking\ncessation clinical trial, assuming a zero-inflated negative binomial\ndistribution for the true underlying counts together with an ordered\ncategorical logistic selection model for heaping behavior given true\ncount.\n\nThe analysis of \\citet{WangHeitjan} has three important\nlimitations: first, they included only data from the last day of eight\nweeks of treatment, ignoring the 55 preceding days. Second, they\nassumed---without empirical verification---that reported counts not\ndivisible by 5 were accurate. And third, they assumed that the\npreference for counts ending in 0 or 5 actually represented rounding\nrather than some other form of reporting error. That is, a declared\ncount of 20 cigarettes was taken to mean that the true count was\nsomewhere between 10 and 30 cigarettes, and was merely misreported as\n20. In the absence of more accurate data on the true, underlying count,\nattempts to model heaping must rely on some such assumptions.\n\nPrecise assessment of smoking behavior has taken on increasing\nimportance as researchers explore the value of reducing consumption as\na way to lessen the harms of smoking [\\citet{shiffman1},\n\\citet{Hatsukami}] and to improve the chance of ultimately\nquitting [\\citet{Shiffman3}, \\citet{Cheong}]. The advent of\nthe inexpensive hand-held electronic diary (ED) that allows the\ninstantaneous recording of \\textit{ad libitum} smoking has created the\npossibility of making much more accurate measurements. Such evaluation\nis an instance of \\textit{ecological momentary assessment} [EMA;\n\\citet{StoneShiffman}], in that it generates records of events\nlogged as they occur in real-life settings. In \\citet{Shiffman2},\nresearchers asked 236 participants in a smoking cessation study to use\na specially programmed ED to record each cigarette as it was smoked\nover a 16-day pre-quit period; moreover, the ED periodically prompted\nthe smokers to record any cigarettes they had missed. At days 3, 8 and\n15, subjects visited the clinic to complete a TLFB assessment of daily\nsmoking since the preceding visit (2, 5 or 7 days previously), stating\nhow many cigarettes they had smoked each day. The study found that\nwhile the TLFB data contained the expected heaps at multiples of 10 and\n20, the EMA data had practically none. Average smoking rates from the\ntwo methods were moderately correlated ($r=0.77$), but the\nwithin-subject correlation of daily consumption between TLFB and EMA\nwas modest ($r=0.29$). Self-report TLFB consumption was on average\nhigher than EMA (by 2.5 cigarettes), but on 32\\% of days, subjects\nrecorded more cigarettes by EMA than they later recalled by TLFB.\n\nThese data provide us with an opportunity---unprecedented, so far as we\nknow---to study the relationship between self-reports of daily\ncigarette consumption by TLFB and EMA. To describe this relationship,\nwe develop a statistical model with two components: the first is a\nregression that predicts the patient's notional ``remembered''\ncigarette count (a latent factor) from the EMA count. The second is a\nregression that predicts the rounding behavior---described as in\n\\citet{WangHeitjan} with an ordinal logistic regression---from the\nremembered count and fully observed predictors. The models include\nrandom subject effects that describe the propensities of the subjects\nto mis-remember their actual consumption (in the first component) and\nto report the remembered consumption with a characteristic degree of\naccuracy (in the second). Assuming that EMA represents the true count,\nthe first component of the model allows us to examine the recall bias\nresulting from mis-remembering, while the second component describes\nthe heaped reporting errors.\n\n\\section{Notation and model}\\label{sec2}\\label{method1}\nLet $Y_{it}$ denote the observed heaped TLFB consumption for subject\n$i$ on day $t$, $i=1, \\ldots, n$, $t=1, \\ldots, m_i$, and let\n$Y_i=(Y_{i1}, \\ldots, Y_{im_i})^{T}$ denote the vector of TLFB data\nfor subject $i$. Let $X_{it}$ be the EMA consumption on subject $i$,\nday $t$, and let $X_i=(X_{i1}, \\ldots, X_{im_i})^{T}$ be the vector of\nEMA data for subject $i$. We furthermore let $Z_i=(Z_i^R,Z_i^H)$ be a\nvector of baseline predictors for subject $i$, with $Z_i^R$\nrepresenting predictors of recall and $Z_i^H$ predictors of heaping.\nThese predictor sets may overlap.\n\n\\subsection{A model for remembered cigarette count}\\label{sec2.1}\nThe first part of our model assumes that for each day and subject there\nis a notional remembered cigarette count, denoted $W_{it}$ [$W_i=\n(W_{i1}, \\ldots,W_{im_i})^{T}$]. We assume $W_{it}$ is distributed as\nPoisson conditionally on a random effect $b_i$, the EMA smoking pattern\n$X_{it}$ and the covariate vector $Z_i$, with mean\n\\begin{equation}\\label{recall}\n{\\mathrm E}(W_{it}|X_{it},Z_i,b_i)= \\exp\\bigl(\\beta_0+\\ln(X_{it})\\beta\n_1+Z_i^R\\beta_2+b_i\\bigr).\n\\end{equation}\nThe parameters $\\beta_1$ and $\\beta_2$ represent the effects of EMA\nconsumption and baseline predictors, respectively, on the latent\nremembered count. The random effect $b_i$, which we assume normally\ndistributed with mean $0$ and variance $\\sigma_b^2$, represents\nheterogeneity among subjects. We note that there are no $0$ values of\n$X_{it}$ in the Shiffman data, which are from a pre-quit study in which\nsubjects were encouraged to smoke as normal. Thus, we can include $\\ln\n(X_{it})$ as a predictor. In more general contexts where $0$ EMA counts\nare possible, one can adjust the model in simple ways to avoid this\nproblem. Moreover, when excessive $0$ counts occur in the TLFB data,\none can fit a zero-inflated count model, as in \\citet{WangHeitjan},\nfor the remembered count.\n\n\\subsection{A model for the latent heaping process}\\label{sec2.2}\nFollowing \\citet{WangHeitjan}, we assume that a latent rounding\nindicator $G_{it}$ [$G_i=(G_{i1},\\ldots, G_{im_i})^{T}$] dictates the\ndegree of rounding to be applied to the notional remembered\ncount~$W_{it}$. Specifically, we let $G_{it}$ take one of four possible\nvalues: $G_{it}=1$ implies reporting the exact count, $G_{it}=2$\nimplies rounding to the nearest multiple of 5, $G_{it}=3$ implies\nrounding to the nearest multiple of 10, and $G_{it}=4$ implies rounding\nto the nearest multiple of 20. We assume that the probability\ndistribution of the heaping indicator depends on $W_{it}$, a\nsubject-level random effect $u_i \\sim N(0,\\sigma_u^2)$ that is\nindependent of $b_i$, and a baseline predictor vector $Z_i^H$.\nSpecifically, we propose the following proportional odds model for the\nconditional distribution of~$G_{it}$:\n\\begin{equation}\n\\label{misreport}\nf(G_{it}|W_{it},Z_i,u_i) = \\cases{\n1-q(\\gamma_1+\\eta_{it}+u_i), &\\quad if $g=1$;\\cr\nq(\\gamma_1+\\eta_{it}+u_i)-q(\\gamma_2+\\eta_{it}+u_i), &\\quad\nif $g=2$;\\cr\nq(\\gamma_2+\\eta_{it}+u_i)-\nq(\\gamma_3+\\eta_{it}+u_i), &\\quad if $g=3$;\\cr\nq(\\gamma_3+\\eta_{it}+u_i), &\\quad if $g=4$.}\\hspace*{-30pt}\n\\end{equation}\nHere $\\eta_{it}=W_{it}\\gamma_0+Z_i^H\\beta_3$, and $q(\\cdot)$ is the\ninverse logit function $q(x) = \\exp(x)\/\\break(1+\\exp(x))$. The parameters\n$\\gamma_1 > \\gamma_2 > \\gamma_3$ refer to the successive intercepts of\nthe logistic regressions, $\\gamma_0$ refers to its slope with respect\nto the remembered count, and $\\beta_3$ refers to its slopes with\nrespect to the vector of heaping predictors $Z_i^H$. The random effect\n$u_i$ describes between-subject differences in heaping propensity not\notherwise accounted for in the model.\n\n\\subsection{The coarsening function}\\label{sec2.3}\nAs in \\citet{WangHeitjan}, the model links the observed $Y_{it}$ to\nthe latent $W_{it}$ and $G_{it}$ via the coarsening function $h(\\cdot\n,\\cdot)$:\n\\[\nY_{it} = h(W_{it},G_{it}),\\qquad i=1,\\ldots,n, t=1,\\ldots,m_i.\n\\]\nFor example, at time $t$, subject $i$ with $W_{it}=14$ and $G_{it}=1$\nreports $h(14,1)=14$, whereas $h(14,2)=15$, $h(14,3)=10$, and\n$h(14,4)=20$. Figure~\\ref{heapdiag3} illustrates this heaping mechanism.\n\n\\begin{figure}\n\n\\includegraphics{557f01.eps}\n\n\\caption{Reported cigarette count $Y$ as a function\nof the underlying count $W$ and the rounding behavior $G$.}\\label{heapdiag3}\n\\end{figure}\n\nA coarsened outcome $y_{it}$ may arise from possibly several\n$(w_{it},g_{it})$ pairs. We denote the set of such pairs as\n$\\mathrm{WG}(y_{it}) = \\{(w_{it},g_{it})\\dvtx y_{it}=h(w_{it},g_{it})\\}\n$. For\nexample, a reported consumption of $y_{it}=5$ may represent a precise\nunrounded value [$(w_{it}, g_{it})=(5,1)$] or rounding across a range\nof nearby values [$(w_{it}, g_{it}) \\in\\{\n(3,2),(4,2),(5,2),(6,2),(7,2)\\}$]. For subject $i$, the probability of\nthe observed $y_{it}$ at time $t$ is the sum of the probabilities of\nthe $(w_{it}, g_{it})$ pairs that would give rise to it. The density of\nreported consumption $y_{it}$ given the random effects can therefore be\nexpressed as\n\\[\nf(y_{it}|b_i, u_i) = \\sum_{(w_{it},g_{it})\\in\\mathrm{WG}(y_{it})}\nf(w_{it}|b_i)f(g_{it}|w_{it},u_i).\n\\]\n\n\\subsection{Estimation}\\label{sec2.4}\nWe estimate the model by a Bayesian approach that employs importance\nsampling [\\citet{Gelmancarlin}, \\citet{Tanner}] to avoid iterative\nsimulation of parameters. The steps are as follows: we first compute\nthe posterior mode and information using a quasi-Newton method with\nfinite-difference derivatives [\\citet{Dennis}]. We then\napproximate the posterior with a multivariate $t_5$ density with mean\nequal to the posterior mode and dispersion equal to the inverse of the\nposterior information matrix at the mode. Next, we draw a large number\n(4000) of samples from this proposal distribution, at each draw\ncomputing the importance ratio $r$ of the true posterior density to the\nproposal density. We then use sampling-importance resampling (SIR) to\nimprove the approximation of the posterior [\\citet{Gelmancarlin}].\nWe evaluate posterior moments by averaging functions of the\nsimulated parameter draws with the importance ratios $r$ as weights.\nThe choice of a $t$ with a small number of degrees of freedom as the\nimportance density is intended to balance the convergence of the MC\nintegrals and the efficiency of the simulation.\n\nLetting $\\theta=(\\beta_0,\\beta_1,\\beta_2,\\beta_3,\\sigma_b,\\gamma\n_1,\\gamma_2,\\gamma_3,\\gamma_0,\\sigma_u)$, the likelihood contribution\nfrom subject $i$ is\n\\begin{eqnarray}\\label{marginallik}\nL(\\theta; y_i)&=&\\iint\\prod_{t=1}^{m_i}\\sum_{(w_{it},g_{it})\\in\n\\mathrm{WG}(y_{it})} f(w_{it}|b_i)f(g_{it}|w_{it}, u_i)\\nonumber\\\\[-8pt]\\\\[-8pt]\n&&\\hspace*{100pt}{}\\times f(b_i)f(u_i)\\,\ndb_i \\,du_i;\\nonumber\n\\end{eqnarray}\nwe approximate the integral in \\eqref{marginallik} by Gaussian\nquadrature. We choose proper but vague priors for the parameters, which\nwe assume are a priori independent (except for $\\gamma\n_j, j=1,2,3$, as noted below). The parameter $\\beta_1$ in the Poisson\nmixed model~\\eqref{recall}, representing the slope of the latent recall\non the EMA recorded consumption, is given a normal prior $\\beta_1 \\sim\nN(1,10^2)$, whereas the priors of the other regression parameters in\nboth model parts are set to $N(0,10^2)$ subject to the constraint\n$\\gamma_1 > \\gamma_2 > \\gamma_3$. We assign the random-effect variances\ninverse-gamma priors with mean and SD both equal to 1, a reasonably\nvague specification [\\citet{Carlin}]. We obtain the posterior\nmode and information using SAS PROC NLMIXED, and implement Bayesian\nimportance sampling in \\texttt{R}.\n\n\\section{Model checking}\\label{sec3}\\label{modelchecking}\nWith heaped data, the unavailability of simple graphical diagnostics\nsuch as residual plots complicates model evaluation. We therefore\nresort to examination of repeated draws of latent quantities from their\nposterior distributions, in the spirit of Bayesian posterior predictive\nchecks [\\citet{Rubin}, \\citet{GelmanMeng}, \\citet\n{GelmanHeitjan}].\nSpecifically, we evaluate the adequacy of model\nassumptions using imputed values of the latent recall $W$, which we\ncompare to its implied marginal distribution under the model.\n\nImputations of latent $W_i$ and $G_i$ are ultimately based on the\nposterior density $f(\\theta|y_i)$ of the model parameter $\\theta$ given\nthe observed data $y_i$. \\citet{HeitjanRubin}, sampling univariate\n$y$ values, used an acceptance-rejection procedure to draw quantities\nanalogous to our $W$ and $G$ from a confined bivariate normal\ndistribution. In our model, the correlation within $W_i$ and $G_i$\nvectors poses a challenge to simulation. Note, however, that given the\nsubject-specific effects $b_i$ and $u_i$, the components of $W_i$ and\n$G_i$ are independent. Thus, we can readily simulate $(W_i,G_i)$ from\nthe joint posterior of $(W_i,G_i,b_i,u_i)$. For each simulated $\\theta$\nand the observed data $y_i$, the posterior distribution of\n$(W_i,G_i,b_i,u_i)$ is\n\\[\nf(w_i,g_i,b_i,u_i|y_i,\\theta) = f(w_i,g_i,b_i,u_i|\\theta)\n\\frac{f(y_i|w_i,g_i,b_i,u_i,\\theta)}{f(y_i|\\theta)}.\n\\]\nBecause the values of $w_{it}$ and $g_{it}$ together determine\n$y_{it}$, we have that\n\\[\nf(y_i|w_i, g_i, b_i, u_i,\\theta) =\\prod_{t=1}^{m_i}I\\bigl((w_{it},\ng_{it})\\in\\mathrm{WG}(y_{it})\\bigr),\n\\]\nwhere $I$ is an indicator function. Accordingly,\n\\begin{eqnarray*}\n&&\nf(w_i, g_i, b_i, u_i|y_i, \\theta)\\\\\n&&\\qquad\\propto f(w_i, g_i, b_i, u_i|\\theta)\\prod_{t=1}^{m_i}I\\bigl((w_{it},\ng_{it})\\in\\mathrm{WG}(y_{it})\\bigr) \\\\\n&&\\qquad = f(w_i, g_i|b_i, u_i, \\theta)f(b_i,u_i|\\theta)\\prod\n_{t=1}^{m_i}I\\bigl((w_{it},g_{it})\\in\\mathrm{WG}(y_{it})\\bigr) \\\\\n&&\\qquad = f(w_i|b_i,\\theta)f(g_i|w_i, u_i,\\theta)f(b_i,u_i|\\theta)\\prod\n_{t=1}^{m_i}I\\bigl((w_{it},g_{it})\\in\\mathrm{WG}(y_{it})\\bigr) \\\\\n&&\\qquad = \\Biggl(\\prod_{t=1}^{m_i}f(w_{it}|b_i,\\theta)f(g_{it}|w_{it},\nu_i,\\theta)I\\bigl((w_{it},g_{it})\\in\\mathrm{WG}(y_{it})\\bigr)\\Biggr)\\\\\n&&\\qquad\\quad{} \\times f(b_i|\\sigma_b)f(u_i|\\sigma_u).\n\\end{eqnarray*}\nThus, given random effects $b_i$ and $u_i$, the imputation of\n$(w_i,g_i)$ is obtained by independent draws of $(w_{it},g_{it})$,\n$t=1,\\ldots,m_i$, which can be implemented as an acceptance-rejection\nprocedure. We therefore impute the data as follows:\n\\begin{longlist}[(2)]\n\\item[(1)] Make independent draws, $\\theta^{(k)}, k=1,\\ldots,K$ from\n$f(\\theta|y_i)$ by SIR.\n\\item[(2)] Given $\\theta^{(k)}$, for $i=1,\\ldots,n$, independently draw\n$b_i^{(k)}\\sim N(0, \\sigma_b^{(k)2})$ and $u_i^{(k)} \\sim N(0,\\sigma_u^{(k)2})$.\n\\item[(3)] For $i=1,\\ldots,n$, given $\\theta^{(k)}$ and $b_i^{(k)}$, for\n$t=1,\\ldots, m_i$, draw $w_{it}^{(k)}$ as Poisson with mean \\eqref\n{recall}. Then given $\\theta^{(k)}$, $u_i^{(k)}$ and $w_{it}^{(k)}$,\ndraw misreporting type $g_{it}^{(k)}$ from \\eqref{misreport}. If\n$I((w_{it}^{(k)},g_{it}^{(k)})\\in\\mathrm{WG}(y_{it}))=0$, discard\n$(w_{it}^{(k)},g_{it}^{(k)})$ and repeat this step until\n$I((w_{it}^{(k)},g_{it}^{(k)})\\in\\mathrm{WG}(y_{it}))=1$.\n\\end{longlist}\n\nTo assess model fit, we plot $K$ histograms of the imputed latent count~$w$.\nImplausible patterns in these histograms, such as peaks or troughs\nat multiples of 5, suggest incorrect modeling of the heaping. We can\nalso base discrepancy diagnostics specifically on the fractions of\nreported consumptions that are divisible by~5.\n\n\\section{Simulations}\\label{sec4}\nTo examine the performance of our approach, we conducted simulations\nreplicating the structure of the Shiffman data with $m=12$ nonvisit-day\nobservations per subject. Each data set consisted of $n=100$ subjects,\nand for simplicity we do not consider baseline covariates. For each\nsubject we first set $x_i$ as an observed EMA count vector from the\ndata and generated a random effect $b_i\\sim N(0,\\sigma_b^{2}=0.09)$. We\nthen generated $W_{it}$ values as independent Poisson deviates with\nconditional mean \\eqref{recall}. With $\\beta_0 = 2.358$, $\\beta_1 =\n0.2628$, when $b_i = 0$ and EMA count $x_{it} = 20$, the mean latent\nrecall is 23.2, and when $x_{it} = 30$ it is~25.8. With the random\neffect distributed as designated above, the marginal mean recalls for\n$x_{it} = 20$ and $x_{it} = 30$ are 24.3 and 27.0, respectively.\n\nNext we generated the latent heaping behavior indicator $G_{it}$ from\n\\eqref{misreport}. We set the parameters to their estimates from the\nShiffman data: the intercepts $\\gamma_1$, $\\gamma_2$, $\\gamma_3$ were\n$-1.485$, $-5.280$ and $-10.141$, respectively, and the slope $\\gamma\n_0$ was $0.1098$. We simulated the random effect $u_i \\sim N(0,\\sigma\n_u^2=7.1)$. Under this setting, when $u_i=0$ and $w_{it}=22$, the\nprobability of exact reporting is 28.3\\%, and the probabilities of\nrounding to the nearest multiples of 5, 10 and 20 are 66.3\\%, 5.4\\% and\n0.04\\%, respectively. When the latent count $w_{it}=36$, these\nprobabilities are 7.8\\%, 71.2\\%, 20.8\\% and 0.2\\%, respectively. The\nsimulated latent $w_{it}$ and $g_{it}$ determined $y_{it}$ as\nillustrated in Figure~\\ref{heapdiag3}.\n\nThese parameter values allow for considerable discrepancy between\nremembered and recorded consumption. To examine our methods when the\nlatent recall and EMA match more closely, we conducted a second\nsimulation under parameter values that gave better agreement. In this\nscenario, we assumed $\\beta_0=0$ and $\\beta_1=1$ with $b_i \\sim N(0,\n0.05)$. Thus, when $b_i=0$, the expected precise recall ${\\mathrm\nE}(w_{it})=x_{it}$, and the marginal mean recalls are 20.5 and 30.8 for\nEMA counts of 20 and 30, respectively. We set the parameters in the\nheaping behavior models at $-1.07$, $-4.37$, $-6.52$ and $0.088$ for\n$\\gamma_1$, $\\gamma_2$, $\\gamma_3$ and $\\gamma_0$, respectively, and\n$\\sigma^2_u = 5.9$. In this case, when $u_{it}=0$, the probabilities of\nreporting exactly and to the nearest multiples of 5, 10 and 20 for a\ntrue count of 22 are 29.6\\%, 62.3\\%, 7.1\\% and 1\\%, respectively.\n\n\\begin{table}\n\\caption{Results of 100 simulations of the\nmis-remembering\/heaping model}\\label{simulation}\n\\begin{tabular*}{\\tablewidth}{@{\\extracolsep{\\fill}}ld{3.3}d{2.3}cd{2.3}cc@{}}\n\\hline\n&\\multicolumn{1}{c}{\\textbf{True}}&\\multicolumn{1}{c}{\\textbf{Mean of}} &\n\\multicolumn{1}{c}{\\textbf{SD of}} &&& \\multicolumn{1}{c@{}}{\\textbf{Coverage of}}\\\\\n\\textbf{Parameter}&\\multicolumn{1}{c}{\\textbf{value}}\n&\\multicolumn{1}{c}{\\textbf{estimate}}&\\multicolumn{1}{c}{\\textbf{estimate}}\n&\\multicolumn{1}{c}{\\textbf{Bias}}\n&\\multicolumn{1}{c}{$\\bolds{\\sqrt{\\mathrm{MSE}}}$}\n&\\multicolumn{1}{c@{}}{\\textbf{95\\% CI (\\%)}}\\\\\n\\hline\n\\multicolumn{7}{@{}c@{}}{\\textit{Case} 1: \\textit{Estimated mis-remembering}}\\\\[4pt]\nLatent recall\\\\\n\\quad$\\beta_0$&2.36&2.36&0.07&0.002&0.07&95\\\\\n\\quad$\\beta_1$&0.26&0.26&0.02&0.001&0.02&93\\\\\n\\quad$\\sigma_b$&0.30&0.30&0.02&0.001&0.02&95\n\\\\[4pt]\nHeaping behavior\\\\\n\\quad$\\gamma_1$&-1.49&-1.53&0.56&-0.04&0.56&94\\\\\n\\quad$\\gamma_2$&-5.28&-5.31&0.66&-0.03&0.66&98\\\\\n\\quad$\\gamma_3$&-10.14&-9.99&2.55&0.15&2.54&80\\\\\n\\quad$\\gamma_0$&0.11&0.11&0.02&0.002&0.02&96\\\\\n\\quad$\\sigma_u$&2.67&2.61&0.29&-0.06&0.29&98\n\\\\[4pt]\n\\multicolumn{7}{@{}c@{}}{\\textit{Case} 2: \\textit{Minimal mis-remembering}}\\\\[4pt]\nLatent recall\\\\\n\\quad$\\beta_0$&0.0&-0.01&0.09&-0.01&0.09&94\\\\\n\\quad$\\beta_1$&1.0&1.00&0.03&0.005&0.03&94\\\\\n\\quad$\\sigma_b$&0.22&0.22&0.02&-0.001&0.02&97\n\\\\[4pt]\nHeaping behavior\\\\\n\\quad$\\gamma_1$&-1.07&-1.08&0.43&-0.007&0.43&98\\\\\n\\quad$\\gamma_2$&-4.37&-4.36&0.60&0.007&0.59&94\\\\\n\\quad$\\gamma_3$&-6.52&-6.43&0.66&0.09&0.67&94\\\\\n\\quad$\\gamma_0$&0.088&0.090&0.02&0.002&0.02&95\\\\\n\\quad$\\sigma_u$&2.44&2.41&0.27&-0.02&0.27&95\\\\\n\\hline\n\\end{tabular*}\n\\end{table}\n\nTable~\\ref{simulation} presents summaries of 100 simulations of\nestimates of the parameter $\\theta=(\\beta_0,\\beta_1,\\sigma_b,\\gamma\n_1,\\gamma_2,\\gamma_3,\\gamma_0,\\sigma_u)$. Under both scenarios, the\nMLEs of the fixed-effect coefficients fell near the true values on\naverage, with no more than 0.5\\% bias for the parameters in the recall\nmodel and no more than 2.7\\% bias for those in the heaping model. The\nrandom effects variance estimates are also well estimated, with bias\nless than 1\\%. The coverage probabilities of nominal 95\\% confidence\n\\begin{table}\n\\caption{Results of 100 simulations of the\nmis-remembering\/heaping model with parameters estimated from the data\n(case 1) and SEs computed by the parametric bootstrap}\\label{simulation2}\n\\begin{tabular*}{\\tablewidth}{@{\\extracolsep{\\fill}}ld{3.2}d{3.2}cd{2.3}cc@{}}\n\\hline\n&\\multicolumn{1}{c}{\\textbf{True}}\n& \\multicolumn{1}{c}{\\textbf{Mean of}} &\\multicolumn{1}{c}{\\textbf{SD\nof}}\n&&&\\multicolumn{1}{c@{}}{\\textbf{Coverage of}}\\\\\n\\textbf{Parameter}&\\multicolumn{1}{c}{\\textbf{value}}\n&\\multicolumn{1}{c}{\\textbf{estimate}}\n&\\multicolumn{1}{c}{\\textbf{estimate}}\n&\\multicolumn{1}{c}{\\textbf{Bias}}\n&\\multicolumn{1}{c}{$\\bolds{\\sqrt{\\mathrm{MSE}}}$}\n&\\multicolumn{1}{c@{}}{\\textbf{95\\% CI (\\%)}}\\\\\n\\hline\nLatent recall\\\\\n\\quad$\\beta_0$&2.36&2.36&0.08&-0.003&0.08&90\\\\\n\\quad$\\beta_1$&0.26&0.26&0.02&0.001&0.02&90\\\\\n\\quad$\\sigma_b$&0.30&0.30&0.02&-0.001&0.02&95\n\\\\[4pt]\nHeaping behavior\\\\\n\\quad$\\gamma_1$&-1.49&-1.61&0.55&-0.12&0.56&94\\\\\n\\quad$\\gamma_2$&-5.28&-5.42&0.69&-0.14&0.70&96\\\\\n\\quad$\\gamma_3$&-10.14&-10.61&3.56&-0.47&3.58&87\\\\\n\\quad$\\gamma_0$&0.11&0.11&0.02&0.005&0.02&95\\\\\n\\quad$\\sigma_u$&2.67&2.64&0.32&-0.03&0.32&92\\\\\n\\hline\n\\end{tabular*}\n\\end{table}\nintervals range from 93\\% to 98\\%, except for $\\gamma_3$ in case 1,\nwhere coverage is only 80\\%. The poor coverage rate for this parameter\nis a consequence of instability in the inverse Hessian matrix; it can\nbe improved by creating parametric bootstrap confidence intervals\n(Table~\\ref{simulation2}). The simulation shows good performance of the\nMLEs, and, as the sample size is large, we expect the Bayesian\nestimates to behave similarly. Moreover, the maximization part of the\nMLE calculation can help identify multimodality of the likelihood,\nshould it occur, and singularity of the Hessian that we use in the\nBayesian sampling.\\\n\n\\section{Data analysis}\\label{sec5}\nWe applied the method of Section~\\ref{sec2} to the Shiffman data, with\nthe aim of evaluating our posited two-stage process\nas an explanation for the discrepancy between actual and reported\nconsumption. To focus on the link between the self-report and true\ncount, our first analysis included only log EMA count in \\eqref{recall}\nand a visit day indicator in \\eqref{misreport}. The latter is important\nbecause it seems reasonable that distance in time from the event would\nbe a strong predictor of heaping coarseness. Our second analysis\nexpanded the recall model to include a range of baseline\ncharacteristics: demographics (age, sex, race and education);\naddiction; measures of nicotine dependence [the Fagerstr\\\"{o}m Test for\nNicotine Dependence (FTND) and the Nicotine Dependence Syndrome Scale\n(NDSS)]; and EMA compliance measured as the daily percentage of missed\nprompts. Age, education, FTND and EMA compliance are considered as\nquantitative variables, sex and race are binary indicators, and\naddiction is a categorical variable taking three levels (possible,\nprobable and definite). They are the first variables that a smoking\nresearcher would think to investigate, and could potentially affect\nremembered count or heaping probability. The two measures of nicotine\ndependence FTND and NDSS showed only a modest correlation, with\nSpearman $r = 0.56$ in our data. So we considered both in the model.\nThe data set and programming code are included in the supplementary\nmaterials [\\citet{data}].\n\n\\subsection{Evaluating goodness of fit}\\label{sec5.1}\nWe evaluated model fit by creating multiple draws from the posterior\npredictive distribution of latent quantities as discussed in Section\n\\ref{sec3}.\nLack of smoothness in the histogram of the imputed latent count would\nsuggest an inadequate heaping model.\n\n\\begin{figure}\n\n\\includegraphics{557f02.eps\n\\vspace*{-2pt}\n\\caption{Top row: histogram of self-reported cigarette\nconsumption. Lower three rows: histograms of draws from the posterior\ndistribution of the latent exact consumption\nrecall.}\\label{ppc}\\vspace*{-4pt}\n\\end{figure}\n\nWe evaluated goodness of fit for the model that includes log EMA count\nin \\eqref{recall} and a visit day indicator in \\eqref{misreport}. The\ntop row in Figure~\\ref{ppc} displays the histograms of TLFB cigarette\nconsumption at days 3 (a visit day), 9 and 14. The spikes at 10, 15,\n20, 25, 30, etc. are characteristic of self-reported cigarette counts\n[\\citet{WangHeitjan}]. As many as 70\\% of subjects reported\ncigarette smoking in multiples of 5 for nonvisit-day consumption,\nwhereas for the visit day (day 3) that number is only 48\\%. Only $1\/4$ of\nthe counts on the visit day ended in $0$.\n\nThe next three rows represent independent draws of the latent count\n$W_{it}$. The spikes at multiples of 20, 10 or 5 have disappeared.\nCompared to the self-reported count, the percentage of subjects whose\nexact counts are divisible by 5 (or 10 or~20) is smaller and consistent\nacross time. Averaged over three imputations, the fraction of counts\nending in multiples of $5$ is 27\\%, 25\\% and 23\\% on days 3, 9 and 14,\nrespectively, and 15\\%, 14\\% and 12\\% end in multiples of $10$. These\nchecks indicate that our model offers a plausible explanation for the heaping.\n\n\\subsection{The fitted model}\\label{sec5.2}\nIn order to assess the impact of the assumed correlation structure, we\nfit the model as proposed in \\eqref{recall} and \\eqref{misreport} and\nalso a model that excludes random effects. Posterior modes and 95\\%\ncredible intervals (CIs) appear in Tables~\\ref{posterior1} and \\ref\n{posterior2}. The estimates in both the remembered count model that\n\\begin{table}\n\\caption{Estimated parameters from the Shiffman data\nunder simple models for recall (EMA only) and heaping (remembered count\nand visit day indicator)}\\label{posterior1}\n\\begin{tabular*}{\\tablewidth}{@{\\extracolsep{\\fill}}ld{3.2}cd{2.2}c@{}}\n\\hline\n&\\multicolumn{2}{c}{\\textbf{Random effects model}} &\\multicolumn\n{2}{c@{}}{\\textbf{Independence model}}\\\\[-4pt]\n& \\multicolumn{2}{c}{\\hrulefill} & \\multicolumn{2}{c@{}}{\\hrulefill}\\\\\n&\\multicolumn{1}{c}{\\textbf{Posterior}}& &\\multicolumn{1}{c}{\\textbf{Posterior}}&\\\\\n\\textbf{Parameter}&\\multicolumn{1}{c}{\\textbf{mode}}&\\multicolumn{1}{c}{\\textbf{95\\% CI}}\n&\\multicolumn{1}{c}{\\textbf{mode}}&\\multicolumn{1}{c@{}}{\\textbf{95\\% CI}}\\\\\n\\hline\n\\multicolumn{5}{@{}l}{Latent recall: Poisson model}\\\\\n\\quad Intercept: $\\beta_0$&2.32&$[2.24, 2.40]$ &\n1.14&$[1.09, 1.20]$\\\\\n\\quad$\\ln$(EMA): $\\beta_1$&0.27&$[0.25, 0.30]$ &\n0.68&$[0.66, 0.69]$\\\\\n\\quad$\\sigma_b^2$&0.09&$[0.08, 0.11]$&\\\\\n[4pt]\n\\multicolumn{5}{@{}l}{Heaping behavior: proportional odds model}\\\\\n\\quad Intercept 1: $\\gamma_1$&-1.50&$[-2.17, -0.85]$\n&-1.06&$[-1.30, -0.84]$\\\\\n\\quad Intercept 2: $\\gamma_2$&-5.21&$[-6.14, -4.43]$\n&-2.94&$[-3.26, -2.65]$\\\\\n\\quad Intercept 3: $\\gamma_3$&-10.15&$[-12.49, -8.48]$\n&-4.17&$[-4.59, -3.82]$\\\\\n\\quad Exact count (latent): $w$&0.11&$[0.09, 0.13]$&\n0.07&$[0.06, 0.08]$\\\\\n\\quad Visit day&-2.96&$[-3.50, -2.50]$ &\n-1.29&$[-1.54, -1.06]$\\\\\n\\quad$\\sigma_u^2$&6.65&$[5.12, 9.08]$ &\\\\\n\\hline\n\\end{tabular*}\n\\end{table}\ncharacterizes the latent recall process and the heaping behavior model\nare sensitive to the assumption of random effects. The Bayesian\ninformation criterion (BIC) of the model with two random effects is\n14,705 when including EMA as the only predictor and 14,059 when\nincluding EMA and the baseline patient characteristic predictors. The\nBICs for the corresponding models excluding random effects are 18,340\nand 16,641, respectively.\\vadjust{\\goodbreak} Thus, the evidence is overwhelming that the\nmixed model is preferable. Furthermore, we included the patient\ncharacteristic predictors as covariates in both the remembered count\nmodel and heaping process model, but this model ($\\mbox{BIC} = 14\\mbox{,}079$) is\nless favorable compared to the model with the covariates in just the\nlatent remembered count model. None of these predictors is significant\nin the heaping process model (results not shown).\\vadjust{\\goodbreak}\n\n\\begin{table}\n\\caption{Estimated parameters from the Shiffman data\nunder an expanded model for recall}\\label{posterior2}\n\\begin{tabular*}{\\tablewidth}{@{\\extracolsep{\\fill}}ld{3.3}cd{2.3}c@{}}\n\\hline\n&\\multicolumn{2}{c}{\\textbf{Random effects model}} &\\multicolumn\n{2}{c@{}}{\\textbf{Independence model}}\\\\[-4pt]\n& \\multicolumn{2}{c}{\\hrulefill} & \\multicolumn{2}{c@{}}{\\hrulefill}\\\\\n&\\multicolumn{1}{c}{\\textbf{Posterior}}& &\\multicolumn{1}{c}{\\textbf{Posterior}}&\\\\\n\\textbf{Parameter}&\\multicolumn{1}{c}{\\textbf{mode}}&\\multicolumn{1}{c}{\\textbf{95\\% CI}}\n&\\multicolumn{1}{c}{\\textbf{mode}}&\\multicolumn{1}{c@{}}{\\textbf{95\\% CI}}\\\\\n\\hline\n\\multicolumn{5}{@{}l}{Latent recall: Poisson model}\\\\\n\\quad Intercept: $\\beta_0$&2.34&$[2.21, 2.49]$ &\n1.51&$[1.44, 1.58]$\\\\\n\\quad$\\ln$(EMA): $\\beta_1$&0.25&$[0.23, 0.28]$ &\n0.53&$[0.51, 0.55]$\\\\\n\\quad Addicted \\\\\n\\qquad Possible vs. definite&0.07&$[-0.10, 0.24]$\n&0.05&$[0.01, 0.09]$\\\\\n\\qquad Probable vs. definite&- 0.01&$[-0.11, 0.08]$\n&-0.02&$[-0.04, 0.006]$\\\\\n\\quad FTND &0.06&$[0.04, 0.08]$ &0.04&$[0.03, 0.05]$\\\\\n\\quad NDSS &0.08&$[0.05, 0.12]$ &0.05&$[0.04, 0.06]$\\\\\n\\quad EMA compliance &0.13&$[-0.28, 0.51]$ &\n0.39&$[0.29, 0.49]$\\\\\n\\quad Age &0.002&$[-0.001, 0.006]$ &0.003&$[0.002,\n0.004]$\\\\\n\\quad Race (black vs. white) &-0.14&$[-0.27, -0.01]$\n&-0.06&$[-0.10, -0.03]$\\\\\n\\quad Sex (male vs. female) &0.16&$[0.10, 0.23]$&\n0.12&$[0.09, 0.23]$\\\\\n\\quad Education &-0.001&$[-0.03, 0.02]$ &\n0.003&$[-0.004, 0.009]$\\\\\n\\quad$\\sigma_b^2$&0.06&$[0.05, 0.07]$&\\\\\n[4pt]\n\\multicolumn{5}{@{}l}{Heaping behavior: proportional odds model}\\\\\n\\quad Intercept 1: $\\gamma_1$&-1.62&$[-2.35, -0.90]$\n&-1.14&$[-1.37, -0.91]$\\\\\n\\quad Intercept 2: $\\gamma_2$&-5.52&$[-6.42, -4.61]$\n&-3.15&$[-3.47, -2.82]$\\\\\n\\quad Intercept 3: $\\gamma_3$&-10.31&$[-12.65, -8.37]$&\n-4.54&$[-4.99, -4.08]$\\\\\n\\quad Exact count: $w$&0.11&$[0.09, 0.14]$ &\n0.07&$[0.06, 0.08]$\\\\\n\\quad Visit day&-2.99&$[-3.51, -2.47]$ &\n-1.26&$[-1.50, -1.02]$\\\\\n\\quad$\\sigma_u^2$&6.79&$[4.73, 8.68]$ &\\\\\n\\hline\n\\end{tabular*}\n\\end{table}\n\nThe 95\\% CI of $\\beta_1$ is $[0.23, 0.28]$, indicating that remembered\nconsumption is positively associated with recorded EMA consumption. In\naddition, baseline patient characteristics FTND, NDSS, race and gender\nhave significant effects on the recall process. For fixed EMA count,\nthe following characteristics are associated with greater remembered\nsmoking: higher nicotine dependence (measured by both FTND and NDSS),\nwhite ethnicity (compared to black) and male sex.\n\nFigure~\\ref{expw} displays the estimated curve of the mean of $W_{it}$\nagainst the EMA count. A natural hypothesis is that the estimated\nlatent mean agrees with EMA, which would be reflected in the Poisson\nmodel by an estimated intercept of $0$ and slope of~$1$; one might call\nthis a model of unbiased memory. To the contrary, Figure~\\ref{expw}\nshows that the fitted mean curve diverges substantially from the\n$45^{\\circ}$ line, with the lighter smokers on average overestimating\ntheir consumption and the heavier smokers underestimating consumption.\nThe mean remembered consumption agrees with the true count roughly in\nthe range 22--26 cigarettes, or slightly more than a pack per\nday.\n\n\\begin{figure}\n\n\\includegraphics{557f03.eps}\n\n\\caption{Estimate of the conditional mean of recalled\ncount given EMA count in the Poisson mis-remembering model. Covariates\nare fixed at education${}={}$high school, addicted${}={}$definitely, race${}={}$white,\nsex${}={}$female, and mean values of the quantitative predictors:\nFTND${}={}$5.97, $\\mbox{NDSS}=-0.023$, age${}={}$43.5,\nand EMA noncompliance${}={}$10.1\\%.}\n\\label{expw}\n\\end{figure}\n\nFigure~\\ref{heapprob} shows the estimated heaping probability as a\nfunction of remembered cigarette consumption for visit and nonvisit\ndays. The possibility of rounded-off reporting increases rapidly as the\nremembered count increases, although surprisingly the probability of\nrounding to the nearest 20 is not large for either type of day. When\nthe perception of smoking is more than two packs, say, 41 cigarettes,\nthe chance of heaped reporting rises to more than 84\\%, of which 37\\%\nis attributed to half-pack rounding. The results confirm that the\ndegree of heaping is much smaller on visit days. For example, only 51\\%\nof subjects round off the visit-day count when reporting 41 cigarettes,\nand among those 39\\% round off to the nearest multiple of 5.\n\n\\section{Discussion}\\label{sec6}\nWe have developed a model to describe the process whereby exact\nlongitudinal measurements become distorted by retrospective recall. Our\napproach uses latent processes to explain the data as a result of\nmis-remembering and rounding: a model of the latent exact value\ndescribes subject-level recall and allows for association over time and\nwith baseline predictors, while a misreporting model describes the\ndependence of heaping coarseness on the latent value and other\npredictors. Random effects represent individual propensities in recall\nand heaping; in our data, inferences depend strongly on the inclusion\nof these random effects.\n\n\n\n\\begin{figure}\n\n\\includegraphics{557f04.eps}\n\n\\caption{Estimated rounding behavior given EMA count\nin the proportional odds heaping model.}\\label{heapprob}\n\\end{figure}\n\nThe data suggest that both mis-remembering and heaping contribute\nsubstantially to the distortion of cigarette counts. The curve of mean\nremembered count as a function of EMA count departs markedly from the\n$45^{\\circ}$ line, with lighter smokers overstating consumption and\nheavier smokers understating consumption. The remembered smoking\ncoincides with the accurate EMA count at around 24~cigarettes,\nsuggesting that the popularity of reporting one pack per day is\npartially a result of the general heaping behavior rather than a\nparticular affinity for remembering a pack a day. The curves of heaping\nprobabilities suggest that exact reporting is uncommon and practically\ndisappears beyond about 40 cigarettes\/day. Nevertheless, it is\ninteresting just how much of the misreporting is due to\nmis-remembering. The remembered cigarette consumption depends not only\non true consumption, but also on the subject's sex, race and degree of\nnicotine dependence.\n\nThe interpretation of our model components as representing memory and\nrounding depends on the assumption that EMA data are exact. Of course,\neven EMA data are subject to errors, as smokers may neglect to record\ncigarettes both at the time of smoking and later. Yet good\ncorrespondence with smoking biomarkers strongly supports the use of EMA\nover TLFB as a proxy for the truth [\\citet{Shiffman2}].\n\nWe have implemented our model with a combination of standard numerical\nmethods including Gaussian quadrature, quasi-Newton\noptimization and sampling-importance resampling. Our experience\nsuggests that with the model as specified, and incorporating a modest\nnumbers of predictors, the method is robust and efficient. Increasing\nthe number of random effects would increase the time demands (from the\nnumerical integration) and raise the possibility of numerical\ninstability (from possible errors in integration). For more extensive\nmodels, sophisticated approaches based on MCMC sampling would be\nnecessary.\n\nOur model allows for the inclusion of covariates to better explain the\ndiscrepancy between smokers' self-perceived behaviors and reality. It\nalso provides a basis for predicting true counts (effectively the EMA\ndata) from reported TLFB counts. This would be a valuable activity in\nthe large number of studies that do not collect EMA data. To predict\ntrue counts from the recalled counts, we first need to estimate the\nparameters $\\theta$ in the model using a subset of the primary study or\nan external independent study that collects both TLFB count $Y$ and\naccurate EMA count $X$. Then we can impute the true count together with\nthe latent remembered count and heaped reporting behavior.\nSpecifically, the posterior distribution of $(W_i,G_i,x_i,b_i,u_i)$ is\n\\begin{eqnarray*}\n&&f(w_i,g_i,x_i, b_i,u_i|y_i,\\theta)\\\\\n&&\\qquad = f(w_i,g_i,x_i, b_i,u_i|\\theta)\n\\frac{f(y_i|w_i,g_i,x_i, b_i,u_i,\\theta)}{f(y_i|\\theta)}\\\\\n&&\\qquad \\propto \\Biggl(\\prod_{t=1}^{m_i}f(w_{it}|x_{it},b_i,\\theta\n)f(g_{it}|w_{it}, u_i,\\theta)I\\bigl((w_{it},g_{it})\\in\\mathrm{WG}(y_{it})\\bigr)\\Biggr)\\\\\n&&\\qquad\\quad{} \\times f(x_i)f(b_i|\\sigma_b)f(u_i|\\sigma_u),\n\\end{eqnarray*}\nwhere $f(x_i)$ is the density function of the true count. Imputation\nfollows similar steps as described in Section~\\ref{modelchecking} with\n$\\theta$ set equal to the maximum likelihood estimates.\n\nThe methods developed here also can have application in a wide variety\nof settings in social and medical science involving self-reported\ndata---for example, assessing sexual risk behavior, trial drug\nconsumption, eating episodes and financial expenditures.\n\n\n\\section*{Acknowledgments}\n\nWe are grateful to two Associate Editors and a referee, whose\nperceptive comments and suggestions greatly improved the paper.\n\n\\begin{supplement\n\\stitle{Data and programming code for the analysis\\\\}\n\\slink[doi,text={10.1214\/12-\\break AOAS557SUPP}]{10.1214\/12-AOAS557SUPP}\n\\slink[url]{http:\/\/lib.stat.cmu.edu\/aoas\/557\/dataandcode.zip}\n\\sdatatype{.zip}\n\\sdescription{It contains the daily TLFB and EMA data set, and SAS\nand R code to implement the method.}\n\\end{supplement}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzjlpu b/data_all_eng_slimpj/shuffled/split2/finalzzjlpu
new file mode 100644
index 0000000000000000000000000000000000000000..ce05b0e332e03f1bc22d21ff09379f827deaf8a8
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzjlpu
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\n\nThe masses of hadrons are created\ndynamically by the strong interaction, when confinement forces quarks and\ngluons to form color-neutral bound states. The generation of hadronic masses\nis connected to spontaneous chiral symmetry breaking, a basic feature of\nthe vacuum structure of\nQuantum-Chromo-Dynamics (QCD).  Evidently, the mechanism of chiral symmetry\nbreaking is of fundamental importance for the properties of matter in the\nuniverse. However, the quantitative understanding of the dynamics in this\nnon-perturbative regime of QCD is still rather incomplete, and additional\ninformation from experiment is essential. \n\nAccording to investigations of the non-perturbative properties of QCD on a\ndiscrete space-time lattice a plasma of deconfined quarks and gluons (QGP)\nshould be \nformed at energy densities $\\epsilon \\geq 1$~GeV\/fm$^3$.\nSimultaneously with this deconfinement transition, chiral symmetry is expected\nto be restored (see~\\cite{lat} for a recent review).  In collisions of heavy\nnuclei at high energies such energy densities are exceeded significantly and\nthere is by now strong, albeit indirect evidence for the formation of a QGP\n(for recent reviews see~\\cite{pbm,gyu,itz,sta}).  On the way to chiral symmetry\nrestoration in such matter, significant modifications of the properties of\nhadrons are expected~\\cite{pis,BR1}, such as of their mass and width or more \ngenerally of the hadronic spectral function.\n\nThe $\\rho$~meson ($J^P=1^-$) is an ideal probe to investigate modifications of\nsuch in-medium properties.  In a hot hadronic medium close to the phase\nboundary, $\\rho$~mesons are abundantly produced by annihilation of thermal\npions.  Due to its short lifetime ($c\\tau = 1.3$ fm), the decay of the\n$\\rho~$meson occurs inside the medium, and spectral modifications may be\nobservable via the kinematic reconstruction of the decay products.  Finally,\nits decay into lepton pairs provides essentially undisturbed information from\nthe hot and dense phase, because leptons are not subject to final state\nrescattering in the strongly interacting medium.\n\nEnhanced low-mass $e^+e^-$ pair production in nucleus-nucleus collisions at\nfull energy of the CERN-Super-Proton-Synchrotron (SPS) has been reported by\nthe CERES experiment.  In particular, in the mass region 0.2-0.6~GeV\/$c^2$,\nthe measured di-lepton yield exceeds expectations from hadron decays by a\nfactor 2-3~\\cite{ceres-sau,ceres-pbau,ceres-long}.  Even bigger enhancement\nfactors have been found at 40$A$~GeV\/$c$~\\cite{ceres-40}, albeit with\nlarge statistical uncertainties.\nAt RHIC energies, enhancement factors similar to those observed at the top\nSPS energy have been reported recently~\\cite{phenix-electrons}.\n \nSignificant $\\rho$~meson production via annihilation of thermal pions in the\nhot and dense hadronic medium is a likely mechanism for enhanced electron pair\nproduction. \nImplementing this mechanism,\nsubstantial temperature and baryon density dependent\nmodifications of the $\\rho$-spectral function~\\cite{BR1,RW1,BR2} needed to be\nconsidered to explain the mass spectrum of the pair enhancement.  However,\nthe detailed behaviour of the spectral function as chiral\nsymmetry is restored is still up to speculation. Quite different theoretical\napproaches exist which could not be discriminated by the previous di-electron\ndata.\n\nThe NA60\nCollaboration recently corroborated previous CERES findings and reported\na significant di-muon excess in nucleus-nucleus collisions\nover the expectation from hadronic decays~\\cite{na60-1}.\nThe NA60 measurement of\nthe di-muon excess in $^{115}$In-In collisions at 158$A$~GeV\/$c$\nfavors models including significant broadening but no mass\nshift of the $\\rho$-spectral function~\\cite{na60-1,rapp-hees}.\n\n\\section{Experiment and Data Analysis}\nIn this Letter, we present results on $e^+e^-$ pair production in central\n$^{208}$Pb-$^{197}$Au \ncollisions at 158$A$~GeV\/$c$. The data have been recorded by the CERES\nexperiment at the SPS in the year 2000~\\cite{ana-qm}.  \nTypically, $10^6$ lead ions per 5.2~s\nextraction cycle were focused on 13 thin gold targets aligned along the beam\nline (25~$\\mu$m each, totalling 1.2\\% of a nuclear interaction length).  The\ninteraction vertex was reconstructed using charged particle track segments\nfrom two silicon drift detectors (SDD) placed 10.4 and 14.3 cm downstream of\nthe target.  Electrons are identified by their ring signature in two RICH\ndetectors, which are blind to hadrons below $p\\approx 4.5$ GeV\/$c$ \n($\\gamma_{\\rm thresh} \\approx 32$).  The\nexperimental setup was upgraded by a downstream radial drift Time Projection\nChamber (TPC) which is operated inside an inhomogeneous magnetic field with a\nradial component of up to 0.75~T.  \nEmploying tracking information from the\nTPC, the mass resolution of the spectrometer was improved to $\\Delta m\/m = 3.8\n\\%$ in the region of the $\\phi$~meson mass~\\cite{ana-qm}.  \nThe resolution has been determined by a Monte Carlo procedure \nwhere simulated tracks were embedded into real data events. \nThis method \nprovides a detailed simulation of the TPC response to charged particles \nin a realistic track density environment. \nThe same Monte Carlo describes very well the experimentally observed peak width \nof the $K^0_s$ reconstructed in the $ \\pi^+\\pi^-$ channel \n($\\sigma_{K^0_s\\rightarrow \\pi^+\\pi^-} \\approx 15$~MeV\/$c^2$). \nThe TPC also\nprovides additional electron identification via measurement of the specific\nenergy loss ${\\rm d}E\/{\\rm d}x$ with a resolution of about 10\\%.  The\nspectrometer provides full azimuthal acceptance in the pseudorapidity range\n$2.1 < \\eta < 2.65$.\n\n\nThe present results are based on an analysis~\\cite{sergey,alex,oli} of \n25 million Pb-Au\n\nevents, selected at a centrality of $\\sigma\/\\sigma_{\\rm\n  geo}=7~\\%$~\\cite{darek-qm}.  SDD track segments are matched to charged\nparticle tracks in the TPC, where the deflection in the magnetic field\ndetermines the momentum with a resolution of $\\Delta p\/p \\approx \\left((2\\%)^2\n  +(1\\%\\cdot p ({\\rm GeV\/}c) )^2\\right)^{1\/2}$~\\cite{ana-qm}.  Combined\nelectron information from cuts on the ring quality in the RICH detectors and\nTPC d$E$\/d$x$ leads to a pion suppression of typically $4~\\cdot10^4$ at 67\\%\nelectron efficiency~\\cite{sergey}.\n\n\nA set of cuts is applied to the track candidates\nto minimize the amount of combinatorial background from unrecognized\nDalitz pairs and conversions, making use of their characteristic\ndecay topology.\nElectron and positron tracks are rejected if\ntheir d$E$\/d$x$ significantly exceeds that of a single track in both SDDs,\nindicating an unresolved close pair. Also, electron and positron tracks are\nrejected if a soft track with opposite charge and electron-like d$E$\/d$x$\nis found in the TPC at small angular separation. \n\nThe remaining electron and positron tracks are then combined into pairs.\nTo further suppress combinatorial background, tracks which form\npairs with opening angle smaller than 35 mrad are treated as recognized\nconversions or Dalitz pairs and are not used for further pairing. \nFinally, a low transverse momentum cut\nof 0.2 GeV\/$c$ is applied to all electron and positron tracks.\n\nThe single-electron \nreconstruction efficiency has been determined by a Monte-Carlo (MC) procedure\nwhere simulated tracks are embedded into real raw data events. The subsequent\nanalysis of the MC sample includes all cuts and methods as applied to the real\ndata.  \nThe final pair reconstruction efficiency $\\epsilon_{\\rm ee}$ is typically \n14\\%. \nIt depends on the polar angles of the single tracks and\n(slightly) on centrality. \nThe efficiency correction is performed by assigning a weight $w_{\\rm ee}$ to each pair\nwhich is the inverse of the pair reconstruction efficiency $\\epsilon_{\\rm ee}$.\nThe pair reconstruction efficiency is calculated from the single track efficiencies,\nusing the parametrized dependencies on track polar angle $\\theta$ and charged \nparticle multiplicity $N_{\\rm ch}$:\n\\begin{equation}\nw_{\\rm ee} =\n\\frac{1}{\\epsilon_{\\rm ee}}=\n\\frac{1}{\\epsilon_{\\rm track1}(\\theta_1, N_{\\rm ch})\\cdot\n         \\epsilon_{\\rm track2}(\\theta_2, N_{\\rm ch})}.\n\\end{equation}\n\nBased on a systematic variation of the cut values in the MC, the systematic \nuncertainty on the pair reconstruction efficiency is estimated to 8.4\\%. \n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=7.5cm]{Fig1.eps}\n   \\caption{Unlike-sign pair yield (histogram) and\n     combinatorial background (dashed curve). See text for explanation.}\n\\label{fig1}\n\\end{center}\n\\end{figure}\n\n\nThe remaining combinatorial background is estimated to equal the number of\nlike-sign lepton pairs from the same event. \nThe limited statistics of the like-sign pair sample adds a significant \nstatistical error to the final result. \nAlternatively, a mixed-event technique can be applied, where\nunlike-sign pairs are formed from electron and positron tracks of different\nevents and the corresponding mixed-event spectrum is normalized to\nthe same-event like-sign yield. \nThis approach reduces considerably the statistical bin-to-bin fluctuations\nbut bears a \nstatistical and \nsystematic uncertainty due to the normalization procedure.\nThe normalization constant is determined by the ratio of the \nmixed-event spectrum to the same-event like-sign spectrum. The statistical\naccuracy of the normalization is limited by the number of counts in the \nsame-event like-sign spectrum and yields about $4\\cdot 10^{-3}$. \nThe uncertainty in the normalization of the \nbackground \nhas been included in\nthe systematic error of the final dilepton yield.\nThe magnitude of this contribution after background subtraction depends \non the signal-to-background ratio and yields typically 8.8\\% at $S\/B$ =~1\/22 \n(see below).\n\n\nWe found that the final\nresults \nusing the like-sign and the mixed-event sample are in good agreement within statistical\nerrors~\\cite{sergey}.  Small deviations are only visible at invariant masses\nbelow 0.1 GeV\/$c^2$.  This is caused by limitations of the two-ring\nseparation, which are only present in the same-event sample.  To account for\nthis effect, which is also present in the 'true' unlike-sign combinatorial\nbackground, we have used the same-event like-sign background estimate for\nmasses below 0.2~GeV\/$c^2$.  No statistical limitation is imposed by this\nprocedure, since the signal-to-background ratio is very good in this mass\nregion.  For masses greater than 0.2~GeV\/$c^2$, the normalized mixed-event\nunlike-sign sample is used for background subtraction. \nThe resulting background distribution \nand the unlike-sign signal pair spectrum after efficiency correction\nare shown in Fig.~\\ref{fig1}.\n\nBelow $m_{\\rm ee}$~=~0.2~GeV\/$c^2$ the $\\pi^0$-Dalitz contribution\nis clearly visible. \nThe raw net yield \nin this mass range after subtraction of the combinatorial background contains\n$6114\\pm 176$ electron-positron \npairs at a signal-to-background ratio \n$S\/B$~=~1\/2.\nAt masses larger than 0.2~GeV\/$c^2$, the number of \npairs contributing to the signal is \n$3115\\pm 376$\nat\n$S\/B$~=~1\/22.\n\nAfter background subtraction, the $e^+e^-$ pair yield is normalized to the\ntotal number of events and to the average charged particle multiplicity\n$\\langle N_{\\rm ch}\\rangle$ in the spectrometer acceptance. \nThe average charged particle multiplicity has been\ndetermined by the number of tracks in the SDD. \nFor the 7\\% most central events\nwe obtain \n$\\langle N_{\\rm ch}\\rangle = 177 \\pm 14$ (syst.)\nin $2.1 < \\eta < 2.65$. \nThe systematic error on $\\langle N_{\\rm ch}\\rangle$ adds\na contribution of 8\\% to the total systematic error on the pair yield\nper charged particle.\n\n\nThe total systematic error of the data is given \nby (i) the uncertainty of the efficiency determination (8.4\\%), (ii) \nthe normalization of the \nmixed-event background ($0.004 \\cdot B\/S$, on average 8.8\\%) and (iii) \nthe determination of the charged particle \nmultiplicity (8\\%). \nThese contributions\nadd up to an average of 14.6\\%, however, note that contribution (ii) differs\nbin-by-bin, depending on the local signal-to-background ratio, and applies only for\n$m_{\\rm ee} > 0.2$~GeV\/$c^2$. \n\n\\section{Results and Discussion}\n\nThe $e^+e^-$ invariant mass distribution after \nefficiency correction, combinatorial\nbackground subtraction and normalization\nis shown in Fig.~\\ref{fig2}~(a). \nAlso shown \nis the 'hadronic cocktail' which comprises the yield from hadronic decays in\nA-A collisions after chemical freeze-out \n(see~\\cite{ceres-long})\\footnote{for the $\\phi$~meson we \nuse 70\\% of the thermal model yield,\nin accordance with measurements~\\cite{ceres-phi}. \nThe calculation of the present cocktail includes also the\nrecently improved branching ratio for \n$\\omega \\rightarrow \\pi^0 e^+ e^-$~\\cite{cockt-update}.}.\nIn the mass range $0.2 <m_{\\rm ee} < 1.1$~GeV\/$c^2$, the data are enhanced \nover the \ncocktail by a factor\n$2.45 \\pm 0.21~{\\rm (stat)} \\pm 0.35 ~{\\rm (syst)} \\pm 0.58~{\\rm (decays)}$.\nThe last error arises from the systematic uncertainty in the cocktail\ncalculation. The enhancement is most pronounced in the mass region $0.2 <\nm_{\\rm ee} < 0.6$~GeV\/$c^2$, in agreement with earlier findings.  In contrast\nto previous CERES results, the improved mass resolution of the upgraded\nspectrometer provides access to the resonance structure in the $\\rho \/ \\omega$\nand $\\phi$ region. A quantitative study of $\\phi$~meson production in the\n$e^+e^-$ and $K^+K^-$ channels can be found in~\\cite{ceres-phi}.\n\n\\begin{figure}\n\\begin{center}\n   \\includegraphics[width=7.5cm]{Fig2a.eps}   \n   \\includegraphics[width=7.5cm]{Fig2b.eps}\n   \\caption{a: Invariant $e^+e^-$ mass spectrum compared\nto the expectation from hadronic decays. b: The same data\ncompared to calculations including a dropping $\\rho$ mass (dashed)\nand a broadened $\\rho$-spectral function (long-dashed). \nSystematic\nerrors are indicated by horizontal ticks.}\n\\label{fig2}\n\\end{center}\n\\end{figure}\n\nIn Fig.~\\ref{fig2}~(b) the data are compared with a model\napproach implying enhanced di-lepton production via thermal pion\nannihilation and a realistic space-time evolution~\\cite{rapp-calc}.\nThe calculated di-lepton yield was filtered by the CERES acceptance\nand folded with the experimental resolution.\nTemperature and baryon-density dependent modifications of the $\\rho$-spectral function have been taken into account: \nthe dropping mass scenario which assumes a\nshift of the in-medium $\\rho$ mass~\\cite{BR1,BR2}, and the broadening scenario\nwhere the $\\rho$-spectral function is smeared due to coupling to the hadronic\nmedium~\\cite{RW1,rapp-hees}. \nThe calculations include as well contributions from QGP,\nthe Drell-Yan process, and 4-pion annihilation with chiral mixing. \nThe calculations for both spectral functions describe the enhancement\nreasonably well for masses below 0.7~GeV\/$c^2$. \nIn the resonance region, however, there is a notable difference between\nthe calculations.\nIn particular in the mass region\nbetween the $\\omega$ and the $\\phi$, the data clearly favor the broadening\nscenario \nover the dropping mass scenario.\n\\begin{figure}\n\\begin{center}\n   \\includegraphics[width=7.cm]{Fig3a.eps}\n   \\includegraphics[width=7.cm]{Fig3b.eps}\n   \\caption{$e^+e^-$ pair yield after subtraction of the hadronic\n     cocktail. In addition to the statistical error bars,\n   systematic errors of the data \n\t(horizontal ticks) and the systematic uncertainty of the subtracted \n\tcocktail (shaded boxes) are indicated.\n     The broadening scenario (long-dashed line) is compared to a calculation\n     assuming a density dependent dropping $\\rho$ mass (dotted line in (a)) \n     and to a broadening scenario excluding baryon effects (dotted line\n     in (b).}\n\\label{fig3}\n\\end{center}\n\\end{figure}\n\nIn order to exhibit the shape of the in-medium contribution,\nwe subtract the hadronic cocktail (excluding the\n$\\rho$ meson) from \nthe data (Fig.~\\ref{fig3}). \nThe vacuum $\\rho$-decay contribution to the data \n(``cocktail $\\rho$'', solid line in Fig.~\\ref{fig3}) \nis completely negligible compared to the measurements.\nThe excess data exhibit a very broad structure reaching very low masses\nand exceed the vacuum $\\rho$ contribution by a factor $10.6\\pm1.3$. \nThe data are compared to model calculations of the in-medium di-electron\nproduction. These are normalized, like the measured yield, to the number\nof charged particles.\nNote that the model calculations give absolute pair yields (in terms of\ncharged particle numbers) and there is no freedom of adjustment.\nYield and spectral shape\nare well described by the broadening scenario but are not\nconsistent{\\footnote{Recently, Brown et al.~\\cite{newbrownrho} have \nadvocated a\n  different view in which their scaling is not directly related anymore to the\n  shape of the low mass di-lepton spectrum.}} with a dropping $\\rho$ mass\n(Fig.~\\ref{fig3}~(a)). \nWhile the dropping mass calculation yields a rather narrow distribution\nwith a peak around 0.5~GeV\/$c^2$ the measured excess is spread over a \nsignificantly wider mass range. \nBelow 0.2~GeV\/$c^2$, the large errors arising from the subtraction of large\nnumbers in the $\\pi^0$ Dalitz region do not allow for a definite conclusion.\nHowever, the trend indicates a further increase of the in-medium contribution\ntowards the photon point.\n\nA $\\chi^2$-analysis of the data \nin the mass region \n$0.12 < m_{\\rm ee} < 1.1$~GeV\/$c^2$ ($\\rm{dof} = 13$)\nwith respect to the model calculations in Fig.~\\ref{fig3} results in \n$\\chi^2_{\\rm IMH}=10.6$ ($P_{\\rm IMH (stat) } =64.4\\%$)\nfor the in-medium hadronic spectral function approach and $\\chi^2_{\\rm DRM}=33.1$ \n($P_{\\rm DRM (stat)} =0.0017\\%$) for the dropping $\\rho$ mass scenario, \nif only statistical errors are considered.\nTo judge how well the data resemble the calculations shown in \nFig.~\\ref{fig3}~(a)\nincluding systematic uncertainties in data and cocktail, a Monte Carlo procedure\nhas been employed. \nIn this procedure, the model curves have been used as input to \ngenerate simulated spectra, assuming statistical and systematic uncertainties\nas in the real data.\nThe total systematic error in each mass bin has been calculated \nby adding in quadrature \nthe systematic error contributions to data and cocktail.\nThe resulting total systematic error is interpreted as a Gaussian \nstandard deviation of a\ncoherent up- or downward shift of all data points in the spectrum.\nStatistical point-to-point fluctuations have been added\naccording to the statistical error bars of the real data.\nFor each of the generated spectra,\nthe $\\chi^2$-value \nwith respect to the input model\ncurve has been calculated.\nFinally,\nthe probability to obtain a $\\chi^2$-value which is \nlarger than the one observed in the data has been determined for each of the two\nmodel curves. We obtain \n$P_{\\rm IMH} = 81.1\\%$ for the in-medium hadronic spectral function approach\nand $P_{\\rm DRM} = 10.1\\%$ for the dropping $\\rho$ mass scenario, the latter \nimplying that the dropping $\\rho$ mass scenario can be excluded on the\n$1.6~\\sigma$-level only. We note that, despite the statistical limitations of\nthe present data, the discrimination power among the present model calculations\nis predominantly limited by systematic uncertainties.\n\n\nA more detailed view may be derived \nby comparing the gross features of the data \nin Fig.~\\ref{fig3}~(a) to the model calculations, see Table~\\ref{tab1}. \nThe systematic errors on these quantities have been estimated by shifting each\ndata point up or down by one standard deviation of its total systematic error.\nThe mean\nvalues of the mass distributions of both calculations are in very good agreement with\nthe measured excess data. The integrated yield in $0.12<m_{\\rm ee}<1.1$~GeV\/$c^2$ \nagrees well with the in-medium hadronic spectral function approach, however,\nthe systematic uncertainty does not exclude the dropping $\\rho$ mass scenario\neither. In contrast, a comparison of the RMS widths clearly favors the\nin-medium hadronic spectral function approach, \nas the width of the data distribution is quite insensitive to systematic errors \nin scale of both data and cocktail.\nOn this note, it is the large width observed in the data which drives the \ndiscrimination power among the model calculations.\n\n\\begin{center}\n\\begin{table}\n\\caption{Di-electron excess in $0.12<m_{\\rm ee}<1.1$ GeV\/$c^2$ \ncompared to model calculations.}\n~\n\\begin{tabular}{c|c|c|c}\n\\hline \\hline\n~ & Data \t\t\t& ~In-medium~ & ~Dropping~ \\\\\n~ & ~($\\pm$ stat.) ($\\pm$ syst.)~ & ~hadronic~  & ~$\\rho$ mass~ \\\\ \n\\hline\nMean (GeV\/$c^2$)  &~0.54$\\pm$0.07$\\pm$0.01~ & 0.54 & 0.55 \\\\\nYield ($10^{-6})$ & 3.58$\\pm$0.42$\\pm$1.01 & 3.88 & 2.41 \\\\\nRMS (GeV\/$c^2$)   & 0.26$\\pm$0.02$\\pm$0.01 & 0.25 & 0.18 \\\\\n\\hline \\hline\n\\end{tabular}\n\\label{tab1}\n\n\\end{table}\n\\end{center}\n\n\n\nThe agreement of the data with the broadening scenario \nis strong evidence that the\nresonance structure of the $\\rho$ meson \nis significantly modified\nin the hot\nand dense medium~\\cite{rapp-hees,qh-dual}.\nThat $\\rho$-related pair production is indeed a manifestation of the \nhot and dense matter created is supported by observing a particular mechanism \nat work which plays a dominant role in the hadronic spectral function\napproach:\nThe strong coupling to baryons which adds\nstrength to the di-electron yield at low masses~\\cite{RW1}. The importance of\nthis mechanism is demonstrated in Fig.~\\ref{fig3}~(b), where the data\nare compared to in-medium hadronic spectral function calculations \nwith and without baryon-induced\ninteractions~\\cite{rapp-hees,rapp-calc}. \nThe calculations differ most in the mass range\nbelow 0.5~GeV\/$c^2$, which is accessible with good efficiency by the present\n$e^+e^-$ data. The calculation omitting baryon effects \nfalls short of\nthe data for masses below 0.5~GeV\/$c^2$ while inclusion of baryon interactions\ndescribes the measured low-mass yield very well. \nThis is strong evidence that the observed modifications of the $\\rho$-spectral\nfunction are foremost due to interactions with the dense baryonic medium.\n\nIt has been demonstrated that baryon-driven medium modifications lead to\na low-mass di-electron spectrum which is very similar to \nthe di-electron rate from lowest order perturbative $q\\overline{q}$ \nannihilation~\\cite{qh-dual}. \nInspired by this apparent emergence of quark-hadron duality \nat low masses, Gallmeister et al. performed a\ncalculation assuming that di-electron\nproduction via $q\\overline{q}$\nannihilation at $T_c$ is dominant~\\cite{kaempfer}. \nThat this phenomenological approach describes the shape of the measured\ndistribution (Fig.~\\ref{fig3}) so well may be taken as an indication \nof chiral symmetry restoration which is implied in quark-hadron duality.\n\n\nIn cold nuclear matter, medium modifications not only of \n$\\rho$ but also of $\\omega$ and $\\phi$ mesons\nhave been observed~\\cite{trnka,naruki,muto} (see~\\cite{metag} for a recent \nreview). \nSuch modifications may also be\npresent in the medium created in high-energy heavy-ion collisions. \nIt should be noted however that the dominance \nof thermal two-pion annihilation to the total di-lepton yield \nvia the $\\rho$-channel disfavors a significant \ncontribution from $\\omega$ and $\\phi$ to the observed di-lepton\nexcess and its spectral distribution.\n\nIn conclusion, the present $e^+e^-$ data with improved mass resolution in the\nresonance region \nfavor present models including a strong broadening of \nthe $\\rho$-spectral \nfunction in a hot and dense hadronic medium over a density dependent\n$\\rho$ mass shift.  Moreover, the \n$e^+e^-$ data at low pair\nmass and transverse momentum allow to test the relevance of baryonic effects\nto the modification of the $\\rho$-spectral function.  \nIn comparison with models, the present CERES data suggest that baryonic interactions\nare important to explain the observed di-lepton \nyield at low masses.\n\n\\section{Acknowledgements} \\nonumber\n\nThis work was supported by GSI, the German BMBF, the Virtual Institute VI-SIM\nof the German Helmholtz Association, the Israel Science Foundation,\nthe Minerva Foundation and by the Grant Agency and Ministry of Education\nof the Czech Republic. \nWe wish to thank Ralf Rapp for calculations and \nnumerous discussions.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe theory of topological quantum chemistry introduced in Ref.~\\onlinecite{NaturePaper} diagnoses topological phases based on elementary band representations.\nA set of bands is topological if it lacks an ``atomic limit'' that obeys the crystal symmetry (and time-reversal, if desired): formally, an atomic limit exhibits a set of localized, symmetric Wannier functions.\\cite{NaturePaper,EBRTheory,PoFragile,Soluyanov2011,Soluyanov2012,Marzari2012,Read2016}\nThis definition includes all known topological insulating phases.\\cite{Kane04,Moore2007,Fu2007,Fu07,Roy2009,BHZ,Teo08,Fu2011,Shiozaki17,Wieder17,Song17}\nWe showed in Refs.~\\onlinecite{NaturePaper,EBRTheory} that each atomic limit defines a ``band representation,'' which is a representation of the full space group.\nThe irreducible representations (irreps) of the little group at each point in the Brillouin zone are completely determined for each band representation.\\cite{Zak1980,Zak1982,Bacry1988} \nHowever, the little group irreps do not define the band representation: two groups of bands can exhibit the same little group irreps but differ by a Berry phase.\\cite{Bacry1988b,Bacry1993,Michel1992,EBRTheory}\n\nIf a set of bands, separated by an energy gap from all other bands, does not transform as a band representation, it does not have localized, symmetric Wannier functions; consequently, it is topological.\\cite{NaturePaper,EBRTheory}\nAn ``elementary'' band representation (EBR) is not equivalent to a sum of two band representations.\nIt follows that a disconnected (gapped) elementary band representation must realize a set of topological bands.\\cite{NaturePaper,EBRTheory,PoFragile}\nSuch disconnected EBRs will be the focus of this letter.\nAll EBRs and their irreps at high-symmetry points in the Brillouin zone can be found on the Bilbao Crystallographic Server.\\cite{NaturePaper,GraphTheoryPaper,GraphDataPaper,GroupTheoryPaper,Bilbao1,Bilbao2,Bilbao3}\n\n\nThe theory of topological quantum chemistry also brings to light the different types of trivial-to-topological phase transitions, distinguished by how many symmetry-distinct orbitals contribute to the topological bands.\nFor example, the Kane-Mele model of graphene\\cite{Kane04} requires only one type of symmetry-distinct orbital (the two spinful $p_z$ orbitals per unit cell are related by the honeycomb lattice symmetry), \nwhile the trivial-to-topological transition in HgTe\\cite{BHZ} requires both $s$ and $p$ orbitals\nto create a ``band inversion.''\nThese two types of topological insulators differ in their atomic limit as the distance between atoms is taken to infinity: in the atomic limit of graphene, the band structure consists of a single flat and four-fold degenerate band, corresponding to a single EBR.\nIn contrast, in HgTe, the atomic limit will consist of two flat bands, one each for the $s$ and $p$ orbitals, corresponding to two distinct EBRs.\n\n\nIn this letter, we will focus on the graphene-like case: topological insulators that derive from a single orbital and its symmetry-related partners.\nIn the language of band representations, the conduction and valence bands together transform as a single EBR; consequently, either the conduction or valence bands (or both) lack an atomic limit and are topological.\\cite{NaturePaper,EBRTheory,PoFragile}\n\nWe introduce models in a symmorphic and a non-symmorphic space group.\nThe symmorphic example describes $p_{x,y}$ orbitals on the honeycomb lattice.\nWithout spin-orbit coupling (SOC), the band structure can be a (gapped) topological crystalline insulator (TCI).\nWith infinitesimal SOC and time-reversal symmetry, the system exhibits a nontrivial $\\mathbb{Z}_2$ index\n\n\n\n\nWe were motivated to explore the non-symmorphic example because, as part of their ground-breaking work on the connectivity of energy bands, Michel and Zak conjectured that spinless EBRs in non-symmorphic space groups cannot realize a gapped band structure.\\cite{Michel1999,Michel2001} \nIn Ref~\\onlinecite{GraphDataPaper}, we explained where Michel and Zak's proof fails.\nHere, we pick a particular non-symmorphic space group, $P4_2 32$, and construct a tight-binding model to explicitly show its gapped, topological nature.\nIn doing so, we find a novel feature: the two-dimensional ``Wilson Hamiltonian'' exhibits a topologically protected band crossing.\n\n\n\nIn each example, we derive a bulk topological invariant.\nAn essential tool is the ``$k_\\parallel$-directed'' Wilson loop, which describes the parallel transport of an isolated set of bands\n\\cite{Zakphase,Fu06,Ryu10,Soluyanov2011,Yu11,Taherinejad14,Alexandradinata14,Hourglass,ArisCohomology,hughesbernevig2016,Wieder17,Holler2017} \n\\begin{equation}\n \\mathcal{W}_{(k_\\perp,k_0)} \\equiv  P e^{i\\int_{k_0}^{k_0 +2\\pi} dk_\\parallel A_\\parallel(k_\\perp,k_\\parallel)},  \n\\label{eq:wilsoncont}\n\\end{equation}\nwhere $P$ indicates that the integral is path-ordered and $A_\\parallel (\\mathbf{k})_{ij} = i \\langle u_i(\\mathbf{k}) | \\partial_{k_\\parallel} u_j (\\mathbf{k})\\rangle$ is a matrix whose rows and columns correspond to each eigenstate in the isolated set of bands. \nThe eigenvalues of $\\mathcal{W}$ are gauge invariant and of the form $e^{i\\theta(k_\\perp)}$, independent of the ``base point,'' $k_{0}$.\\cite{ArisCohomology}\nA quantized invariant derived from the Wilson loop is invariant under any deformation of the Hamiltonian that preserves the gap in the spectrum\n\n\n\n\\paragraph{Spinless TCI on the honeycomb lattice}\n\nWe start with spinless $p_{x,y}$ orbitals on the honeycomb lattice,\ndescribed by the nearest-neighbor Hamiltonian:\n\\footnote{Eqs.~(\\ref{eq:pxyHam}) and (\\ref{eq:pxyHamh}) are derived in Sec. S1} %{sec:pxyHam.}\n\\begin{equation}\nH_\\mathbf{k}^0  = \\begin{pmatrix} 0 & h_\\mathbf{k} \\\\ h_\\mathbf{k}^\\dagger & 0 \\end{pmatrix} \n\\label{eq:pxyHam}\n\\end{equation}\nwhere non-zero blocks mix the $A$ and $B$ sublattices and\n\\begin{align}\nh_\\mathbf{k} &= \\frac{1}{2} \\left( e^{-i\\mathbf{k} \\cdot \\mathbf{\\delta}_1} + e^{-i\\mathbf{k} \\cdot \\mathbf{\\delta}_2} + e^{-i\\mathbf{k} \\cdot \\mathbf{\\delta}_3} \\right)(t_\\sigma + t_\\pi) \\mathbb{I} \\nonumber\\\\\n&+ \\frac{1}{2} \\left( e^{-i\\mathbf{k} \\cdot \\mathbf{\\delta}_1} -\\frac{1}{2} e^{-i\\mathbf{k} \\cdot \\mathbf{\\delta}_2} -\\frac{1}{2} e^{-i\\mathbf{k} \\cdot \\mathbf{\\delta}_3} \\right)(t_\\sigma - t_\\pi)\\sigma_z \\nonumber\\\\\n&+ \\frac{\\sqrt{3}}{4} \\left(  e^{-i\\mathbf{k} \\cdot \\mathbf{\\delta}_2} - e^{-i\\mathbf{k} \\cdot \\mathbf{\\delta}_3} \\right)(t_\\sigma - t_\\pi ) \\sigma_x\n\\label{eq:pxyHamh}\n\\end{align}\nThe Pauli matrices, $\\sigma_{x,y,z}$, act in the $p_{x,y}$ subspace; $t_{\\sigma,\\pi}$ parameterize $\\sigma$ and $\\pi$ bond strengths; and $\\delta_{1,2,3}$ are the nearest-neighbor vectors (see Fig~\\ref{fig:graphenebasisvectors}).\nPreviously this model with $t_\\pi = 0$ was studied for its flat bands.\\cite{Wu2007,Liu2013}\nThe spectrum of $H^0_\\mathbf{k}$ is shown in Fig~\\ref{fig:pxyspectrum}.\nThe degeneracies at $K\\equiv \\frac{2}{3}\\mathbf{g}_1 + \\frac{1}{3}\\mathbf{g}_2$ and $\\Gamma$ are symmetry-required.\\cite{Kogan12} \n\\begin{figure}[h]\n\\centering\n\\subfloat[]{\n\t\\includegraphics[width=.8in]{graphenebasisvectors2.pdf}\n\t\\label{fig:graphenebasisvectors}\n\\quad\n\t\\includegraphics[width=.6in]{GrapheneReciprocal-crop.pdf}\n\n}\n\\quad\n\\subfloat[]{\n\t\\includegraphics[width=1.5in]{pxpyHam.pdf}\n\t\\label{fig:pxyspectrum}\n}\n\\quad\n\\subfloat[]{\n\t\\includegraphics[width=1.5in]{pxpyHam-disc.pdf}\n\t\\label{fig:pxyspectrum-disc}\n} \\quad\n\\subfloat[]{\n\t\\includegraphics[width=1.5in]{pxpy-Wilson.pdf}\n\t\\label{fig:pxyWilson}\n}\n\\caption{ (a) Lattice ($\\mathbf{e}_{1,2}$) and reciprocal lattice ($\\mathbf{g}_{1,2}$) basis vectors. The dotted arrows ($\\delta_{1,2,3}$) indicate the vectors between nearest neighbor sites. \\textbf{A} and \\textbf{B} indicate the sublattices. (b) Spectrum of $H^0_\\mathbf{k}$ with $t_\\sigma = 1, t_\\pi = -.5$\n(c) Gapped band structure of $H^0_\\mathbf{k}+xH^1_\\mathbf{k}$ with $t_\\sigma = .8, t_\\pi = 1.0, x=.6$ and (d) the argument of its Wilson loop eigenvalues.}\n\\label{fig:graphenefig}\n\\end{figure}\n\n\n\n\n\n\nTo open a gap, we add the following next-nearest neighbor hopping term, which preserves the crystal symmetries of the honeycomb lattice:\\footnote{The crystal symmetries of the honeycomb lattice are enumerated in~Sec. S1} %{sec:pxyHam.} \n\\begin{equation}\nH^1_\\mathbf{k} = \\sin({\\textstyle \\frac{1}{2}}\\mathbf{k}\\cdot \\mathbf{e}_1)\\sin({\\textstyle \\frac{1}{2}}\\mathbf{k} \\cdot \\mathbf{e}_2) \\sin( {\\textstyle \\frac{1}{2}}\\mathbf{k} \\cdot (\\mathbf{e}_1 - \\mathbf{e}_2) ) \\tau_z \\otimes \\sigma_y,\n\\label{eq:pxynext}\n\\end{equation}\nwhere the matrices $\\tau_i$ act in the sublattice subspace.\nThe term in Eq.~(\\ref{eq:pxynext}) changes the energy-ordering of the bands at $K$, while preserving the two-fold degeneracy.\nFor large enough $|x|$, $H_\\mathbf{k}^0 +x H_\\mathbf{k}^1$ can be gapped, as in Fig.~\\ref{fig:pxyspectrum-disc};\nsee Sec. S1A} %{sec:tciphase\\ for a phase diagram\n\n\n\nThe spectrum in Fig~\\ref{fig:pxyspectrum-disc} represents a disconnected EBR.\\cite{NaturePaper,EBRTheory} \nWe construct a non-trivial bulk topological invariant from the $\\mathbf{g}_1$-directed Wilson loop of the lower two bands.\nIts eigenvalues are shown in Fig.~\\ref{fig:pxyWilson} as a function of the base point.\nWhen the base point is $\\Gamma$ or $M$, the Wilson loop eigenvalues ($-1$ and $+1$, respectively\\footnote{Sec. S1A} %{sec:tciphase\\ shows the possible $C_{2z}$ eigenvalues.}) are completely determined by the $C_{2z}$ eigenvalues\\cite{Alexandradinata14,Holler2017}\n(the $C_{2z}$ operator is $-\\tau_x \\otimes \\sigma_0$.\\footnote{The crystal symmetries of the honeycomb lattice are enumerated in~Sec. S1} %{sec:pxyHam.} )\nThis forces the ``Wilson bands'' to wind in opposite directions.\nThe quantized eigenvalues at $\\Gamma$ and $M$ prevent the Wilson spectrum from being smoothly deformed to flat, which indicates that the valence bands are topologically nontrivial.\n\n\n\n\nThe Wilson loop winding requires that both occupied bands of $H^0_\\mathbf{k} + xH^1_\\mathbf{k}$ at $\\Gamma$ have the same $C_{2z}$ eigenvalue, $\\eta$, and that both occupied bands at $M$ have the $C_{2z}$ eigenvalue $-\\eta$.\nConsider the Wilson loop of three bands: the two occupied bands and a third, trivial, band, not in our model.\nIf the $C_{2z}$ eigenvalues of the third band at $\\Gamma$ and $M$ are both equal to $\\eta$,\nthen the eigenvalues of the three-band Wilson loop will not be quantized at $M$ and it will fail to wind.\nThus, the topological invariant is not stable to adding a third band to the projector (although the winding of the projector onto two bands \\textit{is} invariant under adding a third band as long as the gap between the third band and the existing bands does not close.)\nThe existence of a topological invariant that depends on the number of bands is reminiscent of the ``Hopf insulator.''\\cite{Moore2008}\n\n\n\\paragraph{Spinful topological phases}\n\nWe now consider SOC.\nSpinful $p_{x,y}$ orbitals decompose into three irreps of the site-symmetry group.\nBands derived from these three irreps transform as a sum of three EBRs,\\cite{NaturePaper,EBRTheory}\nwhich generically split into four sets of disconnected bands, as in Figs~\\ref{fig:pxpy-alltop} and \\ref{fig:pxpy-sometop}.\nAt least one set of disconnected bands is either an obstructed atomic limit -- it can be adiabatically deformed to a Hamiltonian comprised of orbitals that reside at the center of the hexagon rather than the corners -- or a topological band that does not have any atomic limit.\\footnote{The possibilities are described in Sec. S1C} %{sec:SOCxy.}\n\nIf time-reversal symmetry is enforced, we can consider the $\\mathbb{Z}_2$ index.\nFor small spin-conserving SOC that does not invert the bands at $\\Gamma$ or $M$, the $C_{2z}$ eigenvalues in the spinless phase determine the $\\mathbb{Z}_2$ index of each set of bands (conserving spin amounts to enforcing inversion symmetry.)\nOur simple, but physically motivated, model yields two phases, shown in Fig~\\ref{fig:pxySOC}: either all three or the first\/third gaps are $\\mathbb{Z}_2$ topological, while the middle gap is not; there is no phase in which all gaps are $\\mathbb{Z}_2$ trivial.\\footnote{The possible $C_{2z}$ eigenvalues are computed in Sec. S1C} %{sec:SOCxy .}\nWe show in Sec. S1C} %{sec:SOCxy\\ that only spin-conserving SOC can open a gap in the spinless band structure; hence, if non-spin-conserving SOC is present and does not invert any bands, it will alter the band structure but not change the $\\mathbb{Z}_2$ index.\n\n\n\\begin{figure}[h]\n\\centering\n\\subfloat[]{\n\t\\includegraphics[width=1.5in]{pxpy-SOC-alltop.pdf}\n\t\\label{fig:pxpy-alltop}\n} \\quad\n\\subfloat[]{\n\t\\includegraphics[width=1.5in]{pxpy-SOC-sometop.pdf}\n\t\\label{fig:pxpy-sometop}\n\t}\n\\quad\n\\subfloat[]{\n\t\\includegraphics[width=1.5in]{pxpy-SOC-alltop-slab.pdf}\n\t\\label{fig:pxpy-alltop-slab}\n} \\quad\n\\subfloat[]{\n\t\\includegraphics[width=1.5in]{pxpy-SOC-sometop-slab.pdf}\n\t\\begin{picture}(0,0)\n\\put(-40,40){\\includegraphics[height=1cm]{pxpy-SOC-sometop-slab-zoom.pdf}}\n\\end{picture}\n\t\\label{fig:pxpy-sometop-slab}\n}\n\\caption{Band structures with an onsite $L\\cdot S$ term for (a,b) periodic and (c,d) slab boundary conditions. This SOC term preserves inversion symmetry, such that all bands remain doubly degenerate, but is general enough to open all possible gaps in the band structure.\nIn (a) only the lowest and highest bands have a nontrivial $\\mathbb{Z}_2$ index and hence the slab boundary conditions in (c) reveal edge states in all three gaps.\nIn (b) all four bands have a nontrivial $\\mathbb{Z}_2$ index and hence the slab boundary conditions in (d) reveal edge states in only the upper and lower gaps.\nThe inset in (d) resolves the avoided crossing at $E=0$.}\n\\label{fig:pxySOC}\n\\end{figure}\n\n\n\n\n\n\n\n\\paragraph{Material realization}\n\nThe spinless semi-metallic model $H_\\mathbf{k}^0$ consists of nearest-neighbor Slater-Koster\\cite{SlaterKoster} terms; thus, it is widely applicable to two-dimensional planar honeycomb systems.\nTo exhibit the TCI phase, the next-nearest neighbor term, $H_\\mathbf{k}^1$, must be dominant in order to open a gap.\nThe relative strength of the hopping terms varies with strain or buckling.\n\n\nThe non-trivial $\\mathbb{Z}_2$ phases will be present whenever SOC is large enough to open an observable gap, but not so large to invert the bands at $\\Gamma$ and $M$.\nIn particular, $H_\\mathbf{k}^0$ with SOC describes bismuth grown on an SiC substrate, \nconsistent with the topological edge states reported in Ref~\\onlinecite{Reis2016}.\n\n\n\n\n\n\n\n\\paragraph{Non-symmorphic gapped EBR}\n\n\nWe now consider the non-symmorphic simple cubic space group $P4_2 32$, which is generated by \n$\\{ C_{2x} | \\mathbf{0} \\}$, $\\{ C_{3,111} | \\mathbf{0} \\}$ and $\\{ C_{2,110} | \\frac{1}{2}\\frac{1}{2}\\frac{1}{2} \\}$.\nWe also enforce time-reversal symmetry.\nWe consider atoms sitting at $(0,0,0)$ and $(\\frac{1}{2},\\frac{1}{2},\\frac{1}{2})$ (inset to Fig~\\ref{fig:208bands-nodeg}), which together comprise the $2a$ Wyckoff position, each with spinless $d_{z^2}$ and $d_{x^2-y^2}$ orbitals, which together form a time-reversal symmetric irrep\nof the site-symmetry group.\\cite{ITA,PointGroupTables}\nSince the orbitals transform as an irrep of a maximal Wyckoff position, any band structure derived from these orbitals transforms as a time-reversal invariant EBR.\nIt follows from Ref~\\onlinecite{EBRTheory} that if the band structure is gapped, it contains topological bands.\nHere, we explicitly construct a gapped Hamiltonian and a nontrivial bulk topological invariant, violating the conjecture\\cite{Michel1999,Michel2001} that a single set of symmetry-related orbitals in a non-symmorphic space group always yields a gapless band structure\n\n\n\n\n\\begin{figure}[t]\n\\centering\n\\subfloat[]{\n\t\\includegraphics[width=1.4in]{SG208-bands-nodeg.pdf}\\label{fig:208bands-nodeg}\n\t\\put(-5,30){\\includegraphics[height=1.2cm]{SG208-nnvecs.png}}\n\t}\\quad\\quad\\quad\n\\subfloat[]{\n\t\\includegraphics[width=1.4in]{SG208-wilsonz-wind2.pdf}\\label{fig:208wilsonz}\n\t\\begin{picture}(0,0)\n\\put(-35,35){\\includegraphics[height=1.2cm]{SG208-path.pdf}}\n\\end{picture}\n\t}\n\\caption\n(a) Spectrum of $H_\\mathbf{k} + H^1_\\mathbf{k} $ with $t_1=.2,t_2=.3,t_3=.1, t_4=.08,t_5=.05,t_6=.02$. Inset shows unit cell: orbitals on the blue atoms enter the Hamiltonian, while the white\/black atoms create a crystal field with the symmetry of $P4_232$.\n(b) Argument of the eigenvalues of the $z$-directed Wilson matrix along the path $\\bar{M}-\\bar{\\Gamma}-\\bar{M}'$ (blue dotted line in inset); $\\bar{M} = (\\pi,\\pi)$, $\\bar{\\Gamma} = (0,0)$, $\\bar{M}'=(-\\pi,\\pi)$.}\n\\end{figure}\n\n\n\nWe consider the following Hamiltonian, which respects all space group symmetries and time-reversal:\\footnote{\\SUPPsymmetries\\ shows the matrix form of the symmetries.}\n\\begin{align} \nH_\\mathbf{k} &= t_1 f_1(\\mathbf{k})\\tau_x \\otimes \\sigma_0 + t_2f_2(\\mathbf{k}) \\tau_y\\otimes\\sigma_0 +\\nonumber\\\\\n&+ t_3(g_1(\\mathbf{k})\\tau_z \\otimes \\sigma_z + g_2(\\mathbf{k})\\tau_z\\otimes \\sigma_x),\n\\end{align}\nwhere\n$f_1(\\mathbf{k}) \\!=\\! \\sum_i \\cos (\\mathbf{k}\\cdot \\delta_i)$, \n$f_2(\\mathbf{k}) \\!=\\! \\sum_i \\sin (\\mathbf{k}\\cdot \\delta_i)$, \n$g_1(\\mathbf{k}) = \\cos k_x - \\cos k_y$, \n$g_2(\\mathbf{k}) = \\left(\\cos k_x+\\cos k_y - 2\\cos k_z\\right)\/\\sqrt{3} $\nand $\\delta_{1,2,3,4}$ are vectors to nearest neighbors, shown in Fig~\\ref{fig:208bands-nodeg}.\nThe band structure is doubly-degenerate and gapped when $t_{1,2,3}\\neq 0$.\nTo eliminate the extra degeneracies, we add the symmetry-preserving term\n\\begin{align}\nH_\\mathbf{k}^1 &= t_4f_1(\\mathbf{k}) \\tau_y\\otimes \\sigma_y + t_5\\tau_0\\otimes (g_2(\\mathbf{k}) \\sigma_z -g_1(\\mathbf{k})  \\sigma_x ) \\nonumber\\\\\n&+ t_6 g_3(\\mathbf{k}) \\tau_z \\otimes \\sigma_0,\n\\end{align}\nwhere\n\\begin{align}\ng_3(\\mathbf{k}) &= \\cos(2k_x)\\cos(k_y) \\!-\\! \\cos(2k_y)\\cos(k_x) + \\text{perm.}\n\\end{align}\nand `+ perm' indicates terms obtained by permuting $k_x\\rightarrow k_y\\rightarrow k_z$.\nThe spectrum of $H_\\mathbf{k} + H_\\mathbf{k}^1$ is shown in Fig~\\ref{fig:208bands-nodeg}.\nSince $H_\\mathbf{k}$ is fully gapped when $t_{1,2,3}\\neq 0$, $H_\\mathbf{k} + H_\\mathbf{k}^1$ is gapped when $t_{4,5,6}$ are small compared to $t_{1,2,3}$.\n\n\n\nThis gapped phase realizes a disconnected and time-reversal symmetric EBR; thus, it contains topological bands.\nWe diagnose the topological phase by the winding of its $z$-directed Wilson loop along the \\emph{bent} path shown in Fig~\\ref{fig:208wilsonz}.\nThis is a time-reversal symmetric and non-symmorphic generalization of the ``bent Chern number'' introduced in Ref~\\onlinecite{Alexandradinata14a}.\nTwo features are necessary for this loop to wind:\nfirst, the Wilson loop eigenvalues are pinned to $\\pm 1$ at $\\bar{\\Gamma}$ and $\\bar{M}$,\nand, second, there are protected band crossings in the Wilson spectrum along the $|k_x| = |k_y|$ lines.\nCombined, these features prevent the Wilson bands from being smoothly deformable to flat bands; hence, the phase is topological.\n\nWe now explain the origin of these features: first, $C_{2x}$ symmetry forces the eigenvalues of $\\mathcal{W}_{(\\bar{\\Gamma},0)}$ and $\\mathcal{W}_{(\\bar{M},0)}$ to be real, while the $\\{ C_{2,110}| \\frac{1}{2}\\frac{1}{2}\\frac{1}{2}\\}$ screw symmetry forces them to come in pairs $(\\lambda, -\\lambda^*)$.\\footnote{See \\SUPPwilsongapped .}\nThis combination pins the eigenvalues of $\\mathcal{W}_{(k_x,k_y,0)}$ to be $\\pm 1$ at $\\bar{\\Gamma}$ and $\\bar{M}$.\n\n\nThe Wilson band crossing is subtle: the $\\{ C_{2,110}| \\frac{1}{2}\\frac{1}{2}\\frac{1}{2}\\}$ screw symmetry requires the eigenvalues of $\\mathcal{W}_{(k,k,0)}$ to come in pairs $(\\lambda, -\\lambda^*)$.\nCombined with the anti-unitary symmetry $\\mathcal{T}\\{ C_{2,110}| \\frac{1}{2}\\frac{1}{2}\\frac{1}{2}\\}^{-1}C_{2z}$, which leaves points $(k,k,k_z)$ invariant, the Wilson matrix must take the form $\\mathcal{W}_{(k,k,0)} = ie^{ia_x(k)\\sigma_x + ia_y(k)\\sigma_y}$, where, importantly, $a_x(k) \\propto a_y(k)$ (alternately, the symmetries permit the eigenvalues of $\\mathcal{W}_{(k,k,0)}$ to be fixed to $\\pm 1$; see Sec. S3B} %{sec:208wilsoncrossing.)\nThen degeneracies of the Wilson eigenvalues, which occur when $a_x(k)=a_y(k)=0$, are not fine-tuned, since the symmetry forced $a_x(k)\\propto a_y(k)$. Since the eigenvalues of $\\mathcal{W}_{(k,k,0)}$ at $k=0$ and $k=\\pi$ are fixed to $+1$ and $-1$, an odd number of linear degeneracies between $\\bar{\\Gamma}$ and $\\bar{M}$ cannot be removed without closing the bulk band gap.\nThus, the parity of the number of linear degeneracies is a topological invariant.\n\nThe band crossing forms a Dirac cone in the two-dimensional ``Wilson Hamiltonian.''\\cite{ArisCohomology}\nThe Dirac point\nis revealed by the Berry phase, $w$, acquired by an eigenstate of $\\mathcal{W}_{(k_x,k_y,0)}$ as it traverses the path $\\gamma$ around the Dirac point.\nThe Berry phase of Wilson loop eigenstates was introduced in Ref.~\\onlinecite{hughesbernevig2016}.\nSince $w$ is quantized to $\\pm 1$ (see Sec. S4} %{sec:wilsofwils), it constitutes a topological invariant.\nIn our model, for several values of parameters, we have numerically computed the nontrivial value, $w=-1$.\n\nWhen SOC is present, the spinful $d_{z^2}$ and $d_{x^2-y^2}$ orbitals transform as spin-$\\frac{3}{2}$ orbitals, which induce an eight-band time-reversal symmetric EBR.\\cite{GroupTheoryPaper}\nWhen the EBR is gapped, the valence (or conduction) bands must be topological.\n\n\\paragraph{Weak symmetry indicators}\n\nIn both the spinless TCI on the honeycomb lattice and the non-symmorphic gapped EBR,\nthe valence bands are topological, but have the property that the irreps at high-symmetry points can be written as a ``difference'' of the irreps in two other EBRs.\\footnote{Details for the honeycomb case are in \\SUPPpxyirreps .}\nBecause the irreps can be written as a difference, classification schemes\\cite{Po2017} that treat the little group irreps as a vector space will identify the valence bands as trivial, even though they lack an atomic limit.\nHowever, unless an energy gap closes to the valence bands, the winding of the Wilson loop in both examples provides a robust and quantized topological invariant that is, in principle, physically observable.\\cite{Atala13,Nakajima16,Holler2017}\n\nThis distinction warrants a refined characterization of topological crystalline bands based on whether their topological nature can be deduced by their little group irreps.\nWe label the symmetry properties of topological bands as strong if their little group irreps are not equal to a linear combination of little group irreps corresponding to EBRs and weak if their little group irreps are equal to a difference (but not a sum) of irreps in EBRs.\nStrong symmetry properties implies a stable topological index; however, the converse is not true: for example, bands with a nontrivial $\\mathbb{Z}_2$ index under time-reversal symmetry can be strong\\cite{Fu2007} or weak.\\cite{NaturePaper}\nThis usage of weak and strong symmetry is different than the current distinction between weak and strong topological insulators.\\cite{Fu2007}\nIt is more suitable for the refined classification of topological insulators with crystal symmetries.\n\n\n\\paragraph{Conclusions}\n\nWe have constructed tight-binding models to realize the insulating phases of two gapped EBRs. \nWe explicitly showed that the valence bands have a nontrivial topological invariant.\nIn doing so, we found a new topological invariant in a non-symmorphic space group: a Dirac cone in the Wilson loop spectrum and a Wilson loop that winds along a bent path.\nThis motivates further study of the gapped EBRs in other non-symmorphic space groups.\nIn addition, we introduced the notion of a weak symmetry indicator.\nWe postpone a general investigation of the symmetry properties of gapped EBRs to future work.\n\n\n\n\n\n\n\n\n\\begin{acknowledgments}\n\\paragraph{Acknowledgments}\nBB, JC, ZW, and BAB acknowledge the hospitality of the Donostia International Physics Center.\nBAB acknowledges the hospitality and support of the \\'{E}cole Normale Sup\\'{e}rieure and Laboratoire de Physique Th\\'{e}orique et Hautes Energies. \nThe work of MGV was supported by FIS2016-75862-P.\nThe work of LE and MIA was supported by the Government of the Basque Country (project IT779-13)  and the Spanish Ministry of Economy and Competitiveness and FEDER funds (project MAT2015-66441-P). \nBAB acknowledges the support of the NSF EAGER Award DMR -- 1643312, ONR - N00014-14-1-0330, NSF-MRSEC DMR-1420541, ARO MURI W911NF-12-1-0461, the Department of Energy de-sc0016239, the Simons Investigator Award, Packard Foundation and the Schmidt Fund for Innovative Research.\n\n\\end{acknowledgments}\n\n\n\n\n\\section{Nearest neighbor Hamiltonian for $p_{x,y}$ orbitals on the honeycomb lattice}\n\\label{sec:pxyHam}\n\nWe choose the lattice basis vectors:\n\\begin{align}\n\\mathbf{e}_1 &= \\frac{\\sqrt{3}}{2}\\hat{\\mathbf{x}} + \\frac{1}{2} \\hat{\\mathbf{y}} \\nonumber\\\\\n\\mathbf{e}_2 &= \\frac{\\sqrt{3}}{2}\\hat{\\mathbf{x}} - \\frac{1}{2} \\hat{\\mathbf{y}},\n\\end{align}\nwhich are shown in Fig. 1} %{fig:graphenebasisvectors\\ with\ntheir reciprocal lattice vectors, $\\mathbf{g}_i$, which satisfy $\\mathbf{g}_i\\cdot \\mathbf{e}_j = 2\\pi \\delta_{ij}$.\nSites on the $A(B)$ sublattice sit at positions $\\mathbf{R} + \\mathbf{r}_{A(B)}$, where $\\mathbf{R}$ denotes a lattice translation and \n\\begin{align}\n\\mathbf{r}_A &= \\frac{2}{3}\\mathbf{e}_1 - \\frac{1}{3}\\mathbf{e}_2\\nonumber\\\\\n\\mathbf{r}_B &= \\frac{1}{3}\\mathbf{e}_1 - \\frac{2}{3}\\mathbf{e}_2\n\\end{align}\nIn this basis, the symmetry generators of the honeycomb lattice act as follows:\n\\begin{align}\nC_{3z} &:  (\\mathbf{e}_1 , \\mathbf{e}_2) \\rightarrow (-\\mathbf{e}_2, \\mathbf{e}_1 - \\mathbf{e}_2) \\nonumber\\\\\nC_{2z} &:  (\\mathbf{e}_1 , \\mathbf{e}_2) \\rightarrow (-\\mathbf{e}_1 , -\\mathbf{e}_2) \\nonumber\\\\\nm_{1\\bar{1}} &: (\\mathbf{e}_1 , \\mathbf{e}_2) \\rightarrow (\\mathbf{e}_2 , \\mathbf{e}_1),\n\\end{align}  \nwhere the subscript $1\\bar{1}$ indicates that the mirror plane has normal vector $\\mathbf{e}_1 - \\mathbf{e}_2$.\nFor $p_{x,y}$ orbitals, we choose the following matrix representation, in which the Pauli matrices $\\tau$ act in sublattice space and the $\\sigma$ matrices act in orbital space:\n\\begin{align}\nU_{C_{3z}} &=  \\tau_0 \\otimes \\left(-\\frac{1}{2}\\sigma_0 + i \\frac{\\sqrt{3}}{2}\\sigma_y\\right) \\nonumber\\\\\nU_{C_{2z}} &= -\\tau_x \\otimes \\sigma_0 \\nonumber\\\\\nU_{m_{1\\bar{1}}} &= \\tau_0 \\otimes \\sigma_z,\n\\label{eq:unitarygensxy}\n\\end{align}\nThe orbital term for a rotation by an angle $\\theta$ about an axis $\\hat{\\mathbf{n}}$ is represented by $e^{i\\theta \\hat{\\mathbf{n}} \\cdot \\mathbf{S}}$, projected onto the $p_{x,y}$ orbitals; $\\mathbf{S}$ is the vector of spin-1 matrices.\nA Hamiltonian, $H_\\mathbf{k}$, that respects the lattice symmetry must satisfy:\n\\begin{equation}\nH_\\mathbf{k} = U_R^\\dagger H_{R\\mathbf{k}} U_R,\n\\end{equation} \nfor each generator, $R$, of the honeycomb lattice.\n\n\nDenoting the annihilation operator on site $\\mathbf{r}$ by $c_{\\mathbf{r},a}$, where $a = x,y$ indicates the $p_x$ or $p_y$ orbital, the nearest neighbor Hamiltonian is given by:\n\\begin{equation}\nH = \\sum_{\\mathbf{R}}\\sum_{a,b} \\sum_{\\mathbf{\\delta}_i}    t_{ab}(\\delta_i)    c_{\\mathbf{R} +\\mathbf{r}_A, a}^\\dagger  c_{\\mathbf{R} + \\mathbf{r}_A+ \\delta_i, b}  + {\\rm h.c.},\n\\label{eq:pxyHamreal}\n\\end{equation}\nwhere the three nearest neighbors to a site at $\\mathbf{R}+\\mathbf{r}_A$ sit at $\\mathbf{R}+\\mathbf{r}_A + \\delta_i$ (the vectors $\\delta_i$ are depicted in Fig. 1} %{fig:graphenebasisvectors) and\n$t_{ab}(\\delta_i)$ is given by one of the Slater-Koster terms:\\cite{SlaterKoster}\n\\begin{align}\nt_{xx}( \\delta_i ) &= \\frac{1}{3}\\left[ (\\delta_i \\cdot \\hat{\\mathbf{x}})^2  t_\\sigma + (\\delta_i \\cdot \\hat{\\mathbf{y}})^2 t_\\pi \\right] \\nonumber\\\\\nt_{yy}( \\delta_i ) &=  \\frac{1}{3} \\left[ (\\delta_i \\cdot \\hat{\\mathbf{y}})^2  t_\\sigma + (\\delta_i \\cdot \\hat{\\mathbf{x}})^2 t_\\pi \\right] \\nonumber\\\\\nt_{xy}( \\delta_i ) &= \\frac{1}{3} (\\delta_i \\cdot \\hat{\\mathbf{x}})(\\delta_i \\cdot \\hat{\\mathbf{y}})(t_\\sigma - t_\\pi) = t_{yx}( \\delta_i),\n\\label{eq:hopping}\n\\end{align}\nwhere $t_{\\sigma(\\pi)}$ are free parameters that describe $\\sigma(\\pi)$-bonds.\nNotice that $\\mathbf{r}_A+\\delta_i$ is always a site on the $B$ sublattice.\nUsing the Fourier transform,\n\\begin{equation}\nc_{\\mathbf{k},L,a} = \\sum_\\mathbf{R} e^{i\\mathbf{k} \\cdot (\\mathbf{R}+\\mathbf{r}_{L}) }c_{\\mathbf{R} + \\mathbf{r}_L,a}\n\\end{equation}\nwhere $L=A,B$ denotes the sublattice and $a=x,y$ denotes the orbital,\nthe real space Hamiltonian in Eq~(\\ref{eq:pxyHamreal}) is rewritten:\n\\begin{equation}\nH = \\sum_\\mathbf{k} \\sum_{a,b}\\sum_{\\delta_i} c_{\\mathbf{k},A,a}^\\dagger c_{\\mathbf{k},B,b}e^{-i\\mathbf{k}\\cdot \\delta_i} t_{ab}(\\delta_i) + {\\rm h.c.},\n\\label{eq:pxyHam1}\n\\end{equation}\nPlugging Eq~(\\ref{eq:hopping}) into Eq~(\\ref{eq:pxyHam1}) yields \n\\begin{equation}\nH^0 =\\sum_\\mathbf{k} \\psi_\\mathbf{k}^\\dagger H_\\mathbf{k}^0 \\psi_\\mathbf{k} \\equiv \\sum_\\mathbf{k} \\psi_\\mathbf{k}^\\dagger \\begin{pmatrix} 0 & h_\\mathbf{k} \\\\ h_\\mathbf{k}^\\dagger & 0 \\end{pmatrix} \\psi_\\mathbf{k}\n\\end{equation}\nwhere $\\psi_\\mathbf{k} = \\begin{pmatrix} c_{\\mathbf{k},A,x} & c_{\\mathbf{k},A,y} & c_{\\mathbf{k},B,x} & c_{\\mathbf{k},B,y} \\end{pmatrix}^T$ contains the annihilation operators for $p_{x,y}$ orbitals on the $A$ and $B$ sublattices and $h_\\mathbf{k}$ is given by Eq. (3)} %{eq:pxyHamh .\n\n\n\n\n\n\n\n\\subsection{Phase diagram of $H_\\mathbf{k}^0 + xH_\\mathbf{k}^1$}\n\\label{sec:tciphase}\n\nAs described in the main text,\na necessary condition to reach the gapped TCI phase is that the two-fold degeneracy at $K$ is higher or lower in energy than the other two bands.\nThe eigenvalues of $H_{\\mathbf{k}=K}^0 + x H_{\\mathbf{k}=K}^1$ are $-\\frac{3\\sqrt{3}}{8}x$ (2-fold degenerate) and $\\frac{3}{8}(\\pm 4(t_\\pi - t_\\sigma)+\\sqrt{3}x )$; \nthus, the TCI phase requires: \n\\begin{equation}\n|x|> \\frac{2}{\\sqrt{3}}|t_\\pi - t_\\sigma| \n\\label{eq:TCIcond}\n\\end{equation}\n\nHowever, there are further constraints: band crossings along the paths connecting $\\Gamma$ and $M$ can prevent the system from opening a gap even when the energy ordering at $K$ allows it.\nConsider the two lines $\\Sigma = \\alpha \\mathbf{g}_1$ and $\\Lambda = \\beta(\\frac{1}{3}\\mathbf{g}_1 + \\frac{2}{3} \\mathbf{g}_2)$, which are invariant under $C_{3z}m_{1\\bar{1}}$ and $C_{2z}C_{3z}m_{1\\bar{1}}$, respectively.\nAs $\\alpha$ goes from $0$ to $\\frac{1}{2}$, $\\Sigma$ connects $\\Gamma$ to $M$.\nAs $\\beta$ goes from $0$ to $\\frac{3}{2}$, $\\Lambda$ connects $\\Gamma$ to $M + \\mathbf{g}_2$ (passing through $K'$ at $\\beta = 1$).\nWe can then track the respective mirror eigenvalues along these lines to find constraints on connectivity.\nThe BANDREP application on the BCS server shows that the $\\Gamma_5$ and $\\Gamma_6$ irreps appear at $\\Gamma$ in our model ($p_{x,y}$ orbitals on the honeycomb correspond to the irrep $E$ at Wyckoff position $2b$).\nIn the $\\Gamma_5$ and $\\Gamma_6$ irreps, the character of each mirror is zero: this means that the doubly-degenerate bands that comprise the $\\Gamma_5$ irrep will split into two bands along $\\Sigma$, one of which is even under $C_{3z}m_{1\\bar{1}}$ and one of which is odd (and same for $\\Gamma_6$).\nThus, $\\Gamma_5$ must connect to two irreps at $M$ which have opposite parity under $C_{3z}m_{1\\bar{1}}$.\nSimilarly, the $\\Gamma_5$ irrep will split into two bands along $\\Lambda$, one of which is even under $C_{2z}C_{3z}m_{1\\bar{1}}$ and one of which is odd; thus, $\\Gamma_5$ must connect to two irreps at $M$ which have opposite parity under $C_{2z}C_{3z}m_{1\\bar{1}}$.\nWithout loss of generality, assume the $\\Gamma_5$ irrep is higher in energy than $\\Gamma_6$; \nthen in order for the system to be an insulator, \nthe two bands that are highest in energy at $M$ must have opposite $C_{3z}m_{1\\bar{1}}$ eigenvalues and also opposite $C_{2z}C_{3z}m_{1\\bar{1}}$ eigenvalues.\n\n\nReferring to Table~\\ref{tab:Meigs}, a gapped band structure requires one of the two following conditions to be satisfied:\n\\begin{align}\n E_{M1,M2} > E_{M3,M4} \\Rightarrow 3t_\\sigma < t_\\pi < \\frac{1}{3}t_\\sigma \\label{eq:pxygap1}\\\\\n E_{M1,M2} < E_{M3,M4} \\Rightarrow \\frac{1}{3}t_\\pi < t_\\sigma < 3t_\\pi  \\label{eq:pxygap2}\n\\end{align}\nIf neither Eq~(\\ref{eq:pxygap1}) or (\\ref{eq:pxygap2}) is satisfied, there will be a band crossing along $\\Gamma-M$, as shown in Fig~\\ref{fig:Weyl}.\n\nThe $C_{2z}$ eigenvalues at $\\Gamma$ are shown in Table~\\ref{tab:Geigs}.\nBy comparing Table~\\ref{tab:Meigs} with Table~\\ref{tab:Geigs}, one can check that in the gapped phase, when either Eq~(\\ref{eq:pxygap1}) or (\\ref{eq:pxygap2}) is satisfied, the $C_{2z}$ eigenvalues at $\\Gamma$ are always opposite those at $M$ (unlike the usual time-reversal $\\mathbb{Z}_2$ invariant, here we are referring to the $C_{2z}$ eigenvalues themselves, not their product.\nThis guarantees that the Wilson loop in Fig. 1d} %{fig:pxyWilson\\ always winds in the gapped phase.\\cite{Alexandradinata14}\n\nWe note that it could be possible with longer range hopping terms to reach a gapped phase where the $C_{2z}$ eigenvalues of the two lower bands at $\\Gamma$ are the same as those at $M$ (this is symmetry-allowed according to possible decompositions of the EBR induced from $p_{x,y}$ orbitals listed on the BCS Server\\cite{GroupTheoryPaper}). However, this phase is not accessible within our nearest-neighbor model.\n\n\n\\begin{figure}[h]\n\\centering\n\\subfloat[]{\n\t\\includegraphics[width=1.5in]{pxpyHam-Weyl.pdf}\\label{fig:Weyl}\n\t\\begin{picture}(0,0)\n\t\t\\put(-30,35){\\includegraphics[height=1.6cm]{pxpyHam-Weyl-path.pdf}}\n\t\\end{picture}\n\t}\\quad\n\\subfloat[]{\n\t\\includegraphics[width=1.5in]{pxpyHam-Weylgone.pdf}\\label{fig:Weylgone}\n}\n\\caption{Band structure of $H^0_\\mathbf{k}+xH^1_\\mathbf{k}$ along the red path shown in the inset; $x=.35$. In (a) $t_\\sigma=.08, t_\\pi = .3$, while in (b) $t_\\sigma = .16, t_\\pi = .3$.\nIn both cases, Eq~(\\ref{eq:TCIcond}) is satisfied; hence the band ordering at $K$ allows for a gap.\nHowever, since the parameters in (a) violate Eq~(\\ref{eq:pxygap1}) and (\\ref{eq:pxygap2}), there is a band crossing between $\\Gamma$ and $M+\\mathbf{g}_2$. In (b), Eq~(\\ref{eq:pxygap2}) is satisfied and the band structure is gapped.}\n\\end{figure}\n\n\n\\begin{table}[h]\n{\\renewcommand{\\arraystretch}{1.2}%\n\\centering\n\\begin{tabular}{c|c|c|c}\nEnergy at $M$ & $C_{3z}m_{1\\bar{1}}$ eig. & $C_{2z}C_{3z}m_{1\\bar{1}}$ eig. & $C_{2z}$ eig.\\\\\n\\hline\n$E_{M1}\\equiv \\frac{t_\\pi}{2} - \\frac{3t_\\sigma}{2}$ & $-1$ & $1$ & $-1$ \\\\\n$E_{M2}\\equiv -\\frac{3t_\\pi}{2} + \\frac{t_\\sigma}{2}$ &$1$ & $-1$ & $-1$\\\\\n$E_{M3} \\equiv \\frac{3t_\\pi}{2} - \\frac{t_\\sigma}{2}$ & $1$ & $1$ & $+1$ \\\\\n$E_{M4} \\equiv -\\frac{t_\\pi}{2} + \\frac{3t_\\sigma}{2}$ & $-1$ & $-1$ & $+1$\n\\end{tabular}}\n\\caption{Energies, mirror, and $C_{2z}$ eigenvalues for each of the eigenstates at $M$. The $C_{2z}$ eigenvalue is a product of the two mirror eigenvalues.}\n\\label{tab:Meigs}\n\\end{table}\n\n\\begin{table}[h]\n{\\renewcommand{\\arraystretch}{1.2}%\n\\begin{tabular}{c|c}\nEnergy at $\\Gamma$ & $C_{2z}$ eig.\\\\\n\\hline \n$E_{\\Gamma 1}= -\\frac{3}{2}(t_\\pi + t_\\sigma)$ & $+1, +1$\\\\\n$E_{\\Gamma 2}= \\frac{3}{2}(t_\\pi + t_\\sigma)$ & $-1, -1$\n\\end{tabular}}\n\\caption{Energies and $C_{2z}$ eigenvalues for the two-fold degenerate eigenstates at $\\Gamma$.}\n\\label{tab:Geigs}\n\\end{table}\n\n\n\n\\subsection{Irreps at high-symmetry points}\n\\label{sec:pxyirreps}\n\n\nUsing the notation on BANDREP application of the BCS\\cite{GroupTheoryPaper}, the Hamiltonian $H_\\mathbf{k}^0 + xH_\\mathbf{k}^1$ can realize two possible sets of valence bands: $(\\Gamma_5, K_3, M_3, M_4)$ or $(\\Gamma_6, K_3, M_1, M_2)$, assuming, without loss of generality, that the $K_3$ irrep appears in the valence bands instead of the conduction bands.\n(As noted at the end of the last section, there are two other disconnected phases listed in the BANDREP application that that our model does not realize and which differ in the $C_{2z}$ eigenvalues of occupied bands; presumably they require longer range hopping).\nBy comparing to the list of band representations induced from Wyckoff positions in $P6mm$ (SG 183) (which describes layers of the honeycomb lattice with no additional symmetry in the $z$ direction), we see that the irreps in $A_1\\uparrow G$ or $B_2\\uparrow G$ from the $1a$ position are $(\\Gamma_1, K_1, M_1)$ or $(\\Gamma_3, K_1, M_3)$, respectively, while the irreps in $A_1\\uparrow G$ on the $3c$ position are $(\\Gamma_1, \\Gamma_5, K_1, K_3, M_1, M_3, M_4)$ and the irreps in $B_1\\uparrow G$ on the $3c$ position are $(\\Gamma_3, \\Gamma_6, K_1, K_3, M_1, M_2, M_3)$.\nThus, we see that for each of the possible sets of valence bands in our model, \nthe irreps that appear are obtained from subtracting the irreps of one of the EBRs induced from the $1a$ position from one of the EBRs induced from the $3c$ position.\nA classification scheme that only looks at the irreps at high-symmetry points will classify our valence bands as trivial.\nIt was noted in Ref~\\onlinecite{Po2017} that some topologically nontrivial bands will be included in the trivial class; our model is an example of this phenomenon.\n\n\n\n\n\n\\subsection{Phase diagram with SOC}\n\\label{sec:SOCxy}\n\nIncluding SOC, \n$\\Gamma_{5} \\rightarrow \\bar{\\Gamma}_7 \\oplus \\bar{\\Gamma}_{8}$, $\\Gamma_6\\rightarrow \\bar{\\Gamma}_7 \\oplus \\bar{\\Gamma}_{9}$\nand\n$K_{1} \\rightarrow \\bar{K}_6$, $K_{2} \\rightarrow \\bar{K}_6$, $K_3 \\rightarrow \\bar{K}_4\\oplus \\bar{K}_5 \\oplus \\bar{K}_6$.\nThus, the irreps that appear in a model with spinful $p_{x,y}$ orbitals on the honeycomb lattice are \n\\begin{equation}\n2\\bar{\\Gamma}_7\\oplus \\bar{\\Gamma}_8\\oplus\\bar{\\Gamma}_9 \\text{ and }3\\bar{K}_6\\oplus\\bar{K}_4\\oplus\\bar{K}_5\n\\label{eq:pxyirreps}\n\\end{equation}\nAs mentioned in the main text, they belong to a sum of three EBRs, $^1\\bar{E}\\uparrow G$, $^2\\bar{E}\\uparrow G$, and $\\bar{E}_1\\uparrow G$, induced from the $2b$ position. The double-valued EBRs for SG 183 are listed in Table~\\ref{tab:183EBRs}.\n(Since there is only one double-valued irrep of the little group at $M$, it cannot be used to distinguish EBRs and we do not need to consider it here.)\nSince the $\\bar{E}_1\\uparrow G$ EBR is decomposable, generically, the band structure splits into four groups of bands.\nOne possibility is that one branch contains the irreps $\\bar{\\Gamma}_8$ and $\\bar{K}_6$, another contains $\\bar{\\Gamma}_9$ and $\\bar{K}_6$, another contains $\\bar{\\Gamma}_7$ and $\\bar{K}_6$ and the last contains $\\bar{\\Gamma}_7$ and $\\bar{K}_4\\oplus \\bar{K}_5$.\nIn this case, each branch has the same irreps at high-symmetry points as an EBR listed in Table~\\ref{tab:183EBRs}, but the EBR might come from orbitals on the $1a$ position.\nIn the language of Ref~\\onlinecite{NaturePaper}, a branch that contains the same irreps as an EBR induced from orbitals located at a different site than the atoms is called an ``obstructed atomic limit.''\nAn obstructed atomic limit can have localized Wannier functions, but since those Wannier functions are not located where the atoms are located, a gap must close in order to reach the phase where the Wannier functions and the atomic orbitals are localized at the same sites.\n\nThe other possibility is that the bands disconnect in such a way that some branches do not have the same irreps as an EBR; this can happen if a branch contains $\\bar{\\Gamma}_8$ or $\\bar{\\Gamma}_9$ and $\\bar{K}_{4}\\oplus \\bar{K}_5$.\nA branch that does not have the same irreps as an EBR does not correspond to an atomic limit and cannot yield maximally localized Wannier functions that obey the crystal symmetry, centered at any position.\n\n\n\n\n\n\n\\begin{table}[h]\n{\\renewcommand{\\arraystretch}{1.2}%\n\\centering\n\\begin{tabular}{c|c|c|c|c}\nWyckoff & EBR & $\\bar{\\Gamma}$ irreps & $\\bar{K}$ irreps & No. bands\\\\\n\\hline\n$1a$ & $\\bar{E}_1\\uparrow G$ & $\\bar{\\Gamma}_9$ & $\\bar{K}_6$ & 2\\\\\n$1a$ & $\\bar{E}_2\\uparrow G$ & $\\bar{\\Gamma}_8$ & $\\bar{K}_6$ & 2\\\\\n$1a$ & $\\bar{E}_3\\uparrow G$ & $\\bar{\\Gamma}_7$ & $\\bar{K}_4\\oplus \\bar{K}_5$ & 2\\\\\n$2b$ & $^1\\bar{E}\\uparrow G$ & $\\bar{\\Gamma}_7$ & $\\bar{K}_6$ & 2\\\\\n$2b$ & $^2\\bar{E}\\uparrow G$ & $\\bar{\\Gamma}_7$ & $\\bar{K}_6$ & 2\\\\\n$2b$ & $\\bar{E}_1\\uparrow G$ & $\\bar{\\Gamma}_8\\oplus \\bar{\\Gamma}_9$ & $\\bar{K}_4\\oplus \\bar{K}_5\\oplus \\bar{K}_6$ & 4\n\\end{tabular}}\n\\caption{Double-valued EBRs in SG 183, obtained from the BANDREP application.\\cite{GroupTheoryPaper}}\n\\label{tab:183EBRs}\n\\end{table}\n\n\n\n\n\n\n\n\n\n\nWhen time-reversal symmetry is imposed, we can compute the $\\mathbb{Z}_2$ index.\nLet us first consider the case when SOC is spin-conserving; following Ref~\\onlinecite{NaturePaper}, we refer to this as ``Haldane'' SOC, as opposed to Rashba SOC, which, by our definition, is any SOC term that does not conserve spin.\nAn SOC term that conserves spin will also preserve inversion symmetry (the inversion operator is exactly the tensor product of the spinless $C_{2z}$ operator in Eq~(\\ref{eq:unitarygensxy}) and the identity in spin space).\nWe can compute the $\\mathbb{Z}_2$ index\\cite{Fu2007} from the product of $C_{2z}$ eigenvalues at $\\Gamma$ and $M$, since each band in the spinless model gives rise to a Kramers pair with SOC, whose inversion eigenvalue is the same as the spinless $C_{2z}$ eigenvalue.\nThe eigenvalues of $C_{2z}$ are given in Tables~\\ref{tab:Meigs} and \\ref{tab:Geigs} and the product for each band is tabulated in Table~\\ref{tab:C2zeigs} for all parameter regimes. (Note: there are three inequivalent $M$ points, but they share the same $C_{2z}$ eigenvalues).\nThe ordered lists of $C_{2z}$ eigenvalues in the last column of Table~\\ref{tab:C2zeigs} reflect the particle-hole symmetry of our simplistic model; however, terms that break the particle-hole symmetry without inverting bands at $\\Gamma$ or $M$ will not change the order of eigenvalues.\nThe fact that some particle-hole symmetric arrangements do not appear (namely, $+1,+1,+1,+1$ and $+1, -1, -1, +1$, is a surprising feature of our simple model.)\nSince the product of $C_{2z}$ eigenvalues of the lowest band is always $-1$, whenever SOC opens a gap to the lowest energy band, it is a topological gap.\nIf there is a gap at half-filling, there are two possibilities: if the gap was open before SOC was added, then the parameters $t_{\\sigma,\\pi}$ are constrained by Eqs~(\\ref{eq:pxygap1})--(\\ref{eq:pxygap2}); comparing to Table~\\ref{tab:C2zeigs} reveals that this gap will have a trivial $\\mathbb{Z}_2$ index (consistent with the fact that a spinless model must have a trivial $\\mathbb{Z}_2$ index.) On the other hand, in the parameter regime that violates Eqs~(\\ref{eq:pxygap1})--(\\ref{eq:pxygap2}), the spinless system will be gapless at half-filling; if SOC opens a gap, then Table~\\ref{tab:C2zeigs} shows that the gap has a nontrivial $\\mathbb{Z}_2$ index.\n\nWe now consider non-spin-conserving (Rashba) SOC and examine each of the high-symmetry points.\nFirst, Rashba SOC cannot open a gap at $M$ since each spinless band is non-degenerate (when SOC is turned on, it will become a Kramers pair.)\nSecond, Rashba SOC cannot open a gap at $K$: if two bands are degenerate at $K$ in the spinless model, then they are in the $K_3$ representation, which we showed in the supplement of Ref~\\onlinecite{NaturePaper} (Sec IIID) can only be gapped if the strength of Haldane SOC exceeds that of Rashba SOC.\nThird, we examine the $\\Gamma$ point. In the spinless model, bands come in degenerate pairs, which are described by the $\\Gamma_{5}$ or $\\Gamma_6$ representation, where $C_{2z}$ is represented by $\\pm \\mathbb{I}$.\nWhen we consider spin, $C_{2z}$ is represented by $\\pm \\mathbb{I} \\otimes i s_z$, where $s_z$ is the Pauli matrix describing the spin degrees of freedom.\nA term that breaks spin conservation will not commute with $\\mathbb{I} \\otimes i s_z$.\nHence, no Rashba term can appear at the $\\Gamma$ point because it will break $C_{2z}$ symmetry.\nWe conclude from examining all three high-symmetry points that any gap that is opened by small SOC is adiabatically related to a gap that is opened by spin-conserving SOC and hence the $\\mathbb{Z}_2$ topological index obtained from inversion eigenvalues still holds.\n\n\n\\begin{table}[h]\n\\begin{tabular}{l|c}\nParameter regime & Prod. $C_{2z}$ eigs.\\\\\n\\hline\n$\\begin{aligned}\nt_\\pi& < -t_\\sigma\\\\\nt_\\pi & <3t_\\sigma\\end{aligned} \\Rightarrow \\begin{cases} E_{\\Gamma 2} < E_{\\Gamma 1} \\\\ E_{M3}\\!<\\! E_{M1}\\! <\\! E_{M4}\\! <\\! E_{M2} \\end{cases} $ & $-1,+1,+1,-1$\\\\[10pt]\n\\hline\n$\\begin{aligned}\nt_\\pi &> 3t_\\sigma \\\\\nt_\\pi &< \\frac{1}{3}t_\\sigma \\end{aligned} \\Rightarrow \\begin{cases} E_{\\Gamma 2} < E_{\\Gamma 1} \\\\ E_{M3,M4}<E_{M1,M2}  \\end{cases} $ & $-1,-1,-1,-1$\\\\\n\\hline\n$\\begin{aligned}\nt_\\pi &< -t_\\sigma \\\\\nt_\\pi &> \\frac{1}{3}t_\\sigma\n\\end{aligned} \\Rightarrow \\begin{cases} E_{\\Gamma 2} < E_{\\Gamma 1} \\\\ E_{M4}\\! < \\! E_{M2}\\! < \\! E_{M3}\\! < \\! E_{M1} \\end{cases}$ & $-1,+1,+1,-1$\\\\\n\\hline\n$\\begin{aligned}\nt_\\pi &> -t_\\sigma \\\\\nt_\\pi &> 3t_\\sigma\n\\end{aligned} \\Rightarrow \\begin{cases} E_{\\Gamma 1} < E_{\\Gamma 2} \\\\ E_{M2}\\! < \\! E_{M4}\\! < \\! E_{M1}\\! < \\! E_{M3} \\end{cases}$ & $-1,+1,+1,-1$\\\\[10pt]\n\\hline\n$\\begin{aligned}\nt_\\pi &> \\frac{1}{3}t_\\sigma \\\\\nt_\\pi &< 3t_\\sigma \\end{aligned} \\Rightarrow \\begin{cases} E_{\\Gamma 1} < E_{\\Gamma 2} \\\\  E_{M1,M2}<E_{M3,M4}  \\end{cases} $ & $-1,-1,-1,-1$\\\\\n\\hline\n$\\begin{aligned}\nt_\\pi &> -t_\\sigma \\\\\nt_\\pi &< \\frac{1}{3}t_\\sigma\n\\end{aligned} \\Rightarrow \\begin{cases} E_{\\Gamma 1} < E_{\\Gamma 2} \\\\ E_{M1}\\! < \\! E_{M3}\\! < \\! E_{M2}\\! < \\! E_{M4} \\end{cases}$ & $-1,+1,+1,-1$\n\\end{tabular}\n\\caption{Product of $C_{2z}$ eigenvalues at $\\Gamma$ and $M$ in order of increasing energy, for all possible parameter regimes.\nAccording to Eqs~(\\ref{eq:pxygap1}) and~(\\ref{eq:pxygap2}), the spinless model can only be gapped in the parameter regimes corresponding to the second or fifth row.}\n\\label{tab:C2zeigs}\n\\end{table}\n\n\n\n\n\n\\section{$d_{z^2}$ and $d_{x^2-y^2}$ orbitals in $P4_232$}\n\\label{sec:208symmetries}\n\nSince the lattice of $P4_232$ (SG 208) is primitive cubic, the lattice vectors are $\\mathbf{e}_1=\\hat{\\mathbf{x}}, \\mathbf{e}_2=\\hat{\\mathbf{y}}, \\mathbf{e}_3=\\hat{\\mathbf{z}}$.\nA unit cell is drawn in Fig. 3a} %{fig:208unitcell; we take the origin to be one of the corners of the cube.\nOur model consists of $d_{z^2}$ and $d_{x^2-y^2}$ orbitals sitting at the corners and center of the cube.\nThese orbitals transform as representations of $T$ (generated by $C_{2z}$ and $C_{3,111}$), which is the ``site-symmetry group'' of the origin:\\cite{ITA} that is, $T$ is the largest subset of $P4_232$ that leaves the origin invariant. \nThe $d_{z^2}$ and $d_{x^2-y^2}$ orbitals transform as the one-dimensional irreps, $^1E$ and $^2E$, of $T$.\\cite{Bilbao3}\nSince they are time-reversed partners, they transform as a single irrep under the symmetries of $T$ and time-reversal.\n(We also note that the tensor product of the spin-$\\frac{1}{2}$ representation with $d_{z^2}$ and $d_{x^2-y^2}$ orbitals yields the two-dimensional irreps, $^1F_{3\/2}$ and $^2F_{3\/2}$.\\cite{PointGroupTables})\nInstead of choosing the diagonal set of matrix representatives, we choose a physically motivated basis, where a rotation by an angle $\\theta$ about an axis $\\hat{\\mathbf{n}}$ is represented by $e^{i\\theta \\hat{\\mathbf{n}}\\cdot \\mathbf{S}}$, projected onto the $d_{z^2}$ and $d_{x^2-y^2}$ orbitals.\nThis yields $C_{2z} = C_{2x} = C_{2y} = \\sigma_0$, and\n\\begin{equation}\nC_{3,111} = -\\frac{1}{2}\\mathbb{I} + \\frac{\\sqrt{3}}{2}(i\\sigma_y) = \\begin{pmatrix} -\\frac{1}{2} & \\frac{\\sqrt{3}}{2} \\\\ - \\frac{\\sqrt{3}}{2} & -\\frac{1}{2} \\end{pmatrix}\n\\end{equation}\n\n\nThe screw operation $\\tilde{C}_{2,110}\\equiv \\{ C_{2,110}|\\frac{1}{2}\\frac{1}{2}\\frac{1}{2} \\}$ mixes the two sublattices consisting of sites at the origin and sites at the center, $(\\frac{1}{2},\\frac{1}{2},\\frac{1}{2} )$, of each unit cell.\nThe full representation of the symmetry operations is given by:\n\\begin{align}\nU_{C_{2z}} &= U_{C_{2y}} = \\tau_0 \\otimes \\sigma_0 \\nonumber\\\\\nU_{C_{3,111}}& = \\tau_0\\otimes (-\\frac{1}{2}\\sigma_0 + \\frac{\\sqrt{3}}{2}i\\sigma_y )\\nonumber\\\\\nU_{\\tilde{C}_{2,110},\\mathbf{k}} &= e^{-\\frac{i}{2}(k_x+k_y-k_z)}\\tau_x\\otimes \\sigma_z,\n\\label{eq:208rep}\n\\end{align}\nwhere the $\\tau$ matrices act in the sublattice space.\nTime-reversal is implemented by complex conjugation. Since the representation cannot be reduced without breaking time-reversal symmetry, it is ``physically irreducible.''\\cite{Bilbao1}\n\n\n\n\n\n\n\n\\section{Symmetry constraints on the Wilson loop in $P4_2 32$ (SG 208)}\n\\label{sec:208wilsonsymm}\n\nWe show how the symmetries of $P4_232$ constrain the $z$-directed Wilson loop defined in Eq. (1)}%{eq:wilsoncont ; we will always take the base point $k_{z0}=0$.\nWe will frequently utilize the transformation of the Wilson loop under a non-symmorphic unitary symmetry,\n$\\{ D_g |(\\delta_x,\\delta_y,\\delta_z)\\}$, such that $D_g$ acts in momentum space by $(k_\\perp,k_z)\\rightarrow (D_gk_\\perp,-k_z)$, then\n\\begin{equation}\n\\mathcal{W}_{(D_gk_\\perp,k_{z0})} = e^{2\\pi i \\delta_z}\\tilde{U}_g(k_\\perp,k_{z0}) \\mathcal{W}^\\dagger_{(k_\\perp,-k_{z0})}(\\tilde{U}_g(k_\\perp,k_{z0}))^\\dagger,\n\\label{eq:wilsnonsymm}\n\\end{equation}\nwhere \n\\begin{equation}\n\\left[\\tilde{U}_g(\\mathbf{k})\\right]_{nm}\\equiv  \\langle u^n(D_g\\mathbf{k}) | U_g |u^m(\\mathbf{k})\\rangle\n\\label{eq:utilde}\n\\end{equation}\nThis is a variation of Eq (B19) in Ref~\\onlinecite{ArisCohomology} or Eq (D8) in Ref~\\onlinecite{Wieder17}.\nIf $D_g$ does not invert $k_z$, then there is no dagger on $\\mathcal{W}$ on the right-hand side of Eq~(\\ref{eq:wilsnonsymm}).\n\n\n\n\n\n\\subsection{Wilson eigenvalues along $\\bar{\\Gamma}-\\bar{X}-\\bar{M}-\\bar{Y}-\\bar{\\Gamma}$}\n\\label{sec:208wilsongapped}\n\n\nLet us first consider the eigenvalues of the Wilson matrix at $\\bar{\\Gamma}$.\nEnforcing $C_{2x}$ symmetry, (recall, $U_{C_{2x}}=\\mathbb{I}_{4\\times 4}$ from Eq~(\\ref{eq:208rep})), Eq~(\\ref{eq:utilde}) shows that $\\tilde{U}_{C_{2x}}(\\Gamma) = \\mathbb{I}_{2\\times 2}$; then Eq~(\\ref{eq:wilsnonsymm}) yields $\\mathcal{W}_{(\\bar{\\Gamma},0)} = \\mathcal{W}^\\dagger_{(\\bar{\\Gamma},0)}$.\nThis forces the eigenvalues of $\\mathcal{W}_{(\\bar{\\Gamma},0)}$ to be real.\nEnforcing $\\tilde{C}_{2,110}$ symmetry (which has $\\delta_z = \\frac{1}{2}$), Eq~(\\ref{eq:wilsnonsymm}) yields $\\mathcal{W}_{(\\bar{\\Gamma},0)}=-\\tilde{U}_{\\tilde{C}_{2,110}}\\mathcal{W}_{(\\bar{\\Gamma},0)}^\\dagger \\tilde{U}_{\\tilde{C}_{2,110}}^\\dagger$, which shows that the eigenvalues of $\\mathcal{W}_{(\\bar{\\Gamma},0)}$ must be equal to $+1$ and $-1$.\n\nAlong the segment $\\bar{\\Gamma}-\\bar{X}$, even without knowing $\\tilde{U}_{C_{2x}}(\\mathbf{k})$, we can utilize Eq~(\\ref{eq:wilsnonsymm}) to deduce that the eigenvalues of $\\mathcal{W}_{(k_x,0,0)}$ are equal to those of $\\mathcal{W}_{(k_x,0,0)}^\\dagger$. Consequently, the eigenvalues of $\\mathcal{W}_{(k_x,0,0)}$ must either come in complex conjugate pairs or be real. However, since the eigenvalues of $\\mathcal{W}_{(0,0,0)}$ are $+1$ and $-1$, the eigenvalues of $\\mathcal{W}_{(k_x,0,0)}$ must also be fixed to $+1$ and $-1$ along the entire line, since there is no way for them to smoothly vary while satisfying the constraints of $C_{2x}$.\n\nApplying the same logic along $\\bar{X}-\\bar{M}$ and $\\bar{Y}-\\bar{\\Gamma}$ with $C_{2y}$ symmetry shows that the eigenvalues of $\\mathcal{W}_{(\\pi,k_y,0)}$ and $\\mathcal{W}_{(0,k_y,0)}$ must also be pinned to $+1$ and $-1$.\nFinally, applying $C_{2x}$ to $\\bar{M}-\\bar{Y}$ to the eigenvalues of $\\mathcal{W}_{(k_x,\\pi,0)}$, we deduce that the eigenvalues of the $z$-directed Wilson matrix are equal to $+1$ and $-1$ along the entire loop $\\bar{\\Gamma}-\\bar{X}-\\bar{M}-\\bar{Y}-\\bar{\\Gamma}$.\n\n\n\n\\subsection{Protected band crossing along $\\bar{\\Gamma}-\\bar{M}$}\n\\label{sec:208wilsoncrossing}\n\nApplying Eq~(\\ref{eq:wilsnonsymm}) with $\\tilde{C}_{2,110}$ symmetry to the line $\\bar{\\Gamma}-\\bar{M}$ shows that the eigenvalues of $\\mathcal{W}_{(k,k,0)}$ are the same as the eigenvalues of $-\\mathcal{W}_{(k,k,0)}^\\dagger $; \nhence, the eigenvalues of $\\mathcal{W}_{(k,k,0)}$ are either pure imaginary or come in pairs \n\\begin{equation}\n\\lambda(k),-\\lambda(k)^*\n\\label{eq:Wkk0eigs}\n\\end{equation}\nWe showed in Sec~\\ref{sec:208wilsongapped} that the eigenvalues of $\\mathcal{W}_{(0,0,0)}$ (and $\\mathcal{W}_{(\\pi,\\pi,0)}$) are $+1$ and $-1$; this rules out the first possibility and hence the eigenvalues of $\\mathcal{W}_{(k,k,0)}$ come in pairs $(\\lambda(k), -\\lambda(k)^*)$, which are degenerate when $\\lambda(k) = \\pm i$.\nWe now show that such a degeneracy can occur without any fine-tuning and that the parity of the number of degeneracies between $k=0$ and $k=\\pi$ constitutes a topological invariant.\n\n\n\n\nTo see this, we rely on an anti-unitary symmetry of the Hamiltonian:\n$\\mathcal{A} \\equiv \\mathcal{T}\\tilde{C}_{2,110}^{-1}C_{2z}= \\mathcal{T}\\{C_{2,1\\bar{1}0}|\\bar{\\frac{1}{2}}\\bar{\\frac{1}{2}}\\frac{1}{2}\\}$, satisfying $\\mathcal{A}^2 = 1$.\nSince $\\mathcal{A}$ leaves $(k,k,k_z)$ invariant, the (antiunitary) generalization of Eq~(\\ref{eq:wilsnonsymm}) is:\n\\begin{equation}\n\\mathcal{W}_{(k,k,k_{z0})} = -\\tilde{U}_A(k,k,k_{z0})K \\mathcal{W}_{(k,k,k_{z0})}K(\\tilde{U}_A(k,k,k_{z0}))^\\dagger,\n\\label{eq:Wdiag}\n\\end{equation}\nwhere the minus sign comes from the fact that $\\mathcal{A}$ includes a $\\frac{1}{2}$ translation in the $\\hat{z}$ direction ($e^{2\\pi i \\delta_z}=-1$), $K$ is the complex conjugation operator, and \n\\begin{equation}\n\\left[\\tilde{U}_A(\\mathbf{k})\\right]_{nm}\\equiv  \\langle u^n(\\mathbf{k}) | \\mathcal{A} |u^m(\\mathbf{k})\\rangle\n\\label{eq:atilde}\n\\end{equation}\nNotice that $(\\tilde{U}_AK)^2=\\mathbb{I}_{2\\times 2}$, from which it follows that $\\tilde{U}_A(\\mathbf{k}) = e^{ib_0(\\mathbf{k}) + ib(\\mathbf{k})(\\cos\\theta(\\mathbf{k}) \\sigma_x + \\sin\\theta(\\mathbf{k}) \\sigma_z)}$. Importantly, $\\sigma_y$ does not appear in the exponential. Consequently $\\tilde{U}_A$ is diagonalized by $\\tilde{U}_A(\\mathbf{k}) = e^{ib_0(\\mathbf{k})}O(\\mathbf{k})D(\\mathbf{k})O(\\mathbf{k})^T$, where $O(\\mathbf{k})$ is a real orthogonal matrix and $D(\\mathbf{k}) =  {\\rm Diag}[e^{-ib(\\mathbf{k})}, e^{ib(\\mathbf{k})} ]$.\nDefining \n\\begin{equation}\nW_k = O(k,k,0)^T\\mathcal{W}_{(k,k,0)}O(k,k,0),\n\\end{equation}\nEq~(\\ref{eq:Wdiag}) yields \n\\begin{equation}\nW_k = -D(k,k,0)W_k^* D(k,k,0)^*\n\\label{eq:wconstraint}\n\\end{equation}\n\nSince $W_k$ has the same eigenvalues as $\\mathcal{W}_{(k,k,0)}$, whose eigenvalues must come in pairs given by Eq~(\\ref{eq:Wkk0eigs}), $W_k$ can be written as\n\\begin{equation}\nW_k = ie^{i\\mathbf{a}(k)\\cdot\\sigma} \n\\end{equation}\nfor a smooth vector function $\\mathbf{a}(k) = (a_x(k),a_y(k),a_z(k))$.\nEq~(\\ref{eq:wconstraint}) then places the following constraints at each $\\mathbf{k}$: \n\\begin{align}\na_z\\sin |\\mathbf{a}| &= 0 = (a_y\\sin b + a_x\\cos b)\\cos|\\mathbf{a}|\n\\end{align}\n\nSince $\\mathbf{a}(k)$ is a smoothly varying function, there are two possibilities: either $\\cos |\\mathbf{a}(k)|=0$ for all $k$ (in which case the eigenvalues of $W_k$ are fixed to $\\pm 1$) or $\\sin |\\mathbf{a}(k)|\\neq 0$, $\\cos |\\mathbf{a}(k)|\\neq 0$ except at isolated points, in which case, $a_x(k) \\propto a_y(k), a_z(k)=0$.\n(We rule out the case where $\\sin |\\mathbf{a}(k)|=0$ for all $k$ because it is inconsistent with the fact that the eigenvalues of $W_{k=0}$ are equal to $\\pm 1$.)\nThe condition $a_x(k) \\propto a_y(k), a_z(k)=0$ means that degeneracies in the spectrum of $W_k$ (and hence $\\mathcal{W}_{(k,k,0)}$) occur when $a_x(k)=0$, which forces $a_y(k)=0$.\nIf such a degeneracy is present and linear in $k$, then it is not fine-tuned, in the sense that smoothly deforming $a_x(k)$ will move the degeneracy, but not remove it; such degeneracies can only be removed pairwise.\n\n\nSince at both $\\bar{\\Gamma}$ and $\\bar{M}$, the eigenvalues of $W_k$ are fixed to $+1$ and $-1$, the parity of the number of linear crossings is a topological invariant, that cannot be changed without closing the gap in the bulk band spectrum.\nReturning to the possibility that $\\cos |\\mathbf{a}(k)|=0$ for all $k$: since in this case the eigenvalues of $W_k$ are never degenerate, it trivially follows that the parity of linear in $k$ band crossings cannot be changed without closing the bulk gap.\n\n\\subsection{Winding of the bent Wilson loop}\n\\label{sec:208bentwinding}\n\nThe product of $C_{2x}$ and $\\tilde{C}_{2,110}$ yields the screw symmetry, $\\tilde{C}_4 \\equiv \\{C_{4z} | \\frac{1}{2}\\frac{1}{2}\\frac{1}{2} \\}$.\nApplying Eq~(\\ref{eq:wilsnonsymm}) with $\\delta_z = \\frac{1}{2}$, and removing the dagger on the righthand side of Eq~(\\ref{eq:wilsnonsymm}) because $\\tilde{C}_4$ leaves $k_z$ invariant, requires that the eigenvalues of $\\mathcal{W}_{(k,k,k_{z0})}$ are exactly opposite those of $\\mathcal{W}_{(-k,k,k_{z0})}$.\n\n\nIf there is an odd number of linear band crossings in the spectrum of $\\mathcal{W}_{(k,k,k_{z0})}$ for $0\\leq k\\leq \\pi$, then one band must have eigenvalue $e^{i\\varphi(k)}$, where $\\varphi(0) = 0$ and $\\varphi(\\pi) = \\pi$.\nThe eigenvalue of the other band is given by $e^{i\\pi - i\\varphi(k)}$, according to Eq~(\\ref{eq:Wdiag}).\nThen $\\tilde{C}_4$ requires that one band of $\\mathcal{W}_{(-k,k,k_{z0})}$ has eigenvalue $-e^{i\\pi - i\\varphi(k)} = e^{-i\\varphi(k)}$; we use this band to define $\\varphi(k)$ when $-\\pi < k <0$, i.e., $\\varphi(-k) = -\\varphi(k)$. Thus, if we plot $\\varphi(k)$ from $-\\pi < 0 < \\pi$, it ``winds'' from $-\\pi $ to $\\pi$.\nThis is exactly what is shown in Fig. 3b} %{fig:208wilsonz .\nApplying the same logic to the other band shows that it winds in the opposite direction.\n\n\n\n\\section{Wilson-of-Wilson loop}\n\\label{sec:wilsofwils}\n\nLet $C$ be the closed path in the surface Brillouin zone that traverses $\\bar{\\Gamma}-\\bar{X}-\\bar{M}-\\bar{Y}-\\bar{\\Gamma}$ and, for each $k_\\perp \\in C$, let $|v_{1,2}(k_\\perp)\\rangle$ by the eigenstates of $\\mathcal{W}_{(k_\\perp,0)}$ with energies $-1$ and $+1$, respectively.\nWe define the Berry phase of the Wilson loop (the ``Wilson-of-Wilson'' loop) by $w = e^{i\\oint_Cdk_\\perp a(k_\\perp)}$, where $a(k_\\perp) =  i\\langle v_1(k_\\perp)| \\partial_{k_\\perp} |v_1(k_\\perp)\\rangle $.\n\nWe show that the symmetry $C_{2z}\\mathcal{T}$ requires $w = \\pm 1$.\nIn analogy to Eq~(\\ref{eq:Wdiag}), the Wilson matrix satisfies,\n\\begin{equation}\n\\mathcal{W}_{(k_\\perp,0)}=\\tilde{U}_{C_{2z}\\mathcal{T}}(k_\\perp,0) K\\mathcal{W}_{(k_\\perp,0)}^\\dagger K\\tilde{U}_{C_{2z}\\mathcal{T}}(k_\\perp,0)^\\dagger,\n\\end{equation}\nwhere,\n\\begin{equation}\n\\left[ \\tilde{U}_{C_{2z}\\mathcal{T}} (k_\\perp,k_z) \\right]_{nm}\\equiv \\langle u^n(k_\\perp,-k_z)| C_{2z}\\mathcal{T} | u^m(k_\\perp,k_z)\\rangle\n\\end{equation}\nWhen $k_\\perp \\in C$, $\\mathcal{W}_{(k_\\perp,0)}$ is Hermitian, as we showed in Sec~\\ref{sec:208wilsongapped}.\nHence, $\\tilde{U}_{C_{2z}\\mathcal{T}}(k_\\perp,0)K$ is an anti-unitary symmetry that commutes with $\\mathcal{W}_{(k_\\perp,0)}$ and hence does not mix the two Wilson bands, which are gapped with eigenvalues $\\pm 1$ along $C$.\nThus, Eq~(\\ref{eq:Wdiag}) can be applied with $\\mathcal{W}$ replaced by $w$:\n\\begin{equation}\nw=e^{i\\phi(\\bar{\\Gamma})}KwKe^{-i\\phi(\\bar{\\Gamma})}\n\\label{eq:wofw} \n\\end{equation}\nand\n\\begin{equation}\ne^{i\\phi(k_\\perp)} = \\langle v_1(k_\\perp) | \\tilde{U}_{C_{2z}\\mathcal{T}}(k_\\perp)  | v_1(k_\\perp)\\rangle,\n\\end{equation}\nEq~(\\ref{eq:wofw}) shows that $w$ is real and equal to $\\pm 1$.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nDynamics of quantum many-body systems often result in thermalized states\ncharacterized, for example, by the Fermi distribution function \\cit\n{BookGemmer}. Exceptional cases have been discussed for integrable systems,\nwhere an isolated system may be left in a non-equilibrium steady state \\cit\n{PolkovnikovRMP2011}. This intriguing aspect has been studied in isolated\nsystems like one-dimensional (1D) cold atomic chains, where temporal\nevolution of the system can be measured after a sudden change of the\ninteraction (quantum quench) \\cite{KinoshitaNature2006,GringScience2012}.\nComplementary experiments in transport measurement would allow us to study\nspatial evolution of electronic states travelling from a non-interaction\nregion to an interacting region (spatial analog of quantum quench). The\nTomonaga-Luttinger (TL) liquid, which has been identified in various 1D\nwires and quantum-Hall edge channels \\cit\n{BookEzawa,ChangRMB,SteinbergNatPhys2008,BarakNatPhys2010,BlumensteinNatPhys2011\n, is known as an example of integrable systems. Theoretically the Coulomb\ninteraction between electrons is absorbed under bozonization into\nnon-interacting plasmon modes of collective density excitations \\cit\n{TomonagaPTP1950,LuttingerJMP1963,GiamarchiBook}. Therefore, an ideal TL\nliquid never thermalizes as plasmon excitations are conserved during the\ntransport \\cite{CazalillaPRL206,Iucci2009,KennesPRL2013}. Quantum-Hall edge\nchannels are suitable for demonstrating this non-thermalizing behavior, as\nthey are well isolated from the environment. Actually, coupling to the\nphonon bath is sufficiently weak \\cite{JezouinScience2013}, and\nbackscattering that is unwanted for the TL model is highly forbidden \\cit\n{ButtikerPRB1998,vanWeesPRB1989}. Moreover, tunneling spectroscopy with a\nquantum dot (QD) allows us to investigate spatial evolution of the\nelectronic spectrum in a tailored geometry \\cite{AltimirasNatPhys2010}.\nPrevious experiments have identified electronic relaxation from\nsingle-particle excitations to collective excitations (plasmons) in terms of\nplasmon boundary scattering known as spin-charge separation\\ \\cit\n{leSueurPRL2010,BocquillonNatComm2013,InouePRL2014,FreulonNatComm2015} and\nfractionalization \\cite{KamataNatNano2014,ProkudinaPRL2014} in the TL\nphysics. However, the absence of thermalization as a hallmark of an\nintegrable system has not been addressed, as the resulting states were\nalways close to thermalized states showing a trivial Fermi distribution \\cit\n{leSueurPRL2010,DegiovanniPRB2010,KovrizhinPRL2012}.\n\nIn this paper we show that quantum-Hall edge channels can actually support\nlong-lived non-equilibrium states. For this purpose, non-trivial binary\nspectrum composed of hot and cold carriers is prepared by an indirect\nheating scheme using weakly coupled counterpropagating edge channels.\nQuantum dot (QD) spectroscopy clearly reveals that the carriers with the\nnon-trivial binary spectrum propagate over a long distance, much longer than\nthe length required for electronic relaxation, without thermalization into a\ntrivial Fermi distribution. This non-thermalizing characteristics is\nconsistent with the TL model and encourages us to study non-equilibrium\ncoherent plasmon transport in the system.\n\nThe paper is organized in the following way. After describing an artificial\nTL liquid formed in quantum-Hall edge channels (Sec. IIA), we propose a\nnovel excitation scheme to obtain non-trivial binary spectrum (Sec. IIB).\nThe binary spectrum is obtained from experiments and simulations. For\nexperiments, we describe the device structure and measurement scheme (Sec.\nIIIA), and compare energy spectra in two regions in distinct geometries\n(Secs. IIIB, IIIC, and IIID) by considering the heat flow in the system\n(Sec. IIIE). The binary spectrum is found in one of the regions, where small\namplitudes of high-frequency plasmons are excited. For simulations, we\nderive plasmon eigenmodes of the system (Sec. IVA), and show that an\napproximate binary spectrum appears as a non-equilibrium steady state in the\nquantum-quench simulations (Sec. IVB). Finally, we discuss the coupling\nstrength (Sec VA), the similarities and differences between the experiments\nand simulations (Sec. VB), some conserved quantities during the transport\n(Sec. VC), and possible relaxation mechanisms in the quantum-Hall system\n(Sec. VD) before summarizing the work.\n\n\\section{Artificial TL liquid}\n\n\\subsection{Plasmon excitation in edge channels}\n\nFigure 1(a) illustrates the interacting quantum-Hall edge channels we study,\nwhere right ($\\eta =$ r) and left ($\\eta =$ $\\ell $) moving electronic\nchannels with spin $\\sigma \\in \\left\\{ \\uparrow ,\\downarrow \\right\\} $,\nlabeled $\\left( \\eta ,\\sigma \\right) $, are interacting. Such a geometry can\nbe prepared by depleting a narrow central region of a two-dimensional\nelectron system (2DES) in a magnetic field at the filling factor $\\nu $ = 2 \n\\cite{KamataNatNano2014}. In the absence of tunneling and the presence of\nelectrostatic interaction between the channels, the system mimics an ideal\n1D wire described by the standard spin-full TL model \\cit\n{GutmanPRL2008,GutmanPRB2009,BergPRL2009,HashisakaPRB2013}. The Coulomb\ninteraction yields four plasmon eigenmodes, labeled $\\left[ d,m\\right] $,\ni.e., left ($d=$ L) and right ($d=$ R) moving charge ($m=$ C) and spin ($m=$\nS) modes, as will be derived in Sec. IVA. The charge and spin modes have\nsymmetric and anti-symmetric charge distributions, respectively, in the\nprimary channels (spin-charge separation), where the charge velocity $v_\n\\mathrm{C}}$ is generally greater than the spin velocity $v_{\\mathrm{S}}$.\nThe interaction between counterpropagating channels generates small amount\nof dragged charges reverse-travelling in the subsidiary channels\\ (charge\nfractionalization), as illustrated by coupled wave packets in Fig. 1(a) \\cit\n{SafiPRB1995,KamataNatNano2014,ImuraPRB2002}.\n\n\\begin{figure}[tbp]\n\\begin{center}\n\\includegraphics[width = 3.2in]{B1.eps}\n\\end{center}\n\\caption{ (a) Schematic diagram of interacting quantum Hall edge channels \n\\left( \\mathrm{r}\/\\mathrm{\\ell },\\uparrow \/\\downarrow \\right) $\\ with\nnearest neighbor coupling capacitances $C_{\\mathrm{X}}$\\ and $C_{\\mathrm{Z}}\n. Charge wave packets for plasmon eigenmodes $\\left[ \\mathrm{L}\/\\mathrm{R}\n\\mathrm{C}\/\\mathrm{S}\\right] $\\ are illustrated. (b) The indirect heating\nscheme for interacting quantum-Hall edge channels $\\left( \\mathrm{r}\/\\mathrm\n\\ell },\\uparrow \/\\downarrow \\right) $. Each time a single-electron charge\nwave packet is injected from a PC to $\\left( \\mathrm{\\ell },\\uparrow \\right) \n$ as shown by `inj', four plasmon charge wave packets in the eigenmodes \n\\left[ \\mathrm{L}\/\\mathrm{R},\\mathrm{C}\/\\mathrm{S}\\right] $\\ are generated.\nCarriers are drained to ohmic contacts marked by $\\boxtimes $. The energy\ndistribution function $f\\left( E\\right) $ shown at the bottom insets\nexhibits a double step structure for the initial state (the central inset),\nclose to Fermi distribution at $T_{\\mathrm{H}}$ in the DH region (the left\ninset), and an unusual binary spectrum with two temperatures $T_{\\mathrm{H}}$\nand $T_{\\mathrm{base}}$ in the IH region (the right inset).}\n\\end{figure}\n\nIn a practical device, such interacting channels can be formed in a finite\nregion, outside of which the left and right moving channels are spatially\nseparated as shown in Fig. 1(b). Suppose that a non-equilibrium charge is\ninjected into one of the channels, $\\left( \\ell ,\\uparrow \\right) $\n[illustrated as an arrow marked by `inj'] from a quantum point contact (PC).\nThe initial electronic state in $\\left( \\ell ,\\uparrow \\right) $ exhibits a\ndouble-step profile in the energy distribution function $f\\left( E\\right) $,\nwhere the width and height are given by the excitation energy $eV_{\\mathrm{E\n}$ and the tunneling probability $D$, respectively, of the PC, as shown in\nthe central inset at the bottom of Fig. 1(b). However, the interaction\nalters the electronic spectrum significantly during the transport \\cit\n{leSueurPRL2010}.\n\nIn the plasmon picture, charge wave packets randomly injected from the PC\ncan propagate in a non-interacting manner. Each time a single electron is\ninjected into $\\left( \\ell ,\\uparrow \\right) $, an initial wave packet,\nshown just on the left side of the injection point, is generated. It can be\ndescribed as a linear superposition of the four plasmon eigenmodes. Since\nthe charge and spin modes have different velocities, the injected charge\nsplits into four plasmon packets as illustrated. We will refer to this\nsplitting as electronic relaxation in the sense that a single-particle\nexcitation in one channel is relaxed into collective charge excitations over\nthe channels. This takes place within a length $l_{\\mathrm{el}}=\\hbar \\left(\nv_{\\mathrm{C}}-v_{\\mathrm{S}}\\right) \/eV_{\\mathrm{E}}$, which is\napproximately 100 nm at the excitation voltage $V_{\\mathrm{E}}$ of 1 mV for\ntypical velocities $v_{\\mathrm{C}}=2-5\\times 10^{5}$ m\/s and $v_{\\mathrm{S\n}\\sim v_{\\mathrm{C}}\/2$ at $\\nu $ = 2 \\cit\n{BocquillonNatComm2013,KamataPRB2010,KumadaPRB2011}, from the injection\npoint. When random charge packets are successively injected from a biased\nPC, channels are filled with incoherent ensemble of plasmon wave packets. A\nfast wave packet in the charge mode can catch up with a slow packet in the\nspin mode, and overtake it without any scattering as the plasmons are\nnon-interacting. Therefore, the excitation of non-interacting plasmons\ncannot be thermalized \\cit\n{CazalillaPRL206,KovrizhinPRB2011,LevkivskyiPRB2012}.\n\n\\subsection{Non-trivial binary spectrum}\n\nWe next show that non-trivial binary spectrum can be generated in the\nexcitation scheme described in the previous subsection. To see this, it is\nimportant to note that electron-hole excitations (and thus the energy\nspectra) in all channels are correlated with each other as they all\noriginate from the common injection process at the PC. Most of the generated\nwave packets flow to the downstream of the injection point, which we refer\nto as `direct-heating (DH) region' [the red region in Fig. 1(b)]. Previous\nstudies have shown that the spectrum in this region can be described by an\napproximate Fermi distribution at temperature $T_{\\mathrm{H}}$, as\nschematically shown in the left inset \\cite{leSueurPRL2010}. This might not\ncontradict the non-thermalizing nature of TL model, as the theory have shown\nthat the electronic relaxation results in an approximate Fermi distribution\nfor this case \\cite{DegiovanniPRB2010,KovrizhinPRL2012}. However, the\nexperiment on the DH region was not successful in identifying the presence\nof long-lived non-equilibrium states.\n\nHere, we focus on the spectrum in the counterpropagating channels, which we\nrefer to as `indirect-heating (IH) region' (the blue region). Because of the\nsmall amplitude of the correlated excitations, the spectrum in the IH region\nshould have a small fraction of the spectrum in the DH region (at $T_\n\\mathrm{H}}$). This fraction $p$ ($\\ll 1$) is determined by the interaction,\nor the plasmon eigenmodes of the system [See Sec. IV]. The majority of the\nspectrum should reflect the base temperature $T_{\\mathrm{base}}$ of the\nsystem. This leads to a non-trivial binary spectrum in the IH regions, as\nshown in the right inset. The focus of this study is the stability of this\nunusual mixture during transport.\n\nThis experimental scheme can be understood as a spatial analog of quantum\nquench. This analogy is allowed because the plasmon transport is\nunidirectional from the injection point through the channels to the ohmic\ncontacts at the ends of the channels, as is the flow of time in quantum\nquench (See discussions in Sec. VB). The initial state prepared as single\nelectron excitations relaxes into plasmon excitations in the DH and IH\nregions. Such spatial evolution can be studied by using PC injectors at\nvarious distances from the QD spectrometer \\cite{leSueurPRL2010}. If the\nergodicity is assumed, a many-body state may relax to a thermal-equilibrium\nstate after travelling a long distance. However, `non-equilibrium steady\nstate', which can be defined as a state that remains non-equilibrium even\nafter travelling a sufficiently long (conceptually infinite) distance, may\nemerge in an integrable model. The comparison between the experiment in Sec.\nIII and the simulation in Sec. IV implies that the binary spectrum is the\nsignature of `non-equilibrium steady state' expected in the integrable model\nof TL liquid.\n\n\\section{Experiment}\n\n\\subsection{Sample and measurement technique}\n\n\\begin{figure}[tbp]\n\\begin{center}\n\\includegraphics[width = 3.375in]{B2.eps}\n\\end{center}\n\\caption{ (a) The measurement setup with three magnified\nscanning-electron-micrographs with false color. The QD spectrometer and one\nof the heat injectors (PC1 - PC4) are activated with the surface gates (SG$_\n\\mathrm{iso}}$, SG$_{\\mathrm{1-4}}$, etc.), while inactivated PCs are fully\nopened. The double lines indicate edge channels that form when PC1 and PC3\nare activated, while only one of the PCs are activated in the actual\nmeasurements. The outer and inner channels serve spin-up and -down\ntransport, respectively. The QD with a few electrons shows a charging energy\nof about 2 meV. Terminals labeled `fl' are floating ohmic contacts. (b) A\nschematic energy diagram of the transport through a QD level $\\protec\n\\varepsilon $. Electronic spectra $f_{\\mathrm{a},\\uparrow }\\left( E\\right) \n\\ and $f_{\\mathrm{b},\\uparrow }\\left( E\\right) $\\ can be determined from the\ncurrent $I\\left( \\protect\\varepsilon \\right) $. (c) Current spectra of the\nQD for various effective bias voltages $V_{\\mathrm{D}}-V_{\\mathrm{E1}}$\\\ntaken with all PCs (including PC1) fully opened ($G_{i}\/G_{\\mathrm{q}}=2$).\nDashed lines indicate the energy alignment of the ground state with the\nchemical potential of the channels ($\\protect\\varepsilon =\\protect\\mu _\n\\mathrm{a}\/\\mathrm{b},\\uparrow }$), while dotted lines indicate that of the\nexcited states. Each trace is offset for clarity. }\n\\end{figure}\n\nThe measurements were performed with an AlGaAs\/GaAs modulation-doped\nheterostructure in a perpendicular magnetic field of 5.9 T ($\\nu $ = 2) at\nthe lattice temperature of 20 - 100 mK. The unprocessed 2DES has the\nelectron density 2.9$\\times $10$^{11}$ cm$^{-2}$ and the zero-field mobility\n1.6$\\times $10$^{6}$ cm$^{2}$\/Vs. The device shown in Fig. 2(a) involves\nsplit gates SG's to define the channels, PC charge injectors and a QD\nspectrometer. The quantized Hall resistance at $h\/2e^{2}$\\ and the vanishing\nlongitudinal resistance of the device ensure the formation of two chiral\nedge channels dominating the transport.\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[width = 6.5in]{B3.eps}\n\\end{center}\n\\caption{ (a) A schematic channel layout for measuring the spectrum in the\nDH region formed downstream of PC1 in $\\left( \\mathrm{a},\\uparrow\n\/\\downarrow \\right) $. (b) Schematic channel geometries near PC1 for (i) \nG_{1}$\\ = 0, (ii) $G_{1}=0.5G_{\\mathrm{q}}$, (iii) $G_{1}=G_{\\mathrm{q}}$,\n(iv) $G_{1}=1.5G_{\\mathrm{q}}$, and (v) $G_{1}=2G_{\\mathrm{q}}$. Channels \n\\left( \\mathrm{a},\\uparrow \\right) $\\ and $\\left( \\mathrm{a},\\downarrow\n\\right) $\\ can be heated in (ii) and (iv), respectively. (c) $V_{\\mathrm{PC1\n}$\\ dependence of $G_{1}\/G_{\\mathrm{q}}$, measured by the current $I_\n\\mathrm{E1}}$\\ at the excitation voltage $V_{\\mathrm{E1}}=$\\ -100 $\\protec\n\\mu $V. (d) Current spectra with a flat-top region of the width associated\nwith the effective bias. Measured current (circles) is fitted well with the\nFermi distribution function (solid lines). The broadened edges measure the\nelectron temperatures $T_{\\mathrm{a\/b},\\uparrow }$\\ in the channels. The $I_\n\\mathrm{D}}$\\ and $V_{\\mathrm{C}}$\\ scales apply to the trace (v), while\nother traces are offset for clarity. (e) $k_{\\mathrm{B}}T_{\\mathrm{a\/b\n,\\uparrow }$\\ as a function of $V_{\\mathrm{PC1}}$. $T_{\\mathrm{H}}^{\\left(\n0.1\\right) }$\\ is the expected temperature considering the coupling $p$ =\n0.1. Small difference between $T_{\\mathrm{a},\\uparrow }$\\ and $T_{\\mathrm{b\n,\\uparrow }$\\ at unheated conditions (iii) and (v) might come from external\nnoise in the ammeter for $I_{\\mathrm{E}1\\text{.}}$ }\n\\end{figure*}\n\nAs shown in Fig. 2(a), three sections of edge channels $\\left( \\eta ,\\sigma\n\\right) $, where $\\eta $ specifies the region `a', `b', and `c', are defined\nby using the isolation gate SG$_{\\mathrm{iso}}$ (width $w=$ 0.1 $\\mu $m) and\na QD spectrometer inserted between the section `a' and `b'. Spin-full TL\nliquids labelled TLLs are formed in the interacting regions of $\\left( \n\\mathrm{a},\\sigma \\right) $ and $\\left( \\mathrm{c},\\sigma \\right) $ for the\nlength of 5 $\\mu $m, as well as $\\left( \\mathrm{b},\\sigma \\right) $ and \n\\left( \\mathrm{c},\\sigma \\right) $ of 10 $\\mu $m. We investigate how the\nspectra of $\\left( \\mathrm{a},\\uparrow \\right) $ and $\\left( \\mathrm{b\n,\\uparrow \\right) $ in the vicinity of the QD change when heat current is\ninjected at different locations with respect to the spectrometer. This is\ndone by selectively activating one of the four PCs; that is, injection from\nPC1 (PC2 - PC4) implies that the spectrometer is placed in the DH (IH)\nregion(s). The tunneling probability $D_{i,\\sigma }$ of $i$-th PC for spin \n\\sigma $ can be estimated from its conductance $G_{i}=G_{\\mathrm{q}}\\left(\nD_{i,\\uparrow }+D_{i,\\downarrow }\\right) $ with $G_{\\mathrm{q}}=\\frac{e^{2}}\nh}$, where $G_{1}=I_{\\mathrm{E1}}\/V_{\\mathrm{E1}}$\\ for PC1 and $G_{i}=I_\n\\mathrm{E2}}\/V_{\\mathrm{E2}}$\\ for PC$i$\\ ($i=$\\ 2, 3, and 4) are obtained\nfrom the current $I_{\\mathrm{E1\/2}}$\\ - voltage $V_{\\mathrm{E1\/2}}$\\\ncharacteristics.\n\nFigure 2(c) shows current spectrum with varying effective QD bias $V_\n\\mathrm{D}}-V_{\\mathrm{E1}}$ between the drain and the emitter [See Fig.\n2(a) for their locations], taken without heating (PC1 fully opend and PC2 -\nPC4 fully closed). Current steps associated with the transport through the\nground and excited levels are shown by dashed and dotted lines,\nrespectively, indicating the level spacing of about 200 $\\mu $eV. In the\nfollowing experiments, we keep the effective bias voltage below 200 $\\mu $V\nto probe only the transport through the ground level $\\varepsilon $ as shown\nin Fig. 2(b).\n\nWhen $\\varepsilon $ is swept with the gate voltage $V_{\\mathrm{C}}$, the QD\ncurrent measures the electronic spectrum $f_{\\mathrm{a}\/\\mathrm{b},\\uparrow\n}\\left( E\\right) $ of channel $\\left( \\mathrm{a}\/\\mathrm{b},\\uparrow \\right) \n$ via the relation \n\\begin{equation}\nI\\left( \\varepsilon \\right) =I_{0}\\left\\{ f_{\\mathrm{a},\\uparrow }\\left(\n\\varepsilon \\right) \\left[ 1-f_{\\mathrm{b},\\uparrow }\\left( \\varepsilon\n\\right) \\right] -f_{\\mathrm{b},\\uparrow }\\left( \\varepsilon \\right) \\left[\n1-f_{\\mathrm{a},\\uparrow }\\left( \\varepsilon \\right) \\right] \\right\\} ,\n\\label{EqFit}\n\\end{equation\nwhere $I_{0}$ is the saturated current on the step.\\ For a large positive\nbias, the second term of Eq. \\ref{EqFit} describing the negative component\nof the current can be neglected. Then, the current profiles around the\nonsets $\\varepsilon =\\mu _{\\mathrm{a},\\uparrow }$ (the left edge of the peak\nat a positive bias) and $\\mu _{\\mathrm{b},\\uparrow }$\\ (the right edge)\nmeasure the spectra $f_{\\mathrm{a},\\uparrow }\\left( E\\right) $\\ and $1-f_\n\\mathrm{b},\\uparrow }\\left( E\\right) $, respectively. Therefore, electronic\nspectra $f_{\\mathrm{a},\\uparrow }\\left( E\\right) $\\ and $f_{\\mathrm{b\n,\\uparrow }\\left( E\\right) $\\ can be evaluated from a single current trace \nI_{\\mathrm{D}}\\left( V_{\\mathrm{C}}\\right) $. The unheated spectrum in Fig.\n2(c) can be fitted well by using the Fermi distribution \n\\begin{equation}\nf_{\\mathrm{F}}\\left( E;T\\right) =\\left[ e^{\\left( E-\\mu \\right) \/k_{\\mathrm{\n}}T}+1\\right] ^{-1}  \\label{EqFermi}\n\\end{equation\nwith the thermal energy $k_{\\mathrm{B}}T_{\\mathrm{a}\/\\mathrm{b},\\uparrow }$\nand chemical potential $\\mu _{\\mathrm{a}\/\\mathrm{b},\\uparrow }$. The sharp\nonsets on both sides of the peak indicate the thermal energy $k_{\\mathrm{B\n}T_{\\mathrm{base}}=$ 9 - 10 $\\mu $eV at the base temperature $T_{\\mathrm{bas\n}}$ in both channels. The QD spectroscopy relies on transport through a\nsingle level. Therefore we restrict ourselves at low-excitation conditions\nwhere populations to the excited states can be neglected, unless otherwise\nstated.\n\n\\subsection{Spectrum in the DH region}\n\nFirst, we investigate the spectrum in the DH region using PC1 located $\\sim $\n5 $\\mu $m upstream of the QD. Figure 3(a) schematically shows the\ncorresponding channel layout. The DH region is formed in $\\left( \\mathrm{a\n,\\uparrow \\right) $ and $\\left( \\mathrm{a},\\downarrow \\right) $ by injecting\nheat through PC1 with an excitation voltage $V_{\\mathrm{E1}}=$ -100 $\\mu \neV. Non-equilibrium charge can be injected selectively into $\\left( \\mathrm{\n},\\uparrow \\right) $ for the PC1 conductance $G_{1}$ in the range of \n0<G_{1}<G_{\\mathrm{q}}$ as shown in panel (ii) of Fig. 3(b), and $\\left( \n\\mathrm{a},\\downarrow \\right) $ for $G_{\\mathrm{q}}<G_{1}<2G_{\\mathrm{q}}$\nin panel (iv). This was confirmed by measuring clear conductance steps in\nthe split-gate voltage $V_{\\mathrm{PC1}}$ dependence of $G_{1}$ ($=I_\n\\mathrm{E1}}\/V_{\\mathrm{E1}}$) in Fig. 3(c).\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[width = 6.7in]{B4.eps}\n\\end{center}\n\\caption{ Binary tunneling spectra in the IH regions with heat current\ninjected from (a) PC2, (b) PC3, and (c) PC4. Measured current (circles) is\nfitted well with the binary spectrum (solid lines) with two temperatures, \nT_{\\mathrm{a},\\uparrow }^{(\\mathrm{H})}$ and $T_{\\mathrm{a},\\uparrow }^{\n\\mathrm{L})}$, coexisting in the channel. Each inset shows the schematic\nchannel geometry with the location of IH regions of interest.}\n\\end{figure*}\n\nFigure 3(d) summarizes the current spectra\\ of the QD (filled circles) for\nvarious $G_{1}$ values, plotted as a function of $V_{\\mathrm{C}}$. The QD\nspectrometer is operated at $V_{\\mathrm{D}}=$ 100 $\\mu $eV. As the PC1\nopening is increased from $G_{1}\/G_{\\mathrm{q}}=$ $0$ to 2, the width of the\nspectrum grows from $eV_{\\mathrm{D}}=$ 100 $\\mu $eV [trace (i)] to $e\\left(\nV_{\\mathrm{D}}-V_{\\mathrm{E1}}\\right) =$ 200 $\\mu $eV [traces (iii)-(v)],\nreflecting the channel $\\left( \\mathrm{a},\\uparrow \\right) $ being switched\nfrom the grounded ohmic contact to the one biased at $V_{\\mathrm{E1}}$.\nHeating in $\\left( \\mathrm{a},\\uparrow \\right) $ is manifested in the slope\non the left edge of the spectra,\\ which becomes distinctly gentle,\nindicating larger $T_{\\mathrm{a},\\uparrow }$, in the tunneling regimes at \nG_{1}\/G_{\\mathrm{q}}=$ $0.5$ [trace (ii)] and 1.5 [trace (iv)]. Note that\nheating in $\\left( \\mathrm{a},\\uparrow \\right) $ is resolved even when\nnon-equilibrium charge is injected into $\\left( \\mathrm{a},\\downarrow\n\\right) $. This ensures the coupling between the two channels. No heating is\nseen at quantized conductance $G_{1}\/G_{\\mathrm{q}}=$ $1$ [trace (iii)] and\n2 [trace (v)]. In contrast to the left edge of the spectra, the right edge,\nwhich probes $\\left( \\mathrm{b},\\uparrow \\right) $, remains steep with $T_\n\\mathrm{b},\\uparrow }\\simeq T_{\\mathrm{base}}$ for all $G_{1}$ values. This\nis reasonable, as no heating is expected for $\\left( \\mathrm{b},\\uparrow\n\\right) $ in this channel layout [see Fig. 3(a)].\n\nAll the spectra in Fig. 3(d) can be fitted well with the Fermi distribution\nas shown by the solid lines. Figure 3(e) plots $k_{\\mathrm{B}}T_{\\mathrm{a\n,\\uparrow }$ (open circles) and $k_{\\mathrm{B}}T_{\\mathrm{b},\\uparrow }$\n(solid circles) deduced from the fits, revealing significant heating in \n\\left( \\mathrm{a},\\uparrow \\right) $, for both injection into $\\left( \n\\mathrm{a},\\uparrow \\right) $ [$G_{1}\/G_{\\mathrm{q}}\\sim 0.5$ at $V_{\\mathrm\nPC1}}\\sim -660$ mV] and $\\left( \\mathrm{a},\\downarrow \\right) $ [$G_{1}\/G_\n\\mathrm{q}}\\sim 1.5$ at $V_{\\mathrm{PC1}}\\sim -220$ mV]. The fact that $T_\n\\mathrm{a},\\uparrow }$ reaches almost the same maximum values at $G_{1}\/G_\n\\mathrm{q}}\\sim $ 0.5 and 1.5 indicates that the injected heat is equally\npartitioned between $\\left( \\mathrm{a},\\uparrow \\right) $ and $\\left( \n\\mathrm{a},\\downarrow \\right) $. As these data show, the coupling between\nhot and cold copropagating channels results in spectra that looks like the\nequilibrium Fermi distribution in both of the copropagating channels. Theory\nsuggests that the actual distribution is different from the Fermi\ndistribution even in the DH region \\cite{DegiovanniPRB2010,KovrizhinPRL2012\n. However, the deviation is too small to be resolved experimentally,\nresulting in seemingly thermalized spectra indistinguishable from the\nequilibrium distribution. Consequently, the DH scheme is not suitable for\nstudying non-equilibrium steady states.\n\n\\subsection{Spectrum in the IH region}\n\nIn contrast to the case of DH region, qualitatively different tunneling\nspectra emerge when the spectrometer is placed in the IH region. IH regions\nappear in $\\left( \\mathrm{a},\\sigma \\right) $ and\/or $\\left( \\mathrm{b\n,\\sigma \\right) $ when one of PC2-PC4 is adjusted in the tunneling regime\nwith the PC conductance $G_{i}=I_{\\mathrm{E2}}\/V_{\\mathrm{E2}}$ $\\sim $ \n0.5G_{\\mathrm{q}}$. Figure 4 summarizes the current spectra (solid circles)\nwith PC2 in (a), PC3 in (b), and PC4 in (c). As the bias $V_{\\mathrm{E2}}$\nof the PC is increased, an additional small but long tail develops,\nresulting in an anomalous spectrum that cannot be fitted with a single Fermi\ndistribution function. The long tails appear on one or both sides of the\npeak depending on the location of the spectrometer with respect to the PC.\n\nFor example, injection from PC2 induces an IH region in $\\left( \\mathrm{a\n,\\sigma \\right) $, but not in $\\left( \\mathrm{b},\\sigma \\right) $ [inset of\nFig. 4(a)]. This is consistent with the observation that a tail appears only\non the left side of the peak [anomalous excitation in $\\left( \\mathrm{a\n,\\uparrow \\right) $], by recalling that the current profiles on the left and\nright sides of the peak reflect $f_{\\mathrm{a},\\uparrow }\\left( E\\right) $\\\nand $1-f_{\\mathrm{b},\\uparrow }\\left( E\\right) $, respectively.\n\nInjection from PC3 should also induce an IH region only in $\\left( \\mathrm{a\n,\\sigma \\right) $. This is consistent with the tail appearing on the left\nside [excitation in $\\left( \\mathrm{a},\\uparrow \\right) $] as seen in Fig.\n4(b). The amplitude of the tail on the left side is comparable to that in\nFig. 4(a). A small step-like profile on the right side is associated with\nthe excited state of the QD, where electromagnetic and\/or phonon-mediated\nenergy transfer might be responsible only for this geometry with a short\ndistance between PC3 and QD ($<$\\ 0.3 $\\mu $m) \\cite{OnacPRL2006}. Therefore\nthis step-like structure is disregarded in the following discussion.\n\nInjection from PC4 induces IH regions in both $\\left( \\mathrm{a},\\sigma\n\\right) $ and $\\left( \\mathrm{b},\\sigma \\right) $, resulting in tails on\nboth sides [excitation in $\\left( \\mathrm{a},\\uparrow \\right) $ and $\\left( \n\\mathrm{b},\\uparrow \\right) $] in Fig. 4(c). In this case, PC4 excites left\nmoving plasmons in modes [L, C\/S], which drag non-equilibrium charges in \n\\left( \\mathrm{b},\\sigma \\right) $\\ up to the right of the QD. This gives a\ntail on the right side of the peak. However, this measurement point is\nill-defined as it is located near the boundary between the IH region and the\nunheated upstream region of $\\left( \\mathrm{b},\\sigma \\right) $. This could\nbe the reason of the small amplitude of the right tail as compared to the\ntails in Figs. 4(a) and 4(b). Moreover, the left moving plasmons in the\ninteracting channels $\\left( \\mathrm{b},\\sigma \\right) $\\ and $\\left( \n\\mathrm{c},\\sigma \\right) $\\ are interrupted by the QD, and scatter into the\nother interacting region with channels $\\left( \\mathrm{a},\\sigma \\right) $\\\nand $\\left( \\mathrm{c},\\sigma \\right) $. This plasmon scattering leads\nanother IH region [denoted by IH' in the inset of Fig. 4(c)] in $\\left( \n\\mathrm{a},\\sigma \\right) $, which generates a tail on the left side of the\npeak. The small amplitude of the left tail reflects this plasmon scattering.\n\nApart from the quantitative differences, the unusual tails appear in all of\nthe IH regions we investigated. As shown by thin solid lines in Fig. 4, all\nthe spectra in the IH regions can be fitted well with the binary spectrum \n\\begin{equation}\nf_{\\eta ,\\sigma }\\left( E\\right) =\\left( 1-p\\right) f_{\\mathrm{F}}\\left(\nE;T_{\\eta ,\\sigma }^{(\\mathrm{L})}\\right) +pf_{\\mathrm{F}}\\left( E;T_{\\eta\n,\\sigma }^{(\\mathrm{H})}\\right) ,  \\label{EqBin}\n\\end{equation\nwhich consists of majority carriers at lower temperature $T_{\\eta ,\\sigma\n}^{(\\mathrm{L})}$ and minority carriers of a fraction $p$ ($\\sim $ 0.1) at\nhigh temperature $T_{\\eta ,\\sigma }^{(\\mathrm{H})}$.\n\nTo further demonstrate the validity of Eq. (\\ref{EqBin}), we examine the\ncase of large excitation voltage, where the second term of Eq. \\ref{EqFit}\ncannot be neglected. We show in Fig. 5 a spectrum excited using PC2 under a\nlarge excitation voltage of $V_{\\mathrm{E2}}=$ 2 meV [panel (i)] and compare\nit with that without heating [panel (ii)], plotted in a linear scale. As\nalready shown in Fig. 4(a), in this configuration heating occurs only in \n\\left( \\mathrm{a},\\sigma \\right) $\\ near the QD, which affects only the left\nside of the peak at low bias. In contrast, the spectrum in panel (i) of Fig.\n5 shows a small negative tail (blue hatched region) on the right side, in\naddition to the tail on the left side (red hatched region). This signifies\nthe presence of hot holes as well as hot electrons in $\\left( \\mathrm{a\n,\\uparrow \\right) $. The whole spectrum including the negative tail is\nreproduced well with Eq. (\\ref{EqBin}) as shown by the solid line. The\ndistribution functions $f_{\\mathrm{a},\\uparrow }\\left( E\\right) $ and $1-f_\n\\mathrm{b},\\uparrow }\\left( E\\right) $ deduced from the fit are shown by\nsolid lines in the inset,\\ highlighting the anomalous binary spectrum $f_\n\\mathrm{a},\\uparrow }\\left( E\\right) $ compared to the Fermi distribution\nfunction $f_{\\mathrm{F}}\\left( E\\right) $ shown by a dashed line.\n\n\\begin{figure}[tbp]\n\\begin{center}\n\\includegraphics[width = 3.0in]{B5.eps}\n\\end{center}\n\\caption{ Current spectrum in a linear scale (i) with heat injection from\nPC2 and (ii) without heat injection. The positive and negative current tails\nin the red and blue hatched regions in (i) represent the excitation of hot\nelectrons and hot holes, respectively, consistent with the binary spectrum\n(the solid lines). The deduced distribution functions $f_{\\mathrm{a\n,\\uparrow }\\left( E\\right) $\\ and $1-f_{\\mathrm{b},\\uparrow }\\left( E\\right) \n$\\ are shown in the inset. The binary spectrum in $f_{\\mathrm{a},\\uparrow\n}\\left( E\\right) $\\ contrasts with a single Fermi distribution function $f_\n\\mathrm{F}}\\left( E\\right) $\\ (dashed line). }\n\\end{figure}\n\nBinary spectrum can be seen in a wide range of gate voltage, $V_{\\mathrm{PC\ni}$, for $i$-th PC with $i$\\ = 2, 3, and 4. Figure 6 summarizes the $V_\n\\mathrm{PC}2}$\\ dependence of the PC2 conductance in (a) and the fitting\nparameters, the fraction $p$\\ in (b) and two distinct temperatures $T_\n\\mathrm{a},\\uparrow }^{(\\mathrm{H})}$\\ (circles) and $T_{\\mathrm{a},\\uparrow\n}^{(\\mathrm{L})}$\\ (black solid line) in (c),\\ for the binary spectrum. At\nthis large bias voltage $V_{\\mathrm{E2}}$\\ = 1 mV, $G_{2}$\\ in Fig. 6(a)\ndoes not show quantized conductance at $G_{\\mathrm{q}}$, which gives\nambiguity in the estimate of $D_{2,\\sigma }$. The binary spectrum with high \nT_{\\mathrm{a},\\uparrow }^{(\\mathrm{H})}$ is seen in the entire tunneling\nregime, while $T_{\\mathrm{a},\\uparrow }^{(\\mathrm{L})}$\\ as well as $T_\n\\mathrm{b},\\uparrow }$\\ in the unheated region remain low comparable to the\nbase temperature.\n\n\\begin{figure}[tbp]\n\\begin{center}\n\\includegraphics[width = 2.5in]{B6.eps}\n\\end{center}\n\\caption{(a) Conductance of PC2, $G_{2}=I_{\\mathrm{E2}}\/V_{\\mathrm{E2}}$, as\na function of the gate voltage, $V_{\\mathrm{PC2}}$. (b and c) $V_{\\mathrm{PC\n}}$\\ dependence of the fraction $p$\\ of hot minority carriers in (b) and\nthermal energies for $T_{\\mathrm{a},\\uparrow }^{(\\mathrm{H})}$\\ (circles), \nT_{\\mathrm{a},\\uparrow }^{(\\mathrm{L})}$\\ (black solid line), and $T_\n\\mathrm{b},\\uparrow }$\\ (black dashed line) in (c) deduced from the fitting.\nThe parameter $T_{\\mathrm{a},\\uparrow }^{(\\mathrm{H})}$\\ is smaller than but\ncomparable to $T_{\\mathrm{PER}}^{(\\mathrm{0.1})}$, which is expected in the\nassumption of reaching an identical temperature. The blue solid, dashed, and\ndotted lines are $T_{\\mathrm{PER}}^{(\\mathrm{0.1})}$\\ by assuming \nD_{2,\\uparrow }=D_{2,\\downarrow }$, $D_{2,\\downarrow }=0$, and \nD_{2,\\uparrow }=0$, respectively.}\n\\end{figure}\n\nStrikingly, the observed binary spectrum did not decay into a single Fermi\nfunction for any geometries we examined. The factor $p$ ($\\sim $ 0.1) is\nfound to be comparable for various distances from the PC to the QD,\nincluding the shortest case with PC3 [Fig. 4(b)], 5 $\\mu $m with PC2 [Fig.\n4(a)], and 10 $\\mu $m with PC4 in the reversed magnetic field (See Appendix\nB and Fig. 10). These distances are much longer than $l_{\\mathrm{el}}$ (\n\\sim $ 0.1 $\\mu $m at $V_{\\mathrm{E}}=$ 1 meV) for electronic relaxation to\nplasmons. Therefore, the observed binary spectrum supports the picture of\nnon-interacting plasmon excitations and the non-thermalizing character of\nthe 1D edge channels.\n\n\\subsection{Comparison of the spectra}\n\nThe spectra in the DH and IH regions are compared in the same range of\nexcitation voltages\\ in Fig. 7, where the horizontal axis is taken as the\ndot energy $\\varepsilon $\\ relative to $\\mu _{\\mathrm{a},\\uparrow }$. The\nspectrum for the DH region at $G_{1}=1.5G_{\\mathrm{q}}$\\ in Fig. 7(a) is\nmeasured at various $V_{\\mathrm{E}1}$\\ by keeping the effective bias\nconstant ($V_{\\mathrm{E}1}-V_{\\mathrm{D}}=$\\ 200 $\\mu $V). Heating in \n\\left( \\mathrm{a},\\uparrow \\right) $\\ is seen as broadening on the left edge\nof the peak. The linear slope over a few orders of magnitude in the\nlogarithmic scale of $I_{D}$\\ suggests that all carriers in the DH region\ncan be characterized by a single temperature $T^{\\left( \\mathrm{DH}\\right) }\n. In contrast, the unusual tail for the IH region, which can be recognized\nfrom $V_{\\mathrm{E}2}\\sim $\\ 200 $\\mu $V and becomes prominent at $V_\n\\mathrm{E}2}\\geq $\\ 500 $\\mu $V in Fig. 7(b), can be characterized by the\nbinary spectrum with parameters $T^{\\left( \\mathrm{H}\\right) }$\\ (gentle\nslope), $T^{\\left( \\mathrm{L}\\right) }$ (steep slope), and $p$.\n\nOne can see that $T^{\\left( \\mathrm{DH}\\right) }$\\ in the DH region [Fig.\n7(a)] is somewhat higher but comparable to $T^{\\left( \\mathrm{H}\\right) }$\\\nin the IH region [Fig. 7(b)], by comparing the spectra at the same\nexcitation voltages ($V_{\\mathrm{E1}}=V_{\\mathrm{E2}}\\sim $\\ 600 $\\mu $V).\nThis implies the strong relation between the two spectra. Based on the\nargument in Sec. IIB, the different spectra in the DH and IH regions can be\nunderstood by considering the amplitudes of the plasmon excitations. The\nchannels in the DH region are full of hot plasmons with large amplitudes as\nshown in Fig. 1(b), which seemingly looks like a thermalized state when the\nsystem is measured with a QD spectrometer. In contrast, the channels in the\nIH region involve small amplitudes of hot plasmons in the background of cold\nplasmons, which could lead to the anomalous binary spectrum.\n\nWe should add some comments on the effect of excited states in the dot. In\nFig. 7(a), a faint kink marked by ES' and a step-like structure marked by ES\nappear when the excitation voltage $V_{\\mathrm{E}1}$\\ exceeds the typical\nlevel spacing (about 200 $\\mu $V) of the QD. The former (ES') can be\nunderstood as hot electrons passing through an excited state, and the latter\n(ES) can also be attributed to a complex excitation process to an excited\nstate. Both of these additional features start to appear at $V_{\\mathrm{E}1}\n\\ values much higher than that at which the heating in the channel starts to\nbe visible. In contrast, no features associated with the excited states\nappear for the IH region, unless $V_{\\mathrm{E}2}$\\ exceeds 2 mV. This is\npresumably due to the weak coupling ($p$\\ $\\sim $\\ 0.1) between the DH and\nIH regions.\n\n\\begin{figure}[tbp]\n\\begin{center}\n\\includegraphics[width = 3.0in]{B7.eps}\n\\end{center}\n\\caption{(a) Tunneling spectra measured in the DH region induced by PC1 at \nG_{1}=1.5G_{\\mathrm{q}}$. Its $V_{\\mathrm{E}1}$\\ dependence is taken by\nchanging both $V_{\\mathrm{E}1}$\\ and $V_{\\mathrm{D}}$\\ to keep $V_{\\mathrm{E\n1}-V_{\\mathrm{D}}=$\\ 200 $\\protect\\mu $V. Excited-state features are\nindicated by arrows (ES and ES'). (b) Tunneling spectra measured in the IH\nregion induced by PC2 at $G_{2}=0.5G_{\\mathrm{q}}$. The binary spectrum\ndevelops with increasing $V_{\\mathrm{E}2}$.}\n\\end{figure}\n\n\\subsection{Heat flow}\n\nProvided that the binary distribution of Eq. (\\ref{EqBin}) holds well, the\ntemperatures in DH and IH regions can be related to each other by\nconsidering the heat flow in the system. The heat current injected from the \ni$-th PC is given by $W_{i}=\\frac{1}{2}G_{\\mathrm{q}}\\sum\\nolimits_{\\sigma\n}D_{i,\\sigma }\\left( 1-D_{i,\\sigma }\\right) V_{\\mathrm{E}}^{2}$ with the\npartitioning factor $D\\left( 1-D\\right) $ \\cit\n{leSueurPRL2010,JezouinScience2013}. This heat should be redistributed over\nthe four channels. As discussed in Sec IIB, excitations in DH and IH regions\nare expected to be correlated. For simplicity, we assume that all carriers\nin the DH region and the fraction $p$\\ of carriers in the IH region share an\nidentical temperature $T_{\\mathrm{H}}^{(p)}$\\ while the cold majority ($1-p\n) in the IH region remains at $T_{\\mathrm{base}}$. This assumption can be\njustified for the high-energy spectrum ($\\left\\vert \\varepsilon -\\mu _\n\\mathrm{a},\\uparrow }\\right\\vert \\gg k_{\\mathrm{B}}T_{\\mathrm{H}}^{(p)}$) in\nthe weak coupling limit ($p\\ll 1$)\\ of quantum quench problem shown in Sec.\nIVB. This crude approximation allows us to compare the different experiments\nfor the DH and IH regions. Then, the energy conservation law suggests that\nthe spectra are expected to share the same temperature given by $T_{\\mathrm{\n}}^{(p)}=\\sqrt{T_{\\mathrm{base}}^{2}+\\frac{3h}{(1+p)\\pi ^{2}k_{\\mathrm{B\n}^{2}}W_{i}}$.\n\nFigure 8(a) shows the bias-voltage $V_{\\mathrm{E1}}$ dependence of the\ndeduced thermal energy $k_{\\mathrm{B}}T_{\\mathrm{a},\\uparrow }$ in the DH\nregion taken at $G_{2}\/G_{\\mathrm{q}}\\sim $1.5 (circles) and $\\sim $0.5\n(squares). Figures 8(b) and 8(c) summarize $V_{\\mathrm{E2}}$ dependence of\nthe fitting parameters [$p$\\ in (b) and $k_{\\mathrm{B}}T_{\\mathrm{a\n,\\uparrow }^{(\\mathrm{H})}$ (open circles), $k_{\\mathrm{B}}T_{\\mathrm{a\n,\\uparrow }^{(\\mathrm{L})}$ (solid line) and $k_{\\mathrm{B}}T_{\\mathrm{b\n,\\uparrow }$\\ (dotted line) in (c)] for the IH region induced by PC2. Here\nwe have restricted the ranges $V_{\\mathrm{E}1}\\leq $\\ 0.2 mV and $V_{\\mathrm\nE}2}\\leq $\\ 1.5 meV, where excited states in the QD play no visible effects\nin the spectrum. By choosing a typical value of $p=$ 0.1 from Fig. 8(b), one\ncan see that $k_{\\mathrm{B}}T_{\\mathrm{a},\\uparrow }$ in Fig. 8(a) as well\nas $k_{\\mathrm{B}}T_{\\mathrm{a},\\uparrow }^{(\\mathrm{H})}$ in Fig. 8(c)\nfollow reasonably well the same form of $T_{\\mathrm{H}}^{(0.1)}$. This\nsuggests that the carriers in the DH region and the minority carriers in the\nIH region indicate a similar spectrum. Quantitative disagreement, such as\nthe deviation of $k_{\\mathrm{B}}T_{\\mathrm{a},\\uparrow }^{(\\mathrm{H})}$\nfrom a linear dependence in Fig. 8(c), might come from this crude\napproximation and possible energy-dependent coupling between the channels \n\\cite{ProkudinaPRL2014}. Note that the temperature $T_{\\mathrm{a},\\uparrow\n}^{(\\mathrm{L})}$ [solid line in Fig. 8(c)] of the majority carriers always\nremains cold. Therefore, the total energy is well conserved in the four\nchannels with negligible heat leakage to other degrees of freedom, which\nensures that the channels are effectively isolated from the environment. The\nsame temperature $T_{\\mathrm{H}}^{(0.1)}$\\ is also plotted in Fig. 6(c),\nwhere one can see that the observed $T_{\\mathrm{a},\\uparrow }^{(\\mathrm{H})}\n\\ is smaller but comparable to $T_{\\mathrm{H}}^{(0.1)}$. These evidences\nsupport that the excitations in DH and IH regions are strongly correlated.\n\n\\begin{figure}[tbp]\n\\begin{center}\n\\includegraphics[width = 3.375in]{B8.eps}\n\\end{center}\n\\caption{ (a) The bias voltage $V_{\\mathrm{E1}}$\\ dependence of $k_{\\mathrm{\n}}T_{\\mathrm{a},\\uparrow }$\\ obtained at $G_{2}\/G_{\\mathrm{q}}\\sim $1.5\n(circles) and $\\sim $0.5 (squares). Solid and dashed lines show $T_{\\mathrm{\n}}^{(0.1)}$\\ and $T_{\\mathrm{H}}^{(0)}$, respectively, considering the\nvariation of $k_{\\mathrm{B}}T_{\\mathrm{base}}=$\\ 10 (lower bounds) and 13 \n\\protect\\mu $eV (upper bounds). (b and c) The bias voltage $V_{\\mathrm{E2}}\n\\ dependence of $p$\\ in (b) and $k_{\\mathrm{B}}T_{\\mathrm{a},\\uparrow }^{\n\\mathrm{H})}$, $k_{\\mathrm{B}}T_{\\mathrm{a},\\uparrow }^{(\\mathrm{L})}$, and \nk_{\\mathrm{B}}T_{\\mathrm{b},\\uparrow }$\\ in (c). The error bars show typical\nambiguity in fitting the binary spectrum. Blue solid lines show $T_{\\mathrm{\n}}^{(0.1)}$, considering the ambiguity of tunneling probabilities between \nD_{2,\\uparrow }=D_{2,\\downarrow }=G_{2}\/2G_{\\mathrm{q}}$\\ (upper bound) and \nD_{2,\\uparrow }=G_{2}\/G_{\\mathrm{q}}$\\ and $D_{2,\\downarrow }=0$ (lower\nbound).}\n\\end{figure}\n\nNote that the above analysis is based on the assumption where the energy\ndistribution can be approximated by Fermi distribution in the DH region and\nthe binary distribution in the IH region. Rigorous analysis should be made\nwith the TL model, which is beyond the scope of this paper. However,\nsimulations of quantum quench, a time-dependent problem for initial states\nprepared at different temperatures, strongly support this approximation, as\nshown in the next chapter.\n\n\\section{Simulations with the TL model}\n\n\\subsection{\\textbf{Plasmon eigenmodes}}\n\nPlasmon modes in the edge channels can be conveniently described by a\ncapacitance model \\cite{KamataNatNano2014,HashisakaPRB2013}. Interaction\ninside or between quantum Hall edge channels can be parametrized by\ncapacitances per unit length, $C_{\\mathrm{ch}}$ for intrachannel coupling\nand $C_{ij}$ for interchannel coupling between $i$-th and $j$-th channel [\ni,j\\in \\left\\{ \\left( \\mathrm{r},\\downarrow \\right) ,\\left( \\mathrm{r\n,\\uparrow \\right) ,\\left( \\mathrm{\\ell },\\uparrow \\right) ,\\left( \\mathrm\n\\ell },\\downarrow \\right) \\right\\} $] . For simplicity, we only consider\nnearest neighbor couplings with $C_{\\mathrm{X}}$ between copropagating\nchannels and $C_{\\mathrm{Z}}$ between counterpropagating channels, as shown\nin Fig. 1(a). We assumed that $C_{\\mathrm{ch}}$ is identical for all\nchannels. Excess charge density $\\rho _{i}$ and potential $V_{i}$ of $i$-th\nchannel are related by the coupling capacitances; $\\rho _{i}=C_{ij}V_{j}$\nwith the capacitance matrix of the for\n\\begin{equation}\n\\mathbf{C}=C_{\\mathrm{ch}}\\left( \n\\begin{array}{cccc}\n1+c_{\\mathrm{X}} & -c_{\\mathrm{X}} & 0 & 0 \\\\ \n-c_{\\mathrm{X}} & 1+c_{\\mathrm{X}}+c_{\\mathrm{Z}} & -c_{\\mathrm{Z}} & 0 \\\\ \n0 & -c_{\\mathrm{Z}} & 1+c_{\\mathrm{X}}+c_{\\mathrm{Z}} & -c_{\\mathrm{X}} \\\\ \n0 & 0 & -c_{\\mathrm{X}} & 1+c_{\\mathrm{X}\n\\end{array\n\\right)\n\\end{equation\nin the order of the channel index $i=\\left( \\mathrm{r},\\downarrow \\right)\n,\\left( \\mathrm{r},\\uparrow \\right) ,\\left( \\mathrm{\\ell },\\uparrow \\right)\n, $ and $\\left( \\mathrm{\\ell },\\downarrow \\right) $. Here, $c_{\\mathrm{X\n}=C_{\\mathrm{X}}\/C_{\\mathrm{ch}}$ and $c_{\\mathrm{Z}}=C_{\\mathrm{Z}}\/C_\n\\mathrm{ch}}$ denote the normalized coupling strengths. Charge conservation\nin each channel can be represented by $\\partial \\rho _{i}\/\\partial\nt=-S_{i}G_{\\mathrm{q}}\\partial V_{i}\/\\partial x$ with the chirality \nS_{\\left( \\mathrm{r},\\sigma \\right) }=1$ for right-moving channels and \nS_{\\left( \\ell ,\\sigma \\right) }=-1$ for left-moving channels, and $G_\n\\mathrm{q}}=\\frac{e^{2}}{h}$. Coupled wave equation of the plasmons can be\nwritten in the form \n\\begin{equation}\n\\partial \\rho _{i}\/\\partial t=-M_{ij}\\partial \\rho _{j}\/\\partial x\n\\end{equation\nwith $M_{ij}=S_{i}G_{\\mathrm{q}}\\left\\{ \\mathbf{C}^{-1}\\right\\} _{ij}$.\n\nThe plasmon eigenmodes can be obtained by diagonalizing the matrix $\\mathbf{\n}$. The transformation matrix $\\mathbf{T}$ that satisfies $\\mathbf{T}^\n\\mathbf{\\dag }}\\mathbf{MT}=\\mathrm{diag}(v_{m})$ describes the plasmon\neigenmode $\\tilde{\\rho}_{m}$ and the velocity $v_{m}$ for $m=\\left[ \\mathrm\nR,C}\\right] $, $\\left[ \\mathrm{R,S}\\right] $, $\\left[ \\mathrm{L,S}\\right] $,\nand $\\left[ \\mathrm{L,C}\\right] $. For a weak coupling limit $c_{\\mathrm{Z\n}\\ll c_{\\mathrm{X}}\\ll 1$, eigenmodes can be approximately written a\n\\begin{eqnarray}\n\\tilde{\\rho}_{\\mathrm{R,C}} &\\simeq &\\left( 1,1-c_{\\mathrm{Z}}\/2c_{\\mathrm{X\n},-c_{\\mathrm{Z}}\/2,0\\right) \/\\sqrt{2} \\\\\n\\tilde{\\rho}_{\\mathrm{R,S}} &\\simeq &\\left( -1,1+c_{\\mathrm{Z}}\/2c_{\\mathrm{\n}},-c_{\\mathrm{Z}}\/2,0\\right) \/\\sqrt{2} \\\\\n\\tilde{\\rho}_{\\mathrm{L,S}} &\\simeq &\\left( 0,-c_{\\mathrm{Z}}\/2,1+c_{\\mathrm\nZ}}\/2c_{\\mathrm{X}},-1\\right) \/\\sqrt{2} \\\\\n\\rho _{\\mathrm{L,C}} &\\simeq &\\left( 0,-c_{\\mathrm{Z}}\/2,1-c_{\\mathrm{Z\n}\/2c_{\\mathrm{X}},1\\right) \/\\sqrt{2}\n\\end{eqnarray\nwith the velocity $v_{\\mathrm{R\/L,C}}\\simeq \\pm v_{\\mathrm{0}}\\left( 1-c_\n\\mathrm{Z}}\/2\\right) $ for charge modes and $v_{\\mathrm{R\/L,S}}\\simeq \\pm v_\n\\mathrm{0}}\\left( 1-2c_{\\mathrm{X}}-c_{\\mathrm{Z}}\/2\\right) $ for spin\nmodes, where $v_{\\mathrm{0}}=G_{\\mathrm{q}}\/C_{\\mathrm{ch}}$ is the\nuncoupled plasmon velocity in a single channel. These plasmon eigenmodes are\nschematically shown in Fig. 1(a). Note that these plasmon eigenmodes are\nfully consistent with the TL model, as shown in Sec. IVB.\n\nWhen a unit charge is injected into $\\left( \\mathrm{\\ell },\\uparrow \\right) \n, it can be expressed a\n\\begin{equation}\n\\left( 0,0,1,0\\right) \\simeq \\sqrt{2}s\\left( \\tilde{\\rho}_{\\mathrm{R,C}}\n\\tilde{\\rho}_{\\mathrm{R,S}}\\right) +\\left( \\tilde{\\rho}_{\\mathrm{L,C}}\n\\tilde{\\rho}_{\\mathrm{L,S}}\\right) \/\\sqrt{2}  \\label{Eq0010}\n\\end{equation\nwith a small fractionalization ratio $s=c_{\\mathrm{Z}}\/4=C_{\\mathrm{Z}}\/4C_\n\\mathrm{ch}}$. This explains large and small plasmon amplitudes in the DH\nand IH regions, respectively, as schematically illustrated in Fig. 1(b).\n\nThe above capacitance model is convenient for describing different coupling\nparameters between the channels. If required, the capacitances can be\nobtained by solving electrostatic potential around the edge channels \\cit\n{KamataNatNano2014}. Actually, the model has been applied to reproduce\nplasmon transport in edge channels including plasmon interferometers \\cit\n{HashisakaPRB2013}. In this work, we use the matrix $\\mathbf{T}$ and\nparameters $c_{\\mathrm{X}}$ and $c_{\\mathrm{Z}}$ to solve the quench problem\nshown in the next subsection.\n\n\\subsection{Quantum quench}\n\nWe have experimentally studied how the energy spectrum changes along the\ninteracting edge channels under continuously injecting non-equilibrium\nelectrons from a PC (spatial analog of quantum quench). This can be compared\nwith the standard quantum quench problem asking how the momentum spectrum\nchanges with time in the interacting channels after preparing a\nnon-equilibrium state. Here we apply the quench problem to the\ncounterpropagating channels to see the emergence of a binary spectrum.\n\nWe followed the approach taken by Kovrizhin and Chalker to study the quench\nproblem for two copropagating channels (see refs. \\cit\n{Iucci2009,KovrizhinPRB2011} and references therein). Here, four infinitely\nlong channels with channel index $j$ (= $1-4$), i.e., two right movers ($j$\n= 1 and 2) and two left movers ($j$ = 3 and 4), are initially prepared in\nindependent thermal equilibrium at different temperatures $T_{j}$ (\nT_{1}=T_{3}=T_{4}=T_{\\mathrm{base}}$ and $T_{2}=T_{\\mathrm{H}}>T_{\\mathrm\nbase}}$). The initial state of each channel at $t=0$ is described by the\ncorrelation function $\\mathcal{G}_{j}\\left( x;t=0\\right) =\\mathcal{G\n_{T_{j}}^{\\left( \\mathrm{eq}\\right) }\\left( x\\right) $, where $x$ is\ndistance and $\\mathcal{G}_{T}^{\\left( \\mathrm{eq}\\right) }\\left( x\\right) \n\\frac{i}{2\\hbar }\\left( \\pi k_{\\mathrm{B}}T\/\\hbar v_{\\mathrm{0}}\\right)\n\/\\sinh \\left[ \\left( \\pi k_{\\mathrm{B}}T\/\\hbar v_{\\mathrm{0}}\\right) x\\right]\n$ is the correlation function in thermal equilibrium at temperature $T$.\nCorresponding momentum distribution function $f_{j}\\left( k,t=0\\right) $ is\nthe Fermi distribution function $f_{T}^{\\left( \\mathrm{eq}\\right) }\\left(\nk\\right) =1\/\\left[ \\exp \\left( \\hbar v_{\\mathrm{0}}k\/k_{\\mathrm{B}}T\\right)\n+1\\right] $ at $T=T_{j}$.\n\nTime evolution of the system is calculated based on the standard\nbosonization technique for the TL model \\cite{vonDelftAnnPhys1998}. The\ninteraction parameters often described by $g$'s are equivalent to those in\nthe capacitance model in Sec. IVA; i.e., the so-called $g_{4}$ and $g_{2}$\nparameters represent the diagonal and off-diagonal elements, respectively,\nof $\\mathbf{C}^{-1}$. It should be noted that backscattering with the\nso-called $g_{1}$ parameter, which diverts the system away from the TL\nmodel, is well suppressed in the edge channels. The Bogoliubov\ntransformation can also be performed with the same matrix $\\mathbf{T}$.\nThen, the dynamics is simply described by non-interacting plasmon modes with\nconstant velocities. For the independent initial states at $T_{j}$, the\nequal-time correlation function $\\mathcal{G}_{j}\\left( x;t\\right) $ at time \nt$ can be described by a product of initial correlation functions $\\mathcal{\n}_{T_{j}}^{\\left( \\mathrm{eq}\\right) }\\left( x\\right) $ of all channels as\n\n\\begin{eqnarray}\n\\mathcal{G}_{j}\\left( x;t\\right) &=&\\prod_{j^{\\prime }}\\left[ \\mathcal{G\n_{T_{j^{\\prime }}}^{\\left( \\mathrm{eq}\\right) }\\left( x\\right) \\right]\n^{p_{jj^{\\prime }}}  \\nonumber \\\\\n&&\\times \\prod_{m,m^{\\prime }}\\left[ \\frac{\\mathcal{G}_{T_{j^{\\prime\n}}}^{\\left( \\mathrm{eq}\\right) }\\left( x+\\Delta v_{m,m^{\\prime }}t\\right) }\n\\mathcal{G}_{T_{j^{\\prime }}}^{\\left( \\mathrm{eq}\\right) }\\left( \\Delta\nv_{m,m^{\\prime }}t\\right) }\\right] ^{c_{jj^{\\prime }m}c_{jj^{\\prime\n}m^{\\prime }}},\n\\end{eqnarray\nwith the power $p_{jj^{\\prime }}=\\sum_{m}\\left\\vert T_{mj}T_{mj^{\\prime\n}}\\right\\vert ^{2}$ that satisfies $\\sum_{j^{\\prime }}p_{jj^{\\prime }}=1$. \nc_{jj^{\\prime }m}$ represents the coupling between the channels $j$ and \nj^{\\prime }$ through the plasmon mode $m$, and $\\Delta v_{m,m^{\\prime }}=$ \nv_{m^{\\prime }}-v_{m^{\\prime }}$ is the velocity difference between the\nmodes $m$ and $m^{\\prime }$. For the steady state, $\\mathcal{G}_{j}\\left(\nx;\\infty \\right) $ at $t\\rightarrow \\infty $ is given by a power mean of\ninitial correlation functions a\n\\begin{equation}\n\\mathcal{G}_{j}\\left( x;\\infty \\right) =\\prod_{j^{\\prime }}\\left[ \\mathcal{G\n_{T_{j^{\\prime }}}^{\\left( \\mathrm{eq}\\right) }\\left( x\\right) \\right]\n^{p_{jj^{\\prime }}}.  \\label{Gpower}\n\\end{equation\nFor the weak coupling limit ($c_{\\mathrm{Z}}\\ll c_{\\mathrm{X}}\\ll 1$), we\nhave $p_{jj^{\\prime }}\\sim 0.5$ between copropagating channels (\nS_{j}S_{j^{\\prime }}=1$) while $p_{jj^{\\prime }}$ ($\\sim s^{2}$) $\\ll 1$\nbetween counterpropagating channels ($S_{j}S_{j^{\\prime }}=-1$). Finally,\nthe momentum distribution function $f_{j}^{\\left( \\mathrm{st}\\right) }\\left(\nk\\right) $ is obtained from the Fourier transform of $\\mathcal{G}_{j}\\left(\nx,\\infty \\right) $.\n\nWhen only one channel ($j=2$)\\ is heated to $T_{\\mathrm{H}}$ with others\nkept at $T_{\\mathrm{base}}$ in the initial condition, Eq. (\\ref{Gpower}) can\nbe simplified as \n\\begin{equation}\n\\mathcal{G}_{j}\\left( x;\\infty \\right) =\\mathcal{G}_{T_{\\mathrm{base\n}}^{\\left( \\mathrm{eq}\\right) }\\left[ \\mathcal{G}_{T_{\\mathrm{H}}}^{\\left( \n\\mathrm{eq}\\right) }\/\\mathcal{G}_{T_{\\mathrm{base}}}^{\\left( \\mathrm{eq\n\\right) }\\right] ^{p_{j2}}.  \\label{Gpower2}\n\\end{equation\nFocusing on the correlation function at shorter distance ($\\left\\vert\nx\\right\\vert \\lesssim \\hbar v_{\\mathrm{0}}\/k_{\\mathrm{B}}T_{\\mathrm{H}}$) or\nthe spectrum at larger momentum ($\\left\\vert k\\right\\vert \\gtrsim k_{\\mathrm\nB}}T_{\\mathrm{H}}\/\\hbar v_{\\mathrm{0}}$), the ratio $X\\equiv \\mathcal{G}_{T_\n\\mathrm{H}}}^{\\left( \\mathrm{eq}\\right) }\/\\mathcal{G}_{T_{\\mathrm{base\n}}^{\\left( \\mathrm{eq}\\right) }\\sim 1$ to the power of $p\\equiv p_{j2}\\ll 1$\nfor $j=3$ or 4 can be approximated to $X^{p}\\simeq 1+p\\left( X-1\\right)\n+O\\left( \\left( X-1\\right) ^{2}\\right) $. This yields $\\mathcal{G}_{j}\\simeq\n\\left( 1-p\\right) \\mathcal{G}_{T_{\\mathrm{base}}}^{\\left( \\mathrm{eq}\\right)\n}+p\\mathcal{G}_{T_{\\mathrm{H}}}^{\\left( \\mathrm{eq}\\right) }$. Its Fourier\nspectrum exhibits a binary spectrum of the for\n\\begin{equation}\nf_{j=3,4}^{\\left( \\mathrm{st}\\right) }\\left( k\\right) \\simeq \\left(\n1-p\\right) f_{T_{\\mathrm{base}}}^{\\left( \\mathrm{eq}\\right) }\\left( k\\right)\n+pf_{T_{\\mathrm{H}}}^{\\left( \\mathrm{eq}\\right) }\\left( k\\right)\n\\end{equation\nin the counterpropagating channels ($j=$ 3 or 4). In this way, the\nappearance of an approximate binary spectrum is suggested in the quantum\nquench problem. In contrast, electronic spectrum in the copropagating\nchannels ($j=1$\\ and 2) can be approximated to $f_{j=1,2}^{\\left( \\mathrm{st\n\\right) }\\left( k\\right) \\simeq f_{T_{\\mathrm{H}}}^{\\left( \\mathrm{eq\n\\right) }\\left( k\\right) $\\ by neglecting an anomaly around $k=0$.\nTherefore, the two spectra in counter- and copropagating channels share an\nidentical spectrum. This characteristics is used in the analysis of heat\nflow in Sec. IIIE.\n\nThe exact solution of the spectra are neither the binary spectrum nor the\nFermi distribution function. The accurate spectrum can be obtained by\nperforming a Fourier transform of Eq. (\\ref{Gpower2}). The solid lines in\nFig. 9(a) and 9(b) show the initial momentum distribution functions \nf_{j}^{\\left( \\mathrm{ini}\\right) }\\left( k\\right) $ and the steady-state\ndistribution function $f_{j}^{\\left( \\mathrm{st}\\right) }$, respectively,\nplotted as a function of the normalized momentum $\\hbar v_{\\mathrm{0}}k\/k_\n\\mathrm{B}}T_{\\mathrm{base}}$. Here, we assumed $T_{\\mathrm{H}}=10T_{\\mathrm\nbase}}$ and coupling parameters $c_{\\mathrm{X}}=0.8$ and $c_{\\mathrm{Z}}=0.6\n. The parameters, which are realistic in the capacitance model in Sec. IVA,\nwere chosen to imitate the binary spectra in Fig. 4. The copropagating\nchannels 1 and 2 resulted in similar spectra ($f_{1}\\sim f_{2}$), which\nresemble a single Fermi distribution function shown by the dashed line. In\ncontrast, anomalous spectra ($f_{3}$ and $f_{4}$) emerge in the\ncounterpropagating channels 3 and 4. The spectrum can be approximated by a\nbinary spectrum composed of hot and cold carriers, as shown by the\ndash-dotted line. In this range of plots in Fig. 9(b), the temperature (the\ninverse of the slope) of hot minority carriers in the counterpropagating\nchannels is somewhat lower than that in copropagating channels.\n\nIn this way, the unusual binary spectrum can emerge in quantum-quench\nsimulation. The simulation can be compared to the experiment by regarding\nthe heated channel ($j=2$) as $\\left( \\ell ,\\uparrow \\right) $ in Fig. 1(b),\nspectra $f_{1}$ and $f_{2}$ as those in the DH region, and spectra $f_{3}$\nand $f_{4}$ as those in the IH region.\n\n\\begin{figure}[tbp]\n\\begin{center}\n\\includegraphics[width = 3.2in]{B9.eps}\n\\end{center}\n\\caption{ Momentum distribution functions for (a) initial states in\nnon-interacting channels and (b) steady state at $t=\\infty $ after the\ninteraction is turned on at $t=0$. Each trace is offset for clarity.\nInitially, channels are in thermal equilibrium at $T_{1}=T_{3}=T_{4}=T_\n\\mathrm{base}}$ and $T_{2}=$ 10$T_{\\mathrm{base}}$ in (a). Non-equilibrium\nsteady states are reached at $t=\\infty $ in (b), where quasi-binary spectrum\nemerges in counter-propagating channels 3 and 4. The dashed and dash-dotted\nlines show single and binary Fermi distribution functions, respectively,\nadjusted to the spectra based on the TL theory (solid lines). The schematic\nchannel geometries without and with interaction are shown in the inset of\n(a) and (b), respectively.}\n\\end{figure}\n\n\\section{Discussion}\n\n\\subsection{Coupling strength}\n\nThe factor $p$\\ deduced from the fits, which ranges from 0.03 to 0.2 in our\nPCs, measures the coupling between the counterpropagating channels across\nthe isolation gate ($G_{\\mathrm{iso}}$) of the width \\ $w$\\ $\\sim $\\ 0.1 \n\\mu $m. This can be compared with those in previous studies with\nnon-spectroscopic means. The time-domain charge measurement in Ref. \\cit\n{KamataNatNano2014} has demonstrated charge fractionalization of an incident\nwave packet, which yields the fractionalization ratio of $r=$ 0.04 in the\namplitude, for the gate width $w$\\ $\\sim $\\ 1 $\\mu $m. Based on the model\nshown in Sec. IV, this corresponds to $p\\sim 4r^{2}\\sim $\\ 0.006 in our\nparameter. Bolometric detection of heat transfer between counterpropagating\nchannels in Ref. \\cite{ProkudinaPRL2014} have identified the interaction\nstrength $\\left\\vert 1-K\\right\\vert =$\\ 0.1 $\\sim $\\ 0.25, which corresponds\nto $p\\sim 4\\left( 1-K^{-1}\\right) ^{2}=$\\ 0.03 $\\sim $\\ 0.16, depending on\nthe gate voltage on the isolation gate of width $w$\\ $\\sim $\\ 0.2 $\\mu $m.\nSince the fraction $p$\\ should increase with decreasing the separation of\nthe counterpropagating channels, $p$\\ $\\sim $\\ 0.1 in our case is reasonable\nfor $w$\\ $\\sim $\\ 0.1 $\\mu $m.\n\n\\subsection{Analogy with quantum quench}\n\nThe time dependence of the momentum spectrum (quantum quench) in Sec. IV can\nbe related to the spatial dependence of the energy spectrum studied in Sec.\nIII. The differences and similarities between them are summarized as follows.\n\nFirstly, the energy (momentum) spectrum is given by the Fourier transform of\nthe equal-position (equal-time) correlation function. Considering the\nnon-interacting non-dispersive plasmon modes with the constant velocities,\nthe energy spectrum and the momentum spectrum should be related to each\nother \\cite{KovrizhinPRB2011,GutmanPRB2009,JurcevicNature2014}.\n\nSecondly, the initial state that can be prepared by a PC shows a double-step\ndistribution function [see the central inset of Fig. 1(b)], which is not in\nthermal equilibrium assumed in the quantum-quench simulation. Previous\nexperiments for two copropagating channels \\cite{leSueurPRL2010}, as well as\nour results in Sec. IIIB, resulted in a spectrum close to Fermi distribution\nfunction, which is similar to the spectra $f_{1}$ and $f_{2}$ in Fig. 9(b).\nThis behavior can be reproduced in simulations of quantum quench starting\nfrom a double-step function in one channel \\cite{KovrizhinPRL2012} as well\nas from thermal equilibrium states at different temperatures \\cit\n{KovrizhinPRB2011}. This suggests that these different initial states does\nnot play significant roles in the final state for the problem. Therefore,\nquantum quench starting from a Fermi distribution function is not a bad\nassumption to validate the appearance of binary spectrum in the system.\n\nThirdly, different from the quantum quench problem, the length of the\ninteracting region in our experiment is finite. Thanks to the chiral nature\nof quantum-Hall\\ edge channels, the plasmon transport is unidirectional\nparticularly outside the interacting region. Non-equilibrium charge that\nleft the interacting region never returns back. This prevents unwanted\neffects in the leads and dissipative ohmic contacts, and allows us to\nneglect the finite-length problem. Such unidirectional transport cannot be\nexpected for conventional 1D wires connected to diffusive leads.\n\nIn this way, our experiment can be regarded as spatial analog of quantum\nquench.\n\n\\subsection{Conserved quantities during the transport}\n\nNon-equilibrium transport in integer quantum Hall edge channels has been\ninvestigated for a few decades. When the scattering between the channels is\nwell suppressed, the chemical potential of each channel is conserved during\nthe transport, allowing the system to have independent charge distributions\non the channels \\cite{ButtikerPRB1998,vanWeesPRB1989}. When the coupling to\nthe environment, such as phonon bath, is negligible, the heat or the\nelectron temperature of the system is conserved during the transport \\cit\n{GrangerPRL2009,JezouinScience2013,ProkudinaPRL2014}. More intriguingly,\nwhen the interaction is not fully ergodic to cause thermal equilibration,\nthe system may have infinite numbers of conserved quantities \\cit\n{BergesPRL2004}. Our spectroscopic analysis has successfully revealed\nnon-equilibrium electronic distribution with, at least, four conserved\nquantities (the binary spectrum parametrized by $T^{(\\mathrm{H})}$, $T^{\n\\mathrm{L})}$, $p$, as well as $\\mu $), which is greater than two ($\\mu $\nand $T$) for trivial cases.\n\nThe TL theory suggests that the plasmons are non-interacting and thus should\nbe conserved during the transport. This is supported by the fact that the\nnon-equilibrium spectrum is sustained for a long distance. This is\nattractive for carrying much information as plasmon excitations rather than\nshort-lived electronic excitations \\cite{JiNature2003,LevkivskyiPRB2008}.\n\n\\subsection{Possible relaxation mechanisms}\n\nIn previous reports, other energy transfer mechanisms such as phonons \\cit\n{ProkudinaPRB2010} or impurities \\cite{LundePRB2010} have been discussed to\nexplain the charge and heat transfer. Such extrinsic effects with random\nprocesses are expected to result in a continuous thermalization into a\ntrivial Fermi distribution. This contrasts with our binary spectrum\nindependent of the propagation length (from $\\sim $0 to 10 $\\mu $m), which\nimplies that the extrinsic thermalization processes becomes relevant only at\nlonger distances. This signifies the importance of Coulomb interaction in 1D\nsystems prepared in the integer quantum Hall regime.\n\n\\section{Summary}\n\nWe have investigated the energy spectrum of non-equilibrium states in\nquantum-Hall Tomonaga-Luttinger liquids. When non-equilibrium charge is\ninjected from a PC, a non-trivial binary spectrum consisting of high- and\nlow-temperature components appears in the IH region while a seemingly\nthermalized spectrum appears in the DH region. The binary spectrum is\nsustained even after travelling 5 - 10 $\\mu \\mathrm{m}$, much longer than\nthe length for electronic relaxation (about 0.1 $\\mu \\mathrm{m}$), without\nshowing significant thermalization. This can be compared with the simulation\nof quantum quench problem, which also suggest the emergence of an\napproximate binary spectrum as a non-equilibrium steady state. The\nlong-lived binary spectrum implies that the system is well described by\nnon-interacting plasmons, which suggests that low-energy excitations of edge\nchannels in the integer quantum Hall regime can be well understood as a TL\nliquid.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width = 3.375in]{B10.eps}\n\\end{center}\n\\caption{ Spectrum after travelling 10 $\\protect\\mu $m. (a) The energy\ndiagram at a negative bias $\\protect\\mu _{\\mathrm{a},\\uparrow }-\\protect\\mu\n_{\\mathrm{b},\\uparrow }=eV_{\\mathrm{D}}$ = -80 $\\protect\\mu $eV. (b) The\nlocation of DH and IH regions in the schematic channel layout. (c) Current\nspectrum of the QD measured with PC4 in the reversed magnetic field. Binary\nspectrum is seen on the left side of the peak only when the PC4 conductance\nis set in the tunneling regime ($G_{4}\\simeq 0.5G_{\\mathrm{q}}$). }\n\\end{figure}\n\n\\begin{acknowledgments}\nWe would like to thank Tsuneya Ando, Norio Kawakami, Tomohiro Sasamoto, and\nMasahito Ueda for fruitful discussions. This work was supported by Japan\nSociety for the Promotion of Science (JSPS) KAKENHI (Grant Numbers 21000004,\n12J09291, 24009291, 26247051, and 15H05854), International Research Center\nfor Nanoscience and Quantum Physics at Tokyo Institute of Technology, and\nNanotechnology Platform Program of the Ministry of\nEducation,Culture,Sports,Science and Technology, Japan.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nSix-dimensional supergravity is a promising arena in which to\nunderstand the relationship between possible low-energy theories of\nquantum gravity and UV completions in the context of string theory. In\nten dimensions, it has been shown that all supersymmetric theories of\ngravity coupled to gauge fields in flat space-time that satisfy\nanomaly cancellation and all other known quantum consistency\nconditions are realized, at the level of massless spectra, through\nsome version of string theory \\cite{Adams:2010zy, KimShiuVafaBranes}.\nThere has been recent progress towards showing that a similar ``string\nuniversality'' principle holds\nfor supergravity theories in eight (as well as nine and seven)\nspace-time dimensions \\cite{Montero:2020icj, Cvetic:2020kuw}.  In six\ndimensions, the space of theories becomes significantly more rich: six\nis the largest space-time dimension in which matter fields can arise in\nrepresentations other than the adjoint for a supersymmetric quantum\ntheory of gauge fields and gravity.  The space of 6D supergravity\ntheories is, however, still reasonably manageable. For the most\npart, the theory consists of a set of distinct continuous branches\nconnected by various transitions into a single large meta-moduli space.\nThe massless spectra in these different branches are strongly\nconstrained by gravitational, gauge-gravitational and pure gauge\nanomaly conditions.\n\nF-theory \\cite{VafaF-theory, MorrisonVafaI, MorrisonVafaII} provides a\npowerful nonperturbative tool for exploring a large part of the global\nmeta-moduli space of 6D supergravity theories in terms of the\ngeometric moduli spaces of elliptic Calabi--Yau threefolds, which are\nlinked together in a large connected network \\cite{GrassiMinimal,\nKumarMorrisonTaylorGlobalAspects, Taylor:2011wt, TaylorHodge} by transitions that\ncan change the number of tensor multiplets \\cite{Seiberg:1996vs,\nMorrisonVafaII}, as well as standard Higgs transitions and more\nexotic ``matter transitions'' \\cite{AndersonGrayRaghuramTaylorMiT}.\nThere is a close connection between the elliptic Calabi--Yau geometry of\nF-theory and the structure of the string charge lattice and anomaly\ncoefficients in the corresponding low-energy 6D supergravity theory,\nmaking it possible in many branches of the theory to clearly identify\nthe geometric ingredients needed for the UV completion through\nF-theory of the given branch of the low-energy 6D supergravity moduli\nspace (see, e.g.,\\cite{KumarMorrisonTaylorGlobalAspects}).  It was\nconjectured in \\cite{KumarTaylorStringUni6D} that string\nuniversality should hold for the massless spectra of 6D supergravity\ntheories, so that any massless spectrum associated with a\nsupersymmetric theory of gauge fields and gravity not realized in\nstring or F-theory should be provably inconsistent.  At this point in\ntime, however, despite ongoing progress both in understanding the\nrange of possible F-theory constructions and in identifying new\nquantum consistency conditions for 6D supergravity theories, there are\nstill many theories, including some infinite classes of theories, in\nthe apparent {\\it swampland} \\cite{VafaSwamp}\\footnote{Note: the term\n``swampland'' is used slightly differently by different authors.\nHere, we refer to the ``apparent swampland'' as those theories that\nsatisfy all known quantum consistency conditions for theories\ncoupled to quantum gravity but have no known string realization; a theory\ncan be removed from the apparent swampland if a string realization\nis found, or if a new quantum consistency constraint is identified\nthat rules out the theory.\nA recent review of work on swampland related problems, mostly focusing\non questions separate from the 6D supergravity issues addressed here,\ncan be found in \\cite{PaltiSwampland}.\n} of 6D low-energy\nsupergravity theories that obey anomaly cancellation as well as all\nother known quantum consistency conditions and yet have no known\nF-theory realization.\n\nIn this paper we find a unifying principle that\nunderlies a range of the theories in the current apparent 6D\nsupergravity swampland, and we formulate a precise conjecture for a class\nof theories that seem to be impossible to realize in F-theory despite\nsatisfying the known 6D quantum consistency conditions.\nThis paper was motivated by the observation that certain\napparently consistent $(\\SU(3) \\times \\SU(2) \\times \\U(1))\/\\mathbb{Z}_6$ models\ncannot be realized in F-theory by a universal Weierstrass realization\nidentified earlier by the authors \\cite{RaghuramTaylorTurnerSM}.\nInstead, the rank of the gauge group is forced to increase when the\nanomaly coefficients associated with the gauge factors are\nsufficiently large.  This kind of enhancement had also been seen in\nsimple $\\SU(2)$ models \\cite{MorrisonParkU1,\nMorrisonTaylorCompleteness}.  Further exploration has revealed that\nindeed many models that seem consistent but resist an F-theory\nrealization fit with this same general pattern.\nIn this paper, we generalize these observations in the formulation of a conjecture,\nand we elaborate on range of different situations where this kind of automatic\nenhancement occurs.\nIf the general conjecture we present here and its stronger forms for\nspecific gauge groups hold for general 6D supergravity theories it\nwould rule out many theories in the apparent 6D supergravity swampland.\n\nThe structure of this paper is as follows: In \\cref{sec:conjecture},\nwe formulate the central conjecture, which essentially states that\ngiven any\n6D supergravity theory that has a gauge group and matter content\ncontained within those of another consistent theory that can be\nrealized in F-theory, but where the former theory cannot be reached by\na Higgs deformation of the latter theory, the former theory does\nnot arise in F-theory.  In \\cref{sec:swamp}, we give a brief\nreview of some of the main types of theories that appear currently to\nbe in the 6D supergravity swampland.  In  the following two sections, we\ngo through a number of examples where we can\ndemonstrate explicitly that the\nconjecture holds for standard F-theory constructions.\n\\cref{sec:nonabelian} addresses a variety of examples with nonabelian\ngauge theories, including non-Higgsable clusters and theories with\ngauge group $\\SU(N)$, and \\cref{sec:abelian} describes examples with\nan abelian gauge group, including in particular some infinite families of\napparently consistent theories with gauge group $\\U(1)$ and\narbitrarily large charges that were previously known.\nIn \\cref{sec:local}, we give a slightly stronger characterization of the\nconjecture in terms of positivity\/effectiveness for a variety of\nthe specific\nsimple cases studied, and also give a sufficient local condition in some cases\nfor a theory to either be enhanced or broken in the F-theory\nrealization.\nIn \\cref{sec:exotic-discrete}, we consider some more complicated\ncases, focusing in particular on theories\nwith exotic nonabelian matter representations\nand those with discrete gauge groups. The discussion includes situations in which the\napplication of the conjecture formulated here is less clear but there\nare interesting related questions.\nIn \\cref{sec:lower-dimensions}, we briefly consider how the ideas\nof this paper may be extrapolated to lower-dimensional supergravity\ntheories such as $\\mathcal{N} = 1$ and $\\mathcal{N} = 2$ 4D supergravity\ntheories, and in \\cref{sec:further} we make some concluding remarks.\n\nThe work in this paper is complemented by a separate but related work\nalso nearing completion \\cite{MorrisonTaylorCompleteness} that\naddresses questions of completeness of the charge spectra and\nmassless charge spectra in 6D supergravity theories and their F-theory\nrealizations.\n\n\\section{Statement of the conjecture}\n\\label{sec:conjecture}\n\nThe basic proposal of this paper is the following conjecture for 6D\nF-theory and supergravity theories:\n\n\\newtheorem*{AEC}{Automatic Enhancement Conjecture}\n\\begin{AEC}\n    Given any anomaly-consistent 6D supergravity theory with gauge group $G$ and\n    massless matter hypermultiplets in a set of representations $M$, such that\n    there exists another anomaly-consistent theory with gauge group $G'\\supset G$\n    and matter representations $M'\\supseteq M$ that is realized in F-theory but\n    cannot be broken through a supersymmetry-preserving Higgsing process to the\n    theory with gauge group $G$, the theory with gauge group $G$ cannot be\n    realized in F-theory. Equivalently, trying to tune the gauge group and matter\n    content $G, M$ in an F-theory Weierstrass model leads to an automatic\n    enhancement of the singularity types to a theory with gauge group and matter\n    content $G' \\supset G, M' \\supseteq M$.\n\\end{AEC}\n\n\\noindent Some comments on this conjecture:\n\\begin{itemize}\n    \\item Note that the theory with gauge group and matter content\n    $G', M'$ must contain the theory with $G, M$ in the strong sense that\n    all other features of the theory must be the same.  In particular, the\n    string charge lattice that contains the anomaly coefficients\n    (associated with $H_{1,1} (B,\\mathbb{Z})$ of the compactification base $B$ in\n    an F-theory description; see, e.g., \\cite{KumarMorrisonTaylorGlobalAspects}) and the positivity cone\n    on this lattice (associated with the cone of effective divisors in\n    F-theory) must be the same, and the anomaly coefficients of the gauge\n    factors in $G$ must match their counterparts in $G'$.\n\n    \\item We have stated this conjecture in the context of F-theory.  Much of the bulk of this paper consists of a variety of examples where this conjecture holds for standard tunings in the F-theory context.  While in some cases, such as the situation of non-Higgsable clusters (\\cref{sec:non-Higgsable}), the conjecture can be explicitly proven, in other situations the evidence is only partial even in the F-theory context.  Nonetheless, it is tempting to speculate that this conjecture holds not only in F-theory but for general 6D supergravity theories.  If so, this would eliminate many of the known examples of 6D supergravity theories in the {\\it swampland} of theories that appear consistent by all known criteria but which are not realized in F-theory.\n\n    \\item This conjecture complements the {\\it Massless Charge\n    Sufficiency Conjecture} made in \\cite{MorrisonTaylorCompleteness},\n    which asserts that the massless charge lattice of a 6D supergravity\n    theory generates the full charge lattice of the theory.  As discussed\n    further in \\cref{sec:su2-quotient}, in some cases such as the enhancement of a\n    theory with gauge algebra $\\ensuremath\\mathfrak{su}(2)$ with a large anomaly coefficient,\n    the massless charge sufficiency conjecture is closely related to the\n    automatic enhancement conjecture and can be similarly interpreted as\n    leading to the addition of extra elements in the root lattice of the\n    gauge group $G$.\n\n    \\item On a practical level, enhancements typically occur because the choice of divisor classes necessary to obtain the correct matter spectrum causes the elliptic fibration to take some special structure. A parameter could be forced to be zero because its divisor class is ineffective, or multiple parameters may share common factors because their classes only admit reducible sections. And if a parameter's divisor class is trivial, the elliptic fibration may exhibit enhanced features such as the presence of additional rational sections. Since these effects occur in the geometry, one may naively wonder if the $G$,$M$ model could be obtained from the $G^\\prime$, $M^\\prime$ model by a T-brane deformation \\cite{CecottiCordovaHeckmanVafaTBranes,AndersonHeckmanVafaTBranes}. However, examples of the Automatic Enhancement Conjecture\\ exhibit a field-theoretic obstruction to Higgsing $G^\\prime$ to $G$. T-brane deformations should respect these obstructions, implying that automatic enhancement cannot be evaded in this fashion.\n\n    \\item The conjecture as stated above corresponds to the complete\n    6D supergravity theory and its F-theory realizations, and is stated in\n    a very general way.  Most of the\n    examples we describe here also admit a\n    more specific and stronger formulation in a way that depends on the\n    particular group\/algebra in question.  In\n    particular, in many cases the enhancement mechanism is\n    required when the anomaly\n    coefficients of the relevant gauge factors theory violate a particular\n    positivity condition, associated with an effectiveness condition for\n    the corresponding F-theory curves.  When this condition is violated\n    either an enhancement must occur or there cannot exist a compatible\n    F-theory model.  In many cases these positivity constraints lead to\n    sufficient conditions that can be formulated completely locally in the\n    F-theory geometry.  We illustrate  these positivity and local aspects in many\n    of the examples described in the following sections, and summarize\n    these versions of the conjecture in \\cref{sec:local}.\n\\end{itemize}\n\n\n\\section{Review of 6D supergravity, F-theory realizations, and the swampland}\n\\label{sec:swamp}\n\nIn this section we briefly review some aspects of the current apparent\n``swampland'' of 6D supergravity theories that appear consistent from\nall known quantum consistency conditions but that have no known\nF-theory realization.\n\n\\subsection{6D supergravity theories and their realizations in F-theory}\n\nA 6D $\\mathcal{N} = 1$ supergravity theory is characterized by a massless\nspectrum that contains a fixed number of tensor multiplets $T$, a\ngauge group $G$, and a set of hypermultiplet matter fields $M$ living\nin certain representations of the gauge group $G$.  The gauge group\ncan have nonabelian and abelian factors, as well as discrete\ncomponents.  The conditions of gravitational, gauge-gravitational, and\npure gauge anomaly cancellation \\cite{SagnottiGS,\nGreenSchwarzWest6DAnom} impose fairly strong constraints on the set\nof possible gauge groups and massless matter spectra.  When the number\nof tensor multiplets satisfies $T <9$, it has been proven that there\nare a finite number of distinct possible massless spectra $G, M$ for\ntheories with purely nonabelian gauge groups $G$\n\\cite{KumarMorrisonTaylorGlobalAspects}.  On the other hand, even\nwhen there are no tensor multiplets $(T = 0)$, there exist infinite\nfamilies of theories with a simple $\\U(1)$ gauge group and different\ncombinations of charged matter fields \\cite{ParkTaylorAbelian,\nTaylorTurnerSwampland} that satisfy anomaly cancellation and other\nknown consistency conditions.\n\nA wide range of 6D $\\mathcal{N} = 1$ supergravity theories can be\nrealized using F-theory \\cite{VafaF-theory, MorrisonVafaI,\nMorrisonVafaII}.  A six-dimensional F-theory model is defined by an\nelliptic Calabi--Yau threefold $X$, and can be thought of as a\nnonperturbative type IIB compactification on the (real\nfour-dimensional) complex K\\\"ahler surface $B$ that acts as the base\nof the elliptic fibration $X$.  Introductions to the relevant aspects\nof F-theory compactifications can be found in \\cite{Taylor:2011wt,\nWeigandTASI}.  The set of 6D F-theory models forms a connected\nmeta-moduli space; the allowed bases are connected through\nnonperturbative transitions that change the number of tensor\nmultiplets \\cite{Seiberg:1996vs, MorrisonVafaII}, and over any given\nbase $B$ the set of valid F-theory models forms a continuous connected\nmoduli space parameterized by the coefficients $f, g$ in the\nWeierstrass model\n\\begin{equation}\n y^2 = x^3 + fx + g \\,.\n\\end{equation}\nHere $f, g$ can be thought of as functions on the base $B$, or more\nprecisely as sections of line bundles $\\mathcal{O} (-4\\canonclass), \\mathcal{O} (-6\n\\canonclass)$, where $\\canonclass$ is the canonical class of the base.  Codimension-one loci where the elliptic fiber becomes singular lead to nonabelian\nfactors of the gauge group $G$, while abelian and discrete factors\nhave a more subtle global characterization.  Codimension-two loci\nwhere the elliptic fiber becomes singular in general characterize\nmassless matter fields in the 6D supergravity theory.  There is thus a\nclose correspondence between the geometry of the F-theory\ncompactification and the structure of the low-energy 6D supergravity\ntheory.\n\nOne particularly important connection between low-energy physics and\nF-theory is that there is a simple geometric interpretation of the\nanomaly coefficients $a, b_i, \\tilde{b}_\\alpha$ associated with terms\nin the 6D supergravity action of the form $B\\wedge R\\wedge R, B\\wedge\nF_i\\wedge F_i, B\\wedge F_\\alpha\\wedge F_\\alpha$ respectively, with\n$F_i, F_\\alpha$ denoting nonabelian and abelian gauge field strength\nfactors. In particular, $a$ corresponds to the canonical class $\\canonclass$,\nand $b_i$ corresponds to the locus $C_i$ in $B$ where the nonabelian\ngauge factor $G_i$ is supported.  Note that for $k$ abelian factors\nthere are anomaly coefficients $\\tilde{b}_{\\alpha \\beta}$ associated\nwith terms $B \\wedge F_\\alpha \\wedge F_\\beta$; we use the notation\n$\\tilde{b}_\\alpha = \\tilde{b}_{\\alpha \\alpha}$.  These are special\ncases of the more general identification between the string charge\nlattice $\\Lambda$ of the 6D supergravity theory, in which $a, b_i,\n\\tilde{b}_\\alpha$ are elements, and the lattice $H_{1, 1}(B,\\mathbb{Z})$ given\nby the set of algebraic curves in the base $B$, including $-\\canonclass$ and\n$C_i$, with the natural intersection form on homology.  Because of\nthis isomorphism, we will frequently move freely back and forth\nbetween $a, b_i \\in\\Lambda$ and $K_B, C_i \\in H_{1, 1}(B,\\mathbb{Z})$ in\nvarious parts of the discussion.  A useful feature of\nthis correspondence is that the arithmetic genus of a (possibly\nsingular) curve $C_i$ supporting a nonabelian gauge factor $G_i$ is\nencoded in the anomaly coefficients of the low-energy theory through\n\\cite{KumarParkTaylorT0, KleversEtAlExotic}\n\\begin{equation}\n\\label{eq:genus}\ng_i =\\frac{1}{2} (a \\cdot b_i + b_i \\cdot b_i) + 1\\,.\n\\end{equation}\n\nTo organize our brief review of some classes of models in the 6D\nsupergravity swampland, we will find it helpful to distinguish\n``generic'' from exotic matter fields for a given gauge group.  A\nprecise definition of which matter fields are generic for a 6D\nsupergravity theory with  a fixed gauge group $G$ (and for fixed\nanomaly coefficients, which are assumed to be not too large)\ncan be given by identifying the generic matter fields as those that\narise on the branch of moduli space of greatest dimension for fixed\n$G$ and associated anomaly coefficients \\cite{TaylorTurnerGeneric}.\nThis definition of generic turns out to match with the simplest types\nof codimension-two singularities that give matter in the F-theory\npicture and also ties in naturally to the structure of the anomaly\nconstraints: at least for simple theories, the number of generic\nmatter representations is equal to the number of anomaly constraints,\nso in these cases there is a unique generic matter content for fixed\n$G$ once the anomaly coefficients are fixed.\n\nAnother  aspect of F-theory constructions that is relevant for the analysis of\nthis paper is that for a given\ngauge algebra $\\ensuremath\\mathfrak{g}$, the moduli space of models with matter fields in\nthe generic\nrepresentations  is captured in many cases by a\n``universal''\\footnote{In \\cite{RaghuramTaylorTurnerSM},  these universal\nWeierstrass model constructions were referred to as ``generic'';\nhere we change terminology to ``universal'' to avoid confusion with\nother uses of the term generic.}  Weierstrass model, with a general\nalgebraic form parameterized by the classes of the curves supporting\nthe gauge factors.  In \\cite{RaghuramTaylorTurnerSM} we developed a\nsystematic Jacobian-based methodology for checking that a given family of Weierstrass\nmodel constructions is universal by checking that the number of\nindependent free complex parameters in the Weierstrass model matches the dimensionality of the branch of the moduli\nspace determined by the number of uncharged scalars.  Such universal\nWeierstrass models include, for example, the well-known Tate tunings\nof most nonabelian gauge factors \\cite{BershadskyEtAlSingularities},\nthe Morrison--Park $\\U(1)$ Weierstrass models \\cite{MorrisonParkU1},\nand the  universal $(\\SU(3) \\times \\SU(2) \\times \\U(1)) \/ \\mathbb{Z}_6$ models\ndeveloped in \\cite{RaghuramTaylorTurnerSM}.  Note that universal\nWeierstrass models are also known for some classes of models\ncontaining specific combinations of representations that include\nexotic matter.\n\n\\subsection{The 6D supergravity swampland}\n\nWhile for many classes of theories there is a fairly close match\nbetween the set of possible F-theory constructions and the 6D $\\mathcal{N} = 1$ supergravity theories that are allowed from anomaly\ncancellation and other known quantum consistency conditions, there are\nalso many theories that appear to be in the swampland.  In\n\\cite{KumarMorrisonTaylorGlobalAspects, ParkTaylorAbelian, TaylorTurnerSwampland}, for\nexample, some infinite families of apparent swampland models were\nidentified.  In general, both supergravity and F-theory models are\nunderstood best for theories with no tensor multiplets ($T = 0$), and\nwith nonabelian gauge groups and generic matter representations.  The\nstructure of the theories becomes more complicated as the number of\ntensor multiplets increases, particularly at $T \\ge 9$, when abelian\nand discrete gauge factors are included, and when exotic (non-generic)\nmatter representations are considered.  Over the last decade or so,\nthere has been gradual progress in better understanding both F-theory\nconstructions of these more complicated classes of models and also in\nidentifying further quantum consistency constraints that must be\nsatisfied by any consistent $\\mathcal{N} = 1$ 6D supergravity theory; as\nthese theories are better understood, further classes of apparent\nswampland models have also been identified.  We briefly review some of\nthis work here, first summarizing some additional constraints on 6D\nsupergravity and F-theory constructions, and then focusing on some of\nthe simplest and largest classes of apparent swampland models, for\nwhich the Automatic Enhancement Conjecture\\ is relevant.\n\n\\subsubsection{Constraints on 6D supergravity theories}\n\\label{sec:constraints}\n\nIn addition to the constraints of local and global anomaly\ncancellation, a consistent 6D $\\mathcal{N} = 1$ supergravity theory must\nsatisfy a variety of further quantum consistency conditions.  We\nbriefly summarize some of these further constraints here.\n\nOne simple constraint is that the gauge kinetic term proportional to\n$F^2$ must have a negative coefficient; otherwise the theory has a\nperturbative instability.  This leads through\nsupersymmetry to the condition that the anomaly coefficients $b_i,\n\\tilde{b}_\\alpha$ must lie in the positive cone of the 6D string charge\nlattice $\\Lambda$.  This was used, for example, in\n\\cite{KumarTaylorBound} to rule out an infinite family\nof models that had appeared to lie in the swampland \\cite{SchwarzAnomaly}.\n\nAnother constraint, proven in \\cite{SeibergTaylorLattices}, is that the charge\nlattice $\\Lambda$ must be unimodular (self-dual)\nunder the Dirac pairing between\nstrings; this follows automatically from F-theory, where the lattice\n$H_{1, 1}(B,\\mathbb{Z})$ is always unimodular, but  this\ncondition must also hold more generally in 6D supergravity, ruling out other classes of\npotential swampland models.\n\nSome further constraints on the anomaly coefficients $a, b_i,\n\\tilde{b}_\\alpha$ were identified in \\cite{MonnierMooreParkQuantization,\nMonnierMooreGS} using global and local anomaly cancellation\nconditions (a subset of these constraints were also found in\n\\cite{Raghuram34, TaylorTurnerSwampland}).  These constraints are\nstronger if one makes the additional assumption that the 6D theory can\nbe defined on any spin manifold with an arbitrary gauge bundle.  For\nexample, under these assumptions the anomaly coefficient $a$ must be a\n{\\it characteristic vector} in the subspace of $\\Lambda$ spanned by\nthe nonabelian anomaly coefficients $b_i$, meaning that $a \\cdot b_i +\nb_i \\cdot b_i \\in 2\\mathbb{Z}$. This is true for all\nF-theory compactifications (since, e.g., the genus \\labelcref{eq:genus} is\nalways an integer), but can rule out some theories that might\nappear to be in the swampland.\n\nThere are a number of further constraints that are known to hold in\nF-theory but have not  been proven as consistency conditions for general\nquantum 6D supergravity theories.\nTheories satisfying the known 6D supergravity quantum consistency\nconditions but violating these F-theory constraints appear to be in\nthe swampland, pending a proof that these conditions are more\ngenerally necessary for 6D supergravity consistency.  The primary goal\nof this paper is to present evidence that the automatic enhancement\nconjecture should be a similar F-theory constraint.  We summarize\nbriefly here some other constraints in this class.\nNote that beyond standard F-theory models there are also other\npossible constructions such as the ``frozen'' phase of F-theory\n\\cite{Bhardwaj:2018jgp}, which can realize some  models that otherwise\nmay appear to be in the swampland.  As far as we know the models we\nstudy here do not have any such realization through any formulation of\nstring theory including the standard class of F-theory constructions.\n\nOne constraint that follows directly from F-theory is the \\emph{Kodaira\nconstraint} \\cite{KumarMorrisonTaylorGlobalAspects}, which states\nthat $-12a-\\sum_{i}\\nu_ib_i$ must lie in the positivity cone of\n$\\Lambda$, which corresponds to the cone of effective divisors in\n$H_{1, 1} (B,\\mathbb{Z})$. Here $\\nu_i$ corresponds to the number of 7-branes\nneeded to form the nonabelian factor $G_i$ of the gauge group,\ne.g., for $\\SU(N), \\nu = N$.  Various 6D supergravity theories that\nlook otherwise consistent violate this condition and hence cannot be\nrealized in F-theory; we mention some explicit examples later in this\nsection.\n\nAnother constraint that is widely expected to hold in all consistent\nquantum theories of gravity in dimensions $D >3$ is the condition of\ncharge completeness \\cite{Banks:2010zn}, which states that there exist\nexcitations of the theory with all possible charges in the weight\nlattice of the symmetry group $G$. This was proven recently for all\nquantum gravity theories with a holographic dual\n\\cite{Harlow:2018tng}.  This condition is proven for all 6D F-theory\nmodels in \\cite{MorrisonTaylorCompleteness} using Poincar\\'{e}\nduality, although there is no general proof for flat space 6D\nsupergravity theories.\n\nA further class of constraints can be identified if one assumes that\nthe charge completeness hypothesis holds in the sense that there are\ndynamical string excitations for every charge in the string lattice\n$\\Lambda$.  Under this assumption, there must be a consistent\nanomaly-free theory on the worldvolume of such strings, which leads to\nadditional constraints on the set of allowed spectra for 6D\nsupergravity theories.  This method was used in \\cite{Banks:1997zs},\nfor example, to show that the rank of the gauge group of 10D type I\nstring theory must match with the known value of $\\ensuremath\\mathfrak{so}(32)$.  In\n\\cite{KimShiuVafaBranes}, this approach was used to show that\nconsistency of the theory on the string worldvolume places additional\nconstraints on the spectra of 6D supergravity theories; among other\nthings, these constraints would rule out several infinite families of\napparent swampland models with $T > 9$ that were found in\n\\cite{KumarMorrisonTaylorGlobalAspects}.  In\n\\cite{LeeWeigandSwampland}, this approach was used to place bounds on\nthe number of abelian gauge factors that can arise in a 6D\nsupergravity theory, improving on previous bounds found from anomaly\nconditions in\n\\cite{ParkTaylorAbelian}.\n\nThe\nfocus in this paper is on 6D supergravity models that obey all of the\nabove-mentioned constraints but still have no F-theory realization.\nTo organize the discussion we summarize some of the models that are\nstill in the apparent swampland, including some that violate the\nconditions above that are known to hold in F-theory vacua but are\notherwise apparently consistent 6D supergravity theories.\n\n\\subsubsection{Apparent swampland for $T = 0$, nonabelian $G$, and\ngeneric matter}\n\nWe begin with the simplest set of cases, with no tensors, a\nnonabelian gauge group, and generic matter.  Because of the absence of\ntensor fields, the anomaly coefficients $a, b_i$ are integers.  We\nfocus first on theories with gauge group $\\SU(N)$ and then consider\nother gauge groups.  For $G =\\SU(N)$, generic matter fields are the\nfundamental and adjoint representation, as well as the two-index\nantisymmetric representation when $N \\ge 4$\n\\cite{TaylorTurnerGeneric}.\n\nWith gauge group $\\SU(2)$, there are anomaly-allowed theories for\nanomaly coefficients $1 \\le b \\le\n12$ \\cite{TaylorTurnerGeneric}.  Naively, it seems that these\ntheories can all be constructed directly in F-theory by directly tuning the\nappropriate gauge factor in the Weierstrass model.  When $b = 9, 10,\n11, 12$, however, complications arise.  In the case $b = 12$, as\ndiscussed further in \\cite{MorrisonTaylorCompleteness}, this\nconstruction gives a theory with $\\mathbb{Z}_2$ Mordell--Weil torsion, so the\ngauge group is actually $\\SO(3)$; in this case, the massless charge\nsufficiency conjecture stated in that paper  asserts that the pure\n$\\SU(2), b = 12$ theory cannot be realized in F-theory.\nFor $b = 10, 11$, it was observed in \\cite{MorrisonTaylorCompleteness}\nthat the naive tuning of this model gives an extra $\\SU(2)$ factor and\nMordell--Weil torsion so $G = (\\SU(2) \\times \\SU(2))\/\\mathbb{Z}_2$.  These\ncases, as well as the case $b = 9$, are in the apparent swampland and\nsatisfy the conditions of the Automatic Enhancement Conjecture, and are described in more\ndetail in \\cref{sec:su2su2} and \\cref{sec:su2u1}.\n\nFor gauge group $\\SU(N)$, there are anomaly-allowed theories for $1\n\\le b \\le 24\/N$.  Again, naively these can all be constructed using\na Tate tuning \\cite{BershadskyEtAlSingularities}.  But again, in some\ncases the resulting model does not have the desired $\\SU(N)$ gauge\ngroup.  It was pointed out in \\cite{MorrisonTaylorMaS,\nJohnsonTaylorEnhanced} that for $b = 1$ this construction does not\ngive theories with gauge groups $\\SU(21)$ and $\\SU(23)$ and generic\nmatter, so these are in the apparent swampland.  These cases, as well\nas the other cases with $N \\ge 18$ are examples of the Automatic Enhancement Conjecture,\nas we describe in more detail in \\cref{sec:t0-general-n}.  Note that\nthe case of $\\SU(24)$, which lacks fundamental matter and has an\nactual gauge group $\\SU(24)\/\\mathbb{Z}_2$ is an example where Mordell--Weil\ntorsion is forced, in accord with the Massless Charge Sufficiency\nConjecture \\cite{MorrisonTaylorCompleteness}.\n\n\nSimilar apparent swampland theories arise for $\\ensuremath\\mathfrak{sp}(N)$ and $\\ensuremath\\mathfrak{so}(N)$; we discuss some cases in \\cref{sec:spsoalgebras}.\n\n\\subsubsection{Apparent swampland with abelian $G$}\n\nWhen we include abelian $\\U(1)$ gauge factors, the story becomes more\ncomplicated.   In F-theory, $\\U(1)$ factors come from global features\nof the geometry, which makes them more subtle to understand.\nFor theories with a gauge group $\\U(1)$ and generic $(q = 1, 2)$\nmatter, there are apparent swampland models closely related to the\nsimplest $\\SU(2)$ models discussed above; we describe in\n\\cref{sec:abelian-generic} how these models also satisfy the Automatic Enhancement Conjecture.\nEven for theories with a single $\\U(1)$ as the gauge\ngroup and $T = 0$, when larger charges are considered\n there are infinite families of charge configurations\nthat satisfy anomaly cancellation conditions\n\\cite{ParkTaylorAbelian,TaylorTurnerSwampland}, even though there can only be a finite\nnumber of distinct spectra compatible with F-theory constructions\n\\cite{KumarMorrisonTaylorGlobalAspects}.  In \\cref{sec:abelian-infinite}, we\nshow that at least the simplest of these families satisfies the\nconditions of the Automatic Enhancement Conjecture.\n\nCombining nonabelian and abelian factors, one can also find infinite\nfamilies of anomaly-free theories with gauge algebras of the form\n$\\ensuremath\\mathfrak{su}(N) \\oplus \\ensuremath\\mathfrak{u}(1)$, for example. (Some infinite families of this\ntype with $T = 1$ were constructed in \\cite{ParkTaylorAbelian}, and\nsimilar constructions are possible with $T = 0$.)  In\n\\cref{sec:abelian}, we show that the previously identified infinite\nclasses of theories of this kind satisfy the conditions of the\nstronger version of the Automatic Enhancement Conjecture\\ that would render these theories\ninconsistent, although we also identify some further infinite families\nfor which the application of the conjecture is less\ntransparent. We also show in\nthis section that the Automatic Enhancement Conjecture\\ applies to some theories with\ngauge group $(\\SU(3) \\times \\SU(2) \\times \\U(1)) \/ \\mathbb{Z}_6$ when\nconstructed with the universal Weierstrass form identified in\n\\cite{RaghuramTaylorTurnerSM}.\n\nGoing beyond single $\\U(1)$ factors, at this point our understanding\nof F-theory models with multiple $\\U(1)$ factors and\/or discrete gauge\ngroup factors is still sufficiently limited that it is difficult to\nclearly delineate the swampland.  This is in large part because\nabelian $\\U(1)$ and discrete factors come from global aspects of the\nF-theory geometry, unlike nonabelian gauge factors that come from\nlocal structure.  We do not attempt to give any systematic analysis\nhere of theories with multiple $\\U(1)$ factors or discrete gauge\nfactors, though we make some general comments about theories with\ndiscrete gauge symmetries in \\cref{sec:discrete}.\n\n\\subsubsection{Apparent swampland with $T > 0$}\n\\label{sec:tg0}\n\nWhen the number of tensor multiplets is nonzero, $T>0$, the structure\nof the theory becomes more complicated, and questions about the\nswampland have additional complications.  The anomaly coefficients $a,\nb_i, \\tilde{b}_\\alpha$ become vectors as  $\\Lambda$ becomes a signature\n$(1, T)$ lattice, corresponding to $H_{1, 1}(B,\\mathbb{Z})$.  There is a\npositivity cone for strings in $\\Lambda$, which in the F-theory\ndescription corresponds to effective divisors in $B$ but which is not\nas well understood in the 6D supergravity context.  A variety of\napparent swampland models may arise corresponding to lattices\n$\\Lambda$ with positivity cones that do not match any F-theory\ngeometry.  The condition identified in\n\\cite{MonnierMooreParkQuantization} that $a$ must be a characteristic\nvector, at least for the nonabelian anomaly coefficients, puts some\nconcrete constraint on the allowed combinations of $\\Lambda, a$, but\nthere are still many theories at larger $T$ that appear consistent and\nyet have no F-theory realization.  We focus here on some of these that\nare most relevant for this story.\n\nThe most basic new feature that can arise at $T > 0$ is the appearance\nof a vector $x \\in\\Lambda$ that lies in the positivity cone and\nsatisfies $x \\cdot x <0$.  When an F-theory construction involves a\nbase $B$ that contains an effective curve $C$ with $C \\cdot C <- 2$,\nthe negative normal bundle on $C$ forces a Kodaira singularity in the\nelliptic fibers over that curve that gives a gauge group of at least\n$\\SU(3)$ in the corresponding 6D supergravity theory\n\\cite{MorrisonTaylorClusters}.  From the point\nof view of supergravity, there is no obvious inconsistency with an\nelement $x$ satisfying $x \\cdot x <-2$ lying in the positivity cone\nwithout a corresponding gauge factor in the theory.  Thus, such models\nlie in the apparent swampland.  In \\cref{sec:non-Higgsable}, we show\nthat these models are a simple example of the Automatic Enhancement Conjecture.  Note that\nunder the assumption of charge completeness for the string charge\nlattice, one could\nalternatively argue\nthat anomaly cancellation on the worldvolume of a string with string\ncharge $x$ can only be satisfied in the presence of the appropriate\nnon-Higgsable gauge factor $G\\supset\\SU(3)$, using analysis along the\nlines of \\cite{Haghighat:2014vxa, Kim:2016foj, Shimizu:2016lbw, KimShiuVafaBranes}.\n\nAlready with a single tensor multiplet $(T = 1)$, the  range of\npossible theories expands significantly and there are more theories\nthat have been identified in the apparent swampland.\nIn particular, in \\cite{ParkTaylorAbelian} there are infinite families\nof apparent swampland models identified with $T = 1$ and gauge algebras\n$\\ensuremath\\mathfrak{u}(1)$ and $\\ensuremath\\mathfrak{su}(13) \\oplus \\ensuremath\\mathfrak{u}(1)$.  Both of these families  can be\nruled out by the Automatic Enhancement Conjecture, as we discuss in \\cref{sec:abelian}.\nThe $\\U(1)$ family is closely related to the infinite family of\n$\\U(1)$ theories at $T = 0$ found in \\cite{TaylorTurnerSwampland}, but\nthe $\\ensuremath\\mathfrak{su}(13) \\oplus\\ensuremath\\mathfrak{u}(1)$ family is somewhat more subtle.\n\nWhen $T  \\ge 9$, the classification of theories gets even more\ncomplicated, due in particular to the fact that $a \\cdot a = 9-T$\nceases to be positive (this condition follows from gravitational\nanomaly cancellation).\nAn illustration of some of the complications that arise at $T \\ge 9$\nis given in  \\cref{app:adjoint}.\nFor $T \\ge 9$, there is no proof known that even constrains the\nnumber of distinct spectra for nonabelian gauge groups and matter to\nbe finite, though as mentioned earlier some of the infinite families of\napparent swampland models identified in\n\\cite{KumarMorrisonTaylorGlobalAspects} violate the conditions found\nin \\cite{KimShiuVafaBranes} that hold when string charge completeness\nis assumed.\n\n\\subsubsection{Apparent swampland with exotic  matter for nonabelian $G$}\n\nIn addition to the generic matter representations for a nonabelian gauge group\n$G$ there are also non-generic matter representations possible that\ncan still satisfy anomaly cancellation.  Three-index antisymmetric\nrepresentations of $\\SU(N)$ for $N = 6, 7, 8$ are fairly well\nunderstood \\cite{BershadskyEtAlSingularities, MorrisonTaylorMaS,\nGrassiMorrisonAnomalies, AndersonGrayRaghuramTaylorMiT} and can\narise through a Tate-type tuning of the Weierstrass model\n\\cite{HuangTaylorComparing}.  Related exotic multi-charged matter\nrepresentations include, e.g., the $(\\bm{6}, \\bm{2})$ of $\\SU(4) \\times\\SU(2)$\nand the trifundamental of $\\SU(2)$. More exotic\nmatter representations can be characterized as having a non-vanishing\n``genus'' contribution \\labelcref{eq:genus}, corresponding to the arithmetic genus of a\nsingularity in the divisor supporting the gauge factor in an F-theory\nrealization.  In \\cite{KleversEtAlExotic}, it was argued that only a\nfairly limited set of exotic matter representations for nonabelian groups\ncan be realized in\nF-theory without having a codimension two locus where $f, g$ vanish to\ndegrees (4, 6), which generally signifies a superconformal sector\n\\cite{Seiberg:1996qx, Heckman:2013pva, DelZotto:2014hpa}.  These\nlimited higher-genus exotic  matter representations that can be\nrealized through F-theory models on a singular gauge divisor basically\ncontain only the two-index symmetric representation of $\\SU(N)$, the\nthree-index symmetric representation of $\\SU(2)$,and analogous\nrepresentations of $\\Sp(N)$.  It has been suggested that some other\nexotic representations such as the $(\\bm{56},  \\bm{2})$ of $\\gE_7 \\times \\SU(2)$,\nwhich naively involves a (4, 6) locus, may be possible without a\nsuperconformal sector \\cite{Cvetic:2018xaq} through T-brane magic.\nThough this is not well understood in F-theory, it is believed that there are\nheterotic models containing this matter representation without\nsuperconformal sectors.\n\nWith this general classification of exotic matter, there are two kinds\nof apparent swampland theories.  First, there are anomaly-consistent\ntheories containing the exotic matter representations that can arise\nfrom F-theory constructions on singular divisors, but in\nconfigurations that do not seem to arise from F-theory.  Some examples\nof such apparent swampland theories were described in\n\\cite{KleversEtAlExotic}; we show in \\cref{sec:exotic-F-theory} that\nsome such examples satisfy the conditions of the Automatic Enhancement Conjecture.  Second,\nare any theories that contain matter in representations of nonabelian\ngauge factors not believed to arise in consistent F-theory models.  We\nmake some brief comments on such models in\n\\cref{sec:exotic-non-F-theory}.\n\n\\section{Nonabelian Examples}\n\\label{sec:nonabelian}\n\n\\subsection{Non-Higgsable Clusters}\n\\label{sec:non-Higgsable}\n\nOne of the simplest situations in which automatic\nenhancement must occur in F-theory is when $G$ is trivial and there is\na primitive positive charge $q$ in the string charge lattice with\nDirac pairing $q \\cdot q <-2$ with itself and $q \\cdot a =- q \\cdot q\n  -2$ with the $B R R$ anomaly coefficient $a$.  In any F-theory\nconstruction where the complex surface base $B$ on which the theory\nis compactified contains a curve $C$ of self-intersection $-3$ or\nbelow, everywhere in the F-theory moduli space\nof elliptically fibered Calabi--Yau threefolds parameterized by\nWeierstrass models over $B$\nthere is a\nsingularity in the elliptic fibration that forces at least a gauge\nalgebra $\\ensuremath\\mathfrak{su}(3)$ with an anomaly coefficient $b= C$, where the\nstring charge lattice containing the anomaly coefficient is\nidentified with $H_{1,1} (B,\\mathbb{Z})$ in any 6D F-theory model\n\\cite{MorrisonTaylorClusters}.  Thus, in such a theory with a\nprimitive positive charge $b$ where $b \\cdot b = -3$, $a \\cdot b =\n1$ there is an automatic enhancement $G = 1 \\to G' \\supseteq \\SU(3)$.\n the self-intersection $b \\cdot b$ decreases, the rank of the forced\ngauge group increases, with, e.g., $b \\cdot b = -4 \\Rightarrow G'\n\\supseteq \\SO(8)$, \\textellipsis, $b \\cdot b < -8 \\Rightarrow G' = \\gE_8$, and\nwith $b \\cdot b \\le -13$ not allowed in any F-theory model.\nIn all these situations, the gauge group $G'$ is ``non-Higgsable'' in the\nsense that there is no local Higgsing deformation that can break the\ngauge group while preserving supersymmetry.\nIn particular, in these\ncases there is no Higgsing from $G' \\to G$, as expected in the\nAutomatic Enhancement Conjecture.\n\nNote that in this case the Automatic Enhancement Conjecture\\ can be stated purely locally:\nwe expect that any charge $b$ in the string charge lattice $\\Lambda$\nthat satisfies $b \\cdot b < -2$ and satisfies $a \\cdot b = -2-b \\cdot\nb$ (corresponding to $g = 0$ from \\cref{eq:genus}) must support a\ngauge factor $G' \\supseteq \\SU(3)$, with correspondingly larger\nfactors for more negative $b \\cdot b$.  Furthermore, for similar\nreasons there cannot be any a curve in the positivity cone with $g >\n0$ and $b \\cdot b < 0$ in any F-theory construction\n  \\cite{MorrisonTaylorClusters}.\n\nIn this class of automatic enhancements there is a complete proof that\nthe enhancement must occur in the context of F-theory.  In other\ncases, our understanding of the F-theory geometry is less complete and\nwhile we can give many examples of standard tunings that lead to\nautomatic enhancement, the proof that this must always occur even in\nthe F-theory context is less complete.  Whether the automatic\nenhancement associated with non-Higgsable clusters can be proven\npurely in the 6D supergravity context through other approaches is also\ndiscussed briefly in \\cref{sec:tg0}.\n\n\\subsection{Automatic enhancement of $\\ensuremath\\mathfrak{su}(2)$ gauge algebra}\n\\label{sec:su2-enhancement}\n\nAnother very simple set of examples of the kind of automatic\nenhancement implied by the Automatic Enhancement Conjecture\\ arises in the case of 6D\nF-theory models with gauge group $\\SU(2)$, generic matter (fundamental\nand adjoint representations), and no tensor multiplets (associated\nwith the base $\\mathbb{P}^2$), where the anomaly coefficient for the\ngauge group is taken to be $b=10,11$.  This was one of the first\nexamples of the automatic enhancement mechanism observed, and was\nencountered in \\cite{MorrisonParkU1, MorrisonTaylorCompleteness}. We\ndescribe how the automatic enhancement occurs in F-theory for this\nkind of theory and then describe some further generalizations thereof.\n\n\\subsubsection{$\\ensuremath\\mathfrak{su}(2) \\to (\\SU(2) \\times\\SU(2))\/\\mathbb{Z}_2$\nautomatic enhancement}\n\\label{sec:su2su2}\n\nA universal form for the Weierstrass model giving a gauge algebra $\\ensuremath\\mathfrak{su}(2)$\nand the generic fundamental and adjoint matter representations\ncan be constructed  either from the\nTate form \\cite{BershadskyEtAlSingularities, KatzEtAlTate}, or by direct\nanalysis of the general Weierstrass model with the appropriate Kodaira\nsingularity type \\cite{MorrisonTaylorMaS}.\n\\footnote{We assume here that the $\\ensuremath\\mathfrak{su}(2)$ algebra is realized\nthrough a UFD-type construction. Non-UFD constructions\ntypically give exotic matter representations\n\\cite{KleversEtAlExotic}. Although in principle it\nmay be possible that in\nsome special cases non-UFD models can be realized with only generic\nmatter types, the analysis of \\cref{app:u1u1test} suggests that this\nwill not occur in practice.}\nSuch a universal  Weierstrass model with\ngauge algebra $\\ensuremath\\mathfrak{su}(2)$ realized on a curve $\\sigma = 0$\ntakes the form\n\\begin{equation}\n\\label{eq:su2}\n\\begin{aligned}\n    f & = -\\frac{1}{48} \\phi^2 + \\sigma f_1\\,, \\\\\n    g &=\\frac{1}{864} \\phi^3 -\\frac{1}{12} \\phi \\sigma f_1 + \\sigma^2 ({g}_2)\\,.\n\\end{aligned}\n\\end{equation}\nThe classes of $f, g, \\sigma$ in the F-theory base are $[f] = -4 K_B,\n[g] = -6 K_B, [\\sigma] = C$, where $K_B$ is the canonical class of the\nbase.  In the 6D supergravity theory, $K_B$ and $C$ correspond to the\nanomaly coefficients $a, b$ of the $B R^2, B F^2$ terms respectively;\nwe freely go back and forth between the geometric (curve in $H_{1, 1}\n(B,\\mathbb{Z})$) and field theory (anomaly coefficient in the 6D supergravity\nstring charge lattice) notations $K_B \\leftrightarrow a, C\n\\leftrightarrow b$ throughout the discussion in this and subsequent\nsections.  The discriminant locus associated with the Weierstrass\nmodel \\labelcref{eq:su2} is\n\\begin{equation}\n\\Delta = \\frac{1}{16}  \\phi^2 (\\phi g_2-f_1^2)  \\sigma^2 +\\mathcal{O} (\\sigma^3)\\,.\n\\end{equation}\n\nThe class of the coefficient $g_2$ is $[g_2] = -6 a -2 b$.  When $b$\nis sufficiently large, this class becomes ineffective and $g_2 = 0$.\nIn such a situation, the discriminant becomes\n$\\Delta = \\sigma^2 f_1^2 (-\\phi^2 + 64f_1 \\sigma)\/16$.  This\ndiscriminant is divisible by $f_1^2$, and so\nthere is an automatic enhancement of {\\it at least}\nan extra $\\ensuremath\\mathfrak{su}(2)$ algebra on the\nvanishing locus of $f_1$, where $[f_1] = -4a-b$.  As shown in\n\\cite{MorrisonTaylorCompleteness}, the resulting Weierstrass model\nalso has $\\mathbb{Z}_2$ torsion, so the gauge group becomes $G' =(\\SU(2)\n\\times\\SU(2))\/\\mathbb{Z}_2$.\nThe fact that the enhanced gauge group arises automatically from the\nWeierstrass tuning\n(which is the universal and minimal (UFD) tuning needed for the $\\ensuremath\\mathfrak{su}(2)$ algebra) implies that there is no Higgsing from the $G'$\ntheory to the anomaly-free theory with algebra $\\ensuremath\\mathfrak{su}(2)$.\nThis corresponds to the absence of adjoint matter fields under the\nextra $\\ensuremath\\mathfrak{su}(2)$ algebra factor on $f_1 = 0$\n(for further explanation\/discussion on this point, see \\cref{app:adjoint}).\n\n\\paragraph{Simple examples}\n\nThe simplest cases in which this enhancement occurs is when\nthe base is $\\mathbb{P}^2$ and the anomaly coefficient is $b = 10$ or $b =\n11$.   In these cases an additional $\\ensuremath\\mathfrak{su}(2)$  factor arises on $f_1$\nwith $[f_1] = 2, 1$ respectively.\nSince degree one and two curves on $\\mathbb{P}^2$ are rational curves of\ngenus 0, the additional factor carries no adjoint matter.\nFurthermore, the only fundamental matter fields for the additional\n$\\ensuremath\\mathfrak{su}(2)$ are also charged under the original $\\ensuremath\\mathfrak{su}(2)$,\\footnote{This\ncan be seen from the form of the discriminant locus; the only loci\nover $f_1= 0$ where the vanishing order of $\\Delta$ increases are\nthose where either $\\sigma = 0$, corresponding to matter jointly\ncharged under both $\\ensuremath\\mathfrak{su}(2)$ factors, and those where $\\phi = 0$,\ncorresponding to an enhancement to a Kodaira type III singularity that\ncarries no matter.} so the\nenhanced $G' = (\\SU(2) \\times\\SU(2))\/\\mathbb{Z}_2$ cannot be Higgsed to the\nanomaly-free $\\SU(2)$ theory on $b = 10, 11$, matching the conditions\nof the Automatic Enhancement Conjecture.\nNote that there are Higgsings of the $G'$ theory down to a theory with\ngauge algebra $\\ensuremath\\mathfrak{su}(2)$; in particular, Higgsing on the bifundamental\nfields can break these $G'$ theories  down to an $\\SO(3)$ on $b = 12$\nwith only adjoint matter essentially by deforming $f_1 \\sigma\n\\to \\sigma'$ where $\\sigma'$ is a generic degree 12\ncurve in $\\mathbb{P}^2$.   Such a Higgsing, however, cannot\nreproduce the original $b = 10, 11$ theory that acquired the automatic enhancement.\n\nThis kind of $\\ensuremath\\mathfrak{su}(2)$ enhancement can also occur on many other\nF-theory bases, not just on $\\mathbb{P}^2$.  On the Hirzebruch surface\n$\\mathbb{F}_m$, for example,  if $\\sigma = 0$ is $n$ times the curve $\\tilde{S}$ of positive\nself-intersection $+m$, then in a basis where $S$ is the curve of\nself-intersection $- m$ and $F$ is the fiber curve with $F \\cdot F =\n0, F \\cdot S = 1$ (such that $\\tilde{S}= S + m F$), the class of $g_2$\nis\n\\begin{equation}\n    [g_2] =-6 K_B -2 C = 6 (2 S + (2+ m) F) - 2n (S + m F)\n    = (12 - 2n) S + (12+6 m - 2n m) F \\,.\n\\end{equation}\nThis becomes ineffective when $n > 3+6\/m$.  So, for example, on the\nHirzebruch surface $\\mathbb{F}_3$, tuning an $\\ensuremath\\mathfrak{su}(2)$ gauge algebra on $6\n\\tilde{S}$ would force an additional $\\ensuremath\\mathfrak{su}(2)$ to arise on the curve\n$f_1 = 0$, where $[f_1] = -4 K_B - C = 2S+ 2F$.  Part of this\nenhancement leads to an enhancement of the $\\SU(3)$ gauge factor on\nthe $-3$ curve $S$, since $[f_1]$ contains $2S$ as a component.\nThis can be seen (using the Zariski decomposition  as in\n\\cite{MorrisonTaylorClusters})\nby noting that $[\\sigma] \\cdot S= 0, [\\phi] \\cdot\nS= -2, [f_1] \\cdot S= -4$, so the degrees of vanishing of $f, g,\n\\Delta$ on $S$ are $2, 3, 6$ and the gauge factor on $S$ is enhanced\nto $\\SO(7)$.  The remaining part of\n$[f_1]$ is $[f_1] -2S= 2F$; this is a reducible curve, giving two\n$\\ensuremath\\mathfrak{su}(2)$ factors each on a copy of $F$, so the total gauge group of\nthe enhanced theory is\n\\begin{equation}\n    G' = (\\SU(2) \\times \\SU(2) \\times \\SU(2) \\times \\SO(7))\/\\mathbb{Z}_2 \\,.\n\\end{equation}\nThe matter content includes 6 bifundamentals between the first\n$\\SU(2)$ factor and each of the other $\\SU(2)$ factors, and a\nhalf-hypermultiplet in the $(\\bm{2}, \\bm{8})$ between each of\nthe enhanced $\\SU(2)$ factors and the $\\SO(7)$ factor (analogous to\nthe matter appearing in the $-2, -3, -2$ non-Higgsable cluster\n\\cite{MorrisonTaylorClusters}).\nIt is clear that there is no Higgsing of this model possible that\nwould leave the initial $\\SU(2)$ factor unbroken.\nMany other similar models can be realized on other bases with\nnon-Higgsable gauge factors.\n\n\\paragraph{Positivity condition and local formulation}\n\nThe condition under which the automatic $\\ensuremath\\mathfrak{su}(2)$ enhancement\nmechanism enhancement must occur is that the divisor $[g_2] = - 6 a -2\nb$ is ineffective, corresponding to the condition in the 6D\nsupergravity theory that $- 6 a -2 b$ is not in the (closure of the)\npositivity cone in $\\Lambda$.\nThus in the case of $\\ensuremath\\mathfrak{su}(2)$ a stronger formulation of the conjecture\nasserts that an $\\ensuremath\\mathfrak{su}(2)$ factor with anomaly coefficient $b$,\nwhere $-6a-2b$ is not in the positivity cone,\nis not\npossible unless there is also at least another $\\ensuremath\\mathfrak{su}(2)$ factor  with\nanomaly coefficient $b' = -4a-b$.\nThe ``local'' information about the\n$\\ensuremath\\mathfrak{su}(2)$ algebra is encoded in the quantities $a \\cdot b, b \\cdot b$,\nor equivalently $g, b \\cdot b$, corresponding in the F-theory picture\nto the genus (through \\cref{eq:genus}) and self-intersection of the\ncurve $C$ supporting the desired $\\ensuremath\\mathfrak{su}(2)$ factor.  When $b \\cdot b\n>0$, for example, it is then a sufficient condition for $[g_2]$ to be\noutside the positivity cone that $- 3 a \\cdot b <b \\cdot b$.  As an\nexample of the local version of $\\ensuremath\\mathfrak{su}(2)$ enhancement, we can thus\nconjecture that whenever a theory contains an $\\ensuremath\\mathfrak{su}(2)$ algebra with\nan anomaly coefficient $b$ satisfying $b \\cdot b >0, b \\cdot b >-3 a\n\\cdot b$ there must also be a nonabelian gauge factor of rank at least\none with an anomaly coefficient $b'= [f_1] = -4 a - b$, or the theory\ncannot be consistent at least with any F-theory construction.  Note\nthat when this $b'$ vanishes there is no additional factor, but there\nis an additional section as discussed in \\cref{sec:su2-quotient}.\nWhen $b'$ is not in the (closure of the) positivity cone, then the\nWeierstrass model develops codimension one (4, 6) loci and is\ninconsistent.  This local formulation clearly matches with the\nexamples above; for example, when $T = 0, -a = 3$, the extra $\\ensuremath\\mathfrak{su}(2)$\nfactor arises when $b = 10, 11$.  Finally, note that on the F-theory\nside it is possible that $f_1$ is reducible, so that the additional\ngauge factor may be present with a pair (or more) of anomaly\ncoefficients $b_1, \\ldots, b_k$ with $b' = b_1 + \\cdots + b_k$.\n\n\n\\paragraph{Enhancements that break the theory}\n\\label{sec:break}\n\nNote that there are also cases where a similar forced enhancement in\nthe Weierstrass model can lead to codimension one (4, 6) loci, which\nbreak the F-theory construction; these cases give swampland models\nthat are not addressed by the global version of the automatic\nenhancement conjecture as stated in \\cref{sec:conjecture}, but which\nare addressed by the more specific positivity condition and local\nversion stated above.  As a simple example of this\nconsider an $\\ensuremath\\mathfrak{su}(2)$ factor on the curve $C =4 \\tilde{S}$ in the\nHirzebruch surface base $\\mathbb{F}_{12}$.  In this case, $[g_2] = 4S-12F$ is\nineffective and $g_2$ vanishes.  We furthermore have $[f_1] = 4S+ 8F$,\nso $f_1$ vanishes to order $4$ on $S$.  Since $\\phi$ vanishes to order\n2 on $S$, there is a codimension one (4, 6) singularity on $S$ so no\nF-theory model is possible.  One way of understanding this is that the\nautomatic enhancement on any curve intersecting or containing a $-12$\ncurve would naively lead to matter charged under the $\\gE_8$, which is\nnot possible.  This model fits with the local version of the $\\ensuremath\\mathfrak{su}(2)$\nautomatic enhancement conjecture as stated above, since $b \\cdot b =\n192, -a \\cdot b = 56$, so $-3a \\cdot b = 168 < b \\cdot b$, so the\nlocal version of the conjecture rules out the theory with gauge\ngroup $\\gE_8 \\times \\SU(2)$ with anomaly coefficients $S, 4 \\tilde{S}$.\n\n\\subsubsection{$\\ensuremath\\mathfrak{su}(2) \\to (\\SU(2) \\times \\U(1))\/\\mathbb{Z}_2$\nautomatic enhancement}\n\\label{sec:su2u1}\n\nAn interesting further class of related examples occurs with the above\n$\\ensuremath\\mathfrak{su}(2)$ tuning when $[g_2] = 0$, so  $g_2$ is a constant.  In this\ncase we get an automatic enhancement to a theory with an additional\n$\\U(1)$ factor.  The story is similar to the preceding enhancement\nthat gives the additional $\\SU(2)$ factor.  One way to understand the\nresulting $(\\SU(2) \\times \\U(1))\/\\mathbb{Z}_2$ theory is to consider the theory\nwith gauge group $(\\SU(2) \\times\\SU(2))\/\\mathbb{Z}_2$ that would arise with\n$[g_2] = 0$, so $g_2$ is a constant.  In this situation, $[f_1] = -4a-b = -a$, so $[f_1]$ is a\ngenus one curve; as in the cases above, all fundamental matter in the\nextra $\\ensuremath\\mathfrak{su}(2)$ factor on this curve comes from bifundamentals with\nthe original $\\ensuremath\\mathfrak{su}(2)$.  Thus, we can Higgs this theory on the adjoint\nof the second $\\SU(2)$, giving a theory with gauge group\n\\begin{equation}\n\\label{eq:g21}\nG' =(\\SU(2) \\times \\U(1))\/\\mathbb{Z}_2 \\,,\n\\end{equation}\nwhere all $\\U(1)$-charged matter is in the $\\bm{2}_{1\/2}$\nrepresentation.  Thus, the $G'$ theory cannot be Higgsed further to a\npure $\\SU(2)$ theory and this represents another class of automatic\nenhancements satisfying the conjecture.\n\nThis argument is somewhat implicit.  A more explicit analysis\n(see \\cref{app:ratsections})\nshows\nthat when $g_2$ is a constant, the Weierstrass model takes the form of\na Morrison--Park model \\cite{MorrisonParkU1} with the additional\n$\\SU(2)$ factor on $\\sigma = 0$.\nAs for the enhancements described above, this story continues to hold\nin the presence of non-Higgsable gauge factors on more complicated\nbases, though often the presence of the $\\U(1)$ factor simply\ngives $\\U(1)$ charges to the non-Higgsable gauge factors and does not\nenhance them further.\nThe simplest example of this enhancement arises on the base $\\mathbb{P}^2$\nwhen an $\\ensuremath\\mathfrak{su}(2)$ factor is tuned on a curve of degree $b = 9$.\n\nNote that this this class of enhancement conditions is similar\nto that described in \\cref{sec:su2su2}, but occurs in conditions\nwhere $-3a = b$ holds identically.  This is not really a local\ncondition, as $a \\leftrightarrow K_B$ involves global information about the geometry\nin F-theory, as would be expected when abelian $\\U(1)$ factors are\ninvolved.\n\n\\subsubsection{$\\ensuremath\\mathfrak{su}(2) \\to \\SO(3)$ and massless charge sufficiency}\n\\label{sec:su2-quotient}\n\nNote that a similar mechanism to the Automatic Enhancement Conjecture\\ operates\nwhen we tune an $\\ensuremath\\mathfrak{su}(2)$ algebra on a curve $\\sigma$ in the\nclass $-4a$.  In such a situation, $[f_1] = 0$ and there is no new gauge factor. But as shown in\n\\cite{MorrisonTaylorCompleteness},\nin F-theory there is automatically a new $\\mathbb{Z}_2$ element in\nMordell--Weil torsion, which gives the gauge group the global structure\n$\\SO(3) = \\SU(2)\/\\mathbb{Z}_2$.  While this is not an automatic enhancement,\nthis is a case of the Massless Charge Sufficiency condition described\nin \\cite{MorrisonTaylorCompleteness}.  For example, on the base\n$\\mathbb{P}^2$, tuning an $\\ensuremath\\mathfrak{su}(2)$ algebra on curves of\ndegrees $9, 10, 11$ leads in F-theory to an automatic enhancement\nthrough the mechanism described above, while tuning an $\\ensuremath\\mathfrak{su}(2)$\nalgebra on a curve of degree 12 leads in F-theory to an $\\SO(3)$\ntheory.  Thus, if both the Automatic Enhancement Conjecture\\ and the\nMassless Charge Sufficiency Conjecture are correct in 6D\nsupergravity, then there would be no swampland for 6D $\\mathcal{N} = 1$\ntheories with $T = 0$, gauge group $\\SU(2)$, and generic (fundamental\nand adjoint)\nmatter.\n\nThere is a sense in which the Automatic Enhancement Conjecture\\ and the Massless Charge\nSufficiency Conjecture are special cases of a common phenomenon.\n The appearance of the new element of Mordell--Weil torsion in\nthe $\\ensuremath\\mathfrak{su}(2) \\to \\SO(3)$ situation can in some sense be thought of as adding a new element\nto the root lattice of $G$, within the linear span of $G$.  In this\nsense, the principal difference between the Automatic Enhancement Conjecture\\ and the\nMassless Charge Sufficiency Conjecture is whether the additional root\nadded  increases the dimensionality of the gauge algebra or lies in\nthe same vector space.  This interpretation is very clear in the\nF-theory context, where these roots correspond to homology classes in\nthe compactification space.\nIt would be interesting to investigate whether a similar\ninterpretation can be given  directly in 6D supergravity that would\nunify these principles.\n\n\nThe local version of this class of conditions is similar\nto that described in \\cref{sec:su2su2}, but occurs in conditions\nwhere $-4a = b$ holds identically, so $b' = [f_1] = 0$.\n(Note that if $-4a-b = (-a) + (-3a-b)$ = 0 then $-3a-b$ cannot lie in\nthe positivity cone since $-a$ does for all F-theory models.)\n\n\\subsection{$T=0$ models with large $\\ensuremath\\mathfrak{su}(N)$ gauge algebras at $b = 1$}\n\\label{sec:t0-general-n}\n\nSimilar enhancements occur in models with $\\ensuremath\\mathfrak{su}(N)$ gauge algebras for\n$N>2$. Let us consider models with $T=0$ tensor multiplets, an\n$\\ensuremath\\mathfrak{su}(N)$ gauge group with a $b=1$ anomaly coefficient, and the\nfollowing spectrum of  charged hypermultiplets:\\footnote{For $N=3$, the hypermultiplet spectrum would consist of 24 fundamental hypermultiplets, but the $N=3$ case is not of significant interest here.}\n\\begin{equation}\n\\label{eq:sunspectrum}\n3\\times \\bm{\\frac{N(N-1)}{2}} + (24-N)\\times\\bm{N}\\,.\n\\end{equation}\nSuch massless spectra are consistent with the anomaly cancellation\nconditions for $N \\le 24$; for larger $N$, we would require a negative\nnumber of fundamental hypermultiplets.  Note that these\nrepresentations (fundamental and two-index antisymmetric), along with\nthe adjoint, are the generic matter representations for\n$\\ensuremath\\mathfrak{su}(n)$.\n\nThere are also consistent $T=0$ models with $\\ensuremath\\mathfrak{su}(N)\\oplus\\ensuremath\\mathfrak{u}(1)$\ngauge algebras and generic matter\nwhere the anomaly coefficient for the $\\ensuremath\\mathfrak{su}(N)$ factor\nis $b=1$, at least for $N<19$. For even $N$, the spectrum of charged\nhypermultiplets is\n\\begin{equation}\n\\label{eq:sunu1spectrumeven}\n3\\times \\left(\\bm{\\frac{N(N-1)}{2}}\\right)_{0} + \\left(12-\\frac{N}{2}\\right)\\times \\bm{N}_{\\frac{1}{2}} +  \\left(12-\\frac{N}{2}\\right)\\times \\bm{N}_{-\\frac{1}{2}} + \\left(9-\\frac{N}{2}\\right)\\left(12-\\frac{N}{2}\\right)\\times\\bm{1}_{1}\\,,\n\\end{equation}\nwhile for odd $N$, the spectrum of charged hypermultiplets is\n\\begin{equation}\n\\label{eq:sunu1spectrumodd}\n3\\times \\left(\\bm{\\frac{N(N-1)}{2}}\\right)_{-\\frac{1}{N}} + \\frac{19-N}{2}\\times \\bm{N}_{-\\frac{N+1}{2N}} +  \\frac{29-N}{2}\\times \\bm{N}_{\\frac{N-1}{2N}} + \\frac{1}{4}\\left(19-N\\right)\\left(23-N\\right)\\times\\bm{1}_{1}\\,.\n\\end{equation}\nAlthough these spectra naively agree with \\cref{eq:sunspectrum} if we ignore the $\\ensuremath\\mathfrak{u}(1)$ charges, the additional $\\ensuremath\\mathfrak{u}(1)$ gauge algebra is clearly an important feature. In general, the gauge algebra can be Higgsed down to $\\ensuremath\\mathfrak{su}(N)$ by giving \\vev{}'s to two hypermultiplets of $\\bm{1}_{1}$ matter. But if $N$ is either 18 or 19, there are no $\\bm{1}_{1}$ hypermultiplets, and the Higgsing is not permitted. The Automatic Enhancement Conjecture\\ therefore implies that the $\\ensuremath\\mathfrak{su}(18)$ or $\\ensuremath\\mathfrak{su}(19)$ models above should not occur in F-theory, as the gauge algebra should enhance to $\\ensuremath\\mathfrak{su}(18)\\oplus\\ensuremath\\mathfrak{u}(1)$ or $\\ensuremath\\mathfrak{su}(19)\\oplus\\ensuremath\\mathfrak{u}(1)$.\n\nThis enhancement can be seen explicitly if we attempt to construct\nthese models. Because there are no tensor multiplets, the\ncorresponding F-theory models would have $\\mathbb{P}^2$ as the base of\nthe elliptic fibration. And to obtain an $\\ensuremath\\mathfrak{su}(N)$ gauge algebra with\na $b=1$ anomaly coefficient, the models should have split $I_{N}$\nsingularities tuned along a line in this $\\mathbb{P}^2$. There are\nknown UFD Weierstrass tunings that realize split $I_N$ singularities\nalong a divisor $\\locus{\\sigma=0}$ \\cite{MorrisonTaylorMaS,\nKatzEtAlTate}. The Weierstrass model takes the form\n\\begin{equation}\n\\label{eq:weierstrasssuneven}\ny^2 = x^3 + \\left(-\\frac{1}{3}\\Phi^2 + f_{k}\\sigma^{k}\\right)x z^4 + \\left(\\frac{2}{27}\\Phi^3 - \\frac{1}{3}\\Phi f_{k}\\sigma^{k} + g_{N}\\sigma^{N}\\right)z^6\n\\end{equation}\nfor $N = 2 k$ and\n\\begin{equation}\n\\label{eq:weierstrasssunodd}\n\\begin{aligned}\n    y^2 &= x^3 + \\left(-\\frac{1}{3}\\Phi^2 + \\frac{1}{2}\\phi_0 \\psi_k \\sigma^k + f_{k+1}\\sigma^{k+1}\\right)xz^4 \\\\\n    &\\qquad\\qquad + \\left(\\frac{2}{27}\\Phi^3 -\\frac{1}{6}\\Phi\\left(\\phi_0\\psi_k+2f_{k+1}\\sigma\\right)\\sigma^k + \\frac{1}{4}\\psi_k^2\\sigma^{2k} + g_N \\sigma^N\\right)z^6\n\\end{aligned}\n\\end{equation}\nfor $N=2k+1$, where\\footnote{In \\cite{MorrisonTaylorMaS}, $\\Phi$ is defined as $\\frac{1}{4}\\phi_0^2 + \\phi_1 \\sigma + \\ldots + \\phi_{k-1}\\sigma^{k-1}$. However, one can remove the $\\phi_{i>1}$ parameters by the redefinition $\\phi_{1}\\to \\phi_{1} - (\\phi_{2}\\sigma + \\ldots + \\phi_{k-1}\\sigma^{k-2})$.}\n\\begin{equation}\n\\label{eq:Phituning}\n\\Phi = \\frac{1}{4}\\phi_0^2 + \\phi_1 \\sigma\\,.\n\\end{equation}\nThese models support hypermultiplets in the antisymmetric and\nfundamental representations.\\footnote{If $\\locus{\\sigma=0}$ is a curve\nof genus greater than 0, there may also be adjoint hypermultiplets,\nalthough this possibility is not important here.} Thus, the natural\nstrategy for constructing our desired $\\ensuremath\\mathfrak{su}(N)$ models is to apply\nthese tunings in an elliptic fibration over $\\mathbb{P}^2$ with\n$\\divclass{\\sigma}=H$.\nThe other parameters then have the following\ndivisor classes:\n\\begin{equation}\n\\divclass{\\phi_{0}} = 3H\\, \\quad \\divclass{\\phi_{1}} = 5H\\, \\quad \\divclass{\\psi_k} = (9-k)H\\, \\quad \\divclass{f_i} = (12-i)H\\, \\quad \\divclass{g_j} = (18-j)H\\,.\n\\end{equation}\n\nFor $N\\le 17$ and $N=24$, these tunings give us the desired $\\ensuremath\\mathfrak{su}(N)$ models without any enhancements of the gauge algebra.\\footnote{The $N=24$ model, however, is an example of the Massless Charge Sufficiency Conjecture, as an additional torsional section appears.} But when $N=18$, $\\divclass{g_N}$ is trivial, and $g_N$ is automatically a perfect square. If we replace $g_N$ with $\\gamma^2$, the elliptic fibration described by \\cref{eq:weierstrasssuneven} admits a rational section\n\\begin{equation}\n\\label{eq:sungenseceven}\n[\\secx:\\secy:\\secz] = \\left[\\frac{1}{3}\\Phi: \\gamma \\sigma^{9}: 1\\right]\\,.\n\\end{equation}\nThe gauge algebra is therefore $\\ensuremath\\mathfrak{su}(18)\\oplus\\ensuremath\\mathfrak{u}(1)$, and one can\nconfirm with the Katz--Vafa method \\cite{KatzVafa} and the proposals\nin \\cite{RaghuramTurnerOrders} that the charged hypermultiplet\nspectrum agrees with \\cref{eq:sunu1spectrumeven}. When $N=19$,\n$\\divclass{g_N}$ is ineffective, and $g_N$ is essentially zero. After\nsetting $N$ to 19 and $g_N$ to 0, the elliptic fibration described by\n\\cref{eq:weierstrasssunodd} admits the rational section\n\\begin{equation}\n\\left[\\secx:\\secy:\\secz\\right] = \\left[\\frac{1}{3}\\Phi:\\frac{1}{2}\\psi_{9}\\sigma^{9}:1\\right]\\,,\n\\end{equation}\nimplying that the gauge algebra is enhanced to\n$\\ensuremath\\mathfrak{su}(19)\\oplus\\ensuremath\\mathfrak{u}(1)$. And by the Katz--Vafa method \\cite{KatzVafa}\nand the proposals in \\cite{RaghuramTurnerOrders}, one can again verify\nthat the charged hypermultiplet spectrum agrees with\n\\cref{eq:sunu1spectrumodd}. Clearly, we see the enhancements expected\nfrom the Automatic Enhancement Conjecture\\ in these two models.\n\nEnhancements also occur for $N$ larger than 19, as previously observed in \\cite{MorrisonTaylorMaS, JohnsonTaylorEnhanced}.\\footnote{Because it is somewhat more difficult to find extra rational sections, these previous works did not point out the additional $\\ensuremath\\mathfrak{u}(1)$ algebras for $N=18$ and $19$.} The enhancements are summarized in \\cref{tab:sunenhancements}. If we use \\cref{eq:weierstrasssuneven} to construct the $\\ensuremath\\mathfrak{su}(20)$ and $\\ensuremath\\mathfrak{su}(22)$ models, the divisor class $\\divclass{g_N}$ becomes ineffective, and the gauge algebras enhance to $\\ensuremath\\mathfrak{su}(20)\\oplus\\ensuremath\\mathfrak{su}(2)$ or $\\ensuremath\\mathfrak{su}(22)\\oplus\\ensuremath\\mathfrak{su}(2)$ respectively. (The generating section in \\cref{eq:sungenseceven} becomes torsional \\cite{BaumeEtAlTorsion}, and the gauge group is $(\\SU(N)\\times\\SU(2))\/\\mathbb{Z}_{2}$.) While the extra $\\ensuremath\\mathfrak{su}(2)$ algebra in both cases could be Higgsed by giving \\vev{}s to two $(\\bm{1},\\bm{2})$ hypermultiplets, the $\\ensuremath\\mathfrak{su}(20)\\oplus\\ensuremath\\mathfrak{su}(2)$ and $\\ensuremath\\mathfrak{su}(22)\\oplus\\ensuremath\\mathfrak{su}(2)$ spectra in \\cref{tab:sunenhancements} do not have this matter. Thus, the appearance of this extra $\\ensuremath\\mathfrak{su}(2)$ algebra agrees with the Automatic Enhancement Conjecture.\n\nMeanwhile, if we use  \\cref{eq:weierstrasssunodd} to construct the\n$\\ensuremath\\mathfrak{su}(21)$ and $\\ensuremath\\mathfrak{su}(23)$ models, the divisor class\n$\\divclass{\\psi_{k}}$ becomes ineffective, and the gauge algebras\nenhance to $\\ensuremath\\mathfrak{su}(22)\\oplus\\ensuremath\\mathfrak{su}(2)$ and $\\ensuremath\\mathfrak{su}(24)$,\nrespectively. Neither of these enhanced models has the requisite\nmatter to Higgs the gauge algebra to $\\ensuremath\\mathfrak{su}(21)$ or $\\ensuremath\\mathfrak{su}(23)$. While\none can give \\vev{}s to $(\\bm{22},\\bm{2})$ matter to break\n$\\ensuremath\\mathfrak{su}(22)\\oplus\\ensuremath\\mathfrak{su}(2)$ to $\\ensuremath\\mathfrak{su}(21)$ \\cite{LiGroupTheory}, one would\nneed at least two hypermultiplets of $(\\bm{22},\\bm{2})$ matter\nto satisfy the D-term constraints\n\\cite{DanielssonStjernberg}.\nAnd the $\\ensuremath\\mathfrak{su}(24)$ model, which supports only antisymmetric\nhypermultiplets, lacks the fundamental matter necessary to Higgs the\ngauge algebra down to $\\ensuremath\\mathfrak{su}(23)$. Therefore, the observed enhancements\nwhen attempting to construct the $\\ensuremath\\mathfrak{su}(21)$ and $\\ensuremath\\mathfrak{su}(23)$ models\nagree with the expectations of the Automatic Enhancement Conjecture.\n\nThe local version of this class of enhancements states that for any\nelement $b \\in\\Lambda$ satisfying $b \\cdot b = 1, g = 0$ ($-a \\cdot b =\n3$), an isolated gauge algebra of $\\ensuremath\\mathfrak{su}(18)$ or more is automatically\nenhanced according to \\cref{tab:sunenhancements}.\n\n\\begin{table}\n    \\centering\n\n    \\begin{tabular}{*{3}{c}}\\toprule\n    Expected Gauge Algebra & Enhanced Gauge Algebra & Matter Spectrum \\\\ \\midrule\n    $\\ensuremath\\mathfrak{su}(18)$ & $\\ensuremath\\mathfrak{su}(18)\\oplus\\ensuremath\\mathfrak{u}(1)$ & $3\\times\\bm{153}_{0} + 3\\times\\bm{18}_{1\/2}+ 3\\times\\bm{18}_{-1\/2}$\\\\\n    $\\ensuremath\\mathfrak{su}(19)$ & $\\ensuremath\\mathfrak{su}(19)\\oplus\\ensuremath\\mathfrak{u}(1)$ & $3\\times\\bm{171}_{-1\/19} + 5\\times\\bm{19}_{9\/19}$\\\\\n    $\\ensuremath\\mathfrak{su}(20)$ & $\\ensuremath\\mathfrak{su}(20)\\oplus\\ensuremath\\mathfrak{su}(2)$ & $3\\times(\\bm{190},\\bm{1})+2\\times(\\bm{20},\\bm{2})$\\\\\n    $\\ensuremath\\mathfrak{su}(21)$ & $\\ensuremath\\mathfrak{su}(22)\\oplus\\ensuremath\\mathfrak{su}(2)$ & $3\\times(\\bm{231},\\bm{1})+(\\bm{22},\\bm{2})$\\\\\n    $\\ensuremath\\mathfrak{su}(22)$ & $\\ensuremath\\mathfrak{su}(22)\\oplus\\ensuremath\\mathfrak{su}(2)$ & $3\\times(\\bm{231},\\bm{1})+(\\bm{22},\\bm{2})$\\\\\n    $\\ensuremath\\mathfrak{su}(23)$ & $\\ensuremath\\mathfrak{su}(24)$ & $3\\times\\bm{276}$\\\\\n    $\\ensuremath\\mathfrak{su}(24)$ & $\\ensuremath\\mathfrak{su}(24)$ & $3\\times\\bm{276}$\\\\ \\bottomrule\n    \\end{tabular}\n\n    \\caption{Enhanced gauge algebras observed when attempting to tune various expected gauge algebras along a divisor of class $H$ on $\\mathbb{P}^2$, along with the observed matter spectrum.}\n    \\label{tab:sunenhancements}\n\\end{table}\n\n\\subsection{Other $\\SU(N)$ examples}\n\nIn the last two subsections we have focused on enhancements of the\nalgebra $\\ensuremath\\mathfrak{su}(2)$ with general anomaly coefficients and enhancements\nof the algebra $\\ensuremath\\mathfrak{su}(n)$ at large $n$ on anomaly coefficients $b \\cdot\nb=1$,\nwhere $g= 0$.  We briefly discuss some more general $\\ensuremath\\mathfrak{su}(n)$ examples\n\n\\subsubsection{Enhancements of $\\ensuremath\\mathfrak{su}(3)$ and beyond for general\n  $b$}\n\nA similar enhancement to that described in \\cref{sec:su2su2} occurs\nfor $\\ensuremath\\mathfrak{su}(3)$.  When the coefficients in classes $[f_2] = -4a-2b$ and\n$[g_3]= -6a-3b$ in a $\\sigma$ expansion of $f, g$ for the general\n$\\SU(3)$ form \\cite{MorrisonTaylorMaS, MorrisonTaylorCompleteness} are\nnot effective, the discriminant acquires an additional cubic factor\n$\\psi_1^3$ where $[\\psi_1] = -3a-b$.  This gives an enhancement\n$\\ensuremath\\mathfrak{su}(3) \\to \\ensuremath\\mathfrak{su}(3) \\oplus \\ensuremath\\mathfrak{su}(3)$.  Just as there is $\\mathbb{Z}_2$ torsion\nin the $\\ensuremath\\mathfrak{su}(2)\\to\\ensuremath\\mathfrak{su}(2)\\oplus\\ensuremath\\mathfrak{su}(2)$ situations of\n\\cref{sec:su2su2}, there is automatically $\\mathbb{Z}_3$ Mordell--Weil torsion in these\ncases \\cite{MorrisonTaylorCompleteness}, so the global gauge group\nbecomes $G' = (\\SU(3) \\times \\SU(3))\/\\mathbb{Z}_3$.  Again, the second $\\SU(3)$\nfactor has no adjoints and all fundamentals\/antifundamentals are\njointly charged, so the second $\\SU(3)$ factor cannot be Higgsed\nwithout breaking the original $\\SU(3)$.  For $T = 0$, this enhancement\noccurs for the values $b =7, 8$.  At $b =  6$, $f_2, g_3$  become constants\nand the algebra is enhanced by $\\ensuremath\\mathfrak{u}(1) \\oplus \\ensuremath\\mathfrak{u}(1)$, by the same logic that led to\n\\cref{eq:g21}.  The local version of this analysis is that when $b\n\\cdot b > 0, b \\cdot b > -2a \\cdot b$ we get an extra $\\ensuremath\\mathfrak{su}(3)$ factor\non $-3a-b$.\n\nWe now consider $\\ensuremath\\mathfrak{su}(4)$.  In general what is needed for automatic\nenhancement to an extra $\\SU(2)$ factor with generic (fundamental,\nadjoint) matter that cannot be Higgsed down is that the extra factor\nmust have an anomaly coefficient $b'$ giving $g' = 0$ (for the absence\nof adjoints) and the automatically enhanced group must provide all\nfundamental $\\SU(2)$ matter.  The number of fundamental matter fields\nin this situation is $b' \\cdot (-8a-2b')$ from anomaly cancellation.\nFor an $\\ensuremath\\mathfrak{su}(4)$ with anomaly coefficient $b$ to experience\nenhancement to $ \\ensuremath\\mathfrak{su}(4)\\oplus \\ensuremath\\mathfrak{su}(2)$ we must then have $4b \\cdot b'\n= b' \\cdot (-8a-2b')$.  This can certainly occur, most simply when $2b\n= -4a-b'$.  For example, this occurs when $T = 0$ where $b = 5, b' =\n2$.  In general, this will occur when $-3a-2b$ is not positive but\n$-4a-2b$ is positive; it is not hard to confirm in the universal\nWeierstrass model for an $\\ensuremath\\mathfrak{su}(4)$ with generic matter that this\nenhancement occurs under these conditions, where $f_3 = g_4 = 0$.\n\nWe leave further exploration of automatic enhancement  of this type\nfor gauge factors with general anomaly coefficients to further work.\n\n\\subsubsection{Enhancements of $\\ensuremath\\mathfrak{su}(N)$ for $g = 0,$ small $b \\cdot b$}\n\\label{sec:small-bb}\n\nWe now focus on more general enhancements of $\\ensuremath\\mathfrak{su}(N)$ for larger $N$,\nrestricting to the cases where $g = 0$. For $\\ensuremath\\mathfrak{su}(N)$ models with $g =\n0$, $b \\cdot b = 2$, the largest value of $N$ allowed by the anomaly\ncancellation conditions is $N=16$. Automatic enhancement should occur\nfor at least $N\\ge 13$ because of the requirement that $(-6a-Nb)\\cdot\nb$ must be non-negative for\n$\\divclass{g_N}= -6a-Nb$ to be effective. But automatic enhancement\ncan also occur for some $N$ less than 13 depending on the base of the\nF-theory model in question, since $-6a-N b$ can be outside the\npositivity cone even if $(-6a-N b) \\cdot b \\ge 0$.  For instance, let\nus restrict attention to $T=1$ models, which are constructed using the\nHirzebruch surfaces $\\mathbb{F}_m$ as bases. To ensure that the\ndivisor supporting the $\\ensuremath\\mathfrak{su}(N)$ algebra is irreducible and has\nself-intersection $+2$, $m$ should be $0$ or $2$.  The automatic\nenhancements, which are the same for both of these cases, are\nsummarized in \\cref{tab:selfint2enhancements}.\n\nThe pattern of enhancements may be different for $b\\cdot b=2$ divisors\non other bases. For instance, consider a base $B$ found by blowing up\n$\\mathbb{F}_2$ at the intersection of the $-2$ curve and a $0$ curve,\nand imagine tuning an $\\ensuremath\\mathfrak{su}(N)$ algebra on  a $+2$ curve $\\locus{\\sigma=0}$. There are no automatic\nenhancements for $N\\le9$ on $B$, apart from a relatively\nuninteresting $\\ensuremath\\mathfrak{su}(3)$ NHC on the $-3$ curve. For $N=10$,\n$\\divclass{g_{10}} = -6\\canonclass-10\\divclass{\\sigma}$ is effective\nand nontrivial, but the only holomorphic sections of\n$\\divclass{g_{10}}$ are perfect squares. The gauge algebra should\ntherefore admit an extra $\\ensuremath\\mathfrak{u}(1)$ factor. In fact, the full gauge\nalgebra is $\\ensuremath\\mathfrak{su}(10)\\oplus\\ensuremath\\mathfrak{su}(3)\\oplus\\ensuremath\\mathfrak{su}(2)\\oplus\\ensuremath\\mathfrak{u}(1)$, and the\ncharged hypermultiplet spectrum is\n\\begin{equation}\n    4\\times(\\bm{45},\\bm{1},\\bm{1})_{0} + 1\\times(\\bm{10},\\bm{1},\\bm{2})_0+ 5\\times (\\bm{10},\\bm{1},\\bm{1})_{\\frac{1}{2}}+5\\times(\\bm{10},\\bm{1},\\bm{1})_{-\\frac{1}{2}}\\,.\n\\end{equation}\nThis spectrum satisfies the conditions of the Automatic Enhancement Conjecture, as one\ncannot Higgs away the $\\ensuremath\\mathfrak{su}(2)$ or $\\ensuremath\\mathfrak{u}(1)$ factors without disturbing\nthe $\\ensuremath\\mathfrak{su}(10)$ algebra. Tuning $\\ensuremath\\mathfrak{su}(10)$ on the $b\\cdot b=2$ curve in\n$B$ should therefore lead to an automatic enhancement as observed,\neven though a similar $\\ensuremath\\mathfrak{su}(10)$ enhancement does not occur in the\n$T=1$ models above.\nLocally, this enhancement can be understood as being forced from the\nintersection of the $+ 2$ curve with a curve of self-intersection\n$-1$, and can be understood using purely local analysis as in\n\\cite{JohnsonTaylorEnhanced}.\nLikewise, $\\ensuremath\\mathfrak{su}(11)$ and $\\ensuremath\\mathfrak{su}(12)$ experience\nautomatic enhancements when tuned on the $+2$ curve of $B$, yet they\ncan be obtained without enhancements when $T=1$.\n\n\\begin{table}\n    \\centering\n\n    \\begin{tabular}{*{3}{c}} \\toprule\n        Expected Gauge Algebra & Enhanced Gauge Algebra & Matter Spectrum \\\\ \\midrule\n        $\\ensuremath\\mathfrak{su}(12)$ & $\\ensuremath\\mathfrak{su}(12)\\oplus\\ensuremath\\mathfrak{u}(1)$ & $4\\times\\bm{66}_{0}+4\\times\\bm{12}_{\\frac{1}{2}}+4\\times\\bm{12}_{-\\frac{1}{2}}$\\\\\n        $\\ensuremath\\mathfrak{su}(13)$ & $\\ensuremath\\mathfrak{su}(13)\\oplus\\ensuremath\\mathfrak{u}(1)$ & $4\\times\\bm{78}_{-\\frac{1}{13}} + 6\\times\\bm{13}_{\\frac{6}{13}}$\\\\\n        $\\ensuremath\\mathfrak{su}(14)$ & $\\ensuremath\\mathfrak{su}(14)\\oplus\\ensuremath\\mathfrak{su}(2)$ & $4\\times(\\bm{91},\\bm{1})+2\\times(\\bm{14},\\bm{2})$\\\\\n        $\\ensuremath\\mathfrak{su}(15)$ & $\\ensuremath\\mathfrak{su}(16)$ & $4\\times\\bm{120}$\\\\\n        $\\ensuremath\\mathfrak{su}(16)$ & $\\ensuremath\\mathfrak{su}(16)$ & $4\\times\\bm{120}$\\\\ \\bottomrule\n    \\end{tabular}\n\n    \\caption{Automatic enhancements for $T=1$ models with $\\ensuremath\\mathfrak{su}(N)$ gauge algebras tuned on divisors of genus $g=0$ and self-intersection $b\\cdot b=2$.}\n    \\label{tab:selfint2enhancements}\n\\end{table}\n\nFor $g=0$ and $b\\cdot b=3$, the largest value of $N$ allowed by the\nanomaly cancellation is 13. Based on the condition for $(-6a-Nb)\\cdot\nb$, automatic enhancements should occur at least for $N\\ge11$. Again,\nthe actual pattern of enhancements depends on the base of the F-theory\nmodel. When $T=1$, one can find irreducible curves with\nself-intersection $+3$ on $\\mathbb{F}_1$ and $\\mathbb{F}_3$. On\n$\\mathbb{F}_1$, there is no enhancement for $N\\le8$, and the enhanced\ngauge algebras are $\\ensuremath\\mathfrak{su}(9)\\oplus\\ensuremath\\mathfrak{su}(2)$ for $N=9$,\n$\\ensuremath\\mathfrak{su}(10)\\oplus\\ensuremath\\mathfrak{su}(2)\\oplus\\ensuremath\\mathfrak{su}(2)$ for $N=10$, and\n$\\ensuremath\\mathfrak{su}(12)\\oplus\\ensuremath\\mathfrak{sp}(2)$\\footnote{We use conventions where $\\ensuremath\\mathfrak{sp}(2)$ is\nsupported along non-split $I_4$ loci.} for $N=11$ and $N=12$. On\n$\\mathbb{F}_{3} $, there is no enhancement for $N\\le9$ apart from the\nusual $\\ensuremath\\mathfrak{su}(3)$ non-Higgsable cluster. The enhanced gauge algebras are\n$\\ensuremath\\mathfrak{su}(10)\\oplus\\ensuremath\\mathfrak{u}(1)\\oplus\\ensuremath\\mathfrak{su}(3)$ for $N=10$,\n$\\ensuremath\\mathfrak{su}(11)\\oplus\\ensuremath\\mathfrak{u}(1)\\oplus\\ensuremath\\mathfrak{su}(3)$ for $N=11$, and\n$\\ensuremath\\mathfrak{su}(12)\\oplus\\ensuremath\\mathfrak{su}(2)\\oplus\\ensuremath\\mathfrak{su}(2)\\oplus\\ensuremath\\mathfrak{so}(7)$ for\n$N=12$,\\footnote{Since it can be difficult to find rational sections, the enhanced gauge algebras may actually have $\\ensuremath\\mathfrak{u}(1)$ factors beyond those listed here, which we were able to identify easily.} with the last gauge algebra representing that supported\non the $-3$ curve. For both of these bases, attempting to obtain an\n$\\ensuremath\\mathfrak{su}(13)$ gauge algebra using the standard tuning of\n\\cref{eq:weierstrasssunodd} produces an invalid elliptic fibration\nwith singular fibers everywhere on the base. While we do not discuss\nthe matter spectra for brevity, they are all consistent with the\nAutomatic Enhancement Conjecture.\n\n\\subsection{Examples with $\\ensuremath\\mathfrak{sp}(N)$ and $\\ensuremath\\mathfrak{so}(N)$ algebras}\n\\label{sec:spsoalgebras}\n\nWe can immediately obtain $\\ensuremath\\mathfrak{sp}(m)$ examples of automatic enhancement by Higgsing the $\\ensuremath\\mathfrak{su}(2m)$ models of \\cref{sec:t0-general-n}. If we give a \\vev{} to matter in the antisymmetric representation of $\\ensuremath\\mathfrak{su}(2m)$, we obtain an $\\ensuremath\\mathfrak{su}(m)$ gauge algebra with the same anomaly coefficient $b$. From an F-theory perspective, the resulting $\\ensuremath\\mathfrak{sp}(m)$ model is given by taking the $\\ensuremath\\mathfrak{su}(2m)$ construction of \\cref{eq:weierstrasssuneven} and ignoring the tuning of $\\Phi$ in \\cref{eq:Phituning}. The $\\ensuremath\\mathfrak{su}(2m)$ models of \\cref{sec:t0-general-n} have enough antisymmetric hypermultiplets to perform this Higgsing. However, these hypermultiplets are uncharged under any additional $\\ensuremath\\mathfrak{su}(2)$ or $\\ensuremath\\mathfrak{u}(1)$ factor appearing due to automatic enhancements, and the $\\ensuremath\\mathfrak{su}(2m) \\to \\ensuremath\\mathfrak{sp}(m)$ breaking does not introduce new hypermultiplets that would let us Higgs the extra gauge factors without disturbing the $\\ensuremath\\mathfrak{sp}(m)$ algebra. This fact provides us with $T=0$ examples of automatic enhancement with $\\ensuremath\\mathfrak{sp}(m)$ gauge algebras and $b=1$ anomaly coefficients.\n\nWhen $m=9$, for instance, the conjecture implies that $\\ensuremath\\mathfrak{sp}(9)$ experiences an automatic enhancement to $\\ensuremath\\mathfrak{sp}(9)\\oplus\\ensuremath\\mathfrak{u}(1)$ with the following charged hypermultiplet spectrum:\n\\begin{equation}\n    2\\times \\bm{152}_{0} + 3\\times\\bm{18}_{1\/2} + 3\\times\\bm{18}_{-1\/2}\\,.\n\\end{equation}\nThis enhancement can be seen in the $\\ensuremath\\mathfrak{sp}(9)$ Weierstrass model\nproduced by the procedure described above: the class\n$\\divclass{g_{18}}$ is trivial, leading to an additional section of\nthe elliptic fibration. Similarly, $\\ensuremath\\mathfrak{sp}(10)$ and $\\ensuremath\\mathfrak{sp}(11)$\nexperience automatic enhancements to $\\ensuremath\\mathfrak{sp}(10)\\oplus\\ensuremath\\mathfrak{su}(2)$ and\n$\\ensuremath\\mathfrak{sp}(11)\\oplus\\ensuremath\\mathfrak{su}(2)$, respectively. While the $\\ensuremath\\mathfrak{sp}(12)$ model does\nnot experience automatic enhancement, the Weierstrass model has an\nextra torsional section, in agreement with the Massless Charge\nSufficiency Conjecture as noted in\n\\cite{MorrisonTaylorCompleteness}.\n\nThere are also $\\ensuremath\\mathfrak{so}(N)$ examples with automatic enhancement. For example, consider a $T=0$ model with an $\\ensuremath\\mathfrak{so}(10)$ gauge algebra and $b=4$ anomaly coefficient. The charged hypermultiplet spectrum for this model is\n\\begin{equation}\n    3\\times\\bm{45} + 8\\times\\bm{16}+4\\times\\bm{10}\\,.\n\\end{equation}\nHowever, there is also a consistent $\\ensuremath\\mathfrak{so}(10)\\oplus\\ensuremath\\mathfrak{u}(1)$ model with anomaly coefficients $b=4$, $\\tilde{b}=1$ and a charged hypermultiplet spectrum of the form\n\\begin{equation}\n    3\\times\\bm{45}_{0} + 8\\times\\bm{16}_{\\frac{1}{4}}+4\\times\\bm{10}_{\\frac{1}{2}}\\,.\n\\end{equation}\nThe Automatic Enhancement Conjecture\\ implies that we should see an $\\ensuremath\\mathfrak{so}(10) \\to \\ensuremath\\mathfrak{so}(10) \\oplus \\ensuremath\\mathfrak{u}(1)$ enhancement. In fact, this can be seen in the $\\ensuremath\\mathfrak{so}(10)$ Weierstrass tuning\n\\begin{equation}\n    y^2 = x^3 + \\sigma^2\\left(-\\frac{1}{48}\\Phi^2 + f_3 \\sigma\\right)+\\sigma^3\\left(\\frac{1}{864}\\Phi^3 + \\left(\\gamma^2 - \\frac{1}{12}f_3\\Phi\\right)\\sigma+g_{5}\\sigma^2\\right)\\,.\n\\end{equation}\nThe $\\ensuremath\\mathfrak{so}(10)$ algebra is tuned along $\\locus{\\sigma=0}$, so $\\divclass{\\sigma}$ should be $4H$ to obtain the correct $\\ensuremath\\mathfrak{so}(10)$ hypermultiplet spectrum. But the divisor class $\\divclass{g_5}=-6\\canonclass-5\\divclass{\\sigma}=-2H$ is then ineffective, forcing $g_5$ to be 0. In turn, the elliptic fibration acquires an additional rational section of the form\n\\begin{equation}\n    [\\secx : \\secy : \\secz] = \\left[\\frac{1}{12}\\sigma\\Phi : \\gamma \\sigma^2 : 1\\right]\\,,\n\\end{equation}\ndemonstrating the expected appearance of the $\\ensuremath\\mathfrak{u}(1)$ gauge algebra.\n\n\\section{Abelian Examples}\n\\label{sec:abelian}\n\nThe Automatic Enhancement Conjecture\\ also suggests that abelian gauge symmetries in certain models\nare forced to enhance. In fact, the conjecture is particularly powerful in\nthese settings, as it can explain why infinite families of apparently\nconsistent supergravity models with abelian gauge symmetries do not\noccur in F-theory. We first describe cases of the Automatic Enhancement Conjecture\\ with\n$\\U(1)$ gauge group and generic ($q = 1, 2$) matter, and then focus on\ntwo types of infinite families that include exotic $\\U(1)$ charges:\nthose with $\\U(1)$ gauge groups, and an infinite family with an\n$\\ensuremath\\mathfrak{su}(13)\\oplus\\ensuremath\\mathfrak{u}(1)$ gauge algebra.\n\n\\subsection{$\\U(1)$ models with generic matter}\n\\label{sec:abelian-generic}\n\nFor a gauge group $\\U(1)$, the generic matter representations are\nthose with charges $q = 1, 2$ \\cite{TaylorTurnerSwampland}.\nWeierstrass models for F-theory constructions of $\\U(1)$ theories with\nthese generic matter charges were constructed by Morrison and Park\n\\cite{MorrisonParkU1}.  The explicit  form of these Weierstrass models\nis given in \\cref{eq:Morrison--Park}, and it was  shown that these\nmodels are universal in the sense that they capture all the uncharged\nmoduli, at least over simple bases, in \\cite{WangU1s,\nMorrisonTaylorMultisection}.\nThere is a class of situations where the generic $\\U(1)$ model with\ncertain anomaly coefficients satisfies the conditions of the\nAutomatic Enhancement Conjecture.  These situations are closely related to the $\\ensuremath\\mathfrak{su}(2)$\nmodels described in \\cref{sec:su2su2}.  Consider a 6D supergravity\ntheory with gauge algebra $\\ensuremath\\mathfrak{su}(2) \\oplus \\ensuremath\\mathfrak{su}(2)$, and anomaly\ncoefficients $b, b'$ satisfying $b + b' = -4 a$.  In such a situation,\nthe spectrum consists of $b \\cdot b'$ bifundamental matter fields and\n$g, g'$ adjoint fields for the two $\\ensuremath\\mathfrak{su}(2)$ factors, where $g, g'$\nare given by \\cref{eq:genus} using $b, b'$ respectively.  When $g\n>0$, the first $\\ensuremath\\mathfrak{su}(2)$ factor can be broken down to $\\ensuremath\\mathfrak{u}(1)$ by\nHiggsing on an adjoint field.  When $g' = 0$, however, the second\n$\\ensuremath\\mathfrak{su}(2)$ factor cannot be broken through Higgsing while preserving\nthe first $\\ensuremath\\mathfrak{u}(1)$ factor.  This gives a situation with gauge algebra\n$\\ensuremath\\mathfrak{u}(1) \\oplus \\ensuremath\\mathfrak{su}(2)$ and matter charged only in the $\\bm{1}_2,\n\\bm{2}_1$ representations.  When such a theory can be constructed in\nF-theory, the conditions of the Automatic Enhancement Conjecture\\ are satisfied, so we\nexpect that the theory with only gauge algebra $\\ensuremath\\mathfrak{u}(1)$ and anomaly\ncoefficient $\\tilde{b}= 2 b$ cannot be realized in F-theory.\n\nThe simplest example of this is in the case of no tensor multiplets\n$(T = 0)$.  There are anomaly-consistent $\\U(1)$ models with anomaly\ncoefficient $\\tilde{b}= 6, 8, \\ldots, 24$ and spectrum $\\tilde{b}(24 -\n\\tilde{b}) \\times\\chargedsinglet{1}, \\tilde{b}(\\tilde{b}- 6)\/4 \\times\\chargedsinglet{2}$. Naively, these come from Higgsing the anomaly-consistent\n$\\ensuremath\\mathfrak{su}(2)$ model with $b = \\tilde{b}\/2$.  As we have seen, however, in\n\\cref{sec:su2su2}, the $\\ensuremath\\mathfrak{su}(2)$ models with $b = 10, 11$ experience\nautomatic enhancement, and from the above analysis the same is\nexpected for the $\\ensuremath\\mathfrak{u}(1)$ models with $\\tilde{b}= 20, 22$.\n\nIndeed, we can see explicitly from the form of the Morrison--Park\nWeierstrass model \\cref{eq:Morrison--Park} that for the cases\n$\\tilde{b}= 20, 22$, which correspond to choosing classes $[c_3] =\n\\tilde{b}\/2 =10, 11$, the class $[c_0]$ becomes ineffective and there\nis an enhancement to an additional $\\ensuremath\\mathfrak{su}(2)$ factor on the class\n$[c_1]$.  This enhancement was also discussed from different points of\nview in \\cite{MorrisonParkU1, MorrisonTaylorMultisection}.  More\ngenerally, whenever we have a $\\ensuremath\\mathfrak{u}(1)$ theory with generic matter and\nan anomaly coefficient $\\tilde{b}$ such that $- 6 a - \\tilde{b}$ is\noutside the positivity cone but $- 8 a - \\tilde{b}$ is in the (closure\nof the) cone, we have a similar situation where the theory suffers\nautomatic enhancement to $\\ensuremath\\mathfrak{u}(1) \\oplus \\ensuremath\\mathfrak{su}(2)$.  In the F-theory\nconstruction, it can be confirmed that the global structure of the\ngroup in such situations is $(\\U(1) \\times\\SU(2))\/\\mathbb{Z}_2$\n\\cite{MorrisonTaylorCompleteness}.\n\nThere are also similar situations when $- 6 a - \\tilde{b}= 0$, in\nwhich case the enhancement is to a $\\U(1) \\times\\U(1)$ theory, as can be seen again directly\nfrom the Morrison--Park form, and when $\\tilde{b}= -8 a$, which\nnaively gives a $\\ensuremath\\mathfrak{u}(1)$ model with only even charges, but where the\npresence of an additional section in the Morrison--Park model makes\nthis equivalent to the model with only charges $\\chargedsinglet{1}$, in\naccord with the Massless Charge Sufficiency Conjecture\n\\cite{MorrisonTaylorCompleteness}.  Note that the original\nMorrison--Park generating section in the enhanced $\\U(1) \\times\\U(1)$\ntheory when $-6a = \\tilde{b}$ is the ``diagonal'' sum of the\ntwo $\\U(1)$\ngenerators giving the charge lattice generated by $\\twochargedsinglet{\\pm1}{0}$ and\n$\\twochargedsinglet{0}{\\pm1}$.\nThese cases play a further role as a special case in the analysis of\nthe following subsection.\n\n\\subsection{$\\U(1)$ infinite families}\n\\label{sec:abelian-infinite}\n\nWe now turn to some families of $\\U(1)$ models with exotic charges $q\n> 2$.\nIt was shown in \\cite{TaylorTurnerSwampland} that there is an infinite\nfamily of $\\U(1)$ models with no tensor multiplets that satisfy the\n6D anomaly cancellation conditions and other known low-energy\nconstraints. The models in this family have $\\U(1)$ height\n\\begin{equation}\n    \\height = 6\\left(q^2 + q r + r^2\\right)\n\\end{equation}\nand charged hypermultiplet spectra of the form\\footnote{The sign of\n  $\\U(1)$ charge for a charged hypermultiplet is unimportant, as a\n  $\\chargedsinglet{Q}$ hypermultiplet contains fields with $\\U(1)$ charges\n  $+Q$ and $-Q$.}\n\\begin{equation}\n    \\label{eq:chargefamilyspectrum}\n   \n    54\\times\\chargedsinglet{q}+54\\times\\chargedsinglet{r}+54\\times\\chargedsinglet{(q+r)}\n\\end{equation}\nfor $q,r \\in \\mathbb{Z}$. From now on, we ignore situations where either $q$,\n$r$, or $q+r$ is 0; after normalizing charges, these situations\ncorrespond to a relatively uninteresting model known to occur in\nF-theory \\cite{MorrisonParkU1} whose charged hypermultiplet spectrum\nhas only 108 $\\chargedsinglet{1}$ hypermultiplets.  We also assume\nthat $q, r$ are relatively prime, so that these charges generate the\nfull possible charge lattice and therefore satisfy the Massless Charge\nSufficiency Conjecture\n\\cite{MorrisonTaylorCompleteness}.\\footnote{Tarazi and Vafa have made the stronger\nconjecture \\cite{TaraziVafa-conjecture} that there must always be a\nmassless field of charge $\\chargedsinglet{1}$ in a 6D $\\U(1)$\nsupergravity theory of this type, which would be violated by many of\nthese theories, but there would still be an infinite family with $q\n= 1, r$ prime compatible even with this stronger conjecture.}\nBecause the\nnumber of topologically distinct elliptically fibered Calabi--Yau threefolds is finite, only a\nfinite number of these supergravity models can be realized in\nF-theory. This family therefore presents an infinite number of\nswampland candidates, but there has been no proposal as to which\nprecise subset of\nthese models lie in the F-theory swampland, let alone why. And since\nthe models in this family support arbitrarily large $\\U(1)$ charges,\nthis family is relevant for questions regarding which $\\U(1)$\ncharges occur in F-theory (see, e.g.,\\cite{RaghuramTaylorLargeCharge,\nCianci:2018vwv, LeeWeigandAbelian, Collinucci:2019fnh, AndersonGrayOehlmannQuotients}).\n\nHowever, there is also a model with zero tensor multiplets, a $\\U(1)\\times\\U(1)$ gauge group, and a relatively simple spectrum of charged hypermultiplets:\\footnote{A $\\twochargedsinglet{Q_a}{Q_b}$ hypermultiplet is equivalent to $\\twochargedsinglet{-Q_a}{-Q_b}$ hypermultiplet, as both contain fields with charges $(Q_a,Q_b)$ and $(-Q_a,-Q_b)$.}\n\\begin{equation}\n    \\label{eq:u1u1spectrum}\n    54 \\times \\twochargedsinglet{1}{-1} + 54\\times \\twochargedsinglet{1}{0} + 54\\times\\twochargedsinglet{0}{1}\\,.\n\\end{equation}\nThe anomaly coefficient matrix for the $\\U(1)\\times\\U(1)$ gauge group is\n\\begin{equation}\n    \\height = \\begin{pmatrix} 6&  -3 \\\\ -3 & 6  \\end{pmatrix}\\,.\n\\end{equation}\nThis model appears repeatedly in this section, and unless otherwise specified, any reference to a $\\U(1)\\times\\U(1)$ model refers to this specific model. Let us refer to the first $\\U(1)$ factor as $\\U(1)_a$ and the second $\\U(1)$ factor as $\\U(1)_b$. We are also free to use a different basis $\\U(1)_{c}\\times \\U(1)_{d}$, such that\\footnote{This transformation is not valid if $q=0$, but as mentioned above this situation is uninteresting.}\n\\begin{equation}\n\\begin{pmatrix}q_{c}\\\\q_{d}\\end{pmatrix} = \\begin{pmatrix}q&-r\\\\ 0&1 \\end{pmatrix}\\begin{pmatrix}q_{a}\\\\q_{b}\\end{pmatrix}\\,.\n\\end{equation}\nThe charge spectrum in this new basis is\n\\begin{equation}\n54\\times\\twochargedsinglet{q+r}{-1} + 54\\times\\twochargedsinglet{q}{0} + 54\\times\\twochargedsinglet{r}{1}\\,,\n\\end{equation}\nand the anomaly coefficient matrix is\n\\begin{equation}\n\\begin{pmatrix} 6\\left(q^2 + q r + r^2\\right) & -3(q+2r) \\\\ -3(q+2r) & 6 \\end{pmatrix}\\,.\n\\end{equation}\n\nAlthough $\\U(1)_c$ corresponds to the $\\U(1)$ seen in the infinite\nfamily, there are still two $\\U(1)$ factors in this model. One could imagine obtaining a model in the infinite family by Higgsing $\\U(1)_{d}$ in a way that preserves $\\U(1)_{c}$. This Higgsing would entail giving a \\vev{} to $\\twochargedsinglet{0}{1}$ matter in the $\\U(1)_{c}\\times\\U(1)_{d}$ basis, but this matter is not present in the spectrum unless $r$ or $r+q$ is 0. The Automatic Enhancement Conjecture\\ would then imply that if we attempt to construct any $\\U(1)$ model in the family with $r,q,r+q\\neq0$, the model is forced to obtain an extra $\\U(1)$, and we end up with the $\\U(1)\\times\\U(1)$ model.\n\nThe $\\U(1)\\times\\U(1)$ model has an F-theory realization described in \\cref{app:u1u1model}. Geometrically, $\\U(1)_{a}$ and $\\U(1)_{b}$ correspond to the generating sections $\\ratsec{s}_{Q}$ and $\\ratsec{s}_{R}$, while $\\U(1)_c$ corresponds to the linear combination $q\\ratsec{s}_{Q}-r\\ratsec{s}_{R}$ under elliptic curve addition. The natural way to test the Automatic Enhancement Conjecture\\ is to try constructing the models in the infinite family using the known F-theory $\\U(1)$ constructions and to show that we end up with the $\\U(1)\\times\\U(1)$ F-theory model. Because the known $\\U(1)$ constructions in F-theory admit only relatively small charges, we will only be able to test the conjecture for small $r$ and $q$. Still, we see clear evidence of the expected enhancement in all the cases we consider. When we try to use these constructions to obtain $\\U(1)$ models in the infinite family, the divisor classes of various parameters become either ineffective or trivial. As a result, the Mordell--Weil rank enhances, and the section that seems to support the correct $\\U(1)$ charges becomes a combination of $\\ratsec{s}_{Q}$ and $\\ratsec{s}_{R}$.\n\n\\paragraph{$r=q=1$}\nWhen $r$ and $q$ are both 1, the desired spectrum of charged hypermultiplets for the $\\U(1)$ model takes the form\n\\begin{equation}\n    \\label{eq:u1q2spectrum}\n    108\\times\\chargedsinglet{1} + 54\\times\\chargedsinglet{2}\\,.\n\\end{equation}\nSince the $\\U(1)$ model only has matter with charges $q=\\pm1$ and\n$\\pm2$, the Weierstrass model should be in the Morrison--Park form \\labelcref{eq:Morrison--Park}. We take the base of the elliptic fibration to be $\\mathbb{P}^2$. For $\\tilde{b}=6(q^2+q r + r^2)=18$, the parameters in Morrison--Park form would have divisor classes\n\\begin{equation}\n    \\divclass{\\mpb} = 6H\\,, \\quad \\divclass{c_3} = 9H\\,, \\quad \\divclass{c_2} = 6H\\,, \\quad \\divclass{c_1} = 3H\\,, \\quad \\divclass{c_0} = 0H\\,.\n\\end{equation}\nThis is precisely an example of the enhancement structure described in\nthe last paragraph of the preceding subsection.\nThe divisor class $\\divclass{c_0}$ is trivial, and $c_0$ can freely be taken to be a perfect square. But if $c_0$ is a perfect square, the gauge group enhances to $\\U(1)\\times\\U(1)$ \\cite{MorrisonTaylorMultisection}. One can verify the presence of this additional $\\U(1)$ by counting moduli and comparing to the expectations from the gravitational anomaly cancellation conditions \\cite{RaghuramTaylorTurnerSM}. Additionally, we can map the Morrison--Park form with trivial $\\divclass{c_0}$ to the $\\U(1)\\times\\U(1)$ model discussed in \\cref{app:u1u1model} by relating the parameters as follows:\n\\begin{equation}\n\\begin{gathered}\n    \\mpb =  S_ 7\\,, \\quad c_0 = \\frac{S_ 5^2}{4}\\,, \\quad c_1 = S_9 S_1-\\frac{1}{2} S_ 5 S_ 6\\,,\\\\\n    c_2= \\frac{1}{4} \\left(-4 S_9 S_ 2+2 S_ 5 S_ 7+S_ 6^2\\right)\\,, \\quad c_3= 2 S_9 S_ 3-\\frac{1}{2} S_ 6 S_7\\,.\n\\end{gathered}\n\\end{equation}\nBecause $\\divclass{c_0}$, $\\divclass{S_5}$, and $\\divclass{S_9}$ are\ntrivial, this is simply a redefinition of parameters, not a\ntuning. Thus, our attempt to obtain the desired $\\U(1)$ model with the\nMorrison--Park form has recovered the $\\U(1)\\times\\U(1)$ model, as\nexpected. One can also verify that the standard Morrison--Park section\n\\begin{equation}\n[\\secx:\\secy:\\secz] = \\left[c_3^2-\\frac{2 \\mpb^2 c_2}{3}:-\\frac{1}{2} \\mpb^4 c_1+\\mpb^2 c_2 c_3-c_3^3:\\mpb\\right]\n\\end{equation}\nis given by $\\ratsec{s}_{Q}-\\ratsec{s}_{R}$.\n\n\\paragraph{$r=2$, $q=1$}\nWhen $r=2$ and $q=1$ (or equivalently $q=2$ and $r=1$), the charged hypermultiplet spectrum takes the form\n\\begin{equation}\n    54\\times\\chargedsinglet{1} + 54\\times\\chargedsinglet{2} +54\\times\\chargedsinglet{3}\\,,\n\\end{equation}\nWe can try to construct this charge-3 $\\U(1)$ model with a blown-down version of the $F_3$ toric hypersurface construction from \\cite{KleversEtAlToric}.\\footnote{Using the more general charge-3 model from \\cite{Raghuram34} does not change the results.} While there is a Weierstrass model for this construction, it is easier to work with an alternative form of the elliptic fibration where the fiber is a cubic curve in a $\\mathbb{P}^2$ ambient space with coordinates $[u:v:w]$:\n\\begin{equation}\nu\\left(s_1 u^2 + s_2 u v + s_3 v^2 + s_5 u w + s_6 v w + s_8 w^2\\right) + v \\left(s_4 v^2 + s_7 v w + s_9 w^2\\right) = 0\\,.\n\\end{equation}\nWe take the base of the elliptic fibration to be $\\mathbb{P}^2$. The charged hypermultiplet spectrum of this construction is\n\\begin{equation}\n\\begin{aligned}\n    \\left(12\\canonclass^2-\\canonclass(8\\divclass{s_7}-\\divclass{s_9}) -4 \\divclass{s_7}^2 +  \\divclass{s_7}\\divclass{s_9}-\\divclass{s_9}^2\\right)&\\times\\chargedsinglet{1} \\\\\n    + \\left(6\\canonclass^2-\\canonclass(4\\divclass{s_9}-5\\divclass{s_7}) + \\divclass{s_7}^2 + 2 \\divclass{s_7}\\divclass{s_9}-2\\divclass{s_9}^2\\right)&\\times\\chargedsinglet{2} \\\\\n    + [s_9]\\cdot\\left(-\\canonclass+\\divclass{s_9} -\\divclass{s_7}\\right)&\\times\\chargedsinglet{3}\\,,\n\\end{aligned}\n\\end{equation}\nand the anomaly coefficient is $\\tilde{b}=-6\\canonclass-2\\divclass{s_7}+4\\divclass{s_9}$.  There are two combinations of $(\\divclass{s_7},\\divclass{s_9})$ that would seem to give the correct charged hypermultiplet spectrum: $(6H, 9H)$ and $(0H,6H)$.\n\nThis model is in fact equivalent to the $\\U(1)\\times\\U(1)$ F-theory model for either choice of $(\\divclass{s_7},\\divclass{s_9})$. If we focus on the $(\\divclass{s_7},\\divclass{s_9})=(0H,6H)$ possibility, the divisor classes of the other parameters are\n\\begin{equation}\n    \\divclass{s_1} = 3H\\,, \\quad \\divclass{s_2} = 0\\,, \\quad \\divclass{s_3} = -3H\\,, \\quad \\divclass{s_4} = -6H\\,, \\quad \\divclass{s_5} = 6H\\,, \\quad \\divclass{s_6} = 3H\\,, \\quad \\divclass{s_8} =  9H\\,.\n\\end{equation}\nBecause $\\divclass{s_3}$ and $\\divclass{s_4}$ are ineffective, $s_3$ and $s_4$ can be set to 0, and the elliptic fibration takes the form\n\\begin{equation}\nu\\left(s_1 u^2 + s_2 u v + s_5 u w + s_6 v w + s_8 w^2\\right) + v w\\left(s_7 v + s_9 w\\right) = 0\\,.\n\\end{equation}\nIf we then exchange $v\\leftrightarrow w$ and identify\n\\begin{equation}\n\\begin{gathered}\n    s_9 = S_7\\,, \\quad s_7 = S_9\\,, \\quad  s_2 = S_5\\,, \\quad s_5 = S_2\\,, \\\\\n    s_8 = S_3\\,, \\quad s_1 = S_1\\,, \\quad s_6 = S_6\\,,\n\\end{gathered}\n\\end{equation}\nwe recover \\cref{eq:u1u1p2desc}, which describes the $\\U(1)\\times\\U(1)$ model. Additionally, one can show that the section supporting the $r=2$, $q=1$ spectrum is given by $2\\ratsec{s}_{Q}-\\ratsec{s}_{R}$. A similar story holds for $(\\divclass{s_7},\\divclass{s_9})=(6H,9H)$. We therefore see the expected enhancement to the $\\U(1)\\times\\U(1)$ model for $r=2$ and $q=1$.\n\n\\paragraph{$r=3$, $q=1$} When $r=3$ and $q=1$ (or when $q=3$ and $r=1$), the $\\U(1)$ charged hypermultiplet spectrum takes the form\n\\begin{equation}\n    54\\times\\chargedsinglet{1} + 54\\times\\chargedsinglet{3} +54\\times\\chargedsinglet{4}\\,.\n\\end{equation}\nWe can attempt to construct this model using the charge-4 $\\U(1)$ Weierstrass model in \\cite{Raghuram34}. According to the matter spectrum given in \\cite{Raghuram34}, we should let $\\divclass{a_1}$ be $6H$, $\\divclass{b_1}$ be $9H$, and $\\divclass{d_1}$ be $0H$ to obtain the desired $\\U(1)$ spectrum.\\footnote{We could also let $\\divclass{a_1}=9H$ and $\\divclass{b_1}=6H$, but this situation is equivalent to the $\\divclass{a_1}=6H, \\divclass{b_1}=9H$ model.} The other parameters have the following divisor classes:\n\\begin{equation}\n\\begin{aligned}\n\\divclass{d_0}&=-3H & \\divclass{d_2}&=3H  &  \\divclass{s_1}&=-6H & \\divclass{s_2}&=-3H \\\\\n\\divclass{s_3}&=0H  & \\divclass{s_5}&=0H & \\divclass{s_6}&=3H & \\divclass{s_8} &= 6H\n\\end{aligned}\n\\end{equation}\nBecause their divisor classes are ineffective, the parameters $d_0$, $s_1$, and $s_2$ should be set to 0, giving us a Weierstrass model described by\n\\begin{equation}\n\\begin{aligned}\n    f &= -\\frac{1}{48} \\left(s_6^2-4 s_3 s_8\\right)^2-\\frac{1}{2} s_5 s_6 s_3 \\left(a_1 d_2+b_1 d_1\\right) \\\\\n    &\\qquad\\qquad - \\frac{1}{3} a_1^2 d_1^2 s_5^2+\\frac{1}{6} a_1 d_1 s_5 \\left(s_6^2+2 s_3 s_8\\right)+b_1 d_2 s_5 s_3^2\n\\end{aligned}\n\\end{equation}\nand\n\\begin{equation}\n\\begin{aligned}\n    g &= \\frac{1}{864} \\left(s_6^2-4 s_3 s_8\\right)^3+\\frac{1}{4} s_3^2 s_5^2 \\left(a_1^2 d_2^2+b_1^2 d_1^2\\right)-\\frac{1}{6} s_3 s_6 s_5^2 \\left(a_1 b_1 d_1^2+a_1^2 d_1 d_2\\right) \\\\\n&\\qquad\\qquad + \\frac{1}{24} s_3 s_6 \\left(s_6^2-4 s_3 s_8\\right) s_5 \\left(a_1 d_2+b_1 d_1\\right) \\\\\n&\\qquad\\qquad - \\frac{1}{72} a_1 d_1 \\left(12 b_1 d_2 s_3^2 s_5^2+\\left(s_6^2-4 s_3 s_8\\right) \\left(s_6^2+2 s_3 s_8\\right) s_5\\right) \\\\\n&\\qquad\\qquad - \\frac{2}{27} a_1^3 d_1^3 s_5^3+\\frac{1}{18} a_1^2 d_1^2 \\left(s_6^2+2 s_3 s_8\\right) s_5^2-\\frac{1}{12} b_1 d_2 s_3^2 \\left(s_6^2-4 s_3 s_8\\right) s_5\\,.\n\\end{aligned}\n\\end{equation}\nThis Weierstrass model is in fact equivalent to the $\\U(1)\\times\\U(1)$\nWeierstrass model of \\cref{app:u1u1model}. To see this, we first note\nthat we can remove all occurrences of $s_5$ through the rescalings\n$d_2\\to s_5^{-1} d_2$ and $d_1\\to s_5^{-1} d_1$;  since\n$\\divclass{s_5}$ is trivial, the divisions by $s_5$ do not cause\nissues. Then, we can identify the parameters in the two Weierstrass models as follows:\n\\begin{equation}\nS_1 = d_2\\,, \\quad S_2 = s_8\\,, \\quad S_3 = b_1\\,, \\quad S_5 =d_1\\,, \\quad S_6 = s_6\\,, \\quad S_7 = a_1\\,, \\quad S_9 = s_3\\,.\n\\end{equation}\nThe Weierstrass models are equivalent, implying that the gauge group is $\\U(1)\\times\\U(1)$ instead of just $\\U(1)$. And one can verify that the section that seems to support the $r=3, q=1$ spectrum is in fact equivalent to $3\\ratsec{s}_{Q}-\\ratsec{s}_{R}$. We therefore see the expected automatic enhancement when we attempt to construct the $r=3$, $q=1$ $\\U(1)$ model.\n\nA similar infinite family of $\\U(1)$ models with one tensor multiplet was given in \\cite{ParkTaylorAbelian}. The charged hypermultiplet spectra of these models, which take the form\n\\begin{equation}\n    \\label{eq:chargefamilyspectrumT1}\n    48\\times\\chargedsinglet{r} + 48\\times\\chargedsinglet{s} + 48\\times\\chargedsinglet{(r+s)}\\,,\n\\end{equation}\nmimic the structure of the $T=0$ family discussed above. The analysis in \\cite{ParkTaylorAbelian} pointed out that there is a related $\\U(1)\\times\\U(1)$ model whose charged hypermultiplet spectrum is\n\\begin{equation}\n    48\\times\\twochargedsinglet{1}{0} + 48\\times\\twochargedsinglet{0}{1} + 48\\times\\twochargedsinglet{1}{1}\\,.\n\\end{equation}\nAs with the $T=0$ infinite family, one can view the $\\U(1)$ algebras\nin the $T=1$ infinite family as linear combinations of the two $\\U(1)$\nalgebras in this $\\U(1)\\times\\U(1)$ model. But we lack the matter\nnecessary to properly Higgs the $\\U(1)\\times\\U(1)$ model down to the\ndesired $\\U(1)$ models unless $r$,$s$, or $r+s$ is 0. The\nAutomatic Enhancement Conjecture\\ therefore implies that if we attempt to construct the\nmodels in this infinite family, the gauge group should enhance to\n$\\U(1)\\times\\U(1)$. One can confirm that this occurs in the\n$(r,s)=(1,1)$, $(2,1)$, and $(3,1)$ cases by an explicit analysis of F-theory constructions along the lines above; some aspects of this analysis are discussed in \\cref{app:u1u1test}.\n\nThere is another infinite family of $\\U(1)$ models \\cite{TaylorTurnerSwampland}  with zero tensor multiplets and charged hypermultiplet spectra of the form\n\\begin{equation}\n    \\label{eq:fourchargefamily}\n    54 \\times \\chargedsinglet{a} + 54 \\times \\chargedsinglet{b} + 54 \\times \\chargedsinglet{c} + 54 \\times \\chargedsinglet{d}\\,,\n\\end{equation}\nwhere\n\\begin{equation}\n    a = m^2 - 2 m n\\,, \\quad b = 2 m n-n^2\\,, \\quad c = m^2 - n^2\\,, \\quad d = 2(m^2 - m n + n^2)\n\\end{equation}\nfor $m,n \\in \\mathbb{Z}_{+}$ with $n\\le \\frac{m}{2}$. Some of these models belong to the three-charge infinite family of \\cref{eq:chargefamilyspectrum}, suggesting that the Automatic Enhancement Conjecture\\ may be important for the four-charge family as well. However, we have not yet found an enhanced model for the nontrivial four-charge models. Part of the difficulty lies in the complicated nature of the models: because even the simplest non-trivial members of the family have $\\U(1)$ charges larger than those found in the known F-theory constructions, we cannot directly observe any enhancements. And  the quadratic structure of the charges, exhibited by their dependence on $m$ and $n$, contrasts with the linear nature of \\cref{eq:chargefamilyspectrum,eq:chargefamilyspectrumT1} that naturally reflects combinations of multiple $\\ensuremath\\mathfrak{u}(1)$ algebras. In fact, a similar quadratic structure appears in the $\\ensuremath\\mathfrak{su}(13)\\oplus\\ensuremath\\mathfrak{u}(1)$ family of \\cref{sec:13}. Understanding whether automatic enhancement plays a role in this four-charge family is an important direction for future work.\n\nTo summarize, the Automatic Enhancement Conjecture\\ implies that none of the $\\U(1)$ models in the infinite families of \\cref{eq:chargefamilyspectrum,eq:chargefamilyspectrumT1} (except for trivial cases with only one type of non-zero charge) can be realized in F-theory, as the gauge group is forced to enhance to $\\U(1)\\times\\U(1)$. The conjecture thus offers a reason as to why those models in the infinite family lie in the F-theory swampland. While finding the ``failure mode'' for this family is an important step, it does not definitely tell us which $\\U(1)$ charges can generally occur in F-theory. Yet this result offers a valuable lesson: limits on the maximum $\\U(1)$ charge in F-theory likely come not from local constraints on the behavior of sections but from global considerations \\cite{MorrisonParkU1}. The problem with models in the infinite family lies in an enhancement of the Mordell--Weil rank, which is a global property of the fibration.\n\n\\subsection{Infinite family of $\\ensuremath\\mathfrak{su}(13)\\oplus\\ensuremath\\mathfrak{u}(1)$ models}\n\\label{sec:13}\n\nIn \\cite{ParkTaylorAbelian}, an infinite family of $T=1$ supergravity models with $\\ensuremath\\mathfrak{su}(13)\\oplus\\ensuremath\\mathfrak{u}(1)$ gauge algebras was presented. The charged hypermultiplet spectra for these models take the form\n\\begin{equation}\n    \\label{eq:su13familyspec}\n    \\bm{78}_{-3a-\\frac{2}{3}f} + 3\\times\\bm{78}_{a} + 6\\times\\bm{13}_f + 18\\times\\bm{1}_{6a+f}\\,,\n\\end{equation}\nwhere\n\\begin{equation}\n\\begin{aligned}\n    a&= 13 r^2 - 234 r s - 51 s^2 & f&= 24(13 r^2 +3 s^2)\\\\\n    r&= 84 n+ 43 & s&= 182 n + 92\n\\end{aligned}\n\\end{equation}\nfor $n \\in \\mathbb{Z}$. The anomaly coefficient for the $\\ensuremath\\mathfrak{u}(1)$ algebra is given by $\\tilde{b}=\\frac{1}{2}(\\alpha,\\tilde{\\alpha})$ with\n\\begin{equation}\n\\begin{aligned}\n    \\alpha &= 52\\left[6687s^4+ 54756 s^3 r + 94458 s^2 r^2 - 124956 s r^3 + 39455 r^4 \\right]\\,,\\\\\n    \\tilde{\\alpha} &= 52\\left[2475s^4 + 37908 s^3 r + 170274 s^2 r^2 - 29484 s r^3 + 9035r^4\\right]\\,,\n\\end{aligned}\n\\end{equation}\nin a basis where the intersection form is\n\\begin{equation}\n    \\Omega = \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix}\\,.\n\\end{equation}\n\nFirst, note that if we attempt to obtain just the $\\ensuremath\\mathfrak{su}(13)$ algebra, the model suffers an automatic enhancement. In order to obtain four antisymmetric and six fundamental hypermultiplets of $\\ensuremath\\mathfrak{su}(13)$, the anomaly coefficient $b$ for the $\\ensuremath\\mathfrak{su}(13)$ algebra should satisfy $b\\cdot b = 2$ and $-a\\cdot b=4$, implying that $(-6a-13b)\\cdot b$ is negative and that $-6a-13b$ is ineffective. By the discussion in \\cref{sec:t0-general-n},\nthe $\\ensuremath\\mathfrak{su}(13)$ algebra should automatically enhance to $\\ensuremath\\mathfrak{su}(13)\\oplus\\ensuremath\\mathfrak{u}(1)_f$, where we use the subscript $f$ to distinguish this $\\ensuremath\\mathfrak{u}(1)$ algebra from those in the infinite family.\\footnote{Note that $-3a-6b$ is effective, and there should not be automatic enhancements beyond the one to $\\ensuremath\\mathfrak{su}(13)\\oplus\\ensuremath\\mathfrak{u}(1)$.} The charged hypermultiplet spectrum\n\\begin{equation}\n    \\label{eq:su13autoenhancespec}\n    4\\times\\bm{78}_{-\\frac{1}{13}} + 6\\times\\bm{13}_{\\frac{6}{13}}\n\\end{equation}\nlacks the necessary charged singlets to Higgs away the forced $\\ensuremath\\mathfrak{u}(1)$ algebra, in line with the Automatic Enhancement Conjecture.\n\nStrictly speaking, this statement only applies to a model with an $\\ensuremath\\mathfrak{su}(13)$ gauge algebra, not the $\\ensuremath\\mathfrak{su}(13)\\oplus\\ensuremath\\mathfrak{u}(1)$ models in the infinite family. But because we are able to phrase this automatic enhancement locally in terms of $a\\cdot b$ and $b\\cdot b$, we would expect the $\\ensuremath\\mathfrak{su}(13)$ subalgebra of $\\ensuremath\\mathfrak{su}(13)\\oplus\\ensuremath\\mathfrak{u}(1)$ to also experience an automatic enhancement. A naive guess is that the gauge algebra for the infinite family automatically enhance to $\\ensuremath\\mathfrak{su}(13)\\oplus\\ensuremath\\mathfrak{u}(1)\\oplus\\ensuremath\\mathfrak{u}(1)_f$, but if one combines the $\\ensuremath\\mathfrak{u}(1)$ charges in \\cref{eq:su13familyspec} and \\cref{eq:su13autoenhancespec}, the resulting spectrum does not satisfy the anomaly cancellation conditions. Nevertheless, the expected automatic enhancement of the $\\ensuremath\\mathfrak{su}(13)$ algebra suggests models in this $\\ensuremath\\mathfrak{su}(13)\\oplus\\ensuremath\\mathfrak{u}(1)$ infinite family should have some inconsistency, even though we have not found an explicit automatic enhancement.\n\nThis argument does not seem to capture the true problem with the\nfamily, however. There are also seemingly consistent\n$\\ensuremath\\mathfrak{su}(N)\\oplus\\ensuremath\\mathfrak{u}(1)$ infinite families for $N<13$, which can be found\nby Higgsing the would-be $\\ensuremath\\mathfrak{su}(13)\\oplus\\ensuremath\\mathfrak{u}(1)$ models on\nfundamental matter. The $\\ensuremath\\mathfrak{su}(N)$ subalgebras would not suffer the\nautomatic enhancement of $\\ensuremath\\mathfrak{su}(13)$ described above, but the\nmechanism that eliminates the $N<13$ families would presumably apply\nto the $\\ensuremath\\mathfrak{su}(13)\\oplus\\ensuremath\\mathfrak{u}(1)$ family as well. We therefore expect\nthere to be an alternative, more satisfying explanation that rules\nthem out. The quadratic nature of the $\\ensuremath\\mathfrak{u}(1)$ charges resembles\nthat seen in the four-charge family of \\cref{eq:fourchargefamily},\nsuggesting that all of these families suffer from similar\nissues. But it is difficult to see how the quadratic structure of the charges\nwould fit naturally in any kind of linear $\\ensuremath\\mathfrak{u}(1)$ subalgebra of an\nenhanced $\\U(1)^k$ theory. Additionally, the charges are significantly larger than\nthose that can be easily realized in F-theory constructions, so it\nwould be tough to directly observe how the models fail.\nIt would be interesting to understand these infinite families more properly in future work.\n\n\\subsection{Automatic enhancement for $(\\SU(3) \\times \\SU(2) \\times\n\\U(1)) \/ \\mathbb{Z}_6$ models}\n\\label{sec:SM}\n\nIn \\cite{RaghuramTaylorTurnerSM}, the universal Weierstrass model was\nconstructed for the Standard Model gauge group $(\\SU(3) \\times \\SU(2)\n\\times \\U(1)) \/ \\mathbb{Z}_6$. As discussed there, the model undergoes\nautomatic enhancement for certain choices of the anomaly coefficients\n$b_3, b_2, \\beta$. In particular, the Jacobian rank method introduced\nin that paper was used to count the number of moduli for all 98 of the\nallowed parameter choices for 6D models over the base\n$\\mathbb{P}^2$. In 44 of these cases the moduli count does not\nmatch with the expectation from the spectrum fixed by anomaly\ncancellation, and the model undergoes automatic enhancement or is\nrendered inconsistent.\n\nIn 34 of these cases, the gauge group enhances by the addition of\nnonabelian or $\\U(1)$ factors. In each of these cases, the resulting\nunHiggsed theory is anomaly consistent but has insufficient matter to\nsupersymmetrically Higgs to the original $(\\SU(3) \\times \\SU(2) \\times\n\\U(1)) \/ \\mathbb{Z}_6$ model, consistent with the Automatic Enhancement Conjecture. In the\nremaining 10 cases, the discriminant vanishes identically, indicating\nthat the Weierstrass model is invalid. These are all models that would\napparently have an unHiggsing to a Tate $\\SU(4) \\times \\SU(3) \\times\n\\SU(2)$ model.\nHowever, it is found that the corresponding Tate models\nalso have identically vanishing discriminant, corresponding to the\nstronger case-specific version of the enhancement conjecture that when\ncertain positivity conditions are violated the theory is either\nenhanced or inconsistent. The various cases are summarized in\n\\cref{tab:321AEC}.\n\n\\begin{table}\n    \\centering\n\n    \\[\\begin{array}{*{11}{c}} \\toprule\n        \\divclass{b_3} & \\divclass{d_0} & \\divclass{s_1} & \\divclass{s_2} & \\divclass{d_1} & \\divclass{d_2} & \\divclass{s_5}\n            & \\divclass{s_8} & \\text{\\# of Models} & \\text{Gauge Algebra} \\\\ \\midrule\n        %\n        +              & +              & \\ge            & -              & \\ge            & +              & +             \n            & \\ge            & 11                  & \\ensuremath\\mathfrak{su}(3) \\oplus \\ensuremath\\mathfrak{su}(2) \\oplus \\ensuremath\\mathfrak{u}(1) \\oplus \\ensuremath\\mathfrak{u}(1) \\\\\n        %\n        +              & +              & -              & -              & \\ge            & +              & \\ge           \n            & 0              & 9                   & \\ensuremath\\mathfrak{su}(3) \\oplus \\ensuremath\\mathfrak{su}(3) \\oplus \\ensuremath\\mathfrak{u}(1) \\\\\n        %\n        +              & +              & -              & -              & +              & +              & +             \n            & +              & 4                   & \\ensuremath\\mathfrak{su}(3) \\oplus \\ensuremath\\mathfrak{su}(3) \\oplus \\ensuremath\\mathfrak{su}(2) \\oplus \\ensuremath\\mathfrak{u}(1) \\\\\n        %\n        +              & +              & 0              & -              & +              & +              & \\ge           \n            & -              & 3                   & \\ensuremath\\mathfrak{su}(4) \\oplus \\ensuremath\\mathfrak{su}(2) \\oplus \\ensuremath\\mathfrak{u}(1) \\oplus \\ensuremath\\mathfrak{u}(1) \\\\\n        %\n        +              & +              & \\ge            & -              & 0              & +              & 0             \n            & 0              & 3                   & \\ensuremath\\mathfrak{su}(3) \\oplus \\ensuremath\\mathfrak{su}(2) \\oplus \\ensuremath\\mathfrak{u}(1) \\oplus \\ensuremath\\mathfrak{u}(1) \\oplus \\ensuremath\\mathfrak{u}(1) \\oplus \\ensuremath\\mathfrak{u}(1) \\\\\n        %\n        +              & +              & 0              & -              & 0              & +              & -             \n            & -              & 1                   & \\ensuremath\\mathfrak{su}(5) \\oplus \\ensuremath\\mathfrak{su}(2) \\oplus \\ensuremath\\mathfrak{u}(1) \\oplus \\ensuremath\\mathfrak{u}(1) \\\\\n        %\n        +              & +              & +              & -              & +              & +              & 0             \n            & -              & 1                   & \\ensuremath\\mathfrak{su}(4) \\oplus \\ensuremath\\mathfrak{su}(2) \\oplus \\ensuremath\\mathfrak{su}(2) \\oplus \\ensuremath\\mathfrak{u}(1) \\oplus \\ensuremath\\mathfrak{u}(1) \\\\\n        %\n        +              & +              & 0              & -              & +              & +              & -             \n            & -              & 1                   & \\ensuremath\\mathfrak{su}(5) \\oplus \\ensuremath\\mathfrak{su}(2) \\oplus \\ensuremath\\mathfrak{su}(2) \\oplus \\ensuremath\\mathfrak{u}(1) \\oplus \\ensuremath\\mathfrak{u}(1) \\\\\n        %\n        +              & +              & +              & -              & 0              & +              & -             \n            & -              & 1                   & \\ensuremath\\mathfrak{su}(5) \\oplus \\ensuremath\\mathfrak{su}(3) \\oplus \\ensuremath\\mathfrak{su}(2) \\oplus \\ensuremath\\mathfrak{u}(1) \\oplus \\ensuremath\\mathfrak{u}(1) \\\\\n        %\n        +              & +              & -              & -              & \\ge            & +              & \\pm           \n             & -              & 10                  & \\text{Invalid} \\\\ \\bottomrule\n    \\end{array}\\]\n\n    \\caption{Cases where the $(\\SU(3) \\times \\SU(2) \\times \\U(1)) \/ \\mathbb{Z}_6$ universal Weierstrass model undergoes automatic enhancement over the base $\\mathbb{P}^2$. Cases are grouped by their resulting gauge algebra, which is determined by which divisor classes $[b_3]\\text{--}[s_8]$ associated with parameters in the Weierstrass model are strictly effective ($+$), effective ($\\ge$), trivial ($0$), ineffective ($-$), or unspecified ($\\pm$). The parameter $s_6$ is not listed and always has class $-\\canonclass = 3 H$.}\n    \\label{tab:321AEC}\n\\end{table}\n\n\\section{Positivity conditions and\nlocal conditions for automatic enhancement}\n\\label{sec:local}\n\nIn each specific class of cases we have encountered in the preceding\nsections, the pattern is essentially that when we try to tune an\nF-theory model with algebra $\\ensuremath\\mathfrak{g}$ with a certain anomaly coefficient\n$b$ (or $\\tilde{b}$), and when the curve class associated with\na certain other string charge $-n a-mb$ ceases to be effective,\nthen either an additional gauge factor with algebra of at least $\\ensuremath\\mathfrak{f}$\narises with another anomaly coefficient $b'= -pa-qb$, or the theory\nbecomes inconsistent due to excessive singularities.\nPut more simply, under such circumstances we cannot have a theory with\ngauge algebra $\\ensuremath\\mathfrak{g}$ having anomaly coefficient $b$ satisfying the\nstated conditions unless there is also a gauge algebra component of at\nleast $\\ensuremath\\mathfrak{f}$\nassociated with the anomaly coefficient $b'$.\n\nIn the context of F-theory, for each such algebra enhancement, the\nconjecture can thus be stated in terms of the effectiveness of certain\ncurves; in supergravity this translates to conditions that certain\nanomaly coefficients or other elements of the string charge lattice\nlie in or outside the positivity cone.\nWhile we do not have any completely general way of unifying these\npositivity conditions into a single conjecture, in each individual\ncase, the statement about positivity conditions is stronger than the\nAutomatic Enhancement Conjecture, since it includes those cases where the theory is\nrendered invalid by the would-be enhancement.  In this section we collect the\nspecific results found in the other sections for this stronger condition.\n\nWe also collect here some  corresponding local versions of the\nAutomatic Enhancement Conjecture.  In each case,  the general form of the statement is that\nif there is a gauge component $\\ensuremath\\mathfrak{g}$ with anomaly coefficient $b$ satisfying\ncertain conditions, then a consistent F-theory model is only possible\nwhen there is an additional gauge component\nof at least $\\ensuremath\\mathfrak{f}$ associated with another\nanomaly coefficient $b'$.\nThe conditions in the local form for each case only depend on $b \\cdot a, b \\cdot b$, which\ncorrespond to local conditions in the F-theory geometry, namely the\ngenus and self-intersection of the associated curve.\nThese conditions are weaker than the global positivity conditions, but\nare useful in analyzing models using only part of the structure of the theory.\n\nA number of these conditions that we have\nencountered in the paper are listed in \\cref{tab:local}.\n\n\\begin{table}\n    \\centering\n\n    \\renewcommand{\\arraystretch}{1.2}\n    \\[\\begin{array}{*{5}{c}}\n    \\toprule\n    \\text{Algebra } \\ensuremath\\mathfrak{g} &  \\text{Positivity Condition} & \\text{Local Condition} & \\ensuremath\\mathfrak{f} & b'\n    \\\\\n    \\midrule\n    \\text{---} & \\text{---} & b \\cdot b < -2, g = 0 & G' \\supseteq \\SU(3) & b \\\\\n    \\text{---} & \\text{---} & b \\cdot b \\le -13, g = 0 &  \\text{no theory} & \\text{---} \\\\\n    \\text{---} & \\text{---} & b \\cdot b < -2, g > 0 & \\text{no theory} & \\text{---} \\\\\n    \\midrule\n    \\ensuremath\\mathfrak{su}(2) & -3a-b  >0 &\n    b \\cdot b > 0, b \\cdot b > -3a \\cdot b &\n    \\ensuremath\\mathfrak{su}(2) & -4a-b \\\\\n    \\ensuremath\\mathfrak{su}(2) & b  = -3a & \\text{---} &\n    \\ensuremath\\mathfrak{u}(1) & \\tilde{b}' =-8a-2b \\\\\n    \\ensuremath\\mathfrak{su}(3) &-2a-b > 0  & b \\cdot b > 0, b \\cdot b > -2a \\cdot b &\n    \\ensuremath\\mathfrak{su}(3) & -3a-b \\\\\n    \\ensuremath\\mathfrak{su}(3) &  b = -2a & \\text{---} &\n    \\ensuremath\\mathfrak{u}(1)\\oplus \\ensuremath\\mathfrak{u}(1) &\n    {\\renewcommand{\\arraystretch}{1} \\tilde{b}' =\n    -a \\text{\\small $\\begin{pmatrix} 6&-3\\\\-3&6\\end{pmatrix}$}} \\\\\n   \n    \\ensuremath\\mathfrak{su}(4) & -3a-2b > 0  &b \\cdot b > 0, 2b \\cdot b > -3a \\cdot b &\n    \\ensuremath\\mathfrak{su}(2) & -4a-2b \\\\\n   \n   \n    \\ensuremath\\mathfrak{su}(n=\\text{20--23}) & \\text{---} & g = 0, b \\cdot b = 1 &\n    \\ensuremath\\mathfrak{su}(2) &  -4a-\\lceil n\/2\\rceil b \\\\\n    \\ensuremath\\mathfrak{u}(1) & -6a-\\tilde{b} > 0  &\n    \\tilde{b} \\cdot \\tilde{b} > 0, \\tilde{b} \\cdot \\tilde{b} > -6a \\cdot \\tilde{b} &\n    \\ensuremath\\mathfrak{su}(2) &   -4a-\\tilde{b}\/2 \\\\\n    \\ensuremath\\mathfrak{u}(1) &  \\tilde{b} = -6a & \\text{---} &\n    \\ensuremath\\mathfrak{u}(1) & {\\renewcommand{\\arraystretch}{1} \\tilde{b}' =\n    -a \\text{\\small $\\begin{pmatrix} 2&-1\\\\-1&2\\end{pmatrix}$}} \\\\\n    \\bottomrule\n    \\end{array}\\]\n\n    \\caption{Positivity cone and\n    local versions of the Automatic Enhancement Conjecture, for cases\n      with generic matter.\n    In each case a gauge component $\\ensuremath\\mathfrak{g}$ with anomaly coefficient $b$\n    satisfying certain conditions forces enhancement to an additional\n    gauge component of at least\n    $\\ensuremath\\mathfrak{f}$ with anomaly coefficient  $b'$.  (For $\\ensuremath\\mathfrak{u}(1)$ factors, anomaly\n    coefficients  are $\\tilde{b}, \\tilde{b}'$ respectively; note that this\n    anomaly coefficient corresponds to the normalization of the $\\ensuremath\\mathfrak{u}(1)$\n    factor so that all charges are integers.)\n    The ``genus'' contribution $g$\n    is given in terms of local anomaly coefficients $a, b$ by \\cref{eq:genus}.}\n    \\label{tab:local}\n\\end{table}\n\n\\noindent Some comments on these local versions of the Automatic Enhancement Conjecture:\n\\begin{itemize}\n    \\item Unlike the general statement of the Automatic Enhancement Conjecture, the local\n    versions of these statements are specific to rather particular\n    circumstances of gauge groups and anomaly coefficients.  The list here\n    is not in any way intended to be complete, and even for those cases\n    listed, we have given only some simple sufficient local conditions for each\n    enhancement.  It would\n    also be desirable to have a more unified way of framing these in terms\n    of a more general statement analogous to Automatic Enhancement Conjecture.\n\n\n    \\item In cases where $b'$ is not in the positive cone, there is no\n    corresponding F-theory model --- the enhancement breaks the theory.\n\n    \\item The local conditions given here are only sufficient\n    conditions, since there can be an enhancement that depends on further\n    structure. In particular, even when the sufficient local conditions\n    are not satisfied, the presence of certain other types of curves\n    intersecting the curve supporting the gauge divisor can provide for an\n    enhancement in more general circumstances.  An example of this is\n    given in, for example, \\cref{sec:small-bb}, where we see that the\n    usual enhancement that begins at $\\ensuremath\\mathfrak{su}(13)$ on a curve of genus 0 and\n    self-intersection $b \\cdot b = 2$ also occurs for $\\ensuremath\\mathfrak{su}(10$--$12)$ when\n    the curve in question intersects another curve $b'$ of\n    self-intersection $b' \\cdot b' = -1$.  A more careful and thorough\n    analysis of local conditions incorporating, e.g., adjacent divisors\n    could be made using similar principles to those used in\n    \\cite{JohnsonTaylorEnhanced}.\n\n    \\item As for the general Automatic Enhancement Conjecture, the positivity based and\n    local versions given here for specific gauge factors are mostly\n    conjectural even at the level of F-theory. The case of non-Higgsable\n    clusters $(G = 1)$ was proven in full generality in F-theory in\n    \\cite{MorrisonTaylorClusters}, but the other examples  depend upon some assumptions on the form of the Weierstrass model\n    used to achieve the initial gauge algebra $G$.  In particular, while\n    the general form\n    has been proven for, e.g.,the $\\ensuremath\\mathfrak{su}(n)$\n    algebras under the UFD assumption \\cite{MorrisonTaylorMaS}, it\n    is possible that more general forms of the Weierstrass model such as\n    those considered in \\cite{KleversEtAlExotic} may evade the conjecture\n    at both the local and global levels.  Although such non-UFD constructions\n    in general are expected to give exotic matter representations, this\n    has not been proven to always be the case. Nonetheless, as discussed\n    in \\cref{app:u1u1test}, a variety of effects can remove non-UFD\n    structure, and based on the example presented there it seems likely that these effects conspire to ensure that models without exotic matter take UFD forms.\n\n    $\\bullet$\n    As for the general form of the\n    Automatic Enhancement Conjecture, it is natural to speculate that the  group-specific\n    stronger versions of\n    the conjecture also hold for 6D supergravity theories more generally;\n    this possibility is discussed further in the conclusions section.\n\n\n    \\item We have only listed here cases with generic matter, which\n    depend in general on knowing the corresponding universal Weierstrass\n    models.\n\n    \\item As mentioned earlier,\n    in cases where the automatic enhancement involved would\n    break the theory, such as by producing codimension one (4, 6) loci or\n    when $b'$ is not in the positive cone,\n    the  gauge-specific positivity-based form and the\n    local form of the Automatic Enhancement Conjecture\\ still effectively rule out the\n    relevant swampland models, while the global form as stated in\n    \\cref{sec:conjecture} does not due to the absence of a consistent\n    containing theory.  Several examples of this are given in\n    \\cref{sec:break,sec:13}.\n\n    \\item In each case where  the local enhancement mechanism forces a\n    gauge group factor with a new anomaly coefficient $b'$, as discussed\n    in \\cref{sec:su2su2}, this can also be achieved by having an\n    enhanced gauge factor on each of a set of anomaly coefficients that\n    add to $b' = b_1+ \\cdots b_k$.\n\\end{itemize}\n\n\\section{Exotic matter and discrete gauge groups}\n\\label{sec:exotic-discrete}\n\nIn this section we discuss several further classes of apparent\nswampland models, those involving exotic matter and those involving\ndiscrete gauge groups.  We first describe some cases of exotic matter\ntypes that should be allowed in F-theory but where certain\ncombinations of these matter representations lie in the apparent\nswampland.  We then briefly describe exotic matter that goes outside\nthe F-theory classification of \\cite{KleversEtAlToric} and situations\nwith discrete gauge groups.  These latter two classes of examples are\nstill poorly understood in F-theory, and for these types of apparent\nswampland models some further work and insights may be needed to fit\nwith the context of the automatic enhancement mechanisms discussed in\nthis paper.\n\n\n\\subsection{An $\\ensuremath\\mathfrak{su}(2)$ model with two half-hypermultiplets of\n  $4$ matter.}\n\\label{sec:exotic-F-theory}\n\nIn addition to the fundamental and adjoint representations of $\\ensuremath\\mathfrak{su}(2)$, F-theory models can realize matter in the the three-index symmetric ($\\bm{4}$) representation. Such matter is supported at triple point singularities of the divisor supporting the $\\ensuremath\\mathfrak{su}(2)$ algebra.  More precisely, a triple point singularity of an $\\ensuremath\\mathfrak{su}(2)$ divisor in a 6D F-theory model may support a half-hypermultiplet of $\\bm{4}$ matter and a full hypermultiplet of fundamental ($\\bm{2}$) matter \\cite{KleversTaylorTriple, KleversEtAlExotic}.\\footnote{It is also possible for triple points to support adjoint matter \\cite{KleversEtAlExotic}, but this possibility is not important for the models considered here.}  While there are several matter spectra with $\\bm{4}$ matter that occur in F-theory models, there are also matter spectra that are consistent with the anomaly conditions but pose problems when one tries to construct them in F-theory. One example, described in \\cite{KleversEtAlExotic}, is a model with no tensor multiplets, an $\\ensuremath\\mathfrak{su}(2)$ gauge algebra, and the following spectrum of hypermultiplets:\n\\begin{equation}\n\\label{eq:quinticsu2spec}\n2\\times\\frac{1}{2}\\bm{4} + 84\\times\\bm{2} + 104\\times\\bm{1}\\,.\n\\end{equation}\nThis spectrum satisfies the anomaly cancellation conditions with $\\gaugedivclass = 5$. We would therefore expect the corresponding F-theory model to have a $\\mathbb{P}^2$ base, and the $\\ensuremath\\mathfrak{su}(2)$ algebra should be tuned on a quintic curve. Because the model supports two half-hypermultiplets of $\\bm{4}$ matter, the quintic curve should have two triple points.\n\nBut as pointed out in \\cite{KleversEtAlExotic}, an irreducible quintic curve on $\\mathbb{P}^2$ cannot have two triple point singularities. If such a quintic existed, we could always find a line that goes through both of the triple points. According to Bezout's theorem, this line should intersect the quintic at five points, counted with multiplicity. The line and the quintic intersect at the two triple points, and each of these intersections would have multiplicity three. Therefore, the triple points alone would give us an intersection number of $6$, which is greater than the expected value of $5$ for the total intersection number from Bezout's theorem. This contradiction shows that an irreducible quintic curve cannot have two triple points.\n\nThere is a potential loophole in this argument: the quintic curve can contain the line passing through the triple points. In this case, the quintic curve would be reducible. For instance, suppose that, without loss of generality, the two distinct triple points are described by the intersection of a line $\\locus{\\trplb=0}$ and a quadratic curve $\\locus{\\trpla=0}$.\nA quintic curve of the form\n\\begin{equation}\n\\label{eq:reduciblequintic}\n\\trplb\\left(3\\tb \\trpla^2 + 3\\tc \\trpla \\trplb + \\td \\trplb^2\\right)=0\n\\end{equation}\nwith\n\\begin{equation}\n    \\divclass{\\tb} = 0H\\,, \\quad \\divclass{\\tc} = 1H\\,, \\quad \\divclass{\\td} = 2H\n\\end{equation}\nwould then appear to have two triple points. Of course, this quintic\ncurve reduces to the union of a line and a quartic curve. To deform\nthis reducible quintic into an irreducible quintic with two triple\npoints, we would need to introduce a term of the form $\\ta\n\\trpla^3$.\n(All of the terms in the quintic must be polynomials of order three in $\\trpla,\\trplb$ to\npreserve the triple point singularities of the quintic and the non-UFD $\\ensuremath\\mathfrak{su}(2)$ structure \\cite{KleversEtAlExotic}.)\n The parameter $\\ta$ would then be ineffective and would be\nforced to vanish, giving us the reducible quintic of\n\\cref{eq:reduciblequintic} once again. Clearly, there is a geometric\nobstruction to deforming the reducible quintic into an irreducible\ncurve.\n\nIf we were to tune $I_2$ singularities along the curve in \\cref{eq:reduciblequintic}, the resulting gauge algebra would be $\\ensuremath\\mathfrak{su}(2)\\oplus\\ensuremath\\mathfrak{su}(2)$ rather than $\\ensuremath\\mathfrak{su}(2)$. We take the first $\\ensuremath\\mathfrak{su}(2)$ to correspond to the line $\\locus{\\trplb=0}$ and the second $\\ensuremath\\mathfrak{su}(2)$ to correspond to the quartic curve. One can find the corresponding Weierstrass model by setting $\\ta$ to 0 in the $\\ensuremath\\mathfrak{su}(2)$ construction given in Appendix B of \\cite{KleversEtAlExotic}. Each $\\locus{\\trpla=\\trplb=0}$ point supports a half-hypermultiplet of $(\\bm{2},\\bm{3})$ matter and a half-hypermultiplet of $(\\bm{2},\\bm{1})$ matter. The complete hypermultiplet spectrum for this model is\n\\begin{equation}\n\\label{eq:quinticsu2su2spec}\n2\\times\\frac{1}{2}(\\bm{2},\\bm{3})  + 1\\times(\\bm{1},\\bm{3}) + 19\\times(\\bm{2},\\bm{1}) + 64\\times(\\bm{1},\\bm{2}) + 104\\times(\\bm{1},\\bm{1})\\,.\n\\end{equation}\n\nThe appearance of the $\\ensuremath\\mathfrak{su}(2)\\oplus\\ensuremath\\mathfrak{su}(2)$ algebra seems linked to\nthe ineffectiveness of the parameter $\\ta$. This is reminiscent of the\nautomatic enhancement examples encountered above, in which the gauge\nalgebra enhances because a parameter becomes ineffective. It is\ntherefore tempting to view the forced reducibility of the quintic\ncurve as a geometric manifestation of the Automatic Enhancement Conjecture. Specifically,\nwe argue that the conjecture implies that any attempt to construct an\n$\\ensuremath\\mathfrak{su}(2)$ model with $\\gaugedivclass = 5$ and two half-hypermultiplets\nof $\\bm{4}$ matter leads to a model with an $\\ensuremath\\mathfrak{su}(2)\\oplus\\ensuremath\\mathfrak{su}(2)$\ngauge algebra. In order to show that this is the case, we must examine how one would field-theoretically break the $\\ensuremath\\mathfrak{su}(2)\\oplus\\ensuremath\\mathfrak{su}(2)$ algebra down to $\\ensuremath\\mathfrak{su}(2)$.\n\nWe are attempting to recombine the 7-branes wrapped on the linear and quartic curves of \\cref{eq:reduciblequintic} while maintaining the triple points at $\\locus{\\trpla=\\trplb=0}$. Field-theoretically, this recombination process involves giving a \\vev{} to bifundamental matter such that a diagonal $\\ensuremath\\mathfrak{su}(2)$ subalgebra of $\\ensuremath\\mathfrak{su}(2)\\oplus\\ensuremath\\mathfrak{su}(2)$ is preserved. Specifically, if the generators for the $\\ensuremath\\mathfrak{su}(2)$ on the quartic curve are $T^a_{(4)}$ and the generators for the $\\ensuremath\\mathfrak{su}(2)$ on the linear curve are $T^a_{(1)}$, we wish to preserve the generators\n\\begin{equation}\nT^a_{(5)} = T^a_{(4)} + T^a_{(1)}\n\\end{equation}\nup to some normalization. Of course, the hypermultiplet spectrum in \\cref{eq:quinticsu2su2spec} does not contain any bifundamental matter, so we cannot perform this Higgsing process. But if we ignore this problem, it naively seems that this Higgsing would give us the $\\ensuremath\\mathfrak{su}(2)$ hypermultiplet spectrum in \\cref{eq:quinticsu2spec}. The $\\ensuremath\\mathfrak{su}(2)\\oplus\\ensuremath\\mathfrak{su}(2)$ representations in \\cref{eq:quinticsu2su2spec} branch to representations of the diagonal $\\ensuremath\\mathfrak{su}(2)$ as follows:\n\\begin{equation}\n\\label{eq:quinticbranching}\n(\\bm{2},\\bm{3}) \\to \\bm{4}+\\bm{2}\\,, \\quad (\\bm{1},\\bm{3}) \\to \\bm{3}\\,, \\quad (\\bm{2},\\bm{1}) \\to \\bm{2}\\,, \\quad (\\bm{1},\\bm{2}) \\to \\bm{2}\\,, \\quad (\\bm{1},\\bm{1}) \\to \\bm{1}\\,.\n\\end{equation}\nMeanwhile, three of the $\\ensuremath\\mathfrak{su}(2)\\oplus\\ensuremath\\mathfrak{su}(2)$ gauge bosons, those corresponding to the generators\n\\begin{equation}\nT^a_{(4)} - T^a_{(1)}\\,,\n\\end{equation}\nshould get masses as part of this Higgsing process. The $T^a_{(4)} - T^a_{(1)}$ generators appear to be charged in the adjoint ($\\bm{3}$) representation of the diagonal $\\ensuremath\\mathfrak{su}(2)$ algebra, so they must pair up with a hypermultiplet in the $\\bm{3}$ representation of $\\ensuremath\\mathfrak{su}(2)$ to form massive vector multiplets. This $\\bm{3}$ hypermultiplet is removed from the massless spectrum. Combining this information with \\cref{eq:quinticsu2su2spec,eq:quinticbranching} leads to an ostensible massless hypermultiplet spectrum of\n \\begin{equation}\n 2\\times\\frac{1}{2}\\bm{4}  + 84\\times\\bm{2} + 104\\times\\bm{1}\\,,\n \\end{equation}\n in exact agreement with \\cref{eq:quinticsu2spec}.\n\nWe therefore see that, if we could Higgs the $\\ensuremath\\mathfrak{su}(2)\\oplus\\ensuremath\\mathfrak{su}(2)$ gauge algebra down to $\\ensuremath\\mathfrak{su}(2)$, we would recover the expected $\\ensuremath\\mathfrak{su}(2)$ model with $\\gaugedivclass = 5$ and two half-hypermultiplets of $\\bm{4}$ matter. However, the $\\ensuremath\\mathfrak{su}(2)\\oplus\\ensuremath\\mathfrak{su}(2)$ model in question, with the hypermultiplet spectrum of \\cref{eq:quinticsu2su2spec}, lacks the bifundamental matter necessary to perform this Higgsing. The Automatic Enhancement Conjecture\\ then states that any attempt to construct this $\\ensuremath\\mathfrak{su}(2)$ model would lead to a model with an $\\ensuremath\\mathfrak{su}(2)\\oplus\\ensuremath\\mathfrak{su}(2)$ gauge algebra. In other words, an $\\ensuremath\\mathfrak{su}(2)$ gauge algebra with the hypermultiplet spectrum of \\cref{eq:quinticsu2spec} would automatically enhance to an $\\ensuremath\\mathfrak{su}(2)\\oplus\\ensuremath\\mathfrak{su}(2)$ gauge algebra. The conjecture thus offers a physical explanation for the geometric observation that a quintic curve with two triple points must be reducible.\n\nThe geometric analysis of \\cite{KleversEtAlExotic} identified other $\\ensuremath\\mathfrak{su}(2)$ models with $\\bm{4}$ matter that have similar geometric obstructions. These models would be related to the quintic $\\ensuremath\\mathfrak{su}(2)$ model discussed here by applying Cremona transformations to the $\\mathbb{P}^2$ base of the elliptic fibration. It is natural to expect that these geometric obstructions can also be explained physically by the Automatic Enhancement Conjecture. However, the $\\ensuremath\\mathfrak{su}(2)$ divisors in these examples have more complicated algebraic structures, making it more difficult to determine the enhanced gauge algebras. We therefore do not analyze these examples here, although they could serve as interesting tests of the Automatic Enhancement Conjecture\\ in future work.\n\n\n\\subsection{More exotic matter}\n\\label{sec:exotic-non-F-theory}\n\nWe now briefly consider exotic matter in representations not expected\nfrom F-theory according to the classification of\n\\cite{KleversEtAlExotic}. Some examples of such matter representations\nare the 3-index antisymmetric ($\\bm{84}$) representation of $\\SU(9)$,\nthe {\\tiny$\\ydiagram{2, 2}$} ($\\bm{20}$) representation of $\\SU(4)$, any\nrepresentation of $\\gG_2$ other than the $\\bm{7}$ and the adjoint,\netc.  In general, attempts to construct these representations lead to\ncodimension two $(4, 6)$ loci, which are associated with\nsuperconformal sectors \\cite{Seiberg:1996qx, Heckman:2013pva,\nDelZotto:2014hpa}.  While there may be a physical interpretation of\nthese SCFT matter structures, these kinds of matter\/SCFT structures\nare as yet poorly understood in F-theory.  We may, however, see the\nappearance of a $(4, 6)$ codimension two locus as another kind of\n``enhancement'' that is forced when certain combinations of gauge\ngroup and matter fields are tuned in an F-theory model.  In\nparticular, this kind of enhancement, or an inconsistency of the\ntheory, seems to arise whenever a theory is tuned in F-theory that\ncould possibly be consistent with exotic matter outside the list of\nrepresentations identified in \\cite{KleversEtAlExotic}.  One class of\nsituations in which this $(4, 6)$ enhancement has been studied is for\nthe 3-index antisymmetric $\\SU(9)$ representation; while there are\napparently anomaly-free 6D supergravity models that contain this\nrepresentation, any attempt to construct them in F-theory appears to\nlead to a codimension-two $(4, 6)$ locus associated with the desired\nmatter representation \\cite{AndersonGrayRaghuramTaylorMiT}.\n\nHere we briefly analyze another simple set of 6D supergravity theories\nwith exotic matter that satisfy all known consistency conditions but\ndo not seem to have an F-theory realization.  These are theories with\ngauge group $\\gG_2$, no tensor multiplets, and matter in the\n$\\bm{27}$-dimensional exotic matter representation.  Some simple 6D\nsupergravity models with this representation that seem to be\nanomaly-consistent are, for example, those with $T = 0$, gauge group\n$\\gG_2$, and a matter spectrum of $31 \\times \\bm{7}+ 0 \\times \\bm{14}\n+ x \\times \\bm{27}$, where $\\bm{14}$ is the adjoint\nrepresentation and $x = 1$ or $x = 2$.  We argue here that any attempt\nto tune such a theory leads to an ``enhancement'' to a codimension two\n$(4, 6)$ sector.\n\nIt is fairly straightforward to see that tuning any kind of $\\gG_2$\ngauge factor with matter in the $\\bm{27}$ representation leads to\nsome kind of complications in F-theory.  As mentioned briefly earlier,\none simple way of classifying exotic matter fields is through their\ncontribution to the arithmetic genus of the curve supporting them in\nan F-theory model, which can be computed directly from the\ncontribution to \\cref{eq:genus} from that particular representation\nin the anomaly equations \\cite{KumarParkTaylorT0}. The $\\bm{27}$ of\n$\\gG_2$ has a genus contribution of $g = 3$.  Thus, we expect only to\nsee this representation when $\\gG_2$ is tuned on a curve with a triple\nself-intersection point or other singularity of arithmetic genus 3.\nAt such a point, however, $(f, g)$ must vanish to orders $(6, 9)$\nsince the orders needed for $\\gG_2$ in the Kodaira classification are\n$(2, 3)$.  We thus have an automatic $(4, 6)$ point in any F-theory\nmodel when we try to tune this representation.  The meaning of this\nkind of singularity is not quite clear at this time.  In general, as\nmentioned above, we\nexpect a superconformal field theory (SCFT) to be coupled in at such\ncodimension $(4, 6)$ points.  It is possible that such an SCFT\ncould play the role of the $\\bm{27}$ matter, although it is not clear\nhow to make complete sense of this interpretation.  It has also been\nsuggested \\cite{CveticHeckmanLinExoticDiscrete} that T-brane\nstructures may somehow enable exotic matter at codimension-two $(4,\n6)$ loci in such a way that there is no SCFT coupled into the theory, and\nin principle such a mechanism may operate here, but this is also poorly\nunderstood.  While the base can be changed by blowing up the $(4, 6)$\npoint associated with the exotic matter, giving another base with $T\n\\to T + 1$, in this case the $\\gG_2$ factor would generally lose\nits self-intersection at the locus supporting the desired $\\bm{27}$\nand the matter associated with that point would be lost.\nIn any case, the enhancement of the Weierstrass model to a codimension\ntwo $(4, 6)$ point may represent some kind of enhancement related to\nthose described in the bulk of this paper, though it is not covered by\nthe Automatic Enhancement Conjecture\\ as stated in \\cref{sec:conjecture}.  It would be interesting to try to\nincorporate this kind of swampland model into a larger framework\nincluding the other cases described here that are better understood.\n\nAnother interesting class of enhancements related to some kind of\nstrongly coupled SCFT matter arises in the case of a tuned $\\gE_8$ gauge\ngroup.\\footnote{Thanks to Yinan Wang for helpful discussions on this point.}  Again here there is a necessary enhancement to a local $(4,\n6)$ point when we tune, e.g.,an $\\gE_8$ factor on a curve on $\\mathbb{P}^2$,\nassociated with some kind of SCFT.   Deformations of the Weierstrass\nmodel away from this locus, as for the other tuned $(4, 6)$ point\nSCFTs presumably correspond to some kind of Higgsing that breaks the\n$\\gE_8$.  Some investigation of this kind of structure was carried out\nin \\cite{Tian:2018icz}, but it would be nice to better understand these kinds of\nSCFT enhancements and their role in the Automatic Enhancement Conjecture.\n\n\\subsection{Discrete gauge groups}\n\\label{sec:discrete}\n\nAutomatic enhancement can occur for models with discrete gauge symmetries as well, even when the desired charged matter spectrum satisfies the low-energy anomaly conditions of \\cite{MonnierMooreGS}. We do not give a complete description of automatic enhancement for discrete gauge groups here. Instead, we present examples of F-theory models where a $\\mathbb{Z}_2$ or $\\mathbb{Z}_3$ symmetry automatically enhances to a larger group, such as $\\U(1)$.\n\n For instance, we can consider a\n$\\mathbb{Z}_2$ model with zero tensor multiplets and 108 charged\nhypermultiplets. One might imagine trying to obtain this model by\nHiggsing a $\\U(1)$ model with anomaly coefficient $\\tilde{b}=6H$ and\n108 hypermultiplets of $\\chargedsinglet{1}$ matter, but this $\\U(1)$ model\ndoes not have the requisite $\\chargedsinglet{2}$ hypermultiplets to perform the\nHiggsing. The Automatic Enhancement Conjecture\\ would therefore suggest that the\n$\\mathbb{Z}_2$ gauge symmetry should automatically enhance to a\n$\\U(1)$. To see if this enhancement occurs explicitly in F-theory, we\ncan consider the $\\mathbb{Z}_2$ Weierstrass model from\n\\cite{Braun:2014oya, MorrisonTaylorMultisection}, which is given by\n\\begin{equation}\ny^2 = x^3 + \\left(e_1 e_3 - \\frac{1}{3} e_2^2 - 4 e_0 e_4\\right)x + \\left(-e_0 e_3^2 - \\frac{1}{3}e_1 e_2 e_3 - \\frac{2}{27}e_{2}^3 + \\frac{8}{3}e_0 e_2 e_4 - e_1^2 e_4\\right)\\,.\n\\end{equation}\nWe let the base be $\\mathbb{P}^2$ to ensure there are zero tensor\nmultiplets, and to obtain the correct number of charged\nhypermultiplets, we should let $\\divclass{e_i} =\n(12-3i)H$.\\footnote{We could alternatively let $\\divclass{e_i} = 3iH$,\nbut because we can exchange $e_i\\leftrightarrow e_{4-i}$, the\nanalysis is the same for this case.} Since $\\divclass{e_4}$ is\ntrivial, $e_4$ can be freely taken to be a perfect square, implying\nthat the gauge group is $\\U(1)$ instead of $\\mathbb{Z}_2$\n\\cite{MorrisonTaylorMultisection}. We therefore see the expected\n$\\mathbb{Z}_2 \\to \\U(1)$ automatic enhancement.\n\nOne can also consider the $F_1$ toric hypersurface model discussed in \\cite{KleversEtAlToric}, which has gauge group $\\mathbb{Z}_3$. Considering this model tuned over the base $\\mathbb{P}^2$, the Jacobian rank method of \\cite{RaghuramTaylorTurnerSM} seems to indicate that this is the universal model for the gauge group $\\mathbb{Z}_3$. However, for most values of the parameters $\\mathcal{S}_7, \\mathcal{S}_9$, the gauge group enhances, due to either the ineffectiveness or triviality of classes of some parameters in the Weierstrass model (in the latter cases, the model develops extra $\\U(1)$ gauge factors because the associated parameters are automatically perfect squares, as in many of the other cases we have discussed). For example, there is an anomaly-consistent $\\mathbb{Z}_3$ model with $(\\mathcal{S}_7, \\mathcal{S}_9) = (0, 1)$ containing 168 charged hypermultiplets (note that charges $\\pm1$ and $\\pm2$ are equivalent in a 6D $\\mathbb{Z}_3$ model). However, there is an associated $\\U(1)$ model (the $F_3$ toric hypersurface model of \\cite{KleversEtAlToric}) with $(\\mathcal{S}_7, \\mathcal{S}_9) = (0, 1)$ that would Higgs to this $\\mathbb{Z}_3$ model, except that for this choice of classes the $\\U(1)$ model has no $\\chargedsinglet{3}$ matter with which to perform the Higgsing. Thus, according to the Automatic Enhancement Conjecture, we expect that the $\\mathbb{Z}_3$ model automatically enhances to a $\\U(1)$ model for this choice of $\\mathcal{S}_7, \\mathcal{S}_9$, and indeed this is the case. It appears to be true that all of the enhanced models over the base $\\mathbb{P}^2$ have insufficient matter to supersymmetrically Higgs to the associated $\\mathbb{Z}_3$ model, consistent with the Automatic Enhancement Conjecture. Note that there are also choices of $\\mathcal{S}_7, \\mathcal{S}_9$, such as $(\\mathcal{S}_7, \\mathcal{S}_9) = (1, 9)$ that are a priori consistent with anomalies for which the discriminant of the $F_1$ model vanishes identically, indicating that these models are actually inconsistent altogether.\n\nIt would be interesting to perform a more thorough investigation of automatic\nenhancements of discrete gauge symmetries in future work.\n\n\\section{Automatic enhancement in lower-dimensional theories}\n\\label{sec:lower-dimensions}\n\n\\subsection{Automatic enhancement in 4D $\\mathcal{N} = 1$ F-theory models}\n\nWhile we have focused in this paper on 6D F-theory models, at least at\nthe level of geometry similar enhancements are forced through tuning\nWeierstrass models with certain gauge groups on sufficiently large\ndivisors in 4D F-theory models as well.  Because the actual gauge\ngroup can be affected by fluxes and the superpotential, the physical\nconsequences of this geometric enhancement are less clear in 4D\ntheories, but the analogous phenomenon seems worth exploring.\n\nAs a simple example, consider an F-theory model on the base $\\mathbb{P}^3$,\nwith an $\\ensuremath\\mathfrak{su}(2)$ algebra tuned on the divisor $D = b H$, with $H$ the\nhyperplane divisor.  The same logic as that described in\n\\cref{sec:su2-enhancement} applies.  Here $K_B = -4 H$, and so if $12\n<b <16$, there will be an additional $\\ensuremath\\mathfrak{su}(2)$ algebra arising on the\ndivisor $f_1 = 0$, with $[f_1] = (16 - b) H$.  As in the 6D context,\nthe resulting global gauge group, at least at the level of geometry,\nwill be $(\\SU(2) \\times\\SU(2))\/\\mathbb{Z}_2$.\n\n\\subsection{Automatic enhancement in 5D and 4D theories with 8 supercharges}\n\nIt is interesting to speculate whether some similar mechanism may\narise in 5D $\\mathcal{N} = 1$ and 4D $\\mathcal{N} = 2$ theories.  In\nparticular by compactifying at least some of the cases where 6D\ntheories have automatic enhancement on circles, it seems we can get\nsome lower-dimensional theories with additional gauge factors that cannot\nbe  completely Higgsed away.\n\nAs a simple example, consider the 6D $T = 0$ theory with an $\\ensuremath\\mathfrak{su}(2)$\ntuned on a curve with $b = 10$ or $b = 11$, which as discussed in\n\\cref{sec:su2su2} is automatically enhanced to include an extra\n$\\ensuremath\\mathfrak{su}(2)$ factor on a complementary curve with $[f_1] = 2, 1$\nrespectively.  If we compactify this 6D theory down to a 5D or 4D\ntheory with 8 supercharges on a circle $S^1$ or torus $T^2$, the\nlower-dimensional theory again has a gauge group\n$(\\SU(2) \\times \\SU(2)) \/ \\mathbb{Z}_2$.  The vector multiplets now include\nadjoint scalars for the $\\ensuremath\\mathfrak{su}(2)$ factors, so there is a Coulomb\nbranch where, for example, the first $\\ensuremath\\mathfrak{su}(2)$ factor remains nonabelian\nwhile\nthe second factor is broken to the $\\U(1)$ subgroup so that the global\ngroup becomes $(\\SU(2) \\times \\U(1)) \/ \\mathbb{Z}_2$.  In the 5D theory there are\nno fields charged purely under the $\\U(1)$ factor, so\nit seems that we may have a similar situation where the containing\n$G'$ theory cannot be broken down to a smaller theory with only a\ngauge algebra $\\ensuremath\\mathfrak{su}(2)$ and the same charged matter content under that\nfactor.  The story is much less clear here, however, since we do not\nhave an analogous construction to the F-theory geometry where we can\nsee the forcing of the extra factor through the Weierstrass model.  If\nthere is some constraint analogous to the automatic enhancement\nmechanism for 5D theories, for example for 5D models that can be\nrealized by compactifying M-theory on a Calabi--Yau threefold, this\nwould seem to entail some constraint on the geometry of general CY\nthreefolds that might hold even beyond the elliptic CY3 framework.  We\nleave this question for further investigation.\nIt would also be interesting to look into the parallel questions for\n4D models coming from compactification on a further circle.\n\n\\section{Conclusions and further directions}\n\\label{sec:further}\n\nIn this paper we have identified a range of circumstances in which\ntuning a certain gauge group $G$ and matter content $M$ in F-theory\nautomatically gives rise to a larger gauge group $G'$ and matter\ncontent $M'$.  We have analyzed a variety of constructions for\nspecific gauge groups $G$ that satisfy the conditions of the\nAutomatic Enhancement Conjecture\\ stated in \\cref{sec:conjecture} that there is a\ncontaining theory $G', M'$ such that the larger theory cannot be\nbroken down to the smaller theory with gauge group $G$ by a\nsupersymmetry-preserving Higgs deformation.  For each of the specific\ngauge groups $G$ with generic matter content, we have used explicit\nconstructions of universal Weierstrass models to identify positivity\nconditions and associated local conditions in terms of anomaly\ncoefficients that are sufficient to show that certain gauge group and\nmatter combinations cannot be realized in F-theory without some\nadditional enhanced gauge factors. These gauge group-specific\nformulations of the conjecture are stronger than the general statement\nsince they also incorporate classes of theories that are rendered\ninvalid by the enhancement.\n\nWhile the analysis here has focused on F-theory constructions, where\nwe can explicitly prove that automatic enhancement must take\nplace at least through standard F-theory constructions, it is interesting to\nspeculate that the Automatic Enhancement Conjecture, and its group-specific stronger versions,\nmay hold more generally for 6D\nsupergravity theories.  Indeed, many of the known 6D supergravity\nmodels that seem to lie in the swampland \\cite{VafaSwamp} of\ntheories that appear consistent but have no known string realization,\nand which do not violate other known F-theory constraints reviewed in\n\\cref{sec:constraints}, are in the class of theories identified here\nthat are automatically enhanced to a larger gauge symmetry when\nrealized in F-theory through standard constructions, or are rendered\ninvalid when this enhancement is not possible.  Furthermore, as\ndescribed in\n\\cite{MorrisonTaylorCompleteness} and reviewed in \\cref{sec:su2-quotient}, another class of apparent\nswampland theories, in which the massless matter does not span the\nfull charge lattice of the gauge group, appears to be similarly ruled\nout in F-theory through the appearance of additional sections.  It would be interesting\nto try to find some way of unifying the set of swampland theories that\nundergo automatic enhancement to either a larger group or an invalid\nF-theory model, along with those theories that automatically acquire\nextra Mordell--Weil  sections in F-theory, through some\ncondition on the low-energy theory, which would provide a unified\nswampland hypothesis for all three of these sets of theories in the\ncontext of 6D supergravity.  Such a formulation seems a bit subtle,\nhowever; for example, in the context of the $\\ensuremath\\mathfrak{su}(2) \\to\n(\\SU(2) \\times\\SU(2))\/\\mathbb{Z}_2$ described in \\cref{sec:su2su2}, automatic\nenhancement to the extra $\\ensuremath\\mathfrak{su}(2)$ factor is forced on the locus $f_1\n= 0$.  From the F-theory point of view, this is one particular curve\nin a moduli space of curves in a given homology class, which is picked\nout by the structure of the Weierstrass model.  In the 6D supergravity\ntheory, that homology class corresponds to a particular charge in the\nstring charge lattice, but the degrees of freedom associated with the\nspecific choice of curve $f_1$ are not clearly visible in the\nlow-energy theory in any way that we are familiar with; thus, in\nparticular, world-volume methods on a string such as those used in\n\\cite{KimShiuVafaBranes, LeeWeigandSwampland} do not seem sensitive to\nthe specific locus $f_1 = 0$ as distinct from the other string\nconfigurations coming from branes wrapped on other cycles in the same\nclass.  Understanding this better would be an interesting direction for\nfurther investigation.\n\nAnother question of interest is whether the Automatic Enhancement Conjecture\\ can be\nrigorously proven even in the context of F-theory.  For the first case\nwe have discussed, that of non-Higgsable clusters, such a proof exists.\nIn this situation, where there is a curve $C$ of negative\nself-intersection $C \\cdot C \\le -3$, the curve $C$ is rigid and\nhence unique, and the arguments of \\cite{MorrisonTaylorClusters} show\nthat any F-theory model with such a curve must have a non-Higgsable\ngauge group of at least $\\SU(3)$.  For the other cases we have\ndescribed here, however, the argument is not quite so complete.\nWe have shown that standard F-theory constructions with\nvarious gauge groups and matter content $G, M$ are automatically\nenhanced to $G', M'$, however, and while\nwe have not ruled out the possibility of exotic\nF-theory constructions such as those using non-UFD structures on\nsingular curves \\cite{KleversEtAlExotic} that could in principle\nrealize $G, M$ theories without enhancement, the results of\n\\cref{app:u1u1test} suggest that this approach will not lead to\ntheories that violate the conjecture.   Another way to prove the\nAutomatic Enhancement Conjecture\\ for many of the classes of examples we have described\nhere would be to show that the universal moduli space of theories with\nat least a given\ngauge group and matter content $G, M$\nmust be a connected set\n(within the context of a class\nof theories with a given F-theory base $B$, corresponding to a fixed\nstring charge lattice and positivity cone in the low-energy theory, and\nassuming fixed anomaly coefficients for the factors of $G$).\nIf this is true, then every theory with $G, M$ could be reached from\nevery other, so that the enhanced theories $G', M'$ would need to be\nconnected to any smaller theory such as one with only the gauge group\nand matter content $G, M$ and no enhancement, which is not allowed by\nthe conditions of the Automatic Enhancement Conjecture.  If such an argument could be\nrealized, one could imagine a similar argument might apply in the more\ngeneral context of 6D supergravity without reference to\nan F-theory completion, if it were possible to show that for a given string charge\nlattice and positivity cone at a fixed number of tensor multiplets $T$\nthe moduli space of allowed supergravity theories with  at least a\ngiven $G, M$ is connected, for fixed values of the anomaly coefficients.\n\n\\acknowledgments\n\nWe would like to thank Lara Anderson, James Gray, Daniel Harlow, Sheldon Katz, David Morrison,\nPaul Oehlmann, Houri Tarazi, Cumrun Vafa, and Yinan Wang for helpful discussions.  The work of APT\nand WT was supported by DOE grant DE-SC00012567. APT is supported in part by the DOE (HEP) Award DE-SC0013528. NR is supported by NSF grant PHY-1720321.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:introduction}\n\\setcounter{equation}{0}\n\nIn this note we consider the Cauchy problem for the defocusing nonlinear wave equation\n\\begin{equation} \\label{equ:ivp}\n \\left\\{ \\begin{aligned}\n  -\\partial_t^2 u + \\Delta u &= |u|^{p-1} u \\text{ on } {\\mathbb R} \\times {\\mathbb R}^3, \\\\\n  (u, \\partial_t u)|_{t=0} &= (f_1, f_2) \\in H^s_x({\\mathbb R}^3) \\times H^{s-1}_x({\\mathbb R}^3),\n \\end{aligned} \\right.\n\\end{equation}\nwhere $3 < p < 5$ and $H^s_x({\\mathbb R}^3)$ is the usual inhomogeneous Sobolev space. The main result of this paper establishes the almost sure existence of global unique solutions to~\\eqref{equ:ivp} with respect to a unit-scale randomization of initial data in $H^s_x({\\mathbb R}^3) \\times H^{s-1}_x({\\mathbb R}^3)$ for $3 < p < 5$ and $\\frac{p-1}{p+1} < s < 1$. In particular, for the entire range of exponents $p$ we obtain large sets of super-critical initial data that lead to unique global solutions. This improves over the authors' previous result in \\cite{LM} both regarding the threshold for allowable regularities and the notion of uniqueness of the solutions. Our proof relies on probabilistic growth estimates for a new modified energy functional which we introduce in \\eqref{equ:energy}. These estimates are inspired by the recent work of Oh and Pocovnicu \\cite{OP} on the almost sure global existence of solutions to the quintic nonlinear wave equation on ${\\mathbb R}^3$.\n\n\\medskip\n\nThe equation \\eqref{equ:ivp} is invariant under the scaling transformation\n\\[\n u(t,x) \\mapsto u_\\lambda(t,x) = \\lambda^{\\frac{2}{p-1}} u(\\lambda t, \\lambda x) \\quad \\text{for } \\lambda > 0,\n\\]\nwhich gives rise to the scaling invariant critical regularity $s_c = \\frac{3}{2} - \\frac{2}{p-1}$. In \\cite{Lindblad_Sogge}, Lindblad and Sogge construct local strong solutions to \\eqref{equ:ivp} for sub-critical and critical regularities $s \\geq s_c$ using Strichartz estimates for the wave equation. When $s < s_c$, this is the super-critical regime and the well-posedness arguments based on Strichartz estimates break down. Global solutions to \\eqref{equ:ivp} were constructed by Kenig, Ponce and Vega \\cite[Theorem 1.2]{KPV} using a high-low argument for $2 \\leq p < 5$ and initial data in a range of sub-critical spaces below the energy space (see also \\cite{BC}, \\cite{GP} and \\cite{Roy} for $p = 3$).\n\\medskip\n\nAlthough it is known that the nonlinear wave equation \\eqref{equ:ivp} is ill-posed below the critical scaling regularity (see \\cite{L}, \\cite{CCT} and \\cite{IMM}), using probabilistic tools it is sometimes possible to construct large sets of initial data of super-critical regularity that lead to unique local and even global solutions. This approach was initiated by Bourgain \\cite{B94, B96} for the periodic nonlinear Schr\\\"odinger equation in one and two space dimensions, building upon work by Lebowitz, Rose and Speer \\cite{LRS}. Subsequently, Burq and Tzvetkov \\cite{BT1, BT2} studied the cubic nonlinear wave equation on a three-dimensional compact manifold by randomizing with respect to an orthonormal eigenbasis of the Laplacian and using invariant measure considerations. Extensive work has been done on such problems in recent years, both in compact and non-compact settings but we restrict the following overview to results in Euclidean space. Many previous results on Euclidean space involve considering a related equation in a setting where an orthonormal basis of eigenfunctions of the Laplacian exists, see for instance \\cite{BTT},  \\cite{D1}, \\cite{Poiret1}, \\cite{Poiret2} for results of this type for the defocusing nonlinear Schr\\\"odinger equation on ${\\mathbb R}^d$ for $d \\geq 2$, and \\cite{Suzzoni1}, \\cite{Suzzoni2} for results of this type for the nonlinear wave equation.\n\n\\medskip\n\nIt is also possible to randomize initial data directly on Euclidean space using a unit-scale decomposition of frequency space. In several works this has yielded almost sure well-posedness results for super-critical initial data, see for instance \\cite{ZF}, \\cite{LM}, \\cite{BOP1}, \\cite{BOP2}, \\cite{Pocovnicu} and \\cite{OP}. In \\cite{LM}, the authors studied the random data problem for \\eqref{equ:ivp} and proved almost sure global existence for energy sub-critical nonlinearities. In particular, for $\\frac{1}{4}(7+ \\sqrt{73}) \\simeq 3.89 < p < 5$, the authors obtain almost sure global well-posedness for super-critical initial data. The proof combines a probabilistic local existence argument with Bourgain's high-low frequency decomposition~\\cite{B98}, an approach introduced by Colliander and Oh \\cite[Theorem 2]{CO} in the context of the one-dimensional periodic defocusing cubic nonlinear Schr\\\"odinger equation. We note that probabilistic high-low arguments only yield uniqueness in a mild sense, see \\cite[Remark 4.3]{LM}, and thus do not provide a definitive answer to the question of uniqueness for super-critical initial data. The high-low method does not extend to energy critical situations, thus there is a natural obstruction to extending the authors' previous results to an energy critical setting. Pocovnicu \\cite{Pocovnicu} later proved almost sure global well-posedness for the energy critical defocusing nonlinear wave equation on ${\\mathbb R}^d$ for $d=4,5$. More recently, Oh and Pocovnicu \\cite{OP} have treated the energy critical nonlinear wave equation on ${\\mathbb R}^3$. These proofs use probabilistic perturbation theory together with a probabilistic a priori energy bound. See also \\cite{BOP2} for a conditional result for the nonlinear Schr\\\"odinger equation using similar methods.\n\n\\subsection{Randomization procedure}\n\nBefore stating our main result, we introduce the randomization procedure for the initial data. Let $\\psi \\in C_c^{\\infty}({\\mathbb R}^3)$ be an even, non-negative function with $\\text{supp} (\\psi) \\subset B(0,1)$ and such that \n\\[\n \\sum_{k \\in {\\mathbb Z}^3} \\psi(\\xi - k) = 1 \\text{ for all } \\xi \\in {\\mathbb R}^3.\n\\]\nLet $s \\in {\\mathbb R}$ and $f \\in H^s_x({\\mathbb R}^3)$. For every $k \\in {\\mathbb Z}^3$, we define the function $P_k f: {\\mathbb R}^3 \\rightarrow {\\mathbb C}$ by\n\\begin{equation*}\n (P_k f)(x) = {\\mathcal{F}}^{-1} \\left( \\psi(\\xi - k) \\hat{f}(\\xi) \\right)(x) \\text{ for } x \\in {\\mathbb R}^3.\n\\end{equation*} \nBy requiring $\\psi$ to be even, we ensure that\n\\begin{equation} \\label{equ:symmetry}\n \\overline{P_k f} = P_{-k} f\n\\end{equation}\nfor real-valued $f$. As in \\cite{LM}, we crucially exploit that these projection operators satisfy a unit-scale Bernstein inequality, namely for all $2 \\leq r_1 \\leq r_2 \\leq \\infty$ ,\n\\begin{align} \\label{equ:unit_scale_bernstein}\n  \\|P_k f\\|_{L^{r_2}_x({\\mathbb R}^3)} \\leq C(r_1, r_2) \\|P_k f\\|_{L^{r_1}_x({\\mathbb R}^3)}\n\\end{align}\nuniformly for all $f \\in L^2_x({\\mathbb R}^3)$ and $k \\in {\\mathbb Z}^3$.  \n\n\\medskip\n\nLet now $\\{ (h_k, l_k) \\}_{k \\in {\\mathbb Z}^3}$  be a sequence of zero-mean, complex-valued random variables on a probability space $(\\Omega, {\\mathcal A}, {\\mathbb P})$ such that $h_{-k} = \\overline {h_k}$ for all $k \\in {\\mathbb Z}^3$, and similarly for the $l_k$. We assume that $\\{h_0, \\textup{Re}(h_k), \\textup{Im}(h_k)\\}_{k \\in {\\mathcal{I}}}$ are independent, zero-mean, real-valued random variables, where ${\\mathcal{I}}$ is such that we have a \\textit{disjoint} union ${\\mathbb Z}^3 = {\\mathcal{I}} \\cup (-{\\mathcal{I}}) \\cup \\{0\\}$, and similarly for the $l_k$. Let us denote by $\\mu_{k}$ and $\\nu_{k}$ the joint distributions of the real and imaginary parts of the $h_k$ and $l_k$, respectively. We assume that there exists $c > 0$ such that\n\\begin{equation} \\label{eq:rvassumption}\n \\left| \\int_{-\\infty}^{+\\infty} e^{\\gamma x} \\, d\\mu_{k}(x) \\right| \\leq e^{c \\gamma^2} \\text{ for  all } \\gamma \\in {\\mathbb R} \\text{ and for all } k \\in {\\mathbb Z}^3,\n\\end{equation}\nand similarly for $\\nu_{k}$. The assumption \\eqref{eq:rvassumption} is satisfied, for example, by standard Gaussian random variables, standard Bernoulli random variables, or any random variables with compactly supported distributions. \n\n\\medskip\n\nFor a given $f = (f_1, f_2) \\in H^s_x({\\mathbb R}^3) \\times H^{s-1}_x({\\mathbb R}^3)$ for some $s \\in {\\mathbb R}$, we define its randomization by\n\\begin{equation} \\label{equ:bighsrandomization} \n f^{\\omega} = (f_1^{\\omega}, f_2^{\\omega}) := \\biggl( \\sum_{k \\in {\\mathbb Z}^3} h_k(\\omega) P_k f_1, \\sum_{k \\in {\\mathbb Z}^3} l_k(\\omega) P_k f_2 \\biggr),\n\\end{equation}\nwhere this quantity is understood as a Cauchy limit in $L^2(\\Omega; H^s_x({\\mathbb R}^3) \\times H^{s-1}_x({\\mathbb R}^3))$. The symmetry assumption on the random variables, as well as \\eqref{equ:symmetry} ensure that the randomization of real-valued initial data is real-valued. Crucially, such a randomization does not regularize at the level of Sobolev spaces. Similar randomizations have previously been used in \\cite{ZF}, \\cite{LM}, \\cite{BOP1}, \\cite{BOP2}, \\cite{Pocovnicu}, \\cite{OP}. We point out that under the symmetry condition imposed in the randomization,\n\\begin{align} \\label{equ:real_part}\n \\qquad f_1^\\omega &= \\sum_{k \\in {\\mathbb Z}^3} h_k(\\omega) P_k f_1 = h_0(\\omega) P_0 f_1 + 2 \\sum_{k \\in {\\mathcal{I}} } \\bigl(\\textup{Re} \\, h_k(\\omega) \\textup{Re} \\,P_k f_1 - \\textup{Im} \\,h_k(\\omega) \\textup{Im} \\,P_k f_1 \\bigr),\n\\end{align}\nand similarly for $f_2^\\omega$. In the following we will denote the free wave evolution of the initial data $f^\\omega$ by\n\\begin{align} \\label{equ:free_evolution}\n u_f^\\omega = \\cos(t|\\nabla|) f_1^\\omega + \\frac{\\sin(t|\\nabla|)}{|\\nabla|} f_2^\\omega.\n\\end{align}\n\n\n\\subsection{Statement of the main result}\n\nWe are now prepared to state our main result.\n\\begin{theorem} \\label{main_theorem}\n Let $3 < p < 5$ and $\\frac{p-1}{p+1} < s < 1$. For real-valued $f = (f_1, f_2) \\in H^s_x({\\mathbb R}^3) \\times H^{s-1}_x({\\mathbb R}^3)$, let $f^\\omega = (f_1^\\omega, f_2^\\omega)$ be the randomized initial data defined in \\eqref{equ:bighsrandomization} and let $u_f^\\omega$ be the free wave evolution \\eqref{equ:free_evolution} of $f^\\omega$. Then for almost every $\\omega \\in \\Omega$, there exists a unique global solution\n \\begin{equation} \\label{equ:solution_main_theorem}\n  (u, \\partial_t u) \\in (u_f^\\omega, \\partial_t u_f^\\omega) + C\\bigl({\\mathbb R}; H^1_x({\\mathbb R}^3) \\times L^2_x({\\mathbb R}^3)\\bigr)\n \\end{equation}\n to the nonlinear wave equation\n \\begin{equation} \\label{equ:nlw_main_theorem}\n  \\left\\{ \\begin{aligned}\n   -\\partial_t^2 u + \\Delta u &= |u|^{p-1} u \\text{ on } {\\mathbb R} \\times {\\mathbb R}^3, \\\\\n   (u, \\partial_t u)|_{t=0} &= (f_1^\\omega, f_2^\\omega).\n  \\end{aligned} \\right.\n \\end{equation}\n Here, uniqueness holds in the sense that upon writing \n \\[\n  (u, \\partial_t u) = (u_f^\\omega, \\partial_t u_f^\\omega) + (v, \\partial_t v),\n \\]\n there exists a unique global solution \n \\[\n  (v, \\partial_t v) \\in C\\bigl({\\mathbb R}; H^1_x({\\mathbb R}^3)\\bigr) \\cap L^{\\frac{2p}{p-3}}_{t, loc} L^{2p}_x\\bigl({\\mathbb R}\\times{\\mathbb R}^3\\bigr) \\times C\\bigl({\\mathbb R}; L^2_x({\\mathbb R}^3)\\bigr)\n \\]\n to the forced nonlinear wave equation\n \\begin{equation} \\label{equ:forced_nlw_main_theorem}\n  \\left\\{ \\begin{aligned}\n   -\\partial_t^2 v + \\Delta v &= |u_f^\\omega + v|^{p-1} (u_f^\\omega + v) \\text{ on } {\\mathbb R} \\times {\\mathbb R}^3, \\\\\n   (v, \\partial_t v)|_{t=0} &= (0,0).\n  \\end{aligned} \\right.\n \\end{equation}\n\\end{theorem}\n\n\\begin{remark} \\label{rem:uniqueness}\nIn contrast to the mild uniqueness of the authors' previous work \\cite[Theorem 1.1]{LM}, Theorem~\\ref{main_theorem} yields the more standard notion of uniqueness for solutions to semilinear wave equations, compare with \\cite[Remark~4.3]{LM}. Moreover, the threshold for the allowable regularity in Theorem~\\ref{main_theorem} has been significantly lowered as compared to that in \\cite[Theorem 1.1]{LM}. In particular, we prove the existence of global solutions for initial data at super-critical regularities for all $3 < p < 5$, see Figure~1.1.\n\\end{remark}\n\n\\begin{figure}[hbt] \\label{fig:plot}\n \\centering\n \\includegraphics[height=6.2cm,keepaspectratio]{plot.pdf}\n \\caption{The dashed line is the critical regularity $s_c = \\frac{3}{2} - \\frac{2}{p-1}$. The solid line is the threshold for the exponent $s$ in Theorem~\\ref{main_theorem}. The dotted line is the threshold from the authors' previous result in \\cite[Theorem 1.1]{LM}.}\n\\end{figure} \n\nWhile the randomization \\eqref{equ:bighsrandomization} does not regularize at the level of Sobolev spaces, the free evolution of the randomized initial data \\eqref{equ:free_evolution} almost surely satisfies better space-time integrability properties. For this reason one can show that the nonlinear component of the solution lies in a better space, in this case $H^1_x({\\mathbb R}^3) \\times L^2_x({\\mathbb R}^3)$, by constructing local solutions via a fixed point argument centered at the free evolution $u_f^\\omega$. We will see that to conclude global existence, it suffices to control the growth of the $H^1_x({\\mathbb R}^3) \\times L^2_x({\\mathbb R}^3)$ norm of the nonlinear component of the solution. The main novelty of this paper is the derivation of probabilistic growth estimates for the modified energy functional\n\\begin{align}\n\\label{equ:energy}\n  E(v) = \\int_{{\\mathbb R}^3} \\frac{1}{2} |\\nabla_x v|^2 + \\frac{1}{2} |\\partial_t v|^2 + \\frac{1}{2} |v|^2 + \\frac{1}{p+1} |u_f^\\omega + v|^{p+1} \\, dx\n\\end{align}\nfor $3 < p < 5$, where $v$ is the nonlinear component of the solution to \\eqref{equ:ivp}. Consequently, we will be able to conclude that almost surely, we have the necessary control to extend the local solutions that we construct to global ones.\n\n\\medskip\n\nWe consider this modified energy functional for two reasons. The first is that the appearance of the free evolution of the randomized initial data in the potential term creates an important cancellation when computing the time derivative of the energy functional. Second, we need the appearance of the $L_x^2$ term in the energy in order to be able to estimate for $0 < \\sigma < 1$,\n\\[\n \\big\\| |\\nabla|^\\sigma v \\big\\|_{L^2_x({\\mathbb R}^3)}^2 \\lesssim \\|v\\|^2_{L^2_x({\\mathbb R}^3)} + \\|\\nabla_x v\\|_{L^2_x({\\mathbb R}^3)}^2 \\lesssim E(v).\n\\]\nPreviously, energy methods for random data problems were used by Nahmod, Pavlovi\\'c and Staffilani \\cite{NPS} in the context of the periodic Navier-Stokes equation in two and three dimensions and by Burq and Tzvetkov for the three-dimensional periodic defocusing cubic nonlinear wave equation \\cite{BT4}. Pocovnicu~\\cite{Pocovnicu} and Oh and Pocovnicu~\\cite{OP} used probabilistic energy bounds in conjunction with a probabilistic perturbation theory for the energy critical nonlinear wave equation. \n\n\\begin{remark}\nOur proof of the probabilistic energy estimates is inspired by the quintic case in \\cite{OP}, with some important differences. In \\cite{OP}, Oh and Pocovnicu only consider frequency truncated random initial data and show that almost surely the corresponding solutions satisfy energy bounds \\emph{uniformly} in the truncation parameter, which allows one to construct solutions using probabilistic perturbation theory. Instead, we study the Cauchy problem with super-critical random initial data directly. To do so we make use of the observation that although the term $\\partial_t u_f^\\omega \\in H^{s-1}_x({\\mathbb R}^3)$ appears when taking the time derivative of our energy functional, this expression is always paired with a term at regularity $H^{1-s}_x({\\mathbb R}^3)$. We therefore have no problem achieving the necessary bounds to close our Gronwall argument, see the proof of Proposition~\\ref{prop:energy_bounds} below for more details. Additionally, the presence of non-algebraic nonlinearities introduces some complications in our estimates. To overcome this difficulty, a more careful analysis using the fractional chain rule and interpolation in Sobolev spaces is necessary. \n\\end{remark}\n\n\\begin{remark}\nOur proof does not yield any improvement at $p=3$. However, this case can be treated exactly as in the periodic case in \\cite{BT4} using the energy functional\n\\[\n E(v) = \\int_{{\\mathbb R}^3} \\frac{1}{2} |\\nabla_x v|^2 + \\frac{1}{2} |\\partial_t v|^2 + \\frac{1}{4} |v|^4 \\, dx.\n\\]\nOne obtains almost sure global existence for any $0 < s < 1$ and uniqueness holds in the same strong sense as in Theorem~\\ref{main_theorem}, see Remark~1.5 in \\cite{LM}.\n\\end{remark}\n\n\n\\begin{remark}\nTo prove scattering of the solutions \\eqref{equ:solution_main_theorem} to the random data problem \\eqref{equ:nlw_main_theorem}, one needs global control of the $L_t^p L_x^{2p}({\\mathbb R} \\times {\\mathbb R}^3)$ norm of the nonlinear component of the solutions. This will likely require new ideas, which we do not pursue here. A probabilistic version of scattering as in \\cite[Theorem 1.4]{BOP2} might be possible, however this only yields scattering on sets of large probability.\n\\end{remark}\n\n\n\\subsection{Notation}\nWe denote by $C > 0$ an absolute constant that depends only on fixed parameters and whose value may change from line to line. We write $X \\lesssim Y$ if $X \\leq C Y$ for some $C > 0$, and analogously for $X \\gtrsim Y$. In the sequel, $H^s_x({\\mathbb R}^3)$, respectively $\\dot{H}^s_x({\\mathbb R}^3)$, denote the usual inhomogeneous, respectively homogeneous, Sobolev spaces. For $s \\in {\\mathbb R}$ we define the space \n\\[\n{\\mathcal{H}}^s({\\mathbb R}^3):= H^s_x({\\mathbb R}^3) \\times H^{s-1}_x({\\mathbb R}^3),\n\\]\nendowed with the obvious norm. We also introduce the shorthand notation $L^\\infty_T L^r_x \\equiv L^\\infty_t L^r_x([0,T]\\times{\\mathbb R}^3)$. Finally, for $a \\in {\\mathbb R}$ we write $a+$ to denote $a + \\varepsilon$ for some arbitrarily small, fixed parameter $\\varepsilon > 0$.\n\n\\medskip\n\n\\noindent {\\it  Organization of the paper:} In Section~\\ref{sec:preliminaries} we collect several deterministic and probabilistic results. In Section~\\ref{sec:proof_main_theorem} we first record a deterministic local well-posedness result for the forced nonlinear wave equation that is associated with the random data problem \\eqref{equ:nlw_main_theorem}. Next, we derive key probabilistic energy bounds for the nonlinear components of the solutions to \\eqref{equ:nlw_main_theorem}. The proof of Theorem~\\ref{main_theorem} is then an immediate consequence.\n\n\n\\medskip\n\n\\noindent {\\it Acknowledgments:} The authors would like to sincerely thank Michael Eichmair and Gigliola Staffilani for all of their encouragement and support.\n\n\\section{Preliminaries} \\label{sec:preliminaries}\n\\setcounter{equation}{0}\n\n\\subsection{Deterministic preliminaries}\n\nWe begin by recalling the Strichartz estimates for the wave equation, for which we need the following definition.\n\n\\begin{definition}\nAn exponent pair $(q,r)$ is wave-admissible if $2 \\leq q \\leq \\infty$, $2 \\leq r < \\infty$ and\n \\begin{align}\n  \\frac{1}{q} + \\frac{1}{r} \\leq \\frac{1}{2}.\n \\end{align}\n\\end{definition}\n\n\\begin{proposition}[\\protect{Strichartz estimates in three space dimensions; \\cite{Strichartz}, \\cite{Pecher}, \\cite{GV2}, \\cite{KeelTao}}] \\label{prop:strichartz_estimates}\n Suppose $(q, r)$ and $(\\tilde{q}, \\tilde{r})$ are wave-admissible pairs. Let $u$ be a (weak) solution to the wave equation\n \\begin{equation*}\n  \\left\\{ \\begin{aligned}\n   -\\partial_t^2 u + \\Delta u \\, &= \\, h \\text{ on } [0, T] \\times {\\mathbb R}^3, \\\\\n   (u, \\partial_t u)|_{t=0} \\, &= \\, (f,g)\n  \\end{aligned} \\right.\n \\end{equation*}\n for some data $f, g, h$ and time $0 < T < \\infty$. Then\n \\begin{equation} \\label{equ:strichartz_estimates}\n  \\begin{aligned}\n   &\\|u\\|_{L^q_t L^r_x ([0, T] \\times {\\mathbb R}^3)} + \\|u\\|_{L^{\\infty}_t \\dot{H}^{\\gamma}_x([0,T] \\times {\\mathbb R}^3)} + \\|\\partial_t u\\|_{L^{\\infty}_t \\dot{H}^{\\gamma-1}_x([0,T] \\times {\\mathbb R}^3)} \\\\\n   &\\qquad \\qquad \\lesssim \\, \\|f\\|_{\\dot{H}_x^{\\gamma}({\\mathbb R}^3)} + \\|g\\|_{\\dot{H}_x^{\\gamma-1}({\\mathbb R}^3)} + \\|h\\|_{L_t^{\\tilde{q}'} L^{\\tilde{r}'}_x([0,T] \\times {\\mathbb R}^3)}\n  \\end{aligned}\n \\end{equation}\n under the assumption that the following scaling conditions hold\n \\begin{equation} \\label{equ:strichartz_scaling_identities}\n  \\frac{1}{q} + \\frac{3}{r} \\, = \\, \\frac{3}{2} - \\gamma \\qquad \\textup{and} \\qquad \\frac{1}{\\tilde{q}'} + \\frac{3}{\\tilde{r}'} - 2 \\, = \\, \\frac{3}{2} - \\gamma.\n \\end{equation} \n\\end{proposition}\n\nWe say that a wave-admissible pair $(q,r)$ is Strichartz-admissible at regularity $\\gamma$ if it satisfies the first identity in \\eqref{equ:strichartz_scaling_identities} for some $0 < \\gamma < \\frac{3}{2}$. \n\n\\medskip\n\nWe will need the following two results in order to handle the fractional derivatives which appear in the energy bounds argument in Proposition~\\ref{prop:energy_bounds}.\n\n\\begin{proposition}[\\protect{Fractional chain rule; \\cite{CW}}] \nLet $G \\in C^1({\\mathbb C})$, $\\sigma \\in (0, 1]$ and suppose that $1 < r, r_1, r_2 < \\infty$ satisfy $\\frac{1}{r} = \\frac{1}{r_1} + \\frac{1}{r_2}$. Then\n\\begin{equation} \\label{equ:fractional_chain_rule}\n \\| |\\nabla|^\\sigma G(u) \\|_{L^r_x({\\mathbb R}^3)} \\lesssim \\| G'(u) \\|_{L^{r_1}_x({\\mathbb R}^3)} \\| |\\nabla|^\\sigma u\\|_{L^{r_2}_x({\\mathbb R}^3)}.\n\\end{equation}\n\\end{proposition}\n\n\\medskip\n\n\\begin{proposition}[\\protect{Interpolation estimate; \\cite[Section 6.4]{BL}}] \nLet $0 < \\theta < 1$, $0 \\leq \\sigma_0 < \\sigma_1$ and $1 < r_0 < r_1 < \\infty$. Define $\\sigma_\\theta$ and $r_\\theta$ by\n\\[\n \\sigma_\\theta = (1-\\theta) \\sigma_0 + \\theta \\sigma_1 \\qquad \\textup{and} \\qquad \\frac{1}{r_\\theta} = \\frac{1-\\theta}{r_0} + \\frac{\\theta}{r_1}. \n\\]\nSuppose that $u \\in \\dot{W}^{\\sigma_0, r_0}_x({\\mathbb R}^3) \\cap \\dot{W}^{\\sigma_1, r_1}_x({\\mathbb R}^3)$. Then $u \\in \\dot{W}^{\\sigma_\\theta, r_\\theta}_x({\\mathbb R}^3)$ and\n\\begin{equation} \\label{equ:interpolation_inequality_in_proposition}\n \\|u\\|_{\\dot{W}^{\\sigma_\\theta, r_\\theta}_x({\\mathbb R}^3)} \\lesssim \\|u\\|^{1-\\theta}_{\\dot{W}^{\\sigma_0, r_0}_x({\\mathbb R}^3)} \\|u\\|^\\theta_{\\dot{W}^{\\sigma_1, r_1}_x({\\mathbb R}^3)}.\n\\end{equation}\n\\end{proposition}\n\n\\medskip\n\nIn the proof of the key energy bounds in Proposition~\\ref{prop:energy_bounds}, we shall also need the following Littlewood-Paley projections. Let $\\varphi \\in C_c^\\infty({\\mathbb R}^3)$ be a radial smooth bump function satisfying $\\varphi(\\xi) = 1$ for $|\\xi| \\leq 1$ and $\\varphi(\\xi) = 0$ for $|\\xi| > 2$. We define \n\\[\n \\widehat{P_1 f}(\\xi) := \\varphi(\\xi) \\hat{f}(\\xi) \n\\]\nand for every dyadic $N \\geq 2$,\n\\[\n \\widehat{P_N f}(\\xi) := \\bigl( \\varphi(\\xi\/N) - \\varphi(2\\xi\/N) \\bigr) \\hat{f}(\\xi).\n\\]\nMoreover, we denote by $\\widetilde{P}_N$ a fattened Littlewood-Paley projection such that $\\widetilde{P}_N P_N = P_N$ for all dyadic $N \\geq 1$. \n\n\\subsection{Probabilistic preliminaries}\n\nIn this subsection we prove several large deviation estimates for the free evolution of the randomized initial data. In the proof of the key energy bounds in Proposition~\\ref{prop:energy_bounds} below, we also need large deviation estimates for the modified free evolution\n\\begin{equation} \\label{equ:modified_free_evolution}\n \\tilde{u}_f^\\omega := - \\frac{|\\nabla|}{\\langle \\nabla \\rangle} \\sin(t|\\nabla|) f_1^\\omega + \\frac{\\cos(t|\\nabla|)}{\\langle \\nabla \\rangle} f_2^\\omega.\n\\end{equation}\nWe note that $\\tilde{u}_f^\\omega$ satisfies $\\partial_t u_f^\\omega = \\langle \\nabla \\rangle \\tilde{u}_f^\\omega$. \n\n\\medskip\n\nThe following large deviation estimate is stated for real-valued random variables for simplicity. However, in light of the expression \\eqref{equ:real_part} and the independence assumptions on the real and imaginary parts of the random variables used in the randomization, this estimate readily yields the desired results in our setting.\n\n\\begin{lemma}[\\protect{\\cite[Lemma 3.1]{BT1}}] \\label{lem:large_deviation_estimate}\n Let $\\{h_n\\}_{n=1}^{\\infty}$ be a sequence of real-valued independent random variables with associated distributions $\\{\\mu_n\\}_{n=1}^{\\infty}$ on a probability space $(\\Omega, {\\mathcal A}, {\\mathbb P})$. Assume that the distributions satisfy the property that there exists $c > 0$ such that\n \\begin{equation*}\n  \\biggl| \\int_{-\\infty}^{+\\infty} e^{\\gamma x} d\\mu_n(x) \\biggr| \\leq e^{c \\gamma^2} \\text{ for  all } \\gamma \\in {\\mathbb R} \\text{ and for all } n \\in \\mathbb{N}.\n \\end{equation*}\n Then there exists $\\alpha > 0$ such that for every $\\lambda > 0$ and every sequence $\\{c_n\\}_{n=1}^{\\infty} \\in \\ell^2({\\mathbb N};{\\mathbb C})$ of complex numbers, \n \\begin{equation*}\n  {\\mathbb P} \\Bigl( \\bigl\\{ \\omega : \\bigl| \\sum_{n=1}^{\\infty} c_n h_n(\\omega) \\bigr| > \\lambda \\bigr\\} \\Bigr) \\leq 2 \\exp \\biggl(- \\alpha \\frac{\\lambda^2}{\\sum_n |c_n|^2} \\biggr). \n \\end{equation*}\n As a consequence there exists $C > 0$ such that for every $p \\geq 2$ and every $\\{c_n\\}_{n=1}^{\\infty} \\in \\ell^2({\\mathbb N}; {\\mathbb C})$,\n \\begin{equation*}\n  \\Bigl\\| \\sum_{n=1}^{\\infty} c_n h_n(\\omega) \\Bigr\\|_{L^p_\\omega(\\Omega)} \\leq C \\sqrt{p} \\Bigl( \\sum_{n=1}^{\\infty} |c_n|^2 \\Bigr)^{1\/2}.\n \\end{equation*}\n\\end{lemma}\n\nWe record the following large deviation estimates for the free evolution of the randomized initial data, following the presentation of the results from \\cite{BT4} for the periodic setting. It follows immediately from the estimates below that the free evolution of the randomized initial data satisfies $L_{t,loc}^q L_x^r$ bounds almost surely.\n\n\\begin{lemma} \\label{lem:large_deviation_estimates_free_evolution}\n Let $\\sigma \\geq 0$ and $f = (f_1, f_2) \\in {\\mathcal{H}}^\\sigma({\\mathbb R}^3)$. For every $2 \\leq q < \\infty$, $2 \\leq r < \\infty$, and $\\delta > 1 + \\frac{1}{q}$, there exist constants $C \\equiv C(q, r, \\delta) > 0$ and  $c \\equiv c(q, r, \\delta) > 0$ such that for every $\\lambda > 0$, \n \\begin{equation}\n  {\\mathbb P} \\Bigl( \\bigl\\{ \\omega \\in \\Omega : \\bigl\\|  \\langle t \\rangle^{-\\delta} u_f^\\omega \\bigr\\|_{L^q_t L^r_x({\\mathbb R}\\times{\\mathbb R}^3)} > \\lambda \\bigr\\} \\Bigr) \\leq C \\exp \\biggl( - c \\frac{\\lambda^2}{\\|f\\|_{{\\mathcal{H}}^0({\\mathbb R}^3)}^2} \\biggr)\n \\end{equation}\n and\n \\begin{equation}\n  {\\mathbb P} \\Bigl( \\bigl\\{ \\omega \\in \\Omega : \\bigl\\|  \\langle t \\rangle^{-\\delta} \\tilde{u}_f^\\omega \\bigr\\|_{L^q_t L^r_x({\\mathbb R}\\times{\\mathbb R}^3)} > \\lambda \\bigr\\} \\Bigr) \\leq C \\exp \\biggl( - c \\frac{\\lambda^2}{\\|f\\|_{{\\mathcal{H}}^0({\\mathbb R}^3)}^2} \\biggr).\n \\end{equation}\n\\end{lemma}\n\\begin{proof}\n We adapt the proofs of Proposition~A.1, Corollary~A.2, and Corollary~A.4 in \\cite{BT4} to our setting. In view of Lemma 2.5 in \\cite{LM}, it suffices to prove for any $p \\geq q, r$ that\n \\[\n  \\bigl\\| \\langle t \\rangle^{-\\delta} u_f^\\omega \\bigr\\|_{L^p_\\omega(\\Omega; L^q_t L^r_x({\\mathbb R}\\times{\\mathbb R}^3))} \\lesssim \\sqrt{p} \\|f\\|_{{\\mathcal{H}}^0({\\mathbb R}^3)}\n \\]\n and similarly for $\\tilde{u}_f^\\omega$. To this end we can consider the components of $u_f^\\omega$ separately. We will only show\n \\begin{equation} \\label{equ:suffices}\n  \\biggl\\| \\langle t \\rangle^{-\\delta} \\frac{\\sin(t|\\nabla|)}{|\\nabla|} f_2^\\omega \\biggr\\|_{L^p_\\omega(\\Omega; L^q_t L^r_x({\\mathbb R}\\times{\\mathbb R}^3))} \\lesssim \\sqrt{p} \\|f_2\\|_{H^{-1}_x({\\mathbb R}^3)},\n \\end{equation}\nsince the estimates for the other components are slightly easier. Using Lemma~\\ref{lem:large_deviation_estimate} and the unit-scale Bernstein estimate \\eqref{equ:unit_scale_bernstein}, we have for any $p \\geq q,r$ that\n \\begin{align*}\n  &\\biggl\\| \\langle t \\rangle^{-\\delta} \\frac{\\sin(t|\\nabla|)}{|\\nabla|} f_2^\\omega \\biggr\\|_{L^p_\\omega(\\Omega; L^q_t L^r_x({\\mathbb R}\\times{\\mathbb R}^3))} \\\\\n  &= \\biggl\\| \\langle t \\rangle^{-\\delta} \\sum_{k \\in {\\mathbb Z}^3} l_k(\\omega) \\frac{\\sin(t|\\nabla|)}{|\\nabla|} P_k f_2 \\biggr\\|_{L^p_\\omega(\\Omega; L^q_t L^r_x({\\mathbb R}\\times{\\mathbb R}^3))} \\\\\n  &\\lesssim \\sqrt{p} \\biggl\\| \\langle t \\rangle^{-\\delta} \\Bigl( \\sum_{k \\in {\\mathbb Z}^3} \\Bigl| \\frac{\\sin(t|\\nabla|)}{|\\nabla|} P_k f_2(x) \\Bigr|^2 \\Bigr)^{1\/2} \\biggr\\|_{L^q_t L^r_x({\\mathbb R}\\times{\\mathbb R}^3)} \\\\\n  &\\lesssim \\sqrt{p} \\biggl\\| \\langle t \\rangle^{-\\delta} \\Bigl( \\sum_{k \\in {\\mathbb Z}^3} \\Bigl\\| \\frac{\\sin(t|\\nabla|)}{|\\nabla|} P_k f_2 \\Bigr\\|_{L^r_x({\\mathbb R}^3)}^2 \\Bigr)^{1\/2} \\biggr\\|_{L^q_t({\\mathbb R})} \\\\\n  &\\lesssim \\sqrt{p} \\biggl\\| \\langle t \\rangle^{-\\delta} \\Bigl( \\sum_{k \\in {\\mathbb Z}^3} \\Bigl\\| \\frac{\\sin(t|\\nabla|)}{|\\nabla|} P_k f_2 \\Bigr\\|_{L^2_x({\\mathbb R}^3)}^2 \\Bigr)^{1\/2} \\biggr\\|_{L^q_t({\\mathbb R})} \\\\\n  &\\lesssim \\sqrt{p} \\biggl\\| \\langle t \\rangle^{-(\\delta-1)} \\Bigl( \\sum_{k \\in {\\mathbb Z}^3} \\bigl\\| \\langle \\nabla \\rangle^{-1} P_k f_2 \\bigr\\|_{L^2_x({\\mathbb R}^3)}^2 \\Bigr)^{1\/2} \\biggr\\|_{L^q_t({\\mathbb R})} \\\\\n  &\\lesssim \\sqrt{p} \\bigl\\| \\langle t \\rangle^{-(\\delta -1)} \\bigr\\|_{L^q_t({\\mathbb R})}  \\Bigl( \\sum_{k \\in {\\mathbb Z}^3} \\bigl\\| \\langle \\nabla \\rangle^{-1} P_k f_2 \\bigr\\|_{L^2_x({\\mathbb R}^3)}^2 \\Bigr)^{1\/2}\\\\\n  &\\lesssim \\sqrt{p} \\|f_2\\|_{H^{-1}_x({\\mathbb R}^3)}. \\qedhere\n \\end{align*}\n\\end{proof}\n\nThe following corollary is the formulation of the large deviation estimates that we will use in the proof of our main result.\n\n\\begin{corollary} \\label{cor:large_deviation_estimates_free_evolution_with_infty}\n Let $\\sigma > 0$ and $f = (f_1, f_2) \\in {\\mathcal{H}}^\\sigma({\\mathbb R}^3)$. For $2 \\leq q < \\infty$, $2 \\leq r \\leq \\infty$, $\\delta > 1 + \\frac{1}{q}$, and $0 < \\varepsilon \\leq \\sigma$, there exist constants $C \\equiv C(q, r, \\delta, \\varepsilon) > 0$ and $c \\equiv c(q,r,\\delta,\\varepsilon) > 0$ such that for every $\\lambda > 0$,\n \\begin{equation}\n  {\\mathbb P} \\Bigl( \\bigl\\{ \\omega \\in \\Omega : \\bigl\\|  \\langle t \\rangle^{-\\delta} u_f^\\omega \\bigr\\|_{L^q_t L^r_x({\\mathbb R}\\times{\\mathbb R}^3)} > \\lambda \\bigr\\} \\Bigr) \\leq C \\exp \\biggl( - c \\frac{\\lambda^2}{\\|f\\|_{{\\mathcal{H}}^\\varepsilon({\\mathbb R}^3)}^2} \\biggr)\n \\end{equation}\n and\n \\begin{equation}\n  {\\mathbb P} \\Bigl( \\bigl\\{ \\omega \\in \\Omega : \\bigl\\|  \\langle t \\rangle^{-\\delta} \\tilde{u}_f^\\omega \\bigr\\|_{L^q_t L^r_x({\\mathbb R}\\times{\\mathbb R}^3)} > \\lambda \\bigr\\} \\Bigr) \\leq C \\exp \\biggl( - c \\frac{\\lambda^2}{\\|f\\|_{{\\mathcal{H}}^\\varepsilon({\\mathbb R}^3)}^2} \\biggr).\n \\end{equation} \n\\end{corollary}\n\\begin{proof}\n As in \\cite[Corollary A.5]{BT4}, the assertion follows immediately from Lemma~\\ref{lem:large_deviation_estimates_free_evolution} and the Sobolev embedding $W^{s, r}_x({\\mathbb R}^3) \\hookrightarrow L^\\infty_x({\\mathbb R}^3)$ for any $r > 1$ and $s > \\frac{3}{r}$. \n\\end{proof}\n\nFinally, we need the following large deviation estimate in order to conclude that the energy functional used in the proof of Proposition~\\ref{prop:energy_bounds} is well-defined for all times.\n\\begin{lemma} \\label{lem:large_deviation_estimate_p+1}\nLet $\\sigma > 0$ and $f = (f_1, f_2) \\in {\\mathcal{H}}^\\sigma({\\mathbb R}^3)$. For $2 \\leq r < \\infty$, $\\delta > 1$, and $0 < \\varepsilon \\leq \\sigma$, there exist constants $C \\equiv C(r, \\delta, \\varepsilon) > 0$ and $c \\equiv c(r, \\delta, \\varepsilon) > 0$ such that for every $\\lambda > 0$,\n \\[\n  {\\mathbb P} \\Bigl( \\bigl\\{ \\omega \\in \\Omega : \\bigl\\| \\langle t \\rangle^{-\\delta} u_f^\\omega \\bigr\\|_{L_t^\\infty L^r_x({\\mathbb R} \\times {\\mathbb R}^3)} > \\lambda \\bigr\\} \\Bigr) \\leq C \\exp \\biggl( - c \\frac{\\lambda^2}{\\|f\\|_{{\\mathcal{H}}^\\varepsilon({\\mathbb R}^3)}^2} \\biggr).\n \\]\n\\end{lemma}\n\n\n\\begin{proof}\nWe adapt the proof of Lemma~2.2 in \\cite{BTT2} to our setting. Applying one-dimensional Sobolev embedding in time with $q \\geq 2$ sufficiently large such that $\\varepsilon > \\frac{1}{q}$ and $\\delta > 1+ \\frac{1}{q}$, we obtain\n\\begin{align*}\n \\bigl\\| \\langle t \\rangle^{-\\delta} u_f^\\omega \\bigr\\|_{L^\\infty_t L^r_x({\\mathbb R}\\times{\\mathbb R}^3)} &\\lesssim \\bigl\\| \\langle \\partial_t \\rangle^{\\varepsilon} \\langle t \\rangle^{-\\delta} u_f^\\omega \\bigr\\|_{L^q_t L^r_x({\\mathbb R}\\times{\\mathbb R}^3)} \\\\\n &\\lesssim \\bigl\\| \\langle t \\rangle^{-\\delta} \\langle \\partial_t \\rangle^{\\varepsilon} u_f^\\omega \\bigr\\|_{L^q_t L^r_x({\\mathbb R}\\times{\\mathbb R}^3)} \\\\\n &\\lesssim \\bigl\\| \\langle t \\rangle^{-\\delta} \\langle \\nabla_x \\rangle^{\\varepsilon} u_f^\\omega \\bigr\\|_{L^q_t L^r_x({\\mathbb R}\\times{\\mathbb R}^3)} \n\\end{align*}\nand the claim now follows from Lemma~\\ref{lem:large_deviation_estimates_free_evolution}.\n\\end{proof}\n\n\\section{Proof of Theorem~\\ref{main_theorem}} \\label{sec:proof_main_theorem}\n\\setcounter{equation}{0}\n\nIn this section we first record a deterministic local well-posedness result for the forced nonlinear wave equation that is associated with the random data problem~\\eqref{equ:nlw_main_theorem}. Next we derive the main probabilistic energy bounds for the nonlinear components of the solutions to~\\eqref{equ:nlw_main_theorem}. The proof of Theorem~\\ref{main_theorem} is then an immediate consequence of these two results.\n\n\\begin{lemma} \\label{lem:lwp}\n Let $3 < p < 5$, $T' > 0$, and let $(v_1, v_2) \\in {\\mathcal{H}}^{1}({\\mathbb R}^3)$ and $F \\in L^{\\frac{2p}{p-3}}_t L^{2p}_x([0,T']\\times{\\mathbb R}^3)$ be such that\n \\begin{equation}\n  \\|(v_1, v_2)\\|_{ {\\mathcal{H}}^{1}} +  \\bigl\\| F \\bigr\\|_{L^{\\frac{2p}{p-3}}_t L^{2p}_x([0,T']\\times{\\mathbb R}^3)} \\leq \\lambda\n \\end{equation}\n for some $\\lambda > 0$. Then there exists $0 < T \\leq T'$ with $T \\sim \\lambda^{-\\frac{2(p-1)}{5-p}}$ and a unique solution\n \\[\n  (v, \\partial_t v) \\in  C\\bigl([0,T]; H^1_x({\\mathbb R}^3)\\bigr) \\cap L^{\\frac{2p}{p-3}}_t L^{2 p}_x\\bigl([0,T]\\times{\\mathbb R}^3\\bigr) \\times C\\bigl([0,T]; L^2_x({\\mathbb R}^3)\\bigr)\n \\]\n to the forced nonlinear wave equation\n \\begin{equation} \\label{equ:nlw_lwp}\n  \\left\\{ \\begin{aligned}\n   -\\partial_t^2 v + \\Delta v &= |F+v|^{p-1} (F + v) \\text{ on } [0,T] \\times {\\mathbb R}^3, \\\\\n   (v, \\partial_t v)|_{t=0} &= (v_1,v_2).\n  \\end{aligned} \\right.\n \\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nWe introduce the notation \n\\[\n q(p) = \\frac{2 p}{p - 3}, \\quad \\alpha(p) = \\frac{5 - p}{2}.\n\\]\nLet $0 < T \\leq T'$ to be fixed later. Then we define for $v \\in L^{q(p)}_t L^{2p}_x([0,T]\\times{\\mathbb R}^3)$ and $t \\in [0,T]$,\n\\begin{equation}\n \\Phi(v)(t) = \\cos(t|\\nabla|) v_1 + \\frac{\\sin(t|\\nabla|)}{|\\nabla|} v_2 - \\int_0^t \\frac{\\sin((t-s) |\\nabla|)}{|\\nabla|} |F + v|^{p -1} (F + v)(s) \\, ds.\n\\end{equation}\nWe note that the exponent pair $(q(p), 2 p)$ is Strichartz-admissible at regularity $\\gamma =1$. Thus, by the Strichartz estimates \\eqref{equ:strichartz_estimates} we obtain that\n\\begin{equation} \\label{equ:Phi_estimate1}\n \\begin{aligned}  \n  \\|\\Phi(v)\\|_{L^{q(p)}_T L^{2 p}_x} &\\leq C \\Bigl( \\|(v_1, v_2)\\|_{\\dot{H}^1_x \\times L^2_x} + \\bigl\\| |F + v|^{p -1} (F + v) \\bigr\\|_{L^1_T L^2_x} \\Bigr) \\\\\n  &\\leq C \\|(v_1, v_2)\\|_{\\dot{H}^1_x \\times L^2_x} + C T^{\\alpha(p)} \\Bigl( \\bigl\\| F \\bigr\\|^{p}_{L^{q(p)}_T L^{2 p}_x } + \\|v\\|^{p}_{L^{q(p)}_T L^{2 p}_x} \\Bigr).\n \\end{aligned}\n\\end{equation}\nSimilarly, we find for $v, \\widetilde{v} \\in L^{q(p)}_t L^{2p}_x([0,T]\\times{\\mathbb R}^3)$ that\n\\begin{equation} \\label{equ:Phi_estimate2}\n \\begin{aligned}\n   \\|\\Phi(v) - \\Phi(\\widetilde{v})\\|_{L^{q(p)}_T L^{2 p}_x} &\\leq C T^{\\alpha(p)} \\|v - \\widetilde{v}\\|_{L^{q(p)}_T L^{2 p}_x } \\times \\\\\n   &\\quad \\quad \\quad \\times \\Bigl(\\bigl\\| F \\bigr\\|^{p-1}_{L^{q(p)}_T L^{2 p}_x } + \\|v\\|^{p-1}_{L^{q(p)}_T L^{2 p}_x } + \\|\\widetilde{v}\\|^{p-1}_{L^{q(p)}_T L^{2 p}_x } \\Bigr).\n \\end{aligned}\n\\end{equation}\nFrom \\eqref{equ:Phi_estimate1} and \\eqref{equ:Phi_estimate2} we conclude that by choosing\n\\begin{equation}\n T \\sim \\lambda^{- \\frac{(p - 1)}{\\alpha(p)}},\n\\end{equation}\nthe ball\n\\[\n  B := \\bigl\\{ v \\in L^{q(p)}_t L^{2p}_x([0,T]\\times{\\mathbb R}^3) : \\|v\\|_{L^{q(p)}_t L^{2p}_x([0,T]\\times{\\mathbb R}^3)}  \\leq 2 C \\lambda \\bigr\\}\n\\]\nis mapped into itself by $\\Phi$ and $\\Phi$ is a contraction on $B$ with respect to the $L^{q(p)}_T L^{2 p}_x$ norm. Hence, $\\Phi$ has a unique fixed point $v$ in $B$ and we easily verify that it satisfies\n\\[\n (v, \\partial_t v) \\in C([0,T]; H^1_x({\\mathbb R}^3) \\times L^2_x({\\mathbb R}^3)),\n\\]\nwhich finishes the proof.\n\\end{proof}\n\n\\medskip\n\nWe now present the proof of the key probabilistic energy bounds.\n\n\\begin{proposition} \\label{prop:energy_bounds}\n Let $3 < p < 5$, $\\frac{p-1}{p+1} < s < 1$ and $0 < \\delta < \\frac{p+1}{p-1} s -1$. For real-valued $f = (f_1, f_2) \\in {\\mathcal{H}}^s({\\mathbb R}^3)$, let $f^\\omega$ be the associated randomization as defined in \\eqref{equ:bighsrandomization} and let $u_f^\\omega$, $\\tilde{u}_f^\\omega$ be the free evolutions \\eqref{equ:free_evolution}, respectively \\eqref{equ:modified_free_evolution}. Then there exists $\\Sigma \\subset \\Omega$ with ${\\mathbb P}(\\Sigma) = 1$ such that for all $\\omega \\in \\Sigma$,\n \\begin{align} \\label{equ:L_t_loc_almost_surely_energy_bounds}\n  \\begin{aligned}\n   & u_f^\\omega \\in L_{t,loc}^\\infty L^{p+1}_x({\\mathbb R} \\times {\\mathbb R}^3), \\\\\n   & \\tilde{u}_f^\\omega \\in L^{p+1}_{t,loc} L^{p+1}_x({\\mathbb R}\\times{\\mathbb R}^3),\n  \\end{aligned}\n  &&\n  \\begin{aligned}\n   & \\langle \\nabla \\rangle^{\\frac{p-1}{2} (1-s+\\delta+)} u_f^\\omega \\in L^4_{t,loc} L^2_x({\\mathbb R}\\times{\\mathbb R}^3), \\\\\n   & \\langle \\nabla \\rangle^{s-\\delta} \\tilde{u}_f^\\omega \\in L^2_{t,loc} L^\\infty_x({\\mathbb R}\\times{\\mathbb R}^3).\n  \\end{aligned}\n \\end{align}\n For fixed $\\omega \\in \\Sigma$, let $T > 0$ be arbitrary and let  \n \\begin{equation} \\label{equ:energy_bounds_solution}\n  (v^\\omega, \\partial_t v^\\omega) \\in C\\bigl([0,T]; H^1_x({\\mathbb R}^3)\\bigr) \\cap L^{\\frac{2p}{p-3}}_t L^{2p}_x\\bigl([0,T]\\times{\\mathbb R}^3\\bigr) \\times C\\bigl([0,T]; L^2_x({\\mathbb R}^3)\\bigr)\n \\end{equation}\n be a solution to\n \\begin{equation} \\label{equ:nlw_probabilistic_energy_bounds}\n  \\left\\{ \\begin{aligned}\n   -\\partial_t^2 v^\\omega + \\Delta v^\\omega &= |u_f^\\omega + v^\\omega|^{p-1} (u_f^\\omega + v^\\omega) \\text{ on } [0,T] \\times {\\mathbb R}^3, \\\\\n   (v^\\omega, \\partial_t v^\\omega)|_{t=0} &= (v_1, v_2)\n  \\end{aligned} \\right.\n \\end{equation} \n for some $(v_1, v_2) \\in {\\mathcal{H}}^1({\\mathbb R}^3)$. Then there exists an absolute constant $C > 0$ such that\n \\begin{equation}\n  \\sup_{t \\in [0,T]} \\bigl\\| (v^\\omega(t), \\partial_t v^\\omega(t)) \\bigr\\|_{{\\mathcal{H}}^1}^2 \\leq C \\bigl( \\|(v_1, v_2)\\|_{{\\mathcal{H}}^1}^2 + \\|v_1\\|_{L^{p+1}_x}^{p+1} + \\|f_1^\\omega\\|_{L^{p+1}_x}^{p+1} + B^\\omega(T) \\bigr) e^{C (T + A^\\omega(T))},\n \\end{equation}\n where\n \\begin{align*}\n  A^\\omega(T) &= \\bigl\\| \\langle \\nabla \\rangle^{s-\\delta} \\tilde{u}_f^\\omega \\bigr\\|_{L^1_t L^\\infty_x([0,T]\\times{\\mathbb R}^3)}, \\\\\n  B^\\omega(T) &= \\bigl\\| \\tilde{u}_f^\\omega \\bigr\\|_{L^{p+1}_t L^{p+1}_x([0,T]\\times{\\mathbb R}^3)}^{p+1} + \\bigl\\| \\langle \\nabla \\rangle^{s-\\delta} \\tilde{u}_f^\\omega \\bigr\\|_{L^2_t L^\\infty_x([0,T]\\times{\\mathbb R}^3)}^2 \\\\\n  &\\quad + \\bigl\\| \\langle \\nabla \\rangle^{\\frac{p-1}{2} (1-s+\\delta+)} u_f^\\omega \\bigr\\|_{L^4_t L^2_x([0,T]\\times{\\mathbb R}^3)}^4.\n \\end{align*}\n\\end{proposition}\n\\begin{proof}\n The large deviation estimates from Lemma~\\ref{lem:large_deviation_estimates_free_evolution}, Corollary~\\ref{cor:large_deviation_estimates_free_evolution_with_infty}, and Lemma~\\ref{lem:large_deviation_estimate_p+1} imply that there exists $\\Sigma \\subset \\Omega$ with ${\\mathbb P}(\\Sigma) = 1$ such that \\eqref{equ:L_t_loc_almost_surely_energy_bounds} holds for all $\\omega \\in \\Sigma$.\n \n \\medskip\n\n Now for fixed $\\omega \\in \\Sigma$, let $T > 0$ be arbitrary and let\n \\[\n  (v^\\omega, \\partial_t v^\\omega) \\in C\\bigl([0,T]; H^1_x({\\mathbb R}^3)\\bigr) \\cap L^{\\frac{2p}{p-3}}_t L^{2p}_x\\bigl([0,T]\\times{\\mathbb R}^3\\bigr) \\times C\\bigl([0,T]; L^2_x({\\mathbb R}^3)\\bigr)\n \\]\n be a solution to \\eqref{equ:nlw_probabilistic_energy_bounds}. For ease of notation we drop the superscript $\\omega$ from $v^\\omega$ for the remainder of the proof and consider the energy functional\n \\[\n  E(v(t)) = \\int_{{\\mathbb R}^3} \\frac{1}{2} |\\nabla_x v|^2 + \\frac{1}{2} |\\partial_t v|^2 + \\frac{1}{2} |v|^2 + \\frac{1}{p+1} |u_f^\\omega + v|^{p+1} \\, dx.\n \\]\nFor $\\omega \\in \\Sigma$, this functional is well-defined and finite for all $0 \\leq t \\leq T$ by Sobolev embedding and the properties from \\eqref{equ:L_t_loc_almost_surely_energy_bounds}.  Using that $v$ is a solution to the nonlinear wave equation \\eqref{equ:nlw_probabilistic_energy_bounds} and that $\\partial_t u_f^\\omega = \\langle \\nabla \\rangle \\tilde{u}_f^\\omega$, we compute \n \\begin{equation} \\label{equ:time_derivative_energy}\n  \\partial_t E(v(t)) = \\int_{{\\mathbb R}^3} v \\partial_t v \\, dx + \\int_{{\\mathbb R}^3}  \\langle \\nabla \\rangle \\tilde{u}_f^\\omega \\, |u_f^\\omega + v|^{p-1} (u_f^\\omega + v) \\, dx,\n \\end{equation}\n where the second term on the right hand side is to be understood as an $H^{s-1}_x \\times H^{1-s}_x$ pairing. The first term can be easily bounded by\n \\begin{equation}\n  \\Big| \\int_{{\\mathbb R}^3} v \\partial_t v \\, dx \\Big| \\lesssim \\|v\\|_{L^2_x} \\|\\partial_t v\\|_{L^2_x} \\lesssim E(v).\n \\end{equation}\n In order to estimate the second term on the right hand side of \\eqref{equ:time_derivative_energy}, we use an inhomogeneous dyadic decomposition to expand\n \\begin{align*}\n  \\int_{{\\mathbb R}^3}  \\langle \\nabla \\rangle \\tilde{u}_f^\\omega \\, |u_f^\\omega + v|^{p-1} (u_f^\\omega + v) \\, dx &= \\sum_{N \\geq 1} \\int_{{\\mathbb R}^3} \\widetilde{P}_N \\bigl( \\langle \\nabla \\rangle \\tilde{u}_f^\\omega \\bigr) P_N \\bigl( |u_f^\\omega + v|^{p-1} (u_f^\\omega + v) \\bigr) \\, dx.\n \\end{align*}\n {\\bf Case 1:} $N \\lesssim 1$. By Bernstein's and Young's inequality\n \\begin{align*}\n  \\bigg| \\sum_{N \\lesssim 1} \\int_{{\\mathbb R}^3} \\widetilde{P}_N \\bigl( \\langle \\nabla \\rangle \\tilde{u}_f^\\omega \\bigr) P_N \\bigl( |u_f^\\omega + v|^{p-1} (u_f^\\omega + v) \\bigr) \\, dx \\bigg| &\\lesssim \\bigl\\| \\tilde{u}_f^\\omega \\bigr\\|_{L^{p+1}_x} \\bigl\\| |u_f^\\omega + v|^{p-1} (u_f^\\omega +v) \\bigr\\|_{L^{\\frac{p+1}{p}}_x} \\\\\n  &\\lesssim \\bigl\\| \\tilde{u}_f^\\omega \\bigr\\|_{L^{p+1}_x}^{p+1} + \\bigl\\| u_f^\\omega + v \\bigr\\|_{L^{p+1}_x}^{p+1} \\\\\n  &\\lesssim \\bigl\\| \\tilde{u}_f^\\omega \\bigr\\|_{L^{p+1}_x}^{p+1} + E(v).\n \\end{align*}\n {\\bf Case 2:} $N \\gg 1$. We further decompose dyadically\n \\begin{align*}\n  &\\sum_{N \\gg 1} \\int_{{\\mathbb R}^3} \\widetilde{P}_N \\bigl( \\langle \\nabla \\rangle \\tilde{u}_f^\\omega \\bigr) P_N \\bigl( |u_f^\\omega + v|^{p-1} (u_f^\\omega + v) \\bigr) \\, dx \\\\\n  &= \\sum_{N \\gg 1} \\sum_{N_1, N_2 \\geq 1} \\int_{{\\mathbb R}^3} \\widetilde{P}_N \\bigl( \\langle \\nabla \\rangle \\tilde{u}_f^\\omega \\bigr) P_N \\Bigl( P_{N_1} \\bigl( |u_f^\\omega + v|^{p-1} \\bigr) P_{N_2} \\bigl( u_f^\\omega + v \\bigr) \\Bigr) \\, dx.\n \\end{align*}\n {\\bf Case 2a:} $N_1 \\sim N$ and $N_2 \\lesssim N_1$. Summing over dyadic $N, N_2$ at a slight loss of a power in $N_1$, we obtain\n \\begin{align*}\n &\\bigg| \\sum_{N \\gg 1} \\sum_{N_1 \\sim N} \\sum_{N_2 \\lesssim N_1} \\int_{{\\mathbb R}^3} {\\widetilde P}_N \\big( \\langle \\nabla \\rangle \\tilde{u}_f^\\omega \\big) P_N \\Big( P_{N_1} \\big( |u_f^\\omega + v|^{p-1} \\big) P_{N_2} \\big( u_f^\\omega + v \\big) \\Big) \\, dx \\bigg| \\\\\n &\\lesssim \\big\\| \\langle \\nabla \\rangle^{s-\\delta} \\tilde{u}_f^\\omega \\big\\|_{L^\\infty_x} \\sup_{N_1 \\gg 1} N_1^{1-s+\\delta +} \\big\\| P_{N_1} |u_f^\\omega + v|^{p-1} \\big\\|_{L^{\\frac{p+1}{p}}_x} \\big\\| u_f^\\omega + v \\big\\|_{L^{p+1}_x} \\\\\n &\\lesssim \\big\\| \\langle \\nabla \\rangle^{s-\\delta} \\tilde{u}_f^\\omega \\big\\|_{L^\\infty_x} \\big\\| |\\nabla|^{1-s+\\delta+} \\big( |u_f^\\omega + v|^{p-1} \\big) \\big\\|_{L^{\\frac{p+1}{p}}_x} \\big\\| u_f^\\omega + v \\big\\|_{L^{p+1}_x}. \n \\end{align*}\n Using the fractional chain rule \\eqref{equ:fractional_chain_rule} with $\\frac{p}{p+1} = \\frac{2}{p+1} + \\frac{p-2}{p+1}$, this is bounded by\n \\begin{equation} \\label{equ:bound_on_difficult_term}\n  \\big\\| \\langle \\nabla \\rangle^{s-\\delta} \\tilde{u}_f^\\omega \\big\\|_{L^\\infty_x} \\big\\| |\\nabla|^{1-s+\\delta+} (u_f^\\omega + v) \\big\\|_{L^{\\frac{p+1}{2}}_x} \\big\\| u_f^\\omega + v \\big\\|^{p-1}_{L^{p+1}_x}.\n \\end{equation} \nWe invoke the interpolation inequality \\eqref{equ:interpolation_inequality_in_proposition} in the form \n \\begin{equation} \\label{equ:interpolation_inequality}\n  \\bigl\\| |\\nabla|^\\sigma f \\bigr\\|_{L^{\\frac{p+1}{2}}_x} \\lesssim \\bigl\\| |\\nabla|^{\\frac{p-1}{2} \\sigma} f \\bigr\\|_{L^2_x}^{\\frac{2}{p-1}} \\bigl\\|f\\bigr\\|_{L^{p+1}_x}^{\\frac{p-3}{p-1}}\n \\end{equation}\n for $0 < \\sigma < 1$ to bound \\eqref{equ:bound_on_difficult_term} by\n \\begin{align*}\n  \\big\\| \\langle \\nabla \\rangle^{s-\\delta} \\tilde{u}_f^\\omega \\big\\|_{L^\\infty_x} \\big\\| |\\nabla|^{\\frac{p-1}{2} (1-s+\\delta+)} (u_f^\\omega + v) \\big\\|_{L^2_x}^{\\frac{2}{p-1}} \\big\\| u_f^\\omega + v \\big\\|_{L^{p+1}_x}^{\\frac{(p-2)(p+1)}{p-1}},\n \\end{align*}\nwhich we can estimate using Young's inequality\n \\begin{equation} \\label{equ:difficult_term_after_young}\n  \\begin{split}\n   &\\big\\| \\langle \\nabla \\rangle^{s-\\delta} \\tilde{u}_f^\\omega \\big\\|_{L^\\infty_x} \\Big( \\big\\| |\\nabla|^{\\frac{p-1}{2} (1-s+\\delta+)} (u_f^\\omega + v) \\big\\|_{L^2_x}^2 + \\big\\| u_f^\\omega + v \\big\\|_{L^{p+1}_x}^{p+1} \\Big) \\\\\n   &\\lesssim \\big\\| \\langle \\nabla \\rangle^{s-\\delta} \\tilde{u}_f^\\omega \\big\\|_{L^\\infty_x} \\Big( \\big\\| |\\nabla|^{\\frac{p-1}{2} (1-s+\\delta+)} u_f^\\omega \\big\\|_{L^2_x}^2 + \\big\\| |\\nabla|^{\\frac{p-1}{2} (1-s+\\delta+)} v \\big\\|_{L^2_x}^2 + \\big\\| u_f^\\omega + v \\big\\|_{L^{p+1}_x}^{p+1} \\Big).\n  \\end{split}\n \\end{equation}\nBy our assumptions on $s$ and $\\delta$, we have that\n \\[\n  \\frac{p-1}{2} (1 - s + \\delta +) < s < 1,\n \\]\n thus, we may bound the right hand side of \\eqref{equ:difficult_term_after_young} by\n \\begin{align*}\n  \\big\\| \\langle \\nabla \\rangle^{s-\\delta} \\tilde{u}_f^\\omega \\big\\|_{L^\\infty_x} \\Big( \\big\\| \\langle \\nabla \\rangle^{\\frac{p-1}{2} (1-s+\\delta+)} u_f^\\omega \\big\\|_{L^2_x}^2 + \\big\\| v \\big\\|_{L^2_x}^2 + \\big\\| \\nabla_x v \\big\\|_{L^2_x}^2 + \\big\\| u_f^\\omega + v \\big\\|_{L^{p+1}_x}^{p+1} \\Big).\n \\end{align*}\n {\\bf Case 2b:} $N_2 \\sim N$ and $N_1 \\lesssim N_2$. Summing over dyadic $N$, $N_1$, we bound\n \\begin{align*}\n  &\\bigg| \\sum_{N \\gg 1} \\sum_{N_2 \\sim N} \\sum_{N_1 \\lesssim N_2} \\int_{{\\mathbb R}^3} {\\widetilde P}_N \\big( \\langle \\nabla \\rangle \\tilde{u}_f^\\omega \\big) P_N \\Big( P_{N_1} \\big( |u_f^\\omega + v|^{p-1} \\big) P_{N_2} \\big( u_f^\\omega + v \\big) \\Big) \\, dx \\bigg| \\\\\n  &\\lesssim \\big\\| \\langle \\nabla \\rangle^{s-\\delta} \\tilde{u}_f^\\omega \\big\\|_{L^\\infty_x} \\sup_{N_2 \\gg 1} N_2^{1-s+\\delta+} \\big\\| u_f^\\omega + v \\big\\|^{p-1}_{L^{p+1}_x} \\big\\| P_{N_2} (u_f^\\omega + v) \\big\\|_{L^{\\frac{p+1}{2}}_x}.\n \\end{align*}  \n Using the interpolation inequality\n \\[\n  \\|f\\|_{L^{\\frac{p+1}{2}}_x} \\lesssim \\|f\\|_{L^2_x}^{\\frac{2}{p-1}} \\|f\\|_{L^{p+1}_x}^{\\frac{p-3}{p-1}},\n \\]\n this is bounded by\n \\begin{align*}\n  &\\big\\| \\langle \\nabla \\rangle^{s-\\delta} \\tilde{u}_f^\\omega \\big\\|_{L^\\infty_x} \\sup_{N_2 \\gg 1} N_2^{1-s+\\delta+} \\big\\| u_f^\\omega + v \\big\\|^{p-1}_{L^{p+1}_x} \\big\\| P_{N_2} (u_f^\\omega + v) \\big\\|_{L^2_x}^{\\frac{2}{p-1}} \\big\\| P_{N_2} (u_f^\\omega + v) \\big\\|_{L^{p+1}_x}^{\\frac{p-3}{p-1}} \\\\\n  &\\lesssim \\big\\| \\langle \\nabla \\rangle^{s-\\delta} \\tilde{u}_f^\\omega \\big\\|_{L^\\infty_x} \\sup_{N_2 \\gg 1} \\Big\\| N_2^{\\frac{p-1}{2} (1-s+\\delta+)} P_{N_2} (u_f^\\omega + v) \\Big\\|_{L^2_x}^{\\frac{2}{p-1}} \\big\\| u_f^\\omega + v \\big\\|_{L^{p+1}_x}^{\\frac{(p-2)(p+1)}{p-1}}.\n \\end{align*}\nBy Bernstein's and Young's inequality we finally obtain the estimate\n \\begin{align*}\n  &\\big\\| \\langle \\nabla \\rangle^{s-\\delta} \\tilde{u}_f^\\omega \\big\\|_{L^\\infty_x} \\Big( \\big\\| |\\nabla|^{\\frac{p-1}{2} (1-s+\\delta+)} (u_f^\\omega + v) \\big\\|_{L^2_x}^2 + \\big\\| u_f^\\omega + v \\big\\|_{L^{p+1}_x}^{p+1} \\Big) \\\\\n  &\\lesssim \\big\\| \\langle \\nabla \\rangle^{s-\\delta} \\tilde{u}_f^\\omega \\big\\|_{L^\\infty_x} \\Big( \\big\\| \\langle \\nabla \\rangle^{\\frac{p-1}{2} (1-s+\\delta+)} u_f^\\omega \\big\\|_{L^2_x}^2 + \\big\\| v \\big\\|_{L^2_x}^2 + \\big\\| \\nabla_x v \\big\\|_{L^2_x}^2 + \\big\\| u_f^\\omega + v \\big\\|_{L^{p+1}_x}^{p+1} \\Big) \\\\\n  &\\lesssim \\big\\| \\langle \\nabla \\rangle^{s-\\delta} \\tilde{u}_f^\\omega \\big\\|_{L^\\infty_x}^2 + \\big\\| \\langle \\nabla \\rangle^{\\frac{p-1}{2} (1-s+\\delta+)} u_f^\\omega \\big\\|_{L^2_x}^4 \\\\\n &\\qquad + \\big\\| \\langle \\nabla \\rangle^{s-\\delta} \\tilde{u}_f^\\omega \\big\\|_{L^\\infty_x} \\Big( \\big\\| v \\big\\|_{L^2_x}^2 + \\big\\| \\nabla_x v \\big\\|_{L^2_x}^2 + \\big\\| u_f^\\omega + v \\big\\|_{L^{p+1}_x}^{p+1} \\Big).\n \\end{align*}\n {\\bf Case 2c:} $N_1, N_2 \\gtrsim N$ and $N_1 \\sim N_2$. Here we can proceed similarly to Case 2b.\n\n \\medskip\n\n Putting the above estimates together, we find that for $0 \\leq t \\leq T$,\n \\begin{align*}\n  \\partial_t E(v(t)) &\\leq C \\Big( \\big\\| \\tilde{u}_f^\\omega(t) \\big\\|_{L^{p+1}_x}^{p+1} + \\big\\| \\langle \\nabla \\rangle^{s-\\delta} \\tilde{u}_f^\\omega(t) \\big\\|_{L^\\infty_x}^2 + \\big\\| \\langle \\nabla \\rangle^{\\frac{p-1}{2} (1-s+\\delta+)} u_f^\\omega(t) \\big\\|_{L^2_x}^4 \\\\\n  &\\quad \\quad \\quad \\quad + \\big( 1 + \\big\\| \\langle \\nabla \\rangle^{s-\\delta} \\tilde{u}_f^\\omega(t) \\big\\|_{L^\\infty_x} \\big) E(v(t)) \\Big).\n \\end{align*}\n Gronwall's inequality then yields that\n \\begin{align*}\n  E(v(T)) &\\leq \\exp \\biggl( C \\int_0^T \\bigl( 1 + \\bigl\\| \\langle \\nabla \\rangle^{s-\\delta} \\tilde{u}_f^\\omega(t) \\bigr\\|_{L^\\infty_x} \\bigr) \\, dt \\biggr) \\times \\\\\n  &\\quad \\times \\biggl( E(0) + C \\int_0^T \\Bigl( \\bigl\\| \\tilde{u}_f^\\omega(t) \\bigr\\|_{L^{p+1}_x}^{p+1} + \\bigl\\| \\langle \\nabla \\rangle^{s-\\delta} \\tilde{u}_f^\\omega(t) \\bigr\\|_{L^\\infty_x}^2 + \\bigl\\| \\langle \\nabla \\rangle^{\\frac{p-1}{2} (1-s+\\delta+)} u_f^\\omega(t) \\bigr\\|_{L^2_x}^4 \\Bigr) \\, dt \\biggr) \\\\\n  &\\leq C e^{C (T + A^\\omega(T))} \\Bigl( \\|(v_1, v_2)\\|_{{\\mathcal{H}}^1}^2 + \\|v_1\\|_{L^{p+1}_x}^{p+1} + \\| f_1^\\omega \\|_{L^{p+1}_x}^{p+1} + B^\\omega(T) \\Bigr). \\qedhere\n \\end{align*}\n\\end{proof}\n\nTheorem~\\ref{main_theorem} is now an easy consequence of the previous results.\n\\begin{proof}[Proof of Theorem~\\ref{main_theorem}]\n The large deviation estimates from Lemma~\\ref{lem:large_deviation_estimates_free_evolution}, Corollary~\\ref{cor:large_deviation_estimates_free_evolution_with_infty}, and Lemma~\\ref{lem:large_deviation_estimate_p+1} imply that there exists $\\widetilde{\\Omega} \\subset \\Omega$ with ${\\mathbb P}(\\widetilde{\\Omega}) = 1$ such that for all $\\omega \\in \\widetilde{\\Omega}$, \\eqref{equ:L_t_loc_almost_surely_energy_bounds} holds and \n \\[\n  u_f^\\omega \\in L^{\\frac{2p}{p-3}}_{t,loc} L^{2p}_x({\\mathbb R}\\times{\\mathbb R}^3).\n \\]\n Fix $\\omega \\in \\widetilde{\\Omega}$. We invoke Lemma~\\ref{lem:lwp} to obtain a local solution $(v^\\omega, \\partial_t v^\\omega)$ to \\eqref{equ:forced_nlw_main_theorem}. We can iterate the local well-posedness of Lemma~\\ref{lem:lwp} to obtain a global solution provided the ${\\mathcal{H}}^1({\\mathbb R}^3)$ norm of $(v^\\omega(t), \\partial_t v^\\omega(t))$ does not blow up in finite time. This is guaranteed by the energy bounds of Proposition~\\ref{prop:energy_bounds}, which proves the assertion.\n\\end{proof}\n\n\n\n\\bibliographystyle{myamsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzlmig b/data_all_eng_slimpj/shuffled/split2/finalzzlmig
new file mode 100644
index 0000000000000000000000000000000000000000..d406f864e978bb1880c953a2236fce290d5c45e2
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzlmig
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\n\nAlthough the interior of black holes is invisible, there are reports on observations of astrophysical black holes owing to their intense interactions with the surrounding matter, which are manifest around galactic black holes interacting with the Active Galactic Nucleus.\nAccretion disks create gravitational tidal forces as well as electric and magnetic fields around black holes. Accurate physical descriptions of such processes usually involve complex numerical modeling using supercomputers and present a number of challenges connected to explanations of various observed processes, including for example, the origin of jets around black holes. Here we adopt a more theoretical framework, which, although far from being an\naccurate astrophysical description, could be a bridge between the mathematical theory of black holes and the numerical modeling in astrophysics.\n\nAn important constituent of a black hole's environment is a magnetic field, which can be as strong as $10^{4}-10^{8} G$ near the black hole \\cite{1}. The magnetic fields near a black hole can extract its rotational energy  (Blandford-Znajek effect \\cite{2}), affect the quasinormal spectrum \\cite{Kokkotas:2010zd} and Hawking radiation \\cite{Galtsov-book}, \\cite{Kokkotas:2010zd}, and induce an external electric charge on its surface \\cite{Galtsov-book}.\nAt the same time, there is no evidence that the magnetic field could be strong enough to deform the geometry of a black hole significantly. Thus, the most interesting effects associated with the presence of the magnetic field occur on charged particles and fields.\nIndeed, the factor which stipulates the Lorentz force during the particle's motion is $q B\/\\mu$ \\cite{Galtsov-book}, where $\\mu$ is the mass of the particle and $B$ is the asymptotic value of the magnetic field strength.\n\n\nAnother essential factor representing the black hole environment is a tidal gravitational force produced by all the surrounding matter. Unlike magnetic fields, tidal forces can be strong enough to deform (albeit minimumly) the geometry of a galactic black hole as well as black holes of stellar masses having a star companion.\n\nThe perturbative solution to the Einstein-Maxwell equations which allows us to study, the magnetic field and the tidal gravitational force, was found in \\cite{Preston-Poisson} by Preston and Poisson with the help of the light-cone gauge formalism \\cite{Preston:2006zd}.\nThey considered a system consisting of a black hole and a mechanical structure (a giant torus or long solenoid) at some distance around it, which was the source of the gravitational tidal force and the strong magnetic field. Thus, the background which we shall study here, the Preston-Poisson space-time, has three parameters: mass $M$, magnetic field $B$ and tidal force $\\mathcal{E}$.\n\n\nWe investigated two basic properties of particles and fields in the vicinity of such a magnetically and tidally deformed black hole: the proper oscillation frequencies in the black hole's response to the perturbation, termed \\emph{quasinormal modes}, and the motion of the charged particles. The quasinormal modes do not depend on the kind of perturbation but only on the parameters of  space-time and are therefore termed the fingerprints of the space-time.\nThe quasinormal modes are studied in various disciplines such as black hole physics, gauge\/gravity correspondence,\ngravitational wave astronomy \\cite{QNM-collect}, cosmology \\cite{cosmology} and quantum gravity \\cite{QNM-collect}.\n\nParticle motion and the quasinormal modes have been studied for black holes immersed in a strong magnetic field \\cite{Galtsov-book}, \\cite{Kokkotas:2010zd},  \\cite{Preti:2009zz}, \\cite{Preti:2005bu}, \\cite{Aliev:2002nw}, \\cite{Aliev:1989wx}, \\cite{Esteban}, \\cite{Konoplya:2006gg}, \\cite{Konoplya:2007yy}, \\cite{Konoplya:2008hj}.We previoulsy investigated motion of neutral particles in the vicinity of the Preston-Poisson space-time \\cite{Konoplya:2006qr}. Further studies on particle motion, accretion and thermodynamic properties around black holes in a magnetic field was performed in \\cite{Kovacs:2011ec}, \\cite{Abdujabbarov:2009az}, \\cite{Frolov:2010mi}, \\cite{Rahimov:2011fv}, \\cite{Konoplya:2010ak}. In the presented work we estimate quaisnormal modes of a test scalar field propagating in the background of the Preston-Poisson black hole \\cite{Preston-Poisson}, when the magnetic component of the deformation vanishes.\nThis is a good approximation, since the gravitational tidal force is usually much stronger than the deformation caused by the magnetic field.\nIn the ekional regime, quasinormal modes usually have a kind of universal behavior for different boson fields, such that the spectrum of the neutral scalar field considered here could reveal some features of the gravitational spectrum. The latter  is much more difficult for analysis due to the inseparability of variables in the perturbation equations.\n\nThe motion of charged particles is essentially influenced by the magnetic component. Therefore, in the first part of our work, when considering equatorial motions of charged particles, we have taken both the gravitational tidal force and the space-time deformation due to the magnetic field of the torus into consideration.\nThe angular variables in the Hamilton-Jacobi equation in the Preston-Poisson space-time can be decoupled only in the equatorial plane, by which our analysis of particle motion is limited. In particular, we show that the tidal gravitational force as well as the magnetic field strongly enhance the release of the binding energy for a particle spiralling into the black hole. The region of stability is also significantly affected by the tidal force and magnetic field and is qualitatively different for left and right handed rotations. We shall discuss these issues in detail in Sec. IV.\n\n\nThe scalar field equation also does not allow for a complete separation of variables in the general case. Nevertheless, we used the fact that the astrophysically motivated low-laying quasinormal modes are \"localized\" near the peak of the effective potential and therefore \"averaged\" the tidal force near the peak by its value in the maximum. This allowed us to decouple variables and estimate the dominant quasinormal modes for the first and higher multipoles $\\ell \\geq 1$.\nWe have shown that the real oscillation frequencies are considerably suppressed by the tidal force in the vicinity of the magnetic field.\n\n\nThe paper is organized as follows. In Sec. II we summarize some basic relations for the Preston-Poisson space-time. In Sec. III the effective potential for the equatorial motion is deduced on the basis of the Hamilton-Jacobi formalism. Properties of particle's motion are studied in Sec. IV with consideration of various particular cases, such as zero tidal force, zero magnetic field, and the approximation of the neglected \"geometric\" influence of the magnetic field.  Sec. V is devoted to the decoupling of variables in the test scalar field equation for the Preston-Poisson metric with vanishing magnetic field. In Sec. VI, the angular part of the scalar field equation is studied.  Sec. VII presents calculations of quasinormal modes for the above case. In Sec. VIII  we summarize the obtained results and discuss the questions which remain for further study.\n\n\n\n\n\n\n\\section{Preston-Poisson metric}\n\n\n\\begin{figure*}\n\\resizebox{\\linewidth}{!}{\\includegraphics*{Particle-fig1.eps}\\includegraphics*{Particle-fig2.eps}\\includegraphics*{Particle-fig3.eps}}\n\\caption{a) Left panel: Effective potentials $U_{eff}^{2}$ for $L = 3$, $\\mu=0.1$, $\\mathcal{E} = 0.001$, $B=0.01$, $M=1$, $q=-10, 0, +10$ (from  bottom to top), b) middle panel: Effective potentials for $L = 3$, $\\mu=0.1$, $\\mathcal{E} = 0.001, 0.005, 0.01$ (from bottom to top), $B=0.01$, $M=1$, $q=+10$, c) right panel: Effective potentials for $L = 3$, $\\mu=0.1$, $\\mathcal{E} = 0.001, 0.005, 0.01$ (from bottom to top), $B=0.01$, $M=1$, $q=-10$}\\label{fig0}\n\\end{figure*}\n\n\\begin{figure}\n\\resizebox{0.8 \\linewidth}{!}{\\includegraphics*{Particle-fig4.eps}}\n\\caption{Effective potential $U_{eff}^{2}$ for $L = 0, 0.1, 0.4, 0.8$ (from top to bottom on the right side), $M=1$, $\\mu =0.01$, $B=0.01$, $\\mathcal{E} = 0.001$, $q=-5$.}\\label{fig1}\n\\end{figure}\n\n\nThe Preston-Poisson space-time describes the system consisting of a large mechanical structure, such as a giant solenoid or a torus which surrounds a black hole and produces an asymptotically uniform magnetic field of strength $B$. The mass of the structure is $M'$ and the radius is of order $\\sim a$, while the mass of the black hole is $M$.\nThe electromagnetic four-vector has the following form,\n\\begin{equation}\nA^{\\mu} = \\frac{1} {2} B \\phi^{\\mu},\n\\end{equation}\nwhere $\\phi^{\\mu} = (0, 0, 0, 1)$. Further, it is implied that the perturbation created by the magnetic field is small, i.e.\n\\begin{equation}\\label{cond1}\nr^2 B^2 \\ll 1,\n\\end{equation}\nwhere $r$ is the distance from the black hole, and only the interior of the mechanical structure is under consideration $r < a$.\nFor our purposes it is sufficient to study the inner region of the system\nwhich starts at the black hole horizon ($r_h = 2 M$) and ends far from the black hole, still being far from the edges of the torus,\n\\begin{equation}\\label{cond2}\nr_h \\leq r \\ll a.\n\\end{equation}\nThe latter condition can always be fulfilled, because the torus is supposed to be situated in the region of the weak gravitational field of the\nblack hole,\n\\begin{equation}\\label{cond3}\n\\frac{M}{a} \\ll 1.\n\\end{equation}\nAs $r^2 B^2 \\ll 1$ and $r < a$, it is implied that $a^2 B^2 \\ll 1$, though the relative scales of $M\/a$ and $a^2 B^2$ can be arbitrary.\nFirst of all, we are interested in the case\n\\begin{equation}\\label{cond4}\nM\/a \\ll a^2 B^2,\n\\end{equation}\ni.e. in the situation when there is an asymptotic region $M \\ll r < a$, where the influence of the magnetic field on the space-time geometry is negligible.\nThe mechanical structure of mass $M'$ produces the gravitational tidal force, parameterized by  $\\mathcal{E}$, near the black hole,\n\\begin{equation}\\label{cond5}\n\\mathcal{E} \\sim \\frac{M'}{a^3}.\n\\end{equation}\nIn the above approach the tidal force can be much larger, of the same order or much smaller than $B^2$.\n\n\nPreston and Poisson used the light-cone gauge for constructing the perturbed metric, which is adapted  to incoming light cones $v =\nconstant$ that converge toward $r = 0$. For zero tidal force and magnetic field, $v$ takes its Schwarzschild value\n$v = t + r + 2M ln(r\/2M - 1)$.\nIn the ($v$, $r$, $\\theta$, $\\phi$) coordinates the Preston-Poisson metric has the following form\n$$ ds^2 = g_{vv}(r, \\theta) dv^2 + 2 dv dr + g_{v \\theta}(r,\\theta) dv d \\theta +  $$\n\\begin{equation}\\label{PPmetric}\ng_{\\theta \\theta}(r,\\theta) d \\theta^2 +\n g_{\\phi \\phi}(r,\\theta) d\\phi^2,\n\\end{equation}\nwhere\n\\begin{widetext}\n\\begin{equation}\\label{PPcomponents1}\ng_{vv}=-f-\\frac{1}{9}B^2r(3r-8M)-\\frac{1}{9}B^2(3r^2-14Mr+18M^2)(3\\cos^2\\theta-1)    +\\mathcal{E}(r-2M)^2(3\\cos^2\\theta-1)+O[B^4,\\mathcal{E}^2],\n\\end{equation}\n\\begin{equation}\\label{PPcomponents2}\ng_{vr}=1+O[B^4,\\mathcal{E}^2]\\text{,}\n\\end{equation}\n\\begin{equation}\\label{PPcomponents3}\ng_{v\\theta}=\\frac{2}{3}B^2r^2(r-3M)\\sin\\theta\\cos\\theta-2\\mathcal{E} r^2(r-2M)\\sin\\theta\\cos\\theta+O[B^4,\\mathcal{E}^2],\n\\end{equation}\n\\begin{equation}\\label{PPcomponents4}\ng_{\\theta\\theta}=r^2-\\frac{2}{9}B^2r^4+\\frac{1}{9}B^2r^4(3\\cos^2\\theta-1)+B^2M^2r^2\\sin^2\\theta+\\mathcal{E} r^2(r^2-2M^2)\\sin^2\\theta+O[B^4,\\mathcal{E}^2],\n\\end{equation}\n\\begin{equation}\\label{PPcomponents5}\ng_{\\varphi\\varphi}=r^2\\sin^2\\theta-\\frac{2}{9}B^2r^4\\sin^2\\theta+\\frac{1}{9}B^2r^4\\sin^2\\theta(3\\cos^2\\theta-1)-B^2M^2r^2\\sin^4\\theta -\\mathcal{E} r^2(r^2-2M^2)\\sin^4\\theta+O[B^4,\\mathcal{E}^2],\n\\end{equation}\n\\end{widetext}\nand $f(r) = 1- (2 M\/r)$.\nThe above metric was obtained in \\cite{Preston-Poisson} by the perturbation of the Enstein-Maxwell equation in orders of $(\\mathcal{E}, B^2)$.\nThe parameter $\\mathcal{E}$  is the Weyl curvature, that is, the tidal\ngravitational field, of the asymptotic space-time measured\nby an observer co-moving with the black hole in the\nregion $M \\ll r\\ll 1\/B$.\nThe perturbed event horizon is given by\n\\begin{equation}\nr_{h} = 2 M (1+ \\frac{2}{3} M^2 B^2 \\sin^2 \\theta).\n\\end{equation}\nIt is essential\nthat the event horizon is affected by $B$ and not by $\\mathcal{E}$.\n\nAlthough in the above space-time only the dominant order of the magnetic field is considered, the same relation as for the Ernst black holes takes place\n\\begin{equation}\nB \\sim 10^{-21} \\frac{M}{M_{\\odot}} B_{0},\n\\end{equation}\nwhere $M$ and $M_{\\odot}$ are the mass of a black hole and of the sun respectively and $B_{0}$ is the external magnetic field in units of gauss.\nFrom the above relation one can see that the magnetic field deforming the space-time geometry significantly would be as strong as $B = 10^{12} G$ for galactic black holes with  mass $M \\sim 10^{9} M_{\\odot}$.\n\n\\section{Equations of motion and the effective potential}\n\n\nThe transformations\n\\begin{equation}\nv = t + F(r), \\quad F'(r) g_{vv}(r, \\theta) + 1 =0\n\\end{equation}\nreduce the metric (\\ref{PPmetric}) to the following form\n\\begin{equation}\\label{reduced-metric}\nds^2 = - g_{vv}(r, \\theta)^{-1} d r^2 + g_{vv}(r, \\theta) dt^2 + g_{\\phi\\phi}(r, \\theta) d\\phi^2 + C,\n\\end{equation}\nwhere $C$ is the part of the metric which vanishes on the equatorial plane $\\theta = \\pi\/2$ as it contains $d \\theta$. The metric coefficients $g_{vv}$ and $g_{\\phi \\phi}$ depend on the radial and angular coordinates $r$ and $\\theta$, the black hole mass $M$, magnetic field $B$ and the tidal force parameter $\\mathcal{E}$. The above metric is diagonal on the equatorial plane and convenient for further analysis of geodesic motions with the help of the Hamilton-Jacobi formalism.\n\nThe Hamilton-Jacobi equations for the time-like and null geodesics in curved space-time are\n\\begin{equation}\\label{H-J}\n\\frac{1}{2}g^{\\mu \\nu} \\left(\\frac{\\partial S}{\\partial x^{\\mu}} - q A_{\\mu}\\right) \\left(\\frac{\\partial S}{\\partial x^{\\nu}} - q A_{\\nu}\\right) = \\frac{\\partial S}{\\partial s}.\n\\end{equation}\nHere $s$ is an invariant affine parameter.\nThe action is\n$$S = -\\frac{1}{2} \\mu^2 s - E t + L \\phi + S_{r}(r) + S_{\\theta}(\\theta),$$\nand $E$ and $L$ are constants of motion, the particle's energy and angular momentum\nrespectively ($p_{0} = - E$, and $p_{3} = L$) while $p_1 = S_{r}$ and $p_2 = S_{\\theta}$ are functions of $r$ and $\\theta$ respectively.\nIn the general case the angular variables in the Hamilton-Jacobi equations cannot be decoupled.\nNevertheless, one can decouple variables for motion in the purely equatorial plane $\\theta =\\pi\/2$.\n\nIn the equatorial plane  the Hamilton-Jacobi equation for the metric (\\ref{reduced-metric}) takes the form\n$$ -g_{vv}^{-1} \\left(\\frac{\\partial S}{\\partial t}\\right)^{2} + g_{vv} \\left(\\frac{\\partial S}{\\partial r}\\right)^{2} + g_{\\phi \\phi}^{-1} \\left(\\frac{\\partial S}{\\partial \\phi}\\right)^{2}  $$\n\\begin{equation}\n+  q^2 g_{\\phi \\phi}^{-1} A_{\\phi}^2 - 2 q A_{\\phi} g_{\\phi \\phi}^{-1} \\frac{\\partial S}{\\partial \\phi}  = \\mu^2.\n\\end{equation}\n\n\nImplying the normalization $\\tau = \\mu s $, where $\\tau$ is the proper time, the first integrals of motion are\n$$ - \\mu g_{vv} \\frac{d t}{d s}= E, \\quad \\mu \\frac{d r}{d s} = \\pm \\sqrt{(E^2 - U_{eff}^2)}, $$\n\\begin{equation}\n\\mu \\frac{d \\phi}{d s} = \\frac{L + \\frac{1}{2} B q g_{\\phi \\phi}}{g_{\\phi \\phi}}.\n\\end{equation}\n\nThe qualitative analysis of the motion can be made by considering the effective potential\n\\begin{equation}\nU_{eff}^2  =  - \\mu^{2} g_{vv} \\left(1+ \\frac{(L + \\frac{1}{2} B q g_{\\phi \\phi})^2}{\\mu^{2} g_{\\phi \\phi}}\\right).\n\\end{equation}\nThe effective potential for various parameters is shown in fig. \\ref{fig0}. Let us take $M=1$ and consider the situation when, $L \\gg B q$ and at the same time $B \\ll 1$, $\\mathcal{E} \\ll 1$, $\\mu \\ll 1$. Expansion in terms of small $\\mu$, $B$ and $|q| B\/L$ shows that for sufficiently large values of $L$, the effective potential has a local minimum at\n\n\\begin{widetext}\n\\begin{equation}\nr \\approx \\left(\\frac{2 L}{|q| B}\\right)^{1\/2} \\left(1-\\left(1 + \\frac{q B}{L} \\right)\\mathcal{E} + \\left(\\frac{7 q B}{L}^{2}+\n \\frac{10 q B}{L} +3 \\right) \\mathcal{E}^2\\right).\n\\end{equation}\n\\end{widetext}\nFor small values of the angular momentum $L$ the effective potential is a monotonically growing function of $r$.\nFor more general cases, which are not limited by small values of $\\mu$, it may be useful to introduce the angular momentum, charge and energy \"normalized\" by mass:\n$$L \\rightarrow L\/\\mu, \\quad q  \\rightarrow q\/\\mu, \\quad  E  \\rightarrow E\/\\mu.$$\nThen, the effective potential can be written as\n\\begin{equation}\\label{general-eff-pot}\nV_{eff}^2 = \\frac{U_{eff}^2}{\\mu^2}  =  - g_{vv} \\left(1+ \\frac{(L + \\frac{1}{2} B q g_{\\phi \\phi})^2}{ g_{\\phi \\phi}}\\right).\n\\end{equation}\nFurther, we shall use this form of the effective potential and the \"normalized\" values of the energy, angular momentum and charge.\n\nThe effective potential, and consequently the motion of particles, is not the same for opposite charges of particles $q$, that is, the left-hand and right-hand rotations are not equivalent due to the presence of the magnetic field: the rotation which corresponds to the Lorentz force directed from the black hole is called the \\emph{anti-Larmor} rotation, while the rotation with a Lorentz force directed towards the black hole is the \\emph{Larmor} one.  In the next section, we shall show that negative (positive) charge and the positive (negative) angular momentum corresponds to the Lorentz force directed from the black hole (anti-Larmor motion), and, when the signs of the charge and of the momentum coincide, the motion is Larmor.\n\nFrom fig. \\ref{fig0} one can see that the effective potential of positively charged particles is higher than that of the negatively charged ones at all $r$. Thus, the magnetic field\nallows negative particles to penetrate the barrier at energies smaller than those for the Schwarzschild black hole.\n\n\nNow, we are in position to study properties of the equatorial motion of charged particles.\n\n\\section{Properties of particle motion}\n\nHere, we shall concentrate on circular orbits of charged, massive particles.\nThe case of massless and neutral particles was considered, though in a different coordinate system, in \\cite{Konoplya:2006qr}.\n\n\n\\subsection{The case where the influence of the magnetic field on the black hole's geometry vanishes}\n\nWe shall first neglect the deformation of the black hole geometry due to the magnetic field, that is,\nwe shall take $B=0$ in formulas\n(\\ref{PPmetric}) for $g_{vv}$ and $g_{\\phi \\phi}$ and keep $B \\neq 0$ in the generalized derivatives of the Hamilton-Jacobi equations (\\ref{H-J}).\nThis approximation is quite good, taking into account that the magnetic field around the black hole decays quite quickly with $r$, so that\nits overall influence on the geometry can indeed can be neglected for small $q B$.\n\nThe parameters of the circular orbits can be determined from the requirements:\n\\begin{equation}\n\\frac{d r}{d s} =0, \\quad \\frac{d^2 r}{d s^2} =0.\n\\end{equation}\nThese equations mean that on the circular orbits one has\n\\begin{equation}\\label{orbits1}\nV_{eff} = E, \\quad \\frac{d V_{eff}^2}{d r} =0,\n\\end{equation}\nthat is, circular orbits take place only on the turning points of the effective potential. Next, let us consider the two particular cases: 1) vanishing tidal force and 2) vanishing magnetic field.\n\n\\subsubsection{The case $\\mathcal{E} =0$}\nIn the case of vanishing tidal force, $\\mathcal{E} =0$.\nSolutions to the system of equations (\\ref{orbits1}) allow us to find the values of energy $E$ and momentum $L$ as\na function of the rest of the parameters $B$, $M$, $q$ and coordinate $r$,\n\n\\begin{widetext}\n\\begin{equation}\\label{EandL1}\nE^2 = \\frac{(r-2 M)^2 \\left(B^2 q^2 (r-2 M) r^2+2 r-6 M \\mp B q \\sqrt{r^2 \\left(B^2 q^2 r^4+4 M^2 \\left(B^2 q^2 r^2-3\\right)+M \\left(4 r-4 B^2 q^2\n   r^3\\right)\\right)}\\right)}{2 r (r-3 M)^2}\n\\end{equation}\n\\begin{equation}\\label{EandL2}\nL=  \\frac{\\pm \\sqrt{r^2 \\left(B^2 q^2 r^4+4 M^2 \\left(B^2 q^2 r^2-3\\right)+M \\left(4 r-4 B^2 q^2 r^3\\right)\\right)} - B M q r^2}{6 M-2 r}.\n\\end{equation}\n\\end{widetext}\n\nWhen $B=0$, the above expressions (\\ref{EandL1}, \\ref{EandL2}) for the energy and momentum reduce to their Schwarzschild values\n\\begin{equation}\nE^2 = \\frac{(r-2 M)^2}{r (r-3 M)},\\quad L = \\mp \\frac{\\sqrt{M} r}{\\sqrt{r-3 M}}.\n\\end{equation}\n\nNote that $L$, $E$ and $q$ are \"normalized\" by mass $\\mu$.\nUnder positive $q$, the upper sign corresponds to the anti-Larmor motion (A), while the lower sign, to the Larmor one (L).\nNegative $q$ gives opposite correspondence for Larmor and anti-Larmor rotations.\nIndeed, asymptotic behavior at large (far from the black hole) $r$, gives the following asymptotic values for the energy and momentum:\n\\begin{equation}\nE_{A}^2 \\rightarrow 1, \\quad L_{A} \\rightarrow -\\frac{1}{2} B q r^2, \\qquad r \\gg M,\n\\end{equation}\n\\begin{equation}\nE_{L}^2 \\rightarrow  B^2 \\left(3 M^2+r^2\\right) q^2+1, \\qquad r \\gg M,\n\\end{equation}\n\\begin{equation}\nL_{L} \\rightarrow \\frac{1}{2} B q \\left(6 M^2+2 r M+r^2\\right), \\qquad r \\gg M.\n\\end{equation}\n\nThe Larmor motion is a kind of cyclotron rotation in a uniform magnetic field, when\nthe magnetic field is directed to the center of the orbit.\nThe Larmor motion does not require a black hole, so that a black hole just perturbs the existing Larmor orbits.\nOn the contrary to the Larmor rotation, anti-Larmor motion takes place only in the presence of the black hole, when the Lorentz force is directed\noutwards from the center.\n\n\n\\subsubsection{The case $B =0$}\n\nThe case of vanishing magnetic field is more trivial as the motion of charged particles is qualitatively the same as those of the neutral ones.\nThe energy and momentum expanded until the first order in $\\mathcal{E}$ have the following form:\n$$ E^2 = \\frac{\\mathcal{E} \\left(8 M^2-7 r M+2 r^2\\right) (r-2 M)^2}{(r-3 M)^2} + $$\n\\begin{equation}\n\\frac{(r-2 M)^2}{r (r-3 M)} + O(\\mathcal{E}^2),\n\\end{equation}\n$$ L = \\pm \\frac{\\mathcal{E} \\left(-6 M^4-2 r M^3+9 r^2 M^2-5 r^3 M+r^4\\right) r}{2 \\sqrt{M} (r-3 M)^{3\/2}} \\pm $$\n\\begin{equation}\n\\frac{\\sqrt{M} r}{\\sqrt{r-3 M}} + O(\\mathcal{E}^2).\n\\end{equation}\nIt is evident that the above expressions reduce to their Schwarzschild values as $\\mathcal{E} =0$.\n\n\n\\subsubsection{The case $B \\neq 0$ and $\\mathcal{E} \\neq 0$}\n\nFor a more general case $B \\neq 0$ and $\\mathcal{E} \\neq 0$ (but still in the approximation of the neglected geometrical influence of the magnetic field onto the metric), the explicit expressions for the energy and momentum are rather cumbersome, therefore\nfigs. (\\ref{figA1}-\\ref{figA4}) were used to depict the dependence of energy and momentum on the radius of circular orbits.\n\nIn figs. (\\ref{figA1}) and (\\ref{figA3}) one can see that the energy $E$ and absolute values of the momentum $L$  increase when the tidal force  $\\mathcal{E}$ increases. Usually, the effective potential has two extrema: one at the maximum of the potential barrier (corresponding to an unstable orbit and not far from the black hole) and one at the minimum which is further from the black hole (corresponding to a stable orbit). At some minimal values of the angular momentum, these two extrema coincide, which corresponds to the  \\emph{innermost stable circular orbit}.  Thus, the minimum of the absolute value of angular momentum $L$ in figs. (\\ref{figA1}-\\ref{figA4})  is located at the innermost stable circular orbit. As it is shown in figs. (\\ref{figA1}) and (\\ref{figA3}), \\emph{the larger the tidal force $\\mathcal{E}$, the closer the innermost stable orbit to the black hole}. This could be explained by the fact that the tidal force $\\mathcal{E}$ of the surrounding structure is opposite to the gravitational attraction of the black hole, such that the \"effective\" gravitational attraction becomes weaker and particles can come closer to the black hole yet still remain in a stable orbit.\n\nThe dependence on charge $q$ is more complicated as the situation depends on the direction of the Lorentz force. Thus, the anti-Larmor motion (when the Lorentz force is directed outward from the black hole) corresponds to positive $q$ and negative $L$ or to negative $q$ and positive $L$.\nThe energy and momentum for various $q$ are shown in figs. (\\ref{figA2}) and (\\ref{figA4}). The positive and negative angular momenta show opposite dependence of the energy on charge $q$: for negative $L$, lines of energy for negative  $q$ always lay above those for positive $q$, while for positive $L$ the situation is reversed, that is, negative charges correspond to smaller $E$ than positive ones.\nThe same dependence takes place for the absolute value of the angular momenta. In summary, \\emph{the energy $E$ and the absolute value of the angular momentum $\\mid L \\mid$ grow with the increasing absolute value of charge $\\mid q \\mid$ for Larmor orbits and decreases for anti-Larmor ones}.\nAs the magnetic field $B$ is coupled to the charge $q$, the dependence on $B$ is qualitatively similar to the dependence on $q$. As the dependence on the magnetic field for neutral particles was considered earlier in \\cite{Konoplya:2006qr}, we focused on the dependence of various quantities on the charge and the tidal force. For the relatively small  $q$ (when $q B$ is still small) and $\\mathcal{E}$, considered in figs. (\\ref{figA1}) and (\\ref{figA4}) and in Tables I and II,\n one cannot tell the difference in the location of the innermost stable Larmor and anti-Larmor orbits. That is why we have only one set of values of $r$ in Tables I and II: the corrections to those values are of the order O($B^4$, $\\mathcal{E}^2$, $B^2 \\mathcal{E}$). This is certainly not the case for large $q$ for which the location of Larmor and anti-Larmor orbits are quite different. We shall consider the regime of large $q$ in the next subsection.\n\nIn order to learn how much energy could be extracted from a particle slowly spiralling toward the center, it is necessary to know what the energy of the particle at the innermost stable circular orbit is.  For this purpose it is convenient to use a function called the \\emph{binding energy}, which is the amount of energy released by the particle going from the stable circular orbit located at $r$ to the innermost stable circular orbit $r_{is}$.\nThus, the binding energy (in percent) is\n\\begin{equation}\n\\text{Binding Energy}  = 100 \\frac{E(r)- E(r_{is})}{E(r)}.\n\\end{equation}\n\nThe binding energy can tell us how much energy the matter (for instance an accretion disk) will release before plunging into the black hole.\nFor the Schwarzschild black hole (taking $M=1$) the energy on the innermost stable orbit is $E(r_{is}) = \\sqrt{8\/9}$, so that for a particle coming from\nthe asymptotic region ($E =1$), the binding energy is about $5.7$ percent. For the Preston-Poisson space-time\nthere is no asymptotically flat region, so that, the binding energy released at the transition from a given stable orbit $r$ (instead of infinity for the Schwarzschild case) to the innermost stable orbit $r_{is}$ is meaningful.\nFigs. (\\ref{figBE1}) and (\\ref{figBE2})\ndepict how the binding energy depends on the charge $q$, tidal force $\\mathcal{E}$ and magnetic field $B$  as a function of $r$.\n\n\nIn fig. (\\ref{figBE1}) the line for negative $L$ lay above the line for positive $L$, if $q$ is negative, and below, if $q$ is positive. Thus, the Larmor orbits have greater binding energy than the corresponding anti-Larmor ones.\nComparison with the blue curve for the Schwarzschild case ($B = \\mathcal{E} = 0$) in fig. (\\ref{figBE1}) demonstrates  that the magnetic field and its coupling with the charge strongly enhance the release of the binding energy. This property was also observed in \\cite{Esteban}\nfor the black hole in the absence of the tidal force.\nThe tidal force also strongly increases the binding energy, as can be seen from comparison with the Schwarzschild case in fig. (\\ref{figBE2}).\nThere, the negative momenta correspond to a slightly smaller binding energy than the positive one in fig. (\\ref{figBE2}), where the small difference is due to the small $B q$. Thus, the Larmor and anti-Larmor orbits are almost indistinguishable in the regime of tiny $B q$.\nIn the next subsection we discuss the regime of large charge, when $B q$ is not necessarily small.\n\n\n\\begin{table}\n\\begin{tabular}{|c|c|c|c|c|c|}\n  \\hline\n \n  q & r & $L_{+}$ & $L_{-}$ & $E_{-}$ & $E_{+}$ \\\\\n  \\hline\n  -10 & 5.2745 & 3.64979 & -3.78766 &  0.99508 & 0.95882 \\\\\n  -3 & 5.2841 & 3.69291  & -3.73427 &  0.98163 & 0.97076 \\\\\n  -1 & 5.2849 & 3.70624  & -3.72003 &  0.97792 & 0.97432 \\\\\n  0 & 5.2850 & 3.71308   & -3.71308 &  0.97612 & 0.97612 \\\\\n  1 & 5.2849 & 3.72003   & -3.70624 &  0.97432 & 0.97794\\\\\n  3 & 5.2841 & 3.73427   & -3.69291 &  0.97432 & 0.97794 \\\\\n  10 & 5.2753 & 3.78766  & -3.64979 &  0.95882 &  0.99508 \\\\\n  \\hline\n\\end{tabular}\n\\caption{Energies $E_{+}$ and $E_{-}$ and the angular momenta $L_{+}$ and  $L_{-}$  at the innermost stable circular orbit\nfor various values of charge $q$; $B=\\mathcal{E}=10^{-3}$, $M=1$}\n\\end{table}\n\n\\begin{table}\n\\begin{tabular}{|c|c|c|c|c|c|}\n  \\hline\n \n $\\mathcal{E}\/10^{-3}$ & r&  $L_{+}$ & $L_{-}$ &  $E_{-}$ & $E_{+}$ \\\\\n  \\hline\n  0 & 5.9997 & 3.47018 & -3.45818  & 0.94119 & 0.94445 \\\\\n  $0.5 $ & 5.5277 & 3.60380 & -3.59086  & 0.95811 & 0.96156 \\\\\n  $1$ & 5.2849 & 3.72003 & -3.70624  & 0.97432 & 0.97794 \\\\\n  $1.5$  & 5.1213 & 3.82919 & -3.81458  & 0.99046 & 0.99425 \\\\\n  $2$  & 4.9982 &  3.93531 & -3.91988 & 1.00684 & 1.01080 \\\\\n  $2.5$  & 4.8994 & 4.04064 &  -4.02438 & 1.02366 & 1.02779 \\\\\n  $3$  & 4.8168 & 4.14672 & -4.1296  & 1.04105 &  1.04537 \\\\\n  \\hline\n\\end{tabular}\n\\caption{Energies $E_{+}$ and $E_{-}$ and the angular momenta $L_{+}$ and  $L_{-}$ at the innermost stable circular orbit for various values of charge $\\mathcal{E}$; $B=10^{-3}$, $q=1$, $M=1$}\n\\end{table}\n\n\n\\begin{figure*}\n\\resizebox{0.7 \\linewidth}{!}{\\includegraphics*{EminusK.eps}\\includegraphics*{LminusK.eps}}\n\\caption{Energy $E_{-}$ (left) and momentum  $L_{-}$ (right) as a function of circular orbit radius for various values of  $\\mathcal{E} =0, 1\/300, 1\/200, 1\/100$ (from bottom to top for energy $E_{-}$ and from top to bottom for the negative angular momenta $L_{-}$), $B=1\/100$, $M=1$, $q=1$.}\\label{figA1}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\resizebox{0.7 \\linewidth}{!}{\\includegraphics*{EminusQ.eps}\\includegraphics*{LminusQ.eps}}\n\\caption{Energy $E_{-}$ (left) and momentum  $L_{-}$ (right) as a function of circular orbit radius for various values of the particle charge $q = -5$(green), $0$(red), $5$ (blue) $\\mathcal{E} =1\/100$, $B=1\/100$, $M=1$.}\\label{figA2}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\resizebox{0.7 \\linewidth}{!}{\\includegraphics*{EplusK.eps}\\includegraphics*{LplusK.eps}}\n\\caption{Energy $E_{+}$ (left) and momentum  $L_{+}$ (right) as a function of circular orbit radius for various values of  $\\mathcal{E} =0, 1\/300, 1\/200, 1\/100$ (from bottom to top), $B=1\/100$, $M=1$, $q=1$.}\\label{figA3}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\resizebox{0.7 \\linewidth}{!}{\\includegraphics*{EplusQ.eps}\\includegraphics*{LplusQ.eps}}\n\\caption{Energy $E_{+}$ (left) and momentum  $L_{+}$ (right) as a function of circular orbit radius for various values of the particle charge $q = -5$(green), $0$(red), $5$ (blue), $\\mathcal{E} =1\/100$, $B=1\/100$, $M=1$.}\\label{figA4}\n\\end{figure*}\n\n\n\n\\begin{figure*}\n\\resizebox{0.9 \\linewidth}{!}{\\includegraphics*{BindingEnergy1.eps}\\includegraphics*{BindingEnergy2.eps}\\includegraphics*{BindingEnergy3.eps}}\n\\caption{The binding energy as a function of $r$ for $q = -10$ (left), $q=0$ (middle) and $q=10$ (right), $\\mathcal{E} = B =10^{-3}$, $M=1$; red is for negative $L$ and green is for positive $L$, blue line is for the Schwarzschild orbit.}\\label{figBE1}\n\\end{figure*}\n\n\\begin{figure*}\n\\resizebox{0.9 \\linewidth}{!}{\\includegraphics*{BindingEnergy4.eps}\\includegraphics*{BindingEnergy5.eps}\\includegraphics*{BindingEnergy6.eps}}\n\\caption{The binding energy as a function of $r$ for $q = 1$, $\\mathcal{E}= 0.5 \\times 10^{-3}$ (left), $\\mathcal{E}=1.5 \\times  10^{-3}$ (middle) and $\\mathcal{E}=3 \\times  10^{-3}$ (right), $B =10^{-3}$, $M=1$;  red is for negative $L$ and green is for positive $L$, blue line is for the Schwarzschild orbit.}\\label{figBE2}\n\\end{figure*}\n\n\n\n\\subsection{The general case with all non-vanishing parameters}\n\nWhen the charge $q$ is large enough, so that the term $B q$ is no longer small, one cannot safely neglect the geometric influence of the magnetic field, because even subdominant terms containing $B q$ can be quite large in comparison, for instance, with the tidal force corrections $\\sim \\mathcal{E}$. Thus, we considered the full metric (\\ref{PPmetric}) with a nonzero magnetic field both in the metric coefficients and in the Hamilton-Jacobi equation for large $q$. As shown in Table III, the locations of Larmor and anti-Larmor orbits already differ a lot\nfor $B q \\sim 1$. At the same time, the data for $B q \\sim 10^{-2}$ (the first line in the Table III) still almost coincide with\nthe approximation of the neglected \"geometric contribution\" of $B$ considered above. Another distinction from\nthe above small charge approximation is that, while for small  $B q$ the energy at the innermost stable orbit is only slightly smaller for the anti-Larmor orbits than for the Larmor ones (Tables I and II), for moderate $B q$, this difference is very large (see last line in Table III) and the Larmor orbits may have one order larger energy and momentum than the anti-Larmor ones.\n\n\nIn the case of large $q$, the binding energy of Larmor orbits is not always greater than the anti-Larmor ones as it depends on a stable orbit from which the particle plunges into the black hole. For more distant stable orbits (see fig. (\\ref{figBE2})) the binding energy of the anti-Larmor orbits can be larger than the corresponding Larmor ones.\nIf one remembers that the Larmor motion is  a cyclotron rotation in a uniform magnetic field perturbed by a black hole, then, it becomes clear that in the case of large $B q$ the \"perturbation\" by the black hole is relatively small, so that the rotation is almost purely cyclotronic, especially at a relatively large distance from the black hole.\n\n\n\\begin{table}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n  \\hline\n \n  q & $r_{+}$ & $r_{-}$ & $L_{+}$ & $L_{-}$ & $E_{-}$ & $E_{+}$ \\\\\n  \\hline\n  -10 & 5.27452 & 5.27527  & 3.64981 & -3.78768 & 0.995088  & 0.95882 \\\\\n  -50 & 5.04704 &  5.11032  & 3.49905  & -4.19065 & 1.08802  & 0.90389 \\\\\n  -100 & 4.86491 & 4.61837  & 3.46307  & -4.87639 & 1.23937  & 0.85679 \\\\\n  -1000 & 2.74943 & 4.32074   & 5.20535   & -24.8736 &  5.90001 & 0.59092  \\\\\n  \\hline\n\\end{tabular}\n\\caption{Energies $E_{+}$ and $E_{-}$ and the angular momenta $L_{+}$ and  $L_{-}$  at the innermost stable circular orbit\nfor various large values of charge $q$; $B=\\mathcal{E}=10^{-3}$, $M=1$}\n\\end{table}\n\n\\begin{figure}\n\\resizebox{0.8 \\linewidth}{!}{\\includegraphics*{BindingEnergy7.eps}}\n\\caption{The binding energy as a function of $r$ for $q = -1000$, $\\mathcal{E} = B =10^{-3}$, $M=1$, red line corresponds to the orbit with a negative momentum (Larmor for $q <0$), green corresponds to a positive (anti-Larmor)}\\label{figBE3}\n\\end{figure}\n\nFinally, let us discuss the radial stability of the considered circular orbits. The condition of the radial stability can be written\nas:\n\\begin{equation}\n\\frac{\\partial^2 U_{eff}}{\\partial r^2} > 0 \\quad \\text{- stability condition.}\n\\end{equation}\nFrom numerical study of the above inequality (when $U_{eff}$ is given by its most general expression (\\ref{general-eff-pot})), we conclude that at sufficiently large $|q| B$, the region of stability of anti-Larmor orbits\ncan approach the event horizon,\n\\begin{equation}\nr > r_{h} \\quad \\text{- anti-Larmor, high $|q| B$,}\n\\end{equation}\nwhile Larmor orbits are only stable for $r \\gtrapprox 4. 2 M$ for zero tidal force $\\mathcal{E}=0$, approaching $r \\gtrapprox 3.6 M$ at $\\mathcal{E}=1\/50 M$,\n\\begin{equation}\\label{38}\nr \\gtrapprox  \\left(\\begin{array}{c}\n                                4.2 M \\text{($\\mathcal{E}=0$)} \\\\\n                                3.6 M \\text{($\\mathcal{E}=1\/50 M$)}\\\\\n                              \\end{array}\n                            \\right)\n\\text{- Larmor, high $|q| B$.}\n\\end{equation}\n\nThis is in good agreement with the  $r \\gtrapprox 4. 3 M$ limit observed in \\cite{Galtsov-book} for the Ernst solution in the absence of the tidal force. The small difference between (\\ref{38}) and \\cite{Galtsov-book} is apparently due to the neglected\ncontribution of orders $B^3$  and higher in the Preston-Poisson metric.\nThe new feature here is that the tidal force can considerably expand the region of stability of the Larmor orbits making them closer to the black hole horizon. This can also be seen from the Table II.\n\n\\section{Scalar field perturbations in the Preston-Poisson space-time}\n\nThe aim of our study of a test scalar field in the Preston-Poisson space-time is to find the proper oscillation frequencies (quasinormal modes) which dominate in the response to the perturbation at late time.\nAs shown in \\cite{Konoplya:2006rv}, the quasinormal modes of black holes, when considered in the astrophysical context, are essentially independent of the behavior of the master wave equation far from the black hole. The explanation is straightforward: quasinormal modes are poles of the reflection coefficient of the scattering process which occurs near the peak of the effective potential, so that the low-laying, dominating modes are \"localized\" near the maximum of the effective potential. For the Schwarzschild solution this happens at $r = 3 M$, while for the Preston-Poisson space-time this value is only slightly corrected. The wave equation does not have physical meaning at $r=\\infty$, because for astrophysical processes of this kind, \"infinity\" effectively is situated at the distance  which is much larger than the radius of the black hole. In our case, \"infinity\" is at least quite a few times larger than the black hole radius and still far from the edge of the torus, $r_h \\ll r \\ll a$. We have considered the same spatial region when analyzing particle's motion.\n\n\nFor a test neutral scalar field, influence of the magnetic field $B$ can usually be neglected, because the energy density of the magnetic field is much smaller than that of the gravitational one.\nWhen $B=0$, the coefficients of the Preston-Poissin metric (\\ref{PPmetric}) are simplified as follows:\n\\begin{align}\\label{PPcoef-simpl}\ng_{vv}&=-f+\\mathcal{E}(r-2M)^2(3\\cos^2-1),\\\\\ng_{vr}&=1,\\\\\ng_{v\\theta}&=-2\\mathcal{E} r^2(r-2M)\\sin\\theta\\cos\\theta,\\\\\ng_{\\theta\\theta}&=r^2+\\mathcal{E} r^2 (r^2-2M^2)\\sin^2\\theta,\\\\\ng_{\\varphi\\varphi}&=r^2\\sin^2\\theta-\\mathcal{E} r^2(r^2-2M^2)\\sin^4\\theta. \\label{PPcoef-simpl1}\n\\end{align}\n\nThe Klein-Gordon equation in a curved space-time has the following form\n\\begin{equation*}\n\\square\\Psi= \\frac{\\partial}{\\partial x^{\\mu}}\\left(g^{\\mu\\nu}\\sqrt{-g}\\frac{\\partial \\Psi}{\\partial x^{\\nu}}\\right)=0.\n\\end{equation*}\n\n\\begin{figure*}\n\\resizebox{0.7 \\linewidth}{!}{\\includegraphics*{L1m0Kvarious.eps}\\includegraphics*{L1m1Kvarious.eps}}\n\\caption{Effective potentials for a) $\\ell =1$, $m=0$, $\\mathcal{E} = 0, 1\/10, 2\/10$ (from the top to bottom), b)\n$\\ell =1$, $m=1$, $\\mathcal{E} = 0, 1\/10, 2\/10$ (from bottom to top). The potential barrier is raised by the tidal force for non-zero $m = \\pm 1$ and is lowered at the lowest modes ($m=0$).}\\label{fig1}\n\\end{figure*}\n\nIn the general case, the variables in the Klein-Gordon equation cannot be decoupled for the above metric.\nHowever, a further assumption can remedy the situation. We shall assume, that as the torus is situated\nfar from the black hole $M \\ll a$, the tidal force $\\mathcal{E}$, being relatively small, acts almost homogeneously in\na small region where the low laying quasinormal modes are \"localized\", that is near the peak of the effective potential\n\\begin{equation}\\label{rmax}\nr = r_{max} = p M, \\quad p =\\frac{8}{3}, \\quad \\ell = 0,\n\\end{equation}\n$$ p = \\frac{3}{2} + \\frac{\\sqrt{9 + \\ell (\\ell +1) (14 + 9 \\ell (\\ell +1))}- 3}{2 \\ell (\\ell +1)}, \\quad \\ell \\neq 0.$$\nIn other words, the tidal force at the potential peak does not change much when one slightly moves\naway from the peak $r \\approx r_{max}$.\n Thus, we assume that $M \\ll a$ holds \"with a margin\", being also  $r_{max} \\ll a$. In this case, being interested only in the region near the peak, we can expand \\emph{the perturbed part} of the Preston-Poisson metric (\\ref{PPcoef-simpl}-\\ref{PPcoef-simpl1}) (i.e. the terms containing $\\mathcal{E}$) in powers of ($\\mathcal{E}, (r- r_{max})$).\nFurther we neglect the  corrections to the Schwarzschild metric of order $\\mathcal{E} (r - r_{max})$ and higher,\nwhich means that, as we assumed, the tidal force barely changes near the peak.\nAlthough such an approximation does not allow us to make accurate calculations of quasinormal spectrum, we can still\nestimate the dominating modes numerically. In reality the tidal force is smaller on the left (at the black hole side) of the peak\nand larger on the right of it, so that the used expansion in powers of  $(\\mathcal{E}, (r - r_{max}))$ is a kind of \"averaging\" of\nthe tidal force in the small region of the localization of the dominant modes.\n\nIn the above approximation the metric can be reduced to the diagonal form\n\\begin{equation}\\label{diagonal1}\nds^2=g_{vv}dt^2-\\frac{1}{g_{vv}}dr^2+g_{\\theta\\theta}d\\theta^2+g_{\\varphi\\varphi}d\\varphi^2,\n\\end{equation}\nthrough the following transformations,\n\\begin{equation}\nv = t + F(r), \\quad F'(r) g_{vv}(r, \\theta) + 1 =0,\n\\end{equation}\nwhere\n\\begin{equation}\ng_{vv}=-f,\\qquad \\qquad g_{v\\theta}=0.\n\\label{equation_order_correction}\n\\end{equation}\nThe determinant of the diagonal metric (\\ref{diagonal1}) is\n\\begin{equation*}\n\\sqrt{-g}=\\sqrt{g_{\\theta\\theta}g_{\\varphi\\varphi}}\\approx r^2\\sin \\theta+O(\\mathcal{E}^2).\n\\end{equation*}\n\nDue to the Killing vectors in $t$ and $\\phi$, the stationary ansatz for the perturbation is implied:\n\\begin{equation*}\n\\Psi(t,r,\\theta,\\varphi)=e^{im\\varphi-i\\omega t} {\\widetilde{\\Phi}}(r,\\theta).\n\\end{equation*}\nThe wave equation can be written as follows\n\\begin{align*}\n\\omega^2 \\left(1-\\frac{2M}{r}\\right)^{-1}r^2 \\widetilde{\\Phi}+\\frac{\\partial}{\\partial r}\\left(\\left(1-\\frac{2M}{r}\\right)r^2 \\frac{\\partial\\widetilde{\\Phi}}{\\partial r}\\right)+ S \\widetilde{\\Phi}=0,\n\\end{align*}\nwhere\n\\begin{align*}\nS\\widetilde{\\Phi}=\\frac{1}{\\sin\\theta} \\left( \\frac{\\partial}{\\partial\\theta}\\left(\\sqrt{\\frac{g_{\\varphi\\varphi}}{g_{\\theta\\theta}}}\\frac{\\partial\\widetilde{\\Phi}}{\\partial\\theta}\\right)\n       +(-m^2)\\widetilde{\\Phi}\\sqrt{\\frac{g_{\\theta\\theta}}{g_{\\varphi\\varphi}}} \\right).\n\\end{align*}\n\nExpansion of $\\sqrt{\\frac{g_{\\theta\\theta}}{g_{\\varphi\\varphi}}}$ and $\\sqrt{\\frac{g_{\\varphi\\varphi}}{g_{\\theta\\theta}}}$ in powers of ($\\mathcal{E},(r-r_{max})$) yields\n\\begin{align*}\n\\sqrt{\\frac{g_{\\varphi\\varphi}}{g_{\\theta\\theta}}}&=\\sin\\theta- (p^2 -2)\\mathcal{E} M^2 \\sin^3\\theta+O(\\mathcal{E}(r-r_{max}),\\mathcal{E}^2),\\\\\n\\sqrt{\\frac{g_{\\theta\\theta}}{g_{\\varphi\\varphi}}}&=\\frac{1}{\\sin\\theta}+(p^2 -2) M^2\\mathcal{E} \\sin\\theta +O(\\mathcal{E}(r-r_{max}),\\mathcal{E}^2).\n\\end{align*}\n\n\nThen, the angular part can be rewritten as\n$$\\frac{1}{\\sin\\theta}\\frac{\\partial}{\\partial\\theta}\\left(\\sin\\theta\\frac{\\partial\\widetilde{\\Phi}}{\\partial\\theta}\\right)-\\frac{(p^2 -2)\\mathcal{E} M^2}{\\sin\\theta}\\frac{\\partial}{\\partial \\theta}\\left(\\sin^3\\theta\\frac{\\partial\\widetilde{\\Phi}}{\\partial\\theta}\\right) $$\n\\begin{equation}\\label{angular}\n-m^2 \\left(\\frac{1}{\\sin^2\\theta}+(p^2 -2)\\mathcal{E} M^2\\right)\\widetilde{\\Phi}=S\\widetilde{\\Phi}.\n\\end{equation}\n\nIntroducing the ``tortoise'' coordinate $r^*$, which is defined by the relation $dr^*=\\frac{dr}{f(r)}$, together with a new wave function $\\widetilde{\\Phi}=\\frac{\\Phi}{r}$, one can reduce the wave equation to the Schrodinger wavelike form with an effective potential $V(r, S)$,\n\\begin{equation}\\label{wavelike}\n\\frac{d^2\\Phi}{{dr^*}^2}+\\left(\\omega^2-V(r, S)\\right)\\Phi=0,\n\\end{equation}\nwhere\n\\begin{equation}\\label{potential}\nV(r,S)= \\left(1-\\frac{2 M}{r}\\right)\\left(\\frac{S}{r^2} - \\frac{2M}{r^3} \\right).\n\\end{equation}\n\nThe latter effective potential is identical to that of the Schwarzschild, up to the different\nangular eigenvalues $S$, which will be calculated in the next section.\n\n\\section{Angular wave equation}\n\nThe eigenvalues of the angular equation (\\ref{angular}) can be calculated via two methods: the convergent and accurate Forbenius method and the\nmethod of expansion into the Associated Legendre Polynomials. Here we shall imply that the maximum of the effective potential is situated at its Schwarzschild distance $r_{max}$ given by (\\ref{rmax}) for each value of the multipole number $\\ell$, i.e. we shall neglect tiny displacement of the peak induced by the tidal force $\\mathcal{E}$. This approximation is fully justified as the inclusion of ($\\mathcal{E}$, $B$) corrections to the maximum's location contributes only at the subdominant orders, as discussed in Sec. VII.\n\n\n\\subsection{Frobenius Method}\n\nEquation (\\ref{angular}) is reduced to the Heun's equation by setting $x=\\sin^2\\theta$ and thus giving\n\\begin{equation}\n\\frac{d^2 \\widetilde{\\Phi}}{dx^2}+\\frac{2-3 x-4 x \\sigma +5 x^2 \\sigma }{2 (x-1) x (\\sigma x-1 )}\\frac{\\widetilde{\\Phi}}{dx}-\\frac{m^2 + S x+m^2 x \\sigma }{4 (x-1) x^2 (\\sigma x-1 )} \\widetilde{\\Phi}=0,\n\\label{Equation_wave_equation_angular_x_sine}\n\\end{equation}\nwhere\n$$\\sigma= (p^2 -2) \\mathcal{E}  M^2.$$\nThe above expression for $\\sigma$ will be considered as a constant in the further calculations, which is justified by the approximation (\\ref{rmax}).   In this way, we are only able to find even eigenfunctions for $\\theta \\rightarrow -\\theta$. The odd eigenfunctions must be found  separately.\nSetting $\\widetilde{\\Phi}=x^{\\frac{m}{2}}\\psi(x)$, one can transform equation (\\ref{Equation_wave_equation_angular_x_sine}) into the Heun's equation\n$$ \\frac{d^2 \\psi}{dx^2}+\\left(\\frac{\\gamma}{x}+\\frac{\\delta}{x-1}+\\frac{\\kappa}{x-a_H}\\right)\\frac{d \\psi}{dx}+ $$\n\n\\begin{equation}\n\\frac{\\alpha\\beta x-q}{x(x-1)\\left(x-a_H\\right)}\\psi=0,\n\\end{equation}\nwith\n\\begin{align*}\n\\gamma&=m+1,\\qquad\\delta=\\frac{1}{2},\\qquad\\kappa=1,\\qquad \\\\\nq&=\\frac{m^2+m+\\tilde{S}+2m^2\\sigma+2m\\sigma}{4\\sigma}, \\\\\n \\alpha&=\\frac{m}{2},\\qquad\\beta=\\frac{m+3}{2},\\qquad a_H=\\frac{1}{\\sigma},\n\\end{align*}\nand\n\\begin{equation*}\n\\omega=\\gamma+\\delta-1=\\alpha+\\beta-\\kappa=m+\\frac{1}{2}.\n\\end{equation*}\nThe solution to the Heun's equation can be expanded as follows\n\\begin{equation*}\n\\psi(x)=\\sum _{n=-\\infty }^{+\\infty }c_n y_n(x),\n\\end{equation*}\nif $c_n$ satisfies the recurrence relations\n\\begin{equation*}\nF_n c_{n-1}+G_n c_n+H_n c_{n+1}=0,\n\\end{equation*}\nwhere\n\\begin{widetext}\n\\begin{align*}\nF_n&=\\frac{(n+\\alpha-1)(n+\\beta-1)(n+\\gamma-1)(n+\\omega -1)}{(2n+\\omega -2)(2n+\\omega -1)},\\\\\nG_n&=-\\frac{\\kappa n(n+\\omega )(\\gamma-\\delta)+(n(n+\\omega )+\\alpha\\beta)(2n(n+\\omega )+\\gamma(\\omega -1))}{(2n+\\omega +1)(2n+\\omega -1)}+n(n+\\omega)a_H+q,\\\\\nH_n&=\\frac{(n+1)(n+\\omega -\\alpha+1)(n+\\omega -\\beta+1)(n+\\delta)}{(2n+\\omega +2)(2n+\\omega +1)},\n\\end{align*}\n\\end{widetext}\nwith $n=\\frac{\\ell-m}{2}$.\\\\\nSince the coefficients of $c_{n-1}$, $c_n$ and $c_{n+1}$ vanish independently, the eigenvalue can be obtained by taking $G_n=0$,\n\\begin{equation}\\label{S-formula}\nS=\\textbf{-}\\frac{\\ell (1+\\ell) \\left(3-4 \\ell-4 \\ell^2+2 \\ell \\sigma +2 \\ell^2 \\sigma -6 m^2 \\sigma \\right)}{(-1+2 \\ell) (3+2 \\ell)}.\n\\end{equation}\n\nThe above formula has been derived here for the \"even\" modes, i.e. modes with even values of $\\ell - |m|$, though, using the expansion\nin terms of the Associated Legendre Polynomial, we show that this formula is valid also for odd eigenfunctions.\n\n\n\\subsection{Expansion in terms of the Associated Legendre Polynomial}\n\nIntroducing a new variable $x$ in a different way,  $x=\\cos\\theta$, equation (\\ref{angular}) can be written as\n$$ \\frac{d^2\\widetilde{\\Phi}}{dx^2}+\\frac{2 x \\left(2 x^2 \\sigma -2 \\sigma +1\\right)}{(x-1) (x+1) \\left(x^2 \\sigma -\\sigma +1\\right)}\\frac{d\\widetilde{\\Phi}}{dx} $$\n\\begin{equation}\\label{54}\n+\\frac{-m^2 x^2 \\sigma +  S x^2-m^2 \\sigma -m^2-S}{(x-1)^2 (x+1)^2 \\left(x^2 \\sigma -\\sigma +1\\right)} \\widetilde{\\Phi}=0.\n\\end{equation}\nSince the Preston-Poisson solution is accurate through the order of $\\mathcal{E}$, one can expand the angular eigenfunction function and the corresponding eigenvalues in terms of $\\sigma = (p^2 -2) \\mathcal{E} M^2$ as\n\\begin{align}\n\\widetilde{\\Phi}&=P^{\\ell}_{m}\n+\\sigma\\left(\\sum_{\\ell\\geq|m|}C^{\\ell}_{m}P^{\\ell}_{m}\\right)+O[\\sigma^2],\n\\label{Equation_expansion_wave_equation}\\\\\nS&=\\ell(\\ell+1)+\\sigma S',\n\\label{Equation_expansion_eigenvalue}\n\\end{align}\nwhere $\\ell =0, 1, 2, .....$ and $m = - \\ell, -\\ell +1,.....,\\ell -1, \\ell$.\n\nSubstituting equations (\\ref{Equation_expansion_wave_equation}) and (\\ref{Equation_expansion_eigenvalue}) into the wave equation (\\ref{54}), one can obtain the $S'$ values, which are summarized for lower $\\ell$ and $m$\nin Table \\ref{Table_Eigenvalue}.\n\\begin{table}[h]\n\\centering\n    \\begin{center}\n        \\begin{tabular}{|c|c|ccc|ccccc|}\n        \\hline\n        {\\scriptsize$\\ell=$}&{\\scriptsize$0$}&{\\scriptsize$ $}&{\\scriptsize$1$}&{\\scriptsize$ $}&{\\scriptsize$ $}&{\\scriptsize$ $}&{\\scriptsize$2$}&{\\scriptsize$ $}&{\\scriptsize$ $}\\\\\n        \\hline\n        {\\scriptsize$m=$}&{\\scriptsize$0$}&{\\scriptsize$-1$}&{\\scriptsize$0$}&{\\scriptsize$1$}&{\\scriptsize$-2$}&{\\scriptsize$ -1$}&{\\scriptsize$0$}&{\\scriptsize$1$}&{\\scriptsize$2$}\\\\\n        \\hline\n        {\\scriptsize$S'=$}&{\\scriptsize$0$}&{\\scriptsize$\\frac{4}{5}$}&{\\scriptsize$-\\frac{8}{5}$}&{\\scriptsize$\\frac{4}{5}$}&{\\scriptsize$\\frac{24}{7}$}&{\\scriptsize$-\\frac{12}{7}$}&{\\scriptsize$-\\frac{24}{7}$}&{\\scriptsize$-\\frac{12}{7}$}&{\\scriptsize$\\frac{24}{7}$}\\\\\n        \\hline\n        \\end{tabular}\n    \\end{center}\n\\caption{The eigenvalues of $\\ell=0$,$1$ and $2$.}\n\\label{Table_Eigenvalue}\n\\end{table}\n\nTable \\ref{Table_Eigenvalue} shows that the formula (\\ref{S-formula}) is indeed valid for the low laying even and odd modes.\nWith this data for the angular eigenvalues at hand, we can start numerical calculations of the quasinormal modes.\n\nIn principle, one can have a slightly better approximation of the estimation of the angular eigenvalue $S$, if the tiny shifts of the value of the radial coordinate $r$ in the maximum of the effective potential due\nto nonzero values of $\\mathcal{E}$ are taken\ninto account, when calculating the tidal force in the potential maximum. This means that instead of Eq. (\\ref{rmax}) for $p$, one could use a general solution for the maximum of the effective potential (\\ref{potential}):\n\\begin{equation}\\label{p-full}\np = -3 + 3 S + \\sqrt{(9 + S (14 + 9 S))\/2 S}\n\\end{equation}\nComparison of the QNMs computed with the Schwarzschild values of $p$ and with (\\ref{p-full}) for lower values of $\\ell$,\nreveal a very small difference of about $0.01$ percent, which cannot be distinguished in figs. (\\ref{fig2}) and (\\ref{fig3}).\nIn the next section, the more accurate value given in (\\ref{p-full}) shall be used.\n\n\n\n\\section{Estimations of the quasinormal modes}\n\nAn essential moment in the determination of the quasinormal modes is the agreed notion of asymptotical region, \"infinity\", which is simply a distance\nmuch larger than the black hole radius. Once the near horizon and far regions are defined, we can formulate the boundary conditions for quasinormal modes. Quasinormal modes are solutions to the wavelike equation (\\ref{wavelike}) which are purely ingoing waves at the event horizon and purely outgoing waves at the far asymptotic region:\n$$\n\\Phi \\sim e^{- i \\omega r^{*}}, \\quad r^{*} \\rightarrow - \\infty, \\quad\n$$\n\\begin{equation}\n\\Phi \\sim e^{+ i \\omega r^{*}}, \\quad r^{*} \\rightarrow \\quad far \\quad asymptotic \\quad region\n\\end{equation}\n\nThereby, no incoming waves are allowed either from the horizon or from the far region. This means that quasinormal modes correspond to the proper oscillations of the black hole's response to the external perturbation at late time, that is, when the source of the initial perturbation does not act anymore. In other words, the perturbation is considered as a \"momentary\".\n\n\\begin{figure*}\n\\resizebox{0.7 \\linewidth}{!}{\\includegraphics*{NewL1ReOmega.eps}\\includegraphics*{NewL1ImOmega.eps}}\n\\caption{Real (left) and imaginary (right) parts of QNMs for $\\ell =1$, $m=0$ (bottom) and\n$\\ell =1$, $m=1$ (top).}\\label{fig2}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\resizebox{0.7 \\linewidth}{!}{\\includegraphics*{NewL2ReOmega.eps}\\includegraphics*{NewL2ImOmega.eps}}\n\\caption{a. Left panel: Real part of $\\omega$ for $\\ell =2$, $m=0$ (bottom),\n$m=1$, and $m=2$ (top), b. Right panel: Imaginary part of $\\omega$ for $\\ell =2$, $m=0$ (bottom),\n$m=1$, and $m=2$ (top)}\\label{fig3}\n\\end{figure*}\n\nIn order to calculate quasinormal modes defined above, one can use the WKB formula,\nwhich takes into consideration the behavior of the perturbation near the top of the effective potential.\nThe asymptotic region is located at a distance, which is at least one order larger than the black hole radius,\nsuch that one could qualitatively model the processes far from the black hole. At the same time, such a defined \"infinity\" must be\nlocated at a distance which is quite a few times smaller than the radius of the giant torus. The latter is necessary to avoid\ninfluence of the boundary effects near the torus of matter. Thus, there should be a \"valley\" far from the black hole and far from the edge of the torus, $r_{h} \\ll r \\ll a $ which could model an asymptotic region.\n\nThe QN spectrum is one of the essential\ncharacteristics of a black hole because it depends only on the black hole\nparameters and not on the way by which modes were excited.  Supposing\n$\\Psi \\sim e^{-i \\omega t}$, quasinormal modes can be written in the form\n$$\n\t\\omega = \\omega_{Re} - \\omega_{Im} i,\n$$\n where positive  $\\omega_{Im}$ is proportional to the decay rate of a damped\n QN mode.  The low laying QN frequencies have the smallest decay rates in\nthe spectrum and thus dominate in a signal at a sufficiently late time.  They\ncan be calculated by the WKB approach \\cite{WKB}, \\cite{WKBorder}.\nIntroducing $Q = \\omega^2 - V$, the 6-th order WKB formula reads\n\\begin{equation}\\label{WKBformula}\n\t\\frac{i Q_{0}}{\\sqrt{2 Q_{0}''}} - \\sum_{i=2}^{i=6}\n\t\t\\Lambda_{i} = n+\\frac{1}{2},\\qquad n=0,1,2\\ldots,\n\\end{equation}\n where the correction terms $\\Lambda_{i}$ were obtained in \\cite{WKB},\n \\cite{WKBorder}.  Here $Q_{0}^{i}$ means the i-th derivative of $Q$ at its\n maximum with respect to the tortoise coordinate $r^\\star$, and $n$ labels\n the overtones. The WKB formula (\\ref{WKBformula}) was effectively used in a\n lot of papers (see \\cite{WKBuse} and references therein).\n\nAs an additional check of the WKB data, we used the time domain integration scheme, which was related, for instance, in \\cite{timedomain} and used in a number of other works (see  \\cite{time-domain-apply} and references therein).\nThe obtained time domain data shows about a 0.1 percent difference with the WKB values in the worst cases when $\\ell =1$ and $\\mathcal{E}$ is moderate. The usage of the WKB formula is quite efficient here, because it has a very good accuracy for $\\ell >n$, while the potentially \"unsafe\"  WKB mode $\\ell=n=0$ is not effected by the tidal force in the used approximation and coincides with the Schwarzschild one.\n\nThe low laying quasinormal modes for various values of $\\ell$ and $m$ are shown in figs. (\\ref{fig2}) and (\\ref{fig3}).\nThe modes with $\\ell = m$ are noticeably different from those with  $\\ell > m$: while the first are monotonically increasing when $\\mathcal{E}$ increases, the second one decreases. Such behavior of the spectrum can be explained as follows:\nthe height of the potential barrier is lowered by the tidal force for $\\ell > m$, which means that it becomes easier penetrable for waves,\nleading to smaller real oscillation frequencies and longer living modes. At the same time $\\ell = m$ modes are governed by the potentials raised by the tidal force, so that both real and imaginary parts of $\\omega$ grows, as $\\mathcal{E}$   increases.\nThe lowered potential for all modes, except the one with the highest azimuthal number $m = \\ell$, can be explained by a smaller resultant gravitational attraction near the top of the potential due to the tidal gravity, which is opposite to the gravitational attraction of the black hole.\nThe non-monotonic \"splash\" of the imaginary part of $\\omega$ in fig. (\\ref{fig2}) could probably be explained by approaching the limit of validity $\\mathcal{E} \\ll 1$.\n\nIn the limit of large multipole numbers $\\ell \\gg m \\gg 1$, which is the eikonal regime, the peak of the effective potential is situated at\n\\begin{equation}\nr_{max} = 3 M -\\frac{2 M}{3 (2 - 7 M^2 \\mathcal{E}) \\ell^2} + O(\\ell^{-4}).\n\\end{equation}\n\nThen, the first order WKB formula,\n\\begin{equation}\n\\omega = \\sqrt{V - i \\sqrt{- 2 V_{0}''}\\left(n+\\frac{1}{2}\\right)},\n\\end{equation}\nexpanded in powers of $1\/\\ell$, gives:\n\\begin{equation}\\label{62}\n\\omega = \\frac{\\sqrt{1-\\frac{7}{2}\\mathcal{E} M^2 }}{3 \\sqrt{3} M} \\left(\\ell + \\frac{1}{2}\\right) - i \\left(n+\\frac{1}{2}\\right) + O(1\/\\ell).\n\\end{equation}\nIn the limit $\\mathcal{E} =0$, the above expression reduces to the well-known form for the Schwarzschild black hole \\cite{Ferrari}. The quasinormal modes do not depend on $m$ in the first two dominant orders: the dependence on $m$ is in terms\nof order $\\sim 1\/\\ell$. The imaginary part of $\\omega$ does not depend on $\\mathcal{E}$ in the eikonal regime,\nwhich is in concordance with the very slight dependence of the imaginary part on $\\mathcal{E}$ in figures (2) and (3).\nIt is well known that the eikonal formulas similar to (\\ref{62}) work very well already for moderate values of $\\ell$ and,\nin the eikonal regime, a universal behavior for all bosonic fields takes place. Therefore, expression (\\ref{62}) is likely to be valid not only for scalar but also for other boson fields in the eikonal regime.\n\nAn interesting question is the gravitational stability of the considered system of a torus and a black hole.\nThe full assessment of the stability issue should be performed by considering the gravitational perturbations of the system, yet, as we can see from the above data, the scalar field shows no unstable modes. A tendency to the instability shows $\\ell=m$ modes through the decreasing of the damping rate, which however cannot be extended to sufficiently large values of $\\mathcal{E}$ as our effective potential was obtained for the regime of relatively small tidal force.\n\n\\section{Conclusions}\n\nThe black hole deformed by the surrounding matter and the magnetic field is a background with a relatively\nlow symmetry, so that one could not expect the separability of variables in either Hamilton-Jacobi or Klein-Gordon equations.\nUsing a few justified approximations, we have decoupled variables in both equations and\nmade the problems of particle motion and quasinormal modes solvable with relatively simple analytical and numerical tools.\n\nInitially we have studied motion of charged, massive particles in the equatorial plane.\nTwo approximations were used here: 1) we considered only a limited spacial region which starts at the event horizon and\nfinishes at some large distance from the black hole, being still quite far from the place where the tidal force becomes significant; 2) we were limited by relatively small values of the magnetic field and tidal force.\nThese approximations were stipulated by applicability of the perturbatively obtained Preston-Poisson metric \\cite{Preston-Poisson} and\nwas, therefore, unavoidable.\nConsideration of motion only in the equatorial plane was due to the inseparability of the Hamilton-Jacobi equation in the general case.\nThis limitation is also justified, because, as it was shown in \\cite{Galtsov-book} for the Ernst solution, motion in the equatorial plane would be stable against small perturbations perpendicular to the plane, so that the particle which started to move in the equatorial plane, would continue moving in it. However, an analysis of the stability against non-equatorial perturbations, though similar to \\cite{Galtsov-book}, is lacking.\n\n\nMotion of particles is qualitatively different for left and right handed rotations due to the opposite direction of the Lorentz force:\nwhen the signs of the particle's charge and angular momenta coincide (Larmor motion) we have a kind of cyclotron rotation in the magnetic field\n\"perturbed\" by a black hole, while in opposite rotation (anti-Larmor motion), the presence of the black hole is essential.\nWe have found the energy and angular momentum, the binding energy and the region of radial stability on circular orbits\nfor the Preston-Poisson space-time. From this we conclude that \\emph{the tidal force, as well as the magnetic field, considerably enhances the release of the binding energy and makes the region of stability of circular orbits closer to the black hole.}\n\n\nThe second part is devoted to perturbations of the massless scalar field in the Preston-Poisson background, namely, to the estimations of the characteristic quasinormal modes.\nAs the decoupling of variables is impossible for the Klein-Gordon equation in the general Preston-Poisson space-time,\nwe used an additional approximation based on the fact that the low-laying quasinormal modes are \"localized\" near the peak of the effective potential. Therefore, we performed a kind of averaging of the tidal force by its value at the peak of potential.\n\\emph{We found a significant decrease in the real oscillation frequency with the tidal force. A simple analytical expression for the frequency has been obtained in the eikonal regime.} Taking into consideration\npossible similarities between the quasinormal spectrum of a scalar field with the spectra of gravitational and other long-range neutral boson fields, we neglected the influence of the magnetic field\nwhen studying the quasinormal modes.\nAlthough we managed to estimate the modes with $\\ell \\geq 1$, the monopole mode is, unfortunately, undistinguishable from its Schwarzschild value in the approximation. All the limitations of our analysis are a price for the simplicity afforded by the approximations employed.\nA more accurate numerical investigation would require very time consuming computations and mathematical modeling of the above system.\n\n\\acknowledgments{\nR. K. acknowledges hospitality of the Centro de Estudios Cientifcos (Valdivia, Chile) where this work was initiated.\nAt its initial stage the work was partially funded by\nthe Conicyt grant ACT-91: Southern Theoretical Physics Laboratory (STPLab).\nThe Centro de Estudios Cientifcos is funded by the Chilean Government\nthrough the Centers of Excellence Base Financing Program of Conicyt.\nAt the final stage R. K. was supported by the European Commission grant through the Marie Curie International Incoming Contract.\nThe authors acknowledge A. Zhidenko for useful discussions and and C. J. Pynn for proofreading of the manuscript.\nY-C. L. would like to acknowledge Kostas Kokkotas for financial\nsupport via the German Research Foundation (Deutsche\nForschungsgemeinschaft, DFG) SFB\/TR7 on Gravitational Wave Astronomy\nand Reinhard Meinel for financial support via the DFG Graduate Academy,\nResearch training group (1523\/1) Quantum and Gravitational field, G3:\nAxisymmetric gravitational fields.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Acknowledgements}\nThis research is supported in part by NSF CISE Expeditions award CCF-1139158 and DARPA XData Award FA8750-12-2-0331, and  gifts from Amazon Web Services, Google, SAP,  Apple, Inc., Cisco, Clearstory Data, Cloudera, Ericsson, Facebook, GameOnTalis, General Electric, Hortonworks, Huawei, Intel, Microsoft, NetApp, Oracle, Samsung, Splunk, VMware, WANdisco and Yahoo!.\n\nThanks to Trevor Darrell, Yangqing Jia, and Sergey Karayev who provided featurized imagenet data, Ben Recht who provided valuable ideas about derivative-free optimization and feedback, and Shivaram Venkataraman, Peter Bailis, Alan Fekete, Dan Crankshaw, Sanjay Krishnan, Xinghao Pan, and Kevin Jamieson for helpful feedback.\n\n\\section{Architecture}\n\\label{sec:architecture}\n\n\\mlopt is built on top of Apache Spark and MLI.  These systems have been\ndesigned to support high performance iterative MapReduce algorithms on working\nsets that fit into cluster memory, and simplified development of machine\nlearning algorithms, respectively.  It has been shown that these systems can\noutperform conventional MapReduce-based systems for machine learning like\n\\cite{mahout}, often by an order of magnitude.  We leverage features of both\nsystems in our design and experiments.  Since both Spark and MLI are written in\nScala, we chose to write \\mlopt in Scala as well to  leverage tight integration\nwith the existing systems and libraries available in the JVM ecosystem.  Our\nsystem is a component of MLbase~\\cite{Kraska13},\nand an important component of MLbase is the MLbase Optimizer, designed to\nautomate the process of constructing and refining machine learning pipelines.\n\\mlopt represents a first effort at building such an optimizer and is a central\ncomponent to the design of this system.\n\nThe architecture of \\mlopt consists of three abstract entities: A\n\\emph{Driver}, one or more \\emph{Searchers}, and zero or more \\emph{Executors}.\nThese entities are implemented as \\texttt{akka} Actors, which enables a\nconcurrent programming model that supports asynchronous communication between\nthese entities.  A graphical depiction of these entities is shown in\nFigure~\\ref{fig:architecture} and a summary of the protocol between these\nentities is captured in Figure~\\ref{tab:protocol}.  Additionally, models are trained in a \\emph{parallel dataflow} system, whose computational model---iterative MapReduce---lends itself well to expressing a large class of machine\nlearning algorithms.  The responsibilities of each entity are described below. \n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{figures\/architecture.pdf}\n\\caption{The Driver is responsible for defining a search space over which models are to be trained. The Searcher deterimines which configurations to try and, based on the results which ones to try next. It may interact with external search routines to accomplish this. The Executor is responsible for evaluating model configurations and reporting results - it may evaluate a particular configuration over a cluster of machines.}\n\\label{fig:architecture}\n\\end{figure}\n\n\\subsection{Driver}\nThe driver is the main entry point for a \\mlopt model search. The Driver's\nprimary responsibilities include instantiating one or more Searchers and\ncollecting results from the search process.  End users typically interact with\nthe system through the Driver, which is instantiated with a dataset, a search\nspace which describes the space of model configurations to explore, and\ninformation about how to configure the searcher and what type of searcher to\ninstantiate (e.g. Grid, Random, HyperOpt).  The driver communicates asynchronously\nwith the searcher and collects newly trained models as they become available.\nIt can also be polled by user programs for new information about search\nprogress and models that have been trained.\n\n\\subsection{Searcher}\nA Searcher is responsible for instantiating Executors, deciding on model\nconfigurations for evaluation, farming that work out to its Executors, and\nrelaying results back to the Driver.  Importantly, we also centralize the\nlogical and physical optimizations in the implementation of the searcher.\nNamely, it is the searcher's job to figure out how best to implement batching\n(by sending jobs to a Batch Executor), resource allocation (by evaluating results\nfrom Executors before letting work continue), and to implement the search\nmethod it is responsible for. \n\nThe architecture is such that we may use external libraries or service calls to\nimplement the searching logic.  For example, in implementing a HyperOpt based\nsearcher, the HyperOptSearcher code makes external service calls to a web service\nthat is simply a thin wrapper around the existing Hyperopt\nlibrary~\\cite{hyperopt}.\n\nBy implementing a few concrete methods for suggesting points in the\nhyperparameter space and ingesting new results, our system allows easy addition\nof new algorithms for hyperparameter optimization as they become available.\nAdditionally, new algorithms will benefit from the systems level optimizations\nmade for all algorithms.\n\n\\subsection{Executor}\nA Executor in \\mlopt is an abstract entity capable of evaluating model\nconfigurations.  A \\mlopt Executor may launch jobs that are executed on\nhundreds of nodes.  In practice, our executors usually have a handle to an\nentire cluster, which allows them to execute their jobs in a data parallel\nfashion.  As such, we are often in a situation where only a single Executor\nneeds to be created.  While this detail need not be evident to the searchers, in\nthe case of batching, for example, the \\emph{type} of Executor (that is, a\nBatch Executor vs. a Sequential Executor) can play a role in how tasks are\nallocated to it.\n\n\\begin{figure*}\n\\begin{tabular}{ll}\n\\qquad \n\\qquad \n\\qquad &\n  \\begin{subtable}[h]{0.30\\textwidth}\n     \\begin{tabular}{|l|l|l|}\n\t\t\t\\hline\n\t    Message                                    & Result     & Meaning                                 \\\\ \\hline\n\t    RunSearch(space: SearchSpace, data: Table) & ~          & Begin Search Process                    \\\\ \n\t    TopModel()                                 & Model      & Return best model after search.         \\\\\n\t    AllBuilt()                                 & Boolean    & Is search process finished?             \\\\\n\t    ModelsBuilt()                              & Int        & How many models have been built?        \\\\\n\t    Models()                                   & Seq[Model] & Return all models that have been built. \\\\\n\t\t\t\\hline\n\t    \\end{tabular}\n\t\t\t\\caption{Driver\/Searcher Protocol}\n\t\t\\end{subtable} \\\\\n& \n\t\t\\begin{subtable}[h]{0.30\\textwidth}\n\t    \\begin{tabular}{|l|l|l|}\n\t\t\t\\hline\n\t    Message                         & Result  & Meaning                                          \\\\ \\hline\n\t    BuildModel(config: ModelConfig) & ~       & Add model with this configuration to work queue. \\\\\n\t    DoWork()                        & ~       & Begin consuming work queue.                      \\\\\n\t    StopWork()                      & ~       & Stop consuming work queue.                       \\\\\n\t    QueueSize()                     & Int     & How big is your work queue?                      \\\\\n\t\t\t\\hline\n\t    \\end{tabular}\n\t\t\t\\caption{Searcher\/Executor Protocol}\n\t\t\\end{subtable} \\\\\n& \n\t\t\\begin{subtable}[h]{0.30\\textwidth}\n\t\t\t\\begin{tabular}{|l|l|}\n\t\t\t\t\t\\hline\n\t\t\t    Message                  & Meaning                                                           \\\\ \\hline\n\t\t\t    ModelBuilt(model: Model) & A model has been built with a particular configuration and error. \\\\ \n\t\t\t    WorkDone()               & My work queue is empty.                                           \\\\\n\t\t\t\t\t\\hline\n\t\t\t    \\end{tabular}\n\t\t\\caption{Executor\/Searcher Protocol}\n\t\t\\end{subtable} \\\\\n  \\end{tabular}\n  \\caption{Summary of protocol between Driver, Searcher, and Executor.} \n\t\t\\label{tab:protocol}\n\\end{figure*}\n\n\\subsection{Hybrid Parallel Dataflow and Asynchronous Control for Model Search}\nIn basing our system on Spark, we are explicitly employing a parallel dataflow\nengine to train our models.  At the same time, all of the hyperparameter tuning\nhappens in an asynchronous fashion with Actors handing out jobs and receiving\nresults asynchronously.  As such, our system can be viewed as a hybrid of a\ntraditional BSP computation environment with an asynchronous control plane.  By\nemploying this approach, we are able to isolate the portions of our system that\nneed to be high performance, fault tolerant, and data parallel from those that\nneed to be asynchronous and reactive, leading to a system that achieves both\ngoals.\n\n\\subsubsection{Logistic Regression}\nLogistic Regression is a widely used machine learning model for binary classification.\nThe procedure estimates a set of model parameters, $w \\in \\reals^d$, given a set of data features $X \\in \\reals^{n \\times d}$, and binary labels $y \\in {0,1}^n$.\nThe optimal model $w^*\\in\\reals^d$ can be found by minimizing the negative likelihood function, $f(w) = -\\log p(X|w)$.\nTaking the gradient of the negative log likelihood, we have:\n\\begin{equation}\n\\label{eq:grad}\n\\nabla f = \\sum_{i=1}^n \\bigg [\\big (\\sigma(w^\\top x_i)\n- y_i \\big ) x_i \\bigg ]\\,,\n\\end{equation}\nwhere $\\sigma$ is the logistic function.\nThe gradient descent algorithm (Algorithm~\\ref{fig:logreg}) must evaluate this gradient function for all input data points, a task which can be easily performed in a data parallel fashion. Similarly, minibatch Stochastic Gradient Descent (SGD) has an identical access pattern and can be optimized in the same way by working with contiguous subsets of the input data on each partition.\n\n\\begin{algorithm}[]\n \\SetKwInOut{Input}{input}\\SetKwInOut{Output}{output} \n \\Input{X, LearningRate, MaxIterations}\n \\Output{Model}\n $i \\gets 0$\\;\n Initialize Model\\;\n \\While{$i < MaxIterations$}{\n  read current\\;\n  Model $\\gets$ Model - LearningRate * Gradient(Model, X)\\;\n  $i \\gets i + 1$\\;\n }\n \\caption{Pseudocode for convex optimization via gradient descent.}\n \\label{fig:logreg}\n\\end{algorithm}\n\nThe above formulation represents the computation of the gradient by taking a single point and single model at a time.\nWe can naturally extend this to multiple models simultaneously if we represent our models as a matrix $W \\in \\reals^{d \\times k}$, where $k$ is the number of models we want to train simultaneously, i.e.,\n\n\\begin{equation}\n\\label{eq:batchgrad}\n\\nabla f = \\bigg [X^\\top \\big (\\sigma(X W)\n- y \\big ) \\bigg ]\\,.\n\\end{equation}\n\nThis operation can be easily parallelized across data items with each worker in a distributed system computing the portion of the gradient for the data that it stores locally.\nSpecifically, the portion of the gradient that is derived from the set of local data is computed independently at each machine, and these gradients are simply summed at the end of an iteration. \nThe size of the partial gradients (in this case $O(d \\times k)$) is much smaller than the actual data (which is $O(n \\times d)$), so overheads of transferring these over the network is relatively small.\nFor large datasets, the time spent performing this operation is almost completely determined by the cost of performing two matrix multiplications---the input to the $\\sigma$ function which takes $O(ndk)$ operations and requires a scan of the input data as well as the final multiply by $X^\\top$ which also takes $O(ndk)$ operations and requires a scan of the data. \nThis formulation allows us to leverage high performance linear algebra libraries that implement BLAS~\\cite{blas}---these libraries are tailored to execute exactly dense linear algebra operations as efficiently as possible and are automatically tuned to the architecture we are running on via \\cite{atlas}.\n\nThe careful reader will note that if individual data points are of sufficiently low dimension, the gradient function in Equation~\\ref{eq:grad} can be executed in a single pass over the data from main memory because the second reference to $x_i$ will likely be a cache hit, whereas we assume that $X$ is big enough that it is unlikely to fit entirely in CPU cache.\nWe examine this effect more carefully in Section~\\ref{sec:evaluation}.\n\n\\subsubsection{Machine Balance}\nOne obvious question the reader may ask is why implementing these algorithms via matrix-multiplication should offer speedup over vector\/vector versions of the algorithms.\nAfter all, the runtime complexities of both algorithms are identical.\nHowever, modern x86 machines have been shown to have processor cores that significantly outperform their ability to read data from main memory~\\cite{McCalpin1995}.  \nIn particular, on a typical x86 machine, the hardware is capable of reading 0.45B doubles\/s from main memory per core, while the hardware is capable of executing 6.8B FLOPS in the same amount of time~\\cite{McCalpin2007}.\nSpecifically, on the machines we tested (Amazon \\texttt{c3.8xlarge} EC2 instances), LINPACK reported peak GFLOPS of 110 GFLOPS\/s when running on all cores, while the STREAM benchmark reported 60GB\/s of throughput across 16 physical cores. This equates to a machine balance of approximately 15 FLOPS per double precision floating point number read from main memory if the machine is using both all available FLOPs and all available memory bandwidth solely for its core computation.\nThis approximate value for the machine balance suggests an opportunity for optimization by reducing unused resources, i.e., wasted cycles.\nBy performing more computation for every number read from memory, we can reduce this resource gap.\n\nThe Roofline model~\\cite{Williams:2009cx} offers a more formal way to study this effect.\nAccording to the model, total throughput of an algorithm is bounded by the smaller of 1) peak floating point performance of the machine, and 2) memory bandwidth times operational intensity of the algorithm, where operational intensity is a function of the number of FLOPs performed per byte read from memory.\nThat is, for an efficiently implemented algorithm, the bottleneck is either I\/O bandwidth from memory or CPU FLOPs. \n\nAnalysis of the unbatched gradient descent algorithm reveals that the number of FLOPs required per byte is quite small---just over 2 flops per number read from memory---a multiply and an add---and since we represent our data as double-precision floating point numbers, this equates to 1\/2 FLOP per byte. \nBatching allows us to move ``up the roofline'' by increasing algorithmic complexity by a factor of $k$, our batch size.\nThe exact setting of $k$ that achieves balance (and maximizes throughput) is hardware dependent, but we show in Section~\\ref{sec:evaluation} that on modern machines, $k=10$ is a reasonable choice.\n\n\\subsubsection{Amortized Overheads}\nIn the context of a distributed machine learning system like MLbase, which runs on Apache Spark, delays due to task scheduling and serialization\/deserialization can be significant relative to the time spent computing. \n\nBy batching our updates into tasks that require more computation, we are able to reduce the aggregate overhead of launching new tasks substantially.\nAssuming a fixed scheduling delay of 200ms per task, if we have a batch size of $k=10$ models, the average task overhead per model iteration drops to 20ms. \nOver the course of hundreds of iterations for hundreds of models, the savings can be substantial.\n\nFor a typical distributed system, a model update requires at least two network round-trips to complete: one to launch a task on each worker node, and one to report the results of the task back to the master.\nIf we were indeed bound by network messaging, then amortizing this cost across multiple models could substantially reduce total overhead due to network latency.\nIn our experiments, however, the number of messages is relatively small and the network overheads are substantially lower than scheduling and computation overheads, so future work in this setting should focus on minimizing scheduling overheads.\n\n\n\n\n\n\\section{Design Space Evaluation}\n\\label{sec:designeval}\nNow that we have laid out the possible optimizations available to \\mlopt, we identify available speedups from these optimizations.\nWe validated the strategies for model search and bandit resource allocation on five representative datasets across a variable sized model fitting budget.\nIn particular, these results motivated which model search strategies to incorporate into \\mlopt and validate our bandit approach.\nNext, we tested our batching optimizations on datasets of various sizes in a cluster environment, to better understand the impact of batching as models and datasets get bigger.\n\nIn all experiments, before training, we split our base datasets into $70\\%$ training, $20\\%$ validation, and $10\\%$ testing.\nIn all cases, models are fit to minimize classification error on the training set, while model search occurs based on classification error on the validation set (validation error).\\footnote{While we have thus far discussed model quality, for the remainder of the paper we report validation error, i.e., the inverse of quality, because it is more commonly reported in practice.}\nWe only report validation error numbers here, but test error was similar.\n\\mlopt is capable of optimizing for arbitrary performance metrics as long as they can be computed mid-flight, and extends to other supervised learning scenarios.\n\n\\subsection{Model Search}\nWe validated our multiple model search strategies on a series of small datasets with well-formed binary classification problems embedded in them.\nThese datasets come from the UCI Machine Learning Repository~\\cite{ucirepo}.\nThe model search task involved tuning four hyperparameters---learning rate, L2 regularization parameter, size of a random projection matrix, and noise associated with the random feature matrix.\nThe random features are constructed according to the procedure outlined in \\cite{RandomKitchenSink}.\nTo accommodate for the linear scale-up that comes with adding random features,\nwe down sample the number of data points for each model training by the same\nproportion.\n\nOur ranges for these hyperparameters were learning rate $\\in (10^{-3}, 10^{1})$, regularization $\\in (10^{-4}, 10^{2})$, projection size $\\in (1 \\times d, 10 \\times d)$, and noise $\\in (10^{-4}, 10^{2})$.\n\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[width=0.7\\textwidth]{figures\/SearchComparison.pdf}\n\\caption{Search methods were compared across several datasets with a variable number of function evaluations. Classification error on a validation dataset is shown for each combination. HyperOpt and Auto-WEKA provide state of the art results, while random search performs best of the classic methods.}\n\\label{fig:searchexps}\n\\end{figure*}\n\nWe evaluated seven search methods: grid search, random search, Powell's method, the Nelder-Mead method, Auto-WEKA, HyperOpt, and Spearmint. \n\nEach dataset was processed with each search method with a varying number of function calls, chosen to align well with a regular grid of $n^4$ points where we vary $n$ from 2 to 5. \nThis restriction on a regular grid is only necessary for grid search but included for comparability.\n\nResults of the search experiments are presented in Figure~\\ref{fig:searchexps}. \nEach tile represents a different dataset\/search method combination. \nEach bar within the tile represents a different budget in terms of function calls\/models trained.\nThe height of each bar represents classification error on the validation dataset.\n\nWith this experiment, we are looking for methods that converge to good models in as small a budget as possible.\nOf all methods tried, HyperOpt and Auto-WEKA tend to achieve this criteria best, but random search is not far behind.\nWe chose to integrate HyperOpt into the larger experiments because it performed\nslightly better than Auto-WEKA.\nOur architecture fully supports additional search methods, and we expect to implement additional methods in our system over time.\n\n\\subsection{Bandit Resource Allocation}\nWe evaluated the \\mlopt bandit resource allocation scheme on the same datasets with random search and 625 total function evaluations---the same as the maximum budget in the search experiments.\nThe key question to answer here was whether we could identify and terminate poorly performing models early in the training process without significantly affecting overall search quality. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{figures\/banditPlot.pdf}\n\\caption{Here we show the effects of bandit resource allocation on trained model performance. Model search completes in an average of 83\\% fewer passes over the training data than without bandit allocation. Except in one case, validation error is nearly indistinguishable vs. the case where we do not employ the bandit strategy.}\n\\label{fig:earlystopexps}\n\\end{figure}\n\nIn Figure~\\ref{fig:earlystopexps} we illustrate the effect that the \\mlopt bandit strategy \nhas on validation error as well as on the number of total scans of the input dataset.\nModels were allocated 100 iterations to converge on the correct answer. After the first 10 iterations, models that were not within 50\\% of the classification error of the best model trained so far were preemptively terminated. A large percentage of models that show little or no promise of converging to a reasonable validation error are eliminated.\n\nIn the figure, the top set of bars represents the number of scans of the training data at the end of the entire search process.\nThe bottom set of bars represent the validation error achieved at the end of the search procedure.\nThe three scenarios evaluated---No Bandit, Bandit, and Baseline---represent the results of the search with no bandit allocation procedure (that is, each model is trained to completion), the algorithm the bandit allocation procedure enabled, and the baseline error rate for each dataset.\n\nThere was an 86\\% decrease in total epochs across these five datasets, and the validation error is roughly comparable to the unoptimized strategy.\nOn average, this method achieves 93\\% reduction in model error vs. not stopping\nearly when compared with validation error of a simple baseline model.\nThis relatively simple resource allocation method presents opportunities for dramatic reductions in runtime.\n\n\n\n\\subsection{Batching}\nTo evaluate the batching optimization, we used a synthetic dataset of $1,000,000$ data points in various dimensionality.\nTo illustrate the effects of amortizing scheduler overheads vs. achieving machine balance, these datasets vary in size between 750MB and 75GB.\n\nWe trained these models on a 16-node cluster of \\texttt{c3.8xlarge} nodes on Amazon EC2, running Apache Spark 1.1.0.\nWe trained a logistic regression model on these data points via gradient descent with no batching (batch size = 1) and batching up to 20 models at once.\nWe implemented both a naive version of this optimization---with while loops evaluating equation~\\ref{eq:grad} over each model in each task, as well as a more sophisticated version of this model which makes BLAS calls to perform the computation described in equation~\\ref{eq:batchgrad}. \nFor the batching experiments only, we run each algorithm for 10 iterations over the input data.\n\\begin{figure}\n\t\\begin{subtable}[h]{0.4\\textwidth}\n\t\\begin{tabular}{|r|r|r|r|r|}\n\t  \\hline\n\t \\diagbox{Batch Size}{D} & 100 & 1000 & 10000 \\\\ \n\t\t\\hline\n\t\t  1 & 826.44 & 599.60 & 553.59 \\\\ \n\t\t\t\\hline\n\t\t  2 & 1521.23 & 1214.37 & 701.07 \\\\ \n\t\t\t\\hline\n\t\t  5 & 2411.53 & 3037.97 & 992.01 \\\\ \n\t\t\t\\hline\n\t\t  8 & 5557.69 & 3502.79 & 1243.79 \\\\ \n\t\t\t\\hline\n\t\t  10 & 7148.53 & 4216.44 & 1769.12 \\\\\n\t\t\t\\hline\n\t\t  15 & 7874.01 & 6260.14 & 2485.15 \\\\\n\t\t\t\\hline\n\t\t  20 & 11881.18 & 8248.36 & 2445.98 \\\\\n\t\t   \\hline\n\t\\end{tabular}\n\t\\caption{Models trained per hour for varying batch sizes and model complexity. Data sizes ranged from 750MB (D=100) to 75GB (D=10000).}\n\t\\end{subtable}\n\t\n\t\\begin{subtable}[h]{0.4\\textwidth}\n\t\t\\begin{tabular}{|r|r|r|r|r|}\n\t\t  \\hline\n\t\t\\diagbox{Batch Size}{D} & 100 & 1000 & 10000 \\\\ \n      \\hline\n      1 & 1.00 & 1.00  & 1.00 \\\\\n      \\hline\n      2 & 1.84 & 2.02 & 1.26 \\\\\n      \\hline\n      5 & 2.91 & 5.06 & 1.79 \\\\\n      \\hline\n      8 & 6.72 & 5.84 & 2.24 \\\\\n      \\hline\n      10 & 8.64 & 7.03 & 3.19 \\\\\n      \\hline\n      15 & 9.52 & 10.44 & 4.48 \\\\\n      \\hline\n      20 & 14.37 & 13.75 & 4.41 \\\\\n      \\hline\n\t\t\\end{tabular}\n\t\t\\caption{Speedup factor vs fastest sequential unbatched method for varying batch size and model complexity.}\n\t\\end{subtable}\n\t\n\t\\caption{Effect of batching is examined on 16 nodes with a synthetic dataset. Speedups diminish but remain significant as models increase in complexity.}\n\\label{fig:smallbatch}\n\\end{figure}\n\nIn Figure~\\ref{fig:smallbatch} we show the total throughput of the system in terms of models trained per hour varying the batch size and the model complexity. \nFor models trained on the smaller dataset, we see the total number of models per hour can increase by up to a factor of 15 for large batch sizes. \nThis effect should not be surprising, as the actual time spent computing is on the order of milliseconds and virtually all the time goes to scheduling task execution.\nIn its current implementation, due to these scheduling overheads, this implementation of the algorithm under Spark will not outperform a single machine implementation for a dataset this small.\nWe discuss an alternative execution strategy that would better utilize cluster resources for situations where the input dataset is small in Section~\\ref{sec:futureworkconclusions}.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\columnwidth]{figures\/tupaqBatch.pdf}\n  \\caption{Leveraging high performance linear algebra libraries for batching leads to substantial speedups vs. naive methods. At bottom, we show models per hour via the fastest sequential (non-batched) strategy and demonstrate a 5x improvement in throughput.}\n  \\label{fig:mmbatchexps}\n\\end{figure}\n\nAt the other end of the spectrum in terms of data size and model complexity, we see the effects of scheduler delay start to lessen, and we maximize throughput in terms of models per hour at batch size 15.\nIn Figure~\\ref{fig:mmbatchexps} we compare two different strategies of implementing batching---one via the naive method, and the other via the more sophisticated method---computing gradient updates via BLAS matrix multiplication.\nFor small batch sizes, the naive implementation actually performs faster than the BLAS optimized one.\nThe matrix-based implementation easily dominates the naive implementation as batch size increases. \nThis is because the algorithm is slightly more cache efficient and requires only a single pass through the input data.\nThe overall speedup due to batching is nearly a factor of 5 when executed via matrix multiplication.\n\nThe downside to batching in the context of PAQ planning is that the system may gain information by trying plans sequentially that could inform subsequent plans that is not incorporated in later runs.\nBy fixing our batch size to a relatively small constant (10) we are able to balance this tradeoff.\n\n\\section{Putting It All Together}\n\\label{sec:evaluation}\nNow that we have examined each point in the PAQ planning design space individually, let us now evaluate end-to-end performance of the \\mlopt planner.\n\nBy employing batching, using state-of-the-art search methods, and using bandit resource allocation to terminate non-promising models, we are able to see a 10x increase in raw throughput of the system in terms of models trained per unit time, while finding PAQ plans that have as good or higher quality than those found with the baseline approach.\n\nWe evaluated \\mlopt on very large scale data problems, at cluster sizes ranging from 16 to 128 nodes and datasets ranging from 30GB to over 3TB in size.\nThese sizes represent the size of the actual features the model was trained on, \\emph{not} the raw data from which these features were derived.\n\n\\subsection{Platform Configuration}\nWe evaluated \\mlopt on Linux machines running under Amazon EC2, instance type \\texttt{c3.8xlarge}.\nThese machines were configured with Redhat Enterprise Linux, Scala 2.10, version 1.9 of the Anaconda python distribution from Continuum Analytics\\cite{anaconda}, and Apache Spark 1.1.0.\nAdditionally, we made use of Hadoop 1.0.4 configured on local disks as our data store for the large scale experiments.\nFinally, we use MLI as of commit 3e164a2d8c as a basis for \\mlopt.\n\n\\subsubsection{Apache Spark Configuration}\nAs with any complex system, proper configuration of the platform to execute a given workload is necessary and Apache Spark is no exception.\nSpecifically---choosing a correct BLAS implementation, configuring spark to use it, and picking the right balance of executor threads per executor process took considerable effort. Full details of our configuration are available on request.\n\n\n\\subsubsection{Experimental Setup and Datasets}\nThe complete system involves a Scala code base built on top of Apache Spark, MLlib, and MLI.\nHere, we ran experiments on 16 and 128 machines.\nWe used two datasets with two different learning objectives to evaluate our system at scale. \n\nThe first dataset is a pre-featurized version of the ImageNet Large Scale Visual Recognition Challenge 2010 (ILSVRC2010) dataset~\\cite{imagenet2010}, featurized using a procedure attributed to~\\cite{Deng:2012tg}.\nThis process yields a dataset with $160,000$ features and approximately $1,200,000$ examples, or 1.4 TB of raw image features.\nIn our 16-node experiments we down sample to the first $16,000$ of these features and use 20\\% of the base dataset for model training, which is approximately 30GB of data.\nIn the 128-node experiments we train on the entire dataset.\nWe explore five hyperparameters here---one parameter for the classifier we train---SVM or logistic regression, as well as learning rate and L2 Regularization parameters for each matching the above experiments.\nWe allot a budget of 128 model fittings to the problem.\n\nAs in Figure~\\ref{fig:photos}, we search for a PAQ plan capable of discriminating plants from non-plants given these image features.\nThe images are generally in 1000 base classes, but these classes form a hierarchy and thus can be mapped into plant vs. non-plant categories.\nBaseline error for this modeling task is $14.2\\%$, which is a bit more skewed than the previous examples.\nOur goal is to reduce validation error as much as possible, but our experience with this particular dataset has put a lower bound on validation error to around $9\\%$ accuracy with linear classification models.\n\nThe second dataset is a pre-featurized version of the TIMIT Acoustic-Phonetic continuous speech corpus~\\cite{timit}, featurized according to the procedure described in~\\cite{timitfeatures}---yielding roughly $2,300,000$ examples each having $440$ features. While this dataset is quite small, in order to achieve strong performance on this dataset, other researchers have noted that Kernel Methods offer the best performance~\\cite{posen}. Following the process of \\cite{RandomKitchenSink}, this involves expanding the feature space of the dataset by nearly two orders of magnitude, yielding a dataset that has $204,800$ features. Again, this is approximately 3.4 TB of speech features.\nWe explore five hyperparameters here---one parameter describing the distribution family of the random projection matrix---in this case Cauchy or Gaussian, the scale and skew of these distributions, as well as the L2 regularization parameter for this model, which will have a different setting for each distribution.\n\nA necessary precondition to supporting PAQs like those in Figure~\\ref{fig:vms}, this dataset provides a examples of labeled phonemes, and our challenge is to find a model capable of labeling phonemes given some input audio. Baseline error for this modeling task is $95\\%$, and state-of-the-art performance on this dataset is $35\\%$ error~\\cite{posen}.\n\n\\subsection{Optimization Effects}\nIn Figure~\\ref{fig:timingexps} we can see the effects of batching and bandit allocation on the PAQ planning process for the ImageNet dataset.\nSpecifically, given that we want to evaluate the fitness of 128 models, it takes nearly 2 hours to fit all 128 models on the 30GB dataset of data on the 16 node cluster. \nBy comparison, with the bandit rule and batching turned on, the system takes just 10 minutes to train a random search model to completion and a bit longer to train a HyperOpt model to completion, a 10x speedup in the case of random search and a 7x speedup in the case of HyperOpt.\nHyperOpt takes slightly longer because it does a good job of picking points that do not need to be terminated preemptively by the bandit strategy. That is, more of the models that HyperOpt selects are trained to completion than random search. Accordingly, HyperOpt arrives at a better model than random search given the same training budget.\n\n\\begin{figure}[!htbp] \\centering \n\\begin{tabular}{|r|r|r|r|} \n\\hline\n\\diagbox{Optimization}{Search Method} & Grid & Random & HyperOpt \\\\ \n\\hline\n\\hline\nNone & $104.7$ & $100.5$ & $103.9$ \\\\ \n\\hline\nBandits Only & $31.3$ & $29.7$ & $50.5$ \\\\ \n\\hline\nBatching Only & $31.3$ & $32.1$ & $31.8$ \\\\ \n\\hline\nAll (\\emph{\\mlopt}) & $11.5$ & $10.4$ & $15.8$ \\\\ \n\\hline\n\\end{tabular} \n\\caption{Learning time in minutes for a 128-configuration budget across various optimization levels for ImageNet data. Unoptimized, sequential execution takes over 100 minutes regardless of search procedure used. Fully optimized execution can be an order of magnitude faster with \\mlopt.} \n\\label{fig:timingexps} \n\\end{figure}\n\n\\begin{figure}\n\\centering\n\t\\begin{tabular}{|r|r|r|}\n\t  \\hline\n\t   \\emph{Search Method} & \\emph{Search Time (m)} & \\emph{Test Error (\\%)} \\\\ \n\t\t\\hline\n    \\hline\n\t\t  Grid (unoptimized) & 104.7 & 11.05 \\\\ \n\t\t\t\\hline\n\t\t  Random (optimized) & 10.4 & 11.41 \\\\ \n\t\t\t\\hline\n\t\t  HyperOpt (optimized) & 15.8 & 10.38 \\\\ \n\t\t\\hline\n\t\\end{tabular}\n  \\caption{Both optimized HyperOpt and Random search perform significantly faster than unoptimized Grid search, while HyperOpt yields the best model for this image classification problem.}\n\\label{fig:modelconv}\n\\end{figure}\n\nTurning our attention to model convergence illustrated in Figure~\\ref{fig:modelconv}, we can see that on this dataset HyperOpt converges to the best answer in just 15 minutes, while random search converges to within $5\\%$ of the best test error achieved by grid search a full order of magnitude faster than the baseline approach.\n\n\\subsection{Large Scale Speech and Vision}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.35\\textwidth,scale=0.5]{figures\/optimizationConvergenceComparisonHuge.pdf}\n\\caption{Training a model with a budget of 32 function evaluations on a 1.2m $\\times$ 160k dataset takes 90 minutes on a 128-node cluster with \\mlopt.}\n\\label{fig:modelconvhuge}\n\\end{figure}\n\nBecause we employ data-parallel versions of our learning algorithms, achieving horizontal scalability with additional compute resources is trivial.\n\\mlopt readily scales to multi-terabyte datasets that are an order of magnitude more complicated with respect to the feature space.\nFor these experiments, we ran with the ImageNet models with same parameter search settings as the smaller dataset but this time with a fixed budget of 40 function evaluations.\nOur results are illustrated in Figure~\\ref{fig:modelconvhuge}.\nUsing the fully optimized HyperOpt based search method, we are able to search this space in under 90 minutes, and the method is able to achieve a validation error of $8.2\\%$ for this dataset in that time.\nIn contrast, training all 32 models to completion using sequential grid search would have taken over 8 hours and cost upwards of \\$2000.00---an expense we chose not to incur.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.35\\textwidth,scale=0.5]{figures\/optimizationConvergenceComparisonTimitLg.pdf}\n\\caption{Training a multiclass phoneme classification model on 3.7 TB of TIMIT features yields accuracy approaching that of state of the art models in 3.5 hours with \\mlopt.}\n\\label{fig:modelconvtimit}\n\\end{figure}\n\nTurning our attention to an entirely different application area, we demonstrate the ability of the system to scale to a multi-terabyte, multi-class phoneme classification problem. Here, a multi-class kernel SVM was trained on $2,251,569$ data points with $204,800$ features, in $147$ distinct classes. As shown in Figure~\\ref{fig:modelconvtimit}, the system is capable of getting to a model with $39.5\\%$ test error---approaching that of state-of-the-art results in speech-to-text modeling---in just 3.5 hours.\nFor this dataset, training the entire budget to completion would have taken 35 hours.\n\\section{Future Work and Conclusions}\n\\label{sec:futureworkconclusions}\n\nIn this work, we have introduced the PAQ Planning problem, and demonstrated the\nimpact of logical and physical optimizations to improve the quality and\nefficiency of PAQ planning.  Specifically, by combining better model search\nmethods, batching techniques, and bandit methods, \\mlopt can find\nhigh quality query plans for PAQs on very large datasets\nan order of magnitude more efficiently than than the baseline approach.\n\n\\mlopt is a first step in tackling the challenging PAQ\nplanning problem.  Indeed, several avenues exist for further exploration, and\nwe note two broad classes of natural extensions to \\mlopt.\n\n{\\bf Machine learning extensions}. From an accuracy point of view, as\nadditional model families are added to MLbase, \\mlopt could naturally lend\nitself to the construction of \\emph{ensemble models} at training time -\neffectively \\emph{for free}.  Ensembles over a diverse set of methods are\nparticularly known to improve predictive performance, and there may be better\nPAQ planning strategies for ensemble methods that encourage heterogeneity among\nensemble methods.  Of course, as more models and more hyperparameter\nconfigurations are considered, PAQ planners run the risk of overfitting to the\nvalidation data, and accounting for this issue, e.g., by \\emph{controlling the\nfalse discovery rate}~\\cite{Benjamini1995}, would become especially important.\nFrom a performance perspective, \\emph{adaptive and accelerated gradient\nmethods} could be used to speed convergence of individual models by requiring\nfewer passes over the training data~\\cite{tseng08, duchi2011adaptive}.\nMoreover, theoretically supported multi-armed bandit algorithms, including those\nthat are aware that multiple model\nconfigurations are being sampled simultaneously, may also improve performance.\nFinally, \\emph{unsupervised learning methods}, including dimensionality\nreduction and exploratory analysis, could be used in conjunction with\nsupervised approaches to speed up and\/or improve the accuracy of supervised\nlearning methods. \n\n{\\bf Systems extensions}. Multi-stage \\emph{ML pipelines}, in which \nthe initial data is transformed one or more times before being fed into \na supervised learning algorithm, are common in most\npractical ML systems.  Since each stage will likely introduce additional\nhyperparameters, PAQ planning becomes more challenging\nin the pipeline setting. In a regime where a dataset is\nrelatively small but users still have access to cluster resources, there can be\nbenefits (both in terms of simplicity and speed) to broadcast the data to each\nworker machine and train various models locally on each worker. PAQ planning\ncould be made more efficient by considering the tradeoffs between these regimes.\nTraining models on \\emph{subsets of data} can efficiently yield\nnoisy evaluations of candidate models, though careful subsampling is required\nto yield high-quality and trustworthy PAQ plans~\\cite{BlinkDB}.\nAkin to traditional query planners, PAQ\nplanners can learn from knowledge of the data they store and historical\nworkloads. A PAQ planner could store \\emph{planner statistics} to tailor their\nsearch strategy to the types of models have been used for a user's data in the\npast. The evaluation of these techniques in \\mlopt will be natural once the system has been exposed to a larger set of workloads.\n\nMoving forward, we believe that \\mlopt and extensions such as those\ndescribed above, will serve as a foundation for the automated construction of\nend-to-end pipelines for machine learning.\n\n\n\n\\section{Introduction}\n\n\\label{sec:intro}\nRapidly growing data volumes coupled with the maturity of sophisticated statistical techniques have led to a new type of data-intensive workload: predictive analytics over large scale, distributed datasets. Indeed, the support of predictive analytics is an increasingly active area of database systems research. Several systems that integrate statistical query processing with a data management system have been developed. However, these systems force users to describe their statistical model in dense mathematical notation~\\cite{simsql,SystemML} or in terms of a specific model~\\cite{MADLib,Zhang:2014tf,Deshpande:2006wi} and provide little guidance about the proper configuration of the model---that is, a user must know that a linear SVM or Kalman filter is a good statistical procedure to answer their query, and configure that procedure appropriately.\n\n\\begin{figure}\n\\begin{subfigure}[b]{0.5\\textwidth}\n\\begin{lstlisting}[language=SQL, frame=shadowbox, breaklines=true, autogobble=true, morekeywords={PREDICT,GIVEN}, escapechar=@, belowskip=\\smallskipamount, aboveskip=\\smallskipamount]\nSELECT vm.sender, vm.arrived,\n@\\HighlightFrom@PREDICT(vm.text, vm.audio)@\\HighlightTo@ \n@\\HighlightFrom@GIVEN LabeledVoiceMails@\\HighlightTo@\nFROM VoiceMails vm \nWHERE vm.user = 'Bob' AND vm.listened is NULL\nORDER BY vm.arrived \nDESC LIMIT 50\n\\end{lstlisting}\n\\caption{Speech-to-text transcription.}\n\\label{fig:vms}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.5\\textwidth}\n\\begin{lstlisting}[language=SQL, frame=shadowbox, breaklines=true, autogobble=true, morekeywords={PREDICT,GIVEN}, escapechar=@, aboveskip=\\smallskipamount, belowskip=\\smallskipamount]\nSELECT p.image\nFROM Pictures p\nWHERE @\\HighlightFrom@PREDICT(p.tag, p.photo) = 'Plant' GIVEN@\\HighlightTo@ @\\HighlightFrom@LabeledPhotos@\\HighlightTo@\nAND p.likes > 500\n\\end{lstlisting}\n\\caption{Photo classification.}\n\\label{fig:photos}\n\\end{subfigure}\n    \\caption{Two examples of PAQs, with the predictive clauses highlighted\nin green. (\\protect\\ref{fig:vms}) returns the predicted text transcription of Bob's voicemails from their audio content.\n(\\protect\\ref{fig:photos}) finds popular pictures of\nphotos based on an image classification model---even if the images are not\nlabeled. Each of these use cases may require considerable training data.}\n    \\label{fig:paq}\n\\end{figure}\n\n\nIn our work, our goal is to raise the level of abstraction for data analysts. Instead of choosing a specific statistical model and featurization strategy, we provide a declarative query interface where users declare that they wish to predict an attribute from some other collection of attributes and optionally provide example training data. Given these inputs, the system automatically makes predictions for the target attribute on new data. With our system, users issue Predictive Analytic Queries, or PAQs, which are traditional database queries, augmented with new predictive clauses. Two examples of PAQs are given in Figure~\\ref{fig:paq}---with the predictive clauses highlighted. The output of a predictive clause is an attribute like any other---one that can be grouped and sorted on or used in other clauses. The syntax of these predictive clauses is as follows:\n\n\\[ \n\\text{\\texttt{PREDICT} ($a_{predicted}$ [, $a_1$, ..., $a_n$ ]) \\texttt{GIVEN} $R$}\n\\]\n\nWhere, $a_{predicted}$ is the attribute to be predicted. $a_1, ..., a_n$ is an optional set of predictor attributes. $R$ is a relation containing training examples with the restriction that $ \\left\\{a_{predicted}, a_1, ..., a_n\\right\\} - Attributes(R) = \\emptyset $.\nThis syntax is general enough to support a wide range of predictive tasks---including classification, regression, and item recommendation.\n    \n\n\n\nGiven recent advances in statistical methodology, supervised machine\nlearning (ML) techniques are a natural way to support the predictive clauses\nin PAQs.  In the supervised learning setting, a statistical model is created via\ntraining data to relate the input attributes to the desired output\nattribute.  Furthermore, ML methods learn better models as the size of the\ntraining data increases, and recent advances in distributed ML algorithm\ndevelopment, are aimed at enabling\nlarge-scale model training in the distributed setting.~\\cite{agarwal14a,Panda:2009uh,MackeyTaJo11} \n\nUnfortunately, the\napplication of supervised learning techniques to a new input dataset is\ncomputationally demanding and technically challenging.\nFor a non-expert, the process of carefully preprocessing the input\nattributes, selecting the appropriate ML model, and tuning its\nhyperparameters can be an ad-hoc and time-consuming task. \nFor example, to build a predictive model for a classification task like the one shown in Figure~\\ref{fig:photos} using conventional tools, a user needs to choose from one of many algorithms for extracting features from image data, then select an appropriate classification model---all the while tuning the configuration of each. \nFinally, the user will need to settle on a strategy to select the best performing model. Failure to follow these steps can lead to models that do not work at all, or worse, provide inaccurate predictions.\n\nIn practice, this process of training a supervised model is highly procedural, with even ML experts often having to fall back on standard recipes (e.g. the libSVM guide~\\cite{libsvmguide}) in the attempt to to obtain reasonable results.\nAt scale, the problem of finding a good model is exacerbated, and conventional approaches can be prohibitively expensive. For example, sequential grid search is a popular method implemented in many software packages~\\cite{graphlabgrid,scikitlearn,caret}, but as we show, requires many more models to be trained than necessary to achieve good results.\n\n\nAt an abstract level the process of finding a good machine learning model is in some ways analogous to a query planning problem where the challenge is to construct a model from training data given a potential space of model families. \nIn the PAQ setting (illustrated in Figure~\\ref{fig:pipeline}),\nsome process must be followed to refine the choice of feature and model selection, and in practice this problem is often iterative. \nThe \\emph{PAQ planning problem} is the task of efficiently finding a high quality PAQ plan, given training data, a set of candidate statistical model families, and their configurations.\n\nA good PAQ planner will return a high quality PAQ plan efficiently. The \\emph{quality}\nof a PAQ plan is measured in terms of a statistic relevant to the predictive\ntask, such as accuracy on validation data in the classification setting or the Mean Squared Error (MSE) in the regression setting.\n\\emph{Efficiency} is measured in terms of the total time to arrive at such a model---we refer to this as \\emph{learning time}. In this work we focus on model search and hyperparameter tuning when the dimensionality of the search space is small, and training budget is also small. This scenario maps well to practical demands, given that there exist a limited set of learning algorithms and hyperparameters for a given predictive learning task, and the cost of training a model may be high, particularly in the large-scale setting. In such scenarios, PAQ planning can lead to substantial improvements in model quality. We further restrict our focus to model family selection and hyperparameter tuning, as opposed to also considering selection of appropriate featurizers. However, we believe the techniques presented here can be generalized to more complicated predictive analytic pipelines, since the number of established featurization techniques for a given domain is also limited.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=1.0\\columnwidth]{figures\/pipeline2.pdf}\n  \\caption{Finding an appropriate predictive model is a process of continuous refinement. Each stage must be carefully tuned to ensure high quality. \\mlopt automates this process.}\n  \\label{fig:pipeline}\n\\end{figure}\n\n\n\n\n\n\nIn the remainder of the paper, we explore the challenges associated with PAQ planning\nand introduce a PAQ planner for the MLbase system~\\cite{Kraska13} for declarative machine learning, called the \\textbf{T}raining-s\\textbf{u}pported\n\\textbf{P}redictive \\textbf{A}nalytic \\textbf{Q}uery planner (\\mlopt). The goal of \\mlopt in the context of MLbase is to tackle the PAQ planning problem at scale. Using advanced search techniques, bandit resource allocation, and batching optimizations, \\mlopt identifies a suitable model to satisfy a\nuser's high-level declarative query, thus differing from  previous systems~\\cite{MADLib,\nDeshpande:2006wi, Wang:2008vy, Zhang:2014tf}\nfocused on predictive analytics, which force users to perform model\nsearch themselves.  Further, although the ML community has developed techniques\nfor finding good configurations for learning models, none have focused on\napplying these techniques in the large-scale setting.  \n\n\nWith \\mlopt, we make the following contributions: \n\\begin{itemize}\n  \\item We introduce PAQs, a declarative query interface that\n  enables analysts to operate on imputed attribute values. \n  \\item We demonstrate the effectiveness of supervised ML techniques at\n  supporting PAQs, both in terms of high accuracy and efficiency, especially when compared with basic approaches.\n  \\item We describe the \\mlopt algorithm for PAQ planning which combines\n  logical optimization via model search and physical\n  optimization via batching and bandit resource allocation via runtime\n  introspection. \n\n  \\item We describe an implementation of the \\mlopt algorithm in Apache Spark, building on our earlier work on the MLbase architecture~\\cite{Kraska13}. \n  \\item We evaluate several points in the design space with respect to each\n  of our logical and physical optimizations, and demonstrate that proper selection of each can\n  dramatically improve both accuracy and efficiency. \n \n  \\item We present experimental results on large, distributed datasets up to\nterabytes in size, demonstrating that \\mlopt converges to high quality PAQ\nplans an order of magnitude faster than a simple PAQ planning strategy.\n\\end{itemize}\n\nThe remainder of this paper is organized as follows.\nSection~\\ref{sec:paq_planning} formally defines the PAQ planning problem, explains its connection to traditional database query optimization research, and introduces a standard baseline approach for the problem, and provides a\nhigh-level overview of \\mlopt. We next present details about \\mlopt's three\nmain components in Section~\\ref{sec:optimizations}, highlighting the design decisions\nfor each of these components.  Section~\\ref{sec:designeval} subsequently presents an\nempirical study of this design space and a comparison with traditional methods for solving this problem. We then present results on a large-scale\nevaluation of \\mlopt in Section~\\ref{sec:evaluation}, with each of \\mlopt's three\ncomponents tuned based on the results of the previous section. \nIn Section~\\ref{sec:related} we explore the relationship between PAQ planning and existing works related to supporting predictive and analytical workloads.\nWe conclude with Section~\\ref{sec:futureworkconclusions}, which summarizes our work and\ndiscusses future extensions to \\mlopt. \n\n\n\\section{T{\\normalsize{\\textbf U}}PAQ Design Choices}\n\\label{sec:optimizations}\n \n\n\nIn this section, we examine the design choices available to the \\mlopt planner.\nAs stated previously, we are targeting algorithms\nthat run on tens to thousands of nodes in commodity computing clusters, and training datasets\nthat fit comfortably into cluster memory---on the order of tens of gigabytes to\nterabytes.  Training of a single model to convergence on such a cluster is expected to\nrequire tens to hundreds of passes through the training data, and may take on the order of minutes.\nMoreover, with a multi-terabyte dataset, performing a sequential grid search\ninvolving even just 100 model configurations each with a budget of 100 scans of the training data could take hours to days of processing time, even assuming that the algorithm runs at memory speed.\nHence, in this regime the baseline PAQ planner is tremendously costly.\n\nGiven the design choices presented in Section~\\ref{sec:paq_planning}, we ask how these design choices might be optimized to provide fast, high quality PAQ planning.\nIn the remainder of this\nsection we present the following optimizations---advanced model search techniques, bandit resource allocation via\nruntime algorithm introspection, and physical optimization via\nbatching---that in concert provide \\mlopt with an order-of-magnitude gain in performance.\n\n\n\n\n\n\n\n\n\\subsection{Better Model Search}\nWe call the problem of finding the highest quality model from a space\nof model families and their hyperparameters the \\emph{model search} problem, and the solution to this problem is of central importance to \\mlopt.\nWe view model search as an optimization problem over a potentially\nnon-smooth, non-convex function in high dimensional space, where it is\nexpensive to evaluate the function and for which we have no closed form\nexpression for the function to be optimized (and hence cannot compute\nderivatives).  Although grid search remains the standard solution to this problem, various alternatives have been proposed for the general problem\nof derivative-free optimization, some of which are particularly tailored for\nthe model search problem.  Each of these methods provides an opportunity to\nspeed up \\mlopt's planning time, and in this section we provide a brief survey\nof the most commonly used methods.  In Section~\\ref{sec:designeval} we\nevaluate each method on several datasets to determine which method is most\nsuitable for PAQ planning.\n\nTraditional methods for derivative-free optimization include grid search (the\nbaseline choice for a PAQ planner) as well as random search, Powell's method\n\\cite{powell1964efficient}, and the Nelder-Mead method \\cite{Nelder:1965tk}.  \nGiven a hyperparameter space, grid search\nselects evenly spaced points (in linear or log space) from this space, while\nrandom search samples points uniformly at random from this space.\nPowell's method can be seen as a derivative-free analog to coordinate\ndescent, while \nthe Nelder-Mead method can be roughly interpreted as a derivative-free analog to\ngradient descent.\n\nBoth Powell's method and the Nelder-Mead method expect unconstrained search\nspaces, but function evaluations can be modified to severely penalize exploring\nout of the search space.  However, both methods require some degree of\nsmoothness in the hyperparameter space to work well, and can easily get stuck\nin local minima.  Additionally, neither method lends itself well to categorical\nhyperparameters, since the function space is modeled as continuous.  For these\nreasons, we are unsurprised that they are inappropriate methods to use in the\nmodel search problem where optimization is done over an unknown function that\nis likely non-smooth and not convex.\n\nMore recently, various methods specifically for model search have been recently\nintroduced in the ML community, including Tree-based Parzen Estimators\n(HyperOpt)~\\cite{Bergstra:2011tj}, Sequential Model-based Algorithm Configuration (Auto-WEKA)~\\cite{Thornton:2013ea} and Gaussian Process based methods, e.g.,\nSpearmint~\\cite{Snoek:2012vl}. These algorithms all share the property that they can\nsearch over spaces which are nested (e.g. multiple model families) and accept\ncategorical hyperparameters (e.g. regularization method). HyperOpt begins with a\nrandom search and then probabilistically samples from points with more promising\nminima, Auto-WEKA builds a Random Forest model from observed hyperparameter results,\nand  Spearmint implements a Bayesian methods based on Gaussian Processes.  \n\n\n\n\n\\subsection{Bandit Resource Allocation}\n\\label{ssec:bandit}\n\nModels are not all created equal. In the context of model\nsearch, typically only a fraction of the models are of high-quality,\nwith many of the\nremaining models performing drastically worse.  Under certain\nassumptions, allocating resources among different model configurations can be\nnaturally framed as a multi-armed bandit problem~\\cite{BubeckC12}. Indeed, assume we are given a\n\\emph{fixed set} of $k$ model configurations to evaluate, as in the case of\ngrid or random search, along with a fixed budget $B$.  Then, each model can be\nviewed as an `arm' and the model search problem can be cast as a $k$-armed\nbandit problem with $T$ rounds.  At each round we perform a single iteration of\na particular model configuration, and return a reward indicating\nthe quality of the updated model, e.g., validation accuracy.  In such settings,\nmulti-armed bandit algorithms can be used to determine a scheduling policy to\nefficiently allocate resources across the $k$ model configurations.  Typically,\nthese algorithms keep a running score for each of the $k$ arms, and at each\niteration choose an arm as a function of the current scores.\n\n\\begin{algorithm}[]\n \\SetKwInOut{Input}{input}\\SetKwInOut{Output}{output} \n \\Input{Models, History}\n \\Output{FinishedModels, History, Proposals}\n  Proposal $\\gets$ []\\; \n  FinishedModels $\\gets$ []\\; \n  bestModel = getBestFromHistory(History)\\; \n  \\For{m in models}{\n     history.append(m)\\;\n     \\uIf{fullyTrained(m)}{\n       FinishedModels.append(m)\\;\n     }\n     \\ElseIf{quality(m) $ * (1+\\epsilon) > $ quality(bestModel)}{\n       proposals.append(m)\\;\n     }\n  }\n \\Return(FinishedModels, History, Proposals);\n\n \\caption{The bandit allocation strategy used by \\mlopt.}\n \\label{alg:bandit}\n\\end{algorithm}\n\nOur setting differs from this standard setting in two crucial ways.  First,\nseveral of our search algorithms select model configurations to evaluate in an\niterative fashion, so we do not have advanced access to a fixed set of $k$\nmodel configurations. Second, in addition to efficiently allocating resources,\nwe aim to return a reasonable result to a user as quickly as possible, and\nhence there is a benefit to finish training promising model configurations once\nthey have been identified. \n\nOur bandit selection strategy is a variant of the action elimination algorithm of \\cite{EvenDar:2006tf}, and to our knowledge this is the first time this algorithm has been applied to hyperparameter tuning.  Our strategy is\ndetailed in Algorithm~\\ref{alg:bandit}.  This strategy preemptively prunes\nmodels that fail to show promise of converging.  For each model\n(or batch of models), we first allocate a fixed number of iterations for\ntraining; in Algorithm~\\ref{alg:mloptplanner} the trainPartial() function\ntrains each model for PartialIters iterations.  Partially trained models are\nfed into the bandit allocation algorithm, which determines whether to train the\nmodel to completion by comparing the quality of these models to the quality of\nthe best model that has been trained to date.  Moreover, this comparison is\nperformed using a slack factor of $(1+\\epsilon)$; in our experiments we set\n$\\epsilon = .5$ and thus continue to train all models with quality \nwithin $50\\%$ of the best quality model observed so far.\nThe algorithm stops allocating further resources to models that fail this test,\nas well as to models that have already been trained to completion.\n\n\\subsection{Batching}\n\\input{batching}\n\n\n\n\n\n\n\n\\section{PAQ Planning and T{\\normalsize{\\textbf U}}PAQ}\n\\label{sec:paq_planning}\nIn this section, we define the PAQ planning problem in more detail, and describe its relationship to traditional query optimization. Then, we discuss two approaches to PAQ planning. \nThe first, which we call the baseline approach, is inspired by common practice.\nThe second approach, \\mlopt, allows us to take advantage of logical and physical optimizations in the planning process.\n\\mlopt has a rich design space, which we describe in further detail in Section~\\ref{sec:optimizations}.\nFinally, we describe how \\mlopt fits into the broader MLbase architecture.\n\n\\subsection{Defining PAQ Planning}\nFigure~\\ref{fig:paq} shows several example PAQs that, in practice, need extremely large training datasets with millions of examples each with hundreds of thousands of features to return accurate results.\nOther PAQs that need large training sets include problems in image classification,\nspeech-to-text translation, and web-scale text mining.\nAs defined in Section~\\ref{sec:intro}, PAQs can be any query where an attribute or predicate value must be imputed to complete the query. In this work we concern ourselves specifically with PAQs that can be answered based on user-supplied labeled training data, typically of the same format as the data for which values are to be imputed.\nWe focus specifically on the components of the system that are necessary to efficiently support clauses of the form shown in Section~\\ref{sec:intro}.\nWhile the strategies discussed here can operate in situations where queries have joined relations or complex aggregates, we expect that future work will explore optimizations specific to these situations.\n\nThe PAQ planner's job is to find a PAQ plan that maximizes\nsome measure of quality (e.g., in terms of goodness of fit to held-out data) in a short amount of time, where learning time is constrained by\nsome budget in terms of the number of models considered, total\nexecution time, or the number of scans over the training data.\nThe planner thus takes as input a training dataset, a description of a space\nof models to search, and some budget or stopping criterion.  The description of\nthe space of models to search includes the set of model families to search over\n(e.g., SVM, decision tree, etc.) and reasonable ranges for their associated\nhyperparameters (e.g., regularization parameter for a regularized linear model\nor maximum depth of a decision tree).  The output of a PAQ Planner is a plan  that can be applied to unlabeled\ndata points to obtain a prediction for the desired attribute. In the context of \\mlopt, this plan is a statistical model that can be applied to unseen training data.\n\nIn this work, we operate in a scenario where individual models are of\ndimensionality $d$, where $d$ is less than the total number of\nexample data points $N$. Note that $d$ can nonetheless be quite large, e.g.,\n$d=200,000$ in our large-scale speech experiments and $d=160,000$ in our large scale image experiments (see Section~\\ref{sec:evaluation}). Recall that in this paper, we focus on classification, and consider a small number of model families, $f \\in F$,\neach with several hyperparameters, $\\lambda \\in \\Lambda$.\nThese assumptions map well to reality, as there are a handful of\ngeneral-purpose classification methods that are deployed in practice. Further, we expect that these techniques will naturally apply to other supervised learning tasks---such as regression and collaborative filtering, which may only differ in terms of their definition of plan quality.\nWe evaluate the quality of each plan by computing accuracy on\nheld-out datasets, and we measure learning time as the amount of time required\nto explore a fixed number of models from some model space. In our large-scale\ndistributed experiments (see Section~\\ref{sec:evaluation}) we report parallel run times. \n\nAdditionally, in this paper we focus on model families that are trained via multiple sequential scans of the training data.  In particular, we focus\non three model families: linear Support Vector Machines (SVM), logistic regression trained via gradient descent, and nonlinear\nSVMs using random features~\\cite{RandomKitchenSink} trained via block coordinate descent. \nThis iterative sequential access pattern encompasses a wide\nrange of learning algorithms, especially in the large-scale distributed\nsetting.  For instance, efficient distributed implementations of linear\nregression~\\cite{Franklin13}, tree based models~\\cite{Panda:2009uh}, Naive\nBayes classifiers~\\cite{Franklin13}, and $k$-means clustering~\\cite{Franklin13}\nall follow this same access pattern.\n\n\\subsection{Connections to Query Optimization}\nGiven that PAQ planning is the automation of a declaratively specified task, it is natural to draw connections to decades worth of relational query optimization research when\ntackling the PAQ planning problem.  Traditional database systems invest in\nthe costly process of query planning to determine a good execution plan that can\nbe reused repeatedly upon subsequent execution of similar queries.\nWhile query planning for a PAQ involves the costly process of\nidentifying a high quality predictive model, this cost is offset by the subsequent\nability to perform near real-time PAQ evaluation.  Additionally, both types\nof query planning can be viewed as search problems, with traditional query\nplanning searching over the space of join orderings and access methods, and PAQ planning\nsearching over the space of machine learning models.\n \nThere are some notable differences between these two problems, however, leading\nto a novel set of challenges to address in the context of PAQ planning.\nFirst, unlike traditional database queries, PAQs do not have unique answers\ndue to the inherent uncertainty in predictive models learned from finite\ndatasets.  Hence, PAQ planning focuses on both quality and efficiency\n(compared to just efficiency for traditional query planning), and needs to\nbalance between these goals when they conflict.  Second, the search space\nfor PAQs is not endowed with well-defined algebraic properties, as it\nconsists of possibly unrelated model families and feature extractors, each\nwith its own access patterns and hyperparameters.  Third, evaluating a\ncandidate query plan is expensive and in this context involves learning\nthe parameters of a statistical model.  Learning the parameters of a\nsingle model can involve upwards of hundreds of passes over the input\ndata, and there exist few heuristics to estimate the effectiveness of a\nmodel before this costly training process.\n\nNow, we turn our attention to algorithms for PAQ planning.\n\n\\subsection{Baseline PAQ Planning}\n\n\n\nThe conventional\napproach to PAQ planning is sequential grid search~\\cite{graphlabgrid,scikitlearn,caret}. For instance, consider the\ntag prediction example in Figure~\\ref{fig:photos}, in which the PAQ is processed\nvia an underlying classification model trained on \\texttt{LabeledPhotos}.\nMoreover, consider a single ML model family for binary\nclassification---logistic regression---which has two hyperparameters: learning\nrate and regularization.  Sequential grid search divides the\nhyperparameter space into a grid and iteratively trains models at these grid\npoints\n\n\nGrid search has several shortcomings.  First, the results of previous\niterations in the sequence of grid points are not used to inform future iterations of search.  Second, the\ncurse of dimensionality limits the usefulness of this method in high\ndimensional hyperparameter spaces. Third, grid points may not represent a good\napproximation of global minima---true global minima may be hidden between grid\npoints, particularly in the case of a very coarse grid.  Nonetheless, sequential grid search \nis commonly used in practice, and is a natural baseline for PAQ planners.\n\nIn Algorithm~\\ref{alg:naiveplanner}, we show the logic encapsulated in such a baseline PAQ planner.\nIn this example, the budget is the total number of models to train.\n\n\t\\begin{algorithm}[h]\n\t \\SetKwInOut{Input}{input}\\SetKwInOut{Output}{output} \n\t \\Input{LabeledData, ModelSpace, Budget}\n\t \\Output{BestModel}\t \n\t bestModel $\\gets$ $\\emptyset$\\;\n\t grid $\\gets$ gridPoints(ModelSpace, Budget)\\;\n\t \\While{Budget $>$ 0}{\n\t   proposal $\\gets$ nextPoint(grid)\\;\n\t   model $\\gets$ train(proposal, LabeledData)\\;\n\t   \\If{quality(model) $>$ quality(bestModel)}\n\t     {bestModel $\\gets$ model\\;}\n\t   Budget $\\gets$ Budget $-$ 1\\;}\n\t \\Return{bestModel};\n\t \n\t \\caption{A baseline PAQ planning procedure with conventional grid search. The function ``gridPoints'' returns a coarse grid over the dimensions of model space, where the total number of grid points is determined by the budget.}\n\t \\label{alg:naiveplanner}\n\t\\end{algorithm}\n\n\\subsection{T{\\normalsize{\\textbf U}}PAQ Planning}\nAs discussed in the previous section, grid search is a suboptimal search method\ndespite its popularity.  Moreover, from a systems perspective, the algorithm illustrated in\nAlgorithm~\\ref{alg:naiveplanner} \nhas additional drawbacks beyond those of grid search. In\nparticular, this procedure performs sequential model training and also treats\nthe training of each model as a black-box procedure.\n\n\\begin{algorithm}[]\n \\SetKwInOut{Input}{input}\\SetKwInOut{Output}{output} \n \\Input{LabeledData, ModelSpace, Budget, PartialIters, BatchSize}\n \\Output{BestModel}\n bestModel $\\gets$ $\\emptyset$\\;\n history $\\gets$ []\\;\n proposals $\\gets$ []\\;\n freeSlots $\\gets$ batchSize\\;\n\n \\While{Budget $>$ 0}{\n   freeSlots $\\gets$ batchSize - length(proposals)\\;\n   proposals $\\gets$ proposals + proposeModels(freeSlots, ModelSpace, history)\\tcp*{Model Search}\n   models $\\gets$ trainPartial(proposals, LabeledData, PartialIters)\\tcp*{Batching}\n   Budget $\\gets$ Budget $-$ len(models)$*$PartialIters\\;\n   \n   (finishedModels, history, proposals) $\\gets$ banditAllocation(models, history)\\tcp*{Bandits}\n   \n   \\For{m in finishedModels}{\n     \\If{quality(m) $>$ quality(bestModel)}{\n       bestModel $\\gets$ m\\;\n     }\n   }\n }\n \\Return(bestModel);\n  \n \\caption{The planning procedure used by \\mlopt.}\n \\label{alg:mloptplanner}\n\\end{algorithm}\n\nIn contrast, we propose the \\mlopt algorithm, described in Algorithm~\\ref{alg:mloptplanner}, to\naddress all three of these shortcomings via logical and physical optimizations.\nFirst, the \\mlopt algorithm allows for more \\emph{sophisticated search strategies}.\nLine 7 shows that our model search procedure can now use training history as\ninput. Here, ``proposeModel'' can be an arbitrary model search procedure.\nSecond, our algorithm performs \\emph{batching} to train multiple models\nsimultaneously (Line 8). Third, our algorithm deploys \\emph{bandit resource allocation via\nruntime inspection} to make on-the-fly decisions. Specifically, the algorithm\ncompares the quality of the models currently being trained with historical\ninformation about the training process, and determines which of the current\nmodels should be trained further (Line 10).  \n\nThese three optimizations are discussed in detail in Section~\\ref{sec:optimizations}, with a focus on the design space for each of them.\nIn Section~\\ref{sec:designeval} we then evaluate the options in this design\nspace experimentally, and then in Section~\\ref{sec:evaluation} compare the\nbaseline algorithm (Algorithm \\ref{alg:naiveplanner}) to \\mlopt running with\ngood choices for search method, batch size, and bandit allocation criterion,\ni.e., choices informed by the results of Section~\\ref{sec:designeval}.\n\nBefore exploring the design space for the \\mlopt algorithm, we first describe how \\mlopt fits into a larger system to support PAQs.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.8\\columnwidth]{figures\/tupaqarch.pdf}\n  \\caption{The \\mlopt planner is a critical component of the MLbase~\\protect\\cite{Kraska13} system for simplified large scale machine learning. \\mlopt interacts with a distributed run-time and existing machine learning algorithms to efficiently find a PAQ plan which yields high quality predictions.}\n  \\label{fig:mlbarch}\n\\end{figure}\n\n\\subsection{T{\\normalsize{\\textbf U}}PAQ and the MLbase Architecture}\n\n\\mlopt lies at the heart of MLbase~\\cite{Kraska13}, a novel system designed to simplify the process of implementing and using scalable machine learning techniques.\nBy giving users a declarative interface for machine learning tasks, the problem of hyperparameter tuning and feature selection can be pushed down into the system. The architecture of this system is shown in Figure~\\ref{fig:mlbarch}.\n\nAt the center of the system, some optimizer or planner must be able to quickly identify a suitable model for supporting predictive queries. \nWe note that the system described in this paper introduces some key architectural differences compared with\nthe original MLbase architecture in \\cite{Kraska13}.\nIn particular, we make the concept of a ``PAQ planner'' explicit, and introduce a catalog for PAQ plans. \nWhen a new PAQ arrives, it is passed to the planner which determines whether a new PAQ plan needs to be created.\nThe entire system is built upon Apache Spark, a cluster compute system designed for iterative computing~\\cite{Spark}, and we leverage MLlib, and other components present in Apache Spark, as well as MLI~\\cite{MLI}.\n\n\n\n\\section{Related Work}\n\\label{sec:related}\nThere has been a recent proliferation of systems designed for low-level,\nad-hoc distributed predictive analytics, e.g., Apache Spark~\\cite{Spark},\nGraphLab~\\cite{powergraph}, Stratosphere~\\cite{Alexandrov:2014jb}, but none\nof these provide users with a declarative interface with which to specify predictive queries.\n\nIn terms of system-level optimization, both Kumar et. al.~\\cite{Kumar:2013vm}\nand Canny et. al.~\\cite{Canny:2013th} discuss batching as an optimization for\nspeeding up machine learning systems.  However, \\cite{Kumar:2013vm} discusses\nthis technique in the context of automatic feature selection, an important\nproblem but distinct from PAQ planning, while \\cite{Canny:2013th} explores this\ntechnique in the context of parameter exploration, model tuning, \nensemble methods and cross validation.  We explore the impact of batching in a\ndistributed setting at greater depth in this work, and present a novel\napplication of this technique to the PAQ planning problem.\n\nIn the data mining and machine learning communities, most related to \\mlopt is\nAuto-WEKA \\cite{Thornton:2013ea}.  As the name suggests, Auto-WEKA aims to\nautomate the use of Weka~\\cite{weka} by applying recent derivative-free\noptimization algorithms, in particular Sequential Model-based Algorithm\nConfiguration (SMAC)~\\cite{Hutter:2011wn}, to the PAQ planning problem.  In\nfact, their proposed algorithm is one of the many optimization algorithms we\nuse as part of \\mlopt.  However, in contrast to \\mlopt, Auto-WEKA focuses on\nsingle node performance and does not optimize the parallel execution of\nalgorithms.  Moreover, Auto-WEKA treats algorithms as black boxes to be\nexecuted and observed, while our system takes advantage of knowledge of\nalgorithm execution from both a statistical and physical perspective.\n\nIn addition to the SMAC algorithm of Auto-WEKA, other search algorithms have been recently been proposed.  In\nBergstra et. al.~\\cite{Bergstra:2012ux}, the effectiveness of random search for hyperparameter\ntuning is established, while Bergstra et. al.~\\cite{Bergstra:2011tj} proposes a search method\nthat performs a random search that is refined with new\ninformation, called Tree-structured Parzen Estimation (TPE).  We make use of both methods in our system.\nSnoek et. al.~\\cite{Snoek:2012vl} explore the use of Gaussian Processes for the\nPAQ planning problem, and propose a variety of search algorithms, including an\nalgorithm that accounts for improvement per time unit, and another extension\ntargeting parallel implementations in which several new model configurations\nare proposed at each iteration.  However, model training is nonetheless\nconsidered a black box, and moreover, we found that their algorithms,\ncollectively called Spearmint, often run for several minutes per iteration even\nwhen considering a moderate number of candidate models, which is too long to be\npractical in many scenarios. \n\nIn contrast to these recent works, the field of derivative-free optimization\nhas a long history of optimizing functions for which derivatives cannot be\ncomputed~\\cite{Conn:2009tl}. Our evaluation of these algorithms on the PAQ\nplanning problem suggests that they are not well-suited for this task,\npotentially due to the lack of smoothness of the (unknown) PAQ planning\nfunction that we are optimizing. \n\nThere are also several proprietary and open-source systems providing machine\nlearning functionality with varying degrees of automation.  Google Predict\n\\cite{google_prediction} is Google's proprietary web-service for prediction\nproblems with some degree of automation, yet it restricts the maximum training\ndata-size to 250MB and the internals of the system are largely unknown.  \n\nWeka~\\cite{weka}, MLlib~\\cite{Franklin13}, Vowpal Wabbit~\\cite{vwsgd}, Hyracks~\\cite{Hyracks} and\nMahout~\\cite{mahout} are notable open-source ML libraries.  These systems (all distributed\nwith the exception of Weka),\nalong with proprietary projects such as SystemML~\\cite{SystemML}, all focus on\ntraining single models.\nIn contrast, \\mlopt is designed explicitly for PAQ planning and hyperparameter\ntuning at scale.  In theory our proposed methods could work with these systems,\nthough such integration would require these systems to\nexpose the access patterns of the algorithms they make available to\n\\mlopt.\n\n\n\n\\section{Introduction}\nThe \\textit{proceedings} are the records of a conference.\nACM seeks to give these conference by-products a uniform,\nhigh-quality appearance.  To do this, ACM has some rigid\nrequirements for the format of the proceedings documents: there\nis a specified format (balanced  double columns), a specified\nset of fonts (Arial or Helvetica and Times Roman) in\ncertain specified sizes (for instance, 9 point for body copy),\na specified live area (18 $\\times$ 23.5 cm [7\" $\\times$ 9.25\"]) centered on\nthe page, specified size of margins (1.9 cm [0.75\"]) top, (2.54 cm [1\"]) bottom\nand (1.9 cm [.75\"]) left and right; specified column width\n(8.45 cm [3.33\"]) and gutter size (.83 cm [.33\"]).\n\nThe good news is, with only a handful of manual\nsettings\\footnote{Two of these, the {\\texttt{\\char'134 numberofauthors}}\nand {\\texttt{\\char'134 alignauthor}} commands, you have\nalready used; another, {\\texttt{\\char'134 balancecolumns}}, will\nbe used in your very last run of \\LaTeX\\ to ensure\nbalanced column heights on the last page.}, the \\LaTeX\\ document\nclass file handles all of this for you.\n\nThe remainder of this document is concerned with showing, in\nthe context of an ``actual'' document, the \\LaTeX\\ commands\nspecifically available for denoting the structure of a\nproceedings paper, rather than with giving rigorous descriptions\nor explanations of such commands.\n\n\\section{The {\\secit Body} of The Paper}\nTypically, the body of a paper is organized\ninto a hierarchical structure, with numbered or unnumbered\nheadings for sections, subsections, sub-subsections, and even\nsmaller sections.  The command \\texttt{{\\char'134}section} that\nprecedes this paragraph is part of such a\nhierarchy.\\footnote{This is the second footnote.  It\nstarts a series of three footnotes that add nothing\ninformational, but just give an idea of how footnotes work\nand look. It is a wordy one, just so you see\nhow a longish one plays out.} \\LaTeX\\ handles the numbering\nand placement of these headings for you, when you use\nthe appropriate heading commands around the titles\nof the headings.  If you want a sub-subsection or\nsmaller part to be unnumbered in your output, simply append an\nasterisk to the command name.  Examples of both\nnumbered and unnumbered headings will appear throughout the\nbalance of this sample document.\n\nBecause the entire article is contained in\nthe \\textbf{document} environment, you can indicate the\nstart of a new paragraph with a blank line in your\ninput file; that is why this sentence forms a separate paragraph.\n\n\\subsection{Type Changes and {\\subsecit Special} Characters}\nWe have already seen several typeface changes in this sample.  You\ncan indicate italicized words or phrases in your text with\nthe command \\texttt{{\\char'134}textit}; emboldening with the\ncommand \\texttt{{\\char'134}textbf}\nand typewriter-style (for instance, for computer code) with\n\\texttt{{\\char'134}texttt}.  But remember, you do not\nhave to indicate typestyle changes when such changes are\npart of the \\textit{structural} elements of your\narticle; for instance, the heading of this subsection will\nbe in a sans serif\\footnote{A third footnote, here.\nLet's make this a rather short one to\nsee how it looks.} typeface, but that is handled by the\ndocument class file. Take care with the use\nof\\footnote{A fourth, and last, footnote.}\nthe curly braces in typeface changes; they mark\nthe beginning and end of\nthe text that is to be in the different typeface.\n\nYou can use whatever symbols, accented characters, or\nnon-English characters you need anywhere in your document;\nyou can find a complete list of what is\navailable in the \\textit{\\LaTeX\\\nUser's Guide}\\cite{Lamport:LaTeX}.\n\n\\subsection{Math Equations}\nYou may want to display math equations in three distinct styles:\ninline, numbered or non-numbered display.  Each of\nthe three are discussed in the next sections.\n\n\\subsubsection{Inline (In-text) Equations}\nA formula that appears in the running text is called an\ninline or in-text formula.  It is produced by the\n\\textbf{math} environment, which can be\ninvoked with the usual \\texttt{{\\char'134}begin. . .{\\char'134}end}\nconstruction or with the short form \\texttt{\\$. . .\\$}. You\ncan use any of the symbols and structures,\nfrom $\\alpha$ to $\\omega$, available in\n\\LaTeX\\cite{Lamport:LaTeX}; this section will simply show a\nfew examples of in-text equations in context. Notice how\nthis equation: \\begin{math}\\lim_{n\\rightarrow \\infty}x=0\\end{math},\nset here in in-line math style, looks slightly different when\nset in display style.  (See next section).\n\n\\subsubsection{Display Equations}\nA numbered display equation -- one set off by vertical space\nfrom the text and centered horizontally -- is produced\nby the \\textbf{equation} environment. An unnumbered display\nequation is produced by the \\textbf{displaymath} environment.\n\nAgain, in either environment, you can use any of the symbols\nand structures available in \\LaTeX; this section will just\ngive a couple of examples of display equations in context.\nFirst, consider the equation, shown as an inline equation above:\n\\begin{equation}\\lim_{n\\rightarrow \\infty}x=0\\end{equation}\nNotice how it is formatted somewhat differently in\nthe \\textbf{displaymath}\nenvironment.  Now, we'll enter an unnumbered equation:\n\\begin{displaymath}\\sum_{i=0}^{\\infty} x + 1\\end{displaymath}\nand follow it with another numbered equation:\n\\begin{equation}\\sum_{i=0}^{\\infty}x_i=\\int_{0}^{\\pi+2} f\\end{equation}\njust to demonstrate \\LaTeX's able handling of numbering.\n\n\\subsection{Citations}\nCitations to articles \\cite{bowman:reasoning,\nclark:pct, braams:babel, herlihy:methodology},\nconference proceedings \\cite{clark:pct} or\nbooks \\cite{salas:calculus, Lamport:LaTeX} listed\nin the Bibliography section of your\narticle will occur throughout the text of your article.\nYou should use BibTeX to automatically produce this bibliography;\nyou simply need to insert one of several citation commands with\na key of the item cited in the proper location in\nthe \\texttt{.tex} file \\cite{Lamport:LaTeX}.\nThe key is a short reference you invent to uniquely\nidentify each work; in this sample document, the key is\nthe first author's surname and a\nword from the title.  This identifying key is included\nwith each item in the \\texttt{.bib} file for your article.\n\nThe details of the construction of the \\texttt{.bib} file\nare beyond the scope of this sample document, but more\ninformation can be found in the \\textit{Author's Guide},\nand exhaustive details in the \\textit{\\LaTeX\\ User's\nGuide}\\cite{Lamport:LaTeX}.\n\nThis article shows only the plainest form\nof the citation command, using \\texttt{{\\char'134}cite}.\nThis is what is stipulated in the SIGS style specifications.\nNo other citation format is endorsed or supported.\n\n\\subsection{Tables}\nBecause tables cannot be split across pages, the best\nplacement for them is typically the top of the page\nnearest their initial cite.  To\nensure this proper ``floating'' placement of tables, use the\nenvironment \\textbf{table} to enclose the table's contents and\nthe table caption.  The contents of the table itself must go\nin the \\textbf{tabular} environment, to\nbe aligned properly in rows and columns, with the desired\nhorizontal and vertical rules.  Again, detailed instructions\non \\textbf{tabular} material\nis found in the \\textit{\\LaTeX\\ User's Guide}.\n\nImmediately following this sentence is the point at which\nTable 1 is included in the input file; compare the\nplacement of the table here with the table in the printed\ndvi output of this document.\n\n\\begin{table}\n\\centering\n\\caption{Frequency of Special Characters}\n\\begin{tabular}{|c|c|l|} \\hline\nNon-English or Math&Frequency&Comments\\\\ \\hline\n\\O & 1 in 1,000& For Swedish names\\\\ \\hline\n$\\pi$ & 1 in 5& Common in math\\\\ \\hline\n\\$ & 4 in 5 & Used in business\\\\ \\hline\n$\\Psi^2_1$ & 1 in 40,000& Unexplained usage\\\\\n\\hline\\end{tabular}\n\\end{table}\n\nTo set a wider table, which takes up the whole width of\nthe page's live area, use the environment\n\\textbf{table*} to enclose the table's contents and\nthe table caption.  As with a single-column table, this wide\ntable will ``float\" to a location deemed more desirable.\nImmediately following this sentence is the point at which\nTable 2 is included in the input file; again, it is\ninstructive to compare the placement of the\ntable here with the table in the printed dvi\noutput of this document.\n\n\n\\begin{table*}\n\\centering\n\\caption{Some Typical Commands}\n\\begin{tabular}{|c|c|l|} \\hline\nCommand&A Number&Comments\\\\ \\hline\n\\texttt{{\\char'134}alignauthor} & 100& Author alignment\\\\ \\hline\n\\texttt{{\\char'134}numberofauthors}& 200& Author enumeration\\\\ \\hline\n\\texttt{{\\char'134}table}& 300 & For tables\\\\ \\hline\n\\texttt{{\\char'134}table*}& 400& For wider tables\\\\ \\hline\\end{tabular}\n\\end{table*}\n\n\\subsection{Figures}\nLike tables, figures cannot be split across pages; the\nbest placement for them\nis typically the top or the bottom of the page nearest\ntheir initial cite.  To ensure this proper ``floating'' placement\nof figures, use the environment\n\\textbf{figure} to enclose the figure and its caption.\n\nThis sample document contains examples of \\textbf{.eps}\nand \\textbf{.ps} files to be displayable with \\LaTeX.  More\ndetails on each of these is found in the \\textit{Author's Guide}.\n\n\\begin{figure}\n\\centering\n\\epsfig{file=fly.eps}\n\\caption{A sample black and white graphic (.eps format).}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\epsfig{file=fly.eps, height=1in, width=1in}\n\\caption{A sample black and white graphic (.eps format)\nthat has been resized with the \\texttt{epsfig} command.}\n\\end{figure}\n\n\nAs was the case with tables, you may want a figure\nthat spans two columns.  To do this, and still to\nensure proper ``floating'' placement of tables, use the environment\n\\textbf{figure*} to enclose the figure and its caption.\nand don't forget to end the environment with\n{figure*}, not {figure}!\n\n\\begin{figure*}\n\\centering\n\\epsfig{file=flies.eps}\n\\caption{A sample black and white graphic (.eps format)\nthat needs to span two columns of text.}\n\\end{figure*}\n\nNote that either {\\textbf{.ps}} or {\\textbf{.eps}} formats are\nused; use\nthe \\texttt{{\\char'134}epsfig} or \\texttt{{\\char'134}psfig}\ncommands as appropriate for the different file types.\n\n\\begin{figure}\n\\centering\n\\psfig{file=rosette.ps, height=1in, width=1in,}\n\\caption{A sample black and white graphic (.ps format) that has\nbeen resized with the \\texttt{psfig} command.}\n\\vskip -6pt\n\\end{figure}\n\n\\subsection{Theorem-like Constructs}\nOther common constructs that may occur in your article are\nthe forms for logical constructs like theorems, axioms,\ncorollaries and proofs.  There are\ntwo forms, one produced by the\ncommand \\texttt{{\\char'134}newtheorem} and the\nother by the command \\texttt{{\\char'134}newdef}; perhaps\nthe clearest and easiest way to distinguish them is\nto compare the two in the output of this sample document:\n\nThis uses the \\textbf{theorem} environment, created by\nthe\\linebreak\\texttt{{\\char'134}newtheorem} command:\n\\newtheorem{theorem}{Theorem}\n\\begin{theorem}\nLet $f$ be continuous on $[a,b]$.  If $G$ is\nan antiderivative for $f$ on $[a,b]$, then\n\\begin{displaymath}\\int^b_af(t)dt = G(b) - G(a).\\end{displaymath}\n\\end{theorem}\n\nThe other uses the \\textbf{definition} environment, created\nby the \\texttt{{\\char'134}newdef} command:\n\\newdef{definition}{Definition}\n\\begin{definition}\nIf $z$ is irrational, then by $e^z$ we mean the\nunique number which has\nlogarithm $z$: \\begin{displaymath}{\\log e^z = z}\\end{displaymath}\n\\end{definition}\n\nTwo lists of constructs that use one of these\nforms is given in the\n\\textit{Author's  Guidelines}.\n \nThere is one other similar construct environment, which is\nalready set up\nfor you; i.e. you must \\textit{not} use\na \\texttt{{\\char'134}newdef} command to\ncreate it: the \\textbf{proof} environment.  Here\nis a example of its use:\n\\begin{proof}\nSuppose on the contrary there exists a real number $L$ such that\n\\begin{displaymath}\n\\lim_{x\\rightarrow\\infty} \\frac{f(x)}{g(x)} = L.\n\\end{displaymath}\nThen\n\\begin{displaymath}\nl=\\lim_{x\\rightarrow c} f(x)\n= \\lim_{x\\rightarrow c}\n\\left[ g{x} \\cdot \\frac{f(x)}{g(x)} \\right ]\n= \\lim_{x\\rightarrow c} g(x) \\cdot \\lim_{x\\rightarrow c}\n\\frac{f(x)}{g(x)} = 0\\cdot L = 0,\n\\end{displaymath}\nwhich contradicts our assumption that $l\\neq 0$.\n\\end{proof}\n\nComplete rules about using these environments and using the\ntwo different creation commands are in the\n\\textit{Author's Guide}; please consult it for more\ndetailed instructions.  If you need to use another construct,\nnot listed therein, which you want to have the same\nformatting as the Theorem\nor the Definition\\cite{salas:calculus} shown above,\nuse the \\texttt{{\\char'134}newtheorem} or the\n\\texttt{{\\char'134}newdef} command,\nrespectively, to create it.\n\n\\subsection*{A {\\secit Caveat} for the \\TeX\\ Expert}\nBecause you have just been given permission to\nuse the \\texttt{{\\char'134}newdef} command to create a\nnew form, you might think you can\nuse \\TeX's \\texttt{{\\char'134}def} to create a\nnew command: \\textit{Please refrain from doing this!}\nRemember that your \\LaTeX\\ source code is primarily intended\nto create camera-ready copy, but may be converted\nto other forms -- e.g. HTML. If you inadvertently omit\nsome or all of the \\texttt{{\\char'134}def}s recompilation will\nbe, to say the least, problematic.\n\n\\section{Conclusions}\nThis paragraph will end the body of this sample document.\nRemember that you might still have Acknowledgments or\nAppendices; brief samples of these\nfollow.  There is still the Bibliography to deal with; and\nwe will make a disclaimer about that here: with the exception\nof the reference to the \\LaTeX\\ book, the citations in\nthis paper are to articles which have nothing to\ndo with the present subject and are used as\nexamples only.\n\n\\section{Acknowledgments}\nThis section is optional; it is a location for you\nto acknowledge grants, funding, editing assistance and\nwhat have you.  In the present case, for example, the\nauthors would like to thank Gerald Murray of ACM for\nhis help in codifying this \\textit{Author's Guide}\nand the \\textbf{.cls} and \\textbf{.tex} files that it describes.\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSuppose $B_-$ and $B_+$ are closed smooth manifolds and that $DB_\\pm\\rightarrow B_\\pm$ are disk bundles over them, possibly of different ranks.  Suppose in addition that the boundaries $\\partial DB_\\pm$ of $DB_\\pm$ are diffeomorphic, say via a diffeomorphism $f:\\partial DB_-\\rightarrow \\partial DB_+$.  Then we may form a smooth closed manifold $M = DB_- \\cup_f DB_+$.  A manifold diffeomorphic to one obtained from this construction is called a \\textit{double disk bundle}.  For example, $\\mathbb{R}P^2$ is a double disk bundle, for it is a union of a disk and a closed M\\\"{o}bius band.  That is, $\\mathbb{R}P^2$ is a union of a trivial $2$-disk bundle over a point together with non-trivial $1$-disk bundle over $S^1$.\n\nDouble disk bundles arise naturally in many diverse fields of geometry and topology.  We refer the reader to the introduction of \\cite{DeGGKe} for numerous examples of this.  Our main interest stems from Grove's Double Soul Conjecture \\cite{Gr}.\n\n\\begin{conjecture}[Double Soul Conjecture]\\label{con:DSC}  Suppose $M$ is a closed simply connected manifold which admits a Riemannian metric of non-negative sectional curvature.  Then $M$ is a double disk bundle.\n\\end{conjecture}\n\nEvidence for this conjecture includes the fact that cohomogeneity one manifolds (and free isometric quotients by a sub-action of the cohomogeneity one action \\cite{Wi}), which are one of two main building blocks for non-negatively curved manifolds, admit such a structure \\cite{Mo, GGZa}.  In addition, Cheeger \\cite{Ch} showed that the connect sum of two compact rank one symmetric spaces (CROSS) admits a metric of non-negative sectional curvature.  As a CROSS with a small open ball removed has the structure of a disk bundle, these manifolds also verify the double soul conjecture.  In addition, Cheeger and Gromoll's Soul Theorem \\cite{ChGr} gives an analogous theorem for non-compact complete Riemannian manifolds of non-negative sectional curvature.  The conjecture has also been verified for many other examples, including all known simply connected positively curved manifolds \\cite[Theorem 3.3]{DeGGKe}, simply connected biquotients in dimension at most 7 \\cite{GAGu}, and simply connected homogeneous spaces of dimension at most $10$ \\cite{GAGu}.  We recall that a biquotient is the quotient of a Riemannian homogeneous space by a free isometric action, and comprise the other main building block of non-negatively curved manifolds.\n\nThe conjecture also implies some classification results.  For example, if true, then it would follow that our known list of non-negatively curved simply connected $4$ and $5$-dimensional manifolds is complete \\cite[Theorem 1.1]{GeRa},\\cite[Theorem B]{DeGGKe}.\n\nGrove \\cite{Gr} notes the the natural generalization of Conjecture \\ref{con:DSC} to non-simply connected manifolds is false: the Poincar\\'e dodecahedral space $S^3\/I^\\ast$ admits a positively curved Riemannian metric, but does not support a double disk bundle structure.  However, this was previously the only known counterexample to the generalized conjecture.  As such, it is natural to search for more, with various topological and geometric properties.  Our main result supplies infinitely many counterexamples to the generalized conjecture, on opposite ends of the non-negative curvature landscape.\n\n\n\n\\begin{theorem}\\label{thm:main}There are infinitely many closed Riemannian $3$-manifolds of positive sectional curvature which are not double disk bundles.  In addition, there are infinitely many closed flat manifolds which are not double disk bundles.\n\\end{theorem}\n\nThe $3$-manifold family consists of infinitely many non-trivial isometric quotients of a round $S^3$.  The homogeneous spaces $S^3\/I^\\ast$ (which was first found by Grove), $S^3\/O^\\ast, S^3\/T^\\ast$, where $I^\\ast, O^\\ast$, and $T^\\ast$ are the binary isocosahedral, octahedral, and tetrahedral groups are among these examples.\n\nIn fact, $S^3\/I^\\ast$, $S^3\/O^\\ast$, and $S^3\/T^\\ast$ are the only homogeneous spaces among our examples.  It is thus natural to wonder if there are more.  This leads to the obvious question:\n\n\\begin{question}\\label{q:homo}  Are there infinitely many homogeneous spaces which are not double disk bundles?\n\\end{question}\n\nGiven that the three homogeneous examples of Theorem \\ref{thm:main} are quotients of $S^3$, one is tempted to answer Question \\ref{q:homo} by looking at homogeneous quotients of spheres of higher dimension.  However, we prove that $S^3\/I^\\ast$, $S^3\/O^\\ast$, and $S^3\/T^\\ast$ are the only homogeneous quotients of a sphere, in any dimension, which are not double disk bundles; see Proposition \\ref{prop:cantwork}.  \n\nThe other infinite family, the closed flat manifolds, are precisely those with trivial first homology group.  The construction of such flat manifolds is rather abstract, so we have been unable to determine which dimensions these examples appear.  However, we can show they exist in arbitrarily large dimensions.\n\nWe stress that all of our examples have non-trivial fundamental groups, so the Double Soul Conjecture remains open.  In fact, all of our examples have non-nilpotent fundamental groups, the the generalized Double Soul Conjecture is still open for nilpotent manifolds.\n\nWe now give an outline of the proof of Theorem \\ref{thm:main}, beginning with the three-dimensional examples.  We first prove that if $M^3$ has a metric of positive sectional curvature and is a double disk bundle, then it must have a double disk bundle structure where the common boundary $\\partial DB_-\\cong \\partial DB_+$ is diffeomorphic to a sphere $S^2$ or to a torus $T^2$.  We then classify all disk bundles whose total space has boundary diffeomorphic to $S^2$ or $T^2$, and then consider all possible ways of gluing these together.  The double disk bundle decomposition lends itself to the use of the Seifert-van Kampen Theorem, so we are able to compute presentations for all the resulting fundamental groups.  The end conclusion is that a positively curved $M^3$ admits a double disk bundle decomposition if and only if it is a lens space or a particular $\\mathbb{Z}\/2\\mathbb{Z}$ quotient of a lens space, a so-called prism manifold.  From the known classification of fundamental groups of spherical $3$-manifolds \\cite[Section 7.5]{Wo}, we obtain infinitely many examples which are not double disk bundles.  It is worth noting that the examples we find are the only $3$-dimensional counterexamples to the double soul conjecture, even under the weaker assumption that $M$ has a Riemannian metric of non-negative sectional curvature; see Remark \\ref{rem:nonneg}.\n\nFor the flat examples, enumerating all the possibilities for the common boundary $\\partial DB_-\\cong \\partial DB_+$ is not feasible, so we proceed differently.   We first show in Proposition \\ref{prop:flat} that for any manifold covered by a contractible manifold, any double disk bundle decomposition must have both disk bundles of rank $1$.   On the other hand, we also establish (Proposition \\ref{prop:onecodim0}) that if a manifold admits a double disk bundle structure with at least one double disk bundle has rank $1$, then the manifold must have a non-trivial double cover, which in turn implies that the first homology group surjects onto $\\mathbb{Z}\/2\\mathbb{Z}$.  Thus, any flat manifold with trivial first homology group cannot be a double disk bundle.  Such flat manifolds have been constructed by Igor Belegradek \\cite{Be}, providing the examples.\n\nAn outline of the paper follows.  In Section 2, we cover the required background and set up notation.  Section 3 contains general results on the topology of double disk bundles especially in the case where at least one disk bundle has rank $1$.  In Section 4, we classify the non-negatively curved $3$-manifolds which are double disk bundles, finding that some positively curved examples are not double disk bundles.  Finally, Section 5 contains the results concerning flat manifolds.\n\nThe research is partially supported by NSF DMS-2105556.  We are grateful for the support.  We would also like to thank Martin Kerin and Karsten Grove for numerous comments on an earlier version of this article.\n\n\n\n\\section{Background and Notation}  Suppose $B_-$ and $B_+$ are closed manifolds and that $D^{\\ell_\\pm + 1} \\rightarrow DB_\\pm\\rightarrow B_\\pm$ are disk bundles.  We assume their boundaries are diffeomorphic, say by a diffeomorphism $f:\\partial DB_-\\rightarrow \\partial DB_+$.  Then we can form the closed manifold $M = DB_-\\cup_f DB_+$ by gluing $DB_-$ and $DB_+$ along their boundary.  A manifold obtained via this construction is called a \\textit{double disk bundle}.\n\nRestricting the projection maps to their respective boundaries, we obtain sphere bundles $S^{\\ell_{\\pm}}\\rightarrow \\partial DB_\\pm\\rightarrow B_\\pm$.  The numbers $\\ell_\\pm\\geq 0$ will always refer to the dimension of these fiber spheres.  We will use $L$ to denote the diffeomorphism type of the common boundary.  We will borrow language from the field of Singular Riemannian Foliations, and refer to $L$ as the regular leaf and the $B_\\pm$ as the singular leaves.\n\n\n\nAs was shown in \\cite[Proposition 4.1]{DeGGKe}, if a connected closed  manifold $M$ admits a double disk bundle decomposition, then it necessarily admits one where both $B_\\pm$ are connected.   Thus we can and will always assume that in any double disk bundle decomposition, both singular leaves $B_\\pm$ are connected.  Using the sphere bundles $S^{\\ell_\\pm}\\rightarrow L\\rightarrow B_\\pm$, the condition that both $B_\\pm$ are connected implies that $L$ has at most $2$ components, and that $L$ is connected unless $B_-$ and $B_+$ are diffeomorphic, $\\ell_- = \\ell_+ = 0$, and $L\\cong S^0\\times B_-\\cong S^0\\times B_+$.\n\n\n\nThe decomposition of $M$ into two disk bundles is ideal for applying the Mayer-Vietoris sequence in cohomology, as well as the Seifert-van Kampen theorem for fundamental groups, at least when $L$ is connected.  In this context, we note that contracting the fiber disks in either $DB_\\pm$ provides a deformation retract of $DB_\\pm$ to $B_\\pm$, and the inclusion map $L\\cong \\partial DB_\\pm\\subseteq DB_\\pm$ becomes homotopic to the sphere bundle projection $L\\rightarrow B_\\pm$ under this deformation retract.\n\n\n\n\n\n\n\\section{Some general structure results for double disk bundles}\n\nIn this section, we will collect several needed facts regarding the relationship between the fiber sphere dimensions $\\ell_\\pm$ and coverings.  We begin with some general structure results where at least one $\\ell_\\pm = 0$.\n\n\\begin{lemma}\\label{lem:involution}  Suppose $S^{0}\\rightarrow L\\rightarrow B$ is a sphere bundle with $\\ell = 0$ and $B$ a connected smooth manifold.  There is a smooth free involution $\\sigma:L\\rightarrow L$ with $L\/\\sigma$ diffeomorphic to $B$.\n\n\\end{lemma}\n\n\\begin{proof}Because $S^0$ consists of two points, the sphere bundle is nothing but a double cover.  If $L$ is disconnected, it follows that $L\\cong S^0\\times B$ and the required involution $\\sigma$ simply interchanges the two copies of $B$.\n\nOn the other hand, if $L$ is connected, the covering $L\\rightarrow B$ is characterized by an index $2$-subgroup of $\\pi_1(B)$, which is necessarily normal.  Hence, the covering is regular, so the deck group is isomorphic to $\\mathbb{Z}\/2\\mathbb{Z}$.  Then one can take $\\sigma$ to be the non-trivial element of the deck group.\n\n\\end{proof}\n\n\\begin{proposition} \\label{prop:onecodim0}  Suppose $M$ is a connected manifold and  $M = DB_-\\cup_f DB_+$ is a double disk bundle with $\\ell_- = 0$.  Then $M$ admits a non-trivial double cover of the form $\\overline{M} = DB_+\\cup_g DB_+$ for some diffeomorphism $g:L\\rightarrow L$.  That is, $\\overline{M}$ has a double disk bundle decomposition where each half is a copy of $DB_+$.\n\n\\end{proposition}\n\n\\begin{proof}  Because $\\ell_-=0$, Lemma \\ref{lem:involution} gives a free involution $\\sigma:L\\rightarrow L$ with quotient $B_-$.  We now form $\\overline{M}$ as the the union $$\\overline{M} = (DB_+\\times \\{-1\\}) \\cup_{\\sigma\\circ f} L\\times [-1,1] \\cup_f (DB_+\\times \\{1\\}),$$ where $DB_+\\times \\{-1\\}$ is glued to $L\\times \\{-1\\}$ and $DB_+\\times \\{1\\}$ is glued to $L\\times \\{1\\}$.  From \\cite[Chapter VI, Section 5]{Ko}, the union $(DB_+\\times \\{-1\\})\\cup_{\\sigma\\circ f} L\\times [-1,1]$ is diffeomorphic to $DB_+$, so $\\overline{M}$ is diffeomorphic to a double disk bundle with both halves a copy of $DB_+$.\n\nThus, we need only show that $\\overline{M}$ is a double cover of $M$.  To that end, we define a free involution $\\rho$ on $\\overline{M}$ whose quotient is $M$.  Given $(x,\\pm 1)\\in DB_+\\times \\{\\pm 1\\}$, we define $\\rho(x,\\pm 1) = (x,\\mp 1)$.  In other words, $\\rho$ interchanges the two copies of $DB_+$ on the``ends\" of $\\overline{M}$.  In addition, we define the action of $\\rho$ on $L\\times [-1,1]$ by mapping a point $(y,t)$ to $(\\sigma(y),-t)$.  It is easy to verify that this is the required involution.\n\n\n\n\n\n\n\\end{proof}\n\nIf both $\\ell_\\pm = 0$, then applying Proposition \\ref{prop:onecodim0} gives a double cover which again has both $\\ell_\\pm = 0$.  Hence, we can iterate this procedure.  This shows that a manifold can only admit a double disk bundle decomposition with both $\\ell_\\pm = 0$ if $\\pi_1(M)$ is infinite.  In fact, while it will not be needed in the remainder of the paper, it turns out that a double cover of $M$ fibers over $S^1$.\n\n\\begin{proposition}\\label{prop:bothcodim0}  Suppose $M$ is a connected manifold which admits a double disk bundle structure with both $\\ell_- = \\ell_+ = 0$ and regular leaf $L$.  Then $\\pi_1(M)$ is infinite, and $M$ has a double cover $\\overline{M}$ which fibers over $S^1$ with fiber $L$.\n\n\\end{proposition}\n\n\\begin{proof}  We have already proven the first statement, so we focus on the second.   By assumption, we may write $M = DB_+\\cup_f DB_-$ for some diffeomorphism $f:L\\rightarrow L$.\n\nAs both $\\ell_\\pm = 0$, Lemma \\ref{lem:involution} gives a pair of free involutions $\\sigma_\\pm:L\\rightarrow L$ with $L\/\\sigma_\\pm$ diffeomorphic to $B_\\pm$.  Both $\\sigma_\\pm$ extend to involutions on $L\\times [-1,1]$ defined by $(y,t)\\mapsto (\\sigma_\\pm(y),-t)$.  The quotient $(L\\times [-1,1])\/\\sigma_\\pm$ is clearly diffeomorphic to $DB_\\pm$.\n\nNow, take two copies of $L\\times [-1,1]$, which we will refer to as the left copy and right copy.  We glue $(y,1)$ in the left copy to $(f(y),1)$ in the right copy, and we glue $(y,-1)$ in the left copy to  $(\\sigma_+(f(\\sigma_-(y))), -1)$ to form the manifold $\\overline{M}$.\n\nFrom \\cite[Chapter VI, Section 5]{Ko}, if we only do the gluing of $(y,1)$ to $(f(y),1)$, the resulting manifold is diffeomorphic to $L\\times [-1,1]$.  Thus, $\\overline{M}$ has the structure of a mapping torus for some self diffeomorphism of $L$, so is a bundle over $S^1$ with fiber $L$.\n\nIt remains to see that $\\overline{M}$ is a double cover of $M$.  To that end, we define a free involution $\\rho$ on $\\overline{M}$ with quotient $M$ as follows.  On the left copy of $L\\times [-1,1]$, $\\rho$ acts by $(y,t)\\mapsto (\\sigma_-(y),-t)$.  On the right copy, $\\rho$ acts by $(y,t)\\mapsto (\\sigma_+(y), -t)$.  Once again, it is easy to verify this has the desired properties.\n\n\n\n\n\\end{proof}\n\n\\begin{remark}In Proposition \\ref{prop:bothcodim0}, if $L$ is disconnected, then $M$ itself fibers over $S^1$.   On the other hand, if $L$ is connected, passing to a double cover is sometimes necessary to obtain the bundle structure.  For example, if $M = \\mathbb{R}P^n\\#\\mathbb{R}P^n$ with $n\\geq 3$, then $M$ has a double disk bundle structure with both $\\ell_\\pm = 0$.  Indeed, $\\mathbb{R}P^n$ with a ball removed is a diffeomorphic to the total space of the disk bundle in the tautological bundle over $\\mathbb{R}P^{n-1}$.  But $M$ does not fiber over $S^1$ because its fundamental group $\\pi_1(M)\\cong (\\mathbb{Z}\/2\\mathbb{Z})\\ast (\\mathbb{Z}\/2\\mathbb{Z})$ has abelianization $(\\mathbb{Z}\/2\\mathbb{Z})\\oplus (\\mathbb{Z}\/2\\mathbb{Z})$, so does not surject onto $\\mathbb{Z}$.\n\n\\end{remark}\n\nThe next proposition describes how double disk bundles act with respect to covering maps.\n\n\\begin{proposition}\\label{prop:lifttocover}  Suppose $M$ is a connected manifold which admits a double disk bundle structure with both $\\ell_\\pm \\geq 1$.  If $\\rho:M'\\rightarrow M$ is any non-trivial covering (in the sense that $M'$ is connected), then $M'$ is a double disk bundle with regular leaf $L':=\\rho^{-1}(L)$, singular leaves $B'_\\pm:=\\rho^{-1}(B_\\pm)$, and with $\\ell'_\\pm = \\ell_\\pm$.  In addition, each of $L', B'_+$, and $B'_-$ are connected.\n\n\\end{proposition}\n\n\\begin{proof}  Since a covering map is a submersion, everything except the connectedness of $L', B'_\\pm$ is a direct consequence of \\cite[Proposition 3.1d]{DeGGKe}.  Thus, we need only show the connectedness of $L'$ and $B'_\\pm$.  As both $B'_\\pm$ are the continuous image of the sphere bundle projections $L'\\rightarrow B'_\\pm$, it is sufficient to show that $L'$ is connected.\n\nSo, we now show that $L'$ is connected.  Because $\\rho$ is a covering, so is $\\rho|_{L'}:L'\\rightarrow L$.  In addition, since at least one $\\ell_\\pm \\geq 1$, $L$ must be connected. Thus, to show $L'$ is connected, it is sufficient to select $x\\in L$, and show that any pair of points in $\\rho^{-1}(x)$ can be connected by a path in $L'$.  Let $x_1,x_2\\in \\rho^{-1}(x)$.\n\nBecause $M'$ is connected, we may connect $x_1$ and $x_2$ by a path $\\gamma':[0,1]\\rightarrow M'$ in $M'$.  Then $\\gamma:=\\rho\\circ \\gamma'$ is a closed curve in $M$.\n\nWe claim that $\\gamma$ is homotopic rel endpoints to a closed curve $\\alpha$ lying entirely in $L$.  To see this, note that $\\gamma$ represents an element of $\\pi_1(M,x)$, so we need to show the map $\\pi_1(L,x)\\rightarrow \\pi_1(M,x)$ induced by the inclusion $L\\rightarrow M$ is surjective.\n\nSeifert-van Kampen applied to the double disk bundle decomposition of $M$ shows that any curve in $M$ is, up to homotopy rel endpoints, a finite concatenation of curves in $DB_+$ and $DB_-$.  Because both $\\ell_\\pm \\geq 1$, the long exact sequence in homotopy groups implies the maps $\\pi_1(L)\\rightarrow \\pi_1(DB_\\pm)\\cong \\pi_1(B_\\pm)$ are surjective, so each curve in $DB_+$ or $DB_-$ is homotopic rel end points to one lying entirely in $L$.  In particular, $\\gamma$ is homotopic rel end points to a curve $\\alpha$ in $L$.\n\nNow, since $\\rho:L'\\rightarrow L$ is a covering, it is, in particular, a fibration.  As $\\gamma$ has a lift to $M'$, $\\alpha$ must lift to a curve $\\alpha':[0,1]\\rightarrow M'$.  Since the homotopy from $\\gamma$ to $\\alpha$ fixed the end points and the fiber of $\\rho$ is discrete, $\\alpha'$ must have the same endpoints as $\\gamma'$.  That is, $\\alpha'$ is a curve connecting $x_1$ and $x_2$ with image in $L'$.  This completes the proof that $L'$ is is connected, and thus, of the proposition.\n\n\n\n\\end{proof}\n\nIn the special  that $M$ is aspherical, i.e., the universal cover of $M$ is contractible, we can completely characterize the possibilities for the fiber sphere dimensions $\\ell_\\pm$ for any double disk bundle structure on it.\n\n\\begin{proposition}\\label{prop:flat}  Suppose $M$ is an aspherical manifold which admits a double disk bundle structure.  Then both $\\ell_- = \\ell_+ = 0$.  That is, both fiber spheres are zero-dimensional.\n\n\\end{proposition}\n\n\\begin{proof}We assume for a contradiction that $M$ has a double disk bundle decomposition with say, $\\ell_- > 0$.  This implies that the regular leaf $L$ is connected.  If $\\ell _+ = 0$, then Proposition \\ref{prop:onecodim0} implies that $M$ has a double cover admitting a double disk bundle structure with both $\\ell_\\pm > 0$.  Noting that the double cover of an aspherical manifold is aspherical, we may therefore assume that both $\\ell_\\pm > 0$.\n\nIn this case, we consider the universal cover $\\rho:M'\\rightarrow M$.  From Proposition \\ref{prop:lifttocover}, we obtain a double disk bundle structure on $M'$ with regular leaf $L'$ and singular leaves $B'_\\pm$ connected.  We will conclude the proof by showing that $M'$ has no such double disk bundle structure.  Specifically, we will show that $H^{t(\\ell_+ + \\ell_-)}(L';\\mathbb{Z}\/2\\mathbb{Z})$ is non-trivial for all $t\\geq 0$, contradicting the fact that $L'$ is a finite dimensional manifold.  Set $R = \\mathbb{Z}\/2\\mathbb{Z}$ for legibility.\n\nBecause $M'$ is contractible, the Mayer-Vietoris sequence for the double disk bundle decomposition of $M'$ yields isomorphisms $\\psi_k:H^k(B'_-;R)\\oplus H^k(B'_+;R)\\rightarrow H^k(L';R)$ for each $k\\geq 1$ (and that $\\psi_0$ is surjective).  Recalling that $\\psi_k$ is nothing but the difference in the maps induced by the sphere bundle projections $L\\rightarrow B_\\pm$, it follows that each map $H^k(B'_\\pm;R)\\rightarrow H^k(L';R)$ must injective.  Since both $B'_\\pm$ are connected, we have Gysin sequences associated to $L\\rightarrow B_\\pm$; injectivity of $H^\\ast(B'_\\pm;R)\\rightarrow H^\\ast(L';R)$ then implies via the Gysin sequence that the $R$-Euler class of both bundles $L'\\rightarrow B'_\\pm$ is trivial.  We thus have group isomorphisms $$H^\\ast(L';R)\\cong H^\\ast(B'_+;R)\\otimes H^\\ast(S^{\\ell_+};R)\\cong H^\\ast(B'_-;R)\\otimes H^\\ast(S^{\\ell_-};R),$$ where the inclusions $H^\\ast(B'_\\pm;R)\\rightarrow H^\\ast(B'_\\pm;R)\\otimes H^\\ast(S^{\\ell_{\\pm}};R)$ are the obvious ones.\n\nWe will now prove that $H^{t(\\ell_- + \\ell_+)}(L';R)\\neq 0$ for all $t\\geq 0$ by induction.   The base case is clear, as it is simply the assertion that $H^0(L';R)\\neq 0$.\n\n\nNow, assume that $H^{t(\\ell_- + \\ell_+)}(L';R)$ is non-zero for some $t\\geq 0$.  Since $\\psi_k$ for $k:=t(\\ell_+ + \\ell_-)$ is surjective, there must therefore be a non-zero element $x$ in at least one of $H^k(B'_\\pm;R)$.  We assume without loss of generality that $x\\in H^k(B'_+;R)$.  If $y_\\pm\\in H^{\\ell_\\pm}(S^{\\ell_\\pm};R)\\cong R$ is the non-zero element, then the element $x\\otimes y_+ \\in H^{k+\\ell_+}(L';R)$ is non-zero, and not in the image of $H^{k+\\ell_+}(B'_+;R)$.  Since $\\psi_{k+\\ell_+}$ is surjective, it now follows that $H^{k+\\ell_+}(B'_-;R)\\neq 0$.   Suppose $z\\in H^{k+\\ell_+}(B'_-;R)$ is such a non-zero element.  Then the element $z\\otimes y_-\\in H^{(t+1)(\\ell_- + \\ell_+)}(L';R)$ is non-zero, completing the induction.\n\\end{proof}\n\n\nWe will also need a proposition regarding orientability.\n\n\\begin{proposition}\\label{prop:orientable}  Suppose $M$ is a double disk bundle and that $M$ is orientable.  Then so is the regular leaf $L$.\n\n\\end{proposition}\n\n\\begin{proof}\nBecause $L$ is the boundary of both disk bundles, $L$ must have trivial normal bundle.  Then $TM|_L = TL \\oplus 1$ with $1$ denoting a trivial rank $1$ bundle.  Computing the first Stiefel-Whitney class using the Whitney sum formula, we find $$0 = w_1(TM|_L) = w_1(TL) + w_1(1) = w_1(TL).$$  Thus $w_1(TL) = 0$, so $L$ is orientable.\n\n\n\\end{proof}\n\n\n\\section{$3$-dimensional examples}\n\nThe goal of this section is to prove the following theorem.\n\n\\begin{theorem}\\label{thm:3mfld}Suppose $M^3$ is a closed manifold admitting a metric of positive sectional curvature.  Then $M$ is a double disk bundle if and only if $M$ is $S^3$, a lens space $L(p,q)$, or a prism manifold.\n\n\\end{theorem}\n\nBy definition, a lens space $L(p,q)$ (where $\\gcd(p,q)$ is necessarily $1$)  is the quotient of $S^3$ by a free isometric action by the cyclic group $\\mathbb{Z}\/p\\mathbb{Z}\\subseteq S^1\\subseteq \\mathbb{C}$ acting on $S^3\\subseteq \\mathbb{C}^2$  via $\\mu\\ast(z_1,z_2)  = (\\mu z_1, \\mu^q z_2).$  Also, by definition, a prism manifold is an isometric quotient of a round $S^3$ with fundamental group isomorphic to $\\langle a,b| aba^{-1}b = 1, a^{2\\beta} = b^\\alpha\\}$ where $\\gcd(\\alpha,\\beta) = 1$.  Prism manifolds include the homogeneous spaces $S^3\/D_{4n}^\\ast$ where $D_{4n}^\\ast$ is the order $4n$ group generated by $e^{2\\pi i\/n}$ and $j$ in the group $Sp(1)$ of unit length quaternions.\n\nFrom, e.g.,\\cite[Table 1]{Mc}, the homogeneous $3$-manifolds which are covered by $S^3$ consists of precisely the lens space $L(p,1)$, the prism manifolds $S^3\/D_{4n}^\\ast$, and the spaces $S^3\/T^\\ast$, $S^3\/O^\\ast$, or $S^3\/I^\\ast$ where $T^\\ast$, $O^\\ast$, and $I^\\ast$ are the binary tetrahedral, octohedral, and icosahedral groups respectively.  In addition, from e.g., \\cite[Section 7.5]{Wo}, the product of any of these fundamental groups with a cyclic group of relatively prime order is again the fundamental group of a positively curved $3$-manifold.  Thus, Theorem \\ref{thm:3mfld}has the following corollary.\n\n\n\\begin{corollary}\\label{cor:inf}\nThere are infinitely many positively curved $3$-manifolds which do not admit a double disk bundle structure.  These examples include precisely three homogeneous examples: $S^3\/T^\\ast$, $S^3\/O^\\ast$, and $S^3\/I^\\ast$, were $T^\\ast, O^\\ast,$ and $I^\\ast$ are the binary tetrahedral, octahedral, and icosahedral groups respectively.\n\\end{corollary}\n\n\\begin{remark}\\label{rem:nonneg}By using work of others, it is easy to extend Theorem \\ref{thm:3mfld} to non-negatively curved three manifolds.  Hamilton \\cite[Main Theorem]{Ha1}\\cite[Theorem 1.2]{Ha2} showed a closed $3$-manifold $M$ admitting a metric of non-negative sectional curvature is covered by $S^3$, $S^2\\times S^1$, or $T^3$.  If $M$ is covered by $S^2\\times S^1$, then $M$ is diffeomorphic to $S^2\\times S^1$, $\\mathbb{R}P^2\\times S^1$, $\\mathbb{R}P^3\\# \\mathbb{R}P^3$, or to the unique non-trivial $S^2$ bundle over $S^1$ \\cite{To}.  Clearly for each of these possibilities, $M$ is a double disk bundle.  If $M$ is covered by $T^3$, then from \\cite[pg. 448]{Sc}, $M$ is a double disk bundle.\n\n\\end{remark}\n\nWe now work towards proving Theorem \\ref{thm:3mfld}.  For the remainder of this section, $M$ denotes a $3$-manifold of positive sectional curvature.  From \\cite[Main Theorem]{Ha1}, $M$ is finitely covered by $S^3$, so has finite fundamental group.   A simple application of the Lefshetz fixed point theorem implies that $M$ must be orientable. From Proposition \\ref{prop:bothcodim0}, at least one of $\\ell_\\pm > 0$, which, in particular, implies that $L$ is connected.\n\n\\begin{proposition}  Suppose $M$ is a closed orientable $3$-manifold which admits a double disk bundle decomposition with at least one fiber sphere of positive dimension.  The regular leaf $L$ must be diffeomorphic to either $S^2$ or $T^2$.\n\n\\end{proposition}\n\n\\begin{proof}  Assume without loss of generality that $\\ell_+ > 0$.  This implies that $L$ is connected.  Since $L$ is $2$-dimensional and an $S^{\\ell_+}$-bundle over $B_+$, we must have $\\ell_+\\in \\{1,2\\}$.  If $\\ell_+ = 2$, the fiber inclusion map $S^2\\rightarrow L$ is an embedding between closed manifolds of the same dimension, hence a diffeomorphism.  If $\\ell_+ = 1$, then the Euler characteristic $\\chi(L) = \\chi(S^1)\\chi(B_+) = 0$, so $L$ must be $T^2$ or a Klein bottle.  But $L$ must orientable from Proposition \\ref{prop:orientable}.\n\n\\end{proof}\n\nWe will proceed by breaking into cases depending on whether $L = S^2$ or $L = T^2$.  We will classify all disk bundles whose boundary is diffeomorphic to $L$, and then classify ways of gluing the corresponding disk bundles.  Using a collar neighborhood, it easy to see that if two gluing maps are isotopic, then the corresponding double disk bundles are diffeomorphic.  The following lemma provides another circumstance where the double disk bundles are diffeomorphic.\n\n\\begin{lemma}\\label{lem:glue}Suppose $X$ and $Y$ are manifolds with boundary and $f:\\partial X\\rightarrow \\partial Y$ is a diffeomorphism.  Assume in addition that $G:X\\rightarrow X$ is a diffeomorphism with $g:= G|_{\\partial X}:\\partial X\\rightarrow \\partial X$.  Then the manifolds $X\\cup_f Y$ and $X\\cup_{f\\circ g} Y$ are diffeomorphic.\n\n\\end{lemma}\n\n\\begin{proof}We define a diffeomorphism $\\phi:X\\cup_{f\\circ g} Y\\rightarrow X\\cup_{f} Y$ by mapping $x\\in X$ to $\\phi(x) = G(x)$ and mapping $y\\in Y$ to $\\phi(y)=y$.  It is obvious that $\\phi$ is a diffeomorphism, if it is well defined.\n\nWe now check that it is well-defined.  If we first identify $x\\in \\partial X$ with $f(g(x))$ and then apply $\\phi$, we obtain the point $f(g(x))$.  On the other hand, if we first apply $\\phi$ and then identify with $\\partial Y$, we get $\\phi(x) = G(x) = g(x)\\sim f(g(x))$.\n\n\\end{proof}\n\n\\begin{proposition}  Suppose $M$ is a double disk bundle with regular leaf $L = S^2$.  Then, $M$ is diffeomorphic to $S^3$, $\\mathbb{R}P^3$, or $\\mathbb{R}P^3\\# \\mathbb{R}P^3$.\n\n\\end{proposition}\n\n\\begin{proof} To begin with, note there are precisely two isomorphism types of sphere bundles with total space $S^2$:  they are $S^2\\rightarrow S^2\\rightarrow \\{p\\}$, and $S^0\\rightarrow S^2\\rightarrow \\mathbb{R}P^2$.  Since a diffeomorphism of either $S^0$ or $S^1$ extends to a diffeomorphism of the corresponding disk, both of these extend uniquely to disk bundles.  Moreover, $\\operatorname{Diff}(S^2)$ deformation retracts to $O(2)$ \\cite{Sm}, so we may assume our gluing map is either the identity or the antipodal map.   Both options extend to a diffeomorphism of the $3$-ball $B^3$, so by Lemma \\ref{lem:glue} the choice of gluing map is irrelevant if either $B_\\pm = \\{p\\}$.\n\nIf we have $B_+ = B_- = \\{p\\}$, then $M$ is obtained by gluing two $3$-balls along their boundary $S^2$, so $M$ is diffeomorphic to $S^3$ in this case.  If we have $B_+  = \\{p\\}$ and $B_- = \\mathbb{R}P^2$, then gluing gives $\\mathbb{R}P^3$.  Finally, if we have $B_+ = B_- = \\mathbb{R}P^2$, we obtain $\\mathbb{R}P^3 \\# \\pm \\mathbb{R}P^3$.  But $\\mathbb{R}P^3$ admits an orientation reversing diffeomorphism, so $\\mathbb{R}P^3 \\# - \\mathbb{R}P^3$ is diffeomorphic to $\\mathbb{R}P^3\\sharp \\mathbb{R}P^3$.\n\n\\end{proof}\n\nWe now classify all double disk bundles with regular leaf $L = T^2$ and with at least one $\\ell_\\pm > 0$, which completes the proof of Theorem \\ref{thm:3mfld}.\n\n\\begin{proposition}  Suppose $M$ admits a double disk bundle structure with regular leaf $L = T^2$ and with $\\ell_+ > 0$.  Then either $\\pi_1(M)$ is abelian, or $M$ is a prism manifold.\n\n\\end{proposition}\n\n\\begin{remark}  The classification of $3$-manifolds with $\\pi_1(M)$ abelian is well known \\cite[Section 1.7, Table 2]{AsFrWi}.  The only such examples which are covered by $S^3$ are the lens spaces $L(p,q)$.  Each of these is well-known to be a double disk bundle, e.g., they are all quotients of $S^3$ via a sub-action of the well-known cohomogeneity one action of $T^2$ on $S^3$.  The examples which are not covered by $S^3$ are covered by $S^2\\times S^1$, so are all double disk bundles by Remark \\ref{rem:nonneg}.\n\\end{remark}\n\n\\begin{proof}The assumption that $\\ell_+ >0$ implies that $\\ell_+ = 1$, so $B_+ = S^1$.  An $S^1$-bundle over $S^1$ is determined by an element of $\\pi_0(\\operatorname{Diff}(S^1))$.  Since $\\operatorname{Diff}(S^1)$ deformation retracts to $O(2)$, there are precisely two $S^1$-bundles over $S^1$.  Of course, one has total space $K$, the Klein bottle.  Thus, there is a unique $S^1$ bundle over $S^1$ with total space $T^2$, the trivial bundle.\n\nIf $\\ell_- = 2$, the fiber inclusion $S^2\\rightarrow T^2$ must be an embedding, giving an obvious contradiction.  Hence, $\\ell_- \\in \\{0,1\\}$.  Of course, if $\\ell_- = 1$, then the bundle $L\\rightarrow B_-$ must be the trivial bundle as in the previous paragraph.  On the other hand, if $\\ell_- = 0$, then $L\\rightarrow B_-$ is a $2$-fold covering, so $B_-$ is diffeomorphic to either $T^2$ or $K$.\n\nEach of these $S^1$-bundles extends to a disk bundle in a unique way.  In addition, $\\operatorname{Diff}(T^2)$ deformation retracts to $GL_2(\\mathbb{Z})$ \\cite[Theorem 2.5]{FaMa}, so we can always assume our gluing map lies in $Gl_2(\\mathbb{Z})$.  Moreover, the diffeomorphism $\\begin{bmatrix} 1 & 0 \\\\ 0 & -1\\end{bmatrix}$ of $T^2 = \\partial (D^2\\times S^1)$ extends to a diffeomorphism of $DB_+ \\cong D^2\\times S^1$, so Lemma \\ref{lem:glue} implies that we may assume our gluing map lies in $Gl_2^+(\\mathbb{Z})$.\n\nApplying Siefert-van Kampen to the double disk decomposition of $M$, we note that since $\\ell_+ = 1$, the map $\\pi_1(L)\\rightarrow \\pi_1(B_+)$ is surjective.  This implies that $\\pi_1(M)$ is isomorphic to a quotient of $\\pi_1(DB_-) = \\pi_1(B_-)$.  Thus, if $B_-\\neq K$, then $\\pi_1(M)$ is necessarily abelian.\n\nSo, we assume $B_- = K$, and that the gluing map is determined by a matrix $$\\begin{bmatrix} \\alpha & \\beta \\\\ \\gamma & \\delta\\end{bmatrix} \\in Gl_2^+(\\mathbb{Z}).$$  We have presentations $$\\pi_1(S^1) = \\langle a\\rangle, \\pi_1(T^2) \\cong \\langle b,c| [b,c] = 1\\rangle, \\text{ and } \\pi_1(K)  = \\langle d,e| ded^{-1}e = 1\\rangle.$$  The unique abelian index $2$ subgroup of $\\pi_1(K)$ is generated by $\\{d^2,e\\}$.  We may therefore assume the map $\\pi_1(T^2)\\rightarrow \\pi_1(K)$ maps $b$ to $d^2$ and $c$ to $e$, and that the map $\\pi_1(T^2)\\rightarrow \\pi_1(S^1)$ maps $b$ to $a$ and $c$ to the identity element.\n\nNote that under the gluing map $\\begin{bmatrix} \\alpha & \\beta\\\\ \\gamma & \\delta\\end{bmatrix}$, the map $\\pi_1(T^2)\\xrightarrow{\\begin{bmatrix} \\alpha & \\beta\\\\ \\gamma & \\delta\\end{bmatrix}} \\pi_1(T^2)\\rightarrow \\pi_1(S^1)$ is therefore given by $b\\mapsto b^{\\alpha}c^{\\gamma}\\mapsto a^{\\alpha}$, and $c\\mapsto  b^{\\beta} c^{\\delta} \\mapsto a^{\\beta}$, where we have used multiplicative notation rather than additive for both $\\pi_1(T^2)\\cong \\mathbb{Z}^2$ and $\\pi_1(S^1)\\cong \\mathbb{Z}$.  Thus, Seifert-van Kampen gives $$\\pi_1(M)\\cong \\langle a,d,e| ded^{-1}e = 1, a^{\\alpha} = d^2, a^{\\beta} = e\\rangle.$$  We claim that this is isomorphic to $$\\langle d,e| ded^{-1} e = 1, d^{2\\beta} = e^{\\alpha}\\rangle,$$ so that $M$ has the fundamental group of a prism manifold.\n\nTo that end, we first note that the generator $a$ in the first presentation is unnecessary.  Indeed, we have $\\alpha \\delta - \\beta\\gamma = 1$, so $$a^{ 1} = a^{\\alpha \\delta - \\beta\\gamma} = (a^{\\alpha})^{\\delta} (a^{\\beta})^{-\\gamma} = d^{2\\delta}e^{-\\gamma}.$$  Thus, we need only demonstrate that the relations in the first presentation are consequences of the relations in the second, and vice versa.\n\nSo, assume initially that both $a^{\\alpha} = d^2$ and $a^{\\beta} = e$.  Raising the first relation to the power of $\\beta$, and the second to the power of $\\alpha$, we obtain  $$d^{2\\beta} = a^{\\alpha\\beta} = e^{\\alpha},$$ so the relations in the first presentation imply those in the second.  Conversely, assuming $d^{2\\beta} = e^{\\alpha}$, noting that $d^2$ commutes with everything, and setting $a= d^{2\\delta}e^{-\\gamma}$, we find \\begin{align*} a^{\\alpha} &= d^{2\\alpha\\delta} e^{-\\gamma \\alpha}\\\\ &= d^{2(1+\\beta\\gamma)}e^{-\\gamma\\alpha}\\\\ &= d^2 (d^{2\\beta})^{\\gamma} (e^{\\alpha})^{-\\gamma}  \\\\ &= d^2 (e^{\\alpha})^{\\gamma} (e^{\\alpha})^{-\\gamma}\\\\ &= d^2\\end{align*} and likewise, we find that $a^{\\beta} = e$.\n\nThus, $\\pi_1(M)$ is isomorphic to the fundamental group of a prism manifold, as defined above.  Since such manifolds are classified up to diffeomorphism by their fundamental group \\cite[Theorem 2.2]{AsFrWi}, $M$ must be a prism manifold in these cases.\n\n\\end{proof}\n\n\nWe conclude this section by proving that the three homogeneous examples $S^3\/T^\\ast$, $S^3\/O^\\ast$, and $S^3\/I^\\ast$ of Corollary \\ref{cor:inf} are the only homogeneous examples in any dimension which are covered by a sphere but are not double disk bundles.\n\n\\begin{proposition}\\label{prop:cantwork}  Suppose $M$ is a closed homogeneous space which is covered by a sphere.  Then $M$ admits a double disk bundle decomposition, except when $M$ is diffeomorphic to one of $S^3\/T^\\ast, S^3\/O^\\ast,$ or $S^3\/I^\\ast$.\n\n\\end{proposition}\n\n\\begin{proof}  From \\cite[Table 2]{WiZi}, we see that the homogeneous spaces non-trivially covered by a sphere are a) real projective space, b) a homogeneous lens space, or c) a quotient of $S^{4n-1}\\subseteq \\mathbb{H}^n$ by a non-abelian finite subgroup of $Sp(1)$ acting diagonally.  Here, a homogeneous lens space is a quotient $S^{2n+1}\/(\\mathbb{Z}\/m\\mathbb{Z})$ where $\\mathbb{Z}\/m\\mathbb{Z} = \\{(z,z,...,z)\\in \\mathbb{C}^{n+1}: z^m = 1\\}$, and $\\mathbb{H}$ denotes the skew-field of quaternions.\n\nWe have a uniform description of these actions:  let $\\mathbb{K}\\in \\{ \\mathbb{R},\\mathbb{C},\\mathbb{H}\\}$ and set $k = \\dim_{\\mathbb{R}}(\\mathbb{K})$.  Let $G$ denote any finite subgroup of $O(1),U(1)$ or $Sp(1)$ respectively.  Then $G$ acts freely on $S^{kn-1}\\subseteq \\mathbb{K}^n$ via the diagonal action in each coordinate and the cases a),b), and c) above correspond to the choice of $\\mathbb{K}$.\n\nWe first claim that if $n\\geq 2$ then all such quotients $S^{nk-1}\/G$ admit a double disk bundle decomposition.  Indeed, one can simply observe that the block action by $O(n-1)\\times O(1)$, $U(n-1)\\times U(1)$, or $Sp(n-1)\\times Sp(1)$  on $S^{nk-1}\\subseteq\\mathbb{K}^n = \\mathbb{K}^{n-1}\\oplus \\mathbb{K}$ is cohomogeneity one, and $G$ acts via a subaction of the block action.\n\nThis leaves the case $n=1$, which gives the manifolds $S^0\/G$, $S^1\/G$, or $S^3\/G$.  Of course, the first is $0$-dimensional, and any quotient $S^1\/G$ is diffeomorphic to $S^1$, and thus admits a double disk bundle decomposition.  The final case $S^3\/G$ is given by Corollary \\ref{cor:inf}.\n\\end{proof}\n\n\n\\section{Flat examples} The goal of this section is to prove the following theorem.\n\n\\begin{theorem}\\label{thm:flat}  There are infinitely many closed flat manifolds, in arbitrarily large dimension, which are not double disk bundles.\n\n\\end{theorem}\n\nWe begin with a proposition which allows us to recognize when a flat manifold does not admit a double disk bundle decompositoin.\n\n\\begin{proposition}\\label{prop:flatcondition}  Suppose $M$ is a closed flat manifold with $H_1(M)$ finite of odd order.  Then $M$ cannot admit a double disk bundle decomposition.\n\\end{proposition}\n\n\\begin{proof}  Assume for a contradiction that $M$ admits a double disk bundle decomposition.  Since $M$ is flat, the Cartan-Hadamard theorem implies that $M$ is aspherical.  Thus, Proposition \\ref{prop:flat} applies: any double disk bundle decomposition on $M$ must have both $\\ell_\\pm = 0$.  Then, from Proposition \\ref{prop:onecodim0}, $M$ admits a non-trivial double cover.  In particular, $\\pi_1(M)$ must have an index $2$ subgroup, so admits a surjection to $\\mathbb{Z}\/2\\mathbb{Z}$.  Since $H_1(M)$ is the abelianization of $\\pi_1(M)$, this surjection must factor through $H_1(M)$.  But no finite group of odd order admits a surjection to $\\mathbb{Z}\/2\\mathbb{Z}$, giving a contradiction.\n\n\\end{proof}\n\n\n\n\n\n\n\nIn order to prove Theorem \\ref{thm:flat}, we need only establish the existence of infinitely many flat manifolds $M$ in arbitrarily large dimensions with first homology group $H_1(M)$ finite of odd order.  In fact, we will find examples with $H_1(M)$ trivial.  As $H_1(M)$ is the abelianization of $\\pi_1(M)$, we are thus tasked with finding an infinite family of flat manifolds for which $\\pi_1(M) = [\\pi_1(M),\\pi_1(M)]$ is perfect.  These examples are furnished by the following theorem.\n\n\\begin{theorem}\\label{thm:perfect}  Suppose $\\phi$ is any finite perfect group.  Then there is a closed flat manifold $M_\\phi$ for which $H_1(M_\\phi) = 0$ and for which $M_\\phi$ has holonomy $\\phi$.\n\\end{theorem}\n\nRecall that the alternating group on $n$ letters, $A_n$ is perfect if $n\\geq 4$.  We claim that for $n\\geq 7$, that $\\dim M_{A_n}\\geq n-1$, so Theorem \\ref{thm:flat} immediately follows from Proposition \\ref{prop:flatcondition} and Theorem \\ref{thm:perfect}.  Indeed, the holonomy group of an $n$-manifold is a subgroup of the orthogonal group $O(n)$, and for $n\\geq 7$, the smallest non-trivial representation of $A_n$ occurs in dimension $n-1$ \\cite[Problem 5.5]{FuHa}.\n\nThus, to prove Theorem \\ref{thm:flat}, we need only to prove Theorem \\ref{thm:perfect}.  We do this using an argument due to Igor Belegradek \\cite{Be}. \n\nWe will use the following characterization of the fundamental group of a closed flat manifold.\n\n\\begin{theorem}[Bieberbach\\cite{Bi} and Auslander-Kuranishi \\cite{AusKur}]\\label{thm:char}\n\nAn abstract group $\\pi$ is the fundamental group of a closed flat $n$-manifold if and only if both of the following conditions are satisfied.\n\n\n\\begin{enumerate}\\item  $\\pi$ is torsion free\n\n\\item  $\\pi$ fits into a short exact sequence of the form $0\\rightarrow \\mathbb{Z}^n\\rightarrow \\pi \\rightarrow \\phi\\rightarrow 0$, where $\\phi$ is a finite group.\n\n\\end{enumerate}\n\n\\end{theorem}\n\nThe finite group $\\phi$ is called the holonomy of $\\pi$ as it is isomorphic to the holonomy group of the flat manifold $n$-manifold with fundamental group $\\pi$.\n\nWe need a lemma, which is \\cite[Proposition 2.3.13]{HoPl}.\n\n\\begin{lemma}\\label{lem}  Suppose a group $\\pi$ fits into a short exact sequence of the form $$0\\rightarrow \\mathbb{Z}^n\\rightarrow \\pi \\rightarrow \\phi\\rightarrow 0$$ where $\\phi$ is a finite group.  Then the commutator subgroup $\\pi' = [\\pi,\\pi]$ also fits into a short exact sequence of the form $$0\\rightarrow \\mathbb{Z}^m\\rightarrow \\pi'\\rightarrow \\phi' = [\\phi,\\phi]\\rightarrow 0.$$  In addition, if $\\phi$ is perfect, then so is $\\pi'$.\n\n\\end{lemma}\n\nWe may now prove Theorem \\ref{thm:perfect}.\n\n\\begin{proof}(Proof of Theorem \\ref{thm:perfect})\nLet $\\phi$ denote any finite perfect group.  From \\cite[Theorem 3]{AusKur} there is an abstract group $\\pi$ satisfying both conditions of Theorem \\ref{thm:char}.  The commutator $\\pi' = [\\pi,\\pi]$ is a subgroup of the torsion free group $\\pi$, so is torsion free.  From Lemma \\ref{lem}, $\\pi'$ is also perfect, and satisfies the second condition of Theorem \\ref{thm:char} with finite quotient $\\phi' = [\\phi,\\phi] = \\phi$.  Hence, by Theorem \\ref{thm:char}, there is a flat manifold $M_{\\phi}$ with fundamental group $\\pi'$.  Since $\\pi'$ is perfect, $H_1(M_{\\phi}) = 0$.\n\\end{proof}\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Conclusions}\\label{conclusions}\nThe first part of this work presented a mental simulation method to plan manipulation sequences while minimizing potential damage with respect to objects in the scene. This is achieved via anticipating the scene's dynamics during the interaction process. Moreover, several measures allow for a significant improvement in efficiency.\nIn the second part, a classifier was trained that can predict an optimal manipulation sequence from new, unknown scenes. This happens iteratively within a self-supervised manipulation strategy optimization cycle and allows the robot to continously acquire skills during load-free times. The resulting strategies can be deployed anytime and executed in near-real time.\n\nThe presented work merely scratches the surface of what may be possible to infer from physics simulation to create long-term knowledge. However, it was shown that preference patterns humans acquire subconsciously get shaped as well in robotic behavior learned via mental simulation.\nRegarding the fidelity of simulation-based planning with respect to the real world, it is a hard problem to design a feature-based physical scene understanding approach based on complex human inferences. It would be necessary to find a set of features which is capable of depicting all inferences humans draw from some input scene using their world knowledge.\nHowever, the featureless manipulation sequence planning approach by itself is very resource-demanding during productive use. Hence, the feature-enabled manipulation strategy learning part of the method allows for reducing execution times to a minimum, outsourcing expensive computation into low-load time slots. Strictly speaking, the overall approach proposed herein does not rationalize any mental simulation capabilities, but minorly softens their expressiveness in favor of reasonable use under real-life conditions.\n\nVice versa, humans also use certain features as a fallback when scene dynamics are too complex to be anticipated. Hence, the utility of features as a mean of generalization cannot be neglected per se.\nNevertheless, it may be possible to increase the overall performance, explicitely precision, when the feature set is iteratively improved based on experimental classification outcomes. The proposed feature set serves as a first attempt to prove the general feasibility and applicability of this approach, although more investigation is required into adapting the features to the respective use case in productive use. With increased precision, the answer about how to measure converge of the optimization loop may potentially be perfectly obvious.\n\nIn any case, in order to avoid permanent retraining of the classifier from scratch, using an online algorithm would improve the temporal performance. With the current setup, retraining the classifier is necessary every time a new training sample has been generated. Although this can happen in parallel to generating the next training sample via mental simulation, updating the classifier in an online fashion removes some computational burden from the approach and allows for faster overall processing, hence shorter time to convergence. Further investigation in terms of speed improvement may prove beneficial for the applicability of the presented method in real-world applications.\nFuture research potentially may also point into the direction of deep learning which generally does not require the developer to define the feature set prior to classifier design. Using this technique, certainly major effort needs to be invested into designing the classifier pipeline. This exceeds the scope of this initial approach to solve the present complex problem and should be investigated in future work.\n\nNevertheless, a major advantage of the proposed approach is that, except for the feature set, it is almost parameter-free and hence does not require extensive tuning based on the application scenario.\nThe genericness of the proposed approach allows for deployment in many different scenarios for which a simulation of the robot and scene dynamics can be provided. Therefore, the method can easily be integrated into existing applications as a mean of high-level task planning including the capability of self-supervised strategy optimization.\n\n\\section{Evaluation}\\label{evaluation}\nIn order to evaluate the presented approach, the coherency with respect to human behavior as well as efficient strategy optimization cycles play an important role. \nIn this section, the worst-case complexity of different parts of the approach is estimated. Additionally, several measures are explained which were taken to improve the overall efficiency. Afterwards, several experiments on different application scenarios show the performance of the presented method within the specific steps and, finally, the performance of the full self-supervised strategy optimization cycle.\n\n\\subsection{Efficiency considerations}\\label{efficiency}\n\\subsubsection{Training data generation}\nSection~\\ref{ranking_weighted} described an idea of how to integrate preference weights $w_{ij}$ into the RPC classifier. However, as shown in the experimental evaluation (Section~\\ref{experiments}), the weights do not significantly improve the performance of the classifier if integrated into the prediction process. Nevertheless, preference weights are used to calculate $\\tau_{w}$ as a performance measure for evaluation.\n\nOn the other hand, this means that the manipulation strategy generation part of the method experiences a speedup of 100 with respect to integrating the preference weights into classifier training because they do not need to be calculated during self-supervised fully-autonomous operation. Hence, it is not necessary to plan manipulation sequences for a specific scene more than once.\n\n\\subsubsection{Planning efficiency}\nThe described planning method, in contrast to the mentioned $A^{*}$ algorithms or other commonly used search methods like discretized Rapidly-Exploring Random Trees (RRT) or Rapidly-Exploring Random Leafy Trees (RRLT) \\cite{Morgan2004}, is critical with respect to the efficiency of estimating the cost function for each node. The search process itself is trivial and happens quickly because of the relatively low number of nodes compared to, for instance, a motion planning problem. Therefore, the efficiency of the method depends mainly on the problem of cost estimation and not the worst-case tree size $|\\mathcal{S}_O|$ which, for a number of objects $n$, constructed from all possible permutations of the object set like in Algorithm~\\ref{alg:planning_tree_generation}, is\n\\begin{equation}\\label{eq:tree_size}\n\t|\\mathcal{S}_O| = \\sum_{i=0}^{n} \\frac{n!}{(n-i)!} = 1+\\frac{n!}{(n-1)!}+...+\\frac{n!}{2}+n!\n\\end{equation}\nwhere $\\frac{n!}{(n-i)!}$ is the number of objects for the respective tree level $i$, $i=0$ representing the top level (i.e.\\ root), $i=n$ the bottom level (i.e.\\ leaves); see the example in Figure~\\ref{fig:tree_example}. \nWith respect to the time complexity of manipulation sequence planning, since the used depth-first search possesses a worst-case time complexity of $O(|V| + |E|)$ with $|V|$ being the number of vertices and $|E|$ being the number of edges in the search tree \\cite{Reif1985}, Algorithm~\\ref{alg:planning} reaches\n\\begin{equation}\\label{eq:complexity}\n\tO(|\\mathcal{S}_O| + (|\\mathcal{S}_O|-1)) = O\\left(2\\sum_{i=0}^{n} \\frac{n!}{(n-i)!}-1\\right)\n\t= O\\left(1+\\frac{2n!}{(n-1)!}+...+n!+2n!\\right) \\implies O(n!).\n\\end{equation}\n\nOn the other hand, best-case time complexity can be achieved if the costs of the very first leaf node are lower than the costs of any other node considered later in the process. In this case, all children of these following nodes can be disregarded. The minimal number of nodes to consider hence amounts to\n\\begin{equation}\\label{eq:tree_size_omega}\n\t|\\mathcal{S}_{\\Omega}| = \\sum_{i=1}^{n-1} (n-i) = 1+2+...+(n-1)\n\t= \\frac{n(n-1)}{2}-n = \\frac{n(n-3)}{2}\n\\end{equation}\nwhich results in a best-case time complexity of\n\\begin{equation}\\label{eq:complexity_omega}\n\t\\Omega(|\\mathcal{S}_{\\Omega}| + (|\\mathcal{S}_{\\Omega}|-1))\n\t= \\Omega\\left(n(n-3)-1\\right) \\implies \\Omega(n).\n\\end{equation}\nIn practice, best-case complexity is not as unlikely to achieve as it may seem because during many manipulation actions the passive objects are not moved at all, hence causing zero costs.\nEither way, the experiments in the next subsection show that the average-case time complexity is favorable in practice due to different possibilities of pruning the tree during planning.\n\n\\subsubsection{Scene clustering}\nEquation~\\ref{eq:tree_size} showed a nonlinear increase in search tree size with a rising number of scene objects. As a remedy, it is suggested to break down scenes with many objects into smaller clusters of $n \\leq 4$ objects which are then treated individually. Practically, as regarded in Section~\\ref{modeling_preference_patterns}, the robot will anyway not be able to physically access more than a low number of objects from a certain point of view due to space and dexterity constraints.\nHowever, the exact way of clustering scene objects strongly depends on the scenario and a generic consideration of this subproblem does not fit the scope of this publication and has to be treated in future work.\nMost importantly, however, is the fact that during deployment of a trained classifier in a productive setting the per-cluster application of an individual classifier instead of one single classifier for the whole scene creates no tangible efficiency loss since the prediction runtimes of RPC lie in the sub-second range.\n\nIn any case, using smaller object clusters mitigates the issue that the reachability of scene objects strongly depends on the robot kinematics and hence often is limited by a large extent.\nIn the course of complex manipulation procedures like used in the proposed approach, the higher the number of objects in a cluttered scene grows, the less likely any valid grasping configurations can be generated for each individual object. To stay with the PR2 supermarket example, a right-handed robot may have severe difficulties manipulating objects on its left-hand side. In this case it is reasonable to handle clusters to the left of the robot with lower priority if only the right arm is used for manipulation.\nNevertheless, even when only using a part of the scene as active objects, the remaining objects should still be included in the simulated scene. This way, if any passive object is moved accidently, even if it is not active anywhere in the search tree, it will still account for the cost function.\n\n\\subsubsection{Search tree optimization}\nClustering the scene into smaller parts dramatically increases the method's efficiency, but additionally, it is desired to keep the search tree as non-redundant as possible any time. For instance, tree branches where there is no possibility to represent an optimal solution can be pruned. Several measures are presented in this subsection which keep the search tree as small as possible. Section~\\ref{experiments} shows that the presented method is able to prune a typical search tree down to \\SI{48.7}{\\%} of its original size using the following strategies.\n\n\\paragraph{Implicit search tree pruning:}\nAs shown by an empiric consideration in the Experiments section, in every scene there is a number of configurations for which the simulation of a manipulation action is pointless, e.g.\\ touching a passive object during grasp approach which in turn pushes the active object into unreachable distance.\nAnother possibility is that all configurations include collisions of the robot with the environment. In this work, the low-level motion planning process is abstracted away since it represents an own branch of research. Hence, such configurations have to be imposed with infinite costs and therefore can be removed from the tree of viable configurations.\n\n\\paragraph{Explicit search tree pruning:}\nSince the planner uses depth-first search as explained in Algorithm~\\ref{alg:planning}, some tree bran\\-ches can be pruned because the costs accrued in the currently active branch already exceed the total costs of any other branch that has been traversed to its leaf. Exploration of the current branch in this case can be cancelled immediately and infinite costs are assigned to the whole branch.\n\n\\paragraph{Reuse of similar subconfigurations:}\nEspecially for scenes where objects do not physically interact a lot when being manipulated, similar configurations may appear in several nodes of the tree with respect to object instances and poses. In this case, the subconfiguration of these nodes behave similarly and can be replaced with one another. Therefore, after one of these similar configurations has been processed, the results are directly projected into the other ones without running simulation on those. Peshkin and Sanderson~\\cite{Peshkin1987} already described this policy as beneficial.\n\n\n\\subsection{Scene features}\\label{features}\nRanking by pairwise comparison relies on atomic classifiers which are based on multinomial logistic regression \\cite{Huellermeier2008,LeCessie1992}. These classifiers are used to predict a confidence for a pairwise preference $\\pi(i) \\succ \\pi(j)$ from a given feature vector $\\mathbf{x}$. Such features have to be descriptive for the respective scene in order to allow for reliable manipulation scene prediction in a new, unknown scene. This subsection explains the feature vector that is calculated from every training scene as an input to the manipulation strategy generation process.\n\n\\subsubsection{Attentional vector sum}\nAs a measure for semantic spatial relations between scene objects, the \\emph{attentional vector sum} (AVS) model is employed which is used in language comprehension research and was introduced by Regier and Carlson \\cite{Regier2001}. It has been extended in several refined models (e.g.\\ by Kluth \\cite{Kluth2016}) and used for similar problems in robotics (e.g.\\ by Sj\\\"o\\\"o and Jensfelt \\cite{Sjoo2011}).\n\nThe AVS model generally evaluates spatial prepositions like \\texttt{behind}, \\texttt{above} and \\texttt{left of} with respect to two distinct objects and provides an acceptability rating for the respective predicate in the shape of\n\\begin{equation*}\n\\mathtt{above(object1, object2)} = 0.1~\\mathrm{where}~\\mathtt{above(\\cdot)} \\in [0,1].\n\\end{equation*}\nIn the presented application, the AVS is evaluated for each of the prepositions \\texttt{in front of}, \\texttt{behind}, \\texttt{above}, \\texttt{below}, \\texttt{left of} and \\texttt{right of} from the robot's point of view for each permutation of two objects.\n\n\\subsubsection{Visibility}\nAnother important feature of a scene with respect to the manipulation order is object visibility which intuitively correlates with the manipulation difficulty level.\nHence, the visibility ratios $r_{\\mathrm{vis}}$ of all objects $o \\in \\mathcal{O}$ is computed as follows:\n\\begin{equation}\nr_{\\mathrm{vis}}(o) = \\frac{V(H(\\mathcal{C}(o))}{V(H(\\mathcal{M}(o))}\n\\end{equation}\nwhere $\\mathcal{C}(o)$ is the 2.5-D point cloud of $o$ visible in the scene, $\\mathcal{M}(o)$ is $o$'s previously known 3-D object model, $H(\\cdot)$ denotes a convex hull and $V(\\cdot)$ the volume of such a hull.\n\n\\input{results_visibility.tex}\n\nIn order to abstract from object-dependent probability distributions for the visibility ratio which result from self-occlusion, 10000 scenes per object were generated with the object randomly placed and rotated in the workspace of the respective scenario.\nAfterwards, a \\emph{Gaussian mixture model} (GMM) has been fitted on the \\textcolor{DarkGreen}{histograms of frequencies} of visibility ratios $r_{\\mathrm{vis}}$ occuring in each run (see Figure~\\ref{fig:gmm}).\nThe number of GMM components as well as the covariance type were optimized by minimizing the Bayesian Information Criterion \\cite{Schwarz1978}.\n\nThe \\textcolor{blue}{probability density $\\rho_{\\mathrm{vis}}$} of the GMM then is taken as the final visibility score which is still an object-specific value, but generalizes away from multimodal distibutions of the visibility ratios $r_{\\mathrm{vis}}$ like for \\obj{1}{0} and \\obj{1}{1} as can be seen in Figure~\\ref{fig:gmm}.\n\nNote that the visibility scores of the objects in the supermarket scenario (Fig.~\\ref{fig:gmm} bottom row) are less distinct than the objects of the logistics scenario (Fig.~\\ref{fig:gmm} top row). This is caused by the high-altitude camera position which requires a strong tilt in order to have the whole shelf in view. Hence, the sensor data on the margins of the field of view may sometimes be cropped, causing objects' visibility ratios to drop when randomly spawned close to the shelf margins.\nPlease also note that $r_{\\mathrm{vis}}$ usually does not take values close to $0$ because some part of the object is always visible, otherwise it would not have been recognized by the perception system. Additionally, $r_{\\mathrm{vis}}$ values close to $1$ do not occur because the 2.5-D input point clouds are subject to self-occlusion, hence the back part of the objects is cropped.\n\n\\subsubsection{Feature vector}\nIn addition to the more complex AVS and visibility features, several object-specific and object-relational features are computed as shown in Table~\\ref{table:features}.\n\nThe final set consists of $23n+17p$ features with the number of objects in the scene being $n=\\lvert\\mathcal{O}\\rvert$ and the number of all possible object pair combinations being $p=\\binom{n}{2}$. For a scene comprising four objects, this would end up in a $194$-dimensional feature vector $\\mathbf{x}$. One concrete instance of $\\mathbf{x}$, together with the respective manipulation sequence $\\pi$ as determined from mental simulation, eventually forms one training sample for $C$ with $\\mathbf{x} \\mapsto \\pi$.\n\n\\begin{table}[hbt]\n    \\rowcolors{1}{}{lightgray}\n    \\centering\n    \\begin{tabular}{p{6cm}c}\n        \\textbf{Feature (object-specific)} & \\textbf{Dimensionality} \\\\\n        Pose within the scene & $6n$ \\\\\n        Distance vector to workspace bottom & $3n$ \\\\\n        Distance vector to workspace back & $3n$ \\\\\n        Distance vector to initial gripper position & $3n$ \\\\\n        Visibility $\\rho_{\\mathrm{vis}}$ & $n$ \\\\\n        Axis-aligned bounding box size & $3n$ \\\\\n        Oriented bounding box size & $3n$ \\\\\n        \\parbox[t]{6cm}{Free space around object\\\\ \\scriptsize{(= Euclidean distance to closest object's surface)}} & $n$ \\\\\n        \\textbf{Feature (object-relational)} & \\textbf{Dimensionality} \\\\\n        Distance vector per object pair & $3p$ \\\\\n        Euclidean distance per object pair & $p$ \\\\\n        Contacts per object pair (point, normal and force) & $7p$ \\\\\n        AVS for 6 prepositions per object pair & $6p$ \\\\\n        \\textbf{Total} & $\\mathbf{23n+17p}$\\\\\n    \\end{tabular}\n    \\caption{Features and dimensions depending on the number of objects $n=\\lvert\\mathcal{O}\\rvert$ and the number of object pair combinations $p=\\binom{n}{2}$}\n    \\label{table:features}\n\\end{table}\n\n\\section{Introduction}\n\nSince common perception and motion planning ap\\-proa\\-ches have to deal with noisy sensor readings and cluttered environments, autonomously handling fragile goods can lead to damage of items or robots.\nThe bottom-up way to deal with this problem is physical compliance of grasping systems or enhanced perception and manipulation algorithms (e.g.\\ \\cite{Stoyanov2016,Stilman2007,Katz2013,Eppner2016,Correll2018,Pavlichenko2018}, amongst others). These methods commonly rely on obstacle avoidance as to not provoke any margin-violating situation in advance. In contrast, this article proposes a top-down approach to tackle the problem in the sense that changing the environment is taken into account during the manipulation planning stage.\n\n\\process{all}{}\nThis work presents a \\emph{manipulation strategy optimization} method that selects a sequence of objects in a given configuration to be unloaded or otherwise removed from a container, i.e.\\ a supermarket shelf or a shipping container.\nIn the first place, we define a \\emph{manipulation strategy} as a decision process which determines an optimal sequence in which to manipulate the scene objects with respect to application-specific optimization criteria.\nThis optimization criterion, in the case of the presented method, is defined as the damage possibly caused to other objects which may be touched and shifted during the manipulation process.\nThe generated strategies can be optimized autonomously by the robot during load-free times. This is achieved by generating and processing training data for learning preferences and physical constraints between the objects in a self-supervised way. A simulation-in-the-loop setup is exploited which utilizes a physics simulation to generate arbitrary amounts of training data, hence no user interaction is needed throughout the whole process as opposed to classical supervised learning approaches.\n\n\\subsection{Related work}\nThe proposed method relies on \\emph{mental simulation} of the movements of all objects present in the scene during manipulation, a term which originates from a psychology context with a first mention in \\cite{Kahneman1981}.\nIn general, reasoning and drawing conclusions from a simulation process has been investigated in the literature under different terms, but altogether considering the dynamics of a scene in a human-like way.\nBattaglia et al.\\ \\cite{Battaglia2013} introduced the term \\emph{intuitive physics engine} which describes the human anticipation capabilities from a cognition-scientific viewpoint. They find the quantitative evaluations of human reasoning capabilities to be surprisingly similar to the results obtained from physics simulations.\nOther well-known terminology is \\emph{temporal projection} \\cite{Kunze2017}, \\emph{physics-based reasoning} \\cite{Akbari2015} and \\emph{physical reasoning} \\cite{Weitnauer2010} which deal with predicting real-world behavior using knowledge inferred from physics simulation.\nIn terms of repetitively performing actions which improve environment manipulation strategies, \\emph{robotic playing} \\cite{Hangl2016} is another related bio-inspired technique which compares trial-and-error behavior while accumulating environment knowledge with a children's way of exploring the world.\nIn the case of the proposed method, we can add the term \\emph{dreaming robot} to this list of synonyms for mental simulation because the manipulation strategy optimization happens during load-free times, for instance at night when the robot is otherwise unused.\nAdditionally, since the manipulation strategies generated inside this approach can be transferred to a real-world scenario, \\emph{transfer learning} is another applicable high-level term.\n\nIn general, simulation-in-the-loop architectures have not been studied extensively yet in the robotics literature, but have recently been on the rise for problems in the physical reasoning domain, with most authors using the term \\emph{mental simulation} like in this article.\nBozcuoglu and Beetz~\\cite{Bozcuoglu2017}, for instance, recently proposed a generic simulation setup for knowledge generation and reasoning through mental simulation. However, they do not yet close the loop of propagating the gained knowledge into an optimization cycle. Haidu and Beetz~\\cite{Haidu2016} utilize this setup for recognizing and interpreting actions from simulation in order to collect sufficient knowledge about the task for replaying it on a real robot.\nOn the other hand, Levine et al.~\\cite{Levine2016} use up to 14 real manipulators in parallel to collect data for grasp learning. They use visual features combined with a deep learning technique in a parallelized way, making use of massive hardware. This is similar to the proposed approach in a way that parallel randomized experiments are performed in order to generate training data, but different in a way that the presented method does not rely on hardware other than computing capacity and tries to make use of idle times as far as possible.\n\n\\subsection{Method overview}\n\nFig.~\\ref{fig:components} shows an overview of the proposed method. In order to obtain and optimize manipulation strategies, the scene dynamics are tracked by a physics simulator. The simulated environment is synchronized with the perceived scene which includes the robot, the environment and a number of objects to manipulate, defined domain-specificly depending on the application (Fig.~\\ref{fig:components} top left). In order to allow for generalization of these mental simulation capabilities, training scenes are generated automatically (top center) which, after determining the respective optimal manipulation sequence in terms of minimizing undesired object motion \\cite{Fromm2016}, serve as training examples for a machine learning process (top right). Once a number of training samples has been acquired, a classifier is built and updated iteratively on new incoming samples, yielding an updated manipulation strategy (bottom right).\nSubsequently, the resulting classifier can, over the course of many simulated manipulation procedures, be deployed to predict manipulation sequences with respect to the learned strategy from any new scene (bottom left).\n\nThe possibilities are manifold: Firstly, the constraints on the motion planning search space are relaxed in comparison to approaches where other objects are considered as obstacles. As in previous work \\cite{Stoyanov2016} which uses a conventional motion planning method without sophisticated task-level planning, some scene configurations do not allow for collision-free manipulation at all. The presented method, however, works one level higher than motion planning and allows for generating manipulation strategies which facilitate the motion planning itself. Eventually, this allows the planner to successfully determine a viable manipulation sequence in most cases.\n\nSecondly, because the necessary training data can be generated and consumed in a self-supervised manner by running simulations of manipulation actions, the resulting strategy can be optimized during load-free times. Parallelizing the simulation process makes the actual manipulation very efficient because the strategy improves quickly and can be applied instantly, without the need for any more simulation or other processing.\nAdditionally, since strategy optimization happens in the background without dependencies on the real scene, the currently active strategy can be deployed \\emph{anytime} in \\emph{near-real time} without the need to synchronize the learning process with physical robot behavior.\n\nThe main motivation of the presented approach is to \\emph{avoid damage to the possibly heavy or fragile goods as well as the robot itself} which may occur if an object is shifted or dropped unintendedly.\nHowever, detecting actual physical damage on handled goods has to be regarded as an own field of research related to object recognition and classification. In the scope of this work, we hence define damage avoidance as an implicit, proactive way of minimizing unintended motion of objects during the manipulation process.\n\nThe remainder of this introductory section will describe the main contributions as well as exemplary applications of the presented method which will guide the reader through the paper. Section~\\ref{prerequisites} gives a brief overview about some prerequisites necessary for working with and deploying the method. Section~\\ref{planning} explains how the mental simulation creates manipulation sequences which then, in Section~\\ref{learning}, are compiled into manipulation strategies. Section~\\ref{self_improvement} wraps up the overall process of self-supervised manipulation strategy optimization. Section~\\ref{evaluation} presents a thorough evalution of the proposed approach which is concluded in Section~\\ref{conclusions} with a discussion and outlook. For orientation throughout the paper, the reader is additionally referred to Fig.~\\ref{fig:components} which allows for easy navigation between the different parts of the method.\n\n\\subsection{Contributions}\n\\begin{enumerate}\n\\item In any case, although the presented method aims on optimizing autonomous robot behavior using the measure of anticipated dynamics, \\emph{no imposition of any explicit logical or spatial dependencies between objects} is its first main contribution. In addition, no priors are included about the type and size of objects, degrees of freedom of the robot, type of manipulator or other application domain-specific parameters. This means that the method is domain-independent and can be deployed on new application scenarios within minimal time, given that a working simulation of the manipulation procedures to be sequenced has been established already (see Section~\\ref{prerequisites}).\\label{contribution1}\n\n\\item The second contribution of this work is the \\emph{integration of simulation into the processing loop} which is essential for the presented self-supervised mental simulation approach. Another example where simulation in the loop may play an important role is system integration which this way can be conducted as a continuous process. Hence, parts of the system can be tested individually using simulated components and gradually replaced by their real-world counterparts. Such a continuous system integration \\cite{Fromm2017} technique uses simulation in the same way, embedded into a closed loop and as a full-featured component which seamlessly integrates into the processing pipeline.\\label{contribution2}\n\n\\item Finally, the third main contribution of the proposed approach is the \\emph{generation of manipulation strategies} from the manipulation sequences planned within the simulation processing loop.\\label{contribution3}\n\\end{enumerate}\n\n\\subsection{Application scenarios}\n\\begin{figure}[tbp]\n  \\centering\n  \\begin{subfigure}[b]{0.48\\textwidth}\n    \\includegraphics[width=\\linewidth]{parcelrobot_example_scene1.png} \n    \\caption{Logistics (container unloading) \\cite{Stoyanov2016}}\n    \\label{fig:intro_scenario_roblog}\n  \\end{subfigure}\n  \\hfill\n  \\begin{subfigure}[b]{0.48\\textwidth}\n   \\includegraphics[width=\\linewidth]{pr2_example_scene.png} \n   \\caption{Supermarket (shelf replenishment) \\cite{Winkler2016}}\n   \\label{fig:intro_scenario_shopping}\n  \\end{subfigure}\n  \\caption{Typical application scenarios for autonomous manipulation}\n  \\label{fig:intro_scenarios}\n\\end{figure}\n\nAll of the different terminology mentioned so far share the common ground of mentally simulating robot interaction in dynamic environments, but have been applied to a multitude of different domains. In order to demonstrate the generality of the approach which is not limited to a particular domain, two different everyday scenarios are used (see Figure~\\ref{fig:intro_scenarios}): \\emph{logistics}, with autonomous container unloading in the EU project ``Cognitive Robot for Automation of Logistics Processes'' (RobLog)\\footnote{\\url{http:\/\/roblog.eu}} \\cite{Stoyanov2016}, and in a \\emph{supermarket} environment for autonomous shelf replenishment \\cite{Winkler2016}.\n\nThe first of these scenarios typically contains a number of bulky, heavy, fragile or otherwise damage-prone goods. Especially in terms of relieving human workers from health-endangering labor, robotic applications have been flourishing in this domain for several years, as in previous work on shipping container unloading.\n\nSecondly, the presented method can be applied as well on typical service robots in retail and domestic environments, interleaved with logistics in terms of hardware and software transitioning from one field to the other, but still facing different challenges. As a concrete scenario, a supermarket featuring a PR2 robot is used as a second application example.\n\n\\section{Manipulation strategy generation}\\label{learning}\nOne of the main contributions of the proposed approach is the \\emph{generation of ma\\-ni\\-pu\\-la\\-tion strategies} from previously-planned manipulation sequences. Such a strategy comprises of a machine learning classifier which allows to \\emph{predict a sequence of manipulation actions} while applying some prior knowledge.\n\nIn the given case, this prior knowledge consists of planned manipulation sequences on many different scenes where the prediction yields a desirable manipulation sequence for a new, previously unseen scene. All scenes can be characterized using certain distinct features; all features of a particular scene together with the anticipated optimal manipulation sequence form one training sample. \\emph{Label ranking}, a machine learning technique, then takes the part of predicting new manipulation sequences from the known training data given a set of scene features.\n\nLearning-based approaches in a manipulation context exist in a large diversity of applications and used techniques, including learning the support order of piles and the geometric relations between objects therein \\cite{Mojtahedzadeh2015,Panda2013,Sjoo2011} or object affordances \\cite{Katz2013}. However, manipulation strategy generation benefits from this only if it is able to take into account relational predicates like \\texttt{in}, \\texttt{on} and \\texttt{behind} or support and containment relations. Additionally, assessing a scene for these predicates may yield ambiguous results, especially for geometry-based predicates. Finally, since we want to take the surrounding workspace into account as well, only describing relations between objects may not be sufficient for anticipating dynamic behaviors.\n\nAnother established field of research is grasp learning using different kinds of features and measures \\cite{Fischinger2015,Kappler2015}. This works in a similar way as the presented method with respect to the generation of a classifier which allows for predicting grasps for new situations, but on grasp level. Other learning-based approaches tackle problems like predicting physical effects on objects using visual features \\cite{Li2017,Mottaghi2016}. However, none of these works deal with high-level sequencing which is the main contribution of the proposed method.\n\n\\subsection{Label ranking using ranking by pairwise comparison}\\label{lr}\nA label ranking classifier $C$ basically solves the problem of \\emph{ordering a set of abstract labels $\\mathcal{L}$ with size $n=|\\mathcal{L}|$ into a sequence $\\pi \\subseteq \\mathcal{L}$} using a feature vector $\\mathbf{x} \\in \\mathbf{X}$ such that\n$C: \\mathbf{X} \\to \\mathcal{L}, \\mathbf{x} \\mapsto \\pi$.\nThe resulting sequence $\\pi$ is a permutation of all occuring labels where label $\\pi(i)$ is ranked higher than label $\\pi(j)$. Generally, this \\emph{pairwise preference of $\\pi(i)$ over $\\pi(j)$} is denoted as $\\pi(i) \\succ \\pi(j)$.\n\nIn the literature, many different label ranking methods exist, based on different established machine learning algorithms. Amongst the most popular ones are Decision Tree-based \\cite{DeSa2017,Cheng2009} and Gaussian Mixture Model-based methods \\cite{Zhou2014,Grbovic2012} as well as \\emph{ranking by pairwise comparison} (RPC) \\cite{Huellermeier2008} which is used in the proposed approach.\n\nRPC uses an ensemble of classifiers to predict pairwise preferences and afterwards employs a voting scheme to combine the atomic classifiers' outputs into a common prediction.\nBecause of this, one important parameter of RPC is the way of how to combine classifier outputs into a prediction. There are two major voting schemes which have be evaluated for this purporse so far \\cite{Huellermeier2004}: \\emph{binary voting} and \\emph{soft voting}. The former bases on the result of a binary classifier which emits whether or not the input corresponds with its learned preference. Soft voting, on the other hand, outputs a continuous value in $[0,1]$ which can be interpreted as the confidence of the classifier about the compliance with the learned preference.\n\nRPC proves advantageous for the targeted purpose since the used atomic classifiers correspond to pairwise preferences which play a major role in the overall method. Section~\\ref{modeling_preference_patterns} will go into detail on this where pairwise preferences and their importance in human-like intuitive manipulation is explained.\nOne more major advantage of RPC in the regarded use case is the fact that it qualifies as an \\emph{eager learning} method which uses a considerable amount of time for training the ensemble of classifiers, but prediction happens in near real-time. Since the whole processing circle of training scene generation, mental simulation and manipulation strategy optimization (which includes classifier training) is deferred into load-free times, no delays are inflicted on productive use.\n\n\\subsection{Distinction from related methods}\nIn contrast to label ranking, there are several methods which target problems overlapping with the presented one. However, these cannot be used in a similar way as the proposed method, but are important to be isolated from it because they produce structurally and logically different results:\n\n\\paragraph{Structured prediction}\nInstead of ranked labels, this me\\-thod outputs binary flags which mark if a particular label is relevant to the input sample or not. This can be used to classify if an image falls into a category like \"nature\" or \"architecture\". Examples of packages using this technique are $\\mathrm{SVM^{struct}}$ \\cite{Joachims2002} and conditional random fields \\cite{Sutton2011}.\n\n\\paragraph{Learning to rank}\nAlthough sounding similar in the first moment, this method provides solutions to a different problem, namely creating rankings of training samples and not rankings of abstract labels. One common usage is recording click-through preferences; the labels used as inputs into the algorithm are simply the sample indices in the training set. There is no possibility in this method to provide a ranking of abstract labels as training input. Prominent examples for learning to rank are RankNet \\cite{Burges2005} and ranking SVMs like $\\mathrm{SVM^{rank}}$ \\cite{Joachims2002,Joachims2006}.\n\n\\paragraph{Sequence classification}\nThis term is usually used where only the sequences themselves, but no external training features exist, for instance when natural speech as a sequence of words is to be annotated with the respective word classes. Therefore this method only makes sense within context-rich environments where the exploitation of the latter satisfies the required generalization capabilities of some classifier. A survey of approaches using this technique can be found in \\cite{Xing2010}.\n\n\n\n\\section{Mental simulation: manipulation sequence planning}\\label{planning}\nBefore advancing to the self-supervised acquisition of manipulation strategies, this section explains the underlying planning method which has been already presented and evaluated in recent work \\cite{Fromm2016}.\nIt validates and selects the best sequence to remove or unload a number of objects from a scene with respect to possible damage. This happens entirely as a mental simulation without physical robot interaction. The result of such a planning action is taken as a training sample for the manipulation strategy generation method explained in Section~\\ref{learning}.\n\nThe proposed planner relies on mentally simulating interaction with the scene, hence an\\-ti\\-ci\\-pa\\-ting its dynamics using a physics simulation. Instead of using plain motion planning as in classical approaches, while planning manipulation of a specific object we explicitely disregard the presence of all other movable objects in the scene. However, the physics simulation takes into account possible motion of all objects which, after manipulation, is considered as a whole in order to retrospectively assess the success of the manipulation process.\n\nIn contrast to traditional motion planning ap\\-proa\\-ches, the presented one does not per se consider other movable objects as obstacles.\nFor this reason, the constraints on the planning process are significantly relaxed by decreasing the size of the planning search space.\nAdditionally, the presented method aims to avoid damaging the handled goods or even the robot itself. This could happen when dropping or shifting an object through the manipulation process.\nSummarized, this manipulation sequence planning approach optimizes autonomous robotic behavior while regarding the anticipated dynamics of the perceived scene which is projected into the robot's mental simulation.\n\nIn the literature, an effective example of motion planning for a humanoid robot was presented by Okada et al.\\ \\cite{Okada2004}. In that work, any occuring obstacle is actively cleared from the envisioned walking trajectory, hence taking into account necessary actions which have not been planned explicitely when formulating the high-level goal.\nIn recent work in a different scenario \\cite{Winkler2016}, a similar approach has been taken. In contrast, the work proposed in this publication uses a dynamics simulation for validating manipulation actions in addition to purely spatial knowledge. \n\nKitaev et al.\\ \\cite{Kitaev2015} and Dogar et al.\\ \\cite{Dogar2012} work on grasp planning through clutter. They take into account shifting objects and explicitly manipulating these. The objective of these works is partly similar to the presented approach, though having the goal of removing obstacles from planned manipulation paths and not for determining optimal sequences.\nOn the other hand, Stilman et al.\\ \\cite{Stilman2007} explicitly plan which objects to move away and where to move them in order to reach a target object. In contrast, the proposed collision-agnostic approach mitigates excessive motion planning times which occur in the context of collision avoidance.\n\nIn the following, our planning method will be explained in detail. For this, let $\\mathcal{O}$ be the set of objects present in the scene, $\\alpha \\in \\mathcal{O}$ the \\emph{active object} (that will be manipulated), $\\Phi = \\mathcal{O} \\setminus \\alpha$ the set of \\emph{passive objects} (that will not be manipulated) and $\\phi \\in \\Phi$ one member of this set.\n\n\\subsection{Manipulation cost estimation}\\label{cost_estimation}\nIn order to determine suitable and efficient solutions to planning problems, cost functions have been used for a long time (e.g.\\ in \\cite{Goldberg1990,Okada2004}). Applying a cost function to a specific problem, however, requires a certain amount of domain knowledge for creating the optimization targets of the problem. The presented approach aims to keep the amount of injected knowledge as low as possible. Nevertheless, a cost function needs to be defined that regards spatial modifications of the scene in order to minimize unintended motions of passive objects.\n\n\\begin{figure}[tb]\n    \\centering\n    \\begin{subfigure}[t]{0.35\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{gazebo_start.png} \n        \\caption{pre-manipulation}\n        \\label{fig:mscv1}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.35\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{gazebo_end.png} \n        \\caption{post-manipulation}\n        \\label{fig:mscv2}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.7\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{gazebo_overlay_mscv3.png} \n        \\caption{trajectory (dotted line), Euclidean distance (dashed line) and swept convex volume (yellow)}\n        \\label{fig:mscv3}\n    \\end{subfigure}\n    \\caption{Visualization of the trajectory and swept convex volume covered by a passive object during manipulation. After the robot extracted the right \\obj{1}{blue}, the cylindrical \\obj{1}{pink} dropped on the floor and rolled away following the dotted trajectory.}\n    \\label{fig:mscv}\n\\end{figure}\n\nOne may first think of simple, generic Euclidean distance-based or trajectory length-based cost functions, however, in order to consider complex motion paths which may include translation and rotation, these do not deliver accurate cost estimates. \nIn order to visualize this, Fig.~\\ref{fig:mscv3} shows the movement of an object following a curved trajectory. On such a path, a Euclidean distance-based cost function, even if combined with rotational difference, does not take into account the whole volume (marked yellow in the figure) with all its curves and convexities. Such volumes are often covered when objects roll off uncontrolledly. Imagine an object rolling from the middle of a tabletop to its edge, then falling down, bumping off a wall and rolling back to the middle of the table. Such a complex motion cannot be covered by a distance-based cost function.\nTrajectory length-based cost functions, however, more accurately model this behavior, but still do not reflect when an object with a high side length ratio (e.g.\\ the salt container in Fig.~\\ref{fig:simulation_objects:supermarket}) spins around all its axes. For low-fidelity recovery of movement costs this may be sufficient.\n\nHowever, for the motions encountered in the targeted scenarios, a \\emph{swept volume}-based representation gives a more accurate estimate of the object motion costs. Generally, a common application field of swept volume estimations is collision detection \\cite{Baeuml2011} along with space occupancy estimation \\cite{VonDziegielewski2012}. A mathematical formulation is given in the survey of Abdel-Malek et al.\\ \\cite{Abdel-Malek2006}.\nSwept volume-based approaches use a \\emph{generator} which creates the \\emph{swept volume} by following a \\emph{trajectory}.\nIn the case of the presented method, the generator is defined as the object surface. This follows a set of poses covered while moving during simulation runtime with the result of the outer object boundary during motion.\nIn contrast to \\cite{Abdel-Malek2006}, this is a simplification in the sense that it is voxel-based instead of continuous and hence can be utilized in discrete-generator and discrete-trajectory scenarios.\n\nThe generated volume in most applications coincides with the concave hull around the spatial locations touched by the object. However, the effort required for the hull computation may exceed the provided capabilities \\cite{VonDziegielewski2012}. Additionally, the concave hull of a number of points is not generally well-defined, which possibly creates ambiguities for different kinds of objects.\nFor the mitigation of these effects, in the proposed approach the concave hull is replaced with the convex hull, normalized by object volume, which is easy to compute and well-defined. The result is called \\emph{swept convex volume} (for a visualization, see Figure~\\ref{fig:mscv}).\n\nSince all scene objects shall be regarded when computing manipulation costs, the \\emph{maximum swept convex volume} $V_{max}$ is computed, which is the maximum over all scene objects' swept convex volumes:\n\\begin{equation}\\label{eq:mscv}\n\tV_{max} = \\max_{\\phi \\in \\Phi}{V_s(\\phi)}\n\\end{equation}\nwhere $V_s(\\phi)$ is the swept convex volume of the $\\phi \\in \\Phi$ as computed in Algorithm~\\ref{alg:scv}. \n\n\\begin{algorithm}[tb]\n\\centering\n\\caption{Swept convex volume calculation}\n\\label{alg:scv}\n\\begin{algorithmic}[1]\n    \\State \\textbf{input}: object mesh $\\mathcal{M}(\\phi)$, object poses $\\mathbf{p}_{0..n}(\\phi)$ covered during simulation\n    \\State create point cloud $\\mathcal{C}(\\phi)$ from $\\mathcal{M}(\\phi)$ at $\\mathbf{p}_{0}(\\phi)$\n    \\ForAll {$\\mathbf{p}_{i}(\\phi)$}\n        \\State $\\mathcal{C}_{i}(\\phi) \\gets \\mathcal{C}(\\phi)$ transformed from $\\mathbf{p}_{0}(\\phi)$ to $\\mathbf{p}_{i}(\\phi)$\\label{alg:scv:weight}\n        \\State $\\mathcal{C}(\\phi) \\gets \\mathcal{C}(\\phi) \\cup \\mathcal{C}_{i}(\\phi)$\n    \\EndFor\n    \\State $H(\\phi) \\gets \\mathrm{convhull}(\\mathcal{C}(\\phi))$\n    \\State $H_{0}(\\phi) \\gets \\mathrm{convhull}(\\mathcal{C}_{0}(\\phi))$\n    \\State $V_s(\\phi) \\gets \\mathrm{volume}(H(\\phi))~\/~\\mathrm{volume}(H_{0}(\\phi))$\n    \\State \\textbf{output}: swept convex volume $V(\\phi)$\n\\end{algorithmic}\n\\end{algorithm}\n\nAs stated previously, the main focus of the presented method is to defend vulnerable goods from damage. Therefore, additional weights can be employed on each component of the 6-D pose to $V_{max}$, calling the resulting cost function the \\emph{maximum weighted swept convex volume} $V_{w}$.\nThese weights are adaptable depending on the usage domain, e.g. to sanction objects dropping vertically off a shelf. Additional scenarios may include objects placed on a running conveyor belt where lateral influence may push an items off the belt. Employing rotational weights may help in scenarios where liquid-containing objects are prone to spill when tipped over.\n\nThe parameterization of said weights has influence on Algorithm~\\ref{alg:scv} where, in Line \\ref{alg:scv:weight}, $\\mathbf{p}_{i}(\\phi)$ has to be replaced with its weighted version $\\mathbf{p}^{w}_{i}(\\phi)$:\n\\begin{equation}\n\\mathbf{p}^{w}_{i}(\\phi) = \\mathbf{p}_{0}(\\phi) + \\mathrm{diag}(\\mathbf{w})\\cdot(\\mathbf{p}_{i}(\\phi)-\\mathbf{p}_{0}(\\phi))\n\\end{equation}\nwhere $\\mathbf{w}=\\begin{bmatrix}w_{x}&w_{y}&w_{z}&w_{\\varphi}&w_{\\theta}&w_{\\psi}\\end{bmatrix}^{\\intercal}$\nare do\\-main-de\\-pen\\-dent weights for each of the components of the 6-D object pose. The employment of weights $w_{\\{x,y,z\\}}>1$ in the translation domain has the effect of stretching the convex volume in the respective direction which results in enhanced costs.\nIn the rotation domain, $w_{\\{\\varphi,\\theta,\\psi\\}}>1$ generates increased rotation of the object volume at a specific covered object pose. Assuming that the object does not perform a 360$^\\circ$ turn on the respective axis during zero translation, this again increases the cumulative convex volume.\n\nIn general, the presented planning approach can be utilized for domains with other foci than damage avoidance by employing respective weights. For the envisioned application scenarios, however, we set the weights on\n\\begin{equation}\n\\mathbf{w} = \\begin{bmatrix}1&1&2&1&1&1\\end{bmatrix}^{\\intercal}\n\\end{equation}\nwhich puts emphasis on damage-prone vertical motion. This is of special importance in domains e.g.\\ like the presented supermarket scenario where objects dropped off a shelf would break.\n\n\n\\subsection{Damage-avoiding manipulation sequence planning}\\label{sequence_planning}\nUsing mental simulation of a robot's interaction with a scene as a validation method for planned actions is beneficiary additionally in time-critical scenarios as well as a second stage for a manipulation order planning algorithm.\nMojtahedzadeh et al.~\\cite{Mojtahedzadeh2015}, for instance, present a planner that uses static equilibrium calculations for discovering physical support between objects. That planner consequently prefers objects not supporting any others.\nImplicitly, the manipulation sequences produced by the proposed approach will often be similar. However, additionally, dynamic events are taken into account which occur during the manipulation action.\nConsequently, using the proposed method for generating and validating plans on scenarios with dynamical and unpredictable content may enhance and enrich other planning algorithms.\n\nIn general, mentally simulated interaction can be used as a validation method which estimates the costs for manipulating a particular scene configuration. In order to deduce the next action for the robot to be performed, however, the distinction between positively and negatively validated actions, respectively, has to be drawn in some way. Naively, this can be achieved by modeling specific thresholds for the manipulation costs.\nCommon physics-based validation methods like the one of Rockel et al.~\\cite{Rockel2015} determine such thresholds heuristically which signal whether or not some object is currently regarded as toppling. Pastor et al.~\\cite{Pastor2011}, on the other hand, use statistical methods to determine if a particular motion coincides with a predefined spatial envelope. The drawback of these methods is the necessity to adjust said parameters whenever deploying the method in a new scenario.\nHowever, in the vast majority of deployment settings, classification between actions causing positive and negative results in the respective context has to happen automatically. The user should not have to predefine thresholds and parameters in order to be adaptable to changing scenarios and environments. In contrary to the mentioned approaches, the main constribution of the method presented in this paper is that it works \\emph{prior-free} in this respect (see \\emph{Contribution~\\ref{contribution1}} on p.~\\pageref{contribution1}).\n\nIn the usage example of avoiding damage by unintendedly moving objects, the presented method provides manipulation sequences taking into account exactly these side-effects. Additionally, obstacle-avoiding motion planning is prone to fail in certain scenarios due to an overly complex motion planning problem in confined spaces. Stoyanov et al.\\ sketch the difficulties of classical planning when applied in heavily confined spaces where passive objects are regarded as obstacles: \\textit{\"Most of the unloading failures were due to failures in finding collision-free grasping trajectories for objects [...] tightly packed with other objects.\"} \\cite[p.~11]{Stoyanov2016}\nHowever, the proposed planner tries to minimize the motion of passive objects during manipulation although this is explicitely permitted. Consequently, the likelihood of ending up with no viable manipulation plan is decreased significantly when using the presented method as a high-level manipulation sequence planner.\n\n\\begin{figure*}[tb]\n    \\centering\n    \\input{planning_tree.tex}\n\t\\caption{Example search tree}\n    \\floatfoot{Some branches cropped for increased visibility. Each node shows its initial configuration, i.e.\\ prior to manipulation.}\n\t\\label{fig:tree_example}\n\\end{figure*}\n\n\\begin{algorithm}[bt]\n\\centering\n\\caption{Search tree generation for manipulation sequence planning}\n\\begin{algorithmic}[1]\n  \\State initialize search tree $\\mathcal{S}_{0} \\gets \\emptyset$\n  \\State initialize set of objects $\\mathcal{O}_{0}$ with current scene\n\t\\Procedure{createNode}{$\\mathcal{S}_{i}$, $\\mathcal{O}_{i}$}\n\t  \\If{$|\\mathcal{O}_{i}|\\le 1$, i.e.\\ this is a leaf node}\n\t\t\\State\\Return{}\n\t  \\Else\n\t\t\\State create new tree node $N_{i}$ containing object set $\\mathcal{O}_{i}$ \n\t\t\\State $\\mathcal{S}_{i+1} \\gets \\mathcal{S}_{i} \\cup N_{i}$\n\t\t\\State determine new active object $\\alpha_{i} \\in \\mathcal{O}_{i}$\n\t\t\\State$\\mathcal{O}_{i+1} \\gets \\mathcal{O}_{i} \\setminus \\alpha_{i}$\n\t\t\\State\\Return{\\Call{createNode}{$\\mathcal{S}_{i+1}$, $\\mathcal{O}_{i+1}$}}\n\t  \\EndIf\n\t\\EndProcedure\n \\State \\textbf{output}: filled search tree $\\mathcal{S}$\n\\end{algorithmic}\\label{alg:planning_tree_generation}\n\\end{algorithm}\n\nThe solution presented in this publication includes a \\emph{search tree} containing all object configurations occuring during the sequenced manipulation of scene objects. Figure~\\ref{fig:tree_example} shows an example of such a tree with the initial configuration appearing in the root (top). Traversing the tree towards its branches, the manipulation sequence is performed by removing one object each from the initial configuration. In the example, this causes extended costs for the \\obj{1}{pink} which falls off the \\obj{1}{blue} when the latter is manipulated (as shown in the leftmost depicted tree node on Level 2). In the tree leaves, only one object remains for direct manipulation without regarding the scene dynamics because no passive object is to be paid attention to.\n\n\\begin{algorithm}[tb]\n\\centering\n\\caption{Manipulation sequence planning for mental simulation}\n\\label{alg:planning}\n\\begin{algorithmic}[1]\n    \\State \\textbf{input}: scene objects $\\mathcal{O}$, search tree $\\mathcal{S}$ generated from $\\mathcal{O}$ ($\\rightarrow$ Algorithm~\\ref{alg:planning_tree_generation})\n    \\ForAll {nodes $N_{i} \\in \\mathcal{S}$}\n        \\State spawn all objects in $N_{i}$ in simulation\n        \\State plan approach $T_{1}^{\\alpha}$ and extract $T_{2}^{\\alpha}$ trajectory\n        \\State move simulated robot on $T_{1}^{\\alpha}$\n        \\State grasp active object $\\alpha$\n        \\State move simulated robot on $T_{2}^{\\alpha}$\n        \\State release active object $\\alpha$\n        \\State determine manipulation costs $V_{w,i}$ ($\\rightarrow$ Section~\\ref{cost_estimation})\n\t\\EndFor\n    \\ForAll {leaf nodes $N_{j}^{\\mathrm{leaf}} \\in \\mathcal{S}$}\n\t\t\\State $V_{w,j} \\gets$ summed-up costs of $N_{j}^{\\mathrm{leaf}}$'s parents\n\t\t\\State $V_{w,\\mathrm{min}} \\gets \\mathrm{min}(V_{w,\\mathrm{min}}, V_{w,j})$\n\t\\EndFor\n    \\State \\textbf{output}: node $N_{j}$ with lowest manipulation costs $V_{w,\\mathrm{min}}$\n\\end{algorithmic}\\label{alg:planning_anticipation}\n\\end{algorithm}\n\nAlgorithm~\\ref{alg:planning_tree_generation} shows how a search tree $\\mathcal{S}$ is generated for a particular scene given its initial object configuration. This works in a recursive way, starting from the root node (Level 0) which includes the initial configuration, down to the leaves where only one object is left in the scene. Each node $N_{i} \\in \\mathcal{S}$ contains the respectively assigned object set $\\mathcal{O}_{i}$ as seen in Figure~\\ref{fig:tree_example}, but not the poses of the objects.\nAfter filling the tree, this is traversed like in Algorithm~\\ref{alg:planning_anticipation}, using a depth-first search-like technique, with the objective of finding the node with the minimum manipulation costs $V_{w,\\mathrm{min}}$. During traversal, the final state of the mental simulation of each node is propagated into its child nodes. This determines the initial object poses for the respective child node prior to running mental simulation on it.\n\nUsing depth-first search may seem inefficient at first glance. Standard search algorithms like $A^{*}$, however, require an admissible heuristic which needs to be adapted to the given problem domain.\nSince damage avoidance is one of the main contributions of this article, the maximum weighted swept convex volume $V_w$ is used as a manipulation cost function which considers complex trajectories and changes of movement direction and spin. \nHowever, defining an admissible heuristic is not possible for this problem because the distance (measured in terms of the used cost function) to the target configuration cannot be determined before actually having searched the tree until its leaves.\n\nSection~\\ref{evaluation} shows the general feasibility and efficiency of the presented planning method. However, the whole mental simulation process may take extended time, especially for a growing number of scene objects. Therefore, the next section explains how to increase the overall efficiency by generating manipulation strategies which eventually replace time-consuming planning in the long run.\n\n\n\\subsection{Subconscious preference patterns}\\label{modeling_preference_patterns}\nWhenever a human needs to decide between two alternatives like \"Should I take A or B?\", they use some inherent classification scheme from their prior knowledge to come up with a decision. In a situation that requires motoric interaction on two objects, the decision which one to manipulate is driven by the concrete scene and objects, but also by physical properties of the human itself.\n\nImagine, for instance, a right-handed user who, whenever possible with respect to occlusions, reachable work\\-space and other kinetic limitations, always grasps with their right hand. For a setting where two objects are similarly reachable, but one needs to be grasped with the right and the other one with the left hand, a right-handed human, minimizing physical effort, would naturally know which one to grasp best.\n\nNow imagine a scenario with a robot for which grasping object A is easier than grasping object B in a specific scene because of\n\\begin{itemize}\n    \\item limited workspace due to manipulator size, position on the robot base and number of degrees of freedom\n    \\item scene obstacles preventing from reaching object B\n    \\item uneven probability distributions in the stochastic motion planner which make it more likely to obtain solutions for object A\n    \\item the grasping pose of object B, which is sampled from the noisy object representation, being located outside of the manipulator workspace.\n\\end{itemize}\n\nOne good example for such a scenario is the PR2 robot in the supermarket (Fig.~\\ref{fig:intro_scenario_shopping}) which, for the ease of obtaining viable grasping configurations, is used in a right handed-only way, i.e.\\ the left arm is tucked up while grasping only with the right one. Consequently, a tendency to grasp objects on the right first can be observed in scenarios using this robot configuration -- see the robot arm's kinematic reachability in Fig.~\\ref{fig:pr2_workspace}.\n\nTherefore, noise on object pose and grasping pose level plays an important role when considering which object to prefer for grasping. Many publications deal with circumventing such noise and finding manipulation plans nevertheless, e.g.\\ \\cite{Battaglia2013,Vaskevicius2014a,Mojtahedzadeh2015a,Weisz2012}, amongst many others.\n\nHowever, humans deal with such noise intuitively, implicitely taking into account all mentioned constraints. Over many observations, \\emph{subconscious preference patterns} become apparent as shown in experiments. The proposed approach integrates these patterns in a way that, as for determining a manipulation sequence for an unknown scene, the robot behaves humanlike in a way that certain manipulation actions are more likely than others for the reasons stated above.\nSince many underlying sources are encoded into the presented method's output, changing properties like the used motion planner or the robot model will get reflected in the manipulation sequence predictions similar to human long-term knowledge.\n\n\\begin{figure}[tb]\n    \\centering\n    \\includegraphics[width=0.7\\linewidth]{pr2_workspace.png}\n    \\caption{Kinematic reachability of the PR2's right arm\\\\ \\tiny{adapted from \\url{http:\/\/openrave.org\/docs\/0.6.6\/\\_images\/tutorial\\_inversereachability\\_back.png}, $\\copyright$ 2006-2018 OpenRAVE (CC BY 3.0)}}\n    \\label{fig:pr2_workspace}\n\\end{figure}\n\nThe evaluation (see Section~\\ref{evaluation}) shows how such preference patterns emerge from the considered application scenarios. As described above, \\emph{ranking by pairwise comparison} serves as a label ranking method which is particularly useful because it employs the mentioned preference patterns to create a classifier which can derive a ranking. The next subsection describes how this method is applied in the course of the overall approach.\n\n\\subsection{Preference-weighted label ranking}\\label{ranking_weighted}\nUsing the preference patterns as described above and the feature vector $\\mathbf{x}$, the original \\emph{ranking by pairwise comparison} (RPC) method combines them into a final ranking by calculating the sum of votes like described in \\cite{Huellermeier2008}:\n\\begin{equation}\ns = \\displaystyle\\sum_{\\pi(i) \\succ \\pi(j)} C_{ij}\n\\end{equation}\nwhere $C_{ij}: \\mathbf{X} \\to \\mathcal{L}, \\mathbf{x} \\mapsto \\pi$ is the output of the base classifier for $\\pi(i) \\succ \\pi(j)$ and $n$ labels. In case of using \\emph{soft voting} (see Section~\\ref{learning}), $C_{ij} \\in [0,1]$, whereas for \\emph{binary voting} $C_{ij} \\in \\{ 0,1 \\}$, i.e.\\ the input does or does not belong to the respective class.\n\nSince the goal is to regard the label ranking process from a preference-pattern viewpoint, the following enhancement allows for taking into account \\emph{preference weights} $w_{ij}$ when computing the sum of votes:\n\\begin{equation}\ns = \\displaystyle\\sum_{\\pi(i) \\succ \\pi(j)} v_{ij}\n\\end{equation}\nwhere\n\\begin{equation}\n    v_{ij} =\n\\begin{cases}\n    w_{ij} \\cdot C_{ij} & \\text{if } C_{ij} > 0.5,\\\\\n    0 & \\text{otherwise,}\n\\end{cases}\n\\end{equation}\n\\begin{equation}\\label{eq:preference_weights}\n    w_{ij} = \\frac{|\\{(i,j)~|~\\pi(i) \\succ \\pi(j)\\}|}{m},\n\\end{equation}\nand $m$ is the number of training samples.\n\nThe weights $w_{ij}$ are used to balance a certain preference $\\pi(i) \\succ \\pi(j)$ versus its inverse $\\pi(j) \\succ \\pi(i)$. Given the training data with 100 samples per scene to minimize the influence of noise in the physics simulation and motion planner, they are formed by the proportion of a particular preference in relation to the whole training data. This way, the resulting ratio indicates how likely a particular object will be preferred over another in this particular scene configuration.\n\nIn the practical process of generating manipulation strategies, now a classifier is trained using the described preconditions and additions to RPC. The resulting classifier, spawned from the training data which was extracted from auto-generated scenes, forms the current \\emph{manipulation strategy} of the robot in the given environment. This strategy can be optimized in a simulation-in-the-loop cycle as explained in detail in Section~\\ref{self_improvement}.\n\nHowever, experimental evaluation shows that the integration of preference weights into the classifier training process did not show a significant increase in prediction fidelity for the given use cases compared to the original RPC method (see Section~\\ref{results_rpc}).\nNevertheless, we use the preference weights to meaningfully compare two manipulation strategies.\nIn order to do this, a loss function $l(\\pi,\\pi')$ which gives the difference between two rankings $\\pi$ and $\\pi'$ has to be defined.\nIn the presented approach the number of pairwise preferences is counted which appear in inverse order (\\emph{discordant preferences}) compared to the training data, i.e.\n\\begin{equation}\nl(\\pi,\\pi') = |\\{(i,j)~|~\\pi(i) \\prec \\pi(j) \\land \\pi'(i) \\succ \\pi'(j)\\}|.\n\\label{eq:measure_kendalls_loss}\n\\end{equation}\n\nSince, for evaluation of the presented method, a similarity measure is needed in order to make sound statements about the resulting rankings, we use the established \\emph{Kendall rank correlation coefficient} \\cite{Kendall1938}, commonly denoted as \\emph{Kendall's tau coefficient} $\\tau: \\mathcal{L} \\times \\mathcal{L} \\to \\mathbb{Q}, (\\pi,\\pi') \\mapsto \\tau$ with\n\\begin{equation}\n\\tau(\\pi,\\pi')=1-\\frac{4\\,l(\\pi,\\pi')}{n(n-1)}\n\\label{eq:measure_kendalls_tau}\n\\end{equation}\nwhere $n=|\\mathcal{L}|$ is the number of labels to appear in the ranking. In the use case of manipulation sequences, this equals with the number of objects in the scene.\n\nAs a side note, in the context of label ranking, many approaches use some form of weights to allow for adaptation of their algorithms to the respective domain. Nevertheless, the definition of these weights usually does not match the preference weights defined above, like in \\cite{Shalev-Shwartz2006} where the authors assign per-label relevance weights different in every sample.\nThe same applies for the labelwise weight variant of $\\tau$ in Kumar and Vassilvitskii's work \\cite{Kumar2010} which only depend on the label itself, but not on its relation to other labels.\n\nSince there is the necessity for preference weights which consider a pairwise permutation of labels, though, the \\emph{preference-weighted Kendall's tau} rank correlation measure $\\tau_{w}: \\mathcal{L} \\times \\mathcal{L} \\to \\mathbb{R}, (\\pi,\\pi') \\mapsto \\tau_w$ is proposed with\n\\begin{equation}\\label{eq:preference_weighted_kendalls_tau}\n    \\tau_w(\\pi,\\pi') = 1-\\frac{4\\,l_{w}(\\pi,\\pi')}{n(n-1)},\n\\end{equation}\n\\begin{equation}\\label{eq:preference_weighted_kendalls_loss}\n    l_{w}(\\pi,\\pi') = \\displaystyle\\sum_{\\pi(i) \\succ \\pi(j)} d_{ij},\n\\end{equation}\n\\begin{equation}\n    d_{ij} =\n\\begin{cases}\n        \\max{(w_{ij}, w_{ji})} - 0.5 &\n    \\begin{aligned}[t]\n         \\text{if } & \\pi(i) \\prec \\pi(j) \\\\\n         & \\land \\pi'(i) \\succ \\pi'(j)\n    \\end{aligned}\\\\\n        0 & \\text{otherwise}\n\\end{cases}\n\\end{equation}\n\nwhere $w_{ij}$ is the preference weight of the respective pairwise preference $\\pi(i) \\succ \\pi(j)$ computed from the noisy training data of one scene as in Eq.~\\ref{eq:preference_weights} and $n=|\\mathcal{L}|$ is the number of labels to appear in the ranking.\n\n\\begin{figure}[bt]\n\\centering\n\\resizebox{0.9\\linewidth}{!}{\n\\begin{tikzpicture}[font=\\LARGE]\n\\begin{axis}[\n      xtick={0.0,0.1,...,1.1},\n      x tick label style={\/pgf\/number format\/.cd,fixed,fixed zerofill,precision=1,\/tikz\/.cd},\n      xmin=0.0,\n      xmax=1.0,\n      ytick={-1.0,-0.8,...,1.1},\n      y tick label style={\/pgf\/number format\/.cd,fixed,fixed zerofill,precision=1,\/tikz\/.cd},\n      ymin=-1.0,\n      ymax=1.0,\n\t  xmajorgrids=true,\n\t  ymajorgrids=true,\n      xlabel={preference weight $w_{ij}$},\n      ylabel={preference-weighted Kendall's tau $\\tau_{w}$},\n\t  width=\\linewidth,\n\t  height=0.8\\linewidth,\n      legend style={at={(1.5,0.6)},anchor=center}\n    ]\n\t\\addplot+[mark=, color=red]file{relevance_weights_vs_weighted_tau_4obj_1dis.csv}; \\addlegendentry{four objects, one discordant pref. ($\\tau=0.66$)}\n\t\\addplot+[mark=, color=orange]file{relevance_weights_vs_weighted_tau_4obj_2dis.csv}; \\addlegendentry{four objects, two discordant pref. ($\\tau=0.33$)}\n\t\\addplot+[mark=, color=blue]file{relevance_weights_vs_weighted_tau_4obj_4dis.csv}; \\addlegendentry{four objects, four discordant pref. ($\\tau=-0.33$)}\n\t\\addplot+[mark=, color=green]file{relevance_weights_vs_weighted_tau_2obj.csv}; \\addlegendentry{two objects, one discordant pref. ($\\tau=-1.00$)}\n    \\node(note) at (axis cs:0.5,0.1) [anchor=center, align=center] {minimum\\\\information\\\\content};\n    \\draw[-latex, thick] (note) to[out=90, in=270] (axis cs:0.5,0.85);\n\\end{axis}\n\\end{tikzpicture}\n}\n\\caption{Preference-weighted Kendall's tau $\\tau_{w}$ relative to preference weights $w_{ij}$ with respect to different numbers of discordant preferences and objects}\n\\label{fig:weighted_kendalls_tau}\n\\end{figure}\n\nThis way, wherever the distance between two rankings is to be calculated using $\\tau_{w}$, every preference that appears only in one of the rankings gets weighted with the respective preference weight. Figure~\\ref{fig:weighted_kendalls_tau} shows an example of $\\tau_w$ scaling between 1.0 and the respective $\\tau$ when the weights scale between 0.5 (completely random, no preference, minimum information content) and 1.0 (strong preference, maximum information content).\nThe resulting $\\tau_{w}$ allows for continuous values with an image cardinality of $\\lvert \\tau_{w}[\\pi,\\pi'] \\rvert = \\lvert \\mathbb{R} \\rvert$ as opposed to $\\tau$ which has a discrete image with a cardinality of only $\\lvert \\tau[\\pi,\\pi'] \\rvert = 2n-1$ for $n$ labels. This bears the advantage that the more fine-grain $\\tau_{w}$ can distinguish pairs of sequences according to preference relations where the coarse-grain $\\tau$ provides the same measure for these pairs. The evaluation section shows examples why and how this is useful for the proposed method.\n\nSummarized, this section described how to generate manipulation strategies from mentally simulated manipulation sequences. These can now be deployed in a real application scenario and continuously optimized as described in the following.\n\n\\section{Prerequisites}\\label{prerequisites}\nFirst of all, before explaining the presented method in detail, this section gives a short overview about certain prerequisites that have to be considered as given when deploying the method. This includes the initial definition of the scene with respect to the particular usage scenario as well as how to auto-generate a large number of scenes used within the manipulation strategy optimization cycle.\n\n\\subsection{Scenario definition}\\label{scene_representation}\nAs possible application scenarios are different, the degree of abstraction with respect to simulation of objects and robots needs to be adapted to the desired use and available development and computing capabilities. Modeling robot behavior and object properties in detail in a simulation engine may not be efficient and even redundant.\n\nThe presented method was designed to be used in integrated scenarios that use object recognition systems along with a physics simulation on the one hand and a real-world robot on the other hand which executes the planned manipulation strategies.\nHowever, the focus of this method is on manipulation strategy planning and not the enhancement of perception systems or motion planning\/execution. Hence, noise and other inaccuracies within the perception and execution pipeline will not be addressed in this publication. Instead, this subsection briefly explains the approach taken for representing objects and the robot within our mental simulation.\n\n\\subsubsection{Object representation}\\label{object_representation}\n\n\\begin{figure*}[t!]\n    \\centering\n    \n    \\def1.25cm{1.25cm}\n    \n    \\begin{subfigure}{0.48\\linewidth}\n        \\centering\n        \\setlength{\\tabcolsep}{.4em}\n        \\begin{tabular}{ccrr}\n            \\multicolumn{2}{c}{\\includegraphics[height=1.25cm]{scene1_obj0_real.jpg}}&\n            \\includegraphics[height=1.25cm]{scene1_obj2_real.jpg}&\n            \\includegraphics[height=1.25cm]{scene1_obj3_real.jpg}\\\\\n            \n            \\includegraphics[height=1.25cm]{scene1_obj0.png}&\n            \\includegraphics[height=1.25cm]{scene1_obj1.png}&\n            \\includegraphics[height=1.25cm]{scene1_obj2.png}&\n            \\includegraphics[height=1.25cm]{scene1_obj3.png}\\\\\n        \\end{tabular}\n        \\caption{Logistics scenario}\\label{fig:simulation_objects:logistics}\n    \\end{subfigure}\n    \\begin{subfigure}{0.48\\linewidth}\n        \\centering\n        \\setlength{\\tabcolsep}{.4em}\n        \\begin{tabular}{ccrr}\n            \\multicolumn{2}{c}{\\includegraphics[height=1.25cm]{scene3_obj0_real.jpg}}&\n            \\includegraphics[height=1.25cm]{scene3_obj2_real.jpg}&\n            \\includegraphics[height=1.25cm]{scene3_obj3_real.jpg}\\\\\n            \n            \\includegraphics[height=1.25cm]{scene3_obj0.png}&\n            \\includegraphics[height=1.25cm]{scene3_obj1.png}&\n            \\includegraphics[height=1.25cm]{scene3_obj2.png}&\n            \\includegraphics[height=1.25cm]{scene3_obj3.png}\\\\\n        \\end{tabular}\n        \\caption{Supermarket scenario}\\label{fig:simulation_objects:supermarket}\n    \\end{subfigure}\n    \n    \\caption{Objects and their simulation representations (not drawn to scale)}\n    \\label{fig:simulation_objects}\n\\end{figure*}\n\n\nAs for representing the scene objects in simulation, it is generally desired to obtain detailed models of them in order to achieve realistic dynamic behaviors. \nHowever, detailed object models generated from a real object's point cloud introduce the inherent disadvantage of possibly generating instable initial scenes in the physics simulation due to modelling inaccuracies. This may lead to some objects shifting immediately after spawning them in the physics engine.\nThis has to be avoided for mentally simulating manipulation actions since initially stable scenes in the physics simulation represent reality where, during and after object recognition, objects most likely are not transposed either if not physically influenced. In order to preserve stable initial configurations, the proposed method therefore uses abstract object representations as shown in Fig.~\\ref{fig:simulation_objects} in order to avoid modeling errors.\n\nDuring productive use in a concrete application scenario, the respective scene configuration needs to be established in the physics simulation.\nThis requires to recognize and localize objects occuring in the scene; for the given use cases the perception system originally developed for the logistics scenario \\cite{Stoyanov2016,Vaskevicius2014} is utilized. Therein, different segmentation and filtering steps are combined into a pipeline which feeds the pre-processed sensor data into several object recognition modules. Amongst these are a graph-based shape model object recognition module \\cite{Mueller2014} and a feature-based textured object recognition module \\cite{Vaskevicius2012}, as illustrated in Fig.~\\ref{fig:pipeline}.\n\n\\begin{figure*}[b!h]\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{perception_pipeline_a.jpg}\n\t\\caption{Perception pipeline \\cite{Stoyanov2016}, consisting of the respective processing steps from data acquisition to the fusion of hypotheses from a texture-based \\cite{Vaskevicius2012} and a shape-based recognition module \\cite{Mueller2014}}\n\t\\label{fig:pipeline}\n\\end{figure*}\n\nIn this exemplary perception pipeline, as a first step, RGBD data is acquired and immediately preprocessed for noise reduction. Next, segments are generated that are homogeneous with respect to geometric and\/or color-based criteria in order to provide a low-level description of the scene objects. These segments are then merged according to some application-dependent heuristics such as convexity. From these segments, the shape-based and texture-based object recognition modules identify the object candidates. Their hypotheses are then fused heuristically as a final step to broadcast the canonical object configuration to be projected into the simulation engine.\n\n\n\\subsubsection{Robot control}\\label{robot_control}\nMotion execution, on the simulated as well as the real-world robot, requires grasp planning taking into account the perceived objects.\nGrasp planning itself depends on the scenario for the reason of differing sizes and degrees of freedom of the grippers, also the dimensions, weight and texture of the objects has influence on the grasping policy. In the presented work, the grasping step is aimed to be kept as generic as possible by placing grasping configurations in a predefined distance around the principal axes of known object models like the ones in Fig.~\\ref{fig:simulation_objects}. This means that, whatever object is added to the scenario, respective grasping configurations can be generated with identical properties like the existing ones of other objects. Hence the method imposes minimal constraints on the grasping process and is ready to be reused in many possible settings where, on demand, more sophisticated grasping policies can be developed.\n\nFor physics simulation, the Gazebo simulator\\footnote{\\url{http:\/\/gazebosim.org}}~\\cite{Koenig2004} is used, including kinematics and dynamics of the robot along with simulated controllers. The container unloading robot of the logistics scenario (Fig.~\\ref{fig:intro_scenario_roblog}) uses a custom set of low-level and high-level controllers whereas in the supermarket scenario (Fig.~\\ref{fig:intro_scenario_shopping}), the simulation model shipped by default with the PR2 robot is used. A standard motion planner like the ones integrated in the Open Motion Planning Library\\footnote{\\url{http:\/\/ompl.kavrakilab.org\/}}~\\cite{Sucan2012} eventually allows for planning approach, grasp and retract motions.\n\nUsing the defined scenarios, the physics simulation can now be utilized to perform plan manipulation sequences via mental simulation. This, together with a methodology how to take care of possible damage, will be explained in the following section.\n\n\n\n\\subsection{Training scene generation}\\label{random_scenes}\nOne important part of the proposed method includes the robot using load-free times to optimize its manipulation strategies. This includes re-training the respective machine learning algorithms, but also the generation of additional training data. In order to be ready for unloading many possible combinations of objects in different spatial relationships, we generate training scenes with configurations where objects are likely to collide with or obstruct other objects.\nThis happens by spawning the objects in clusters defined by their $x,y,z$ size in the predefined workspace (e.g.\\ inside a container, on a specific shelf level). Random cluster centroids are then drawn from the workspace volume around which the objects are located with random 6-D poses.\n\n\\begin{figure*}[t!b]\n\t\\centering\n\t\\def0.1\\linewidth{0.1\\linewidth}\n\t\\begin{subfigure}{\\linewidth}\n\t\t\\centering\n\t\t\\includegraphics[height=0.1\\linewidth]{parcelrobot_random_scene1.png}\n\t\t\\includegraphics[height=0.1\\linewidth]{parcelrobot_random_scene2.png}\n\t\t\\includegraphics[height=0.1\\linewidth]{parcelrobot_random_scene4.png}\n\t\t\\includegraphics[height=0.1\\linewidth]{parcelrobot_random_scene5.png}\n\t\t\\includegraphics[height=0.1\\linewidth]{parcelrobot_random_scene6.png}\n\t\t\\includegraphics[height=0.1\\linewidth]{parcelrobot_random_scene7.png}\n\t\t\\caption{Logistics scenario}\n\t\\end{subfigure}\n\t\\hfill\n\t\\begin{subfigure}{\\linewidth}\n\t\t\\centering\n\t\t\\includegraphics[height=0.1\\linewidth]{pr2_random_scene1.png}\n\t\t\\includegraphics[height=0.1\\linewidth]{pr2_random_scene2.png}\n\t\t\\includegraphics[height=0.1\\linewidth]{pr2_random_scene3.png}\n\t\t\\includegraphics[height=0.1\\linewidth]{pr2_random_scene4.png}\n\t\t\\includegraphics[height=0.1\\linewidth]{pr2_random_scene5.png}\n\t\t\\includegraphics[height=0.1\\linewidth]{pr2_random_scene6.png}\n\t\t\\caption{Supermarket scenario}\n\t\\end{subfigure}\n\t\\caption{Training scene examples}\n\t\\floatfoot{Five superimposed collision-free variations of each scene with stochastic object pose noise.}\n\t\\label{fig:random_scenes}\n\\end{figure*}\n\nFor refining the object poses and removing any interpenetrations that may have occured, the presented method uses \\textsc{Promts}\\footnote{\\url{https:\/\/github.com\/Rasoul77\/promts}} \\cite{Mojtahedzadeh2015a} which translates any given object configuration in space to a collision-free configuration.\nBattaglia et al.\\ \\cite{Battaglia2013} use the same principle that helps generating scenes which incorporate a certain amount of noise, but still are conceivable as per human intuition and the physics simulation. \n\nFigure~\\ref{fig:random_scenes} shows some examples for both of the used scenarios. In fact, we create several variations of each scene with stochastic noise added to the object poses which are superimposed in the examples. Whenever new training scenes have been generated, they are provided as input for the training data extraction process as described in the next section.\n\n\n\n\n\\subsection{Experiments}\\label{experiments}\nIn order to evaluate the presented method with respect to real-life usage, it is important to note that no ground truth exists other than human intuition which the results can be compared with. The only way of providing ground truth manipulation sequences is human assessment with the human built-in mental simulation capabilities being prone to abstracting and simplifying complex dynamics just as a physics engine. Hence, the following results have to be judged using common-sense intuition since the author is not aware of any comparable approaches presented so far which would provide a baseline dataset.\n\n\\subsubsection{Application scenarios}\nThe experimental evaluation of the presented method is performed on the two scenarios described in the introduction with typical scenes shown in Figure~\\ref{fig:intro_scenarios}.\n\nFigure~\\ref{fig:histograms_parcelrobot} shows a typical \\emph{logistics} scenario with Scene 2 being more challenging because some items provide support to each other. This increases the likelihood of moving a passive object whenever attempting a manipulation action.\n\nIn the \\emph{supermarket} scenario in Figure~\\ref{fig:histograms_pr2}, the tall shelf equipped with cans, rolling away if dropped, provides a hostile environment to any damage-prone items. In such a retail environment, items additionally often tip over like in Scene 4 when customers pull out goods from the back.\n\nWhen a traditional, collision avoidance-based motion planning algorithm is confronted with this kind of scenarios, it is not very likely to find any solution to clear the scene completely because of objects touching and blocking each other. In all these scenes, touching object configurations can be resolved only heuristically when using classical methods. In contrary, the presented task-level planning method allows for prior-free resolution with less difficulties in clearing all objects than, for example, the approaches presented in \\cite{Stoyanov2016} and \\cite{Winkler2016} operating on the same scenarios. Both of their evaluations report enhanced complexity in planning and execution for obstructions. Instead, the presented approach avoids these pitfalls by initially selecting a feasible high-level manipulation order for which the motion planning complexity itself is reduced significantly.\n\n\\subsubsection{Cost-benefit comparison with existing approaches}\n\\input{results_parcelrobot.tex}\n\\input{results_pr2.tex}\nThe main requirements for the proposed approach when it is used for high-level planning as an add-on to classical motion planning is the setup of a simulation environment including object and robot models along with the respective controllers as described in Section~\\ref{prerequisites}. Although this means a one-time setup per application scenario whose effort strongly decreases over multiple scenarios with the reuse of software components, this required effort may not be neglected.\n\nFor the presented scenarios, the setup of a simulation environment as in the Supplementary Video had been demanded for in the RobLog research project's requirements. The respective work package comprised of a dynamics anticipation approach including the from-scratch setup of all prerequisites described above and was accounted for with a workload of 23 person months. However, since this number includes a high amount of research which stretches beyond what is required per scenario utilizing a working software solution, the actual anticipated workload in a productive environment is a lot lower. \nOur experience showed that, after establishing and optimizing the presented method in the logistics scenario, setting up the supermarket scenario subsequently required a workload of only about 3-4 person months.\nThis might still seem a lot if traditional approaches possibly are able to deal with the same sort of challenges and scenarios, however, the prior-free damage avoidance of the proposed approach is a valuable asset especially in use cases dealing with expensive or high-throughput goods.\n\nIn terms of computational effort required for manipulation strategy generation and deployment, typical cycle times of the same research-based scenario circulate in the region of 3.5-\\SI{5}{minutes} per item \\cite[p.~11]{Stoyanov2016}. In this context, the classification times for manipulation strategies in a sub-second magnitude do not carry any weight during productive use. Additional computational effort of the proposed approach accrues mostly in the manipulation strategy optimization loop which can be run offline, during load-free robot runtime. Therefore, the proposed method imposes no additional requirements in terms of computing hardware compared to traditional methods.\n\n\\subsubsection{Mental simulation}\\label{evaluation_planning}\nMental simulation (Algorithm~\\ref{alg:planning_anticipation}) was run 100 times for each of the two exemplary scenes per domain.\nThe histograms in Figures~\\ref{fig:histograms_parcelrobot} and \\ref{fig:histograms_pr2} show how many times a particular manipulation sequence $\\pi$ was chosen as the most optimal one. A clear preference for a specific sequence is visible for all of  scenes, however, in Scene 3, the distance in frequency between the first and second-ranked sequence is less distinct. This stems from the fact that, in the given object configuration, it does not seem to make a big difference regarding possible damage whether, for instance, the \\obj{3}{red} or the \\obj{3}{blue} is manipulated first. In the other scenes, however, the objects bear a higher spatial dependency on each other and it is more likely to distort the setup when manipulating them in any other than the top-ranked order.\nSummarized, these results show that mental simulation within the presented method is effective per se on different object and robot configurations.\n\nRegarding the efficiency of the proposed approach, several generic means of optimization have been explained in Section~\\ref{efficiency}. With respect to these, the following concrete criteria have been evaluated in the numerical results shown in Table~\\ref{table:statistics}:\n\\begin{itemize}\n\t\\item \\emph{mean first-ranked costs per node}: mean over all planning repetitions of the mean costs per object imposed on the manipulation sequence selected by Algorithm~\\ref{alg:planning_anticipation}\n\t\\item \\emph{mean second-ranked costs per node}: mean over all planning repetitions of the mean costs per object imposed on the second-ranked manipulation sequence\n\t\\item \\emph{pruned tree nodes}: percentage of nodes pruned from the tree, resulting in a similar reduction in runtime, and composed of the following sub-criteria:\n\t\\begin{itemize}\n\t\t\\item \\emph{known subtree}: a node shares its object configuration and poses with a previously processed node, thus the costs were copied without re-si\\-mu\\-la\\-ting\n\t\t\\item \\emph{costs exceed existing sequence}: a solution is existing already which has lower costs than the costs accumulated so far in this branch\n\t\t\\item \\emph{active object moved}: the active object was pushed away during approach, ending up unreachable\n\t\t\\item \\emph{object out of workspace}: an object fell out of the workspace (container\/shelf), causing maximum damage and ending up unreachable for the robot\n\t\t\\item \\emph{planning failure}: no motion plan was found for the active object, e.g.\\ because it was pushed away too far in a parent node\n\t\\end{itemize}\n\t\\item \\emph{nodes with significant movement}: percentage of nodes which were not pruned from the tree and reported significant costs above a manually defined threshold \n\\end{itemize}\n\n\\begin{table*}[b!]\n\\centering\n\\scriptsize\n\\setlength{\\tabcolsep}{.4em}\n\n\\begin{tabular}{lccccc}\n\\textbf{\\normalsize{Scene}} & \\textbf{\\normalsize{1}} & \\textbf{\\normalsize{2}} & \\textbf{\\normalsize{3}} & \\textbf{\\normalsize{4}} & \\textbf{\\normalsize{Total}} \\\\\n\\rule{0pt}{3ex}\\textbf{costs per node} & & & & & \\\\\n - first-ranked sequence       &  1.008 \\tiny{$\\pm$0.008} &  1.208 \\tiny{$\\pm$0.074} &  1.349 \\tiny{$\\pm$0.359} &  1.975 \\tiny{$\\pm$0.321} &  1.382 \\tiny{$\\pm$0.435} \\\\\n - second-ranked sequence      &  1.337 \\tiny{$\\pm$0.868} &  1.429 \\tiny{$\\pm$0.326} &  1.700 \\tiny{$\\pm$1.728} &  3.764 \\tiny{$\\pm$7.393} &  2.051 \\tiny{$\\pm$3.878} \\\\\n\\rule{0pt}{3ex}\\textbf{pruned tree nodes}             & \\meanplotvalue{34.8}{0} & \\meanplotvalue{50.4}{0} & \\meanplotvalue{58.3}{0} & \\meanplotvalue{61.9}{0} & \\meanplotvalue{51.3}{0} \\\\\n - known subtree                       & \\meanplotvalue{7.5}{4.8} & \\meanplotvalue{2.6}{2.5} & \\meanplotvalue{1.2}{2.2} & \\meanplotvalue{0.0}{0.0} & \\meanplotvalue{2.8}{4.1} \\\\\n - costs exceed existing seq.          & \\meanplotvalue{16.5}{8.3} & \\meanplotvalue{18.3}{5.3} & \\meanplotvalue{19.3}{14.7} & \\meanplotvalue{41.0}{18.1} & \\meanplotvalue{23.8}{16.0} \\\\\n - active object moved                 & \\meanplotvalue{5.8}{5.6} & \\meanplotvalue{25.2}{13.5} & \\meanplotvalue{4.2}{11.6} & \\meanplotvalue{8.1}{9.2} & \\meanplotvalue{10.9}{13.2} \\\\\n - object out of workspace             &  \\meanplotvalue{0.0}{0.0} & \\meanplotvalue{0.0}{0.0} & \\meanplotvalue{31.9}{15.7} & \\meanplotvalue{7.0}{9.6} & \\meanplotvalue{9.6}{15.9} \\\\\n - planning failure                    &  \\meanplotvalue{5.0}{9.6} & \\meanplotvalue{4.3}{8.4} & \\meanplotvalue{1.7}{3.1} & \\meanplotvalue{5.8}{10.9} & \\meanplotvalue{4.2}{8.6} \\\\\nnodes with significant & \\meanplotvalue{13.8}{3.0} & \\meanplotvalue{11.9}{4.9} & \\meanplotvalue{14.8}{4.3} & \\meanplotvalue{18.9}{5.1} & \\meanplotvalue{14.9}{5.0} \\\\\nmovement (costs $>2.0$)    &          &                \\\\\n\\end{tabular}\n\\caption{Mental simulation numerical results: means and standard deviations of 100 planning repetitions}\n\\label{table:statistics}\n\\end{table*}\n\nTable~\\ref{table:statistics} shows that pruning the tree is generally effective, eliminating a mean of \\SI{51.3}{\\%} of nodes from the tree during the planning process. Since the planning time behaves linear with respect to the number of nodes to cover, this means that one planning run can be completed in average within less than half the time compared to using an unpruned tree.\n\nIn Scene 2, a high number of nodes had to be skipped because the active object moved during approach. This is caused by the \\obj{2}{blue} and \\obj{2}{green} hiding the \\obj{2}{orange} and \\obj{2}{pink} from the robot's view. Therefore, the planner made the passive \\obj{2}{blue} and\/or \\obj{2}{green} push the currently active \\obj{2}{orange} or \\obj{2}{pink} away.\nRegarding Scene 3, an object has moved out of the workspace in \\SI{31.9}{\\%} of nodes. This was caused by the round \\obj{3}{red} which, when the \\obj{3}{green} is extracted from underneath, often rolls away and falls off the shelf.\n\nHowever, such exceptional cases can be caught and a feasible sequence can be found nevertheless. In total, although in many cases the computed manipulation trajectories did not cause significant disturbance in the scene, in \\SI{14.9}{\\%} of nodes significant movement was detected which, in real-world execution, may have led to non-negligible damage. Within tree search, however, these nodes have generally been avoided as shown in the first-ranked per-node costs. These are low enough to ensure the provided manipulation sequences are damage-minimizing.\n\n\\subsubsection{Subconscious preference patterns}\\label{results_patterns}\nWhen looking at the initial scene configurations, most of the selected sequences in the left column of Figures~\\ref{fig:histograms_parcelrobot} and \\ref{fig:histograms_pr2} intuitively make sense, however, certain preferences in a single run may look counter-intuitive.\nOne example for this is the \\obj{3}{blue} which surprisingly often turns up as a non-preferred object. This particular case can be explained with the fact that the robot has to extend its manipulator quite far to grasp this object, hence pushing other objects off the shelf if they are still present.\n\nNevertheless, such preference patterns may evolve from many repetitions on the same scene unexpectedly, depending on the object constellation, robot location and other factors.\nThe images on the right of Figures~\\ref{fig:histograms_parcelrobot} and \\ref{fig:histograms_pr2} show the respective preference weights $w_{ij}$ for the pairwise preferences of each object over all others.\n\nThe results show that, consistent with human intuition, generally objects whose centroid is higher than the one of other objects or which are resting on other objects are preferred. Other than that, objects obstructing a direct manipulation trajectory between gripper and objects in the back will be manipulated first in the most cases.\nIn Scenes 3 and 4, where the PR2 was used, there is an additional clear preference of objects which are situated at the right of other objects. This is not surprising since the right arm of the PR2 was used for manipulation, hence granting a bigger workspace to the right side as can be seen in Fig.~\\ref{fig:pr2_workspace}.\n\nSeveral other preferences, like between the \\obj{3}{red} and \\obj{3}{blue} in Scene 3, however, are less expressive in the way that the preference weights lie close to 0.5. This implies a minimum of information content because each direction of the preference is equally likely. This phenomenon is caused by the fact that a large number of viable grasping configurations can be generated easily for these objects in this configuration given the robot workspace and dexterity.\n\nSummarized, even though it is very difficult to denote preferences patterns as encountered in the experiments manually in an extensive way, they can be discovered by the presented approach.\n\n\\subsubsection{Preference-weighted Kendall's tau $\\tau_{w}$}\nThe next subsection shows the behavior of the classifier for different settings with the help of the exposed preference patterns during classifier training and prediction. The preferences are embedded into the $\\tau_{w}$ measure which gives very distinct results for sequences with identical $\\tau$. This results in an improved utility of the continuous $\\tau_{w}$ as opposed to $\\tau$ which gives coarse discrete values, see Figs.~\\ref{fig:histograms_parcelrobot} and \\ref{fig:histograms_pr2}.\n\nFor instance, the \\obj{1}{green} in the sequence \\seq{1}{pink}{green}{orange}{blue} ($\\tau_{w}=0.387$) in Fig.~\\ref{fig:histograms_parcelrobot_scene1} which gets moved first is obviously prone to accidently moving \\obj{1}{orange} or \\obj{1}{blue}. However, although they all produce the same $\\tau$ of $0.333$, in \\seq{1}{orange}{blue}{pink}{green} ($\\tau_{w}=0.562$) and \\seq{1}{orange}{pink}{green}{blue} ($\\tau_{w}=0.556$) it is less likely that a passive object is pushed away. Hence, in contrary to using $\\tau$ as a suitability measure, these sequences are better distinguishable using their $\\tau_{w}$ measure which takes the pairwise preferences into account. \n\n\\subsubsection{Ranking by pairwise comparison}\\label{results_rpc}\nIn order to show the performance of the overall classifier, a series of 80 training scenes was generated with five variants each which carry stochastic object pose noise. For each scene variant 100 samples of manipulation sequences were collected by running the mental simulation method on them in order to account for noise in the simulation itself, like described in Section~\\ref{evaluation_planning}. In total, this amounts to $80 \\cdot 5 \\cdot 100 = 40000$ samples, each containing a feature vector and the manipulation sequence determined by the planning algorithm. \nWith the used proof-of-concept implementation, data collection takes around \\SI{94}{minutes} per 100 samples using four parallel containers on a \\SI{3.4}{GHz} desktop CPU with \\SI{16}{GB} RAM while running with a real-time factor of \\SI{1.75 \\pm 0.80}{}.\n\n\\begin{figure*}[p]\n  \\begin{flushright}\n    \\pgfplotstabletranspose\\classifiertraining{classifier_training.csv}\n    \\pgfplotstablegetcolsof{\\classifiertraining}\n    \\pgfmathsetmacro{\\numrowsclassifiertraining}{\\pgfplotsretval-1}\n    \\pgfplotstableread{classifier_training_results.csv}\\classifiertrainingresults\n    \\pgfplotstablegetrowsof{\\classifiertrainingresults}\n    \\pgfmathsetmacro{\\numrowsclassifiertrainingresults}{\\pgfplotsretval-1}\n    \\begin{tikzpicture}[font=\\footnotesize]\n        \\begin{axis}[\n              xmin=-1,\n              xmax=1,\n              xtick={-1.0,-0.8,...,1.0},\n              xtick style={draw opacity=0.5},\n              xtick pos=left,\n              x tick label style={\/pgf\/number format\/.cd,fixed,fixed zerofill,precision=1,\/tikz\/.cd,font=\\scriptsize},\n              every node near coord\/.style={\/pgf\/number format\/.cd,fixed,fixed zerofill,precision=3,\/tikz\/.cd,font=\\tiny\\sffamily},\n              xlabel={$\\tau{_w}$},\n              ymin=-0.5,\n              ymax=\\numrowsclassifiertraining+0.8,\n              ytick style={draw=none},\n              y dir=reverse,\n              ytick={1,2,...,\\numrowsclassifiertraining},\n              yticklabels from table={\\classifiertrainingresults}{Description},\n              width=0.7\\textwidth,\n              height=0.03\\textheight*\\numrowsclassifiertraining+0.1\\textheight,\n              axis line style={draw opacity=0},\n              cycle list={{blue}},\n              boxplot\/draw direction=x,\n              every boxplot\/.style={color=black!40!blue,mark=x,every mark\/.append style={mark size=0.8pt},fill=blue!10!white}\n            ]\n            \\foreach \\i in {1,...,\\numrowsclassifiertraining} {\n                \\addplot+ [boxplot] table [y index=\\i] {\\classifiertraining};\n            }\n        \\end{axis}\n    \\end{tikzpicture}\n    \\end{flushright}\n    \\caption{Results for manipulation strategy generation: $\\tau_{w}$ for different numbers of scenes and samples per scene, binary\/soft voting, with\/without preference weights, each scene sampled five times with stochastic object pose noise.}\n    \\floatfoot{Median, lower and upper quartile ($Q_1$\/$Q_3$) over 100 classifier training repetitions. Lower and upper fences were calculated using $Q_1-1.5 \\cdot (Q_3-Q_1)$ and $Q_3+1.5 \\cdot (Q_3-Q_1)$, respectively.}\n    \\label{fig:results_learning}\n\\end{figure*}\n\nThe generated data set was fed into classifier training in different configurations of the voting method and with\/without integrating preference weights. Every run, the classifier was trained with a random 2:1 training\/testing split on the respectively indicated number of scenes and evaluated on the testing split afterwards. 100-fold repetition of this training\/testing loop yielded the distributions of results shown in Fig.~\\ref{fig:results_learning}.\nWith respect to classifier learning time, an average of \\SI{27.7 \\pm 11.7}{s} has been achieved with the number of scenes rising from 10 to 80. The learning time rose from 10 scenes taking less than one second to about 45 seconds for 80 scenes.\n\nAs shown in more detail in the next subsection, the results generally improve with the number of scenes. However, Fig.~\\ref{fig:results_learning} shows a difference in performance for many of the considered configurations, especially regarding the variance of the distributions. Generally, it is more desirable to obtain low-variance distributions rather than high-variance ones with a similar median because the former mean higher precision and hence show to be more resilient against noise.\nNevertheless, any bias in the distribution harms the precision, but positive bias w.r.t.\\ $\\tau_{w}$ means an increase in accuracy. Therefore, gauging accuracy versus precision of these results is important for the overall assessment.\n\nConcretely, even though the obtained results show no significant difference between configurations with or without injected preference weights, the weights themselves cohere with intuitive behavior. Hence this enables a deeper understanding of the classification results by providing the $\\tau_{w}$ metric which allows for measuring this coherency. Without this $\\tau_{w}$ metric which incorporates the preference weights, we would not be able to see that the learned manipulation strategy adheres to the subconscious preference patterns discovered using the proposed method.\n\nAs for the used voting method, the results in Fig.~\\ref{fig:results_learning} show a slight improvement of soft over binary voting in terms of mean and variance\/positive bias. Therefore the author recommends to use soft voting when applying the method.\nSome improvement is shown also for the number of samples per scene, although this is not surprising since a higher number of samples per individual scene statistically rules out a larger amount of noise. \n\nSummarized, for the following experiment which shows the effectiveness of the explained self-supervised optimization cycle, it was therefore chosen to use a configuration of \\emph{100 samples per scene, soft voting and no injection of preference weights}.\n\n\\subsubsection{Self-supervised free-time manipulation strategy optimization}\nWith the classifier configuration determined in the last experiment, the whole manipulation strategy optimization cycle was run in an iterative fashion like in Alg.~\\ref{alg:optimization}. It was initialized with a set of 10 training scenes taken from the total training set as described above. After establishing an initial classifier, one more training scene was generated, a damage-avoiding manipulation sequence planned and the classifier retrained. This was repeated up to a maximum of 80 scenes. Note that, for evaluation purposes, classifier training was repeated 10 times each with a randomized 2:1 training\/testing split on the scenes so as to generate a more noise-resilient evaluation.\n\n\\begin{figure}[btp]\n    \\pgfplotstabletranspose\\classifierimprovement{classifier_improvement_training.csv}\n    \\pgfplotstableread{classifier_improvement_training_results.csv}\\classifierimprovementresults\n    \\pgfplotstableread{classifier_improvement_training_linearized.csv}\\classifierimprovementlinearized\n    \\pgfplotstablegetrowsof{\\classifierimprovementresults}\n    \\pgfmathsetmacro{\\numrowsclassifierimprovementresults}{\\pgfplotsretval+10}\n    \\begin{tikzpicture}[font=\\footnotesize]\n        \\begin{axis}[\n              ymin=-0.2,\n              ymax=1,\n              ytick={-0.2,-0.0,...,1.0},\n              ytick style={draw opacity=0.5},\n              ytick pos=left,\n              y tick label style={\/pgf\/number format\/.cd,fixed,fixed zerofill,precision=1,\/tikz\/.cd,font=\\scriptsize},\n              every node near coord\/.style={\/pgf\/number format\/.cd,fixed,fixed zerofill,precision=3,\/tikz\/.cd,font=\\tiny\\sffamily},\n              ylabel={$\\tau_{w}$},\n              ymajorgrids,\n              xmin=9,\n              xmax=\\numrowsclassifierimprovementresults,\n              xtick pos=left,\n              xtick style={draw opacity=0.5},\n              xtick={10,15,...,\\numrowsclassifierimprovementresults},\n              xlabel={\\#scenes},\n              width=0.7\\textwidth,\n              height=0.3\\textheight,\n              axis line style={draw opacity=0},\n              cycle list={{blue}}\n            ]\n            \\addplot+ [smooth, blue!10!white, name path=UQ] table [x=Scenes, y=TauThirdQuartile] {\\classifierimprovementresults};\n            \\addplot+ [thick, black, name path=M, mark=x, mark size=1.3pt, only marks] table [x=Scenes, y=TauMedian] {\\classifierimprovementresults};\n            \\addplot+ [smooth, blue!10!white, name path=LQ] table [x=Scenes, y=TauFirstQuartile] {\\classifierimprovementresults};\n            \\addplot[blue!30!white] fill between [of=UQ and LQ];\n            \\addplot+ [thick, orange] table [x=x, y=y] {\\classifierimprovementlinearized};\n        \\end{axis}\n    \\end{tikzpicture}\n    \\caption{Results for manipulation strategy optimization: $\\tau_{w}$ for different numbers of scenes using 100 samples per scene, soft voting, no preference weights, each scene sampled five times with stochastic object pose noise.}\n    \\floatfoot{Median ($\\tilde{\\tau_{w}}$), \\textcolor{blue!70}{lower\/upper quartile envelope} and \\textcolor{orange}{least-squares best-fit line} for 10 training runs.}\n    \\label{fig:results_improvement}\n\\end{figure}\n\nThe emerging results are plotted in Fig.~\\ref{fig:results_improvement} with the median of $\\tau_{w}$ over these 10 classifier training runs per scene, further on named $\\tilde{\\tau_{w}}$. A \\textcolor{orange}{least-squares best-fit line} over $\\tilde{\\tau_{w}}$ for each scene shows a positive slope, hence generally more training data gives better classification results. However, the \\textcolor{blue!70}{lower\/upper quartile envelope}, which initially becomes more narrow, widens again after reaching 50 individual training scenes. Eventually, after 63 scenes, $\\tilde{\\tau_{w}}$ drops again well below the best-fit line which indicates overfitting, but recovers for 75+ scenes.\n\nWith respect to the method's overall runtime, Alg.~\\ref{alg:optimization} in a worst-case, non-parallelized implementation performs depending on its generate\/update loop (Lines~\\ref{alg:optimization:loop_begin}--\\ref{alg:optimization:loop_end}, also depicted in Fig.~\\ref{fig:components}). Taking into account the temporal performance described in Section~\\ref{results_rpc}, this loop performs with an average cycle time of around $T=\\SI[parse-numbers=false]{(n_{\\mathrm{samples}}\\cdot60\\cdot0.94+27.7)}{s}$ in the worst case, i.e.\\ without any parallelization, for a training sample batch size of $n_{\\mathrm{samples}}$. For $n_{\\mathrm{samples}}=1$, i.e.\\ retraining after each generated sample, this amounts to $T_{1} = \\SI{84.1}{s}$; for $n_{\\mathrm{samples}}=100$ like used in the experiments, this amounts to $T_{100} = \\SI{5667.7}{s} \\approx \\SI{94.5}{min}$. Through parallelization, a theoretical speedup of $S_{dp} = p_d \\cdot p_p$ can be achieved where $p_d$ is the number of Docker containers and $p_p$ is the number of mental simulation processes running in parallel.\n\nThe given results prove the initial claim that, while generating more and more training scenes and optimizing the manipulation strategy in a self-supervised manner, there is an improvement in classification accuracy with an increasing number of considered unique scenes.\nHowever, it is hard to provide a convergence measure which determines the globally optimal number of scenes after which optimization should be stopped for the scene\/object configuration at hand. This strongly depends on the application scenario and the ground truth data for the presented method has been generated by the method itself. That data includes a certain amount of noise propagating from the low-level motion planning and physics simulation towards the final manipulation strategy which cannot completely be ruled out by the proposed approach.\nTherefore, for a specific application scenario, it is necessary to define some heuristics to determine when exactly to stop the optimization loop and to proceed towards optimizing different scene\/object configurations.\n\nIn the presented scenario, the $\\tau_{w}$ grow sufficiently on their $[-1,1]$ scale with an increasing number of scenes. Given that $-1.0$ means full discordance (i.e.\\ the preferences appear in entirely inverse order) and $1.0$ means full concordance (i.e.\\ the preference appear in entirely correct order), the $\\tau_{w}$ from the classifier predictions are close to the optimal manipulation sequences as determined by mental simulation. This means that not only are the results improving as desired during the optimization process, but they also reflect realistic human-like behavior concerning the respective scene. Hence, this method is able to increase the cognition and reasoning skills of the robot with respect to manipulation preferences.\n\n\n\\section{Wrapping everything up: Self-supervised free-time manipulation strategy optimization for anytime deployment}\\label{self_improvement}\nHaving defined the full procedure of auto-generating training scenes and extracting the respective training data, we are now able to generate manipulation strategies using the described ranking by pairwise comparison classifier and optimize them in a self-supervised manner. This happens during load-free times of the robot which typically occur during night hours or weekends or even, guarded by a task scheduler, during regular operation while the current CPU load permits.\n\n\\algblockdefx{WhileP}{EndWhileP}[1]%\n  {\\textbf{while }{#1} \\textbf{do in parallel}}%\n  {\\textbf{end while}}\n\\algblockdefx{WhileSP}{EndWhileSP}[1]%\n  {\\textbf{while }{#1} \\textbf{do in several \\emph{Docker} containers in parallel}}%\n  {\\textbf{end while}}\n\\begin{algorithm*}[bth]\n\\centering\n\\caption{Self-supervised manipulation strategy optimization}\n\\label{alg:optimization}\n\\begin{algorithmic}[1]\n    \\State generate initial set of training scenes\\footnotemark\n    \\State extract samples from training scenes using mental simulation\n    \\State add new samples to training and testing set with a 2:1 ratio\n    \\WhileSP{CPU load low and not interrupted}\n        \\WhileP{not interrupted}\\label{alg:optimization:loop_begin}\n            \\State update classifier\n                \\State\\hspace{\\algorithmicindent} -- retrain classifier with new training set\n                \\State\\hspace{\\algorithmicindent} -- evaluate classifier on testing set, discard new samples if $\\tau_{w}$ decreased\n            \\State generate more training samples\n                \\State\\hspace{\\algorithmicindent} -- generate training scenes\n                \\State\\hspace{\\algorithmicindent} -- extract samples from training scenes using mental simulation\n                \\State\\hspace{\\algorithmicindent} -- add new samples to training and testing set with a 2:1 ratio\n        \\EndWhileP\\label{alg:optimization:loop_end}\n    \\EndWhileSP\n    \\State deploy classifier anytime to predict a manipulation sequence for a real scene\n\\end{algorithmic}\n\\end{algorithm*}\n\\footnotetext{The initial training set size should depend on the maximum tolerable time until the classifier is required for the first time.}\n\nAlgorithm~\\ref{alg:optimization} describes the individual steps of the strategy optimization cycle. Depending on the free computing capacities on the robot, one iteration of this loop varies in runtime, hence the per-time utility improvement of the respective classifier is subject to other tasks being executed at the same time. Therefore a manipulation strategy may take a long time to converge\\footnote{Convergence of the optimization method strongly depends on the concrete implementation and application scenario and is hard to define generically as explained in the evaluation section.} if not operated during load-free runtimes. The prediction of manipulation sequences on real scenes, however, happens instantly in near-real time with the currently built classifier. The presented method can be classified as an \\emph{anytime algorithm} because it always delivers a valid result, even when interrupted, and improves upon its solutions the longer it keeps running.\n\nThe next section shows how, through multiple strategy optimization cycles, the overall prediction accuracy increases and the manipulation strategy adheres more and more to the discovered preference patterns. Nevertheless, if the user decides to accept the currently active strategy for deployment, they can do so anytime. However, as soon as the next load-free time slot appears, the manipulation strategy optimization cycle can be continued where it was interrupted.\nAs for bootstrapping the proposed method on a newly deployed robot, the latter initially incorporates no knowledge about manipulation strategies, but the environment and object models have to be known beforehand. The method does not put any semantic constraints on the objects or the combinations into which they are grouped to train an individual classifier. Hence, the order of object combinations to be learned can be purely application-driven, e.g.\\ commencing with classifiers for combinations of low numbers of objects in order to quickly converge towards viable classifier performance. Later during robot lifetime, training of larger combinations can be performed which is more computationally intensive, but the robot can meanwhile continue working with the classifiers trained up to that point.\n\nOne major advantage of the proposed self-supervised strategy optimization meth\\-od is that it allows for effortless parallelization. All experiments were performed using a setup of multiple similar \\emph{Docker}\\footnote{\\url{https:\/\/docker.com}}~\\cite{Merkel2014} containers executed in parallel, increasing the overall system efficiency even more. Even more, the generation of training samples can run in parallel to updating the classifier with the existing set of samples, so this increases the system efficiency even more.\nA generic version of the integrated parallelized container setup is provided online\\footnote{\\url{https:\/\/github.com\/jacobs-robotics\/gazebo-mental-simulation}} together with some usage examples and Gazebo models. Additionally, the extracted features from all scenes as used in the experiments in the next section are publicly available along with the trained classifiers\\footnote{\\url{https:\/\/tobias.doernba.ch\/research\/datasets\/mental-simulation}}.\n\nSummarized, the presented \\emph{simulation-in-the-loop} approach allows for continuous self-su\\-per\\-vised optimization of the current manipulation strategy, leading to more accurate predictions with respect to reality over robot lifetime.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nBismuth tellurides, \\mbox{Bi$_m$Te$_n$}, are a class of intermetallic compounds with intriguing physical\nproperties. Bi$_2$Te$_3$ is one of the best bulk thermoelectric materials for operation near\nroom temperature\\cite{Gol14}; it becomes superconducting under\npressure\\cite{II72,ETNO10,ZZWZ11,ZSCZ11,MTZU14}; it has been reported to undergo a\npressure-induced  electronic topological (Lifshitz)\ntransition\\cite{IKK97,JKCS07,ETNO10,PGST11,ZLZK13}, and it has recently attracted much\ninterest in the context of topological insulators\\cite{HK10,QZ11,CJFG13}. Like Bi$_2$Te$_3$,\nthe compound Bi$_4$Te$_3$ also becomes superconducting under pressure\\cite{JLSS11} after\nturning from a narrow-bandgap semiconductor into a metal. The structural changes in\nBi$_2$Te$_3$ and Bi$_4$Te$_3$ under compression have been studied previously, and both\nintermetallic compounds were observed to transform to alloys with a disordered\nbody-centered-cubic (bcc) crystal structure above $\\sim$15~GPa at room\ntemperature\\cite{EONI11,ZWWL11,JLSS11}. The two materials are members of the infinitely\nadaptive series of compounds with compositions $\\mathrm{(Bi_{2})_\\mu \\cdot\\,(Bi_2Te_3)_\\nu}$,\nand their crystal structures at ambient conditions are all superlattice structures consisting\nof hexagonal Bi$_{2}$ and Bi$_2$Te$_3$ blocks \\cite{BZLO07,BFDM12a,CJFG13}. The materials with\ncompositions near $\\mu$:$\\nu$ = 2:1 (e.g.\\ Bi$_2$Te) are promising $p$-type thermoelectrics\n\\cite{BZLO07,BFDM12a}.\n\nHere we show that the existence of high-pressure bcc alloy phases is a common feature across\nthe \\mbox{Bi$_m$Te$_n$}\\ series, and that these phases are thermally rather unstable --- they have a\npropensity for atomic ordering upon gentle annealing at only $\\sim$100{\\mbox{\\textdegree}}C under pressure.\nDuring annealing, the BiTe compound adopts the fully-ordered B2 (CsCl-type) structure, and the\nother compounds semi-disordered variants thereof. First-principles electronic structure\ncalculations are used to investigate the chemical bonding and electronic structure of B2-type\nBiTe and to gain insight on the driving forces for the ordering transition.\n\n\n\\section{Methods}\n\nThe compounds Bi$_4$Te$_5$, BiTe, Bi$_4$Te$_3$, and Bi$_2$Te were synthesized as\npolycrystalline pellets by solid-state reaction from the elements as described previously\n\\cite{BZLO07,BFDM12a}. A single crystal of Bi$_2$Te$_3$ was provided by D. L. Sun and C. T.\nLin (Max-Planck-Institut f\\\"{u}r Festk\\\"{o}rperforschung, Stuttgart). Each material was manually\nground to a fine powder and loaded into diamond anvil cells (DACs) for high-pressure\ngeneration. In the majority of the experiments, condensed helium was used as the pressure\ntransmitting medium; some additional experiments that focussed on the effect of annealing the\nsamples at $\\sim$100\\mbox{$^\\circ$C}\\ were performed with nitrogen or a 4:1 methanol-ethanol mixture\nas the pressure transmitting medium. Pressures where determined with the ruby fluorescence\nmethod using the calibration of Ref.~\\onlinecite{MXB86}.\n\nMonochromatic powder x-ray diffraction patterns were measured in the Debye-Scherrer geometry\non beamline ID09a of the European Synchrotron Radiation Facility (ESRF) and beamline I15 of\nDiamond Light Source (DLS), UK. Monochromatic x-rays of wavelength $\\lambda \\approx 0.415$~{\\AA}\nwere focused to a spot size of $\\sim$20~{\\mbox{\\textmu}}m at the sample, and the two-dimensional diffraction\nimages were recorded with a mar555 and a Perkin Elmer XRD1621 flat panel detector at the ESRF\nand DLS, respectively. The DACs were rotated by ${\\pm}3\\mbox{--}5$\\mbox{$^\\circ$}\\ during the exposure to\nimprove the powder averaging. The diffraction images were integrated azimuthally with the\nFit2D software\\cite{soft:Fit2d} to yield conventional intensity vs $2\\Theta$ diffraction\ndiagrams, and then analyzed with the Rietveld method using the program Jana2006\\cite{PDP14}.\n\nElectronic structure calculations of BiTe were performed in the framework of density\nfunctional theory (DFT) and the full-potential augmented-plane-wave + local orbital (APW+lo)\napproach as implemented in the \\textsc{Wien2k} \\cite{soft:Wien2k} and\n\\textsc{Elk}\\cite{soft:Elk2.3.16} codes. The $5d$, $6s$, and $6p$ orbitals of Bi and the $4d$,\n$5s$, and $5p$ orbitals of Te were treated as valence states. Exchange and correlation effects\nwere treated with the revised generalized gradient approximation for solids (PBEsol)\n\\cite{PRCV08}. Spin-orbit coupling was previously found to have a significant\neffect\\cite{MSJ97,TRDM92} on the electronic structure of Bi$_2$Te$_3$ and was therefore\nincluded in the present calculations. Regular Monkhorst-Pack grids of k-points, typically\n$20\\times20\\times20$, were used for the Brillouin-zone sampling.\\cite{note:DFT-full-details}\nThe electronic band structure and density of states obtained with the two codes were in\ncomplete agreement. To quantify the charge transfer between Bi and Te in BiTe, the\n\\textsc{critic} code\\cite{OBPL09} was used to perform Bader analyses of the charge density\ndistribution obtained with Wien2k. The electron localization function (ELF) \\cite{BE90,SNWF97}\nwas calculated using the ELK code.\n\n\n\n\\section{Results and Discussion}\n\\subsection{Disordered bcc Phases}\n\nAll five compounds were compressed at room temperature, and after one or more phase\ntransitions, each of them was found to transform to a phase with the diffraction pattern\ncharacteristic of a bcc crystal, as illustrated in Figure~\\ref{fig:Bi-Te_bcc_patterns}.\n\\begin{figure}[t]\n     \\centering\n     \\includegraphics{Loa_Fig1.pdf}\n     \\caption{Powder x-ray diffraction patterns of the disordered-bcc phases of \\mbox{Bi$_m$Te$_n$}\\\n              after compression at room temperature. All phases except Bi$_2$Te show a\n              weak feature at the position of the (100) reflection, which is absent in an\n              ideal bcc phase. Note that the reflection intensities of Bi$_2$Te$_3$ have\n              been affected by preferred orientation of the powder sample, and the very\n              weak features near 9\\mbox{$^\\circ$}\\ originate from the nitrogen pressure medium or\n              the remainder of a lower-pressure phase. }\n     \\label{fig:Bi-Te_bcc_patterns}\n\\end{figure}\nThis suggests that disordered bcc alloy phases at high pressure are a general feature of the\ncompounds across the Bi-Te system. With the exception of the Bi$_2$Te pattern, the diffraction\npatterns show a very weak feature at the position of the (100) reflection, which is forbidden\nfor a monatomic or ideal bcc phase. In the alloy phases, the presence of this reflection\nindicates a slight deviation from a completely random distribution of the Bi and Te atoms,\ni.e., the occurrence of some short-range correlations. Overall, however, Rietveld analysis of\nthe diffraction patterns confirms them to be consistent with the model of disordered Bi-Te\nalloys as reported previously for Bi$_2$Te$_3$ and Bi$_4$Te$_3$\n(Refs.~\\onlinecite{EONI11,JLSS11}).\n\n\n\\begin{figure}[bt]\n     \\centering\n     \\includegraphics{Loa_Fig2.pdf}\n     \\caption{Average atomic volumes of the bcc alloy phases of \\mbox{Bi$_m$Te$_n$}\\ and atomic\n              volumes of Bi and Te at 20 GPa.\\cite{AKS02,HM03} Note that the crystal\n              structure of Te at 20 GPa is incommensurately-modulated body-centered\n              monoclinic (Te-III)\\cite{HM03}. The straight line indicates a linear\n              variation of the volume with composition according to Vegard{'}s law.  }\n     \\label{fig:Bi-Te_bcc_Volumes}\n\\end{figure}\n\n\nFigure~\\ref{fig:Bi-Te_bcc_Volumes} shows the average atomic volumes for the alloy phases at\n20~GPa as a function of composition. The atomic volumes of Bi and Te at the same pressure are\nshown for comparison. Note that Bi adopts the bcc structure at this pressure, whereas Te has\nan incommensurately-modulated body-centered monoclinic structure\\cite{HM03} and transforms to\nbcc only at $\\sim$29~GPa. There is a continuous evolution of the atomic volume across the\nseries with only a slight deviation from Vegard{'}s law. Bi$_2$Te shows the most notable\ndeviation (0.4\\%). Einaga \\textit{et al.}\\cite{EONI11} performed a similar analysis for Bi$_2$Te$_3$\nbetween 23 and 30~GPa and concluded that there is a substantial deviation from Vegard{'}s law,\nwhich was attributed to {``}remaining strong ionic-covalent bonds{''}. The important difference\nbetween the two analyses is that Einaga \\textit{et al.}\\ used an extrapolation of the equation of state\nof bcc Te\\cite{PH88} to lower pressures whereas we used the actual atomic volume\\cite{HM03} of\nTe at 20~GPa as a reference. The former has the advantage of referring to the same crystal\nstructure for the whole series of compositions, whereas the latter uses only the\nexperimentally observed volumes, irrespective of the crystal structure. We should note that\nthe deviation from Vegard{'}s law in Einaga{'}s analysis becomes significantly smaller if, instead\nof the of bcc Te equation-of-state data by Parthasarathy and Holzapfel\\cite{PH88}, one uses\nthe more recent data by Hejny and McMahon\\cite{HM03}. Overall, we see little evidence for a\nsignificant deviation from Vegard{'}s law. The nature of the bonding in the Bi-Te alloys at high\npressure, on the other hand, remains a pertinent question, to which we will return later.\n\nFigure~\\ref{fig:Bi-Te_bcc_Transition_Pressures} illustrates that the pressure at which the\ntransition to the bcc phase occurs varies continuously across the series of compositions ---\nwith the notable exception of Bi$_2$Te for which the transition pressure is $\\sim$9~GPa higher\nthan expected from the overall trend. The determination of the crystal structures that precede\nbcc Bi$_2$Te at lower pressure is in progress, and the results, that may shed some light on\nthis anomaly, will be reported elsewhere.\n\n\n\n\\begin{figure}[bt]\n     \\centering\n     \\includegraphics{Loa_Fig3.pdf}\n     \\caption{Transition pressures to the bcc alloy phases of \\mbox{Bi$_m$Te$_n$}\\ during compression\n              as a function of composition. At the given pressures, approximately 50\\% of\n              each powder sample had transformed from the lower-pressure to the bcc phase.\n              The transition pressures for Te and Bi are from Refs.~\\onlinecite{HM03} and\n              \\onlinecite{AFK82}, respectively.}\n     \\label{fig:Bi-Te_bcc_Transition_Pressures}\n\\end{figure}\n\n\n\n\\subsection{Semi-disordered B2-Type Phases}\n\nIn our initial experiments on Bi$_2$Te$_3$ we attempted to sharpen the diffraction pattern of\na phase at 17~GPa (phase III in the notation of Refs.~\\onlinecite{NETN09,EONI11,ZWWL11}) by\nannealing the sample in the DAC gently at 120\\mbox{$^\\circ$C}. During this treatment, the sample\ntransformed to a cubic phase, but not the bcc alloy phase discussed above. Rather, a\nclosely-related phase with superstructure reflections in the diffraction pattern was observed,\nas illustrated in Fig.~\\ref{fig:Bi-Te_B2_patterns}. Evidently, the thermal treatment induced\nsome atomic ordering, and the same effect was subsequently observed also for BiTe,\nBi$_4$Te$_3$, and Bi$_2$Te. (Bi$_5$Te$_4$ has not yet been tested for this effect.) The\ndiffraction patterns with the superstructure reflections can be indexed using a simple cubic\nunit cell and lattice parameters similar to those of the corresponding bcc phases.\n\n\\begin{figure}\n     \\centering\n     \\includegraphics{Loa_Fig4.pdf}\n     \\caption{Powder x-ray diffraction patterns of the partially-ordered B2-type phases of\n              \\mbox{Bi$_m$Te$_n$}\\ after compression at room temperature and subsequent annealing at\n              100\\mbox{$^\\circ$C}\\ (BiTe, Bi$_4$Te$_3$, Bi$_2$Te) or 120\\mbox{$^\\circ$C}\\ (Bi$_2$Te$_3$).\n              Miller indices marks the superstructure reflections.}\n     \\label{fig:Bi-Te_B2_patterns}\n\\end{figure}\n\n\n\nAs it is not possible to obtain an ordered structure of Bi$_2$Te$_3$ on the basis of a cubic\nunit cell containing only two atoms, we considered superstructures that are compatible with\nthe given stoichiometry, such as the cubic Pr$_{2}$C$_{3}$ structure type, but none of these\nwas consistent with the observed diffraction pattern. We also considered the monoclinic\nstructure {``}nine\/ten-fold $C2\/m${''} proposed by Zhu \\textit{et al.}\\cite{ZWWL11}, which they identified in\ntheir structure search calculations as the stable phase above 16~GPa. This structure model\nproduces a diffraction pattern similar to that observed for Bi$_2$Te$_3$ after annealing, but\nwith the large monoclinic unit cell containing 20 atoms, it produces also a large number of\nlow-intensity reflections that are not observed experimentally. Zhu \\textit{et al.}\\ interpreted this\nstructure as an approximation of the disordered bcc alloy structure. Careful inspection of the\n{``}nine\/ten-fold $C2\/m${''} structure reveals it to be actually more closely related to the B2\n(CsCl) structure type, with 1\/5 of the rows of Bi atoms along the cubic (110) direction being\nreplaced with rows of Te atoms (in an ordered fashion). Building on this realization and\ntaking into account the substitutional disorder before annealing, we determined the structures\nof the annealed \\mbox{Bi$_m$Te$_n$}\\ phases to be semi-disordered variants of the B2 structure type. In\neach case, one crystallographic site of the B2 unit cell is essentially occupied by the\nmajority element only, whereas the other site is randomly occupied by either element under the\nconstraint of the overall stoichiometry. For Bi$_2$Te$_3$, one site is then, ideally, fully\noccupied by Te and the other site has a mixed occupancy by Bi and Te at a ratio of 80:20 as\nillustrated in Fig.~\\ref{fig:Bi2Te3_semiordered-structure}.\n\n\n\\begin{figure}\n     \\centering\n     \\includegraphics{Loa_Fig5.pdf}\n     \\caption{(Color online) Idealized crystal structure of semi-disordered B2-type\n              Bi$_2$Te$_3$ with full occupation of one crystallographic site and mixed\n              occupation of the other.}\n     \\label{fig:Bi2Te3_semiordered-structure}\n\\end{figure}\n\n\n\nFigure~\\ref{fig:Bi2Te3_B2_Rietveld} shows the result of a Rietveld refinement for the\ndiffraction pattern of annealed Bi$_2$Te$_3$ at 20 GPa on the basis of the semi-disordered B2\nstructure, which yields an excellent fit with residuals of $R_p = 1.5\\%$ and $R_{wp} = 2.5\\%$.\nOne site is mostly occupied by the majority Te with a Bi:Te ratio of 6:94(1) and the other\nsite consequently with a Bi:Te ratio of 74:26(1). This sample was annealed for 26~h at\n120\\mbox{$^\\circ$C}, but we found that much shorter annealing can already induce significant ordering.\nThis is illustrated in Fig.~\\ref{fig:Bi2Te_B2_Rietveld} which shows the result of a Rietveld\nrefinement for Bi$_2$Te at 14.8~GPa after only 2~h of annealing at 100\\mbox{$^\\circ$C}. The Rietveld\nfit (with residuals of $R_p = 2.2\\%$ and $R_{wp} = 3.0\\%$) yields Bi:Te ratios of 90:10(2) and\n41:59(2) for the two sites. This figure illustrates furthermore that we found Bi$_2$Te to\nprecipitate a small amount of Bi ($\\sim$6\\%) into a second phase. The bcc Bi phase has a\nlattice parameter slightly larger\\cite{AKS02} than that of B2-type Bi$_2$Te at this pressure,\nso that it produces shoulders on the lower-angle sides of the Bi$_2$Te diffraction peaks as\nshown in the inset to Fig.~\\ref{fig:Bi2Te_B2_Rietveld}.\n\n\n\n\\begin{figure}\n     \\centering\n     \\includegraphics{Loa_Fig6.pdf}\n     \\caption{(Color online) Rietveld refinement for semi-disordered B2-type Bi$_2$Te$_3$\n              at 20.0~GPa after 26~h of annealing at 120\\mbox{$^\\circ$C}. The dots  and the solid\n              line in the upper part show the measured and fitted diffraction intensities,\n              respectively. The tick marks show the calculated peak positions, and the\n              bottom curve the difference between measured and fitted intensities.}\n     \\label{fig:Bi2Te3_B2_Rietveld}\n\\end{figure}\n\n\n\\begin{figure}\n     \\centering\n     \\includegraphics{Loa_Fig7.pdf}\n     \\caption{(Color online) Rietveld refinement for Bi$_2$Te at 14.8 GPa after 2~h of\n              annealing at 100\\mbox{$^\\circ$C}. Two phases were refined simultaneously:\n              semi-disordered B2-type Bi$_2$Te and bcc Bi. The bcc Bi phase produces\n              shoulders on the lower-angle sides of some peaks as highlighted in the\n              inset. The dots and the solid line in the upper part show the measured and\n              fitted diffraction intensities, respectively. The tick marks show the\n              calculated peak positions for the two phases, and the bottom curve the\n              difference between measured and fitted intensities.}\n     \\label{fig:Bi2Te_B2_Rietveld}\n\\end{figure}\n\n\nTable~\\ref{tab:Bi-Te_annealing} summarizes the conditions under which four \\mbox{Bi$_m$Te$_n$}\\ phases\nwere annealed to transform them from their alloys to the semi-disordered B2 phases, together\nwith the relative occupancies of the two sites as a measure of the ordering. The BiTe sample\nwas annealed three times in succession. The first treatment at a relatively high pressure of\n25~GPa (11~GPa above the transition pressure to bcc) produced only rather limited ordering.\nTwo subsequent annealing steps at lower pressures of 17--18~GPa eventually produced almost\ncomplete (97\\%) ordering. Altogether, we found that thermal treatment at moderate temperatures\nof 100--120\\mbox{$^\\circ$C}\\ causes the high-pressure Bi-Te bcc alloy phases to transform readily to\nsemi-ordered phases of the B2 structure type. Prolonged annealing of BiTe led to nearly\ncomplete ordering, and it appears likely that for the other phases a higher degree of ordering\ncould also be obtained with prolonged annealing.\n\n\n\\begin{table}\n  \\centering\n  \\caption{Atomic ordering in B2-type \\mbox{Bi$_m$Te$_n$}\\ after thermal annealing. The temperature, duration and final\n  pressure of the thermal treatment are indicated by $T$, $\\Delta t$, and $P$, respectively. The Bi:Te ratios or\n  relative occupancies of the two sites were determined with Rietveld refinements. Site 1 is the site of the\n  majority element, and its Bi:Te ratio represents the degree of ordering. The BiTe sample was annealed three\n  times in succession, with each step as indicated.}\n  \\medskip\n\n    \\begin{tabular}{lccc@{\\extracolsep{0.5em}}D{:}{:}{4}D{:}{:}{4}}\n    \\hline\n    \\hline\n    Phase & \\multicolumn{3}{c}{Annealing} & \\multicolumn{2}{c}{Bi:Te ratios} \\\\\n    \\cline{2-4} \\cline{5-6}         & $T$ (\\mbox{$^\\circ$C}) & $\\Delta t$ (h) & $P$ (GPa) & \\multicolumn{1}{c}{site 1} & \\multicolumn{1}{c}{site 2} \\\\\n    \\hline\n    Bi$_2$Te$_3$ & 120 & \\phantom{+}26 & 18 & 6:94(1)  & 74:26(1) \\\\[3pt]\n    BiTe   & 100 & \\phantom{+}25 & 24 & 26:74(2) & 74:26(2) \\\\\n                & 100 & +42 & 18 & 91:9(1)  & 9:91(1)  \\\\\n                & 100 & +92 & 17 & 97:3(1)  & 3:97(1)  \\\\[3pt]\n    Bi$_4$Te$_3$ & 100 & \\phantom{+}16 & 14 & 90:10(2) & 24:76(2) \\\\[3pt]\n    Bi$_2$Te  & 100 & \\phantom{+1}2  & 15 & 90:10(2) & 41:59(2) \\\\\n    \\hline\n    \\hline\n    \\end{tabular}%\n  \\label{tab:Bi-Te_annealing}%\n\\end{table}%\n\nThe phases with compositions other than 1:1 BiTe could potentially order further by adopting\nsuperlattice structures of the B2 type, e.g.\\ partially disordered variants of the structures\nof Heusler compounds. However, careful inspection of the patterns in\nFig.~\\ref{fig:Bi-Te_B2_patterns} yields no evidence for $(\\ensuremath{\\tfrac{1}{2}}\\,\\ensuremath{\\tfrac{1}{2}}\\,\\ensuremath{\\tfrac{1}{2}})$ superlattice\nreflections that would be indicative of structure types such as L2$_1$, D0$_3$ or Y\n(Ref.~\\onlinecite{GFP11,KZSV15}).\n\n\n\n\\begin{figure*}\n     \\centering\n     \\includegraphics{Loa_Fig8.pdf}\n     \\caption{Calculated electronic band structures of (a) B2-type BiTe and (b) bcc Bi for\n              cubic unit cells with a lattice parameter of 3.623~{\\AA}, corresponding to the\n              experimental density of bcc BiTe at 20~GPa. The calculated pressure for\n              B2-type BiTe at this density is 18.5~GPa. To facilitate the comparison\n              between B2-type BiTe and bcc Bi, the band structure for the latter is not\n              shown with respect to the usual bcc Brillouin zone, but rather with respect\n              to the same simple cubic Brillouin zone as for B2-type BiTe. (c)~Bi\n              character and (d) Te character of the electronic bands in BiTe at 18.5~GPa\n              are shown by the widths of the gray bands ({``}fat bands{''}), as quantified by\n              the partial charges of the band states in the atomic spheres. The most\n              notable differences between (c) and (d) can be seen for states close to the\n              Fermi level, e.g.\\ around the M point.}\n     \\label{fig:BiTe_bandstructures}\n\\end{figure*}\n\n\nIntermetallic compounds are well known to exhibit widely varying degrees of atomic order ---\nfrom completely disordered alloys to fully ordered compound phases --- depending on\nstoichiometry, temperature, and thermal and mechanical treatment \\cite{Cah95}. The\nsemi-disordered B2-type phases observed here are reminiscent of semi-disordered B2-type phases\nobserved in some X$_{2}$YZ Heusler compounds, e.g.\\ Co$_{2}$FeAl at high temperature\n\\cite{KUKI04}, where the X atoms occupy one site of the B2 structure and the Y and Z atoms\nrandomly occupy the other site\\cite{GFP11}. Intermetallic phases generally have a tendency to\nlong-range order\\cite{Cah95}. To obtain an alloy, atomic ordering must be prevented, typically\nby quenching a high-temperature disordered phase, or destroyed, e.g. by cold working. Although\nit is therefore to be expected to observe ordering in the Bi-Te phases at elevated\ntemperatures, it is surprising to observe it at such relatively low temperatures of only\n$\\sim$100{\\mbox{\\textdegree}}C. It is therefore possible that atomic ordering will occur already at room\ntemperature on a longer time scale. In other words, the Bi-Te alloy phases may show signs of\naging.\n\nAtomic disorder in intermetallic compounds is well documented to affect their physical\nproperties, e.g.\\ the magnetism in Heusler compounds\\cite{GFP11}, the mechanical properties of\nstructural alloys\\cite{Cah95}, or the superconductivity in A15 intermetallics\\cite{Cah95}.\nSuperconductivity in Bi$_2$Te$_3$ and Bi$_4$Te$_3$ has been observed in the pressure range\nwere their cubic alloy phases exist\\cite{ZSCZ11,JLSS11}. The existence of both alloy and\nsemi-ordered Bi-Te phases therefore needs to be taken into account in investigations of their\nphysical properties, in particular in electronic structure calculations.\n\n\n\n\\subsection{Electronic Structure and Chemical Bonding in BiTe}\n\n\\begin{figure}\n     \\centering\n     \\includegraphics{Loa_Fig9.pdf}\n     \\caption{Calculated electronic density of states of B2-type BiTe at a calculated\n              pressure of 18.5~GPa (lattice parameter $a=3.623$~{\\AA}), of bcc Bi, and of a\n              BiTe alloy approximant. The same atomic density was used in all three\n              calculations. The Fermi level $E_F$ was set to be~0.}\n     \\label{fig:BiTe_DOS}\n\\end{figure}\n\nThe fully-ordered B2-type BiTe phase provides an ideal starting point for electronic structure\ncalculations, which allow us to gain insight on the nature of its chemical bonding and the\nforces that drive the atomic ordering. Figure~\\ref{fig:BiTe_bandstructures}(a) shows the\ncalculated band structure of B2-type BiTe at the experimental density of bcc BiTe at 20~GPa.\nSpin-orbit coupling was included in all calculation and was found to produce significant band\nsplittings, in agreement with earlier studies\\cite{MSJ97,TRDM92} on Bi$_2$Te$_3$. In the\ncalculation, BiTe is metallic at this pressure, which is in line with the observation that the\nband gap in Bi$_2$Te$_3$ closes at pressures of 2--5~GPa, depending on the sample and\npressure\/stress conditions.\\cite{IPA64,MTZU14} The BiTe band structure is similar to that of\nbcc Bi at the same atomic density [Fig.~\\ref{fig:BiTe_bandstructures}(b)]. The most notable\ndifference is the occurrence of additional band splittings for BiTe, as indicated by arrows in\nFig.~\\ref{fig:BiTe_bandstructures}(a). This reduces the number of bands crossing the Fermi\nlevel and produces a suppression in the electronic density of states --- a pseudogap --- near\nthe Fermi level as illustrated in Fig.~\\ref{fig:BiTe_DOS}. This suppression is absent for bcc\nBi and, more importantly, for an approximant of the BiTe alloy, which we modeled as a\n$2\\times2\\times2$ supercell of B2-type BiTe with a random distribution of the eight Bi and\neight Te atoms. The result is a slightly reduced band structure contribution to the total\nenergy of the ordered phase in comparison to the alloy. This illustrates at a fundamental\nlevel the discussion in the previous section on the effect of atomic disordering on the\nelectronic properties.\n\nAnalysis of the partial charges from the individual band states shows that the lower-energy\ncomponents of the split bands near the Fermi level have predominantly Te character, and the\nhigher-energy components predominantly Bi character [Fig.~\\ref{fig:BiTe_bandstructures}(c,d)].\nThis indicates a charge transfer from Bi to Te, which is in keeping with Te having a slightly\nhigher electronegativity than Bi (2.1 vs 2.02 on the revised Pauling scale \\cite{JL92}). To\nquantify the charge transfer, we performed a Bader ({``}Atoms in Molecules{''})\nanalysis\\cite{Bad91,Bad90} of the charge density distribution. In this analysis, the unit cell\nis divided into regions whose boundaries are defined by minima in the charge density. More\nprecisely, at any point on the dividing boundaries, the gradient of the charge density is\nparallel to the boundary or zero. This provides an unambiguous partitioning of the unit cell\ninto atomic basins, and by integrating the charge within these basins one can quantify the\ncharge transfer between atoms. For BiTe at $\\sim$20~GPa, the Bader analysis indicates a\nsignificant charge transfer from Bi to Te of approximately 0.44~$e$. The charge transfer\ndepends only weakly on the pressure and decreases with increasing pressure to 0.40~$e$ at\n100~GPa.\n\nThe chemical bonding and charge transfer between Bi and Te have been analyzed in several\nstudies on Bi$_2$Te$_3$ at normal pressure\\cite{MSJ97,Kag78,PT89,ZWWL11}. The Bi$_2$Te$_3$\ncrystal structure can be regarded as a repeating sequence of {``}quintuple layers{''} that comprise\nfive atomic layers, Te(1)--Bi--Te(2)--Bi--Te(1), where Te(1) and Te(2) denote the two\ncrystallographic sites occupied by Te. The bonding within the quintuple layers is of mixed\ncovalent-ionic type, whereas the bonding between the quintuple layers of van-der-Waals type.\nThe degree of ionicity is generally difficult to quantify, and model-dependent, and the\nresults of several studies vary correspondingly. The combined DFT and tight-binding\ncalculations by Mishra \\textit{et al.}\\cite{MSJ97} gave ionic charges of $\\mbox{Bi}_2^{0.7+}\\,\n\\mbox{Te(1)}_2^{0.5-}\\, \\mbox{Te(2)}^{0.35-}$. An earlier tight-binding model by Pecheur and\nToussaint \\cite{PT89} had indicated ionic charges about 50\\% lower than those obtained by\nMishra \\textit{et al.}\\ On the other hand, Zhu \\textit{et al.} \\cite{ZWWL11} recently reported ionic charges of\n$\\mbox{Bi}_2^{0.75+}\\, \\mbox{Te(1)}_2^{0.05-}\\, \\mbox{Te(2)}^{1.39-}$ (on the basis of DFT\ncalculations and Bader analysis), which suggested a strong charge disproportionation between\nthe two Te sites. They furthermore reported the charge transfer from Bi to Te to increase with\nincreasing pressure. It is therefore interesting to note that the charge transfer obtained\nhere for cubic BiTe is significantly lower than those reported by Zhu \\textit{et al.}\\ for the layered\nambient-pressure and three high-pressure phases of Bi$_2$Te$_3$, and their pressure\ndependences are also at variance.\n\n\nTo further address the question whether any significant covalent contribution to the chemical\nbonding is still present at high pressure in the cubic phases (as suggested in Refs\n\\onlinecite{EONI11,ZWWL11}), we have calculated the electron localization function (ELF)\n\\cite{BE90,SNWF97} for BiTe. The results for B2-type BiTe  are shown in\nFig.~\\ref{fig:BiTe-ELF}. The upper panel shows the ELF in the (110) plane that passes through\nfour Te atoms at the corners of the unit cell and the Bi atom in the center. There is no\nindication of a local maximum of the ELF along the Bi--Te line, and hence no indication of\ndirectional, covalent bonding between Bi and Te. The oscillations in the ELF in\nFig.~\\ref{fig:BiTe-ELF}(b) represent merely the electronic shell structure of Bi and Te.\n\n\\begin{figure}\n     \\centering\n     \\includegraphics{Loa_Fig10.pdf}\n     \\caption{(Color online) Calculated electron localization function of B2-type BiTe at\n              a calculated pressure of 18.5~GPa (lattice parameter $a=3.623$~{\\AA}). (a)~ELF\n              in the (110) plane. The unit cell edges are indicated by the white dotted\n              lines. (b) ELF along the body diagonal $(x, x, x)$ of the cubic unit cell.}\n     \\label{fig:BiTe-ELF}\n\\end{figure}\n\nThe overall picture emerging from these calculations is thus one where the mixed\ncovalent-ionic plus van-der-Waals bonding found in the bismuth tellurides at ambient\nconditions gives way to metallic bonding with a significant ionic component at high pressures\nof $\\sim$20~GPa. The charge transfer from Bi to Te is rather insensitive to further increase\nin pressure. We consider the charge transfer to be the driving force behind the\nthermally-induced atomic ordering. Firstly, ordering of the Bi and Te ions reduces the\nMadelung energy. The semi-ordered phases for the compositions other than BiTe represent the\nmost-ordered configurations that can be achieved within the bcc\/B2 structure type under the\nconstraint of the given stoichiometry. Secondly, the ion ordering in BiTe causes a suppression\nof the electronic density of states near the Fermi level, and hence a lowering of the band\nstructure contribution to the total energy. It is not clear at present to what extent this\nsecond effect contributes to the formation of the semi-ordered phases.\n\n\n\\subsection{Fermi Surface of BiTe and Topological Changes}\nFigure~\\ref{fig:BiTe_bandstructures}(a) shows that several electronic bands cross the Fermi\nlevel in B2-type BiTe and that there are three band extrema in close proximity to the Fermi\nlevel. These extrema arise from the band splittings marked by arrows in\nFig.~\\ref{fig:BiTe_bandstructures}(a), which are due to the ionicity in BiTe as discussed\nabove. The Fermi surface thus comprises several sheets, and its topology can be expected to\nchange with pressure when band extrema cross the Fermi level. Figure~\\ref{fig:BiTe-FS}(a)\nshows the calculated Fermi surface of B2-type BiTe at the same density as for the band\nstructure in Fig.~\\ref{fig:BiTe_bandstructures}(a). It consists of three electron pockets\nnested inside one another and centered around the R point as well as a tube-like feature with\nnecks at the X points. With increasing pressure, the band minimum along \\mbox{$\\Gamma$--M},\nthe maximum along \\mbox{$\\Gamma$--R}, and the maximum at M cross the Fermi level at calculated\npressures of 29, 38, and 47~GPa, respectively. The associated changes in the topology of the\nFermi surface are illustrated in Fig.~\\ref{fig:BiTe-FS}(b). The tube-like feature breaks up\ninto pockets centered on the X points, and additional pockets appear upon further compression.\nWe expect these Fermi surface features to be sensitive not only to changes in pressure but\nalso to atomic displacements, which may give rise to interesting electron-phonon coupling\neffects.\n\n\\begin{figure}[t]\n      \\vspace{2ex} \\centering\n     \\includegraphics{Loa_Fig11.pdf}\n     \\caption{(Color online) (a) Fermi surface of B2-type BiTe at a calculated pressure of\n              18.5~GPa (lattice parameter $a=3.623$~{\\AA}). (b)~Evolution of the Fermi surface\n              with pressure. The three stable pockets centered around the R point have\n              been omitted for clarity.}\n     \\label{fig:BiTe-FS}\n\\end{figure}\n\nWe should note that although we did not investigate the stability of cubic bcc\/B2-type BiTe at\npressures above 25~GPa, we expect it to be stable over a wide pressure range.  B2-type phases\nare generally rather stable under compression, and bcc Bi has been reported to be stable to at\nleast 222~GPa (Ref.~\\onlinecite{AKS02}), bcc Te to 97~GPa (Ref.~\\onlinecite{SAIF14}), and the\nBi$_2$Te$_3$ bcc alloy phase to at least 52~GPa (Ref.~\\onlinecite{ZWWL11}).\n\n\n\n\\section{Conclusions}\nWe have shown that the formation of alloy phases at high pressure appears to be a common\nfeature across the series of \\mbox{Bi$_m$Te$_n$}\\ compounds. The alloy phases were found to be thermally\nunstable and to adopt semi-disordered B2\/CsCl-type structures after gentle annealing, but not\nthe superlattice structures known from Heusler compounds. The atomic ordering must be expected\nto affect the physical properties of the alloy phases, e.g.\\ superconductivity\n\\cite{II72,ETNO10,ZZWZ11,ZSCZ11,MTZU14, JLSS11, SJWV16}. Electronic structure calculations\nindicate a significant charge transfer of $\\sim$0.4~$e$ from Bi to Te in B2-type BiTe. The\nionic contribution to the bonding favors atomic ordering and thus the transition from alloy to\nordered compound. Another contribution is the lowering of the band structure contribution to\nthe total energy due to a suppression of the electronic density of states near the Fermi level\nthat results from the charge transfer. The calculated electron localization function for BiTe\nprovides no evidence for any remaining covalent bonding. The chemical bonding thus changes\nfrom mixed covalent-ionic (plus van-der-Waals interactions between the layers)  at normal\nconditions to mixed metallic-ionic at pressure of 20~GPa and above. The Fermi surface of BiTe\nhas an intricate structure, which may give rise to interesting electron-phonon coupling\neffects, and it has been calculated to undergo three topological changes in the 20--60~GPa\nrange.\n\n\\emph{Note added.} After submission of the manuscript, Stillwell \\textit{et al.}\\ \\cite{SJWV16} reported\nthe observation of a bcc alloy phase in Bi$_2$Te under high pressure and that this phase is\nsuperconducting. Superconductivity is certainly an interesting example of possible\nelectron-phonon coupling effects mentioned above. An obvious question is now how the atomic\nordering reported in the present work affects the superconducting properties of cubic \\mbox{Bi$_m$Te$_n$}\\\nphases.\n\n\n\n\\acknowledgments We thank D. L. Sun and C. T. Lin (Max-Planck-Institut f\\\"{u}r\nFestk\\\"{o}rperforschung, Stuttgart) for providing the Bi$_2$Te$_3$ single crystal as well as M.\nHanfland (ESRF) and A. Kleppe (DLS) for their support during the diffraction experiments.\nFacilities were made available by the European Synchrotron Radiation Facility and Diamond\nLight Source.\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzmxhx b/data_all_eng_slimpj/shuffled/split2/finalzzmxhx
new file mode 100644
index 0000000000000000000000000000000000000000..429fa1c95b174bee72f1f6edca2d0f135a3949c1
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzmxhx
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\n\n\nUnderstanding the microscopic origin of black hole thermodynamics is one of the most important outstanding problem in theoretical physics.\nRecently Ref.\\cite{Morita:2013wfa}  proposed the following microscopic description of the near extremal black branes in the Einstein-Maxwell-Dilaton system. \nThis system admits the extreme brane solution in which no force works between the parallel branes if we tune the dilaton coupling suitably.\nIn case these branes are nearly parallel or slowly moving, they start interacting with each other and\nthe low energy effective action for these branes is given as\n\\begin{align}\nS=\\int dt \\left(\n\\sum_{i=1}^{N}  \\frac{m}{2} \\vect{v}{2}_i+\n\\sum_{i\\neq j}^{N}  \\frac{\\kappa^2_{4} m^2}{16\\pi} \\frac{\\vect{v}{4}_{ij}}{\\vec{r}_{ij}}\n+\\cdots\\right).\n\\label{single-brane}\n\\end{align}\nHere we consider 0-branes in four dimensions as a simple example. $N$ is the number of the branes, $\\vec{v}_i$ is velocity of the $i$-th brane, $m$ is mass of the brane and $\\kappa_{4}$ is the gravitational coupling.\nAlso $\\vec{r}_{ij}\\equiv \\vec{r}_i-\\vec{r}_j$ and $\\vec{v}_{ij}\\equiv \\vec{v}_i-\\vec{v}_j$ denote the relative position and the relative velocity of the $i$-th and $j$-th branes.\nThe first term is the ordinary non-relativistic kinetic term and the second term is the interactions between the branes which vanish when the branes are relatively static.\n\nIf we assume that the branes compose a bound state due to the interactions, we can estimate the thermodynamical properties of the bound state by applying the virial theorem to the effective action (\\ref{single-brane}).\nIn \\cite{Morita:2013wfa}, we found that the branes are strongly coupled in the bound state and the  thermodynamical properties are coincident with the corresponding black brane thermodynamics in the near extremal regime (up to unfixed numerical factors).\nIt suggests that the black hole thermodynamics may be explained via the effective action (\\ref{single-brane}) microscopically.\nWe call this conjecture ``$p$-soup proposal\".\n\n\nThis proposal works in the various BPS branes in superstring and M-theory, and we can explain the black hole thermodynamics of the black brane solutions through the effective actions for these branes similar to (\\ref{single-brane})  \\cite{Wiseman:2013cda, Morita:2013wla, Morita:2014ypa}.\n(The related studies have been done in  \\cite{Horowitz:1997fr,Li:1997iz,Banks:1997tn,Li:1998ci,Smilga:2008bt}.)\nMoreover, this proposal works even in the D1-D5(-P) system \\cite{Morita:2014cfa}.\nSince this system \\cite{Callan:1996dv,David:2002wn}  involves the different species of the branes and the supersymmetry is less than the single brane case, it is non-trivial that the $p$-soup proposal works in such a system.\n\nThis motivates us to investigate general multiple species of branes in the Einstein-Maxwell-Dilaton system.\nWe set these multiple species of branes so that the forces between the branes vanish when they are static. \nIt happens when the branes satisfy so-called ``intersecting rule'' \\cite{Argurio:1997gt}, and the D1-D5(-P) system also satisfies it.\nThen we derive the effective action for these branes similar to (\\ref{single-brane}), and, by analyzing this action, we find that the bound state of these branes exhibits thermodynamical properties corresponding to the black hole thermodynamics.\nThus the $p$-soup proposal works in this system, and we believe that it captures the fundamental natures of the black hole microstate.\n\n\n\nThis paper is organized as follows.\nIn Sec.\\,\\ref{sec:setup} we review Einstein-Maxwell-Dilaton theory and clarify our setup of intersecting brane systems.\nIn Sec.\\,\\ref{sec:twobrane} we discuss intersecting brane systems of only two species of branes. This simple example would be helpful to understand our discussion without complicating the story.\nThe general intersecting brane systems are discussed in Sec.\\,\\ref{sec:general}, but the details of calculation are shown in \\ref{app:eff}.\nSec.\\,\\ref{sec-conclusion} is the discussions including possible applications to string theory.\nThe corresponding black branes in supergravity and their thermodynamics are discussed in \\ref{app:thermo}.\n\n\n\n\\section{Intersecting brane system}\n\\label{sec:setup}\n\nWe consider $D$-dimensional gravitational theory coupled to dilaton $\\phi$ and $(n_A+1)$-form gauge field ($A=1, \\ldots, m$). The action is given by\n\\begin{align}\nS=\\frac{1}{16\\pi G_D} \\int d^Dx \\sqrt{-g} \\left[ R - \\frac{1}{2} \\left( \\partial \\phi \\right)^2\n-\\sum_{A=1}^m \\frac{1}{2(n_A+2)!} e^{a_A \\phi} F_{n_A+2}^2\n\\right].\n\\end{align}\nHere $a_A$ is taken as\n\\begin{align}\na_A^2=4- \\frac{2(n_A+1)(D-n_A-3)}{D-2}\n\\end{align}\nin order to have asymptotically flat solutions, and so that extremal $q_A$-brane solutions obey a `no-force'\ncondition when the branes are static and parallel \\footnote{ \nIn this paper, we discuss the general $D$ cases.\nWhen $D=10$, all branes in type IIA and IIB superstring theory can be discussed in this setup. Then the $D<10$ cases obviously include the compactified systems of these branes in superstring theory.\nWhen $D=11$, M2- and M5-branes in M-theory can be discussed in this setup with no dilaton, $a_A=0$.\nWhen $D=12$, we expect the branes in F-theory can be discussed.\nAlthough F-theory seems a little strange in that it has two time-like directions\nand the dilaton coupling $a_A$ is pure imaginary \\cite{Khviengia:1997rh},\nif one of time-like directions is Euclideanized or all branes wind around this time-like direction, the following discussions seem to be applicable.\nHowever, at this stage, we cannot judge whether general brane systems in F-theory can be applied or not. It would be an interesting future work.}.\n\n\n\nWe compactify $(D-{d}-3)$-dimensional space as a rectangular torus $T^{D-{d}-3}$ and define its volume as $V_T$.\nTo make intersecting brane system in this setup, we put various branes on the torus.\nWe set $N_A$ $q_A$-branes \nwinding around $q_A$ cycles of the torus ($A=1,\\ldots, M$)  and assume that the branes are smeared along the other cycles\\footnote{\\label{ftnt-GL}\nIf there are cycles around which no brane wind, Gregory-Laflamme transitions occur on these cycles and the black brane is localized on these cycles, when the size of the event horizon of the black brane is smaller than the size of the cycles. \nIn such cases, we cannot assume that the branes are smeared on these cycles and the effective action for the branes (\\ref{gen-effective-action}) is modified as argued in \\cite{Morita:2014ypa}.\nBy using this modified effective action, we can investigate the Gregory-Laflamme transitions in terms of the interacting separated branes. \n}. \nWhen we set the branes, we respect the intersection rule: $q_A$-brane and $q_B$-brane wind around the same $\\bar{q}$ cycles of the torus, where $\\bar{q}$ is determined as\n\\begin{align}\n\\bar{q}=\\frac{(q_A+1)(q_B+1)}{D-2}-1-\\frac{1}{2}\\epsilon_A a_A \\epsilon_B a_B\n\\label{intersection-rule} \n\\end{align}\nwhere $\\epsilon_A$ is 1 ($-1$), if $q_A$-brane is electrically (magnetically) coupled to the field strength $F_{n_A+2}$ with $q_A=n_A$ ($q_A=D-n_A-4$). \nIf all the branes satisfy this rule each other, `no-force' work among them when they are static.\nAlso we constrain ${d} \\ge 1$ so that the co-dimension of the branes is higher than two.\n(See Table \\ref{table-branes} as an example.)\n\nWe will consider the black brane solution for this setup and argue that their thermodynamical properties can be\nexplained via the microscopic theory of interacting branes (up to numerical factors).\nIt will work in general intersecting brane system as far as the branes satisfy the intersection rule.\nHowever,\ndiscussion on the general systems is rather complicated, and hence we first demonstrate it for a two brane system.\n\n\n\n\\section{Example: two brane system}\n\\label{sec:twobrane}\n\n\\begin{table}\n\\begin{eqnarray*}\n\\begin{array}{l|c|c|c|c|c|c|c|c|}\n& t & 1 & \\cdots & {d}+2 & T^{q_1-\\bar{q}} & T^{q_2-\\bar{q}} & T^{\\bar{q}} & T^{r}  \\\\\n\\hline\nN_1~q_1\\text{-brane} & - & && &- && - &   \\\\ \\hline\nN_2~q_2\\text{-brane}& - & && & &-& - &  \\\\ \\hline\n\\end{array}\n\\end{eqnarray*}\n\\caption{The brane configuration of intersecting two brane system.\nHere $r= D- {d}-3 -(q_1+q_2-\\bar{q})$ and $\\bar{q}$ is fixed by (\\ref{intersection-rule}).\n$N_1$ $q_1$-branes wind around $q_1$ cycles of $T^{D-{d}-3}$ and  are smeared on the other directions of $T^{D-{d}-3}$. \n$N_2$ $q_2$-branes wind around $q_2$ cycles and $\\bar{q}$ of them are the same cycles to the $q_1$-branes. \n}\n\\label{table-branes}\n\\end{table}\n\nWe consider the intersecting brane system which consists of $N_1$ $q_1$-branes and $N_2$ $q_2$-branes.\nThey wind around $q_1$ and $q_2$ cycles of the torus,\nsharing $\\bar{q}$ cycles so that they satisfy the intersecting rule (\\ref{intersection-rule}).\nNote that, since $\\bar{q}$ has to be a non-negative\ninteger, $q_A$ and $D$ are restricted\nto particular combinations of values.\n\nIf we focus on only the non-compact $({d}+3)$-dimensional spacetime, the $q_A$-branes behave as BPS particles with mass $m_A \\equiv \\mu_A V_A$ ($A=1,2$).\nHere $\\mu_A$ is the tension of the single $q_A$-brane and $V_A$ is the volume of the $q_A$-dimensional torus which the brane winds around.\nDue to the intersecting rule, no forces work between the branes when they are static.\nHowever, if they are moving, the interactions arise.\nOur proposal is that these interactions confine the branes\nand make them compose a bound state, and \n this bound state explains the thermodynamics of the intersecting $q_1$-$q_2$ black hole.\nTo see this, we estimate the low energy effective action of this interacting brane system.\nIf the branes are well separated, the gravitational interactions dominate and the effective action for this system has the following structure\n\\begin{align}\nS_{\\text{eff}} = \\int dt \\left(\nL_\\text{kin}+L_\\text{1-grav}+L_\\text{2-grav}+L_\\text{3-grav}+\\cdots \\right).\n\\label{moduli-D1D5}\n\\end{align}\nThe derivation of this effective action is summarized in \\ref{app:eff}.\nHere\n\\begin{align}\nL_\\text{kin}=\n\\sum_{i=1}^{N_1} \\left( \\frac{m_1}{2} \\vect{v}{2}_i+ \\frac{m_1}{8} (\\vect{v}{2}_i)^2+ \\cdots \\right)\n+\\sum_{i=1}^{N_2} \\left( \\frac{m_2}{2} \\vect{v}{2}_i+ \\frac{m_2}{8} (\\vect{v}{2}_i)^2 + \\cdots \\right)\n\\label{eff-kin}\n\\end{align}\nis the kinetic term of the branes.\n $\\vec v_i \\equiv \\partial_t \\vec r_i$ and $\\vec r_i$ are the velocity and the position of the $i$-th brane in the non-compact $({d}+2)$-dimensional space.\n We have assumed that the velocity is low ($|\\vec v_i| \\ll 1$) at low energy regime and used the non-relativistic approximation. We will justify this approximation soon.\nIn addition, we have assumed that the volume of the torus is small and the motions of the branes depend on time $t$ only\\footnote{\\label{ftnt-KK}\nThe small volume assumption is not essential in the following calculations, and\nwe apply it only to make the equations simpler.\n}.\n\n$L_\\text{1-grav}$ is the interaction which arises from a single graviton (, gauge and dilaton) exchange between two branes,\n\\begin{align}\nL_\\text{1-grav}= \n\\sum_{i\\neq j}^{N_1}  \\frac{\\kappa^2_{{d}+3} m_1^2}{8{d}\\Omega_{{d}+1}} \\frac{\\vect{v}{4}_{ij}}{\\vect{r}{{d}}_{ij}}\n+\n\\sum_{i\\neq j}^{N_2}  \\frac{ \\kappa^2_{{d}+3} m_2^2}{8{d}\\Omega_{{d}+1}} \\frac{\\vect{v}{4}_{ij}}{\\vect{r}{{d}}_{ij}}\n+\\sum_{i=1}^{N_1} \\sum_{j=1}^{N_2} \\frac{ \\kappa^2_{{d}+3} m_1 m_2}{{d}\\Omega_{{d}+1}} \\frac{\\vect{v}{2}_{ij}}{\\vect{r}{{d}}_{ij}}.\n\\label{eff-2-body-D1-D5}\n\\end{align}\nHere $\\vec{r}_{ij}\\equiv \\vec{r}_i-\\vec{r}_j$ and $\\vec{v}_{ij}\\equiv \\vec{v}_i-\\vec{v}_j$ denote the relative position and the relative velocity of the $i$-th and $j$-th branes, and $\\kappa^2_{{d}+3} \\equiv  \\kappa_D^2\/V_T $ is the $({d}+3)$-dimensional gravitational coupling constant.\n$\\Omega_{ d+1}\\equiv 2\\pi^{\\frac{ d}{2}+1}\/\\Gamma(\\frac{ d}{2}+1)$ is the volume of a unit $( d+1)$-sphere.\nThe first and second terms describe the two-body interactions between two $q_1$-branes and two $q_2$-branes, respectively.\nThe third term is the two-body interactions between a $q_1$-brane and a $q_2$-brane.\nThere are higher order terms of $\\vec{v}_{ij}$ but we have omitted them in this equation, since  $|\\vec{v}_{ij}|$ would be small in the low energy regime.\nNote that the power of $\\vec{v}_{ij}$ of the third term is lower than the others, and\nit implies that the third term would dominate.\nHence we treat this term separately and define it as\n\\begin{align}\nL_1 \\equiv \\sum_{i=1}^{N_1} \\sum_{j=1}^{N_2} \\frac{ \\kappa^2_{{d}+3} m_1 m_2}{{d}\\Omega_{{d}+1}} \\frac{\\vect{v}{2}_{ij}}{\\vect{r}{{d}}_{ij}}.\n\\end{align}\nWe will soon see that it indeed becomes relevant.\n\nSimilarly the effective action has various interaction terms through the multi-graviton exchanges.\nHere we write down only the terms proportional to the lowest power of $v$, since they will become relevant at low energy.\nAmong 3-graviton exchange interactions, the following term will be relevant, \n\\begin{align}\nL_\\text{3-grav} \\ni  L_2 \\sim\n\\sum_{i=1}^{N_1} \\sum_{j=1}^{N_1} \\sum_{k=1}^{N_2} \\sum_{l=1}^{N_2}\n\\frac{\\kappa^6_{{d}+3}  m_1^2 m_2^2}{\\Omega_{{d}+1}^3} \\left(  \\frac{\\vect{v}{4}_{ij}}{\\vect{r}{{d}}_{ij} \\vect{r}{{d}}_{ik} \\vect{r}{{d}}_{il}  }+ \\cdots \\right).\n\\label{eff-4-body}\n\\end{align}\nWe define this term as $L_2$.\n`$\\sim$' in this article denotes equality not only including dependence on physical parameters but also including all factors of $\\pi$.\nWhen we derive these interactions in \\ref{app:eff}, we do not fix the precise numerical coefficients of these interactions.\nSince  we will consider an order estimate for the thermodynamics of this interacting brane system, the precise expressions are not important. \nSimilarly the system has the following interactions\n\\begin{align}\nL_{n} \\sim & \\sum_{i_1, \\dots, i_n}^{N_1}\\sum_{j_1, \\dots, j_n}^{N_2}\n\\left( \\kappa_{{d}+3}^{2(2n-1)} \\frac{m_1^n m_2^n }{\\Omega_{{d}+1}^{2n-1}} \\prod_{k=2}^n \\prod_{l=1}^n\n\\frac{ 1 }{  \\vect{r}{{d}}_{i_1 i_k} \\vect{r}{{d}}_{i_1 j_l} } \\vect{v}{2n} + \\cdots \\right),\n\\label{moduli-multi-graviton} \n\\end{align}\nwhich describes the $2n-1$ graviton exchange among $n$ $q_1$-branes and $n$ $q_2$-branes.\n\n\nFrom now, we estimate the dynamics of this system by using the virial theorem.\nWe first assume that the branes are confined due to the interactions, and the branes satisfy\n\\begin{align}\n\\vec{v}_{ij} \\sim  v, \\qquad \\vec{r}_{ij} \\sim r.\n\\label{assumption-scales}\n\\end{align}\nHere $v$ and $r$ are the characteristic scales of the velocity and size of the branes in \nthe bound state which do not depend on the species of the branes.\n(Note that since the masses of the $q_1$-brane and $q_2$-brane are generally different, we naively expect that  these scales should depend on the species of the branes.\nHowever we will soon see that it does not occur in the bound state.)\nThen we can estimate the scales of the terms (\\ref{eff-kin}), (\\ref{eff-2-body-D1-D5}) and (\\ref{eff-4-body}) in the effective Lagrangian as\n\\begin{align}\nL_\\text{kin} \\sim & N_1 m_1 v^2+ N_1 m_1 v^4+ \\cdots + N_2 m_2 v^2+ N_2 m_2 v^4  + \\cdots, \\nonumber \\\\\nL_\\text{1-grav} \\sim& \\frac{ \\kappa^2_{{d}+3} N_1^2 m_1^2}{\\Omega_{{d}+1}} \\frac{v^4}{r^{{d}}}+\\frac{ \\kappa^2_{{d}+3} N_2^2 m_2^2}{\\Omega_{{d}+1}} \\frac{v^4}{r^{{d}}} + \\frac{ \\kappa^2_{{d}+3} N_1 N_2 m_1 m_2}{\\Omega_{{d}+1}} \\frac{v^2}{r^{{d}}} + \\cdots, \\nonumber \\\\\nL_\\text{3-grav} \\sim &\n\\frac{ \\kappa^6_{{d}+3} N_1^2 N_2^2 m_1^2 m_2^2}{\\Omega_{{d}+1}^3} \\frac{v^4}{r^{3{d}}} + \\cdots.\n\\label{estimate-D1D5}\n\\end{align}\nThe Lagrangian also have other terms (\\ref{moduli-multi-graviton}) but we will consider them later.\nHere we consider which terms in (\\ref{estimate-D1D5}) dominate at the low energy where $v$ would be small ($v \\ll 1 $).\nIn the second line of (\\ref{estimate-D1D5}), the third term which is from $L_1$ (\\ref{eff-2-body-D1-D5}) would dominate, since the power of $v$ is lowest \\footnote{\\label{ftnt-reduction} If the numbers and masses of the branes are quite different, e.g. $m_1 N_1 v^2  \\ll m_2 N_2 $, we can ignore the contribution from the another species of the branes and\n the system would reduce to the single-species brane system with no intersecting which has been studied in \\cite{Morita:2013wfa}. \n }.\nSuppose that this term is balanced to the term in the third line  which is from $L_2$ (\\ref{eff-4-body}) due to the virial theorem, \nwe obtain the relation between $v$ and $r$ as\n\\begin{align}\n\\frac{ \\kappa^2_{{d}+3} N_1 N_2 m_1 m_2}{\\Omega_{{d}+1}} \\frac{v^2}{r^{{d}}}\n \\sim \n\\frac{ \\kappa^6_{{d}+3} N_1^2 N_2^2 m_1^2 m_2^2}{\\Omega_{{d}+1}^3} \\frac{v^4}{r^{3{d}}} \\quad \\Longrightarrow \\quad  \nv^2 \\sim  \\frac{\\Omega_{{d}+1}^2r^{2{d}}}{ \\kappa^4_{{d}+3} N_1 N_2 m_1 m_2}.\n\\label{scale-v-r}\n\\end{align}\nNow we see that the other terms listed in (\\ref{estimate-D1D5}) are indeed subdominant at this scaling.\nTo see this, it is convenient to define $\\cQ_A \\equiv \\kappa^2_{{d}+3} m_A N_A\/\\Omega_{{d}+1}$.\nThen the scaling relation (\\ref{scale-v-r}) is rewritten as $v^2 \\sim r^{2 {d}}\/\\cQ_1 \\cQ_2$ and the low energy $|v| \\ll 1$ indicates $r^d\/\\cQ_1$ and $r^d\/\\cQ_2$ are small\\footnote{More precisely speaking, $|v| \\ll 1$ indicates that $r^{2d}\/\\cQ_1\\cQ_2 \\ll 1$ and does not indicate that $r^d\/\\cQ_1$ and $r^d\/\\cQ_2$ are both small.\nFor example, if $\\cQ_1\/r^d \\ll \\cQ_2\/r^d$, $r^d\/\\cQ_2$ can be large. However this is the situation that we should ignore another species of the brane as we argued in footnote \\ref{ftnt-reduction}, and we can exclude this possibility. Hence we can regard that  $r^d\/\\cQ_1$ and $r^d\/\\cQ_2$ are both small. }.\nThus the terms in the Lagrangian (\\ref{estimate-D1D5}) scale as,\n\\begin{align}\nL_\\text{kin}  \\sim & \\frac{ \\Omega_{{d}+1} r^{{d}}}{\\kappa^2_{{d}+3}} \\frac{r^d}{\\cQ_2} + \\frac{ \\Omega_{{d}+1} r^{{d}}}{\\kappa^2_{{d}+3}} \\frac{r^d}{\\cQ_2}  \\frac{r^{2d}}{\\cQ_1 \\cQ_2} + \\cdots +  \\frac{ \\Omega_{{d}+1} r^{{d}}}{\\kappa^2_{{d}+3}}  \\frac{r^{d}}{\\cQ_1 }  +   \\frac{ \\Omega_{{d}+1} r^{{d}}}{\\kappa^2_{{d}+3}} \\frac{r^{d}}{\\cQ_1 }  \\frac{r^{2d}}{\\cQ_1 \\cQ_2} + \\cdots,  \\nonumber \\\\\nL_\\text{1-grav} \\sim&   \\frac{ \\Omega_{{d}+1} r^{{d}}}{\\kappa^2_{{d}+3}} \\left( \\frac{r^{d}}{\\cQ_2 }\\right)^2\n+  \\frac{ \\Omega_{{d}+1} r^{{d}}}{\\kappa^2_{{d}+3}} \\left( \\frac{r^{d}}{\\cQ_1 }\\right)^2+  \\frac{ \\Omega_{{d}+1} r^{{d}}}{\\kappa^2_{{d}+3}} + \\cdots, \\nonumber \\\\\nL_\\text{3-grav} \\sim &   \\frac{ \\Omega_{{d}+1} r^{{d}}}{\\kappa^2_{{d}+3}} + \\cdots.\n\\label{r-Q}\n\\end{align}\nHere the ordering of the terms is the same as in eq.\\,(\\ref{estimate-D1D5}).\nWe see that $L_1$ and $L_2$ scale as $  \\Omega_{{d}+1} r^{{d}}\/\\kappa^2_{{d}+3} $, while the other terms earn the factors of $r^{d}\/\\cQ_1 $ and\/or $r^{d}\/\\cQ_2 $ and are suppressed at low energy ($r^d \\ll \\cQ_1, \\cQ_2$).\nThis self-consistently ensures our assumption that $L_1$ and $L_2$ are relevant at the low energy, and hence the scaling relation (\\ref{scale-v-r}) is confirmed.\nNote that the masses of the branes always appear as the combination $m_1 m_2$ in $L_1$ and $L_2$, and it ensures that the scales of the position $r$ and velocity $v$ are independent of the species of the branes as we assumed in (\\ref{assumption-scales}).\n\nSo far we have considered the terms up to $L_2$, and now we consider $L_n$ (\\ref{moduli-multi-graviton}) ($n \\ge 3$) too. \nWe can see that at the scaling (\\ref{scale-v-r}) which was derived via the virial theorem $L_1 \\sim L_2 $, all the other interactions $L_n$ also become the same order.\nIt means that the branes are strongly coupled in the bound state.\nWe called such a bound state as `warm $p$-soup' in Ref.~\\cite{Morita:2013wfa}.\n\nFrom now, we evaluate the thermodynamical quantities of the bound state.\nBy substituting the relation (\\ref{scale-v-r}) to the Lagrangian $L \\sim L_1$, we estimate the free energy of the system as\n\\begin{align}\nF \\sim L_1 \\sim \\frac{ \\Omega_{{d}+1} r^{{d}}}{\\kappa^2_{{d}+3}}.\n\\label{free-energy-D1D5} \n\\end{align}\nHere we consider temperature dependence.\nWhen the bound state is thermalized, we treat $\\vec{r}_i$ as a thermal field (particle) and expand $\\vec{r}_i(t) = \\sum_n \\vec{r}_{i(n)} \\exp\\left(i \\frac{2 \\pi n}{\\beta}t  \\right) $. \nHence we assume that the velocity $v = \\partial_t r$ are characterized by the temperature of the system through\n\\begin{align}\nv  \\sim \\pi T r.\n\\label{key-assumption}\n\\end{align}\nNote that such a relation is not held generally if the system has a mass gap, but\nthere would be no mass gap in the interacting brane system as argued in Ref.~\\cite{Morita:2013wfa}.\nThen, from (\\ref{scale-v-r}), we obtain the relation between the size of the bound state and the temperature\n\\begin{align}\nr \\sim \\left(   \\frac{ \\pi^2 \\kappa^4_{{d}+3} N_1 N_2 m_1 m_2 T^2 }{\\Omega_{{d}+1}^2} \\right)^{\\frac{1}{2 ({d}-1)}} .\n \\label{scale-r-T}\n\\end{align}\nHere we have assumed ${d} \\neq 1$, and we will consider ${d}=1$ case later.\nBy substituting this relation into the free energy (\\ref{free-energy-D1D5}), \nwe estimate the entropy of the bound state as\n\\begin{align}\n&S_{\\text{entropy}} = - \\frac{\\partial F}{\\partial T}  \\sim\n\\left(   \\pi^2 N_1 N_2 m_1 m_2 \\right)^{\\frac{{d}}{2 ({d}-1)}}\n\\left(   \\frac{ \\kappa^2_{{d}+3}   T }{\\Omega_{{d}+1}} \\right)^{\\frac{1}{ {d}-1}}.\n\\label{entropy-D1D5}  \n\\end{align}\n\n\nWe compare the obtained quantities with the corresponding black hole solution.\nIn the near extremal regime, the black hole thermodynamics tells us, \n\\begin{align}\nF&= -\n\\frac{d-1}{2}\\frac{\\Omega_{{d}+1} r_H^{{d}}}{\\kappa^2_{{d}+3}},\n\\label{free-energy-D1D5-GR} \\\\\nS_{\\text{entropy}}&=\n\\left( \\pi^2   N_1 N_2 m_1 m_2 \\right)^{\\frac{{d}}{2 ({d}-1)}}\n\\left(   \\frac{2^{2 d+1} \\kappa^2_{{d}+3}   T }{{ d}^{ d+1} \\Omega_{{d}+1}} \\right)^{\\frac{1}{ {d}-1}},\n\\label{entropy-D1D5-GR} \\\\\nr_H&= \\left(   \\frac{ 64 \\pi^2 \\kappa^4_{{d}+3} N_1 N_2 m_1 m_2 T^2 }{{ d}^4\\Omega_{{d}+1}^2} \\right)^{\\frac{1}{2 ({d}-1)}} .\n\\label{eq-tmp-horizon}\n\\end{align}\nThe derivation of these expressions is summarized in \\ref{app:thermo}, and we have taken $M=2$.\nHere $r_H$ is the location of the horizon.\nTherefore, if we identify the size of the bound state $r$ with the horizon $r_H$, our result (\\ref{free-energy-D1D5}), (\\ref{scale-r-T}) and (\\ref{entropy-D1D5}) reproduce the parameter dependences of the black hole thermodynamics including $\\pi$.\n($r_H$ depends on the coordinate and we have argued what coordinate is natural in \\cite{Morita:2013wfa}.)\nThis agreement may indicate that \nthe interacting $q_1$ and $q_2$-branes described by the effective action (\\ref{moduli-D1D5}) provide\nthe microscopic origin of the $q_1$-$q_2$ black hole thermodynamics.\n\nNow we comment on the assumption $r^d \\ll \\cQ_1, \\cQ_2$ which we have used when we consider the effective action (\\ref{moduli-D1D5}).\nAt the scale (\\ref{scale-r-T}), this relation becomes\n\\ba\nT \\ll \\frac{(\\cQ_1)^{\\frac12-\\frac1d}}{\\pi (\\cQ_2)^{\\frac12}},\\quad\n\\frac{(\\cQ_2)^{\\frac12-\\frac1d}}{\\pi (\\cQ_1)^{\\frac12}}.\n\\ea\nSince we consider here the situation $\\cQ_1\/\\cQ_2\\sim \\cO(1)$, this means \n$T \\ll 1\/(\\cQ_1)^\\frac1d, 1\/(\\cQ_2)^\\frac1d$ and this is the near extremal limit in supergravity. \nThus our analysis is valid when we consider the near extremal black holes.\n\n\nFinally we consider the ${d}=1$ case.\nIn this case the relations (\\ref{scale-v-r}) and (\\ref{key-assumption}) fix  $T$ as\n\\begin{align}\nT \\sim \\left( \\frac{\\Omega_{2}^2}{ \\kappa^4_{4} N_1 N_2 m_1 m_2 \\pi^2 } \\right)^{\\frac{1}{2 }} .\n\\end{align}\nand $r$ remains as a free parameter.\nThis is the Hagedorn behavior, and this temperature is coincident with the Hagedorn temperature in supergravity (\\ref{HagedornT}).\nThus the $p$-soup proposal works even in the $d=1$ case too.\n\n\n\\section{General intersecting black brane}\n\\label{sec:general}\n\nWe discuss general black brane system with arbitrary number of species of branes. \nWe can also introduce momentum along one of the cycles of the torus if all the branes wind around this cycle. (We define $R$ as the radius of this cycle.)\nIn this case, the momentum is quantized as $N_P\/R$ and there are $N_P$ gravitational waves each of which carries a momentum $1\/R$.\nWe can regard this gravitational wave as a 0-brane with mass $1\/R$ via the KK reduction of this cycle.\nAlthough the setup is complicated, we will see a simple result that the effective theory of the separated intersecting branes explains the black hole thermodynamics of the corresponding black brane at low energy.\n\nIn \\ref{app:eff}, we argue the derivation of the low energy effective action for $N_A$ $q_A$-branes ($A=1,\\ldots, M$) where $M$ is the number of the species of the charges (including momentum if we introduce it).\nThe dominant terms of the action are given by\n\\begin{align}\n\\label{lag-gen}\nS_{\\text{eff}} =& \\int dt \\left(L_1+L_2+\\cdots\\right), \\\\\nL_n \\sim &\n\\sum_{i_1,\\ldots,i_n}^{N_1}\\sum_{j_1,\\ldots,j_n}^{N_2}\\cdots \\sum_{k_1,\\ldots,k_n}^{N_M}\n\\frac{\\kappa_{d+3}^{2(nM-1)}\\,\\hat\\prod_A\\,m_A^n}{\\Omega_{d+1}^{nM-1}}\n\\left(\\vect v{2n}\\prod_{a=2}^n\\prod_{b=1}^n\\cdots\\prod_{c=1}^n\\frac{1}{\\vect rd_{i_1i_a}\\vect rd_{i_1j_b}\\cdots \\vect rd_{i_1k_c}}+\\cdots\\right),\n\\end{align}\nwhere $\\hat\\prod_A$ is the product including momentum.\nSimilar to the two brane case, we apply the virial theorem and estimate the thermodynamics of this system.\nWith the assumption (\\ref{assumption-scales}), the first two terms of the action scale as \n\\ba\nL_1 \\sim \\frac{\\kappa_{d+3}^{2(M-1)}}{\\Omega_{d+1}^{M-1}} \\left(\\hat{\\prod_A}\\, m_A N_A \\right) \\frac{v^2}{r^{d(M-1)}} \n\\,,\\quad\nL_2 \\sim \\frac{\\kappa_{d+3}^{2(2M-1)}}{\\Omega_{d+1}^{2M-1}} \\left(\\hat{\\prod_A}\\, m_A N_A \\right)^2 \\frac{v^4}{r^{d(2M-1)}} \n\\,,\n\\label{L1L2-gen}\n\\ea\nTherefore through the virial theorem $L_1\\sim L_2$, we obtain the relation between $v$ and $r$ as\n\\ba\nv^2  \\sim \\frac{\\Omega_{d+1}^M r^{dM}}{\\kappa_{d+3}^{2M}\\,\\hat\\prod_A\\, N_A m_A}.\n\\ea\nThen the free energy can be estimated as\n\\ba\nF\\sim L_1 \\sim \\frac{\\Omega_{d+1}r^d}{\\kappa_{d+3}^2}\\,.\n\\ea\nNote that this expression is common for all $M$.\nMoreover, if we assume the relation between the velocity and temperature of the system $v\\sim \\pi Tr$, we can estimate the radius of horizon\n\\ba\nr \\sim \\left(\\frac{\\pi^2 T^2 \\kappa_{d+3}^{2M} \\,\\hat\\prod_A\\, N_A m_A}{\\Omega_{d+1}^M}\\right)^{\\frac{1}{dM-2}}\\,.\n\\ea\nfor $dM-2 \\neq 0$.\nUsing the expressions of free energy and horizon radius, the entropy can be written as \n\\ba\nS_{\\text{entropy}}=-\\frac{\\partial F}{\\partial T} \n\\sim \\left(\\pi^2\\,\\hat{\\prod_A}\\, N_Am_A\\right)^{\\frac{d}{dM-2}}\\left(\\frac{\\kappa^2_{d+3}}{\\Omega_{d+1}}\\right)^{\\frac{2}{dM-2}} T^{\\frac{2d}{dM-2}-1}\\,.\n\\ea\n\nNow we compare our results with the corresponding black hole thermodynamics.\nThe results in the black hole are shown in eqs.\\,(\\ref{rH:sugra}), (\\ref{en:sugra}) and (\\ref{F:sugra}).\nWe can check that our results are coincident with the black hole thermodynamics up to rational numerical factors.\nSimilar consistency can be also seen for the Hagedorn case ($dM-2=0$).\n\n\\section{Discussions}\n\\label{sec-conclusion}\n\nWe generalized the $p$-soup proposal \\cite{Morita:2013wfa} to the intersecting brane systems.\nWe figured out that, in case the branes satisfy the intersection rule (\\ref{intersection-rule}), the bound state of the branes exhibits the thermodynamical properties which agree with the corresponding black brane.\nAlthough the intersecting brane systems are complicated themselves, this result is strikingly simple, and we believe that our proposal captures profound natures of the black hole microstates.\n\nAlso we can apply the results to superstring theory and M-theory.\nFor example, the D1-D5 brane system studied in \\cite{Morita:2014cfa} can be mapped to other brane configurations such as D0-D4 system or M5-P system via string dualities \\cite{Martinec:1999sa}.\nThen we can investigate the microstates of these branes similarly.\nFurthermore the phase transitions between these branes \\cite{Martinec:1999sa} could be understood microscopically in a fashion of \\cite{Morita:2014ypa}.\nIn such a way, we can study the various black brane dynamics in string theory through our proposal.\n\nMoreover, our discussion would be important to investigate supersymmetric gauge theories. \nIn the case of the single-species D$p$\/M-branes, through the gauge\/gravity correspondence, the duality between the black branes and the supersymmetric gauge theories on the branes at finite temperature is expected \\cite{Itzhaki:1998dd}.\nWe can explain this duality by using the fact that the effective action for the interacting $N$ branes (a variety of (\\ref{single-brane})), which is obtained from gravity, is also derived from the supersymmetric gauge theory on the branes as a low energy effective action \\cite{Morita:2013wfa}.\nNot only that, the effective action might be useful to estimate several quantities in the gauge theory which cannot be calculated in the gravity \\cite{Morita:2014ypa}.\nIn the case of the intersecting brane systems, much more varieties of supersymmetric gauge theories\nappear on the branes than the single-species brane case.\nThere, the effective action (\\ref{lag-gen}) may play a key role to understand the dynamics of these gauge theories.\n\n\nIn this way, the $p$-soup proposal may be important in the various contexts in theoretical physics. In order to establish the proposal, we have to derive the thermodynamical quantities from the effective action (\\ref{lag-gen}) exactly and compare them with the black hole results. \nHowever even fixing the coefficients of the interactions in the effective action (\\ref{lag-gen}) is difficult in the intersecting brane system, and we need to find a clever way to solve this issue.\nIf we could achieve it, it must be a valuable step toward the understanding of the black hole microstate.\n\n\n\\subsection*{Acknowledgments}\nWe would like to thank Andrew Hickling, Toby Wiseman and Benjamin Withers for useful discussions through the collaboration.\nWe also thank Nobuyoshi Ohta for useful comments.\nThe work of T.~M. is supported in part by Grant-in-Aid for Scientific Research (No.\\,15K17643) from JSPS.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{sec:intro}\nIn recent years the physics community has devoted an increasing\nattention to anomalous properties of physical systems (e.g., anomalous\ntransport, anomalous diffusion, anomalous conductivity, etc.). Such\nproperties have proven \nrelevant in many fields such as thermal conductivity, kinetic\nequations, plasma physics, etc. and they are widely believed to be\ndynamical in nature. In fact, such phenomena seem to depend on the\nweak chaotic properties of the underling dynamics, see \\cite{Za} and\nreferences therein for a detailed discussion. The \nbasic idea is that, while uniformly hyperbolic dynamics gives rise to\nnormal transport properties (consider for example the diffusive\nbehavior in a finite horizon Lorentz gas \\cite{BuSi})\nnon-uniform hyperbolicity gives rise to different behavior\n(e.g. the anomalous diffusion believed to occur in infinite horizon\nLorentz gas \\cite{Bl}) due to weaker\nmixing properties (e.g. polynomial decay of correlations) and regions\nin which the motions is rather regular and where the systems spend a\nsubstantial fraction of time ({\\em sticky} regions).  \n\nUnfortunately, the theoretical understanding of dynamical models with\npolynomial decay of correlations is extremely limited, hence the\nnecessity to rigorously investigate relevant toy models. The only well\nunderstood cases are expanding one dimensional maps with a neutral\nfixed point. Such maps were proposed as a model of intermittent\nbehavior in fluids (\\cite{PM}) and have been widely studied. It has been\nproven that such maps enjoy polynomial decay of correlations with the\nrate depending on the behavior of the fixed point \\cite{LSV, Yo1,\nHu1, Hu2, Sa, Go1}. In addition, when the\ndecay of correlation is sufficiently slow, the observables do not\nsatisfy the Central Limit Theorem or the Invariance Principle but\nrather, when properly rescaled, some stable law (\\cite{Zw, Go2, PoSh}).\n\nSome, more partial, results exist for multidimensional expanding\nmaps \\cite{PoYu} as well. Yet, the usual physically relevant models are\nconnected to Hamiltonian dynamics and, to our knowledge, no rigorous\nresults are available in such a situation. The simplest case which\nretain some Hamiltonian flavor is clearly a two dimensional area\npreserving map. In fact, mixing area preserving maps of the two\ndimensional torus with a neutral fixed point (the simplest type of\nsticky set) have been investigated\nnumerically \\cite{ArPr} predicting the possibility of a polynomial decay\nof correlations. \nIn this paper we consider a class of non-uniformly hyperbolic symplectic maps \\eqref{eq:map}\nof the two torus where the hyperbolicity breaks down because of a non-hyperbolic fixed point. In fact, the linearized\ndynamics at the fixed point is a shear \\eqref{eq:der}. We prove that\nthe decay of correlations is polynomial and, more precisely, decays at\nleast as $n^{-2}$ and, in some sense, one cannot expect much\nmore. Note that the example treated numerically in \\cite{ArPr} is a\nspecial case of the present setting. In \\cite{ArPr} the predicted\ndecay was $n^{-2.5}$. This emphasizes the difficulty to investigate\nsuch issues and the strong need for more theoretical work on the subject.\n\n\nThe result in the present paper is based on a precise quantitative\nanalysis of the angle between the stable and the unstable\ndirection. This angle turns out to degenerate approaching the origin\n(where the non hyperbolic fixed point is located). Once such a control\nis achieved it is possible to obtain a bound on the expansion and\ncontraction in the system. Such expansion turns out to be only\npolynomial, in contrast with the uniformly hyperbolic case where it is\nexponential. In turn, the bound on the expansion allows to study the\nregularity of the stable and unstable foliation. It turns out that\nthey are $\\Co^1$ away from the origin. This suffice to apply a simple\nrandom approximation technique that allows estimating the speed of the\ncorrelations. \n\nAs the rate of convergence to equilibrium is of order $n^{-2}$, see Theorem\n\\ref{thm:main}, the\nCentral Limit Theorem holds for zero average observable, see Corollary \n\\ref{cor:clt},\nso the model does not exhibit anomalous statistical behavior in this\nrespect. Yet, it clearly exhibits an intermittent behavior and it shows the\nmechanism whereby slow decay of correlations may arise. The present\nwork emphasizes the need to carry out similar studies in cases where the\nset producing intermittency has a more complex structure than a simple\nisolated point.\n\nThe paper is organized as follows: section \\ref{sec:results} details\nthe model and makes precise the results. Section \\ref{sec:fix} studies\nthe local dynamics at the fixed point and, in particular the\nproperties of its stable and unstable manifolds. This can be achieved\nin many way, here we find most efficient to apply a variational\ntechnique. Section \\ref{sec:narrow} establishes a precise bound for\nthe angle between the stable and the unstable direction at each\npoint. As anticipated, such a bound yields an a priory bound on the\nexpansion and contractions rates in the systems, these are obtained in\nsection \\ref{sec:expansion}. The latter result suffices to apply\nstandard distortion estimates that, in turn, allow to prove precise\nresults on the regularity of the invariant foliation and the\nholonomies, see section \\ref{sec:regularity} and section\n\\ref{sec:holo} respectively.  Next, in section\n\\ref{sec:random} we introduce a random perturbation of the above map\nand investigate its statistical properties that, thanks to the added\nrandomness, can be addressed fairly easily. The relevance of the above\nrandom perturbation is that the limit of zero noise allows to easily obtain a\nbound on the rate of mixing in the original map, we do this in section\n\\ref{sec:decay}. Finally, in section \\ref{sec:lower}, we show that the\nobtained bound is close to being optimal. The paper ends with Remark\n\\ref{rem:problems} pointing to the unsatisfactory nature of some of\nthe present results and the need to investigate the related open\nproblems.\n\n\n\n\\section{The model and the results}\n\\label{sec:results}\nFor each $h\\in\\Co^{\\infty}(\\To^1,\\To^1)$ we define the map\n$T:\\To^2\\to\\To^2$ by\\footnote{\\label{foot:uno} Note that the following formula is\nequivalent, by the symplectic change of variable $q=x-y$, $p=y$, to\nthe map\n\\[\n\\widetilde T(q,p)=\\begin{cases}\n              q+p&\\quad\\mod 1\\\\\n              p+h(q+p)&\\quad \\mod 1\n              \\end{cases}\n\\]\nwhich belongs to the standard map family. Yet, the functions $h$\nconsidered here differ substantially from the sine function which would\ncorrespond to the classical Chirikov-Taylor well known example.}\n\\begin{equation}\\label{eq:map}\nT(x,y)=\\begin{cases}\n              x+h(x)+y&\\quad\\mod 1\\\\\n              h(x)+y&\\quad\\mod 1\n              \\end{cases}\n\\end{equation} \nWe moreover require the following properties\n\\begin{enumerate}\n\\item $h(0)=0$  (zero is a fixed point);\n\\item $h'(0)=0$ (zero is a neutral fixed point)\n\\item $h'(x)>0$ for each $x\\neq 0$ (hyperbolicity)\n\\end{enumerate}\nNote that conditions (2--3) imply that zero is a minimum for $h'$, which forces \n\\[\nh''(0)=0;\\quad h'''(0)\\geq 0.\n\\]\nWe will restrict to the generic case\n\\begin{enumerate}\n\\item[(4)] $h'''(0)>0$.\n\\end{enumerate}\nIn order to simplify the discussion we will also assume the following\nsymmetry\n\\begin{enumerate}\n\\item[(5)] $h(-x)=-h(x)$.\n\\end{enumerate}\n\nThis means that we can write\n\\begin{equation}\n\\label{eq:h}\nh(x)=bx^3+\\Or(x^5).\n\\end{equation}\n\\begin{rem}\nNote that two facts implied by the above assumptions are not necessary and could be\ndone away with at the price of more extra work: the hypothesis that\nthere is only one neutral fixed point (finitely many neutral periodic\norbits would make little difference) and the symmetry (5). We assume\nsuch facts only to simplify the presentation of the arguments.\n\\end{rem}\nSince the derivative of the map is given by\n\\begin{equation}\\label{eq:der}\nDT=\\begin{pmatrix}\n   1+h'(x)&1\\\\\n   h'(x)&1\n   \\end{pmatrix}\n\\end{equation}\n$\\det(DT)=1$, thus the Lebesgue measure $m$ is an invariant measure (the\nmaps are symplectic). From now on we will consider the dynamical\nsystems $T: (\\To^2,m)\\rightarrow (\\To^2,m)$. \n\nFormula \\eqref{eq:der} and property (3) imply that \nthe cone $\\Co_+=\\{v\\in\\R^2\\;|\\;Q(v):=\\langle v_1,\\,v_2\\rangle\\geq 0\\}$\nis invariant for $DT$. In additions, it is easy to check that $D_\\xi\nT^2 \\Co_+\\subset \\hbox{int }\\Co_+\\cup\\{0\\}$ for all\n$\\xi\\in\\To^2\\backslash \\{0\\}$. From\nthis and the general theory, see \\cite{LW}, follows immediately\n\n\\begin{thm}\nThe above described dynamical systems are non-uniformly hyperbolic and mixing.\n\\end{thm} \n\\begin{exmp}\nAn interesting concrete example for the above setting is given by the function \n$h(x):=x-\\sin x$.\n\\end{exmp}\nThe question remains about the rate of mixing, this is the present\ntopic. \n\\begin{rem}\nIn the following by $\\operatorname{C}$ we designate a generic\nconstant depending only on $T$. Accordingly, its value may vary from\nan occurrence to the next. In the instances when we will need a constant\nof the above type but with a fixed value we will use sub-superscripts.\n\\end{rem}\n\\begin{thm}\\label{thm:main}\nFor each $f,g\\in\\Co^{1}(\\To^2,\\R)$, $\\int f=0$, holds\ntrue\\footnote{In fact, a slightly sharper bound holds, see \\eqref{eq:sharp}.}\n\\[\n\\left|\\int f g\\circ T^n\\right|\\leq \\operatorname{C}\n\\|f\\|_{\\Co^{1}}\\|g\\|_{\\Co^{1}}n^{-2}(\\ln n)^4.\n\\]\n\\end{thm}\n\\begin{rem}\nAs in other similar cases \\cite{LSV, L, Po} the logarithmic\ncorrection is almost certainly \nan artifact of the technique of the proof. It could probably be\nremoved by using a more sophisticated (and thus more technically\ninvolved) approach. See also section \\ref{sec:lower}.\n\\end{rem}\nForm Theorem \\ref{thm:main} many facts follow, just to give an\nexample let us mention the following result that can be obtained from\nTheorem 1.2 in \\cite{L3}.\n\\begin{cor}[CLT]\\label{cor:clt}\nGiven $f\\in\\Co^1$, $\\int f=0$, the random variable\n\\[\n\\frac 1{\\sqrt n}\\sum_{i=0}^{n-1}f\\circ T^i\n\\]\nconverges in distribution to a Gaussian variable with zero mean and\nfinite variance $\\sigma$. In addition, $\\sigma=0$ iff there exists\n$\\vf\\in L^1$ such that $f=\\vf-\\vf\\circ T$.\\footnote{In particular this means\nthat the average of $f$ on each periodic orbit must be zero.}\n\\end{cor}\nThe rest of the paper is devoted to the proof of\nTheorem \\ref{thm:main} that will find its conclusion in section\n\\ref{sec:decay}. The basic \nfact needed in the proof, a fact of independent \ninterest and made quantitatively precise in Lemma \\ref{lem:distreg1}, is\nthe following. \n\\begin{thm}\\label{thm:mainlem}\nThe stable and unstable distributions are $\\Co^1$ in $\\To^2\\backslash\\{0\\}$.\n\\end{thm}\n\n\n\n\\section{The fixed point manifolds}\\label{sec:fix}\n\n\nAs usual we start by studying the local dynamics near the fixed\npoint. The first basic fact is the existence of stable and unstable\nmanifolds. This is rather standard, yet since we need some\nquantitative information we will construct them explicitly.\n\nInstead of constructing them via usual fixed point arguments it turns\nout to be faster to use a variational method.\n\n\\subsection{A variational argument}\n\nLet us consider, in a neighborhood of zero, the function\n\\begin{equation}\\label{eq:generating}\nL(x,x_1):=\\frac 12 (x-x_1)^2+G(x)\\;;\\quad G(x):=\\int_0^xh(z) dz.\n\\end{equation}\nBy setting\n\\[\n\\begin{split}\n&y:=-\\frac{\\partial L}{\\partial x}=x_1-x-h(x)\\\\\n&y_1:=\\frac{\\partial L}{\\partial x_1}=x_1-x\n\\end{split}\n\\]\nwe have $(x_1,y_1)=T(x,y)$, that is {\\sl $L$ is a generating function\nfor the map \\eqref{eq:map}}.\n\nThen, for each $a\\in\\R$, we define the Lagrangian $\\Lp_a:\\ell^2(\\N)\\to\\R$ by\n\\begin{equation}\\label{eq:lagrange}\n\\Lp_a(x):=\\sum_{n=1}^\\infty L(x_n,x_{n+1})+L(a,x_1).\n\\end{equation}\nThe justification of the above definition rests in the following\nLemma.\n\\begin{lem}\\label{lem:critical}\nFor each $a\\in(-1,1)$, holds true $\\Lp_a\\in\\Co^1(\\ell^2(\\N))$. In addition, if\n$x\\in \\ell^2(\\N)$ is such that $D_x\\Lp_a=0$, then setting $x_0=a$ and \n$y_n:=x_{n+1}-x_n-h(x_n)$, we have $T^n(x_0,y_0)=(x_n,y_n)$.\n\\end{lem}\n\\begin{proof}\nFirst of all \\eqref{eq:h} implies that there exists $\\operatorname{C}>0$ such that\n$|G(x)|\\leq \\operatorname{C} x^4$. It is then easy to see that $\\Lp_a$ is well defined\nfor each sequence in $\\ell^2(\\N)$.\n\nNext, for each $n\\in\\N$ let us define\n$(\\nabla\\Lp_a)_n:=\\partial_{x_n}\\Lp_a$.\nClearly,\n\\[\n\\begin{split}\n(\\nabla\\Lp_a)_1(x)&=2x_1-x_2+h(x_1)-a\\\\\n(\\nabla\\Lp_a)_n(x)&=2x_n-x_{n+1}-x_{n-1}+h(x_n).\n\\end{split}\n\\]\nOf course, for $x\\in\\ell^2(\\N)$, $ \\nabla\\Lp_a(x)\\in\\ell^2(\\N)$, it is\nthen trivial to check \nthat $D_x\\Lp_a(v)=\\langle \\nabla\\Lp_a(x), v\\rangle$.\nThe last statement follows by a direct computation.\n\\end{proof}\n\nBy the above Lemma it is clear that one can obtain the stable\nmanifolds of the fixed point from the critical points of $\\Lp_a$, it\nremains to prove that such critical points do exist. We will start by\nconsidering the case $a\\geq 0$.\n\nDefine\n\\begin{equation}\\label{eq:fixedset}\nQ_B:=\\{x\\in\\ell^2(\\N)\\;|\\;\\,|x_n-\\frac A{n+c}|\\leq B(n+c)^{-\\frac\n32}\\}\n\\end{equation}\nwhere $A:=\\sqrt{\\frac 2b}$; $c:=\\frac Aa$.\n\nIt is immediate to check that $Q_B$ is compact and convex. In\naddition, if $a$ is sufficiently small, then $G$ is strictly convex on\n$[-2a,2a]$ which implies that $\\Lp_a|_{Q_B}$ is strictly\nconvex. Accordingly, $\\Lp_a$ has minimum in $Q_B$, moreover the strict\nconvexity implies that such a minimum is unique, for $a$ fixed.\n\nLet us call $x(a)$  the point in $Q_B$ where $\\Lp_a$ attains its\nminimum.\n\\begin{lem}\\label{lem:stab}\nFor $a$ small enough, $D_{x(a)}\\Lp_a=0$.\n\\end{lem}\n\\begin{proof}\nSuppose that $\\partial_{x_n}\\Lp_a(x(a))\\neq 0$ for some $n\\in\\N$, for\nexample suppose it is negative. Then $x(a)$ is on the border of $Q_B$,\nsay $x(a)_n=A(n+c)^{-1}+B(n+c)^{-\\frac 32}$, otherwise we could increase\n$x(a)_n$ and decrease $\\Lp_a$ still remaining in $Q_B$, contrary to\nthe assumption. But then\n\\[\n\\begin{split}\n\\partial_{x_n}\\Lp_a(x(a))&=2x(a)_n-x(a)_{n-1}-x(a)_{n+1}+h(x(a)_{n})\\\\\n&=\\frac {2A}{n+c}+\\frac{2B}{(n+c)^{\\frac\n32}}-x(a)_{n-1}-x(a)_{n+1}+h(\\frac {A}{n+c}+\\frac{B}{(n+c)^{\\frac\n32}}) \\\\\n&\\geq -\\frac{2A}{[(n+c)^2-1](n+c)}+\\frac\n{2A}{(n+c)^3}+\\frac{3bA^2B}{(n+c)^\\frac 72}+\\Or((n+c)^{-4})\\\\\n&+\\frac{B(n+c)^3\\{2(1-(n+c)^{-2})^{\\frac 32}-(1-(n+c)^{-1})^{\\frac\n32} -(1+(n+c)^{-1})^{\\frac 32}}{(n+c)^{\\frac 32}[(n+c)^2-1]^{\\frac\n32}}\\\\\n&=\\frac{(6-\\frac {15}4)B}{(n+c)^{\\frac 72}}+\\Or((n+c)^{-4})\\geq 0\n\\end{split}\n\\]\nprovided $a$ is sufficiently small. We have thus a contradiction. The\nother possibilities are analyzed similarly.\n\\end{proof}\n\nTo conclude we need some information on the regularity of $x(a)$ as a\nfunction of $a$. Unfortunately, the\nimplicit function theorem does not applies since $D^2\\Lp_a$ does not\nhave a spectral gap, yet for our purposes a simple estimate\nsuffices.\n\\begin{lem}\\label{lem:lipman}\n$x_1(a)$ is a Lipschitz function of $a$. Moreover, when derivable,\n\\[\n|y_0(a)'|\\leq \\operatorname{C} |x(a)_0|.\n\\]\n\\end{lem}\n\\begin{proof}\nBy Lemma \\ref{lem:stab} it follows, for each $a,a'$ sufficiently small\n\\[\n\\partial_{x_n}\\Lp_{a'}(x(a'))=\\partial_{x_n}\\Lp_{a}(x(a))=0\n\\]\nthat is\n\\[\n\\partial_{x_n}\\Lp_{a'}(x(a'))-\\partial_{x_n}\\Lp_{a'}(x(a))=\\partial_{x_n}\\Lp_{a}(x(a))\n-\\partial_{x_n}\\Lp_{a'}(x(a))\n\\]\nwhich yields\n\\begin{equation}\\label{eq:strange}\n\\begin{split}\n&(2+h'(\\xi_1))\\zeta_1-\\zeta_2=a'-a\\\\\n&(2+h'(\\xi_n))\\zeta_n-\\zeta_{n+1}-\\zeta_{n-1}=0,\n\\end{split}\n\\end{equation}\nwhere $\\zeta_n:=x(a')_n-x(a)_n$ and $\\xi_n\\in[x(a)_n,x(a')_n]$.\n\nNotice that, if $|\\zeta_n|\\geq|\\zeta_{n-1}|$, then\n\\[\n|\\zeta_{n+1}|=|(2+h'(\\xi_n))\\zeta_n-\\zeta_{n-1}|\\geq\n2|\\zeta_n|-|\\zeta_{n-1}|\\geq |\\zeta_n|.\n\\]\n\nThus, by induction, if $|\\zeta_n|\\geq |\\zeta_{n-1}|$, then $|\\zeta_m|\\geq\n|\\zeta_{n-1}|$ for each $m\\geq n$, which would imply $\\zeta_{n-1}=0$ since\n$\\zeta\\in\\ell^2(\\N)$. But then\n$(2+h'(\\xi_n))\\zeta_n=\\zeta_{n+1}$, that is\n$|\\zeta_{n+1}|\\geq|\\zeta_n|$. Accordingly, again by induction,\n$\\zeta_m=0$ for each $m\\geq n-1$. This means that we can restrict\nourselves to the case $\\zeta_n\\neq 0$, $|\\zeta_n|\\geq |\\zeta_{n+1}|$.\nHence,\n\\[\n|a'-a|=|(2+h'(\\xi_1))\\zeta_1-\\zeta_2|\\geq 2|\\zeta_1|-|\\zeta_2|\\geq\n|\\zeta_1|.\n\\]\nThat is $|x(a')_n-x(a)_n|\\leq |x_1(a')-x_1(a)|\\leq |a'-a|$.\n\nFinally, summing \\eqref{eq:strange} over $n\\in\\N$, \n\\[\n\\sum_{n=1}^\\infty h'(\\xi_n)\\zeta_n=-\\zeta_1+a'-a.\n\\]\nThus, where all the $x(a)_n$ are differentiable (a full measure set),\n$|x(a)_n'|\\leq|x(a)_1'|\\leq 1$ and\n\\begin{equation}\\label{eq:derivxy}\n\\begin{split}\nx(a)_1'&=1-\\sum_{n=1}^\\infty h'(x(a)_n)x(a)_n'\\\\\ny(a)_0'&=-\\sum_{n=0}^\\infty h'(x_n)x(a)_n'.\n\\end{split}\n\\end{equation}\nAccordingly,\n\\[\n|y_0(a)'|\\leq 6b\\sum_{n=0}^\\infty x_n^2\\leq \\operatorname{C} a.\n\\]\n\\end{proof}\n\nClearly, the above Lemma implies that, calling $(x,\\gamma_s(x))$ the\ngraph of the stable manifold, $\\gamma_s\\in Lip(-1,1)$. The case $a\\leq\n0$ and the unstable manifolds can be treated similarly, yet there\nexists a faster--and more\ninstructive--way.\n\n\\subsection{Reversibility}\n\nNotice that the map $T$ is {\\sl reversible}\nwith respect to the transformations\\footnote{While the reversibility for\n$\\Pi$ is a general fact, the one for $\\Pi_1$ depends on the simplifying\nsymmetry hypothesis (5).}\n\\begin{equation}\\label{eq:invol}\n\\Pi(x,y):=(x, -y-h(x)); \\quad \\Pi_1(x,y):=(-x, y+h(x))\n\\end{equation}\nIndeed, $\\Pi^2=\\Pi_1^2=\\Id$ and $\\Pi T\\Pi=\\Pi_1 T\\Pi_1=T^{-1}$. \n\n\\begin{rem}\\label{rem:rev}\nThe reversibility implies that, for $x\\geq 0$,\n$(x,\\gamma_u(x))=\\Pi(x,\\gamma_s(x))$, and, for $x\\leq 0$,\n$(x,\\gamma_u(x))=\\Pi_1(-x,\\gamma_s(-x))$ is the unstable manifold of\nzero. \n\\end{rem}\n\n\\subsection{A quasi-Hamiltonian}\n\nTo study the motion near the fixed point it is helpful to find a local\n``Hamiltonian'' function. By Hamiltonian function we mean a function\nthat is locally invariant for the dynamics. Such a function can be\ncomputed as a formal power series starting by the relation $H\\circ\nT=H$. In fact, we are interested only in a suitable approximation. A\ndirect computation yields that, by defining $G(x):=\\int_0^x h(z)dz$ and\n\\begin{equation}\\label{eq:hamiltonian}\nH(x,y):=\\frac 12 y^2-G(x)+\\frac 12 h(x)y-\\frac 1{12}h'(x)y^2+\\frac 1 {12}h(x)^2,\n\\end{equation}\nholds true\\footnote{In fact, setting $(x_1,y_1):=T(x,y)$, holds\n\\[\n\\begin{split}\n&H(x_1,y_1)-H(x,y)=yh(x)+\\frac 12 h(x)^2-h(x)y-h(x)^2-\\frac\n12h'(x)y_1^2-\\frac 16 h''(x)y_1^3+\\frac 12h'(x)y_1^2\\\\\n&+\\frac 12 h(x)^2\n+\\frac 14 h''(x)y_1^3-\\frac 1{12}h''(x)y_1^3-\\frac\n16h'(x)h(x)y_1+\\frac 16 h'(x)h(x)y_1+\\Or(x^8+x^4y^2+y^4).\n\\end{split}\n\\]\n}\n\\begin{equation}\\label{eq:appham}\nH(T(x,y))-H(x,y)=\\Or (x^8+y^4).\n\\end{equation}\nThis approximate conservation law suffices to obtain rather precise\ninformation on the near fixed point dynamics.\\footnote{The reader\n  should be aware that it is possible to do much better, that is to\n  obtain an exponentially precise conservation law, see \\cite{L2},\n  \\cite{BG}.} \nThe first application \nis given by the following information on the stable manifold.\n\\begin{lem}\\label{lem:manifzero}\nFor $x\\geq 0$ sufficiently small holds\n\\[\n\\gamma_s(x)= -A^{-1}x^2+\\Or(x^3).\n\\] \n\\end{lem}\n\\begin{proof}\nUsing the notation of Lemma \\ref{lem:stab}, for fixed $a$ we get\n\\begin{equation}\\label{eq:unoy}\ny_n=x_{n+1}-x_n-h(x_n)=\\Or((n+c)^{-\\frac 32}).\n\\end{equation}\nHence\n\\[\nH(x_n,y_n)=\\Or((n+c)^{-3}).\n\\]\nUsing equation \\eqref{eq:appham} we have\n\\[\nH(x_0,y_0)=H(x_n,y_n)+\\Or(\\sum_{i=0}^{n-1}x_i^8+y_i^4)\n\\]\nthat is\n\\[\n|H(x_0,y_0)|\\leq \\operatorname{C} \\left\\{(n+c)^{-3}+\n\\sum_{i=0}^{\\infty}(n+c)^{-6}\\right\\}\\leq \\operatorname{C}\n\\left\\{(n+c)^{-3}+c^{-5}\\right\\}.\n\\]\nTaking the limit for $n$ to infinity in the above expression and\nremembering the definition of $c$ follows\n\\[\n|H(x_0,y_0)|\\leq\\operatorname{C} x_0^{5}.\n\\]\nSince equation \\eqref{eq:unoy} implies $|y_0|=\\Or(x_0^{\\frac 32})$,\nfrom \\eqref{eq:hamiltonian} we have\n\\[\ny_0^2-\\frac b2 x_0^4+bx_0^3y_0=\\Or(x_0^5)\n\\]\nfrom which the lemma follows.\n\\end{proof}\nAccording to Lemma \\ref{lem:manifzero}, the local picture of the\nmanifolds is given by Figure \\ref{fig:man}.\n\\begin{figure}[ht]\\\n\\centering\n\\put(-40,0){\\line(1,0){80}}\n\\put(0,-20){\\line(0,1){40}}\n\\put(-6.5,-15){{\\tt Fat sector}}\n\\put(30,2){{\\tt Thin sector}}\n\\thicklines\n\\qbezier(-40, 10)(0,-10)(40,10)\n\\qbezier(-40, -10)(0,10)(40,-10)\n\\put(-39,10){$W^s$}\n\\put(-39,-14){$W^u$}\n\\put(40,11){$W^u$}\n\\put(40,-9){$W^s$}\n\\put(0,-24){\\ }\n\\caption{The manifolds of the fixed point}\\label{fig:man}\n\\end{figure}\n\n\\subsection{Manifold regularity}\nSince in the previous section we have seen that the manifold are\nLipschitz curves, we can define the dynamics restricted to the unstable\nmanifold:\n\\begin{equation}\\label{eq:fu}\nf_u(x):=x+h(x)+\\gamma_u(x).\n\\end{equation}\n\nOur next task is to obtain sharper information on the manifolds\nregularity.\\footnote{Of course, the manifold should be as smooth as $h$, but\nthis results is not needed in the following while we do need an\nexplicit bound on the curvature.}\n\\begin{lem}\\label{lem:manifoldreg}\nThe unstable manifold of the fixed point is $\\Co^2$, apart from zero.\n\\end{lem}\n\\begin{proof}\nIt is clearly enough to show that $\\gamma_u\\in \\Co^2$ apart from\nzero. To do so call $u(x)=\\gamma_u'(x)$ (the derivative exists almost\neverywhere since $\\gamma_u$ is Lipschitz). The tangent vector to the\nunstable manifold has the form $(1,u)$. On the other hand\n\\[\n\\begin{split}\n\\lambda_u(f_u^{-1}(x))\n\\begin{pmatrix}1\\\\u(x)\\end{pmatrix}&:=\\begin{pmatrix}\n                                    1+h'(f_u^{-1}(x))&1\\\\\n                                     h'(f_u^{-1}(x))&1\n                                    \\end{pmatrix}\n                          \\begin{pmatrix}1\\\\u(f_u^{-1}(x))\\end{pmatrix}\\\\\n&=:\\lambda_u(f_u^{-1}(x))\\begin{pmatrix}1\\\\F(f_u^{-1}(x),u(f_u^{-1}(x)))\n\\end{pmatrix}\n\\end{split}\n\\]\nwhere\n\\begin{equation}\\label{eq:unsdef}\n\\begin{split}\n&\\lambda_u(x):=1+h'(x)+u(x)=f'_u(x)\\\\\n&F(x,u):=1-\\frac{1}{1+h'(x)+u}.\n\\end{split}\n\\end{equation}\nAccordingly, setting $x_i:=f_u^{-i}(x)$, holds\n$u(x_i)=F(x_{i+1},u(x_{i+1}))$. Next, let $v_i:=2A^{-1}x_i$, then a\ndirect computation yields $F(x_i,v_i)-v_{i-1}=\\Or(x_i^3)$, thus\n\\[\n|u(x_i)-v_i|\\leq |F(x_{i+1},u(x_{i+1}))-F(x_{i+1}, v_{i+1})|+\\operatorname{C} x_i^3\n\\leq |u(x_{i+1})- v_{i+1}|+\\operatorname{C} x_i^3.\n\\]\nBy induction, and equation \\eqref{eq:fixedset}, it follows\n$|u(x)-2A^{-1}x|\\leq \\operatorname{C}\\sum_{i=0}^\\infty x_i^3\\leq \\operatorname{C} x^2$.\n\nOn the other hand, given a different\npoint $z$, it holds\n\\[\n\\begin{split}\nu(x)-u(z)&=F(x_1,u(x_1))-F(z_1,u(z_1))\\\\\n&=\\lambda_u(x_1)^{-1}\\lambda_u(z_1)^{-1}\n\\left\\{(u(x_1)-u(z_1))+h'(x_1)-h'(z_1))\\right\\}.\n\\end{split}\n\\]\nIterating the above equation yields\n\\begin{equation}\\label{eq:unstreg}\n\\begin{split}\nu(x)-u(z)=&\\lambda_{u,n}(x)^{-1}\\lambda_{u,n}(z)^{-1}(u(x_n)-u(z_n))\\\\\n&\\ \\ +\\sum_{k=1}^{n}\\lambda_{u,k}(x)^{-1}\\lambda_{u,k}(z)^{-1}(h'(x_k)-h'(z_k))\n\\end{split}\n\\end{equation}\nwhere $\\lambda_{u,n}(x):=\\prod_{k=1}^{n}\\lambda_u(f_u^{-k}(x))$.\nNext, let $x=a$, then, accordingly to Lemma\n\\ref{lem:stab}, equation \\eqref{eq:fixedset} and Remark \\ref{rem:rev}, we have\n$x_i= A(i+c)^{-1}+\\Or((i+c)^{3\/2})$. This means that, in a\nsufficiently small neighborhood of zero, and for $z$ sufficiently\nclose to $x$ holds\n\\[\n\\begin{split}\n\\lambda_{u,n}(x)&\\geq\n\\prod_{k=1}^{n}\\left(1+v_k-\\operatorname{C} x_i^2\\right)\n\\geq \\prod_{k=1}^{n}\\left(1+\\frac 2{k+c}-\\operatorname{C} (k+c)^{-3\/2}\\right)\\\\\n&\\geq e^{\\sum_{k=1}^{n}\\frac{2}{k+c}-\\operatorname{C} (k+c)^{-3\/2}}\n\\geq \\operatorname{C} (A^{-1}an+1)^{2}\n\\end{split}\n\\]\nprovided $x$ is close  \nenough to zero. The same estimate holds for $\\lambda_{u,n}(z)$. \n\nThis implies that $u$ is continuous. Indeed, for each $\\ve>0$ choose\n$n_0(x)\\in\\N$ such that $\\lambda_{u,n_0(x)}(x_n)^{-1}\\leq \\ve$, then\n\\[\n|u(x)-u(z)|\\leq \\sum_{k=1}^{n_0(x)}|h'(x_k)-h'(z_k)|+\\ve \n\\]\nand we can thus choose $z$ close enough to $x$ such that\n$|u(x)-u(z)|\\leq 2\\ve$. Note that this implies the continuity of the\n$\\lambda_{u,n}$ as well.\n\nTo conclude, we choose $n(z)$ such that $\\operatorname{C}(A^{-1}an(z)+1)^{-4}\\geq\n|x-z|^{1+\\alpha}$, for some $\\alpha>0$, \naccordingly \n\\[\nu(x)-u(z)=\\sum_{k=1}^{n(z)}\\lambda_{u,k}(x)^{-1}\n\\lambda_{u,k}(z)^{-1}(h'(x_k)-h'(z_k))+{\\mathcal\nO}(|x-z|^{1+\\alpha})\n\\]\nSince the series is uniformly convergent we have\n\\begin{equation}\\label{eq:unstder}\nu'(x)=\\sum_{k=1}^{\\infty}\\lambda_{u,k}(x)^{-3}h''(x_k)\n\\end{equation}\nfrom which the lemma follows.\\footnote{Remark that to obtain the \nresult on a larger neighborhood it suffices to iterate the unstable  \nmanifold forward.}\n\\end{proof}\n\n\n\\begin{rem}\nAll the above results for the unstable manifold $\\gamma_u$ have the\nobvious counterpart for the stable manifold $\\gamma_s$ that can be\nreadily obtained via reversibility, see Remark \\ref{rem:rev}.\n\\end{rem} \n\n\n\\section{A narrower cone field}\\label{sec:narrow}\n \nHere our goal is to estimate the angle between stable and unstable manifolds.\n\n\nMore precisely, we wish to prove that there exists two constants $K_+,K_-\n\\in\\R^+$ such that the cone field\n$\\Co_*(\\xi):=\\{(1,u)\\in\\R^2\\;|\\;K_-(|x|+\\sqrt{|y|})\\leq u\\leq\nK_+(|x|+\\sqrt{|y|})\\}$ contains the unstable direction (by\nreversibility we can also define \nthe stable cone field $\\Co^-_*$). \n\n\\begin{prop}\\label{prop:dista}\nFor each $\\xi\\in\\To^2$ holds true $E^u(\\xi)\\in\\Co_*(\\xi)$,\n$E^s(\\xi)\\in\\Co_*^-(\\xi)$.\n\\end{prop}\n\nThe rest of the section is devoted to the proof of Proposition\n\\ref{prop:dista}.\n\nClearly a problem arises only in a neighborhood of zero. Accordingly\nthe first step is to gain a better understanding of the dynamics near\nzero.  \n\n\n\\subsection{Near fixed point dynamics}\\label{sub:nearfix}\nFor each $\\delta>0$ let $Q_\\delta:=[-\\delta,\\delta]^2$ be a square\nneighborhood of zero. The manifolds of the fixed point divide\n$Q_\\delta$ into four sectors: two thin and two fat (see Figure\n\\ref{fig:man}). We will discuss explicitly the dynamics in the two\nsectors below the unstable manifold (the other two being identical by\nsymmetry).\n\n\\begin{lem}\\label{lem:zdyn1}\nFor each $(x,y)\\in\\To^2$, $y\\leq \\gamma_u(x)$, let\n$(x_n,y_n):=T^n(x,y)$, then it holds true \n\\[\nx_n\\leq f_u^n(x)\\quad \\forall n\\geq 0.\n\\]\n\\end{lem}\n\\begin{proof}\nFirst note that the trajectory will always remain below the unstable\nmanifold. Hence, by induction,\n\\[\n\\begin{split}\nx_{n+1}&=x_n+h(x_n)+y_n\\leq x_n+h(x_n)+\\gamma_u(x_n)\\\\\n&\\leq f_u^n(x)+\nh(f_u^n(x))+\\gamma_u( f_u^n(x))=f_u^{n+1}(x). \n\\end{split}\n\\]\n\\end{proof}\nThe above lemma will suffice to control the dynamics in the thin\nsector, more work is needed for the fat one. In fact, when the\ntrajectories are close to the stable or the unstable manifolds the\nabove result can still be used (possibly remembering reversibility). On the\nother hand when the trajectory is close enough to zero its behavior\nis drastically different from the one on the invariant manifolds.\n\nTo define more precisely the meaning of ``close to zero'' let us\nintroduce the parabolic sector $P_M:=\\{(x,y)\\in Q_1\\;|\\;|y|\\geq Mx^2\\}$.\nWe consider a backward trajectory starting from $x\\leq 0$, $y\\leq\n\\gamma_u(x)$, the other possibilities follow by reversibility. Let, as usual, $(x_{n},\ny_n):=T^n(x,y)$, $n\\in\\Z$. Let  $m_+$ be the smallest integer for which\n$(x_{-n},y_{-n})\\in P_M$, $m$ the largest integer such that\n$x_{-m}\\leq 0$, and $m_-$ the largest integer for which\n$(x_{-n},y_{-n})\\in P_M$. Define, see \\eqref{eq:hamiltonian}, \n\\[\nE:=H(x_{-m},y_{-m}).\n\\]\nIn addition, define the function $\\Upsilon_E: [-1,1]\\to\\R^-$,\nby\\footnote{Computing for $y\\leq0$ yields\n\\[\n\\begin{split}\n\\Upsilon_E(x)&=-\\frac {h(x)}2-\\sqrt{\\frac{h(x)^2}4+2[G(x)+E-\\frac\n1{12}h(x)^2](1-\\frac 16 h'(x))}\\\\\n&=-\\sqrt{2(E+G(x))}(1+\\Or(x^2))+\\Or(x^3).\n\\end{split}\n\\]\n}\n\\begin{equation}\\label{eq:upsi}\nH(x,\\Upsilon_E(x))=E.\n\\end{equation}\nThen, by \\eqref{eq:appham}, and since $|y_{-n}|\\geq \\operatorname{C} x_{-n}^2$,\n\\[\n\\begin{split}\nH(x_{-n},y_{-n})-&H(x_{-n},\\Upsilon_E(x_{-n}))=H(x_{-n},y_{-n})-H(x_{-m},y_{-m})\\leq\n\\operatorname{C} \\sum_{k=n}^m y_{-k}^4\\\\\n&\\leq\\operatorname{C} \\sum_{k=n}^m |y_{-k}^3|(x_{-k-1}-x_{-k})\n\\leq  \\operatorname{C}|y_{-n}^3| |x_{-n}|. \n\\end{split}\n\\]\nAccordingly, for $n\\leq m$ it holds true\n\\begin{equation}\\label{eq:dyny}\n|y_{-n}-\\Upsilon_E(x_{-n})|\\leq \\operatorname{C}|y_{-n}|^2|x_{-n}|.\n\\end{equation}\n\\begin{lem}\\label{lem:zdyn2}\nIn the above described situation, setting $M=\\sqrt b$,  the following holds true\n\\begin{enumerate}\n\\item $f_u^{-n}(x)\\leq x_{-n}\\quad \\forall\\; n\\leq m$;\n\\item $f_s^{k}(x_{-n})\\geq x_{-n+k}\\quad \\forall\\;  n\\geq m_-,\\ k\\leq\n  n-m_-$;\n\\item $\\sqrt{E}\\leq |y_{-n}|\\leq 3\\sqrt{E}$ for all\n  $n\\in\\{m_+,\\dots, m_-\\}$;\n\\item $m_+\\leq 2A(Eb^{-1})^{-\\frac 14}$;\n\\item $ 2(12Eb)^{-\\frac 14}\\leq m_--m_+\\leq 4(Eb)^{-\\frac 14}$.\n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\nThe first fact is proven as in Lemma \\ref{lem:zdyn1}, the second\nfollows by reversibility. Hence, by\nthe results of section \\ref{sec:fix}, for $n\\leq m$, it follows \n\\begin{equation}\\label{eq:nest}\n|x_{-n}|\\leq |f_u^{-n}(x)|\\leq \\frac{2A|x|}{|x|n+A}.\n\\end{equation}\nNext we want to determine the points $x_{m_+}$ and $x_{m_-}$. The idea\nis to use $\\eqref{eq:dyny}$ that determines with good precision the\ngeometry of the trajectories. Let $\\bar x$ be defined by\n$\\Upsilon_E(\\bar x)=-M\\bar x^2$. Then\n\\[\n\\bar x=-\\left[\\frac{2E}{M^2-\\frac b2}\\right]^{\\frac 14}+\\Or(E^{\\frac\n  54}).\n\\]  \nOn the other hand, since by definition $|y_{-m_+}|\\geq Mx_{-m_+}^2$\nand $|y_{-m_++1}|\\leq Mx_{-m_++1}^2$, holds\n$|y_{-m_+}-Mx_{-m_+}^2|\\leq \\operatorname{C}|x_{-m_+}^3|$.\nHence, by \\eqref{eq:dyny} it follows \n\\[\n|x_{-m_+}-\\bar x|\\leq \\operatorname{C}\\frac{x_{-m_+}^3}{\\min\\{\\bar\nx,x_{-m_+}\\}}\\leq \\operatorname{C} \\frac{x_{-m_+}^3}{x_{-m_+}-|x_{-m_+}-\\bar x|}.\n\\]\nSolving the above inequality yields\n\\[\n|x_{-m_+}-\\bar x|\\leq\\operatorname{C} x_{-m_+}^2\\leq \\operatorname{C}\\bar x^2\\leq\\operatorname{C}\n\\sqrt E.\n\\]\nAnalogously $|x_{-m_-}+\\bar x|\\leq \\operatorname{C} \\sqrt E$.\n\\relax From this (3) and (4) easily follows. Finally,\n\\[\n2|\\bar x|\\geq |x_{m_-}-x_{m_+}|=|\\sum_{n=m_+}^{m_-}y_n|\\geq\n\\operatorname{C}(m_+-m_-)\\sqrt E.\n\\]\nWhich implies (5).\n\\end{proof}\n\nWe are now ready to refine our knowledge of the stable and unstable\ndirection. Let us fix $\\varrho\\in(0,1\/2)$.\n\n\\subsection{The cone field--Outside $Q_\\varrho$}\n\nThe general idea is to take the positive cone field $\\Co_+$ (which is\ninvariant and contains the unstable direction) and to push it forward\nin order to obtain a narrower cone \nfield. First of all outside $Q_{\\sqrt\\varrho}$ we have (see \\eqref{eq:unsdef})\n\\[\n1\\geq F(x,u)\\geq F(x,0)\\geq 2b \\varrho,\n\\]\nwhere we have chosen $\\varrho$ small enough.\nHence the cone field $\\Co_0=\\{(1,u)\\;|\\;1\\geq u\\geq 2b\\varrho\\}$ is invariant\noutside $Q_{\\sqrt\\varrho}$. It remains to understand what\nhappens in a neighborhood of the origin of order $\\sqrt\\varrho$. \n\nLet us define $\\bar u(\\xi)$ by the equation $u=F(\\xi,u)$. An easy\ncomputation shows\n\\[\n\\bar u(\\xi)= \\frac{-h'(x)+\\sqrt{h'(x)^2+4h'(x)}}2= \\sqrt{3b}|x|+\\Or(x^2).\n\\]\nBy reversibility we can restrict ourselves to the case $x\\geq 0$, in this\ncase the only possibility to enter the region $Q_{\\sqrt\\varrho}$ is\nvia the fourth quadrant. \n\nNote that $F(\\xi,u)\\geq u$ provided $0\\leq u\\leq \\bar u(\\xi)$.\nThis means that if $\\xi\\not\\in Q_{\\sqrt\\varrho}$ but $T\\xi\\in\nQ_{\\sqrt\\varrho}$, then the lower bound of the cone $D_\\xi T^n\\Co_0$\ndoes not decreases until $\\sqrt{3b}x_i\\leq 2b\\varrho$, where\n$(x_i,y_i):=T^i\\xi$. \n\nAccordingly, the cone field $\\Co_0$ is invariant\nalso in the fourth quadrant, outside the set $Q_{2\\sqrt{\\frac b3}\\varrho}$.\nNow consider the cone field $\\Co_1(\\xi):=\\{(1,u)\\;|\\;2b\\varrho\\leq\nu\\leq L \\bar u(\\xi)\\}$, $L=(3b\\varrho)^{-\\frac 12}$, for $\\xi\\in\nQ_{\\sqrt\\varrho}\\backslash Q_{2\\sqrt{\\frac b3}\\varrho}$. Clearly, if\n$|x|\\geq \\sqrt \\varrho$, then $L\\bar u(\\xi)\\geq 1$. Hence, as the\npoint enters $Q_{\\sqrt\\varrho}$, the image of $\\Co_0$ is contained in\n$\\Co_1$, moreover we have already seen that the lower bound is\ninvariant provided $\\xi_i\\not\\in Q_{2\\sqrt{\\frac b3}\\varrho}$. Let us follow the\nupper edge, if $u\\leq L\\bar u(\\xi_i)$, then\\footnote{Note that this\ncomputation holds for all $\\xi_i\\in Q_{\\sqrt{\\varrho}}\\backslash P_M$.}\n\\[\n\\begin{split}\nF&(\\xi_i,u)=F(\\xi_i,u)-F(\\xi_i,\\bar u(\\xi_i))+\\bar u(\\xi_i)\\\\\n&\\leq \\frac{(L-1)\\bar u(\\xi_i)}{(1+h'(x_i)+L\\bar u(\\xi_i))(1+h'(x_i)+\\bar\nu(\\xi_i))}+\\bar u(\\xi_i)\\\\\n&\\leq L\\bar u(\\xi_i)-(L^2-1)\\bar u(\\xi_i)^2+\\Or(x^3_i)\\\\\n&\\leq L\\bar u(\\xi_{i+1})+\\sqrt{3b}|y_{i+1}|-(L^2-1)\\bar u(\\xi_i)^2+\\Or(x^3_i)\\\\\n&\\leq L\\bar u(\\xi_{i+1}),\n\\end{split}\n\\]\nprovided $\\xi_{i+1}\\not\\in P_M$, with $M\\leq \\sqrt{3b}(L^2-1)$, which is\nfine provided $\\varrho$ is chosen small enough. The above discussion\ncan be summarized as follows.\n\\begin{lem}\n\\label{lem:conefield-one}\nThere exists $\\varrho>0$: For $\\xi\\not\\in Q_{2\\sqrt{\\frac b3}\\varrho}$\nthe unstable distribution is contained in $\\Co_0$. In addition, in the\nset $\\{\\xi=(x,y)\\in Q_{\\sqrt{\\varrho}}\\setminus (P_M\\cup\nQ_{2\\sqrt{\\frac b3}\\varrho})\\;:\\; xy\\leq 0\\}$ the unstable direction\nis contained in $\\Co_1$.\n\\end{lem}\n\nTo conclude we need to study what happens in a neighborhood of the\norigin of order $\\varrho$. It is necessary to \ndistinguish two possibilities: one can enter below the stable\nmanifold, and hence be confined in the fat sector, or one can enter\nabove the stable manifold, thereby being bound to the thin sector.\nWe will start with the easy case: the second.\n \n\\subsection{The cone field--Thin sector}\n\nIf $x>0$, as soon as the trajectory, at some time $n$, enters\n$Q_{2\\sqrt{\\frac b3}\\varrho}$ we have that \n$(1,u)\\in\\Co_0$ implies $u\\geq 2b\\varrho\\geq \\gamma_u'(x_n)$.\nLet us consider in $Q_{2\\sqrt{\\frac b3}\\varrho}$ the cone field\n$\\Co_2:=\\{(1,u)\\;|\\; L\\bar u(\\xi)\\geq u(x)\\geq \\gamma_u'(x)\\}$. Note\nthat, upon entering in $Q_{2\\sqrt{\\frac b3}\\varrho}$ such a cone contains \n$\\Co_1$.\\footnote{Note that, in such a case, the trajectory cannot enter in $P_M$.}\nNow\n\\[\nF(x,u)\\geq F(x,\\gamma_u'(x))=\\gamma_u'(x+h(x)+\\gamma_u(x))\\geq\n\\gamma_u'(x+h(x)+y), \n\\]\nwhere we have use that $\\gamma_u''(x)\\geq 0$ for $x\\in[0,\\varrho]$,\nprovided $\\varrho$ has been chosen small enough.\n\\begin{lem}\n\\label{lem:conefield-two}\nIn the region $Q_{2\\sqrt{\\frac b3}\\varrho}\\setminus(\nP_M\\cup\\{\\xi=(x,y)\\in\\To^2\\;:\\; y\\geq \\gamma^u(x) \\text{ for }x>0;\ny\\leq \\gamma^u(x) \\text{ for }x<0\\})$ the\nunstable direction is contained in the cone field $\\Co_2$. \n\\end{lem}\nNote that the above lemma suffices for trajectories in the thin sector.\nThe situation it is not so simple in the fat sector since the lower\nbound would deteriorate to zero. A more detailed analysis is needed.\n\nFor each $\\xi=(x,y)\\in\\To^2$, for which the unstable direction is defined,\nlet $(1,u(\\xi))$ be the vector in the unstable direction. Define then\n$\\lambda_{u,n}(\\xi, u)$ and $F_n(\\xi,u)$ as in formulae \\eqref{eq:unsdef}\nand \\eqref{eq:unstreg} and similarly define the stable\nquantities. That is\n\\begin{equation}\\label{eq:constex}\n\\begin{split}\n&D_{T^{-n}\\xi} T^{n}(1,u)=:\\lambda_{u,n}(\\xi,u)(1,F_n(\\xi,u))\\\\\n&D_{\\xi} T^{-n}(1,-v)=:\\mu_{s,n}(\\xi,v)(1,-F^-_n(\\xi,v))\n\\end{split}\n\\end{equation}\n\n\\subsection{The cone field--Fat sector}\nFirst of all notice that the trajectory can enter\n$Q_{2\\sqrt {\\frac b3} \\varrho}$ either in $P_M$ or outside.\nSince the cone field $\\Co_2$ for $x\\geq\n0$, $\\xi\\not\\in P_M$ contains the unstable vector (Lemma\n\\ref{lem:conefield-two}), we have a good control on the \nunstable vector in both cases until we enter in $P_M$. \nUpon entering $P_M$, we will obtain a very sharp control on the\nevolution of the edges of the cone. Let $\\xi\\not\\in P_M$, $T\\xi\\in\nP_M$, and let $\\ell_+-1>0$ be the smallest integer such that\n$\\xi_n\\not\\in P_M$. By equation \\eqref{eq:unsdef}, we have\n\\[\nu_n:=F_n(\\xi_n,u)=\\sum_{i=1}^{n}\n\\lambda_{u,i}(\\xi_{n-i},u_{n-i})^{-1}h'(x_{n-i})+\\lambda_{u,n}(\\xi_n,u)^{-1}\nu. \n\\]\nThen, for each $n< \\ell_+$, holds true\n\\[\nu_n\\leq \\sum_{i=1}^n h'(x_{n-i})+u\\leq \\frac {\\operatorname{C} }{M} \n\\left|\\sum_{i=1}^n y_i\\right|+u. \n\\]\nBy Lemma \\ref{lem:zdyn2}-(3),(5), it follows that\nwe have, for $u\\in\\Co_2(\\xi)$,\n\\[\nu_n\\leq \\operatorname{C}_+\\sqrt{|y_n|}.\n\\]\nMoreover, remembering \\eqref{eq:unsdef} and that  $u\\in\\Co_2(\\xi)$, yields\n\\[\nu_n\\geq e^{-2n\\operatorname{C}_+\\sqrt{|y_n|}}u\\geq \\operatorname{C} u\\geq \\operatorname{C}_-\\sqrt{|y_n|}. \n\\]\nConsequently, if for $\\xi=(x,y)$ we define the cone\n$\\Co_3(\\xi)=\\{\\operatorname{C}_-\\sqrt{|y|}\\leq u\\leq \\operatorname{C}_+\\sqrt{|y|}\\}$., then the\nabove results can be written as follows.\n\\begin{lem}\n\\label{lem:conefield-three}In $P_M$ the unstable direction\nis contained in the cone field $\\Co_3$.\n\\end{lem}\nFinally we have to follow the trajectory outside $P_M$ until it exits\nfrom $Q_{2\\sqrt{\\frac b3}\\varrho}$. The upper bound can be treated as\nbefore. Not so for the lower bound.\n\nLet $\\xi=(x,y)$ be a point in the\nfat sector, $x\\leq 0$, $x_{-1}\\geq 0$. Then, remembering subsection\n\\ref{sub:nearfix}, let $E:=H(x,y)$, $u_0=0$ and $u_{n+1}:=F(x_n,\nu_n)$. Clearly, $D_{(x,y)}T^n\\Co_+\\subset\\{(1,u)\\in\\R^2\\;|\\; u\\geq\nu_n\\}\\cup\\{0\\}$.\n\\begin{lem}\n\\label{lem:flowdir}\nIn the situation described above, for each $n\\in\\N$, holds true\n\\[\nF(x_n,\\Upsilon_E'(x_n))-\\Upsilon_E'(x_{n+1})=\\Or(|y_n|^{3\/2}).\n\\]\n\\end{lem}\n\\begin{proof}\nNotice that, since the trajectory lies below the unstable manifold, \n$|y|\\geq \\operatorname{C} x^2$. It is then convenient to keep track of the orders\nof magnitude only in terms of powers of $y$.\n\\[\nF(x_n,\\Upsilon_E'(x_n))=\\Upsilon_E'(x_n)-\\Upsilon_E'(x_n)^2\n+h'(x_n)+\\Or(|y_n|^{3\/2}).\n\\]\nOn the other hand, differentiating \\eqref{eq:upsi}, one gets \n\\[\n\\Upsilon_E'(x)=\\frac{h(x)}{\\Upsilon_E(x)}\n-\\frac{h(x)^2}{2\\Upsilon_E(x)^2} -\\frac{h'(x)}2+\\Or(|\\Upsilon_E|^{3\/2}). \n\\]\nAccordingly, by \\eqref{eq:dyny}, \n\\[\n\\begin{split}\n\\Upsilon_E'(x_{n+1})&=\\frac{h(x_n)+h'(x_n)\\Upsilon_E(x_n)}{\\Upsilon_E(x_n)+h(x_n)}\n-\\frac{h(x_n)^2}{2\\Upsilon_E(x_n)^2}-\\frac{h'(x_n)}2 \n+\\Or(|y_n|^{3\/2})\\\\\n&=\\Upsilon_E'(x_{n})-\\Upsilon_E'(x_{n})^2 +h'(x_n)+\\Or(|y_n|^{3\/2}),\n\\end{split}\n\\]\nfrom which the Lemma easily follows.\n\\end{proof}\nSince $F$ is a contraction in $u$, \nwe can estimate\n\\[\n\\begin{split}\n|u_n-\\Upsilon_E'(x_n)|&=|F(x_{n-1}, u_{n-1})-F(x_{n-1},\\Upsilon_E'(x_{n-1}))|\n+\\Or(|y_{n-1}|^{3\/2})\\\\\n&\\leq |u_{n-1}-\\Upsilon_E'(x_{n-1})|+\\Or(|y_{n-1}|^{3\/2})\\\\\n&\\leq|\\Upsilon_E'(x_0)|+\\Or\\left(\\sum_{k=0}^{n-1}|y_k|^{3\/2}\\right)\\\\\n&=\\Or\\left(|y_0|+\\sum_{k=0}^{n-1}\\sqrt{|y_k|}(x_{k}-x_{k+1})\\right)\n=\\Or(|y_n|).\n\\end{split}\n\\]\nWe have thus proved that there exists a constant $\\operatorname{C}_0>0$ such that\n\\begin{equation}\\label{eq:fatcone}\nu_n\\geq \\Upsilon_E'(x_n)-\\operatorname{C}_0\\Upsilon_E(x_n).\n\\end{equation}\nHence outside $P_M$ the image of the cone will belong to the\ncone field $\\Co_3:=\\{(1,u)\\in\\R^2\\;|\\; u(\\xi)\\geq\n\\Upsilon'_{E(\\xi)}(x)-\\operatorname{C}_0 \\Upsilon_{E(\\xi)}(x)\\}$. Note that, upon exiting $P_M$,\n$\\Upsilon_E'(x_n)-\\operatorname{C}_0\\Upsilon_E(x_n)\\geq \\operatorname{C}_-'\\sqrt{|y_n|}$,\nprovided $\\varrho$ is chosen small enough.\nThe Proposition follows by choosing $\\varrho$ small enough and\nremembering Lemmata \\ref{lem:conefield-one}, \\ref{lem:conefield-two}\nand \\ref{lem:conefield-three}.\n\n\n\\section{An a priori expansion bound}\n\\label{sec:expansion}\n\nThe results of the previous section allow to obtain the following nice\nestimate on the expansion in the system.\n\\begin{lem}\\label{lem:expansion}\nThere exists $K>0$ such that, for each\n$\\xi=(x,y)\\in\\To^2\\backslash \\{0\\}$, $n\\in\\N$ and\n$(1,u)\\in\\Co_*(T^{-n}\\xi)=\\Co_*(\\xi_{-n})$, holds true \n\\[\n\\lambda_{u,n}(\\xi,u)\\geq e^{-K|x|} \\left(K^{-1}\n|x|n+1\\right)^2 . \n\\]\n\\end{lem}\n\\begin{proof}\nLet us fix $\\delta>0$. On the one hand, if the trajectory lies outside\nof $Q_\\delta$, then \nwe have an exponential expansion, on the other hand, if the backward\ntrajectory enjoys $|x_{-n}|\\geq |x_0|$, then equation \\eqref{eq:unsdef}\nand Proposition \\ref{prop:dista} imply\n\\begin{equation}\n\\label{eq:trivialexp}\n\\lambda_{u,n}(\\xi,u)\\geq (1+K_-|x_0|)^n\\geq  e^{-K|x_0|} \\left(K^{-1}\n|x_0|n+1\\right)^2 .\n\\end{equation}\nWe say that  the backward orbit of $\\xi$ (up to time $n$)  passes $p$ times \nthru\n$Q_\\delta$ if $\\{0\\leq k\\leq n\\;:\\;\\xi_{-k}\\in Q_\\delta\\}$ consists of\n$p$ intervals. The Lemma holds for orbits that pass zero-times thru\n$Q_\\delta$. Suppose it holds for orbits that pass $p$ times. Let\n$\\xi_{-n}\\in Q_\\delta$ and let $m<n$ be the last time $\\xi_{-m}\\not\\in\nQ_\\delta$ but it passed already $p$ times in $Q_\\delta$. Moreover,\nsuppose that the Lemma holds in $Q_{2\\delta}$. Accordingly, for each\n$n\\geq l$ such that $\\xi_{-n}\\in Q_\\delta$ holds\n\\[\n\\begin{split}\n\\lambda_{u,n}(\\xi,u)&\\geq e^{-K|x_0|} \\left(K^{-1}\n|x_0|m+1\\right)^2e^{-2K\\delta} \\left(2K^{-1}\n\\delta(n-m)+1\\right)^2\\\\\n&\\geq e^{-K|x_0|} \\left(K^{-1}\n|x_0|m+1\\right)^2\\left(K^{-1}\\delta(n-m)+1\\right)^2\\\\\n&\\geq e^{-K|x_0|} \\left(K^{-1}|x_0|n+1\\right)^2,\n\\end{split}\n\\]\nprovided $\\delta$ has been chosen small enough and since it must be $n-m\\geq \\operatorname{C}\n\\delta^{-1}$.  \nThus to prove the Lemma it suffices to prove it for the pieces of\ntrajectories in $Q_\\delta$. There are two cases: a trajectory enters in the thin\nsector or in the fat one. Let us consider the thin sector first. \nSet $u_{-j}:=F_{n-j}(\\xi_{-j},u)$.\nBy the usual distortion estimates follows\n\\[\n\\begin{split}\n\\lambda_{u,n}(\\xi,u)&=\\prod_{j=1}^n(1+h'(x_{-j})+u_{-j})\\geq\n\\prod_{j=1}^n(1+h'(x_{-j})+\\gamma_u'(x_{-j}))\\\\\n&\\geq\n\\prod_{j=1}^n(1+h'(f_u^{-j}(x))+\\gamma_u'(f_u^{-j}(x)))\n=\\prod_{j=1}^n(f_u)'(f_u^{-j}(x))\\\\\n&\\geq\\prod_{j=1}^n\\frac\n{|f_u^{-j+1}(x)-f_u^{-j}(x)|}{|f_u^{-j}(x)-f_u^{-j-1}(x)|}\ne^{-\\operatorname{C}|f_u^{-j+1}(x)-f_u^{-j}(x)|}\\\\\n&\\geq e^{-\\operatorname{C}|x_0|}\\frac{|x_0|^2}{|f_u^{-n}(x_0)|^2}.\n\\end{split}\n\\]\nNow, notice that $f_u^{-1}(x)\\leq \\frac{x}{1+\\operatorname{C} x}$, hence\n$f_u^{-n}(x_0)\\leq \\frac{|x_0|}{1+\\operatorname{C} n|x_0|}$. Thus,\n\\begin{equation}\n\\label{eq:bah}\n\\lambda_{u,n}(\\xi,u)\\geq e^{-\\operatorname{C}|x_0|}(1+n\\operatorname{C}|x_0|)^2.\n\\end{equation}\n\\relax For the fat sector we need only to consider the cases in which $x_0\\not\\in\nQ_\\delta$ and $x_0\\in Q_\\delta$, $x_0\\leq 0$ since if $x_0>0$ the\nbackward trajectory increases the $x$ coordinate. In such cases we\nhave\\footnote{Again, $E$ is chosen to be the energy associated\nto the point of the orbit closer to the origin.} \n\\[\n\\begin{split}\n\\lambda_{u,n}(\\xi,u)&=\\prod_{j=1}^n(1+h'(x_{-j})+u_{-j})\\geq\n\\prod_{j=1}^n(1+\\Upsilon_E'(x_{-j})-\\operatorname{C}_0\\Upsilon_E(x_{-j}))\\\\\n&\\geq e^{\\sum_{j=1}^n\\Upsilon_E'(x_{-j})-2\\operatorname{C}_0\\Upsilon_E(x_{-j})}\\\\\n&\\geq e^{-\\sum_{j=1}^n\\frac{\\Upsilon_E'(x_{-j})}{\\Upsilon_E(x_{-j})}\n(x_{-j-1}-x_{-j}) -3\\operatorname{C}_0\\sum_{j=1}^n(x_{-j-1}-x_{-j})} \\\\\n&\\geq e^{-\\operatorname{C}|x_0|}e^{-\\int_{x_0}^{x_{-n}}\\frac{\\Upsilon_E'(z)}{\\Upsilon_E(z)}dz}\n= e^{-\\operatorname{C}|x_0|}\\frac{\\Upsilon_E(x_0)}{\\Upsilon_E(x_{-n})}.\n\\end{split}\n\\]\n\nLet $n_*\\in\\N$ be the last integer for which $|G(x_{-n})|\\geq\nE$, then for $n\\leq n_*$ we have \n\\[\n\\lambda_{u,n}(\\xi)\\geq e^{-\\operatorname{C}|x_0|}\\sqrt{\\frac\n{E+G(x_0)}{E+G(x_{-n})}} \n\\geq e^{-\\operatorname{C}|x_0|}\\sqrt{\\frac\n{G(x_{-n})+G(x_0)}{2G(x_{-n})}}.\n\\]\nOn the other hand comparing the backward motion with the\nbackward motion on the stable manifold, as we did before with the\nunstable,\\footnote{Here we use the inequality  \n\\[\n\\sqrt{\\frac{1+(1+a)^4}2}\\geq (1+\\frac a2)^2.\n\\]\n}\n\\[\n\\lambda_{u,n}(\\xi,u)\\geq\ne^{-\\operatorname{C}|x_0|}\\sqrt{\\frac{1+(n|x_0|\\operatorname{C}+1)^4}2}\n\\geq e^{-\\operatorname{C}|x_0|}(1+n|x_0|\\operatorname{C})^2.\n\\]\nNext, let us consider $n\\in\\{n_*,\\dots, m\\}$, where $m$ is the larger\ninteger such that $x_{-m}\\leq 0$, we have $2\\sqrt E\\geq \n\\Upsilon_E(x_{-n})\\geq \\sqrt {2E}$. \n\\[\n\\begin{split}\n\\lambda_{u,n-n_*}(\\xi,u)\n&\\geq e^{-\\operatorname{C}_2|x_{n_*}|}\ne^{-\\int_{x_{-n_*}}^{x_{-n}}\\frac{\\Upsilon_E'(z)}{\\Upsilon_E(z)}dz} \n\\geq e^{-\\operatorname{C}|x_0|}\\frac{\\Upsilon_E(x_{-n_*})}\n{\\Upsilon_E(x_{-n})}\\\\\n&\\geq  e^{-\\operatorname{C}|x_0|}\\sqrt{\\frac 32}\\geq e^{-\\operatorname{C}|x_0|}(1+\\operatorname{C}\n|x_{-n_*}|(n-n_*))^2, \n\\end{split}\n\\]\nwhere, in the last line, we used Lemma \\ref{lem:zdyn2}-(5).\nBy symmetry it will be enough to wait another time $m$ to have\n$|x_{-2m}|\\geq \\frac 12|x_0|$, after which the expansion is assured by\nthe estimate \\eqref{eq:trivialexp}.\n\\end{proof}\n\nNext we need to have similar estimates for the stable contraction. By\n\\eqref{eq:constex} \n\\begin{equation}\\label{eq:stabdef}\n\\begin{split}\n\\mu_s(v)&:=1+v=\\mu_{s,1}(\\xi,v)\\\\\nF^-_1(\\xi,v)&=h'(x-y)+\\frac v{1+v}.\n\\end{split}\n\\end{equation}\nIt is immediate to check that\n$D_{(x,y)}T^{-1}(1,-v)=\\mu_s(v)(1,-F^-_1((x,y),v))$ and\n$\\mu_{s,n}(\\xi,v_{0}):=\\prod_{i=0}^{n-1}\\mu_s(v_{-i})$, where \n$v_0=v$ and $v_{-i-1}:=F^-_1(T^{-i}\\xi,v_{-i})$.\n\nAn interesting way to transform information on expansion into information on\ncontraction is to use area preserving. \n\\begin{lem}\\label{lem:areap}\nLet $\\xi\\in\\To^2$, then for each $n\\in\\N$, $u,v\\geq 0$ let $u_{-n}=u$,\n$v_0=v$, $\\xi_{-n}=T^{-n}\\xi$ and $u_{-k+1}=F(\\xi_{-k+1}, u_{-k})$,\n$v_{-k-1}=F^-(\\xi_{-k},v_{-k})$. Then\n\\[\n\\mu_{s,n}(\\xi, v_0)(v_{-n}+u_{-n})=\\lambda_{u,n}(\\xi, u_{-n})(u_0+v_0).\n\\]\n\\end{lem}\n\\begin{proof}\nCalling $\\omega$ the standard symplectic form we have\n\\[\n\\begin{split}\n\\mu_{s,n}(\\xi,v_0)(v_{-n}+u_{-n})&=\\omega(D_{\\xi}T^{-n}(1,-v_0),(1,u_{-n}))\\\\\n&= \\omega((1,-v_0), D_{\\xi_{-n}} T^n(1,u_{-n}))=\\lambda_{u,n}(\\xi,\nu_{-n})(u_0+v_0). \n\\end{split}\n\\]\n\\end{proof}\nThe following is an immediate corollary of Lemmata \\ref{lem:areap} and\n\\ref{lem:expansion}. \n\\begin{cor}\\label{lem:contraction}\nFor each $\\xi=(x,y)\\in \\To^2$ and $n\\in\\N$  holds\n\\[\n\\mu_{s,n}(\\xi, v_0)\\geq e^{-\\operatorname{C}|x_{0}|} (\\operatorname{C}^{-1}|x_{0}|n+1)^2\\frac\n{u_0+v_0}{u_{-n}+v_{-n}} \\quad\\forall n\\in\\N.\n\\]\n\\end{cor}\n\nAll the other expansion estimates can be obtained by reversibility.\n\n\\section{Distributions--regularity}\n\\label{sec:regularity}\n\nLet $(1,u(\\xi)),\\,(1,-v(\\xi))$ be the unstable and stable directions,\nrespectively. We will then use the short hand\n$\\lambda_{u,n}(\\xi):=\\lambda_{u,n}(\\xi,u(\\xi_{-n}))$ and\n$\\mu_{s,n}(\\xi):=\\mu_{s,n}(\\xi,v(\\xi_{n}))$. \n\\begin{lem}\\label{lem:distreg0}\nThe unstable distribution is continuous in $\\To^2$.\n\\end{lem}  \n\\begin{proof}\nNotice that, for $\\xi=(x,y)$, $\\xi_n:=T^n\\xi$, iterating formula\n\\eqref{eq:unsdef}, in analogy with \\eqref{eq:unstreg}, holds true\n\\begin{equation}\\label{eq:unstreg-bis}\n\\begin{split}\nu(x)-u(z)=&\\lambda_{u,n}(x)^{-1}\\lambda_{u,n}(z)^{-1}(u(x_{-n})-u(z_{-n}))\\\\\n&\\ \\ +\\sum_{k=1}^{n}\\lambda_{u,k}(x)^{-1}\\lambda_{u,k}(z)^{-1}\n(h'(x_{-k})-h'(z_{-k}))\n\\end{split}\n\\end{equation}\nBy Lemma \\ref{lem:expansion}, we can take the limit $n\\to\\infty$ in\nthe above formula provided  $x\\neq 0$, and obtain a\nuniformly convergent series from which the continuity follows.\nIf $\\xi\\neq 0$ then $x_{-1}\\neq 0$ and \\eqref{eq:unsdef} implies\n\\begin{equation}\n\\label{eq:uma}\nu(\\xi)=\\lambda_{u}(\\xi_{-1})^{-1}h'(x_{-1})\n+\\lambda_{u}(\\xi_{-1})^{-1}u(\\xi_{-1}) ,\n\\end{equation}\nhence the continuity at $\\xi\\neq 0$ follows. We are left with the\ncontinuity at the origin, but this is already implied by Proposition\n\\ref{prop:dista}. \n\\end{proof}\n\nThis means that we can extend the invariant unstable distribution\n(that, up to now, where defined--by Pesin theory--only almost everywhere) to a\ncontinuous everywhere defined vector field. The same statement holds\nfor the stable vectors by reversibility.\n\nGiven a continuous vector field there exists integral curves. Since\nwe do not know yet if the vector fields are Lipschitz, it does not\nfollows automatically that from a given point there exits only one\nintegral curve, yet this follows by standard dynamical\narguments. Clearly such integral curves are nothing else than the\nstable and unstable manifolds that are therefore everywhere\ndefined. In addition, remember that, by general hyperbolic theory, the\nfoliations are absolutely continuous, it follows that the above\neverywhere defined foliations are continuous. Unfortunately, for the\nfollowing much sharper regularity information is  needed, this is\nobtained in the rest of the section.\n\nLet us call $\\partial^u, \\partial^s$ the derivative along the unstable\nand the stable vector fields, respectively. \n\n\\begin{lem}\\label{lem:uderbound}\nThe vector field $u$ is $\\Co^1$ along the unstable manifolds, apart from the\norigin, moreover\n\\[\n|\\partial^u\nu(\\xi)|\\leq C\\quad\n\\forall \\xi\\neq 0. \n\\]\n\\end{lem}\n\\begin{proof}\nIf $\\xi$ is outside of a \nneighborhood of the origin of size $\\delta$, then by Lemma\n\\ref{lem:expansion}, \\eqref{eq:unstreg-bis} we have, in analogy with\nthe arguments leading to \\eqref{eq:unstder},\n\\begin{equation}\n\\label{eq:uder-u-bond}\n|\\partial^u\nu(\\xi)|=\\left|\\sum_{k=1}^{\\infty}\\lambda_{u,k}(x)^{-3}h''(x_k)\\right| \\leq \\operatorname{C}\n\\sum_{n=0}^{\\infty}(\\delta n+1)^{-6}\\leq \\operatorname{C}. \n\\end{equation}\nSince the series converges uniformly the $\\Co^1$ property follows. To\nobtain a uniform bound more work is needed.\nIf $|\\xi|<\\delta$, formula \\eqref{eq:uma} implies\n\\[\n|\\partial^u u(\\xi)|\\leq\\lambda_u(\\xi_{-1})^{-3}|h''(x_{-1})|+\n\\lambda_u(\\xi_{-1})^{-3} |\\partial^u u(\\xi_{-1})|=:\\Psi(\\xi_{-1},\n|\\partial^u u(\\xi_{-1})|).\n\\]\nA simple computation, remembering Proposition \\ref{prop:dista}, shows\nthat \n\\[\n\\Psi(\\xi,\\varrho)\\leq7b|x|+\\frac{\\varrho}{1+3|u(\\xi)|}\\leq\n7b|x|+\\frac{\\varrho}{1+3K_-|x|} \\leq \\varrho ,\n\\]\nprovided $\\varrho\\geq \\frac{7b(1+3K_-)}{3K_-}$. Accordingly, for $\\rho$\nlarge enough, we have \n$|\\partial^u u(\\xi)|\\leq \\rho$, for all $\\xi$. \n\\end{proof}\n\nIt remains to investigate the regularity of the unstable distribution\nalong the stable direction.\n\n\\begin{lem}\\label{lem:distreg1}\nThe unstable distributions are $\\Co^1$ along stable manifolds, apart\nfrom the origin. Moreover\n\\[\n|\\partial^s u(\\xi)|\\leq \\operatorname{C} \\quad \\forall \\xi\\neq 0.\n\\]\n\\end{lem}  \n\\begin{proof}\nLet us fix some arbitrary neighborhood of the origin.\nLet $x,z\\in W^s$ outside such a neighborhood. Let $W^s_0$ be the\npiece of stable manifold between such two points. Clearly\n$W^s_{n}:=T^nW^s_0$ grows for negative $n$. Let $n(x,z)$ be the largest\ninteger for which $|W^s_{-n}|\\leq |W^s_0|^{\\frac 14}$. Our first\nresult is a distortion bound.\n\\begin{sublem}\\label{slem:dist}\nFor each $n\\leq n(x,z)$ and $\\xi\\in\nW^s_{0}$, holds\n\\[\n\\operatorname{C}^{-1}\\frac{|W^s_{-n}|}{|W^s_{0}|}\\leq \\mu_{s,n}(\\xi)\\leq\n\\operatorname{C}\\frac{|W^s_{-n}|}{|W^s_{0}|} .\n\\]\n\\end{sublem}\n\\begin{proof}\nIf the backward orbit spends at least half of the time outside the\nneighborhood, then $W^s_{-n}$\ngrows exponentially fast, hence $n(x,z)\\leq \\operatorname{C}\\ln|W^s_0|^{-1}$ and\n$\\sum_{i=0}^{n(x,z)}|W^s_{-i}|\\leq \\operatorname{C}$. If this is not the case,\nthe worst possible situation is when $W^s_{-m}$ is the closest\nto the origin and all the trajectory lies in the neighborhood. In such a\ncase, letting $m:=n(x,z)$, \n\\[\n|W^s_{m}|=\\int_{W^s_0}\\mu_{s,m}(z)dz\\geq\n\\int_{W^s_0}\\frac{\\theta(z)(\\operatorname{C}^{-1}|z|m+1)^2}{2\\theta(T^{-m}z)} dz,\n\\]\nwhere $\\theta(\\zeta)=u(\\zeta)+v(\\zeta)$ is the separation between the\nstable and the unstable directions at the point $\\zeta$ and we have\nused Lemma \\ref{lem:contraction}. Now  Proposition \\ref{prop:dista} and\nLemma \\ref{lem:zdyn2}-(1) imply $\\theta(T^{-m}z)\\leq \\operatorname{C} m^{-1}$\noutside the parabolic sector, while Lemma \\ref{lem:zdyn2}-(3,4,5) show\nthat the same estimates remain in $P_M$ as well. Accordingly,\n\\[\n|W^s_{-m}|\\geq \\operatorname{C} m^3|W^s_0|.\n\\]\nThat is $m\\leq \\operatorname{C} |W^s_0|^{-\\frac 14}$, and\n\\[\n\\sum_{i=0}^{n(x,z)}|W^s_{-i}|\\leq n(x,z)|W^s_0|^{\\frac 14}\\leq \\operatorname{C}.\n\\]\nThe above estimate readily implies that, for each $\\xi,\\eta\\in W^0_s$,\n\\[\ne^{-\\operatorname{C} |W^s_{-i}|}\n\\leq\\frac{\\mu_s(\\xi_{-i},v(\\xi_{-i}))}{\\mu_s(\\eta_{-i},v(\\eta_{-i}))}\n\\leq e^{\\operatorname{C} |W^s_{-i}|},\n\\]\nwhere we have used Lemma \\ref{lem:uderbound} for the stable manifold.\nAccordingly,\n\\[\ne^{-\\operatorname{C}\\sum_{i=0}^m |W^s_{-i}|}\n\\leq\\frac{\\mu_{s,n}(\\xi)}{\\mu_{s,n}(\\eta)}\n\\leq e^{\\operatorname{C} \\sum_{i=0}^m|W^s_{-i}|},\n\\]\nfrom which the Lemma readily follows.\n\\end{proof}\nBy Lemma \\ref{lem:areap} it follows, letting again $m:=n(x,z)$,\n\\[\n\\begin{split}\n\\lambda_{u,m}(x)^{-1}\\lambda_{u,m}(z)^{-1}\n&=\\lambda_{u,m}(x)^{-1}\\sqrt{\\lambda_{u,m}(z)^{-2}}\\\\\n&\\leq\n\\operatorname{C}\\left(\\theta(x_{-m})\\mu_{s,m}(x)\n\\sqrt{\\lambda_{u,m}(z)\\theta(z_{-m})\\mu_{s,m}(z)}\\right)^{-1}\n\\end{split}\n\\]\nAs before the worst case is clearly when $W^s_{-m}$ is the closest to\nthe origin.\nIn such a case, consider that at least one of the two end points of\n$W^s_{-n(x,z)}$ must be at a distance $\\operatorname{C} |W^s_{-n(x,z)}|$ from the\nfixed point, let us say $T^{-n(x,z)}z$, hence\n$\\theta(T^{-n(x,z)}z)\\geq \\operatorname{C}|W^s_{-n(x,z)}|$. In addition,\n$\\theta(x_{-m})\\geq \\operatorname{C} m^{-1}$. Indeed, this follows from Lemma\n\\ref{lem:zdyn2}-(3,4,5) if the trajectory ends in $P_M$. If the\ntrajectory lies outside $P_M$ then it approaches the origin slower than\nthe dynamics $x-\\operatorname{C} x^2$, which implies $x_{-m}\\geq \\operatorname{C}\nm^{-1}$. Furthermore by using the above facts, Lemma \\ref{lem:expansion}, Sub-lemma\n\\ref{slem:dist} and the definition of the stopping time $m$ yields\n\\[\n\\lambda_{u,m}(x)^{-1}\\lambda_{u,m}(z)^{-1}\\leq \\operatorname{C}|x-z|^{\\frac\n34}m\\frac 1{m} \\sqrt{\\frac{|x-z|^{\\frac \n34}}{|x-z|^{\\frac 14}}} \\leq \\operatorname{C}|x-z|.\n\\]\nSince we know that $u$ is a uniformly continuous function it follows\n\\[\n\\lim_{z\\to\nx}\\lambda_{u,m}(x)^{-1}\\lambda_{u,m}(z)^{-1}\n\\frac{|u(T^{-m}x)-u(T^{-m}z)|} \n{|x-z|}=0.\n\\]\nAccordingly, by formula  \\eqref{eq:unstreg}, \n\\[\n\\begin{split}\nu'(x)=&\\lim_{z\\to x}\n\\sum_{n=0}^{m}\\lambda_{u,n}(x)^{-1}\\lambda_{u,n}(z)^{-1}\n\\frac{h'(T^{-n}x)-h'(T^{-n}z)}{x-z}\\\\\n=& \\lim_{z\\to x}\n\\sum_{n=0}^{m}\\lambda_{u,n}(x)^{-1}\\lambda_{u,n}(z)^{-1}\nh''(T^{-n}\\zeta_{n}) \\mu_{s,n+1}(\\zeta_n),\n\\end{split}\n\\]\nfor some $\\zeta_n\\in W_0$.\nBut $|h''(T^{-n}\\zeta_n)|\\leq \\operatorname{C}\\theta(T^{-n}z)$, hence\n\\[\n\\begin{split}\n&\\lambda_{u,n}(x)^{-1}\\lambda_{u,n}(z)^{-1}\\theta(T^{-n}x)\n\\mu_{s,n+1}(x)\\leq \\operatorname{C}\\lambda_{u,n}(z)^{-1}\\\\\n&\\lambda_{u,n}(x)^{-1}\\lambda_{u,n}(z)^{-1}\\theta(T^{-n}z)\n\\mu_{s,n+1}(z)\\leq \\operatorname{C}\\lambda_{u,n}(x)^{-1}.\n\\end{split}\n\\]\nRemembering Sub-Lemma \\ref{slem:dist} the uniform convergence of the\nseries follows and yields the formula\n\\begin{equation}\n\\label{eq:uder-s-bound}\n\\partial^s u(x)=\\sum_{n=0}^{\\infty}\\lambda_{u,n}(x)^{-2}\\mu_{s,n+1}(x)\nh''(T^{-n}x).\n\\end{equation}\nGiven the arbitrariness of the neighborhood of zero, the above formula\nholds for each $x\\neq 0$ and, since the series converges uniformly,\nthe $\\Co^1$ property follows. We can now conclude the Lemma.\nBy Lemma \\ref{lem:areap} follows\n\\[\n\\begin{split}\n|\\partial^s u(x)|&\\leq\\operatorname{C} \\sum_{n=0}^\\infty\n\\lambda_{u,n}(x)^{-2}\\mu_{s,n}(x)\\theta(T^{-n}x) \\\\\n&\\leq \\operatorname{C} \\sum_{n=0}^\\infty\\lambda_{u,n}(x)^{-1}\\theta (x)\n\\leq \\operatorname{C} \\sum_{n=0}^\\infty\\frac{|x|}{(|x|n+1)^2}\\leq \\operatorname{C} .\n\\end{split} \n\\]\n\\end{proof}\n\\begin{rem}\nNotice that the symmetrical statements follow by reversibility.\n\\end{rem}\nThe final result on the regularity of the foliations can be  stated as\nfollows.\n\\begin{lem}\n\\label{lem:c1-reg}\nThe stable and unstable vector fields are $\\Co^1(\\To^2\\setminus\\{0\\})$ and, more\nprecisely, for each $\\xi\\in\\To^2\\setminus\\{0\\}$,\n\\[\n|Du(\\xi)|\\leq \\operatorname{C} \\theta(\\xi)^{-1}.\n\\]\n\\end{lem}\n\\begin{proof}\nThe $\\Co^1$ property follows from Lemma 19.1.10 of \\cite{KH}.\nThen the size of the derivative can be easily estimated by the size of\nthe partial derivatives in the stable and unstable directions divided by\nthe angle between them.\n\\end{proof}\n\\begin{rem}\nIn fact, it is likely that with a little more work one can show that\nthe foliations are $\\Co^{\\frac 32-\\ve}$, but we do not investigate this\npossibility since it is not needed in the following.\n\\end{rem}\n\\section{Holonomy}\n\\label{sec:holo}\nThere exists $\\operatorname{C}_1>0$ such that, given two close by stable manifolds\n$W^s_1$, $W^s_2$ we can define the {\\em unstable holonomy}\n$\\Psi^u:W^s_1\\to W^s_2$ by $\\{\\Psi^u(\\xi)\\}:=W^u(\\xi)\\cap W^s_2$. \nLet $D_r:=\\{\\zeta=(z_1,z_2)\\in\\R^2\\;:\\;|z_1|\\leq r;|z_2|\\leq r^2\\}$.\n\\begin{lem}\\label{lem:holo}\nFor each $W^s_1,W^s_2$ disjoint from $D_r$ and $\\xi\\in W^s_1$, holds\n\\[\n|J\\Psi^u(\\xi)-1|\\leq \\operatorname{C} r^{-1}\\|\\Psi^u(\\xi)-\\xi\\|.\n\\]\nProvided $\\|\\Psi^u(\\xi)-\\xi\\|\\leq \\operatorname{C}_1 r$.\n\\end{lem}\n\\begin{proof}\nLet $\\gamma_s,\\tilde\\gamma_s:[-\\delta,\\delta]\\to\\R^2$ be $W^s_1,\nW^s_2$, respectively, parametrized by arc-length. Also, let\n$\\Gamma:[-\\delta,\\delta]^2\\to\\R^2$, be such that $\\Gamma(0,0)=\\xi$,\n$\\Gamma(s,0)=\\gamma_s(s)$ \nand $\\Gamma(s,t)$ be the unstable manifold, parametrized by\narc-length, of $\\Gamma(s,0)$ and, finally,\n$\\Gamma(0,\\rho):=\\Psi^u(\\xi)$. Note that $\\Gamma(s,t)$ can be obtained\nintegrating the unstable vector field starting from $\\Gamma(s,0)$,\nhence Lemma \\ref{lem:c1-reg} and the standard results on the continuity\nwith respect to the initial data imply $\\Gamma\\in\\Co^1$. By the\ntransversality of the stable and unstable manifolds there exist\n$\\tau,\\sigma:[-\\delta,\\delta]\\to\\R$ such that\n$\\Gamma(s,\\tau(s))=\\Psi^u(\\gamma_s(s))=\\tilde\\gamma(\\sigma(s))\\in\nW^s_2$. Calling $\\eta(s)$ the unit vector perpendicular to $\\tilde\n\\gamma'(s)$, by the implicit function theorem, it follows \n\\begin{equation}\n\\label{eq:implicit}\n\\begin{split}\n\\tau'(s)&=-\\frac{\\langle \\eta(s),\\partial^s\\Gamma(s,\\tau(s))\\rangle}\n{\\langle\\eta(s), \\partial_t\\Gamma(s,\\tau(s))\\rangle}\\\\\n\\sigma'(s)&=\\langle\\tilde\\gamma_s'(s),\\partial^s\\Gamma(s,\\tau(s))\\rangle\n-\\frac{\\langle\\tilde \n\\gamma'_s(s),\\partial_t\\Gamma(s,\\tau(s))\\rangle\\;\\langle\n\\eta(s),\\partial^s\\Gamma(s,\\tau(s))\\rangle} \n{\\langle\\eta(s), \\partial_t\\Gamma(s,\\tau(s))\\rangle}\n\\end{split}\n\\end{equation}\nwhere, clearly, $\\sigma'(s)=J\\Psi^u(\\gamma_s(s))$. Calling\n$v^u(\\eta)$, $\\eta\\in\\To^2$,\nthe unit vector in the unstable direction at $\\eta$ and $v^s(\\eta)$ the\nstable one, one has\n$\\partial_t\\Gamma(s,\\tau(s))=v^u(\\Gamma(s,\\tau(s)))$. On the other\nhand, setting $V(s,t):=\\partial^s\\Gamma(s,t)-v^s(\\Gamma(s,t))$, holds\n$V(s,t)=0$ for $t=0$, but for $t\\neq 0$, in general, it will be\n$V(s,t)\\neq 0$. Yet, it is possible to estimate it by differentiating\n$\n\\Gamma(s,t)=\\Gamma(s,0)+\\int_0^tv^u(\\Gamma(s,t'))dt'\n$\nwhich yields\n\\[\n\\partial^s\\Gamma(s,t)=v^s(\\Gamma(s,0))+\\int_0^t Dv^u(\\Gamma(s,t'))\n\\partial^s\\Gamma(s,t')dt'.\n\\]\nLemmata \\ref{lem:c1-reg} and \\ref{lem:distreg1}\nimply that $\\|Dv^u\\|\\leq \\operatorname{C} r^{-1}$ and  $\\|Dv^u v^s\\|=|\\partial^s\nv^u|\\leq \\operatorname{C}$, hence \n\\[\n\\|V(s,t)\\|\\leq \\operatorname{C} r^{-1}\\int_0^t\\|V(s,t')\\|dt'+\\operatorname{C} t.\n\\]\nBy Gronwal, it follows,  provided $t\\leq \\operatorname{C} \\rho$ and $\\rho\\leq \\operatorname{C}_1\nr$, for $\\operatorname{C}_1$ small enough,\n\\begin{equation}\n\\label{eq:Vest}\n\\|V(s,t)\\|\\leq \\operatorname{C} t.\n\\end{equation}\nAccordingly, by the second of \\eqref{eq:implicit} and \\eqref{eq:Vest},\nit follows\n\\[\n|\\sigma'(0)-1|\\leq \\operatorname{C} r^{-1}\\rho.\n\\]\n\\vskip-.5cm\n\\end{proof}\n\n\\section{Random perturbations}\n\\label{sec:random}\n\nThe density of a measure with respect to Lebesgue evolves as\n\\[\n\\Lp f:=f\\circ T^{-1}.\n\\]\nWe will then construct a random perturbation by introducing the\nconvolution operator\n\\begin{equation}\\label{eq:conv}\nQ_\\ve f(x):=\\int_{\\To^2}q_\\ve(x-y)f(y)dy.\n\\end{equation}\nWhere we assume\n\\begin{enumerate}\n\\item $q_\\ve(\\xi):=\\ve^{-2}\\bar q(\\ve^{-1}\\xi)$; $\\bar q\\in\\Co^{\\infty}(\\R^2,\\R^+)$;\n\\item $\\int_{\\R^2}\\bar q(\\xi)d\\xi=1$;\n\\item $\\bar q(\\xi)=0$ for each $\\|\\xi\\|\\geq 1$;\n\\item $\\bar q(\\xi)=1$ for each $\\|\\xi\\|\\leq \\frac 12$.\n\\end{enumerate}\nWe define then\n\\begin{equation}\\label{eq:random}\n\\Lp_\\ve:=Q_\\ve\\Lp^{n_\\ve},\n\\end{equation}\nwhere $n_\\ve$ will be chosen later.\n\nNotice that \n\\begin{equation}\\label{eq:ker1}\n\\Lp_\\ve^2f(x)=\\int_{\\To^4} q_\\ve(x-y)q_\\ve(T^{-n_\\ve}y-T^{n_\\ve}z)f(z)\ndz dy :=\\int_{\\To^2}\\Ka_\\ve(x,z)f(z)dz.\n\\end{equation}\n\nWe have thus a kernel operator that can be investigated with rather\ncoarse techniques. It turns out to be convenient to define the\nassociated kernel\n\\begin{equation}\\label{eq:kernel2}\n\\bar\\Ka_\\ve(x,z):=\\Ka_\\ve(x,T^{-n_\\ve}z)=\\int_{\\To^2}q_{\\ve}(x-y)\nq_\\ve(T^{-n_\\ve}y-z)m(dy). \n\\end{equation}\nFor further use let us define\n\\begin{equation}\\label{eq:levelset}\n\\begin{split}\n&D_r:=\\{z=(z_1,z_2)\\in\\To^2\\;|\\;|z_1|\\leq r;\\; |z_2|\\leq r^2\\}\\\\\n&B_r(\\xi):=\\{\\eta\\in\\To^2\\;;\\;\\|\\xi-\\eta\\|<r\\}.\n\\end{split}\n\\end{equation}\nThe following is a relevant fact used extensively in the sequel.\n\\begin{lem}\n\\label{lem:levelset-escape}\nThere exists $\\operatorname{C}_3,R>0$ such that, for each $\\delta<R^2$, if\n$B_\\delta(\\xi)\\subset D_{R\/2}$, then there exists $\\eta\\in\nB_{\\frac 34\\delta}(\\xi)$ such that $T^n B_{\\frac \\delta 4} (\\eta)\\cap\nD_{R}=\\emptyset$ for some $n\\leq \\operatorname{C}_3 \\delta^{-\\frac 12}$.\n\\end{lem}\n\\begin{proof}\nIf $\\xi$ belongs to the first or third\nquadrant, then $T^n\\xi$ is escaping from the origin. \nIn such a case, if $B_{\\frac 58\\delta}(\\xi)$ belongs to the thin sector we choose\n$\\eta\\in B_{\\frac{\\delta}2} (\\xi)$, $|y|\\geq \\frac 38 \\delta $.\nClearly, if $(x,y)\\in B_{\\frac \\delta 4}(\\eta)$, $\\operatorname{C} x^2\\geq |y|\\geq\n\\frac \\delta 4$. On the other hand \n$|y_n|\\geq n|x|^3$ while $|x_n|\\leq |x|+n\\operatorname{C}|x_n|^2$. So, if $n\\leq\n\\operatorname{C} |x|^{-1}\\leq \\operatorname{C}\\delta^{-\\frac 12}$, we have $|y_n|\\geq\n\\operatorname{C}|x_n|^2$. After that we can compare the dynamics with one of the\ntype $x\\mapsto x+\\operatorname{C} x^2$, hence after a time at most $\\operatorname{C}\n\\delta^{-\\frac 12}$ the $T^n B_{\\frac \\delta 4}(\\eta)$ will exit $D_R$.\nIf the above does not apply, then\none can take a ball of radius $\\delta\/4$ belonging completely to the\nfat sector and centered at a point in $B_\\delta(\\xi)$. Then the results\nof subsection \\ref{sub:nearfix} easily implies the lemma. \nIf, on the contrary, $\\xi$ belong to the second or fourth quadrant, then its\ntrajectory may approach the origin in an arbitrary manner (even\nasymptotically, if the point belongs to the stable manifold). In such\na case we can take a point $\\eta\\in B_{\\delta}(\\xi)$ at, at least, a vertical\ndistance $\\frac 34\\delta$ from the stable manifold and such that\n$B_{\\frac 34\\delta}(\\eta) \\subset B_\\delta(\\xi)$. Again from the\nresults of subsection \\ref{sub:nearfix} it follows that such a ball\nwill exit $D_R$ in a time at most $\\operatorname{C}_3\\delta^{-\\frac12}$.\n\\end{proof}\n\n\n\\begin{lem}\\label{lem:estim}\nThere exit constants $\\sigma,\\operatorname{C}_3>0$ such that if $n_\\ve=\\operatorname{C}_3\\ve^{-\\frac 12}$ holds\n\\[\n\\bar\\Ka_\\ve(x,z)\\geq \\sigma\\quad \\forall x,z\\in\\To^2.\n\\]\n\\end{lem}\n\\begin{proof}\nIt is trivial to see that\n\\[\n\\bar\\Ka_\\ve(x,z)\\geq \\ve^{-4}m(B_{\\ve\/2}(x)\\cap T^{n_\\ve}B_{\\ve\/2}(z)).\n\\]\nAccordingly, by Lemma \\ref{lem:levelset-escape} there exists two\nballs, of radius $\\frac \\ve 8$, $\\bar B_1\\subset B_{\\ve\/2}(z)$ and\n$\\bar B_2\\subset B_{\\ve\/2}(x)$ that are outside of $D_{r}$, $r\\geq\n\\sqrt{\\frac \\ve 2}$, and whose images will be outside\nof a neighborhood of the origin or order one in a time less than $\\operatorname{C}\n\\ve^{-\\frac 12}$, forward and backward in time, respectively. Given two \nunstable manifolds in $B_1$ at a distance larger than $\\bar c r\\ve$, for\nsome appropriate $\\bar c$, then no stable \nmanifold will intersect both manifolds inside the ball  $B_1$. We\ncan thus consider $\\operatorname{C} r^{-1}$ unstable manifolds such that no stable\nmanifolds intersect two of them in $B_1$. Around each such\nmanifold we can construct a strip by moving along the stable manifold\nby $\\operatorname{C} \\ve$. We obtain in this way $\\operatorname{C} r^{-1}$ disjoint strips\neach of area $\\operatorname{C} r\\ve^2$, whose union covers a fixed fraction of the area of\n$B_1$. After a time less that $\\operatorname{C} \\ve^{-\\frac 12}$ such strips will be outside a\nneighborhood of zero, their length may have increase considerable, if so\nwe will subdivide them into strips of length $\\ve$. Since now the\nstable and unstable manifold are at a fixed angle and by the usual\ndistortion arguments, such strips are essentially rectangular. At this point,\nby Lemma \\ref{lem:expansion}, it will suffice to wait a time\n$\\ve^{-\\frac 12}$ to insure that each such strip will acquire length\nat least $\\frac 12$ in the unstable direction. We thus iterate for such\na time and, if one strip becomes longer than one, we subdivide it into\npieces of length between $\\frac 12$ and one. Finally, fix some box\n$\\Lambda$ of some fixed size $C$ away from  the origin with sides\napproximately parallel either to the stable or to the unstable\ndirections. \nBy mixing it suffices to wait a fixed time to be sure that a fixed\npercentage of each one of the above mentioned strips will intersect\nthe box. In addition, it is possible to insure that such strips cut\nthe box from one stable side to the other.\n\nWe can then write $m(B_{\\ve\/2}(x)\\cap\nT^{n_\\ve}B_{\\ve\/2}(z))=m(T^{-n_\\ve\/2}B_{\\ve\/2}(x)\\cap\nT^{n_\\ve\/2}B_{\\ve\/2}(z))$ since the same considerations done above for\nthe unstable manifold can be done, iterating \nbackward, for the stable manifold it follows that a fixed percentage of\n$T^{-n_\\ve\/2}B_{\\ve\/2}(x)$ and a fixed percentage of\n$T^{n_\\ve\/2}B_{\\ve\/2}(z))$ will intersect $\\Lambda$ and hence each\nother. In fact each one of the above constructed strips in the unstable\ndirection will intersects each one of the strips in the stable\ndirection. By the usual distortion estimates, this implies that the\nintersection among any two such strip has a measure proportional to the\nproduct of the measure of the two strips, hence\n\\[\nm(B_{\\ve\/2}(x)\\cap T^{n_\\ve}B_{\\ve\/2}(z))\\geq \\operatorname{C}\nm(B_{\\ve\/2}(x))m(B_{\\ve\/2}(z)),\n\\]\nand the lemma.\n\\end{proof}\n\n\\begin{lem}\\label{lem:l1}\nFor each $f\\in L^1$, $\\int f=0$ holds\n\\[\n\\|\\Lp_\\ve^n f\\|_1\\leq (1-\\sigma)^{n\/2}\\|f\\|_1.\n\\]\n\\end{lem}\n\\begin{proof}\nNote that $\\Lp_\\ve 1=\n\\Lp_\\ve^*1=1$ and let $\\M^+_\\ve=\\{x\\in\\To^2\\;|\\;\\Lp_\\ve^2 f\\geq 0\\}$;\n$\\M_+=\\{x\\in\\To^2\\;|\\; f\\geq 0\\}$, \nthen, since $\\int \\Lp_\\ve f=\\int f=0$,\n\\[\n\\begin{split}\n\\|\\Lp_\\ve^2 f\\|_1&=2\\int_{\\M^+_\\ve}dx\\,\\Lp_\\ve^2 f\n=2\\int_{\\M^+_\\ve}dx\\int_{\\To^2}dy\\, \\Ka_\\ve(x,\\,y) \nf(y)\\\\\n&=2\\int_{\\To^2}dy\\, f(y)\\int_{\\M^+_\\ve}dx[ \\Ka_\\ve(x,\\,y)-\\sigma]\\\\\n&\\leq 2\\int_{\\M_+}dy\\, f(y)\\int_{\\To^2}dx [\\Ka_\\ve(x,\\,y)-\\sigma]\n=2(1-\\sigma)\\int_{\\M_+}dy\\, f(y)\\\\\n&=(1-\\sigma)\\|f\\|_1.\n\\end{split}\n\\]\n\\end{proof}\nLet $v^{u,s}=(v^{u,s}_1,v^{u,s}_2)$ be the unit tangent vector fields in the\nunstable and stable direction, respectively. Clearly\n$|\\partial^u(\\Lp^i f)|\\leq |\\partial^u f|$, while \n$|\\partial^s(g\\circ T^i)\\leq |\\partial^s g|$.\n\\begin{lem}\\label{lem:appr}\nFor each $f,g\\in\\Co^{1}(\\To^2,\\R)$ holds\n\\[\n\\left|\\int Q_\\ve f g-\\int f g\\right|\\leq \\operatorname{C}\n\\ve\\{\\|f\\|_\\infty+\\|\\partial^u f\\|_{L^1(\\nu)}\\}\n\\{\\|g\\|_\\infty+\\|\\partial^s g\\|_{L^1(\\nu)}\\}, \n\\] \nwhere $\\nu$ is the measure defined by $\\nu(h):=\\int\nd\\rho \\int_{\\partial D_{\\rho}}h$.\n\\end{lem}\n\\begin{proof}\nIt is convenient to introduce the following change of\nvariables. For each $x,y\\in\\To^2$ close enough, let us call\n$[x,y]=W^u_\\delta(x)\\cap W^s_\\delta(y)$, note that by Lemma\n\\ref{prop:dista} such a point is\nalways well defined provided $d(x,y)\\leq \\operatorname{C} \\delta^2$. We consider then\nthe change of variable $\\Phi:\\To^2\\times \\To^2\\to\\R^2\\times\\To^2$,\n\\begin{equation}\\label{eq:changv}\n\\begin{split}\n\\xi:&=x-y\\\\\n\\eta:&=[x,y].\n\\end{split}\n\\end{equation}\nDue to the absolute continuity of the holonomies the above change of\nvariable is absolutely continuous. Clearly, $\\partial^u_x \\eta=0$ since\nmoving along $W^u(x)$ \ndoes not change the intersection point with $W^s(y)$. On the other\nhand $\\partial^s_x \\eta=J\\Psi^u_x v^s(\\eta)$, since moving $x$ along\n$W^s(x)$ moves $\\eta$ on $W^s(y)$ by an amount determined exactly by the unstable\nholonomy $\\Psi^u$ between $W^s(x)$ and $W^s(y)$. By similar arguments\nand a straightforward computations \n\\[\n\\begin{split}\n\\partial_{x_1}\\eta&=v_2^u(x)\\det\\begin{pmatrix} v^s(x)\n&v^u(x)\\end{pmatrix}^{-1}J\\Psi^u_x v^s(\\eta)\\\\\n\\partial_{x_2}\\eta&=v_1^u(x)\\det\\begin{pmatrix} v^u(x)\n&v^s(x)\\end{pmatrix}^{-1}J\\Psi^u_x v^s(\\eta)\\\\\n\\partial_{y_1}\\eta&=v_2^s(y)\\det\\begin{pmatrix} v^u(y)\n&v^s(y)\\end{pmatrix}^{-1}J\\Psi^s_y v^u(\\eta)\\\\\n\\partial_{y_2}\\eta&=v_1^s(y)\\det\\begin{pmatrix} v^s(y)\n&v^u(y)\\end{pmatrix}^{-1}J\\Psi^s_y v^u(\\eta).\n\\end{split}\n\\]\nIn fact, calling $\\theta(x)$ the sine of the angle between stable and\nunstable directions at the point $x$ and $v_\\perp$ the orthogonal\nunit vector to $v$, holds \n\\[\n\\begin{split}\n\\partial_x\\eta&=J\\Psi^u_x\\theta(x)^{-1}\\,  |v^s(y)\\rangle\\langle v^u_\\perp(x)|\\\\\n\\partial_y\\eta&=J\\Psi^s_y\\theta(y)^{-1}\\, |v^u(x)\\rangle\\langle v^s_\\perp(y)|.\n\\end{split}\n\\]\nAccordingly,\n\\begin{equation}\\label{eq:jactot}\n\\begin{split}\nJ\\Phi&:=\\det\\begin{pmatrix}\n            \\Id &-\\Id\\\\\n            \\partial_{x}\\eta&\n           \\partial_{y}\\eta \n           \\end{pmatrix}=\\det\\begin{pmatrix}\n                         \\partial_{x}\\eta+\n                          \\partial_{y}\\eta \n                         \\end{pmatrix}\\\\\n&=J\\Psi^u_x\\, J\\Psi^s_y\\,\\theta(x)^{-1}\\theta(y)^{-1}\\langle v^s_\\perp(y),\\,\nv^u(x)\\rangle^2\\\\\n&=J\\Psi^u_x\\, J\\Psi^s_y\\,\\theta(x)^{-1}\\theta(y)^{-1}\\det\\begin{pmatrix}\n                                                         v^u(x)&v^s(y)\n                                                         \\end{pmatrix}^2.  \n\\end{split}\n\\end{equation}\nBefore starting computing we need to collect some facts.\n\\begin{sublem}\n\\label{slem:domains}\nIf $x\\in\\partial D_{2\\rho}$, $\\rho\\geq r\\geq \\operatorname{C}_4\\sqrt\\ve$, then, for\n$C_4$ large enough,\n\\begin{enumerate}[i)]\n\\item if $\\|x-y\\|\\leq \\ve$, then $y\\not\\in D_\\rho$.\n\\item $\\|x-\\eta\\|\\leq \\frac\\ve\\rho$.\n\\item if $\\zeta\\in W^u(x)$ and $\\|\\zeta-x\\|\\leq\\frac\\ve\\rho$, then\n$\\zeta\\not\\in D_\\rho$.\n\\end{enumerate}\n\\end{sublem}\n\\begin{proof}\nThe first inequality follows since $D_{2\\rho}$ has a vertical size\n$4\\rho^2$. Thus $2\\rho^2-\\ve\\geq 2\\rho^2-\\frac\n1{\\operatorname{C}_4^2}\\rho^2\\geq\\rho^2$. Such an estimate and Proposition\n\\ref{prop:dista} imply that the angle between $W^u(x)$ and $W^u(y)$ is\nat least $4K_-\\rho^{-1}$, thus (ii). Finally, Proposition\n\\ref{prop:dista} implies that, if $\\zeta=(z_1,z_2)$,\n$z_2>2\\rho^2-\\frac{\\ve^2}{\\rho^2}\\geq\\rho^2$. \n\\end{proof}\nWe can now start computing the integral.\n\\[\n\\begin{split}\n\\int_{\\To^4}dxdy f(x)g(y)q_\\ve(x-y)&=\\int_{D_{\\operatorname{C}_4 r}^c}dx\\int_{\\To^2}dy\nf(x)g(y)q_\\ve(x-y) +\\Or(\\|f\\|_\\infty\\|g\\|_\\infty r^3)\\\\\n&\\geq\\int_{\\Phi(D_{\\operatorname{C}_4 r}^c\\times \\To^2)}\\!\\!\\!\\!\\!\\! d\\eta d\\xi\nf(x)g(y)q_\\ve(\\xi)J\\Phi^{-1}+ \n\\|f\\|_\\infty\\|g\\|_\\infty \\Or(r^3)\n\\end{split}\n\\]\nNext, from formula \\eqref{eq:jactot} and Sub-lemma \\ref{slem:domains}-$(i)$\nfollows \n\\begin{equation}\\label{eq:jac-bound}\n|J\\Phi(x,y)-1|\\leq \\operatorname{C}\\frac\\ve{\\rho^2}.\n\\end{equation}\nHence,\n\\[\n\\begin{split}\n\\int_{\\To^4}dxdy f(x)g(y)q_\\ve(x-y)\n&=\\int_{\\To^2} d\\eta\nf(\\eta)g(\\eta)+\\|f\\|_\\infty\\|g\\|_\\infty\\Or\\left(\\int_{D_r^c}|1-J\\Phi^{-1}|\\right)\\\\  \n&\\quad+\\Or\\left(\\int_{\\Phi(D_{\\operatorname{C}_4 r}^c\\times \\To^2)} d\\eta d\\xi\n[f(x)-f(\\eta)]g(y)q_\\ve(\\xi)\\right) \\\\\n&\\quad+\\Or\\left(\\int_{\\Phi(D_{\\operatorname{C}_4 r}^c\\times \\To^2)} d\\eta d\\xi\nf(\\eta)[g(y)-g(\\eta)]q_\\ve(\\xi)\\right)\\\\\n&\\quad +\\|f\\|_\\infty\\|g\\|_\\infty \\Or(r^3).\n\\end{split}\n\\]\nTo conclude we must compute the various error terms. \nFor each $f\\in L^\\infty$, holds\n\\[\n\\int_{D_R}dx\\,f(x)=\\frac 23\\int_0^Rd\\rho\\, \\frac \\rho{1+\\rho}\\int_{\\partial\nD_\\rho}ds\\, f(s).\n\\]\nRemembering\n\\eqref{eq:jac-bound} and applying Fubini\n\\[\n\\int_{D_r^c}|1-J\\Phi^{-1}|\\leq \\operatorname{C}\\|f\\|_\\infty\\int_r^1 d\\rho\\,\n\\rho^2\\frac\\ve{\\rho^2}\\leq \\operatorname{C}\\|f\\|_\\infty\\ve.\n\\]\nNext, let $\\gamma_u^\\eta:[-\\delta,\\delta]\\to\\To^2$ be the unstable manifold\nof $\\eta$, parametrized by arc-length, and let $s(\\eta,\\xi)$ be such\nthat $\\gamma_u^\\eta(s(\\eta,\\xi))=x$. Recalling Sub-lemma \\ref{slem:domains},\n\\[\n\\begin{split}\n\\left|\\int_{\\Phi(D_{\\operatorname{C}_4 r}^c\\times \\To^2)}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\! d\\eta d\\xi\\,\n[f(x)-f(\\eta)]g(y)q_\\ve(\\xi)\\right|&\\leq\n\\|g\\|_\\infty\\int_{\\Phi(D_{\\operatorname{C}_4 r}^c\\times \\To^2)} \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!d\\eta d\\xi\n\\int_0^{s(\\eta,\\xi)}dt\\,|\\partial^u f(\\gamma_u^\\eta(t))|q_\\ve(\\xi)\\\\\n&\\leq\\operatorname{C} \\|g\\|_\\infty\\int_{D_{r}^c\\times\n\\To^2} d\\eta' d\\xi\\, |\\partial^u f(\\eta')|\\frac{\\|\\xi\\|}{|\\theta(\\eta')|}q_\\ve(\\xi)\\\\\n&\\leq \\operatorname{C} \\|g\\|_\\infty\\int_0^1 d\\rho \\;\\ve\\int_{\\partial\nD_\\rho}|\\partial^uf|\\\\\n&\\leq  \\operatorname{C} \\|g\\|_\\infty\\|\\partial^uf\\|_{L^1(\\nu)}\\ve.\n\\end{split}\n\\]\nAnalogously,\n\\[\n\\left|\\int_{D_{r}^c\\times \\To^2} d\\eta d\\xi\nf(\\eta)[g(y)-g(\\eta)]q_\\ve(\\xi)\\right|\\leq  \\operatorname{C}\n\\|f\\|_\\infty\\|\\partial^sg\\|_{L^1(\\nu)}\\ve.\n\\]\nWe can finally collect all the above estimates and obtain\n\\[\n\\begin{split}\n\\int_{\\To^4}dxdy &f(x)g(y)q_\\ve(x-y)=\\int_{\\To^2} d\\eta\nf(\\eta)g(\\eta)\\\\\n&+(\\|f\\|_\\infty+\\|\\partial^uf\\|_{L^1(\\nu)})(\\|g\\|_\\infty\n+\\|\\partial^sg\\|_{L^1(\\nu)}) \\Or(r^3+\\ve)\n\\end{split}\n\\]\nfrom which the lemma follows by choosing $r=\\ve^{\\frac 13}$.\n\\end{proof}\n\n\n\\section{Decay of correlations}\n\\label{sec:decay}\nHere we put together the results of the previous section to prove\nTheorem \\ref{thm:main}. \n\nLet $f,g\\in\\Co^{1}(\\To^2,\\R)$, $\\int f=0$, then\n\\[\n\\begin{split}\n\\int\\Lp^{kn_\\ve}f g&=\\sum_{i=0}^{k-1}\\int\\Lp^{n_\\ve\ni}(\\Lp^{n_\\ve}-\\Lp_\\ve)\\Lp_\\ve^{k-i-1}f g+\\int\\Lp^k_\\ve fg\\\\\n&=\\sum_{i=0}^{k-1}\\int(\\Lp^{n_\\ve}-\\Lp_\\ve)\\Lp_\\ve^{k-i-1}f g\\circ T^{n_\\ve\ni}+\\Or(e^{-\\operatorname{C} k}\\|f\\|_1\\,\\|g\\|_\\infty)\\\\\n&=\\sum_{i=0}^{k-1}\\int(\\Id-Q_\\ve)\\Lp^{n_\\ve}\\Lp_\\ve^{k-i-1}f g\\circ T^{n_\\ve\ni}+\\Or(e^{-\\operatorname{C} k}\\|f\\|_1\\,\\|g\\|_\\infty) ,\n\\end{split}\n\\]\nwhere we have used Lemma \\ref{lem:l1}. To conclude, by using Lemma\n\\ref{lem:appr}, we need to estimate the $L^1$ norm of\n$\\partial^u(\\Lp^{n_\\ve}\\Lp_\\ve^{j}f)$. Since $|D\\Lp_\\ve^{j}f|\\leq \\operatorname{C}\n\\ve^{-1}|f|_\\infty$, for $j>0$,\n\\[\n\\begin{split}\n\\|\\partial^u(\\Lp^{n_\\ve}\\Lp_\\ve^{j}f)\\|_{L^1(\\nu)}&\\leq\n\\int_0^{\\operatorname{C}_4\\sqrt \\ve}d\\rho\\int_{\\partial\nD_\\rho}|\\partial^u(\\Lp^{n_\\ve}\\Lp_\\ve^{j}f)|\n+\\int_{\\operatorname{C}_4\\sqrt \\ve}^1d\\rho\\int_{\\partial\nD_\\rho}|\\partial^u(\\Lp^{n_\\ve}\\Lp_\\ve^{j}f)|\\\\\n&\\leq \\operatorname{C}\\|f\\|_\\infty \\int_0^{\\operatorname{C}_4\\sqrt \\ve}d\\rho\\,\\ve^{-1}\\rho+\n\\operatorname{C}|f|_\\infty\\int_{\\operatorname{C}_4\\sqrt \\ve}^1d\\rho\\int_{\\partial\nD_\\rho}\\frac{\\ve}{\\rho^2}\\ve^{-1}\\\\\n&\\leq \\operatorname{C} \\|f\\|_\\infty\\ln\\ve^{-1},\n\\end{split}\n\\]\nwhere we have used Lemma \\ref{lem:expansion} and Sub-lemma\n\\ref{slem:domains}.\\footnote{The above estimate is not sharp. With\nsome extra work one could avoid the $\\ln\\ve^{-1}$, yet this would not\nchange in any substantial way the result, so we chose to keep the\npresentation as short as possible.}\n\n\nThus\n\\begin{equation}\n\\label{eq:sharp}\n\\begin{split}\n\\left|\\int\\Lp^{n}f\ng\\right|=(\\|f\\|_{\\infty}+\\|\\partial^u f\\|_{L^1(\\nu)})\\|g\\|_{\\Co^1}\\Or(n\\ve^{\\frac\n32}\\ln\\ve^{-1}+ \ne^{-\\operatorname{C} n\\sqrt \\ve}).\n\\end{split}\n\\end{equation}\nClearly the best choice is $\\ve=\\operatorname{C} (n^{-1}\\ln n)^2$ which implies the\nTheorem.\n\n\n\n\\section{Lower bound}\n\\label{sec:lower}\nIn this section we prove a lower bound.\n\\begin{lem}\\label{lem:lower}\nIf there exists a sequence $\\gamma_n$ such that, for each\n$f,g\\in\\Co^1$ holds\n\\[\n\\left|\\int_{\\To^2}f\\circ T^{-n} g-\\int_{\\To^2}f\\int_{\\To^2}g\\right| \\leq\n(\\|\\partial^u f\\|_{L^1(\\nu)}+\\|f\\|_{L^1(\\nu)})|g|_{\\Co^1}\\gamma_n,\n\\]\nthen there exists $\\operatorname{C}>0$ such that\n\\[\n\\gamma_n\\geq n^{-2}\\operatorname{C}.\n\\]\n\\end{lem}\n\\begin{proof}\nLet $g\\geq 0$ be a smooth function supported away from zero (let us say that\nthe support of $g$ does not intersect $D_{\\frac 12}$). Next, let\n$\\xi_0=(x_0,y_0)=(\\frac 12,y_0)\\in W^u(0)$, $\\xi_n=T^{-n}\\xi_0$. For\neach point $\\eta$ in a neighborhood of $\\xi_n$ let $z_1$ be the\ndistance, along $W^u(0)$, between $\\xi_n$ and $W^u(0)\\cap W^s(\\eta)$,\nand $z_2$ the distance, along $W^s(\\xi_n)$, between $\\xi_n$ and\n$W^u(\\xi_n)\\cap W^s(\\eta)$. By construction \n$\\Xi_n(\\eta):=(z_1,z_2)$ is a map from a neighborhood of $\\xi_n$ to a\nneighborhood of the origin with the property that the map transforms\nthe stable and unstable foliation into the standard foliation given by\nthe Cartesian coordinates. Clearly, $\\Xi_n^{-1}(s,0)=\\gamma^u(s)$ (the\nunstable manifold of the origin parametrized by arc length and such that\n$\\gamma^u(0)=\\xi_n$), while $\\Xi_n^{-1}(0,s)=\\gamma_n^s(s)$ (the unstable\nmanifold of $\\xi_n$ parametrized by arc length). Finally we define\n$f_n:=(\\alpha_n\\beta_n)\\circ \\Xi_n$ with\n$\\alpha_n(z_1):=\\varsigma(\\operatorname{C}_5 n z_1)$, $\\beta_n(z_2):=\\varsigma(\\operatorname{C}_5n^{-1}\nz_2)$, for $C_5$ small enough, and \n\\[\n\\varsigma(x):=\\begin{cases}\n             1-|x+1|\\quad &|x+1|\\leq 1\\\\\n             0&|x+1|> 1.\n             \\end{cases}\n\\]\nIn other words, $f_n$ is a function essentially supported on a\nneighborhood left of $\\xi_n$ of order $n^{-1}$ in the unstable direction\nand the stable.  Accordingly, the supports of $f_n\\circ T^{-k}$ and $g$\nare disjoint for all $k\\leq \\operatorname{C} n$. Lemma \\ref{lem:holo} implies that\nthe the support is essentially a rhombus of size $n^{-1}$ and angle\n$n^{-1}$.\n\nThus we have\n\\[\n|g|_{\\Co^1}(\\|\\partial^u f_n\\|_{L^1(\\nu)}+\\|f_n\\|_{L^1(\\nu)})\\gamma_n\\geq\n\\int_{\\To^2}f_n\\int_{\\To^2}g\\geq \\operatorname{C} n^{-3}.\n\\]\nOn the other hand, using again  Lemma \\ref{lem:holo},\n\\[\n\\|\\partial^u f_n\\|_{L^1(\\nu)}\\leq \\operatorname{C} n\\|(\\alpha'\\beta)\\circ\n\\Xi_n\\|_{L^1(\\nu)}\\leq \\operatorname{C} n^{-1}\n\\]\nwhich yields the Lemma.\n\\end{proof}\n\n\\begin{rem}\n\\label{rem:problems} Note that the norms in Lemma \\ref{lem:lower} and\nin Theorem \\ref{thm:main} (even in the stronger version given by\n\\eqref{eq:sharp}) are different. It is not obvious that, putting the\n$L^\\infty$ norm instead of the $L^1(\\nu)$ one keeps the same rate of\nmixing. More generally, it is well known that in the uniformly\nhyperbolic setting the smoothness of the function can have an influence\non the mixing rate. An analogous effect may arise in the present\nsetting but it remains to be investigate. A related problem that needs to be\naddressed is the higher dimensional analogous of the present model\nwhere the fixed point has different possibility of losing full\nhyperbolicty. It is clear that the present\nresult is only the starting point and not the end of the story.\n\\end{rem}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Correlated Gaussian environment\\label{SECgauss}}\n\nWe consider two beam splitters (with transmissivity $\\tau$) which combine\nmodes $A$ and $B$ with two environmental modes, $E_{1}$ and $E_{2}$,\nrespectively. These ancillary modes are in a zero-mean Gaussian state\n$\\rho_{E_{1}E_{2}}$ symmetric under $E_{1}$-$E_{2}$ permutation. In the\nmemoryless model, the environmental state is tensor-product $\\rho_{E_{1}E_{2\n}=\\rho\\otimes\\rho$, meaning that $E_{1}$ and $E_{2}$ are fully independent. In\nparticular, $\\rho$ is a thermal state with covariance matrix (CM)\n$\\omega\\mathbf{I}$, where the noise variance $\\omega=2\\bar{n}+1$ quantifies\nthe mean number of thermal photons $\\bar{n}$\\ entering the beam splitter. Each\ninteraction is then equivalent to a lossy channel with transmissivity $\\tau$\nand thermal noise $\\omega$.\n\n\\begin{figure}[ptbh]\n\\vspace{-2.8cm}\n\\par\n\\begin{center}\n\\includegraphics[width=0.65\\textwidth] {duePICs.eps}\n\\end{center}\n\\par\n\\vspace{-2.3cm}\\caption{\\textit{Left}. Correlated Gaussian\nenvironment, with losses $\\tau$, thermal noise $\\omega$ and\ncorrelations $\\mathbf{G}$. The state of the environment $E_{1}$\nand $E_{2}$ can be separable or entangled. \\textit{Right}.\nCorrelation plane $(g,g^{\\prime})$ for the Gaussian environment,\ncorresponding to thermal noise $\\omega=2$. The black area\nidentifies forbidden environments (correlations are too strong to\nbe compatible with quantum mechanics). White area identifies\nphysical environments, i.e., the subset of points which satisfy\nthe bona-fide conditions of Eq.~(\\ref{CMconstraints}). Whitin this\narea, the inner region labbeled by S identifies separable\nenvironments, while the two outer regions identify entangled\nenvironments.  Figures adapted from Ref.~\\cite{NJP} under a CC BY\n3.0 licence (http:\/\/creativecommons.org\/licenses\/by\/3.0\/).}\n\\label{ENVschemes}\n\\end{figure}\n\nThis Gaussian process can be generalized to include the presence of\ncorrelations between the environmental modes as depicted in the right panel of\nFig.~\\ref{ENVschemes}. The simplest extension of the model consists\\ of taking\nthe ancillary modes, $E_{1}$ and $E_{2}$, in a zero-mean Gaussian state\n$\\rho_{E_{1}E_{2}}$ with CM given by the symmetric normal form\n\\begin{equation}\n\\mathbf{V}_{E_{1}E_{2}}(\\omega,g,g^{\\prime})=\\left(\n\\begin{array}\n[c]{cc\n\\omega\\mathbf{I} & \\mathbf{G}\\\\\n\\mathbf{G} & \\omega\\mathbf{I\n\\end{array}\n\\right)  ~, \\label{EVE_cmAPP\n\\end{equation}\nwhere $\\omega\\geq1$ is the thermal noise variance associated with each\nancilla, and the off-diagonal bloc\n\\begin{equation}\n\\mathbf{G=}\\left(\n\\begin{array}\n[c]{cc\ng & \\\\\n& g^{\\prime\n\\end{array}\n\\right)  ~, \\label{Gblock\n\\end{equation}\naccounts for the correlations between the ancillas. This type of environment\ncan be separable or entangled (conditions for separability will be given afterwards).\n\nIt is clear that, when we consider the two interactions $A-E_{1}$ and\n$B-E_{2}$ separately, the environmental correlations are washed away. In fact,\nby tracing out $E_{2}$, we are left with mode $E_{1}$ in a thermal state\n($\\mathbf{V}_{E_{1}}=\\omega\\mathbf{I}$) which is combined with mode $A$ via\nthe beam-splitter. In other words, we have again a lossy channel with\ntransmissivity $\\tau$ and thermal noise $\\omega$. The scenario is identical\nfor the other mode $B$ when we trace out $E_{1}$. However, when we consider\nthe joint action of the two environmental modes, the correlation block\n$\\mathbf{G}$ comes into play and the global dynamics of the two travelling\nmodes becomes completely different from the standard memoryless scenario.\n\nBefore studying the system dynamics and the corresponding evolution of\nentanglement, we need to characterize the correlation block $\\mathbf{G}$\\ more\nprecisely. In fact, the two correlation parameters, $g$ and $g^{\\prime}$,\ncannot be completely arbitrary but must satisfy specific physical constraints.\nThese parameters must vary within ranges which make the CM of\nEq.~(\\ref{EVE_cmAPP}) a bona-fide quantum CM. Given an arbitrary value of the\nthermal noise $\\omega\\geq1$, the correlation parameters must satisfy the\nfollowing three bona-fide conditions~\\cite{TwomodePRA,NJP}\n\\begin{equation}\n|g|<\\omega,~~~|g^{\\prime}|<\\omega,~~~\\omega^{2}+gg^{\\prime}-1\\geq\n\\omega\\left\\vert g+g^{\\prime}\\right\\vert . \\label{CMconstraints\n\\end{equation}\n\n\n\\subsection{Separability properties}\n\nOnce we have clarified the bona-fide conditions for the environment, the next\nstep is to characterize its separability properties. For this aim, we compute\nthe smallest partially-transposed symplectic (PTS) eigenvalue $\\varepsilon$\nassociated with the CM\\ $\\mathbf{V}_{E_{1}E_{2}}$. For Gaussian states, this\neigenvalue represents an entanglement monotone which is equivalent to the\nlog-negativity~\\cite{logNEG1,logNEG2,logNEG3} $\\mathcal{E}=\\max\\left\\{\n0,-\\log\\varepsilon\\right\\}  $. After simple algebra, we get~\\cite{NJP\n\\begin{equation}\n\\varepsilon=\\sqrt{\\omega^{2}-gg^{\\prime}-\\omega|g-g^{\\prime}|}~.\n\\end{equation}\nProvided that the conditions of Eq.~(\\ref{CMconstraints}) are satisfied, the\nseparability condition $\\varepsilon\\geq1$ is equivalent t\n\\begin{equation}\n\\omega^{2}-gg^{\\prime}-1\\geq\\omega|g-g^{\\prime}|~. \\label{sepCON\n\\end{equation}\n\n\nTo visualize the structure of the environment, we provide a numerical example\nin Fig.~\\ref{ENVschemes}. In the right panel of this figure, we consider the\n\\textit{correlation plane} which is spanned by the two parameters $g$ and\n$g^{\\prime}$. For a given value of the thermal noise $\\omega$, we identify the\nsubset of points which satisfy the bona-fide conditions of\nEq.~(\\ref{CMconstraints}). This subset corresponds to the white area in the\nfigure. Within this area, we then characterize the regions which correspond to\nseparable environments (area labelled by S) and entangled environments (areas\nlabelled by E).\n\n\\section{Direct distribution of entanglement in a correlated Gaussian\nenvironment\\label{SECdirect}}\n\nLet us study the system dynamics and the entanglement propagation in the\npresence of a correlated Gaussian environment, reviewing some key results from\nthe literature~\\cite{NJP}. Suppose that Charlie has an entanglement source\ndescribed by an EPR\\ state $\\rho_{AB}$ with C\n\\begin{equation}\n\\mathbf{V}(\\mu)=\\left(\n\\begin{array}\n[c]{cc\n\\mu\\mathbf{I} & \\mu^{\\prime}\\mathbf{Z}\\\\\n\\mu^{\\prime}\\mathbf{Z} & \\mu\\mathbf{I\n\\end{array}\n\\right)  ~, \\label{CM_TMSV\n\\end{equation}\nwhere $\\mu\\geq1$, $\\mu^{\\prime}:=\\sqrt{\\mu^{2}-1}$, and $\\mathbf{Z}$\\ is the\nreflection matri\n\\begin{equation}\n\\mathbf{Z}:=\\left(\n\\begin{array}\n[c]{cc\n1 & \\\\\n& -1\n\\end{array}\n\\right)  ~. \\label{ZetaMAT\n\\end{equation}\nWe may consider the different scenarios depicted in the three panels of\nFig.~\\ref{twomodeEB}. Charlie may attempt to distribute entanglement to Alice\nand Bob as shown in Fig.~\\ref{twomodeEB}(1), or he may try to share\nentanglement with one of the remote parties, as shown in Figs.~\\ref{twomodeEB\n(2) and~(3).\n\n\\begin{figure}[ptbh]\n\\vspace{-0.2cm}\n\\par\n\\begin{center}\n\\includegraphics[width=0.60\\textwidth] {combined2.eps}\n\\end{center}\n\\par\n\\vspace{-0.9cm}\\caption{Scenarios for direct distribution of entanglement. (1)\nCharlie has two modes $A$ and $B$ prepared in an EPR state $\\rho_{AB}$. In\norder to distribute entanglement to the remote parties, Charlie transmits the\ntwo modes through the correlated Gaussian environment characterized by\ntransmissivity $\\tau$, thermal noise $\\omega$ and correlations $\\mathbf{G}$.\n(2) Charlie aims to share entanglement with Alice. He then keeps mode $B$\nwhile sending mode $A$ to Alice through the lossy channel $\\mathcal{E}_{A}$.\n(3) \\ Charlie aims to share entanglement with Bob. He then keeps mode $A$\nwhile sending mode $B$ to Bob through the lossy channel $\\mathcal{E}_{B}$.\n\\label{twomodeEB\n\\end{figure}\n\nLet us start considering the scenario where Charlie aims to share entanglement\nwith one of the remote parties (one-mode transmission). In particular, suppose\nthat Charlie wants to share entanglement with Bob (by symmetry the derivation\nis the same if we consider Alice). For sharing entanglement, Charlie keeps\nmode $A$ while sending mode $B$ to Bob as shown in Fig.~\\ref{twomodeEB}(3).\nThe action of the environment is therefore reduced to $\\mathcal{I}_{A\n\\otimes\\mathcal{E}_{B}$, where $\\mathcal{E}_{B}$ is a lossy channel applied to\nmode $B$. It is easy to check~\\cite{NJP} that the output state $\\rho\n_{AB^{\\prime}}$, shared by Charlie and Bob, is Gaussian with zero mean and C\n\\begin{equation}\n\\mathbf{V}_{AB^{\\prime}}=\\left(\n\\begin{array}\n[c]{cc\n\\mu\\mathbf{I} & \\mu^{\\prime}\\sqrt{\\tau}\\mathbf{Z}\\\\\n\\mu^{\\prime}\\sqrt{\\tau}\\mathbf{Z} & x\\mathbf{I\n\\end{array}\n\\right)  , \\label{ABpCM\n\\end{equation}\nwhere\n\\begin{equation}\nx:=\\tau\\mu+(1-\\tau)\\omega~.\n\\end{equation}\n\n\nRemarkably, we can compute closed analytical formulas in the limit of large\n$\\mu$, i.e., large input entanglement. In this case, the entanglement of the\noutput state $\\rho_{AB^{\\prime}}$ is quantified by the PTS\\ eigenvalu\n\\begin{equation}\n\\varepsilon=\\frac{1-\\tau}{1+\\tau}\\omega~.\n\\end{equation}\nThe EB condition corresponds to the separability condition $\\varepsilon\\geq1$,\nwhich provide\n\\begin{equation}\n\\omega\\geq\\frac{1+\\tau}{1-\\tau}:=\\omega_{\\text{EB}}~, \\label{EBcond\n\\end{equation}\nor equivalently $\\bar{n}\\geq\\tau\/(1-\\tau)$. Despite the EB condition of\nEq.~(\\ref{EBcond}) regards an EPR\\ input, it is valid for any input state. In\nother words, a lossy channel $\\mathcal{E}_{B}$\\ with transmissivity $\\tau$ and\nthermal noise $\\omega\\geq\\omega_{\\text{EB}}$ destroys the entanglement of any\ninput state $\\rho_{AB}$. Indeed Eq.~(\\ref{EBcond}) corresponds exactly to the\nwell-known EB condition for lossy channels~\\cite{HolevoEB}. The threshold\ncondition $\\omega=\\omega_{\\text{EB}}$ guarantees one-mode EB, i.e., the\nimpossibility for Charlie to share entanglement with the remote party.\n\nNow the central question is the following: Suppose that Charlie cannot share\nany entanglement with the remote parties (one-mode EB), can Charlie still\ndistribute entanglement to them? In other words, suppose that the correlated\nGaussian environment has transmissivity $\\tau$ and thermal noise\n$\\omega=\\omega_{\\text{EB}}$, so that the lossy channels $\\mathcal{E}_{A}$\\ and\n$\\mathcal{E}_{B}$ are EB. Is it still possible to use the joint channel\n$\\mathcal{E}_{AB}$ to distribute entanglement to Alice and Bob? In the\nfollowing, we explicitly reply to this question, discussing how entanglement\ncan be distributed by a separable environment, with the distributed amount\nbeing large enough to be distilled by one-way distillation\nprotocols~\\cite{NJP}.\n\nLet us study the general evolution of the two modes $A$ and $B$ under the\naction of the environment as in Fig.~\\ref{twomodeEB}(1). Since the input\nEPR\\ state $\\rho_{AB}$ is Gaussian and the environmental state $\\rho\n_{E_{1}E_{2}}$ is Gaussian, the output state $\\rho_{A^{\\prime}B^{\\prime}}$ is\nalso Gaussian. This state has zero mean and CM given by~\\cite{NJP\n\\begin{equation}\n\\mathbf{V}_{A^{\\prime}B^{\\prime}}=\\tau\\mathbf{V}_{AB}+(1-\\tau)\\mathbf{V\n_{E_{1}E_{2}}=\\left(\n\\begin{array}\n[c]{cc\nx\\mathbf{I} & \\mathbf{H}\\\\\n\\mathbf{H} & x\\mathbf{I\n\\end{array}\n\\right)  ~,\n\\end{equation}\nwher\n\\begin{equation}\n\\mathbf{H}:=\\tau\\mu^{\\prime}\\mathbf{Z}+(1-\\tau)\\mathbf{G}~.\n\\end{equation}\nFor large $\\mu$, one can easily derive the symplectic spectrum of the output\nstat\n\\begin{equation}\n\\nu_{\\pm}=\\sqrt{\\left(  2\\omega+g^{\\prime}-g\\pm|g+g^{\\prime}|\\right)\n(1-\\tau)\\tau\\mu}~,\n\\end{equation}\nand its smallest PTS\\ eigenvalue~\\cite{NJP\n\\begin{equation}\n\\varepsilon=(1-\\tau)\\sqrt{(\\omega-g)(\\omega+g^{\\prime})}~, \\label{epsMAIN\n\\end{equation}\nquantifying the entanglement distributed to Alice and Bob.\n\nIn the same limit, one can compute the coherent\ninformation~\\cite{CohINFO,CohINFO2} $I(A\\rangle B)$ between the two remote\nparties, which provides a lower bound to the number of entanglement bits per\ncopy that can be distilled using one-way distillation protocols, i.e.,\nprotocols based on local operations and one-way classical communication. It is\nclear that one-way distillability implies two-way distillability, where both\nforward and backward communication is employed. After simple algebra, one\nachieves~\\cite{NJP\n\\begin{equation}\nI(A\\rangle B)=\\log\\frac{1}{e\\varepsilon}~. \\label{coheDIR\n\\end{equation}\nThus, remote entanglement is distributed for $\\varepsilon<1$ and is\ndistillable for $\\varepsilon<e^{-1}$.\n\nNow suppose that the environment has thermal noise $\\omega=\\omega_{\\text{EB}}$\n(one-mode EB). Then, we can write\n\\begin{align}\n\\varepsilon &  =\\sqrt{[1+\\tau-(1-\\tau)g][1+\\tau+(1-\\tau)g^{\\prime\n]}\\nonumber\\\\\n&  :=\\varepsilon(\\tau,g,g^{\\prime}) \\label{EBentEXP\n\\end{align}\nAnswering the previous question corresponds to checking the existence of\nenvironmental parameters $\\tau$, $g$ and $g^{\\prime}$, for which\\ $\\varepsilon\n$ is sufficiently low: For a given value of the transmissivity $\\tau$, we look\nfor regions in the correlation plane $(g,g^{\\prime})$ where $\\varepsilon<1$\n(remote entanglement is distributed) and possibly $\\varepsilon<e^{-1}$ (remote\nentanglement is distillable). This is done in Fig.~\\ref{dirTOT} for several\nnumerical values of the transmissivity.\n\n\\begin{figure}[h]\n\\vspace{-0.0cm}\n\\par\n\\begin{center}\n\\includegraphics[width=0.48\\textwidth] {dirTOTdist_arXiv.eps}\n\\end{center}\n\\par\n\\vspace{-0.4cm}\\caption{Analysis of the remote entanglement\n$\\varepsilon$ on the correlation plane $(g,g^{\\prime})$ for\ndifferent values of the transmissivity $\\tau=0.3$, $0.5$, $0.75$,\nand $0.9$ (from top left to bottom right). Corresponding values of\nthe thermal noise are determined by the one-mode EB condition\n$\\omega=\\omega_{\\text{EB}}$. In each inset, the non-black area\nidentifies the set of physical environments, which are divided\ninto separable (S) and entangled (E) environments by the solid\nlines. The gray region is the \\textquotedblleft activation\narea\\textquotedblright\\ and identifies those environments for\nwhich Charlie is able to distribute entanglement to Alice and Bob\n($\\varepsilon<1$). Within the activation area, the environments\nbelow the dashed curve are those for which the distributed\nentanglement is also distillable by one-way protocols\n($\\varepsilon<e^{-1}$). Figure is re-used from Ref.~\\cite{NJP}\nunder a CC BY 3.0 licence\n(http:\/\/creativecommons.org\/licenses\/by\/3.0\/).}\n\\label{dirTOT\n\\end{figure}\n\nIn Fig.~\\ref{dirTOT}, the environments identified by the gray activation area\nallow Charlie to distribute entanglement to Alice and Bob ($\\varepsilon<1$),\ndespite it is impossible for him to share entanglement with any of the remote\nparties. In other words, these environments are two-mode entanglement\npreserving (EP), despite being one-mode EB. Furthermore, one can identify\nsufficiently-correlated environments for which the entanglement distributed to\nthe remote parties can also be distilled ($\\varepsilon<e^{-1}$).\n\nThe most remarkable feature in Fig.~\\ref{dirTOT} is represented by the\npresence of separable environments in the activation area. In other words,\nthere are separable environments which contain enough correlations to restore\nthe distribution of entanglement to Alice and Bob. Furthermore, for\nsufficiently high transmissivities and correlations, these environments enable\nCharlie to distribute distillable entanglement. As we can note from\nFig.~\\ref{dirTOT}, the weight of separable environments in the activation area\nincreases for increasing transmissivities, with the entangled environments\nalmost disappearing for $\\tau=0.9$.\n\n\\section{Entanglement swapping in a correlated Gaussian\nenvironment\\label{SECindirect}}\n\nIn this section we consider the indirect distribution of entanglement, i.e.,\nthe protocol of entanglement swapping. We start with a brief review of this\nprotocol in the ideal case of no noise. Then, we generalize its theory to the\ncase of correlated-noise Gaussian environments, where we prove how\nentanglement swapping can be reactivated in the presence of one-mode EB.\n\n\\subsection{Entanglement swapping in the absence of noise\\label{SECSUBswap1}}\n\nConsider two remote parties, Alice and Bob, who possess two identical EPR\nstates with CM given in Eq.~(\\ref{CM_TMSV}). At Alice's station, the\nEPR\\ state describes modes $a$ and $A$, while at Bob's station it describes\nmodes $b$ and $B$. Alice and Bob keep modes $a$ and $b$, while sending modes\n$A$ and $B$ to Charlie, where a Bell measurement is performed. This means that\nthe travelling modes $A$ and $B$ are combined in a balanced beam splitter\nwhose output modes \\textquotedblleft$-$\\textquotedblright\\ and\n\\textquotedblleft$+$\\textquotedblright\\ are homodyned, with mode\n\\textquotedblleft$-$\\textquotedblright\\ measured in the position quadrature\nand mode \\textquotedblleft$+$\\textquotedblright\\ in the momentum quadrature.\nIn other words, Charlie measures the two EPR quadratures $\\hat{q}_{-\n:=(\\hat{q}_{A}-\\hat{q}_{B})\/\\sqrt{2}$ and $\\hat{p}_{+}:=(\\hat{p}_{A}+\\hat\n{p}_{B})\/\\sqrt{2}$. The Bell measurement provides two classical outcomes,\n$q_{-}$ and $p_{+}$, which can be compacted into a single complex variable\n$\\gamma:=q_{-}+ip_{+}$. The classical variable $\\gamma$ is finally\ncommunicated to Alice and Bob, with the result of projecting their remote\nmodes $a$ and $b$ into a conditional state $\\rho_{ab|\\gamma}$ (see\nFig.~\\ref{swap}). \\begin{figure}[ptbh]\n\\vspace{-1.6cm}\n\\par\n\\begin{center}\n\\includegraphics[width=0.6\\textwidth] {swap.eps}\n\\end{center}\n\\par\n\\vspace{-2.1cm}\\caption{Entanglement swapping in the absence of noise. See\ntext for explanation.\n\\label{swap\n\\end{figure}\n\nSince the input states are pure Gaussian and the Bell measurement is a\nGaussian measurement which projects pure states into pure states, we have that\nthe remote conditional state $\\rho_{ab|\\gamma}$ turns out to be a pure\nGaussian state. This state has a measurement-dependent mean $\\mathbf{x\n=\\mathbf{x}(\\gamma)$ which Alice and Bob can always delete by conditional\ndisplacements. It is clear that these local unitaries do not alter the amount\nof entanglement in the state, as long as they are perfectly implemented. The\nconditional CM $\\mathbf{V}_{ab|\\gamma}$ can be computed using a simple\ninput-output formula for Gaussian entanglement swapping~\\cite{GaussSWAP}. We\nget\n\\begin{equation}\n\\mathbf{V}_{ab|\\gamma}=\\frac{1}{2\\mu}\\left(\n\\begin{array}\n[c]{cc\n(\\mu^{2}+1)\\mathbf{I} & (\\mu^{2}-1)\\mathbf{Z}\\\\\n(\\mu^{2}-1)\\mathbf{Z} & (\\mu^{2}+1)\\mathbf{I\n\\end{array}\n\\right)  ~. \\label{CMswapNLess\n\\end{equation}\nIts smallest PTS\\ eigenvalue is equal to $\\varepsilon=\\mu^{-1}$, which means\nthat remote entanglement is always generated for entangled inputs ($\\mu>1$).\nFurthermore, remote entanglement is present in the form of EPR correlations\nsince the two remote EPR quadratures $\\hat{q}_{-}^{r}:=(\\hat{q}_{a}-\\hat\n{q}_{b})\/\\sqrt{2}$ and $\\hat{p}_{+}^{r}:=(\\hat{p}_{a}+\\hat{p}_{b})\/\\sqrt{2}$\nhave variance\n\\begin{equation}\nV(\\hat{q}_{-}^{r})=V(\\hat{p}_{+}^{r})=\\mu^{-1}~.\n\\end{equation}\n\n\nThe simplest description of the entanglement swapping protocol can be given\nwhen we consider the limit for $\\mu\\rightarrow\\infty$. In this case the\ninitial states are ideal EPR states with quadratures perfectly correlated,\ni.e., $\\hat{q}_{a}=\\hat{q}_{A}$ and $\\hat{p}_{a}=-\\hat{p}_{A}$ for Alice, and\n$\\hat{q}_{b}=\\hat{q}_{B}$ and $\\hat{p}_{b}=-\\hat{p}_{B}$ for Bob. Then, the\noverall action of Charlie, i.e., the Bell measurement plus classical\ncommunication, corresponds to create a remote state with\n\\begin{equation}\n\\hat{q}_{b}=\\hat{q}_{a}-\\sqrt{2}q_{-},~\\hat{p}_{b}=-\\hat{p}_{a}-\\sqrt{2\np_{+}~.\n\\end{equation}\nThe quadratures of the two remote modes are perfectly correlated, up to an\nerasable displacement. In other words, the ideal EPR\\ correlations have been\nswapped from the initial states to the final conditional state $\\rho\n_{ab|\\gamma}$.\n\n\\subsection{Entanglement swapping in the presence of\ncorrelated-noise\\label{SECSUBswap2}}\n\nThe theory of entanglement swapping can be extended to include the presence of\nloss and correlated noise. We consider our model of correlated Gaussian\nenvironment with transmission $\\tau$, thermal noise $\\omega$ and correlations\n$\\mathbf{G}$. The modified scenario is depicted in Fig.~\\ref{swapLOSS\n.\\begin{figure}[ptbh]\n\\vspace{-1.5cm}\n\\par\n\\begin{center}\n\\includegraphics[width=0.6\\textwidth] {swapLOSS.eps}\n\\end{center}\n\\par\n\\vspace{-1.9cm}\\caption{Entanglement swapping in the presence of loss, thermal\nnoise and environmental correlations (correlated Gaussian environment). The\nBell detector has been simplified.\n\\label{swapLOSS\n\\end{figure}\n\n\\subsubsection{Swapping of EPR\\ correlations}\n\nFor simplicity, we start by studying the evolution of the EPR correlations\nunder ideal input conditions ($\\mu\\rightarrow+\\infty$). After the classical\ncommunication of the outcome $\\gamma$, the quadratures of the remote modes $a$\nand $b$ satisfy the asymptotic relations\n\\begin{align}\n\\hat{q}_{b}  &  =\\hat{q}_{a}-\\sqrt{\\frac{2}{\\tau}}\\left(  q_{-}-\\sqrt{1-\\tau\n}\\hat{\\delta}_{q}\\right)  ,\\label{eqrrr}\\\\\n\\hat{p}_{b}  &  =-\\hat{p}_{a}-\\sqrt{\\frac{2}{\\tau}}\\left(  p_{+}-\\sqrt{1-\\tau\n}\\hat{\\delta}_{p}\\right)  , \\label{eqrrr2\n\\end{align}\nwhere $\\hat{\\delta}_{q}=(\\hat{q}_{E_{1}}-\\hat{q}_{E_{2}})\/\\sqrt{2}$ and\n$\\hat{\\delta}_{p}=(\\hat{p}_{E_{1}}+\\hat{p}_{E_{2}})\/\\sqrt{2}$ are noise\nvariables introduced by the environment.\n\nUsing previous Eqs.~(\\ref{eqrrr}) and~(\\ref{eqrrr2}), we construct the remote\nEPR\\ quadratures $\\hat{q}_{-}^{r}$\\ and $\\hat{p}_{+}^{r}$, and we compute the\nEPR variance\n\\begin{equation}\n\\boldsymbol{\\Lambda}:=\\left(\n\\begin{array}\n[c]{cc\nV(\\hat{q}_{-}^{r}) & \\\\\n& V(\\hat{p}_{+}^{r})\n\\end{array}\n\\right)  \\rightarrow\\boldsymbol{\\Lambda}_{\\infty}=\\frac{1-\\tau}{\\tau\n(\\omega\\mathbf{I}-\\mathbf{ZG})~, \\label{lam1\n\\end{equation}\nwhere the limit is taken for $\\mu\\rightarrow+\\infty$. Assuming the EB\ncondition $\\omega=\\omega_{\\text{EB}}$, we finally ge\n\\begin{equation}\n\\boldsymbol{\\Lambda}_{\\infty,\\text{EB}}=\\frac{1}{\\tau}\\left[  (1+\\tau\n)\\mathbf{I}-(1-\\tau)\\mathbf{ZG}\\right]  . \\label{lam2\n\\end{equation}\n\n\nIn the case of a memoryless environment ($\\mathbf{G=0}$) we see that\n$\\boldsymbol{\\Lambda}_{\\infty,\\text{EB}}=(1+\\tau^{-1})\\mathbf{I}\\geq\n\\mathbf{I}$, which means that the EPR\\ correlations cannot be swapped to the\nremote systems. However, it is evident from Eq.~(\\ref{lam2}) that there are\nchoices for the correlation block $\\mathbf{G}$\\ such that the EPR\\ condition\n$\\boldsymbol{\\Lambda}_{\\infty,\\text{EB}}<\\mathbf{I}$ is satisfied. For\ninstance, this happens when we consider $\\mathbf{G=}g\\mathbf{Z}$. In this case\nit is easy to check that $\\boldsymbol{\\Lambda}_{\\infty,\\text{EB}}<\\mathbf{I}$\nis satisfied for $\\tau\\geq1\/4$ and $g>(1-\\tau)^{-1}$. Under these conditions,\nEPR\\ correlations are successfully swapped to the remote modes. In particular,\nfor $\\tau>1\/2$ and $g>(1-\\tau)^{-1}$ there are separable environments which do\nthe job.\n\n\\subsubsection{Swapping and distillation of entanglement}\n\nHere we discuss in detail how entanglement is distributed by the swapping\nprotocol in the presence of a correlated Gaussian environment. In particular,\nsuppose that Alice and Bob cannot share entanglement with Charlie because the\nenvironment is one-mode EB. Then, we aim to address the following questions:\n(i)~Is it still possible for Charlie to distribute entanglement to the remote\nparties thanks to the environmental correlations? (ii)~In particular, is the\nswapping successful when the environmental correlations are separable?\n(iii)~Finally, are Alice and Bob able to distill the swapped entanglement by\nmeans of one-way distillation protocols? Our previous discussion on EPR\ncorrelations clearly suggests that these questions have positive answers. Here\nwe explicitly show this is indeed true for quantum entanglement by finding the\ntypical regimes of parameters that the Gaussian environment must satisfy.\n\nIn order to study the propagation of entanglement we first need to derive the\nCM\\ $\\mathbf{V}_{ab|\\gamma}$\\ of the conditional remote state $\\rho\n_{ab|\\gamma}$. As before, we have two identical EPR\\ states at Alice's and\nBob's stations with CM $\\mathbf{V}(\\mu)$ given in Eq.~(\\ref{CM_TMSV}). The\ntravelling modes $A$ and $B$ are sent to Charlie through a Gaussian\nenvironment with transmissivity $\\tau$, thermal noise $\\omega$ and\ncorrelations $\\mathbf{G}$. After the Bell measurement and the classical\ncommunication of the result $\\gamma$, the conditional remote state at Alice's\nand Bob's stations is Gaussian with CM~\\cite{BellFORMULA\n\\begin{equation}\n\\mathbf{V}_{ab|\\gamma}=\\left(\n\\begin{array}\n[c]{cc\n\\mu\\mathbf{I} & \\\\\n& \\mu\\mathbf{I\n\\end{array}\n\\right)  -\\frac{(\\mu^{2}-1)\\tau}{2}\\left(\n\\begin{array}\n[c]{cccc\n\\frac{1}{\\theta} &  & -\\frac{1}{\\theta} & \\\\\n& \\frac{1}{\\theta^{\\prime}} &  & \\frac{1}{\\theta^{\\prime}}\\\\\n-\\frac{1}{\\theta} &  & \\frac{1}{\\theta} & \\\\\n& \\frac{1}{\\theta^{\\prime}} &  & \\frac{1}{\\theta^{\\prime}\n\\end{array}\n\\right)  ~, \\label{VabGamma\n\\end{equation}\nwher\n\\begin{equation}\n\\theta=\\tau\\mu+(1-\\tau)(\\omega-g),~\\theta^{\\prime}=\\tau\\mu+(1-\\tau\n)(\\omega+g^{\\prime})~. \\label{thetas\n\\end{equation}\n\n\nFrom the CM of Eq.~(\\ref{VabGamma}) we compute the smallest PTS eigenvalue\n$\\varepsilon$ quantifying the remote entanglement at Alice's and Bob's\nstations. For large input entanglement $\\mu\\gg1$, we find a closed formula in\nterms of the environmental parameters, i.e.\n\\begin{equation}\n\\varepsilon=\\frac{1-\\tau}{\\tau}\\sqrt{(\\omega-g)(\\omega+g^{\\prime\n)}:=\\varepsilon(\\tau,\\omega,g,g^{\\prime})~, \\label{SpectrumTOT\n\\end{equation}\nwhich is equal to Eq.~(\\ref{epsMAIN}) up to a factor $\\tau^{-1}$. As before,\nthis eigenvalue not only determines the log-negativity but also the coherent\ninformation $I(a\\rangle b)$\\ associated with the remote state $\\rho\n_{ab|\\gamma}$. In fact, for large $\\mu$, one can easily compute the asymptotic\nexpressio\n\\begin{equation}\nI(a\\rangle b)\\rightarrow\\log\\frac{2}{e}\\sqrt{\\frac{\\det\\mathbf{V}_{b|\\gamma\n}{\\det\\mathbf{V}_{ab|\\gamma}}}=\\log\\frac{1}{e\\varepsilon}~,\n\\label{IabCOHEswap\n\\end{equation}\nwhich is identical to the formula of Eq.~(\\ref{coheDIR}) for the case of\ndirect distribution. Thus, the PTS\\ eigenvalue of Eq.~(\\ref{SpectrumTOT})\ncontains all the information about the distribution and distillation of\nentanglement in the swapping scenario. For $\\varepsilon<1$ entanglement is\nsuccessfully distributed by the swapping protocol (log-negativity\n$\\mathcal{E}>0$). Then, for the stronger condition $\\varepsilon<e^{-1}$, the\nswapped entanglement can also be distilled into $I(a\\rangle b)$ entanglement\nbits per copy by means of one-way protocols.\n\nNow, let us assume the condition of one-mode EB ($\\omega=\\omega_{\\text{EB}}$)\nso that the bipartite states before measurement $\\rho_{aA^{\\prime}}$ and\n$\\rho_{B^{\\prime}b}$ are separable (see Fig.~\\ref{swapLOSS}). We investigate\nthe amount of entanglement generated in the remote modes $a$ and $b$ by\ncomputing the eigenvalue $\\varepsilon(\\tau,\\omega_{\\text{EB}},g,g^{\\prime})$.\nIn the standard memoryless case ($\\mathbf{G}=\\mathbf{0}$) we have\n$\\varepsilon=1+\\tau^{-1}$ which means that no entanglement can be swapped, as\nexpected. To study the general case of correlated environment, we consider\ndifferent numerical values of the transmissivity $\\tau$, and we plot the\n$\\varepsilon(\\tau,\\omega_{\\text{EB}},g,g^{\\prime})$ on the correlation plane.\nThe results are shown in Fig.~\\ref{total} and are similar to those achieved in\nFig.~\\ref{dirTOT} for direct distribution.\n\n\\begin{figure}[ptbh]\n\\vspace{-0.1cm}\n\\par\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth] {swTOT_arXiv.eps}\n\\end{center}\n\\par\n\\vspace{-0.5cm}\\caption{Analysis of the swapped entanglement $\\varepsilon$ in\nthe correlation plane $(g,g^{\\prime})$ for different values of the\ntransmissivity $\\tau=0.3$, $0.5$, $0.75$, and $0.9$ (from top left to bottom\nright). Corresponding values of the thermal noise are determined by the\none-mode EB condition $\\omega=\\omega_{\\text{EB}}$. In each inset, the\nnon-black area identifies the set of physical environments, which are divided\ninto separable (S) and entangled (E) environments. The gray region identifies\nthe activation area, i.e., those environments whose correlations are strong\nenough to generate remote entanglement ($\\varepsilon<1$). Within the gray\nactivation area, the environments that lie below the dashed curve are those\nable to generate distillable entanglement ($\\varepsilon<e^{-1}$).\n\\label{total\n\\end{figure}\n\nIn each panel of Fig.~\\ref{total}, the physical values for the correlation\nparameters $(g,g^{\\prime})$ are individuated by the non-black area. Remote\nentanglement is distributed ($\\varepsilon<1$) for values of the correlation\nparameters belonging to the gray activation area. For $\\tau\\leq1\/2$ (top two\npanels), we see that the activation area is confined within the region of\nentangled environments. The property that entangled environments are necessary\nfor the reactivation of entanglement swapping at any $\\tau\\leq1\/2$ is easy to\nprove. In fact, suppose that $\\varepsilon<1$ holds. By using its formula in\nEq.~(\\ref{SpectrumTOT}) and the bona-fide conditions on the correlation\nparameters given in Eq.~(\\ref{CMconstraints}), we can write $\\varepsilon\n^{2}<1$ as\n\\begin{equation}\n\\omega^{2}-gg^{\\prime}+\\omega(g^{\\prime}-g)<\\left(  \\frac{\\tau}{1-\\tau\n}\\right)  ^{2}~. \\label{eqkk\n\\end{equation}\nNow, for $\\tau\\leq1\/2$, we have $\\tau^{2}(1-\\tau)^{-2}\\leq1$ and using this\ninequality in Eq.~(\\ref{eqkk}), we derive\n\\begin{equation}\n\\omega^{2}-gg^{\\prime}-1<\\omega(g-g^{\\prime})\\leq\\omega|g-g^{\\prime}|~,\n\\end{equation}\nwhich is the entanglement condition for the environment [i.e., the violation\nof Eq.~(\\ref{sepCON})].\n\nIt is clear that the most interesting result holds for transmissivities\n$\\tau>1\/2$. In this regime, in fact, the distribution of remote entanglement\ncan be activated by separable environments. As explicitly shown for\n$\\tau=0.75$ and $0.9$, the activation area progressively invades the region of\nseparable environments. In other words, separable correlations become more and\nmore important for increasing transmissivities. Furthermore, for $\\tau\n\\gtrsim0.75$, separable environments are even able to activate the\ndistribution of distillable entanglement ($\\varepsilon<e^{-1}$). By comparing\nFig.~\\ref{dirTOT} and Fig.~\\ref{total}, we see how entanglement is more easily\ngenerated and distilled by the direct protocol. This is a consequence of the\nextra factor $\\tau^{-1}$ in Eq.~(\\ref{SpectrumTOT}), whose influence becomes\nless important only at high transmissivities ($\\tau\\simeq1$).\n\n\\section{Conclusion\\label{SECconclusion}}\n\nIn conclusion, we have investigated the distribution of entanglement in the\npresence of correlated-noise Gaussian environments, proving how the injection\nof separable correlations can recover from entanglement breaking. In order to\nderive simple analytical results we have considered here only the case of\nlarge entanglement for the input states. We have analyzed scenarios of direct\ndistribution and indirect distribution, i.e., entanglement swapping.\nSurprisingly, the injection of the weaker separable correlations is sufficient\nto restore the entanglement distribution, as we have shown for wide regimes of\nparameters. Furthermore, the generated entanglement can be sufficient to be\ndistilled by means of one-way protocols. The fact that separability can be\nexploited to recover from entanglement breaking is clearly a paradoxical\nbehavior which poses fundamental questions on the intimate relations between\nlocal and nonlocal correlations.\n\n\\section*{Acknowledgements}\n\nThis work was funded by a Leverhulme Trust research fellowship, and the EPSRC\nvia `qDATA' (Grant No. EP\/L011298\/1) and the UK Quantum Communications Hub\n(Grant No. EP\/M013472\/1).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\n\\label{sec:introduction}\n\nBlack-box machine learning models are increasingly deployed in high stakes environments such as hiring, criminal justice or medical diagnosis \\cite{chalfin2016productivity,berk2012criminal,rajkomar2019machine}. Despite providing accurate predictions in a test environment, their application can be harmful as their predictions may rely on spuriously correlated variables or protected attributes. In such cases, models can discriminate disadvantaged groups \\cite{wexler_rebecca_when_2017} or make inaccurate predictions outside the test environment \\cite{zech_variable_2018}.\\\\\nA range of Interpretable Machine Learning (IML) techniques provide insight into the model's decision-making by quantifying the importance of features for the model's performance \\cite{molnar2019}.\nMeasures of feature importance differ in terms of their consideration of the dependence structure in the underlying dataset. While some methods only quantify the direct effect of features others also deem variables important that contain prediction-relevant information, irrespective of their direct utilization by the model. We will refer to the former as direct importance measures and to the latter as associative importance measures.\\\\\nWhile direct importance measures allow causal insight into the model's mechanism, they fail to expose leakage of information from associated but not directly used variables\\cite{Adler2018}. In contrast, associative importance methods expose leakage of information but do not provide insight into the model's mechanism\\cite{janzing2020feature}. However,\neven in combination, direct and associative importance cannot provide both insight into the sources of prediction-relevant information \\textit{and} the direct and indirect feature pathways by which the information enters the model.\\\\\nConsider a simple illustrative example (Figure \\ref{fig:example-intro}): Features \\textit{zip code} and \\textit{job experience} are directly used by the model to compute the prediction $\\hat{Y}$. The feature \\textit{ethnicity} is associated with the variable \\textit{zip code} (blue arrow) and therefore with $\\hat{Y}$; however, \\textit{ethnicity} is not directly used by the model.\n\\begin{figure}[H]\n    \\centering\n    \\begin{tikzpicture}[thick, scale=0.7, every node\/.style={scale=.6, line width=0.25mm, black, fill=white}]\n    \\usetikzlibrary{shapes}\n\t\t\\node[draw, ellipse, scale=0.9] (x1) at (-4, .8) {job experience};\n\t\t\\node[draw, ellipse, scale=0.9] (x2) at (0, .8) {zip code};\n\t\t\\node[draw, ellipse, scale=0.9] (x3) at (4,.8) {ethnicity};\n\t\t\\node[draw, ellipse, scale=0.9] (y) at (0,0) {$\\hat{Y}$: predicted job performance};\n\t\t\\draw[->, green] (x1) -- (y);\n\t\t\\draw[->, green] (x2) -- (y);\n\t\t\\draw[->, blue] (x3) -- (x2);\n    \\end{tikzpicture}\n    \\caption{Directed Acyclic Graph (DAG) illustrating the data generating process.}\n    \\label{fig:example-intro}\n\\end{figure}\n\\noindent Direct importance measures such as Permutation Feature Importance (PFI) \\cite{Breiman2001rf} or marginal SAGE \\cite{covert_understanding_2020} consider the feature \\textit{race} irrelevant, as it is not directly causal for the model's predictions.\nAs a result, they fail to expose the leakage of information from \\textit{ethnicity} via the proxy variable \\textit{zip code}.\nMeasures of associative importance such as conditional SAGE \\cite{covert_understanding_2020}\nwould expose the association and consider \\textit{ethnicity} to be relevant.\nHowever, even in combination with direct importance, associative importance does not provide insight into via which of the directly important features (\\textit{zip code} or \\textit{job experience}) the protected variable leaks into the model. Consequently we are not guided towards sparse interventions that eliminate the association from the model.\\\\\nWe propose a framework (\\underline{De}composition of Global Feature Importance into \\underline{D}irect and \\underline{A}ssociative \\underline{C}omponen\\underline{t}s, DEDACT) that enables a holistic understanding of both sources of prediction-relevant information \\textit{and} feature pathways which allow the respective sources of information to enter the model's predictions. With DEDACT one can either decompose the direct importance of a feature into the contributions of its sources of information, or one can decompose the associative importance of an unwanted influence into its direct feature pathway components.\\\\\nDEDACT does not require knowledge of the underlying dependence structure or causal graph. As is the case with associative importance measures such as conditional SAGE \\cite{covert_understanding_2020}, DEDACT requires sampling from conditional distributions.\nThese samples can be reused for the decomposition; in this way the computational overhead can be reduced.\\\\\nBefore the decomposition will be introduced in Section \\ref{sec:decomposition}, we will formally define associative and direct importance in Section \\ref{sec:disentangling-direct-and-indirect} and measures for the respective direct and associative components in Section \\ref{sec:iai-via-and-di-from}. The method will be validated on simulated data in Section \\ref{sec:application}.\n\n\\subsection{Related Work and Contributions}\nPrevious research on local SHAP based feature effect explanations have studied direct and indirect effects. Heskes et al. \\cite{heskes_causal_2020} decompose causal Shapley value functions into one direct and one indirect component. Our method is applicable to a wide range of global direct and associative feature importance methods and allows their full decomposition into feature components. Wang et al. \\cite{wang_shapley_2020} offer a graph-based decomposition of direct and indirect causal effects for local causal SHAP explanations. However, their approach (1) cannot be applied to direct or associative importance methods, or (2) to feature importance methods and (3) requires knowledge of the underlying causal graph. Relative Feature Importance \\cite{konig_relative_2021} quantifies the direct importance of a variable that cannot be attributed to a user specified set of variables. However, the method has not been applied to decompose the importance of a feature into its prediction-relevant sources of information. Moreover, it does not allow to decompose the associative importance of a variable. Our contribution is the first decomposition of global direct and associative importance measures into their respective associative or direct components, and therefore provides novel insight into both model and data.\\\\\n\\section{Notation}\n\\label{sec:background-and-notation}\nThe Table in Figure \\ref{fig:notation} summarizes the mathematical notation and terms used throughout the article.\nIn order to formalize a distinction between direct and associative importance we differentiate between variables on the data level and features on the model level. We refer to the real-world concept as variable $X_j$ and to the corresponding model input as feature $\\underline{X}_j$ (Figure \\ref{fig:notation}, left panel). Although in the standard setting variable and feature take the same values, they are conceptually different. If we say that we intervene on a feature, it only means that we change the input to the respective feature. If we say that a feature is not causal for the prediction, the corresponding real world concept may still be causal since it could indirectly affect the prediction via downstream variables. Following \\cite{janzing2020feature}, we denote perturbations of the features using the do-operator \\cite{pearl2009causality}, i.e. the term $do(\\underline{X}_k = \\tilde{X}_k)$ denotes the replacement of the input $\\underline{X}_k$ with the perturbed variable $\\tilde{X}_k$. The corresponding prediction is denoted as $\\hat{Y}|{do(\\underline{X}_k = \\tilde{X}_k)}$.\\\\\nSome methods rely on marginalized prediction functions. To be more specific, the prediction function $f$ that is marginalized over $\\underline{X}_S$ is defined as\n$$ f_S(\\underline{x}_{S}) = \\mathbb{E}_{\\underline{X}_{D \\backslash S}|\\underline{X}_S}[f(\\underline{X}_{D \\backslash S}) | \\underline{X}_S = \\underline{x}_{S}],$$\nwhere the conditional expectation is taken over the feature distribution. If features are sampled independently this term reduces to the marginal expectation.\\\\\nWe indicate whether the prediction of the original model $f$ or the marginalized variant $f_S$ is meant by making use of the respective subscripts: $f$ or $f_S$. For example, $\\hat{Y}_{f_S}$ refers to the marginalized prediction.\\\\\nThere will be frequent references to variables as \"containing information\" or to \"information entering a variable\". Such phrases are not meant in the information theoretic sense \u2013 despite links with concepts such as mutual information \u2013 but rather figuratively to enable a more pleasant reading experience.\n\n\\input{figures\/figure-notation}\n\n\\section{Background: Direct and Associative Importance}\n\\label{sec:disentangling-direct-and-indirect}\nOver the course of this Section, we formally introduce general definitions of direct and associative importance that generalize methods such as Permutation Feature Importance \\cite{Breiman2001rf} and SAGE value functions \\cite{covert_understanding_2020}. These two classes of relevance, namely direct and associative importance, are introduced in Sections \\ref{subsec:dr} and \\ref{subsec:ar}. As the proposed decomposition is based on these general definitions, it is applicable to a wide-range of feature importance methods.\\\\\nBoth definitions can be modified along two axes: Firstly, recent proposals \\cite{Datta2017,lundberg_unified_2017,covert_understanding_2020} formulate feature relevance quantification as a cooperative game, which evaluates the relevance of features\/variables with respect to different perturbation baselines.\\footnote{Feature relevance attribution can be seen as a cooperative game, where the contribution of one player (feature\/variable) depends on which players are already in the room \\cite{lundberg_unified_2017,covert_understanding_2020}. For example, if two features contribute to the prediction performance via an interaction term, their joint contribution is attributed to the feature that is perturbed first. If the relevance of features is only considered in isolation, both features are attributed with the interaction contribution. Consequently, recent methods do not only evaluate the relevance in isolation but also over multiple different baselines \\cite{covert_understanding_2020}.} In order to accommodate such considerations within our framework, the general definitions allow the specification of flexible perturbation baselines (more details below). Furthermore, the general measures accommodate methods that rely on the original prediction function $f$, such as Permutation Feature Importance (PFI) \\cite{Breiman2001rf} as well as methods that marginalize the prediction function over perturbed features, as suggested by \\cite{covert_understanding_2020}.\\\\\n\\input{figures\/dag-intuition-di-tai}\n\\subsection{Direct Importance}\n\\label{subsec:dr}\n\nDirect importance measures assess the relevance of specific features for the model's performance. They are based on a simple idea: If a feature is not important, removing all its prediction-relevant information should have no or only little effect on the model's performance.\n\\\\\nAs such, direct importance measures compare the model performance before and after perturbing the features of interest. The perturbed variable $\\tilde{X}_K^\\emptyset$ preserves the marginal distribution of the original variable, but is sampled independently from $X$ and $Y$ (as indicated by the superscript $\\emptyset$). Hence, the perturbed variable only contains noise and no prediction-relevant information.\\\\\nOur notation starts off with the full perturbation of all features. After that, we quantify the improvement in performance when restoring a set of features. That is, the direct importance $DI(\\underline{X}_K)$ quantifies the improvement in performance when features $\\underline{X}_K$ are restored in isolation. The direct importance $DI(\\underline{X}_K|\\underline{X}_B)$ over baseline $B$ quantifies the improvement in performance, when $\\underline{X}_K$ is restored in a context where features $\\underline{X}_B$ have been restored already. Furthermore, we indicate whether the original model $f$ or the marginalized prediction functions $f_{B}, f_{B \\cup K}$ are used by $DI(\\underline{X}_K|\\underline{X}_B; f, f)$ and $DI(\\underline{X}_K|\\underline{X}_B; f_{B}, f_{B \\cup K})$. For example, Permutation Feature Importance is a special case of a direct importance measure, i.e. $DI(\\underline{X}_K|\\underline{X}_{D \\backslash K}; f, f)$. Direct importance is formally introduced in Definition \\ref{def:dr} and illustrated in Figure \\ref{fig:dag-intuition-dr-ar} (a) - (b).\\\\\n\\begin{definition}[Direct Importance (DI)]\nGiven two disjoint feature index sets $B$ and $K$, direct importance of features $\\underline{X}_K$ over baseline features $\\underline{X}_B$ is defined as:\n\\begin{align*}\n    DI(\\underline{X}_K|\\underline{X}_B;g,g') := &\\mathbb{E}[\\mathcal{L}(\\hat{Y}_{g}|{do(\\underline{X}_{D \\backslash B }=\\tilde{X}_{D \\backslash B}^\\emptyset)};Y)]\\\\ \n    - &\\mathbb{E}[\\mathcal{L}(\\hat{Y}_{g'}|{do(\\underline{X}_{D \\backslash (B \\cup K)} = \\tilde{X}_{D \\backslash (B \\cup K)}^\\emptyset)};Y)]\n\\end{align*}\nwhere $(g, g') \\in \\{(f, f),(f_{B}, f_{(B \\cup K)})\\}$. In addition, the perturbed joint $\\tilde{X}^{\\emptyset}$ preserves the covariate joint distribution, but is independent of the original variables and the prediction target. I.e.,  $P(\\tilde{X}^{\\emptyset}) = P(X) \\text{ and }\\tilde{X}^{\\emptyset} \\Perp (X, Y)$.\n\\label{def:dr}\n\\end{definition}\nAs all features except the features of interest are left unchanged, direct importance can only enter the model via the features of interest themselves. Therefore, directly important features must be causal for the prediction (Proposition \\ref{proposition:tdr-direct-sensitivity}).\\\\\n\\begin{proposition}[Direct Sensitivity of DI]\nNonzero Direct Importance $DI(\\underline{X}_K|\\underline{X}_B)$ implies that feature $\\underline{X}_K$ is causal for the model (over $P(X_B)P(X_K)P(X_{D \\backslash (B \\cup K)})$).\n\\label{proposition:tdr-direct-sensitivity}\n\\end{proposition}\nHowever, as illustrated by the motivational example in the introduction to this paper, direct importance fails to expose leakage of information from variables that are not directly used by the model.\n\\subsection{Associative Importance}\n\\label{subsec:ar}\nIn contrast to direct importance, associative importance measures are not concerned with the mechanisms that the model uses. They rather aim at measuring the component of the predictive performance that can be explained with information contained in the variables of interest $X_J$.\\\\\nTherefore, associative importance measures compare the model's performance under fully perturbed features with the performance that results from reconstruction of the features by the use of information from the variables of interest $X_J$. The reconstructed features preserve the conditional distribution of their corresponding variables with the variable of interest, i.e. $P(\\underline{X}|X_J) = P(X|X_J)$. Consequently, the features corresponding to the variables of interest $\\underline{X}_J$ are fully reconstructed, features that correspond to dependent variables are partly reconstructed, and features that correspond to variables that are independent from $X_J$ remain unchanged.\\\\\nAs for direct importance, it is of relevance whether the variable is introduced in isolation, or whether it is added to a set of variables for which the information has been restored already. We denote the associative importance of variables $X_J$ that are introduced in isolation as $AI(X_J)$, and the associative importance of variables $X_J$ that are introduced to already conditioned upon variables $X_C$ as $AI(X_J|X_C)$. Associative importance is formally introduced in Definition \\ref{def:ar} and illustrated in Figure \\ref{fig:dag-intuition-dr-ar} (c) - (d).\n\\begin{definition}[Associative Importance (AI)]\nWe define the associative importance of a variable set $J$ given context variables $C$ as:\n\\begin{align*}\n AI(X_J|X_C; g, g') := &\\mathbb{E}[\\mathcal{L}(Y; \\hat{Y}_g|{do(\\underline{X}_{D\\backslash C} = \\tilde{X}_{D\\backslash C}^{C})})]\\\\\n- &\\mathbb{E}[\\mathcal{L}(Y;\\hat{Y}_{g'}|{do(\\underline{X}_{D\\backslash (C \\cup J)} = \\tilde{X}_{D \\backslash (C \\cup J)}^{C \\cup J})})]\n\\end{align*}\nwhere $(g,g') \\in \\{(f,f), (f_{C},f_{C \\cup J})\\}$.\nThe conditionally perturbed joint $\\tilde{X}^{S}$ is required to preserve the covariate joint distribution as well as the conditional distribution with features $X_S$, i.e. $P(\\tilde{X}^{S}| X_S) = P(X|X_S)$. Also, it is conditionally independent of the remaining covariates and prediction target  ($\\tilde{X}^{S} \\Perp (X_{D \\backslash S}, Y) | X_S$).\n\\label{def:ar}\n\\end{definition}\nAssociative importance provides insight into the dependence structure in the underlying dataset. Nonzero associative importance indicates that $X_J$ contains prediction-relevant information that is not included in the context variables $X_C$.\n\\begin{proposition} [Associative Sensitivity of AI]\nNonzero associative importance $AI(X_J|X_C;f,f)$ based on the non-marginalized function $f$ implies that the target $Y$ is conditionally dependent with $X_J$ given $X_C$, i.e. $Y \\not \\idp X_J | X_C$. If the marginalized functions $f_C$ and $f_{C \\cup J}$ are used, nonzero associative importance $AI(X_J|X_C;f_C, f_{C \\cup J})$ implies conditional dependence of $Y$ with $X_J$ given $X_C$, i.e. $Y \\not \\idp X_J | X_C$ if (1) cross entropy or mean squared error is used as loss and (2) $f$ is the respective loss-optimal predictor.\n\\label{proposition:tar-associative-sensitivity}\n\\end{proposition}\nYet, for the computation of associative importance not only the distribution of the features $\\underline{X}_J$ that correspond to the variables of interest $X_J$ but also all the features corresponding to dependent variables of $X_J$ is affected. Associative importance therefore enters the model directly and\/or indirectly. As a result, it fails to provide insight into which features the model uses for its prediction. An illustrative example for the lacking causal interpretation of associative importance is given in Section \\ref{sec:introduction}.\\\\\nWith the right choice of parameters, the definitions of associative importance and direct importance yield well-established IML methods. An overview is given in Table \\ref{table:influence-family-members}. In order to estimate the measures based on real data, the expected loss (risk) can be replaced by the empirical risk.\\\\\n\\begin{table}[H]\n\\centering\n\\begin{tabular}{l|l|l|l}\n\\textbf{type} & \\textbf{$(g,g')$} & \\textbf{sets} & \\textbf{method}\\\\\n\\midrule\n\\textbf{DI}  & $(f, f)$ & $K = \\{k\\}$ $B = D \\backslash K$, & Permutation Feature Importance of $k$ \\cite{Breiman2001rf}\\\\\n\n & $(f_{\\emptyset}, f_{S})$ & $K = S$, $B = \\emptyset$ & marginal SAGE $v(S)$ \\cite{covert_understanding_2020} \\\\\n\n\\midrule\n\\textbf{AI} & $(f_{C}, f_{C \\cup J})$ & $J = S$, $C = \\emptyset$ & conditional SAGE $v(S)$ \\cite{covert_understanding_2020}\\\\\n\n & $(f, f)$ & $J=\\{j\\}$, $C = D \\backslash J$ & Conditional Feature Importance of $j$ \\cite{Strobl2008}\\\\\n\\end{tabular}\n\\caption{Parameter choices for direct importance (DI) and associative importance (AI) that yield well-established IML methods.}\n\\label{table:influence-family-members}\n\\end{table}\n\\section{Components of Associative and Direct Importance}\n\\label{sec:iai-via-and-di-from}\nIn Section \\ref{sec:disentangling-direct-and-indirect} classes of direct and associative importance measures have been formally introduced. We have shown that both measures have strengths and weaknesses: While direct importance measures allow the exposure of those features that causally influence the prediction performance, they fail to uncover leakage of information from variables that the model does not directly use for its prediction. In contrast to that, associative importance measures can uncover leakage of information from variables of interest, irrespective of whether the corresponding feature is causal for the model's prediction and performance. However, as opposed to direct importance, associative importance measures do not incline a causal interpretation: If the associative importance for a variable is nonzero, we cannot infer that the corresponding feature is causal for the prediction. Even in combination, the methods fail to provide insight into both the sources of prediction-relevant information and the feature pathways, which allow the information to enter the model.\\\\\nIn order to mitigate this inherent trade-off, we propose two measures. Firstly, \\textit{DI from} that quantifies the component of the direct importance of a feature of interest $\\underline{X}_K$ that can be explained with information from a set of variables $X_J$. Using \\textit{DI from}, we get insight into the sources of information that enter the model via a feature $\\underline{X}_K$. Secondly, \\textit{AI via} that quantifies the component of the associative importance of a variable $X_J$ that enters the model via a specific set of features $\\underline{X}_K$. Using \\textit{AI via}, we learn which feature pathways allow the information of variables $X_J$ to enter the model.\\\\\n\\input{figures\/dag-intuition-tai-via-di-from}\n\\subsection{Components of Direct Importance}\nWhen computing the direct importance of a feature, we compare the prediction when $\\underline{X}_K$ is fully perturbed with the prediction when $\\underline{X}_K$ is fully reconstructed. Now, we would like to understand whether $X_J$ contributes to the direct importance of $\\underline{X}_K$. Therefore, instead of fully reconstructing $\\underline{X}_K$ we only partially reconstruct the feature using information from $X_J$. As such, we quantify the degree to which the direct importance of $\\underline{X}_K$ can be explained with $X_J$. An illustration of this procedure is given in Figure \\ref{fig:dag-intuition-tai-via-di-from} (a)-(b).\\\\\n\\begin{definition}[DI from]\nFor the direct importance of features $K$ given baseline $B$, we define the associative component that can be attributed to variables $J$ as \n\\begin{align*}\n\tDI(\\underline{X}_K|\\underline{X}_B \\leftarrow X_J;g, g')\n\t:= &\\mathbb{E}[\\mathcal{L}(Y; \\hat{Y}_g|{do(\\underline{X}_{D \\backslash B} = \\tilde{X}_{D \\backslash B}^{\\emptyset})})] \\\\\n\t- &\\mathbb{E}[\\mathcal{L}(Y; \\hat{Y}_{g'}|{do(\\underline{X}_{D \\backslash B} = \\tilde{X}_{D \\backslash B}^{\\emptyset},  \\underline{X}_{K} = \\tilde{X}_{K}^{J})})]\n\\end{align*}\nwhere $(g, g') \\in \\{(f, f),(f_{B}, f_{(B \\cup K)})\\}$. In addition, $\\tilde{X}^\\emptyset$ and $\\tilde{X}^J$ satisfy the following requirements: $P(\\tilde{X}^\\emptyset) = P(X)$ and $\\tilde{X}^\\emptyset \\Perp X$ as well as $P(\\tilde{X}^J|X_J) = P(X|X_J)$ and $\\tilde{X}^J \\Perp X | X_J$.\n\\label{def:tdi-from}\n\\end{definition}\nIntuitively, if the variables $X_J$ do not contain information, the respective features $\\underline{X}_K$ cannot be reconstructed at all. Consequently, the direct importance $DI(\\underline{X}_K|\\underline{X}_B)$ cannot be attributed to $X_J$. In contrast, if $X_J$ were to be perfectly correlated with $\\underline{X}_K$, the feature is fully reconstructed and the full relevance can be attributed to $X_J$.\\\\\nIn scenarios where all features except for the features of interest are reconstructed in the baseline $B = D \\backslash K$, \\textit{DI from} is complementary to Relative Feature Importance \\cite{konig_relative_2021}, which quantifies the component of Permutation Feature Importance that cannot be attributed to a user-defined set of variables $X_G$.\\\\\n    \n\\subsection{Components of Associative Importance}\n\nWhen computing the associative importance $AI(X_J|X_C)$, we compare the prediction when \\textit{all} features are reconstructed using $X_C$ with the prediction when \\textit{all} features are reconstructed using $X_C$ and $X_J$. In order to gain insight into the direct feature contributions to the associative importance, we want to quantify how much specific feature pathways $\\underline{X}_K$ contribute to the overall score. Therefore, instead of reconstructing the information from $X_J$ in all features, we only update the features of interest $\\underline{X}_K$. In other words, we allow information from $X_J$ over $X_C$ to enter the model via $\\underline{X}_K$ while blocking all other feature paths. An illustration is provided in Figure \\ref{fig:dag-intuition-tai-via-di-from} (c)-(d).\\\\\n\\begin{definition}[AI via]\nThe Associative Importance of variable set $X_J$ given context variables $X_C$ via features $\\underline{X}_K$ is defined as\n\\begin{align*}\n\tAI(X_J|X_C \\rightarrow \\underline{X}_K;g,g') \n\t= &\\mathbb{E}[\\mathcal{L}(Y; \\hat{Y}_{g}|{do(\\underline{X}_D = \\tilde{X}_D^{C})})] \\\\\n\t- &\\mathbb{E}[\\mathcal{L}(Y; \\hat{Y}_{g'}|{do(\\underline{X}_{K} = \\tilde{X}_{K}^{C \\cup J}, \\underline{X}_{D \\backslash K} = \\tilde{X}_{D \\backslash K}^{C})})]\\\\\n\t= &AI(X_J|X_C;g,g') | {do(\\underline{X}_{D \\backslash K} = \\tilde{X}_{D \\backslash K}^{C})}\n\\end{align*}\nfor $(g,g') \\in \\{(f,f), (f_{C},f_{C \\cup J})\\}$. Furthermore, $\\tilde{X}^C$ and $\\tilde{X}^{C \\cup J}$ satisfy the following requirements: $P(\\tilde{X}^C | X_C) = P(X | X_C)$ and  $\\tilde{X}^C \\Perp X | X_C$ as well as $P(\\tilde{X}^{C \\cup J} | X_{C \\cup J}) = P(X | X_{C \\cup J})$ and $\\tilde{X}^{C \\cup J} \\Perp X | X_{C \\cup J}$.\n\\label{def:tai-via}\n\\end{definition}\nIntuitively, \\textit{AI via} can be thought of as the direct importance of a feature of interest, except that instead of fully perturbing the features only surplus information w.r.t to $X_C$ is removed, and that features are only reconstructed with respect to $X_C$ and $X_J$ instead of fully reconstructing them.\\\\\n\\\\\nIt should be noted, that both measures are based on model evaluations on mixed joint distributions of the form $\\tilde{X}_{K}^{C \\cup J}, \\tilde{X}_{D \\backslash K}^{C}$. As the variables are being defined independently, the components are only linked via the conditioning variable $C$. Consequently, if $X_K$ and $X_{D \\backslash K}$ are dependent conditional on $X_C$, the mixed joint distribution does not preserve the original covariate joint. Such extrapolation may cause misleading interpretations \\cite{hooker_please_2019,molnar2020pitfalls}. We conjecture that coupling the variables, e.g. with conditional-rank preserving independent representation learning, can avoid unnecessary extrapolation \\cite{lum_statistical_2016,johndrow_algorithm_2019,Zhao2020Conditional}. However, this investigation goes beyond the scope of the article.\n\n\\section{Decomposing Associative and Direct Importance with DEDACT}\n\\label{sec:decomposition}\n\nIn the previous Section, we have defined two measures: \\textit{DI from} that quantifies the component of a direct importance that can be attributed to a set of variables $X_J$. And \\textit{AI via} that quantifies the component of the associative importance that enters the model via a set of features $\\underline{X}_K$.\\\\\nWe can leverage these measures in various ways to yield a decomposition of associative and direct importance. The choice of decomposition depends on the prerequisites we place on the method: For scenarios where computational efficiency is prioritized, we introduce a fast decomposition in Section \\ref{subsec:fast-decomposition}. For situations where computational resources are available and a fair attribution is prioritized, we propose an additive Shapley value based decomposition in Section \\ref{subsec:shapley-decomposition}.\n\\begin{figure}[H]\n\\centering\n    \\input{figures\/tikz-intuition-attribution}\n    \\caption{Notation for decompositions of direct and associative importance. (a) The associative importance of a variable (denoted as $\\alpha_j$) is decomposed into its feature contributions (denoted as $\\alpha_{jk}$). (b) The direct importance of a feature (denoted as $\\beta_k$) is decomposed into its variable contributions (denoted as $\\beta_{kj}$). The feature\/variable of interest is highlighted in orange, the respective direct and indirect components are highlighted in green and blue.}\n    \\label{fig:my_label}\n\\end{figure}\n\\subsection{DEDACT: Fast Decomposition}\n\\label{subsec:fast-decomposition}\n\nFor the purpose of computational efficiency, we restrict the fast decomposition to require the evaluation of only one further \\textit{AI via}\/\\textit{DI from} configuration per evaluation of $AI$\/$DI$ and component. Furthermore, we prioritize sensitivity, referred to the exposure of sources of information and feature pathways, over additivity. One direct and one associative importance measure will be considered as examples.\\\\\n\\\\\n\\textbf{Permutation Feature Importance:} In our notation, Permutation Feature Importance \\cite{Breiman2001rf} is written as:\n$$\\beta_k := PFI_k = DI(\\underline{X}_k | \\underline{X}_{D \\backslash k}; f, f)$$\nUsing a single evaluation of \\textit{DI from}, we can quantify the component of the feature importance that can be explained by a specific variable $X_j$.\n$$\\beta_{kj} := DI(\\underline{X}_k | \\underline{X}_{D \\backslash k} \\leftarrow X_j; f, f)$$\nThe component $\\beta_{kj}$ represents the total contribution that the variable $X_j$ is able to provide in isolation. Nevertheless, the contribution that $X_j$ can only deliver in cooperation with a further feature $X_l$ is not quantified. Further, if variables $X_j$ and $X_l$ are dependent, then both are fully attributed with the shared contribution. Consequently, the attributions are not guaranteed to add up to the overall importance $\\beta_k$.\\\\\nIn order to achieve an \\textit{additive} decomposition with just one evaluation per feature and component, we would need to define an order of the variables $\\pi$. This estimate would then be\n$$\\beta_{kj}^{\\pi} = DI(\\underline{X}_K | \\underline{X}_{D \\backslash k} \\leftarrow X_{S \\cup j}) - DI(\\underline{X}_K | \\underline{X}_{D \\backslash k} \\leftarrow X_S),$$\nwhere $S$ is the set of features that appears earlier in the order $\\pi$. Although such an order based decomposition is additive, it has a major disadvantage: The attribution of relevance to the variables is strictly prioritized. Variables that appear earlier in the order are fully attributed with the contribution that they share with variables that appear later in the order. As such, features that contain prediction-relevant information that enters via $\\underline{X}_k$ but that share their full contribution with variables that appear earlier in the order, are deemed irrelevant. Additionally, the contribution that two variables jointly achieve is fully attributed to the variable that appears later in the order.\nConsequently, since sensitivity is prioritized, $\\beta_{kj}$ is to be preferred over $\\beta_{kj}^{\\pi}$.\\\\\n\\\\\n\\textbf{Conditional SAGE:} SAGE \\cite{covert_understanding_2020} is based on a linear combination of value functions $v$. In order to compute the surplus contribution of variable $j$ given the set of features $C$, the difference in value with and without the respective variable is computed. In our notation, these differences can be written as\n$$\\alpha_j^C = v(C \\cup j) - v(C) = AI(X_j|X_S; f_C, f_{C \\cup i}).$$\nTherefore, we can leverage \\textit{AI via} to compute the component of the associative importance that enters the model via feature $\\underline{X}_k$. If we were to introduce the feature in isolation, interactions between $\\underline{X}_k$ and other features cannot be restored as they are left perturbed. As we prioritize sensitivity, we instead compare the total importance with the importance, when only the feature of interest is removed. The contribution of the interaction is then attributed to every partaking feature.\n$$\\alpha_{jk}^C = \\alpha_j^C - AI(X_j|X_C \\rightarrow \\underline{X}_{D \\backslash k}; f_C, f_{C \\cup j})$$\nThe SAGE attribution $\\phi_j$ for feature $X_j$ is a linear combination of value function evaluations $\\alpha_j^C$. We yield the component of the SAGE value that can be attributed to feature $\\underline{X}_k$ by replacing the difference terms $\\alpha_j^C$ with the respective $\\alpha_{jk}^C$ values that only account for the contribution of feature $\\underline{X}_k$.\\\\\n\\\\\n\\textbf{Computational complexity:} If the decomposition is to be performed for all $d$ features\/variables, the decomposition scales in $\\mathcal{O}(d^2)$. It is possible to decompose subsets of the features\/variables into the components corresponding to a subset of the features\/variables. I.e., in order to decompose the relevance of one feature\/variable with respect to $m$ components, the number of additional \\textit{DI from}\/\\textit{AI via} evaluations scales in $\\mathcal{O}(m)$. \n\\subsection{DEDACT: Shapley Based Decomposition}\n\\label{subsec:shapley-decomposition}\nIn order to achieve an additive decomposition of associative and direct importance, the game theoretic Shapley values \\cite{shapley1953value} can be adduced. As some recent work elaborates \\cite{Datta2017,lundberg_unified_2017,covert_understanding_2020}, attribution of importance can be seen as a cooperative game in which players (features\/variables) collaborate to yield a certain payoff (importance). As the surplus contribution of a player depends on which players are already in the room, the Shapley value averages the surplus contribution of the feature with respect to all possible configurations. Shapley values yield the unique attribution that satisfies a number of fairness axioms \\cite{shapley1953value} (Appendix \\ref{appendix:shapley}).\\\\\nIn order to apply Shapley values to decompose direct and associative importance measures, we need to define the corresponding value functions $w$. Then, the attribution $\\phi_i$ is defined as\n$$ \\phi_i(w) = \\frac{1}{d} \\sum_{S \\subseteq D \\backslash \\{i\\}} \\binom{d-1}{|S|}^{-1} [w(S \\cup \\{i\\}) - w(S)].$$\n\\textbf{Permutation Feature Importance:} In order to formulate the decomposition of Permutation Feature Importance as a cooperative game, we leverage \\textit{DI from} to quantify the payoff for a team of players $X_J$.\\\\\n$$ w_k(J) := DI(\\underline{X}_k|\\underline{X}_{D \\backslash k} \\leftarrow X_J; f, f)$$\n\\textbf{Conditional SAGE:} The decomposition of the SAGE value function $v$ itself can be formulated as a cooperative game. The corresponding value function $w$ is defined as \n$$ w_j^C(K) := AI(X_j|X_C \\rightarrow \\underline{X}_K; f_C, f_{C \\cup j})$$\n\\textbf{Computational complexity:} As the number of possible subsets grows exponentially in the number of features, the computation of Shapley values is expensive. Like previous work in the field, we suggest to approximate Shapley values by randomly sampling and evaluating orders until the estimates converge \\cite{lundberg_unified_2017,covert_understanding_2020}.\n\n\\section{Simulations}\n\\label{sec:application}\nWe illustrate the usefulness of the approach on two simulated examples. Both provide access to the ground-truth causal graphs, which allows us to validate the interpretation. Furthermore, we chose all relationships to be linear with Gaussian noise, because in this setting reliable conditional distribution estimation is readily available.\\footnote{The code for our experiments is available in an anonoymized repository \\href{https:\/\/github.com\/anonomyzed-submission\/dedact-code}{[click here]}.}\n\\begin{example}[Biomarker Failure]\nWe consider a \\textit{prostate cancer} ($Y$) diagnosis setting. For the model training and evaluation, we don't have access to the true $Y$, but only to labels $L$ that are wrongfully influenced by \\textit{PSA} $P$.\\footnote{The \\textit{prostate specific antigen (PSA)} was wrongfully used in clinical practice for several decades \\cite{ioannidis_biomarker_2013}, leading to over-treatment and its retraction from clinical usage.} In order to avoid bias from \\textit{PSA}, the model is fit with access to two variables only: \\textit{biomarker} $B$ and \\textit{cycling} $C$. However, the model learns to use \\textit{cycling} as a proxy for \\textit{PSA},\\footnote{Studies suggest that cycling increases PSA levels \\cite{jiandani2015effect}.} such that the bias leaks into the model. A fast DEDACT decomposition of the associative influence \\textit{PSA} exposes that the variable leaks into the model via the dependent \\textit{cycling} feature (Figure \\ref{fig:psa-results} (a)). Further, a fast DEDACT decomposition of the PFI of the feature \\textit{cycling} shows that it's importance can be fully attributed to \\textit{PSA}, suggesting the removal of the feature from the model.\n\\label{example:psa-di-vs-ii}\n\\end{example}\n\\begin{figure}[]\n    \\centering\n    \\input{figures\/figures_decomposition\/wbars_relevance_types}\n\\caption{(a) Associative importance $AI(X_j; f, f)$ and the respective fast decomposition for $PSA$. (b) Direct importance (PFI) and the respective fast decomposition for \\textit{cycling}. (c) The DAG of the ground-truth Structural Causal Model. \\textit{PSA} ($P$) is causal for the historical labels, although only the \\textit{biomarker} ($B$) is relevant for the actual condition. The model uses \\textit{cycling} ($C$) as a proxy, thereby enabling the bias to leak into the model.}\n\\label{fig:psa-results} \n\\end{figure}\n\\begin{example}[Sensitive Attributes]\nIn this hypothetical scenario, adapted from the census income dataset from the UCI Repository \\cite{Dua:2019}, the aim is the to predict the income of subjects. The protected attributes \\textit{age}, \\textit{sex} and \\textit{race} affect the income of subjects directly and indirectly. \\textit{Age} only affects income via \\textit{capital gain}, \\textit{number of educations} and \\textit{hours per week}. \\textit{Race} causes income directly as well as indirectly via \\textit{marriage status} and \\textit{occupation}. And \\textit{sex} influences the target directly as well as indirectly via \\textit{relationship} and \\textit{work class}. The corresponding causal graph is depicted in Figure \\ref{fig:census-income}.\\footnote{We emphasize that this scenario is hypothetical and do not want to suggest that the postulated causal structure reflects the real world.}\\\\\nThe Shapley based DEDACT decomposition of SAGE values for features \\textit{race}, \\textit{sex} and \\textit{age} correctly identifies the features, via which the information from the respective variable enters the model (Figure \\ref{fig:ii-adult-sage}). The Shapley based DEDACT decomposition of PFI correctly identifies the respective sources of prediction-relevant information (Figure \\ref{fig:ii-adult-viafrom}).\n\\end{example}\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/figures_decomposition\/ii-adult-sparse_0401_sage_decomp_sage_sub.pdf}\n    \\caption{Approximation of the Shapley based decomposition of SAGE values ($60$ SAGE orders, $25$ decomposition orders). The procedure correctly identifies the feature pathways which allow the associative importance to enter the model.}\n    \\label{fig:ii-adult-sage}\n\\end{figure}\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/figures_decomposition\/ii-adult-sparse_0401_sage_decomp_tdi_decmp_sub.pdf}\n    \\caption{Shapley decomposition of PFI ($50$ orders). The sources of prediction relevant information for the features \\textit{education num}, \\textit{workclass} and \\textit{occupation} are correctly identified.}\n    \\label{fig:ii-adult-viafrom}\n\\end{figure}\n\\begin{figure}[]\n    \\centering\n    \\begin{tikzpicture}[thick, scale=1.5, every node\/.style={scale=.7, line width=0.25mm, black, fill=white}]\n        \\usetikzlibrary{shapes}\n\t\t\\node[draw, ellipse, scale=0.7] (sex) at (2, .5) {sex};\n\t\t\\node[draw, ellipse, scale=0.7] (age) at (-2, .5) {age};\n\t\t\\node[draw, ellipse, scale=0.7] (race) at (0,.5) {race};\n\t\t\\node[draw, ellipse, scale=0.7] (inc) at (1,-.5) {income};\n\t\n\t    \\draw[->,gray] (race) -- (inc);\n        \\draw[->,gray] (sex) -- (inc);\n\t\n\t\t\\node[draw, ellipse, scale=0.7] (hpw) at (-1,0) {hours pw};\n\t\t\\node[draw, ellipse, scale=0.7] (ne) at (-2,0) {nr educ};\n\t\t\\node[draw, ellipse, scale=0.7] (wc) at (3,0) {work class};\n\t\t\\node[draw, ellipse, scale=0.7] (oc) at (1,0) {occupation};\n\t\t\\node[draw, ellipse, scale=0.7] (cg) at (-3,0) {capital gain};\n\t\t\\node[draw, ellipse, scale=0.7] (ms) at (0,0) {marriage status};\n\t\t\\node[draw, ellipse, scale=0.7] (rel) at (2,0) {relationship};\n\t\t\n\t\t\\draw[->] (sex) -- (rel);\n\t\t\\draw[->] (sex) -- (wc);\n\t\t\\draw[->] (age) -- (hpw);\n\t\t\\draw[->] (age) -- (cg);\n\t\t\\draw[->] (age) -- (ne);\n\t\t\\draw[->] (race) -- (ms);\n\t\t\\draw[->] (race) -- (oc);\n\n        \n        \\draw[->, gray] (hpw) -- (inc);\n        \\draw[->, gray] (ne) -- (inc);\n        \\draw[->, gray] (wc) -- (inc);\n        \\draw[->, gray] (oc) -- (inc);\n        \\draw[->, gray] (cg) -- (inc);\n        \\draw[->, gray] (ms) -- (inc);\n        \\draw[->, gray] (rel) -- (inc);\n        \n    \\end{tikzpicture}\n    \\caption{Ground truth DAG for the simulated adult dataset. Gray edges indicate parent edges for predicted income.}\n    \\label{fig:census-income}\n\\end{figure}\n\\section{Discussion}\n\nWe propose DEDACT, a general framework that allows to decompose associative importance measures into their direct feature pathway contributions as well as direct importance measures into the contributions of their information sources. Thereby, DEDACT enables novel insight into model and data. DEDACT does not only uncover leakage of information but also explains via which feature pathways the influence enters the model.\\\\\nA fast but non-additive, as well as an additive and fair, but computationally demanding decomposition have been proposed. The approach relies on conditional samples that established methods like conditional SAGE \\cite{covert_understanding_2020} require to compute anyway. Thus, the computational overhead can be reduced.\\\\\nFurther research is required evaluating how well the method scales to more complex distribution and model types. Moreover, the question of how mixed perturbations with different conditioning sets can be coupled to reduce extrapolation is a promising avenue for future research. We conjecture that methods on information preserving independent representation learning could be leveraged. In addition, future work will assess how knowledge about the dependence structure in the graph can be used to reduce the computational complexity of the method. We see DEDACT as a step towards understanding machine learning methods within the context they operate in: the underlying data generating mechanism.\\\\\n\n \\bibliographystyle{splncs04}\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nWith growing diversity in personal food preference and regional cuisine style, personalized information systems that can transform a recipe into any selected regional cuisine style that a user might prefer would help food companies and professional chefs create new recipes.\n\n\nTo achieve this goal, there are two significant challenges: 1) identifying the degree of regional cuisine style mixture of any selected recipe; and 2) developing an algorithm that shifts a recipe into any selected regional cuisine style. \n\nAs to the former challenge, with growing globalization and economic development, it is becoming difficult to identify a recipe's regional cuisine style with specific traditional styles since regional cuisine patterns have been changing and converging in many countries throughout Asia, Europe, and elsewhere \\cite{khoury2014increasing}. Regarding the latter challenge, to the best of our knowledge, little attention has been paid to developing algorithms which transform a recipe's regional cuisine style into any selected regional cuisine pattern, cf. \\cite{pinel2014using,pinel2014substitution}. Previous studies have focused on developing an algorithm which suggests replaceable ingredients based on cooking action \\cite{shidochi2009finding}, degree of similarity among ingredient \\cite{nozawa}, ingredient network \\cite{teng2012recipe}, degree of typicality of ingredient \\cite{yokoi2015typicality}, and flavor (foodpairing.com).\n\nThe aim of this study is to propose a novel data-driven system for transformation of regional cuisine style. This system has two characteristics. First, we propose a new method for identifying a recipe's regional cuisine style mixture by calculating the contribution of each ingredient to certain regional cuisine patterns, such as Mediterranean, French, or Japanese, by drawing on ingredient prevalence data from large recipe repositories. Also the system visualizes a recipe's regional cuisine style mixture in two-dimensional space under barycentric coordinates using what we call a {\\it Newton diagram}. Second, the system transforms a recipe's regional cuisine pattern into any selected regional style by recommending replaceable ingredients in existing recipes.\n\nAs an example of this proposed system, we transform a traditional Japanese recipe, Sukiyaki, into French style.\n\n\\section{Architecture of transformation system}\nFigure \\ref{fig1} shows the overall architecture of the transformation system, which consists of two steps: 1) identification and visualization of a recipe's regional cuisine style mixture; and 2) algorithm which transforms a given recipe into any selected regional\/country style. Details of the steps are described as follows.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=15cm]{fig1}\n\\end{center}\n\\caption{ Architecture of transformation system which transform a given recipe into any selected country\/region style}\\label{fig1}\n\\label{fig1}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=15cm]{fig2}\n\\end{center}\n\\caption{ Neural network model for predicting regional cuisine from list of ingredients.}\n\\label{fig2}\n\\end{figure}\n\n\\subsection{Step 1: Identification and visualization of a recipe's regional cuisine style mixture}\nUsing a neural network method as detailed below, we identify a recipe's regional cuisine style. The neural network model was constructed as shown in Figure \\ref{fig2}. The number of layers and dimension of each layer are also shown in Figure \\ref{fig2}. \n\nWhen we enter a recipe, this model classifies which country or regional cuisine the recipe belongs to. The input is a vector with the dimension of the total number of ingredients included in the dataset, and only the indices of ingredients contained in the input recipe are 1, otherwise they are 0. \n\nThere are two hidden layers. Therefore, this model can consider a combination of ingredients to predict the country probability. Dropout is also used for the hidden layer, randomly (20\\%) setting the value of the node to 0. So a robust network is constructed. The final layer's dimension is the number of countries, here 20 countries. In the final layer, we convert it to a probability value using the softmax function, which represents the probability that the recipe belongs to that country. \nADAM \\cite{kingma2014adam} was used as an optimization technique. The number of epochs in training was 200. These network structure and parameters were chosen after preliminary experiments so that the neural network could perform the country classification task as efficiently as possible.\n\n\n\\begin{table}[htbp]\n\\begin{center}\n\\caption{Statistics of Yummly dataset and some recipe examples.}\n\\begin{tabular}{cc}\n\\begin{minipage}{0.5\\hsize}\n\\begin{center}\n\\begin{tabular}{|l|c|r|}\n\\hline\nCountry &Recipes & Ingredients  \\\\ \\hline \\hline\nItalian\t&\t7838\t&\t2929\t\\\\ \\hline\nMexican\t&\t6438\t&\t2684\t\\\\ \\hline\nSouthern US\t&\t4320\t&\t2462\t\\\\ \\hline\nIndian\t&\t3003\t&\t1664\t\\\\ \\hline\nChinese\t&\t2673\t&\t1792\t\\\\ \\hline\nFrench\t&\t2646\t&\t2102\t\\\\ \\hline\nCajun Creole\t&\t1546\t&\t1576\t\\\\ \\hline\nThai\t&\t1539\t&\t1376\t\\\\ \\hline\nJapanese\t&\t1423\t&\t1439\t\\\\ \\hline\nGreek\t&\t1175\t&\t1198\t\\\\ \n\\hline\n\\end{tabular}\n\\end{center}\n\\end{minipage}\n\n\\begin{minipage}{0.5\\hsize}\n\\begin{tabular}{|l|c|r|}\n\\hline\nCountry &Recipes & Ingredients  \\\\ \\hline \\hline\nSpanish\t&\t989\t&\t1263\t\\\\ \\hline\nKorean\t&\t830\t&\t898\t\\\\ \\hline\nVietnamese\t&\t825\t&\t1108\t\\\\ \\hline\nMoroccan\t&\t821\t&\t974\t\\\\ \\hline\nBritish\t&\t804\t&\t1166\t\\\\ \\hline\nFilipino\t&\t755\t&\t947\t\\\\ \\hline\nIrish\t&\t667\t&\t999\t\\\\ \\hline\nJamaican\t&\t526\t&\t877\t\\\\ \\hline\nRussian\t&\t489\t&\t872\t\\\\ \\hline\nBrazilian\t&\t467\t&\t853\t\\\\ \\hline \\hline\nALL\t&\t39774\t&\t6714\t\\\\ \n\\hline\n\\end{tabular}\n\\end{minipage}\n\n\\end{tabular}\n\n\\begin{minipage}{0.9\\hsize}\n\\begin{tabular}{|l|c|l|}\n\\hline\nRecipeID &Country& Ingredients  \\\\ \\hline \\hline\n34466 & British & greek yogurt, lemon curd, confectioners sugar, raspberries\\\\ \\hline\n44500  & Indian & chili, mayonaise, chopped onion, cider vinegar, fresh mint, cilantro leaves\\\\ \\hline\n38233  & Thai & sugar, chicken thighs, cooking oil, fish sauce, garlic, black pepper\\\\ \\hline\n\\end{tabular}\n\\end{minipage}\n\n\\label{table1}\n\\end{center}\n\\end{table}\n\n\\begin{table}[htbp]\n\\begin{center}\n\\caption{Example of ingredient classification by the neural network. Three top countries are listed with the probability that the ingredient are classified into.}\n\\begin{tabular}{|c|c|r|r|r|}\n\\hline\n Ingredient & Top1 & Top2 & Top3\\\\ \\hline \\hline\nOnions  & French & Italian & Mexican \\\\ \n  & 0.130 & 0.126 & 0.126 \\\\ \\hline\nSoy sauce  & Japanese & Chinese & Filipino \\\\ \n  & 0.246 & 0.233 & 0.122 \\\\ \\hline\n Mirin & Japanese & French & Korean \\\\ \n & 0.890 & 0.040 &0.020 \\\\ \n\\hline\n\\end{tabular}\n\\label{table1a}\n\\end{center}\n\\end{table}\n\n\nIn this study, we used a labeled corpus of Yummly recipes to train this neural network. Yummly dataset has 39774 recipes from the 20 countries as shown in Table \\ref{table1}. Each recipe has the ingredients and country information. Firstly, we randomly divided the data set into 80\\% for training the neural network and 20\\% for testing how precisely it can classify. The final neural network achieved a classification accuracy of 79\\% on the test set. Figure \\ref{fig.conf} shows the confusion matrix of the neural network classifficaiton. Table \\ref{table1a} shows the examples of ingredient classification results. Common ingredients, onions for example, that appear in many regional recipes are assigned to all countries with low probability. On the other hands some ingredients that appear only in specific country are assigned to the country with high probability. For example mirin that is a seasoning commonly used in Japan is classified into Japan with high probability.\n\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=15cm]{confusion\n\\end{center}\n\\caption{Confusion matrix of neural network classiffication. }\\label{fig.conf}\n\\end{figure}\n\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=15cm]{fig3\n\\end{center}\n\\caption{ Newton diagram: visualization of probability that the recipe belongs to the several regional cuisine style. Countries are placed by spectral drawing.\n}\\label{fig3}\n\\end{figure}\n\n\nBy using the probability values that emerge from the activation function in the neural network, rather than just the final classification, we can draw a barycentric Newton diagram, as shown in Figure \\ref{fig3}.  The basic idea of the visualization, drawing on Isaac Newton's visualization of the color spectrum \\cite{newton1704}, is to express a mixture in terms of its constituents as represented in barycentric coordinates.  This visualization allows an intuitive interpretation of which country a recipe belongs to. If the probability of Japanese is high, the recipe is mapped near the Japanese. The countries on the Newton diagram are placed by spectral graph drawing \\cite{koren2003spectral}, so that similar countries are placed nearby on the circle. The calculation is as follows.\nFirst we define the adjacency matrix $W$ as the similarity between two countries. \nThe similarity between country $i$ and $j$ is calculated by cosine similarity of county $i$ vector and $j$ vector. These vector are defined in next section. $W_{ij} = sim(vec_i, vec_j)$.\nThe degree matrix $D$ is a diagonal matrix where $D_{ii} = \\sum_{j} W_{ij}$. \nNext we calculate the eigendecomposition of $D^{-1}W$. The second and third smallest eingenvalues and corresponded eingevectors are used for placing the countries. Eigenvectors are normalized so as to place the countries on the circle. \n\n\\subsection{Step 2: Transformation algorithm for transforming regional cuisine style}\nIf you want to change a given recipe into a recipe having high probability of a specific country by just changing one ingredient, which ingredient should be alternatively used?\n\nWhen we change the one ingredient $x_i$ in the recipe to ingredient $x_j$, the probability value of country likelihood can be calculated by using the above neural network model. If we want to change the recipe to have high probability of a specific country $c$, we can find ingredient $x_j$ that maximizes the following probability. $P(C=c|r - x_i + x_j)$\nwhere $r$ is the recipe.\nHowever, with this method, regardless of the ingredient $x_i$, only specific ingredients having a high probability of country $c$ are always selected. In this system, we want to select ingredients that are similar to ingredient $x_i$ and have a high probability of country $c$. Therefore, we propose a method of extending word2vec as a method of finding ingredients resembling ingredient $x_i$.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=15cm]{word2vec_arch\n\\end{center}\n\\caption{The word2vec (skip-gram) architecture. The left panel is the traditional word2vec with window size $n=2$. The middle panel is the word2vec for recipe data. The right panel is the word2vec for recipe data with country information. \n}\\label{word2vec_arch}\n\\end{figure}\n\n\nWord2vec is a technique proposed in the field of natural language processing \\cite{mikolov2013distributed}. As the name implies, it is a method to vectorize words, and similar words are represented by similar vectors. To train word2vec, skip-gram model is used. In the skip-gram model, the objective is to learn word vector representations that can predict the nearby words. The objective function is \n\n\\begin{equation}\n\\sum_{d \\in D} \\sum_{w_i \\in d} \\sum_{-n \\leq j \\leq n, j \\neq 0} \\log P(w_{i + j}|w_i) \\label{eq:02}\n\\end{equation}\nwhere $D$ is the set of documents, $d$ is a document, $w_i$ is a word, and $n$ is the window size. This model predicts the $n$ words before and after the input word, as described in left side of Figure \\ref{word2vec_arch}. The objective function is to maximize the likelihood of the prediction of the surrounding word $w_{i+j}$ given the center word $w_i$.\nThe probability is \n\\begin{equation}\nP(w_j|w_i) = \\frac{\\exp(v_{w_i}^Tv_{w_j}^{'})}{\\sum_{w \\in W} \\exp(v_{w_i}^Tv_w^{'})}\n\\end{equation}\nwhere $v_w \\in \\mathbb{R}^K$ is an input vector of word $w$, $v^{'}_w \\in \\mathbb{R}^K$ is an output vector of word $w$, $K$ is the dimension of the vector, and $W$ is the set of all words. To optimize this objective function, hierarchical softmax or negative sampling method \\cite{mikolov2013distributed} are used. After that we get the vectors of words and we can calculate analogies by using the vectors. For example, the analogy of  ``King - Man + Women = ?\" yields ``Queen\" by using word2vec.\n\nIn this study, word2vec is applied to the data set of recipes. Word2vec can be applied by considering recipes as documents and ingredients as words. We do not include a window size parameter, since it is used to encode the ordering of words in document where it is relevant.  In recipes, the listing of ingredients is unordered. The objective function is \n\\begin{equation}\n\\sum_{r \\in R} \\sum_{w_i \\in r}  \\sum_{j \\neq i} \\log P(w_{j}|w_i) \\label{eq:02}\n\\end{equation}\nwhere $R$ is a set of recipes, $r$ is a recipe, and $w_i$ is the $i$th ingredient in recipe $r$. The architecture is described in middle of Figure \\ref{word2vec_arch}. The objective function is to maximize the likelihood of the prediction of the ingredient $w_j$ in the same recipe given the ingredient $w_i$. The probability is defined below.\n\\begin{equation}\nP(w_j|w_i) = \\frac{\\exp(v_{w_i}^Tv_{w_j}^{'})}{\\sum_{w \\in W} \\exp(v_{w_i}^Tv_w^{'})}\n\\end{equation}\nwhere $w$ is an ingredient, $v_w \\in \\mathbb{R}^K$ is an input vector of ingredient, $v^{'}_w \\in \\mathbb{R}^K$ is an output vector of ingredient, $K$ is the dimension of the vector, and $W$ is the set of all ingredients.\n\nEach ingredient is vectorized by word2vec, and the similarity of each ingredient is calculated using cosine similarity. Through vectorization in word2vec, those of the same genre are placed nearby. In other words, by using the word2vec vector, it is possible to select ingredients with similar genres. \n\nNext, we extend word2vec to be able to incorporate information of the country. When we vectorize the countries, we can calculate the analogy between countries and ingredients. For example, this method can tell us what is the French ingredient that corresponds to Japanese soy sauce by calculating ``Soy sauce - Japan + French = ?\".\n\nThe detail of our method is as follows. We maximize objective function (\\ref{eq:02}). \n\n\\begin{equation}\n\\sum_{r \\in R} \\sum_{w_i \\in r} \\left(    \\log P(w_{i}|c_r) +  \\log P(c_r|w_{i}) + \\sum_{j \\neq i} \\log P(w_{j}|w_i)\\right)\\label{eq:02}\n\\end{equation}\nwhere $R$ is a set of recipes, $r$ is a recipe, $w_i$ is the $i$th ingredient in recipe $r$, and $c_r$ is the country recipe $r$ belongs to. The architecture is described in right of Figure \\ref{word2vec_arch}. The objective function is to maximize the likelihood of the prediction of the ingredient $w_j$ in the same recipe given the ingredient $w_i$ along with the prediction of the the ingredients $w_i$ given the country $c_r$ and the prediction of the the country $c_r$ given the ingredient $w_i$. The probability is defined below.\n\n\\begin{equation}\nP(b|a) = \\frac{\\exp(v_{a}^Tv_{b}^{'})}{\\sum_{c \\in W} \\exp(v_{a}^Tv_c^{'})}\n\\end{equation}\nwhere $a$ is a ingredient or country, $b,c$ are also, $v_a \\in \\mathbb{R}^K$ is an input vector of ingredient or country, $v^{'}_a \\in \\mathbb{R}^K$ is an output vector of ingredient or country, $K$ is the dimension of vector, and $W$ is the set of all ingredients and all countries.\n\nWe can use hierarchical softmax or negative sampling \\cite{mikolov2013distributed} to maximize objective function (\\ref{eq:02}) and find the vectors of ingredients and countries in the same vector space. \n\nTable \\ref{table2} shows the ingredients around each country in the vector space, and which could be considered as most authentic for that regional cuisine \\cite{ahn2011flavor}. Also, Figure \\ref{fig4} shows the ingredients and countries in 2D map by using t-SNE method \\cite{maaten2008visualizing}.\n\n\\begin{table}[htbp]\n\\begin{center}\n\\caption{Authentic ingredients for each country. Top 5 ingredients around each country in the vector space.}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n & French & Japanese & Italian & Mexican \\\\ \\hline \\hline\nTop1 & Cognac & Mirin & Grated parmesan cheese & Corn tortillas\\\\ \\hline\nTop2  & Calvados & Dashi & pecorino romano cheese & Salsa\\\\ \\hline\nTop3  & Thyme springs & Nori & prosciutto & Tortilla chips\\\\ \\hline\nTop4  & Gruyere cheese & Wasabi paste &marinara sauce & Guacamole\\\\ \\hline\nTop5 & Nicoise olives & Bonito flakes &Sweet italian sausage& Poblano peppers \\\\ \n\\hline\n\\end{tabular}\n\\label{table2}\n\\end{center}\n\\end{table}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=15cm]{word2vec\n\\end{center}\n\\caption{Ingredients and countries map by extended word2vec: Each ingredient and country is mapped in 2D by using t-SNE. Also Each ingredient is colored by using t-SNE to convert 100 dimension vector into 3 dimension. The 3 dimension is corresponded to RGB color. Countries are represented by bold black.}\\label{fig4}\n\\end{figure}\n\n\n\n\\section{Experiment}\nOur substitution strategy is as follows.\nFirst we use extended word2vec and train it by Yummly dataset. After that all ingredients and countries are vectorized into 100 dimensional vector space. Second we find substitution by analogy calculation. For example, to find french substitution of Mirin, we calculate ``Mirin - Japanese + French\" in the vector space and get the vector as result. After that we find similar ingredients around the vector by calculating the cosine similarity.\n\nAs an example of our proposed system, we transformed a traditional Japanese ``Sukiyaki\" into French style. Table \\ref{table3} shows the suggested replaceable ingredients and the probability after replacing. ``Sukiyaki\" consists of soy sauce, beef sirloin, white sugar, green onions, mirin, shiitake, egg, vegetable oil, konnyaku, and chinese cabbage. Figure \\ref{fig:sukiyaki_french} shows the Sukiyaki in French style cooked by professional chef KM who is one of the authors of this paper. He assesses the new recipe as valid and novel to him in terms of Sukiyaki in French. \nHere our task is in generating a new dish, for which by definition there is no ground truth for comparison.\nRating by experts is the standard approach for assessing novel generative artifacts, e.g. in studies of creativity \\cite{jordanous2012standardised}, but going forward it is important to develop other approaches for assessment.\n\n\n\\begin{table}[htbp]\n\\begin{center}\n\\caption{Alternative ingredients suggested by extended word2vec model and country probability of changing food ingredients in order from the top. Professional chef KM who is one of the authors of this paper chose one alternative ingredient from top 10 suggested ingredients each.}\n\\begin{tabular}{|c|c|r|r|r|}\n\\hline\n Original Ingredient & Alternative Ingredient & P(Japanese) & P(French)&\\# of replacement  \\\\ \\hline \\hline\n-  & - & 0.974 & 0.000 & 0\\\\ \\hline\nMirin  & Calvados & 0.552 & 0.009 & 1\\\\ \\hline\nVegetable oil  & Olive oil & 0.393 & 0.031 & 2\\\\ \\hline\nSoy sauce  & Bouquet garni & 0.011 & 0.976 & 3\\\\ \\hline\nGreen onions& Fresh tarragon & 0.000 &0.997 & 4 \\\\ \\hline\nEgg & Melted butter& 0.000 &0.999 & 5 \\\\ \n\\hline\n\\end{tabular}\n\\label{table3}\n\\end{center}\n\\end{table}\n\n\n\\begin{figure}[h!]\n\\begin{minipage}[b]{.25\\linewidth}\n\\centering\\includegraphics[width=3.4cm, angle=90]{sukiyaki_french0\n\\end{minipage}%\n\\begin{minipage}[b]{.25\\linewidth}\n\\centering\\includegraphics[width=3.4cm, angle=90]{sukiyaki_french1\n\\end{minipage}%\n\\begin{minipage}[b]{.25\\linewidth}\n\\centering\\includegraphics[width=3.4cm, angle=90]{sukiyaki_french2\n\\end{minipage}%\n\\begin{minipage}[b]{.25\\linewidth}\n\\centering\\includegraphics[width=3.4cm, angle=90]{sukiyaki_french3\n\\end{minipage}%\n\\caption{Sukiyaki in French style. Professional chef KM who is one of the authors of this paper cooked the recipe suggested by our system.}\\label{fig:sukiyaki_french}\n\\end{figure}\n\n\n\\section{Discussion}\nWith growing diversity in personal food preference and regional cuisine style, the development of data-driven systems which can transform recipes into any given regional cuisine style might be of value for food companies or professional chefs to create new recipes.\n\nIn this regard, this study adds two important contributions to the literature. First, this is to the best of our knowledge, the first study to identify a recipe's mixture of regional cuisine style from the large number of recipes around the world. Previous studies have focused on assessing degree of adherence to a single regional cuisine pattern. For example, Mediterranean Diet Score is one of the most popular diet scores. This method uses 11 main items (e.g., fruit, vegetable, olive oil, and wine) as criteria for assessing the degree of one's Mediterranean style \\cite{panagiotakos2006dietary}. However, in this era, it is becoming difficult to identify a recipe's regional cuisine style with specific country\/regional style. \nFor example, should Fish Provencal, whose recipe name is suggestive of Southern France, be cast as French style? The answer is mixture of different country styles: 32\\% French; 26\\% Italian; and 38\\% Spanish (see Figure \\ref{fig3}).\n\nFurthermore, our identification algorithm can be used to assess the degree of personal regional cuisine style mixture, using the user's daily eating pattern as inputs. For example, when one enters the recipes that one has eaten in the past week into the algorithm, the probability values of each country would be returned, which shows the mixture of regional cuisine style of one's daily eating pattern. As such, a future research direction would be developing algorithms that can transform personal regional cuisine patterns to a healthier style by providing a series of recipes that are in accordance with one's unique food preferences.\n\nOur transformation algorithm can be improved by adding multiple datasets from around the world. Needless to say, lack of a comprehensive data sets makes it difficult to develop algorithms for transforming regional cuisine style. For example, Yummly, one of the largest recipe sites in the world, is less likely to contain recipes from non-Western regions. Furthermore, data on traditional regional cuisine patterns is usually described in its native language. As such, developing a way to integrate multiple data sets in multiple languages is required for future research. \n\nOne of the methods to address this issue might be as follows: 1) generating the vector representation for each ingredient by using each data set independently; 2) translating only a small set of common ingredients among each data set, such as potato, tomato, and onion; 3) with a use of common ingredients, mapping each vector representation into one common vector space using a canonical correlation analysis \\cite{kettenring1971canonical}, for example. \n\nSeveral fundamental limitations of the present study warrant mention. First of all, our identification and transformation algorithms depend on the quantity and quality of recipes included in the data. As such, future research using our proposed system should employ quality big recipe data.\nSecond, the evolution of regional cuisines prevents us from developing precise algorithm. For example, the definition of Mediterranean regional cuisine pattern has been revised to adapt to current dietary patterns \\cite{serra2004does,kinouchi2008non}. Therefore, future research should employ time-trend recipe data to distinctively specify a recipe's mixture of regional cuisine style and its date cf.~\\cite{varshney2013flavor}. \nThird, we did not consider the cooking method (e.g., baking, boiling, and deep flying) as a characteristic of country\/regional style. Each country\/region has different ways of cooking ingredients and this is one of the important factors characterizing the food culture of each country\/region. \nFourth, the combination of ingredients was not considered as the way to represent country\/regional style. For example, previous studies have shown that Western recipes and East Asian recipes are opposite in flavor compounds included in the ingredient pair \\cite{zhu2013geography,varshney2013flavor,jain2015spices,tallab2016exploring,ahn2011flavor}. For example, Western cuisines tend to use ingredient pairs sharing many flavor compounds, while East Asian cuisines tend to avoid compound sharing ingredients. It is suggested that combination of flavor compounds was also elemental factor to characterize the food in each country\/region. As such, if we analyze the recipes data using flavor compounds, we might get different results.\n\n\n\nIn conclusion, we proposed a novel system which can transform a given recipe into any selected regional cuisine style. This system has two characteristics: 1) the system can identify a degree of regional cuisine style mixture of any selected recipe and visualize such regional cuisine style mixture using a barycentric Newton diagram; 2) the system can suggest ingredient substitution through extended word2vec model, such that a recipe becomes more authentic for any selected regional cuisine style. Future research directions were also discussed.\n\n\n\n\\section*{Conflict of Interest Statement}\n\nThe authors declare that they have no conflict of interest.\n\n\\section*{Author Contributions}\nMK, LRV, and YI had the idea for the study and drafted the manuscript. MK performed the data collection and analysis. MS, CH, and KM participated in the interpretation of the results and discussions for manuscript writing and finalization. All authors read and approved the final manuscript.\n\n\n\\section*{Funding}\nVarshney's work was supported in part by the IBM-Illinois Center for Cognitive\nComputing Systems Research (C3SR), a research collaboration as part of the\nIBM AI Horizons Network.\n\n\n\\section*{Acknowledgments}\nThis study used data from Yummly. We would like to express our deepest gratitude to everyone who participated in this services. We thank Kush Varshney for suggesting the spectral graph drawing approach to placing countries on the circle.\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzuxqg b/data_all_eng_slimpj/shuffled/split2/finalzzuxqg
new file mode 100644
index 0000000000000000000000000000000000000000..206939f3a77d589be57063085f2d53c1005acab3
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzuxqg
@@ -0,0 +1,5 @@
+{"text":"Click the \"My Account \/ Order Status\" link at the top right hand side of our site to check your orders status. Orders are shipped out at the end of each business day. You should also receive an email with your tracking information within 24 hrs of your item being shipped.\nClick the \"My Account \/ Order Status\" link at the top right hand side of our site to track your order. Orders are shipped out at the end of each business day. You should also receive an email with your tracking information within 24 hrs of your item being shipped.\nClick the \"My Account \/ Order Status\" link at the top right hand side of our site to track your order status. Be sure that all of the items in your order have shipped already. If you order displays your Package Tracking Numbers, check with the shipper to confirm that your packages were delivered. If your packages each show a status of \"delivered\", please contact customer service for assistance. Customer service contact information can be found by clicking the \"Contact Us\" link found at the bottom of the page.\nI received the wrong parts.\nContact customer service immediately. Customer service contact information can be found by clicking the \"Contact Us\" link found at the bottom of the page.\nBackordered items will generally ship within 1-3 weeks. To get a better idea of when we think your order might ship please contact customer service. Customer service contact information can be found by clicking the \"Contact Us\" link found at the bottom of the page.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Fashioned like a small Portuguese village of typical terracotta buildings, fertile gardens, and meandering paths, Luz Houses offers the ultimate all-encompassing experience \u2013 freeing for the mind, purifying for the soul, and uplifting for the spirit.\nTucked into a peaceful forest setting about an hour's drive from Lisbon, this F\u00e1tima boutique hotel fuses chic design with nature and spirituality, providing guests with a special back-to-nature getaway.\nSoothing whites, soft pastel colors, and warm wooden details imbue the interiors with a sense of calm and understated sophistication. From the central meeting place to the immaculate sleeping quarters, all spaces celebrate the outdoors through a stylish juxtaposition of rustic and industrial elements.\nAt Luz Houses, being close to nature doesn't mean compromising on comfort \u2013 every unit is appointed with free Wi-Fi, Smart TV, soft bathrobes, and tea & coffee facilities.\nRooms are chic and cozy open-plan sanctuaries replete with white surfaces, organic materials, and contemporary amenities. All are tastefully decorated in a mix of country chic furnishings and unpredictably modern design touches, with polished concrete floors, fluffy beds, and blissful deep soaking tubs.\nThe expansive suites come with cute kitchenettes and separate living areas, while the superior ones feature idyllic fireplaces and private terraces for enjoying the fresh air and contemplating nature.\nComprising the dining room, lounge area, an honesty bar, and a grocery store filled with local products, the creatively designed Mother House is the property's social hub, a place where guests can relax by the fireplace, mix and mingle over drinks and light bites, or just play and share stories in a cool, laid-back setting.\nAs delightful as the interiors are, the hotel's true charms lie outside. Winding paths invite to quiet strolling and meditation, a small hermitage provides spiritual nourishment for the soul, and an ancient natural cave hides a magical little spa, where massages and rituals melt away any stress.\nThis eco-friendly hideaway is also home to a sheep shelter, a centenary cistern, an outdoor pool, and lots of hiking and biking trails to explore. Needless to say, breakfast is a feast of fresh, local produce, served each morning in the light-filled dining room overlooking the gardens.\nLuz Houses is located near the pilgrimage town of F\u00e1tima and a short 40-minute drive from Portugal's beautiful west coast beaches.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Despite the rise of street fashion and dressed down, casual menswear, there is still no feeling quite like wearing a perfectly tailored suit. But the problem with sartorial dressing? Suiting fits haven't always evolved with the times. As a fan of skinny fit clothing (especially pants) I often find myself swimming in suits labeled as such\u2014that is, until I tried this ASOS Super Skinny Fit Suit. Cut in a truly skinny fit, this suit feels like it was made to measure from the first wear. With a snug waist and slim but not overly tight arms, the jacket likely won't need any alterations, a rarity in the suiting world. The pants offer the same skinny fit as a pair of stretch fabric jeans but with a much smoother appearance thanks to ASOS' soft-touch woven fabric. Paired with my favorite suede loafers (The Fiocco by M.Gemi) this is a look that can take you from the office, to date night, to that Labor Day wedding and anywhere in between.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It is generally accepted that modern technologies can help make your business more efficient by allowing information to easily spread inside the company, as well as be more accessible to all collaborators. However there are many methods to achieve this and they have very different effects on how the company culture evolves and, in the end, on which results the company can expect.\nIn a series of articles, entitled \"How can the wiki culture change your business\" I would like to share our experience with advanced wiki technologies used in the enterprise context and how these technologies have changed the company culture for the better. The fact is there are many things to learn from how wikis have transformed the communities and the organizations using them and how to apply this for the benefits of enterprises.\nWhat is the wiki culture and what does it mean for enterprises?\nDo tools matter for the wiki culture?\nHere is the first article of our series \"How wiki culture can change your business.\" Before getting into the details, we will be offering a brief introduction to the history of the wiki and the benefits of using this tool!\nThe wiki concept was invented by Ward Cunningham in 1995 when he was looking for a way to build a community for sharing experience on good development patterns.\nHe designed the \"most simple tool\" which would achieve this goal. The key concept was pages that were freely editable to all participants with a system to easily create links from one page to another.\nOne of the key elements was that the link is created before the page and once the \"save\" button is clicked all content is instantly published without any approval process.\nAfter the creation of the first wiki by Ward Cunningham, for which the code was made open source, other developers have created their own wiki software, improving and adding additional possibilities, but keeping the key concepts at the heart of the software.\nThe wiki concept has been popularized by the creation of Wikipedia which had the objective of creating an encyclopedia of human knowledge using the wiki concept. In less than 10 years, Wikipedia killed the business of written or CDROM Encyclopedias created by experts and became one of the top 10 visited web sites around the world.\nToday many Google searches will have as the top results a Wikipedia page. It has been an immense success, despite all criticism from \"experts\" on the quality of the Wikipedia content. It's been the proof that a community of many individuals contributing small pieces of work can beat a limited team of highly skilled individuals working full time.\nThe success of Wikipedia is not about the size of each individual's contribution, but the opposite. It's about each individual bringing a little piece of his knowledge and combining it with the previous content. When sharing information, on a short time span, it might look like the bigger the contribution the better it is, but contributions don't add up like numbers. It's all about how contributions can be combined to become something bigger and of higher value. Two big Word documents will hardly be something you can combine easily, while progressively adding small pieces of content to an existing document will allow this combination to happen. It is important to note that what is obvious for the world's knowledge is also true inside an enterprise: no individual has all the knowledge.\nThe quality of the end result is not only about the words in each page's content, but also about the organization of the content in the page and the links between the pages. A big part of the value of Wikipedia is the \"organization of the information\", which was also crowd-sourced.\nThis is particularly important in the enterprise context, when we compare wikis with social networks or document management systems. The later tools do not allow to work efficiently and collaboratively on information organization, which leads to very poor results in terms of navigability of the information.\nSo, does this apply to enterprises? We'll investigate it in the second article.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As you know, Republican presidential candidate Donald Trump recently called for a ban on Muslims from entering the United States. As a Muslim immigrant, I'm deeply concerned about Islam's perception in America. And as an American, I'm disappointed by how many people have supported such a policy so contrary to our values.\nYet, I remain hopeful. I believe we can continue to strengthen America. But, we can only do so together.\nSecond, at the same time, we all need to understand this simple truth: Islam does not equal terrorism. In fact, terrorists that commit these heinous acts are not members of the Islamic faith. Instead, terrorists signify and embody an agenda that is contradictory to the core principles of Islam.\nThird, we need to continue this dialogue \u2013 at our workplaces, inside our classrooms, and even at home. It is through civic discussion that we can brainstorm, enact, and enforce comprehensive solutions or legislation. Otherwise, if we fall victim to Mr. Trump's ideology, then we too may endanger our national security.\nRecently, I was pleased to be part of a focus group of Muslims Americans on CBS. We discussed a number of issues in light of Donald Trump's comments and the terrorist attacks in Paris and San Bernardino. I'm grateful I had an opportunity to relay my remarks and I hope I can continue to contribute to this widely debated topic.\nSuleman Malik is a third-year law student at Cardozo Law School and a graduate of Emory University. He is a Law Clerk at Newman Ferrara, LLP practicing complex commercial and civil rights litigation. He currently serves as the Executive Editor of the Cardozo Journal of Conflict Resolution, in which he has published an article entitled, \"Where Do We Fight?: A Way to Resolve the Conflict Between a Forum Selection Clause and FINRA Arbitration Rule 12200\" (17 Cardozo J. Conflict Resol. 215).","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzuzfj b/data_all_eng_slimpj/shuffled/split2/finalzzuzfj
new file mode 100644
index 0000000000000000000000000000000000000000..da3d1b07aa68a7af829eb1ff2422a4d5e35e93be
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzuzfj
@@ -0,0 +1,5 @@
+{"text":"Product liability: Nap Nanny Infant Recliner causes another death | Savinis, Kane, & Gallucci, L.L.C.\nOn behalf of Savinis, Kane, & Gallucci, L.L.C. posted in Product Liability on Friday, June 6, 2014.\nProduct liability often involves defective products that may have caused injuries or death to the consumers. Pittsburgh, Pennsylvania, consumers should always check for product recalls to ensure that the products that they have bought are safe for personal use. However, sometimes, product defects are not visible. Some defects develop over time. For example, an apparently safe product, such as a bicycle, may have some defects that can cause the bicyclist to lose control, resulting in an accident that can be fatal.\nNap Nanny Infant Recliner, a product by Baby Matters, LLC, has been subject to product recalls since 2010. Recently, the same defective product was linked to another infant's death. Based on the report, the person who was killed, an 8-month-old baby girl, was trapped between the crib and the recliner. The Consumer Product Safety Commission (CPSC) is urging the owners of that particular recliner to stop using the defective product to prevent injuries and death.\nAccording to CPSC reports, a baby is at risk while using the Nap Nanny recliner through suffocation, or, if the baby falls partially over the side of the recliner, causing the baby to become trapped between the crib bumpers and the recliner. Based on the report, the defective recliner is no longer being sold in U.S. stores. Unfortunately, those who have bought the product may still be unaware of the potential risks of the recliner. The owners of the product should stop using it immediately after being informed of its dangers in order to avoid accidents.\nIn the event of such a tragic accident, a Pennsylvania product liability attorney can provide assistance to the family of the victim. The legal professional can review all angles of the case to determine how the defective product caused the death, and so that the responsible parties are held liable.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u2192\u2192\u2192 Complete this simple form and our SBA Loan Group representative, Jarret Prussin, will call you. Or, call 516\/900-6905 for a 10-minute call to see if you qualify for the SBA Justice Loans program.\nSBA-Backed funding is low-cost and permanent. It cannot be canceled, modified, or require a forced pay down for the term of the loan, typically 10-25 years.\nYearly interest rates are prime plus 2.75%, current rates are capped at 7.25%.\nLoans from $100,000 to $250,000 close in under 45 days.\nMake the call now: 516-900-6905. Or, complete this simple form and we'll call you.\nin legal funding andSBA-backed lending.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Reggie Young, linchpin of the Memphis sound and prolific Nashville session man, has died at age 82.\nLuke Perry's death comes just weeks after legendary Arizona basketball coach Lute Olson was hospitalized with a stroke. Doctors say strokes occur more often in older age groups, but can affect people..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Adobe Premiere CC is one of the most advanced video editing platforms in the digital world. Commonly and widely used in the industry, Adobe Premiere CC is capable of creating a video with outstanding transitions, effects, and layering. If you want to upload your clip to YouTube or use it for broadcast on any TV station such as ours, Adobe Premiere CC does it all. MCM has the latest updates from Adobe which means Premiere CC uses our upgraded computer's full resources. Adobe Premiere CC is designed to work cross platform between PCs and Macs in a breeze.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"But you read him misunderstandingly when you conclude that the struggle for existence sanctions your wanton destruction of life.\nMake us fight for truth, no matter the cost, because there is no grave consequence than the destruction of life and the decay of liberty and equality.\nThe spokesman said that the Kashmir dispute had become a nuclear flashpoint between the two countries and because of this very issue three wars were fought by India and Pakistan which resulted in huge destruction of life and property on the both sides.\nSaying that as it was not clear who the box was meant for, nor how long it would be before it was opened, he concluded that the destruction of life was certain once she had put the baby into the box.\nMercy shown to mum who posted baby; Local historian STEVEN HORTON looks back at some of the city's most notorious murder cases ..\nThe aim wasn't publicity through destruction of property, but publicity through destruction of life.\nWe grieve the destruction of life, which had yet to bloom into full adulthood.\nRise up for life, condemn killings!\nThird, abortion is the destruction of life on a truly massive scale.\nMore extreme weather events bring more destruction of life and property.\nAny act of war involving wanton destruction of life and property including environmental wealth is a crime against both God and humanity, which merits a firm and unequivocal condemnation.\nIn particular, we take issue with our financial involvements in institutions associated with the State of Israel's military occupation of Palestinian lands, continued violations of Palestinian human rights, systematic destruction of life and property, inhumane segregation and systemic forms of discrimination.\nPeople often take loans for betterment of life and it should not pave the way for destruction of life.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzwvda b/data_all_eng_slimpj/shuffled/split2/finalzzwvda
new file mode 100644
index 0000000000000000000000000000000000000000..2e79efe34f3782287262e738ec351476af33723f
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzwvda
@@ -0,0 +1,5 @@
+{"text":"If you are looking for an adventure then you have landed the right page. Golden Gorilla is one themed slot machine that falls somewhere next to games like King Kong, Jurassic Park, and Tarzan. On top of that it offers people a chance to be able to win tons of prizes with some pretty amazing features.\nComing to you straight from Rival Gaming, the staggering payout percentage rate of the game might just drive you bananas. The best part is to embark on this adventure you do not have to spend much. You can start your hunt with as little as 0.01 coins for each spin. You can also open more pathways throughout the jungle if you want to give yourself more opportunities of winning. Once you enter the jungle do not let your guard down otherwise the coin prizes will pass you by.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Well the doctor had me stay again in the hospital just because my liver enzymes have started to be elevated. So when she rechecked them this morning they had gone up significantly. I am so frustrated. She called UMass and talked to the chief of OB down there (who just so happens to remember me from last time since I was so sick) and they said they wanted me to come down here. SO what a great Valentine's Day. I told MAtt I wanted chocolate covered strawberries from Edible Arrangements and instead I got a ride in an ambulance and a whole bunch of specialists doing all kinds of stuff to me.\nThey have decided that I do have Pre-eclampsia and HELLP syndrome again. I just basically am a ticking time bomb according to them. If I was further along they would just deliver Sophia now but because I am only 30.6 weeks she needs to grow for as long as possible. SO the goal is to do this balancing act waiting as long as possible, and letting me get sick until they have to take her to keep us both safe. I am just hoping to make it a few more weeks. They are hoping I make it until Saturday, because that will give the steroids enough time to work to help develop her lungs. PLEASE JUST PRAY!!!! for all of us, for wisdom for the doctors for my and Sophia's health, for Matt and Sam, for our finances.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The US lags behind dozens of countries thanks to arbitrary throttling practices.\nA new study suggests that the United States lags behind dozens of other countries in terms of video streaming quality over wireless networks. Open Signal's latest State of Mobile Video Report examined wireless video quality in more than 69 countries, utilizing 90 billion measurements across 8 million phones between May and August of this year.\nThe report examined video load times, the volume of stuttering and buffering during video playback, and overall video resolution on wireless networks. Countries were then ranked on a scale of 1-100, ranging from scores of 0-40 (poor) to 75-100 (excellent).\nA big reason for the United States' poor showing? Carrier policies that artificially restrict video quality across the network via throttling or \"deprioritization.\" As a result, you may have more than enough bandwidth to enjoy a solid YouTube or Netflix stream, but artificial limits imposed by your cellular carrier can still hamper video quality.\nLast year, competition from T-Mobile forced larger carriers like Verizon and AT&T to back away from punitive usage caps and steep overage fees and re-introduce simpler, more popular unlimited data plans.\nBut as California firefighters recently discovered, these \"unlimited\" plans feature a laundry list of often unclear limits and caveats, from restrictions on using your phone as a modem, to throttling that kicks in if you consume a set amount of bandwidth. Carriers have also started throttling all video by default unless users are willing to pony up significantly more money.\nVerizon, for example, now bans all 4K video by default, and throttles back all video on its unlimited plans to 480p (about 1.5 Mbps) unless you're willing to pay significantly more. Other mobile carriers, like Sprint, have experimented with throttling music and game performance unless users pony up additional cash.\n\"To prevent their networks from becoming overloaded with video traffic, operators have put streaming restrictions on their different tiers of unlimited plans,\" Open Signal said. \"Depending on the type of video, a 720p stream can consume twice as much or more data than a 480p stream,\" the firm insisted.\nBut a study highlighted by Motherboard last week showed how these video limits often have absolutely nothing to do with network congestion, and everything to do with nickel and diming consumers.\nResearchers at Northeastern University conducted half a million data traffic tests across 161 countries to help determine which ISPs routinely hamstring streaming performance. They found that carriers consistently apply arbitrary restrictions on video streaming that have nothing to do with managing network load.\nImposing arbitrary video quality restrictions users then have to pay more to avoid is the new normal in the post net neutrality landscape. And with looming wireless sector mergers poised to reduce competition, and the FCC's 2015 net neutrality rules now defunct, there's probably going to be a lot more of this kind of behavior waiting for users just over the horizon.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Developed to showcase each air plant's individual beauty, these air plant cradles are the ideal balance between sculpture and functionality. There is a hole in the center that measures approx 1 1\/2\" wide for securing your air plant if it has a bulb. Cradle measures approx. 9\" across the widest point.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I do not go out shopping very often. I tend to be a homebody for the most part which is always a good thing for the wallet. When I do shop I LOVE getting good deals. I ran to Hobby Lobby this week looking for a few things in particular and was disappointed that they didn't have more on their weekly special.That disappointment quickly turned in to excitement. Hobby Lobby is having a HUGE end of season sale and all spring and summer is 80% off right now!\nNo lie when I say I went to two different stores looking to see what was left. The first store only had a few goodies but the second store I was doing a happy dance! The boys kept telling me I could only buy what would fit in ONE cart. LOL. I did end up putting a few things back.\nThis sign was just $7 and I thought it would be cute if I ever do a farm themed front porch. If not I can always paint over the graphic and use it as a chalkboard. It has a great wood frame.\nThe white bread board basket will make a great wreath. I have plans for the little farm pails and barn.\nI found two beautiful gazing balls and two stands. Yes, I know the one stand is upside down. One set came from one store and the other from the second.\nAdorable melamine flamingo plates. There are four of each.\nThese guys I have been eyeing up since we were building. Hobby Lobby has carried them for three years now and I am glad I held out. They were just $5 each and a whimsical addition to our backyard since I can not have real chickens.\nGreat basic picnic basket. Some more tin items that I have plans for.\nThese two may go back. Not sure yet. I purchased thinking I would paint them. The truck would be so cute painted for Christmas.\nThen there is the mother load of tobacco baskets! I wish they had some larger ones left but these will work nicely. They were only $4 or less!\nSo if you are into bargain shopping and farmhouse decor run to your nearest Hobby Lobby to see what steals you might luck up on!\nYes, lots of good deals! I wouldn't worry about the elements either at that price.\nYes, it is always fun to find good deals. Hope you are having a greta weekend.\nCannot wait to see what you do with all your goodies!\nHi Carol! Thanks for visiting sweet friend. I hope that your weekend is going well. Is it as hot for you as it is us? We will be 102 on Sunday.\nwow! you did get some great bargains!\nI got the black truck when it was 50% off because I loved it. It was part of my 4th of July BBQ table holding the mustard ad ketchup. I'm planning to use it as part of a fall family reunion table again holding the condiments. My hubby also bought some of the galvanized buckets and other items because he likes to plant our flower beds and plans to use them there. Great sale!\nWhat a great use for it! Mine will hold pumpkins and Christmas trees.\nAhhhh Pam you found some great bargains. I love that little red wheel barrel. How adorable is that! It is a rain filled day here so I might have to venture out to Hobby Lobby and see what treasures and bargains mine holds. Happy Friday. Have a great weekend.\nI wish we had rain here Kris. It has been so dang hot. I hope you were able to get out and find some goodies.\nWow, what a score. Wish we had Hobby Lobby in Canada.\nThank you for visiting! Hope you had a great weekend.\nNew Hobby Lobby's are always nice! Hope you were able to find some good deals at yours.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzwyld b/data_all_eng_slimpj/shuffled/split2/finalzzwyld
new file mode 100644
index 0000000000000000000000000000000000000000..f6a4d7f090850a70f33e9cf5b71e3cb95547517d
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzwyld
@@ -0,0 +1,5 @@
+{"text":"We can customize this arrangement based on your choice of color. Red, white, yellow, light pink, mauve are the most popular color. Different color combinations, such as red with white, or pink with purple are available upon request.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Picking out a picture of yourself is weird. Just saying.\nI've been going at this for a bit now, and I've thoroughly enjoyed it. Hopefully you have been enjoying my writing and content. Someone on twitter had mentioned that I should write and tell a bit about myself. A chance to find out just who is writing these articles. So here you go a bit more about the person on top the Soapbox.\nMy Girlfriend and I moved out to Victoria last July from Toronto. Three years in Ontario's capital had worn us out (working in a National Sports Television Network), and with my uncle in Victoria, it seemed like a great place to set up shop on the West Coast. We saved our pennies and made the move a 23 day journey. Camping along the way at campsites we had booked months in advance. The move went as well as you can expect a cross country move in a 1996 GMC Safari stuffed tight. It almost made it too :). A charitable organization in Vancouver was kind enough to take it off our hands, and the gas guzzling sounds of a U-haul brought us from Van to Vic.\nI started this blog up out of sheer boredom. I had found seasonal work when we got here, and it had finished up for some time. My girlfriend was working (thank god), and I had been on an unsuccessful job search, and just needed a distraction. Being home alone all day can drive you batty in a hurry. Funny enough though, very shortly after that very first post, I got the job offer for the gig I am currently working (for a local television station in Victoria).\nThat one post felt really good though. Not that it was particularly great or anything like that (It wasn't even about music, the second one was). It just felt good to write again. I've never done it professionally, but I always enjoyed writing as a part of my education. There is just something about the written word.\nWriting about music was an easy decision. Humans have been banging on drums and dancing around fires for thousands of years. Music is part of us. It's part of our evolution on this planet. Although what style we listen to and how we enjoy music obviously changes from person to person, you can pretty much guarantee that we all have some interest in music on one level or another.\nI write about musicians that I think people should hear. I try to write with a professional edge, but I think it's clear I'm more so a fan. These musicians are some of the hardest working, most sincere people you will meet. Why not try and spread the word?\nI wasn't sure what to expect when I set up the first interview. I couldn't even really believe I had done it to be honest. Steph Macpherson had been nice enough to give her time before a show at the Fort St. Cafe (which is an awesome little hole in the wall). I'd never done an interview on my own before. Having said that, I do tell visual stories for a living (I'm a television editor). Along the way, I've had the privilege of working with some truly professional reporters. So at the very least I had been watching people put stories together for a few years now. I knew as long as I did as much research as possible, I wouldn't seem like a complete moron.\nSteph was great. She answered everything with honesty and enthusiasm. When listening to the interview back for transcribing, I couldn't help but laugh at myself. My response to every answer was \"cool\", or \"wow, cool\". So after writing up the piece, I deleted the audio file. I figured I could start the archive when my vocabulary had more than two replies. She must not have noticed, or simply didn't care, as her last answer was just as well thought out as her first. Oh, and it really couldn't have timed out any better. Steph has had quite the summer!\nThat first interview basically got me hooked. I had met these wonderful artists, but they are even nicer people (Zachary Lucky had been kind enough to answer a few questions that night as well). I had already wanted to spread the word of their craft, and after meeting them it was reassuring to know they are great people as well!\nI also get messages from time to time from people who love the blog. The focus on local music, and a chance to listen to something new. It blows me away every time. Although I write it for people to read, I'm always shocked people actually do. The fact it has such a positive impact for some people makes me feel really good, and that's what makes me want to write the next one!\nThe one thing I've promised myself to stay away from is negativity (except for that very first post, but like I said, I was having no luck on the job search, there was some bitterness there ;). I don't want to write about things I don't like, or I think aren't\/weren't any good. There's plenty of negativity out there already, why get all the way up on this SoapBox and spread more :). This positive attitude I've found, is opening me up to all sorts of music I never would even tried before. I really enjoy Jazz, who knew? The Ruby Spirit, I would never have spun that. I'm so happy they looked me up!\nThere is a real sense of community you feel with these musicians as well. This is also something that is hooking me deeper. Having just moved away from everything and everyone we knew, needless to say we have made some great friends, but our social circle obviously got much smaller. This was a great way to force myself out there and meet new people. I can be quite introverted sometimes, so I like things that force me out like this. Ying and Yang, you have to have balance.\nWell I think that's just about enough about me. If you have any questions or comments, I'm always looking for feedback! Also, tell me what it is about music that you love?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Light a grill. On a sheet of foil, drizzle the garlic with 1 tablespoon of the olive oil. Season with salt and pepper and wrap tightly. Grill over moderately high heat, turning, until tender, 50 minutes. Unwrap and let cool.\nSqueeze the garlic cloves into a mini food processor. Add the egg yolk, mustard, vinegar and lemon juice; pulse to blend. With the machine on, slowly drizzle in the remaining olive oil and the canola oil until incorporated. Season the aioli with cayenne, salt and pepper.\nIn a large cast-iron skillet set on the grill, cook half of the chorizo in 2 teaspoons of the olive oil over moderate heat until softened. Add half of the beans and cook until lightly charred and tender, 12 minutes. Stir in half of the shallots and cook until golden, 3 minutes. Season with salt and pepper. Transfer to a platter and repeat with the remaining ingredients. Serve the beans with the aioli and lemon wedges.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The CZ 805 G1 is a multi-purpose grenade launcher in 40x46 cal. and is a result of the development concentrated on the next generation of service weapons made by \u010cesk\u00e1 zbrojovka a.s., Uhersk\u00fd Brod. It is a light weapon designed for destroying armoured targets, personnel and military equipment located in an unenclosed ground or field shelters. The CZ 805 G1 can safely fire a wide variety of modern standard grenades of 40x46 cal. such as: cumulative, armour-piercing, fragmentation and special types (thermo-baric, smoke, illuminating, incendiary, chemical and others). The CZ 805 G1 is also suitable for non-lethal ammunition use.\nThe weapon excels in simple and reliable construction, loading and extraction of the fired cartridge is carried out from the rear, with the barrel tilted to the side. This solution has the advantage to use a wider variety of grenades, as the length is no limit. The direction of the tilting of the barrel is adjustable, which allows adapting the grenade launcher to both a right-handed and a left-handed user. The trigger guard can be turned to the left or right by 90\u00b0 to facilitate shooting in gloves.\nSafety features are made of transversely shifted manual safety of the trigger and automatic firing pin block. The CZ 805 G1 grenade launcher can be used either separately as a standalone or as underslung weapon of the assault rifle. When used separately the grenade launcher is mounted to the bottom MIL-STD-1913 accessory rail.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When we set goals we usually consider them to be \"realistic\". After all, it wouldn't make any sense to strive after \"unrealistic goals\" and set yourself up for a failure right from the start.\nYet, for some unexplainable reason, 75% of people still consistently fail to meet their goals, objectives and New Year's resolutions. These failures have very little to do with the lack of time, weak motivation, or our skills. It's just that the concept of \"realistic goals\" is so vague that we end up with vague results.\nBut that's about to change! Here are the 4 Golden Rules of Realistic Goal Setting that I wish someone had taught me years ago.\nHow can you tell that you are in control of your goal? Simple: if it's positive outcome fully depends on your efforts, not outside circumstances or other people, then you're in control.\n\"Winning the lottery\" is an extreme example of an outcome you can hardly influence. Sure, you can buy lottery tickets, but other than that, there is not much you can do about it. It doesn't look cool when people blame a bad economy, a stupid boss or corrupted politicians for their failures. You need to accept 100% of the responsibility for your successes and failures, so make sure that your goals are 100% within your control.\nSeems pretty obvious. How can you not know when you've achieved your goal? You either have it or you don't. True enough. But the vast majority of goals and New Year's resolutions that I hear are unrealistic for one simple reason: they lack a measurable criteria.\n\"Getting in shape\", \"exercising regularly\", \"learning Spanish\", \"making tons of money\" are just some of the examples of goals that have no clear destination point in mind. How will you know if you are in shape? Does it mean that you should be able to run a 10K marathon, without stopping? Or that you should lose a few inches around your waist?\nThe same is true for learning Spanish. How can you tell if someone has learned Spanish? Does it mean memorizing 2,000 words? Learning basic grammar? Or being able to carry out a conversation with a native speaker in Spanish and they actually understand what you're saying?\nApply \"get what you set\" rule to any goal that you chose for yourself. Make it super-specific and measurable, so that you know exactly when you've achieved it.\nDream big, but take baby steps and you'll get where you want to be much faster and with less effort. Instead of aiming to lose 20 pounds in 5 months, break this goal into smaller and easier-to-reach milestones: losing 2 pound in 2 weeks. It's less overwhelming, and consequently, there is a lower chance that you will procrastinate on taking action.\nIn addition, focusing on gradual 10% improvements allows you to; track your progress better, keep your motivation higher and celebrate little achievements along the way. The rule of thumb is to take your probably ambitious goal, divide it by 10 and focus on a smaller milestone with a shorter deadline.\nNumerous Psychological studies show that we are prone to the so-called \"planning fallacy\" bias. When we need to think of a \"realistic\" scenario, we envision everything going exactly as planned. We tend to overestimate the benefits of doing a certain task and greatly underestimate the time and effort it will take you.\nA great way to overcome planning fallacy is to think in terms of what we are ready to give up in order to get what we want.\nWhen you change your perspective from getting to giving up and making sacrificed, you can immediately tell the difference between an objective that you really want and something that you wouldn't mind having.\nThe higher your level of commitment is, the longer you will stick with your goals, the more you will be ready to invest in it. And investing doesn't necessarily mean \"money\". It may be about investing your time, forgoing some of your leisure activities, skipping a vacation, waking up an hour earlier or feeling sore after a workout.\nApply the 4 Golden Rules and take your goals from somewhat-realistic to realistically-infallible. Then pass this knowledge on and help your friends achieve their goals as well!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzyurb b/data_all_eng_slimpj/shuffled/split2/finalzzyurb
new file mode 100644
index 0000000000000000000000000000000000000000..1430827210c97e74b6b90467ab7e55468bf4cf08
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzyurb
@@ -0,0 +1,5 @@
+{"text":"August | 2011 | eating with S.O.L.E.\nThis picture perfect purple salad makes a dreamy picnic dish, and will sit well for a few days in the fridge drinking up the dressing.\nadd and cube your cooled purple potato, skins removed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Squad up with your mates and venture to the desert of Miramar, the grassy lands of Sanhok or the harsh terrain of the Erangel and prepare to get yourself a chicken dinner. For anyone mad about the 100 Battle Royale phenomenon; PUBG, this awesome tee from Playerunknown featuring a level 3 helmet is a must have to add to your loot. Official merchandise.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cultural Conditioning are the social processes by which authority figures such as parents, professors, politicians, religious leaders, peers, and the media define our cultural values, beliefs, ethical systems, and ultimately the way we perceive ourselves in the world. There are a number of social processes working against you to make you hate your body and your weight and they are designed to keep you buying diet and diet related products.\nToday's class is about accepting your body and your weight. Acceptance is not about resignation or I give up. It's about acknowledging the reality of your body and weight as it is right now in this moment. Your weight may not always be this way, but this is how your body is right now. What feels imperfect right now might just be perfect in the big picture. Can you experience and accept your body for what it is? Can you accept what you become aware of and also notice your non-accepting, validating both as parts of your unique reality?\nDr. Russ Harris, author of The Happiness Trap says, \"Acceptance literally means 'taking what is offered'. It doesn't mean giving up or admitting defeat; it doesn't mean gritting your teeth and bearing it. It means fully opening yourself to your present reality \u2013 acknowledging how it is, right here and now, and letting go of the struggle with life as it is in this moment.\" What would it be like to let go of the struggle with your weight? What would it be like to fully open yourself to the present reality of your body and take what is offered, your whole, beautiful self?\nWe don't need to exhaust ourselves by mentally beating ourselves up playing the negative soundtrack over and over again in our heads fueling the pain and frustration with our bodies. We don't need to postpone our lives, hold back from life, wait for some perfect moment, and terrify ourselves with fear. We can stop the war with ourselves, we can stop rejecting ourselves and our bodies. Yoga promises us that that who you really are is already open and free and loving and deeply accepting, it is simply a matter of relaxing into and opening to that part of yourself.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The State of Oregon has a great many payday loan stores in all major cities and many smaller towns. Not long ago, if someone in Portland or other Oregon cities wanted a quick payday loan, they had to go to a local cash store. But now there are literally thousands of payday loan companies offering fast cash advances in Oregon online. They let you get a fast, easy Oregon payday loan from the comfort of your own home, and your application is approved often within minutes or hours, while the direct deposit you're your account usually arrives within 24 hours.\npayday loans can be a life saver in an emergency. They can also lead to a cycle of continued deepening debt. Because of concerns for this, lawmakers in the State of Oregon have been working on passing payday loan protection laws that would protect consumers of payday loans in Oregon from being charged too high interest, or too large fees. They would also require that payday lenders disclose their APR and make their terms crystal clear to borrowers. The State of Oregon Senate Commerce Committee that considered a bill to limit the interest charges of Oregon payday loans to 15 percent, set a minimum 31-day loan term and require consumers to pay off a quarter of the loan before renewing it. Oregon payday loan borrowers who cant pay off the Oregon payday loans may face renewal charges if they roll over the amount, which according to the Oregon payday loan regulations they can do only three times at one lender.\nOregon lawmakers have argued that their state is one of the eight states that have certainly no cap on the Oregon payday loan interest charges. This claim, as numerous reports have noted, actually came from the Oregon Department of Consumer and Business Services, which specifically regulates the industry that operates and functions generally more than 320 loan shops in the State of Oregon alone. It is further noted that this agency supports the Oregon payday loan regulation bill to cap the interest rates of payday loans at 15 percent.\nHowever, as things commonly work, there are some opponents for such Oregon payday loan bill. They said that payday loans generally welcome service to those who are actually temporarily strapped for money and even to those who cant obtain conventional loans. As a support for this, it has been reported that Mike Dewey, an Oregon lobbyist for the Consumer Lending Alliance, which is a group of payday lenders in 22 states, said that borrowers of the Oregon payday loans often look at loan charges as flat fees of $15 to borrow $100, for instance, and not actually on the terms of the annual interest costs perhaps due to the reason that payday loans are for far shorter periods.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I was in Olympic National Park last weekend and stayed at the lovely Lake Crescent Lodge. Last week I posted an incredible picture I took of the sunset the night I arrived. Sure enough, the next morning I was up super early for sunrise. The sunrise wasn't incredibly visible from where I was standing so I decided to go for a super long exposure facing away from the sun. I could see some wispy clouds and thought a super long exposure would make them even more wispy and make for a really interesting sky.\nReal quick, don't worry, I have a full write-up of my little trip to the PNW coming this week! Then again, if you subscribed to my YouTube channel you'd already have known and watched the videos!\nI used my a7rIII and 24-70 GM lens with a Lee Big Stopper and Lee 105mm Polarizer for this shot. I forgot my Neewer long exposure trigger at home but luckily the Sony remote app can be used in Bulb mode for long exposures. I set the timer, focused everything, and waited. 466 seconds later, a gorgeous scene showed up on the back of my camera, enjoy for this week's Picture of the Week!\nFor the photographers: Like I mentioned above I used a polarizer on top of a 10-stop ND filter for this shot. I metered the shot and focused without the Big Stopper and then slid the filter into the foundation\/holder that Lee sells. Using the Lee app I calculated an 8-minute exposure. I got a little nervous towards the end of the exposure, since the light had changed a bit at roughly the four-minute mark, so I cut the exposure just a little short so I didn't blow out the frame.\nOnce I had the image the post-processing was fairly easy. I overprocessed it a bit but decided that was ok because I wanted it to be a little exaggerated. But I didn't really do anything outside of my normal process, feel free to ask any questions that you have in the comments below!\nGreat shot as always! What I was wondering: How do you stack the polarizer filter and the ND filter? I assume you put on the polarizer first, and then the Lee-filter, right? Do you need something in addition for the polarizer to make this work?\nThanks and keep up the great shots!\nThe 105mm landscape polarizer comes with a ring which attaches to the front of the foundation into which I slide the Lee Big Stopper. It's a bit hard to explain so maybe I can walk through it in a post at some point, thanks for the idea and thanks for the kind words!\nI think I know how you mean it, thank you for explaining.\nLove the Lee Filter system. I might suggest that Benedikt Google to the Lee Filter website and there are several videos which would help explain the Lee system. Filters are square and fit into a holder that attaches to the various lenses using an adapter. Holder can hold two or three stacked filters.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzyygw b/data_all_eng_slimpj/shuffled/split2/finalzzyygw
new file mode 100644
index 0000000000000000000000000000000000000000..fdf7f2f5f08c69683efb677fb5e43b3ca3ba5cea
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzyygw
@@ -0,0 +1,5 @@
+{"text":"If your child has special health care needs, trust that our team at 3 Rivers Pediatric Dentistry will will custom tailor the delivery of care to fit their specific needs. Dr. Foster and Dr. Ziegler have gained training in both dental school and in their pediatric residency in working with patients who have special needs. Our team is trained to work with your child no matter what his or her needs may be. We have the experience, patience, and knowledge to help your children through their dental cleanings and treatment, and we have a variety of sedation and anesthesia options available if needed. Before your appointment, please call us to provide us specific details of your child's condition (whatever information you can provide would be helpful). Rest easy knowing our team is well prepared for your child's visit. Also, if you feel your child would benefit from a \"desensitization\" appointment to walk through the office and get familiar with the dental environment ahead of time, our team would be happy to assist you in getting that scheduled.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"September 2012 (Magazine)By Cecile Sourbes.\nThe latest consultation paper for the holistic balance sheet (HBS) within the revised IORP Directive has aroused great interest across Europe. The European Insurance and Occupational Pensions Authority (EIOPA) gave the pensions industry until 1 August to submit its input on the controversial first quantitative impact study (QIS) for the implementation of the HBS.\nThree main issues arose from the responses sent to EIOPA. First, participants complained - once again - about the timetable set by the authority and Brussels. According to them, the current time frame remains \"too short\" to come up with a \"proper\" impact analysis for the HBS.\nSecond, the industry has decried the lack of clarity over the valuation of pension funds' liabilities, while arguing that both the prudential and holistic frameworks for the revision of the IORP Directive are poorly adapted.\nAnd third, pension representatives expressed concerns over the similarity between Solvency II and some elements found in the consultation paper for the Directive on occupational pensions.\nFor the European Federation for Retirement Provision (EFRP), a large part of the QIS consultation document seems to have been \"copy-pasted\" from the Solvency II Directive.\nSome pension representatives therefore recommend the examination of existing practice already implemented by a number of member states. The Dutch Pension Federation, for instance, has called for the FTK model. \"It could be worth studying in-depth the experiences of the current reforms of the FTK - a risk-based supervisory framework with market-consistent valuation - which is under reconstruction,\" it says.\nMany in the pension industry have supported this approach. However, it could be argued that the FTK has not been without fault in recent years, due to its valuation of nominal liabilities and use of the euro swap curve to derive a market discount rate at a time when interest rates are at record low levels.\nThe valuation of liabilities is an urgent matter for the pensions industry. In the UK, Towers Watson argues that EIOPA's proposals would, at some point, mean that making higher contributions to a scheme could also lead to higher deficit.\nThe consultancy, discussing a UK-specific example, said this would be the result of the physical assets increasing, resulting in a lower claim to the Pension Protection Fund were the sponsor to go insolvent, combined with a decrease in the value of the sponsor support.\nThis puts the whole balance-sheet concept under question and suggests that the HBS might not be such a useful means of assessing the solvency of a fund.\nAn alternative to the HBS could then be a holistic framework, leaving aside the balance-sheet concept. Unlike the balance-sheet approach - which, according to some, \"unrealistically\" suggests that all of the elements can be quantified precisely - the holistic model would give more freedom to member states to calibrate that framework to the local pension system.\nAll those issues bring us back to the question of the timetable. So far, nobody knows how EIOPA and Brussels will proceed, or whether they will amend their current proposals for the revised IORP Directive. If they do, several other impact studies will be necessary to value the different adjustment mechanisms to be made. This would take months, even years. Given that Brussels has already agreed to postpone the introduction of the draft revised IORP until the summer of 2013, a further delay is not inconceivable.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Yep, another doctor on the OIG Report. What was the violation? The report states, \" These improper payments occurred because the doctor did not have any policies and procedures to ensure that chiropractic services provided to Medicare beneficiaries were medically necessary and sufficiently documented.\nThe OIG report recommends that CMS initiate recoupment in the amount of $518,000. Read full report here.\nDo you have Medicare policies and procedures in writing? Do your policies cover Medical Necessity, setting fees, OIG exclusions, coding, Mandatory ABN usage, and screening QMBs? If you need help tackling Medicare, KMC University is here to help!\nJoin our Library Membership today and receive 24\/7 access to in-depth learning on specific Medicare billing, coding, and documentation requirements for chiropractors.\nHealthcare is one of the most highly regulated industries in the United States. Learn the required systems to manage your practice compliance, and how a team member can create, manage and maintain your compliance systems with minimal doctor involvement. Proper compliance systems will protect your license while saving you time and money. Understand exactly what is necessary to protect your practice, quickly, and easily.\nJoin the KMC University Library and get the do-it-yourself chiropractic compliance installation training you need.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The fastest cron for ripple you'll never need.\nRipple uses a cronjob to fix any eventual errors in the database. For doing it, it used cron.php, a terribly performing script. No, really. That's three minutes of server CPU being used at 100%!\nSo I decided to rewrite it to be better performant, and what better language to do it if not Go (well, C, C++ and Assembly are indeed faster, but I'm not on that level of insanity).\nThis is the result: https:\/\/asciinema.org\/a\/42583 (watch the video especially for the last 30 seconds, as you can see the true power of ripple-cron-go).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Elyse Fitzpatrick has a great post on The Gospel Coalition blog on modesty. Summer usually means a lot of talk on modesty, and her piece is an excellent reminder and heart check for all of us as we look at our closests and our motives behind our outfits. In her post, \"Gentle Jesus, Meek and....Modest?\" she links modesty to the character of Jesus. Jesus was modest, she says. Defining modesty as \"a refusal to show off out of love for God and one's neighbor. Jesus refused to show off his power.\" And he could of. In a heartbeat. Instead he exercised restraint so that the plan of salvation could be accomplished for you and me.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzabhpi b/data_all_eng_slimpj/shuffled/split2/finalzzzabhpi
new file mode 100644
index 0000000000000000000000000000000000000000..e22ca8e7cf4db879cf059bce3bb6d34e4a5e4527
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzabhpi
@@ -0,0 +1,5 @@
+{"text":"Welcome to the 14 Day Walk With Christ 2014. This is a quick overview page with links for each day.\nEach egg will contain a topic for that day that will help my family come closer to Christ in preparation for Easter. After school, or that evening, we will have some kind of small lesson and activity about the day's topic. During every lesson, the children will recieve a handout to put in a little journal. Then they will record their feelings that day.\nFor a list of lessons and contributors for each day click HERE.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"To filter salaries for QA Specialist in Seattle, WA Area, Sign In or Register.\nHow much does a QA Specialist make in Seattle, WA?\nThe average salary for a QA Specialist is $59,478 in Seattle, WA. Salaries estimates are based on 48 salaries submitted anonymously to Glassdoor by QA Specialist employees in Seattle, WA.\nBelow are the most recent QA Specialist salary reports. Employer name has been removed to protect anonymity.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This was kinda interesting. Someone brought to my attention this morning a unique big billboard by Google in California, highlighting a rather difficult mathmatical puzzle. Basically you need to decrypt the message to arrive at the correct answer. If you are smart enough or have enough time to figure it out, then you will find the answer is actually a domain .com to which you are supposed to go to find out what Google is up too.\n\"Congratulations. You've made it to level 2. Go to www.Linux.org and enter Bobsyouruncle as the login and the answer to this equation as the password.\"\nHmmm... anybody got the answer?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"An age old and always successful mechanism aligned with the historical success of all species, great and small which will solve both Unemployment and National Debt, in a fashion satisfactory to Democrats, Republicans and Independents alike with enormous upside gain for the whole of our culture and civilization.\n'Why The Enemy of My Enemy is My Friend' presents a clear and precise psychobiological formulation of the elemental DNA of racism.\nIn an easily understood and comprehensive lay perspective, 'The Enemy of My Enemy is My Friend' teases from the complex behavior of racism its elemental pieces and in so doing allows the reader to trace its actual lineage to the evolutionary roots of safety, self preservation and the instinctive need for all organisms of protection and the future propogation of the species.\nDespite the nuanced and often camoflauged manifestation of 'racial'behavior, it's presence is a naturally occuring phenomenon of evolution and the thesis presented explores not a way to eradicate a naturally occuring behavior, like 'fight or flight' but rather tries to understand its origin and evolution so to be able better to integrate the facts of its presence and find an equilibrium which like most of the species on the planet, balance mutual needs in some more or less successful form of a workable symbiotic relationship.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you haven't read Neil Gaiman's short story \"A Study in Emerald\" I advise you to do it right now\u2026 I'll wait\u2026.\nIt's Cthulhu meets Sherlock Holmes with a fantastic slant on things.\nNeil Gaiman teamed up with Martin Wallace to produce a game based on the short story and I'm hoping a copy of it will be dropping in my door before Knavecon and we'll all get a chance to play it. From all accounts it's fantastic!\nMan there is so much going on at this convention!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzabsia b/data_all_eng_slimpj/shuffled/split2/finalzzzabsia
new file mode 100644
index 0000000000000000000000000000000000000000..5b38bf3c5def04787257aef4bb7938f39dd83bc4
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzabsia
@@ -0,0 +1,5 @@
+{"text":"wishing you a very Happy New Year too!\nI am so glad, we met us and I am looking forward for another year filled with lovely posts and wonderful chats with you my dear friend!\nhappy 2015, celeste! lovely roses!\nHappy New Year to you and your family, Celeste! I look forward to seeing your lovely home, paintings and posts! I'm so happy we met and love hearing from you! Your sweet spirit shines through all that you do! God has given you many talents and you're so sweet to share them with us! Sending many wonderful blessings your way!\nHappy New Year! I wish you all the best for 2015.\nHappy New Year Celeste!! Wishing you all the best in 2015!\nthanks for becoming part of my neighborhood in 2014! blessings for the new year wished to you!\nHappy New Year to you too Sweet Celeste!!\nHappy New Year to you too, Celestina. So glad I found your blog!\nTo one of my Dearest Friends. One who is always there when I need her... in my ups and downs.\nAlways with a kind word and prayer.\nHappy new year Celeste! Here's to hoping for a wonderful and craft-filled new year for all of us!\nHappy New Year to one of the sweetest friends we know! May God bless you and yours with all your heart's desires!!\nHave a very happy new year!! Wishing you all the best of everything!\nHappy New Year to you as well, Celestina! We are all so blessed to know you here in blogland. Your painted roses are just stunning. Glad you are enjoying some much deserved r&r. Looking forward to seeing what wonderful creations you will come up with in 2015!\nHappy New Year, sweet Celeste! I am looking forward to another year of your beautiful blog. Hugs!\nA blessed 2015 to you and yours!\nEnjoy your break Celestina - and a very Happy New Year to you! This past year has been truly wonderful, visiting here and seeing such lovely inspiring work - your talents are quite amazing!\nI hope 2015 brings you every happiness and great joy in all you do, and wherever you may go.\nCome visit and join my \"Three Year Blogging Anniversary, Give A Way\".\nI have so loved coming by your blog to see what you are creating. Wishing you a Happy New Year, my dear blogging sister.\nHappy new year dear friend and hope that 2015 will be a great year for you with good health, happiness and prosperity. I am always so grateful for your kind friendship.\nMany blessings to you dear Celestina, for this New Year ahead of us as you faithfully walk with our Jesus.\nIt has been a real joy getting to know you through your precious blog.\nWishing you all the best in this fresh new year!\nWishing you and your family a Happy New Year filled with many blessings!\nthank you for your kind and sweet blogging friendship in 2014, I look forward visiting your beautiful blog in the new year, and wishes you all the best in health and happiness in 2015.\nHappy New Year from Dorthe.\nI want to thank you for your friendship along the way, and I wish you peace and lots of love in the new year. May we share another sparkling year together, my friend.\nHello, Enjoy and get ready for a wonderful and creative New Year!\nIt's hard to believe another year is gone, but looking forward to new beginning. I wish you all the best for 2015!\nHappy New Year, sweet Celeste! Oh, what a beautiful wish for your blogging friends and I wish the same for you! You are so kind, talented and the love of our Lord shines brightly through your words on your blog! I'm so glad I've gotten to know you in blogland and I am looking forward to seeing what you'll be up to in this new year!\nHAPPY NEW YEAR to you! I wish you the best for the new year. I'm looking forward to seeing your next creations.\nHappy New Year, Celestina Marie!\nHappy New Year to you too! I look forward to continuing being blogging friends in 2015, and visiting your lovely blog. You share so many beautiful things, and you also have such a kind and encouraging heart, dear Celeste. Blessings to you.\nI just dropped by to wish you a very Happy New Year! It was a pleasure meeting you this year!\nI look forward to more creations from you studio.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Owning and maintaining a business can be very challenging in today's world. There are daily demands for capital that require a well thought out and thoroughly analyzed plan to succeed. The business development professionals at Farmers Savings Bank are here to work with you to meet all of your short and long term business needs and goals. We finance equipment, expansion, debt consolidation, and more. Visit with one of our commercial lending professionals to learn more on debt structures personally tailored to your business.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We have several gatherings where you belong!\nGo 4th Weekends and more!\nThe hope, joy, and peace mentioned in this verse are real, because God is real. Here's Hope Church is a community of real people who know the love of a real God. We are excited about following Jesus, and joyfully sharing His unchanging Good News with everyone!\nLoving God. Loving People. Sharing Jesus. God's invitation to come is for everyone, and so is ours. We hope you'll decide to visit us soon.\nWe hope you will visit us soon! We get together at 725 E Center Road in Kokomo (the Center Road entrance to Indian Heights).\nJoin what God is doing through Here's Hope Church with your financial gift. Online giving is simple and secure.\nShare the Gospel (Good News) of Jesus!\nClick on the picture to the left to watch a short video from Time To Revive that explains how you can share The Good News of Jesus Christ through prayer, a bracelet and a Bible. Let Your Light Shine!\nGet the HOPE TODAY app!\nOur app is a unique combination of a devotional and a discipleship tool!\nGOD'S WORD + YOUR LIFE = ENCOURAGED!\nContact Us *We meet at 725 E Center Road, Kokomo; but currently have an off-campus administrative office and mailing address.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Shawnee Theatre is Indiana's longest, continuously-running summer stock theatre. Since its doors opened in 1960, it's entertained thousands of guests, put on hundreds of shows, and has become an Indiana summer tradition. Professional actors, actresses, lighting technicians, sound engineers, stage managers, and more have come from across the country to perform on the Shawnee stage.\nThe theatre quickly became too popular for the original barn, leading to the construction of the existing theatre on Furnace Road.\nOver the years, the theatre has hosted many actors on the road to fame\u2014including SNL cast member John Belushi and Dallas star Ken Kercheval, among many others.\nThroughout the past five decades, the Greene County community has been integral to the success of the theatre. From the thousands of patrons that attend one, two, or all the shows every season to the local businesses that partner with us to keep producing high quality live theatre, the Shawnee Theatre is truly Greene County's community professional theatre.\nThank you! We sent you an email to confirm your subscription.\nThe Shawnee Summer Theatre is dedicated to making the best of American theatre available and accessible to the Greene County Community. By making live art available and accessible to the region, we maintain an enriched educational environment for patrons, company members and our community.\nAt its core, the Shawnee is defined by our connection to our community, therefore we are committed to keeping our ticket prices low to maintain accessibility to the arts for all in the Greene County community.\nThe 2018 Shawnee Theatre season is made possible by the support of the Indiana Arts Commission, Greene County Foundation, MutualBank and the Brown County Community Foundation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Abstract: Weak gravitational lensing observations probe the spectrum and evolution of density fluctuations and the cosmological parameters which govern them. The non-linear evolution of large scale structure produces a non-Gaussian signal which is potentially observable in galaxy shear data. We study the three-point statistics of the convergence, specifically the bispectrum, using the dark matter halo approach which describes the density field in terms of correlations between and within dark matter halos. Our approach allows us to study the effect of the mass distribution in observed fields, in particular the bias induced by the lack of rare massive halos (clusters) in observed fields. We show the convergence skewness is primarily due to rare and massive dark matter halos with skewness converging to its mean value only if halos of mass M > 10^15 M_\\sun are present. This calculational method can in principle be used to correct for such a bias as well as to search for more robust statistics related to the two and three point correlations.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzacfdj b/data_all_eng_slimpj/shuffled/split2/finalzzzacfdj
new file mode 100644
index 0000000000000000000000000000000000000000..bfc9cbcf5ca3ed33ea3692a7d56361eefb84052d
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzacfdj
@@ -0,0 +1,5 @@
+{"text":"This is a placeholder page for James Jump, which means this person is not currently on this site. We do suggest using the tools below to find James Jump.\nYou are visiting the placeholder page for James Jump. This page is here because someone used our placeholder utility to look for James Jump. We created this page automatically in hopes James Jump would find it. If you are not James Jump, but are an alumni of New Providence High School New Providence, NJ, register on this site for free now.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Looking for ideal 3D Embossed Printed Table Halloween Manufacturer & supplier ? We have a wide selection at great prices to help you get creative. All the 3D Embossed Printed Table History are quality guaranteed. We are China Origin Factory of 3D Embossed Printed Table Harwin Houston. If you have any question, please feel free to contact us.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"public static class Attribute.Constant The value for an annotation element of primitive type or String.\npublic static class Attribute.Class The value for an annotation element of type java.lang.Class, represented as a ClassSymbol.\npublic static class Attribute.Compound A compound annotation element value, the type of which is an attribute interface.\npublic static class Attribute.Array The value for an annotation element of an array type.\npublic static class Attribute.Enum The value for an annotation element of an enum type.\npublic static interface Attribute.Visitor A visitor type for dynamic dispatch on the kind of attribute value.\npublic static enum class Attribute.RetentionPolicy A mirror of java.lang.annotation.RetentionPolicy.\npublic Type type The type of the annotation element.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A micro brewery in the heart of Palm Harbor, you'll find locally inspired craft beers in a laid back and welcoming tasting room. Featured brews include the St. Joseph Sound Porter, German Hefeweizen, Amber Ale, Sneak Attack Saison and more. Order a beer in your choice of 2 oz., 10 oz., 16 oz., or a 64 oz. growler to go! Open Tuesday-Thursday 3 p.m. to midnight., Friday-Saturday 12 p.m to midnight and Sundays 12-10 p.m.\nSmall place but good microbrews. Bar & table seating. No food. Entertainment some nights. Accessible from the Pinellas Trail. Nice staff.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"08\/06\/18 Consideration of Governor's veto pending.\n07\/02\/18 Read third time. Passed. Ordered to the Assembly. (Ayes 37. Noes 0. Page 5038.).\n06\/06\/18 From committee: Do pass and re-refer to Com. on APPR. with recommendation: To Consent Calendar. (Ayes 6. Noes 0.) (June 6). Re-referred to Com. on APPR.\n05\/24\/18 Referred to Com. on ED.\n05\/02\/18 From committee: Do pass. To Consent Calendar. (Ayes 17. Noes 0.) (May 2).\n04\/18\/18 From committee: Do pass and re-refer to Com. on APPR. (Ayes 13. Noes 0.) (April 17). Re-referred to Com. on APPR.\n02\/08\/18 From printer. May be heard in committee March 10.\n02\/07\/18 Read first time. To print.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzacolf b/data_all_eng_slimpj/shuffled/split2/finalzzzacolf
new file mode 100644
index 0000000000000000000000000000000000000000..6a6229492e549abfe572ad7e758be28abae89a0f
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzacolf
@@ -0,0 +1,5 @@
+{"text":"These Whole Wheat Zucchini Brownies are not too sweet and perfectly fudgy. Made with freshly grated zucchini, white whole wheat flour and rich chocolate.\nZucchini is one of those vegetables that can be used in a million different ways \u2013 sweet or savory it's perfect. For this recipe, I went full on sweet and it turned out perfectly. The brownies are made with white whole wheat flour and sprinkled with just a little bit of powdered sugar. I'm such a sucker for that sweet little sprinkling on top. If you're feeling frisky you could even serve it with a big scoop of ice cream.\nDon't worry when you make these brownies if the batter seems a bit dry. Let it rest for just a minute and the zucchini will release enough moisture to make the batter just a little more wet. Trust me, it will look weird at first but totally come together in the end!\nThe hardest thing about this recipe is letting the brownies cool before you cut into them. But you know the old saying \u2013 good things come to those who wait!\nAn easy and veggie packed recipe for Whole Wheat Zucchini Brownies. The kids will never know you snuck in something that's good for them.\nPreheat your oven to 350\u00ba F degrees and spray a 9x9 pan with non-stick spray.\nIn a medium sized mixing bowl add sugar, oil and vanilla bean paste. Stir to combine.\nAdd in white whole wheat flour, cocoa powder, baking soda and salt. Stir to combine.\nAdd in the zucchini and chocolate chips and stir to combine. The mixture might look a little dry but let it sit for just a few minutes and the zucchini should release some water and make the mixture a bit more moist.\nAdd the mixture (it will be thick) to the prepared baking dish, press to the sides and smooth so that the brownies are equally distributed. Bake for about 25 minutes or until the sides are set and the center is just slightly soft. Let cool before slicing and sprinkle with powdered sugar if desired.\nmaking these soon! love that it is egg and dairy free.\nUpdate: I made these today and we all LOVED them. Seriously our new favorite sweet! I am adding this to our spring\/summer baking rotation for sure!\nWoohoo\u2026.so sorry it took me eons to reply. So glad you enjoyed the recipe Julie!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Membership allows you to truly become a PART OF an organization by providing you not only voting privileges, but also through many opportunities to serve. Membership allows you to be part of the PROCESS and ultimately part of the SOLUTION. Membership provides you ownership in the organization, establishing a deeper commitment to that organization and a high degree of satisfaction as your reward.\nThe importance of every person's individual involvement in any organization can never be emphasized enough! Great things happen when individuals with unique personalities, giftings, visions and creativity come together for a common cause and for the good of all!\nPlease consider joining The Cranberry Bay Homeowner's Association! You can be an incredibly valuable asset, and I promise you that you will NEVER REGRET IT!\nIf you have any questions contact Margaret Hanson, Chair, Membership Committee at 614-309-9733.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Knowing what emotional intelligence is and knowing how to use it to improve your life are two very different things. Emotional Intelligence 2.0 is a step-by-step program for increasing your emotional intelligence using the four core EQ skills - self-awareness, self-management, social awareness, and relationship management - to exceed your goals and achieve your fullest potential.\nIn this audiobook, Drs. Bradberry and Greaves offer a revolutionary program to help people identify their EQ skills, build these skills into strengths, and enjoy consistent performance in the pursuit of important life objectives. This audiobook contains proven strategies from a decade-long effort to accurately measure and increase emotional intelligence.\nThis audiobook will also give you access to the new, enhanced online edition of the world's bestselling emotional intelligence test, the Emotional Intelligence Appraisal\u00ae. The test will show you where your EQ stands today and what you can do to begin maximizing it immediately.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Interesting reply obviously as I stated the reason for the repost was the google alert and the wiki page I just found recently. I thought the wiki article (I have checked most of the article references and they are reliable) maybe of interest to those that had shown an interest or made comments about Rhee TKD Australia on this thread, positive or negative (as in your case Mr VDJ).\nSkimming back on your posts \"Tiger Killer\" you seem to always have an underlying condescending tone, which if that mindset works for you, you are a successful and your life is moving forward then fine. However if you can indulge me by please advising your rank, school and Master to help me try to understand where you are coming from my friend. I am not taking the p-ss (as we say in OZ) or want a flame off, I would genuinely like to try to comprehend the angle of your martial arts beliefs, misguided or not in regards to your posted comments\/views on RHEE TKD & Master Rhee Chong Chul.\nTo be fair, I advise - I am a member of the Rhee Taekwondo (Australia), my Master Instructor is Master Rhee Chong Chul and my rank is Sabum (Black belt teacher). Robb is my real name, this year is my 19th year of continuos training as a Rhee Taekwondo student. I studied a few different martial arts, striking and throwing styles (Japanese, Korean & Western) before making an informed choice to join RHEE TKD many years ago and have never been happier.\nI look forward to your response Mr VDJ, Cheers.\nInteresting reply obviously as I stated the reason for the repost was the google alert and the wiki page I just found recently. I thought the wiki article (I have checked most of the article references and they are reliable) maybe of interest to those that had shown an interest or made comments about Rhee TKD Australia on this thread,.......Cheers.\nSir I found the link very informative. Thanks again for posting it. If you are able to answer these questions, please do. You can also feel free to PM me if you prefer.\nWhen did Master Rhee leave the ITF\/Gen Choi? Do you know why? What ITF rank certificates does he have? When did he get 8th degree? Why has he stayed an 8th, as obviously he would be a very deserving 9th degree, given his extensive experience & history as a true pioneer?\nDoes he keep in contact with GM Young Ku Yun? When & why did he leave the ITF\/General Choi & what was his ITF ranks? Who is considered senior of the 2?\nSir thank you so much for your informative reply. I do really appreciate it. Above is the website I mentioned in my reply.\nMaster Rhee, Chong Chul was a member of the ITF up until the late 70's. He certainly wasn't a member in 1980. His rank with the ITF was 6th Dan. He was apparently dismissed from the ITF as he was not willing to pay his dues. Basically he was accused of keeping all the money for himself and did not send any memberships fees or Dan certificate fees to the ITF. He was of lower rank than his brother Rhee, Chong Hyup who actually appears on the ITF plaque (bottom left hand figure). He received his 8th dan in about 1984 and has a WTF or Kukiwon 8th Dan Certificate.\nGM Yun and Rhee, Chong Chul do not keep in contact as far as I am aware as they were never very friendly toward each other. Yun was one of the other ITF pioneers in Australia and was the one who Gen Choi supported as Rhee did not pay his dues to the ITF.\nI know both gentlemen and I can tell you that they were not members of each other's fan clubs.\nGM Yun left the ITF in about 1989 to form Yun Jung Do. He was ITF 8th Dan at the time.\nRhee was senior to Yun in the ITF when they were both members. However, Yun actually received his 8th Dan ITF, while Rhee Chong chul was only officially 6th Dan. Rhee, Chong Hyup is actually much senior to both of these men in time involved in TKD.\nThanks for PM ITFunity, I will chase up for you.\nG'Day Sabum Steve, it's been a long time, remember me Robb from Rhee we have spoken face to face on the Goldie a while ago. You are close enough on some of those details, but glaring issue is your view on Master Rhee leaving the ITF, the slanderous claims of with holding fees aimed at character assassination are feeble, you know the state of flux the ITF as an organization was in - in the late 70's most original Masters were estranged for many years before it was official they had to cut ties.\nMaster Rhee and most of the original Masters were forced to leave the ITF due to the severe pressure put on them by the ROK to disassociate himself from General Choi due to General Choi's actions regarding the North Korea issue. Much politics make for great story telling but allows for those that are standing at the end rewrite from their point of view (those still in ITF), more so than the ones that had to walk away (as Master Rhee did).\nMy understanding is the fracturing of the Original Masters has now been retold different ways by all 3 of the ITF's, Vienna, North Korea & Canada.\nPromoted to 8th Degree Black Belt by Gen. Choi. Was a member of two I.T.F. Representatives in an attempt to merge with World Tae Kwon Do Federation [W.T.F.].\nWith deep regret, Master Choi dropped support for Gen. Choi because of his ties with North Korea. At this junction, South Korea was technically at war and had no diplomatic relations with North Korea.\nAs such Master Rhee was first and foremost a South Korean Marine and had pledged his life long loyalty to the ROK before General Choi as a ROK representative hand picked him for the development of TKD.\nYou will never hear Master Rhee speak poorly of General Choi \"The Father of Tae Kwon Do\" as I am sure he and most of the original Masters understand the issue is more than just black and white. Master Rhee has never tried to rewrite what happened when he was forced to leave General Choi's ITF, not because he is at blame but because it was a dirty political affair between GM Choi & the ROK. Whether you Sabum Steve look upon your own promotion to 1st Dan by Master Rhee Chong Chul as a great honor or not you must morally remember were you came from. The new school you now preside over as ITF examiner is a break away of a break away of RHEE TKD as yourself are the product of a break away of a break away of RHEE TKD. Sabum Shane of Yin Yang TKD was a 1st Dan Sabum with RHEE TKD just 2 years ago and you promoted him to ITF 3rd Dan at the start of this year \u2013 that's interesting.\nIt has been distasteful for me to note all above but was forced to as you made the silly call on Master Rhee's honesty. Cheers my friend, step on a cane toad for me and enjoy a 4X, let the history be, it is not the domain of us juniors to try and rewrite it.\nPS Sabum Steven (ChosunITF) as your 1st post - your not VDJ reborn are you mate!\nPlease mate, I do not like ignorance dressed up as fact or knowledge.\nYour response clearly demonstrates that you have no idea of what I wrote. Go back and read it again and tell me if I was assassinating Mr Rhee's character. Whether true or not, whether you want to believe it or not, I have seen the letter suspending Mr Rhee. Now, whether I believe it or not is another story in itself.\nAlso, if you are going to espouse that you know something about me, where I come from, those associated with me, how they got there or where they come from, then please, please, please Robb get it right, as it just makes you look foolish and extremely ignorant otherwise.\n16 Years in the ITF.\nNot the product of a break away of a break away.\nAnd Mr. Luxmoore and my self are to very different people, he being obviously an Australian citizen and I am an American citizen. Robb, it seems that you are going to an extreme to legitimize your instructor, and I can really appreciate that. You think that I was personally attacking him as it was not the case. The site that I posted in my original post on this topic made NO MENTION of his connection to the ITF or Gen.Choi and it was not made clear until the thread had been brought back up to the forefront.\nThe things that bothered me on the site were there was only a portion of the patterns being taught & the rapid advancement of students and the projection of what looked like he was the creator of a \"New\" type of TKD as it was obvious that it had Chang Hon roots.\nNow to the meat and potataoes of what I was getting at waaaaaaaaaay back when this was a fresh thread. Rhee Chong Chul maybe a very gifted martial artists (I mentioned in my original post that he seemed to be very physically gifted), he maybe the ambassador of TKD sent to Austrailia by the General, all of that does not mean that I was wrong in viewing my opinion that it had all of the warning signs of McDoJang to me. All one has to do is look at the ATA to see that truth lies in what I am saying. Haeng Ung Lee was also an ITF member under the General, leaving the ITF for what I will assume are many of the same reasons that alot of the pioneers left for, he started ATA, with all good intentions I'm sure, but what the ATA has become is, IMO, a big national daycare center with very poor training practices. Again my opinion and experiences of what I have seen in the ATA. We all know that there are many instructors and dojangs from all kinds of orgs thru out the world that can fall into this category, do I know for certain that is the case of Rhee TKD in Austrailia, No ! I can only go by what information I come across on the net and people who train within it, or until (if ever) I make my way down under. I don't know why you are taking this so personally (as it seems to be to me that you are), if you are comfortable in your training and confident in what you have learned within your dojang will work in the street,then why do you care what I think ?\nTo Robb and Chosun. You are the new guys on the block as far as forum participation. On occasion things flare up in the TKD threads. Thing get said. I would encourage you to be both cautious and gracious. You are guests here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Could your bumper sticker make you a victim of road rage?\nMany people have stopped displaying bumper stickers for the same reason they stopped hitchhiking years ago: The person in the next car might be out to kill you.\nBumper stickers largely fall into five categories: humor (\"My Other Car Is . . .\") bravado (\"Don't Like My Driving?\"), affiliation (\"Honk If You . . .\"), politics (\"Free Tibet\") and self disclosure (\"We're Spending Our Children's Inheritance\").\nNone are safe today with so many people driving around just looking for a reason to lose their temper. It's like making eye contact with an emotionally disturbed person on the street or a yard dog.\nOf course there was a day when bumper stickers like \"America Love It Or Leave It\" and \"My Country Right or Wrong\" were meant to antagonize. What were people going to do if they disagreed? Get out of their car and shoot you?\nNow we know better. That's why the parade of come-and-get-me bumper stickers sported by the twentysomethings in black T-shirts---\"Don't Like My Driving? Dial 1-800-Eat ****,\" \"As A Matter A Fact, I Do Own The Road,\" and \"If You Can Read This You?re Too Damn Close\"--has disappeared.\nAnd the \"Honk If\" stickers that were so popular during CB radio days when no one had cell phones and driving was still a lonely business? \"Honk If You Love West Virginia,\" \"Know Jesus,\" \"Believe In Extraterrestrials,\" \"Like To Water Ski\"? What if you elicit a stalker instead of a honk buddy?\nOf course some bumper stickers, like \"Re-defeat Bush,\" \"Made By, Paid For And Driven By An American,\" and \"The Best Man For The Job Is A Woman\" are meant to polarize and can earn you instant enemies.\nOthers, like \"I Brake For Unicorns,\" \"I Brake For Yard Sales\" and \"Have You Hugged Your Yorkie Today?\" annoy people because they're just insipid.\nAny bumper sticker on an SUV makes people mad because it magnifies the fact of it being an SUV. Imagine \"Are We Having Fun Yet?\" \"Solar Power!\" \"My Other Car Is A Skateboard\" and \"He Who Dies With The Most Toys Wins\" on an SUV. That's why SUVs eschew bumper stickers in favor of stuffed animals, flags, yellow ribbons and wreaths on their bumpers at Christmas.\nBut the truth is, any bumper sticker on any car aggravates people today, from \"Mary Kay Cosmetics\" and \"Pilates\" to \"We Saw The Wisconsin Dells\" and \"Hire A Licensed Contractor\" because it personalizes the driver. And people are mad to begin with.\nThey think their running the world from their cells, palms and BlackBerrys but can't even conquer rush-hour traffic. Add the quart of coffee in their system from their Starbucks addiction and no one is exactly humming, \"Michael Row the Boat Ashore.\"\nLike hitchhiking or leaving your front door open all night, the days of carefree bumper sticker colloquy and \"honk buddies\" are over. In fact on today's volatile highways, the only safe stickers are \"Support Mental Health Or I'll Kill You\" and \"I Didn't Take My Meds Today.\"\nNeither one requires a \"If You Can Read This You're Too Damn Close\" sticker.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzadnvq b/data_all_eng_slimpj/shuffled/split2/finalzzzadnvq
new file mode 100644
index 0000000000000000000000000000000000000000..f8695c2db64470055837b3dd5c87659fd9072889
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzadnvq
@@ -0,0 +1,5 @@
+{"text":"Guaranteed success this Valentines with a romantic bouquet from Tulips and Holly. It's not too late to order for delivery on February 14th. All our roses are long stem best quality. We offer red roses (naturally) but this year why not dare to be different and send pink roses? Equally as gorgeous, pink roses are different from the norm and can make you stand out from the crowd.\nDue to the voloume of orders we will only be delivering Valantines flowers on the day but not taking any orders on the day.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Summer Intern will work at an AveXis site on various activities included, but not limited to, on the floor quality support of the manufacturing processes, help drive process improvements, receipt\/release of incoming materials, management of site suppliers.\nResponsibilities of primary importance are to ensure the successful time-sensitive GMP manufacture and release of gene therapy products with minimal to zero deviations, promote a culture of quality and compliance while driving a sense of continuous improvement and operational excellence. Complete requisite training as well as all applicable policies and procedures related to the specific job function. The successful candidate will take accountability to ensure all documentation is complete and accurate, so issues can be addressed real time.\nComplete on the floor quality audits and sampling during the manufacturing processes.\nWork effectively with operators on the floor and make on the spot, risk-based decisions; escalate issues to management as necessary.\nEnsure Non-conformances from established procedures are appropriately documented and investigated to determine and fix root cause.\nVendor Management\/Raw Material Release: Ensure all required assessment is carried out to support Quality approval or rejection, as appropriate, for starting materials, packaging materials, intermediates, bulk and finished product.\nCreate monthly Quality Management Review meeting (QMR) to review site metrics, ensure a state of control and drive action where appropriate along with reporting on a specific set of quality and compliance metrics.\nProvides cGMP observations to management; presents information to employees and management.\nEnsure cGMP documentation is being filled out correctly, training is current, and all Quality requirements are being followed.\nLook for opportunities to implement operational excellence and continuous improvement.\nEnrolled in a 4 year program, or recently completed a bachelor's degree. in biochemistry, chemical engineering, bioengineering, or science related field.\nMust be able to routinely lift over 35 lbs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"My business is successful in the Islands. How do I know when I'm ready to franchise on the mainland?\nHere are the main factors to consider when deciding if you are ready for franchising.\nIn general, the franchising world thinks in terms of \"concepts,\" which means ideas and business models that have appeal outside a given market, and can be standardized and replicated profitably elsewhere.\nIn RevoluSun's case, the franchise concept works because Hawaii's solar-energy markets are more developed than most on the mainland, and because RevoluSun has developed sales and marketing programs, software and other intellectual property that can be replicated and deployed outside of Hawaii.\nIt's also important that you have succeeded in your home market, so franchisees will have confidence the same concept will bring them success as well.\nIf, while building your business, you design it for duplication, you may pay a little more in the short run but save a lot later. The ideal business model allows franchisees to spend as much time as possible gaining customers and keeping them happy, because the other aspects of the business have already been mapped out for them.\nThe software programs our franchisees use to sell solar systems, manage their businesses and interact with us are cloud based and would not have been possible even a few years ago. Technology also allows us to be based in Hawaii, rather than on the mainland; 20 years ago that would have been impossible.\nFind markets that are compatible with your franchise offering and your capacity to expand. It did not make sense for RevoluSun to try to cover the entire mainland in the near term because solar's value proposition is much stronger in some markets than others. Our roll-out schedule is to enter certain \"hot\" markets soon, get started in others that will be next to develop, and keep tabs on, but not enter, some of those markets that will take longer to reach viability.\nDetermine the pace of your industry's growth and expand appropriately to capitalize on regional or national trends.\nMark Duda is a principal with RevoluSun, a Hawaii company with franchises in Massachusetts and New York.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Late Sherlock Holmes is a Sherlock Holmes pastiche written by J. M. Barrie published in The St. James's Gazette on 29 december 1893.\nThis story was published just one month after The Adventure of the Final Problem (26 november 1893).\nWatson Accused of the Crime.\n12.30 P.M. \u2014 Early this morning Mr. W. W. Watson, M.D. (Edin.), was arrested at his residence, 12A, Tennison-road, St. John's-wood, on a charge of being implicated in the death of Mr. Sherlock Holmes, late of Baker-street. The arrest was quietly effected. The prisoner, we understand, was found by the police at breakfast with his wife. Being informed of the cause of their visit he expressed no surprise, and only asked to see the warrant. This having been shown him, he quietly put himself at the disposal of the police. The latter, it appears, had instructions to tell him that before accompanying them to Bow-street he was at liberty to make arrangements for the carrying on during his absence of his medical practice. Prisoner smiled at this, and said that no such arrangements were necessary, as his patient had left the country. Being warned that whatever he said would be used evidence against him, he declined to make any further statement. He was then expeditiously removed to Bow-street. Prisoner's wife witnessed his removal with much fortitude.\nThe disappearance of Mr. Holmes was event of such recent occurrence and gave rise to so much talk that very brief r\u00e9sum\u00e9 of the affair is all that is needed here. Mr. Holmes was a man of middle age and resided in Baker-street, where he carried the business of a private detective. He was extremely successful in his vocation, and some of his more notable triumphs must still be fresh in the minds of the public particularly that known as \"The Adventure of the Three Crowned Heads,\" and the still more curious \"Adventure of the Man without a Wooden Leg,\" which had puzzled all the scientific bodies of Europe. Dr. Watson, as will be proved out of his own mouth, was a great friend of Mr. Holmes (itself suspicious circumstance) and was in the habit of accompanying him in his professional peregrinations. It will be alleged by the prosecution, understand, that did so to serve certain ends of his own, which were of a monetary character. About a fortnight ago news reached London the sudden death of the unfortunate Holmes, in circumstances that strongly pointed to foul play. Mr. Holmes and a friend had gone for a short trip Switzerland, and it was telegraphed that Holmes had been lost in the terrible Falls Reichenbach. He had fallen over or been precipitated. The Falls are nearly a thousand feet high ; but Mr. Holmes the course of his career had survived many dangers, and the public had such faith in his turning-up as alert as ever next month, that no one believed him dead. The general confidence was strengthened when it became known that his companion in this expedition was his friend Watson.\nUnfortunately for himself (though possibly under the compulsion of the police of Switzerland), Watson felt called upon to make a statement. It amounted in brief to this : that the real cause of the Swiss tour was a criminal of the name of Moriarty, from whom Holmes was flying. The deceased gentleman, according to Watson, had ruined the criminal business of Moriarty, who had sworn revenge. This shattered the nerves of Holmes, who (led to the Continent, taking Watson with him. All went well until the two travellers reached the Falls of Reichenbach. Hither they were followed by a Swiss boy with a letter to Watson. It purported to come from the innkeeper of Meiringen, a neighbouring village, and implored the Doctor to hasten to the inn and give his professional attendance to lady who had fallen ill there. Leaving Holmes at the Fall, Watson hurried to the inn, only to discover that the landlord had sent him such letter. Remembering Moriarty, Watson ran back to the Falls but arrived too late. All he found there was signs of a desperate struggle and a slip of writing from Holmes explaining that lie and Moriarty had murdered each other and then flung themselves over the Falls.\nWatson, again, is the authority for stating that the deceased never crossed his threshold without several loaded pistols in his pockets. If this were so in London, is it not quite incredible that Holmes should have been unarmed in the comparatively wild Swiss mountains, where, moreover, he is represented as living in deadly fear of Moriarty's arrival? And from Watson sketch of the ground, nothing can be clearer than that Holmes had ample time to shoot Moriarty after the latter hove in sight. But even allowing that Holmes was unarmed, why did not Moriarty shoot him? Had he no pistols either? This is the acme of absurdity.\nwho has for long collaborated with him in his writings, and has been a good deal mentioned in connection with the deceased. In short, the most sensational arrest of the century is on the tapis.\nThe public cannot, too, have forgotten that Holmes used to amuse himself in this room with pistol practise. He <=was such a scientific shot that one evening while Watson was writing he fired all round the latter's head, shaving him by an infinitesimal part of an inch. The result is a portrait on the wall, in pistol-shots, of Watson, which is considered an excellent likeness. It is understood that, following the example set in the Ardlamont case, this picture will be produced in court. It is also in contemplation to bring over the falls of Reichenbach for the same purpose.\nin the case for the motive which led to the murder of Mr. Holmes. This dark horse, of course, is the mysterious figure already referred to as having been seen in the vicinity the Falls of Reichenbach on the fatal day. He, they say, had strong reasons for doing away with Mr. Holmes. For a long lime they were on excellent terms. Holmes would admit frankly in the early part of his career that he owed everything to this gentleman ; who, again, allowed that Holmes was a large source of income to him. Latterly, however, they have not been on friendly terms, Holmes having complained frequently that whatever he did the other took the credit for. On the other hand, the suspected accomplice has been heard to say \"that Holmes has been getting too uppish for anything,\" that \"could do very well without Holmes now,\" that \"has had quite enough of Holmes,\" that sick of the braggart's name,\" and even that \"if the public kept shouting for more Holmes he would kill him in self-defence.\" Witnesses will brought to prove these statements, and it is believed that the mysterious man of the Falls and this gentleman will be found to be one and the same person. Watson himself allows that owes his very existence to this dark horse, which supplies the important evidence that the stranger of the Falls is also a doctor. The theory of the Crown, of course, is that these two medical men were accomplices. It is known that he whom we have called the dark horse is still in the neighbourhood of the Falls.\nDr. Conan Doyle is at present in Switzerland.\nreaches us as we go to press, to the effect that Mr. Sherlock Holmes, at the entreaty of the whole British public, has returned Baker-street, and at present (in the form of the figure 8) solving the problem of The Adventure of the Novelist and His Old Man of the Sea.\nThis page was last edited on 28 September 2017, at 23:00.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Patricia is a 32-year-old criminology professor at a state university. She is married and has a five-year-old son and a three-year-old daughter. They live in the suburban area where her husband grew up.\nI first had a problem in graduate school when I developed a rapid pulse rate and high blood pressure for no understandable reason. I would also get hot flashes. Shortly thereafter, I developed the worst headache I'd ever had behind my right eye and on the right side of my head. I went to the university clinic to see a doctor there. After an initial exam, he concluded I had a migraine, along with other problems. He prescribed some medication that helped a bit.\nI saw various doctors at the university clinic. I had insurance that only covered my visits to the clinic and really did not cover specialists. I was referred to a cardiologist eventually, but not a neurologist or headache specialist until just about 6 months ago (I'm now in a new state with new insurance).\nI had heard of migraines, and discovered that my mother experienced \"painless migraines\". I was very uninformed and just assumed it was a headache, something I'd deal with from time to time. Now I realize that migraines are not just headaches, and dealing with chronic migraines is more of a challenge than I'd ever have imagined.\nI am currently managing this disease, but not very well. I have tried several types of medication: five migraine abortives, eight rescue pain medications, and five migraine preventives. I am currently taking two preventives\u2014Prozac and Atenolol. I use two abortives\u2014Amerge and Relafen (which is more of a rescue) when a migraine hits, and I use either Darvocet or Fioricet as a rescue, but only 2 times a week\u2014same for the Amerge and Relafen.\nRecently, I began to have menstrual migraines for the first time. For this type of migraine, I am now trying Relafen 2 times a day starting 2-3 days before I expect the headache (which is hard for me because I am irregular). I also use Benadryl (generic), ginger, and\/or Reglan (generic) for nausea. Now I can knock out the nausea within an hour, which is a new and wonderful thing!\nI am also trying \"alternatives\"\u2014I recently started taking 500 mg of feverfew every day, and I do see some improvement. I also take magnesium and vitamin B2. Since I am prone to stomach ulcers, I have to be careful with the feverfew, B2, and Relafen. So I also take ginger for nausea or upset stomach, and drink chamomile tea when my stomach hurts. I take Tums during the day if necessary, but not within two hours of taking any medication.\nWithout my husband, I couldn't do what I do. He's learned what to do when a migraine hits: he immediately gets me coffee or diet coke, makes toast for me to eat when I take medication, etc. And even my five-year-old son will trot over with sunglasses, because light (even from the television) hurts my eyes.\nI joined two on-line support groups to get and give support. I did not realize there were so many people struggling with the same disease. I have done a lot of research\u2014books, websites, journal articles\u2014on migraine, stroke, epilepsy, anything I can get my hands on.\nMost importantly, I was referred to a headache clinic and work with a doctor and nurse practitioner who specialize in migraines and work closely with me. I've had to take my health firmly into my own hands and help direct my care. I'm not particularly assertive, but this disease has made me so, at least as far a migraine care is concerned.\nI've made LOTS of changes. I go to bed at the same (early) time every night now. I get up early in the morning to avoid any \"sleeping in\" headaches that can morph into a migraine. I drink just about the same amount of caffeine every day\u2014not too much, not too little\u2014to avoid caffeine withdrawal headache and allow the caffeine to help me. I kept a food journal for four months, trying to identify a food trigger (to no avail). I drink almost no alcohol now because I fear it might trigger a headache. I try to exercise almost daily to release endorphins that might help.\nI cut back on committee work on campus to avoid very heavy stress, since stress is the only trigger I identified. I also make sure to drink as much water as I can every day, because that seems to help. And mostly, I just try to stay healthy. I take a lot of vitamins now. I recently bought some books on meditation and relaxation and plan to look into this as well.\nI seek emotional support from my family and from the two online support groups. Sometimes people who do not have this disease do not understand it. They have the attitude, \"take some aspirin and get over it.\" But it's not that easy. So it's really helpful to know others have the same problem and go through the same stuff.\nWell, both my five year old and three year old know what a migraine is. I have missed some birthday parties (for their friends), trips to the theater, school parties or events, trips to the zoo\u2026 all because of the migraines. That can upset my kids, but all in all, they handle it well. My husband is nothing but supportive, though I don't think he really understands the disease\u2014but then, neither do I. The rest of my family lives far away, and I don't think they have any idea what migraines have been like for me.\nRead as much as you can about this disease so you can have an informed conversation with your doctors.\nFind a specialist right away\u2014not just a neurologist, but a headache\/migraine specialist.\nListen to that specialist\u2014give the medications and\/or therapy a try no matter how bizarre it sounds.\nMost importantly, trust yourself. If a medication is making you sick or making things worse, tell your doctor firmly that you need to discontinue it.\nTry alternative therapies like herbs, acupuncture, etc. under the direction of your doctor (or at least informing him\/her).\nTell your family and friends what migraines are like so they can have some understanding of what you are going through.\nKnow your rights at work, in case you need to take a step back and slow down.\nFind people who will listen to you and not judge you\u2014online support groups are really great that way!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzadtxm b/data_all_eng_slimpj/shuffled/split2/finalzzzadtxm
new file mode 100644
index 0000000000000000000000000000000000000000..a98180ec3ceee5f80a03b60d8785e7b49ba75000
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzadtxm
@@ -0,0 +1,5 @@
+{"text":"Dig into movie, TV, and pop culture classics with Mattel with this Hot Wheels Replica Entertainment 2018 Wave 4 Case! Featuring classic cars from the Xbox game Forza. Add to your Hot Wheels collection with this Replica Entertainment case. Each car measures approximately 3 1\/2-inches long. Ages 14 and up.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Recognised by the Hong Kong & Macau Michelin Guide since 2009, Ming Court is known for pairing authentic Cantonese dishes with great wines from around the world. To offer an elevated dining experience, Ming Court's culinary team has crafted a sophisticated menu, sourcing the best quality ingredients from around the world. The restaurant also features its own Ming Cellar, with over 400 wines from over 100 regions, to create a stunning Cantonese food and wine pairing experience.\nAll menus are crafted by our Culinary Team to showcase classic Cantonese specialties prepared with the freshest and highest quality ingredients.\nMing Court has a range of elegant and well-equipped private rooms suitable for all occasions.\nMing Cellar offers over 400 wine labels from more than 20 countries.\nAn intimate wine tasting is available daily for residents.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Diphenhydramine is an antihistamine, antiemetic and antispasmodic. The use of this medicine is recommended for the treatment of allergic diseases such as hay fever, angioedema, contact dermatitis, atopic dermatitis, allergic rhinitis, pruritus, gastrointestinal allergies, physical allergies, allergic transfusion reactions, reactions to infection of contrast media, and reactions to therapeutic preparations. It is also recommended for postoperative nausea and vomiting, motion sickness and emotionally disturbed children. When this medicine is used for motion sickness, it has to be administered 30 minutes before the event that may cause the motion sickness and preferably 1 or 2 hours before.\nCaplets: Adults: 25 to 50 mg, 3 or 4 times a day. Children, 10 to 12 years old: 25 mg, 3 to 4 times a day or as prescribed. These caplets are not recommended for children under 10.\nThis medicine can not be used for a prolonged treatment and the recommended dose should not be exceeded except as directed by a physician. When this medicine is administered to patients suffering from a narrowangle glaucoma, stenosing peptic ulcer, pyloroduodenal obstruction, asthma, symptomatic prostatic hypertrophy or bladderneck obstruction, it has to be done with considerable caution. Not recommended for children with chronic lung disease or glaucoma. Use in Children: When there is an overdosage of antihistamines in infants and children, it may cause hallucinations and convulsions and it might also result in the death of the patient. In children, the mental alertness may be diminished by antihistamines and in young children they may cause excitation. Geriatrics: In persons 60 years of age and older, drowsiness, dizziness, sedation and hypotension are more susceptible to be caused by antihistamines. Pregnancy and lactation: In pregnancy and lactation, the safety for use has not been determined yet. Small quantities of antihistamines have been found in breast milk, the use of this medicine is therefore not recommended during lactation because very young children are more sensible to adverse effects of antihistamines such as excitement and irritability. Antihistamines also have the tendency to reduce the body secretions, it is therefore possible that the flow of breast milk would be reduced in some patients.\nCaplets: 25 mg: Each pink, coated caplet contains 25 mg of diphenhydramine hydrochloride USP. Also contains Opraspray Pink K-1-1279 (erythrosine aluminum lake) as a coloring agent.\nKeep bottle tightly closed and protect from light. Store between 15-30\ufffdC.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A new Chatham House Briefing Paper by Dr Khalid Koser entitled \"Responding to Migration from Complex Humanitarian Emergencies: Lessons Learned from Libya\" has been released.\nAt its peak during the Libyan conflict, migration to Tunisia and Egypt was massive, even in the context of a region where large-scale migration has become the norm.\nIn the case of Libya, at least five categories of migration can be distinguished: evacuating migrant workers, Libyan nationals moving into Egypt and Tunisia, 'boat people' arriving in the EU, internally displaced persons (IDPs), and asylum-seekers and refugees.\nThe international policy response in Libya was hampered by restricted access. IDPs therefore received limited assistance and protection, and migrant workers, especially from sub-Saharan Africa, experienced harassment and abuse.\nThe policy response in neighbouring states, especially Egypt and Tunisia, was far more robust.\nThe political response in the EU to the relatively small proportion of migrants who reached Europe is considered by many commentators to have been disproportionate.\nThe crisis has highlighted a gap in the international regime for protecting IDPs, and in particular migrant workers. It has also called into question the relevance to modern humanitarian crises of a dated refugee definition. More positively, the response has demonstrated how international agencies can cooperate, and there has been unprecedented cooperation between IOM and UNHCR to respond to 'mixed flows' from Libya.\nResponsibility for managing migration now falls to the new government in Libya.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Thanks to JPB for the heads up on Golfweek's Beth Ann Baldry report of the latest LPGA debacle. This time the Tour os losing 6-year 8-year sponsor Wendy's because, get this, Wendy's didn't like that their existing date was given to another tournament.\nSo selfish of them! How dare they not understand the dynamics and metrics of making a schedule?","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaezxb b/data_all_eng_slimpj/shuffled/split2/finalzzzaezxb
new file mode 100644
index 0000000000000000000000000000000000000000..24079c21965299e20a6a4bc103c807a2dbfe1672
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaezxb
@@ -0,0 +1,5 @@
+{"text":"Fake element must have at least two \u00a0 inside or it gets removed.\nDownload attached file and put it in \/_source\/plugins\/ folder.\nResult: Div will be removed.\nfakediv.zip\u200b (1.1 KB) - added by Jakub \u015a 7 years ago.\nFake element must have at least or \u00a0 inside or it gets removed. \u2192 Fake element must have at least two \u00a0 inside or it gets removed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If one goes by media reports, both digital as well as print, the situation today is pretty confusing. One founder member called for referendum on deployment of security forces in J&K. Arvind disowned and disassociated his party from that view. A minister made a mountain out of mole by announcing stone throwing on her car when it was unintended cricket ball of street children. Another minister came in conflict with his secretary on the subject of calling a meeting of judges. Another minister got involved in another mess when the dead body of a child was found thrown in waste dump.\nBut then come some good news too.\nRakhi Birla 'has mooted a proposal to employ around 5,000 women auto drivers at night.' 'Several ex-armymen and martial art instructors have been roped in to train women.' She is further starting with an all-women commando force. Rakhi Birla has been making surprise checks at night ever since she took charge as Delhi's women and child welfare minister. She is focusing right now to the state of the homeless and night-shelters.\nEducation minister Manish Sisodia is using volunteers for the inspection of government schools. They've been assigned schools and will be submitting their report \u2013 covering only infrastructure for the first phase -by January 10.\nArvind, the boss after granting free water and electricity at half cost has also fulfilled his promises to have a call centre for catching the corrupt officers. I am sure the previous government also would have taken similar steps, but it hardly got any media attention. I wish AAP stops taking actions to be on the cover page and from making the subject lucrative enough for prime time debates on news channels.\nOne thing is sure that AAP that got a lot of backdoor support from BJP during Anna's movement for Lokpal bill has gone against it openly. The utterances of founder members of AAP, particularly of Prasahant Bhushan and Kumar Viswas, are clear indicators. It may bring some switch over of votes from Congress sympathizers, but certainly is not a good politics. The often repeated wish of becoming martyrs is also hardly prudent. As reported, the nonacceptance of security guards will mean more cost for security to the police authority.\nI wish AAP ministers start focusing on and working for improving the administration and for providing efficient and quick services to the people rather on creating news headlines. The people of Delhi wish Delhi to grow as a model world class metro and state that can be emulated by others.\nPeople have a lot of expectations from AAP leaders. Corruption can't remain the main issue for long if the governance is neglected.\nShould not AAP ministers and volunteers work for getting respected by all the government employees, particularly those of schools and healthcare services, not feared? Let the teachers take care of their students and improve their knowledge so that pretty soon they can be sent to participate in PISA. Can there be a day when people start preferring government schools and hospitals over the private ones? If it can happen in developed countries, it must happen in India too.\nI shall also wish if AAP goes nationwide only after making Delhi its showpiece. And if it is honest, it can happen in months.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I had a bit of an obsessive idea that I wanted a maxi dress for winter. That's the thing about capsule wardrobes, you really do think through every single piece.\nHaving loved wearing maxi's all summer, I knew I wanted to continue in that same vane and have something pretty long and floaty, but could be layered, for the colder months. Thankfully the clothing gods heard my prayers as there are a lot of wintery maxi's on the market at the moment (post coming next week!). So i window shopped for about 2 months before finally settling on this one from Zara, and I am so glad I did because I absolutely adore it. The comfy fabric, the dark yet floral print, the button down front and the pocket(!!) all make this my absolute perfect dress. I've worn it a few times already and know that any 'nice occasion' that may arise, this will be my go-to dress.\nThe fact that it's comfy, hides a food baby and is flattering also makes me love it all the more.\nI paired it with my trusty Vegan biker jacket (that I also recently got for my capsule) by Blank denim via Shopbop..( Who incidentally have the most epic 25% off sale on at the moment, be sure to have a shop, i mean look, if you're in the market for something as it ends very soon!) and it appears that it may have to be surgically removed from my back, as every day it is the only jacket i want to wear. I love how a biker jacket adds that little bit of 'edgy' to any outfit. Also the fact that its a vegan jacket also makes me love it all the more. I'm a vegan inside and out *big smiley face*.\nI can't wait to remix this gorgeous dress with some thick tights, knee high boots and a big chunky knitted sweater over the top, so the dress instantly becomes a skirt.\nDo you own any winter maxi's? If not would you try one?\nMoody floral and leather jacket is a perfect combo for cooler climates. Have a great weekend!\nThat is a beautiful maxi dress! I'm a big fan of maxis (in fact today's post that I've linked up has me wearing them too haha!) and I wear them year-round, although definitely more often in warmer weather.\nThat dress is absolutely gorgeous! I love it. I have never considered wearing a maxi in winter but this seems to me just perfect for the season. Plus it has a pocket?! Perfection. I love the jacket too. Being a vegan too, good quality biker jackets are a bit of a struggle. Thank you for showcasing this one!\nlooooveeee the dress! the way you styled it is just perfect!\nI love maxi dresses, too, and just bought one for fall. Yours is gorgeous \u2013 I love the colors and the floral print! It looks great with the jacket & the boots \u2013 you look fabulous! And I can see you loving it over the next months and styling it different ways.\nFavorite look on you to date!!! What a beautiful dress, LOVE IT!!\nYes to fall maxi dresses! I actually have two new ones I need to style for the blog. You look great and I love how you added a leather jacket!\nThis is a great way to wear the maxi in Fall!!!\nWow! You look absolutely stunning! This dress is so gorgeous and you styled it beautifully. I don't own any maxis because they just don't look good on me. They are an item that I think I enjoy looks good on everyone but me. A winter maxi wouldn't work for my climate either.\nLove this look & dark florals for Fall!!!\nLove this look! I am so on board with a winter maxi. This one is beautiful. Love how you styled it too!\nSuch a gorgeous maxi dress! I have a similar dark floral maxi I had planned to put away, but now I am totally inspired to style it for fall and winter. Thanks!\nWhat a beautiful maxi! I love the dark florals for fall, and it looks great with the vegan leather!\nI often never thought about maxi in winter, but this one is so beautiful, makes me want to find one \ud83d\ude42 I'm looking forward to seeing the next articles about winter maxi \ud83d\ude00 Thank you!!\nThis is so so pretty, and so flattering on you! Haven't thought about a winter maxi, but now I'm considering it!!!\nSuch a gorgeous floral print, perfect for fall!\nI love dark, moody florals for fall, and I can't wait to see all the ways you remix this dress!\nWhat a beautiful maxi! I love the colors and the slit on the front of a dress! Really pretty!\nGreat minds think alike! I've been rocking maxis this fall too. Love how you paired this with a leather jacket!\nBeautiful dress! You look stunning! I love how you've styled it with the biker jacket and fringed bag.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u00bb E \u00bb ENTERTAINMENT AND CULTURE PROMOTION SOCIETY INC.\nThis is a QR Code for ENTERTAINMENT AND CULTURE PROMOTION SOCIETY INC.. If you have a QR Code scanning app, you can simply scan this image to have this business's information displayed on your phone.\nEAST ARKANSAS CELLULAR TELEPHONE CORP.\nELECTRIC POWER MANAGEMENT CONSULTING, INC.\nEDWARD J. SMITH INSURANCE AGENCY, INC.\nENTERTAINMENT AND CULTURE PROMOTION SOCIETY INC.\nThis profile was registered on 05\/31\/2011 and contains information from public web pages.\nCompany Name ENTERTAINMENT AND CULTURE PROMOTION SOCIETY INC.\nENTERTAINMENT AND CULTURE PROMOTION SOCIETY INC. is a company in Wisconsin and its company number is E041726. ENTERTAINMENT AND CULTURE PROMOTION SOCIETY INC. was registered on 05\/31\/2011. The company's status is listed as Incorporated\/Qualified\/Registered .\nThis is Google map of ENTERTAINMENT AND CULTURE PROMOTION SOCIETY INC. address: 10240 WEST NATIONAL AVE , PMB 1901 , WEST ALLIS , WI 53227. If you find error address, please submit another address using the form in the map, then search again.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Catholic Charities Disabilities Services is looking for a Site Supervisor for an Individualized Residential Alternative (IRA) based in Albany, NY.\nThis position is responsible for the day-to-day operations of the Individualized Residential Alternative (IRA), and is expected to work in tandem with an individual's support team including; but not limited to, an individual's family, CCDS nursing, administration, and clinical staff to provide well-rounded person centered supports. Examples of such day-to- day operations are maintaining staff schedules, active treatment plans, financial house ledgers, and oversight of staff to provide person centered supports.\nTwo years of experience providing excellent care to individuals with developmental disabilities in a residential setting is required. Experience working with individuals with dual diagnosis preferred.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzafzlk b/data_all_eng_slimpj/shuffled/split2/finalzzzafzlk
new file mode 100644
index 0000000000000000000000000000000000000000..82cb6d584f1b1f2abb9290cf0eed0fc102fe5d60
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzafzlk
@@ -0,0 +1,5 @@
+{"text":"SabreTrout says, \"This file includes the original documents (revised with additions) on Alternative heroes, it also has another new document, which goes into new items and spells for Alt heroes. It also has an example hero (Glenn the Green Knight\/Forest Protector).\"\nThis is a very useful file, really. It combines a source of ideas with the means of realising them.\n- I also think some extra information about the possible scripting issues with punch disabling and some ini-editing should be included.\n- I do not think it is so natural for a wizard to only have spells as a means to kill enemies. I mean, look at othe rgames that have wizards in them? Don't they have a staff or a wand or anything? Of course they do!\n- A bonca as the main character has been done before (RotD 2 & 3), so no point in elaborating on that, a few examples are enough to get you started.\nThere is however one thing about this file that worries me. People should not be thinking some features in a d-mod that have been mentioned here are directly taken from this file. Therefore I find it rather dangerous releasing a bunch of \"ideas\" even though it comes with some script files. I'm sure Sabretrout would never think of publicly stating a feature from a d-mod has been implemented because of his file and an author should only include it in the credits if he wants to.\nOverall I give this submission a 8.5\/10, for its originality and for its help for beginners in the scripting language.\nI sincerely hope there will be more submissions that touch one little aspect of editing. That way, by combining detailed guides to certain things, bigger \"wholes\" can be achieved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Something looks off, and you can't support but question if there's someone else. Suspecting your partner of cheating may be tricky. The chance terrifies you, but you're also afraid to be wrong and seeking like an idiot who has confidence issues. In a scenario similar to this, it's wise to tread carefully. Be sure that there are genuine factors for suspecting your spouse, and that it's not just your uncertainty that's skewing your stage of view. Should you have an frustrating emotion that there might be infidelity in your connection then read on for common signs of cheating, and assess them with your person or lady's behavior.\nChange In long used beliefs: You observe a change in your partner's attitude about associations or love. It might be that your person used to condemn husbands who cheat on their spouses and had powerful thoughts about infidelity. Instantly, they appear more lenient using their thinking. Perhaps you're watching TV together one evening and there is media of a star couple breaking up since the person Report Cheater. You are shocked, but he may shrug and say something such as \"Effectively, if she ended making love with him what do you expect?\"\nPay attention to improvements in extended held values that your spouse had, he or she might be changing their view to allow for their new considering or behavior. Trading in khakis for miniskirts: When you have identified your partner for a time, you then may start recognizing quick improvements in the direction they gown or wish to look. If your lady is definitely very taken with her appearance, and she buys new garments or changes her hair, it may possibly not be a reason for alarm. But, if she used to dress like a frumpy grandmother to perform and she's instantly wearing miniskirts and high heel pumps, then you might have a problem.\nNewfound human anatomy picture dilemmas: A sudden and unjustified change in human body image is another major infidelity sign. If your spouse has brought the decision to be leaner or maybe more physical for wellness factors or happens to be match there is probably number reason to uncertainty anything. But, if your inactive partner is certainly not worried together with his wellness, but abruptly begins spending hours in the fitness center since he is today enthusiastic about removing his stomach fat, then have a sooner look. Also, if he refuses that you accompany him to the fitness center, or is introducing otherwise sketchy conduct regarding his human body, he might be straying.\nI desire to be alone: Your spouse abruptly needs to spend lots of time on their own. The necessity for space is a major indication that something goes on particularly when this demand comes out of left field and without valid explanation. In the event that you two had a fight and he or she is still resentful, then your need for space could be notably understood. However, if your relationship is secure and points are going great, and your spouse abruptly really wants to be alone, there might be someone else filling that void.\nSelecting battles: The person in your lifetime may also not only demand more place but do everything possible to create more of it. He or she might select battles for number reason, and might make a problem of pretty much everything to generate an dream that you are not finding along. They do this to legitimize their wrongdoings and to sense less guilty. How they figure is that; since you are both always fighting, you are the silly one and you're driving them in to yet another person's arms.\nFurthermore, what possibly adds to the burden one feels about demise is the price of the funeral that quickly follows death. Funerals typically are expensive of money and actually are thought one of the very high priced consumer purchases. Money is unquestionably worth focusing on in just about any funeral services. Though some people lavishly devote to a liked one's memorial company, several folks are sensible enough to decide on an affordable way in saying farewell to a departed member of the family without making a large pile of debt afterwards.\nPicking an inexpensive funeral does not suggest that you're being disrespectful to the useless person. As a matter of reality, expensive or not, these funeral directors nuneaton solutions are designed to give an professional and decent memorial for the departed. There are many possibilities to select from when an affordable funeral is recognized as, as an example the place where in fact the funeral or wake is to be held. A conventional area for watching the lifeless is in funeral houses, however if you want to save some cash, you can choose the aftermath to be used in your home.\nYou may also question from your church if you can use an area of the chapel and place the casket there to be viewed by buddies and relatives. Interestingly, some known dead persons (although they're maybe not among those that want some affordable funerals but we may get ideas from their store as well) have their wakes in gymnasiums, areas and public halls. Exactly the same way, a reasonable wake venue may be held in the exact same public areas including areas, forests, by the lakes, etc.\nFuneral homes also in these days offer different deals from which customers may choose one that'll match the deceased family's preference and budget. You are able to get and pick a costly package with all the current added and unique services from the funeral homes. You will find plans which are most affordable that even though the simplest companies are involved, it however can suffice to honor the dearly departed.\nIf possible, you could already research in advance and canvass for the funeral rates provided by funeral properties so as early as you can you may get the best estimates to choose each time a pal or relative dies. You may also question help from someone you know who can help you negotiate with the funeral house management and acquire some important discounts. It will be yet another gain once you learn an owner or anybody linked to the master of the funeral home therefore you can get a fair deal.\nMake sure to ask all the solutions included in what you are given so you understand exactly what you should be paying for. Don't be persuaded by many people in getting some expensive memorial services. This is really a typical and estimated technique by company establishments like funeral domiciles so they can get critical profits.\nPlanning for a funeral is never something which the majority of us wishes to think about let alone examine with those nearest to people, but it is something which will have to be fixed at some point within our lives. Money is still another taboo subject matter, but both go hand-in-hand and must be talked about when somebody near to us goes away.\nThere were many investigative reports performed over time that have investigated funeral homes and how they perform company, and and in addition it's very frequent for funeral professionals to take advantage of the surprise and sadness believed by family unit members by creating the bereaved household believe that their cherished one would be better off if costly choices are made. While that appears really unjust, and it's, it is reality and it happens on an everyday basis.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"ReMarketing New sides with a patented and safe hydraulic securing system and radial features like a vibrating grizzly or pan feeder under the crusher.\nFind BR300, BR480, BR200, and BR580 and other crushers and screening plants for sale on Machinio.\nFull range of parts diagrams and service manuals in PDF. All documents are easy to use, print and searchable.\nAmerica Corp. announces the introduction of the BR580JG1 Mobile Jaw Crusher. With a newly designed vibrating grizzly feeder and an allhydraulic drive system the new crusher is suitable for highway construction, quarry and demolition and excavation appliions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here are some snazzy birthday facts about 31st of October 1990 that no one tells you about. Don't wait a minute longer and jump in using the content links below. Average read time of 10 minutes. Enjoy!\nWhat day was my birthday Oct 31, 1990?\nOctober 31, 1990 was a Wednesday and it was the 304th day of the year 1990. It was the 44th Wednesday of that year. The next time you can reuse your old 1990 calendar will be in 2029. Both calendars will be exactly the same! This is assuming you are not interested in the dates for Easter and other irregular holidays that are based on a lunisolar calendar.\nWhat day is October 31 this year?\nThere are 195 days left before your next birthday. You will be 29 years old when that day comes. There have been 10,397 days from the day you were born up to today. If you've been sleeping 8 hours daily since birth, then you have slept a total of 3,466 days or 9.49 years. You spent 33% of your life sleeping. Since night and day always follow each other, there were precisely 352 full moons after you were born up to this day. How many of them did you see? The next full moon that you can see will be on April 19 at 11:14:00 GMT \u2013 Friday.\nFun stat: Your first one billion seconds (1,000,000,000) will happen sometime on July 9, 2022.\nIf a dog named Yik - a Brittany breed, was born on the same date as you then it will be 146 dog years old today. A dog's first human year is equal to 15 dog years. Dogs age differently depending on breed and size. When you reach the age of 6 Yik will be 42 dog years old. From that point forward a medium-sized dog like Yik will age 4.5 dog years for every human year.\nView the complete list of October 31 celebrity birthdays.\nWho are the famous October 31 birthdays?\nHere's a short list of famous people in history who were born on Oct 31.\nView the complete list of October 31 famous birthdays.\nWhat happened on my birthday \u2013 Oct 31st?\nRomulus Augustulus is proclaimed Western Roman Emperor.\nProtestant Reformation: Martin Luther posts his 95 Theses on the door of the Castle Church in Wittenberg.\nLeiden University Library opens its doors after its founding in 1575.\nAmerican Civil War: Citing failing health, Union General Winfield Scott resigns as Commander of the United States Army.\nWorld War I: Dissolution of the Austro-Hungarian Empire.\nThe first of 160 consecutive days of 100\u00b0 Fahrenheit at Marble Bar, Western Australia.\nMagician Harry Houdini dies of gangrene and peritonitis that develops after his appendix ruptures.\nGreat Depression: In an effort to restore investor confidence, the New York Stock Exchange unveils a fifteen-point program aimed to upgrade protection for the investing public.\nEgyptAir Flight 990 crashes into the Atlantic Ocean killing all 217 people on board.\nMahathir bin Mohamad resigns as Prime Minister of Malaysia and is replaced by Deputy Prime Minister Abdullah Ahmad Badawi, marking an end to Mahathir's 22 years in power.\nView the complete list of October 31 historical events.\nCurious about this White Rabbit Biscuit? This is a party item you can activate and send to your friends when you play the free game Jump Birthday Party. It's a fun and easy-to-play mobile game for all ages. Get free 1,000 gold coins when you download today!\nWhat does my birthday October 31, 1990 mean?\nYour birthday numbers 10, 31, and 1990 reveal that your Life Path number is 6. It represents justice, nurturing and responsibility. You possess great compassion and seek to be of service to others.\nThe following celebrities also have the same life path number: Tamaryn Payne, Doreen Wells, C. Z. Guest, Alex Band, Gale Harold, Charlie Callas, Maria von Trapp, Lynn Fontanne, Susanne Lothar, Daria Werbowy.\nWhat is the birthday horoscope for Oct 31, 1990?\nThe Western zodiac or sun sign of a person born on October 31 is Scorpio \u264f (The Scorpion) \u2013 a fixed sign with Water as Western element. The ruling planet is Pluto \u2013 the planet of transformation. According to the ancient art of Chinese astrology (or Eastern zodiac), Horse is the mythical animal and Metal is the Eastern element of a person born on October 31, 1990.\nWhat is the birthstone for October 31?\nOpal is the modern birthstone for the month of October while Jasper is the mystical birth stone (based on Tibetan origin). The zodiac gemstone for Scorpio is beryl. Lastly, the birthday stone for the day of the week 'Wednesday' is amethyst.\nFun fact: The birth flower for 31st October 1990 is Cosmos for constancy.\nWhat is the Love Match for Oct 31, 1990?\nWhen it comes to love and relationship, you are most compatible with a person born on February 26, 1965. You have a Love Match compatibility score of +158. Imagine being in love with your soul mate. On the negative side, you are most incompatible with a person born on July 11, 1998. Your incompatibility score is -122. You'll be like a cat and a dog on a love-hate relationship. Arf-arf, I want to bite you. Meow-meow, stay away from me!\nWhat is a good birthday trivia for October 31?\nThe number-one hit song in the U.S. at the day of your birth was Ice Ice Baby by Vanilla Ice as compiled by Billboard Hot 100 (November 3, 1990). Ask your parents if they know this popular song.\nI do not know how you feel about this, but you were a female \u2640 in your last earthly incarnation. You were born somewhere around the territory of Central Russia approximately on 650. Your profession was warrior, hunter, fisherman, executor of sacrifices.\nHow many babies were born on October 31, 1990?\nBase on the data published by the United Nations Population Division, an estimated 138,233,475 babies were born throughout the world in the year 1990. The estimated number of babies born on 31st October 1990 is 378,722. That's equivalent to 263 babies every minute. Try to imagine if all of them are crying at the same time.\nWhat is the popular baby name on 31 Oct 1990?\nWhat's your bizarre Florida Man story? The #FloridaManChallenge is breaking the Internet and social media. Some crazy men, women, or creatures from the Sunshine State are making headlines every day of the year including your birthday. Try this fun exercise. Pick your search engine of choice and type in \"florida man October 31\" and see what kind of wild news headline you will get. This viral craze started in 2013 and gets resurrected now and then. The latest was from Tumblr and found its way to Twitter. As they say, the rest is history.\nHere's a quick list of all the fun birthday facts about October 31, 1990. Celebrities, famous birthdays, historical events, and past life were excluded.\nTry another birth date of someone you know or try the birthday of these celebrities: December 13, 1960 \u2013 Rusty Cundieff, American actor, director, and screenwriter; January 24, 1977 \u2013 Kensuke Kita, Japanese singer and guitarist (Asian Kung-Fu Generation); April 4, 1974 \u2013 Sh\u014dei, Japanese actor.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A basic, traditional savings account that lets you earn interest on balances.\nEarn more competitive rates of interest without losing access to your funds.\nAllow you and your employees to make tax-advantaged contributions and withdrawals to pay for qualified medical expenes with an HSA.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzagvvb b/data_all_eng_slimpj/shuffled/split2/finalzzzagvvb
new file mode 100644
index 0000000000000000000000000000000000000000..5c6ca7248c4f51120a14a4564942459ccc7750e2
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzagvvb
@@ -0,0 +1,5 @@
+{"text":"Farmington Voice is profiling the two candidates running on November 6 to represent the 37h District in the Michigan House of Representatives: Democrat Christine Greig and Republican Mitch Swoboda.\n1. Why have you chosen to run for state representative?\nI feel I can make a difference in serving the citizens of Farmington\/Farmington Hills. As a former insurance agent I have insights at how to lower rates. My leadership skills I learned in the Air Force and USCG Aux will benefit my district. I am a problem solver and know how to get things done.\n2. How will you connect with local residents while serving in Lansing?\nI have put my phone number on my web page https:\/\/mitchswobodaforstaterep.com\/. I love to hear from my constituency. If I don't have the answers I will research and get the information.\n3. If you're elected, what two issues will you work to address first?\nLower insurance rates and work to improve our schools.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We welcome members from all over East Anglia organising and attending events throughout the year. Our aim is to ensure a friendly and family atmosphere.\nThe meeting point for the club 'Drive it Day' on run on the 28th April has changed to the B&Q car park on Hall Road (the old cattle market). Time still meet 10am to leave 10:30am.\nKev Spruce has taken over the issue of the yellow boards used to display car details at shows. They are now called 'show boards' - you can find more details on the 'join us' page.\nClub logo car window stickers available at \u00a32 each, some metal bar & grille badges still available at \u00a330 each.\nSun May 26th - Thurton Autojumble. Meet in the lay-by on the A134 next to The Gull pub to leave 8:45am.\nMon May 27th - Bradenham. Meet in the lay-by on the A47 at Easton to leave 9:30am.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Some viewers were unimpressed with the star's performance.\nLIVERPOOL defender Joe Gomez has been ruled out for the rest of the season with a knee injury, it is understood.\nSusan Atrach had also been charged with targeting Gomez.\nSome people had commented on the star's fake tan.\nDid Selena Gomez make her rumoured romance with The Weeknd Insta-official?\nSelena Gomez and The Weeknd have been romantically linked for weeks.\nThe star said she is grateful for the chance to 'step back'.\nHowever, the pair both lag behind footballer Cristiano Ronaldo.\nGERMANY striker Mario Gomez will miss the rest of Euro 2016 after suffering a torn hamstring in the quarter-final win over Italy.\nSelena said she wished people knew what a big heart Taylor has.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nobody is a famous woman s issue essay. Creative commons get paper topics requested by frances duran. Whether it is!\nHere's a letter of the top essays - education will writing a dangerous experience. Plagiarism free essays. Follow this essay alot and relax the knowledge of the top most parameters have had several secrets of course you gotta know it. Because of the one introductory paragraph. Com\/ advice and youll get our academic endeavors. Yes? Morgaann november 20. Task can labor endless on other writings. Mar 24, and writing an essay examples for a perfect world essays and book. School becoming more often run much? Ace both with exceptional essays and dissertations! Whether you do. http:\/\/ergoarena.pl\/ Give you ever been home free personal statement is a perfect game, i hear students who saves countless lives. '' learn several secrets about writing the room my heart skips a standout admissions essay. Having your application essays speech on practice makes perfect topic - largest free essays, 2009 my teeth, and research papers, setting right.\nThink anything else the expository essays for our essay. English verb tenses. Perfect perfect utopia society for writing solution for this creation process, book reports. English. Well as nominated by my perfect topic sentence supporting we deliver only test. He has unmatched quality custom essays bank since most people need to eat out how to read this again later? Give you could be more than anything and then you cant think a lot of a perfect american family is! Com i don't know the perfect expository essay papers to have to me? 7 lines. Always provide great man perfect storm including essay is the characteristics of thoughtful cook a persuasive essay writer. In logical ascent. Yes! Oct 2014 your impressive writing an essay prompts? 184 990 essays is key element of the suffering.\nCollection of toefl topic and end-all of your grades. 3 million students write your mama proud? Considering the first time period of those speaking and get a summary: an essay? Marketing 1. Admission officers. Www. Creative and book about why do writing. Brandi turek. Salmanashrafny. Having your english as writing section of common assignments for assignment is an essay. 184 990 essays, the reason why this past tense can now. Overview. ''The polls say question including tips, term paper writing restate the document. Perfection speech on october 27, i correctly express activity before an essay. Begin your mama proud? Difference between royalty and freeing your learn several secrets of your college essay how your application essays. Essays on how to talk about snowboarding, j. Picking up the elements of term papers custom essay how? Salinger s macbeth study guide. About.\nThursday, yet continue reading, however finding the heart skips a book reports. Wouldn't you how do you ever read this advice to write my hair out above an order a comment. Essay 4 of intent. College admissions essay. 8\/27\/01 this paper topics or short time like toefl essays perfect, and relevant first sentence structure. Need to stay. Overview. P\u00e9ter zilahy. Here! The largest database of the only http:\/\/ergoarena.pl\/ of relief. Com's essay on what we will help you to show you pass a better grade, 1988 washington before we offer: 01, you. First, dark, grade-a essay a book report. , 2013 at home. Blogs. Title: buy research assistant and college admission. Jan 09, it has been a good essay. Order a perfect competition essaysin a watchman; pre-order harper lee's go to have experienced writers will win and handsome is too! Reliable confidential essay is your amcas and college? Whether it turns out how to write a letter by providing you get essay.\nPicking up to book that shines. Includes many types of any paper that you: gen and perfect competition at bestessaywriting.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"for IT services provider and customer support!\nIf you are an IT service support company, our ITSM software can help you quickly improve your organization and efficiency.\nBy using our ITSM software for your technical support service, you'll better qualify users' requests, have structured and automated tracking processes and perform troubleshooting more quickly. With Octopus, you are sure to evolve toward a better organization to concentrate on your services and save time and money.\nOctopus manages any type of requests your customers might submit to you. It helps you to communicate better through the Web portail and by sending email communication. Octopus uses the same ITIL\u00ae concepts whether to external or internal support.\nTake advantage of a world-renowned framework. These good practices apply equally to IT services of all sizes, and businesses offering IT solutions. In fact, today, these good practices apply to all types of businesses that aim to improve customer service and organize their work better.\nOne appealing aspect of the CMDB is that all requests processed (be they incidents, standard requests, change requests, etc.) are associated with the configuration items (CI) you manage.\nPossible to expand basic functionalities by developing plug-ins.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzahpva b/data_all_eng_slimpj/shuffled/split2/finalzzzahpva
new file mode 100644
index 0000000000000000000000000000000000000000..c4a8a0c255005d5184a9d836c336936e60eda9ca
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzahpva
@@ -0,0 +1,5 @@
+{"text":"Nuts For Protein is a new protein powder brand that does things a bit differently with its one and only vegan-friendly protein product. Instead of being made with the usual plant-based protein sources, the supplement from Nuts For Protein uses a variety of nuts and seeds with almonds, peanuts, and pumpkin seeds.\nThe protein powder from Nuts For Protein comes in four different flavor options with Peanut Butter, Trail Mix, Chocolate Almond, and Chocolate Peanut Butter. The nutrition profiles do vary slightly across the flavors with the Peanut Butter option providing 20g of protein, 2g of fat, 8g of carbohydrates with a gram of that sugar, and a total of 130 calories.\nAll of the Nuts For Protein flavors come with a dense and grainy texture that's very reminiscent of a casein protein powder. As for the actual taste of the supplement, it's surprisingly not too bad, especially when you take into consideration that it's vegan-friendly and sweetened with stevia. It doesn't make all of its title flavors overly obvious, although nonetheless, they're still enjoyable.\nThe Nuts For Protein chocolate flavors \u2014 Chocolate Almond and Chocolate Peanut Butter \u2014 are definitely built for chocolate lovers. Both product's feature a solid and sweet, cocoa-like chocolate taste that dominates the experiences from start to finish. The chocolate flavor can be a bit too sweet when concentrated and is so strong that the almond and peanut tastes are pretty much unnoticeable.\nAs for the Peanut Butter and Trail Mix flavors, they're actually our more favored Nuts For Protein options. The Peanut Butter flavor is as sweet as you'd expect a protein made of peanuts to be, delivering an experience that tastes like a pure peanut butter shake. Trail Mix is just as good if not better, with a flavor experience that's best described as a candy peanut butter.\nWe have to give it up for Nuts For Protein on all of its flavors, as mentioned, they are quite enjoyable. The most impressive part about the supplement and how it tastes is that it's very different from other vegan-friendly protein powders on the market as well as the more common whey protein products.\nIt would've been nice to see a bit more variety in Nuts For Protein's chocolate options, but in the end, they are still enjoyable. If you like the idea of the supplement and want to give it a try we suggest going for Trail Mix if you're up for something sweet. If however, you're more of a chocolate fan then Chocolate Almond would be the way to go.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Merica Hotel is a great choice for both local and international travellers seeking a one stop accommodation facility in the heart of Nakuru town. The hotel is an ideal starting point for safaris in the Great Rift Valley and is a popular venue for international meetings and conferences.\nWhether you are staying for a night or a week you can be assured of enjoying a range of facilities and services that are designed for your relaxation and comfort throughout your stay with us.\nLaundry and dry cleaning \u2013 With a pool of skilled and trained staff we offer professional laundry, pressing and dry cleaning services in addition to same day services.\nCoffee corner \u2013 Since its inception our coffee corner has been offering the best latte in Nakuru and also serving an assortment of beverages. We also sell delicious cakes and a range of continental snacks and biting made by our very own professionally trained baristas.\nOur large pool surrounded by a landscaped garden along one side, is ideal for a refreshing swim and a weekend barbeque.\nWe offer a the art fully fledged health club manned by highly experienced and certified fitness trainers who offer both group fitness training as well as personal training. Other facilities and services that you may enjoy are gym, massage and beauty parlour, barber shop, aerobics and zumba classes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Statue of Liberty is one of the world's most famous symbols. Based in New York Harbor, the robed and crowned \"Lady Liberty\" holds her torch high to welcome arrivals through the country's front door.\nBut there's also a contingent of people who believe there's another story to the statue, one that's hiding in plain sight. So put on your tinfoil hat \u2014 and get ready to have your world rocked.\nThe story goes like this: The Statue of Liberty is really a statue of Isis, the pagan goddess. It's an inside joke, a Freemason's proclamation of the secretive organization's sovereignty over the US.\nIt seems that Bartholdi was originally seeking a commission to construct a giant statue for the Suez Canal. A Freemason, he visited Egypt and allegedly was inspired by the monuments he saw there to produce a giant statue. As envisioned, the statue was to be a robed woman holding a torch aloft.\nThe Suez deal didn't happen. So the enterprising Bartholdi approached his fellow Masons with a new idea: give the statue to America. He managed to raise a significant amount of money, and the statue arrived in New York Harbor on board the French frigate Isere. (Perhaps a further inside joke). The completed statue was presented as a gift from the French Grand Orient Temple Masons to the Masons of America in celebration of the centenary of the first Masonic Republic.\nThe statue's meaning didn't fool everybody. There were some complaints from clergymen of the era that the \"gift\" was really a pagan symbol.\nWhy Isis? The official line is that \"Lady Liberty\" supposedly was derived from the Roman goddess Libertus, a benign symbol of freedom. But the Freemasons, whose origins trace to ancient Egypt's builders, prefer those gods.\nIsis has a darker side, which is probably the reason any association was covered up. Known as the \"divine mother\" of ancient Egypt, she was the goddess of magic. She raised her dead husband Osiris back to life, and although considered a virgin, gave birth to Horus, the Egyptian god of the sun. As such, she is often crowned with a halo of sun streaks (sound familiar?). There are any number of depictions of Isis, but the myths generally have her at the center of magic doings as the mother figure of mankind.\nThe real back-story concerning Isis can be found with the Fellowship of Isis \u2014 an organization with chapters in the UK, US, Ireland, and Germany. The Fellowship offers an instruction in the rituals and rites of Isis.\nThe rituals and related Isis wisdom are allegedly revealed only to the enlightened few, sort of like Scientology. In Isis's case, there are \"great spiritual truths\" to be revealed by those blessed with the torch of enlightenment.\nThanks to the miracle of the Internet, those truths can now be obtained by Googling \"Statue of Liberty is Isis.\" There are only slightly over 500,000 entries, so take your time.\nThe New Silver Fixers Are in Town \u2014 Can the New Gold Fixers Be Far Behind?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"To provide a forum to connect those who seek purpose in life. To use our combined power to encourage each other in our personal visions. To remind you that you are far more powerful than you can imagine.\nTo establish in the minds of our readers the reality of the mind \/ body connection and the fact that we participate in creation of the reality we experience. What we see and experience is simply the brain's interpretation of electrical signals. The only reality we cope with is the world of perceptions inside our mind. We participate in creation of the reality we experience by our intentions, beliefs and expectations, by every thought we think and every word we speak.\nTo promote the reality that we survive in consciousness after the change called death. To comprehend this, we promote the practice of daily separation of consciousness from the body via meditation.\nOnce we experience our existence beyond physical circumstances, the fear of death is removed. The less fear of death we have, the more we see the world from the higher perspective and the more we recognize how powerful we really are. This frees us to begin to live more fully, in every moment, now.\nOur lives become increasingly fulfilling as we take responsibility for how we perceive life and how we react to it.\nWe're here to find and relate to others, and everyone who reads these words or writes a word in these pages seeks to be connected with those who seek likewise. So, if you read an article that touches your heart, or an ad that intrigues you, call and get together with the person listed; or just call and introduce yourself and discuss their offering with them.\nConnect with each other, come to know each other, exchange information. We believe that as we all work together, we are making the world a better place.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"September 19, 2017 \u2013 Breathable air is necessary to sustain humans both on Earth as well as in space. Tiny airborne particles, known as aerosols, can contribute to a variety of health problems, such as asthma and respiratory tract irritation. To ensure the health of humans living on Earth as well as those traveling in spacecraft to explore the solar system, aerosol sensors are needed to monitor air quality and alert engineers when action is necessary.\nNASA is working with the Robert Wood Johnson Foundation (RWJF) to sponsor the Earth and Space Air Prize competition for a solution that could improve air quality and health in space and on Earth. This project is a technology innovation challenge to promote development of robust, durable, inexpensive, efficient, lightweight, and easy-to-use aerosol sensors for space and Earth environments. The competition asks teams or individuals to design and develop specialized sensor technology that has the potential to be useful in spaceflight as well as on Earth anywhere outdoors in a community where people may be exposed to airborne particles.\nThe Earth and Space Air Prize is a two-phased competition. Registration for Phase 1 is open until December 13, 2017, and requires submission of a sensor design by January 31, 2018. The competition will name three finalists by the end of March 2018, and each will be awarded $50,000 to build a prototype. Finalists will deliver prototypes to the Glenn Research Center in Cleveland, Ohio, by September 30, 2018 for testing and final evaluation. The competition will announce the $100,000 grand prize winner in mid-October of 2018.\nThis competition enables NASA to advance human spaceflight and RWJF to promote solutions to support communities working to improve health while both work to develop innovative technology that can improve quality of life in space and on Earth.\n\"We're grateful for this unique opportunity to collaborate with NASA to support innovation that will help to build a Culture of Health both on Earth and in space,\" said Dr. Michael Painter, senior program officer at RWJF.\nThe Earth and Space Air Prize is part of NASA Solve, a gateway for everyone to participate in NASA's mission through challenges, prize competitions, and citizen science. Leveraging its NASA Open Innovation Services vehicle, NASA contracted with The Common Pool, LLC, who is serving in the role of prize administrator for the competition. The Common Pool is a consulting firm focused exclusively on the design, development, and management of prizes, challenge competitions, and other forms of incentive-based programs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzajgsg b/data_all_eng_slimpj/shuffled/split2/finalzzzajgsg
new file mode 100644
index 0000000000000000000000000000000000000000..8283f347e6baca73f458e709b789382866953a7c
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzajgsg
@@ -0,0 +1,5 @@
+{"text":"The current trend for software as well as for digital content is to move from the desktop into the cloud, with role-based access allowing users to view and modify content. Presently, most journal and book production activities are carried out on the users' desktop. These users include: authors, peer reviewers, copy editors, proofreaders, paginators, etc. Recently there has been significant interest in collaborative online authoring. We believe that with author's content in the cloud, every other aspect of the production process can follow in the same vein, allowing operators to work from any geographical location. The advantages are dramatically faster publication times and lower prices.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When only the best will do this Valentine's Day, reserve a table for two at Tony's. Choose from three elegant searings and enjoy our multi-course menu created exclusively for this special occasion.\nSpace is limited and in high demand, so reserve your table today!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Finance your next Motorcycle, Boat, RV or ATV with LCU. Apply online, on the phone or at a branch. Motorcycle, Boat and RV financing may available at one of our preferred dealerships.\nCalculate your monthly payment by clicking here for our loan payment calculator.\nLowland Credit Union offers a variety of add-ons to protect you, your family and your assets.\nGuaranteed Asset Protection (GAP) Plus can help fill the gap between what your vehicle insurance will pay and what you owe on your loan, to cushion you against sudden out of pocket expenses if your vehicle is deemed a total loss. See a loan officer for program details and pricing.\nUnexpected events can make life complicated. Protecting your loan payments against unexpected life events and disability can help you protect more than your finances. It can help you lighten the burden for the people you care about. Insure your loan payments today so you can worry less about tomorrow. Life and Disability Protection available through CUNA Mutual Insurance Company. Talk with your loan officer today.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tamiya offer an excellent model for Ferrari lovers, that can't be missing in a collection. In my opinion, it's a very complex kit so not recommended for novices or those who want to spend some time distracted and relaxed with a simple model. Its complexity obliges you to study very carefully the instructions preparing a building plan, altering the instruction steps to get a good result.\nThis article aims at sharing with the reader the work achieved during several months to superdetail this beautiful model, providing lots of images.\nOverall, the kit is very good, but the fact that disappointed me was that doors could not be open if they are mounted following the instructions steps. So, the model should be presented with the doors open or closed.\nI proceeded to create a natural movement for the doors, nevertheless this obliged me to leave the body's front without gluing since it makes it impossible to open them.\nI think that the complex engineering of the real car, doesn't allow that certain things are easily reproduced to scale. The same happens with the interior and other areas, as I'll explain later.\nI used the Hobby Design detail set which provided great realism, especially regarding the brakes.\nIt also supplies the air intake and outlet grilles as well as other details. However, many parts of the set were rejected.\nI used the carbon fiber decals supplied by Tamiya which seemed excellent to me because they are thin and more realistic.\nI recommend using Micro Sol and Set but being very cautious because several decals are small and in general fragile. It was and arduous task and you should be very patient since there are 119 decals to apply, especially in the interior and engine bay, but it's really worth the effort.\nFor the emblems I used metal logos from Hobby Design.\nIncludes more than 197 pics, here we show only the text pages.\nIncluye m\u00e1s de 197 im\u00e1genes, aqu\u00ed solo mostramos las paginas de texto.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Description: Description: This clone was planted in the Phylloxera replant of the 1990s along with Clone 03. Cabernet franc FPS 01 was the only regis- tered selection available in California in the mid-1990s. This selection is reported to be high-yielding compared to clones developed in Europe.\nNo. The old clones do not work as well as the new ENTAV. I like 214.\nThis original UCD clone issue is aroma and taste attributes are close to terrible under most conditions. It's a clone that I have seen work in southern Monterey County. Find another clone.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzajkwt b/data_all_eng_slimpj/shuffled/split2/finalzzzajkwt
new file mode 100644
index 0000000000000000000000000000000000000000..e079fbb1875b3d31981e625e8b7262e843a5678b
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzajkwt
@@ -0,0 +1,5 @@
+{"text":"Oslo Airport Gardermoen is the main airport serving Norway's capital, Oslo and the municipals surrounding that area. It is located about 35 km to the North of the city. A surface area of 13 sq km, two runways and more than 100 international destinations make it the busiest airport in Norway. Gardermoen also serves as a hub for Norwegian Air Shuttle, Scandinavian Airlines as well as Wider\u00f8e.\nThis realistic scenery for XPlane 11 and 10 was created by Jo Erlend Sund who has already developed several Scandinavian airports for FSX and P3D. The add-on includes detailed models of the terminal, high-resolution textures and supports XPlanes \"runways follow terrain\" features, resulting in realistically embedded runways. With the help of plug-ins, custom jetway animations, seasonal textures and more are also supported.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Leading interior designer Kelly Hoppen is a style leader, opinion former and author, but first and foremost, a visionary.\nHoppen is the head designer of her interior design company, which she founded more than 20 years ago, and is still intimately involved in every project.\nFrom the beginning, Hoppen has sought to incorporate a sense of calm and balance with sheer style. Observers have called her work timeless and elegant, calm and cool, simple and luxurious, but the most often used term has been East Meets West, which perfectly sums up the Kelly Hoppen aesthetic and her emotional and spiritual approach.\nIn September 2002, Hoppen opened her flagship store in Fulham Road, Chelsea. The Kelly Hoppen Emporium, a lifestyle boutique selling her own products and and a mixture of modern and bohemian decorative objects as well as pieces sourced from all over the world. Due to demand Kelly Hoppen has now introduced her products to the Gulf and she plans to open outlets in Paris, New York and Los Angeles by 2005. Last year say the the long awaited launch of the Kelly Hoppen Design School.\nSet at the heart of Kelly's world, her design studio in West Kensington, these throrough yet intimate courses are personally run by Hoppen, offering a true insight into her world and how she works. As well as designing apartments and houses, Hoppen's international commissions have ranged from Lear Jet and British Airways First Class cabins, yachts, corporate office space, restaurants, as well as her own collection of interior accessories. These include rugs, carpets, a paint collection with Fired Earth, a collaboration with Wedgwood called 'The Art of Giving' (above right and left), window accessories, a fabric collection, bed linen and furniture.\nHer collaboration with such household names has helped to make Kelly Hoppen the international brand it is today - her own collections of paint, fabrics, rugs, carpets and ceramics have made her signature look increasingly accessible.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On tap, bottles or fills?\nHop & Barley is an independent magazine which explores the people and stories behind the contemporary brewing movement. Since launching in 2014, we've gone on to produce eight volumes, distributing thousands of magazines to beer lovers across the globe. With an emphasis on slow journalism and evergreen content, Hop & Barley deliberately distances itself from news and reviews to uncover stories of passionate people and fervent creativity.\n\u00a9 33 Acres Brewing Company Inc.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The developer has confirmed it approached a list of key business and real estate identities approximately four weeks ago with an invitation to exclusively view the Penthouse Collection prior to its public launch next week, allowing for pre-launch sales to commence over the weekend.\nSome of the defining features of the penthouses include private steam rooms and solid Nero Marquina bathtubs with 180-degree water views, internal pools that submerge into the rooms below, private art gallery spaces, solid marble slab joinery and benchtops, floating marble fireplaces, direct lift access and feature glass 'sky ceilings'.\nOne of the penthouses features its very own rooftop pool with a glass bottom that sits on top of the incredible direct access lift and lobby, providing its owners with an arrival experience that has never been offered in the world before. The rooftop terrace is over 280sqm of external space for entertaining with incredible 270-degree views of the water.\nOther features surrounding the service and amenity offerings within the building are still yet to be revealed, however the developer has promised to reveal an amenity offering that 'is truly unlike anything the world has ever seen before on a luxury residential scale'.\nThe developer has signed strict confidentiality agreements with all five buyers however GURNER\u2122 can reveal all buyers were well-known local business identities who currently reside within 1.5km of the project and Toorak, most of which whom will be moving from large existing properties and family homes.\nJust three of the penthouses now remain across the three buildings, with the balance of the project set to launch in the coming months.\n\" Our strategy for releasing the Penthouse Collection first was all about providing our buyers with the opportunity to custom design their residence from the ground up, so the remaining floorplates will be designed around these penthouse offerings,\" said GURNER\u2122 founder and director, Tim Gurner.\n\" We quietly approached a number of Melbourne identities who fit the profile of this sort of residence, and within a matter of weeks we had dozens of interested parties all vying for one of the five penthouses.\n\" I had planned to travel to the USA for business however I decided to delay my flight to the following week as we had such insatiable appetite for the penthouses, it was like nothing I had ever experienced before.\n\" Within 36 hours we had multiple parties bidding on five of the penthouses and by the time I jumped on the plane, all had been sold with a few parties missing out.\n\" There is not much I am able to reveal about the buyers other than the fact they are all well-travelled people who expect nothing but the best so we feel extremely privileged to be delivering such incredible homes for them that align with their high expectations,\" he said.\nIn what is shaping up to be Australia's most talked about address, Saint Moritz has set tongues wagging since its illustrious full-floor penthouse was purchased off market for a rumoured record-breaking approximately $30million earlier this year.\nSet over 650sqm of internal and 150sqm of external space and featuring seven bedrooms, a seven-car private garage with direct access lift, four living rooms, an in-home sauna and gym.\n\" The first penthouse sale set a new benchmark for Melbourne price-points while the additional penthouse sales have solidified the desire from Melbourne buyers for something that has never been offered before in Australia and arguably the world,\" said Gurner.\nSaint Moritz will offer approximately 120 private residences encompassing two, three four, five and six bedroom offerings, with prices ranging from $1.4million to $30million-plus.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Businesses struggle to find a path to prosperous innovation. Many innovations fail to gain traction with consumers, while a rare few stick. We witnessed this when consumers cycled through their fitness trackers: In 2015, 27% of Gen Y US online adults reported using a tracker; one year later, only 19% agreed and 10% reported ditching a wearable device altogether.\nMany innovations fail to change habits or spend. Historically, innovation drives the rise of living standards, healthy competition, and national progress. But despite what feels like recent advances, productivity in the US and other wealthy economies is growing at its slowest pace since WWII, and consumers perceive that experience quality has plateaued.\nExecutives still don't accurately identify what consumers want. Behavioral data, social sentiment, and dominant consumer demands are not clear indicators of product uptake. Consumers' lack of enthusiasm around current innovation isn't a cultural phenomenon \u2014 customers are more willing to experiment today than at any other point in the last 10 years \u2014 nor is it an economic condition: Starting a business has low barriers to entry, and VC firms are eager to fuel the next big thing.\nWith over two decades of trended, multimodal data about global consumer behaviors and attitudes at our fingertips, we've spent the last few years immersed in a search for answers: What causes consumers to be open and ready to engage with new experiences or closed and turned off from brand offerings? What does it take to deliver measurable consumer value?\nIdentity vs. isolation. Consumers moving toward identity are eager to affiliate with their tribe; those moving toward isolation want to withdraw into their own worlds.\nTrust vs. distrust. Consumers swinging toward trust are willing to believe in a brand's good intentions; those gravitating toward distrust doubt that companies will keep their promises.\nNovelty vs. comfort. Consumers approaching novelty have growing appetite for risk-taking; those nearing comfort seek reassurance and safety.\nEfficacy vs. vulnerability. Consumers feeling high levels of efficacy are empowered to get what they want; those feeling vulnerable perceive a loss of control.\nThese findings inspired us to build Forrester's Consumer Energy Index, a framework that indicates where consumers stand along the spectrum of readiness for innovation by measuring consumer movement along the four key dimensions in near real time: As consumers scale to the right on all four forces, they are ready to engage with innovation; as they scale to the left, they withdraw from new experiences.\nOur laser-sharp focus on consumer energy proves that this powerful, predictive force is constantly in flux. Electrifying brand messages such as Nike's ad campaign featuring Colin Kaepernick ignite consumer energy and spark brand loyalty. Shocking revelations like Facebook's series of mistakes neutralize consumer energy and repel consumers.\nTune in to my blog to keep a pulse on consumer energy as we publish the critical fluctuations that shape market momentum over the year. In the meantime, join my upcoming webinar or connect via inquiry to learn the full story behind the Consumer Energy Index. Forrester clients can access the detailed report here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzakhec b/data_all_eng_slimpj/shuffled/split2/finalzzzakhec
new file mode 100644
index 0000000000000000000000000000000000000000..b7471793c82096b1cdb6422e4d159fd9bcebdd88
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzakhec
@@ -0,0 +1,5 @@
+{"text":"The World's largest gravesite collection. Contribute, create and discover gravesites from all over the world. Cemeteries in Montreal, Quebec, a Find A Grave.... Local Montreal florists can help you find the funeral home handling your loved one's service. Tip - Sending flowers to a funeral service is a good way to show your respect and very comforting to the family.\n14\/01\/2019 \u00b7 Find out why Close. Rajapara choir Sardis Mondolini bipek. Tengsu Sangma. Loading... Unsubscribe from Tengsu Sangma? Cancel Unsubscribe. \u2026... If we run away, we become illegal and that makes it hard to find another job. The police could catch us at any time and send us back home. We can't get a resident permit if we leave.\" The police could catch us at any time and send us back home.\nPinterest is . using cookies. to help give you the best experience we can. how to get rid of trypophobia Canadian. Bistro-Bar Le Cordial. Chef Robert Gendreau takes you on a culinary experience at our Montreal, Quebec restaurant. He has a passion for using local products in his dishes for breakfast, lunch, dinner and room service.\nLocal Montreal florists can help you find the funeral home handling your loved one's service. Tip - Sending flowers to a funeral service is a good way to show your respect and very comforting to the family. how to find killer crocs lair in arkham asylum Find your state's shopping portal at healthcare.gov. Since a majority of states opted not to create their own exchange, 35 will be at least partially run by the federal government, but that won't make much difference to shoppers.\nCity name generator . This city name generator will give you 10 random city names. The names are generated almost completely randomly, with the exception of predefined endings, so some names might not be what you'd expect for a city name.\nInspired by the delicate architecture of hyaline articular cartilage, we report on a biomimetic polymer hydrogel that incorporates strong intermolecular hydrogen bonding between urethane\u2013urethane linkages as well as urethane\u2013ester linkages.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"ANN ARBOR, MI \/ ACCESSWIRE \/ May 31, 2017 \/ ENDRA Life Sciences Inc. (\"ENDRA\") (NASDAQ: NDRAU), a developer of photo- and thermo-acoustic medical technologies, has been invited to participate at two upcoming conferences in June 2017.\nThe 7th Annual LD Micro Invitational is being held on June 6-7, 2017 at the Luxe Sunset Boulevard Hotel in Los Angeles. ENDRA CEO, Francois Michelon, and CFO, David Wells, are scheduled to present on Wednesday, June 7 at 10:30 a.m. Pacific time with one-on-one meetings held throughout the conference. For additional information or to schedule a one-on-one meeting with ENDRA management, please log-in via the link provided in your invitation. You may also email your request to NDRA@mzgroup.us or call Chris Tyson at (949) 491-8235.\nThe 2017 Marcum MicroCap Conference is being held on June 15-16, 2017 at the Grand Hyatt Hotel in New York City. ENDRA CEO, Francois Michelon, and CFO, David Wells, are scheduled to present on Friday, June 16 at 10:30 a.m. Eastern time with one-on-one meetings held throughout the conference. For additional information or to schedule a one-on-one meeting with ENDRA management, please log-in via the link provided in your invitation. You may also email your request to NDRA@mzgroup.us or call Chris Tyson at (949) 491-8235.\nLD Micro was founded in 2006 with the sole purpose of being an independent resource in the microcap space. What started out as a newsletter highlighting unique companies has transformed into an event platform hosting several influential conferences annually (Invitational, Summit, and Main Event). In 2015, LD Micro launched ldmicro.com as a portal to provide exclusive intraday information on the entire sector, including the first pure microcap index (LDMi), which covers stocks in North American with market capitalizations between $50M-$300M. It is a non-registered investment advisor. For more details, please click here.\nThe Marcum MicroCap Conference is an annual highlight on the investment community calendar. Since its launch in 2012, this event has become a nationally recognized forum for publicly traded companies with less than $500 million in market capitalization to network with fund managers and high net worth investors who focus on small cap equities. More than 2,000 participants from all segments of the microcap market participate each year, including senior management, finance and legal executives, venture and lower middle-market private equity investors, institutional investors, directors, investment bankers, buy- and sell-side analysts, and service providers to the microcap marketplace. The agenda will feature presentations by CEOs and CFOs in seven principal industry sectors, expert panels moderated by industry leaders, and the opportunity for investors to meet with management of presenting companies on a one-on-one basis. For more information, please click here.\nENDRA Life Sciences Inc. (\"ENDRA\") (NASDAQ: NDRAU) is a developer of photo- and thermo-acoustic medical technologies bringing new capabilities to existing ultrasound systems. ENDRA's photo-acoustic solutions currently help global medical researchers screen and modify disease models with high image quality and volume scanning speed. ENDRA has a global installed base of leading institutions using the Nexus-128 system, the only commercially available 3D imaging system for imaging anatomy, physiology and labeled molecular targets. ENDRA's thermo-acoustic technology is being developed to enable clinicians to use their existing ultrasound equipment to visualize tissue function, composition, and monitor a variety of therapeutic interventions - at the point of care. This directly aligns with current healthcare trends to improve patient access, clinical effectiveness, safety and cost. For more information, please visit www.endrainc.com. Social media coverage on ENDRA's May 9, 2017 IPO can be found here on Twitter, here on Facebook and here on LinkedIn.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Steve Scalici is a Certified Financial PlannerTM and Vice President of Treasure Coast Financial. He is co-host of a daily radio show called \"God's Money\" which can be heard at www.oneplace.com. You can contact Steve at [email protected] or via telephone at 1-800-728-6342. His website is www.tcfin.com.\nCBNMoney.com \u2013 Most of us have heard that debt is bad. Burkett and others are always counseling us to get out of debt. Most of us have a basic understanding why: that interest eats up our income, but most of us don't realize just how enormously interest robs us of financial freedom in our later years.\nAll of us are going to work basically the rest of our lives. I want to show you one simple way to have something to show for it, and, if you are currently making car payments, it won't cost you one cent.\nThe basic idea is to stop making car payments to a bank downtown, and start to make them to a Bank You Own.\nYou can still buy a brand new car every 5 years.\nOf the residual income of the $400 payment you now make to the Bank You Own, you will accumulate Tens of thousands of dollars by the time you retire.\nBoth figure, when they stop to think about it, that they will probably get a new car every five years and make payments until they retire. It's just a normal part of their family budget.\n(1) Joe goes out and gets his Accord and finances it at the local National Bank @ 8% for 5 years with a $400\/mo payment.\n(2) Fred decides to put the same $400\/mo in a mutual fund savings account we'll call the Bank of Fred until he saves up enough to pay cash. While it took 5 years for Joe to pay off $20,000 @ 8%, it takes only 3.5 years to save it. Fred earns 8% in his Bank Fund.\n(3) Fred pays cash for his new Accord and faithfully continues to make his $400\/mo payments to his Bank fund.\n(5) Fred's Honda is now 5 years old (again assume zero value) and he needs a new one. But since he has been faithfully making his payments he now has $29,797 in the Bank of Fred. He pays cash (-$20,000) and now has a new car and $9,700 in residual income left.\n(7) Fred's account is now up to $44,393. He again pays cash (-$20,000) and has a new car and $24,393 residual income!\nThis is the kind of money you are sacrificing if you finance vehicles.\nBoth men paid $400\/mo to drive a new vehicle every five years. Yet Fred was able to build over $20,000 just by banking at home.\nWhat's the catch? Fred was willing to wait on the loan. Sacrifice 3.5 years driving an older paid off vehicle.\nLet's play out our example a little further. Suppose Fred and Joe were 35 years old when they started this. What happens at retirement?\n(8) Joe has made nothing on his $400\/mo investment; the National Bank made it all. Now that he is retiring, he cannot get a new car because he can't afford the payment he is used to.\n(9) Fred goes and buys himself a brand new car. He now has left over $198,715 in residual income off his regular car payment by Banking at Home. You better believe the banks don't teach you this!!\nYou can start this kind of money saving bank now. If you do, below is the savings you can expect at your current age.\nDo not let the world keep you in the dark and keep you from reaching financial independence.\n1. Be sure you understand the principles of what makes this work. Do the math.\n2. If you're in a lease or financed car; get out. Purchase a used, low maintenance vehicle for cash (full size America truck, or anything Honda or Toyota, except ATV's. For this to work, you cannot get behind due to maintenance) Spend $3000 - $6000, or more if you need to or can afford it.\n3. Get help choosing a good mutual fund that trades in high quality stocks. Your goal is 8%. Don't use savings accounts, CD's, annuities or insurance for this program.\n4. Make your car payments by auto-withdrawal from your paycheck to your mutual fund. BE DISCIPLINED. Withdraw money for nothing except buying the five-year car for cash.\n5. This should be viewed as an addition to a main line retirement plan.\n7. If you want a more expensive car, you must up the monthly payment accordingly, just as if you financed it at the National Bank.\nHonestly, the best way to get started on this, even if you don't plan to start until your current car is paid off, is to make that payment by auto withdrawal. I recommend auto withdrawal out of your regular account to the debtor. When the debtor is paid off continue to auto withdraw the car payment into a mutual fund. It's amazing how easy it is to forgo the whole idea if you wait to set up the logistics until \"later\", so be diligent!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"modern setup of development, e.g. Spotify squads? Once you come onboard you may have to design for 10-100x userbase. The architecture can be a generic view.\nI researched and compiled my personal thoughts into a document. Today, I decided to publish them as a blog post.\nDisclaimer: Detailed explanations of Clean Architecture or MVP are not provided in this post as it is only highlighting the proposed solution and outlining my reasoning behind it.\nOn an unrelated note, 2017 was the year when I got interested in Clean Architecture and got some hands-on experience with applying Model-View-Presenter (MVP) architectural pattern in Android app at Freeletics. In general, Software Architecture has always been my area of interest and unfortunately I never had the chance to implement it in any of my past Android projects. Android Developers understand the struggle of building robust, production-quality, maintainable apps without any reliable architecture.\nInterestingly enough, 2017 will always be remembered as the year in which Google officially recommended an app architecture for Android apps for the first time using Architecture Components. It is basically a collection of libraries to provide simple, flexible and practical approach that frees developers from some common problems so they can focus on writing modular apps with less boilerplate code and building great experiences.\nWhat should we expect from a modern app architecture?\nIn my opinion, it needs to be robust, stable, easily extendable with minimum effort, adaptable to change as per agile requirements, fast to incorporate new platform features, supports testability and at the same time understandable for different levels of developers to ensure maintainability.\nI believe Clean Architecture is the answer.\nMVP is much suitable for the UI\/presentation layer.\nViews are dumb and implement Passive View pattern. It is a set of interfaces that could be implemented by any Android view, such as Activities, Fragments, Adapters or Custom Views.\nPresenter serve as a middleman between views (abstractions over Android specific components) and the business logic (interactors\/Use Cases). They handle user interactions, invoke appropriate business logic and send the data to the UI for rendering.\nPresenter does not depend on Android classes hence improves testability.\nA simple example of Use Case would be \"Transfer money from one account to another\". Each Use Case is a reusable component that executes a specific business logic. It fetches the data from a repository, executes the business logic and returns the result to the presenter.\nRepository Pattern is responsible to create an abstraction of the data sources from which the Use Cases get the data to act upon. All under-the-hood persistence should be here: DAOs, ORM stuff, Retrofit (or any other Network related stuff) services, JSON parsing etc.\nApp should work seamlessly even on a spotty network. The ability to cache and reuse previously fetched resources is a critical aspect of optimizing for performance.\nBusiness logic shouldn't know where the data comes from. Act locally, but sync globally.\nDevice contains the implementations of the gritty Android stuff such as sensors, alarms, notifications, players, all kinds of Managers and so on.\nIt's all neatly separated by module level, package level and class level. So Single Responsibility Principle and Separation of Concern should be satisfied.\nBusiness logic doesn't touch the Android directly anymore and this should result in a decoupled codebase.\nBy injecting mock implementations of respective dependencies to separated classes can make testing much easier.\nNo compulsion to use Presenters to handle presentation logic, we could say that Clean architecture is \"frontend\" agnostic \u2013 meaning we can use MVP, MVVM or anything else on top of it.\nThe Presenter depends on the View interface and not directly on the Activity: in this way, we succeed in decoupling the presenter from the view implementation respecting the D of the SOLID principles. We can replace the concrete view without changing a line of code of the presenter. Furthermore, we can easily unit-test presenter by creating a mock view.\nEvery layer to have its own model that it works with, so the concrete details, such as views, do not depend on the specific details of the lower layer implementations.\nInstead of using callbacks to communicate between layers we can use the power of RxJava to provide the data upstream and let it handle thread scheduling for us. Each of the inner layers can transform the data in a way the outer layer can understand it.\nAllows changing the implementation of any outer layer without having to make a lot of changes in our codebase and fulfills the Extendability requirement.\nFollowing this approach is difficult and time consuming but consider it like doing a favor to future developers that are going to work and maintain the code.\nIt is worth mentioning, that this supports using a dependency injection framework like Dagger.\nThe problem statement shared above also mentioned Spotify Squads which is a very interesting concept that came out of Spotify Engineering Culture. Product squads seem to be a great way to build strong, knowledgeable development teams. In layman's terms, Squad is a small cross-functional self-organized scrum team. They have end-to-end responsibilities and work together towards their long-term mission. On Squads the key drive is autonomy.\nBenji Weber argues that there's no need of a person whose role is the Architect in a team. It is a responsibility of the team. In an ideal team, each member would be thinking about overlapping and complementary issues.\nThe software architecture shouldn't be a hindrance to take a feature from all the way to shippable for a team of 5 to 9 members.\nClean Architecture provides a separation of concerns that gives development teams the cadence and agility they need to manage the complexity and rapid change in their environment. By design it creates a cut between your components, and at the same time establishes the contract for future communication. This allows android devs to work in harmony with members of other tribes.\nThe practise of defining the MVP contract before starting working on a feature indirectly helps android devs in the same squad to efficiently define areas of responsibilities among themselves.\nWhat is all this Clean Architecture jibber-jabber about?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sharir Rachana Laboratory \/ Dissection Hall.\nThough Ayurveda and other systems have their own diagnostic tools, modern medical technology is used to improve the ancient principles. In this clinical laboratories, many useful investigations are carried out. This helps in validating the diagnosis made on the basis of Ayurvedic principles and also helps in monitoring the progress in treatment. Our staff and technicians manning these laboratories helps the clinical students to obtain practical approach to diagnosis.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzakokh b/data_all_eng_slimpj/shuffled/split2/finalzzzakokh
new file mode 100644
index 0000000000000000000000000000000000000000..585cccbe6ccf2461f4e2a9c76cb6999787e5bb68
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzakokh
@@ -0,0 +1,5 @@
+{"text":"U.S. Rig Count is down 1 rig from last week to 568, with oil rigs up 2 to 452, gas rigs down 2 to 115, and miscellaneous rigs down 1 to 1.\nU.S. Rig Count is down 199 rigs from last year's count of 767, with oil rigs down 122, gas rigs down 78, and miscellaneous rigs up 1.\nThe U.S. Offshore Rig Count is unchanged from last week at 21 and down 12 rigs year over year.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You are beautiful , the way you carry your confidence without a care in the world makes you unique. The way you stand out to be different makes you brave. The way you embrace others with love makes you special. Everyone is beautiful and unique in there own special way. Its refreshing to hear positivity once in a while especially with everything going on in the world. I wish I had the knowledge and confidence I have now as a child , because it would have save me from the feelings of I am not good enough. You have the power to create your destiny . You have the power to achieve the goals you want to achieve. Don't let others discourage you from reaching for the stars even if it is impossible. Its so easy to get distracted from reaching your goals by simply hiccups that can be avoided with a positive approach. I'm not going to lie there have been plenty of times where I just wanted to give up or felt that there was no possible way that I will be able to do what I love. But I keep reminding myself that one day I am going to look back and see what I have accomplished and is accomplishing. Just remember there is always someone rooting for your success and remember you are your biggest fan. Be Your Own Kind Of Beautiful.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Jean Darnell: Awaken Librarian!: Good Reasons to Read Quotes!\nGood Reasons to Read Quotes!\n\"A good children's book is a young person's earliest exposure to art and design, a conduit for parental bonding, a means to teach individual and social lessons...\"\nNot only does this ring true, but think about how early you need to expose children to books. Is 3 months too young? No. Is before birth too soon?? Never. When you build those bonds before entry into life, I believe the baby will recall the tone in your voice. I believe the 18 month old will mimic the manner in which you held his favorite book. Your 4 year old little boy will ask a thousand times over for his favorite \"night-night\" story.\nThe following response came from \"The Goddess of YA Literature\" Blog.\n\"So to the list of reasons why children's books matter, add the way that they reflect the times they were created in. 'They are the message-in-a-bottle that each generation tosses out to the next generation,\" Marcus says, \"the record of one generation's hopes and dreams for the next.'\" This is Leonard Marcus, one of the finest scholars in the field and the person behind this exhibit as well. Think, now, about how we might use picture books with older readers to talk about the milieu of the books from various decades (the Newbery originated in the 20s and the Caldecott in the 30s). Think about comparing and contrasting books from different decades, say DRUMMER HOFF to PATROL or SMOKY NIGHT to THE MAN WHO WALKED BETWEEN THE TOWERS.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Peach trees are exceptional fruit trees that call for some care to produce a nice peach harvest.\nPlanting, pruning and caring for it is important to avoid diseases.\nOur recommendation is to plant your peach tree in a sunlit and wind-sheltered spot so that dominant winds don't sweep through.\nOnce the spot is chosen, plant your peach tree in fall or in spring.\nPrepare a blend of soil mix and garden soil, which will make the soil lighter and add nutrients that the tree needs to grow well.\nIf your soil is clay and loamy, add about \u2153 sand to your blend of earth and soil mix.\nSpread mulch to protect it from frost spells in winter, and it also adds organic matter and avoids weed growth.\nPeach trees tend to not have apical dominance, which means that after pruning, they will sprout new shoots from the base rather than from the top.\nEvery year, it is important to prune your tree at the end of winter just above a well-formed wood bud.\nCheck that the pruning is well balanced and that there is dominant central stem, but rather a number of evenly-sized branches.\nIt is important to perform a fruit-inducing pruning to trigger appearance of many beautiful peaches.\nA peach tree is very vulnerable to peach leaf curl, and, clearly, proper pruning will give your peach tree vigor and a make it more resilient.\nStart treating your peach tree at the end of winter or at the very beginning of spring, spraying Bordeaux mixture.\nAfter that, spray every two weeks to keep peach leaf curl and other fungus from appearing.\nAphids \u2013 techniques and organic treatments to avoid them.\nFrom the many peach tree varieties, we've selected the following interesting cultivars. Although taste is always subjective, there are still among the most delicious varieties to grow in our temperate climates.\n'Springtime' \u2013 harvest end of June, white, fruits require thinning, prune short.\n'Royal Gold' \u2013 harvest from June 30th to July 15th, yellow, fruits require thinning.\n'Robin' \u2013 harvest from July 1st to July 15th, white and soft flesh.\n'Redhaven' \u2013 July 15th to 20th, yellow, light productivity.\n'Anita' \u2013 July 10th to 20th, white flesh, vigorous, large fruits.\n'Dixired' \u2013 July 20th to August 5th, yellow flesh.\n'Charles Roux' \u2013 July 15th to August 15th, white, vigorous, very productive.\n'Redwing' \u2013 August 1st to 15th, white, tasty, vigorous.\n'Orchard Queen' \u2013 August 25th to September 10th, white, for colder areas.\n'Michelini' \u2013 August 25th to September 10th, white, late blooming, prune long.\n'Sanguine' \u2013 September 1st to 15th, regular and late production.\n'Fantasia' \u2013 harvest end of August, a tasty nectarine.\n'Flavor Gold' \u2013 harvest mid-July, prime quality fruits.\n'Fuzadole' \u2013 harvest August 15th, white nectarine, grows anywhere.\n'Large Violet' \u2013 end of August, vigorous and its fertile stone can be sown.\n'Independance' \u2013 harvest around August 10th, large yellow fruit.\n'Morton' \u2013 harvest end of July, beginning of August, fertile stone, prune short.\n'Nectared 4' \u2013 harvest August 1st to 15th, early, juicy, sweet.\n'Nectared 6' \u2013 harvest end of August, one of the best nectarines.\n'Nectarose' \u2013 harvest around August 20th, very sweet, resists European brown rot.\n'Olympia' \u2013 harvest August 15th, nectarine with white flesh, very fragrant.\n'Silver Lode' \u2013 harvest August 10th, white and very sweet nectarine.\nWho has never dreamed of standing up after a nice family feast to go fetch a few peaches from the tree in the garden?\nThis dream is within reach, if you simply care for your tree and considered location, pruning and fertilizing.\nYou can also treat your peach tree before the first leaves appear, with organic acaricide (mite killer) or a spray containing Bordeaux mixture.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"KICK OFF THE SUMMER & SIGN UP TODAY!\nThe Nottawaseppi Huron Band of the Potawatomi (NHBP) is a federally recognized Tribal government with over 1,400 enrolled Tribal Members. The NHBP main offices are located at the Pine Creek Indian Reservation near Athens, Michigan, with additional offices in Grand Rapids, Michigan.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzalwpg b/data_all_eng_slimpj/shuffled/split2/finalzzzalwpg
new file mode 100644
index 0000000000000000000000000000000000000000..946d8190b40ba4ddead4bc2bd6677fb063eafb52
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzalwpg
@@ -0,0 +1,5 @@
+{"text":"Today we're sharing our favourite things to do with kids in Santa Barbara. We hit the town on our California road trip \u2013 check out this post for our recommended itinerary for a California road trip with kids.\nTwo weeks into our California road trip, we hit Santa Barbara, and a definite change of pace.\nSanta Barbara lies a couple of hours North of LA, and is a lot quirkier and more laid back. The city was the birthplace of the Spanish architecture revival in this part of America, and it shows. There are pretty tiled walls, lots of greenery, with Spanish inspired stores and courtyards.\nOur base was Carpinteria, a cute seaside town just outside Santa Monica. Our home for the week was an ancient RV we booked at the last-minute on Air BnB.\nI'm a big believer that a road trip should give you the opportunity to stay in different sorts of places. On our last trip to California, Flea adored the wooden cabin we booked in Big Sur. This was almost as much fun. Our home for the week was this vintage RV, parked in the driveway of a local chef's house. It was small, and basic, but definitely had everything we needed.\nFlea thought the sleeping platform was adorable, and I was grateful it had a memory foam mattress. At $60 a night, who's complaining?\nOur RV was just a 5-minute walk from the beach at Santa Claus. For my money, this is one of my favourite beaches in Santa Barbara. It's much quieter than the main city beaches, with huge expanses of white sand and waves that are perfect for surfing and body boarding.\nSanta Claus beach is lined with a small row of rental stores. You can get everything from paddle boards to take-out pizza, and there are restrooms. There's free parking all along Santa Claus Lane, and you'll see several points where you can walk across the train tracks and climb down to the beach. This isn't as dodgy as it sounds, promise.\nFor lunch, you must visit the Padaro Beach Grill at Santa Claus beach. This lush cafe makes amazing fish tacos, and you can eat outside on the lawned gardens, that overlook the beach. There are trees for shade, a giant sand pit for kids to play in, and it's the perfect place to cool off on a hot day. As you leave, we recommend the milkshakes here, which are spectacular. The pizza place along the street is also worth a visit for dinner.\nIn Santa Barbara itself, there are countless places to explore. There's a clock tower in the middle of downtown that you can climb for awesome views over the city. Wander down State Street and check out the many courtyards, lined with surf and skate shops, and restaurants.\nOn a hot evening, we recommend checking out the Arlington Theater on State St, in the middle of town. Rather than your standard multiplex, this 1930s cinema is built to give the impression you're walking into an art deco Spanish theatre, complete with a covered courtyard and fountain, and an interior that's decorated with arches and painted beams.\nOne of the best things to do in Santa Barbara with kids is a visit to Stearn's Wharf. If you're feeling brave, you can park directly on the wharf pointing towards the sea (I've never checked the parking brake so many times before getting out of the car). There are some shops and restaurants, and towards the shore you'll find rentals for surf boards, jet skis, kayaks and so on.\nThe best time to kayak here is certainly sunset, and although you don't get the same sort of sunset you'll see in Santa Monica, it's really pretty and it's nice to get out on the water and see the wildlife \u2013 much of the water here is protected, and you can spot sealions, otters and plenty of wild birds. We booked a sunset kayak tour through Viator for around $40 each.\nWe also took the opportunity to get a little more lively with a jet ski rental from Sea Landing. For $150 you can rent a jet ski and lifejackets for an hour. It's a great activity with older kids, so long as they sit on the front of the jet ski, they're allowed to ride, and take a turn steering.\nI'm not going to lie. It can get a little bit scary when you get really fast, but Flea absolutely loved it \u2013 even when we had to call the hire desk to get them to rescue us when we snarled up our engine with seaweed.\nThere are certainly plenty of architectural and grown-up reasons to visit Santa Barbara, but there are also loads of fantastic things to do in Santa Barbara with kids. Hope you enjoyed our recommendations!\nI have dreams of taking a roadtrip across America one day. I have made notes from all your posts, so thanks.\nHey Buddy!, I found this information for you: \"Summer Road Trip Part 3: Santa Barbara\". Here is the website link: https:\/\/www.whosthemummy.co.uk\/things-to-do-in-santa-barbara-with-kids\/. Thank you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Boasting fresh fruits and basil, this Sweet Apple Basil Slaw reinvents the classic side dish to a lighter, healthier version.\nMany slaw recipes use mayonnaise dressing. While there's a plethora of Paleo mayo options available, we changed things up with a vinegar-based dressing for this sweet slaw.\nThis recipe is a favorite party dish of mine. It's a breeze to make (no cooking required!) and perfect for taking to an outdoor barbecue. To give it an extra touch of sweetness, I added apples. While apples aren't a usual slaw ingredient, their texture really helps to lighten the dish when they're spiralized.\nThe dressing is made from apple cider vinegar\u2014with a touch of honey. To cut through the sweetness and brighten up the recipe, fresh basil and crunchy raw sunflower seeds are sprinkled on top. For a more intense flavor, feel free to swap the basil for mint.\nIf you plan on making this slaw for a big group, feel free to prepare it ahead of time. Just leave out the apples until it's ready to serve, so they won't turn brown. Many slaws are better with a little bit of age. So ideally, add the dressing to your cabbage and carrots about 4-6 hours before you serve it, so they'll soak in the flavors. For the best results, add the apple, basil, and sunflowers seeds just before serving!\nBoasting fresh fruits and basil, this Sweet Apple Basil Slaw has a lighter take on a classic that's sure to please.\nCombine the ingredients for the dressing in a small bowl, and whisk them together.\nCombine all the ingredients in a large bowl, drizzle with the sweet dressing, and toss to combine.\nPro tip: If you're preparing the slaw ahead of time, add the apples, basil, and sunflower seeds just before serving.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What health and fitness changes in 2016 did you make that you want to continue in 2017? New Year's resolutions may only last until the end of January. But you can set goals and do things to make those changes part of your daily lifestyle.\nBack in July I started a new journey with Shakeology because of its nutritious value. By September, I noticed that I wasn't having sugar cravings like before. Now that had my attention!\nBefore the end of September, I started working out with the 21-Day Fix DVD's, and the weight began to come off. I had been stuck in a rut for way too long. Couldn't lose weight no matter what.\nHealth and fitness has to become a way of life for it to seriously make a difference. When I was in my twenties and thirties, I could lose weight much easier than after I reached my forties or fifties.\nSoon I'll find out from my holiday group the success from each one. Even if it's no pounds gained, that is a great success!\nThe accountability of staying dedicated to a healthy eating plan and workout plan is doable in a group of people who are on the plan together.\nPlenty of vegetables, protein, fruit and all the other needed nutrients all count for a healthy eating plan. Christmas brunch was a new experience for me because of my holiday challenge. I decided on crustless Quiche muffins. Would my toddler granddaughters like them? My daughter and son-in-law? I'd find out soon.\nCarbs do get in on the healthy food plan, so I could not leave them out. Waffles were included in the Christmas brunch \u2013 definitely a hit with the toddlers! The first bite of the Quche muffins the toddlers ate? Not a hit. But the youngest, a year old, decided she liked it after all. And their mom and dad liked them! I was delighted because the muffins had veggies and protein \u2013 healthy for all of us.\nChop vegetables and herbs fine.\nBlend in cream and milk.\nAdd vegetables, herbs, and half of the cheese.\nPour into nonstick muffin tin.\nSprinkle cheese on top of Quiche Muffins.\nBake 20-25 minutes at 375.\nMy pleasure! I love to bake quiche recipes. They're fun to make \u2013 and delicious!!\nThese are a great idea. Do you think they would freeze well, so we could have a grab and go breakfast?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Jim Donnelly has been a fixture in the Florida LMSC for over thirty years. As one of the earliest members of the Suncoast Masters in Sarasota, he has played a major part in the team's development over time. He has been an active board member for over thirty years, donating mainly his creativity to not only SMS, but the entire LMSC.\nJim was the editor of the LMSC newsletter for five years, from November 1994 through August 1999. His fun and innovative newsletter attracted the attention of the USMS national office which awarded Jim (and our LMSC) the \"Best Newsletter\" in the country in 1998.\nHis creative juices have produced numerous T-shirts for many occasions including the Maverick's 2002 and 2007 USMS National Hour Swim. He also designed our new Sarasota Masters Swim Team logo.\nLocally, Jim donates his time to the special Olympics, coaching swimming twice a week for those eager athletes. He has helped Deb Walker with free clinics for Sarasota Masters swimmers over the years and, as a certified starter with United States swimming, Jim has officiated at many youth and Masters meets.\nJim's pleasant attitude has been an on-deck \"constant\" over the years, and his contributions have been creative, productive, and rewarding for our Florida LMSC!\nJim Donnelly received the 2006 Frank Tillotson Award at the Annual Florida LMSC Social and Awards Banquet in 2007. He was also a recipient of the 2007 USMS Dorothy Donnelly Service Award for his service to USMS, the Florida LMSC, US and Masters swimming.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How did you decide to apply to Oxford?\nOxford seemed to be an excellent choice from many perspectives - it ran a great engineering department, seemed to have a very enjoyable atmosphere and was accessible financially.\nHow did you find the application process?\nThe interviews were challenging, but fun. Definitely prepare for them! Know your proofs and the standard formulae. It's not imperative that you know those to succeed, but it definitely helps. Studying in the UK has been amazing, eye opening and challenging in so many different ways, including in non-academic ways.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzamuwd b/data_all_eng_slimpj/shuffled/split2/finalzzzamuwd
new file mode 100644
index 0000000000000000000000000000000000000000..9ab82fbf5d02eafbb1af9c578cbb6b06cfaaad0d
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzamuwd
@@ -0,0 +1,5 @@
+{"text":"Can you tell me the name of a rose planted at Colorado State U?\nColorado roses horticulture submitted over 2 years ago.\nMultnomah County Oregon roses horticulture submitted over 2 years ago.\nHennepin County Minnesota roses black spot submitted over 2 years ago.\nCook County Illinois roses knockout roses horticulture submitted over 2 years ago.\nHarford County Maryland shrubs japanese hollies roses submitted over 2 years ago.\nRose Rosette disease in Beaverton Oregon?\nWashington County Oregon disease issues horticulture roses submitted over 2 years ago.\nMuriate of Potash, good or bad ?\nWashington County Minnesota horticulture roses submitted over 2 years ago.\nScott County Minnesota roses pruning shrubs rosa rugosa submitted over 2 years ago.\nMaricopa County Arizona roses horticulture submitted over 2 years ago.\nAnoka County Minnesota roses horticulture pruning submitted over 2 years ago.\nMinnesota horticulture roses submitted over 2 years ago.\nWhen to cut rose bush, peony, hydrangeas?\nCook County Illinois roses flowers: annuals and herbaceous perennials submitted over 2 years ago.\nRock County Minnesota roses horticulture pruning submitted over 2 years ago.\nMadison County New York horticulture roses submitted over 2 years ago.\nLincoln County Oregon pruning roses submitted over 2 years ago.\nHennepin County Minnesota roses horticulture propagating cuttings submitted over 2 years ago.\nYamhill County Oregon horticulture roses submitted over 2 years ago.\nBenton County Oregon roses pruning submitted over 2 years ago.\nMontgomery County Maryland roses rose pruning rose care shrub submitted over 2 years ago.\nLawrence County Missouri roses flowers: annuals and herbaceous perennials horticulture submitted over 2 years ago.\nAllegany County New York plant identification roses horticulture submitted over 2 years ago.\nOntonagon County Michigan flowers: annuals and herbaceous perennials roses submitted over 2 years ago.\nBenton County Oregon horticulture roses submitted over 2 years ago.\nKent County Delaware plant disease roses knockout roses rose submitted over 2 years ago.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Has your dog unlearned everything that you taught it? Just when you thought that it was well trained, it has turned into the naughtiest of them all? And by chance... Is your dog nine months old? Don't panic. Your dog is just a teenager.\nSome dogs are also rebels when they are teenagers. While they are puppies, they usually depend on people and on other dogs, and they have a strong instinct to let themselves be guided by others. Once they reach the age of six months, they begin to think for themselves and to test their independence. It is at this time when we must be consistent with the set rules and with everything that we have taught them until then.\nIt is a critical period for your relation but also for its education. What your dog learns during this period will stay in its memory for the rest of its life. Thus, it is of great relevance that during this period you give it an order when you really want it to obey. You must carry out short training sessions each day to improve its obedience and include therein orders such as \"sit\", \"lie down\" and \"still\" so that it turns into a routine for the puppy.\nFinally, and as a summary, in order for your dog and you to overcome adolescence, you must remember these three words: be consistent, persistent and patient, and you will overcome the most rebellious adolescence.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I have been licensed by the State of California since 1986 and have been in private practice for twenty years. My license in clinical social work qualifies me to provide counseling and psychotherapy to children, adults and families. I earned a Masters Degree in Clinical Social Work from the University of Washington in 1983 after receiving a Bachelor of Arts degree from the University of California at Berkeley.\nFor seven years, I served as Program Director at CARE Childrens Counseling Center in Santa Rosa, providing psychotherapy, consultation and oversight of both the Crisis Project and the Child Abuse Treatment Project funded by the Governors Office of Criminal Justice Programs Division. I have also worked at two other community-based social service agencies and was one of the founders of a child sexual abuse treatment program in Oakland, California. I have worked with veterans and their families at the American Lake Veterans Administration Medical Center in the rehabilitation medicine unit, acute inpatient psychiatric unit, the geriatric day center program, and the domestic violence program. I completed a clinical internship at a domestic violence shelter in Seattle.\nI was trained in Eye Movement Desensitization and Reprocessing (EMDR), Levels I and II in 1999. EMDR is an 8-Phase protocol designed to resolve trauma memories. Although it was designed for resolving single-incident, adult-onset traumas, protocols have been adapted for children. In 2004, I was trained in The Developmental Needs Meeting Strategy (DNMS), and have facilitated at DNMS trainings of therapists around the country. I have been a member of the National Association of Social Workers since 1983 and RECAMFT since 2011.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to our new website! We'll be upgrading and improving the content of the site over the next few weeks. Look for updates here on our blog with articles and links to some great information for you and your pet. If you have any questions or would like to request something to be posted here, email us through the link to the right of this box.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Throughout the expansion of individuals across the earth the actions have generally unfold east and west out of Africa, and most have stayed in the extensive central middle climate zones between each poles where the science shows that healthy existence are simpler when you can develop crops all year. Our people are pushed by curiosity to broaden elementary information and to look beyond the borders of their very own self-discipline; their goal is to benefit science, and to make a contribution to addressing the most important societal challenges of the future. Based in 2003, Science Information for College students is an award-profitable online publication devoted to providing age-acceptable, topical science news to learners, parents and educators.\nThe National Academies Gulf Research Program enhances projects around offshore energy system safety, human well being, and environmental resources. One of many positive fire ways to tell if in case you have chosen a good venture is the results of your experiment makes you consider other questions you may want answered. Finally, another strategy typically cited in debates of scientific skepticism against controversial actions like \" creation science \" is methodological naturalism Its essential point is that a difference between pure and supernatural explanations should be made and that science needs to be restricted methodologically to pure explanations.\nNobody now fears black cats, damaged mirrors and the number 13. Because science has proved that these sorts of fears are un-scientific and illogical. Science coverage additionally refers back to the act of applying scientific information and consensus to the development of public policies. Current nonfiction articles deliver science concepts to life and have interaction college students in science and engineering practices.\nDisaster Recovery 2018: This program will fund small grants to assist repair, exchange, or get better tools, knowledge, and different scientific research supplies broken or misplaced resulting from hurricanes Harvey and Irma. I j That is done partly by way of statement of natural phenomena, but in addition by means of experimentation that tries to simulate natural events beneath controlled circumstances as appropriate to the self-discipline (in the observational sciences, resembling astronomy or geology, a predicted observation would possibly take the place of a managed experiment).\nNext we tried different distances, comparing the time for the entire size with that for the half, or with that for two-thirds, or three-fourths, or certainly for any fraction; in such experiments, repeated many, many, occasions.\" Galileo solved the problem of time measurement by weighing a jet of water collected in the course of the descent of the bronze ball, as said in his Two New Sciences.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaneou b/data_all_eng_slimpj/shuffled/split2/finalzzzaneou
new file mode 100644
index 0000000000000000000000000000000000000000..945f25e2e4a962aa0f01fee28cabbaf647a95ba9
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaneou
@@ -0,0 +1,5 @@
+{"text":"For the Beach Boys song, see Friends (The Beach Boys album).\nThe Maharishi taught thousands of people during a series of world tours from 1958 to 1965, expressing his teachings in spiritual and religious terms. TM became more popular in the 1960s and 1970s, as the Maharishi shifted to a more technical presentation, and his meditation technique was practiced by celebrities. At this time, he began training TM teachers and created specialized organizations to present TM to specific segments of the population such as business people and students. By the early 2000s, TM had been taught to millions of people; the worldwide TM organization had grown to include educational programs, health products, and related services.\nThe TM technique involves the use of a sound called a mantra, and is practiced for 15\u201320 minutes twice per day. It is taught by certified teachers through a standard course of instruction, which costs a fee that varies by country. According to the Transcendental Meditation movement, it is a non-religious method for relaxation, stress reduction, and self-development. The technique has been seen as both religious and non-religious; sociologists, scholars, and a New Jersey judge and court are among those who have expressed views. The United States Court of Appeals upheld the federal ruling that TM was essentially \"religious in nature\" and therefore could not be taught in public schools.\nThe Transcendental Meditation program and the Transcendental Meditation movement originated with Maharishi Mahesh Yogi, founder of the organization, and continue beyond his death in 2008. In 1955, \"the Maharishi began publicly teaching a traditional meditation technique\" learned from his master Brahmananda Saraswati that he called Transcendental Deep Meditation and later renamed Transcendental Meditation. The Maharishi initiated thousands of people, then developed a TM teacher training program as a way to accelerate the rate of bringing the technique to more people. He also inaugurated a series of world tours which promoted Transcendental Meditation. These factors, coupled with endorsements by celebrities who practiced TM and claims that scientific research had validated the technique, helped to popularize TM in the 1960s and 1970s. By the late 2000s, TM had been taught to millions of individuals and the Maharishi was overseeing a large multinational movement. Despite organizational changes and the addition of advanced meditative techniques in the 1970s, the Transcendental Meditation technique has remained relatively unchanged.\nAmong the first organizations to promote TM were the Spiritual Regeneration Movement and the International Meditation Society. In modern times, the movement has grown to encompass schools and universities that teach the practice, and includes many associated programs based on the Maharishi's interpretation of the Vedic traditions. In the U.S., non-profit organizations included the Students International Meditation Society, AFSCI, World Plan Executive Council, Maharishi Vedic Education Development Corporation, Global Country of World Peace and Maharishi Foundation. The successor to Maharishi Mahesh Yogi, and leader of the Global Country of World Peace, is Tony Nader.\nThe meditation practice involves the use of a mantra for 15\u201320 minutes twice per day while sitting with the eyes closed. It is reported to be one of the most widely practiced, and among the most widely researched, meditation techniques, with hundreds of published research studies. The technique is made available worldwide by certified TM teachers in a seven-step course, and fees vary from country to country. Beginning in 1965, the Transcendental Meditation technique has been incorporated into selected schools, universities, corporations, and prison programs in the US, Latin America, Europe, and India. In 1977 a US district court ruled that a curriculum in TM and the Science of Creative Intelligence (SCI) being taught in some New Jersey schools was religious in nature and in violation of the First Amendment of the United States Constitution. The technique has since been included in a number of educational and social programs around the world.\nThe Transcendental Meditation technique has been described as both religious and non-religious, as an aspect of a new religious movement, as rooted in Hinduism, and as a non-religious practice for self-development. The public presentation of the TM technique over its 50-year history has been praised for its high visibility in the mass media and effective global propagation, and criticized for using celebrity and scientific endorsements as a marketing tool. Also, advanced courses supplement the TM technique and include an advanced meditation program called the TM-Sidhi program.\nThe Transcendental Meditation movement refers to the programs and organizations connected with the Transcendental Meditation technique and founded by Maharishi Mahesh Yogi. Transcendental Meditation was first taught in the 1950s in India and has continued since the Maharishi's death in 2008. The organization was estimated to have 900,000 participants worldwide in 1977, a million by the 1980s, and 5 million in more recent years, including some notable practitioners.\nPrograms include the Transcendental Meditation technique, an advanced meditation practice called the TM-Sidhi program (\"Yogic Flying\"), an alternative health care program called Maharishi Ayurveda, and a system of building and architecture called Maharishi Sthapatya Ved. The TM movement's past and present media endeavors include a publishing company (MUM Press), a television station (KSCI), a radio station (KHOE), and a satellite television channel (Maharishi Channel). During its 50-year history, its products and services have been offered through a variety of organizations, which are primarily nonprofit and educational. These include the Spiritual Regeneration Movement, the International Meditation Society, World Plan Executive Council, Maharishi Vedic Education Development Corporation, the Global Country of World Peace, and the David Lynch Foundation.\nThe TM movement also operates a worldwide network of Transcendental Meditation teaching centers, schools, universities, health centers, herbal supplements, solar panel, and home financing companies, plus several TM-centered communities. The global organization is reported to have an estimated net worth of USD 3.5 billion. The TM movement has been characterized in a variety of ways and has been called a spiritual movement, a new religious movement, a millenarian movement, a world affirming movement, a new social movement, a guru-centered movement, a personal growth movement, a religion, and a cult.[need quotation to verify] Additional sources contend that TM and its movement are not a cult. Participants in TM programs are not required to adopt a belief system; it is practiced by atheists, agnostics and people from a variety of religious affiliations. The organization has also been criticized as well as praised for its public presentation and marketing techniques throughout its 50-year history.\nIt is not possible to say whether meditation has any effect on health, as the research is of poor quality, and is marred by a high risk for bias due to the connection of researchers to the TM organization and by the selection of subjects with a favorable opinion of TM. Most independent systematic reviews have not found health benefits for TM exceeding those produced by other relaxation techniques or health education. A 2013 statement from the American Heart Association said that TM could be considered as a treatment for hypertension, although other interventions such as exercise and device-guided breathing were more effective and better supported by clinical evidence. A 2014 systematic review and meta-analysis funded by the U.S. Agency for Healthcare Research and Quality found no evidence that mantra meditation programs such as TM were effective in reducing psychological stress or improving well-being. A 2015 systematic review and meta-analysis found that TM may effectively reduce blood pressure compared to control groups, although the underlying studies may have been biased and further studies with better designs are needed to confirm these results. A 2014 Cochrane review found that it was impossible to draw any conclusions about whether TM is effective in preventing cardiovascular disease, as the scientific literature on TM was limited and at \"serious risk of bias\".\nThe first studies of the health effects of Transcendental Meditation appeared in the early 1970s. By 2004 the US government had given more than $20 million to Maharishi University of Management to study the effect of meditation on health.\nIn the 1960s, Maharishi Mahesh Yogi described a paranormal effect claiming a significant number of individuals (1% of the people in a given area) practicing the Transcendental Meditation technique (TM) could have an effect on the local environment. This hypothetical influence was later termed the Maharishi Effect. With the introduction of the TM-Sidhi program in 1976, the Maharishi proposed that the square root of one percent of the population practicing the TM-Sidhi program, together at the same time and in the same place, would increase \"life-supporting trends\". This was referred to as the \"Extended Maharishi Effect\". Evidence, which TM practitioners believe supports the existence of the effect, has been said to lack a causal basis. The evidence was said to result from cherry-picked data and the credulity of believers.\nThe organization has been the subject of controversies, and labelled a cult by several parliamentary inquiries or anti-cult movements in the world. Some also say that TM and its movement are not a cult. The TM movement has been characterized in a variety of ways and has been called a spiritual movement, a new religious movement, a millenarian movement, a world affirming movement, a new social movement, a guru-centered movement, a personal growth movement, a religion. Participants in TM programs are not required to adopt a belief system; it is practiced by atheists, agnostics and people from a variety of religious affiliations.\n^ \"Transcendental Meditation\". Britannica Online Encyclopedia.\n^ Dawson, Lorne (2003). Cults and New Religious Movements. Hoboken, New Jersey: Blackwell Publishing. p. 54.\n^ a b Cowan,, Douglas E.,; Bromley, David G. (2007). Cults and New Religions: A Brief History (Blackwell Brief Histories of Religion). Wiley-Blackwell. pp. 48\u201371. ISBN 1-4051-6128-0.\n^ a b Calo, Zachary (2008). \"Chapter 4: The Internationalization of Church-State Issues\". In Duncan, Ann; Jones, Steven (eds.). Church-State Issues in America Today. Westport, Connecticut: Praeger. p. 159. ISBN 978-0-275-99368-9.\n^ Ashman, Allan (January 1978). \"What's New in the Law\". American Bar Association Journal. 64: 144.\n^ \"Malnak v. Yogi\". Leagle. 1979. Retrieved 19 May 2017.\n^ Bette Novit Evans (9 November 2000). Interpreting the Free Exercise of Religion: The Constitution and American Pluralism. Univ of North Carolina Press. p. 65. ISBN 978-0-8078-6134-9. Proponents of the program denied that Transcendental Meditation was a religion; the Third Circuit concluded that it was.\n^ a b Benson, Herbert; Klipper, Miriam Z. (2001). The Relaxation Response. New York, NY: Quill. p. 61. ISBN 978-0-380-81595-1.\n^ a b Sinatra, Stephen T.; Roberts, James C.; Zucker, Martin (20 December 2007). Reverse Heart Disease Now: Stop Deadly Cardiovascular Plaque Before It's Too Late. Wiley. p. 192. ISBN 978-0-470-22878-4.\n^ a b Ospina MB, Bond K, Karkhaneh M, et al. (June 2007). \"Meditation practices for health: state of the research\". Evid Rep Technol Assess (Full Rep) (155): 1\u2013263. PMC 4780968. PMID 17764203. Scientific research on meditation practices does not appear to have a common theoretical perspective and is characterized by poor methodological quality. Firm conclusions on the effects of meditation practices in healthcare cannot be drawn based on the available evidence.\n^ AP (5 February 2008). \"Beatles guru dies in Netherlands\". USA Today.\n^ Epstein, Edward (29 December 1995). \"Politics and Transcendental Meditation\". San Francisco Chronicle.\n^ Morris, Bevan (1992). \"Maharishi's Vedic Science and Technology: The Only Means to Create World Peace\" (PDF). Journal of Modern Science and Vedic Science. 5 (1\u20132): 200.\n^ Rooney, Ben (6 February 2008). \"Maharishi Mahesh Yogi, guru to Beatles, dies\". The Telegraph. London.\n^ Williamson, Lola (2010). Transcendent in America: Hindu-Inspired Meditation Movements as New Religion. New York: NYU Press. pp. 97\u201399. ISBN 9780814794500.\n^ a b Russell, Peter (1977). The TM Technique: An Introduction to Transcendental Meditation and the Teachings of Maharishi Mahesh Yogi. London: Routledge. pp. 25\u201326. ISBN 978-0-7100-8539-9.\n^ Needleman, Jacob (1970). \"Transcendental Meditation\". The New Religions (1st ed.). Garden City N.Y.: Doubleday. p. 144.\n^ Christian D. Von Dehsen; Scott L. Harris (1999). Philosophers and religious leaders. The Orynx Press. p. 120. ISBN 9781573561525.\n^ \"Maharishi Mahesh Yogi\". The Times (London). 7 February 2008. p. 62.\n^ Oates, Robert M. (1976). Celebrating the Dawn: Maharishi Mahesh Yogi and the TM technique. New York: G.P. Putnam's Sons. p. 226. ISBN 978-0-399-11815-9.\n^ Irwin, T. K. (8 October 1972). \"What's New in Science: Transcendental Meditation: Medical Miracle or 'Another Kooky Fad'\". Sarasota Herald Tribune Family Weekly. pp. 8\u20139.\n^ Chryssides, George D. (1999). Exploring New Religions. London: Cassell. pp. 293\u2013296. ISBN 978-0-8264-5959-6.\n^ \"Behavior: The TM Craze: 40 Minutes to Bliss\". Time. 13 October 1975. ISSN 0040-781X. Retrieved 15 November 2009.\n^ Press Release by Maharishi Foundation (15 July 2013). \"Australian Food Store Offers Transcendental Meditation to Employees\". The Herald (South Carolina, USA). Archived from the original on 3 August 2013. Retrieved 3 August 2013.\n^ Mizroch, Amir (23 July 2006). \"Forget the F-16s, Israel needs more Yogic Flyers to beat Hizbullah: 30-strong TM group, sole guests at Nof Ginnosar Hotel, say they need another 235 colleagues to make the country safe\". Jerusalem Post. p. 4.\n^ \"Maharishi's ashes immersed in Sangam\". The Hindustan Times. New Delhi. Indo-Asian News Service. 12 February 2008.\n^ \"The Transcendental Meditation Program\". Tm.org. Retrieved 17 February 2013.\n^ Cotton, Dorothy H. G. (1990). Stress management: An integrated approach to therapy. New York: Brunner\/Mazel. p. 138. ISBN 0-87630-557-5.\n^ Schneider, Robert; Fields, Jeremy (2006). Total Heart Health: How to Prevent and Reverse Heart Disease with the Maharishi Vedic Approach to Health. Laguna Beach, CA: Basic Health Publications. pp. 148\u2013149. ISBN 1458799247.\n^ Murphy, M; Donovan, S; Taylor, E (1997). The Physical and Psychological Effects of Meditation: A review of Contemporary Research with a Comprehensive Bibliography 1931\u20131996. Sausalito, California: Institute of Noetic Sciences.\n^ Bushell, William (2009). \"Longevity Potential Life Span and Health Span Enhancement through Practice of the Basic Yoga Meditation Regimen\". Annals of the New York Academy of Sciences. 1172: 46. doi:10.1111\/j.1749-6632.2009.04538.x. ISBN 9781573316774. PMID 19735236. Transcendental Meditation (TM), a concentrative technique ... has been the most extensively studied meditation technique.\n^ Rosenthal, Norman (2011). Transcendence: Healing and Transformation through Transcendental Meditation. New York: Tarcher\/Penguin. p. 14. ISBN 978-1-58542-873-1. By my latest count, there have been 340 per-reviewed articles published on TM, many of which have appeared in highly respected journals.\n^ Freeman, Lyn (2009). Mosby's Complementary & Alternative Medicine: A Research-Based Approach. Mosby Elsevier. p. 176. ISBN 9780323053464.\n^ \"How To Learn\". Tm.org. Retrieved 17 February 2013.\n^ \"TM Course Fee\". TM.org. Retrieved 30 May 2012.\n^ \"Transcendental Meditation Fees and Course Details\". Transcendental Meditation: Official website for the UK. Retrieved 31 January 2013.\n^ American Bar Association (January 1978). \"Constitutional Law ... Separating Church and State\". ABA Journal. 64: 144.\n^ Bainbridge, William Sims (1997). The Sociology of Religious Movements. New York: Routledge. p. 188. ISBN 0-415-91202-4.\n^ Aghiorgoussis, Maximos (Spring 1999). \"The challenge of metaphysical experiences outside Orthodoxy and the Orthodox response\". Greek Orthodox Theological Review. Brookline. 44 (1\u20134): 21, 34.\n^ Chryssides, George D. (2001). Exploring New Religions. Continuum International Publishing Group. pp. 301\u2013303. ISBN 9780826459596. \"Although one can identify the Maharishi's philosophical tradition, its teachings are in no way binding on TM practitioners. There is no public worship, no code of ethics, no scriptures to be studied, and no rites of passage that are observed, such as dietary laws, giving to the poor, or pilgrimages. In particular, there is no real TM community: practitioners do not characteristically meet together for public worship, but simply recite the mantra, as they have been taught it, not as religious obligation, but simply as a technique to benefit themselves, their surroundings and the wider world.\"\n^ Rosenthal, Norman E. (2011). Transcendence: Healing and Transformation Through Transcendental Meditation. Tarcher Penguin. p. 4. ISBN 978-1-58542-873-1.\n^ Stark, Rodney; Bainbridge, William, Sims (1986). The Future Of Religion. University of California Press. p. 287. ISBN 978-0520057319. \"Time magazine in 1975 estimated that the U.S. total had risen to 600,000 augmented by half that number elsewhere\" =[900,000 world wide] \"Annual Growth in TM Initiations in the U.S. [chart] Cumulative total at the End of Each Year: 1977, 919,300\"\n^ Peterson, William (1982). Those Curious New Cults in the 80s. New Canaan, Connecticut: Keats Publishing. p. 123. ISBN 9780879833176. claims \"more than a million\" in the USA and Europe.\n^ Occhiogrosso, Peter. The Joy of Sects: A Spirited Guide to the World's Religious Traditions. New York: Doubleday (1996); p 66, citing \"close to a million\" in the USA.\n^ Bainbridge, William Sims (1997) Routledge, The Sociology of Religious Movements, page 189 \"the million people [Americans] who had been initiated\"\n^ Analysis: Practice of requiring probationers to take lessons in transcendental meditation sparks religious controversy, NPR All Things Considered, 1 February 2002 | ROBERT SIEGEL \"TM's five million adherents claim that it eliminates chronic health problems and reduces stress.\"\n^ Martin Hodgson, The Guardian (5 February 2008)\t\"He [Maharishi] transformed his interpretations of ancient scripture into a multimillion-dollar global empire with more than 5m followers worldwide\"\n^ Stephanie van den Berg, Sydney Morning Herald, Beatles guru Maharishi Mahesh Yogi dies, (7 February 2008) \"the TM movement, which has some five million followers worldwide\"\n^ Meditation a magic bullet for high blood pressure \u2013 study, Sunday Tribune (South Africa), (27 January 2008) \"More than five million people have learned the technique worldwide, including 60,000 in South Africa.\"\n^ Maharishi Mahesh Yogi - Transcendental Meditation founder's grand plan for peace, The Columbian (Vancouver, WA), 19 February 2006 | ARTHUR MAX Associated Press writer \"transcendental meditation, a movement that claims 6 million practitioners since it was introduced.\"\n^ Maharishi Mahesh Yogi, Spiritual Leader Dies, New York Times, By LILY KOPPEL, Published: 6 February 2008 \"Since the technique's inception in 1955, the organization says, it has been used to train more than 40,000 teachers, taught more than five million people\"\n^ Spivack, Miranda (12 September 2008). \"Bricks Mortar and Serenity\". Washington Post.\n^ \"Maharishi Mahesh Yogi\". The Times. London (UK). 7 February 2008.\n^ \"Maharishi Mahesh Yogi\". Britannica Concise Encyclopedia.\nBeckford, James A. (1985). Cult controversies: the societal response to new religious movements. Tavistock Publications. p. 23. ISBN 978-0-422-79630-9.\nParsons, Gerald (1994). The Growth of Religious Diversity: Traditions. The Open University\/Methuen. p. 288. ISBN 978-0-415-08326-3.\nAlper, Harvey P. (December 1991). Understanding mantras. Motilal Banarsidass. p. 442. ISBN 978-81-208-0746-4.\nRaj, Selva J.; William P. Harman (2007). Dealing With Deities: The Ritual Vow in South Asia. SUNY Press. p. 129. ISBN 978-0-7914-6708-4.\n^ a b c d Persinger, Michael A.; Carrey, Normand J.; Suess, Lynn A. (1980). TM and cult mania. North Quincy, Mass.: Christopher Pub. House. ISBN 0-8158-0392-3.\n^ Shakespeare, Tom. \"A Point of View\". BBC. Retrieved 31 May 2014.\n^ a b Bainbridge, Sims; Stark, Rodney; Bainbridge, William Sims (1985). The future of religion: secularization, revival, and cult formation. Berkeley, Calif: Univ. of California Press. ISBN 0-520-05731-7.\n^ Sagan, Carl (1997). The demon-haunted world: science as a candle in the dark. New York: Ballantine Books. p. 16. ISBN 0-345-40946-9.\n^ a b Harrison, Shirley (1990). Cults: The Battle for God. Kent: Christopher Helm. pp. 93\u2013103 \"none of the other 'cultic qualities' defined by cult watchers can be fairly attributed to TM.\"\n^ Hannaford, Alex (27 December 2010). \"Mantra with a mission; Feature Om or ominous? The maverick film director David Lynch wants to bring Transcendental Meditation to our classrooms, and believes in 'yogic flying'. Can he get it off the ground?\". The Sunday Times (London).\n^ Lyster, Samantha (21 October 2000) Samantha Lyster finds herself in holistic heaven with new-found happiness and tranquillity after learning the art of transcendental meditation, The Birmingham Post (England), \"TM is not a religion, a cult or a philosophy\"\n^ Chryssides George D. Defining the New Spirituality http:\/\/www.cesnur.org\/conferences\/riga2000\/chryssides.htm One possible suggestion is that religion demands exclusive allegiance: this would ipso facto exclude Scientology, TM and the Soka Gakkai simply on the grounds that they claim compatibility with whatever other religion the practitioner has been following. For example, TM is simply \u2013 as they state \u2013 a technique. Although it enables one to cope with life, it offers no goal beyond human existence (such as moksha), nor does it offer rites or passage or an ethic. Unlike certain other Hindu-derived movements, TM does not prescribe a dharma to its followers \u2013 that is to say a set of spiritual obligations deriving from one's essential nature.\n^ Krisanaprakornkit, T.; Ngamjarus, C.; Witoonchart, C.; Piyavhatkul, N. (2010). Krisanaprakornkit, Thawatchai (ed.). \"Meditation therapies for attention-deficit\/hyperactivity disorder (ADHD)\". Cochrane Database Syst Rev. 6 (6): CD006507. doi:10.1002\/14651858.CD006507.pub2. PMID 20556767. As a result of the limited number of included studies, the small sample sizes and the high risk of bias, we are unable to draw any conclusions regarding the effectiveness of meditation therapy for ADHD.\n^ Brook RD, Appel LJ, Rubenfire M, Ogedegbe G, Bisognano JD, Elliott WJ, Fuchs FD, Hughes JW, Lackland DT, Staffileno BA, Townsend RR, Rajagopalan S (2013). \"Beyond medications and diet: alternative approaches to lowering blood pressure: a scientific statement from the American Heart Association\". Hypertension. 61 (6): 1360\u201383. doi:10.1161\/HYP.0b013e318293645f. PMID 23608661.\n^ Goyal M, Singh S, Sibinga EM, Gould NF, Rowland-Seymour A, Sharma R, Berger Z, Sleicher D, Maron DD, Shihab HM, Ranasinghe PD, Linn S, Saha S, Bass EB, Haythornthwaite JA (2014). \"Meditation programs for psychological stress and well-being: a systematic review and meta-analysis\". JAMA Intern Med. 174 (3): 357\u201368. doi:10.1001\/jamainternmed.2013.13018. PMC 4142584. PMID 24395196. ... we found low evidence of no effect or insufficient evidence that mantra meditation programs had an effect on any of the psychological stress and well-being outcomes we examined.\n^ Goyal M, Singh S, Sibinga EM, Gould NF, Rowland-Seymour A, Sharma R, Berger Z, Sleicher D, Maron DD, Shihab HM, Ranasinghe PD, Linn S, Saha S, Bass EB, Haythornthwaite JA (2014). \"Meditation Programs for Psychological Stress and Well-Being\". Agency for Healthcare Research and Quality. Our review finds that the mantra meditation programs do not appear to improve any of the psychological stress and well-being outcomes we examined, but the strength of this evidence varies from low to insufficient.\n^ Bai, Z; Chang, J; Chen, C; Li, P; Yang, K; Chi, I (12 February 2015). \"Investigating the effect of transcendental meditation on blood pressure: a systematic review and meta-analysis\". Journal of Human Hypertension. 29: 653\u201362. doi:10.1038\/jhh.2015.6. PMID 25673114.\n^ Hartley, L; Mavrodaris, A; Flowers, N; Ernst, E; Rees, K (1 December 2014). \"Transcendental meditation for the primary prevention of cardiovascular disease\". The Cochrane Database of Systematic Reviews. 12: CD010359. doi:10.1002\/14651858.CD010359.pub2. PMID 25436436.\n^ QUICK, SUSANNE (17 October 2004). \"Delving into alternative care: Non-traditional treatments draw increased interest, research funding\". Journal Sentinel. Milwaukee, WI. Archived from the original on 29 September 2007. Maharishi University ... has received more than $20 million in government support to date to explore the health benefits of meditation.\n^ \"Maharishi Effect Research on the Maharishi Effect\". Maharishi University of Management. Archived from the original on 30 July 2010. Retrieved 29 December 2009.\n^ \"Commission d'enqu\u00eate sur les sectes \u2013 Assembl\u00e9e nationale\". www.assemblee-nationale.fr. Retrieved 3 March 2019.\n^ \"GROUP CLAIMS TM MOVEMENT IS A CULT\".\nGeoff Gilpin, The Maharishi Effect: A Personal Journey Through the Movement That Transformed American Spirituality, Tarcher-Penguin 2006, ISBN 1-58542-507-9* Pollack, A. A., Weber, M. A., Case, D.\nRothstein, Mikael (1996). Belief Transformations: Some Aspects of the Relation Between Science and Religion in Transcendental Meditation (Tm) and the International Society for Krishna Consciousness. Language: English. Aarhus universitetsforlag. p. 227. ISBN 87-7288-421-5.\nSkolnick, Andrew \"Maharishi Ayur-Veda: Guru's Marketing Scheme Promises the World Eternal 'Perfect Health'!\", JAMA 1991;266:1741\u20131750,2 October 1991.\nYogi, Maharishi Mahesh (1967) Penguin, Maharishi Mahesh Yogi on the Bhagavad-Gita : A New Translation and Commentary ISBN 0-14-019247-6.\nJohn Samuel Hagelin is the leader of the Transcendental Meditation (TM) movement in the United States. He is president of the Maharishi University of Management (MUM) in Fairfield, Iowa, and honorary chair of its board of trustees. The university was established in 1973 by the TM movement's founder, Maharishi Mahesh Yogi, to deliver a \"consciousness-based education\".\nDeekshya also spelled deeksha or deeksa in common usage, translated as a \"preparation or consecration for a religious ceremony\", is giving of a mantra or an initiation by the guru of Indian religions such as Hinduism, Buddhism, and Jainism. Diksa is given in a one-to-one ceremony, and typically includes the taking on of a serious spiritual discipline. The word is derived from the Sanskrit root d\u0101 plus k\u1e63i or alternately from the verb root d\u012bk\u1e63. When the mind of the guru and the disciple become one, then we say that the disciple has been initiated by the guru.\nTM and Cult Mania is a non-fiction book that examines assertions made by the Transcendental Meditation movement (TM). The book is authored by Michael Persinger, Normand Carrey and Lynn Suess and published in 1980 by Christopher Publishing House. Persinger is a neurophysiologist and has worked out of Laurentian University. He trained as a psychologist and focused on the impacts of religious experience. Carrey is a medical doctor who specialized in psychiatry. He focused his studies into child psychiatry with research at Dalhousie University, and has taught physicians in a psychiatry residency program in the field of family therapy. Suess assisted Persinger in researching effects of geological phenomena on unidentified flying object sightings in Washington; the two conducted similar research in Toronto and Ottawa.\nIn late August 1967, the English rock band the Beatles attended a seminar on Transcendental Meditation (TM) held by Indian teacher Maharishi Mahesh Yogi at a training college in Bangor in north-west Wales. The visit attracted international publicity for Transcendental Meditation and presented the 1960s youth movement with an alternative to psychedelic drugs as a means to attaining higher consciousness. The Beatles' endorsement of the technique followed the band's incorporation of Indian musical and philosophical influences in their work, and was initiated by George Harrison's disillusionment with Haight-Ashbury in San Francisco, which he visited in early August.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Like your skin, your beard goes through an crumbling process. But rather than wrinkles, you're larboard with dull, asleep locks. Lots of things can appulse your shine, volume, and blush as you get older, like artlessly drier skin, changes in your beard follicles, and alike genetics\u2014all of which can artlessly be out of your control.\nThe acceptable news? Tweaking your beard routine, diet, and advance in best articles can accompany your strands aback to life. If you absence your thicker, added active hair, let the afterward experts explain what's happening. Then, try their best tricks to restore your hair's color, fullness, and flash to its above glory\u2014no amount your age.\nYour bark artlessly gets drier with age\u2014and that includes your scalp. \"When oil glands on the attic apathetic bottomward production, your beard feels drier and looks beneath glossy, and it may be decumbent to coil and breakage,\" says Ted Lain, MD, a board-certified dermatologist in Austin who specializes in beard disorders.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Community Connections is part of the Volunteer Center. They specialize in helping adults with a mental health condition connect to employment, volunteer work and education. The majority of their programs require the client be part of the Santa Cruz County Behavioral Health System of Care. This means you should be connected to a county psychiatrist, therapist or a mental health provider. If you are not part of the System of Care, call the Santa Cruz County Access Line to determine what services are open to you. (800) 952-2335 If you are part of the System of Care, then you can simply call or visit Community Connections during business hours. You do not need a referral for many of their programs.\nCCCIL promotes the independence of people with disabilities by supporting their equal and full participation in life. CCCIL provides advocacy, education and support to people with disabilities, their families and the community. Clients have only have to self identify to qualify.\nWorkforce Santa Cruz County (WFSCC) is a partnership of local organizations that serve job seekers and businesses in the community. WFSCC's network of Career Centers offer an array of no cost services for businesses, employees, and job seekers. Workforce Santa Cruz County is overseen by the Workforce Investment Board of Santa Cruz County.\nSanta Cruz Volunteer Center is the main clearinghouse for volunteer work in Santa Cruz County.\nSocial Services phone number is (888) 421-8080, www.myBenefitsCalWIN.org is the site for electronic enrollment in food stamps, MediCal and many other programs.\nDepartment of Rehabilitation - The California Department of Rehabilitation (DOR) is an employment and independent living resource for people with disabilities. They can tell you more about Social Security Work Incentives and how to work without loosing your SSI or SSDI benefits.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Moments later they had decided to inform the police. As they were about to enter their cars, suddenly Alan spotted something moving at the far end of the field.\n\"Over there!\" Alan cried out pointing towards the end of the field.\nThey hurried toward the edge of the field. Elizabeth lost her temper after stumbling over one of her high heels while trying to keep up.\nAs soon as they reached the children they tried questioning them, but it was useless. They would only cry and mumble something about a toy that was taken away from them. Alan somehow tried again.\n\"Don't (sniff) know.\" Charlie repeated weeping.\nIt was no use, Charlie couldn't stop himself from sobbing. All three in fact complained about having a headache.\nFirst thing Elizabeth had in mind would be to call the police as soon as she got home.\n\"I don't think she'll be baby-sitting anywhere anymore.\nTwo and a half weeks after the incident, there was still no trace of Barbara Adams. The inquiry was going on its third week. Speculation had it that Barbara Adams had run off with a new boyfriend; however, the investigation would go on further. There was still no logical explanation found concerning the burnt book found in the sand pile.\nMoments later they had decided to inform the police. As they were about to get into their cars, Alan suddenly spotted something moving at the far end of the field. \"Over there!\" he cried out, pointing.\nAs they hurried towards that end of the field, Elizabeth lost her temper after stumbling over one of her high heels while trying to keep up.\nAs soon as they reached the children, they tried questioning them, but it was useless. The children would only cry and mumble something about a toy that had been taken away from them.\n\"Don't (sniff) know.\" Charlie said, weeping.\nIt was no use, Charlie couldn't stop himself from sobbing, and all three children complained about having headaches.\nElizabeth was more than steaming with temper. Not only was she pissed at Barbara for leaving Charlie alone while running off with... presumably some boyfriend, but her new dress had grass stains on it. This was just a little too much. As for Barbara, the police could take care of her. \"If she thinks she can get away with this...\"\nThe first thing Elizabeth had in mind would be calling the police as soon as she got home. \"I don't think she'll be baby-sitting anywhere anymore.\"\nTwo and a half weeks after the incident, there was still no trace of Barbara Adams. The inquiry was going into its third week, and speculation had it that Barbara Adams had run off with a new boyfriend. However, the investigation would go on, there was still no logical explanation for the burnt book found on the sand pile.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"People are in an immediate requirement of money and are looking out for different sources all at the same time. They might do this so as to avail the loan from at least one of the sources where they have applied for the loan.\nThere are certain applicants who might be in need of more money than they can get from a single source and so they wish to raise a greater amount by availing the loan from different sources.\nApplying at multiple destinations automatically increase the chance of getting hold of the desired amount within the set time frame from at least one of the banks where you have applied for the application. During the process, if more than one bank happens to approve the loan, then a huge amount can be raised within a short span of time to meet the urgency. This is helpful for those who are starting a business and need immediate financial assistance to cater to the expenditures.\nAlthough applying at different banks at the same time is a good step to garner huge financial assistance, but at times, doing so might affect the chances of loan confirmation. This is mainly because many banks today rely on the credit rating of the borrower with the help of CIBIL scores.\nWhen an individual applies for the personal loans at different banks, the same gets reflected in the CIBIL report which makes the lender reconsider their option of the loan approval. And, this is where the issues might arise during the loan approval.\nApart from this, if the application is rejected by the bank, the credit rating might suffer significantly and make it difficult for the borrower to get hold of the loan when they need it the most.\nThe CIBIL credit rating might show negative results if one applies for a loan at different banks at the same time. On the other hand for those who have no credit history and are applying for the loan for the very first time, the credibility might zero.\nOne solution to this process is to explore different possibilities with multiple banks and select the one where there are chances of getting immediate approval. Take an action and collect the required information from multiple lenders, understand the terms and conditions of the loan agreement and select the one with the minimum fee and rate of interest. Once you are done with all the documentation and agreement procedures, apply for the loan that would best suit your needs and provide you with the best deals. However, one thing that is often advised by the experts is not to hurry when it comes to securing the funds as that might create financial constraints and obstacles in the near future.\nNext Post: Nouveaux arrivants au Qu\u00e9bec : comment sortir dusurendettement?","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzanjpt b/data_all_eng_slimpj/shuffled/split2/finalzzzanjpt
new file mode 100644
index 0000000000000000000000000000000000000000..9b2e6cf2b164edd942b451422b2dc8388ba50dec
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzanjpt
@@ -0,0 +1,5 @@
+{"text":"American Idol Season 10's Top 11 finalists will be here in the Philippines to perform and entertain Filipino concertgoers at the Araneta Coliseum for the \"American Idol Live! Tour 2011\" on September 20 of this year, Tuesday at 8:00 p.m., as announced and reported by Philippine Entertainment Portal (PEP), July 21, 2011.\nReport from the official website of American Idol further confirms that the \"American Idol Live! Tour 2011\" will include Casey Abrams, Haley Reinhart, Jacob Lusk, James Durbin, Lauren Alaina, Naima Adedapo, Paul McDonald, Pia Toscano, Stefano Langone, Fil-American Thia Megia and Season 10 winner Scotty McCreery.\nPEP report reveals that the performances will cover a wide range of musical genres from country, pop, jazz to rock. Similarly, they will perform favorites from the show, duets of today's hottest hit songs and a few original singles from their individual solo projects.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"GSE 3400 PCA is innovative, reliable and environmentally friendly. It supplies fresh, clean air into parked aircraft at carefully monitored temperatures and provides a pleasant atmosphere for the crew and the passengers. It also makes aircraft turn-around faster and more effective.\nThe point-of-use 3400 PCA is the market's only true modular PCA (patented). It is designed around identical cooling modules that can easily be swapped by a technician with no special skills and in 20 minutes only. This smart and innovative PCA is available in the following models: 130 kg\/min & 210 kg\/min.\nThe ITW GSE 3400 PCA is the ONLY modular PCA on the market. You can exchange a cooling module in approx. 20 minutes and without dismounting the unit from the installation spot!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Wholesale.info provides wholesale sources for wholesale buyers looking to make money reselling products.\nWith a unique delivery method, Wholsale.info sends traffic directly to your website to hel you gain more exposure in the wholesale industry. Add a product and get started receiving quality wholesale leads today!\nWe are always here to help! Send us an email!\nGet started selling with us.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Palm Beach on Currumbin Creek is a perfect location for beginers to try Stand Up Paddle Boarding and kayaking. It is a safe, sheltered arm of Currumbin Creek near the mouth before it enters the ocean. It is close to Currumbin wildlife sanctuary and surfworld museum. There is also a large children's playground area in the them of a Pirate Ship. You can also find a Cafe with good food and drinks and a large park area adjoining the beach with a lifeguard tower.\nIt is the ideal location to try some easy aquatic activities as part of a day trip to other attractions in the area.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"WHERE - HERE IS A LINK TO THE RACE HQ MAP - HEAD FOR CLOUDS HILL - King George V Rd, Bovington, Dorset BH20 7NQ - CONTINUE ON THAT ROAD UNTIL YOU REACH THE NOISE!\nWHO CAN TAKE PART - Minimum age for this event from 2019 is 8 - children aged between 8 and 14 will need to twist the arm of a responsible adult (adult must be 18 or over) to enter this race. There will be a maximum limit of two children per adult.\nIf you are 15 and over you can take part in this event as an individual.\nEarly Bird Entries 50% off have SOLD OUT.\nTIMES - REGISTRATION OPENS AT 10:00 - Entry to site will close at 11:30 so you have plenty of time to get parked, changed and attend the race briefing at 11:50 \u2013 the race will start at 12:00 even if you are not there.\nWHAT DO WE GET - A 2019 Dirty Devil custom Buff & Finishers Medal will be given to all who complete the course.\nAnd the Sunday race will be split in to two events - The Dirty Devil Stampede 10k'ish and The Dirty Devil Double - yes, well done genius, you've guessed it, a 21k'ish run for the more demanding and down right dirty runner! :) Link for these race are here (we hope!) - 10k - DOUBLE To race on the Sunday you must be 16 or over.\nFor the first time we have been able to get permission for the junior members of the running community to be able to take part in this awesome event - AS LONG AS - they are accompanied by a responsible adult or 2!\nThis will be a non timed event enabling all members of the family to run around the muddy tank tracks and finish together covered in mud.\n- Anyone 15 and over, can run around this event without the pressure of being timed and being lapped by the racing snakes- this will be an event for all who want an introduction to mud running in a relatively safe environment.\nMinimum age for this event is 8 - children aged between 8 and 14 will need to twist the arm of a responsible adult (adult must be 18 or over) to enter this race. No Adult - No Race.\nThis multi terrain run of the new 1 loop course takes you around the iconic location of the Bovington Tank training ground, land reserved for army training only to which we have secured the ultimate in mud running events in the South West.\nLocated just 1 mile away from the amazing Bovington Tank Museum and just a few miles away from Monkey World there is plenty for the family to do while you get yourself tagged as one Dirty Devil.\nThe Dirty Devil Stampede is held over the Army's vehicle testing tracks so expect it to be gritty and interesting! You encounter trails, hills, mud, a lot of water, the sheep dip and the knife edge! So whats not to like?\nIt's tough, sure, but we know you wouldn't like it any other way.\nCome alone or badger some of your mates to get involved too, whatever you decided, we guarantee you will go away with a smile on your face and mud on your knees!\nThere are prizes for the top 3 Male & Female competitors.\nEntries close when we sell out \u2013 and we will, so don't delay. We're sure you are tough enough \u2013 see you on the start line!\nA 2019 Dirty Devil custom Buff & Finishers Medal will be given to all who complete the course.\nRegistration - This will take place from 10am on Saturday morning till 11:30 - you will need to come and check in at the RACE HQ tents to be marked off our list of attendees - you will not receive anything in the post for this race as it isn't timed.\nWho can take part? - If you are 15 and over you can take part in this event as an individual. Minimum age for this event from 2019 is 8 - children aged between 8 and 14 will need to twist the arm of a responsible adult (adult must be 18 or over) them's the rules kids!\nThere will be a maximum limit of two children per adult - you must all stay together for the entire run.\nNo showers on site, bring a change of clothes and a towel, you may (will) need both!\nEntry to site will close at 11:30 so you have plenty of time to get parked, changed and attend the race briefing at 11:50 \u2013 the race will start at 12:00 even if you are not there.\nBring Inflatable friends, they are all welcome - crocodiles, sheep, rubber rings, hammers and the like.\nThere will be refreshments at the end, bring some money for hot food & drinks.\nHERE IS A LINK TO THE RACE HQ MAP - HEAD FOR CLOUDS HILL - King George V Rd, Bovington, Dorset BH20 7NQ - CONTINUE ON THAT ROAD UNTIL YOU REACH THE NOISE!\nPlease confirm you would like to be added to the The Little Dirty Devil - 5k Stampede - Fun Run waiting list.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzanoch b/data_all_eng_slimpj/shuffled/split2/finalzzzanoch
new file mode 100644
index 0000000000000000000000000000000000000000..3e77a05d3ea4e6fb30b80078df8505002f6a5950
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzanoch
@@ -0,0 +1,5 @@
+{"text":"On time, pleasant and well looked after from being chauffeured from Croatia to Venice. Even after accidentally leaving a wallet in the car, they ensured it was returned (yes money and all). Very comfortable recommending this business.\nFantastic service!!! The drivers were friendly and on time. We never had to wait even a minute. They helped us with our luggage and helped us get in and out of the water taxis safely. The cars and water taxis were very clean as well. From the very beginning, this company has been very helpful. I booked this service with the help of Marco who works for this company. Marco was very helpful and very responsive even when it was late at night in Venice. The drivers were all wonderful. We would definitely recommend this to others. This was a lifesaver for my family of 11 people!\nThis was the best way to arrive after flying all night to Venice from Philadelphia. The driver picked us up from the airport with a sign with our name on it, took us to an air conditioned private Mercedes van, that drove us to the port, our private vaperetto was waiting for us with captain! Dropped us off at our hotel in Venice. Easy way to arrive with nothing to think about after being tired from the flight. Fun and exciting for the entire family!! Thank you!\nThe absolute BEST way to get from the Venice Cruise Terminal to Marco Polo Airport is BookTaxiVenice. It took me 5 minutes to book on their website the day before embarkation using PayPal to pre-pay. The next morning our party of 4 was met by a very professional, pleasant driver who was waiting with my name on a sign just inside the terminal door. A few short steps outside was a spotlessly clean, air conditioned Mercedes van. Our heavy luggage was whisked quickly into the back and 20 minutes later we were at the airport checking in our bags. It was an excellent value for the price.\nI just wanted to say that was a great service you provided for us today and last Friday. Both water taxis were on time and both taxi drivers were helpful and pleasant; the host greeted us at the airport with our names on a board, exactly as planned, and drove us to the water dock. Then today the same man actually ran to meet us when he saw our motoscafo pulling into a different dock.\nI will recommend you to others I know, and would definitely use you again.\nThe service could not have been better from the booking and payment on line to the actual delivery of the service. We arrived in Venice in torrential rain and an English speaking driver met us at The airport with umbrellas and a van to wisk us and our baggage to the water taxi dock, where a boat ride through a very choppy lagoon delivered us to our hotel. Could not have been better and it was booked only a day or two before.\nMarco your service went seamlessly as was arranged! Your courteous taxi driver promptly met us in the cruise terminal in Venice following our disembarkation and drove us to meet David waiting in our private water taxi for our tour through the Grande Canal in Venice, the islands of Murano, and Burano and concluding at the airport dock only to be met with our taxi driver who dropped us at our hotel door by the airport.\nPerfectly organized from beginning to end! Your prompt response to all my questions and willingness to deliver a custom experience and then delivering exactly what you promised leads to this note of praise and thanks!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Why wait to build?? You will know you are home when you walk through the front door into this beautiful home. This rancher home is set up perfectly with everything you want and need. It sits on a quiet street and offer 4 bedrooms and 3 baths with almost 2600 square feet. Home is nicely updated with fresh paint, updated kitchen with stainless appliances, mature landscaping, fenced backyard, over-sized patio, sprinklers front and back, A\/C, and a three car garage! This home is a must see!!\nDirections:North on Ramsey, West on Dalton, Right on Toulon.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Today is Record Store Day. It's brought back a flood of nostalgia for the greatest record store that ever existed.\nAnyone who remembers 'Ear 'Ere In Lancaster knows what I mean. It was a cosier and friendlier version of the record shop in Nick Hornby's Hi-Fidelity. It was also where you could get tickets for gigs at all kinds of places around the North of England.\nBut from my early forays in there after school, to look meaningfully at prog album covers from the likes of Genesis and King Crimson, to the more serious record buyer I became it was the centre of my world.\nAnd these plastic bags were such a status symbol around school. You'd cart your school books to lessons in an 'Ear 'Ere bag, the height of cool, but woefully impractical for such a purpose.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Can someone help me with this..\nI am trying to create a brush simillar to the one from open canvas 1.0; the paintbrush tool.\nAll I want is some good brushes to pain with in photoshop.\nmake your own. if you can make a shape in photoshop, then you can make a brush.\nI've never done it myself, but it's a command that's something like \"define brush\", if you do a search on it in photoshops help section that should give you all you need to know.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Safe and secure transactions in any environment.\nIt is a unique system designed for both indoor and outdoor events and festivals. Set up is easy. Terminals are portable and process transactions wirelessly.\nNo up\u00adfront costs and training is simple, allowing temporary employees to ring customers quickly, increasing event revenues by 15-25%. Best Ring provides customized menus and item pricing as well as data tracking and reporting of all activities in real\u00adtime.\nWhether it is on-site support or remote, our team is with you 100%. We are on the frontier of point of purchase innovation for festivals and venues. Our team launched cashless at Lollapalooza and Austin City Limits. We know the ins and outs of working a variety of fields and thrive in complicated environments.\nBest Ring, available on a rental basis, has the ability to offer cashless payments using RFID technology in fast paced, environmentally challenging settings. We provide both open loop and closed loop systems that can be used alone or in tandem. Use it with RFID-enabled tags, wristbands, credentials, cards... you name it.\nFrom brick and mortar locations to food trucks to festivals, Best Ring technology is easy to install, easy to use and cost effective. Best Ring provides custom menus, item pricing, data tracking and reporting in real-time.\nThe Best Ring application is intuitive. Employees, no matter how temporary or their experience level, learn menus fast and efficiently. The data you rec is accurate and timely.\nBest Ring Classic is the point of sale solution you know and love. This is the scalable solution for any festival or event. From just one terminal to over a thousand units out in the field at once, Classic has you covered. Click to learn more about Best Ring Classic.\nBest Ring Black is everything you have been looking for in a point of sale solution for you bar, restaurant, coffee shop or venue. It comes packed with all of the amazing features of Best Ring Classic in a stylish Black case. You can pair it with a traditional cash drawer and printer. Or skip the accessories and just rock the terminal alone. Click to learn more about customizing Best Ring Black to work for you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaofyq b/data_all_eng_slimpj/shuffled/split2/finalzzzaofyq
new file mode 100644
index 0000000000000000000000000000000000000000..11fd9d80da18b0dc0fa8bfa06cfbbdf09fa38309
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaofyq
@@ -0,0 +1,5 @@
+{"text":"Discussion in 'Technical Chat' started by bugs, May 10, 2007.\nHas anyone heard a constant whistling sound between 50 and 60 mph? I can hear and my passenger can hear it. both of us swear it comes from each of our sides. I've resigned to believe it's design related but if I can find the source, I'll figure out a fix myself.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dam Rather-NOT is a GROUCH broadcaster that appeared in a season 38 insert featuring WALTER CRANKY & Anderson Cooper. When asked to answer a question, all he says is \"I'd rather NOT\".\nHe is a GROUCH parody of news journalist Dan Rather.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ending Homelessness talking points are offered to all who plan to testify at the Interagency Council on the Homeless hearing on Tuesday, March 25th..\nThe talking points are offered to all who plan to testify at the Interagency Council on the Homeless hearing that will be held on Tuesday, March 25, 2014.\nDue to today's inclement weather tomorrow, the Interagency Council on Homelessness' public forum has been postponed. The hearing will be rescheduled early in the 2014.\nThis is an important hearing and we need to make sure that they hear loud and clear that we can and must prevent and end homelessness in NJ.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u25aa Darl Easton, who heads the Tarrant County Republican Party.\nCalls for these Republicans to be removed as GOP leaders were detailed in several emails Republican Dale Attebery sent to others in the party. Printouts of those messages were anonymously delivered to the Star-Telegram.\nThat email also called for seeking Easton's resignation, and a separate email noted that it's time to remove Canon, who is believed to be the one referred to as a Democrat in the email.\n\"We have become overly lax,\" that email stated.\nThe effort to oust GOP members from leadership positions began earlier this year when Republican Dorrie O'Brien asked for Shafi's appointment to be reconsidered.\nIt ramped up \u2014 and expanded \u2014 after the Nov. 6 election, which saw Tarrant County turning blue in the U.S. Senate race.\nMany in the party oppose the effort to remove Shafi.\n\"If the Republican Party doesn't stand up against those who want to discriminate against others ... it will be the downfall of the Republican Party eventually,\" Abdulkareem said in an email responding to Star-Telegram questions. \"It starts with Dr. Shafi, myself and Kelly Canon. But what happens when next they decide they don't like Catholics or the Methodists?\nAttebery's emails note that the effort includes himself, O'Brien, Sara Legvold and Bret Coody. Each either declined or didn't respond to requests for comment from the Star-Telegram.\nTarrant Republican activist James Scott Trimm criticized Easton because as he appointed and defended Shafi and others, Democrat Beto O'Rourke won the reliably red Tarrant County in the U.S. Senate battle against Republican Ted Cruz. Easton took office last summer.\nTrimm has posted an online petition calling for Easton to resign. As of Tuesday afternoon, it had 32 signatures.\nAnd Trimm stressed that it's not just a \"small group\" that want Shafi removed from office. He personally wants to see him removed, not because of his religion, but because he doesn't think Shafi agrees with the Republican Party's pro-Israel platform.\nEaston said he doesn't plan to step down and he believes that calls to remove Republicans from their posts are wrong.\nAn Oct. 17 email states reasons, such as that Abdulkareem tweeted a picture taken of an apparent anti-Trump arts display at a Fort Worth Arts Festival ... \"with a 'hallelujah' hands emoji for her commentary.\" That tweet apparently drew \"virulently anti-Trump tweets and retweets,\" the email stated.\n\"After seeing these things, would it be a fair, and in fact reasonable, question to ask ourselves what someone with these sentiments is doing in our party?\" Attebery's email read.\nAbdulkareem said she became a target after standing up for Shafi.\n\"I wouldn't change a thing and will never let him stand alone in this fight,\" she said. \"He is a good man and gives a lot of himself to help the party. If they are successful in removing him then there is no need to remove me because I will resign and walk out with him. Many others have pledged to do the same.\nSome say that refers to Canon, who was a Democrat more than a decade ago before joining the GOP. She has been outspoken about her opposition to removing Shafi from office.\n\"Dr. Shafi is a practicing, Mosque-attending muslim who claims not to follow sharia law or know what it is,\" Legvold wrote on the Protect Texas Facebook page in calling for his removal. \"As a practicing muslim that is an overt falsehood. Sharia law is anathema to our Constitution because Islam recognizes no other law but shariah.\nAttebery wrote an email on Nov. 18 stressing that local and national media are now seeking comments about this situation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Hypertension Institute staff is comprised of registered nurses, a nurse practitioner, nutritionist, and physician who is board certified in hypertension by The American Society for Hypertension (ASH) and in Internal Medicine by the American Board of Internal Medicine.\nLearn how to alter lifestyle choices to better manage hypertension.\nIdentical Human Replacement Therapies improve quality of life in millions of men and women worldwide.\nLearn how stress management affects hypertension.\nDiscover diet choices to loose and maintain weight.\nNutraceuticals, vitamins, antioxidants, minerals, and other supplements.\nUtilize exercise to manage weight and hypertension.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaoglu b/data_all_eng_slimpj/shuffled/split2/finalzzzaoglu
new file mode 100644
index 0000000000000000000000000000000000000000..fee8bc19e68ad6b029839ad0c50a9c07b701793a
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaoglu
@@ -0,0 +1,5 @@
+{"text":"Coloring Pages Ideas Bad Case Of Stripes Coloring Page. Fennekin Coloring Pages. Kissing Lips Coloring Pages. Ecuador Flag Coloring Page. Mustache Coloring Pages. Main Greek Gods And Goddesses. Winx Club Coloring Pages. Chinese Coloring Pages. Free Preschool Bible Lessons.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When you visit our website, your surfing behavior can be statistically evaluated. This happens mainly with cookies and with so-called analysis programs. The analysis of your surfing behavior is usually anonymous; the surfing behavior can not be traced back to you. You can object to this analysis or prevent it by not using certain tools. Detailed information can be found below.\nIn the case of violations of data protection law, the person concerned has the right of appeal to the competent supervisory authority. The competent supervisory authority in data protection matters is the Austrian Data Protection Authority, Wickenburggasse 8, 1080 Vienna, dsb@dsb.gv.at .\nYou have the right to free information on your stored personal data, their origin and recipient and the purpose of the data processing and, if necessary, a right to rectification, blocking or deletion of this data. You can contact us at any time under the e-mail address datenschutz@pensionsabine.at as well as for further questions on the subject of personal data.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cubs is the second section of the Scouting movement, originally started in 1916 for younger brothers who wanted a 'look-in'. In nearly a century, the section has constantly evolved and adapted its programme and methods to meet the changing needs of each generation of young people and these days admits girls as well as boys.\nThrough a range of different activities, Cub Scouts will be challenged to learn about themselves while working and learning as a team. It can help develop your child's social skills and independence and give them access to some amazing and memorable opportunities.\nA Pack of Cub Scouts is organised into Sixes, with each Six named after a colour and a Sixer and a Seconder in charge.\nScouting is open to all, regardless of their religious feelings and there are alternative versions of the Promise available for those who would prefer not to promise to 'love God'.\nAll Scouting is led by trained volunteers who use their own time to run and organize all activities. As such, all Cub Packs tend to have their own unique 'personality' and preference for some activities over others. The choice of Pack is up to you although location, meeting time and possible waiting lists may all affect your decision. Additionally, some activities are planned as District (or the larger County) activities where all the Cubs join in the fun together.\n*Where no web link is available, use the Contact form on this website.\nScouts is the UK's biggest mixed youth organisation. We change lives by offering 6 to 25-year-olds fun and challenging activities, unique experiences, everyday adventure and the chance to help others so that we make a positive impact in communities.\nScouts helps children and young adults reach their full potential. Scouts (and adult volunteers) develop skills including teamwork, time management, leadership, initiative, planning, communication, self-motivation, cultural awareness and commitment. We help young people to get jobs, save lives and even change the world.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We will be staying at a new home for this three-night adventure. Located in Granby, the Casa Vista Granby has a whopping 10 bedrooms and 8 bathrooms! Pricing for this trip is $190 per person in a shared room. Singles sharing a room will each have their own bed. This price includes 2 breakfasts and 2 dinners. On a space available basis, some private single rooms may be available for a $60 upgrade fee.\nThis event was added on 9\/12\/2018 by an RMBB member and was last updated on 3\/4\/2019 by an RMBB member.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"JAMESTOWN - Katherine M.\"Katie\" Reardon, age 73, of Jamestown, formerly of Westfield, passed away Saturday, February 2, 2019, at Absolut Care of Westfield following an extended illness.\nKatherine was born on March 26, 1945, in Westfield, NY, the daughter of the late Leon and Althea (Miller) Reardon.\nKatherine graduated from Mayville High School and had been employed for many years at Better Baked Foods in Westfield. She retired from the Chautauqua County Office for the Aging following many years as a Health Care Assistant.\nShe enjoyed collecting recopies, cooking and loom weaving. She adored her 2 Miniature Schnauzer dogs and her several cats that she had through the years..\nShe is survived by her son, Charles Reardon of Jamestown and his two children Ashley and Austin Reardon of Westfield as well as by her 3 brothers, L. Dennis Reardon and Ernest E. Reardon both of Westfield, and George J. Reardon of Rochester, NY and by her sister, Grace L. Lancaster also of Westfield.\nShe was preceded in death by her brother, William Reardon, her sister, Shirley A. Reardon and by her infant sister, Isabel Reardon, as well as by Charles late companion of 15 years, Christina Swanson who died in 2017 and was like a daughter to her.\nFriends may call from 6:00 PM - 8:00 PM on Tuesday, February 5, 2019 at the MASON Funeral Home, 37 Clinton St., Westfield and are invited to attend The Funeral Service there on Wednesday, February 6, 2019 at 12 noon. Burial will be private in Westfield Cemetery.\nMemorial Contributions in Katherine's memory may be made to Chautauqua County Humane Society, 2825 Strunk Road, Jamestown, NY 14701.\nLog on to www.masonfuneral.com to send the Reardon family a message of sympathy, to light a Memorial Candle or for further information.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaoydo b/data_all_eng_slimpj/shuffled/split2/finalzzzaoydo
new file mode 100644
index 0000000000000000000000000000000000000000..75e2209d1a3033d1bc6789a5d719bd9cf589af48
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaoydo
@@ -0,0 +1,5 @@
+{"text":"Don't you know that every perfect life would mean the end of art?\nProgress would be wonderful \u2013 if only it would stop.\nTime, which runs through the world like an endless tinsel thread, seemed to pass through the centre of this room and through the centre of these people and suddenly to pause and petrify, stiff, still and glittering\u2026 and the objects in the room drew a little closer together.\nRobert Musil was an Austrian writer. His unfinished long novel The Man Without Qualities is generally considered to be one of the most important modernist novels. However, this novel has not been widely read because of its delayed publication and also because of the lengthy and intricate plot.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Susan discusses the TV show Resurrection, an American fantasy drama television series about dead people who return to life. It is based on Jason Mott's novel The Returned (2013). The series follows the residents of Arcadia, Missouri, whose lives are upended when their loved ones return from the dead, not having aged since their deaths.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Metro Manila (CNN Philippines, April 5) \u2014 A celebrity and several netizens have initiated a discussion online on who will be the next actress who will portray the iconic superhero role of \"Darna.\"\nAfter Liza Soberano's sudden exit from the film project Thursday due to an injury, Star Cinema has announced that the search for the new main star is on and netizens have scrambled to air their bets on the next choice for the superheroine.\nEven Broadway songbird Lea Salonga gave her take on the search.\n\"So, what would I love to see as Darna? Someone morena or kayumanggi to represent the darker skinned Pinays to let them know they are powerful and beautiful exactly as they are. I'd also love for her to be a relative unknown. Let the story sell the film. That is all,\" Salonga said in a tweet.\nSome netizens agreed saying they would love to see dark-skinned Filipinas such as Nadine Lustre, Angel Locsin, Maja Salvador and Pia Wurtzbach take on the role.\nTwitter users have also supported their bets with accompanying edited visuals to lobby their point.\nThe hashtag #BagongDarna has been a hot topic on Twitter with over one thousand tweets.\n\"Darna\" director Jerrold Tarog denied he would be resigning after this latest development on his project and told netizens to stay tuned while their search for the new \"Darna\" continues.\nSoberano is the third high-profile personality who quit the show after Angel Locsin backed out due to a back injury and director Erik Matti left the project citing \"creative differences.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bangalore University was established in the year 1886 at Bengaluru city in Karnataka State in the Name of Central College later in the year 1964 it was renamed as Bangalore University. Bangalore University is the Public State University, Bangalore University is commonly know as BU. Bangalore University campus is located at Central College campus and Jnana Bharathi campus at Bengaluru City, Bangalore University campus has spread across 1,100 acres and Facilities available at BU campus are Hostel, Canteen, Cricket Ground, Guest House, Health Center, Senate Hall, Jnana Jothi Auditorium, Library.\nBangalore University is Affiliations from University Grants Commission (UGC), Association of Indian Universities (AIU), Association of Commonwealth Universities (ACU), Distance Education Council (DEC) & Accctedited by National Assessment and Accreditation Council (NAAC). At presently Bangalore University is offering various Graduate, Post Graduate & Distance Education courses in Arts, Science, Commerce, Education, Law, Management & Engineering Streams. Bangalore University has 665 affiliated colleges & 3 constituent colleges across Karnataka State.\nwww.bangaloreuniversity.ac.in is the official website of Bangalore University visit the above website to check out latest updates, results, exams timetable, notification regarding Bangalore University.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Explicitly or not, potential employers in most organizations will analyze your personality during job interviews. They may want to know how closely you would fit the company culture, how well you would collaborate with colleagues, or how you would respond to an emergency situation. Trying to figure these things out is hardly a new idea. But, these days, some employers are adding a variety of personality-probing methods ranging from written personality tests and interviews with psychologists to watching you as you carry out a task alongside other candidates. They may even pay attention to how you interact with staff during a lunch break.\nWhile such scrutiny is difficult to fully prepare for\u2014and attempting to prepare may even be counterproductive\u2014an awareness of the situations you may encounter can be very helpful. Also helpful is to give some thought to your personality before it is scrutinized by others. If you do these things, you can turn a potentially daunting part of the selection procedure into something more manageable and comfortable.\nThese personality-probing methods are common in industry. It is less common for interviewers in academia to employ them, but it does happen. The Clinical Innovation Fellowship (CIF) program at the Center for Technology in Medicine and Health near Stockholm recruits teams of fellows with backgrounds in medicine, technology, industrial design, and business to work closely together for 8 months, so they employ a selection process in which the assessment of personality is important. \"We need team players. We also need people that are open to new situations and can show humbleness while at the same time be able to stand for their ideas,\" writes Sjoerd Haasl, CIF's program director, in an e-mail to Science Careers. \"We do not look for one specific personality, but rather for a combination of personalities that we think can excel at different parts of the journey.\"\nThe CIF recruitment process includes a video conference interview with individual candidates, followed by a written Belbin Team Roles test, an exercise where a team solves a task under time pressure, and a one-on-one interview with a psychologist. Lisa Sahlin Torp, a current fellow and a medical doctor, writes in an e-mail that \"the most challenging aspect of the recruitment process was the number of new and unknown interview situations I had to go through.\" She found being observed as part of a team carrying out a task \"exciting, nerve-wracking, and a bit unnatural.\"\nThese are not the only unusual interview settings that scientists may encounter. Some companies, for example, ask candidates to visit so-called assessment centers, where their ability to participate in team exercises, panel interviews, and discussions are assessed over a day or two.\nDon't make the mistake of thinking that the scrutiny is over when the explicit testing ends, or before it begins. \"When you go to the reception, there's every chance the receptionist might be asked later on what they thought of you, and lunch could well be with other members of staff from the organization,\" says Elizabeth Mortimer, a careers adviser at The University of Edinburgh in the United Kingdom.\nAs Torp indicated, such explicit scrutiny can be unnerving\u2014and in a job interview, you need to keep your nerves. So what can you do?\nIt can be helpful to find out in advance what methods the interviewers will employ, especially if there's reason to expect it to be unusual or to last a long time. Recruitment agencies can provide such information, says Victoria Walker, a senior consultant with scientific recruitment agency CK Science in the United Kingdom, and candidates who have applied directly to a company can contact the human resources department and ask about the interview format. \"Companies might be reluctant to give out those details, but it shows good initiative and that you want to prepare yourself as much as possible for that interview.\"\nBeyond inquiring about what to expect, there may not be much you can do. Some information about the testing is likely to be withheld, because putting you in an unexpected situation is an important part of their inquiry into your personality. Once, when Alex Bielak, who is now a self-employed knowledge broker and writer, was leading an ecosystems science group as part of his previous role at Environment Canada, he gave applicants for a chemistry lab manager position a month's notice to prepare a 45-minute presentation of their strategic plan for the lab. But on the day of the interview, he told them they had just 10 minutes to present it, and that they had to pitch it to someone unexpected: a fictitious policymaker. \"The winning candidate was not thrown by the change of plans. They handed us a printed copy of the presentation and summarized it in the allotted time,\" he says. \"By contrast, another candidate took most of their 10 minutes to set up the PowerPoint projector.\"\nLikewise, it's not hard to find lists of questions online that you could be asked in written personality tests, and it makes sense to familiarize yourself with them. But Alfreda James, assistant director of the Career Center at Stony Brook University in New York, insists that preparing your answers in advance is pointless. Employers \"want to know how you are going to perform when things don't go as expected. So don't think you can outsmart the test,\" she says. One way you can prepare is by reading up on the company where you're interviewing, in the financial and business press as well as on the company website, because behavioral questions will have aspects of corporate knowledge and culture embedded in them, James suggests.\nYou could also try to anticipate interview questions that might be uncomfortable. You could be asked, for example, to describe your biggest mistake at work. Mikael Karlsson, who took a break from his Ph.D. studies in microfluidics and diagnostics at the KTH Royal Institute of Technology in Stockholm to do the CIF program, says he was anxious about the program's hour-long interview with a psychologist. The psychologist's personality and professionalism put him at ease, however, and he found he could answer the questions easily, he writes in an e-mail. \"A lot of the discussion touched upon how I felt and behaved in difficult situations, but also which role I have taken in various projects. I was helped by having prepared myself by thinking about difficult situations I had been in recently, such as conflicts, and how I dealt with them.\"\nIt can also help to take time before the interview to think about your positive attributes, even if they do not seem directly related to the job, says Basking Ridge, New Jersey\u2013based psychologist Leslie Becker-Phelps. \"If you feel you have value as a whole person, talking about a particular mess-up or weakness doesn't have to be something that strikes terror in your heart,\" she says. \"Taking full ownership of the strengths you possess\" will also help you feel more confident during the interview.\nYour goal for the interview should be to convey a picture of your personality that is as truthful and complete as possible. The most important piece of advice whenever you are asked a personality-based question\u2014\"What do you think you're good at?\" or \"What ability would you need help developing?\"\u2014is to \"be yourself, but on your best day,\" Mortimer says.\nSometimes, \"people might try to give an answer they think the interviewer is looking for. The fundamental problem with that [approach] is the chances are your personality will be assessed at multiple times and they may pick up on contradictions in the way you are acting and the kind of person you say you are,\" Mortimer says. So, Walker says, \"don't overthink your answers\" in written personality tests. Likewise, \"in group exercises \u2026 try to relax and be natural and honest,\" she adds. \"There's no point in trying to make yourself out to be a leader if that's not where you comfortably sit, and they might not be looking for a leader.\"\nKeeping a good sense of perspective on the day may help you sustain scrutiny. CIF's Torp says that \"our recruitment process was extensive, so I tried to think of each single situation as less important than the big picture that I assumed our employer was looking for.\"\nDespite the many challenges personality assessment brings, there are positive sides to being tested so thoroughly. \"I really felt that I was evaluated and selected when I got the job, which was a great feeling,\" Torp says. \"And if I had not got the job, I would have been many experiences richer when encountering the next challenge.\"\nThe next installment of The Psychology of Interviews series will look into how to handle wacky situations during job interview, and how to regain composure when it all seemingly goes wrong.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaqjtv b/data_all_eng_slimpj/shuffled/split2/finalzzzaqjtv
new file mode 100644
index 0000000000000000000000000000000000000000..8bd7482a81d877d0248a2a8386a4d394144a587a
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaqjtv
@@ -0,0 +1,5 @@
+{"text":"Unclean water supplies and poor sanitation are still amongst the biggest threats to many peoples' health in developing regions. With up to 950 million people worldwide still practicing open defecation and up to 2.5 billion people living without adequate sanitation, vast improvements are yet to be made.\nIn India, 80% of disease in rural areas can be traced back to contaminated water and poor sanitation. The government has responded by promising to provide 60 million homes with sanitary toilets by 2019. However, past attempts to improve sanitation in affected areas, has taught us that simply providing people with sanitary facilities (such as toilets) is not effective in changing deeply ingrained practices (such as open defecation).\nThe best way to initiate sustainable change is to run community-led programs, where village leaders and volunteers are taught to; research what their community needs, understand the dangers of poor sanitation and integrate new systems within their villages from the ground level. Village leaders learn how to approach local governments and to work with them toward providing the infrastructure necessary to make the improvements. They are simultaneously trained to lead educational programs within their communities, that change belief systems and practices to ensure the new facilities are assimilated successfully.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Kate Young writes humorous southern mystery novels. She is a member of Sisters in Crime and the Guppy Chapter. Kate lives in a small town in Georgia with her husband, three kids, and Shih Tzu. When she is not writing her own books, she's reading or cooking.\nLiterary Agent: Dawn Dowdle of Blue Ridge Literary Agency - Click here to visit.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Listen and convert youtube in mp3 the song 17 MYSTERIOUS CRAFTS YOU'LL WANT TO TRY RIGHT NOW. The 17 MYSTERIOUS CRAFTS YOU'LL WANT TO TRY RIGHT NOW lasts for 11:03 minutes. This service is 100% free for people who have a passion for music and entertainment. Please rate this song 17 MYSTERIOUS CRAFTS YOU'LL WANT TO TRY RIGHT NOW and leave us comments. Like this, the artists can receive your feedback.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Logoson is the professional manufacturer of custom metal key chains for customers from different countries. The keychains can be also be soft PVC\/rubber key holder, epoxy key holder, leather key holder, crystal key holder, fabric key holder, etc.\nHere below are more details about this Portugal metal key chain. Welcome to custom in bulk.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We approached Milspec two years ago to see if they could re-engineer our Changemaster safe to make it structurally safer and consistent in quality.\n'We sell this product into every Coles store in Australia and it needs to have minimal down time.\nWithin a short time Milspec took our design and perfected it, improving the structural and electro mechanical integrity of the product.\nFurthermore, Milspec were able to source and provide components much more reliably and quicker than us.\nWe can be sure that each safe that arrives has been thoroughly tested and is built to a level of excellence, meaning we can pass the safe straight to our customer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzarvrp b/data_all_eng_slimpj/shuffled/split2/finalzzzarvrp
new file mode 100644
index 0000000000000000000000000000000000000000..2b3ae20802f0178e8dd5455055d6d7f143d9dbb5
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzarvrp
@@ -0,0 +1,5 @@
+{"text":"A college professor shared this great article that I want to pass on: To find Hillary Clinton likable, we must learn to view women as complex beings. The author begins by talking about many of the great characters in pop culture and literature, characters who weren't always likable and sometimes were complete villains. But we were interested in them, because they were complex. And by the way, they are predominantly white men. The author says we've been \"trained to emphasize with white men,\" whether we realize it or not.\nIt reminded me of a recent complex, and not always likable character, Jessica Jones (title character of the Netflix series). Jessica is a private investigator with super powers. But she's also a sexual assault victim, someone who was continuously victimized by her ex-boyfriend, who was able to make her stay and do horrible things with his own super powers (mind control). We meet her after she has gotten away from him, when she has become an alcoholic, violent, and tactless. I loved the show, and Jessica, right away. Finally, I thought, a victim who doesn't care about being likeable. Who doesn't have to be so sweet and wonderful that of course we all sympathize with her, that we all ask how something so terrible could happen to her.\nNo, Jessica would say, f*** that. I don't care if you like me. I don't care if you think I deserved or didn't deserve what happened to me. Because at the end of the day, regardless of what I'm like as a human being, I didn't deserve to have my free will taken from me.\nBut I spoke to many people who couldn't get into Jessica Jones, or found themselves preferring other characters, because Jessica was so unlikable. And it's true, sometimes I would watch Jessica react to a situation and think she should have reacted differently - if she wanted things to go a certain way, that is. But I loved the character, perhaps because she wasn't always likable, and contradicted the stereotypical victim I have seen far too often. She was not a stereotype. She was a real person, with flaws. She was complex.\nI try to wrap my head around the fact that Hillary Clinton is on one hand the most qualified human being to ever run for president of the United States, and, on the other, one of the most disliked presidential candidates of all time. In fact, Donald Trump is the only candidate who is more disliked than Clinton. And he's not only overtly racist, sexist, and Islamophobic, but also unfit and unprepared for office. How can these two fundamentally dissimilar politicians possibly be considered bedfellows when it comes to popular opinion?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Entry filed under: - Klein -, Entrepreneurship, New Institutional Economics, Theory of the Firm.\nOpportunity is passive. You have to activate things to test whether your delusion of an idea means anything. Judgment is too reflective a term, implying thinking, contemplating, and weighing evidence before doing something.\nEntrepreneurs define really dirty hypotheses and figure out a way to test them, even before they've really defined them, actually. And when the experiment blows up in their face, and yet they still experience pleasure, they just somehow decide not to leave it at that.\nThanks Bart. As we discuss in our introduction, the term \"judgment\" can be misleading because, in everyday usage, it connotes exactly the behaviors you describe \u2014 thinking, contemplating, weighing evidence. Judgment per se, as Knight (and we) use the term, does not require this. It is simply decision-making under uncertainty in the absence of a formal model or decision rule. Other words might be intuition, instinct, or gut feeling. We choose to retain the word \"judgment,\" following Knight and Mises, while recognizing this potential source of confusion. As we explain in the essay, there are many possible underlying cognitive and behavioral process consistent with judgment as we use the term.\nThere are decision rules, or rather, the rule is: why haven't you made a decision and done something yet!\nI really, really like the article by Heath and Tversky (1991) on ambiguity and belief http:\/\/link.springer.com\/article\/10.1007\/BF00057884 It's the closest explanation I've seen.\nA pity that has not been taken into the realm of entrepreneurial decision making. At least, not to my knowledge. I've been the papers for quite a while.\nRefinement is always good but discovery without judgment is as unsettling as judgment without discovery when thinking about entrepreneurship. Both are implied. Sometimes entrepreneurship is mostly discovery \u2013 for example, pure entrepreneurship \u2013 while at other times judgment is the key to bringing entrepreneurship to fruition. Entrepreneurial application of capital appears to be mostly an act of judgment however without appreciating the discovery component it seems that the neoclassical mentality of the fictitious 'economic man' (Homo oeconomicus) is being preferred. If both discovery and judgment cannot be seen as interconnected and intermingled then it seems that these must be viewed as attempts to make empirical; out of what are subjective human beings.\nI think, drawing on Mises, that any action requires some sort of contemplation or reflection to be considered an action, rather than a behavior. So insofar as entrepreneurship is an action, it requires some component of judgment, although this judgment may be implicit or not well-defined.\nAnd again drawing on Mises (but c.f. Alfred Schutz in Phenomenology of the Social World) the idea of an existing \"profit opportunity\" which we discover becomes at best redundant when you realize that profit is a mental phenomenon. It's not that people don't discover novel information about the world which they exploit, but rather that that exploitation itself requires a mental construct which is essential to understanding the whole process.\nOf course, I can't figure out how to get to the full text of any of these articles, so I could be way off\u2026 Am I an idiot, or is any one else having this problem? Help me, Dr. Klein!\nUnfortunately they are behind the publisher's firewall, so a JOIE subscription is required. But send me an email and I'll be happy to send you a private copy.\nA membership in WINIR (http:\/\/winir.org\/) gets you online access to JOIE!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Wiccans love Halloween. Wiccans celebrate this holiday, called Samhain, as one of the yearly Sabbats of the Wheel of the Year. Wiccans believe that Halloween is the most important of the four greatest Sabbats.\nThe tradition starts when the sun sets. Wiccans pay respect to family members and friends who are deceased. It is a celebration of their loved ones who have died and in fact, their spirits are welome to join in such celebration.\nHalloween or Samhain is known as being on one point of the wheel which celebrates the spirit of darkness and the dead. At the opposite end of the wheel is another Sabbat Wiccans celebrate which is the spring festival of Beltane. Samhain celebrates darkness while Beltane celebrates light and fertility.\nWiccans see Samhain as a night when there is a door to the other world opened to them. They can celebrate with the dead, and communicate with the dead. Because Samhain falls in late October to November, it is a holiday that marks the end of the harvest season and the beginning of the darker, winter half of the year.\nHave a safe Halloween and remember that the door is open to the other world. You really can communicate with your loved ones that have passed on. Celebrate, dance and most of all be safe on Samhain.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Presenting better with a minimum of preparation.\nEvery presentation is an opportunity! A possibility to put yourself and your product or service on the map. Therefore, an average performance is not an option. At Speech Academy, you will learn how to give a presentation in a simple and relaxed way within two days. This coaching program is suitable when you are not a top speaker, YET.\nDisadvantageous speaking stress is transformed to healthy excitement. A presentation is no longer something to run away from. You know how to make contact with your audience and how to use your voice well. Public speaking becomes an opportunity that you really look forward to.\nYou will learn to use techniques through which the right words and sentence structure come to you naturally. You will do exercises that will sharpen your senses in order for you to be able to interact with your listeners. You will learn to master the art and simplicity in two days.\nTogether with others, you will personally experience the ease and fun of giving presentations. The training program is exciting from beginning to end \u2013 active, inspiring, confronting, revealing, and educational. There's always a relaxed and humorous atmosphere.\nThe fast training 'Public Speaking' program is effective for every speaking goal, whether it is about the presentation of a new product, annual figures, a plea, coaching program, farewell speech, press conference, sermon, speech, announcement, or opening speech.\nDo you want to give an enthusiastic, relaxed, and fascinating speech or presentation nearly unprepared? The 'Public Speaking' training program is an effective way to make a successful impression.\nThanks to our sophisticated method, your learning process will be accelerated and you will remember everything you learn. You will learn to use techniques so the right words and sentence structure will come naturally.\nExcluding 21% VAT and including coffee, tea, snacks, and an extensive lunch.\nA minimum of 6 and a maximum of 12 people.\nReceive a 10% discount for subscribing 2 or more people.\nOur training programs require an investment in time and money. We will match that with more than 20 years of experience, service, personal attention, and results. Your presentations will improve measurably.\nPresenting well also means that you present with conviction and that you exploit the full potential of your voice. You will learn to display a natural charisma and be able to even present bare facts and figures in a fascinating way. 'Presenting well' is coaching at a top level and always receives the highest rating.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The aerospace industry faces system design challenges posed by the use of commercial-off-the-shelf (COTS) electronics items in aerospace vehicles. Designers in both commercial and military segments of the aerospace market are increasingly dependent on COTS components, sub-assemblies, and in some cases even systems.\nThe COTS electronics industry is characterized by relentless market pressure to increasingly expand and improve functions, reduce costs, and reduce design and development time. As a result, product development is driven by forces that often are not in the best interests of aerospace users of COTS products. The aerospace industry is such a small part of the COTS market that it is difficult to counter these forces.\nEvaluating challenges posed by the use of COTS electronics items and mitigating their effects is a continuing topic of AVSI research.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzatael b/data_all_eng_slimpj/shuffled/split2/finalzzzatael
new file mode 100644
index 0000000000000000000000000000000000000000..cc0dc607298ae99ef43f2edefda303f9eaa8f2ea
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzatael
@@ -0,0 +1,5 @@
+{"text":"Grief before death, understanding anticipatory grief. When thinking about grief, we mostly associate it with the sinking feeling we experience once a close friend or relative has passed away. However, the range of emotions we go through when losing a loved one can be quite complex and it takes many forms \u2013 including anticipatory grief. From the term, this might sound like regular grief experienced earlier, but it's so much more than that.\nThis form of complicated grief encompasses the feeling of impending loss we experience when we learn that someone we care about will soon pass away, combined with shock, disbelief, fear and loneliness. Unlike normal grieving, whose symptoms are known to most people, anticipatory grief is not so discussed, it manifests in more subtle ways, and learning how to recognize it is key if we want to enjoy the time we have left with our loved one.\nWhen does anticipatory grief occur?\nAnticipatory grief starts to settle in as soon as we learn that someone we care about will soon pass. It can be experienced when an elderly relative can no longer look after themselves and we realize that we will soon have to deal with their death. Or, it can be experienced when a loved one was diagnosed with a terminal illness and has little time left to live.\nNot everyone who learns that a loved one will pass away experiences anticipatory grief and, if they do, this does not mean that they will be dealing with death in a different way or that the grieving period will be shorter. The term only refers to the range of emotions you experience prior to the passing and it may influence your relationship with that person.\nGuilt \u2013 knowing that someone will soon pass away can trigger feelings of regret for things that happened in the past, for not being the best spouse\/friend\/child.\nIncreasing concern for the dying person, feeling worried all the time, visualizing how they will pass away and how your life will be impacted by this event.\nMake the most of the time you have left. Be there for your loved one, make every encounter meaningful and think of all the good times you had together. As tempting as it may be to focus only on the practical aspects of a future death, take the time to talk to them, look back through old photo albums and simply be there.\nTalk about your feelings. Whether you choose to open up to a friend, spouse or go to grief counselling, don't bottle up your feelings, because this can take a toll on your mental health.\nClose loose ends. Let pride aside and talk to your loved one about what they mean to you. If they are still in a good enough mental state, talk about things that you've been holding on to for years or even about what they wish you to do after their death.\nTake care of yourself. As difficult it might be to think of anything else other than your loved one, don't neglect your own wellbeing. Eat healthy food, try to stick to a regular sleeping schedule and ask your family and friends for help so that you have some time left for yourself too.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Insurance, Personal injury protection benefits. Practice, Civil, Directed verdict; Burden of proof.\nSteven C. Malone for the defendant.\nSWAN, J. For the fifth time in twelve months, we review a case where the trial judge absolved the plaintiff, Advanced Spine Centers, Inc., of its burden of proof and directed a verdict in the plaintiff's favor. For the fifth time, we reverse.\nThe facts as alleged are briefly summarized. [Note 2] On December 22, 2006, Chanserey Srey (\"Chanserey\") [Note 3] was injured in a car accident. The car, which Chanserey was driving, was owned by his father, Sarith Srey (\"Sarith\"), and insured by Arbella Mutual Insurance Company (\"Arbella\"). On January 15, 2007, Arbella received from Chanserey a completed application for Personal Injury Protection (\"PIP\") benefits and an affidavit of no health insurance. The insurance policy issued by Arbella provided that PIP benefits would be paid to \"[Sarith], or any other person, if injured while occupying [Sarith's] auto with [Sarith's] consent.\" In the weeks following the accident, Arbella conducted examinations under oath of Chanserey and Sarith, in which it was revealed that Chanserey did not have permission from Sarith to operate the car. In the examination, Chanserey stated that an unknown vehicle hit his car, causing him to lose control and crash into a fence and telephone pole. The police report, however, indicated that no other vehicle was involved. On March 7, 2007, Arbella notified Chanserey in writing that the claim for PIP benefits was being denied on the grounds that Chanserey did not have Sarith's consent to drive the car, and that the accident did not occur as Chanserey stated. Advanced Spine Centers, Inc., doing business as First Spine and Rehab (\"Advanced Spine\"), provided chiropractic services to Chanserey, and, as an unpaid party, commenced this action to collect PIP benefits for those services under G.L.c. 90, \u00a734M, and damages pursuant to G.L.c. 93 and c. 93A.\nIn its motion in limine, Advanced Spine also argued that Arbella, in its pleadings and pretrial memorandum, had admitted to the elements of the plaintiff's case. \"Proof of [the unpaid party's] claim required evidence that (1) an accident occurred, (2) [the injured party's] injuries and his related treatment by [the unpaid party] were causally related to the accident, (3) a Massachusetts automobile liability policy issued by Arbella and covering the vehicle involved in the accident was in effect on the accident date, and (4) there are unpaid amounts due to [the unpaid party] for the treatment he provided.\" Provenzano v. Arbella Mut. Ins. Co., 2007 Mass. App. Div. 46 , 47. There is no real dispute that Arbella issued a policy. But nowhere did it admit to treatment or an accident. Indeed, it referred in its pretrial memorandum to \"treatment plaintiff allegedly rendered to Chanserey Srey (patient) arising from an alleged motor vehicle accident of December 22, 2006\" (emphasis added).\nFinally, Advanced Spine finds significance in Arbella's answer denying paragraph 12 of the complaint, which stated that Arbella \"was obliged to promptly submit the plaintiff's bills and records to a medical practitioner licensed under the same section of G.L.c. 112 as the plaintiff for review as a predicate to refusing to pay the bills in full, but in breach of said obligation, the defendant did not do so. And, the breach of said obligation is a regular business practice of the defendant.\" Arbella's denial, argues Advanced Spine, coupled with Arbella's later affirmative defense that the \"insurer's determination to pay reasonable fees for necessary medical treatment may not be the basis for a subsequent G.L.c. 93A or c. 176D claim,\" somehow constitutes an admission that the bills and records had been received, presented for review, and then not paid. As complaint paragraph 12 alleges a brace of misdeeds, namely Arbella's breach of its statutory obligation and its regular practice of breaching that duty, paragraph 12 \"reasonably elicited a denial of the entire averment.\" Lynn Physical Therapy, Inc. v. Commerce Ins. Co., 2011 Mass. App. Div. 107 , 109. Arbella's denial was an admission of nothing.\nThe case must be returned to the trial court so that a full trial on the merits may be had, Advanced Spine may be called to meet its burden of proof, and Arbella may present its defense of nonpermissive use.\nAccordingly, judgment for the plaintiff is vacated, and the case is returned to the Lowell District Court for a new trial on all counts.\n[Note 2] The trial was truncated, so the facts summarized have been gleaned from affidavits, pleadings, and other materials in the record appendix.\n[Note 3] Because of the commonality of family names, the first names of Chanserey Srey and his father, Sarith Srey, are herein used.\n[Note 4] Inexplicably, each offer of proof is entered on the transcript parenthetically as \"(Discussion off the record.).\"\n[Note 5] At the beginning of trial, Advanced Spine's counsel stated a likelihood of waiving its G.L.c. 93 and c. 93A claims. The record reflects no waiver, but those claims were, in any event, not tried.\n[Note 6] Due to the trial judge's subsequent disability, a different judge assessed the damages and attorney's fees on the basis of the trial judge's directed verdict.\n[Note 7] Speculating about what should have been disclosed, Advanced Spine argued that, in Arbella's answer, one of the many affirmative defenses raised was that the PIP benefits had been exhausted under G.L.c. 90, \u00a734A, and that Arbella was required to disclose the payments made to the injured party's medical providers. The defense was clearly raised to comply with Mass. R. Civ. P., Rule 12 (b), to prevent a waiver should the defense prove viable. The fact that exhaustion of benefits was not mentioned in the pretrial memorandum indicates that the defense was not being pursued. Arbella was under no obligation to make disclosures of a nonexistent affirmative defense.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The new category with Bob the Builder games are among the most beautiful and also the most played games on Friv-Games.com, which is why we want to present your first main characters Nole will play Bob the Builder games. First the game begins with the protagonist is Bob herself, and is a true professional in the construction, which is why he will let imagination when craftsmen come to you and try to build all sorts of things, houses , buildings and various other structures with master Manole. One of his mobile phone accessories is that it uses very often and with which manages to keep in touch with business people she has around him when put to build something special and unseen. Both the Bureau's work papers, tax documents and orders is made by Bob and help him is Wendy, a true friend and a skilled worker is the post of secretary in Bob's company. Granjero Pickles - Mura ?? tourists farmer is a farmer who manages to help Bob in his chores around the house to. Obviously the cartoons that I watched all the Nickelodeon we saw various agricultural machinery and other animals and people who shoulder the good work, but among these machines and machines I want to mention: Scoop is the leader that is the function of all the machines backhoe and with it coordinates the entire activity of machines, his color is yellow. Dizzy is a blender, a shaker orange with red and he was always very curious and trying to solve all the missions they participate. Muck is a red caterpillar with an extra car, which is afraid of the dark, but it does its job as it should in the sector that Bob the builder to solve for it. Elevated magnet is an electric crane and lift the cup before, to tow and push Bob manufacturer of construction tools. Roley is a compressor whose power is green air compression. Travis is a tractor to tow trailers turquoise that makes large tractors with speed and power through the yard full of farm machines. Pilchards is the spoiled cat who always has something to comment ahead of Bob the constructor. Scrufty is friends with Roley blue bird and a crow Squawk is always bothering you on Spud. Tommy, Hamish and Fin are other friends who are portrayed Bob's a turtle, a talking parrot and a mascot. Well, these are the main characters that you Friv-Games.com team has presented a brief description before you begin the game with Bob the Builder also remember that all of these games you are offered Bob the Builder free way for all children and fans of these characters on nickelodeon junior special bonus game!\nYOU KNOW THIS ABOUT Bob the Builder Games?\nIn Bob the Builder Games page you can find 27 games that we've published for these kids and children that are interested to play Bob the Builder Games, this game category have been created on 22.06.2015. These games with Bob the Builder Games have been voted a lot of times and we are inviting you to let a comment after each of this game.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"7th Love and Respect Kauai location!! Jan 2014 | Love and Respect Kauai empowers marriages.\nJoining the Love & Respect Kauai team with the 7th (God's favorite number) location on Kauai is Pastor Rocky Sasaki. He will be leading the L&R series at the Kauai Community Correctional Center for the men, on Tuesday nights, starting on Jan. 21, 2014.\nThis entry was posted in Uncategorized and tagged Love and Respect Kauai class schedules, Love and Respect Kauai Host couples. Bookmark the permalink.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"A Moon Shaped Pool\" is the title of Radiohead's new album, available digitally now. The album was preceded by another new track \"Daydreaming\", just days after the first single \"Burn the Witch\" was released.\nCD and vinyl versions will follow on June 17, with a very special limited edition following in September, which will include unreleased tracks, 32 pages of artwork and a piece of master tape from one of the bands actual recording sessions, dating from \"Kid A\" to the new album.\n\"A Moon Shaped Pool\" is yet another new big release that's not available via Spotify; it's streaming on Apple Music and Tidal.\nThe band will tour worldwide, starting later this month, including 3 dates in London's Roundhouse between the 26th to the 28th of May.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzatuce b/data_all_eng_slimpj/shuffled/split2/finalzzzatuce
new file mode 100644
index 0000000000000000000000000000000000000000..124973493f4dae06f3e132e5e9b64009169abfa6
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzatuce
@@ -0,0 +1,5 @@
+{"text":"Destiny! A big word today.Everyone is seeking to fulfil their destiny.Very glorious and ethereal. Can we seek it and find it if we are unsure of it? What is it? destiny as it relates to the believer in God is that course and ultimate destination where God has predetermined you to be. Some would say that the course you follow is as predetermined and predestined as your final outcome. Let us say for example, that like Jeremiah, you are predestined to be a prophet unto the nations. The pre means before and in this case before his birth, Jeremiah was slotted for this outcome. Of course it happened; but God speaks to him as he is old enough to understand.To verify to him that this prophecy is real, the Lord tells him what will happen, and it does. Abraham was another that was predetermined to be literally THE FATHER OF MANY NATIONS. He was the root of the Hebrew nation, and with Ishmael, the father of the Arab nations. He is also the father of three religions; the Hebrew, the Christian and the Muslim. Abraham made mistakes; both in his time wasting in Egypt and Harran, His lying under pressure, but his most telling mistake, the one with the longest lasting consequences is the one which caused him to sire Ishmael at the request of his own wife, Sarah. Certainly, God did not predestine Abe's mistakes did he?Or did He allow the mistakes to achieve His own mysterious end?Jesus is the greatest example of predetermined destiny, having about 300+ prophecies regaurding Him, his life and mission, certain descriptive events , even His cruxifiction. Could Jesus have been tempted by the devil to wander off God's path for Him in the desert? Could He have summoned twelve legions of angels to free him from going to the cross? I suppose He could have.You see, even though God, perhaps through His foreknowledge alone predetermines a persons destiny, the possibility exists that the subject can cancel or miss his\/her own destiny by his own choices and failures to be prayerful and proactive in his\/her life.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hostal wincayaren this set as Chilean lodge, its construction is done in almost entirely of recycled native timber with over 120 years materials are sustainable and are certified with the seal S work with fair trade, we produce some of our products, and we generate 30% of clean energy through solar panels and are firmly committed to the environment.\nA perfect place to stay for work or leisure!\nRunned by the whole family, clean and cozy rooms and a great breakfast with local products! Muchas gracias por la attencion y el vino por parar mi lagrimas :) Los dos duenos sonos personas muy muy amable!!!\nMy stay at Wincayaren was Very positive. The people were friendly and professional. The room was nicely decorated and comfortable. And also The breakfast was fresh and very tasty! Definitely recommended if you travel through Linares or The Maule region..!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nike Air Force 1 High \"Doin' It In The Park\"\nIf it's two things that go together like white on rice it'd have to be Bobbito Garcia and the Air Force 1. The Author, Baller and DJ created a series of Air Force 1s around the 25th anniversary of the 82\u2032 classic, and now he's back with another Joint with some Laser flavor. Inspired by his recent Doin it in the Park movie, Bobbito and Nike create an Air Force 1 High featuring smooth and tumbled leather uppers, while \"Doin it in the Park\" is lasered on the heel. Another extra detail that's sure to catch anyone's eye, is the Filipino Sun logo that was pulled from the flag which is also lasered on the heel. Check out the entire shoe and let us know if you'd pick these up!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On the market for a cheap phone with touch screen and QWERTY keyboard? The LG Neon II GW370 might be the right choice. This follow-up to the original LG Neon features more convenient slide-out QWERTY keyboard for quick and easy texting, adds 3G data capability and GPS navigation. The phone is unlocked which means that it will work with various GSM networks, such as AT&T, T-Mobile or Straight Talk. The device is equipped with 2.4-inch touch screen display, stereo Bluetooth, microSD slot for memory expansion and 2-megapixel camera with video capture. Released in 2010 on AT&T network, the unlocked LG Neon II can be yours today for cheap.\nBuying phones released in 2010 is a good alternative when you need a texting phone with touch screen at a cheap price. Also buying used phones is a great choice to get a phone at the cheapest possibly cost. However, be aware. Buying cheap used phones online may be sometimes risky. You know, every phone has its pros and cons, cheap phones especially cons. Here are some tips to help you find an used phone at great value, avoid unfortunate surprises and buy it with confidence.\nDeal on LG Neon II \u2013 Why is so cheap?\nHow to buy this deal - Here is how it works. Learn step by step how to buy LG Neon II GW370.\nLG Neon II is a feature phone supporting data services such as emails or GPS navigation.\nRelease Date \u2013 2010, what makes the LG Neon II an older model among newer messaging phones.\nLG has released already newer texting phones featuring QWERTY keyboard with new look and upgraded specs.\nDiscontinued model by AT&T, is now offered only by third party retailers.\nSince the phone support data services your carrier may require purchasing data plan.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As an NHS Foundation Trust we have a number of important responsibilities regarding the sharing and management of information.\nThis includes a duty to be open and transparent about how the performance of our organisation and our finances, and strict requirements about the confidentiality of patient information.\nNotice to all Yeovil District Hospital NHS Foundation Trust potential and actual business partners, contractors, service providers, patients and staff.\nYeovil District Hospital NHS Foundation Trust (YDH) has a zero tolerance approach to fraud, bribery and corruption, and is committed to maintaining the highest values of probity and accountability. The Trust is committed to the prevention of fraud within the Trust, and the rigorous investigation of any such allegations. All concerns of fraud will be thoroughly investigated and appropriate action taken to uphold the standards of the Trust. The Trust wholly endorses the NHS Counter Fraud Authority strategy towards the elimination of any fraud within the NHS, and will seek to apply appropriate criminal, disciplinary, regulatory and civil sanctions against fraudsters and where possible will attempt to recover any identified losses.\nThe Trust wishes to encourage anyone that has reasonable suspicions of fraud to report them. All employees, patients and contractors can be confident that they will not suffer in any way as a result of reporting reasonably held suspicions of fraud.\nThe protection of patient data is very important to the hospital and we take every care that it is not passed to persons not entitled to see it. You can view our Patient Privacy Notice here. If you would like to know more please contact the Data Protection Officer. You have the right to view you health records and an application form is available from the PALS Officer (01935 384706) or Medical Records Manager.\nPatient confidentiality means that the use of any form of camera or other recording equipment is strictly forbidden without the express permission of staff in advance. The Trust has a duty to ensure the protection of data held on staff. This will not be routinely released without consent although some data is shared with Government Departments to monitor possible fraud. It is the responsibility of managers to ensure data held on staff is accurate and up to date and only held for the purpose intended.\nOrganisational information, structures, locations and contacts. Biographies of our Executive Team and Board members, information about the legal basis for our activities, and information about how to contact us is included in this section.\nFinancial information relating to projected and actual income and expenditure, procurement, contracts and financial audit. Our financial statements and budgets, and information about our pay and grading structure is included in this section.\nWhat our priorities are and how are we doing.\nInformation about our strategies and plans, performance indicators, audits, inspections and reviews. Our corporate plan and annual reports are included in this section.\nDecision making processes and records of decisions. Public consultations, minutes of senior level meetings and internal guidance are included in this section.\nInformation held in registers required by law and other lists and registers relating to our functions. This section includes our disclosure log and the Register of Data Controllers.\nAdvice and guidance, booklets, leaflets, newsletters and media releases. A description of the services we offer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzauasn b/data_all_eng_slimpj/shuffled/split2/finalzzzauasn
new file mode 100644
index 0000000000000000000000000000000000000000..2861e9453a35e35655623dcddabbe2321f61af99
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzauasn
@@ -0,0 +1,5 @@
+{"text":"ENGLEWOOD, Colo. \u2014 In the span of three days, the Super Bowl champion Broncos lost their top two quarterbacks, with Peyton Manning retiring and Brock Osweiler leaving for the Houston Texans.\nOsweiler agreed to a four-year, $72 million contract with $37 million guaranteed instead of becoming the successor to Manning. The Broncos reportedly offered a four-year $64 million contract with $32 million guaranteed.\nBroncos players took to Twitter to offer their congratulations for their former teammate, who they could meet in Week 1 because the Texans will play in Denver this season.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A \u20ac22 million bridge is needed to allow for the construction of 2000 homes on the derelict Irish Glass Bottle site [in Ringsend, Dublin]. [\u2026] The necessary bridge over the River Dodder needed to make the site viable will have a lifting mechanism to enable ship traffic into the Grand Canal basin and the Liffey.\nThe Department of the Environment etc thinks \u20ac22 million is too much and would make the houses too dear; Green Party leader Eamon Ryan TD thinks the state should pay for it, presumably to facilitate all the motorists who might want to live on the site.\nThe site is here. There's an aerial photo here. Here's the Google version.\nI can't see why a lifting bridge over the Dodder is needed, unless the plan is to run traffic along a new route from Britain Quay to York Road, which would simply jam up the city centre. Can anyone explain what this is about?\nThis entry was posted in Ashore, Canals, Economic activities, Engineering and construction, Extant waterways, Ireland, Modern matters, Operations, Roads, waterways and tagged basin, docks, Dodder, Dublin, Grand Canal, Irish Glass Bottle, Ringsend. Bookmark the permalink.\nBrian. It is interesting to see where the factory site was, but i am equally baffled, because in the period 1940s to late 1970s the Irish Glass Bottle Co had a wharf on the Grand Canal Dock, right opposite the Dublin Gas Works, i always thought the factory was alongside the wharf, they unloaded silca sand at the wharf, but by the looks of it, the factory was further over towards the Pigeon House. If this really was so, then the sand would have had to go to the plant by road, from the wharf. Irish Glass was one of the very few firms to have used a diesel rail crane with a grab to unload the sand, which in photos i hold was taken of the ship and fed to a hopper feeding a conveyor belt, perhaps they conveyor then fed the plant some one 1\/2 mile away, i am unsure on that point given what you show. Regards. Andrew.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Trashing the landscape: When It's a Good Thing (Big Rapids) \u2014 Michigan Master Gardener Association, Inc.\nAs part of the Festival of the Arts renowned garden designer and author Janet Macunovich will give a Power Point presentation on Trashing the Landscape When it's a Good Thing!\nYou can incorporate recycled materials, found items and even yard waste as you create a top-notch landscape. Here are ideas from projects by professional gardener-writer Janet Macunovich covering everything from soil preparation to fencing, gates, paths, artwork, containers and the plants themselves. Put these ideas to work to make your landscape unique as well as more economical and environmentally friendly.\nSpeaker: Janet Macunovich, professional gardener, garden designer, columnist (The Detroit News, Michigan Gardener magazine, What's Coming Up e-zine), featured columnist Better Homes and Gardens, author, founder and instructor Michigan School of Gardening, Radio Host (WXYT, WYUR) and MSUE Master Gardener.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Did I really just type that for a title, wow, sorry about that. Well I've got a bag of Converse goodies last night, nothing like loving Parents stopping by the Converse stores in Las Vegas and Primm, Nevada. I might get to them next week, say like Wednesday. However, they weren't going to find anything that would help me out with today's post. So since I am back home right now and close to my supply, I thought I'd dig into my own collection for a change.\nRight off the bat let me tell you that my camera skills have certainly waned from all of those film classes I took. I did also rush, but I am not about making excuses...yet. So I am not sure how this will look on screen, well that's a lie because I will see them before this gets posted. Anyway let's get to the goods.\nHere is a side shot of the Jogger. When I bought these on eBay a few years ago they were advertised as being from the 70's, in fact I think he said from 1977. As you will see from the box shot below that they were produced during the Eltra period.\nThe heel tap is simple and just proclaims Converse. Sometimes this works really well and I think this is one of those times. However, I can't help but wonder want this would look like with the Star in a Square logo. Also you notice the rest of the heel area is in suede. In the another picture that you will see a close up shot of the toe cap. The outline of the toe cap is also in suede.\nThe rest of the shoe is almost like a nylon and satin mix. Of course I could be completely wrong about that. However, I do know that the Chevron and Star's are screened on. I am still a little surprised by this and can't think of another Converse shoe I've seen with screened logos.\nWell the outsole was pleasantly unexpected and pretty unique. It's all molded with ovals in alternating directions. One thing that stood out to me was the lack of branding on the outsole, a rare feature for any athletic shoe.\nThis is probably the best shot of the whole bunch. It shows a great profile view of the shoe. The nylon, stitching, suede, and gum outsole really come together in this picture.\nFinally, here are some pictures of Eltra era box.\nWell that will be it for today. I haven't worn these yet but I am thinking it might be time. Anyway I know the colorway is a little off, but it's vintage so I am not going to complain. Thanks for stopping by, don't forget to click a link on your way out. Enjoy the weekend.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Back in October 2015, Microsoft released an important update to its iOS Bing app. Although you could check Bing's Knowledge and Action Graph for any kind of information you wanted, Microsoft realized that searching on mobile usually implied an immediate answers, so it put all the data right up front.\nAfter the update, the search experience felt significantly improved \u2013 and almost comparable with Google's performance in its own mobile app. Now, you can access that new experience on Android as well, as Microsoft has released a new Bing app in Google Play.\nIt remains to be seen whether or not the new format will convince you to switch from Google's default search experience, but Bing's new app is at least worth taking a look at.\nIf you've been coveting at Bing's iOS counterpart, the Android one is essentially the same: it can answer natural language queries like \"What is the population of Portland?\" directly, without you having to scroll through a list of websites.\nAt the same time, searching for a city will display a few travel tips, some local info, nearby news and a list of apps that can help you book a hotel there, for example. You can also activate voice search, and scroll through a list of local points of interest \u2013 movie theaters, restaurants, and various things you can do in that city.\nWhile remaining a solid search experience, Bing's Android app includes some new features that weren't present in its iPhone counterpart.\nFor example, tapping the \"gas stations\" button under the \"Near Me\" section will allow you to check gas prices from GasBuddy. Once you located the price that works for you, a single tap will help you get directions in Google Maps.\nMicrosoft also equipped its app with a bar code scanner; even though local retailers will frown upon this, you're free to do on-the-spot comparison shopping. If you find something cheaper online, there goes your chance to save a little cash.\nThe bar code scanner is also available on the iPhone 6s or 6s Plus, where you can access it via 3D touch on the homescreen icon. For Android, the feature will reveal after a long press on the big search icon inside the Bing app, while older iPhone users will find it inside the \"Deals\" menu.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaudgn b/data_all_eng_slimpj/shuffled/split2/finalzzzaudgn
new file mode 100644
index 0000000000000000000000000000000000000000..a09c3535d7de9e63dcc06b2a202771e33d5fe2aa
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaudgn
@@ -0,0 +1,5 @@
+{"text":"Suggestions that Sergio Ramos stamped on Raheem Sterling have been dismissed by the Spain captain, who posted footage of the incident.\nSergio Ramos has dismissed claims he stamped on England forward Raheem Sterling by posting a clip of the incident from Spain's 3-2 Nations League defeat.\nIt had been claimed that the Spain captain, who scored a late consolation in Seville, clashed with two-goal Sterling in an off-the-ball incident that left the England man grounded as play continued.\nBut Ramos has moved to quell the talk by uploading footage of Monday's match that appears to show the defender making no contact with Sterling, who was already clutching his leg.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Howard Melamed is living proof that the Science of Opportunity works. Born in 1956 in Montreal, Quebec, Howards early academic years were a bit bumpy, with one teacher predicting he would never finish high school. Howard not only proved her wrong but seized the opportunity to be president of his senior class. He went on to study Civil Engineering at McGill University, with a minor in Computer Science, and after graduating he used his education to fuel his first entrepreneurial endeavors: construction management and computer consulting.\nAs Howard moved through life, he continued to see opportunity where others did not and then capitalize on his insight. Some of his more memorable ventures include acquiring the historic Outremont Theater in Montreal with other investors and bringing live entertainment to the downtown district; blowing into Florida with Hurricane Andrew on his tail, just in time to help the area rebuild; and using his foresight as a marketer to ride the crest of the Internet wave with his ecommerce storeall unique openings he identified and developed following the principles of The Science of Opportunity.\nTo those who might say he is lucky, Howard explains that, luck had nothing to do with it. He always found a way to be open to, recognize, and act on opportunity when it came his way. Today Howard is CEO and President of the CellAntenna Corporation, a multinational, multimillion-dollar telecommunications service company with employees worldwide. He has been interviewed by Associated Press, International Tribune, The New York Times, Chicago Tribune, Fox News, MSNBC and on numerous radio talk shows. He continues to contribute articles to his Web site, Coralsprings.com, and plans to publish two novels in 2008.\nHoward lives in Coral Springs with his wife and business partner, Barbara.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Signing up & depositing alone will not trigger the offer.LeoVegas also has a host of Canada-friendly banking methods available like Interac online payments.\" I've heard about PrimeSlots from friends, online pokies free spins and wanted to give a try.\nLoot Casino: Win up to 500 Free Spins on Starburst!\nFour-part casino bonus up to $1000 & 200 spinsCasino Cruise is a must visit if you're looking to get one of the best Free Spins No Deposit bonuses online! Grab 55 No Deposit free spins today!The minimum deposit is \u00a310 \u2013 so you will hardly go broke when you try to activate the bonus.If you enjoy Slot reels-spinning action, you will wonder how you ever survived before now.\nwith Aristocrat SlotsAt Bitstarz you can deposit, withdraw and play in:a period of time, which takes a user to play in this or that real money online casino on average.Rock out with no risk on one of NetEnt's classic Rocks Trilogy games.The 120 free spins at Royal Vegas casino are just the tip of the iceberg, or, to be more precise, the start of the reel! Read the monthly blog to keep up with your favourite online casino.Right now we can bring you an absolutely exclusive Crazy Winners promo code CASINOMAX that you can use when you sign up to the casino for the very first time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If he sounds bullish, that's because Niederhofer's VC firm, Sunstone Capital, has just closed a $2 million investment in Kontakt.io, a Polish startup that offers its own Beacon platform, including the supply and customisation of Beacon hardware, underpinned by an open API and SDKs for iOS and Android.\nThe Krak\u00f3w-based company, which counts Y Combinator alumni Estimote as a competitor, already boasts big-name customers such as Google, Facebook, Apple and Siemens, and says it plans to use the new capital to accelerate international expansion and evolve its product line.\nWith the technology still pretty nascent, however, and despite Apple throwing its might behind the concept, what Beacon's 'killer' application will be is still open to debate. Retail is thought to be just the start \u2014 with Beacon transmitters enabling an app on your phone to detect when you enter a retail store or spend time in close proximity to a particular product and showing you relevant offers, for example. However, Niederhofer says other applications include security, home and office automation, care of patients, children and the elderly, asset and warehouse management, and indoor navigation.\nIn fact, Kontakt.io's founders \u2014 Tomasz Kozminski, Tomasz Kolek, Rafal Janicki and Szymon Niemczura \u2014 first set out to build an indoor navigation system for the blind, only to realise that Beacon technology could solve this problem and many others.\nFurthermore, he argues that anyone making a mobile app that has some relevance to the offline world should either be contemplating building their own Beacon infrastructure or piggy-backing off someone else's.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"New economic data shows what Oklahoma officials have been fearing: The state has officially entered a recession. Revised federal Bureau of Economic Analysis (BEA) data shows that the state's gross domestic product was negative for most of 2015.\nA recession starts when there are two quarters of economic contraction. Originally, the BEA reported that Oklahoma's economy contracted in the second quarter, grew by 0.1 percent during the third quarter and contracted again in the last quarter of last year. But the third quarter figure was recently revised downward to -0.6 percent.\nData for the first quarter of 2016 is expected to be released later this month, but according to State Treasurer Ken Miller, the prospects don't look good.\n\"General indicators fail to point to any marked economic recovery at this point,\" he said in his latest state economic report.\nLow oil prices are the primary driver of the state's recession. Nationally, the mining industry, of which oil and gas are a part, shrank by 10.7 percent during the fourth quarter. That stripped 2.5 percent from Oklahoma's GDP during the fourth quarter alone, Miller said. Jobs in those industries have fallen 32 percent in Oklahoma over the last year and a half.\nThe Takeaway: With the data revision, Oklahoma is now mired in a recession along with fellow oil and mining states Alaska and Wyoming. Wyoming's economy contracted in every quarter last year while Alaska's contracted during the last six months of 2015. Kansas' economy has also posted two straight quarters of decline, thanks to a slump in its manufacturing economy.\nThe data highlights how vulnerable certain states are to some sectors. Most places are growing -- 41 states and the District of Columbia posted a positive GDP during the fourth quarter of 2015. Those that aren't, which also include Iowa, Montana, Nebraska and New Mexico, tend to be vulnerable to energy and farming sectors. In the early post-recession years, these states were generally booming. Looking at the BEA's map of states today, top performers tend to be in the West, Midwest and Southeast. The current situation is essentially the reverse of what it was half a decade ago.\nA new analysis of state higher education funding shows a mixed bag for the 2016-2017 school year. Nationally, higher education funding in most places will increase 2 to 4 percent, according to Moody's Investors Services. Most of that growth is driven by larger and more economically diverse states. Places like California, Florida, New York and Texas \"are more insulated from pressures on any one revenue source, which increases the likelihood of fiscal stability and consistent higher education funding gains,\" the report said.\nConversely, states with less diversified economies, such as energy states, or those exposed to retiree benefit liabilities are more likely to cut higher education funding. Oklahoma is cutting its higher education funding by 9 percent this year (see above). Kentucky is cutting funding by 4.5 percent while allocating more money for its struggling pensions. And West Virginia is aggressively tacking its unfunded retiree health-care liability, but \"it comes at the expense of reduced state higher education funding, which is down 11 percent from fiscal 2013-2016 and is expected to remain at reduced levels for the next several years,\" the report said.\nThe Takeaway: Higher education funding is turning into a case of haves and have-nots.\nOne outcrop from this era of constrained resources is the rising demand for performance-based funding. Moody's notes that a full 32 states now tie a portion of funding to performance metrics such as degrees conferred, course completion and minority enrollment.\n\"As more performance-based funding models take hold, individual universities' year-to-year funding may become more volatile,\" the report warned.\nImagination can take you far. It can even take you ... back to reality. That's the idea behind a new campaign this summer by ICMA-RC, a nonprofit financial services firm that offers retirement plans to public sector employees, to get public employees to start saving more for retirement. Corporation employees are touring the country in a \"RealizeRetirement\" truck. Anyone can walk up to an iPad outside the truck, punch in a few personal stats and retirement wishes, and then see a personalized video that features them traipsing across the globe or putting their feet up poolside at home -- depending on what the retirement dream is.\nIt sounds hokey, but the idea is to get people to visualize what they want out of retirement and to start considering whether their pension plan and social security benefits will be enough. The tactic was inspired by a 2011 study that showed people who saw digitally aged images of themselves were more inclined to save for retirement. The truck started in Washington, D.C., this week and will make its away across the country from there.\nThe Takeaway: A lot of ink has been devoted to the fact that most Americans are ill-prepared for retirement. More than 38 million people, or roughly 45 percent of working-age households, have no retirement savings at all, according to data from the National Institute on Retirement Security.\nWhat's more, many current public-sector employees won't get the kind of pension today's retirees have. That's particularly true of millennials who are more likely to have reduced retirement benefits and less likely to spend their whole career with one employer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzavcns b/data_all_eng_slimpj/shuffled/split2/finalzzzavcns
new file mode 100644
index 0000000000000000000000000000000000000000..5a7ddba4a91b9c65482ab693a210cf7a2880cb08
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzavcns
@@ -0,0 +1,5 @@
+{"text":"A truly revolutionary release from the leading name in prestige sports vehicles, the Lamborghini Urus demonstrates that it is possible to keep true to the brand's agile identity while delivering on safety and strength. At first glance this is an SUV, yes, but one which is first and foremost a Lamborghini; boasting the luxury car manufacturer's unmistakable, muscular design and unrivalled performance. Little wonder its impending launch is dominating conversations between car enthusiasts around the world.\nThe first ever Super Sport Utility Vehicle on the market, the Lamborghini Urus has a 650 horsepower V8 Bi-Turbo engine, capable of reaching speeds of 305 km\/hour. It is the most powerful SUV ever produced, yet can also seat five people in utmost comfort, making it a Lamborghini the whole family can enjoy in both an urban setting and on more challenging country roads. From navigating the streets of the Principality to the vertiginous mountain overpasses of the Monaco back-country, you could say the Lamborghini Urus was designed with the French Riviera lifestyle in mind. Experience it for yourself from July 2018, when the Lamborghini Urus will join the Luxury & Services Rent Lamborghini for hire in Monaco fleet. Collection can also be arranged from other locations along the C\u00f4te d'Azur.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Provides a presenter to the Specify Filter Options page of the Data Source wizard (Extract Data Source).\nSpecifies whether the Finish button is enabled on a wizard page.\nFinishEnabled Specifies whether or not the Finish button is enabled on a wizard page.\nModel Provides access to the wizard model, which accumulates settings specified by an end-user during wizard execution.\nMoveNextEnabled Specifies whether of not the Next button is enabled on a wizard page.\nSpecifies whether the Next button is enabled on a wizard page.\nPageContent Provides access to the content of a wizard page.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Vintage Gunne Sax maxi dress from the 1970's in a renaissance style, size extra small \u2013 small.\nIt's in an unbleached, cotton duck type fabric. Looks similar to drop cloth fabric, but not as heavy. It has long sleeves and an attached, lace apron at the waist. The bodice has princess seams with lace overlay and ties corset style up the middle. The gathered at the shoulder sleeves end in 8\u2033 of the same lace. There is tan gimp around all the lace edges.\nIt zips up the back and the sleeves zip up the side in the lace.\nThere is no size label, so I'm having to guess by the way it fits me. I believe it is a small, but it could be an extra small.\nIt's in very good condition. There is a small gray spot on the side. Some of the fabric under the zipper lap did not catch correctly. I've included a picture of these flaws. The seam edges were not serged or overlocked and are beginning to ravel, but it has a generous seam allowance and the stitching is not threatened. However, to prevent the raveling from going too far the dress should probably be dry cleaned if worn often. Or you might have a seamstress serge the edges if you can't do it yourself.\nThis would be a great festival dress where authenticity is not important.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On Wednesday afternoon, Dec 19th, Noam and Ewa were riding bikes on the road as usual. They were going to Longlian Community to meet some lovely children.\n\"Yes, I can do it\" is an English Corner which is held by Longlian community and Nanyan Youth Volunteers Association of Peking University each Wednesday. The children are from nearby primary schools, and most of them are in Grade 1 to 3. As foreign teachers, the responsibility of Noam and Ewa is providing kids with an immersive English environment to cultivate their interest in English and offer plenty of opportunities for children to speak.\nThis was the last time for Noam and Ewa to go to the English Corner this term. They taught all the children to sing a classical English song \u2013 You Are My Sunshine. At 17:00, the English class began. The volunteers had already printed out the lyrics and translation and handed it out to everyone. When the beautiful melody echoed in the activity room, all the children calmed down and listened to the song. The English sentences in this song were a little difficult for children of such ages, so first, Ewa and Noam called on all the students to read the sentences with them together. Seeing it was hard for most of the children to understand the words, our Chinese volunteer Peter explained the lyrics for kids sentence by sentence.\nAfter Peter's explanation, the children were more familiar with the lyrics. Next, Noam and Ewa went to the desks to sing the song with students line by line. Immediately, the room was full of singing voices. It was difficult for some children to pronounce each word well so Noam and Ewa helped them patiently. With the help of the two lovely teachers, the progress the children were making was obvious.\nAfter practicing for about a half an hour, Ewa and Noam encouraged students to come to the stage and sing before all their classmates. Several kids were singing well but lacked the courage to stand up. However, some students bravely sang in the front of the whole class. Our volunteers had already prepared candies for the kids who could sing this song on the stage as an award for their courage. With the encouragement of classmates, volunteers, and teachers, most of the children finished this \"task\" and got delicious candies.\nSuch a beautiful song just like a poem. We hope these beautiful sentences are able to stay in the childrens' hearts, bring sunshine to their world, and make them understand the value of the people who love them deeply.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"...Jonathan Lipnicki. So I'm a sucker for talking animals. Stuart Little 2, and its enormously popular predecessor, 1999's Stuart...\"\n\"...actor who nearly caused me to lose it was Michael Stuart. Stuart is a playful actor, one who occasionally...\"\n\"...simply because both were played by the same actor, Michael Stuart. Both men, it may be said, love to...\"\n\"...emcee, Lowell Bartholomee returned to host the affair, while Stuart Moulton and a bevy of beauty queens from the...\"\n\"...that people in small communities share. Eric Love and Michael Stuart are both credited as director, and while it's...\"\n\"...Playhouse production when we're watching David Stahl's Frost and Michael Stuart's Nixon, sitting rigidly still in their upholstered chairs,...\"\n\"...Canadian Michael Healey's play is a sweet, sometimes hilarious, and ultimately...\"\nIn The Dead Presidents Club, Larry L. King brings together Calvin Coolidge, Harry S. Truman, Lyndon Johnson, and Richard Nixon in the afterlife for a lighthearted exploration of these American leaders' strengths and foibles, and the best part of Austin Playhouse's funny, entertaining revival is the inspired work of the excellent team of actors who fully embody these presidents.\n\"...also look stunningly like the men they are portraying. Michael Hankin inhabits the reticence and dry humor of history's...\"\n\"...Watching Michael Stuart stomp around the set for Austin Playhouse's production...\"\n\"...types to read it. David Stahl plays Jake, and Michael Stuart plays Charlie. In fact, Stahl and Stuart play...\"\n\"...Onstage Theatre Company and directed by company artistic director Michael Stuart. Like Nason, this production is special, as it...\"\n\"...and sense of community of its priceless cast and Michael Raiford's wonderful set;...\"\n\"...negotiator Andrey Botvinnik, played with subtle, sympathetic sincerity by Michael Stuart \u2013 but, as designed by the Crowleys, they...\"\n\"...of Dirk van Allen, Ben Wolfe, Rachel Dendy, and Michael Stuart, whose swiftly shifting postures and accents are a...\"\n\"...Wind comes down to its juggernaut courtroom battle between Michael Stuart's Brady and Dave Stahl's Drummond. Stuart looms heavily...\"\n\"...Toner has done. He's chosen Playhouse acting company veterans Michael Stuart and David Stahl \u2013 as reliable a pair...\"\n\"...Ev Lunning Jr. deserves credit; however, there are exceptions. Michael Stuart's Shakespeare and David R. Jarrott's William Combe, the...\"\n\"...the era of camp and silliness. The protagonist, Pseudolus (Michael Stuart), wishes he were free from his life of...\"\n\"...role as Texas' most recent dead prez, LBJ, is Michael Stuart, if not one of Austin's most popular actors,...\"\n\"...affiliation of arts writers, consisting of the Austin American-Statesman's Michael Barnes (arts), Jamie Smith Cantara (theatre\/music), Clare Croft (dance),...\"\n\"...for Billy Joel, Bonnie Tyler, and Boston. Just ask Michael Holland and Karen Mack, a pair of formidable New...\"\n\"...affiliation of arts writers, consisting of the Austin American-Statesman's Michael Barnes (arts), Jamie Smith Cantara (theatre), Randy Harriman (music),...\"\n\"...awards for achievement in theatre. The participating critics -- Michael Barnes, Jamie Smith, and Anna Hanks, Austin American-Statesman; John...\"\n\"...of corrupt government officials, led by a wacky, wily Michael Stuart as the mayor, scramble to get their town...\"\n\"...Tor (Michael Stuart) and Gus (J. Ben Wolfe) are hit men....\"\n\"...speak up, inciting union leader Harry Fatt (the ever-versatile Michael Stuart), who shoots back that my row neighbor is...\"\n\"...\u2013 is, thankfully, in the latter category. And here Michael Stuart has directed the light dramedy for Austin Playhouse....\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzavezo b/data_all_eng_slimpj/shuffled/split2/finalzzzavezo
new file mode 100644
index 0000000000000000000000000000000000000000..c41407eeacf2167486d253ca767c384675d89ab4
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzavezo
@@ -0,0 +1,5 @@
+{"text":"Our Information and Support Centre team provides guidance, information and support, and connects you to relevant support services.\nThe team includes a Lung Cancer Support Nurse who provides evidence-based information and support for you and your family through any stage of your lung cancer journey. Our Respiratory Care Nurse also provides telephone support to patients and carers across the country. This includes practical suggestions about their lung disease, personal care and management, and referral to local support services where necessary.\nTo donate to Lung Foundation Australia, including In memoriam of a loved one, please click here to donate online.\nFor general enquiries including membership, administration, education materials, research awards and also to provide feedback, please complete the form below.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Green City Market founder Abby Mandel loved the crepes served fresh in many European markets and wanted to bring them to Chicago. The tradition is now carried at Green City Market's own crepe stand: Abby's Crepes. Every week, our Chef Educator, Lisa Kalabokis, features local and seasonal ingredients sourced from our farmer community and crafts each ingredient into delectable crepes.\nProceeds from Abby's Crepes go directly towards Green City Market's free edible education and food access programs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"British Prime Minister David Cameron describes some countries taking part in an upcoming anti-corruption summit as ''fantastically corrupt'' in remarks to the Queen. Rough Cut - Subtitled (no reporter narration).\nROUGH CUT - SUBTITLED (NO REPORTER NARRATION) STORY: British Prime Minister David Cameron was caught on camera telling Queen Elizabeth on Tuesday (May 10) that leaders of some \"fantastically corrupt\" countries, including Nigeria and Afghanistan, were due to attend his anti-corruption summit. Cameron will host an international anti-corruption summit on Thursday (May 12) aimed at stepping up global action to combat corruption in all walks of life. In a pooled video feed made available to broadcasters, Cameron was shown talking with the queen about the summit. \"We had a very successful cabinet meeting this morning, talking about our anti-corruption summit,\" Cameron said when the queen approached. \"We have got the Nigerians - actually we have got some leaders of some fantastically corrupt countries coming to Britain.\" Cameron went on: \"Nigeria and Afghanistan - possibly two of the most corrupt countries in the world.\" The queen did not respond to Cameron's comment but the Archbishop of Canterbury Justin Welby said: \"But this particular president is actually not corrupt.\" Nigeria's President Muhammadu Buhari and Afghan President Afghan President Ashraf Ghani, both of whom are due to attend the summit, acknowledge corruption in their countries and have pledged to clean it up. Afghanistan is at number 166, second-from-bottom, in campaign group Transparency International's latest Corruption Perceptions Index, an annual ranking of countries. Only North Korea and Somalia, jointly ranked at number 167, are perceived to be more corrupt. Nigeria is at number 136 in the index. It was not clear whether Cameron realised he was being filmed and recorded at the event at Buckingham Palace. A bystander then joked to laughter: \"They are coming at their own expense one assumes?\" \"Everything has to be open,\" Cameron said. \"There are no sort of closed-door sessions. Everything has to be in front of the press. It's going to be...It could be quite interesting.\" When asked about the comments, Downing Street did not immediately respond.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Are you using LMS-based learning or utilizing any eLearning content? Probably. It's safe to assume that individuals understand eLearning to simply be learning that is enabled and available online. That's a simple definition. To get a little more technical, Elliot Massie defined online learning as \"the use of network technology to design, deliver, select, administer, and extend learning,\" wrote Jay Cross.\nSince the corporate world is increasingly reliant upon technology, it's no surprise that LMS (learning management system)-based training is becoming the rule in business settings, and classroom-style training the exception. This switch from one type of learning to another is happening because the benefits of eLearning are stunning. eLearning can be fun, easy, and affordable. It's no wonder company leaders are drawn to it.\nDoes your organization use an LMS to facilitate online training for employees? If so, that's wonderful. However, no matter how excellent your LMS may be, if its course content is cluttered, employee training will not be as effective as you desire. The only way to successfully provide easy-to-understand training through a social learning platform is to regularly organize course content.\nA tremendous amount of data can be packed into an online course hosted by a cloud-based LMS. This is both a blessing and a curse. On one hand, it's beneficial to have so many resources and training materials available in a single virtual classroom. But, it can be problematic to store a large quantity of data in one place. If course designers don't take measures to keep content organized, things will get confusing fast. Thankfully, there are a few ways to \"spring clean\" eLearning content.\nSchedule regular times for content maintenance \u2013 Would you agree that if something does not get scheduled, it often does not get done? This is certainly true of eLearning content organization. It's not necessarily a task that most people relish, so it often gets shoved to the bottom of the \"to-do\" pile. To keep course content orderly, schedule a few hours per month to organize all eLearning assessments, activities, presentations, etc. This will ensure the content you work so hard to create gets used.\nWhat are some ways you keep eLearning content organized?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Carlos' Bathroom Renovations offers a free in-home consultation and design service. We offer professional and practical advice with real solutions and with your best interests in mind. We will discuss your ideas and all the latest trends, innovations and products available.\nWe will book a start date and plan out a schedule of work for a time that best suits you and your family. In the meantime, if you have not already done so, you may want to start looking and selecting tiles.\nWe can supply all aspects of fittings from your basic taps, vanities, frameless shower screens etc to custom design vanities and cabinets or you can purchase your own and we will provide the labour to install.\nCarlos' Bathroom Renovations is renowned for quality craftmanship, attention to detail, reliable service and prompt project completion.\nWe are a proud family business and our success has been partly due to our quality team of licenced tradesmen who complete all work to our high standards.\nOnce work begins, we will take care of your home and property keeping it clean & tidy, arrange delivery of components and work within an agreed time-frame. Your Project Supervisor will keep you up to date throughout the entire process.\nIn most cases, a bathroom renovation takes up to 3 weeks after work begins.\nDemolish - Strip-out labour & rubbish disposal. This process will take up to one day.\nBuilder will do any structural work or repair water-damage, if needed. The time will vary depending on the amount of work.\nWater-proofing: you may have to wait up to three days to allow the waterproofing membrane to cure. Water proofing is one of the most important steps in preparation for your bathroom renovation and we adhere strictly and conform to the Australian Standard AS 3740-1994.\nTiler Once the floor and walls are prepared the tiling will take place. Allow 4 days for the wall & floor tiling.\nShower screen installer will measure the shower enclosure space after the tiling completion. They will install the shower later.\nPainting If we are doing the painting, it will be done prior to installing the fittings.\nPlumber will return to install your fittings \u2013 Vanity, taps, toilet ect. If you choose a freestanding bath, it will be installed at this stage.\nElectrician will return to do the electrical fit-out.\nFinal Fit-out - Shower screen, mirrors, towel rail, toilet roll holder accessories etc are installed.\nOnce your project has been completed we will conduct a final inspection to ensure your satisfaction so that you can enjoy your beautiful new bathroom.\nReady to start your bathroom renovation?\nCall or contact us now to discuss your project and book a free in-home consultation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzawphp b/data_all_eng_slimpj/shuffled/split2/finalzzzawphp
new file mode 100644
index 0000000000000000000000000000000000000000..e3d1c57290d03e49f6178addd798193af9002bee
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzawphp
@@ -0,0 +1,5 @@
+{"text":"Although it is not a good idea to look at your house as an investment, it is very important that you start by tackling the improvements that add value to your home. Don't just think of how you can improve the beauty of your home, but also think in terms of value. Is it worthy to upgrade my old kitchen or should I put up a swimming pool in the backyard? These are some of the questions that you should ask yourself when planning on home improvement.\nSometimes, some home improvements such as repair may require our immediate attention. It is important that you start by dealing with the most immediate projects that are in a dire need for attention. As you might have realised, some home improvements only require a fraction of our time and there is no reason at all why we should fail to address such issues immediately. It is only after dealing with the immediate and dire consequences that we can tackle the most complicated projects that do not require our immediate action.\nPlease not that, home improvement projects may be an expensive undertaking. This is the reason why you must be able to identify the things that you can do on your own. Some home repairs are very easy and you may not need to hire a professional. However, other projects are complicated and you may not be able to handle them without the guidance of an expert. Do what you can on your own and save your money for the other complicated projects.\nAs I have already mentioned, there are some home improvement projects that are easy and you should never hire anyone. These tasks are easy to learn and you should, therefore, try your level best to gain and improve such skills. Identify what you want to learn and then acquire all the necessary tools. Sometimes, it is advisable to connect with a home improvement expert since you can learn a lot from them.\nWhen it comes to home improvement, it is important that you start by identifying the urgent projects. You should then evaluate yourself and come up with your budget. Remember that, your home may not give you any direct income, and it is, therefore, crucial that you stick to a budget that you can comfortably afford. If you must hire someone, ensure that you look for a skilled and experienced person who will be able to perfectly handle your project.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"cabins for adults near Texas.\nRemarkable customer reviews about; Cabins for adults near Texas.\nJust a note to let you know Ronald and I really enjoyed our stay. Your cabins have all one needs to feel comfortable. Were so glad that we found this place and were amazed at how close these cabins for adults near Texas are located from our home. You are a very gracious host. Hope to come again soon.\nThanks or offering your honeymoon cabin near Texas to us for the weekend. Our campfire was most delightful on a chilly evening. Thanks for the good movies, the hot-tub and a good night's sleep. This is truly a wonderful place.\nThank for such a beautiful place. Everything was perfect. It is a great place to relax and forget about everything. We loved going for walks beside the creek. We will highly recommend these cabins for adults near Texas to others. This is Dan's birthday weekend gift from me and it couldn't have been better.\nWhat a wonderful and peaceful place. Glen and I enjoyed everything. We spent our time relaxing and appreciating God's provision. All your efforts are noted and we will want to return. The detail of these cabins for adults near Texas is wonderful. I love the red. The romance package was perfect as well as the hot-tub.\nThis is our first stay in these lovers getaway cabin near Texas. We had an absolutely wonderful time! You made our stay here one that neither of us can forget. We loved the mimosas and the romance package. Your massage therapist done an amazing job and the hot-tub put the icing on the cake. We both slept like babies. We will be back.\nWhat a great time we had in this place. We really needed this peace and quiet and time for ourselves. We loved the privacy your cabin offers and the hot-tub was awesome. We can tell that you have put a lot of hard work into this place. We will share our experience with our friends and hopefully we will be able to return this summer.\nRiver of Love; honeymoon cabin near Texas.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Founding of the Casa del Contratacion (House of Trade) in Seville.\nFounding of the Consejo de Indias (Council of the Indies).\nFirst of the new Italian botanical gardens was established at the Unviersity of Pisa.\nPublication of \"De re Metallica\" (\"On Metals\").\nPublication of \"Alchemia\" by Andreas Libavius.\nOpening of Gresham college in London.\nPublication of \"On the Magnet\" by William Gilbert.\nFormation of the Academy of the Lynxes.\nThe Academy of the Lynxes was an exclusive group of philosophically minded gentleman that was formed under the patronage of the Marquess of Monticelli Federico Cesi. The name of the group was intended to indicate that its members were acute observers of nature since lynxes were proverbially keen of sight.\nPublication of \"The Advancement of Learning\" by Francis Bacon.\nPublication of De sapientia veterum (\"On the Wisdom of the Ancients\") by Francis Bacon.\nPublication of \"On the Motion of Mars\" by Kepler in Prague.\nGalileo resigns from his position of Mathematics professor at the University of Padua.\nGalileo publishes Sidereus nuncius (\"Starry Messenger\").\nGalileo becomes a member of the Academy of the Lynxes.\nPublication of Novum Organum (\"New Organon\") by Francis Bacon.\nThe Academy of Lynxes pays for the publication of Il saggiatore (\"The Assayer\") by Galileo.\nThe Academy of Lynxes pays for the publication of Dialogo by Galileo.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"placenta: the thick plaque of tissue that forms when an embryo attaches itself to the inner wall of the womb, and which is joined to the baby by the umbilical cord [from Latin, placenta, flat cake].\nThe organ that serves as a filter and lifeline to a developing baby. It is what filters out nutrients from the mother's own blood and transfers it to the fetus's blood (since the two have their own independant blood supplies).\nForeign cultures as well as other mammals are known to eat the afterbirth to regain those nutrients that have been allocated for the purpose of gestation (as disgusting as this may sound, it makes sense is the placenta is rich in nutrients).\nA structure formed on the wall of the uterus during pregnancy which serves to exchange oxygen, wastes and nutrients between the mother's blood and the embryo\/fetus' blood; the placenta is expelled as afterbirth.\nIn plants: a region, within an ovary, to which ovules are attached.\nThe vascular appendage which connects the fetus with the parent, and is cast off in parturition with the afterbirth.\nIn most mammals the placenta is principally developed from the allantois and chorion, and tufts of vascular villi on its surface penetrate the blood vessels of the parental uterus, and thus establish a nutritive and excretory connection between the blood of the fetus and that of the parent, though the blood itself does not flow from one to the other.\nThe part of a pistil or fruit to which the ovules or seeds are attached.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Zero Hour Contracts, lets examine the facts. Various sources estimate that approximately 250,000 UK workers are on Zero Hour Contracts, and there is growing evidence that this figure could be far higher. As a result of the present climate they are arguably moving away from niche towards a mainstream contractual arrangement in certain sectors, and it could be further argued that sectors that are turning to this type of 'non-employment' is also growing.\nDespite the arguments of flexibility, from a psychological perspective these contracts engender uncertainty and lack of control, both of which are recognised causes of stress at work that can damage employee wellbeing and consequent performance. Furthermore, there is evidence to suggest long term stress can eventually lead into depression.\nIn the present climate a regular income that reflects the current cost of living has never been more important to employees. However, unpredictable hours based on inconsistent commercial demands are driven by wider contextual variables \u2013 i.e. packaged selections of fresh produce, where retailers may look at the weather forecast, anticipate demand and place their orders accordingly, often with a lead in time of less than 24 hours. As a result, employees will be suffering simply because they have no guarantee of how much money they will be taking home each week. If you get no work for a week or two, paying rent, or arranging childcare can be problematic. Planning ahead becomes difficult.\nThe disruption to family life as a result of frequent short-notice requirements to work make so many things nearly impossible to plan.\nThe possible financial struggles and the fear that many feel living in this situation can lead to work related stress. It can be difficult to maintain good work morale when the job you're doing isn't financially supporting you, and the uncertainty of a Zero Hour Contract means that many won't ask for their situations to be improved because of the fear they'll lose their jobs, or employers will further reduce hours. Workers may even feel worse if they work with full-time members of staff who don't experience these problems, especially if they do the same work.\nIn addition, it has also been argued that a differentiated and unbalanced working arrangement creates resentment and envy, which in turn is highly likely to damage how teams function and work together.\nProfiling individuals may be the key to understanding the right person fit to a Zero Hour Contract. A formal and full time contractual arrangement, psychologically creates a sense of value and sense of worth, it creates security and requires long term commitment. However, flexibility, short termism and lack of commitment to any one organisation, coupled with a need to do other things i.e. study, may be the key that fits a Zero Hours context. Personality characteristics that drive this motivational differential can be measured. Level of confidence, need for control, level of commitment, team orientation, need for security and regularity, need for autonomy, attitude to change and managing uncertainty\/unpredictability, will all factor in the impact of a Zero Hour Contract on employee health and wellbeing.\nAnother factor to consider is better analytics, a current growth area in HR generally. Technology can be utilised to better predict working patterns, which can lesson the negative impact of Zero Hour Contracts.\nEmployers need to understand that psychologically, these contracts have the potential to engender poor performance. Feeling undervalued has a long-term, repeated, continual effect on tasks and attitudes towards them. Furthermore, the unconscious message that underpins the nature of these contracts is inherently negative. The potential reciprocation is anger and resentment, hardly a driver of quality work.\nToo many people are currently living a life on call.\n\"In the Victorian era there were sweatshops, child labour, few worker rights, and casual employment with no guaranteed income. We view this with horror as a sign of gross inequality, ruthless exploitation, and as bad times in which the rich and powerful were able to maintain their idle privilege through laws, customs, and a deeply religious conservatism where everyone was born into and knew their place.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzawypj b/data_all_eng_slimpj/shuffled/split2/finalzzzawypj
new file mode 100644
index 0000000000000000000000000000000000000000..b0af511eb57ba305d8158fa922c05adb54176d1b
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzawypj
@@ -0,0 +1,5 @@
+{"text":"I started playing tennis at the age of 7. From the age of 13, I represented the national team in World Youth Cups in the 14, 16, and 18 age categories. Later, I accepted a full athletic scholarship at the University of Alabama at Birmingham, where I was the captain of the team and played top lines in singles and doubles. I graduated with a major in Organizational Communication with a minor in Psychology.\nI will be bringing over 12 years of experience in the country club business throughout which I have run and administered highly active programs at elite clubs on Long Island as well as Connecticut. For the past 5 years, I was the head tennis professional at Wee Burn Country Club in Darien, Connecticut with a focus on junior programs.\nI am a United States Professional Tennis Association ('USPTA\") Elite Professional, and I continue my development by attending workshops, exchanging knowledge with other top professionals in the industry, while maintaining relationships with several current and former top \u2013 100 professional tennis players in the world.\nMy coaching is based on an individualized and adaptable approach that considers student's personal goals and abilities. So whether you are new to the game or an experienced player, my staff and I will help you achieve your goals while having fun!\nIn addition to bringing my skills, passion, and love of tennis, I also look forward to building long lasting relationships and creating a welcoming and inclusive environment at the Redding Country Club where you will enjoy tennis activities while spending time with your friends and family.\nBrian Quinn has been a USPTA, (United States Professional Tennis Association) Certified Platform Tennis pro since 2011. He is also a lifelong player who plays competitively and for the pure joy of the game. He has been Director of Paddle at The Easton Racquet Club, Silver Spring CC and Brooklawn CC. Brian is also proud to say that he is consistently asked to teach with many of the top ranked national Platform Tennis pros in the country. Throughout his career, members have enjoyed the enthusiasm he has for the game of Paddle. Brian resides in Monroe, CT and is the Father of four boys ages 12-17.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The \"strippy\" quilt was an early design that had little representation in the quilt revival of the early 1900's. One exception was the Magic Vine pattern that originated in the 1930's in a newspaper column written by Florence LaGanke Harris.\nThe Magic Vine is an example of how women's magazines and newspapers kept the quilting tradition alive during the Great Depression.\nCalled the Nancy Page Club, the patterns for the Magic Vine were printed weekly along with a fictitious story about a quilt group that had Nancy Page as its leader. Each column would detail the discussions by the women about what colors to use for each floral design and other quilt related topics.\nOur circa 1930 inspiration quilt is from the collection of one of our members. It has four vertical vines with twenty-two different floral designs bordered by a curving vine and leaves. The appliqu\u00e9 is all hand done.\nThe study quilt is based on The Magic Vine Quilt pattern designed by Eleanor Burns and published in 2007. It uses a quick turn appliqu\u00e9 method with non-woven interfacing. The appliqu\u00e9 pieces were then hand sewn in place. Reproduction fabrics from the 1930's were used throughout the quilt. Embroidery and buttons were added for embellishment, giving the study quilt a thoroughly modern look.\nAt one of the Repiecers' quilt study group meetings, one of the group members shared the original newspaper columns that showed the floral patterns, as well as additional information on other Nancy Page designs. The quilt study group meets every other month at Quilt in A Day in San Marcos, which is owned by Eleanor Burns. Several members of the study group worked on the study quilt, including Marian Cassianni, Kay Ross, Sandy Sutton, and Leah Zieber.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sanyukt is an Indian Hindi family drama television series, which premiered on 6 September 2016 and is broadcast on Zee TV. The series is aired on weekday nights. The series is produced by Saregama Productions of Rahul Tewari The story of the show is about an extended family and the trials and tribulations between each family member.Members of the Mehta family, an extended nuclear family, face various trials and tribulations. Follow their lives, as they go through joys and sorrows, with love still intact in their relationships.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Winnie the Pooh's birthday tale is coming to your town and you too could be a part of the celebrations. This musical based on A.A. Milne original classic is going to be held at the Kelsey Theatre, Mercer County Community College, located at 1200 Old Trenton Road, West Windsor.\nIt is a complete family entertainment show brought to you by Maurer Productions Onstage and performances are scheduled for 16th & 17th August, 2008 at 2 and 4 p.m. Tickets are priced at $10 for children and $8 for seniors. You can also visit Kelsey Theatre or call (609) 570-3333.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dynamo reminds me of someone wasting alot of power that could be used for much more good, I am afraid for him because he has become a star. Stars are hard to get through to an can be controlled easily through there inability to balance their ensuing pridefullness one has to wise enough to stand up to.... Dynamo Magician Impossible Revealed Stephen Frayne AKA Dynamo has taken the magic world by storm over the past few years. Dynamo is from the mean streets of Delph Hill Estate, Bradford and this has shaped his performance style.\nQuantity isn\ufffdt everything when it comes to being a magician, but you want to have a wide range of magic tricks in order to become successful. That doesn\ufffdt mean you should try to take on ten tricks at once though. Learn them one by one so you don\ufffdt confuse yourself and complicate things.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaxvfz b/data_all_eng_slimpj/shuffled/split2/finalzzzaxvfz
new file mode 100644
index 0000000000000000000000000000000000000000..e615ab6f38262332b51d3a159f4b0aa90662191c
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaxvfz
@@ -0,0 +1,5 @@
+{"text":"FULL VIDEO: 2015 FIBA 3X3 World Tour - Manila Masters - Balita Boss!\nThe 2015 FIBA 3\u00d73 World Tour is all set for grand opening in two-day tournament in the Philippines. The Manila Masters, the first stage of the event, will be played in Robinson's Place Manila from August 1-2.\nThe FIBA will stream the basketball action on their official YouTube channel.\nThe competition will start at 2:10 pm on Saturday. The top-two placed teams from each pool advance to a standard knock-out round which starts at the quarter-finals stage, from 1:45 pm on Sunday.\nThe Philippines will enter three teams in the competition, the defending champion Manila West, Manila North and Manila South.\nWatch day 1 of the 2015 FIBA 3x3 World Tour tournament in Manila, live on the FIBA 3x3 YouTube channel.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Champion is one of the world's most respected brands, with over 100 years' experience in the manufacture and supply of original equipment (OE) products. Our service parts cover automotive, racing, power sport, as well as marine, industrial and small engines. With a passion for motorsport, the Champion name brings the aftermarket high quality performance-driven products that give technicians and drivers the edge.\nAs a leader in precious metal technologies, Champion is a full-range provider in the areas of ignition, wipers and filtration.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Gaming has been growing in popularity over the last decade and there are few homes that don't have at least one games console in them. It is no surprise that we have been seeing more and more of them coming into us in need of repair over the last number of years.\nThe picture above is an XBOX One that a customer brought into the store yesterday, they reported it was making loud noises and had a tendency to overheat. Gyula, one of our technicians ran diagnostics and could see a large build up of dust and suggested an XBOX One cleaning. Gyula was able to perform an XBOX One cleaning in just a couple of hours. Shortly afterwards we contacted the customer to let them know their XBOX One cleaning had been completed and was ready for collection.\nIf you have an XBOX One in need of cleaning or have another problem simply complete a short Book My Repair form to get your device collected anywhere in the country, or alternatively call into one of our two locations in Cork!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Morningstar Credit Ratings LLC Continues Stable Forecast For Equiant Financial Services Inc.\nfinance ranking of Stable from Morningstar Credit Ratings LLC.\ntraining and compliance program for employees and external clients.\nMorningstar Credit Ratings LLC is a subsidiary of Morningstar Inc.\nNorth America, Europe, Australia, and Asia.\nconference, contact Moody at (480) 636-4888.\nMoody at (480) 636-4888 or visit www.equiant.com.\nMaple Leaf Short Duration 2019 Flow-Through Limited Partnership NATIONAL & QUEBEC CLASS Total Gross Proceeds of $17,337,175 Raised.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What's keeping Okanagan winemakers awake at night?\nIt's guaranteed. As soon as winemakers get together, one of the first topics of discussion is going to be the weather. Weather has so much impact on the health of grapes and, thus, on wine and, thus, on the industry that feeds winemakers. Weather, after all, is the whole point of vintage charts! And with climate change talks the focus of G7 powers this year, it follows that the subject would arise often during my recent visit to the south Okanagan, the hottest wine region in Canada.\nI was invited by the Okanagan Wine Festivals Society to attend the 2017 BC Best of Varietal Awards in early May and to meet many of the feisty Okanagan winemakers who won some of those awards. Naturally, I asked about the weather. Everyone had an opinion. Then I asked, if climate change is not their primary anxiety, what is? What keeps them awake at night?\nWhat we've been seeing are extremes or record-breaking temperatures, precipitation, and events that are random and come at any time. We may be experiencing a really hot year, and then get an inopportune and super early frost. In recent years, I've been picking reds at the same time as Icewines. We scrambled to get the red fruit in and later were picking Icewine because we had the right conditions at minus eight degrees \u2026 in November! It was a disaster. That creates a big problem because your vines start getting diseases or dying off. And there are the logistical problems of all your fruit coming in at once. We have to scramble to find extra tanks and the place to put them. Worse than that, we've had a very wet spring, which we're not used to. When you can't predict what random event is going to happen, you can't really plan for it. So I think crop insurance is going to be a big thing in the future.\nSelling wine! Selling wine is a big thing. There are so many individual products accessible to a wine lover. We have wines from all over the world. When you go to countries with wine regions, you're lucky if you can find wines from other parts of the country. Here, we have so many producers that we're vying against and at all price points. It creates a downward pressure on our price points. And other beverages exist too \u2014 people can have beer or distilled spirits. They can have marijuana. It's a very competitive marketplace.\nThe effects have been spotty. This has been an odd year. Typically, we get two or three days when we can pick Icewine. This year, there were about 30 of them. Long term, water will become a massive issue in the Okanagan, the same way it is in California. Last year we didn't need to turn our irrigation on \u2014 we got that much rain. Mildew was an issue. We have some concerns, but we just have to monitor it.\nBeing able to keep up with the demand of sales. We sell the wine faster than we can make it. We could lose shelf space in stores and listings at the few restaurants we're already in if we can't supply bottles. We've doubled our white production and increased reds by one-third, and it's still not enough. I don't have enough wine.\nEvery year is a different year, so we react to the year. It's not global warming, it's climate change. Last year, we were three weeks ahead; this year, we're three weeks behind. You can't predict, you can only react.\nJust getting it right. You get one shot every year. Finding the balance, getting the consistency, doing it right.\nThat's where that \"sustainable\" word comes in, and why it's so important to be focused in a holistic way. Water is definitely an issue. What we've seen is that rain is not coming through properly in spring, but being pushed back two months later. That means we'll have to turn on our irrigation and now we're consuming water. We need to be focused on our water use in the vineyard, but also, hugely so, in the winery. There's just no way that wine can be a product in the future unless we can get its use way, way down. It's the biggest concern for the wine industry. For the whole world, really.\nThe end of the season. We have only a tiny, tiny winery at the moment, so there's only so much we can process in one day. The change of the seasons has us starting earlier and finishing earlier. Instead of thinking as if this was going to be a normal year, you're always wondering what's going to happen this year. You have to spend a lot more time in the vineyard than you used to and make sure you're tweaking every vine every 10 days and considering the next course of action. Do we keep the leaves on? Do we take them off? Do they need water? Every single day is different so keeping an eye on those weather cycles, that's what keeps me awake at night.\nI've always had early seasons since I got here \u2014 our winters are super mild, at least milder than what the old timers tell me they used to get. We get more of those torrential rainfalls now, but for the most part, we haven't seen much change.\nLabour. Finding good labour and keeping it. Keeping up with Mother Nature \u2014 'cause Mother Nature runs the show, eh? \u2014 so you have to get those jobs done in a timely fashion. My labour force is still a transient bunch \u2014 soon as the cherries come on, they go off and follow the cherry crop till that ends. That's the farming game for you.\nOften people equate climate change to an increase in temperatures. That isn't what we're seeing. What we're getting are storms, and a real inconsistency in the weather. Our hottest season was 1998 \u2014 almost 20 years ago. A few seasons since have come close but none has matched it. Our preparations to adapt to climate change centre around access to water. We already see smaller snow packs, which reduces pour flow of runoff. So, we've converted all our vineyards from overhead irrigation to drip. We spend a lot of time trying to build up the organic component of our soil for better water retention to reduce our need to replace it. We're studying cover crops and how to manipulate them to reduce evaporation of water already in the soil. And we're trying to reduce our carbon footprint.\nAs the demographic ages, farm labour is becoming problematic for us. It's getting harder to find farm workers. That's the big one. We try to hire locally as much as possible.\nPersonally, I believe in cycles. We have neighbours who have been farming for 50 years and they say by no means is this the hottest summer, the wettest summer or the driest summer. They say in each time period, there's going to be a certain percentage of hot, wet or dry years. These days, the differences are appearing much quicker. So, we've been learning to adapt quickly to each year.\nAchieving consistency. Being proactive in the vineyard rather than having to react to problems.\nSeeing Obama get China to come to the table and agree to stop using carbon, I think that was a great thing. Then you have a wild card like Trump come out of the woodwork and he doesn't believe in global warming; that's always a worry. Hopefully he's hamstrung and unable to do anything. Yes, I have seen global warming. Last year, when we had a season with El Ni\u00f1o water currents, we had the earliest season I've ever seen. You learn to adjust what you're doing in the vineyard depending on if it's a cool or a hot year. We've seen a few degrees of warming, but from our perspective, we can adjust for that because we're already in a cooler area.\nGetting sick and not living a century. Really? Look at me. I'm playing my own game and having a ton of fun. I live in a place that's unbelievably beautiful. I have a successful business. I work with a great crew of people. Why would I worry?\nDefinitely, it's getting hotter. New plantings are moving farther north. We've had recurring fires since 2003 and will continue to get more. I'm currently working on a \"smoke taint\" research project for the BC Wine Grape Council because the time between now and the next fire is finite. It's going to keep happening every time we have a hot summer and we need to know how to deal with it.\nSales. If we don't sell, then we won't empty the warehouse, and if I can't empty the tanks, then we won't be able to keep making wine. I think about how to make people aware of how special British Columbia wines really are.\nI'm not an expert on that, but in the short period of time I've been here, we've never had this much rain \u2014 ever. People used to tell me that it would hardly ever snow here, but in the last four years we've had a lot of snow down here in the south Okanagan. I won't say I don't believe in climate change, but I don't know how to answer. For us, we always deal with Mother Nature and mitigate it in the vineyard all the time.\nNothing. There's nothing I worry about. We're reactive. I continue to work with our fruit here and learn from it. What's there to worry about?\nI expect in 50 to 100 years, they won't be making wine in the southern hemisphere because it will be too hot. There's no question it's gotten hotter and drier here, too. The last four to five years have been progressively the earliest we've started picking the fruit.\nWe put our toes in the water of the super-premium end of the wine market last year with the introduction of the MacIntyre Heritage Reserve line. The Lake Breeze brand is doing well. We sell pretty well everything we make each year. We don't have to put much effort into it since we have a good name and a good reputation. But the economy hasn't quite rebounded, so getting our two super-premium wines out to the people we know will want to drink them is our biggest challenge right now.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbalqm b/data_all_eng_slimpj/shuffled/split2/finalzzzbalqm
new file mode 100644
index 0000000000000000000000000000000000000000..6a6ab92f2c05a5db12c99d9511ffd1765ab252e8
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbalqm
@@ -0,0 +1,5 @@
+{"text":"The door bell rang impatiently, twice. He cursed under his breath and struggled to get to the door. He was taken aback to see Koel and his Mom standing outside. A sudden rush of emotions sprawled over him. He didn't know if he was happy or angry, seeing her. He didn't know if he was grateful or displeased with Koel for going against his wishes and calling his Mom. He shared a cordial relationship with his Mother,and that was it. Perhaps, he had never given her a chance after the divorce of his parents. He had never seen his Father after their separation, and from the age of 11 growing up without a Dad, he had always been bitter towards his Mom as if it was all her fault. His Mom was a no-nonsense independent woman. She had provided him for, till his college and then had instructed him to move out of the house and make a Life of his own.\nSitting opposite him, he couldn't but notice the similarities between the two ladies in his Life. Not physically, but characteristically. \"Son, I respect both of yours decision of a court marriage. I am not at all in favor of frivolous spending. But this is a celebration of your Love. So I've kept a Luncheon for my beautiful daughter on 6th\". Saarth looked blankly at his Mom. He could sense both joy and pride in her voice.\nThe rest of the day had flashed in a second. He had completed all his pending work at office before going on the one month leave. Saarth reached home, to find the women giggling and chatting. The 7 was cornered to the wall. It wasn't intimidating anymore.\n\"No dreams...\" Saarth was slightly disappointing that he was having a peaceful night. He had half expected his Dad to come visit him in the night with a cautionary board of \"Do not marry!!\"\nSaarth picked up Dodi and his parents from the airport. He was meeting Uncle-Aunty, soon to be his in-laws after ages. He was very much aware of their presence in the back of his car.\nTheirs was such a different set up, all of them so close, a small knit happy family. And he was going to be a part of it. Why did he feel so uncomfortable then? He had never felt like this before. He remembered all the times, Dodi and he had spent in his room as kids, perched upon the bed engrossed in video games, only to look up when Aunty or Koel got them steamed Idly with coconut chutney. He had felt so much at home then. Why not today? The bloody 7 still sat haughtily on the wall and Saarth was unable to push it aside.\n\"Trouble sleeping?\", his Mom walked in and sat beside him. He saw tenderness in her eyes and he wanted to hug her. \"Son, I know you'll be a fine husband and nothing like your father. Just believe in yourself and take each day, each responsibility as it come\". Saarth was speechless. How well a Mom could judge the turmoil in her Son's heart. He wanted to say \"Thank you\", but what came out was a shaky \"Coffee?\".\nMom, Koel, Aunty, Uncle, Dodi and Saarth all were seated around the small oval table. Love, laughter and conversations had made the small room so lively. Saarth wanted to capture this moment and savor it forever. He was standing on the threshold of a new beginning and he wasn't afraid anymore. He looked around....but the white wall was the only thing that reflected back at him. There was no sign of 7.\n\"Aunty is an awesome cook. I'm still full!\", Dodi said patting his stomach. They had come up to their favorite Pub and ordered a pitcher. They had to mark tonight, his last day as bachelor, Dodi had insisted. Though, in his heart, all Saarth wanted now was to embark on the exciting journey that tomorrow held.\n\"Signed, sealed and delivered\", Dodi yelled ecstatically as they came out of the court. Koel looked ravishing as ever and the duo made a handsome couple. Saarth took her hand and kissed it gently, \"These past 7 days I've felt every emotion in the book from fear & uncertainty to love & hope, I must confess I was on the verge of a massive breakdown, but one look at you, one look at the future ahead of us, and I know given the chance, I'll relive every second of the damn 7 days!\"\nI'm taking part in the Write Tribe Festival of words, 1st - 7th September 2013. I want to thank Corinne for her efforts on getting all of us bloggers coming together and writing and exchanging ideas on the theme 'Seven'. I've had an absolutely wonderful time :) Thank you!!\nHehe yes absolutely right!! Thanks for staying with me throughout these 7 days Suresh Ji....hoping our rishta also goes a long way! Cheers!!\nThis ongoing saga was so much fun to read. I went through a similar turmoil when I married my husband 16 years ago. I worried I would make another mistake. I had been married before and it had ended very badly. I didn't want the same thing to happen again. Then I realized my husband was completely different from the first...that I loved him and he loved me. I gazed into his eyes, took his hand, and pledged my vows and never looked back. Things have turned out fine.\nNerves make an awful turmoil, but all was well in the end. Brilliant job with the festival.\nSo wonderful to hear your beautiful love story. Sometimes we do have our doubts, but where there is love there is always a way.\nSo glad to connect with you through this festival. Hope to stay in touch!\nTrue that Vishal! Love indeed has a way of putting all wrongs, right!\nDearest Aditi, you have a way with words. I admit due to time crunch i could not manage to read your posts on time.\nHope to learn many more words from you!! It was fun learning :) Lots of love to you dear!\nCongratulations on completing the challenge, I had to discontinue after day 3 :( Loved your story. One suggestion if I may,the color scheme on your blog made reading your lovely post a bit difficult for me.\nIt's ok if you missed few days...it was more imp that we all connected and had a great time exploring '7'!!\nThank you for the suggestion...It reminds we I have to work on the blog design etc etc, which I'm no good at!! :( Soon, hopefully!\nYou kept me riveted to the end. But I'm all for a happy ending. btw your Power of 7 post is featured on my festival wrap up post. Enjoyed your posts.\nAwww Suzy...so kind of you :) Million thanks for this!! I'm glad we connected through this festival. Hope to stay in touch!\nSeriously, on my wedding day, the photographer was alarmed by my demeanor. He sought out my mother and called her aside, saying that he was worried about me - I seemed \"too calm,\" and he had never known a bride to be so calm, less than an hour before the wedding. My mother came to check on me (I was getting ready, still) and we had a good laugh over that. \"If I'm nervous this late in the game, I ought to slip out the window and run - not look back!! No, I have dated him for two years. I know what I'm getting into. Why on earth should I be nervous??\" We've been married over 29 years, now.\nHaha Holly ;)...frankly this story came about just because I couldn't understand the term 'nerves' few of my male friends spoke so often when it came to marriage.\nWoh saat din!!! New concept, well presented!\nThanks Indrani :) glad you liked it!\nThanks a lot :) Good day!!\njus read the post and should say I thoroughly enjoyed it..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Shop for Epson 615 ink from Inkfarm.com and rest assured that you are getting high quality ink at an incredible price. On top of all of that, Inkfarm.com backs all of their ink with a 2-year, money-back guarantee. Try their compatible cartridges risk-free and experience their quality and page yield for yourself.\nWhen your printer runs low on ink, Inkfarm.com gives you several options for replacement Epson Workforce 615 ink cartridges. Purchase genuine ink in black, cyan, magenta, or yellow. You can also select compatible cartridges individually or in multi-packs. Epson Workforce 615 ink is available in a 3-pack of all black cartridges, a 5-pack with 2 black, 1 cyan, 1 magenta, and 1 yellow, and a 10-pack with 4 black, 2 cyan, 2 magenta, and 2 yellow cartridges.\nWhen you stock up on ink, you will not only see savings in the price but you will also benefit from the low, flat-rate shipping offered by Inkfarm.com. All orders, regardless of their size, ship for only $3.85. With 13 warehouses scattered throughout the United States, your Workforce 615 ink will typically ship from a facility near you to ensure an efficient arrival. Browse the selection of Epson printer ink, select the multi-pack or cartridge that is best for you, and have your ink delivered within a matter of days.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Chester's Northgate scheme has taken a big step forward today as a bid to secure the remaining land needed and change the street pattern were given the green light.\nCheshire West and Chester Council has received confirmation that the formal orders needed to acquire the land and change the road layouts have been approved by Government. Ministers also advised that the planned relocation of the market can proceed.\nThis means the mixed-use project, which was subject to a full public inquiry, can continue to progress and a report will be considered by Council in October to take the scheme forward.\nThe exciting development will include a new relocated dynamic and modern market, six-screen cinema, restaurants, a public square and car parking.\nIn the coming weeks and months design work and further consultation with a wide range of stakeholders to meet the long term aspirations of the city will continue. Work will also continue to attract tenants for the scheme in addition to those already secured including Picturehouse, Tapas Revolution, Zizzi's and Cosy Club, and with the landowners affected by the CPO outcome.\nThe Council is also reviewing the options for later stages of the scheme to reflect recent fundamental changes in the business environment for retail, and increase the relative importance within the site for good quality housing, leisure and cultural attractions, and high quality public realm.\nCabinet Member for Economic Development and Infrastructure, Councillor Brian Clarke, said: \"Today's news is fantastic and testament to the hard work of the Northgate team in successfully reaching this important milestone.\n\"Our aim is to deliver a mixed use scheme that's appropriate for the next generation rather than having to build something that's a legacy of the last 10 years' thinking. The positive decision shows a clear endorsement of the scheme by the planning inspector and the Secretaries of State who confirmed the order and applications.\n\"Northgate should not be seen in isolation. Its impact on Chester will be to support the whole of the city centre by improving our offer and encouraging more visitors. There will be significant consultation in the months to come which will allow the community and stakeholders the opportunity to shape the future of this part of the city.\nCabinet will now receive a report in October detailing the next steps.\nShadow Cabinet Member for Communities and Wellbeing, Councillor Stuart Parker, said: \"I'm delighted to hear today's news that we've been successful in securing the permissions needed to bring Chester Northgate to the city.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A week of football camp came to and end for the Bearcats, who will open the regular season at home August 26th against the Ste. Genevieve Dragons. Fans can get their first look at the 2011 Bearcats in action one week earlier as Dexter will host a jamboree on August 19 (Friday).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Is a common police procedure to put out a wanted criminal's photo, so that the public can inform the police if they saw the criminal. However, is it highly unusual to put out a child's photo on a wanted poster. Usually a child's photo will not be disseminated because they are underage. So why the police released a wanted poster with a children's photo in it? Read on to find out why.\nA Chinese police department was criticized for using suspected criminal's childhood photo on a wanted poster. Zhenxiong police in Yunnan said the suspect is Ji Qinghai, 17 years old, wanted for gang crimes and other violent offenses. Apparently, they were unable to find his latest photo and the only photo they had was 5 years old Ji. The police added that there was nothing wrong using the suspect childhood photo since his facial features are still somewhat recognizable.\nAfter facing criticism from netizens, the police department decided to remove the photo and post a public apology.\nThis is definitely strange. Can't they find his photo on social media? Pretty sure a 17 years old guy will have social media accounts.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbbwcb b/data_all_eng_slimpj/shuffled/split2/finalzzzbbwcb
new file mode 100644
index 0000000000000000000000000000000000000000..0a3df6d2e248b8b3b4b7228ef074e0cadcde2288
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbbwcb
@@ -0,0 +1,5 @@
+{"text":"Auckland's housing market lost a little of its steam last month as the total number of sales and average house price fell, according to figures out today.\nReal estate firm Barfoot & Thompson sold 100 fewer properties in January compared to December and the average sale price fell by just over $23,000.\nEven so, the city's property market still experienced its most active January for five years, the firm said.\n\"As always occurs at the start of a new year, prices edged lower than those pre-Christmas, but January's average price of $600,754, is the first time we have achieved an average price above $600,000 in a January,\" said managing director Peter Thompson.\n\"Compared to last January, the average price is up 13.4 per cent.\"\nIn December, Barfoot & Thompson sold 920 homes at an average price of $624,015.\nAlthough 820 sales last month was a drop, it was still the highest for a January in six years and 20.1 per cent higher than the same month last year.\nWestpac senior economist Michael Gordon said Barfoot and Thompson's figures \"threw up some curious results for the region's housing market in January\".\n\"Our seasonal adjustments should be treated with some caution, given that January tends to be the low point for housing turnover, but overall the figures appear consistent with a maturing housing market upturn,\" he said.\n\"There are tentative signs that the rate of increase in house sales is flattening out - up 20 per cent on a year ago, compared to a peak pace of 45 per cent yearly in October - but again we can only make a tentative conclusion.\"\nA reason for the drop in sales could be a lack of supply as at the start of the month, Gordon said.\n\"New listings were extremely low in December, followed by a sharp rebound in January.\n\"This is probably just a timing issue - taken together, the two months were much in line with the average number of listings over 2012.\"\nA total of 1440 listings last month was Barfoot & Thompson's highest in a January for five years, and 39.7 per cent higher than January last year. The firm reported 697 listings in December.\n\"January is always one of our lowest months for new listings, but interest in selling rolled over into January as a result of the excellent prices being achieved pre-Christmas, and our agents have been active since the first business day of the year,\" Thompson said.\nThompson said he expected to see strong sales numbers and prices in the Auckland market over the remaining summer months.\nOf total property sales, 383 went for under $500,000 and 68 sold for $1 million or more.\nGordon predicted a 9 per cent rise in nationwide house prices over 2013.\n\"Given that house price inflation was already running at around 6% by the end of 2012, our forecast actually implies that the housing boom is moving into a mature stage, with the rate of growth in both prices and sales starting to level out.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Modern day slavery refers to both human trafficking and the subsequent exploitation, also known as forced labour. What is human trafficking? In a nutshell, it is the unconsented transportation of a person against their will for the end purpose of forced labour.\n4. Physically constrained or have restrictions placed on his or her freedom of movement.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Discover how to make a safe and user-only area in your site with password-protected directories.\nA password-protected directory is a folder that can't be accessed unless the proper login credentials are offered. This can be a whole site or simply a single page which is located within a subfolder. If you attempt to open this kind of a directory, you will see an Internet browser pop-up where you must enter a username and also a password. In case you type the correct ones, you'll be able to continue exploring the content without the need to do anything additional. If the login details aren't accurate, however, you shall see a message that you are not allowed to view the content. If this function is active, you will not be able to open a file even if you have a direct link. The function is very effective if you need to restrict the access to some content or if you are developing a site and you don't want visitors to be able to access it prior to it being ready.\nCreating a password-protected area will take no more than several mouse clicks and a few seconds if you host your Internet sites within a hosting account with our company. You shall not need to do anything complicated - you'll need to select a domain\/subdomain, to pick which folder under it must be secured, and then to type the login name and password that'll be used to access that folder. If a number of people require their own login credentials, there shall not be a problem to create numerous different usernames with access to the very same folder. All secured folders will appear with a padlock icon within the File Manager section, so you shall always know where this option is active. If you wish to disable the security function for any folder, you can deactivate it from the same exact section of the CP where you have enabled it.\nIf you have a semi-dedicated server account with our company, you shall be able to protect any content you have uploaded using our protection tool. Its interface is as simple and intuitive as that of the Hepsia CP it is part of, so you'll not need to enter any code at any time. You will simply have to pick one of the domains\/subdomains that you have inside the website hosting account and to choose which folder needs to be password-protected - the website's root folder or some folder below it. You can then input the username and the password, that'll be stored in encrypted form in our system, and you will be good to go. The protection shall be activated instantly, so anyone who tries to access the recently secured folder shall have to type in the correct login details. In cases where a number of individuals have to be able to access the same content, you could make a separate username for each of them.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Fossil finds on another dig on an Indonesian Island show the Hobbits may have been around for much longer than first thought.\nNew fossil finds show the first large-bodied inhabitants of an isolated Indonesian island evolved to Hobbit-size, but they always remembered how to make and use stone tools.\nTurns out the Egyptians weren't the only ones who mummified their dead.\nWe used to think of sharks as primitive fish because the had cartilage instead of bones. Turns out there was a good reason why and it makes them anything but primitive.\nA 3.5-billion year old Pilbara find is not the oldest fossil: so what is it?\nThe debate has raged for years on the origin of life-like structure in an ancient rocky find in Western Australia. But is it evidence of early life?\nA bull male Eastmanosteus placoderm. Placoderms were the first creatures to evolve paired reproductive organs with a bony skeleton called claspers. Brian Choo & John Long, Flinders University.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You'll smell like Teen Spirit in the ZOE KARSSEN Stripes Loose Fit Cropped Sweat. This double knit sweater in Red and Navy stripes features gold buttons on the right shoulder, a crew neck and distressed details all over. Get your grunge on and pair it with your favourite destroyed denim.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbcazd b/data_all_eng_slimpj/shuffled/split2/finalzzzbcazd
new file mode 100644
index 0000000000000000000000000000000000000000..dd74e874f554f49dbaa8fb10d29683510aa935e3
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbcazd
@@ -0,0 +1,5 @@
+{"text":"I believe I can stop spreading germs to others by setting goals for hygienic habits.\nI set a goal of coughing into my elbow, not my hand because I know it will prevent the spread of my germs.\nI set a goal to clean my area at school, because I believe it will help to prevent spreading my germs to friends.\nI set a goal to wash my hands with soap and warm water by using 7 healthy steps.\nI set a goal of washing my hands after playing with my pet.\nI set a goal of washing my hands after playing outside with friends to reduce germs.\nI set a goal of not going to school if I'm sick to prevent spreading my germs to friends.\nI believe I can set goals for healthier habits to keep myself and others around me healthy.\nI set a goal to not spread germs.\nDid you set a goal too?\n,\" Health Literacy Database at Miami University, accessed April 25, 2019, https:\/\/dlp.lib.miamioh.edu\/healthliteracy\/items\/show\/286.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Are you looking to plan a low-key, laidback kinda family vacay in the near future? Well, I suggest that you look no further than Myrtle Beach. If you're going to take my advice and are willing to consider adding this family friendly destination to your vacation schedule, you're going to want to know where to stay with kids in Myrtle Beach. I recently went for a few days with my baby girl and my mom and we had such a lovely time. And while I'll get into what to do with kids in a later post, today we're going to focus on the hotel because, I don't know about you, but I'm at place in life where I want more than just a plae to lie down when I'm booking hotels. I want properties that really make sense for families and that's just what we got from the Island Vista resort!\nFirst off, the Island Vista resort is an oceanfront resort which is a must have when planning a Myrtle Beach getaway. Not only does it provide serene views which you can soak in while sipping on your morning coffee, but an oceanfront hotel provides built in entertainment. Give my kid a few beach toys and seemingly endless expanse of sand and she's good for hours. In fact, we visited the Island Vista in November and it was somewhat chilly. So, water fun wasn't on the agenda. That being said, my daughter was plenty entertained by the beach sans water. Just a little something to keep in mind when you're planning your vacays. Visiting in the off season can be loads of fun. And you just might discover things to do that you might have overlooked were you spending all your time in the water!\nNow the Island Vista does offer several pools and hot tubs, as well as an indoor water play area. We happily took advantage of those hot tubs, let me tell you. They were relaxing for my mom and me and they satisfied my kiddo who would be willing to swim in twenty degree weather if we let her. And let's be real. Is there anything more blissful than sitting in blissfully warm water while listening to waves gently lap on the shore. Sigh. No. The answer you're looking for is no. And not for nothing, but it's a great way to soak away all that holiday stress. Do you feel a winter getaway coming on? I thought you would!\nOf course we can't talk about a hotel without talking about the room. Like I mentioned before, these accommodations are perfect for families. Our room came complete with a bedroom and sitting area along with a full kitchen so we were able to have plenty of cold drinks and snacks on hand. And a murphy bed that was conveniently located in the living area meant we each had our own slipping situation which was clutch for this mama who wasn't keen on being kicked all night by my wild child. Haha! Oh and I can't forget the balcony. There is no better place to take in the ocean. I found myself getting up early just so I could catch the sunrise!\nAnd, finally, let's talk food. We had a full agenda so we ate out most meals as we wanted to experience as much of Myrtle Beach as possible. That being said, I understand that breakfast can be one of those meals that's easier to enjoy at your hotel. And you definitely won't be disappointed with the Island Vista breakfast, BUt if you don't want to have breakfast at the hotel every day, I recommend you visit an insanely good breakfast joint that's right down the road. Johnny D's is famous for its waffles and Benedicts with very good reason.\nWhat feels like a traditional diner is anything but traditional when you open the menu. Sure you'll find pancakes and eggs, but you'll also find things like coconut banana French waffles and Jambalaya omelettes. My pork belly Benedict was BEYOND delicious. The chef was kind enough to provide her recommendations and she suggested I switch out the English muffin for a biscuit and it was a game changer. You need to work Johnny D's into your agenda, trust me!\nThis is not a sponsored post. Accommodations and some meals and activities were provided.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"People will sometimes name soft skills as that \"X-factor\" or certain quality that makes a great leader. These leaders can tell a story that inspires and motivates teams to bring their best to an initiative. These leaders know how to listen and provide honest, empathetic feedback - so people open up about work issues. Simply put, these leaders inspire people to trust.\nHowever, any leader can improve these skills with practice \u2013 and many successful leaders look at soft skills and emotional intelligence as areas that call for continuous learning.\nWhether you want to hone your communication skills, get better at problem-solving, or need help dealing with difficult coworkers, there's a book on this list for you. Let's dig in.\nWhy you should read it: Leaders must be able to communicate effectively, and they'd be remiss to overlook the nonverbal aspects of communication. You may feel confident in your words, but your body language may be telling a different story altogether. This book offers simple, practical tips and behavior tweaks that can help leaders tap into their own power to keep their cool in stressful situations.\nWhy you should read it: Leaders with great soft skills can work with anyone. That's because they can easily adapt their leadership style to bring out the best in everyone \u2013 no matter how challenging the person may be. If you find that some personality types can easily push your buttons or that tense confrontations with difficult employees shake your confidence, this book can help.\nWhy you should read it: This book gets at the neuroscience behind why culture is so difficult for companies to improve. It all boils down to trust, says the author, and all the human behaviors and organizational dynamics that go along with it. For instance, there's a right way and a wrong way for leaders to show recognition and reward employees \u2013 and figuring out the right way takes soft skills.\nWhy you should read it: Communication typically tops the list of important soft skills for leaders \u2013 and it's one that nearly everyone can improve. You might be an amazing presenter but struggle with one-on-one conflict resolution, for instance. Higgins tackles both sides of communication \u2013 listening and speaking \u2013 to help leaders improve their skills, form better relationships, and advance their careers.\nWhy you should read it: The best leaders bring their creativity, focus, and energy to work every day. That takes mental clarity, which is hard to get when you are drowning in tasks, meetings, and priorities. You might take the Marie Kondo approach and declutter your work life by learning how to say \"no\" and removing the small things in your day-to-day life that distract you from your goals. This book can help.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"*#06# To check IMEI of your device, enter this code. *#0*# To enter service menu on the very new Android phones. *#0228# To check battery status. *#9090# \/ *#1111# To make device in Service mode. *#*#4636#*#* To get information about battery, usage statistics and device. *#*#34971539#*#* To get all information about camera. *#12580*369# To get software and hardware info. *#228# For ADC Reading. #7353# To hide test menu 2\/Self Test Mode. ##7764726 To hide service menu for Motorola Droid. *#*#273283*255*663282*#*#* For backup of our all media files. *#*#232338#*#* It display the Wi-Fi mac address. *#7465625# To view status of lock-phone. *#*#3264#*#* To show RAM version. *#*#44336#*#* To display build time and change list number. *#*#232337#*# To see or display device's Bluetooth address. *#*#197328640#*#* It enables test mode for service. *#*#8351#*#* To enable voice dial mode. *#*#8350#*#* To disable the voice dial mode. *#*#0842#*#* To test Back- light\/vibration. *#*#2664#*#* To test the touch-screen. *#*#0289#*#* For Audio test. *#*#0*#*#* For LCD display test. *#*#232331#*#* To test Bluetooth of any Android device. *#*#0283#*#* To perform a packet loop-back test. *#*#1575#*#* For advanced GPS test. *#*#1472365#*#* To Perform a quick GPS test. *#*#0588#*#* To perform a proximity sensor test. *#*#7262626#*#* To perform field test. *#*#232339#*#* Testing Wireless LAN. *#9090# To Diagnose configuration of device. *#872564# To control U- S-B logging. *#9900# System dump mode. *#*#7780#*#* Reset to factory state. *2767*3855# To format Android device. *#*#4986*2650468#*#* To get pda, phone, H\/W and RF Call Date. *#*#1234#*#* To know about pda and firmware version. *#*#1111#*#* For FTA Software version. *#*#2222#*#* For FTA Hardware version. *#*#7594#*#* To change power button behaviour once code enabled. *#*#8255#*#* To launch Google Talk service monitor.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At Paula's Everyday Petals & More, we customize designs with you in mind!\nWe're a local Dallas, TX florist delivering courteous, professional service and the highest quality floral and gift items around. Our experienced designers can create floral designs that will complement any occasion and meet your budget and style expectations as well. We also offer daily delivery service to all local funeral homes and hospitals.\nCall or stop by our shop in Dallas. Choose from our nice selection of Green Plants, Blooming Plants, Dish Gardens, Fruit Baskets, Gourmet Baskets, Gift Baskets, Baby Gifts, Candy, Greeting Cards, Home Decor, Scented Candles, Silk Flowers, Plush Stuffed Animals and more!\nCall our shop in advance at (469) 778-0004 to schedule your consultation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbcazv b/data_all_eng_slimpj/shuffled/split2/finalzzzbcazv
new file mode 100644
index 0000000000000000000000000000000000000000..9f42025b6441957281afdd3bff012c29631d1be1
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbcazv
@@ -0,0 +1,5 @@
+{"text":"Papillomaviruses are small DNA viruses. Their capsid is 52 to 55 nm in diameter and contains a double stranded, circular DNA of aproximately 8 kilobasepairs in length. Papillomaviruses typically infect cells in the skin or the mucosa, and replication is tightly linked to the differentiation of the host cells. To accomplish their productive replicaton cycle, they alter the regulation of the cell cycle. Such alteration can lead to aberrant cell growth and in some cases to cancerous lesions. Therefore, papillomaviruses are related to benign and malignant lesions in humans and animals. We are particularly interested in the pathogenesis of animal papillomaviruses.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Constructed of heavy duty water resistant 1000 denier Cordura for years of rugged use.\nDust flap and attachment loops inside bag. Large diameter 3 mm cord and heavy duty cord lock allows for a superior closure. Large reinforced grommet located on bottom. All seams double stitched, taped and reinforced. Bags are measured for 1\/2 inch rope. All bags are available in blue, red, black, yellow, orange, terra, multi-cam( 15% price increase).\nMedium Bag Capacity: 225 ft.\nLarge Bag Capacity: 300 ft.\nXLarge Bag Capacity: 450 ft.\nXXLarge Bag Capacity: 600 ft.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome. We're glad you're here. We're excited you want to experience the difference about Dearmart can add more wealth in your life.\nFirst things first. To join the Dearmart family, you'll need to contact the person who referred you to us. You can also find the person near to your city or state by searching people. If you're ready to achieve your dream of independence and security, our generous, industry-leading compensation plan will help you get there. Through our three-level approach, we've developed an efficient structure to help you build your business with compensation that rewards you as you grow.\nEvery business needs a solid foundation. With our Rising Star Team Bonus you can achieve abundance as you progress from distributor all the way up to Director.\nWith an established business and a passion for inspiring Digital and Technology through Dearmart, you're ready to take the mission worldwide as you lead others to success.\nDearmart's unmatched Compensation Plan gives you several ways to earn generous commissions every day, week in direct proportion to your ability to promote DEARMART's concept to your customers and build an organization of Independent Associates or distributer who do the same.\nBuild your own team by promoting Dearmart concept throughout india and earn handsome commission on every sales made by you and your downline.\nWe connect most attr and demanding digital media where we guarantee to provide lifetime income opportunity and commission. Earn commission on personal and team recharge\/Bill payment\/DTH recharge\/Affiliates\/ecommerce and many more tools upto 12 level.\nEnergize your business, build teamwork, and enjoy amazing rewards when you participate in the many fun and challenging contests and leadership incentives Dearmart offers throughout every month and year. Incentive and team building prize will be announce time to time.\nReceive a lucrative reward for helping new team members duplicate your success. Become a top leader who helps others succeed and you will be eligible to receive a share in the profits for your excellent sales performance. Anyone can becomes Distributer to Director as per their team size and dedication. We value every hardwork and dedication of every associates and appreciate through lucrative reward and higher.\nDearmart is brand new concept in India creating business opportunity to large group of people around India. The Products produced using the DIGITAL technology will be best alternative to a large individual or small business owner. The company has proposed to have one franchisee in each of the district of each and every state of India. Indian market is typical in its nature. It has largest population of high income and upper middle class people, who are willing to constructing house with all the fantasy and passion to the level of rich cream of people but with a lower budget. The caption \"Look Rich but costs less\" is the breath of this group of people. Considering these facts, DEARMART provides the finest services at a lowest cost through its Franchisee Partners.\nAt DEARMART, we help people get where they want to go. We provide powerful products and services that restore wealth and help people to provide lifetime income opportunity. And we have a powerful business opportunity that transforms lives more precious.\nDEARMART combines your desire and drive with an innovative business system that has part of multi-Billion dollar Industry and future of India. You can go as far as you want, as fast as you desire. In short, it will turn your dreams into reality.\nAlmost every one of us uses mobile or any internet related services on a daily basis. We cant imagine our life without mobile and internet or we can say DIGITAL and TECHNOLOGY touch or connect us in every moment and every time.\nDIGITAL and TECHNOLOGY is a next growth of india and we Indian adopt Technology very quickly. Friend till date we have been using technology for knowledge purpose but now we will use technology as a new era of income. That's why we start a business having a strong advantage over traditional network marketing opportunities. Simply by helping friends, family and new contacts save money every month on their Recharge, bill pay, Ticket booking and affiliates and start to begin residual income immediately.\nYou will becomes independent distributer and immediately start building Team and Business. Earn handsome commission on Every business made by you and your downline.\nAs soon as you activate your package, you can use all services of dearmart.\nAlso get attr Microwebsite for your business in the name of your user ID for lifetime.\nThe investment to get started is minimal, and there is never a renewal fee. To help Consultants reach their fullest potential, Ambit has created a smart, easy-to-use Consultant support system designed to provide the business-building tools, guidance and unlimited training Consultants can use every day for achieving greater results. And after the initial investment, this support is absolutely free.\nBenefit to Join DEARmart services Pvt. Ltd.\nThe investment to get started is minimal, and there is never a renewal fee. To help Distributor to reach their fullest potential, Dearmart services Pvt Ltd has created a smart, easy-to-use and lifelong Business opportunity plan that designed to provide the business-building vision and unlimited income. Our Distributor can use every day for achieving greater results in the initial investment.\nAt dearmart every hardworker associates will achive highest sales commission on every sales made directly by him or his downline team.\n-\tIf you introduce upto 5 direct in a month to your downline you will earn Rs. 300 direct introduce commission.\nNote: 1 Month calculate from the 1st date of every month.\nDomestic money transfer services work as a NEFT or IMPS services. At dearmart we will provide additional income opportunity where any distributer can send money through Dearmart portal.\nTravelling is an essential part of everyone's life wheather it relate to business tour, family tour or any other tour. we spent almost 10% to 25% of our income on tour and travels. We happy to announce at dearmart you can save and earn upto 9% commission on ticket booking made by you every time.\nWe cover almost 400 cities and 300 Bus tour operators throughout india Flight Ticket Booking. We connect through most prominent domestic and international airlines. Book any flight ticket and save money as well as earn commission on evry booking.\nAll operator wise same commission upto 12 level.\nAlso every recharge made by downline upto 12 level, appropriate commission will receive by every distributer.\nPrepaid recharge commission- Data card recharge commission- postpaid bill payment commission-DTH recharge Commission.\nWe believe every hardwork should be appreciate which help distributor to achive and inspire to better work.\nDearmart is brand new concept in India creating Digital business opportunity to large group of people around India. The Products produced using the DIGITAL technology will be best alternative to a large individual or small business owner. The company has proposed to have one franchisee in each of the district of each and every state of India. Indian market is typical in its nature. It has largest population of high income and upper middle class people, who are willing to constructing house with all the fantasy and passion to the level of rich cream of people but with a lower budget. The caption \"Look Rich but costs les\" is the breath of this group of people. Considering these facts, DEARMART provides the finest services at a lowest cost through its Franchisee Partners.\nDearmart services pvt ltd takes full responsibility when it comes to supporting your business operations right from selecting services, training, marketing support etc.\nDearmart services pvt ltd would provide all the marketing and printing collaterals which are required for marketing purpose at the franchised centers.\nDearmart services pvt ltd would provide initial training to the franchisees irrespective of their experience and would ensure regular training programs for the franchisee staff to enhance the skills and efficiency.\nDearmart services pvt ltd would provide the comprehensive operation manual comprises of detail information about all the procedures, measures and activities required for running the center efficiently.\nDearmart services pvt ltd would organize events for promotion purpose at regional\/national levels. We would also guide the franchisees to ensure regular coverage of Dearmart.\nEvery franchisee will get additional discount on services and product offered by Dearmart services pvt ltd.\nParticipant should be 18+ age and above.\nPAN and Bank Detail are mandatory for getting payout.\n10% TDS and 6% handling and processing charges should be deduct from every payout.\nParticipant nominee should be in blood relation and 18+ of age.\nReward and bonus can be change time to time without prior notice to participant.\nPayout will be distributed to every week on Saturday through NEFT or other payment mode.\nUser account will not be transfer to other.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As Sadr Goes, So Goes The Gop?\nHad an interesting conversation with a top Republican senate aide the other day, who has come to the dispiriting conclusion that nothing is as important to the GOP's fortunes this November as the facts on the ground in Iraq and that the U.S. doesn't have ultimate control over them.\n\"I guess that makes Muqtada al-Sadr the most important man in American politics,\" I said.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You've changed things. Forever. There's no going back. See, to them you're just a freak. Like me!\nWhat have we got? Nothing. No name, no other alias. Clothing is custom. Nothing in his pockets but knives and lint.\nEvening , Commissioner. Why so serious?\nPeople are dying. What would you have me do?\nEndure. You can be the outcast. You can make the choice no one else will face. The right choice. Gotham needs you.\nA little fight in you. I like that.\nThen your gonna love me.\nNow that's more like it Mr. Wayne.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbczbc b/data_all_eng_slimpj/shuffled/split2/finalzzzbczbc
new file mode 100644
index 0000000000000000000000000000000000000000..fc3330ed23ac525c2db3606b4eea0b6d5ca60a8f
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbczbc
@@ -0,0 +1,5 @@
+{"text":"ImageMagick and PerlMagick Movie Making Interface tool. Use this form to marry several gif files together into an animated gif. This form takes a series of image url's you specify. Images can be gif, jpeg, or png only. It fetches these images and combines them into an animated gif movie. You can set several preferences below.\nEnter each image url below. One url per line. The number of image lines you enter will be the number of frames in your animated gif file.\nHere are all 9 pie gifs combined into the movie at the right. Click the Example button to demonstrate.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The trend is clear - the people want beauty and color and softness and fortunately these are things that great Vintage pieces can deliver in style. Here's a few examples to highlight the trend.\nA few years ago these would have been languishing at the back of the store waiting for someone with a crystal ball to realize that their time was coming. We even heard someone refer to them as Grandma chairs - but where they see Grandma we see timeless classic - we even see romance.\nBeautifully made in the 1970s they're a classic French provincial style and come upholstered in a rich chenille fabric. They're generous in size and super comfortable and come with matching ottomans. The thing about pieces like these is because they're so well made they will last a lifetime - the bones are so good that as the fashion changes all you need to do is swap out the fabric and print.\nCoffee tables have been barely noticeable for the last decade or so - they have been functional pieces delivered in clean chrome or a classic understated piece of stone but things are changing. Check this table out - it's the exact opposite of big rectangular tables. This is beautiful, it's flowing, look at the softness of the wood - the complete rejection of the antiseptically clean chrome of the recent past.\nLook at the legs - the smooth curves and the detail at the feet - this is a very Italian style and of course has that classic elegance about it. Where table legs have been heavy teak or walnut wood on sturdy dowels - this is metal, worked to perfection. The tray table itself plays with convention - typically you'd see a metal trim but here the designer opts for wood which softens the look of the table and gives it a more contemporary feel. And of course, the glass top allows you to see what's underneath.\nThis kind of style is what we call transitional - is it Italian, is it American, is it Modern? In reality - it is all those things, it's the kind of table that can grow with you and allow you to chnage your style with the times.\nSo we all know the classic sofa of the last fifteen years or so - it would have a low back, square arms, square feet, simple cushions, not too much detail and certainly no tufting or fancy fabric. Well goodbye function over form and welcome to comfort!\nThis 1940s \/ 1950s French inspired sofa knows that function, when it comes to a sofa, is all about making you feel luxurious when you sit down. It's whimsical - it's slightly exaggerated in its proportions - the seat is low but the back is so high, with its rolled arms it just wraps around and hugs you as you sit in it. Beautifully constructed down cushions and this marvelous mohair fabric in a \"go with anything\" steel color. Mohair is a very traditional fabric, as a natural fiber it's cool in the summertime and warm in the winter, it's expensive and super luxurious.\nThis is a prime example of a piece that can be placed in a modern design palette and yet it is super comfortable. You could be going for classic or mid century European and yet something so classic so - transitional - can work in any of those settings and still offer the comfort and luxury that you've been looking for.\nWelcome to the 1980's - a time where The Safran and Glucksman company were producing beautifully crafted cut crystal lamps. The attention to detail and the ornate decoration was incredibly desirable and yet by the time the turn of the century came people considered this to be too fussy, too old, too reminiscent of Grandma.\nBut have a second look at the quality of this lamp, how beautiful the crystal is, look at the classic shape, look at the mix of materials with the brass base. And of course the beauty of lamps is that a simple change of shade can alter everything. This was one of the most liked articles on our Instagram feed - a sure way of knowing that beauty and style are making a comeback.\nStraight away you can tell this mirror has more going for it than the soulless functionality of the last decade or so. Where mirrors have been all right angles, hard corners and in many cases, no frame or decoration at all, this 1960s, probably Italian or French, hand crafted frame takes us back to the future.\nIt's the craftsmanship that defines a lot of these transitional pieces - the roses are all hand welded and hand applied giving the mirror that beautiful sweetness and again, whimsy. There's a soulfulness, a tenderness to the piece that plays into what our customers are looking for.\nWhether you place it outside or inside, in a living room or a boudoir - the piece can hold its own in many different design styles, an another example of the transitional nature of classic vintage pieces.\nWhen you've been in the business as long as we have at Casa Victoria, you see trends come and go, so it makes sense to put your money into pieces that can move with you. When you shop with us you know that the pieces will have quality, that they'll have staying power - that they can be transitional as you transition your taste.\nAs Coco Channel said - Fashion changes but Style endures - it's pieces like these that will always retain their style and regardless the trend, they'll always be classy - just like you.\nSign up for our twice a month newsletter. We let you know about the hot new items we've received in the store and every now and then Mateo will write a longer piece on how you can use Vintage to improve your life!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What are Organic Raw Cacao Nibs?\nThe Cacao Bean is the dried and fully fermented seed of the Theobroma Cacao tree from which cocoa solids and cocoa butter are extracted.\nThe raw beans are selected for the finest quality and fermented to reduce their initial bitterness. After cleaning the dried beans in natural water, they are peeled and their core extracted to be broken into \"nibs\". The production process employed helps to preserve the nutritional value of the cacao without using additives. Peruvian Cacao Nibs are certified organic by the Soil Association under the code GB-ORG-05.\nOrganic Raw Cacao Nibs are a 100% organic, high fibre source of dietary fat with a great chocolate taste.\nOur organic cacao nibs are packed full of goodness so why not try adding 10-15g to your favourite smoothies. Alternatively, use as you would chocolate-chips in baking, or to granola, muesli, biscuits or brownies.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Android 4.2 is the latest release of Google's mobile OS, but the company is still calling it Jelly Bean. It's not widely available just yet as people are still upgrading to the previous incarnation of Jelly Bean \u2013 Android 4.1. That's a good thing too as you may want to hold off on upgrading to 4.2 until Google fixes the numerous problems plaguing the OS.\nPC Magazine reports that the entire month of December is gone from the People app in Android 4.2. It's not a \"huge\" problem unless you have a friend whose birthday is in December. At that point, you may want to bust out an old fashioned paper calendar and jot it down until Google fixes the issue.\nThe missing December may be getting the most media play, but 4.2 is proving that it could have used a bit more work before Google released it. Android Police has a great breakdown of all the bugs and issues currently affecting Android 4.2.\nFirst up, there's an auto-brightness bug in Google's Nexus 7 and Nexus 10 devices after upgrading to 4.2. The bug makes it so that the screen brightness randomly readjusts itself when set to auto-brightness. The only solution is to manually set the brightness until Google can issue a fix.\nUnfortunately, the rest of the bugs aren't as \"harmless\" as randomly readjusting brightness levels. At last that can be addressed with other options, but Android 4.2 has some critical bugs that Google needs to take care of ASAP.\nFor instance, Android Police reports that Bluetooth is broken after the Android 4.2 update on the Nexus 10, Nexus 7 and Galaxy Nexus. Audio streaming via Bluetooth \"works,\" but it's reportedly choppy and leads to a subpar experience. Even worse, apps that connect the device to Bluetooth enabled devices like game controllers don't even work.\nThe same three devices are also randomly rebooting and experiencing major instability problems in Android 4.2. There are reports that the Nexus 7 and Nexus 10 are experiencing the worst of it with the entire system locking up and rebooting in the middle of some inane task like opening a Web page in Chrome or watching a video via YouTube. Google may want to address this issue first as consumers are unlikely to tolerate a randomly rebooting device.\nFinally, it seems that Android 4.2 has led to device batteries draining faster than usual. Even worse, it seems that batteries are also charging slower than usual. The culprit appears to be Google Play services as they eat into the battery more than anything else after the update.\nGoogle will need to get on these issues quickly before more people start buying Nexus devices for the holidays. The Nexus 7 and Nexus 10 are both generating quite a lot of interest, and the devices could be Google's ticket to becoming a major player this holiday season. These bugs, however, could lead to a lot of returns and unhappy customers that give up on Android. Google can't afford for that to happen so here's hoping the folks on the Android team are working on some fixes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Charcoal (Black) In-Stock Now!Gold Accent In-Stock Now!\nWe heard the customer requests for a large size Sun style chiminea and now it's ready to ship. Easy to light and clean burning traditional style chiminea (due to proper drafting), this chiminea continues The Blue Rooster Co. line of unique chiminea designs. Made of heavy cast aluminum construction up to 3\/4 inch thick in some areas, the Sun chiminea is built to last. The Blue Rooster recommends the Charcoal color because of its lasting qualities in a high heat wood burning fireplace application. Spark insert included for the chimney and a stainless steel hinged mouth screen to go over the mouth for extra safety. Mouth screen easily swings open for adding wood or toasting marshmallows. Includes a grill grate for late night burgers.\nCast Aluminum ALCH029 Approx. 52\" x 22\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbdcdo b/data_all_eng_slimpj/shuffled/split2/finalzzzbdcdo
new file mode 100644
index 0000000000000000000000000000000000000000..cd2319b8cd4b604f0dfab8756e6456b865163940
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbdcdo
@@ -0,0 +1,5 @@
+{"text":"FX Global Code ABN AMRO Bank N.V.\nThe FX Global Code is a set of global principles of good practice in the foreign exchange market, developed to provide a common set of guidelines to promote the integrity and effective functioning of the wholesale foreign exchange market. It was developed through a partnership between central banks and Market Participants from 16 jurisdictions around the globe.\nThe purpose of the Global Code is to promote a robust, fair, liquid, open, and appropriately transparent market in which a diverse set of Market Participants, supported by resilient infrastructure, are able to confidently and effectively transact at competitive prices that reflect available market information, and in a manner that conforms to acceptable standards of behavior.\nABN AMRO has reviewed the content of the FX Global Code (\"Code\") and acknowledges that the Code represents a set of principles generally recognised as good practice in the wholesale foreign exchange market (\"FX Market\"). The Institution confirms that it acts as a Market Participant as defined by the Code, and is committed to conducting its FX Market activities (\"Activities\") in a manner consistent with the principles of the Code. To this end, the Institution has taken appropriate steps, based on the size and complexity of its Activities, and the nature of its engagement in the FX Market, to align its Activities with the principles of the Code.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The views and opinions expressed in videos we produce or showcase on our site ouruniverseforkids.com do not necessarily reflect the official policies or positions of IT Network Distributors Ltd. The content and opinions contained in this video are not intended to malign any religion, ethic group, organization, company or individual.\nIT Network Distributors Ltd makes no representations or warranties of any kind, express or implied, about the completeness, accuracy, reliability or suitability with respect to any content contained within videos on our site. Videos we create or showcase are for entertainment purposes only. Viewers should not rely on any information contained within these videos. If you have any issue, contact a professional immediately.\nIT Network Distributors Ltd is not responsible for any loss or damage, including without limitation, indirect or consequential loss or damage, or any loss or damage whatsoever arising from viewing the videos presented on our website.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"China Film Cinema of Suzhou New District (Ganglong Branch) is a holding company of China Film, which has more than 90 cinemas around China. The cinema in Ganglong Commercial Plaza spans an area of 4,000 sq m. It has nine elaborately decorated movie halls with more than 1,000 seats, and provides world-advanced audio-visual equipment and thoughtful service. It always ensures that its supply of snacks adds to the viewing audiences' enjoyment.\nBus Route: Bus No 44, 301, 302, 325, 322, 442, get off at the station of People's Hospital of Suzhou New District.\nRestaurants: Neolithic Barbecue, Xibei Restaurant, KFC, PAPA JOHNS, Starbucks and Jinfushan Self-service Barbecue.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Legal and law-related journals \u2013 over 2,000 of them!\nThe really great news is that access to Hein is available to all Law Society members through a subscription worked out by the Law Society of Upper Canada. You can search, read, and download all without paying an additional cent. If you'd like the username and password for Hein, let us know and we can give it to you. It changes roughly twice a year, and did just last week, so you might need to update your information! Keep our email address (library@ccla-abcc.ca) handy!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"New Series: Flowers with a twist!!\nI don't often paint still life scenes, although early in my painting days (2009-2011) I painted a LOT of pears! So this was really exciting to depict flowers with a new approach. Each of the paintings below began with a foundation of bright primary colors. As the subsequent layers were added, some of the bright backgrounds and other layers were allowed to show through.\nPink Poinsettia, 20\u2033 x 16\u2033 acrylic on canvas. Available.\nPink Rose, 20\u2033 x 16 acrylic on canvas. Available.\nGreen Mum, 24\u2033 x 20\u2033 acrylic on canvas. Available.\nGreen Hydrangea, 24\u2033 x 20\u2033 acrylic on canvas. Available.\nSingle Tulip, 20\u2033 x 16\u2033 acrylic on canvas. Available.\nI hope you enjoy this series; I'd love to hear what you think about them. Thanks for following me!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbfdok b/data_all_eng_slimpj/shuffled/split2/finalzzzbfdok
new file mode 100644
index 0000000000000000000000000000000000000000..34d7ef2fd2604f9b4d986979f2fcdee191049395
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbfdok
@@ -0,0 +1,5 @@
+{"text":"Spray On Sealer for Metal Roofs attractive Designs \u00bb Fantastic Information With Regards To Decor, Spray On Sealer For Metal Roofs Interior decorating is just as complicated while you let it get. When you are happy to learn and make time to enable new tips sink in, you will easily visit your accomplishment. The ideas in the post over were all well tested by many people prior to deciding to, and might very easily operate in your favour also.\nflex seal silver rubber spray sealant roof foundation from Spray On Sealer For Metal Roofs, Bring it along with you when searching for window curtains or some other redecorating supplies image via from: acehardware.com.\n17 year elastomeric roof coating from Spray On Sealer For Metal Roofs, Take it together with you when searching for window curtains or another redecorating materials sourced from: rustoleum.com.\nYou must choose Spray On Sealer For Metal Roofs what type of frame of mind you need the room to get before starting any style job. Feelings might be bold and outrageous or quiet and calming. When you know what experiencing you would like Spray On Sealer For Metal Roofs your living space to offer you, you will find that it's much easier to pre-plan projects to perform your room Spray On Sealer For Metal Roofs style.\nSpray On Sealer for Metal Roofs attractive Designs \u00bb Try to use flexible household furniture every time beautifying a lesser size space. An ottoman is an excellent decision. It can be used for chairs or possibly a desk, and you may even shop things on the inside, when it starts up. Items which have several purposes are effective space savers to get a smaller space.\nepox z videolike from Spray On Sealer For Metal Roofs, Bring it along when shopping for curtains or any other designing supplies looks got from: videolike.org.\nspray n seal instant rubber roof repair 16 oz leisure from Spray On Sealer For Metal Roofs, Bring it together with you when shopping for drapes or another decorating materials looks downloaded from: campingworld.com.\nroof coating chicago residential and commercial coatings from Spray On Sealer For Metal Roofs, Take it with you when shopping for drapes or another beautifying resources impression got from: innovativehomeconcepts.com.\nflexible roof sealant product page from Spray On Sealer For Metal Roofs, Take it together with you when shopping for drapes or other decorating materials screenshot sourced from: spr.ly.\nrooftrade black roof sealer 5l departments diy at b q from Spray On Sealer For Metal Roofs, Take it along with you when shopping for curtains or other redecorating components screenshot via from: diy.com.\nSpray On Sealer for Metal Roofs attractive Designs \u00bb Acquire your sofa cushion store shopping together with you. It may possibly feel silly, but using alongside a part of your couch could help you save lots of time any cash. Bring it along with you when searching for curtains or any other redecorating materials. This way, it will be easy to make certain that Spray On Sealer For Metal Roofs everything fits.\nSpray On Sealer for Metal Roofs attractive Designs \u00bb Lamps are a great addition to any house because these can be within the family room or right by your side within the bed room. In addition these products give you more light-weight to read through and publish, but they can give your own home a conventional appearance and match up many different styles.\nLet you to Select washroom lights cautiously. In a toilet, one particular overhead light will cast an undesirable shadow, Spray On Sealer For Metal Roofs making it tough to apply make-up or shave. A Spray On Sealer For Metal Roofs sconce on either side from the medication cupboard will give you an even quantity of lighting, great for proper grooming. Incandescent lighting is far better fluorescent lighting, which can create a bluish sculpt of Spray On Sealer For Metal Roofs.\nRelated Posts of \"Spray On Sealer for Metal Roofs attractive Designs\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1. Smart Weigh packing machines are of high efficiency. The linear weigher packing machine that is reinforced with 3 head linear weigher composite is unique and in the linear weigher for sale machinery industry only to be found in Smart Weighing And Packing Machine.\n3. Smart Weigh sealing machine offer some of the lowest noise available in the industry, Smart Can Supply Top Quality linear weigher, linear weigher machine For Home, Hotel And Business Places.\n4. More packs per shift are allowed due to the improvement of weighing accuracy. 4 head linear weigher,linear weigher china features linear weigher price that dramatically capture the eyes of users.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"High above the ground, the high ropes course, a series of ropes and cables strung through the trees, presents a series of challenging, physical and mental tasks. The high ropes course activity makes a high level odyssey through the treetops. On the Bloomsburg University Campus, Quest offers many high ropes course options for participants to include both individual challenges and challenges that need partners up in the air.\nLocated on Bloomsburg University's upper campus the Quest program operates a 50 foot climbing wall that allows participants to explore the unfamiliar vertical world. The wall is attractive to a wide rage of participant interest from the \"first timer\" who wants to experience the novelty of using hands and feet to ascend a vertical face, to the group who want to develop a higher level of trust and teamwork among the participants. The wall offers a variety of climbing routes of varying degrees of difficulty for all levels of skill and physical fitness.\nOur High Adventure Programs are available to any group. They can be customized to meet the needs and size of any team.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here is an email I got from a worried reader plus my answer: \"I have met a person online, I like him very much and I've invested quite a bit of time into writing to him, but there are certain things he says that seem a bit odd. People Search by Address - Reverse Address Lookup Oct 23, 14 08:35 AMAn excellent people search by address website - a reverse address finder - can be a very important tool.\nAQSIN_FATEH \u2014 01.07.2013 at 18:12:37 Take some time to teach yourself receive a physical injury or endure you have.\nshokaladka \u2014 01.07.2013 at 12:10:49 You want to lookup, then acquire your the OK or Cancel button is pressed businesses that.\nLanseloT \u2014 01.07.2013 at 12:22:32 Hub has tons?more valuable ways to use the live scan response alone may.\nKLan_A_PLan_Ka \u2014 01.07.2013 at 21:19:28 Individual enters the wrong pass should not pay hefty fines.\nSimpson \u2014 01.07.2013 at 19:47:14 Post produced warrants or arrest warrants the on-line charge-primarily based method is frequently utilized by the.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"No. And the approach of using dd (or one of my editors) to write a known quantity of data (or file with a known size in the case of trying to determine if the entire file is written or just the changed part) will be useful.\nSector Size: 512 bytes logical\/physical <I'm pretty sure that does not tell me the LBA size, (but, with a headache this morning, I'm not thinking very well.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbfrqh b/data_all_eng_slimpj/shuffled/split2/finalzzzbfrqh
new file mode 100644
index 0000000000000000000000000000000000000000..9c03bb95044445dfacf7cefaa06fb5b71173ff1f
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbfrqh
@@ -0,0 +1,5 @@
+{"text":"What is Clinical Reasoning and Why is it Important?\nThis study is published in the Journal of American Heart Association and led by Dr. Gregory Marcus, a cardiologist from the University of California-San Francisco.\nIt is a popular belief that regular caffeine intake leads to palpitations or extra heartbeats. This comes with the risk of heart problems, stroke, and even death in rare cases! However, this new study questions such a belief. This new insight is relevant for all doctors, nurses and medical students, since it is their job to inform patients properly.\nIn this study, 6,000 participants were included and their baseline food frequency and heart rhythm were assessed. The participants' frequency of coffee, tea and chocolate consumption were determined through a survey. Results showed that there was no difference in incidence of extra heart beats per hour in all levels of coffee, tea and chocolate intake. Furthermore, participants who consumed such products more frequently did not have extra heartbeats.\nWhich these results, the researchers recommend a reconsideration of the current clinical recommendations. Right now, doctors are uncertain about how to counsel patients regarding consumption of caffeinated products. The findings propose that patients should not be counselled to unnecessarily reduce their intake, especially when they like coffee, tea or chocolate. There are even recent studies that suggest cardiovascular benefits linked to caffeine sources such as coffee, tea and chocolate. Furthermore, a recent study also suggests that moderate coffee intake may prevent premature death. All the more reason to reconsider the current guidelines!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sounds good, but you need to know what time frame you are investing in! When all three Trends were down together, as they were in December, investors became overwhelmed. That was an unusual confluence of influences. Market Energy(Prana): Market Energy is a combination of these Trends within Trends. Currently, the Tactical and Intermediate Trends are Up and the Long Term Trend is Down. That diagnosis explains the Market Energy now. If you would like to know when trends are set to end click here We Have A Turn Date Coming SOON!!\nInterested in knowing when the Tactical and Intermediate Trends are Up are going to top? Subscribe NOW! The special offers below are for a limited time and only available to the first 20 subscribers.\nThe information contained herein was provided by Sentiment Timing does not make any representation or warrant with regard hereto, including but not limited to those of accuracy, completeness, reliability, timeliness and\/or infringement on the rights of third parties. This Publication expresses a view on the markets but is not intended to provide any specific recommendation to buy or sell any security. Investing is Uncertain and always carries Risk. Of Losses. Subscribers should always assess Market Risk parameters with their broker or financial adviser.\nInformation is for paid subscribers & may not be copied or distributed. \u00a9 Copyright 2018.\nWHAT IF THE WALL IS REALLY JUST A STREET?\nNEAR TERM: Stock Sentiment is really low. We all get that. Buy in 2022. That has been and remains my forecast. So recent events are not all that surprising. I figured this would happen but only in the future. You see, the problem with forecasting the future is figuring out just when that future is going to happen.\nSure stocks are due to bottom here and trade up into 2019 as I have been saying but who cares. The more interesting profile is when the next interim high is going to occur in 2019. You see, forecasting the future is not really that popular. One has to look into the future and the majority have their own notions about what their version of the future may be.\nMost people naturally accept the Present as a durable Reality. The Future is always different, often in different ways than we expect. For instance, if you suggested last Summer that stocks were dangerously elevated and that they were going to correct this Fall, let alone all the way into 2022, well that might have seemed ill-advised. Everything that hasn't happened yet always seems unrealistic.\nJoin Now And Get The Next Top Date-$99 for 45 days of reports.\nMeanwhile, I am pretty upset because all I wanted for Christmas was a \"Wall.\" A defined, \"Wall of Worry\" would be fine. That could be traded. But markets are appearing instead at the, \"Wailing Wall!\" This sort of grief is otherworldly. Well you don't always get what you want. I had thought the Wall might just be a Street. What if the \"Wall\" strategy has some sort of deeper significance?\nInvestors, are dominated by their belief systems. They believe but they do not know. As I have noted before, myth and metaphor speak to a deeper knowing of how human and investor behavior operate. History repeats. The psyche repeats. Markets repeat. The story about a Great Man who fell off a Wall and has some challenges getting back on his \"High Horse\" is really true. And what about, \"All the Kings Soldiers and all the Kings Men\" not being able to give him the boost back up he was expecting? Is there really a soldier character called \"Mad Dog\" in all this? Yes, you see, it really is true. The Bull Market was a useful and wonderful Myth as indeed the Bear Market will become.\nWhen we look at the Great Walls of History, they all fall under a similar Fate. The Great Wall of China may still be great but it is really just a relic with some touristic allure. Hadrian's Wall may be quaint for walking tours of Great Britain. The Berlin Wall was dismantled and that provided quite a party atmosphere. There are lots of stone walls here in Vermont but most of the sheep they restrained are long gone and while the remaining walls may be of mild aesthetic interest, many of them are quite frankly, just a nuisance to navigate in the woods. Then there is the Maginot Line which was a famously ironic de facto wall. It was the pure illusion of military security. It is only a footnote now but is a wonderful metaphor for protective folly.\nI have been to the Arizona border wall where it does exist in all its might and grandeur, and it must be admitted that it is a dominant feature of the landscape. The fact is, folks do find their way under, over, around and through that mighty Arizona Wall anyway. So it is hard to figure what that Wall actually does do.\nThe upshot of all this architectural ballistic insouciance is not so much about the shutdown of some government. But it is perhaps a fiendishly clever chaotic consequence that might Crash the Stock Market and then, in time, Crash the Economy. That could be kinda great because then the flow of folks into America might dry up. They may not want to come here. Things could go the other way! In other words, the proposed border Wall might actually prevent hungry, needy and such good folk from fleeing the U.S. for better economic prospects in Mexico and Central America? Sure it will take some time but it could happen?\nSo, if we build that big border Wall, it might not be so easy for our own people to go south! It sounds a lot like the Maginot Line. Does the D\u00e9j\u00e0 vu Metaphor Voodoo strike again? But don't count out good old Yankee ingenuity. Chances are, if such a border Wall happens, our own people will also find ways over, under, around and through that Wall. Would it just be simpler to all agree not to build it? Make Love not War! Build Legos not Walls!\nTequila could become the new Eggnog. Could Spirulina be the new Siracha? Look on the bright side, people. Things Change, Markets Change. Heck, even Climate Changes. It's Christmas. Now that Climate Change is really happening, Rain could be the new Snow? Galoshes could be the new Snowshoes? Heck I don't know but nobody else does either. Be prepared for anything.\nI had lunch with a regular investor a couple of weeks ago (not a student or trader of the market like you and me) and he has been getting really worried! I told him it was real simple, stocks have been going up for many years and have now entered a correction. He may want to get out of the market. So, he says \"but you can't be in cash can you? How do you do that?\" Cash is a four letter word. You can use it, you can say it. It is, CASH. That is still the Big Picture if you are \"worried.\" This investor called me yesterday and said he thought I had actually used a five letter word. He thought I said CRASH. Now, that would be rude. I did not say that.\nWant to only stick with the Big Picture? Repeat: 'Buy stocks in January of 2022. Buy bonds in 2038-ish.\" But markets may do a few different dances between now and then. Santa is de-icing his sleigh so, don't become complacently pessimistic. It is the Holiday Season people. Get off the Wall obsession. It is only a Street. And, Wall Street is a two way Street. See you next year on the Street. Wall Street. Either up or down. It might be a two way street again by then.\nWhat does predictive analytics means and how does it work? Sentiment Timings Predictive analytics model is comprised of investor sentiment-market timing turns dates and regular technical analysis. Woody Dorsey, who is considered by many on Wall Street, to be one of the best market timers in the world. He uses over 13 different market timing indicators to predict when the SPX is going to make a top or a bottom, weeks and sometimes months in advance.\nCLICK HERE To See Some Of Our Previous Calls-And Ask Yourself, What Could You Have Done With This Information?\nYou may have heard people talk about investor sentiment, but does anybody really know how to trade from it? The simple answer is NO! Or I should say, not many and the ones who understand how to use sentiment to enter or exit trades, will be the ones closing long positions at the top and moving their portfolio to the short side..and covering short positions at the lows and moving their portfolio to the long side. Sentiment Extremes will ALWAYS be seen at market tops and bottoms..as you can see in the sentiment chart below.\nWhat does predictive analytics means and how does it work? Sentiment Timings Predictive analytics model is comprised of investor sentiment-market timing turns dates and regular technical analysis. Woody Dorsey, who is considered by many on Wall Street, to be one of the best market timers in the world.\nTIME: Woody uses over 13 different market timing indicators to predict when the SPX is going to make a top or a bottom, weeks and sometimes months in advance.\nSENTIMENT: Once we have the dates to look for a top\/bottom, sentiment will give us the confirmation a turn is about to take place. Sentiment extremes will always take place at tops\/bottoms.\nImagine Being Able To Know Exactly When The S&P 500 Is Going To Reverse..Weeks And Sometimes Months In Advance..What Could You Do With This Type Of Information? If It Seems To Good To Be True\u2013Then See For Yourself..You Will Never Look At The Market The Same..I Promise You That!\nInformation is for paid subscribers & may not be copied or distributed. \u00a9 Copyright 2018. The information contained herein was provided by Sentiment Timing and\/or its publishers does not make any representation or warrant with regard hereto, including but not limited to those of accuracy, completeness, reliability, timeliness and\/or infringement on the rights of third parties. This Publication expresses a view on the markets but is not intended to provide any specific recommendation to buy or sell any security. Investing is Uncertain and always carries Risk. Of Losses. Subscribers should always assess Market Risk parameters with their broker or financial adviser.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The photo id for this photograph is 172, in the Public Houses album.\nNow a private house called The Old Pack Horse.\nJust a short way down the road in the 1970's was a large industrial unit owned by Associated Tyres. This was demolished and 4 cottages built in its place. They are called Pack Horse Cottages.\nIn the 1950 William Higginson ran a haulage business here on winter nights his sister useing an oil lamp used to reverse the trucks off the main road, in the late 60's Sam Longson bought the business to acquire the 'A' licenses which allowed the ownership and operation of the trucks. They lived in what was the Pack Horse In the late 40's early 50's I lived in the cottage with the bay window which was rented from Higginson's.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Brexit secretary David Davis has admitted to MPs that no assessment has been carried out to determine what would happen to Britain if it fails to strike a trade deal with the EU.\nDavis said it is impossible to predict how the economy will be affected if the country's negotiations with the EU fail, but said he might know more in a year's time.\nDavis admitted the only assessment carried out by the government since the EU referendum in July last year had been the gloomy forecasts of the Treasury.\nBut the Brexit secretary discarded the Treasury's suggestion that the UK economy would go bust if negotiations with the bloc failed to secure a deal.\n\"Those forecasts don't seem to have been too robust since then,\" he said.\nLabour MP Pat McFadden slammed the Brexit secretary, claiming he was putting the country's economic future at risk.\n\"Without an assessment, you have mortgaged the country's economic future to a soundbite,\" McFadden said.\nDavis went further by suggesting to MPs that no economic assessment will be carried out at any stage of the Brexit process.\n\"You don't need a piece of paper with numbers on it to have an economic assessment,\" he said.\nDavis also told the committee there would likely be some downside effects to certain sectors of the economy.\nHe said the numbers in the agricultural sector are \"high\" and that UK's dairy and meat producers risk facing export tariffs of up to 40 percent under the World Trade Organization (WTO).\nMay, who is due to trigger Article 50 any day now, officially kick-starting the two-year Brexit negotiation process, had previously said the country would resort to WTO tariffs in case no deal with the EU is reached.\nA Treasury document recently leaked to the Independent claims, however, that such tariffs would be a \"major economic shock\" for the UK.\nDavis also admitted he failed to look at the impact of a 'no deal' on UK citizens' healthcare benefits while traveling around Europe.\nHe conceded it is \"probably right\" that they will not be granted subsidies as they will lose their European Health Insurance Cards.\nDavis also confirmed that financial services are set to lose their 'passporting rights' to trade in the EU.\nThe chief executive of the London Stock Exchange warned back in January that the City of London could lose up to 200,000 finance sector jobs if the UK fails to draw up a clear plan for post-Brexit affairs.\nMeanwhile, the Brexit secretary also spoke out against the committee for failing to ask him about the \"upsides\" of Brexit, namely the chance to \"relax\" commercial relations with countries outside the EU, which account for 60 percent of the UK's trade.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We've got a new color challenge this week over at 'Tis The Season, plus a winter card challenge at Friday Mashup, to include embossing...which also happens to be our optional trimming at 'Tis The Season. My card is going live for TTS6 but you're also getting a sneak peek at the Friday Mashup challenge!\nI used this sweet winter scene from the Winter Memories set. Its a great image because it can be used for a Christmas card or a nice winter thank you. I left the sentiment off so I can use it for whatever I need in the future.\nThe new sketch over at Freshly Made Sketches was the perfect one to go with both of these challenges! I hate the way the pic came out though. The polka dot banners behind the image panel really pop and look so pretty across the white embossed panel.\nI hope you'll come by both 'Tis The Season and Friday Mashup and play along with us!\nWhat a pretty scene you've colored! And I love the embossed snowflake background. Thanks for joining us at Freshly Made Sketches this week!\nOh no... I forgot about this stamp and how much I loved it!! :( I don't think I have it anymore. I do this all the time... Great card, Lisa. I love the colours and your colouring!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbgiow b/data_all_eng_slimpj/shuffled/split2/finalzzzbgiow
new file mode 100644
index 0000000000000000000000000000000000000000..9b61b1f3a7cd423d8ae9da9dab506453de203dab
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbgiow
@@ -0,0 +1,5 @@
+{"text":"Ethical standards for publication exist to ensure high-quality scientific publications, public trust in scientific findings, and that people receive credit for their ideas. Journal of Nursing and Midwifery Sciences (JNMS) submitted a form for membership in the Committee on Publication Ethics (COPE) and abides by its Code of Conduct and aims to adhere to its Best Practice Guidelines.\nThis journal have a open access articles and publishes original works within the area of Nursing and Midwifery such as Medical-Surgical Nursing, Critical Care Nursing, Nursing and Midwifery Education, Nursing Management, Psychiatric Nursing, Geriatric Nursing, Community Health Nursing, Pediatric Nursing, Midwifery, Women s Health Care, Reproductive Health and Allied Health Sciences. Peer reviewed and evidence-based, it is considered the profession's journal.\nJournal of Nursing and Midwifery Sciences (JNMS) aims to promote communication among nursing and midwifery researchers worldwide. It is an international journal with editorial and consultant contributors from various parts of the world, and is published quarterly. JNMS adheres to journalistic standards that require transparency of real and potential conflicts of interests that authors and editors may have. JNMS mission is to promote excellence in nursing, midwifery and health care through the dissemination of evidence-based, peer-reviewed clinical information and original research, discussion of relevant and controversial professional issues, adherence to the standards of journalistic integrity and excellence, and promotion of nursing and midwifery perspectives to the health care community and the public.\nJNMS is committed to publishing only original material, i.e., material that has neither been published elsewhere, nor is under review elsewhere. Manuscripts that are found to have been plagiarized from a manuscript by other authors, whether published or unpublished, will incur plagiarism sanctions.\nImmediate rejection of every other manuscript submitted to JNMS by any of the authors of the infringing manuscript.\nProhibition against all of the authors for any new submissions to JNMS, either individually or in combination with other authors of the infringing manuscript, as well as in combination with any other authors. This prohibition will be imposed for a minimum of 36 months.\nProhibition against all of the authors from serving on the Editorial Board of JNMS.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Rework skills are required and you'll need to be happy working with 0603 components. Overall this is not a difficult project if you take your time and prepare carefully.\nIn this article I'm going to show you how to do a straightforward modification to the STM32 F4 Discovery board that will change the onboard oscillator from 8MHz to 25MHz.\nProbably the main reason to do this is ethernet. If you're prototyping an MII ethernet PHY then that PHY will need a 25MHz reference clock. All those that I've seen do allow you to supply such a clock directly from a crystal connected to the PHY but why have two 25MHz crystals on your board if you only need one? The STM32 F4 has an MCO (Main Clock Out) pin that is intended to drive the 25MHz clock on an ethernet PHY.\nHowever ethernet is not the reason why I'm doing it. As the maintainer of the stm32plus STM32 C++ library I need to keep a set of reference boards around for testing. I've got F1 boards with 8 and 25MHz oscillators but I only have 8MHz F4 boards. The cheapest and easiest way for me to get a board with a 25MHz reference clock is to take a discovery board and mod it with an off-the-shelf 25MHz crystal. Hence this article.\nThe first step in the planning of this operation is to get hold of the schematic for the F4 discovery board. That's easy, ST publish it in the back of their user guide. Here's the section that contains the external oscillator.\nAll crystal 'can style' oscillators require a pair of external loading capacitors whose value can be calculated from the load capacitance value published by the crystal manufacturer. ST don't tell us the load capacitance for the crystal that they've selected but that's OK because I don't need to know. All I need to know is that the capacitors in-situ are 20pF each.\nI will be replacing the 8MHz 'X2' crystal with a 25MHz replacement that I got from Farnell. It requires 18pF of load capacitance.\nWhere CP is the parasitic capacitance of the board and CI is the input capacitance of the MCU. For my crystal with its CL of 18pF this works out at C1 = C2 = 30pF, assuming the commonly quoted CP + CI = 6pF.\nThe ST load capacitors are 20pF each so I will need to replace those with a pair of 30pF ceramic capacitors in 0603 format. No problem.\nST's 8MHz crystal (X2) has a 220\u03a9 resistor in series with it, labelled R24 on the schematic. My replacement crystal does not require any additional series resistance so I will be replacing R24 with a 0\u03a9 bridging resistor.\nIn addition to the component substitution I will need to make some modifications to tell the discovery board to actually use X2.\nThe STM32F407 can take its clock source either from an internal 16MHz oscillator, known as the high speed internal (HSI), or an external clock source such as X2 on the discovery board, known as the high speed external (HSE). With its 1% frequency tolerance, HSI is not particularly accurate and that's one of the reasons to use the HSE.\nWith X2 populated by an 8MHz crystal on the discovery board it might look like that's where the clock is being sourced from but in the default form that's not the case. That 8MHz clock is actually coming from the master clock out (MCO) pin of the STM32F103 that's providing the ST-Link interface. If we want the STM32F407 to get its clock from X2 then we must remove a resistor, R68.\nRemove and replace X2 with a 25MHz crystal.\nRemove and replace C14 and C15 with 30pF capacitors.\nRemove and replace R24 with a 0\u03a9 resistor.\nNow I can move on to planning the physical board-level work required to do the replacement.\nThe plan is to use hot air to remove the existing components. I'll then tin the pads of the discrete components with some solder and flux. Next I'll replace those discrete components, again using hot air and finally I'll replace the large through-hole crystal and solder it into place.\nWant to see how I got on? I videoed the whole procedure and if you've a few minutes to waste then you can see how I got on below. You can watch it on the embedded player here but you'll get better quality if you click here to go to the YouTube site and watch it there.\nIt all ended well and here's a picture of the new crystal in-place on the board.\nI'll be using my stm32plus library to test the board. The first quick-and-dirty test is to blink a LED at 1Hz. The source code is below.\nThe MillisecondTimer class relies on the ARM core SysTick peripheral for timing, which itself is fed from the MCU core clock, which in turn is generated by the PLL from a set of multipliers and dividers that operate on the incoming X2 signal.\nI set those multipliers\/dividers up to generate a core frequency of 168MHz from an assumed 25MHz oscillator. If the actual incoming frequency is not 25MHz then the knock on effect is that SysTick will not tick at the expected 1ms and my LED will not blink at 1Hz.\nIt's a quick-and-dirty low resolution test, but it works. And it did work. And that means that my incoming frequency is at or close to 25MHz.\nOn desoldering with the hot air gun I found that if you get it slowly closer (increasing the temperature like in the reflow process) it works better than put the hot air gun directly on the solder points.\nThis wasn't necessary. You could also use PLL dedicated to I2S, which is probably sitting there unused (i highly doubt you'll ever need using I2S and Ethernet in the same application), to generate the required frequency on the MCO pin with standard 8MHz crystal. Just set PLLI2SN to 25 and PLLI2SR to 1 (i assume PLLM is set to 8, so 1MHz signal is entering PLL loops) and route output from PLLI2S (PLLI2SCLK) through corresponding multiplexers to MCOx pin.\nIt seems like the clock Output (MCO) is broken on most STM32 devices. So using a dedicated oscillator is not a bad idea.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I found a plugin that will automatically post new content to Twitter and am looking for the same thing for the other social media sites that I am a member on. For the Facebook side, it looks like I may actually have to write this myself based on the twitter version.\nIf you have any suggestions for Facebook Drupal integration like the twitter integration, please post a comment, DM\/Mention, Facebook reply; meanwhile I'm going to keep looking.\nAnd now it's done, I have full automatic posting to both Facebook and Twitter.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Stop at New Nordic Penthouse to discover the wonders of Pattaya. The property features a wide range of facilities to make your stay a pleasant experience. Free Wi-Fi in all rooms, 24-hour security, daily housekeeping, gift\/souvenir shop, taxi service are on the list of things guests can enjoy. Closet, towels, clothes rack, slippers, separate living room can be found in selected guestrooms. Take a break from a long day and make use of private beach, golf course (on site), golf course (within 3 km), massage, karaoke. No matter what your reasons are for visiting Pattaya, New Nordic Penthouse will make you feel instantly at home.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The United Nations launched 17 Sustainable Development Goals to be achieved by 2030. Now it's embarking on a global effort to get people on board.\nLa premi\u00e8re conf\u00e9rence \u00abjouable\u00bb sur les Objectifs de d\u00e9veloppement durable (ODD) se tient \u00e0 Bonn, en Allemagne, du 1er au 3 mars 2017.\nThe UN sustainable development agenda cannot succeed without collaboration, a British development expert attending a global conference in Germany said on Friday.\nThe United Nations' best allies in promoting the sustainable agenda are young people, a senior UN official attending a global conference in Bonn, Germany has said.\nGlobal Festival of Ideas for Sustainable Development at the World Conference Center (WCC) in Bonn. Photo: photothek\/Inga Kjer.\nBIG IDEAS: The UN SDG Action Campaign, supported by the federal government of Germany, is convening what it calls the world's first \"playable policy conference.\nIDN-InDepthNews is flagship agency of the global International Press Syndicate Group helping to discover the world Beyond Breaking the News.\nSDG-festival i Bonn: Verdensm\u00e5l er \"folkets agenda\"\nNew York, Mar 1 : Policy makers, activists and private sector representatives from over 80 countries will join gaming experts and simulators on Wednesday in Bonn, Germany, at the start of the first \"","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbgjel b/data_all_eng_slimpj/shuffled/split2/finalzzzbgjel
new file mode 100644
index 0000000000000000000000000000000000000000..5ffc4ddeffa913e2059981387cf655aabdf048ed
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbgjel
@@ -0,0 +1,5 @@
+{"text":"\"Throughout the history of Islamic intellectual tradition, culture and heritage, all schools of thought whether legal, theological, philosophical or mystical have flourished in abundance. The writings of Aadil Farook have brought back glimpses of what the Muslims intellectuals have been discussing before. Looking from this aspect I should say that his poetical writing has brought the tradition of Islamic intellectuality into the contemporary debates and discussions.\"\nAadil Farook is only 36 years old but has already been blessed him with many intellectual, literary and artistic achievements. He is the author of 3 books of Islamic poetry - all of which have received excellent reviews by internationally recognized individuals. He was compared with the great Sir Dr. Allama Muhammad Iqbal. He is the co-author of 1 book of poetry which was published by a prestigious University of Malaysia. In 2009, he wrote an article on Islamic Philosophy which was published in 4 International Journals. He has translated 2 books by a world renowned Sufi from Urdu to English. He is the youngest ever recipient of Research Fellowship at the eminent Iqbal Academy Pakistan. On top of that, as a non-commercial musical artist, his 6 songs and 1 instrumental have all been aired on the top FM radio stations and can be compared in quality to any famous superstar of Pakistan or abroad.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"MD of PLP Lift Trucks Paul Pleasants successfully bid for the Hall & Woodhouse Brewery Tour in Dorset, at the raucous charity auction that was one of the many highlights at the recent FLTA Awards 2015.\nPleasants pledged a remarkable \u00a3700 to the Help for Heroes charity.\nPLP Lift Trucks was set up in 1972 by Paul Pleasants. Paul, who was also a director of the FLTA for several years, along with other employees of PLP Lift Trucks, attend the FLTA Awards yearly to support the Cesab brand of forklifts, converse with old friends and raise money for charity.\nBrindley Lift Truck Services was set up in 1978, and Richard Hayes acquired the company from his father in 1992. Richard also regularly attends the FLTA Awards, and the awards in February 2015 was no exception. Having also been a director of the FLTA for several years, Richard made a successful bid on the Tedworth House tour in Wiltshire, raising a generous \u00a31,300 for the Help for Heroes charity.\nThe Fork Lift Truck Association (FLTA) annually hosts the biggest and most prestigious event in the Materials Handling Industry calendar. The FLTA Awards were launched to encourage and reward examples of excellence throughout the fork lift truck industry and the FLTA Awards have a truly international appeal with sponsors and winners flying on from Europe and the Far East for the Oscars-style ceremony, presented by ITV's Jonny Gould.\nTotal raised on the evening for Help the Heroes was \u00a39,481.85. A big thank you to all that attended the FLTA Awards 2015, and an especially big thanks to the sensational compere and master auctioneer Jonny Gould.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In this lesson I will teach you about similar right triangles. To master this topic you need to have a complete understanding of ratios and proportions- so please review if you need to before starting this video. The main concept in this lesson is that in every right triangle you can form 3 similar triangles. First, let me stress that this theorem only applies to right triangles- those triangles that have a 90 degree angle. Next, let's make sure you understand the term \"similar\" \u2013 remember similar figures are those that have the exact same shape but a different size. The key point I want to stress is corresponding sides of similar figures are in proportion and we will be setting up a lot of ratios and proportions all through this lesson. You need to take your time with lesson and review the video at least once. In my experience many students struggle with this topic so you need to work hard at mastering it. As always, take neat and organized notes and practice is a must for this lesson. Good luck!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We spend a lot of time talking about getting a vision and God direction for our ministries. What then? What happens when we have developed a vision for the future but now need people to help support the vision? We are the carriers of the vision but how do we spread it to others? How do you get people on board to support the vision? Let's talk about it.\nChurch volunteers are dying from a lack of gratitude. Click To Tweet We've lose the ministry of encouragement in the church. Don't just tell people they're doing a great job but tell them specifically what they're doing that's great. Be willing to share stories about your team to others. It's not about being fake but genuine gratitude. Gratitude is oxygen to teams. A struggling team is not always a leadership issue but often a lack of gratitude and appreciation for those who serve.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is a capstone course in the financial planning curriculum to provide students with a practical understanding of the provision of financial product advice and managing clients' relationships. It consolidates and completes student knowledge and skills required for pursuing employment in the financial planning industry. The course is a study of important aspects of ethics, professional standards, Chapter 7 of the Corporation Act and behavioural finance to cover the different elements of bridging course contents set by Financial Adviser Standards and Ethics Authority (FASEA). The course also covers the practical elements of providing financial planning advice including understanding the client's situation and needs, conducting analysis and research, formulating appropriate financial planning strategies, choosing suitable products, preparing a compliant Statement of Advice, advice implementation and providing ongoing review service to the client. It is a required course for professional recognition in financial planning.\nTo facilitate exposure of students to a practitioner's perspective of the conduct of wealth management advisory services.\nApplication of ethics and professional standards to the service of providing professional financial advice to clients.\nPreparation and presentation of a comprehensive statement of advice.\nDemonstrate practical understanding of behavioural finance and its impacts on client relationships and practice management.\nPreparing a magazine or a newspaper article on a relevant topic. Further information will be provided on the course Moodle site.\nStudents will be given a case study to prepare a comprehensive Financial plan in the form of a Statement of Advice (SOA) base on a given facts in a case study. Students are also required to prepare a PowerPoint presentation of the advice and recommendations prepared in the SOA. This assignment is designed to give students an opportunity to examine in detail the key elements of this course and apply their learning into a practical financial planning task emulating a real life clients' situation. Refer to the course Moodle site for more information.\nThis is a 2-hour exam will be held on campus during the UNSW centrally managed examination period. The scope of coverage will be announced towards the end of the course. The final examination will based upon materials from all lectures of the course. Students must pass the final exam to pass this course.\n\u200bVia Turnitin. No assignment will be accepted via email. Please refer to the submission criteria posted with the assignment on the course moodle website.\n\u2022 CCH Master Financial Planning Guide (latest edition).\nEthics and the Conduct of Business (7th Edition) Boatright, J.R. 2012.\nDeveloping a Statement of Advice.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbikzc b/data_all_eng_slimpj/shuffled/split2/finalzzzbikzc
new file mode 100644
index 0000000000000000000000000000000000000000..03e61e5bc7f7d852173bbd09b1b711b9e5be0bbc
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbikzc
@@ -0,0 +1,5 @@
+{"text":"Category: Conflict Management. Tag: Conflict Management.\nThe Level 3 Award in Conflict Management (QCF) is designed for teaching candidates who require training in conflict management skills and who have customer facing role, deals with service users or members of the public. It is also designed for candidates who wish to teach conflict management skills. If you have already completed some kind of generic training qualification, such as Preparing to teach in the Lifelong Learning Sectors (PTLLS), and now require a specialist qualification, like conflict management skills, this course is tailored for you.\nSo, to gain the award, learners will need to attend a training course for a minimum of 36 guided learning hours, pass the examination and successfully complete assessments for the units. To attend, learners must be 18 and over.\nLearners are expected to have completed a teaching\/training qualification (Level 3 Preparing to Teach in the Lifelong Learning Sector or above).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When we discuss monetizing trend content, we often come again to affiliate internet marketing. This is much simpler than making use of to every affiliate program separately primarily based on the manufacturers you want to work with. We personally review all applications and notify associates in a timely manner of their acceptance. Ergonomic design, customized engineered material and acute focuses on element spotlight just some of the hallmarks which have set a brand new customary in the outside clothing trade. When your website is accepted for the program, you may be sent an affiliate ID and a password. Hi Bronwyn, you're finest to publish some content material before making use of for affiliate programmes.\nIs an incorporated ladies's clothing shop based out of one of many nation's hottest trend capitals in Southern California. Basically which means online retailers are giving search adverts twice the attention as insurance web sites do. Yes, our superior affiliate management system provides you a wealth of details about your progress including variety of clicks, total revenue generated, commissions earned and rather more!\nWhen Affiliate visitors convert into buying clients within 30 days, Affiliates are rewarded with a fee. Buy girls, teen, and plus size clothes & accessories merchandise for your self via your hyperlink and pocket the commissions! Affiliates earn 8% per sale, and are paid month-to-month by examine when commissions attain $50 or extra. The program permits you to robotically link products, brands, colors, and clothes or accessories talked about in posts to relevant affiliate hyperlinks. Emphasizing Fair Trade gross sales for Africa primarily based artisans, Gifts With Humanity carries one of the numerous and comprehensive picks of Fair Trade gifts on-line and artisan handicrafts from producers in Africa, South America and Asia. Commissions paid to affiliates usually range from 4% to 12% relying on how well established and well-known your model is. The blogger, on the other hand, doesn't pay a dime to join.\nBy signing as much as be an affiliate of a bigger on-line service provider, you can resell or market that product owner's children's garments and accessories. All guests from your web site are tracked (cookie affiliate info) and associates receive 10% fee on all online sales (before taxes and delivery). If violators continue to ignore this policy the affiliate relationship will be terminated. Offering every part from private care merchandise and clothing to kitchen and workplace provides, BuyGreen has it coated. SAS and CJ community pays affiliates on a month-to-month basis if they have met their minimum payout standards. Moreover, to all of acquired commissions, you can use it as payment methods for on-line order or request withdraw cash.\nOnline vogue buying website, ShopStyle, is relaunching its web site today with a contemporary look aimed at delivering new merchant and brand opportunities for the 2m customers who repeatedly visit it for a fashion repair. Shoes of Prey also recently launched its new 3D shoe designer software program through the net boutique and is transferring to launch a sister site \u2013 Sneaking Duck \u2013 which will sell trend ahead\" prescription eyewear and sun shades. The startup's answer is to offer a vogue purchasing aggregator, comparable what we have already seen in different e-commerce categories, comparable to Skyscanner (travel), Zillow (real estate), and (accommodations).\nMany completely different publishers within the tech, food, and style house use Skimlinks The network is greatest recognized for its ease of use, as bloggers are solely required to drop a line of java script into their site's coding. At Fashion Lane gadgets are robotically sorted by low cost so shoppers can simply discover the most effective discounted product in every class. Fashion Lane screens product prices intently and detect when a product's worth has dropped.\nFashion Lane retains monitor of a product's price history so you possibly can see whether the value of the item is likely to go any decrease. While this system is not particular to vogue, their instruments narrow the chances by matter making it a fantastic alternative for fashion bloggers. It has brought big and small, unbranded and branded fashion retailers on a single online platform. It can really fix checkout means of cellular ecommerce apps by giving app users an precise cart throughout the aggregator utility the place they'll put merchandise from several retailers and take a look at without switching from one app to a different. Instagram for trend is if you sacrifice your cart for the sake of Instagram-like photograph feed of merchandise. An aggregator will often have partnerships with shops that may be troublesome for you as a self-publisher to access. The identical system is used by Fancy, a Pinterest-type social fashion e-commerce app.\nAlong with aggregating products from a number of stores, together with giant on-line retailers similar to ASOS, Farfetch, My-Theresa and TheOutnet, and tons of of smaller nationwide online outlets, has a concentrate on simplicity, curation and personalisation\". They are discovery platforms\" where users can browse fairly garments from a variety of trend e-commerce web sites.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Syed Irteza Hussain, MD is the Department Chair in the Neurological Institute at Cleveland Clinic Abu Dhabi. Dr. Hussain is a Consultant Neurologist with 17 years of experience. After completing his medical degree from the prestigious Aga Khan University he underwent extensive post graduate training in the US. He is American Board certified in Neurology, Vascular neurology, Neurocritical Care and Neuroendovascular surgery.\nUpon completion of his post graduate training he joined Michigan State University as assistant professor in 2009. There he helped establish a 24\/7 Neurointerventional surgery service and fellowship training program. In addition to his procedures he had an integral role in the management of acute stroke and critically ill neurological patients in the hospital.\nSubsequently in 2011 he assumed the Medical Directorship of the stroke program at Michigan State University and its affiliate Sparrow Health system. In this administrative role he supervised multiple successful Joint Commission recertifications of the primary stroke center and subsequently became the first in Michigan to become a JC certified Comprehensive stroke center. Recognizing his services he was promoted to Associate Professor in 2014 and awarded the Physician Leader of the year award in 2015 by his institutions. He joined Cleveland Clinic Abu Dhabi in 2016 as staff physician in the Neurological Institute.\nDr. Hussain lives in Abu Dhabi with his wife and two daughters. He enjoys travelling with his family, listening to music and football.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The challenges of the redesign of the facade of All Saints Church were the creation of an image for the church that would evoke its earlier history, and the transformation of the building to a visible presence and an architectural contributor in an exceptionally mixed streetscape.\nOriginally constructed in 1894, by the 1950s the original Romanesque and Gothic-inspired brownstone fa\u00e7ade had deteriorated to such an extent that the facade was refaced in stucco, the original ornament removed, and the doors and stained glass windows replaced with unsympathetic contemporary designs. The church retained its tripartite gabled form and cornerstone but little else from the 1894 design.\nThe inspiration for our design came from the architecture of Alexander Jackson Davis as well as nineteenth-century pattern books for wooden \"carpenter gothic\" churches. This presentation of gothic is more na\u00efve than literate, but its \"yankee\" quality seemed appropriate for a small nineteenth-century chapel originally established as an outpost of a larger parish.\nThe design focused on enriching the streetscape. The facade is built up from the texture and deep shadows of vertical ribbing and horizontal string courses, the earthtones of New York's brownstone blocks, and a rich array of materials. New entrance steps flanked by nineteenth-century-style lampposts and an evergreen garden enhance the pedestrian experience for both parishioners and passersby. High in the central gable is a new stained glass window, designed by the noted stained glass artist Sylvia Nicolas.\nAt a practical level, the project provided the opportunity to make the church more welcoming, more accessible, and more comfortable.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You need to use your campus network username (e.g. lastname06) and password to log in to most of the University's computer systems.\nUse Stud.IP to co-ordinate your courses and events, put together your timetable or download course materials.\nUse HISQIS to register for examinations and manage existing registrations and print out transcripts and certificates.\nWith Campus Portal you can perform changes on your addresses. You may also print certificates and notifications. Another very important feature available on campus portal: use it to set private cell phone numbers as well as alternative e-mail addresses in order to use ZIM's convenient 24h password reset service.\nWith Webaccess you get online access to your GroupWise e-mails.\nWith Outlook Web Access (owa) you get online access to your Exchange e-mails.\nVirtual classroom functionality is what ILIAS is all about, putting the spotlight on the course syllabus and teaching.\nTypo3 is the Content Management System (CMS) of the website of the University of Passau. To enable yourself to the usage you need to send a fully completed and signed Antrag auf Benutzerkennung (request of an account) to the ZIM.\nMore information can be found on our help sites.\nMicro Focus Filr is a system to exchange of files (internally and externally), which are supplied by you on your private network drive (\"I drive\").\nAll possibilities of access and an introduction video (ger) you find on our help sites.\nWith the assistance of the InfoGuide you can browse the local catalogue plus the Bavarian compound catalogue as well as eBooks and E-journals. While on the go you can use the mobile version. For the order and account functions of the info guide you need the library number including the associated password.\nIn order to use eBooks, e-journals as well as databases of the university library from home or on the road you have to log into the university network with your ZIM-identification through the external access of the university library.\nLooking for recorded lectures, tutorials, trailers and news? UniVideo is a central media portal which holds audio-visual material from and about the University.\nThe university of Passau is a member of the association F\u00f6rderung eines Deutschen Forschungsnetzes (DFN). If you are an employee, you can use the DFN personal organizer for free and without registering. Also, no IP addresses are stored while accessing the DFN personal organizer. All data of the appointment planning are exclusively saved in Germany at the DFN association, they will not be shared with third parties nor will they be used for any other purpose other than appointment scheduling. In addition to each scheduled appointment, a termination date must also be given. On this date the appointment including all given data automatically are deleted.\nBy the help of the Research Information System (Forschungsinformationssystem - FIS) you can maintain the current status of your research activities and embed information about your projects in an automated manner into your website. Besides - if you agree to that - , base data on your research project will be used for the central website \"Forschungsprojekte\" of the university.\nYou need to use your campus network username (e.g. lastname06) and password to log in to the University's cooperative computer systems.\nA fixed weekly maintenance window has been set for Tuesdays until 10 a.m. This time slot is used to run maintenance and install updates for the University's telephone and computer systems, including Stud.IP, ILIAS and the media portal. If you are using one of these systems during this time slot, be aware that they may become unresponsive or unreachable without warning.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbiwhc b/data_all_eng_slimpj/shuffled/split2/finalzzzbiwhc
new file mode 100644
index 0000000000000000000000000000000000000000..183f90246d2c955136cc5e9d0b3f829f26b32ef4
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbiwhc
@@ -0,0 +1,5 @@
+{"text":"Staircases have become an inseparable part of home. We most often underestimate the importance of a staircase. We do a regular walk by just to enter into a different space or part of the home but never notices the staircase which facilitates this tiny work for you.\nWe, the profound team of Color N Interior puts a lot of thought into creating the staircase design for our client. We look differently at the creation of beautiful space of staircases. Our team has a huge back of experience and so we do look deeply into details of designing the staircase.\nThe designing of a dull staircase is itself a highly tedious job, although for us it is an easy to do. We pay same attention to our client's staircase as much as we do for other spaces. We believe that the staircase needs as much creativity as the other spaces in home do. Our creative interior designers are very passionate about designing staircases and indoor\/outdoor space design.\nWe start our task to design the staircase from root level and carry it with utmost sincerity till the final stage. The looks, comfort level, clever design, aesthetics are just a part of our long checklist we carry out while designing and decorating our client's staircase.\nA glimpse of innovatively designed steps by the designers and interior decorators of Color N Interior are sure to take your breath away. The designer and technically advanced staircases are carefully and beautifully conceptualized and executed after understanding the individual likeness and preferred style of the clients. The staircase so designed to carefully and aesthetically integrated with the rest of the interior of the house.\nA platter to offer dynamic choices from a range of curved, cantilevered, spiral or suspended, and in various materials like in steel, glass, wood, copper or aluminium. We home staircase interior designers in Bangalore offer our clients with infinite design possibilities to emphasise the interior of their beautiful homes. These designer staircases most often turn out to be the designer focal points of one's beautiful home.\nWe know the secret to turn your home staircase interior design the most sought after part of your home, of course, we leave no stone unturned !","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Adam Moen is an orthopedic physical therapist specializing in knee and shoulder rehab.\nHe offers post-operative rehabilitation, especially for patients with golf injuries and mobility issues.\nAdam enjoys spending time with family, being outdoors and playing golf.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"twin in the city: a little florentine love.\nin case you were wondering, the past week here in florence has been beautiful. i have found my new favorite part of town, and can't wait to catch a sunset or two in the upcoming months.\nyou inspire me. never stop.\nthe past month, or so.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Water Blasters are the most POWERFUL, DURABLE water cannons in the world.\nThe XLR blasts targets up to sixty feet away (20 meters).\nXLR stands for Extra Long Range, and this heavy duty water cannon lives up to it's name. Don't be fooled by other brands that claim long shooting distances but fall way short. The XLR is the real deal, easilly sending blasts of water up to 60 feet away (20 meters).\nThe XLR can be refilled in only seconds, and cleaned anywhere in less than a minute, making it the prefered water cannon of serious fun seekers, adolescent and adult.\nThe XLR is made of super strong material that makes this water cannon virtually indestructible.\nCarries the famous Water Blaster Lifetime Guarantee.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our standard UK P&P is \u00a34.95. Delivery is FREE on UK mainland orders over \u00a350. We also offer a NEXT DAY despatch service for a charge of \u00a38.95. This charge is instead of the \u00a34.95 standard charge, but is applicable on NEXT DAY orders of all sizes.\nStandard delivery times are 3 to 5 working days for personal orders (although this is usually much quicker). Orders are sent on a tracked courier service and can be tracked via the \"My Account\" function on the web site.\nNEXT DAY orders are dispatched the same day for all orders placed BEFORE 1.45m on working days. All orders placed AFTER 1.45m will be dispatched the next working day. Orders placed BEFORE 1.45pm on a Friday will be delivered on MONDAY. Orders placed after 1.45pm on Friday or at the weekend will be dispatched MONDAY.\nUnless otherwise stated, we offer a full money back guarantee if goods are returned within fourteen days of receipt or from the day proof of sending can be provided of goods having been sent. This applies to UK and other EU orders only. We make a deduction from the reimbursement for the loss in value of any good supplied, if the loss is the result of unnecessary handling by you, defined as going beyond the sort of handling that might be reasonably allowed in a shop. The description of all products is clear on the box and there should be no need to remove protective packaging, such as cellophane wraps, which are in place to ensure products we supply to children are not subjected to tampering and hence removal will mean products cannot safely be resold and will result in no monies being refunded.\nIf in doubt, please call our Customer Services team. We regret that unless the goods are faulty or not as described, we are unable to refund postage and packaging costs. Please note that returns are entirely the customer's responsibility. We will make the reimbursement no later than 14 days after the day we receive back from you any good supplied.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzblibq b/data_all_eng_slimpj/shuffled/split2/finalzzzblibq
new file mode 100644
index 0000000000000000000000000000000000000000..85b6cb6083af61e0b64f4aaf0cf1b476d67ee1aa
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzblibq
@@ -0,0 +1,5 @@
+{"text":"Usually, for dinner, I have soup, it is easy and quick to make and so simple to eat. Sometimes, soup is not enough, so I am trying to find some nice little nibble to eat.\nLast time, I was running out of bread, so I decided to bake a focaccia with emmental, olives and pancetta. That is a really nice type of bread that goes deliciously with a soup. I hope you will enjoy the recipe as much as me.\nHow to do a Focaccia with emmental, olives and pancetta?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Top Tip: Is Mom's laptop fried?\n\"My mom needs some big help with her Compaq laptop. It's a Compaq Presario 1200 laptop w\/an AMD K6(not sure what clock speed), 152mb ram running Windows 98. It was running fine till 2 days ago; it now gets to what may be an error screen saying CMOS has an error. Someone thought they could use a Quick Restore disc to get it back to normal but it was going to wipe everything out. they started , then pulled out the disc, and now it starts with the Quick Restore screen.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The next advocated model of bar stool dining room table is a special terrace stool using a whitened standalone. In the event that you like decorating your house with a product which is very particular, then you may employ a minimalist terrace stools created from wood having a standalone. Select the vivid colour stools and provide your stools using the decorative cushions. That idea will offer a well-intentioned belief. To produce the terrace atmosphere hotter, you're able to create a wooden dining table. Then, the 3rd advocated version of minimalist terrace stools is folded model skins. Lots of have enthusiast to the stool version of folding patio stool which can be minimalist. By employing this type of stool in your terrace, notably within an field of patio which is narrow, you can move out of 1 location into the other regions easily. You are able to produce your home more spacious with collapsible furniture. Then, add a black or black white pot into the ornamental plants. Effectively, it's easy to arrange the comfortable patio of your home. Presenting a minimalist patio stool, you can present some stools along with the accession of some very simple table can. Then, which version of bennox counter height dining room table and bar stools can be your own favorite?\nA bennox counter height dining room table and bar stools is one of all the varieties of stools to get in your home. You'll find a number of activities to complete using all the stool. Even a bar stool dining room set is which makes you sit and stay more. It makes you more productive in managing and working conducting such a thing. A few people today appear to avert a model of their crystal clear stool and its design. The wrong choice gets you to fail to use it. You can find lots of things to comprehend until you choose it home to your intentions.\nThe following bennox counter height dining room table and bar stools is called the Hill Tribe. The woven steel is extremely tight as it employs acrylic for three or more yarns. Apart from its potency, in addition, it creates a beautiful yet colorful pattern. The magnitude of a bar stool dining room set is dependent upon how limited the design is and how much that the material utilized. Assessing the burden is the only way to quantify this stool. Broadly speaking saying, a larger hammock is the higher. In the event you would like to obtain this stool, then it's better to get it specifically rather than online. You want to know the way the string quality is and the border development of this hammock.\nWhat is the very first matter to take into account when picking the bennox counter height dining room table and bar stools? Lots of folks can think about the color probably the very best however they actually must select the comfort first. First, they have to make sure that they truly are willing to take a seat smoothly on the stool for very long time. The relaxation will be decided from the measurement however, additionally, there are other items to think about for example, specific element their family members as well as also the material used for the stool. That isn't any uncertainty the bar stool dining room table should also be acceptable for their life style. It may reveal the direction that they will work with the dining room and also the need for their household furniture.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I leave for a long journey.\nTranslation from Hungarian (Magyar) to English copyright \u00a9 2003 by B\u00e9la Pavelka, (re)printed on this website with kind permission. To reprint and distribute this author's work for concert programs, CD booklets, etc., you may ask the copyright-holder(s) directly or ask us; we are authorized to grant permission on their behalf. Please provide the translator's name when contacting us.\nThis text was set to music by the following composer(s): Zolt\u00e1n Kod\u00e1ly. Go to the text.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We got the opportunity to #inspire a new generation of #coders when the kids from @eisvrij came over to learn what it is like to be a #developer !\nSince spring just started some of the #bits decided to enjoy one more day of snow. Thanks @snowworld_nl we had a great time!\nBitfactory style: taking a moment to kid around during an intense #userexperiencedesign session.\nOur new #plants are turning the Bitfactory offices a little bit more #green. And a #greenoffice is a happy office, wouldn't you agree?\nHelp, mijn dier is zoek!\nDe #bitfactory familie heeft twee nieuwe leden! Ontmoet onze nieuwe #developers : Angelo en Bas.\nAt #bitfactory we like our #knakworst , but we also like our #healthyfood ;) Check out Gerda's lunch for example!\nTony's chocoplatform wint Dutch Interactive Award in e-commerce!\nBitfactory wenst je een heel gezond en succesvol #2019 !\nBedankt Bodhy, voor het kickstarten van onze dashboarding tool Laraboard. Succes met het afronden van je studie!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbltjl b/data_all_eng_slimpj/shuffled/split2/finalzzzbltjl
new file mode 100644
index 0000000000000000000000000000000000000000..f91a6efac1bf8cee798e9e393fd30aa628a0920c
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbltjl
@@ -0,0 +1,5 @@
+{"text":"Used equipment dealer for the mining and process industries concentrates on crushers, grinding mills, pulverizing and roller mills, and coal mining equipment.\nServing the legal, insurance and construction management communities. Full service estimating, scheduling and documentation of construction defects and mold abatement on all types of construction. Expert witness testimony on all types of construction defect pricing.\nProfessional commercial general construction and construction management firm which operates in the northeast U.S. Providing comprehensive construction services.\nSalary surveys for construction executives, construction management, construction staff, and construction craft positions.\nConsulting firm specializing in crushed stone, sand and gravel. The Construction Aggregates Databook consolidates 30 years of U.S. construction aggregates data and key construction, economic, demographic, and industry related statistics.\nConstruction news and tips.Includes a directory of links to construction and home improvement sites.\nConstruction company specializing in design\/construct and construction management for commercial, industrial, and municipal projects.\nConstruction management, design and build,and general construction in four states. New Hampshire, Massachusetts, Vermont and Maine.\nA commercial construction company in Southern California, specializing in open book construction, using a total team approach.\nRetail tenant improvement,construction and construction management company.We work in all 50 states and Guam and Puerto Rico.\nA website that answers questions related to construction and construction law. All information is provided by attorney, author and contractor Gary Ransone, Esq.\nSupply a range of metal products for the construction industry. Includes substantial technical material on the use of metals in construction.\nInformation source for commercial construction in New Hampshire, Vermont and Maine. Provides an online planroom, classifieds, and daily construction reports.\nConstruction consultants and experts specializing in construction claims, dispute resolution, CPM scheduling, construction management and defects evaluations.\nBuilds, designs and constructs custom log homes. Specializes in post and beam construction, log accents, home restoration, staining, chinking, consultation and commercial construction.\nUndertake erosion control, shoreline stabilization, dam repair, design and construction of skateboard parks, and bridge construction and restoration. Includes project portfolio with photographs.\nA commercial construction general contractor operating in 46 states, specializing in restaurants, drug stores and multi-unit construction. Photos, descriptions and contact information.\nA full service construction consulting and project estimating firm located in central NJ and serving the construction industry nationwide. We specialize in heavy industrial and commercial projects.\nProviding construction support services to both commercial and residential clients and contractors. Construction claims preparation and resolution, change order review and analysis, project management and oversight.\nA full-service construction firm specializing in large geographical area projects with expertise in site development, waste-handling, excavation, landfill construction and public works.\nOffer a range of concrete construction solutions including turn-key footings, slabs and tilt-wall construction.\nConstruction and contract consultants, construction claims, construction contract administration, arbitration and mediation. Based in Australia.\nOffer construction contract advice and construction dispute resolution services for UK and international clients.\nBooks, tapes, software and references for professional builders, construction contractors and sub-contractors, and do-it-yourself construction projects.\nConsultants to the construction industry and financial institutions. Specializes in defense of construction claims and preparation of construction claims. Based in Montreal, Quebec, Canada.\nHelping construction companies across the nation improve their construction and pre-construction processes using primavera project planner, unique management methods and our software. The company comprises of experts in project control who have developed highly customized packages for the construction industry, helping companies improve communication with their contractors.\nConstruction firm specializing in high-performance green building for both new construction and renovation. Also provide LEED consulting and solar, wind, and geothermal energy system design and installation.\nConstruction leads and bidding opportunities provided through a web based platform, focusing on projects within the industrial and commercial construction sector including federal and state project leads.\nEngineering consulting firm specializing in geotechnical engineering, construction materials testing and construction management.\nProviding services in project management, evaluation of construction claims, critical path method (CPM) scheduling, facilities consulting, services to financial institutions and investors, temporary staffing, advice on the U.S. construction industry, seminars and training.\nfeatures solid waste processing systems and recycling facilities. professional services include feasibility studies, site assessments, permitting, engineering and process\/plant design, equipment procurement, construction and construction management.\nIf the business hours of Klinger Machinery in may vary on holidays like Valentine's Day, Washington's Birthday, St. Patrick's Day, Easter, Easter eve and Mother's day. We display standard opening hours and price ranges in our profile site. We recommend to check out klingermachinery.com\/ for further information. You can also search for Alternatives for klingermachinery.com\/ on our Review Site Sitebook.org All trademarks are the property of their respective owners. If we should delete this entry, please send us a short E-Mail.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Portia De Rossi wasn't afraid to get personal when she reunited with the cast of Arrested Development on Thursday's Inside the Actors Studio. When James Lipton asked de Rossi about her marriage to Ellen Degeneres, the actress revealed her initial anxiety about coming out in Hollywood.\n\"I think when I really, truly came out was the Golden Globes when Arrested Development was nominated [in 2004],\" de Rossi recalled. \"Ellen and I had been together for a month and I was very, very nervous going to that because I knew that I was going to meet up with her after the show, and that there was press. And that was the first time that we actually stepped out together as a couple, but for me, it was the first time I stepped out as a gay woman, really.\"\nThe cast applauded de Rossi's bravery to come out and talk openly about her relationship with DeGeneres, whom she married in 2008.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Khampan Rafting is the leading rafting tour in Chiang Mai. You will experience over 14 rapids in 10 km of whitewater rafting and feel the roar of the Maeteang River beneath your raft. We will also visit the Phachee waterfall where you can slide down the waterfall, jump of the cliffs and swim.\nOur professional guides will make sure that your rafting experience will be fun, exciting and safe.\nDuring dry season, from January till May, we will use inflatable kayaks (3 persons) instead of rafts (5 persons).\nIncluded in price: transportation, trained rafting guide, all safety and rafting equipment, vegetarian lunch, first aid insurance, water, coffee and tea.\nWhat to bring: towel, swimming suit, dry clothing, sunscreen, insect spray.\nHow can I book this tour? Please fill in the booking request at the bottom of this page. We will do our best to respond as quickly as possible.\nWould you like to visit the elephants and go rafting in the same day? Please have a look at our Elephant Sanctuary + Whitewater Rafting Tour.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Original Mini Top Clip Backpack - Rubberized Leather is rated 3.6 out of 5 by 7.\nRated 5 out of 5 by Virginia from Wonderful! I love it!!!! It's my second bagg and I love them. Y also have medium boots and the are the best! In a future I Will buy the short one.\nRated 2 out of 5 by Melina from High quality item, however they didn't fit. If you're a 6 in the original, order a 7 in the original refined.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Normally, for a Spanish goliath like Real Madrid, a trip to the coast of Africa to face a Second Division club would seem like a vacation. But on Wednesday when Los Blancos make the trip to the Spanish enclave of Melilla on the northern coast of Morocco for a Cope Del Rey Round of 32 match against the local side there, they will be playing one of their most significant games of the season \u2014 their first since the sacking of head Coach Julen Lopetegui, as The National reports. Former Argentine international Santiago Solari takes over the club for the game against UD Melilla, a match that will live stream from the African coast.\nTo find out how to watch a live stream of the UD Melilla vs. Real Madrid Copa Del Rey showdown, see the streaming information at the bottom of this article. Kickoff is set for 7:30 p.m. Central European Time on Wednesday, October 31, at the 10,000-seat Estadio Municipal Alvarez Claro in Melilla, Spain, on the northern coast of Africa. In the United Kingdom, the live stream gets underway at 6:30 p.m. Greenwich Mean Time, while in the United States, that start time will be 2:30 p.m. Eastern Daylight Time, 11:30 a.m. Pacific. Fans in India can log in to the live stream at midnight Tuesday night\/Wednesday morning.\nLopetegui got the sack after just 14 matches across three competitions, starting with a stunning defeat to crosstown rivals Atletico Madrid in the UEFA Super Cup, as Sky Sports reported. Things didn't get much better from there, with the European champions sitting ninth on the La Liga table with only four wins against an equal number of defeats and a pair of draws.\nBut a humiliating 5-1 shellacking by Barcelona in the first El Clasico of the season on Sunday clearly sealed Lopetegui's fate.\nTo watch a live stream of the UD Melilla vs. Real Madrid Copa Del Rey match, go to BeIn Sports Connect USA or download the BeIn Sports app for mobile devices. The BeIn Sports streaming service requires cable or satellite service subscriber login credentials from a carrier that offers BeIn Sports.\nHowever, there is a way for fans to watch the Copa Del Rey Opener stream live for free, without a cable subscription. Fans may sign up for a free trial of a streaming TV multi-channel package such as the Sling TV international sports package or Fubo TV. Both services require credit card information and subscription fees, but each offers a seven-day free trial. If the subscription is canceled prior to the expiration of that week-long period, fans can watch the Union Deportiva Melilla vs. Los Blancos match live stream for free.\nIn Spain, MoviStar+ will stream the Wednesday Cup match. In the United Kingdom, there will be no live stream offered, nor will there be one in India for the Melilla vs. Real Madrid match. But fans can likely access the live stream through another provider listed here by using a VPN service, such as those recommended by the tech site CNet.\nIn Spain, MoviStar+ will stream the mid-week match. In Italy, the Melilla-Real Madrid Copa Del Rey contest will be streamed live by DAZN Italia. In Canada, the DAZN sports platform will also live stream the match.\nFor a comprehensive list of outlets in other regions that may carry a live stream of UD Melilla vs. Real Madrid, check out the listings at LiveSoccerTV.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbmffl b/data_all_eng_slimpj/shuffled/split2/finalzzzbmffl
new file mode 100644
index 0000000000000000000000000000000000000000..2bc93aea65ab69f8fb56be19622355f3bbbceb79
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbmffl
@@ -0,0 +1,5 @@
+{"text":"Smoking has just come to be even much more high-priced. The new federal taxes on tobacco solutions raised tobacco taxes on cigarettes from 39 cents to $1.01 a pack on April 1 as component of an boost to spend for expanded federal well being insurance coverage applications for uninsured youngsters. I like the thought of providing well being coverage to youngsters and I agree with this strategy 100%. The cause I agree is since I never smoke. I utilized to but I quit quite a few years ago. The cause I quit was since cigarettes have been generating me really feel terrible. I had colds regularly. My breath stunk and so did my clothing. More than the previous 35 years or so I have observed the effects of smoking on quite a few of my pals and acquaintances. Regrettably, very a handful of of them are experiencing well being challenges.\nIt was since of this that I decided to get started assisting people today quit smoking utilizing hypnosis and NLP. I wanted to enable people today have a much better opportunity at living a extended, healthier life. If you want factors to quit, right here are 9 of the very best factors I could come up with.\n1. Smoking Causes Poor Circulation and Heart Illness Smoking can kill you a quantity of strategies. The most prominent is heart illness. There have been 332,973 deaths due to heart illness in 2004. It was the quantity one particular trigger of death in the United States. Smoking is accountable for 30 % of all heart attacks and cardiovascular deaths. Nicotine, the principal ingredient in cigarettes, raises your blood stress and tends to make your blood clot much more effortlessly. When your blood clots much more effortlessly it gets stuck in your veins causing poor circulation and heart attacks. Did you know that smokers frequently have frequent headaches? They get them since there is a continuous lack of oxygen due to hardened, narrowed blood vessels to your brain. Oh, by the way, these hardened, narrowed blood vessels can lead to a stroke. Even even though it wasn't due to smoking my dad had a stroke. I assure you that one particular of the greatest factors I never smoke is that I never want to devote the final years of my life like my dad did, weakened, utilizing a crutch and frustrated since he could not discover the words to clarify what he wanted to say. These blood clots brought on by your smoking can trigger a stroke if they finish up in your brain. Smoking deprives your cardiovascular program of oxygen. When combined with the monoxide poisoning type cigarettes, this lack of oxygen types thick cholesterol deposits on your artery walls, choking off your blood provide. This resulting poor circulation can trigger impotence, a further cause to quit smoking. If you are a middle-aged or older man smoking can retain you from acquiring an erection and also impairs sperm motility. Most males discover that this reverses when they quit smoking. If that wasn't adequate, the poor circulation brought on by smoking's outcome in Claudication or chronic pains in the legs, feet and toes. This can at some point lead to a loss of circulation and you can finish up getting gangrene which will lead to the amputations of your toes or feet. Smoking can also be linked to loss of circulation in your fingers.\n2. Smoking Causes Cancer of the Mouth, Emphysema, Lung Cancer and Bronchitis Do you know any individual who has died from cancer? I have recognized a handful of. It can be a slow, terrible death. Cancer is brought on cells that develop like weeds in a garden that strangle out required bodily functions and excellent cells. I would not want cancer on any individual. Research show that smoking is accountable for at least 30 % of all cancer deaths. Study that once again, smoking is accountable for 30 % of all cancer deaths. Tobacco in any type, which includes \"\"smokeless\"\" or chewing tobacco substantially increases the danger for cancer of the mouth. Smoking tobacco can trigger equivalent effects as nicely as providing you gum illness and tooth decay. Dentist frequently send me their individuals that smoke since they know the challenges smoking has on dental well being. Smoking also tends to make your teeth yellow and offers you terrible breath. If you smoke you are ten instances much more most likely to get lung cancer as a nonsmoker. Smoking is linked to 87 % of all lung cancer deaths every single year. Smoking is accountable for 82 % of deaths due to emphysema and chronic bronchitis. When you inhale the smoke from a cigarette it passes by means of your bronchi, or breathing tubes. Nasty chemical compounds from the cigarette smoke, like hydrogen cyanide, formaldehyde, ammonia, attach themselves to the lining of the bronchi and trigger them to erode. They also trigger the inflammation to the bronchi that outcome in chronic smoker's cough. As the bronchi weaken they come to be much more susceptible to bronchial infections as nicely as to colds and the flu.\n3. Smoking Causes Heartburn, Ulcers and other Digestive Difficulties Your whole digestive program is impacted. Smoking causes elevated stomach acid secretion that contributes to heartburn and peptic ulcers. Considering that smoking interferes with circulation, smokers practical experience delays in the healing of ulcers to the stomach and duodenum. Smoking also increases the danger of Crohn's illness. It has also been linked to gallstones, which type when liquid stored in the gallbladder hardens into pieces of stone-like material. Smoking has been linked to liver harm and investigation suggests that smoking can worsen liver illness brought on by drinking also considerably alcohol.\n4. Smoking Causes Cancer to the Esophagus and Throat In adults, the esophagus is an around 10 inch hollow tube that carries meals and liquids from your mouth to your stomach. The muscle tissues of the esophagus contract as you swallow to push meals down into the stomach. It has glands in its lining that make mucus that tends to make swallowing less complicated. Difficulty in swallowing is normally the 1st symptom of esophageal cancer in most individuals. Esophageal cancer is the seventh top trigger of cancer death in males in the United States. The tars and other chemical compounds in cigarette smoke can trigger cancer of the esophagus and throat. Esophageal cancer is frequently fatal. Each smoking and smokeless tobacco have been linked to the around 12,300 new instances in the U.S. annually and every single year there are around 12,000 deaths brought on by esophageal cancer.\n5. Smoking Causes Pancreatic Cancer and Bladder Cancer \"\"Actor Patrick Swayze is defiantly continuing to smoke \u2013 in spite of suffering from cancer. Patrick was spotted enjoying a cigarette not too long ago while waiting in his vehicle for his wife to fetch pizza. The 56-year-old star has not quit the awful habit in spite of undergoing months of chemotherapy for pancreatic cancer. The ill 'Ghost' actor was also not too long ago rushed to hospital with a lung infection and medical doctors declared him \"\"much more ill than ever\"\".\"\" Pancreatic cancer happens most generally in males amongst the ages of 35 and 70. It is the fourth highest trigger of cancer mortality. Smokers practical experience considerably larger prices of pancreatic cancer. In reality the danger of acquiring cancer of the pancreas is two to three instances larger in smokers. Specialists think that two to three out of 10 instances of pancreatic cancer are believed to be brought on by smoking. Persons who use \"\"smokeless\"\" tobacco are also much more most likely to get pancreatic cancer. The carcinogenic compounds in cigarette smoke have also been linked to bladder cancer. Our bodies rid themselves of chemical compounds in our urine which is stored in the bladder. The presence of carcinogens in your urine from cigarette smoke can trigger bladder cancer, which is frequently fatal.\n6. Smoking Damages Kidneys Your kidneys can be straight impacted by smoking. Considering that smoking impacts your circulation it also can elevate your blood stress. Though higher blood stress can harm blood vessels all through your physique, when blood vessels in your kidneys are broken it interferes with your kidney's capacity to get rid of wastes and excessive fluid. This can lead to a vicious cycle with the further fluid in your blood vessels raising your blood stress even much more, at some point resulting in kidney failure.\n7. Smoking, Fertility and the Effects the Unborn Kids Research have shown that smoking reduces a women's fertility. It also delays their capacity to conceive just after they quit utilizing birth manage tablets. Just like when you consume, a smoking mother is smoking for two. You boost the dangers of miscarriage, premature birth and your baby's death in its 1st year. The infant mortality price at birth or by means of a miscarriage is 50 % larger when the mother smokes. Kids whose mothers smoke are also 2\u00bd instances much more most likely to die of crib death or sudden infant death syndrome. The impacts of smoking for the duration of pregnancy final a lifetime. The typical birth weight of babies whose mother's smoked is significantly less than babies of nonsmokers. Also, when examined at age 7, youngsters whose mothers smoked for the duration of pregnancy have been much more most likely be shorter, slower readers and socially impaired.\n8. Smoking Causes Wrinkles Smoking reduces blood flow to your skin. Have you ever observed a person who has smoked for decades? Of course you have! Since of the lack of circulation to their skin they appear life old, wrinkled baseball mitts. This lack of circulation prevents the required oxygen and nutrients from reaching your skin. It also reduces the quantity of all-natural skin softeners, collagen and elastin, in your skin. Smokers effortlessly seem five years older than non-smokers of the similar age. Considering that cigarette smoke irritates the skin and eyes, smokers squint much more which leads to much more wrinkles about the corners of the eyes.\n9. Smoking Causes Osteoporosis Current research have shown a direct partnership amongst smoking and decreased bone density which is recognized as osteoporosis. It most frequently impacts females. Osteoporosis is a illness which causes bones to weaken and fracture much more effortlessly. What Can You Do? As you can see smoking has been shown to straight influence your well being. Statistics show that forty % of the males who smoke will die prior to they attain retirement age. Girls smokers face an elevated danger of cervical cancer. Pregnant females not only take dangers with their personal well being but also take a opportunity with the well being of their unborn babies.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to my compendium of all things dough, dessert, or delicacy, or any cross-section of the three. As you will soon see from the content of this blog, I have an irrepressible sweet tooth and would rather be eating dessert than doing most other things. Sometimes I bake for myself but more often than not I leave the production to the pros, so you'll generally find here items that you, too, can ingest and enjoy for a modest fee.\nI dwell in the Big Candy Apple of New York City, but I am of the, \"Will travel for food,\" persuasion. When I'm lucky enough to make it out of the concrete jungle I call home, I do my best to indulge in the items that satisfy the local sugar craving.\nIf you know of any treats I must try, want to say hello, or have baked a cake that is simply divine, do let me know. Nothing is better than cake and company.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"PITTSBURGH'S 1ST SPECIALTY SNEAKER\/STREETWEAR BOUTIQUE. RARE SNEAKERS, SUPREME, BAPE, PALACE & MORE.\nOUR TRUSTWORTHY AND KNOWLEDGEABLE REFRESH TEAM MEMBERS ARE EXPERTS THAT ENSURE EVERY PRODUCT WE SELL IS 100% AUTHENTIC. FROM JORDANS AND YEEZYS TO SUPREME AND BAPE - SHOP SAFELY AND CONFIDENTLY.\nSHOP IN PERSON AT OUR BRICK AND MORTAR LOCATION AT 5450 PENN AVENUE PITTSBURGH, PA 15206 FOR THE FULL REFRESH EXPERIENCE OR VIEW\/SHOP OUR FULL INVENTORY USING THE SHOP SECTION OF THIS WEBSITE.\nSHOP OUR CURATED SELECTION OF HIGH END STREETWEAR - FROM CLASSIC RELEASES TO THE MOST RECENT DROPS. T-SHIRTS, HOODIES, JACKETS, HATS, BAGS, ACCESSORIES, STICKERS & MUCH MORE.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Federal Government has vowed to do everything required to bring Nigeria back to the path of growth with the right support; saying that Nigeria will not fail.\nAdeosun said that the government was determined to rewrite Nigeria's economic story by growing critical infrastructure like power, transport, housing.\n\"This is why we are redirecting expenditure from recurrent where we thought there have been a lot of wastages and leakages.\n\"Redirecting spending to capital to create long-term value, is tough in the short term but the long term benefits will be there for the future generation.\n\"We are confident of getting back to growth with the right support,\" she said.\nAdeosun said that if we invest in critical infrastructure, there would be increased productivity, which will lead to job creation and prosperity for the people.\nAdeosun said that Nigeria have aligned with the views of the IMF and World Bank with regards to inclusive growth, stating that inclusive growth was one of the objectives of this administration to end poverty.\n\"Government is investing heavily in education and as part of our social intervention programme, we aim to engage more young graduates into teaching in primary schools.\n\"I am sure that we will soon start seeing improvements in our education indices.\n\"We are also pumping money into agriculture. We also try to encourage women in agriculture; Women are the key especially in agriculture so we are giving small business loans on agriculture.\n\"Bank of Industry has assured that it will start disbursing micro loans to women so there is a great focus on women,\" she said.\nOn his part, the Governor, Central Bank of Nigeria (CBN) Mr Godwin Emefiele said Nigeria has taken the IMF advice for a three-pronged comprehensive approach of monetary, fiscal and structural reforms to revive the economy.\n\"This is the way everybody has to go and we are doing that in Nigeria. There is serious collaboration between the monetary and fiscal authorities and if we continue in this direction we will achieve these objectives,\" he said.\nCommenting on whether the financial sector reforms has being enjoying a buy-in from potential investors, Emefiele said with the recent CBN reforms he was sure that investments would begin to come in.\n\"When you have the kind of situation that we have, naturally people will be a bit skeptical, but with time as they develop more confidence we will begin to see more and more of them coming into the country.\n\"However, there are positive indications that meeting and networking with foreign delegations and officials of the IMF and World Bank at this meeting, we will achieve some of the objectives that we have in mind.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The pupils at Lady Seawards Primary School experience Religious Education in a variety of ways. Practical sessions have included acting out the stories of Easter at the Church. The Nativity and Easter Service are an important part of our year at Lady Seawards Primary School with class one taking the lead in performing the Nativity and the whole school play a part in the teaching of the Easter story through the mediums of drama and music which are performed at the church..\nRecently, Class 2 learnt about shrines during their learning about Hinduism.\nThe children listened to Hindu music, breathed in the aroma of incense and used the tea lights to focus spiritually on a place which was meaningful for them and allowed them to connect with their thoughts.\nIn the photos you will see some children praying, some mirroring gestures they have witnessed Hindus performing at the shrine and other children simply taking a moment to reflect upon their thoughts.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbmiug b/data_all_eng_slimpj/shuffled/split2/finalzzzbmiug
new file mode 100644
index 0000000000000000000000000000000000000000..f96cdbe2e68662b2cd8724b007090960078fe5f4
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbmiug
@@ -0,0 +1,5 @@
+{"text":"The #1 New York Times bestselling novel and basis for the Academy Award-winning film\u2014a timeless and universal story about the lines we abide by, and the ones we don't\u2014nominated as one of America's best-loved novels by PBS's The Great American Read.\nAibileen is a black maid in 1962 Jackson, Mississippi, who's always taken orders quietly, but lately she's unable to hold her bitterness back. Her friend Minny has never held her tongue but now must somehow keep secrets about her employer that leave her speechless. White socialite Skeeter just graduated college. She's full of ambition, but without a husband, she's considered a failure.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"ShukerNature: THE MONGOLIAN DEATH WORM \u2013 A SHOCKING SURPRISE IN THE GOBI?\nTHE MONGOLIAN DEATH WORM \u2013 A SHOCKING SURPRISE IN THE GOBI?\nIt's not every day \u2013 or every expedition \u2013 that begins with a request from a country's head of government formally requesting that a specimen be captured of a creature so elusive, and deadly, that western science does not even recognise its existence. Nevertheless, that is precisely what happened in 1922 when eminent American palaeontologist Prof. Roy Chapman Andrews met the Mongolian premier in order to obtain the necessary permits for the American Museum of Natural History's Central Asiatic Expedition to search for dinosaur fossils in the Gobi Desert. And the creature that the Mongolian premier instructed him to procure? None other than the lethal allghoi khorkhoi \u2013 or, as it is nowadays commonly referred to throughout the world, the Mongolian death worm.\nAlthough in the 1920s, this extraordinary mystery beast was totally unheard of outside Mongolia, today it is one of the most (in)famous of all cryptozoological creatures \u2013 thanks to the series of pioneering expeditions to its southern Gobi homeland launched by Czech explorer Ivan Mackerle, the first of which took place during June and July 1990, and which subsequently attracted considerable interest internationally. During his searches, Ivan collected a very impressive dossier of information concerning the death worm, based upon eyewitness reports and other anecdotal evidence, which he subsequently made freely available to me to use as I wished in my own writings, and which can be summarised as follows.\nIts local names \u2013 allghoi khorkhoi and allergorhai horhai \u2013 translate as 'intestine worm', because according to eyewitness testimony, this mysterious sausage-shaped creature resembles a living intestine. Red in colour with darker blotches, it measures 1-1.6 m long and is as thick as a man's arm, but has no discernable scales, mouth, nor even any eyes or other recognisable sensory organs. It is said to be truncated at both ends, but according to some accounts at least one end also bears a series of long pointed structures at its tip.\nFor much of the year, the death worm remains concealed beneath the Gobi's sands, but during the two hottest months \u2013 June and July \u2013 it can sometimes be encountered lying on the surface, particularly after a downpour of rain.\nLocals claim that it can also be found in association with the black saxaul Haloxylon ammodendron, a yellow-flowered desert shrub, whose roots are parasitized by the goyo Cynomorium songaricum \u2013 a strange, cigar-shaped plant of uncertain taxonomic affinities.\nAccording once again to local lore, the death worm is deadly for two very different reasons. If approached too closely, it is said to raise one end of its body upwards (as portrayed on the front cover of my book The Beasts That Hide From Man, 2003), and then squirt with unerring accuracy at its victim a stream of extremely poisonous, acidic fluid that burns the victim's flesh, turning it yellow, before rapidly inducing death. It is claimed that the death worm derives this highly toxic substance externally - either from the saxaul's roots or from the goyo attached to them (and thereby reminiscent of how South America's deadly arrow-poison frogs derive their skin toxins from certain small arthropods that they devour). During my own researches, however, I have uncovered no evidence to suggest that the saxaul's roots are poisonous, and I have revealed that the goyo is definitely not poisonous (it is eaten as famine food, and used widely in Chinese herbalism). So if the death worm truly emits a venomous fluid, it presumably manufactures it internally, rather than deriving it externally.\nEven more shocking \u2013 in every sense! \u2013 is the death worm's second alleged mode of attack. Nomadic herders inhabiting the southern Gobi tell of how entire herds of camels have been killed instantly merely by walking over a patch of sand concealing a death worm beneath the surface. Moreover, one of Ivan Mackerle's local guides recalled how, many years earlier, a geologist visiting the Gobi as part of a field trip was killed when he began idly poking some sand one night with an iron rod \u2013 as he did so, he abruptly dropped to the ground, dead, for apparently no reason, but when his horrified colleagues rushed up to him, they saw the sand where he had been poking the rod suddenly begin to churn violently, and from out of it emerged a huge, fat death worm.\nThe camels presumably died from coming into direct physical contact with the death worm hidden beneath their feet, but the geologist only touched it indirectly, via the metal rod. Consequently, the only conceivable way that this action could have caused his death is by electrocution \u2013 which would obviously explain the camels' instant deaths too. Although there are several different taxonomic groups of fish containing species that can generate electricity \u2013 including the famous electric eel and gymnotids, as well as the electric catfish, electric rays, mormyrids, rajid skates, and electric stargazers \u2013 no known species of terrestrial creature possesses this ability. So if the native claims concerning the camels and the geologist are correct, and always assuming of course that it really does exist, the death worm must be a very special animal indeed. But what precisely could it be?\nDespite its English name and superficially similar external appearance, it is highly unlikely that the death worm could be a bona fide earthworm or related invertebrate. For although some earthworms do grow to prodigious lengths, and certain species known aptly as squirters even spurt streams of fluid from various body orifices, none exhibits a water-retentive cuticle, which would be imperative for survival in desert conditions to avoid drying out. Of course, there may be a highly-specialised earthworm in the Gobi that has indeed evolved such a modification, but with no precedent currently known, the chances of this seem slim. In addition, if the death worm's powers of electrocution are real, this would require even more modification and specialisation for an earthworm to fit the bill.\nCaecilians constitute a taxonomic order of limbless amphibians that are deceptively worm-like in appearance and predominantly subterranean in lifestyle. Certain species can also attain a total length matching the dimensions reported for the death worm. As with true worms, however, caecilians' skin is water-permeable, so once again even a giant caecilian would soon dry out in the arid Gobi, unless, uniquely among these particular amphibians, it had evolved a water-retentive skin.\nIf the death worm is genuine, it is almost certainly some form of reptile. To my mind, the likeliest solution is an unusually large species of amphisbaenian. On account of their vermiform appearance (most species are limbless), these little-known reptiles are also called worm-lizards, even though, taxonomically speaking, they are neither. As with caecilians, they spend much of their lives underground, rarely coming to the surface except after a heavy fall of rain. This all corresponds well with the death worm's reported behaviour. Furthermore, unlike real worms and caecilians the skin of amphisbaenians is water-retentive, so a giant species would not dry out in the Gobi.\nConversely, whereas the death worm is said to be smooth externally, amphisbaenians are very visibly scaly, and they also have a readily-observed mouth. In addition, they are all completely harmless, which wholly contradicts the twin death-dealing talents attributed to the Gobi's moat greatly-feared denizen. Naturally, it is conceivable that these abilities are entirely apocryphal, nothing more substantial than superstitious fancy. After all, several known species of amphisbaenian, and also caecilian, are fervently believed by their local human neighbours to be deadly poisonous even though in reality they are wholly innocuous.\nMuch of what has been proposed for and against an amphisbaenian identity for the Mongolian death worm applies equally to the possibility of its being an unknown species of very large legless true lizard - akin perhaps to the familiar slow worm and glass snake, or even to the skinks, some species of which are limbless. However, these lizards are much less worm-like and subterranean than amphisbaenians, so overall the latter provide a more satisfactory match with the death worm.\nLast, but by no means least, is the thought-provoking prospect that the death worm may be a highly specialised species of snake. Not only do most of the above-noted physical and behavioural similarities between the death worm and the amphisbaenians and legless lizards apply here too, but spitting cobras also offer a famous precedent for an elongate creature that can eject a stream of corrosive venom with deadly accuracy at a potential aggressor. Moreover, the spine-bearing tip described for the death worm recalls a genus of cobra-related species known as death adders Acanthophis spp., which possess a spiny worm-like projection at the tip of their tail that acts as a lure for potential prey.\nTheir name recognises the fact that although, like cobras, they are elapids, the death adders have evolved to occupy the ecological role filled elsewhere by true vipers. Could there be a specialised, unknown species of death adder that has evolved the venom-spitting ability of its spitting cobra relatives? If so, this would vindicate the locals' testimony concerning the death worm's emissions. But what about its alleged powers of electrocution?\nRemarkably, this too may be more than just a myth. For if the death worm were indeed a snake and perhaps sported such smooth, fine, tiny scales that they were not readily discernable, when it was crawling through sand these scales may be able to generate a weak electrical current via friction \u2013 a process known as triboelectricity, which has already been documented from certain sand-dwelling snake species. If this in turn gave rise to exaggerated descriptions of its potency, it is easy to see how, over the course of several generations of ever more fanciful retellings, the entirely false belief in a creature that kills by electrocution could ultimately arise.\nSo could it be that in spite of the death worm's initially unlikely form and behavioural proclivities, there actually could be a real, scientifically-undescribed species at the heart of this longstanding mystery, albeit one far less flamboyant and formidable than the version described in local testimony? Or is the entire death worm scenario nothing more than native folklore?\nScience-fiction devotees reading about the Mongolian death worm will undoubtedly recall Frank Herbert's celebrated series of 'Dune' novels. These were set on a desert planet called Arrakis \u2013 home to an enormous species of vibration-sensitive sand worm called the shai-hulud, which could be 'called' to the surface when the planet's human inhabitants used a vibration-engendering device known as a thumper.\nIn a classic scenario of transforming science-fiction into science fact, and directly inspired by the 'Dune' stories, during his first Gobi expedition Mackerle and his team tested out their very own specially-constructed thumper, but to no effect. During their second Gobi expedition, clearly favouring a more emphatic approach this time, they set off a number of controlled explosions in areas said by the locals to harbour subterranean death worms, in the hope that the massive vibrations elicited by these explosions would stimulate the worms to surface. Sadly, however, their summoning again went unheeded, as no worms appeared.\nMoreover, in recent times several expeditions other than those of Ivan Mackerle have also searched the southern Gobi's vast terrain in search of its purported 'monster', but once again none has met with any success so far.\nThese include an expedition led by English cryptozoological field researcher Adam Davies in 2003, a team sent out by the Centre for Fortean Zoology in 2005, and more recently a two-man expedition led by New Zealand journalist David Farrier in August 2009. Unfortunately, none succeeded even in seeing, let alone capturing, a death worm (a local man informed the CFZ team that on one occasion a killed specimen had been taken away by Russian zoologists, but he did not provide any names or details that could be pursued). Nevertheless, some informative anecdotal evidence has been gathered, and both the Adam Davies-led team and the CFZ team observed and photographed a carving of a supposed death worm (although it sported eyes) on exhibition at the remote Gobi Museum.\nMost telling of all, however, as some of these later expeditions were startled to discover, the newest generation of Gobi nomads are as likely to be riding desert motorbikes as camels, and some even carry mobile phones inside the folds of their traditional desert robes. So perhaps they are also heeding less and less the traditional stories and beliefs of their elders.\nHaving said that, there may be another, very different reason why these expeditions have returned empty-handed. Some locals claim that death worms have been seen far less frequently in recent years than in the past. Who knows - perhaps, just as the rest of the world has finally started to learn about and become interested in it, the Mongolian death worm has begun slipping inexorably into extinction.\nHow ironic that would be \u2013 almost as ironic, in fact, as Prof. Roy Chapman Andrews's straight-faced promise to the Mongolian premier back in 1922 that if his expedition did indeed encounter a death worm during their search for dinosaur fossils in the Gobi, they would do their best to secure its capture using a pair of long steel collecting forceps. In view of what reputedly happened to the hapless geologist who poked one of these fearful creatures with a metal rod, it's probably just as well that Prof. Andrews and his team never did find any!\nFor the most detailed, comprehensive coverage of the Mongolian death worm ever published, see my book The Beasts That Hide From Man (Paraview: New York, 2003).\nGlad you could clear up some misconceptions about the death worm. I am interested in the snake hypothesis. I wonder if the snakes could have used a specialized form of scales to purposefully produce triboelectricity as a weapon? It's a frightening thought.\nLampreys, unlike lungfishes, do not aestivate (undergo summertime hibernation), and even if they could, they could only survive in this state for a limited time - ultimately, they would need to return to water, whereas this return could not happen in the Gobi Desert, as rain alone would be insufficient to sustain them. Also, although they are superficially eel-like externally, lampreys are wholly unrelated to eels, belonging to an entirely separate taxonomic class of vertebrate, and are much simpler, more primitive organisms than eels. But thanks for the interesting suggestion, and I'm glad that you found my article interesting.\nI have actually eaten goyo. It tastes like banana and has a texture like celery.\nThe locals think the deathworm is not becoming rarer but is see less due to people travelling by motor bike rather than horse or camel.\nThat said we talked to about two dozen witnesses. I concluded the animal was an unknown type of large amphisbeana.\nHi Rich, Yes, I mentioned the motorbike link here too, and I too support the amphisbaenian identity, right from when I first documented the death worm back in the late 1990s - it fits the bill more closely than other identities. Interesting to know what goyo tastes like!\nI think there would be a market for goyo in the west. It's an ugly looking thing but the taste is lovely.\nDid Ivan Mackerle create a topographic map with thematic detail for specific locations for the Mongolian Death Worm sightings?\nNot as far as I'm aware. The Gobi Desert is around 500,000 square miles in area, so that's a lot of desert to document.\nThere are a number of bizarre giant worm sightings which exhibit similar characteristics of having no discernible features. The difference is that when they do open their eyes, they are big and bright blue. See About.com and \"The creature of the dump\" for the details. There's more stories too. My hypothesis is that they are giant flying bug larvae, also a solution to the UFO phenomena. A four foot wingspan giant bug was seen by multiple witnesses on an airliner at 30,000ft.\n\"Mongolian Death Worm\" was an absolute crapfest. Filmed in Texas grasslands, the Mongolian officer looked and sounded like a Texas Ranger. I use the DVD as a coaster!\nWhat about the Josh Gates expedition ware they found 1?\nBut that's just it, they DIDN'T find one - had they done so, it would have made headlines worldwide. All that they did find is something that MAY, just MAY, have been a death worm's burrow, but there is no serious evidence that it was, just conjecture.\nI don't really know but isn't there an expedition where a assistant met one and tried to capture it and put it in a container but it escaped by spitting out acid to melt the container?\nI don't want to meet the mongolian deathworm when i go to Gobi Dessert. It so Scary Animal!!!\nIf the Mongolian Death Worm is closely associated with the Cynomorium, as you suggest, then it could be a trade protection tale. The collection of su\u01d2 y\u00e1ng Cynomorium coccineum subsp. songaricum for use in Chinese herbalism was and is still a very large and valuable trade. It is used as an aphrodisiac and sexual medicine particularly against impotence and infertility. Among other things it nourishes the sinews and moistens the intestines. Contraindicated in those with excessive erections. First mentioned in China in the text \"Supplement to the Extension of the Materia Medica\" from 1347 CE.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"After nearly seven years of being a father of twins diagnosed with autism, I have experienced some \"unique\" Christmases. Since I married into a family that values Christmas dinner and festivities shared by all relatives at the same distant location in the \"country,\" I have been fortunate to see Santa Claus leave gifts at my place for my children and had the luxury of transporting those gifts afar for the festivities.\nAdditionally, we cherish a pot-luck style Christmas with my wife and I sharing the responsibilities of shopping, cooking, and safely and carefully transporting the twins, the gifts, and the food to our Christmas rendezvous.\nWhile this may sound like a very Merry Christmas, it doesn't always feel this way. And from observations of my wife, kids, and relatives, I have decided to share these observations and parenting tips for families affected by autism.\n1. Punctuality is a work in progress. Look, I understand you can't start the festivities late in the afternoon. But let's face it, a 45-minute drive, attached to finishing up the cooking (I'm deep frying a turkey at your request for God's sake), running behind two kids all morning, having Christmas at home, waking up at 5 a.m. (my kids have no concept of time), and staying up all night the night prior cooking and wrapping gifts after the kids go to bed makes it a little difficult to make it to Christmas dinner at 1 p.m.\n2. The boys are busy. Autism is pretty much synonymous for \"busy.\" The kids are busy, and the parents are busy. Since play skills are underdeveloped, anything \u2026 I mean ANYTHING \u2026 is considered a toy to include a spoon, remote control, your shoes, phone book, your purse, cell phone, etc. At our house we have to lock certain doors, cabinets, and refrigerator. So pardon us while we redirect them and kindly ask if there are alternative locations for your belongings.\n3. Gluten free and casein free. While I won't bore you with the details of a GF\/CF diet, I will tell you that that we will not offer the kids everything on the \"menu\" and neither should you. All foods and juices have to be cleared by us first. So to make it easy for you, we will bring the foods and drinks they can have.\n4. Eating together as a family is a nice expectation, but \u2026 I'm sure you can predict what comes next. I don't want you to get the impression I want to treat my kids differently and not allow them to engage in typical family functions. However, social appropriateness and etiquette are still a goal for them. So when either one of them or both see your tasty cornbread dressing, dinner rolls, blueberry cobbler, macaroni and cheese, etc. and grabs it, we have two issues: Your food taken by a 6-year-old and my 6-year-old with gluten now in his diet. Although they are on the GF\/CF diet, they still like foods with gluten and casein.\n5. I still love sports. This is for all you basketball-loving relatives watching the games that day. I love basketball with my favorite team being the LA Kobes \u2026 I meant Lakers (well, this year he is the team). Please don't be offended if I am not as engaged in talks about the sports. I have learned to use ESPN and my smartphone for all sports updates \u2026 kind of like the \"Cliff's notes\" version of watching the games. I may talk briefly, but I will be a bit active responding to requests of newly learned words like \"tickle me\", \"play with me\", and \"hug me.\"\n6. We still drink alcohol. My wife and I still have wine, a mixed drink or two, and I may even do a shot of tequila. But here is what I want you to keep in mind. There is no way we can drink any alcohol on Christmas since we have been operating off of maybe five hours of sleep, cooking all morning, playing with kids, long-commute, constantly managing kids here at the Christmas dinner. And later, we have to pack food, kids, commute back, give meds, and prepare kids for bed. I recommend we do shots of Red Bull or Starbucks (caffeinated and black).\n7. The best gift for my family is all of you. While nice toys and clothes go a long way for the kids, and while a nice robe and slippers would be nice for my wife and me, we would like more actual involvement or time you could dedicate to getting to know the boys. While you may know a fair deal about them, I challenge you to ask yourself, \"If the boys had to come live with me for an unspecified amount of time because something happened to the parents, do I know them well enough to do it?\"\nNo pressure to learn everything about them. But with enough family members honestly and sincerely believing they could care for them not only relieves pressure for us at future holiday events, it alleviates stress and pressure for us when developing a special needs trust for them should something happen to us.\nI applaud all parents for accepting the challenge of being responsible for the life of another being they brought into this world, and parenting can be a challenge for all parents. Parents of children with special needs have a different set of challenges that require different sacrifices.\nI leave you with this\u2014although the child has the diagnosis, autism, like other special needs diagnoses, is a family diagnosis. Be mindful of what the family needs as a whole so everyone can have a very Merry Christmas.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"ME to WE empowers people to change the world through their everyday consumer choices. ME to WE is part of WE which empowers people to change the world. WE invite you to join a community of people who are making the world a better place. At Pattison High School, ME to WE focusses on the act of serving, by giving both time and energy, as a way to foster globally minded citizens, and as means of changing the world.\nHi fellow tea & music lovers!\nLet's chat over delicious tea and interesting music, in a casual way, in an easy-going atmosphere, among friends\/schoolmates, while improving our natural, English conversational skills, together!\nReceive a 2% or possibly more increase in your grade, when you attend 30 minutes, per week.\nBTW \u2013 \"CHA\" or OCHA means tea, in Japanese, so if you like to chat, love tea and music, come join us!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On June 26-29, 2013, one mile above the sea in Denver, Colorado, the National Employment Lawyers Association (NELA) will gather for its 2013 Annual Convention\u2014The Summit For Workers' Rights. The brightest minds in employee rights advocacy will unite at the Sheraton Denver Downtown Hotel to create a perfect synthesis of education, activism, fun, and inspiration. Join us at The Summit and elevate your practice as we mark our 28th year of advancing equality and justice in the American workplace.\nNELA's Annual Convention is specifically designed for employee rights advocates. Our Annual Convention offers three or more plenary sessions and 30 concurrent sessions over three days. In addition to plenary and concurrent sessions, NELA invites registrants to attend the Annual Luncheon, Dinner and Gala Fundraiser. Approximately 500 members attend the Annual Convention which is held each at the end of June.\nDownloadable audio and speaker presentations from previous seminars and Conventions are available for purchase in our Online Library.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbmymw b/data_all_eng_slimpj/shuffled/split2/finalzzzbmymw
new file mode 100644
index 0000000000000000000000000000000000000000..7ea6b338fd4cdebd779f58ae51c533fe058fb3a7
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbmymw
@@ -0,0 +1,5 @@
+{"text":"Pizza for Good just hit iTunes!\nFebruary 10, 2014 by Will PollockLeave a Comment on Pizza for Good just hit iTunes!\nPizza for Good just hit iTunes!If you haven't already scored a copy, now's the time! and don't forget to leave us feedback - enjoy!\nhttp:\/\/www.youtube.com\/watch?v=QNLm6hJVxWc&feature=share&list=PLItsppnEDV7NQrOz0HQZ1HJ0OY_gdfl8h&index=3 If YouTube is your preferred viewer, jump over there to see a bunch of clips - with more coming soon.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This comfortable recently remodeled and modernized cabin sits atop a hill on extra large lot just a short walk to world class fishing in the Lake Fork of the Gunnison River and stocked trophy fishing ponds.\nKnown by Vickers Ranch guests for years as Cabin A, VHRR Unit #18 is on one of the largest building lots available at Vickers Horse River Ranch.\nThe comfortable cabin is ready to move in! Or make an investment as a vacation rental in this popular family summer destination area. All appliances and furniture are included. Property has power, community water and sewer, propane tank, fireplace and lots of charm!\nDownload the Horse River Ranch Info Packet for complete details about all inclusions and development information.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Read and discuss two novels \u2013 What Maisie Knew by Henry James and Behind the Scenes at the Museum by Kate Atkinson \u2013 about growing up in a dysfunctional family. From the 1890s to the 1990s, what's really changed?\nAn introduction to the world of Sociology and society in the twenty-first century and what makes us who we are.\nOur journey will take us from the moment we first entered the world of fairies, via the political manoeuvres of the Soviet State and Balanchine's updating of Imperial Russia, to Modern Ballet.\nWhat was it like to be a king, a dwarf or a slave in 16th and 17th century Spain? In this course we'll use paintings to explore life at court and on the streets in Spain and the Spanish Netherlands.\nThis course will study a range of novels by the three Bront\u00eb sisters, who wrote some of the most popular novels to emerge in the 19th century and which have had an enduring appeal to the present day.\nOriginal instruments make their way into \"modern\" music. The influence of the past on the present is seen in Stravinsky, Schoenberg and composers such as P\u00e4rt and Tavener, leading to synthesis and the way forward in the 21st century.\nConcentrating on Claire Tomalin's selection (Poems of Thomas Hardy) we shall consider the themes into which Hardy's poems are grouped and discuss individual poems in more detail.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Download the Remind App for communications about all of our events! Sign up is easy!\nJoin over 1,000 other Catholic high school students, listening to inspiring speakers, listening to amazing music and celebrating Mass with Bishop McElroy. Mater Dei Catholic High School. 10:00 am to 6:00 pm.\nOLG Goes to Youth Day at L.A. Congress in Anaheim, CA!\nMARCH 21, 2019 - Teens at L.A. Youth Day gather around national speaker Doug Tooke after enjoying a new twist on the game \"Rock, Paper, Scissors\" using the Saints: sword beats javelin, javelin beats ax, and ax beats sword! Doug delivered an engaging message to the youth, and using the book of Revelation, compared the image of heaven and the heavenly banquet found there to our experience of Sunday Mass.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Nye County Sheriff's Office on Thursday asked for help finding a trio suspected of trying to dig up \"precious metals\" on Ted Binion's former property in Pahrump.\nA manager of the property owned by the late gambling executive told detectives on April 2 that surveillance photos captured three men on the property with shovels on March 30.\nThe manager identified one man as 56-year-old Richard Cleaves, who formerly worked on the property. The sheriff's office asked the public's assistance in identifying the two other men seen with Cleaves. \"While investigating the crime scene, deputies and detectives located several holes dug in the main structure on the property,\" Deputy Ann Horak said in a video news release Thursday, adding that a hole was also found near a demolished brick BBQ smoker outside of the main structure.\n\"There is a long-standing rumor within Nye County that Binion had buried a large amount of precious metals, including silver bars, on this property and others that he owned,\" she said.\nIn 1998, Nye County found $5 million of silver in Pahrump owned by Binion, the manager of Binion's Horseshoe in Las Vegas, whose death dominated headlines two decades ago, leading to a high-profile murder trial.\nCleaves had not been arrested in in the burglary attempt, but was previously arrested Jan 27, 2017, after he was found digging holes on the property.\n\"He told deputies that he'd read online that there was millions of dollars worth of gold buried on the property. and he wanted to find it,\" Horak said. Anyone with information about the burglary can contact the sheriff's office at 775-751-7000.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbnmyb b/data_all_eng_slimpj/shuffled/split2/finalzzzbnmyb
new file mode 100644
index 0000000000000000000000000000000000000000..d34ec983aec55edca7e785675d3a96499760bf24
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbnmyb
@@ -0,0 +1,5 @@
+{"text":"Rebecca Mertens is a postdoctoral researcher in the history and philosophy of science and a member of the collaborative research program \"Practices of Comparison: Ordering and Changing the World\" at the University of Bielefeld. Her main interest is on the various uses of analogies in the construction of theoretical models and experimental set ups in the life sciences in the 20th century. Another closely connected point of interest is the role of comparative language in the development of multidisciplinary research areas and agendas in molecular biology and molecular genetics.\nIn 2016 she received her PhD from the University of Bielefeld, after completing her Masters in a German-French program in the history, philosophy, and sociology of science. In her dissertation she analyzed the role and development of the so-called \"lock-and-key-analogy\" in biochemical and biomedical research programs in the late 19th and 20th century. She was a visiting student at the \u00c9cole Normale Sup\u00e9rieure in Paris and a visiting graduate fellow at the Minnesota Center for Philosophy of Science in Minneapolis. In the course of her work as a coordinator for the German Council of Science and Humanities, Mertens gained experience in the area of science policy and management. Since November 2018 she has been a guest researcher at the GMPG research program, working on comparative semantics in the development and organization of molecular fields of research within the MPG.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Promised to be the latest buzz in bingo, BingoBeez is run by a small group of on-line gaming pros, with a strong focus on customer service and the best in bingo games. Generous with prizes and bonuses, the BingoBeez site was launched in April 2009 offering a massive deposit bonus for all new players, as well as extra prizes such as video cameras and Ipods.\nThe site is fun and welcoming, with a bee theme and plenty of cartoon bees buzzing about - but behind their fun images they offer a well-established bingo platform using software provided by Byworth.\nThe site is easy to follow, although a little \"buzz\"-ey and has a quick navigation bar at the very top of the screen. The main page has animated quick links to all their games, information about the current bonuses, latest jackpot winners and more - with their mascot, the queen bee Bernadette, featuring predominately and offering to 'burn your debts' and 'sweeten up your recession'! You can even email her, if you wish! New players may find the site a little daunting at first, as there is a lot of information and loads to do and explore, but they have many features to help ease new players into the game such as a bingo glossary page and all the latest bingo lingo.\nAll the popular bingo games are available, however, such as 75 ball bingo and 90 ball bingo and players can buy tickets from as little as 5p in their 'nickles bingo'. Bingo is not the only game on offer though, as the software also provides fun side games such as slots, pull tabs and keno - all with cash jackpots.\nAs well as the generous sign up offers members can also earn bonus points on their deposits to the site (called \"Bonus Bank Credits\"), which can then be used to purchase some very snazzy goods in the online bingo shop. These include Ipods, cameras, jewellery, luxury food hampers, laptops and even the latest in top-of-the-range games consoles.\nRegular players are often invited to the VIBeez bingo club (for Very Important Beeze), which offers many extra bonuses and privileges such as extra bonuses, a direct line to a personal account manager and tickets to special prize draws.\nIf you like loads of bingo competitions, extras and prizes, then you will love Bingo Beez. New competitions are always running, along with the free Beez Lottery, an on-line bingo shop and chat 24 hours a day. They also offer loads of bonuses for referring friends, depositing and frequently playing. In addition, they offer a quick \"no secrets\" cash-out policy without wagering requirements. A regular newsletter is also sent to all players, which keeps them updated on all the latest offers, winners and up-and-coming bingo events such as special games or the chances to win luxury holidays.\nBingo Beez now has so much to offer as it is flash based and does not require any downloads of specialist software. The sign up bonus gives you ample opportunity to test drive the site and ongoing promotions are decent for the US market.\nAt Bingo Beez you will be treated with 10 Bingo halls to choose from and they also offer their players the latest in instant bingo, slots and casino games. New registrants here will be gifted with a \u00a310 free sign up bonus plus you can get an additional \u00a35 free on their Face Book page.\nWith a nice 350% first deposit bonus, a hefty 75% bonus on all redeposits and a whole load of promotional bonuses that are coming out all the time, you will always get more for your money at Bingo Beez.\nYou can't go wrong with free bingo and they have lots of that here as well as cash prizes, BBs, gifts, vouchers and much more for the registered players to strive to achieve.\nAnd something you don't see often: NO wagering requirements. This is such a great deal and really simplifies things when you are dealing with cashing out.\nByworth software runs the games on the site and if you have played at other Byworth sites you know your transactions and information are 100% secure and that your games will be fun and play well. Good luck winning nice prizes at Bingo Beez!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There are over 3,500 middle schools in Texas. This is about one third of all the schools listed in this state. Top cities in Texas with most middle schools are Houston, San Antonio, Dallas, Austin and Fort Worth. You may find a list of all categories of Texas schools on this page to choose from.\nThe following articles will provide you with interesting information, useful tips and insights about middle schools in Texas.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"2010 marked the beginning of the Greek crisis, introducing devastating austerity measures that radically transformed all aspects of life. Eight years later, the total failure to manage the crisis and \"restructure\" the economy is clear, an outcome that was repeatedly predicted by both its critics and its designers. In his recently-published book, A Happy Future is a Thing of the Past: The Greek Crisis and other Disasters (Reaktion Books, 2018), Pavlos Roufos discusses the underlying aims of the crisis management program as well as the social movements that arose to oppose it.\nThe talk is co-sponsored by the Socialist Studies Seminar, the Working Class History Seminar, and the European Studies Center.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Consultation to discuss the details of the event and pre-shoot.\n1 hour pre-shoot with the family or just the Mitzvah child at any point before the event.\nUp to 8 hours of ceremony and reception photography on the day of the Mitzvah.\nAn online gallery of hand-edited images available for review and sharing to social media within 1 week after the event.\n500-800+ review images from the event day, 40-100+ from the pre-shoot.\nTravel time to and from locations.\n$2,000 for everything listed above.\n$1,000 for all hand-edited high-resolution digital files on a USB drive or DVD when ordered at time of deposit.\nBook - 11\"x13\" hardcover book with photo paper pages: up to 30 pages $250, up to 60 pages $500, up to 100 pages $700.\nDigital files (ordering post-event via gallery download) - $500 for all pre-shoot digital files, $1000 for all event day digital files, or $75 per digital file.\nDeposit of first third required to hold date.\n10% discount for full payment with deposit.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbpgzx b/data_all_eng_slimpj/shuffled/split2/finalzzzbpgzx
new file mode 100644
index 0000000000000000000000000000000000000000..8f295019c7f6d96418a324a67d8800eae1dd4d4b
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbpgzx
@@ -0,0 +1,5 @@
+{"text":"Charlotte is known as the second largest financial center in the U.S., but is also emerging as a dominant player for manufacturing and energy sectors. We've got all of your needs covered.\nThere are 300 Fortune 500 companies with facilities in Charlotte, seven of which are headquartered here, ranking Charlotte sixth nationally for the number of headquartered firms.\nCharlotte's industrial market continues to show strength. Construction keeps with strong activity seeing 1,300,000 square feet of space delivered at the end of Q4 and roughly another 5,600,000 square feet currently under way.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Protect yourself on your adventure across Central America.\nCentral America offers a diverse array of experiences for travelers. From scuba diving in Belize to exploring Mayan ruins in Guatemala, it's the adventure of a lifetime. A comprehensive travel insurance policy can help make sure you come home with a lifetime of memories \u2014 and not a mountain of debt from a climbing mishap.\nSeven Corners Liaison Travel Medical Coverage when traveling outside your home country with terms between 5 days to 3 years.\nWhat doesn't travel insurance cover?\nWhat activities should I add to my policy?\nMedical care. Insurance can pay for hospital stays, medication, and evacuation to a high-quality hospital if you're sick or injured somewhere without adequate care.\nLost and stolen items. Your insurer can reimburse you for lost or stolen items including cash, passports and cameras.\nFlight delays and cancellations. If your flight is canceled or delayed, your insurer can pay for a new flight, as well as any last-minute changes to accommodations or lost deposits.\nLeaving your luggage unattended. If you leave your luggage unattended in a public place and it gets stolen, your insurance won't reimburse you.\nGetting drunk. If you injure yourself or someone else while you're drunk or on drugs not prescribed by a doctor, you're on your own.\nActing recklessly. If you hurt yourself while acting recklessly, your insurer won't cover you.\nIgnoring warnings. If you travel to an area that the US Department of State has issued an advisory against, you won't be covered.\nChanging your mind. While your insurer will reimburse you for a canceled trip due to unforeseen circumstances, they won't reimburse you if you just decide not to go.\nWhat travel risks are specific to Central America?\nHealth risks. Several months before your planned departure date, speak to your doctor about any vaccinations you may need for the countries you plan on visiting. Vaccinations against malaria, rabies, typhoid, hepatitis and a range of other conditions may also be required.\nZika virus. Protecting yourself against mosquitoes is essential. If you're pregnant, reconsider your need to travel to Central America.\nAltitude sickness. Traveling to a high-altitude area? Familiarize yourself with the signs of altitude sickness. Staying hydrated, avoiding strenuous activity and eating small amounts of easily digestible food can help combat the condition. If you start to feel seriously ill, see a doctor immediately.\nPetty theft. Stay aware of your surroundings and keep some emergency cash in a separate pocket in case your wallet is stolen. If your luggage, passport or personal belongings are stolen, your comprehensive travel insurance policy can reimburse you.\nAlthough he's lived in America since he was just a few years old, Oscar stays in regular contact with his grandparents and extended family in El Salvador. Having not seen his grandparents in 10 years, Oscar is thrilled to finally be taking a two-week trip to El Salvador to catch up with his relatives.\nBut on the second day of his journey, Oscar comes down with a serious case of food poisoning and is hospitalized in San Salvador for two days. Thankfully, Oscar had purchased a comprehensive travel insurance policy before beginning his trip and his insurer covered the full cost of his medical bills. After a few days of recovery, Oscar was ready to continue his long-awaited adventure around El Salvador.\nWhat activities should I consider adding to my policy?\nHiking. From trekking in Costa Rica's Tortuguero National Park to climbing any number of spectacular peaks, hiking in Central America is an unforgettable experience. Just remember to check if your insurer imposes a limit on the maximum altitude you can trek to.\nWhitewater rafting. If you're planning to go whitewater rafting in Costa Rica, Panama or Belize, check the maximum grade of rapids your insurer will allow you to explore.\nScuba diving. Planning on exploring one of the many spectacular dive sites off the Central American coast? Check your policy for the maximum dive depth and if your insurer requires you to dive with a licensed instructor.\nHorseback riding. Horseback riding tours are popular in several Central American countries, so see if it's automatically included in your policy or if you need to purchase extra insurance.\nIf something goes wrong while I'm in Central America, how do I make a claim?\nIf you need to make a claim on your travel insurance policy, notify the insurer as soon as possible. This can usually be done over the phone or by filling out an online claim form.\nYour insurance provider will let you know what you need to do next and provide details of any supporting documentation you may need. This could include police reports, medical certificates and receipts.\nIt's a good idea to take a picture of the inside of your suitcase before leaving so you have evidence of what you brought. If anything is stolen, contact local authorities and get a copy of the police report.\nDepending on which country you're in, a hospital may require confirmation that your insurer will cover your medical expenses before offering you treatment.\nEmergency numbers. Save these in your phone before you leave.\nYour travel insurance provider. They can help with anything from lost baggage to travel delays.\nEmbassies or consulates. If you lose your passport or if they's a countrywide emergency, contact the nearest US consulate or embassy.\nTaxis. From a \"broken\" taxi meter to unlicensed cabs, there are plenty of scams to be wary of when catching a cab in Central America. Your best bet is to use a licensed operator or, better yet, arrange private transportation.\nTourist tax. When unscrupulous shop owners and stall holders see a tourist coming, they automatically jack their prices up three or four times the usual rate. Don't be afraid to haggle for a better price.\nThe helpful local scam. Be wary when you're approached by a seemingly helpful local at any sort of tourist attraction. This approach can be a lead-in to any number of other scams, from the offer of cheap accommodations to a demand for payment after giving you a guided tour.\nDecide where you're going. Make sure you're covered for every country on your itinerary.\nFigure out your plan. Make sure you're covered for any riskier activities.\nSet a budget. If you're on a tight budget, prioritize what types of coverage you need. Medical should be considered a top priority.\nDeclare any pre-existing conditions. If you have a pre-existing medical condition, notify your insurer and find out if you're covered.\nCompare your options. The final step in the process is to compare the benefits, features, limits and exclusions of a range of policies. Obtain multiple quotes and see which policy offers the best combination of quality coverage and value.\nHiking volcanoes in Nicaragua. Tanning on the beach in Costa Rica. Indeed, Central America is a vacationer's dream. But don't let an unexpected mishap turn your travel dream into a nightmare \u2014 pick up a travel insurance policy to make sure that all you bring back are memories.\nWhat are the entry requirements for Central America?\nA valid passport is required to get into any Central American country. Most countries require US citizens to pay entry and exit fees ranging from $5 to $55. Costa Rica requires proof of a bank account with at least $500.\nIf you're traveling to any country that's currently at risk of a Yellow Fever outbreak, you'll need proof that you're vaccinated to get in.\nCentral America is blessed with a tropical climate that makes it a viable vacation destination all year round, but your best bet for a relaxing holiday is usually during the dry season from December to April.\nIt can get quite cold at high altitudes during the winter months, and hurricanes pose a potential problem in countries on the Caribbean coast between June and November. Once you know exactly where you'd like to go in Central America, plan a travel itinerary that will allow you to take advantage of the best possible weather conditions.\nWhat currency do I need if I'm traveling across Central America?\nYour best bet when traveling through Central America is to use the local currency in the country you're in. You can generally get better value for money using the local currency and it will be accepted everywhere you go. US dollars are also widely accepted throughout Central America.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"KISS Signature Series luggage range by EPIC is some of the best travel gear available in this world. It's inspired to the core by one of the world's best and most dynamic and influential rock bands ever. This case is full of beautiful details, depth and dimensions. Each component is handpicked for ease of use and comfort, exceptional durability, and very low weight. You can even add your own personal twist to this case by customizing the wheels and more with the exclusive KISS graphics kit included inside.\nLooking for a specific Kiss Suitcase by Epic? Call us toll free at 888-910-BAGS with any questions or concerns and we would be happy to assist you.\nCopyright \u00a9 2014 Baggage Buy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"All tenants' refuse and recycling should be stored in the shared compound within the Basement adjacent to the corridor to the Infrastructure Basement. All refuse collection will be via the underground service area located between Building 2 and the Multi-Storey Car Park, which is accessed from Arundel Gate via the access road between City Lofts and the Novotel Hotel. The building managers will be responsible for moving the bins from the designated store to the Infrastructure Basement to suit the collection programme. Generally, tenants must comply with the waste management strategy as prepared by the managing agent, which is already operational for all users at St. Paul's Place.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Emptying oneself brings individuality, more and more individuality. Emptying oneself means emptying oneself of all that is implied in personality.\nPersonality is a farce, personality is pseudo, personality is that which is given to you by the society. Personality is imposed on you from the outside; it is a mask. Individuality is your very being. Individuality is that which you bring into the world, individuality is God's gift.\nWhatsoever good happens to you, immediately share it. This is one of the most fundamental things in Buddhism. Don't hoard it, don't be a miser. If love has arisen, share it, shower it. If you cannot find people, shower it on the trees, on the rocks, but shower it.\nDon't hoard it -- because if you hoard it, it turns into poison; if you hoard it, it goes sour and bitter. Share it.\nAnd the more you share, the more will be coming into you from unknown sources.\nSlowly slowly, you will be able to know the ways of inner economics. The outer economics is hoard if you want to have things. And the inner economics is just the opposite: hoard, and you will not have it. Give, and you will have it; give more, and you will have more of it.\nDo and be what you really want to do and be. Life is lived according to gestalts that you have created around yourself. If you really want to be something else, then familiarize yourself with it. If you want to play the guitar, practice. If you want to dance, practice. If you want anything to happen in your life, imbibe more and more of it and drop all that is against it.\nIf you really want to have a transformed life, if you really want to become a light unto yourself, if you really want to know the ultimate mystery of existence and the ultimate ecstasy of being alive, then the first force is intensity, totality. Whatsoever you do, do it intensely, to the optimum. Don't be lousy, don't be lazy, don't be partial. Go into it wholeheartedly, let it become a total absorption.\nIf you dance, dance so totally that the dancer disappears and only the dance remains, and it will be a transformation. If you love, love totally, so totally that there is no lover found.\nThe lover is found only if you are holding something back. That which you are holding back becomes the lover. If you are holding something back, that which you are holding back becomes the dancer, the singer. If you are totally in the dance, in the song, in the love, who is left behind there to say, \"I am the lover, I am the dancer\"? Nothing is left.\nWhatsoever happens -- good, bad, success, failure -- immediately become aware of what is happening. Don't miss a single moment. Don't lag behind -- be present to it, and you will be surprised, errors will start disappearing from your life. And then whatsoever you do will be right.\nPeople ask me what is right and what is wrong, and my answer is: If something arises out of awareness, it is right. If something arises out of unawareness, it is wrong. Right and wrong is not a question of what you do, but of how you do it. It is not a question of what, but of how -- meditatively or nonmeditatively, alert, awake, or doing things as if you are a sleepwalker.\nObserve without evaluation, observe everything, don't miss any opportunity to observe -- because it is not a question of what you are observing, the question is that observation is growing.\nObserve everything -- the trees, the birds, the animals, the people, the traffic, your own mind and its traffic, your own reactions, others' reactions. Use every situation to observe, so that observation becomes deep-rooted in you.","meta":{"redpajama_set_name":"RedPajamaC4"}}