{"text":"\\section{Introduction}\n\nIt is a long standing conjecture that for every Jordan curve, there is an inscribed rectangle of every aspect ratio.  Even for smooth Jordan curves, little is known in general about the set of aspect ratios for inscribed rectangles. For instance, smooth Jordan curves always have inscribed rectangles of aspect ratio $1$ and $\\sqrt{3}$, but no other aspect ratios have been proven to always be present \\cite{Me}\\cite{survey}.  We seek to give a nontrivial lower bound for the measure of the set of aspect ratios.  To this end, we will now present the important definitions and theorems of this paper. \n\n\\begin{dfn}\nLet $S^1$ be the unit norm complex numbers. We define $T(n,k)$, the $(n,k)$ torus link, to be the subset of $\\mathbb{C}\\times S^1$ consisting of points of the form $(g,r)$ such that $r^k = g^n$. Thus, it wraps around the unit circle in the $\\mathbb{C}$ coordinate $k$ times and the $S^1$ coordinate $n$ times. \n\\end{dfn}\n\n\\begin{dfn}\nLet $M_1$ and $M_2$ be two disjoint smoothly embedded M\\\"obius strips in $\\mathbb{C}\\times S^1 \\times [0,\\infty) $ such that $\\partial M_1\\cup \\partial M_2 $ is isotopic to $ T(4,2)\\times\\{0\\}$ in $\\mathbb{C}\\times S^1 \\times\\{0\\}$. Let $Y$ denote the one-point compactification of $\\mathbb{C}\\times S^1\\times [0,\\infty)$. Note that $H_1(Y\\setminus M_1;\\mathbb{Z}\/2\\mathbb{Z}) \\simeq \\mathbb{Z}\/2\\mathbb{Z}$. We say $M_1\\prec M_2$ if and only if the map $$H_1(M_2;\\mathbb{Z}\/2\\mathbb{Z})\\to H_1(Y\\setminus M_1;\\mathbb{Z}\/2\\mathbb{Z})$$ induced by inclusion is trivial. \n\\end{dfn}\n\n\\begin{thm}\nLet $M_1,...,M_n$ be pairwise disjoint M\\\"obius strips smoothly embedded in $\\mathbb{C}\\times S^1 \\times [0,\\infty) $ with $\\partial M_1 \\cup  ... \\cup \\partial M_n$ isotopic to $ T(2n,n)\\times\\{0\\}$  in $\\mathbb{C}\\times S^1 \\times\\{0\\}$. Then the reflexive relation on $\\{M_1,...,M_n\\}$ which extends $\\prec$ is a total ordering. \n\\end{thm}\n\nConsider the $S^1$ action on $ \\mathbb{C}\\times S^1 \\times [0,\\infty)$ which comes from multiplying an element of $S^1$ with the $S^1$ coordinate. If $X\\subseteq  \\mathbb{C}\\times S^1 \\times [0,\\infty)$ and $u\\in S^1$, then $u\\cdot X$ is defined to be $ \\{(x,u \\cdot y, z): (x,y,z)\\in S\\}$.\n\n\\begin{thm}\nIf $M$ is a M\\\"obius strip smoothly embedded in $ \\mathbb{C}\\times S^1 \\times [0,\\infty) $ with $\\partial M = T(2,1)\\times\\{0\\}$, and $u$ is a uniformly random element of $S^1$, then $P(M\\cap\\;u\\cdot M \\neq \\varnothing) \\geq 1\/3$. \n\\end{thm}\n\nNote that the condition for this theorem to hold is the actual equality $\\partial M = T(2,1)\\times\\{0\\}$. If they are merely isotopic, it will not necessarily be the case that $\\partial M \\cup u\\cdot \\partial M$ is isotopic to $T(4,2)\\times\\{0\\}$, which is necessary for the proof to work. It is also sufficient for $\\mathbb{C}\\times S^1\\times\\{0\\}$ to admit an $S^1$-action preserving isotopy that takes $\\partial M$ to $T(2,1)\\times \\{0\\}$.\n\n\\begin{thm}\nThere exists a M\\\"obius strip $M$ smoothly embedded in $ \\mathbb{C}\\times S^1 \\times [0,\\infty) $ with $\\partial M = T(2,1)\\times\\{0\\}$ such that $M\\cap (e^{i\\theta})\\cdot M \\neq \\varnothing$ if and only if $\\theta\\in [2\\pi\/3, 4\\pi\/3]$. Thus, the bound in Theorem 2 is tight. \n\\end{thm}\n\n\\begin{crl}\nLet $\\gamma: S^1\\to \\mathbb{C}$ be a smooth Jordan curve, and let $X$ be the set of all $r \\in [0,1]$ so that there is an inscribed rectangle in $\\gamma$ of aspect ratio $\\tan(r\\cdot \\pi\/4)$. Then the Lebesgue measure of $X$ is at least $1\/3$. \n\\end{crl}\n\n\\begin{proof}\nLet $M$ be the M\\\"obius strip in $\\mathbb{C}\\times\\mathbb{C}$ parameterized by unordered pairs of elements of $S^1$ with the formula $$\\{x,y\\}\\mapsto \\left(\\frac{\\gamma(x) + \\gamma(y)}{2}, (\\gamma(x) - \\gamma(y))^2 \\right)$$ Let us take a small tubular neighborhood $N = \\mathbb{C}\\times D_\\varepsilon$ around $\\mathbb{C}\\times \\{0\\}$ where $D_\\varepsilon$ is some small circular open disk  around $0$ in $\\mathbb{C}$. Since the disk we removed is circular, the $S^1$ action on the second coordinate of $\\mathbb{C}\\times \\mathbb{C}$ extends to an $S^1$ action on $(\\mathbb{C}\\times \\mathbb{C})\\setminus N$. Furthermore, we can identify $(\\mathbb{C}\\times \\mathbb{C})\\setminus N$ with $\\mathbb{C}\\times S^1\\times[0,\\infty)$ in such a way that preserves the $S^1$ action. Finally, by smoothness of $\\gamma$, we see that for sufficiently small $\\varepsilon$, the image of $M$ in $\\mathbb{C}\\times S^1\\times[0,\\infty)$ forms a M\\\"obius strip $M'$ such that there is an $S^1$-action preserving isotopy of $\\mathbb{C}\\times S^1\\times[0,\\infty)$ which takes $\\partial M'$ to $T(2,1)\\times\\{0\\}$. Therefore, we can apply Theorem 2 to see that for at least one third of all $u\\in S^1$, we have $M' \\cap u\\cdot M' \\neq \\varnothing$, and thus  $M \\cap u\\cdot M \\neq \\varnothing$. If $u = e^{i\\theta}$, such intersections correspond to inscribed rectangles of aspect ratio $\\tan(\\theta\/4)$, because these intersections give two pairs of points with the same midpoint and with the angle between their line segments equal to $\\theta\/2$. Furthermore, $u^{-1}$ gives the same aspect ratio as $u$, so by symmetry we have at least 1\/3 of the $\\theta\\in [0,\\pi]$. Substituting $r = \\theta\/\\pi$ gives the corollary. \n\\end{proof}\n\n\\section{Ordering M\\\"obius strips}\n\n\\begin{lem}\nLet $M_1$ and $M_2$ be disjoint M\\\"obius strips smoothly embedded in $\\mathbb{C}\\times S^1\\times [0,\\infty)$ such that $\\partial M_1\\cup \\partial M_2 = T(4,2)\\times\\{0\\}$, and let $u$ be a regular value of the canonical projection $M_1\\cup M_2\\to S^1$. Let $L_i = M_i\\cap (\\mathbb{C}\\times\\{u\\}\\times[0,\\infty))$ for $i = 1,2$. These will be 1 dimensional manifolds which each have two boundary points.  Let $P$ be the straight line segment in $\\mathbb{C}\\times \\{u\\}\\times\\{0\\}$ between the two boundary points of $L_1$.  Let $\\Sigma$ be a compact surface in $\\mathbb{C} \\times \\{u\\}\\times [0,\\infty)$ so that $\\partial \\Sigma = L_1\\cup P$, and which intersects $L_2$ transversely. Then $M_1 \\prec M_2 $ if and only if $|\\Sigma \\cap L_2|$ is even. \n\\end{lem}\n\n\\begin{proof}\nLet $Q$ be a path in $\\partial M_2$ between the two boundary points of $L_2$. Then $Q\\cup L_2$ represents a generator for $H_1(M_2;\\mathbb{Z}\/2\\mathbb{Z})$. Let $R$ be the union of the pair of rays that emanate from the two boundary points of $L_2$ and lie in the line between the two points. Then let $\\Sigma_1$ be a properly embedded surface of finite genus in $\\mathbb{C}\\times\\{u\\}\\times[0,\\infty)$ with $\\partial \\Sigma = R\\cup L_2$ which intersects $\\Sigma$ transversely, and let $\\Sigma_2$ be a properly embedded surface of finite genus in $\\mathbb{C}\\times S^1\\times\\{0\\}$ which is disjoint from $M_1$ and has $\\partial \\Sigma_2 = Q\\cup R$. Then let $\\Sigma' = \\Sigma_1\\cup \\Sigma_2$. We see that $\\Sigma'$ intersects $M_1$ transversely, and $\\Sigma'$ compactifies to a surface in $Y$ (from Definition 1) with boundary which generates $H_1(M_2;\\mathbb{Z}\/2\\mathbb{Z})$. Thus, the parity of the number of intersections between $\\Sigma'$ and $M_1$ determines the nontriviality of the map $H_1(M_2;\\mathbb{Z}\/2\\mathbb{Z})\\to H_1(Y\\setminus M_1;\\mathbb{Z}\/2\\mathbb{Z})$, which determines the truth value of $M_1\\prec M_2$. More specifically, $M_1\\prec M_2$ if and only if $|\\Sigma'\\cap M_1|$ is even. Now, consider the compact 1-manifold $\\Sigma\\cap \\Sigma'$. This must have an even number of boundary components because it is a 1-manifold. Furthermore, $ \\partial(\\Sigma\\cap \\Sigma') = (\\Sigma'\\cap M_1) \\cup (\\Sigma \\cap L_2) \\cup (R\\cap P)$, and $R\\cap P = \\varnothing$, so $|\\Sigma'\\cap M_1|$ is even if and only if $|\\Sigma \\cap L_2|$ is even. Therefore, we finally conclude that $|\\Sigma\\cap L_2| \\equiv 0 \\pmod 2$ is equivalent to $M_1\\prec M_2$.\n\\end{proof}\n\n\\begin{lem}\nLet $M_1$ and $M_2$ be disjoint M\\\"obius strips smoothly embedded in $\\mathbb{C}\\times S^1\\times [0,\\infty)$ such that $\\partial M_1\\cup \\partial M_2 $ is isotopic to $ T(4,2)\\times\\{0\\}$  in $\\mathbb{C}\\times S^1 \\times\\{0\\}$. Then $$ M_1\\prec M_2 \\iff \\neg (M_2\\prec M_1)  $$\n\\end{lem}\n\n\\begin{proof}\nWithout loss of generality, we can assume $\\partial M_1\\cup \\partial M_2 = T(4,2)\\times\\{0\\}$. Let $u$ be a regular value of the canonical projection $M_1\\cup M_2\\to S^1$. Let $L_i = M_i\\cap (\\mathbb{C}\\times\\{u\\}\\times[0,\\infty))$ for $i = 1,2$. Let $P_1$ be the straight line segment between the boundary points of $L_1$, and let $P_2$ be the straight line segment between the boundary points of $L_2$. Note that $P_1$ and $P_2$ intersect transversely at a single point. Let $\\Sigma_1$ and $\\Sigma_2$ be transversely intersecting compact surfaces with $\\partial \\Sigma_i = L_i\\cup P_i$ for $i = 1,2$. Finally, we see that $\\Sigma_1\\cap \\Sigma_2$ is a compact 1-manifold with  $\\partial (\\Sigma_1\\cap \\Sigma_2) = (L_1\\cap \\Sigma_2) \\cup (L_2\\cap \\Sigma_1) \\cup (P_1\\cap P_2)$. Since $|\\partial (\\Sigma_1\\cap \\Sigma_2)|$ is even and $|P_1\\cap P_2|$ is odd, we see that $|L_1\\cap \\Sigma_2|$ and $|L_2\\cap \\Sigma_1|$ have opposite parities. Therefore, by the previous lemma, $M_1\\prec M_2$ and $M_2\\prec M_1$ have opposite truth values. \n\\end{proof}\n\n\\begin{lem}\nThere exists a triple of punctured M\\\"obius strips $M_1',M_2',M_3'$ smoothly embedded in the manifold $(\\mathbb{C}\\times S^1\\times [0,\\infty))\\setminus B^4$, where $B^4$ is a small open 4-ball around $(0,1,1)$, so that the puncture boundary components form Borromean rings in $\\partial B^4$, and the other boundary components comprise $T(6,3)\\times \\{0\\}$. Furthermore, if we let $(i,j) = (1,2), (2,3)$ or $(3,1)$, and we fill in the puncture boundary components of $M_i'$ and $M_j'$ with disjoint disks in $\\partial B_4$ to make M\\\"obius strips $\\overline{M_i'}$ and $\\overline{M_j'}$ respectively, then $\\overline{M_i'}\\prec \\overline{M_j'}$. \n\nSuch a triple can be taken to have the boundary components of the M\\\"obius strips in any permutation of the components of $T(6,3)$. \n\\end{lem}\n\n\\begin{proof}\nConsider an alternating tangle which projects to three arcs that each cross the other two once, leaving a triangle in the middle. If we rotate this picture 180 degrees, we get a different tangle. There is a homotopy from this tangle to it's 180 degree rotation which has a single triple point, which simply consists of shrinking the central triangle to a point then opening it back up. If we trace out a singular surface in four dimensions by letting the homotopy evolve with one coordinate, and we take a small 4-ball around the triple point, the intersection between the boundary of the ball and surface will be the Borromean rings. The M\\\"obius strips we describe in the lemma then come from rotating this tangle 180 degrees and then applying the homotopy with the triple point to get back where we started, and excising a small $B^4$ around the triple point.  \n\n\\begin{figure}[h]\n\\caption{The tangle yields its rotation after a singularity.}\n\\centering\n\\includegraphics[scale = 0.8]{Singularity}\n\\end{figure}\n\n\\begin{figure}[h]\n\\caption{Attaching the tangle to its rotated mirror image yields the Borromean rings. This will be the boundary of a neighborhood of the singularity.}\n\\centering\n\\includegraphics[scale = 1.2]{Rings}\n\\end{figure}\n\nTo see that $\\overline{M_i'}\\prec \\overline{M_j'}$ for $(i,j) = (1,2), (2,3)$ or $(3,1)$, we use Lemma 1. For a fixed time away from the triple point, each strand goes over the next strand and under the previous strand, so a disk bounding one strand and the line segment between its endpoints will intersect the next strand once and the previous strand zero times. If we flip all the crossings in the tangle and to the same thing, the cyclic order we get from $\\prec$ is reversed. Therefore, we can get any permutation of $\\partial M_1, \\partial M_2,$ and $\\partial M_3$ in $T(6,3)\\times\\{0\\}$ by flipping the crossings and relabeling. \n\\end{proof}\n\n\\begin{lem}\nLet $M_1, M_2,$ and $M_3$ be disjoint M\\\"obius strips smoothly embedded in $\\mathbb{C}\\times S^1\\times [0,\\infty)$ such that $\\partial M_1\\cup \\partial M_2 \\cup \\partial M_3 $ is isotopic to $ T(6,3)\\times\\{0\\}$ in $\\mathbb{C}\\times S^1 \\times\\{0\\}$. Then it is impossible to have $$M_1\\prec M_2 \\prec M_3 \\prec M_1$$ \n\\end{lem}\n\n\\begin{proof}\nWe will derive a contradiction by assuming that we have M\\\"obius strips $M_1\\prec M_2 \\prec M_3 \\prec M_1$. Without loss of generality, we can assume $\\partial M_1\\cup \\partial M_2 \\cup \\partial M_3 = T(6,3)\\times\\{0\\}$.\n\nWe can glue $M_1\\cup M_2\\cup M_2 \\subseteq \\mathbb{C}\\times S^1\\times [0,\\infty)$ to $M_1'\\cup M_2'\\cup M_3'(\\mathbb{C}\\times S^1\\times [0,\\infty))\\setminus B^4$ (from the previous lemma) along their mutual boundary to get a triple of smoothly embedded punctured Klein bottles $K_1,K_2,K_3$ in $(\\mathbb{R}^3\\times S^1 )\\setminus B^4$ which bound Borrommean rings. We glue such that $K_i = M_i\\cup M_i'$ for each $i$.\n\nWe claim that each $K_i$ is $\\mathbb{Z}\/2\\mathbb{Z}$-null-homologous relative its boundary in the complement of the other two Klein bottles. It suffices to take a surface $\\Sigma\\subseteq \\partial B^4$ disjoint from $K_2$ and $K_3$ with $\\partial \\Sigma = \\partial K_1$, and a 3-manifold $Z$ with $\\partial Z = \\Sigma \\cup K_1$ which intersects $K_2$ and $K_3$ transversely, and prove that the 1-manifold $Z\\cap (K_2\\cup K_3)$ is $\\mathbb{Z}\/2\\mathbb{Z}$-null-homologous in $K_2\\cup K_3$. (Although, it is possible to have a nonorientable surface embedded in $\\mathbb{R}^4$ which does not bound an embedded 3-manifold, we are in a situation where cutting along $\\mathbb{C}\\times\\{u\\}\\times[0,\\infty)$ leaves us with an orientable surface, so such complications do not arise here.) Let $P$ be the surface in $\\mathbb{C}\\times S^1 \\times\\{0\\}$ that, for each $u\\in S^1$ cross section, consists of the straight line between the two boundary points of $M_1$. We can then select $Z$ by selecting a 3-manifold $Z_1$ in  $\\mathbb{C}\\times S^1\\times [0,\\infty)$ that bounds $M_1\\cup P$, and a 3-manifold $Z_2$ in $(\\mathbb{C}\\times S^1\\times [0,\\infty))\\setminus B^4$ that bounds $M_1'\\cup P\\cup \\Sigma$, then letting $Z = Z_1\\cup Z_2$. We can assume $Z_1$ and $Z_2$ only intersect $\\mathbb{C}\\times S^1 \\times\\{0\\}$ at $P$, and therefore that $Z$ is disjoint from $\\partial M_2$ and $\\partial M_3$. Therefore for $i = 1,2$ we know that $Z\\cap K_i$ is disjoint from $\\partial M_i$, and from this we can deduce that the homology class represented by $Z\\cap K_i$ is in the image of $H_1(M_i ;\\mathbb{Z}\/2\\mathbb{Z})\\to H_1(K_i;\\mathbb{Z}\/2\\mathbb{Z})$. Therefore, it suffices to take a regular value $u$ that is not in the image of $B^4$, and show that the number of points in $Z\\cap K_i \\cap (\\mathbb{R}^3\\times\\{u\\})$ for $i  =1,2$ is even. By Lemmas 1 and 2, the parity of the number of points on one side of $\\mathbb{C}\\times S^1\\times\\{0\\}$ will equal the parity of the number of points on the other, since $M_1\\prec M_2 \\prec M_3 \\prec M_1$ and $\\overline{M_i'}\\prec \\overline{M_j'}$ for $(i,j) = (1,2), (2,3)$ or $(3,1)$. This proves our claim.\n\n Thus, we can take a triple of transversely intersecting surfaces $\\Sigma_1,\\Sigma_2,\\Sigma_3$ in $\\partial B^4$ and a triple of transversely intersecting 3-manifolds $Z_1,Z_2,Z_3$ in $(\\mathbb{R}^3\\times S^1 )\\setminus B^4$ so that  $\\partial \\Sigma_i = \\partial K_i$,  $\\partial Z_i = \\Sigma_i\\cup K_i$, and $Z_i\\cap K_j = \\varnothing$ for $i\\neq j$. Then since the Milnor invariant of the Borrommean rings is odd, there is an odd number of points in the set $\\Sigma_1\\cap \\Sigma_2\\cap \\Sigma_3$. This leads to a contradiction because this set is the boundary of the compact 1-manifold $Z_1\\cap Z_2\\cap Z_3$.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem 1]\nBy Lemmas 2 and 4, the relation is a total ordering. \n\\end{proof}\n\n\\section{Bounding the intersection probability}\n\n\\begin{proof}[Proof of Theorem 2]\nLet $X$ be the subset of $S^1$ given by $$ X = \\{u\\in S^1: M\\cap\\; u \\cdot M = \\varnothing \\;\\; \\text{and} \\;\\; M\\prec u\\cdot M\\} $$  Using $u^{-1}$ instead of $u$ has the effect of swapping the M\\\"obius strips being compared by $\\prec$, so, by Lemma 2, we can conclude that $X$ is disjoint from $X^{-1}$ and $$X\\cup X^{-1} = \\{u\\in S^1: M\\cap\\; u \\cdot M = \\varnothing \\}$$  \nNow, we proceed by means of contradiction. Let $\\mu$ be the Haar probability measure on $S^1$. We will assume that $P(M\\cap \\;u\\cdot M \\neq \\varnothing) < 1\/3$ for uniformly random $u$, and thus we have that $\\mu(X) > 1\/3$. \n\nNow, we know that $\\mu(X \\cdot X) > 2\/3$ by Kemperman's theorem in $S^1$. Since $\\mu(X^{-1}) + \\mu(X \\cdot X) > 1$, we know that $(X\\cdot X) \\cap (X^{-1})$ is nonempty. Thus, we have $a,b,c\\in X$ such that $a \\cdot b \\cdot c = 1$.  Let $M_1 = M$, $M_2 = a\\cdot M$, and $M_3 = (a\\cdot b)\\cdot M = (c^{-1})\\cdot M$. Finally, since $a,b,$ and $c$ are all in $X$, we have $M_1\\prec M_2 \\prec M_3 \\prec M_1$. This yields a contradiction by Lemma 4. \n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Theorem 3]\nFirst, consider a punctured copy of Boy's surface in $\\mathbb{C}\\times[0,\\infty)$ with its boundary on $\\mathbb{C}\\times\\{0\\}$. We will associate an element of $S^1$ to each point in this surface to obtain the desired M\\\"obius strip. Boy's surface has one triple point, and three arcs of double points that travel from that triple point to itself. In addition to this, the surface can be built by attaching three disks to each arc of double points, and a cylinder that includes the boundary attaches to the remaining double points. We will now specify how we associate an element of $S^1$ to each point of this surface.  At the triple point, the associated points in $S^1$ will be cubic roots of unity. As we go around one of the arcs of double points the difference in $S^1$ at the double point will go across one third of $S^1$, along the shortest path from one nontrivial cubic root of unity to the other. This can be done in such a way that the paths in $S^1$ corresponding to the disks to be attached are nullhomotopic, and the path to which the cylinder is to be attached goes around $S^1$ twice. We can therefore extend the map to $S^1$ to the rest of the surface so that the boundary maps to $T(2,1)\\times\\{0\\}$.   \n\nIn Figure 3, we show a sequence of cross sections of this M\\\"obius strip, where each image represents a cross section in $\\mathbb{C}\\times S^1 \\times \\{t\\}$ projected down to $\\mathbb{C}$. We use the conventions of Turaev's shadow links \\cite{Turaev} to represent the knot. The numbers in the regions represent the number of times you will wind around $S^1$ if you move along the boundary of the region counterclockwise, following the knot, and moving in the positive direction along a fiber of $S^1$ at each crossing until you hit the next arc of the knot. The arrows with angles denote the angular distance in $S^1$ as we go from the tail of the arrow to the head of the arrow in the positive direction along an $S^1$ fiber. We see from this diagram that the angular distance at each crossing is always in the interval $[(2\/3)\\pi, (4\/3)\\pi]$. \n\\end{proof}\n\n\\begin{figure}[h]\n\\caption{The evolution of a cross section of the M\\\"obius strip.}\n\\centering\n\\includegraphics[scale = 0.7]{Movie}\n\\end{figure}\n\n\n\n\\nocite{*}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\nGamma-ray bursts (GRBs) are subdivided on grounds of\nboth their durations and spectra into short-hard and long-soft bursts\n(Kouveliotou et al. 1993).  While a lot has been learned from the \ndetection of long GRB afterglows since 1997 (Costa et al. 1997, van Paradijs\net al. 1997, Frail et al. 1997, Bloom et al. 1999, Stanek et al. 2003, Hjorth\net al. 2003), for example that long GRBs are related to death of massive stars\nin star-forming regions, the afterglows of short GRBs remained elusive \nuntil the early summer of 2005. The recent detections of afterglows from short\nGRBs point to a central engine that is related to an old stellar\npopulation. In particular, it has been argued (Bloom et al. 2006, Berger et\nal. 2005, Barthelmy et al. 2005, Villasenor et al. 2005) that the observations\nare consistent with being the result of compact binary mergers: they occur\nsystematically at lower redshifts\\footnote{It has recently been suggested \n(Berger et al. 2006b) that\n  at least 1\/4 of the short bursts could lie at redshifts $z>0.7$ and would\n  therefore imply substantially larger isotropised energies than was inferred\n  for the first set of detected short GRBs with afterglows.} than their \nlong-duration cousins (e.g. Fox\net al. 2005), both in galaxies with (Hjorth et al. 2005) and\nwithout star formation (Berger et al. 2005, Barthelmy et al. 2005) and they\nare not accompanied by a detectable supernova explosion (Bloom et al. 2006, \nFox et al. 2005, Hjorth et al. 2005).\\\\ \n\\begin{figure}[!t]\n \\includegraphics[angle=-90,width=0.75\\columnwidth]{dens_cut_fd36.ps}\n  \\includegraphics[angle=-90,width=0.75\\columnwidth]{dens_cut_fd45.ps}\n  \\includegraphics[angle=-90,width=0.75\\columnwidth]{dens_cut_fd120.ps}\n \\includegraphics[angle=-90,width=0.77\\columnwidth]{gwave_amplitudes_ns14_mag_1e6part.ps}\n  \\caption{Merger of a double neutron star system with 1.4 ${\\rm M_{\\odot}} \\;$\n  each (snapshots at 4.41, 5.54 and 15.0 ms). \nThe lowest panel shows the gravitational wave amplitudes times the distance\nto the source, $d$ (in units of 1.5 km).}\n  \\label{fig:DNS_1.4}\n\\end{figure}\nThere are, however, puzzling observations of late-time flaring activity in\nseveral short bursts Berger et al. 2006a). For example, GRB050724 showed long-lasting ($\\sim 100\n$s) X-ray flaring activity after a delay of $\\sim 30$ s. These long-lasting\nflares motivated MacFadyen et al. (2006) to suggest an accretion-induced\ncollapse of a neutron star in a low-mass X-ray binary as an alternative \ncentral engine. This central engine would produce the flaring by interaction \nof the GRB outflow with the non-compact companion star.\\\\ \nAs the final identification of the central engine will probably only come from\nan integral view of various features from a large sample of events, we will\ngive an overview over the properties and signatures that can be expected from\ncompact binary mergers. They will include gravitational waves, the structure of\nthe forming accretion disk, the neutrino emission and the burst\nitself. Moreover, we will discuss possibilities to produce the observed\nlate-time flaring activity.\\\\ \nThe discussion is mostly based on own simulations, some of which have been\nreported previously  (Rosswog and Davies 2002, Rosswog and Ramirez-Ruiz 2002\nand 2003, Rosswog and Liebend\\\"orfer 2003, Rosswog et al. 2003, Rosswog et\nal. 2004, Rosswog 2005) and several new simulations that are introduced here, \nsee Table \\ref{runs}. The physics included in our models and the numerical \ntechniques that we use have been described in a series of papers (Rosswog \nand Davies 2002, Rosswog and Liebend\\\"orfer 2003, Rosswog et al. 2003, \nRosswog et al. 2004).\n\\begin{figure}\n\\includegraphics[angle=-90,width=0.75\\columnwidth]{dens_cut_ns11_ns16_fd13.ps}\n\\includegraphics[angle=-90,width=0.75\\columnwidth]{dens_cut_ns11_ns16_fd50.ps}\n\\includegraphics[angle=-90,width=0.75\\columnwidth]{dens_cut_ns11_ns16_fd120.ps}\n\\includegraphics[angle=-90,width=0.77\\columnwidth]{gwave_ns11_ns16.ps}\n  \\caption{DNS merger with 1.1 and 1.6 ${\\rm M_{\\odot}} \\;$ (at 1.51, 6.17 and 15.0 ms).\n The lowest panel shows the gravitational wave amplitudes times the distance\nto the source, $d$ (in units of 1.5 km), as a function of time.} \n  \\label{fig:DNS_1.1_1.6}\n\\end{figure}\n\\begin{figure}\n\\includegraphics[angle=-90,width=0.99\\columnwidth]{algol_impact.ps}\n\n  \\caption{Merger of 1.1 and 1.6 ${\\rm M_{\\odot}} \\;$ system (t=1.63 ms): Impact of accretion\n    stream onto \n    surface of accreting neutron star. Color-coded is density in the orbital\n  plane, the arrows indicate velocities.} \n  \\label{fig:algol_impact}\n\\end{figure}\n\nThe new neutron star merger calculations have been produced with a new Smoothed\nParticle Hydrodynamics (SPH) code that benefits from a slew of numerical\nimprovements. It incorporates \n\\bi\n\\ii an artificial viscosity oriented at Riemann solvers, \n    see Chow and Monaghan (1997),\n\\ii a consistent accounting of the effects of the so-called ``grad-h''-terms, \n    see Springel and Hernquist (2002), Monaghan (2002) and Price (2004),\n\\ii a consistent implementation of adaptive gravitational softening \n    lengths, see Price and Monaghan (2006),\n\\ii the option to  evolve magnetic fields together with the fluid, \n    either via so-called Euler potentials (Stern 1970) or via a SPH \n    discretisation of the MHD-equations (Price and Monaghan 2005).\n\\ei\nFirst results obtained with this code have been published (Price and Rosswog\n2006), a detailed code documentation can be found in Rosswog and Price (2006).\n \n\n\n\n\\section{Gravitational waves as probes of the coalescence dynamics}\n\\label{sec:dynamics} \n\n\\begin{table*}\n\\begin{flushleft}\n\\caption{New runs: DNS stands for double neutron star systems, NSBH for\n  neutron star black hole systems. M$_1$ and M$_2$ are the component masses, \n  $q$ is the mass ratio,\n  a$_0$ is the initial separation, N stands for Newtonian, PW for the\n  \\Pacz-Wiita pseudo-potential, $R_{\\rm abs}$ is the radius of the absorbing \n  boundary, in units of $G M_{\\rm bh}\/c^2$, the next column gives the SPH particle number.} \\label{runs}\n\\begin{tabular}{rccccccrccccccccc} \nrun & type & M$_1$, M$_2$, $q$ & a$_0$ [km] & grav. & R$_{\\rm abs}$ & SPH  part. &\\\\ \\hline \\\\\nDA  & DNS  & 1.4,  1.4 ,  1      & 48    & N  & - &$2\\cdot 10^6$ &\\\\\nDB  & DNS  & 1.1,  1.6 ,  0.6875 & 48    & N  & - &$5\\cdot 10^5$ &\\\\\nNA  & NSBH & 1.4,  3.0 ,  0.4667 & 60    & N  & 6 &$6\\cdot 10^5$ &\\\\\nNB  & NSBH & 1.4,  4.0 ,  0.35   & 72    & N  & 6 &$2\\cdot 10^5$ &\\\\\nNC  & NSBH & 1.4,  7.0 ,  0.2    & 90    & N  & 6 &$2\\cdot 10^5$ &\\\\\nND  & NSBH & 1.4, 10.0 ,  0.14   & 112.5 & PW & 3 &$2\\cdot 10^5$ &\\\\\n\\end{tabular}\n\\end{flushleft}\n\\end{table*} \n\nGravitational waves can serve as a direct probe of the merger dynamics and, if\ndetected coincident with a short GRB, they would provide the ultimate proof of\nthe compact binary nature of the central engine. The dynamics of the\nmerger process is sensitive to the mass ratio of \nthe involved components, therefore, the signal from the merger of a double \nneutron star system (DNS) with a mass ratio close to unity can be very \ndifferent from a neutron star black hole (NSBH), where the black hole can, \nin principle, be much more massive than the neutron star. \\\\\nIn the last minutes before the coalescence the gravitational wave signal\nwill slowly sweep through the frequency range that is accessible to\nground-based gravitational wave-detectors \nsuch as GEO600 (Grote et al. 2005), LIGO (Abramovici et al. 1992), TAMA (Ando\net al. 2004) or VIRGO (Spallici et al. 2005, Freise 2005). \nThe detection of a gravitational wave signal coincident with a short GRB would\nallow to determine the distance to the source, the luminosity and the beaming\nangle (Kobayashi and \\Mesz 2003).\\\\\nThe coalescence of a double neutron star system\ncan be divided into three phases: the secular inspiral due to\ngravitational wave emission, the actual merger and, finally, the ``ring-down''\nphase, in which the death-struggle of the freshly formed super-massive neutron\nstar will take place. After this phase the remnant will have settled into its\nfinal state, probably a black hole (an alternative is discussed in\nSec.~\\ref{sec:magnetar}).\\\\  \nDuring the inspiral, the gravitational wave forms can be accurately described \nvia post-Newtonian expansions for point masses. To date, post-Newtonian\nformalisms exist up to 3.5PN order, see Blanchet (2006) for a recent\nreview, and lowest order spin-spin and spin-orbit couplings can be accounted\nfor (e.g. Will 2005).  In the\ninspiral phase, both the frequency and the amplitude of the waves will \nincrease, the system is said to ``chirp''. The orbit decays secularly until \nthe last stable orbit is reached, where the binary enters a ``plunging\nphase''. In this phase the stars fall nearly radially towards each other \nand merge within about one orbital period. This dynamical instability \n(Chandrasekhar 1975, Tassoul 1975) is the result a steepening of the \ngravitational potential due to both purely Newtonian tidal (Lai et al. 1993) \nand general relativistic effects, for a recent review on relativistic binaries\nsee Baumgarte and Shapiro (2003).  For the last two phases, the merger and \nringdown, three-dimensional hydrodynamic simulations are required to \npredict the gravitational wave signal.\\\\ \nDouble neutron star merger wave forms has been predicted by several groups. \nThe calculations started with Newtonian calculations\n(e.g. Oohara and Nakamura 1989, Rasio and Shapiro 1994, Ruffert et al. 1997,\n2001).\nPost-Newtonian approaches (Ayal et al. 2001, Faber and Rasio 2000) \nturned out not to be particularly successful due to the importance of \nhigher order PN-corrections.\nIn the conformal flatness approach (Isenberg 1978, Wilson et al. 1996) it\nis assumed that the dynamical degrees of freedom of the gravitational\nfields, i.e. gravitational waves, can be neglected and that the spatial \npart of the metric is (up to a conformal factor) flat and remains so during the\nfurther evolution. This approximation has been used in several compact binary\nsimulations (e.g. Wilson et al. 1996, Oechslin et al. 2002, 2004, \nFaber et al. 2004, Oechslin 2006) and it could be shown (Cook et al. 1996)\nto be provide reasonable accuracy in several cases.\nFinally, fully relativistic simulations have been performed by Shibata and\ncollaborators (Shibata 1999, Shibata and Uryu 2002, Shibata \net al. 2005). For a recent review on numerical relativity\nand compact binaries we refer to Baumgarte and Shapiro (2003).\\\\\nThe simulation of neutron star black hole binaries has been somewhat lagging\nbehind the DNS status. Again, first simulations were  Newtonian\n(e.g. Lee 1999a,b, Janka 1999, Lee 2000, 2001, Rosswog et al. 2004), slightly\nlater followed simulations that used a \\Pacz-Wiita (1980) pseudo-potential\n(Lee and Kluzniak 1999a, Rosswog 2005, Setiavan et al. 2004, 2006).\nVery recently, progress has been made with relativistic approaches. \nFor example, Taniguchi et al. (2005) were able to construct general \nrelativistic quasi-equilibrium sequences for black holes that are much \nmore massive than the neutron star. Faber et al. (2006) used an exact metric\ntogether with a conformal flatness approach in which the black hole \nposition was artifically kept fixed in space.\nL\\\"offler et al. (2006) were able to treat binary components of comparable\nmass, but their approach was restricted to a head-on collision. Recently, \nShibata and Uryu (2006) reported on first results of neutron star black hole \nbinaries in full general relativity.\\\\\nHere, we will report mainly on our own, either Newtonian or pseudo-Newtonian\nresults. These calculations focused on the microphysics rather than the\nstrong-field gravity aspect, therefore, the results concerning gravitational\nwaves should be taken with a grain of salt. Details about the calculation of\nthe wave forms can be found in Rosswog et al. (2004), we will mention in the\nappropriate places in which direction fully relativistic calculations are\nexpected to change the results. \n\n\\subsection{Double neutron stars}\nFor many years, neutron star masses, at least those in double neutron star\nsystems, were thought to be tightly clustered around a mass of  m=\n1.35 ${\\rm M_{\\odot}} \\;$ (Thorsett and Chakrabarty 1999). Recent data, see e.g. Stairs\n(2004), show considerable deviations from this value for several of the known \nbinary systems. The recently discovered double neutron star system \nPSR J1756-2251 (Faulkner et al. 2005), for example, has a mass ratio of \nq= 0.84 ($m_1= 1.40$ and $m_2= 1.18$ ${\\rm M_{\\odot}}$) and the double neutron star \nJ1518+4904 even has $q= 0.67$ ($m_1= 1.56$ and $m_2= 1.05$ ${\\rm M_{\\odot}}$), but \nstill relatively large errors (see Stairs 2004).\\\\ \nFor our new DNS simulations, we take masses of twice 1.4 ${\\rm M_{\\odot}} \\;$ as the \ntypical case, but we also consider the more extreme case with a 1.1 and a 1.6\n${\\rm M_{\\odot}} \\;$ neutron star. The dynamical evolution during the last moments of \na binary system with twice 1.4 ${\\rm M_{\\odot}} \\;$ is shown in Fig.~\\ref{fig:DNS_1.4}.\nThe neutron stars have no initial spin, their initial separation is just\noutside the plunging regime for our equation of state at 48 km and each star\nis modelled with slightly more than a million SPH particles (run DA in \nTab.~\\ref{runs}). Once the plunging sets in, the stars merge within about\none orbital revolution. Excess angular momentum is shed into symmetric spiral\narms that subsequently spread into an extended accretion disk, see \nSec.~\\ref{sec:accretion}, around a super-massive, neutron star-like object.\nThe evolution of the 1.1 and 1.6 ${\\rm M_{\\odot}} \\;$ system (run DB in Tab.~\\ref{runs}), \nsimilar to J1518+4904, is shown in Fig.~\\ref{fig:DNS_1.1_1.6}. In this case\nthe lighter star starts to transfer mass towards and then completely\nengulfs the more massive component. The mass transfer leads to a complete \ntidal disruption of the 1.1 ${\\rm M_{\\odot}} \\;$ star, parts of which form a tidal tail, \nsee panel three of Fig.~\\ref{fig:DNS_1.1_1.6}. For the chosen system \nparameters, the circularization radius of the accreted material is smaller \nthan the accretor radius, therefore, like in an Algol system, the accretion \nstream directly impacts on the accretor crust, see panel one in \nFig.~\\ref{fig:DNS_1.1_1.6} and Fig.~\\ref{fig:algol_impact}. For less \nextreme mass ratios the accreted material slides more smoothly around the \nsurface of the heavier companion, otherwise the evolution is similar.\\\\\nIn the last panels of Figs.~\\ref{fig:DNS_1.4} and \\ref{fig:DNS_1.1_1.6} \nwe show the retarded gravitational wave amplitudes, $h_+$ and $h_\\times$, \nas seen along the binary axis by a distant observer at distance $d$.\nIn both cases one sees the last stages of the ``chirp''-signal up to \nabout 2.5 ms, when the stars come into contact.\nFor the chosen initial separation of the 2 x 1.4 ${\\rm M_{\\odot}} \\;$ case the peak \nfrequency of $\\approx$ 1 KHz is reached 2.5 ms after the simulation start. \nIf $h_{\\rm min}$ labels the minimum detectable amplitude, such a system\nwould be visible out to a distance $d= 18 \\; {\\rm Mpc} \\; \n(10^{-21}\/h_{\\rm min})$. During the chirp phase up to about\n2.5 ms the signal is the result of the binary orbital motion, after that, up \nto about 6 ms it is determined by the elongated central object and the spiral\narms. The central object does not become rotationally symmetric by the end \nof the simulation and, therefore, keeps emitting quasi-periodic \ngravitational waves at about a tenth of the amplitude. This is behaviour\nis expected for a stiff equation of state (EOS) such as the \nShen et al. (1998) EOS that we use. It had been noted very early on (Rasio \nand Shapiro 1994) that this post-merger gravitational wave signal is a way \nto constrain the equation of state.\\\\\nFor the 1.1-1.6 ${\\rm M_{\\odot}} \\;$ system the peak frequency is slightly lower, \n$\\nu_{\\rm peak} \\approx 780$ Hz, the same statement is true for \nthe peak amplitude.\\\\\n\\begin{figure}\n\\includegraphics[angle=-90,width=1.1\\columnwidth]{exponent_mass_radius_v3.ps}\n  \\caption{Exponent of the mass-radius relationship for a polytropic star.}  \n  \\label{fig:m-r-expon}\n\\end{figure}\n\n\n\\begin{figure*}\n\\centerline{\\includegraphics[angle=-90,width=1.15\\columnwidth]{position_fd29_nsbh_q046.ps}\n            \\includegraphics[angle=-90,width=1.15\\columnwidth]{position_fd30_nsbh_q01.ps}}\n\\centerline{\\includegraphics[angle=-90,width=1.15\\columnwidth]{position_fd36_nsbh_q046.ps}\n            \\includegraphics[angle=-90,width=1.15\\columnwidth]{position_fd45_nsbh_q01.ps}}\n\\centerline{\\includegraphics[angle=-90,width=1.15\\columnwidth]{position_fd162_nsbh_q046.ps}\n            \\includegraphics[angle=-90,width=1.2\\columnwidth]{position_fd300_nsbh_q01.ps}}\n  \\caption{Dynamical evolution of a neutron star black hole system.\n           Left column: example of a system that undergoes episodic mass transfer\n          (M$_{\\rm ns}$= 1.4 ${\\rm M_{\\odot}}$, M$_{\\rm bh}$= 3 ${\\rm M_{\\odot}}$). A mini-neutron star\n          keeps orbiting the hole, thereby transfers mass directly into\n          the hole and sheds mass which subsequently spreads into a dilute disk\n          surrounding the orbiting binary system. It is only after 47 orbital\n          revolutions ($t\\approx 220$ ms) that the mini-neutron star is finally \n          disrupted, see panel 4, and its remains form an accretion\n          disk of $\\approx 0.05$ ${\\rm M_{\\odot}}$, see Fig.~\\ref{fig:bound_debris}. This\n          episodic mass transfer is imprinted on the gravitational wave\n          signal, see Fig.~\\ref{fig:GW_comparison}.\n          Right column: binary in which the neutron star is directly disrupted\n          (M$_{\\rm ns}$= 1.4 ${\\rm M_{\\odot}}$, M$_{\\rm bh}$= 14 ${\\rm M_{\\odot}}$). The figures in\n          each column refer to 3, 5 and 37 ms, respectively, after the onset\n          of mass transfer.}   \n  \\label{fig:nsbh_dynamics}\n\\end{figure*}\n\n\\begin{figure}\n\\centerline{\\includegraphics[angle=-90,width=1.15\\columnwidth]{pos3_t127ms.ps}}\n\\centerline{\\includegraphics[angle=-90,width=1.15\\columnwidth]{pos5_t217ms.ps}}\n\\centerline{\\includegraphics[angle=-90,width=1.2\\columnwidth]{pos6_t285ms.ps}}\n  \\caption{Dynamical evolution of a neutron star black hole system that\n          undergoes episodic mass transfer; continuation of\n          Fig.~\\ref{fig:nsbh_dynamics}, left column.}   \n  \\label{fig:nsbh_dynamics_cont}\n\\end{figure}\n\n\nThis basic picture of the dynamical evolution carries over to general\nrelativistic calculations. The stronger gravity, however, produces\nneutron stars of higher compactness and a more compact remnant. Apart from the\nmost obvious consequence, the possible gravitational collapse to a black hole,\na general relativistic merger yields higher peak frequencies and gravitational\nwave amplitudes and therefore ensures detectability out to larger distances. \nFor more details we refer to the recent calculations of Shibata et al. (2005).\n\n\\subsection{Neutron star black hole}\nThe accretion and merger dynamics of neutron star black hole systems is,\nmainly due to the larger possible mass ratios, substantially more complicated. \nIt is a result of the interplay between gravitational wave emission, mass\ntransfer and the nuclear EOS  (Rosswog et al. 2004). Gravitational wave\nemission will drive the binary towards coalescence, the mass transfer from the\nlighter neutron star to the heavier black hole has the opposite tendency.\nThe influence of the EOS is\ntwofold. On the one hand it determines the radius of the star, $R_{\\rm ns}$,\nand therefore the tidal radius, at which the neutron star will be disrupted. \nA simple estimate for this radius is given by\n\\be\nR_{\\rm tid}= \\left(\\frac{{M}_{\\rm bh}}{M_{\\rm ns}}\\right)^{1\/3} R_{\\rm ns}.\n\\ee\nOn the other hand the EOS is responsible for the way the star reacts to \nmass loss, i.e. whether it expands or contracts. For a polytropic star \nthe mass-radius relationship is\n$R \\propto M^\\frac{\\Gamma-2}{3\\Gamma-4}$, where $\\Gamma$ is the polytropic \nexponent of the EOS (e.g. Kippenhahn and Weigert 1990). The exponent of the \nmass-radius relation of a polytropic star, $(\\Gamma-2)\/(3\\Gamma-4)$,\nis plotted in Fig.~\\ref{fig:m-r-expon}. For $\\Gamma<2$ the star expands\non mass loss and for $\\Gamma>2$ it shrinks on mass loss. For a realistic EOS,\nhowever, the situation is complicated by the fact that $\\Gamma$ changes with\ndensity. For the EOS we use (Shen et al. 1998), the neutron star will shrink \non mass loss for masses above 0.4 ${\\rm M_{\\odot}} \\;$ and expand otherwise.\\\\ \nTo build up an accretion disk a large ratio of tidal radius to innermost stable\ncircular orbit, $R_{\\rm isco}$,\n\\be\n\\frac{R_{\\rm tid}}{R_{\\rm isco}}= \\frac{c^2}{6 G} \\frac{R_{\\rm ns}}{(M_{\\rm\n    ns} M_{\\rm bh}^2)^{1\/3}}, \n\\ee\nis favorable. Thus, this simple estimate suggests that low masses for \nboth the black hole and the neutron star are favorable.\\\\  \nTaking together the calculations described in Rosswog (2005) and those of \nTable \\ref{runs} the  whole plausible black hole mass range from 3 to 20 ${\\rm M_{\\odot}} \\;$ \n(Fryer and Kalogera 2001) has been covered. For black holes\nbelow 10 ${\\rm M_{\\odot}} \\;$ Newtonian potentials and for more massive BHs \\Pacz-Wiita\n(PW) pseudo-potentials have been used. To mimic the presence of a last stable\norbit in the Newtonian calculations, we set up an absorbing boundary at\n$R_{\\rm isco}= 6 GM_{\\rm BH}\/c^2$. For the PW-potential matter is absorbed at\n$3 GM_{\\rm BH}\/c^2$, for details see appendix in Rosswog (2005), but this\nmaterial has passed the last stable orbit already and falls \nfreely towards the hole, so the exact location of the boundary is\nunimportant. In all calculations a non-spinning neutron star mass with\n1.4 ${\\rm M_{\\odot}} \\;$ was used.\\\\\nFor illustration, we want discuss two of the runs in more detail: \nA run with a 3 ${\\rm M_{\\odot}} \\;$ black hole (run NA in Tab.~\\ref{runs}) as an \nexample for a long-lived, episodic mass transfer, and a 14 ${\\rm M_{\\odot}} \\;$ run in \nwhich the neutron star is disrupted at during the first encounter.\n\n\\subsubsection{Episodic mass transfer}\n\\label{sec:episodic}\nIn this calculation with a 3 ${\\rm M_{\\odot}} \\;$ black hole (run NA in Tab.~\\ref{runs}) the\nneutron star undergoes a long-lived mass transfer without being disrupted, see\nFig.~\\ref{fig:nsbh_dynamics}, left column. After a {\\em primary mass transfer\n  episode} with peak mass transfer rates beyond 100 ${\\rm M_{\\odot}}$\/s, see\nFig.~\\ref{fig:mdot}, left panel, the neutron star recedes again and quenches\nthe mass transfer. During this episode its mass is reduced to about 0.5 ${\\rm M_{\\odot}}$,\nsee right panel of Fig.~\\ref{fig:episodic_orbit}.\nSubsequently, it undergoes several more mass transfer episodes, after each\nepisode mass transfer practically shuts off, leaving the neutron star constant\nfor a short time (see ``step-like'' behaviour in the right panel of\nFig.~\\ref{fig:episodic_orbit}). After about 25 episodes the mass transfer \nsettles into a stationary state in which the neutron star transfers \napproximately 1 ${\\rm M_{\\odot}}$\/s directly into\nthe hole without forming an accretion disk. Its general tendency during this\nepisode is to move further away from the black hole, see left panel of\nFig.~\\ref{fig:episodic_orbit}. Some material is tidally shed from the\nneutron star into a dilute disk around the still orbiting binary. It is only\nafter 47 orbital periods, at t$\\approx$ 218 ms, that the orbiting mini-neutron \nstar becomes tidally disrupted and forms an accretion disk around the hole, see\npanels two and three in Fig.~\\ref{fig:nsbh_dynamics_cont}. This final tidal\ndisruption sets in when the mini-neutron star reaches its lower mass limit of\nabout 0.18 ${\\rm M_{\\odot}}$, see right panel of Fig.~\\ref{fig:episodic_orbit} (for the mass radius\nrelationship of the Shen et al. EOS see Fig. 1 in Rosswog et al. (2004)).\\\\\nThis long-lived binary phase is imprinted on the gravitational wave signal, see\nFig.~\\ref{fig:GW_comparison}, left panel. The signal ``chirps'' until mass\ntransfer sets in. The primary mass transfer episode reduces the gravitational\nwave amplitude by more than a factor of two. While the neutron star is\nconstantly stripped by the black hole, the gravitational wave amplitude\ndecreases until the minimum neutron star mass is reached. During this phase\nthe binary emits at a frequency of $\\approx 500$ Hz, close to LIGO's \nsensitivity peak. This phase should be detectable out to a distance of\n$d= 10 \\; {\\rm Mpc} \\; (10^{-21}\/h_{\\rm min})$. Once this stage is\nreached, the gravitational wave signal shuts off abruptly as the matter \nsettles quickly into an axisymmetric disk. During this final disruption, \nthe mass transfer rate increases for a last time beyond 10 ${\\rm M_{\\odot}}$\/s, see the\nlast pronounced peak around 200 ms in the left panel of Fig.~\\ref{fig:mdot}.\n\n\n\\subsubsection{Direct disruption} \n\\label{sec:direct}\n\nFor more massive black holes, the self-gravity of the neutron star can be\novercome already during the first, primary mass transfer. An example of such a\ncase with a 14 ${\\rm M_{\\odot}} \\;$ black hole is shown in Fig.~\\ref{fig:nsbh_dynamics},\nright column.\nAlthough the neutron star is disrupted, a density peak remains visible in the\ndebris spiral arm until the end of the simulation. In this case an \naccretion disk forms, but\nlarge parts of this disk are inside the innermost stable circular orbit of the\nblack hole and therefore falling with large radial velocities into the\nhole. This disks are geometrically thin and, apart from a spiral shock from the\nself-interaction of the accreted matter, essentially cold. A detailed\ndiscussion of this calculation can be found in Rosswog (2005). During the first\napproach mass is transferred into the black hole at a peak rate of more than \n1000 ${\\rm M_{\\odot}}$\/s, see right panel of Fig.~\\ref{fig:mdot}.\\\\\nAgain the dynamics leaves a clear imprint on the gravitational wave signal:\nafter the chirp stage the signal immediately dies off, see\nFig.~\\ref{fig:GW_comparison} right panel. At peak amplitude, the\nsignal should be visible out to \n$d= 250 \\; {\\rm Mpc} \\; (10^{-21}\/h_{\\rm min})$.\\\\ \nFor the higher mass black holes, we do not find accretion disks that are\npromising to produce energetic GRBs. While some systems do form low-mass \ndisks, see Fig.~\\ref{fig:nsbh_dynamics}, right column, for a 14 ${\\rm M_{\\odot}} \\;$ BH, \nfor black holes with $M\\ge 18$ ${\\rm M_{\\odot}}$, nearly the whole neutron\nstar is directly fed into the hole, only a small fraction ($<$ 0.08 ${\\rm M_{\\odot}}$)\nreceives enough angular momentum to be ejected in form of a half-ring (see\nFigure 3 in Rosswog 2005) of neutron-rich debris material.\\\\\n\n\n\n\\subsection{Constraints on the nuclear equation of state from gravitational\n               waves} \n\nAs mentioned earlier, the neutron star equation of state (EOS) could be \nseriously constrained by a detected gravitational wave signal. We want to\nbriefly summarize gravitational wave signatures that carry information on the\nnuclear EOS.\n\\bi\n\\item A simple estimate for the neutron star radius can be obtained using\n  Newtonian gravity and $\\nu_{\\rm GW}= 2 \\nu_{\\rm orb}$, where $\\nu_{\\rm GW}$\n  and $\\nu_{\\rm orb}$ are the gravitational wave and orbital frequencies.  If\n  one assumes an equal mass binary, that the peak occurs once the neutron\n  star surfaces touch and that changes due to tides are negligible, one finds\n\\begin{eqnarray}\nR_{\\rm ns}&=& \\left( \\frac{G M}{8 \\pi^2 \\nu_{\\rm p}^2} \\right)^{1\/3}\\nonumber\\\\\n          &=& \\hspace*{-0.3cm}16.8 \\; {\\rm km} \\; \\left(\\frac{M}{2.8 \n{\\rm M}_\\odot}\\right)^{\\frac{1}{3}} \n  \\left(\\frac{1 {\\rm MHz}}{\\nu_{\\rm p}}\\right)^{\\frac{2}{3}}. \n\\end{eqnarray}\nHere, $\\nu_{\\rm p}$ is the peak GW-frequency at the end of the chirp phase\nand $M$ the total mass of the binary.\\\\\nMore accurately, using quasi-equilibrium sequences of neutron star binaries,\n  the compactness $GM\/Rc^2$ can be determined from the final deviation of the\n  gravitational wave energy spectrum from a point mass binary signal (Faber et\n  al. 2002). If additionally the neutron star masses are known from the\n  inspiral signal (Cutler and Flanagan 1994), the neutron star radius can be\n  derived and severe constraints on the equation of state can be imposed.\n\\item The stiffness of the EOS influences the post-merger behaviour of the\n  remnant. If the EOS is stiff enough, the central object will keep a triaxial,\n  ellipsoidal shape for many orbital periods and thus will continue to emit \n  gravitational waves (e.g. Rasio and Shapiro 1994), \n  before it (probably) finally  collapses into a black hole. For a soft EOS it\n  will quickly become axi-symmetric and the GW-signal will die off fast.\\\\\n  The influence of quark matter on the gravitational wave\n  signal has recently been investigated by Oechslin et al. (2004).\n\\item As was shown in Sec.\\ref{sec:episodic}, NSBH systems with low black \n  hole masses can undergo a\n  long-lived (in comparison to orbital time), mostly episodic mass transfer\n  activity. Test calculations with a soft ($\\Gamma=2$) polytrope (Rosswog et\n  al. 2004) indicate that this behaviour is specific for stiff equations of\n  state. Moreover, from a gravitational wave signal such as the one shown\n  Fig.~\\ref{fig:GW_comparison}, left panel, the time when the neutron star\n  reaches its minimum mass can be directly read off: at this point the \n  gravitational wave signal suddenly shuts off. \n\\ei\n\n\n\n\\begin{figure*}\n\\center{\\includegraphics[angle=-90,width=1.0\\columnwidth]{mdot.ps}\n        \\includegraphics[angle=-90,width=1.0\\columnwidth]{mdot_t_nsbh_q01_cor_PW_noFD_smooth.ps}}\n  \\caption{Left panel: mass transfer (in ${\\rm M_{\\odot}}$\/s) into the black hole for \n    the 1.4 ${\\rm M_{\\odot}} \\;$ and 3 ${\\rm M_{\\odot}} \\;$ system with episodic mass transfer (run NA in Tab.~\\ref{runs}). \n    Right panel: dito  for the 1.4 ${\\rm M_{\\odot}} \\;$ and 14 ${\\rm M_{\\odot}} \\;$ system in which the\n    neutron star is directly disrupted.}   \n  \\label{fig:mdot}\n\\end{figure*}\n\n\\begin{figure*}\n\\center{\\includegraphics[angle=-90,width=0.9\\columnwidth]{cm_trajectory_nsbh_q0466.ps}\n \\vspace*{-0.5cm}       \\includegraphics[angle=-90,width=1.19\\columnwidth]{m13_evolution_nsbh_q0466.ps}}\n  \\caption{Left panel: orbit of the mini-neutron star in the neutron star \n                        black hole system (M$_{\\rm ns}$= 1.4 ${\\rm M_{\\odot}} \\;$, \n                        M$_{\\rm bh}$= 3 ${\\rm M_{\\odot}}$) that undergoes episodic\n                        mass transfer. After mass transfer \n                        has set in, the neutron star orbits the black hole \n                        for 47 orbital revolutions before it is finally \n                        disrupted.\n           Right panel: elvolution of the neutron star mass (defined as the\n                        mass with $\\rho > 10^{13}$ \\gcc) of the same system.}   \n  \\label{fig:episodic_orbit}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\center{\\includegraphics[angle=-90,width=1.7\\columnwidth]{GW_comparison_episodic_direct.ps}\n}\n  \\caption{Left panel: gravitational wave signal of a binary system that\n    undergoes episodic mass transfer (M$_{\\rm ns}$= 1.4 ${\\rm M_{\\odot}}$, M$_{\\rm bh}$= 3 ${\\rm M_{\\odot}}$).\n    Right panel: gravitational waves for the 1.4 ${\\rm M_{\\odot}} \\;$ and 14 ${\\rm M_{\\odot}} \\;$ system in which the\n    neutron star is directly disrupted (run II of Rosswog (2005)).}   \n  \\label{fig:GW_comparison}\n\\end{figure*}\n\\section{Accretion disks}\n\\label{sec:accretion}\nThe accretion disks produced in a compact binary merger are in several \nrespects\ndifferent from a standard Shakura-Sunyaev (1973) accretion disk. Having just\nbeen built up in a violent disruption, they are far from a steady state and\ncompletely opaque to photons. Cooling is only possible via neutrino\nemission. The corresponding processes, however, are not generally fast enough\nto cool the disks on a timescale comparable to the dynamical timescales. The\ndisks often span the full range from neutrino opaque to completely transparent\nwith relatively large semi-transparent transition regions making an analytical\ntreatment without restrictive assumptions difficult. Over the last years\nseveral groups have worked in different approximations on the neutrino-cooled\ndisk problem (e.g. Popham et al. 1999, Narayan et al. 2001, Kohri and\nMineshige 2002, DiMatteo et al. 2002). Recently, equilibrium disks around\nrotating black holes have been constructed by Chen and Beloborodov (2006).\nNeutrino opacity effects have been included in several of the recent \nnumerical simulations (Ruffert et al. 1997, Rosswog and Liebend\\\"orfer 2003, \nLee et al. 2005).\n\\subsection{Double neutron star mergers}\nWe show in Fig.~\\ref{fig:DNS_disk} disk properties of run DA that we consider\nrepresentative for the DNS case. The dominant neutrino-emitting parts of the \ndisks have\ndensities in the range from $\\sim 10^{13}$ down to $\\sim 10^{10}$ \\gcc, see\npanel one, the temperatures in these regions are $\\sim 3$ MeV. \nAt the time of the merger, the neutron stars are still very close to cold\n$\\beta$-equilibrium as the tidal interaction is too short to \ncause substantial compositional changes at the prevailing temperatures\n($T<10^8$ K; Lai 1995). The weak interaction rates during the merger are \nsufficiently slow that the matter in the remnant is at $\\sim$ 15 ms \nstill close to its initial electron fraction, $Y_e\\sim 0.05$, see panel \nthree in Fig.~\\ref{fig:DNS_disk}. Therefore the assumption of\n$\\beta$-equilibrium is not justified.\\\\\nThe central object is apart from its surface layers completely opaque to the\nneutrinos. The inner parts of the debris torus still have large optical\ndepths, see Fig. 8-10 in Rosswog and Liebend\\\"orfer (2003), only the outer\nparts are neutrino transparent. As the disks cannot cool on a dynamical \ntimescale they are puffed-up and advection dominated. Inflow towards the \ncentral object proceeds mainly along the equatorial plane (see arrows \nin panel two and three) and along the disk surface, the flow inside the \ndisks shows convective circulation. For a further discussion of the\ncirculation patterns see Lee and Ramirez-Ruiz (2002), \nRosswog and Davies (2002) and Lee et al. (2005).\\\\  \n\\subsection{Neutron star black hole mergers}\nIn the neutron star black hole case we find it much harder to build up massive\naccretion disks. In the low-mass cases the episodic mass transfer prevents \ndisk formation until the neutron star is finally disrupted once it has\nbeen stripped down to its minimum mass of about 0.18 ${\\rm M_{\\odot}} \\;$. Such a case was \nshown in detail in Figs.~\\ref{fig:nsbh_dynamics} and \n\\ref{fig:nsbh_dynamics_cont}. The resulting disk as seen in panel six of\nFig.~\\ref{fig:nsbh_dynamics_cont} has about 0.05 ${\\rm M_{\\odot}}$. For the cases with\nlarger black hole mass, the neutron star is completely disrupted early on, see \nSec.~\\ref{sec:direct}, but close to the innermost stable circular orbit.\nAs a consequence, matter falls with large radial velocities into the hole \nwithin about one orbital period. These disks are essentially cold (apart \nfrom a spiral shock that occurs due to self-interaction of the accretion \nstream, see Fig. 6 in Rosswog 2005), low in density and, contrary to the \nneutron star merger disks, they are geometrically thin.\nFor even higher black hole masses the neutron star is nearly swallowed\ncompletely without any disk formation. For a further discussion we refer to\nRosswog (2005).\\\\\nThese result seems to depend on the nuclear equation of state. \nJanka et al. (1999) who use the softer Lattimer-Swesty EOS\nfind disks that are more promising for GRBs. \\\\\n\\begin{figure}[!t]\n  \\includegraphics[angle=-90,width= \\columnwidth]{disk_dens_global_fd135.ps}\n  \\includegraphics[angle=-90,width= \\columnwidth]{disk_T_velocity_fd135_zoom1.ps}\n  \\includegraphics[angle=-90,width= \\columnwidth]{disk_Ye_velocity_fd135_zoom1.ps}\n  \\caption{Vertical structure of the debris disk resulting from an \n   equal mass (1.4 ${\\rm M_{\\odot}}$) double neutron star merger (run DA, see Table 1, \n  at t= 16.88 ms). \n  Panel one shows the global mass density distribution in the XZ-plane; \n  panel two zooms into inner disk region, color-coded is the temperature. To\n  enhance the temperature contrasts in the disk the upper limit of the\n  colorbar has been fixed to 6 MeV.\n  Panel three shows the distribution of the electron fraction, $Y_e$; in\n  panels two and three the velocity field is overlaid.}\n  \\label{fig:DNS_disk}\n\\end{figure}\n\nTo quantify the amount of present debris material for both the DNS and the\nNSBH cases, we use the mass that is gravitationally bound and has a \ndensity $\\rho<10^{13}$ \\gcc.\nThe evolution of this debris mass is shown in Fig.~\\ref{fig:bound_debris}. \nIn the DNS cases the debris mass is around 0.3 ${\\rm M_{\\odot}}$, for the black hole cases\nit is about one order of magnitude smaller, at maximum $\\approx 0.05$ ${\\rm M_{\\odot}}$. \nIn run NA, which is described in detail in Sec.~\\ref{sec:episodic}, the mass \ntransfer is constantly driving neutron star oscillations, see \nFig.~\\ref{fig:bound_debris}, right panel. The debris mass only increases\nbeyond 0.01 ${\\rm M_{\\odot}} \\;$ after the neutron star is completely disrupted at t$\\approx\n218$ ms.\\\\\nTo further illustrate the different thermodynamic condition prevailing in the\ndebris of DNS and NSBH remnants, we show in \nFig.~\\ref{fig:DNS_rho_T_trajecories}\ntrajectories in the $\\rho-T$-plane. The left panel refers to the two\nneutron star cases, the right panel shows typical trajectories of NSBH\nsystems. The solid lines always refer to the average of the hottest \n10 \\% of the debris, the dashed lines show the average conditions of all\nmatter. While in the neutron star case the hottest material is always at very\nhigh densities ($>10^{14}$ \\gcc), the hottest regions in the NSBH cases \nmove during the evolution down to densities of $\\le 10^{10}$\\gcc and below. In\nthe direct disruption case that was discussed in Sec.~\\ref{sec:direct}, \nthe average temperature of even the hottest 10 \\% of the material quickly drops\nbelow 1 MeV (solid green curve).\n\n\\begin{figure*}\n\\center{\\includegraphics[angle=-90,width=1.03\\columnwidth]{debris_masses_DNS.ps}\n        \\includegraphics[angle=-90,width=1.\\columnwidth]{debris_masses_NSBH.ps}}\n  \\caption{Comparison between double neutron star and neutron star black hole mergers.\n        Left panel: debris mass (gravitationally bound, \n        $\\rho < 10^{13}$ \\gcc) for the double neutron star case. \n        Right panel: debris mass for the neutron star black hole cases. Note\n        the different scales on the axes of both panels.}   \n  \\label{fig:bound_debris}\n\\end{figure*}\n\\begin{figure*}\n  \\centerline{\\includegraphics[angle=-90,width=1.1\\columnwidth]{rho_T_trajectories_DNS.ps}\n\\includegraphics[angle=-90,width=1.1\\columnwidth]{rho_T_trajectory_NSBH_q0466_q035.ps}}\n  \\caption{Comparison between double neutron star and neutron star black hole\n    mergers. Left panel: Trajectories in the $\\rho-T$-plane for\n     double neutron star cases (run DA and DB), right panel: typical neutron\n star black hole cases. For each case, the trajectories of the hottest 10\\% of\n    the material and an average over all matter are shown.}\n  \\label{fig:DNS_rho_T_trajecories}\n\\end{figure*}\n\n\n\\section{Neutrino emission}\n\\label{sec:neutrinos}\n\n\\subsection{Double neutron stars}\nThe debris in the DNS case is very neutron-rich and hot, so the\nneutrino luminosities are generally dominated by electron anti-neutrinos\nfrom positron captures onto free neutrons, $e^+ + n \\rightarrow p +\n\\nu_e$. Electron neutrinos, mainly from electron captures,  \n$e + p \\rightarrow n + \\bar{\\nu}_e$, are second most important and \nfollowed by the heavy lepton neutrinos \n$\\nu_\\mu,\\bar{\\nu}_\\mu,\\nu_\\tau,\\bar{\\nu}_\\tau$ that we generally refer to as\n$\\nu_x$. While the $\\nu_e$ and $\\bar{\\nu}_e$ are emitted predominantly from\nthe inner disk regions or the surface layers of the central object, the heavy\nlepton neutrinos $\\nu_x$ are mainly produced in the hot high-density regions\ninside the central object. Since the heavy lepton neutrinos are not \nabsorbed by nucleons, the surrounding matter is more transparent to them. \nTherefore, they can escape from the hotter, high density regions more easily,\nwhich is why they have the largest average energy. The average energies of \nthe neutrino species are relatively \nrobust against changes in the system parameters, typically $\\langle \nE_{\\nu_e} \\rangle \\approx 7$ MeV, $\\langle E_{\\bar{\\nu}_e} \\rangle \\approx 12$\nMeV and $\\langle E_{\\nu_x} \\rangle \\approx 23$ MeV. In the unlikely case\nthat neutrinos from a neutron star merger would be detected, the substantial \ndrop in the $\\nu_x$, but only a milder drop in the $\\nu_e$ and\n$\\bar{\\nu}_e$ luminosity would indicate the moment when the central objects\ncollapses into a black hole.\\\\\nThe neutrino luminosities after a DNS merger increase \nsmoothly until a stationary state is reached, see Fig.~\\ref{fig:nu_lum_DNS}. \nAt this stage the luminosities of the numerical models can be roughly fit by\n\\be\nL_{\\nu}^{tot}= L_1 \\left(\\frac{M}{2.6 \\; {\\rm M}_\\odot}\\right)^\\alpha\n\\label{eq:lnu_tot}\n\\ee\nwith $L_1= 6.8\\cdot10^{52}$ erg\/s,  $\\alpha=4.5$ and $M$\nbeing the total binary mass. This formula does not apply to cases where an\nAlgol-like impact occurs (run DB), in such a case the neutrino-luminosities\nare higher due to the very strong shock heating at moderate densities.\nThe contribution of the different neutrino flavors to the total\nluminosity changes with the details of the system under consideration, but \nthey are approximately given by \n$L_{\\nu_e} \\approx 0.3 \\cdot L_{\\nu}^{tot}$, $L_{\\bar{\\nu}_e}\n\\approx 0.5 \\cdot L_{\\nu}^{tot}$ and $L_{\\nu_x} \\approx 0.2 \\cdot\nL_{\\nu}^{tot}$.\\\\\nThe disks that were produced in neutron star black hole calculations were only\nof moderate masses and temperatures and therefore produced much lower\nluminosities than in the neutron star case. Calculations with the softer EOS\nof Lattimer and  Swesty (1991) (Janka et al. 1999), find results not too\ndifferent from the DNS case.\n\n\\begin{figure}\n\\center{\\includegraphics[angle=-90,width=1.1\\columnwidth]{Lnu_DNS.ps}}\n  \\caption{Neutrino luminosities of the double neutron star merger\n    cases. Solid lines refer to the case with 2 x 1.4 ${\\rm M_{\\odot}}$, the dashed lines \nto the 1.1 and 1.6 ${\\rm M_{\\odot}}$ case.}   \n  \\label{fig:nu_lum_DNS}\n\\end{figure}\n\n\n\\section{GRBs}\n\\label{sec:GRBs}\nCompact binary mergers have been recognized as possible\ncentral engines of GRBs since many years (Blinnikov et al. 1984, \nEichler et al. 1998, \\Pacz 1991, Narayan et al. 1992). For general \nreviews on the manifold aspects of gamma-ray bursts we refer to recent reviews\n(e.g. Piran 2005 or \\Mesz 2006).\\\\\nTo attain the ultra-relativistic motion required to explain both the\nshort-time variability and the non-thermal spectrum of GRBs, large amounts\nof energy have to be deposited in a region of space that is practically devoid\nof baryons. Two popular mechanisms to achieve this are\nthe annihilation of neutrino anti-neutrino pairs (Eichler et al. 1989,\nMochkovitch et al. 1993, Ruffert et al. 1997, Popham et al. 1999, Asano and\nFukuyama 2003, Rosswog and Ramirez-Ruiz 2002, Rosswog et al. 2003) and \nmechanisms that rely on (ultra-)strong magnetic fields (Narayan\net al. 1992, Usov 1992, Duncan and Thompson 1992, Thompson 1994, \\Mesz and\nRees 1997, Kluzniak and Ruderman 1998, Lyutikov et al. 2003, Rosswog et\nal. 2003).\\\\ \n\\subsection{$\\nu_i-\\bar{\\nu}_i$-annihilation}\nThe energy deposition rate per volume via neutrino anti-neutrino annihilation\n(see Ruffert et al. 1997) can be written is discretised form as\n\n\n\\begin{eqnarray}\n&&Q_{\\nu \\bar{\\nu}}(\\vec{r})=\\sum_{i= e,\\mu,\\tau} Q_{\\nu_i \\bar{\\nu}_i}(\\vec{r})\\nonumber\\\\\n&=&\\sum_{i= e,\\mu,\\tau} A_{1,i} \n\\sum_k \\frac{L^k_{\\nu_i}}{d_k^2} \\sum_{k'} \\frac{L^{k'}_{\\bar{\\nu}_i}}{d_{k'}^2}\n[ \\langle E_{\\nu_i}\\rangle^k +  \\langle E_{\\bar{\\nu}_i}\\rangle^{k'}] \n\\mu_{kk'}^2 \\nonumber\\\\\n &+&\\sum_{i= e,\\mu,\\tau} A_{2,i} \\sum_k \\frac{L^k_{\\nu_i}}{d_k^2} \\sum_{k'} \n\\frac{L^{k'}_{\\bar{\\nu}_i}}{d_{k'}^2}\n \\frac{\\langle E_{\\nu_i}\\rangle^k +  \\langle E_{\\bar{\\nu}_i}\\rangle^{k'}}\n{\\langle E_{\\nu_i}\\rangle^k  \\langle E_{\\bar{\\nu}_i}\\rangle^{k'}}\n\\mu_{kk'}\\label{eq:ann}\n\\end{eqnarray}\nwhere the index $i$ labels the type of neutrino.  Here $L^k$ is the\nneutrino luminosity of grid cell $k$, $d_k$ is the distance from the\ncentre of grid cell $k$ to the point $\\vec{r}$, $d_k=\n|\\vec{r}-\\vec{r}_k|$, $\\langle E_{\\nu_i}\\rangle^k$ is the average\nneutrino energy in grid cell $k$, $\\mu_{kk'}= 1-\\cos \\theta_{kk'}$ and\n$\\theta_{kk'}$ is the angle at which neutrinos from cell $k$ encounter\nanti-neutrinos from cell $k'$ at the point $\\vec{r}$. The constants \nare given by\n$A_{1,e}=\\frac{1}{12\\pi^2}\\frac{\\sigma_0}{c(m_ec^2)^2}\n[(C_V-C_A)^2+(C_V+C_A)^2] $,\n$A_{1,\\mu}=A_{1,\\tau}=\\frac{1}{12\\pi^2}\\frac{\\sigma_0}{c(m_ec^2)^2}\n[(C_V-C_A)^2+(C_V+C_A-2)^2] $, $A_{2,e}=\n\\frac{1}{6\\pi^2}\\frac{\\sigma_0}{c} [2 C_V^2-C_A^2]$,\n$A_{2,\\mu}=A_{2,\\tau}= \\frac{1}{6\\pi^2}\\frac{\\sigma_0}{c} [2\n(C_V-1)^2-(C_A-1)^2] $, where $C_V= 1\/2+2 \\sin^2 \\theta_W$, $C_A=\n1\/2$, $\\sin^2\\theta_W= 0.23$ and $\\sigma_0= 1.76 \\times 10^{-44}$\ncm$^2$.\n\nThe thick-disk geometry that is a natural outcome of a double neutron star \nmerger, see Fig.~\\ref{fig:DNS_disk}, is favorable for an efficient annihilation\n as neutrinos have enhanced probability to collide head-on. The most\ninteresting regions for the energy deposition are the centrifugally\nbaryon-cleaned regions along the original binary rotation axis.\nContours of the matter density and the annihilation energy deposition rate can\nbe found in Fig. 2 in Rosswog et al. (2003).\nThe deposition of a large amount of energy in a photon-pair plasma between the \nfunnel walls results in a bipolar outflow along the binary rotation axis. \nThe ratio of deposited annihilation and rest mass energy determines the \nasymptotic Lorentz-factors of the resulting outflow. For a double neutron star \nmerger these Lorentz-factors lie between a few dozens and a few times $10^4$\n(see Rosswog et al. 2003, Figs. 4-6). The total energy contained in \nthis relativistic outflow is $\\sim 10^{48}$ erg \n(Rosswog and Ramirez-Ruiz 2003).\\\\\n\\begin{figure*}\n\\center{\\includegraphics[angle=-90,width=1.03\\columnwidth]{nu_lum_t_NSBH_0466_035.ps}\n\\includegraphics[angle=-90,width=1.\\columnwidth]{nu_lum_t_nsbh_q02_q014.ps}}\n  \\caption{Neutrino luminosities of the neutron star black hole cases. Left\n  panel: black holes with 3 (dashed) and 4 ${\\rm M_{\\odot}} \\;$ (solid curves). \n  Right panel: black holes with 7 (solid) and 10 ${\\rm M_{\\odot}} \\;$ (dashed curves).}   \n  \\label{fig:nu_lum_nsbh}\n\\end{figure*}\nSimilar to the case of a new-born proto-neutron star,\nthe huge neutrino luminosity will also drive an energy baryonic winds off \nthe remnant. The interaction with the baryonic\nmaterial collimates the neutrino-annihilation-driven bipolar outflow. Using\nsimple estimates Rosswog and Ramirez-Ruiz (2003) found a broad \ndistributions of opening angles and isotropised luminosities, centered \naround 7$^\\circ$ and a few times $10^{50}$ erg, respectively.\\\\\nMore recently, Aloy et al. (2005) have performed detailed, axisymmetric \nrelativistic hydrodynamic simulations in which they parametrised the energy \ndeposition above remnant accretion disks. Some of\ntheir models did not produce ultra-relativistic outflows and instead lead to\nlow-luminosity UV-flashes, others were able to produce ultra-relativistic, \nstructured outflows that are promising GRBs sources. While the\nultimate outcome seems to depend quite sensitively on the details of the\nconsidered system (disk structure, energy deposition rates, injection\nduration), their most promising systems produced ultra-relativistic\noutflows with half-opening angles of 5-10$^\\circ$ and apparent isotropic\nenergies of up to $10^{51}$ erg.\\\\\nIt has to be stressed that to date no self-consistent calculations of a full\nmerger with neutrino emission, annihilation, launch of winds and relativistic\noutflow could be performed. Only individual aspects have been simulated that\nare a posteriori patched together into a global picture. Nevertheless, the \ntheoretical numbers found in the last two investigations are consistent with\nrecent observations of short GRBs that find values of $E_\\gamma$ of a \nfew times $10^{48}$ erg and opening half-angles from about 9 to about \n14$^\\circ$ (Fox et al. 2005, Berger et al. 2005). Note however, that there \nare recent observations (Berger et al. 2006b) that suggest that a fraction of \nshort GRBs may have substantially larger energies than the first few short \nbursts that were detected.\\\\\nThus, neutrino annihilation from compact merger remnants, at least in the DNS\ncase, seems to be in reasonable agreement with recent observations of short \nGRBs.\\\\\n\n\\subsection{Magnetic fields}\nNevertheless, it is hard to see how magnetic fields should not play an\nimportant role. Most neutron stars are endowed with strong magnetic fields:\nyoung pulsars typically have $\\sim 10^{12}$ G and  magnetars (Duncan and \nThompson 1992)  are thought to have magnetic fields between\n$10^{14}$ and $10^{15}$ G. While still more speculative than\nthe neutrino annihilation mechanism in the sense that the existing\ncalculations are less detailed, there are good  arguments to expect a\ndramatic growth of existing seed fields.\\\\\nThe merger debris is differentially rotating, both the central object before\ncollapse, see Fig.~9 in Rosswog and Davies (2002), and the surrounding disk.\nThus the initial seed fields can grow either just by a linear winding up the\nfield lines or, more likely, via the magnetorotational instability (MRI)\n(e.g. Balbus and Hawley 1998). Recent magnetohydrodynamics simulations (Price\nand Rosswog 2006) suggest that magnetic seed fields are rapidly amplified in\nthe Kelvin-Helmholtz instability at the shear interface between the two\nneutron stars, see Fig.~\\ref{fig:shear_KH}. This amplification occurs on a\ntimescale of only $\\sim$ 1 ms, long before the collapse \ncan set in. As the shortest modes grow fastest in a Kelvin-Helmholtz\ninstability, the reached maximum field strengths are limited by the finite\nnumerical resolution. At the highest affordable resolution, about\n$2\\cdot10^{15}$ G are reached, but very plausibly {\\em much} higher \nfield strengths\nwill be realized in nature. The kinetic energy of the central object is large,\nabout $8\\cdot10^{52}$ erg, and if say 10 \\% of this energy can be transformed\ninto magnetic field energy, the field strength averaged over the central\nobject will be in excess of $10^{17}$ G. Locally, the field could be\nsubstantially higher.\\\\\n\\begin{figure}[!t]\n  \\includegraphics[angle=-90,width=0.95\\columnwidth]{shear_at_contact_contour.ps}\n  \\includegraphics[angle=-90,width=0.95\\columnwidth]{shear_at_fd21_contour_v3.ps}\n  \\caption{Velocity fields during the merger of two 1.4 ${\\rm M_{\\odot}} \\;$ neutron\n  stars (run DA). The shear interface, see panel one, becomes \n  Kelvin-Helmholtz unstable and forms a string of vortex rolls \n  (panel two, velocities in corotating frame) in which the magnetic field is\n  curled up.}   \n  \\label{fig:shear_KH}\n\\end{figure}\n\\begin{figure}[!t]\n  \\includegraphics[angle=-90,width=1.1\\columnwidth]{equipartition_field_of_rho.ps}\n  \\caption{Field strength (in G), where the magnetic pressure, \n           equals the matter pressure (calculated from the Shen et\n           al. EOS (1998) with $T=0.5$ MeV and $Y_e=0.1$).}   \n  \\label{fig:Beq}\n\\end{figure}\nTo become dynamically important, the fields have to become extremely strong.\nThe field strengths at which the magnetic pressure balances the nuclear matter \npressure are shown in Fig.~\\ref{fig:Beq}. In the central object of a merger \nthese field strengths are in excess of $10^{17}$ G, in the accretion disk \nthey are still between $10^{14}$ and $10^{16}$ G.\nIf the central object remains stable for long enough, it could provide a\nPoynting-dominated, scaled-up relativistic pulsar wind. In this case \nenergy is released at a rate of \n\\be\n\\left(\\frac{dE}{dt}\\right)_{\\rm md} \\sim 10^{51} B_{16}^2 P_{\\rm ms}^{-4} R_6^6\n\\; {\\rm erg\/s}\n\\ee\nand the central object will spin down on a timescale given by\n\\be\n\\tau_{\\rm sd} \\sim  2 {\\rm s} \\; B_{16}^{-2} P_{\\rm ms}^2 \\left(\\frac{20 \\; {\\rm\n      km}}{R_{\\rm co}}\\right)^4  \\left(\\frac {M_{\\rm co}}{2.5 \\; {\\rm M}_\\odot}\\right),\n\\label{eq:dipole_spindown}\n\\ee\nwhere $B_{16}$ is $B\/10^{16}$G and $P_{\\rm ms}= P\/$1 ms.\n\nFluid instabilities, such as the Kelvin-Helmholtz \ninstability illustrated for run DA in Fig.~\\ref{fig:shear_KH}, will locally\ncurl up the magnetic field lines. This had been suggested earlier\n(Rosswog et al. 2003) and has recently been confirmed by first MHD-merger\nsimulations (Price and Rosswog 2006). Once such high-field pockets reach field\nstrengths close the equipartition, see Fig.~\\ref{fig:Beq}, they will become\nbuoyant, float up and produce explosive reconnection events (e.g. Kluzniak and\nRuderman 1998, Rosswog et al. 2003). This may also occur in the disk\n(e.g. Narayan et al. 1992), though at a lower field strength, see\nFig.~\\ref{fig:Beq}. \nOnce the central object has collapsed, the ``standard\nGRB-engine'', a black hole plus accretion torus, remains, for which various\nenergy extraction mechanisms have been suggested  (e.g. Blandford and Znajek\n1977, Blandford and Payne 1982 or McKinney 2005). For a discussion of these \nmechanisms we refer to the literature.\\\\\nAs discussed in the previous sections in our neutron star black hole\ncalculations we find it difficult to form massive disks. The neutrino\nluminosities are even in the more optimistic cases two orders of magnitude\nlower than in the DNS case, see Figs.~\\ref{fig:nu_lum_DNS} and\n\\ref{fig:nu_lum_nsbh}. This is mainly due to the much smaller debris mass, see\nFig.~\\ref{fig:bound_debris}. As the neutrino annihilation roughly scales with\nthe square of the neutrino luminosity, see Eq.~(\\ref{eq:ann}), neutrino\nannihilation deposits substantially less energy than in the DNS case.  \nNSBH binaries would therefore not produce energetic GRBs, but rather a \nlow-luminosity tail to the short GRB distribution. Possible observable \nsignals are discussed in Rosswog (2005).\\\\\nWe want to stress again that these results are sensitive to the nuclear \nEOS. With the softer Lattimer-Swesty-EOS Janka et al. (1999) find more \npromising conditions.\n\n\\section{Formation of a magnetar?}\n\\label{sec:magnetar}\nThere are essentially two proposed formation mechanisms for pulsars: i) the\ncollapse of very rapidly rotating stars with ordinary magnetic fields or ii)\nthe collapse of extraordinarily magnetized stars with ordinary rotational\nspeeds. In the first case the new-born proto-neutron star has to be\nrotating rapidly enough so that rotational periods are smaller than the\noverturn times, $\\sim 3$ ms, of convective eddies (Duncan and Thompson 1992,\nThompson and Duncan 1993, Thompson and Duncan 1995), so that an efficient\n$\\alpha$-$\\omega$-dynamo can become active. In this way, fields as strong as\n$\\sim 10^{15}$ G can be plausibly generated. Such a new-born neutron star will\nbe spun down quickly by magnetic dipole radiation (and possibly mass loss) and\nthis will fuel the \nsupernova explosion with an additional $2 \\pi^2 I\/P^2\\sim 2 \\cdot 10^{52}\nI_{45}\/P_{\\rm ms}^2$ ergs stemming from rotational energy. Here we have used\n$I_{45}$, the moment of inertia in units of $10^{45}{\\rm g cm}^2$ and $P_{\\rm\n ms}$, the rotation period as measured in milliseconds. Therefore,\nparticularly bright supernova explosions going along with magnetar formation\nare a natural prediction of this formation path. However, the three supernova\nremnants that seem to be associated with magnetars are remarkably \nunremarkable: rather than being particularly energetic they only have \nkinetic energies of $\\sim 10^{51}$ ergs (Sasaki et al. 2004, Vink 2006). \nAn alternative formation scenario could be that the magnetar field \nstrengths are obtained via magnetic flux conservation alone during the \ncollapse of highly magnetized progenitor stars (Ferrario and \nWickramasinghe 2005).\\\\\nHere, we want to discuss a third possible formation channel, namely \na neutron star coalescence that produces produces a stable, magnetar-like \nremnant. \n\nThe first point to clarify is whether the DNS merger rates are large enough to\npossibly contribute a non-negligible fraction of magnetars.\nThe estimated rates at which double neutron systems (DNS) merge have been\nconstantly increasing over the last years. The discovery of the double pulsar\nPSR J0737-3039A (Burgay et al. 2003) alone has increased the estimated merger\nrate based on observational data by an order of magnitude. Current estimates\n(Kalogera et al. 2004) are in the range from 4 to 224 $\\cdot10^{-6}$ per year\nand Galaxy. Nakar et al. (2005) based their estimates on the\nassumption that short-hard gamma-ray bursts result from DNS mergers. They find\na ``best guess'' value that is higher than the above upper limit by more than\nan order of magnitude. This last rate may be somewhat optimistic, but it is\nfair to state that the DNS merger rate is comparable to the magnetar \nbirthrate in our Galaxy, about 1-10 $\\cdot10^{-4}$ per year \n(Duncan and Thompson 1992). DNS mergers could therefore plausibly contribute to\nthe magnetar production.\\\\\nThe main question that needs to be addressed is whether the final remnant of \na DNS merger can have a mass that can be permanently sustained by the\nequation of state. The maximum neutron star mass is unfortunately still only\npoorly constrained. An upper limit based on very general physical \nprinciples such as causality is 3.2 ${\\rm M_{\\odot}}$ (Rhoades and Ruffini \n1974)\\footnote{See Psaltis (2004) for a critical analysis\nof the underlying assumptions.}. In a review of realistic models of \nnuclear forces\nAkmal et al. (1998) find that realistic models of nuclear forces limit the\nmaximum mass of neutron stars to be below 2.5 ${\\rm M_{\\odot}}$. The masses\nwith the smallest error bars come from the observation of DNS and they\nare consistent with a remarkably narrow underlying Gaussian mass distribution\nwith M= 1.35 $\\pm$ 0.04 ${\\rm M_{\\odot}} \\;$ (Thorsett and Chakrabarty 1999). Observations of\nneutron stars in binary systems with a white dwarf, however, yield \nconsistently higher neutron star masses, but the error bars are to date \nstill much larger than in DNS. The most extreme case is PSR J0751+1807 \nwith an estimated pulsar mass of $2.1\\pm 0.2$ ${\\rm M_{\\odot}} \\;$ (Nice et al. 2005). \nRecent observations of the neutron star EXO 0748-676 also seem to support \nlarge possible neutron star masses and have been interpreted as hints \nthat neutron stars may have a conventional neutron-proton composition \n(\\\"Ozel 2006). Here, we want to explore the possibility that the final \noutcome of a DNS coalescence is, after some mass loss, a very massive and\nhighly magnetised, but stable neutron star.\\\\ \nLet us first estimate the gravitational mass of a merger remnant.\nLattimer and Yahil (1989) found empirically that \n\\be\nB= \\alpha \\left( \\frac{M}{M_{\\odot}}\\right)^2,  \n\\ee\nwith $\\alpha=0.084$ ${\\rm M_{\\odot}}$, yields a good fit of the binding energy, $B$, as a\nfunction of the gravitational mass $M$. Thus, a standard binary system of\ntwice 1.35 ${\\rm M_{\\odot}} \\;$ corresponds to a baryonic mass of about 3.0 ${\\rm M_{\\odot}}$, or, a\nmerger remnant made of all the baryons of the initial binary system would have\n2.48 ${\\rm M_{\\odot}} \\;$ of gravitational mass.\\\\\nWe plot in Figure \\ref{fig:baryonic_mass} the baryonic mass that needs \nto become unbound in order for the final remnant to be \nabove a specified maximum neutron star mass M$_{\\rm max}$:\n\\be\nm_{\\rm loss}= 2(\\alpha M_{\\rm ns}^2 +  M_{\\rm ns}) -  (\\alpha M_{\\rm max}^2 +  M_{\\rm max}),\n\\ee\nwhere we have restricted ourselves to the case of equal mass neutron\nstars. For example, if we \nassume the maximum neutron star mass to be 2.2 ${\\rm M_{\\odot}}$, so just beyond the\nestimated mass of PSR J0751+1807, a system with two 1.2 (1.4) ${\\rm M_{\\odot}} \\;$ neutron\nstars would need to loose a baryonic mass of 0.035 (0.52) ${\\rm M_{\\odot}}$.\\\\\nMass can be lost by dynamical ejection or by the combined action of neutrinos,\nrotation and strong magnetic field as a neutrino-magnetocentrifugally driven \noutflow. The latter mechanism has been discussed recently (Thompson et al. \n2004) in the context of the formation of magnetars in supernovae.\\\\\nThe amount of mass that is {\\em dynamically} ejected is sensitive to the \nnuclear equation of state (EOS) with stiffer EOSs ejecting more than \nsofter ones (Rosswog et al. 2000). With the EOS of Shen et al. (1998), \nwhich is at the stiffer end of the EOS spectrum and therefore consistent with \nthe assumption of a relatively large maximum neutron star mass, we find that \ntypically $m_{ej}\\approx 0.03$ ${\\rm M_{\\odot}} \\;$ are dynamically ejected.\\\\\nThe combined action of neutrino-heating, strong magnetic field and rapid\nrotation is hard to quantify as details of the \n(unknown) magnetic field geometry enter and the mass loss rates depend\nsensitively on the neutrino luminosity and the rotational period (Duncan et\nal. 1986, Qian and Woosley 1996; Thompson et al. 2001 and Thompson et\nal. 2004). \nStrong magnetic fields enforce near-corotation of the wind with the stellar\nsurface out to the Alfven radius, $R_A$, where the kinetic enery, $\\rho v^2\/2$\napproximately equals the magnetic energy density $B^2\/8\\pi$. Thus, wind mass\nelements are, like spokes on a bike wheel, forced to corotate with the\nremnant, leading much faster loss of angular momentum and spin-down. Thompson\net al. (2004) find that a strong magnetic field together \nwith the rapid rotation will {\\em drastically} increase the mass loss rate, in\nthe fast-rotation limit they find a functional dependence of\n\\be\n\\dot{M} \\propto \\exp(\\omega^2),\n\\ee\nwhere $\\omega$ is the angular frequency. This indicates that a substantial\nmass loss is possible.\nFor example, for a central remnant period of 0.8 ms (see Figure 10 in \nRosswog and Davies 2002)\nand a neutrino luminosity of $L_{\\bar{\\nu}_e}= 8\\cdot 10^{51}$ erg\/s they find\na mass loss rate $\\dot{M} \\approx 0.1$ ${\\rm M_{\\odot}} \\;$\/s. If we assume that the results\nscale like $\\dot{M} \\propto L_{\\nu}^{2.5} M^{-2}$ (Qian and Woosley 1996) and\nuse the empirical relation Eq.~(\\ref{eq:lnu_tot}), we can estimate $\\dot{M}$ \nas a function of the total mass of the DNS, $M$. This is shown as the solid \nblack line in Fig. \\ref{fig:baryonic_mass} for a duration of $\\tau= 0.5 s$, \nthe total unbound mass being $m \\sim \\dot{M} \\cdot \\tau + m_{ej}$. Taking \nthese numbers at face value would mean that, if the maximum neutron star \nmass is $M_{\\rm max}=2.2$ ${\\rm M_{\\odot}} \\;$ (violet line in Fig.\\ref{fig:baryonic_mass}), \nthen even a standard DNS with twice 1.35 ${\\rm M_{\\odot}} \\;$ would loose enough\nmass to leave a stable remnant. For $M_{\\rm max}= 2.0$ ${\\rm M_{\\odot}} \\;$ (blue), \nneutron stars with or below 1.15 ${\\rm M_{\\odot}} \\;$ would still produce a stable \nneutron star.\\\\\nTo summarize, although hard numbers are difficult to obtain, \nthe lower end of the DNS mass distribution could plausibly produce stable, \nhighly magnetized neutron stars rather than black holes.\\\\ \n\\begin{figure}\n{\\includegraphics[angle=0,width=0.75\\columnwidth,angle=-90]{baryonic_mass.ps}}\n\\caption{Shown is the baryonic mass that has to be lost from a double neutron\n  star system with twice $M_{ns}$ if the maximum gravitational neutron star\n  mass is $M_{\\rm max}$. All masses are in solar units.\n  The black solid curve is the estimate of the mass that is lost by the\n  neutrino-magnetocentrifugal wind plus dynamic ejection, see text.\n \\label{fig:baryonic_mass}\n} \n\\end{figure}\n\n\\section{Late-time central engine activity}\n\\label{sec:late_signals}\nIt was one of the surprises of the recent afterglow detections of short GRBs\nthat GRB 050709 and GRB 050724 exhibited long-lasting X-ray flares that\noccurred at least 100 s after triggering the bursts. While the observations\nhave been claimed to be suggestive for compact mergers, it is not immediately\nclear whether the merger model can accommodate this late time X-ray flaring\nactivity. The standard central engine, black hole plus disk, has --at least in\nits most simple form-- difficulties to accommodate these long time scales.\nThe dynamical timescale for an accretion disk is\n\\be\n\\tau_{\\rm d} \\sim \\frac{1}{\\alpha \\Omega_K} \\sim 0.05 {\\rm s} \n\\left(\\frac{R}{200\\; {\\rm km}}\\right)^{\\frac{3}{2}}\\left(\\frac{0.1}{\\alpha}\\right)\\left(\\frac{2.5\n    M_{\\odot}}{M_{\\rm bh}}\\right)\n\\ee\nwhere $\\alpha$ is the Shakura-Sunyaev viscosity parameter (Shakura and Sunyaev\n1973) and $\\Omega_K$ the angular frequency of a ring of matter at distance\nR. Obviously, the estimated accretion timescale is much shorter than the\ntimescales at which the X-ray flaring activity is observed. \\\\\nThe X-ray activity might be caused by either a long-lived central engine, \nor by a wide distribution of Lorentz factors, or else by the deceleration of a\nPoynting flux dominated flow. Various possibilities are discussed in detail\nin Zhang et al. (2005). Here we want to address two possibilities:\ni) that the central object from a DNS merger survives for long enough to\nproduce the flaring and ii) we also address the fallback accretion luminosity\nin the aftermath of a compact binary merger.\\\\ \n\n\\subsection{Survival of the central object in a double neutron star merger}\n\\label{sec:survival}\nThe continued central engine activity could be due to \nthe survival of the central object, either in a meta-stable state or --as\ndiscussed above-- the remnant may in some cases loose enough mass to produce\na stable, highly magnetised neutron star.\\\\\nA meta-stable state is likely to occur, stabilisation could come for example\nfrom differential rotation (e.g. Morrison et al. 2004) or trapped neutrinos \ntogether with non-leptonic negative charges (Prakash et al. 1995), but the\ntime scale until collapse is uncertain. This is because\nall the imprecisely known aspects of the problem such as the EOS at \nsupra-nuclear densities, the remnant radius and magnetic field, ellipticity \nof the central object and so on, enter with large powers in the estimate of \nthe time scale. Therefore in the meta-stable case a large range \nof time scales until collapse can be expected for different initial binary\nsystems. With reasonable parameters even time scales as long as weeks are possible\n(see Rosswog and Ramirez-Ruiz 2002).\\\\\nIf the GRB is indeed caused by the combined action of neutrino annihilation\nand buoyant pockets of ultra-strong magnetic fields, this will plausibly also\ncause late-time flaring activity, provided that the remnant is stable for long\nenough. The magnetic field amplification acts on a time scale of about a\nmilli-second (Price and Rosswog 2006) and thus large magnetic fields can be\nbuilt up before the neutrino luminosity has reached its peak at $\\approx 20$\nms after the merger, see Fig.~\\ref{fig:nu_lum_DNS}. In the early stages \na buoyant high-field bubble can  expand into a practically baryon-free\nenvironment and thus reach large Lorentz-factors. At neutrino\nluminosities in excess of $10^{52}$ erg\/s the remnant also drives a very\nenergetic baryonic wind. If the right conditions are met, the baryonic\nmaterial can help to collimate the outflow (Aloy et al. 2005), but\nit also poses a potential threat to the emergence of an ultra-relativistic\noutflow. Initially, as the rapidly spinning remnant produces high-field\npockets in rapid succession, magnetic pressure may help to keep the funnel\nabove the central object baryon-clean. But as the remnant is constantly braked\nby magnetic fields and gravitational wave emission, it takes\nlonger and longer to reach buoyancy field strength, while more and more\nbaryons are ablated from the remnant by neutrinos. The first explosive reconnection events\nescape into a very clean environment, those produced later will be released \ninto an envelope of neutrino-magneto-centrifugally expelled baryons. This \ndifferent baryon loading may be responsible for the difference between \nbursts and X-ray flares.\n\n\\subsection{Fallback accretion}\nSome of the matter expelled by gravitational torques is still gravitationally\nbound to the central object, and will  fall back towards the remnant (see for\nexample Figure \\ref{fig:DNS_1.1_1.6}, panel three). Direct numerical\ncalculations over the interesting timescales are not feasible, therefore we\nfollow a simplified analytical approach. We assume that the motion of this\nmaterial can be approximated as Keplerian motion in a central potential\nproduced by the remnant. From the total energy, $E_i$, and the angular\nmomentum, $J_i$, of each SPH-particle the eccentricity of the particle orbit,\n$e_i$, can be calculated \n\\be\ne_i^2= 1 + \\frac{2 E_i J_i^2}{G^2 m_i^3 M^2}.\n\\ee\nHere, $m_i$ is the particle and $M$ the enclosed mass. For the particles with\n$e_i<1$, the semi-major axes\nare calculated, $a_i= -G  m_i M\/(2 E_i)$, from which the distances of the \nclosest and farthest approach follow: $R_{\\rm min}= a_i (1-\\epsilon_i)$ and\n$R_{\\rm max}= a_i (1+\\epsilon_i)$. A particle with a velocity $\\vec{v}_i$\nthat is currently located at radius $r_i$ will reach the radius $R_{\\rm dis}$ \nafter a time of \n\n\\be\n\\tau_{i}=\n\\left\\{\\begin{array}{cl}\nI_{r_i,r_{{\\rm max},i}} + I_{r_{{\\rm max},i},R_{\\rm dis}}  \n\\quad {\\rm for} \\quad \\vec{v}_i\\cdot\\vec{r}_i > 0\\\\\nI_{r_i,R_{\\rm dis}} \\hspace*{2.5cm} {\\rm for} \\quad \n\\vec{v}_i\\cdot\\vec{r}_i < 0\n\\end{array}\\right. \\label{fallback_time},\n\\ee\nwhere $I_{r_1,r_2}$ is given by\n\n\\begin{eqnarray}\nI_{r_1,r_2}=&&\n\\left[ \\frac{\\sqrt{A r^{2}+ B r + C}}{A} \\right. \\nonumber \\\\\n&+& \\left. \\frac{B}{2A\n    \\sqrt{-A}} \\arcsin \\left( \\frac{2 A r + B}{\\sqrt{-D}} \\right)\n\\right]_{r_1}^{r_2}\n\\label{I_ri_R}\n\\end{eqnarray}\nwith $D = 4 A C - B^{2}$ and $A= \\frac{2 E_{i}}{m_{i}}, \\; B= 2 G M$ and \n$C=-\\frac{J_{i}^{2}}{m_{i}^{2}}$ (Rosswog 2006).\n\nFor the radius, where the fallback energy is dissipated, $R_{\\rm dis}$, \nwe take in the double neutron star case the disk radius at the end of the \nnumerical simulation. This is conservative in the sense that the disk will\nshrink, see below, and therefore the assumption of a radius fixed at \n$R_{\\rm dis}$ yields shorter timescales and lower accretion luminosities. \nFor the black hole cases, I choose $R_{\\rm dis}= 10 GM\/c^2$, so just outside \nthe innermost stable circular orbit of a non-rotating (Schwarzschild-) black\nhole at $R_{\\rm ISCO}= 6GM\/c^2$. The details of the fallback times and energies\nchange slightly with $R_{\\rm dis}$, but none of the conclusions\ndepends on the exact numerical value of $R_{\\rm dis}$.\\\\\nFig.~\\ref{fig3} shows the accretion luminosities, \n$L_{\\rm acc}= dE_{\\rm fb}\/dt$, derived for various DNS and NSBH systems, for\ndetails see Rosswog (2006).\nHere, $E_{\\rm fb}$ denotes the difference between the potential\nplus kinetic energy at the start radius, $r_i$, and the potential energy at\nthe dissipation radius, $R_{\\rm dis}$. The curves have been obtained \nby binning the energies contained in the fallback material, $E_{\\rm fb}$,  \naccording to the corresponding fallback times, $\\tau_{i}$, see \nEq.~(\\ref{fallback_time}). A fraction $\\epsilon$ of this energy is channeled \ninto X-rays, $L_X = \\epsilon L_{\\rm acc}$.\\\\\nThe double neutron star cases are rather homogeneous with respect\nto their fallback accretion, in all cases the fallback material is\napproximately 0.03 ${\\rm M_{\\odot}}$\\footnote{General relativistic effects might reduce\n  this number somewhat.}. After an initial, short-lived plateau the luminosity\nfalls off with time close to the expected 5\/3-power law\n\\citep{rees88,phinney89}.  \nThe last point in these curves is determined by the numerical mass \nresolution in the hydrodynamics simulations and should therefore be \ninterpreted with some caution. All other points should be a fair \nrepresentation of the overall fallback activity. Typically, the X-ray \nluminosity about one hour after the coalescence is \n$L_X  \\sim \\left(\\frac{\\epsilon}{0.1}\\right) \\cdot 10^{44}$ erg\/s.\nFor the investigated mass range the spread in the luminosities one hour after\nthe coalescence is about one order of magnitude.\\\\\nThe neutron star black hole cases show a larger diversity. The mass in the\nfallback material of different mass ratios varies by about a factor of 500,\nsee Rosswog (2006), an hour after the merger the accretion \nluminosities of the different NSBH systems differ by about two orders of \nmagnitude. The involved time scales change strongly with the binary mass\nratio. For example, the 1.4 and 4 ${\\rm M_{\\odot}} \\;$ NSBH case does not produce much\neccentric fallback material. Accretion, at least to the resolvable level, is\nover in $\\sim 0.2$ s. This accretion period may produce a short GRB, but\nprobably not much X-ray activity. The 1.4 and 18 ${\\rm M_{\\odot}} \\;$ NSBH system is at the\nother extreme: its peak luminosity is lower by three orders orders of\nmagnitude but extends (at a resolvable level) up to about one hour. The mass\nratios in-between could possibly produce a (weak) GRB and extended X-ray\nactivity up to about one day after the burst. \n\\begin{figure}\n\\hspace*{-0.4cm}\\psfig{file=fallback_luminosities.ps,width=0.95\\columnwidth,angle=-90} \n\\caption{Fallback accretion luminosity for various compact binaries, taken\n  from Rosswog (2006).\nCircles refer to double neutron star systems (DNS), triangles to neutron star\nblack hole mergers (NSBH). Note that the rightmost point is determined by the \nmass resolution of the simulation. For reference a straight line with slope\n5\/3 is shown.}\n\\label{fig3}\n\\end{figure}\n\n\\section{Summary}\nWe have discussed various aspects of the last stages in the life of a \ncompact binary. In particular we have addressed\n\n\\bi\n\\item Gravitational waves as a probe of the merger dynamics\n\nWe have shown how the merger dynamics is imprinted on the gravitational wave\nsignal. Examples include the peak gravitational wave frequency of a \ndouble neutron star merger and  the subsequent ringdown emission which both\nare sensitive to the equation of state. For a stiff EOS, low-mass \nblack hole neutron star systems undergo a long-lived, mostly episodic \nmass transfer phase. We show an example where the mass transfer \ncontinues for as long as 47 orbital periods. This last binary phase is \nresponsible for a gravitational wave signal with slowly decaying amplitude \nand increasing frequency. The signal only shuts off when the\nneutron star is finally disrupted upon reaching its minimum mass.\\\\\nThe coincident of detection of a short GRB together with a ``chirping''\ngravitational wave signal would be the ultimate proof of a compact binary\ncentral engine.\n\n\\item Accretion disks\n\nCompact binary mergers produce in some (but not all) cases\naccretion disks from the neutron star debris. These disks differ from the\nstandard thin disk accretion model in various ways. Due to insufficient\ncooling by neutrinos they are often thick and puffed up, they are usually\nneither in a dynamical nor in $\\beta$-equilibrium. For double neutron star\nmergers they are generally massive, of order 0.25 ${\\rm M_{\\odot}}$, with temperatures of\nseveral MeV and an electron fraction of close to 0.1 (at least initially). \nFor neutron star black hole systems we find it substantially more difficult to\nform massive disks. For low-mass black hole systems, the initial episodic mass\ntransfer with a surviving neutron star prevents the build up of a massive\ndisk. Only after a complete disruption of the neutron star a disk of $\\approx\n0.05$ ${\\rm M_{\\odot}} \\;$ forms. A comparison of the debris masses in double neutron star\nand neutron star black hole mergers is shown in Fig.~\\ref{fig:bound_debris}.\nWe find the neutron star black hole systems to produce disks masses that \nare lower by at least a factor six. For higher mass black holes the neutron \nstar is disrupted early on, but very close to the innermost stable circular\n orbit. The resulting accretion disks are geometrically thin and essentially \ncold. For black hole masses larger that about 16 ${\\rm M_{\\odot}} \\;$ no disk forms at \nall, most of the neutron star is fed directly into the hole, the rest is \ndynamically ejected.\\\\\nThese results are sensitive to the equation of state, for a softer equation of\nstate the neutron star is disrupted more easily. \n\n\\item Gamma-ray bursts\n  \nNeutrino annihilation from the remnant of a double neutron star\nmerger can plausibly provide the driving stresses to launch highly\nrelativistic, bipolar outflows, but a large diversity in the observable burst\nproperties and moderate burst energies of $\\sim 10^{48}$ are expected. \nIt is conceivable that a good fraction of systems\nfails to provide the right conditions and instead produces X-ray or\nUV-flashes.\\\\\nMHD simulations of double neutron star mergers show that the magnetic field\ngrows within the first millisecond to (probably much) beyond magnetar\nstrength, possibly allowing for a burst production $\\grave{\\rm a}$ la \nKluzniak and Ruderman (1998). The burst is most likely a result of the \ncombined action of both neutrino annihilation and ultra-strong magnetic \nfield.\\\\\n Of the neutron star black hole binaries only systems with low black\nhole masses could possibly form disks that are hot and dense enough to launch \nGRBs (unless the true nuclear equation of state is much softer than the one we\nuse here). Among them, rapidly spinning holes are preferred as both the last\nstable orbit and the event horizon move closer to the hole. \nThe high-mass end of neutron star black hole binaries probably produces a\nlow-luminosity tail of the short GRB-distribution.\n\\item Magnetar formation\n\nIf the maximum neutron star mass should indeed be as high as indicated by\nrecent observations the amount of mass that would need to be lost \nafter a merger to evade the final collapse to a black hole\nis not implausibly large. Therefore we want to point out the possibility\nthat binary systems from the lower end of the mass distribution could \nproduce a highly magnetised, but stable neutron star rather than a black \nhole. Magnetars outside normal supernova remnants\nwould be obvious candidates for this formation channel. If true, this channel \nshould also produce large amounts of neutron-rich nuclei synthesised in \nboth the dynamically ejected neutron star material and in the \nneutrino-magnetocentrifugal wind that made the survival of the remnant\npossible in the first place. Being synthesised in a neutrino bath dominated\nby electron anti-neutrinos the wind material will to be more\nproton-rich than the dynamically ejected material. The exact $Y_e$ is\ndetermined by both the expansion time scale and the ratio of neutrino and\nanti-neutrino luminosities, but is roughly (Qian and Woosley 1996)\n\\be\nY_e \\sim 0.28 \\left[1 + \\frac{L_{\\rm \\bar{\\nu}_e}\/\n5 \\cdot 10^{52} {\\rm erg}}{L_{\\rm \\nu_e}\/\n2 \\cdot 10^{52} {\\rm erg}}\\right]^{-1},\n\\ee\nwhere $L_{\\rm \\nu_e}$ and $L_{\\rm \\bar{\\nu}_e}$ are the electron neutrino\nand anti-neutrino luminosities, respectively. The more \nneutron-rich dynamical ejecta with $Y_e\\approx 0.1$ would be expected to \nbe located at larger distances from the magnetar than the wind material.  \n\n\\item Late time X-ray activity\n\nIn the speculative case that the central object of a neutron star \nmerger survives, late-time flaring would be caused by the same \nmechanism that produced the burst. If the burst is (at least in part) produced\nby explosive reconnection events of buoyant high-field bubbles as suggested\nby Kluzniak and Ruderman (1998), parts of the remnant that become buoyant at\nlater times have to expand into an environment that is already heavily\nbaryon-polluted by neutrino-driven winds. Due to the higher baryon loading\nthe emission would be mainly in the X-ray band of the spectrum.\n\nMuch less speculative is the accretion of fallback material which occurs\nnaturally in the aftermath of a compact binary merger. A few percent of \na solar mass are ejected into eccentric, but still bound orbits. The \nfallback of this material produces extended X-ray activity up to many hours \nafter the main burst.\n\\ei\n\n\\acknowledgements\n\\noindent Some of the figures have been produced using the \nsoftware SPLASH kindly provided by Daniel Price.\\\\\nThe calculations shown in this paper were in part performed on the JUMP system\nof the H\\\"ochstleistungsrechenzentrum J\\\"ulich.\\\\\nIt is a pleasure to thank Joe Monaghan and Andrew Melatos for their\nhospitality during the last stages of writing this paper.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\nA knowledge of the precise positions of cosmic gamma--ray bursts (GRBs)\nis important for many studies.  For example, recent observations of the optical counterparts to \nbursts have provided evidence that at least two of them are at cosmological\ndistances (Metzger et al. 1997, Kulkarni et al. 1998) and that, by implication, most or all of\nthem may also be.  One optical counterpart has been associated\nwith a galaxy, and the majority of the models for the energy\nrelease in bursts have long invoked neutron star-neutron star or neutron star-\nblack hole mergers (e.g. Paczynski 1991).  If this is correct, GRBs should originate in or close\nto galaxies, and correlations between burst positions and the large scale structure \nof the universe might be expected.  In this context, it has been debated whether\nthe positions of bursts are correlated with those of Abell clusters (Kolatt \\& Piran 1996; \nHurley et al. 1997a; Struble \\& Rood 1997) and radio-quiet quasars (Hurley et al. 1997b;\nSchartel, Andernach, and Greiner, 1997).  In both cases, increased precision in the\nlocation accuracy leads to more stringent tests.  Similarly, precise positions greatly improve\nsearches for quiescent counterparts to GRBs in other wavelength ranges (e.g. Tokanai et al. 1997),\nand can help confirm proposed associations between GRBs and fading counterparts (Hurley et al.\n1997c).  Finally, burst recurrence from a single source is excluded in most cosmological models.  \nSuch recurrence would reveal itself through small-scale clustering in the angular distribution\nof burst locations, and the sensitivity of the distribution function to repeating sources is\ngreater when the burst location accuracy is improved (Kippen, Hurley, \\& Pendleton 1998). \n\nWe present here a catalog of 218 gamma--ray bursts from the detector aboard the \\it Ulysses \\rm\nspacecraft, which were also observed by the Burst and Transient Source Experiment\n(BATSE), and cataloged in the 3rd BATSE catalog (Meegan et al. 1996).  In the\npresent catalog, their positions have been improved in accuracy by arrival time analysis, or\n``triangulation''.  Detection of an event by these two widely\nseparated instruments leads to a narrow annulus which intersects the BATSE\nerror circle and reduces its area\nby a factor of $\\approx$ 30.  Such annulus\/error circle intersections have\nplayed important roles in the examples cited above.\n\n\\section{Instrumentation}\n\nThe \\it Ulysses \\rm GRB detector (Hurley et al. 1992) is one instrument in the 3rd\nInterplanetary Network of burst detectors.  It consists of two 3 mm thick\nhemispherical CsI scintillators\nwith a projected area of about 20 cm$^2$ in any direction.  The detector is mounted\non a magnetometer boom far from the body of the spacecraft, and therefore has a practically\nunobstructed view of the full sky.  Because the \\it Ulysses \\rm mission is in interplanetary\nspace, the instrument also benefits from an exceptionally stable background.  The\nGRB detector operates continuously, and over 97\\% of the data are recovered.\n\nA unique feature of the mission is the fact that it is in heliocentric\norbit with aphelion $\\approx$ 5 AU.  Thus it reaches an apogee of $\\approx$\n6 AU or 3000 light-seconds, and during portions of the orbit, the\nGRB experiment is farther than any burst detector\nhas ever been from Earth, resulting in greatly improved location accuracies\nfor gamma-ray bursts.\n\nThe GRB instrument observes bursts in both a triggered and a real-time mode.\nThe triggered mode is initiated when the number of counts in one of two possible\ntime intervals exceeds a preset value.  Because the background is stable, this\nnumber is defined in terms of an absolute number of counts, rather\nthan a number of standard deviations above the background.  The two time intervals\nmay be selected among values from 0.125 to 4 s.  Each of the two time intervals is associated\nwith an independently selectable energy window, whose lower and upper thresholds lie\nbetween $\\approx$ 15 and 150 keV.  The data collected in the triggered mode consist\nof energy spectra, and either 16 s of 8 ms count rates, or 64 s of 32 ms count rates, in the $\\approx$\n25 to 150 keV energy range; $\\approx$ 1.8 and 7.4 s of the data stream record the\ntime histories prior to trigger, respectively.  Following a trigger, data are\nread out for $\\approx$ 40 m, during which time the experiment cannot re-trigger.\nAn example is shown in Figure 1.\n\nIndependent of the trigger, a continuous stream of real time data is transmitted.\nThis consists of count rates taken with 0.25, 0.5, 1, or 2 s time resolution.  The\nresolution depends on the spacecraft telemetry rate, and the energy range is $\\approx$\n25 to 150 keV, as for the trigger.  Data are compressed in this mode.  An example\nis shown in Figure 2.\nThe real time data serve many purposes.  For\nexample, numerous gamma-ray bursts which are not intense\nenough to trigger the\ndetector can be reliably identified in the real time data, as we demonstrate below.\nBursts arriving during trigger readout may also be identified.\nFinally, bursts whose duration exceeds the duration of the triggered memory can\nbe fully recorded in this mode.  \n\nBATSE consists of eight detector modules situated at the corners of the \\it Compton\nGamma-Ray Observatory \\rm spacecraft.  Each contains a Large Area Detector (LAD), a \n50.8 cm diameter by 1.27 cm thick\nNaI scintillator.  The trigger algorithm examines the count rates of two or more\ndetectors in the 50 - 300 keV energy range, over 64, 256, and 1024 ms intervals.  The\nexperiment is described in more detail in Meegan et al. (1996).  For most of the\nevents in this catalog, we have utilized the DISCSC and PREB data types summed over\ntwo, three, or four detectors.  This gives a 0.064 s resolution time history from 2 seconds prior to\nthe trigger to 240 s or more after the trigger.  We select discriminator channels 1\nand 2, corresponding to the energy range 25 - 100 keV, to match the \\it Ulysses \\rm\nenergy range as closely as possible.  Rarely, this data type is not available; in that\ncase we use MER, TTS, TTE, or 1 s resolution data.  As BATSE is in low Earth\norbit, the time history data are occasionally interrupted by data gaps and Earth occultation.\n\n\\section{Search Procedure}\n\nEvery cosmic burst detected by BATSE is systematically searched for in the\n\\it Ulysses \\rm data as soon as the BATSE data are available for it.\nThis is done by using the approximate arrival direction\nfrom BATSE and the position of the \\it Ulysses \\rm spacecraft to calculate\na range of possible arrival times at \\it Ulysses \\rm.  The calculation assumes generous uncertainties\nin the arrival direction to assure that the crossing window always includes the\nburst.  Typical window lengths are 300 - 500 s (e.g., Figure 2).  \nIn cases where the BATSE positions undergo substantial revision, a\nrevised search window is calculated and the search is redone.\n\nTo carry out the search, the \\it Ulysses \\rm real time count rates are extracted for\nthe crossing window and plotted.  The data are searched both automatically\nand by eye.  In the automatic procedure, the background count rate is typically\ncalculated over ten time intervals, and a $>4 \\sigma$ increase is searched\nfor over 2 or more consecutive time intervals.  The search by eye confirms the\npresence of the increase, and in some cases, identifies increases that were\nnot detected in the automatic procedure due, for example, to a long rise time.   \nIf the burst is intense enough to trigger the \\it Ulysses \\rm GRB detector,\nburst detection is usually immediately apparent in both the real time and\nthe triggered data streams, with one noteworthy exception.  Very short bursts \n(durations $<$ 0.25 s) can trigger, but remain difficult to identify in the\nlower time resolution real time data.  Conversely, long bursts which do not\nreach large peak intensities can be identified in the real time data, but may\nnot trigger regardless of their time-integrated fluxes.  \n\n\n\n\\section{Deriving Annuli by Comparing BATSE and \\it Ulysses \\rm \\bf Time Histories}\n\nWhen a GRB arrives at two spacecraft with a delay $\\rm \\delta$T, it may be\nlocalized to an annulus whose half-angle $\\rm \\theta$ with respect to the\nvector joining the two spacecraft is given by cos$\\rm \\theta$=c$\\rm \\delta$T\/D,\nwhere c is the speed of light and D is the distance between the two\nspacecraft.  (This assumes that the burst is a plane wave, i.e. that its\ndistance is much greater than D.)  The annulus width d$\\rm \\theta$, and thus one dimension of\nthe resulting error box, is c $\\rm \\sigma(\\delta$T)\/Dsin$\\rm \\theta$ where\n$\\rm \\sigma(\\delta$T) is the uncertainty in the time delay.  This term has\ntwo components, one statistical and one systematic.  The first is\nassociated with the uncertainty in comparing the time histories of a burst\nrecorded by two different instruments; the second is due to the uncertainty\nin the clock calibrations of the two spacecraft.  A third source of error\nis the uncertainty in the knowledge of the spacecraft positions.  We\ndiscuss each in turn.\n\nIn the simple case of identical detectors with identical backgrounds,\nthe most probable delay $\\rm \\delta$T and its error $\\rm \\sigma(\\delta$T) may\nbe estimated as follows.  Denote the two background-subtracted time\nhistories by X$_i$, Y$_i$, i=1,...m, that is, X$_1$ is the number of\ncounts in the first time bin for detector X, and Y$_1$ the number of\ncounts in the first time bin for detector Y, and so on.  Assume that the\nbackground can be estimated with arbitrary accuracy, and further that each value of X$_i$ and\nY$_i$ is an independent normal random variable with\nvariance X$_i$ and Y$_i$ respectively.  For simplicity of notation, assume\nthat the two time histories recorded by detectors X and Y are perfectly\naligned when time bin 1 of detector X corresponds to time bin 1 of detector Y.\nThen for this perfect alignment, corresponding to zero lag between the two\ntime histories, the quantity $R_i=(X_i-Y_i)\/ \\sqrt{X_i+Y_i}$\n is a random\nvariable with mean zero and variance unity for every i.  Thus\n\\begin{equation}\n\\sum_{i=1}^{n}R_i^2\n\\end{equation}\nis distributed as $\\chi^2$ with n degrees of freedom.  Here the sum\nis performed over some number n of time bins, where n $\\le$ m.  Now allow the lag\nto vary slightly about zero, so that time bin i of detector X corresponds to time bin\nj of detector Y, and form the random variable $R_{ij}=(X_i-Y_j)\/ \\sqrt{X_i+Y_j}$\nwhere the lag ij is now variable.\nThe quantity\n\\begin{equation}\nr_{ij}^2=\\sum R_{ij}^2\n\\end{equation}\nwhere the summation is over the intervals which X and Y have in common, \nis similarly distributed as $\\chi^2$.  In practice we do not\nknow \\it a priori \\rm the lag at which the best correlation will occur, so we test\nvarious lags ij until r$_{ij}^2$ reaches a minimum, r$_{min}^2$.  Then the lag or\ntime delay corresponding to a 3$\\sigma$ equivalent confidence level is \nr$_{ij}^2$=r$_{min}^2$+9.  (All annuli in this catalog have widths corresponding\nto 3$\\sigma$ equivalent confidence.)\n\nAlthough this simple example describes the essential features of the method, there\nare numerous complicating factors in practice.  For BATSE, the background varies\nover the orbit, and its value may not always be known accurately; the\nassumption of a normal distribution may also be violated when Cyg X-1 is\nin the field of view.  The BATSE and \\it Ulysses \\rm detectors\nhave different areas, burst and background count rates, and\ntime resolutions.  We treat this by first, normalizing the BATSE background-\nsubtracted time history to the \\it Ulysses \\rm one so that two time histories have the\nsame total number of counts.  We also generate a BATSE time history with identical\ntime resolution to the \\it Ulysses \\rm one by scaling the counts in the original BATSE\ntime bins.  These adjustments complicate the above expressions\nconsiderably.  However, through Monte-Carlo simulations, as well as time history\ncomparisons using a completely different method (Laros et al. 1998), we have\nverified that the procedure gives correct results.    In\naddition to calculating the r$_{ij}^2$ statistic, we also compute the\ncorrelation coefficient as a function of lag, namely\n\\[\n\\rho_{ij}=\\frac{ \\sum(X_i-\\langle X \\rangle)(Y_j-\\langle Y \\rangle)}\n{\\left\\{ \\sum(X_i-\\langle X \\rangle)^2 \\sum(Y_i-\\langle Y \\rangle)^2)\n\\right\\}^{0.5}}\n\\]\nwhere the summations are over the intervals that X and Y have in common,\nand find its maximum value.\nThe purpose of this calculation is to assure that the r$_{ij}^2$ statistic has\nindeed identified the best lag; we require that the maximum $\\rho_{ij}$ and the\nminimum r$_{ij}^2$ correspond to identical lags, or at worst, that the difference\nbetween the lags be much\nless than the 3$\\sigma$ confidence interval.  Finally, parabolic fits are done to\nboth statistics to identify the best lag and the confidence intervals.  An example\nis shown in Figures 3 and 4.\n\n\n\n\nThe second uncertainty, due to the clock calibrations, is in principle much\nsmaller than the 3$\\sigma$ confidence interval.  The clock on the \\it Compton\nGamma-Ray Observatory \\rm is accurate to 100 $\\mu$s, and this accuracy is verified\nthrough pulsar timing.  Onboard software increases the uncertainty in the BATSE trigger times to\n$\\approx$ 1 ms.  The clock on the \\it Ulysses \\rm spacecraft is calibrated during\neach daily tracking pass, and the clock drift is extrapolated until the next pass to obtain the\ncorrect time; the calibration is verified at\nsix month intervals by sending\ncommands to the GRB experiment at precisely known times.  The exact procedure\nis described in Hurley (1994).  Due to spacecraft buffering of the commands, the\nresult is that \\it if \\rm it can be assumed that no errors larger than 125 ms\nare present, \\it then \\rm the clock can be shown to be accurate to several\nmilliseconds.  For the purposes of triangulation, however, we take the\nextremely conservative approach that no 3 $\\sigma$ cross-correlation uncertainty\nis less than 125 ms.  Larger timing errors have been identified in the past for both missions.\nThe \\it CGRO \\rm clock was found to have a consistent 2 s offset in the early\npart of the mission, and random timing errors between 0.25 and several hundred\nseconds occurred in certain types of \\it Ulysses \\rm data.  In all cases the\ndata were reprocessed to obtain the results in this paper. \n\n\nFinally, the uncertainties in the spacecraft ephemerides are truly negligible.\nThe \\it Ulysses \\rm range is known to $\\approx$ 10 km, and the right ascension and\ndeclination of the spacecraft are accurate to $\\approx$ 0.00003 degrees.  The\nGRO range is known to better than 50 km, and the right ascension and declination\nto better than 0.01 degrees.  The error that these uncertainties produce in\nthe knowledge of the interspacecraft vector (i.e., the center of the annulus) is\nof order 0.0005 degrees, and therefore usually two orders of magnitude smaller\nthan the timing uncertainties.  As a conservative measure, this uncertainty is added\nlinearly to the uncertainties in the annulus parameters caused by the timing.  It\ntypically increases the annulus width by several tenths of an arcsecond.\n\nThe radius of each annulus and its right ascension and declination are\ntransformed to a heliocentric frame.  This is equivalent to an aberration\ncorrection (maximum value $\\approx$ 20 arcseconds), and involves two adjustments.  \nThe first is to the spacecraft time.\nThe signal from a distant spacecraft may travel several thousand seconds on its\nway to Earth.  Depending on the Earth's velocity vector relative to the spacecraft,\nthe relative motion of the Earth and the spacecraft may not be negligible\nduring this period.  ``Geocentric'' clock times are\nrelated to the position of the Earth when the signal is \\it received \\rm from the\nspacecraft.  ``Heliocentric'' clock times, i.e., times related to the position\nof the Earth when the signal was \\it transmitted, \\rm must be used.  The\ndifference between the two may be\nas large as several hundred milliseconds, and is often comparable to or\ngreater than the statistical uncertainty of the time history comparison.\nThe second adjustment is to the inter-spacecraft vector.  Two observers moving\nwith the Earth and with the sun see different vectors (i.e., annulus centers)\nas a result of their motions during the burst's transit time between the\ntwo spacecraft.  This correction, too, may not be negligible.  The result of\ntriangulating the 1993 January 31 burst is shown in Figure 5.\n\nAlthough there have never been more than three widely separated spacecraft\nin the 3rd IPN, which would have provided redundant determinations and\ntherefore a verification of burst\npositions, there have been numerous instances where a burst source with\na known position was triangulated with \\it Ulysses \\rm, BATSE, and other\nspacecraft.  These include the soft gamma repeater SGR1806-20 and the bursting\npulsar GRO J1744-28 (Hurley et al. 1996, 1998a), the 1997 January 11 burst localized by\nBeppoSAX (Galama et al. 1997), the 1997 February 28 burst, with both a BeppoSAX location and an\noptical transient (Hurley et al. 1997c), and numerous other BeppoSAX bursts (e.g. Hurley et al.\n1998b).  In each case the triangulated position was in good agreement with the known\nposition of the source.\n\n\\section{Burst Selection Criteria}\n\nWe have used several selection criteria for the bursts in this catalog.\nThe first is that the burst must have been detected by \\it Ulysses \\rm and\nBATSE.  Because of the very different detector sizes, all the bursts satisfying\nthis criterion are observed to be rather intense by BATSE, and therefore\ndetection by BATSE is unambiguous.  However, the weaker events (as observed\nby \\it Ulysses \\rm in the real time mode) may be difficult to identify.  In these cases,\nwe rely on several supplementary semi-empirical criteria.  One is that the  r$_{min}^2$ statistic,\ndivided by the number of degrees of freedom, be less than unity.  Another is\nthat the maximum value of the correlation coefficient $\\rho$ be greater than\n0.5.  Yet another is that the ratio of \\it Ulysses \\rm counts to BATSE counts lie\nin the range 0.019 $\\pm$ 0.009.  It might seem unusual that the average value of\nthis ratio is 0.019, whereas the ratio of the the \\it Ulysses \\rm\nprojected area in any direction to a single BATSE LAD area is closer to 0.01, \nand the BATSE counts are derived from at least two LAD's in this procedure.  \nSeveral factors account\nfor this.  First, the \\it Ulysses \\rm energy range is wider than BATSE's:\n25 - 150 keV \\it vs. \\rm 25 - 100 keV.  Second, the \\it Ulysses \\rm lower\nenergy threshold actually allows photons with energies as low as 15 keV to\nbe counted, albeit with lower efficiency.  Third, the two \\it Ulysses \\rm hemispherical\ndetectors are thin, so photons which do not interact in one hemisphere may\ninteract in the other, increasing the effective area.  Finally, the BATSE\ncount rates are not corrected for dead time, which is known to affect the\nbrighter bursts (which are in general among the bursts we observe).  \n\nThis set of criteria is not an absolute one.  To a good approximation, every\ngamma-ray burst is different from every other one.  In addition, data gaps,\nEarth occultation, \ntelemetry noise, unusual background behavior, and different\ndata modes complicate some analyses.  In some cases, bursts\nsatisfying all the criteria above are rejected because they are simply too\nweak to allow reliable triangulation.  In other cases, bursts which do not\nsatisfy one or more criteria may be accepted.\n\nA final criterion for inclusion in this catalog is that the burst must\nnot have been observed by a third interplanetary spacecraft.  The \\it Ulysses \\rm\/\nBATSE bursts also observed by \\it Mars Observer \\rm have been published in Laros et al.\n(1997).  The \\it Ulysses \\rm\/BATSE bursts also observed by \\it Pioneer Venus Orbiter \\rm are\nin press (Laros et al. 1998).\n\n\n\n\\section{A Few Statistics}\n\nOver the period covered by the 3B catalog, \\it Ulysses \\rm observed 346 confirmed\ncosmic gamma-ray bursts (i.e., bursts that were observed by at least one\nother spacecraft).  Of these, 274 were observed by \\it Ulysses \\rm and BATSE and\nin some cases, other spacecraft as well.  \\footnote{A list of all cosmic bursts and the spacecraft which detected them may be found\nat http:\/\/ssl.berkeley.edu\/ipn3\/index.html.}\nSince BATSE observed 1122 bursts during this period, \\it Ulysses \\rm observes\napproximately one out of every 4 BATSE bursts.\nOf the 274, 64 were observed by \n\\it Ulysses \\rm in triggered mode, and 210 in real time mode.  Finally, 220 were\nobserved by \\it Ulysses \\rm and BATSE, but not by a third interplanetary spacecraft;\ntwo of these could not be triangulated according to the criteria explained\nabove.  \n\nThe histogram of Figure 6 shows the distribution of annulus half-widths for the\n218 bursts localized.  The smallest is about 7 \\arcsec, the\nlargest 32 \\arcmin, and the average is 4.5 \\arcmin.  158 of the annuli,\nor 72\\%, intersect the BATSE 1 $\\sigma$ error circles, whose radii are defined by \n$\\rm r_{1\\sigma}=\\sqrt{\\sigma_{stat}^2 + \\sigma_{sys}^2}$,\nwhere $\\sigma_{sys}$ is the systematic error, 1.6$^{\\rm o}$,\nand $\\sigma_{stat}$ is the statistical error.  Figure\n5 shows one example.  This is less than\nthe number which would be predicted (87\\%).  An analysis of a preliminary\nversion of the IPN catalog describes several more complicated BATSE error\nmodels that are consistent with the BATSE-IPN separations (Briggs et al. 1998a).\nA more extensive analysis, utilizing the Ulysses supplement to the\n4B catalog, is in progress (Briggs et al. 1998b).\nOne quantity of interest is how close the\nannulus passes to the center of the error circle.  Let \n$\\alpha_1, \\delta_1$ be the right ascension and declination of\nthe center of a BATSE error circle, and let $\\alpha_2, \\delta_2, \\theta_2$\nbe the right ascension, declination, and radius of an annulus.\nThen the minimum distance between the error circle and the annulus\nis given by\n$d=\\mid \\theta_2 - \\cos^{-1}(\\sin(\\delta_1) \\sin(\\delta_2) +\n\\cos(\\delta_1) \\cos(\\delta_2) \\sin(\\alpha_1 - \\alpha_2) ) \\mid $.\nA histogram of the distribution of minimum distances\nbetween the annuli and the centers of the BATSE error circles\nis given in Figure 7.\n\nIn general, the annuli obtained by triangulations are small circles on the celestial\nsphere, so their curvature, even across a relatively small BATSE error circle, is\nnot always negligible, and a simple, four-sided error box cannot be defined. \nThis curvature can be seen in Figure 5. \nFor this reason, we do not cite the intersection points of\nthe annulus with the error circle.  However, we give here the formulas for finding\nthese points for those cases where it may be useful.  Let $\\alpha_1, \\delta_1, \\theta_1$\nand $\\alpha_2, \\delta_2, \\theta_2$ be the right ascension, declination, and radii of\nthe two small circles.  Let $\\alpha, \\delta$ be the right ascension and declination\nof the intersection points.  Then\n\n$\\sin\\delta=\\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}$\n\nwhere\n\na=$-\\cos^2\\delta_1 \\cos^2\\delta_2 \\sin^2(\\alpha_1-\\alpha_2) + \n2\\sin\\delta_1 \\sin\\delta_2 \\cos\\delta_1 \\cos\\delta_2 \\cos(\\alpha_1-\\alpha_2) -\n\\sin^2\\delta_2 \\cos^2\\delta_1 -\n\\sin^2\\delta_1 \\cos^2\\delta_2$,\n\nb=$-2(\\cos\\theta_1 \\sin\\delta_2 + \\cos\\theta_2 \\sin\\delta_1) \\cos\\delta_1 \\cos\\delta_2 \\cos(\\alpha_1 - \\alpha_2)+\n2\\cos\\theta_2 \\sin\\delta_2 \\cos^2\\delta_1 +\n2\\cos\\theta_1 \\sin\\delta_1 \\cos^2\\delta_2$,\n\nand\n\nc=$\\cos^2\\delta_1 \\cos^2\\delta_2 \\sin^2(\\alpha_1-\\alpha_2) +\n2\\cos\\theta_2 \\cos\\theta_1 \\cos\\delta_1 \\cos\\delta_2 \\cos(\\alpha_1 - \\alpha_2) -\n\\cos^2\\theta_2 \\cos^2\\delta_1 -\n\\cos^2\\theta_1 \\cos^2\\delta_2$,\n\nFor each of the two values of the declination $\\delta$ the corresponding right ascension\n$\\alpha$ is given by\n\n$\\alpha=\\alpha_1 + \\cos^{-1}\\frac{\\cos\\theta_1 - \\sin\\delta \\sin\\delta_1}\n{\\cos\\delta \\cos\\delta_1}$\n\nFigure 8 shows the BATSE peak fluxes and fluences for 162 of the 218 bursts\nwith flux and fluence entries in the 3B catalog.  There is a slight tendency for \nsome of the smaller\nfluence events to have relatively large peak fluxes over a time interval comparable\nto that of the \\it Ulysses \\rm real time data, which helps to explain why these bursts were\ndetected by \\it Ulysses \\rm.  To estimate the completeness of this catalog, we have\ncalculated the ratios of the number of bursts detected by \\it Ulysses \\rm to the number\ndetected by BATSE above several\nflux and fluence thresholds starting with the weakest events in the BATSE catalog; \nwe have used the 25-100 keV fluences in erg cm$^{-2}$ and\nthe peak 25 - 300 keV fluxes over 256 ms in photons cm$^{-2}$ s$^{-1}$.  Since events\ndetected by a third spacecraft are not included in the present catalog, the numbers\ngiven represent lower limits to the completeness.  Table 1 summarizes the results.\n\n\\clearpage\n\\begin{deluxetable}{cc}\n\\tablecaption{Completeness above various thresholds}\n\\tablehead{\n\\colhead{BATSE Threshold}&\\colhead{Approximate completeness (lower limit)}\n}\n\\startdata\n\n1.8 x 10$^{-9}$ erg cm$^{-2}$ & .25  \\nl\n10$^{-6}$ erg cm$^{-2}$ & .49 \\nl\n10$^{-5}$ erg cm$^{-2}$ & .74 \\nl\n.27 photons cm$^{-2}$ s$^{-1}$ & .25 \\nl\n1 photon cm$^{-2}$ s$^{-1}$ & .34 \\nl\n10 photons cm$^{-2}$ s$^{-1}$ & .65 \\nl\n\\enddata\n\\end{deluxetable}\n\\clearpage\n\n\nFinally, we have analyzed these annuli for intersections which might provide evidence\nof repeating bursts from a single source.  Although there was a large, but statistically\ninsignificant number of\ntwo annulus intersections, we found only one case where a three-way\nintersection almost occurred (BATSE bursts 451, 2286, and 2619); the data are therefore\nconsistent with no burst repetition. \n\n\\section{Table of Annuli}\n\nThe ten columns in table 1 give:\n1) the date of the burst, in ddmmyy format, \n2) the Universal Time of the burst at Earth,\n3) the BATSE number for the burst,\n4) the BATSE right ascension of the center of the error circle (J2000), in degrees,\n5) the BATSE declination of the center of the error circle (J2000), in degrees,\n6) the total 1 $\\sigma$ statistical BATSE error circle radius, in degrees, (the \napproximate total\n1$\\sigma$ radius is obtained by adding 1.6$^{\\rm o}$ in quadrature, but see\nBriggs et al. 1998a,b for an improved error model), \n7) the right ascension of the center of the IPN (BATSE\/\\it Ulysses \\rm)\nannulus, epoch J2000, corrected to the heliocentric frame, in degrees,\n8) the declination of the center of the IPN (BATSE\/\\it Ulysses \\rm)\nannulus, epoch J2000, corrected to the heliocentric frame, in degrees, \n9) the angular radius of the IPN (BATSE\/\\it Ulysses \\rm) annulus, corrected to the heliocentric\nframe, in degrees, and\n10) the half width of the IPN (BATSE\/\\it Ulysses \\rm) annulus, in degrees; the 3 $\\sigma$\nconfidence annulus is given by R$_{\\rm IPN}$ $\\pm$ $\\delta$ R$_{IPN}$.\n\n\nThis list supercedes previous ones which have been circulated in the past, but\nnot published.  The changes to it include:\n1) the use of final time history and spacecraft ephemeris data; earlier lists\noften used preliminary and predict data, with correspondingly larger timing uncertainties,\n2) transformation of annuli to heliocentric coordinates; earlier lists often gave\ngeocentric coordinates, with correspondingly larger uncertainties in the annulus widths,\n3) removal of weak events according to the criteria in section 5,\n4) reprocessing of data to eliminate known clock errors, and\n5) careful examination of bursts whose BATSE time histories were truncated\nby data gaps and Earth occultation.   \n\nFigures 9 and 10 compare the BATSE error circles with the IPN annulus\/error circle\nintersections for the bursts in this catalog.  To generate this plot, it was \nassumed that all annuli pass through the centers of their corresponding BATSE error \ncircles.\n \nWe stress that while the BATSE data are for the bursts listed in the 3B\ncatalog, the locations and error circle radii are in fact the updated ones\nwhich will appear in the 4B catalog.  They have been taken from the latest online catalog,\nand are given here for convenience only. \\footnote{\nThe catalog (http:\/\/www.batse.msfc.nasa.gov\/data\/grb\/4bcatalog\/)\nshould be considered to be the ultimate source of the\nmost up-to-date BATSE data.   Table 1 is available electronically from\nhttp:\/\/ssl.berkeley.edu\/ipn3\/index.html.}  Finally, we add that 150 \\it Ulysses \\rm\/BATSE\nbursts were detected over the period between the end of the 3B catalog\nand the end of the 4B catalog.  Analysis of these events has been completed and\nis being published in conjunction with the 4B catalog. \n\n\\clearpage\n\\begin{deluxetable}{cccccccccc}\n\\tablecaption{\\it Ulysses \\rm\/BATSE annuli}\n\\tablehead{\n\\colhead{Date}&\\colhead{UT}&\\colhead{N$_B$}&\\colhead{$\\alpha_{2000, B}$}&\n\\colhead{$\\delta_{2000, B}$}&\\colhead{$\\sigma_{stat, B}$}&\\colhead{$\\alpha_{2000, IPN}$}&\n\\colhead{$\\delta_{2000, IPN}$}&\\colhead{R$_{IPN}$}&\\colhead{$\\delta R_{IPN}$}\n}\n\\startdata\n250491 & 00:37:46 & 109 & 91.29 & -22.77 & 1.02 & 115.140 & 23.171 & 51.784 & 0.051 \\nl\n290491 & 03:11:50 & 121 & 174.75 & 14.33 & 1.36 & 116.426 & 22.919 & 59.157 & 0.136 \\nl\n020591 & 22:37:23 & 142 & 46.22 & -52.71 & 0.62 & 297.624 & -22.677 & 87.126 & 0.023 \\nl\n030591 & 07:04:13 & 143 & 87.45 & 38.74 & 0.87 & 117.731 & 22.655 & 30.360 & 0.009 \\nl\n110591 & 02:11:48 & 179 & 266.48 & 58.30 & 1.66 & 300.193 & -22.137 & 83.666 & 0.018 \\nl\n230591 & 19:03:25 & 222 & 106.99 & 0.36 & 1.28 & 124.202 & 21.228 & 26.847 & 0.051 \\nl\n010691 & 19:22:14 & 249 & 310.12 & 32.34 & 0.17 & 307.035 & -20.537 & 52.942 & 0.008 \\nl\n020691 & 22:55:01 & 257 & 142.76 & 53.99 & 2.06 & 127.387 & 20.448 & 34.526 & 0.163 \\nl\n260691 & 07:15:13 & 444 & 133.38 & 7.72 & 1.27 & 134.557 & 18.501 & 10.164 & 0.020 \\nl\n300691 & 07:37:02 & 469 & 304.70 & 35.13 & 0.91 & 315.764 & -18.148 & 52.439 & 0.085 \\nl\n210791 & 19:30:12 & 563 & 315.52 & -14.29 & 1.31 & 322.022 & -16.198 & 6.769 & 0.436 \\nl\n090891 & 06:15:00 & 659 & 195.43 & 19.37 & 1.85 & 147.088 & 14.483 & 50.860 & 0.092 \\nl\n090891 & 19:35:45 & 660 & 281.96 & 64.74 & 2.01 & 327.238 & -14.431 & 85.747 & 0.016 \\nl\n050991 & 23:48:55 & 761 & 360.00 & -81.05 & 0.82 & 334.075 & -11.942 & 70.787 & 0.017 \\nl\n070991 & 01:05:23 & 764 & 136.55 & -28.23 & 1.30 & 154.322 & 11.849 & 43.573 & 0.131 \\nl\n260991 & 18:25:57 & 824 & 346.07 & 21.36 & 2.70 & 338.688 & -10.171 & 32.155 & 0.294 \\nl\n270991 & 23:26:53 & 829 & 50.16 & -39.65 & 0.34 & 338.937 & -10.074 & 68.870 & 0.036 \\nl\n300991 & 11:55:30 & 841 & 131.80 & -20.62 & 1.27 & 159.447 & 9.874 & 39.201 & 0.013 \\nl\n061091 & 09:01:54 & 871 & 33.30 & -45.44 & 3.38 & 340.603 & -9.419 & 50.544 & 0.087 \\nl\n221091 & 04:13:58 & 914 & 123.91 & 70.34 & 1.03 & 163.395 & 8.312 & 68.583 & 0.017 \\nl\n261091 & 13:07:01 & 938 & 74.68 & -22.27 & 2.40 & 344.080 & -8.038 & 80.644 & 0.068 \\nl\n061191 & 03:42:01 & 1008 & 343.03 & -35.16 & 0.83 & 345.570 & -7.445 & 29.223 & 0.023 \\nl\n231191 & 08:15:33 & 1114 & 22.81 & 30.34 & 1.14 & 347.386 & -6.730 & 49.421 & 0.015 \\nl\n271191 & 04:22:08 & 1122 & 269.14 & 49.73 & 0.38 & 347.680 & -6.617 & 88.466 & 0.005 \\nl\n071291 & 09:50:45 & 1150 & 307.79 & 29.30 & 1.59 & 348.248 & -6.408 & 53.876 & 0.059 \\nl\n091291 & 18:35:58 & 1157 & 259.98 & -45.08 & 0.40 & 348.333 & -6.380 & 82.576 & 0.003 \\nl\n171291 & 08:22:06 & 1190 & 6.96 & 22.67 & 1.02 & 348.480 & -6.340 & 35.915 & 0.032 \\nl\n211291 & 22:03:26 & 1200 & 215.82 & -42.72 & 1.98 & 168.475 & 6.356 & 61.764 & 0.111 \\nl\n241291 & 11:02:58 & 1212 & 45.43 & -10.79 & 4.68 & 348.444 & -6.377 & 57.290 & 0.117 \\nl\n271291 & 01:05:53 & 1235 & 350.44 & -68.25 & 0.67 & 348.389 & -6.408 & 61.852 & 0.064 \\nl\n100192 & 09:17:58 & 1288 & 58.08 & -20.80 & 0.42 & 347.700 & -6.748 & 69.190 & 0.018 \\nl\n210292 & 06:14:32 & 1425 & 184.14 & 48.48 & 0.65 & 161.368 & 8.446 & 42.142 & 0.008 \\nl\n240292 & 02:43:39 & 1432 & 169.65 & -66.78 & 2.73 & 160.726 & 8.515 & 75.175 & 0.539 \\nl\n270292 & 06:00:51 & 1443 & 68.36 & 66.84 & 1.33 & 160.016 & 8.591 & 83.353 & 0.005 \\nl\n070392 & 00:18:04 & 1467 & 355.57 & -45.15 & 1.16 & 338.049 & -8.795 & 39.316 & 0.066 \\nl\n150392 & 04:19:27 & 1484 & 320.58 & -20.89 & 0.46 & 336.280 & -8.960 & 16.873 & 0.013 \\nl\n070492 & 12:03:40 & 1543 & 341.44 & -10.68 & 2.04 & 332.036 & -9.187 & 9.927 & 0.437 \\nl\n080492 & 12:05:56 & 1545 & 138.96 & 40.83 & 0.58 & 151.890 & 9.190 & 34.380 & 0.022 \\nl\n290492 & 18:59:32 & 1571 & 179.08 & -14.33 & 2.85 & 149.602 & 8.963 & 39.318 & 0.091 \\nl\n020592 & 06:02:59 & 1577 & 33.05 & -17.73 & 1.04 & 329.440 & -8.908 & 60.781 & 0.026 \\nl\n130592 & 16:52:39 & 1606 & 211.16 & -44.77 & 0.65 & 148.956 & 8.591 & 74.250 & 0.005 \\nl\n300592 & 22:59:55 & 1630 & 162.68 & 18.13 & 1.67 & 148.993 & 7.901 & 17.618 & 0.070 \\nl\n170692 & 05:26:51 & 1652 & 37.66 & 77.51 & 0.51 & 149.807 & 6.977 & 89.064 & 0.012 \\nl\n220692 & 07:05:05 & 1663 & 162.10 & 47.17 & 0.31 & 150.169 & 6.667 & 41.336 & 0.005 \\nl\n220692 & 12:54:21 & 1664 & 346.81 & 10.01 & 0.55 & 330.187 & -6.649 & 18.892 & 0.018 \\nl\n270692 & 13:02:36 & 1676 & 166.02 & -2.58 & 0.65 & 150.593 & 6.327 & 23.504 & 0.017 \\nl\n010792 & 19:38:19 & 1683 & 314.64 & 33.44 & 0.63 & 330.973 & -6.033 & 42.560 & 0.017 \\nl\n140792 & 20:23:26 & 1700 & 140.36 & -44.64 & 3.28 & 152.298 & 5.079 & 50.262 & 0.099 \\nl\n180792 & 14:40:36 & 1708 & 22.44 & -3.98 & 1.14 & 332.718 & -4.782 & 49.196 & 0.032 \\nl\n180792 & 21:32:43 & 1709 & 296.39 & -55.98 & 0.48 & 332.751 & -4.760 & 58.948 & 0.004 \\nl\n210792 & 18:17:48 & 1717 & 37.55 & 33.49 & 0.73 & 333.083 & -4.530 & 72.485 & 0.033 \\nl\n230792 & 01:00:34 & 1721 & 85.79 & 7.78 & 0.48 & 153.234 & 4.427 & 68.971 & 0.008 \\nl\n010892 & 01:16:40 & 1733 & 157.86 & 68.90 & 1.12 & 154.338 & 3.667 & 64.462 & 0.021 \\nl\n130892 & 03:33:48 & 1807 & 206.22 & -28.34 & 1.75 & 155.907 & 2.581 & 63.921 & 0.051 \\nl\n140892 & 06:09:53 & 1815 & 258.68 & -44.63 & 0.79 & 336.053 & -2.478 & 78.143 & 0.006 \\nl\n240892 & 10:53:03 & 1872 & 228.59 & -51.79 & 0.33 & 157.427 & 1.505 & 76.788 & 0.003 \\nl\n300892 & 01:45:18 & 1883 & 258.53 & -73.99 & 0.54 & 338.192 & -0.949 & 86.267 & 0.010 \\nl\n020992 & 00:29:05 & 1886 & 279.15 & -20.70 & 2.29 & 338.594 & -0.651 & 61.607 & 0.015 \\nl\n030992 & 00:49:17 & 1888 & 61.97 & -68.37 & 0.75 & 338.732 & -0.549 & 88.165 & 0.010 \\nl\n250992 & 21:45:16 & 1956 & 160.26 & 67.38 & 1.34 & 161.773 & -1.864 & 68.233 & 0.020 \\nl\n031092 & 06:35:40 & 1974 & 150.32 & -14.71 & 0.55 & 162.694 & -2.674 & 19.765 & 0.021 \\nl\n091092 & 06:41:11 & 1983 & 120.41 & -29.10 & 0.39 & 163.411 & -3.345 & 46.240 & 0.003 \\nl\n151092 & 01:36:32 & 1989 & 123.04 & 41.69 & 0.95 & 164.066 & -3.998 & 57.677 & 0.040 \\nl\n221092 & 15:20:59 & 1997 & 253.78 & -12.42 & 0.41 & 164.862 & -4.861 & 88.377 & 0.002 \\nl\n291092 & 12:37:56 & 2018 & 34.14 & -1.57 & 1.67 & 345.514 & 5.651 & 50.169 & 0.045 \\nl\n301092 & 01:16:31 & 2020 & 86.16 & 6.86 & 1.75 & 165.561 & -5.710 & 79.293 & 0.042 \\nl\n181192 & 22:11:44 & 2061 & 350.45 & 50.32 & 1.07 & 346.930 & 7.990 & 41.232 & 0.128 \\nl\n231192 & 06:18:31 & 2067 & 334.24 & -53.25 & 0.72 & 347.111 & 8.481 & 64.897 & 0.008 \\nl\n031292 & 21:44:22 & 2074 & 233.62 & 21.03 & 0.74 & 167.337 & -9.664 & 68.339 & 0.119 \\nl\n061292 & 18:24:19 & 2080 & 176.48 & 45.72 & 0.77 & 167.341 & -9.975 & 51.667 & 0.029 \\nl\n071292 & 16:00:48 & 2083 & 306.33 & -42.93 & 0.44 & 347.336 & 10.074 & 62.709 & 0.003 \\nl\n091292 & 02:40:56 & 2089 & 172.98 & 61.45 & 0.51 & 167.325 & -10.228 & 68.392 & 0.048 \\nl\n091292 & 11:35:38 & 2090 & 332.97 & -52.02 & 0.68 & 347.320 & 10.269 & 63.995 & 0.009 \\nl\n181292 & 02:30:02 & 2102 & 355.62 & -55.31 & 1.76 & 347.091 & 11.168 & 66.903 & 0.076 \\nl\n271292 & 12:44:08 & 2106 & 206.33 & -25.65 & 1.04 & 166.535 & -12.089 & 39.698 & 0.113 \\nl\n301292 & 08:57:51 & 2110 & 19.52 & 17.21 & 0.33 & 346.301 & 12.353 & 32.939 & 0.112 \\nl\n301292 & 14:05:46 & 2111 & 150.54 & 28.63 & 1.71 & 166.282 & -12.370 & 45.110 & 0.086 \\nl\n060193 & 15:37:39 & 2121 & 6.08 & 2.00 & 0.53 & 345.556 & 12.984 & 21.048 & 0.021 \\nl\n060193 & 19:57:15 & 2122 & 124.79 & -32.62 & 0.68 & 165.536 & -12.996 & 42.302 & 0.149 \\nl\n080193 & 02:28:55 & 2123 & 94.62 & -30.44 & 0.99 & 165.383 & -13.100 & 65.116 & 0.069 \\nl\n120193 & 03:47:31 & 2127 & 226.32 & 27.74 & 1.22 & 164.853 & -13.415 & 72.911 & 0.021 \\nl\n120193 & 15:18:02 & 2128 & 210.10 & 51.92 & 1.02 & 164.786 & -13.451 & 76.849 & 0.066 \\nl\n160193 & 02:47:02 & 2136 & 246.99 & 15.91 & 1.02 & 164.275 & -13.700 & 88.748 & 0.026 \\nl\n270193 & 16:14:25 & 2149 & 310.95 & -39.63 & 1.16 & 342.242 & 14.382 & 60.557 & 0.155 \\nl\n310193 & 18:57:12 & 2151 & 182.04 & -8.24 & 2.16 & 161.406 & -14.563 & 22.249 & 0.009 \\nl\n010293 & 16:41:55 & 2156 & 333.53 & 41.15 & 1.26 & 341.214 & 14.601 & 27.869 & 0.023 \\nl\n050293 & 10:10:31 & 2166 & 278.00 & -45.74 & 5.98 & 340.403 & 14.729 & 83.936 & 0.277 \\nl\n170293 & 14:44:38 & 2197 & 117.78 & 0.98 & 2.27 & 157.517 & -14.945 & 41.809 & 0.234 \\nl\n020393 & 21:33:17 & 2213 & 90.33 & 5.43 & 0.62 & 154.166 & -14.821 & 66.384 & 0.016 \\nl\n090393 & 03:07:50 & 2228 & 325.41 & 51.73 & 0.72 & 332.605 & 14.648 & 36.594 & 0.012 \\nl\n100393 & 07:19:20 & 2232 & 284.35 & -52.39 & 0.89 & 332.316 & 14.607 & 79.362 & 0.007 \\nl\n180393 & 12:29:53 & 2255 & 150.48 & -18.85 & 1.09 & 150.361 & -14.267 & 5.323 & 0.320 \\nl\n290393 & 03:09:25 & 2273 & 179.25 & -8.65 & 0.94 & 148.090 & -13.723 & 35.582 & 0.010 \\nl\n310393 & 03:11:16 & 2276 & 43.00 & 47.27 & 2.03 & 327.701 & 13.613 & 77.070 & 0.078 \\nl\n050493 & 08:54:49 & 2286 & 198.18 & -8.80 & 0.55 & 146.754 & -13.315 & 52.136 & 0.005 \\nl\n060493 & 07:16:25 & 2289 & 299.81 & 29.01 & 0.51 & 326.595 & 13.263 & 31.383 & 0.011 \\nl\n090493 & 04:50:10 & 2295 & 295.83 & 43.18 & 2.04 & 326.125 & 13.098 & 37.466 & 0.366 \\nl\n090493 & 21:25:40 & 2297 & 176.91 & -39.75 & 1.25 & 146.019 & -13.056 & 39.155 & 0.088 \\nl\n140493 & 02:27:15 & 2303 & 311.42 & 9.92 & 0.91 & 325.406 & 12.819 & 14.314 & 0.061 \\nl\n180493 & 14:38:18 & 2307 & 311.65 & 21.37 & 1.27 & 324.831 & 12.570 & 14.596 & 0.415 \\nl\n240493 & 19:19:00 & 2315 & 54.22 & 41.60 & 1.67 & 324.179 & 12.245 & 86.680 & 0.134 \\nl\n250493 & 10:17:29 & 2316 & 18.00 & -35.19 & 0.31 & 324.122 & 12.214 & 70.285 & 0.034 \\nl\n260493 & 12:40:32 & 2319 & 67.82 & 10.13 & 1.10 & 144.025 & -12.159 & 75.515 & 0.028 \\nl\n280493 & 01:07:12 & 2320 & 129.85 & -55.45 & 1.68 & 143.900 & -12.085 & 46.202 & 0.005 \\nl\n300493 & 15:01:18 & 2321 & 106.33 & 20.97 & 0.36 & 143.709 & -11.967 & 50.803 & 0.009 \\nl\n020593 & 13:52:52 & 2323 & 254.72 & -46.53 & 0.78 & 323.580 & 11.882 & 87.103 & 0.039 \\nl\n060593 & 14:52:51 & 2329 & 64.45 & -5.97 & 0.24 & 143.364 & -11.717 & 77.827 & 0.004 \\nl\n180593 & 03:20:25 & 2346 & 349.17 & -37.51 & 1.04 & 323.086 & 11.354 & 53.387 & 0.019 \\nl\n230593 & 19:40:53 & 2350 & 14.80 & 4.21 & 2.64 & 323.121 & 11.238 & 44.153 & 0.124 \\nl\n310593 & 15:53:38 & 2362 & 340.04 & 13.25 & 2.16 & 323.341 & 11.152 & 15.895 & 0.044 \\nl\n090693 & 10:07:24 & 2383 & 208.73 & -16.38 & 1.62 & 143.799 & -11.158 & 64.377 & 0.132 \\nl\n140693 & 03:40:30 & 2393 & 168.14 & 23.95 & 1.15 & 144.132 & -11.210 & 42.800 & 0.008 \\nl\n050793 & 12:39:08 & 2429 & 283.08 & -39.24 & 0.99 & 326.244 & 11.861 & 65.410 & 0.124 \\nl\n060793 & 08:09:10 & 2432 & 210.09 & -1.36 & 1.31 & 146.342 & -11.898 & 63.498 & 0.045 \\nl\n080793 & 05:12:38 & 2433 & 86.55 & -69.02 & 6.25 & 146.572 & -11.989 & 62.867 & 0.076 \\nl\n090793 & 06:14:22 & 2436 & 194.41 & -46.73 & 0.67 & 146.700 & -12.042 & 54.525 & 0.028 \\nl\n140793 & 16:12:53 & 2446 & 257.70 & 23.70 & 0.97 & 327.399 & 12.344 & 68.720 & 0.048 \\nl\n210793 & 01:17:22 & 2452 & 78.66 & -33.15 & 1.52 & 148.275 & -12.751 & 66.262 & 0.085 \\nl\n210793 & 12:45:35 & 2453 & 298.58 & -40.79 & 1.79 & 328.342 & 12.785 & 61.079 & 0.101 \\nl\n270793 & 08:08:29 & 2466 & 159.99 & -23.89 & 1.28 & 149.185 & -13.209 & 16.546 & 0.136 \\nl\n310793 & 03:16:31 & 2475 & 116.53 & 45.49 & 2.15 & 149.754 & -13.514 & 66.345 & 0.036 \\nl\n050893 & 12:39:28 & 2482 & 334.55 & -15.60 & 0.83 & 330.584 & 13.983 & 30.305 & 0.208 \\nl\n090893 & 05:11:04 & 2486 & 37.97 & -74.63 & 1.79 & 151.166 & -14.323 & 80.857 & 0.088 \\nl\n220893 & 14:44:51 & 2500 & 213.09 & -0.68 & 2.28 & 153.348 & -15.724 & 65.872 & 0.232 \\nl\n310893 & 09:16:05 & 2507 & 223.29 & -3.63 & 0.47 & 154.815 & -16.774 & 67.536 & 0.007 \\nl\n030993 & 00:19:29 & 2511 & 88.67 & -10.42 & 4.20 & 155.258 & -17.109 & 64.672 & 0.109 \\nl\n050993 & 03:26:30 & 2514 & 310.95 & -1.22 & 0.63 & 335.615 & 17.389 & 28.629 & 0.006 \\nl\n100993 & 12:12:08 & 2522 & 167.59 & -66.69 & 0.54 & 156.521 & -18.114 & 48.810 & 0.044 \\nl\n140993 & 02:49:11 & 2530 & 34.26 & 34.29 & 1.53 & 337.128 & 18.628 & 50.198 & 0.173 \\nl\n160993 & 20:19:23 & 2533 & 280.71 & 65.39 & 0.22 & 337.586 & 19.026 & 57.392 & 0.013 \\nl\n220993 & 06:24:46 & 2537 & 271.20 & 55.89 & 0.45 & 338.488 & 19.847 & 67.999 & 0.003 \\nl\n270993 & 04:18:07 & 2542 & 108.61 & -9.52 & 1.62 & 159.296 & -20.624 & 52.084 & 0.115 \\nl\n061093 & 21:31:50 & 2566 & 66.31 & 65.28 & 1.70 & 340.850 & 22.262 & 70.461 & 0.049 \\nl\n081093 & 07:18:14 & 2570 & 42.98 & 21.37 & 0.85 & 341.069 & 22.510 & 58.202 & 0.082 \\nl\n081093 & 11:09:10 & 2571 & 165.40 & 33.93 & 0.86 & 161.094 & -22.537 & 60.104 & 0.027 \\nl\n141093 & 17:01:01 & 2586 & 272.50 & 9.34 & 0.50 & 342.042 & 23.672 & 67.956 & 0.010 \\nl\n171093 & 01:17:49 & 2590 & 181.44 & -83.11 & 0.65 & 162.389 & -24.110 & 58.979 & 0.003 \\nl\n191093 & 18:22:34 & 2593 & 242.92 & -18.70 & 1.66 & 162.779 & -24.628 & 73.476 & 0.025 \\nl\n241093 & 13:29:09 & 2603 & 137.63 & -8.49 & 0.53 & 163.447 & -25.567 & 30.574 & 0.083 \\nl\n261093 & 11:35:57 & 2606 & 51.36 & -10.56 & 0.48 & 343.704 & 25.955 & 75.482 & 0.087 \\nl\n301093 & 08:08:42 & 2609 & 326.41 & 58.24 & 0.55 & 344.202 & 26.745 & 31.863 & 0.058 \\nl\n311093 & 04:06:35 & 2611 & 325.10 & 62.73 & 0.32 & 344.306 & 26.918 & 37.073 & 0.005 \\nl\n031193 & 16:25:42 & 2617 & 87.88 & 65.05 & 0.31 & 344.732 & 27.657 & 72.278 & 0.008 \\nl\n061193 & 20:32:04 & 2619 & 196.75 & -3.13 & 1.11 & 165.094 & -28.337 & 40.406 & 0.232 \\nl\n121193 & 18:45:53 & 2628 & 311.51 & 55.95 & 0.53 & 345.711 & 29.654 & 34.291 & 0.010 \\nl\n261193 & 04:46:46 & 2660 & 157.28 & 70.08 & 3.02 & 346.750 & 32.795 & 80.134 & 0.109 \\nl\n261193 & 19:31:39 & 2661 & 3.29 & -17.40 & 0.22 & 346.785 & 32.945 & 50.498 & 0.004 \\nl\n041293 & 09:48:10 & 2676 & 250.15 & 38.23 & 0.18 & 347.083 & 34.824 & 76.909 & 0.004 \\nl\n051293 & 14:59:51 & 2679 & 15.56 & 66.49 & 2.22 & 347.108 & 35.133 & 38.364 & 0.018 \\nl\n081293 & 04:37:19 & 2682 & 198.73 & -29.93 & 0.56 & 167.142 & -35.783 & 34.384 & 0.050 \\nl\n081293 & 09:34:40 & 2683 & 234.11 & -40.29 & 3.06 & 167.143 & -35.836 & 51.960 & 0.143 \\nl\n211293 & 02:09:50 & 2700 & 91.02 & -43.17 & 0.50 & 166.799 & -39.143 & 56.073 & 0.036 \\nl\n221293 & 21:21:33 & 2703 & 192.52 & 28.65 & 0.66 & 166.675 & -39.621 & 73.261 & 0.251 \\nl\n251293 & 23:20:12 & 2709 & 4.85 & 9.87 & 0.79 & 346.415 & 40.440 & 35.141 & 0.212 \\nl\n261293 & 20:11:09 & 2711 & 205.95 & 21.63 & 2.82 & 166.329 & -40.669 & 74.178 & 0.127 \\nl\n010194 & 22:42:00 & 2729 & 189.27 & 62.84 & 3.47 & 345.578 & 42.289 & 78.226 & 0.064 \\nl\n020194 & 02:45:52 & 2732 & 10.90 & 45.11 & 1.27 & 345.554 & 42.335 & 18.951 & 0.195 \\nl\n030194 & 22:12:56 & 2736 & 267.90 & 7.14 & 0.65 & 345.274 & 42.810 & 76.721 & 0.016 \\nl\n130194 & 16:47:54 & 2756 & 209.39 & -23.16 & 9.65 & 163.289 & -45.332 & 42.651 & 0.049 \\nl\n190194 & 15:44:40 & 2773 & 131.41 & 39.29 & 1.39 & 161.654 & -46.793 & 89.606 & 0.051 \\nl\n280194 & 16:50:58 & 2790 & 224.02 & -15.22 & 0.96 & 158.510 & -48.816 & 64.647 & 0.059 \\nl\n290194 & 10:43:17 & 2793 & 127.46 & 28.59 & 0.82 & 158.217 & -48.971 & 81.612 & 0.072 \\nl\n010294 & 07:40:40 & 2794 & 268.61 & 22.37 & 1.67 & 337.027 & 49.542 & 59.277 & 0.125 \\nl\n030294 & 15:46:56 & 2797 & 184.16 & -40.95 & 0.58 & 156.002 & -49.973 & 20.512 & 0.052 \\nl\n060294 & 00:08:37 & 2798 & 144.20 & -59.96 & 0.15 & 154.922 & -50.380 & 13.687 & 0.039 \\nl\n100294 & 19:13:16 & 2812 & 152.29 & 82.13 & 0.66 & 332.566 & 51.121 & 48.298 & 0.007 \\nl\n170294 & 23:02:42 & 2831 & 29.07 & 4.55 & 0.69 & 328.739 & 51.947 & 69.334 & 0.005 \\nl\n180294 & 19:32:28 & 2833 & 349.71 & -19.69 & 0.35 & 328.264 & 52.023 & 73.939 & 0.010 \\nl\n280294 & 11:29:00 & 2852 & 127.98 & -12.36 & 0.44 & 142.713 & -52.495 & 42.173 & 0.012 \\nl\n010394 & 20:10:37 & 2855 & 103.51 & 64.35 & 0.35 & 321.919 & 52.509 & 59.400 & 0.029 \\nl\n020394 & 05:08:31 & 2856 & 12.49 & -23.97 & 0.21 & 321.700 & 52.508 & 88.572 & 0.005 \\nl\n120394 & 11:28:22 & 2877 & 221.50 & -56.17 & 0.61 & 135.895 & -52.124 & 48.964 & 0.052 \\nl\n190394 & 23:57:20 & 2889 & 68.74 & -12.60 & 0.39 & 132.014 & -51.403 & 62.552 & 0.090 \\nl\n210394 & 22:05:07 & 2890 & 160.33 & 3.10 & 0.85 & 131.093 & -51.168 & 58.918 & 0.233 \\nl\n230394 & 22:04:38 & 2891 & 213.89 & 8.37 & 0.85 & 310.173 & 50.907 & 85.032 & 0.016 \\nl\n280394 & 12:05:34 & 2894 & 97.18 & -32.35 & 0.58 & 128.216 & -50.236 & 26.692 & 0.028 \\nl\n290394 & 05:49:26 & 2895 & 237.51 & -68.89 & 0.50 & 127.917 & -50.121 & 51.112 & 0.006 \\nl\n290394 & 18:15:37 & 2897 & 204.91 & -10.33 & 1.00 & 127.712 & -50.040 & 69.434 & 0.026 \\nl\n300394 & 20:44:39 & 2899 & 193.80 & -33.63 & 1.72 & 127.286 & -49.861 & 48.714 & 0.320 \\nl\n060494 & 01:10:54 & 2913 & 225.44 & -39.10 & 1.24 & 125.133 & -48.806 & 69.435 & 0.043 \\nl\n100494 & 15:45:02 & 2919 & 254.89 & 30.80 & 0.93 & 303.788 & 47.974 & 42.261 & 0.047 \\nl\n120494 & 01:40:31 & 2922 & 217.17 & -38.35 & 1.12 & 123.422 & -47.710 & 58.846 & 0.303 \\nl\n140494 & 16:46:25 & 2929 & 181.26 & -26.15 & 0.50 & 122.794 & -47.222 & 49.154 & 0.026 \\nl\n190494 & 19:10:58 & 2940 & 359.86 & -47.95 & 0.58 & 121.768 & -46.270 & 73.314 & 0.062 \\nl\n290494 & 00:43:53 & 2953 & 35.10 & -57.52 & 0.32 & 120.524 & -44.594 & 51.727 & 0.006 \\nl\n030594 & 05:07:44 & 2958 & 161.00 & 10.06 & 0.60 & 120.195 & -43.873 & 64.039 & 0.057 \\nl\n120594 & 17:37:50 & 2974 & 232.97 & 55.88 & 2.65 & 299.935 & 42.376 & 44.482 & 0.007 \\nl\n200594 & 00:21:38 & 2984 & 323.50 & 6.93 & 0.88 & 300.146 & 41.389 & 40.428 & 0.019 \\nl\n260594 & 10:55:26 & 2993 & 80.41 & -30.39 & 0.96 & 120.590 & -40.650 & 33.571 & 0.347 \\nl\n260594 & 20:20:05 & 2994 & 131.85 & 34.20 & 1.69 & 120.622 & -40.611 & 76.690 & 0.012 \\nl\n290594 & 03:17:02 & 2998 & 163.64 & -25.33 & 3.45 & 120.834 & -40.382 & 35.002 & 0.334 \\nl\n290594 & 16:20:36 & 3002 & 66.19 & -23.75 & 3.30 & 120.889 & -40.330 & 47.443 & 0.021 \\nl\n290594 & 21:18:56 & 3003 & 214.84 & 58.93 & 0.46 & 300.908 & 40.314 & 56.125 & 0.083 \\nl\n090694 & 16:29:06 & 3024 & 284.14 & 23.34 & 1.01 & 302.269 & 39.513 & 18.866 & 0.118 \\nl\n190694 & 21:31:20 & 3035 & 299.05 & -29.89 & 0.34 & 303.996 & 39.141 & 69.575 & 0.026 \\nl\n220694 & 13:18:16 & 3039 & 299.58 & -31.24 & 0.82 & 304.505 & 39.108 & 72.495 & 0.012 \\nl\n230694 & 18:46:23 & 3042 & 108.37 & 75.78 & 0.46 & 304.747 & 39.101 & 65.666 & 0.013 \\nl\n010794 & 21:46:38 & 3055 & 145.18 & -6.43 & 0.81 & 126.479 & -39.199 & 37.139 & 0.157 \\nl\n020794 & 09:28:11 & 3056 & 29.73 & -23.53 & 0.57 & 126.586 & -39.214 & 78.876 & 0.116 \\nl\n030794 & 04:40:46 & 3057 & 131.50 & 27.39 & 1.25 & 126.771 & -39.239 & 67.508 & 0.005 \\nl\n040794 & 23:32:07 & 3060 & 212.49 & 47.25 & 0.64 & 307.181 & 39.305 & 67.688 & 0.082 \\nl\n060794 & 19:24:33 & 3062 & 265.11 & 26.40 & 1.86 & 307.614 & 39.384 & 36.694 & 0.287 \\nl\n080794 & 20:42:06 & 3067 & 301.58 & 24.66 & 1.02 & 308.112 & 39.490 & 17.427 & 0.022 \\nl\n140794 & 12:54:21 & 3075 & 333.58 & -42.10 & 1.59 & 309.555 & 39.862 & 85.495 & 0.067 \\nl\n170794 & 03:24:29 & 3087 & 109.80 & 12.93 & 0.71 & 130.246 & -40.076 & 54.328 & 0.006 \\nl\n280794 & 02:55:32 & 3099 & 42.57 & -49.08 & 0.76 & 133.354 & -41.280 & 60.269 & 0.045 \\nl\n280794 & 14:06:40 & 3100 & 291.22 & 4.45 & 1.29 & 313.491 & 41.344 & 44.189 & 0.181 \\nl\n280794 & 23:58:54 & 3101 & 85.49 & -39.87 & 1.44 & 133.620 & -41.397 & 36.263 & 0.066 \\nl\n060894 & 04:38:48 & 3108 & 276.21 & -31.33 & 10.69 & 316.177 & 42.651 & 86.962 & 0.079 \\nl\n060894 & 09:32:56 & 3109 & 242.38 & 13.65 & 1.22 & 316.242 & 42.686 & 69.105 & 0.126 \\nl\n100894 & 02:22:42 & 3115 & 212.01 & -17.47 & 0.53 & 137.468 & -43.347 & 67.696 & 0.009 \\nl\n120894 & 00:27:42 & 3119 & 340.49 & -46.12 & 2.23 & 138.120 & -43.715 & 86.692 & 0.263 \\nl\n170894 & 08:40:15 & 3128 & 280.84 & 3.47 & 0.40 & 319.994 & 44.823 & 53.907 & 0.010 \\nl\n210894 & 21:51:27 & 3131 & 294.69 & 43.48 & 1.77 & 321.670 & 45.864 & 16.142 & 0.191 \\nl\n260894 & 20:55:20 & 3138 & 93.59 & -34.63 & 0.45 & 143.593 & -47.103 & 37.845 & 0.013 \\nl\n300894 & 08:59:56 & 3143 & 193.66 & 25.66 & 1.58 & 145.011 & -48.048 & 89.313 & 0.042 \\nl\n100994 & 19:31:24 & 3163 & 332.01 & -34.11 & 0.94 & 330.099 & 51.525 & 87.032 & 0.065 \\nl\n150994 & 06:52:19 & 3168 & 219.42 & 47.70 & 4.53 & 332.318 & 53.052 & 62.836 & 0.393 \\nl\\nl\n\n\\enddata\n\\end{deluxetable}\n\\clearpage\n\n\n\\section{Acknowledgments}\n\nSupport for the \\it Ulysses \\rm GRB experiment is provided by JPL Contract 958056.  Joint\nanalysis of \\it Ulysses \\rm and BATSE data is supported by NASA Grant NAG 5-1560.  CK acknowledges\nsupport from NASA Grant NAG5-2560.\n\n\\clearpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\\label{sec1}\n\nFunctional data analysis has received growing attention in recent decades, owing to its great flexibility and widespread application in complex data, refer to a comprehensive introduction in \\cite{ramsaysilverman2005}. Functional regression models feature prominently in functional data analysis literature, see \\cite{morris2015}. A large amount of work has been devoted to regression models with functional predictors, of which the most widely used are functional linear models (FLM). In FLM, a scalar response is associated with the inner product of a functional predictor and an unknown coefficient function, refer to \\cite{caihall2006, caiyuan2012, cardotetal1999, cardotetal2003, yaoetal2005a}.\nFunctional data can be viewed as elements from a functional space such as Hilbert space and reproducing kernel Hilbert space (RKHS). Therefore, dimension reduction is required to address the infinite dimensionality issue in functional data analysis. The popular strategy is to project the functional data into a low-rank functional subspace and take their projections as predictors in regression models. One of the most well-studied dimension reduction tool for functional data is functional principal component analysis (FPCA), discussed in \\cite{ricesilverman1991, yaoetal2005b, halletal2006}. Denote $X(t)$ a random function of $L^2(\\mathcal{T})$, $t\\in\\mathcal{T}$, with mean function $\\nu(t)$ and covariance function $\\mathcal{C}(s,t) = cov\\{X(s),X(t)\\}$. Classical FPCA take eigen-decomposition of the corresponding covariance operator as $(\\mathcal{C} \\psi_k)(t) = \\lambda_k \\psi_k(t)$, $k=1,2,\\ldots$, where $\\lambda_1 \\geq \\lambda_2 \\geq \\cdots$ are eigenvalues and $\\{\\psi_1(t),\\psi_2(t),\\ldots\\}$ is a set of eigenfunctions. Thus, $X(t)$ has the Karhunen-Lo\\`eve expansion\n\\[\nX(t) = \\nu(t) + \\sum_{k=1}^{\\infty}\\zeta_k\\psi_k(t),\n\\]\nwhere $\\zeta_k = \\int_{\\mathcal{T}}(X(t)-\\nu(t))\\psi_k(t)dt$ represents the score associate with the $k$-th eigenfunction, which is called functional principal component (FPC) score.\nAnd researchers use FPC scores associated with leading eigenfunctions as predictors in regression models to specify the effect of functional predictor.\n\nAlthough widely used, linear models can be restrictive in terms of general applications and many researchers have investigated nonlinear functional regression models, such as \\cite{james2002, mullerstadtmuller2005, crainiceanuetal2009, lietal2010, mulleryao2008}\nSome researchers \\cite{sangetal2018, wongetal2019} incorporated the effects of both the trajectories and scalar covariates on the prediction of the response.\nIn these models, the effect of functional predictors is represented by its transformed FPC scores whereas scalar predictors are modeled linearly. The estimation for such models typically truncates the nonparametric part to several leading FPCs,\nsee \\cite{mulleryao2008, zhuetal2014} also.\nThe estimation procedure above can be seen as a method for model selection because it results in a parsimonious model by truncating or imposing regularized penalties. Thus, model uncertainty arises from deciding which components are retained in candidate models.\n\nThis study considers the partially linear functional score (PLFS) model which describe the connection of a functional predictor and scalar predictors to a scalar response variable of interest, while follows a different estimation strategy namely model averaging. Recall that model selection methods aim to pick out one best model among a set of candidate models, and in this regard, various model selection criteria have been studied, such as Akaike information criterion by \\cite{akaike1973} and Bayesian information criterion in \\cite{schwarz1978}.\nModel averaging, as an alternative to model selection, combines all candidate models by assigning weights to different models to address model uncertainty. Bayesian model averaging has been a popular approach. And Hoeting et~al. (1999) provides a thorough overview of this direction \\cite{hoetingetal1999}.\nA rapidly growing body of literature with the frequentist paradigm for model averaging has been developed, like \\cite{yang2001, yang2003, hansen2007, liangetal2011, hansenracine2012}. The choice of weight plays a fundamental and crucial role in model averaging because it determines the performance of the resulting estimator. Information criterion-based weighting was advocated by \\cite{bucklandetal1997, hjortclaeskens2003}, which suggested taking weights based on AIC, BIC, or focused information criterion scores of candidate models. Hansen and Racine (2012) proposed jackknife model averaging \\cite{hansenracine2012} and similar weighting procedures based on cross-validation were developed by \\cite{chenghansen2015, gaoetal2016}. Both jackknife and cross-validation model averaging may become computationally complex when processing a large sample.\nHansen (2007) proposed Mallows criterion, which suggests weights that minimize this criterion, and established the asymptotic optimality for the model averaging estimator \\cite{hansen2007}. Following their work, corresponding Mallows-type\ncriterions for weight selection in linear mixed-effects model, partially linear model (PLM), and varying-coefficient PLM were established in \\cite{zhangetal2014, zhangwang2019, zhuetal2019}, respectively.\n\nTo our best knowledge, the literature on functional data contains few works in which the technique of model averaging is applied to regression models. For example, Zhu et~al. (2018) proposed optimal model averaging for partially linear FLM based on Mallows-type criterion \\cite{zhuetal2018}. Zhang et~al. (2018), Zhang and Zou (2020) developed a cross-validation model-averaging estimator based on FLM and generalized FLM, respectively \\cite{zhangetal2018, zhangzou2020}. In this study, we investigated Mallows-type model averaging for PLFS model. As mentioned above, because of the intrinsically infinite-dimensionality of functional data, a dimension reduction procedure is required, and therefore the components retained for scalar predictors and FPC scores could be expected to have an impact on prediction performance. Because model selection methods pose a risk of selecting an inferior model, we take advantage of model averaging method. This method assigns model weights such that Mallows-type criterion associated with the squared error loss is minimized.\n\nOur work differs from that of \\cite{zhangwang2019}, which considered optimal model averaging for PLM. As we use FPC scores to represent the effect of functional predictor, which are unobservable and needed to be estimated first, this situation is more complicated than that of the ordinary PLM. The theoretical derivation of asymptotic optimality for PLFS model is quite different from the previous work owing to the estimated FPC scores. Furthermore, the model uncertainty associated with PLM results from both the choice of covariates and the decision to which part, parametric or nonparametric, the covariate should enter. However, for PLFS model, the uncertainty mainly arises from deciding which scalar covariates and FPC scores should be included in the list of candidates because of the inherent division between scalar predictors and the functional predictor. Besides, this study is also different from \\cite{zhuetal2018} in that we handles nonparametric effect whereas they only deal with linear effects.\n\nThe remainder of this paper is organized as follows. Section 2 presents model setup and model averaging estimator. The asymptotic optimality of the model averaging estimator is also established in Section 2. Section 3 compares the finite sample performance of the proposed estimator with several information criterion-based model selection and averaging estimators. The proposed procedure is subsequently applied to real data in Section 4. Section 5 concludes our work with a discussion. All proofs are given in Appendix.\n\n\n\n\n\\section{Methodology}\\label{sec2}\n\n\n\\subsection{Model and estimator}\\label{sec21}\n\nLet $Y$ be a scalar response variable associated with a scalar predictor vector ${\\mathbf Z}$ and a functional predictor $X(t)$, $t\\in \\mathcal{T}$, and let $\\{Y_i, {\\mathbf Z}_i, X_i(\\cdot)\\}_{i=1}^n$ be independent identically distributed (iid) copies of $\\{Y, {\\mathbf Z}, X(\\cdot)\\}$. The relationship between the response and predictors is modeled as\n\\[\nY_i = m({\\mathbf Z}_i, X_i) + \\varepsilon_i,\n\\]\nwhere $\\varepsilon_i$ are random errors.\nDirect modeling $m(\\cdot)$ is adversely affected by the ``curse of dimensionality''. Thus, many popular alternatives are developed, such as FLM and functional additive model \\cite{mulleryao2008}, modelling the effect of $X(t)$ through its FPC scores based on FPCA. We follow a similar strategy and simplify the modelling as follows.\n\nDenote by $\\bzeta_{i}=(\\zeta_{i1}, \\zeta_{i2},\\ldots)$ the sequence of FPC scores of $X_i(t)$ associated with eigenvalues $\\{\\lambda_1,\\lambda_2,\\ldots\\}$ satisfying $\\lambda_1\\geq \\lambda_2 \\geq \\cdots > 0$. In addition, $\\bxi_i=(\\xi_{i1},\\xi_{i2},\\ldots)^T$ represents the sequence of transformed FPC scores, i.e. $\\xi_{ik}=\\Phi\\big(\\lambda_k^{-1\/2}\\zeta_{ik}\\big)$, where $\\Phi(\\cdot)$ is a continuously differentiable map from $\\mathbb{R}$ to $[0,1]$. The transformed FPC scores $\\bxi_{i}$ can help to avoid possible scale issues. For simplicity, we take $\\Phi(\\cdot)$ as a suitable cumulative distribution function (CDF), such as standard Gaussian CDF. If $\\zeta_{ik}$ approximately follows Gaussian distribution, $\\xi_{ik}$ will almost be uniform in $[0,1]$.\nNow consider our PLFS model\n\\begin{equation}\\label{model}\n\tY_i = \\mu_i + \\varepsilon_i = {\\mathbf Z}_i^T\\btheta + {\\mathbf f}(\\bxi_i) + \\varepsilon_i,\n\\end{equation}\nwhere $\\varepsilon=(\\varepsilon_1,\\ldots,\\varepsilon_n)^T$ is random error with conditional mean 0 and variance matrix $\\boldsymbol{\\Omega}=\\mbox{diag}(\\sigma_1^2,\\ldots,\\sigma_n^2)$.\n\n\nWe use $M$ candidate models to approximate the true PLFS model, where $M$ is allowed to diverge to infinity as $n\\to \\infty$. The $m$-th candidate PLFS model includes $p_m$ regressors in ${\\mathbf Z}_i$ and $q_m$ regressors in $\\bxi_i$ ($m=1,\\ldots,M$),\n\\[\n\\begin{aligned}\n\tY_i &= \\mu_{(m),i} + \\varepsilon_{(m),i} \\\\\n\t&= {\\mathbf Z}_{(m),i}^T\\btheta_{(m)} + {\\mathbf f}_{(m)}\\big(\\bxi_{(m),i}\\big) + \\varepsilon_{(m),i},\n\\end{aligned}\n\\]\nwhere ${\\mathbf Z}_{(m),i}$ is a $p_m\\times 1$ vector, $\\btheta_{(m)}$ is the corresponding unknown coefficients, $\\bxi_{(m),i}$ is a $q_m\\times 1$ vector, ${\\mathbf f}_{(m)}$ is an unknown function from $[0,1]^{q_m}$ to $\\mathbb{R}$. And $\\varepsilon_{(m),i}$ contains the approximation error of the $m$-th candidate and random error.\n\nThe kernel smoothing method \\cite{speckman1988} is used in estimation.\nDenote $\\mathcal{K}_{h_m}(\\cdot) = \\prod_{l=1}^{q_m} k_{h_{m,l}}(\\cdot\/h_{m,l})$ a product kernel function, where $k_{h_{m,l}}$ is a univariate kernel function and $h_{m,l}$ is the scalar bandwidth. We take $h_{m,l}=h_m$ for simplicity, $l=1,\\ldots,q_m$. Furthermore, let ${\\mathbf K}_{(m)} = \\big(K_{(m),ij}\\big)$ be a $n\\times n$ smoother matrix with\n\\[\nK_{(m),ij} = \\frac{\\mathcal{K}_{h_m}\\big(\\bxi_{(m),i}-\\bxi_{(m),j}\\big)}{\\sum_{j'=1}^n\\mathcal{K}_{h_m}\\big(\\bxi_{(m),i}-\\bxi_{(m),j'}\\big)}.\n\\]\nThen, the suggested kernel smoothing estimators of $\\btheta_{(m)}$ and ${\\mathbf f}_{(m)}\\big(\\bxi_{(m)}\\big)$ are as follows,\n\\[\n\\begin{aligned}\n\t& \\widetilde{\\btheta}_{(m)} = \\big(\\widetilde{{\\mathbf Z}}_{(m)}^T\\widetilde{{\\mathbf Z}}_{(m)}\\big)^{-1}\\widetilde{{\\mathbf Z}}_{(m)}^T({\\mathbf I}-{\\mathbf K}_{(m)})Y, \\\\\n\t& \\widetilde{{\\mathbf f}}_{(m)}(\\bxi_{(m)}) = {\\mathbf K}_{(m)}(Y-{\\mathbf Z}_{(m)}\\widetilde{\\btheta}_{(m)}),\n\\end{aligned}\n\\]\nwhere $\\widetilde{{\\mathbf Z}}_{(m)} = ({\\mathbf I}-{\\mathbf K}_{(m)}){\\mathbf Z}_{(m)}$. Obviously, $\\widetilde{\\btheta}_{(m)}$ is actually a least square estimate and $\\widetilde{{\\mathbf f}}_{(m)}$ is a Nadaraya-Watson estimator.\nTherefore, the estimator of $\\mu$ under the $m$-th candidate is given by\n\\[\n\\begin{aligned}\n\t\\widetilde{\\mu}_{(m)} &= {\\mathbf Z}_{(m)}\\widetilde{\\btheta}_{(m)} + \\widetilde{{\\mathbf f}}_{(m)}(\\bxi_{(m)}) \\\\\n\t&= \\widetilde{{\\mathbf Z}}_{(m)}\\big(\\widetilde{{\\mathbf Z}}_{(m)}^T\\widetilde{{\\mathbf Z}}_{(m)}\\big)^{-1}\\widetilde{{\\mathbf Z}}_{(m)}^T({\\mathbf I}-{\\mathbf K}_{(m)})Y + {\\mathbf K}_{(m)}Y \\\\\n\t&\\equiv {\\mathbf P}_{(m)}Y.\n\\end{aligned}\n\\]\nLet $\\widetilde{{\\mathbf P}}_{(m)} \\equiv \\widetilde{{\\mathbf Z}}_{(m)}\\big(\\widetilde{{\\mathbf Z}}_{(m)}^T\\widetilde{{\\mathbf Z}}_{(m)}\\big)^{-1}\\widetilde{{\\mathbf Z}}_{(m)}^T$ which is idempotent, and ${\\mathbf P}_{(m)} \\equiv \\widetilde{{\\mathbf P}}_{(m)}\\big({\\mathbf I}-{\\mathbf K}_{(m)}\\big)+{\\mathbf K}_{(m)}$.\n\n\n\\subsection{Weight choice criterion}\\label{sec22}\n\nLet $\\boldsymbol{\\omega}=(\\omega_1, \\ldots, \\omega_M)^T$ be a weight vector in the unit simplex of $\\mathbb{R}^M$,\n\\[\n\\mathcal{H}_n = \\Big\\{\\boldsymbol{\\omega}\\in [0,1]^M: \\sum_{m=1}^M \\omega_m = 1\\Big\\}.\n\\]\nThen the model averaging estimator of $\\mu$ follows as\n\\[\n\\widetilde{\\mu}(\\boldsymbol{\\omega}) = \\sum_{m=1}^M \\omega_m\\widetilde{\\mu}_{(m)} = \\sum_{m=1}^M \\omega_m{\\mathbf P}_{(m)}Y = {\\mathbf P}(\\boldsymbol{\\omega})Y,\n\\]\nwhere ${\\mathbf P}(\\boldsymbol{\\omega}) = \\sum_{m=1}^M \\omega_m{\\mathbf P}_{(m)}$. Define the square error loss function and corresponding conditional risk function as\n\\[\nL_n(\\boldsymbol{\\omega}) = \\|\\widetilde{\\mu}(\\boldsymbol{\\omega})-\\mu\\|^2 = \\|{\\mathbf P}(\\boldsymbol{\\omega})Y-\\mu\\|^2\n\\]\nand\n\\[\n\\begin{aligned}\n\tR_n(\\boldsymbol{\\omega}) &= \\mathbb{E}(L_n(\\boldsymbol{\\omega})|{\\mathbf Z}, X) \\\\\n\t&= \\|({\\mathbf P}(\\boldsymbol{\\omega})-{\\mathbf I})\\mu\\|^2 + tr({\\mathbf P}^T(\\boldsymbol{\\omega}){\\mathbf P}(\\boldsymbol{\\omega})\\boldsymbol{\\Omega}),\n\\end{aligned}\n\\]\nrespectively. We may select the optimal weights based on the following Mallows-type criterion\n\\[\nC_n(\\boldsymbol{\\omega}) = \\|Y-\\widetilde{\\mu}(\\boldsymbol{\\omega})\\|^2 + 2tr({\\mathbf P}(\\boldsymbol{\\omega})\\boldsymbol{\\Omega}).\n\\]\nIt is observed that $\\mathbb{E}(C_n(\\boldsymbol{\\omega})|{\\mathbf Z}, X) = R_n(\\boldsymbol{\\omega}) + tr(\\boldsymbol{\\Omega})$. Thus, $C_n(\\boldsymbol{\\omega})$ is an unbiased estimator of the expected in-sample squared error loss plus a constant, which is similar to the Mallow's criterion proposed in \\cite{hansen2007}. Because $tr(\\boldsymbol{\\Omega})$ is unrelated to $\\boldsymbol{\\omega}$, the optimal weights can be obtained by minimizing $C_n(\\boldsymbol{\\omega})$ if $\\boldsymbol{\\Omega}$ is known.\n\n\nHowever, $\\bzeta$ and $\\bxi$ are unobservable, so the estimation procedure above cannot be implemented directly. For the sake of practical applicability, we replace the original $\\bxi_{(m)}$ with its estimator $\\widehat{\\bxi}_{(m)}$, which is common practice. That is, we first estimate the FPC scores using the previously proposed FPCA method as. That is, suppose the discrete noisy measurements of $X_i(t)$ are available,\n\\[\nX_{ij} = X_i(t_{ij}) + e_{ij},\\:i=1,\\ldots,n,\\:j=1,\\ldots,N_i,\n\\]\nwhere $e_{ij}$'s are independent measurement errors with mean 0 and variance $\\sigma_e^2$. We focus on the densely observed trajectories such that $X_i(t)$ can be effectively recovered from $\\{(t_{ij}, X_{ij}):j=1,\\ldots,N_i\\}$ by a smoother operator \\cite{kongetal2016,wongetal2019}. The recovered function is denoted by $\\widetilde{X}_i(t)$. Then the mean and covariance functions of $X(t)$ can be estimated by\n\\[\n\\begin{aligned}\n\t\\widehat{\\nu}(t) &= \\frac{1}{n}\\sum_{i=1}^n\\widetilde{X}_i(t), \\\\\n\t\\widehat{\\mathcal{C}}(s,t) &= \\frac{1}{n}\\sum_{i=1}^n\\Big(\\widetilde{X}_i(s)-\\widehat{\\nu}(s)\\Big)\\Big(\\widetilde{X}_i(t)-\\widehat{\\nu}(t)\\Big)^T.\n\\end{aligned}\n\\]\nThe spectral decomposition $\\widehat{\\mathcal{C}}(s,t) = \\sum_{k=1}^{n-1} \\widehat{\\lambda}_k\\widehat{\\psi}_k(s)\\widehat{\\psi}_k(t)$ yields sample eigenvalues $\\{\\widehat{\\lambda}_k\\}$ and eigenfunctions $\\{\\widehat{\\psi}_k\\}$. The estimates for FPC scores are subsequently obtained by\n\\[\n\\begin{aligned}\n\t\\widehat{\\zeta}_{ik} &= \\int_{\\mathcal{T}}(\\widetilde{X}_i(t)-\\widehat{\\nu}(t))\\widehat{\\psi}_{k}(t)dt,\\\\\n\t\\widehat{\\xi}_{ik} &= \\Phi\\big(\\widehat{\\lambda}_k^{-1\/2}\\widehat{\\zeta}_{ik}\\big).\n\\end{aligned}\n\\]\nOnce we get $\\widehat{\\bxi}_{(m)}$, the original quantities listed above have their substitutes in practice, as shown below.\n\n\nThe smoother matrix is now denoted as $\\widehat{{\\mathbf K}}_{(m)}$ with $i,j$-element\n\\[\n\\widehat{K}_{(m),ij} = \\frac{\\mathcal{K}_{h_m}\\big(\\widehat{\\bxi}_{(m),i}-\\widehat{\\bxi}_{(m),j}\\big)}{\\sum_{j'=1}^n\\mathcal{K}_{h_m}\\big(\\widehat{\\bxi}_{(m),i}-\\widehat{\\bxi}_{(m),j'}\\big)}.\n\\]\nThe final kernel smoothing estimators of $\\btheta_{(m)}$ and ${\\mathbf f}_{(m)}$ are given by\n\\[\n\\begin{aligned}\n\t& \\widehat{\\btheta}_{(m)} = \\big(\\widehat{{\\mathbf Z}}_{(m)}^T\\widehat{{\\mathbf Z}}_{(m)}\\big)^{-1}\\widehat{{\\mathbf Z}}_{(m)}^T({\\mathbf I}-\\widehat{{\\mathbf K}}_{(m)})Y, \\\\\n\t& \\widehat{{\\mathbf f}}_{(m)}(\\widehat{\\bxi}_{(m)}) = \\widehat{{\\mathbf K}}_{(m)}\\big(Y-{\\mathbf Z}_{(m)}\\widehat{\\btheta}_{(m)}\\big),\n\\end{aligned}\n\\]\nwhere $\\widehat{{\\mathbf Z}}_{(m)} = \\big({\\mathbf I} - \\widehat{{\\mathbf K}}_{(m)}\\big){\\mathbf Z}_{(m)}$. Besides, the $m$-th estimator and the model averaging estimator of $\\mu$ are\n\\[\n\\begin{aligned}\n\t\\widehat{\\mu}_{(m)} &= {\\mathbf Z}_{(m)}\\widehat{\\btheta}_{(m)} + \\widehat{{\\mathbf f}}_{(m)}(\\widehat{\\bxi}_{(m)}) \\\\\n\t&= \\widehat{{\\mathbf Z}}_{(m)}\\big(\\widehat{{\\mathbf Z}}_{(m)}^T\\widehat{{\\mathbf Z}}_{(m)}\\big)^{-1}\\widehat{{\\mathbf Z}}_{(m)}^T\\big({\\mathbf I}-\\widehat{{\\mathbf K}}_{(m)}\\big)Y + \\widehat{{\\mathbf K}}_{(m)}Y \\\\\n\t&\\equiv \\widehat{{\\mathbf P}}_{(m)}Y, \\\\\n\t\\widehat{\\mu}(\\boldsymbol{\\omega}) &= \\sum_{m=1}^M \\omega_m\\widehat{\\mu}_{(m)} = \\sum_{m=1}^M \\omega_m\\widehat{{\\mathbf P}}_{(m)}Y = \\widehat{{\\mathbf P}}(\\boldsymbol{\\omega})Y,\n\\end{aligned}\n\\]\nwhere $\\widehat{{\\mathbf P}}(\\boldsymbol{\\omega}) = \\sum_{m=1}^M \\omega_m \\widehat{{\\mathbf P}}_{(m)}$.\n\nDenote $\\overline{{\\mathbf P}}_{(m)} \\equiv \\widehat{{\\mathbf Z}}_{(m)}\\big(\\widehat{{\\mathbf Z}}_{(m)}^T\\widehat{{\\mathbf Z}}_{(m)}\\big)^{-1}\\widehat{{\\mathbf Z}}_{(m)}^T$ which is still idempotent, and $\\widehat{{\\mathbf P}}_{(m)} \\equiv \\overline{{\\mathbf P}}_{(m)}\\big({\\mathbf I}-\\widehat{{\\mathbf K}}_{(m)}\\big)+\\widehat{{\\mathbf K}}_{(m)}$.\nThe modified loss, conditional risk, and Mallows-type criterion are transformed into\n\\[\n\\begin{aligned}\n\t\\widehat{L}_n(\\boldsymbol{\\omega}) &= \\|\\widehat{\\mu}(\\boldsymbol{\\omega}) - \\mu\\|^2 = \\|\\widehat{{\\mathbf P}}(\\boldsymbol{\\omega})Y - \\mu\\|^2, \\\\\n\t\\widehat{R}_n(\\boldsymbol{\\omega}) &= \\mathbb{E}(L_n(\\boldsymbol{\\omega})|{\\mathbf Z}, X), \\\\\n\t\\widehat{C}_n(\\boldsymbol{\\omega}) &= \\|Y - \\widehat{\\mu}(\\boldsymbol{\\omega})\\|^2 + 2tr(\\widehat{{\\mathbf P}}(\\boldsymbol{\\omega})\\boldsymbol{\\Omega}).\n\\end{aligned}\n\\]\n\n\nLet $\\widetilde{\\boldsymbol{\\omega}} = \\arg\\min_{\\boldsymbol{\\omega}\\in \\mathcal{H}_n} \\widehat{C}_n(\\boldsymbol{\\omega})$. However, the covariance matrix $\\boldsymbol{\\Omega}$ is unknown and the criterion $\\widehat{C}_n(\\boldsymbol{\\omega})$ is therefore still computationally infeasible. Hence, we should estimate $\\boldsymbol{\\Omega}$ to obtain a feasible criterion. Following Hansen (2007) \\cite{hansen2007}, we estimate $\\boldsymbol{\\Omega}$ based on the largest candidate model indexed by $M^{*} = \\arg\\max_{1\\leq m\\leq M}(p_m+q_m)$, leading to an estimator\n\\begin{equation}\\label{whOmg}\n\t\\widehat{\\boldsymbol{\\Omega}} = \\mbox{diag}\\big(\\hat{\\epsilon}_{(M^*),1}^2, \\ldots, \\hat{\\epsilon}_{(M^*),n}^2\\big),\n\\end{equation}\nwhere $\\big(\\hat{\\epsilon}_{(M^*),1}, \\ldots, \\hat{\\epsilon}_{(M^*),n}\\big)^T = Y - \\widehat{\\mu}_{(M^*)}$.\n\nWhen $\\boldsymbol{\\Omega}$ is replaced by $\\widehat{\\boldsymbol{\\Omega}}$, we select the optimal weights by\n\\begin{equation}\\label{whCn}\n\t\\begin{aligned}\n\t\t\\widehat{\\boldsymbol{\\omega}} &= \\arg\\min_{\\boldsymbol{\\omega}} \\widehat{C}_n(\\boldsymbol{\\omega})|_{\\boldsymbol{\\Omega} = \\widehat{\\boldsymbol{\\Omega}}} \\\\\n\t\t&= \\arg\\min_{\\boldsymbol{\\omega}}\\|Y-\\widehat{\\mu}(\\boldsymbol{\\omega})\\|^2 + 2tr(\\widehat{{\\mathbf P}}(\\boldsymbol{\\omega})\\widehat{\\boldsymbol{\\Omega}}),\n\t\\end{aligned}\n\\end{equation}\nwhich can be treated as a feasible counterpart of $\\widehat{C}_n(\\boldsymbol{\\omega})$.\nLet ${\\mathbf H} = \\big(Y-\\widehat{\\mu}_{(1)},\\ldots,Y-\\widehat{\\mu}_{(M)}\\big)$ and $b = \\big(tr(\\widehat{{\\mathbf P}}_{(1)}\\widehat{\\boldsymbol{\\Omega}}),\\ldots,tr(\\widehat{{\\mathbf P}}_{(M)}\\widehat{\\boldsymbol{\\Omega}})\\big)^T$. It is clear that (\\ref{whCn}) is a standard quadratic programming problem of the form\n\\[\n\\begin{aligned}\n\t& \\min_{\\boldsymbol{\\omega}} \\widehat{C}_n(\\boldsymbol{\\omega})|_{\\boldsymbol{\\Omega} = \\widehat{\\boldsymbol{\\Omega}}} = \\min_{\\boldsymbol{\\omega}} \\boldsymbol{\\omega}^T{\\mathbf H}^T{\\mathbf H}\\boldsymbol{\\omega} + 2\\boldsymbol{\\omega}^T b \\\\\n\t& \\quad \\mbox{subject to}\\quad \\mathbf{1}^T\\boldsymbol{\\omega}=1 \\: \\mbox{and}\\: \\boldsymbol{\\omega}\\geq 0,\n\\end{aligned}\n\\]\nwhere $\\mathbf{1}$ is a vector with all entries equal to 1. The problem can be efficiently optimized by the R package {\\it quadprog}\\footnote{S original by Berwin A. \\& Turlach R port by Andreas W. (2019). quadprog: Functions to Solve Quadratic Programming Problems. R package version 1.5-7. https:\/\/CRAN.R-project.org\/package=quadprog.}.\n\n\n\n\\subsection{Asymptotic optimality}\\label{sec3}\n\nDefine $\\eta_n = \\inf_{\\boldsymbol{\\omega}} R_n(\\boldsymbol{\\omega})$ and let $\\lambda_{\\max}(\\cdot)$ denote the largest singular value of a matrix. Let $\\boldsymbol{\\omega}_m^0$ be a weight vector in which the $m$-th component is one and the others are zero. Let $\\widetilde{p} = \\max_m p_m$, $\\widetilde{q} = \\max_m q_m$, and $h = \\min_m h_m$.\n\nThe following regularity conditions are required for the model averaging estimator to achieve asymptotic optimality.\n\n(C1).\n\\[\n\\begin{aligned}\n\t& c_{\\lambda}^{-1}k^{-\\alpha}\\leq \\lambda_k \\leq C_{\\lambda}k^{-\\alpha}, \\\\\n\t& \\lambda_k - \\lambda_{k+1} \\geq C_{\\lambda}^{-1}k^{-1-\\alpha}, \\quad k=1,2,\\ldots\n\\end{aligned}\n\\]\nAssume that $\\alpha>1$ to ensure $\\sum_{k=1}^{\\infty} \\lambda_k < \\infty$.\n\n(C2). $\\mathbb{E}(\\|X\\|^4)<\\infty$ and there exists a constant $C_{\\zeta}>0$ such that $\\mathbb{E}(\\zeta_k^2\\zeta_j^2)\\leq C_{\\xi}\\lambda_k\\lambda_j$ and $\\mathbb{E}(\\zeta_k^2-\\lambda_k)^2<C_{\\zeta}\\lambda_k^2$, $\\forall k\\neq j$.\n\n\nCondition (C1) assumes that the eigenvalues decay at a polynomial rate, which is a relatively slow rate and allows $X(t)$ to be flexibly modeled as a $L^2$ process. The second part of Condition (C1) requires the spacings among eigenvalues not to be too small to ensure the identifiability and consistency of sample eigenvalues and eigenfunctions. Condition (C1) is widely used in the FLM literature \\cite{caihall2006,caiyuan2012}. Condition (C2) is a weak moment restriction on $X(t)$, which is satisfied if $X(t)$ is a Gaussian process. Note that the fourth-order moment condition is commonly used when a good convergence property of the second-order moment is desired. As we use FPC scores to model the effect of $X(t)$ in the proposed method, it is reasonable to use Condition (C2) to ensure the estimated FPC scores are effective.\n\n\n(C3). The kernel function $k(\\cdot)$ is a bounded symmetrical density with compact support and continuous and bounded first derivative function.\n\n(C4). $\\max_i \\sum_{j=1}^n |K_{(m),ij}| = O(1)$ and $\\max_j \\sum_{i=1}^n |K_{(m),ij}| = O(1)$ uniformly for $m = 1, \\ldots, M$, a.s.\n\n\nConditions (C3) and (C4) place certain restrictions upon the kernel method.\nCondition (C3) is a common assumption on kernel functions. Condition (C4) bounds the elements of the smoother matrix, which has been discussed in \\cite{speckman1988,zhangwang2019}. The smoother matrix constructed by Epanechnikov kernel in simulation study naturally satisfies this condition.\n\n\n(C5). For some integer $G\\geq 1$, $\\max_i \\mathbb{E}(\\varepsilon_i^{4G}|{\\mathbf Z}_i, X_i)<\\infty$ for all $i=1,\\ldots,n$, a.s.\n\n(C6). $M\\eta_n^{-2G}\\sum_{m=1}^M \\big(R_n(\\boldsymbol{\\omega}_m^0)\\big)^G = o_p(1)$.\n\n(C7). $\\lambda_{\\min}\\big({\\mathbf K}_{(m)}\\big)\\geq c_{K} >0$, where $c_{K}$ is a constant, $m=1,\\ldots, M$.\n\n(C8). $\\widetilde{q}=O(n^{1\/(2+2\\alpha)})$ where $\\alpha$ relates to Condition (C1). $n^{-1\/2}\\widetilde{q}=o_p(1)$, $n^{1\/2}\\eta_n^{-1}\\widetilde{q} = o_p(1)$, $\\eta_n^{-1}\\widetilde{q}^2 = o_p(1)$.\n\n\n(C9). $\\|\\mu\\|^2\/n = O(1)$, a.s.\n\n\nConditions (C5), (C6) and (C9) are standard conditions for model averaging in the literature. Condition (C5) constrains the conditional moment of random errors, see \\cite{hansen2007, zhangetal2014} also. Condition (C6) is commonly used to prove the optimality of model averaging under the scenario that all candidate models are misspecified, which requires $\\eta_n$ goes to infinity and constrains rates of the number of candidate models $M$ and the risk of each candidate model; see \\cite{wanetal2010, zhangwang2019, zhuetal2019}, among others. Condition (C9) concerns the sum of $\\mu_i^2$ and is commonly used in the context of linear regression \\cite{liangetal2011}. Condition (C7) is a technical condition in quantifying the order of $\\lambda_{\\max}\\big({\\mathbf P}_{(m)} - \\widehat{{\\mathbf P}}_{(m)}\\big)$, which requires ${\\mathbf K}_{(m)}$ not being ill-conditioned. Condition (C8) constrains the growth rate of the number of FPC scores, which guarantees an effective estimation accuracy. And we show that under conditions (C7) and (C8), $n\\eta_n^{-1}\\max_{1\\leq m\\leq M} \\lambda_{\\max}\\big({\\mathbf P}_{(m)} - \\widehat{{\\mathbf P}}_{(m)}\\big)=o_p(1)$ holds, which is commonly assumed in the literature \\cite{zhangetal2014, zhangyu2018}.\n\n\nThe theorem provides the asymptotic optimality of the model averaging estimator when $\\boldsymbol{\\Omega}$ is known.\n\\begin{theorem}\\label{th1}\n\tUnder Conditions (C1)--(C9), it holds that\n\t\\begin{equation}\\label{A1}\n\t\t\\frac{L_n(\\widetilde{\\boldsymbol{\\omega}})}{\\inf_{\\boldsymbol{\\omega}\\in \\mathcal{H}_n}L_n(\\boldsymbol{\\omega})}\\to 1\n\t\\end{equation}\n\tin probability as $n\\to \\infty$.\n\\end{theorem}\n\nTheorem \\ref{th1} illustrates the asymptotic optimality of $\\widetilde{\\boldsymbol{\\omega}}$ in the sense that the squared loss based on the weight vector $\\widetilde{\\boldsymbol{\\omega}}$ is asymptotically identical to that obtained using the infeasible optimal weight vector if $\\boldsymbol{\\Omega}$ is known. The proof of Theorem \\ref{th1} is shown in the Appendix.\n\n\nFollowing \\cite{liuokui2013}, we process $tr(\\widehat{{\\mathbf P}}(\\boldsymbol{\\omega})\\boldsymbol{\\Omega})$ as one entity rather than considering $\\boldsymbol{\\Omega}$ in isolation, and estimate it by $\\sum_{i=1}^n \\hat{\\epsilon}_{(M^*),i}^2\\rho_{ii}(\\boldsymbol{\\omega})$ where $\\rho_{ii}(\\boldsymbol{\\omega})$ is the $i$-th diagonal element of $\\widehat{{\\mathbf P}}(\\boldsymbol{\\omega})$. Denote $\\rho_{ii}^{(m)}$ as the $i$-th diagonal element of $\\widehat{{\\mathbf P}}_{(m)}$. When $\\boldsymbol{\\Omega}$ is replaced by its estimate $\\widehat{\\boldsymbol{\\Omega}}$ given in $(\\ref{whOmg})$, provided that the following additional conditions are satisfied, it can be shown that the model averaging estimator based on $\\widehat{\\boldsymbol{\\omega}}$ shares same asymptotic optimality as $\\widetilde{\\boldsymbol{\\omega}}$ in Theorem \\ref{th1}.\n\n\n(C10). There exists a constant $c$ such that $\\max_{i}\\rho_{ii}^{(m)}\\leq cn^{-1}|tr\\big(\\widehat{{\\mathbf P}}_{(m)}\\big)|$, $\\forall m = 1, \\ldots, M$.\n\n(C11). $tr\\big({\\mathbf K}_{(m)}\\big) = O(h^{-\\widetilde{q}})$ uniformly for $m\\in \\{1,\\ldots,M\\}$.\n\n(C12). $\\eta_n^{-1}\\widetilde{p} = o_p(1)$ and $\\eta_n^{-1}h^{-\\widetilde{q}} = o_p(1)$.\n\n\nCondition (C10) means that there should not be any dominant or strongly influential subjects as shown in \\cite{li1987} and \\cite{andrews1991}. Condition (C11) is similar to Condition (h) of \\cite{speckman1988} and Condition 4 of \\cite{zhangwang2019}. Condition (C12), similar to Condition (C.9) of \\cite{zhuetal2019} and Condition 3 of \\cite{zhangetal2018}, places additional restrictions on the growth rate of the number of scalar predictors and the number of FPC scores.\n\n\\begin{theorem}\\label{th2}\n\tUnder Conditions (C1)--(C12), we have that\n\t\\begin{equation}\\label{B1}\n\t\t\\frac{L_n(\\widehat{\\boldsymbol{\\omega}})}{\\inf_{\\boldsymbol{\\omega}\\in\\mathcal{H}_n} L_n(\\boldsymbol{\\omega})}\\to 1\n\t\\end{equation}\n\tin probability as $n\\to \\infty$.\n\\end{theorem}\n\nTheorem \\ref{th2} shows that Theorem \\ref{th1} remains valid when $\\boldsymbol{\\Omega}$ is replaced by $\\widehat{\\boldsymbol{\\Omega}}$. Thus, the practically feasible $\\widehat{\\boldsymbol{\\omega}}$ also enjoys the asymptotic optimality. The Appendix provides the detailed proof of Theorem \\ref{th2}.\n\n\n\n\n\n\\section{Simulation study}\\label{sec4}\n\nIn this section, we compare the finite sample performance of the proposed Mallows-type model averaging (MMA) estimator with several model selection and averaging estimators based on information criteria.\n\nThe data are generated from the following PLFS model,\n\\begin{equation}\\label{mod:1}\n\tY_i = \\mu_i + \\varepsilon_i = \\sum_{j=1}^{M_0}\\theta_j Z_{ij} + {\\mathbf f}(\\bxi_i) + \\varepsilon_i, \\: i = 1, \\ldots, n,\n\\end{equation}\nwhere $\\bxi_i$ is transformed FPC scores vector from $\\bzeta_i$ with $\\zeta_{ik}$ being generated independently from $N(0, \\lambda_k)$ and standard Gaussian CDF being the transformation $\\Phi(\\cdot)$. The following scenarios are considered.\n\n\n\\noindent\\textbf{Design 1.} $M_0 = 50$ and $\\theta_j = j^{-2\/3}$. ${\\mathbf Z}_i\\sim MN(0, \\Sigma)$, generated independently from the functional predictor $X_i(t)$, with the $a,b$-th element $\\Sigma_{ab}$ being $0.5^{|a-b|}$. $X_i(t)$ is obtained by\n\\[\nX_i(t) = \\sum_{k=1}^{40} \\zeta_{ik}\\psi_k(t),\\quad t\\in [0,1],\n\\]\nwhere $\\zeta_{ik}\\sim N(0, k^{-3\/2})$, $\\psi_k(t) = \\sqrt{2}\\sin(k\\pi t)$, $k = 1,\\ldots,40$. $\\varepsilon_i$'s are homoscedastic as $\\varepsilon_i\\sim N(0, \\eta^2)$. Varying $\\eta$ such that $R^2 = var(\\mu_i)\/ var(Y_i)$ varies between 0.1 and 0.9, where $var(\\mu_i)$ and $var(Y_i)$ are variances of $\\mu_i$ and $Y_i$ respectively. And\n\\[\n{\\mathbf f}(\\bxi) = \\exp\\Big\\{\\sum_{k=1}^{40}\\xi_{k}\/k\\Big\\}.\n\\]\n\n\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\\includegraphics[width=0.8\\textwidth]{nmse1.pdf}\n\t\\caption{ Normalized mean squared error (NMSE) comparison for Design 1. }\n\t\\label{nmse1}\n\\end{figure}\n\n\n\\noindent\\textbf{Design 2.} $M_0 = 50$ and $\\theta_j = j^{-1\/2}$. Consider ${\\mathbf Z}$ and $X(t)$ being correlated. Simulate $({\\mathbf Z}_i, \\zeta_{i1})\\sim MN(0, \\Sigma)$ where the $a,b$-th element $\\Sigma_{ab}=0.5^{|a-b|}$. The functional predictor $X_i(t)$ is obtained by\n\\[\nX_i(t) = \\sum_{k=1}^{20} \\zeta_{ik}\\psi_k(t),\\quad t\\in [0,10],\n\\]\nwhere $\\zeta_{ik}\\sim N(0, k^{-2})$, $k = 2,\\ldots,20$. $\\psi_k(t) = \\cos(k\\pi t\/5)\/\\sqrt{5}$, $k = 1,\\ldots,20$. $\\varepsilon_i$'s are heteroscedastic as $\\varepsilon_i\\sim N\\big(0, \\eta^2(u_{i}^2+0.01)\\big)$, where $u_i$ is $U[-1,1]$. Still varying $\\eta$ such that $R^2$ varies between 0.1 and 0.9. And\n\\[\n{\\mathbf f}(\\bxi) = \\xi_1\\xi_2 + \\xi_3^2 + \\sum_{k=4}^{20} \\frac{1}{k}\\Big(\\xi_k-\\frac{1}{2}\\Big).\n\\]\n\n\n\\noindent\\textbf{Design 3.} Design 3 is close to Design 1 except that ${\\mathbf Z}_i$ is correlated to $X_i(t)$ as Design 2, and random errors are heteroscedastic as $\\varepsilon_i\\sim N\\big(0, \\eta^2(Z_{i1}^2+0.01)\\big)$. Still varying $\\eta$ such that $R^2$ varies between 0.1 and 0.9.\n\n\nFor each design, $X(t)$ is observed at 100 equally-spaced grids on $\\mathcal{T}$ with measurement errors. Denote the $i$-th observation of $X$ at $t_{j}$ by $X_{ij} = X_i(t_j) + e_{ij}$, where measurement errors $e_{ij}$'s are independent $N(0,0.2)$ variables.\nThe sample size is set to $n=50$, $100$, $200$ and $400$. Consider three kinds of candidate model setting corresponding to each design as follows.\n\n\n\\noindent \\textbf{M15a}. For Design 1, a nested setting is considered, that is, containing the first $s$ components of ${\\mathbf Z}$ and $\\bxi$. A candidate model contains at least one of $\\{Z_1,\\ldots,Z_5\\}$ and at least one of $\\{\\xi_1,\\xi_2,\\xi_3\\}$, which leads to $M = 5\\cdot 3 = 15$ candidates.\n\n\n\\noindent \\textbf{M15b}. For Design 2, we only restrict the nested mode in nonparametric part. The parametric part contains at least one of $\\{Z_1,Z_2\\}$. For nonparametric part, the first $s$ transformed FPC scores of $\\{\\xi_1,\\ldots,\\xi_5\\}$ are contained. It results in $ M = \\left[\\binom{2}{2}+\\binom{2}{1}\\right]\\cdot 5 = 15\n$ candidates.\n\n\n\\noindent \\textbf{M21}. For Design 3, assume at least one of $\\{Z_1,Z_2,Z_3\\}$ and at least one of $\\{\\xi_1,\\xi_2\\}$ are included in a candidate model. Thus, there are $M = \\left[\\binom{3}{3}+\\binom{3}{2}+\\binom{3}{1}\\right]\\cdot \\left[\\binom{2}{2}+\\binom{2}{1}\\right] = 21$ candidates.\n\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\\includegraphics[width=0.8\\textwidth]{nmse2.pdf}\n\t\\caption{ Normalized mean squared error (NMSE) comparison for Design 2. }\n\t\\label{nmse2}\n\\end{figure}\n\n\nThe construction of \\textbf{M21} is based on the observation that the effects of FPCs on the response does not necessarily coincide with their magnitudes \\cite{zhuetal2007,hadiling1998, bairetal2006, zhuetal2014}. Therefore, the structure regarding FPC scores in \\textbf{M21} is not restricted to the nested one.\n\n\nWe use Epanechnikov kernel $k(u) = \\frac{3}{4}(1-u^2)I(|u|\\leq 1)$ for all candidate models with bandwidth $h_m$ being $n^{-1\/(1+q_m)}$ based on rule-of-thumb method, $m=1,\\ldots,M$. In addition, we compare the finite sample performance of MMA estimator with five alternative methods - AIC, BIC, equally weighting, smoothed AIC (SAIC) and smoothed BIC (SBIC) suggested by Buckland et~al. (1997) \\cite{bucklandetal1997}. For the $m$-th candidate model,\nAIC and BIC select the model with the smallest scores, defined as $AIC_m = \\log(\\widehat{\\sigma}_m^2) + 2tr(\\widehat{{\\mathbf P}}_{(m)})\/n$ and $BIC_m = \\log(\\widehat{\\sigma}_m^2) + \\log(n)tr(\\widehat{{\\mathbf P}}_{(m)})\/n$, where $\\widehat{\\sigma}_m^2 = \\frac{1}{n}\\|Y-\\widehat{\\mu}_{(m)}\\|^2$.\nSAIC and SBIC assign weights to the $m$-th candidate as $\\omega_m^{AIC} = \\exp(-AIC_m\/2) \/ \\sum_{m=1}^M \\exp(-AIC_m\/2)$ and $\\omega_m^{BIC} = \\exp(-BIC_m\/2) \/ \\sum_{m=1}^M \\exp(-BIC_m\/2)$, respectively. Equally weighting just assigns equal weights to each candidates. Mean squared error (MSE) of each methods is compared,\n\\[\nMSE = \\frac{1}{nD}\\sum_{d=1}^{D}\\|\\widehat{\\mu}^{(d)}-\\mu^{(d)}\\|^2,\n\\]\nwhere $D = 200$ denotes the number of repetitions and $d$ represents the $d$-th trial. For easy comparison, all MSE's are normalized by dividing by the MSE of AIC model selection estimator. Thus, a normalized MSE (NMSE) smaller than 1 indicates the corresponding estimator is superior to AIC estimator, and vice versa.\n\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\\includegraphics[width=0.8\\textwidth]{nmse3.pdf}\n\t\\caption{ Normalized mean squared error (NMSE) comparison for Design 3. }\n\t\\label{nmse3}\n\\end{figure}\n\n\nFigures~\\ref{nmse1}--\\ref{nmse3} present corresponding results of Design 1--3.\nFor Design 1, MMA and SAIC in Figure~\\ref{nmse1} exhibits superiorities for large and medium $R^2$ values, whereas for small $R^2$ value, equally weighting performs the best. BIC and SBIC cannot provide comparable results in Design 1. Besides, MMA performs slightly better than SAIC for small sample size or small $R^2$ values. Also, it is shown in Figure~\\ref{nmse1} that SAIC and SBIC outperform their model selection counterparts --- AIC and BIC, where the differences decrease as $R^2$ or $n$ grows.\n\nFor Design 2, as shown in Figure~\\ref{nmse2}, MMA dominates the other methods for large and medium $R^2$ values. Similar to the results for Design 1, we can also observe that BIC, SBIC and equally weighting have a marginal advantage for small $R^2$ values. As $R^2$ increases, the differences between these six methods decrease.\n\nFor Design 3, Figure~\\ref{nmse3} illustrates that MMA shows an edge over AIC, BIC, SAIC and SBIC for small and medium $R^2$ values, and the differences between these five methods become smaller as $R^2$ grows. Equllay weighting outperforms the other methods for small $R^2$ values but deteriorates rapidly as $R^2$ grows. MMA, AIC and SAIC behave more closely as $n$ grows.\n\n\nIn summary, the proposed MMA estimator delivers more satisfactory outcomes than the other competing estimators in most cases. The superior performance of MMA estimator in finite sample is partly attributed to the merit that its optimality does not depend on the correct specification of candidate models, which means that the true model is not necessarily included in the candidate set.\nMoreover, equally weighting method performs well for small and $R^2$ values and SBIC usually yields rather competitive results when $R^2$ is small, whereas their performances generally worsen as $R^2$ increases. This shows that equally weighting and SBIC methods are not capable in selecting optimal weights with minor noise level in our simulation settings.\nIn addition, model averaging estimators, SAIC and SBIC, always outperform their model selection counterparts --- AIC and BIC, and the differences between AIC and SAIC, or BIC and SBIC generally decrease as $R^2$ increases. SAIC typically shows a moderate advantage over AIC in most cases.\nFurthermore, it is observed that MMA has a growing edge over other methods when the structure of candidate model becomes unrestricted. This directly reveals that the optimality of MMA in finite sample relies on candidate models on hand. Therefore, combining various types of candidate model is appropriate practice.\n\n\n\n\n\n\\section{Application to real data}\\label{sec5}\n\nIn this section, we illustrate the application of the proposed method to two data sets, both consisting of near-infrared (NIR) spectra data and some reference variables.\n\n\\subsection{NIR shootout 2002 data set}\nThe {\\it NIR shootout 2002} data set was published by the International Diffuse Reflectance Conference (IDRC) in 2002 and is available from Eigenvetor Research Inc, USA\\footnote{Conference International Diffure Reflectance. (2002). NIR Spectra of Pharmaceutical Tablets from Shootout. {\\it Eigenvector Research}. http:\/\/\n\twww.eigenvector.com\/data\/tablets\/index.html}. It contains NIR spectra for 655 pharmaceutical tablets (functional predictor $X$), measured at two spectrometers over the spectral region from 600 to 1898 nm with 2 nm increments on the wavelength scale. Some quantities for reference analysis, such as weight of the active ingredient (response variable $Y$), weight of each tablet (scalar predictor $Z_1$), and hardness of each tablet (scalar predictor $Z_2$), are also provided. The data have already been divided into training (155), validation (40) and test (460) subsets. Here, the spectra from instrument 1 were used. And the sample data of $Y$, $Z_1$, and $Z_2$ were standarized for simplicity. We trained PLFS models on training subset and evaluated the performances on test subset.\nMean squared prediction error (MSPE) was used to compare the predictive efficiency.\n\\[\nMSPE = \\frac{1}{n_{test}}\\sum_{i=1}^{n_{test}}\\big(Y_{i} - \\widehat{\\mu}_{i}\\big)^2,\n\\]\nwhere $n_{test}$ is the size of test set.\nAs for candidate models, we considered $\\{Z_1, Z_2\\}$ for parametric components, and $\\{\\xi_1,\\xi_2\\}$, $\\{\\xi_1,\\xi_2,\\xi_3\\}$, $\\{\\xi_1,\\ldots,\\xi_4\\}$ as three kinds of candidate sets for FPC scores. Every candidate model comprised at least one component of parametric and nonparametric parts, respectively. Therefore, $M = 9, 21, 45$ corresponding to three settings.\n\n\\begin{table}\n\t\\renewcommand\\arraystretch{1.5}\n\t\\caption{MSPE for {\\it NIR shootout 2002} data.}\n\t\\label{tab:shootout}\n\t\\centering\n\t\\begin{tabular}{lrrrrrr}\n\t\t\\hline\n\t\t& $AIC$& $BIC$& $Equal$& $SAIC$& $SBIC$& $MMA$\\\\\n\t\t\\hline\n\t\t$M=9$ & \\textbf{0.7702}& 0.7957& 0.7825& 0.7796& 0.7933&\t\t\t 0.7762 \\\\\n\t\t$M=21$& \t\t  0.7301 & 0.7301& 0.7379& 0.7203& 0.7494& \\textbf{0.7177}\\\\\n\t\t$M=45$& \t\t  0.6956 & 0.7267& 0.7349& 0.6908& 0.7160& \\textbf{0.6906}\\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\nTable~\\ref{tab:shootout} presents the MSPE results of different procedures for {\\it NIR shootout 2002} data set. We can observe from the table that for this data set, AIC generally yields smaller MSPEs than BIC does. Comparison between SAIC and SBIC also shows a similar pattern. Second, the relatively small MSPEs produced by MMA estimator indicates that MMA performs the best among these four model averaging methods. It is shown that equally weighting yields relatively large MSPEs under all settings, which suggests naive equally weighting precedure suffer severely in practice. Moreover, Table~\\ref{tab:shootout} shows that MMA has an advantage over AIC in $M = 21, 45$ settings. Hence, MMA is likely to handle it well when encountered with diversity of candidate models.\n\n\n\\subsection{Equine articular cartilage data set}\nThis data set\\footnote{Sarin, J. K. et~al. (2019). Dataset on equine cartilage near infrared spectra, composition, and functional properties. {\\it figshare. Collection.} https:\/\/doi.org\/10.6084\/m9.figshare.c.4423139.v2} contains NIR spectroscopy measurements (functional predictor $X$) within the spectrum region of 700--1050 nm from 869 different locations across the articular surfaces of five equine fetlock joints, paired with comprehensive reference measurements from biomechanics, chemical composition and internal structure of the tissue, such as, instantaneous moduli ($Y$), collagen contents ($Z_1$), proteoglycan contents ($Z_2$), cartilage thickness ($Z_3$), and calcified layer thickness ($Z_4$), etc. More details are available \\cite{sarinetal2019}. \nThe sample data with size 530 were retained after removing all incomplete records. To evaluate the performance of our proposed procedure, we randomly selected 80\\% of records as training set and constructed the test set using remaining records. Furthermore, we standardized sample data of $Z_1,\\ldots, Z_4$, and performed successively logarithmic transformation and centralization on data of $Y$ to facilitate computation.\nTwo nested kinds of candidate models were considered: one included $\\{Z_1, Z_2, Z_3\\}+\\{\\xi_1, \\ldots, \\xi_4\\}$, the other comprised\n$\\{Z_1, \\ldots, Z_4\\}+\\{\\xi_1, \\ldots, \\xi_4\\}$. Each candidate model contained respectively at least one component of parametric and nonparametric parts, resulting in $M = 12, 16$. We conducted $D = 200$ runs. For each repetition, we still evaluated MSPE.\n\\[\nMSPE^{(d)} = \\frac{1}{n_{test}}\\sum_{i=1}^{n_{test}}\\big(Y_{i}^{(d)}-\\widehat{\\mu}_{i}^{(d)}\\big)^2, \\quad d=1,\\ldots,D,\n\\]\nwhere $n_{test}$ is the size of test set, $Y_{i}^{(d)}$ is the $i$-th response of the $d$-th test set, and $\\widehat{\\mu}_{i}^{(d)}$ is the prediction for $Y_{i}^{(d)}$. The average MSPEs with their standard error across $D$ repetitions were compared.\n\n\n\\begin{table}\n\t\\renewcommand\\arraystretch{1.5}\n\t\\caption{The average MSPE with standard error on test set for Equine articular cartilage data.}\n\t\\label{tab:equine}\n\t\\centering\n\t\\begin{tabular}{lrrrrrr}\n\t\t\\hline\n\t\t& $AIC$& $BIC$& $Equal$& $SAIC$& $SBIC$& $MMA$\\\\\n\t\t\\hline\n\t\t$M=12$\t& 0.37908& 0.37908& 0.44997& 0.37908& 0.37908& \\textbf{0.37856}\\\\\n\t\t& (0.00057)& (0.00057)& (0.00050)& (0.00057)& (0.00058)& (0.00055)\\\\\n\t\t$M=16$\t& 0.63542& 0.63694& 0.69663& 0.63516& 0.63565& \\textbf{0.63494}\\\\\n\t\t& (0.00001)& (0.00002)& (0.00000)& (0.00000)& (0.00002)& (0.00000)\\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\n\nTable~\\ref{tab:equine} illustrates the average MSPEs with their standard errors across replications on test set.\nFirst, it can be seen that MMA delivers the smallest results in terms of MSPE, which demonstrates the superiority of MMA to other model averaging and selection estimators and verifies the better prediction accuracy of MMA.\nSAIC performs the second with larger MSPEs, slightly inferior to MMA. And AIC shares no smaller  average MSPEs than SAIC and MMA.\nSecond, BIC and SBIC do not perform well on this data set. The average MSPEs of BIC and SBIC are of larger scales compared to other methods.\nEqually weighting method performs the worst with much larger mean MSPEs, which again emphasizes prudent use of it in practice.\nThe average MSPEs of model averaging estimators are always smaller than that of their model selection counterpart, which indicates that model averaging is a satisfactory alternative to model selection when prediction effect is of primary interest.\nTo sum up, these results show that the proposed MMA procedure is able to effectively deliver competitive outcomes.\n\n\n\n\\section{Conclusion and discussion}\\label{sec6}\n\nWe presented a Mallows-type model averaging approach for PLFS model in which a scalar response depends both on scalar covariates and a functional predictor.\nWe verified the asymptotic optimality of MMA estimator when the function predictor is densely measured with error.\nAdditionally, a finite sample simulation was used to demonstrate the performance of the proposed estimator is either superior or comparable to that of classic competing model selection and averaging methods. Also, real data analysis manifested that the proposed estimator generally facilitated modification of the prediction results and reduced the possibility of producing poor outcomes when using a single model.\n\nMany aspects deserve future research. In practice, if lots of variables are available, it would be reasonable to derive a suitable model averaging estimation for high-dimensional regression problems.\nMoreover, there is room for studying the situation in which functional data are sparsely or irregularly observed, similar to cases in longitudinal studies. Besides, if more than one functional predictor were to exist, it would be interesting to determine how to effectively and efficiently conduct model averaging.\nFinally, our asymptotic optimality is derived on the base of that all candidate models are misspecified. A consistent estimator is more desired if the correct model exists in our candidate set. Therefore, considering a consistent model averaging approach would be an avenue for future research.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\nA rich diversity in network latency and packet loss rates have become\nessential for experimental evaluation of BitTorrent and other\ncommunication protocols used in the Internet. The need for such\na diversity in network latency and packet loss rates is because of the\nheterogeneous nature of the Internet~\\cite{Floyd_2001_DiffSimInternet,\n  Floyd_2008_ToolsTestbeds, Choffnes_2010_PitfallTestbed}. The\nheterogeneity of the Internet is the primary motivation for the\ncreation of testbeds such as PlanetLab~\\cite{PLANETLAB,\n  Dischinger_2008_SatelliteLab}. However, due to the shared nature of\nthe PlanetLab platform, the results of the experiments performed on\nPlanetLab are not reproducible~\\cite{Spring_2006_PlanetLabMyths}. In\ncontrast, dedicated clusters, such as Grid'5000~\\cite{Grid5000}, not\nonly offer a reproducible environment but also enable the scaling of\nBitTorrent experiments by supporting a large number of BitTorrent\ninstances on a single machine. The primary shortcoming of experiments\nperformed on clusters is the absence of the diverse network latency\nand packet loss rates that can be found in the Internet. As \\emph{the\n  impact of the network latency and packet loss on BitTorrent\n  performance is not known}, there exists \\emph{a dilemma while\n  selecting a testbed for the BitTorrent experiments}.\n\nThe BitTorrent protocol uses TCP to efficiently distribute the\n\\emph{pieces} of a file to a large number of peers using peer-to-peer\n(P2P) connections~\\cite{Cohen_2008_BitTorrentProtocol}. The peers\ncontribute to the file distribution by uploading the pieces that have\nbeen downloaded. During the file download, the peers exchange control\nmessages to select the pieces of the file to upload and the peers to\nwhom these pieces are to be uploaded. BitTorrent also allows the users\nto limit the rate at which data is uploaded and downloaded. These rate\nlimits allow the users to restrict the network bandwidth that\nBitTorrent can compete for during a BitTorrent session. As the users\ncan control the upload process by limiting the upload rates, and\nBitTorrent uses control messages to decide the connections to upload\nto, BitTorrent is inherently different from other TCP based file\ntransfer protocols such as HTTP and FTP.\n\nOver the years BitTorrent performance has received considerable\nattention~\\cite{Qui_2004_ModelingP2P,Bharambe_2006_AIBNP,Chiu_2008_MinFDT}.\nAs these studies do not evaluate the interaction between TCP and\nBitTorrent, the impact of network latency and packet loss rates\non BitTorrent performance is not known. Network latency and packet\nloss introduce a \\emph{ramp-up} period which is required by TCP to\nattain the maximum upload rate that can be\nachieved~\\cite{RFC5681_2009_TCPCC, Mathis_1997_MBTCP}. BitTorrent\nusers that limit the upload rate can therefore limit the impact of\nTCP ramp-up. Apart from the TCP throughput, the BitTorrent performance is\nalso dependent on the control messages exchanged by the peers. The\ncontrol messages generated by a peer can generate a delayed\nresponse at a remote peer because of the network latency and the packet\nlosses. The delayed response can affect the various algorithms used by\nBitTorrent, such as the peer selection algorithm which is used to\ndecide the peer to upload to. Due to the above reasons, the analytical\nmodels for TCP cannot be directly used to provide the impact of network\nlatency and packet loss on BitTorrent performance.\n\nIn this paper, we use the download completion time, i.e, the time\nrequired to download a file using BitTorrent, as a metric to study the\nimpact of network latency and packet loss. In our experiments, we\nobserve that network latency and packet loss have a marginal impact\n(less than 15\\%) on the download completion time of a file. We first\nstudy the impact of network latency without packet loss. Network\nlatency causes not only delays in receiving control messages but also\nTCP ramp-up. We therefore study the impact of network latency by\nstudying the impact of the delays in receiving control messages and\nTCP ramp-up on the download completion time. We observe that the\ndownload completion time, when the round-trip time (RTT) between\n\\emph{any two peers} in the torrent is 1000~ms, is not more than 15\\%\nof the download completion time when the RTT is 0~ms.\n\nWe then study the impact of network latency with packet loss. We observe\nthat an RTT of 400~ms \\emph{between any two peers} and a packet loss\nrate of 5\\% does not increase the download completion time by more\nthan 15\\% of the download completion time observed when the RTT is\n0~ms and the packet loss rate is 0\\%. We also observe that an RTT of\n1000~ms \\emph{between any two peers} with a packet loss rate of\n5\\% can increase the download completion time by more than 15\\% of the\ndownload completion time observed when the RTT is 0~ms and the packet\nloss rate is 0\\%. As an RTT greater than 400~ms between \\emph{any two\n  peers} is unrealistic, our results show that for upload rates\ntypically seen in the Internet, the download completion time is not\nsensitive to the network conditions that can be found in the Internet;\n\\emph{dedicated clusters, such as Grid'5000, can be used to perform\n  BitTorrent experiments}.\n\nThe remainder of this paper is structured as follows. The network\ntopologies and the technique used to emulate the latency and\npacket loss are presented in \\refSec{sec:Methodology}. We first\npresent the impact of latency without packet loss on download\ncompletion time. In \\refSec{sec:HomogeneousLatency}, we emulate the\nsame latency between any pair of peers in the torrent. We use this\ntopology to study the impact of TCP ramp-up and the impact of the\ndelays in receiving the BitTorrent control messages on the download\ncompletion time. In \\refSec{sec:HeterogeneousLatency}, we emulate\ntorrents to study the impact of network latency when the condition of\nsame network latency between any two peers is relaxed. We then study the\nimpact of network latency and packet loss in\n\\refSec{sec:ImpactLosses}. We finally conclude in\n\\refSec{sec:Conclusion}.\n\n\n\\section{Methodology}\n\\label{sec:Methodology}\n\nIn this paper, we use the terminology used by the BitTorrent\ncommunity. A \\emph{torrent}, also known as a BitTorrent session or a\nswarm, consists of a set of peers that are interested in having a\ncopy of the given file. A peer in a torrent can be in two states:\nthe \\emph{leecher} state when it is downloading the file, and the\n\\emph{seed} state when it has a copy of the file being\ndistributed. The peers distribute the file in chunks called\n\\emph{pieces}; a piece is further split into \\emph{blocks} to\nfacilitate the piece upload and download. A peer is said to\n\\emph{unchoke} a remote peer if it is uploading the blocks of\na piece. A \\emph{tracker} is a server that keeps track of the peers\npresent in the torrent. For our experiments, we use a private torrent\nwith one tracker, one initial seed (henceforth called the seed), and\n300 leechers. We use the \\emph{download completion time}, the time\nrequired by the leechers to download the file distributed using\nBitTorrent, as the metric to study the impact of network latency and\npacket loss. \n\nAll the experiments were performed using an instrumented\nversion of the \\emph{BitTorrent mainline client}~\\cite{BTInstru} on\nmachines running Linux as the host operating system. The BitTorrent\nmainline client internally uses TCP. We used TCP\nCubic~\\cite{Ha_2008_Cubic}, the default TCP implementation for the\ncurrent series of the Linux kernel, for our experiments. The latest\nversion of uTorrent, a BitTorrent client, is based on uTP which uses\nUDP as the transport layer protocol~\\cite{Norberg_2010_uTP,\n  LEDBAT}. As in the case of TCP, uTP has a window based congestion\ncontrol mechanism. The design of uTP also ensures that uTP ramp-up is\nnot faster than TCP~\\cite{Norberg_2010_uTP}. The control messages \ngenerated by uTorrent using uTP are similar to those that are present\nin BitTorrent clients that use TCP. Hence, we believe that the results\npresented in this paper are valid for BitTorrent clients using uTP.     \n\nIn this section, we first present the various scenarios used in the\nexperiments. We then present the procedure to emulate the network\nlatency and packet loss. This is followed by an\noverview of Grid'5000, the experimental platform we used. Finally, we\npresent the parameters used while performing the experiments.\n\n\\subsection{Experiment Scenarios}\n\\label{sec:ExpScen}\n\nWe now enumerate the scenarios used to study the impact of network\nlatency and packet loss on the download completion time of a file.\n\\begin{enumerate}\n\\item \\emph{Scenario of Homogeneous Latency.} In this scenario, we\n  emulate \\emph{a fixed network latency between any two peers} in the\n  torrent. The fixed network latency, though unrealistic, provides a\n  controlled environment to study the impact of network latency on the\n  download completion time of a file. We use this scenario to get an\n  insight on the threshold of network latency beyond which the network\n  latency affects the download completion time. The scenario also\n  gives the download completion time when the maximum latency between\n  any two peers in a torrent in known. The results of this study can also\n  be used to give the impact of network latency when the hosts are\n  geographically distributed and are connected with links that have\n  negligible packet loss.  \n\\item \\emph{Scenario of Heterogeneous Latency.} In this scenario we\n  relax the condition of fixed network latency to emulate a more realistic\n  network topology. We use this scenario to confirm that the results\n  obtained in the scenario of homogeneous latency are valid even when\n  the condition of fixed network latency between any two peers is relaxed. For\n  this scenario, we group peers that have the same network latency\n  among themselves in an emulated Autonomous System (AS). We assume\n  these ASes to be fully meshed and that the inter-AS latency is\n  greater than the intra-AS latency; we assume the network latency between the\n  peers in a given AS is the same.  \n\\end{enumerate}\n\nBitTorrent allows its users to limit the upload and download\nrate. These rate limits restrict the network bandwidth that BitTorrent\ncan compete for with other applications such as Web browsers. In\nthis paper, we do not place any restrictions on the download rate. We\nset the upload rate limit of the peers from a wide range of values,\nfrom 10~kB\/s to 100~kB\/s. We use this range of upload rates as\nChoffnes~\\emph{et~al.}~\\cite{Choffnes_2010_PitfallTestbed} show that 90\\% of\nthe hosts present in public torrents upload at rates that are smaller\nthan 100~kB\/s. In our experiments, we assume the same upload rate \nlimit at all the leechers while studying the impact of network latency\nand packet loss. We use the following torrent configurations to set\nthe limit on the upload rate of the peers.   \n\\begin{enumerate}\n\\item \\emph{Slow Seed and Slow Leechers.}  In this scenario, we assume all\n  the peers in the torrent have a low upload rate. We also assume the\n  same upload rate at the seed and leechers. We performed two\n  experiments for this scenario; we limit the upload to 10~kB\/s for\n  the first experiment and 20~kB\/s for the second experiment.\n\\item \\emph{Fast Seed and Slow Leechers.} Some torrents have seeds\n  that upload faster than the leechers. Public torrents are also known\n  to have leechers that are capable of high upload and download\n  rates. In torrents where the seed favors fast\n  leechers~\\cite{Legout_2007_IncentiveBt}, these leechers are able to\n  download the file faster than their slower peers. As these leechers\n  are capable of downloading pieces faster than their slower peers,\n  they act like a fast seed to the slow peers in their peer-set. We\n  emulate torrents that have a fast seed by limiting the upload rate\n  of the seed to 50~kB\/s and the upload rate of the leechers to\n  20~kB\/s.\n\\item \\emph{Fast Seed and Fast Leechers.} We perform these experiments to\n  emulate torrents where all the peers are capable of high upload\n  rates. We performed two experiments for this scenario; we limit the\n  upload to 50~kB\/s for the first experiment and 100~kB\/s for the\n  second experiment.\n\\end{enumerate}\n\n\\subsection{Emulation of Network Latency and Packet Loss}\n\\label{sec:EmulLatency}\n\n\\begin{figure}\n\\begin{centering}\n\\subfloat[][Network latency observed in the\nInternet.]{\\label{fig:LatencyInternet}\\includegraphics[width=0.45\\columnwidth]{latency-cdf.eps}} \n\\hspace{0.02\\columnwidth}\n\\subfloat[][Loss rate observed in the Internet.]{\\label{fig:LossRateInternet}\\includegraphics[width=0.45\\columnwidth]{loss-rate-cdf.eps}}\n\\caption{iPlane measurements of network latency and loss\n  rate observed in the Internet. \\scap{98\\% of the links have an RTT\n    less than 400~ms and 99.8\\% of the links have an RTT less than\n    1000~ms. 95\\% of the links have a loss rate less than 0.05, i.e.,\n    5\\%.}}  \n\\label{fig:iPlaneMeasurements}\n\\end{centering}\n\\end{figure}\n\nWe use the Network Emulation (NetEm) module for the Linux\nkernel~\\cite{Hemminger_2005_NETEM} to emulate the network latency and\npacket loss between the peers.  The NetEm module operates between the\nTCP\/IP implementation and the device driver for the network device. In\nour experiments, the peers used the ethernet and the loopback device\nto communicate with each other. The NetEm module emulates network\nlatency by en-queuing the packets at the ingress interface and the\negress interface of a network device; packets losses are emulated by\ndropping the packets at the ingress and egress interfaces. When the\nloss is on the egress interface, due to cross-layer optimizations\nperformed by the TCP implementation in Linux, TCP retransmits the\npacket without considering it as a loss in the network. To avoid such\nretransmissions, we introduce packet losses on the ingress interface\nof the network device.\n\nA shortcoming of NetEm is that the network latency and packet losses\nare \\emph{inherently} bound to the network device. NetEm can be used\nto emulate network latency and losses for each connection, however such\ntopologies are hard to verify and manage. In this paper we set\nthe network latency and packet loss rates for a particular\nnetwork device. The network latency between a pair of peers is the sum\nof the network latency at the devices used; the loss rate seen by a\npacket is the loss rate at the ingress interface of the destination.\n\nWe use the publicly available iPlane\nmeasurements~\\cite{iPlane,Madhyastha_2006_IPlane} to obtain the values\nof the network latency and loss rate to\nemulate. \\refFig{fig:iPlaneMeasurements} shows the distribution of\nRTT and loss rate observed in 5 iPlane samples; each sample contains\nthe measurement of about $6*10^5$ links. In\n\\refFig{fig:LatencyInternet}, we observe that 98\\% of the links have an\nRTT less than 400~ms, and 99.8\\% of the links have an RTT less than\n1000~ms. We use these results to emulate a wide range of RTT values,\nfrom 0~ms to 1000~ms, between a pair of peers in the\ntorrent. In \\refFig{fig:LossRateInternet}, we observe\nthat 95\\% of the links probed have a loss rate less than 0.05, i.e,\n5\\% packet loss. We emulate a 5\\% packet loss on the ingress of each \ninterface of the machines while studying the impact of packet\nlosses.\n\n\\subsection{Experiment Setup}\n\\label{sec:ExpParam}\n\nWe performed our experiments on the Grid'5000 experimental\ntestbed~\\cite{Grid5000}. Grid'5000 consists of a grid of clusters\nthat are geographically distributed across France. Each cluster\nconsists of several machines connected by a dedicated and\nhigh speed LAN; these clusters are inter-connected by high speed\nlinks. We performed our experiments on \\emph{a single cluster of\n  Grid'5000}. In this cluster, we observe an RTT of less than 1~ms\nbetween a pair of machines when network latency is not emulated. Due\nto the very small network latency and the absence of packet loss \nbetween the machines in the LAN, the cluster does not reflect the\nnetwork conditions present in the Internet. Unlike other testbeds such\nas PlanetLab, the machines in the cluster are not shared. Grid'5000\nthus provides a reliable and robust platform for performing\nreproducible experiments.     \n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=\\columnwidth]{impact-num-nodes-rtt.eps}\n\\caption{Impact of the number of leechers running on a machine on the RTT\n  estimate of TCP. \\scap{The machines used can be used to run up to\n    100 leechers.}}\n\\label{fig:ImpNumNodes}\n\\end{centering}\n\\vspace{-0.1in}\n\\end{figure}\n\nWe scale our experiments by running 100 leechers on a single\nmachine of the cluster. We performed the following test to ensure that\nthe machines can support up to 100 leechers while emulating the\ndesired network latency. We distribute a 50~MB file in a private torrent with\na single seed and a single tracker. The seed and tracker ran on one of\nthe machines of the cluster. We used another machine in the cluster to\nrun the leechers. We limit the upload rate of each of the peers to\n100~kB\/s and we vary the number of leechers running on a machine \nfrom 2 to 100. To study the impact of the number of leechers, we monitor \nthe RTT estimate of TCP in the following manner. All the peers in the\ntorrent use the socket interface of TCP to communicate with each\nother. The \\verb,send, method of this interface is used by the peers\nto send data to other peers in the torrent. On each call of\n\\verb,send,, we sample the RTT estimate of TCP by using the\n\\verb,TCP_INFO, option of the \\verb,getsockopt,\nmethod. \\refFig{fig:ImpNumNodes} shows the average RTT estimate of TCP  \nwhen we emulate an RTT of 0~ms, 400~ms, and 1000~ms, between the peers;\nthe error bars represent the minimum and maximum RTT estimate observed\nin five iterations. We observe that the number of leechers running on\na given machine has a less than 15\\% impact on the average RTT\nestimate of TCP.   \n\nThe NetEm module buffers the TCP frames which are in flight for a time\nperiod equal to the RTT being emulated. An RTT of 1 second\n(1000~ms) for an upload rate of 100~kB\/s would require 100~frames of\n1~kB to be in flight. For our experiments, we use a buffer size of\n$10^5$ frames to support 1000 frames of each of the 100 peers to be in\nflight.\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=0.85\\columnwidth]{Topology-Machines.eps}\n\\caption{Experiment setup. \\scap{One machine for the tracker and the\n    initial seed, and three machines each with 100 leechers.}}\n\\label{fig:Grid5KTopology}\n\\end{centering}\n\\end{figure}\n\nWe perform our experiments in a private torrent with one\ntracker, one initial seed (henceforth called the seed), and a\nflash crowd of 300 leechers. We distribute a 50 MB file in this\ntorrent. For our experiments, we assume that the peers remain in the\ntorrent until all the leechers have finished downloading the file. As\nshown in~\\refFig{fig:Grid5KTopology}, we use one machine for  \nthe tracker and the seed. We use three machines for the 300 leechers;\neach machine runs 100 instances of the leechers. A pair of peers in the\ntorrent use either the loopback interface or the ethernet interface to\ncommunicate with each other. \n\n\\subsection{Impact of TCP Segmentation Offloading}\n\\label{sec:ImpactTSO}\n\nThe Maximum Segment Size (MSS) of TCP specifies the maximum payload\nlength that can be exchanged over a\nconnection~\\cite{RFC5681_2009_TCPCC}. One factor that contributes to\nthe MSS is the Maximum Transmission Unit which is typically 1500 bytes\nfor devices used in LANs. The MTU is set to 16436 bytes for the\nloopback interface on many Linux distributions. To ensure that the\npayload length exchanged by the peers does not depend the interface\nused by the connection, we set the MTU on the loopback interface to\n1500 bytes. Despite this limit, we observed that a significant number\nof TCP segments have a size larger than the MSS negotiated during\nconnection establishment. We observe these large segments because of\nTCP Segmentation Offloading (TSO) which is enabled by default in the\n2.6 series (the current series) of the Linux kernel. \n\nTCP Segmentation Offloading (TSO) enables the host machine to offload\nsome of the TCP\/IP implementation, such as segmentation, and\ncalculation of IP checksum, to the network device. TSO also supports\nthe exchange of data in frames of sizes that can be greater that the\nunderlying MTU size~\\cite{Mogul_2003_TCPOFFLOAD}. The increase in the\nframe size can result in significant improvement in throughput; the\nimprovement depends on various factors such as CPU processing power\nand the amount of data being\ntransferred~\\cite{Freimuth_2005_SERVERTSO}. Clusters, such as  those\npresent in Grid'5000, include hosts that support TSO. \n\nWe now study the impact of RTT on the TCP payload length when TSO is\nenabled. We use four machines to create a private torrent with 300\nleechers, one tracker, and one initial seed, to distribute a 50~MB\nfile. We assume the same RTT between any two peers for the results\npresented in \\refFig{fig:PktLenTSOEnabled} and\n\\refFig{fig:PktLenTSODisabled}. We also limit the upload rate to\n20~kB\/s. \\refFig{fig:PktLenTSOEnabled} shows the distribution of \npayload length for different RTT values when TSO is enabled; the\nresults present the distribution of the payload lengths observed in 5\niterations. We observe that an increase in the network latency between\nthe peers results in an increase in the number of TCP segments with a\nlarge payload length. This was observed when the MTU was set to 1500\nbytes indicating that TSO can result in payload lengths greater than\nthe MTU value specified. \\refFig{fig:PktLenInternet} shows the payload\nlengths from  the publicly available traces of the Internet traffic\nobserved in the WIDE backbone~\\cite{MAWI}; the values presented are\nfrom the sample taken on the WIDE backbone on November 29, 2009. In\n\\refFig{fig:PktLenTSOEnabled} and \\refFig{fig:PktLenInternet}, we\nobserve that the maximum payload length of the packets sent over the\nInternet is smaller than the maximum payload length observed in\nGrid'5000 when TSO is enabled. We observe small payload lengths in\n\\refFig{fig:PktLenInternet} because hardware support on all the\nintermediate devices is essential for exchange of large segments. When\nTSO is disabled and the MTU is set to 1500 bytes,\n\\refFig{fig:PktLenTSODisabled} shows the distribution of the payload \nlength for different RTT values; the results present the\ndistribution of the payload lengths observed in 5 iterations. We\nobserve that when TSO is disabled and the MTU is set to 1500 bytes,\nthe maximum payload length is similar to that observed in the\nInternet. \n\nWe set the MTU to 1500 bytes on the loopback interfaces and\ndisabled TSO on the ethernet and loopback interface for the\nexperiments presented in the \\refSec{sec:HomogeneousLatency} and\n\\refSec{sec:HeterogeneousLatency}.  The outcome of the experiments\nwith TSO enabled are discussed with the impact of packet  loss in\n\\refSec{sec:ImpactLosses}. \n\n\\begin{figure}\n\\begin{centering}\n\\subfloat[][Payload length with TSO\nEnabled. Increasing the RTT causes an increase in\npayload length.]{\\label{fig:PktLenTSOEnabled}\\includegraphics[width=\\columnwidth]{pkt-len-TSO-enabled.eps}}\\\\\n\\vspace{-0.15in}\n\\subfloat[][Payload lengths observed in the WIDE\nbackbone. Maximum payload length less than 1500 bytes as TSO requires\nhardware support on all the links on a connection.]{\\label{fig:PktLenInternet}\\includegraphics[width=\\columnwidth]{pkt-len-Internet.eps}}\\\\\n\\vspace{-0.15in}\n\\subfloat[][Payload length with TSO Disabled and MTU set to 1500 bytes.]{\\label{fig:PktLenTSODisabled}\\includegraphics[width=\\columnwidth]{pkt-len-TSO-disabled.eps}}\n\\caption{Impact of TSO on TCP payload length. \\scap{Setting the MTU to 1500\n  bytes and disabling TSO ensures that the maximum payload length is\n  similar to that observed in the Internet.}}\n\\label{fig:ImpactTSO}\n\\end{centering}\n\\vspace{-0.2in}\n\\end{figure}\n\n\\section{Homogeneous Latency}\n\\label{sec:HomogeneousLatency}\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=\\columnwidth]{sym-dct-tso-dis.eps}\n\\caption{Impact of latency on the average download completion\n  time. The error bars indicate the minimum and maximum download\n  completion time. \\scap{The latency increases the download completion\n    time by at most $15\\%$.}}\n\\label{fig:SymmetricDownloadCompletion}\n\\vspace{-0.15in}\n\\end{centering}\n\\end{figure}\n\nIn this section, we assume the same network latency between any two\npeers in the torrent because \\emph{this scenario gives the worst case\n  impact of a given network latency for a given upload rate\n  limit}. The same network latency results in the same \nRTT between any two peers in the torrent because the sum of the other\ndelays, such as queuing delays at the intermediate routers, is less\nthan 1~ms in a Grid'5000 cluster. In each experiment, we choose an\nRTT to emulate from a wide range of values, from 0~ms to 1000~ms. In\n\\refSec{sec:EmulLatency}, we observed that 99.8\\% of the Internet\nlinks have an RTT less than 1000~ms. \\emph{An RTT of 1000~ms between\nany two peers thus gives a worst case impact of the network latency\nthat can be observed in the Internet}. We use a wide range of values,\nfrom 10~kB\/s to 100~kB\/s, to set the upload rate limit of the peers in\nthe torrent; we assume the same upload rate limit for all the leechers\nin the torrent. For a given upload rate limit,\n\\refFig{fig:SymmetricDownloadCompletion}  \nshows the impact of the RTT between any two peers on the download\ncompletion time. Each point on the plot is the average download\ncompletion time in 10 iterations; the error bars indicate the minimum\nand maximum download completion time observed in 10 iterations. The\ntwo factors that can be affected by network latency and cause an\nincrease in the download completion time are TCP ramp-up and the\ndelays in receiving the control messages. We study the impact of\nnetwork latency on the download completion time by studying the impact\nof network latency on these two factors in \\refSec{sec:ImpactRampUp}\nand \\refSec{sec:ControlMessagesImpact} respectively.\n\nIn \\refFig{fig:SymmetricDownloadCompletion}, for a given upload rate\nlimit from 10~kB\/s to 100~kB\/s, we observe that an RTT of up to\n1000~ms between any two peers does not increase the average download\ncompletion time by more than 15\\% of the average download completion\ntime when the RTT is 0~ms. When the maximum upload rate of the seed\nand leechers is limited to 10~kB\/s,\n\\refFig{fig:SymmetricDownloadCompletion} shows that the download\ncompletion time increases when the RTT is greater than 200~ms. In\n\\refSec{sec:ImpactRampUp}, we show how TCP ramp-up is responsible for\nthis increase in the download completion time. When the upload rate of\nall the peers is limited to 20~kB\/s, in\n\\refFig{fig:SymmetricDownloadCompletion}, we observe that the download\ncompletion time is not a monotonous function of the RTT. Peers having\nan RTT of 1000~ms have a lower download completion time compared to\npeers having an RTT of 400~ms. We show that this is the impact of\nnetwork latency on the delays in receiving BitTorrent control messages\nin \\refSec{sec:ControlMessagesImpact}.\n\nAn RTT of 1000~ms between any two peers gives the worst case impact\nof network latency. \\refFig{fig:SymmetricDownloadCompletion} thus\nshows that \\emph{network latency has a marginal impact, less than\n  15\\%, on the average download completion time for the upload rates\n  observed in public torrents}.\n\n\\subsection{Impact of TCP Ramp-Up}\n\\label{sec:ImpactRampUp}\n\n\\begin{figure}\n\\begin{centering}\n\\subfloat[][Distribution of the time between successive send\nsystem calls at the \\emph{leechers} when RTT between peers is 0~ms.]{\\label{fig:Leecher0}\\includegraphics[width=\\columnwidth]{socketIAT-Large-peer-0ms.eps}}\n\\hspace{0.05\\columnwidth}\n\\subfloat[][Distribution of the time between successive send\nsystem calls at the \\emph{leechers} when RTT between peers is 400~ms.]{\\label{fig:Leecher400}\\includegraphics[width=\\columnwidth]{socketIAT-Large-peer-400ms.eps}}\n\\caption{Distribution of the time between successive send system calls\n  at the leechers (semi-log scale). \\scap{The RTT does not affect the upload\n  process at the seed and the leechers for high upload rates.}}\n\\label{fig:IATLeecher}\n\\end{centering}\n\\vspace{-0.25in}\n\\end{figure}\n\nThe BitTorrent application uses the \\verb,send, system call of the\nsocket interface of TCP to upload the pieces of a file. BitTorrent\nlimits the upload rate for a given TCP connection by periodically\ncalling the \\verb,send, system call. The time between successive calls\nof \\verb,send, is high when the upload rate is low. Each call of\n\\verb,send, typically results in creation of a TCP\nsegment which is transmitted. If the acknowledgment for the\ntransmitted segment arrives before the subsequent call of \\verb,send,,\nthen the upload rate of the application does not require a congestion\nwindow ramp-up. \\emph{A ramp-up is required on the TCP connections\n  where the time between successive calls of} \\verb,send, \\emph{is\n  smaller than the RTT}. BitTorrent also enables a peer to simultaneously\nupload pieces of the file to many peers of a torrent in parallel. An\nincrease in the number of parallel unchokes results in an increase in\nthe time between successive calls of \\verb,send,. This is because the\nupload rate limit is divided over the connections used for the\nunchoke.\n\nWe present the distribution for the time between successive calls of\nthe \\verb,send, at all the leechers (over 5 iterations) in\n\\refFig{fig:IATLeecher}; the RTT between any two peers is 0~ms in\n\\refFig{fig:Leecher0} and 400~ms in \\refFig{fig:Leecher400}.\nFor an upload rate of 10~kB\/s, in \\refFig{fig:Leecher0} and\n\\refFig{fig:Leecher400}, we observe that the time between\nsuccessive \\verb,send, calls is greater than 200~ms for a significant\nnumber of calls. For these calls, when the RTT between the peers is\nless than or equal to 200~ms a ramp-up shall not be required. Due to\nthe absence of ramp-up, in \\refFig{fig:SymmetricDownloadCompletion}\nfor an upload rate limit of 10~kB\/s, we observe that an RTT less than\nor equal to 200~ms does not increase the average download completion\ntime by more than 5\\% of the average download completion time observed\nwhen the RTT is 0~ms. For the upload rate limit of 10~kB\/s, an\nRTT larger than 200~ms may require a TCP ramp-up. Similarly, for the\nupload rate limits of 20~kB\/s, 50~kB\/s, and 100~kB\/s, in\n\\refFig{fig:IATLeecher}, we observe that ramp-up may be   \nrequired for an RTT greater than 120~ms, 20~ms, and 20~ms\nrespectively. Hence, for an upload rate limit in the range of 10~kB\/s\nto 100~kB\/s, TCP ramp-up will be required for an RTT greater than\n200~ms. In \\refFig{fig:SymmetricDownloadCompletion} for an RTT \nof up to 1000~ms, we observe that TCP ramp-up does not increase the\naverage download completion time by more than 15\\% of the download\ncompletion time observed the RTT is 0~ms. Thus, \\emph{TCP ramp-up has\n  a marginal impact, less than 15\\%, on the average download\n  completion time for the upload rates observed in public torrents}.  \n\n\\subsection{Impact of Latency on Control Messages}\n\\label{sec:ControlMessagesImpact}\n\nIn \\refFig{fig:SymmetricDownloadCompletion} for an upload rate of\n20~kB\/s, we observe that the download completion time is not a\nmonotonous function of the RTT. The download completion time when the\nRTT is 1000~ms is smaller than the download completion time when the\nRTT is 400~ms. We now present the reasons for this behavior. We would\nlike to comment that this discussion is specific to the implementation\nof the BitTorrent client we used in the experiments and that\n\\emph{these observations may not be true for other BitTorrent\n  clients that are available}.\n\nOnce every 10 seconds, a seed selects from its peer-set a set of\nleechers to unchoke. The leechers are selected based on their download\nrate and time since the start of their last unchoke. During an\nunchoke, the leecher downloading the piece requests for multiple\nblocks of a piece to pipeline the blocks for the unchoke. The number\nof blocks requested is a function of the estimated download rate at\nthe leecher. This estimate of the download rate is a moving average\nand it grows slowly to attain the rate at which the seed uploads to\nthe given leecher. This growth is even slower if the connection\nrequires a TCP ramp-up. The slow growth in the estimated download rate\nthus results in a slow increase in the number of blocks that are in\nthe pipeline at the seed. We observe that this slow increase, along\nwith the latency in receiving the block requests, results in time\nperiods during which the seed unchoking the leecher is idle, i.e.,\nawaiting block requests. If the leecher selection algorithm of the\nseed is invoked during these \\emph{idle periods}, then the leecher\n\\emph{will not be selected} for an unchoke resulting in an abrupt\ntermination of the unchoke. We observe that this abrupt termination of\nthe unchoke occurs frequently when the upload rate limit is 20~kB\/s\nand when the RTT between the peers is greater than 120~ms and less\nthan 800~ms. \n\n\\begin{figure}\n\\begin{centering}\n\\subfloat[][Number of pieces available with the leechers when the\nupload rate is limited to 20~kB\/s.]{\\label{fig:Have20}\\includegraphics[width=\\columnwidth]{sym-have-20kBs.eps}}\n\\hspace{0.05\\columnwidth}\n\\subfloat[][Number of pieces available with the leechers when the\nupload rate of the leechers is limited to 20~kB\/s and that of the\nseed is limited to 50~kB\/s.]{\\label{fig:Have5020}\\includegraphics[width=\\columnwidth]{sym-have-50kBs-20kBs.eps}}\n\\caption{Evolution of pieces available at the leechers. \\scap{Latency has\n  marginal impact on the pieces available with the leechers.}}\n\\label{fig:RTTHaveImpact}\n\\end{centering}\n\\vspace{-0.15in}\n\\end{figure}\n\nWe now show how the abrupt termination of an unchoke affects the\navailability of pieces at the leechers. Once a piece is downloaded,\nthe leecher can then upload this piece to other leechers in its\npeer-set. After the piece download, the leecher sends a \\emph{HAVE}\nmessage with the piece identifier to the peers in its peer-set. The\n\\emph{HAVE} message indicates that the piece can be downloaded from\nthis leecher. In \\refFig{fig:RTTHaveImpact}, we use the \\emph{HAVE}\nmessages sent by the leechers to show the evolution of the average\nnumber of pieces (over 10 iterations) that are available at the\nleechers. For reasons mentioned above, when the upload rate limit of\nthe seed and leechers is 20~kB\/s, an RTT of 400~ms between the seed\nand a leecher typically results in the abrupt termination of the\nunchoke. The abrupt termination results in an incomplete download of a\npiece and affects the availability of pieces as shown in\n\\refFig{fig:Have20}. We do not observe abrupt termination of the\nunchoke when the RTT is 0~ms and when the RTT is 1000~ms. Hence, the\naverage download completion time when the RTT is 400~ms is greater\nthan the average download completion observed when the RTT is\n1000~ms. \n\nWe do not observe abrupt termination of unchokes when we have\na fast seed (upload rate limited to 50~kB\/s) in a torrent with slow\nleechers (upload rate limited to 20~kB\/s); \\refFig{fig:Have5020} shows\nthe evolution of pieces for this torrent configuration. We observe a\nsimilar evolution for the \\emph{HAVE} messages when the upload rate at\nthe peers is limited to 50~kB\/s and 100~kB\/s.\n\n\\subsection{Summary}\n\nIn this section, we emulated a large range of RTT values, from 0~ms to\n1000~ms, to study the impact of network latency on the download\ncompletion time. We use an RTT of 1000~ms to give the worst case\nimpact of network latency for a given upload rate. We observe that an\nRTT of up to 1000~ms between any two peers has a marginal impact, less\nthan 15\\%, on the average download completion time of a file. For an\nupload rate of 20~kB\/s, we observe that the download completion time is\nnot a monotonous function of the RTT. This behavior emphasizes\n\\emph{that the models for TCP throughput cannot be directly used to\n  study the impact of network latency on the time required to download\n  a file using BitTorrent}. \n\nIn the next section, we relax the condition of same latency between any\ntwo peers in the torrent to confirm that the scenario of same latency\ngives us the worst case impact of network latency.\n\n\\section{Heterogeneous Latency}\n\\label{sec:HeterogeneousLatency}\n\n\nWe now present results to confirm that the observations made in\n\\refSec{sec:HomogeneousLatency} are valid even when the condition of\nfixed latency is relaxed. We relax the condition of same latency\nbetween any two peers by emulating ASes in the following\nmanner. Public torrents have peers that are spread out\ngeographically. A pair of peers in the same AS typically have a\nsmaller network latency compared to a pair of peers that are present\nin different ASes. For our experiments, we use a private torrent with\npeers distributed in emulated ASes. For our experiments, we\nemulate an AS using one machine. We assume the same intra-AS\nlatency and we also assume that the intra-AS latency is less than the \ninter-AS latency. We assume that all the ASes are fully meshed.\n\n\\subsection{Emulation of ASes}\n\n\\begin{table}\n\\begin{centering}\n\\begin{tabular}{|c|c|c|}\n\\hline\nAS & Latency on  & Latency on \\\\\n   & Loopback (ms) & Ethernet (ms)  \\\\\n\\hline\n$AS_{1}$ & 2 & 5 \\\\\n\\hline\n$AS_{2}$ & 5 & 15 \\\\\n\\hline\n$AS_3$ & 10 & 25 \\\\\n\\hline\n$AS_4$ & 25 & 100 \\\\\n\\hline\n$AS_5$ & 50 & 100\\\\\n\\hline\n\\end{tabular}\n\\caption{Latency values on the loopback and ethernet device while\n  emulating an AS on a machine.}\n\\label{tab:LatencyAS}\n\\end{centering}\n\\end{table}\n\\begin{table}\n\\begin{centering}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\n     & $AS_{1}$ & $AS_{2}$ & $AS_3$ & $AS_4$ & $AS_5$\\\\\n\\hline\n$AS_{1}$ & 8 ms  & 40 ms  &   60 ms &  210 ms & 210 ms\\\\\n\\hline\n$AS_{2}$ & 40 ms & 20 ms  &   80 ms &  230 ms & 230 ms\\\\\n\\hline\n$AS_3$ & 60 ms  & 80 ms  &   40 ms &  250 ms & 250 ms \\\\\n\\hline\n$AS_4$ & 210 ms & 230 ms &  250 ms&  100 ms & 400 ms\\\\\n\\hline\n$AS_5$ & 210 ms & 230 ms &  250 ms &  400 ms & 200 ms\\\\\n\\hline\n\\end{tabular}\n\\caption{RTT between a pair of leechers. \\scap{ RTT between a leecher in $AS_1$\n  and a leecher in $AS_5$ is 210 ms.}}\n\\label{tab:RTTAS}\n\\end{centering}\n\\end{table}\n\nAs in the case of homogeneous latency, we use four machines in each\nof the experiments. We use a private torrent with 300 leechers, one\ntracker, and one initial seed. We emulate four ASes: three ASes each\nwith 100 leechers, and the fourth AS to emulate the AS of the seed and\nthe tracker. The four ASes used in these experiments were chosen from\na set of five ASes ($AS_{1}$, $AS_{2}$, $AS_{3}$, $AS_{4}$, and\n$AS_{5}$).\n\nWe now present an explanation for emulating these five\nASes. In \\refFig{fig:SymmetricDownloadCompletion},\nfor an upload rate of 20~kB\/s, we observe that an RTT smaller than\n120~ms between any two peers has a smaller impact on the download\ncompletion time as compared to an RTT larger than 120~ms. In three of\nthe five ASes, namely $AS_{1}$, $AS_{2}$, and $AS_{3}$, the RTT\nbetween a pair of peers in these three ASes is less than 120~ms. The\nRTT between a pair of peers in $AS_{4}$ is less than 120~ms; the RTT\nbetween a peer in $AS_{4}$ and any other peer is greater than\n120~ms. Finally, a peer in $AS_{5}$ has an RTT greater than 120~ms\nwith any other peer. As we use one machine to emulate an AS, peers in \nthe same AS use the loopback device to communicate with each other;\nthe peers use the ethernet device to communicate with all the other\npeers in the torrent. We emulate the latency values given in\n\\refTab{tab:LatencyAS} for the ethernet and loopback device while\nusing a machine to emulate an AS.\n\nWe now give an example to show how \\refTab{tab:LatencyAS} can be used\nto find the RTT between a pair of leechers. The RTT between a leecher\nin $AS_1$ and a leecher in $AS_2$ is 40~ms (5+15+15+5) as the leechers\nuse the ethernet device to communicate with each other. The RTT\nbetween a pair of leechers in $AS_1$ is 8~ms (2+2+2+2) as the\nleechers use the loopback device to communicate with each\nother. \\refTab{tab:RTTAS} gives the RTT values between all such pairs\nof leechers.\n\n\\begin{table}\n\\begin{centering}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\n     & $AS_{1}$ & $AS_{2}$ & $AS_3$ & $AS_4$ & $AS_5$\\\\\n\\hline\n$AS_{1}'$ & 20 ms  & 40 ms  &   60 ms &  210 ms & 210 ms\\\\\n\\hline\n$AS_{2}'$ & 40 ms & 60 ms  &   80 ms &  230 ms & 230 ms\\\\\n\\hline\n$AS_3'$ & 60 ms  & 80 ms  &   100 ms &  250 ms & 250 ms \\\\\n\\hline\n$AS_4'$ & 210 ms & 230 ms &  250 ms&  400 ms & 400 ms\\\\\n\\hline\n$AS_5'$ & 210 ms & 230 ms &  250 ms &  400 ms & 400 ms\\\\\n\\hline\n\\end{tabular}\n\\caption{RTT between the seed and the leechers in the torrent. \\scap{$AS_{i}'$\n  indicates that seed is placed in as AS with latency values similar\n  to $AS_{i}$. RTT between the seed in $AS_1'$ and a peer in $AS_1$ is\n  20 ms.}}\n\\label{tab:RTTASSeed}\n\\end{centering}\n\\end{table}\n\nFor our experiments, we assume that the seed and the tracker are\nplaced in a dedicated AS with no leechers. We use $AS_i'$ to denote\nthat the seed and the tracker are placed in an AS with the same\nlatency values as $AS_i$. For example, $AS_{1}'$ implies that the seed\nand tracker are placed in an AS having the same latency values as\n$AS_{1}$. \\refTab{tab:RTTASSeed} gives the RTT values between the seed\nand the leechers.\n\n\n\\subsection{Presentation and Discussion of Results}\n\n\\begin{figure}\n\\begin{centering}\n\\subfloat[][Download completion time for leechers present in $AS_{1}$, $AS_{2}$, and\n$AS_{3}$. The difference in the average download completion time is\nless than 15\\%.]{\\label{fig:Asym20AS123}\\includegraphics[width=\\columnwidth]{asym-up-20-dct-AS-1-2-3.eps}}\n\n\\subfloat[][Download completion time for leechers present in $AS_{1}$, $AS_{3}$, and\n$AS_{4}$. An RTT of 400~ms between a leecher in $AS_4$ and the seed\nin $AS_4'$ does not increase the average download completion time by\nmore than 15\\%.]{\\label{fig:Asym20AS134}\\includegraphics[width=\\columnwidth]{asym-up-20-dct-AS-1-3-4.eps}}\n\n\\subfloat[][Download completion time for leechers present in $AS_{1}$, $AS_{3}$, and\n$AS_{5}$. An RTT of 400~ms between a leecher in $AS_4$ and the seed\nin $AS_4'$ does not increase the average download completion time by\nmore than 15\\%.]{\\label{fig:Asym20AS135}\\includegraphics[width=\\columnwidth]{asym-up-20-dct-AS-1-3-5.eps}}\n\\caption{Download completion time of a 50 MB file by leechers in a\n  given AS when the maximum upload rate of all the peers is\n  20~kB\/s. \\scap{Despite the wide range of RTT values emulated, the\n  difference in the download completion time is less than 15\\%.}}\n\\label{fig:AsymUp20}\n\\end{centering}\n\\end{figure}\n\n\\refFig{fig:AsymUp20} show the impact of heterogeneous latency on the\ndownload completion time of a 50 MB file when the upload rate of the\npeers is limited to 20~kB\/s. The impact of network latency when the\nupload rate limit is 50~kB\/s is presented in \\refFig{fig:AsymUp50}. In\n\\refFig{fig:AsymUp20} and \n\\refFig{fig:AsymUp50}, the X-axis represents the AS of the leechers\npresent in the torrent, and the Y-axis represents the download\ncompletion time in seconds. The figures present the average download\ncompletion time over 10 iterations; the error bars indicate the\nminimum and maximum download completion time observed in 10\niterations.\n\n\\refFig{fig:Asym20AS123} shows the outcome of three experiments\nwith leechers in $AS_{1}$, $AS_{2}$, and $AS_{3}$. For a given\nexperiment, the seed was either in $AS_{1}'$, $AS_{2}'$, or\n$AS_{3}'$. Despite the different RTT values between the peers, in the\nthree experiments presented in \\refFig{fig:Asym20AS123}, we observe\nthat the difference in the average download completion time is less\nthan 15\\%. According to \\refTab{tab:RTTAS}, \nthe RTT between any two peers in these experiments was less than\n120~ms. For the experiments presented in \\refFig{fig:Asym20AS134} and\n\\refFig{fig:Asym20AS135}, a peer in $AS_1$ or $AS_3$ and \nanother peer in $AS_4$ or $AS_5$ have an RTT greater than 120~ms. \nDespite the wide range of RTT values, we observe that\nthe average download completion time in \\refFig{fig:Asym20AS134} and\nin \\refFig{fig:Asym20AS135} is not more than $15\\%$ of the average\ndownload completion time observed in \\refFig{fig:Asym20AS123}, i.e.,\nwhen the RTT between a pair of peers is less than 120~ms. \n\nAccording to \\refTab{tab:RTTAS}, a peer in $AS_5$ and the seed in\n$AS_5'$ have an RTT of 400~ms. We observe that the average download\ncompletion time in \\refFig{fig:Asym20AS135} is smaller than the\naverage download completion time observed in\n\\refFig{fig:SymmetricDownloadCompletion} for an upload rate limit of\n20~kB\/s and an RTT of 400~ms. This shows that \\emph{the scenario of\nhomogeneous latency can be used to give a worst case impact of network\nlatency for a given upload rate}. When the upload rate limit is set to\n50~kB\/s, in \\refFig{fig:AsymUp50} we observe that the difference\nin the average download completion time is less than 15\\%.     \n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=\\columnwidth]{asym-up-50-dct-AS-1-3-5.eps}\n\\caption{Download completion time of a 50 MB file by leechers present\n  in a given AS when the maximum upload rate of all the peers is\n  50~kB\/s. \\scap{Despite the wide range of RTT values emulated, the\n  difference in the download completion time is less than 15\\%.}}\n\\label{fig:AsymUp50}\n\\end{centering}\n\\end{figure}\n\n\\subsection{Summary}\n\nIn this section, we relax the condition of fixed latency between any\ntwo peers in a torrent. We observe that an RTT of up to 400~ms has a\nmarginal impact, less than 15\\%, on the average download completion\ntime. These observations show that the \\emph{upload process at the \n  peers is not sensitive to the variations in the TCP throughput due\n  to the change in latency}. These observations also confirm that, for\na given upload rate among the peers, \\emph{the scenario of homogeneous\nlatency provides an upper bound on the download completion time of a\nfile when the maximum latency between any two peers in a torrent is\nknown}. \n\n\n\\section{Impact of Packet Loss}\n\\label{sec:ImpactLosses}\n\nWe now present the impact of packet loss on the download completion\ntime of a file. For our experiments, we emulate a 5\\% packet loss on\nthe ingress interface of the loopback and ethernet devices of the\nmachines.\n\nIn \\refTab{tab:ImpactLossTSO}, we present the average download\ncompletion time of a 50~MB file for a given upload rate limit and a\ngiven RTT between any two peers. As in the case of homogeneous\nlatency, we consider a torrent consisting of a one tracker, one seed,\nand a flash crowd of 300 leechers. We observe the download completion\ntime for the following network conditions.\n\\begin{enumerate}\n\\item \\emph{Homogeneous latency and TSO is disabled}. We do not emulate\n  packet losses in this scenario. As packet losses are not\n  emulated, this setting gives the outcome of experiments performed on\n  clusters that do not support TSO. The results of this scenario are\n  discussed in detail in \\refSec{sec:HomogeneousLatency}.\n\\item \\emph{Homogeneous latency and TSO is enabled}. We do not emulate\n  packet losses in this scenario. The results present the outcome of\n  experiments performed without emulating packet loss on clusters that\n  support TSO.\n\\item \\emph{Homogeneous latency with a loss rate of 5\\% and TSO\n  disabled}. We use this scenario to emulate network conditions present\n  in the Internet.\n\\end{enumerate}\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c||c|c|c|}\n\\cline{3-5}\n\\multicolumn{2}{}{ }& \\multicolumn{3}{|c|}{\\bf Average Download Completion Time}\\\\\n\\hline\nUpload& RTT & {TSO disabled} & { TSO enabled} & { TSO disabled}\\\\\nRate & & { Loss Rate 0\\%} & { Loss Rate 0\\%} & { Loss Rate 5\\%}\\\\\n\\hline\n10 &  0 ms & 7314.7 s & 7268.4 s & 7359.1 s \\\\\n\\cline{2-5}\nkB\/s & 400 ms & 8006.0 s & 7823.4 s & 8183.6 s\\\\\n\\cline{2-5}\n & 1000 ms & 8274.19 s  & 8060.6 s & {8827.6 s}\\\\\n\\hline\n20 &  0 ms & 3634.9 s & 3728.3 s & 3711.2 s\\\\\n\\cline{2-5}\nkB\/s & 400 ms &4023.9 s & 3985.3 s & 4034.3 s \\\\\n\\cline{2-5}\n & 1000 ms & 3768.3 s & 3796.6 & 4102.7 s \\\\\n\\hline\n50 &  0 ms & 1437.96 s& 1433.7 s & 1432.7s \\\\\n\\cline{2-5}\nkB\/s & 400 ms & 1457.2 s& 1463.9 s& 1476.9\\\\\n\\cline{2-5}\n & 1000 ms & 1466.9 s& 1470.5 s & 1638.2 s\\\\\n\\hline\n100 &  0 ms & 838.9 s & 828.9 s & 832.7 s\\\\\n\\cline{2-5}\nkB\/s &  400 ms & 863.4 s & 860.4 s & 940.20 s\\\\\n\\cline{2-5}\n &  1000 ms & 844.5 s & 865.4 s & {1619.87 s} \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Impact of RTT and loss rate on download completion\n  time. \\scap{An RTT of up to 400~ms and the loss rate of up to 5\\%\n    does not increase the average download completion time by more\n    than 15\\% of the average download completion time observed when\n    the RTT is 0~ms and the loss rate is 0\\%.}}\n\\label{tab:ImpactLossTSO}\n\\vspace{-0.15in}\n\\end{table}\n\nIn \\refSec{sec:HomogeneousLatency}, we observed the impact of network\nlatency when TSO is disabled and packet loss is not emulated. We observe\nthat for an RTT of up to 1000~ms, TCP ramp-up and the delays in\nreceiving the control messages have a marginal impact, less than 15\\%,\non the average download completion time of a file.\n\nWhen TSO in enabled and packet losses are not emulated, we observe\nthat the impact of network latency on the download completion time is\nsimilar to that observed in the case of homogeneous latency, which has\nbeen presented in \\refSec{sec:HomogeneousLatency}. These results show\nthat \\emph{BitTorrent experiments performed on clusters that support TSO \nshall produce results that are similar to those performed on clusters\nthat do not support TSO}.\n\nWe now discuss the scenario of homogeneous latency with packet loss\nwhen TSO is disabled. While emulating a packet loss, each packet\nen-queued by NetEm is dropped with a probability controlled by the\nloss rate. A packet loss result in retransmission of the packet by\nthe source. The source reduces the congestion window in response\nto the loss and then ramps-up its congestion window to attain the\ndesired upload rate. We observe that for an upload rate limit of\n20~kB\/s and 50~kB\/s, an RTT of 1000~ms and a loss rate of $5\\%$\nbetween any two peers does not increase the average download completion time\nby more than $15\\%$ of the average download completion time observed\nwhen \\emph{neither latency nor packet loss was emulated}. However,\nwhen the upload rate limit is 10~kB\/s or 100~kB\/s, the RTT between any\ntwo peers is 1000~ms, and when the loss rate is 5\\%, we observe that the\naverage download completion time is more than 15\\% of the download\ncompletion time when the RTT is 0~ms and the loss rate is 0\\%. An RTT\nof 1000~ms between any two peers is unrealistic as 99.8\\% links probed\nby iPlane have an RTT less than 1000~ms. However, \\emph{an RTT of up\nto 400~ms is realistic} as 98\\% of the links probed by iPlane have an\nRTT less than 400~ms. For the upload rate limit from 10~kB\/s to\n100~kB\/s, when the RTT between any two peers is 400~ms and the loss\nrate is 5\\%, we observe that the download completion time is not more\nthan 15\\% of the download completion time observed when the RTT is\n0~ms and loss rate is 0\\%. In \\refSec{sec:HeterogeneousLatency} we\nobserve that the scenario of homogeneous latency gives an upper bound\non the impact of a given network latency. Therefore, the results\npresented in \\refTab{tab:ImpactLossTSO} show that, for an upload rate\nlimit from 10~kB\/s to 100~kB\/s, \\emph{a loss rate of up to 5\\% and an\n  RTT of up to 400~ms between any two peers in the torrent has a\n  marginal impact, less than 15\\%, on the average download completion\n  time of a file.} \n\n\\section{Concluding Remarks}\n\\label{sec:Conclusion}\n\nIn this paper we present the impact of network latency and packet\nloss on the download completion time of a file distributed using\nBitTorrent. We use the download completion time as the metric for\nevaluation because the BitTorrent users are primarily interested in\nthe download completion time of a file.\n\nWe first studied the impact of network latency on the download\ncompletion time. For a given upload rate limit, we emulated the same\nlatency among the peers to give a worst case impact of network latency\non the download completion time. The download completion time can be\naffected by TCP ramp-up and the delays in receiving the BitTorrent\ncontrol messages. We therefore studied the impact of network latency\non the download completion time by studying the impact of network\nlatency on TCP ramp-up and the delays in receiving the control\nmessages. For our experiments, we varied the upload rate limit from\n10~kB\/s to 100~kB\/s and the RTT from 0~ms to 1000~ms. We observe that\nthe TCP ramp-up and the delays in receiving the BitTorrent control\nmessages only have a marginal impact, less than 15\\%, on the average\ndownload completion time. The high RTT values used in our experiments\nalso emulate torrents with peers that are not only geographically\napart but also connected with high capacity links that support the\nBitTorrent upload rate without causing congestion. Our results show\nthat \\emph{experiments performed on Grid'5000 give results similar to\nthose performed on testbeds such as PlanetLab that have geographically\ndistributed hosts that are connected by high capacity links}. We also\nstudy the impact of network latency on the delays in receiving\nthe control messages; this impact cannot be captured using the \ntraditional models for TCP throughput.\n\nWe then studied the impact of packet loss by emulating a loss rate of\n5\\%. For the upload rates seen in public torrents, from 10~kB\/s to\n100~kB\/s, we observe that realistic RTT values of up to 400~ms and a\npacket loss rate up to 5\\%, have a marginal impact, less than 15\\%, on\nthe average download completion time.  We performed our experiments\nover a wide range of RTT values, from 0~ms to 1000~ms, and a wide\nrange of packet loss rates, from 0\\% to 5\\%, to study the impact of\nnetwork latency and packet loss. Our results show that\n\\emph{experiments can be performed on dedicated clusters, such as\n  those present in Grid'5000, without explicitly emulating latency and\n  packet loss between the peers in a torrent}.    \n\nWe also studied the impact of using devices that support TSO on the\noutcome of BitTorrent experiments performed on clusters. Our results\nshow that BitTorrent experiments performed on clusters that support TSO\nproduce results that are similar to those performed on dedicated\nclusters that do not support TSO.  \n\nOur main conclusion is that, for upload rates seen in public torrents,\nnetwork latency and packet loss have a marginal impact on the download\ncompletion time of a file, hence, dedicated clusters such as Grid'5000\ncan be safely used to perform realistic and reproducible BitTorrent\nexperiments.   \n\n\\section{Acknowledgment}\n\nExperiments presented in this paper were carried out using the Grid'5000\nexperimental testbed, being developed under the INRIA ALADDIN development\naction with support from CNRS, RENATER and several Universities as well\nas other funding bodies (see https:\/\/www.grid5000.fr).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}