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+{"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"}}