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{"url":"https:\/\/johncarlosbaez.wordpress.com\/category\/quantum-technologies\/page\/2\/","text":"Time Crystals\n\n26 September, 2012\n\nWhen water freezes and forms a crystal, it creates a periodic pattern in space. Could there be something that crystallizes to form a periodic pattern in time? Frank Wilczek, who won the Nobel Prize for helping explain why quarks and gluons trapped inside a proton or neutron act like freely moving particles when you examine them very close up, dreamt up this idea and called it a time crystal:\n\n\u2022 Frank Wilczek, Classical time crystals.\n\n\u2022 Frank Wilczek, Quantum time crystals.\n\n\u2018Time crystals\u2019 sound like something from Greg Egan\u2019s Orthogonal trilogy, set in a universe where there\u2019s no fundamental distinction between time and space. But Wilczek wanted to realize these in our universe.\n\nOf course, it\u2019s easy to make a system that behaves in an approximately periodic way while it slowly runs down: that\u2019s how a clock works: tick tock, tick tock, tick tock\u2026 But a system that keeps \u2018ticking away\u2019 without using up any resource or running down would be a strange new thing. There\u2019s no telling what weird stuff we might do with it.\n\nIt\u2019s also interesting because physicists love symmetry. In ordinary physics there are two very important symmetries: spatial translation symmetry, and time translation symmetry. Spatial translation symmetry says that if you move an experiment any amount to the left or right, it works the same way. Time translation symmetry says that if you do an experiment any amount of time earlier or later, it works the same way.\n\nCrystals are fascinating because they \u2018spontaneously break\u2019 spatial translation symmetry. Take a liquid, cool it until it freezes, and it forms a crystal which does not look the same if you move it any amount to the right or left. It only looks the same if you move it certain discrete steps to the right or left!\n\nThe idea of a \u2018time crystal\u2019 is that it\u2019s a system that spontaneously breaks time translation symmetry.\n\nGiven how much physicists have studied time translation symmetry and spontaneous symmetry breaking, it\u2019s sort of shocking that nobody before 2012 wrote about this possibility. Or maybe someone did\u2014but I haven\u2019t heard about it.\n\nIt takes real creativity to think of an idea so radical yet so simple. But Wilczek is famously creative. For example, he came up with anyons: a new kind of particle, neither boson nor fermion, that lives in a universe where space is 2-dimensional. And now we can make those in the lab.\n\nUnfortunately, Wilczek didn\u2019t know how to make a time crystal. But now a team including Xiang Zhang (seated) and Tongcang Li (standing) at U.C. Berkeley have a plan for how to do it.\n\nActually they propose a ring-shaped system that\u2019s periodic in time and also in space, as shown in the picture. They call it a space-time crystal:\n\nHere we propose a space-time crystal of trapped ions and a method to realize it experimentally by confining ions in a ring-shaped trapping potential with a static magnetic field. The ions spontaneously form a spatial ring crystal due to Coulomb repulsion. This ion crystal can rotate persistently at the lowest quantum energy state in magnetic fields with fractional fluxes. The persistent rotation of trapped ions produces the temporal order, leading to the formation of a space-time crystal. We show that these space-time crystals are robust for direct experimental observation. The proposed space-time crystals of trapped ions provide a new dimension for exploring many-body physics and emerging properties of matter.\n\nThe new paper is here:\n\n\u2022 Tongcang Li, Zhe-Xuan Gong, Zhang-Qi Yin, H. T. Quan, Xiaobo Yin, Peng Zhang, L.-M. Duan and Xiang Zhang, Space-time crystals of trapped ions.\n\nAlas, the press release put out by Lawrence Berkeley National Laboratory is very misleading. It describes the space-time crystal as a clock that\n\nwill theoretically persist even after the rest of our universe reaches entropy, thermodynamic equilibrium or \u201cheat-death\u201d.\n\nNO!\n\nFirst of all, \u2018reaching entropy\u2019 doesn\u2019t mean anything. More importantly, as time goes by and things fall apart, this space-time crystal, assuming anyone can actually make it, will also fall apart.\n\nI know what the person talking to the reporter was trying to say: the cool thing about this setup is that it gives a system that\u2019s truly time-periodic, not gradually using up some resource and running down like an ordinary clock. But nonphysicist readers, seeing an article entitled \u2018A Clock that Will Last Forever\u2019, may be fooled into thinking this setup is, umm, a clock that will last forever. It\u2019s not.\n\nIf this setup were the whole universe, it might keep ticking away forever. But in fact it\u2019s just a small, carefully crafted portion of our universe, and it interacts with the rest of our universe, so it will gradually fall apart when everything else does\u2026 or in fact much sooner: as soon as the scientists running it turn off the experiment.\n\nClassifying space-time crystals\n\nWhat could we do with space-time crystals? It\u2019s way too early to tell, at least for me. But since I\u2019m a mathematician, I\u2019d be happy to classify them. Over on Google+, William Rutiser asked if there are 4d analogs of the 3d crystallographic groups. And the answer is yes! Mathematicians with too much time on their hands have classified the analogues of crystallographic groups in 4 dimensions:\n\nSpace group: classification in small dimensions, Wikipedia.\n\nIn general these groups are called space groups (see the article for the definition). In 1 dimension there are just two, namely the symmetry groups of this:\n\n\u2014 o \u2014 o \u2014 o \u2014 o \u2014 o \u2014 o \u2014\n\nand this:\n\n\u2014 > \u2014 > \u2014 > \u2014 > \u2014 > \u2014 > \u2014\n\nIn 2 dimensions there are 17 and they\u2019re called wallpaper groups. In 3 dimensions there are 230 and they are called crystallographic groups. In 4 dimensions there are 4894, in 5 dimensions there are\u2026 hey, Wikipedia leaves this space blank in their table!\u2026 and in 6 dimensions there are 28,934,974.\n\nSo, there is in principle quite a large subject to study here, if people can figure out how to build a variety of space-time crystals.\n\nThere\u2019s already book on this, if you\u2019re interested:\n\n\u2022 Harold Brown, Rolf Bulow, Joachim Neubuser, Hans Wondratschek and Hans Zassenhaus, Crystallographic Groups of Four-Dimensional Space, Wiley Monographs in Crystallography, 1978.\ufeff\n\nQuantizing Electrical Circuits\n\n2 February, 2012\n\nAs you may know, there\u2019s a wonderful and famous analogy between classical mechanics and electrical circuit theory. I explained it back in \u201cweek288\u201d, so I won\u2019t repeat that story now. If you don\u2019t know what I\u2019m talking about, take a look!\n\nThis analogy opens up the possibility of quantizing electrical circuits by straightforwardly copying the way we quantize classical mechanics problems. I\u2019d often wondered if this would be useful.\n\nIt is, and people have done it:\n\n\u2022 Michel H. Devoret, Quantum fluctuations in electrical circuits.\n\nMichel Devoret, Rob Schoelkopf and others call this idea quantronics: the study of mesoscopic electronic effects in which collective degrees of freedom like currents and voltages behave quantum mechanically.\n\nI just learned about this from a talk by Sean Barrett here in Coogee. There are lots of cool applications, but right now I\u2019m mainly interested in how this extends the set of analogies between different physical theories.\n\nOne interesting thing is how they quantize circuits with resistors. Over in classical mechanics, this corresponds to systems with friction. These systems, called \u2018dissipative\u2019 systems, don\u2019t have a conserved energy. More precisely, energy leaks out of the system under consideration and gets transferred to the environment in the form of heat. It\u2019s hard to quantize systems where energy isn\u2019t conserved, so people in quantronics model resistors as infinite chains of inductors and capacitors: see the \u2018LC ladder circuit\u2019 on page 15 of Devoret\u2019s notes. This idea is also the basis of the Caldeira\u2013Leggett model of a particle coupled to a heat bath made of harmonic oscillators: it amounts to including the environment as part of the system being studied.\n\nA Quantum Hammersley\u2013Clifford Theorem\n\n29 January, 2012\n\nI\u2019m at this workshop:\n\nSydney Quantum Information Theory Workshop: Coogee 2012, 30 January \u2013 2 February 2012, Coogee Bay Hotel, Coogee, Sydney, organized by Stephen Bartlett, Gavin Brennen, Andrew Doherty and Tom Stace.\n\nRight now David Poulin is speaking about a quantum version of the Hammersley\u2013Clifford theorem, which is a theorem about Markov networks. Let me quickly say a bit about what he proved! This will be a bit rough, since I\u2019m doing it live\u2026\n\nThe mutual information between two random variables is\n\n$I(A:B) = S(A) + S(B) - S(A,B)$\n\nThe conditional mutual information between three random variables $C$ is\n\n$I(A:B|C) = \\sum_c p(C=c) I(A:B|C=c)$\n\nIt\u2019s the average amount of information about $B$ learned by measuring $A$ when you already knew $C.$\n\nAll this works for both classical (Shannon) and quantum (von Neumann) entropy. So, when we say \u2018random variable\u2019 above, we\ncould mean it in the traditional classical sense or in the quantum sense.\n\nIf $I(A:B|C) = 0$ then $A, C, B$ has the following Markov property: if you know $C,$ learning $A$ tells you nothing new about $B.$ In condensed matter physics, say a spin system, we get (quantum) random variables from measuring what\u2019s going on in regions, and we have short range entanglement if $I(A:B|C) = 0$ when $C$ corresponds to some sufficiently thick region separating the regions $A$ and $B.$ We\u2019ll get this in any Gibbs state of a spin chain with a local Hamiltonian.\n\nA Markov network is a graph with random variables at vertices (and thus subsets of vertices) such that $I(A:B|C) = 0$ whenever $C$ is a subset of vertices that completely \u2018shields\u2019 the subset $A$ from the subset $B$: any path from $A$ to $B$ goes through a vertex in a $C.$\n\nThe Hammersley\u2013Clifford theorem says that in the classical case we can get any Markov network from the Gibbs state\n\n$\\exp(-\\beta H)$\n\nof a local Hamiltonian $H,$ and vice versa. Here a Hamiltonian is local if it is a sum of terms, one depending on the degrees of freedom in each clique in the graph:\n\n$H = \\sum_{C \\in \\mathrm{cliques}} h_C$\n\nHayden, Jozsa, Petz and Winter gave a quantum generalization of one direction of this result to graphs that are just \u2018chains\u2019, like this:\n\no\u2014o\u2014o\u2014o\u2014o\u2014o\u2014o\u2014o\u2014o\u2014o\u2014o\u2014o\n\nNamely: for such graphs, any quantum Markov network is the Gibbs state of some local Hamiltonian. Now Poulin has shown the same for all graphs. But the converse is, in general, false. If the different terms $h_C$ in a local Hamiltonian all commute, its Gibbs state will have the Markov property. But otherwise, it may not.\n\nFor some related material, see:\n\n\u2022 David Poulin, Quantum graphical models and belief propagation.\n\nProbabilities Versus Amplitudes\n\n5 December, 2011\n\nHere are the slides of the talk I\u2019m giving at the CQT Annual Symposium on Wednesday afternoon, which is Tuesday morning for a lot of you. If you catch mistakes, I\u2019d love to hear about them before then!\n\nAbstract: Some ideas from quantum theory are just beginning to percolate back to classical probability theory. For example, there is a widely used and successful theory of \u201cchemical reaction networks\u201d, which describes the interactions of molecules in a stochastic rather than quantum way. If we look at it from the perspective of quantum theory, this turns out to involve creation and annihilation operators, coherent states and other well-known ideas\u2014but with a few big differences. The stochastic analogue of quantum field theory is also used in population biology, and here the connection is well-known. But what does it mean to treat wolves as fermions or bosons?\n\nLiquid Light\n\n28 November, 2011\n\nElisabeth Giacobino works at the Ecole Normale Sup\u00e9rieure in Paris. Last week she gave a talk at the Centre for Quantum Technologies. It was about \u2018polariton condensates\u2019. You can see a video of her talk here.\n\nWhat\u2019s a polariton? It\u2019s a strange particle: a blend of matter and light. Polaritons are mostly made of light\u2026 with just enough matter mixed in so they can form a liquid! This liquid can form eddies just like water. Giacobino and her team of scientists have actually gotten pictures:\n\nPhysicists call this liquid a \u2018polariton condensate\u2019, but normal people may better appreciate how wonderful it is if we call it liquid light. That\u2019s not 100% accurate, but it\u2019s close\u2014you\u2019ll see what I mean in a minute.\n\nHere\u2019s a picture of Elisabeth Giacobino (at right) and her coworkers in 2010\u2014not exactly the same team who is working on liquid light, but the best I can find:\n\nHow to make liquid light\n\nHow do you make liquid light?\n\nFirst, take a thin film of some semiconductor like gallium arsenide. It\u2019s full of electrons roaming around, so imagine a sea of electrons, like water. If you knock out an electron with enough energy, you\u2019ll get a \u2018hole\u2019 which can move around like a particle of its own. Yes, the absence of a thing can act like a thing. Imagine an air bubble in the sea.\n\nAll this so far is standard stuff. But now for something more tricky: if you knock an electron just a little, it won\u2019t go far from the hole it left behind. They\u2019ll be attracted to each other, so they\u2019ll orbit each other!\n\nWhat you\u2019ve got now is like a hydrogen atom\u2014but instead of an electron and a proton, it\u2019s made from an electron and a hole! It\u2019s called an exciton. In Giacobino\u2019s experiments, the excitons are 200 times as big as hydrogen atoms.\n\nExcitons are exciting, but not exciting enough for us. So next, put a mirror on each side of your thin film. Now light can bounce back and forth. The light will interact with the excitons. If you do it right, this lets a particle of light\u2014called a photon\u2014blend with an exciton and form a new particle called polariton.\n\nHow does a photon \u2018blend\u2019 with an exciton? Umm, err\u2026 this involves quantum mechanics. In quantum mechanics you can take two possible situations and add them and get a new one, a kind of \u2018blend\u2019 called a \u2018superposition\u2019. \u2018Schr\u00f6dinger\u2019s cat\u2019 is what you get when you blend a live cat and a dead cat. People like to argue about why we don\u2019t see half-live, half-dead cats. But never mind: we can see a blend of a photon and an exciton! Giacobino and her coworkers have done just that.\n\nThe polaritons they create are mostly light, with just a teeny bit of exciton blended in. Photons have no mass at all. So, perhaps it\u2019s not surprising that their polaritons have a very small mass: about 10-5 times as heavy as an electron!\n\nThey don\u2019t last very long: just about 4-10 picoseconds. A picosecond is a trillionth of a second, or 10-12 seconds. After that they fall apart. However, this is long enough for polaritons to do lots of interesting things.\n\nFor starters, polaritons interact with each other enough to form a liquid. But it\u2019s not just any ordinary liquid: it\u2019s often a superfluid, like very cold liquid helium. This means among other things, that it has almost no viscosity.\n\nSo: it\u2019s even better than liquid light: it\u2019s superfluid light!\n\nThe flow of liquid light\n\nWhat can you do with liquid light?\n\nFor starters, you can watch it flow around obstacles. Semiconductors have \u2018defects\u2019\u2014little flaws in the crystal structure. These act as obstacles to the flow of polaritons. And Giacobimo and her team have seen the flow of polaritons around defects in the semiconductor:\n\nThe two pictures at left are two views of the polariton condensate flowing smoothly around a defect. In these pictures the condensate is a superfluid.\n\nThe two pictures in the middle show a different situation. Here the polariton condensate is viscous enough so that it forms a trail of eddies as it flows past the defect. Yes, eddies of light!\n\nAnd the two pictures at right show yet another situation. In every fluid, we can have waves of pressure. This is called\u2026 \u2018sound\u2019. Yes, this is how ordinary sound works in air, or under water. But we can also have sound in a polariton condensate!\n\nThat\u2019s pretty cool: sound in liquid light! But wait. We haven\u2019t gotten to the really cool part yet. Whenever you have a fluid moving past an obstacle faster than the speed of sound, you get a \u2018shock wave\u2019: the obstacle leaves an expanding trail of sound in its wake, behind it, because the sound can\u2019t catch up. That\u2019s why jets flying faster than sound leave a sonic boom behind them.\n\nAnd that\u2019s what you\u2019re seeing in the pictures at right. The polariton condensate is flowing past the defect faster than the speed of sound, which happens to be around 850,000 meters per second in this experiment. We\u2019re seeing the shock wave it makes. So, we\u2019re seeing a sonic boom in liquid light!\n\nIt\u2019s possible we\u2019ll be able to use polariton condensates for interesting new technologies. Giacobimo and her team are also considering using them to study Hawking radiation: the feeble glow that black holes emit according to Hawking\u2019s predictions. There aren\u2019t black holes in polariton condensates, but it may be possible to create a similar kind of radiation. That would be really cool!\n\nBut to me, just being able to make a liquid consisting mostly of light, and study its properties, is already a triumph: just for the beauty of it.\n\nScary technical details\n\nAll the pictures of polariton condensates flowing around a defect came from here:\n\n\u2022 A. Amo, S. Pigeon, D. Sanvitto, V. G. Sala, R. Hivet, I. Carusotto, F. Pisanello, G. Lemenager, R. Houdre, E. Giacobino, C. Ciuti, and A. Bramati, Hydrodynamic solitons in polariton superfluids.\n\nand this is the paper to read for more details.\n\nI tried to be comprehensible to ordinary folks, but there are a few more things I can\u2019t resist saying.\n\nFirst, there are actually many different kinds of polaritons. In general, polaritons are quasiparticles formed by the interaction of photons and matter. For example, in some crystals sound acts like it\u2019s made of particles, and these quasiparticles are called \u2018phonons\u2019. But sometimes phonons can interact with light to form quasiparticles\u2014and these are called \u2018phonon-polaritons\u2019. I\u2019ve only been talking about \u2018exciton-polaritons\u2019.\n\nIf you know a bit about superfluids, you may be interested to hear that the wavy patterns show the phase of the order parameter \u03c8 in the Landau-Ginzburg theory of superfluids:\n\nIf you know about quantum field theory, you may be interested to know that the Hamiltonian describing photon-exciton interactions involves terms roughly like\n\n$\\alpha a^\\dagger a + \\beta b^\\dagger b + \\gamma (a^\\dagger b + b^\\dagger a)$\n\nwhere $a$ is the annihilation operator for photons, $b$ is the annihilation operator for excitons, the Greek letters are various constants, and the third term describes the interaction of photons and excitons. We can simplify this Hamiltonian by defining new particles that are linear combinations of photons and excitons. It\u2019s just like diagonalizing a matrix; we get something like\n\n$\\delta c^\\dagger c + \\epsilon d^\\dagger d$\n\nwhere $c$ and $d$ are certain linear combinations of $a$ and $b$. These act as annihilation operators for our new particles\u2026 and one of these new particles is the very light \u2018polariton\u2019 I\u2019ve been talking about!\n\nIs Life Improbable?\n\n31 May, 2011\n\nMine? Yes. And maybe you\u2019ve wondered just how improbable your life is. But that\u2019s not really the question today\u2026\n\nHere at the Centre for Quantum Technologies, Dagomir Kaszlikowski asked me to give a talk on this paper:\n\n\u2022 John Baez, Is life improbable?, Foundations of Physics 19 (1989), 91-95.\n\nThis was the second paper I wrote, right after my undergraduate thesis. Nobody ever seemed to care about it, so it\u2019s strange\u2014but nice\u2014to finally be giving a talk on it.\n\nMy paper does not try to settle the question its title asks. Rather, it tries to refute the argument here:\n\n\u2022 Eugene P. Wigner, The probability of the existence of a self-reproducing unit, Symmetries and Reflections, Indiana University Press, Bloomington, 1967, pp. 200-208.\n\nAccording Wigner, his argument\n\npurports to show that, according to standard quantum mechanical theory, the probability is zero for the existence of self-reproducing states, i.e., organisms.\n\nGiven how famous Eugene Wigner is (he won a Nobel prize, after all) and how earth-shattering his result would be if true, it\u2019s surprising how little criticism his paper has received. David Bohm mentioned it approvingly in 1969. In 1974 Hubert Yockey cited it saying\n\nfor all physics has to offer, life should never have appeared and if it ever did it would soon die out.\n\nAs you\u2019d expect, there are some websites mentioning Wigner\u2019s argument as evidence that some supernatural phenomenon is required to keep life going. Wigner himself believed it was impossible to formulate quantum theory in a fully consistent way without referring to consciousness. Since I don\u2019t believe either of these claims, I think it\u2019s good to understand the flaw in Wigner\u2019s argument.\n\nSo, let me start by explaining his argument. Very roughly, it purports to show that if there are many more ways a chunk of matter can be \u2018dead\u2019 than \u2018living\u2019, the chance is zero that we can choose some definition of \u2018living\u2019 and a suitable \u2018nutrient\u2019 state such that every \u2018living\u2019 chunk of matter can interact with this \u2018nutrient\u2019 state to produce two \u2018living\u2019 chunks.\n\nIn making this precise, Wigner considers more than just two chunks of matter: he also allows there to be an \u2018environment\u2019. So, he considers a quantum system made of three parts, and described by a Hilbert space\n\n$H = H_1 \\otimes H_1 \\otimes H_2$\n\nHere the first $H_1$ corresponds to a chunk of matter. The second $H_1$ corresponds to another chunk of matter. The space $H_3$ corresponds to the \u2018environment\u2019. Suppose we wait for a certain amount of time and see what the system does; this will be described by some unitary operator\n\n$S: H \\to H$\n\nWigner asks: if we pick this operator $S$ in a random way, what\u2019s the probability that there\u2019s some $n$-dimensional subspace of \u2018living organism\u2019 states in $H_1$, and some \u2018nutrient plus environment\u2019 state in $H_1 \\otimes H_2$, such that the time evolution sends any living organism together with the nutrient plus environment to two living organisms and some state of the environment?\n\nA bit more precisely: suppose we pick $S$ in a random way. Then what\u2019s the probability that there exists an $n$-dimensional subspace\n\n$V \\subseteq H_1$\n\nand a state\n\n$w \\in H_1 \\otimes H_2$\n\nsuch that $S$ maps every vector in $V \\otimes \\langle w \\rangle$ to a vector in $V \\otimes V \\otimes H_2$? Here $\\langle w \\rangle$ means the 1-dimensional subspace spanned by the vector $w$.\n\n$\\mathrm{dim}(H_1) \\gg n$\n\nthen this probability is zero.\n\nYou may need to reread the last few paragraphs a couple times to understand Wigner\u2019s question, and his answer. In case you\u2019re still confused, I should say that $V \\subseteq H_1$ is what I\u2019m calling the space of \u2018living organism\u2019 states of our chunk of matter, while $w \\in H_1 \\otimes H_2$ is the \u2018nutrient plus environment\u2019 state.\n\nNow, Wigner did not give a rigorous proof of his claim, nor did he say exactly what he meant by \u2018probability\u2019: he didn\u2019t specify a probability measure on the space of unitary operators on $H$. But if we use the obvious choice (called \u2018normalized Haar measure\u2019) his argument can most likely be turned into a proof.\n\nSo, I don\u2019t want to argue with his math. I want to argue with his interpretation of the math. He concludes that\n\nthe chances are nil for the existence of a set of \u2018living\u2019 states for which one can find a nutrient of such nature that interaction always leads to multiplication.\n\nThe problem is that he fixed the decomposition of the Hilbert space $H$ as a tensor product\n\n$H = H_1 \\otimes H_1 \\otimes H_2$\n\nbefore choosing the time evolution operator $S$. There is no good reason to do that. It only makes sense split up a physical into parts this way after we have some idea of what the dynamics is. An abstract Hilbert space doesn\u2019t come with a favored decomposition as a tensor product into three parts!\n\nIf we let ourselves pick this decomposition after picking the operator $S$, the story changes completely. My paper shows:\n\nTheorem 1. Let $H$, $H_1$ and $H_2$ be finite-dimensional Hilbert spaces with $H \\cong H_1 \\otimes H_1 \\otimes H_2$. Suppose $S : H \\to H$ is any unitary operator, suppose $V$ is any subspace of $H_1$, and suppose $w$ is any unit vector in $H_1 \\otimes H_2$ Then there is a unitary isomorphism\n\n$U: H \\to H_1 \\otimes H_1 \\otimes H_2$\n\nsuch that if we identify $H$ with $H_1 \\otimes H_1 \\otimes H_2$ using $U$, the operator $S$ maps $V \\otimes \\langle w \\rangle$ into $V \\otimes V \\otimes H_2$.\n\nIn other words, if we allow ourselves to pick the decomposition after picking $S$, we can always find a \u2018living organism\u2019 subspace of any dimension we like, together with a \u2018nutrient plus environment\u2019 state that allows our living organism to reproduce.\n\nHowever, if you look at the proof in my paper, you\u2019ll see it\u2019s based on a kind of cheap trick (as I forthrightly admit). Namely, I pick the \u2018nutrient plus environment\u2019 state to lie in $V \\otimes H_2$, so the nutrient actually consists of another organism!\n\nThis goes to show that you have to be very careful about theorems like this. To prove that life is improbable, you need to find some necessary conditions for what counts as life, and show that these are improbable (in some sense, and of course it matters a lot what that sense is). Refuting such an argument does not prove that life is probable: for that you need some sufficient conditions for what counts as life. And either way, if you prove a theorem using a \u2018cheap trick\u2019, it probably hasn\u2019t gotten to grips with the real issues.\n\nI also show that as the dimension of $H$ approaches infinity, the probability approaches 1 that we can get reproduction with a 1-dimensional \u2018living organism\u2019 subspace and a \u2018nutrient plus environment\u2019 state that lies in orthogonal complement of $V \\otimes H_2$. In other words, the \u2018nutrient\u2019 is not just another organism sitting there all ready to go!\n\nMore precisely:\n\nTheorem 2. Let $H$, $H_1$ and $H_2$ be finite-dimensional Hilbert spaces with $\\mathrm{dim}(H) = \\mathrm{dim}(H_1)^2 \\cdot \\mathrm{dim}(H_2)$. Let $\\mathbf{S'}$ be the set of unitary operators $S: H \\to H$ with the following property: there\u2019s a unit vector $v \\in H_1$, a unit vector $w \\in V^\\perp \\otimes H_2$, and a unitary isomorphism\n\n$U: H \\to H_1 \\otimes H_1 \\otimes H_2$\n\nsuch that if we identify $H$ with $H_1 \\otimes H_1 \\otimes H_2$ using $U$, the operator $S$ maps $v \\otimes w$ into $\\langle v\\rangle \\otimes \\langle v \\rangle \\otimes H_2$. Then the normalized Haar measure of $\\mathbf{S'}$ approaches 1 as $\\mathrm{dim}(H) \\to \\infty$.\n\nHere $V^\\perp$ is the orthogonal complement of $V \\subseteq H_1$; that is, the space of all vectors perpendicular to $V$.\n\nI won\u2019t include the proofs of these theorems, since you can see them in my paper.\n\nJust to be clear: I certainly don\u2019t think these theorems prove that life is probable! You can\u2019t have theorems without definitions, and I think that coming up with a good general definition of \u2018life\u2019, or even supposedly simpler concepts like \u2018entity\u2019 and \u2018reproduction\u2019, is extremely tough. The formalism discussed here is oversimplified for dozens of reasons, a few of which are listed at the end of my paper. So far we\u2019re only in the first fumbling stages of addressing some very hard questions.\n\nAll my theorems do is point out that Wigner\u2019s argument has a major flaw: he\u2019s choosing a way to divide the world into chunks of matter and the environment before choosing his laws of physics. This doesn\u2019t make much sense, and reversing the order dramatically changes the conclusions.\n\nBy the way: I just started looking for post-1989 discussions of Wigner\u2019s paper. So far I haven\u2019t found any interesting ones. Here\u2019s a more recent paper that\u2019s somewhat related, which doesn\u2019t mention Wigner\u2019s work:\n\n\u2022 Indranil Chakrabarty and Prashant, Non existence of quantum mechanical self replicating machine, 2005.\n\nThe considerations here seem more closely related to the Wooters\u2013Zurek no-cloning theorem.\n\nQuantum Information Processing 2011 (Day\u00a02)\n\n12 January, 2011\n\nHere are some very fragmentary notes on the second day of QIP 2011. You can see arXiv references, slides, and videos of the talks here. I\u2019ll just give links to the slides, and again I\u2019ll only mention 3 talks.\n\nStephanie Werner gave a talk on the relation between the uncertainty principle and nonlocality in quantum theory. There\u2019s a general framework for physical theories, called \u201cgeneralized probabilistic theories\u201d, which includes classical and quantum mechanics as special cases. In this framework we can see that while quantum theory is \u201cnonlocal\u201d in the sense made famous by John Bell, even more nonlocal theories are logically possible!\n\nFor example, while quantum theory violates the Clauser\u2013Horn\u2013Shimony\u2013Holt inequality, which is obeyed by any local hidden variables theory, it doesn\u2019t violate it to the maximum possible extent. There is a logically conceivable gadget, the Popescu\u2013Rohrlich box, which violates the CHSH inequalities to the maximum extent allowed by a theory that prohibits faster-than-light signalling. However, such a gadget would give us implausibly godlike computational powers.\n\nIn Werner\u2019s talk, she explained another reason not to hope for more nonlocality than quantum theory provides. Namely, given the \u201csteering\u201d ability we have in quantum theory \u2014 that is, our ability to prepare a state at one location by doing a measurement at another \u2014 the theory cannot be more nonlocal than it is while still obeying the uncertainty principle.\n\nJ\u00e9r\u00e9mie Roland gave a talk on generalizations of Grover\u2019s search algorithm. Grover\u2019s algorithm is one of the implausibly godlike powers that quantum computers might give us, if only we could build the bloody things: it\u2019s a way to search a list of N items for a given item in a time that\u2019s only on the order of N1\/2. This algorithm assumes we can freely jump from place to place on the list, so instead of a linearly ordered \u201clist\u201d it\u2019s probably better to visualize this data structure as a complete graph with N vertices. Roland\u2019s talk explained a way to generalize this idea to arbitrary graphs.\n\nHe began by considering a Markov chain on the graph \u2014 that is, a way to blunder randomly from vertex to vertex along the graph, where you can only go from one vertex to another in one step if there\u2019s an edge connecting them. He assumed that it\u2019s reversible and ergodic. Then, starting from this, he described how to fashion a quantum process that finds the desired vertex (or vertices) in about the square root of the time that the Markov chain would take.\n\nRobin Kothari gave a talk on using quantum computation to efficiently detect certain properties of graphs. He considered \u201cminor-closed properties\u201d of graphs. Let me just tell you what those properties are, and tell you about a fascinating older result about them.\n\nThe word graph means many slightly different things, but in this blog entry I mean a finite set $V$ whose elements are called vertices, together with a set $E$ of unordered pairs of distinct vertices, which we call edges. So, these are undirected finite graphs without self-loops or multiple edges connecting vertices.\n\nA minor of a graph is another graph that can be obtained from the first one by repeatedly:\n\n1) removing edges,\n\n2) removing vertices that have no edges connected to them, and\n\n3) contracting edges, that is, shrinking them to nothing and then identifying the vertices at both ends, like this:\n\nFor example, this graph:\n\nis a minor of this one:\n\nA property of graphs is minor-closed if whenever one graph has it, all its minors also have it.\n\nWhat are some minor-closed properties? An obvious one is lacking cycles, that is, being a forest. You can get rid of cycles by getting rid of edges and vertices, or contracting edges, but you can\u2019t create them!\n\nA more interesting minor-closed property is planarity. If you can draw a graph on the plane, you can clearly also draw the graph you get by removing an edge, or removing a lone vertex, or contracting an edge.\n\nNow, Kuratowski showed that planar graphs as precisely those that don\u2019t have this graph:\n\nor this one:\n\nas minors.\n\nSimilarly, graphs that lack cycles are precisely those that don\u2019t have a triangle as a minor!\n\nSo, we could ask if this pattern generalizes. Given a minor-closed property of graphs, is it equivalent to a property saying that there\u2019s some finite set of graphs that don\u2019t show up as minors?\n\nThis is called Wagner\u2019s conjecture, after Klaus Wagner. He published it in 1970.\n\nWagner\u2019s conjecture is true! It was proved by Neil Robertson and Paul Seymour in 2004. But the really interesting thing, to me, is that their proof takes about 400 or 500 pages!\n\nI find this quite surprising\u2026 but then, I wouldn\u2019t have guessed the four-color theorem was so hard to prove, either.\n\nTo make sure you understand Wagner\u2019s conjecture, check that if we dropped the word \u201cfinite\u201d, it would have a one-sentence proof.","date":"2019-05-22 18:09:38","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 102, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6646244525909424, \"perplexity\": 729.4533261310726}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-22\/segments\/1558232256887.36\/warc\/CC-MAIN-20190522163302-20190522185302-00053.warc.gz\"}"} | null | null |
\section{Introduction}
Mathematical models have proven to be successful in understanding infectious disease dynamics. Often, the focus is on (controlling) the beginning of an epidemic. In many of those models branching process approximations and the concept of the basic reproduction number $R_0$ (which corresponds to the offspring mean of the approximating branching process) play an important role. In the current paper we consider the entire epidemic outbreak instead. We do so in a setting where there are multiple types of infected individuals. Suppose that a large epidemic outbreak has taken place in the population. Then a certain fraction of the population will have become infected. The question that we concern ourselves with is: what fraction of the infected population of a certain type $j$ was infected by individuals of type $i$? Here $i,j=1,\ldots K$, where $K$ is the number of types in the population. In other words, who is the infector?
This question is not so straightforward to answer. Timing of events plays an essential role. We approach this question using an epidemic graph construction, which is used as a tool in proving the two main theorems of this paper. But, the construction proves to be interesting in itself, and a substantial part of this paper is devoted to this construction. Using the graph representation of the epidemic, we consider susceptibility processes. Susceptibility \emph{sets} were introduced in infectious disease modelling by Ball and co-authors \cite{Ball01,Ball02}. The susceptibility set of a vertex $v$ consists of all vertices $u$ in the vertex set from which there is a path from $u$ to $v$ in the (restricted) epidemic random graph. The epidemic process and epidemic random graph can be coupled in such a way that $u$ is in the susceptibility set of $v$ if and only if $v$ is infected during the epidemic conditioned on $u$ being initially infectious. Susceptibility sets have proven to be important tools in proving results concerning e.g.\ the final size of different epidemic models \cite{Ball02,Ball09} (see also \cite{Ande99}). However, for the research question in this paper we need to consider the susceptibility process instead, i.e.\ we need to take timing into account.
In this paper, we also consider a multi-type (backward) branching process. This branching process is constructed in such a way that the distribution of the tree-like graph corresponding to it, is the same as the susceptibility process (up to a certain time). The coupling between the susceptibility process and the branching process enables us to prove the main results of this paper formulated in Theorems~\ref{firstmain} and \ref{secondmain}. Using existing theory for branching processes we find an answer in Theorem~\ref{firstmain} for the question ``who is the infector?'' by means of an expression for the expected fraction $\rho_{ij}$ of infected individuals of type $j$ that are infected by individuals of type $i$, conditioned on that there is a large outbreak in a population (as the population size tends to infinity), $i,j=1,\ldots K$.
In general, this expression remains rather implicit. However, for a special class of models, we are able to obtain upper and lower bounds for the quantities $\rho_{ij}$ of interest (Theorem~\ref{secondmain}) if we keep the fractions of individuals of the different types and the expected number of infectious contacts between different types of individuals fixed. As the $\rho_{ij}$ of Theorem~\ref{secondmain} are rather implicit, these bounds allow us to gain more insights in the importance of different types of infected individuals in the transmission dynamics in the population.
The class of models that allows for identifying upper and lower bounds are the topic of interest of our twin paper \cite{Leun18}. This paper \cite{Leun18} is motivated by infectious diseases, such as influenza and chlamydia, for which we can categorise infected individuals as symptomatic or asymptomatic (showing no apparent signs of the disease) giving rise to two types of infected individuals. Asymptomatically infected individuals are generally hard to detect by public health authorities. Therefore, we would like to gain insights in their role in the transmission process and determine whether asymptomatically infected individuals often play the role of the infector.
The structure of this paper is as follows. In Section~\ref{secmod} we introduce the model, the notation and the two main theorems of the paper. Next, in Section~\ref{sec:construction}, we discuss the construction that enables us to prove the desired results. The proofs are then presented in Section~\ref{sec:infector}. We end with a short discussion in Section~\ref{sec:discussion}.
\section{Model, notation and main results}
\label{secmod}
\subsection{Model and notation}\label{sec:model_notation}
We denote the number of elements in a set $\mathcal{A}$ by $|\mathcal{A}|$. For $k \in \mathbb{N}$ we use the notation $[k] = \{1,2, \cdots, k\}$. We use $\mathbb{N}$ for the strictly positive integers and $\mathbb{N}_0 = \mathbb{N} \cup \{0\}$ for the non-negative integers.
Unless specified otherwise, limits are for population size $n \to \infty$. We say that an event happens with high probability (w.h.p.) if the probability of the event converges to 1 as $n \to \infty$. We adhere to the usual order notation, i.e.\ $f =O(g)$ means that $\limsup_{n \to \infty} |f(n)/g(n)| < \infty$ and $f =o(g)$ means that $\lim_{n \to \infty} |f(n)/g(n)| =0$. In addition, for a sequence of random variables $\{X^{(n)}; n\in \mathbb{N}\}$, we write $X^{(n)} = O_p(g)$ if $|X^{(n)}/g(n)|$ is bounded in probability and $X^{(n)}=o_p(g)$ if $X^{(n)}/g(n) \to 0$ in probability. See \cite[Section 1.2]{Jans11} for a discussion of this notation.
We consider a population of $n$ individuals where $V^{(n)}$ denotes the set of all individuals. For some of our results we consider a sequence of models in growing populations, i.e.\ for $n \to \infty$. If no confusion is possible we write $V= V^{(n)}$. We assume that there are $K$ types of individuals. For $i \in [K]$, let $V_i = V_i^{(n)}$ be the set of vertices of type $i$ and $n_i = |V_i|$ the number of vertices of type $i$.
Within this sequence of populations we consider the spread of an infectious disease in which individuals are either susceptible or infected. When individual $v \in V_i^{(n)}$ is infected it makes contacts to different individuals of type $j$ ($i,j \in [K]$) according to a point process $\xi_v^j = \{\xi_v^j(t);t\geq 0\}$ on the interval $[0,\infty)$. If there are more than $n_j$ points in the process, then only the first $n_j$ points represent contacts. The time parameter in the definition of $\xi_v^j$ represents the time since infection of individual $v$. The processes $\xi_v^1$, $\xi_v^2$, $\cdots$ and $\xi_v^K$ may be dependent, and their joint distribution may depend on the type $i$ of $v$. However, the point processes associated to different individuals are independent, i.e.\ the vectors of processes $\{(\xi_v^1, \cdots \xi_v^K);v \in V\}$ are independent. For convenience we introduce the vector of stochastic process $(\xi_{i1}, \xi_{i2},\cdots, \xi_{iK})$, which is distributed as $(\xi_v^1, \cdots \xi_v^K)$ for $v \in V_i$, $i \in [K]$. We make the following assumptions.
\begin{assumption}\label{nipiass}
$n^{-1} n_i \to p_i>0$ for all $i \in [K]$.
Furthermore,
\begin{equation*}
\max_{i,j\in [K]} \left|\frac{p_i}{p_j}-\frac{n_i}{n_j}\right|=O(1/n).
\end{equation*}
\end{assumption}
\begin{assumption}\label{finass}
For all $i \in [K]$, the distribution of $(\xi_{i1}, \xi_{i2},\cdots, \xi_{iK})$ is independent of population size $n$ and for all $i,j \in [K]$,
\begin{equation}\label{eq:mij}
m_{ij}=\mathbb{E}[\xi_{ij}(\infty)]<\infty,
\end{equation}
i.e.\ the expected number $m_{ij}$ of contacts that an individual of type $i$ makes with individuals of type $j$ is finite, for all $i,j \in [K]$. Furthermore, we assume that $|(\xi_{i1}, \xi_{i2},\cdots, \xi_{iK})|$ does a.s.\ only contain jumps of size 1 and its distribution has no atoms.
\end{assumption}
\begin{assumption}\label{boundass}
There exists a constant $\kappa <1$ such that for all $i \in [K]$, $$\mathbb{P}\left(\max_{v \in V} \xi_v^i(\infty) < n^{\kappa}\right) \to 1 \qquad \mbox{as} \qquad n \to \infty.$$
\end{assumption}
For future reference, we let $M=\{m_{ij}\}_{i,j\in[K]}$ denote the matrix with the $m_{ij}$ (as defined in Assumption \ref{finass}) as elements.
Assumption \ref{finass} guarantees that w.h.p.\ all non-trivial paths in the epidemic graph defined below have different lengths.
Note that Assumption \ref{boundass} is easily met, e.g.\ if for all $v \in V$ and $i \in [K]$ there exists $\epsilon>0$ and $\ell_0 \in (0,\infty)$, for which $\mathbb{P}[\xi_v^i(\infty)>\ell] < \ell^{-(1+\epsilon)}$ for all $\ell>\ell_0$, then for $\kappa \in (1/(1+\epsilon),1)$
\begin{align*}
\mathbb{P}\left(\max_{v \in V} \xi_v^i(\infty) < n^{\kappa}\right) &= \prod_{v \in V} [1-\mathbb{P}\left( \xi_v^i(\infty) > n^{\kappa}\right)] \\
&\geq 1 - \sum_{v \in V}\mathbb{P}\left( \xi_v^i(\infty) > n^{\kappa}\right) \\
&\geq 1 - n \max_{j \in [K]}\mathbb{P}\left( \xi_{ji}(\infty) > n^{\kappa}\right)\\
& >1 -n \times n^{-(1+\epsilon)\kappa} \\
& \to 1.
\end{align*}
At the points of $\xi_v^j$ ($j \in [K]$), $v$ contacts an individual from $V_j$. The individuals that are contacted are uniformly chosen without replacement. If $v$ is of type $j$, we allow for $v$ to be among the contacted individuals in $\xi_v^j$. If the individual that is contacted is still susceptible at the time of the contact then it becomes infected.
Note that we may assign the point processes $\{\xi_v^j;v \in V, j \in [K]\}$ and decorate the points with the labels of the individuals these points represent contacts to, already before the epidemic starts. This allows us to create a new set of random variables $\{\eta(u,v);u,v \in V\}$. Assume that individual $v$ is of type $j \in [K]$. If there is a point in $\xi_u^j$ with label $v$, then $\eta(u,v)$ takes the value of this point. If there is no such point in $\xi_u^j$, set $\eta(u,v)= \infty$. Observe that the distribution of $\{\eta(u,v);u,v \in V\}$ depends on $n$ in this construction, because the probability that $v$ is chosen as a label is decreasing in $n$. Note that there is a broad class of models that satisfy Assumptions~\ref{nipiass}-\ref{boundass}, see Remark~\ref{rk:SEIR} for an example of a specific class of epidemic models.
This construction provides us with a graph representation of the population and the epidemic on it. We construct the weighted random graph $G=(V,E)$ as follows. The edge set $E$ consists of all directed pairs $(u,v) \in V \times V$, with $u \neq v$. For edge $(u,v) \in V \times V$, we say that $u$ is the tail of $(u,v)$ and $v$ is its head. The weight of edge $(u,v)$ is given by $\eta(u,v)$ for all $(u,v) \in E$. For some of our arguments we restrict to the weighted edge set $E' \subset E$ of all edges $(u,v) \in E$ with finite weight, i.e.\ $(u,v) \in E'$ if and only if $\eta(u,v)< \infty$. The corresponding random graph is denoted by $G'=(V, E')$. For $i,j \in [K]$, $u \in V_i$ and $v \in V_j$, we also introduce the random variable $\eta_{ij}$, which is distributed as $\eta(u,v)$ conditioned on $\eta(u,v)<\infty$ (i.e.\ conditioned on $(u,v) \in E'$).
Observe that
\begin{equation}\label{disteta}
\begin{aligned}
\mathbb{P}(\eta_{ij} \leq t) &= \mathbb{P}(\eta(u, v) \leq t|\eta(u, v) < \infty) \\
&= \frac{\mathbb{P}(\eta(u,v) \leq t)}{\mathbb{P}(\eta(u,v) < \infty)} = \frac{\frac{1}{n_j}\mathbb{E}[\xi_{ij}(t)]}{\frac{1}{n_j}\mathbb{E}[\xi_{ij}(\infty)]}= \frac{\mathbb{E}[\xi_{ij}(t)]}{m_{ij}},
\end{aligned}
\end{equation}
which is independent of $n$. We mainly consider the random graph $G'=(V,E')$. See Figure \ref{fig:forward} for an example of $G'$.
\begin{figure}
\centering
\includegraphics[scale=0.7]{forwardguilty.pdf}
\caption{An example of $G'=(V,E')$ with $n=8$ and $K=2$. The vertices are labeled $a,b,\cdots,h$. Vertices of type 1 are represented by circles and vertices of type 2 by boxes. Contacts made by vertices of type 1 are represented by solid directed edges and contacts made by vertices of type 2 by dashed directed edges. The numeric values next to the edges reflect the time since infection $\eta(u,v)$ of the tail $u$ of the edge until a contact with the head $v$ takes place, $u,v \in \{a,b,\cdots,h\}$.}
\label{fig:forward}
\end{figure}
An ordered set of distinct vertices $\pi =(v_1, v_2, \cdots, v_m)$ is a path in $G'$ if $(v_i,v_{i+1}) \in E'$ for all integers $i \in [m-1]$. With some abuse of terminology, we sometimes refer to the set of edges connecting the vertices of $\pi$ as the path $\pi$ and speak of a path in $E'$. The length of a path $\pi =(v_1, v_2, \cdots, v_m)$ is $\ell(\pi) = \sum_{i=1}^{m-1} \eta(v_i,v_{i+1})$. Assumption \ref{finass} implies that all non-zero paths in $E'$ have different lengths with probability 1. Let $\Pi_{uv}$ be the set of all paths from $u$ to $v$ in $E'$ and define the (quasi) distance from $u$ to $v$ as $d(u,v)=\min_{\pi \in \Pi_{uv}}\ell(\pi)$. As an example, in Figure \ref{fig:forward} the distance from $a$ to $d$ is given by $$d(a,d) =\min\left(\eta(a,b)+\eta(b,d), \eta(a,c)+\eta(c,d)\right)=1.8.$$
In general, $d(u,v)\neq d(v,u)$ if $u\neq v$ since the graph $G'$ is directed. Therefore $d$ is actually a quasi-distance. We say that $d(u,v)=\infty$ if $\Pi_{uv} = \emptyset$. Furthermore, $d(v,v)=0$ for all $v\in V$.
Next we formulate an ``irreducibility'' assumption:
\begin{assumption}
\label{assirred}
For every $i,j\in[K]$, there is w.h.p.\ a path from a vertex in $V_i$ to a vertex in $V_j$ in $E'$.
\end{assumption}
An epidemic process is reproduced from $G'$ as follows.
Let $V_{\text{init}}=V_{\text{init}}^{(n)}$ be the set of vertices that are initially infected. This set may be predetermined or randomly selected and satisfies the following assumption.
\begin{assumption}
\label{assvinit}
The set $V_{\text{init}}^{(n)}$ of vertices that are initially infected satisfies $|V_{\text{init}}^{(n)}|=O_p(1)$.
\end{assumption}
We set $\sigma_v$ to be the time between the start of an epidemic until $v$ becomes infected, $v \in V$, i.e.\ $
\sigma_v = \inf \{d(u,v);u \in V_{\text{init}}\}$. In particular, $\sigma_v =0$ for $v \in V_{\text{init}}$.
So, suppose we let $V_{\text{init}}=\{a\}$ in Figure \ref{fig:forward}, then $\sigma_a=0$, $\sigma_b=0.3$, $\sigma_c = 1.3$, $\sigma_d = 1.8$ etc.
Moreover, if $\sigma_v= \infty$, then $v$ will never become infected. Note that $G'$ contains some redundant information regarding the epidemic: (i) there are edges with finite weights with infection time of the tail being $\infty$, i.e.\ the tails of those edges will never get infected, (ii) if $\sigma_v < \sigma_u + \eta(u,v)$, then the edge $(u,v)$ represents a contact between two already infected individuals.
\begin{remark}[The SEIR epidemic model]\label{rk:SEIR}
The framework of this section includes the SEIR (Susceptible $\to$ Exposed $\to$ Infectious $\to$ Recovered) epidemics as follows. In the SEIR framework, exposed, infectious and recovered individuals are all counted as infected. Assign to every vertex $v \in V$ a random latent period $L_v$ and a random infectious period $\iota_v$, which might be dependent on $L_v$. The vectors $\{(L_v,\iota_v);v \in V\}$ are independent and their distribution functions only depend on the type of the vertex. Suppose that $v$ is of type $i$. Conditioned on $(L_v,\iota_v)$, let the processes $\{\hat{\xi}_v^j; j \in [K]\}$ be independent homogeneous Poisson processes on the interval $(L_v, L_v + \iota_v)$ with intensity $p_j \lambda_{ij}$. At the points of this point process, $v$ makes contacts to vertices in $V_j$ chosen uniformly with replacement. By keeping only the points in $\{\hat{\xi}_v^j; j \in [K]\}$ whose label did not appear before in this process, we obtain $\{\xi_v^j; j \in [K]\}$. If $L_v$ and $\iota_v$ are exponentially distributed, then the SEIR epidemic is a Markov process that is often referred to as the Markov SEIR epidemic.
At time $t$, a vertex $v$ is susceptible if $t< \sigma_v$, Exposed if $t \in [\sigma_v, \sigma_v + L_v)$, while it is Infectious if $t \in [\sigma_v + L_v, \sigma_v + L_v+ \iota_v)$. Finally $v$ is Recovered if $t \geq \sigma_v + L_v+ \iota_v$.
If $\mathbb{P}(L_v =0)=1$, then we are in the so-called SIR (Susceptible $\to$ Infectious $\to$ Recovered) epidemics framework. In addition, if $\iota_v$ is exponentially distributed then the process is called a Markov SIR epidemic.
\end{remark}
Finally, to conclude this section, we introduce the basic reproduction number $R_0$. As this is possibly the most studied quantity in mathematical modelling of the spread of infectious diseases, no epidemic modelling paper would be complete without at least mentioning $R_0$. In a single-type epidemic model $R_0$ is defined as the expected number of infectious contacts made by a newly infected individual in an otherwise susceptible population. The multi-type equivalent of $R_0$ is given by the dominant eigenvalue of the $K \times K$ matrix $M=\{m_{ij}\}_{i,j\in[K]}$ (see Assumption~\ref{finass}). $R_0$ is always real and strictly positive \cite[Chapter 4]{Jage75},\cite[Chapter 7]{Diek12}. We say that the epidemic process is supercritical (resp.\ critical, resp.\ subcritical) if $R_0>1$ (resp.\ $R_0=1$, resp.\ $R_0<1$). As $n \to \infty$, the epidemic becomes large with positive probability if and only if the process is supercritical \cite{Diek12}. In the remainder of the paper we assume that $R_0>1$.
\subsection{Main results}
Because we assume that $R_0>1$, a large outbreak occurs with positive probability and at the end of such a large outbreak, a fraction of the infected individuals is of type $j$ w.h.p. However, the question that we are interested in is: what fraction of those individuals was infected by individuals of type $i$, for $i,j\in[K]$?
So, if we denote set of infected individuals of type $j$ by $\mathcal{I}_j$, and the subset of those individuals, which are infected by individuals of type $i$ by $\mathcal{I}_j^i$, then we are interested in $|\mathcal{I}_j^i|/|\mathcal{I}_j|$.
For example, in Figure \ref{fig:forward}, if $a$ represents the initially infectious individual,
then there are two individuals of type 1 that are infected by an individual of type i (namely the individuals represented by vertices $c$ and $g$) and one individual of type 1 that is infected by an individual of type 2 (namely the individual represented by vertex $d$).
So, the fraction of individuals infected by an individual of type 1 among all infected individuals of type 1 is $2/3$ (we exclude vertex $a$ because it was initially infected, and not infected by another individual in the population).
The question posed above leads to the main results that are formulated in Theorems~\ref{firstmain} and~\ref{secondmain} below. For our results, we need to define a multi-type branching process $\{\mathcal{Z}(t); t \geq 0\}$. The branching process is defined as follows. Let $Z^i(t)$ be the number of particles in the branching process at time $t$ if the process is started by a single particle of type $i$, $i \in [K]$. For $i,j \in [K]$, particles of type $j$ in $\mathcal{Z}(t)$ give birth to particles of type $i$ according to a Poisson process with intensity $\frac{p_i}{p_j} \mathbb{E}[\xi_{ij}(da)]$, where $a$ is the age of the particle of type $j$. All of these Poisson processes are independent.
The definition of the branching process $\mathcal{Z}(t)$ is such that we can apply theory from $\cite{Iksa15}$. In particular, there exists a Malthusian parameter $\alpha>0$ and a random variable $W^i$, such that
\begin{equation*}
e^{-\alpha t}Z^i(t) \to W^i \qquad \mbox{ a.s.\ as $t \to \infty$}
\end{equation*}
and
\begin{equation*}
\mathbb{P}(W^i \in (0, \infty))=1 -\mathbb{P}(W^i =0) = \mathbb{P}(Z^i(t) \to \infty).
\end{equation*}
Let $W^i(r)$ for $r \in \mathbb{N}$ be independent copies of $W^i$. We now state our first main result.
\begin{theorem}
\label{firstmain}
Conditioned on the occurrence of a large outbreak,
the fraction of infected individuals of type $j$ that are infected by an individual of type $i$ during an outbreak in a population of size $n$ converges in probability to $\rho_{ij}$. Here $\sum_{i=1}^K\rho_{ij}=1$, and
\begin{equation}
\label{mainequ}
\rho_{ij}=\frac{1}{\mathbb{P}(Z^j(t) \to \infty)}
\mathbb{E}\left[\frac{\sum_{r=1}^{X_{ij}}e^{-\alpha \tau_{ijr}} {W}^i(r)}{\sum_{k =1}^K \sum_{r=1}^{X_{kj}} e^{-\alpha \tau_{kjr}} {W}^k(r)} 1\hspace{-2.5mm}{1}\left(\sum_{k =1}^K \sum_{r=1}^{X_{kj}} {W}^k(r) >0\right)
\right],
\end{equation}
where, for $k \in [K]$ and $r \in \mathbb{N}$, the random variables $\tau_{kjr}$ are independent with distribution function $\mathbb{P}(\tau_{kjr} <a) = {\mathbb{E}[\xi_{kj}(a)]}/{m_{kj}}$ and $X_{kj}$ is Poisson distributed with expectation $\frac{p_k}{p_j}m_{kj}$.
\end{theorem}
Note that, in general, it is hard to give a more explicit expression for $\rho_{ij}$ than~\eqref{mainequ}. Often, there is no explicit description of the distribution of ${W}^k(r)$. We are able to obtain bounds for~\eqref{mainequ} for the important special case discussed in \cite{Leun18}, leading to Theorem~\ref{secondmain}.
The model of \cite{Leun18} is as follows. We consider $K=2$, and $(\xi_v^1,\xi_v^2)$, $v \in V$, obtained from a single marked point process $\xi_v$. In this process $\xi_v$ the points get independently mark 1 with probability $p_1$ and mark 2 otherwise. Then the process $\xi_v^1$ consists of the points with mark 1, while $\xi_v^2$ consists of the points with mark 2. By construction of the process, the probability that an ultimately infected vertex of type $i$ is infected by a vertex of type 1 is the same for $i=1$ and $i=2$, i.e.\ $\rho_{11}=\rho_{12}=\rho_1$. With some abuse of notation we write $\xi_i$ for a point process distributed as $\xi_v$ for $v \in V_i$, $i \in [2]$. Furthermore, note that $m_{ij}=\mathbb E[\xi_{ij}(\infty)]=p_j\mathbb E[\xi_i(\infty)]$, i.e.\ we can write $m_{ij}=p_j \tilde m_i$ with $\tilde m_i=\mathbb E[\xi_i(\infty)]$. Here $\tilde m_i$ can be interpreted as the expected number of secondary cases caused by a newly infected individual of type $i$ in an otherwise susceptible population (one can think of the $\tilde m_i$ as the type-specific reproduction numbers). Then the basic reproduction number is $R_0=p_1\tilde m_1+p_2\tilde m_2$, with $p_2=1-p_1$. Indeed, a newly infected individual is of type $i$ with probability $p_i$ and the expected number of secondary cases it produces is $\tilde m_i$, $i=1,2$.
For this model we can compute $\rho_1^-$ and $\rho_1^+$, the minimum and maximum fraction of the infected vertices that are infected by type 1 vertices, if the matrix $M$ and $p_i$ are held fixed for $i \in [K]$. We let $\tilde{q}_1$ be the smallest positive solution in $(0,1]$ of
\begin{equation}\label{qsequa}
x= e^{-p_1 \tilde m_{1}(1-x)}
\end{equation}
and $\tilde q_2$ the smallest positive solution in $(0,1]$ of
\begin{equation}\label{qaequa}
x= e^{-p_2 \tilde m_{2}(1-x)}.
\end{equation}
Furthermore, we let $q$ be the unique solution in $(0,1)$ of
\begin{equation}\label{qequa}
x = e^{-(1-x)\left(p_1 \tilde m_{1}+ p_2 \tilde m_{2}\right)}=e^{-(1-x)R_0}.
\end{equation}
Note that we assume that $R_0=p_1\tilde m_{1}+ p_2\tilde m_{2}>1$, so the unique solution $q\in(0,1)$ exists. We can interpret $1-\tilde q_1$ (resp.\ $1-\tilde q_2$) as the final fraction of the population that ultimately gets infected when only individuals of type 1 (resp.\ type 2) are able to transmit, conditional on a large outbreak. Furthermore, $1-q$ can be interpreted as the final fraction of the population that ultimately gets infected, conditional on a large outbreak (or conversely, $q$ is the fraction of the population that remains susceptible throughout the epidemic).
\begin{theorem}
\label{secondmain}
Consider the two-type model described above. In the limit as population size $n\to\infty$ and for fixed $p_1$, $\tilde{m}_1$ and $\tilde{m}_2$, the fraction of ultimately infected vertices that is infected by type 1 vertices is bounded from above by
\begin{equation*}
\rho_1^+ = \left( 1 - \frac{p_1 \tilde m_1 (\tilde{q}_1 + q)}{2} \right) \frac{\tilde{q}_1-q}{1-q}.
\end{equation*}
\end{theorem}
By interchanging the role of the types $1$ and $2$, we also obtain the lower bound $\rho_1^-$. Indeed, note that $\rho_1^-=1-\rho_2^+$, where
\begin{equation*}
\rho_2^+ = \left( 1 - \frac{p_2 \tilde m_2 (\tilde{q}_2 + q)}{2} \right) \frac{\tilde{q}_2-q}{1-q}.
\end{equation*}
In other words, for any point process $\{(\xi_v^1,\xi_v^2); v\in V\}$ that satisfies the assumptions of Section~\ref{sec:model_notation} and that can be obtained from independently marking points of a one-dimensional point process, the fraction $\rho_1$ of infected individuals that are infected by individuals of type 1 is bounded by $\rho_1^-$ and $\rho_1^+$, i.e.\ $\rho_1^-\leq \rho_1 \leq \rho_1^+$.
\begin{remark}
In Section~\ref{sec:proofsecondmain} we consider a more general setting than the one in Theorem~\ref{secondmain}. Instead of assuming a single marked process $\xi_i$, one can consider general distributions $(\xi_{i1}, \xi_{i2})$ and obtain bounds~\eqref{finaleq} for $\rho_{21}^-$ and~\eqref{eq:rho11_final} for $\rho_{11}^+$ (and, by interchanging the roles of types 1 and 2, bounds $\rho_{12}^-$ and $\rho_{22}^+$). As this is somewhat more involved, we choose to present the bounds in the form of Theorem~\ref{secondmain} instead.
\end{remark}
\section{Susceptibility process, backward branching process and the coupling}\label{sec:construction}
Throughout we assume that all random variables and processes are defined on a suitable rich enough probability space, which we do not specify.
\subsection{The susceptibility process}\label{sec:susceptibility}
In this subsection we use the idea of susceptibility sets \cite{Ball01,Ball02,Ball09}, and construct this set through a stochastic process: the susceptibility process.
We define the susceptibility process $\{\mathcal{S}^{(n)}_v(t);t \geq 0\}$ as
\begin{equation*}
\mathcal{S}^{(n)}_v(t) = \{u \in V; d(u,v) \leq t\}.
\end{equation*}
Note that $\mathcal{S}^{(n)}_v(t)$ is non-decreasing in $t$. The susceptibility set of vertex $v$ is defined as $\mathcal{S}^{(n)}_v = \lim_{t \to \infty} \mathcal{S}^{(n)}_v(t)$, i.e.\ the susceptibility set $S_v^{(n)}$ of $v$ consists of all vertices $u\in V$ for which there is a path from $u$ to $v$ in $G'$. As an illustration, in Figure \ref{fig:forward}, the susceptibility set of vertex $f$ is given by $\mathcal{S}^{(n)}_f = \{a,b,c,d\}$, while $\mathcal{S}^{(n)}_f(t=1.1) = \{c,d\}$. Note that $\mathcal{S}^{(n)}_v \cap V_ {\text{init}} = \emptyset$ if and only if $v$ remains uninfected throughout the epidemic, i.e.\ if and only if there is no path in $G'$ from $V_ {\text{init}}$ to $v$. Also note that the susceptibility set may contain vertices of different types.
Define for $a \geq 0$ and $j \in [K]$,
\begin{equation*}
\mathcal{S}^{(n)}_v(t;a,j) = \{u \in \mathcal{S}^{(n)}_v(t)\cap V_j; d(u,v) > t- a\}.
\end{equation*}
That is, $\mathcal{S}_v(t;a,j)$ consists of the vertices of type $j$ in $\mathcal{S}^{(n)}_v(t)$, that are not yet part of $\mathcal{S}^{(n)}_v(t-a)$.
Let $v \in V \setminus V_{\text{init}}$ be a randomly chosen vertex. We derive the susceptibility process $\{\mathcal{S}^{(n)}_v(t);t \in (0,t_*)\}$ by constructing part of the random graph $G'$ around vertex $v$ by means of an exploration process $\{\hat{G}(\ell)\}= \{\hat{G}(\ell); \ell \in \mathbb{N}_0\}$ in which vertices in the susceptibility process are explored one at a time. We note that $\{\hat{G}(\ell)\}$ depends on $v$. Here $t^*= t^*(n)$ is a given time (we defer the specification of $t^*$ until~\eqref{eq:t*} below). The process $\{\hat{G}(\ell)\}$ allows us to couple $\{\mathcal{S}^{(n)}_v(t)\}$ with an appropriate branching process. In this way, we can make the coupling between the susceptibility process and the backward branching process that is needed to prove Theorems~\ref{firstmain} and~\ref{secondmain} in Section \ref{sec:infector}.
Before we define the exploration process $\{\hat{G}(\ell)\}$ around vertex $v$, we introduce some additional variables and terminology. $\hat{G}(\ell)$ is a 4-tuple:
\begin{equation*}
\{\hat{G}(\ell)= (\hat{V}^{a}(\ell),\hat{V}^{p}(\ell),\hat{V}^{e}(\ell),\hat{E}(\ell));\ell \in \mathbb{N}_0\}.
\end{equation*}
Here $\hat E(\ell)$ denotes the edge set of $\hat{G}(\ell)$. Vertices in $\hat{G}(\ell)$ can be `active', `passive', or `explored'. The sets of these vertices are denoted by $\hat{V}^{a}(\ell)$, $\hat{V}^{p}(\ell)$, and $\hat{V}^{e}(\ell)$, respectively. The sets $\hat{V}^{a}(\ell)$, $\hat{V}^{p}(\ell)$, $\hat{V}^{e}(\ell)$, and $\hat{E}(\ell)$ are defined in the construction below (where also their names will become apparent). Finally, before we explain the construction, we mention that throughout the process we may `flag' the process (see step 4). This flagging plays a role when coupling the exploration process with a branching process to represent the susceptibility process. The construction is as follows.
\begin{itemize}
\item[1)]
To set the initial conditions of the construction, let $\hat{V}^a(0) = v$, $\hat{V}^e(0) = \emptyset$, and $\hat{E}(0)$ be the set of all edges in $E'$ for which $v$ is the tail, whereas $\hat{V}^p(0)$ is the set of all heads of edges in $\hat{E}(0)$ that are in $V\setminus v$.
\item[2)]
For $\ell \in \mathbb{N}_0$, assume that $\hat{V}^a(\ell)\neq \emptyset$ and that there exists a vertex in $\hat{V}^a(\ell)$ from which there is a path in $\hat{G}(\ell)$ to $v$ of length at most $t^*$. In step $\ell+1$ pick (according to some rule) one of the vertices from $\hat{V}^a(\ell)$, from which there is a path in $\hat{G}(\ell)$ to $v$ of length at most $t^*$. Say that this vertex is $v' \in V_j$. Move $v'$ to the set of explored vertices, i.e.\ $\hat{V}^e(\ell+1) = \hat{V}^e(\ell) \cup v'$. Assign to $v'$ independently a binomial random number $x(v';i)$ with parameters $n_i$ and $m_{ij}/n_j$.
\item[3)]
The remainder of step $\ell+1$ is split up in $\sum_{i=1}^K x(v';i)$ sub-steps as follows. We introduce
\begin{equation*}
\left\{{G}^*(\ell,\ell');\ell \in \mathbb{N}_0, \ell' \in \{0,1,\cdots, \sum_{i=1}^K x(v';i)\}\right\},
\end{equation*}
where
\begin{equation*}
{G}^*(\ell,\ell')= ({V}^{*,a}(\ell,\ell'),\hat{V}^{*,p}(\ell,\ell'),\hat{V}^{*,e}(\ell,\ell'),\hat{E}^*(\ell,\ell')).
\end{equation*}
Furthermore, set ${G}^*(\ell,0)=\hat{G}(\ell)$. Next, let $\ell' \in \left(\sum_{i'=1}^{i-1} x(v';i'),\sum_{i'=1}^i x(v';i')\right]$, where the empty sum is $0$.
\item[4)]
In the $\ell'$-th sub-step of step $\ell+1$ pick uniformly a vertex from $V_i$.
If we pick a vertex we have picked in one of the $\ell'-1$ sub-steps before we say that the exploration process is \textit{flagged}.
In that case we choose new vertices from $V_i$ until we obtain a vertex that was not chosen in the previous $\ell'-1$ sub-steps. This step is equivalent to picking the vertices without replacement.
Say that the vertex that is picked is $v''$.
\begin{itemize}
\item [4a)] If $v'' \in \hat{V}^{a}(\ell) \cup \hat{V}^{e}(\ell)$, then nothing changes in the exploration graph, i.e.\ ${G}^*(\ell+1,\ell')= {G}^*(\ell+1,\ell'-1)$. This is because if $v'' \in \hat{V}^{a}(\ell) \cup \hat{V}^{e}(\ell)$, then we already have explored the edges with tail $v''$.
\item [4b)] If $v'' \in V \setminus \left(\hat{V}^{a}(\ell) \cup \hat{V}^{e}(\ell)\right)$, assign to $v''$ the vector of point processes $(\xi_{v''}^{j'}, j' \in [K])$. The distribution of $(\xi_{v''}^{j'}, j' \in [K])$ is equal to the distribution of $(\xi_{ij'}, j' \in [K])$, given that $\xi_{ij}$ contains a vertex with label $v'$. Assign label $v'$ to a uniformly chosen point in $\xi_{v''}^{j}$, and assign uniform labels without replacement from $V_j \setminus v'$ to the other points in $\xi_{v''}^{j}$ and uniform labels from $V_{j'}$ to the points in $\xi_{v''}^{j'}$ for $j' \in [K]\setminus j$.
If none of the newly assigned labels correspond to vertices in $\hat{V}^{e}(\ell)$ then ${E}^*(\ell+1,\ell')$ contains all edges in ${E}^*(\ell+1,\ell'-1)$ plus the edges with tail $v''$ and heads corresponding to the labels of the points in $(\xi_{v''}^{j'}, j' \in [K])$, with the obvious edge lengths. In addition, all heads of those edges which were not in ${V}^{*,a}(\ell+1,\ell'-1)$ move to ${V}^{*,p}(\ell+1,\ell')$ (if they were not in that set already). Furthermore, $v'' \in {V}^{*,a}(\ell+1,\ell')$.
If any of the newly assigned labels correspond to vertices in $\hat{V}^{e}(\ell)$ then we flag the process and we return to the start of step $4)$. This last part of step 4b) is equivalent to conditioning on the event that there are no edges with tail $v''$ and an already explored vertex as head.
\item[4c)] Set $\hat G(\ell+1)=\hat G\left(\ell+1, \sum_{i=1}^Kx(v',i)\right)$.
\end{itemize}
\item[5)] Continue this process by increasing $\ell$ until there are no active vertices having a path of length less than $t^*$ towards $v$ in $\hat{G}(\ell)$. Say that $\ell^*$ is the smallest $\ell$ for which this is the case.
\end{itemize}
We note the following:
\begin{itemize}
\item The edge set $\hat{E}(\ell)$ is a subset of $E'$ and contains all edges in $\hat{G}(\ell)$.
\item All edges in $E'$ with tails in $\hat{V}^{a}(\ell)$ are in $\hat{E}(\ell)$, but there might still be edges in $E'$ with heads in $\hat{V}^{a}(\ell)$ that are not in $\hat{E}(\ell))$.
\item Every passive vertex is the head of an edge in $\hat{E}(\ell)$, but the passive vertices themselves are not explored, and none of the edges in $E'$ with tail in $\hat{V}^{p}(\ell)$ are in $\hat{E}(\ell)$.
\item All edges in $E'$ with head or tail in $\hat{V}^{e}(\ell)$ are in $\hat{E}(\ell)$.
\item The tails of edges in $\hat{E}(\ell)$ are in $\hat{V}^{a}(\ell) \cup \hat{V}^{e}(\ell)$ and their heads are in $\hat{V}^{a}(\ell) \cup \hat{V}^{p}(\ell) \cup \hat{V}^{e}(\ell)$.
\end{itemize}
The construction of the exploration process yields $\hat{V}^e(\ell^*) = \mathcal{S}^{(n)}_v(t^*)$. Furthermore, if the process $\{\hat G(\ell)\}$ is not flagged until step $\ell^*$, then the construction of $\hat{V}^e(\ell^*)$ (and the distances from the vertices in this set to $v$) is equivalent to constructing a branching process $\{\mathcal{\tilde Z}^{(n)}(t);t \geq 0\}$ up to time $t^*$. In this branching process, particles of type $j$ give birth to a binomial distributed number of particles of type $i$, $i \in [K]$, where the parameters of the binomial distributed random variable are $n_i$ and $(n_j)^{-1}m_{ij}$. The number of children of the different types of particles are independent. Furthermore, the ages of the mother particles of type $j$ at birth of a child of type $i$ are independent and have density $\mathbb{E}[\xi_{ij}(da)]/m_{ij}$.
\subsection{The (backward) branching process}
\label{subsecbp}
We create a multi-type branching process $\{\mathcal{Z}(t);t \geq 0\}$ that can be coupled to $\{\mathcal{S}^{(n)}_v(t);t \geq 0\}$. The coupling is performed in Section~\ref{sec:coupling}. The branching process $\{\mathcal{Z}(t);t \geq 0\}$ is constructed in such a way that the distribution of the corresponding tree-like graph is the same as that of $\{\mathcal{S}^{(n)}_v(t);t \geq 0\}$ up to time $t^*$ with $t^*$ defined by~\eqref{eq:t*}. We leave out some of the details in the arguments. Those details can be filled in analogous to \cite{Ball95, Ball14, Barb13} for related models.
Without loss of generality we assume that $v \in V_1$. The multi-type (backward) branching process is as follows. The (single) ancestor is of type $1$. Particles of type $j$ give birth to particles of type $i$ according to a Poisson process with intensity $\frac{p_i}{p_j} \mathbb{E}[\xi_{ij}(da)]$, where $a$ is the age of the type $j$ particle, $i,j\in[K]$. All Poisson processes are independent.
The branching process $\{\mathcal{Z}(t);t \geq 0\}$ is analysed using existing theory from \cite{Iksa15}. First of all, the mean offspring measure of this backward branching process is defined through
\begin{equation}
\label{meanmeas}
\mu^{(b)}_{ji}(dt)= \frac{p_i}{p_j} \mathbb{E}[\xi_{ij}(dt)].
\end{equation}
Note that
\begin{equation*}
m^{(b)}_{ji}= \int_0^{\infty} \mu^{(b)}_{ji}(dt)= \int_0^{\infty} \frac{p_i}{p_j} \mathbb{E}[\xi_{ij}(dt)]= \frac{p_i}{p_j}m_{ij}
\end{equation*}
is the expected number of children of type $i$ of a particle of type $j$. Here $m_{ij}$ is given by~\eqref{eq:mij} (and the corresponding matrix is $M$). Let $M^{(b)}=\{m^{(b)}_{ji}\}_{j,i \in [K]}$. Straightforward matrix theory gives that $M$ and $M^{(b)}$ have the same dominant eigenvalue $R_0$, which by assumption is strictly larger than 1, i.e.\ the branching process is supercritical).
Define
\begin{equation}
\label{laplback}
\hat{m}^{(b)}_{ji}(x)= \int_{0}^{\infty} e^{-x t} \frac{p_i}{p_j} \mathbb{E}[\xi_{ij}(dt)]
\end{equation}
and let $\hat{M}^{(b)}(x)$ be the matrix with elements $\hat{m}^{(b)}_{ji}(x)$. Finally, let $\alpha$ be such that
\begin{equation}
\label{maltdef}
\hat{M}^{(b)}_{ji}(\alpha)=1.
\end{equation}
Note that $K<\infty$, all elements of $M^{(b)}$ are finite and the dominant eigenvalue $R_0$ of $M^{(b)}$ is real and larger than 1. Therefore, $\alpha$ exists and is positive.
We define the random vector $Z^i(t)$ as $Z^i(t) = (Z_1^i(t),Z_2^i(t),\cdots,Z_K^i(t))$, where $Z_j^i(t)$ is the number of particles of type $j \in [K]$ in $\mathcal{Z}(t)$ if the process starts with one newborn particle of type $i$. With some abuse of notation, let $\sigma(x)$ be the time of birth of particle $x$ in the branching process. Note that particles in the branching process $\mathcal{Z}(t)$ never die.
We know that there exists an $\alpha >0$ such that
\begin{equation}\label{bpeq1}
e^{-\alpha t} Z^{i}(t) \to W^{i} =(W^{i}_1, W^{i}_2, \cdots W^{i}_K) \qquad \mbox{a.s.\ as $t\to \infty$},
\end{equation}
where $W^{i}$ is a random vector that has, with probability 1, strictly positive elements on the set $\sum_{j=1}^K Z^{i}_j(t) \to \infty$ as $t \to \infty$~\cite{Iksa15}.
For particle $x \in \{\mathcal{Z}(t);t \geq 0\}$, define
\begin{equation*}
\varphi_x(t) = \varphi_x(t;a) = 1\hspace{-2.5mm}{1}(t-\sigma(x)< a).
\end{equation*}
Let $\hat{Z}_j^{i}(t; a) = \sum_x \varphi_x(t;a)$, where the sum is taken over all particles of type $j$ in $\{\mathcal{Z}(t);t \geq 0\}$, i.e.\ $\hat{Z}_j^{i}(t; a)$ is the number of particles of type $j$ that have age less than $a$ in the branching process at time $t$. Note that $\hat{Z}_j^{i}(t; a)$ is increasing in $a$.
If $Z_j^{i}(t) \to \infty$ for all $j \in [K]$ (i.e.\ if new particles keep on being born in the branching process), then
\begin{equation}
\label{bpeq2}
\frac{\hat{Z}_j^{i}(t; a)}{\sum_{j=1}^K Z_j^i(t)} \to c(a,j) \qquad \mbox{a.s.\ as $t \to \infty$,}
\end{equation}
where $c(a,j)$ is a constant independent of $i$ (Theorem 2.7 of \cite{Iksa15}). For our purposes we do not need to specify $c(a,j)$ further.
Although the branching process is not dependent on $n$, we want to have some bound on the number of vertices born in the branching process as a function of $n$. This is used in the coupling in Section~\ref{sec:coupling}. We set
\begin{equation}\label{eq:t*}
t^*= \frac{1-\kappa}{4}\log[n]/\alpha
\end{equation}
(where $\kappa$ is as in Assumption \ref{boundass}). We obtain by (\ref{bpeq1}) that, for all $i,j \in [K]$,
\begin{equation}
\label{martbound}
e^{-\alpha t^*}Z^{i}_j(t^*) = n^{-(1-\kappa)/4} Z^{i}_j(t^*) \to W^i_j \in (0,\infty) \qquad \mbox{ a.s.\ as $n \to \infty$}
\end{equation}
on the survival set of $\{\mathcal{Z}(t);t \geq 0\}$. In particular, this implies that
\begin{equation}
\label{martbound2}
Z^{i}_j(t^*) =o\left(n^{(1-\kappa)/3}\right) \qquad \mbox{w.h.p.}
\end{equation}
\subsection{The coupling}\label{sec:coupling}
Note that the only difference between the branching process $\{\mathcal{\tilde Z}^{(n)}(t)\}$ associated to the susceptibility process of Section~\ref{sec:susceptibility} and the multi-type branching process $\{\mathcal{Z}(t)\}$ of Section~\ref{subsecbp} is the distribution of the number of particles of type $i$ a particle of type $j$ gives birth to (binomially distributed with parameters $n_i$ and $m_{ij}/n_j$ and Poisson distributed with expectation $p_j/p_im_{ij}$, resp.)
We know from \cite[eq.\ (1.23)]{Barb92} (see also \cite{Barb13}) that the total variation distance between a binomial distributed random variable with parameters $n_i$ and $\frac{1}{n_i}\frac{n_i}{n_j} m_{ij}$ and a Poisson random variable with parameter $\frac{n_i}{n_j} m_{ij}$ is $O(\frac{1}{n_i})$. Moreover, the total variation distance between a Poisson distributed random variable with parameter $\frac{n_i}{n_j} m_{ij}$ and a Poisson distributed random variable with parameter $\frac{p_i}{p_j} m_{ij}$ is $O(\sqrt{|\frac{n_i}{n_j}-\frac{p_i}{p_j}|}) = O(\frac{1}{\sqrt{n}})$ \cite[Theorem 1.C]{Barb92} (see also \cite{Barb13}). Here we have also used Assumption \ref{nipiass}. By the triangle inequality this implies that the total variation distance between a Poisson distributed random variable with expectation $\frac{p_i}{p_j} m_{ij}$ and a binomial distributed random variable with parameters $n_i$ and $\frac{1}{n_i}\frac{n_i}{n_j} m_{ij}$ is $O(\frac{1}{\sqrt{n}})$. Hence, as long as the number of particles born in any of the two branching processes is $o(\sqrt{n})$, the two processes can be perfectly coupled w.h.p. In the remainder of this section we show that the coupling is w.h.p.\ perfect up to time $t^*$ with $t^*$ given by~\eqref{eq:t*}.
By construction, if the first $|\hat{V}^a(\ell^*)| + |\hat{V}^e(\ell^*)|$ vertices that we ``try to include'' in $\{\hat{V}^{a}(\ell) \cup \hat{V}^{e}(\ell);\ell \in \mathbb{N}_0\}$ are all different and none of the first $|\hat{V}^p(\ell^*)|$ that we ``try to include'' in $\{\hat{V}^{p}(\ell);\ell \in \mathbb{N}_0\}$ are in $\hat{V}^{a}(\ell^*) \cup \hat{V}^{e}(\ell^*)$, then the process is not flagged.
The law of large numbers yields
\begin{equation}
\label{setbound}
|\hat{V}^a(\ell^*)| \leq 2 \max_{i,j \in [K]}\frac{n_i}{n_j}m_{ij} |\hat{V}^e(\ell^*)|, \qquad \mbox{w.h.p.}\end{equation}
Here we use that the expected number of edges in $E'$ with any given vertex in $V'$ as head is bounded from above by $\max_{i,j \in [K]}\frac{n_i}{n_j}m_{ij}$. Equation (\ref{setbound}) implies that
\begin{equation}
\label{setbound1a}
|\hat{V}^e(\ell^*)| + |\hat{V}^a(\ell^*)| = O(|\hat{V}^e(\ell^*)|), \qquad \mbox{w.h.p.}
\end{equation}
Using Assumption~\ref{boundass} we obtain
\begin{equation}
\label{setbound2}
|\hat{V}^p(\ell^*)| \leq n^{\kappa} (|\hat{V}^a(\ell^*)|+ |\hat{V}^e(\ell^*)|), \qquad \mbox{w.h.p.}
\end{equation}
Combining inequalities (\ref{setbound1a}) and (\ref{setbound2}) yields
\begin{equation}
\label{setbound3}
|\hat{V}^p(\ell^*)| = O_p(n^{\kappa}|\hat{V}^e(\ell^*)|),
\end{equation}
Since $\{\mathcal{Z}(t);t \geq 0\}$ can be perfectly coupled with $\{\mathcal{S}^{(n)}_v(t);t \geq 0\}$ w.h.p.\ until $o(\sqrt{n})$ particles are born in $\{\mathcal{Z}(t);t \geq 0\}$, we obtain by using~\eqref{martbound2}, that $|\mathcal{Z}(t^*)| = o_p(n^{(1-\kappa)/3})$. Therefore, we also know that $|\mathcal{Z}(t^*)| = o(n^{(1/2})$ and $|\hat{V}^e(\ell^*)| = o_p(n^{(1-\kappa)/3})$.
Combined with (\ref{setbound3}) this gives
\begin{equation}
\label{setbound4}
|\hat{V}^e(\ell^*)| + |\hat{V}^a(\ell^*)| = o_p(n^{(1-\kappa)/3})
\end{equation}
and
\begin{equation}
\label{setbound5}
|\hat{V}^p(\ell^*)| = o(n^{(1+2\kappa)/3}) = o_p(n).
\end{equation}
Using birthday-problem-like arguments \cite[p.24]{Grim01} the probability that the first $|\hat{V}^e(\ell^*)| + |\hat{V}^a(\ell^*)|$ activated vertices in $\{\hat{G}(\ell); \ell \in \mathbb{N}_0\}$ are not all different is bounded from above by
\begin{equation*}
\left(\min_{j \in [K]}n_j\right)^{-1}(|\hat{V}^e(\ell^*)| + |\hat{V}^a(\ell^*)|)^2= o_p(n^{-(1+2\kappa)/3}),
\end{equation*}
while the probability that among the first $|\hat{V}^p(\ell^*)|$ vertices that we ``try to include'' in $\{\hat{V}^{p}(\ell);\ell \in \mathbb{N}_0\}$ there are vertices in $\hat{V}^{a}(\ell^*) \cup \hat{V}^{e}(\ell^*)$ is bounded from above by
\begin{align*}
|\hat{V}^p(\ell^*)| \frac{|\hat{V}^e(\ell^*)| + |\hat{V}^a(\ell^*)|}{\min_{j \in [K]}n_j} &= o_p(n^{(1+2\kappa)/3}) \times o_p(n^{(1-\kappa)/3}) \times O(n^{-1}) \\
&=o_p(n^{(\kappa-1)/3})\\
&=o_p(1).
\end{align*}
We conclude that the probability that the exploration process $\{\hat{G}(\ell); \ell \in \mathbb{N}_0\}$ is flagged up to and including step $\ell^*$ goes to 0 as $n \to \infty$, as we desired. We summarise the main result of this section in the following lemma.
\begin{lemma}
\label{couplelemma}
There exists a probability space on which we can define the branching process $\{\mathcal{Z}(t);t \geq 0\}$ and the susceptibility process $\{\mathcal{S}^{(n)}_v(t);t \geq 0\}$ for all $n$, such that
\begin{equation*}
\mathbb{P}\left(|\mathcal{S}^{(n)}_{v}(t;a,j)| = \hat{Z}_j^i(t; a) \mbox{ for all $t \leq t^*$, $a \in (0,t^*)$ and $j \in [K]$}\right) \to 1,
\end{equation*}
where $t^*= t^*(n)$ is given by~\eqref{eq:t*}.
\end{lemma}
\section{Proofs}\label{sec:infector}
\subsection{Proof of Theorem~\ref{firstmain}}
We are interested in the expected fraction of vertices infected by vertices of type $i$ among the ultimately infected vertices of type $j$ during a major outbreak. By exchangeability this expected fraction is given by
\begin{equation}
\label{infprob}
\mathbb{P}\left(\mbox{$v$ is infected by a type $i$ vertex}|\mbox{$v$ is ultimately infected},v \in V_j\right).
\end{equation}
Consider $\mathcal{S}^{(n)}_v(t^*)$, where we assign to each vertex $v' \in \mathcal{S}^{(n)}_v(t^*)$ the value $\sigma'_{v'} = d(v',v)$. In Section~\ref{sec:susceptibility} we constructed part of the graph $G'$ by looking backward in time. To analyse (\ref{infprob}), we construct another part of the graph $G'$, by looking forward in time and starting at $V_{\text{init}}$. By Assumption \ref{assvinit}, $\mathcal{S}^{(n)}_v(t^*)$ does not overlap with $V_{\text{init}}$ w.h.p. We condition on this event.
We construct the relevant part of $G'$ through a series of subgraphs $\{\tilde{G}(\ell); \ell \in \mathbb{N}_0\}$ as follows.
\begin{itemize}
\item[1)] Construct $\mathcal{S}^{(n)}_v(t^*)$ and the edges that connect these vertices in $G'$ as in Section~\ref{sec:susceptibility}. Let $\tilde{G}(0)$ be this graph together with (the isolated) vertices in $V_{\text{init}}$.
\item[2)] The vertices in $V_{\text{init}}$ are active in $\tilde{G}(0)$.
\item[3)] Assume that we know $\tilde{G}(\ell)$. Let $\tilde{\sigma}_v(\ell)$ be the distance from $V_{\text{init}}$ to $v$ in $\tilde{G}(\ell)$. If there is no path from $V_{\text{init}}$ to $v$ in $\tilde{G}(\ell)$ then we set $\tilde{\sigma}_v(\ell) = \infty$. If there are no active vertices in $\tilde{G}(\ell)$ with distance from $V_{\text{init}}$ less than $\tilde{\sigma}_v(\ell)-t^*$, then the shortest path from $V_{\text{init}}$ to $v$ is the same in $G'$ as in $\tilde{G}(\ell)$, and we set $\tilde{G}(k) = \tilde{G}(\ell)$, for all $k \geq \ell$.
\item[4)] If there are active vertices in $\tilde{G}(\ell)$ with distance from $V_{\text{init}}$ less than $\tilde{\sigma}_v(\ell)-t^*$, then we construct $\tilde{G}(\ell+1)$ as follows. Pick the active vertex with the lowest distance from $V_{\text{init}}$ in $\tilde{G}(\ell)$ (in case of a tie, which occurs if there are still several active vertices in $V_{\text{init}}$, make a uniform choice among those vertices). Say that this vertex is $u \in V_i$. Then assign to $u$ the point processes $(\xi_u^1, \cdots \xi_u^K)$, having the correct distribution (see Section \ref{secmod}) and label the points of $\xi_u^j$ ($j\in [K]$) with independent uniform vertices from $V_j$ without replacement.
First we check whether some of the labels chosen correspond to vertices in $\mathcal{S}^{(n)}_v(t^*)$. If there are such labels, we add ``preliminary'' edges with tail $u$ and heads equal to the respective vertices in $\mathcal{S}^{(n)}_v(t^*)$ (whether the preliminary edges become ``actual'' edges in the graph is determined in step 5 of the construction). The lengths of such edges correspond to the points in the point processes.
Say that $u' \in \mathcal{S}^{(n)}_v(t^*) \cap V_j$ is one of the labels chosen and that $u$ is chosen at ``age'' $t'$.
If $u' \not\in \mathcal{S}^{(n)}_v(t^*;t',j)$ for at least one of the vertices that $u$ connects to, then the distance from $u$ to $v$ becomes less than $t^*$ in the preliminary graph. We set $\tilde{G}(\ell+1) = \tilde{G}(\ell)$. Note that we have identified all vertices from which there is a path of length at most $t^*$ to $v$ in the exploration of $\mathcal{S}^{(n)}_v(t^*)$. However, since we know that $u \not\in \mathcal{S}^{(n)}_v(t^*)$, we have to condition on the event that no edges with tail $u$ and heads in the set $\mathcal{S}^{(n)}_v(t^*-t')$ is shorter than $t'$.
\item[5)] Finally, in deciding whether the preliminary edges become edges in the graph $\tilde G(\ell)$, we consider the following. If the distance from $u$ to $v$ does not become less than $t^*$ in the preliminary graph, then $u$ becomes passive in $\tilde{G}(\ell+1)$, and the edge set of $\tilde{G}(\ell+1)$ consists of all edges in $\tilde{G}(\ell)$ plus the edges with tail $u$ and heads corresponding to the labels of the points in $(\xi_u^1, \cdots \xi_u^K)$. The edges that are added have the obvious lengths. The heads of the edges in $\tilde{G}(\ell+1)$ are part of the vertex set of $\tilde{G}(\ell+1)$. The active vertices in $\tilde{G}(\ell+1)$ are the active vertices in $\tilde{G}(\ell)$ apart from $u$ plus the newly added vertices in $\tilde{G}(\ell+1)$.
\end{itemize}
In the construction above, vertices of type $j$ are added in an interchangeable way, $j\in[K]$. If a preliminary edge of length $a$ has a head in $\mathcal{S}^{(n)}_v(t^*)\cap V_j$, then every vertex in $\mathcal{S}^{(n)}_v(t^*;a,j)$ has the same probability of being the head of this edge and becoming part of the constructed graph.
Next, for the coupling, we analyse $\mathcal{S}^{(n)}_v(t^*;a,j)$ even further. Denote the vertices in $\{u \in V_i; (u,v) \in E'\}$, i.e.\ the vertices in $V_i$, which are in the first generation of the susceptibility set of $v$, by $u_{i1},u_{i2}, \cdots, u_{i, X_{i}(v)}$. Here $$X_{i}(v) = |\{u \in V_i; (u,v) \in E'\}|$$ is the number of vertices of type $i$ in this first generation of the susceptibility set. Note that $X_{i}(v)$ is distributed as $X_{ij}$ where $X_{ij}$ is
binomially distributed with parameters $n_i$ and $m_{ij}/p_j$. So, $X_{ij}$ converges in distribution to a
Poisson distributed random variable with expectation $(p_i/p_j)m_{ij}$. For $r \in [X_{i}(v)]$, let $\tau_{ir}(v)= \eta(u_{ir},v)$ be the length of edge $(u_{ir},v) \in E'$. Here $\tau_{ir}(v)$ is distributed as $\tau_{ijr}$ with distribution function $\mathbb P(\tau_{ijr}\leq a)=\mathbb E[\xi_{ij}(a)]/m_{ij}$. For convenience, we let $X_i=X_i(v)$ and $\tau_{ir}=\tau_{ir}(v)$.
We consider the susceptibility processes of the vertices in $\{u_{ir}; i \in [K], r \in [X_{i}]\}$ up to distance $t^*$ from $v$ separately, i.e.\ we consider the susceptibility set of $u_{ir}$ up to time $t^*- \tau_{ir}$. Those susceptibility sets are (following the arguments in Section~\ref{subsecbp}) w.h.p.\ not overlapping and independent. Also note that
\begin{equation}
\label{susuni}
\mathcal{S}^{(n)}_{v}(t^*;a,j)= v \cup \bigcup_{i =1}^K \bigcup_{r=1}^{X_{i}} \mathcal{S}^{(n)}_{u_{ir}}(t^*-\tau_{ir};a,j).
\end{equation}
By Lemma~\ref{couplelemma}, we know that
\begin{equation}
\label{whpeq}
|\mathcal{S}^{(n)}_{u_{ir}}(t^*-\tau_{ir},a,j)| = \hat{Z}^i_j(t^*-\tau_{ir}; a) \qquad \mbox{w.h.p.}
\end{equation}
Furthermore, by equations (\ref{bpeq1}) and (\ref{bpeq2}), we obtain that
\begin{equation}
\label{infectconv}
e^{-\alpha t} \hat{Z}^{i}_j(t; a) \to c(a,j) W^{i} \qquad \mbox{a.s.\ as $t \to \infty$}
\end{equation}
on $\{\hat{Z}^{i}(t) \to \infty\}$. Since the sets $\mathcal{S}^{(n)}_{u_{ir}}(t^*-\tau_{ir})$ are w.h.p.\ not overlapping (and do not contain $v$), it follows from (\ref{susuni}) that
\begin{equation*}
|\mathcal{S}^{(n)}_{v}(t^*;a,j)\setminus v|= \sum_{i =1}^K \sum_{r=1}^{X_{i}} |\mathcal{S}^{(n)}_{u_{ir}}(t^*-\tau_{ir};a,j)| \qquad \mbox{w.h.p.}
\end{equation*}
Then, by (\ref{whpeq}) and (\ref{infectconv}), for all $i \in [K]$ and $r \in [m_i]$,
\begin{equation*}
e^{-\alpha t^*} |\mathcal{S}^{(n)}_{u_{ir}}(t^*-\tau_{ir};a,j)| \to e^{-\alpha \tau_{ir}} c(a,j) W^{i}(r)
\end{equation*}
in probability on $\{\hat{Z}^{i}_j(t) \to \infty\}$ as $n \to \infty$. Here, the random variables $W^i(1)$, $W^i(2)$, $\ldots$ are independent copies of $W^i$, $i\in[K]$. Let $\hat{W}^i$ be distributed as $W^i1\hspace{-2.5mm}{1}(\hat{Z}^{i}(t) \to \infty)$, and let $\hat{W}^i(r)$ be defined analogously. Note that $\{\hat{Z}^{i}(t) \to \infty\}$ implies that the branching process survives and new particles are born w.h.p.\ in the interval $[t^*/2,t^*]$. If $\{\hat{Z}^{i}(t) \not\to \infty\}$, then there is a last birth in the process. Because $t^*/2 \to \infty$ as $n \to \infty$, there is then no particle born in in the interval $[t^*/2,t^*]$ w.h.p. It follows from the coupling arguments above that, as $n \to \infty$,
\begin{equation*}
\mathbb{P}\left(\mathcal{S}^{(n)}_{v}(t^*) \neq \mathcal{S}^{(n)}_{v}(t^*/2)\right) \to \mathbb{P}\left(\hat{Z}^{i}(t) \to \infty\right).
\end{equation*}
We are interested in the fraction of vertices (possibly specified by type and age) at time $t^*$ in $\mathcal{S}^{(n)}_v(t^*)$ that are connected to $v$ through a path of vertices in $\mathcal{S}^{(n)}_v(t^*)$ that include $u_{ir}$, $i \in [K]$ and $r \in [X_{ij}]$. That is, we want to analyse
\begin{equation*}
\frac{|\mathcal{S}^{(n)}_{u_{ir}}(t^*-\tau_{1r};a,j)|}{\sum_{j =1}^K \sum_{r=1}^{X_{j}} |\mathcal{S}^{(n)}_{u_{jr}}(t^*-\tau_{jr};a,j)|},
\end{equation*}
on the set $\sum_{j =1}^K \sum_{r=1}^{X_{j}} |\mathcal{S}^{(n)}_{u_{jr}}(t^*-\tau_{jr};a,j)| \neq 0$. Lemma \ref{couplelemma} allows us to couple the epidemic process with a branching processes $\{\mathcal{Z}(t);t \geq 0\}$ such that w.h.p.
\begin{equation}\label{eq:fraction}
\frac{|\mathcal{S}^{(n)}_{u_{ir}}(t^*-\tau_{1r};a,j)|}{\sum_{j =1}^K \sum_{r=1}^{X_{j}} |\mathcal{S}^{(n)}_{u_{jr}}(t^*-\tau_{jr};a,j)|} =
\frac{Z_j^{i,r}(t^*-\tau_{1r};a)}{\sum_{k =1}^K \sum_{r=1}^{X_{kj}} Z_j^{k,r}(t^*-\tau_{kr};a)},
\end{equation}
where $Z_j^{i,r}(t;a)$ are independent copies of $Z_j^{i}(t;a)$, $r \in \mathbb{N}$. Multiplying numerator and denominator of~\eqref{eq:fraction} by $e^{-\alpha t^*}$ and using (\ref{infectconv}) we obtain that (\ref{eq:fraction}) is equal to
\begin{equation*}
\frac{e^{-\alpha \tau_{ir}} (e^{-\alpha (t^*-\tau_{ir})}Z_j^{i,r}(t^*-\tau_{1r};a))}{\sum_{k =1}^K \sum_{r=1}^{X_{kj}} e^{-\alpha \tau_{kr}} (e^{-\alpha (t^*-\tau_{kr})} Z_j^{k,r}(t^*-\tau_{kr};a)},
\end{equation*}
which converges a.s.\ to
\begin{equation}\label{eq:fraction2}
\frac{e^{-\alpha \tau_{ir}} c(a,j) \hat{W}^i(r)}{\sum_{k =1}^K \sum_{r=1}^{X_{kj}} e^{-\alpha \tau_{kr}} c(a,j) \hat{W}^k(r)} = \frac{e^{-\alpha \tau_{ir}} \hat{W}^i(r)}{\sum_{k =1}^K \sum_{r=1}^{X_{kj}} e^{-\alpha \tau_{kr}} \hat{W}^k(r)}
\end{equation}
as $n \to \infty$ (i.e.\ as $t^* \to \infty$). Note that the right hand side of~\eqref{eq:fraction2} does not depend on $a$ or $j$. Since vertices of type $j$, $j \in [K]$, are added in such a way that every vertex in $\mathcal{S}^{(n)}_v(t^*;a,j)$ has the same probability of being the head of this edge, every vertex in $\mathcal{S}^{(n)}_v(t^*;a,j)$ has the same probability of becoming part of the constructed graph. So,
$\mathbb{P}\left(\mbox{$v$ is infected by a type $i$ vertex}|\mbox{$v$ is ultimately infected}\right)$ is equal to
$$
\frac{1}{\mathbb{P}(\mathcal{S}^{(n)}_{v}(t^*) \neq \mathcal{S}^{(n)}_{v}(\frac{t^*}{2}))} \mathbb{E}\left[\frac{\sum_{r=1}^{X_i} |\mathcal{S}^{(n)}_{u_{ir}}(t^*-\tau_{ir};a,j)|}{\sum_{k =1}^K \sum_{r=1}^{X_k} |\mathcal{S}^{(n)}_{u_{kr}}(t^*-\tau_{kr};a,j)|} 1\hspace{-2.5mm}{1}\left(\mathcal{S}^{(n)}_{v}(t^*) \neq \mathcal{S}^{(n)}_{v}\left(\frac{t^*}{2}\right)\right)
\right]
$$
and converges to $\rho_{ij}$, with $\rho_{ij}$ given by~\eqref{mainequ}. This completes the proof for Theorem \ref{firstmain}.
\subsection{Towards to proof of Theorem~\ref{secondmain}: Bounds for the $\boldsymbol{\rho_{ij}}$}\label{sec:bounds}
Theorem~\ref{firstmain} provides us with an expression for the asymptotic fractions $\rho_{ij}$ of infected individuals of type $j$ that were infected by individuals of type $i$. However, often there is no explicit description of the distribution of ${W}^k(r)$. In Theorem~\ref{secondmain} we consider bounds for the $\rho_{ij}$ for a special class of models, as specified in Section \ref{secmod}. In order to obtain those bounds, in this subsection and Section \ref{sec:proofsecondmain}, we discuss, for the general setting, how to obtain the bounds for $\rho_{ij}$ using the epidemic random graph $G'$. In most of the subsequent analysis we analyse the graph $G'$ without taking the lengths of edges in into account.
\begin{figure}
\centering
\includegraphics[scale=0.7]{backwardguilty2.pdf}
\caption{Illustration of (a part of) the susceptibility set of $v_*$ in $G'=(V,E')$. Vertices of type 1 are represented by circles and vertices of type 2 are represented by boxes. The part of the susceptibility set illustrated in the figure is the set connected to $v_*$ through paths of edges with heads of type 1.}
\label{fig:backward}
\end{figure}
Note that Assumption \ref{finass} guarantees that, with probability 1, all paths in $E'$ have different lengths. In order to obtain the maximum and minimum of the probability $\rho_{ij}$ for fixed mean offspring matrix $M$, we first investigate the susceptibility set of $v_*$ ($v_* \in V_j$) restricted to the graph $G'_{ij}=(V,E'_{ij})$. We denote this susceptibility set by $\mathcal{S}_{v_*,ij}$. Here $E'_{ij} \subset E'$ is the subset of $E'$ that consists of the edges that either have tail vertex in $V_i$ or head vertex not in $V_j$, i.e.\ $E \setminus E'_{ij}$ is the set of edges with tails in $V \setminus V_i$ and heads in $V_j$.
In Figure \ref{fig:backward}, the set $\mathcal{S}_{v_*,11}$ consist of all vertices of type 1 (the circles).
Let $\mathcal{S}^{j}_{v_*,ij}= V_j \cap \mathcal{S}_{v_*,ij}$. If $\mathcal{S}_{v_*,ij} \cap V_{\text{init}} = \emptyset$, i.e.\ if there is no path from $V_{\text{init}}$ to $v_*$ in $G'_{ij}$, then, by the definition of the epidemic process, every vertex of type $j$ in $\mathcal{S}_{v_*,ij}$ has the same probability to be the first one to be infected in the epidemic, i.e.\ $u_*=\argmin_ {u \in \mathcal{S}^j_{v_*,ij}}d(V_{\text{init}},u)$ is uniform in $\mathcal{S}^j_{v_*,ij}$.
Condition on $\mathcal{S}_{v_*} \cap V_{\text{init}} \neq \emptyset$. If $u_* =v_*$ and $\mathcal{S}_{v_*,ij}\cap V_{\text{init}} = \emptyset$, then $v_*$ is infected by a vertex that is not of type $i$. On the other hand, if $u_* \neq v_*$ or $\mathcal{S}_{v_*,ij} \cap V_{\text{init}} \neq \emptyset$, then $v_*$ might be infected by a type $i$ vertex. Hence
\begin{align*}
1-\rho_{ij} &= \mathbb{P}(\mbox{$v_*$ is not infected by a type $i$ vertex}|\mathcal{S}_{v_*} \cap V_{\text{init}} \neq \emptyset)\\
& \geq \mathbb{P}(u_* = v_*|\mathcal{S}_{v_*} \cap V_{\text{init}} \neq \emptyset).
\end{align*}
Now assume that $\mathbb{P}(\eta_{i',j'}> n^{-1}) =1$, for $i' \in [K]\setminus i$ and $j'=j$, with $j$ the type of $v_*$, and $\mathbb{P}(\eta_{i',j}< n^{-2}) =1$ otherwise. This implies that the lengths of any path with only edges in $E'_{ij}$ is less than any edge in $E'\setminus E'_{ij}$. The assumptions guarantee that, for $u_* \neq v_*$ or $\mathcal{S}_{v_*,ij} \cap V_{\text{init}} \neq \emptyset$, conditioned on $\mathcal{S}_{v_*} \cap V_{\text{init}} \neq \emptyset$, $v_*$ is infected by a vertex of type $i$ (the tail of the edge with head $v_*$ in the shortest path in $E_{ij}'$ from $u_*$ or $V_{\text{init}}$ to $v_*$). Therefore, for this model,
\begin{equation*}
\mathbb{P}(\mbox{$v_*$ is not infected by a type $i$ vertex}|\mathcal{S}_{v_*} \cap V_{\text{init}} \neq \emptyset) = \mathbb{P}(u_* = v_*|\mathcal{S}_{v_*} \cap V_{\text{init}} \neq \emptyset).
\end{equation*}
Hence, for a given distribution of $E'$, models with $\mathbb{P}(\eta_{i',j'}> n^{-1}) =1$, for $i' \in [K]\setminus i$ and $j'=j$ and $\mathbb{P}(\eta_{i',j}< n^{-2}) =1$ otherwise, are among the models for which the fraction of the ultimately infected vertices of type $j$, infected by a type $i$ vertex is maximised.
Using similar arguments we obtain that $\rho_{ij}$, the fraction of ultimately infected vertices of type $j$ that are infected by vertices of type $i$, is minimal if the edge lengths of vertices with tail in $i$ and head in $j$ are much longer than the other edges. This will be used in Section \ref{sec:proofsecondmain}.
\subsection{Proof of Theorem~\ref{secondmain}}\label{sec:proofsecondmain}
We consider the so-called symptom-response SEIR epidemic model that is introduced in detail in \cite{Leun18}, see also Remark~\ref{rk:SEIR}.
That is, consider the model introduced in Section \ref{secmod} with $K=2$ and consider general distributions $(\xi_{i1}, \xi_{i2})$.
For this model, using the arguments from Section \ref{sec:bounds}, we can compute the maximal probabilities $\rho_{11}^+$, $\rho_{22}^+$ (and minimal probabilities $\rho_{21}^-$ and $\rho_{11}^-$) explicitly.
In this subsection we consider the model in which edges from $V_1$ to $V_1$ are infinitesimally short and all other edges in $E'$ are relatively long.
From Section \ref{sec:bounds} we know that this is the model for which the fraction of vertices of type $1$ infected by type $1$ vertices is maximised.
For reasons of convenience we assume that $v_*$ is of type 1 with $\mathcal S_{v_*}\cap V_{\text{init}}\neq\emptyset$. Note that Theorem~\ref{secondmain} considers the special case that $(\xi_{i1}, \xi_{i2})$ is obtained from the independent labelling of a one-dimensional point process $\xi_i$. In this special case we have $\rho_{11}=\rho_1=\rho_{12}$ and $\rho_{21}=\rho_2=\rho_{22}$. We treat Theorem~\ref{secondmain} in Remark~\ref{rk:prf_thm2} at the end of this section.
First, we compute the upper bound $\rho_{11}^+$ (and lower bound $\rho_{21}^-$) for the general setting.
We note that it is harder (if not impossible) to obtain an explicit expression for $\rho_{12}^+$ or $\rho_{21}^+$. As will become clear in the computation below, the difficulty with $\rho_{12}^+$ or $\rho_{21}^+$ is that one would need to consider paths in $G_{12}'$ and $G_{21}'$ that contain vertices of both type 1 and type 2. In contrast, for computing $\rho_{11}^+$, paths in $G_{11}'$ that end in $v^*$ contain only vertices of type 1.
If we ignore the lengths of edges in $G'$ in the general model introduced in Section \ref{secmod}, then
the approximating (backward) branching process describing the generation-based growth of $\mathcal{S}_{v_*}(t)$ is defined through the following offspring distributions. The number of children of type $i$ of a particle of type $j$ is Poisson distributed with expectation $m_{ji}^{(b)} = \frac{p_i}{p_j}m_{ij}$, $i,j\in[2]$. For different $i$ and $j$ the distributions are independent of each other.
Let $Y= |\mathcal{S}_{v_*,11}|$. It is easily seen that $Y$ is approximated by the size of a branching process with Poisson offspring distribution that has expectation $m_{11}^{(b)}$. Then $Y$ is Borel distributed with parameter $m_{11}^{(b)}$, i.e.\ for $\ell \in \{1,2,\cdots\}$,
\begin{equation}
\label{Boreldist}
\mathbb{P}(Y=\ell) = \frac{(m_{11}^{(b)} \ell)^{\ell-1} e^{-m_{11}^{(b)} \ell}}{\ell!}
\end{equation}
(see~\cite{Aldo98}). If $m_{11}^{(b)}>1$, then $\mathbb{P}(Y=\infty) > 0$. Standard results on Borel distributions \cite{Aldo98} give that, for $m_{11}^{(b)} \leq 1$,
\begin{equation}
\label{Borelinv}
\mathbb{E}[1/Y] = 1-m_{11}^{(b)}/2.
\end{equation}
Define
\begin{equation*}
\rho_{11} = \mathbb{P}(\mbox{$v_*$ is infected by by a type 1 vertex}|\mathcal{S}_{v_*} \cap V_{\text{init}} \neq \emptyset),
\end{equation*}
for $v_* \in V_1$. From the arguments in Section~\ref{sec:bounds} we know that
\begin{equation}
\label{rho1eq}
\rho_{21}^- = \mathbb{E}[1/Y|\mathcal{S}_{v_*} \cap V_{\text{init}} \neq \emptyset.]
\end{equation}
Consistency then yields
\begin{equation}\label{eq:rho11}
\rho_{11}^+=1-\rho_{21}^-.
\end{equation}
We use the following extinction probabilities in the backward branching process:
\begin{itemize}
\item[$q_1$:] the probability that the backward branching process starting with a single type 1 particle goes extinct,
\item[$q_2$:] the probability that the backward branching process starting with a single type 2 particle goes extinct,
\item[$\tilde{q}_1$:] the probability that the backward branching process restricted to type 1 particles dies out, i.e.\ $\tilde{q}_1 = \mathbb{P}(Y<\infty)$.
\end{itemize}
From theory on multi-type supercritical branching processes \cite[Chap.\ 4]{Jage75} we know that $(q_1, q_2)$ is the unique solution in $(0,1)^2$ of
\begin{eqnarray}
x & = & \sum_{k=0}^{\infty} \frac{(m_{11}^{(b)})^k}{k!}e^{-m_{11}^{(b)}} x^k
\sum_{\ell=0}^{\infty} \frac{(m_{12}^{(b)})^{\ell}}{\ell!}e^{-m_{12}^{(b)}} y^{\ell}
= e^{-[m_{11}^{(b)}(1-x)+ m_{12}^{(b)}(1-y)]} \label{q1eq}\\
y & = & \sum_{k=0}^{\infty} \frac{(m_{21}^{(b)})^k}{k!}e^{-m_{21}^{(b)}} x^k
\sum_{\ell=0}^{\infty} \frac{(m_{22}^{(b)})^{\ell}}{\ell!}e^{-m_{22}^{(b)}} y^{\ell}
= e^{-[m_{21}^{(b)}(1-x)+ m_{22}^{(b)}(1-y)]}. \label{q2eq}
\end{eqnarray}
Furthermore, $\tilde{q}_1$ is the smallest positive solution of~\eqref{qsequa} which might be 1 or strictly smaller than 1, depending on whether or not $p_1m_{11} = m_{11}^{(b)} \leq 1$.
We distinguish between $m_{11}^{(b)} \leq 1$ and $m_{11}^{(b)}>1$. This distinction is not necessary, but we think the argument becomes clearer, by treating the case $\tilde{q}_1= 1$ separately.
\subsubsection{The case $\boldsymbol{m_{11}^{(b)} \leq 1}$}
Assume that $m_{11}^{(b)} \leq 1$. Let $\mathcal{A}$ be the event that the backward branching process with ancestor of type 1, involving both type 1 and type 2 individuals, survives. We explore the backward branching process of a particle of type 1 on $\mathcal{A}$ as follows.
\begin{itemize}
\item Explore the backward process of particles of type 1. If we ignore the conditioning on $\mathcal{A}$, the process can be described by a subcritical branching process with Poisson offspring distribution with expectation $m_{11}^{(b)}$. The random variable $Y$ is the total size of this branching process, including the initial individual. We know that $Y$ is Borel($m_{11}^{(b)}$) distributed.
\item Condition on event $\mathcal{A}$: the probability that a particle of type 1 has infinitely many descendants is $\mathbb{P}(\mathcal{A})= 1-q_1$, where $q_1$ is defined through~\eqref{q1eq} and \eqref{q2eq}. We also use the probability that a particle has infinite offspring, conditioned on having no children of type 1. This probability is $1-q$, where
\begin{equation}\label{eq:qq1}
q= \sum_{\ell =0}^{\infty} \frac{(m_{12}^{(b)})^{\ell}}{\ell !} e^{-m_{12}^{(b)}} (q_2)^{\ell} = e^{-m_{12}^{(b)}(1-q_2)}=q_1 e^{m_{11}^{(b)} (1-q_1)}.
\end{equation}
Using Bayes' rule we obtain
\begin{align}
\mathbb{P}(Y=\ell|\mathcal{A}) & = \frac{\mathbb{P}(\mathcal{A}|Y=\ell) \mathbb{P}(Y= \ell)}{1-q_1}\nonumber\\
& = \frac{1-q^{\ell}}{1-q_1} \frac{(m_{11}^{(b} \ell)^{\ell-1} e^{-m_{11}^{(b)} \ell}}{\ell!}\nonumber\\
&= \frac{1}{1-q_1} \frac{(m_{11}^{(b)} \ell)^{\ell-1} e^{-m_{11}^{(b)} \ell}}{\ell!} - \frac{q}{1-q_1} \frac{(m_{11}^{(b)} \ell q)^{\ell-1} e^{-m_{11}^{(b)} \ell}}{\ell!}\nonumber\\
&=\frac{1}{1-q_1} \left( \frac{(m_{11}^{(b)} \ell)^{\ell-1} e^{-m_{11}^{(b)} \ell}}{\ell!} - \frac{q_1(m_{11}^{(b)} q_1\ell)^{\ell-1} e^{-m_{11}^{(b)} q_1 \ell}}{\ell!} \right).\label{Borelcond}
\end{align}
where we used~\eqref{eq:qq1} in the last equality.
\item Using~\eqref{Borelinv}, \eqref{rho1eq} and (\ref{Borelcond}) we find that
\begin{align}
\label{hitprob}
\rho_{21}^-&= \mathbb{E}[Y^{-1}|\mathcal{A}]\nonumber\\
&= \frac{1}{1-q_1} \left(1- \frac{m_{11}^{(b)}}{2} -q_1\left(1-\frac{m_{11}^{(b)} q_1}{2} \right) \right)\nonumber\\
&= \frac{1}{1-q_1} \left(1- q_1 - (1-(q_1)^2) \frac{m_{11}^{(b)}}{2} \right)\nonumber\\
&= 1 - (1+q_1) \frac{m_{11}^{(b)}}{2}.
\end{align}
\item Using~\eqref{eq:rho11} we find the desired expression $\rho_{11}^+$ for $m_{11}^{(b)}$:
\begin{equation}\label{eq:upper}
\rho_{11}^+=1-(1+q_1) \frac{m_{11}^{(b)}}{2}.
\end{equation}
\end{itemize}
\subsubsection{The case $\boldsymbol{m_{11}^{(b)} > 1}$}
Assume that $m_{11}^{(b)} > 1$. Then it is possible that the backward process restricted to particles of type 1 is already large, i.e.\ the approximating branching process with Poisson($m_{11}^{(b)}$) offspring distribution already survives (call this event $\mathcal{A}_1$). In that case the probability that vertex $v$ is infected by a vertex of type 1 approaches 1 as the population size tends to infinity.
The other possibility is that the backward process restricted to particles of type 1 stays small. Call this event $\mathcal{A}_1^C$, the complement of $\mathcal{A}_1$. The probability of this event is $\mathbb{P}(\mathcal{A}_1^C)=\tilde{q}_1$, where $\tilde{q}_1$ is the unique solution in $(0,1)$ of equation (\ref{qsequa}) (note that $m_{11}^{(b)}>1$ ensures that $\tilde q_1$ exists). From the theory of branching processes we know that, conditioned on $\mathcal{A}_1^C$, the approximating branching process is still a branching process with Poisson distributed offspring distribution, but now with offspring expectation $m_{11}^{(b)} \tilde{q}_1$. We still condition on the event that the approximating branching process (including both types of particles) survives.
Conditioned on $\mathcal{A}_1^C$, the total size of the backward process restricted to particles of type 1 is Borel distributed with parameter $m_{11}^{(b)} \tilde{q}_1$. We use Bayes' rule,
\begin{equation*}
\mathbb{P}(Y=\ell|\mathcal{A},\mathcal{A}_1^C) = \frac{\mathbb{P}(\mathcal{A}|Y=\ell,\mathcal{A}_1^C) \mathbb{P}(Y= \ell|\mathcal{A}_1^C)}{\mathbb{P}(\mathcal{A}|\mathcal{A}_1^C)}.
\end{equation*}
Note that
\begin{equation*}
\mathbb{P}(\mathcal{A}|\mathcal{A}_1^C) = \frac{ \mathbb{P}(\mathcal{A},\mathcal{A}_1^C)}{ \mathbb{P}(\mathcal{A}_1^C)}= \frac{\mathbb{P}(\mathcal{A}_1^C)-\mathbb{P}(\mathcal{A}^C,\mathcal{A}_1^C)}{ \mathbb{P}(\mathcal{A}_1^C)}= \frac{\mathbb{P}(\mathcal{A}_1^C)-\mathbb{P}(\mathcal{A}^C)}{ \mathbb{P}(\mathcal{A}_1^C)}= \frac{\tilde{q}_1-q_1}{\tilde{q}_1}.
\end{equation*}
Then, with the same arguments as those leading to (\ref{hitprob}) but with $m_{11}^{(b)}$ replaced by $m_{11}^{(b)} \tilde{q}_1$, yield
\begin{equation*}
\mathbb{P}(\mathcal{A}|Y=\ell,\mathcal{A}_S^C) =1- q^{\ell} = 1-(q_1)^{\ell} e^{m_{11}^{(b)} (1-q_1)\ell}.
\end{equation*}
and
\begin{equation*}
\mathbb{P}(Y= \ell|\mathcal{A}_1^C)=
\frac{(m_{11}^{(b)} \tilde{q}_1 \ell)^{\ell-1} e^{-m_{11}^{(b)} \tilde{q}_1 \ell}}{\ell!}.
\end{equation*}
Combining these identities with~\eqref{qsequa} yields
\begin{align*}
& \mathbb{P}(Y=\ell|\mathcal{A},\mathcal{A}_1^C)\\
&= \frac{\tilde{q}_1}{\tilde{q}_1-q_1} \left(1-(q_1)^{\ell} e^{m_{11}^{(b)} (1-q_1)\ell}\right)\left(\frac{(m_{11}^{(b)} \tilde{q}_1 \ell)^{\ell-1} e^{-m_{11}^{(b)} \tilde{q}_1 \ell}}{\ell!} \right)\\
&= \frac{\tilde{q}_1}{\tilde{q}_1-q_1} \left(\frac{(m_{11}^{(b)} \tilde{q}_1 \ell)^{\ell-1} e^{-m_{11}^{(b)} \tilde{q}_1 \ell}}{\ell!} \right) - \frac{q_1 (\tilde{q}_1)^{\ell}}{\tilde{q}_1-q_1} \left(\frac{(m_{11}^{(b)} q_1 \ell)^{\ell-1} e^{-m_{11}^{(b)} (\tilde{q}_1+q_1-1) \ell}}{\ell!} \right)\\
& = \frac{\tilde{q}_1}{\tilde{q}_1-q_1} \left(\frac{(m_{11}^{(b)} \tilde{q}_1 \ell)^{\ell-1} e^{-(m_{11}^{(b)} \tilde{q}_1 \ell}}{\ell!} \right) - \frac{\tilde{q}_1}{\tilde{q}_1-q_1}\left(\frac{(m_{11}^{(b)} q_1 \ell)^{\ell-1} e^{-m_{11}^{(b)} q_1 \ell}}{\ell!} \right).
\end{align*}
By standard results on Galton Watson branching processes we know that $m_{11}^{(b)} \tilde{q}_1 \leq 1$. Moreover, since $q_1 \leq \tilde{q}_1$, also $m_{11}^{(b)} q_1 \leq 1$ holds. Then, using (\ref{Borelinv}), we obtain
\begin{align*}
\mathbb{E}[Y^{-1}|\mathcal{A},\mathcal{A}_1^C] &= \frac{\tilde{q}_1}{\tilde{q}_1-q_1} \left(1-\frac{m_{11}^{(b)} \tilde{q}_1}{2}\right) -
\frac{q_1}{\tilde{q}_1-q_1} \left(1-\frac{m_{11}^{(b)} q_1}{2}\right)\\
& = 1 - \frac{m_{11}^{(b)} (\tilde{q}_1 + q_1)}{2}.
\end{align*}
This leads to
\begin{align}
\rho_{21}^- &= \mathbb{E}[Y^{-1}|\mathcal{A}]\nonumber\\
&= \mathbb{E}[Y^{-1}|\mathcal{A},\mathcal{A}_1^C]\mathbb{P}[\mathcal{A}_1^C|\mathcal{A}] + \mathbb{E}[Y^{-1}|\mathcal{A},\mathcal{A}_1] \mathbb{P}[\mathcal{A}_1|\mathcal{A}]\nonumber\\
&= \mathbb{E}[Y^{-1}|\mathcal{A},\mathcal{A}_1^C]\frac{\mathbb{P}[\mathcal{A}_1^C,\mathcal{A}]}{\mathbb{P}[\mathcal{A}]} + 0 \nonumber\\
&= \mathbb{E}[Y^{-1}|\mathcal{A},\mathcal{A}_1^C]\frac{\mathbb{P}[\mathcal{A}_1^C]-\mathbb{P}[\mathcal{A}_1^C,\mathcal{A}^C]}{\mathbb{P}[\mathcal{A}]}\nonumber\\
&= \left( 1 - \frac{m_{11}^{(b)} (\tilde{q}_1 + q_1)}{2} \right) \frac{\tilde{q}_1-q_1}{1-q_1}.\label{finaleq}
\end{align}
Note that this expression~\eqref{finaleq} is consistent with the result~\eqref{hitprob} for $m_{11}^{(b)}\leq 1$, where $\tilde{q}_1=1$. Finally, using~\eqref{eq:rho11}, we find the expression for $\rho_{11}^+$:
\begin{equation}\label{eq:rho11_final}
\rho_{11}^+=1-\left( 1 - \frac{m_{11}^{(b)} (\tilde{q}_1 + q_1)}{2} \right) \frac{\tilde{q}_1-q_1}{1-q_1}.
\end{equation}
\begin{remark}[Symptom-response SEIR epidemic model: proof of Theorem~\ref{secondmain}]\label{rk:prf_thm2}
In \cite{Leun18} we interpret vertices of type 1 as individuals that show symptoms when infectious, while vertices of type 2 are asymptomatic throughout their infectious period. We assume that whether infected individuals become symptomatic or not, does not depend on who infected them or when they were infected. So, we may assign i.i.d.\ types to the vertices before the epidemic. Note that the type of a vertex that does not become infected during the epidemic has no epidemiological relevance. Type 1 and type 2 vertices are equally susceptible. This implies that $\{(\xi_v^1(t),\xi_v^2(t)); t \geq 0\}$ can be obtained by considering a one-dimensional point process $\{\xi_v(t); t \geq 0\}$, for all $v\in V$. One can assign the types (type 1 with probability $p_1$ or type 2 with probability $p_2=1-p_1$) independently to the points of this process. Then $\rho_{11}=\rho_1=\rho_{12}$ and $\rho_{22}=\rho_2=\rho_{21}$. Furthermore $m_{ji}^{(b)}=p_i \tilde m_j$, with $\tilde m_j=\mathbb{E}[\xi_j(\infty)]$ the expected number of secondary cases generated by a newly infected type $j$ individual in an otherwise susceptible population, $i,j=1,2$. Note that~\eqref{eq:rho11_final} reduces to the fraction $\rho_1^+$ of the ultimately infected vertices, that are infected by symptomatic vertices:
\begin{equation*}
\rho_1^+ = 1-\left( 1 - \frac{p_1\tilde m_1(\tilde{q}_1 + q)}{2} \right) \frac{\tilde{q}_1-q}{1-q},
\end{equation*}
where $\tilde{q}_1$ and $q$ are solutions of~\eqref{qsequa} and~\eqref{qequa}. This proves Theorem~\ref{secondmain}.
\end{remark}
\section{Discussion}\label{sec:discussion}
In this manuscript we couple a (fairly) general multi-type stochastic epidemic process with a weighted random graph. We use this random graph to obtain a characterisation of the (large population limit) fraction of individuals in the population that had an infector of (say) type 1 given a large outbreak.
The results of this paper are applied (and in more detail in \cite{Leun18}) to a model where the types of individuals represent whether the infected individual will show symptoms at some moment if infected, or he or she stays asymptomatic if infected.
From a public health perspective, a relevant variant of the model would be to consider a population with only one type of individuals, where infectious individuals may start off asymptomatic after which they become symptomatic (see \cite{Fras04}, where the asymptomatic phase is also referred to as ``presymptomatic''). The related question for this model would be: ``what fraction of the infected population had a symptomatic infector, given a large outbreak?''. In order to analyse this model we need to assign types to the edges in the epidemic graph instead of to the vertices. We can use the techniques of this paper to characterise the answer to the above question. We can still define a susceptibility process from a vertex $v$. The susceptibility process can be approximated by a multi-type branching process. The type of a particle in the branching process should then corresponds to the type of the edge through which the particle is added to the susceptibility process. That is, the type of the particle (say $u$) in the branching process depends on whether the first edge in the shortest path from $u$ to $v$ in the epidemic graph represents a contact that was made while $u$ was symptomatic or asymptomatic. As in Theorem \ref{firstmain} the answer will be implicit through its dependence on the distribution of the martingale limits $W$.
\section*{Acknowledgements}
The authors are supported by Vetenskapsr{\aa}det (Swedish Research Council), grants 2015-05015
(TB and KYL) and 2016-04566 (PT).
\bibliographystyle{abbrv}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,575 |
Q: 2 users but 0 sessions? The screenshot below is showing pages reached via google/organic.
I'm confused by the highlighted row below. How can I have 2 users for this page with 0 sessions (and 0 everything else).
I'm assuming it's not a bot if the source/medium is googleorganic?
A: This is normal as the "sessions" metric is only incremented on the FIRST hit of the session. https://support.google.com/analytics/answer/2934985?hl=en
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,886 |
Q: how to set a the minimal height of a div to adjust it's content I'm trying to make a webpage with the following structure:
1 big div (main), and 3 divs inside it, a left shadow, content, and a right shadow.
these is the css code for them, mleft and mright are the shadows.
body,html{height:100%;}
.main {
width:900px;
height:100%;
}
.mleft, .mright {
width:25px;
height:100%;
float:left;
}
.mleft { background-image: url("shadowleft.jpg"); }
.mright { background-image: url("shadowright.jpg"); }
.content {
width:850px;
float:left;
background-color:red;
}
And the html is like this:
<div class="main">
<div class="mleft"></div>
<div class="mcontent">
(content, some text and images)
</div>
<div class="mright"></div>
</div>
I want this to be viewable in big and small screens, the problem is that when viewing in small screens or making the window small, the main div height goes below the height of content div, so the shadow is too short to cover content div.
I've been playing with min-height, but min-height:auto, doesn't work, and none of the values of "overflow" does what I want.
Any clean way of solving this that works on any browsers?
Should I use javascript?, redo everything another way?
Update:This is an image of how it looks
Update2: The height of main seems to be directly the height of the window (100%) so I main is always the size of the window, which if small it's less than the content inside it, I tried playing with min-height with no success. The expected result is that it resizes until it reaches the size of it's contents, when it should stop.
A: OK, I've deleted all the old stuff... found a solution using positioning :)
http://jsfiddle.net/Damien_at_SF/AtX4A/
Basically, the shadows sit inside the content div and with absolute positioning are placed at 0,0 left and 0,0 right (or you could move them outside the content using negative positioning)
UPDATE: put the main div back in and applied margin:auto to it's style in order to center the whole lot :)
HTML
<div class="main">
<div class="mcontent">
<div class="mleft"></div>
<div class="mright"></div>
(content, some text and images)>
<div style="clear:both;"></div>
</div>
</div>
CSS
body,html{
height:100%;
margin:0px;
padding:0px;
}
.main {
width:900px;
height:100%;
margin:auto;
}
.mleft, .mright {
width:25px;
height:100%;
}
.mleft {
background:green;
position:absolute;
top:0px;
left:0px;
}
.mright {
background:blue;
position:absolute;
top:0px;
right:0px;
}
.mcontent {
width:850px;
background-color:red;
position:relative;
padding-left:25px;
}
Hope that helps :)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,795 |
Stig Verner Fyring, född 7 december 1934 i Stockholm, död 7 mars 2015 i Stockholm, var en svensk marinmålare, illustratör och krögare.
Stig Fyring var uppväxt på Stora Essingen i Stockholm. Han studerade på Konstfackskolan åren 1956–1960 och arbetade därefter som textilfabrikör och krögare innan han på 1990-talet övergick helt till konstnärlig verksamhet. Fyring arbetade i akvarell, i olja och med stenciler och glaskonst och ofta i blandteknik på drivved, på förpackningar, på segelduk och på bottenplåtar. Han sökte sina motiv främst i skärgården, där han hade släkt på Möja och runt Östersjön, men även annorstädes i marina miljöer i Spanien och på Kanalöarna. Hans målningar tar ofta avstamp i Fyrings tidigare verksamhet som krögare och kombinerar bilder med recept, menyer och med fågelägg eller måltidsdetaljer. Fyring är också känd för kryddat brännvin. Fyrings konst är representerad på Vasamuseet, Sjöhistoriska museet, Skärgårdsstiftelsen och Stockholms läns landsting. Han är begravd på Värmdö kyrkogård.
Illustrerade publikationer
Källor
Svenska marinmålare
Svenska illustratörer
Svenska målare under 1900-talet
Konstnärer från Stockholm
Födda 1934
Avlidna 2015
Män | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,282 |
Q: Disable selected checkbox on button click Just started working with the MVVM design pattern and I'm stuck.
When my application launches, I have a treeview populated with a list of objects names. I've setup the IsChecked Binding, and it works fine. I'm trying to setup the IsEnabled Binding.
I want the user to select the items in the treeview he wants, then click one of three buttons to perform an action. On click, I want the selected items to remain in the treeview, but be disabled, so the user cannot perform another action on those items.
I'm using a RelayCommand class in the application.
private ICommandOnExecute _execute;
private ICommandOnCanExecute _canExecute;
public RelayCommand(ICommandOnExecute onExecuteMethod,
ICommandOnCanExecute onCanExecuteMethod)
{
_execute = onExecuteMethod;
_canExecute = onCanExecuteMethod;
}
#region ICommand Members
public event EventHandler CanExecuteChanged
{
add { CommandManager.RequerySuggested += value; }
remove { CommandManager.RequerySuggested -= value; }
}
public bool CanExecute(object parameter)
{
return _canExecute.Invoke(parameter);
}
public void Execute(object parameter)
{
_execute.Invoke(parameter);
}
#endregion
My object model class uses this
private bool _isEnabled;
public bool IsEnabled
{
get { return true; }
set { _isEnabled = value};
}
Then within my button method I have
if (interfaceModel.IsChecked)
{
//Does Something
MyObjectName.IsEnabled = false;
}
And here is my xaml
<CheckBox IsChecked="{Binding IsChecked}" IsEnabled="{Binding IsEnabled, Mode=TwoWay}">
<TextBlock Text="{Binding MyObjectName}" Margin="5,2,1,2" HorizontalAlignment="Left" />
</CheckBox>
A: You need a setup like this:
// Your ViewModel should implement INotifyPropertyChanged
class ViewModel : INotifyPropertyChnaged
{
private bool _isEnabled;
public bool IsEnabled
{
get { return _isEnabled; }
set
{
_isEnabled = value;
SetPropertyChanged("IsEnabled"); // Add this to your setter.
}
}
// This comes from INotifyPropertyChanged - the UI will listen to this event.
public event PropertyChangedEventHandler PropertyChanged;
private void SetPropertyChanged(string property)
{
if (PropertyChanged != null)
{
PropertyChanged( this, new PropertyChangedEventArgs(property) );
}
}
}
Note that PropertyChanged comes from having your ViewModel implement INotifyPropertyChanged. To notify the UI, you have to raise that event, and tell it what property was changed (usually in the setter - see above).
Alternatively, if you don't like raw strings (I don't, personally), you can use generics and expression trees to do something like this:
public void SetPropertyChanged<T>(Expression<Func<T, Object>> onProperty)
{
if (PropertyChanged != null && onProperty.Body is MemberExpression)
{
String propertyNameAsString = ((MemberExpression)onProperty.Body).Member.Name;
PropertyChanged(this, new PropertyChangedEventArgs(propertyNameAsString));
}
}
Where in your setter you can say:
public bool IsEnabled
{
set
{
_isEnabled = value;
SetPropertyChanged<ViewModel>(x => x.IsEnabled);
}
}
And now it's strongly typed, which is kinda nice.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,568 |
{"url":"https:\/\/math.stackexchange.com\/questions\/1639461\/on-groups-with-presentations-langle-a-b-ca2-b2-c2-abp-bcq-car-a","text":"# On groups with presentations $\\langle a,b,c|a^2=b^2=c^2=(ab)^p=(bc)^q=(ca)^r=(abc)^s=1\\rangle$\u2026\n\n$$\\langle a,b,c|a^2=b^2=c^2=(ab)^p=(bc)^q=(ca)^r=1\\rangle =\\Delta(p,q,r)$$ This is a presentation of a triangle group $\\Delta(p,q,r)$, a special kind of Coxeter group.\n\nEDIT In fact, these are called extended triangle groups, by G. Jones and D. Singerman in Maps, hypermaps and triangle groups...\n\nWhat about the following presentation: $$\\langle a,b,c|a^2=b^2=c^2=(ab)^p=(bc)^q=(ca)^r=(abc)^s=1\\rangle$$ Do these groups have a name and where are they treated?\n\nThe presentation in question are motivated by this and that...\n\n\u2022 Have you come across generalised triangle groups? What I call a triangle group is the group of transformations of the tiled plane (no reflections); Wikipedia calls these von Dyck groups. Such groups have presentation $\\langle x, y, x; x^p, y^q, z^r, xyz\\rangle$, and here $ab=x$, $bc=y$ and $ca=z$. A generalised triangle group is a group with presentation $\\langle x, y, x; x^p, y^q, z^r, W(x, y, z)\\rangle$. Fine and Rosenberger wrote a whole book motivated by these groups and their generalisations (one-relator products). Jim Howie has also written a lot about them. \u2013\u00a0user1729 Jul 8 '16 at 10:36\n\u2022 @user1729 you mean $<x,y,{\\bf z};\\dots>$? Do you have some explicite references or links to them? \u2013\u00a0draks ... Jul 8 '16 at 13:17\n\u2022 Yes, $x, y, z$. When I google the first link is to a paper of Jim Howie (macs.hw.ac.uk\/~jim\/preprint27.pdf). The references look extensive. I am pretty sure he gave a talk on these at a conference in 2012 I was at. The book of Fine and Rosenberger is \"Algebraic generalizations of discrete groups: a path to combinatorial group theory through one-relator products\". \u2013\u00a0user1729 Jul 8 '16 at 14:02\n\u2022 @user1729 why not posting this as an answer... \u2013\u00a0draks ... Jul 12 '16 at 5:11\n\nI haven't come across a name for this family in full generality, but the special case in which $p=2$ was defined and studied by Coxeter in his paper\n\nH. S. M. Coxeter, The abstract groups $G^{ m, n, p}$, Trans. Amer. Math. Soc. 45 (1939), 73-150.\n\nwhere (in your notation) the group is called $G^{q,r,s}$.\n\nAlso, when $s$ is even, your group has a subgroup of index $2$ with presentation $\\langle x,y \\mid x^p=y^q=(xy)^r=[x,y]^{s\/2} \\rangle$.\n\nThese groups were studied in the same paper by Coxeter, and denoted $(p,q,r;s\/2)$.\n\nBoth of these families have been extensively studied since then, in particular concerning their finiteness. They are generally infinite for sufficiently large values of the parameters, and there is just a handful of remaining cases for which their finiteness is still unknown.\n\nA few years ago Havas and I showed, using a big computer calculation, that $(2,3,13;4)$ is finite of order $358\\,848\\,921\\,600$. So your group with $(p,q,r,s) = (2,3,13,8)$ has twice that order.\n\n\u2022 +1 great the paper is available online and for free \u2013\u00a0draks ... Feb 4 '16 at 14:57\n\u2022 hmmm: when I replace $A,B,C$ in the definition of $G^{m,n,p}$ by $A=x\/y, \\; B=y\/z,\\; C=z\/x$ and use that $x^2=y^2=z^2=1$ (so they are self-inverse) I get: $\\langle (xy)^m=(yz)^n=(zx)^p=x^2=y^2=z^2=(\\frac xy \\frac yz \\frac zx)^2=1 \\rangle$, where the triple product is trivial. Now it looks just like the triangle group. What have I done (wrong)? \u2013\u00a0draks ... Feb 4 '16 at 19:24\n\u2022 Maybe you are interested: The presentation in question are motivated by this and that... \u2013\u00a0draks ... Feb 4 '16 at 19:50\n\u2022 Help you with what? \u2013\u00a0Derek Holt Feb 9 '16 at 9:51","date":"2019-07-22 03:22:30","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5538988709449768, \"perplexity\": 516.1342690950262}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-30\/segments\/1563195527474.85\/warc\/CC-MAIN-20190722030952-20190722052952-00114.warc.gz\"}"} | null | null |
Q: Hard coded Firebase Auth I have an app which is connected to Firebase.
I have set the rules in the database so that one user (lets name it testUsr) has the permission to read the database but not write to it.
In the iOS app I implemented the login function to the firebase, BUT I hard coded the username and password for the testUsr.
So my question is that will my app get rejected for that hard coded user authentication if I publish it someday?
A: Yes, your application will be rejected by Apple because you are using a login service without implementing login with Apple (According to the most recently issued guidelines). They might make an exception though if the whole process happens in code alone—I'm not sure but I would avoid the hassle.
Why is a login required at all? Can't you just make your database on the firebase side read-only without the need of a sign-in at all? You can alter the rules in your database I'm thinking something like. Let me know if it doesn't work and ill do some more researching.
// Allow read/write access on all documents to any user signed in to the application
service cloud.firestore {
match /databases/{database}/documents {
match /{document=**} {
allow write: if request.auth.uid == "**YOUR MAC ACCOUNT**";
allow read;
}
}
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,745 |
package spec
import (
"io"
"github.com/google/go-containerregistry/pkg/v1"
imagespec "github.com/opencontainers/image-spec/specs-go/v1"
)
// ImageSource interface to an image. It can have its config read, and a its containers
// can be read via an io.ReadCloser tar stream.
type ImageSource interface {
// Config get the config for the image
Config() (imagespec.ImageConfig, error)
// TarReader get the flattened filesystem of the image as a tar stream/
TarReader() (io.ReadCloser, error)
// Descriptor get the v1.Descriptor of the image
Descriptor() *v1.Descriptor
// V1TarReader get the image as v1 tarball, also compatibel with `docker load`
V1TarReader() (io.ReadCloser, error)
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,945 |
\section{Introduction}
The field of generalized inverses has grown much in the last few decades. We shall study the group generalized inverse, which appears to have a central role in some applications \cite{MEYER, GOLUB, SOARES, MEYER2}, as well as in characterizations of some other generalized inverses \cite[section 3.3]{DJR}. Our main task will be to generalize some known results about bounded linear operators that appear in \cite{DENG}. The question of the invertibility of $p-q$, $p,q$-idempotents, is especially important since it is connected with various other questions. For details, see introductory section in \cite{DENG} and references mentioned there.\\[3mm]
Throughout this text let $\mathcal{R}$ always stand for an arbitrary ring with unit. Its unit is denoted typically, by 1.
For $a\in\mathcal{R}$, the group inverse of $a$ \cite{DJR} is every element $c\in\mathcal{R}$ that satisfies
$$aca=a,\ cac=c,\ ac=ca.$$
Such element $c$ need not exist, but if it does, it is unique and we write $c=a^\#$. Moreover, $a^\#$ double commutes with $a$ and we say that $a$ is group invertible. This definition represents a natural extension of the definition of the group inverse for square matrices, as introduced in \cite{BENISRAEL} or \cite{CAMPBELL}, to arbitrary rings with unit. Group inverse can also be considered as a special case of the Drazin inverse. Namely, if $a^D$ exists with $ind(a^D)\leq 1$, then $a$ is group invertible and $a^\#=a^D.$ \\[3mm]
In \cite{DENG} Deng has proved many interesting results concerning bounded linear idempotents $P, Q\in\mathcal{B}(\mathcal{H})$. Almost all of his proofs use the method of decomposing Hilbert space as $\mathcal{H}=\mathcal{R}(P)\oplus\mathcal{N}(P)$ for some idempotent $P$, and then studying the matrix decomposition of certain operators with respect to this decomposition. An appropriate analogue for this space decomposition in rings with unit is described as follows. If $a\in\mathcal{R}$ is group invertible, by $a^\pi=1-aa^\#$ we denote the so called spectral idempotent of $a$. It is easily checked that this idempotent generates the ring decomposition $\mathcal{R}=(a^\pi)^\circ\oplus a^\pi\mathcal{R}$, where $(a^\pi)^\circ$ and $a^\pi\mathcal{R}$ are the right annihilator and the right ideal of $a$, respectively. A quick reminder:
$$
x^\circ=\lbrace y\in\mathcal{R}:\ xy=0\rbrace,\
x\mathcal{R}=\lbrace xy:\ y\in\mathcal{R}\rbrace.
$$
Notice that with respect to this ring decomposition we can write $a$ in the unique way as $a=a_1\oplus 0$, where $a_1$ is invertible in the subring $(1-a^\pi)\mathcal{R}(1-a^\pi)$ with $a^\#$ being its inverse. Another way for writing this representation is $a=\bmatrix a_1 & 0\\0 & 0\endbmatrix_{1-a^\pi}$, and this matrix representation will appear the most appropriate for our study. We introduce this form of representation in the subsequent section.
\section{Decompositions with respect to idempotent}
Very important tool in working with generalized inverses is the representation of an arbitrary element $a\in\mathcal{R}$ in the matrix form using idempotent(s). There are
two standard ways to represent an arbitrary element of a ring with unit in the matrix form: using one or two idempotents. Matrix representation using two idempotents $p$ and $q$ is the most often used in $C^*-$algebras and unital rings with involution, see for example the novel work of Djordjevi\'{c} and Mihajlovi\'{c} \cite{MIHAJLOVIC}. Since our setting is much coarser than this, we choose using only one idempotent and describe this in the sequel. But first, let us remind ourselves how the usual representation using an idempotent looks like for the algebra of linear bounded operators.\\[3mm]
If $\mathcal{H}$ is a Hilbert space, by $\mathcal{B}(\mathcal{H})$ we denote the algebra of linear bounded operators on $\mathcal{H}$. It is an algebra with the unit $I$, where $I$ is the identity operator. Let $P\in\mathcal{B}(\mathcal{H})$ be an idempotent, $P^2=P.$ Then the matrix representation of $P$ associated with the space decomposition $H=\mathcal{R}(P)\oplus\mathcal{N}(P)$ is
\begin{equation}\label{Operatorska}
P=\begin{bmatrix}
I & 0 \\
0 & 0
\end{bmatrix},
\end{equation} that is $P=I\oplus 0$. Arbitrary idempotent $Q\in\mathcal{B}(\mathcal{H})$, $Q^2=Q,$ has matrix representation \begin{equation}\label{Operatorska2}Q=\begin{bmatrix}
Q_1 & Q_2 \\
Q_3 & Q_4
\end{bmatrix}
\end{equation} with respect to the space decomposition $\mathcal{H}=\mathcal{R}(P)\oplus\mathcal{N}(P).$ Since $Q^2=Q$ we obtain
$$\begin{bmatrix}
Q_1^2+Q_2Q_3 & Q_1Q_2+Q_2Q_4 \\
Q_3Q_1+Q_4Q_3 & Q_3Q_2+Q_4^2
\end{bmatrix}=\begin{bmatrix}
Q_1 & Q_2 \\
Q_3 & Q_4
\end{bmatrix}.$$\\
Analogously, let $p\in\mathcal{R}^\bullet$. It is easily checked that if $p$ is an idempotent, then $1-p$ is an idempotent as well. We call it complementary idempotent related to $p$ and sometimes denote it by $\overline{p}$. Let $a\in \mathcal{R}$. Then we write
$$
a=pap+pa(1-p)+(1-p)ap+(1-p)a(1-p)
$$
and use the notations
$$
a_{11}=pap,\quad a_{12}=pa(1-p),\quad a_{21}=(1-p)ap,\quad a_{22}=(1-p)a(1-p) .
$$
Hence, every projection $p\in\mathcal{R}$ induces a representation of an arbitrary element
$a\in\mathcal{R}$ given by the following matrix
$$ a=\bmatrix pap & pa(1-p)\\(1-p)ap &
(1-p)a(1-p)\endbmatrix_p = \bmatrix a_{11}(p)&
a_{12}(p)\\a_{21}(p)&a_{22}(p)\endbmatrix_p.
$$
As a very important feature, we must say that multiplication in the ring $\mathcal{R}$ agrees with multiplication of matrices. That is, if $a$ and $b$ are represented by matrices $\bmatrix a_{11}&
a_{12}\\a_{21}&a_{22}\endbmatrix_p$ and $\bmatrix b_{11}&
b_{12}\\b_{21}&b_{22}\endbmatrix_p$, than product $ab$ can be calculated by the usual matrix multiplication, namely $$ab=\bmatrix a_{11}b_{11}+a_{12}b_{21}&a_{11}b_{12}+a_{12}b_{22}\\a_{21}b_{11}+a_{22}b_{21}& a_{21}b_{12}+a_{22}b_{22}\endbmatrix_p.$$
\noindent If $p$ is well-known, then we use $a_{ij}$ instead of $a_{ij}(p),\
i,j\in\{1,2\}.$\\[3mm]
If $q\in\mathcal{R}^\bullet$ is an arbitrary idempotent, then we obviously have
\begin{equation}\label{glavna}
p=\begin{bmatrix}p&0\\0&0\end{bmatrix}_p,\quad q=\begin{bmatrix}q_{1}&q_{2}\\q_{3}&q_{4}\end{bmatrix}_p,\quad p,q_1,q_2,q_3,q_4\in\mathcal{R},
\end{equation}
and this is the desired representation of an element in a $\mathcal{R}$ that is an appropriate analogue of (\ref{Operatorska}), (\ref{Operatorska2}). By analogy with the similar situation for operators, from $q^2=q$ we get \begin{equation}\label{glavna2}\begin{bmatrix}
q_1^2+q_2q_3 & q_1q_2+q_2q_4 \\
q_3q_1+q_4q_3 & q_3q_2+q_4^2
\end{bmatrix}_p=\begin{bmatrix}
q_1 & q_2 \\
q_3 & q_4
\end{bmatrix}_p,
\end{equation} where $q$ is given as in (\ref{glavna}).\\[1mm]
We will give a very important observation in a connection with the
representation (\ref{glavna}). Namely, if we compare $P$ from (\ref{Operatorska}) with $p$ from $(\ref{glavna}$), then we see an important difference. In (\ref{Operatorska}) we have an instance of the identity operator, which is the unit for the algebra $\mathcal{B}(H)$. However, in (\ref{glavna}), instead of $1$, the unit of the ring $\mathcal{R}$, we have $p$ at the corresponding position. This will have a huge impact on our work in a way that some of the analogues of theorems appearing in \cite{DENG} will have proofs that no longer hold without some additional assumptions. For that reason, we will modify those assumptions so that they exclude some of the idempotents $q$ represented as in (\ref{glavna}) (this is different from \cite{DENG} where we had corresponding results holding \textit{for all} idempotents $Q$ of the appropriate representation). Since $p$ is the unit of the subring $p\mathcal{R}p$, the easiest way to accomplish this is to leisurely suppose that idempotents $q\in p\mathcal{R}p$. In other words, instead of (\ref{glavna}), we would suggest
\begin{equation}\label{glavna3}
p=\begin{bmatrix}p&0\\0&0\end{bmatrix}_p,\quad q=\begin{bmatrix}q_{1}&q_{2}\\q_{3}&q_{4}\end{bmatrix}_p,\quad p,q_1,q_2,q_3,q_4\in p\mathcal{R}p.
\end{equation}
However, this solution appears to be too strict in a sense that assuming it all of the results in the following two sections trivially hold. So, in the sequel we relax this condition $q\in p\mathcal{R}p$ in an appropriate way (Theorems \ref{teorema2.1}, \ref{teorema2.2} and their corollaries...)\\[3mm]
As a supplement, let us just say that this matrix representation of elements in a ring is useful in many other areas of this theory, and not only for the group inverses. For example, the generalized Drazin inverse can be finely represented using this tool. It is well-known (\cite{DJR}) that $a\in \mathcal{R}^d$ can be represented in the
following matrix form
$$
a=\bmatrix a_1&0\\0&a_2\endbmatrix_p
$$
with respect to $p=aa^d=1-a^\pi$, where $a^\pi$ is the spectral idempotent of $a$,
$a_1$ is invertible in the subring $p\mathcal{R} p$ and $a_2$ is quasinilpotent in the subring
$(1-p)\mathcal{R} (1-p)$. Notice that $a_2$ is quasinilpotent in the ring $\mathcal{R}$ as well. Then the generalized Drazin
inverse of $a$ is given by
$$
a^d=\bmatrix [a_1]_{p\mathcal{R}p}^{-1}&0\\0&0\endbmatrix_p.
$$
\section{Group invertibility of some expressions depending on two idempotents}
First we state the following result:
\begin{Lema}\label{lema2.1} If $a$ is an idempotent, then $a^{\#}$=$a$. If $a$ and $b$ are group invertible and $ab=ba$, then
$$(ab)^\#=b^\#a^\#=a^\#b^\#,\ a^\#b=ba^\#\ and\ b^\#a=ab^\#.$$
If $ab$ and $ba$ are group invertible, then\\
\begin{equation}\label{Klajn}(ab)^\#=a[(ba)^\#]^2b.\end{equation}
\end{Lema}
Proof of this lemma is familiar in the literature and it follows from the fact $ab=ba$, so we omit it. See \cite{DENG} for the operator case.\\[2mm]
If $A$ and $B$ are complex matrices of dimension $n$, the Cline's formula \cite{CLINE} is $ (AB)^D=A[(BA)^D]B$. Lemma \ref{lema2.1} implies that if $AB$ and $BA$ are group invertible, then
\begin{equation*}
\begin{split}
(AB)^\#A=A[(BA)^\#]^2BA=A(BA)^\#,\\
\ B(AB)^\#=BA[(BA)^\#]^2B=(BA)^\#B.
\end{split}
\end{equation*}
In \cite{CAO,OLESKY}, some necessary and sufficient conditions for the existence of the group inverse for some block square matrices are given, and Deng \cite{DENG} gave a generalization to bounded linear operators. Here, we will show some results still hold in rings with unit.
\begin{Lema}
\label{lema2.2}
(\cite{OLESKY}, Theorem 2.2 for the matrix case, and \cite{DENG}, Lemma 2.2 for the operator case]) Let $m\in\mathcal{R}$ have the matrix form
$$m=\begin{bmatrix}
0 & a \\
b & 0
\end{bmatrix}_p.$$
Then $m$ is group invertible if and only if $ab$ and $ba$ are group invertible and
$$(ab)^\pi a=0,\quad b(ab)^\pi=0.$$
In this case,
$$m^\#=\begin{bmatrix}
0 & (ab)^\#a \\
b(ab)^\# & 0
\end{bmatrix}_p=\begin{bmatrix}
0 & a(ba)^\# \\
(ba)^\#b & 0
\end{bmatrix}_p.$$
\end{Lema}
\textbf{Proof. }Assume that $m$ is group invertible. Then $m^2=\begin{bmatrix}
ab & 0 \\
0 & ba
\end{bmatrix}_p$ is group invertible and, therefore, $ab$ and $ba$ are group invertible. Moreover, the group invertibility of $m$ implies that $m\mathcal{R}=m^2\mathcal{R}$, that is $ ab\mathcal{R}=a\mathcal{R}$ and $ba\mathcal{R}=b\mathcal{R}$. The equality $ab(ab)^\#ab=ab$ implies that $ab(ab)^\#x=x$ for all vectors $x\in ab\mathcal{R}$. Thus $ab(ab)^\#x=x$ for all $x\in a\mathcal{R}$, and therefore $ab(ab)^\#a=a$.\\
Similarly, we obtain the second desired equality.\\[3mm]
Assume that $ab$ and $ba$ are group invertible and $(ab)^\pi a=0, b(ab)^\pi=0$. Let $x=\begin{bmatrix}
0 & (ab)^\#a \\
b(ab)^\# & 0
\end{bmatrix}_p.$ From $b(ab)^\#a=(ba)^\#ba$ we get
$$mx=xm=(ab)^\#ab\oplus b(ab)^\#a=(ab)^\#ab\oplus(ba)^\#ba.$$
So,
$$xmx=\begin{bmatrix}
0 & (ab)^\#a \\
b(ab)^\# & 0
\end{bmatrix}_p\begin{bmatrix}
(ab)^\#ab & 0 \\
0 & b(ab)^\#a
\end{bmatrix}_p=x.$$
From $(ab)^\pi a=0$ and $b(ab)^\pi=0$ we get
$$mxm=\begin{bmatrix}
(ab)^\#ab & 0 \\
0 & b(ab)^\#a
\end{bmatrix}_p\begin{bmatrix}
0 & a \\
b & 0
\end{bmatrix}_p=\begin{bmatrix}
0 & a \\
b & 0
\end{bmatrix}_p=m.$$
Hence, $m$ is group invertible and $m^\#=x$. $\square$\\[3mm]
Let $p$ and $q$ be two idempotents. Next, we provide some equivalent conditions for the existence of the group inverses od $p - q$, $\overline{p} - q$ and $pq - qp$ under the assumption that $p$ and $q$ are given by (\ref{glavna}). As already said, we demand that $p$ and $q$ satisfy some extra conditions (whose analogues do not exist in \cite{DENG}) that we shall call \textit{additional assumptions}.
\begin{Teorema}\label{teorema2.1}\textit{Let $p$ and $q$ be idempotents given by (\ref{glavna}).}\\[1mm]
(i) Let us assume that $pq_1=q_1p=q_1,\ pq_2=q_2$ and $q_3p=q_3$. Then $p - q$ is group invertible if and only if $p - q_1$ and $q_4$ are group
invertible and
\begin{equation}\label{sixth}
\begin{split}
\left(p - q_1\right)^\# q_2 = q_2 q_4^\#, & \quad q_2 q_4^{\pi} = \left(p - q_1\right)^{\pi} q_2 = 0,\\
q_3\left(p - q_1\right)^\# = q_4^\# q_3, & \quad q_4^{\pi} q_3 = q_3\left(p - q_1\right)^{\pi} = 0.
\end{split}
\end{equation}
In this case,
\begin{equation}\label{seventh}
\left(p - q\right)^\# = \begin{bmatrix}
\left(p - q_1\right)^\#\left(p - q_1\right) & - \left(p - q_1\right)^\# q_2 \\ - q_4^\# q_3 & - q_4^\# q_4
\end{bmatrix}_p.
\end{equation}
(ii) Let us assume that $\overline{p}q_4=q_4\overline{p}=q_4,\ q_2\overline{p}=q_2$ and $\overline{p}q_3=q_3$. Then $\overline{p} - q$ is group invertible if and only if $q_1$ and $\overline{p} - q_4$ are group
invertible and
\begin{equation}\label{eight}
\begin{split}
\left(\overline{p} - q_4\right)^\# q_3 = q_3 q_1^\#, & \quad q_3 q_1^{\pi} = \left(\overline{p} - q_4\right)^{\pi} q_3 = 0,\\
q_2\left(\overline{p} - q_4\right)^\# = q_1^\# q_2, &\quad q_1^{\pi} q_2 = q_2\left(\overline{p} - q_4\right)^{\pi} = 0.
\end{split}
\end{equation}
In this case,
\begin{equation}
\left(\overline{p} - q\right)^\# = \begin{bmatrix}
- q_1^\# q_1 & - q_1^\# q_2 \\
- \left(\overline{p} - q_4\right)^\# q_3 & \left(\overline{p} - q_4\right)^\# \left(\overline{p} - q_4\right)
\end{bmatrix}_p.
\end{equation}
(iii) Let us assume that additional assumptions from $(i)$ and $(ii)$ hold:
\begin{equation*}
\begin{split}
pq_1=q_1p=q_1,\ & pq_2=q_2\ and\ q_3p=q_3,\\
\overline{p}q_4=q_4\overline{p}=q_4,\ & q_2\overline{p}=q_2\ and\ \overline{p}q_3=q_3.
\end{split}
\end{equation*}
If $p - q$ and $\overline{p} - q$ are group invertible, then $pq - qp$ is group invertible and
\begin{equation}\label{ten}
\left(pq - qp\right)^\# = \begin{bmatrix}
0 & - q_1^\# \left(p - q_1\right)^\# q_2 \\
q_3 q_1^\# \left(p - q_1\right)^\# & 0
\end{bmatrix}_p.
\end{equation}
\end{Teorema}
\textbf{Proof.} (i) Assume that $p - q$ is group invertible. Then $\left(p - q\right)^2$ is group invertible. Since $\left(p - q\right)^2 = p + q - pq - qp$, by using representation (\ref{glavna}) and additional assumptions, we get $\left(p - q\right)^2 = \left(p - q_1\right) \oplus q_4$. We will not stress the use of additional assumptions any further. Hence $p - q_1$, $q_4$ are group invertible and
$$
\left(p - q\right)^\# = \left[ \left(p - q\right)^2\right]^\# \left(p - q\right) = \left(p - q\right) \left[ \left(p - q\right)^2\right]^\#
$$
$$
= \begin{bmatrix}
p - q_1 & 0 \\
0 & q_4
\end{bmatrix}_p^\#
\begin{bmatrix}
p - q_1 & - q_2 \\
- q_3 & - q_4
\end{bmatrix}_p =
\begin{bmatrix}
p - q_1 & - q_2 \\
- q_3 & - q_4
\end{bmatrix}_p
\begin{bmatrix}
p - q_1 & 0 \\
0 & q_4
\end{bmatrix}_p^\#
$$
$$
= \begin{bmatrix}
\left(p - q_1\right)^\#\left(p - q_1\right) & - \left(p - q_1\right)^\# q_2 \\ - q_4^\# q_3 & - q_4^\# q_4
\end{bmatrix}_p = \begin{bmatrix}
\left(p - q_1\right)\left(p - q_1\right)^\# & - q_2 q_4^\# \\ - q_3 \left(p - q_1\right)^\# & - q_4 q_4^\#
\end{bmatrix}_p.
$$
Comparing the two sides of the above equation, we have
\begin{equation}\label{eleventh}
\left(p - q_1\right)^\# q_2 = q_2 q_4^\# \quad and \quad q_3\left(p - q_1\right)^\# = q_4^\# q_3.
\end{equation}
From $\left(p - q\right)\left(p - q\right)^\#\left(p - q\right) = \left(p - q\right)$ and
$$
\left(p - q\right)\left(p - q\right)^\# = \left(p - q\right)^2\left[ \left(p - q\right)^2\right]^\# = \left(p - q_1\right)\left(p - q_1\right)^\# \oplus q_4 q_4^\#,
$$
we get
$$
\begin{bmatrix}
\left(p - q_1\right)\left(p - q_1\right)^\# & 0 \\ 0 & q_4 q_4^\#
\end{bmatrix}_p
\begin{bmatrix}
p - q_1 & - q_2 \\
- q_3 & - q_4
\end{bmatrix}_p = \begin{bmatrix}
p - q_1 & - q_2 \\
- q_3 & - q_4
\end{bmatrix}_p.
$$
Hence,
$$
q_2 = \left(p - q_1\right)\left(p - q_1\right)^\# q_2, \quad\quad q_3 = q_4 q_4^\# q_3.
$$
The first equality of (\ref{eleventh}) yields $\left(p - q_1\right)\left(p - q_1\right)^\# q_2 = \left(p - q_1\right) q_2 q_4^\#$. Moreover, (\ref{glavna2}) leads to $\left(p - q_1\right) q_2 q_4^\# = q_2 q_4 q_4^\#$. The second equality of (\ref{eleventh}) yields $q_4 q_4^\# q_3 = q_4 q_3\left(p - q_1\right)^\#$. Moreover, (\ref{glavna2}) leads to $q_4 q_3\left(p - q_1\right)^\# = \left(q_3 - q_3 q_1\right)\left(p - q_1\right)^\# = q_3\left(p - q_1\right)\left(p - q_1\right)^\#$.
Assume that $p - q_1$, $q_4$ are group invertible and expressions in (\ref{sixth}) hold. Let us denote by $x$ the right side of (\ref{seventh}). Then
$$
\left(p - q\right)x = \begin{bmatrix}
p - q_1 & - q_2 \\
- q_3 & - q_4
\end{bmatrix}_p \begin{bmatrix}
\left(p - q_1\right)^\#\left(p - q_1\right) & - \left(p - q_1\right)^\# q_2 \\ - q_4^\# q_3 & - q_4^\# q_4
\end{bmatrix}_p
$$
$$
= \begin{bmatrix}
\left(p - q_1\right)\left(p - q_1\right)^\# & 0 \\ 0 & q_4 q_4^\#
\end{bmatrix}_p
$$
because $\left(p - q_1\right)\left(p - q_1\right)^\# q_2 = q_2 = q_2 q_4^\# q_4$ and $q_3\left(p - q_1\right)\left(p - q_1\right)^\# = q_3 = q_4 q_4^\# q_3$ in view of (\ref{sixth}), $p - q_1 + q_2 q_4^\# q_3 = p - q_1 + \left(p - q_1\right)^\# q_2 q_3 = \left(p - q_1\right)\left(p - q_1\right)^\#\left(p - q_1\right) + \left(p - q_1\right)^\#\left(p - q_1\right) q_1 = \left(p - q_1\right)^\#\left(p - q_1\right)$ and $q_3\left(p - q_1\right)^\# q_2 + q_4 = q_3 q_2 q_4^\# + q_4 = \left(p - q_4\right)q_4 q_4^\# + q_4 = q_4 q_4^\#$ in view of (\ref{sixth}) and (\ref{glavna2}). In a similar way we get
$$
x\left(p - q\right) = \begin{bmatrix}
\left(p - q_1\right)^\#\left(p - q_1\right) & - \left(p - q_1\right)^\# q_2 \\ - q_4^\# q_3 & - q_4^\# q_4
\end{bmatrix}_p \begin{bmatrix}
p - q_1 & - q_2 \\
- q_3 & - q_4
\end{bmatrix}_p
$$
$$
= \begin{bmatrix}
\left(p - q_1\right)\left(p - q_1\right)^\# & 0 \\ 0 & q_4 q_4^\#
\end{bmatrix}_p
$$
because $ \left(p - q_1\right)^\# \left(p - q_1\right) q_2 = q_2 = q_2 q_4^\# q_4 = \left(p - q_1\right)^\# q_2 q_4$ and \\
$q_4^\# q_3 \left(p - q_1\right) = q_3 \left(p - q_1\right)^\# \left(p - q_1\right) = q_3 = q_4 q_4^\# q_3$ in view of (\ref{sixth}), $p - q_1 + \left(p - q_1\right)^\# q_2 q_3 = p - q_1 + \left(p - q_1\right)^\# \left(p - q_1\right) q_1 = \left(p - q_1\right)^\# \left(p - q_1\right)$ and $q_4^\# q_3 q_2 + q_4 = q_4^\# q_4 \left(p - q_1\right) + q_4 = q_4 q_4^\#$ in view of (\ref{sixth}) and (\ref{glavna2}). So
\begin{equation}\label{twelve}
x \left(p - q\right) = \left(p - q\right) x = \left(p - q_1\right)\left(p - q_1\right)^\# \oplus q_4 q_4^\#.
\end{equation}
Now,
$$
\left(p - q\right) x\left(p - q\right) = \begin{bmatrix}
p - q_1 & - q_2 \\
- q_3 & - q_4
\end{bmatrix}_p \begin{bmatrix}
\left(p - q_1\right)\left(p - q_1\right)^\# & 0 \\ 0 & q_4 q_4^\#
\end{bmatrix}_p
$$
$$
= \begin{bmatrix}
p - q_1 & - q_2 \\
- q_3 & - q_4
\end{bmatrix}_p
$$
and
$$
x \left(p - q\right) x = \begin{bmatrix}
\left(p - q_1\right)^\#\left(p - q_1\right) & - \left(p - q_1\right)^\# q_2 \\ - q_4^\# q_3 & - q_4^\# q_4
\end{bmatrix}_p \begin{bmatrix}
\left(p - q_1\right)\left(p - q_1\right)^\# & 0 \\ 0 & q_4 q_4^\#
\end{bmatrix}_p
$$
$$
= \begin{bmatrix}
\left(p - q_1\right)^\#\left(p - q_1\right) & - \left(p - q_1\right)^\# q_2 \\ - q_4^\# q_3 & - q_4^\# q_4
\end{bmatrix}_p .
$$
Hence, $p - q$ is group invertible and $\left(p - q\right)^\#$ has the representation (\ref{seventh}).\\[1mm]
(ii) Observe that $\overline{p} - q = 1 - p - q = - \left[p - \left(1 - q\right)\right]$. Thus the group invertibility of $\overline{p} - q$ is equivalent to the group invertibility of $p - \left(1 - q\right)$. If $p$ and $q$ are represented as in (\ref{glavna}), then $p - q$ has an obvious representation by means of (\ref{glavna}), and we can apply item (i) in an evident manner.\\[1mm]
(iii) If $p - q$ and $\overline{p} - q$ are group invertible, by (\ref{glavna2}) and Lemma \ref{lema2.1},
$$
\left(q_2 q_3\right)^\# = q_1^\#\left(p - q_1\right)^\# = \left(p - q_1\right)^\# q_1^\#, \quad
$$
$$
\left(q_3 q_2\right)^\# = q_4^\#\left(p - q_4\right)^\# = \left(p - q_4\right)^\# q_4^\#.
$$
By (\ref{sixth}) and (\ref{eight}),
$$
\left(q_2 q_3\right) \left(q_2 q_3\right)^\# q_2 = \left(p - q_1\right)\left(p - q_1\right)^\# q_1 q_1^\# q_2 = \left(p - q_1\right)\left(p - q_1\right)^\# q_2 = q_2
$$
and
$$
\left(q_3 q_2\right) \left(q_3 q_2\right)^\# q_3 = \left(\overline{p} - q_4\right)\left(\overline{p} - q_4\right)^\# q_4 q_4^\# q_3 = \left(\overline{p} - q_4\right)\left(\overline{p} - q_4\right)^\# q_3 = q_3.
$$
Hence, $q_2 q_3$, $q_3 q_2$ are group invertible, $\left(q_2 q_3\right)^{\pi} q_2 = 0 $ and $\left(q_3 q_2\right)^{\pi} q_3 = 0$. By Lemma \ref{lema2.2}, $pq - qp = \begin{bmatrix}
0 & q_2 \\
- q_3 & 0
\end{bmatrix}_p$ is group invertible and (\ref{ten}) follows directly by Lemma \ref{lema2.2}. $\square$\\[3mm]
According to (\ref{glavna2}) and Theorem \ref{teorema2.1}, we get the following result easily.
\begin{Posledica}\label{posledica2.1} \textit{Let $p$ and $q$ be idempotents given by (\ref{glavna})}.\\[1mm]
(i) If $p - q$ is group invertible with $$pq_1=q_1p=q_1,pq_2=q_2\ and\ q_3p=q_3,$$ then $\left(p - q\right)^{\pi} = \left(p - q_1\right)^{\pi} \oplus q_4^{\pi}$.\\[1mm]
(ii) If $\overline{p} - q$ is group invertible with $$\overline{p}q_4=q_4\overline{p}=q_4,\ q_2\overline{p}=q_2\ and\ \overline{p}q_3=q_3,$$ then $\left(\overline{p} - q\right)^{\pi} = q_1^{\pi} \oplus \left(\overline{p} - q_4\right)^{\pi}$.\\[1mm]
(iii) \textit{If $p - q$ and $\overline{p} - q$ are group invertible with
\begin{equation*}\begin{split}pq_1=q_1p=q_1,pq_2=q_2&\ and\ q_3p=q_3,\\
\overline{p}q_4=q_4\overline{p}=q_4,\ q_2\overline{p}=q_2&\ and\ \overline{p}q_3=q_3,
\end{split}
\end{equation*} then $\left(pq - qp\right)^{\pi} = \left(q_1^2 - q_1\right)^{\pi} \oplus \left(q_4 - q_4^2\right)^{\pi}$}.
\end{Posledica}
\textbf{Proof.} We only prove item (i). By (\ref{twelve}),
$$
\left(p - q\right)\left(p - q\right)^\# = \left(p - q_1\right)\left(p - q_1\right)^\# \oplus q_4 q_4^\#.
$$
Hence $\left(p - q\right)^{\pi} = \left(p - q_1\right)^{\pi} \oplus q_4^{\pi}$. $\square$
The next topic we are interested in is the product of idempotents. We state the second main theorem of this article, again, of course, under some additional assumptions.
\begin{Teorema}\label{teorema2.2} \textit{Let $p$ and $q$ be idempotents given by (\ref{glavna})}. Suppose that in (i-ii) $p$ commutes with $q_1, q_2$, and in (iii-iv) $p$ commutes with $q_3,q_4$. Then:\\[1mm]
(i) $pq$ is group invertible if and only if $pq_1$ is group invertible and $(pq_1)^{\pi}pq_2 = 0$.\\[1mm]
(ii) $\overline{p}q$ is group invertible if and only if $pq_4$ is group invertible and $(pq_4)^{\pi}pq_3 = 0$.\\[1mm]
(iii) $p\overline{q}$ is group invertible if and only if $p-pq_1$ is group invertible and $(p-pq_1)^{\pi}pq_2 = 0$.\\[1mm]
(iv) $\overline{p}\hspace{1mm}\overline{q}$ is group invertible if and only if $p - pq_4$ is group invertible and $\left(p - pq_4\right)^{\pi}pq_3 = 0$.
\end{Teorema}
\textbf{Proof.} We only prove item (i), items (ii)-(iv) can be proved in the same way. Let $p$ and $q$ be given by (\ref{glavna}).
Then
$$
pq = \begin{bmatrix}
pq_1 & pq_2 \\
0 & 0
\end{bmatrix}_p, \quad \left(pq\right)^2 = \begin{bmatrix}
pq_1^2 & pq_1q_2 \\
0 & 0
\end{bmatrix}_p, \quad and \quad \left(pq\right)^3 = \begin{bmatrix}
pq_1^3 & pq_1^2q_2 \\
0 & 0
\end{bmatrix}_p,
$$
with respect to the ring decomposition $\mathcal{R} = p\mathcal{R} \oplus p^\circ$. Since $pq$ is group invertible, ind$(pq) = 1$. So
$$
pq\mathcal{R} = (pq)^2\mathcal{R} = (pq)^3\mathcal{R}=\cdots\quad and \quad (pq)^\circ = ((pq)^2) ^\circ= ((pq)^3)^\circ=\cdots
$$
For every $x \in(pq_1^2)^\circ, x \oplus 0 \in((pq)^2) ^\circ= (pq)^\circ$. So $pq_1 x = 0$ and $(pq_1^2)^\circ \subset (pq_1)^\circ$. Since $(pq_1)^\circ \subset (pq_1^2)^\circ$ is trivial, we get $(pq_1^2)^\circ = (pq_1)^\circ$. For every $z \in pq_1\mathcal{R}$ there exists $x \in p\mathcal{R}$ such that
$$
\begin{bmatrix}
pq_1 &p q_2 \\
0 & 0
\end{bmatrix}_p \begin{bmatrix}
x \\
0
\end{bmatrix}_p = \begin{bmatrix}
z \\
0
\end{bmatrix}_p \in pq\mathcal{R} = (pq)^3\mathcal{R}.
$$
So, there exist $u \in p\mathcal{R}$ and $v \in p^\circ$ such that
$$
\begin{bmatrix}
pq_1^3 & pq_1^2q_2 \\
0 & 0
\end{bmatrix}_p \begin{bmatrix}
u \\
v
\end{bmatrix}_p = \begin{bmatrix}
z \\
0
\end{bmatrix}_p.
$$
Therefore, $z = pq_1^3u +pq_1^2 q_2 v = pq_1^2\left(q_1 u + q_2 v\right) \in pq_1^2\mathcal{R}$ and $pq_1\mathcal{R} \subset pq_1^2\mathcal{R}$. Since $pq_1^2\mathcal{R} \subset pq_1\mathcal{R} $ is trivial, we get $pq_1^2\mathcal{R} = pq_1\mathcal{R}$. Hence, ind$(pq_1) \leq 1$, i.e. $(pq_1)^\#$ exists. Now, let
\begin{equation}\label{thirteen}
x = \begin{bmatrix}
pq_1^\# & \left((pq_1)^\#\right)^2pq_2 \\
0 & 0
\end{bmatrix}_p.
\end{equation}
It is trivial to check that $pqx = xpq$, $xpqx = x$ and $(pq)^3x = (pq)^2$. These imply that $x$ is the Drazin inverse of $pq$ and ind($pq) \leq 2$. Since $pq$ is group invertible, $(pq)^2x = pq$. We get
$$
\begin{bmatrix}
pq_1^2 &p q_1q_2 \\
0 & 0
\end{bmatrix}_p \begin{bmatrix}
pq_1^\# & \left((pq_1)^\#\right)^2pq_2 \\
0 & 0
\end{bmatrix}_p = \begin{bmatrix}
pq_1 & pq_2 \\
0 & 0
\end{bmatrix}_p,
$$
which implies that $pq_1 (pq_1)^\# pq_2 = pq_2$, i.e. $(pq_1)^{\pi} pq_2 = 0$.
On the other hand, if $pq_1$ is group invertible and $(pq_1)^{\pi}pq_2 = 0$, it is easy to check that (\ref{thirteen}) is the group inverse of $pq$. $\square$
Note that, Cao in \cite{CAO} gave that, if $A \in \mathds{C}^{r \times r}, C \in \mathds{C}^{s \times s}$ and $M = \begin{bmatrix}
A & B \\
0 & C
\end{bmatrix}$, then there exists $M^\#$ if and only if there exists $A^\#$, $C^\#$, and $\left(1 - AA^\#\right) B \left(1 - C^\# C\right) = 0$. Furthermore, when $M^\#$ exists, it is given by $\begin{bmatrix}
A & B \\
0 & C
\end{bmatrix}^\# = \begin{bmatrix}
A^\# & Y \\
0 & C^\#
\end{bmatrix}$, where $Y = (A^\#)^2 BC^{\pi} + A^{\pi}B(C^\#)^2 - A^\# BC^\#$.
It is worth pointing out, in \cite[Theorem 2.1]{RAKOCEVIC}, Koliha et al. proved that $p - q$ is invertible if and only if $p + q$ and $1 - pq$ are invertible. For the set of square matrices, Theorems \ref{teorema2.1} and \ref{teorema2.2} show that, if $P - Q$ is group invertible, then $P\overline{Q}$ and $\overline{P}Q$ are group invertible. If $\overline{P} - Q$ is group invertible, then $PQ$ and $\overline{P}\hspace{1mm}\overline{Q}$ are group invertible. \\[3mm]
Here is another theorem which is trivial for $q$'s as in (\ref{glavna3}), and holds for $q$'s as in (\ref{glavna}) under additional assumptions.
\begin{Teorema}\textit{Let $p$ and $q$ be idempotents given by (\ref{glavna}). Suppose that $p$ commutes with $p_i$,\ $i=1,2,3$. Then $pq - qp$ is group invertible if and only if $pq_i\left(1 - q_i\right)p, i = 1,4$ are group invertible, $[pq_1\left(1 - q_1\right)p]^{\pi} pq_2 = 0$ and $pq_3[pq_4\left(1 - q_4\right)p]^{\pi}= 0$.}
\end{Teorema}
\textbf{Proof.} By
(\ref{glavna}), $pq - qp = \begin{bmatrix}
0 & q_2p \\
- q_3p & 0
\end{bmatrix}_p$. By (\ref{glavna2}), $pq_2 q_3p = pq_1\left(1 - q_1\right)p$ and $pq_3 q_2p = pq_4\left(1 - q_4\right)p$. The result follows immediately by Lemma \ref{lema2.2}. $\square$
\textbf{Remark.} Note that
$$
pq + qp = - (p + q) (\overline{p} - q) = - (\overline{p} - q)(p + q).
$$
By Lemma \ref{lema2.1}, if $p + q$ and $\overline{p} - q$ are group invertible, then $pq + qp$ is group invertible. $\triangle$
\section{Formulae for $\left(p - q\right)^\#$ and $\left(p + q\right)^\#$ when $p - q$ is group invertible}
In this section, we will obtain explicit representations for the group inverses of products and differences of projections. First, we give the following definition.
\begin{Definicija}\label{definicija3.1}Let $p$ and $q$ be idempotents such that $p - q$ is group invertible. Define elements $f,g$ and $h$ as
\begin{equation}\label{fourteen}
f = p\left(p - q\right)^\#, \quad g = \left(p - q\right)^\#p, \quad h = \left(p - q\right)^\#\left(p - q\right).
\end{equation}
\end{Definicija}
In what follows, we will have many statements concerning not only idempotents $p$ and $q$, but also their complementary idempotents $\overline{p}$ and $\overline{q}$. as well. For the algebra of linear bounded operators $\mathcal{B}(H)$, representations of $P$ and $\overline{P}$ used in \cite{DENG} are the following:
\begin{equation*}
P=\begin{bmatrix}I & 0\\0 & 0\end{bmatrix},\quad\overline{P}=\begin{bmatrix}0 & 0\\0 & I\end{bmatrix}.\\
\end{equation*}
For the ring $\mathcal{R}$, however, representations of $p$ and $\overline{p}$ are:
\begin{equation*}
p=\begin{bmatrix}p & 0\\0 & 0\end{bmatrix},\quad\overline{p}=\begin{bmatrix}0 & 0\\0 & \overline{p}\end{bmatrix}.\\
\end{equation*}
Notice the difference between $\overline{P}$ and $\overline{p}$. In the bottom right corner of the matrix representation, $\overline{P}$ has operator $I$, and $\overline{p}$ has element $\overline{p}$ at the same position. $I$ is an identity in the algebra $\mathcal{B}(H)$, but $\overline{p}$ is not an identity in the ring $\mathcal{R}$ nor in the ring $p\mathcal{R}p$. This will leave some significant consequences in the structure of this section. Namely, not all of the statements in \cite{DENG} related to complementary idempotents will have their analogues in the setting $\mathcal{R}$. This is the reason why some of the mentioned analogues will be left out, and we will not especially emphasize this fact. For example, in part (i) of the next theorem one would want to have $f=(p-q)^\#\overline{q}$ (the analogue of \cite[Theorem 3.1 (i)]{DENG}). However, this equality does not hold not even under additional assumptions of the Theorem \ref{teorema3.1}. If we add extra conditions $q_2p=q_4^\#q_4p=0$ then we would get this equality for $f$. However, for $g=\overline{q}(p-q)^\#$ \cite[Theorem 3.1 (ii)]{DENG} we would need some other extra conditions and so on... This is the reason for us to go no further with additional conditions than the ones stated in the Theorem \ref{teorema3.1}. However, in the part (ii) of the Theorem \ref{teorema3.3} we will be forced to make an exception.
\begin{Teorema}\label{teorema3.1} \textit{Let $p$ and $q$ be idempotents given by (\ref{glavna}) such that $p - q$ is group invertible with $pq_1=q_1p=q_1,\ pq_2=q_2,\ q_3p=p$, and let $f,g$ and $h$ be given by Definition \ref{definicija3.1}. Then $f,g$ and $h$ are idempotents and}\\[1mm]
(i) $ f\mathcal{R} = ph\mathcal{R} = (p-q_1)\mathcal{R}$.\\[1mm]
(ii) $g^\circ = (ph)^\circ = p^\circ \oplus (p - q_1)^\circ$.
\end{Teorema}
\textbf{Proof.} From Theorem \ref{teorema2.1}(i) it follows
\begin{equation}\label{fifteen}
f = \begin{bmatrix}
\left(p - q_1\right)^\#\left(p - q_1\right) & - \left(p - q_1\right)^\# q_2 \\ 0 & 0
\end{bmatrix}_p, \quad g = \begin{bmatrix}
\left(p - q_1\right)^\#\left(p - q_1\right) & 0 \\ - q_3\left(p - q_1\right)^\# & 0
\end{bmatrix}_p.
\end{equation}
and
\begin{equation}\label{sixteen}
h = \left(p - q_1\right)^\#\left(p - q_1\right) \oplus q_4^\# q_4.
\end{equation}
So, $f,g$ and $h$ are idempotents,
$$
f\mathcal{R} = \left[\left(p - q_1\right)^\#\left(p - q_1\right)\right]\mathcal{R} = \left[p - q_1\right]\mathcal{R} = ph\mathcal{R},
$$
and
$$
g^\circ= \left[\left(p - q_1\right)^\#\left(p - q_1\right)\right]^\circ \oplus p^\circ = \left(p - q_1\right)^\circ \oplus p^\circ= (ph)^\circ.
$$
$\square$\\[3mm]
Let $\mathcal{S}$ and $\mathcal{T}$ be subrings of $\mathcal{R}$ such that $\mathcal{R}=\mathcal{S}\oplus\mathcal{T}.$ If there exists idempotent $p\in\mathcal{R}$ with $p\mathcal{R} = \mathcal{S},\ p^\circ = \mathcal{T}$, we denote it by $p_{\mathcal{S},\mathcal{T}}$. As a particular case in the following corollary, if $p - q$ is invertible we will have that $p - q_1$, $q_4$ are invertible and $h = 1$. So we get the following special results.
\begin{Posledica}\label{posledica3.1} Let $p$ and $q$ be given by (\ref{glavna}) with $pq_1=q_1p=q_1,\ pq_2=q_2,\ q_3p=p$, and let $f,g$ and $h$ be given by Definition \ref{definicija3.1}. If $p - q$ is invertible, then\\[1mm]
(i) $f = p \left(p - q\right)^{-1} = p_{p\mathcal{R},q\mathcal{R}}$.\\[1mm]
(ii) $g = \left(p - q\right)^{-1}p = p_{q^\circ,p^\circ}$.\\[1mm]
(iii) $\overline{p}f = 0$.\\[1mm]
(iv) $g \overline{p} = 0$.
\end{Posledica}
Moreover, by (\ref{glavna2}), (\ref{fifteen})-(\ref{sixteen}), we have the following result.
\begin{Posledica}\label{posledica3.2} Let $p$ and $q$ be idempotents given by (\ref{glavna}) such that $p - q$ is group invertible with $pq_1=q_1p=q_1,\ pq_2=q_2,\ q_3p=p$, and let $f$ and $g$ be given by Definition \ref{definicija3.1}. Then
\begin{equation}\label{seventeen}
fg = \left(p - q_1\right)^\# \oplus 0.
\end{equation}
\end{Posledica}
By simple algebraic techniques, we can deduce more relations among $f,g$ and $h$.
\begin{Teorema} Let $p$ and $q$ be idempotents given by (\ref{glavna}) such that $p - q$ is group invertible with $pq_1=q_1p=q_1,\ pq_2=q_2,\ q_3p=p$, and let $f,g$ and $h$ be given by Definition \ref{definicija3.1}. Then\\[1mm]
(i) $fp = pg = ph = hp$.\\[1mm]
(ii) $qhq = qh = hq = hqh$.\\[1mm]
\end{Teorema}
\textbf{Proof.} (i) By (\ref{glavna}), (\ref{fifteen}) and (\ref{sixteen}), we get that
$$
fp = pg = ph = hp = \left(p - q_1\right)\left(p - q_1\right)^\# \oplus 0.
$$
(ii) If $p - q$ is group invertible, by Theorem \ref{teorema2.1}(i), $q_2q_4^{\pi} = \left(p - q_1\right)^{\pi} q_2 = 0$ and $q_4^{\pi} q_3 = q_3\left(p - q_1\right)^{\pi} = 0$. Then
$$
qh = \begin{bmatrix}
q_1 & q_2 \\ q_3 & q_4
\end{bmatrix}_p \begin{bmatrix}
\left(p - q_1\right)\left(p - q_1\right)^\# & 0 \\ 0 & q_4 q_4^\#
\end{bmatrix}_p = \begin{bmatrix}
\left(q_1 - q_1^2\right)\left(p - q_1\right)^\# & q_2 \\ q_3 & q_4
\end{bmatrix}_p
$$
and
$$
hq = \begin{bmatrix}
\left(p - q_1\right)\left(p - q_1\right)^\# & 0 \\ 0 & q_4 q_4^\#
\end{bmatrix}_p \begin{bmatrix}
q_1 & q_2 \\ q_3 & q_4
\end{bmatrix}_p = \begin{bmatrix}
\left(q_1 - q_1^2\right)\left(p - q_1\right)^\# & q_2 \\ q_3 & q_4
\end{bmatrix}_p.
$$
So $qh = hq $ and
$$
hqh = \begin{bmatrix}
\left(p - q_1\right)\left(p - q_1\right)^\# & 0 \\ 0 & q_4 q_4^\#
\end{bmatrix}_p\begin{bmatrix}
\left(q_1 - q_1^2\right)\left(p - q_1\right)^\# & q_2 \\ q_3 & q_4
\end{bmatrix}_p = qh.
$$
Moreover, from
$$
q_1 \left[\left(q_1 - q_1^2\right)\left(p - q_1\right)^\#\right] + q_2q_3 = q_1^2 \left[\left(p - q_1\right)\left(p - q_1\right)^\#\right] + q_1\left(p - q_1\right) \quad by\ (\ref{glavna2})
$$
$$
= q_1^2 \left[\left(p - q_1\right)\left(p - q_1\right)^\#\right] + q_1\left(p - q_1\right) \left[\left(p - q_1\right)\left(p - q_1\right)^\#\right]
$$
$$
= \left(q_1 - q_1^2\right)\left(p - q_1\right)^\#
$$
and
$$
q_3 \left[\left(q_1 - q_1^2\right)\left(p - q_1\right)^\#\right] + q_4q_3 = q_3q_1 \left[\left(p - q_1\right)\left(p - q_1\right)^\#\right] + q_3\left(p - q_1\right) \quad by\ (\ref{glavna2})
$$
$$
= q_3q_1 \left[\left(p - q_1\right)\left(p - q_1\right)^\#\right] + q_3\left(p - q_1\right) \left[\left(p - q_1\right)\left(p - q_1\right)^\#\right]
$$
$$
= q_3 \left[\left(p - q_1\right)\left(p - q_1\right)^\#\right] \quad by\ (\ref{sixth})
$$
$$
= q_3.
$$
we have
$$
qhq = \begin{bmatrix}
q_1 & q_2 \\ q_3 & q_4
\end{bmatrix}_p \begin{bmatrix}
\left(q_1 - q_1^2\right)\left(p - q_1\right)^\# & q_2 \\ q_3 & q_4
\end{bmatrix}_p
$$
$$
= \begin{bmatrix}
\left(q_1 - q_1^2\right)\left(p - q_1\right)^\# & q_2 \\ q_3 & q_4
\end{bmatrix}_p = hq.
$$
$\square$\\[3mm]
Subsequently, we present some group inverse formulae related to the sum and the difference of two idempotents $p$ and $q$.
\begin{Teorema}\label{teorema3.3} Let $p$ and $q$ be idempotents given by (\ref{glavna}) such that $p - q$ is group invertible with $pq_1=q_1p=q_1,\ pq_2=q_2,\ q_3p=p$, and let $f,g$ and $h$ be given by Definition \ref{definicija3.1}. Then\\[1mm]
(i) $\left(p - q\right)^\# = f+g-h.$\\[1mm]
(ii) If we demand $q_2p=q_4p=0$, then $$\left(p + q\right)^\# = \left(p - q\right)^\#\left(p + q\right)\left(p - q\right)^\# = \left(2g - h\right)\left(f + g - h\right)$$ if and only if $ph = p$.
\end{Teorema}
\textbf{Proof.} The item (i) follows directly by Theorem \ref{teorema2.1}(i) and relations in (\ref{fifteen})-(\ref{sixteen}). So we only need to prove the item (ii). Here we have not only the additional assumptions ($pq_1=q_1p=q_1,\ pq_2=q_2,\ q_3p=q_3$), but also extra conditions $q_2p=0,\ q_4p=0$. These extra conditions ensure that $(p-q)^\#(\overline{p}+\overline{q})=(p+q)(p-q)^\#$, and in this way we provide that the proof of this theorem follows the same path as the one presented in \cite[Theorem 3.3 (ii)]{DENG}.
Denote by $x = \left(p - q\right)^\#\left(p + q\right)\left(p - q\right)^\#$. By the previous argument we have
$$
\left(p + q\right)x = \left(p + q\right)\left(p - q\right)^\#\left(p + q\right)\left(p - q\right)^\#
$$
$$
= \left(p - q\right)^\#\left(\overline{p} + \overline{q}\right)\left(p + q\right)\left(p - q\right)^\#
$$
$$
= \left(p - q\right)^\#\left(p - q\right)^2\left(p - q\right)^\#
$$
$$
= \left(p - q\right)\left(p - q\right)^\# = h,
$$
$$
x \left(p + q\right) = \left(p - q\right)^\#\left(p + q\right)\left(p - q\right)^\#\left(p + q\right)
$$
$$
= \left(p - q\right)^\#\left(p + q\right)\left(\overline{p} + \overline{q}\right)\left(p - q\right)^\#
$$
$$
= \left(p - q\right)^\#\left(p - q\right)^2\left(p - q\right)^\#
$$
$$
= \left(p - q\right)\left(p - q\right)^\# = h,
$$
and
$$
x \left(p + q\right)x = x\left(p - q\right)\left(p - q\right)^\#
$$
$$
= \left(p - q\right)^\#\left(p + q\right)\left(p - q\right)^\#\left(p - q\right)\left(p - q\right)^\#
$$
$$
= \left(p - q\right)^\#\left(p + q\right)\left(p - q\right)^\# = x.
$$
Hence, $\left(p + q\right)^\# = x$ if and only if $\left(p + q\right)x\left(p + q\right) = p + q$. Next, we prove that $\left(p + q\right)x\left(p + q\right) = p + q$ if and only if $ph = p$. We divide the proof in two steps:
First, if $ph = p$, by (\ref{sixth}) and (\ref{sixteen}), we have that $p - q_1$ is invertible, $qh = q$ and
$$
\left(p + q\right)x\left(p + q\right) = \left(p + q\right)h = p + q.
$$
Second, note that
$$
\left(p + q\right)x\left(p + q\right) = \left(p + q\right)h
$$
$$
= \begin{bmatrix}
p + q_1 & q_2 \\ q_3 & q_4
\end{bmatrix}_p\begin{bmatrix}
\left(p - q_1\right)^\#\left(p - q_1\right) & 0 \\ 0 & q_4^\# q_4
\end{bmatrix}_p
$$
$$
= \begin{bmatrix}
\left(p + q_1\right)\left(p - q_1\right)^\#\left(p - q_1\right) & q_2 \\ q_3 & q_4
\end{bmatrix}_p. \quad by\ (\ref{sixth})
$$
If $\left(p + q\right)x\left(p + q\right) = p + q$, then $\left(p+ q_1\right)\left(p - q_1\right)^\#\left(p - q_1\right) = p+ q_1$. Since $p - q_1$ as an element acting on $p\mathcal{R}$ is group invertible,
$$
p\mathcal{R} = \left(\left(p - q_1\right)^{\pi}\right)^\circ
\oplus \left(\left(p - q_1\right)^{\pi}\right)\mathcal{R} = \left(p - q_1\right)\mathcal{R} \oplus\left(p - q_1\right)^\circ.
$$
For every $x \in \left(p - q_1\right)^\circ$, we have $q_1 x = x$ and $2x = \left(p + q_1\right)x = \left(p + q_1\right)\left(p- q_1\right)^\#\left(p - q_1\right)x = 0.$
It follows that $\left(p - q_1\right)^\circ = \{0\}$ and so $(p-q_1)\mathcal{R}=\mathcal{R}$ and $p - q_1$ is invertible. Hence $ph = p$.
Now, by the definition of the group inverse, we know $$x = \left(p - q\right)^\#\left(p + q\right)\left(p - q\right)^\# = \left(p + q\right)^\#.$$ Moreover, by item (i) of this Theorem,
$$
\left(p + q\right)^\# = \left(p - q\right)^\#\left(p + q\right)\left(p - q\right)^\#
$$
$$
= \left(p - q\right)^\# p \left(p - q\right)^\# + \left(p - q\right)^\# q \left(p - q\right)^\#
$$
$$
= g\left(f + g - h\right) + \left(g - h\right)\left(f + g - h\right)
$$
$$
= \left(2g - h\right)\left(f + g - h\right).
$$
$\square$\\[3mm]
\textbf{Remark}
(i) Let $x = \left(p - q\right)^\#\left(p + q\right)\left(p - q\right)^\#$. The proof of Theorem \ref{teorema3.3}, item (ii) shows that if $p - q$ is group invertible and without assuming $ph = p$, then we have that $x\left(p + q\right) = \left(p + q\right)x$ and $x\left(p + q\right)x = x$, in the other words, $x$ is a $\{1,5\}-$ inverse of $p + q$.
(ii) The condition $ph = p$ in Theorem \ref{teorema3.3}, item (ii) is necessary to assure $\left(p + q\right)x\left(p + q\right) = p + q$.
This can be demonstrated in the following example.
\begin{Primer} Define idempotents $p,q$ on $\oplus_{i = 1}^4 \mathds{R}$ ($\mathds{R}$ is the set of real numbers) by
$$
p = 1 \oplus 1 \oplus 0 \oplus 0 \quad and \quad q = 1 \oplus 0 \oplus 1 \oplus 0.
$$
Then
$$
\left(p - q\right)^\# = 0 \oplus 1 \oplus - 1 \oplus 0,
$$
$$
\left(p + q\right)^\# = \frac{1}{2} \oplus 1 \oplus 1 \oplus 0,
$$
$$
h = \left(p - q\right)\left(p - q\right)^\# = 0 \oplus 1 \oplus 1 \oplus 0,
$$
$$
x = \left(p - q\right)^\#\left(p + q\right)\left(p - q\right)^\# = 0 \oplus 1 \oplus 1 \oplus 0.
$$
In this case, $ph \neq p, \left(p + q\right)x\left(p + q\right) \neq p + q$ and, hence $\left(p + q\right)^\# \neq x$. $\triangle$
\end{Primer}
\begin{Posledica} \label{posledica3.4}\cite[Theorem 2.2]{RAKOCEVIC}. \textit{Let $p$ and $q$ be idempotents given by (\ref{glavna}) such that $pq_1=q_1p=q_1,\ pq_2=q_2,\ q_3p=q_3$, $f = p_{p\mathcal{R},q\mathcal{R}}$ and $g = p_{q^\circ ,p^\circ}$. If $p - q$ is invertible with $q_2p=q_4p=0$ in addition, then}\\[1mm]
(i) $\left(p + q\right)^{-1} = \left(p - q\right)^{-1}\left(p + q\right)\left(p - q\right)^{-1}$.\\[1mm]
(ii) $\left(p - q\right)^{-1} = \left(p + q\right)^{-1}\left(p - q\right)\left(p + q\right)^{-1}$.\\[1mm]
(iii) $\left(p - q\right)^{-1} = f+g - 1$.\\[1mm]
(iv) $\left(p + q\right)^{-1} = \left(2g - 1\right)\left(f + g - 1\right)$.
\end{Posledica}
If $p - q$ is invertible, Harte gave that \cite[Theorem 7.5.1]{HARTE}
$$
\left(1 - pqp\right)^{-1} = 1 - p + p\left(p - q\right)^{-2}.
$$
In fact, we can obtain some similar results on the group inverses.
\begin{Teorema}\label{teorema3.4} Let $p$ and $q$ be idempotents given by (\ref{glavna}) such that $p - q$ is group invertible with $pq_1=q_1p=q_1,\ pq_2=q_2,\ q_3p=p$, and let $f,g$ and $h$ be given by Definition \ref{definicija3.1}. Then\\[1mm]
(i) $\left(p - pqp\right)^\# = fg$.\\[1mm]
(ii) $\left(p - pq\right)^\# = [fg]^2\overline{q}$.\\[1mm]
(iii) $\left(p - qp\right)^\# = \overline{q}[fg]^2$.
\end{Teorema}
\textbf{Proof.} (i) From (\ref{glavna}), $p - pqp = \left(p - q_1\right) \oplus 0$, hence by Corollary \ref{posledica3.2},
$$
\left(p - pqp\right)^\# = \left(p - q_1\right)^\# \oplus 0 = fg.
$$
(ii) - (iii) Note that
$$
\left(p - pq\right)^\# = \begin{bmatrix}
\left(p - q_1\right)^\# & -\left[\left(p - q_1\right)^\# \right]^2 q_2 \\ 0 & 0
\end{bmatrix}_p
$$
$$
= \begin{bmatrix}
\left[\left(p - q_1\right)^\# \right]^2 & 0 \\ 0 & 0
\end{bmatrix}_p \begin{bmatrix}
p - q_1 & -q_2 \\ -q_3 & \overline{p} - q_4
\end{bmatrix}_p
$$
$$
= \left(fg\right)^2 \overline{q},
$$
and
$$
\left(p - qp\right)^\# = \begin{bmatrix}
\left(p - q_1\right)^\# & 0 \\ -q_3\left[\left(p - q_1\right)^\# \right]^2 & 0
\end{bmatrix}_p
$$
$$
= \begin{bmatrix}
p - q_1 & -q_2 \\ -q_3 & \overline{p} - q_4
\end{bmatrix}_p \begin{bmatrix}
\left[\left(p - q_1\right)^\# \right]^2 & 0 \\ 0 & 0
\end{bmatrix}_p
$$
$$
= \overline{q} \left(fg\right)^2 .\quad \square
$$
\noindent\textbf{Acknowledgments}\\[3mm]
\hspace*{6mm}This work was supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia under Grant No. 451-03-9/2021-14/ 200125\\[3mm]
\textbf{Declaration of interest}\\[3mm]
\hspace*{6mm}I wish to confirm that there are no known conflicts of interest associated with this publication and
there has been no significant financial support for this work that could have influenced its outcome other than listed above. I confirm that there are
no other persons who satisfied the criteria for authorship but are not listed.
I confirm that there are no impediments to publication with respect to intellectual property.\\[3mm]
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,087 |
package info.u_team.u_team_core.gui.elements;
import java.util.ArrayList;
import java.util.List;
import com.mojang.blaze3d.vertex.PoseStack;
import net.minecraft.client.Minecraft;
import net.minecraft.client.gui.components.AbstractSelectionList;
import net.minecraft.client.gui.components.ObjectSelectionList;
import net.minecraft.client.gui.components.events.GuiEventListener;
public abstract class ScrollableListEntry<T extends ScrollableListEntry<T>> extends ObjectSelectionList.Entry<T> {
protected final Minecraft minecraft;
private final List<GuiEventListener> children;
public ScrollableListEntry() {
minecraft = Minecraft.getInstance();
children = new ArrayList<>();
}
protected <B extends GuiEventListener> B addChildren(B listener) {
children.add(listener);
return listener;
}
@Override
public boolean mouseClicked(double mouseX, double mouseY, int button) {
children.forEach(listener -> listener.mouseClicked(mouseX, mouseY, button));
return true;
}
@Override
public boolean mouseReleased(double mouseX, double mouseY, int button) {
for (final GuiEventListener listener : children) {
if (listener.mouseReleased(mouseX, mouseY, button)) {
return true;
}
}
return false;
}
@Override
public boolean mouseDragged(double mouseX, double mouseY, int button, double dragX, double dragY) {
for (final GuiEventListener listener : children) {
if (listener.mouseDragged(mouseX, mouseY, button, dragX, dragY)) {
return true;
}
}
return false;
}
@Override
public abstract void render(PoseStack poseStack, int index, int top, int left, int width, int height, int mouseX, int mouseY, boolean hovered, float partialTicks);
@SuppressWarnings("deprecation")
protected AbstractSelectionList<T> getList() {
return list;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,640 |
{"url":"https:\/\/socratic.org\/questions\/58f24b3db72cff72ff66ad49","text":"Apr 15, 2017\n\n$\\log {6}^{\\frac{1}{3}}$\n\n#### Explanation:\n\nUsing the $\\textcolor{b l u e}{\\text{law of logarithms}}$\n\n$\\textcolor{red}{\\overline{\\underline{| \\textcolor{w h i t e}{\\frac{2}{2}} \\textcolor{b l a c k}{\\log {x}^{n} \\Leftrightarrow n \\log x} \\textcolor{w h i t e}{\\frac{2}{2}} |}}}$\n\n$\\Rightarrow \\frac{\\log 6}{3} = \\frac{1}{3} \\log 6 = \\log {6}^{\\frac{1}{3}}$\n\nApr 15, 2017\n\n#### Explanation:\n\nGiven: $\\frac{\\log \\left(6\\right)}{3}$\n\nThe above can be written as:\n\n$\\left(\\frac{1}{3}\\right) \\log \\left(6\\right)$\n\nUse the property of logarithms $\\left(c\\right) \\log \\left(a\\right) = \\log \\left({a}^{c}\\right)$:\n\n$\\log \\left({6}^{\\frac{1}{3}}\\right)$\n\nWe know that the $\\frac{1}{3}$ power is the same as the cube root:\n\n$\\log \\left(\\sqrt[3]{6}\\right) \\leftarrow$ the answer.","date":"2020-04-05 11:11:56","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 10, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9876375198364258, \"perplexity\": 2637.3605483933125}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-16\/segments\/1585371576284.74\/warc\/CC-MAIN-20200405084121-20200405114121-00122.warc.gz\"}"} | null | null |
Home»Author: Bella Rich
Author: Bella Rich
The New York Stock Exchange Bull
By Bella Rich November 21, 2022 0
The New York stock exchange bull stands just north of Bowling Green on Broadway in the Financial District. The statue represents the power and influence of financial power. It is also known as the Charging Bull or the Bowling Green Bull. This statue was sculpted by Italian artist Arturo Di Modica and was dedicated to the exchange in 1893. Arturo Di Modica's Charging Bull The Charging Bull stands just north of Bowling Green, on Broadway in the Financial District of Manhattan. It is also known as the Bull of Wall Street or the Bowling Green Bull. The sculpture is an…
If you are looking to invest in property in Chicago, you can choose between renting an apartment or buying a single family home. Whether you are renting or buying, you should consider the neighborhood in which you plan to live. This article will give you some insight into what makes a great investment in Chicago property. Renting an apartment in Chicago is a good investment If you are looking for a long-term investment, renting an apartment in Chicago is a great choice. It is a good choice for many reasons. Many people rent in Chicago for the flexibility it offers.…
Future Price of Coinbase,With the rise of crypto, Coinbase has become a top choice for retailers looking for the best exchanges. The company is focused on retail and institutional trading, which is expected to contribute to a big part of its revenue after the recent BlackRock deal. The company has many areas for future growth, and it is just scratching the surface of the cryptocurrency market. Coinbase's profitability The profitability of Coinbase depends largely on its transaction fees, which account for about 80% of its revenues. Depending on the funding method, Coinbase charges between 1.5% and 4% of the value…
Bitcoin mining is a process of creating new bitcoins by solving complex math problems. In return, the miner is rewarded with a fixed amount of bitcoins. The value of bitcoin has risen rapidly since its introduction in 2009, resulting in a huge interest in mining. The high costs and lack of good prospects have discouraged most people from taking on the task, though. Resource-intensive Bitcoin mining is a resource-intensive industry, requiring massive amounts of electricity. Many miners have been moving to areas with cheap energy, such as Kazakhstan, where coal is plentiful. Some governments have become wary of the cryptocurrency,…
Top 100 Crypto by Market Cap
By Bella Rich October 10, 2022 0
Top 100 Crypto by Market Cap If you are looking to invest in crypto, you may want to check out the top 100 coins by market cap. However, this information will not necessarily tell you which ones are the best investments. You should also take into account that the top 100 coins will vary in price. Therefore, it is important to choose the right coin based on your investment goals and risk tolerance. Binance Coin (BNB) Binance Coin (BNB) is a digital asset that is native to the Binance blockchain. It was launched by the online cryptocurrency exchange Binance in…
Pros and Cons of Outsourcing the Sales Function
By Bella Rich October 7, 2022 0
Pros and Cons of Outsourcing the Sales Function Outsourcing the sales function can be a great way for a small business to grow fast. It offers many benefits but also poses some challenges. Before you start outsourcing, you need to consider the pros and cons of this strategy. These advantages and disadvantages are determined by economic, organizational, and human factors. If done correctly, outsourcing the sales function can lead to an immediate competitive advantage for your company and increase the speed at which it can grow. Costs Outsourcing the sales function has many benefits, but costs must be considered carefully.…
What is the Underlying ISIN of the Poly London Stock Exchange?
What is the Underlying ISIN of the Poly London Stock Exchange? If you're curious about the underlying ISIN for the poly london stock exchange, you've come to the right place. The underlying company is an anglo-russian or Cyprus-based precious metals mining group that is listed on the London Stock Exchange. To understand how the stock is priced, you need to know the underlying company. underlying ISIN of poly london stock exchange starts with JE The underlying ISIN of poly londen stock exchange starts with JE, which is not the same as the ticker symbol. The ticker symbol is used to…
Apple Announces the Next Generation of AirPodsPro
Apple Announces the Next Generation of AirPodsPro The next generation of Apple's AirPods will offer enhanced battery life and personalized spatial audio. A new H2 chip will help the headphones produce better audio quality, and Apple has said that the active noise cancelation will be even better than before. The company has also added a new adaptive transparency mode that helps reduce sudden loud noises around you while listening to your music. These features should make the AirPods Pro a worthy upgrade for any Apple fan. AirPods Max The next generation of Apple AirPods will feature improved high-bandwidth connectivity and…
GM's Return-to-office Mandate Causes Mass Exodus
By Bella Rich September 27, 2022 0
GM Delays Return-To-Office Mandate GM executives issued a memo announcing their company would delay the return-to-office mandate until the first quarter of next year. This represents a major shift from the flexible "work the way you want" policies the company announced and approved in April. GM executives described the new policy as "flexible" and said the time spent at the office will vary based on the employee, the project, and the week. GM's return-to-office mandate causes mass exodus A new return-to-office mandate is forcing GM workers to go back to the office for three days a week. The new system…
The Latest Casualty of the Crypto Winters
The latest casualty of the Crypto Winters is data center firm Compute North, which has lost between $100 million and $500 million in assets. The company had tried to secure alternative funding through Generate Capital and other prospective investors, but a combination of factors pushed the company into bankruptcy. Energy costs, supply-chain disruptions, and a decline in bitcoin prices slashed the firm's liquidity. Compute North files for Chapter 11 bankruptcy Compute North, the largest crypto mining data center in the world, has filed for voluntary Chapter 11 bankruptcy. The company had a presence in Texas, North Carolina, and Nebraska, and… | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,207 |
Q: I having compatibilty issues of WordPress theme with woo-commerce I just updated WordPress and all the plugin.
Seems that the theme I was using is not compatible with the latest version of woo commerce, and I just need to fix the issue. Any idea or alternative to solve this bug?
A: add_theme_support( 'woocommerce' );
Write above statement in functions.php
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,247 |
Join the Mountain Community Women in Business (MCWinBiz) for our monthly gathering. Each month, we feature a different speaker on a topic of interest.
There is NO CHARGE for the meeting. Come early to network.
Please feel free to provide a Raffle Prize for our drawing.
All are welcomed to bring samples, promo items, brochures, business cards, etc. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,381 |
Q: Как разделить строку на список слов разделённых пробелами? Например у меня есть :
"I like apple."
Как сделать так чтобы было :
["I", "like", "apple"]
A: s = "I like apple."
lst = s.split()
print(lst) # ['I', 'like', 'apple.']
Если нужно без точки, то
s = "I like apple."
lst = s.replace('.', '').split()
print(lst) # ['I', 'like', 'apple']
Метод split() (без параметров) у строки разделяет строку на список, для разделения separator по умолчанию None
что означает любое количество пробельных символов
Про None поправил @pank
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,360 |
The fruit and vegetable pulping machine is an ideal machine for making jam, fruit juice, vegetable juice, suitable for a variety of fresh fruit and vegetables in pulping separation. such as: orange flesh, grapes, kiwi, mulberry, waxberry, peach (to nuclear) and so on,it can change them into jam juice, it can also pulping green vegetables , tomatoes, peppers, celery. the vegetables bar isolated, is an important equipment for food processing .
The machine can separate fruit juice and tomato skin and seed.
Our company has engaged in producing and exporting best selling carrot vegetable /grape juicer/orange/ strawberry/ kiwi/pineapple/ papaya / orange flesh Juicer 0086-18703616827 for many years, and until now, our machine has exported to overseas market, such as Brazil, India, Indonesia, Russia, Thailand and so on, and get the foreign customers' unanimous praise.
We believe that you purchase Shandong Leader Machinery Co.,ltd. best selling carrot vegetable /grape juicer/orange/ strawberry/ kiwi/pineapple/ papaya / orange flesh Juicer 0086-18703616827 is your best and sensible choice. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,479 |
{"url":"https:\/\/socratic.org\/questions\/58cf3bcab72cff11ef61df71","text":"# Question #1df71\n\nMar 20, 2017\n\nGiven that first term of geometric series $a = - 11$,\nthe common ratio $r = - 4$,\nthe last or 8 th term ${t}_{8} = a {r}^{7} = 180224$\n\nNow\n\n${S}_{8} = a + a r + a {r}^{2} + a {r}^{3} + \\ldots \\ldots . a {r}^{7} - - - - \\left(1\\right)$\n\n$r {S}_{8} = a r + a {r}^{2} + a {r}^{3} + \\ldots \\ldots + a {r}^{7} + a {r}^{8} - - - - \\left(2\\right)$\n\nsubtracting (2) from (1) we get\n\n$\\left(1 - r\\right) {S}_{8} = a - a {r}^{8} = a - r \\times a {r}^{7}$\n\n$\\implies {S}_{8} = \\frac{a - r \\times {t}_{8}}{1 - r} = \\frac{- 11 - \\left(- 4\\right) \\times 180224}{1 - \\left(- 4\\right)}$\n\n$= \\frac{- 11 + 720896}{5} = 144177$","date":"2019-08-21 05:19:13","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 8, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8596958518028259, \"perplexity\": 3878.1334855396535}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-35\/segments\/1566027315809.69\/warc\/CC-MAIN-20190821043107-20190821065107-00468.warc.gz\"}"} | null | null |
{"url":"http:\/\/www.deater.net\/weave\/vmwprod\/appleiibot\/part5.html","text":"# AppleIIbot 280 char Applesoft Demos -- Part 5\n\n## Hi-res Moving Waves\n\nUnlike the previous ones that are just effectively doing fake palette shifting, this one the sinewave pattern is actually actively moving around. This ends up being a bit slower.\n\n1FORI=0TO139:POKE1013+I,4*PEEK(2126+I)-204+(PEEK(2266+I\/3)-35)\/4^(I-INT(I\/3)*3):NEXT\n2&\"\/foL:Be;569e7W4>qM5\\bO4=Q2EqY3KegilrZb;3o\\a[EA@ln\\a\/JVA4[<9YDp]b3KQ3KDhE22]aM4DU<[C4ARVC4U-hdbZkbrgZZle;g+SbAaEBRl]\\;O^rW[;(14)36\/6879:,7::;SW$@#@;#;*#,04M533&&D$$HP(I04&KI0IDAGK]MOTD+\\& Link to machine code source: moving.s ## Ocean Scene (LOGO) link I was working on the Demosplash LOGO demo so was doing a lot of experimenting with compact LOGO code. This scene is similar to one in the final version. LOGO sizecoding is a bit different than the BASIC and 6502 variants. {L} SETPC 4 SETPOS[90 80] CLEAN REPEAT 36[RT 5 FD 2 RT 5] REPEAT 9[RT 40 PU SETPOS[100 80] FD 25 PD BK 10] SETPC 5 PU HOME PD RT 180 FD 119 RT 90 FD 280 RT 90 FD 119 RT 45 REPEAT 20[FD 10 RT 90 FD 10 LT 90] FILL BK 50 SETPC 1 REPEAT 40[FD 1 RT 3] RT 90 REPEAT 40[FD 1 RT 3] HT ## Thick Sine link Was trying to draw some circles but ended up making this cool looking sine pattern. Eventually got a different version that nearly fits in 64 bytes. Getting sines in a small space is difficult. The Applesoft BASIC ROM does have a cosine table for shape tables but it's missing a term and it's not really much smaller to use it rather than just including the proper table that we want. 1FORI=0TO102:POKE1013+I,4*PEEK(2126+I)-204+(PEEK(2229+I\/3)-35)\/4^(I-INT(I\/3)*3):NEXT 2&\"\/foL:Be;569e7W4>qM5\\bE4=S3EqM3Y3KYdjS1Tr[3Ip\\qb3C75EI\\r[38Hpbpljeml94Vq=5bBa3\\_BrTlF943-35\/5768959:+7:;SW@#@;&3*SE%O)CD&\/MF-<Z;W$$>?+5a# Link to machine code source: thick_sine.s ## Turkey Time link I've done a few holidays before with the lo-res block drawing code, this time I made one for Thanksgiving. 0REM\/ G O-(>-?!$C<@)%B5:)*-'.),\/-4):=&.)8;-4!07%,(16*0(3414(,;5G(*=7D(()EG(>?EG)\/\/GL)88GL).0LL)79LL 25!' 07()\/25*, 25,,(34*,)34.0)25..(25&'-45&'\n1DEFFNP(X)=PEEK(2054+I*5+X)-32:GR:POKE49234,0:FORI=0TO27:COLOR=FNP(0):FORY=FNP(3)TOFNP(4):HLINFNP(1),FNP(2)ATY:NEXTY,I:GETA\n\n\n## Hi-res Doom Fire\n\nThis is the \"Doom Fire\" effect once again, but trying to do it for hi-res mode. Trickier than you might think. Partly because there aren't good colors, partly because of the non-linear hi-res memory layout. Also generating random numbers compactly is a challenge as well. In the end it uses a large lookup table for colors, and uses the ROM contents for \"randomness\".\n\n1FORI=0TO139:POKE1013+I,4*PEEK(2126+I)-192+(PEEK(2266+I\/3)-35)\/4^(I-INT(I\/3)*3):NEXT\n2&\"*el,cl;EQ8Z2WdEb4U_N74dW9M8Y4P9QfbN74aU9[0h01[80B3R<0\\9Vk0\\QT7k8)b2j81_0Z_R81R4dYhfUd]R80lY8Q9R8BCP9Q8\\3Y_H0T9R4kYhjhbglU0QOOTXbfdeoblodo[S#A*.]5G4:E5S$347'5=-C)77#++,4)@7?=4#,#+&7GH:2 Link to machine code source: flame.s ## Hi-res Screen Wipe link I was working on a hi-res screen wipe for the Peasant's Quest demake and this was an unintended side effect. It goes a bit too fast but I couldn't find the bytes needed to add a delay. 1FORI=0TO138:POKE1013+I,4*PEEK(2126+I)-204+(PEEK(2265+I\/3)-35)\/4^(I-INT(I\/3)*3):NEXT 2&\"\/foT5XleggbqcmdcX<Qq[5Srcr;52le8=2lr8=4aq;93e6gmGkF0ffVq=-\\nTq\\nTgjrQn535;3o\\7SpW4\\cQ<Vo]IU<Y<9I3T:egk8\\im6FoTgbqi7dim7g\\kKqModjm4cro'daKSA.NIV\/3)FU+\/16#]3<>(4LP-4:5X;J04.5C<I;F?:@C(U# Link to machine code source: rectangle.s ## Christmas Scene (BASIC) link I guess this ended up being my Christmas Demo for 2021. All in BASIC. It was tricky getting a decent looking tree as well as an OK looking house. Snow was a pain too. If you let this run forever it will completely cover the screen in snow. 0Q=38:GR:FORI=0TO9:X(I)=RND(1)*Q:Y(I)=I:COLOR=1:VLIN33,QAT16+I:COLOR=4:HLIN9-I\/2,9+I\/2AT28+I:COLOR=8:VLIN29,33AT16+I:NEXT:VLIN36,QAT20:HLIN0,QAT39:PLOT9,Q 3FORI=0TO9:V=X(I):W=Y(I):COLOR=15:PLOTV,W:PLOTV,W+1:COLOR=0:IFSCRN(V,W+2)>0THENX(I)=RND(1)*Q:W=0 6PLOTV,W:Y(I)=W+1:NEXT:GOTO3 ## Edgarware Intro (BASIC) link Another lo-res boxdrawing, this one because I was re-watching some old Homestar Runner stuff. This one got a lot of views because someone told Strongbad about it. Drop a train on them, Edgar! 0REM&*<DG&,:@C&-9??&\/7=>\/6:DF 78BC 88BE!::BC%33@A%33DE+15(=+\/61:+.715(66(.(15''(\/3(*(\/1+,(\/0-.%\/6\/\/'\/600\/\/113\/4613#2366\/159:\/35;< 1GR:?\" PROGRAMMED IN EDGAR'S MOM'S BASEMENT\":DEFFNP(X)=PEEK(2054+I*5+X)-32 2FORI=0TO24:COLOR=FNP(0):FORY=FNP(3)TOFNP(4):HLINFNP(1),FNP(2)ATY:NEXTY,I ## Cosine Wave Effect 3 link When trying to get the Thick Sine demo down to 64-bytes for Lovebyte I ran into a lot of interesting failure modes. This half-height almost greyscale effect was an interesting one. 1FORI=0TO68:POKE1013+I,4*PEEK(2125+I)-204+(PEEK(2194+I\/3)-35)\/4^(I-INT(I\/3)*3):NEXT 2&\"\/foT5aXnEDMKYS3TNEn@M2Y[2XW]3eYll5kmapU9rO&BrPk\\e:B\\aO3[38Hpap\\kmgbljSAJ)$46''$U+Q\/Y4Z'#\/Q#S Link to machine code source: oops3_cos.s ## Cosine Wave Effect 4 link Yet another cosine mistake when working on Thick Sine for Lovebyte. Note that these are so short they break the encoder I have (it assumes an offset to start of data with a 3-digit length in the FOR loop), I really need to fix the code to handle small code properly. 1FORI=0TO68:POKE1013+I,4*PEEK(2125+I)-204+(PEEK(2194+I\/3)-35)\/4^(I-INT(I\/3)*3):NEXT 2&\"\/foT5aXnEDYS2XOBrAI2YW2XV]3ceom7l_irI9rI2EqTlUq=>\\aK2MK[38Hpap\\kmginjSAJ)'043'+$[U;*@63#\/Q#5\n\nLink to machine code source: oops4_cos.s\n\n## Orb Horror\n\n1FORI=0TO134:POKE1013+I,4*PEEK(2126+I)-204+(PEEK(2261+I\/3)-35)\/4^(I-INT(I\/3)*3):NEXT","date":"2023-03-21 16:57:02","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.48002588748931885, \"perplexity\": 7100.636806792674}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-14\/segments\/1679296943704.21\/warc\/CC-MAIN-20230321162614-20230321192614-00381.warc.gz\"}"} | null | null |
Election Forecast: "13 Days Out"
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Debriefing on Gov. Rick Synder's last State of the State Recap 1/24
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Detroit is recovering from its 2013 bankruptcy by focusing on improving schools and economic development, Chad Livengood, reporter for Crain's Detroit Business, told Capitol Issues Forum Sept. 20 in his talk "Covering Detroit's Resurgence." As city leaders tackle blight, expensive auto insurance and workforce challenges, Detroit's comeback story is still being written. Investments by entrepreneurs like Tom Gores, Dan Gilbert and Shahid Khan have brought business and jobs int
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Entrepreneurship creates jobs and drives Michigan's economy said Neil Sheridan, chair of the Small Business Association of Michigan (SBAM) Entrepreneurship Task Force, and Rob Fowler, President and CEO of SBAM, at the April Capitol Issues Forum "Accelerating Economic Growth through Entrepreneurship." More than 650,000 registered small businesses in Michigan rely on economic growth and entrepreneurship opportunities. To help better align public and private support of entrepre
Rehabilitation, training focus of corrections system
Inmates now have a better chance of success outside of prison thanks to programs focused on self-improvement, said Chris Gautz, Michigan Department of Corrections, at the March Capitol Issues Forum "Rehabilitation & Reentry." Although prisoners at the Cooper Street Correctional Facility in Jackson, for example, only made up five percent of the Jackson College student body, they were 57 percent of the Dean's List this past semester, said Gautz, who is MDOC's public information
"Pipes and Pavement" Recap Summary 2/16
Michigan could be losing out on jobs and economic growth thanks to inadequate investment in infrastructure, said Mike Nystrom, Michigan Infrastructure and Transportation Association and Janice Beecher, MSU's Institute of Public Utilities, who spoke at Capitol Issues Forum Feb. 16 on findings and recommendations of Gov. Snyder's 21st Century Infrastructure Commission. Potential businesses looking to relocate or expand to the area are concerned about Michigan's i | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,180 |
{"url":"https:\/\/medicus.gda.pl\/hot-fudge-kxvmqjh\/euler%27s-theorem-pdf-74c2c2","text":"Let X = xt, Y = yt, Z = zt Theorem. The selection of pressure and temperature in (15.7c) was not trivial. Thus n\u2212s is uniquely de\ufb01ned \ufffd\ufffd. Euler\u2019s Theorem Theorem If a and n have no common divisors, then a\u02da(n) 1 (mod n) where \u02da(n) is the number of integers in f1;2;:::;ngthat have no common divisors with n. So to compute ab mod n, rst nd \u02da(n), then calculate c = b mod \u02da(n). Euler (pronounced \"oiler'') was born in Basel in 1707 and died in 1783, following a life of stunningly prolific mathematical work. Leonhard Euler. I \u2026 Euler\u2019s theorem: Statement: If \u2018u\u2019 is a homogenous function of three variables x, y, z of degree \u2018n\u2019 then Euler\u2019s theorem States that x del_u\/del_x+ydel_u\/del_y+z del_u\/del_z=n u Proof: Let u = f (x, y, z) be the homogenous function of degree \u2018n\u2019. to the Little Theorem in more detail near the end of this paper. Euler\u2019s theorem offers another way to \ufb01nd inverses modulo n: if k is relatively prime to n, then k.n\/1 is a Z n-inverse of k, and we can compute this power of k ef\ufb01ciently using fast exponentiation. Fermat\u2019s Little Theorem Review Theorem. &iF&\u0370+\ufffdE#\u072bq\ufffdB}\ufffdt}c\ufffdbm\ufffd\u04ed\ufffd\ufffd\ufffdYq\ufffd\ufffdn\u06b1\ufffd\ufffd Then all you need to do is compute ac mod n. For n\u2208N we set n \u2212s= e logn, taking the usual real-valued logarithm. We will also discuss applications in cryptog-raphy. x\u06b5VK\ufffd\ufffd4\ufffd\u03ef\ufffd G\ufffdM\ufffdJb\ufffd;h\ufffdH4\ufffd\ufffd\ufffd\ufffd\ufffdvw\ufffdI'M\ufffd\ufffd\ufffd\ufffd\ufffd\ufffdr93\ufffd;\ufffd !.\ufffd].\ufffd\ufffd\ufffd\ufffd|\ufffd\ufffd\ufffd\ufffdN\ufffdLT\\ Introduction Fermat\u2019s little theorem is an important property of integers to a prime modulus. An important property of homogeneous functions is given by Euler\u2019s Theorem. 4\ufffd\ufffdKM\ufffd\ufffd\ufffd\ufffd\ufffd\ufffdb%6s\ufffdR\ufffd\ufffd\ufffd\u027c\ufffdqkG\ufffd=\ufffd\ufffdG\ufffd\ufffdE\/\ufffd'X\ufffd\ufffd\ufffd\ufffd\ufffdL\u069a\ufffd]\ufffd0z\ufffd\ufffd+\ufffd\ufffd_\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd2\ufffdo\ufffd_\ufffd\u03f6\u051eoBvOF\ufffdz\ufffdf\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\\.7'\ufffd\ufffd~(\ufffdUr=dR\ufffd\u03f6\ufffd\ufffdh\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd9\ufffd\/W\u0115\u02edi\ufffd\ufffd7\ufffd\ufffd\ufffd\ufffd\u02b7\ufffd\ufffd\ufffd\ufffd1R}\ufffd\ufffd>\ufffd\ufffdh\ufffd\ufffdy\ufffd\u07fe\ufffd\ufffd\ufffd\u0504\u0663\ufffdv\uf78e\ufffdf*\ufffd\ufffd=\ufffd .\ufffd\u39a4\\\ufffd\ufffd+boJJtwk\ufffdX\ufffd\ufffd\ufffd4\ufffd\ufffd:\ufffd\/\ufffd\ufffdB\ufffd\ufffd\ufffd\ufffd.\u05edI\ufffd\ufffd;\ufffd\/\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd7Ouuz\ufffdx\ufffd(\ufffd\ufffd\ufffd\ufffd2\ufffdV\ufffd\ufffd\ufffd\ufffd(\ufffdT\ufffd\ufffd6\ufffdo\ufffd\ufffd Returns to Scale, Homogeneous Functions, and Euler's Theorem 161 However, production within an agricultural setting normally takes place with many more than two inputs. i\ufffd\ufffdi\ufffd:8!\ufffdh\ufffd>\ufffd\ufffd{\u05c4\ufffd4]Lb\ufffd\ufffd\ufffd\ufffd^\ufffdx#XlZ\ufffd\ufffd9\ufffd\ufffd\ufffd,\ufffd9N\u0128Q\ufffd\ufffd\u0153\ufffd*i}MEv\ufffd\ufffd\ufffd\ufffd#}bp\u058f\ufffdd\ufffd\ufffd\ufffd\ufffdm>b\ufffd\ufffd\ufffd\ufffdO. last edited March 21, 2016 Euler\u2019s Formula for Planar Graphs The most important formula for studying planar graphs is undoubtedly Euler\u2019s formula, \ufb01rst proved by Leonhard Euler, an 18th century Swiss mathematician, widely considered among the greatest mathematicians that ever lived. EULER\u2019S THEOREM KEITH CONRAD 1. We can now apply the division algorithm between 202 and 12 as follows: (4) 5 0 obj As seen in Example 5, Euler's theorem can also be used to solve questions which, if solved by Venn diagram, can prove to be lengthy. Euler theorems pdf Eulers theorem generalizes Fermats theorem to the case where the. Left: distinct parts \u2192odd parts. The Euler\u2019s theorem on Homogeneous functions is used to solve many problems in engineering, science and finance. Homogeneous Function ),,,( 0wherenumberanyfor if,degreeofshomogeneouisfunctionA 21 21 n k n sxsxsxfYs ss k),x,,xf(xy = > = [Euler\u2019s Theorem] Homogeneity of degree 1 is often called linear homogeneity. 7.1 The Theorem of Euler-Fermat Consider the unit group (Z\/15Z)\u00d7 of Z\/15Z. Euler\u2019s theorem gave birth to the concept of partial molar quantity and provides the functional link between it (calculated for each component) and the total quantity. Cosets-Lagrange's Theorem-Euler's Theorem (For the Course MATH-186 \"Elementary Number Theory\") George Chailos. In this paper we have extended the result from Example input: partition of n =100 into distinct \u2026 Fermat\u2019s Little Theorem is considered a special case of Euler\u2019s general Totient Theorem as Fermat\u2019s deals solely with prime moduli, while Euler\u2019s applies to any number so long as they are relatively prime to one another (Bogomolny, 2000). >> Euler\u2019s theorem is a general statement about a certain class of functions known as homogeneous functions of degree $$n$$. (By induction on the length, s, of the prime-power factorization.) As a result, the proof of Euler\u2019s Theorem is more accessible. However, in our presentation it is more natural to simply present Fermat\u2019s theorem as a special case of Euler\u2019s result. Ifp isprimeandaisanintegerwithp- a,then ap\u22121 \u22611 (modp). This is because clocks run modulo12, where the numbers Euler\u2019s theorem 2. Euler's Theorem We have seen that a spherical displacement or a pure rotation is described by a 3\u00d73 rotation matrix. However, this approach requires computing.n\/. \u0153\ufffd\ufffd\ufffd\/\ufffd\ufffd\ufffdH6\ufffdPUS\ufffd? Let n n n be a positive integer, and let a a a be an integer that is relatively prime to n. n. n. Then 1 Fermat.CALIFORNIA INSTITUTE OF TECHNOLOGY. The key point of the proof of Fermat\u2019s theorem was that if p is prime, {1,2,...,p \u2212 1} are relatively prime to p. This suggests that in the general case, it might be useful to look at the numbers less than the modulus n which are relatively prime to n. 1. Finally we present Euler\u2019s theorem which is a generalization of Fermat\u2019s theorem and it states that for any positive integer $$m$$ that is relatively prime to an integer $$a$$, $a^{\\phi(m)}\\equiv 1(mod \\ m)$ where $$\\phi$$ is Euler\u2019s $$\\phi$$-function. The solution (positive and negative) of generalized Euler theorem (hypothesis) are shown, for arbitrary x, y, z, t and the exponents of the type (4 + 4m) is provided in this article. There is another way to obtain this relation that involves a very general property of many thermodynamic functions. We start by proving a theorem about the inverse of integers modulo primes. \ufffd\ufffd\ufffd>u\u024bBe\ufffd0\\Y\ufffdmw\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd)\u07e8B\ufffd\ufffd\ufffd\ufffd\ufffd0\ufffdrY\ufffd\ufffds$t\ufffd\ufffd&[\ufffd\ufffd\ufffd\ufffd'\ufffd\ufffd\ufffd\ufffd\ufffdG\ufffdQfBpk\ufffdDV\ufffdJ\ufffdl#k^[A.~As>\ufffd\ufffd\u0212\ufffd\ufffd\u05c2 \ufffdm@\ufffdF\ufffd \/Filter \/FlateDecode The Theorem of Euler-Fermat In this chapter we will discuss the generalization of Fermat\u2019s Little Theorem to composite values of the modulus. Theorem. Euler\u2019s totient is defined as the number of numbers less than \u2018n\u2019 that are co-prime to it. 1.3 Euler\u2019s Theorem Modular or \u2019clock\u2019 arithmetic appears very often in number theory. Proof. }H]\ufffd\ufffdeye\ufffd Nonetheless, it is a valuable result to keep in mind. This property is a consequence of a theorem known as Euler\u2019s Theorem. Theorem 4.1 of Conformable Eulers Theor em on homogene ous functions] Let \u03b1 \u2208 (0, 1 p ] , p \u2208 Z + and f be a r eal value d function with n variables de\ufb01ned on an op en set D for which stream This theorem is credited to Leonhard Euler.It is a generalization of Fermat's Little Theorem, which specifies it when is prime. If f is a multiplicative function and if n = p a1 1 p a 2 2 p s s is its prime-power factorization, then f(n) = f(p a1 1)f(p a 2 2) f(p s s). \/Length 1125 \u016d\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffdp\ufffd=tr\ufffd\ufffd\ufffd\ufffdGr\ufffdm\ufffd\ufffdQR\ufffd[\ufffd\ufffd\ufffd1\ufffd\ufffd\u0591\ufffd}\ufffde\ufffd\ufffd8\ufffd+\u0108\ufffd\ufffd\ufffd(!D\u0175.\ufffd\u06ef\ufffdm\ufffdU\u0241,\ufffd\ufffd\ufffd\ufffdr\ufffdYnKYb\ufffd}\ufffdk\ufffd\ufffdeJy{\ufffd\ufffd\ufffd7\ufffd\ufffd\u030di2j4\ufffd\ufffd'\ufffd*\ufffd\ufffdz\ufffd\ufffd\ufffd#&\ufffdw\ufffd\ufffd#MN\ufffd\ufffd3\ufffd\ufffd\ufffdLv\ufffdd!\ufffdn]\ufffd\ufffd\ufffdi #V.apHhA\u00ba\ufffd\ufffd\ufffd\ucbf9m\ufffdZ\ufffd\ufffds\ufffdz@~\ufffdI-\ufffd6\ufffd\ufffd\ufffdDB\ufffdB\ufffd\ufffd\ufffd?$\ufffd-\ufffdkt$\\R)j\ufffdS\ufffdh\ufffd$61\ufffd\"El(\ufffd\ufffdCr In this article, I discuss many properties of Euler\u2019s Totient function and reduced residue systems. Euler's theorem is the most effective tool to solve remainder questions. Historically Fermat\u2019s theorem preceded Euler\u2019s, and the latter served to generalize the former. ... Theorem 2.2: a is a unit in n n if and only if gcd (a, n) 1 . \ufffdyl\u1034\ufffd\ufffdh \ufffdO\ufffd\ufffd\ufffdkY\ufffd\ufffd\ufffdP\ufffdD\ufffd\\\ufffdi\ufffd\ufffd\ufffd\ufffd>\ufffd\ufffd\ufffdx\ufffd\ufffd\ufffdu\ufffd\ufffd\"HC\ufffdC\ufffdN^\ufffd \ufffdV\ufffd\ufffd\ufffd}\ufffd\ufffdM\ufffd\ufffd\ufffd\ufffdW\ufffd\ufffd7\ufffd\ufffd\ufffdj\ufffd*\ufffd\ufffdJ\ufffd\" stream 4 0 obj Hence we can apply Euler's Theorem to get that $29^{\\phi (13)} \\equiv 1 \\pmod {13}$. x\ufffd\ufffd\u03ef\ufffd=\ufffd%\ufffd\ufffdK\ufffd\ufffd\ufffd\ufffdW\ufffdJn\ufffd\ufffdl\ufffd1hB\ufffd\ufffdb\ufffd\ufffdk\ufffd\ufffdL3M\ufffd\ufffd\ufffdd>>\ufffd8O\ufffd\ufffdVu\ufffd^\ufffdB\ufffd\ufffd\ufffd\ufffd\ufffdM\ufffdd\ufffd\ufffd\ufffdp\ufffd\ufffd\ufffd~|\ufffd\ufffd?>\ufffdk\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd^\ufffd\u057f\ufffd\ufffd\ufffd\ufffd_\ufffd\ufffd\ufffd~\ufffd?\ufffd\ufffdG\ufffd\ufffd\u03ef\ufffd\ufffd\ufffd Euler's theorem is a generalization of Fermat's little theorem dealing with powers of integers modulo positive integers. Idea: The key point of the proof of Fermats theorem was that if p is prime.EULERS THEOREM. Dirichlet in 1837 to the proof of the theorem stating that any arithmetic progression with di\ufb00erence k PROCEEDINGS OF THE STEKL OV INSTITUTE OF MATHEMATICS Vo l. \u2026 Since 13 is prime, it follows that $\\phi (13) = 12$, hence $29^{12} \\equiv 1 \\pmod {13}$. ]#u\ufffd?\ufffd\ufffd\u054b\ufffdE\ufffd\ufffd\\\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffdM\ufffd\ufffd\ufffd\ufffdT\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffdO\ufffd\ufffd\ufffd\ufffdw'\ufffd\u01eea7\ufffd\ufffd\ufffd+{N#\ufffd\ufffd7\ufffd\ufffdb\ufffdP\ufffdn\ufffd>\ufffd\ufffd\ufffd\ufffdIz\"\ufffd;\ufffd+{\ufffd\ufffd4\ufffd\ufffd\ufffdx>h'\ufffd=\ufffdS\ufffd_=\ufffdYf\ufffd\ufffd?\ufffd\ufffd[\ufffd\ufffdv8\ufffd\ufffdOU\ufffd\ufffd_[\ufffd\ufffd\ufffd\ufffd\ufffdVwR\ufffdY\ufffd\ufffdq\ufffd\ufffdi\ufffdi\ufffdq\ufffd\ufffdu\ufffd\ufffdf\ufffd>>\ufffd\ufffd\ufffd\u06bfe\ufffd\u069f\ufffdk#\ufffdE \ufffd\ufffdf\ufffdz_\ufffd\ufffd\ufffd\ufffd w>\ufffdQ~>|\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffdV}\ufffdN\ufffdl9\ufffdu\u02e2\ufffd\ufffd\ufffd\\. euler's rotation theorem pdf Fermats little theorem is an important property of integers to a prime modulus. %PDF-1.7 Corollary 3 (Fermat\u2019s Little Theorem\u2026 According to Euler's theorem, \"Any displacement of a rigid body such that a point on the rigid body, say O, remains fixed, is equivalent to a rotation about a fixed axis through the point O.\" Remarks. CAT Previous Papers PDF CAT Previous Papers PDF E uler\u2019s totient Euler\u2019s theorem is one of the most important remainder theorems. After watching Professor Robin Wilson\u2019s lecture about a Euler\u2019s Identity, I am finally able to understand why Euler\u2019s Identity is the most beautiful equation. Jan 02, 2021 - Partial Differential Part-4 (Euler's Theorem), Mathematics, CSE, GATE Computer Science Engineering (CSE) Video | EduRev is made by best teachers of Computer Science Engineering (CSE). THEOREM OF THE DAY Euler\u2019s Partition Identity The number of partitions of a positive integer n into distinct parts is equal to the number of partitions of n into odd parts. Alternatively,foreveryintegera,ap \u2261a (modp). Let be Euler's totient function.If is a positive integer, is the number of integers in the range which are relatively prime to .If is an integer and is a positive integer relatively prime to ,Then .. Credit. , where a i \u2208C. Justin Stevens Euler\u2019s Theorem (Lecture 7) 3 \/ 42 Euler\u2019s Formula and Trigonometry Peter Woit Department of Mathematics, Columbia University September 10, 2019 These are some notes rst prepared for my Fall 2015 Calculus II class, to Download Free PDF. %\ufffd\ufffd\ufffd\ufffd euler's theorem 1. Home \u00bb Courses \u00bb Electrical Engineering and Computer Science \u00bb Mathematics for Computer Science \u00bb Unit 2: Structures \u00bb 2.3 Euler's Theorem 2.3 Euler's Theorem Course Home With usual arithmetic it would seem odd to say 10+5 = 3 but when considering time on a clock this is perfectly acceptable. Theorem 1.1 (Fermat). %PDF-1.5 << In the next section, we\u2019ll show that computing .n\/ is easy if we know the This video is highly rated by Computer Science Engineering (CSE) students and has been viewed 987 times. It arises in applications of elementary number theory, including the theoretical underpinning for the RSA cryptosystem. If n = pa 1 1 then there is nothing to prove, as f(n) = f(pa 1 1) is clear. It is usually denoted as \u0278 (n). TheConverter. I also work through several examples of using Euler\u2019s Theorem. It is imperative to know about Euler\u2019s totient before we can use the theorem. First, they are convenient variables to work with because we can measure them in the lab. If n = pa 1 1 p a 2 Euler\u2019s theorem generalizes Fermat\u2019s theorem to the case where the modulus is composite. <> \ufffd\ufffd\ufffd\ufffdr\ufffd\ufffd~\ufffd\ufffd\/Y\ufffdp\ufffd\ufffd\ufffdq\u071d.\ufffd\ufffd\ufffd\ufffd\ufffd\ufffdx\ufffd\ufffd_\ufffd\ufffd_\ufffd\ufffd\ufffd\u061d\ufffd\ufffd\ufffd\ufffd\ufffd\ufffdo\ufffd\u06cf\ufffd\ufffdt\ufffd\ufffd\ufffd\ufffdl\ufffd\ufffdC\ufffds\/\ufffdy\ufffd\ufffd\ufffd\ufffd\ufffdX:\ufffd\ufffdkZ\ufffd\ufffdrx\ufffd\u4dc7\ufffd\ufffd\ufffdQ?~\ufffd_\ufffdwx\ufffd\ufffd\u0487\ufffdh\ufffdz]\ufffdn\ufffd\ufffdX>>\ufffd.\ufffd_\ufffdl\ufffdp;\ufffdN\ufffd\ufffd\ufffd\ufffd\ufffd\ufffdmi\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffdo\ufffd\ufffd\ufffd\ufffd|\ufffd\ufffd\ufffd\ufffdg\ufffd\ufffd\ufffdv;\ufffd\ufffd\ufffd\ufffd1\ufffdO\ufffd\ufffd7\ufffd\ufffd\/\/\ufffd\ufffd\u07caO\ufffd\ufffd\ufffd\u05ef\ufffd\/O\ufffd\ufffd~\ufffd6}\ufffd\ufffd_\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffdq\ufffd\u0716>?\ufffds]F\ufffd\ufffd\ufffd\ufffd\u1eb6|\ufffd|\\?.\ufffd\ufffd\ufffdo~\ufffd\ufffd}\\N\ufffd\ufffd\ufffdBUyt\ufffdx\ufffd\ud3f7_\ufffd\ufffdg\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd}\ufffdD\ufffd)\ufffd\ufffdz\ufffd\ufffd\ufffd]\ufffd\ufffd\ufffd\ufffd>p\ufffd\ufffdWRY\ufffd\ufffd[\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd;\/\ufffd\u04bf\ufffd?\ufffdt\ufffd\ufffd\ufffd\ufffd\ufffdO\ufffdP\ufffd\ufffd\ufffdy\ufffd\u02ef\ufffd\ufffdon\ufffd\ufffd\ufffdz\ufffdl} \ufffdV\ufffd\ufffdV>\ufffdN>\ufffdE\ufffd5\ufffdo\ufffd\ufffd\ufffd\ufffd?\ufffd:\ufffdO\ufffd7\ufffd?\ufffd\ufffd\ufffd\ufffd\ufffdm\ufffd\ufffd\ufffd*\ufffd}\ufffd\ufffd\ufffdm\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd|\ufffd\ufffd\ufffd\ufffd\ufffdn?-\ufffd\ufffd\ufffdT\ufffdT\ufffd\ufffd\ufffd\ufffd\u0497]:\ufffd.Og\ufffd\ufffdu!sX\ufffde\ufffd\ufffd\ufffdU\ufffd\u6c37\ufffdSa\ufffd\ufffd\ufffdz\ufffdrx\ufffd\ufffd\ufffdV\ufffd{'\ufffd'S\ufffdn\ufffd\ufffd^\u06bf\ufffd.\u03ef\ufffdW\ufffd_\ufffd\ufffdh\ufffdM;\ufffd\ufffd\ufffd\ufffd~\ufffd\/\ufffd'\ufffd\ufffd\ufffd\uf5bf\ufffd\ufffdu\ufffdq\ufffd\ufffd\ufffd7\ufffdY\ufffd\ufffd\ufffdU0\ufffd\ufffd\ufffdp\ufffd?n\ufffd\ufffd\ufffd\ufffdU{\ufffd\ufffd\ufffd\ufffd}~\ufffd\ufffd\ufffdt\ufffd\ufffd\ufffd\ufffdog]\ufffd\/\ufffd\u03fa\ufffdO\/ \ufffd\ufffd\ufffd\ufffd\ufffd4\u05cb\u0578h6[\u0330\ufffd\ufffd\ufffd\ufffdf\ufffd\ufffd?\ufffdx\ufffd=\ufffd^\ufffd \ufffd\ufffd\ufffd\ufffd\ufffdL\ufffd\ufffdY\ufffd\ufffd\ufffd2\ufffd\ufffd1\ufffdl\ufffdY\ufffd\/e\ufffdj\ufffdAO\ufffd\ufffdew\ufffd\ufffd1\u079e\ufffd_o\ufffd\ufffd\u05bc\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffdr.\ufffd\ufffd\ufffd[\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffdo\u4fd4Ol\ufffd=\ufffd\ufffdO\ufffd\ufffda\ufffd\ufffdK\ufffd\ufffdR_O\ufffd\ufffd\/\ufffd3\ufffd\ufffd\ufffd2|xQ\ufffd\ufffd\ufffd\ufffd\ufffd>yq\ufffd}\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffda\ufffd_\ufffd,\ufffd\ufffd\ufffd\ufffd7U\ufffdY\ufffdr:m}#\ufffd\ufffd\ueeac\ufffd\ufffd\ufffd\ufffd\ufffdQ\ufffd\ufffdH\ufffd\ufffdi\ufffd\ufffd\ufffd9\ufffdO\ufffd\ufffd+9\ufffd\ufffd\ufffd_\ufffd\ufffd\ufffd\ufffd8\ufffd\ufffd.\ufffdFf63g\/\ufffd\ufffdS\ufffdx\ufffd\ufffd\ufffd\ufffd3\ufffd\ufffd=_\u03cd\ufffdq\ufffd\ufffd\ufffd\ufffd\ufffd#\ufffdq\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffdr\ufffd\/\ufffd\ufffd\ufffd\ufffd\ufffd\ufffdg=\\H@\ufffd\ufffd.\u01d3\ufffd\ufffd\ufffds8\ufffd\ufffdp\ufffd\ufffd\ufffd\\\\d\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\u00c5\ufffd\u04540 In number theory, Euler's theorem (also known as the Fermat\u2013Euler theorem or Euler's totient theorem) states that if n and a are coprime positive integers, then a raised to the power of the totient of n is congruent to one, modulo n, or: {\\displaystyle \\varphi (n)} is Euler's totient function. 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Fermat \u2019 s is!\n\nBlacksmith Repair Wow, G37 Sedan Headlights Custom, Withdrawal Defense Mechanism, Deep Fried Shrimp, Kenya Currency To Usd, Advantus Flea Soft Chews For Small Dogs, Where Can I Donate A Suitcase To Foster Care, Toto Meaning In Nigeria, Top Destination Management Companies 2019, Onion Images Hd, Please Let Me Know If This Works For You Formal, Full Body Whitening Cream For Men's,","date":"2022-07-04 03:13:39","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 2, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8514118790626526, \"perplexity\": 1042.6934888206733}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-27\/segments\/1656104293758.72\/warc\/CC-MAIN-20220704015700-20220704045700-00655.warc.gz\"}"} | null | null |
{"url":"https:\/\/tex.stackexchange.com\/questions\/142957\/shortcut-for-variable-taking-command","text":"# Shortcut for Variable-Taking Command\n\nI'm working with vectors, which I'm denoting with $\\overrightarrow{a}$, where a is just the name of the vector, and thus is a variable that can take any value. I would like to redefine this as $\\v{a}$, to save time and effort. How do I do this?\n\nApologies if this question is a duplicate - I did find some similar questions, but none that I thought directly answered this.\n\n\u2022 \\newcommand*{\\V}[1]{$\\overrightarrow{#1}$} Use it as \\V{a} \u2013\u00a0user11232 Nov 8 '13 at 9:42\n\u2022 Remark: if I'm already in math-mode, it's better to have the command as \\newcommand*{\\V}[1]{\\overrightarrow{#1}} \u2013\u00a0Newb Nov 8 '13 at 9:49\n\u2022 @HarishKumar how would I do that? \u2013\u00a0Newb Nov 8 '13 at 9:57\n\u2022 If I end up only using it in one mode, why would the latter option be better than the first? \u2013\u00a0Newb Nov 8 '13 at 10:02\n\nLaTeX provides the semantic macro \\vec to typeset a vector. This is short, clear and easy to understand for others. As Martin pointed out in Short names for macros there are several one letter macros that do accent stuff. It is also harder to read for others, once you submit your file for editing or proof reading.\nAs has been pointed out in the comments, you can do \\newcommand*{\\V}[1]{\\overrightarrow{#1}} without automatic selecting math mode if necessary. Be sure to be the one who decides when math should be involved. You can read more about this in When not to use \\ensuremath for math macro?","date":"2019-06-26 17:52:18","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7891040444374084, \"perplexity\": 791.378985008715}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-26\/segments\/1560628000414.26\/warc\/CC-MAIN-20190626174622-20190626200622-00001.warc.gz\"}"} | null | null |
\section{Introduction}
\label{sec:Introduction}
In software engineering, static analyses are commonly used in order to analyze a software system and to identify potential defects. A well established form of static analyses are software metrics \cite{FentonBieman14}, which are used for the prediction of faults \cite{RadjenovicHerickoTorkar+13} or maintainability issues \cite{RiazMendesTempero09}.
In Software Product Lines (SPLs), variability information is an important part, which is not covered by traditional software metrics.
The SPL research community developed new variability-aware metrics to address this issue, which received increasing attention over the last decade \cite{El-SharkawyYamagishi-EichlerSchmid19, BezerraAndradeMonteiro+15, MontagudAbrahaoInsfran12}.
In a previous study \cite{El-SharkawyYamagishi-EichlerSchmid19}, we identified 147 variability-aware metrics to measure qualitative characteristics of variability models and code artifacts, which partly influence each other \cite{BergerGuo14}.
While traditional software metrics for single systems are well analyzed with respect to their ability to draw qualitative conclusions \cite{RadjenovicHerickoTorkar+13}, there are only very few evaluations available regarding the application of variability-aware metrics for SPLs \cite{El-SharkawyYamagishi-EichlerSchmid19}. Further, there are no comparisons between well-established single system and variability-aware metrics available. The lack of available tools for measuring variability-aware metrics aggravates the situation.
In this paper, we present a concept for efficiently parsing code files of SPLs that stores sufficient information for the realization of single system metrics from traditional software engineering as well as variability-aware code metrics designed for the needs of SPLs. In addition, our concept allows the arbitrary combination of variability-aware code metrics with feature metrics, which was not investigated so far.
Thus, the presented parsing concept provides the foundation for the realization and evaluation of new SPL metric suites like MetricHaven\footnote{\label{fn:MetricHaven}Available at \url{https://github.com/KernelHaven/MetricHaven}}. Here, we present the concepts behind the tool, which was presented in \cite{El-SharkawyKrafczykSchmid19}.
We pursue the following research questions:
\begin{enumerate}[label={\RQ{\arabic*}},leftmargin=*]
\item \label{rq:Classical vs SPL Metrics} What are the requirements to support a flexible measurement of single system and variability-aware code metrics?
\item \label{rq:Combination of Metrics} How can existing variability-aware metrics for code and variability models be combined?
\item \label{rq:Efficient} What abstraction is required to support a scalable analysis of large-scale SPLs?
\end{enumerate}
We implemented our concept in the publicly available tool MetricHaven \cite{El-SharkawyKrafczykSchmid19}, which provides practitioners and researchers with a foundation for the flexible definition and measurement of code metrics for SPLs implemented in C. MetricHaven is also designed as a highly configurable software product line and provides re-implementations of traditional and variability-aware code metrics from different research groups. Its design supports the highly efficient measurement of more than 42,000 metric combinations on large-scale product lines.
Overall, we make the following contributions:
\begin{itemize}
\item We present the concept of Reduced Abstract Syntax Trees (RASTs) that contain sufficient information for the definition of most traditional and variability-aware code metrics, while minimizing resource overhead.
\item A concept that allows a flexible combination of variability-aware feature and code metrics.
\item A discussion of the limitations of the presented approach.
\end{itemize}
\section{Related Work}
\label{sec:Related Work}
The research community developed a huge variety of variability-aware metrics, designed for the needs of SPLs \cite{El-SharkawyYamagishi-EichlerSchmid19, BezerraAndradeMonteiro+15, MontagudAbrahaoInsfran12}. Below, we discuss the related work on variability-aware metrics based on four characteristics: \textit{Tool support}, \textit{applicability}, \textit{flexibility}, and \textit{scalability}.
\textbf{Tool support.} In 2012, Montagud et al.\ \cite{MontagudAbrahaoInsfran12} investigated to which extend authors of variability-aware metrics provide tool-support. Their study included metrics for all life cycles of SPLs and, thus, was not limited to implementation. They conclude that only 52\% of 35 identified papers provide (partial) tool support for the computation of metrics.
We address this issue by providing a concept together with a publicly available tooling for the flexible realization of variability-aware code metrics. The presented approach supports a broad variety of single system as well as variability-aware code metrics of different research groups \cite{El-SharkawyKrafczykSchmid19}.
\textbf{Applicability.} An important aspect is the applicability of the available metrics. We categorized implementation-related metrics according to four categories \cite{El-SharkawyYamagishi-EichlerSchmid19}: Metrics for \textit{variability models} (this was included, because variability models are used to manipulate all artifacts of SPLs), \textit{annotation-based code}, \textit{composition-based code}, and the combination of \textit{code and variability model metrics}. We discovered that available concepts and their realizations are limited either to one of the aforementioned categories or are further restricted to certain file types. For instance, \texttt{S.P.L.O.T.}\ \cite{MendoncaBrancoCowan09} and \texttt{DyMMer} \cite{BezerraBarbosaFreires+16} provide various metrics for variability models saved in the \texttt{S.P.L.O.T.}\ file format (XML files). \texttt{FEATUREVISU} \cite{ApelBeyer11} was used for the measurement of code artifacts from composition-based SPLs, using different feature-oriented implementation techniques. In the context of annotation-based code, many authors implemented their metrics to operate directly on the XML output of \texttt{srcML}\footnote{\label{note:srcML}\url{https://www.srcml.org/}} \cite{LiebigApelLengauer+10, HunsenZhangSiegmund+16}. Thus, their measurement is limited to a specific set of implementation languages and require a re-implementation for the measurement of SPLs using a different annotation technique. Passos et al.\ \cite{PassosQueirozMukelabai+18} do not specify an implementation for the measurement of scattering degree metrics, but their appendix\footnote{\url{https://github.com/Mukelabai/featurescattering18/}} contains a set of \texttt{Bash} scripts explicitly designed for the analysis of Linux. This approach requires a re-implementation of their metrics for the measuring of other SPLs, even if they use a similar implementation technique.
We present a measurement concept for the analysis of annotation-based code artifacts of SPLs. In our implementation we decoupled parsing, data model, and the metrics computation from each other. Consequently, only a new parser is required for the analysis of SPLs realized with different programming languages.
\textbf{Flexibility.} Even if the variability model is often used for the configuration of code artifacts \cite{CzarneckiGruenbacherRabiser+12}, there are very few metrics available that include the complexity of the variability model when measuring code artifacts \cite{El-SharkawyYamagishi-EichlerSchmid19}. More precisely, we know only one study providing an evaluation for such a measure \cite{KolesnikovRothApel14}. Further, we do not know any comparisons of variability-aware code metrics with traditional metrics for single system metrics.
We present a concept that allows measuring of traditional and variability-aware code metrics in a single pass. For the use of variability-aware code metrics, we further allow the flexible integration of feature metrics to consider the complexity of the variability model.
\textbf{Scalability.} According to \cite{El-SharkawyYamagishi-EichlerSchmid19}, only 36\% of published metrics have been evaluated whether these metrics are sufficient to draw any qualitative conclusions. While some metrics have been applied on large-scale product lines from industry or publicly available SPLs, we did not discover any detailed examination of their runtime in general.
Our concept stores the information required for measuring different code metrics. We demonstrate the scalability of our approach by the application of 29,976 different metric variations on the Linux Kernel with more than 20,356 code files resulting in 53 GiB of measurement data. This is the first published performance analysis of SPLs metrics to the best of our knowledge.
\section[Tradeoffs in Designing Static Analysis Tools]{Tradeoffs in Designing Static\\Analysis Tools}
\label{sec:Tradeoffs}
Different parsing approaches exist for the static analysis of software, which result in different forms of Abstract Syntax Trees (ASTs). These parsing approaches come with different tradeoffs. In the context of SPLs, there also exist different analysis strategies: Product-based, family-based, and feature-based analysis approaches~\cite{ThumApelKastner+14}. Below we discuss \mbox{(dis-)}advantages of these concepts and show why we choose a partial parsing approach in combination with a family-based analysis technique. Figure~\ref{fig:Categorization} provides an overview of the considered analysis strategies and parsing approaches together with a classification of our approach and existing analysis tools.
\begin{figure}[tb]
\centering
\includegraphics[trim={0cm 4.5cm 10.5cm 0cm}, width=.8\columnwidth]{figures/CategorizationOfStaticAnalyses.pdf}
\Description{Categorization of static analysis approaches with respect to the used parsing approach.}
\caption{Categorization of static analysis approaches with respect to the used parsing approach.}
\label{fig:Categorization}
\end{figure}
\input{041b-TableOfElements}
\subsection{SPL Analysis Strategies}
\label{sec:SPL Strategies}
Thüm et al.\ \cite{ThumApelKastner+14} surveyed analysis approaches for SPLs and identified three categories of analysis strategies:
\textit{Product-based analysis techniques} operate on instantiated products of the SPL. This strategy allows the usage of standard analysis techniques from traditional software engineering, since the variability information is resolved \cite{ThumApelKastner+14}. For instance, professional metric tool suites like the Axivion Bauhaus Suite\footnote{\url{https://www.axivion.com/en/products-60\#produkte_bauhaussuite}}, Teamscale from CQSE\footnote{\url{https://www.cqse.eu/en/products/teamscale/landing/}}, and SonarQube\footnote{\url{https://www.sonarqube.org/}} may be utilized for the measurement of instantiated code artifacts. However, for a high coverage of the original SPL, this strategy requires redundant computations as the products share code and, thus, is very time-consuming. Further, the analysis of all supported product variants of the SPL is often not feasible in practice as the number of products is typically exponential in the number of features.
\textit{Family-based analysis techniques} operate on product line artifacts containing variability information and take advantage of a variability model to limit the analysis to valid configurations only. This strategy allows analysis of the code for all possible product configurations, without the need of generating any products. However, this strategy does not work with available tools developed for the analysis of single systems. Since family-based analysis techniques consider all product line artifacts as a whole, the size of the analysis problem can easily exceed physical boundaries such as the available memory \cite{ThumApelKastner+14}.
\textit{Feature-based analysis techniques} analyze product line artifacts containing variability information, too. Contrary to family-based approaches, this strategy analyzes each feature in isolation and ignores all other features as well as the variability model. This reduces the potentially exponential number of analysis tasks. However, this kind of analyses cannot detect any problems caused by feature interactions \cite{ThumApelKastner+14}.
Most of the surveyed variability-aware metrics operate on product line artifacts containing variability information and consider all features, but ignore the variability model \cite{El-SharkawyYamagishi-EichlerSchmid19}. Thus, they can be classified somewhere in between family-based and feature-based analysis approaches. We designed our analysis approach so that it can reproduce the current state-of-the-art in variability-aware metrics but may also incorporate information from the variability model.
\subsection{AST Parsing Strategies for SPL Analyses}
\label{sec:AST Strategies}
We observed two fundamentally different parsing strategies for family-based analysis approaches. Sincero et al.\ \cite{SinceroTartlerLohmann+10} focus on parsing only \textit{preprocessor blocks} to extract variability information of product line artifacts. This approach takes advantage of the strong abstraction and allows the extraction of variability information in $\mathcal{O}(n)$ with the number of variation points. According to \cite{SinceroTartlerLohmann+10}, Undertaker\footnote{\url{https://vamos.informatik.uni-erlangen.de/trac/undertaker}} requires about half an hour to parse all 25,844 source code files (*.c, *.h, *.S) of the Linux Kernel Version 2.6.33 with a quad core CPU and 8 GB RAM. While this strategy is very fast compared to more detailed data representations, the analysis capabilities of this approach are very limited. The authors designed this approach for the analysis of (un-)dead code with respect to the implemented variability \cite{TartlerLohmannSincero+11}. This approach does not support any code analysis, since the parser does not parse any elements of the programming language.
\begin{figure*}[bt]
\centering
\includegraphics[width=0.95\textwidth]{figures/ClassDiagram.pdf}
\vspace*{-10pt}
\Description{Class diagram of presented RAST contains elements of annotation language (e.g., C-preprocessor statements) and programming language (e.g., C language) in one data model.}
\caption{Simplified class structure used for parsing single system and variability-aware metrics (yellow:\ related to syntax elements of the programming language, blue:\ elements of the annotation language, green:\ related to both languages).}
\label{fig:ClassDiagram}\vspace*{-10pt}
\end{figure*}
Kästner et al.\ \cite{KastnerGiarrussoRendel+11} use a more sophisticated parsing strategy consisting of a \textit{variability-aware lexer} and a \textit{variability-aware parser}, implemented as part of TypeChef\footnote{\url{https://ckaestne.github.io/TypeChef/}}. The lexer annotates all tokens of the programming language with its presence conditions, i.e., the condition of the enclosing variation point used for the selection of the token. It also includes all header files and expands macros. The parser creates for each supported configuration of the parsed code an alternative subtree as part of the resulting \textit{variable AST}. The authors use a SAT-solver during lexing and parsing to reason about code parts that belong together or may be skipped. The very detailed code representation in conjunction with annotated variability information allows a broad range of family-based analysis techniques, like variability-aware type checking, variable control-flow graphs, and variability-aware liveness analysis \cite{LiebigRheinKastner+13}.
The creation of the very detailed variable AST requires much more effort than the previous approach. Parsing of the x86 architecture of the Linux Kernel version 2.6.33.3 with 7,665 C-files (*.h are included through the variability-aware lexer) requires roughly 85 hours on dual/quad-core lap computers with 2 to 8 GB RAM (the authors do not precisely specify their measurement system) \cite{KastnerGiarrussoRendel+11}. This parsing approach has an additional downside beside the massive time consumption. Through the macro expansion and the treatment of statements belonging to different configurations, the variable AST does not represent the developers view on the code anymore.
We surveyed existing traditional and variability-aware code metrics in order to design a Reduced AST. On the one hand, our RAST
contains more information than the approach by Sincero et al., which stores only information about the variation points used in code artifacts. On the other hand, our approach stores less information than the variable AST and, thus, does not facilitate the same code analyses as supported by the TypeChef infrastructure. However, our concept provides an efficient measurement of a large variety of traditional and variability-aware code metrics, which can not be done by any of the previously discussed parsing strategies.
\section{Concept}
\label{sec:Concept}
Here, we present the concept of parsing Reduced Abstract Syntax Trees (RASTs). This was motivated by designing a tailored parsing approach which is able to extract the information needed for the desired static analyses. In our case, we planned a flexible definition of single system and variability-aware code metrics to allow comparisons of them. Based on our survey \cite{El-SharkawyYamagishi-EichlerSchmid19} on variability-aware code metrics and an informal literature study on metrics from traditional software engineering, we came up with the following requirements for parsing RASTs (cf.\ \ref{rq:Classical vs SPL Metrics}):
\begin{enumerate}[label={\textbf{Req\arabic*}},leftmargin=*]\label{req:list}
\item \textit{Parsing of un-preprocessed code}. While established metric analysis tools from commercial vendors usually resolve preprocessor statements before conducting metrics, variability-aware metrics analyze the preprocessor statements. Thus, we require a common data representation for code annotations (in our case C-preprocessor statements) and for elements of the programming language (in our case AST elements of the C-language). This is a challenging task, since the used preprocessor is not part of the programming language and can be used at arbitrary positions inside a code file, independently of any syntax definitions.
\item \label{rq:No syntactically correct AST}\textit{No syntactically correct AST needed}. An important aspect is to which extent the resulting AST-structure needs to support only syntactically correct programs. Contrary to compilation tasks and type checking analyses, we do not need a syntactical correct AST for the computation of code metrics. However, the AST structure should be as close as possible to the actual code structure to simplify the definition of code metrics. Thus, it is still a challenging task to enhance a traditional AST structure with variability annotations, since these annotations may be inserted at arbitrary positions intertwined with AST elements of the programming language.
\item \textit{Granularity of RAST}. For optimization as well as for practical reasons it is important to assess the required granularity of parsed elements. A very fine grained AST, containing representations for all syntax elements of the annotation and programming language, conceptually supports every code metric. On the other hand, this requires much more effort to develop a very comprehensive parsing approach and leads to higher resource consumption. Due to limited development resources, we designed a Reduced Abstract Syntax Tree (RAST), which is sufficient for measuring all planed metrics and may be easily extended to support further metrics, if desired. The granularity of the RAST is driven by the measured elements of surveyed metrics, which we present in Table~\ref{tab:Supported Metrics}.
\end{enumerate}
\subsection{Reduced Abstract Syntax Tree (RAST)}
\label{sec:Concept:AST}
Based on our SLR on variability-aware code metrics \cite{El-SharkawyYamagishi-EichlerSchmid19} and an informal literature study on metrics for single systems, we designed a Reduced Abstract Syntax Tree (RAST) for the efficient measurement of the most relevant traditional and variability-aware code metrics. Our goal is the measurement of C-based SPL implementations.\footnote{The concepts we propose here could also be applied well beyond C.}
Thus, we limited the scope of our analysis to the measurement of metrics on a per-function basis. Figure~\ref{fig:ClassDiagram} presents the main elements of our RAST:
\input{044-controlflow}
\begin{itemize}
\item \texttt{SourceFile}s represent the RAST representation of code files.
\item The \texttt{CodeElement} is the super class of all RAST elements. It stores the line numbers to trace parsed elements back to their location in code files and facilitates LoC-metrics. Further, we store for each element two representations of the condition of surrounding variation points: The \texttt{condition} stores the condition of the innermost variation block, considering conditions of siblings for \texttt{\#elif}/\texttt{\#else}-blocks. For instance, we store the condition \texttt{$A$} of the \texttt{while} statement in Line~\ref{lst:condWhile} of the listing in Figure~\ref{fig:Concept:ControlFlow}. This allows the computation of feature-based metrics on all parsed elements, e.g., \textit{Scattering Degree} metrics. Second, \texttt{presenceCondition} provides an alternative as it stores the full presence condition for the inclusion of the element, also considering all surrounding variation points. For code elements that are not surrounded by any variation points, we set \texttt{condition} and \texttt{presenceCondition} to \texttt{TRUE}.
\item The \texttt{SingleStatement} is the most fine-grained element of the RAST. We do not provide RAST representations for expressions of statements, but we store these elements as unstructured text (\texttt{UnparsedCode}). For instance, in Line~\ref{lst:stmt} we store a \texttt{SingleStatement} with the text ``\texttt{stmt;}'', not knowing whether this is a function call, a variable declaration, or anything else.
\item The \texttt{IUnparsed} element facilitates the storage of preprocessor elements at arbitrary positions inside the RAST. This is required since preprocessor directives are not syntactical elements of the C programming language and may be used at arbitrary positions inside a code file, independently of any syntax definitions. The \texttt{IUnparsed} element is the parent of all (parsed) \texttt{CppBlock}s and \texttt{UnparsedCode} expressions of \texttt{SingleStatement}s.
\item We use \texttt{CppBlock} to store conditional blocks, i.e., variation points. This means, we store \texttt{\#if}, \texttt{\#ifdef}, \texttt{\#ifndef}, \texttt{\#elif}, and \texttt{\#else}-blocks in separate instances, referring to all siblings of the same block structure. The \texttt{type} attribute is used to distinguish between the different preprocessor elements and to allow a differentiation during the computation of metrics, if required. \texttt{CppBlock} inherits from \texttt{IUnparsed}, which is used for elements of \texttt{SingleStatement}s, and inherits from \texttt{CodeElementWithNesting}, which is used as a container inside our RAST. The multiple inheritance allows a nesting of preprocessor directives at arbitrary positions inside the RAST.
\item We use \texttt{BranchStatement}s similar to \texttt{CppBlock}s to store the \texttt{if} and \texttt{else} statements of the programming language. This class also stores the siblings of the same if/else-structure. Again, the \texttt{type} denotes which specific syntax element was used to allow a differentiation during the metrics computation, if necessary.
\item \texttt{LoopStatement}s represent any loop of the programming language. Contrary to \texttt{BranchStatement}s they do not have siblings. Again, we support different loop \texttt{type}s.
\item \texttt{Function}s represent function definitions. The function's signature is stored as \texttt{UnparsedCode}, while the function body is composed of previously described elements.
\item \texttt{Reference} elements are special as they neither represent syntax elements of the programming language nor of the annotation language. They are used in case that syntactical elements of the presented RAST, like loops or control structures, are split into multiple parts by C-preprocessor statements. The listing of Figure~\ref{fig:Concept:ControlFlow} shows an example in which the C-preprocessor is used for the conditional compilation of a loop statement, while the statements of the loop are always present. In this case, a \texttt{LoopStatement} with one \texttt{Reference} is stored inside a \texttt{CppBlock}. This can be seen on the right side of Figure~\ref{fig:Concept:ControlFlow}. The actual statements are stored outside of the \texttt{CppBlock}. This way it is possible to simultaneously define metrics on the same parsed data structure, that consider the nested statements as variable as well as metrics that do not treat this statement as variable.
\end{itemize}
\subsection{Application of RAST}
\label{sec:Concept:Application}
The RAST is designed to preserve the code structure in order to facilitate the computation of variability-aware code metrics according to their original definitions, while reducing the performance overhead by omitting elements that are not required for the metrics.
For this we neither resolve the variability as usually done by commercial metric tool suites nor do we duplicate parsed code elements of alternative variants as done by TypeChef \cite{KastnerGiarrussoRendel+11}, as this would lead to modified metric values.
As a consequence, our RAST contains a 150\% representation of the parsed code.
Parsing of C-code requires the ability to cope with \textit{undisciplined annotations} \cite{LiebigKastnerApel11, MedeirosRibeiroGheyi+18}, which are conditional compilation directives that do not align with the underlying syntactic structure of the code.
Our RAST provides two concepts to support these annotations as described above:
In cases that elements of a statement are conditional, a \texttt{CppBlock} inside a \texttt{SingleStatement} may be used to store the conditional elements.
Further, \texttt{Reference}s may be used to represent conditional control structures.
Based on the RAST, new metrics may be implemented as a visitor,\footnote{According to the visitor design pattern.} which may be parameterized to represent different variations of a metric family.
For instance, MetricHaven uses one visitor to compute different variations of McCabe's Cyclomatic Complexity measure \cite{McCabe76}, which counts the linear independent paths of the (variable) control graph.
For the single system version, we completely ignore the variability of the code and add 1 to the number of visited control structures (\texttt{while}, \texttt{for}, \texttt{if}, \texttt{case}).
According to \cite{ConteDunsmoreShen86} this counting approach is equivalent to the original definition of McCabe.
However, this approach provides support for control structures of undisciplined annotations, since we do not need to compute a syntactically correct control graph.
The left side of Figure~\ref{fig:Concept:ControlFlow} provides an example on how we compute the cyclomatic complexity of a conditional loop.
By ignoring the annotations, we detect two linear independent paths of the resulting control graph.
The variability-aware version of this metric considers only paths created by variation points \cite{Lopez-HerrejonTrujillo08}, i.e., \texttt{\#ifdef}-statements.
Finally, we provide a superimposition of both variants by counting the number of control structures of the programming language and the annotation language.
The right side of Figure~\ref{fig:Concept:ControlFlow} visualizes the resulting control graph, which contains three linear independent paths:
The loop may be present but omitted completely at run-time \textcircled{1}, the loop may be executed \textcircled{2}, and the loop may be removed through conditional compilation but the statements are kept \textcircled{3}.
\section{Realization}
\label{sec:Realization}
Our concept is implemented as a prototype for the analysis of C-based SPLs like the Linux Kernel. This is realized in several plug-ins for the \texttt{KernelHaven}\ infrastructure \cite{KroeherEl-SharkawySchmid18a,KroeherEl-SharkawySchmid18b}. This infrastructure supports three types of extractors to read information from the product line to be analyzed:
\begin{itemize}
\item Code extractors extract variability information from the source code implementing the software product line. For C source code, this typically involves parsing \ifDef-block{}s of the C-preprocessor.
\item Build model extractors extract variability information from the build process of the product line. This typically involves presence conditions that define in which configurations a given source code file is compiled into the product line.
\item Variability model extractors extract the variability model of the software product line. This contains a list of all features and constraints between them.
\end{itemize}
In \texttt{KernelHaven}, the result from these extractors is represented in models that are agnostic to implementation details of specific product lines, while still being extensible to additional information. This allows the following analysis components, which access these models, to be implemented independently from implementation details of specific product lines. The following paragraphs will first introduce which extractors were used in our prototype implementation and then explain the analysis process that implements our approach.
The parsing of the product line source code is implemented in the \texttt{srcMLExtractor}\footnote{\url{https://github.com/KernelHaven/srcMLExtractor}} plug-in for \texttt{KernelHaven}. It is based on the \texttt{srcML}\textsuperscript{\ref{note:srcML}} tool which parses source code to an XML format \cite{collard2011lightweight}. The \texttt{srcMLExtractor}{} parses this XML and converts it into a model compatible with \texttt{KernelHaven}. The extension mechanism of the general code model in \texttt{KernelHaven}{} is used to model the Reduced Abstract Syntax Tree (RAST), as introduced in Section~\ref{sec:Concept:AST}. The parsing process of the \texttt{srcMLExtractor}{} does not parse the full AST output of \texttt{srcML}, but only descends to a granularity that provides sufficient information to build the RAST. See Figure~\ref{fig:ClassDiagram} for a simplified class diagram of the resulting RAST structure.
For the build and variability models, we use the \texttt{KbuildMiner\-Extractor}\footnote{\url{https://github.com/KernelHaven/KbuildMinerExtractor}} and \texttt{KconfigReader\-Extractor}\footnote{\url{https://github.com/KernelHaven/KconfigReaderExtractor}}.
The former extracts variability information from the \texttt{Kbuild} build process of the Linux Kernel, the latter reads the \texttt{Kconfig} variability model present in the Linux Kernel source tree. These plug-ins and their underlying tools were already used in previous analyses of the Linux Kernel, and there were no changes done to these plug-ins when implementing the approach presented in this paper.
The analysis process in \texttt{KernelHaven}{} typically consists of multiple analysis components that are combined to an analysis pipeline. The output of the previous component(s) is used as the input for the following component(s). The initial input for the first analysis component(s) are the models supplied by the three extractors (see above). This structure allows for simple re-use of analysis components when creating new analysis pipelines.
The calculation of metrics is implemented as such an analysis pipeline in the \texttt{MetricHaven}$^{\ref{fn:MetricHaven}}$ plug-in for \texttt{KernelHaven}. Figure~\ref{fig:MetricHavenPipeline} shows an overview of this pipeline structure. The coloring of the lines indicate the flow of the three models extracted from the product line, as described above. The actual metric computation happens in the rightmost component at the end of the pipeline. The input for this component are the three extracted models (the code model went through the \texttt{FunctionFilter} first) and the output of three preprocessing components.
\begin{itemize}
\item The \texttt{FunctionFilter} component splits the code model into individual functions, and removes any elements that are outside of functions (such as global variables). This component does not compute any values for metrics; it is only used for convenient data organization. The result is a stream of code functions.
\item The \texttt{FunctionCallMap} component analyses calls between functions. Since the Reduced Abstract Syntax Tree (RAST) is not fully parsed (cf.~Section~\ref{sec:Concept:AST}), function calls inside statements are identified heuristically: If the unparsed code string of a statement contains a function name followed by an opening parenthesis, we consider that statement to contain a call to this function. For each identified function call, the calling function (caller), the called function (callee), and the presence condition and location of the statement containing the function call are stored. This information is for example used in the Fan-In/-Out and Degree Centrality metrics.
\item The \texttt{ScatteringDegree} component calculates the Scattering Degree metric for all features of the variability model. The result is a map of all features and their values for the different scattering degree types (\SD$_\text{VP}$\ \cite{LiebigApelLengauer+10,CoutoValenteFigueiredo11,JbaraFeitelson13, PassosQueirozMukelabai+18} and \SD$_\text{File}$ \cite{ZhangBeckerPatzke+13, HunsenZhangSiegmund+16}).
\item The \texttt{FeatureSize} component calculates the Feature Size metric for all features in the variability model. The result is a map of all features and the number of statements controlled by the feature.
\end{itemize}
The preprocessing components are executed before the final metric computation component, because they require a full overview of the complete code model. In contrast, the metric calculation component calculates the metric values on a per-function basis. This reduced view on the code model allows to reduce the complexity of the metric calculation component and also helps to mitigate a memory problem. Since our implementation scales to a large number of metrics to be calculated per function, the amount of resulting metric values can grow quickly. In practice, the memory required to store this are several gigabytes. With the per-function approach in the metric calculation component the results of a single function can directly be written to disk, freeing the main memory.
Our implementation offers a number of configuration options. Most importantly, it allows for free selection of metrics to calculate. The user can select anything from 1 up to 42,796 metrics and metric combinations to be calculated per function. Additionally, the extractors at the beginning of the pipeline can be exchanged. This enables our infrastructure to run on different software product lines, while the analysis components require no adaptation. This is because the models used to represent the extraction result are agnostic to specifics of single product lines. Finally, the number of threads used in the code extraction plug-in and the metric calculation component can be configured independently. See the evaluation in Section~\ref{sec:Evaluation} for details on the potential performance improvements.
\section{Discussion}
\label{sec:Discussion}
In the development of any kind of analysis tool, one needs to make a number of tradeoffs. These relate, in particular, to generality (what range of metrics to support), speed, and range of supported artifacts.
The key principles we used for developing the solution, described here, are:
\begin{itemize}
\item \textit{Genericity:} It should support a large range of product line metrics.
Basically, it should support all metrics that were documented as product line metrics so far and at the same time, it should support a very large (though not necessary complete) range of single system metrics.
\item \textit{Scalability:} It should be able to deal with very large product lines and very large numbers of metrics simultaneously.
\item \textit{Performance:} It should perform these tasks very efficiently.
\end{itemize}
The key innovation was to introduce the solution of a reduced abstract syntax tree (RAST) and to tailor it very well to the task at hand. This avoids the representation of language details that are not relevant to the metrics analysis and abstracting those that are relevant as far as possible. A detailed analysis of the representational needs of the metrics provided the basis of our design.
At this point also some trade-offs had to be made. As a result, we do not support all kinds of metrics. For example, we do not support Halstead metrics \cite{Halstead77}.
In order to achieve high performance, we create the metrics values nearly completely in a single pass. This leads to significant performance improvements as technical properties like CPU caching are used in an optimal way and the generation of the data structures does not need to be made multiple times.
The exception is the approximation of Eigenvalue Centrality, which requires a two-pass approach.
Overall, we managed to get an extremely high performance, using our approach along with very good scalability properties, both in terms of the number of metrics and analyzed code size. For example, in our evaluation we found 6 hours 20 minutes for analyzing the complete Linux Kernel (cf.\ Section~\ref{sec:Evaluation}), which means that about 0.76 seconds were required per metric. Or, to put it differently, for producing 29,976 metrics, we needed about 0.06 seconds per function. We regard this as an extremely strong performance, although it is very difficult to compare as there is no other metrics tool that supports a similarly wide range of metrics.
\section{Evaluation}
\label{sec:Evaluation}
We ran our prototype implementation on the x86 architecture of the Linux Kernel version 4.15\footnote{https://mirrors.edge.kernel.org/pub/linux/kernel/v4.x/linux-4.15.tar.xz} to evaluate the implementation of our concept. Based on the RAST presented in Figure~\ref{fig:ClassDiagram}, we were able to realize various single system and variability-aware code metrics \cite{El-SharkawyKrafczykSchmid19}. Through the combination of variability-aware code metrics with feature metrics we support 42,796 metric variations (as of Summer 2019), which can be measured in a single pass (cf.\ \ref{rq:Efficient}). While most metrics can be implemented in a straightforward manner, some implementations require adaptations for our RAST. For instance, the detection of function calls for the Fan-In/-Out metric can only be implemented heuristically, since it would require full parsing of the expression syntax (cf.\ Section~\ref{sec:Realization}).
The x86 architecture in the Linux Kernel has 20,356 C-source files that are evaluated by us.
106 ($\approx 0.5\%$) of these files, cannot be handled by the current implementation of the \texttt{srcMLExtractor}, which translates the XML output of \texttt{srcML}{} to \texttt{KernelHaven}'s code model.
This is mostly related to very special corner cases of \textit{undisciplined} C-preprocessor usage \cite{MedeirosRibeiroGheyi+18}, i.e., conditional annotations that do not align with the syntactic structure of the code.
\texttt{srcML}{} marks up the C-syntax independently of C-preprocessor directives.
In conjunction with \textit{undisciplined} C-preprocessor directives, this can lead to incorrect markups provided by \texttt{srcML}{}.
Proper handling of those structures requires adaptations of the \texttt{srcML}{} parser.
Some special cases are detected and fixed by our \texttt{srcMLExtractor}.
However, this approach cannot repair all of these cases and requires significant development effort.
Further, a minority of these corner cases cannot be mapped to our RAST at all as they violate the few structural assumptions of the RAST.
An even more lenient RAST that can model these cases, however, would significantly complicate the definition and computation of metrics.
We ran two sets of metrics on the Linux Kernel, to evaluate scalability of our approach (cf.\ \ref{rq:Efficient}): First, a selected subset of metric combinations that we also used in practice for our own work with the Linux Kernel. We call this set \textit{atomic metrics}. It contains all basic code metrics and all possible combinations of code metrics combined with a single feature metric. Code metrics combined with multiple feature metrics are not included. This results in a set of 648 metrics. Second, we allowed all metric combinations. However, the implementation of one metric family (approximation of Eigenvector Centrality) requires significantly more memory than the other implementations and is not optimized for the provided parallelization capabilities of MetricHaven. For this reason, we executed only the 148 metric variations of this metric, which were already executed as part of the atomic metrics, while we executed all variations of the remaining metrics. This results in a set of 29,976 metrics.
For the performance measurements, we ran our implementation in a virtual machine running Ubuntu 16.04 with 40 logical CPU cores of an Intel Xeon E5-2650v3 @ 2.3 GHz and 314 GiB RAM. For the analysis, we limited the JVM once to 50 GiB memory\footnote{via the command line switches: -Xmx50g -Xms50g} and once to 24 GiB. However, it must be noted that the extractors run in separate processes as they execute external tools and, thus, allocate additional memory. Accurate timings for specific phases are hard to measure since KernelHaven makes heavy use of parallelization. For example, the preprocessing components already start to run while code parsing is still running. However, the actual metric calculation component can only start when the complete code model has been passed through the preprocessing components. That means that we identify two distinct execution phases: code parsing and metric calculation. The preprocessing phase, which partially happens in parallel to the code parsing only takes a few seconds, which is insignificant compared to the total runtime. Thus, we do not supply measurements for this phase. The code parsing and metric calculation components can also be independently configured to use a specified amount of threads. We measured the runtime of these components for different numbers of configured threads.
For the experiments, we used a range of 1 to 10 threads to cover a spectrum which is supported by most workstation computers.
Running the 648 \textit{atomic metrics} on all 409,253 functions that we parse from the x86 Linux Kernel architecture produces 265,195,944 measures. In CSV format, this is about 1.2 GiB. The metric calculation step takes about 27.5 minutes (54 minutes with 25 GiB memory for the JVM) to run on a single thread and can be decreased to about 13.75 minutes (38.5 minutes) on 10 computation threads.
Running the large set of 29,976 metrics on all 409,253 functions that we parse from the x86 Linux Kernel architecture produces 12.2 billion measures.
In CSV format, this is about 53 GiB.
The metric calculation step takes from 11 hours and 26 minutes on one thread to 6 hours 15 minutes on 10 threads (6 hours 20 minutes total runtime).
The parsing performance stays the same as described in the previous paragraph as it is independent from the number of computed metrics.
The parallelization benefit of metric computations is slightly less pronounced compared to the \textit{atomic metrics}.
However, our prototype is not fully optimized with regard to parallelization.
\begin{figure}[tb]
\centering
\includegraphics[trim={0.04cm 8.5cm 20.95cm 0cm},clip,width=\columnwidth]{figures/Timings.pdf}
\Description{Benchmark shows performance of MetricHaven to measure 648 metric variations on the Linux Kernel.
The parsing of code files takes from about 16 minutes and 20 seconds on 1 thread to about 2 minutes on 10 threads.
The metrics runtime takes about 27 minutes and 39 seconds on 1 thread to about 13 minutes and 14 seconds on 10 threads.
The total runtime takes about 44 minutes and 11 seconds and goes down to 16 minutes on 10 threads.}
\vspace*{-18pt}
\caption{Parsing/computation/total runtime of 648 \textit{atomic metrics} with different number of threads.}
\label{fig:ExtractorTimings}
\end{figure}
Figure~\ref{fig:ExtractorTimings} visualizes the runtime of running the 648 \textit{atomic metrics} on a different number of threads with 50 GiB memory.
The number of threads for the parsing and the metric computation phase were both modified together on a range of 1 to 10.
For each run, we present the parsing time (lower, blue bars), the metrics computation time (upper, orange bars), and the total execution time (numbers above), which requires some additional time for loading the plug-ins and writing the results.
The gray, dotted line indicates the trend line of the total execution time.
Running the experiments with less memory results in a similar behavior.
However, the performance gain of configuring more threads is much smaller compared to the experiment with 50 GiB memory, because of the increased workload of the JVM garbage collector.
Please note that MetricHaven keeps the parsed RAST of the complete Linux Kernel in memory when it starts its computation.
\section{Conclusion}
\label{sec:Conclusion}
In this paper, we presented the concept of MetricHaven, for simultaneously evaluating a large number of metrics on very large product lines in a highly efficient manner. MetricHaven supports more than 42,000 metrics, requiring less than 0.06 seconds for computing a selection of 29,976 metrics per function of the Linux Kernel leading to a total execution time of about 6 hours and 20 minutes for analyzing the whole Linux Kernel, yielding more than 53 GB of metrics data.
The approach is highly customizable (achieving significant speed-ups when reducing the number of metrics to process).
In particular, beyond analyzing product line metrics, it is also capable of creating a significant range of single system metrics.
The key to achieving these capabilities, was first to identify key realization requirements as required in \ref{rq:Classical vs SPL Metrics}. In Section~\ref{sec:Concept}, we introduced three core requirements that enabled us to create this approach: we had to directly parse un-preprocessed code (1), yielding an AST which is not a syntactically correct representation (2). Further, we had to minimize the required information by abstracting the information to a significant extent, leading to a rather coarse-grained AST with reduced information (RAST) (3).
For \ref{rq:Combination of Metrics}, the answer was actually rather simple: by having an integrated representation that does not replicate basic code elements (as, for example, some approaches to handling variable code do \cite{KastnerGiarrussoRendel+11}) and integrating the variability given by the preprocessor information, we could simply handle the subset of non-variable information also from a metrics point of view.
We addressed \ref{rq:Efficient} by creating the notion of a reduced abstract syntax tree (RAST). The abstraction level of this is tailored to exactly the level of detail required for handling all relevant product line metrics. All further information is skipped, respectively, not parsed in detail. We described this in detail in Section~\ref{sec:Concept:AST}.
In future work, we plan to further extend this framework in terms of the range of supported metrics, improve its performance and apply it to study numerous properties of product line implementations. We are particularly interested in the prediction of defects based on product line metrics.
\section{Introduction}
\label{sec:Introduction}
In software engineering, static analyses are commonly used in order to analyze a software system and to identify potential defects. A well established form of static analyses are software metrics \cite{FentonBieman14}, which are used for the prediction of faults \cite{RadjenovicHerickoTorkar+13} or maintainability issues \cite{RiazMendesTempero09}.
In Software Product Lines (SPLs), variability information is an important part, which is not covered by traditional software metrics.
The SPL research community developed new variability-aware metrics to address this issue, which received increasing attention over the last decade \cite{El-SharkawyYamagishi-EichlerSchmid19, BezerraAndradeMonteiro+15, MontagudAbrahaoInsfran12}.
In a previous study \cite{El-SharkawyYamagishi-EichlerSchmid19}, we identified 147 variability-aware metrics to measure qualitative characteristics of variability models and code artifacts, which partly influence each other \cite{BergerGuo14}.
While traditional software metrics for single systems are well analyzed with respect to their ability to draw qualitative conclusions \cite{RadjenovicHerickoTorkar+13}, there are only very few evaluations available regarding the application of variability-aware metrics for SPLs \cite{El-SharkawyYamagishi-EichlerSchmid19}. Further, there are no comparisons between well-established single system and variability-aware metrics available. The lack of available tools for measuring variability-aware metrics aggravates the situation.
In this paper, we present a concept for efficiently parsing code files of SPLs that stores sufficient information for the realization of single system metrics from traditional software engineering as well as variability-aware code metrics designed for the needs of SPLs. In addition, our concept allows the arbitrary combination of variability-aware code metrics with feature metrics, which was not investigated so far.
Thus, the presented parsing concept provides the foundation for the realization and evaluation of new SPL metric suites like MetricHaven\footnote{\label{fn:MetricHaven}Available at \url{https://github.com/KernelHaven/MetricHaven}}. Here, we present the concepts behind the tool, which was presented in \cite{El-SharkawyKrafczykSchmid19}.
We pursue the following research questions:
\begin{enumerate}[label={\RQ{\arabic*}},leftmargin=*]
\item \label{rq:Classical vs SPL Metrics} What are the requirements to support a flexible measurement of single system and variability-aware code metrics?
\item \label{rq:Combination of Metrics} How can existing variability-aware metrics for code and variability models be combined?
\item \label{rq:Efficient} What abstraction is required to support a scalable analysis of large-scale SPLs?
\end{enumerate}
We implemented our concept in the publicly available tool MetricHaven \cite{El-SharkawyKrafczykSchmid19}, which provides practitioners and researchers with a foundation for the flexible definition and measurement of code metrics for SPLs implemented in C. MetricHaven is also designed as a highly configurable software product line and provides re-implementations of traditional and variability-aware code metrics from different research groups. Its design supports the highly efficient measurement of more than 42,000 metric combinations on large-scale product lines.
Overall, we make the following contributions:
\begin{itemize}
\item We present the concept of Reduced Abstract Syntax Trees (RASTs) that contain sufficient information for the definition of most traditional and variability-aware code metrics, while minimizing resource overhead.
\item A concept that allows a flexible combination of variability-aware feature and code metrics.
\item A discussion of the limitations of the presented approach.
\end{itemize}
\section{Related Work}
\label{sec:Related Work}
The research community developed a huge variety of variability-aware metrics, designed for the needs of SPLs \cite{El-SharkawyYamagishi-EichlerSchmid19, BezerraAndradeMonteiro+15, MontagudAbrahaoInsfran12}. Below, we discuss the related work on variability-aware metrics based on four characteristics: \textit{Tool support}, \textit{applicability}, \textit{flexibility}, and \textit{scalability}.
\textbf{Tool support.} In 2012, Montagud et al.\ \cite{MontagudAbrahaoInsfran12} investigated to which extend authors of variability-aware metrics provide tool-support. Their study included metrics for all life cycles of SPLs and, thus, was not limited to implementation. They conclude that only 52\% of 35 identified papers provide (partial) tool support for the computation of metrics.
We address this issue by providing a concept together with a publicly available tooling for the flexible realization of variability-aware code metrics. The presented approach supports a broad variety of single system as well as variability-aware code metrics of different research groups \cite{El-SharkawyKrafczykSchmid19}.
\textbf{Applicability.} An important aspect is the applicability of the available metrics. We categorized implementation-related metrics according to four categories \cite{El-SharkawyYamagishi-EichlerSchmid19}: Metrics for \textit{variability models} (this was included, because variability models are used to manipulate all artifacts of SPLs), \textit{annotation-based code}, \textit{composition-based code}, and the combination of \textit{code and variability model metrics}. We discovered that available concepts and their realizations are limited either to one of the aforementioned categories or are further restricted to certain file types. For instance, \texttt{S.P.L.O.T.}\ \cite{MendoncaBrancoCowan09} and \texttt{DyMMer} \cite{BezerraBarbosaFreires+16} provide various metrics for variability models saved in the \texttt{S.P.L.O.T.}\ file format (XML files). \texttt{FEATUREVISU} \cite{ApelBeyer11} was used for the measurement of code artifacts from composition-based SPLs, using different feature-oriented implementation techniques. In the context of annotation-based code, many authors implemented their metrics to operate directly on the XML output of \texttt{srcML}\footnote{\label{note:srcML}\url{https://www.srcml.org/}} \cite{LiebigApelLengauer+10, HunsenZhangSiegmund+16}. Thus, their measurement is limited to a specific set of implementation languages and require a re-implementation for the measurement of SPLs using a different annotation technique. Passos et al.\ \cite{PassosQueirozMukelabai+18} do not specify an implementation for the measurement of scattering degree metrics, but their appendix\footnote{\url{https://github.com/Mukelabai/featurescattering18/}} contains a set of \texttt{Bash} scripts explicitly designed for the analysis of Linux. This approach requires a re-implementation of their metrics for the measuring of other SPLs, even if they use a similar implementation technique.
We present a measurement concept for the analysis of annotation-based code artifacts of SPLs. In our implementation we decoupled parsing, data model, and the metrics computation from each other. Consequently, only a new parser is required for the analysis of SPLs realized with different programming languages.
\textbf{Flexibility.} Even if the variability model is often used for the configuration of code artifacts \cite{CzarneckiGruenbacherRabiser+12}, there are very few metrics available that include the complexity of the variability model when measuring code artifacts \cite{El-SharkawyYamagishi-EichlerSchmid19}. More precisely, we know only one study providing an evaluation for such a measure \cite{KolesnikovRothApel14}. Further, we do not know any comparisons of variability-aware code metrics with traditional metrics for single system metrics.
We present a concept that allows measuring of traditional and variability-aware code metrics in a single pass. For the use of variability-aware code metrics, we further allow the flexible integration of feature metrics to consider the complexity of the variability model.
\textbf{Scalability.} According to \cite{El-SharkawyYamagishi-EichlerSchmid19}, only 36\% of published metrics have been evaluated whether these metrics are sufficient to draw any qualitative conclusions. While some metrics have been applied on large-scale product lines from industry or publicly available SPLs, we did not discover any detailed examination of their runtime in general.
Our concept stores the information required for measuring different code metrics. We demonstrate the scalability of our approach by the application of 29,976 different metric variations on the Linux Kernel with more than 20,356 code files resulting in 53 GiB of measurement data. This is the first published performance analysis of SPLs metrics to the best of our knowledge.
\section[Tradeoffs in Designing Static Analysis Tools]{Tradeoffs in Designing Static\\Analysis Tools}
\label{sec:Tradeoffs}
Different parsing approaches exist for the static analysis of software, which result in different forms of Abstract Syntax Trees (ASTs). These parsing approaches come with different tradeoffs. In the context of SPLs, there also exist different analysis strategies: Product-based, family-based, and feature-based analysis approaches~\cite{ThumApelKastner+14}. Below we discuss \mbox{(dis-)}advantages of these concepts and show why we choose a partial parsing approach in combination with a family-based analysis technique. Figure~\ref{fig:Categorization} provides an overview of the considered analysis strategies and parsing approaches together with a classification of our approach and existing analysis tools.
\begin{figure}[tb]
\centering
\includegraphics[trim={0cm 4.5cm 10.5cm 0cm}, width=.8\columnwidth]{figures/CategorizationOfStaticAnalyses.pdf}
\Description{Categorization of static analysis approaches with respect to the used parsing approach.}
\caption{Categorization of static analysis approaches with respect to the used parsing approach.}
\label{fig:Categorization}
\end{figure}
\input{041b-TableOfElements}
\subsection{SPL Analysis Strategies}
\label{sec:SPL Strategies}
Thüm et al.\ \cite{ThumApelKastner+14} surveyed analysis approaches for SPLs and identified three categories of analysis strategies:
\textit{Product-based analysis techniques} operate on instantiated products of the SPL. This strategy allows the usage of standard analysis techniques from traditional software engineering, since the variability information is resolved \cite{ThumApelKastner+14}. For instance, professional metric tool suites like the Axivion Bauhaus Suite\footnote{\url{https://www.axivion.com/en/products-60\#produkte_bauhaussuite}}, Teamscale from CQSE\footnote{\url{https://www.cqse.eu/en/products/teamscale/landing/}}, and SonarQube\footnote{\url{https://www.sonarqube.org/}} may be utilized for the measurement of instantiated code artifacts. However, for a high coverage of the original SPL, this strategy requires redundant computations as the products share code and, thus, is very time-consuming. Further, the analysis of all supported product variants of the SPL is often not feasible in practice as the number of products is typically exponential in the number of features.
\textit{Family-based analysis techniques} operate on product line artifacts containing variability information and take advantage of a variability model to limit the analysis to valid configurations only. This strategy allows analysis of the code for all possible product configurations, without the need of generating any products. However, this strategy does not work with available tools developed for the analysis of single systems. Since family-based analysis techniques consider all product line artifacts as a whole, the size of the analysis problem can easily exceed physical boundaries such as the available memory \cite{ThumApelKastner+14}.
\textit{Feature-based analysis techniques} analyze product line artifacts containing variability information, too. Contrary to family-based approaches, this strategy analyzes each feature in isolation and ignores all other features as well as the variability model. This reduces the potentially exponential number of analysis tasks. However, this kind of analyses cannot detect any problems caused by feature interactions \cite{ThumApelKastner+14}.
Most of the surveyed variability-aware metrics operate on product line artifacts containing variability information and consider all features, but ignore the variability model \cite{El-SharkawyYamagishi-EichlerSchmid19}. Thus, they can be classified somewhere in between family-based and feature-based analysis approaches. We designed our analysis approach so that it can reproduce the current state-of-the-art in variability-aware metrics but may also incorporate information from the variability model.
\subsection{AST Parsing Strategies for SPL Analyses}
\label{sec:AST Strategies}
We observed two fundamentally different parsing strategies for family-based analysis approaches. Sincero et al.\ \cite{SinceroTartlerLohmann+10} focus on parsing only \textit{preprocessor blocks} to extract variability information of product line artifacts. This approach takes advantage of the strong abstraction and allows the extraction of variability information in $\mathcal{O}(n)$ with the number of variation points. According to \cite{SinceroTartlerLohmann+10}, Undertaker\footnote{\url{https://vamos.informatik.uni-erlangen.de/trac/undertaker}} requires about half an hour to parse all 25,844 source code files (*.c, *.h, *.S) of the Linux Kernel Version 2.6.33 with a quad core CPU and 8 GB RAM. While this strategy is very fast compared to more detailed data representations, the analysis capabilities of this approach are very limited. The authors designed this approach for the analysis of (un-)dead code with respect to the implemented variability \cite{TartlerLohmannSincero+11}. This approach does not support any code analysis, since the parser does not parse any elements of the programming language.
\begin{figure*}[bt]
\centering
\includegraphics[width=0.95\textwidth]{figures/ClassDiagram.pdf}
\vspace*{-10pt}
\Description{Class diagram of presented RAST contains elements of annotation language (e.g., C-preprocessor statements) and programming language (e.g., C language) in one data model.}
\caption{Simplified class structure used for parsing single system and variability-aware metrics (yellow:\ related to syntax elements of the programming language, blue:\ elements of the annotation language, green:\ related to both languages).}
\label{fig:ClassDiagram}\vspace*{-10pt}
\end{figure*}
Kästner et al.\ \cite{KastnerGiarrussoRendel+11} use a more sophisticated parsing strategy consisting of a \textit{variability-aware lexer} and a \textit{variability-aware parser}, implemented as part of TypeChef\footnote{\url{https://ckaestne.github.io/TypeChef/}}. The lexer annotates all tokens of the programming language with its presence conditions, i.e., the condition of the enclosing variation point used for the selection of the token. It also includes all header files and expands macros. The parser creates for each supported configuration of the parsed code an alternative subtree as part of the resulting \textit{variable AST}. The authors use a SAT-solver during lexing and parsing to reason about code parts that belong together or may be skipped. The very detailed code representation in conjunction with annotated variability information allows a broad range of family-based analysis techniques, like variability-aware type checking, variable control-flow graphs, and variability-aware liveness analysis \cite{LiebigRheinKastner+13}.
The creation of the very detailed variable AST requires much more effort than the previous approach. Parsing of the x86 architecture of the Linux Kernel version 2.6.33.3 with 7,665 C-files (*.h are included through the variability-aware lexer) requires roughly 85 hours on dual/quad-core lap computers with 2 to 8 GB RAM (the authors do not precisely specify their measurement system) \cite{KastnerGiarrussoRendel+11}. This parsing approach has an additional downside beside the massive time consumption. Through the macro expansion and the treatment of statements belonging to different configurations, the variable AST does not represent the developers view on the code anymore.
We surveyed existing traditional and variability-aware code metrics in order to design a Reduced AST. On the one hand, our RAST
contains more information than the approach by Sincero et al., which stores only information about the variation points used in code artifacts. On the other hand, our approach stores less information than the variable AST and, thus, does not facilitate the same code analyses as supported by the TypeChef infrastructure. However, our concept provides an efficient measurement of a large variety of traditional and variability-aware code metrics, which can not be done by any of the previously discussed parsing strategies.
\section{Concept}
\label{sec:Concept}
Here, we present the concept of parsing Reduced Abstract Syntax Trees (RASTs). This was motivated by designing a tailored parsing approach which is able to extract the information needed for the desired static analyses. In our case, we planned a flexible definition of single system and variability-aware code metrics to allow comparisons of them. Based on our survey \cite{El-SharkawyYamagishi-EichlerSchmid19} on variability-aware code metrics and an informal literature study on metrics from traditional software engineering, we came up with the following requirements for parsing RASTs (cf.\ \ref{rq:Classical vs SPL Metrics}):
\begin{enumerate}[label={\textbf{Req\arabic*}},leftmargin=*]\label{req:list}
\item \textit{Parsing of un-preprocessed code}. While established metric analysis tools from commercial vendors usually resolve preprocessor statements before conducting metrics, variability-aware metrics analyze the preprocessor statements. Thus, we require a common data representation for code annotations (in our case C-preprocessor statements) and for elements of the programming language (in our case AST elements of the C-language). This is a challenging task, since the used preprocessor is not part of the programming language and can be used at arbitrary positions inside a code file, independently of any syntax definitions.
\item \label{rq:No syntactically correct AST}\textit{No syntactically correct AST needed}. An important aspect is to which extent the resulting AST-structure needs to support only syntactically correct programs. Contrary to compilation tasks and type checking analyses, we do not need a syntactical correct AST for the computation of code metrics. However, the AST structure should be as close as possible to the actual code structure to simplify the definition of code metrics. Thus, it is still a challenging task to enhance a traditional AST structure with variability annotations, since these annotations may be inserted at arbitrary positions intertwined with AST elements of the programming language.
\item \textit{Granularity of RAST}. For optimization as well as for practical reasons it is important to assess the required granularity of parsed elements. A very fine grained AST, containing representations for all syntax elements of the annotation and programming language, conceptually supports every code metric. On the other hand, this requires much more effort to develop a very comprehensive parsing approach and leads to higher resource consumption. Due to limited development resources, we designed a Reduced Abstract Syntax Tree (RAST), which is sufficient for measuring all planed metrics and may be easily extended to support further metrics, if desired. The granularity of the RAST is driven by the measured elements of surveyed metrics, which we present in Table~\ref{tab:Supported Metrics}.
\end{enumerate}
\subsection{Reduced Abstract Syntax Tree (RAST)}
\label{sec:Concept:AST}
Based on our SLR on variability-aware code metrics \cite{El-SharkawyYamagishi-EichlerSchmid19} and an informal literature study on metrics for single systems, we designed a Reduced Abstract Syntax Tree (RAST) for the efficient measurement of the most relevant traditional and variability-aware code metrics. Our goal is the measurement of C-based SPL implementations.\footnote{The concepts we propose here could also be applied well beyond C.}
Thus, we limited the scope of our analysis to the measurement of metrics on a per-function basis. Figure~\ref{fig:ClassDiagram} presents the main elements of our RAST:
\input{044-controlflow}
\begin{itemize}
\item \texttt{SourceFile}s represent the RAST representation of code files.
\item The \texttt{CodeElement} is the super class of all RAST elements. It stores the line numbers to trace parsed elements back to their location in code files and facilitates LoC-metrics. Further, we store for each element two representations of the condition of surrounding variation points: The \texttt{condition} stores the condition of the innermost variation block, considering conditions of siblings for \texttt{\#elif}/\texttt{\#else}-blocks. For instance, we store the condition \texttt{$A$} of the \texttt{while} statement in Line~\ref{lst:condWhile} of the listing in Figure~\ref{fig:Concept:ControlFlow}. This allows the computation of feature-based metrics on all parsed elements, e.g., \textit{Scattering Degree} metrics. Second, \texttt{presenceCondition} provides an alternative as it stores the full presence condition for the inclusion of the element, also considering all surrounding variation points. For code elements that are not surrounded by any variation points, we set \texttt{condition} and \texttt{presenceCondition} to \texttt{TRUE}.
\item The \texttt{SingleStatement} is the most fine-grained element of the RAST. We do not provide RAST representations for expressions of statements, but we store these elements as unstructured text (\texttt{UnparsedCode}). For instance, in Line~\ref{lst:stmt} we store a \texttt{SingleStatement} with the text ``\texttt{stmt;}'', not knowing whether this is a function call, a variable declaration, or anything else.
\item The \texttt{IUnparsed} element facilitates the storage of preprocessor elements at arbitrary positions inside the RAST. This is required since preprocessor directives are not syntactical elements of the C programming language and may be used at arbitrary positions inside a code file, independently of any syntax definitions. The \texttt{IUnparsed} element is the parent of all (parsed) \texttt{CppBlock}s and \texttt{UnparsedCode} expressions of \texttt{SingleStatement}s.
\item We use \texttt{CppBlock} to store conditional blocks, i.e., variation points. This means, we store \texttt{\#if}, \texttt{\#ifdef}, \texttt{\#ifndef}, \texttt{\#elif}, and \texttt{\#else}-blocks in separate instances, referring to all siblings of the same block structure. The \texttt{type} attribute is used to distinguish between the different preprocessor elements and to allow a differentiation during the computation of metrics, if required. \texttt{CppBlock} inherits from \texttt{IUnparsed}, which is used for elements of \texttt{SingleStatement}s, and inherits from \texttt{CodeElementWithNesting}, which is used as a container inside our RAST. The multiple inheritance allows a nesting of preprocessor directives at arbitrary positions inside the RAST.
\item We use \texttt{BranchStatement}s similar to \texttt{CppBlock}s to store the \texttt{if} and \texttt{else} statements of the programming language. This class also stores the siblings of the same if/else-structure. Again, the \texttt{type} denotes which specific syntax element was used to allow a differentiation during the metrics computation, if necessary.
\item \texttt{LoopStatement}s represent any loop of the programming language. Contrary to \texttt{BranchStatement}s they do not have siblings. Again, we support different loop \texttt{type}s.
\item \texttt{Function}s represent function definitions. The function's signature is stored as \texttt{UnparsedCode}, while the function body is composed of previously described elements.
\item \texttt{Reference} elements are special as they neither represent syntax elements of the programming language nor of the annotation language. They are used in case that syntactical elements of the presented RAST, like loops or control structures, are split into multiple parts by C-preprocessor statements. The listing of Figure~\ref{fig:Concept:ControlFlow} shows an example in which the C-preprocessor is used for the conditional compilation of a loop statement, while the statements of the loop are always present. In this case, a \texttt{LoopStatement} with one \texttt{Reference} is stored inside a \texttt{CppBlock}. This can be seen on the right side of Figure~\ref{fig:Concept:ControlFlow}. The actual statements are stored outside of the \texttt{CppBlock}. This way it is possible to simultaneously define metrics on the same parsed data structure, that consider the nested statements as variable as well as metrics that do not treat this statement as variable.
\end{itemize}
\subsection{Application of RAST}
\label{sec:Concept:Application}
The RAST is designed to preserve the code structure in order to facilitate the computation of variability-aware code metrics according to their original definitions, while reducing the performance overhead by omitting elements that are not required for the metrics.
For this we neither resolve the variability as usually done by commercial metric tool suites nor do we duplicate parsed code elements of alternative variants as done by TypeChef \cite{KastnerGiarrussoRendel+11}, as this would lead to modified metric values.
As a consequence, our RAST contains a 150\% representation of the parsed code.
Parsing of C-code requires the ability to cope with \textit{undisciplined annotations} \cite{LiebigKastnerApel11, MedeirosRibeiroGheyi+18}, which are conditional compilation directives that do not align with the underlying syntactic structure of the code.
Our RAST provides two concepts to support these annotations as described above:
In cases that elements of a statement are conditional, a \texttt{CppBlock} inside a \texttt{SingleStatement} may be used to store the conditional elements.
Further, \texttt{Reference}s may be used to represent conditional control structures.
Based on the RAST, new metrics may be implemented as a visitor,\footnote{According to the visitor design pattern.} which may be parameterized to represent different variations of a metric family.
For instance, MetricHaven uses one visitor to compute different variations of McCabe's Cyclomatic Complexity measure \cite{McCabe76}, which counts the linear independent paths of the (variable) control graph.
For the single system version, we completely ignore the variability of the code and add 1 to the number of visited control structures (\texttt{while}, \texttt{for}, \texttt{if}, \texttt{case}).
According to \cite{ConteDunsmoreShen86} this counting approach is equivalent to the original definition of McCabe.
However, this approach provides support for control structures of undisciplined annotations, since we do not need to compute a syntactically correct control graph.
The left side of Figure~\ref{fig:Concept:ControlFlow} provides an example on how we compute the cyclomatic complexity of a conditional loop.
By ignoring the annotations, we detect two linear independent paths of the resulting control graph.
The variability-aware version of this metric considers only paths created by variation points \cite{Lopez-HerrejonTrujillo08}, i.e., \texttt{\#ifdef}-statements.
Finally, we provide a superimposition of both variants by counting the number of control structures of the programming language and the annotation language.
The right side of Figure~\ref{fig:Concept:ControlFlow} visualizes the resulting control graph, which contains three linear independent paths:
The loop may be present but omitted completely at run-time \textcircled{1}, the loop may be executed \textcircled{2}, and the loop may be removed through conditional compilation but the statements are kept \textcircled{3}.
\section{Realization}
\label{sec:Realization}
Our concept is implemented as a prototype for the analysis of C-based SPLs like the Linux Kernel. This is realized in several plug-ins for the \texttt{KernelHaven}\ infrastructure \cite{KroeherEl-SharkawySchmid18a,KroeherEl-SharkawySchmid18b}. This infrastructure supports three types of extractors to read information from the product line to be analyzed:
\begin{itemize}
\item Code extractors extract variability information from the source code implementing the software product line. For C source code, this typically involves parsing \ifDef-block{}s of the C-preprocessor.
\item Build model extractors extract variability information from the build process of the product line. This typically involves presence conditions that define in which configurations a given source code file is compiled into the product line.
\item Variability model extractors extract the variability model of the software product line. This contains a list of all features and constraints between them.
\end{itemize}
In \texttt{KernelHaven}, the result from these extractors is represented in models that are agnostic to implementation details of specific product lines, while still being extensible to additional information. This allows the following analysis components, which access these models, to be implemented independently from implementation details of specific product lines. The following paragraphs will first introduce which extractors were used in our prototype implementation and then explain the analysis process that implements our approach.
The parsing of the product line source code is implemented in the \texttt{srcMLExtractor}\footnote{\url{https://github.com/KernelHaven/srcMLExtractor}} plug-in for \texttt{KernelHaven}. It is based on the \texttt{srcML}\textsuperscript{\ref{note:srcML}} tool which parses source code to an XML format \cite{collard2011lightweight}. The \texttt{srcMLExtractor}{} parses this XML and converts it into a model compatible with \texttt{KernelHaven}. The extension mechanism of the general code model in \texttt{KernelHaven}{} is used to model the Reduced Abstract Syntax Tree (RAST), as introduced in Section~\ref{sec:Concept:AST}. The parsing process of the \texttt{srcMLExtractor}{} does not parse the full AST output of \texttt{srcML}, but only descends to a granularity that provides sufficient information to build the RAST. See Figure~\ref{fig:ClassDiagram} for a simplified class diagram of the resulting RAST structure.
For the build and variability models, we use the \texttt{KbuildMiner\-Extractor}\footnote{\url{https://github.com/KernelHaven/KbuildMinerExtractor}} and \texttt{KconfigReader\-Extractor}\footnote{\url{https://github.com/KernelHaven/KconfigReaderExtractor}}.
The former extracts variability information from the \texttt{Kbuild} build process of the Linux Kernel, the latter reads the \texttt{Kconfig} variability model present in the Linux Kernel source tree. These plug-ins and their underlying tools were already used in previous analyses of the Linux Kernel, and there were no changes done to these plug-ins when implementing the approach presented in this paper.
The analysis process in \texttt{KernelHaven}{} typically consists of multiple analysis components that are combined to an analysis pipeline. The output of the previous component(s) is used as the input for the following component(s). The initial input for the first analysis component(s) are the models supplied by the three extractors (see above). This structure allows for simple re-use of analysis components when creating new analysis pipelines.
The calculation of metrics is implemented as such an analysis pipeline in the \texttt{MetricHaven}$^{\ref{fn:MetricHaven}}$ plug-in for \texttt{KernelHaven}. Figure~\ref{fig:MetricHavenPipeline} shows an overview of this pipeline structure. The coloring of the lines indicate the flow of the three models extracted from the product line, as described above. The actual metric computation happens in the rightmost component at the end of the pipeline. The input for this component are the three extracted models (the code model went through the \texttt{FunctionFilter} first) and the output of three preprocessing components.
\begin{itemize}
\item The \texttt{FunctionFilter} component splits the code model into individual functions, and removes any elements that are outside of functions (such as global variables). This component does not compute any values for metrics; it is only used for convenient data organization. The result is a stream of code functions.
\item The \texttt{FunctionCallMap} component analyses calls between functions. Since the Reduced Abstract Syntax Tree (RAST) is not fully parsed (cf.~Section~\ref{sec:Concept:AST}), function calls inside statements are identified heuristically: If the unparsed code string of a statement contains a function name followed by an opening parenthesis, we consider that statement to contain a call to this function. For each identified function call, the calling function (caller), the called function (callee), and the presence condition and location of the statement containing the function call are stored. This information is for example used in the Fan-In/-Out and Degree Centrality metrics.
\item The \texttt{ScatteringDegree} component calculates the Scattering Degree metric for all features of the variability model. The result is a map of all features and their values for the different scattering degree types (\SD$_\text{VP}$\ \cite{LiebigApelLengauer+10,CoutoValenteFigueiredo11,JbaraFeitelson13, PassosQueirozMukelabai+18} and \SD$_\text{File}$ \cite{ZhangBeckerPatzke+13, HunsenZhangSiegmund+16}).
\item The \texttt{FeatureSize} component calculates the Feature Size metric for all features in the variability model. The result is a map of all features and the number of statements controlled by the feature.
\end{itemize}
The preprocessing components are executed before the final metric computation component, because they require a full overview of the complete code model. In contrast, the metric calculation component calculates the metric values on a per-function basis. This reduced view on the code model allows to reduce the complexity of the metric calculation component and also helps to mitigate a memory problem. Since our implementation scales to a large number of metrics to be calculated per function, the amount of resulting metric values can grow quickly. In practice, the memory required to store this are several gigabytes. With the per-function approach in the metric calculation component the results of a single function can directly be written to disk, freeing the main memory.
Our implementation offers a number of configuration options. Most importantly, it allows for free selection of metrics to calculate. The user can select anything from 1 up to 42,796 metrics and metric combinations to be calculated per function. Additionally, the extractors at the beginning of the pipeline can be exchanged. This enables our infrastructure to run on different software product lines, while the analysis components require no adaptation. This is because the models used to represent the extraction result are agnostic to specifics of single product lines. Finally, the number of threads used in the code extraction plug-in and the metric calculation component can be configured independently. See the evaluation in Section~\ref{sec:Evaluation} for details on the potential performance improvements.
\section{Discussion}
\label{sec:Discussion}
In the development of any kind of analysis tool, one needs to make a number of tradeoffs. These relate, in particular, to generality (what range of metrics to support), speed, and range of supported artifacts.
The key principles we used for developing the solution, described here, are:
\begin{itemize}
\item \textit{Genericity:} It should support a large range of product line metrics.
Basically, it should support all metrics that were documented as product line metrics so far and at the same time, it should support a very large (though not necessary complete) range of single system metrics.
\item \textit{Scalability:} It should be able to deal with very large product lines and very large numbers of metrics simultaneously.
\item \textit{Performance:} It should perform these tasks very efficiently.
\end{itemize}
The key innovation was to introduce the solution of a reduced abstract syntax tree (RAST) and to tailor it very well to the task at hand. This avoids the representation of language details that are not relevant to the metrics analysis and abstracting those that are relevant as far as possible. A detailed analysis of the representational needs of the metrics provided the basis of our design.
At this point also some trade-offs had to be made. As a result, we do not support all kinds of metrics. For example, we do not support Halstead metrics \cite{Halstead77}.
In order to achieve high performance, we create the metrics values nearly completely in a single pass. This leads to significant performance improvements as technical properties like CPU caching are used in an optimal way and the generation of the data structures does not need to be made multiple times.
The exception is the approximation of Eigenvalue Centrality, which requires a two-pass approach.
Overall, we managed to get an extremely high performance, using our approach along with very good scalability properties, both in terms of the number of metrics and analyzed code size. For example, in our evaluation we found 6 hours 20 minutes for analyzing the complete Linux Kernel (cf.\ Section~\ref{sec:Evaluation}), which means that about 0.76 seconds were required per metric. Or, to put it differently, for producing 29,976 metrics, we needed about 0.06 seconds per function. We regard this as an extremely strong performance, although it is very difficult to compare as there is no other metrics tool that supports a similarly wide range of metrics.
\section{Evaluation}
\label{sec:Evaluation}
We ran our prototype implementation on the x86 architecture of the Linux Kernel version 4.15\footnote{https://mirrors.edge.kernel.org/pub/linux/kernel/v4.x/linux-4.15.tar.xz} to evaluate the implementation of our concept. Based on the RAST presented in Figure~\ref{fig:ClassDiagram}, we were able to realize various single system and variability-aware code metrics \cite{El-SharkawyKrafczykSchmid19}. Through the combination of variability-aware code metrics with feature metrics we support 42,796 metric variations (as of Summer 2019), which can be measured in a single pass (cf.\ \ref{rq:Efficient}). While most metrics can be implemented in a straightforward manner, some implementations require adaptations for our RAST. For instance, the detection of function calls for the Fan-In/-Out metric can only be implemented heuristically, since it would require full parsing of the expression syntax (cf.\ Section~\ref{sec:Realization}).
The x86 architecture in the Linux Kernel has 20,356 C-source files that are evaluated by us.
106 ($\approx 0.5\%$) of these files, cannot be handled by the current implementation of the \texttt{srcMLExtractor}, which translates the XML output of \texttt{srcML}{} to \texttt{KernelHaven}'s code model.
This is mostly related to very special corner cases of \textit{undisciplined} C-preprocessor usage \cite{MedeirosRibeiroGheyi+18}, i.e., conditional annotations that do not align with the syntactic structure of the code.
\texttt{srcML}{} marks up the C-syntax independently of C-preprocessor directives.
In conjunction with \textit{undisciplined} C-preprocessor directives, this can lead to incorrect markups provided by \texttt{srcML}{}.
Proper handling of those structures requires adaptations of the \texttt{srcML}{} parser.
Some special cases are detected and fixed by our \texttt{srcMLExtractor}.
However, this approach cannot repair all of these cases and requires significant development effort.
Further, a minority of these corner cases cannot be mapped to our RAST at all as they violate the few structural assumptions of the RAST.
An even more lenient RAST that can model these cases, however, would significantly complicate the definition and computation of metrics.
We ran two sets of metrics on the Linux Kernel, to evaluate scalability of our approach (cf.\ \ref{rq:Efficient}): First, a selected subset of metric combinations that we also used in practice for our own work with the Linux Kernel. We call this set \textit{atomic metrics}. It contains all basic code metrics and all possible combinations of code metrics combined with a single feature metric. Code metrics combined with multiple feature metrics are not included. This results in a set of 648 metrics. Second, we allowed all metric combinations. However, the implementation of one metric family (approximation of Eigenvector Centrality) requires significantly more memory than the other implementations and is not optimized for the provided parallelization capabilities of MetricHaven. For this reason, we executed only the 148 metric variations of this metric, which were already executed as part of the atomic metrics, while we executed all variations of the remaining metrics. This results in a set of 29,976 metrics.
For the performance measurements, we ran our implementation in a virtual machine running Ubuntu 16.04 with 40 logical CPU cores of an Intel Xeon E5-2650v3 @ 2.3 GHz and 314 GiB RAM. For the analysis, we limited the JVM once to 50 GiB memory\footnote{via the command line switches: -Xmx50g -Xms50g} and once to 24 GiB. However, it must be noted that the extractors run in separate processes as they execute external tools and, thus, allocate additional memory. Accurate timings for specific phases are hard to measure since KernelHaven makes heavy use of parallelization. For example, the preprocessing components already start to run while code parsing is still running. However, the actual metric calculation component can only start when the complete code model has been passed through the preprocessing components. That means that we identify two distinct execution phases: code parsing and metric calculation. The preprocessing phase, which partially happens in parallel to the code parsing only takes a few seconds, which is insignificant compared to the total runtime. Thus, we do not supply measurements for this phase. The code parsing and metric calculation components can also be independently configured to use a specified amount of threads. We measured the runtime of these components for different numbers of configured threads.
For the experiments, we used a range of 1 to 10 threads to cover a spectrum which is supported by most workstation computers.
Running the 648 \textit{atomic metrics} on all 409,253 functions that we parse from the x86 Linux Kernel architecture produces 265,195,944 measures. In CSV format, this is about 1.2 GiB. The metric calculation step takes about 27.5 minutes (54 minutes with 25 GiB memory for the JVM) to run on a single thread and can be decreased to about 13.75 minutes (38.5 minutes) on 10 computation threads.
Running the large set of 29,976 metrics on all 409,253 functions that we parse from the x86 Linux Kernel architecture produces 12.2 billion measures.
In CSV format, this is about 53 GiB.
The metric calculation step takes from 11 hours and 26 minutes on one thread to 6 hours 15 minutes on 10 threads (6 hours 20 minutes total runtime).
The parsing performance stays the same as described in the previous paragraph as it is independent from the number of computed metrics.
The parallelization benefit of metric computations is slightly less pronounced compared to the \textit{atomic metrics}.
However, our prototype is not fully optimized with regard to parallelization.
\begin{figure}[tb]
\centering
\includegraphics[trim={0.04cm 8.5cm 20.95cm 0cm},clip,width=\columnwidth]{figures/Timings.pdf}
\Description{Benchmark shows performance of MetricHaven to measure 648 metric variations on the Linux Kernel.
The parsing of code files takes from about 16 minutes and 20 seconds on 1 thread to about 2 minutes on 10 threads.
The metrics runtime takes about 27 minutes and 39 seconds on 1 thread to about 13 minutes and 14 seconds on 10 threads.
The total runtime takes about 44 minutes and 11 seconds and goes down to 16 minutes on 10 threads.}
\vspace*{-18pt}
\caption{Parsing/computation/total runtime of 648 \textit{atomic metrics} with different number of threads.}
\label{fig:ExtractorTimings}
\end{figure}
Figure~\ref{fig:ExtractorTimings} visualizes the runtime of running the 648 \textit{atomic metrics} on a different number of threads with 50 GiB memory.
The number of threads for the parsing and the metric computation phase were both modified together on a range of 1 to 10.
For each run, we present the parsing time (lower, blue bars), the metrics computation time (upper, orange bars), and the total execution time (numbers above), which requires some additional time for loading the plug-ins and writing the results.
The gray, dotted line indicates the trend line of the total execution time.
Running the experiments with less memory results in a similar behavior.
However, the performance gain of configuring more threads is much smaller compared to the experiment with 50 GiB memory, because of the increased workload of the JVM garbage collector.
Please note that MetricHaven keeps the parsed RAST of the complete Linux Kernel in memory when it starts its computation.
\section{Conclusion}
\label{sec:Conclusion}
In this paper, we presented the concept of MetricHaven, for simultaneously evaluating a large number of metrics on very large product lines in a highly efficient manner. MetricHaven supports more than 42,000 metrics, requiring less than 0.06 seconds for computing a selection of 29,976 metrics per function of the Linux Kernel leading to a total execution time of about 6 hours and 20 minutes for analyzing the whole Linux Kernel, yielding more than 53 GB of metrics data.
The approach is highly customizable (achieving significant speed-ups when reducing the number of metrics to process).
In particular, beyond analyzing product line metrics, it is also capable of creating a significant range of single system metrics.
The key to achieving these capabilities, was first to identify key realization requirements as required in \ref{rq:Classical vs SPL Metrics}. In Section~\ref{sec:Concept}, we introduced three core requirements that enabled us to create this approach: we had to directly parse un-preprocessed code (1), yielding an AST which is not a syntactically correct representation (2). Further, we had to minimize the required information by abstracting the information to a significant extent, leading to a rather coarse-grained AST with reduced information (RAST) (3).
For \ref{rq:Combination of Metrics}, the answer was actually rather simple: by having an integrated representation that does not replicate basic code elements (as, for example, some approaches to handling variable code do \cite{KastnerGiarrussoRendel+11}) and integrating the variability given by the preprocessor information, we could simply handle the subset of non-variable information also from a metrics point of view.
We addressed \ref{rq:Efficient} by creating the notion of a reduced abstract syntax tree (RAST). The abstraction level of this is tailored to exactly the level of detail required for handling all relevant product line metrics. All further information is skipped, respectively, not parsed in detail. We described this in detail in Section~\ref{sec:Concept:AST}.
In future work, we plan to further extend this framework in terms of the range of supported metrics, improve its performance and apply it to study numerous properties of product line implementations. We are particularly interested in the prediction of defects based on product line metrics.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,364 |
Text Copyright © 2009 Hatherleigh Press
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Library of Congress Cataloging-in-Publication Data
How to hug a porcupine: Easy ways to love the difficult people in your life.
p. cm.
ISBN 978-1-57826-293-9 (alk. paper)
eBook ISBN: 978-1-57826-537-4
1. Interpersonal conflict. 2. Love. I. Hatherleigh Press. BF637.I48H69 2008
158.2–dc22
2008040086
v3.1
# CONTENTS
_Cover_
_Title Page_
_Copyright_
Foreword
A Note from the Publisher
[PART I
THE NATURE AND BEING OF THE PORCUPINE](Elli_9781578265374_epub_p01_r1.htm)
The Porcupine in Nature
[PART II
A BASIC GUIDE TO UNDERSTANDING PORCUPINES](Elli_9781578265374_epub_p02_r1.htm)
[PART III
WHERE PORCUPINES DWELL](Elli_9781578265374_epub_p03_r1.htm)
Porcupines at Work
Porcupines at Home
Porcupines Out and About
[PART IV
THE PORCUPINE WITHIN US ALL](Elli_9781578265374_epub_p04_r1.htm)
A Last Word
Resources for Porcupines
About the Author
# FOREWORD
By Dr. Debbie Joffe Ellis
This book is a powerful tool.
In fact, it is a tool so powerful that, if we practice its recommendations, it can lead us towards harmony, compassion and a better world.
How?
This book reminds us that we humans have the capacity to create our attitudes and emotions, as well as our actions. If we choose to, we can maintain inner peace despite difficult outer circumstances. We can prevent unnecessary suffering.
If you allow your quills of judgment to turn inwards and pierce you, this book can help you to foster self-acceptance. If you make yourself angry at others who prick you by acting in harmful and provocative ways, the strategies offered in this book will enable you to calm and stabilize yourself. And if you mistakenly allow yourself to be pierced by others, the wisdom of the words in the book reminds you not to take it too personally!
With thoughtful, easy to follow strategies, _How to Hug a Porcupine_ shows you how to embrace the Porcupine tendency, while still respecting the quills. You can learn to unconditionally accept yourself as well as others.
The brilliant pioneering psychologist, Albert Ellis PhD, taught that no one can upset us unless we allow them to. Instead we can choose to think rationally, feel calm, and act with consideration, kindness and empathy. We can learn not to think and act irrationally and not to automatically stick our quills out during threatening or provocative situations. Even if the Porcupinely-acting person who you are dealing with resists, your open-hearted approach will not be wasted. Every time we practice kindness, compassion and unconditional acceptance of others, we are reinforcing it within and for ourselves.
_How to Hug a Porcupine_ makes change for a lifetime possible. This is a book that can be picked up, enjoyed and implemented at any time. In fact, it will benefit us tremendously to consult it regularly. We all benefit from reminders, and repeating these exercises will erode persistent negative tendencies. Keep practicing these principles over and over again. Share this book with others, and set an example by modeling its principles.
Life is brief. Time is precious. Wasting it in defense and attack, or in anger and fear, is regretful. Choose instead to practice patience, empathy, compassion, kindness, understanding and unconditional acceptance. Work towards creating greater harmony within yourself and in your relationships, and you will contribute to creating a healthier, saner world.
This book can be used as a dose of preventative medicine, a first-aid kit, and a healing balm, but more than anything else, it is an effective tool for emotional and mental health and well-being. Turn to it to encourage greater awareness of heart and mind.
Use it in your own life, and you will bring stability, peace and joy to yourself, others and the world around you.
— DR. DEBBIE JOFFE ELLIS
# A NOTE FROM THE PUBLISHER
We all know people who are difficult to be around. We may run into them at work, at home, through friends, or simply whenever we are going through our day.
Unfortunately, those "porcupines" are not going to go away!
On the following pages you will find strategies for dealing with those prickly personalities. This is important not only because we simply don't have a choice but to deal with porcupines, but also because it is actually _good_ for us to find ways to get along with those who are different from us.
Other people challenge us. Other people improve us. Other people make us better human beings.
— ANDREW FLACH, PUBLISHER
# PART I
THE NATURE AND BEING OF THE PORCUPINE
"The supreme happiness of life is the conviction that we are loved — loved for ourselves, or rather, loved in spite of ourselves."
— VICTOR HUGO
# WHY PORCUPINES?
For centuries, cultures from around the world have used animal characteristics to identify and describe human personality traits. In Native American culture, for example, everybody in the community must undergo a mystical rite of passage to identify their animal spirit. This animal then becomes part of their name (examples include "Sitting Bull," and "Little Turtle"). The Chinese Zodiac assigns one of 12 animals to every year, and an individual's personality is believed to be reflected in the attributes of the animal that was appointed the year that one was born. The animal characters in Aesop's fables embody different attributes of human behavior, and their stories and predicaments offer lessons on human nature. Today, expressions like "stubborn as a mule," "wise as an elephant," and "sly as a fox" are still sprinkled throughout conversation.
Whether these expressions, stories, or themes are accurate or not, the way mankind draws links between animal behavior and human behavior reveals a great deal about our struggle to understand ourselves, and each other. This book seeks to help you, the reader, understand and cope with a certain type of person: those individuals who are difficult and challenging. We call these people "porcupines."
Why? To answer that, we need to look at the actual behavior and habits of the porcupine.
## THE PORCUPINE IN NATURE
Porcupines are rodents whose coats feature modified, spiny hairs known as quills. These quills are embedded in the skin, and a single porcupine may have over 30,000 quills in his coat. There are 27 species of porcupine, divided into two families: those found in the Old World (Europe, Asia, and Africa) and the New World (the Americas and Australia). Porcupine ancestors date from 30 million years ago. Old World porcupines live on the ground, while New World porcupines are avid tree-climbers. Although some species of porcupine eat small reptiles and insects in addition to a diet of bark, roots, fruit, meadow grass, and tubers, most species of porcupines are strictly herbivores (they eat only vegetables). Unlike most herbivores, who have to seek safety in numbers, the effectiveness of the porcupine's quills as a defense against predators allows it to lead a solitary life.
Even today, porcupines are generally misunderstood. Contrary to a legend which dates back to the Greek philosopher Aristotle, porcupines cannot throw or fire their quills. Nor are their quills poisonous. Their name itself is a misnomer; it derives from the Latin words _porcus_ and _spina_ , which together mean "thorny pig"—porcupines are neither thorny, nor are they related to pigs (they are, in fact, most similar to beavers in habitat, diet and temperament).
Porcupine babies, called porcupettes, are born with soft, pliable quills. Within a matter of hours, these harden into defensive weapons.
## THE PORCUPINE, THREATENED
When porcupines are threatened, tiny muscles at the base of each quill tighten, causing their quills to stand up and the porcupine to appear much larger than it really is. Then the porcupine will rattle its quills, grunt, growl, and stamp its feet, all in an effort to scare off its attacker. If the predator persists, the porcupine will turn its back, raising its tail and crouching defensively. Finally, the porcupine will charge backwards, whack the predator with its tail or throw its body at its attacker. This often leaves a handful of quills embedded, painfully, in its opponent. Over time, the quills can work their way into the flesh of the attacker and cause infection.
Despite its unappetizing quills and highly developed counter-attack, the porcupine is still vulnerable to larger, carnivorous predators. One predator, the American fisher, has evolved a technique for getting around the porcupine's quills; it flips the porcupine onto its back, exposing its tender—and undefended—underbelly.
## HOW ARE PEOPLE LIKE PORCUPINES?
Human porcupines would probably be indistinguishable from anyone else, except for their reaction to any kind of threat or incursion. It's then that, like the porcupine in nature, they puff themselves up to try to scare off their adversary.
Unfortunately, this defensive behavior in the human porcupine is "too little, too late" in many ways. That is, the human porcupine only reveals himself when you've already stepped into his bubble—and he is already on the defensive. How do you make up for encroaching on his territory, fix the situation, and determine the best way to avoid future confrontation?
Learning more about the human porcupine, and adapting your behavior, is the first step.
# PART II
A BASIC GUIDE TO UNDERSTANDING PORCUPINES
"Love one another and you will be happy. It's as simple and as difficult as that."
— MICHAEL LEUNIG
#
What follows are some specific techniques to help you not only survive, but thrive, in your encounters with human porcupines. No matter where you encounter a porcupine, whether you are meeting him for the first time or have known her for years, having some reliable strategies in your back pocket can make all the difference in the world. With knowledge and the right attitude on your side, you can not only make the best out of a bad situation, but, with practice, learn how to avoid painful spots altogether. Here are ways to cope with any prickly porcupine with humor, wisdom and practicality.
##
**Learn Your Porcupine's Warning Signs**
For human porcupines, their "quills" are most often their words, delivered aggressively with fierce language, and often in a loud voice. Learn to recognize these warning signs so that you can be one step ahead when your porcupine gets aggressive.
##
**Keep Your Quills In**
A porcupine's defensive behavior can be contagious. During a disagreement with a porcupine, many people will resort to their own defensive mode. Don't! There is a _very_ big difference between trying to understand what is bothering your porcupine and getting into a big fight.
_So, take a step back. Take a deep breath. And try again_.
_Don't let your own quills get the best of you_.
##
**Respect Your Porcupine's Boundaries**
The attack of a porcupine is a last resort of a frightened, cornered creature trying to keep a threat at bay. We should always remember this. Because a porcupine often attacks out of fear, respecting his boundaries will keep him from lashing out.
##
**Consider Your Porcupine's Real Needs and Fears**
It's important to remember that human porcupines, like porcupettes, are born soft. But bad experiences, fears, and failed relationships have forced them to harden their exteriors and sharpen their quills.
We should keep in mind that a porcupine's quills—the result of past injuries—are a part of who the porcupine is, and they aren't necessarily meant to hurt us. This perspective helps us relate to the porcupine and provides us with the understanding we need to successfully approach her.
##
**Find that Soft Spot!**
Even porcupines have soft spots—their belly! Keep in mind that although it may require some close attention and careful strategy, you can find your human porcupine's emotional "soft spot." This will be a topic that, no matter what, brings a smile to his face and makes him feel good whenever he talks about it. It could include a passion, a favorite hobby, or anecdotes about a loved one.
Learn what subjects make your porcupine feel joyful and bring them up—you'll make him feel special and he'll see that you really care about how his life is going.
##
**Be Empathic**
Loving a difficult person requires a great deal of empathy. Take the time to ask yourself: What is that person feeling? How would I feel if I were inside those quills? What might it be like to deal with someone like _me_? Being responsible and loving towards a porcupine requires emotional maturity and the flexibility to think empathically.
##
**Pay Attention**
Pay attention... a lot of attention! If you know your porcupine doesn't like loud music or spicy foods, then exposing her to either one is just asking for trouble. Sometimes, avoiding conflict can make a world of difference!
##
**Get to Know Your Porcupine's Likes**
Everybody wants to feel special; porcupines are no different. If you know your porcupine likes to relax with a glass of wine after a hard day at work, try to have one ready for her when she gets home. Does your porcupine love to watch Sunday afternoon football uninterrupted? Make plans to steer clear of the TV so he can enjoy this time to himself.
Thinking of a porcupine's needs ahead of time is half the battle. Nothing disarms a porcupine faster than showing her that you care!
##
**Make Your Likes and Dislikes Clear, Too**
Loving a porcupine is a two-way street. It's a process of communication, education and awareness—on behalf of both parties. This means paying attention to ourselves, too. It's our responsibility to make our needs clear to our loved ones, just as it is their responsibility to do the same with us.
##
**Try to "Speak Porcupine"**
The best way to deal with a defensive person is to try to get him to talk to you about what gets his quills up. Often the best way to do that is to tell your porcupine what gets _your_ quills up. "Speaking porcupine" in this way will show him that you can relate to what it is he is going through and the frustration he may be experiencing.
##
**Share with Your Porcupine**
Sometimes, exposing your own vulnerabilities can be your best defense. By admitting to your fears and apprehensions, your porcupine will see that you both have weaknesses. This will open up a door of communication.
_Sharing will help your porcupine feel safe enough to share her own vulnerabilities with you_.
##
**Keep a Safe Distance**
Don't rush towards a porcupine's quills unless you are prepared to handle the encounter (remember, quills can't hurt you unless you run up against them!) Learn how to avoid rushing onto or against your porcupine's quills until you are ready, or until he has calmed down a little.
##
**Don't Take it Personally**
Chances are, your porcupine's sour mood, though _triggered by_ you, has little or nothing to do _with_ you. The more you recognize that a porcupine's aggressive behavior is really about your porcupine figuring out her own issues, the more energy you will have to find a solution that works for the both of you.
##
**Avoid Porcupine Strangers so You Can Focus on Your Porcupine**
When you are working hard to understand a porcupine who is close to you, avoid those that you don't know. You simply won't have enough energy leftover to handle your porcupine.
So, step one: _avoid porcupine strangers if you can_. If you are caught in a sneak attack (these porcupines include an angry driver you encounter in traffic, or the surly store attendant who is really having a bad day), do your best to ignore their bad mood. _Save your own energy for the porcupines closest to you!_
##
**Deflect an Attack with Kindness**
Porcupines attack when they feel threatened, so in order to deflect that defensiveness, try a little kindness instead. By using kind words and expressing a generous and caring attitude, we can defuse the porcupine's apprehensions.
Once he knows you're not a threat, your porcupine will retract his quills and peace can be restored.
##
**Use Your Porcupine's Name**
Everybody is calmed by the sound of their own name. In soothing tones, reassure your porcupine it's going to be all right.
##
**Plan Ahead**
As with most things in life, it helps to be prepared. Before a potential confrontation, decide how you are going to deal with your porcupine and stick with the plan. Having a course of action will help keep you from becoming defensive, too.
##
**Don't Get Angry**
Patience is a virtue. So is understanding. Dealing with a porcupine requires a great deal of both. Resist the temptation to get angry and, instead, approach any confrontation or dilemma with patience and understanding above all else.
##
**Don't Get Frustrated**
Avoid bringing your own issues "into the ring" with your porcupine. If you do, you risk becoming very frustrated, and frustration means your porcupine's gotten you hooked—in other words, it is just a matter of time before you start acting as out of control as your porcupine.
_So take a deep breath and rebalance. Let your porcupine rant—you've got calm and wisdom on your side_.
##
**Be Thick-Skinned—and Cold-Blooded!**
If you are thin-skinned, you will feel every prick of your porcupine's anxiety. Try to be thick-skinned instead. And don't react off-the-cuff. Instead, go slow. Be a tortoise! Your porcupine will burn through her energy quickly, and once he does, you will be on top. Slow and steady wins the race!
##
**Stay in Control**
The stresses, anxieties and bad habits you bring to any encounter with your porcupine are a huge factor in determining how things will work out between the two of you. Remember, your relationship with your porcupine is a two-way street, and you must be a careful negotiator. How you behave, how you speak, react—even smile—can shape the outcome. So, no matter what, stay in control, and don't let your emotions get the best of you.
##
**Stop!**
All right, so you are locked in an angry phone conversation with your porcupine. You're each saying things you don't really mean. What should you do?
STOP!
Would you run at a bull waving a red flag? Practice your backstroke in a pool filled with piranhas? No. So why would you go _mano a mano_ with a porcupine? If your porcupine is backed up, quills quivering, back off.
_Pause and take a moment. Wait until things calm down before continuing_.
##
**Get Advice**
One of the best resources for help dealing with difficult or defensive people is—well, other people! Ask friends, co-workers, or neighbors for their perspective. Chances are, each and every one of them has a porcupine in their life. Talking will help you get some things off your chest and perhaps provide you with some strategies, too.
##
**Let Your Porcupine Vent**
Everybody needs to vent. So let your porcupine rave if she needs to. In fact, encouraging her to "get it all out" can be an important first step towards change. After all, part of your porcupine's reason for being so spiny could be because she hits a roadblock whenever she has to express her feelings. It can be frightening to engage in a dialogue with someone else when you don't understand emotional language. So help her learn by encouraging her to get everything off her chest.
_Remember: you can help. Your love and attention are the crucial first steps to calming your porcupine_.
##
**Give Your Porcupine Time**
Your porcupine's needs and fears go way back. Uncovering them and dealing with them is a time-consuming process. Don't rush things. Take the time that is needed. Your patience will be reassuring.
##
**First Deal with Your Porcupine's Feelings... Then with the Problem**.
Too often, we focus on behavior, instead of the motivations _behind_ that behavior. This is a crucial mistake, for it's in fact the motivation behind action that most needs to be addressed.
When confronting your porcupine, consider his motivation first. Was he feeling afraid prior to acting out? Is so, what made him feel that way? Thinking along these lines will help you ascertain how you can make him feel less threatened in the future.
Don't forget: the porcupine's defensiveness is an _emotional reaction;_ defuse the emotions and you'll defuse the reaction.
##
**Make Your Porcupine Feel Safe**
The best way to get your porcupine to retract her quills is to reassure her that she can trust you. Communicate with your porcupine openly, honestly, and empathically. This will help her understand that if there is an issue, that its roots must lie elsewhere.
_Remind your porcupine that she is safe with you_.
##
**Be Specific**
Usually when we generalize about another's behavior ("You're always late!" "You never tell me how you feel!"), it's because we're missing the point. It's easier to tackle your porcupine's issues if you don't speak in grand, sweeping terms. Instead, try being more specific. What has your porcupine done that bothered you? Start with that. Be clear about why and how it bothered you. Feel free to use examples, as long as you don't get critical.
Now, it may be impossible for your porcupine not to feel defensive in this situation. But being specific and concise about how a given exchange made you feel will give your porcupine plenty to chew on. Your clarity will limit your porcupine's reaction.
##
**Think: Compromise!**
Any heart-to-heart with your porcupine involves emotional work on your part. This is a good thing. A relationship that requires effort on the part of only one party is not a true relationship. For better or for worse, this means that you'll have to make some compromises. A compromise might mean making some changes to your own behavior. This will communicate to your porcupine that you are as good as your word, and that you are claiming responsibility for your 50% of the relationship.
##
**Ask, Don't Assume**
Fears and phobias are matters of perspective. So if you are unsure about where your porcupine might be coming from, don't make the mistake of coming to the table with assumptions about what's driving her behavior. Instead, allow her to explain her behavior in her own words. She will be far less defensive (your curiosity is a good thing) and you might just learn something!
##
**Don't Blame**
Nothing brings out the porcupine in each of us faster than being blamed. This is because blame implies that one person has to accept sole responsibility, and this, in turn, eliminates any chance for understanding and compromise. To a porcupine, blame is the opposite of trust and communication. For this reason, blame is one of the things porcupines fear most.
So, when confronting your porcupine, avoid blaming him. This means never using "you" or "your" ("when _you_ did this," etc); otherwise he will get defensive immediately. _Be very specific about how your porcupine's behavior affects you, but be expressive about your feelings, and don't place blame or pass judgment_.
##
**Don't Try to Win**
There are no winners and losers in a relationship. The same goes for discussions with your porcupine. If you're trying to "win," you're sure to lose.
This doesn't mean that there is no such thing as a "victory" when it comes to relationships with porcupines. Any discussion that promotes honesty, openness and a willingness to talk and change, is a "victory" for you and your porcupine. After all, you are a _team_.
##
**Respect the Porcupine, if not the Behavior**
Anyone approaching a human porcupine has to be properly prepared. An essential part of that preparation is the ability to differentiate between your porcupine and his behavior. They are not the same!
While we may not always agree with what our porcupine does, we should never lose sight of why we care for him. We can disagree with or be hurt by our porcupine's behavior, but the porcupine remains front and center in our concerns. We may be disappointed by the deed, but we cherish and respect the doer.
_Show care and respect in all your interactions with your porcupine_.
##
**Don't Be Manipulated**
When cornered, who hasn't been defensive, blamed someone else, lied, or created a diversion? This strategy is, unfortunately, human nature, and this means of behavior is especially common in porcupines. So, when dealing with your porcupine, be prepared for psychological strategies you haven't faced since the playground. " _I'm_ difficult? What about _you_?" (This kind of cyclical argument really isn't all that different from "I'm rubber and you're glue"). When this starts, be ready for it, and resist. Don't let your porcupine pull you into this counterproductive back and forth. These tactics are fear-filled ways to dodge the question, and they will get you nowhere.
Instead, stay calm and in control.
##
**Set an Example**
This is another way of saying that, when dealing with your porcupine, you need to set a good example. So, just as you should avoid being defensive, you should also try to be as communicative, understanding and generous as you'd like your porcupine to be.
Setting an example shows that you are committed to change for the both of you.
##
**Think: Socrates**
The "Socratic method" of teaching (named after the Greek philosopher Socrates) depends not on lectures, but questions. By asking the right questions, you can lead porcupines around to your point of view... and make it seem like they got there all by themselves!
Instead of railing about your porcupine's behavior or reactions, ask her for his version of things. Questions like: "What did it feel like when you did this?" or "How did you feel when this happened?" show that you are paying attention (this will be welcome), and, by not limiting the discussion to your take on things, you will encourage him to open up. The next thing you know, the two of you will have revealed aspects of your porcupine's personality that may not have been known to either one of you.
##
**Be as Good as You Can Be**
As should be clear by now, dealing with your porcupine will require your very best effort. You will need to be patient, kind, and loving. You will have to be generous, understanding and empathic. In short, to love a porcupine, we must be the best that we can be.
##
**Don't Be Dismissive**
It's not your place to agree or disagree with someone else's feelings. So don't dismiss your porcupine's fears, anxieties or concerns. If she feels them—so much so that it affects their behavior—then those feelings must be respected, and dealt with.
##
**Don't Demand**
Your way of dealing with a certain situation may not be right for your porcupine, and you must accept that. In other words, don't confuse "we're sharing" with _demanding_ that your porcupine share. Solutions may seem crystal clear to you—but that doesn't matter. _What matters is what is best for your porcupine. Accepting this limitation is essential to finding a way to successfully communicate with your porcupine_.
##
**Don't Interrupt**
Interruptions are power-plays. They're ways of suggesting that what you have to say is more important than what someone else has to say. This can be particularly dangerous when it comes to dealing with a porcupine—not to mention inconsiderate.
Let your porcupine speak. Be polite and hold your tongue until it is your turn to share.
##
**Talk, Talk, and More Talk...**
Let your porcupine speak and encourage them to keep speaking by asking questions. A talking porcupine is too busy thinking and feeling to get defensive. Even better, a talking porcupine will give you clues about what makes him tick. You can use this information in the future to relate even more fully to your porcupine.
##
**Accept Responsibility, Then Apologize**
Apologies can be suspicious. After all, there is really no commitment behind the words, "I'm sorry." What's really required of an apology is responsibility. That is, we need to first show that we understand what we did or said, and then "own" up to it. Then, and only then, can we give an apology that is truly heartfelt. We may have to face some consequences as a result, but meaningful apologies are the only way to move forward.
##
**Communicate Your Needs Clearly**
You can't accuse your porcupine of "stepping over the line" if you've never taken the time to draw that line. In other words, until you clearly state your needs or boundaries, your porcupine cannot respect them. The first step is yours: state your limits.
##
**Abandon the Useless Ambition to Be Right**
No one is right all the time. In fact, most of us aren't right much of the time. Imperfect as we are, we nevertheless take great pride in suggesting to those around us that we have everything in working order—and we further insist on our own perfection by declaring that the problem isn't with _us_ , it's with _someone else_.
This kind of defensive thinking is a recipe for disaster when it comes to your relationship with your porcupine. Remember: there's plenty about ourselves for each and every one of us to work on and change; plenty of bad habits to overcome.
_Stay humble. Obsessing about being "in the right" will lead nowhere_.
##
**Be a Good Companion**
The greatest gift you can give your porcupine is the sense that she is not alone. If she is facing a tough time or is confronted with something she's not prepared for, let your porcupine know you'll always be there to help her. This will give her the reassurance she needs to deal with difficult issues and will strengthen the bond you have with her.
_Remember: you are on the same team!_
# PART III
WHERE PORCUPINES DWELL
"I always prefer to believe the best of everybody, it saves so much trouble."
— RUDYARD KIPLING
We've laid out some general strategies for loving your porcupine. Now it's time to get much more specific. A porcupine's behavior changes with his or her environment, so the right way to deal with a porcupine at work, for example, is very different from coping with a porcupine at home.
In this next section, we will offer strategies aimed at porcupines in their various surroundings.
# PORCUPINES AT WORK
Monday through Friday, most of us spend more time with our co-workers than we do with family and friends. That can make going to work extremely interesting... and potentially exasperating.
Why? Because it's the rare workplace that doesn't have a porcupine or two among its employees. These include the shouting bosses, chronic complainers, non-communicative co-workers and those pessimists who seem incapable of seeing the upside of anything. So what do you do when it feels like a workplace porcupine is making every working hour miserable? How do you handle a porcupine who you have to work with, like it or not?
Don't give up. There are ways to make the workplace more manageable, so you can enjoy your job—and even the people you work with—again.
##
**Stay Firm**
It is the nature of a porcupine, as we have seen, to escalate disagreements into full-blown conflicts. Most often, this occurs when your porcupine feels he needs to defend his position. It follows that, when your porcupine is most defensive, it probably means it is because you've hit a nerve. In other words, you have raised a workplace issue that needs to be addressed.
So the next time the boss is shouting and his quills are up, don't back away! If your complaint or concern is legitimate, then you have every right to air it. What's more, you have a responsibility. No matter how high the noise level gets, if you are in the right, stay firm.
##
**... But Don't Be Stubborn**
Remember to be firm, not stubborn. Firmness means using all your diplomatic skills to find a way around your porcupine's defenses to a solution that will work for everyone (not just you). If you are steadfast, confident, an open-minded to all possible solutions, you will not only survive the encounter, but you will also provide your porcupine and your co-workers with an excellent example of the best way to be considerate towards each other.
##
**Ask Good Questions...**
What is a good question? Well, if your boss explodes when you mention a minor problem, it may be because a bigger issue is looming, such as a deadline. Set the small stuff aside and get to the heart of the matter with a smart question. Why not say, "You seem anxious about the deadline. I am, too. What can I do to get us back on schedule?"
_A direct question can cut through your porcupine's anxiety and help the two of you move forward_.
##
**... And Stick with the Topic**
Once you ask the question, porcupines, being true to their nature, may give you a defensive, accusatory answer. Remember: stay firm. Fight the urge to get defensive in your own right. Instead, ask another question. It will soon be apparent that you are trying to help—not accuse or expose—and any good boss will soon recognize this and back down.
A good question, posed carefully, will lead to a good solution.
##
**Listen**
By listening, you are transformed from a potential adversary to an ally. By listening, you become someone to be trusted. By listening, you cease to be part of the problem and become, instead, someone who has the information to offer a solution.
##
**Try to See Things Through Your Porcupine's Eyes**
Chances are, your porcupine's anxiety is legitimate. Once again, your empathy is your best strategy. Try to imagine how you would feel, whatever the challenge. Share that feeling with your porcupine. When she understands that she's not alone, her defensiveness will lose some of its steam.
##
**Be Supportive**
Workplaces and jobsites can be fiercely competitive environments and bring out the worst in any porcupine. Avoid repeating a pattern of accusations followed by self-defense. Instead, use empathy, firmness and sensitivity to defuse workplace competitiveness. This is the best way to move beyond combative behavior and step into a new, more fruitful, mode of co-existing.
##
**Look for Common Outside Interests**
There is a good reason for holiday events, bowling teams or company picnics—they help employees meet and get to know one another. So, if your company doesn't have extracurricular events such as these, plan some. Go see a ballgame, relax at a local park, or catch movies and a coffee on a Friday after work. Odds are, you will find that you share many things with your co-workers. By discovering and sharing these in an informal setting, you will be paving the way for more successful, efficient encounters in the future, and you will be building the trust necessary for cooperation.
_When conflict arises, interpersonal tools can really help. Work to know more about your co-workers_.
##
**Let's Make a Deal**
Negotiation is a fine art. To truly negotiate, the disputing parties must understand each others' needs and demands, avoid getting emotional, and really concentrate on the facts of the matter. This is key in dealing with porcupines.
So, go ahead! Make a deal with your porcupine.
##
**Be Strong Enough to Admit When You Are Wrong**
Admitting fault (which is not the same as apologizing) is a healthy, essential emotional act. By "owning" your actions, you will be setting an example for your porcupine to do the same. If there is no shame in admitting fault then there is no fear of blame, and this gives your porcupine one more reason to feel at ease with you.
# PORCUPINES AT HOME
Family life is the building block of society, the primary arena of social interaction. In this way, life at home can provide us with the tools we need to prosper as we grow, or cripple us with insecurities and bad habits. It is within the family network that we learn who we are and how to deal with others.
Family life, therefore, is ground zero for learning how to deal with your porcupine. We all must deal with difficult spouses, problem children, or grouchy parents. How we do so will help shape the quality of our daily lives.
Here are some hints.
YOUR PORCUPINE SPOUSE
Living with a porcupine spouse can be particularly difficult. Not only do we live with our spouses and share our lives with them, but they also live in our hearts. This means their prickly quills can really hurt. These kinds of injuries not only tempt us to act defensively—releasing our own inner porcupine—but also cause a lot of damage to our relationships in the long term. The good news is, with preparation and patience we can protect ourselves and start working on finding solutions so that life with our porcupine can be enjoyable again.
In this section we lay out some ways to deal with your prickly spouse while still loving them as they are. With these tips, you can build a closer relationship and your porcupine spouse may even let down his guard!
##
**Shed Some Light on the Problem**
A common element in advice for dealing with human porcupines is _rationality_. The porcupine's defenses are triggered by passion and irrationality; shed some cold, hard light on the situation, and you take away some of the justification for your porcupine's reaction.
_By calmly and raionally analyzing the situation that is unsettling your porcupine, you will go a long way towards defusing his orneriness_.
##
**To Love Your Porcupine, You Must Love Yourself**
Your confrontation with the porcupine within you will better equip you to deal with the porcupines in your life. How?
_First_ , it will help you build up emotional muscle. See it as a sort of training camp before the big game.
_Second_ , it will make you more empathic. Having dealt with your own fears, you'll know how hard it can sometimes be to confront the weakest parts of yourself. It will enhance your patience.
_Third_ , it will allow you to lead by example. Your courage and clarity will be inspirational to the porcupines in your life.
##
**Call it a Disagreement**
Instead of calling it an "argument," try calling it a "disagreement." After all, the word "argument" has no promise of an end point; it implies a never-ending battle of words. The word "disagreement," however, contains the possibility of a solution in its very definition: "agreement." Instead of being viewed as a point of no return, a disagreement can be viewed as a temporary stage or a stepping stone on the way towards eventual accord.
##
**Recognize and Avoid Stalemates**
Work _together_ towards a solution. This will require calm, rational communication. Begin by isolating the issue at hand, striping all your emotional reactions away from it, and then get to the heart of the matter. Avoid accusation, crying, cajoling, guilt-tripping, or any other stalemates. Focus on cooperation.
##
**Carefully Choose a Time to Talk**
It is crucial that conversation does not take place in the middle of a fight. Instead, schedule a time to have a family meeting, or a one-on-one, about the issue. Then, in calm, empathic terms, ask your porcupine what is bothering her.
Be warned: porcupines, being the defensive creatures that they are, will expect defensive denials and illogical counter-accusations from you. But don't give in! Resist the temptation to play the porcupine's game. Instead, in soothing tones, continue to work at unearthing the problem.
Chances are, once you know what's really going on, a solution won't be so hard to find.
_It is key to begin by carefully choosing an ideal time to communicate, so you can take all the time you need_.
##
**Suggest Alternatives**
Let's say that, after talking, you and your porcupine come to the conclusion that she needs some time alone. Now it's time to find a solution that will work for the both of you.
Explore ways in which she can be helpful and participate around the house that doesn't conflict with her "alone time." This might include doing the dishes after the kids are in bed, or walking the dog after she has gotten to relax on her own. In this way, your porcupine can still participate, while getting the time she needs by herself.
##
**Work Together**
Once you have come up with a plan, be sure to execute it.
After your porcupine has shared his feelings with you, he needs to know that you are partners. Show this is true by carefully working together so that your porcupine will see that his new and improved behavior is supported, and appreciated, by you. The best way to show this is through action.
_So put a plan to work—together_.
##
**Be Persistent and Consistent**
There is nothing more confusing than resolutions that only last a day or two! Be persistent and consistent. Diets, the Egyptian pyramids, and healthy relationships have one thing in common: they all take time.
##
**Be a Role Model**
Set a good example for your porcupine as well as the other members of your family. Your willingness to talk openly about relationships, and to accept criticism of your own behavior, will help shape how your porcupine learns to handle uncomfortable situations.
##
**Hang in There**
At times, it may feel like you are the only one who is really trying. But know that your porcupine is always watching. Demonstrate your love by practicing patience and empathy and by being supportive. Love is what will teach your porcupine to keep his quills down.
_Just when you are about the give up... your porcupine will surprise you_.
##
**Play Together**
Don't get stuck in a rut. Play together! This is the quickest way to beat grinding routine. This can mean a family vacation, a neighborhood game night, an outing to the movies, or a barbecue. By stepping out of the hum-drum, your porcupine just might have so much fun that he or she forgets all about those quills!
##
**Stop and Smell the Roses**
In times of stress, taking the time to enjoy the little things is more important than ever. Worry can turn anyone into a porcupine, so remember to set aside your concerns for a few minutes every day in order to enjoy something small and simple. This is also important to keep your own energy up so that you can deal with your porcupine.
##
**Keep Your Sense of Humor**
Sometimes, this may seem like the hardest thing in the world! But don't give in to a snarly mood. Try not to get too frustrated when it seems your porcupine has a difficult time changing. Instead, try to see the humor in the situation... people are curious creatures, after all.
##
**Laugh, Don't Yell**
If a fight is looming, try to steer the course away from harsh words. As we've learned, the act of yelling is one that is chock-full of fear. Laughter, on the other hand, is filled with confidence, assurance and vitality.
_So try something funny... just for fun. It might be just the breath of fresh air that you both needed_.
"Love is an act of endless forgiveness, a tender look which becomes a habit."
— PETER USTINOV
##
**Take a Rain-check**
If the timing is not right (you're exhausted from work, the baby's crying, or the pasta is boiling over), be flexible and table your scheduled discussion for another time. This will defuse some of the passion from the moment and give you both a chance to reflect on the disagreement. There is no reason to push the conversation when the mood isn't right. Still, be sure to reschedule promptly for a time that works for you both.
YOUR PORCUPINE CHILD
Nowadays, with text messaging, instant messaging and on-line chatting, our children are preoccupied with relationships and social issues long before they have a firmly established sense of who they are and their values. It is the responsibility of parents to make sure that children develop a strong sense of self and don't squander it on frivolous and sometimes dangerous digital exchanges. With the barrage of distractions our children face today—video and computer gaming, films with age-inappropriate language or subject matter, instant communications and a raft of consumer goods being marketed directly to them—it is essential that parents rise to the challenge.
Despite today's cyber-challenge, in many ways the porcupine child of today is no different than the porcupine child of ten years ago. Consistency, honesty and clarity are still vital parenting skills. Here are some ways to be a good parent when your child gets prickly.
##
**Explain Your Values**
Too often, our moral values emerge after the fact, in a sort of default mode. For example, we don't inform our children we don't accept lying until they are caught in a fib. Needless to say, it's too late at that point.
Be forthright and clear about your family values—especially in regards religious faith, tolerance of others, and issues like honesty, trustworthiness, and generosity. This will make it easier for your child to understand the next time you offer her a lesson, and, ultimately, will help her determine what's right and wrong on her own as she grows.
##
**Explore Your Values**
Part of communicating your family's values to your children is discussing them—in some cases, even defending them. This is an important step in your porcupine child's moral development, as well as your growth as a parent. When your child asks, "Why?" you should either have an answer or be willing to search for one. Sometimes there is no easy answer, but by sitting down with your child and talking through a situation with him, you can teach him a better path towards respecting your family's values and those of others.
##
**Stick With the Subject**
Porcupines (both the little ones with quills and the larger ones with attitude) depend on distracting others as a form of self-defense. What are accusations and explanations but ways of shifting attention away from the offending behavior and onto something or someone else? That's a classic distraction technique! Don't fall for it. Instead, remember that the best way to bring a disagreement with your porcupine to a swift conclusion is to stay on topic.
_Never let yourself get distracted_.
##
**Don't Lecture**
Explain. Explore. But don't lecture. In dealing with porcupine children, you are attempting to draw their feelings out into the open. A lecture will shut them up. Instead of giving a speech, use a conversational tone which invites participation.
##
**Be Vulnerable**
Many parents feel they have to be faultless super-beings in order for their children to respect them; in fact, the opposite is true. It is by being vulnerable and by showing your children that you are affected by the same rules, disappointments and conflicts as they are, that you can show your porcupine child that she isn't alone. Show her you struggle, too, and you will show her that you understand why she sometimes sticks her quills out. Then you can move to the next step: showing her that there is a better way to go about things than getting prickly.
##
**Be as Good as Your Word**
It's simple, but powerful.
_Conduct yourself as you would have your child conduct him or herself_.
##
**Get to Know What Your Porcupine Likes**
Your relationship with your porcupine child requires sharing and exchange. As we have seen, a relationship that is all one-way is not a dialogue but a diatribe—and it's bound to be frustrating for everyone involved.
No parent should expect that his or her preferences should be their children's, too. Instead, expand your mind a little and try to understand your child's likes and dislikes. For example, if your porcupine son likes listening to one particular hard rock radio station, let him put it on the next time you are both in the car. Feel free to comment—you don't have to just grin and bear it! If you like something, let him know. If you don't, let him know that as well. Expressing your likes and dislikes in a calm, unaggressive manner illustrates to your child the right way to communicate—and just might lead to an interesting discussion.
##
**Visit Your Porcupine on His or Her Turf**
In the wild, the porcupine goes into its defensive posture when its territory is encroached upon. Our children are no different. If a porcupine child's room is "off limits" to her parents, any visit is going to seem like an invasion.
Respect your child's need for privacy and only enter her room if she gives you permission. If she is OK with it, spend time together in her room during homework time. This will help her realize that you are a friend, not an enemy.
##
**Unplug Yourselves**
It's difficult to regulate your children's text messaging if you are texting as much as they are! So why not plan non-digital hours or days into your family schedule? You can start out with a "no texting" rule at mealtimes, after 8 on weeknights, or on Saturdays or Sundays. Less time in the digital world means more time in the real world, and more opportunities to get to know one another.
_Resist the temptation to use technology to escape a difficult situation_.
##
**Enjoy Yourselves... For Free!**
Competing with TV, video games, cell phones, computers, and other means of entertainment can make spending time with your porcupine child a real challenge.
So be sure to turn off the television, the computer, the Wii, and the X-Box every once in a while, and break out a deck of cards or favorite board game. If your kids don't know the rules, teach them poker, gin rummy, war, Old Maid, or any other game you know. Chances are you'll have fun, and get a chance to see a side of your kids you'd miss if you were all staring at a TV screen. What's more, it will be _your_ family's fun... home-made, real-time fun that the best memories are made of!
"Love is the condition in which the happiness of another person is essential to your own."
— ROBERT HEINLEIN
##
**Listen as Much as You Speak**
A verbal exchange in which one person speaks more than they listen is not called a conversation; it's called a monologue. Resist the temptation to listen to the sound of your own voice! Instead, ask questions and engage your porcupine. When he speaks, get him to explain, not just complain. And listen for as long as he needs you to.
THE PORCUPINE PARENT
The relationship between parent and child is a tricky one. As individuals, a parent and their offspring may not always agree, but as family members you want to be there for each other. This dynamic can make for a lot of friction.
The situation is especially difficult when, as adults, we want to be independent, yet still seek to have a connection to our parents. We love our parents, but we can be overly conscious of their faults and shortcomings. We love our parents, but it may seem that the difference between their life experiences and our own is too vast to overcome.
Still, we all want to rise above old habits and establish meaningful relationships with our parents. This can be especially hard with a porcupine parent. How do we learn to live with a porcupine parent that might be grouchy, judgmental, critical, or just about impossible, let alone appreciate him or her? It can seem like a hopeless task. Fortunately, there is a solution.
First, take a deep breath. Now, read on.
##
**Understand "That Was Then, This Is Now"**
For most of us, childhood is a time when we develop self-awareness and self-reliance. Often, our growing sense of self collides with our parent's sensibilities, expectations, and values. Although such conflict is often painful, it is vital to attaining independence.
Some children and their parents, however, cannot forgive or forget the specifics of that conflict. Things like screaming fits, slammed doors, or hurtful words continue to sting, even into adulthood.
When dealing with a porcupine parent, remember: that was then, this is now. Our "now" may contain echoes of the past, but it can also be a brand new start. In other words, although we may strive to understand the past, we do not have to be a slave to it.
_The direction of our present, and our future, is up to us_.
##
**Meet Your Parent as a Stranger**
Try this experiment. Pretend you are on an airplane, and you have just taken your seat for a transatlantic flight next to an older stranger... in this case, your parent. How would you start a conversation? What might you learn about him or her? How would you relate to one another? Be as specific as you can when you imagine the conversation; think about how, as you discovered more and more about each other, the conversation would start to flow. Maybe you would even make plans to meet again after the trip.
There is no reason that kind of mutually satisfying, mutually engaging conversation can't take place at your kitchen table. There is much about each other that you are your parent don't know; the more you share, the more you may discover you have in common.
##
**Put Yourself in Your Porcupine's Shoes**
Sometimes it can seem that, no matter how hard you try, your porcupine parent is still being difficult. No matter what you do, something always seems to be wrong—with your cooking, your child-rearing skills, the way you dress—everything! This can be very frustrating.
Keep in mind that you were once reliant on your porcupine parent; now, that is no longer the case. Your parent may be drawing comparisons—between their parenting and yours, their house and yours, their lifestyle and yours—as a way to maintain their influential role in your life. This isn't malicious; she is just operating the way she used to. Appreciate what she is dealing with. Try to understand why changing might take time for her.
_Your calm and empathic manner will defuse most situations_.
##
**Allow Your Parent a Legacy**
Everyone, especially a porcupine, has an image of themselves as they wish to be seen and remembered by others. _An amazing dancer. An expert fisherman. The life of the party. A prize-winning baker_.
Let your porcupine parent define his legacy and explore it with you and your children. His concept of himself may not coincide with what you cherish him for. But it is how he thinks of himself and would like to be remembered. Honor this. And think: you might also learn about aspects of your parent's life that you never knew.
##
**Keep Your Porcupine Parent Involved**
A parent may be acting like a prickly porcupine simply because she is bored! Involve your parent in your daily life. Invite her to join you on trips to the movies, errands in town, light lunches or barbecues at a friend's or neighbor's house. Let her plan some outings for the two of you as well. If your schedule seems too hectic for her, try to slow down, or encourage her to get involved in other activities. Then, regroup at the end of the day and let her tell you all about what she experienced that day.
##
**Calm with Care**
Openness trumps defensiveness. Care beats mistrust. Attention cures fear.
_By staying present and engaging with your parent, you are communicating love to your porcupine_.
# PORCUPINES OUT AND ABOUT
How often have you run across someone who is clearly having a rotten day, if not a lousy week? These individuals can include a surly phone customer service representative, a checkout clerk with attitude, or a grouchy waitress.
What is the best way to deal with them? After all, we don't know them that well, and it can be hard to know how much time and energy to invest in negotiating their moods and struggles. But just because we may never run across this porcupine again doesn't mean we shouldn't try to make the best of the situation.
Here are some strategies for making the best of those run-ins with life's unexpected porcupines.
##
**Try a Kind Word**
Some days you just don't have to time to really talk to someone and guide them past whatever is bothering them (and, even if you did, some porcupines just stay prickly no matter what).
_So, keep things simple, and just offer a kind word. Sometimes, that's all that's needed to change a porcupine's outlook_.
##
**Don't Blame Yourself**
If a situation quickly becomes awkward when you encounter a strange porcupine, remember, it isn't your fault! The awkwardness is emanating from the porcupine. You are not to blame.
##
**Know When to Walk Away**
Being compassionate doesn't mean being a pushover. Your porcupine may have some problems he wants to get off his chest. Encourage him to express it in constructive ways. But if his vocalizations get personal, you have every right to walk away.
##
**Consult a Manager**
Asking to see a manager may seem like an embarrassing last resort. But you may, in fact, be providing the porcupine in question with an opportunity: it is quite possible that her employer has needed to address a long-simmering situation for some time now. So, if it seems necessary to you, calmly and politely ask for the manager. With the right attitude, you just might lead all parties involved towards mutual understanding and change things for the better—for good.
# PART IV
THE PORCUPINE WITHIN US ALL
"Make it a practice to judge persons and things in the most favorable light at all times and under all circumstances."
— SAINT VINCENT DE PAUL
#
It should be clear by now that everyone has an inner porcupine that springs to life whenever we are challenged or criticized—especially when we are faced with a habit or behavior we are self-conscious about.
We all have some aspect of ourselves that we wish could be a little different; some of us wish we were more productive at work, that we went to the gym, or that we kept the living room a little cleaner. These are our sore spots, and, lo and behold, whenever someone brings them up, our porcupine springs into action. Instead of considering someone else's thoughts or conserving our energy, the quills stand up, and our porcupine defends our work performance, physical stamina, or the cleanliness of the living room with fire and fury. By the time our inner porcupine retreats, we have caused a fight, wasted a lot of energy, and moved ourselves even farther from the opportunity to change.
Remember: when it comes to successfully interacting with the porcupines in your life, it is crucial that you start with the one you know best...
YOU.
##
**Be Brutally Honest with Yourself**
We can't expect our loved ones to be honest and responsible about their actions if we can't do it ourselves. So take stock of your faults and weaknesses—that's the first step towards change.
##
**Recognize Your Defensiveness**
Everyone manifests their anxieties and defensiveness in different ways. Some people get whiny. Others become short-tempered and irritable. Some find they eat too much, or not enough. Others tune out.
_Learn to spot the clues that signal your defensiveness. That way, you can get to the root of what's really wrong, rather than continuing to act out in unproductive ways_.
##
**Acknowledge Your Shortcomings**
What are you defensive about? Why? These are crucial questions that you may or may not be ready to answer. But you must ask them, if you are ever going to get to a solution. If you need help, ask a loved one, friend or therapist.
##
**Don't Cut Corners**
Confronting yourself is the hardest, most essential challenge you will ever have to face. Be kind to yourself during this process, but be honest about what you need to accomplish and commit yourself to the work that needs to be done. Acknowledge your shortcomings. Make a plan to deal with them. And stick with the plan—no matter how much you may try to talk yourself out of it!
# A LAST WORD....
You have come to the final page of this little book. We sincerely hope these tips have helped prepare you for dealing with all of the porcupines in your life. Most of all, we hope these words have shown you that all people (even the prickly ones!) need and deserve love. Bring these tips into your own life, and don't be afraid to hug a porcupine!
# RESOURCES FOR PORCUPINES
Ellis, Albert. _How To Stubbornly Refuse To Make Yourself Miserable About Anything – Yes Anything!_ (New York: Kensington Publishing Corp., 2006).
Ellis, Albert. _The Myth of Self Esteem_. (Amherst, New York: Prometheus Books, 2005).
Lama, Dalai, H.C. Cutler. _The Art of Happiness_. (New York: Riverhead Books, 1998).
Leunig, Michael. _When I Talk to You: A Cartoonist Talks to God_. (Andrews McMeel Publishing, 2006).
# ABOUT DR. DEBBIE JOFFE ELLIS
DR. DEBBIE JOFFE ELLIS is a licensed psychologist in Australia and mental health counselor in New York. She is affiliated with several major psychological associations and societies including being a Member of the Australian Psychological Society, and an International Affiliate Member of the American Psychological Association. For several years, she worked with her husband, Dr. Albert Ellis, giving public presentations and professional trainings in Rational Emotive Behavior Therapy (REBT), as well as collaborating on writing and research projects, until his death in 2007. She now continues to present, practice and write about his groundbreaking psychotherapeutic approach of REBT. She has also co-authored several forthcoming books with Dr. Albert Ellis. She currently has a private practice in New York City, and also delivers lectures, workshops and seminars throughout the U.S.A. and across the globe.
| {
"redpajama_set_name": "RedPajamaBook"
} | 9,813 |
using namespace ad_patres::messages;
payload_t&
ad_patres::operator<<(payload_t& payload, const block_message& obj)
{
for (const auto& byte : itobl(obj.data.size()))
payload.push_back(byte);
for (const auto& byte : obj.data)
payload.push_back(byte);
return payload;
}
std::istream&
ad_patres::operator>>(std::istream& is, block_message& obj)
{
uint32_t size;
is.read(reinterpret_cast<char*>(&size), sizeof(size));
obj.data = payload_t(size);
is.read(reinterpret_cast<char*>(obj.data.data()), size);
return is;
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,498 |
{"url":"https:\/\/gmatclub.com\/forum\/is-x-98148-20.html","text":"GMAT Question of the Day - Daily to your Mailbox; hard ones only\n\n It is currently 17 Feb 2019, 12:14\n\n### GMAT Club Daily Prep\n\n#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.\n\nCustomized\nfor You\n\nwe will pick new questions that match your level based on your Timer History\n\nTrack\n\nevery week, we\u2019ll send you an estimated GMAT score based on your performance\n\nPractice\nPays\n\nwe will pick new questions that match your level based on your Timer History\n\n## Events & Promotions\n\n###### Events & Promotions in February\nPrevNext\nSuMoTuWeThFrSa\n272829303112\n3456789\n10111213141516\n17181920212223\n242526272812\nOpen Detailed Calendar\n\u2022 ### Free GMAT Algebra Webinar\n\nFebruary 17, 2019\n\nFebruary 17, 2019\n\n07:00 AM PST\n\n09:00 AM PST\n\nAttend this Free Algebra Webinar and learn how to master Inequalities and Absolute Value problems on GMAT.\n\u2022 ### Valentine's day SALE is on! 25% off.\n\nFebruary 18, 2019\n\nFebruary 18, 2019\n\n10:00 PM PST\n\n11:00 PM PST\n\nWe don\u2019t care what your relationship status this year - we love you just the way you are. AND we want you to crush the GMAT!\n\n# Is |x|<1?\n\nAuthor Message\nTAGS:\n\n### Hide Tags\n\nSenior Manager\nJoined: 08 Nov 2010\nPosts: 331\nRe: Inequalities, Is |X| < 1 ?\u00a0 [#permalink]\n\n### Show Tags\n\n14 Sep 2011, 07:08\nmustu wrote:\nscbguy wrote:\nI see the answer as A, obviously I'm wrong but I don't see how x is 1\/3 in statement 1\n\nPosted from GMAT ToolKit\n\n(1) |x + 1| = 2|x \u2013 1|\n\nThis has 2 cases.. X>0 and X<0\nIf X>0 , then X+1 = 2(x-1)\nIf X<0 , then X+1 = -2(x-1)\n\nSolving these equations we get X= 3 or X= 1\/3. Since we have YES and NO situation => Not sufficient\n\n(2) |x \u2013 3| > 0\n\nSolving this equation , we get x>3 or X<3, in either cases, X<> 3. So not sufficient.\n\n(1) + (2) ==> X= 1\/3 . Since X<> 3.\n\nRegards,\nMustu\n\nMustu - if im not wrong - u should check not for X><0 but X<>1,-1\n_________________\nVeritas Prep GMAT Instructor\nJoined: 16 Oct 2010\nPosts: 8882\nLocation: Pune, India\nRe: Inequalities, Is |X| < 1 ?\u00a0 [#permalink]\n\n### Show Tags\n\n15 Sep 2011, 21:58\n5\nmustu wrote:\nIs |x| < 1 ?\n\n(1) |x + 1| = 2|x \u2013 1|\n\n(2) |x \u2013 3| > 0\n\nAs I have said before, most mod questions are best tackled using a number line. You don't need to do many calculations then.\n|x| means the distance from 0.\n|x-3| means the distance from 3.\netc. For details of this approach, check out:\n\nLet's go on to this question now.\nIs |x| < 1 i.e. Is the distance of point x from 0 less than 1?\n\nStatement 1: |x + 1| = 2|x \u2013 1|\nThis means 'distance of x from -1 is twice the distance of x from 1'. Draw the number line now. There will be 2 points where the distance from -1 will be twice the distance from 1.\nAttachment:\n\nQues6.jpg [ 5.12 KiB | Viewed 3250 times ]\n\nFor one of these points, distance from 0 is less than 1, for the other it is more than 1. So not sufficient.\n\nStatement 2: |x \u2013 3| > 0\nThis statement tells us that distance of x from 3 is more than 0 i.e. x does not lie at 3. It can lie anywhere else.\nYou can look at it in another way: Mods are always more than or equal to 0. All this statement tells us is that this mod is not equal to zero i.e. x is not equal to 3.\nFor some of these points, distance from 0 will be less than 1, for the others it will be more than 1. So not sufficient.\n\nUsing both statements together, statement 1 says that x is either 3 or a point between 0 and 1 (which I don't really need to calculate). Statement 2 tells us that x is not 3. So together, x must be a point between 0 and 1 and its distance from 0 must be less than 1. Sufficient.\n_________________\n\nKarishma\nVeritas Prep GMAT Instructor\n\nIntern\nJoined: 17 Aug 2011\nPosts: 5\nRe: Inequalities, Is |X| < 1 ?\u00a0 [#permalink]\n\n### Show Tags\n\n15 Sep 2011, 22:23\nmustu wrote:\nIs |x| < 1 ?\n\n(1) |x + 1| = 2|x \u2013 1|\n\n(2) |x \u2013 3| > 0\n\n(1) There are 2 cases:\n1\/ x+1= 2(x-1) ---> x=3\n2\/ x+1= -2(x-1) ---> x=1\/3\n(2) X is not 3\n\nIntern\nJoined: 20 Aug 2011\nPosts: 3\nRe: Inequalities, Is |X| < 1 ?\u00a0 [#permalink]\n\n### Show Tags\n\n23 Sep 2011, 08:10\nKarishma,\n\netc. For details of this approach, check out:\nVeritas Prep GMAT Instructor\nJoined: 16 Oct 2010\nPosts: 8882\nLocation: Pune, India\nRe: Inequalities, Is |X| < 1 ?\u00a0 [#permalink]\n\n### Show Tags\n\n23 Sep 2011, 20:24\nshikari wrote:\nKarishma,\n\netc. For details of this approach, check out:\n\nI apologize. Here you go:\n\nhttp:\/\/www.veritasprep.com\/blog\/2011\/01 ... edore-did\/\n_________________\n\nKarishma\nVeritas Prep GMAT Instructor\n\nIntern\nStatus: Knerd\nJoined: 15 Feb 2011\nPosts: 28\nGMAT 1: 700 Q45 V41\nWE: Marketing (Consumer Products)\nRe: Inequalities, Is |X| < 1 ?\u00a0 [#permalink]\n\n### Show Tags\n\n24 Sep 2011, 17:40\nAm I doing this right:\n\nStatement 1:\n\n(1) |x + 1| = 2|x \u2013 1|\n\n(x+1)=2(x-1)\nx+1=2x-2\n3=x (NO)\n&\n(x+1)=-2(x-1)\nx+1=-2x+2\n3x=1\nx=1\/3 (YES)\n\nInsufficient\nB,C,orE\n\nStatement 2\n\n(2) |x \u2013 3| > 0\n\n(x-3)>0\nx>3 (NO)\n\n-(x-3)>0\n-x+3>0\n-x>-3\nx<3 (MAYBE)\n\nNot sufficient\n\nC or E\n\n(\n\nCombined:\n\nStatement 1: x=1\/3,3\nStatement 2: x <> 3\n\nSince x CANNOT equal 3, x = 1\/3\n\nSince |1\/3| < 1, both statements are sufficient to answer the prompt.\n\nC\n\n_________________\n\nKnewton Knerd\n\nVeritas Prep GMAT Instructor\nJoined: 16 Oct 2010\nPosts: 8882\nLocation: Pune, India\nRe: Inequalities, Is |X| < 1 ?\u00a0 [#permalink]\n\n### Show Tags\n\n25 Sep 2011, 22:13\nThis is fine as long as you know why you are doing this:\n\n4LEX wrote:\nAm I doing this right:\n\nStatement 1:\n\n(1) |x + 1| = 2|x \u2013 1|\n\n(x+1)=2(x-1) (When both the terms are positive, x > 1)\nx+1=2x-2\n3=x (NO) (Valid value for x since 3 >1)\n&\n(x+1)=-2(x-1) (When -1 < x < 1, (x+1) is positive but (x-1) is negative so you are put a negative sign here)\nx+1=-2x+2\n3x=1\nx=1\/3 (YES) (Valid value since -1 < 1\/3 < 1)\n\nThere would be another case x < -1. In that case both the terms will be negative.\n-(x+1)=-2(x-1)\ngiving x = 3 (Not a valid value since 3 is not less than -1)\nI am assuming that you saw the two negatives will get canceled out and give x = 3 which will not be valid so you skipped this step. In some questions, you could get a valid value here.\nSo you have only 2 values for x (3 and 1\/3).\n\nInsufficient\nB,C,orE\n\nStatement 2\n\n(2) |x \u2013 3| > 0\n\n(x-3)>0\nx>3 (NO)\n\n-(x-3)>0\n-x+3>0\n-x>-3\nx<3 (MAYBE)\n\nNot sufficient\n\nC or E\n\nCombined:\n\nStatement 1: x=1\/3,3\nStatement 2: x <> 3\n\nSince x CANNOT equal 3, x = 1\/3\n\nSince |1\/3| < 1, both statements are sufficient to answer the prompt.\n\nC\n\n_________________\n\nKarishma\nVeritas Prep GMAT Instructor\n\nVP\nStatus: Been a long time guys...\nJoined: 03 Feb 2011\nPosts: 1101\nLocation: United States (NY)\nConcentration: Finance, Marketing\nGPA: 3.75\nRe: Is |x| < 1 ?\u00a0 [#permalink]\n\n### Show Tags\n\n31 Dec 2012, 19:57\nmonir6000 wrote:\nIs |x| < 1 ?\n(1) |x + 1| = 2|x \u2013 1|\n(2) |x \u2013 3| > 0\n\nFrom statement 1, we are able to get two values of x; they are $$x=3$$ and $$x=1\/3$$. Two values of x, hence insufficient.\nFrom statement 2, all we know is that the distance from x is more than 0 or it indirectly implies that x is not 0. Not enough information. Hence insufficient.\n\nOn combining these two statements, we come to know that x cannot be 3 and x=1\/3. Since $$1\/3$$ < 1, hence $$|x|<1$$.\n+1C.\n\n_________________\nIntern\nJoined: 08 Dec 2012\nPosts: 1\n\n### Show Tags\n\n07 Feb 2013, 12:22\nThis is my approach:\nIs |x|<1?\n1st start from statement 2, cause it is easier,\n|x \u2013 3| > 0 just tell us x is note equal to 3, so it is insufficient to solve the target question\n2nd for statement 2: |x + 1| = 2|x \u2013 1|\nwe have to separate the condition to x<-1, -1<x<1, x>1 that is , |x|<1 and |x|> 1 to do further thinking\n1) when |x|<1, we could know we will get the solution in this range after solving equation, thus get the answer \"YES\" for question |x|<1\n2) when |x|>1, we could know we will get the same answer in x<-1 and x>1 condition, and we could assure the answer is \"NO\" for target question\nso based on above, statement 2 is insufficient to solve the target question\n\nwe only left option C and E now.\nTo test whether statements together will help to solve target question, we could use the denied solution x=3 in statement 2 to statement 1 to see whether it is one of the two solutions of equation.\n\nIf it is one of the solutions, then statement 2 will help to reduce the two solutions to one, thus, support the target question. We could feel free to choose option C\nIf it is not one of the solution, then statement 2 will not help to reduce the number of solutions, thus, we could feel free to choose option E.\n\nLet us test now.\nLS 3+1|=4 RS:2*|3-1|=4,\nwe could know x=3 is one of the two answers.\nThus we could choose C\nMagoosh GMAT Instructor\nJoined: 28 Dec 2011\nPosts: 4486\nRe: Inequality - Data Sufficiency Problem 3\u00a0 [#permalink]\n\n### Show Tags\n\n26 Feb 2014, 14:02\nfaceharshit wrote:\nIs |x| < 1 ?\n1. |x+1| = 2|x-1| 2. |x-3| > 0\n\nHow to approach and solve this kind of problem ..\n\nDear faceharshit,\nI'm happy to respond. I dare say, this problem is a little bit harder than what the GMAT will ask of you.\n\nStatement #1: |x+1| = 2|x-1|\nIf we are given |P| = |Q|, this means: P = Q OR P = -Q. Notice that the word \"or\" is not a piece of garnish there: rather, it is an essential piece of mathematical equipment.\n\n|x + 1| = 2|x - 1|\n\nCase I\n(x + 1) = 2(x - 1)\nx + 1 = 2x - 2\nx = 3\n\nCase II\n(x + 1) = -2(x - 1)\nx + 1 = -2x + 2\n3x = 1\nx = 1\/3\n\nThis, from statement #1, we have x = 3 or x = 1\/3. With this, we do not have sufficient information to answer the prompt question. This statement, by itself, is insufficient.\n\nStatement #2: |x-3| > 0\nForget about everything we did in statement #1. Here, x could equal 10, in which case |x| is not less than 1, or x could equal 0, in which cases |x| is less than 1. We can pick different values that satisfy |x-3| > 0, x = 10 and x = 0, that give two different answers to the prompt question. Therefore, we do not have sufficient information to answer the prompt question. This statement, by itself, is insufficient.\n\nCombined:\n#1 gives us x = 3 or x = 1\/3\nThe value x = 3 does not satisfy the second statement, so we reject that value.\nThe value x = 1\/3 is only value that satisfies both statements, and with this, |x| < 1.\n\nCombined, the statements are sufficient.\n\nDoes all this make sense?\nMike\n_________________\n\nMike McGarry\nMagoosh Test Prep\n\nEducation is not the filling of a pail, but the lighting of a fire. \u2014 William Butler Yeats (1865 \u2013 1939)\n\nVeritas Prep GMAT Instructor\nJoined: 16 Oct 2010\nPosts: 8882\nLocation: Pune, India\nRe: Inequality - Data Sufficiency Problem 3\u00a0 [#permalink]\n\n### Show Tags\n\n26 Feb 2014, 20:50\nfaceharshit wrote:\nIs |x| < 1 ?\n1. |x+1| = 2|x-1| 2. |x-3| > 0\n\nHow to approach and solve this kind of problem ..\n\nUse the number line to solve it quickly. Check: http:\/\/www.veritasprep.com\/blog\/2011\/01 ... edore-did\/\n\n'Is |x| < 1' implies 'Is distance of x from 0 less than 1?' i.e. does x lie within -1 and 1 (excluding the points -1 and 1)?\n\n1. |x+1| = 2|x-1|\n\nThis tells you that distance of x from -1 is twice the distance of x from 1. There are two values of x for which this is possible:\nAttachment:\n\nQues3.jpg [ 8.77 KiB | Viewed 1224 times ]\n\nThe red line is twice the length of the blue line in both the cases. For the first case, x lies somewhere between 0 and 1 but for the second case, x lies at 3. Hence we can't answer whether x will lie between -1 and 1 from this statement alone.\n\n2. |x-3| > 0\nThis tells us that x is a point whose distance from 3 is more than 0. That means it is not at 3 but on its left or right. This statement alone doesn't tell us whether x lies between -1 and 1.\n\nBoth statements together: Stmnt 1 tells us that x lies between -1 and 1 or at 3. Stmnt 2 tells us that x doesn't lie at 3. Then there is only one option left: x must lie between -1 and 1.\n\n_________________\n\nKarishma\nVeritas Prep GMAT Instructor\n\nNon-Human User\nJoined: 09 Sep 2013\nPosts: 9837\n\n### Show Tags\n\n06 Sep 2018, 09:43\nHello from the GMAT Club BumpBot!\n\nThanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).\n\nWant to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.\n_________________\nRe: Is |x|<1? \u00a0 [#permalink] 06 Sep 2018, 09:43\n\nGo to page \u00a0 Previous \u00a0 \u00a01\u00a0\u00a0\u00a02\u00a0 \u00a0[ 32 posts ]\n\nDisplay posts from previous: Sort by","date":"2019-02-17 20:14:36","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.678648829460144, \"perplexity\": 2277.216353963054}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-09\/segments\/1550247482478.14\/warc\/CC-MAIN-20190217192932-20190217214932-00055.warc.gz\"}"} | null | null |
# View clearing command for Laravel 5.
[](https://github.com/Kyslik/column-sortable/releases)
[](LICENSE.md)
[](https://packagist.org/packages/Kyslik/view-clear)
Simple artisan command to clear view folder in <strike>Laravel 5.1</strike> and [Laravel-5.0](https://github.com/Kyslik/view-clear/tree/Laravel-5.0)
Simply put: clears `storage/framework/views` folder.
# Laravel 5.1 UPDATE
If you are using Laravel 5.1.* you do **NOT** need this package since it is part of Laravel base commands.
`$ php artisan view:clear`
## Setup
### Version(s)
- For Laravel 5.0 use version **~1.0.0**
- <strike>For Laravel 5.1 use version **~2.0.0**</strike> Laravel 5.1 has its own view:clear command, this package is not needed.
### Composer
Pull this package in through Composer.
```
{
"require-dev": {
"kyslik/view-clear": "~2.0.0"
}
}
```
$ composer update
Add the package to your application service providers in `config/app.php`
```
'providers' => [
App\Providers\RouteServiceProvider::class,
/*
* Third Party Service Providers...
*/
Kyslik\ViewClear\ViewClearServiceProvider::class,
],
```
## Usage
$ php artisan view:clear --help
### Use case
When developing blade extensions it is useful to clear view cache in process.
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,578 |
In May 2017 Radleys, UK based chemistry equipment and laboratory tools manufacturer, launched Mya 4, a brand new reaction station that has since benefited many businesses and laboratories with improved flexibility, efficiency and space-saving.
Meeting the demands of the modern lab in over 14 countries, Mya 4's introduction to the market has been hugely successful, making chemistry more productive than ever.
In the past year and a half, multiple businesses have reaped success from using Mya 4 to improve their laboratory operations, such as the R&D department at Purolite Life Sciences. So much so that a year after the installation of their first Mya 4 station, Purolite has just invested in a second machine.
Dr Patrick Gilbert from Purolite, was looking for better temperature control of temperatures close to room temperature, as well as data logging of experiments, and he commented "The fact that the unit is so versatile is a strong advantage – on previous systems I have looked at, the flask size was limited to 1 or 2 options.
Within a few weeks of being installed, Mya 4 was already making a huge difference in the Purolite lab. Used every day for agarose R&D work, Mya 4 enables projects Purolite could not do before. Experiments are now directly scalable to pilot plant scale but also performed in smaller batches, which means less cost and less waste.
Mya 4 is a 4-zone reaction station that offers precise heating, active cooling, software control and data logging for 24/7 unattended chemistry. The versatility of Mya 4 allows users to have more control over their chemistry than ever before.
The Reaction Station allows for multiple experiments with individual control in a single compact benchtop system. Mya 4 provides ultimate flexibility: independent control over 4 zones, with active cooling and heating from -30 °C to +180 °C, a wide choice of vessels from 2 ml to 400 ml, and magnetic or powerful overhead stirring.
Mark Radley, Managing Director of Radleys, is looking forward to seeing how Mya 4 can continue to dramatically improve chemistry productivity. He commented "Our team of expert engineers and chemists have sought to solve the problem of multi-tasking chemistry with Mya 4. We have been blown away by all the positive feedback surrounding Mya 4, and all the success stories of our happy customers. | {
"redpajama_set_name": "RedPajamaC4"
} | 311 |
File #: Int 0910-2012 Version: * A Name: Use of bicycles for commercial purposes.
Committee: Committee on Transportation
Enactment date: 10/25/2012 Law number: 2012/056
Title: A Local Law to amend the administrative code of the city of New York, in relation to the use of bicycles for commercial purposes.
Sponsors: James Vacca, Gale A. Brewer, Daniel R. Garodnick, Jessica S. Lappin, Margaret S. Chin, Peter A. Koo, Karen Koslowitz, Rosie Mendez, Ruben Wills, Ydanis A. Rodriguez, James G. Van Bramer, G. Oliver Koppell, Inez E. Dickens, James F. Gennaro, Vincent J. Gentile, Robert Jackson, Jumaane D. Williams
Attachments: 1. Int. No. 910 - 7/25/12, 2. Proposed Int. No. 910-A - 9/6/12, 3. Committee Report 9/6/12, 4. Hearing Testimony 9/6/12, 5. Hearing Transcript 9/6/12, 6. Committee Report 10/11/12, 7. Hearing Transcript 10/11/12, 8. Committee Report - Stated Meeting, 9. Fiscal Impact Statement, 10. Hearing Transcript - Stated Meeting 10-11-12, 11. Mayor's Letter, 12. Local Law
10/25/2012 A James Vacca Mayor Signed Into Law by Mayor Action details Meeting details Not available
10/25/2012 A James Vacca City Council Recved from Mayor by Council Action details Meeting details Not available
10/25/2012 A James Vacca Mayor Hearing Held by Mayor Action details Meeting details Not available
10/11/2012 A James Vacca City Council Sent to Mayor by Council Action details Meeting details Not available
10/11/2012 A James Vacca City Council Approved by Council Pass Action details Meeting details Not available
10/11/2012 * James Vacca Committee on Transportation Hearing Held by Committee Action details Meeting details Not available
10/11/2012 * James Vacca Committee on Transportation Amendment Proposed by Comm Action details Meeting details Not available
10/11/2012 * James Vacca Committee on Transportation Amended by Committee Action details Meeting details Not available
10/11/2012 A James Vacca Committee on Transportation Approved by Committee Pass Action details Meeting details Not available
9/6/2012 * James Vacca Committee on Transportation Hearing Held by Committee Action details Meeting details Not available
9/6/2012 * James Vacca Committee on Transportation Amendment Proposed by Comm Action details Meeting details Not available
9/6/2012 * James Vacca Committee on Transportation Laid Over by Committee Action details Meeting details Not available
7/25/2012 * James Vacca City Council Introduced by Council Action details Meeting details Not available
7/25/2012 * James Vacca City Council Referred to Comm by Council Action details Meeting details Not available
Int. No. 910-A
By Council Members Vacca, Brewer, Garodnick, Lappin, Chin, Koo, Koslowitz, Mendez, Wills, Rodriguez, Van Bramer, Koppell, Dickens, Gennaro, Gentile, Jackson and Williams
A Local Law to amend the administrative code of the city of New York, in relation to the use of bicycles for commercial purposes.
Section 1. Section 10-157 of the administrative code of the city of New York is amended to read as follows:
§ 10-157 Bicycles used for commercial purposes. a. [Every] For purposes of this section, the following terms shall have the following meanings:
(1) "bicycle" shall have the same meaning as in section 19-176 of this code, and shall also mean any wheeled device propelled exclusively by human power as well as any motor-assisted device that is not capable of being registered by the New York state department of motor vehicles;
(2) "business using a bicycle for commercial purposes" shall mean a person, firm, partnership, joint venture, association [or], corporation, or other entity which engages in the course of its business, either on behalf of itself or others, in delivering packages, parcels, papers or articles of any type by bicycle. Nothing contained in this section shall be construed as applying to persons under the age of sixteen who use a bicycle only to deliver daily newspapers or circulars.
b. A business using a bicycle for commercial purposes shall provide identification of [the] such business by requiring every bicycle [or bicycle operator] to be identified by[:
(1)] affixing to the rear of each bicycle, bicycle seat or both sides of the delivery basket, a metal or plastic [or other] sign [of a type approved by the police commissioner] measuring no less than three inches by five inches, with the name of the business and [a three digit identification number which identifies the bicycle operator in lettering and numerals] an identification number unique to that particular bicycle in lettering and numerals no less than one inch in height so as to be plainly readable at a distance of not less than ten feet and maintaining same in good condition thereon. A sign is no longer in good condition if it is missing any of its component parts or is otherwise damaged so as to impair its functionality.[; and
(2) by requiring each bicycle operator to wear a jacket, vest, or other wearing apparel on the upper part of the cyclist's body while making deliveries, or otherwise riding a bicycle on behalf of the business, the back of which shall indicate the business name and the bicycle operator's individual identification number in lettering and numerals so as to be plainly readable at a distance of not less than ten feet.
b] c. Every [person, firm, partnership, joint venture, association or corporation engaged in providing a service as authorized herein] business using a bicycle for commercial purposes must assign to every bicycle operator employed by such business a three digit identification number. Such business must issue to every bicycle operator [a numbered] an identification card which contains the name, [residence address] three digit identification number and photo of the bicycle operator and the name, address and telephone number of the [company for whom] business by which the bicycle operator is employed. Such business shall ensure that such identification card [must be] is carried by the bicycle operator while [the cyclist]such bicycle operator is making deliveries, or otherwise riding a bicycle on behalf of the business, and [must be produced] such bicycle operator shall carry such identification card while operating a bicycle on behalf of such business. Such bicycle operator shall be required to produce such identification upon the demand of [a police officer or any other law enforcement officer] an authorized employee of the police department or department of transportation or any other person authorized by law.
[c] d. Every [person, firm, partnership, joint venture, association or corporation engaged in providing a service as authorized herein] business using a bicycle for commercial purposes shall maintain [in a log book to be kept for such purpose,] a roster of bicycle operators employed by such business. Such roster shall include the name and place of residence address of every employee operating a bicycle on behalf of such business, the date of employment and discharge of each such [person in said service] employee, [and] every [messenger or delivery] such [person's] employee's three digit identification number, and whether such employee has completed the bicycle safety course required by paragraph 3 of subdivision e of this section. The owner of any business [engaged in providing a service as authorized in this section] using a bicycle for commercial purposes shall be responsible for maintaining [in the log book a daily trip record in which all entries shall be made legibly in ink and each entry shall be dated and include the bicycle identification number, the operator's name and the name and place of origin and destination for each trip. No entry shall be rewritten either in whole or in part except in such manner as may be provided by regulation of the commissioner; any such unauthorized rewriting shall give rise to a rebuttable presumption of an act of fraud, deceit or misrepresentation] such roster. Such [log book] roster shall be made available for inspection during regular and usual business hours or any other such time that such entity is open for business upon request of an [agent] authorized employee of the police [commissioner] department or department of transportation or any [police officer or any] other person authorized by law.
[d. The owner of any business engaged in providing a service as authorized in this section shall file an annual report in such form as shall be designated by the police commissioner by rule or regulations. Said report shall include, inter alia, the number of bicycles it owns and the number and identity of any employees it may retain. Any business engaged in providing a service as authorized in this section shall be responsible for the compliance with the provisions of this section of any employees it shall retain. Nothing contained in this section shall be construed as applying to persons under the age of sixteen who use a bicycle to deliver daily newspapers or circulars.]
e. (1) The owner of any business [engaged in providing a service as authorized in this section] using a bicycle for commercial purposes shall provide, at its own expense, protective headgear suitable for each bicycle operator. Such headgear shall:
(i) meet the standards set forth by the consumer product safety commission in title 16, part 1203 of the code of federal regulations;
(ii) be readily available at each [employment] site of the business using a bicycle for commercial purposes for use by each bicycle operator; and
(iii) be replaced if such headgear is no longer in good condition. Headgear is no longer in good condition if it is missing any of its component parts or is otherwise damaged so as to impair its functionality.
(2) Each bicycle operator shall wear protective headgear that meets the requirements of paragraph 1 of this subdivision while making deliveries or otherwise operating a bicycle on behalf of such business. The term "wear such protective headgear" means having the headgear fastened securely upon the head with the headgear straps.
f. The owner of any business [engaged in providing a service as authorized in this section] using a bicycle for commercial purposes, notwithstanding that a bicycle may be provided by an employee thereof, shall provide at its own expense and ensure that each bicycle is equipped with a lamp; a bell or other device capable of giving an audible signal from a distance of at least one hundred feet, provided however that a siren or whistle shall not be used; brakes; reflective tires or, alternately, a reflex reflector mounted on the spokes of each wheel; as well as other [reflective] devices or material[,] in accordance with section 1236 of the vehicle and traffic law.
g. Any business using a bicycle for commercial purposes shall be responsible for the compliance with the provisions of this section of any employees it shall retain. [Except as otherwise provided in subdivision h of this section, the violation]Violation of any of the provisions of this section by any such business, or of any of the rules or regulations that may be promulgated pursuant hereto, shall be a violation triable by a judge of the criminal court of the city of New York and upon conviction thereof shall be punishable by a fine of not less than one hundred dollars nor more than two hundred [and] fifty dollars or imprisonment for not more than fifteen days or both such fine and imprisonment. In addition, any business using a bicycle for commercial purposes that violates any of the provisions of this section or any of the rules promulgated pursuant hereto shall be subject to a civil penalty of one hundred dollars. Any such business that violates a provision of this section or rule promulgated pursuant hereto more than thirty days after such business has already violated the same provision or rule shall be subject to an additional civil penalty of two hundred fifty dollars. Such civil penalties may be in addition to any criminal penalty imposed, and shall be recoverable against such business in an action or proceeding in any court or tribunal of competent jurisdiction or the environmental control board.
h. Any person who makes deliveries or otherwise operates a bicycle on behalf of a business using a bicycle for commercial purposes without carrying the identification sign required by subdivision b of this section or without carrying the identification card required by subdivision c of this section or who fails to produce such identification sign or card upon demand [as required by] pursuant to such [subdivision] subdivisions, or who fails to wear protective headgear required by subdivision e of this section or the retro-reflective apparel required by subdivision i of this section, shall be guilty of a traffic infraction and upon conviction thereof shall be liable for a fine of not less than twenty-five dollars nor more than fifty dollars. It shall be an affirmative defense to such traffic infraction that [the] such business did not provide the protective headgear, the identification or the retro-reflective apparel required by [subdivision] subdivisions b, c, e or i of this section. Such traffic infraction may be adjudicated by such an administrative tribunal as is authorized under article two-A of the vehicle and traffic law.
§ 2. Subdivisions a and d of section 10-157.1 of the administrative code of the city of New York, as added by local law number 10 for the year 2007, are amended to read as follows:
a. Every [person, firm, partnership, joint venture, association or corporation subject to the provisions of] business using a bicycle for commercial purposes, as defined in subdivision a of section 10-157 of this chapter, shall post one or more signs at each [employment] business using the bicycle for commercial purposes site summarizing:
(1) the responsibilities of bicycle operators and businesses pursuant to section 10-157 of this chapter; and
(2) the provisions of the vehicle and traffic law, administrative code of the city of New York and department of transportation traffic rules and regulations that the commissioner of transportation in his or her discretion determines are most important for the safe operation of bicycles in New York city. Not less than fifteen days prior to the effective date of this section, the department of transportation shall post on its website the provisions required to be posted under this subdivision.
d. The violation of any provision of subdivision a or b of this section, or of any of the rules or regulations that may be promulgated pursuant hereto, shall be a violation triable by a judge of the criminal court of the city of New York and upon conviction thereof shall be punishable by a fine of not less than one hundred dollars nor more than two hundred [and] fifty dollars or imprisonment for not more than fifteen days or both such fine and imprisonment. In addition, any [person] business using a bicycle for commercial purposes, as defined in subdivision a of section 10-157 of this chapter who violates any provision of subdivision a or b of this section or any of the rules or regulations promulgated pursuant hereto shall be [liable for a civil penalty of three hundred dollars] subject to a civil penalty of one hundred dollars. Any such business that violates a provision of this section or rule promulgated pursuant hereto more than thirty days after such business has already violated the same provision or rule shall be subject to an additional civil penalty of two hundred fifty dollars. Such civil penalties may be in addition to any criminal penalty imposed, and shall be recoverable against such business in an action or proceeding in any court or tribunal of competent jurisdiction or the environmental control board.
§ 3. This local law shall take effect one hundred eighty days after it shall have become law, except that the commissioners of the department of transportation and the police department shall take all actions necessary, including the promulgation of rules, if necessary, to implement this local law on or before the date upon which it shall take effect.
LS#3709 | {
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} | 1,741 |
3 students injured after Chino cop's…
3 students injured after Chino cop's assault rifle discharged during safety demo
Three students were injured after a Chino police officer's weapon discharged during a safety demonstration for Red Ribbon Week at Newman Elementary School on Wednesday. (Jennifer Cappuccio Maher/Staff Photographer)
By Canan Tasci | canan.tasci@langnews.com |
PUBLISHED: October 23, 2013 at 12:00 am | UPDATED: August 30, 2017 at 8:14 am
CHINO >> Police are investigating how and why an officer's AR-15 assault rifle was accidently discharged, injuring three students, during Red Ribbon Week activities Wednesday morning at Newman Elementary School.
The weapon was mounted to a police motorcycle that was on display during the activities, said Tamrin Olden, Chino police spokeswoman.
The gun went off at about 11:14 a.m. at the elementary school, 4150 Walnut Ave.
"The event and activities did not involve the display or use or demonstration of any weapons," Olden said during a press conference Wednesday evening.
During the morning activity, a child was able to approach the motorcycle and pull the trigger of the rifle that was locked in to a weapons mount on the side of the vehicle, police said.
The rifle was never removed from the locked mounting device, police officials said.
Olden said it is likely metal debris struck two children resulting an extremely minor injuries. They were transported to a local hospital for treatment.
A third student was assessed at the school and released to parents, she said.
Chino Valley Unified officials said the incident occurred during recess.
"I do not know what grades were out on the playground at the time," said Julie Gobin, district spokeswoman.
Gobin said students were taken to nearby classrooms and cafeteria after the incident.
A fail-safe measure with the rifle was effective in preventing further injuries in that the bullet disintegrated by striking a metal plate where the barrel rests against the weapons mount, Olden said.
The 5.56mm bullet is the same used by the armed forces in their assault weapons.
"We are looking into exactly what led to the accidental discharge," Olden said. "There are security measures in place to prevent that, but that's part of the investigation."
Olden did not have information on where the motorcycle was in relation to the Red Ribbon activities. She also did not release any information on the officer involved in the incident.
"Depending on the variables involved in the incident, such as this, or discharge of a firearm, are all handled differently," Olden said. "We do have someone who has been assigned to looking into that as part of the investigation."
Olden said the officers who went to Newman were on duty and the motorcycle was there as part of the "activity area where we were doing the Red Ribbon Week demonstration.
Chino resident Scott Londo, who lives a block away from Newman Elementary, said the officer should have known better.
"To me it's an ignorant move," said Londo, 22. "Knowing that this bike was on display, with a firearm – it should have been unloaded and taken care of before you let the children around them."
Red Ribbon week is an annual event where police personnel visit various school sites to provide safety information related to drug and alcohol abuse, Olden said.
Canan Tasci | {
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} | 9,610 |
Der Kootenay Pass, lokal auch als Salmo-Creston bekannt, ist ein Gebirgspass in den Selkirk Mountains in British Columbia, Kanada.
Der Passes liegt im Stagleap Provincial Park, in der Nähe des Bridal Lakes. Er liegt auf der Wasserscheide zwischen dem Einzugsgebiet des Pend d'Oreille Rivers im Westen (über die Nebenflüsse Stagleap Creek, South Salmo River und Salmo River) von dem des Kootenay Rivers/Kootenay Lakes im Osten (über den Nebenfluss Summit Creek). Der British Columbia Highway 3, auch als Crowsnest Highway bezeichnet, durchquert die Selkirks und verbindet die Gemeinden Salmo und Creston. Eine Webcam, die das ganze Jahr über läuft, zeigt jeweils die aktuellen Verkehrsbedingungen des Passes. Bei seiner Eröffnung wurde die Highway-Route auch als Kootenay Skyway bezeichnet.
Der Kootenay Pass ist einer der höchstgelegenen Straßenpässe Kanadas, der das ganze Jahr über geöffnet ist, auch wenn er bei schlechtem Wetter häufig wegen Lawinengefahr und Trümmerräumung geschlossen wird. Der Bow Summit am Icefields Parkway im Banff-Nationalpark, Alberta, ist mit 2088 m höher. Der Highwood Pass in Kananaskis Country, Alberta, ist mit 2206 m sogar noch höher, wird aber vom Highway 40 überquert, der jedes Jahr vom 1. Dezember bis zum 15. Juni gesperrt ist.
Lawinenkontrolle
Die Lawinenkontrolle am Kootenay Pass erfolgt hauptsächlich durch eine künstliche Lawinenauslösung über ein System der Lawinenauslösung durch Gasgemischzündung, mit dem die Lawinentechniker von ihrem Büro auf dem Gipfel aus Lawinen auslösen können, die von der Gipfel-Webcam aus sichtbar sind. Dieses System hat es ermöglicht die Artilleriestellungen zu entfernen, von denen aus bisher ballistische Granaten in die Auslösezonen geschossen wurden. Gelegentlich explodierten Artilleriegranaten nicht und mussten in den Sommermonaten unter erheblichen Kosten und Gefahren gefunden und zerstört werden.
Weblinks
Topographische Karte des "Kootenay Pass" www.opentopomap.org
Eintrag auf Bivouac.com
Einzelnachweise
Selkirk Mountains
Pass in British Columbia | {
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} | 1,330 |
Home » ROCKFON Breaks Ground With U.S. Acoustic Ceiling Panel Manufacturing Facility
ROCKFON Breaks Ground With U.S. Acoustic Ceiling Panel Manufacturing Facility
KEYWORDS acoustic panels / acoustics / manufacturing / Rockfon
ROCKFON LLC hosted a celebration event on March 16, 2015, for the groundbreaking of its first North American acoustic ceiling panel manufacturing facility, under construction in Marshall County, Miss. ROCKFON is the leading supplier of stone wool acoustic solutions, a subsidiary of Denmark-based ROCKWOOL International A/S and an affiliate to ROXUL Inc.
Company leaders from ROCKFON, ROCKWOOL and ROXUL were joined at the construction site by Speaker of the Mississippi House of Representatives, Philip Gunn, in addition to ROCKFON customers, and other leaders from the community, company and industry.
The Mississippi facility will be ROCKFON's fifth manufacturing facility in the world, extending global capacity and meeting the growing demand for ROCKFON's stone wool acoustic ceiling products in North America. The new facility represents an initial investment of approximately $40 million U.S. dollars by ROCKWOOL.
"The investment in this new facility demonstrates the strong commitment of our company to support the North American market, and of the State of Mississippi and Marshall County to support economic development in the area," said John Medio, ROCKFON's president of the Americas.
"Today is a great day for Marshall County as ROCKFON prepares to begin construction of its new acoustic ceiling tile facility. Once complete, the company's significant investment and creation of 90 new jobs will have a strong positive affect on the local community and economy," said Mississippi Governor Phil Bryant. "I congratulate the ROCKFON team and everyone involved in bringing this great company to Mississippi on this milestone event."
Located approximately 31 miles from Memphis, Tenn., the new ROCKFON facility in Marshall County will span 130,000 square feet with room for future expansion. Production is expected to begin mid-2017. ROCKFON's new manufacturing facility in Mississippi, and its strategically positioned U.S. distribution centers, will provide for comprehensive coverage and servicing of the North American market.
In North America, ROCKWOOL operates under the name ROXUL. ROXUL has been in North America since 1988. ROCKFON's new facility will be adjacent to ROXUL's existing facility in Marshall County, which manufactures its full line of residential, commercial, industrial and roof board products.
ROCKFON has been operating in North America since Jan. 2013. With the acquisition of Chicago Metallic in Oct. 2013, ROCKFON provides customers with a complete ceiling system. Its product offering combines ROCKFON stone wool and specialty metal ceiling panels with Chicago Metallic suspension systems.
The new facility in Mississippi will manufacture ROCKFON stone wool acoustic ceiling products. ROCKFON will continue to manufacture its specialty metal ceiling panels and Chicago Metallic suspension systems in its Chicago and Baltimore facilities. Chicago Metallic suspension systems also are manufactured in Belgium, Malaysia and China. ROCKFON's other stone wool manufacturing facilities are located in the Netherlands, Poland, France and Russia.
SOPREMA Breaks Ground on Wadsworth Facility Expansion
ROCKFON to Build First North American Manufacturing Facility
Acoustic Engineering Offices choose Rockfon Ceiling Systems
Acoustic Ceiling Panels with 0.95 NRC | {
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Now that the fps filter has been added (you're my heroes for adding it), it would be great if an fps filter would be auto-inserted before the overlay filter, where framerate conversion is required if the framerates of the two input vary (the way a scale filter is often auto-inserted wherever it's required).
To do this, whenever an overlay filter is used in a filter chain, the framerate of each of its inputs should be detected (using the same function used by ffprobe to populate the avg_frame_rate property), then insert an fps filter to upsample the input with the lower framerate to match the framerate of the other one.
Currently I do this manually: run ffprobe on each video file before performing an overlay, then insert an fps filter between the lower-fps file and the overlay filter. This means 3 procedure calls. It would be great if it could all be done in 1.
This does not sound like a good general solution. IMHO the overlay filter should create a new output frame whenever any of its inputs changes. I don't know if that would work for your use-case though?
I'm not sure I know what you mean by "whenever any of its inputs change" but I don't think that's the current behavior anyway.
The overlay filter takes every frame of the first (main) input in turn and copies it to the output with the same timestamp while overlaying the next frame from the second input (ignoring the timestamps / framerate of the second input).
If it runs out of frames for the second input before the first one, it simply overlays the last frame of the second input on each remaining frame of the first input.
If it runs out of frames for the first input before the second one, the rest of the frames of the second input are ignored.
Therefore, the framerate and duration of the output is always equal to the framerate and duration of the first input. This can lead to a situation where the framerate of the second input (the overlaid video) is not the same as the framerate of the output file, so it appears to be running in slow/fast motion.
Why wouldn't it be a good general solution to equalize the frame rates? Do you think a user might want one of the input streams to appear slowed down / sped up in the output just because it's been overlaid on top of another stream?
At the very least, I think this would be useful to many users if it were included as an optional behavior (i.e. triggered by a config parameter).
The most obvious reason why it is not a general solution is that not all inputs have a (fixed) framerate or something that is a suitable substitute.
Even in a very simple case of overlaying a 24 fps video with a 25 fps one, "equalizing" the fps will not lead to a good result. If you change the 24 fps one to 25 fps that will result in noticeable stutter once a second, for exact results you'd have to go to an insane 600 fps.
If the overlay filter were taking both timestamps into account you could get an _exact_ (though variable fps) output with at most 49 frames per second (though those frames would be at very variable intervals and a lot of players would have problems displaying it accurately I admit, so the win may be questionable in real-world use).
Fair enough, but as I said even now the overlay filter isn't taking both streams' timestamps into account, so what I'm suggesting is the most easily obtainable "fix" for this use case, and I've seen many postings by people using the overlay filter the way I am.
The current default behavior of the overlay filter (ending up with a fast-motion/slow-motion overlay) is, to me, not as good as what I'm suggesting, and can be very confusing.
BTW, I'm using the overlay filter for overlaying (PiP style) two recorded streams of a video conference (one stream per party, as recorded by Flash Media Server).
I think Reimar explained pretty well why your suggestion is not a good idea and is, in fact, not properly defined in the first place. Please remember that filters must be designed for general use, not for any specific use case.
Currently, the overlay filters produces one output frame for each input frame on its "main" input. It uses the timestamps to determiner which overlay frame must go on the main frame.
Adding an option to output a frame also when the overlay changes, like Reimar suggests, is of course possible.
As far as I can tell your claim that it ignores the timestamps on the overlay input is just not true, and I can confirm that two video at different frame rates with visible synchronization stay on sync. The worst that happens is an off-by-one bug that is already underway of being fixed.
Please, be more specific about the actual problems you experience and are trying to solve.
1) the "main" video fps always wins for the output.
What happens if a higher frame-rate was manually specified? Does ffmpeg just repeat the same combination of main and overlay or does it also look for every repeated main frame which overlay frame is used?
You mean with -i ... -i ... -filter_complex ...overlay... -r something output?
AFAIK, the effect of -r comes completely after the effects of filters, so depending on the value of -vsync it will result on the frame being duplicated as you describe.
IMHO, there are too many possible circumstances to be able to "Do What I Mean" always, so it is best to keep everything simple so accurate results can be achieved with the correct combination of filters without bad surprises. | {
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\section{Introduction}
\label{intro}
Minor bodies such as comets and asteroids in the solar system are remnants of the planet formation process \citep{Kokubo2002,Kenyon2006,Wyatt2008,Johansen2017}. These objects play an important role in the evolution of their planetary system \citep{Nesvorny2018,Torres2019,Cai2019a}. In particular, the dynamical evolution of these bodies in any planetary system is dominated by the gravitational interaction with major bodies such as the planets. As the comets come close to planetary regions, planets become the main influence for these objects.
Thus, gravitational interactions with planets, such as close encounters and collisions, may have influenced the planets' history, composition, structure, and evolution \citep{Asphaug2006,Brasser2020,Morgan2021}. Examples of these processes include cometary impacts that may be responsible for the dawn-dusk asymmetry of Mercury's exosphere \citep[e.g.,][]{Benz1988,Pokorny2017}, the changes of the surface and atmosphere on Mars \citep[e.g.,][]{Carr1989,Melosh1989,Woo2019}, and the dynamical evolution of the gas giants and trans-neptunian objects (\citep{Gomes2005,MunosGutierrez2021}, for more examples see \cite{Stern1995,MarovRickman2001,Charnoz2003}. Furthermore, collisions with these remnants may have a dramatic effect on a planet's orbit. For example, repeated collisions may have resulted in the tilt of Uranus \citep[e..g,][]{Brunini1995,Parisi1997,Rogoszinski2021}. Lastly, the Chicxulub impact on Earth is suspected to be the main cause of the extinction of the dinosaurs \citep[e.g.,][]{Alvarez1980,Schulte2010}.
Cometary (and other minor body) impacts in the solar system have been extensively studied in the literature \citep[e.g.,][]{Opik1951,Kessler1981,Greenberg1988,Bottke1993,MarovRickman2001,Muinonen2001,Valsecchi2005,Rickman2014a}. Of particular interest is the impact rate of comet collisions with planets. These calculations are often based on the \"Opik's analytic method \citep{Opik1951}. However, these methods are often tuned to model cometary impacts in the inner part of the solar system. As a result, it is challenging to calculate collision rates in other planetary systems with different architectures, arbitrary configurations and geometries than the solar system.
Here we present a succinct and accurate model to calculate the collisions rate and timescale for a minor body to impact a planet. Our methodology is applicable for all geometries and configurations and is consistent with direct numerical calculations. In \secref{sec2}, we present our model for collisional timescales. In \secref{sec3} we test our model by comparing our predictions with a well known analytic method for collision rates in the solar system and detailed N-body simulations. Finally, we discuss our results in \secref{discussion}.
\section{Collisional timescales for particle impacts on planets}
\label{sec2}
Here we present a general analytical approach, to calculate the collision rate of a minor body with a planet for arbitrary geometry of interaction. Hereafter we refer to a minor body as a particle to highlight the wide range of application.
Consider the collision rate, $\Gamma_{\rm coll}$ (\equref{eq:Gamma}), of a particle with a planet. This rate can be calculated by assuming a population of planet-orbital-crossing particles a with number density $n$ that will eventually collide with the planet. The relative velocity between the planet and the particle ($v_{\rm rel}$), and the cross section of interaction ($\sigma$). We assume an arbitrary configuration for the planet and particle; see \figref{geometry} for illustration. Thus, the collision rate can be approximated as \citep[e.g.,][]{BinneyTremaine2008,Nesvorny2020,Rose2020}
\begin{equation}\label{eq:Gamma}
\Gamma_{\rm coll} = n\,v_{\rm rel}\,\sigma.
\end{equation}
\begin{figure}
\centering
\includegraphics[width=1.\columnwidth]{geometry.png}
\caption{Orbital motion of a particle about it's host star with respect to the planet's orbital plane. The semi-major axis of the planet is given by $a_p$.}
\label{geometry}
\end{figure}
The number density is then simply $n=N_c/V$, where $N_c$ is the number of colliding particles and $V$ is the volume where collision can take place, and is given by,
\begin{equation}\label{volume}
V= 2\pi^{2}a_{p}R_{p}^{2}\,sin\,i_{c},
\end{equation}
where $a_{p}$ and $R_{p}$ are the semi-major axis and radius of the planet respectively and $i_{c}$ is the inclination of the particle.
The relative velocity magnitude between the particle and the planet is given by the magnitude difference between the velocity vector of the planet ${\bf v}_p$ and the particle ${\bf v}_c$, in other words, $v_{\rm rel}=| {\bf v}_{\rm rel}| =| {\bf v}_p -{\bf v}_c |$. To obtain the velocity vectors of the planet and the particle, we first need to establish the geometry of the encounter. We considered a reference plane in which the planet is at the center. The orbital motion of the particle about the star with respect to the center is in three-dimensional space.
The position vector of the particle in the frame of it's bound orbit about the host star is: ${\bf r}_c $=$x\,\hat{\mathbf{x}}+ y\,\hat{\mathbf{y}}+ z\,\hat{\mathbf{z}}$, where $x=r_{c}\,cos\,f_{c}, y=r_{c}\,sin\,f_{c}, z=0$. Where $f_c$ is the true anomaly of the particle, and $r_c$ is given by:
\begin{equation}\label{rc}
r_{c}= a_{c}\,\frac{1-e_{c}^{2}}{1+e_{c}\,\cos\,f_{c}} \ .
\end{equation}
Then, the position vector projected to the plane of the planet is given by ${\bf R}_{1}{\bf R}_{2}{\bf R}_{3} {\bf r}_c$, where ${\bf R}_j$, $j=1,2,3$ are the rotation matrices given by \citep[e.g.,][]{MurrayDermott2000},
\begin{center}
\begin{eqnarray}
R_{1}=
\begin{pmatrix}
cos\,\omega & -sin\,\omega & 0 \\
sin\,\omega & cos\,\omega & 0 \\
0 & 0 & 1
\end{pmatrix}\ \ ,
R_{2} =
\begin{pmatrix}
1 & 0 & 0 \\
0 & cos\,i & -sin\,i\\
0 & sin\,i & cos\,i
\end{pmatrix}\ \ ,\nonumber\\
R_{3}=
\begin{pmatrix}
cos\,\Omega & -sin\,\Omega & 0 \\
sin\,\Omega & cos\,\Omega & 0 \\
0 & 0 & 1
\end{pmatrix}. \
\end{eqnarray}
\end{center}
Therefore, the components of the position vector of the particle projected on the planet's can be calculated by,
\begin{equation}
\begin{pmatrix}
X\\
Y\\
Z
\end{pmatrix}
=
R_{1}R_{2}R_{3}
\begin{pmatrix}
x\\
y\\
z
\end{pmatrix}
\label{vector}\end{equation}
Consequently, the velocity vectors of the particle ($\mathbf{v}_{c}$) and the planet ($\mathbf{v}_{p}$) in their own individual orbital planes at any give time are given by
\begin{equation}\label{vc}
\mathbf{v}_{c,p}= \left( - \frac{h_{c,p}\,\sin\,f_{c,p}}{a_{c,p}\,(1-e_{c,p}^{2})},\frac{h_{c,p}\,(e_{c,p}+\cos\,f_{c,p})}{a_{c,p}(1-e_{c,p}^2)}, 0\right) \ ,
\end{equation}
where, the subscript $c$ and $p$ stands for the particle and the planet respectively. The specific angular momentum of the particle (planet) $h_{c}$ ($h_p$) is given by,
\begin{equation}\label{hc}
h_{c,p}=\sqrt{G\,(M_s + M_p)\,a_{c,p}\,(1-e_{c,p}^2)} \ ,
\end{equation}
where $a_c~(a_{p})$, $e_c~(e_{p})$ and $f_c~(f_{p})$, are the semi-major axis, eccentricity and true anomaly of the particle (planet). Additionally, $M_s$ and $M_p$ are the mass of the star and the planet respectively. Thus, the relative velocity is given by:
\begin{equation}
v_{\rm rel}=|{\bf v}_p -R_{1}R_{2}R_{3} {\bf v}_c| \ .
\label{velrel}\end{equation}
Where recall that we are rotating the particle velocity vector to the planet frame using $R_{1}R_{2}R_{3}$. The relative velocity between the particle and the planet is calculated at the moment of collision. The collision of the particle with the planet takes place at the node, where the cometary orbital plane and the planet's plane coincide.
Finally, the cross section $\sigma$, enhanced by gravitational focusing, is given by:
\begin{equation}\label{eq:sigma}
\sigma=\pi\left (R_p^2+R_p\,\frac{2\,G\,M_p}{v_{\rm rel}^2}\right) \ ,
\end{equation}
Substituting equations \ref{volume}, \ref{velrel}, and \ref{eq:sigma} into \equref{eq:Gamma} we obtain a final expression for the collision rate per year as a function of the particle's orbital elements $\Gamma_{\rm coll}(a_{c},e_{c},i_{c},\Omega_c,\omega_c,f_c)$:
\begin{equation}\label{gamma}
\Gamma_{\rm coll} =\frac{N_c}{2\pi^{2}a_{p}R_{p}^{2}\,sin\,i_{c}}\,\left (v_{\rm rel}\,\pi\,R_p^2+\pi\,R_p\,\frac{2\,G\,M_p}{v_{\rm rel}}\right)
\end{equation}
Equation \ref{gamma} represents the most general expression for a collision between a minor body and a planet in an arbitrary geometry.
\section{Comparison with analytic and numerical methods}
\label{sec3}
In this section we tested our model by comparing it with the often use \"Opik method for collisions in the solar system (see also Appendix \ref{opikvscoll}) and with detailed N-body simulations (Sec.\ref{comparison}).
\subsection{The \"Opik Method}
\label{sec3.1}
The \"Opik method \citep{Opik1951} in its original form provides an expression for the collision rate of particles (asteroids or comets) with planets. In this 1951 method, the planet is assumed to be fixed in space in a circular orbit and the colliding particle on an arbitrary orbit. The collision happened when the orbit of the two bodies intersect. The \"Opik method assumes a restricted 3-body problem, considering the small body massless and moving on an unperturbed heliocentric Keplerian orbit.
The \"Opik method considers two main parts to calculate the collision rate: the relative velocity between the particle and the planet and the collisional area or cross-section. The \"Opik method often uses units $G=1$ and assumes the star's mass $M_{s}$=1\,$M_{\odot}$.
The reference frame is set so that the particle is at one of the nodes of its orbit when the encounter with the planet occurs. Therefore, the relative velocity $U_{\rm opik}$ can be expressed in terms of the Tisserand parameter with respect to the planet and is given by \citep[e.g.,][]{Opik1951,Carusi1990},
\begin{equation}
U_{\rm opik}=\sqrt{3-T} \ ,
\label{vel_T}\end{equation}
where T, is the Tisserand parameter which is defined as,
\begin{equation}
T =\frac{a_{p}}{a_{c}} + 2\,cos\,i_{c} \sqrt{ \frac{a_{c}}{a_{p}} (1-e_{c}^{2})} \ .
\label{T}\end{equation}
The relative velocity components, $U_x$, $U_y$, and $U_z$ are given by \citep[see e.g.,][]{Carusi1990},
\begin{eqnarray}
U_{x} &=& \pm\sqrt{2-1/a_{c} - a_{c}(1-e_{c}^{2})}, \nonumber\\
U_{y} &=& \sqrt{a_{c}(1-e_{c}^{2})}\,cos\,i_{c} -1,\nonumber\\
U_{z} &=& \pm\sqrt{a_{c}(1-e_{c}^{2})}\,sin\,i_{c}\,.
\label{velxyz}\end{eqnarray}
When a particle reaches the Hill's sphere of a planet, the Sun's perturbations can be neglected, and the trajectory of the particle can be modeled as a planetocentric. Once inside of the Hill's region, a particle can collide with the planet if the pericenter distance of the particle $q_{c}$ is smaller or similar to the radius of the planet $R_{p}$, i.e., $q_{c}\leqslant R_{p}$. Therefore the the cross-section for interaction can be expressed by $\sigma_{\rm opik}$ \citep[e.g.,][]{Opik1951,Opik1976},
\begin{equation}
\sigma_{\rm opik}= R_{p}\,\sqrt{1+\frac{2\,G\,M_{p}}{{U_{\rm opik}}^{2}\,R_{p}}}\ .
\label{eq:sigmaopik}
\end{equation}
With the expression for the encounter velocity $U_{\rm opik}$ and the cross-section $\sigma_{\rm opik}$, the collision rate per year as a function of the particle's orbital elements $\Gamma_{\rm opik}(a_{c},e_{c},i_{c})$ is calculated as followed.
The \"Opik method considers a particle in a heliocentric orbit, which crosses two times an sphere of radius $r=1$. The radial velocity of the particle is $U_{x}$. Then the time spent of the particle in the sphere is $dt=2\,dr/U_{x}$. The number of particles $N_c$ in the sphere per orbital revolution can be calculated as $N_c=dt/P$, where $P$ is the particle's orbital period. \cite{Opik1951} showed that a particle could only be found inside a band with two parallel latitudes $\pm i$ and a volume $dV=4\pi\,sin\,i\,dr$. Thus, the collision rate of a particle can be expressed by \citep[see e.g.,][for a detailed derivation]{Opik1976},
\begin{equation}
\Gamma_{\rm opik} =\frac{\sigma_{\rm opik}^{2}\,U_{\rm opik}}{\pi\,sin\,i_c |U_{x}|}.
\label{pc}
\end{equation}
Despite the simplicity, the \"Opik's method yields consistent results for Jupiter-family comets \citep[e.g.,][]{Greenberg1988,Nakamura1998,Dones1999}. However, it fails to accurately model the collision rate for those particles with Tisserand parameter $T\geqslant3$.
Many improvements in \"Opik original theory have been done by several authors \citep[e.g.,][]{Nakamura1998,Manley1998,Dones1999,Levison2000,Zahnle2001,Vokrouhlicky2012,Pokorny2013,Rickman2014a,JeongAhn2017,Vokrouhlicky2019,Abedin2021}, creating a variety of \"Opik-like models that address some of the existing issues of the classic method. These \"Opik like-methods represent a quick (but at times less accurate) alternative to more robust numerical simulations. However, these expansions are mainly tuned for objects in the inner parts of the solar systems and they lack the flexibility to model minor bodies in exo-planetary systems for a wide range of configurations.
\subsection{Numerical Method}
\label{nbody}
We used the N-body package \texttt{REBOUND} \citep{Rein2012} with the WH-Fast integrator \citep{Rein2012} to calculate the collisional history of minor bodies with a planet. We considered a system compose by a solar mass star and a Jupiter-like planet with semi-major axis $a_{p}=5.2$\,au, eccentricity $e_{p}=0.05$, mass $M_{p}=0.001$\,$M_{\odot}$. We used an inflated collisional radius $R_{p}$=$(M_{p}/3M_\star)^{1/3}\,a_p$ (to assure more collisions in shorter time). We added $5000$ test particles representing the minor bodies. To compare the numerical simulation with the analytical calculation we construct two representative runs (see Appendix \ref{sim}). In one, {\bf R-inc}, we vary only the initial inclination but keep all of the other orbital parameters constant, and in the other, {\bf R-ecc} we vary only the initial eccentricity of the particle. The full set of initial conditions are described in \tabref{table1}. We model the collision of the particles as inelastic encounters. For simplicity, every particle that collided with the planet was removed from the simulation. The simulation was run up to $10^4$~yrs. We note that the number of collisions does not converge on this timescale. As a function of time, the number of particles that undergo collisions increases, as expected. We performed a series of tests using a simulation time of $10^5$~yrs, and we did not find qualitatively change our results. In Appendix\,\ref{sim} we show the results of the simulations.
\begin{table}[h!]
\centering
\caption{Input orbital elements of the particle and planet in the numerical simulations: $a_c$, $e_c$, $i_c$, $\Omega_c$, $\omega_c$, and $f_c$, $f_p$ represents the semi-major axis, eccentricity, inclination, longitude of the ascending node, argument of periapsis, and true anomaly of the particle and the particle, respectively.}
\begin{tabular}{l|ccccccc}
name & $e_c$ & $a_c$ & $i$ & $\Omega_c$& $\omega_c$ &$f_c$ &$f_p$ \\
& & [au] & [deg] & [deg] & [deg] &[deg] &[deg] \\
\hline
{\bf R-inc} & 0.5 & 5.5 & $0-180$ & 0 & 0 & 0 & 0 \\
{\bf R-ecc} & $0-1$ & 5.5 & $27.5$ & 0 & 0 & 0 & 0 \\
\label{table1}
\end{tabular}
\end{table}
\subsection{Comparison with Analytic and Numerical Methods}
\label{comparison}
In \figref{fig:nbody_gamma_opik}, we show the average collision rate from the simulation, for different eccentricity and inclination bins (orange solid line). We compare the numerical result with the calculated $\Gamma_{\rm coll}$ and $\Gamma_{\rm opik}$ (red dash-dot and blue dotted lines, receptively). We note that in both cases we use the orbital parameters of the particles at the onset of collision. Before the particles collided their orbit evolves as expected from three-body evolution \citep[e.g.,][]{Naoz2017}, thus, their initial conditions from Table \ref{table1} differs from their orbital parameters when they collide.
As depicted in \figref{fig:nbody_gamma_opik}, our analytical rate calculation, $\Gamma_{\rm coll}$ is consistent with the N-body rate in both its functional form and value.
Note that $\Gamma_{\rm opik}$, is at times few orders of magnitude different than the numerical results. Furthermore, as clearly seen in \figref{fig:nbody_gamma_opik}, $\Gamma_{\rm opik}$ estimated higher collision rate for circular orbits, at odds with the numerical and $\Gamma_{\rm coll}$ estimations.
\begin{figure}
\includegraphics[width=1.\columnwidth]{nbody_gamma_opik_v2.png}
\includegraphics[width=1.\columnwidth]{nbody_gamma_opik_ecc_V2.png}
\caption{Comparison between the N-body simulation (solid orange line), the \"Opik method (dotted blue line) and the $\Gamma_{\rm coll}$ (dash-dot red line). The top panel shows the collision rate as a function of the inclination using the values for {\bf R-inc} listed in Table\,\ref{table1}. While the bottom panel shows the distribution for {\bf R-ecc} and the collision rate as a function of the eccentricity.}
\label{fig:nbody_gamma_opik}
\end{figure}
\section{Summary and Discussion}
\label{discussion}
Here we present an analytical model to determine the collision rate of a minor body (particle) with a planet for any type encounter geometry and orbit (\equref{gamma}). We tested our formulation by comparing with the \"Opik method (\secref{sec3.1}) and detailed N-body simulations (\secref{nbody}). As a proof of concept we choose two representative examples, one for which we vary the colliding particles eccentricities, and the other, by varying their inclinations. Our prediction for the collision rate of a particle impacting a planet is consistent with the simulations, but differ with the \"Opik method.
The inconsistency between our model and the \"Opik method are mainly due to the singularities produced by the \"Opik method not present in our model. The \"Opik method fails to estimate the collision rate for those particles with small values of inclination and $|U_x|$, producing a singularity since \equref{opik_unit} goes to infinity (see \figref{tcollvspcoll}). Therefore bodies similar to the centaurs, nearly isotropic and long-period comets, can not be accurately model following \equref{opik_unit}. These objects are expected to be abundant in exo-planetary systems as a consequence of planet formation \citep[e.g.,][]{Wyatt2008,Johansen2017}. On the other hand, $\Gamma_{\rm coll}$ produces better estimations for the collision rate for all varieties of orbital elements. This is because we allow for arbitrary geometry. As a result, the function $\Gamma_{\rm coll}$ allows us to model any type of minor body orbits given the flexibility to estimate collisional rates in exo-planetary systems accurately.
We note that the \"Opik-like methods might provide better estimations than the original one \citep[e.g.,][]{Valsecchi2005,JeongAhn2017,Vokrouhlicky2019,Abedin2021}. However, a detailed comparison with the variations of the method is beyond the scope of our paper. We omit these comparisons because our intention is not to adapt or extend the \"Opik theory for collisions to any exo-planetary system. Therefore, we focus on a simple comparison with the backbone of the theory, the classic \"Opik method \citep{Opik1976}.
Our formulation $\Gamma_{\rm coll}$ provides a succinct solution to determine the collision rate of a particle as a function of its orbital elements ($\Gamma_{\rm coll}(a_{c},e_{c},i_{c},\Omega_c,\omega_c,f_c)$). These allowed us to model the collision of particles with planets for any encounter geometry and orbit, providing an accurate alternative to costly N-body simulations.
\section*{Acknowledgments}
ST expresses his gratitude to Ylva G\"otberg and Erez Michaely, for their helpful discussions and comments to the present work. ST, SN, GL thank NASA-ATP: AWD-000836-G1. Furthermore, ST and SN thank partial support from the NSF through grant No. AST-1739160 and Howard and Astrid Preston for their generous support. SR thank NASA-ATP grant number 80NSSC20K0505, as well as Nina Byers Fellowship and Michael A. Jura Memorial Graduate Award for support.
\section*{Data Availability}
The python scripts used to generate the data for this work can be accessed here: https://santiago-torres.com/Research
\bibliographystyle{mnras}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,707 |
Paramount Studios presents
Friday The 13th�Part VI: Jason Lives (1986)
"Look, you got to do something. Jason's alive. He killed my friend and now he's coming for me!"
- Tommy Jarvis (Thom Matthews)
Review By: Rich Rosell
Stars: Thom Matthews, Jennifer Cooke, C.J. Graham
Other Stars: David Kagen, Ron Palillo, Kerry Noonan, Renee Jones, Tom Fridley
Director: Tom McLoughlin
MPAA Rating: R for (horror violence)
Run Time: 01h:27m:02s
Audio Transfer
D D B- B- D-
For a series of films that has spawned at least nine (!) sequels as of this review, it never ceases to amaze me how truly crappy the Friday The 13th follow ups have been. I will admit that in 1980 the first Friday The 13th, directed by Sean Cunningham, spooked me, caught me off guard and offered up what would become a literal horror franchise poster boy in the form of the seemingly unstoppable Jason Vorhees and his machete. Purists, of course, will be quick to point out that Jason didn't really make his first full-length debut until the 1981 release Friday The 13th—Part II, but for all practical purposes it was the first film that anchored his legend in celluloid. In what would become a veritable wave of "dead teenager" flicks that would pop up during the period from 1980 through 1986, it was good old Jason that had some kind of cinematic sea legs to keep on killing through an endless wave of really rotten sequels.
Tom McLoughlin wrote and directed this 1986 installment, Friday The 13th—Part VI: Jason Lives, and he did not bother to actually create much in the way of horror, and very little in the way of entertainment. I realize that he wasn't asked to direct the sequel to Citizen Kane, but it just seems that this chapter is nothing more than a series of uneventful killings, topped off with a predictable open-ended climax. Considering that another four Jason films—and counting—have been cranked out since Part VI, it would seem that not even the lack of a substantial story can stop the hockey-masked killer.
This type of film doesn't require much in the way of in-depth understanding of the previous Jason chapters, because in the world of Friday The 13th, all characters speak in those handy "explain everything up 'til now" monologues. In this film, we learn that the town of Crystal Lake has been renamed Forest Green, which apparently was done to make everyone forget the assorted Jason-induced massacres that have occurred there in recent years. That's like saying that if a crazed killer murdered a pile of teenagers in YOUR town, simply changing the name would make everything better.
Jumping a few years forward after the events of Part V, twenty-something Tommy Jarvis (Thom Matthews), who "killed" Jason (yeah, right) in the last go-round, returns here after spending years in therapy. (Part of his therapy is no doubt due to the fact that he was portrayed by Corey Feldman in Part V, but that's another story.) The opening sequence is the "resurrection" scene, where Tommy and his pal Allen (Welcome Back Kotter's Ron "Horshack" Palillo) dig up Jason's grave in some attempt at closure. (Note to self: if a crazed killer that tormented me is dead, as a result of me killing him, and said killer has been buried for years, consider that closure.)
To no one's surprise, the slightly decomposing Jason is accidently brought back to life during Tommy's half-witted attempt at making a point: Hence, the title. During the course of the next 87 minutes, Jason is responsible for 18 kills, which averages out to one violent death scene almost every 5 minutes. It is rather disturbing to think that even with that much carnage, a film like this could be so dull.
Rating for Style: D
Rating for Substance: D
Aspect Ratio 1.85:1 - Widescreen
Original Aspect Ratio yes
Anamorphic yes
Image Transfer Review: Paramount has released Friday The 13th—Part VI: Jason Lives in its original 1.85:1 anamorphic widescreen format. Overall, the color field is bright, and the saturation level is a good, without any annoying bloom; fleshtones remain consistently lifelike throughout, though. There are minimal compression artifacts and general contrast is very good, too. Shadow detail and depth is far better than I would have expected on a cheapo film from 1986, and it would seem that Paramount may have put a little effort into this release.
Nice transfer of a very bad film.
Image Transfer Grade: B-
DS 2.0 English no
Audio Transfer Review: Originally released in 1986 in "Ultra-Stereo", Friday The 13th—Part VI: Jason Lives has been remastered in a 2.0 surround track for this DVD release. I was acutally quite pleased with the dynamic range of the 2.0 track, and the stronger than expected use of the surround channels for cues. Dialogue, which is not a necessity here, is always clean, while Harry Manfredini's memorable theme sounds deep and enveloping.
The viewing experience for this film is enhanced significantly via this track, and stands as major improvement over earlier installments.
Audio Transfer Grade: B-
Disc Extras
Static menu
Scene Access with 16 cues and remote access
Subtitles/Captions in English with remote access
1 TV Spots/Teasers
Packaging: Amaray
1-Sided disc(s)
Layers: dual
Extras Review: A grainy, 1.85:1 anamorphic widescreen theatrical teaser trailer, English subtitles and 16 chapter stops is all there is.
Extras Grade: D-
Final Comments
Friday The 13th—Part VI: Jason Lives. It's Part VI, for chrissakes! Just how good can it be? However, I'll bet that if you love the whole Jason series, then I imagine you will probably find something likeable here. Jason kills a lot of people. That should make you happy, right?
Films like this make me glad I have plenty of tequila in the house. It helps to numb the pain. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,180 |
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Lee Min Ho is a young actor, singer and model from South Korea. He's probably best known for his role in the 2009 Boys Over Flowers. In this 25-episode Korean adaption of the popular Japanese manga, he plays the volatile billionaire's son and leader of the F4 (Flower Four), Gu Jun Pyo. The actor is also an award-winning advertisement model for Cass Beer and other brands, and has recently been named as an ambassador for UNICEF's worldwide "Love Net" campaign, meant to eradicate malaria in African countries. Very popular with his fans, the actor was born in Seoul in 1987, and started his acting career as a young boy at the age of ten. His career began in earnest around 2006 with his appearances in the film Arang, Humming (2007), Public Enemy Returns (2008), and Our School E.T. (2008). In television series he has appeared in Secret Campus in 2006, Mackerel Run and I Am Sam both from 2007, and 2008's Get Up.
Become a moderator for Lee Min Ho's profile - contact us to apply! | {
"redpajama_set_name": "RedPajamaC4"
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Huda joined The Breakfast Club this morning to talk about those beautiful handmade candles. The Candles inspired from the Kurdish culture, they can provide anyone as a costume and made candles for special events and your loved one.
Last chance for Rwanga Awards registration, Goran talked about the Rwanga Awards Deadline, for more information visit www.rwanga.org and check the interview below. The Rwanga Awards Deadline is on 10.Nov.2017 in Saad Abdulla hall, Hurry up register and show your talent. See you all there!
What is Kurdcoin – كوردكۆین? Is bitcoin the future!? People are making major money off of it. KurdCoin is your number one choice for buying and trading electronic currencies (cryptocurrencies) in Kurdistan and Iraq. They buy and sell Bitcoin, Etherium, Litecoin, Ripple and other currencies at the best rates.
Ali performs Christina Aguilera's "Hurt" on the Breakfast Club and he's only 13 years-old!
Satar from Erbil Marathon talks to the Breakfast Club about how you can buy your tickets (3,000IQD) and register for the charity marathon this Friday!
Susakan and Mirna from TEDX Nishtiman Women event talk to the Breakfast Club about the exciting lineup of speakers they will have on November 3. Make sure to register to the event before it's too late. | {
"redpajama_set_name": "RedPajamaC4"
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Tsewang Dorji Namjal (1732 - 1750) was de heerser van het kanaat Dzjoengarije.
Achtergrond
In 1745 overleed Galdan Tseren. Het kanaat was toen in zowel politiek, militair als economisch opzicht nog een machtsfactor van betekenis in Centraal-Azië. Vanaf 1739 was er sprake geweest van een vorm van vrede met het China van de Qing-dynastie. Als gevolg daarvan waren de handelsmogelijkheden tussen Dzjoengarije en China aanzienlijk uitgebreid met positieve gevolgen voor de economische basis van het kanaat. Galdan Tseren had in de vruchtbare oasegebieden van zijn rijk in Oost-Turkestan moderne irrigatiesystemen laten implementeren. Er was sprake van een groei van ambachtelijke bedrijvigheid, vernieuwing in methoden voor mijnbouw en een bescheiden metallurgische industrie.
In de laatste jaren van de periode van Galdan Tseren was er wel sprake van oplopende spanningen met Rusland,maar de verhoudingen waren zeker niet onherstelbaar beschadigd. Er was ook een invloedrijke factie binnen de Russische politieke elite, die in de handhaving van een onafhankelijk kanaat Dzjoengarije een belangrijk tegenwicht zag ten opzichte van de Chinese invloed aan de meer zuidoostelijke grens met Siberië.
In de periode na de dood van Galdan Tseren brak een ongekende strijd uit om de macht binnen het kanaat die resulteerde in een volstrekte politieke anarchie. Dit leidde tot de vernietiging van de Dzjoengaren.
Opvolging van Galdan Tseren
Galdan Tseren had drie zoons en een dochter. Bij zijn dood was de oudste zoon Lama Darja 17 jaar oud, de middelste zoon, Tsewang Dorji Namjal, 13 jaar oud en de jongste zoon, Dashi Dawa, 7 jaar oud.
In zijn testament had Galdan Tseren bepaald dat hij opgevolgd zou moeten worden door zijn middelste zoon Tsewang Dorji Namjal. In 1746 werd hij dan ook door de adel van de Dzjoengaren tot hun nieuwe leider geproclameerd.
Er is weinig over Tsewang Dorji Namjal bekend. De schaarse bronnen beschrijven hem echter als iemand die op zijn jeugdige leeftijd al gezien werd als een perverse, wrede en paranoïde man vooral geïnteresseerd in het drinken van grote hoeveelheden alcohol. Zijn zuster, Ulan Bayar, trachtte hem te beperken in dat gebruik. Zij werd door Tsewang Dorji Namjal vervolgens gevangengezet uit vrees dat anderen hem van zijn macht zouden kunnen beroven.
Die vrees was niet ongegrond, want er vele facties in Dzjoengarije en met name zijn oudere broer Lama Darja die daarnaar streefden.
Relaties met ontwikkelingen in Tibet
In 1747 was de regent van Tibet Pholhanas (1689-1747) overleden en opgevolgd door zijn zoon Gyurme Namgyal (overleden 1750). Het regentschap van Pholhanas had Tibet na de burgeroorlogen in de 17e eeuw, de invallen en bezetting van het land door achtereenvolgens Khoshut-Mongolen en Dzjoengaren in de 18e eeuw en weer daarop volgende burgeroorlogen een periode van 25 jaar rust en stabiliteit gebracht. Pholhanas had altijd een politieke lijn gevolgd in nauw overleg met de Chinese keizers. De Chinese ambans in Lhasa waren ervan overtuigd, dat Gyurme Namgyal een beleid wilde voeren dat tegen de Qing-dynastie was gericht. Zij verdachten hem er ook - terecht - van hiervoor te streven naar een alliantie met het kanaat Dzjoengarije.
Sinds het vredesverdrag met China van 1739 konden de Dzjoengaren weer zogenaamde gekookte thee-missies (Ch. aocha, T. manja) naar Tibet over Chinees grondgebied zenden via de route over Xining. Dat waren in de eerste plaats missies van spirituele aard om eerbetoon aan Tibetaanse tulku's te betuigen,maar werden gecombineerd met intensieve handelscontacten. Voor de Dzjoengaren waren deze missies de gelegenheid om weer relaties te herstellen met boeddhistische instituties in Tibet en met de Khoshut-Mongolen in Kokonor.
Juist in 1747 vond de derde missie sinds 1739 plaats, waarbij de Dzjoengaren begin 1748 een grote klomp goud aan de dalai lama gaven en ook cadeaus aan Gyurme Namgyal. De Chinese autoriteiten waren eerdere allianties tussen Dzjoengaren en dalai lama's uiteraard niet vergeten en nu hun vertrouwenspersoon Pholhanas niet meer aanwezig was, zagen zij dit soort ontwikkelingen met groot wantrouwen.
Dit zou ook voorlopig de laatste missie worden die werd toegestaan.
Het eind van Tsewang Dorji Namjal
Het moeten beëindigen van de missies naar Tibet betekende ook een verzwakking van de positie van Tsewang Dorji Namjal binnen de Dzjoengaarse elite. Hoewel zijn oudere broer Lama Darja lang niet de steun had van alle facties, slaagde hij er in 1749 in om via een staatsgreep aan de macht te komen.
Tsewang Dorji Namjal werd eerst verbannen en blind gemaakt. In 1750 laat Lama Darja Tsewang Dorji Namjal vermoorden.
Geschiedenis van Mongolië
Mongolen | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,369 |
title: Encounter
date: '2016-01-01 00:00:00'
subtitle: Interactive Art Installation
quick-summary: Interactive Artificial Ecosystem
published: true # Use this for drafts!
collaborators:
- name: Youhan Guan
url: "http://youhanguan.com/"
- name: Rose Mengmei Zhou
url:
layout: default
modal-id: 7
img: ~
thumbnail: encounter.jpg
alt: encounter
project-date: 2016
software:
- Unity3D (c#)
- Cinema4D
hardware:
- Microsoft Kinect for Windows
- Android/iOS Smartphone and tablets
exhibitions:
- York University Undergraduate Research Fair (February 2016)
- Digital Media Showcase (April 2016)
- Computational Beauty exhibition(April 2016)
- Digifest Festival (April 2016)
category:
- Interaction Design
- Creative Coding
youtube_url: "https://www.youtube.com/embed/zEDyT4MUois"
vimeo_url: ~
description:
- Encounter is an installation where participants can interact with artificial creatures and environments using mixed reality modes of interaction. Participants act as city planners by physical objects on the map to shape the virtual world.
- Using a smartphone/tablet device and a custom app participants can find out more information about the environment using augmented reality. The app recognizes various objects and images and communicates any changes the 'city planners' make.
- In the projection zone, participants control their digital avatars using their body movements. The environment and creatures of this ecosystem respond to their gestures. Any changes made by the city planners are reflected in the projected view in real time.
- For this Project, my primary focus were the participant's interactions in the projection space, projection design and world modelling.
---
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,816 |
DEFENDING FUNNY CAR WORLD CHAMP RON CAPPS ENJOYS SPECIAL ATMOSPHERE OF LUCAS OIL NHRA NATIONALS
BRAINERD, Minn. (Aug. 16, 2022) – There are several reasons why Brainerd International Raceway is a special place for reigning Funny Car world champion Ron Capps, but the incredible atmosphere at the facility is certainly among them. He's eager to experience it at this weekend's 40th annual Lucas Oil NHRA Nationals, where a legion of fans will again be extra excited to cheer on the longtime standout.
The special "Capps Corner" portion of "The Zoo" campgrounds is filled with Capps aficionados, and they'll be out in full force this weekend as the two-time NHRA Camping World Drag Racing Series world champ looks to build momentum for the Countdown to the Championship.
He's put together a solid season in his 11,000-horsepower NAPA Auto Parts Toyota GR Supra, winning twice, advancing to three final rounds, and picking up four No. 1 qualifiers. Capps also has no trouble getting amped up for what will be another special weekend in Brainerd, where he's put together an impressive six victories at the track.
"We do something pretty cool for a living and sometimes you forget that while you're competing, and you really feel like a rockstar when you go there to 'The Zoo,'" Capps said. "The fans love it. They come out to the race for you and when you go out there to see them, it's so cool. There's a fine line to enjoying the Brainerd race like our fans do and also keeping your game face on and we've found a way to walk that line. We'll all go out there together, but just that mojo and feeding off the fans, it's so fun and always a fun race for us."
Steve Torrence (Top Fuel) and Matt Hagan (Funny Car) won the 2021 event, and this year's race will be broadcast on Fox Sports 1 (FS1) and the FOX broadcast network, including eliminations coverage starting at 3:30 p.m. ET on FOX on Sunday, Aug. 21. It is the 15th of 22 races during the 2022 season and penultimate race in the regular season.
Capps won three straight at the track from 2012-2014 and his latest victory in Brainerd came in 2019, which is a victory he'll always remember, mostly due to the post-race scene in "The Zoo" with the Wally, something he'll try to repeat this weekend.
"We cruised around every campground with the Wally and people were so excited to get their pictures with it and with our team," Capps said. "It was so much fun. There were so many pictures floating around that next week, and I think that Sunday night made a lot of people's weekends."
It would certainly make Capps' weekend to add another victory this season leading into Indy and the Countdown to the Championship. Of course, it won't be easy against a loaded Funny Car lineup that features points leader Robert Hight, who has an incredible six wins this year, defending event winner Hagan, longtime rival John Force, who has 11 Brainerd wins, Topeka winner Bob Tasca III and Cruz Pedregon, but Capps likes how his team is performing heading to crunch time.
With ideal temperatures slated for Brainerd, it could also be a weekend of massive runs, something that suits Capps, who has 70 career victories, just fine.
"Every driver and crew chief are looking at that forecast, and Brainerd is always fast," Capps said. "Conditions like this, you could see some records fall. As a Funny Car driver, it's a great, great surface there and we're all pumped up. I feel good about our team and we're doing a good job of implementing some different things within qualifying. But you have to be Countdown ready when you leave Brainerd. You go to Indy trying to win the race, so Brainerd is the last spot to get the car right and ready for the Countdown."
Torrence became a first-time winner at Brainerd last year in Top Fuel, which also gave him a victory at every track on the NHRA Camping World Drag Racing Series circuit. The Texan finished the year with his fourth straight world title, but Torrence will need to be on his game against a loaded Top Fuel field. It's led by Brittany Force, who has four wins and the points lead, and also features a litany of stars like Mike Salinas, another four-time winner in 2022, three-time Brainerd winner Tony Schumacher, Justin Ashley, who has two wins this season, Doug Kalitta, Denver winner Leah Pruett, Josh Hart, Clay Millican, Topeka winner Antron Brown, and Austin Prock | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,284 |
Luxurious Laniwai Spa at the Aulani Resort, A Disney Resort & Spa
Recently I met Lucia Rodriguez Amasio, the Laniwai Spa Director at the Aulani, a Disney Resort and Spa for a tour of her 23,000-square-foot luxurious spa. As she greeted me in the lobby, I learned Laniwai in Hawaiian means freshwater heaven. The backsplash at the reception counter is wavy and white with a hue of…
Hawaiian Wedding at the Four Seasons Oahu in Ko Olina
Finishing up lunch with Yvonne Hunter, the Director of Public Relations at the Four Seasons Oahu Ko Olina Resort, she took me on a tour of the newer resort. Almost three years ago the JW Marriott Ihilani (built in 1998), became the Four Seasons Oahu. Set among green lawns and a dramatic coastline with the…
Hershey Felder Unveils Beethoven at The Wallis
My review was featured in OnStage Blog – http://www.onstageblog.com/reviews/2018/7/30/review-hershey-felder-unveils-beethoven. At the age of nine while attempting to play one of Beethoven's most recognized and beloved piece's Fur Elise, Hershey Felder developed an interest in one of the world's greatest composers. Not only is Felder a brilliant actor, concert pianist, storyteller, he also is a historian. Right…
BOLD Summer in Beverly Hills
Summer brings more daylight to spend outside and enjoy the city of Beverly Hills. BOLD, which stands for Beverly Hills Open Later Days, begins Aug. 2 with a kickoff on Rodeo Drive from 5 to 9 p.m. BOLD continues every Thursday, Friday and Saturday through Aug. 25. Shop the summer nights featuring live entertainment, special…
Fun Birthday Venues in Los Angeles
Planning a birthday is easy in LA. The city has so many fun celebratory venues for brunch, roof top soirees and festive dinner affairs. When friends ask for my advice about a fun birthday destination, I always give them a list of three. Now my list has grown. Here are my favorite restaurants to celebrate…
Authentic Farm-to-Table at FARMHOUSE
Earlier this year I was invited to a preview of Farmhouse at the former Grand Luxe space, located on the ground floor of the newly renovated Beverly Center. The 7,000 square foot space has been transformed to an upscale farmhouse with wood flooring, three dining areas with tall wood ceilings, and lots of green plants. … | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 683 |
my f4i was runnng great untill this week i smell too much fuel and like i feel like fuel is cutting off on certain gears then back on i got scared and now am freaking out where should i check? I just got the bike in a month it was perfect i ride it everyday new oil everything is topped up except this rich smell of fuel and now cutting off but it has power and tourque just like when i bought it it has 7400 miles now what should i do ?
Start by checking the FPR, fuel pressure regulator. See if it's leaking into the vacuum line.
Last edited by Luqmaan; 05-24-2018 at 12:42 AM.
Did you disconnect the vacuum line and prime the pump or start the engine?
You can get one at a local dealer or search online.
I've bought OEM parts from places like bikebandit.com and hondaparts-direct.com before. | {
"redpajama_set_name": "RedPajamaC4"
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{"url":"https:\/\/courses.lumenlearning.com\/precalctwo\/chapter\/solutions-10\/","text":"Solutions to Try Its\n\n1.\u00a0The amplitude is $\\text{ }3$, and the period is $\\text{ }\\frac{2}{3}$.\n\n2.\n\nx $3\\sin \\left(3x\\right)$\n0 0\n$\\frac{\\pi }{6}$ 3\n$\\frac{\\pi }{3}$ 0\n$\\frac{\\pi }{2}$ $-3$\n$\\frac{2\\pi }{3}$ 0\n\n3.\u00a0$y=8\\sin \\left(\\frac{\\pi }{12}t\\right)+32$\nThe temperature reaches freezing at noon and at midnight.\n\n4.\u00a0initial displacement =6, damping constant = -6, frequency = $\\frac{2}{\\pi }$\n\n5.\u00a0$y=10{e}^{-0.5t}\\cos \\left(\\pi t\\right)$\n\n6.\u00a0$y=5\\cos \\left(6\\pi t\\right)$\n\nSolutions to Odd-Numbered Exercises\n\n1.\u00a0Physical behavior should be periodic, or cyclical.\n\n3.\u00a0Since cumulative rainfall is always increasing, a sinusoidal function would not be ideal here.\n\n5.\u00a0$y=-3\\cos \\left(\\frac{\\pi }{6}x\\right)-1$\n\n7.\u00a0$5\\sin \\left(2x\\right)+2$\n\n9.\u00a0$4\\cos \\left(\\frac{x\\pi }{2}\\right)-3$\n\n11.\u00a0$5 - 8\\sin \\left(\\frac{x\\pi }{2}\\right)$\n\n13.\u00a0$\\tan \\left(\\frac{x\\pi }{12}\\right)$\n\n15.\u00a0Answers will vary. Sample answer: This function could model temperature changes over the course of one very hot day in Phoenix, Arizona.\n\n17.\u00a09 years from now\n\n19.\u00a0$56^\\circ \\text{F}$\n\n21.\u00a0$1.8024$\u00a0hours\n\n23.\u00a04:30\n\n25.\u00a0From July 8 to October 23\n\n27.\u00a0From day 19 through day 40\n\n29.\u00a0Floods: July 24 through October 7. Droughts: February 4 through March 27\n\n31.\u00a0Amplitude: 11, period: $\\frac{1}{6}$, frequency: 6 Hz\n\n33.\u00a0Amplitude: 5, period: $\\frac{1}{30}$, frequency: 30 Hz\n\n35.\u00a0$P\\left(t\\right)=-15\\cos \\left(\\frac{\\pi }{6}t\\right)+650+\\frac{55}{6}t$\n\n37.\u00a0$P\\left(t\\right)=-40\\cos \\left(\\frac{\\pi }{6}t\\right)+800{\\left(1.04\\right)}^{t}$\n\n39.\u00a0$D\\left(t\\right)=7{\\left(0.89\\right)}^{t}\\cos \\left(40\\pi t\\right)$\n\n41.\u00a0$D\\left(t\\right)=19{\\left(0.9265\\right)}^{t}\\cos \\left(26\\pi t\\right)$\n\n43.\u00a0$20.1$ years\n\n45.\u00a017.8 seconds\n\n47.\u00a0Spring 2 comes to rest first after 8.0 seconds.\n\n49.\u00a0500 miles, at ${90}^{\\circ }$\n\n51.\u00a0$y=6{\\left(5\\right)}^{x}+4\\sin \\left(\\frac{\\pi }{2}x\\right)$\n\n53.\u00a0$y=8{\\left(\\frac{1}{2}\\right)}^{x}\\cos \\left(\\frac{\\pi }{2}x\\right)+3$","date":"2020-04-06 03:31:31","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4187487065792084, \"perplexity\": 1702.9920319485898}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-16\/segments\/1585371612531.68\/warc\/CC-MAIN-20200406004220-20200406034720-00503.warc.gz\"}"} | null | null |
Q: Prove that $(f, g, h)$ is a linearly independent list of vectors in $\mathbb{R}[x]^S$ "Recall that $\mathbb{R}[x]$ is a vector space. Suppose that $f, g, h \in \mathbb{R}[x]^S$ and that there is $q \in S$ such that $f(q) = 1$, $g(q) = x^2 + 1$, and $h(q) = x^2 + x$. Prove that $(f, g, h)$ is a linearly independent list of vectors in $\mathbb{R}[x]^S$"
My attempt:
Suppose that $a_1f + a_2g + a_3h = 0$ $\forall s \in S$
Then $a_1f + a_2g + a_3h = 0$ for $s=q$
Hence $a_1(1) + a_2(x^2+1) + a_3(x^2+x) = 0$
Suppose x = 0. Then $a_1(1) + a_2(1) + a_3(0) = 0 \iff a_1 = -a_2$
Suppose x = -1. Then $a_1(1) + a_2(2) + a_3(0) = 0 \iff a_1 = -2a_2$
Thus $a_1 = -a_2 = -2a_2 \iff a_2 = 2a_2 \iff a_2 = 0 \iff a_1 = 0$
Now suppose x = 1. Then $0(1) + 0(2) + a_3(2) = 0 \iff a_3(2) = 0 \iff a_3 = 0$
Because the only solution to the equation is $a_1 = a_2 = a_3 = 0$ it follows that (f,g,h) is a linearly independent list of vectors in $\mathbb{R}[x]^S$
I have no idea if I'm on the right track or not...
| {
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ACCEPTED
#### According to
International Plant Names Index
#### Published in
null
#### Original name
Ardisia bifaria C.T.White & W.D.Francis
### Remarks
null | {
"redpajama_set_name": "RedPajamaGithub"
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\section{Introduction}
Polymer confinement in nano-scale geometries has become a problem of interest in recent years. This phenomenon has applications in the design of nanotechnology devices, such as polymer separation, DNA sequencing, and protein sensing \cite{separation,rant,sequencing,barcode}. It is also a ubiquitous phenomenon in biological environments. Packaging of the viral genome in the capsid \cite{packaging} and its ejection into the host cell \cite{ejection1,ejection2}, translocation of RNA through nuclear pores and protein translocation across the endoplasmic reticulum are known examples \cite{alberts}. Advances in the fabrication of nano-structures for polymer confinement have also lead to considerable achievements in polymer physics. Entropic effects on polymers arisen from confinement were first observed in 1999 \cite{entropic-trapping,craighead1,craighead2}.
Polymer translocation is the passage of a polymer through a nano-pore in a membrane. To date, biological nano-channels have been used for DNA sequencing purposes; while solid-state nano-pores are of interest as a tool for single-molecule studies. The two protein channels used for DNA sequencing are $\alpha$-hemolysin and MspA \cite{dekker}. When DNA passes through the cylindrical part of the $\alpha$-hemolysin channel, the ionic current through the channel depends on 10-15 nucleotides that reside inside the channel. However, the protein channel MspA has a conical shape and the ionic current depends on few nucleotides near the cone tip. Indeed, the electric field is focused in the tip of a cone shaped channel, which also pronounces the difference between the ionic current signals received from the four nucleotides \cite{MspA1,MspA2}.
For interaction between polymers and cone-shaped structures, other biological and technological examples can be counted: The HIV-1 capsid which confines its viral genome has a cone-shaped structure \cite{HIV}, coating of the conical nano-pores with DNA is used for tuning their ionic properties \cite{cone-polymer}, and asymmetric conical pores are shown to act as Brownian ratchets for colloids \cite{colloid} and polymers \cite{unpublished}. In addition, there exist cases that the polymer is confined inside a cone as a result of interaction with the surrounding polymers \cite{werner} or a hydrodynamic flow \cite{sakaue2007}.
Polymers confined near surfaces and inside different geometries such as cylindrical and cone-shaped nano-channels, nano-slits and nano-spheres have been studied (Ref. \cite{hoseinpoor} and references therein). Generally, confinement reduces the number of possible configurations of the polymer and causes its entropy to decrease. For a polymer near a surface, confinement leads to entropic forces on the polymer which are measurable with the enhanced sensitivity of current experimental devices (e.g. AFM and optical tweezers) \cite{kardar1,kantor}. Entropic forces are also the main driving force for polymer escape from confined geometries \cite{luijten}. For a polymer in an asymmetric confinement such as a cone, difference in the entropy of the polymer in the two sides results in an entropic force on the polymer toward the larger side \cite{khalilian}.
\begin{figure*}
\includegraphics[scale=0.23]{fig1.eps}
\caption{(a) Schematics of the polymer inside a long cone-shaped channel (Case I). $\alpha$ is the apex angle and $D_0$ is the opening diameter of the channel, where the polymer end is fixed. The blob size $\xi(x)$ is equal to the local diameter of the channel at position $x$. The polymer is stretched along the channel to size $R_{||}$. The asymmetric shape of the channel exerts an entropic force on the polymer toward the base of the channel.
(b) For a finite channel, a long enough polymer extends to outside the channel (Case II). $L$ defines the channel length along it symmetry axis. The length, the apex angle and the opening diameter of the channel determines the number of monomers that lie inside the channel and the entropic force. (c) A second method to define the blobs. The blobs are tangent to the cone surface and cannot penetrate each other. The first blob is tangent to the beginning of the channel. The number of monomers inside the first blob and the size of the first blob are used to find the limits of applicability of the theory in the cases I and II, respectively.} \label{fig1}
\end{figure*}
In a previous work, the authors studied the translocation of a flexible polymer through a cone-shaped nano-channel, using MD simulations. It was shown that the entropic force from the cone results in a forced polymer translocation. The passage time was shown to have a non-monotonic dependence on the cone angle and a slight dependence on the cone length. A theoretical description was developed for a flexible polymer inside a cone. Small and large cone angles were studies separately, because of the crowding effect in front of the nano-channel in small angles. The number of monomers inside the nano-channel and the entropic force was obtained for a polymer inside a closed cone and inside an open cone-shaped channel. It was shown that the theory can explain the translocation time dependence on the cone angle and length, qualitatively \cite{khalilian}.
In this paper, we use Molecular Dynamics (MD) simulations to further investigate a polymer confined inside a cone-shaped nano-channel. A flexible polymer that one of its ends is fixed inside the channel is studied by two different sets of simulations. In the first set, the channel is so long that the polymer is completely confined inside it (Case I). Extension of the polymer along the channel axis and the entropic force on the polymer are studied in these simulations. It is seen that the polymer extension along the channel rapidly decreases with the cone angle. However, the force has no considerable dependence on the channel angle and the polymer length. In the second set of simulations, the polymer is so long that more than half of the monomers lie outside the channel (Case II). The number of monomers that remain inside the channel and the force on the polymer are investigated, in this case. The number of monomers inside the channel is found to increase with the channel angle, especially for longer channels. However, the force dependence on the channel length is ignorable. Despite the case I, dependence of the force on the channel angle is considerable, in these simulations. These results on the dependence of the force on the properties of the cone-shaped channel can be important in the design of nano-pores for practical purposes. The force arisen from the asymmetric shape of the channel on the polymer is around four times the force from thermal fluctuations. So, its effects on the polymer behavior in experimental conditions can be significant which is explained in details in this manuscript.
The results of MD simulations are also used to check the correctness of the theoretical framework developed in Ref. \cite{khalilian}, quantitatively. It is shown that the four sets of data obtained from the cases I and II can be fitted with the theoretical functions, with only three fit parameters. The theory also explains the distribution of monomers inside the channel. The excellent agreement between the theory and the simulations confirms the strength of the theory in description of a polymer in conical confinement.
The manuscript is organized as follows. In the next section, a brief review of the theory of Ref. \cite{khalilian} and some new notes are presented. In section \ref{method}, the simulation method is described. The simulation results and their agreement with the theory are explained in section \ref{results}. The last section contains a summary of the manuscript, as well as discussions on the correspondence of the results with related experiments and the previously published results.
\section{Theory} \label{theory}
As case I, consider a flexible polymer that one of its ends is fixed at the tip of an infinitely long cone-shaped channel (Fig. \ref{fig1}(a)). In polymer physics, the blob approach is commonly used to study polymer confinement. In length scales smaller than the blob size, the polymer does not feel that it is confined; while, at length scales larger than the blob size, the confinement is dominant. For a polymer inside a cone, the blob size depends on the position along the axis. The blob size is proportional to the local diameter of the channel; $\xi(x) \sim D_0+2(x+a)\tan\alpha$. $D_0$ and $\alpha$ are the diameter of the beginning of the channel and the apex angle of the channel, respectively. $a$ is the distance along the channel axis between the fixed end of the polymer and the beginning of the channel. Here, $a$ is a mathematical tool and is set equal to zero, after calculations. For clarity, this parameter is not shown in Fig. \ref{fig1}.
Number of monomers inside each blob is $g(x) \sim \left(\frac{\xi(x)}{b}\right)^{\frac{1}{\nu}}$ . $\nu$ is the Flory exponent and $b$ is the monomer size. The number of monomers inside a volume of thickness $dx$ along the channel is $dn(x) \sim \frac{g(x)}{\xi(x)} dx$. So the linear density of the monomers along the channel axis $\lambda(x) = \frac{dn(x)}{dx}$ becomes
\begin{equation} \label{density}
\lambda(x) \sim \frac{1}{b}\left(\frac{\xi(x)}{b}\right)^{\frac{1}{\nu}-1}.
\end{equation}
Accordingly, the linear density of the monomers changes with the local diameter of the channel to the power 0.7.
The polymer extension along the channel, $R_{||}$, is obtained by equating the integral over the number density of the monomers with the total number of monomers of the polymer; $N \sim \int_0^{R_{||}} \lambda(x) dx$. The relation between $N$ and $R_{||}$ becomes
\begin{equation} \label{N}
N \sim \frac{b^{-\frac{1}{\nu}}}{\tan\alpha} \left[\left(D_0+2(a+R_{||})\tan\alpha\right)^{\frac{1}{\nu}}-\left(D_0+2a\tan\alpha\right)^{\frac{1}{\nu}}\right].
\end{equation}
On the other hand, the free energy of confining the polymer is calculated from $\frac{F}{k_BT} \sim \int_0^{R_{||}} \frac{dx}{\xi(x)}$
\begin{equation} \label{FE}
\frac{F}{k_BT} \sim \frac{1}{\tan\alpha} \left[\ln\left(D_0+2(a+R_{||})\tan\alpha\right)-\ln\left(D_0+2a\tan\alpha\right)\right].
\end{equation}
The entropic force on the polymer originates from the tendency of the polymer toward the channel base, where it has more space available and larger entropy. As a result, the force is proportional to the changes in the free energy of the polymer when the polymer moves toward the channel base. For a given value of $N$, the force is calculated from the derivative of the free energy with respect to the position of the fixed end of the polymer, $a$. Equation \ref{FE} describes the free energy as a function of the polymer extension $R_{||}$, which is itself a function of the parameter $a$. Considering these points, the force becomes $\frac{f}{k_BT} \sim -\left(1+\left(\frac{\partial R_{||}}{\partial a}\right)_{a=0}\right)\frac{1}{D_0+2R_{||}\tan\alpha}+\frac{1}{D_0}$. The derivative of $R_{||}$ with respect to $a$ is found from eq. \ref{N}. After calculation of the derivatives, $a$ is set equal to zero. The force is obtained as \cite{khalilian}
\begin{equation} \label{fL}
\frac{f}{k_BT} \sim \frac{1}{D_0}\left[1-\left(\frac{D_0}{D_0+2R_{||}\tan\alpha}\right)^{\frac{1}{\nu}}\right]
\end{equation}
Constant $B$ is multiplied into equation \ref{N} to convert it into equality.
Then, this equation is rearranged to give the polymer extension $R_{||}$ as a function of the total length of the polymer $N$ and the channel angle $\alpha$ and tip diameter $D_0$;
\begin{equation} \label{L}
\tilde{R}_{||}\tan\alpha =\left[\left(\frac{N}{B}\tan\alpha+\tilde{D}_0^{\frac{1}{\nu}}\right)^{\nu}-\tilde{D}_0\right].
\end{equation}
$A$ is introduced as constant of proportionality into equation \ref{fL}.
Then, equation \ref{L} is substituted into equation \ref{fL} to obtain the force as a function of the polymer and the channel parameters;
\begin{equation} \label{fN}
\tilde{f} = \frac{A}{\tilde{D}_0} \frac{\frac{N}{B}\tan\alpha}{\frac{N}{B}\tan\alpha+\tilde{D}_0^{\frac{1}{\nu}}}.
\end{equation}
The tilde sign shows that the lengths and the force are scaled with $b$ and $\frac{k_BT}{b}$, respectively.
For case II, consider the polymer confined inside a cone-shaped channel of finite length $L$ (the length of the cone-shaped channel along its symmetry axis, see Fig. \ref{fig1}(b)). Trivially, some of the monomers remain outside the channel, for long enough polymers. So, the channel length determines the force and the number of monomers of the polymer that are inside the channel. In this case, $N$ represents the number of monomers inside the channel. Indeed, the monomers outside the channel do not contribute to the entropic force on the polymer, because they do not feel any confinement. In other words, free-energy of the polymer segment outside of the channel does not depend on $a$.
Taking $a=0$ and substituting $L$ for $R_{||}$, equation \ref{N} describes the number of monomers inside the channel versus the channel parameters $L$, $\alpha$ and $D_0$;
\begin{equation} \label{N2}
N\tan\alpha = B \left[\left(\tilde{D}_0+2\tilde{L}\tan\alpha\right)^{\frac{1}{\nu}}-\tilde{D}_0^{\frac{1}{\nu}}\right].
\end{equation}
The force exerted on the polymer as a function of the channel parameters is obtained from equation \ref{fL} ($R_{||}$ is replaced by $L$);
\begin{equation} \label{fL2}
\tilde{f} = \frac{A}{\tilde{D}_0}\left[1-\left(\frac{\tilde{D}_0}{\tilde{D}_0+2\tilde{L}\tan\alpha}\right)^{\frac{1}{\nu}}\right].
\end{equation}
$A$ and $B$ are the previously introduced proportionality constants.
There is another method to define the confinement blobs for a polymer inside a cone-shaped channel \cite{khalilian} (Fig. \ref{fig1}(c)). The blobs can be defined as spheres that are tangent to the internal surface of the channel and cannot penetrate each other. The size of the blob at each point becomes $\xi(x) = D_0\cos\alpha + 2x\sin\alpha$. The first blob is restricted to be tangent to the beginning of the channel, $x_1 = \frac{\xi(x_1)}{2}$. Using these two relations, the size of the first blob is obtained as $\xi(x_1) \sim \frac{D_0\cos\alpha}{1-\sin\alpha} $. The number of monomers inside the first blob is obtained $g(x_1) \sim \left(\frac{\tilde{D_0}\cos\alpha}{1-\sin\alpha} \right)^{\frac{1}{\nu}}$. $\xi(x_1)$ and $g(x_1)$ can be used to find the limits of applicability of the theory of a polymer confined inside a cone-shaped nano-channel in the two cases.
If the total number of the monomers is smaller than $g(x_1)$ in case I, the infinite channel would have no confining effect on the polymer. In case II, $\xi(x_1)$ gives the shortest length of the cone-shaped channel that exerts an entropic force on the polymer.
For example, with $D_0=1.4b$ and $\alpha=50\,^{\circ}$, one obtains $g(x_1)=10$ and $\xi(x_1)=4b$.
\section{Simulation method} \label{method}
Coarse-grained MD simulations using ESPResSo \cite{espresso} are employed to study a flexible polymer confined inside a cone-shaped nano-channel. The polymer is modeled by a bead-spring chain. The monomers interact with each other and the channel walls, via the shifted and truncated Lennard-Jones potential $U_{LJ}=4\epsilon\left[\left(\frac{b}{r}\right)^{12}-\left(\frac{b}{r}\right)^{6}+\frac{1}{4}\right]$. The cut-off radius for the potential is $2^{\frac{1}{6}}b$. $\epsilon$ and $b$ are the energy and length scales of the interaction, and $r$ is the distance between the monomers.
Adjacent monomers along the polymer are attached by the FENE potential $U_{FENE}=-\frac{1}{2}KR_0^2\ln\left[1-\left(\frac{r}{R_0}\right)^2\right]$. $K=100\frac{\epsilon}{b^2}$ and $R_0=1.5b$ are the spring constant and the maximum distance between the adjacent monomers. The simulations are performed under constant temperature $T=1.0\frac{\epsilon}{k_B}$ using the langevin thermostat with the damping constant $1.0\tau_{MD}^{-1}$ \cite{thermostat}. $\tau_{MD}=b\sqrt{\frac{\epsilon}{m}}$ is the time unit of the simulations, where $m$ is the mass of monomers. Equations of motion are integrated using the Velocity-Verlet algorithm, with a time step equal to $0.01\tau_{MD}$.
At the beginning of the simulations, the polymer is arranged on the cone axis. The first monomer at the tip of the channel is fixed during the simulations. All simulations are continued for $5N^2$ time units. The polymer relaxation time is of the order of the polymer length to the power two. As a result, averages of different quantities are measured after the time $N^2$. These quantities are calculated at each time unit (or equally after each 100 time steps) of the simulations. Totally, $4N^2$ configurations of the system are used to calculate the averages.
Prior to calculation of the total average of the force, moving-average is used to reduce the noise in the force data. The noise results from random motions of the polymer on short time scales that the polymer does not feel the confinement. To this end, the force value at each time is replaced by the average of the force over a time interval starting from the specified time and spanning over $\frac{1}{5}$ of the relaxation time of the polymer.
The final error-bars are smaller than the size of the symbols, in all plots.
The smallest diameter of the channel $D_0$ is taken equal to $1.4b$. The channel angle is changed from $1\,^{\circ}$ to $50\,^{\circ}$.
Two sets of simulations are performed. In case I, the channel length is taken to be two times the total length of the polymer. Three different values for the number of monomers of the polymer, 100, 200 and 300 are examined. In case II, the number of monomers of the polymer is determined such that more than half of the monomers lie outside the channel. Test simulations show that further increasing the total number of monomers do not affect the results in this case.
Channel lengths $10b$, $15b$ and $20b$ are tested. In the first case, the radius of gyration and the end-to-end vector of the polymer parallel to the channel axis are calculated. In the second case, the number of monomers inside the channel is counted.
To find the entropic force on the polymer, sum of the Lennard-Jones forces that the monomers exert on the channel wall is calculated. The result of the force summation is then averaged over time. It is observed that only the component of the average force parallel to the channel axis is nonzero.
The average entropic force exerted by the channel to the polymer is equal in magnitude but opposite in direction to the above-mentioned forces.
\begin{figure}
\includegraphics[scale=0.8]{distlc2.eps}
\caption{Log$_{\text{10}}$-log$_{\text{10}}$ plot of the simulation results for linear density of the monomers along the channel axis versus local diameter of the channel. The simulation results are shown with dots. The polymer has 300 monomers and three different values for the apex angle $1\,^{\circ}$, $5\,^{\circ}$ and $20\,^{\circ}$ are tested. Data points corresponding to some of the monomers of the two polymer ends are not shown, to remove the ends effect. According to the theory, the linear density of the monomers along the channel axis is proportional to the local diameter of the channel to the power 0.7 (Eq. \ref{density}). The slope of the solid line is 0.7 for comparison with the theory.
Inset: Simulation results for the linear density of the monomers along the channel axis. Solid lines show the theoretical prediction; $\lambda(\tilde{x})=P(\tilde{D}_0+2\tilde{x}\tan(\alpha))^{\frac{1}{\nu}-1}$. $\tilde{D}_0=1.4$ and $\nu=0.588$ are used and the proportionality constant $P$ is taken as the fit parameter. For all the curves, $P=0.7$ is obtained.} \label{distlc}
\end{figure}
\begin{figure}
\includegraphics[scale=0.8]{lcone.eps}
\caption{Simulation results for a polymer inside a long channel (Case I). Three different polymer lengths N = 100, 200 and 300 are examined. In simulation of each polymer, the length of the channel is taken equal to two times of the contour length of the polymer. The radius of gyration of the polymer along the channel axis (a) and the entropic force on the polymer (b) are shown versus the channel angle. Although the radius of gyration of the polymer changes rapidly with the channel angle, the entropic force has a weak dependence on the channel angle. The force also depends weakly on the polymer length. This shows that the entropic force originates mainly from the narrow tip of the channel. A schematic of the polymer inside the long channel is also shown in the panel (b). } \label{lcone}
\end{figure}
\section{Simulation results} \label{results}
\subsection*{Case I: Long channel}
The linear density of the monomers along the channel axis obtained from simulations of a polymer containing 300 monomers is shown in the inset of Fig. \ref{distlc}. Three different angles for the channel $1\,^{\circ}$, $5\,^{\circ}$ and $20\,^{\circ}$ are examined.
Log-log plot of the linear density of the monomers versus the local diameter of the channel is shown in the main panel of Fig. \ref{distlc}. Data points related to the two ends of the chain are eliminated. The slope of the solid line is 0.7, the exponent predicted by Eq. \ref{density} of the theory. The data is irregular at small values of the local diameter affected by the fixed end of the polymer.
\begin{figure}
\includegraphics[scale=0.8]{lcone2.eps}
\caption{Simulation results for a polymer inside a long channel (Case I). Data points of the previous figure are shown in rescaled axes, according to equations \ref{L} and \ref{fN} of the theory.
As can be seen, data points related to different polymer lengths follow a master curve in the rescaled axes. This shows a good agreement between the theory and the simulation results.
The solid lines show equations \ref{L} and \ref{fN} in panels (a) and (b), respectively. The values $\nu=0.588$, $D_0=1.4b$, $A=5.29$ and $B=0.35$ are used in the equations. The values of the constants $A$ and $B$ are obtained from fitting the theory to the simulation results of case II. Another constant $C=3.5$ is multiplied in the radius of gyration in panel (a). This constant is equal to the ratio of the end-to-end vector of the polymer to its radius of gyration, from the simulation results. } \label{lcone2}
\end{figure}
The radius of gyration of the polymer parallel to the channel axis versus the channel angle is shown in Fig. \ref{lcone}(a). The channel angle is changed from $1\,^{\circ}$ to $50\,^{\circ}$. Three different polymer lengths N = 100, 200 and 300 are examined. The radius of gyration decreases rapidly with the channel angle. Its dependence on the polymer length is considerable in small angles. However, the difference between the three polymers becomes negligible in larger angles.
The entropic force on the polymer versus the channel angle is shown in Fig. \ref{lcone}(b). Despite the radius of gyration of the polymer, the force has a week dependence on the channel angle. Although, the force on the longer polymer is larger for all angles, there is only a slight difference between the forces acting on the polymers of different lengths. This is because the entropic force on the polymer is exerted mainly by the narrow parts of the channel. Overall, the force is around $3.8k_BT/b$ for different channel angles and polymer lengths. Indeed, the magnitude of the entropic force mainly depends on the diameter of the channel in its tip side, $D_0$. The important note is that this entropic force is larger than $k_BT$ and its effect can be considerable in practical situations.
Equations \ref{L} and \ref{fN} describe the polymer in the long channel corresponding to case I. These equations give the polymer extension along the channel and the force on the polymer as a function of the channel angle and the polymer length. It can be deduced from these equations that if $R_{||}\tan\alpha$ and $f$ are plotted versus $N\tan\alpha$, all data points would fall on a master curve. Figures \ref{lcone2}(a) and \ref{lcone2}(b) show the previous figures of case I, with modified axes. It is observed that the curves related to different polymer lengths collapse onto a single curve, according to one's expectation.
\subsection*{Case II: Long polymer}
Simulation results for linear density of the monomers along the channel axis are sketched in Fig. \ref{distlp}, for monomers inside the channel. Four different angles for the channel $1\,^{\circ}$, $5\,^{\circ}$, $20\,^{\circ}$ and $50\,^{\circ}$ are examined. The log$_{\text{10}}$-log$_{\text{10}}$ plot of the scaled linear density versus the local diameter for points away from the tip of the channel is shown in the inset of Fig. \ref{distlp}. The slope is close to the theoretical value, 0.7.
\begin{figure}
\includegraphics[scale=0.8]{distlp2.eps}
\caption{Simulation results for linear density of the monomers along the channel axis. The channel length is $L=20b$ and four different channel angles $1\,^{\circ}$, $5\,^{\circ}$, $20\,^{\circ}$ and $50\,^{\circ}$ are examined. Solid lines show the theoretical prediction; $\lambda(\tilde{x})=P(\tilde{D}_0+2\tilde{x}\tan(\alpha))^{\frac{1}{\nu}-1}$. $\tilde{D}_0=1.4$ and $\nu=0.588$ are used and the proportionality constant $P$ is taken as the fit parameter. The obtained values for $P$ change from 1.7 to 0.9, with increasing the apex angle.
Inset: Log$_{\text{10}}$-log$_{\text{10}}$ plot of the simulation results for the scaled linear density of the monomers versus the local channel diameter away from the tip of the channel. The value of the linear density at each angle is divided by the parameter $P$. As is seen, all data collapse on a single line. The black solid line has the slope of 0.7 in agreement with Eq. \ref{density} of the theory. Data points related to $\alpha=1\,^{\circ}$ are not shown, considering the nearly constant value of the local diameter along the channel axis. } \label{distlp}
\end{figure}
The number of monomers that are inside the channel is plotted in Fig. \ref{lpolymer}(a). The number of monomers inside the channel increases with the channel angle. This increase is especially considerable for the longer channel with $L=20b$.
The entropic force on the polymer versus the channel angle is shown in Fig. \ref{lpolymer}(b). The force dependence on the channel angle is stronger than case I, which is due to the finite length of the channel. However, despite the growing number of monomers inside the channel (Fig. \ref{lpolymer}(a)), the force on the polymer does not change with the channel length. This again results from the fact that the force originates mainly from the narrow sections of the channel.
In Fig. \ref{lpolymer}(b), the angles between $1\,^{\circ}$ and $5\,^{\circ}$ are also included to compare the results with those of Ref. \cite{khalilian}. It is seen that the force is a monotonic function of the channel angle. This contradicts with functionality of the translocation time of a polymer through a cone-shaped channel, which is a non-monotonic function of the channel angle. It confirms the assumption of Ref. \cite{khalilian} that this non-monotonic behavior is a consequence of the non-equilibrium nature of the translocation process. As the polymer passes through the channel, the monomers crowd at the channel exit and the cone-shaped channel becomes similar to a closed cone-shaped space, for small apex angles.
\begin{figure}
\includegraphics[scale=0.8]{lpolymer.eps}
\caption{Simulation results for a long polymer in a cone-shaped channel (Case II). Three different channel lengths $L=10b$, $15b$ and $20b$ are tested. Total number of monomers is taken such that more than half of the polymer remains outside the channel.
The number of monomers of the polymer that are inside the channel (a) and the entropic force on the polymer (b) are shown versus the channel angle. The number of monomers inside the channel depends on both the channel length and angle. However, the entropic force increases and plateaus with the channel angle and does not depend on the channel length. A schematic of the long polymer and the channel is also shown in panel (b).} \label{lpolymer}
\end{figure}
\begin{figure}
\includegraphics[scale=0.8]{lpolymer2.eps}
\caption{Simulation results for a long polymer in a cone-shaped channel (Case II).
Data points of the previous figure are shown in rescaled axes, according to equations \ref{N2} and \ref{fL2} of the theory.
As can be seen, data points related to different polymer lengths follow a master curve in the rescaled axes. This shows a good agreement between the theory and the simulation results.
The solid lines are the theoretical fit to the simulation results. Equations \ref{N2} and \ref{fL2} are fitted to the simulation results in panels (a) and (b), respectively. The values $\nu=0.588$ and $D_0=1.4b$ are used, and $A=5.29$ and $B=0.35$ are obtained from the fit. } \label{lpolymer2}
\end{figure}
Equations \ref{N2} and \ref{fL2} describe the long polymer in the channel corresponding to case II. These equations give the number of monomers inside the channel and the force on the polymer versus the channel length and angle. Here, one should plot $N\tan\alpha$ and $f$ versus $L\tan\alpha$ to have all the data points collapsed on a master curve. Figs. \ref{lpolymer}(a) and \ref{lpolymer}(b) are shown with appropriate axes in Figs. \ref{lpolymer2}(a) and \ref{lpolymer2}(b). The data points related to different channel lengths merge into a single curve. This shows an excellent agreement between the theory and the simulations.
\subsection*{Fitting theory to simulation results}
The equations \ref{N2} and \ref{fL2} are used to fit the theory to the simulation results for the case II. The proportionality constants $A$ and $B$ are taken as the fit parameters. The values $\nu=0.588$ and $D_0=1.4$ are assumed, according to the simulation conditions. The theoretical results for the number of monomers inside the channel and the entropic force exerted by the channel are shown in Figs. \ref{lpolymer2}(a) and \ref{lpolymer2}(b). It can be seen that the theoretical function follows the simulation results, very well. The fit parameters are obtained as $A=5.29$ and $B=0.15$.
The values obtained for $A$ and $B$ in case II are substituted in equation \ref{fN}, to describe the force exerted by the cone in case I. The resulting function is sketched in Fig. \ref{lcone2}(b). It is seen that the theory describes the simulation results very well.
Equation \ref{L} describe the extension of the polymer $R_{||}$ inside the cone-shaped nano-channel. However, the radius of gyration of the polymer $R_g$ is measured in the simulations. These two quantities are proportional to each other, for long channels. As a result, it is necessary to define a third proportionality constant $C$. Equations \ref{L} and $R_{||}=CR_g$ are fitted to the simulation results. The theoretical function is in excellent agreement with the simulation results. The fit parameter is obtained as $C=3.5$. This value is equal to the ratio of the end-to-end vector of the polymer to its radius of gyration, from simulation results.
\subsection*{Force dependence on the tip diameter}
\begin{figure}
\includegraphics[scale=0.8]{force1d.eps}
\caption{Entropic force versus the tip diameter of the channel in case I. The symbols are the simulation results for a polymer with 100 monomers inside a long channel with the angle $\alpha=20\,^{\circ}$. The values $\nu=0.588$, $A=5.29$ and $B=0.35$ are used in Eq. \ref{fN} to obtain the theoretical prediction shown with the solid line.
The force dependence on the tip diameter is considerable, comparing with its dependence on the polymer length and the channel angle shown in Fig. \ref{lcone}(b). A good qualitative agreement can be seen between theory and the simulation data. } \label{force1d}
\end{figure}
In the simulations of case I, the force does not depend on the polymer length. This means that the force originates mainly from the narrow part (the tip side) of the channel. This is also observed in further simulations; by changing $D_0$ from $1.26b$ to $3.5b$, the force drops from $4.4\frac{k_BT}{b}$ to below $\frac{k_BT}{b}$. The simulation results for a polymer with 100 monomers and a channel with $\alpha=20\,^{\circ}$ are shown with symbols in Fig. \ref{force1d}. This strong dependence of the force on the tip diameter of the channel can be independently obtained from eq. \ref{fN} of the theory. The force however depends weakly on the other parameters of the system in case I (the channel angle and the polymer length), according to both the theory and the simulations. For case II, the theory also predicts a sharp change of the force with the tip diameter of the channel.
\section{Summary and discussion} \label{discuss}
In summary, we have studied polymer confinement inside a cone-shaped channel, theoretically and using MD simulations. The problem is considered in two cases: (I) the channel is longer than the polymer, and (II) the polymer is longer than the channel. The polymer extension along the channel and the number of monomers of the polymer that lie inside the channel are examined in cases I and II, respectively. The entropic force originated from the asymmetric shape of the channel is investigated in both cases. It was shown that the polymer extension along the channel and the number of monomers inside the channel change rapidly with the channel angle and length. However, the force from the channel in case I does not depend on the polymer length and the channel angle. In case II, the entropic force depends on the channel angle for small values of the apex angle. In both cases, the force depends strongly on the opening diameter of the channel.
It is also instructive to have a comparison between the results presented here and those of Ref. \cite{khalilian}. Polymer translocation through a cone-shaped nano-channel under no external forces has been consiered in Ref. \cite{khalilian}, while the entropic force and the static properties of the polymer are studied in equilibrium condition in the present study. In Ref. \cite{khalilian}, the entropic force from the nano-channel is the driving force for the translocation process.
According to the present study, the entropic force increases and saturates with the apex angle. Because the entropic force is the driving force of the translocation process, one expects the translocation time to decrease and then plateau with the angle. This trend is generally observed in the results of ref. \cite{khalilian}. The translocation time reduces sharply from that of a cylindrical channel in small apex angles.
However, a slight local maximum is observed afterwards in the middle values of the apex angle. This is related to the non-equilibrium crowding of the monomers behind the nano-channel at forced polymer translocation. The crowding effect increases with the driving force and decreases with the area of the channel exit. According to the present study, the entropic force reaches to its highest value at $\alpha=20\,^{\circ}$ and then plateau. The local maximum of the translocation time is observed around this angle and then there is a plateau in larger angles.
In the previous study, the translocation time had a weak dependence on the channel length, in agreement with the results of the present study.
According to the present simulation results, the force from the channel has a maximum around $4\frac{k_BT}{b}$, where $b$ is the monomer size. If we consider the translocating polymer to be single-stranded DNA (ssDNA), the size of the coarse-grained monomers is around $1nm$ \cite{rapid}. So, the entropic force from the cone can be as large as $16pN$, which can be considerable in practical situations.
In experiments, the MspA protein channel has an apex angle around $10\,^{\circ}$ and an opening diameter of $D_0=1.2b$ \cite{trends}. ssDNA is usually entered into the MspA through its base, using an applied electric voltage.
It is possible to immobilize the strand inside the nano-channel, using a hairpin construct at one end \cite{MspA1} or a DNA polymerase \cite{MspA2}.
The entropic force on the immobilized ssDNA from the MspA channel can be calculated theoretically from eq. \ref{fL2}, which is approximately equal to $16pN$.
Although a detailed knowledge of the electric field inside the cone-shaped channel is required to find the electric force, it can be estimated using the relation $F_{elc}=q\frac{\Delta V}{\ell}$.
Most of the applied electric voltage $\Delta V$ falls on the narrow constriction of the cone-shaped nano-channel and one can assume an effective length for the nano-channel $\ell=2nm$.
$q$ is the total charge of the nucleotides that are inside this effective length. Each nucleotide has 0.3nm length and one electron charge, $e$.
The applied voltage difference that gives an electric force equal to the entropic force from the channel can be obtained as $\Delta V=30mV$.
Experimentally, the threshold voltage required to keep ssDNA inside the MspA channel can be measured. The threshold voltage depends on the entropic contributions both from the nano-channel and the membrane containing the nano-channel. An immobilized polymer inside a nano-channel can be divided into three segments: the segment inside the channel and the two arms outside the channel. Each of the two outside arms of the polymer is nearly fixed on the wall containing the nano-channel. The wall reduces the available configurations for the polymer and thus exerts an entropic force on the polymer \cite{kardar2}.
Comparison between the threshold voltage of MspA and that of a cylindrical nano-channel can be used to separate the two entropic contributions. ssDNA fixed inside a cylindrical nano-channel (such as $\alpha$-hemolysin) feels only the entropic force from the wall and can be used as a reference to find the entropic force from the MspA channel.
In the experiments of ssDNA translocation through MspA, the voltage difference $\Delta V=180mV$ is often used which gives an electric force equal to $96pN$ on the strand. The polymer is often entered from the base of the cone-shaped channel, to achieve higher capture rates \cite{entry,capture}. In this condition, the entropic force from the channel acts against the direction of the polymer entry and can reduce the capture rate. However, non-equilibrium effects are determining in polymer translocation. The polymer becomes stretched when it enters the pore \cite{sakaue2007,farahpour} and the monomers crowd close to the pore after the passage. As ssDNA enters from the base of the MspA channel, crowding does not alter the entropic force but stretching can reduce interaction of the polymer with the channel walls and decrease the entropic force.
Finally, the theory and the simulation results for the polymer configuration and the entropic force on the polymer inside a cone-shaped channel are in good agreement. The simulation results are well fitted with theoretical functions. It was shown that the entropic force from the channel can be significant in ssDNA translocation through MspA channel. In these experiments, the entropic force depends on the angle and the tip diameter of the channel, and not on the channel length. It means that changing the angle of the cone-shaped channel may have a determining role in polymer translocation but changing the channel length has no noticeable effects. This point can be considered in the design of cone-shaped channels through protein engineering \cite{MspA2}, or through solid state methods in fabricating solid-state cone-shaped channels \cite{solid}. In some experiments, the polymers are used to alter transport properties of cone-shaped nano-channels \cite{cone-polymer}. Distribution of monomers and the polymer extension inside the cone-shaped nano-channel are important in these experiments. The obtained results give a detailed view on these quantities.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,245 |
\section{INTRODUCTION}
The infrared ultraluminous galaxies are recent galaxy mergers in which
much of the gas in the former spiral disks has fallen into the center.
The gas re-establishes rotational support at a radius of a few hundred
parsecs. This is the same size scale as the Narrow Line Region in
active galactic nuclei, and the rotating nuclear disks in some
elliptical galaxies. During its infall from 5\,kpc to 0.5\,kpc, the
high-density gas forms many massive stars, as in the early starbursts
that formed the bulge stars in protogalaxies at high redshifts. The
ultraluminous galaxies we observe in the local universe are
re-enacting this primeval process, with a large part of the central
mass in gas ready to turn into stars --- and ready to fall into a
black hole.
The large nuclear concentration of molecular gas in ultraluminous
galaxies has been detected in the millimeter lines of CO by many
groups during the past decade. Scaling to the signal strengths from
Milky Way molecular clouds, however, soon led to a paradox for many of
the sources --- the estimated gas mass was equal to or larger than the
dynamical mass indicated by the linewidths. For Arp~220, for example,
Scoville et al. (1991) found that nearly all of the mass in the
central few hundred parsecs was in the form of molecular gas. To
resolve this dilemma, we showed that in the extreme environment in the
central 600\,pc of ultraluminous galaxies, much of the CO luminosity
must come from an intercloud medium that fills the whole volume,
rather than from clouds bound by self gravity, and therefore the CO
luminosity traces the geometric mean of the gas mass and the dynamical
mass, rather than just the gas mass (Downes, Solomon, \& Radford
1993). This allows the nuclear disk gas to be overluminous in CO,
relative to self-gravitating clouds in galactic spiral arms. The
important linewidth for this calculation is the dynamical linewidth of
the galaxy, not that of individual self-gravitating GMCs. The
physical basis for estimating gas mass from the CO emission is
therefore different from that in Milky Way GMCs. The Milky Way
conversion factor is relevant for an ensemble of GMCs in an ordinary
spiral galaxy, but not for the center of an ultraluminous galaxy.
In a subsequent survey paper on 37 ultraluminous galaxies observed
with the IRAM 30\,m telescope (Solomon et al. 1997), we derived the
molecular gas mass by several methods. The minimum estimates of the
gas mass, for optically thin CO, did indeed reduce the gas mass to
much lower fractions of the dynamical mass. The true gas mass must be
between the optically thin CO estimate and the dynamical mass, {\it
both} of which are less than the gas mass derived with the Milky Way
conversion factor for self-gravitating molecular clouds.
The significance of the lower molecular gas mass is twofold. First,
it makes the gas mass smaller than the dynamical mass, as it must be.
Second, it means the total molecular gas mass in ultraluminous
galaxies is similar to, and not greater than, the molecular mass in
the disk of a gas-rich spiral. It is thus not necessary to force into
the nuclear region all the gas in the galaxy --- H$_2$ and H~I. The
nuclear starbursts can be fed by pre-existing molecular gas that falls
into the central few hundred parsecs from an original radius of a few
kiloparsecs, and not from the outer H~I disk.
These ideas show that the kinematics of the nuclear gas is critical to
understanding both the evolution of ultraluminous galaxies and the
molecular line formation. The CO emissivity per unit gas mass surface
density is not constant, but varies with position in the nuclear disk.
A more correct treatment requires radiative transfer modeling, with
kinematic data from millimeter interferometers.
To carry out this next step in our study of molecular gas disks at the
centers of IR ultraluminous galaxies, we made high-resolution CO maps
with the IRAM interferometer. We chose sources from our CO survey,
concentrating on the nearest ultraluminous galaxies in the northern
sky, for which we already had accurate measurements of the total flux,
linewidth, and radial velocity. The new observations give accurate
positions, line and continuum fluxes at the interferometer resolution,
images, directions of the kinematic major axes, and radii of the
nuclear disks.
\footnote
{In this paper, distances are for
$H_0 = 75$\,km\thinspace s$^{-1}$ \,Mpc$^{-1}$ and $q_0 = 0.5$.}
These radii and the velocity extrema give the enclosed dynamical mass.
The velocity data allow modeling of the kinematics and the CO
brightness temperature versus radius. For most sources, the CO data
indicate the gas kinetic temperature is higher than in typical
Galactic molecular clouds, but the CO is subthermally excited,
turbulent, and warm, quite unlike the molecular clouds in galaxy-scale
disks. At the positions of maximum CO intensity in the disks, the CO
lines are moderately opaque, with optical depth $\tau$(1--0) $\approx
$ 4 to 10, but in other parts of the disk, the CO lines can be
optically thin. The result is that the molecular gas mass is about a
factor of five lower than it would be if the CO emission came from
self-gravitating clouds. The rest of the dynamical mass is accounted
for by young stars that provide the luminosity and older stars that
were already in the central bulge before the merger.
We argue that the bright spots in the ultraluminous
galaxies are compact extreme starburst regions that occur in the dense gas
traced by HCN and CS emission. They produce
most of the HCN luminosity, and 1mm dust continuum while the more diffuse gas in the
nuclear disks dominates the CO luminosity.
These compact extreme starburst regions which we identify in Arp~193, Arp~220 and Mrk~273 are
the most prodigious star formation events in the local universe. There are undoubtedly many more
too compact to be resolved.
The plan of this paper is as follows. Section~2 describes the
observations and summarizes the parameters obtained by direct
measurement. Section~3 introduces our model of a turbulent rotating
disk and discusses the derivations of the rotation curve, disk radius,
disk height, turbulent velocity, gas mass, and dynamical mass obtained
by fitting the model disk to the data. Sections~4 through 8 present
the results on the galaxies with CO(2--1) data, obtained with
sub-arcsecond resolution. These galaxies are VII~Zw~31, Mrk~231,
Arp~193, Mrk~273, and Arp~220. Section~9 gives the results on five
additional ultraluminous galaxies mapped in CO(1--0) only. Section~10
discusses the source sizes, gas masses, the ratio of gas mass to
dynamical mass, and the mass of young stars needed to power the
starburst.
Section~11 contains our
conclusions.
\section{OBSERVATIONS AND RESULTS}
CO(1--0) and (2--1) were observed with the IRAM interferometer on
Plateau de Bure, France (Guilloteau et al. 1992). Three to four
configurations of the four 15\,m antennas gave baselines from 24 to
410\,m for VII~Zw31, Mrk~231, Mrk~273, and Arp~220, and 24 to 288\,m
for the other sources. The SIS mixers had receiver temperatures of 60
-- 80\,K, and operated single sideband at 3\,mm and double sideband at
1.3\,mm with system temperatures of 200 and 400\,K, respectively. The
spectral correlator covered a 500\,MHz band at 2.5\,MHz resolution,
giving velocity ranges of 1300\,km\thinspace s$^{-1}$\ at 7\,km\thinspace s$^{-1}$\ resolution at 3\,mm,
and 700\,km\thinspace s$^{-1}$\ at 3.4\,km\thinspace s$^{-1}$\ resolution at 1.3\,mm. For analysis, we
smoothed the data to 20 and 40\,km\thinspace s$^{-1}$ . Amplitude and phase were
calibrated relative to quasars and the flux scale was adopted from
same-epoch measurements of quasars and planets with the interferometer
and the IRAM 30\,m telescope. The beamwidths (FWHM) were 1$''$ to
3$''$ at CO(1--0) and $0''.5$ to 1$''$ at CO(2--1). Table~1
summarizes the observing parameters.
Table~2 lists CO source positions and integrated CO line fluxes
from the interferometer. We also list the CO line fluxes from the
30\,m telescope (Radford, Solomon, \& Downes 1991a; Solomon et
al. 1997). Table~3 lists sizes from Gaussian fits to the CO
visibilities. Sources observed in CO(1--0) and (2--1) had the same
sizes in both lines. The visibility fits are independent of the
synthesized beam or the CLEAN algorithm, and there is no need to
deconvolve an apparent size on a map. The phase calibration can
broaden a source due to the baseline error, varying as $2\pi \lambda
/D$ times the angle between the source and its phase calibrator
($\lambda =$ wavelength, $D=$ baseline). For sources and calibrators
in this paper, the broadening is $< 0.1$ beamwidth. After correcting
for atmospheric decorrelation, we obtained sizes $<0''.2$ for the
quasar calibrator sources.
\section{DISK MODELS}
\subsection{Geometry of Nuclear Disks from Kinematic Data}
CO surveys of our Galaxy show most of the gas in the inner $R<5$\,kpc
is in $\sim 6000$ giant molecular clouds (GMCs) of diameter $\sim
50$\,pc and H$_2$ density 150\,cm$^{-3}$. In an ultraluminous IR
galaxy merger, these clouds fall into the central $R\sim 500$\,pc,
forming a disk of height $\sim 50$\,pc. Because the GMCs' density is
too low to stabilize them against tidal shear at the merger's center,
and because the new volume is ten times smaller than the original
volume of all the GMCs in the galactic disk, the GMCs loose their
identity and blend into a continuous medium with a mean H$_2$ density
$\sim 10^3$\,cm$^{-3}$. We therefore modeled the nuclear disks as a
continuous medium rather than an ensemble of individual clouds.
We used the CO spectral data to estimate how much of the gas was in a
high-density, inner ring or disk and how much in a lower-density,
outer disk. To simulate the observed spectra and position-velocity
diagrams, we modified a model for rotating disks developed by
A.~Dutrey (see Dutrey, Guilloteau, \& Simon 1994). This model has an
inner disk of high-density gas between radii $R_{\rm min}$ and $R_1$ =
$R_{\rm min}+\Delta R$, and an outer disk of lower-density gas between
radii $R_{\rm min}$ and $R_{\rm max}$. The H$_2$ density between
$R_{\rm min}$ and $R_{\rm max}$ is
\begin{equation}
n(R) =
n_0\, A\,\exp \bigg[ - 4 \ln 2
\bigg({ {R-R_{\rm min}}\over {\Delta R}} \bigg)^2\ \bigg]
\ \ + \ \ n_0\, R^\alpha \ \ \ \ ,
\end{equation}
where the inner disk is the Gaussian of width $\Delta R$ (FWHM), and
the outer disk is the power law. We fit the outer disks with constant
density ($\alpha = 0$). Our beams cannot distinguish between a ring
and a filled disk, so for most sources, we took $R_{\rm min}$ = 0.
Because of the radiative transfer through the rotating, inclined disk,
the predicted CO map looks like a ring, even if there is gas in to $R
= 0$. For the galaxies VII~Zw~31, Arp~193, and IRAS 10565+2448, the
position-velocity diagrams are better fit with ring models, with
$R_{\rm min}$ = $R_0$, rather than filled disk models. Table~4
gives the source geometry, derived as follows:
{\it Line of nodes:}
The visibility fits in individual spectral channels gave position
offsets vs.\ velocity, which showed the direction of the kinematic
major axis. We checked the angle of the line of nodes from channel
maps and isovelocity contour maps. Errors are $\pm 10^\circ$.
{\it Rotation curve turnover radius $R_0$:}
In the data, the position-velocity diagrams along the kinematic major
axis often have two peaks near the source center. In our models, twin
peaks occur near the points where the rotation curve turns over and
becomes flat. The observed offsets of the two intensity peaks on the
major axis were tried as first guesses for the rotation curve turnover
radii $R_0$. Model fits converged close to these values. In angular
units, the uncertainties are $\sim\pm 0''.1$. In our models, $R_0$ is
the projected radius with the largest gas column density at the same
line-of-sight velocity, and the highest flux. For most of the
sources, the solid angle inside this zone is too small to affect the
maps, even if the gas is really in a filled disk rather than a ring.
{\it Inner disk half-intensity radius, $R_1$ = $R_{\min}+\Delta R$:}
In our models, most of the CO flux comes from the high-density inner
disk. The extent of the central double peak in the position-velocity
diagrams constrained the radial width $\Delta R$, which we varied
until the model position-velocity diagrams matched the data. The
uncertainty in $\Delta R$ is $\sim\pm 0''.2$
The {\it outer radius, $R_{\max}$,} of the low-density disk was taken
to be the observed CO maximum extent on the line of nodes.
{\it Disk thickness $H$:}
Along the $z$-axis perpendicular to the equatorial plane, the gas
volume density $n(z)$ near $z=0$ was approximated by a gaussian with
full width to half-maximum $H$. For a disk with a flat rotation
curve, the disk's thickness can be estimated with the Mestel formula
(Mestel 1963; see also Binney \& Tremaine 1987):
\begin{equation}
H(R) = 1.4 \ \sigma (R) \bigg( {R\over V_{\rm rot}} \bigg) \
\bigg[1+ {\rho_{\rm gas}\over \rho_1 } \bigg(1- {H \over 2R}\bigg)\bigg]^{-0.5}
\ \ \ ,
\end{equation}
where $\sigma$ is the 1-D velocity dispersion (assumed
independent of $z$) and $\rho_1(R)$ is the total mass density (gas plus stars)
in an equivalent sphere with a flat rotation curve.
This equivalent density, in M$_\odot$ pc$^{-3}$, is related to
the rotation velocity, in km\thinspace s$^{-1}$ ,
through the usual integral for dynamical mass,
$\rho_1(R) = 18.5 (V_{\rm rot}/R)^2$.
We dropped the second-order term and estimated the disk thickness as
\begin{equation}
H \approx 1.4 \ { \sigma (R) \bigg( {R\over V_{\rm rot}} \bigg) }
\bigg[1+ {\rho_{\rm gas}\over 18.5}\bigg( {R\over V_{\rm rot}} \bigg)^2 \
\bigg]^{-0.5}
\ \ \ ,
\end{equation}
where $H$ and $R$ are in pc, $V_{\rm rot}$ is in km\thinspace s$^{-1}$ , $\rho_{\rm
gas}$ is in M$_\odot$ pc$^{-3}$. Because the velocity dispersion
decreases with distance from the center, our model disk thickness
tends to a constant value. In reality, the molecular disks are warped
and twisted, and their shape depends on the merger's infall history.
\subsection{Rotation Curves}
We combined our density models with rotation curves that
reproduced the observed channel maps, position-velocity diagrams,
spectral profiles, interferometer visibilities, and line intensities.
We took the rotation velocity to be
\begin{equation}
V_{\rm rot}(R) = V_0\, \bigg({R \over R_0 } \bigg)^\beta \ \ \ \ ,
\end{equation}
with $\beta = 1$ for $R<R_0$ and $\beta= 0$ for $R_0 \leq R \leq R_{\rm max}$,
that is, the curve rises from the center and flattens
after $R_0$. Most of the derived values of $R_0$ are
smaller than our beam, so more elaborate rotation curves are
not justified for now.
For the {\it inclination, $i$}, we started from the major/minor axis
ratio of the integrated CO source, with $i=\cos^{-1}$(minor/major),
and then varied the inclination to match the position-velocity data
and to ensure the derived gas mass was less than the dynamical mass.
The nuclear rings or disks may not be circular; they may be
elliptical, analogous to $x_2$ orbits at the center of a barred
galaxy. The inclination $i$ listed in Table~5 should be
interpreted in the sense $\sin i = \sin I \cos \phi$, where $I$ is the
true inclination of the disk relative to face-on, and $\phi$ is the
azimuth of the major axis of the ellipse, if there is one. With our
current beams, we cannot tell the difference between circular and
elliptical orbits.
For the {\it rotation velocity, $V_0$}, we first guessed the apparent
speed, $V_0 \sin i$, to be half the velocity range of the twin CO
peaks on the major axis. The best fits were close to these values,
giving 200 to 300\,km\thinspace s$^{-1}$\ (corrected for inclination) on the flat part
of the curve (Table~5). This is an equivalent circular velocity
if the orbits are elliptical.
The {\it turbulent velocity, $\Delta V$,} of the gas was taken from
the model fit, not from the observed line profiles, which include both
turbulent broadening and the rotational velocity gradient in the
beam. Rotation alone cannot explain the observed profiles. We need a
local, turbulent line broadening as well. The models actually
constrain this parameter rather well. We assumed the local line
broadening had the form
\begin{equation}
f(V-V_0) =
\exp \bigg[ -\bigg({ {V-V_0}\over {\Delta V}} \bigg)^2\ \bigg]
\end{equation}
where $\Delta V$ is the local line halfwidth to the $1/e$ level
($\Delta V$ = $0.6 \times$ FWHM = 1.4$\sigma$, where $\sigma$ is the
r.m.s.\ velocity dispersion along the line of sight). The radiative
transfer model included the local linewidth at each point and the
rotational velocity versus radius. We iterated the model fits until
the local turbulence and the rotational velocity gradient ---
convolved with our beam --- matched the observed position-velocity
diagrams and the observed line profiles. Turbulent velocities $\Delta
V$ that fit the data were 30 to 140\,km\thinspace s$^{-1}$ , with some of the
position-velocity diagrams indicating lower turbulence at greater
radii from the nucleus. At the rotation curve turnover point $R_0$,
the values of $\Delta V$ correspond to local FWHM linewidths of 70 to
230\,km\thinspace s$^{-1}$ , so the nuclear disks are highly turbulent.
\subsection{CO line parameters}
After deriving sizes and velocities from the data, we adjusted
our models to match the observed CO line intensities. As a
first guess, we set the gas kinetic temperature equal to the
dust temperature derived from blackbody fits to the IRAS fluxes.
At the gas densities in our models, collisions
cannot raise the CO excitation temperature to the gas kinetic
temperature, nor populate the CO levels thermally.
Instead, there are different excitation temperatures for each
transition. The line brightness temperatures, excitation temperatures,
and gas kinetic temperatures are in the sense $T_b<T_{\rm ex}<T_{\rm kin}$,
with $T_{\rm ex}(2-1) < T_{\rm ex}(1-0)$.
We therefore assumed the CO level populations
were determined by the local gas kinetic temperature, density, and
velocity gradient, and we calculated, with an escape probability program,
CO column densities per unit velocity width
for a local velocity gradient of 1\,km\thinspace s$^{-1}$\,pc$^{-1}$ and a
[CO/H$_2$] abundance of $8\times 10^{-5}$, both typical of
Milky Way molecular clouds.
For these values, the escape probability method yields
the following non-LTE excitation temperatures $T_{\rm ex}$:
\begin{equation}
T_{\rm ex}(1 - 0) = 0.186 \ n^{0.622} \ T^{0.240}_{\rm kin}
\end{equation}
\begin{equation}
T_{\rm ex}(2 - 1) = 0.117 \ n^{0.678} \ T^{0.247}_{\rm kin} \ \ \ ,
\end{equation}
where $n$ is the H$_2$ number density in cm$^{-3}$, and
$T_{\rm kin}$ is the gas kinetic temperature in K.
These formulae are valid for
$50<T_{\rm kin}<150$\,K and $100<n<3000$\,cm$^{-3}$.
We then calculated the radiative transfer on the lines of sight
through the disk (eqs.\ 7 \& 8 of Dutrey et al. 1994), convolved the
computed map with our beam, and adjusted the parameters to match the
observed brightness temperatures. Highest CO opacity occurs at the
rotation curve turnover radius, $R_0$, which yields the largest column
density at the same line-of-sight velocity. Table~6 gives the
best-fit model CO(1--0) excitation temperatures and gas densities, and
Tables~7 and 8 give the model CO(1--0) and (2--1) line opacities
and Rayleigh-Jeans brightness temperatures. We also ran models for
higher density gas, with excitation temperatures closer to the kinetic
temperatures, but these models gave too much CO and dust flux, and did
not reproduce the position velocity diagrams.
At the rotation curve turnover radius $R_0$, our models yield
CO(1--0) opacities of 2 to 8. These mean opacities apply to smoothly
distributed gas. The model of subthermal excitation yields the
observed CO(2--1)/(1--0) ratios of 0.6 to 1.0 (Radford et al. 1991a).
In the lower-density, outer disks detected in these sources, the
volume-averaged densities in the model are too low to yield detectable
CO lines. For these outer disks, our model must be corrected with an
area filling factor for molecular clouds, as in CO maps of disks of
normal galaxies. For the mean values for the outer disks in
Table~6, we adopted a volume filling factor of 0.1, an area filling
factor of 0.3, and local gas densities 10 times higher than the
volume-averaged values listed in the Table.
\subsection{Gas Mass and Dynamical Mass}
We used the CO parameters to estimate the gas mass and the dynamical
mass {\it vs.} radius (Table~9). After we found a density law
that reproduced the observed brightness temperatures, we integrated
over the source to get the gas mass. These masses are for a
[CO/H$_2$] abundance of $8\times 10^{-5}$, as in molecular clouds in
our Galaxy. For the sources observed here, it is difficult to lower
the model CO abundance below the Milky Way value. Keeping the same CO
line intensity at a lower CO abundance would force us to make the gas
mass bigger than the dynamical mass, invalidating the model.
The dynamical masses were taken to be $R\,V_{\rm rot}^2/G$, where
$V_{\rm rot}$ is the rotation speed listed in Table~5. Most of
the sources have flat rotation curves, so the dynamical mass increases
linearly with radius. Table~9 lists the ratio of gas mass to
dynamical mass in the inner, high-density disk alone (to the radius
$R_1$ in Table~4) and in the inner and outer disks together (to the
radius $R_{\rm max}$ in Table~4). Relative to our assumed Hubble
constant, the gas mass scales as $H_0^{-2}$, the dynamical mass as
$H_0^{-1}$ and their ratio as $H_0^{-1}$.
We also list in Table~9 the maximum ratio of gas to total (gas +
stars) surface density in our models. We calculated the ratio of gas
to total mass surface density, $\mu/\mu_{\rm tot}$, in 15 bins in
radius, and found maximum values of $\sim $1/3 to 1/4, midway between
the rotation curve turnover radius $R_0$ and the half-intensity radius
$R_1$. The models in this paper are for the distributed gas, but
there is also denser gas in star-forming cores that give rise to
HCN and CS lines. Most of the CO flux comes from the distributed
medium, but our gas masses may have to be corrected upward to allow
for the cores, depending on future interferometer results in the
dense-gas tracer lines. We return to this point in the Discussion section.
\subsection{Dust Continuum Flux}
We used the model gas mass to predict the dust flux, from
\begin{equation}
S(\nu_{\rm obs}) = (1+z) \kappa(\nu_r)M D^{-2}_L B(\nu_r, T_d) \ \ \ \ \ ,
\end{equation}
where $S$ is the dust continuum flux density,
$\nu_{\rm obs}$ and $\nu_r$ are the observed and rest-frame frequencies,
$M$ is the {\it gas} mass, $D_L$ is the luminosity
distance, $B$ is the Planck function, and $T_d$ is the dust temperature.
We took the dust temperatures from our fits to
the IRAS fluxes (Solomon et al. 1997) --- the same
as the gas temperatures in Table~6 --- and
we used a dust mass absorption coefficient
$\kappa (\nu_r ) = 0.1 \times \nu^n$,
where $\nu_r$ is in THz, $\kappa$ is in
cm$^2$\,gm$^{-1}$ of interstellar matter, and we took the index $n =1.5$.
At 230\,GHz, this absorption coefficient is
$\kappa$ $=$ 0.011\,cm$^2$ gm$^{-1}$ of interstellar matter, or, for
a gas-to-dust mass ratio of 100, $\kappa_d$ $=$ 1.1\,cm$^2$ gm$^{-1}$
of dust, as in previous estimates for dense molecular clouds
(see Kr\"ugel \& Siebenmorgen 1994, their Fig.~12).
The predicted thermal fluxes from dust are then
\begin{equation}
S(\nu_{\rm obs}) = 6.4\times 10^{-7} (1+z) M \ D^{-2}_L \ T_d \
\nu^{2+n}_r \ \ \ \ \ ,
\end{equation}
where flux density $S$ is in Jy, gas mass $M$ is in M$_\odot$ ,
luminosity distance $D_L$ is in Mpc, dust temperature $T_d$ is in K,
and rest frequency $\nu_r$ is in THz.
The predicted thermal dust fluxes at 2.6 and 1.3\,mm are compared
with the observed fluxes in Table~10.
Four sources, VII~Zw~31, Mrk~231, Mrk~272, and Arp~220, were observed
in both the CO(1--0) and (2--1) lines, and on the longest baselines of
410\,m. Because we have better angular resolution for these sources,
we discuss them in more detail than our other sources.
\section{ {\bf VII Zwicky 31} }
This source, catalogued by Zwicky (1971), was identified by Fairclough
(1986) as an ultraluminous IR galaxy. Its optical surface brightness
profiles resemble those of elliptical galaxies and its optical line
ratios fit a starburst rather than an AGN (Djorgovski, de Carvalho, \&
Thompson 1990). Sage \& Solomon (1987) found this source had one of
the highest known CO luminosities and deduced that the molecular gas
was a large part of the dynamical mass. Scoville et al. (1989) showed
the CO source had a radius $< 3.8\,$kpc.
The CO maps of the galaxy VII Zw 31 are the best evidence in
our sample for a rotating ring. Models that best fit the data are
those with a minimum ring radius $R_{\rm min}$ in eq.(1) equal to
$R_0$ = 290\,pc. Filled-disk models, with $R_{\rm min}$ = 0, give
poorer fits, with much less contrast. In 40\,km\thinspace s$^{-1}$ -wide channel maps,
the CO peak migrates from south to north with increasing velocity
(Fig.~1). The source is resolved east-west, perpendicular to
the kinematic major axis. This indicates the source is inclined, but
more face-on than edge-on. Our modeling gives a good fit to the data
for a ring inclined at 20$^\circ$ to face-on. Figure~2 shows
the maps of integrated intensity and isovelocity contours across the
source in both CO lines. The total CO(1--0) flux measured with the
interferometer is equal to the flux obtained at the 30\,m telescope
(Radford et al. 1991a). At CO(2--1), the source is partially resolved
out, and the flux from the interferometer is only half the single-dish
flux. In channels off the CO line, the continuum flux is $<2$\,mJy at
109\,GHz and $<10$\,mJy at 221\,GHz. This limit on any thermal dust
flux is consistent with the gas mass of $1.1\times 10^{10}$ deduced
from the CO luminosity, for a dust temperature $<50$\,K.
The CO velocity contours in Fig.~2 indicate the north-south
velocity gradient. In the position-velocity diagrams (Fig.~3)
there is a 200\,km\thinspace s$^{-1}$\ velocity shift over 4$''$ north-south. The
model in the Tables reproduces well the observed diagrams, the CO
profiles across the source, the channel maps, and the observed line
intensities. The CO spectra (Fig.~4) have twin peaks separated
by $\pm 70$\,km\thinspace s$^{-1}$ . The blue and redshifted CO peaks are respectively
$0''.65$ south and north of the source centroid. The line profiles
are remarkably symmetric along the north-south kinematic major axis.
The CO line of nodes differs from the northwest-southeast dust lane
found by Djorgovski et al.\ (1990) on a 10$''$ scale by subtracting a
model from their optical images.
\section{ {\bf Markarian 231 } }
The Seyfert~I galaxy Markarian 231 has an IR luminosity of $3.3\times
10^{12}$\,L$_\odot$\ (e.g., Sanders et al. 1987). Most of the power is
emitted at the center of a major galaxy merger which has tidal tails
extending over 75\,kpc (Cutri, Rieke, \& Lebofsky 1984; Hutchings \&
Neff 1987; Sanders et al. 1987). Optical spectroscopy indicates
stars formed in the past 1~Gyr over a widespread region
(10--15\,kpc). In this large extranuclear region the H$\alpha$
emission comes from shocks rather than stellar photoionization (e.g.,
Hamilton \& Keel 1987; Lipari, Colina, \& Macchetto 1994). The best
previous CO study is that by Bryant \& Scoville (1996), who found an
east-west velocity gradient in the CO source. Our higher-sensitivity
results have a factor of two better resolution than their study, and
not only confirm the velocity gradient, but also distinguish the
inner and outer disks in the position-velocity diagrams.
\subsection{CO in Mrk 231's molecular disk}
In contrast to the large optical extent, the CO source is very
compact, on a scale twenty times smaller in radius. In 40\,km\thinspace s$^{-1}$\
channel maps in CO(1--0) and (2--1), the line peak shifts from west to
east with increasing velocity (Fig.~5). The gradient is clearly
shown on the CO isovelocity maps (Fig.~6). The CO(2--1)
intensity contours (Fig.~6) are symmetric and only slightly
broader than the beam. The CO(2--1) position-velocity diagram along
the line of nodes (Fig.~7) shows an inner nuclear disk of
diameter $1''.2$ (radius 460\,pc) and a 3$''$ outer disk with a lower
velocity gradient. The asymmetric CO(2--1) line profiles $0''.4$ from
the center of the galaxy (Fig.~8) are a characteristic signature
of a rotating nuclear disk. The maximum linewidth in both CO lines is
190\,km\thinspace s$^{-1}$\ FWHM, which is narrow for ultraluminous galaxies.
The CO data imply the molecular disk is face-on, for three reasons.
{\it 1)} The CO major and minor axes are nearly equal, indicating an
inclination $i \leq 20^\circ$. {\it 2)} To reproduce the
single-peaked CO profile, narrow CO linewidth, and observed velocity
gradients, our model disks must be within $20^\circ$ of face-on. For
$i \geq 20^\circ$ to face-on, the gas mass would exceed the dynamical
mass. {\it 3)} The CO does not absorb the nuclear continuum source at
any velocity. This agrees with the CO disk being face-on, leaving the
Seyfert~I nucleus unobscured. For $i= 10^\circ$ to face-on, the true
rotation velocity would be 345\,km\thinspace s$^{-1}$ , the same as we deduce for
Arp~220. If the disk were even more face-on, the observed velocity
gradient would imply a rotation speed $> 400$\,km\thinspace s$^{-1}$ , higher than in
all the other galaxies in our sample, which seems unlikely. We adopt
$i= 10^\circ$ and $V_{\rm rot}$ = 345\,km\thinspace s$^{-1}$ , although the kinematic
data alone would also allow a rotation velocity of 250\,km\thinspace s$^{-1}$ .
In CO, we do not detect the gas emitting in the 1667\,MHz OH megamaser
with 760\,km\thinspace s$^{-1}$\ linewidth (Staveley-Smith et al. 1987). nor do we
detect anything at the IR source $3''.5$ south of the nucleus, to a
limit of 5\,mJy in 40\,km\thinspace s$^{-1}$\ channels (see also Bryant \& Scoville
1996). Armus et al. (1994) interpreted the southern IR source as the
nucleus of the merger partner, but recent HST images show this feature
to be a dense arc of star-forming knots (Surace et al. 1998). In any
case, it is very weak in both CO and the mm continuum.
In the center of Mrk~231, the AGN is detected by centimeter wavelength
VLBI as a variable, nonthermal radio continuum source with a size $<
1$~pc (Preuss \& Fosbury 1983; Neff \& Ulvestad 1988; Lonsdale,
Smith, \& Lonsdale 1993; Taylor et al. 1994). New VLBA images at
1.4\,GHz resolve this nonthermal source into a nuclear core with
pc-scale lobes. Carilli, Wrobel \& Ulvestad (1998) subtracted their
VLBA map from a VLA map and found an $0''.4$ nonthermal nuclear disk
source that emits half the nuclear flux, the rest coming from the
core and the pc-scale lobes.
At millimeter wavelengths, outside the CO line channels, we detect an
unresolved (size $<0''.1$) nonthermal, mm continuum source at the
nucleus, which is probably the nonthermal VLBA core. The mm continuum
source and the CO centroid both coincide with the 8.44\,GHz continuum
source (Condon et al.\ 1991).
\subsection{Mrk~231's Luminosity: 2/3 Starburst, 1/3 AGN}
The CO kinematic data for the central 1 kpc of Mrk~231 and our model
of a face-on disk show the far-IR luminosity comes from a starburst,
not a black hole accretion disk. We assume the central UV-to-FIR
continuum has three components: the visible and UV continuum is
blackbody flux from the AGN accretion disk, the 2$\mu$m peak is
blackbody flux from a dusty AGN torus, and the far-IR peak is
blackbody flux from the molecular disk. The extinction to the
accretion disk is low, $A_V$ = 2\,mag, because both the dusty torus
and the larger-scale molecular disk are face-on.
\noindent
{\it The accretion disk's luminosity} may be derived from the UV and
visible fluxes and the extinction to the Seyfert~I nucleus. The UV
fluxes measured by IUE and HST are $2\times 10^{-15}$\,erg s$^{-1}$
cm$^{-2}$ \AA$^{-1}$ (Schmidt \& Miller 1985; Hutchings \& Neff 1987;
Smith et al. 1995). A blackbody with this flux, corrected for a UV
extinction of 5\,mag (Smith et al. 1995) and a distance of 170\,Mpc,
has a UV luminosity of $4\times 10^{11}$\,L$_\odot$ . For an accretion
disk temperature of 20000 to 30000\,K, as in the models of Malkan \&
Sargent (1982) and Sanders et al. (1989), the blackbody peaks at 2000
to 1000\,\AA, and has a radius of $1\times 10^{-3}$\,pc. This result
is consistent with the optical continuum, which has $A_V$ = 2\,mag and
an intrinsic magnitude $M_V = -25.1$ (luminosity $5\times
10^{11}$\,L$_\odot$ ; Boksenberg et al. 1977).
\noindent
{\it The dusty torus' luminosity} may be derived from the near-IR
continuum. As in other Seyfert~I galaxies, the 2$\mu$m bump is well
fit by a 1470\,K blackbody, close to the dust sublimation point
(Kobayashi et al. 1993). The flux at the 2$\mu$m bump, $9.3\times
10^{-11}$\,erg~s$^{-1}$ cm$^{-2} \mu$m$^{-1}$, corrected for an IR
extinction of 0.2 to 0.6\,mag (Krabbe et al. 1997), implies a
blackbody luminosity of (3 to 4) $\times 10^{11}$\,L$_\odot$ , and a
radius of 0.18\,pc. The compact 10 to 25\,$\mu$m source in Mrk~231
(Matthews et al. 1987; Keto et al. 1992; Miles et al. 1996) arises
farther out in the dusty torus, where the dust temperature is
$\sim$200\,K, at a radius of $\sim$10\,pc from the AGN. This dusty
torus is heated by the AGN. If its thickness is equal to its radius,
then the torus absorbs half the AGN's power, so the AGN's total
luminosity is 6 to $9\times 10^{11}$\,L$_\odot$ , the same as estimated
from the optical and UV flux.
\noindent
{\it The molecular disk's FIR luminosity} may be derived from the IRAS
fluxes, and is $2\times 10^{12}$\,L$_\odot$\ (e.g., Solomon et al. 1997).
The molecular disk has a dust optical depth close to unity at
100\,$\mu$m and causes the deep 10$\mu$m silicate absorption (Allen
1976; Rieke 1976; Roche, Aitken, \& Whitmore 1983). The blackbody
radius calculated from the FIR luminosity is 200\,pc, half the
measured size of the molecular disk. The molecular disk receives some
heat from the dusty torus. With the thickness-to-radius ratio in our
CO model, it would intercept half of the power, or $2\times
10^{11}$\,L$_\odot$\ of re-processed radiation from the AGN. Since this
is only 10\% of the FIR luminosity, most of the FIR luminosity of
Mrk~231 must come from the starburst in the molecular ring or disk. A
similar model was used by Rigopoulou, Lawrence, \& Rowan-Robinson
(1996) to fit the FIR continuum spectrum.
Our CO model fit gives a gas mass of $3.1\times 10^9$\,M$_\odot$\ ---
equal to the mass of all the molecular gas in the Milky Way --- in a
high-density disk between $R=76$ and 850\,pc. There is another
$0.9\times 10^9$\,M$_\odot$\ of gas in a lower density outer disk
extending to a radius of 1.7\,kpc. The high-density molecular disk is
the region observed in H$_2$ at 2\,$\mu$m by Krabbe et al. (1997).
The brightest H$_2$ line is 200\,pc from the nucleus, in the molecular
disk, so the low 2\,$\mu$m extinction on the line of sight to the
nucleus itself does not apply to the H$_2$ flux.
Our conclusion that most of Mrk~231's FIR luminosity comes from a
starburst can be checked for consistency (Table~11). We assume
the new stars in the burst, $M_{\rm new\star}$, yield $L_{\rm
FIR}/M_{\rm new\star}$ $\approx$ 500\,L$_\odot$ \, M$_\odot$$^{-1}$, a
luminosity ratio that can be attained in a fast starburst (Leitherer
\& Heckman 1995).
Dividing this ratio into the FIR luminosity yields
$M_{\rm new\star}$, from which we then calculate
$M_{\rm old\star}= M_{\rm dyn}-M_{\rm gas}-M_{\rm new\star}$,
where $M_{\rm old\star}$ is the mass of old stars
in the nuclear bulge before the merger.
For the region $R \leq 460$\,pc in Mrk~231,
the rotation velocity inferred from our
CO data yields a dynamical mass
$M_{\rm dyn}$($<460$\,pc) = $12.7\times 10^9$\,M$_\odot$\ , and
our CO model yields $M_{\rm gas}=$
$1.8\times 10^9$\,M$_\odot$ .
The consistency check then yields
$M_{\rm old\star}= 7.1 \times 10^9$\,M$_\odot$ ,
which is about the same as the
mass of old stars in a similar radius in the
Milky Way (e.g., Oort 1977). Because of our assumed $L/M$,
the derived stellar masses in the starburst and the bulge
are uncertain by at least a factor of two. Our point is simply that
with a starburst powering the FIR luminosity,
all the gas and stellar masses are quite plausible.
In summary, we think Mrk~231's black hole accretion disk emits
$\sim 1\times 10^{12}$\,L$_\odot$ , or $\sim 30$\%
of the total bolometric luminosity. Most of the luminosity,
$\sim 2\times 10^{12}$\,L$_\odot$ , comes from
the starburst in the molecular ring or disk.
\section{ {\bf Arp 193} }
The ultraluminous merger Arp~193 shows two tidal tails in the visible
and the near IR (Smith et al. 1995, 1996). The IR images show a
single, elongated nucleus, but cm-radio continuum maps show two
sources separated by 1$''$. Strong lines of Br$\gamma$ and H$_2$ are
seen in $K$ band (Goldader et al. 1995). The near-IR continuum is
dominated by young supergiants formed in the merger-induced starburst.
Previous CO line ratio studies indicated sub-thermal CO excitation and
moderate densities (Radford et al. 1991). Our detection of HCN in
Arp~193 indicated, however, that large amounts of dense gas were also
present (Solomon, Downes, \& Radford 1992).
The CO(2--1) emission is nearly unresolved
in individual 20\,km\thinspace s$^{-1}$\ channels, with temperatures up to 8\,K (Fig.~9).
The CO intensity and isovelocity contour maps (Fig.~10) reveal a
rotating disk with
a line of nodes at the same orientation as the optical isophotes in
the center of the merger. The
position-velocity diagrams and CO spectra (Figs.~11 and 12)
along the line of nodes have the characteristic signature of a
rotating ring, with two peaks near the rotation curve turnover radius,
as in VII~Zw~31 and IRAS 10565+24. Models that best fit the data are
those with a minimum ring radius $R_{\rm min}$ in eq.(1) equal to
$R_0$ = 220\,pc. As in VII~Zw~31, filled-disk models, with $R_{\rm
min}$ = 0, give poorer fits, with much less contrast. The centroid of
the CO source coincides with the main radio continuum peak on the 8.4~GHz
map by Condon et al. (1991).
The strongest CO peak is 1$''$ to the southeast of the centroid,
and peaks at $-$120\,km\thinspace s$^{-1}$\
(relative to 225.282\,GHz). It is nearly unresolved on our maps, coincides with
a secondary radio continuum peak. This
hot, compact, southeast CO core in Arp~193 is responsible for the large
difference in the CO(2--1)/(1--0) ratio in the blue and redshifted
sides of the line profiles we observed at the 30\,m telescope (Radford
et al. 1991) . This region is hotter and denser than the disk and has
properties similar to a huge
molecular cloud core.
This compact southeast core is one of several sources
we identify in this study as {\it Extreme
Starburst Regions} (see Section 8 on Arp~220 for a detailed discussion).
Using the observed CO
luminosity, linewidth, radius, and temperature,
we can estimate the gas mass, dynamical mass and
far IR luminosity of the core. The results, summarized in Table 12,
indicate an object with a gas mass
$\sim 6 \times 10^{8}$M$_\odot$ , a luminosity of $\sim 2\times 10^{11}$\,L$_\odot$ ,
and about $\sim 1\times 10^{9}$\,M$_\odot$\ of newly formed stars.
\section{ {\bf Markarian 273 } }
The galaxy Markarian 273 has at least one Seyfert~2 nucleus with
optical linewidths of 700\,km\thinspace s$^{-1}$\ (e.g., Sargent 1972; Koski 1978).
There is evidence for two Seyfert 2 nuclei (Asatrian, Petrosian, \&
B\"orngen 1990), and two nuclei are seen on IR images (Armus et
al. 1992; Majewski et al. 1993; Knapen et al. 1997). Centimeter radio
maps show a $0''.3$ northwest peak coinciding with the CO, and a
weaker $0''.2$ southeast peak with no CO or IR counterpart (Ulvestad
\& Wilson 1984; Schmelz, Baan, \& Haschick 1988; Sopp \& Alexander
1991; Condon et al. 1991). There is also an 18\,cm VLBI source of
16\,mJy (Lonsdale et al. 1993; Smith, Lonsdale, \& Lonsdale 1998a).
The best previous CO study is by Yun \& Scoville (1995), who observed
the source with a 2$''$ beam and found a component extended
north-south as well as an unresolved nuclear component. Our
higher-resolution CO maps also show the extended structure, but now
resolve the nuclear source into a nuclear disk oriented east-west, and
a very compact core embedded in the nuclear disk.
\subsection {Extended molecular gas and nuclear disk in Mrk~273}
The extended gas is best traced in the CO(1--0) channel maps,
blueshifted to the south and redshifted to the north (Fig.~13a).
The CO at $+$250 to $+$350\,km\thinspace s$^{-1}$\ runs in an arc from 3$''$ to 7$''$
(2 to 5\,kpc) north of the nucleus (Fig.~14). This extended gas
also appears in the maps of CO(1--0) integrated intensity and
isovelocity contours (Fig.~15a). The CO(1--0) lines are weaker
and narrower in the northern arc than in the nuclear disk (Fig.~16).
The arc has 4\% of the CO(1--0) flux of Mrk~273, so if gas
mass is proportional to CO flux, then the northern arc has a gas mass
of $1\times 10^8$\,M$_\odot$ . Figure~17a is the CO
position-velocity cut in declination, through the nucleus. The
narrow-line emission extends 7$''$ (5\,kpc) south at 0\,km\thinspace s$^{-1}$\, and
7$''$ north at $+$300\,km\thinspace s$^{-1}$ . It is tempting to speculate that the
streamers are bringing molecular gas into the center. The north-south
extent of the streamers on our CO map agrees with that on the map by
Yun \& Scoville (1995). While those authors deduced a line of nodes
at p.a. 30$^\circ$, our isovelocity contour maps show there are two
perpendicular kinematic systems. In the extended gas the velocity
gradient is north-south, and in the nuclear disk it is east-west.
The $2''$ nuclear disk with its compact core are best seen in the
CO(2--1) channel maps, made with a $0''.6$ beam (Fig.~13b). The
nuclear disk kinematics are shown in the CO(2--1) map (Fig.~15b),
where the isovelocity contours trace the east-west
velocity gradient of the nuclear disk over a 1\,kpc diameter.
Figure~17b is the CO position-velocity diagram of the nuclear disk in
right ascension. The broad nuclear disk line has a full width to half
maximum of 380\,km\thinspace s$^{-1}$ , and its velocity centroid changes from $-300$
to $+200$\,km\thinspace s$^{-1}$\ over $0''.5$ from west to east. This is a projected
velocity gradient of 1.5\,km\thinspace s$^{-1}$\,pc$^{-1}$, the same as deduced by
Schmelz, Baan, \& Haschick (1988) from H~I absorption. Table~11
lists the stellar and gas masses that would be needed in a starburst
in the Mrk~273 molecular disk if the $L_{\rm FIR}/M_{\rm new\star}$
ratio is the same as we adopted in our starburst model of Mrk~231.
\subsection{An Extreme Starburst Region in Mrk~273}
The high-resolution CO(2--1) maps show a remarkable
molecular-line source in the Mrk~273 nuclear disk --- a bright,
$0.35''\times <0.2''$ CO core (Fig.~15b), that
resembles the west nucleus of Arp~220.
This is the most luminous extreme starburst region
in our sample of 10 galaxies.
It has an IR luminosity of about $6 \times 10^{11}$\,L$_\odot$\ ,
generated in a region with a radius of only 120\,pc and
a current molecular mass of $1 \times 10^9$\,M$_\odot$\
(Table~12).
To put this in perspective,
the entire molecular core has a radius about 5 times that of
an IR luminous Milky Way GMC (for example, W51)
but with about 3,000 times the molecular mass and
$\approx \, 10^5$ times the IR luminosity from OB stars.
This core has a broad CO line
with a zero-intensity width of 1060\,km\thinspace s$^{-1}$ , the same
as the OH megamaser (Staveley-Smith et al. 1987).
It coincides with the northwest extended continuum peak
at 8.44\,GHz, which has a size $0''.32 \times 0''.18$
(Condon et al. 1991), the same as the compact CO.
The radio spectrum is nonthermal, with a spectral index of $-0.6$.
The thermal dust emission dominates the spectrum above 350\,GHz
(Chini et al. 1989; Rigopoulou et al. 1996).
At the CO core, we detect continuum fluxes of
11$\pm 2$\,mJy at 111\,GHz and $8\pm 2$\,mJy at 225\,GHz.
Extrapolation of the synchrotron and thermal dust spectra
indicates the dust contributes 50\% of the flux at 1.3\,mm
and $<10$\% at 3\,mm. The extended nonthermal continuum emission coincident with a high
mass of dust and gas leaves little doubt that this region is powered by star formation.
\section{{\bf Arp 220} }
The center of the ultraluminous galaxy Arp~220 has
two radio continuum and two IR sources
$1''.0$ apart that are interpreted as merger nuclei.
These are not exactly the same objects in the radio and the IR, because
both of the east and west IR nuclei are four times larger ($0''.8$)
than the radio sources ($0''.2$) (Condon et al. 1991;
Graham et al. 1990; Majewski et al. 1993; Miles et al. 1996; Scoville
et al. 1998). The $K$-band continuum is starlight associated with the nuclei,
and is best seen on the HST NICMOS images (Scoville et al. 1998).
These images resolve the eastern IR nucleus into a strong NE and weaker SE
component, the latter of which coincides with the eastern radio continuum
component and the eastern OH masers (Diamond et al. 1989).
The two radio sources are extended and nonthermal, and are produced by
supernovae in the most active star-forming regions.
The 1.3 and 1.6\,$\mu$m [Fe II] lines in the west source indicate iron
evaporated from dust in shocks (Armus et al. 1995a;
van der Werf \& Israel 1998).
The best previous CO study is that
by Scoville, Yun, \& Bryant (1997), who derived the kinematics of
the nuclear disk with a resolution of 1$''$.
\subsection{CO and dust emission from the two nuclei of Arp 220}
Our CO(2--1) and 1.3\,mm continuum maps (Fig.~18a) show
two compact sources embedded in more extended emission.
The continuum fluxes at 1.3\,mm are due to dust,
because they are well above the extrapolated radio synchrotron spectra
(Fig.~20).
The different appearance of the dust continuum and the CO maps is
a temperature effect. At 1.3\,mm the dust is optically thin
and its flux varies as the column density and the dust temperature, while
the CO flux is partly opaque, with flux varying with temperature at the
surface, and partly optically thin, with flux
varying inversely with CO excitation temperature.
The Arp~220 molecular disk contributes strongly to the CO maps,
while the warmer compact peaks dominate the dust continuum maps.
From the 1.3\,mm dust flux alone, and for dust at
100~K and solar metallicity, we derive gas masses for Arp~220-west and -east
of 0.6 and $1.1\times 10^9$\,M$_\odot$ .
In {\it Arp~220-west}, the $0''.3$ (100\,pc) mm continuum dust source coincides
within $0''.1$ of the cm-continuum west peak, and
the $K$-band west peak (Scoville et al. 1998).
In the CO(2--1) channel maps (Fig.~19b), Arp~220-west appears
as a strong source in the range $-$310 to $+$210\,km\thinspace s$^{-1}$ , peaking at
$-$110\,km\thinspace s$^{-1}$\ ($cz_{\rm lsr} =$ 5340\,km\thinspace s$^{-1}$ ), with an
{\it observed} CO(2--1) brightness temperature of 27\,K,
of which 10\,K is from the more extended Arp~220 disk.
The positive-velocity emission near Arp~220-west is centered at $+$80\,km\thinspace s$^{-1}$ ,
and appears in both the CO(2--1) channel maps and in a
position-velocity cut at p.a. 300$^\circ$ (Fig.~22b). This positive-velocity
emission has about 25\% of the CO integrated intensity of the negative-velocity
emission of Arp~220-west.
The dust continuum is centered between the
negative and positive velocity emission, suggesting that Arp~220-west is
a composite structure, with a kinematic axis nearly perpendicular to that
of Arp~220's main disk.
At the negative velocity west core, the emission extends
asymmetrically from $-$50 \, to $-$ 400 \, km\thinspace s$^{-1}$\ (Fig.~22b).
At the positive velocity core, it runs from $+$50 \, to $+$350
\, km\thinspace s$^{-1}$. These extreme, asymmetric line profiles,
confined to a small region, suggest possible radial
flows along a {\it bar--like} structure, or immense high-velocity molecular
{\it outflows} similar
to those observed in high mass star formation regions.
Molecular outflows are also the likely sites
of the strong H$_2$ lines (Sturm et al. 1996) .
In {\it Arp~220-east}, the CO covers
a wide velocity range. The negative-velocity gas at $-$230 to $-$30\,km\thinspace s$^{-1}$\ is
separated from Arp~220-west by $0.''85$ at p.a. 110$^\circ$ --- the
same angle as the cm-radio continuum sources, the 18\,cm OH masers,
and the $K$-band SE peak (Scoville et al. 1998).
The center velocity of this gas is $cz_{\rm lsr} =$ 5330\,km\thinspace s$^{-1}$ ,
close to that of the OH megamasers and
H$_2$CO emission at the cm-radio east source (Baan \& Haschick 1995;
Lonsdale et al. 1998). There is also
more extended, positive-velocity gas ($0''.9$ FWHM) at 130 to 290\,km\thinspace s$^{-1}$ .
It is $1''.3$ from Arp~220-west at p.a. 85$^\circ$ --- the same
orientation as the $K$-band NE source (Scoville et al. 1998).
This positive-velocity gas in Arp~220-east,
at $\sim +$200\,km\thinspace s$^{-1}$\ ($cz_{\rm lsr} =$ 5650\,km\thinspace s$^{-1}$ ), has
the same velocity offset from Arp~220-west as the ionized gas seen
in Br$\gamma$ and Pa$\beta$ (Larkin et al. 1995),
and the weak OH masers at 1612~MHz (Baan \& Haschick 1987).
Its {\it observed} maximum CO(2--1) brightness temperature
is 19\,K, of which 4\,K is from the extended Arp~220 disk.
The 1.3\,mm continuum from dust in Arp~220-east has a diameter of $0''.6$ (205\,pc)
and is displaced by $0''.3$ from the cm-radio east source. It
falls in between the $K$-band NE and SE peaks on the images by
Scoville et al. (1998), and in between the positive and negative-velocity CO.
The kinematic data suggest that Arp~220-east has a steady progression
of velocity between the CO peaks at 5330\,km\thinspace s$^{-1}$\ and 5650\,km\thinspace s$^{-1}$ , along
the same position angle (p.a. 50$^\circ$), and in the same sense of rotation
as the main, larger-scale molecular disk of Arp~220 (Fig.~22c).
\subsection{CO in the Arp 220 molecular disk}
The molecular disk is best seen as an extended source in the CO(1--0) maps
(Fig.~18b). The CO(2--1) maps also show this same extended source,
when the compact peaks are subtracted.
The disk has CO halfwidths of $2''.0\times 1''.6$ at p.a. 50$^\circ$,
with a line of nodes at this same position angle,
blueshifted in the southwest and redshifted in
the northeast, as seen in the CO spectra along
the major axis (Fig.~21).
Our model fits yield a rotation curve turnover radius of 200\,pc,
a disk outer radius of 480\,pc, and
a disk rotation velocity of 330\,km\thinspace s$^{-1}$\ on the flat part of the rotation curve.
The CO emission centroid of the
molecular disk is $\sim 0''.3$ east of the western nucleus.
These parameters reproduce the
position-velocity diagrams along the kinematic major axis (Fig.~22),
and are similar to the values derived by Scoville et al. (1997) by a
different algorithm. On a larger scale, CO can be traced in the
8$''$ diameter {\it outer disk} at the same position angle. The inner and
outer disks both contain shocked gas, detected in the near-IR vibrational
lines of H$_2$ (van der Werf 1996).
The CO(1--0) maps also show an eastern streamer extending 7$''$ (3\,kpc)
perpendicular to the disk, in the range $+$45 to $+$245\,km\thinspace s$^{-1}$ ,
curving in to the east nucleus
with increasing velocity offset (Fig.~19a). This east streamer
is most intense at 85 to 165\,km\thinspace s$^{-1}$\ (Fig.~18b).
It may be material that is still falling into the center.
Surprisingly, this CO streamer appears on the HST
$V$-band image by Shaya et al. (1994).
The CO inner and outer disks coincide with the
prominent optical dust lane at 50$^\circ$ (Fig.~23a), while
the CO east streamer coincides with the
perpendicular dust lane in the optical image (Fig.~23b).
Our model fits to the CO position-velocity diagrams of the Arp~220 disk
(Fig.~22) yield a much higher dynamical mass
than estimates from the IR (e.g., Doyon et al. 1994;
Shier, Rieke, \& Rieke 1994, 1996; Larkin et al. 1995).
From the masses in Table~9,
we estimate $M(<350$\,pc) $=8.8\times 10^9$\,M$_\odot$ ,
and $M(<600$\,pc) $=1.5\times 10^{10}$\,M$_\odot$ , three to four times
higher than the IR estimates. The millimeter CO data and the
2.3\,$\mu$m CO bandhead absorption data yield different masses
because the two IR nuclei do not lie on the line of nodes traced by the
CO maps and because the near IR starlight is obscured and
does not trace the full extent of the molecular disk.
The gas mass derived from both the dust and the CO implies a
column density of $1\times 10^{24}$\,cm$^{-2}$ and $A_V \sim 1000$\,mag
through the disk, consistent with the dust having an opacity
of unity at 180\,$\mu$m (Emerson et al. 1984; Scoville et al. 1991).
The high opacity
in the far IR may partly explain why the C$^+$ 158\,$\mu$m line is
weak in Arp~220 (Fischer et al. 1998.).
The ratios of fine structure lines observed by the $ISO$ satellite
yield an equivalent screen $A_V \sim 45$\,mag (Genzel et al. 1998).
As is well known, a screen model gives only an extreme lower limit.
The corresponding extinction when the emitting gas and dust are
completely mixed is $A_V \sim 1000$\,mag, the same as we deduce from
the column density derived from the CO maps.
Figure~24 shows our model of the molecular disk and
the two ``nuclei".
The CO disk is inclined 40$^\circ$ from face-on, Arp~220-west
has a radius of 68\,pc, and Arp~220-east has a radius of 110\,pc.
The disk thickness is 90\,pc, which means the path to the near surface
of the east and west K-band sources is 0 to 20\,pc,
which is why the
two nuclei are visible at all at $K$ band. That is, although
the visible extinction through the entire molecular disk is 1000\,mag, the
visible extinction to the near surfaces of the east and west sources
is only about 50\,mag, because of the shorter path.
Our diagram is similar to that of Scoville et al. (1997; their Fig.~7),
except our presentation is a view from the pole of the disk.
In our model, the near side is north,
the far side is south. This agrees with the optical colors, which are
bluer on the north side of the dust lane (Shaya et al. 1994).
Since the CO east nucleus is south of the major axis, it is on
the far side. The CO disk or ring includes both nuclei
but the nuclei are oriented east-west, and hence the difference in
their line of sight velocities does not give the
full rotation speed of the disk.
Our data and model have some important consequences for the interpretation
of the Arp~220 nuclear sources.
1) The rotation curve of the CO disk indicates a dynamical mass of
$12 \times 10^9$\,M$_\odot$\ interior to 480\,pc, which corresponds to
the central bulge mass of a large spiral like the Milky Way.
2) The gas mass in each of the two extreme starburst ``nuclei'' is only $6\times
10^8$\,M$_\odot$ . Their individual luminosities are $\sim 3\times 10^{11}$\,L$_\odot$ .
(About half of the Arp~220 FIR luminosity comes from the molecular disk,
not the two nuclei).
With an L/M ratio of 1000, corresponding to a super-starburst in its
initial phase, the mass of new stars in each nucleus
would be only $3\times 10^8$\,M$_\odot$ ,
sufficient to explain all the $K$-band continuum luminosity.
The velocity dispersion of the CO in each of the two nuclei implies
a dynamical mass of $\sim 1\times 10^9$\,M$_\odot$ , about the same as our
estimate of the sum of the gas and new stars.
3) Hence, there is no room left over for old stars in the two ``nuclei"
--- they cannot be the relicts of the old nuclei of the pre-merger
galaxies. Furthermore, there is no observational evidence --- radio,
infrared, or optical --- that they contain old stars.
4) In any case, the masses of the two ``nuclei'' are negligible in comparison
with the mass that controls the motion of the molecular disk.
Furthermore,
the two ``nuclei" of Arp~220 have radial velocities indicating that
they take part in the general disk rotation, i.e., they share the
general rotation in the potential of the old bulge, and are
dominated by {\it its} gravity, not their own.
5) In our interpretation, the two ``nuclei" of Arp~220 are not the
pre-merger nuclei at all. We think they are just large ensembles of
pre-existing GMCs that have been strongly compressed on their infall
into the old bulge potential as a result of the merger.
The masses of these large, compressed gas structures can reach
10$^9$\,M$_\odot$ , and the strong compression, in about one orbital period,
leads to the extreme starbursts, with luminosities of a few times
10$^{11}$\,L$_\odot$ ,
from each compact 100-pc structure. These extreme starbursts power
the ultraluminous galaxies. We are not seeing the birth of quasars.
\subsection{Arguments for a starburst}
What produces Arp~220's luminosity? We think it's stars, for three reasons:
there is no obvious AGN, there are enough ionizing photons for a starburst,
and there is enough dense molecular gas to make stars.
\noindent
1. {\it Arp~220 has no AGN lines.} The mid-IR lines of [O~IV] and [Ne~V]
are {\it not} detected (Sturm et al. 1996; Lutz et al. 1996).
The Br$\alpha$ width of 1300\,km\thinspace s$^{-1}$\ reported by Depoy, Becklin, \&
Geballe (1987) as evidence for an AGN was not confirmed in later measurements
(Goldader et al. 1995; Larkin et al. 1995).
\noindent
2. {\it Arp~220 has no AGN in the radio continuum.}
New VLBA maps (Smith et al. 1998b) indicate all the VLBI point sources are
young radio supernovae in a dense medium, with radio luminosities
comparable to SN1986J in NGC\,891.
\noindent
3. {\it The radio nuclei are extended starburst regions.}
At centimeter wavelengths, their diameters are: west, $0''.21\times 0''.14$
($72\times 48$\,pc) and east, $0''.32\times 0''.19$ ($110\times 64$\,pc)
(Condon et al. 1991). The source extent, the low brightness at 5\,GHz, and
the high FIR-to-radio flux ratio all show the
nonthermal radio continuum is starburst-dominated, not AGN-dominated
(Sopp \& Alexander 1991; Condon et al. 1991; Baan \& Haschick 1995).
\noindent
4. {\it Arp~220 is not a ``warm" ultraluminous galaxy.}
IR ultraluminous galaxies with an AGN, like Mrk~231, Mrk 1014, and
08572+3915, have warm mid-IR colors ($f_{25}/f_{60} > 0.2$) and a
bright, symmetric nucleus resembling a reddened QSO (Surace et al. 1998).
Unlike these sources, Arp~220 has strong far-IR and weaker mid-IR flux
($f_{25}/f_{60} = 0.08$). It has been argued that a QSO is
hidden in Arp~220, but the silicate absorption depth at 10\,$\mu$m in
Arp~220 (e.g., Smith, Aitken, \& Roche 1989; Dudley \& Wynn-Williams 1997)
has been greatly overestimated, due to the strong PAH lines
directly adjacent to the silicate feature (Genzel et al. 1998), so
a deeply embedded AGN is not required.
\noindent
5. {\it The diffuse OH megamasers do not indicate black holes.}
A new VLBI study (Lonsdale et al. 1998)
shows Arp~220 has two OH megamaser components in each nucleus,
one diffuse and the other
compact. The {\it diffuse} megamaser component arises in the
extended starburst medium, and can be explained by IR pumping via photons
absorbed in the 35 and 53\,$\mu$m lines of OH. The observed 9-Jy
depth of the 35\,$\mu$m OH line can only be explained by absorption against an
{\it extended} source (Skinner et al. 1997). The {\it compact} OH megamasers,
probably collisionally pumped shocks, are currently the only evidence for
possible AGNs in Arp~220 (Lonsdale et al. 1998), but it is hard to
understand why such AGNs would
have no associated radio continuum core-jet sources.
\subsection{The ratio of bolometric flux to ionizing flux resembles
that in Sgr~B2.}
From the observed Br$\gamma$ flux, Armus et al. (1995b)
and Shier, Rieke, \& Rieke (1996) estimated
that $\leq 10$\% of Arp~220's luminosity came from a starburst.
Armus et al.\ noted, however, if they had underestimated the
$K$ band extinction, then a starburst could power the source.
The higher near-IR extinction is indeed
required by the high column densities found from the CO maps.
From the radio free-free continuum,
Scoville et al. (1991; 1997) claimed there were not enough
ionizing photons for a starburst to power Arp~220.
We think this expectation was wrong, because galactic star-forming regions
like W49, W51, and Sgr~B2 have FIR luminosities
greatly exceeding the Lyman luminosities derived from their radio fluxes.
In compact HII regions like W3(OH), or in our galactic center, the FIR excess
is a factor of $\sim 20$ (e.g., Zylka et al. 1995).
In H~II regions with densities $>10^3$\,cm$^{-3}$,
most of the Lyman photons heat dust rather
than ionizing the gas (Jennings 1975;
Panagia 1977; Fazio 1978; Mezger 1985), so
the Lyman photon rate derived from the radio flux is only a lower limit.
We may estimate this lower limit to Arp~220's Lyman continuum photon flux
from the continuum at 113\,GHz, which includes 15\,mJy from the
synchrotron spectra and 13\,mJy from dust (Fig.~20).
This leaves 13\,mJy as optically thin free-free flux, and implies a
Lyman continuum photon rate $> 1\times 10^{55}$ s$^{-1}$.
Similar lower limits come from the H92$\alpha$ radio recombination line
(Zhao et al. 1996), and the low-frequency turnover of the radio
spectra of the two nuclei (Fig.~20), due to
free-free absorption in ionized gas with an emission measure of
10$^8$\,cm$^{-6}$\,pc
and a mass of $5\times 10^{7}$\,M$_\odot$\ (Sopp \& Alexander 1991).
For comparison, in the galactic giant H~II region and
molecular cloud Sgr~B2, the FIR power is 10 times the ionizing luminosity
derived from the radio (Gatley et al. 1978). Sgr~B2 resembles
Arp~220's molecular gas in column density, strong molecular lines,
and far-IR opacity,
but no one has ever claimed Sgr~B2's FIR excess proves it is powered by a
black hole.
Arp~220 and Sgr~B2 both have the same ratio of
FIR flux (in W m$^{-2}$) to radio free-free flux
(in Jy), and hence the same FIR excess. So our lower
limit on Arp~220's ionizing flux, scaled by the Sgr~B2 template,
yields exactly Arp~220's FIR luminosity of $1.2\times 10^{12}$\,L$_\odot$ ,
and is indeed compatible with a starburst.
Similar conclusions are reached by Genzel et al. (1998) from
$ISO$ line data on the ionizing flux.
\subsection{Arp~220 has plenty of gas for a starburst.}
Our models yield gas masses of
$0.6\times 10^9$\,M$_\odot$\ for Arp~220-west,
$1\times 10^9$\,M$_\odot$\ for Arp~220-east, and
$3\times 10^9$\,M$_\odot$ \ for the disk to $R_2=1.2$\,kpc (Table~9).
The total gas mass is $5\times 10^9$\,M$_\odot$ , about 6 times lower than
previous estimates with a standard ratio of gas mass to CO luminosity.
Our value agrees with the gas mass of $3.5\times 10^9$\,M$_\odot$\
obtained by Sturm et al. (1996) from the H$_2$ lines at 6.9 and 17\,$\mu$m,
although this latter value depends sensitively on the assumed
temperature.
Our method differs from that of
Scoville, Yun, \& Bryant (1997), who assumed all
the dynamical mass is in gas, rather than stars, thereby
forcing their model CO disk to be only 16\,pc thick at $R=$250\,pc,
and to have a {\it mean} density of $1.5\times 10^4$\,cm$^{-3}$.
Because we do not assume all the mass is gas,
we derive a disk thickness of 80\,pc and a
lower mean density than Scoville et al.
Our higher-resolution data indicate the gas density is highest
at the two nuclei, not at the center of the molecular disk as in
the model of Scoville et al., and our value for the total gas mass is therefore a factor
of two lower than their estimate.
Even the lower molecular gas mass, however, is about the same as in
the entire disk of a gas-rich spiral!
It is enough gas for a huge nuclear starburst.
In the Arp~220 inner disk,
the gas surface density is 100 times the
peak value in the Milky Way 4-kpc molecular ring
and more than 20 times the surface density
in the inner few hundred pc of our Galaxy.
To estimate the {\it stellar} mass in the nuclear bulge, we
adopted a global luminosity to starburst mass ratio in the Arp~220 disk
of 300\,L$_\odot$ \,M$_\odot$$^{-1}$, based on
starburst $L/M$ ratios from the models of Leitherer \& Heckman (1995).
This gives a estimate of
the mass in newly-formed stars from the
current merger-induced starbursts. Subtracting this mass and the gas mass
from the dynamical mass then gives an rough estimate of
the mass in old, pre-merger stars.
The resulting bulge mass is $8\times 10^9$\,M$_\odot$\ within $R= 480$\,pc
(Table~11), about the same as in a large galaxy like our own
(e.g., Oort 1977). The starburst
interpretation thus implies that old stars are a
significant part of the dynamical mass.
\subsection{Arp~220-west has 25\% of the total
luminosity.}
The CO lines, the 1.3\,mm dust flux, and the cm-radio continuum
all suggest the currently most active starburst is
Arp~220-west. Its radius of 68\,pc
and dust temperature of 75 to 80\,K imply a blackbody luminosity of
$3\times 10^{11}$\,L$_\odot$ , or about 25\% of
Arp~220's total IR luminosity of $1.3\times 10^{12}$\,L$_\odot$ , as expected
from the $R^2 T_d^4$ ratio of the west and disk sources.
This estimate also agrees with the fraction of the
mid-IR flux absorbed by the 35\,$\mu$m OH line at 5320\,km\thinspace s$^{-1}$\
(Skinner et al. 1997), the velocity of Arp~220 west.
If the Lyman continuum scales
as FIR luminosity, then Arp~220-west has 1000 times the ionizing flux of
30~Doradus, which is 10$^{52}$ photons s$^{-1}$ (Kennicut \&
Chu 1994), from 2400 OB stars in a region of comparable size
(Parker 1993). Because Arp~220-west is small and the dynamical time scale is
short, its starburst must be younger (age $5\times 10^6$\,yr) than
that in the larger Arp~220 disk. The initial phase of a compact starburst
can be highly luminous, reaching $L/M$ ratios of
1000 -- 3000\,L$_\odot$ \,M$_\odot$$^{-1}$ (see models in Leitherer \& Heckman 1995).
Our estimate of the total mass (from the CO linewidth)
and the gas mass (from the CO luminosity), suggests that in
Arp~220-west, 50 to 60\% of the original gas mass has already turned
into the new stars in the current, extreme starburst
(Table~12).
\section{SOURCES OBSERVED IN CO(1--0) ONLY}
We observed these sources with lower resolution and
thus our kinematic models are less precise.
All these sources show the same phenomena as the previous group of sources,
however, with velocity gradients
and line profiles that suggest rotating rings or disks.
\subsection{ {\bf 00057$+$4021} }
The galaxy IRAS 00057$+$4021 has not been well studied,
despite its high luminosity
of $L_{\rm IR} = 4\times 10^{11}$\,L$_\odot$ , and its OH megamaser
(Kaz\`es, Mirabel, \& Combes 1988).
The $R$-band image by Armus, Heckman, \& Miley (1987)
shows a 60$''$ (48\,kpc) southeast-northwest disk with
a tidal tail stretching a further 45$''$ (36\,kpc) southeast.
In contrast to the optical light, the CO source is quite compact.
Most of the molecular gas is in a $1''.1\times <0''.6$ core source.
The CO has wider lines (140\,km\thinspace s$^{-1}$ \ FWHM) and
is more intense on the southeast side of the line of nodes
than on the northwest (Fig.~25).
The continuum flux from the CO source is
$<$10\,mJy at 110\,GHz. This limit on the thermal dust flux is
consistent with the gas mass of $9\times 10^8$\,M$_\odot$\
deduced from our model fits to the CO.
In the position-velocity diagram along the line of nodes at p.a. 135$^\circ$
the CO line has
a velocity range of 300\,km\thinspace s$^{-1}$\ over a diameter of 3$''$ (Fig.~25).
The CO lines are blue-shifted in the northwest
and redshifted in the southeast. In strips parallel to the kinematic major
axis the CO spectra show the double-peak behavior of a rotating
disk or ring. To the southeast, the CO profile is steep
toward the red, and gently sloping toward the blue. To the northwest,
the profile is reversed. The nuclear velocity gradient in CO has the same
major axis as the larger-scale optical isophotes.
The position-velocity data and the spectra both indicate
another, lower-intensity (0.5~K) disk
with a broad linewidth (200\,km\thinspace s$^{-1}$ ) in the same region as the brighter,
high-density disk.
\subsection{ {\bf 02483$+$4302} }
The galaxy IRAS 02483+4302
is a merger with a tidal tail extending 90$''$ to the west.
The merger has two nuclei separated by $3''.8$
east-west. In optical lines, nucleus $A$
(west) has a Seyfert~2 spectrum, nucleus $B$ (east) has a LINER spectrum.
The optically more intense nucleus $A$ appears to belong to an
elliptical galaxy plowing through the disk of a former spiral containing
nucleus $B$ (Kollatschny et al. 1991).
The optical continuum of nucleus $B$ comes partly
from hot stars (Womble et al. 1990).
The CO (Table~2) coincides with nucleus $B$,
supporting the idea that this was originally a gas-rich spiral.
Our CO position also agrees within 0.5$''$ of the 13\,mJy source seen at
5~GHz (Crawford et al. 1996).
At 109\,GHz, the continuum flux from the CO source is
$<10$\,mJy. This limit on the thermal dust flux is
consistent with the gas mass deduced from the CO.
In velocity channels 40\,km\thinspace s$^{-1}$\ wide, the CO source is very compact.
Figure~26 shows the maps of CO integrated intensity and
isovelocity contours across the source.
The CO has a north-south velocity gradient, with
a kinematic major axis perpendicular to the optical tidal
tail extending west from the merger nuclei.
In strips parallel to the kinematic major
axis the CO spectra show the symmetric behavior of a rotating
disk or ring. In the south, the CO profile is steep
toward the blue and gently sloping toward the red. In the north,
the profile is reversed.
The position-velocity diagram along the line of nodes shows
a gradient of 200\,km\thinspace s$^{-1}$\ over $0''.9$ in declination (Fig.~26).
The narrow CO linewidth and the low optical extinction to the
nucleus both suggest the molecular disk around
nucleus $B$ is face-on.
The CO source, at $z=0.05144,$ ($cz_{\rm lsr} = 15420$\,km\thinspace s$^{-1}$ )
and the quasar Q0248+430 at $z=1.311$ are separated on the sky
by $16''.9$ (the galaxy-quasar separation is incorrectly listed as
$3''.5$ by Burbidge (1996) and by Hoyle \& Burbidge (1996)).
The tidal tail that extends $>80$\,kpc from the two merger nuclei
crosses the quasar at a projected distance of 15.4\,kpc from the CO source.
At this position, low-density atomic gas in the tidal
tail is seen in absorption against the quasar in the
Na~I~D and Ca~II~H and K lines at $z=0.0515$ and 0.0523,
with linewidths $< 150$\,km\thinspace s$^{-1}$\ (Womble et al. 1990).
There are two other absorption systems, in Mg~II, at $z = 0.394$ and 0.451,
from the halos of other, more distant galaxies on
the line of sight to the quasar (Womble et al. 1990;
Sargent \& Steidel 1990; Borgeest et al. 1991).
After correcting for the primary beam, we derived the
quasar's continuum flux to be 190~mJy at 110~GHz in June-October 1994.
Toward the quasar, our data at 20\,km\thinspace s$^{-1}$\ resolution
do not show any CO(1--0) absorption within
$\pm 600$\,km\thinspace s$^{-1}$\ of the center frequency (Table~1),
to a limit of 30\,mJy, or 15\% of the quasar's millimeter continuum flux,
This range includes the $z=0.0523$
($cz = 15300$\,km\thinspace s$^{-1}$ ) redshift seen in absorption against the quasar
in Na~I and Ca~II.
If the absorbing atomic gas extends 0.3 to 3\,kpc
on the line of sight, the width of the tidal tail on optical images,
then the column densities derived from the Na~I and Ca~II
lines by Womble et al. (1990) imply H~I densities of 1 -- 10\,cm$^{-3}$,
too low for any gas to be in molecular form.
\subsection{ {\bf 10565$+$2448} }
The main optical and near IR peak of IRAS 10565+2448 has strong Br~$\gamma$
lines (Goldader et al. 1995) and H~II region-type line ratios (Armus,
Heckman, \& Miley, 1989, 1990; Veilleux et al.\ 1995).
Murphy et al. (1996) suggest this merger is a triple galaxy system.
On their $r$ band image there is a secondary source on a tidal tail,
26$''$ (20\,kpc) northeast of the main peak, at the same redshift.
A third object with unknown redshift is 7.9$''$ (6.2\,kpc) southeast
of the main peak. In our 500\,MHz band
we detect no CO toward the other two $r$-band sources,
to a limit of 20\,mJy~beam$^{-1}$ in 20\,km\thinspace s$^{-1}$\
channels. At centimeter wavelengths, there is extended emission and an
unresolved, nonthermal core (Condon et al. 1991; Crawford et al. 1996).
Figure~27 shows the maps of CO integrated intensity, isovelocity
contours, and CO linewidth. The peak of the $2''.3\times 1''.7$ CO source
coincides with the compact cm-radio continuum core.
Along the line of nodes, the CO line profiles are very symmetric,
with well-defined blueshifted peaks in the east and redshifted peaks in the
west (Fig.~28). Together with VII~Zw~31 and Arp~193, the galaxy
10565$+$2448 is one of the best examples of a rotating
ring in our sample.
In the position-velocity diagram (Fig.~29), there are two
prominent peaks, and a
velocity shift of 180\,km\thinspace s$^{-1}$\ over the central $1''$ in R.A.
Models that best fit the data are
those with a minimum ring radius $R_{\rm min}$ in eq.(1) equal to
$R_0$ = 230\,pc.
As with VII~Zw~31 and Arp~193,
filled-disk models, with $R_{\rm min}$ = 0, give
poorer fits, with much less contrast.
The CO ring must be nearly face-on, because of the small
separation (80\,km\thinspace s$^{-1}$ ) of the twin peaks,
the narrow CO linewidth (140\,km\thinspace s$^{-1}$\ FWHP),
and the small extent of the integrated CO intensity on the
kinematic major axis.
Unlike Arp\,220 or VII\,Zw 31, there is not much CO in an outer
disk beyond the nuclear ring of 10565+2448.
\subsection{ {\bf 17208$-$0014} }
The ultraluminous galaxy 17208$-$0014
has $L_{\rm IR} =2.2\times 10^{12}$\,L$_\odot$\ and
an optical and near-IR line spectrum
that indicates H~II region-type excitation and high reddening
(Martin et al. 1989; Kim et al. 1995; Goldader et al. 1995;
Veilleux et al.\ 1995).
Images at $r$-band (6550\,\AA) show two tidal tails
from a merger (Melnick \& Mirabel 1990; Murphy et al.
1996). The innermost parts of the two tails, extending north and east to
radii $>20$ kpc are nicely shown in the $r$-band image by Sanders \& Kim
(in Solomon et al. 1997). At $2''.9$ (2.2\,kpc) southeast of the nucleus,
the $r$ band and $I$ band (0.82\,$\mu$m) images show
a secondary peak on a tidal tail.
The $K$-band (2.2\,$\mu$m) surface brightness has an $r^{1/4}$
profile out to a radius of 10\,kpc, as in elliptical galaxies, probably
due to the merger.
The nucleus has a size of $1''.8 \times 1''.4$ in $K$ band and appears
single, which suggests a completed merger
(Zenner \& Lenzen 1993; Murphy et al. 1996).
The nucleus also has one of the strongest known OH megamasers,
with 10$^3$\,L$_\odot$ \ in the 1667\,MHz OH line,
and a radio continuum source of size $0''.32 \times 0''.26$
(220$\times 270$\,pc; Martin et al. 1989).
Figure~30 shows the maps of CO(1--0) integrated intensity and
isovelocity contours. The CO source has a
size of $1''.8 \times 1''.6$, smaller than
measured by Planesas, Mirabel, \& Sanders (1991), and about the same size
as the $K$~band source. The centroid of the CO
source (Table~2) agrees to within $0''.2$ with the
cm-radio continuum source (Martin et al. 1989).
The isovelocity contours and the CO spectra show the molecular gas is
blueshifted in the northwest and redshifted in the southeast.
The position-velocity diagram (Fig.~30)
indicates a strong velocity gradient
with a change of 400\,km\thinspace s$^{-1}$\ over $1''.5$ at p.a. 120$^\circ$.
We interpret this angle
as the line of nodes of the nuclear disk. The CO spectrum at the
peak of the source (Fig.~30) has a linewidth of
375\,km\thinspace s$^{-1}$\ FWHM, and 700\,km\thinspace s$^{-1}$ \ to zero intensity. At the CO
peak, there appears to be a weak continuum at the 5\,mJy
level.
\subsection{ {\bf 23365$+$3604} }
On $R$-band images, the ultraluminous galaxy 23365$+$3604
has two tidal tails, but only a single, blue nucleus
(Klaas \& Els\"asser 1991; Murphy et al. 1996).
The optical spectrum (e.g., Veilleux et al.\ 1995)
is of the LINER type, with line ratios
implying shock velocities of 80 -- 90\,km\thinspace s$^{-1}$\ and pre-shock electron
densities of 1 -- 10\,cm$^{-3}$ in tenuous gas, as in old
supernova remnants (Klaas \& Els\"asser 1991). In the nucleus,
The [O III] line has a width of 260\,km\thinspace s$^{-1}$\ FWHM (Kim et al. 1995).
The source has strong Br$\gamma$ and H$_2$ lines at 2\,$\mu$m
(Goldader et al. 1995). In the 6\,cm radio continuum, there is a secondary
source extending 1$''$ south of the nucleus (Crawford et al. 1996).
In our 40\,km\thinspace s$^{-1}$\ channel maps, the CO source is compact in all channels
over a range of 320\,km\thinspace s$^{-1}$ .
The CO source has a size of $1''.0\times 0''.9$, with a peak
2$''$ south of the optical position listed by Klaas \& Els\"asser (1993).
The gradient in the isovelocity contours is at p.a. 135$^\circ$, and
the CO spectra along this line of nodes
show asymmetric profiles characteristic of a rotating ring or disk.
Along this kinematic major axis, the position-velocity diagram
shows a shift of 300\,km\thinspace s$^{-1}$ \ over 4$''$ (Fig.~31).
\section {DISCUSSION}
\subsection{Source sizes, gas masses from CO imaging, and the ratios
$M_{\rm gas}$/$L^\prime_{\rm CO}$
and $M_{\rm gas}$/$M_{\rm dyn}$
}
The main results of this study are the CO source
sizes --- the halfwidths of the integrated CO emission (Table~3),
and the inner and outer disk radii from the model fits to the
position-velocity diagrams and the channel maps (Table~4).
The measured radii of the integrated CO correspond to the
half-power radii $R_1$ derived from the kinematic data.
We had previously noted that for ultraluminous galaxies the CO(1--0)
flux (Jy\,km\thinspace s$^{-1}$ ) was $\sim 4$ times the 100\,$\mu$m flux (Jy), as expected
from a blackbody model for the far IR (Downes, Solomon, \& Radford 1993).
We derived the dust temperature from the FIR fluxes, and
took this to be the intrinsic CO brightness temperature. From the CO
luminosity, we then predicted the mean radius, $R_{\rm CO}$, for a spherical
CO source (Solomon et al. 1997). Interestingly, the sizes measured with
the interferometer are within a factor of two of the blackbody sizes.
The radii $R_0$ and $R_1$ from our fits to the
kinematic data (Table~4) bracket our single-dish estimate:
$R_0 < R_{\rm CO} < R_1$. The CO radii measured with the interferometer
now allow us to better estimate the
gas and dynamical masses (Table~9).
In the centers of ultraluminous
galaxies, we find $M_{\rm gas}$/$L^\prime_{\rm CO} \approx 0.8$
M$_\odot$\,(K\,km\thinspace s$^{-1}$\,pc$^2$)$^{-1}$, about 5 times lower than the standard value
for self-gravitating molecular clouds.
We had anticipated this result from single-dish data (Downes et al. 1993;
Solomon et al. 1997), and it is now confirmed by the
interferometer maps.
We had earlier used the CO linewidths and radii from our blackbody model
to obtain dynamical mass, which in turn led us to revise downward the
gas masses from single-dish CO luminosities. We estimated
a minimum gas mass by assuming optically thin CO.
These estimates, which we labeled $M_{\rm thin}$ (Solomon et al. 1997),
are within a factor 1.5 of the gas masses
derived from our model fits to the interferometer data.
Table~9 also lists the ratio of gas mass to dynamical mass,
which is $\sim 1/6$ in the inner, high-density disk,
and 1/10 in the outer, low-density disks.
We showed earlier (Downes et al. 1993) that
the $M_{\rm gas}$/$M_{\rm dyn}$ and
$M_{\rm gas}$/$L^\prime_{\rm CO}$ ratios are related by
\begin{equation}
\bigg({ M_{\rm gas} \over M_{\rm dyn} }\bigg) =
\bigg( {M_{\rm gas} \over L^\prime_{\rm CO}} \bigg)^2
{1 \over \alpha^2 } \ \ \ ,
\end{equation}
where, letting $f\equiv M_{\rm gas}/M_{\rm dyn}$, we have
\begin{equation}
M_{\rm gas} = f^{0.5} \alpha L^\prime_{CO} \ \ \ .
\end{equation}
The factor $\alpha = C n^{0.5}/T_b$, where $n$ is the mean H$_2$ number
density over the whole volume and $T_b$ is the CO brightness temperature.
In the units used here, the constant $C=2.6$ for a sphere
and $\sim $1.0 for a flared disk with thickness/radius ratio of
0.15, as in some of the disks in our sample.
These equations depend only on gravity, and are independent of
whether the gas is self-gravitating or not, or whether the CO lines are
optically thick or thin.
The derived values for $M_{\rm gas}$/$M_{\rm dyn}$ and
$M_{\rm gas}$/$L^\prime_{\rm CO}$ are compatible with starbursts.
Assume a starburst powers an ultraluminous IR galaxy, with
$L/M_{\rm new\star}$ = 300\,L$_\odot$ M$_\odot$$^{-1}$ for the new stars in
the burst, all the power emerging in the IR.
Such galaxies typically have
$L_{\rm IR}/L^\prime_{\rm CO}$ = 200\,L$_\odot$ (K\,km\thinspace s$^{-1}$\ pc$^2$)$^{-1}$.
Combining the two luminosity ratios gives
\begin{equation}
M_{\rm new\star} = (2/3) L^\prime_{\rm CO} \ \ \ ,
\end{equation}
or $\sim 3\times 10^9$\,M$_\odot$\ as the mass of new stars
in the burst. Most galaxies prior to the merger would have had
a rotation velocity $\sim $250\,km\thinspace s$^{-1}$\ at a radius of 500\,pc, which
implies an old stellar bulge about twice as massive as the newly-formed
population. Since $M_{\rm dyn}$ = $M_{\rm gas}$ + $M_{\rm new\star}$
+ $M_{\rm old\star}$, we have from the previous equation that
\begin{equation}
M_{\rm dyn} = M_{\rm gas} + 2 L^\prime_{\rm CO}.
\end{equation}
Solving for the molecular gas mass yields
\begin{equation}
{ M_{\rm gas} \over L^\prime_{\rm CO} } = (1+\alpha^2)^{1/2} - 1 \ \ \ .
\end{equation}
This equation can be generalized to arbitrary mass ratios of the old
and new stellar populations. For the Arp~220 disk, with a {\it mean}
H$_2$ density $\sim 500$\,cm$^{-3}$ and $<T_b>$ = 20\,K, these
equations predict $\alpha\sim 1$, $M_{\rm gas}/M_{\rm dyn}$ = 0.2, and
$M_{\rm gas}/L^\prime_{\rm CO}$ = 0.4\,M$_\odot$\,(K\,km\thinspace s$^{-1}$\,pc$^2)^{-1}$,
close to the values derived from our model fits to the interferometer
data. In other words, the starburst scenario is compatible with the
derived ratios.
\subsection{Dense gas traced by HCN and extreme starburst regions}
There is another important gas component that is better traced by
emission from high dipole moment molecules like HCN, rather than CO.
The HCN emission requires densities $n$(H$_2$)$ >10^4$\,cm$^{-3}$. In
giant molecular clouds in the disks of the Milky Way and normal
spirals, the average HCN emission is weak with a typical ratio of
{\hbox {$L_{\rm HCN}$}}/{\hbox {$L_{\rm CO}$}} $= 1/20$ to 1/40 (Gao \& Solomon 1998). In the Milky Way,
HCN emission is strong only in cloud cores that form high-mass stars.
We showed previously that ultraluminous galaxies have abnormally high
HCN luminosities (Solomon, Downes, \& Radford 1992), with {\hbox {$L_{\rm HCN}$}}/{\hbox {$L_{\rm CO}$}}
$= 1/4$ to 1/8, indicating a higher fraction of the total molecular gas
is in
dense, $ >10^4$\,cm$^{-3}$, star-forming cores than in normal
galaxies. The densities we derive for the smoothly distributed gas in
the rotating disks (see Table 6) are typically 10 times lower than that
needed for HCN excitation, except
for the extreme starburst regions Arp220-west, Arp220-east and the
disk of Mrk231. The denser gas responsible for the HCN emission (and
CS emission) is thus not adequately accounted for in most sources by
the ``diffuse" CO emitting component that fills the disk.
In Arp~220, the HCN lines we observed have the same velocities as
the east and west ``nuclei'',
so it seems that the east and west ``nuclei'' alone account for most of
the HCN emission. These dense, compact sources have a hydrogen column
density of $0.6 \times 10^{25}$\,cm$^{-2}$ and mean density of 20,000
cm$^{-3}$, enough to thermalize the lower rotational levels of HCN by
a combination of collisions and radiative trapping. Within the
Arp~220 east and west sources, the HCN emission may thus have the same
intrinsic brightness temperature as the CO(2--1) emission, namely,
50\,K (Table~8). Using the sizes and linewidths from Tables~4 and 5,
we estimate the HCN luminosity of these two regions alone to be
{\hbox {$L_{\rm HCN}$}} $\approx 7 \times 10^8 $ K\,km\thinspace s$^{-1}$\,pc$^2$, which is 3/4
of the observed total (Solomon, Downes, \& Radford 1992).
These two regions thus emit only 1/4
of the CO luminosity but most of the HCN luminosity.
Their total gas mass of 1.7 $\times 10^9$\,M$_\odot$\ is
already accounted for in the mass budget in Table~9.
Thus for Arp~220, only a small increase in
the mass budget would be enough to account for the HCN emission.
While some of the HCN emission undoubtedly comes
from dense star forming molecular cores embedded
in the (relatively) diffuse CO disks of ultraluminous galaxies,
it is likely that high density, extreme starburst regions
similar to Arp~220 east and west exist in
most of the other ultraluminous galaxies in our sample, and are the
real sources of most of the HCN emission.
Arp~220 is the closest
ultraluminous galaxy and has the best resolved structure.
The second closest
galaxy, Arp~193 (see section 6 and Figs.~10 and 11),
also shows evidence of an extreme starburst in the
southeast, that appears similar
to the strong sources in Arp~220.
We suggest that the high-density, extreme starburst regions are the source of
much of the HCN luminosity but only a fraction of the CO luminosity.
Most are beyond detection at our
current resolution in CO emission since their size of $\approx$100 pc
would be $< 0''.14$
for all but two of the galaxies (see Table~3).
These objects may then add about $1 \times 10^9$\,M$_\odot$ ,
or 25\%, to the total gas mass in galaxies with very high HCN emission.
This is less than we estimated in our 1992 paper.
In line with this re-interpretation, we note that
the galaxy VII~Zw~31 has much weaker HCN emission than most of the others
in our sample. It also has
the largest CO disk, a lower ratio of {\hbox {$L_{\rm FIR}$}}/{\hbox {$L_{\rm CO}$}}\
(less star formation per solar mass of gas), and may not
have any extreme starburst regions like Arp~220 east or west.
\subsection{Stability of the molecular disks}
The standard parameter for characterizing
the stability against local, axisymmetric perturbations
of a disk that is supported
by differential rotation and random motion is
\begin{equation}
Q= {{\sigma_v \kappa}\over {\pi G \Sigma} } \ \ \ \ ,
\end{equation}
where $\sigma_v$ is the one-dimensional random
velocity dispersion, $\kappa$ is the local epicyclic
frequency, and $\Sigma$ is the mass surface density.
(Safronov 1960; Toomre 1964; Goldreich \& Lynden-Bell 1965;
for a recent review of the criterion applied to gravitationally coupled
stars and gas in a disk, see Jog 1996).
If $Q$ is $<1$, the structure is unstable, and
large massive star clusters may form.
We used our interferometer data on the sizes, turbulent velocities,
rotation velocities, and mass surface densities to estimate the
stability of the nuclear disks of the ultraluminous galaxies.
In our model rotation curves, the epicyclic frequencies are
$\kappa \approx 2V/R =$ constant on the rising part of the rotation curve,
and $\kappa \approx \sqrt 2 V/R$ on the flat part of the rotation curve.
The objects in our sample
have high gas mass fractions and
typically have $Q_{\rm gas}\leq 2$, but $Q_{s+g} <1$ for stars plus gas
(the regime in Fig.~1e of Jog 1996).
For Arp~220, we obtain $Q_{\rm gas}$ = 2.2 at $R=480$\,pc for the gas alone,
but
$Q\sim 1$ for the gas plus stars.
For VII~Zw~31, at a radius of $R=1100$\,pc,
we obtain $Q$ = 1.1 and 0.9 for these two masses.
These values suggest the molecular disks
of Arp~220 and VII~Zw~31 are globally unstable against
axisymmetric perturbations
and will form massive star clusters.
Empirically, this instability in the central disks appears to
produce one or more
large clumps of dense molecular gas
--- the compact, nearly unresolved peaks on the CO maps.
Table~12 lists some of these
compact regions of dense molecular gas,
which
we identify with extreme starbursts.
Their typical radius of 70 to 100\,pc may be the scale on which
the two-component (stars + gas) system becomes unstable. Their mean
H$_2$ density can reach
$2\times 10^4$\,cm$^{-3}$, and their mass appears to be
$\sim 1\times 10^9$\,M$_\odot$\ of gas
initially, which then forms
$\sim 10^6$ OB-type stars over 10$^7$\,yr,
or $\sim 1000$ times the number of OB stars in 30~Doradus. The total
luminosity of one of these extreme starburst regions is $3\times 10^{11}$
to $1\times 10^{12}$\,L$_\odot$ . The central disk of an ultraluminous IR
galaxy typically contains two or three such extreme starburst
regions at any given time.
It is mainly these regions that provide the input
power to the ultraluminous galaxies.
They heat the dust in the central disks to typical temperatures of 75\,K,
(the blackbody fits to the colors measured by IRAS). Since the dust
is opaque at 100\,$\mu$m, the disk radiates as a black body. The
typical disk radius is 200 to 300\,pc, and the Stefan-Boltzmann formula
for the disk as a whole yields a luminosity of the order of 10$^{12}$\,L$_\odot$ ,
the typical output power of the ultraluminous galaxies. If these
starbursts occur in the old, pre-merger nuclei, they change them
considerably, creating a new cusp with a much greater density of
stars. If the extreme starbursts occur slightly outside of the old
pre-merger nuclei, they will form new nuclei -- new cusps of high
stellar density.
\subsection{Why stars outshine black holes}
The CO disks' radii of $\sim$500\,pc and
rotation speeds of $\sim$300\,km\thinspace s$^{-1}$\ yield orbital periods of
10\,Myr. If their FIR luminosity of $\sim 10^{12}$\,L$_\odot$\ comes
from new stars, then our estimates of
$L_{\rm FIR}/M_{\rm new\star}$ = 300\,L$_\odot$ M$_\odot$$^{-1}$
imply star forming rates of 50\,M$_\odot$\,yr$^{-1}$. The mass of gas plus
new stars is $\sim 6\times 10^9$\,M$_\odot$ , so
in 10 rotations of the molecular disks, half the gas turns into new stars.
If part of the $\sim 10^{12}$\,L$_\odot$\ luminosity came from a black hole
accretion disk radiating at
$L\sim 0.1\ \dot m c^2$, the accretion rate would be
$\sim 1$\,M$_\odot$\,yr$^{-1}$, so in 10 molecular disk rotations
the black hole could only accrete $\sim 1$\% of the molecular gas.
The other 99\% of the gas would continue to form stars over this long period,
outshining the black hole.
Alternatively, if the gas fell in faster and created
a $6\times 10^9$\,M$_\odot$\ black hole in one or two orbital periods, then
the accretion rate would be 60\,M$_\odot$\,yr$^{-1}$, and a
standard Shakura \& Sunyaev (1973) optically thick
accretion disk, radiating at one-tenth the Eddington luminosity, would have a
luminosity shooting up to 10$^{15}$\,L$_\odot$\ --- 1000 times
more luminous than the IR ultraluminous galaxies and
quasars in the local universe.
If the gas fell in rapidly at an accretion rate of 60\,M$_\odot$\,yr$^{-1}$,
but the
luminosity stayed at 10$^{12}$\,L$_\odot$ , then the accretion disk's
radiative efficiency would be only
$L = 10^{-4}\ \dot m c^2$ --- not any more efficient than producing
energy from starbursts. Although it is unlikely that such a massive,
high-density, accretion flow would be advection dominated (e.g., Rees 1982),
if the flow were in this regime, then almost by definition, the accretion
disk would not be a luminous object.
The model of rapid accretion to a black hole thus poses more problems
than it solves. With a very high accretion rate of 60\,M$_\odot$\,yr$^{-1}$,
why would the IR ultraluminous galaxies only have an output of
10$^{12}$\,L$_\odot$\ if their power source were a black hole? Why aren't
they as luminous as the powerful quasars at high redshifts,
since there is no shortage of fuel? Inversely, if the power source were
a black hole accreting at a modest rate of 1\,M$_\odot$\,yr$^{-1}$,
it would last for $6\times 10^9$\, years and
there would be many more ultraluminous
galaxies than are observed. Since the source statistics indicate
the ultraluminous
phase in mergers lasts for $\sim 10^8$ years, the more likely answer is
that the gas is used up in forming stars -- not quasars.
In summary, even in 10 rotations of the molecular disk, a black hole
radiating 10$^{12}$\,L$_\odot$\ at high efficiency
could accrete only 1\% of the molecular disk's mass.
So even though new stars convert matter into energy less
efficiently than a standard accretion disk,
they make up for this in ultraluminous
IR galaxies by occurring in burst,
in a 2000 times larger volume, with a 2000 times
larger total mass than is available within the Bondi radius $G\,M_{\rm bh}/V^2$
of a supermassive black hole. The molecular disks probably
survive for at least 10 rotations. During this time, at most
10$^7$\,M$_\odot$\ is accreted to a black hole. The rest of the molecular
gas forms stars.
\section{CONCLUSIONS}
\noindent
{\it 1.) Rotating disks:} At sub-arcsecond resolution,
our CO maps of ultraluminous galaxies
show rotating disks of molecular gas that has been
driven into the centers of the mergers.
The maps of Mrk~273 and Arp~220 also show large-scale
streamers or tidal tails roughly perpendicular to the nuclear disks.
\noindent
{\it 2.) CO only moderately opaque:}
Model fits show that the observed double-peaked CO spectra, and the
peak-to-center contrast in the twin-peaked patterns in
CO position-velocity diagrams, are produced by radiative transfer
through the rotating disks, in CO that is only moderately opaque.
The CO(1--0) opacities are 4 to 10 at the peaks, and lower elsewhere in
the disks.
\noindent
{\it 3.) High turbulence in the molecular disks:} In the Arp~220
disk, the 1-D velocity dispersion of the molecular gas is
$\sigma = 100$\,km\thinspace s$^{-1}$\
(FWHM 230\,km\thinspace s$^{-1}$ ). This high turbulence is one of the reasons why the
CO is less opaque than in quiescent Milky Way molecular clouds.
The turbulence also determines the disk
thickness, and the heat input to the gas.
The turbulent heat input, $M_{\rm gas} \Delta V^3/R$, is
$\sim 10^9$\,L$_\odot$ , the same as the output from
the 17\,$\mu$m S(1) line of H$_2$, a principal cooling line (Sturm et al.
1996). The line luminosity is that expected from $10^9$\,M$_\odot$\ of gas
at $\sim 100$\,K.
\noindent
{\it 4.) Most of the CO luminosity comes from relatively ``low-density" gas:}
Except for the compact cores
in Mrk~231, Mrk~273, and Arp~220, the molecular disks have
true CO brightness temperatures $\sim 20$\,K. This is
rather lower than the dust and gas kinetic temperatures, which are
$\sim$65 to 100\,K.
This implies that the density over most of the molecular disk is relatively
low (300 to 2000\,cm$^{-3}$), giving subthermal excitation.
The CO(2--1)/(1--0) ratios are also consistent with
the CO being subthermally excited.
\noindent
{\it 5.) The CO lines in the molecular disks
come from a continuous medium, not from
self-gravitating (``virialized") clouds:}
At the densities of 10$^3$\,cm$^{-3}$
needed to explain the observed flux, low-density molecular clouds
would be unstable
against tidal shear in the rotating disks. Because we also detect
high density tracers like HCN and CS, we estimate
$\sim 10$\% of the gas is in highly opaque,
high-density (10$^5$\,cm$^{-3}$), self-gravitating
clouds that are stable against tidal forces.
In the outer disks, however, the CO emission
may come from ``normal" molecular clouds. In this outer
disk gas, the volume filling factor would be $\sim 0.1$, the area filling
factor 0.3, as in normal galactic disks.
\noindent
{\it 6.) Low dust flux consistent with lower molecular mass:}
Only in Arp~220 is there significant thermal continuum flux from dust
at 1.3\,mm.
The continuum in Mrk~231 at 3 and 1.3\,mm is nonthermal.
At 3\,mm, no thermal continuum is detected from dust,
to limits of $\sim $2 to 10\,mJy, in accord with the gas masses
obtained from CO, corresponding to a CO luminosity to gas mass conversion
factor about 5 times lower than in self-gravitating clouds in the Milky Way.
\noindent
{\it 7.) Still lots of gas, nevertheless:}
In spite of the ``low" gas densities and CO line opacities,
the derived gas mass is high; the mass of $\approx 5 \times 10^9 $\, M$_\odot$\ is
equal to the mass of molecular clouds in a large gas rich spiral galaxy.
Within
the molecular disks, the ratio of gas mass to the enclosed dynamical mass is
$M_{\rm gas}/M_{\rm dyn}$ = 1/6. The ratio of gas to total mass surface
density, $\mu/\mu_{\rm tot}$, reaches a maximum value of 1/3 within the
molecular disks. The ratio $M_{\rm gas}/L^\prime_{\rm CO}$ of gas mass to CO
luminosity is about one-fifth of its value in self-gravitating molecular clouds.
For the galaxies in this sample, typical ratios are
$M_{\rm gas}/L^\prime_{\rm CO}$ =
0.8\,M$_\odot$ (K\,km\thinspace s$^{-1}$\,pc$^2)^{-1}$.
\noindent
{\it 8.) Extreme Starburst Regions:}
Four extreme starbursts are identified in the 3 closest galaxies in the sample
including Arp~220, Arp~193 and Mrk~273. They are the most prodigious star
formation events in the local universe, each representing about 1000 times as
many OB stars as 30~Doradus. They have a
characteristic size of only 100 pc, with about $10^9$ M$_\odot$\ of gas and an IR
luminosity of $\approx 3 \times 10^{11}$ L$_\odot$\ from recently formed OB stars.
Arp~220 has 2 extreme starbursts.
The integrated CO and 1.3\,mm continuum maps of Arp~220
show two compact peaks.
The west nucleus, at $cz_{\rm lsr}$ = 5340\,km\thinspace s$^{-1}$ , is the same as the
cm-radio and IR west peak.
the CO east gas at 5330\,km\thinspace s$^{-1}$ , is associated with the cm-radio east
peak and its OH megamasers.
The CO east gas at 5650\,km\thinspace s$^{-1}$\ is associated with the ionized gas in
the $K$-band east nucleus.
The IR luminosity of the compact peaks in Arp~220 can be
explained by extreme starbursts in their early phases.
The cm-radio continuum, the CO intensity, and the 1.3\,mm dust flux
all suggest the currently most active starburst is the west peak.
Arp~220 west also shows a complex velocity
structure which may indicate a bar in formation or a huge molecular outflow.
The mass of
these compact regions is dominated by molecular gas and young stars, not by a
bulge population.
\noindent
{\it 9.) Regions of dense molecular gas are regions of extreme starbursts:}
We suggest that the HCN emission,
which is very strong in most ultraluminous galaxies,
originates in the high-density, extreme starburst regions
similar to Arp~220 east and west.
These regions
are larger than ordinary GMCs, but are
filled with molecular gas at a density usually found only in small cloud cores.
They do not produce most
of the CO luminosity, but they do emit most of the HCN luminosity. This explains
the high HCN luminosity and directly relates the HCN emission to star
formation, as we suggested previously (Solomon et al. 1992).
\noindent
{\it 10.) Ultraluminous galaxies are powered by starbursts, not AGNs:}
The CO data show the gas in ultraluminous
galaxies is in extended disks that cannot intercept all the power of
central AGNs, if they exist. As a rule of thumb, if you can see the
AGN in the UV/visible, then it is not heavily absorbed, and
cannot be responsible for the far-IR/sub-mm luminosity.
Furthermore,
if the rotating molecular disks were very thin (30\,pc) and extended
all the way in to central AGNs, then their dust would be very hot, emitting
most of their power at 10 to 20\,$\mu$m. In fact, most of the dust (and
gas) observed in the ultraluminous IR galaxies is cool (70\,K),
emitting at 60 to 100\,$\mu$m --- the
usual temperature of dust in molecular clouds heated by starbursts.
We conclude that in
ultraluminous galaxies --- even in Mrk~231 that hosts a
quasar --- the far IR luminosity is powered by extreme
starbursts in the molecular disks, not by dust-enshrouded quasars.
\acknowledgments
We thank the telescope operators at Plateau de Bure and the
IRAM staff astronomers for their help in taking the data,
A.\ Dutrey for use of her disk modelling program, C.M.\ Walmsley
for the escape probability program, and S.\ Guilloteau, R.\ Lucas,
and J.\ Wink for help in data reduction.
We also thank H.\ Ungerechts for re-measuring some
CO fluxes at the IRAM 30\,m telescope, and P.\ van der Werf, R.\ Genzel, and
N.\ Scoville for helpful talks.
We thank the referee for many helpful comments. P.M.S.\ is grateful for a
Research Award from the Alexander von Humboldt Foundation and for
a senior scientific fellowship from the North Atlantic Treaty
Organization.
\clearpage
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,195 |
Jan. 12, 2023 — The January get-togethers
January 12, 2023 at 4:24 am · Filed under Uncategorized and tagged: Field hockey, Lacrosse, Omnibus
Over the next two weeks, there are three major get-togethers involving sports and youth sports. The United Soccer Coaches' Convention in Philadelphia, the U.S. Lacrosse Convention in Baltimore, and the National Field Hockey Coaches' Association Convention in Lake Mary, Fla. allow major figures in the game, vendors, sponsors, and others to meet, mingle, and share ideas.
Many of these ideas revolve around the mental health of athletes, something which I feel is long overdue in youth sports. All three of these conventions have some sort of seminar or panel on this topic, and address them in various ways.
As youth sports transitions out of pandemic-era controls on the participants, coaches, spectators, and the sport itself, coaching and scholastic administration look like they are playing a game of catch-up. I'm seeing, in many spaces, a different relationship between coach and player from what I saw three decades ago.
I've seen first-hand a pool of potential walkons walk right back off again; the pool of 64 wound up being somewhere between six and eight to add to the varsity program which was coached by a Hall-of-Famer. But this wasn't a group which was verbally brow-beaten, or made to work past physical exhaustion, or thrown into mid-July heat. As a coach once told me of hometown walk-ons, "Often, they'll self-select themselves out of the team."
I'll always remember a couple of players who were on my list of published all-stars who went to a local college in order to try to make the team. One never made it to campus because of a family situation which kept her out of university, even though she was a multi-tooled player with great speed. Another player I remember had a horrific hip socket injury of the same kind that ended the football career of Bo Jackson.
I've gotten some pushback from some folks who suggest that putting the needs of the athlete first, rather than the team, means that the culture of sport has somehow gotten "soft."
But I can't help but think that the story of Kory Stringer, the professional football player who died of heatstroke in August 2001, was an enormous rallying cry when it came to how far a coach can push an athlete. Sure, the lore of coaching over the years is full of stories of hard and physical training camps which are meant to whittle down the potential varsity player pool.
Many folks who endure these camps and make the team, and find success on their chosen field of endeavor, look back over the years and wouldn't change a thing.
Yeah, I get it: it's only human nature to posit that a team culture based upon hard work will lead to success, even if the training is of such intensity that the mental or physical health of players is imperiled.
It's a fine balance, one which only a precious few coaches have ever found. | {
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} | 2,638 |
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\section{Introduction}
A widely observed and yet still striking phenomenon is that modern
machine learning models (e.g., deep neural networks) trained by stochastic
gradient descent often generalize while also
achieving near-zero training error (i.e., despite being
overparameterized and under-regularized~\footnote{By ``under-regularized'', we mean that the empirical training loss is near to $0$, such as with OLS when $N \gg d$.}. See~\cite{belkin2020two} for further discussion).
There is reason to believe that characterizing these effects even in conceptually simpler (e.g. linear model) settings will also help our understanding of more complex settings, because many high dimensional effects are also observed even in simple linear models.
For example, this \textit{benign overfitting} effect is also observed
for the ordinary least square (OLS) estimator, where it is widely observed
that OLS generalizes in the overparameterized regime.
For OLS in particular, the recent work of
\citet{bartlett2020benign} established non-asymptotic
generalization guarantees of the \emph{minimum-norm interpolator} for
overparameterized linear regression (the minimum-norm solution that
\emph{perfectly fits} the training samples
\citep{zhang2016understanding,bartlett2020benign}).
More generally, there is a growing body of work studying generalization in basic linear models in the overparameterized regime \citep{nakkiran2019deep,bartlett2020benign,belkin2020two,hastie2019surprises,tsigler2020benign,muthukumar2020classification,chatterji2020finite,nakkiran2020optimal}. In contrast, for \emph{stochastic gradient descent} (SGD) for least squares regression, the algorithmic aspects of generalization are far less well understood, where we lack a sharp characterization of it and when benign overfitting occurs. This is the focus of this work.
\iffalse
A widely observed and yet still striking phenomenon is that modern
machine learning models (e.g., deep neural networks) trained by stochastic
gradient descent often generalize while also
achieving near-zero training error (i.e., despite being
overparameterized and under-regularized~\footnote{By ``under-regularized'', we mean that the underlying empirical optimization near to $0$ training error, such as with ordinary least squares with when $N \gg d$.}. See~\cite{belkin2020two} for further discussion.)
More recently,
there is an increasing realization that many of these effects can be
observed even in simple linear models, and there is a reason to believe that
sharply characterizing these effects even in conceptually simpler
settings will also help our understanding of more complex settings.
For example, this \textit{benign overfitting} effect is also observed
for the ordinary least square (OLS) estimator, where it is widely observed
that OLS generalizes in the overparameterized regime. Recently,
\citet{bartlett2020benign} recently established non-asymptotic
generalization guarantees of the \emph{minimum-norm interpolator} for
overparameterized linear regression (the minimum-norm solution that
\emph{perfectly fits} the training samples
\citep{zhang2016understanding,bartlett2020benign}).
More generally,
there is growing body of work in sharply
understanding generalization in basic linear models in the
overparameterized regime
\citep{nakkiran2019deep,bartlett2020benign,belkin2020two,hastie2019surprises,tsigler2020benign,muthukumar2020classification,chatterji2020finite,nakkiran2020optimal}.
In contrast, even for \emph{stochastic gradient descent} (SGD) for least squares regression,
the algorithmic aspects of benign overfitting are far
less well understood, where we lack an
sharp characterization of if and when benign overfitting occurs. This is the
focus of this work.
\fi
With regards to SGD in the classical underparameterized regime, the
seminal work of~\cite{polyak1992acceleration} showed that iterate
averaged SGD achieves, in the limit as the sample size goes to
infinity, the statistically optimal rate, even up to problem dependent
constant factors~\footnote{\cite{polyak1992acceleration} provided a
stronger distributional limit theorem showing that the distribution
of the averaged iterate (provided by SGD) precisely matches the
distribution of the empirical risk minimizer.}; this optimality
crucially relies on the dimension being held finite, along with
regularity assumptions that make the problem locally strongly
quadratic. For the case of \emph{finite} dimensional, linear regression, there are a number of more
modern proofs which provide finite, non-asymptotic
rates~\citep{defossez2015averaged,bach2013non,dieuleveut2017harder,jain2017parallelizing,jain2017markov}.
With regards to the overparameterized regime, there is far less work
~\citep{DieuleveutB15,berthier2020tight} being notable exceptions.
(See Section~\ref{sect:Related} for further discussion on these related works.)
\iffalse
these results show that, in the finite dimensional case, that
convergence is rapid under either margin based
assumptions~\cite{bach2013non} or strong convexity
assumptions~\citep{jain2017markov,jain2017parallelizing}. With regards
to the overparameterized regime, there is far less work
(~\citep{DieuleveutB15,berthier2020tight} being notable exceptions,
which we discuss later).
\fi
The focus of this work is on the overparameterized regime: we seek a
sharp characterization of if and when generalization
can occur for constant stepsize, \emph{unregularized} SGD.
\iffalse
From a
technical standpoint, our work also develops new proof techniques for
iterate averaged SGD, where we further build on the tools developed in~\cite{jain2017markov} (which
utilized a characterization of the stationary distribution of the
iterates of SGD).
\fi
\paragraph{SGD for linear regression.}
The classical linear regression problem of interest is:
\begin{equation}\label{eq:least_square}
\min_\wb L(\wb) ,\ \textrm{where} \,\, L(\wb) = \frac{1}{2}\EE_{(\xb,y)\sim\cD}\big[(y - \la\wb,\xb\ra)^2\big],
\end{equation}
where $\xb\in\mathcal{H}$, is the feature vector, where, $\mathcal{H}$ is some (finite $d$-dimensional or countably infinite dimensional) Hilbert space; $y\in\RR$ is the
response; $\cD$ is an unknown distribution over $\xb$ and $y$; and $\wb \in \mathcal{H}$ is the weight vector to be optimized.
We consider the stochastic approximation approach
using constant
stepsize SGD, with iterate averaging: at each iteration $t$, an i.i.d.
example $(\xb_t, y_t) \sim \cD$ is observed, and the
weight is updated according to SGD as follows:
\begin{equation}\label{eq:sgd}
\wb_t = \wb_{t-1} + \gamma \rbr{ y_t - \abr{\wb_{t-1}, \xb_t} } \xb_t,\qquad t=1,\dots, N,
\end{equation}
where $\gamma > 0$ is a constant stepsize, $N$ is the number of
samples observed, and the weights are initialized at $\wb_0 \in
\cH$. The final output will be the average of the iterates:
\begin{equation*}
\overline{\wb}_{N} := \frac{1}{N} \sum_{t=0}^{N-1} \wb_t.
\end{equation*}
In the underparameterized setting with finite dimension $d$ ($d\ll
N$), as mentioned earlier (also see Section~\ref{sect:Related}), a rich body of work has established that
$\overline{\wb}_{N}$ enjoys the optimal risk (up to constant factors)
of $\bigO{{d\sigma^2}/{N}}$, for sufficiently large $N$.
The focus of this work is on the overparameterized regime, where $d\gg
N$ (or possibly countably infinite).
\paragraph{Our contributions.}
Our main result can be viewed as a counterpart to
the classical analysis of iterate averaged SGD to the
overparameterized regime for linear regression:
we provide a sharp excess risk bound showing how
(unregularized) SGD can generalize even in the infinite-dimensional setting.
Our bound is stated in a general manner, in terms of the full
eigenspectrum of the data covariance matrix along with a functional dependency on the
initial iterate; our lower bound shows our characterization is
tight. As a corollary, we see how the
benign-overfitting phenomenon can be observed for SGD, provided certain spectrum decay
conditions on the data covariance are met. We also extend our results to SGD with tail-averaging \citep{jain2017markov,jain2017parallelizing}, where we run SGD for $s$ iterations and then take average over the subsequent $N$ iterates as the output. (see Section \ref{sec:tail_averaging} for more details.)
\iffalse
Our main result establishes an excess risk bound for constant stepsize
SGD for overparameterized linear regression, showing how
SGD can generalize well even in the infinite-dimensional setting.
Our bound is stated in a general manner, in terms of the full
eigenspectrum of the covariance matrix along, with a functional dependency on the
initial iterate. As corollary, this bound exhibits
benign-overfitting phenomenon, provided certain spectrum decay
conditions on the data covariance are met.
Furthermore, our results complement the existing theory of SGD solving
least square problems in the underparameterized setting
\citep{jain2017parallelizing,jain2017markov,dieuleveut2017harder}. In
fact, by properly choosing the \emph{effective dimensions} (let $k^*$
be the surrogate dimension parameter and let $\lambda_i=0$ for $i>k^*$
in Theorem \ref{thm:generalization_error}), our results match the
previously obtained convergence rates for constant stepsize SGD with
iterate averaging in the underparameterized regime
\citep{bach2013non}. We remark on the strongly convex case in the
Discussion (see Section~\ref{sect:Related}).
\fi
Some additional notable contributions are:
\begin{enumerate}
\item
The sharpness of our bounds permits us to make comparisons to OLS
(the minimum-norm interpolator) and ridge regression.
Notably, in a contrast to the variance of OLS \citep{bartlett2020benign},
the variance contribution to SGD is well behaved under substantially
weaker assumptions on the spectrum of the data covariance.
This shows how inductive bias of SGD, in comparison to the
minimum-norm interpolator, can lead to better generalization
with no regularization. We also constrast our results to
ridge regression based on the recent work by~\citet{tsigler2020benign}.
\item One notable aspect of our work is a sharp
characterization of a ``bias process'' in SGD. In particular, consider the
special case where $y=\wb^\star \cdot \xb$ (with probability
one), for some $\wb^\star$. Here, SGD still differs from
gradient descent on $L(\wb)$. Our characterization gives a novel
characterization of how the variance in this process contributes
to the final excess risk bound.
\item From a technical standpoint, our work develops new proof techniques for
iterate averaged SGD.
Our analysis tools are based on the operator view of averaged
SGD~\citep{DieuleveutB15,jain2017parallelizing,jain2017markov}.
A core idea in the proof is in connecting the finite sample (infinite dimensional) covariance
matrices of the variance and bias stochastic processes to those of
their corresponding (asymptotic) stationary covariance matrices ---
an idea that was introduced
in~\cite{jain2017markov} for the finite dimensional, variance analysis.
\iffalse
The analysis tools are based on the operator view of averaged SGD
\citep{jain2017markov,jain2017parallelizing,dieuleveut2017harder}, and
the core step is a refined \emph{dimension-free} analysis of the
finite-step error which connects the spectrum decaying properties of
the data covariance matrix. This may be of broader interest.
\fi
\end{enumerate}
\paragraph{Notation.}
We use lower case letters to denote scalars, and we use lower and upper
case bold face letters to denote vectors and matrices
respectively. For a vector $\xb\in \mathcal{H}$,
$\|\xb\|_2$ denotes the norm in the Hilbert space $\cH$, and $\xb[i]$
denotes the $i$-th coordinate of $\xb$. For a matrix $\Mb$, its spectral
norm is denoted by $\|\Mb\|_2$. For a PSD matrix $\Ab$, define
$\|\xb\|_\Ab^2 := \xb^\top\Ab\xb$.
\iffalse
\paragraph{Organization.}
The remainder of the paper is organized as follows: Section
\ref{sec:main_theory} provides the main results, the excess risk bound
for SGD, and we also make comparisons to the recent work on OLS and ridge regression in the
overparameterized regime~\citep{bartlett2020benign,tsigler2020benign}. Section~\ref{sect:Related}
provides a more detailed comparison to the work on iterate averaged SGD for
least squares regression. The proof
techniques are outlined in Section~\ref{sec:general}, and we
conclude this paper in Section~\ref{sec:conclusion}.
\fi
\section{Main Results}\label{sec:main_theory}
We now provide upper and lower excess risk bounds for iterate averaged SGD. We then compare these rates to those of OLS
and ridge regression, where we see striking similarities and notable differences.
\subsection{Benign Overfitting of SGD}
We first introduce relevant notation and our assumptions.
Our first assumption is mild regularity conditions on the moments of the data distribution.
\begin{assumption}[Regularity conditions]\label{assump:second_moment}
Assume $\EE[\xb \xb^\top]$,
$\EE[ \xb \otimes \xb \otimes \xb \otimes \xb ]$,
and $\EE[y^2]$ exist and are all finite.
Furthermore, denote the second moment of $\xb$ by
\[\Hb := \EE_{\xb\sim\cD}[\xb\xb^\top],\]
and suppose that $\mathop{\text{Tr}}(\Hb)$ is finite. For convenience,
we assume that $\Hb$ is strictly positive definite and that $L(\wb)$ admits a unique global
optimum, which we denote by $\wb^* := \argmin_{\wb} L(\wb)$.
\footnote{This is not necessary. In the case where $\Hb$ has
eigenvalues which are $0$, we could instead choose $\wb^*$ to
be the minimum norm vector in the set $\argmin_{\wb} L(\wb)$, and
our results would hold for this choice of $\wb^\star$.
For example, see~\cite{scholkopf2002learning} for a rigorous treatment
of working in a reproducing kernel Hilbert space.}
\iffalse
and that $\Hb$ is strictly
positive definite. Note that assuming $\Hb$ is
strictly positive definite is without loss of generality due to that
we may restrict our analysis to the projection onto the eigensubspace corresponding to the non-zero
eigenvalues of $\Hb$.
\fi
\end{assumption}
\iffalse
This assumption
implies the objective $L(\wb)$ admits a unique global
optimum\footnote{This can be seen by considering the Karhunen–Lo\`eve
theorem and working in the
eigenbasis. If the dimension is countably infinite,
then $\wb^*$ exists as a sequence. See~\cite{someRKHSboklikeScholkopf} for formal results
for working in the RKHS.}, which we denote by $\wb^* := \argmin_{\wb} L(\wb)$.
\fi
Our second assumption is on the behavior of the fourth moment, when viewed as a linear operator on PSD matrices:
\begin{assumption}[Fourth moment condition]\label{assump:bound_fourthmoment}
Assume there exists a positive constant $\alpha > 0$, such that for any PSD
matrix $\Ab$, it holds that
\[
\EE_{\xb\sim\cD}[\xb\xb^\top\Ab\xb\xb^\top]\preceq \alpha \mathop{\text{Tr}}(\Hb\Ab)\Hb.
\]
For Gaussian distributions, it suffices to take
$\alpha=3$. Furthermore, it is worth noting that
this assumption is implied if the distribution over $\Hb^{-\half}\xb$ has sub-Gaussian
tails (see Lemma~\ref{lemma:sub-gaussian} in the Appendix for a
precise claim).
Also, it is not difficult to verify that $\alpha\geq 1$.\footnote{This is due to that the square of the second moment is less than the fourth moment.}
\end{assumption}
Assuming sub-Gaussian tails over $\Hb^{-\half}\xb$ is standard assumption
in regression
analysis (e.g. ~\citealt{HsuKZ14,bartlett2020benign,tsigler2020benign}),
and, as mentioned above, this assumption is substantially weaker.
The assumption is somewhat stronger than what is often assumed for
iterate averaged SGD in the underparameterized regime
(e.g., ~\citealt{bach2013non,jain2017parallelizing}) (see
Section~\ref{sect:Related} for further discussion).
Our next assumption is a noise condition, where it is helpful to interpret $y -
\la\wb^*,\xb\ra$ as the additive noise. Observe that the first order
optimality conditions on $\wb^*$ imply
$\EE_{(\xb,y)\sim\cD}[(y-\la\wb^*,\xb\ra)\xb] = \nabla L(\wb^*) = \boldsymbol{0}
$.
\begin{assumption}[Noise condition]\label{assump:noise}
Suppose that:
\[
\bSigma := \EE \sbr{ (y - \la\wb^*,\xb\ra)^2 \xb\xb^\top }, \quad
\sigma^2 := \norm{\Hb^{-\half} \bSigma \Hb^{-\half}}_2
\]
exist and are finite. Note that $\bSigma$ is the covariance matrix of the gradient noise at $\wb^\star$.
\end{assumption}
This assumption places a rather weak requirement on the
additive noise (due to that it permits model mis-specification) and is often made in the average SGD literature (e.g.,
\citealt{bach2013non,dieuleveut2017harder}).
Observe that for \emph{well-specified models}, where
\begin{equation}\label{eq:well}
y = \la\wb^\star, \xb\ra
+\epsilon, \quad \epsilon\sim\cN(0,\sigma^2_{\mathrm{noise}}),
\end{equation}
we have that $\bSigma = \sigma^2_{\mathrm{noise}}\Hb$ and so $\sigma^2 = \sigma^2_{\mathrm{noise}}$.
Before we present our main theorem, a few further definitions are in
order: denote the eigendecomposition of the Hessian as
$\Hb = \sum_{i}\lambda_i\vb_i\vb_i^\top$, where
$\{\lambda_i\}_{i=1}^\infty$ are the eigenvalues of $\Hb$ sorted in
non-increasing order and $\vb_i$'s are the corresponding
eigenvectors. We then denote:
\begin{gather*}
{\Hb}_{0:k} := \textstyle{\sum_{i=1}^k}\lambda_i\vb_i\vb_i^\top,\quad\mbox{and}\quad
{\Hb}_{k:\infty} := \textstyle{\sum_{i> k}}\lambda_i\vb_i\vb_i^\top.
\end{gather*}
Also, we define
\[
\|\wb\|^2_{\Hb_{0:k}^{-1}}
\sum_{ i\le
k}\frac{(\vb_i^\top\wb)^2}{\lambda_i}, \quad
\|\wb\|^2_{{\Hb}_{k:\infty}}=\sum_{i> k}\lambda_i (\vb_i^\top\wb)^2,
\]
where we have slightly abused notation in that ${\Hb}_{0:k}^{-1}$
denotes a pseudo-inverse.
We now present our main theorem:
\begin{theorem}[Benign overfitting of SGD]\label{thm:generalization_error}
Suppose Assumptions \ref{assump:second_moment}-\ref{assump:noise} hold
and that the stepsize is set so that $\gamma\leq 1/(\alpha\mathop{\text{Tr}}(\Hb))$.
Then the excess risk can be upper bounded as follows,
\begin{align*}
\EE [L(\overline{\wb}_{N})] - L(\wb^*)
&\le 2\cdot \mathrm{EffectiveBias}+2\cdot \mathrm{EffectiveVar},
\end{align*}
where
\begin{align*}
\mathrm{EffectiveBias} & = \frac{1}{ \gamma^2N^2}\cdot\norm{\wb_0 - \wb^*}^2_{\Hb^{-1}_{0:k^*}} + \norm{\wb_0 - \wb^*}^2_{\Hb_{k^*:\infty}} \\
\mathrm{EffectiveVar} & = \frac{\alpha\gamma \|\wb_0-\wb^*\|_2^2}{1-\gamma \alpha\mathop{\text{Tr}}(\Hb)}\cdot \rbr{\frac{k^*}{N^2\gamma^2} + \sum_{i>k^*}\lambda_i^2 } + \frac{ \sigma^2}{1-\gamma \alpha\mathop{\text{Tr}}(\Hb)}\cdot \rbr{\frac{k^*}{N} + N \gamma^2 \sum_{i>k^*}\lambda_i^2 }
\end{align*}
with $k^* = \max \{k: \lambda_k \ge \frac{1}{\gamma N}\}$.
\end{theorem}
The interpretation is as follows: the ``effective bias'' precisely corresponds to the rate of convergence had we run gradient descent directly on $L(\wb)$ (i.e., where the latter has no variance due to sampling).
The ``effective variance'' error stems from both the additive noise $y - \la\wb^*,\xb\ra$, i.e., the second term of the EffectiveVariance error, along with that even if there was no additive noise (i.e.
$y - \la\wb^*,\xb\ra=0$ with probability one), i.e., the first term of the EffectiveVariance error, then SGD would still not be equivalent to GD.
The cut-off index $k^*$, which we refer to as the ``effective dimension'', plays a pivotal role in the excess risk bound, which separates the entire space into a $k^*$-dimensional ``head'' subspace where the bias error
decays more quickly than that of the bias error in the complement ``tail'' subspace.
To obtain a vanishing bound, the effective dimension must be $\smallO{N}$ and the tail summation $\sum_{i> k^*}\lambda_i^2$ must be $\smallO{1/N}$.
In terms of constant factors, the above bound can be improved by a factor of $2$ in the effective bias-variance decomposition (see \eqref{eq:bias_var_decomposition}). We now turn to lower bounds.
\paragraph{A lower bound.}
We first introduce the following assumption that states a lower bound on the fourth moment.
\begin{assumption}[Fourth moment condition, lower bound]\label{assumption:lowerbound_fourthmoment}
Assume there exists a constant $\beta\ge0$, such that for any PSD matrix $\Ab$, it holds that
\begin{align*}
\EE_{\xb\sim\cD}[\xb\xb^\top\Ab\xb\xb^\top]-\Hb\Ab\Hb\succeq \beta\mathop{\text{Tr}}(\Hb\Ab)\Hb.
\end{align*}
For Gaussian distributions, it suffices to take
$\beta=2$.
\end{assumption}
The following lower bound shows that our upper bound is near to unimprovable:
\begin{theorem}[Excess risk lower bound]\label{thm:lowerbound_var}
Suppose $N\geq 500$.
For any well-specified, data distribution $\cD$ (see \eqref{eq:well}) that also satisfies Assumptions \ref{assump:second_moment} and \ref{assumption:lowerbound_fourthmoment}, for any stepsize such that $ \gamma < 1/\lambda_1$, we have that:
\begin{align*}
\EE [L(\overline{\wb}_{N})] - L(\wb^*) &\ge\frac{1}{16\gamma^2N^2}\cdot \|\wb_0-\wb^*\|^2_{\Hb_{0:k^*}^{-1}} + \frac{1}{16}\cdot\|\wb_0-\wb^*\|^2_{\Hb_{k^*:\infty}}\notag\\
&\qquad+ \frac{\beta\gamma \|\wb_0-\wb^*\|_{\Hb}^2}{128e^3 }\cdot \min\bigg\{\frac{1}{\lambda_1}, N\gamma\bigg\}\cdot\sum_{i>k^*}\lambda_i^2+\frac{ \sigma^2_{\mathrm{noise}} }{50} \rbr{\frac{k^*}{N} + N\gamma^2 \sum_{i>k^*}\lambda_i^2 }
\end{align*}
with $k^* = \max \{k: \lambda_k \ge \frac{1}{N \gamma}\}$.
\end{theorem}
Similar to the upper bound stated in Theorem \ref{thm:generalization_error}, the first two terms represent the EffectiveBias and the last two terms represent the EffectiveVariance, in which the third and last terms are contributed by the model noise and variance in SGD. By looking at both lower and upper bounds, two notable observations are as follows: (1) The effect of the variance in SGD is non-negligible if the data covariance has a relatively large component in the tail subspace (i.e., $\sum_{i>k^*}\lambda_i^2$ is non-negligible); (2) Our upper bound nearly matches our lower bound up to some constant, except the term that corresponds to the variance in SGD.
In particular, in our upper bound this term is proportional to $\gamma\|\wb_0-\wb^*\|_2^2\cdot\big(k^*/(N^2\gamma^2)+\sum_{i>k^*}\lambda_i^2\big)$, while in the lower bound this term is roughly in the order of $\gamma\|\wb_0-\wb^*\|_{\Hb}^2/\lambda_1\cdot\sum_{i>k^*}\lambda_i^2$ for stepsize $\gamma \geq 1/(\lambda_1 N)$. Thus there is a gap between our lower and upper bounds if $\|\wb_0-\wb^*\|_{\Hb}^2/\lambda_1\ll \|\wb_0-\wb^*\|_2^2$ or $\sum_{i>k^*}\lambda_i^2\ll k^*/(N^2\gamma^2)$. Our conjecture is that the upper bound is improvable in this regard, which is a direction for further study.
\iffalse
Note that if $y-\wb^\star \cdot
\xb=0$ (with probability one), i.e., the additive noise is $0$, SGD
is still not equivalent to gradient descent on $L(\wb)$. Here,
it is the variance in SGD (in comparison to gradient descent) that contributes to this dependence on $\|\wb_0-\wb^*\|_2^2$ in $\mathrm{EffectiveVariance}$;
our conjecture is that our lower bound is improvable in this regard, which is a direction for further study.
\fi
\paragraph{Special cases.}
\iffalse
Theorem \ref{thm:generalization_error} demonstrates the algorithmic benign overfitting behavior of constant stepsize SGD for overparameterized linear regression. We will compare it with the benign overfitting phenomenon of minimum-norm interpolator \citep{bartlett2020benign} and ridge regression \citep{tsigler2020benign} in detail in Section \ref{sect:Related}.
\fi
It is instructive to consider a few special cases of Theorem \ref{thm:generalization_error}. We first show the result for SGD with large stepsizes.
\begin{coro}[Benign overfitting with large stepsizes]\label{thm:large_stepsize}
Suppose Assumptions \ref{assump:second_moment}-\ref{assump:noise} hold and
that the stepsize is set to $\gamma = 1/(2\alpha\sum_i\lambda_i)$.
Then
\begin{align*}
\mathrm{EffectiveBias} & = \frac{4 \alpha^2 (\sum_i\lambda_i)^2}{ N^2}\cdot\norm{\wb_0 - \wb^*}^2_{\Hb^{-1}_{0:k^*}} + \norm{\wb_0 - \wb^*}^2_{\Hb_{k^*:\infty}} \\
\mathrm{EffectiveVar} & = \bigg(2\sigma^2+\frac{\alpha^2 \cdot (\sum_i\lambda_i) \cdot \|\wb_0-\wb^*\|_2^2}{N}\bigg) \rbr{\frac{k^*}{N}
+ \frac{N \sum_{i>k^*}\lambda_i^2}{4\alpha^2 (\sum_i\lambda_i)^2} },
\end{align*}
where $k^* = \max \{k: \lambda_k \ge \frac{2\alpha \sum_i\lambda_i}{ N}\}$.
\end{coro}
\iffalse
\begin{align*}
\mathrm{EffectiveBias} & = \frac{4 \alpha^2 \mathop{\text{Tr}}(H)^2}{ N^2}\cdot\sum_{i\le k^*}\frac{\big(\wb_0[i]-\wb^*[i]\big)^2}{\lambda_i}+\sum_{i> k^*}\lambda_i\big(\wb_0[i]-\wb^*[i]\big)^2 \\
\mathrm{EffectiveVar} & = \bigg(2\sigma^2+\frac{\alpha^2 \mathop{\text{Tr}}(H) \|\wb_0-\wb^*\|_2^2}{N}\bigg) \rbr{\frac{k^*}{N}
+ \frac{N \sum_{i>k^*}\lambda_i^2}{4\alpha^2 \mathop{\text{Tr}}(H)^2} }
\end{align*}
where $k^* = \max \{k: \lambda_k \ge \frac{2\alpha \mathop{\text{Tr}}(H)}{ N}\}$.
\fi
\iffalse
\begin{remark}
Without lose of generality, $\Hb$ can be set as diagonal, then the effective bias error bound takes the following form:
\begin{align*}
\mathrm{EffectiveBias} & = \frac{1}{ \gamma^2N^2}\cdot\sum_{1\le i\le k^*}\frac{\big(\wb_0[i]-\wb^*[i]\big)^2}{\lambda_i}+\sum_{i> k^*}\lambda_i\big(\wb_0[i]-\wb^*[i]\big)^2.
\end{align*}
It can be observed that the bias error decays in different rates at different coordinates (or eigenvectors of $\Hb$ in general cases). In particular, in the ``head'' eigenspace (spanned by the eigenvectors corresponding to large eigenvalues) the bias error decays in a faster $\bigO{1/N^2}$ rate, while in the remaining ``tail'' eigenspace, the bias error decays in a slower $\bigO{1/N}$ rate.
\end{remark}
\fi
Note that the bias error decays at different rates in different subspaces. Crudely, in the ``head'' eigenspace (spanned by the eigenvectors corresponding to large eigenvalues) the bias error decays in a faster $\bigO{1/N^2}$ rate (though there is weighting of $\lambda_i$ in the head), while in the remaining ``tail'' eigenspace, the bias error decays at a slower $\bigO{1/N}$ rate (due to that all the eigenvalues in the tail are less than $\bigO{1/N}$).
The following corollary provides a crude bias bound, showing that bias never decays more slowly than $\bigO{1/N}$.
\begin{coro}[Crude bias-bound]\label{thm:simplied_theory}
Suppose Assumptions \ref{assump:second_moment}-\ref{assump:noise} hold and
that the stepsize is set to $\gamma = 1/(2\alpha\sum_i\lambda_i)$.
Then
\begin{align*}
\EE [L(\overline{\wb}_{N})] - L(\wb^*)\le \frac{8\alpha\|\wb_0-\wb^*\|_2^2\cdot \sum_{i}\lambda_i}{N}
+4 \sigma^2 \cdot \rbr{\frac{k^*}{N} + \frac{N\sum_{i>k^*}\lambda_i^2}{4\alpha^2 (\sum_{i}\lambda_i )^2} },
\end{align*}
where $k^* = \max \{k: \lambda_k \ge \frac{2\alpha \sum_i\lambda_i}{ N}\}$.
\end{coro}
Theorem \ref{thm:generalization_error} suggests that the excess risk achieved by SGD depends on the spectrum of the covariance matrix. The following corollary gives examples of data spectrum such that the excess risk is diminishing.
\begin{coro}[Example data distributions]\label{thm:example_spectrum}
Under the same conditions as Theorem \ref{thm:generalization_error}, suppose $\norm{\wb_0 - \wb^*}_2$ is bounded.
\begin{enumerate}
\item For $\Hb \in \RR^{d\times d}$, let $s=N^r$ and $d=N^q$ for some positive constants $0<r\le 1$ and $q\ge 1$. If the spectrum of $\Hb$ satisfies
\begin{align*}
\lambda_k =
\begin{cases}
1/s, & k\le s,\\
1/(d-s), & s+1\le k\le d,
\end{cases}
\end{align*}
then $\EE[L(\overline{\wb}_N)]-L(\wb^*) = \bigO{N^{r-1}+N^{1-q}}$.
\item If the spectrum of $\Hb$ satisfies $\lambda_k = k^{-(1+r)}$ for some $r> 0$, then $\EE[L(\overline{\wb}_N)]-L(\wb^*) = \bigO{N^{-r/(1+r)}}$.
\item If the spectrum of $\Hb$ satisfies $\lambda_k = k^{-1}\log^{-\beta}(k+1)$ for some $\beta>1$, then $\EE[L(\overline{\wb}_N)]-L(\wb^*) =\bigO{\log^{-\beta}(N)}$.
\item If the spectrum of $\Hb$ satisfies $\lambda_k = e^{-k}$, then $\EE[L(\overline{\wb}_N)]-L(\wb^*) = \bigO{\log(N)/N}$.
\end{enumerate}
\end{coro}
\subsection{Comparisons to OLS and Ridge Regression}
\label{sect:Compare}
We now compare these rates to those obtained by OLS or
ridge regression.
\paragraph{SGD vs. minimum-norm solution of OLS.}
In a somewhat more restrictive setting, \citet{bartlett2020benign} prove that the minimum $\ell_2$ norm interpolator for the linear regression problem on $N$ training examples, denoted by $\hat{\wb}_N$, gives the following excess risk lower bound:
\begin{align*}
\EE[L(\hat{\wb}_N)]-L(\wb^*) &\ge c\sigma^2\bigg(\frac{k^\star}{N}+\frac{N\sum_{i>k^\star}\lambda_i^2}{(\sum_{i>k^*}\lambda_i)^2}\bigg),
\end{align*}
where $c$ is an absolute constant, $\sigma^2$ is the variance of model noise, and $k^\star = \min\{k\ge 0: \sum_{i>k}\lambda_i/\lambda_{k+1}\ge bN\}$ for some constant $b>0$.
It is clear that in order to achieve benign overfitting, one needs to ensure that $k^\star=\smallO{N}$ and $\sum_{i>k^\star}\lambda_i^2/(\sum_{i>k^\star}\lambda_i)^2=\smallO{1/N}$. The first requirement prefers slow decaying rate of the data spectrum since one hopes to get a large $\sum_{i>k}\lambda_i/\lambda_{k+1}$ for small $k$. On the contrary, the second requirement suggests that the spectrum should decay fast enough since we need to ensure that the tail summation $\sum_{i>k^\star}\lambda_i^2$ is small. Consequently, as shown in Theorem 6 in \citet{bartlett2020benign}, if the data spectrum decays in a rate $\lambda_k = k^{-\alpha}\log^{-\beta}(k+1)$, the minimum $\ell_2$-norm interpolator can achieve vanishing excess risk only when $\alpha = 1$ and $\beta\ge 1$. In contrast, our results show that SGD can achieve vanishing excess risk for any $\alpha>1$ and $\beta\geq 0$ (as well as the case of $\alpha=1$ and $\beta> 1$, see Corollary \ref{thm:example_spectrum} for details) since a fast decaying spectrum can ensure both small $k^*$ (the effective dimension) and small tail summation $\sum_{i>k^\star}\lambda_i^2$.
\paragraph{SGD vs. ridge regression.}
\citet{tsigler2020benign} show that the ridge regression estimator, denoted by $\hat\wb_N^\lambda$, has the following lower bound on the excess risk:
\begin{align*}
\EE[L(\hat{\wb}_N^\lambda)]-L(\wb^*) &\ge \max_k\Bigg\{c_1\sum_i\frac{\lambda_i\wb^*[i]^2}{(1+\lambda_i/(\lambda_{k+1}\rho_k))^2} +\frac{c_2}{n}\sum_{i} \min\bigg(1,\frac{\lambda_i^2}{\lambda_{k+1}^2(\rho_k+2)^2}\bigg)\Bigg\},
\end{align*}
where $\lambda$ is the regularization parameter, $c_1$ and $c_2$ are absolute constants and $\rho_k=\big(\lambda+\sum_{i>k}\lambda_i\big)/(N\lambda_{k+1})$. \citet{tsigler2020benign} further show that the lower bound nearly matches the following upper bound of the excess risk:
\begin{align*}
\EE[L(\hat{\wb}_N^\lambda)]-L(\wb^*) &\le c_1'\bigg(\|\wb^*\|_{\Hb^{-1}_{0:k^\star}}^2\cdot\bigg(\frac{\lambda+\sum_{i>k}\lambda_i}{N}\bigg)^2+\|\wb^*\|_{\Hb_{k^\star:\infty}}^2\bigg)\notag\\
&\qquad + c_2'\sigma^2\bigg(\frac{k^\star}{N}+\frac{N\sum_{i>k^\star}\lambda_i^2}{(\lambda+\sum_{i>k^*}\lambda_i)^2}\bigg),
\end{align*}
where $c_1'$ and $c_2'$ are absolute constants, and $k^\star = \min\{k\ge 0: (\sum_{i>k}\lambda_i+\lambda)/\lambda_{k+1}\ge bN\}$ for some constant $b>0$.
Comparing this to Corollary \ref{thm:large_stepsize} suggests that SGD (using a constant stepsize with iterate averaging) may exhibit an implicit regularization effect that performs comparably to ridge regression with a constant regularization parameter (here we assume that $\mathop{\text{Tr}}(\Hb)$ is of a constant order). A more direct problem-dependent comparison (e.g., consider the optimal learning rate for SGD and optimal $\lambda$ for ridge regression) is a fruitful direction of further study, to more accurately gauge the differences between the implicit regularization afforded by SGD and the explicit regularization of ridge regression.
\iffalse
Without loss of generality, assuming $\mathop{\text{Tr}}(\Hb) =\sum_{i}\lambda_i=1$ and setting $\lambda=1$,
we can spell out their bound as follows
\begin{align*}
\EE[L(\hat{\wb}_N)]-L(\wb^*) &\ge c_1\bigg(\frac{1}{N^2}\cdot\|\wb^*\|_{\Hb^{-1}_{0:k^\star}}^2+\|\wb^*\|_{\Hb_{k^\star:\infty}}^2\bigg)+c_2\sigma^2\bigg(\frac{k^\star}{N}+\frac{N\sum_{i>k^\star}\lambda_i^2}{4}\bigg),
\end{align*}
where $k^\star \simeq \min\{k\ge 0: \lambda_{k+1}\le 1/N\}$, which matches our upper bound on the excess risk achieved by SGD up to some constant factor (see Corollary \ref{thm:large_stepsize}). This suggests that SGD (using constant stepsize with iterate averaging) may exhibit implicit regularization property and performs no worse than ridge regression with constant regularization parameter.
\fi
\section{Further Related Work}
\label{sect:Related}
We first discuss the work on iterate averaging in the finite
dimensional case before turning to the overparameterized regime. In the underparameterized regime, where $d$ is assumed to
be finite, the behavior of
constant stepsize SGD with iterate average or tail average has been
well investigated from the perspective of the \emph{bias-variance
decomposition}
\citep{defossez2015averaged,dieuleveut2017harder,lakshminarayanan2018linear,jain2017markov,jain2017parallelizing}.
For iterate averaging from the beginning,
\citet{defossez2015averaged,dieuleveut2017harder} show a
$\bigO{{1}/{N^2}}$ convergence rate for the bias error and a
$\bigO{{d}/{N}}$ convergence rate for the variance error, where $N$ is
the number of observed samples and $d$ is the number of parameters.
The bias error rate can be further improved by considering averaging
only the tail iterates \citep{jain2017markov,jain2017parallelizing,pmlr-v75-jain18a},
provided that the minimal eigenvalue of $\Hb$ is bounded away from
$0$. We note that the work in~\citet{jain2017markov,jain2017parallelizing,pmlr-v75-jain18a} also
give the optimal rates with model misspecification.
These results all have dimension factors $d$ and do not apply to
the overparameterized regime, though our results recover the finite
dimensional case (and the results for delayed tail averaging from~\citet{jain2017markov,jain2017parallelizing} can
be applied here for the bias term). We further develop on the proof techniques
in~\citet{jain2017markov}, where we use properties of asymptotic
stationary distributions for the purposes of finite sample size analysis.
We now discuss related works in the overparameterized regime~\citep{DieuleveutB15,berthier2020tight} and our assumptions. One
notable difference in our work is that
Assumption~\ref{assump:bound_fourthmoment} (which is implied by
sub-Gaussianity, see Lemma~\ref{lemma:sub-gaussian}) is somewhat stronger than what is often assumed for
iterate average SGD analysis, where $\EE[\xb\xb^\top\xb\xb^\top]
\preceq R^2\Hb$, as adopted in
\citet{bach2013non,defossez2015averaged,dieuleveut2017harder,jain2017markov,jain2017parallelizing}. Our assumption implies
an $R^2$ bound with $R^2 = \alpha \mathop{\text{Tr}}(\Hb)$. In terms of analysis, we
note that our variance analysis only relies on an $R^2$ condition,
while our bias analysis relies on our stronger sub-Gaussianity-like assumption.
Our variance analysis is sharper than~\citet{DieuleveutB15} in
that we provide a bound in terms of the full spectrum (with the same
$R^2$ assumption) along with a lower bound, while~\citet{DieuleveutB15} assume decay conditions on the spectrum. The
bias analysis in~\citet{DieuleveutB15,berthier2020tight} relies on a stronger
assumption in that $\norm{\Hb^{-\alpha} \wb^* }_2$ must be finite, where
$\alpha > 0$ is a constant (e.g., A4 in \citet{bach2013non} and
Theorem 1 condition (a) in \citet{berthier2020tight}); our conjecture
is that without relying on stronger fourth moment assumption (such
as those consistent with sub-Gaussians), such dependencies are not
avoidable. Our fourth moment assumption is a natural starting
point for analyzing the over-parameterized regime because it also allows for direct comparisons to OLS and ridge regression, as discussed above.
\iffalse
As for the results of SGD in the non-parametric setting
\citep{DieuleveutB15,berthier2020tight}, our work distinguishes from
them in many aspects. First, they make strong assumptions on the
optimal, e.g, $\norm{\Hb^{-\alpha} \wb^* }_2$ is finite, where $\Hb$
is the data covariance, $\wb^*$ is the optimal solution, and
$\alpha > 0$ is a constant (e.g., A4 in \citet{bach2013non} and
Theorem 1 condition (a) in \citet{berthier2020tight}), while our
results require no such assumption. Moreover,
\citet{berthier2020tight} cannot address variance error while our
results can, and \citet{DieuleveutB15} involves stronger spectrum
decay assumption (A3) than ours. Finally, it is hard to interpret the
role of dimension from the results obtained in the non-parametric
setting \citep{DieuleveutB15,berthier2020tight}.
In comparison, our bound is based on a consideration on the \emph{effective dimension} specified by the spectrum decaying property of the data covariance, and it clearly indicates the effects of dimension on the generalization of SGD.
Moreover, the derived bound in Theorem \ref{thm:generalization_error}
can be reduced to the underparameterized case where $d\ll N$. In the
non-strongly convex case, according to Corollary
\ref{thm:simplied_theory}, we show that the bias error enjoys a upper
bound $\bigO{\|\wb_0-\wb^*\|_2^2/N}$ and he variance error can be
upper bounded by $\bigO{d/N}$ since $k^*\le d $ and the
$N\sum_{i>k^*}\lambda_i^2\le \sum_{i>k^*}\lambda_i\le d/N$ (we have
$\lambda_i\le 1/(\gamma N)$ for $i>k^*$). Combining the bias and
variance gives a $\bigO{d/N}$ generalization error bound, which
matches the rate in \citet{bach2013non} in the same setting. In the
strongly convex case we have $\lambda_i=\bigTht{1}$ for all $i$ and
thus $k^*=d$. Then according to Corollary \ref{thm:large_stepsize} it
can be seen that the bias and variance error can be bounded by
$\bigO{\|\wb_0-\wb^*\|_2^2/N^2}$ and $\bigO{d/N}$ respectively, which
matches the corresponding bounds proved in
\citet{jain2017parallelizing} when performing the iterate averaging
from the initialization.
As for the results of SGD in the non-parametric setting
\citep{DieuleveutB15,berthier2020tight}, our work distinguishes from
them in many aspects. First, they make strong assumptions on the
optimal, e.g, $\norm{\Hb^{-\alpha} \wb^* }_2$ is finite, where $\Hb$
is the data covariance, $\wb^*$ is the optimal solution, and
$\alpha > 0$ is a constant (e.g., A4 in \citet{bach2013non} and
Theorem 1 condition (a) in \citet{berthier2020tight}), while our
results require no such assumption. Moreover,
\citet{berthier2020tight} cannot address variance error while our
results can, and \citet{DieuleveutB15} involves stronger spectrum
decay assumption (A3) than ours. Finally, it is hard to interpret the
role of dimension from the results obtained in the non-parametric
setting \citep{DieuleveutB15,berthier2020tight}.
In comparison, our bound is based on a consideration on the \emph{effective dimension} specified by the spectrum decaying property of the data covariance, and it clearly indicates the effects of dimension on the generalization of SGD.
\fi
Concurrent to this work, \citet{chen2020dimension} provide dimension
independent bounds for averaged SGD; their excess risk bounds for linear regression are not as sharp as those provided here.
\section{Proof Outline}\label{sec:general}
We now provide the high level ideas in the proof. A key idea is relating the finite sample (infinite dimensional) covariance
matrices of the variance and bias stochastic processes to those of
their corresponding (asymptotic) stationary covariance matrices ---
an idea developed
in~\cite{jain2017markov} for the finite dimensional, variance analysis.
This section is organized as follows:
Section \ref{sec:proof-preliminary} introduces additional notation and relevant linear operators; Section \ref{sec:proof-decomp} presents a refined bound on a now standard bias-variance decomposition; Section \ref{sec:proof-variance} outlines the variance error analysis, followed by
Section \ref{sec:proof-bias} outlining the bias error analysis. Complete proofs of the upper and lower bounds are provided in the Appendix ~\ref{append-sec:proof-upper-bound} and Appendix~\ref{sec:lower_bound}, respectively.
\subsection{Preliminaries}\label{sec:proof-preliminary}
For two matrices $\Ab$ and $\Bb$, their inner product are defined as $\la \Ab, \Bb \ra := \mathop{\text{Tr}}\rbr{\Ab^\top \Bb}$.
The following properties will be used frequently: if $\Ab$ is PSD, and $\Bb \succeq \Bb'$, then
\(
\la \Ab, \Bb \ra \ge \la \Ab, \Bb' \ra.
\)
We use $\otimes$ to denote the kronecker/tensor product.
We define the following linear operators:
\begin{gather*
\cI = \Ib \otimes \Ib,\quad
\cM = \EE [ \xb \otimes \xb \otimes \xb \otimes \xb ],\quad
\tilde{\cM} = \Hb \otimes \Hb, \\
\cT = \Hb \otimes \Ib + \Ib \otimes \Hb - \gamma\cM, \quad
\tilde \cT = \Hb \otimes \Ib + \Ib \otimes \Hb - \gamma\Hb\otimes\Hb.
\end{gather*}
We use the notation $\mathcal{O}\circ \Ab$ to denotes the operator
$\mathcal{O}$ acting on a symmetric matrix $\Ab$.
For example, with these definitions, we have that for a symmetric matrix $\Ab$,
\begin{gather}
\cI \circ \Ab = \Ab, \ \ \ \cM\circ \Ab = \EE [ (\xb^\top \Ab \xb) \xb \xb^\top ], \ \ \ \tilde{\cM} \circ \Ab = \Hb \Ab \Hb, \notag \\
(\cI - \gamma \cT) \circ \Ab = \EE [ (\Ib - \gamma \xb \xb^\top)\Ab (\Ib - \gamma \xb \xb^\top) ], \ \ (\cI-\gamma\tilde\cT)\circ\Ab = (\Ib-\gamma\Hb)\Ab(\Ib-\gamma\Hb). \label{eq:0005}
\end{gather}
Now observe that the covariance matrix of the centered iterate obeys the following update rule:
\begin{equation}\label{eq:sgd-tensor}
\textrm{Cov}_{t} = (\cI - \gamma \cT)\circ \textrm{Cov}_{t-1} + \gamma^2 \bSigma, \,
\textrm{ where } \, \,\, \textrm{Cov}_t := \EE[(\wb_t -\wb^*)(\wb_t -\wb^*)^\top],
\end{equation}
where the SGD update rule \eqref{eq:sgd} implies this recursive form.
We conclude by summarizing a few technical properties of these operators (see Lemma \ref{lemma:operators2} in Appendix).
\begin{lemma}\label{lemma:operators}
An operator $\cO$ defined on symmetric matrices is called PSD mapping, if $\Ab \succeq 0$ implies $\cO\circ \Ab \succeq 0$.
Then we have
\begin{enumerate}
\item $\cM$ and $\tilde\cM$ are both PSD mappings.
\item $\cI-\gamma\cT$ and $\cI-\gamma\tilde\cT$ are both PSD mappings.
\item $\cM - \tilde\cM$ and $\tilde \cT - \cT$ are both PSD mappings.
\item If $0 < \gamma \le 1/\lambda_1$, then $\tilde{\cT}^{-1}$ exists, and is a PSD mapping.
\item If $0 < \gamma \le 1/(\alpha\mathop{\text{Tr}}(\Hb))$, then $\cT^{-1}\circ \Ab$ exists for PSD matrix $\Ab$, and $\cT^{-1}$ is a PSD mapping.
\end{enumerate}
\end{lemma}
\subsection{The Bias-Variance Decomposition}\label{sec:proof-decomp}
It is helpful to consider the bias-variance decomposition for averaged SGD, which has been extensively studied before in the underparameterized regime ($N\gg d$) \citep{DieuleveutB15,jain2017parallelizing,jain2017markov}.
For convenience, we define the centered SGD iterate as $\betab_t := \wb_t - \wb^*$. Similarly we define $\bar{\betab}_{N} := \frac{1}{N}\sum_{t=0}^{N-1} \betab_t$.
\noindent (1) If the sampled data contains no label noise, i.e., $y_{t} = \la\wb^*, \xb_t\ra$, then the obtained SGD iterates $\{\betab^{\bias}_t\}$ reveal the \emph{bias error},
\begin{equation}\label{eq:bias_iterates}
\betab^{\bias}_t = \rbr{\Ib-\gamma\xb_t\xb_t^\top} \betab^{\bias}_{t-1}, \qquad \betab^{\bias}_0 = \betab_0.
\end{equation}
\noindent (2) If the iterates are initialized from the optimal $\wb^*$, i.e., $\wb_0 = \wb^*$, then the obtained SGD iterates $\{\betab^{\var}_t\}$ reveal the \emph{variance error},
\begin{equation}\label{eq:variance_iterates}
\betab^{\var}_t = \rbr{\Ib-\gamma\xb_t\xb_t^\top} \betab^{\var}_{t-1} + \gamma \xi_t \xb_t, \qquad \betab^{\var}_0 = \boldsymbol{0},
\end{equation}
where $\xi_t := y_t - \la\wb^*, \xb_t\ra$ is the inherent noise.
Note the ``bias iterates'' can be viewed as a stochastic process of SGD on a consistent linear system;
similarly, the ``variance iterates'' should be treated as a stochastic process of SGD initialized from the optimum.
Using the defined operators, the update rule of the iterates \eqref{eq:bias_iterates} imply the following recursive form of $\Bb_t : = \EE [\betab_t^\bias \otimes \betab_t^\bias ]$:
\begin{equation}\label{eq:update_Bt}
\Bb_t = (\cI - \gamma\cT)\circ \Bb_{t-1} \qquad \text{and} \qquad \Bb_0 = \betab_0\otimes \betab_0,
\end{equation}
and the update rule \eqref{eq:variance_iterates} imply the following recursive form of $\Cb_t := \EE [\betab_t^\var \otimes \betab_t^\var]$:
\begin{equation}\label{eq:update_Ct}
\Cb_t = (\cI-\gamma\cT) \circ \Cb_{t-1} + \gamma^2\bSigma,\qquad \Cb_0 = \boldsymbol{0}.
\end{equation}
Recall \eqref{eq:sgd-tensor}, we can verify that
\[
\textrm{Cov}_t = \Bb_t + \Cb_t.
\]
We define the averaged version of $\betab^{\bias}_t$ and $\betab^\var_t$ in the same way as $\overline{\wb}_N$, i.e., $\bar{\betab}_{N}^{\bias} := \frac{1}{N}\sum_{t=0}^{N-1} \betab_t^{\bias}$ and $\bar{\betab}_{N}^{\var} := \frac{1}{N}\sum_{t=0}^{N-1} \betab_t^{\var}$.
With a little abuse of probability space, from \eqref{eq:sgd}, \eqref{eq:bias_iterates} and \eqref{eq:variance_iterates} we have that
\[
\betab_t = \betab_t^\bias + \betab_t^\var,
\]
then an application of Cauchy–Schwarz inequality leads to the following \emph{bias-variance decomposition} on the excess risk (see \citet{jain2017parallelizing}, also Lemma \ref{lemma:bias_var_decomposition} in the appendix):
\begin{gather}
\EE [L(\overline{\wb}_{N})] - L(\wb^*) = \frac{1}{2}\la\Hb,\EE[\bar\betab_{N}\otimes \bar\betab_{N}]\ra\le \rbr{ \sqrt{\bias} + \sqrt{\var} }^2, \label{eq:bias_var_decomposition} \\
\text{where }\ \
\bias := \frac{1}{2} \langle \Hb, \EE[{\bar\betab}^{\bias}_{N} \otimes {\bar\betab}^{\bias}_{N}] \rangle, \ \
\var := \frac{1}{2} \langle \Hb, \EE[{\bar\betab}^{\var}_{N} \otimes {\bar\betab}^{\var}_{N}] \rangle. \notag
\end{gather}
In the above bound, the two terms are usually referred to as the \emph{bias error} and the \emph{variance error} respectively.
Furthermore, expanding the kronecker product between the two averaged iterates, and doubling the squared terms, we have the following upper bounds on the bias error and the variance error (see Lemma \ref{lemma:bias_var_decomposition_bound} in the appendix for the proof):
\begin{gather}
\bias := \half \langle \Hb, \EE[{\bar\betab}^{\bias}_{N} \otimes {\bar\betab}^{\bias}_{N}] \rangle
\le \frac{1}{N^2}\sum_{t=0}^{N-1}\sum_{k=t}^{N-1}\big\la (\Ib-\gamma\Hb)^{k-t}\Hb,\Bb_t\big\ra, \label{eq:formula_bias} \\
\var := \half \langle \Hb, \EE[{\bar\betab}^{\var}_{N} \otimes {\bar\betab}^{\var}_{N}]
\le \frac{1}{N^2}\sum_{t=0}^{N-1}\sum_{k=t}^{N-1}\big\la (\Ib-\gamma\Hb)^{k-t}\Hb,\Cb_t\big\ra. \label{eq:formula_var}
\end{gather}
Note that in the above bounds, we keep both summations in finite steps, and this makes our analysis sharp as $N \ll d$. In comparison, \citet{jain2017markov,jain2017parallelizing} take the inner summation to infinity, which yields looser upper bounds for further analysis in the overparameterized setting.
Next we bound the two error terms \eqref{eq:formula_bias} and \eqref{eq:formula_var} separately.
\subsection{Bounding the Variance Error}\label{sec:proof-variance}
We would like to point out that in the analysis of the variance error \eqref{eq:formula_var}, Assumption \ref{assump:bound_fourthmoment} can be replaced by a weaker assumption: $\EE[\xb\xb^\top\xb\xb^\top]\preceq R^2\Hb$, where $R$ is a positive constant \citep{jain2017parallelizing,jain2017markov,dieuleveut2017harder}. A proof under the weaker assumption can be found in Appendix~\ref{append-sec:proof-variance}.
Here, for consistency, we sketch the proof under Assumption \ref{assump:bound_fourthmoment}.
To upper bound \eqref{eq:formula_var}, noticing that $(\Ib-\gamma\Hb)^{k-t}\Hb$ is PSD, it suffices to upper bound $\Cb_t$ in PSD sense.
In particular, by Lemma 5 in \citet{jain2017markov} (restated in Lemma \ref{lemma:monotonicity_phit} in the appendix), the sequence $\{\Cb_t\}_{t=0,\dots}$ has the following property,
\begin{equation}\label{eq:Ct_crude_bound}
0 = \Cb_0 \preceq \Cb_1\preceq \cdots\preceq
\Cb_\infty\preceq \frac{\gamma \sigma^2}{1-\gamma \alpha\mathop{\text{Tr}}(\Hb)}\Ib.
\end{equation}
This gives a uniform but crude upper bound on $\Cb_t$ for all $t\ge 0$.
However, a direct application of this crude bound to \eqref{eq:formula_var} cannot give a sharp rate in the overparameterized setting.
Instead, we seek to refine the bound of $\Cb_t$ based on its update rule in \eqref{eq:update_Ct} (see the proof of Lemma~\ref{lemma:upper_bound_phit} for details):
\begin{align}
\Cb_t &= (\cI- \gamma\cT )\circ \Cb_{t-1} + \gamma^2 \bSigma \notag\\
& = (\cI - \gamma \tilde\cT) \circ \Cb_{t-1} + \gamma^2(\cM - \tilde\cM)\circ \Cb_{t-1}+\gamma^2\bSigma\notag\\
&\preceq (\cI - \gamma \tilde\cT) \circ \Cb_{t-1} + \gamma^2\cM\circ \Cb_{t-1}+\gamma^2\bSigma \qquad (\text{since $\tilde\cM$ is a PSD mapping}) \notag\\
&\preceq (\cI - \gamma \tilde\cT) \circ \Cb_{t-1} + \frac{\gamma^3\sigma^2}{1-\gamma \alpha\mathop{\text{Tr}}(\Hb)}\cM\circ \Ib + \gamma^2\bSigma, \quad (\text{by \eqref{eq:Ct_crude_bound} and $\cM$ is a PSD mapping}) \notag \\
& \preceq (\cI - \gamma \tilde\cT) \circ \Cb_{t-1} + \frac{\gamma^3\sigma^2\alpha\mathop{\text{Tr}}(\Hb) }{1-\gamma \alpha\mathop{\text{Tr}}(\Hb)}\Hb + {\gamma^2\sigma^2}\Hb, \qquad (\text{by Assumptions \ref{assump:bound_fourthmoment} and \ref{assump:noise}}) \notag \\
&= (\cI - \gamma \tilde\cT) \circ \Cb_{t-1} + \frac{\gamma^2\sigma^2 }{1-\gamma \alpha\mathop{\text{Tr}}(\Hb)}\Hb. \notag
\end{align}
Solving the above recursion, we obtain the following refined upper bound for $\Cb_t$:
\begin{align}
\Cb_t
&\preceq \frac{\gamma^2\sigma^2 }{1-\gamma \alpha\mathop{\text{Tr}}(\Hb)} \sum_{k=0}^{t-1} (\cI-\gamma\tilde\cT)^k\circ\Hb \notag\\
&= \frac{\gamma^2\sigma^2 }{1-\gamma \alpha\mathop{\text{Tr}}(\Hb)} \sum_{k=0}^{t-1} (\Ib-\gamma\Hb)^{k}\Hb (\Ib-\gamma\Hb)^{k} \qquad (\text{by the property of $\cI-\gamma\tilde\cT$ in \eqref{eq:0005}})\notag \\
&\preceq \frac{\gamma^2\sigma^2 }{1-\gamma \alpha\mathop{\text{Tr}}(\Hb)} \sum_{k=0}^{t-1} (\Ib-\gamma\Hb)^{k} \Hb
= \frac{\gamma\sigma^2}{1-\gamma\alpha\mathop{\text{Tr}}(\Hb)}\cdot\big(\Ib-(\Ib-\gamma\Hb)^t\big). \label{eq:upperbuond_ct_sketch
\end{align}
Now we can plug the above refined upper bound \eqref{eq:upperbuond_ct_sketch} into \eqref{eq:formula_var}, and obtain
\begin{align}
\var
&\le \frac{ \sigma^2}{ N^2 (1-\gamma\alpha\mathop{\text{Tr}}(\Hb))} \sum_{t=0}^{N-1} \big\la \Ib - (\Ib - \gamma\Hb)^{N-t} , \Ib - (\Ib - \gamma \Hb)^{t} \big \ra\notag \\
&= \frac{ \sigma^2}{N^2 (1-\gamma\alpha\mathop{\text{Tr}}(\Hb))} \sum_{t=0}^{N-1} \sum_{i}\rbr{ 1 - (1 - \gamma\lambda_i)^{N-t}}\rbr{ 1 - (1 - \gamma\lambda_i)^{t}} \notag\\
&\le \frac{ \sigma^2}{N^2 (1-\gamma\alpha\mathop{\text{Tr}}(\Hb))} \cdot N \cdot \sum_{i}\rbr{ 1 - (1 - \gamma\lambda_i)^{N}}^2. \label{eq:finalbound_variance_sketch}
\end{align}
The remaining effort is to precisely control the summations in \eqref{eq:finalbound_variance_sketch} according to the scale of the eigenvalues: for large eigenvalues $\lambda_i\ge \frac{1}{N\gamma}$, which appear at most $k^*$ times, we use $1-(1-\gamma\lambda_i)^N\le 1$; and for the remaining small eigenvalues $\lambda_i< \frac{1}{N\gamma}$, we use $1-(1-\gamma\lambda_i)^N \le \bigO{N\gamma\lambda_i}$.
Plugging these into \eqref{eq:finalbound_variance_sketch} gives us the final full spectrum upper bound on the variance error (see the proof of Lemma \ref{lemma:upperbound_var} for more details). This bound contributes to part of $\mathrm{EffectiveVar}$ in Theorem \ref{thm:generalization_error}.
\iffalse
\begin{lemma}\label{lemma:monotonicity_phit} (See \citep{jain2017markov} Lemma 5)
Under Assumptions \ref{assump:second_moment}, \ref{assump:noise} and \ref{assump:R2}, if the stepsize satisfies $\gamma < 1/R^2$, it holds that
\begin{align*}
0 = \Cb_0 \preceq \Cb_1\preceq \cdots\preceq
\Cb_\infty \preceq \frac{\gamma \sigma^2}{1-\gamma R^2}\Ib.
\end{align*}
\end{lemma}
\begin{lemma}\label{lemma:upper_bound_phit}
Under Assumptions \ref{assump:second_moment}, \ref{assump:noise} and \ref{assump:R2}, if the stepsize satisfies $\gamma < 1/R^2$, it holds that
\begin{equation*}
\Cb_t \preceq \frac{\gamma \sigma^2}{1-\gamma R^2}\cdot \rbr{\Ib - (\Ib - \gamma\Hb)^{2t} } .
\end{equation*}
\end{lemma}
\begin{lemma}\label{lemma:upperbound_var}
Under Assumptions \ref{assump:second_moment}, \ref{assump:noise} and \ref{assump:R2}, if the stepsize satisfies $\gamma\le 1/R^2$, then it holds that
\begin{equation*}
\var \le \frac{ \sigma^2}{1-\gamma R^2} \rbr{\frac{k^*}{N} + \gamma^2 N \cdot \sum_{i>k^*}\lambda_i^2 },
\end{equation*}
where $k^* = \max \{k: \lambda_k \ge \frac{1}{N \gamma}\}$.
\end{lemma}
\fi
\subsection{Bounding the Bias Error}\label{sec:proof-bias}
Next we discuss how to bound the bias error \eqref{eq:formula_bias}.
A natural idea is to follow the same way in analyzing the variance error, and derive a similar bound on $\Bb_t$.
Yet a fundamental difference between the variance sequence \eqref{eq:update_Ct} and the bias sequence \eqref{eq:update_Bt} is that: $\Cb_t$ is increasing, while $\Bb_t$ is ``contracting'',
hence applying the same procedure in the variance error analysis cannot lead to a tight bound on $\Bb_t$.
Instead, observing that $\Sbb_t := \sum_{k=0}^t \Bb_t$, the summation of a contracting sequence, is increasing in the PSD sense. Particularly, we can rewrite $\Sbb_t$ in the following recursive form
\begin{align}\label{eq:update_St}
\Sbb_t &= (\cI-\gamma\cT)\circ\Sbb_{t-1}+\Bb_0,
\end{align}
which resembles that of $\Cb_t$ in \eqref{eq:update_Ct}.
This motivates us to:
(i) express the obtained bias error bound \eqref{eq:formula_bias} by $\Sbb_t$,
and (ii) derive a tight upper bound on $\Sbb_t$ using similar analysis for the variance error.
For (i), by some linear algebra manipulation (see the proof of Lemma \ref{lemma:upperbound_bias} for details), we can bound \eqref{eq:formula_bias}
as follows:
\begin{align}\label{eq:upperbuond_bias_sketch}
\bias
\le \frac{1}{\gamma N^2}\bigg\la\Ib-(\Ib-\gamma\Hb)^{N},\sum_{t=0}^{N-1}\Bb_t\bigg\ra
= \frac{1}{\gamma N^2}\big\la\Ib-(\Ib-\gamma\Hb)^{N},\Sbb_{N-1}\big\ra.
\end{align}
For (ii),
we first show the following properties for the sequence $\{\Sbb_t\}_{t=0,\dots}$ (see Lemmas \ref{lemma:properties_St} and \ref{lemma:T_inv}),
\begin{align}\label{eq:St_crude_bound}
\Bb_0 = \Sbb_0\preceq \Sbb_1\preceq\cdots\preceq \Sbb_\infty, \quad \mbox{and}\quad \cM\circ\Sbb_{\infty}\preceq \frac{\alpha\mathop{\text{Tr}}(\Bb_0)}{\gamma(1-\gamma\alpha\mathop{\text{Tr}}(\Hb))}\cdot\Hb.
\end{align}
Then similar to our previous procedure in bounding $\Cb_t$, we can tighten the upper bound on $\Sbb_t$ by its recursive form
\eqref{eq:update_St} and the crude bound ($\Sbb_\infty$ in \eqref{eq:St_crude_bound}), and obtain the following refined bound (see Lemma \ref{lemma:upperbound_St}):
\begin{align}\label{eq:upperbuond_st_sketch}
\Sbb_t \preceq
\sum_{k=0}^{t}(\Ib-\gamma\Hb)^{k}\Bb_0(\Ib-\gamma\Hb)^k+\frac{\gamma\alpha\mathop{\text{Tr}}(\Bb_0)}{1-\gamma \alpha \mathop{\text{Tr}}(\Hb)}\sum_{k=0}^{t}(\Ib-\gamma\Hb)^{2k}\Hb.
\end{align}
The remaining proof will be similar to what we have done for the variance error bound:
substituting \eqref{eq:upperbuond_st_sketch} into \eqref{eq:upperbuond_bias_sketch} gives an upper bound on the bias error with respect to the summations over functions of eigenvalues. Then by carefully controlling each summation according to the scale of the corresponding eigenvalues, we will obtain a tight full spectrum upper bound on the bias error (see the proof of Lemma \ref{lemma:bound_bias_final} for more details).
As a final remark, noticing that different from the upper bound of $\Cb_t$ in \eqref{eq:upperbuond_ct_sketch}, the upper bound for $\Sbb_t$ in \eqref{eq:upperbuond_st_sketch} consists of two terms.
The first term will contribute to the $\mathrm{EffectiveBias}$ term in Theorem \ref{thm:generalization_error}, while the second term will be merged to the bound of the variance error and contribute to the $\mathrm{EffectiveVar}$ term in Theorem \ref{thm:generalization_error}.
\iffalse
Based on this we can further bound the bias error in the following lemma.
\begin{lemma}\label{lemma:upperbound_bias}
Under Assumptions \ref{assump:second_moment} and \ref{assump:bound_fourthmoment}, if the stepsize satisfies $\gamma\le 1/\lambda_1$, it holds that
\begin{align*}
\bias \le \frac{1}{\gamma N^2}\left\la\Ib-(\Ib-\gamma\Hb)^{N},\sum_{t=0}^{N-1}\Bb_t\right\ra.
\end{align*}
\end{lemma}
\begin{lemma}\label{lemma:properties_St}
Let $\Sbb_t =\sum_{k=0}^t\Bb_k$, if $\gamma\le 1/(\alpha\mathop{\text{Tr}}(\Ab))$, we have
\begin{align*
\Sbb_t &= (\cI-\gamma\cT)\circ\Sbb_{t-1}+\Bb_0.
\end{align*}
Moreover, it holds that
\begin{align*}
\Bb_0 = \Sbb_0\preceq \Sbb_1\preceq\cdots\preceq \Sbb_\infty.
\end{align*}
\end{lemma}
\begin{lemma}\label{lemma:T_inv}
Under Assumptions \ref{assump:second_moment}, and \ref{assump:bound_fourthmoment}, if the stepsize satisfies $\gamma\le 1/(\alpha\mathop{\text{Tr}}(\Hb))$, for any PSD matrix $\Ab$ it holds that
\begin{align*}
\cM\circ\Sbb_\infty
\preceq
\frac{\alpha \mathop{\text{Tr}}(\Bb_0)}{\gamma(1- \gamma R^2)}\cdot\Hb
\end{align*}
\end{lemma}
We now put these lemmas together and provide our upper bound on the bias error:
\begin{lemma}\label{lemma:bound_bias_final}
Under Assumptions \ref{assump:second_moment} and \ref{assump:bound_fourthmoment},
if the stepsize satisfies $\gamma\le 1/(\alpha\mathop{\text{Tr}}(\Hb))$, it holds that
\begin{align*}
\bias\le \frac{\alpha \|\wb_0-\wb^*\|_2^2}{\gamma(1-\gamma \alpha\mathop{\text{Tr}}(\Hb))}\cdot\bigg(\frac{k^*}{N^2} + \gamma^2 \sum_{i> k^*}\lambda_i^2\bigg) + \frac{1}{\gamma^2 N^2}\cdot\|\wb_0-\wb^*\|_{\Hb_{0:k^*}^{-1}}^2+\|\wb_0-\wb^*\|_{\Hb_{k^*:\infty}}^{2},
\end{align*}
where $k^* = \max \{k: \lambda_k \ge \gamma^{-1}/N\}$.
\end{lemma}
\fi
\iffalse
This paper studies the generalization error of constant-stepsize SGD (with iterate averaging) for the (unregularized) linear regression problem. In particular, we provide a sharp excess risk bound, which is stated in terms of the full eigenspectrum of the data covariance matrix. In the overparameterized regime, our results reveal the benign-overfitting phenomenon of SGD by proving the vanishing generalization error under certain spectrum decay conditions on the
data covariance.
\fi
\section{The Effect of Tail-Averaging}\label{sec:tail_averaging}
We further consider benign overfitting of SGD when \emph{tail-averaging} \citep{jain2017parallelizing} is applied, i.e.,
\begin{align*}
\overline \wb_{s:s+N} = \frac{1}{N}\sum_{t=s}^{s+N}\wb_t.
\end{align*}
We present the following theorem as a counterpart of Theorem \ref{thm:generalization_error}. The proof is deferred to Appendix \ref{append-sec:tail-average}.
\begin{theorem}[Benign overfitting of SGD with tail-averaging]\label{thm:generalization_error_tail}
Consider SGD with tail-averaging.
Suppose Assumptions \ref{assump:second_moment}-\ref{assump:noise} hold
and that the stepsize is set so that $\gamma\leq 1/(\alpha\mathop{\text{Tr}}(\Hb))$.
Then the excess risk can be upper bounded as follows,
\begin{align*}
\EE [L(\overline{\wb}_{s:s+N})] - L(\wb^*)
&\le 2\cdot \mathrm{EffectiveBias}+2\cdot \mathrm{EffectiveVar},
\end{align*}
where
\begin{align*}
\mathrm{EffectiveBias} & = \frac{1}{\gamma^2N^2}\cdot\big\|(\Ib-\gamma\Hb)^s(\wb_0-\wb^*)\big\|_{\Hb_{0:k^*}^{-1}}^2 + \big\|(\Ib-\gamma\Hb)^s(\wb_0-\wb^*)\big\|_{\Hb_{k^*:\infty}}^2 \\
\mathrm{EffectiveVar} &= \frac{2\alpha\gamma \|\wb_0-\wb^*\|_2^2}{1-\gamma \alpha\mathop{\text{Tr}}(\Hb)}\cdot \rbr{\frac{k^*}{N^2\gamma^2} + \sum_{i>k^*}\lambda_i^2 }\notag\\
&\qquad + \frac{ \sigma^2}{ 1-\gamma \alpha \mathop{\text{Tr}} (\Hb)} \cdot\bigg(\frac{k^*}{N} + \gamma\cdot \sum_{k^*< i\le k^\dagger}\lambda_i + \gamma^2(s+N)\cdot\sum_{i>k^\dagger}\lambda_i^2\bigg),
\end{align*}
where $k^* = \max \{k: \lambda_k \ge \frac{1}{\gamma N}\}$ and $k^\dagger = \max\{k:\lambda_k\ge \frac{1}{\gamma(s+N)}\}$.
\end{theorem}
Theorem \ref{thm:generalization_error_tail} shows that tail-averaging has improvements over iterate-averaging.
This agrees with the results shown in \citet{jain2017parallelizing}: in the underparameterized regime ($n\gg d$) and for the strongly convex case ($\lambda_d > 0$), one can obtain substantially improved convergence rates on the bias term.
However, in our considered overparameterized case, this improvement is limited due to that the
$\|\wb_0-\wb^*\|_2^2$ dependence in the EffectiveVariance cannot be
improved, even with the delayed tail averaging.
We also provide a lower bound on the excess risk for SGD with tail-averaging to show that our upper bound is nearly tight.
\begin{theorem}[Excess risk lower bound, tail-averaging]\label{thm:generalization_error_tail_lowerbound}
Consider SGD with tail-averaging.
Suppose $N\ge500$.
For any well-specified, data distribution $\cD$ (see \eqref{eq:well}) that also satisfies Assumptions \ref{assump:second_moment}, \ref{assump:bound_fourthmoment} and \ref{assumption:lowerbound_fourthmoment}, for any stepsize such that $ \gamma < 1/\lambda_1$, we have that:
\begin{align*}
\EE [L(\overline{\wb}_{N})] - L(\wb^*) &\ge\frac{1}{16\gamma^2N^2}\cdot \|(\Ib-\gamma\Hb)^s(\wb_0-\wb^*)\|^2_{\Hb_{0:k^*}^{-1}} + \frac{1}{16}\cdot\|(\Ib-\gamma\Hb)^s(\wb_0-\wb^*)\|^2_{\Hb_{k^*:\infty}}\notag\\
&\qquad + \frac{\beta\gamma \|\wb_0-\wb^*\|_{\Hb}^2}{128e^3 }\cdot \min\bigg\{\frac{1}{\lambda_1}, (2s + N)\gamma\bigg\}\cdot\sum_{i>k^\dagger}\lambda_i^2 \\
&\qquad + \frac{ \sigma_{\mathrm{noise}}^2 }{600} \rbr{\frac{k^*}{N} + \gamma\cdot\sum_{k^* < i \le k^{\dagger}}\lambda_i + (s+N)\gamma^2 \cdot \sum_{i>k^{\dagger}}\lambda_i^2 },
\end{align*}
where $k^* = \max \{k: \lambda_k \ge \frac{1}{ N \gamma}\}$ and $k^{\dagger} = \max \{k: \lambda_k \ge \frac{1}{(s+N)\gamma }\}$.
\end{theorem}
It can be observed that our upper bound matches the lower bound in most of the error terms, including the effective bias and the component of effective variance contributed by the model noise, up to some constant factors, while the gap only appears in the term contributed by the variance in SGD, i.e., $\gamma \|\wb_0-\wb^*\|_2^2\cdot\big(k^*/(N^2\gamma^2)+\sum_{i>k^*}\lambda_i^2\big)$ vs. $\gamma\|\wb_0-\wb^*\|_\Hb^2/\lambda_1\cdot\sum_{i>k^\dagger}\lambda_i^2$. Obtaining the matching upper and lower bounds for SGD with tail-averaging is also a direction for future work.
\section{Discussion}\label{sec:discussion}
This work considers the question of how well constant-stepsize SGD
generalizes for the linear regression problem in the overparameterized
regime. Our main result provides a sharp excess risk bound, stated in
terms of the full eigenspectrum of the data covariance matrix. Our
results reveal how a benign-overfitting phenomenon can occur under
certain spectrum decay conditions on the data covariance.
There are number of more subtle points worth reflecting on:
\paragraph{Moving beyond the square loss.} Focusing on linear
regression is a means to understand phenomena that are exhibited more
broadly. One natural next step here would be understand the analogues
of the classical iterate averaging
results~\citep{polyak1992acceleration} for locally quadratic models,
where decaying stepsizes are necessary for vanishing risk.
\paragraph{Sharper upper bounds.} While our upper bound nearly matches
our lower bound up to constant factors, there is notable gap in the EffectiveVariance error. Particularly, the upper bound of the
EffectiveVariance has a dependence on $\|\wb_0-\wb^*\|_2^2\cdot\big(k^*/(N^2\gamma^2)+\sum_{i>k^*}\lambda_i^2\big)$, while the lower bound has a dependence on $\|\wb_0-\wb^*\|_{\Hb}^2/\lambda_1\cdot\sum_{i>k^*}\lambda_i^2$, which could be much smaller that the upper bound if $\wb_0-\wb^*$ has a large component in the tail space.
This gap is due to that we use different approaches (in the proofs of upper and lower bounds) to characterize the contribution of the variance in SGD (noiseless case) to the excess risk. Our conjecture is that our upper bound can be potentially improved to match that of our lower bound by using a sharper characterization of the covariance of bias iterates.
\iffalse\paragraph{Sharper lower bounds.}
While our lower bound nearly matches
our upper bound up to constant factors, there is notable gap in that the
EffectiveVariance has a dependence on $\|\wb_0-\wb^*\|_2^2$.
This term is due to that even if $y-\wb^\star \cdot
\xb=0$ with probability one (i.e. the inherent noise is $0$), then SGD
is still not equivalent to gradient descent; here,
it is the variance in SGD that contributes to this dependence in the EffectiveVariance.
Our conjecture is that our lower bound can be improved to match that of our upper bound.
\fi
\paragraph{Delayed tail averaging.} It is interesting to reflect on the
improvements are to be had with delayed tail averaging. In the strongly convex
case (e.g. see ~\citet{jain2017parallelizing} when $n\geq
d$ and where $\lambda_d \gg 0$),
we can obtain substantially improved
convergence rates on the bias term; namely, the convergence occurs at
a geometric rate with regards to the dependence on $\|\wb_0-\wb^*\|_2^2$. In the overparameterized
case, one interesting observation is that
even if initial iterate, $\wb_0$, has all the weight on the coordinate with
largest eigenvalue (in which case we may hope that bias decays
geometrically);
our lower bound
(Theorem~\ref{thm:generalization_error_tail_lowerbound}) shows the
improvement due to delayed tail averaging is more limited in this case due to that the
EffectiveVariance still depends on $\|\wb_0-\wb^*\|_2^2$ with a
$1/N^2$ rate. In
particular, a geometric decay for the bias is not possible even if
$\wb_0$ has all the weight in the ``head'' subspace; this is due to
the SGD noise, where the bias in the head, in effect, contributes to
error in the tail after a constant number of SGD updates.
\iffalse
It is interesting to reflect on the
improvements are to be had with delayed tail averaging. In the strongly convex
case (e.g. see ~\citet{jain2017parallelizing} when $n\geq
d$ and where $\lambda_d \gg 0$),
we can obtain substantially improved
convergence rates on the bias term. In the overparameterized case, we
see that our upper bound for delayed tail averaging (see
Theorem~\ref{thm:generalization_error_tail}) only improves the ``head'' term in
the EffectiveBias. However, our conjecture is that this
improvement may be somewhat limited due to that the
$\|\wb_0-\wb^*\|_2^2$ dependence in the EffectiveVariance may not be
substantially improvable.
\fi
\paragraph{Relaxing the data distribution assumption.}
While our data distribution assumption (Assumption
\ref{assump:bound_fourthmoment}) can be satisfied if the whitened data
is sub-Gaussian, it still cannot cover the simple one-hot case (i.e.,
$\xb= \eb_i$ with probability $p_i$, where
$\sum_{i}p_i=1$). Here, we conjecture that
modifications of our proof can be used to establish
the theoretical guarantees of SGD under the following relaxed assumption on the
data distribution: assume that
$\EE[\xb\xb^\top\Ab\xb\xb^\top]\le a \mathop{\text{Tr}}(\Hb\Ab)\cdot \Hb +
b\|\Hb\|_2\cdot \Hb^{1/2}\Ab\Hb^{1/2}$ for all PSD matrix $\Ab$ and
some nonnegative constants $a$ and $b$, which is weaker than
Assumption~\ref{assump:bound_fourthmoment} in the sense that we can
allow $a=0$; this assumption captures the case where $\xb$ are
standard basis vectors, with $a=0$ and $b=1$. Another notable
observation is that Assumption \ref{assump:bound_fourthmoment}
requires $\EE[\xb\xb^\top\Ab\xb\xb^\top]\le \alpha \mathop{\text{Tr}}(\Hb\Ab)\cdot
\Hb$ for all PSD matrix $\Ab$, which may not be necessary (e.g., only
making this assumption for $\Ab=\Ib$ reduces to the assumption in
\citet{bach2013non,jain2017markov}). In this sense, we conjecture that
it suffices to consider the PSD matrix $\Ab$ that is commutable to
$\Hb$, and all of current theoretical results hold.
\bibliographystyle{ims}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,733 |
ABSSAC employs some of the latest machining techniques to the lead screws, ball screws and satellite roller screws it supplies, to ensure that a high journal concentricity product is delivered. That same capability can be applied to all screws regardless of diameter but also promotes the ability to provide a dynamic approach to all associated nut designs. Let ABSSAC work with you to reduce your scrap rates and supply leadscrew, ball screw and satellite roller screw solutions that fit into your application every time. ABSSAC can work directly from your dimensional drawings or if they do not exist, a physical sample to reverse engineer from.
Interestingly, a quick turnaround by Abssac has recently aided the restoration of two pre WW11 machine tools being restored to their former glory in a local woollen mills maintenance workshop. Despite the customer being without the original dimensional drawings for the ball screws and the fact that they were very much damaged, meant that Abssac had to rely on its extensive ball screw knowledge to satisfy the customer. First a careful measurement of the last remaining ball screw was undertaken. It was found that the ball screw was imperially sized, with a 1.00 inch diameter by 0.250 inch lead, Gothic arch ball screw format.
Abssac was able to utilise a cold rolled formed ball screw format, which were cut to a length and then machined at the ends to allow them to fit a thrust bearing to the whole assembly. The traditional square section nuts that were used by the old machines were replaced by adapted standard product range nuts and were chosen specifically for authenticity due to their single ball return design. The whole assembly was then low temperature black Chrome plated, which gave a corrosion resistant and exceptionally smooth and scratch resistant surface finish, hopefully giving the machine at least another 50 years of service.
Mr Phil Jones of Abssac says, "We have had great success in reverse engineering original designs of lead and ball screws where no formal drawing exists. Customers have shipped the parts for us to dimensionally check and copy into a new physical part. It is particularly nice to get involved with these types of project were we are part of a restoration. Abssac has a huge range of linear products, so nine times out of ten we can help with these more unusual requirements".
Multiple Piece Lead Screws – Product designs that comprise several components.
Multiple Component Lead Nuts – Custom drive nut assemblies in plastics or metals. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,689 |
Q: AS3 blurring lines using graphics.linestyle I am a rookie to AS3 and i'm creating a small painting application. If I wanted to apply a blur affect to lines drawn, how would i go about that?
graphics.lineStyle(size, color, alpha);
as you can see there is no parameter for it, any help? Thanks
A: You got to use the BlurFilter in order to blur graphics.
var myFilter:BitmapFilter = new BlurFilter(10, 10, BitmapFilterQuality.HIGH);
var lineFilters:Array = new Array();
lineFilters.push(myFilter);
lineContainer.filters = lineFilters;
As far as I know, it is not possible to apply filter to lines or graphics, only containers.
EDIT:
Test program:
var lineContainer:Sprite = new Sprite();
lineContainer.graphics.beginFill(0x000000);
lineContainer.graphics.lineStyle(5);
lineContainer.graphics.lineTo(150,150);
lineContainer.graphics.endFill();
addChild(lineContainer);
var myFilter:BitmapFilter = new BlurFilter(10,10,BitmapFilterQuality.HIGH);
var lineFilters:Array = new Array();
lineFilters.push(myFilter);
lineContainer.filters = lineFilters;
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,902 |
Q: Getting and error while building flutter project I am trying to install my flutter app into my device. It was working fine till I added a class named UserDb(model class). It then started giving me the following error:
The weird thing is that I have renamed the class to another name and removed any object named userDb but it is still showing me that error. I even removed that model class and all its usage from my project.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 623 |
December 14, 2020 December 14, 2020 Editor Trending
Senator Suleman Abdu Kwari, representing Kaduna North District at the National Assembly has praised President Muhammadu Buhari for "graciously approving the sum of N38.7 billion for the Kaduna-Pambegua-Saminaka-Jos road."
The lawmaker, in a media release on Thursday described the project as a historic move by the President in more effectively connecting Plateau, Kaduna and parts of Bauchi State.
Senator Kwari, who is the Senate Committee Chairman on Anti-corruption further stated that the approval could not have come at a better time, "seeing the kind of difficulty confronting motorists plying such roads, especially those travelling with their goods for trades and businesses".
He explained that "good road plays pivotal role in connecting communities for trades, with a potential for very positive impact on the national economy and by extension living standards of the people."
Senator Kwari also commended Kaduna State Governor, Nasir Elrufai for "working tirelessly to ensure that the State enjoys federal presence in such a way that it improves economic activities and help the upkeep of people in areas where such projects are executed". He also extolled the governor, over another Federal Government gas project worth $2.8 bn.
It would be recalled that the Federal Executive Council( FEC) at its meeting presided over by President Muhammadu Buhari approved N58.5 billion for reconstruction and rehabilitation of roads in some parts of the country. This was revealed by Minister of Works, Raji Fashola while briefing State House Correspondents in Abuja on Wednesday.
The Minister said: "The Ministry of Works and Housing presented one memorandum for the award of contracts for the construction of two roads; the Yakasai-Bagume-Damagun road in Kano, for N12.157billion and the rehabilitation of the Kaduna-Pambeguwa-Jos road linking Kaduna and Plateau States for N38.701 billion and the proposals were approved by the Federal Executive Council". | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,593 |
British Council Funded GREAT Scholarship for Studies in the UK
January 3, 2023 January 3, 2023 by foilus.com
For the academic session 2023-24, up to 200 scholarships are on offer from 49 universities across England, Wales, Scotland and Northern Ireland. British Council Funded MAJOR Scholarships offers students from 14 countries the chance to have £10,000 towards their tuition fees for a wide range of one-year postgraduate courses.
Each scholarship is jointly funded by the UK Government's GREAT Britain Campaign and the British Council with participating UK higher education institutions.
Scholarship Sponsors: British Council
Host institutions: Institutions in the United Kingdom
Scholarship Worth: £10,000
Number of awards: Enough
Level of education: Postgraduate (Master)
See also: British Council Full Degree Scholarships 2023 | Application Guide
Funded by the British Council GREAT Scholarship 2023 | Eligibility criteria
To be considered for the GREAT Scholarship funded by the British Council, applicants must meet the following requirements:
Be a citizen of an eligible country
Have an undergraduate degree, be motivated and have an interest in the proposed subject area
Satisfy the UK HE English language requirement
Establish a commitment to the UK as a Scholar, through personal and academic fulfillment
Be keen to attend a networking event of all UK-based SENIOR scholars to discuss experiences and capture perceptions about studying in the UK
Be willing to liaise with the British Council and their Higher Education Institution and act as an ambassador for the BIG Scholarships
As a GREAT Scholarships graduate, please be willing to occasionally speak to potential applicants about their own experience of studying in the UK.
MAJOR Scholarships are available to students from the following countries: Bangladesh, China, Egypt, Ghana, Kenya, India, Indonesia, Malaysia, Mexico, Nigeria, Pakistan, Thailand, Turkey, Vietnam.
How to apply: Interested applicants for the GREAT Scholarships 2023/24 funded by the British Council should visit the participating university's page to find out more information and then follow the link on the university's website.
Applicants should then proceed to apply for individual scholarships following the instructions given on each university's scholarship websites.
Application deadline: The deadline varies by institution | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,914 |
**I N THE YEARS** of 1914 to 1918 over 330,000 Australians served their country in a war far from their homeland, more than 60,000 of them died.
Five of these Australians were brothers; three of them were destined to never return to the home they loved.
_George (Geordie) Tennyson Marlow, born 8 October 1892, was the first son to enlist and later joined the 2nd Light Trench Mortar Battery._
_Charles (Charlie) Edward Marlow, born 29 June, 1891, was the fourth son to enlist. He joined the 38th Battalion._
_Percy Place Marlow, born 10 December 1895, was Allan's twin brother and joined the 38th Battalion._
_Allan Sharp Marlow, born 10 December 1895, was Percy's twin brother. They signed up together to the 38th Battalion._
_Albert Marlow, born 25 November 1897, was the youngest and last son to enlist. He also joined the 38th Battalion._
**W ORLD WAR I** broke out in July 1914. At the time it was considered to be "the war to end all wars". Today we know that it did not put an end to war.
It was one of the greatest tragedies and deadliest conflicts in history.
**F AMILIES ALL OVER** the world suffered during the war, sons rushed to enlist and daughters went to nurse them.
The story of the Marlow brothers in World War I begins in 1882 when their father migrated from England to Australia. He married Sarah Mahoney who was born on the goldfields at Castlemaine. The young couple purchased a small farm at a town called Mologa in northern Victoria.
Sarah and Charles had seven sons: Jim, Charlie, George, Frederick, twins Allan and Percy and the youngest was Albert. Frederick died as a baby.
The family was very proud of the new home they built in 1912. Decades later, found in a time-worn suitcase and in cupboards in this crumbling old house, were over five hundred letters and postcards. The yellow, stained letters tell the tale of a family torn apart by war.
_Sarah Marlow circa 1918._
_James (Jim) Marlow born 30 September, 1889. The eldest son, he attempted to enlist but was rejected._
_Charles Marlow date unknown._
_Sarah Marlow and Albert outside their new home 1912._
**1914**
**T HE MARLOW BROTHERS** did not join the rush to enlist when war was declared in 1914. They were either employed or working on the family farm. The enlistment rules at the time also required that men be at least five feet and six inches tall. Jim and Percy were not tall enough. With the common view that the conflict would be over before Christmas, the brothers stayed at home where the wheat harvest was about to begin.
_1904 version of the 'Rising Sun' badge Proudly worn by soldiers of the 1st and 2nd AIF in both World Wars. It was worn on the upturned brim of a slouch hat._
**Anzac Timeline of WWI**
**28 July 1914**
Austria declares war on Serbia
**4 August 1914**
Britain declares war on Germany
**1 November 1914**
The first Anzac troops leave
**25 April 1915**
Anzac troops land at Gallipoli
**March 1916**
Australian troops arrive in France
**1 July 1916**
The Battle of the Somme begins
**19 July 1916**
Battle of Fromelles begins
**24 July 1916**
Anzacs capture the village of Pozières
**28 October 1916**
The Conscription Referendum is defeated
**6 April 1917**
The United States declares war on Germany
**11 April 1917**
The First Battle of Bullecourt begins
**20 September 1917**
The Battle of Menin Road begins
**4 October 1917**
3rd AIF Division captures Broodseinde Ridge
**12 October 1917**
The Battle for Passchendaele
**31 October 1917**
Australian Light Horse captures Beersheba (Palestine)
**20 December 1917**
The 2nd Conscription Referendum is defeated
**25 April 1918**
Australians capture Villers-Bretonneux
**4 July 1918**
Australian-American infantry attack capture Le Hamel
**8 August 1918**
The Battle of Amiens begins
**5 October 1918**
Australians fight their last battle at Montbrehain
**11 November 1918**
The Armistice is signed and the war ends
**1915**
**W HEN 1915 ARRIVED** it became clear that the war was not about to end quickly. A complex maze of trenches stretched for over 700 kilometres from the coastline of Belgium through France to the Swiss border. This region became known as the Western Front. The invading German forces held much of the high ground on the hills and ridges upon which they began to build concrete fortifications and intricate trench systems. The soldiers of the French, Belgian and British forces dug in on the low ground in front of the German troops to prevent any further regions of France and Belgium from being captured. This frontline was to remain largely deadlocked for the next three years. The opposing forces faced each other across no man's land and neither side could make any significant breakthrough.
After the April 25 landing of the troops of the Australian Imperial Force (AIF) at Gallipoli, the demand for more men to replace those who had been killed or injured increased. In July, the height requirement was reduced to five feet two inches.
_Gallipoli_
_The objective of the Gallipoli campaign was for Allied troops to reach the capital of Turkey (Constantinople) through the Dardanelles Strait, knock Turkey out of the war and clear the access to Russia through the Black Sea._
_From the landing at Anzac Cove on 25 April 1915 to when the Australians were withdrawn in December, around 50,000 men had served on the peninsula. Casualty figures from major battles and campaigns do not always agree but over 8,700 Australians had died and around 19,000 were wounded. In total over 46,000 Allied lives were lost._
_'Trench Warfare' - artwork adapted from Australians on the Western Front, Department of Veteran Affairs (2006)._
**T HE THREE ELDEST** Marlow brothers caught the train to Bendigo to enlist. Jim was turned away, he had poor eyesight. Charlie was also rejected; he had dental problems which had to be fixed. George was to go to war without his brothers.
The local community gathered to farewell George before he boarded the ship to take him to Egypt. The following day the family hitched the wagon and drove George into town where friends and neighbours assembled to say their last goodbyes; Sarah and Charles farewelled their son with a mixture of apprehension and pride. Perhaps George promised he would return; it was a promise young men were making all over the world.
_George Marlow_
_Invitation to George to his farewell_
_1915 recruitment poster._
**G EORGE ARRIVED IN EGYPT** where the soldiers of the AIF were training. The many letters and postcards George sent made it clear that he soon tired of military drill and the sands of Egypt; he was keen to be sent to Gallipoli. This was not to be, there was no longer any point in remaining on the peninsula. The Australians were withdrawn in December.
_George on left, George Downie standing centre, the other men are unknown_
_George's Postcard_
_Christmas card_
_Percy in centre, Allan on right at the rear._
_The embroidered card George sent to his father._
_Allan and Percy's identification discs._
**1916**
**A T HOME IN MOLOGA**, Allan and Percy were anxious to serve their king and country. At the age of 20, they signed up to the new 3rd Division being formed in Australia and were appointed to the 38th Battalion of the 10th Brigade. Charlie made another attempt to enlist but was once again turned away.
As his brothers adjusted to their new life in the training camps of Bendigo and Melbourne, on the other side of the world George was on his way to the Western Front.
**Western Front**
**B Y APRIL GEORGE** had experienced life in the trenches as he wrote home to his family. The Australians were positioned in the frontline near the border between France and Belgium, an area known as Flanders. George volunteered to transfer from the 7th Battalion to the 2nd Light Trench Mortar Battery of the 1st Division.
_George had now experienced the warfare of the Western Front as he wrote home to Allan._
_The Stokes Light Trench Mortar Gun was a rapid fire 3 inch gun used with the infantry in the frontline._
**Australia**
**T WO MONTHS LATER** in Australia, Allan and Percy were preparing for their departure. Their youngest brother Albert had now made at least two attempts to enlist while underage. Until he turned 21, he required the approval of his parents.
On June 20, the twins and their mates from Mologa left Australia's shores on board the _HMAT Runic_.
Many of these young men would not come home.
_A photo taken in Bendigo with Allan and Percy in the front._
_Australian soldier.
Artwork by Jeff Isaacs._
_German Soldier._
_Artwork by Jeff Isaacs._
_Allan's Postcard._
**The Somme**
**A LLAN AND PERCY** were settling into their long six week journey when British troops were ordered to attack the German frontline in an area of France known as the Somme.
On the first day of July, 19,240 British soldiers were killed and over 38,000 were injured. In the following devastating fortnight over 100,000 men were listed as killed, wounded or missing.
The 1st, 2nd and 4th Australian divisions were rushed to the Somme to aid the embattled Allied soldiers. George was with the 1st Division as they waited for their orders. The 5th Division remained near the border close to a village called Fromelles.
_Cousin Harry Marlow of England who was injured on the first day of The Somme._
_The Somme_
_The Allied attack on the Somme on 1 July involved an advance of over 120,000 soldiers on close to a 30 kilometre front. The objective was to capture the high ground on the ridges including the village of Pozières and then to advance to Bapaume._
_The German frontline consisted of three rows of barbed wire and deep trenches with concrete fortifications. After a seven day bombardment of the German lines, the defences had not been broken. There were close to 60,000 Allied casualties on the first day._
_By the end of 1916 there were around one million total casualties, including German soldiers. Of these, 600,000 were Allied troops. One hundred and forty-six thousand of these men were listed as killed or missing._
**Fromelles**
**I T WAS AT FROMELLES** where the Australians made their first attack on the Western Front. On July 19 the soldiers of 5th Division were ordered to storm the well-fortified German lines in full daylight. The attack was a disaster. Australia suffered 5,533 casualties in 24 hours, over 1,900 men were killed, close to 1,300 had no known grave and 470 men became prisoners of war. Today the Battle of Fromelles is regarded as the worst 24 hours in Australia's military history.
_Fromelles Today_
_After the Battle of Fromelles, 1,335 Australian soldiers remained 'missing', they had no known grave. In 2007, a retired Melbourne teacher, Lambis Englezos, encouraged the Australian Army to investigate the possibility that some of these soldiers had been buried in a mass grave at Pheasant Wood by German troops in 1916. The remains of some 200 Australian soldiers were uncovered. Between 30 January and 19 February 2010, the remains of 249 soldiers were reinterred with full military honours in Fromelles (Pheasant Wood) Military Cemetery._
_By 2014, 144 Australian soldiers had been identified by name using forensic science, including DNA samples provided by the ancestors of the missing._
_The Cobbers Memorial at Fromelles depicts Sergeant Simon Fraser (KIA Bullecourt 17.05.1917) carrying a wounded soldier, another cried to him, "Don't forget me, cobber."_
_Fleurbaix trenches near Fromelles AWM P00437_017_
**Pozières**
**T O THE SOUTH** of Fromelles, the German-held village of Pozières sat perched high on a ridge in the Somme Valley. On the night of July 23 the 1st Division attacked the enemy positions. By dawn they had reached the centre of the village. The big guns of the German artillery responded with a severe and relentless bombardment to recapture the lost stronghold. The village was turned to rubble as the Australians staunchly held on to their positions. Four days later, when George was withdrawn with the surviving Australians, the 1st Division had suffered over 5,000 casualties.
_Pozières_
_The British had planned to capture the village of Pozières on July 1, the first day of the Battle of the Somme. The plan failed. The Australian troops made their first attack on the village on July 23. The 1st, 2nd and 4th Australian Divisions all fought desperate battles at Pozières and nearby Mouquet Farm until they were withdrawn on September 5. During this time, the Australians were involved in 19 separate attacks; they suffered 23,000 casualties of which 6,741 men had been killed. Mouquet Farm was later captured by English soldiers on September 26. Many survivors of Pozières suffered from what was called shell shock, a term used in World War One to describe an illness brought on from the trauma of fighting and artillery bombardments. Soldiers suffering shell shock may forget their past or not be able to think clearly, they could be severely anxious or perhaps they could not walk, talk or sleep. They often had dreadful nightmares. Today, the equivalent effects of shell shock is referred to as post-traumatic stress._
_Pozières months after battle. The town and surrounds were reduced to rubble by artillery fire. AWM E00532._
_The main street of the French village of Pozières in 1914 before the battle. AWM GO15341._
_The AIF memorial at the entrance to Mouquet Farm. This German stronghold was eventually captured by British troops in late September._
_The site of an old windmill at Pozières. Charles Bean, the official Australian war historian, wrote of this battlefield that it "...marks a ridge more densely sown with Australian sacrifice than any other place on earth." Close to 7,000 Australians were killed here in just under seven weeks. Mouquet Farm is on the horizon in the middle of the picture._
**T HE BATTLE RAGED** at Pozières for seven weeks as the Australian divisions pushed on in an attempt to capture the fortified bastion of Mouquet Farm. It was heavily defended with pillboxes, tunnels and concrete bunkhouses deep beneath the ground. Eventually, when the troops had reached the point of exhaustion, they were withdrawn. Six thousand, seven hundred and forty-one Australians had been killed. George had been lucky.
**A S GEORGE WAS FIGHTING** for his life at Pozières and the twins were steaming towards England, Charlie broke the news to the family at home that he had enlisted. He was appointed to the 38th Battalion; he would be joining his brothers as a member of the 3rd Division.
_The invitation Charles and Sarah received to the wedding of Charlie and Pearl._
Just weeks before he departed Charlie proposed to and married his sweetheart. As he boarded the ship, and as the _Shropshire_ pulled away from the pier, Charlie was not aware that his young wife was expecting their baby.
_Charlie and Pearl Marlow on their wedding day._
**T HREE DAYS AFTER** Charlie sailed, Albert convinced his parents to give their permission for him to enlist. Sarah and Charles may have believed that Australian men would soon be conscripted to the AIF and be forced to become soldiers, as was the case in other Allied countries. The Federal Government asked Australians to vote to change the laws which prevented compulsory overseas service. The majority of Australians voted against conscription. The AIF was to remain one of the few Allied fighting forces to be dependent upon volunteers, they were citizen soldiers with little time for pettiness and polish.
_Like the young man in the picture, Albert pleaded with his parents, not for a kiss, but to sign his enlistment papers._
_Albert's parents finally agreed to allow their youngest son to join the AIF._
_The postcard Albert sent before he embarked._
**O NLY FOUR WEEKS LATER** Albert sailed for England on board the _Port Lincoln_ to also join the 38th Battalion.
_With just an hour to spare before he boarded the ship, Albert wrote to his mother._
**England**
**O NCE IN ENGLAND,** Allan, Percy and their mates began training at the Larkhill military camp, within walking distance of the prehistoric monument of Stonehenge. Like their fellow Australians they delighted in seeing the famous sites of England. The twins also took the opportunity to meet their English relatives.
_Percy expresses his surprise to his parents regarding Albert's enlistment._
_Allan in London._
While Allan and Percy looked forward to the arrival of Charlie they were shocked to discover that Albert had enlisted. Allan wrote to Albert:
_... Well Albert I heard that you went and enlisted again. What did I tell you before I left. Albert you were foolish you know. There is enough of us here now..._
**A LBERT ARRIVED AT LARKHILL** in the final days of 1916, but his brothers had already left for the Western Front. He was bitterly disappointed and wrote to his parents:
_I expected to see Charlie, 50 of the 3/38 were left behind, and they told me that Charlie had gone to France. I was never so disappointed in all my life. He left a note behind..._
_Albert's cable to home letting his parents know that he had safely arrived in England._
Visiting his relatives and the sights he had read about in school soon cheered Albert. Better still was the moment when George, on 10 days leave in England, arrived at the training camp. The brothers' hope of spending more precious time together was crushed when Albert was hospitalised with the mumps.
_George sent his mother his Christmas wishes from the trenches of France._
_The first of the Field Service cards sent from the trenches._
**A S WINTER APPROACHED** on the Western Front the autumn rains transformed the powdered battlefields into a quagmire. Trenches and shell holes filled with mud, roads and routes to the frontline were impassable, exhausted troops sank with every step. Standing to their knees in the mud of the trenches the troops were about to endure a fearsome winter. George was positioned near Pozières on the Somme where the bitterly cold conditions could freeze a pannikin of hot tea in one minute. It would be the coldest winter recorded in 40 years.
_Field Service cards were both a means of censorship and rapid communication._
_Allan sent copies of this card to his family._
**1917**
**A FTER BRIEFLY MEETING** Charlie at Larkhill, Allan and Percy had arrived in France in time for Christmas. By late January the three brothers were together on the Western Front near a town called Armentières. Allan wrote home:
_The trenches are in a bad state. In places one goes up to his hips in muck and water. It is pretty cold, but not as cold as Lark Hill. We get splendid tucker and plenty of it. The only trouble is the rats, they are in 1 000's. The other night I went to bed for a couple of hours and as soon as I got to sleep they hauled me out and had taken full charge of the dug out. So you see that we have more than Fritz to fight..._
_The text of Allan's personalised Christmas greetings._
_Percy's Christmas card to Jim._
_Charlie's Christmas greeting._
**A LLAN WAS PROMOTED** to lance-corporal in April and was placed in charge of a Lewis gun team.
Charlie, Allan, Percy and Albert were all Lewis gunners.
_A Lewis gun team walking on duckboards on the Ypres battlefield. Wooden duckboards lined the base of trenches and provided solid footing along the routes to the frontlines. AWM E01087._
_Lewis Machine Gun Mk1. Lewis machine gunners worked in a team of five with supplies of ammunition and spare parts being carried into the field where the gunner and the team would move frequently to avoid being sighted by enemy artillery fire._
**I N MID-APRIL** the Marlow family at home received their first official telegram... Allan had been injured. As Allan had made his way along the trenches under cover of darkness he came under artillery fire. As shells thundered down, the trench was hit and exploded above him. Allan was buried under sandbags, soil and duckboards. Fellow soldiers fought to pull him from the debris to save his life. He was hospitalised with internal injuries, but soon recovered. With the delay in communication to Australia it was some five weeks before the family received word that Allan had survived his injuries.
_Red Cross report._
_Allan's identification bracelet._
_Left to right standing John Charles Lewis, Charlie Marlow, Earnie Butler, William Jones; seated Jack Lockett, Edmond Jones, Wren Teale. By the end of the war three of these men had been wounded and another two had lost their lives. Jack Lockett survived and became Australia's oldest living person at the age of 111. He passed away in 2002._
**Bullecourt**
**B Y NOW THE MARLOWS** had lost many of their mates from the Mologa district and several of their new friends. Some had been killed, others wounded. Most devastating was the news that Albert's old schoolmate, Jack Price of the 46th Battalion, had lost his life in a disastrous Allied attack at a village called Bullecourt. He was nineteen.
_Jack Price died at an ill-fated attack on the fortified German defensive line known as the Hindenburg Line at Bullecourt, France in April, 1917. He had just turned 19._
_The Hindenburg Line and the First Battle of Bullecourt_
_In March the German army withdrew to a defensive line made of a trench system of solid concrete fortifications and pillboxes fronted by rolls of barbed wire. In places, the wire towered above the heads of the tallest soldiers. The line stretched 160 kilometres and was up to seven kilometres wide. The withdrawal shortened the German front and made their positions easier to defend._
_Bullecourt, a village in France, was embedded in the Hindenburg Line. The 4th and 12th Brigades of the 4th Division and the 62nd British Division were to capture the village on April 11. The advance was poorly planned, supporting tanks broke down or were destroyed and the Australians did not receive artillery support when it was needed. The soldiers reached Bullecourt but were forced to retreat. The Australians suffered over 3,300 casualties and 1,170 men were taken prisoner - the largest number captured in any battle during the war._
**Messines**
**A LBERT WAS REUNITED** with his brothers of the 38th Battalion in May as the 3rd Division was preparing for its first major action.
They were to capture the Belgian village of Messines.
_Before leaving England, Albert sent home newspaper clippings of the King's visit and a photo of himself with his mates. Albert is standing second from the right. Seven months later three of these men had been killed and another two wounded._
_A brief newspaper report which Albert sent home._
_The King's visit - Albert sent this report home, he has marked the group in which he was marching (bottom left)._
**W HEN THE ALLIES** detonated nineteen mines under the German frontlines, the deafening roar of the explosions shattered the countryside. The 3rd Division attacked the dazed and injured survivors and captured their objectives. The Battle of Messines was declared a great success. Two days later, when the men were relieved from their new positions, they discovered that more of their friends and neighbours were listed as killed, wounded or missing.
_Australian Tunnellers at the Catacomb Entrance, Messines Sector, Belgium. AWM E04486._
_Unexploded Mine at Messines, Belgium. AWM H15258._
_Messines_
_Allied Tunnellers had dug mines beneath German trenches in the Messines area and packed them with explosives. Before dawn on 7 June 1917, 19 mines were blown, destroying the German positions. It is thought that 10,000 German soldiers were killed._
_British, New Zealand and Australian troops advanced to take the devastated ridge. Messines was the first major battle for the 3rd Australian Division. Despite being regarded as a triumph, the battle cost over 26,000 Allied casualties, with over 6,000 Australians killed or wounded._
_Allan and his mate Spuddy (Edward) Kerr – a photo taken just weeks before the Battle of Passchendaele in which Spuddy was wounded. He lay in mud on the battlefield for two days before being rescued. His wounded leg became infected and was later amputated._
_An embroidered card that George sent to Sarah._
_The much-valued green envelope was issued to soldiers for private letters they did not want read and censored by other members of the unit. Green envelopes might be opened at the base to ensure that soldiers were not revealing important information. The brothers often sent their letters home together in one very bulky green envelope._
**B Y MID-JULY** the 38th Battalion were back in the line holding the trenches near Messines. They had enjoyed a short break, taking time to help the local farmers, devour the parcels of food sent from Australia and answer their mail. Charlie wrote:
_... we can buy a decent feed here this last two nights we went to a place, and had coffee, three eggs and some bread and butter..._
_Allan sent this light-hearted photo to his mother._
**N INETEEN-YEAR-OLD ALBERT** wrote to home from the trenches, he had been on the Western Front just over two months. He rested the paper on his knee as he pencilled his letter and then tucked it into his tunic.
The next day Albert was positioned in support lines behind the main frontline when a heavy German artillery barrage rained down upon the Australians. The dugout in which Albert and four others sheltered was directly hit.
All five men were instantly killed.
_Albert._
_Albert's final letter July 16, 1917 p. 1_
_Albert's final letter July 16, 1917 p. 2_
**C HARLIE WAS THE ONLY brother** who was nearby. He rushed to Albert, but there was nothing he could do. He took his youngest brother's final letter from Albert's pocket. The following day Charlie hurried to the cemetery where Albert was to be buried. When he arrived it was too late, the graveside service was over. Charlie wrote an anguished letter home to his parents, he had promised to keep his youngest brother safe, but promises were no match for the might of the German army.
_Charlie's letters to his mother p1-4_
_Albert's grave at Kandahar Farm Cemetery, Belgium today._
_The cross which Charlie organised to mark Albert's grave._
_Kandahar Farm Dressing station. A 3rd Australian Divisional Field Dressing Station at Kandahar Farm, on the afternoon of 7 June 1917, during the Battle of Messines. Albert was buried on the other side of this building. AWM E00482._
**J UST WEEKS AFTER ALBERT'S DEATH,** George moved north with the 1st Division toward Belgium. For the first time since leaving Australia he met Charlie, Allan and Percy. They spent an afternoon together catching up on their news and enjoying the comfort of being with family. George walked miles to visit Albert's grave.
_George is sitting in the centre and his mates of the 2nd Light trench Mortar Battery. Top row left to right Bert Hawkesworth, Harry Hoskins, Charlie Edwards, Syd James, George, Joe Steel, Buff Braithwaite, George Brittain._
**Ypres Salient**
**A S THE 38TH BATTALION** left the area for a rest, the 1st Division was being prepared for another major attack, to take the high ground to the east of the city of Ypres in Belgium. The objective was to eventually capture the village of Passchendaele from where enemy observers had a clear view of the activities of Allied troops, but there were other villages and protective German strongholds which needed to be cleared first.
**Battle of Menin Road**
**O N SEPTEMBER 20** the 1st and 2nd Australian Divisions went into battle. George was in reserve but was called in at the last minute to replace a soldier who suddenly became ill.
In the early hours of the morning 24-year-old George rushed forward in an attack on the German frontline near Polygon Wood. He fell from a gunshot wound to the stomach. He was wounded but alive.
George was carried to a dressing station and then taken by cart to a hospital.
_George._
_Dressing Station Menin Road 1917 AWM E00714._
**G EORGE DIED THE FOLLOWING DAY.**
The surviving brothers received word that George had been injured, not that he had lost his life. Despite their best attempts to find their brother and always hoping that they would soon receive a letter from him, his brothers did not discover that George had been killed for another six weeks.
_George._
_George's grave today at Lijssenthoek Military Cemetery, Poperinghe. Close to 10,000 Commonwealth soldiers are buried in the cemetery._
_The original cross placed over the grave of George._
_The letter Charlie received confirming the death of George._
**Passchendaele**
**I N OCTOBER,** the 1st, 2nd and 3rd Australian divisions and the New Zealand Division advanced on the German stronghold of Broodseinde Ridge. It was the next step in the advance toward the village of Passchendaele. Allan and Percy went over the top with the 38th. Although the attack was described as "the most complete yet won", the sacrifice had been great. The three Australian Divisions suffered 6,500 casualties.
_Troops of the 1st Australian Division at Hooge on the Ypres Salient. AWM E00833._
_The remains of one of two German blockhouses inside the cemetery gates._
_Tyne Cot Cemetery, Broodseinde Ridge. The cemetery sits on the ridge and is the largest Commonwealth cemetery in the world with close to 12,000 graves. The Cross of Sacrifice also marks the memorial to the 3rd Australian Division and was erected above the remains of a German pillbox._
_Broodseinde Ridge_
_The attack began before dawn on October 4, 1917. The Australian troops were shelled heavily and a seventh of them became casualties before the attack began. When it did, the attacking troops saw a line of enemy soldiers advancing towards them; the Germans had chosen the same time to launch an attack of their own. The Australians gained all their objectives along the ridge and 5,000 German soldiers were taken prisoner. It was described in official German records as "the black day of October 4th"._
**R AIN DRENCHED THE BATTLEFIELD** over the following week and the shelled ground turned to thick mud. Despite the conditions, the advance to Passchendaele continued. Allan and Percy clambered from the trenches; the objective of the 3rd Division was to reach a line just beyond the village. No ground was captured. The Australians became bogged in the quagmire as the enemy troops fired down from the ridges. The Allied casualties were severe. Sixty-two percent of the men of the 38th Battalion who fought on that day were listed as killed, wounded or missing.
Allan and Percy had survived. Many of their mates had lost their lives.
_Wounded in the railway cutting Broodseinde Ridge 12 October 1917. On the right facing the camera is Austin Henderson of the 38th Battalion. The man also facing the camera looks like Allan Marlow; however, a positive identification has not been made. AWM E03864._
_The railway cutting and the remains of the line today._
_Passchendaele_
_The taking of Passchendaele and beyond was the objective of the 3rd and 4th Australian divisions combined with the New Zealand Division and five British divisions. The Australian 3rd Division suffered its highest casualties of the entire war. Of the men of the 38th Battalion who went over the top on that day, 381 men were killed, wounded or missing. The 3rd Division lost over 3,000 men. The New Zealanders also lost around 3,000 soldiers while the 4th Division reported 1,000 casualties._
_Passchendaele was eventually captured by Canadian forces in November, 1917 but was lost to the advancing German troops in 1918._
**1918**
**T HE WINTER MONTHS STALLED** the fighting on the Western Front as the Australians remained in the sector of the line about Messines. There were few reinforcements arriving from Australia to replace the casualties. Percy became ill with Trench Fever and in February was sent to England for treatment. Allan was promoted to the rank of lieutenant while Charlie had been made a sergeant.
_Allan on leave in England. A photo his aunt sent to Sarah and Charles._
_Percy front, second from right in hospital in England._
**Germany Strikes**
**I N MARCH SOME** half a million German soldiers began a series of attacks on the British and French lines in the Somme Valley. Their objective was to capture the cities of Amiens and Paris and march onward to the ports along the French coast. Initially the German troops had great success and much of the land that for two years the Allied forces had gallantly claimed and defended was recaptured in five days.
The AIF was rushed to the Somme in the hope they could stop the rapid advance of the German forces.
_54th Battalion soldiers after Pèronne had been captured, September 1918. AWM E03183._
_The Third Division Memorial at Sailly-le-Sec, France where the division courageously repelled the advancing enemy forces._
_The quadrangle at Villers-Bretonneux School._
_Villers-Bretonneux School. The school children of Victoria, Australia raised money to help rebuild the school after the war._
**W ITH ACCLAIMED DETERMINATION,** the Australians held off strong attacks by the German troops across the Somme Valley. They halted the enemy advance towards Amiens at a village called Villers-Bretonneux. Just a few weeks later German troops captured the village from British soldiers. That night Australian soldiers stormed the village and recaptured it in a fierce and courageous battle that was crucial to the protection of Amiens. The date was April 25... Anzac Day.
**A S THE BATTLED RAGED** at Villers-Bretonneux, Charlie and Allan were some 15 kilometres away to the north, protecting the frontline trenches about the villages of Buire and Ribemont.
In the early hours of the morning of April 26, Charlie set off to collect and deliver breakfast to his platoon. On his way, Charlie spoke with Allan at company headquarters and chatted with his mates along the trenches. Just for a moment he forgot to keep his head down.
Charlie was shot by a sniper and died instantly. He was 26 years old.
With a sorrow for which he struggled to find words, Allan wrote to Jim.
_Charlie._
_Charlie's card to his mother._
_Charlie's note._
_Allan's distress is clear as he writes to Jim._
**S ERGEANT CHARLIE MARLOW** had died as the Australians courageously held the line that protected Amiens. In the desperate defence of the city, Sarah and Charles lost another much-loved son. Allan, Percy and Jim lost another brother. Pearl would never see her husband again and baby Beatrice Eva would never know her father.
_The final resting place of Charlie Marlow._
_The simple cross that marked Charlie's original burial place at Heilly before being relocated to Ribemont Communal Cemetery Extension._
_Pearl Marlow and baby Eva._
**B Y JUNE THE GERMAN FORCES WERE WEARY.** To the north in Flanders the Allied soldiers had held the line against enemy attacks. South of the Somme, the American forces had arrived and fought off a German attack on the Marne River, while French forces held firm against the German advance.
_Officers at Le Havre, Allan is in the back row 3rd from right._
_Allan._
Allan was promoted to Battalion Lewis Gun Officer. Just two days later he was sent to the AIF base on the French coast at Le Havre. Allan did not want to leave the men with whom he had been through so much, but his parents had lost three sons. Commanding officers wanted to reduce the chance of yet another Marlow son becoming a casualty. Percy recovered from his illness and returned to France in August but was no longer a combat soldier; he was placed on duty as a guide. The family had sacrificed enough.
**A LLAN AND PERCY'S FRIENDS** and old neighbours were not as fortunate. Some were killed and several were wounded in major battles at Le Hamel, Amiens, Mont St Quentin and Peronne before the Australians fought their last battle in October. Since August they had fought consistently for two months, had advanced some fifty kilometres, liberated many villages and taken thousands of prisoners. The cost was heavy, close to 27,000 Australians were listed as casualties.
_Allan described Les Townsend as being like a brother. Les enlisted with Allan and Percy in the 38th Battalion and was a machine-gunner in D Company. He had sailed with the twins on the Runic. At Mologa they had been at school together and played football in the same team. Les had fought with the Marlow brothers at Messines and had been wounded at Passchendaele but recovered and returned to the front three months later. He was killed by a sniper at Curlu on August 28._
_The Australian Corps Memorial Park, Le Hamel 2012._
_A portion of a photo taken on Armistice Day near le Havre. It reads Taken on 11.11.18 at Rouelles France 1st ACD. Allan is seated far left with his arms folded._
_Allan standing far left, taken on Armistice Day._
**The Armistice**
**O N NOVEMBER 11,** in a train carriage at a forested railway siding at Compeigne in France, German leaders agreed, amongst other conditions, to give up the land they had invaded. The armistice was signed. It was 5.00 am. In six hours, at 11.00 am, all hostilities were to abruptly cease. The war was over.
This is the day we commemorate as Remembrance Day.
**I T TOOK MANY MONTHS** for the Australians to return home, there were simply not enough ships to transport the troops that had come from all over the world. Percy returned to Australia in July, 1919. Allan arrived home in November, 1919, but not before he visited the graves to say farewell to each of the brothers he left behind.
_The Dreadful Cost of World War One_
_Counting the cost of World War One is difficult and figures do not always agree. Some historians suggest that the Allied and Central Powers combined lost 12 million military and civilian lives, others say 16 million; nine million of these were soldiers. Another 20 million were listed as wounded._
_The official Australian war historian Charles Bean, wrote that of the 416,809 men who enlisted, 331,781 fought and of these 59,342 were killed with another 152,171 men wounded. Some were wounded time and again. Australia suffered 64.8% killed or wounded; the highest of all Allied fighting forces._
_Australian WW1 Army soldiers disembarking from a troopship after returning from overseas. A line of private motor cars is waiting to pick some of them up. AWM H13025._
_Allan standing beside George's grave. Allan visited the resting place of each of his brothers prior to leaving France._
**We Will Remember Them**
**I N 1920, THE MOLOGA COMMUNITY** erected a monument to their 22 courageous young men who fought in the Great War. Ten of these men did not come home. Allan helped his mother unveil the memorial. Four years later, Allan married Eva Jones, a young neighbour who sent parcels of cakes to the trenches for the Marlow brothers and to whom Allan would write long thank you letters. They are my grandparents. Grandpa built his own home using handmade mud bricks. He named it _Passchendaele_.
_Allan and Sarah Marlow unveiling the Mologa War Memorial._
_Allan and Eva Marlow on their wedding day 24.09.1924._
_Passchendaele, Mologa 1985, the home that Allan built from handmade mud bricks._
**W HEN MY GREAT-GRANDMOTHER** Sarah passed away, returned soldiers formed up at the cemetery gates and marched ahead of the pallbearers. Family and friends said that Sarah had died of a broken heart.
**O NE HUNDRED YEARS** on it is not by chance that Australia is a safe and prosperous country. Australian soldiers have continued to fight for our freedom in the century that has passed with the same courage, sense of mateship and loyalty of the first Anzacs.
| {
"redpajama_set_name": "RedPajamaBook"
} | 131 |
\section{Introduction} A set $S$ of vertices in a graph $G=(V,E)$ is
called a {\it dominating set\/} if every vertex $v\in V$ is either in
$S$ or adjacent to a vertex in $S$. The domination number of $G$,
$\gamma(G)$, is the minimum size of a dominating set.
Let $P_m$ denote the path on $m$ vertices and $C_n$ the cycle on $n$
vertices; the {\it complete cylindrical grid graph\/} or
{\it cylinder} is the product
$C_n\square P_m$. That is, if we denote the vertices of $C_n$ by
$u_1,u_2,\ldots,u_n$ and the vertices of $P_m$ by $w_1,\ldots,w_m$,
then $C_n\square P_m$ is the graph with vertices $v_{i,j}$, $1\le
i\le n$, $1\le j\le m$, and $v_{i,j}$ adjacent to $v_{k,l}$ if
$i=k$ and $w_j$ is adjacent to $w_l$ or if
$j=l$ and $u_i$ is adjacent to $u_k$.
It will be useful to think of this graph as $P_n\square P_m$,
with the edge paths of length $m$ glued together, that is, connected
with new edges.
P. Pavli{\v c} and J. {\v Z}erovnik\cite{pavlic-zerovnik:dom-no-cyl-graphs}
established upper bounds for
the domination number of $C_n\square P_m$, and Jos\'e Juan Carre{\~n}o
et al.\cite{carreno-et-al:lower-bound-dom-no-cyl-graphs} established
non-trivial lower bounds. For $n\equiv 0\pmod 5$ the bounds agree, so
the domination number is known exactly. Here we improve the lower
bounds, except of course in the case that $n\equiv 0\pmod 5$. The
method is similar, based on a technique first used in
Guichard\cite{drg:dom-grid-digraph-lower-bound}, and later in
Gon\c{c}alves, et al.\cite{goncalves-et-al:domination_number_of_grids},
but we use a different programming technique than that of
\cite{carreno-et-al:lower-bound-dom-no-cyl-graphs}.
\section{Getting a lower bound}
A vertex in $C_n\square P_m$ dominates at most five vertices,
including itself, so certainly $\gamma(C_n\square P_m)\ge nm/5$. If we
could keep the sets dominated by individual vertices from overlapping,
we could get a dominating set with approximately $nm/5$ vertices, and
indeed we can arrange this for much of the graph, with the exception
of the the top and bottom copies of $C_n$ in which the vertices have
only 3 neighbors, and, except when $n\equiv 0\pmod 5$, in the leftmost
and rightmost columns of $P_n\square P_m$ where each vertex in the
leftmost column is adjacent to the corresponding vertex in the
rightmost column. Figure~\ref{fig:12x10} shows one of the nice
examples, when $n$ is divisible by 5.
\begin{figure}[htb]
\hbox to \hsize{\hss\beginpicture
\setcoordinatesystem units <4mm,4mm>
\setplotarea x from 0 to 9, y from 0 to 11
\grid 9 11
\multiput {$\bullet$} at 2 0 4 0 7 0 9 0 /
\multiput {$\bullet$} at 0 1 5 1 /
\multiput {$\bullet$} at 3 2 8 2 /
\multiput {$\bullet$} at 1 3 6 3 /
\multiput {$\bullet$} at 4 4 9 4 /
\multiput {$\bullet$} at 2 5 7 5 /
\multiput {$\bullet$} at 0 6 5 6 /
\multiput {$\bullet$} at 3 7 8 7 /
\multiput {$\bullet$} at 1 8 6 8 /
\multiput {$\bullet$} at 4 9 9 9 /
\multiput {$\bullet$} at 2 10 7 10 /
\multiput {$\bullet$} at 0 11 3 11 5 11 8 11 /
\endpicture\hss}
\caption{The cylinder $C_{10}\square P_{12}$ has
domination number 28. (Vertices on the right side are adjacent to the
corresponding vertices on the left side.)}
\label{fig:12x10}
\end{figure}
Suppose $S$ is a subset of the vertices of $C_n\square P_m$. Let $N[S]$ be
the set of vertices that are either in $S$ or adjacent to a member of
$S$, that is, the vertices dominated by $S$. Define the {\it wasted
domination\/} of $S$ as $w(S)=5|S|-|N[S]|$, that is, the number of
vertices we could dominate with $|S|$ vertices in the best case, less
the number actually dominated. When $S$ is a dominating set,
$|N[S]|=mn$, and if $w(S)\ge L$ then $|S|\ge (L+mn)/5$. Our goal now
is to find a lower bound $L$ for $w(S)$.
\begin{figure}[tb]
\hbox to \hsize{\hss
\beginpicture
\setcoordinatesystem units <2.5mm,2.5mm>
\setplotarea x from 0 to 15, y from 0 to 25
\axis left /
\axis right /
\axis top /
\axis bottom /
\setdashes
\putrule from 0 5 to 15 5
\putrule from 0 10 to 15 10
\putrule from 0 15 to 15 15
\putrule from 0 20 to 15 20
\putrule from 0 25 to 15 25
\put {$G_1$} at 7.5 22.5
\put {$G_2$} at 7.5 17.5
\put {$\vdots$} at 7.5 12.5
\put {$G_{t-1}$} at 7.5 7.5
\put {$G_t$} at 7.5 2.5
\endpicture\hss}
\caption{Partitioned cylinder.}
\label{fig:partitioned}
\end{figure}
Suppose a cylinder $C_n\square P_m$ is partitioned into
subgraphs as indicated in Figure~\ref{fig:partitioned}, where each $G_i$
is a subgraph $C_n\square P_{m_i}$.
Let $S$ be a
dominating set for $G$ and $S_i=S\cap V(G_i)$. Then
\begin{equation}\label{eq:fundamental inequality}
w(S)\ge \sum_{i=1}^t w(S_i).
\end{equation}
Note that in computing $w(S_i)$ we consider $S_i$ to be a subset of $V(G)$,
not of $V(G_i)$ (this affects the computation of $N[S_i]$). To verify
the inequality, note that the following inequalities are equivalent:
\begin{align*}
w(S)&\ge \sum_{i=1}^t w(S_i)\\
5|S|-|N[S]|&\ge \sum_{i=1}^t (5|S_i|-|N[S_i]|)\\
5|S|-|N[S]|&\ge \sum_{i=1}^t 5|S_i|-\sum_{i=1}^t |N[S_i]|\\
|N[S]| &\le \sum_{i=1}^t |N[S_i]|.
\end{align*}
The last inequality is satisfied, since each vertex in $N[S]$ is
counted at least once by the expression on the right.
Note that $S_i$ is a set that dominates all the vertices of $G_i$
except possibly some vertices in the top or bottom row of $G_i$ (or in
the cases of $G_1$ and $G_t$, in the bottom row and top row,
respectively). Let us say that a set that dominates a
cylinder $G$, with the exception of some vertices on the
top or bottom edges, {\it almost dominates\/} $G$. Given a
cylinder $H=C_n\square P_{m_i}$ (namely, one of the $G_i$),
What we want to know is the value of
\begin{equation}\label{eq:desired minimum}
\min_A w(A),
\end{equation}
taking the minimum over sets $A$ that almost dominate
$H$ and computing $w(A)$ as if $A$ were a subset of a larger graph
$C_n\square P_{m_i+2}$ in which $H$ occupies the middle $m_i$ rows, or in
the case of $G_1$ or $G_t$, $A$ is a subset of $C_n\square P_{m_i+1}$ in
which $H$ occupies the top $m_i$ rows. If we can compute this minimum
for (small) fixed $m_i$ and any $n$, we can choose $G_1$ through $G_t$
with a small number of rows and get lower bounds on $w(S_i)$ for any
dominating set $S$ of the original $C_n\square P_m$.
\section{The algorithm}
We describe the algorithm for $G_1$ and $G_t$ (which of course are
isomorphic); the algorithm for the other graphs $G_i$ is nearly
identical, and we describe it more briefly. Imagine a
cylinder $C_n\square P_m$ with a designated subset $S$
of the vertices. Recall that the vertices are denoted by $v_{i,j}$,
$1\le i\le n$, $1\le j\le m$ (say, numbering left to right and bottom
to top). We describe a column, say column number $i$, in such a
diagram by a state vector $\bf s$, in which ${\bf s}_j$ is $0$ if
vertex $v_{i,j}$ is in $S$, $1$ if vertex $v_{i,j}$ is adjacent to a
member of $S$ in column $i$ or column $i-1$, and $2$ otherwise. For
example, the second column from the right in Figure~\ref{fig:12x10}
has state vector $(1,1,0,1,2,1,1,0,1,2,1,0)$. Let $|{\bf s}|$ denote
the number of zeros in $\bf s$.
Given a state vector $\bf s$, we append a column 0 at the left of
$P_n\square P_m$. Let $X$ be the set of vertices in this column
corresponding to the 0 entries in $\bf s$, and let $Y$ be the set of
vertices corresponding to the 2 entries in $\bf s$. An
{\it $({\bf s,t})$-almost-domination\/} of $P_n\square P_m$ is a subset $S$ of
the vertices such that $X\cup S$ dominates the first $n-1$ columns of
$P_n\square P_m$ and the elements of $Y$,
except possibly vertices in the first (i.e., bottom)
row, and for which the state vector of the final column is $\bf t$.
Suppose $S$ is a subset of the vertices of $P_i\square P_j$ and denote
by $w_{i,j}(S)$ the value of $w(S)$ computed in $P_{i+1}\square
P_{j+1}$, in which $P_i\square P_j$ occupies the top $j$ rows and
leftmost $i$ columns. Let
$$w_{i,j}({\bf s,t})=\min_S w_{i,j}(S),$$
taking the minimum over all $({\bf s,t})$-almost-dominations of
$P_i\square P_j$.
If there is no $({\bf s,t})$-almost-domination of
$G_{i,j}$, let $w_{i,j}({\bf s})=\infty$.
Finally, to compute the desired minimum (equation~\ref{eq:desired
minimum}), we compute
$$\min_{\bf s} w_{i,j}({\bf s,s}),$$
since an $({\bf s,s})$-almost-domination of $P_i\square P_j$ almost
dominates $C_i\square P_j$.
Let ${\cal P}({\bf t})$ be the set of state vectors $\bf u$ such that $\bf
u$ is the state vector of the next to last column in an
$({\bf s,t})$-almost-domination of $P_n\square P_m$.
Then
$$w_{n,m}({\bf s,t})=\min_{{\bf u}\in{\cal P}({\bf t})}
\left( 5|{\bf t}| - \mathop{\hbox{\rm nd}}({\bf u},{\bf t})+w_{n-1,m}({\bf s,u}) \right),
$$
where $\mathop{\hbox{\rm nd}}({\bf u},{\bf t})$, the number of newly dominated vertices,
may be computed as follows.
\newcounter{ctr}
\begin{list}{\arabic{ctr}.}{\usecounter{ctr}}
\item $\mathop{\hbox{\rm nd}} = 0$
\item For each $j=1,\dots,m$ for which ${\bf t}_j=0$ and
${\bf u}_j=2$, add 1 to $\mathop{\hbox{\rm nd}}$. This counts the newly dominated vertices
$v_{n-1,j}$.
\item For each $j=1,\dots,m$ for which ${\bf t}_j\le 1$ and
${\bf u}_j\ge 1$, add 1 to $\mathop{\hbox{\rm nd}}$. This counts the newly
dominated vertices $v_{n,j}$.
\item For each $j=1,\dots,m$ for which ${\bf t}_j=0$, add 1 to
$\mathop{\hbox{\rm nd}}$. This counts the newly dominated vertices $v_{n+1,j}$.
\item If ${\bf t}_1=0$, add 1 to $\mathop{\hbox{\rm nd}}$. This counts the newly
dominated vertex below vertex $v_{n,1}$, recalling that we compute
$w(S)$ in $P_n\square P_m$ with an extra bottom row.
\end{list}
\noindent
Now, given some $n$, the algorithm to compute $w_{n,m}({\bf s},{\bf
t})$, $i=1,\dots,n$, is:
\begin{list}{\arabic{ctr}.}{\usecounter{ctr}}
\item {\bf Initialization.} Set $w_{0,m}({\bf s},{\bf u})=0$ if
${\bf u}= {\bf s}$, and $\infty$ otherwise.
\item {\bf Iteration.} Suppose that $i\le n$ and that
$w_{i-1,m}({\bf s},{\bf u})$ has been computed for all ${\bf u}$.
Then for each $\bf t$, set
$$ w_{i,m}({\bf s},{\bf t})=\min_{{\bf u}\in{\cal P}({\bf t})}
\left( 5|{\bf t}| - \mathop{\hbox{\rm nd}}({\bf u},{\bf t})+
w_{i-1,m}({\bf s},{\bf u}) \right).
$$
\end{list}
Thus, for fixed $m$ and any $n$, we can compute $\min_{\bf s}
w_{n,m}({\bf s},{\bf s})$, by computing $w_{i,m}({\bf s},{\bf t})$ for
all ${\bf s}$, ${\bf t}$, and $1\le i\le n$.
Of course, what we want is to know this
value for any $n$ without an infinite amount of work. Livingston and
Stout~\cite{ls:constant-time-dominating-sets} and
Fisher~\cite{dcf:domination-grid-graphs} independently thought of
looking for a sort of periodicity in the values of $\gamma(P_n\square
P_m)$ for fixed $m$. Since they succeeded, we might hope that for
fixed $m$, there are $N$, $p$, and $q$ so that for $n\ge N$ and all
$\bf s$ and $\bf t$,
$$w_{n,m}({\bf s},{\bf t})=w_{n-p,m}({\bf s},{\bf t} )+q.$$
In this case, after a finite amount of computation, we could determine
$\min_{\bf s} w_{n,m}({\bf s},{\bf s})$ for all $n$.
It is easy to modify the algorithm so to check for this
periodicity. When we do this, we find that for $n\ge 65$,
$$\min_{\bf s} w_{n,10}({\bf s},{\bf s}) =
\min_{\bf s} w_{n-1,10}({\bf s},{\bf s})+1 =
n.$$
Thus, for $m\ge 20$ and $n\ge 64$, if $S$ is a dominating set
in $C_n\square P_m$,
$$w(S)\ge \sum_{k=1}^t w(S_k) \ge w(S_1)+w(S_t) \ge 2n,$$
using the inequality (\ref{eq:fundamental inequality}), and so
$$|S|\ge (mn + 2n)/5.$$
Jos\'e Juan Carre{\~n}o
et al.\cite{carreno-et-al:lower-bound-dom-no-cyl-graphs} have independently
arrived at the same conclusion, using a substantially different
algorithm. When $n\cong 0\pmod{5}$, $(mn + 2n)/5$ is also known to be
an upper bound, so that $\gamma(C_n\square P_m)=(mn + 2n)/5$ (in fact,
this is known to be correct for $n\ge 5$).
The
implication, of course, is that for optimal $S$,
$w(S_k)=0$ for $1<k<t$, when $n\cong 0\pmod{5}$.
This is not true in general, so we improve our lower bound by
computing a lower bound on $w(S_k)$, $1<k<t$.
The only change required is to redefine an $({\bf s,t})$-almost-domination as follows:
Given a state vector $\bf s$, we append a column 0 at the left of
$P_n\square P_m$. Let $X$ be the set of vertices in this column
corresponding to the 0 entries in $\bf s$, and let $Y$ be the set of
vertices corresponding to the 2 entries in $\bf s$. An
$({\bf s,t})$-almost-domination of $P_n\square P_m$ is a subset $S$ of
the vertices such that $X\cup S$ dominates the first $n-1$ columns of
$P_n\square P_m$ and the elements of $Y$,
except possibly vertices in the top and bottom rows,
and for which the state vector of the final column is $\bf t$.
Corresponding to this change, in the computation of $nd$, we add a
sixth step:
\begin{list}{\arabic{ctr}.}{\usecounter{ctr}}{\setcounter{ctr}{5}}
\item If ${\bf t}_m=0$, add 1 to $\mathop{\hbox{\rm nd}}$. This counts the newly dominated
vertex above $v_{n,m}$, recalling that we compute $w(S)$ in
$C_n\square P_m$ as if it occupies the middle $m$ rows of a copy of
$C_n\square P_{m+2}$.
\end{list}
Proceeding as before, we find that $\min_{\bf s} w_{n,10} ({\bf s},{\bf s})
=\min_{\bf s} w_{n-5,10} ({\bf s},{\bf s})$, when $n\ge
12$. Specifically, we find that
$\min_{\bf s} w_{n,10}({\bf s},{\bf s})$ is 0, 6, 5, 9,
or 6 as $n$ is 0, 1, 2, 3, or 4 $\pmod 5$.
Thus, with $a$ equal to 0, 6, 5, 9, or 6 as appropriate, we find that
$$|S|\ge {1\over 5}((m+2)n+\lfloor {m-20\over 10}\rfloor\cdot a).$$
That is, lower bounds for the domination number of $C_n\square P_m$,
when $m\ge 20$ and $n\ge 64$,
are:
\begin{align*}
{(m+2)n\over 5},&\qquad n\equiv 0\pmod 5\\
{(m+2)n\over 5}+{6\over 5}\lfloor {m-20\over 10}\rfloor,&\qquad n\equiv 1\pmod 5\\
{(m+2)n\over 5}+\lfloor {m-20\over 10}\rfloor,&\qquad n\equiv 2\pmod 5\\
{(m+2)n\over 5}+{9\over 5}\lfloor {m-20\over 10}\rfloor,&\qquad n\equiv 3\pmod 5\\
{(m+2)n\over 5}+{6\over 5}\lfloor {m-20\over 10}\rfloor,&\qquad n\equiv 4\pmod 5.
\end{align*}
Known upper bounds (see \cite{pavlic-zerovnik:dom-no-cyl-graphs})
for the domination number of $C_n\square P_m$ are:
\begin{align*}
{(m+2)n\over 5},&\qquad n\equiv 0\pmod 5\\
{(m+2)n\over 5}+{7\over 40}(m+2),&\qquad n\equiv 1\pmod 5\\
{(m+2)n\over 5}+{1\over 10}(m+2),&\qquad n\equiv 2\pmod 5\\
{(m+2)n\over 5}+{2\over 5}(m+2),&\qquad n\equiv 3\pmod 5\\
{(m+2)n\over 5}+{1\over 5}(m+2),&\qquad n\equiv 4\pmod 5.
\end{align*}
For $n\equiv 2\pmod 5$ the lower and upper bounds are quite close, but
for the other non-zero values of $n\bmod 5$ there is considerable room
for improvement. It seems likely that the upper bounds are closer to
the true values, as our computation allows vertices on the boundary
(that is, the top and bottom rows) of
the subgraphs $G_k$ to remain undominated. A small increase in the
value of $a$ in each case would eliminate most of the gap.
When $m\bmod 10$ is non-zero, we have effectively ignored one of the
$G_i$, that is, used zero as a lower bound for one of the $w(S_i)$.
We can improve our lower bounds very slightly (by a small constant) by
correcting this. For example, for $m>20$ and
$m\equiv 8\pmod{10}$, we can let all
but one of the $G_i$ have height 10, and the remaining (interior)
graph, say $G_2$, have height 8. Then we run the algorithm again for
height 8 graphs. While we have in fact done the additional
computations, the improvement is very slight, so we omit the results.
Our approach gives us lower bounds for $m\ge 20$;
Crevals~\cite{crevals:domination_cylinder_graphs} computes exact
values for $m\le 22$ and all $n$. He also computes exact values for
$n\le 30$ and all $m$. In the course of our computations, we also
obtain lower bounds for $12\le n<64$ (with only $n>30$ of interest due
to the Crevals results), but they do not seem sufficiently
illuminating to include here.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,777 |
Gooch Palms' L. M. Queen drops debut solo single 'S.M.A.'
The Gooch Palms announced they will be splitting last month
By Jackson Langford
Credit: Press
It's been just under a month since Gooch Palms announced that the band was ending, and frontman Leroy MacQueen has already emerged with his debut solo single 'S.M.A.'
My debut single S.M.A. is out today 🥀 head to link in bio to listen/buy 🖤 @ Melbourne, Victoria, Australia https://t.co/4fzWtvHJnh
— L.M. QUEEN (@l_m_queen) April 7, 2020
Under the stage name L.M. Queen, 'S.M.A' – which stands for Saddest Man Alive – sees MacQueen take more of a dark synth-pop direction as opposed to the garage rock he became known for with Gooch Palms. Listen to it below.
Gooch Palms announced that they will be calling it quits at the beginning of last month, after a decade of making music together.
"Playing in the band forced us out of our introverted shells into the world and through touring we have had so many unimaginably wonderful moments and met some great people, a lot of whom we now call our closest friends," they wrote in a statement.
"We've both had multiple nervous breakdowns and experienced some devastating lows, but thankfully, overall, mostly dizzying highs! But to quote Kenny Rogers 'You've got to know when to hold 'em, know when to fold 'em'. And it's time for us to fold 'em."
The band had planned to embark on a final tour together throughout March, April and May, but they were forced to postpone due to the coronavirus outbreak.
L. M. Queen
The Gooch Palms | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,131 |
Q: Aggregating test results for multiple upstream jobs I have a parametrized Jenkins job that gets executed every time any of the upstream jobs have a stable build; where the number of upstream jobs can vary, but at least there is always one.
(From here on, I will refer to an upstream job as upstream and to the downstream job as downstream)
Graphically, it's something like this:
Upstream_1 ... Upstream_N
| |
| |
\ /
\ /
\ /
| |
Downstream_Parametrized_Job
downstream executes in a special environment the unit tests for the upstream that triggered the execution (all upstreams must be tested in this specific environment)
I am able to aggregate the tests results in upstream for a specific job, but I'm having a hard time to generalize this for any number of upstream jobs.
For a single job, I've followed this solution, specifying in downstream the name of the upstream where to copy the fingerprinted artifact.
But this is a problem, since I can have several upstreams and I can't specify all the names directly.
Is there a way to tell Jenkins to take the artifact from the specific upstream that triggered the job, without specifying a name?
So, in the picture, instead of typing UPSTREAM_1, would it be possible to use an environment variable, something like ${UPSTREAM_JOB_NAME}?
(here, I don't see any built-in variable like that, but I wouldn't be surprised if it weren't documented)
A: The Copy Artifact Plugin's page has a section:
Specifying project dynamically (with variables)
Define a parameter UPSTREAM_JOB_NAME in the downstream and set it accordingly when triggering from the upstream.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,537 |
It's crunch time in election season, and in Florida, people on both sides of the medical marijuana issue Floridians will vote on in November are ramping up their efforts. As Matt Galka tells us, a strong voice in the Florida legislature is saying no.
Pinellas County Senator Jack Latvala wants to make sure everyone knows he's against the Medical Marijuana Amendment 2 measure on the ballot in November.
"I don't want candy like this to be out in candy dishes and people's homes where children can take advantage of it. I just see too many opportunities for abuse," he said.
The Senator – who is set to be that chamber's main money man – joined former Florida Supreme Court Justice Kenneth Bell in opposing putting pot in the state's constitution.
"You create a fundamental right but the legislature is very constrained in the ability to deal with unintended consequences," said Bell.
Sen. Latvala also launched an ad urging people to vote "no" on the amendment.
But supporters are pushing back. We reached Ben Pollara via Skype. He's leading United for Care's push for the Amendment.
"Most people who would benefit under medical marijuana under amendment 2 simply don't have access to it at all under current law," he said.
St. Petersburg Senator Jeff Brandes also chimed in. He supports the amendment, and doesn't agree the legislature has done enough so far.
"Now that the legislature has declared and the Governor has agreed that it has medical properties, I think we should allow doctors to practice medicine," said Sen. Brandes.
Latvala's ad cost him $100,000, but Yes on 2 people are optimistic it will be too little, too late.
A recent Florida Chamber of Commerce poll showed nearly 75 percent of Floridians are in support of the medical marijuana amendment. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,448 |
An agricultural cooperative, also known as a farmers' co-op, is a cooperative in which farmers pool their resources in certain areas of activity.
A broad typology of agricultural cooperatives distinguishes between agricultural service cooperatives, which provide various services to their individually-farming members, and agricultural production cooperatives in which production resources (land, machinery) are pooled and members farm jointly.
Examples of agricultural production cooperatives include collective farms in former socialist countries, the kibbutzim in Israel, collectively-governed community shared agriculture, Longo Maï co-operatives and Nicaraguan production co-operatives.
The default meaning of "agricultural cooperative" in English is usually an agricultural service cooperative, the numerically dominant form in the world. There are two primary types of agricultural service cooperatives: supply cooperatives and marketing cooperatives. Supply cooperatives supply their members with inputs for agricultural production, including seeds, fertilizers, fuel, and machinery services. Marketing cooperatives are established by farmers to undertake transportation, packaging, pricing, distribution, sales and promotion of farm products (both crop and livestock). Farmers also widely rely on credit cooperatives as a source of financing for both working capital and investments.
Purpose
Cooperatives as a form of business organization are distinct from the more common investor-owned firms (IOFs). Both are organized as corporations, but IOFs pursue profit maximization objectives, whereas cooperatives strive to maximize the benefits they generate for their members (which usually involves zero-profit operation). Agricultural cooperatives are therefore created in situations where farmers cannot obtain essential services from IOFs (because the provision of these services is judged to be unprofitable by the IOFs), or when IOFs provide the services at disadvantageous terms to the farmers (i.e., the services are available, but the profit-motivated prices are too high for the farmers). The former situations are characterized in economic theory as market failure or missing services motive. The latter drive the creation of cooperatives as a competitive yardstick or as a means of allowing farmers to build countervailing market power to oppose the IOFs. The concept of competitive yardstick implies that farmers, faced with unsatisfactory performance by IOFs, may form a cooperative firm whose purpose is to force the IOFs, through competition, to improve their service to farmers.
A practical motivation for the creation of agricultural cooperatives is related to the ability of farmers to pool production and/or resources. In many situations within agriculture, it is simply too expensive for farmers to manufacture products or undertake a service. Cooperatives provide a method for farmers to join in an 'association', through which a group of farmers can acquire a better outcome, typically financial, than by going alone. This approach is aligned to the concept of economies of scale and can also be related as a form of economic synergy, where "two or more agents working together to produce a result not obtainable by any of the agents independently". While it may seem reasonable to conclude that larger the cooperative the better, this is not necessarily true. Cooperatives exist across a broad membership base, with some cooperatives having fewer than 20 members while others can have over 10,000.
While the economic benefits are a strong driver in forming cooperatives, it is not the sole consideration. In fact, it is possible for the economic benefits from a cooperative to be replicated in other organisational forms, such as an IOF. An important strength of a cooperative for the farmer is that they retain the governance of the association, thereby ensuring they have ultimate ownership and control. This ensures that the profit reimbursement (either through the dividend payout or rebate) is shared only amongst the farmer members, rather than shareholders as in an IOF.
As agricultural production is often the main source of employment and income in rural and impoverished areas, agricultural cooperatives play an instrumental role in socio-economic development, food security and poverty reduction. They provide smallholder farmers with access to natural and educational resources, tools, and otherwise inaccessible marketplaces. Producer organisations can also empower smallholders to become more resilient; in other words, they build the capacity of farmers to prepare for and react to economic and environmental stressors and shocks in a way that limits vulnerability and promotes their sustainability. Research suggests that membership in a producer organisation is more highly correlated with farmer output or income than other standalone investments such as training, certification, or credit.
In agriculture, there are broadly three types of cooperatives: a machinery pool, a manufacturing/marketing cooperative, and a credit union.
Machinery pool: A family farm may be too small to justify the purchase of expensive farm machinery, which may be only used irregularly, say only during harvest; instead local farmers may get together to form a machinery pool that purchases the necessary equipment for all the members to use.
Manufacturing/marketing cooperative: A farm does not always have the means of transportation necessary for delivering its produce to the market, or else the small volume of its production may put it in an unfavorable negotiating position with respect to intermediaries and wholesalers; a cooperative will act as an integrator, collecting the output from members, sometimes undertaking manufacturing, and delivering it in large aggregated quantities downstream through the marketing channels.
Credit Union: Farmers, especially in developing countries, can be charged relatively high interest rates by commercial banks, or credit may not even be available for farmers to access. When providing loans, these banks are often mindful of high transaction costs on small loans, or may refuse credit altogether due to lack of collateral – something very acute in developing countries. To provide a source of credit, farmers can group together funds that can be loaned out to members. Alternatively, the credit union can raise loans at better rates from commercial banks due to the cooperative having a larger associative size than an individual farmer. Often members of a credit union will provide mutual or peer-pressure guarantees for repayment of loans. In some instances, manufacturing/marketing cooperatives may have credit unions as part of their broader business. Such an approach allows farmers to have a more direct access to critical farm inputs, such as seeds and implements. The loans for these inputs are repaid when the farmer sends produce to the manufacturing/marketing cooperative.
Origins and history
The first agricultural cooperatives were created in Europe in the seventeenth century in the Military Frontier, where the wives and children of the border guards lived together in organized agricultural cooperatives next to a funfair and a public bath.
The first civil agricultural cooperatives were created also in Europe in the second half of the nineteenth century. They spread later to North America and the other continents. They have become one of the tools of agricultural development in emerging countries.
Farmers also cooperated to form mutual farm insurance societies.
Also related are rural credit unions.
They were created in the same periods, with the initial purpose of offering farm loans.
Some became universal banks such as Crédit Agricole or Rabobank.
Supply cooperatives
Agricultural supply cooperatives aggregate purchases, storage, and distribution of farm inputs for their members. By taking advantage of volume discounts and utilizing other economies of scale, supply cooperatives bring down the cost of the inputs that the members purchase from the cooperative compared with direct purchases from commercial suppliers. Supply cooperatives provide inputs required for agricultural production including seeds, fertilizers, chemicals, fuel, and farm machinery. Some supply cooperatives operate machinery pools that provide mechanical field services (e.g., plowing, harvesting) to their members.
Examples
Australia
Co-operative Bulk Handling Limited
Westralian Farmers Co‐operative Limited
Canada
Farmers' Storehouse Company
United Farmers of Alberta
Farmers of North America
France
Agrial (Normandy)
Terrena (Pays de la Loire)
Vivescia
Israel
Granot central cooperative
Japan
Japan Agricultural Cooperatives
Ukraine
Ukrainian cooperative movement
United States
Landisville Produce Co-op, established 1914
Rockingham Cooperative, established in 1921
MFA Incorporated
Darigold
Organic Valley
National Council of Farmer Cooperatives
Southern States Cooperative
Farmers Cooperative Association, Inc.; Frederick, Maryland
Ocean Spray (cooperative)
Land O'Lakes
Michigan Sugar
Sunkist
Wilco stores (Oregon)
Grange Cooperative
Netherlands
Avebe
Agrico
Agrifirm
Marketing cooperatives
Agricultural marketing cooperatives are cooperative businesses owned by farmers, to undertake transformation, packaging, distribution, and marketing of farm products (both crop and livestock.)
New Zealand
New Zealand has a strong history of agricultural cooperatives, dating back to the late 19th century. The first was the small Otago Peninsula Co-operative Cheese Factory Co. Ltd, started in 1871 at Highcliff on the Otago Peninsula. With active support by the New Zealand government, and small cooperatives being suitable in isolated areas, cooperatives quickly began to dominate the industry. By 1905, dairy cooperatives were the main organisational structure in the industry. In the 1920s–'30s, there were around 500 co-operative dairy companies compared to less than 70 that were privately owned.
However, after World War II, with the advent of improved transportation, processing technologies and energy systems, a trend to merge dairy cooperatives occurred. By the late 1990s, there were two major cooperatives: the Waikato-based New Zealand Dairy Group and the Taranaki-based Kiwi Co-operative Dairies. In 2001 these two cooperatives, together with the New Zealand Dairy Board, merged to form Fonterra. This mega-merger was supported by the New Zealand Government as part of broader dairy industry deregulation, which allowed other companies to directly export dairy products. Two smaller cooperatives did not join Fonterra, preferring to remain independent – the Morrinsville-based Tatua Dairy Company and Westland Milk Products on the West Coast of the South Island.
The other main agricultural co-operatives in New Zealand are in the meat and fertiliser industries. The meat industry, which has struggled at times, has proposed various mergers similar to the creation of Fonterra; however, these have failed to gain the necessary member support.
Canada
In Canada, the most important cooperatives of this kind were the wheat pools. These farmer-owned cooperatives bought and transported grain throughout Western Canada. They replaced the earlier privately and often foreign-owned grain buyers and came to dominate the market in the post-war period. By the 1990s, most had demutualized (privatized), and several mergers occurred. Now all the former wheat pools are part of the Viterra corporation.
Former wheat pools include:
Alberta Wheat Pool
Manitoba Pool Elevators
Saskatchewan Wheat Pool
United Grain Growers
Other agricultural marketing cooperatives in Canada include:
Organic Meadow Cooperative (organic dairy)
Gay Lea Foods Co-operative Limited (dairy)
Agropur
Ecuador
The Amazon region of Ecuador is known for producing world-renowned cacao beans. In the Napo region 850 Kichwa families have come together with help from American biologist, Judy Logback, to form an agricultural marketing cooperatives, Kallari Association. This cooperative has helped increase benefits for the families involved as well as to protect and defend their Kichwa culture and the Amazon rainforest.
India
In India, there are networks of cooperatives at the local, regional, state and national levels that assist in agricultural marketing. The commodities that are mostly handled are food grains, jute, cotton, sugar, milk and nuts
Dairy farming based on the Anand Pattern, with a single marketing cooperative, is India's largest self-sustaining industry and its largest rural employment provider. Successful implementation of the Anand model has made India the world's largest milk producer. Here small, marginal farmers with a couple or so heads of milch cattle queue up twice daily to pour milk from their small containers into the village union collection points. The milk after processing at the district unions is then marketed by the state cooperative federation nationally under the Amul brand name, India's largest food brand. With the Anand pattern three-fourths of the price paid by the mainly urban consumers goes into the hands of millions of small dairy farmers, who are the owners of the brand and the cooperative. The cooperative hires professionals for their expertise and skills and uses hi-tech research labs and modern processing plants & transport cold-chains, to ensure quality of their produce and value-add to the milk.
Production of sugar from sugarcane mostly takes place at cooperative sugar cane mills owned by local farmers. The shareholders include all farmers, small and large, supplying sugarcane to the mill. Over the last sixty years, the local sugar mills have played a crucial part in encouraging rural political participation and as a stepping stone for aspiring politicians. This is particularly true in the state of Maharashtra where a large number of politicians belonging to the Congress party or NCP had ties to sugar cooperatives from their respective local areas. Unfortunately, mismanagement and manipulation of the cooperative principles have made a number of these operations inefficient.
Israel
Tnuva Central Cooperative for the Marketing of Agricultural Produce in Israel Ltd.
Netherlands
Coöperatieve Nederlandse Bloembollencentrale (CNB)
Coforta
Royal Cosun
ZON
FloraHolland
FrieslandCampina
Ukraine
Ukrainian cooperative movements
United States
American Legend Cooperative (mink fur) "Blackglama" brand
Blue Diamond Growers (almonds)
Cabot Creamery (dairy)
Darigold
Diamond of California (nuts), formerly a cooperative
Dairylea Cooperative Inc. (Dairy), formerly Dairymen's League
Dairy Farmers of America
Edible Garden
Florida's Natural Growers (citrus fruit)
GreenStone Farm Credit Services [financial products and services]
Humboldt Creamery (dairy), formerly a cooperative
Land O'Lakes (dairy and farm supply)
Maine's Own Organic Milk Company (dairy)
Michigan Milk Producers Association (dairy)
Michigan Sugar Company (sugar beets)
Ocean Spray (cranberries and citrus fruit)
Organic Valley (organic milk, cheese, eggs, soy, butter, yogurt, snack items)
Riceland Foods (rice, soybeans, corn and wheat)
Snokist Growers (pears, apples, cherries)
Sunkist Growers, Incorporated (citrus fruit)
Sun-Maid (raisins)
Sunsweet Growers Incorporated (dried fruit, especially prunes)
Tillamook County Creamery Association (dairy)
Lone Star Milk Producers (dairy)
United Egg Producers
Welch Foods Inc. (Welch's)
Mexico
Zapatista coffee cooperatives
Production cooperatives
These are cooperative farms, jointly owned or managed by a cooperative society.
Cuba
See also
Winemaking cooperative
References
Further reading
McBride, Glynn (2014), Agricultural Cooperatives: Their Why and Their How
Derr, Jascha (2013), The cooperative movement of Brazil and South Africa
Zvi Galor
Cooperative
Agricultural production
Rural community development
Cooperative | {
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\def\Secref#1{Section~\ref{#1}}
\def\Figref#1{Fig.~\ref{#1}}
\def\Ref#1{Ref.~\cite{#1}}
\newcommand*\chem[1]{\ensuremath{\mathrm{#1}}}
\graphicspath{{Figures/}}
\begin{document}
\sf
\coloredtitle{Deriving
approximate functionals with asymptotics}
\author{\color{CITECOL} Kieron Burke}
\affiliation{Departments of Physics and Astronomy and of Chemistry,
University of California, Irvine, CA 92697, USA}
\date{\today}
\begin{abstract}
Modern density functional approximations achieve moderate accuracy at low
computational cost for many electronic structure calculations.
Some background is given relating the gradient expansion of density functional
theory to the WKB expansion in one dimension, and modern approaches
to asymptotic expansions.
A mathematical framework for analyzing asymptotic behavior for the
sums of energies unites both
corrections to the gradient expansion of DFT and hyperasymptotics of sums.
Simple examples are given for the model problem of orbital-free DFT in one
dimension. In some cases, errors can be made as small as 10$^{-32}$ Hartree
suggesting that, if these new ingredients can be applied, they might
produce approximate functionals that are much more accurate
than those in current use.
A variation of the Euler-Maclaurin formula generalizes
previous results.
\end{abstract}
\maketitle
\coltableofcontents
\def\text{floor} #1{{\lfloor}#1{\rfloor}}
\def\sm#1{{\langle}#1{\rangle}}
\def_{disc}{_{disc}}
\newcommand{\mathbb{Z}}{\mathbb{Z}}
\newcommand{\mathbb{R}}{\mathbb{R}}
\def^{\rm WKB}{^{(0)}}
\def^{\rm WKB}{^{\rm WKB}}
\def^{\rm II}{^{\rm II}}
\def\hd#1{\noindent{\bf\textcolor{red} {#1:}}}
\def\hb#1{\noindent{\bf\textcolor{blue} {#1:}}}
\def\epsilon{\epsilon}
\def\epsilon\w{\epsilon^{\rm WKB}}
\def\epsilon_j{\epsilon_j}
\def^{(\eta)}{^{(\eta)}}
\def\ej\upet{\epsilon_j^{(\eta)}}
\deft_j\upet{t_j^{(\eta)}}
\def{\bar \epsilon}_j{{\bar \epsilon}_j}
\def\epsilon\w_j{\epsilon^{\rm WKB}_j}
\deft_j{t_j}
\defv_j{v_j}
\def_{\sss F}{_{\scriptscriptstyle\rm F}}
\defx_{\sss T}{x_{\scriptscriptstyle\rm T}}
\def^{\rm sc}{^{\rm sc}}
\def\alpha{\alpha}
\def\al_e{\alpha_e}
\def\bar j{\bar j}
\def\bar\zeta{\bar\zeta}
\def\eq#1{Eq.\, (\ref{#1})}
\def{\cal N}{{\cal N}}
\sec{Introduction}
\label{intro}
Kohn-Sham density functional theory (KS-DFT) is a very popular electronic
structure method, being used in tens of thousands of papers each year\cite{PGB16}.
However, all such calculations use some approximation to the unknown
exchange-correlation functional of the (spin) densities\cite{KS65}, and most
standard codes allow choices among hundreds (or more) of different
approximations\cite{ED04}, belying claims of a first-principles theory.
There is an {\em exact theory} of DFT exchange-correlation\cite{DG90}, which
is well-developed,
but logically subtle. This theory shows which properties the exact functional
must have, and which it does not. Exact conditions are then often
used to determine parameters in approximations\cite{PCVJ92}. This exact theory
is crucially important in understanding DFT\cite{BW13}, but is not the subject
of this paper.
In elementary quantum mechanics\cite{Gb05}, a standard set of tools is particularly
useful for approximations, such as the variational principle, expansion in
a basis, and perturbation theory. These are used extensively in
traditional {\em ab initio} quantum chemistry\cite{SO82}. In particular, the
repulsion between electrons is considered as weak, and Hartree-Fock
is the starting point of most methods\cite{F30,H35}. Most important, in almost
all treatments, a series of equations can be derived of increasing
computational cost to evaluate which (in the case of convergence) will
yield increasingly accurate results. Similar approaches centering on
the Green's function have been highly successful for calculating responses
of materials, but much less so when used to find ground-state energies\cite{MRC16}.
No such procedure currently exists for density functional theory.
We show here (and in earlier work) that in fact the corresponding
chapter in elementary quantum mechanics is simply that dealing
with semiclassical approximations. In theoretical chemistry, such
methods were tried long ago in
electronic stucture (e.g., Refs \cite{M68b,M68c,CKM86}), but are now
more commonly applied
to treating nuclear motion in quantum dynamics\cite{MF97}.
Their exploration for electronic structure withered once modern self-consistent
approximations\cite{S51} could be implemented numerically
with reasonable accuracy.
That this is the unique
perspective from which density functional approximations can
be understood begins with the work of Lieb and
Simon from 1973\cite{LS73,LS77}. Their
work ultimately shows that, for any atom, molecule,
or solid, the relative error in a total energy in a TF calculation must
vanish under a very precise scaling to a high-density, large particle
number limit. In this limit, the system is weakly correlated, semiclassical,
and mean-field theory dominates\cite{L76,L81}. This has been argued to
be true also in KS-DFT for the XC energy\cite{C83,EB09,CCKB18}.
The gradient expansion is the starting
place for most modern approximations in DFT (generalized
gradient expansions\cite{B88,LYP88,PBE96}, and is used in some form in most calculations today.
The first asymptotic correction to the local density approximation for densities that are
slowly varying\cite{LP77,CCKB18} is that of the
gradient expansion. But a recent paper (hereafter called A\cite{B20}) used an unusual
construction to find the leading correction to the local approximation more generally,
i.e., for finite systems with turning points, for the kinetic energy in one dimension.
It was found that, for such finite systems, the gradient expansion misses a vital contribution,
without which it is much less accurate.
The work of A focuses on the leading corrections to the local approximation. But these
are just the first corrections in an asymptotic series that, in principle, could be
usefully extended to much higher orders.
Recently, asymptotic methods were developed for sums over eigenvalues for bound
potentials\cite{BB20}, hereafter called B.
In a simple case, $v(x)=x$ in the half-space $x>0$ in Hartree atomic units,
the sum over the
first 10 eigenvalues was found to be about 81.5 Hartrees, with
an error of about 10$^{-32}$ Hartree. This extreme accuracy is far beyond any current
computational methods for solving the Schr\"odinger equation.
The way in which this accuracy was achieved employed methods rarely used in
modern electronic structure calculations, involving hyperasymptotics\cite{BH93}. Such methods
are difficult to generalize, and often are applied only to very simple, shape-invariant potentials\cite{CKS95},
where specific formulas for the $M$-th order contribution to an asymptotic expansion
can be found explicitly. It would be wonderful if even a tiny fraction of this powerful
methodlogy could be applied to modern electron structure calculations.
The present work is designed as a further step toward this ultimate goal,
as well as a summary of previous work in this direction.
Section \ref{back} summarizes background material from several different
fields. Section \ref{theory} lays out a general approach to
summations using the Euler-Maclaurin formula, and shows how
the summation techniques of A and
B are special cases of a this general summation formula. That
formula yields the key results in both
A and B, and extends each beyond its original domain of applicability.
I also find a variation that produces the results of A and B simultaneously
and clearly identifies the role of the Maslov index.
I close with a discussion of the relevance of this work to realistic
electronic structure calculations in Section \ref{DFT}.
\sec{Background}
\label{back}
\ssec{Asymptotics}
\label{asymp}
We begin with some simple points about asymptotic expansions, which we illustrate
using the Airy function\cite{A38,DLMF,VS04}. Consider an infinite sequence
of coefficients ${c_n}$, and
the partial sums
\begin{equation}
S_M(x) = \sum_{n=0}^M \frac{c_n}{x^n}.
\label{SM}
\end{equation}
Consider then $R_M(x)=x^M(S_M(x)-f(x))$. If
\begin{equation}
\lim_{x\to\infty} R_M(x)=0,~~~\lim_{M\to\infty} R_M(x)=\infty
\label{lim}
\end{equation}
then $S_M(x)$ is the asymptotic expansion of $f(x)$ as $x\to\infty$, and we write
\begin{equation}
f(x)\approx S_\infty (x)
\end{equation}
Some important well-known points are that the $c_j$, if they exist, are unique, but infinitely
many different functions have the same asymptotic expansion\cite{C08}.
We shall say that $S_M(x)$ is the $M$-th order asymptotic expansion of $f(x)$.
\begin{figure}[htb]
\includegraphics[scale=.6]{AiM.pdf}
\caption{$Ai(-x)$ (black) and its asymptotic expansion to zero (blue), first
(red), second (purple), and third (green) orders.}
\label{AiM}
\end{figure}
A simple example is provided by the Airy function of negative argument. In Fig \ref{AiM}, we plot
this exactly and using its asymptotic expansion of ever increasing order:
\begin{equation}
Ai(-x) = \frac{1}{{\sqrt{\pi}}x^{1/4}}
\Im\{e^{i(z+\pi/4}\, w(z)\}
\label{Aiasy}
\end{equation}
where $z=2 x^{3/2}/3$ and
\begin{equation}
w(z)= \sum_{j=0}^\infty w_j(z)=1-\frac{5i}{72z}-\frac{385}{10368z^2}+..,
\label{wzasy}
\end{equation}
where $w_0=1$, and
\begin{equation}
w_{j+1}=-\frac{i}{2z}
\left(j+
\frac{5}{36(j+1)}\right)\ w_j.
\label{wjasy}
\end{equation}
From Fig. \ref{AiM} we see that, for $x$ sufficiently large (here about 1.5), the asymptotic expansion
is extremely accurate.
On the other
hand, for $x$ sufficiently small, succesive orders worsen the approximation, and zero-order
is least bad. Moreover, inbetween, such as at $x=1$, addition of orders at first improves the result
and then worsens it.
\begin{figure}[htb]
\includegraphics[scale=.6]{dAi.pdf}
\caption{Errors at each order, labelling same as Fig \ref{AiM}.}
\label{dAi}
\end{figure}
In Fig. \ref{dAi}, we plot the errors of the successive asymptotic approximatons. First note
that the scale is 30 times smaller than Fig. \ref{AiM}, and we have begun at $x=1$. We
see that even at $x=2$, the asymptotic behavior has kicked in, and incredibly tiny errors are
made even for $M=2$.
To be certain that this is truely an asymptotic expansion, even though the terms in Eq. \ref{wzasy}
appear to be getting smaller, Eq. \ref{wzasy} shows that as $j\to\infty$, $w_j \approx j!/z^j$
which diverges for any value of $z$.
So, suppose we wish to approximate $Ai(-x)$ for all $x$
starting at some finite value, such as $x=1$.
We define $M_o(x)$ as the value of $M$ with the least error, which we refer to as the optimal
truncation. Then, if we want a `best' approximation to our function by truncating our
asymptotic expansion, we truncate at $M_o(1)$. We know that as $x$ increases (at least in the
asymptotic regime), the error of this truncated expansion will reduce. Thus we expect our
maximum error to be at our lowest $x$, and this truncation will minimize our worst error.
In the top half of Table \ref{T1},
we illustrate this with several orders and several values of $x$.
\begin{table}[htb]
\begin{tabular}{|c|rrrr|c|}
\hline
Order&0&1&2&3&$M_o$\\
\hline
$x$&\multicolumn{4}{c}{Errors}&\\
0.5& 0.0964& -0.0069& -0.3893& 0.1192&1\\
1.0& 0.0247& 0.0177& -0.0291& -0.0248&1\\
1.5& -0.0029& 0.0094& -0.0020& -0.0042&2\\
2.0& -0.0123& 0.0033& 0.0010& -0.0002&3\\
\hline
$x$&\multicolumn{4}{c}{Additions}&\\
0.5& 0.5721& -0.1033& -0.3824& 0.5085&1\\
1.0& 0.5602& -0.0070& -0.0468& 0.0043&1\\
1.5& 0.4614& 0.0123& -0.0114& -0.0022&2\\
2.0& 0.2151& 0.0156& -0.0022& -0.0012&3\\
\hline
\end{tabular}
\caption{Errors in asymptotic expansion of $Ai(-x)$, and contributions
added at each order.}
\label{T1}
\vskip -0.3cm
\end{table}
But hold on. We have surely cheated here, because we used our knowledge of the error to
choose where to truncate, which required knowing the function in the first place! However,
a simple heuristic that usually works is to simply look at the magnitude of the terms that
are being added in each increase in order. These will typically reduce at first, and then
eventually increase. The pragmatic optimal truncation procedure is to simply stop when the
next addition is larger in magnitude than the previous one. We see in Table \ref{T1} that this
indeed corresponds to optimal truncation.
Of greater interest for our purposes will be the asymptotic expansion for the zeroes of $Ai(-x)$,
defined by
\begin{equation}
Ai(-a_j)=0,~~~~~j=1,2,3...
\label{ajdef}
\end{equation}
in order of increasing magnitude. Later, we will show that these are the
eigenvalues of a potential. Each order of truncation of the expansion of
$Ai(-x)$ in Eq. \ref{Aiasy} implies
an asymptotic expansion of $a_j$ to the same order, yielding
\begin{equation}
a_j = y_j^{2/3} \sum_{n=0}^M \frac{T_n}{ y_j^{2n}},~~~~y_j=\frac{3\pi}{2} (j-1/4)
\label{aj}
\end{equation}
where the $T_n$ are found and listed in B (appendix B), the first few being
$1,5/48,-5/35,...$.
Because the lowest zero is about 2.34, the asymptotic behavior already dominates for every zero.
So far we have covered basics in most methods books, such as Arfken\cite{A67}. But now we approach
this from a more modern viewpoint, which holds that often, with the right procedure,
much more useful information can be extracted from such an expansion, especially in cases
that occur in physical problems, i.e., functions that are solutions to relatively simple
differential equations\cite{BH93}.
These methods might generically be called hyperasymptotics\cite{BH99,C08}, and often
begin with the `asymptotics of the asymptotics', i.e., asking what is the behavior of $c_n$
for large $n$ in Eq. \ref{SM}.
Knowing this, one can use a variety of techniques to approximate the
rest of the sum to infinity, and extract features that are entirely missed in the definition
given above. However, to take advantage of such techniques, one must be able to write the
expansion to arbitrary order, and then deduce its behavior.
\begin{figure}[htb]
\includegraphics[scale=.6]{A.pdf}
\caption{Expansion to many orders of $a_1$, the lowest zero of $Ai(-x)$:
the additions (black) and errors (blue).}
\label{A}
\end{figure}
In Fig. \ref{AiM},
we see the first two zeroes, at about 2.34 and 4.09.
In Fig. \ref{A}, we plot both the magnitude of the correction and the
magnitude of the error, on a log (base 10) scale, as a function
of the order of the approximation, $M$, in Eq. \ref{aj}. We see the generic nature of the asymptotic
expansion. For small $M$, the additions are quite large. To zero order, the error is of order 0.02.
As more terms are added, the magnitude of the additions becomes smaller, as does the error.
But at $M=4$, the magnitude of the correction is larger than that of $M=3$, so 3 is the optimal
truncation point. We see that indeed the error also begins to grow. For large $M$, the
additions become so big that they dominate the error, so the two curves merge.
Thus, with simple optimal truncation, our best possible estimate for the lowest zero is with
$M=3$.
\begin{figure}[htb]
\includegraphics[scale=.6]{Asy6.pdf}
\caption{Errors of Fig. \ref{A}, but now for first 6 zeroes,
the first being black, the 6th being brown.}
\label{Asy6}
\end{figure}
In Fig. \ref{Asy6}, we show what happens for higher zeroes. Now the blue curve shows the magnitude
of the error for the second zero. Because it is at a higher value, the optimal truncation occurs
at larger $M$, here about 6.
In fact, the analysis of B shows $M_o(n)\to\text{floor} {\pi*n}$ for the $n$-th zero as $n\to\infty$.
The brown curve is for the sixth level, where the lowest
error is at $M$ about 18, and is of order $10^{-18}$. This demonstrates the insane levels of
accuracy that can be achieved with very elementary means using asymptotic expansions.
Even the lowest order asymptotic expansion is often rather accurate,
once the asymptotic parameter does not come close to 0. To write an
approximate formula and apply it to
all zeroes, one should optimally truncate for the lowest level: All higher levels will then have
lower errors (no lines cross in Fig. \ref{Asy6}). For any level above the lowest, much greater accuracy
can be achieved by optimal truncation for that level (at a much higher order), but including
those higher orders would be disastrous for the approximation of the lower levels.
For example, for the 6th level, truncation at 18th order yields errors of order $10^{-18}$,
but errors of order $10^{-14}$ for the fifth level, $10^{-10}$ for the 4th, and errors
greater than a Hartree for the lowest two levels. To get the lowest error for
every level with a given truncation, Fig. \ref{Asy6} requires truncation at 3rd order.
\ssec{Notation and potentials}
\label{nota}
We choose units with $m=\hbar=1$, so the 1d Schr\"odinger equation is
\begin{equation}
\left\{ -\frac{1}{2} \frac{d^2}{dx^2} + v(x) \right\} \phi_j(x) = \epsilon_j\, \phi_j(x)
\end{equation}
where $j=1,2,..M$, if only $M$ states are bound.
We will consider a variety of shapes of potential and boundary conditions.
A hard-wall boundary condition is one where the wavefunction vanishes
identically, and nothing exists beyond the wall. An asymptotically bound
potential is one where the potential diverges as $x\to\infty$, so that
the system has only discrete states. Hard walls are a subset of these.
Finally, there is the situation that is closer to realistic, where the
potential is asymptotically free, i.e., tends to a finite constant.
We assume $v(x)$ has a minimum which we choose to be
at the origin, and set the constant to make $v(0)=0$. Thus such potentials
tend to $D$ as $x\to\infty$, where $D$ is the well-depth.
\def^{\rm PIB}{^{\rm PIB}}
Specific examples in this paper include the particle in a box, where
$v=0$ between hard walls at $x=\pm L/2$, with eigenvalues
\begin{equation}
\epsilon^{\rm PIB}_j = \frac{\pi^2\, j^2}{2L^2}~~~~~(PIB).
\end{equation}
and the harmonic oscillator $\omega^2 x^2/2$,
\begin{equation}
\epsilon_j = \omega (j - \frac{1}{2})~~~~~~(HO).
\end{equation}
(The unfamiliar minus sign is because the index $j$ begins at 1.)
Another analytically solvable case is the Poschl-Teller (PT) well of depth $D$
\begin{equation}
v(x)=D - D/\cosh^2(x),~~\epsilon_j = D -(\al_e+\frac{1}{2}-j)^2/2~~~(PT),
\end{equation}
where $\al_e={\sqrt{2D+1/4}}$ and $j < \al_e+1/2$. Our last (and most interesting) example
is the linear well $F|x|$, with $F$ a positive constant whose eigenvalues are
\begin{equation}
\epsilon_{j} = \left(\frac{F^2}{2}\right)^{1/3} d_j~~~~(LW),
\label{epslin}
\end{equation}
where $d_{2j+1}$ is the $j$-th zero of $Ai'(-x)$ and $d_{2j}$ is the $j$-th
zero of $Ai(-x)$.
For a symmetric potential $v(x)$, one can always place
a hard wall at the origin. Then the states of odd parity (even number
with our indexing) become the only eigenstates. We call these half wells.
For example, for the linear half-well with $F={\sqrt{2}}$, only the even
levels survive, and are given precisely by the zeroes of $Ai(-x)$ shown in Fig. \ref{AiM},
i.e., $d_{2j}=a_j$ of Eq. \ref{aj}.
\ssec{Non-interacting (Kohn-Sham) fermions}
\label{NIFs}
In text books, one usually solves these 1d problems for individual eigenstates.
But we consider these as KS potentials of some many-body problem, presumably with
some approximate XC functional. As such, we occupy the lowest $N$ levels.
If we keep all spins the same, the total energy is then
\begin{equation}
E_N=\sum_{j=1}^N \epsilon_j.
\label{EN}
\end{equation}
For our simple examples,
\begin{eqnarray}
E_N&=&\frac{\pi^2}{6 L^2}\left( N^3+\frac{3}{2}N^2+\frac{1}{2} N\right)~~(PIB)\nonumber\\
&=&\omega \frac{N^2}{2}~~~~(HO)\nonumber\\
&=&\frac{\al_e}{2}N^2 -\frac{N^3}{6}-\frac{N}{12}~~~(PT)\nonumber\\
&=&\sum_{j=1}^N a_j~~~(LHW).
\label{ENsimp}
\end{eqnarray}
In the last case, there is no simple exact closed form.
The central problem of orbital-free density functional
theory\cite{T27,F28,M57,T62,LPS84,PY89,DG90,WC00} is to find
sufficiently accurate approximations for $T_{\sss S}[n]$, the
kinetic energy of non-interacting electrons as a functional
of their single particle density, $n({\bf r})$. Functional differentiation
and insertion into an Euler equation yields an equation to be
solved directly for the density, avoiding the need
to solve the KS equations. Here, the required
level of accuracy is substantially higher than for XC, as the kinetic energy
is comparable to the entire KS energy. Moreover, since the density of
a given problem will be found by minimizing the energy with the approximate
$T_{\sss S}[n]$, the functional derivative must also be sufficiently so that
the approximate density also does not produce an unacceptable error\cite{KSB13}.
In fact, the original Thomas-Fermi (TF) theory\cite{T27,F28} has precisely this form
for the kinetic energy,
but it is not very accurate, its underlying density has many peculiarities,
and it does not even bind
atoms in molecules\cite{T62}. The form of the TF kinetic energy for spin-unpolarized
electrons in 3d is simply:
\begin{equation}
T_{\sss S}^{\rm TF}[n] = \frac{3(3\pi^2)^{2/3}}{10} \int d^3r\, n^{5/3}({\bf r}).
~~~({\rm 3D,~unpol.})
\end{equation}
We will call electrons in a KS potential NIFs, meaning non-interacting fermions.
The effect of the Pauli principle is simply to make them occupy the lowest $N$
orbitals. Moreover, to avoid keeping track of endless factors of 2\cite{OP79},
we simply choose them all to have
the same spin. For such spin-polarized NIFS in 1D, the analog of the above is
\begin{equation}
T^{\rm TF}[n] = \frac{\pi^2}{6} \int_{\infty}^\infty dx\, n^3(x).
~~~({\rm 1D,~pol.})
\label{TTF1D}
\end{equation}
This is of course the local density approximation for the kinetic energy, and is
exact for a fully polarized uniform electron gas. (As densities scale with
inverse volume, and the kinetic energy operator is a square gradient, in $d$ dimensions
the local density approximation to $T_{\sss S}$ always has power $n^{(d+2)/d}$, and
its prefactor is determined by the uniform gas or the large $N$ limit of any
system.)
Thus if one could achieve very high accuracy in an approximate $T_{\sss S}$
without incurring much computational
cost beyond TF, orbital-free DFT could make solving the KS equations obsolete\cite{WC00}, and
reduce the cost of DFT calculations to that of solving Poisson's equation.
From a regular quantum viewpoint, this is all a very elaborate approach to approximating
the sum in Eq. \ref{EN}.
\ssec{Semiclassical approximations}
\label{semi}
Semiclassical approximations are ubiquitous in physics and chemistry, but are rarely used
directly in electronic structure calculations at present\cite{H18}. All such expansions involve
powers of $\hbar$ that become relatively accurate in the small $\hbar$ limit.
Our interest will be in finding eigenstates of the Schr\"odinger equation, specifically in one
dimension. In this case, the
WKB approximation\cite{W26,K26,B26,D32} is well-known and appears in many introductory text on quantum
mechanics\cite{Gb05}. The WKB formula for eigenvalues is the implicit formula
\begin{equation}
{\cal A}(\epsilon) = 2\pi (j-\beta/4),~~~~~j=1,2,...,
\label{WKB}
\end{equation}
where ${\cal A}$ is the classical action at energy $\epsilon$
over a complete closed orbit and $\beta$
is the Maslov index\cite{MF01}. The Maslov index distinguishes between
hard wall reflections and true turning points, i.e., those where the slope of
the potential is finite. There is no contribution for a hard wall, but for
each true turning point, $\beta$ increases by 1. For our 1d examples,
each full orbit yields a contribution equal to double a transit from left to right.
Thus we write
\begin{equation}
I^\text{WKB} (\epsilon) = \int_{-\infty}^\infty dx\, p(x,\epsilon) = j -\nu,
\end{equation}
where $p=\Re{\sqrt{2(\epsilon-v(x))}}$ is the classical momentum at energy $\epsilon$ in the well,
and $\nu=0$ if there are only hard walls, and increases by $1/4$ for each true
turning point.
We can apply the WKB approximation to each of our wells. For the PIB,
$v=0$,
$p={\sqrt{2\epsilon}}$ and $I^\text{WKB}=L p$, yielding the exact answer as $\nu=0$.
Similarly, for the
HO, and $I^\text{WKB}=\epsilon/\omega$, again yielding the exact answer, as $\mu=1/2$. The first
can be attributed to the equivalence of semiclassical and exact quantum motion for
a constant potential, the second to the exactness of semiclassical results in harmonic
potentials. For the PT well,
\begin{equation}
I^\text{WKB}(\epsilon) = {\sqrt{2D}}-{\sqrt{2D-\epsilon}}~~~~(PT),
\end{equation}
which recovers the dominant semiclassical approximation, using $\nu=1/2$:
\begin{equation}
\epsilon^\text{WKB} (x)={\sqrt{2D}}x - x^2/2~~~~(PT).
\end{equation}
Finally, for the linear half-well, with $\nu=1/4$:
\begin{equation}
I^\text{WKB}(\epsilon) = \frac{2}{3\pi} \epsilon^{3/2},~~~
\epsilon^\text{WKB}(x)= \left(\frac{3\pi x}{2}\right)^{2/3}.
\end{equation}
In fact, for $F={\sqrt{2}}$, these are precisely the zeroes of the leading order expansion
of $Ai(-x)$ of Sec. \ref{asymp}. For the linear half-well, the exact expression for $I$ is
\begin{equation}
I(\epsilon)= \frac{z +\Im{(\log w(z))}}{\pi},~~~~z= \frac{2}{3} \epsilon^{3/2}~~~(LHW),
\label{ILHW}
\end{equation}
where
\begin{equation}
w (z(x))= {\sqrt{\pi}}x^{1/4}\, e^{-i(z+\pi/4)}\left(Bi(x)+i Ai(-x)\right),
\end{equation}
and $Bi(x)$ is the other independent solution of the Airy equation\cite{AS72,DLMF}.
For every well in Sec. \ref{nota},
\begin{equation}
\epsilon_j \to \epsilon^\text{WKB} (j-\nu),~~~~~j\to\infty.
\label{WKBasy}
\end{equation}
But the WKB approximation is just the first term in a delicate asymptotic expansion in $\hbar$,
as shown by Dunham\cite{D32}. We can define an expansion in powers of $\hbar$,
here represented by a dimensionless parameter $\eta$, via
\begin{equation}
I_\eta(\epsilon)=\sum_{k=0}^\infty \eta^{2k}\, I^{(2k)}(\epsilon),
\end{equation}
where $I^\text{WKB}$ is just the leading term. Then
\begin{equation}
I_\eta(\epsilon) = \eta( j-\nu),
\label{Ieta}
\end{equation}
determines the eigenvalues implicitly, which can be inverted power by power to yield
an expansion for the energy levels which becomes more accurate as $j$ increases.
This expansion is well-known\cite{BO78}, but is subtle for systems with
turning points. The naive corrections formally diverge at the turning points,
but these divergences are exactly cancelled by other terms in the wavefunction,
yielding finite contributions in every order.
This is the semiclassical expansion we are interested in, but we wish to find
sums over levels, not individual eigenenergies.
Back in the 1950's and 1960's, there was considerable activity attempting to
use semiclassical approximations to do electronic structure calculations,
especially by the Miller group\cite{M68b,M68c}. In fact, Kohn and Sham
developed a remarkably insightful approach\cite{KSb65} just months before
their most popular paper\cite{KS65}, whose ultimate success for numerical
computation overwhelmed interest in semiclassical approaches.
There was much interest in semiclassical methods for more than one dimension
in the area of quantum chaos\cite{M83}.
One can also deduce, e.g., approximate
wavefunctions (see Sec. \ref{unif} below) in the WKB expansion\cite{Gb05}. Using WKB wavefunctions
to find approximate energies, e.g., by evaluating the Hamiltonian on them,
yields different results\cite{LR91} than those for the eigenvalues, Eq. \ref{WKB}.
All semiclassical methods\cite{BM72} require extreme care in defining
precisely the nature of the expansion and which quantities should be held fixed.
\ssec{Semiclassical limit}
\label{semilim}
We next consider specifically the semiclassical limit for the sum of the energies. In this
case, one simply integrates the WKB energies over the required number of levels.
For almost any potential, it can be shown numerous ways that\cite{MP56}
\begin{equation}
E_N \approx \int_0^N dx\, \epsilon^\text{WKB} (x) \approx E^{\rm TF}_N,~~~~\hbar\to 0,
\end{equation}
and none of the details of the corrections matter. The TF approximation can be treated
as a functional of either the density or the potential (see Sec \ref{pot} below) and the
results are the same.
Alternatively, it is a straightforward matter to
extract the kinetic contribution alone\cite{MP56} and find the local density approximation
to the kinetic energy. Thus, the local approximation (here TF) becomes relatively exact for all
problems in this semiclassical limit. This is a simple case of the much harder proof
by Lieb and Simon of the same statement for all Coulomb-interacting matter\cite{LS77}.
In the language of Sec \ref{asymp}, the TF theory yields the dominant contribution in an
asymptotic series for all matter, which implies that its relative error vanishes
in the limit, Eq. \ref{lim}.
\ssec{Gradient expansion in DFT}
\label{grad}
We focus here on the non-interacting kinetic energy, whose gradient expansion was
performed by Kirzhnits\cite{Kc57}, for a slowly-varying gas, using the Wigner-Kirkwood
expansion\cite{W32,K33}. Ideas of gradient expansions
permeate the HK\cite{HK64} and KS papers\cite{KS65} that created modern DFT,
for both the full functional and
its XC contribution. The first generalized gradient expansion for correlation was from
Ma and Bruckner\cite{MB68}, from which many modern GGA's are descended.
Here we consider only the non-interating kinetic energy in one dimension. In that case,
Samaj and Percus did a thorough job\cite{SPb99}, showing how to generate the expansion to arbitrary
order. We focus on several key points. First, they expand both the density and the
kinetic energy density as functionals of the {\em potential}. (This is, after all, how
quantum mechanics normally works.) Given a potential, the expansion for the density is
\begin{equation}
n(x)=\frac{k_{\sss F}(x)}{\pi} \sum_{j=0}^\infty \frac{ a_j(x)}{k_{\sss F}^{2j}(x)\beta_j},
\label{nv}
\end{equation}
where $\beta_j=1-2j$, $k_F = {\sqrt{2(\mu-v(x))}}$ and $\mu$ is
determined by normalization. The analogous formula for a
kinetic energy density $t(x)$ is found by multiplying by $k_F^2(x)/2$
and replacing $\beta_j$ by $\beta_{j+1}$, where $t(x)$ is a function
whose integral yields $T$.
The coefficients in the expansion are\cite{SPb99}
\begin{equation}
a_0=1,~~~a_1=0,~~~a_2=({{k_F}'}^2+k_F k_F'')/4,...
\label{gradcoefs}
\end{equation}
Inserting Eq. \ref{nv} into $t(x)$, and expanding in small gradients, yields:
\begin{equation}
T^{\rm GEA}[n]=\frac{\pi^2}{6}\int dx\ n^3 (x) - \frac{1}{24}\int dx \left( \frac{dn}{dx}\right)^2+..
\end{equation}
This is the exact analog of the usual expansion in 3D\cite{DG90},
except this is the spin-polarized
form, and the coefficient of the von Weisacker contribution is $1/9$ in 3D but $-1/3$ in 1D.
The gradient expansion
is known to 6-th order in 3D\cite{M81} and has been numerically validated under conditions
where gradient expansions apply\cite{YPKZ97}. It has also been noticed that, for non-analytic
potentials, evaluation of the higher-order terms depends sensitively on
the boundary conditions\cite{PSHP86}.
\ssec{Potential functionals versus density functionals}
\label{pot}
The creation of modern DFT and the KS equations has clearly been very successful.
However, standard approaches to quantum mechanics yield algorithms that predict,
for example, the energy as a functional of a given potential, $v(x)$ here. In the
context of KS-DFT, Yang, Ayers, and Wu\cite{YAW04} first clearly showed the relation between potential
functionals and density functionals. But semiclassical approximations yield
results for a given potential, not density. Thus Cangi et al\cite{CLEB11,CGB13,CP15} revisited the
entire framework
of density functional theory, from the HK viewpoint, and showed that a logical alternative
was to create a potential functional that also satisfied a minimum principle,
namely
\begin{equation}
F_v=F[n_v],
\end{equation}
where $F[n]$ is the universal part of the energy functional and
$n_v({\bf r})$ is the ground-state density of potential $v({\bf r})$. Then
\begin{equation}
E_0= \min_{v'} \left\{ F_{v'} + \int\, v\, n_{v'} \right\},
\end{equation}
yields the exact ground-state energy and $v=v'$ at the minimum\cite{CGB13}.
Given an expression for $n_v$,
various strategies can be used to construct a corresponding $F_v$
and so the entire energy can be found. In the specific case of TF theory, Eq. \ref{TTF1D}
yields exactly the same results for any system whether expressed as a density functional
or a potential functional.
More sophistacted approximations for the
density including higher-order expansions in $\hbar$ (see next section)
are typically not designed
to be variational\cite{GP09}, and minimization might worsen results.
Such minimizations are not needed if direct application to the external potenial
already yields highly accurate results\cite{CLEB10}.
\ssec{Uniform approximations for the density}
\label{unif}
To find semiclassical expansions for the density as a functional of the potential, one can start
from WKB wavefunctions. With hard walls, the wavefunctions are simple
\begin{equation}
\phi_j(x) \approx \sqrt{\frac{2\omega_j}{p_j(x)}}\sin{\theta_j(x)}\,,
\label{phiWKB}
\end{equation}
where $p_j(x)$ is the classical momentum in the $j$-th WKB eigenstate,
$\omega_j$
is the frequency of its
orbit, and $\theta_j(x)$ is the phase accumulated from the left wall\cite{ELCB08}.
Using a variation on the standard Euler-Maclaurin formula (Eq. \ref{EMnorm} below)
in asymptotic form, this yields
a uniform approximation to the density. For any
value of $x$, WKB provides the dominant contribution as $\hbar\to 0$, and its leading corrections
provide the next order in the asymptotic series.
Early on, it was shown how to extract an accurate approximation to both the density and the kinetic
energy density with hard-wall boundary conditions, by evaluating the next order in the WKB
expansion for the wavefunctions, and summing the result\cite{ELCB08,CLEB10}.
The leading corrections to the kinetic energy density integrated to yield the
leading correction to the kinetic energy as an expansion in $\hbar$.
However, the WKB wavefunctions are well-known to diverge at a true turning point.
Langer\cite{La37} found a semiclassical wavefunction that remains uniformly accurate
through the turning point, by replacing $\sin \theta_j$ in Eq. \ref{phiWKB} with
\begin{equation}
z_j^{1/4}(x)\, Ai\left[-z_j(x)\right],
\label{phiLang}
\end{equation}
where $z_j= \left[3\theta_j(x)/2\right]^{2/3}$.
Some years later\cite{RLCE15}, and with considerable difficulty\cite{RB18}, it was deduced
how to repeat the same procedure for a well with real turning points, creating a uniform
approximation for the density in a well with turning points,
i.e., one whose error, relative to the local approximation, vanishes for all $x$
as $\hbar$ vanishes. Note that the expansion in $\hbar$ is in different
orders depending on how close $x$ is to the turning point.
The resulting approximations are exceedingly accurate for both the density and the kinetic energy
density pointwise, but surprisingly do not yield more accurate energies\cite{RB17}. When analyzed,
it was found that the expansion in wavefunctions yields energetic corrections of order $\hbar^{1/3}$,
but the leading corrections are of order $\hbar$\cite{RB17}. This was explained by showing that the
coefficient of
$\hbar^{1/3}$ vanishes identically! Thus, this expansion would have to be continued to
two more orders to yield the leading correction to the energy, which might take generations to derive.
Instead, analysis of the simple potentials\cite{RB17} used to test the uniform appoximations showed that
direct sums over eigenvalues can yield the leading corrections to sums over $N$ occupied orbitals,
bypassing (for now) the density and any other real-space quantity entirely.
\sec{Theory}
\label{theory}
\ssec{Summation formulas}
\label{sum}
We begin our theoretical development with an unusual form of
the Euler-Maclarin formula from Hua\cite{H12}
\begin{equation}
\sum_{a\leq j \le b} f(j) = \int_a^b dx \left( f(x) + P_1(x) f'(x) \right)
- \left[ P_1 (x) f(x) \right]_a^b
\label{EMHau}
\end{equation}
where $a,b$ are real numbers, $f'(x)=df/dx$ must be continuous, and $P_1(x)$ is the
first periodized Bernouilli polynomial\cite{DLMF}. The periodized Bernouilli polynomials
are
\begin{equation}
P_k(x)= B_k (x-\text{floor} {x}),
\end{equation}
where $B_k(x)$ is a Bernouilli polynomial, and
where $\text{floor} {x}$ is the integer part of $x$.
The Bernouilli polynomials, of order $k$, satisfy many simple conditions, with
the lowest few being
\begin{equation}
B_0(x)=1,~~~B_1(x)=x-\frac{1}{2},~~~B_2(x)=x^2-x+\frac{1}{6}.
\end{equation}
The famous Bernouilli numbers are then
\begin{equation}
B_k = B_k(1),
\end{equation}
which vanish for all odd $k$, except $B_1=1/2$.
Eq. \ref{EMHau} is an unusual form because $a$ and $b$ are continuous.
We next perform the standard trick of repeated integration by parts, leading to
\begin{equation}
\sum_{a\leq j \le b} f(j) = \sum_{k=0}^p D_k + R_p,
\label{EEM}
\end{equation}
where the end-point contributions are
\begin{equation}
D_k=\frac{(-1)^k}{k!} \left[ P_k(x) f^{(k-1)}(x) \right]_a^b,
\end{equation}
and the remainder is
\begin{equation}
R_p=\frac{(-1)^{p+1}}{p!} \int_a^b dx\, P_p(x)\, f^{(p)}(x).
\end{equation}
Eq. \ref{EEM} is true for any $p \geq 1$, so long as the $p$-th derivative of $f(x)$
is continuous. Here, the term $f^{(-1)}(x)$ is simply the antiderivative, so the
integral is $D_0$. The $p=1$ case is Eq. \ref{EMHau}.
We call Eq. \ref{EEM} the extended Euler-Maclaurin form, and we use several
variations in what follows.
We note some remarkable features of Eq. \ref{EEM}.
First, any sufficiently smooth function of $x$ that matches $f_j$ at the integers
yields exactly the same sum, so that all differences in the integral on the right
must be cancelled by the other terms.
Thus, there are many allowed choices
for $f(x)$ that yield the exact sum. Adding any sufficiently smooth
$g(x) \sin(\pi x)$ to an acceptable $f$ does not change the sum.
Next, we note that for any range of $a$ and
$b$ between integers, the sum does not change but the integral does, so again such
changes must be absorbed by the remaining terms.
Lastly, we note that the formula is exact for every $p$. Choosing a low
$p$ requires less derivatives, but often the remainder term is more difficult
to evaluate.
Of course, there are many different ways to write this formula that are useful in different
contexts.
The special case $a=1_-$
and $b=N_+$ recovers the commonly given form of Euler-Maclaurin,
\begin{eqnarray}
\sum_{j=1}^N f_j&=&\int_1^N dx\, f(x)
+\frac{f(N)+f(1)}{2} + R_p[f]\nonumber\\
&+& \sum_{k=1}^{{\text{floor} {p/2}}} \frac{B_{2k}}{(2k)!}\left(f^{(2k-1)}(N)-f^{(2k-1)}(1)\right),
\label{EMnorm}
\end{eqnarray}
where the remainder term is the same as above.
The plus sign in the second term on the right occurs because of the discontinuity in
$P_1(x)$ across an integer, and the vanishing of even derivatives is because all odd
Bernouilli numbers are zero, except $B_1$.
This form is perhaps most familar when
approximating an integral by a sum, but we never use it here.
Next, consider the special case $0 < a < 1$ and $b=N+a$. Then the sum becomes
specifically that of the first $N$ terms:
\begin{equation}
S_N=\sum_{j=1}^N f(j),
\end{equation}
while the end contributions simplify to
\begin{equation}
D_k(a)= \frac{(-1)^k}{k!} B_k(a) \left[f^{(k-1)}(x)\right]_a^{N+a}.
\end{equation}
We will have use for two special cases. The first is $a=1/2$, and since\cite{DLMF}
\begin{equation}
B_k(1/2)=-(1-2^{n-1}) B_k,
\end{equation}
then only even terms contribute to the end-points.
The other case we will use is the limit as $a\to 1$, so that $B_k(a)=B_k$,
and
\begin{equation}
D_k(1)= \frac{(-1)^k}{k!} B_k\, \left[f^{(k-1)}(x)\right]_1^{N+1}
\label{Dbmax}
\end{equation}
no longer vanishes for $k=1$.
Both these special cases will be of value: The first yields some of the key
the results of A,
the second of B.
\ssec{Sums of eigenvalues}
\label{sumeig}
We first use Eq. \ref{EMHau} to derive a general exact formula for the sum of $N$
energy levels in terms of $\epsilon(x)$. From Sec \ref{semi}, we know $\epsilon_j=\epsilon(j-\nu)$.
Using this, we find for the sum of the first $N$ energies:
\begin{equation}
E_N = \sum_{j=1}^p D_k(a) + R_p(a),
\end{equation}
where
\begin{equation}
D_k(a)=\frac{(-1)^k}{k!} B_k(a) \left[\epsilon^{(k-1)}(x)\right]_\alpha^{N+\alpha},
\end{equation}
and
\begin{equation}
R_p(a)=\frac{(-1)^{p+1}}{p!} \int_\alpha^{N+\alpha} dx\, P_p(x+\nu)\, \epsilon^{(p)}(x),
\end{equation}
with $\alpha=a-\nu$. In the special case $a=\nu$, all integrals and evaluations
run from 0 to $N$.
Finally, for the standard case of two real turning points, insert $\nu=1/2$
in both $D_k(\nu)$ and $R_p(\nu)$ to yield
\begin{equation}
E_N=\int_0^N dx\, \epsilon(x)+\sum_{k=1}^{p} D_k(\frac{1}{2}) + R_p(\frac{1}{2}).
\end{equation}
For the choice $p=1$, because $B_1(1/2)=0$ the end-term vanishes, yielding
the elegant result
\begin{equation}
E_N^{2TP} = \int_0^N dx\, \left( \epsilon(x) + \sm{x} \epsilon'(x) \right)~~~(\nu=\frac{1}{2},p=1)
\label{E2TP}
\end{equation}
where $\sm{x}=P_1(x+1/2)$, and $2TP$ denotes two turning points.
This is recognizable as Eq. 14 of A, but was derived here by more
elementary means.
The analysis of A is confined to $\nu=1/2$ and $p=1$. The current formulas
apply to all possible potentials, i.e., any Maslov index, and allow higher choices
of $p$. For example, the $p=1$ result for arbitrary $\nu$ yields
\begin{equation}
E_N = E_N^{2TP}
+(\frac{1}{2}-\nu)\left[\epsilon(x)\right]_0^N,~~~(p=1)
\label{EN1nu}
\end{equation}
i.e., there is a simple correction whenever $\nu$ differs from 1/2.
It is straightforward to check that Eq. \ref{EN1nu} produces the exact sum
when used correctly. For example, for a half-harmonic oscillator, $\nu=1/4$ and
$\epsilon(x)= 2\omega x$. In this case, the 2TP contribution is easy to calculate
as the second term vanishes, due to the constancy of $\epsilon'$ and the periodicity
of $\sm{x}$, yielding $\omega N^2$. But there is also a finite addition of
$\epsilon(N)/4$ to produce the exact
$E_N = \omega N (N+1/2)$. Similar corrections are also needed to
recover the exact sum for the particle in a box.
This illustrates the significance of the correct Maslov index in all such calculations.
\ssec{Leading correction to local approximation}
\label{lead}
In this section, we use Eq. \ref{EEM} to
examine just the leading correction to the local approximation.
Because classical action is a monotonically increasing function
of $\epsilon$ as one climbs up a well, then $\epsilon(x)$ also grows
monotonically, so its integral grows even more rapidly. On the
other hand, its derivative will be less rapidly growing, and the
periodic term $\sm{x}$ averages to zero with a constant function.
Thus this term is smaller than the dominant term.
Expanding $I$ in even powers of $\eta$ as in Eq. \ref{Ieta}, we find two leading
corrections to the local approximation to second order:
\begin{equation}
\Delta E^{(2)}_N = \int_0^Ndx\, \left( \epsilon^{(2)}(x) + \sm{x} \frac{d\epsilon^{(0)}}{dx} \right).
\label{E2}
\end{equation}
Thus
there are two corrections: Those due to the next order in the WKB expansion inside the
dominant integral
while others are the error made in approximating the sum over WKB eigenvalues by
an integral. In the case of extended systems where there are no turning points, i.e.,
slowlying varying densities, the spacing between levels goes to zero in the thermodynamic
limit, and the latter correction vanishes. Thus the gradient expansion of Sec. \ref{grad}
misses such
terms completely.
In principle, Eq. \ref{EN1nu} also applies to the linear well, but its expansion is more
difficult
than the previous case. In particular, the asymptotic expansion diverges
at $x=0$, the start of our integral, making it impossible to work with.
We thus use a different version, as developed in B.
\ssec{Hyperasymptotics}
\label{hyper}
We now turn to the work of Ref B. We see immediately that Eq. \ref{E2TP} is not useful
for asymptotic expansions in powers of $\hbar$, as it includes energies down to zero,
where asymptotic expansions like that of Eq. \ref{SM} diverge.
In fact, we use Eq. \ref{Dbmax}, in which both $a$ and $b$ have been maximized, for a given
sum from 1 to $N$. This idea already appeared in the contour chosen in Ref \cite{KSb65},
which circles a pole in the Green's function at $\epsilon_{N+1}$. Thus we choose our second
variation to explore asymptotic expansions:
\def{\bar\nu}{{\bar\nu}}
\begin{equation}
D_k(1)=\frac{(-1)^k}{k!} B_k \left[\epsilon^{(k-1)}(x)\right]_{{\bar\nu}}^{N+{{\bar\nu}}},
\end{equation}
and
\begin{equation}
R_p(1)=\frac{(-1)^{p+1}}{p!} \int_{{\bar\nu}}^{N+{\bar\nu}} dx\, P_p(x+\nu)\, \epsilon^{(p)}(x),
\end{equation}
where ${\bar\nu}=1-\nu$.
The first three $D$'s are:
\begin{equation}
\left[E(x)\right]_{\bar\nu}^{N+{\bar\nu}}
-\frac{1}{2}\left[\epsilon(x)\right]_{\bar\nu}^{N+{\bar\nu}}
+\frac{1}{12}\left[\epsilon'(x)\right]_{\bar\nu}^{N+{\bar\nu}},
\end{equation}
where $E(x)$ is the antiderivative of $\epsilon(x)$. These forms apply to all wells for any
$p > 0$, but have the advantage of being evaluated at the largest possible energies.
It is trivial to check that these forms yield both the exact results for all
the simple potentials we have encountered so far, for any choice of $p$.
They also recover the leading correction to the semiclassical expansion for the PT
well, producing two corrections, one from the 2nd-order WKB, and the other either
from $D_1$ or $R_1$, just as in Eq. \ref{E2}.
But the real use is in hyperasymptotics, i.e., performing asymptotic expansions to high
orders. We apply our formulas to the half linear well, so that $\nu=1/4$ and ${\bar\nu}=3/4$.
We perform the WKB expansion to find an asymptotic series in even powers of $\eta$
for the energies:
\begin{equation}
\epsilon_m(x)=\sum_{p=0}^m\epsilon^{(2p)}(x).
\end{equation}
Inserting this in the summation formula, we chose $p=m$, which guarantees the remainder
term is of order $m+1$ or greater. The $k$-th end-point term contains orders
$k-1$ to $m+k-1$ due to the derivatives, but the terms beyond $m$ can be discarded
to find the asymptotic approximant of order $m$. We can write the result very simply as:
\begin{equation}
E_N \approx \sum_{m=0}^M \left( S_m(N+{\bar\nu}) - S_m({\bar\nu})\right),
\label{ENfromS}
\end{equation}
where
\begin{equation}
S_m(x) = \sum_{k=0}^m \frac{B_{2k)}}{(2k)!}
\epsilon^{(2(m-k),2k-1)}(x)
-\frac{1}{2} \epsilon^{(2m)}(x),
\label{sm2}
\end{equation}
where the first superscript indicates the power of expansion in $\eta$ and the second
denotes the number of derivatives taken.
This recovers exactly the expansion in Eq. (6.4) of Ref B.
The asymptotic expansion is evaluated at $N+1$ rather than at $N$ in the regular
EM formula. This confers two distinct advantages: For a given order, our errors are typically
much smaller when the index is increased by 1, and secondly, since the order of optimal truncation
is $\text{floor} {\pi N}$, by evaluating at $N+1$, three additional orders are added to the optimally
truncated series, with their concommitant improvement in accuracy.
We note that one need only evaluate each contribution in Eq. \ref{ENfromS}
at the upper end. Taking $N=0$
and subtracting then yields $S_N$, guaranteeing correctly its vanishing for $N=0$.
Thus we have recovered the main one-dimensional result of Ref B without need for
(but also missing the elegance of) regularizing sums as $N\to\infty$.
Eq. \ref{ENfromS} can be applied directly to finite wells, such as PT or the truncated linear half-well
of Ref. \cite{BB19}.
\begin{figure}[htb]
\includegraphics[scale=.6]{Sum1.pdf}
\caption{Errors of Fig. \ref{A} (black)
and from summation formula, Eq. \ref{Sm} (blue).}
\label{Sum1}
\end{figure}
We show some results from the summation formula for the linear half well
in Fig. \ref{Sum1} for $N=1$,
where $E_1=\epsilon_1$. The summation formula is less accurate than the original
formula for $M=0$, but is much more accurate even for $M=3$ (by two orders of
magnitude). More importantly, its optimal truncation is at 6, producing
almost 3 orders of magnitude in improvement, i.e., going from milliHartrees
to microHartrees errors!
\begin{figure}[htb]
\includegraphics[scale=.6]{Sum2.pdf}
\caption{Same as Fig. \ref{Sum1}, but adding curves
for first excited state (red) and the sum of lowest two energies (purple).}
\label{Sum2}
\end{figure}
To see that this is due to our evaluation at $N+1$, in Fig. \ref{Sum2} we add in
the second eigenvalue and the second sum, $E_2=\epsilon_1+\epsilon_2$.
Its error curve is almost identical to that of the
summation formula for the {\em first level}. Of course, the summation formula
for the 2nd level has leaped ahead again, with errors of nanoHartrees at the
optimal truncation of $M=9$!
\begin{figure}[htb]
\includegraphics[scale=.6]{Sum6.pdf}
\caption{Same as Fig. \ref{Sum1}, but for the 6th level and its sum.}
\label{Sum6}
\end{figure}
Finally, we attempt to show the error in the 6th level in Fig. \ref{Sum6}. The black
line here is for the error in individual level, and so matches the purple line of
Fig. \ref{Sum1}. But the blue line is the error in the summation formula, which
appears to be least at optimal truncation of about $M=21$, where the error is
about 10 zeptoHartrees. (The noise in the curve is caused by numerical imprecision.)
\ssec{Alternative summation formula}
So far, our analysis has shown that the key formulas of A and B are special cases
of the extended Euler-Maclaurin formula, Eq. \ref{EEM}. The formulas of A apply only
to two turning points, and cannot be used as a basis for asymptotic expansion, because
the energy function must be evaluated at 0. The formulas of B require an infinite
set of eigenvalues, but our Eq. \ref{sm2} allows them to be applied to a finite number.
Eq. \ref{sm2} contains the Maslov index explicitly and has no difficulties at the lower-end,
which does not vanish, even in the two turning point case.
But can we find a single formula that covers all cases? The primary aim is to
generate an expansion for large $N$, in which one can write exact expressions for
the error. For any monontonically increasing function of $x$, as our eigenvalues
are defined to be, the large-$N$ limit of the sum is dominated by the integral.
The leading correction is always given by the end-point contribution near $x=N$.
Thus, choosing our upper end-point as $N+1/2$ always eliminates that contribution,
simplifying the result. Equally, we choose $a=1$ always, so that the lower
energy, even in the presence of two turning points, does not vanish. This yields
the ungainly but practical
\begin{equation}
E_N = \int_a^b dx \epsilon(x) - \sum_{k=1}^{\text{floor} {p/2}}
\frac{B_{2k}}{(2k)!} (1-\frac{2}{4^k})\, \epsilon^{(2k-1)}(b) + \Delta_p,
\label{ENfin}
\end{equation}
where $b=N+1/2-\nu$, $a=1-\nu$,
and $\Delta_p$ is of order $\epsilon^{(2p)}(N)$ and is given exactly by
\begin{equation}
\Delta_p=-\sum_{k=1}^{p}\frac{B_k}{k!} \epsilon^{(k-1)}(a) +R_p.
\end{equation}
Since the integration interval is no longer an integer, $R_p$ does
not vanish beyond a maximum $p$ for simple powers.
We emphasize that this is an exact formula for all potentials that are
sufficiently smooth (the $p$-th derivative must be
continuous), and can be applied with any $p \geq 1$, and to any boundary
conditions. Curiously, almost the same form (but with $a=1/2$) was used in
Eq. 22 of Ref. \cite{CLEB10} to perform the summation correctly, but without
explanation for why it had this form, or the role of the Maslov index.
For the linear half well, Eq. \ref{ENfin} yields the simple closed-form asymptotic expansion
\begin{equation}
E_N \approx \sum_{j=0}^\infty d_j y_N^{5/3-2j},~~~~y_N=\frac{3\pi}{2}\tilde N,
\end{equation}
where $\tilde N = N +1/4$, and
\begin{equation}
d_j= \frac{2}{3\pi} \frac{T_j} {5/3-2j}-\frac{\pi}{16} T_{j-1} (8/3-2j).
\end{equation}
This generates exactly the same asymptotic expansion in $N$ as Eq. 6.4 of B, but in a simpler form
and with terms that differ only by even powers of $z_n$.
The first two terms are:
\begin{equation}
E_N\approx \left(\frac{3\pi}{2}\right)^{2/3}
\left(\frac{3}{5}\tilde N^{5/3}-\frac{5+\pi^2}{36\pi^2\tilde N^{1/3}}+...\right).
\label{altasy}
\end{equation}
This is identical to, but simpler than, Eq. (6.4) of B.
The GEA of Sec. \ref{grad} includes only contributions from the integral in Eq. \ref{ENfin}.
In the 2nd term, GEA
does not include the $\pi^2$ contribution in the numerator, reducing the overall coefficient
by a factor of about 3, and so misses the correct asymptotic expansion.
\sec{Relation to DFT}
\label{DFT}
\ssec{Error in gradient expansion}
\label{graderr}
While these are impressive ways to sum $N$ eigenvalues, what do they mean for DFT calculations?
We focus on the relation to orbital-free DFT in one dimension (not a very practical
application, admittedly).
We first consider the direct potential functional form of the gradient expansion, given
in Eq \ref{nv} of Sec \ref{grad}. All terms can be combined to yield the gradient expansion
for the total energy. This yields formulas identical to those we find from the
WKB expansion inserted into the integral term, $D_0$, and totally misses the corrections
from the rest of the expansion for any system with discrete levels, as are atoms and molecules.
If this term is included, the results are much more accurate (see Table I of A),
because the correct asymptotic expansion has been included to the given order.
As shown throughout these works, the sums are much more accurate than the original expansion for the
individual levels. Without this term, one has only part of the correction, and can
make at best crude guesses (possibly using exact conditions) that yield moderate improvements
at best over the excellent zero-th order contribution.
In Ref A, the correction was first isolated, but only for $\nu=1/2$. We can now give
the corrections in all cases to every order. In Eq. \ref{ENfromS}, the gradient expansion
accounts only for those terms of order $m$ in the WKB expansion that occur with the
same order in the summation, i.e., only the $D_0$ contribution to the sum. Thus
\begin{equation}
S^{GEA}_m(x) = E^{(2m)}(x)
-\frac{1}{2} \epsilon^{(2m)}(x).
\label{Sm}
\end{equation}
for $m>0$, and the missing terms are
\begin{equation}
\Delta S^{GEA}_m(x)=
\sum_{k=1}^m \frac{B_{2k)}}{(2k)!}
D_{2k-1} \epsilon^{(2(m-k))}(x).
\end{equation}
For example, the leading-order missing correction is
\begin{equation}
\Delta S^{GEA}_1(x)=
\frac{1}{12} \frac{d \epsilon^{(0)}}{dx}.
\end{equation}
This is the term that was identified in A. Without it, the 2nd-order gradient
expansion approximation was, at best, an erratic correction to the local approximation.
Including it gave accuracies about 30 times better than the dominant term.
Again, adding the next order led to errors of microHartree order.
Our formulae allow one to extract this missing term to any Maslov index, and so allow it
to be computed, e.g., for the linear half-well. More importantly, they can be (in principle)
applied to all orders (once the WKB expansion has been performed to a similar order).
\ssec{Understanding aspects of practical KS-DFT}
\label{under}
This 1D world may seem very far from the real world of realistic, practical electronic
structure with Coulomb-repelling electrons being Coulomb-attracted to nuclei,
but many of the difficulties and problems with practical approximate functionals show
up in simpler forms here.
For example, many semilocal functionals perform worst for one particle. We see
here that all our results are worst for the lowest level, because the expansion
is asymptotic in the particle number, $N$. Hence self-interaction error\cite{PZ81}
is a chief source of error in semilocal XC calculations.
Semilocal approximations to XC fail when a bond is stretched, often breaking symmetry at
a Coulson-Fischer point\cite{CF49}. We see here that, as a bond is stretched, there is a critical
distance in which the well goes from a single well to two. Beyond that point,
one can perform the expansion in the separate wells, but it is a different expansion
from the one that applies to a single well. Thus the asymptotic expansion relevant
at equilibrium becomes irrelevant (and so highly inaccurate) as the well splits in two.
For interacting electrons, this effect is accompanied by a multi-reference character
to the interacting wavefunction.
A third insight, not explored here, is the derivative discontinuities in the energies
as a function of continuous particle number $N$\cite{PPLB82,Pb85}.
The methodology of A demonstrates
this explicitly,
and the present techniques will be expanded to include this in the future.
\ssec{Importance for practical calculations}
\label{import}
Insights from studying these one-dimensional situations have already contributed to
understanding and creating modern functional approximations. For example,
the parameter in B88 exchange GGA\cite{B88} was derived in Ref. \cite{EB09}, a
mere 21 years after it was first proposed.
The derivation yields a value within
10\% of the fitted value of B88 (and is less accurate for real systems).
One of two crucial conditions in constructing PBEsol\cite{PRCV08}, namely the
restoration of the second-order gradient expansion for exchange, came from these
insights. In fact, the current work may lead to insight into the
second condition, which is the restoration of the LDA surface energy for jellium.
Eq. \ref{altasy} contains a correction missed by the GEA due to the surface
of a linear potential, just the kind of correction being extracted from
the edge electron gas\cite{KMd98,AM05,LMA14}.
Moreover, Ref. \cite{CCKB18} showed that even the correlation energy of finite systems
finally tends to its LDA value (at least for atoms, but logarithmically slowly).
Several of these asymptotic conditions
for atoms as $N\to\infty$ were built into SCAN\cite{SRP15} and other
approximate functionals\cite{CFLD11}. Finally, we mention that all chemical
and materials properties depend on energy differences, not total energies.
Ref. \cite{CSPB11} showed that, for atoms with certain plausible assumptions,
the ionization potential is given exactly by KS-LDA calculations in the
asymptotic limit.
It is tantalizing to note that Ref. \cite{EB09} found that the asymptotic
correction to the local density approximation for exchange was almost exactly
double that of the gradient expansion. This could only be done by numerical
extraction of the coefficient from a sequence of large atom Hartree-Fock calculations.
Eq. \ref{altasy} finds analytically that the correction to the local density approximation
for the total energy of the linear half-well is almost exactly triple that of
the gradient expansion.
\sec{Conclusions}
\label{conc}
I have presented an appropriate mathematical
tool for understanding the successes of modern density functional theory and the
centrality of the local density approximation. In this framework,
the continuum limit achieved as $\hbar\to 0$ in a certain, very well-defined sense
is the reason behind the success of semilocal density approximations.
This framework unites two (apparently) distinct approaches in previous papers, and
generalizes key results from both those works. More importantly, it shows that,
at least in principle, DFT approximations need not be of low accuracy. In the simple
case studied here, a well-defined correction has been identified that is missing from
the starting point of most modern approximate schemes, i.e., the gradient expansion,
and its recovery has greatly improved accuracy in model cases.
Further work will follow.
This research was supported by NSF (CHE 1856165). Kieron Burke thanks Bob Cave,
Attila Cangi, and Raphael Ribeiro
for critical readings of the manuscript.
\bibliographystyle{apsrev}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,922 |
\section{Introduction}
The pattern of galaxy clustering in three dimensions, and its evolution,
encodes abundant information on the cosmological parameters affecting
matter growth. Ongoing and next generation spectroscopic galaxy surveys
will vastly increase our measurements of this clustering, and our
knowledge of cosmology if we can accurately interpret the results in terms
of theory. Measurements accrue an extra contribution to the redshift, and
hence apparent position along the sight, from the galaxy peculiar velocities
induced by the inhomogeneous density field; this gives rise to an anisotropy
in the observed clustering known as redshift space distortions (RSD).
These distortions carry information on the growth rate, as opposed to
just the growth amplitude, and so are valuable for probing cosmology, as
well as the gravitational strength driving the growth. However, linear
theory is insufficient for accurate relation of the redshift space galaxy
power spectrum to the true (real space) matter density power spectrum,
even on quite large scales, or wavenumbers $k>0.05\,h$/Mpc, where the vast
majority of the statistical leverage lies
\cite{percwhite,okujing,jennings1,jennings2,tang,kll}.
Numerous corrections involving higher order perturbation theory have been
employed \cite{scoccimarro,taruya,reidwhite,seljakmoment}
that extend the validity but the region $k>0.1\,h$/Mpc is still
problematic, especially for quantities involving the growth rate and
the gravitational growth characterization. For example, \cite{kll}
demonstrates that these first principles approaches deliver results for
the growth rate that are biased by several standard deviations when using
modes out to $k=0.2\,h$/Mpc.
Here we investigate a basic question: how accurately does one actually need
to know the redshift space distortion mapping in order to extract the
cosmological and gravitational parameter information without substantial
bias or degradation? Similar questions have been considered for weak
gravitational lensing, for example, where one asks how well the nonlinear
matter power spectrum needs to be known to estimate cosmology from the
lensing shear power spectrum \cite{huttak,hearin}.
In Section~\ref{sec:method} we introduce the correction, or reconstruction,
function for the redshift space power spectrum and review the KLL \cite{kll}
form for it. Section~\ref{sec:bias} uses the Fisher bias method to
compute both the individual parameter bias and joint confidence contour
bias due to misestimated RSD, thus giving criteria for the accuracy to
which the RSD effects must be known. Adding fit parameters for uncertainties
in the reconstruction function in Sec.~\ref{sec:marg}, we assess the
impact of marginalizing over them on the cosmological parameters, in
particular for tests of dark energy and gravity. Section~\ref{sec:concl}
summarizes the results and conclusions.
\section{Galaxy Power Spectrum} \label{sec:method}
\subsection{Redshift Space Power Spectrum} \label{sec:model}
In real space the matter density power spectrum is expected to be
isotropic, and the linear power spectrum grows in a scale independent
manner through the growth factor $D(z)$, where $z$ is the redshift.
The observed, redshift space galaxy power spectrum involves a transformation
to redshift space due to the velocity effects, and a bias relation $b(z)$,
usually taken to be scale independent, converting the dark matter overdensity
to galaxy overdensity, and the effects of nonlinear structure formation.
Each of these is modeled in various ways, with attendant uncertainties.
We write the anisotropic redshift space galaxy power spectrum as
\begin{equation}
P(k,\mu,z)=P^r(k,z)\,M(k,\mu,z)\,F(k,\mu,z) \label{eq:pfactors}
\end{equation}
where $P^r$ is the isotropic real space matter power spectrum, $M$ is an
approximate model for redshift space distortions (including galaxy bias),
and $F$ is the reconstruction function accounting for nonlinearities and
more exact velocity effects.
The linear mass power spectrum $P^r$ is given by
a Boltzmann numerical code such as CAMB \cite{camb}. It depends on
the cosmological parameters through its shape as a function of wavenumber
$k$ and through the growth factor $D(z)$ giving its amplitude evolution.
Since we will correct the RSD modeling by the reconstruction function, we choose
$M$ to be simply given by the linear theory prediction, the Kaiser approximation
\cite{kaiser87},
\begin{equation}
M(k,\mu,z)=[b(z)+f(z)\,\mu^2]^2 \ ,
\end{equation}
where $b$ is the galaxy bias, $f=d\ln D/d\ln a$ the growth rate of
density perturbations that in the linear regime grow as $\delta\sim D(a)$,
where the scale factor $a=1/(1+z)$, and $\mu$ is the cosine of the angle made
by the perturbation wavevector $\vec k$ with respect
to the line of sight. Beyond the linear regime, $b$ could be scale
dependent, i.e.\ $b(k)$, but we will absorb that possibility into the
reconstruction function. The reconstruction function is fitted to N-body
simulations by the analytic form of Kwan, Lewis, \& Linder (KLL; \cite{kll}),
\begin{equation}
F(k,\mu,z)=\frac{A(k,z)}{1+B(k,z)k^2\mu^2}+C(k,z)k^2\mu^2 \ . \label{eq:fform}
\end{equation}
This form has been found to reproduce accurately results of N-body
simulations over a wide range of redshifts, and for halos of various
masses as well as dark matter; see \cite{kll,julithesis} for details.
Note that $A$, $B$, $C$ may be cosmology dependent, just as $f$ and $P^r$
are, and their universality is a subject of ongoing research, but here
we treat them as independent parameters (as an analogy, recall how people
treat coefficients within Halofit also as universal, though here we let the
values of $A$, $B$, $C$ float; also see Sec.~\ref{sec:halofit}).
The factor $A$ characterizes nonlinearity of
the real space power spectrum, $B$ describes velocity effects such as
damping from a Lorentzian velocity dispersion but also higher order multipole
terms, while $C$ describes nonlinear enhancement for large $k\mu$ and
possibly breaks the degeneracy in the two roles of $B$.
\subsection{Galaxy Clustering Information}
The cosmological information inherent in the galaxy power spectrum can
be estimated through the Fisher information matrix. The full set of
parameters $\{p_i\}$ includes the cosmological parameters, astrophysical
parameters such as galaxy bias, and parameters for the reconstruction
function. Sensitivity
to cosmology enters through the derivatives $\partial P/\partial p_i$
and the error covariance matrix for the redshift space galaxy power
spectrum $P$.
We follow the usual prescription \cite{fkp,seoeis03,stril} where the
covariance
matrix comes from sample variance (finite volume) and shot noise (finite
resolution of the density field by sparse galaxies). Taken together, the
error can be thought of as depending on the number of modes that the galaxy
redshift survey samples. Treated as Poisson sampling of the density field,
the statistical error is
\begin{equation}
\sigma_P=P+n^{-1}
\end{equation}
from these two effects. The number of Fourier modes is the volume of
a Fourier cell times the number of cells,
\begin{equation}
N_{\rm modes}=2\pi k^2 dk\,d\mu \times V_{\rm survey}/(2\pi)^3 \ .
\end{equation}
Therefore the error covariance matrix $C$ is
\begin{equation}
C=P^2\,\left(\frac{1+nP}{nP}\right)^2 \, \frac{8\pi^2}{V_{\rm survey}k^2 dkd\mu} \ .
\end{equation}
Since the Fisher information matrix is constructed from $C^{-1}$ multiplied
by the sensitivity derivatives $\partial P/\partial p_i$, we can use the
$P^2$ factor to convert the derivatives to involve $\ln P$, which will
be useful in treating the multiplicative factors in Eq.~(\ref{eq:pfactors}).
In summary, the Fisher matrix is
\begin{equation}
F_{ij}=\sum_{z}\sum_{\mu}\sum_{k}\,\frac{\partial\ln P}{\partial p_i}
\frac{\partial\ln P}{\partial p_j}\, V_{\rm eff}(k,\mu,z)
\frac{k^2\Delta k\,\Delta\mu}{8\pi^2} \ ,
\end{equation}
where the survey volume is reduced by the shot noise to a $z$, $k$, and
$\mu$ dependent effective volume
\begin{equation}
V_{\rm eff}(k,\mu,z)=V_{\rm survey}(z)\,
\left[\frac{n(z)P(k,\mu,z)}{n(z)P(k,\mu,z)+1}\right]^2 \ . \label{eq:veff}
\end{equation}
When the galaxies densely sample the underlying field, the effective
volume approaches the survey volume, otherwise modes are lost, diluting
the effective volume due to increased noise.
Note that the logarithmic derivatives can be written as
\begin{eqnarray}
\frac{\partial\ln P}{\partial p_i}\,\frac{\partial\ln P}{\partial p_j}&=&
\frac{(\partial\ln P^r+\partial\ln M+\partial\ln F)}{\partial p_i}\notag\\
&\qquad&\times\frac{(\partial\ln P^r+\partial\ln M+\partial\ln F)}{\partial p_j}\,
\end{eqnarray}
so only the $\partial\ln F$ term depends on $A$, $B$, and $C$.
The reconstruction function derivatives are
\begin{eqnarray}
\frac{\partial F}{\partial A}&=&\frac{1}{1+Bk^2\mu^2} \\
\frac{\partial F}{\partial B}&=&\frac{-Ak^2\mu^2}{(1+Bk^2\mu^2)^2} \\
\frac{\partial F}{\partial C}&=&k^2\mu^2 \ ,
\end{eqnarray}
and are otherwise taken not to depend on cosmology. This is because
we use $A$, $B$, $C$ purely as fiducial values, and investigate how
their variation (from astrophysics or cosmology) impacts the cosmological
parameter estimation.
Our fiducial case attempts to match $F$ to the simulation
results in \cite{kll}, with estimated
\begin{eqnarray}
A(k)&=&1+\left(\frac{k}{0.39\,h/{\rm Mpc}}\right)^{1.58} \label{eq:aform}\\
B(k)&=&20\,({\rm Mpc}/h)^2 \label{eq:bform}\\
C(k)&=&8\,e^{-k/(0.176 h/{\rm Mpc})}\,({\rm Mpc}/h)^2 \ . \label{eq:cform}
\end{eqnarray}
The resulting redshift space distortion reconstruction function of
Eq.~(\ref{eq:fform}) is shown in Fig.~\ref{fig:fplot}.
We emphasize that these are merely the fiducials; we allow
the values of $A$, $B$, $C$ to float freely in bins of wavenumber.
This provides a model independent variation of the power spectrum (within
the reconstruction form) and we can then investigate the influence of such
variations on the cosmological parameter estimation. Conversely, the
question can be phrased as ``what is the accuracy
required on knowledge of the galaxy power spectrum in order to deduce the
cosmology with confidence?''
We later contrast this fiducial with fiducial $(A,B,C)=(1,0,0)$, i.e.\
assuming that perturbation theory (for example linear theory in the Kaiser
case, although $F$ also works with higher order perturbation theory
\cite{kll}) fully captures RSD effects in the model $M$.
\begin{figure}[htbp!]
\includegraphics[width=\columnwidth]{fplot.ps}
\caption{The redshift space distortion reconstruction function $F(k,\mu)$
is plotted for the fiducial expressions for $A$, $B$, $C$ for three values
of angle $\mu$.
}
\label{fig:fplot}
\end{figure}
The analysis is carried out in the next sections in two ways: in
Sec.~\ref{sec:bias} we compute the effect that a given level of unrecognized
power spectrum deviation in some $k$ bin, i.e.\ a systematic error in
modeling, has in biasing the cosmological conclusions, and in
Sec.~\ref{sec:marg} we recognize the existence of systematic uncertainties
and treat them by marginalizing
over the $A$, $B$, $C$ values for each $k$ bin.
\subsection{Survey Characteristics and Parameters} \label{sec:survey}
For the galaxy redshift survey data we consider a next generation
spectroscopic survey of the quality proposed for BigBOSS \cite{bigboss},
covering 14000 deg$^2$ from $z=0.1-1.8$, with a galaxy number density $n$ of
approximately $3\times 10^{-4}\,h^3\,{\rm Mpc}^{-3}$. For the exact
distribution adopted see Table~\ref{tab:nz}. There are actually two
populations of galaxies: luminous red galaxies (LRG) and emission line
galaxies (ELG), each with their own galaxy bias value. These galaxy
biases are taken as free parameters to be marginalized over, for each
redshift bin of width $0.1$. Their fiducials are $b(z)=b_0 D(z=0)/D(z)$
with $b^{\rm ELG}_0=0.8$ and $b^{\rm LRG}_0=1.6$, which provide good fits
to observations. Galaxy populations
with different biases can help reduce sample variance \cite{selmc}, with
the Fisher matrix involving a sum over populations, i.e.
\begin{equation}
\sum_{XY}
\frac{\partial\ln P_X}{\partial p_i}\frac{\partial\ln P_Y}{\partial p_j}
\,\left[\frac{n_X P_X}{n_XP_X+1}\right] \left[\frac{n_Y P_Y}{n_YP_Y+1}\right]
\ .
\end{equation}
Note that for multiple populations the factor $V_{\rm eff}$ in
Eq.~(\ref{eq:veff}) involves the shot noise, i.e.\ $nP$, of each population.
\begin{table}[!htb]
\begin{tabular}{ccc}
$z$ & $\ n_{\rm ELG}\ $ & $\ n_{\rm LRG}\ $\\
$\ $0.15$\ $ & 22.6 & 30.1\\
0.25 & 8.45 & 3.04\\
0.35 & 4.02 & 3.07\\
0.45 & 2.65 & 3.09\\
0.55 & 2.99 & 3.10\\
0.65 & 3.99 & 3.11\\
0.75 & 5.15 & 3.12\\
0.85 & 5.36 & 1.89\\
0.95 & 5.02 & 0.33\\
1.05 & 4.80 & 0.04\\
1.15 & 4.49 & --\\
1.25 & 4.04 & --\\
1.35 & 3.02 & --\\
1.45 & 2.00 & --\\
1.55 & 1.15 & --\\
1.65 & 0.43 & --\\
1.75 & 0.12 & --\\
\end{tabular}
\caption{Spectroscopic survey number densities adopted for emission line
galaxies and luminous red galaxies, in units of $10^{-4}\,h^3/{\rm Mpc}^3$,
for each redshift shell.
}
\label{tab:nz}
\end{table}
For the cosmological parameters we use the physical baryon density
$\Omega_b h^2$ and physical cold dark matter density $\Omega_c h^2$,
reduced Hubble constant $h$, scalar perturbation tilt $n_s$ and
amplitude $A_s$, dark energy equation of state parameters $w_0$
and $w_a$, and gravitational growth index $\gamma$. The gravitational
growth index gives an accurate description of the growth rate for both
general relativity and a range of modified gravity models, and looking
for deviations from its general relativistic value of $\gamma=0.55$ acts
as a test of gravity \cite{groexp,lincahn}. The growth index is
treated as an independent parameter (not a function of $w_0$, $w_a$) and
determines the growth factor at scale factor $a=1/(1+z)$,
\begin{equation}
D(a)=e^{\int_0^a (da'/a')\,\Omega_m(a')^\gamma} \ ,
\end{equation}
that in this ansatz is used to convert the linear power spectrum
delivered by CAMB at $z=0$ to another redshift $z$, to account for
the effects of modified gravitational growth. Note that the growth
rate $f=\Omega_m(a)^\gamma$, and redshift space distortions were highlighted
as a test of gravity in \cite{linrsd}.
For the central question of RSD uncertainties we employ up to 12
free parameters, taking $A$, $B$, $C$ with independent values in each
bin of width $0.1$ in wavenumber above $k=0.1\,h$/Mpc out to some $\kmax$.
This corresponds
to uncertainties $\Delta P_k$. For the current work we follow
\cite{huttak,hearin} and consider the uncertainties only as a function
of wavenumber, not redshift, except in Sec.~\ref{sec:reddep};
we also take the KLL form to be accurate while allowing freedom in the
parameters $A$, $B$, $C$. In summary we fit
for 8 cosmological parameters and up to 39 systematics parameters.
\section{Fisher Bias} \label{sec:bias}
The first question we are interested in answering is what is the
sensitivity of the cosmological parameter determination to errors in
modeling RSD. One might have $M$ or $F$ wrong, but if this does not
mimic a change in cosmology then no harm is done. The Fisher bias
formalism (see, e.g., \cite{fisbias1,fisbias2}) propagates misestimation
of the
observable or theoretical prediction, in this case the redshift space
power spectrum, into biases on the fit parameters. Specifically, we
consider the effect of errors in the $k$ bin values of $A$, $B$, $C$.
The Fisher bias on a parameter $p_i$ from misestimating parameter $p_a$ is
\begin{equation}
\delta p_i=\delta p_a \, \sum_{j} (F^{\rm sub})^{-1}{}_{ij}\,
(F^{\rm full})_{ja} \ ,
\end{equation}
where $\delta p_a=p_a({\rm true})-p_a({\rm fiducial})$. The superscript
``sub'' denotes the Fisher submatrix without entries for the parameters
whose misestimation we are studying. (For the specific case here, the
submatrix will be $35\times 35$ for the cosmology and galaxy bias parameters,
and the full matrix adds the reconstruction parameters one at a time.
We later consider all the reconstruction parameters at once.)
By evaluating the ratio $dp_i/dp_a$ for
$a=A,B,C$ we can assess the sensitivity of the parameter estimation to the
RSD modeling. To take a weak lensing example, \cite{berkwl} found that
a particular form of matter power spectrum distortion with amplitude
$A_{NL}$ at high $k$ distorted estimation of $w_a$ derived from shear power
spectrum measurement by a leverage factor of 18: a misestimation of 10\%
in $A_{NL}$ yielded a $1.8\sigma$ bias in $w_a$.
The bias $\delta p_i$ can be compared to the statistical uncertainty
$\sigma(p_i)$ on the parameter, either directly or through the risk
statistic
\begin{equation}
R_i\equiv \sqrt{\sigma^2(p_i)+\delta p_i^2} \ .
\end{equation}
Treating the bias as a systematic error in this way, to restrict the
degradation in the statistical error to less than 20\%, say, requires
$\delta p_i/\sigma(p_i)<0.66$, thus putting a constraint on the allowable
modeling error $\delta p_a$ etc. We examine two, converse statistics:
the cosmological degradation caused by a certain fractional misestimation
of the reconstruction parameters $\delta p_a/p_a$, and the requirement
on the reconstruction parameter to bound the cosmological parameter bias
to less than a given factor of the statistical uncertainty,
$\delta p_i/\sigma_i$. These are respectively
\begin{eqnarray}
\frac{R_i}{\sigma_i}&=&\sqrt{1+\left(\frac{\delta p_i}{\delta p_a}\frac{\delta p_a}{p_a} \frac{p_a}{\sigma_i}\right)^2} \\
\frac{\delta p_a}{p_a}&=&\left(\frac{\delta p_i}{\delta p_a}\right)^{-1}
\frac{\delta p_i}{\sigma_i}\frac{\sigma_i}{p_a} \ .
\end{eqnarray}
Figure~\ref{fig:degr}, left panel, shows the matrix of
degradations $R_i/\sigma_i$ for fixed $\delta p_a/p_a=0.01$ (i.e.\ 1\%
uncertainty on the reconstruction parameters), where the columns are the
dark cosmological parameters and the rows are the reconstruction parameters.
The right panel gives a similar matrix of the reconstruction
requirements $\delta p_a/p_a$ for fixed $\delta p_i/\sigma_i=1$ (which
corresponds to $R_i/\sigma_i=1.41$). One can scale the results for
different fixed values according to the above equations.
The stripe structure arises because the bias from $A_{0.45}$, where the
subscript indicates the center of the $k$ bin, is of
opposite sign from that of $A_{0.25}$, and $A_{0.35}$ lies in between
near null effect, and similar for $B$ and $C$.
The degradations in determination of the dark parameters $w_0$, $w_a$,
$\gamma$ are less than 22\% for 1\% shifts in the reconstruction parameters
in all cases except $A_{0.25}$, $A_{0.45}$, and $B_{0.45}$.
For $A_{0.25}$ and $A_{0.45}$ the risk error on $w_a$ can exceed the
statistical uncertainty by a factor 3. For the $B$ parameters the worst
case is degradation by 1.5. These results offer indications of what
physics must be most accurately understood, i.e.\ the nonlinearity from
$A$ and, somewhat less critically, the velocity effects from $B$.
In the converse analysis of what accuracy is required on the reconstruction
parameters to ensure that a bias is restricted to below $1\sigma$, we
find that 5\% accuracy is sufficient for all parameters except for the
$A_k$, plus $B_{0.25}$ and $B_{0.45}$. Knowledge of $A_{0.25}$ and
$A_{0.45}$ are needed to 0.3\%, $B_{0.45}$ to 0.9\%, $B_{0.25}$ to 1.4\%,
$A_{0.35}$ to 1.5\%, and $A_{0.15}$ to 3.1\%.
\begin{figure}[htbp!]
\includegraphics[width=\columnwidth]{Risi_dpapa.eps}
\caption{[Left panel] The ratio of the root mean squared error, or risk, to
the statistical uncertainty, $R_i/\sigma_i$, is plotted for each dark
cosmological parameter in the case of a 1\% deviation in a reconstruction
parameter. [Right panel] The fractional requirement on each reconstruction
parameter $\delta p_a/p_a$ needed to ensure bias less than $1\sigma$, i.e.\
$\delta p_i/\sigma_i<1$ is plotted. Dark red indicates danger (high risk
or tight requirement), with lighter colors showing reduced impact. For
the left panel the color scale is $R_i/\sigma_i>2$ (dark red), 1.4--2
(medium orange), 1.05--1.4 (light yellow), and 1--1.05 (white). The right
panel has $|\delta p_a/p_a|<0.01$ (dark red), 0.01--0.05 (medium orange),
0.05--0.2 (light yellow), $>0.2$ (white). Here $\kmax=0.5\,h$/Mpc.
}
\label{fig:degr}
\end{figure}
While these approaches give indications of sensitivity, they
treat the cosmological parameters one by one while a power
spectrum misestimation will generally impact several at once. This can
either tighten or loosen overall requirements, depending on the
covariances. To take this
into account we use the $\Delta\chi^2$ method \cite{dodelsonbias,shapiro}.
This describes the fuller impact of biasing cosmology through quantifying
how far from the fiducial the best fit cosmology is shifted relative to the
confidence contour, taking into account degeneracies between the
reconstruction and cosmological parameters. This measure is given by
\begin{equation}
\Delta\chi^2=\sum_{ij} \delta p_i\,F^{({\rm red})}_{ij}\,\delta p_j \ ,
\end{equation}
where the sum runs only over the reduced parameter set whose bias we
are interested in, e.g.\ $w_0$ and $w_a$ for a 2D $w_0$--$w_a$ joint
likelihood
contour plot. The reduced Fisher matrix $F^{({\rm red})}$ is marginalized
over all other cosmological and galaxy bias parameters (the reconstruction
parameters have already been taken into account in obtaining $\delta w_0$
etc.). The bias $\Delta\chi^2$ accounts for the property that biases in,
say, the direction of the narrow axis of the Fisher ellipse are more
detrimental than those along the degeneracy direction.
Figure~\ref{fig:chi2} illustrates the 2D bias induced in the dark energy
and growth parameters, here for a 1\% misestimation in the reconstruction
parameters one by one. Most such reconstruction parameter errors do not
significantly affect the joint parameter likelihood. In the $w_0$--$w_a$
plane, none of the $C$ parameters and three of the $B$ parameters do not
bias the best fit outside the $1\sigma$ contour, and
$B_{0.45}$ remains within the $2\sigma$ contour. Only errors on the
$A_{0.25}$ and $A_{0.45}$ parameters are particularly damaging, causing a
bias of up to $\Delta\chi^2=39$ (approximately at the $6\sigma$ level).
The covariance between the shifts in $w_0$ and $w_a$ is crucial; if the
same bias in $w_a$ and an even larger bias in $w_0$ occurred such that
the joint shift lay along the degeneracy axis, then the 2D bias would be
scarcely outside the $2\sigma$ uncertainty contour.
For the $w_a$--$\gamma$ plane, the biases are less severe, with only
$A_{0.25}$ and $A_{0.45}$ causing more than a $2\sigma$ joint bias, at
$\Delta\chi^2\approx 16$.
Treating the reconstruction errors one by one effectively takes a
localized bump in the reconstruction function. A smooth variation would
instead affect several of the reconstruction parameters
at once; this has a different effect on the cosmological parameter bias.
As an example, we simultaneously vary all four $A$ parameters by 1\%. Since
$A_{0.25}$ and $A_{0.45}$ have nearly opposite effects this reduces
substantially the $\Delta\chi^2$ due to varying just one of them, e.g.\ from
39 to 7.7. The 2D bias due to such smooth variation is shown in the figures
by the magenta squares.
\begin{figure}[htbp!]
\includegraphics[width=\columnwidth]{w0wabias.ps} \\
\includegraphics[width=\columnwidth]{wagambias.ps}
\caption{The biases in the $w_0$--$w_a$ and $w_a$--$\gamma$ planes
due to 1\% misestimation in the 12 reconstruction parameters, one by one,
are shown by x's (along with the $\Delta\chi^2$ if larger than 2.8).
The contours give the joint 2D $1\sigma$ and $2\sigma$ confidence levels
on the cosmological parameters when the reconstruction parameters take
their fiducial (``true'') values. Magenta squares show the biases when
varying all bins of $A$ simultaneously; such smooth variations are much
less damaging, e.g.\ reducing the individual $\Delta\chi^2=39$ and 34
biases to a joint $\Delta\chi^2=7.7$ offset (or the 16 and 15 in the
$w_a$--$\gamma$ panel to 2.8).
}
\label{fig:chi2}
\end{figure}
While we have thus far been model independent in taking $A$, $B$, $C$
to be independent from one $k$ bin to the next, we can now consider
the difference between two fiducial models for the overall reconstruction
function $F$. This then includes the effects of variations at all $k$'s
simultaneously, and allows a study of bias as a function of $\kmax$.
As we consider
each successive bin at higher $k$, we increase the number of modes,
reducing the statistical uncertainty, but also often increase the deviation
in the power spectra, increasing the bias in the cosmological parameters
if we assume the wrong fiducial as the truth. The truth is taken to be
$F$ as given by Eqs.~(\ref{eq:aform})--(\ref{eq:cform}) in
Eq.~(\ref{eq:fform}), while the incorrect assumption is pure linear theory,
i.e.\ simply the Kaiser form for redshift space distortions, equivalent to
$A=1$, $B=C=0$.
This misestimation of the redshift space galaxy power spectrum causes a
bias in cosmological parameters of
\begin{eqnarray}
\!\!\delta p_i&=&\left (F^{\rm sub}\right)^{-1}_{ij} \sum_z \sum_\mu \sum_k \,
\frac{P(A,B,C)-P(1,0,0)}{P(A,B,C)} \notag\\
&\qquad&\times\frac{\partial\ln P}{\partial p_j}\, V_{\rm eff}(k,\mu,z)
\frac{k^2\Delta k\,\Delta\mu}{8\pi^2} \ .
\end{eqnarray}
The systematic biases tend to worsen with increasing $\kmax$, reaching
$1.4$ in $w_0$, $-8$ in $w_a$, and 0.2 in $\gamma$ for $\kmax=0.5\,h$/Mpc,
and are much larger than the statistical uncertainties for all
$\kmax$. This is not surprising since $F_{\rm Kaiser}$ can deviate
by a factor 2 from the KLL form. Thus, neglecting the uncertainties in
the reconstruction parameters is not a viable option: we must take them
into account.
\section{Marginalization and Selfcalibration} \label{sec:marg}
As an alternative to requiring the power spectrum to subpercent accuracy
and computing the bias from misestimated reconstruction
parameters, we can fit for those parameters and calculate the increased
uncertainty in cosmological parameters due to marginalization over the
extra inputs. The basic question is how well the model needs to be
known for precision determination of cosmology with RSD. This is similar
to what \cite{huttak,hearin} did for matter power
spectrum uncertainties applied to weak lensing cosmology. They used
fractional power spectrum uncertainties in wavenumber bins, assumed
constant with redshift, and applied some level of priors.
\subsection{Global Fit} \label{sec:global}
Now our quantities $A_k$, $B_k$, $C_k$ in each wavenumber bin become
fit parameters. Again, we can study the effects as we extend $\kmax$,
using more bins and hence more parameters. Including these parameters
means that we will not be biased any more with respect to the fiducial,
but the enlarged parameter space will lead to some level of degradation
of the statistical uncertainties, relative to fixing the reconstruction
parameters, at the same $\kmax$.
Table~\ref{tab:sigkmax} shows the effect of extending the data to
higher $\kmax$, while simultaneously allowing for the additional
reconstruction parameters in each bin. Despite the additional degrees
of freedom in the fit, the cosmological parameter estimation sharpens
with increasing $\kmax$. As long as the {\it form\/} of the reconstruction
function holds, we obtain an accurate and unbiased cosmology even allowing
for fitting variation in the amplitudes of $A$, $B$, $C$ in each $k$ bin.
This is an extremely promising initial result for use of the reconstruction.
\begin{table*}[!htb]
\begin{tabular}{l|ccccccccc}
&$\Omega_b h^2$ &$\Omega_c h^2$ &$h$&$n_s$&$10^9 A_s$&$w_0$&$w_a$&$\gamma$&
$\Omega_m$\\
\hline
Fiducial& $\ $0.0226& 0.112& 0.7& 0.96& 2.47& -0.99& 0& 0.55& 0.275\\
$\sigma(\kmax=0.1)$\ & $\ $ 0.00524 & 0.0189 & 0.0542 & 0.0524 & 0.538 &
0.599 & 2.23 & 0.177 & 0.0302\\
$\sigma(\kmax=0.2)$\quad & $\ $0.00284 & 0.0102 & 0.0284 & 0.0288 & 0.325 & 0.197 & 0.779 & 0.0519 & 0.0159\\
$\sigma(\kmax=0.3)$\ & $\ $0.00219 & 0.00760 & 0.0219 & 0.0198 & 0.248 &
0.112 & 0.478 & 0.0272 & 0.0122\\
$\sigma(\kmax=0.4)$\ & $\ $ 0.00148 & 0.00508 & 0.0150 & 0.0130 & 0.170 &
0.0824 & 0.347 & 0.0193 & 0.00834\\
$\sigma(\kmax=0.5)$\ & $\ $0.00141 & 0.00478 & 0.0142 & 0.0119 & 0.158 & 0.0713 & 0.306 & 0.0163& 0.00794\\
\end{tabular}
\caption{$1\sigma$ constraints from future galaxy power spectrum data on
cosmological parameters, marginalized over galaxy bias and redshift space
distortion reconstruction. Note $\Omega_m$ is a derived parameter; $\kmax$ is
in units of $h$/Mpc. Despite the addition of more fit parameters when
increasing $\kmax$, the cosmological parameters can be better determined.
}
\label{tab:sigkmax}
\end{table*}
To better understand why the added fit parameters do not cause an overall
degradation, we look at the correlation matrix of the 47 parameters in
Fig.~\ref{fig:corrmat}. The block of parameters 36--47, representing the
reconstruction parameters, is not highly correlated with other parameters:
correlation coefficients are below 0.58 (0.38 for parameters other than
$n_s$).
(Even within the block, only $B_{0.15}$ and $C_{0.15}$, other than
between the $A_k$, have a correlation coefficient exceeding 0.8,
reaching 0.90; this is expected
since for a low $k$ expansion both $B$ and $C$ contribute as $\mu^2$.)
This means that the change in power spectrum shape due to adjusting the
amplitudes of these parameters in $F$ is not degenerate with a change due
to $w_0$ or other such parameters. That is, the influence of these
parameters have different $k$ and $\mu$ dependences than those of
cosmological parameters and so we find they can be separately fit.
\begin{figure}[htbp!]
\includegraphics[width=\columnwidth]{CORRmatrix.eps}
\caption{Correlation matrix of the 47 parameters for $\kmax=0.5\,h$/Mpc is
shown with color shading reflecting the absolute value of the correlation
coefficient $r_{ij}=C_{ij}/\sqrt{C_{ii} C_{jj}}$. The correlation matrix is
mostly block diagonal and the cosmological parameters are not strongly
correlated with the reconstruction (or galaxy bias) parameters, so
marginalization does not badly degrade cosmological parameter estimation.
}
\label{fig:corrmat}
\end{figure}
Moreover, the reconstruction parameters are selfcalibrated by the data
to good precision. All are determined to better than 3\% (except
$C_{0.15}$, to 8\%) and most to subpercent level. These propagate into
determination of the power spectrum to the subpercent level for variation
of each one individually by $1\sigma$, except for the extreme cases of
$\mu=1$ and $B_{0.15}$
($C_{0.15}$) which gives 1.1\% (1.2\%) uncertainty. Most combinations,
however, give subpercent precision. Adding all their
uncertainties in the most unfavorable way generates an extreme of 2.6\%
power spectrum uncertainty.
Thus unlike the weak lensing probe analyzed by \cite{huttak,hearin},
redshift space distortions do not require any priors to be placed on the
power spectrum
parameters (assuming that the KLL reconstruction form is valid).
Remarkably, in addition
to selfcalibration, the additional fit parameters have little impact on
the cosmological parameter estimation, enlarging the uncertainties by
only 9\%, 22\%, and 7\% on $w_0$, $w_a$, and $\gamma$. And of course
including the extra parameters removes any cosmology bias as suffered
in the previous section (modulo model validity).
Regarding the 12 extra reconstruction parameters, from Fig.~\ref{fig:fplot}
we see that $F$ is smooth in $k$ so taking bins of width 0.1 in $k$ is
reasonable. For completeness, for bins of width 0.02 (and hence 60 extra
parameters) we find that cosmological parameter estimation is mildly
degraded, with uncertainties on $w_0$, $w_a$, $\gamma$ increasing relative
to 0.1 width by 16\%, 33\%, 17\%.
\subsection{Maximum Wavenumber} \label{sec:kmax}
Let us examine the dependence of the results on the maximum wavenumber
$\kmax$ used.
Note that for the $\kmax=0.1$ case, the cosmology parameters are not
determined particularly well even though no reconstruction parameters
are used for $k\le0.1\,h$/Mpc. This is due to strong covariance with
the 27 galaxy bias parameters. Once beyond the linear regime, this
degeneracy is broken and the correlation coefficients drop, greatly
improving the cosmological parameter determination (e.g.\ by factors of
2.9--3.4 on the dark parameters, for $\kmax=0.2$ relative to $\kmax=0.1$).
This continues for higher $\kmax$,
despite the addition of further reconstruction parameters, but gradually
saturates. For example, while relative to the $\kmax=0.5$ case the
uncertainties on $w_0$, $w_a$, or $\gamma$ at $\kmax=0.2$ are greater by
a factor $\sim3$, at $\kmax=0.3$ the factor is 1.6, and at $\kmax=0.4$
is 1.15, as seen in Fig.~\ref{fig:kmaxratio}. Thus, having an accurate
reconstruction form out to $\kmax\approx 0.4-0.5$ is sufficient for
robust cosmological parameter estimation, while selfcalibration obviates
the need for any prior knowledge of the values of the reconstruction
parameters.
\begin{figure}[htbp!]
\includegraphics[width=\columnwidth]{kmaxratio.ps}
\caption{Extending $\kmax$ to values above $0.1\,h$/Mpc breaks
degeneracies, leading to improvements in cosmological parameter estimation
as shown here, even given the addition of reconstruction parameters to
marginalize over. Reconstruction to $\kmax=0.4-0.5\,h$/Mpc is sufficient
to plateau the cosmology estimation precision.
}
\label{fig:kmaxratio}
\end{figure}
Binning such as we use is model independent and closest to the weak
lensing work.
This model independence is important since in the absence of a large suite of
simulations we may have no particular confidence in parametrizations such
as those in Eqs.~(\ref{eq:aform}-\ref{eq:cform}). Recall that those
equations merely give the
fiducial values in each $k$ bin, and then we allow the bin values to
float freely and marginalize over them. Given simulations we might be
able to adopt specific forms and fit for a reduced set of parameters,
e.g.\ the coefficients in those equations.
\subsection{Redshift Dependence} \label{sec:reddep}
To give a first indication of whether adding redshift dependence to
$A$, $B$, $C$ affects the conclusions we include a variation of the
characteristic wavenumber scale entering in the nonlinearity amplitude
$A$ in Eq.~(\ref{eq:aform}), i.e.\ the $0.39\,h$/Mpc, writing this as
\begin{equation}
k_\star(z)=0.39\,(1+z)^{\alpha/1.58}\,h/{\rm Mpc} \ ,
\end{equation}
and adding this evolution parameter $\alpha$ to the fit. The simulation
results in \cite{julithesis} indicate that relative to $A$, the parameters
$B$ and $C$ have negligible additional redshift dependence. Therefore we
scale $B$ and $C$ by the same factor as $A$, i.e.\
\begin{eqnarray}
B(k,z)&=&B(k,0)\,A(k,z)/A(k,0) \\
&=&B(k,0)\,\frac{1+[A(k,0)-1](1+z)^{-\alpha}}{A(k,0)} \ ,
\end{eqnarray}
and the same for $C$.
The introduction of redshift dependence through $\alpha$ has little
impact on the cosmological parameter estimation; the largest correlation
coefficient of $\alpha$ cosmologically is 0.31, with $\gamma$, and overall
0.83 with $B_{0.45}$, while $\alpha$ itself is determined to within 0.025.
Figure~\ref{fig:ellalpha} shows the influence on dark cosmology parameter
estimation of marginalization over the reconstruction
parameters with and without redshift dependence, and fixing the
reconstruction parameters (i.e.\ with a total of 48, 47, or 35
parameters). Uncertainties on $w_0$, $w_a$, and $\gamma$
increase by only 0.8\%, 0.2\%, 5\% respectively upon including
$\alpha$. Other forms of redshift dependence may have different
cosmological impact, and this deserves further analysis through
simulations, but the scaling of the characteristic wavenumber as used
here should give a reasonable first indication.
\begin{figure}[htbp!]
\includegraphics[width=\columnwidth]{w0wa4748.ps}\\
\includegraphics[width=\columnwidth]{wagam4748.ps}
\caption{Joint 2D $1\sigma$ confidence contours on the dark cosmology
parameters are
shown for the cases of all reconstruction parameters being fixed
(dotted red), marginalized over without redshift dependence as
in most of the article (solid black), and additionally marginalizing
over a redshift evolution parameter $\alpha$ (dashed blue). Here
$\kmax=0.5\,h$/Mpc. Note that the fixed $F$ case, shown here centered
on the true cosmology, could be strongly biased if $F$ was misestimated
(see Sec.~\ref{sec:bias}).
}
\label{fig:ellalpha}
\end{figure}
\subsection{Nonlinear Power Spectrum} \label{sec:halofit}
The greatest effect of uncertainty in the reconstruction function comes
from the parameters $A_k$, as seen in Fig.~\ref{fig:fplot} and in
Sec.~\ref{sec:bias} regarding the parameter bias. Recall that $A(k)$
arises from the nonlinear effects in the density field, and even exists
when $\mu=0$. In this limit $A(k)$ acts to map the linear real space
density power spectrum to the nonlinear regime. Therefore, if we had a
robust nonlinear (or quasilinear) real space power spectrum then we would
have no need of a separate parameter as then $A(k)=1$ (this has been tested
and found
accurate to subpercent level by \cite{julithesis}). Substantial
effort is going into developing cosmic emulators \cite{emu} that could
provide accurate nonlinear power spectra, eventually including the
full set of cosmological parameters and redshifts considered here.
Since that is still in the future, we consider the nonlinear prescription
of Halofit \cite{halofit} to get an indication of the potential impact on
our conclusions.
The linear power spectrum at a given redshift is fed into Halofit to
give the approximate nonlinear form. This removes the need for $A(k)$,
setting this equal to one for all $k$ and $z$. As indicated earlier in
this section, the simulation results from \cite{julithesis} imply that
for such a normalized $A$ then the quantities $B$ and $C$ become
substantially redshift independent. Therefore we do not need any
hypothetical model such as the $\alpha$ parametrization, making the
entire analysis more robust. Furthermore, Halofit includes cosmology
dependence and so no assumption about universality of $A$ is needed.
We show the results for cosmological parameter estimation in
Table~\ref{tab:halofit}, for $\kmax=0.5\,h$/Mpc, for the three cases of
using the model
independent approach of fitting for $A(k)$ in bins, using Halofit and $A=1$,
and using the revised version of Halofit from \cite{12082701} and
$A=1$. In all cases we still fit for the binned values of $B$ and $C$.
The use of functional forms for the nonlinear real space power spectrum
allows better determination of the cosmological parameters, by 12\%,
28\%, 12\% for $w_0$, $w_a$, $\gamma$ respectively (15\%, 33\%, 15\% for
revised Halofit, which has slightly more quasilinear power). This offers some
promise for the future use of cosmic emulators, but in this paper we
prefer to be conservative in the estimations and use the model independent
approach.
\begin{table}[!htb]
\begin{tabular}{l|cccc}
&$w_0$&$w_a$&$\gamma$&$\Omega_m$\\
\hline
Fit $A(k)$& $\ $0.0713$\ $ & $\ $0.306$\ $ & $\ $0.0163$\ $& $\ $0.00794$\ $\\
Halofit& 0.0624 & 0.220 & 0.0143 & 0.00678\\
New Halofit$\ $& 0.0603 & 0.206 & 0.0139 & 0.00608\\
\end{tabular}
\caption{$1\sigma$ constraints as in Table~\ref{tab:sigkmax}, using
$\kmax=0.5\,h$/Mpc, but for three different methods of treating
nonlinearities.
}
\label{tab:halofit}
\end{table}
\section{Conclusions} \label{sec:concl}
With the ability to map galaxy clustering in three dimensions over large
volumes of the universe comes the necessity for accurate theoretical
interpretation. This entails linking the isotropic, linear theory real space
density power spectrum to the observed anisotropic, nonlinear redshift space
galaxy power spectrum. We have investigated here some of the relevant issues
involving nonlinearities in the density field and velocity effects, using
the Kwan-Lewis-Linder analytic redshift space reconstruction function
calibrated from numerical simulations.
The main question addressed is to what accuracy the anisotropic redshift
space power spectrum must be known in order to achieve robust cosmological
conclusions. We propagate uncertainties in the power spectrum through a
model independent binning of reconstruction amplitudes with wavenumber and
assess the effects of deviations from fiducial values. To avoid
biasing cosmological parameters such as the dark energy equation
of state and gravitational growth index requires down to 0.3\% accuracy
on the reconstruction parameters in the most stringent cases, while smoother
deviations give more tractable requirements. Note that it is only those
deviations that mimic cosmological variations that are most important.
A more flexible and robust approach is to carry out a global fit for the
binned reconstruction parameters simultaneously with the cosmological
parameters, which avoids biasing the results so long as the {\it form\/}
of the KLL function is a good approximation. With 8 cosmology parameters
and 39 systematics parameters we find that a next generation galaxy redshift
survey such as BigBOSS can tightly and accurately constrain cosmology, for
example determining the equation of state time variation $w_a$ to 0.3 and
testing gravity through the growth index $\gamma$ to 3\%. No external
priors on the reconstruction parameters are necessary as they are
selfcalibrated by the survey, most at the subpercent level. This also
corresponds to subpercent calibration of the redshift space power spectrum.
We have tested the robustness of the conclusions by adding redshift
evolution, which has little effect, varying the number of wavenumber bins,
and exploring the leverage from increasing the maximum wavenumber used,
$\kmax$. Cosmological leverage plateaus by $\kmax=0.4-0.5\,h$/Mpc so the
KLL form need only apply up to this scale. We made no assumptions about
the departure from linearity, allowing the nonlinearity amplitude to float in
a model independent manner in bins of $k$, but also then analyzed the impact
of adopting a nonlinear prescription such as Halofit (or its revision).
This improved the cosmology estimation and offers a promising sign to
motivate continued development of cosmic emulators for the nonlinear
power spectrum.
Several areas exist for further development. The KLL form has been
tested for dark matter, but \cite{julithesis} indicates it is successful
for halos as well. Eventually this must be extended to galaxies, a major
undertaking. On the positive side, we achieved excellent results using
reconstruction starting from simple linear theory (Kaiser approximation);
higher order perturbation theory approaches extend the range where
reconstruction is milder. Universality, i.e.\ cosmology dependence, of
the reconstruction is a major topic for future investigation, requiring
large suites of cosmological simulations, again suited for emulators.
We have taken a first step toward addressing this effect by exploring
the influence of using Halofit and new Halofit cosmological dependences for
the nonlinearity. Simulations may also enable us to compress the information
in bins down to a smaller set of parameters.
Redshift space distortions provide a powerful tool for measuring the
growth rate of cosmic structure, and delivering insights on the competition
between the gravitational laws driving clustering and accelerated expansion
suppressing it. The results here give encouraging indications, and
quantitative measures, that theoretical analysis can take into account
robustly the nonlinear and velocity effects to extract accurate cosmology
from the forthcoming large volume redshift surveys.
\acknowledgments
We thank Sudeep Das, Juliana Kwan, and Alberto Vallinotto for helpful
discussions.
This work has been supported by DOE grant DE-SC-0007867 and the Director,
Office of Science, Office of High Energy Physics,
of the U.S.\ Department of Energy under Contract No.\ DE-AC02-05CH11231.
EL acknowledges World Class University grant R32-2009-000-10130-0 through the
National Research Foundation, Ministry of Education, Science and Technology
of Korea; JS is supported by the Dark Cosmology Centre, funded by the
Danish National Research Foundation, and thanks LBNL for additional support.
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\section{Introduction}
One of recent important developments in study of the AdS/CFT
correspondence is the duality between
gluon scattering amplitudes in ${\cal N}=4$ super
Yang-Mills theory and the area of the minimal surface in AdS spacetime
surrounded by the closed light-like Wilson loops.
From the duality one can compute the gluon scattering amplitudes
at strong coupling.
In the case of the 4-point amplitude, Alday and Maldacena \cite{AlMa}
showed that
the dimensionally regularized area agrees with the formula conjectured by
Bern, Dixon and Smirnov (BDS) \cite{BeDiSm}
based on the perturbative analysis.
See \cite{Al, AlRo,gluons,AlMa2, ItMiMo, AsDoItNa, DoItIw}
for further developments.
The duality between gluon scattering amplitudes and Wilson loops is
shown to hold at weak coupling \cite{Wilson}, which implies that
the amplitude is
invariant under dual conformal symmetry in momentum space
\cite{DrHeKoSo}.
In superstring theory,
this symmetry is interpreted as the symmetry of AdS spacetime and
invariance of the action under the combination of bosonic and fermionic
T-duality\cite{BeMa}.
The anomalous conformal Ward identity
constrains the structure of the 4 and 5-point amplitudes, which agrees
with the BDS formula.
But for higher $n(\geq 6)$-point amplitudes there arises some
ambiguities in the
finite remainder part of the amplitude which can be written in terms of
the conformal invariant cross-ratio of the external gluon momenta
\cite{DrHeKoSo}.
In fact, the explicit calculations of
the two-loop 6-point gluon scattering
amplitude
and the hexagon Wilson loop \cite{BeDiKoRoSpVeVo}
shows that they agree with each
other but differ from the BDS formula by finite term, which depends on
three independent cross-ratios of the Mandelstam variables.
This discrepancy from the BDS ansatz was also observed at strong
coupling by studying zigzag rectangular \cite{AlMa2} and a wavy
circular Wilson lines \cite{ItMiMo}.
But the precise evaluation of the finite deviation from the BDS formula
is difficult
to obtain since the exact solution of the minimal surface
for higher-point amplitudes is not yet known.
In a previous paper \cite{DoItIw}, we constructed the
minimal surfaces corresponding to
the 4, 6 and 8 point amplitudes numerically and evaluate the area
in the radial
cut-off regularization.
The light-like segments of the boundary is the same as
the cut and glue type surface \cite{AsDoItNa}.
We showed that the numerical solutions differ from the cut and glue type
surface and the area is consistent with the IR behavior of the
amplitude.
In this paper we will study the area of the discretized surfaces for
the 6 and 8-point amplitudes by applying conformal transformation and
compare the area to the conjectured BDS formula numerically.
This analysis gives a test of
the duality between gluon scattering amplitudes
and the Wilson loops at strong coupling.
This paper is organized as follows:
In section 2, we review the radial cut-off regularization and
a numerical approach to the construction of the minimal surface.
In section 3, we apply the conformal transformation to the 4-point
amplitudes and compare it with the exact formula of the area in the
radial cut-off regularization. We propose
a method to compare the numerical data with the BDS formula
without using the exact formula of the area in the radial cut-off
regularization.
In section 4, we apply this method to the minimal surfaces
corresponding to the 6 and 8-point amplitudes and compare the numerical
solutions with the BDS formula.
Section 5 is devoted to discussion.
\section{Radial cut-off regularization and discretized minimal surface}
In this section we review the radial cut-off regularization of the
minimal surface in AdS spacetime and a numerical approach to
get discretized version of minimal
surface.
We consider the surface which is
surrounded by the curve $C_n$ made of
light-like segments $\Delta y^\mu=2\pi p_i^\mu$.
This corresponds to the $n$-point gluon amplitude
with on-shell momenta $p_i$ ($p^2_i=0$, $i=1,\cdots, n$).
The coordinates $y^\mu$ ($\mu=0,1,2,3$) and the radial coordinate
$r$ are the Poincar\'e coordinates
in AdS$_5$ spacetime with the metric
\begin{equation}
ds^2=R^2{dy^\mu dy_\mu+dr^2\over r^2},
\end{equation}
and $R$ is the radius of AdS$_5$.
The Nambu-Goto action in the static gauge
$y_3=0$
is given by
\begin{equation}
S={R^2\over 2\pi}
\int dy_1 dy_2 \frac{\sqrt{1+(\partial_i r)^2-(\partial_i y_0)^2
-(\partial_1 r \partial_2 y_0-\partial_2 r \partial_1 y_0)^2}}{r^2}.
\label{eq:ngaction}
\end{equation}
Here $\partial_i$ is the derivative with respect to $y_i$ ($i=1,2$).
The Euler-Lagrange equations become
\begin{eqnarray}
&& \partial_i
\left(
\frac{\partial L}{\partial(\partial_i y_0)}
\right)=0,
\quad \partial_i
\left(
\frac{\partial L}{\partial (\partial_i r)}
\right)-\frac{\partial L}{\partial r}=0,
\label{eq:euler1}
\end{eqnarray}
where $L$ is the Lagrangian of the action.
By solving these non-linear partial differential equations,
one obtains the minimal surface $r=r(y_1,y_2)$ and $y_0=y_0(y_1,y_2)$.
We consider the $4$-point amplitude
for two incoming particles
with momenta $p_1$
and $p_3$ and outgoing particles with momenta $p_2$ and $p_4$.
For the momentum configuration in the $(y_0,y_1,y_2)$-space
\begin{eqnarray}
&& 2\pi p_1=(2,2,0), \quad 2\pi p_2=(-2,0,2), \quad 2\pi p_3=(2,-2,0),
\quad
2\pi p_4=(-2,0,-2), \nonumber\\
\label{eq:4ptmom1}
\end{eqnarray}
the Wilson loop is represented by
the square with corners at $y_1,y_2=\pm 1$.
The boundary condition for the Euler-Lagrange equations is given by
\begin{equation}
r(\pm 1, y_2)=r(y_1,\pm 1)=0,\quad
y_{0}(\pm 1, y_2)=\pm y_2, \quad
y_{0}(y_1,\pm 1)=\pm y_1.
\label{eq:4ptbc}
\end{equation}
Alday and Maldacena \cite{AlMa} found
the exact solution of the nonlinear differential equations
(\ref{eq:euler1}),
which is given by
\begin{equation}
y_0(y_1,y_2)=y_1 y_2,\quad r(y_1,y_2)=\sqrt{(1-y_1^2)(1-y_2^2)}.
\label{eq:4pt}
\end{equation}
The above solution corresponds to the $s=t$ solution, where
$s$ and $t$ are
the Mandelstam variables defined by $s=-(p_1+p_2)^2$
and $t=-(p_2+p_3)^2$.
The general $(s,t)$ solution is obtained by scale and
boost transformation of the $s=t$ solution:
\begin{equation}
r'={a r\over 1+b y_0},\quad
y'_0={a\sqrt{1+b^2}y_0\over 1+b y_0},
\quad
y'_i={a y_i\over 1+b y_0},
\label{eq:conf1}
\end{equation}
where $a$ is a parameter for the scale transformation and $b$ is a boost
parameter.
After the conformal transformation, the momenta become
\begin{eqnarray}
2\pi p_1&=&({2a\sqrt{1+b^2}\over 1-b^2},{2a\over 1-b^2}, -{2ab\over
1-b^2}),
\nonumber\\
2\pi p_2&=&(-{2a\sqrt{1+b^2}\over 1-b^2},-{2ab\over 1-b^2}, {2a\over
1-b^2}),
\nonumber\\
2\pi p_3&=&({2a\sqrt{1+b^2}\over 1-b^2},-{2a\over 1-b^2}, {2ab\over
1-b^2}),
\nonumber\\
2\pi p_4&=&(-{2a\sqrt{1+b^2}\over 1-b^2},{2ab\over 1-b^2}, -{2a\over
1-b^2}).
\end{eqnarray}
The Mandelstam variables $s$ and $t$ are given by
\begin{equation}
(2\pi)^2 s=-{8a^2\over (1-b)^2},\quad (2\pi)^2t=-{8a^2\over (1+b)^2}.
\end{equation}
Using the dimensional regularization for the $Dp$-brane
($p=3-2\epsilon$),
the area is shown to
agree with the BDS formula at strong coupling \cite{AlMa}.
In this paper we will use the radial cut-off regularization instead.
\subsection{Radial cut-off regularization}
In the radial cut-off regularization scheme we introduce a cut-off $r_c$ in the
radial direction\cite{Al,DoItIw}.
For general $(s,t)$ solution,
the regularized area is
surrounded by the cut-off curve $C$ in the $(y_1,y_2)$-plane:
\begin{equation}
r_c^2=a^2 (1-y_1^2)(1-y_2^2){1\over (1+b y_1 y_2)^2}.
\label{eq:cutoffc2}
\end{equation}
The action is evaluated by substituting the solution (\ref{eq:4pt})
into (\ref{eq:ngaction}).
The result is
\begin{equation}
S_4[r_c,b]=\int_S dy_1 dy_2 {1\over (1-y_1^2)(1-y_2^2)}.
\label{eq:int1}
\end{equation}
where $S$ is the region surrounded by the curve $C$.
We can put $a=1$ by rescaling $r_c\rightarrow r_c a$.
For fixed $y_1$, $y_2$ takes the value in the range
$y_{2}^{c-} \leq y_2\leq y_2^{c+}$,
where
\begin{eqnarray}
y_2^{c\pm}&=&{-b r_c^2 y_1\pm \sqrt{(1-y_1^2)(1-y_1^2-r_c^2 +b^2 r_c^2 y_1^2)}
\over 1-y_1^2 +b^2 r_c^2 y_1^2}.
\end{eqnarray}
In (\ref{eq:int1}), the integral over $y_2$ yields
\begin{equation}
S_4[r_c,b]=\int_{-\sqrt{{1-r_c^2\over 1-b^2 r^2_c}}}^{\sqrt{{1-r_c^2\over
1-b^2 r^2_c}}}
dy_1 f(y_1,r_c),
\label{eq:int4rcb}
\end{equation}
where
\begin{equation}
f(y_1,r_c)={1\over 1-y_1^2}{1\over2}\log\left({1+y^{c+}_2\over
1-y^{c+}_2}
{1-y^{c-}_2\over 1+y^{c-}_2}\right).
\end{equation}
Expanding $f(y_1,r_c)$ in $r_c$ we get
\begin{eqnarray}
f(y_1, r_c)=-{1\over 1-y_1^2}\log (r_c^2 {1-b^2 y_1^2\over 4(1-y_1^2)})
+O(r_c^2).
\label{eq:frc1}
\end{eqnarray}
After the integral over $y_1$ in (\ref{eq:int4rcb}), we
obtain the action $S[r_c,b]$ in the radial cut-off regularization.
We note that O($r_c^2$) terms in (\ref{eq:int4rcb})
also contribute a constant term. We then obtain
\begin{eqnarray}
S_4[r_c,b]&=& {1\over4}\log^2\left({r_c^2\over -8\pi^2 s}\right)
+ {1\over4}\log^2\left({r_c^2\over -8\pi^2 t}\right)
-{1\over4 }\log^2({s\over t})
+a_0
+O(r_c^2\log r_c^2).
\nonumber\\
\label{eq:rad4pt1}
\end{eqnarray}
Evaluating the constant $a_0$
numerically up to $O(r_c^{n})$ ($n=500$) terms in (\ref{eq:frc1}), we
get $a_0=-3.28977.$
This shows that the
finite term numerically agrees with the BDS formula \cite{BeDiSm}
\begin{equation}
F_4=-{1\over2}F^{BDS}_4,\quad
F^{BDS}_4={1\over2}\log^2({s\over t})+{2\pi^2\over3},
\end{equation}
since ${\pi^2\over 3}=3.28987...$.
Motivated from the analysis of the 4-point amplitude, the $n$-point amplitude is expected to have the structure \cite{Al,DoItIw}
\begin{equation}
S_n[r_c]={1\over8}\sum_{i=1}^{n}\left(\log
{r_c^2\over -8\pi^2 s_{i,i+1}}\right)^2
+F_n(p_1,\cdots, p_n)+O(r_c^2\log r_c^2),
\label{eq:nptactrc1}
\end{equation}
where
$s_{i,i+1}=-(p_i+p_{i+1})^2$ and $p_{n+1}=p_1$.
We have factored out the cusp anomalous dimension in the above formula.
The first term in (\ref{eq:nptactrc1})
characterizes infra-red divergences of the
amplitude.
The function $F_n(p_1,\cdots, p_n)$
is a finite remainder part of the amplitude and takes the form
\begin{equation}
F_n=-{1\over2}F^{BDS}_n+R_n.
\label{eq:finbds1}
\end{equation}
The term $F^{BDS}_n$ is given by the BDS formula which is written
in terms of the Mandelstam variables
\begin{equation}
x_{ij}^2=t_i^{[j-i]}=(p_i+\cdots+p_j)^2.
\end{equation}
The explicit formula
for $n\geq 5$
\cite{BeDiSm} is
\begin{equation}
F^{BDS}_{n}={1\over2}\sum_{i=1}^{n}g_{n,i},
\end{equation}
where
\begin{equation}
g_{n,i}=-\sum_{r=2}^{[n/2]-1}
\log\left({-t^{[r]}_{i}\over -t^{[r+1]}_{i}}\right)
\log\left({-t^{[r]}_{i+1}\over -t^{[r+1]}_{i}}\right)
+D_{n,i}+L_{n,i}+{3\over2}\zeta_{2}.
\label{eq:bds1}
\end{equation}
Here $D_n$ and $L_n$ are defined by
\begin{eqnarray}
D_{2m+1,i}&=&
-\sum_{r=2}^{m-1}{\rm Li}_2\left(1-{t^{[r]}_{i} t^{[r+2]}_{i-1}\over
t^{[r+1]}_i t^{[r+1]}_{i-1}}\right), \nonumber\\
L_{2m+1,i}&=& -{1\over2}
\ln\left({-t^{[m]}_{i}\over -t^{[m]}_{i+m+1}}\right)
\ln\left({-t^{[m]}_{i+1}\over -t^{[m]}_{i+m}}\right),
\end{eqnarray}
for $n=2m+1$ and
\begin{eqnarray}
D_{2m,i}&=&
-\sum_{r=2}^{m-2}{\rm Li}_2\left(1-{t^{[r]}_{i} t^{[r+2]}_{i-1}\over
t^{[r+1]}_i t^{[r+1]}_{i-1}}\right)
-{1\over2}{\rm Li}_2\left(1-{t^{[m-1]}_{i} t^{[m+1]}_{i-1}\over
t^{[m]}_i t^{[m]}_{i-1}}\right),
\nonumber\\
L_{2m,i}&=& -{1\over4}
\log\left({-t^{[m]}_{i}\over -t^{[m]}_{i+m+1}}\right)
\log\left({-t^{[m]}_{i+1}\over -t^{[m]}_{i+m}}\right),
\end{eqnarray}
for $n=2m$.
${\rm Li}_2(z)$ denotes the dilogarithm function.
The term $R_n$, called the remainder function, is a
function of cross-ratios in momentum space:
\begin{eqnarray}
u_{ijkl}={x_{ij}^2x_{kl}^2\over x_{ik}^2 x_{jl}^2},
\quad
u_{ijlk}= {x_{ij}^2x_{kl}^2\over x_{il}^2 x_{jk}^2},
\end{eqnarray}
which represents a deviation from the BDS formula.
The $F_n$ satisfies the anomalous dual conformal identities but the
function $R_n$ is itself conformally invariant and is not determined by
conformal symmetry.
For the 6-point amplitude \cite{DrHeKoSo}, the remainder part
$R_6=R_6(u_1,u_2,u_3)$ is a function of
the cross-ratios
\begin{eqnarray}
u_1&=&{x_{13}^2 x_{46}^2\over x_{14}^2 x_{36}^2}
={t_1^{[2]}t_{4}^{[2]}
\over t_{1}^{[3]} t_{3}^{[3]}},
\nonumber\\
u_2&=&{x_{24}^2 x_{15}^2\over x_{25}^2 x_{14}^2}
={t_2^{[2]} t_1^{[4]}\over t_2^{[3]}t_1^{[3]}},
\nonumber\\
u_3&=&{x_{35}^2 x_{26}^2\over x_{36}^2 x_{25}^2}
={t_3^{[2]} t_2^{[4]}\over t_3^{[3]}t_2^{[3]}}.
\end{eqnarray}
The BDS formula of the 6-point amplitude for specific momentum
configurations will be discussed in sect. 4.
\subsection{Discretized minimal surface}
Although exact formula for the
minimal surface for $n(\geq5)$-point amplitudes is not yet known so far,
we can study minimal surface
by solving numerically the Euler-Lagrange equations on
the square lattice with spacing $h=\frac{2}{M}$
where $M$ is a positive integer.
At each site $(-1+h i,-1+h j)$ ($i,j=0,\cdots, M$), we assign the variables
\begin{equation}
y_0[i,j]=y_0(-1+h i,-1+h j),\quad
r[i,j]=r(-1+h i,-1+h j).
\end{equation}
For the 4-point amplitude, we discretize the differential equations by
the central difference method with
the boundary conditions
\begin{eqnarray}
&& y_0[i,0]= y_0(-1+hi,-1), \quad y_0[i,M]=y_0(-1+hi,1),
\nonumber\\
&& y_0[0,j]= y_0(-1,-1+hj), \quad y_0[M,j]=y_0(1,-1+hj),
\nonumber\\
&&r[i,0]=r[i,M]=r[0,j]=r[M,j]=0.
\end{eqnarray}
Then we obtain $2\times (M-1)^2$ nonlinear simultaneous equations for
$y_0[i,j]$ and $r[i,j]$ and
use Newton's method to find a numerical solution.
In this paper we use the $M=520$ lattice
data \cite{DoItIw}, where
the Newton method is repeatedly applied until
the discrete equation is satisfied
up to ${\cal O}(10^{-16})$ and
the area of the obtained surface
does not change up to ${\cal O}(10^{-6})$.
The area is approximately evaluated as $S=\sum L[i,j]h^2$,
where $L[i,j]$ and $h$ are the discretized Lagrangian at
a lattice point $(i,j)$ and the lattice spacing, respectively.
The area $S$ becomes large as $M$ increases, which is due to the
IR divergent behavior near cusps.
In \cite{DoItIw}
we have defined the area of the surface in the
radial cut-off regularization
\begin{equation}
S_4^{dis}[r_c]=\sum_{(i,j)\in A[r_c]}L[i,j] h^2,
\end{equation}
where $A[r_c]$ denotes the set of lattice points $(i,j)$
satisfying $r_c[i,j]<r_c$.
In this paper,
we calculate the area of the conformally boosted minimal surface
by evaluating
\begin{equation}
S_4^{dis}[r_c,b]=\sum_{(i,j)\in A[r_c,b]}L[i,j] h^2,
\label{eq:discs4b}
\end{equation}
where $A[r_c,b]$ is made of the points $(i,j)$ satisfying
\begin{equation}
r'[i,j]={r[i,j]\over 1+b y_0[i,j]}<r_c.
\end{equation}
Since it is difficult to estimate the finite $r_c$ correction
from the integral formula (\ref{eq:int4rcb}) in the case of
the 4-point amplitude,
we will compare
the area (\ref{eq:discs4b}) with numerically evaluated integral
(\ref{eq:int4rcb}).
We can also construct the minimal surface for the 6 and 8-point
amplitudes whose boundary conditions are the same as the cut and glue
type surface obtained from the 4-point amplitude \cite{DoItIw}.
\section{Numerical check of the minimal surface for the four-point amplitude}
In this section, in order to confirm the validity of our numerical
approach,
we compare the integral formula $S_4[r_c,b]$
for the
4-point amplitude
with the area $S^{dis}_4[r_c,b]$ of the discretized minimal surface.
We evaluate the area by using
$M=520$ lattice data \cite{DoItIw}.
The exact radial cut-off area $S_4[r_c,b]$ is evaluated numerically
by using Mathematica.
We plot $S[r_c,b]$ and $S^{dis}[r_c,b]$ at
some values of $b$ (Fig. \ref{fig:b00}),
where we can see that our numerical approach agrees with
the exact formula in the region $r_c>0.2$.
In Tables \ref{table:4ptarea1} and \ref{table:4ptarea2},
we compare the values of $S^{dis}[r_c,b]$, $S[r_c,b]$
and $S^{BDS}[r_c,b]$ at $b=0$, $0.4$
by evaluating the ratios $(S_{4}^{dis}[r_c]-S_{4}[r_c])/
S^{dis}_{4}[r_c]$ and $(S^{BDS}_4[r_c]-S_{4}^{dis}[r_c])/
S_{4}^{dis}[r_c]$.
Here $S_4^{BDS}[r_c,b]$ is given by
dropping the $O(r_c^2\log r_c^2)$
corrections in (\ref{eq:rad4pt1}), which is equal to the BDS formula up to the
constant term
\begin{eqnarray}
S^{BDS}_4[r_c,b]&=&{1\over4}\log^2\left( {r_c^2 (1-b)^2\over 16}\right)
+{1\over4}\log^2\left( {r_c^2 (1+b)^2\over 16}\right)
\nonumber\\
&& -{1\over4}\left\{\log\left({1+b\over 1-b}\right)^2\right\}^2
-{\pi^2\over3}.
\label{eq:bds4pt2}
\end{eqnarray}
We
can see that the discretized minimal surface area
agrees with the exact formula for
$r_c\geq 0.2$ within $0.2\%$.
It also differs from the BDS formula
(\ref{eq:bds4pt2}) about $2\%$ .
The $M=520$ data numerically reproduces the analytical result of the
4-point amplitude for $r_c\geq 0.2$.
{}From the ratio $(S^{BDS}_4-S^{dis})/S^{dis}_4$, it is found that
that
finite $r_c$ corrections become small when $r_c$ decreases.
\begin{figure}[t]
\begin{center}
\resizebox{80mm}{!}{\includegraphics{4pt_M520_b_rcSS1v2.eps}}
\end{center}
\caption{$S_4[r_c,b]$ (lines) and $S_4^{dis}[r_c,b]$ (points) at $b=0$,
$0.4$ and $0.8$ }
\label{fig:b00}
\end{figure}
\begin{table}[t]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$r_c$ & $S^{dis}_4[r_c]$ & $S_{4}[r_c]$ & $S^{BDS}_4[r_c]$ &
${S_{4}^{dis}[r_c]-S_{4}[r_c]\over S^{dis}_{4}[r_c]}$
& ${S^{dis}_4[r_c]-S_{4}^{BDS}[r_c]\over S_{4}^{dis}[r_c]}$
\\ \hline
0.2 & 14.6675 & 14.6086 &
14.6590
& 0.004017 & 0.00058
\\
0.3 & 10.0349 & 10.0331&
10.1291
& 0.000179 & -0.00939
\\
0.4 & 7.17606 & 7.16437 & 7.3139
& 0.001630 &
-0.019211
\\
\hline
\end{tabular}
\end{center}
\caption{
the area
of the discretized surface, the integral formula,
the BDS amplitudes and their differences (divided by $S^{dis}$) at $b=0.0$}
\label{table:4ptarea1}
\end{table}
\begin{table}[bthp]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$r_c$ & $S^{dis}_4[r_c]$ & $S_{4}[r_c]$ & $S^{BDS}_4[r_c]$ &
${S_{4}^{dis}[r_c]-S_{4}[r_c]\over S^{dis}_{4}[r_c]}$
& ${S^{BDS}_4[r_c]-S_{4}^{dis}[r_c]\over S_{4}^{dis}[r_c]}$
\\ \hline
0.2 & 15.3943 & 15.2994 & 15.3598
& 0.006166 &
0.002238
\\
0.3 & 10.5848 & 10.5726 & 10.6886
&
0.001145 &
-0.009808
\\
0.4 & 7.60405 & 7.59157 & 7.7731
& 0.001641 &
-0.022231
\\
\hline
\end{tabular}
\end{center}
\caption{the area
of the discretized surface, the integral formula,
the BDS amplitudes and their differences (divided by $S^{dis}$) at
$b=0.4$}
\label{table:4ptarea2}
\end{table}
\subsection{Difference of two areas with different $b$}\label{sec:dta}
Since the exact integral formula is known only for the 4-point amplitude,
the previous comparison between the numerical result and the analytical
expression of the area is only applicable to the case of the 4-point amplitude.
We need to find a different approach to estimate
the deviation from the BDS formula by reducing the possible finite $r_c$
corrections from the numerical result.
In this paper we will consider the difference of two areas with
different boost parameter $b$.
Namely we define the
function
\begin{equation}
G^{dis}_4[r_c,b]=S^{dis}_4[r_c,b]-S^{dis}_4[r_c,0].
\end{equation}
Both terms $S^{dis}_4[r_c,b]$ and $S^{dis}_4[r_c,0]$
include finite $r_c$ correction.
But by taking their difference,
some terms of two corrections would cancel each other.
In particular $b$-independent contribution completely vanishes.
Then we will
compare $G^{dis}_4[r_c,b]$ with the difference of the corresponding BDS formulas
\begin{equation}
G^{BDS}_4[r_c,b]=S^{BDS}_4[r_c,b]-S^{BDS}_4[r_c,0].
\end{equation}
which is also expected to have smaller $r_c$ correction.
In Fig. \ref{fig:4ptdiff1}, we can see the numerical data is
consistent with the BDS formula roughly about $2-20\%$ at $r_c=0.3$
and $0.4$ (see Table \ref{tab:4ptdiff4}).
At $b=0.2$, the ratio $(G^{dis}_4-G_4^{BDS})/G^{dis}_4$ is large.
This is because the ratio is enhanced due to the small value of $G_4$.
Although we can see that there still exist finite $r_c$ corrections,
the difference of two areas is a useful method to compare the numerical
data with the BDS formula.
There are some numerical
errors at small $r_c$ due to finite lattice spacing.
This error would be improved if we can do more precise calculation at
larger $M$.
\begin{figure}[t]
\begin{center}
\resizebox{80mm}{!}{\includegraphics{4pt_M520_diff_a1v2.eps}}
\end{center}
\caption{$G^{dis}_4[r_c,b]$ at $b=0.2$, $0.4$, $0.8$ ($M=520$) and $G^{BDS}_4[r_c,b]$}
\label{fig:4ptdiff1}
\end{figure}
\begin{table}[t]
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
& $r_c=0.2$ & $0.3$ & $0.4$
\\
\hline
$b=0.2$& -0.327895 & 0.156186 & -0.217681 \\
$0.4$& 0.035675 & -0.017397 & -0.072860 \\
$0.6$& 0.024418 & -0.047297 & -0.069778\\
$0.8$& -0.03092 & -0.012591 & -0.061267 \\
\hline
\end{tabular}
\end{center}
\caption{ ${G^{dis}_{4}[r_c,b]-G_4^{BDS}[r_c,b]\over G^{dis}_{4}[r_c,b]}$}
\label{tab:4ptdiff4}
\end{table}
\section{Numerical test of the six and eight-point amplitudes}
We now compare numerical results with the BDS formula for higher-point
amplitudes as we did in the end of the previous section.
In \cite{DoItIw} we constructed numerically the minimal surfaces corresponding
to the 6-point and 8-point amplitudes with the same boundary
conditions as the surface in \cite{AsDoItNa}.
Their boundaries are characterized by the following momenta:\\
{\bf 6-point function solution 1:}
\begin{eqnarray}
2\pi p_1&=&(2,0,-2),\quad
2\pi p_2=(-1,0,1),\quad
2\pi p_3=(1,1,0),\nonumber\\
2\pi p_4&=&(-1,0,1),\quad
2\pi p_5=(1,1,0),\quad
2\pi p_6=(-2,0,-2).
\end{eqnarray}
{\bf 6-point function solution 2:}
\begin{eqnarray}
2\pi p_1&=&(1,1,0),\quad
2\pi p_2=(-1,-1,0),\quad
2\pi p_3=(2,0,2),\nonumber\\
2\pi p_4&=&(-1,1,0),\quad
2\pi p_5=(1,1,0),\quad
2\pi p_6=(-2,-2,0).
\end{eqnarray}
{\bf 8-point function:}
\begin{eqnarray}
2\pi p_1&=&(-1,-1,0),\quad
2\pi p_2=(1,-1,0),\quad
2\pi p_3=(-1,0,1),\nonumber\\
2\pi p_4&=&(1,0,1),\quad
2\pi p_5=(-1,0,1),\quad
2\pi p_6=(1,0,1),\nonumber\\
2\pi p_7&=&(-1,1,0),\quad
2\pi p_8=(1,-1,0).
\label{eq:mom8pt0}
\end{eqnarray}
We apply the conformal transformation (\ref{eq:conf1}) with the boost
parameter $b$ and the scale factor $a$.
\subsection{six-point amplitude solution 1}
Firstly we consider the solution 1 of the 6-point amplitude.
After the conformal transformation,
the Mandelstam variables are given by
\begin{eqnarray}
&&t^{[2]}_1=\frac{4 a^2}{1-b},\;\;
t^{[2]}_2=\frac{4 a^2}{(b+1)^2},\;\;
t^{[2]}_3=2 a^2,\;\;
t^{[2]}_4=\frac{4 a^2}{(b+1)^2},\;\;
t^{[2]}_5=\frac{4 a^2}{1-b},\;\;
t^{[2]}_6=\frac{8 a^2}{(b+1)^2}, \nonumber\\
&&t^{[3]}_1=\frac{4 a^2}{1-b^2},\;\;
t^{[3]}_2=\frac{4 a^2}{b+1},\;\;
t^{[3]}_3=\frac{4 a^2}{b+1},\;\;
t^{[3]}_4=\frac{4 a^2}{1-b^2},\;\;
t^{[3]}_5=\frac{4 a^2}{b+1},\;
t^{[3]}_6=\frac{4 a^2}{b+1}.\nonumber\\
\end{eqnarray}
Then the cross-ratios are evaluated as
\begin{eqnarray}
u_1&=&
{t_1^{[2]}t_{4}^{[2]}
\over t_{1}^{[3]} t_{3}^{[3]}}
=1,
\quad
u_2
{t_2^{[2]} t_1^{[4]}\over t_2^{[3]}t_1^{[3]}}
=1,
\quad
u_3
{t_3^{[2]} t_2^{[4]}\over t_3^{[3]}t_2^{[3]}}
=1.
\end{eqnarray}
The cross-rations are independent of $b$.
From the BDS formula (\ref{eq:finbds1}), the amplitude becomes
\cite{AsDoItNa}
\begin{eqnarray}
S^{(1)BDS}_6[r_c,b]
&=&{1\over8}
\Bigl\{
2\log^2( {r_c^2 (1-b)\over 8})
+2\log^2({r_c^2 (1+b)^2\over 8})
+\log^2{r_c^2\over4}
+\log^2({r_c^2 (1+b)^2\over 16})
\Bigr\}
\nonumber\\
&&-{1\over2}\Bigl\{
\log2\log(1-b)-2\log2\log(1+b)-2\log(1-b)\log(1+b)
\nonumber\\
&&
+{1\over2}\log^2(1-b)+3\log^2(1+b)
\Bigr\}-{3\pi^2\over 16}.
\end{eqnarray}
Adding the remainder function $R_6$, the BDS formula is modified as
\begin{eqnarray}
S^{(1)}_6[r_c,b]&=& S^{(1)BDS}_6[r_c,b]+R_6(1,1,1; r_c).
\end{eqnarray}
Here the remainder function depends on the cut-off parameter $r_c$.
We evaluate $S^{(1)dis}_6[r_c,b]$ from the discretized minimal surface
of $M=520$, which is shown in Fig. \ref{fig:6pt1rcSS}.
\begin{figure}[t]
\begin{center}
\resizebox{80mm}{!}{\includegraphics{6pt1_M520_b_rcSSv2.eps}}
\end{center}
\caption{$S_6^{(1)dis}[r_c,b]$ at $b=0.2$, $0.4$, $0.8$ ($M=520$)}
\label{fig:6pt1rcSS}
\end{figure}
Firstly we check whether
the BDS conjecture without $R_6$ term
is consistent with the numerical data.
\begin{table}[bthp]
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
$r_c$ & $S^{(1)dis}_6[r_c]$ & $S^{(1)BDS}_6[r_c]$ &
${S^{(1)dis}_6[r_c]-S_{6}^{(1)BDS}[r_c]\over S_{6}^{(1)dis}[r_c]}$
\\ \hline
0.2 & 16.9788 & 18.1246 & -0.067486
\\
0.3 & 11.1309 & 12.3751 & -0.11178
\\
0.4 & 7.61016 & 8.89403 & -0.168705
\\
\hline
\end{tabular}
\end{center}
\caption{
$S^{(1)dis}_6$, $S^{(1)BDS}_6$
and their difference (divided by $S^{(1)dis}_6$) at
$b=0.4$}
\label{table:6ptarea1}
\end{table}
In Table \ref{table:6ptarea1}, we can see small $r_c$ correction from
the BDS formula which is 10 times
larger than that of the 4-point amplitude.
This seems to imply that the remainder function $R_6$ is non zero.
But at this moment we do not have enough numerical data in order
to establish the discrepancy from the BDS formula.
Instead we study the difference of two areas with different $b$,
since the effect of the constant factor $R_6$ is canceled.
We define
\begin{equation}
G^{(1)dis}_6[r_c,b]=S^{(1)dis}_6[r_c,b]-S_6^{(1)dis}[r_c,0], \quad
G^{BDS(1)}_6[r_c,b]=S^{BDS(1)}_6[r_c,b]-S_6^{BDS(1)}[r_c,0].
\end{equation}
In Figs. \ref{fig:6pt1diffa1} and \ref{fig:6pt1diffa2},
we compare these two functions
and find that they behave in a
similar
manner as we expect
{}from finite $r_c$ corrections in the case of the 4-point amplitude,
which is about $10\%$
(Table \ref{tab:6pt1diff}).
This table shows that the solution is
numerically consistent with the fact that $R_6$ is independent of $b$.
\begin{figure}[t]
\begin{tabular}{cc}
\begin{minipage}{0.5\hsize}
\begin{center}
\resizebox{80mm}{!}{\includegraphics{6pt1_M520_diff_a1v2.eps}}
\end{center}
\caption{$G_6^{(1)dis}$ and $G^{(1)BDS}_6$ at $b=0.2$, $0.4$ }
\label{fig:6pt1diffa1}
\end{minipage}
\begin{minipage}{0.5\hsize}
\begin{center}
\resizebox{80mm}{!}{\includegraphics{6pt1_M520_diff_a2v2.eps}}
\end{center}
\caption{$G_6^{(1)dis}$ and $G^{(1)BDS}_6$ at $b=0.6$, $0.8$ }
\label{fig:6pt1diffa2}
\end{minipage}
\end{tabular}
\end{figure}
\begin{table}[t]
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
& $r_c=0.2$ & $0.3$ & $0.4$
\\
\hline
$b=0.2$& 0.199346 & -0.068327 & 0.016099\\
$0.4$& 0.079949 & -0.015012 & 0.016734\\
$0.6$& 0.055443 & 0.025344 & 0.036281\\
$0.8$& 0.225821 & 0.044057 & 0.133266\\
\hline
\end{tabular}
\end{center}
\caption{${G_{6}^{(1)dis}[r_c,b]-G_6^{(1)BDS}[r_c,b]\over
G_{6}^{(1)dis}[r_c,b]}$}
\label{tab:6pt1diff}
\end{table}
\subsection{six-point amplitude: solution 2}
Secondly we consider the solution 2 of the 6-point amplitude.
After the conformal transformation,
the Mandelstam variables are
\begin{eqnarray}
&&t^{[2]}_1=\frac{4 a^2}{(1-b)^2},\;\;
t^{[2]}_2=\frac{4 a^2}{b+1},\;\;
t^{[2]}_3=\frac{4 a^2}{1-b},\;\;
t^{[2]}_4=\frac{4 a^2}{(b+1)^2},\;\;
t^{[2]}_5=\frac{4 a^2}{1-b},\;\;
t^{[2]}_6=\frac{4 a^2}{b+1},\nonumber\\
&&t^{[3]}_1=\frac{4 a^2}{1-b^2},\;\;
t^{[3]}_2=4 a^2,\;\;
t^{[3]}_3=\frac{4 a^2}{1-b^2},\;\;
t^{[3]}_4=\frac{4 a^2}{1-b^2},\;\;
t^{[3]}_5=4 a^2,\;\;
t^{[3]}_6=\frac{4 a^2}{1-b^2}.
\end{eqnarray}
The cross-ratios are given by
\begin{eqnarray}
u_1&=&
{t_1^{[2]}t_{4}^{[2]}
\over t_{1}^{[3]} t_{3}^{[3]}}
=1,
\quad
u_2
{t_2^{[2]} t_1^{[4]}\over t_2^{[3]}t_1^{[3]}}
=1,
\quad
u_3
{t_3^{[2]} t_2^{[4]}\over t_3^{[3]}t_2^{[3]}}
=1,
\end{eqnarray}
which are constants.
The BDS formula is given by
\begin{eqnarray}
S^{BDS(2)}_6[r_c,b]&=&
{1\over8}
\Bigl\{
\log^2({r_c^2 (1-b)^2\over8})+\log^2({r_c^2 (1+b)^2\over8})
+2\log^2 ({r_c^2 (1+b)\over 8})+2\log^2 ({r_c^2 (1-b)\over 8})
\Bigr\}
\nonumber\\
&&
-{1\over2}\Bigl\{
{3\over2}\log^2(1-b)+{3\over2}\log^2(1+b)-2\log(1-b)\log(1+b)
\Bigr\}
-{3\pi^2\over 16}.
\nonumber\\
\end{eqnarray}
This BDS formula is modified by adding the remainder function
$R_6(1,1,1;r_c)$,
which is independent of $b$.
The area $S^{(2)dis}_6[r_c,b]$
from the discretized minimal surface at $M=520$,
which is shown in Fig. \ref{fig:6pt2rcSS}.
For example, the values
of $S^{(2)dis}_6[r_c,b]$ and $S^{(2)BDS}_6[r_c,b]$
at $r_c=0.2$ and $b=0.4$ is $18.9343$ and $19.9554$, respectively.
The ratio $(S^{(2)dis}_6-S^{(2)BDS}_6)/S^{(2)dis}_6$ becomes
$-0.057218$, which is the same order as the case of the 6-point solution 1.
We define the difference functions
\begin{equation}
G^{(2)dis}_6[r_c,b]=S^{(2)dis}_6[r_c,b]-S_6^{(2)dis}[r_c,0], \quad
G^{(2)BDS}_6[r_c,b]=S^{(2)BDS}_6[r_c,b]-S_6^{(2)BDS}[r_c,0].
\end{equation}
In Fig. \ref{fig:6pt2diffa3}
and Table \ref{tab:6pt2diff}, we compare $G^{(2)dis}_6[r_c,b]$ with
$G^{(2)BDS}_6[r_c,b]$.
Contribution from the remainder function $R_6$ disappears in
$G^{(2)dis}_6$. The difference function from the
numerical data is consistent with the BDS formula within 10\% at
$r_c=0.4$, which is the same order as we expect from the case of
the 4-point solution.
This is also consistent with the fact that $R_6$ for this momentum
configuration is
independent of $b$.
\begin{figure}[t]
\begin{tabular}{cc}
\begin{minipage}{0.5\hsize}
\begin{center}
\resizebox{80mm}{!}{\includegraphics{6pt2_M520_b_rcSSv2.eps}}
\end{center}
\caption{$S_6^{(2)dis}[r_c,b]$ at $b=0.2$, $0.4$, $0.8$ ($M=520$)}
\label{fig:6pt2rcSS}
\end{minipage}
\begin{minipage}{0.5\hsize}
\begin{center}
\resizebox{80mm}{!}{\includegraphics{6pt2_M520_diff_a3v2.eps}}
\end{center}
\caption{$G_6^{(2)dis}$
and $G^{(2)BDS}_6$ at $b=0.2$, $0.4$, $0.6$, $0.8$ }
\label{fig:6pt2diffa3}
\end{minipage}
\end{tabular}
\end{figure}
\begin{table}[t]
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
& $r_c=0.2$ & $0.3$ & $0.4$
\\
\hline
$b=0.2$& -0.173343 & 0.406764 & -0.084989\\
$0.4$& 0.040559 & 0.076990 & -0.097516 \\
$0.6$& -0.140757 & -0.016108 & 0.010283\\
$0.8$& -0.145579 & 0.049971 & -0.047002\\
\hline
\end{tabular}
\end{center}
\caption{
${G_{6}^{(2)dis}[r_c,b]-G_6^{(2)BDS}[r_c,b]\over G_{6}^{(2)dis}[r_c,b]}$}
\label{tab:6pt2diff}
\end{table}
\subsection{eight-point amplitude}
Finally we discuss the 8-point amplitude.
After the conformal transformation,
the Mandelstam variables are
\begin{eqnarray}
t^{[2]}_{\text{odd}}=\frac{4 a^2}{(b+1)^2},\;\;\;
t^{[2]}_{\text{even}}=2 a^2,\;\;\;
t^{[3]}_{i}=\frac{4 a^2}{b+1},\;\;\;
t^{[4]}_{\text{odd}}=\frac{8 a^2}{(b+1)^2},\;\;\;
t^{[4]}_{\text{even}}=4 a^2.
\end{eqnarray}
It is shown that all the values of the cross-ratios
$u_{ijkl}$ are independent of $b$.
For example, the cross-ratio
\begin{eqnarray}
u_{1346}&=&
{t_1^{[2]}t_{4}^{[2]}
\over t_{1}^{[3]} t_{3}^{[3]}}
={
{4a^2\over (1+b)^2} {2a^2}
\over
{4a^2\over 1+b}{4a^2\over 1+b}
}={1\over2}
\end{eqnarray}
is constant.
The BDS formula for the 8-point amplitude is
\begin{eqnarray}
S^{BDS}_8[r_c,b]&=&
{1\over8}\Bigl\{
4\log^2({r_c^2(1+b)^2\over8})+4\log^2({r_c^2\over4})
\Bigr\}
\nonumber\\
&&
-{1\over2}\Bigl\{
4\log^2(1+b)-4\log2 \log(1+b)-{\pi^2\over6}
\Bigr\}
-{\pi^2\over2}.
\end{eqnarray}
$S_8[r_c,b]$ is obtained by adding the remainder function
$R_8$, which is independent of $b$.
The area $S^{dis}_8[r_c,b]$ obtained from the discretized minimal
surface at $M=520$,
is shown in Fig. \ref{fig:8ptrcSS}.
For example,
at $r_c=0.2$ and $b=0.4$ , $S^{dis}_8[r_c,b]=18.8265$ and
$S^{BDS}_8[r_c,b]=17.4285$.
The ratio $(S^{dis}_8-S_8^{BDS})/S^{dis}_8=0.074256$, which is the same
order deviation as we observed in the case of 6-point amplitudes.
In Fig. \ref{fig:8ptdiffa3} and
Table \ref{tab:8ptdiff}, we compare two difference functions:
\begin{equation}
G^{dis}_8[r_c,b]=S^{dis}_8[r_c,b]-S^{dis}_8[r_c,0], \quad
G^{BDS}_8[r_c,b]=S^{BDS}_8[r_c,b]-S_8^{BDS}[r_c,0].
\end{equation}
We see
that $R_8$-independent $G^{dis}_8$ obtained from the 8-point
discretized minimal surface is
consistent with the BDS
formula up to finite $r_c$
corrections.
This is also consistent with $b$-independence of
the remainder function $R_8$.
\begin{figure}[t]
\begin{tabular}{cc}
\begin{minipage}{0.5\hsize}
\begin{center}
\resizebox{80mm}{!}{\includegraphics{8pt_M520_b_rcSSv2.eps}}
\end{center}
\caption{$S^{dis}_8[r_c,b]$ at $b=0.2$, $0.4$, $0.8$ ($M=520$)}
\label{fig:8ptrcSS}
\end{minipage}
\begin{minipage}{0.5\hsize}
\begin{center}
\resizebox{80mm}{!}{\includegraphics{8pt_M520_diff_a3v2.eps}}
\end{center}
\caption{$G^{dis}_8$ and
$G^{BDS}_8$ at $b=0.2$, $0.4$, $0.6$, $0.8$ }
\label{fig:8ptdiffa3}
\end{minipage}
\end{tabular}
\end{figure}
\begin{table}[t]
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
& $r_c=0.2$ & $0.3$ & $0.4$
\\
\hline
$b=0.2$& 0.179417 & -0.052221 & 0.005237 \\
$0.4$& 0.081445 & 0.011274 & -0.002584 \\
$0.6$& 0.050192 & -0.002827 & -0.001135\\
$0.8$& 0.042021 & 0.002411 & -0.000863\\
\hline
\end{tabular}
\end{center}
\caption{${G^{dis}_{8}[r_c,b]-G_8^{BDS}[r_c,b]\over G^{dis}_{8}[r_c,b]}$}
\label{tab:8ptdiff}
\end{table}
\section{Conclusions and discussion}
In this paper we studied the area of the minimal surfaces in AdS
spacetime surrounded by the light-like boundary which corresponds
to the 4, 6 and 8-points gluon scattering amplitudes with
specific momentum configurations \cite{AsDoItNa}.
For all the solutions,
it is found that the remainder function $R_n$ is independent
of $b$
and the $R_n$-independent difference of the areas with
different boost parameters obtained from
the discretized minimal surface is
consistent with the BDS formula up to
finite $r_c$ corrections.
It would be interesting to study the 6-point solutions with
various momentum configuration (hexagon for example).
We can determine numerically the remainder function $R_6$
as a function of $u_1$, $u_2$ and $u_3$, where at some values we could
compare this with the result obtained in \cite{BeDiKoRoSpVeVo}.
The present numerical approach will be helpful to determine the
exact functional form of the remainder function
$R_n$ via the AdS/CFT correspondence.
It would be also
an interesting problem to estimate the finite $r_c$ correction
analytically.
The integral formula (\ref{eq:int4rcb}) of the 4-point solution
can be expanded in $r_c$ and
be evaluated by using hypergeometric function as
\begin{eqnarray}
S_4[r_c,b]&=& -{2\sqrt{\pi}\over \sqrt{1-b^2 r_c^2}}
\sum_{n=0}^{\infty}{1\over 2n+1} (1-r_c^2)^{n+1}
{\Gamma(n+{3\over2})\over \Gamma(2+n)}
{}_2F_1({1\over2}, {3\over2}+n; 2+n; {1-r_c^2\over 1-b^2 r_c^2}).
\nonumber\\
\label{eq:ef2}
\end{eqnarray}
Because of complexity of this formula,
it is difficult to estimate the finite $r_c$
corrections at this moment.
\subsection*{Acknowledgments}
The authors would like to thank Koh Iwasaki for collaboration in early
stage of this work.
The work of K.~I. is supported in part by the Grant-in-Aid for Scientific
Research from Ministry of Education, Science,
Culture and Sports of Japan.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,875 |
package com.tvd12.ezyfox.bean.testing.prototype;
import com.tvd12.ezyfox.bean.annotation.EzyPrototype;
@EzyPrototype
public class ClassE {}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,322 |
{"url":"https:\/\/math.stackexchange.com\/questions\/367774\/show-that-a-function-has-directional-derivatives-at-a-point-but-is-not-different","text":"# Show that a function has directional derivatives at a point but is not differentiable there. Need help understanding proof\n\nagain!\n\nI simply need help understanding this example.\n\nQUESTION: Show that the function $$g(x) = \\begin{cases} \\frac{xy^2}{x^2 +y^4}, & \\text{if (x,y) \\neq 0}\\\\ 0, & \\text{if (x,y) = (0,0)} \\\\ \\end{cases}$$\n\nhas directional derivatives at $0 = (0,0)$ but is not differentiable at $0 = (0,0)$\n\n(What does this even mean?)\n\nIn any case, the answer is given as:\n\nANSWER Direct computation yields for every $v=(x,y) \\in \\Bbb R^2$, $$D_v(g) = \\lim_{t\\to0} \\frac {g(0+tv) - g(0)}{t} = \\lim_{t\\to0} \\frac {txy^2}{x^2 + t^2y^4} = 0$$ Thus $g$ has directional derivatives in all directions at $(0,0)$\n\nOn the other hand we notice that $g(x^2,x) =\\displaystyle \\frac{1}{2} \\not\\to 0 = g(0,0)$ as $g(x^2,x) \\to 0$ thus g is not continuous at $0$ and is there not differentiable.\n\nNow I understand the first part.\n\nBut my concern is with the second one. Why are we discussing $g(x^2,x)$ in the matter of continuity? I understand the idea is to prove indifferentiablity but why $g(x^2,x)$ to be exact? I know it's simply an example. But I may be thrown off by the use of x in both vector components...\n\n\u2022 \"What does this even mean?\" Maybe you need to re-read some of your definitions and try again? \u2013\u00a0Pedro Tamaroff Apr 21 '13 at 0:01\n\u2022 $v$ should be unit vector --otherwise dir. derivative definition above would be wrong.. \u2013\u00a0Halil Duru Apr 21 '13 at 0:42\n\u2022 @HalilDuru No, that is not necessary. It varies from author to author. \u2013\u00a0Pedro Tamaroff Apr 21 '13 at 1:00\n\u2022 then same direction with different magnitude will give different answers ,right? \u2013\u00a0Halil Duru Apr 21 '13 at 1:03\n\nI take the question to be \"How did someone come up with the particular choice of $x^2$ and $x$ to plug into $g$ and discover discontinuity?\" Of course, not being a mind-reader, I don't know how it was actually done, but here's how it might have been done. The key observation is that, in the fraction that defines $g(x,y)$, the numerator $xy^2$ is the geometric mean of the two terms in the denominator, $x^2$ and $y^4$. (That might sound complicated, but it really just means that the exponents of $x$ and $y$ in the numerator are the averages of their exponents in the two terms of the denominator: For the exponents of $x$, $1$ is the average of $2$ and $0$, and for the exponents of $y$, $2$ is the average of $0$ and $4$.) So if we give $x$ and $y$ values that make the two denominator terms equal, then the numerator will automatically be equal to them also. So the numerator will match each of the terms in the denominator, and the fraction will be $1\/2$. If we can find such values for $x$ and $y$ arbitrarily close to $0$, then that will make $g$ discontinuous at $(0,0)$. The easiest way to achieve this, meaning to make $x^2=y^4$, is to give $y$ an arbitrary value, say $t$, and to set $x=t^2$. So you set $(x,y)=(t^2,t)$ to find points, as close to $(0,0)$ as you like if $t$ is very small, where $g$ takes the value $1\/2$.\n\nFinally, if you're in a nasty mood, you re-name the variable $t$ as $x$, even though it serves as the value to substitute for $y$, just to confuse readers.\n\nP.S. If you know how to use a program like Mathematica, you can have it plot the graph of $g$, and you'll probably be able to see a sort of a ridge in the graph, over the parabola $x=y^2$, at height $z=1\/2$. So this might be another way to \"guess\" the substitution that proves discontinuity of $g$ at $(0,0)$.\n\n\u2022 This makes sense. Thank you! \u2013\u00a0Siyanda Apr 21 '13 at 13:21\n\nYou lost a factor $\\frac{1}{t}$ in your calculation of $D_v(g)$. The actual result is $$D_v(g)=\\lim_{t\\to 0}\\frac {g(0+tv) - g(0)}{t} = \\lim_{t\\to0} \\frac{xy^2}{x^2 + t^2y^4} = \\begin{cases}\\frac{y^2}{x} & x\\neq 0 \\\\ 0 & x=0 \\end{cases}.$$\n\nAs per your question: A function is differentiable only if its directional derivatives are a continuous function of direction, which is not satisfied in this case.\n\nIn a sense $g$ is differentiable in every direction, but these directional derivatives don't fit together to make an actual derivative. That's why we have to choose a strange path like $(x^2,x)$ as a counterexample, since a path that approaches from one specific direction would never work.\n\n\u2022 Try again. I think you'll find that, that isn't the case. \u2013\u00a0Siyanda Apr 20 '13 at 23:49\n\u2022 I'm sorry, but I really think it is. \u2013\u00a0Abel Apr 20 '13 at 23:58\n\nIf a function is differentiable , it is continuous. So if it is not continuous , it is not differentiable. We only need to find one path of approach to the origin along which the limit as we go to the origin does not equal the value of the function at the origin. Since a limit, if it exists, is unique regardless of the path of approach, we can show by one such counter example that the limit does not equal the function and thus that the function is not continuous and not differentiable.\n\nIn this case, we chose the parabola $x=y^2$ as the path of approach because this substitution makes the powers of the terms in the denominator equal.\n\nAs for the definition of differentiability, consider single variable functions: $$\\lim_{x\\to a}\\frac{f(x)-f(a)}{x-a}=f'(a)$$ $$\\lim_{x\\to a}\\frac{f(x)-f(a)}{x-a}=\\lim_{x\\to a}\\frac{f'(a)(x-a)}{x-a}$$ $$\\lim_{x\\to a}\\frac{f(x)-f(a)-f'(a)(x-a)}{x-a}=0$$ $$\\lim_{x\\to a}\\frac{f(x)-(f(a)+f'(a)(x-a))}{x-a}=0$$ Note that $f(a)+f'(a)(x-a)$ is the tangent line to $f$ at $a$.\n\nFor two variables, we use the tangent plane rather than the tangent line to approximate $f$, so we need $$\\lim_{(x,y)\\to (a,b)}\\frac{f(x,y)-(f(a,b)+\\frac{\\partial{f}}{\\partial{x}}(a,b)(x-a))+\\frac{\\partial{f}}{\\partial{y}}(a,b)(y-b))}{||(x,y)-(a,b)||}=0$$","date":"2019-09-22 07:57:55","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9185470938682556, \"perplexity\": 142.98234702701188}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-39\/segments\/1568514575402.81\/warc\/CC-MAIN-20190922073800-20190922095800-00311.warc.gz\"}"} | null | null |
from __future__ import absolute_import
import os
import time
import zipfile
from six import BytesIO, text_type
from django.core.files.uploadedfile import SimpleUploadedFile
from django.core.urlresolvers import reverse
from sentry.testutils import APITestCase, TestCase
from sentry.models import debugfile, File, ProjectDebugFile, DifMeta
# This is obviously a freely generated UUID and not the checksum UUID.
# This is permissible if users want to send different UUIDs
PROGUARD_UUID = text_type("6dc7fdb0-d2fb-4c8e-9d6b-bb1aa98929b1")
PROGUARD_SOURCE = b"""\
org.slf4j.helpers.Util$ClassContextSecurityManager -> org.a.b.g$a:
65:65:void <init>() -> <init>
67:67:java.lang.Class[] getClassContext() -> getClassContext
65:65:void <init>(org.slf4j.helpers.Util$1) -> <init>
"""
class DebugFileTest(TestCase):
def test_delete_dif(self):
dif = self.create_dif_file(
debug_id="dfb8e43a-f242-3d73-a453-aeb6a777ef75-feedface", features=["debug", "unwind"]
)
dif_id = dif.id
dif.delete()
assert not ProjectDebugFile.objects.filter(id=dif_id).exists()
assert not File.objects.filter(id=dif.file.id).exists()
def test_find_dif_by_debug_id(self):
debug_id1 = "dfb8e43a-f242-3d73-a453-aeb6a777ef75"
debug_id2 = "19bd7a09-3e31-4911-a5cd-8e829b845407"
debug_id3 = "7d402821-fae6-4ebc-bbb2-152f8e3b3352"
self.create_dif_file(debug_id=debug_id1)
dif1 = self.create_dif_file(debug_id=debug_id1)
dif2 = self.create_dif_file(debug_id=debug_id2)
difs = ProjectDebugFile.objects.find_by_debug_ids(
project=self.project, debug_ids=[debug_id1, debug_id2, debug_id3]
)
assert difs[debug_id1].id == dif1.id
assert difs[debug_id2].id == dif2.id
assert debug_id3 not in difs
def test_find_dif_by_feature(self):
debug_id1 = "dfb8e43a-f242-3d73-a453-aeb6a777ef75"
debug_id2 = "19bd7a09-3e31-4911-a5cd-8e829b845407"
debug_id3 = "7d402821-fae6-4ebc-bbb2-152f8e3b3352"
self.create_dif_file(debug_id=debug_id1, features=["debug"])
dif1 = self.create_dif_file(debug_id=debug_id1, features=["debug"])
self.create_dif_file(debug_id=debug_id1, features=["unwind"])
dif2 = self.create_dif_file(debug_id=debug_id2)
difs = ProjectDebugFile.objects.find_by_debug_ids(
project=self.project, debug_ids=[debug_id1, debug_id2, debug_id3], features=["debug"]
)
assert difs[debug_id1].id == dif1.id
assert difs[debug_id2].id == dif2.id
assert debug_id3 not in difs
def test_find_dif_by_features(self):
debug_id1 = "dfb8e43a-f242-3d73-a453-aeb6a777ef75"
debug_id2 = "19bd7a09-3e31-4911-a5cd-8e829b845407"
debug_id3 = "7d402821-fae6-4ebc-bbb2-152f8e3b3352"
dif1 = self.create_dif_file(debug_id=debug_id1, features=["debug", "unwind"])
self.create_dif_file(debug_id=debug_id1, features=["debug"])
self.create_dif_file(debug_id=debug_id1, features=["unwind"])
dif2 = self.create_dif_file(debug_id=debug_id2)
difs = ProjectDebugFile.objects.find_by_debug_ids(
project=self.project,
debug_ids=[debug_id1, debug_id2, debug_id3],
features=["debug", "unwind"],
)
assert difs[debug_id1].id == dif1.id
assert difs[debug_id2].id == dif2.id
assert debug_id3 not in difs
def test_find_legacy_dif_by_features(self):
debug_id1 = "dfb8e43a-f242-3d73-a453-aeb6a777ef75"
self.create_dif_file(debug_id=debug_id1)
dif1 = self.create_dif_file(debug_id=debug_id1)
# XXX: If no file has features, in a group, the newest one is chosen,
# regardless of the required feature set.
difs = ProjectDebugFile.objects.find_by_debug_ids(
project=self.project, debug_ids=[debug_id1], features=["debug"]
)
assert difs[debug_id1].id == dif1.id
def test_find_dif_miss_by_features(self):
debug_id = "dfb8e43a-f242-3d73-a453-aeb6a777ef75"
self.create_dif_file(debug_id=debug_id, features=[])
difs = ProjectDebugFile.objects.find_by_debug_ids(
project=self.project, debug_ids=[debug_id], features=["debug"]
)
assert debug_id not in difs
class CreateDebugFileTest(APITestCase):
@property
def file_path(self):
return os.path.join(os.path.dirname(__file__), "fixtures", "crash.dsym")
def create_dif(self, fileobj=None, file=None, **kwargs):
args = {
"file_format": "macho",
"arch": "x86_64",
"debug_id": "67e9247c-814e-392b-a027-dbde6748fcbf",
"data": {"features": ["debug"]},
"path": "crash.dsym",
}
args.update(kwargs)
return debugfile.create_dif_from_id(
self.project, DifMeta(**args), fileobj=fileobj, file=file
)
def test_create_dif_from_file(self):
file = self.create_file(
name="crash.dsym", checksum="dc1e3f3e411979d336c3057cce64294f3420f93a"
)
dif, created = self.create_dif(file=file)
assert created
assert dif is not None
assert dif.file.type == "project.dif"
assert "Content-Type" in dif.file.headers
assert ProjectDebugFile.objects.filter(id=dif.id).exists()
def test_create_dif_from_fileobj(self):
with open(self.file_path, "rb") as f:
dif, created = self.create_dif(fileobj=f)
assert created
assert dif is not None
assert dif.file.type == "project.dif"
assert "Content-Type" in dif.file.headers
assert ProjectDebugFile.objects.filter(id=dif.id).exists()
def test_keep_disjoint_difs(self):
file = self.create_file(
name="crash.dsym", checksum="dc1e3f3e411979d336c3057cce64294f3420f93a"
)
dif1, created1 = self.create_dif(file=file, data={"features": ["unwind"]})
file = self.create_file(
name="crash.dsym", checksum="2b92c5472f4442a27da02509951ea2e0f529511c"
)
dif2, created2 = self.create_dif(file=file, data={"features": ["debug"]})
assert created1 and created2
assert ProjectDebugFile.objects.filter(id=dif1.id).exists()
assert ProjectDebugFile.objects.filter(id=dif2.id).exists()
def test_keep_overlapping_difs(self):
file = self.create_file(
name="crash.dsym", checksum="dc1e3f3e411979d336c3057cce64294f3420f93a"
)
dif1, created1 = self.create_dif(file=file, data={"features": ["symtab", "unwind"]})
file = self.create_file(
name="crash.dsym", checksum="2b92c5472f4442a27da02509951ea2e0f529511c"
)
dif2, created2 = self.create_dif(file=file, data={"features": ["symtab", "debug"]})
assert created1 and created2
assert ProjectDebugFile.objects.filter(id=dif1.id).exists()
assert ProjectDebugFile.objects.filter(id=dif2.id).exists()
def test_keep_latest_dif(self):
file = self.create_file(
name="crash.dsym", checksum="dc1e3f3e411979d336c3057cce64294f3420f93a"
)
dif1, created1 = self.create_dif(file=file, data={"features": ["debug", "unwind"]})
file = self.create_file(
name="crash.dsym", checksum="2b92c5472f4442a27da02509951ea2e0f529511c"
)
dif2, created2 = self.create_dif(file=file, data={"features": ["debug"]})
file = self.create_file(
name="crash.dsym", checksum="3c60980275c4adc81a657f6aae00e11ed528b538"
)
dif3, created3 = self.create_dif(file=file, data={"features": []})
# XXX: dif2 and dif3 would actually be redundant, but since they are more
# recent than dif1 we keep both of them. This assumes that newer uploads
# might contain more specific debug information and should therefore
# receive precedence over older ones.
assert created1 and created2 and created3
assert ProjectDebugFile.objects.filter(id=dif1.id).exists()
assert ProjectDebugFile.objects.filter(id=dif2.id).exists()
assert ProjectDebugFile.objects.filter(id=dif3.id).exists()
def test_skip_redundant_dif(self):
with open(self.file_path, "rb") as f:
dif1, created1 = self.create_dif(fileobj=f)
with open(self.file_path, "rb") as f:
dif2, created2 = self.create_dif(fileobj=f)
assert created1
assert not created2
assert dif1 == dif2
def test_remove_redundant_dif(self):
file = self.create_file(
name="crash.dsym", checksum="dc1e3f3e411979d336c3057cce64294f3420f93a"
)
dif1, created1 = self.create_dif(file=file, data={"features": ["debug"]})
file = self.create_file(
name="crash.dsym", checksum="2b92c5472f4442a27da02509951ea2e0f529511c"
)
dif2, created2 = self.create_dif(file=file, data={"features": ["debug"]})
assert created1 and created2
assert not ProjectDebugFile.objects.filter(id=dif1.id).exists()
assert ProjectDebugFile.objects.filter(id=dif2.id).exists()
class DebugFilesClearTest(APITestCase):
def test_simple_cache_clear(self):
project = self.create_project(name="foo")
url = reverse(
"sentry-api-0-dsym-files",
kwargs={"organization_slug": project.organization.slug, "project_slug": project.slug},
)
self.login_as(user=self.user)
out = BytesIO()
f = zipfile.ZipFile(out, "w")
f.writestr("proguard/%s.txt" % PROGUARD_UUID, PROGUARD_SOURCE)
f.writestr("ignored-file.txt", b"This is just some stuff")
f.close()
response = self.client.post(
url,
{
"file": SimpleUploadedFile(
"symbols.zip", out.getvalue(), content_type="application/zip"
)
},
format="multipart",
)
assert response.status_code == 201, response.content
assert len(response.data) == 1
assert response.data[0]["headers"] == {"Content-Type": "text/x-proguard+plain"}
assert response.data[0]["sha1"] == "e6d3c5185dac63eddfdc1a5edfffa32d46103b44"
assert response.data[0]["uuid"] == PROGUARD_UUID
assert response.data[0]["objectName"] == "proguard-mapping"
assert response.data[0]["cpuName"] == "any"
assert response.data[0]["symbolType"] == "proguard"
difs = ProjectDebugFile.difcache.fetch_difs(
project=project, debug_ids=[PROGUARD_UUID], features=["mapping"]
)
assert len(difs) == 1
assert os.path.isfile(difs[PROGUARD_UUID])
# if we clear now, nothing happens
ProjectDebugFile.difcache.clear_old_entries()
assert os.path.isfile(difs[PROGUARD_UUID])
# Put the time into the future
real_time = time.time
time.time = lambda: real_time() + 60 * 60 * 48
try:
ProjectDebugFile.difcache.clear_old_entries()
finally:
time.time = real_time
# But it's gone now
assert not os.path.isfile(difs[PROGUARD_UUID])
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,399 |
A 24.ª Paris-Roubaix teve lugar a 1 de abril de 1923 e foi vencida pela suíço Henri Suter. Pela primeira vez na história não teve um francês no pódium da Paris-Roubaix.
Classificação final
Referências
Ligações externas
Site Oficial
Resultados completos da corrida
1923
1923 no ciclismo
1923 na França | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,888 |
{"url":"https:\/\/socratic.org\/questions\/599e2d097c01492349784c98","text":"# Question 84c98\n\n##### 1 Answer\nAug 24, 2017\n\n$2$\n\n#### Explanation:\n\nThe formula unit of calcium bicarbonate is \"Ca\"(\"HCO\"_3)_2#.\n\nThis means that there are $2$ bicarbonate ions.\n\nSince there is $1$ carbon atom per bicarbonate ion, there are $1 \\times 2 = 2$ carbon atoms per formula unit.","date":"2020-01-21 02:39:58","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 5, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9285407066345215, \"perplexity\": 13396.708211443674}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-05\/segments\/1579250601241.42\/warc\/CC-MAIN-20200121014531-20200121043531-00069.warc.gz\"}"} | null | null |
{"url":"https:\/\/socratic.org\/questions\/how-do-you-simplify-sqrt4-sqrt3","text":"# How do you simplify sqrt4*sqrt3?\n\nMar 6, 2018\n\n2$\\sqrt{3}$\n\n#### Explanation:\n\nTwo ways to do this are as follows:\n\n1. simplify $\\sqrt{4}$ to 2 and multiply to get 2$\\sqrt{3}$\n\n2. multiply the two numbers inside the square roots together and take the root of the product\nex. $\\sqrt{12}$\nthis can be further broken down into 2$\\sqrt{3}$","date":"2020-04-08 03:22:39","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 5, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8435472846031189, \"perplexity\": 544.7786358542148}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-16\/segments\/1585371807538.83\/warc\/CC-MAIN-20200408010207-20200408040707-00150.warc.gz\"}"} | null | null |
Q: Rails 4, delayed_job, rspec: setting an attribute to Delayed::Job.last.id...how to test? I enqueue a mailer for future delivery when a user clicks a button. If the user cancels the original action, I want to un-queue the to-be-delivered mailer
To do so, I store the ActiveJob id as an attribute on the enqueued job's payload object (the alternative is parsing YAML stuff along the lines of this to locate the payload object).
This line does the work (located in the payload object's model file)
self.update_attributes enqueued_job_id: Delayed::Job.last.id
In my model specs, since mailers are not actually enqueued, Rspec keeps thinking Delayed::Job is nil, so all of the payload model's specs fail.
Question: If (a big 'if') my approach to this problem is 'correct,' how can I make get my model specs to pass?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,023 |
\section{Introduction}
\label{sec:intro}
A rapidly growing number of functional magnetic resonance imaging (fMRI) studies have given us important insights into the mental processes that underpin behavior. However, individual studies are often
power-restricted~\citep{carp2012secret,button2013power}, since the number of subjects and mental processes that can be interrogated in a single experiment is limited~\citep{church2010task}.
One approach to digest the vast literature and synthesize across many studies is to perform a meta-analysis of the reported results, such as the coordinates of the most significant effects (e.g., 3D location of peak brain activation in response to a task).
These meta-analyses usually require expert curation of relevant experiments (e.g. \citep{costafreda2008predictors,minzenberg2009meta,shackman2011integration}).
A critical technical challenge here is the consolidation of synonymous terms.
Importantly, over time, different denominations might be used in different contexts or invented to refine existing ideas.
For instance, ``self-generated thought'', one of the most highly studied functional domains of the human brain~\citep{smallwood2013distinguishing}, can be referred to by varying terms, such as ``task-unrelated thought''~\citep{andrews2014default}.
The selection of reported results for meta-analysis can be automated on data scraped from the published literature~\citep{yarkoni2011large,dockes2020neuroquery,rubin2017decoding}.
Two popular examples of this direction are Neurosynth~\citep{yarkoni2011large} and more recently Neuroquery~\citep{dockes2020neuroquery}.
Neurosynth utilizes automated keyword search to retrieve relevant studies and statistical tests to find summary brain activation maps corresponding to the keywords.
Unlike Neurosynth, Neuroquery is a predictive model that synthesizes activation maps from keywords in the input query.
Despite their differences in modeling, both Neurosynth and Neuroquery only support queries consisting predefined keywords.
Furthermore, Neurosynth does not explicitly handle long queries, while Neuroquery relies on superficial lexical similarity via word co-occurences for inference of longer or rarer queries.
We propose an alternative approach named Text2Brain, which builds on recent neural language models and permits more flexible free-form text queries.
Text2Brain captures a more fine-grained and implicit semantic similarity via vector representations from the neural language model in order to retrieve more relevant studies.
Furthermore, in contrast to tools like Neuroquery, our method computes synthesized activation maps via a 3D convolutional neural network (CNN) model, which we empirically demonstrate, can capture coarse and fine details.
We compare Text2Brain's predictions with those from Neurosynth and Neuroquery, where we used article titles as free-form queries.
Furthermore, we assess model predictions on independent test datasets, including reliable task contrasts and meta-analytic activation maps of well-studied cognitive domains predicted from their descriptions.
Our analysis shows that Text2Brain generates activation maps that better match the target images than the baselines tools.
Given its flexibility in taking input queries, Text2Brain can be used as an educational aid as well as a tool for synthesizing maps based on published results or generating novel hypotheses for future research.
Compared to our conference article~\citep{ngo2021text2brain}, we have extensively expanded our results and analysis.
Specifically, we have expanded on the model validation on article titles with a different test set (section~\ref{subsec:exp_title_setup} and~\ref{subsec:title_results}), added additional evaluation on the contrast maps predicted from their descriptions (section ~\ref{subsec:contrast_results}).
New results and discussion have also been added to this paper, including a high-level conceptual comparison of models (section~\ref{subsec:high_level_comparison}), new experiments on predicting representative meta-analytic results (section~\ref{subsec:contrast_methods} and~\ref{subsec:contrast_results}), and quantitative analysis of the models' robustness to input queries (section~\ref{subsec:synonym_methods} and~\ref{subsec:synonym_results}).
\section{Datasets and Methods}
\subsection{Model overview}
Figure~\ref{fig:model} shows the overview of this work, including data generation, model architecture, and model training.
The Text2Brain model has an encoder-decoder architecture that maps text sequences into brain activation maps~(Section~\ref{subsec:model}).
Its transformer-based encoder uses self-attention to encode a snippet of text input into vector representation~\citep{vaswani2017attention,devlin2018bert}.
Text2Brain's 3D convolutional decoder (CNN) then translates the vector representation into a 3D brain activation map.
The Transformer is currently the most effective approach for modeling text since it can capture long-distance dependency between words and can learn efficiently through self-supervision from massive text corpora~\citep{jawahar2019does,raffel2020exploring}.
On the other hand, 3D CNNs are the most dominant architectural design in medical imaging~\citep{milletari2016v,kamnitsas2017efficient}.
In our proposed approach, we first extract full text and activation coordinates from each research article to create data samples.
Each sample consists of an input snippet from the full text and an output 3D activation map created using the coordinates (Section~\ref{subsec:data}).
Text2Brain is trained to associate the input text to activations at various spatial locations.
Since Text2Brain's transformer-based encoder is context-sensitive, it can better extract information from free-form query by refining the vector representation depending on the specific phrasing of the text inputs~\citep{tenney2019you}.
In contrast, the classical keyword search mainly exploits co-occurrence of keywords regardless of context and therefore may struggle on more nuanced queries~\citep{salton1988term}.
Furthermore, keyword search approaches store one activation map for each supported keyword, which are in turn linearly combined for queries. This approach can limit how many keywords are supported~\citep{yarkoni2011large,dockes2020neuroquery}.
On the other hand, Text2Brain stores the text and activation maps content in its parameters and can scale better to diverse input queries~\citep{petroni2019language}.
We use data augmentation to encourage Text2Brain to construct and store rich many-to-one mappings between textual description and activation maps (Section \ref {subsec:training}).
This allows Text2Brain to better map semantically similar text queries to similar activation maps.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{model.pdf}
\caption{Overview of data preprocessing, the Text2Brain model, and training procedure. All activation maps are 3D volumes, but projected to the surface for visualization.}
\label{fig:model}
\end{figure}
\subsection{Implementation}
\label{subsec:model}
Figure~\ref{fig:model} bottom left corner shows the Text2Brain model with its text encoder and 3D CNN image decoder.
Text2Brain's text encoder is based on SciBERT, a BERT model that has been trained using scientific articles~\citep{beltagy2019scibert}.
BERT is a transformer-based model with bidirectional self-attention trained via self-supervision to learn semantic representations of textual input~\citep{devlin2018bert}.
The text encoder outputs a vector representation of dimension $768$.
This vector is projected using a fully-connected layer and then reshaped to a 3D volume of dimension $4\times5\times4$ voxels with $64$ channels at each voxel.
The image decoder consists of 3 transposed 3D convolutional layers with 32, 16, 8 channels respectively.
Text2Brain was trained using the Adam optimizer~\citep{loshchilov2018decoupled} and the mean-squared error with a batch size of 24 for 2000 epochs.
The learning rate for the text encoder and image decoder are set at $10^{-5}$ and $3\times{10^{-2}}$ respectively.
The model's source code is available at \url{https://github.com/sabunculab/text2brain}.
\subsection{Data Preprocessing}
\label{subsec:data}
We used the same set of 13,000 neuroimaging articles previously released in~\citep{dockes2020neuroquery} in our experiments.
Each article contains one or more tables of results that reported coordinates of peak activation in MNI152 coordinate system~\citep{lancaster2007bias}.
The activation foci are also publicly released by Neuroquery~\citep{dockes2020neuroquery}.
Following the same procedure as \citep{dockes2020neuroquery}, the set of activation foci associated with each table is used to generate an activation map by placing a Gaussian sphere with full width at half maximum (FWHM) of 9mm at each of the coordinates of peak activation.
The chosen FWHM allows a fair comparison with Neuroquery\citep{dockes2020neuroquery} in our experiments, and is consistent with previous work~\citep{wager2009evaluating,yarkoni2011large,yeo2015functional}.
Supplemental section~\ref{sec:ablation_fwhm} shows an analysis of the effect of the Gaussian kernel's FWHM used for preprocessing on Text2Brain's predictive accuracy on an independent test set.
This comparison confirms that the choice of the kernel's FWHM is reasonable.
An article-average activation map is also generated by averaging the activation maps of all the tables in the article.
The text associated with the activation maps are extracted from the articles' full text.
The articles' full text are scraped using their PubMedID via the NCBI API~\footnote{\url{https://www.ncbi.nlm.nih.gov/books/NBK25501/}} and the Elsevier E-utilities API~\footnote{\url{https://dev.elsevier.com/}}.
As there may be multiple text snippets corresponding to the same activation map, the next section (Section~\ref{subsec:training}) shows how the corresponding text of an activation map is selected.
\subsection{Training}
\label{subsec:training}
Each training sample consists of a text-activation map pair and correspond to an neuroimaging article.
The activation map is sampled uniformly at random from the union set of table-specific maps and article-average map.
For each table-specific map, the first sentence of the corresponding table caption is chosen as the map's associated text.
Our initial data exploration suggested that the first sentence to be the most relevant description of the activation map.
For each article-average map, the associated text that describes the activation map is sampled uniformly at random from the following four sources: (1) the article's title; (2) one of the article's keywords; (3) the article's abstract; and (4) a randomly chosen subset of sentences from the discussion section of the article.
This data augmentation strategy encourages Text2Brain to generalize over input texts of different lengths.
Furthermore, matching the same activation pattern with multiple different text snippets encourages the model to recognize important words common across the snippets and to learn the association between different but synonymous words.
Supplemental Figure~\ref{fig:supp_ablation} shows our ablation study on the sampling strategy.
The liberal (and likely noisy) construction of image-text pairs appears to perform better than more deliberate coupling of image-text snippets strategies (not reported) that we tried in our preliminary experiments. We surmise that simply presenting different text snippets to a target brain image is analogous to another augmentation strategy that allows the neural network to pool across samples and learn the relevant words and their weights with respect to the target brain maps.
Training with the set up in~\ref{subsec:model} takes approximately 75 hours on one NvidiaRTX GPU while one inference pass with an input query of up to 140 characters takes less than 1 second.
\subsection{Baselines}
We compare Text2Brain to 2 different baselines: Neurosynth~\citep{yarkoni2011large} and Neuroquery~\citep{dockes2020neuroquery}.
For a keyword, Neurosynth first finds all neuroimaging articles that mention that keyword.
Then, one statistical test per voxel is performed across the activation maps corresponding to those studies to determine a significant association.
Since Neurosynth was not formulated to handle multiple-word queries, for such query, we performed statistical test using activation maps from all articles that contain at least one of the keywords in the query.
Neuroquery extends Neurosynth's vocabulary of keywords by including more curated keywords from lexicons such as MeSH, NeuroNames, and NIF~\citep{lipscomb2000medical,bowden1995neuronames,gardner2008neuroscience}.
The keyword encoding is obtained after performing non-negative matrix factorization of the articles' full text (as a bag of keywords) represented with term frequency - inverse document frequency (TF-IDF) features~\citep{salton1988term}.
A ridge regression model was trained to map the text encoding to the activation.
The inference of a keyword is smoothed by a weighed average of its most related keywords (in the TF-IDF space).
For multiple-word queries, the predicted activation map is obtained by averaging the activation maps from all keywords in the input, weighed by the coefficients learnt during training.
\subsection{Evaluation Metrics}
\label{subsec:metrics}
For thresholded target activation maps such as those computed by ALE~\citep{eickhoff2009coordinate}, the predicted brain maps are thresholded to retain the same number of most activated voxels as the target.
For example, given an estimated activation map by ALE with statistically significant clusters of activation that cover 25\% of the the brain volume, the brain maps predicted by Text2Brain, Neuroquery, and Neurosynth are also thresholded to retain the top 25\% most activated voxels in each map.
The accuracy of prediction is measured by Dice score~\citep{dice1945measures} which quantifies the extent of overlap between the predicted and target brain maps (details are in Supplemental Section~\ref{subsec:supp_metrics}).
Furthermore, we use Dice scores at different thresholds to estimate the similarity between predicted and target activation maps at different levels of detail~\citep{ngo2022predicting}.
This evaluation procedure is similar to that used in \citep{dockes2020neuroquery} for a thresholded target map, but we apply the same thresholding to both the target and predicted map.
For example, at 5\% threshold (considering the 5\% most activated voxels), the Dice score measures the correspondence of the fine-grained details between the target and predicted activation maps.
At higher thresholds (e.g. 25\%), the score captures the gross agreement between activation clusters.
We also estimated the area under the Dice curve (AUC) as a summary measure using approximated integration of Dice scores across all thresholds from 5\% up to 30\%.
Supplemental Figure \ref{fig:supp_dice} shows the Dice curve for an example pair of target-predicted activation maps.
Note that the range of thresholds in the x-axis also conveys the maximum percentage of the gray matter mask that has an activation in the target brain map.
For example, if only a proportion of gray matter mask has activation, such as the case of Neuroquery prediction that mostly extends up to 30\% of the gray matter mask or a sparse target activation pattern from the coordinate-based meta-analysis, the x-axis range will not be extended up to 1.
In our experiments, all evaluation is performed in the MNI152 volumetric space, which is the original space of all predicted maps.
For visualization, with activation maps that mostly concentrate in the cerebral cortex, the original volumetric images are transformed from MNI152 space to fs\_LR surface space using Connetome Workbench~\citep{van2013wu} via the FreeSurfer surface space~\citep{buckner2011organization,fischl2012freesurfer}, with isolated surface clusters of less than 20 vertices being removed~\citep{wu2018accurate}.
Activation maps with significant activation in the non-cortical parts of the brain are visualized by cross-sectional slices with significant activation using Nilearn~\citep{abraham2014machine}.
\subsection{High-level model comparison}
\label{subsec:high_level_comparison}
\begin{table}[]
\centering
\begin{tabular}{|l|l|l|l|}
\hline
& Neurosynth & Neuroquery & Text2Brain \\ \hline
Vocabulary & Fixed & Fixed & Unlimited \\ \hline
\makecell[l]{Handle of\\complex query} & None & \makecell[l]{Lexical\\similarity} & \makecell[l]{Semantic\\similarity} \\ \hline
\makecell[l]{Predictive\\models} & None & \makecell[l]{TF-IDF,\\linear\\regression} & \makecell[l]{Transformer,\\ convolution} \\ \hline
\end{tabular}
\caption{High-level comparison of approaches to meta-analytic brain maps generation}
\label{tab:comparison}
\end{table}
Text2Brain can better handle input text than prior approaches because its vocabulary is not limited to a fixed pre-defined set of words.
In contrast, Neurosynth and Neuroquery rely on fixed word vocabularies and cannot predict for queries consisting of out-of-vocabulary words.
Besides, Neurosynth's and Neuroquery's vocabularies are not sufficiently extensive, covering only a fraction (under 10\%)~\citep{dockes2020neuroquery} of terms in relevant neuroimaging lexicons such as Cognitive Atlas~\citep{poldrack2016brain} and NeuroNames~\citep{bowden1995neuronames}.
Text2Brain's usage of byte-pair encoding enables the model to handle infrequent and out-of-vocabulary words more gracefully, by breaking down those words into digestable sub-word tokens~\citep{sennrich2016neural}.
Hence, Text2Brain's vocabulary is open ended and can scale with training data to be unlimited in theory.
Besides, Text2Brain's training is not limited to only training set data.
Text2Brain can leverage self-supervised learning from non-neuroimaging scientific articles, as well as neuroimaging articles that do not report activation coordinates to learn a better text-to-activation-map transformation.
By finetuning a SciBERT text encoder pretrained on the larger dataset of scientific articles (including non-neuroimaging articles), Text2Brain seems to converge on an optimum with a more useful representational space of the input text. Supplemental section~\ref{sec:ablation_pretraining} shows the comparison between the Text2Brain model that uses pretrained SciBERT text encoder versus a randomly initialized text encoder. Evaluation on predicting article-average activation maps from both sets of test articles in the Neuroquery dataset (similar to section~\ref{subsec:exp_title_setup}) suggests that pretraining benefits the Text2Brain's performance.
Furthermore, Text2Brain uses contextualized text embeddings to model semantic relationship between words so it can deal with nuanced queries more effectively.
Methods such as Neurosynth and Neuroquery may have difficulty dealing with complex expressions.
By simply averaging the keywords' activation maps to arrive at the prediction for a complex query, these methods may fail to account for relationship between words in the query, such as order and semantic.
Lastly, while the predictive approach of Neuroquery constructs the predicted activation map by modelling voxels' activation independently, Text2Brain generates the whole-brain activation with a 3D convolutional decoder that takes in the text encoding produced by the language model.
By upsampling and computing the whole-brain activation from a bottleneck, Text2Brain can better model both the short and long-distance relationship between voxels.
\section{Experimental Setup}
\subsection{Predict activation maps from article title}
\label{subsec:exp_title_setup}
Two test sets were created from the Neuroquery dataset of 13,000 studies.
The first test set consists of 1000 randomly sampled articles.
The second test set also consists of 1000 articles but was randomly sampled such that the keywords (defined by the articles' authors) do not appear in the training and validation articles.
The two test sets are labeled as easy and hard test sets respectively.
Of the remaining articles, 1000 are randomly held out as a validation set for parameters tuning.
For each article, the article-average activation map is predicted from its title using Text2Brain, as well as the Neurosynth and Neuroquery baselines.
Both Text2Brain and Neuroquery were trained on the 10,000 articles in the training set.
The Text2Brain model is trained using both the articles' titles and samples from the full-text, while Neuroquery is trained on the articles' full-text.
We use predictions from the publicly available Neurosynth model at \href{https://neurosynth.org/}{https://neurosynth.org}, which was trained on the articles' abstracts.
Note that Neurosynth is not a predictive model meant for out-of-sample prediction, but for performing automated statistical testing of associations between terms and brain locations.
\subsection{Predict activation maps from contrast descriptions}
\subsubsection{Individual Brain Charting (IBC) task contrasts}
The Individual Brain Charting (IBC) project~\citep{pinho2020individual} estimates an extensive functional atlas of the human brain via fMRI data of subjects measured under a large number of task conditions.
In particular, the IBC dataset consists of 180 task contrasts measured on 12 subjects.
We use the activation maps provided by the IBC project to measure the predictive accuracy of Text2Brain and the two baselines over a wide range of functional domains, given the contrast descriptions from IBC.
\subsubsection{Human Connectome Project (HCP) task contrasts}
While the IBC dataset offers a large number of reference brain maps, the small number of subjects might make some results less reliable.
We also utilized the Human Connectome Project (HCP) data both for reference and a measure of reliability of target maps.
The HCP dataset consists of neuroimaging data from over 1200 subjects, including task fMRI (tfMRI) of 86 task contrasts from 7 domains~\citep{barch2013function}, which overlap with 43 contrasts under the IBC dataset.
We evaluate the model prediction of HCP task contrasts from their descriptions.
While HCP provides detailed descriptions of task contrasts, we opt for the more concise contrast descriptions provided by the Individual Brain Charting (IBC) as they are more succinct and thus more favorable to the baselines.
The IBC contrast descriptions are extracted from the metadata of the activation maps released on Neurovault \href{https://neurovault.org/images/360528}{https://neurovault.org/images/360528}.
The list of all IBC description of HCP contrasts are included in Supplemental Table~\ref{tab:ibc_description}.
On the other hand, the target (ground-truth) activation maps are the HCP group-average contrast maps, as the large number of subjects provides more reliable estimates of the contrast maps.
In the analyses of this experiment, we use the agreement between the IBC and HCP maps as a measure of reliability.
Despite using similar protocols, there are subtle differences between the IBC and HCP experiments.
For instance, the original HCP language task was conducted in English but the corresponding language task in the IBC project was conducted in French.
\subsection{Predict representative meta-analytic brain maps}
\label{subsec:contrast_methods}
The automated approach to brain map generation of Text2Brain and the 2 baselines are compared against published brain maps created from a manually curated set of meta-analyses.
In particular, 5 cognitive concepts and their corresponding activation maps of 5 representative meta-analytic studies from ANIMA database~\citep{reid2016anima} were selected.
The 5 meta-analytic studies were selected for having the most number of experiments and their different coverage of the human brain.
The cognitive processes of interest are visual processing, auditory processing, motor execution~\citep{heckner2021delineating}, working memory~\citep{rottschy2012modelling}, and pain~\citep{xu2020convergent}.
Each study searches for published neuroimaging studies that contain a set of texts queries relevant to the cognitive concept of interest.
For example, in~\cite{rottschy2012modelling}, the phrases to search for working memory-related studies are ``working memory'' and ``short-term memory''.
The same text queries for discovering relevant studies in the original meta-analysis were used as input to Neurosynth, Neuroquery, and Text2Brain.
Table~\ref{tab:domain_exp} shows the search queries and the number of experiments included in the original meta-analysis of the 5 chosen cognitive concepts.
Activation maps generated from all text input queries corresponding to each cognitive concept are averaged to yield a single brain map for each model.
The reference brain images for comparison are the activation maps released by the studies and made publicly available on ANIMA.
The reference activation maps are produced by Activation Likelihood Estimation (ALE)~\citep{turkeltaub2002meta,laird2005ale,eickhoff2009coordinate} and thresholded to retain only the statistically significant clusters of activation.
For all reference ALE maps, the cluster-level forming threshold at voxel-level is $p<0.001$ and cluster-level corrected threshold is set at $p<0.05$ by the original authors~\citep{eickhoff2012activation}.
For comparison, the generated brain maps are thresholded to keep the same number of survived voxels as those in the reference activation maps.
The accuracy of each model's generated brain map is evaluated as the Dice score between the (thresholded) generated brain map and the target (thresholded) brain map (see Section~\ref{subsec:metrics}).
\begin{table}[h!]
\centering
\begin{tabular}{ | l | l | l | }
\hline
Functional domain & \#Exp & Search queries \\ \hline
\makecell[l]{Visual processing\\
(Heckner 2021)} & 114 & \makecell[l]{visual processing\\
face monitor\\
face discrimination\\
film viewing\\
fixation\\
flashing checkerboard\\
passive viewing\\
visual object identification\\
visual pursuit\\
visual tracking\\
visuospatial attention} \\\hline
\makecell[l]{Auditory processing\\
(Heckner 2021)} & 122 & \makecell[l]{auditory processing\\
divided auditory attention\\
music comprehension\\
oddball discrimination\\
passive listening\\
phonological discrimination\\
pitch monitor\\
pitch discrimination\\
tone monitor\\
tone discrimination} \\\hline
\makecell[l]{Motor execution\\
(Heckner 2021)} & 251 & \makecell[l]{motor execution\\
writing\\
chewing\\
swallowing\\
drawing\\
isometric force\\
motor learning\\
grasping\\
finger tapping\\
button press\\
flexion\\
extension\\ } \\ \hline
\makecell[l]{Working memory\\
(Rottschy 2012)} & 189 & \makecell[l]{working memory\\
short-term memory\\ }
\\\hline
\makecell[l]{Pain\\
(Xu 2020)} & 222 & \makecell[l]{
pain\\
noxious\\
nociception }
\\\hline
\end{tabular}
\label{tab:domain_exp}
\caption{Meta-analytic studies of representative functional domains. The studies were selected from the ANIMA dataset~\citep{reid2016anima} that have the most number of experiments and covere a diverse set of brain regions.}
\end{table}
\subsection{Evaluate robustness of model prediction to semantically-equivalent queries}
\label{subsec:synonym_methods}
With the continual improvement of our understanding of the human brain and mind, neuroscientific knowledge is also an ever evolving repertoire.
Several neuroimaging concepts have also been changing, adapting and broadening over time.
Thus, we were interested in examining if our approach is robust to semantically equivalent queries.
For example, ``self-generated thought'', one of the most intensively examined cognitive domains in neuroscience, has had its definition refined and assigned different denominations over the years.
As a cognitive paradigm, different names have been used to refer to the set of inward-oriented psychological processes, such as ``self-generated thought''~\citep{smallwood2013distinguishing}, or ``task-unrelated thought''~\citep{andrews2014default}.
Both terms are associated with ``default network''~\citep{buckner2008brain}, the set of brain regions with elevated activation when subjects are not subjected to any external stimulus.
To assess models' prediction of synonymous queries, we utilized the ontology from the Cognitive Atlas~\citep{poldrack2011cognitive,bilder2009cognitive}.
The Cognitive Atlas is a collaborative knowledge base for neuroscience with content such as cognitive concepts, their description and synonyms (aliases) contributed by the project's voluntary participants~\citep{miller2010cognitive}.
At the time of our experiments, Cognitive Atlas includes 885 concepts with definition, 108 of which have at least one alias.
We considered a model to be robust with respect to a specific cognitive concept's definition if the activation map predicted from the description matches the predicted map from the concept's name.
In particular, given a model's predicted brain maps from all 885 Cognitive Atlas concept names and their description, we assess if the model's brain map predicted from a concept's definition is one of the $k$ maps (out of 886 possible maps) most similar to the model's brain map predicted from the concept's name.
In our experiments, top-1, top-5 and top-10 matching accuracy were evaluated using Dice AUC metrics.
The different values of $k$'s account for the uncertainty of the concepts' natural language text, e.g., different contributors might use different names to refer to the same concept.
Similarly, models' robustness with respect to a cognitive concept's alias is measured by the accuracy of matching the activation maps predicted from the text of a concept's alias and its name.
\section{Results}
\label{sec:results}
\subsection{Validation of activation maps predicted from article title}
\label{subsec:title_results}
\begin{figure}
\centering
\includegraphics[width=0.9\linewidth]{article_title_AUC__combined.pdf}
\caption{Evaluation of article-average activation maps predicted from their titles measured in area under the Dice curve (AUC) score. The left and right graph show the Dice AUCs of samples from the easy and hard test sets, respectively (Section~\ref{subsec:exp_title_setup}). The p-values are computed from paired-sample t-tests between Text2Brains and each of the 2 baselines.}
\label{fig:article_AUC}
\end{figure}
Figure \ref{fig:article_AUC} shows the quality of activation maps predicted from the titles of 1000 articles in each of the two test sets (section~\ref{subsec:exp_title_setup}).
In the easy test set (the test articles' keywords can overlap with the training articles'), the proposed Text2Brain model (mean Dice AUC = $0.0636$) outperforms Neuroquery (mean Dice AUC = $0.0523$) and Neurosynth (mean Dice AUC = $0.0453$).
In the hard test set (the test articles' keywords are not present in the training set), the Text2Brain model (mean Dice AUC = $0.0609$) also performs better than Neuroquery (mean AUC = $0.0499$) and Neurosynth (mean AUC = $0.0457$).
Paired-sample t-tests show that the performance differences in both test sets are statistically very significant.
The p-values when comparing Neuroquery and Neurosynth are $p=5.25\times10^{-27}$ and $p=2.40\times10^{-12}$.
Fig.~\ref{fig:article_AUC} also indicates how the different models handle out-of-sample input text.
Text2Brain can make a prediction for all input texts, evident with positive Dice AUCs for all samples.
On the other hand, Neurosynth fails to make prediction for some article titles in both test sets, resulting in zero Dice AUCs for such samples.
Similarly, Neuroquery fails to make prediction for some samples in the hard test set.
These failure cases are caused by the limited vocabularies of Neurosynth and Neuroquery that cannot cover the words in the test input queries.
On the other hand, the language model of Text2Brain is finetuned from SciBert, which has been pretrained on a broader lexicon and utilizes sub-word tokens to extend the vocabulary to unseen words (more details in Section~\ref{subsec:high_level_comparison}).
\subsection{Prediction of task contrast maps from description}
\label{subsec:contrast_results}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{IBC_all_AUC.pdf}
\caption{Dice AUCs of predicted IBC task activation maps from contrasts' description. The p-values are estimated from paired-sample t-tests between Text2Brain against the two baselines.}
\label{fig:ibc_auc}
\end{figure}
Fig.~\ref{fig:ibc_auc} shows the Dice AUC scores for the prediction of Text2Brain, Neuroquery and Neurosynth against the IBC group-average task contrast maps.
Text2Brain (mean Dice AUC = 0.0507) improves upon both Neuroquery (mean Dice AUC = 0.0457, $p = 4.11\times{10^{-4}}$), and Neurosynth (mean Dice AUC = 0.0404, $p=1.58\times{10^{-9}}$).
The p-values are measured by 2-tail paired-sample t-test between Text2Brain and the two baselines.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{IBC_HCP_AUC.pdf}
\caption{Dice AUCs of predicted HCP task activation maps from contrasts' description. The graph includes 22 contrasts with the highest HCP-IBC's Dice AUC scores and sorted in decreasing order.}
\label{fig:hcp_auc}
\end{figure}
Fig.~\ref{fig:hcp_auc} shows the AUC scores for the prediction of the three models and the IBC average contrasts, against the HCP target maps.
The 22 contrasts with above-average HCP-IBC's AUC scores, considered to be the reliable contrasts, are shown.
Across all 43 HCP contrasts, Text2Brain (mean AUC = $0.082$) performs better than the baselines, i.e. Neuroquery (mean AUC = $0.0755$, $p = 0.08$), Neurosynth (mean AUC = $0.047$, $p = 1.5\times10^{-5}$), where $p$-values are computed from the paired t-test between Text2Brain's and the baselines' prediction.
As reference, IBC contrasts yield a mean AUC = $0.094$ when compared to the corresponding HCP maps (Statistical comparison with Text2Brain, $p = 0.077$).
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{IBC_HCP.pdf}
\caption{Task activation maps predicted from contrasts' description. Each row shows both the thresholded maps of the top 25\% most activated voxels (top) and the overlap between predicted and target binarized brain maps. Blue is activation in the target contrast, red is the predicted activation and yellow is the overlap.}
\label{fig:hcp_example}
\end{figure*}
Figure \ref{fig:hcp_example} shows the prediction for three contrasts correspond to different HCP task groups, namely ``MOTOR'', ``LANGUAGE'', ``RELATIONAL'' thresholded at the top 25\% most activated voxels.
The three task groups were chosen to show results for a range of target images with different levels of reliability.
The two task groups ``MOTOR'' and ``LANGUAGE'' are the two most reliable task (having the highest average HCP-IBC AUC across all contrasts), while ``RELATIONAL'' has the lowest average HCP-IBC AUC.
Text2Brain's prediction improves over the baselines for the three contrasts.
Neurosynth was not able to generate activation maps for one of the contrast descriptions (``Move tongue'').
On the other hand, for the ``Move tongue'' contrast, Neuroquery predicts activation in the primary cortex, but the peak is in the wrong location, shifted more toward the hand region of the homunculus. Additionally, there is a false positive prediction in the occipital cortex, which might be an artifact from modeling brain activation coupled with visual stimuli-related words describing the motor experiments.
\subsection{Prediction of brain maps from representative meta-analytic studies}
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{example_meta_maps.pdf}
\caption{Prediction of brain maps from meta-analytic studies of representative functional domains. The information of the investigated functional domains are listed in Table~\ref{tab:domain_exp}. Reference and predicted activation maps of the first 4 function domains are visualized on the brain surface. The last domain (``pain'') is visualized in the volume as most activation concentrates in the non-cortical parts of the brain. For all functional domains, Text2Brain generates reasonable activation maps and comparable with the baselines for the common functional domains.}
\label{fig:example_meta}
\end{figure*}
Figure~\ref{fig:example_meta} shows the prediction of activation maps for 5 representative meta-analytic studies with the most number of experiments from ANIMA~\citep{reid2016anima}.
Among the three models, Neuroquery has the lowest Dice score on average, with prediction on ``Visual processing'', ``Working memory'', and ``Pain'' that significantly deviates from the target maps.
On the other hand, Neurosynth-derived brain maps consistently match well against the target maps.
The high accuracy of Neurosynth prediction is expected since the five chosen cognitive concepts are among the most commonly studied concepts with the most number of experiments reporting activation coordinates in the literature.
Given high number of available experiments and the input queries mostly exist in Neurosynth's predefined keyword set, the activation coordinates scraped by automated method by Neurosynth would be very similar to the manually curated data in the original meta-analysis.
Lastly, Text2Brain also predicts consistently reasonable brain maps for all five cognitive concepts, and matches the target maps better than Neurosynth for ``Visual Processing'' and ``Pain''.
Results in Figure~\ref{fig:example_meta} shows that Text2Brain could learn appropriate relationship between common search phrases and the activation pattern of a diverse set of functional domains.
\section{Robustness of models to input queries}
\subsection{Example of ``self-generated thought'' synonyms}
We examine the prediction for ``self-generated thought'', which is one of the most extensively investigated functional domains, due to its involvement in a wide range of cognitive processes that do not require external stimuli~\citep{andrews2014default},
and is associated with the default network~\citep{buckner2008brain}.
The ground-truth map for self-generated thought, taken from~\citep{ngo2019beyond}, is estimated using activation likelihood estimation (ALE)~\citep{eickhoff2009coordinate} applied on activation foci across 167 imaging studies of 7 tasks selected based on strict criteria~\citep{spreng2009common,mar2011neural,sevinc2014contextual}.
The resulting ALE map is thresholded with the cluster-level forming threshold at voxel-level $p<0.001$, and cluster-level corrected threshold $p<0.05$~\citep{eickhoff2012activation}.%
\begin{figure*}[h!]
\centering
\includegraphics[width=\linewidth]{self-generated_thought_Dice.pdf}
\caption{Prediction of self-generated thought activation map using synonymous queries. While Text2Brain generates consistent prediction across the similar queries, Neurosynth and Neuroquery's prediction deteriorate on the ``internally-directed thought'' and ``task-unrelated thought'' queries.}
\label{fig:self_generated_thought}
\end{figure*}
Figure \ref{fig:self_generated_thought} shows the prediction of self-generated thought activation map using four different query terms, thresholded to retain the same number of activated voxels as the target map.
Across all four queries, Text2Brain's prediction best matches the ground-truth activation map compared to the baselines.
For the ``self-generated thought'' and ``default network'' queries, all approaches generate activation maps that are consistent with the ground-truth, which includes the precuneus, the medial prefrontal cortex, the temporo-parietal junction, and the temporal pole.
Text2Brain and Neuroquery both make reasonable prediction from the ``internally-directed thought'' query while Neurosynth's prediction is largely scattered and does not match the target map.
Lastly, Text2Brain can also replicate a similar activation pattern to the target from the query ``task-unrelated thought'', evident by only a slight drop in the Dice score.
However, Neuroquery and Neurosynth both generate activation maps that differ from the typical default network's regions, such as activation in the prefrontal cortex, and also result in a large drop of the Dice scores.
\subsection{Prediction of Cognitive Atlas concepts from synonymous queries}
\label{subsec:synonym_results}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{cognitive_atlas_accuracy.pdf}
\caption{Accuracy of matching Cognitve Atlas concept names with their description and aliases using models' predicted brain maps.}
\label{fig:cognitive_atlas_acc}
\end{figure}
Figure~\ref{fig:cognitive_atlas_acc} shows the accuracy of matching cognitive concept names from the Cognitive Atlas~\citep{poldrack2011cognitive} with their definitions and atlases using the different models' predicted brain maps.
Prediction by Text2Brain is more robust than both Neuroquery and Neurosynth with respect to the concept definition and alias.
In particular, Text2Brain has the same top-1 accuracy of matching the brain map predicted from a concept's alias with the prediction from the concept name compared to Neurosynth.
This result is expected given that Neurosynth can yield accurate brain map for keywords that are included in their vocabulary.
In contrast, Text2Brain improves over Neurosynth for top-1 accuracy of matching concept name with the longer text of concept definition.
Text2Brain is more robust than both Neurosynth and Neuroquery baselines in terms of top-5 and top-10 matching accuracies for both concept aliases and definitions.
Figure~\ref{fig:cognitive_atlas_acc} indicates that Text2Brain prediction is robust to natural language text queries of different length and complexity.
\section{Conclusion}
In this work, we present a model named Text2Brain for generating activation maps from free-form text query.
By finetuning a high-capacity SciBert-based text encoder to predict coordinate-based meta-analytic maps, Text2Brain captures the rich relationship in the language representational space, allowing the model to generalize its prediction for synonymous queries.
This is evident in the better performance of Text2Bran in predicting the self-generated thought activation map using different descriptions of the functional domain.
Text2Brain's capability to implicitly learn relationships between textual terms and images ensures the model can remain relevant and useful even as neuroimaging literature continues to evolve with new discoveries and rephrasing of existing concepts.
We also show that Text2Brain accurately predicts most of the task contrasts included in the IBC and HCP dataset, validating its capability to make prediction for longer, arbitrary queries.
Text2Brain also preempts failure cases in Neurosynth and Neuroquery, where they cannot predict input queries undefined in the vocabulary list, even though these queries are relevant to neuroscience research (e.g. title of an article).
On the other hand, we also observed that Text2Brain had difficulties handling queries that involve logical reasoning, such as the direction of a contrast.
For example, while queries such as ``A vs B'' and ``B vs A'' can be inferred by human to correspond with inverted activation maps, Text2Brain sometimes treats one direction to be the same as the other.
We suspect that this type of error is likely due to the model's inability to generalize ``vs'' as an ``subtractive'' operator. Resolving such limitation will likely require modifications to the language model.
Furthore, in the future, we plan to enhance the interpretability of our approach, such as to attribute regions of activations in the generated map to specific words in the input query, as well as to efficiently match activation maps and scientific descriptions most relevant to the synthesized images.
We believe that the flexibility of Text2Brain can significantly lower the barrier for researchers at all stages of their careers to synthesize brain activation maps needed for their research.
For example, the ability of Text2Brain to generate meaningful neural activation patterns of synonymous queries for a functional domain can improve the accuracy of delineating region-of-interests (ROIs) relevant to the functional process, as well as to assess the reliability of each ROI.
Discovery of these ROIs is useful for several applications such as meta-analytic connectivity modeling (MACM)~\citep{laird2013networks}.
We look forward to such application of Text2Brain in aiding future neuroscientific research.
\section*{Acknowledgement}
This work was supported by NIH grants R01LM012719, R01AG053949, the NSF NeuroNex grant 1707312, the NSF CAREER 1748377 grant and Jacobs Scholar Fellowship.
\bibliographystyle{model2-names.bst}\biboptions{authoryear}
\section{Introduction}
\label{sec:intro}
A rapidly growing number of functional magnetic resonance imaging (fMRI) studies have given us important insights into the mental processes that underpin behavior. However, individual studies are often
power-restricted~\citep{carp2012secret,button2013power}, since the number of subjects and mental processes that can be interrogated in a single experiment is limited~\citep{church2010task}.
One approach to digest the vast literature and synthesize across many studies is to perform a meta-analysis of the reported results, such as the coordinates of the most significant effects (e.g., 3D location of peak brain activation in response to a task).
These meta-analyses usually require expert curation of relevant experiments (e.g. \citep{costafreda2008predictors,minzenberg2009meta,shackman2011integration}).
A critical technical challenge here is the consolidation of synonymous terms.
Importantly, over time, different denominations might be used in different contexts or invented to refine existing ideas.
For instance, ``self-generated thought'', one of the most highly studied functional domains of the human brain~\citep{smallwood2013distinguishing}, can be referred to by varying terms, such as ``task-unrelated thought''~\citep{andrews2014default}.
The selection of reported results for meta-analysis can be automated on data scraped from the published literature~\citep{yarkoni2011large,dockes2020neuroquery,rubin2017decoding}.
Two popular examples of this direction are Neurosynth~\citep{yarkoni2011large} and more recently Neuroquery~\citep{dockes2020neuroquery}.
Neurosynth utilizes automated keyword search to retrieve relevant studies and statistical tests to find summary brain activation maps corresponding to the keywords.
Unlike Neurosynth, Neuroquery is a predictive model that synthesizes activation maps from keywords in the input query.
Despite their differences in modeling, both Neurosynth and Neuroquery only support queries consisting predefined keywords.
Furthermore, Neurosynth does not explicitly handle long queries, while Neuroquery relies on superficial lexical similarity via word co-occurences for inference of longer or rarer queries.
We propose an alternative approach named Text2Brain, which builds on recent neural language models and permits more flexible free-form text queries.
Text2Brain captures a more fine-grained and implicit semantic similarity via vector representations from the neural language model in order to retrieve more relevant studies.
Furthermore, in contrast to tools like Neuroquery, our method computes synthesized activation maps via a 3D convolutional neural network (CNN) model, which we empirically demonstrate, can capture coarse and fine details.
We compare Text2Brain's predictions with those from Neurosynth and Neuroquery, where we used article titles as free-form queries.
Furthermore, we assess model predictions on independent test datasets, including reliable task contrasts and meta-analytic activation maps of well-studied cognitive domains predicted from their descriptions.
Our analysis shows that Text2Brain generates activation maps that better match the target images than the baselines tools.
Given its flexibility in taking input queries, Text2Brain can be used as an educational aid as well as a tool for synthesizing maps based on published results or generating novel hypotheses for future research.
Compared to our conference article~\citep{ngo2021text2brain}, we have extensively expanded our results and analysis.
Specifically, we have expanded on the model validation on article titles with a different test set (section~\ref{subsec:exp_title_setup} and~\ref{subsec:title_results}), added additional evaluation on the contrast maps predicted from their descriptions (section ~\ref{subsec:contrast_results}).
New results and discussion have also been added to this paper, including a high-level conceptual comparison of models (section~\ref{subsec:high_level_comparison}), new experiments on predicting representative meta-analytic results (section~\ref{subsec:contrast_methods} and~\ref{subsec:contrast_results}), and quantitative analysis of the models' robustness to input queries (section~\ref{subsec:synonym_methods} and~\ref{subsec:synonym_results}).
\section{Datasets and Methods}
\subsection{Model overview}
Figure~\ref{fig:model} shows the overview of this work, including data generation, model architecture, and model training.
The Text2Brain model has an encoder-decoder architecture that maps text sequences into brain activation maps~(Section~\ref{subsec:model}).
Its transformer-based encoder uses self-attention to encode a snippet of text input into vector representation~\citep{vaswani2017attention,devlin2018bert}.
Text2Brain's 3D convolutional decoder (CNN) then translates the vector representation into a 3D brain activation map.
The Transformer is currently the most effective approach for modeling text since it can capture long-distance dependency between words and can learn efficiently through self-supervision from massive text corpora~\citep{jawahar2019does,raffel2020exploring}.
On the other hand, 3D CNNs are the most dominant architectural design in medical imaging~\citep{milletari2016v,kamnitsas2017efficient}.
In our proposed approach, we first extract full text and activation coordinates from each research article to create data samples.
Each sample consists of an input snippet from the full text and an output 3D activation map created using the coordinates (Section~\ref{subsec:data}).
Text2Brain is trained to associate the input text to activations at various spatial locations.
Since Text2Brain's transformer-based encoder is context-sensitive, it can better extract information from free-form query by refining the vector representation depending on the specific phrasing of the text inputs~\citep{tenney2019you}.
In contrast, the classical keyword search mainly exploits co-occurrence of keywords regardless of context and therefore may struggle on more nuanced queries~\citep{salton1988term}.
Furthermore, keyword search approaches store one activation map for each supported keyword, which are in turn linearly combined for queries. This approach can limit how many keywords are supported~\citep{yarkoni2011large,dockes2020neuroquery}.
On the other hand, Text2Brain stores the text and activation maps content in its parameters and can scale better to diverse input queries~\citep{petroni2019language}.
We use data augmentation to encourage Text2Brain to construct and store rich many-to-one mappings between textual description and activation maps (Section \ref {subsec:training}).
This allows Text2Brain to better map semantically similar text queries to similar activation maps.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{model.pdf}
\caption{Overview of data preprocessing, the Text2Brain model, and training procedure. All activation maps are 3D volumes, but projected to the surface for visualization.}
\label{fig:model}
\end{figure}
\subsection{Implementation}
\label{subsec:model}
Figure~\ref{fig:model} bottom left corner shows the Text2Brain model with its text encoder and 3D CNN image decoder.
Text2Brain's text encoder is based on SciBERT, a BERT model that has been trained using scientific articles~\citep{beltagy2019scibert}.
BERT is a transformer-based model with bidirectional self-attention trained via self-supervision to learn semantic representations of textual input~\citep{devlin2018bert}.
The text encoder outputs a vector representation of dimension $768$.
This vector is projected using a fully-connected layer and then reshaped to a 3D volume of dimension $4\times5\times4$ voxels with $64$ channels at each voxel.
The image decoder consists of 3 transposed 3D convolutional layers with 32, 16, 8 channels respectively.
Text2Brain was trained using the Adam optimizer~\citep{loshchilov2018decoupled} and the mean-squared error with a batch size of 24 for 2000 epochs.
The learning rate for the text encoder and image decoder are set at $10^{-5}$ and $3\times{10^{-2}}$ respectively.
The model's source code is available at \url{https://github.com/sabunculab/text2brain}.
\subsection{Data Preprocessing}
\label{subsec:data}
We used the same set of 13,000 neuroimaging articles previously released in~\citep{dockes2020neuroquery} in our experiments.
Each article contains one or more tables of results that reported coordinates of peak activation in MNI152 coordinate system~\citep{lancaster2007bias}.
The activation foci are also publicly released by Neuroquery~\citep{dockes2020neuroquery}.
Following the same procedure as \citep{dockes2020neuroquery}, the set of activation foci associated with each table is used to generate an activation map by placing a Gaussian sphere with full width at half maximum (FWHM) of 9mm at each of the coordinates of peak activation.
The chosen FWHM allows a fair comparison with Neuroquery\citep{dockes2020neuroquery} in our experiments, and is consistent with previous work~\citep{wager2009evaluating,yarkoni2011large,yeo2015functional}.
Supplemental section~\ref{sec:ablation_fwhm} shows an analysis of the effect of the Gaussian kernel's FWHM used for preprocessing on Text2Brain's predictive accuracy on an independent test set.
This comparison confirms that the choice of the kernel's FWHM is reasonable.
An article-average activation map is also generated by averaging the activation maps of all the tables in the article.
The text associated with the activation maps are extracted from the articles' full text.
The articles' full text are scraped using their PubMedID via the NCBI API~\footnote{\url{https://www.ncbi.nlm.nih.gov/books/NBK25501/}} and the Elsevier E-utilities API~\footnote{\url{https://dev.elsevier.com/}}.
As there may be multiple text snippets corresponding to the same activation map, the next section (Section~\ref{subsec:training}) shows how the corresponding text of an activation map is selected.
\subsection{Training}
\label{subsec:training}
Each training sample consists of a text-activation map pair and correspond to an neuroimaging article.
The activation map is sampled uniformly at random from the union set of table-specific maps and article-average map.
For each table-specific map, the first sentence of the corresponding table caption is chosen as the map's associated text.
Our initial data exploration suggested that the first sentence to be the most relevant description of the activation map.
For each article-average map, the associated text that describes the activation map is sampled uniformly at random from the following four sources: (1) the article's title; (2) one of the article's keywords; (3) the article's abstract; and (4) a randomly chosen subset of sentences from the discussion section of the article.
This data augmentation strategy encourages Text2Brain to generalize over input texts of different lengths.
Furthermore, matching the same activation pattern with multiple different text snippets encourages the model to recognize important words common across the snippets and to learn the association between different but synonymous words.
Supplemental Figure~\ref{fig:supp_ablation} shows our ablation study on the sampling strategy.
The liberal (and likely noisy) construction of image-text pairs appears to perform better than more deliberate coupling of image-text snippets strategies (not reported) that we tried in our preliminary experiments. We surmise that simply presenting different text snippets to a target brain image is analogous to another augmentation strategy that allows the neural network to pool across samples and learn the relevant words and their weights with respect to the target brain maps.
Training with the set up in~\ref{subsec:model} takes approximately 75 hours on one NvidiaRTX GPU while one inference pass with an input query of up to 140 characters takes less than 1 second.
\subsection{Baselines}
We compare Text2Brain to 2 different baselines: Neurosynth~\citep{yarkoni2011large} and Neuroquery~\citep{dockes2020neuroquery}.
For a keyword, Neurosynth first finds all neuroimaging articles that mention that keyword.
Then, one statistical test per voxel is performed across the activation maps corresponding to those studies to determine a significant association.
Since Neurosynth was not formulated to handle multiple-word queries, for such query, we performed statistical test using activation maps from all articles that contain at least one of the keywords in the query.
Neuroquery extends Neurosynth's vocabulary of keywords by including more curated keywords from lexicons such as MeSH, NeuroNames, and NIF~\citep{lipscomb2000medical,bowden1995neuronames,gardner2008neuroscience}.
The keyword encoding is obtained after performing non-negative matrix factorization of the articles' full text (as a bag of keywords) represented with term frequency - inverse document frequency (TF-IDF) features~\citep{salton1988term}.
A ridge regression model was trained to map the text encoding to the activation.
The inference of a keyword is smoothed by a weighed average of its most related keywords (in the TF-IDF space).
For multiple-word queries, the predicted activation map is obtained by averaging the activation maps from all keywords in the input, weighed by the coefficients learnt during training.
\subsection{Evaluation Metrics}
\label{subsec:metrics}
For thresholded target activation maps such as those computed by ALE~\citep{eickhoff2009coordinate}, the predicted brain maps are thresholded to retain the same number of most activated voxels as the target.
For example, given an estimated activation map by ALE with statistically significant clusters of activation that cover 25\% of the the brain volume, the brain maps predicted by Text2Brain, Neuroquery, and Neurosynth are also thresholded to retain the top 25\% most activated voxels in each map.
The accuracy of prediction is measured by Dice score~\citep{dice1945measures} which quantifies the extent of overlap between the predicted and target brain maps (details are in Supplemental Section~\ref{subsec:supp_metrics}).
Furthermore, we use Dice scores at different thresholds to estimate the similarity between predicted and target activation maps at different levels of detail~\citep{ngo2022predicting}.
This evaluation procedure is similar to that used in \citep{dockes2020neuroquery} for a thresholded target map, but we apply the same thresholding to both the target and predicted map.
For example, at 5\% threshold (considering the 5\% most activated voxels), the Dice score measures the correspondence of the fine-grained details between the target and predicted activation maps.
At higher thresholds (e.g. 25\%), the score captures the gross agreement between activation clusters.
We also estimated the area under the Dice curve (AUC) as a summary measure using approximated integration of Dice scores across all thresholds from 5\% up to 30\%.
Supplemental Figure \ref{fig:supp_dice} shows the Dice curve for an example pair of target-predicted activation maps.
Note that the range of thresholds in the x-axis also conveys the maximum percentage of the gray matter mask that has an activation in the target brain map.
For example, if only a proportion of gray matter mask has activation, such as the case of Neuroquery prediction that mostly extends up to 30\% of the gray matter mask or a sparse target activation pattern from the coordinate-based meta-analysis, the x-axis range will not be extended up to 1.
In our experiments, all evaluation is performed in the MNI152 volumetric space, which is the original space of all predicted maps.
For visualization, with activation maps that mostly concentrate in the cerebral cortex, the original volumetric images are transformed from MNI152 space to fs\_LR surface space using Connetome Workbench~\citep{van2013wu} via the FreeSurfer surface space~\citep{buckner2011organization,fischl2012freesurfer}, with isolated surface clusters of less than 20 vertices being removed~\citep{wu2018accurate}.
Activation maps with significant activation in the non-cortical parts of the brain are visualized by cross-sectional slices with significant activation using Nilearn~\citep{abraham2014machine}.
\subsection{High-level model comparison}
\label{subsec:high_level_comparison}
\begin{table}[]
\centering
\begin{tabular}{|l|l|l|l|}
\hline
& Neurosynth & Neuroquery & Text2Brain \\ \hline
Vocabulary & Fixed & Fixed & Unlimited \\ \hline
\makecell[l]{Handle of\\complex query} & None & \makecell[l]{Lexical\\similarity} & \makecell[l]{Semantic\\similarity} \\ \hline
\makecell[l]{Predictive\\models} & None & \makecell[l]{TF-IDF,\\linear\\regression} & \makecell[l]{Transformer,\\ convolution} \\ \hline
\end{tabular}
\caption{High-level comparison of approaches to meta-analytic brain maps generation}
\label{tab:comparison}
\end{table}
Text2Brain can better handle input text than prior approaches because its vocabulary is not limited to a fixed pre-defined set of words.
In contrast, Neurosynth and Neuroquery rely on fixed word vocabularies and cannot predict for queries consisting of out-of-vocabulary words.
Besides, Neurosynth's and Neuroquery's vocabularies are not sufficiently extensive, covering only a fraction (under 10\%)~\citep{dockes2020neuroquery} of terms in relevant neuroimaging lexicons such as Cognitive Atlas~\citep{poldrack2016brain} and NeuroNames~\citep{bowden1995neuronames}.
Text2Brain's usage of byte-pair encoding enables the model to handle infrequent and out-of-vocabulary words more gracefully, by breaking down those words into digestable sub-word tokens~\citep{sennrich2016neural}.
Hence, Text2Brain's vocabulary is open ended and can scale with training data to be unlimited in theory.
Besides, Text2Brain's training is not limited to only training set data.
Text2Brain can leverage self-supervised learning from non-neuroimaging scientific articles, as well as neuroimaging articles that do not report activation coordinates to learn a better text-to-activation-map transformation.
By finetuning a SciBERT text encoder pretrained on the larger dataset of scientific articles (including non-neuroimaging articles), Text2Brain seems to converge on an optimum with a more useful representational space of the input text. Supplemental section~\ref{sec:ablation_pretraining} shows the comparison between the Text2Brain model that uses pretrained SciBERT text encoder versus a randomly initialized text encoder. Evaluation on predicting article-average activation maps from both sets of test articles in the Neuroquery dataset (similar to section~\ref{subsec:exp_title_setup}) suggests that pretraining benefits the Text2Brain's performance.
Furthermore, Text2Brain uses contextualized text embeddings to model semantic relationship between words so it can deal with nuanced queries more effectively.
Methods such as Neurosynth and Neuroquery may have difficulty dealing with complex expressions.
By simply averaging the keywords' activation maps to arrive at the prediction for a complex query, these methods may fail to account for relationship between words in the query, such as order and semantic.
Lastly, while the predictive approach of Neuroquery constructs the predicted activation map by modelling voxels' activation independently, Text2Brain generates the whole-brain activation with a 3D convolutional decoder that takes in the text encoding produced by the language model.
By upsampling and computing the whole-brain activation from a bottleneck, Text2Brain can better model both the short and long-distance relationship between voxels.
\section{Experimental Setup}
\subsection{Predict activation maps from article title}
\label{subsec:exp_title_setup}
Two test sets were created from the Neuroquery dataset of 13,000 studies.
The first test set consists of 1000 randomly sampled articles.
The second test set also consists of 1000 articles but was randomly sampled such that the keywords (defined by the articles' authors) do not appear in the training and validation articles.
The two test sets are labeled as easy and hard test sets respectively.
Of the remaining articles, 1000 are randomly held out as a validation set for parameters tuning.
For each article, the article-average activation map is predicted from its title using Text2Brain, as well as the Neurosynth and Neuroquery baselines.
Both Text2Brain and Neuroquery were trained on the 10,000 articles in the training set.
The Text2Brain model is trained using both the articles' titles and samples from the full-text, while Neuroquery is trained on the articles' full-text.
We use predictions from the publicly available Neurosynth model at \href{https://neurosynth.org/}{https://neurosynth.org}, which was trained on the articles' abstracts.
Note that Neurosynth is not a predictive model meant for out-of-sample prediction, but for performing automated statistical testing of associations between terms and brain locations.
\subsection{Predict activation maps from contrast descriptions}
\subsubsection{Individual Brain Charting (IBC) task contrasts}
The Individual Brain Charting (IBC) project~\citep{pinho2020individual} estimates an extensive functional atlas of the human brain via fMRI data of subjects measured under a large number of task conditions.
In particular, the IBC dataset consists of 180 task contrasts measured on 12 subjects.
We use the activation maps provided by the IBC project to measure the predictive accuracy of Text2Brain and the two baselines over a wide range of functional domains, given the contrast descriptions from IBC.
\subsubsection{Human Connectome Project (HCP) task contrasts}
While the IBC dataset offers a large number of reference brain maps, the small number of subjects might make some results less reliable.
We also utilized the Human Connectome Project (HCP) data both for reference and a measure of reliability of target maps.
The HCP dataset consists of neuroimaging data from over 1200 subjects, including task fMRI (tfMRI) of 86 task contrasts from 7 domains~\citep{barch2013function}, which overlap with 43 contrasts under the IBC dataset.
We evaluate the model prediction of HCP task contrasts from their descriptions.
While HCP provides detailed descriptions of task contrasts, we opt for the more concise contrast descriptions provided by the Individual Brain Charting (IBC) as they are more succinct and thus more favorable to the baselines.
The IBC contrast descriptions are extracted from the metadata of the activation maps released on Neurovault \href{https://neurovault.org/images/360528}{https://neurovault.org/images/360528}.
The list of all IBC description of HCP contrasts are included in Supplemental Table~\ref{tab:ibc_description}.
On the other hand, the target (ground-truth) activation maps are the HCP group-average contrast maps, as the large number of subjects provides more reliable estimates of the contrast maps.
In the analyses of this experiment, we use the agreement between the IBC and HCP maps as a measure of reliability.
Despite using similar protocols, there are subtle differences between the IBC and HCP experiments.
For instance, the original HCP language task was conducted in English but the corresponding language task in the IBC project was conducted in French.
\subsection{Predict representative meta-analytic brain maps}
\label{subsec:contrast_methods}
The automated approach to brain map generation of Text2Brain and the 2 baselines are compared against published brain maps created from a manually curated set of meta-analyses.
In particular, 5 cognitive concepts and their corresponding activation maps of 5 representative meta-analytic studies from ANIMA database~\citep{reid2016anima} were selected.
The 5 meta-analytic studies were selected for having the most number of experiments and their different coverage of the human brain.
The cognitive processes of interest are visual processing, auditory processing, motor execution~\citep{heckner2021delineating}, working memory~\citep{rottschy2012modelling}, and pain~\citep{xu2020convergent}.
Each study searches for published neuroimaging studies that contain a set of texts queries relevant to the cognitive concept of interest.
For example, in~\cite{rottschy2012modelling}, the phrases to search for working memory-related studies are ``working memory'' and ``short-term memory''.
The same text queries for discovering relevant studies in the original meta-analysis were used as input to Neurosynth, Neuroquery, and Text2Brain.
Table~\ref{tab:domain_exp} shows the search queries and the number of experiments included in the original meta-analysis of the 5 chosen cognitive concepts.
Activation maps generated from all text input queries corresponding to each cognitive concept are averaged to yield a single brain map for each model.
The reference brain images for comparison are the activation maps released by the studies and made publicly available on ANIMA.
The reference activation maps are produced by Activation Likelihood Estimation (ALE)~\citep{turkeltaub2002meta,laird2005ale,eickhoff2009coordinate} and thresholded to retain only the statistically significant clusters of activation.
For all reference ALE maps, the cluster-level forming threshold at voxel-level is $p<0.001$ and cluster-level corrected threshold is set at $p<0.05$ by the original authors~\citep{eickhoff2012activation}.
For comparison, the generated brain maps are thresholded to keep the same number of survived voxels as those in the reference activation maps.
The accuracy of each model's generated brain map is evaluated as the Dice score between the (thresholded) generated brain map and the target (thresholded) brain map (see Section~\ref{subsec:metrics}).
\begin{table}[h!]
\centering
\begin{tabular}{ | l | l | l | }
\hline
Functional domain & \#Exp & Search queries \\ \hline
\makecell[l]{Visual processing\\
(Heckner 2021)} & 114 & \makecell[l]{visual processing\\
face monitor\\
face discrimination\\
film viewing\\
fixation\\
flashing checkerboard\\
passive viewing\\
visual object identification\\
visual pursuit\\
visual tracking\\
visuospatial attention} \\\hline
\makecell[l]{Auditory processing\\
(Heckner 2021)} & 122 & \makecell[l]{auditory processing\\
divided auditory attention\\
music comprehension\\
oddball discrimination\\
passive listening\\
phonological discrimination\\
pitch monitor\\
pitch discrimination\\
tone monitor\\
tone discrimination} \\\hline
\makecell[l]{Motor execution\\
(Heckner 2021)} & 251 & \makecell[l]{motor execution\\
writing\\
chewing\\
swallowing\\
drawing\\
isometric force\\
motor learning\\
grasping\\
finger tapping\\
button press\\
flexion\\
extension\\ } \\ \hline
\makecell[l]{Working memory\\
(Rottschy 2012)} & 189 & \makecell[l]{working memory\\
short-term memory\\ }
\\\hline
\makecell[l]{Pain\\
(Xu 2020)} & 222 & \makecell[l]{
pain\\
noxious\\
nociception }
\\\hline
\end{tabular}
\label{tab:domain_exp}
\caption{Meta-analytic studies of representative functional domains. The studies were selected from the ANIMA dataset~\citep{reid2016anima} that have the most number of experiments and covere a diverse set of brain regions.}
\end{table}
\subsection{Evaluate robustness of model prediction to semantically-equivalent queries}
\label{subsec:synonym_methods}
With the continual improvement of our understanding of the human brain and mind, neuroscientific knowledge is also an ever evolving repertoire.
Several neuroimaging concepts have also been changing, adapting and broadening over time.
Thus, we were interested in examining if our approach is robust to semantically equivalent queries.
For example, ``self-generated thought'', one of the most intensively examined cognitive domains in neuroscience, has had its definition refined and assigned different denominations over the years.
As a cognitive paradigm, different names have been used to refer to the set of inward-oriented psychological processes, such as ``self-generated thought''~\citep{smallwood2013distinguishing}, or ``task-unrelated thought''~\citep{andrews2014default}.
Both terms are associated with ``default network''~\citep{buckner2008brain}, the set of brain regions with elevated activation when subjects are not subjected to any external stimulus.
To assess models' prediction of synonymous queries, we utilized the ontology from the Cognitive Atlas~\citep{poldrack2011cognitive,bilder2009cognitive}.
The Cognitive Atlas is a collaborative knowledge base for neuroscience with content such as cognitive concepts, their description and synonyms (aliases) contributed by the project's voluntary participants~\citep{miller2010cognitive}.
At the time of our experiments, Cognitive Atlas includes 885 concepts with definition, 108 of which have at least one alias.
We considered a model to be robust with respect to a specific cognitive concept's definition if the activation map predicted from the description matches the predicted map from the concept's name.
In particular, given a model's predicted brain maps from all 885 Cognitive Atlas concept names and their description, we assess if the model's brain map predicted from a concept's definition is one of the $k$ maps (out of 886 possible maps) most similar to the model's brain map predicted from the concept's name.
In our experiments, top-1, top-5 and top-10 matching accuracy were evaluated using Dice AUC metrics.
The different values of $k$'s account for the uncertainty of the concepts' natural language text, e.g., different contributors might use different names to refer to the same concept.
Similarly, models' robustness with respect to a cognitive concept's alias is measured by the accuracy of matching the activation maps predicted from the text of a concept's alias and its name.
\section{Results}
\label{sec:results}
\subsection{Validation of activation maps predicted from article title}
\label{subsec:title_results}
\begin{figure}
\centering
\includegraphics[width=0.9\linewidth]{article_title_AUC__combined.pdf}
\caption{Evaluation of article-average activation maps predicted from their titles measured in area under the Dice curve (AUC) score. The left and right graph show the Dice AUCs of samples from the easy and hard test sets, respectively (Section~\ref{subsec:exp_title_setup}). The p-values are computed from paired-sample t-tests between Text2Brains and each of the 2 baselines.}
\label{fig:article_AUC}
\end{figure}
Figure \ref{fig:article_AUC} shows the quality of activation maps predicted from the titles of 1000 articles in each of the two test sets (section~\ref{subsec:exp_title_setup}).
In the easy test set (the test articles' keywords can overlap with the training articles'), the proposed Text2Brain model (mean Dice AUC = $0.0636$) outperforms Neuroquery (mean Dice AUC = $0.0523$) and Neurosynth (mean Dice AUC = $0.0453$).
In the hard test set (the test articles' keywords are not present in the training set), the Text2Brain model (mean Dice AUC = $0.0609$) also performs better than Neuroquery (mean AUC = $0.0499$) and Neurosynth (mean AUC = $0.0457$).
Paired-sample t-tests show that the performance differences in both test sets are statistically very significant.
The p-values when comparing Neuroquery and Neurosynth are $p=5.25\times10^{-27}$ and $p=2.40\times10^{-12}$.
Fig.~\ref{fig:article_AUC} also indicates how the different models handle out-of-sample input text.
Text2Brain can make a prediction for all input texts, evident with positive Dice AUCs for all samples.
On the other hand, Neurosynth fails to make prediction for some article titles in both test sets, resulting in zero Dice AUCs for such samples.
Similarly, Neuroquery fails to make prediction for some samples in the hard test set.
These failure cases are caused by the limited vocabularies of Neurosynth and Neuroquery that cannot cover the words in the test input queries.
On the other hand, the language model of Text2Brain is finetuned from SciBert, which has been pretrained on a broader lexicon and utilizes sub-word tokens to extend the vocabulary to unseen words (more details in Section~\ref{subsec:high_level_comparison}).
\subsection{Prediction of task contrast maps from description}
\label{subsec:contrast_results}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{IBC_all_AUC.pdf}
\caption{Dice AUCs of predicted IBC task activation maps from contrasts' description. The p-values are estimated from paired-sample t-tests between Text2Brain against the two baselines.}
\label{fig:ibc_auc}
\end{figure}
Fig.~\ref{fig:ibc_auc} shows the Dice AUC scores for the prediction of Text2Brain, Neuroquery and Neurosynth against the IBC group-average task contrast maps.
Text2Brain (mean Dice AUC = 0.0507) improves upon both Neuroquery (mean Dice AUC = 0.0457, $p = 4.11\times{10^{-4}}$), and Neurosynth (mean Dice AUC = 0.0404, $p=1.58\times{10^{-9}}$).
The p-values are measured by 2-tail paired-sample t-test between Text2Brain and the two baselines.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{IBC_HCP_AUC.pdf}
\caption{Dice AUCs of predicted HCP task activation maps from contrasts' description. The graph includes 22 contrasts with the highest HCP-IBC's Dice AUC scores and sorted in decreasing order.}
\label{fig:hcp_auc}
\end{figure}
Fig.~\ref{fig:hcp_auc} shows the AUC scores for the prediction of the three models and the IBC average contrasts, against the HCP target maps.
The 22 contrasts with above-average HCP-IBC's AUC scores, considered to be the reliable contrasts, are shown.
Across all 43 HCP contrasts, Text2Brain (mean AUC = $0.082$) performs better than the baselines, i.e. Neuroquery (mean AUC = $0.0755$, $p = 0.08$), Neurosynth (mean AUC = $0.047$, $p = 1.5\times10^{-5}$), where $p$-values are computed from the paired t-test between Text2Brain's and the baselines' prediction.
As reference, IBC contrasts yield a mean AUC = $0.094$ when compared to the corresponding HCP maps (Statistical comparison with Text2Brain, $p = 0.077$).
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{IBC_HCP.pdf}
\caption{Task activation maps predicted from contrasts' description. Each row shows both the thresholded maps of the top 25\% most activated voxels (top) and the overlap between predicted and target binarized brain maps. Blue is activation in the target contrast, red is the predicted activation and yellow is the overlap.}
\label{fig:hcp_example}
\end{figure*}
Figure \ref{fig:hcp_example} shows the prediction for three contrasts correspond to different HCP task groups, namely ``MOTOR'', ``LANGUAGE'', ``RELATIONAL'' thresholded at the top 25\% most activated voxels.
The three task groups were chosen to show results for a range of target images with different levels of reliability.
The two task groups ``MOTOR'' and ``LANGUAGE'' are the two most reliable task (having the highest average HCP-IBC AUC across all contrasts), while ``RELATIONAL'' has the lowest average HCP-IBC AUC.
Text2Brain's prediction improves over the baselines for the three contrasts.
Neurosynth was not able to generate activation maps for one of the contrast descriptions (``Move tongue'').
On the other hand, for the ``Move tongue'' contrast, Neuroquery predicts activation in the primary cortex, but the peak is in the wrong location, shifted more toward the hand region of the homunculus. Additionally, there is a false positive prediction in the occipital cortex, which might be an artifact from modeling brain activation coupled with visual stimuli-related words describing the motor experiments.
\subsection{Prediction of brain maps from representative meta-analytic studies}
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{example_meta_maps.pdf}
\caption{Prediction of brain maps from meta-analytic studies of representative functional domains. The information of the investigated functional domains are listed in Table~\ref{tab:domain_exp}. Reference and predicted activation maps of the first 4 function domains are visualized on the brain surface. The last domain (``pain'') is visualized in the volume as most activation concentrates in the non-cortical parts of the brain. For all functional domains, Text2Brain generates reasonable activation maps and comparable with the baselines for the common functional domains.}
\label{fig:example_meta}
\end{figure*}
Figure~\ref{fig:example_meta} shows the prediction of activation maps for 5 representative meta-analytic studies with the most number of experiments from ANIMA~\citep{reid2016anima}.
Among the three models, Neuroquery has the lowest Dice score on average, with prediction on ``Visual processing'', ``Working memory'', and ``Pain'' that significantly deviates from the target maps.
On the other hand, Neurosynth-derived brain maps consistently match well against the target maps.
The high accuracy of Neurosynth prediction is expected since the five chosen cognitive concepts are among the most commonly studied concepts with the most number of experiments reporting activation coordinates in the literature.
Given high number of available experiments and the input queries mostly exist in Neurosynth's predefined keyword set, the activation coordinates scraped by automated method by Neurosynth would be very similar to the manually curated data in the original meta-analysis.
Lastly, Text2Brain also predicts consistently reasonable brain maps for all five cognitive concepts, and matches the target maps better than Neurosynth for ``Visual Processing'' and ``Pain''.
Results in Figure~\ref{fig:example_meta} shows that Text2Brain could learn appropriate relationship between common search phrases and the activation pattern of a diverse set of functional domains.
\section{Robustness of models to input queries}
\subsection{Example of ``self-generated thought'' synonyms}
We examine the prediction for ``self-generated thought'', which is one of the most extensively investigated functional domains, due to its involvement in a wide range of cognitive processes that do not require external stimuli~\citep{andrews2014default},
and is associated with the default network~\citep{buckner2008brain}.
The ground-truth map for self-generated thought, taken from~\citep{ngo2019beyond}, is estimated using activation likelihood estimation (ALE)~\citep{eickhoff2009coordinate} applied on activation foci across 167 imaging studies of 7 tasks selected based on strict criteria~\citep{spreng2009common,mar2011neural,sevinc2014contextual}.
The resulting ALE map is thresholded with the cluster-level forming threshold at voxel-level $p<0.001$, and cluster-level corrected threshold $p<0.05$~\citep{eickhoff2012activation}.%
\begin{figure*}[h!]
\centering
\includegraphics[width=\linewidth]{self-generated_thought_Dice.pdf}
\caption{Prediction of self-generated thought activation map using synonymous queries. While Text2Brain generates consistent prediction across the similar queries, Neurosynth and Neuroquery's prediction deteriorate on the ``internally-directed thought'' and ``task-unrelated thought'' queries.}
\label{fig:self_generated_thought}
\end{figure*}
Figure \ref{fig:self_generated_thought} shows the prediction of self-generated thought activation map using four different query terms, thresholded to retain the same number of activated voxels as the target map.
Across all four queries, Text2Brain's prediction best matches the ground-truth activation map compared to the baselines.
For the ``self-generated thought'' and ``default network'' queries, all approaches generate activation maps that are consistent with the ground-truth, which includes the precuneus, the medial prefrontal cortex, the temporo-parietal junction, and the temporal pole.
Text2Brain and Neuroquery both make reasonable prediction from the ``internally-directed thought'' query while Neurosynth's prediction is largely scattered and does not match the target map.
Lastly, Text2Brain can also replicate a similar activation pattern to the target from the query ``task-unrelated thought'', evident by only a slight drop in the Dice score.
However, Neuroquery and Neurosynth both generate activation maps that differ from the typical default network's regions, such as activation in the prefrontal cortex, and also result in a large drop of the Dice scores.
\subsection{Prediction of Cognitive Atlas concepts from synonymous queries}
\label{subsec:synonym_results}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{cognitive_atlas_accuracy.pdf}
\caption{Accuracy of matching Cognitve Atlas concept names with their description and aliases using models' predicted brain maps.}
\label{fig:cognitive_atlas_acc}
\end{figure}
Figure~\ref{fig:cognitive_atlas_acc} shows the accuracy of matching cognitive concept names from the Cognitive Atlas~\citep{poldrack2011cognitive} with their definitions and atlases using the different models' predicted brain maps.
Prediction by Text2Brain is more robust than both Neuroquery and Neurosynth with respect to the concept definition and alias.
In particular, Text2Brain has the same top-1 accuracy of matching the brain map predicted from a concept's alias with the prediction from the concept name compared to Neurosynth.
This result is expected given that Neurosynth can yield accurate brain map for keywords that are included in their vocabulary.
In contrast, Text2Brain improves over Neurosynth for top-1 accuracy of matching concept name with the longer text of concept definition.
Text2Brain is more robust than both Neurosynth and Neuroquery baselines in terms of top-5 and top-10 matching accuracies for both concept aliases and definitions.
Figure~\ref{fig:cognitive_atlas_acc} indicates that Text2Brain prediction is robust to natural language text queries of different length and complexity.
\section{Conclusion}
In this work, we present a model named Text2Brain for generating activation maps from free-form text query.
By finetuning a high-capacity SciBert-based text encoder to predict coordinate-based meta-analytic maps, Text2Brain captures the rich relationship in the language representational space, allowing the model to generalize its prediction for synonymous queries.
This is evident in the better performance of Text2Bran in predicting the self-generated thought activation map using different descriptions of the functional domain.
Text2Brain's capability to implicitly learn relationships between textual terms and images ensures the model can remain relevant and useful even as neuroimaging literature continues to evolve with new discoveries and rephrasing of existing concepts.
We also show that Text2Brain accurately predicts most of the task contrasts included in the IBC and HCP dataset, validating its capability to make prediction for longer, arbitrary queries.
Text2Brain also preempts failure cases in Neurosynth and Neuroquery, where they cannot predict input queries undefined in the vocabulary list, even though these queries are relevant to neuroscience research (e.g. title of an article).
On the other hand, we also observed that Text2Brain had difficulties handling queries that involve logical reasoning, such as the direction of a contrast.
For example, while queries such as ``A vs B'' and ``B vs A'' can be inferred by human to correspond with inverted activation maps, Text2Brain sometimes treats one direction to be the same as the other.
We suspect that this type of error is likely due to the model's inability to generalize ``vs'' as an ``subtractive'' operator. Resolving such limitation will likely require modifications to the language model.
Furthore, in the future, we plan to enhance the interpretability of our approach, such as to attribute regions of activations in the generated map to specific words in the input query, as well as to efficiently match activation maps and scientific descriptions most relevant to the synthesized images.
We believe that the flexibility of Text2Brain can significantly lower the barrier for researchers at all stages of their careers to synthesize brain activation maps needed for their research.
For example, the ability of Text2Brain to generate meaningful neural activation patterns of synonymous queries for a functional domain can improve the accuracy of delineating region-of-interests (ROIs) relevant to the functional process, as well as to assess the reliability of each ROI.
Discovery of these ROIs is useful for several applications such as meta-analytic connectivity modeling (MACM)~\citep{laird2013networks}.
We look forward to such application of Text2Brain in aiding future neuroscientific research.
\section*{Acknowledgement}
This work was supported by NIH grants R01LM012719, R01AG053949, the NSF NeuroNex grant 1707312, the NSF CAREER 1748377 grant and Jacobs Scholar Fellowship.
\bibliographystyle{model2-names.bst}\biboptions{authoryear}
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{"url":"https:\/\/kittipatkampa.wordpress.com\/category\/research\/","text":"### Archive\n\nArchive for the \u2018Research\u2019 Category\n\n## A good Introduction on\u00a0MapReduce\n\nMapReduce is a framework to efficiently process a task that can be parallelized using cluster or grid. A good introduction can be found in the link below.\n\nhttp:\/\/en.wikipedia.org\/wiki\/MapReduce#Example\n\nIn a sense, MapReduce framework is very similar to message-passing algorithm in graphical models where the Map and Reduce are comparable to building (tree) structure and marginalization of the messages respectively. So, I think MapReduce can make an inference plausible for large-scale graphical models.\n\n## Neuroscience talks\n\nhttp:\/\/www.neuroscience.cam.ac.uk\/talks\/\n\nhttp:\/\/talks.cam.ac.uk\/\n\nInformation theory, pattern recognition, and neural networks\nDraft videos (not yet edited):\nhttp:\/\/www.inference.phy.cam.ac.uk\/itprnn_lectures\/\n\n## Awesome seminars at\u00a0UW\n\nApril 3, 2012 1 comment\n\nThere are some fascinating seminars sponsored by UW, and most of them are recorded:\n\nCSE Colloquia:\nEvery Tuesday 3:30 pm\nhttps:\/\/www.cs.washington.edu\/htbin-post\/mvis\/mvis\/Colloquia#current\n\nYahoo! Machine Learning Seminar\nEvery Tuesday from 12 \u2013 1 pm\nhttp:\/\/ml.cs.washington.edu\/seminars\n\nUWTV: Research\/Technology\/Discovery Channel\nhttp:\/\/www.uwtv.org\/\n\n## Install MATLAB r2010a on Ubuntu\u00a010.04\n\nHere is the step by step how to install MATLAB on Ubuntu\n\n1. Mount the matlab iso file. Let\u2019s say the matlab installation files are in directory \/tmp\/mat2010a\n2. First, install using\n\\$ sudo sh install\nHowever, you might get the error, which looks like this\n\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014-\u00a0\u00a0\u00a0 An error status was returned by the program \u2018xsetup\u2019,\nthe X Window System version of \u2018install\u2019. The following\nmessages were written to standard error:\n\n\/home\/bot\/tmp\/matu20Xa\/update\/install\/main.sh: 178: \/home\/bot\/tmp\/mat2010a\/update\/bin\/glnxa64\/xsetup: Permission denied\n\nAttempt to fix the problem and try again. If X is not available\nor \u2018xsetup\u2019 cannot be made to work then try the terminal\nversion of \u2018install\u2019 using the command:\n\ninstall* -t\u00a0\u00a0\u00a0 or\u00a0\u00a0\u00a0 INSTALL* -t\n\n\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014-\nThe problem occurs because of the permission of the file \u2026\/xsetup is not set properly. So, the easy way is to go to the directory and change the permission by using the command\n\n..\/glnxa64\\$ chmod 777 xsetup\n\nNow, you can go back to the normal installation\n\n3. Next step, create a root matlab folder, and it is suggested that you create the folder in\/usr\/local\/matlabR2010a\n\nby using the command line\n\n`sudo mkdir \/usr\/local\/matlabR2010a`\n4. The rest is easy..you can do it yourself\n\nhttps:\/\/help.ubuntu.com\/community\/MATLAB\/R2010a\n\nCategories: Research, Tutorials Tags: ,\n\n## Cluster Evaluation using Adjusted Rand Index\u00a0(ARI)\n\nHere is the 2 partitions mentioned in the example1 in the tutorial paper \u201cDetails of the Adjusted Rand index and Clustering algorithms\nSupplement to the paper \u201cAn empirical study on Principal Component Analysis for clustering gene expression data\u201d (to appear in Bioinformatics)\u201d pdf\n\nPartition U (ground truth) and V (predicted)\n\nAnd I think they did in the example is exactly the same as the following\n\na = |(4,5) ; (7,8)\u00a0 (7,9) (7,10) (8,9) (8,10) (9,10)| = (2 choose 2) + (4 choose 2) =\u00a0 7\n\nb=|(1,2) (3,4) (3,5) (6,4) (6,5) (3,6)| = 6\n\nc = |(1,3) (2,4) (2,5) (6,7) \u2026 (6,10)| = 7\n\nd = |(1,4)\u2026(1,10) (2,3) (2,6) \u2026(2,10) (3,7) \u2026(3,10) (4,7)\u2026(4,10) (5,7)\u2026(5,10)| = 25\n\nwhere (i,j) denotes the pair (or edge) between node i and node j. Then they use this a, b, c and d to evaluate Rand index and adjusted Rand index.\n\n## How to remove white-border from a\u00a0figure?\n\nWhen adding a figure to your publication, you might want to remove the undesired white-border off your figures. I believe that the best way is to create figures without the border if it is possible. In MATLAB, I think you can do so. However, if you have the figures already, you might want to have a program to remove the borders automatically, wisely and controllably. I developed a toolbox in MATLAB for this purpose. Please refer to the URL below.\n\nThe overview of white-border removal toolbox\n\nCategories: iDea, iDo, Research\n\nApparently, adding the loading factor to the covariance matrix does impact the log-likelihood value. I made some experiments on the issue, and let me share the results with you as seen in the learning curve (log-likelihood curve) of ITSBN with EM algorithm below. The factor is applied to the matrix only when the determinant of the covariance matrix is smaller than $10^{-6}$. There are 5 different factors used in this experiment listed as follows; $10^{-8}, 10^{-6}, 10^{-4}, 10^{-3}, 10^{-2}$. The results show that the learning curves are still monotonically increasing* and level off near the end. Furthermore, we found that the level-off value are highly associated with the value of the factor. The bigger the factor, the smaller the level-off value. This suggested that we should pick smallest value of factor as possible in order to stay as close as the ideal learning curve as possible. Note that the loading factor is not added to the covariance matrix until the second iteration.","date":"2017-02-24 08:05:57","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 2, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4882252514362335, \"perplexity\": 1211.791433409576}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-09\/segments\/1487501171418.79\/warc\/CC-MAIN-20170219104611-00236-ip-10-171-10-108.ec2.internal.warc.gz\"}"} | null | null |
Edward Blom (born 22 October 1970 in Ekerö, Sweden) is a Swedish archivist, trade historian, writer and a television personality.
Biography
Blom was born in Ekerö (Sweden) and is the brother of Anna Dunér. He studied humanities at the universities of Trier, Stockholm, Uppsala and Freiburg, where he became a member of Corps Suevia Freiburg. In 1996, he earned a degree in archival studies. He is one of the founders of the Verein Corpsstudenten in Schweden (Students' Club Corp in Sweden), a board member of the Concordia Catholica (the Catholic Union) and an active member, among other things, of the Sällskapet Emil Hildebrands Vänner (Society of Emil Hilde's Friends) and Par Bricole. He was previously vice president of the Katolsk historisk förening i Sverige (Catholic Historical Association in Sweden).
He works as a boss at särskilda projekt (head of special projects) at the Centrum för Näringslivshistoria (Centre for Business History) in Stockholm. He is the editor of the Centrum för Näringslivshistoria's business magazine Företagsminnen (Corporate History) He is very much involved in the history of trade, food and beverages. He performed the song "Livet på en Pinne" at the Swedish Melodifestivalen 2018.
Blom has taken part in several programs about the history of the trade, especially for the Swedish TV channel TV8, including the series Fredag med Edward (Friday with Edward). In 2009, he became known to a wider audience with the series Mellan skål och vägg med Edward Blom (Between the Bowl and the Wall with Edward Blom), where he travels around Sweden together with economics journalist Peter Andersson, showcasing gastronomy and drinks and telling stories from the places they visited. He appears regularly on radio, especially, on Sveriges Radio P3's programme Morgonpasset.
He has contributed articles to a great number of journals, including Arv och Minne (Heritage and Memory); Den lille Fascikeln (The Little Fascicles); Katolskt magasin (Catholic Magazine), and Tema Arkiv( Theme File).
Blom is a well-known and active member in the Catholic community, and in 2008 was ordained as a Knight of the Order of the Holy Sepulchre of Jerusalem by the Catholic Bishop Anders Arborelius.
Discography
Singles
Featured singles
Bibliography
Books and eBooks
Deutsche, die das Stockholmer Brauereiwesen industrialisierten (2009) (Germans who industrialized the Stockholm brewery industry)
Handelsbilder – 125 år med Svensk Handel (2008) (Business Pictures - 125 years with the Swedish Trade)
Tyskarna som industrialiserade Stockholms bryggerinäring: en studie i till Stockholm invandrade tyska bryggare och bryggeriarbetare under 1800-talets senare hälft, med inriktning på nätverk och personer (2007) (The Germans Who Developed Stockholm's Brewery Industry: a study of the immigrant German brewers and brewery workers during the second half of the 1800s, with a focus on networks and people)
Den svenska handelns historia (2006) (History of Swedish trade (2006)).
Familjeföreningen Concordia Catholica 1995-2005 – Jubileumsskrift över föreningens senaste decennium med anledning av dess 110-årsjubileum (2005), tillsammans med James Blom (Concordia Catholic Family Association 1995-2005 - Anniversary of the Society's Last Decade on the Occasion of its 110th Anniversary, with James Blom)
ICA-historien – i parti och detalj (2003). (ICA history - in the wholesale and retail)
Jehanders – 125 år i Stockholmarnas tjänst (1999), tillsammans med Lars Lundqvist (Jehanders - Jehanders - 125 Years in the Service of Stockholmers, with Lars Lundqvist)
Articles
"Agape", novell i: Anna Braw (red) Tillsammansmat (2007) ("Agape", short story in: Anna Braw (red) Tillsammansmat)
"Säljande gudar och livbojar" i Identitet : om varumärken, tecken och symboler (2002). (Selling gods and lifebuoys in Identity: Brand, Signs and Symbols).
"Berömda arkivspex och arkivrevyer" i Tjugo år med Arkivrådet AAS (redaktörer Olle Ebbinghaus & Ulrika Grönquist) ("Famous Archive Reports and Archive Revues" for Twentieth Years with the Archives Council AAS (editors Olle Ebbinghaus & Ulrika Grönquist)
Flera artiklar i cd/dvd:erna Söder i våra hjärtan (1998). (Several articles in the CD / DVDs South of Our Heart)
The Ericsson Files (2001),
Gamla Stan under 750 år (2002). (Old Town for 750 Years)
References
External links
Edward Blom's blog
"Edward om baksmälla" (Edward on hangover), clip)
"Svensk ölhistoria" (swedish beer history
"Om Gluntarne" (About the Gluntarne"
"Menytips till Kronprinsessan i DITV"(Menu Tips for Crown Princess in DITV)
"Smakprov ur Mellan skål och vägg"(Teasers from the Middle bowl and wall)
"Edward Blom berättar punschens historia i TV8:s Finansnytt" (Edward Blom tells the history of the arrak punch in TV8's Finance news.)
Swedish Roman Catholics
21st-century Swedish historians
Swedish television personalities
Swedish male writers
Swedish archivists
1970 births
Living people
Knights of the Holy Sepulchre
Melodifestivalen contestants of 2018 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,245 |
Protesters now planning to block A2 and A20 in Dover if Brexit is delayed
Farmers and fisherman have reported agreed to help 'bring the country to its knees'
Vicky Castle
Pro-Brexit protesters have threatened to block the A20 in Dover (Image: Charlie Elphicke)
Brexit supporters are furious with Theresa May's decision to extend negotiations beyond the March 29 deadline.
Before she'd even announced plans to ask for an extension to Article 50 yesterday evening (March 20), truckers and car drivers announced they planned to bring all the country's major roads to a standstill.
The pro-Brexit group threatened to bring the M25 to a standstill with blockades of lorries and cars, among other key roads across the country.
And now farmers and fisherman have reported agreed to join the "go slow" on the A2/A20 Jubilee Way, in Dover.
The group suggested there was "significant demand for a truckers go slow" and hinted "some farmers and fishermen keen to join in".
The Brexit Direct Action group was formed on February 28 by multiple Leave supporters on Twitter and has already amassed more than 6,000 followers on the social media platform.
Active on the account @ActionBrexit , said they intend to "organise and coordinate action against those whose intention is to delay/stop Brexit".
They want to see a "clean Brexit from the EU in accordance with the 2016 EU referendum" and are trying to build a large base of pro-Brexit supporters to provide a united front.
In a statement on their Twitter account, the group states their plans to "bring the country to its knees by blocking the main arterial routes".
All about Brexit in Kent
We ate like a Brexit stockpiler
11 ways Brexit will screw Kent
How Kent is the Brexit battleground
The 1 thing Kent agrees on about Brexit
Road bosses share 'worst case scenario'
'My band will be destroyed by Brexit'
Operation Brock cost increases by £15m
'Good riddance to EU funding'
It lists the M1, M6, M25, M62, A1, A55, M5, M4, M42, M55, M61, A66 as its targets.
Another Tweet by the group suggested that there was "significant demand for a truckers go slow" on the A2/ A20 Jubilee Way in Dover, with "some farmers and fishermen keen to join in".
One follower (@realitycheckout) replied to one of the group's tweets saying: "You HAVE to try and cover the M2 and M20.
"The main Kentish routes through to the continent. Also any other arterials around civil and military airport bases."
As well as these protests on the roads, they are also helping to coordinate movements such as March To Leave and pioneering petitions to local political party associations.
You can visit their Twitter account here . | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,808 |
STOP HERE
a novel
Beverly Gologorsky
Seven Stories Press
New York
Copyright © 2013 by Beverly Gologorsky
A Seven Stories Press First Edition
An earlier version of Chapter 3 "Imaginary Friends" was published in Hamilton Stone Review, February, 2010.
All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, including mechanical, electronic, photocopying, recording, or otherwise, without the prior written permission of the publisher.
Seven Stories Press 140 Watts Street New York, NY 10013 www.sevenstories.com
College professors and middle and high school teachers may order free examination copies of Seven Stories Press titles. To order, visit www.sevenstories.com/textbook or send a fax on school letterhead to (212) 226-1411.
ISBN 978-1-60980-505-0
As always, for
Charlie Wiggins,
Georgina, Dónal,
And especially Maya
In Memory of
Frieda Trestman
Dave Gologorsky
Billy Capozzi
The Best of It
However carved up
or pared down we get,
we keep on making
the best of it as though
it doesn't matter that
our acre's down to
a square foot. As
though our garden
could be one bean
and we'd rejoice if
it flourishes, as
though one bean
could nourish us.
—Kay Ryan
Contents
Usable Truths
Reunion
Imaginary Friends
The Way Things Work
In the Silence
Butter and Ketchup
How We Know Before We Know
About Time
Happiness Exists Somewhere
The Things in Between
She was Definitely Here
Stop Here
Acknowledgments
About the Author
About Seven Stories Press
A Seven Stories Press Reading Group Guide
1
Usable Truths
There's no way to ignore the warmongering on Fox News, though Ava is trying. The screen takes up half the wall. Hours ago, it seems, she searched for the remote to lower the volume, but no luck. She went so far as to ask Murray to turn it off, but instead he began jabbering about our brave boys, keeping the country safe, on and on like he knew something no one else did. That sent her mind reeling back to the evening her son pulled her onto the couch to watch the invasion of Iraq. Shock and Awe, he said, repeating what he'd heard the newscasters call it. Then, too, she wanted to close her eyes. She told him the sights were frightening, nothing to celebrate, but it didn't dampen Bobby's childish excitement. That scared her, too.
No one seems bothered by the TV. The party is in this room where the food table and bar are set up, where wraparound windows allow only blueberry darkness, where vaulted ceilings create echoes as people talk and gesture and take up space. If she could find a corner to be alone—to hide, actually—it would help, but no luck there either. She avoids parties, fears the expectations, the false gaiety, and worse, strangers' idle curiosity.
Maybe another glass of wine, but her head feels spacey, her fingers tingle, and when she tries to breathe deep her body tenses. It's been a while since anxiety dogged her, though it happened a lot after her husband was killed. Damn TV. It's boring through her senses.
Wending her way to the bedroom, she switches on a lamp. A ghostly amber light slides across a bed larger than hers by far; piles of bulky winter coats cover the pale satiny spread. It will take forever to find her jacket.
She hurries out, finds her friends at the makeshift bar. "I need to leave. Right now. Could you get my jacket with yours?" From the way they look at her she hasn't disguised the desperation.
"Absolutely," Rosalyn says. "Wait in the foyer."
"We've been here long enough," Mila agrees.
"Murray, I'm leaving. Early shift tomorrow," her voice ridiculously high. Before he can say a word, she heads for the vestibule, but not fast enough. The eerie feeling of nothingness begins to sink her. She takes hold of the doorknob, something solid.
• • •
Picking her way over gravel, she glances back at the enormity of the house with its walls full of windows. Overwhelming. Murray insisted on a never-ending tour, room by room, pointing out each piece of furniture as if she didn't know what a couch or a chair looked like. The décor was beautiful, it's true, but cold; maybe if more people lived there. . . . But just Murray and Sylvie . . . How will they care for it? She beeps open the car doors. Home, she wants to be home.
Rosalyn slides into the passenger seat. Mila, in a puffy jacket as iridescent as pigeon feathers, climbs in back. The doors slam, the sound magnified by the ocean. She flicks on the high beams. Sand dunes loom up like headless figures. She turns the key and the ignition makes a dreadful sound. Praying to gods she doesn't believe in, she counts to five, inserts the key again and turns it. Nothing. Her foot on the gas, she tries again and again and again.
"Ava, stop, don't flood it. I'll call Triple A." Rosalyn pulls a slim phone from her velvet purse. They listen to her give directions for a tow truck. "At least half an hour, probably longer."
"I'll pay for it," Ava promises, wondering which part of her budget to raid. She sees the neat pile of envelopes on her dresser labeled food, gas, telephone, utilities; there's one for emergencies. Also one marked fun, in which she feels compelled by a force she doesn't understand to add a dollar or two, depending on tips. It's grown thick.
"Don't worry, I'm a member," Rosalyn counters generously.
"I can't wait back at Murray's." There's alarm in her voice.
"God forbid," Rosalyn's large eyes examine her as if for the first time. "Let's walk to the shore line, kill some time."
"Are you drunk? It's freezing out there," Mila says.
"What if I am? I'm going to be forty."
"What does that have to do with anything?" Mila mumbles.
"I know Murray's watching us." Ava peers into the darkness. He'll insist they come inside.
"Who cares? The stars are beautiful and ours. Come on, we'll hear the truck when it gets here." Rosalyn is out the door, her shapely body hidden in a red wool coat that reaches her ankles.
Ava hesitates; the car feels safe.
"Rosalyn's right, let's kill some time," Mila is out the door as well.
Does she want to sit here alone? Reluctantly stepping into a vastness she can sense but not see, a faint shiver rides through her. She breathes in the salty air. It's a cold night of stars.
"Little ol' Murray, can you believe it?" Mila's shoes crunch the gravelly sand. "The house dwarfs him. Ava, did you see the height of those ceilings?"
"Who could miss it," her tone too harsh. These are her friends. Tell them she's anxious. Yet how to describe nothingness? She tried once, didn't she, to a military doctor who looked as young as her husband had been. What could he understand? In the end he said bad things happen to everyone as if she didn't know that. It wearies her to correct and explain what feels beyond language. At home she'll be alone. Bobby's sleeping at Dina's. She'll undress, wrap herself in the old robe, maybe flip through a magazine or sit looking out the window. The anxiety will lift, her mood will change. It always has.
Slowly, they follow the dimming cone of headlight toward the wind-driven sound of breaking waves.
"Rosalyn, you do know how to needle Murray. Your warnings about keeping up such a big house . . ." Mila begins, her tone more joyful than critical.
"He expects Sylvie to be his maid, his cook, god knows what else. In Murray's eyes, that's what a wife's for. Anyway, he kind of enjoys my defiance. I certainly do," Rosalyn muses.
Ava steps gingerly over pockets of sand, careful not to trip. It must be nearly midnight. If the tow truck arrives here in thirty minutes it'll be an hour getting home, another half-hour to drop off her friends. She's due at the diner by five. She doesn't mind leaving home before dawn, the houses still dark, no cars or people, the world silent, and she there to witness the morning.
"Nick and Bruce are stuck working the diner all night," Rosalyn says, reading her thoughts as she often does. "Bruce isn't going to be that helpful. There's another man whose wife I wouldn't want to be."
"Shelly's a strong woman. They'll get through this," Mila counters.
Ava's reminded of Shelly's early morning calls to the diner. The flat, powerful voice tinged with irritation as if the person listening were at fault and not Bruce's lateness.
"I'm sure Nick's happy not to go to Murray's housewarming. I could've done without it," Rosalyn admits.
"Then why go?" Mila whips a hat from her pocket, tugging it low over her thick, reddish hair. Only her small, determined face peeks out, which doesn't look much older than her daughter's.
"Because we three rarely get a chance to play . . ."
"I wouldn't call it a fun party," Ava mumbles, watching Rosalyn trek easily over the sand in heeled boots.
"Ava, what's going on with you?" Mila's coal-dark eyes try to read her.
"The joyous occasion lasted too long, that monster TV running the same awful war video over and over. And the things people come out with would drive anyone crazy." The words come quickly but suddenly have little meaning.
"The TV was a bit much, but after a while I didn't notice." Rosalyn can do that, shrug off what she considers unimportant, something she wishes she could do.
"Except for a few of Sylvie's friends, we were the only ones Murray knew to invite. How depressing is that?" Mila sounds gleeful.
"He's a lousy host, lousy boss, and he'll be a lousy husband, too," Rosalyn declares as if she'd never said as much before.
"Are you maligning the man whose money helped buy your condo?" Mila teases.
"Not on Murray's salary."
"Then whose?" Mila asks.
Rosalyn hesitates. "Oh . . . previous, more lucrative jobs. Besides, I'm quite frugal." She's not, but neither of them says so.
"Murray has so much," Mila says.
"Is there something you need?" Rosalyn wants to know.
"Cash. Not stocks, bonds, or plastic. Twenties and fifties, this thick." Mila holds her fingers inches apart.
"Money for what?" Rosalyn asks.
"Know how much a pair of Darla's boots cost?"
"What do you want for yourself?" Rosalyn persists.
"A lot of things."
"For example?"
"Is this a quiz?"
"Sort of."
"If I had lots of money, I wouldn't waste it on a big house. I'd stop working."
"And do what?" Ava asks, faintly alarmed.
"Oh, this and that," Mila's voice drifts off the way it does when questions become too specific.
Ava too frets about money; but such worries are personal. Shapely clouds move across the sky, obscuring the stars, darkening the space around them even more. It reminds her of another night at the beach. She and her husband-to-be wrapped in each other's arms, the cold wind howling; how they laughed. Nothing could touch them, not wind, cold, or anything out there. It was years ago, when she assumed pleasure was her due.
"Ava?" Rosalyn tugs her arm. "I'm talking to you."
"What?" her voice low. The wind whistles past her frozen ears, the thrashing water close too close.
"I said unlike Mila, I don't mind being a waitress. Do you?"
"I'm grateful for the work." It's the closest to the truth she can trust just now. Being busy, moving resolutely, hurrying to get a task done suits her, may even bring her somewhere better faster.
"Sylvie left her job, and why not? Murray isn't as stingy with her as he is with us," Mila says.
"Did you see how Murray sidled up to listen whenever I talked to Sylvie? It's sort of sweet he's afraid of my influence," Rosalyn seems pleased.
She, too, chatted briefly with Sylvie, who revealed tidbits about her decision to leave the theater, her decision to get an ordinary job, to let go of the familiar . . . to try new things . . . have new experiences . . . Words that made her edgy, as if Sylvie was a pioneer whose risks paid off.
At the water's edge, the horizon sucked into the black sky, she wonders . . . If she wades in, experiences the danger of sinking, faces the fear, would the sensation vanish? But what if she can't find the shore again?
Rosalyn teases the water with the tip of her patent leather boot.
"Murray was watching Sylvie everywhere she went tonight," Mila says definitively.
"I'd rather be alone than with a man who didn't trust me," Rosalyn's tone serious.
So often her friends' words provide sustenance, proof of what it means to go on. Tonight, their chatter is useless, something her cop father wouldn't abide. To him conversation had to be useful. He believed usable truths could be shared. Anyway, usable would give her a method, a handle, something to grab on to.
"None of what you say is usable," she says in a soft voice, surprising herself.
"What is it, Ava? You're somewhere else tonight. It's not like you," Mila says impatiently.
Mila's right, it isn't like her, she's generally as grounded as a flat road. She gets things done and rarely complains. "Let's go back to the car," her voice close to a whisper.
They trudge back, sandy wind in their faces. Walking ahead, her friends' voices lost in the crash of waves, she focuses on tomorrow's tasks. First thing, she'll call a taxi to drive her to work. If it takes more than a day to fix her car she'll have to rent one, won't she? Pick up the rental after her morning shift, then stop at the hardware store for lightbulbs, an extension cord, soap pads, new broom, the list on the fridge etched in her head. She'll have to visit the supermarket as well. Bobby wants those precooked potpie things for his school lunches, no doubt expensive, no doubt unhealthy. Ordinary thoughts any mother would have. It's where she needs to be now.
She slides into the car, Rosalyn beside her. Mila settles noisily in back. Doors slam. The dunes look pitifully like what they are, piles of sand. Glancing at the time, she calculates the tow truck should be there any minute. They'll scrunch in beside the driver. Rosalyn will keep up a stream of chatter. Mila will ask a thousand questions, answering none. She'll sit there quietly.
"Call Triple A again, it's been more than thirty minutes," Mila says.
Rosalyn obediently pulls out her cell phone. They listen. Clicking off, she tells them, "Busy night. The truck won't be here for another half hour."
She's making her friends wait in the car when they could . . . but she can't go back in there. "I appreciate your staying with—"
"We're not doing it for you. We're not that pure," Rosalyn says.
"Isn't that the truth? What now?" Mila asks. "Want to share fantasies?" A game they once played after a few drinks.
"You first."
"I buy a lottery ticket every day and fantasize what'd I'd do with all that money."
"Hmmm, well . . . heavens . . . mine is weirder. I fantasize the discovery of a pill to preserve my body . . ." Rosalyn's tone subdued. "You're quiet, Ava. Does that mean you won't reveal your fantasy or that you refuse to have one?"
"Not sure." Her plaintive tone saddens her. Her fantasies, if that's what they are, seem stuck in the past, which makes no sense, nothing can change there, but everything's known.
Rosalyn pats her arm comfortingly, then stares out the window.
Mila sighs loudly.
It's her fault, souring her friends' mood. "Is there something else we can talk about?" she asks softly.
"How about a Bobby or Darla story we haven't heard before?" Mila offers with little enthusiasm.
"No offense, but that's duller than looking through a stranger's photo album." Rosalyn flicks open her cell phone to search her messages.
"Then you come up with an idea. Or we'll sit here listening to our breathing," Mila scolds.
"Fine." Rosalyn shuts the phone, drums her fingers on the dashboard. "Let's tell a story. One of us starts, another continues, and on it goes."
"Like a once-upon-a-time thing?" Mila asks.
"No, no. Not a fairy tale. It has to be real, important, dramatic, revealing, something that'll hold our attention," Rosalyn tells her.
What in her story would be revealing, important? A shameful event from her teen years? Everyone has one, but which is hers? Something dramatic? Stealing a lipstick? Lying to her mother? Who cares? What would hold their attention? Maybe something about her dead husband; or that she hasn't had sex in years; or that she can't feel much. Or how her . . . Searching the night sky wishing her story were scripted there because whatever's ahead can only be imagined, yet wishing, too, she could scroll down dates and events and discover what led to here.
2
Reunion
Between five and six a.m. the weirdos arrive for coffee or handouts or just a booth for a nap. It's too early for real people, but somewhere out there, beyond the still-dark parking lot, Ava can hear the first few cars in what will become the morning rush. Nick is in the kitchen checking inventory. Bruce will show up late; Mila won't get there till ten; Rosalyn is probably still asleep. And Murray . . . well, he'll arrive at eight sharp to check on everyone.
The first pale light skips over the missing floor tiles illuminating the ashy black linoleum beneath. Soon the sun will burnish the white Cape Cod houses that dot the nearby shallow hills. Even on cloudy days, a narrow strip of pink appears in the sky for a few seconds. It's the play of light that tells her the hour, not the bold round clock on the wall.
For the only time all day, the marbled Formica counter is free of dirty dishes. On the back ledge there is an array of pies and cakes that arrive when she does. Taped on the wall over a tray of water pitchers and glasses is the list of blue-plate specials. Murray insists on varying them daily and it drives Nick crazy. How many different ways can you serve a piece of meat? The gold-edged mirror reflects empty red vinyl booths and small black tables. The last sugar container is filled and, bored, she's propped on a stool leafing idly through an old Newsday when the door chimes its ridiculous tune, and he walks in.
"Morning." Clean blue eyes seem to seek her out.
"Coffee?"
"I'm starving." He straddles the stool next to hers.
"What can I get you?" she asks as she stands up and moves behind the counter. She's already pouring coffee and laying out silverware as she hands him a menu, which she normally does without thinking. This time she feels different, questioning herself.
He points to number three, Lumberjack Special. Ham, eggs, hash browns, rye toast, and pancakes.
She gives herself a second cup of coffee.
"My name is Mark."
She waits for a second. Clearly he isn't from around here. "Ava. You passing through?" She registers the blue denim shirt tight across square shoulders, and guesses he's entering his fifties. A thin man carrying weight. Her husband was barrel-chested.
"I'm the new owner of Cross Country Trucks, Long Island and Denver. You must've seen the green and white billboard on Sunrise Highway. I'll be working both places now. I'm renting an apartment in Wantagh."
It's where she lives. Her small house is one of many look-alikes, the highway close enough to hear the traffic, but she hasn't seen the billboard. She piles two orders of toast and serves his eggs, then waits at the open ledge between the kitchen and the restaurant for Nick to deliver the pancakes. For a moment Nick's dark eyes fasten on her. It seems he has something to say, but he doesn't, so she makes busywork at the other end of the counter till she hears the plate slide toward her.
"Is this the end of your night shift?" Mark asks.
"I work four hours dawn and four noon. It gets me time to sleep."
"That your son?" He points at Bobby's photo, stuck into the side of the mirror. She never wants to be far from him. She nods, unsurprised since Bobby looks just like her.
"How old is he?"
"Soon to be eleven." For a moment she too gazes at Bobby's thin, sharp-featured face, his blond hair a bit long and shaggy, a sociable child who is curious about foreign places.
"His father must be proud of him?"
"He was killed in an army helicopter crash. He never met Bobby." The words leave her mouth before she can check them.
He puts down the fork. "How awful for you."
She looks at him, a customer, not even a regular. There's no way he can understand the unceasing pain that was her husband's death, which finally faded to an indifference she's vowed never to disturb.
"I was lucky," Mark says. "Too young for Vietnam, too old for Iraq."
Luck is for people with money, she doesn't say.
"I imagine you never get over that kind of loss," he offers softly.
"You go on," she says, more to herself. When Bobby was a baby and threw his toys on the floor, she'd retrieve them—again and again—to teach him that what leaves comes back. Except she doesn't believe it herself. Or in the power of love to last either.
"Breakfast is fuel," he declares as if suddenly embarrassed, and mops up what's left of the eggs.
She was never good at small talk. She begins filling napkin holders and hears the swiveling stool, senses him watching her.
"You're good at that."
"I can eat fast too, but what does that get me?"
She stuffs the last container with too many napkins, thinks about wiping the counter even though it doesn't have a speck on it.
"No insult meant, truly." He catches her eye with his surefire blue gems.
"None taken. Is this your first time in New York?" she's surprised to hear herself ask.
"Lord, no. I lived in Manhattan for seven years a decade ago. Still miss it."
"Why?"
He takes a sip of his coffee. "People, I'd say. I like people, all kinds. It's what you have there. Turn a corner, hear one language, next corner, another. It's exciting."
"It's refreshing to hear the city described lovingly for a change."
"You ever been to Colorado?" He leans toward her with his elbows planted on the counter. She notes the grime-free nails, odd for someone working trucks. Then again, he's the owner.
"No." When was the last time she left Long Island?
"Imagine a hammock in a field of wildflowers, spectacular cloud formations. Think you'd like that?" He smiles, two dimples.
"Sounds like a resort."
"I suppose . . . You ever travel?" His expression is inquisitive as if he's discovered something in her that sets her apart.
"I don't see travel in my near future."
"It's close by plane, my house in the cup of the mountains. You'd fit right in." He smiles once more. Men flirt with her all the time, though she can't imagine why. Her narrow face with its too-thin nose is intense, unsmiling. Some try to coax a smile, which makes her self-conscious, tighter, as if giving in would be a loss. Dina tells her to rouge her pale cheeks. Maybe she will, someday. Once in a rare while she has a drink with a regular, usually Mila or Rosalyn with her. A mother can't allow strange men into her bedroom.
She gathers up his dishes, sweeps the counter clean, and notices Nick watching her before he glances at the clock. Because of Mark she's missed the sunrise and quickens her pace for the early rush. Mostly it's men with faces weathered from outdoor work in the new construction sites near Jones Beach. They arrive famished in SUVs and vans, dropping half-lit cigarettes on the diner steps, which Murray picks up each morning. Women come on later after leaving their kids at school or day care. She lifts a two-pound bag of coffee off a nearby shelf; fills three urns with water. Then she twists her hair into a ponytail, fastens it with a barrette, and sticks two pencils in her pocket. Mark's eyes are on her all the while.
"I'll be using the diner's hospitality for my meals, unless you know a better place." Only it isn't a question.
• • •
Two weeks of diner breakfasts, two weeks of conversations, and she lets him visit Bobby and her at home for the first time on a Sunday morning. Most people talk about how a boy needs a father, but who discusses how a man needs a son? She's seen men play nice with kids to get to their mothers, but his interest isn't feigned. Bobby is the missing ingredient in his life, and Bobby knows it immediately.
On a few of her days off he drives them to places they've never been: a strip of beach at the end of Long Island with edge-of-the-world rock formations; a park where the scent of cherry blossoms is thick enough to bottle; a seafood restaurant where he teaches Bobby how to crack open a lobster's shell with a knife blow to the belly. After two months he asks to stay over. To her surprise, she doesn't demur, except to say she hasn't slept with a man in years, which probably makes her a virgin again. He chuckles, even though she hadn't meant it as a joke.
That first time he makes love hesitantly. As he does, the memory of her husband enters the bed along with the long-forgotten certainty of her husband's arms, his repeated promise to please her like no one else could, which he had. It was eons since she'd heard his voice and it felt like a warning.
Mornings she has to be up by four and she makes Mark leave with her. Although Dina comes by each day to wake Bobby and help him prepare for school, she isn't ready to have her meet Mark.
By May, they're at her place most of the time, where he spends hours teaching Bobby the mysteries of the outdoors: the difference between a boat and a ship, how to read trail markers, what flies lure trout. He describes the privilege of meeting ravens, hearing birdsong, discerning wind direction, a life invisible to city boys.
In bed together it's easier, although she blames herself and her dead husband for never fully relaxing. Time, she needs more, she tells herself, except he's preparing to leave in a few weeks. The Denver piece of his business needs tending to, and he won't return until September.
• • •
On a June night as sticky as August, the three of them are in her living room playing Scrabble.
"That's it. My game," she declares.
"Mom always wins."
"We'll have to work harder to change that." Mark winks at her.
"Very hard, indeed," she says.
"Bobby, would you like a summer in Colorado?" Mark asks.
For a moment her son is as stunned as she is, then he grabs her hand. "Mom, please, can I, please? I'd really love it. Mom?"
"We'll camp in the mountains. There are some old caves to explore. I have a canoe, a sailboat, too. We'll hike; we'll trout fish . . ." Whatever Mark sees in her face checks his eagerness. He folds her hand in his. "I should have asked you first. It kind of came on me. If you don't agree . . ."
"Mom? Listen. It's hot in the city. I'll hardly have anyone to hang out with. It's such a good idea. And you have to work."
True, and Murray's about to change her schedule yet again, straight shift, ten to dawn, beginning in July. She's already taken over a chunk of the ordering. Now he's showing her the books, bills, lists of salespeople. He isn't a good teacher; she has to concentrate. She'll have to sleep some during the day. What will Bobby do? Languish in this old living room watching TV, the couch too shabby for words, the walls in need of painting? Is that the kind of summer she wants for him?
"Will he have his own room?" That's not what she needs to know, but it's a start.
"Yes. And don't worry about expenses. Believe me, the fare's no big deal."
Bobby's pulling on one hand, Mark pressing the other, and King Solomon comes to mind.
"He's not used to country life." Okay, that's closer, but to what?
"He'll go to the warehouse with me now and then, but we'll have plenty of time for everything else." He releases her hand but his eyes remain steady on her.
"Let me think," she says, but thinking is impossible. Bobby's fingers are cuffing her wrist like she's his prisoner. It's clear what he wants. And yet . . .
"Mom? Listen. If I don't like it I'll come home. But Mom . . . listen? I'm going to love it. Sailing? Where can we go sailing here?"
Her son's excitement is infectious. She glances at Mark. He's been reliable, consistent, even devoted, she'd say. He has a good effect on Bobby. On her, too. The last months have been positive for the three of them. Isn't that enough to hold on to for seven weeks?
• • •
Bobby's face is still glowing in her head when she arrives at the diner. He was so eager to get on the plane, his hug and kiss so quick. It was all she could do not to grab him before he disappeared past security.
Dina is having an early lunch at the counter, as usual. "You let him take Bobby for the summer. A stranger? I don't care if you slept with him. I don't care how much they like each other." Mila, who never keeps a thought to herself, promptly starts talking about those priests: You know, the kindest men on the planet. Who would have guessed! Her friends' words fire her imagination. Rosalyn slips an arm around her. "Listen, trust yourself. You know the guy, don't you?"
"Of course," she shoots back. "Mark isn't a stranger. I know where he lives, where he was born, his upbringing, where he went to school, his past jobs, what he wants for this new business."
"So?" Rosalyn says, "What more do you want? Unless . . . something in you isn't sitting right."
"That's not the point. The man hasn't passed the test of time," Dina says.
"It'd be different if you went with him, but that's expensive," Mila offers.
"Hey," Murray calls from the kitchen. "What is this?"
• • •
The lunchtime crowd is heavy, demanding. She won't miss it when she starts her new hours. She moves fast from table to counter to kitchen, all the time listening for the ring of her cell phone in her pocket. He's only been gone a few hours, but her nerves are shot and her friends' warnings are corrosive. She tries to put their words aside, but it's no use. She apologizes twice for giving the wrong check to a customer. She hurries the hours. Bobby will call as soon as they land. When the phone rings she runs out to the parking lot even though Murray is watching. Bobby's voice sounds so near. He's at the Denver airport; elated, he says they're leaving early the next morning for a few days of camping. The two of them alone in the mountains. That spooks her too.
• • •
Two days later, she finds herself at the Port Authority bus terminal, ticket in hand, a small travel bag over her shoulder, her adrenaline pumping. Dina, Mila, and finally Rosalyn, too, agreed that she had to do this or never sleep again. She weaves her way past food kiosks, panhandlers, and vendors to the Cruiser Line gate. Murray's displeasure follows her. She told him she'd only be gone several days; it was family business. "Who would you know west of Long Island?" Yeah, she said, that far.
• • •
The bus plows through the night. Occasional headlights streak the darkness. The wide aisle is now littered with paper bags, candy wrappers; an empty soda can rolls desolately past her feet. No one sits beside her and she's grateful. A talker would have shattered whatever is holding her together.
She keeps glancing at his photograph, her son's face caught in the light of a waning sun. Is he wishing she'd come or is he too busy, too happy to remember her at all? For several weeks after her husband was killed, she took Bobby into bed with her, kept her arms around his tiny sleeping body. Protecting him made her feel safe.
The driver's voice interrupts her thoughts. "In a few minutes we'll be arriving at the bus terminal. You have fifteen minutes to stretch your legs, get a cup of coffee," which his tired voice sounds in need of.
The bus pulls up in front of a small depot with a dirty plate glass front. She stares out the window. She can just make out a ticket counter inside and the uncertain flicker of fluorescent lights. Two soldiers bring their coffee outdoors and watch the bus as if it might take off without them. She wonders if they're bound for Afghanistan. The last time she saw her husband he was in uniform. They'd spent that week in San Francisco huffing and puffing up the hills, eating and drinking and making love like there was no tomorrow.
Two magazines are stuffed in the mesh pocket of the seat ahead. There is no way can she digest other people's stories now. She remembers Mila saying that diners are a better source of gossip than beauty salons because salons don't include men's input. Rosalyn disagreed, saying men talk half as much as women, and even then you can't believe a quarter of it.
She checks her purse for the tenth time, one hundred dollars and a credit card. She also has her checkbook and a roll of quarters in case her cell phone doesn't work.
Back on the road, the bus picks up speed. But for one or two reading lights, it's dark again. Outside, though, there are glimmers of light in the sky. Across the aisle the two soldiers are asleep, something tender in the way their heads nearly touch. Almost two days, now just another few hours and she'll be there, and she still doesn't know if what she's doing is right.
Loneliness, she'll say, not used to Bobby being gone. Mark will understand her decision to take him back with her. By next summer her trust will be complete.
• • •
For a while the taxi drives along the same highway as the bus, then turns sharply to begin a gradual climb on rutted dirt roads. Lemony sunlight opens the morning and the beauty of it all silences her. She'll arrive there at breakfast time and wonders what Bobby's face will do when he sees her.
The cab drops her off at the foot of a long driveway leading to a white stucco ranch house. Blue wildflowers march uphill like toy soldiers. She walks slowly between trees in summer glory. A dog barks. Maybe she'll say hello and go home. If she feels Bobby's safe, why rob him of this?
Through a picture window she sees a woman. No sign of Bobby or Mark. What if he gave her the wrong address?
She knocks. The woman is in her early fifties, sturdy build, blond hair pulled back from a broad-boned face. Jeans, T-shirt, sandals, her arms deeply tanned. Two rings circle her fingers, one pearl, the other a band of gold.
"Hi, I'm looking for the Dobson house."
"This is it," the woman says huskily, like a smoker.
"I'm Bobby's mother."
Just for a second, the woman's expression freezes. "Oh my goodness, hello. I'm Lydia, Mark's wife." She opens the door wider and yells out, "Mark! Bobby!" Her eyes settle into a gaze, a calculation. "Come in," she finally says in a near whisper. They stand in absolute silence. Not a bird sings, not a dog barks. She feels eerily calm. She already knows the problem here isn't her son's. It's been hard enough for her, why should she make anything easy for any of them?
Suddenly Lydia begins speaking far too quickly.
"What a great son. I know you've heard that before, but he's so smart, so easy to talk to, such a pleasure. He and Mark just came back from camping. The three of us plan to go sailing later. Will you join us?" Her words are friendly, but her gray eyes are hesitant. And why shouldn't they be? No doubt Mark told her Bobby is some needy kid whose poor, overworked mother couldn't give him anything. Certainly not that she's the woman he bedded down with all these months.
Lydia pours her a cup of coffee at the table. She can't remember the last time anyone did that for her.
Feet drum across wooden floors. Entering the kitchen, they both stop.
"Well . . . hello . . ." Mark says. He licks his lips and manages a smile. "You're a long way from home."
"Mom, did something happen?" He's alarmed as if caught somewhere he shouldn't be.
"No, honey. I had some days off and decided to see some of the country. I was near enough to save Mark a trip and pick you up."
Mark leans against the counter, ankles crossed, arranging himself in a pose—no doubt familiar to his wife—she's never seen.
"But I've only been here a few days." His words are half-apologetic, half-accusing.
"Yes, I know. We never did decide how long the vacation would last, did we?" And she touches his cheek.
Sensible words but her brain flashes another headline: duped, betrayed, his sweet talk, endearments, promises, all lies. Never mentioned a wife, did he? Her skin stings. Man needs a son to play with and takes hers. This weird kidnap, isn't that what it is? Her roiling mind searches for a way to upend this ludicrous reunion. She's a cop's daughter, taught to take action. She won't allow Mark to violate two women. Only Bobby's puzzled gaze causes her to hesitate.
"But Mom, we have so many plans. Me and Mark, I mean . . ."
"Won't you reconsider," Lydia asks with little enthusiasm. She must be wondering who this younger woman is.
"No, but thank you. Bobby, pack your stuff. We'll talk more on the bus." Or not, she thinks, because he's so upset by now that his lips are quivering, his eyes narrowing against the tears. She can't allow his disappointment to reach her. If he stayed the summer, he wouldn't be mistreated. But it would be like stealing, wouldn't it? Stealing her trust and then her son. Stealing what only money can't buy. Why should Mark get the pleasure of her son?
"Mom, listen, I have an idea," Bobby's jerking her arm as if to shake some sense into her. With his pale skin and wheat-colored hair, he could disappear into any cornfield.
"I'm listening," she says gently.
"How about if I stay for July? Then you and me can have August together. How about that?"
She can feel it and she's strangely touched. He's trying to negotiate her happiness as well as his own. She stands there in a circle of calmness that nothing in this situation justifies or explains. She knows her job as well as she would if she were working the diner. She has to reassure Bobby that none of this is his fault. And Mark hasn't said a word, doesn't dare to influence the moment one way or the other. If they go on much longer, Mark's reticence will hurt Bobby even more than her insistence.
"I have some plans for us, a surprise, but you need to pack up now." And what would that be, she wonders, but it doesn't matter. Surprises are the easy part. She'll send him to sports camp and worry about how to pay later. He's about to try one last time, but she adds with all the emphasis a mother can bring to bear, "Bobby, go do it, please."
After he's clumped out of the room, she sips her coffee. The fury of a bird's flapping wings speeds past the window.
Mark stands there, a poster of the good husband. Why shouldn't his wife know the truth?
Lydia wipes the table and places the milk carton in the fridge. Nothing left out. Dishes, cups, tumblers in glass-fronted cabinets. A rack with every kind of condiment. A microwave, a Krup's coffeemaker, a Magic Chef stove, all shiny new and ready to come to life at the press of a button. The micro-pearl lights above the white sink sparkle. A room with a view approved by Good Housekeeping. It's nothing like the bare-bones kitchen where Bobby eats his breakfast without her or the diner where she dishes up the eggs and hash each morning; where not so long ago she dished them up to Mark and took him into her home where except for the small window above the cracked sink, there was no view. Now Bobby has a comparison. And just like that she realizes it isn't to Lydia she owes the truth.
She'll make it simple. Mark lied to her to spend time with him. Mark lied to his wife to spend time with her, which makes him untrustworthy. Mark was good to him, which convinced her that he was a good man, but anybody who'd lie so easily is without a conscience and a man she should never have left him with. Bobby won't like hearing it, but he'll get it.
"Thank you for your hospitality," she says to Lydia. "Would you mind calling a taxi? I'll wait outside for Bobby." Without a glance at Mark, she gathers up her purse and traveling bag and heads for the door.
Facing the alley of trees that leads to the road, she remembers a story about a slave child whose mother beat her methodically for five nights; on the sixth morning the mother escaped. When the master took a cane to the child to learn where her mother had gone, the child hardly felt the blows. That's her. Her disappointment must be profound, yes, but it doesn't hurt like when her husband died. And isn't that a relief?
Orange suede plateaus surround her. A bittersweet scent pierces the thin, crisp air. She's never seen the depth of an open sky before. It does make her wonder where she might be next summer.
3
Imaginary Friends
Just the way he likes it. Packed booths, a cacophony of voices, clanking dishes, the sizzling grill, the breakfast smells. Any time of day, Murray knows what's cooking in the kitchen. Ava moving smoothly from counter to tables, wonderful. He's changed her hours several times; he likes her there for the morning rush. He hangs up the wet coat, takes off his boots. Everything has a trade-off, though; he learned that from his father. Even marrying Sylvie?
He remembers it like it was yesterday. She appeared at the diner asking for directions back to the city and then lingered at the counter with coffee and a muffin for a good hour. He was intrigued by how she explained things, so energetically. Her light brown hair shot through with golden strands swept around her face whenever she moved. He remembers being amazed that she wasn't married—and then wondering how come.
He glances outside; snowflakes fall rapidly. Still, he worries: a man of fifty-two has his habits, and marrying late has its inconveniences. It's not like he never dated. He enjoys women; they're good listeners, natural nurses for whatever ails you, and a man needs caring. Until Sylvie, though, the women he met . . . well . . . too soon the cream dissolved.
He deposits several rolls of quarters in the register. He counts the bills, jots down the amount and the time. Okay, he's not thrilled with her theatrical past—the looseness of actors. Her appearance, though, that's something else. Those wild green eyes. The woman needs no makeup. She adds to his presence. Five seven, the last time he checked, with shamefully small hands and feet, although it'd take a boulder to fell him.
Of course it's not quite a year; they're still in the honeymoon phase. Beginnings are like that, filled with talk and a little mystery, but everything becomes routine, and how she'll behave when it does, he's not sure.
He notices the regulars are here despite the storm. A good sign.
Never a generous man—who did he have to spend it on?—he's now the owner of a home in East Hampton, a mansion by anyone's standards. With Ava taking over some of his chores, he's easing up on evening hours.
Black watermarks pool near the diner's entrance. If anyone slips it's a legal problem. He hands the mop to Ray, who is young enough to be his son. It is too late for children of his own—and he does regret that—though he's devoted to his Dobermans.
Ava has the pies out on the counter the way he likes it. He sorts and arranges the bread in several bins, then heads for the kitchen. The dishes are stacked properly; sponges lined up on the lip of the sink, the surfaces clean. A good cook and kitchen person is the heart and soul of a restaurant. Nick is the best, much cleaner than Bruce. Where is Bruce anyway?
"How's it going?" he asks Nick.
"Fine."
Nick talks to him the least. Rosalyn says he probably believes bosses are the enemy. Well, this is America; he can believe what he wants as long as he keeps it to himself. He hands Nick the empty paper bread sack, watches him discard it. He could've done it himself but the order of things is important, and Nick's in charge of the kitchen.
• • •
"Sumptuous" is the word Sylvie comes up with because labeling a thing is as important to her as a handle on a teacup. Still, what's she doing in a house with cathedral ceilings? A velour couch—gray like an impending storm—stretches ridiculously long across one living room wall, green club chairs protect each side of a teak coffee table; the lamps have silk shades. It's all too fresh, too precious. New paintings waiting to be hung, prints she could never have afforded. Was it wrong to leave her job when there's nothing more for her to do but walk the spacious floors and marvel at the strangeness of the architecture?
One windowed wall faces the beach, but she can only watch the wheeling gulls for so long. She eyes the dogs near her feet. Wherever she goes, they tail her. It's oppressive. She dislikes their names as well as their menacing snouts.
She told Murray she isn't fond of animals. They're like children, he responded, give them love and food and they'll offer unqualified devotion. Is that true of her marriage too?
She may not be totally honest with her husband but she is with herself. A woman of forty-one, still single, no savings, moving through life at a decent but unremarkable pace, meets a man who wants to give her everything. How could she not? She flashes on a passing remark from Shelly outside the diner. It was one of those late autumn afternoons when the last rage of sun drenches houses in a golden light. With eyes closed she lifted her face to the warmth. Take what you can when you can where you can, Shelly said, the words odd but Shelly's tone neither mean nor sarcastic, just certain.
The dogs follow her to the kitchen, a room crammed with devices she has no use for. The trash disposal makes a loud sucking sound that upsets her and the dishwasher fills so quickly she's sure it will flood. Where she grew up, clothing was washed in the bathtub, and the scruffy field next door was where she spent her time with imaginary friends, who unlike her dreamy, alcoholic mother, attended to her every wish.
Taking a bowl from the cabinet she begins kneading ground beef. Some of it spatters on the floor. The dogs lap it up, then one licks her hand. That's a first. She tosses each a meat patty, watches the food disappear, feels their heavy warm bodies sidle past her the way cats would. Raw meat, she thinks, the key to their miserable hearts.
Suddenly she's not in the mood to cook and shoves the bowl in the fridge. She grabs her jacket and, wrapped in a scarf, slips on boots and fur mittens, ready to brave the weather. The dogs wait at the door. Murray warned her not to leave them home alone. She has a vision of them wading into the ocean and being carried away.
Together they trek the wet beach, snow-blown wind in her face, gulls screaming into the roaring of the waves. In the snow-veiled distance she can just make out a figure. Curiosity or plain boredom keeps her moving toward what turns out to be a man dressed in a long coat and woolen hat. A backpack is slung over one shoulder.
"Hi!" she calls into the wind. And the dogs begin barking. Damn! She shouts at them to stop but they don't listen. Undaunted, the man lets the dogs sniff his fingers and offers them crackers from his pocket, which quiets them. A youthful handsomeness is apparent in his craggy, ancient face. He's tall and lanky, white hair streaming from under his hat.
"Can't hold a conversation here!" he shouts. "I have a lean-to up the beach!" Without another word, he leads her to a tarp barely held aloft by shivering poles and battened down with bricks. An unzipped sleeping bag covers the sand; an easel is set up nearby. Coals smolder in a fire pit.
"My winter lair. Have a seat." And to her amazement she does. He tucks the edges of the sleeping bag around her. It's ridiculously cozy. After tossing the dogs more crackers, he seats himself in front of her as if to block the wind with his delicate body.
"Liam, here, I live on Jessup Road." Outside the howling wind is but a breath. His voice is soft, courtly, so unlike Murray's rough-edged tones.
"I'm Sylvie. We're new here and I haven't gotten my bearings yet."
"Yes," he muses, "bearings, very important . . . I must admit it took me years."
"So you've been here a while?"
"Since I retired from business and . . . well, so many other things . . ." Something confessional lurks here and she finds herself unexpectedly embarrassed. She points to his backpack.
"Can I see?"
He hands her a bunch of paintings as easily as if they were sandwiches at a picnic. Winter beach scenes. White, gray, silver without a drop of color, yet they shimmer. Could these be the landscape she finds so forbidding, cold and untouchable? She catches him staring at her.
"Too bleak for you?" he asks.
"No. The opposite. Is that how you really see what you see out there?"
"There's no metaphor for the ocean, only how I feel when I try to capture it."
"In this one the waves are ferocious. They're filled with warning . . ."
"Because my fingers were stiff and my knees hurting, the waves spoke to me of what's impending."
"Was that depressing?" Is she probing?
"At my age death is a comrade, a way of leaving, an exit."
"I don't believe everyone your age feels like that." Nothing about him seems tired or worn, though he must be near eighty.
"Maybe not. But there isn't much I'll miss. I love the beach, but I'm alone now. Do you have children?"
"No."
"I had a son killed in 1970, in that dirty war."
The sea, the sky, his death, his son's. He says it all in the same matter-of-fact way.
"How awful," she finally says.
"It was worse than that."
"I'm so sorry."
"You're born, you die. Everything in between is mostly illusion, but there are still sins. The avoidable deaths of young people is one." He gathers up the canvasses and slides them into the worn backpack. "It's no mystery how the dinosaurs disappeared. War kills the young and it's the beginning of extinction."
"You sound pretty certain." She's thinking of Murray, who believes war is a way to keep what you have.
"I did go on, didn't I?"
"Oh no, not at all. Not in the least."
• • •
For the third time Murray punches in the number, his eyes on the snow mounting outside the diner in the empty parking lot. She's supposed to drive in but it's not a good idea. He'll take the train home. Buy some flowers from the guy on the platform. He enjoys bringing her presents; it's a new sensation. So is knowing she'll be waiting for him. Sharing space is easier than he thought. Each night her chatter and that creamy body. It couldn't be finer. He adjusts the thermostat. The windows are steaming up, the snowy world vanishing. He does what he can but the weather is beyond him. He dials the number again.
"Hi?" She sounds breathless.
He imagines her anxious to get to the phone.
"The wind is something else."
Is the woman crazy? Yesterday, too, she was out walking. "Where'd you go?"
"Just along the water. I can't always tell where I am until I look back and see our house."
"The dogs must be frozen." The coffee urn, he notices, is low, the floor beneath the corner stool gummy.
"Do you think so?"
"Yeah, you'd be surprised how delicate they are." He wiggles a few fingers at Rosalyn, who's stamping her boots on the rubber mat. Her dark, piercing eyes beneath heavy brows take in the scene.
"Okay, I'll keep an eye on them."
"So pick me up at the station at eight, it's too dangerous to drive in." He hangs up and draws water for the urn. Who goes for walks in the middle of winter? It's not like anything out there changes.
"Work half-day," he says to Rosalyn, without looking at her. "It's going to be slow." Paying hourly help to sit around irritates him.
"Murray, I drove here in a storm. Unless you close shop, I'm doing full-day." Rosalyn enjoys combat more than a marine.
"Well, don't blame me if you're bored." He wonders if she'll mop the gummy floor without being asked.
"So how's Sylvie?"
"You should see how beautifully she rearranged the living room, a real show."
"Nice of her to pick you up every night. I wouldn't."
He laughs. "Me or anyone?"
"Any man who could drive himself, to be exact."
"Can you take care of that spot on the floor?"
• • •
The train station looks quaint beneath a mist of falling snow. He spots the dogs in the backseat and then kisses Sylvie's cheek, cold and smooth. The promise of a warm house and dinner excites him. "How's it going," he asks, taking over the wheel, not really wanting an answer. It's been a long trip.
"Murray, you never told me why you weren't in the army."
She surprises him constantly. "What brought that up?" Cars are pulling out of the station at a slow, careful pace. He waits his turn impatiently.
"Reading about Iraq, Afghanistan . . ."
"Yeah, I wanted to go . . . badly, but they found a TB spot on my lung."
"TB?"
"I caught the bug somewhere but it never infected me. Happens, I'm told." Actually it was flat feet that did him in.
"How strange. TB may have saved your life."
"It was frustrating. My father thought I was some kind of misfit." He waits for a word of sympathy.
"You must've been relieved. I mean, who wants to go to—"
"Sylvie, war's a man's sport like hunting, no more no less." He maneuvers the car onto the road. The snowplow ahead forces him to reduce his speed to a crawl. Hell.
"So it has nothing to do with patriotism."
What's she nattering on about? Something challenging in her tone annoys him. "It has everything to do with it, everything to do with this great country of ours. And what's all this about anyway? Why are you so interested in stuff that happened a million years ago?"
"Listening to the news . . ."
"Well, read a book."
• • •
Just as he expected, the house is warm and cozy.
"Sylvie," he calls from the bedroom, "let's eat where we can watch the storm." She doesn't respond. "Sylvie," he calls even louder.
"Please don't shout." She appears in the doorway, startling him.
"Let's eat—"
"I heard, okay, we'll do that." She turns to leave.
"Wait." He grabs her arm, nuzzles her ear, sniffs vanilla or maybe something else, he can't tell. He wishes the flower guy had been at the station. "You smell delicious."
"Must be the cooking oil."
Watching her walk out, he pats the bed and both dogs jump on; after wrestling with them awhile, he says, "Okay, boys, down. Now!" They obey, which makes him stupidly happy.
He changes into his sweats, ready for the evening.
She's uncorking the wine, her silvery kimono shimmering in the lamplight. He inhales the luxurious surroundings, nothing like the faded furniture and cracked plaster of his old apartment.
Two dishes of pasta with vegetables steam on the low coffee table. He was hoping for fish or meat, real food.
"Great, baby," he fills their glasses. The wine is so tasty; she knows what to buy. He refuses to ponder how many intimate dinners she had before this one. He leans back, his fingers caressing the velour.
"So how was your day?" she asks.
"Rosalyn pisses me off by the minute, a dyke if ever I saw one."
"Please don't use that word."
"Don't be so sensitive, Sylvie. People are people. I know that. I run a damn restaurant, don't I? All kinds." It's not the first time she's called him on his language. Reminds him of his stiff-assed second grade teacher warning him to talk right or no one would respect him. He's halfway through his first glass and refills it. Sylvie's hardly touched hers. "So you went into town today?"
"I signed a petition against the war, first one ever. There were hundreds of names on it."
"Jesus," he mumbles, "bad move."
"Murray, I've a right to express my beliefs."
"Men are dying over there for your rights."
"That's ridiculous. I want them home alive."
"Sylvie, you don't understand."
"Of course I do."
She's hell-bent on ruining the evening. "Let's not argue, please baby?" For her, he's willing to forget the whole deal. "What else did you do today?"
"Walked, cleaned, read, talked on the phone with Jenny, who's gotten a part in an out-of-town play."
"Oh yeah, where's that?" Jenny is one friend he wishes she'd forget. He'd have to stand on his head to get a smile from her and even then . . .
"Kentucky. Real pretty place. We should visit."
"Oh, you've been?"
"A play in Louisville eons ago."
"No kidding? That's impressive."
"Murray, I've told you about that episode. How the agent saw me in a little commercial and invited me to audition . . ."
". . . And you got the part, yeah I remember." But actually he doesn't like remembering. God knows the people she slept with . . . "On my day off we should hang those paintings. Three in the living room, two in the bedroom, one in the vestibule. You pick."
"Okay, Tuesday?"
"Not sure. I'm still teaching Ava to do some of the ordering. The woman's lonely, I can tell. She ought to marry again. Marriage is good, right?" He can hear himself slurring a bit, but so what, they're home. Before long they'll be in bed and her sweet body will be his.
• • •
As they've done for the past week, she and the dogs go straight to the lean-to. Armed with baggies of raw meat, she keeps the dogs content. Liam pours two cups of sweetened coffee from a thermos. The afternoon's chill demands heat of a sort. With Liam beside her, she watches the ocean swallow the snowflakes.
"If anyone had told me I'd enjoy sitting outdoors in this weather, I'd have said they were loony . . . I mean it wouldn't be something I could do with Murray. He likes his comforts. He's very ritualized. The thing is, when I imagined a husband it was someone more . . . audacious."
"My wife always knew disaster awaited her outdoors. She was terrified of slipping on ice, she wouldn't consider getting on a plane, kept her distance from tall buildings. After our son was killed, though, her fears vanished, which scared me silly. You never know what'll provoke change in a marriage."
Is that what she's waiting for? "It's not that Murray isn't loving. He is, and he's giving to me, but he doesn't spread goodness easily. The people who work for him . . ." Why does that bother her?
His voice drops falling on a dark memory. "My son's death refocused my work life."
"I'm sure it did," she murmurs, staring at the horizon. Her father's death upended everything.
"What is it?" He touches her arm.
What indeed, she wonders. Was she happier in her studio apartment, coming and going with no one to care where or when?
• • •
As she plows home through the wet sand, the dogs loping ahead, Liam's concern warms her. She wonders how he sees her. Daughter, neighbor, intelligent woman? In the theater she would analyze each role: Who is she before she walks onstage? Who now? What's to become of her? The dogs begin barking, then dash into the curtain of white and disappear. Murray must be there, though it's early. An urge to run back down the beach sweeps over her. The front door swings open. "Hi baby! When it started snowing again I decided to leave work. Mila was a little too happy to see me go."
"Want something to eat?"
"Yeah," he says, but he's already unwinding her scarf, removing her coat, embracing her waist, and she knows just what he wants.
Oh lord, not now, she prays. But refusing is as beyond her as an excuse that would convince.
He undresses quickly, perhaps sensing resistance. He's spread-eagle on the bed, his cock stiff; his arms reach for her.
"Haven't I said a hundred times I don't want the dogs in the bedroom with us?"
"Rummy, Cheney, out, now," he orders.
She slams the door. Wishing to sustain the anger but having no reason to, she slips off her sweater, her sweatpants.
"Hey baby," he whispers, "come here, speak to me. But it isn't words he wants; she knows that. She slides in beside him, his leg capturing her thigh, his head pinning her chest. Her fingers bite into his shoulders; the need to push him away is so strong. But she can't bear the thought that he'll be furious, that he'll sulk all night, or worse, demand to know how come, what's wrong, where she's been. And what would she say then?
"Something the matter," his tone soft, far away.
"No," she offers in her breathy just-walked-onstage voice, wondering who is she now? And now, as she draws him closer. And now as she strokes his back. And now as she closes her eyes on a memory: seventeen, in rehearsal, the director pulling her into an empty room, and his first kisses and her thought that rebuffing him will come at a price and—as any good actress would—that last thought: what's a little kiss or two.
• • •
When she and the dogs arrive at the lean-to, he isn't there. She tucks the sleeping bag around her to wait. If Murray discovered her sitting shoulder-to-shoulder with Liam he'd be a lot more than unhappy.
The purple-tinged sky promises more snow. The forces of the universe are sending a message: inclement weather as punishment. An idea Liam would appreciate. Usually he arrives before her; morning light is what he's after. Could he have been and gone? He did say, see you tomorrow. She searches the beach for his figure but windblown sand impedes her view; seagulls glide overhead, the endless waves.
• • •
She drops the dogs at home and begins the trek over the opalescent sand toward the dunes where an exit leads to his road. The tiny A-shaped house squats on a postage-stamp lawn obliterated by snow. The blinds are drawn on the few small windows. She might regret this; it's none of her business. They've only known each other two weeks; he must have other friends who look in on him, but she's already knocking on the door with an insane urgency and shouting his name into the swirling wind. Trudging through the snow, she finds no back entrance, no porch, just two small windows, also with shades. Apprehension fills her. She flashes on her father hanging from the barn rafter. The chair beneath kicked halfway across the floor, sunlight in a cracked mirror flooding the space.
Even if she phoned the police, what would she say? Liam didn't show up at the beach on a blustery day? Liam's not at home? She pounds on the door once more and nearly stumbles forward when he opens it; shoeless, his open shirt revealing a mat of white hair.
She can't tell if he's sad or glad to see her. "Are you okay?"
"I didn't sleep well. Not at all actually." He ushers her into a small living room: bare walls, couch, rocking chair, easel, TV.
"Does something hurt?" she asks because his face is pale, his eyes bloodshot. Strands of hair fall untidily across his wide forehead.
"At my age, things hurt."
"What's wrong, then?"
"I can't explain." And he sighs.
"Of course you can." Jesus, she sounds manic. She takes a breath. "I expected to see your paintings everywhere."
"They're in the closet. Choose a few if you want."
"I will, thank you." And wonders what exactly she'll tell Murray.
"There's coffee in the thermos on the counter."
It's a closet of a kitchen; hardly enough room to turn around. She's about to offer Liam a cup, but he's leaning against the wall as if standing were too difficult. Without a coat he looks thin, fragile. "Liam, why don't you take a nap now?"
"Will you stay?"
"Yes." It comes out before she can think and when she does, she knows this is where she wants to be. She wonders is he afraid to be alone the way she is? When he's alone does he wonder: Who am I? The way she wonders about herself as if she had just wandered on stage, a seventh character in search of an author? But she knows he has no such thoughts. He doesn't seem frightened, only weak.
She follows him to the bedroom and sits down on a folding chair while he stretches out that matchstick frame on the full-size bed. "I am exhausted," he admits. For a few minutes, he stares at the ceiling while she gazes at a photograph of a boy in uniform that sits on the small chest of drawers, the only photo in the room. He's the one to break the silence. "Would you lie next to me?" he asks, his gentle tone almost apologetic.
He's fully dressed; so is she and who's to know? She slips in beside him, inhales the scent of lavender soap. She almost holds her breath, listening for his to even out. She'll leave once he's asleep. He takes her hand, his fingertips icy. His breathing has a raspy edge now. Should she call 911? If she'd been home with her father that morning, what then?
"I'm being visited by the end," he whispers as if reading her thoughts.
"It's your imagination."
"No, death sends a message before he arrives."
"What message?"
"A calming one," he says with quiet assurance. "All weariness to be erased along with pain, regret, ambition, and a thousand other burdens."
Is that what her father felt? She slides her arm around his waist, his words strangely comforting.
• • •
The train crowded with commuters is so raucous he can barely think. He's tried calling her all afternoon. He wondered if she was walking in this weather. He left a message: Pick him up at six. What if she's not at the station?
The man next to him reading a newspaper closes his eyes. He closes his but it's no use. She did seem distant this morning. He probed, but her responses were vague. If she ever left him . . . God, her soft warm body removed from his life? Can a person die of a broken heart? If this is love it's getting difficult. Is he panicking? Worry isn't new to him. But Sylvie gone, that would be something else.
The man, awake now, has resumed reading the paper.
"Some weather," he says because he'd rather talk than think.
"It's going to get worse."
He looks at his watch, another ten minutes. What's the matter with him? Of course she'll be at the station. He peers out the dirt-streaked window; it's snowing heavily.
• • •
It must be nearly six, the darkness ashy; a flashlight would help. Gusts of cold wind burn her face. A veil of snow obscures her view; the unplowed road is hard to navigate. But she's not bothered, her body energized, a reprieve offered. Her arm around Liam went numb but she didn't remove it until he woke. It was as if she were hanging on to him instead of holding him.
She picks her way carefully along the shoulder of the road. Snow-laden trees and bushes hide vacant summerhouses. If she shouts no one will hear but the aloneness rejuvenates. A neighbor, an elderly gentleman she met on the beach, was ill and she helped him, the paintings a thank-you for her ministrations. None of it a lie.
Glancing at the darkened house, she sees her car beneath a cape of snow. Wet flakes hit her face. She's reluctant to call it a day. The dogs bark, and she fishes in her bag for the keys.
Switching on the living room light, she freezes. Chewed couch, torn chairs, gnawed tables, toppled lamps, ripped shades, clawed paintings, scratched walls, splintered floor, ruined, all of it ruined. "Bad, bad dogs," she cries. They look at her dolefully.
Murray will have a fit; he'll curse her and the universe, maybe worse. Better pack a few things; be gone before he arrives. Lord knows what he'll do to the dogs; still, she can't take them with her. What motel will let her in with them in tow? "Bad dogs," she mutters, and peers out the window. It'll be treacherous driving, no visibility at all.
Ignoring the phone's pulsing message button, she hurries to the kitchen, pristine, untouched, appliances gleaming. She quickly fills the dogs' bowls with food and water. They sidle past her; she doesn't want their affection; they could be dead tomorrow.
Lights sweep the window. Christ! A headline flashes: Man in temper kills dogs, wife.
She takes a last look at the animals, meets him at the door. He throws his arms around her. He's mumbling something she can't make out.
Her words, though, are precise. "I left the dogs home alone."
He strides into the living room. He runs his hand along the damaged couch, studies the ravaged coffee table, stares at the paintings, traces the scarred wall with his fingers. "Rummy, Cheney, you bastards . . ." his voice breaks.
"It's my fault," she offers.
"God," he says, "God . . ."
"Leave," she stage-whispers; the dogs sit.
"Miserable, stupid curs," he grabs the back of a chair.
"Murray—" she begins.
"Everything's wrecked, everything."
"Murray?"
"How could you?"
Is he talking to her? "Murray . . . Look at me."
He turns; eyes wide, astonished, wet, skin blue-white and taut, mouth slack. She's never seen him this way, it's scary, a whipped man resigned to the next lash. He's no youngster, a seizure, stroke, anything is possible. Prying his hands from the chair, she walks him to the bedroom, lies down beside him.
"Murray," she whispers, "they're things. They can be replaced . . . we can reupholster the couches . . . and the chairs. The paintings, okay, they're destroyed . . . Are you listening? . . ." She looks into his nearly unblinking eyes. "It's okay, it doesn't matter, it's not important . . ." But he's not responding.
"Say something, please, you're frightening me."
"I never lived this well . . ."
"I know." But does she? She was expecting anger, not collapse.
"You can't imagine what this house, the furniture, means to me," his voice a whimper.
Her eyes slide to the mahogany dresser, the floor-to-ceiling mirror, wicker rocker, all of it inanimate. How can he be so emotional; no one's died.
Suddenly he grabs her arm. "We'll make it beautiful. We did it once, we can do it again." The volume loud, he could be screaming.
She sees herself in one of the large suburban shops, salesmen circling her every move, furniture arranged by room, pillows galore, tables set for eight, peering at materials until her head aches.
"Money's not a problem. We can spend," his voice revelatory, gleeful, which should reassure her, except her energy's gone, a pricked balloon, the air seeping away. She takes a deep breath, but it doesn't help.
"I'll get rid of the dogs. Everything has a trade-off."
"A trainer can teach them to behave." Does she care?
"Are you sure?" his tone childlike, high-pitched.
"Yes."
"It'll all be the same as before, I promise." He's breathing heavily.
Once she watched an actor playing King Lear suffer a heart attack during his last scene. The audience applauded.
"You believe me, don't you?" His voice shaky, he takes her hand. "Are you okay with me?"
She looks past him at the darkness and visualizes the moon's tangerine light above the storm.
"Yes," she replies because the truth now would do him in.
4
The Way Things Work
Shelly stares out the kitchen window, cell phone in hand, as a few cars pass by. Across the road her elderly neighbor wearing a sweater over a robe is watering the lawn as she does every morning. Maybe it's penance, or a pact with the devil in exchange for something. If the devil approached her offering relief, what would she agree to in return? she wonders. She glances at her cell phone. God knows she doesn't want to make this call. Bruce . . . he's killing her. It's the fourth time this week she had to call to say he'd be late. If Murray happens to arrive earlier, if he picks up before Ava, if . . . it won't happen. Sylvie told her he goes to sleep and wakes at the same time every day, no exceptions. Sylvie sounded disappointed, but isn't it a good thing? He functions at a high level, doesn't he? Runs the damn diner like he was born to do it, doesn't make business mistakes that she's heard about and can be counted on to get to work on time. None of which she said to Sylvie. Once more she glances at the phone. It's no use, either she calls or . . . The diner is on speed dial and she listens to it ring. When no one picks up, she exhales and leaves a voice message.
Her cold coffee on the table, she eyes the muffin. Can't eat that. God! Even her oldest, after he sees his father, asks what's happening. She could give him an earful but all she says is, Dad's getting on and it's not easy in a diner kitchen. Bruce was never tidy, but now it's impossible. Leaves things wherever, but, okay, she's used to men doing that. She has three sons for god's sake. But at least he used to shower every day. It's making her crazy. They've shared a bed for twenty-seven years, ever since she was nineteen. She can't do it anymore.
Her eyes slide to the newly painted kitchen walls. Apricot. Lord, what possessed her? She hoped it would cheer her baby before he took off. He did say he'd remember the color, something bright in the desert. It's painful to think of Michael there. She can't watch TV, either, though since Michael left, Bruce is a news junkie. What she can't say to her sons or to Bruce is that fear for Michael's safety has her by the throat, though she confided as much to Ava and Mila during breakfast at the diner. Women with children, they understand the terror.
She glances at the clock. Three times she's told him to get up. He'll lose the damn job. Murray isn't the type you want to piss off too often. She strides out of the kitchen and smashes open the bedroom door. "Bruce, I swear, you don't get out of bed now I'm leaving for good." She hates shouting; it's so crude. Even when the boys were little, she didn't raise her voice. Now she's beginning to sound like her mother who screamed everything.
He rolls slowly toward the side. He's gained weight. He used to have a good build, he jogged and pumped. Now he does none of it. She waits till his feet touch the floor, then walks out.
Leaving Bruce has become a daily fantasy. Then she'd have only herself to care for. Her sons, too, of course, but with Michael away and her oldest married, already saddled with a baby, and the middle guy in Seattle doing whatever with computers, it's just Bruce, isn't it? What would she tell her sons? I can't live with your father anymore. He doesn't bathe, doesn't talk. They're not going to be sympathetic to that. Even if they are, they'll want him to get help; you can't leave a man when he's down. It's immoral. Well, let them come live with him.
Bruce shuffles in, dressed in baggy jeans and a sweater.
"Want breakfast?"
He nods.
"Coffee and a muffin? Because you don't have time for a big one."
He nods, again.
"Why couldn't you get up?" She cuts up the muffin the way he likes. Sweeps the crumbs off the smooth surface of the counter, which she planed and stained herself.
"It's hard."
"Tell me something I haven't heard."
"Just is." She pours the coffee and he gulps it down, though she's sure it's too hot.
"Bruce, be careful, you'll burn your tongue." Habit. She shouldn't bother when the man asks nothing about her well-being.
"Jam?"
"On the table there. I was thinking before how Michael likes the color of the kitchen, I mean he said so in his last—"
"He's a baby. He shouldn't be fighting in this frigging war. He didn't even sign up . . . what's the National Guard doing there anyway," his face reddening.
"Jesus, Bruce, calm down. I agree with you, but what can we do? He's there."
"Do?" He looks at her like she's posed the sixty-four-thousand-dollar question.
"I know how you feel. But Bruce, there's nothing we . . ." She stops, no point repeating herself like an idiot.
"I'm too jittery to drive this morning," he says. "You take me."
"Why not. I have nowhere special to go."
"What about work?"
"I told you twice, they cut my days to three. After twelve years, it's a shame. People are saying they'll call me back. Who knows?" She's one of two assistants to the head bookkeeper. They check accounts, payment errors, and merchandise received. It's satisfying, the order of it, the repetition, the predictability. She watches the colorful bustle on the supermarket floor from the little glass cage of an office. If she slides open one of the panels, the cacophony rises up to remind her there's a world out there.
• • •
As she drives through familiar streets, her eyes flit past houses with extensions not half as good-looking as the one Bruce's brother built onto their place. His brother is creative but can't stop drinking no matter how many programs he attends. Once, sitting at the kitchen table crowded with empty beer bottles, he said AA meetings leave him so desolate only a drink can help. Bruce laughed. It wasn't funny. Later, in bed, Bruce mumbled people choose the way they die. When did he make his choice, she wonders, glancing at him gazing ahead, his face empty of clues.
"Heard anything from your brother?" she asks.
He shakes his head.
"He must be on a bender," she offers.
"Why do you care?" He sounds bothered, as if she's taking something from him.
"I'm just talking, that's all."
"Things on the news now that no one's seeing."
"Bruce, millions of people watch TV."
"Watching and seeing are different."
"Well . . . that's true." Is this the beginning of a conversation? "It's hard for people to really . . ." but he's turned away to stare out the window with the same intense look that comes over him when he watches TV, as if there's something he has to catch before it disappears.
In the diner parking lot, she waits for him to begin his slow climb up the few steps, her eyes glued to the front door till it closes behind him. She could go in too, have a cup of coffee and chat with Mila, but it's the morning rush; Mila will be busy. The sun too bright by far, splashes the front windows. She flips down the visor, steps on the gas, and wonders where she's headed.
• • •
Could be too early to drop in, she thinks, pulling up in front of her son's garage. Well, she's here, isn't she? Ricky rented the small Cape Cod with a deal to buy in two years. He's working his heart out, but he can't control construction. It happens when it happens, he'd be the first to say. He's on a site now, thank god.
Ricky holds open the door for her. "What's up?" He sounds concerned. Firstborns are like that, always waiting for the shoe to drop.
"Hi son. Drove your dad to work and thought I'd pop in to kiss the baby. Hope it's okay."
"Joni still has the coffee hot."
Joni is sipping coffee at the table, her thin body hidden in a cotton robe. Shelly notices the mess in the sink, must be two days' worth of dishes. Should she offer to wash them? She doesn't want to. Besides, Joni might take offense.
"What time do you go in?"
"In a few minutes. We're on weird shifts now. It's a big piece of property near Jones Beach. How's Dad?"
"He felt too jittery to drive this morning."
"Poor guy's beaten down, is what I think."
And what about her? she'd like to ask, eyeing the baby swing next to the high chair, neither of which has been used yet. They were presents from the baby shower where she and Joni's mom were the only people over thirty. Damn thing lasted for hours. Then the men arrived, including Joni's father, but no Bruce.
"Dad needs to get checked out. A doctor might give him something."
"I've suggested it a million times. He looks at me like I'm asking him to climb Mount Everest."
"Ricky, you talk to him," Joni says, which surprises her. Joni's a quiet girl who's loved her son since junior high. And why not? He's earnest, handsome, and energetic. He used to polka his mom around the living room, lift her right off the floor, and laugh. Or was that Bruce? There was a time when Bruce was the man. They were married, but they were lovers, four or five times a week, which is no easy feat with a bunch of kids in the house. Bruce always had his quirks, he's a vet, isn't he? When he first returned all those years ago, he'd open up after a few drinks and talk about the war. She never liked what she heard, but comforted herself that it was over and done with. Except maybe things don't go away, maybe they go into hiding like bears and come out when you're too old to fight them. Bruce will be leaving his fifties real soon.
"Want something to eat?" Joni asks, her tone less than inviting. She can't blame her; these kids don't have much. Besides, Joni's got her hands full with the house, the baby, her telephone canvas job.
She shakes her head. "Where's the baby?"
"Asleep in the carriage. He was up all night. I had to wheel him around. Don't wake him."
"No, I wouldn't think of it." She turns to Ricky, "Joni's right, talk to him. Maybe he'll listen to you."
"I'll take him out for a few beers."
"Like he needs the calories."
"What?"
"Never mind. Pick him up at the diner after his shift."
He nods. "Gotta go." He grabs his jacket. She's about to tell him it's pretty warm out there, but it's none of her business. It's Joni's business now. Joni kisses him hard on the mouth. The girl's still crazy about him, but just wait. Such negativity . . . it's not like her. Where's the bright-eyed, perky Shelly, a woman determined to get what she wanted? A woman who said yes to whatever it took to make it happen, and god help any who stood in her way. Bruce used to laugh at her combativeness, said it would put a grunt to shame.
She's careful not to slam the car door and wake the baby. No point hanging around without Ricky there. Maybe if she'd had a daughter . . . but her sons, they're men, they feel with their dad. Oh they love her, but sometimes she's on a planet by herself. Joni's sweet, but the only thing they have in common is Ricky. She doesn't want to hear Shelly's problems. Does she want to hear Joni's? She could find a therapist who'd listen, or maybe she could go to confession for free. Except you have to believe to receive solace. She could drive till the car runs out of gas and see where she ends up. Her fantasies of flight are beginning to scare her. But what scares her more are the thoughts piling up in her head like so much garbage she can't get rid of.
• • •
Searching the mall for a shady spot to park, she pulls in between a pickup truck and an SUV. With keys in hand, she slings the bag over her shoulder and hurries toward the bakery. As usual, the mall seems endless and unrewarding, but where else to go? A hotel in the city for a few days, not too expensive, with room service, a bar, a restaurant? A temporary escape. She'll take in a show, something deep, not a musical. Ricky will fire how-come questions at her she won't be able to answer. Her sister, too, will be calling to ask if she's okay. Well, she's not.
The well-dressed, stout woman who owns the bakery sits on a stool behind the cash register. The warm scent of fresh bread fills the shop. A gorgeous display of cakes and cookies offer themselves behind a domed glass. She spies a man near the large oven icing cakes on a flat table, his apron a palette of colors.
"Shelly?" From behind the counter, her sister throws her a questioning look.
"Take lunch early, please," she whispers. "I'll be at that café across from the shoe store." She's out the door before Patti can refuse.
Slipping the keys in her bag, she heads toward the café taking in people's expressions on the way. What does her face say? she wonders. The first few years of their marriage, Bruce would study her for long minutes, then try to guess her thoughts. She didn't like it, said it was intrusive; he was invading her head. Some of that would go a long way now.
The café isn't crowded. A waiter loiters near the counter looking bored. He follows her as she finds a table away from the window. Her watch reads eleven. It's too early for anything but coffee, which she orders black. Then she decides on a scone. They're going to be here awhile, is what she thinks. A huge mural covers one wall, a French countryside, she thinks. A trip to Europe by herself would be exciting. She'd go places if Bruce weren't around. But where would he be?
Patti hurries in as she always does, a kind of tic with her, rushing.
"Hi sweetie. What's up?"
"Order something," she says.
"I can't look at cake."
The waiter arrives, pad and pencil in hand. Her sister orders a cappuccino.
"Time of day tells me this isn't a how-are-you visit," Patti says.
She gazes at Patti's long wavy hair dyed the same honey-blond color since she was fifteen; her eye makeup hasn't changed either. Her sister doesn't look different from ever before. "It's Bruce. I don't love him. I can't live with him." The words run out before she can test them, and they surprise her.
"After so many years it's not unusual. There are months I can't stand the sight of Peter. Then, I don't know, some little thing happens, the way the light hits his bald spot, the way he rubs his eyes, it brings it all back. You have to wait for those moments to rekindle." Patti talks fast.
"They're not here, they won't come back." Peter is jovial. He cooks. He loves the house. In their worst periods, he brings her flowers every Friday night.
"How can you be so sure?" Patti asks.
"It's been too long, more than a year. I'm reaching the edge."
"Then talk to him."
"It's no use."
"Why not?"
"He's shut down. Either he takes me to the bottom with him or I let him hit it alone. How can I get out of the marriage without being hated by my sons, you, Bruce, everyone else in our lives? You don't leave a drowning man, and Bruce is drowning."
Two teenage girls make a noisy entrance, laughing like they own the future. Why aren't they in school?
"Shelly, you're going through that time, not yet fifty but closer than further, when we start asking ourselves what is this life? I've been there. I turned fifty last year."
Patti's right, before long she'll be looking back at fifty, then how quick to sixty. There'll be limits that aren't here yet. "So stick it out and one day all will be fine? Is that what you're saying?" She can't help the sarcasm.
"It's the way things work," Patti asserts in that voice of hers that claims to know everything. "You can't hold on to dreams that promise another life, because there isn't any. Maybe for rich folks who travel the world, own mansions. Not us. Think of the struggle, how long it took to create the homes we wanted. You won't be able to do that again. Why give it up because Bruce is going through hard times? You'd be sorry."
"I didn't expect you to agree, but you're not getting it." She should've waited to catch Mila on break. Single mothers know a thing or two about struggle that women who live with men don't. What exactly she'd be hard pressed to explain.
"What's there to get, Shelly? A marriage falls into a pit. Who climbs out first? The woman does. Remember Dad's depressions, what did Ma do? She walked around them till he snapped back. He always did. It never occurred to her to leave."
"We don't know that. It isn't something she would've told us. Anyway, Ma didn't expect anything better." She's not her mom, won't be.
"Shelly, what part in this do you play? What should you be doing that you're not? Is it all Bruce? It might be. But do you know that for sure?" Patti's bright blue eyes, so like her own, remain steady on her. Then she glances at her watch and takes another quick sip of coffee. "I couldn't take lunch, it's too early. I'm on break. It's over. Sorry. I'll call you later." Her sister drops three singles and rushes out. The door slams, she and the teenagers the only customers there.
Is Patti right? Is she contributing to Bruce's behavior? She badgers him constantly. Wear the new boots, take off the hat, pick that up, don't drop it, too hot, too cold, not right, a thousand ways to control. He lets her, Bruce does. Actually, she does pretty much as she pleases, always has, with her sons, the house, her hours at work.
• • •
In the pharmacy next door, she buys a box of Epsom salts and a packet of bubble bath. Then she picks up a steak at the food market. She'll hash brown potatoes the way he likes, though he doesn't need the starch. Wine, yes, that too, and a candle for the table. Maybe some unexpected shaft of light will illuminate his face and rekindle her.
She takes out her cell phone and leaves Ricky a message to pick up Bruce another night.
• • •
Shoving the pillows deep into cases so they're plump, she turns down the duvet, exposing the pretty blue hem of the sheet. She remembers how the bed gave itself up to them as if it too was a star performer, the naughty pleasures that followed her into the next day. Remembers too the years of Bruce's whispered words, always the same endearments because they belonged to her. It's what's left of their marriage. Memories.
She scrubs the bathtub till it's as white as one in a soap commercial. He used to like it pristine. Peering into the crystal clear mirror, the face that stares back hasn't begun to reveal the truth. Strong-boned like her mother's with skin that promises to age gently. A few laugh lines would be good. Her touched-up dark hair is graying at the temples. She doesn't care the way she used to. There was a time she'd dress up on a Saturday night, heels and all, even if they weren't going out. He always noticed, Bruce did. He called her his dark Irish beauty, his one and only. He was her mirror, her happiness reflected there. Now her makeup sits unused on a shelf near the sink. She unzips the pouch, takes out foundation, eyeliner, blue shadow, frosty pink lipstick, all the while thinking this is even crazier than some of her fantasies.
• • •
She places the unlit candle in the center of the table set now with matching plates, stemware, and cloth napkins. She won't broil the steak till he gets out of the bath, then a few glasses of wine. It's strange, the fussing without the excitement that used to rise in her as naturally as her energy. Still . . . if she doesn't try . . .
Hearing the front door, she hurries toward it in the high-slitted long purple dress he brought her from overseas. Her eyes made up, she's wearing earrings and perfume. "Bruce? Honey?" When was the last time she called him honey?
"Something happen?" His face drains of what little color it has.
"No. Everything's fine. Let's have us an evening." She touches his shoulder. "I bought salts and bubble bath for you. I'll fill the bath for you. You'll feel refreshed. Then we'll have some wine. What do you think?"
"Too tired. I'll watch TV in bed."
She sees herself walk past him out the door.
"I need to say a few things. Come in the kitchen. Just for a minute. I'll make you a steak. You can eat it now or later."
She pours wine for both of them. "Sit, Bruce. I can't talk with you standing there like someone's outside waiting for you."
He slips into the armchair and watches her warily. She sits across from him. His curly black hair peppered with gray flattened now from the ridiculous cap he wears, though it's no longer winter. All of her sons have his large, round eyes and long lashes. Always a strapping guy, the kind who could pick up a woman and carry her over the threshold without losing a breath, and he did. Not anymore, though. She wonders if recalling the good times or the better times or just the times when he wasn't the way he is now would be helpful.
He drains the glass in two long gulps, and she refills it. "Just today I had an image of you with your strong arms pushing the stuff that bothers you into a carton. After so many years, the carton splits apart. The stuff floats around looking for a new hideout but it can't find one. I think it's what's happening to you, to us, Bruce." Okay, it's not what she actually thought, but the bear-in-the-cave thing won't work for him.
He looks interested. He puts down the glass and seems to be figuring out something to say. She doesn't want to pressure him, so she goes to the stove and turns the potatoes, only the sound of spitting grease. She waits a beat, then returns to the table. "I was also remembering some of your war stories. They were horrible, particularly . . . and there must be many you never told me. Not that I want to hear them, god help me, but I was thinking . . . if you could vent to . . ." She's taking a chance here, trying to get him to open up when he doesn't want to. The last time she did that, after 9/11, he mumbled, why bother, said it was the beginning of the end anyway. "Bruce, what do you think about what I'm saying?"
"Why do you care?"
She tells herself let him go to bed. She can't. "Bruce, I care, and I'm waiting for an answer."
"Michael's carrying forty pounds, plodding through the mud, afraid of any noise in the trees. He's only nineteen. He doesn't want to die." He leans toward her, breathing hard, his face tense, eyes wide, the tendons swelling in his neck. A cold hand squeezes her gut. It's himself he sees slogging through the mud, being pursued, his old demons creeping up his back and what can she say to prove it.
"Our son is nineteen, yes, Bruce, but he's in the desert, not the jungle. It was you in the jungle."
He gazes at her but lord knows what he's remembering. "Yesterday, a helicopter was shot down. It was on the news. Everyone killed." Then, scraping back the chair, he walks out slowly.
Once Ricky brought home a stray dog that kept growling at them. Why keep a dog that might bite the children? She called the animal shelter and asked for someone to come get it. When the man arrived, the dog stopped growling. He knew it was over.
She turns off the oven and takes the pot of potatoes to the sink. She carries her wineglass to the bathroom. Starts the bathwater, shakes in a cup of salts, adds the bubble stuff, and waits for it to fill. Heat dampens her face, the aroma gardenias, she thinks. Someone brought home flowers for one of her birthdays. Was it her thirty-sixth? Bruce ordered a gigantic cake and Ricky insisted on putting all the candles on it. "Make a wish," they all shouted, even solemn Michael. What did she wish for? Was there something she wanted just for herself? A piece of clothing, a trip, a better job, she can't remember now.
What she does remember is what she feared even then, that Bruce wouldn't last through the marriage, that she'd be left alone with three young children. There were always clues she was good at ignoring. Things can turn on you when you refuse to pay attention.
She undresses, the purple dress a dark puddle on the floor. Then lowers herself in the bath, slowly stretches out. She reaches for the wineglass. No use letting good stuff go to waste. Faint voices reach her from the always-too-loud TV, sounds that will follow her into the den where she sleeps on the pullout couch. Where she gazes at the painting of three little girls in a field of daisies, cherubs with smiling faces lit by the sun.
The phone rings. No doubt it's Patti. Bruce won't pick up. She'll ask Patti for two days' work at the bakery, save enough for a vacation. When's the last time she left home? Bruce will have to fend for himself. He'll never get out of bed. He'll eat a bunch of crap, gain more weight. He'll develop heart problems. She'll return to find him in the ICU with just enough time left for her to say sweet nothings before his eyes close forever. The water tickles the back of her neck. Her body relaxes despite her thoughts as the bubbles quietly disappear.
5
In the Silence
Wiping his hands on the damp towel, Nick peers over the divider. Amazing the junk customers leave behind. Ava's clearing tables with the haste of someone stalking free time. She seems more energetic since that lying-faced guy vanished. Bursting past the kitchen door, she deposits a load of trays and turns to go.
"How's your son?" He can do better than that.
"Frisky. Girls are different. How's Glory?"
"She's looking for a job, not sure about college, moping, sharing little."
"Yeah . . . well . . . give her time."
"As much as she wants."
"Nice beard. I'm going to get the newspapers."
How about a drink? What's so hard about that?
His ear picks up the incessant drip of the sink tap. He'll fix it tomorrow. Charge Murray a plumber's fee. Yeah, right. Bruce shuffles in through the back door, an hour late. A man so worn he makes Nick feel chipper. Bruce wasn't always this way. He used to pay attention to whatever crossed his path, kept an eye on unsavory possibilities. He would think nothing of taking a heavy bin of dirty dishes out of anyone's hands. A helper. Now, well . . .
"What's happening?" he gathers his gear.
"Nothing." Bruce speaks even slower than he moves.
He ought to stay a minute and make conversation. They're buddies, sort of, or would be if they were on a desert island together. Instead he heads out the back door.
Except for Ava and Bruce, he has little to do with fellow employees. His run-in with Murray still tastes sour. He tried to enlighten him about the war. Murray insisted in that loud voice of his, the boys over there are saving New York from another attack. Why did he bother? No matter the facts, the man has an opinion about everything. Yesterday it was Nick's beard, but he's not about to conform to some cockamamy dress code to work in a kitchen.
• • •
Early morning driving. He loves it—no traffic, Glory still asleep. She'd better be. A girl of eighteen isn't always where you want her. He's headed toward Jones Beach, his usual stop before home. No one around but a few male shapes sprawled on the sand. From their garb, he'd guess they have little reason to wake up. The boardwalk is shuttered, light splattering the dark horizon like a cracked egg. He locates his bench facing the ocean. If it weren't for Glory, he'd stretch out here, count a few sheep. But she'd know he didn't come home.
He breathes in the sea air, conjures up his usual vision: China Beach, nurses frolicking in the water. Joyce, his big-boned mate who wasn't yet his wife. Inhaling those weed killers took her lungs as sure as any soldier's. At the end she moaned because anything louder would've put him in a box beside her. One tough lady who kept his mind focused, his body functioning, which was no easy feat. She had a mantra: life's a jigsaw puzzle.
• • •
When he peeks into Glory's room, the computer screen is silver bright stars into the stratosphere. Asleep, her curly red hair is bright against the white pillow, one freckled cheek hidden. He tiptoes out and leaves the door open, the signal he's home. In the four years since Joyce died he and Glory have worked out a routine of sorts broken by his occasional bad days. Which reminds him . . . he follows the worn runner to the bathroom where an array of pill bottles summon him. A handful each morning to ease blood pressure, scare away migraines, lower the decibel of voices. A tiny aspirin to keep him alive. It's a joke.
Glory comes up behind him. "Skip one and Mom will clobber you."
"I hate these damn things, get stuck in my throat, turn my piss orange. Jesus, even if I want to forget mortality, I can't."
"Some people are deaf, blind, and paralyzed, but they still manage to smile if it's sunny out."
"I smile."
"When?"
"You mean that?"
"Totally."
He gazes at her, trying to remember where it was Joyce offered him ten dollars to laugh.
"I need to brush my teeth, and other personal things. You going to be long in here?" she asks.
"I'm out of here to sleep. What about you?"
"What worthy tasks am I about to undertake for the day?"
"Yeah, something like that."
"How about job interview. Or a college visit?"
"Sounds good."
"Well, truthfully, that's not it. I've been meeting with people who started a local antiwar collective. I bumped into them online a few weeks ago. They're very interesting. We meet in one of their houses, not far from here. But after, I really do have a job interview at IHOP."
He groans.
"It's just temporary, till I figure something or everything or a little bit of everything out." She kisses his cheek, making a loud sucking sound the way she did as a kid.
• • •
For reasons he can't locate, sleep eludes him. Is it the start of one of his bad days? He's tired but not edgy, his mind blank, not whispery. He imagines Ava beside him, her lovely hair unpinned, flowing. His affairs have been brief, itch-fulfilling but nothing to ruminate about, no one to bring home. Anyway with Glory here, it's not particularly lonely.
• • •
He opens his eyes, the room is hot, his mouth dry. A dreamless sleep, thank god. Hoisting himself off the bed, he stares at his feet, but remembering depresses him. The old wall clock reads 6:20. Yesterday he conked out for two or three hours, today ten. His whole life lacks regulation. Glory's in the kitchen noisily preparing dinner. The girl can't cook but whatever she serves, he eats.
After a shower, he pads to the kitchen; the table set for two.
Glory seems nervous, excited even, as if she can barely contain news.
"What is it?" he asks.
She studies him for a second. "Let's eat first."
Tension grips his body. "I don't think so."
"The group I told you about, the one I spent the afternoon with, they're amazing. They're part of an organization that's global. People from lots of different countries go together to become witnesses for peace in the Middle East. It's a way to stop the killing and the torture, to show the rest of the world all the evils that are going on so we can eventually stop them." She's nearly breathless, her eyes shining.
"Out of the question."
"Dad, I don't need your permission. I thought I'd have your approval. You believe in peace."
"I'm not sacrificing you for it."
"I'm not going to die. Honestly, dad . . ." She gestures impatiently.
"I can't process this now . . ." His hand flies up to swat it away. The colors of a migraine seem close.
Her light blue eyes take him in. Her translucent skin tight around that tiny nose, her rosebud mouth. She's barefoot in jeans and a T-shirt, her costume of the month. But there's nothing petite about her broad shoulders, her sturdy body, as solid and shapely as Joyce's ever was.
"Later, then," her tone upbeat. She's handling him.
After a one-two, hurry-up dinner of pass-the-salt talk, he drives to the beach. Sits in the parking lot, a blanket of darkness about to fall. Even if he demands Glory not go, she'll do what she thinks is right. It's how kids her age navigate the world, impulsively. It's his fault and Joyce's too. They brought her up to believe everything is possible. Glory standing in some no man's land between tanks? It's insane, suicidal. Shit.
A cluster of young drinkers saunter by, beer cans in hand; their voices compete with the buzz in his head. He pulls out his cell phone.
"Ava, it's Nick. I'm tending a headache. Can you cover till I get there?"
"Sure. You take it easy," her voice curious but restrained.
He could ask what she'd do in his shoes but hangs up instead. Her input might increase his anxiety and he doesn't have any pills for that.
He inhales sharply, warns himself not to drive fast. Useless.
• • •
It's late, but Glory is staring at the computer like it has the last word. Something in him wants to smash the screen.
"Hi sweets. We should settle the deal now." Otherwise it's farewell to sleep and work and sanity.
"Okay." She presses a few buttons and the screen is back to saving stars. "Have a seat. You're looming."
He sits on the bed. She rolls her desk chair around to face him. Only one lamp is on and the dresser fan whirs annoyingly. It's been hot since the last snow melted. He offered her a small A/C but she said it would pollute the air.
"I'm not going to say witness for whatever is a bad thing. Except there's so much turmoil in that part of the world . . ."
"But Dad . . ."
"Wait."
She leans back, stretches out her long legs.
"Your values are no different than mine or your mother's. But neither of us would put ourselves in jeopardy."
"But Dad . . ."
"There are safer ways to protest. In the chaos of Gaza, the West Bank, or wherever the fuck . . . you'll be an ant, a tiny bump, an annoyance, and nothing more."
Her expression closes down. Her eyes lower, her mouth tightens. "Look, I'm going. I need you not to worry. It's three months. I'll e-mail as often as I can."
He begins pacing, wonders if he should threaten her, but how? "Glory, you just met these guys. I'm sure they're sincere, but Christ, who the hell are they? I mean what makes them witnesses? And the money, where's it coming from?"
"Rich people, older people, people who agree with us but can't participate physically."
"Wonderful. They remain in their cream-colored houses while you sleep in the sand."
"Dad . . . please . . . stop pacing, you're not in an emergency room. I need to be involved with something bigger than me," she explains as if he were the child.
"I don't want you going." He's trying to keep the volume down.
"Dad, even if I chose this for pleasure, or out of curiosity . . ."
"I don't want you going." He'll never budge.
". . . It's my life, my time to do such things."
He stares at his daughter, at the band of freckles crossing her forehead, the clarity of her eyes, the strong jaw, the length of her fingers, and realizes he's memorizing her.
• • •
In the silence of her room the computer stands as a gateway to the rest of his life. Her messages were coming two a day, one every two days, once a three-day interval, but it's been six days. Jesus, God, Crap! How's he supposed to survive that? Rereading previous e-mails, which he never deletes, the information is lodged in his brain. She arrived; she's fine; it's too hot. She needs another couple of pairs of shorts; don't send them, no real address. She's bunking in a camp-like situation, electricity on and off. The beauty can feel like an insult: clay-colored dunes, sky so blue it hurts her eyes, stars so bright they light up the night. The misery, the poverty . . . the kids are killing her; hopeless, depressed by age eight. Her last e-mail bragged she could say words in Arabic, some in Hebrew. Tan, Dad, she wrote, I'm so tan, it's amazing.
He picks up the bottle, takes a long swig of bourbon. It'll be a month tomorrow. Two more to go. He's crashing in Glory's bed now. Without the sleep pills, dreams and crawly, creepy stuff wake him every hour or two. Even now his feet feel itchy. He peels off his socks and checks for fungus. Clean.
Then he remembers . . . begins rummaging through the desk drawer, slides a piece of paper out from under others; three names: Josh Towns, Emanuel Walker, Robert Messenger. He takes another long drink from the diminishing contents, finds himself on the faded couch in the living room punching in the first number. It rings until voicemail picks up. He leaves a message. He dials Walker's number.
"Hi." Sounds like a child.
"Is your Mom or Dad there?"
"Who should I say is calling?" Her little voice is prissy.
Who indeed? "A friend of . . ." —and here he takes a chance— "your brother's."
"I don't have a brother."
"Your father's Emanuel Walker?" The decibels rise.
"I can't say."
"Let me talk to your mom, now, please." Jesus! Damn, the bourbon's still in Glory's room.
"Hello." It's a voice so dull he wants to hang up.
He introduces himself. ". . . and I haven't had an e-mail in nearly a week, so I wondered . . ."
"My husband and I . . . we're not communicating much. If I hear anything . . . leave me your number."
He does, but she'll never call.
Staring at the third name, he decides he needs hope; he'll try it tomorrow.
• • •
Two groups of noisy young people take over several diner booths. It's the middle of the night and they're ready to eat everything including the inventory. He works the orders nonstop. A few strays come in as well, probably because there's nowhere else to go. The kitchen is steamy; he's sweating. He stares at a chit but the letters scramble, so he blinks a few times, tries again, but the words still blur. He cups some cold water in his hands and splashes his face, tamps it dry with a paper towel and tries deciphering it again. Better. Except there's a pull in his stomach even though he visited the bathroom minutes ago. "Ava," he stage-whispers over the divider.
She turns her perfectly shaped head to look at him and hurries into the kitchen.
"Have to go home . . . need to check the machine . . . I've been away for hours . . . haven't had an e-mail in days . . ." his voice trails off.
"That doesn't mean a thing, Nick. It's not New York. It can't be easy to find a computer."
"All the orders are done. I'll be back. Can you cover?" Before she can answer he heads out. Great, she thinks he's crazy. Maybe he is. So what? Crazy or sane isn't the issue. Something happens to Glory, he's fucked. In the parking lot, he thumbs down hard on the car keypad; damn door won't unlock. The car begins beeping loudly. Shit, shit. Take a breath. Try again. The noise is deranging. Someone help. Did he say that out loud, because Ava's running out the back?
He hands her the device. She makes the beeping stop, bless her beautiful soul. She's had her tragedy, losing a husband, but it was a long time ago. She can't still be mourning. "How about going for a decent dinner tonight before our shift? On me." See what happens when drink collides with crisis?
• • •
Sunlight brightens the adjacent wall, illuminating the nearly empty bottle. Across the road is a house like his, except it's yellow with blue trim. Inside there is an intact family, a husband, wife, two children, very American. His house is white with green trim chosen by Glory before she was old enough to determine her life. Glory would not want him going to the State Department for help. Then again, the State Department wouldn't be sympathetic. Is he going to sit in front of the screen all day? Ava walks into his head. She's home getting some shut-eye. He wouldn't mind having an afternoon drink with her. But what would that mean for dinner tonight?
A distant ambulance siren penetrates the silence. Someone's life is about to change. It's what he thought as he lay in a makeshift hospital tent listening to the docs talk about amputating his infected foot.
He drags himself to the living room and tries Messenger's number.
"Hello," a woman's annoyed voice. Does she understand where her son has gone?
"I'm a friend of Robert's—"
"I clean. He's not back for months. You call later."
But he won't call later, and he can't sit in front of the computer another minute without losing it.
• • •
In the car in front of Ava's house, he tries to talk himself out of what he's about to do. Fails. He rings the bell with short staccato stabs.
She opens the door, a queen in a purple robe.
"I'm freaking out."
She leads him to the living room where the furniture looks almost as old as his, and he thinks to console her with this observation but finds himself unable to speak. She's pushing something that smells like whiskey under his nose. Should he drink or sniff it? He drinks.
"What happened?" She stands arms crossed, wavy hair falling past her delicate shoulders; her light eyes weary but concerned.
"I could go to the State Department except they'd probably arrest me. It's a lot of money to fly over, and I don't even know where she is exactly. It's stupid of me to barge in like this, but—"
"We're friends, right?" She sits beside him, her clean soapy scent instantly calming. "If something bad happened, you'd be contacted. You're next of kin." She sounds certain. Except his stories don't have happy endings.
"She was e-mailing, so what changed? The thing is, Glory's never on the same computer. If I'm at the machine, I reply instantly. I told her I have to hear from her."
"Children," she sighs. "Have you slept?"
"Couple hours. I should go home, but I'm too wrecked to drive." Is that true? Does she believe him? Her face reveals nothing.
"Do you want the couch for a while?" He wants to close his hand around her long, thin fingers.
"Sure," but to his surprise he follows his body to her bedroom. He takes in the yellow daisy wallpaper, the yellow lampshades, sunshine in the dark. A portrait of her husband is on the wall, the face gigantic. He recognizes the R&R quality, charcoaled too quickly in some godforsaken place; and wants to ask if it's how her husband really looked but decides not to.
The bed is queen-size, the pillows numerous, waiting. He slips out of his shoes, turns down the yellow-and-green-flowered cover. He stretches out. He's lost it.
When she slides in beside him, he remembers hearing about men so exhausted they begin dreaming before their eyes close. But once more he inhales the sweet scent of her.
"Who are you? Why do you care, or something along those lines?" he asks.
"Who says I care. You're suffering."
"It's a motherish thing?"
"You're older than me, I can't be your mother."
"So?"
"It's probably a mistake, but I need . . . I want to be adventurous, a little."
He palms the cool smoothness of her cheek. "You're wonderful, hauntingly . . ." But a weariness he doesn't want shoots lead through his body. "I can't believe this," he mumbles—not that she wouldn't notice his limp prick. He strokes her hair, the silky strands tangle between his fingers, then subsides like a ship in harbor.
• • •
Opening his eyes at an unfamiliar ceiling, the shades drawn, he remembers. Is he mortified or contrite? She's not beside him. Should he call her name? Her husband stares at him without affection.
He walks into the living room. The TV is on, the volume low. She's dressed in a long black skirt, silky, with a white blouse open deep at the neck.
"Drink before dinner?" she asks.
"Dinner?"
"It's nearly six."
"You're kidding."
"No," she says so seriously he's embarrassed.
"Where's your son?"
"Helping Dina next door. I told him you collapsed, and why. He offered to look online and see what he could find out about Witnesses for Peace."
"You're extraordinary." He drops on the couch beside her. Slides an arm around her slim shoulders, dips his chin in her soft hair, done up in some fancy knot. His fingers wander inside her blouse, find her velvety breast . . . With the heat of her throat against his lips, he cradles her head, maneuvers her legs onto the couch; she curls her body to make room for his. He thinks to say a few lovely words, but her eyes are closed, her limbs wrapping his. He enters a land where only distraction and satisfaction exist.
• • •
He watches her attempt to organize the mess he's made of her outfit. "I could offer to have it cleaned, but it'll only happen again."
"Why was that so exciting?" She sounds genuinely surprised.
"Unexpected love. It's the best kind."
"How would you know?" She searches his face.
"I wouldn't."
"Is dinner still happening?" she asks, but he can sense her withdrawing.
"Tonight and tomorrow, if you want?" He means it.
"Bobby's not always going to be conveniently busy."
"Let's take him with us tomorrow."
"No."
"He knows me from the diner, remember?"
"I'm going to change."
• • •
His time with Ava yesterday gave him courage. His car glides into a space beneath a huge tree, which he wants to identify as oak, but he wouldn't know. Hidden by afternoon shadows, they can't see him. Then again these people don't gaze out windows; they have security do that. He stares at the estate. If the house were any nearer the water it would float. The last time he saw this much property he was mustering out of the Marines. But the base was a flat, ugly, brown expanse, dotted with huts that passed for barracks. That anything this lush exists a mere twenty miles from the cheek-by-jowl places he sees daily is staggering.
He replays the voice message in his head. "Come by, we're home all day on Thursdays." What can he possibly say to the rich and famous? They probably own a piece of the Middle East. If someone opens the door, he'll be chased away. A bearded man dressed like a panhandler? He's not wearing his good shoes. Find a bar in town, two drinks to dissolve the edge, then return here.
Finding his cell phone, he calls Ava. Probably napping. Well he needs to talk to her. As the machine picks up he hears her real but tired voice.
"I'm parked outside a mansion. It's me, Nick."
"Yes?" Her plaintive voice, so girl-like, encourages him.
"Parents of one of Glory's friends invited me for tea."
"And?"
"I'm not dressed for it."
"I see . . . You mean your clothes are dirty?"
"Not exactly." He eyes the beige pants and blue shirt.
"Holes in your shoes?"
"Why does everyone focus on shoes?"
"You want to make a particular impression?" Is she being sarcastic?
"I'm Glory's dad."
"Could be they're idiots."
"Not with this estate."
"Nick, who cares? They have information. You'd walk through fire for Glory, so?"
"I didn't mean to wake you."
"Yeah well, it's becoming a habit."
"I wish you were with me."
"You'd better go in."
He clicks off. He does wish she were with him, though it makes no sense. What makes even less sense is the way she's invading his thoughts. These flings work a certain way. Sleep together a few times, then she discovers his faults. Many. She begins to make excuses. Too busy to meet . . . not good for employees to fraternize . . . If Murray catches us . . . Murray? Who cares about Murray? His mind is bending, that's what. Glory's killing him. In his state how can he think about Ava?
Yet her non-dismissive voice helps him out of the car. He lopes up a driveway to what he hopes is the front door, feels the first drops of rain. He discovers a brass knocker. Is that an ornament or real? No bell. He uses the knocker. A maid with brown satin skin, dressed in old movie-style black and white, opens the door. She can't like her costume. So soft-spoken he can barely catch the words. It's his name she wants and he tells her. Then he waits in a vestibule larger than his living room.
The maid ushers him into a den or library or maybe it's a living room. Mr. and Mrs. Towns stand together in front of a fireplace. She's tall, thin, wearing black slacks, a gray blouse. In her early fifties, he'd say. Her husband, too, is tall, thin, in his fifties. He decides Mr. Towns in khakis and polo shirt just returned from playing golf. They look like brother and sister.
When the maid leaves the room he feels alone.
"Please have a seat." Mr. Towns takes the wing chair facing the couch, his wife the chair next to his.
Nick sinks into a leather sofa so soft he may never be able to get up.
"We haven't heard from our son recently. No doubt they're all on some exciting outdoor mission, traveling who knows where." Mrs. Towns seems charmed by the idea.
Maybe he should enlighten her about the outdoors there. Trying to sound matter-of-fact, he says, "When last did you have an e-mail?"
"Josh phoned us ten days ago, right dear?" the husband asks but doesn't wait for an answer. "We spoke a very short time. We agreed, well, he's so involved he can't think about keeping us up to date. At any rate, his voice was gleeful." He reaches over to touch his wife's arm.
Gleeful? How does that word apply to the Middle East? Would Josh tell his father if one of his friends were kidnapped? Wounded? Killed?
The maid brings in a tray with tea and a square plate with flat white crackers that look as if they'd crumble at the touch. He wants bourbon.
"Mr. Towns is certain everything's fine. Josh is our youngest, more of a challenge than Chad or Douglas. Eighteen and already curious about complicated places."
Mr. Towns? Complicated places? What language is this? Mrs. Towns hands him an empty teacup. Christ, he doesn't want to deal with tea stuff.
"If we don't hear from our son in the next two weeks—" Mr. Towns pours himself tea.
Two weeks? He'll be certifiable.
"—Martin will take a look?"
"Martin?" He says, relieved to speak.
"Mr. Town's business associate is somewhere near Saudi Arabia, certainly closer than we are." She gives him a crisp smile. "Martin will investigate, discreetly, make sure all is as it should be. You know how children are? They become so invested in the moment they can't remember to contact us. We'll let you know either way."
Either way? They don't sound a bit worried. Why should they? Their problems are delegated, resolved by others. The green slime of envy fills his veins. Not the mansion or the money, though a little of the latter would get him on a plane. It's Mr. and Mrs. working in tandem, reassuring, affirming each other, refusing disaster.
Carefully, he replaces the unused cup on the tray. With muscles he didn't know he had, he hoists himself off the sofa. "Thanks so much. If I receive a message from Glory, I'll let you know." Do they care? So he adds, "And yes, I'd appreciate hearing anything you learn."
• • •
"I've never met people like them, cool, not as in hip, as in distant. Not snobby . . . almost innocent. Of course Josh's okay. I could've pointed out the guns and tanks, but why do that?"
Ava looks at him tenderly the way she does when they're alone together, except they're seated at a deeply scarred wooden table in the rear of Sully's bar. Far from fancy, the pub has the right amount of noise at this early evening hour to set problems aside, albeit with a little help from booze. The beer is cheap, the wine not cold enough, but they've decided on bourbon and Coke.
"Tell me everything." Her long fingers splayed on the table, no wedding band, only a small reddish gem embedded in a silver ring. He wonders if it's her birthstone. He folds her hand in his.
"His wife calls him 'Mr. Towns.' Jesus, what does she call him in bed? They were a pair of . . . I don't know"—and he doesn't although the experience is still fresh in his mind—"birds squawking a language I couldn't grasp."
"Rich and secretive, I've read about people like them. She cries in bed without making a sound. He drinks too much and wouldn't hear anyway."
"Even if the Towns worry in private, they aren't fixed on disaster. Otherwise Josh would be on a plane home tonight, Martin no doubt the escort. Are they stupid, can't they see the misery there, or do they know something I don't?" He downs the last of the drink.
"Did they manage to reassure you at all?"
"They're in la-la land."
"And you?" Her eyes still on him. He's starting to like being watched, at least by her.
"Me? I'm an ex-marine; reality's my specialty." He wants to tell her how lovely she looks, that her red blouse is exciting, but she's focused on the details of his day. And how often does that happen?
"You get in your own way," she says a bit too soberly, as if she is about to reveal some gruesome truth about him.
"I'll get us refills."
At the bar, he catches the bartender's eye, holds up the empty glasses. Who else in this place is agonizing over a son or daughter? He studies a few faces that reveal nothing but lines. Is it that he can't bear being apart from Glory? He understands the order of things. It's time for her to leave him, but not gunned down or kidnapped by some lunatic. "It's a worthy deed, Dad, better than serving pancakes. There's something big at stake here." It's what she'd said, and not just once.
With a cold glass in each hand, he studies the very real woman waiting for him, has an urge to say thank you, but she won't know what he's talking about.
Setting down the drinks, he asks, "My place or yours, as the saying goes. We have two hours before our shift."
"Yours . . . but—"
Is this the first of the excuses? He waits.
"After, I want to spend time alone with Bobby at home, arrive at the diner in my car."
"I couldn't agree more."
• • •
Sheets of rain splash against the sides of the car, hard pellets of hail drum the roof, the windows fogging. It feels like he's inside a plastic bag. The meteorologist drones on about high meeting slow-moving front, and sounds baffled. He switches off the radio. Only a few miles from his house to the diner but the deepening puddles force the car to a crawl.
The imprint of Ava's slim, strong body still warms him. They made a decision together. If there's no e-mail from Glory in the next three days, he'll phone the State Department, find out the protocol for locating a missing person. Not a solution but it's something—waiting is driving him under. He's on the two-a-day antidepressant now, his mouth dry, nightmares, and worse, sometimes he can't get it up. Thank god that didn't happen this evening.
Is it mist circling a tree? An apparition? He can just make out a figure walking the edge of the road. He swipes the side window with his forearm, cracks it an inch. Peers out. Christ, what's he doing? "Hey," he calls, sliding the window all the way down, water splashing his face. "Bruce!" He stops the car. "Get in, for god's sake."
With a black woolen cap pulled low on his forehead, his round, dark eyes blazing, Bruce stares like he doesn't know him. Water drips off his nose, chin, everywhere. "Get in," he orders, leaning over to open the passenger door. Bruce delivers his bulk.
"Where are you going?"
"That way." Bruce points in the opposite direction.
He's wearing an army jacket, backpack in his lap now. "What, you running away?" He tries for the smile it's impossible to get from Bruce on a good day, and this isn't one of them, of that he's certain.
"Need to get to the National Guard office."
"It's not open this time of night."
"That's what I want."
"Bruce? What the fuck?" Silently, he ticks off possibilities: flashback, sleepwalk, hallucination. Does he want to know?
"National means in the US," Bruce spits out each word as if it's laced with poison.
Drive him home. Let Shelly deal with him.
"I'm going to set fire to the office." Bruce pats the backpack.
"Yeah, well . . . not a great idea with all the water coming down, is it?"
Suddenly he feels nauseous.
"Take me there now." Bruce sounds determined, and worse, he's wearing that mind-blanking expression that hears no reason.
"It won't be raining tomorrow night. If you still want to do this, I'll help you. It's always easier with two, remember?"
"They lifted him out of his life like a stuffed animal."
"Who?"
"They took my boy."
"Michael's in Iraq?"
"Just a baby. Sent him to be killed." He's seen broken and it looks like Bruce.
"Michael's in Iraq?" he asks again.
Bruce doesn't answer. So he puts the car in gear, begins driving him home.
• • •
He nearly walks past the hospital. A small gray brick building with three floors that could be anyone's house, except for the barred windows. He drove here straight off his shift. A few hours' sleep might've helped. His head feels spacey; he's not too thrilled with his balance either. Damn pills. Ava offered to drive him to Manhattan—he was touched. But he can think of better places to take her. The front door is the first of several leading to reception, the architect offering a change of mind at any turn. A baby-faced receptionist who has to be in her fifties gives him a paranoid stare before releasing information. Then she tells him outer doors remain unlocked only during visiting hours. In short, he'd better watch the time.
Shelly's in one of the dayroom's orange plastic chairs, a handsome woman with enough energy to run a country. Now she's thinner than ever; dark pouches beneath her usually curious eyes. It seems as if she's crying without tears. She offers up a ragged face and he pecks a quick kiss.
"I got your message. What happened?" he asks. A man sidles along the wall. Three women stretch their necks to watch a mounted TV. He can't figure if they're in or out. Again, he spots the barred window. The tic on his eyelid is back.
"After you dropped him off, wetter than a seal, right, I got him to bed. Wouldn't get out next morning. Kept saying, I don't care . . . isn't worth it . . . what's the difference . . . things like that. Had to get my oldest to help bring him here. The psychiatrist will evaluate him for seventy-two hours. They mentioned shock treatments. I said, wait. Bruce said it wouldn't make a difference. It's as if he made a conscious decision to stop caring—about anyone. Michael, our baby, is in Iraq, you know, but the way Bruce talks, it's not our son but himself he's seeing, young soldier that he was. The memories of then filling him now, god knows what it is he fears."
"They'll feed Bruce antidepressants. They work." He wonders if he should just lie down like Bruce.
"Yeah, hope so. Do you think Murray will keep Bruce's shift open till he's back on his feet?"
He doubts it, but nods because the desperation in her voice alarms him.
"I've brought up three children. A kitchen holds no surprises. I could work two days of his shift till he returns. Would you put in a word?"
"Sure." Murray won't allow her in his overmanaged kitchen, though he'd welcome Shelly's help.
"He's down the hall, last room on the right. I came out here for a break. Wanted my first cigarette in years, but you can't smoke anywhere. Damn."
• • •
The hallway is too long, too narrow. He passes a man in a robe talking to something invisible in his hand. The guy reminds him of the time Glory asked him to serve Thanksgiving dinner at a shelter. She found the experience uplifting. The scene depressed him for days. Now, too, the wish to turn around and depart is strong.
No bigger than a walk-in closet with one small barred window and a twin-size bed, which is way too small for Bruce, who's curled up facing the wall.
"Bruce, hey. It's Nick." The pajama-clad backside and bare feet scare him.
"Hey," the voice barely audible.
"So . . . how do you feel?"
Nothing.
"We all go through these dark patches . . . a couple days, you're up, better than new."
Bruce shifts around slowly to give him a who-are-you-kidding look. His face is pale, waxy, lips in permanent frown. "I'm not in the mood for chat."
"I thought you'd have a couple of words for me." He glances out the window at a small square of gray sky. Could they make these places any more discouraging?
"Nick, go home."
"Yeah, in a minute."
Bruce closes his eyes, which is a relief.
He pulls a chair from the wall to the bedside. A few more words, then he's out of here. "I know you're worried about Michael. His tour will end and—"
"He could be dead right now." The sharp words stab his gut. He takes a deep breath.
"You and me, we made it out. So will he."
"You don't know that. Horrible things happen there every day."
"You have two other kids waiting for you to pull yourself together."
"Their lives are their own," his voice barely audible.
"You mean if they're not in danger you don't care about them?" His own voice louder than necessary.
"Something like that."
He flashes on three-year-old Glory riding his shoulders. Each time they reached a doorway, she'd yell, "duck." And he'd quack. She thought that was hilarious.
"My daughter's in the Middle East."
Nothing.
"She's a witness . . . for peace." Sounds ridiculous, like some religious calling. Even with Ava he doesn't talk about what Glory's actually doing there. Does he know? He wants to believe in her bravery, her expectations, but how can something good hurt him so much?
"She wants to make a difference," he continues, but this too sounds stupid. It doesn't matter. Bruce isn't listening, his eyes still closed, his face expressionless. He could be dead, but he's alive. No mistaking the stone finality of dead: it's the first thing that hits, even before the smells.
"I'll see you soon. Try and get it together." But even as he says this he wonders if Bruce is finished trying. He wants to give his arm a brotherly pat but is afraid to touch him, a man stalked by doom. He heads out but doesn't stop to speak to Shelly, because what can he say?
The receptionist eyes him suspiciously. "Forget it," he calls over his shoulder. "I'm not moving in."
• • •
It's a few streets to reach his car. He walks quickly, Bruce heavy in his head, the hospital's sour smell with him as well. Not a hint of sun, the air thick, punishing. Heat and traffic noise follow him. He could do with a cold beer. The whole world sucks is what.
Triborough Bridge or Midtown Tunnel, which one to take? He's already having trouble breathing, being underwater won't help. Before he can stick the key in the ignition, Bruce's open-eyed face appears in the windshield. Not a good sign. He switches on the radio. Bruce's face refuses to disappear. He fiddles with the dial to get conversation. Bruce stares at him.
He shuts his eyes. It's a visitation, a warning, the kind Scrooge received. Only it isn't about time past, it's about now, maybe the future. "Anything can happen to anyone anywhere, we know that, man." He's talking out loud. To Bruce. That's crazy. But he can't stop himself. "Our kids want to live. They'll take care same as we did. You and me, Bruce, we share the doom, but that's it, man. Things are changing for me. I'm beginning to get a life here."
He opens his eyes: the grimy windshield holds only a vision of a narrow street of small shops, garbage cans along the curb. Taxis whiz through changing lights. People rush by. Destination is all. He has one, too. He's taking Ava to a late afternoon movie. When was the last time he'd done that? The film could make Ava late for her shift. Rosalyn will stay an extra hour, friend that she is.
The voice on the radio sounds serious. He ratchets up the volume. The man's selling prepaid funerals. He laughs.
6
Butter and Ketchup
Getting out of the shower, she hears the phone, grabs the towel robe, and hurries to the living room.
"Yes? Hello?" A little breathless.
"Dina? It's Rosalyn. You sound funny."
"It's unusual to get a call this early." Actually, phone calls alarm her whatever the time.
"I knew you'd be leaving for Ava's and—"
"Yes, well, what is it?"
"You do sound strange."
"An incident in the shower. It's nothing."
"Did you fall?"
"I didn't." A woman enters her sixties and it's the first question.
"I have to get a pair of shoes for a wedding. I hate going to the mall alone. Do you need anything?"
Since leaving her job she buys only essentials. "No, but I'll keep you company."
"Meet me in front of Baker's shoe store at ten-thirty."
"Okay." An ER nurse, an ICU supervisor, a world within which she functioned for years at high speed, now she is a woman with time on her hands.
Returning to the steamy bathroom—her mug of coffee cooling on the rim of the sink—she stands for a moment remembering. It was nothing. But there's a tug at her insides, not a stomach problem. No, it's a tug of panic, the second one this morning. The first was brought on by seeing the new rubber mat in her tub, which she placed there last night. It's a surprising state, getting older, the limitations, bodily insults, odd sense of both urgency and mortality. But no one can stop the process. The fading beauty thing bothers her least. When she peers in the mirror, the face of yesteryear still meets her eyes. What she can't hold onto is that step-lightly kind of go. The way Ava wills her limber body to comply without complaint, Mila's seemingly never-ending energy, Rosalyn's jaunty step with no thought of tripping.
The clock on the shelf tells her she has thirty minutes to wake Bobby. Her black slacks and pink blouse hang outside the closet door. Pancakes, Bobby loves them. Now that she has time, he's growing up and soon won't need her. Caring for Bobby—so different than her son—is never a chore. Tim insisted on attention. As soon as she stepped through the door he was all chatter and need. Maybe if her husband had lived . . . her son was so young . . . but who knows? She saw Tim during the last snowstorm. He arrived wearing sneakers. She offered to buy him boots and he wanted cash. She gave him what she could. He stayed less than an hour. She was relieved to see him go, something she can barely admit to herself.
Taking a sweater, though it's a strangely hot spring, she checks for her car and house keys, confirms the toaster and coffeemaker are unplugged.
• • •
Driving to the mall from Ava's house, Bobby's on her mind. He was quiet during breakfast, unusual for him. Moodiness is a given at his age. Maybe he's upset about the kitchen guy his mother is dating, not that he'd say so. He's double-digits now, things are happening to his body. Would he talk to her about it? She passes a row of renovated houses with new roofs, landscaped lawns. Beginnings.
Pulling into a parking space, she notices the indoor mall is bustling. Shops, restaurants, and offices occupy two tiers that circle up and around. People dressed for work hurry by, reminding her she's a lady of leisure. Not quite. Still, her recent scheduled-by-the-minute life is done with. The alarm clock is no longer set, but she wakes early anyway. She bought lots of plants, a tomato box she tends daily. There must be more to retirement. Maybe she'll buy a book about it, though she doesn't believe in experts.
Rosalyn waves, a gremlin all wire and vigor, jeans and a short-sleeved shirt like it's already summer. Rosalyn's thick, dark hair frames a face that will always contain beauty. Some faces are like that but she'd never noticed before.
The shoe store is surprisingly crowded for a weekday. Balancing boxes, salesmen scurry back and forth, making her seasick. She finds a seat while Rosalyn studies the display shelves.
"I want to dance, so the heels can't be too high. On the other hand, I need fancy." Rosalyn holds up a black suede pump for her to see.
"Nice, try them." The salesmen interest her more. She searches for one who'd be around thirty, Tim's age. It wouldn't be a job he'd consider. He isn't a server.
Rosalyn, wearing two shoes with different-sized heels, limps across the carpeted floor to sit beside her. "Which one?"
"The left." Even if Tim took a job selling, she doubts he'd hold on to it past the first paycheck. And she remembers all those years ago when he first disappeared, she and the principal searching the empty classrooms. Bobby would never disappear that way.
"How well do you know Nick?" she asks.
"He's not much of a talker. Why?" Rosalyn slips on another pair of shoes, raising one leg to admire the fit.
"Ava's dating him."
"She's a big girl."
"A kitchen guy's not exactly a model for her son." Again she flashes on Tim, wonders if he's working anywhere.
"Model?" Rosalyn laughs a harsh sound. "A cushy job makes a noteworthy man, is that it?"
"A man Nick's age should have a more relevant position."
"What's relevant? Cop? Pencil pusher? Stockbroker? Like that guy Mark, the big business owner from Colorado?"
"Oh don't play that game. You know what I mean. I guess Ava's tired of being alone," she hears herself concede, though she isn't sure she believes it.
"Poor lonely Ava. Comes home from the diner and doesn't have to deal with a man's moods, criticisms, demands. Peace."
"A relationship is more than a list of problems."
"Companion, lover, hand-holding in the dark? How much of that is real, Dina?"
"How cynical," she says.
"It's experience."
"You've closed down, is what."
"With all due love and respect, my past isn't written on my face."
"There are truths in life," she insists.
"The trouble is they keep changing." Rosalyn slips off the shoes.
"You always have a quip."
"Sorry, but I can't sympathize with looking to a man to change life for the better. It's never that simple."
Rosalyn's words resonate, but something in her won't give in. "Of course not. One has to work at it, together." Is that what she did? Filling the few short hectic years of her marriage with all she wanted to accomplish: a new house, furniture, child. Then Howie dies, just like that, and she, too stunned to grieve.
"Dina. You've been a widow how long? Aren't you lonely? Why didn't you join Parents Without Partners like so many people around here? Have you slept with anyone since?"
She did have a brief affair with a kind man who sold medical supplies, but it was complicated. Having to build a relationship, meet Tim's needs, work a high-powered job. It was too much. She ended up wanting simplicity more than companionship. "My true love died," is all she says.
"I see." Rosalyn lines up four pairs of shoes.
"You don't see a thing."
"Are we arguing?"
"Of course not. We're just two women talking."
"I'm sharing, you're talking." Rosalyn gazes at the shoes.
"You're goading me."
"If you say so. Listen, Nick's a sweet, respectable guy who's done well as a single dad in these last years, even if he is too quiet."
"Ava talks to you about the relationship? She hasn't said a word to me." Last year she would've been too busy to notice.
"Well, you know. We share different things with different friends."
"How very kind of you to explain," she mumbles, with no attempt to hide the sarcasm.
Rosalyn glances at her. "I need your help. Please tell me which of these shoes I should buy."
She points to the suede pumps.
• • •
The sun has ducked behind a cloud revealing the grimy glass dome overhead. The first time Tim went missing, she covered every inch of the mall looking for him. The police were sure he'd return once he saw how miserable the streets could be. He did, but not for a month, a month in which she barely slept, traipsing the neighborhood peering into boys' faces. She blamed herself for his absence. But he didn't come home to stay. Money, he needed as much as she could offer. He cajoled, cried, swore he'd go to rehab. Now when he comes he always wants something from her. She began to pray he'd stay away. What kind of mother would do that?
"Dina, it's nearly twelve. Let's have a drink."
"And forfeit my free lunch at the diner?"
"Ava's not there. She's on full night shift now, though Murray feels no shame in shifting her hours whenever he wants. If I were Ava—"
"You're not. But it's weird, no one gives me a check anymore. I've become a fixture of sorts."
"Murray can afford to be generous. Sylvie's gone back to work, you know. Mila told me gleefully that Murray's not happy. Smart move, I say. A woman needs to have her own money. I said as much to Murray. He looked at me like it was my fault. Anyway, he's fond of Ava and knows what you do to help her."
"That's not the reason for the free meal."
"What then?"
"Older woman, invisible or stand-in for Mom. It's revolting."
"No one sees you that way."
"Not yet," she murmurs. "A drink it is. Where?"
Rosalyn turns her dazzling eyes in her direction. "I know a café." They cross the main floor of the mall, a buzz in the air like dying neon lights.
• • •
The café is blessedly quiet. They sit at a small round table near the window. As usual she takes in the ketchup in its easy-squeeze dispenser. Tim added butter and ketchup to everything he ate. It nauseated her. Sometimes he'd make a sandwich of the two ingredients. And she'd have to leave the room to contain her disgust. She wonders now if it indicated some chemical imbalance, perhaps a lack of potassium or sodium? Even as a nurse she'd never thought of it before. It was simply a stupid, even outrageous combination, the way children can pick out clothes that don't match.
The waiter slogs toward them. He seems exhausted, bloodshot eyes, swollen fingers, pasty skin—either a hangover or untreated diabetes. They order two glasses of wine, a grilled cheese sandwich for her, warm apple pie with ice cream for Rosalyn.
"Strange being served . . . I leave huge tips. Ruined by my profession." Rosalyn glances out the window.
"What did you do before being a waitress?"
"You don't want the list."
"I bet it's colorful." She's fond of this woman's spunky refusal to conform; fond, too, of their talks about anything and everything.
The waiter brings their wine, setting each glass down carefully. She notices the slight tremor in his hand, decides his symptoms are alcohol-related.
"To the good life," Rosalyn says, and takes a long drink.
"So?" she persists.
"File clerk for Revlon, very young . . . free makeup, boring, boring. Go-go dancer . . . had its moments, definitely more lucrative. Affiliated escort service and travel agency."
"Who did you escort?"
"Foreign visitors. Men." Rosalyn takes another big swallow, nearly emptying the glass.
"Exotic?"
Rosalyn gives her a half-smile. "Depends how you define the word. And you, always a nurse," but it's not a question.
"Yes, interesting but no spontaneity. The job was about order and control. The right dose, not just of medicine, but of time with patients. Everything doled out with the next task in mind."
"And grateful people? And the god-docs, they were a trip, I bet."
"True." So many years carrying out duties without making any major mistake. She wonders now whether that counts as success.
• • •
Her house sits between two identical small white clapboard structures with black trim, one belongs to Ava, the other to a family newly arrived from India. She sees a light in her upstairs room. Bobby has a key. Why did he lock the door? "Bobby?" She walks past the orderly kitchen to the living room. Why would he be upstairs? "Bobby," she calls again, and climbs the well-worn steps. Before reaching the top, Tim appears.
"I thought I heard you." His voice is deeper than she remembers.
"Oh my." Her hand presses her chest.
"Didn't mean to scare you. I have a key." But the smirk on his face doesn't reassure her. He looks awful, just awful: skinny as a pole, pale, too, shabby clothes, torn sneakers. Has he been sleeping in the streets?
"I put my gear in my room."
"Yes, good," and she turns to go back down because a sudden dizziness threatens her balance.
He follows her to the living room, drops into the club chair, his feet up on the chipped leather ottoman. "I'm in trouble, I need to hang out here. My partner's picking me up tomorrow."
"What kind of trouble?" Her jaw so tense a pain shoots up the side of her cheek.
"You'd be an accomplice if I told you."
"An accomplice? Tim, what have you done?" She's not shouting, but her voice echoes in her head, the way it sometimes does when she's at the beach treading water.
"Don't get wormy. Stay calm."
"Where have you been since I last saw you?" He seems tired, his eyes red-rimmed. But he's not high, which is something.
"Around. You're looking good, Ma. How's the job?" He reaches up, switches on the floor lamp. In the circle of light, his skin pulled tight over delicate bones has a bluish tinge. There's red in his dirty-blonde hair. Has he been in a sunny climate?
"I retired." That's a word she rarely uses. Left, finished, no more nursing, is her usual description.
"What do you do for money?"
"I have a pension. I get along. You needn't worry." His question, though, is self-serving, and a spark of anger ignites inside her.
He gazes at her with opaque eyes.
"What is it, Tim?"
"Remembering living here."
"Not as long as you could have." Does she want to rake up old ashes? Will there be anything new beneath? A difficult child who slept little, wouldn't play by himself, clung to her with such tenacity she froze.
"I was in your way," he says simply.
The boy knew. The boy felt her impatience. Be honest. Own up to it. But she can't. "Don't be ridiculous."
"Yeah, right, ridiculous, that's me."
"Are you hungry? I have chicken. I can order Chinese. Whatever you want."
"Any beer?"
"No."
"Call Ava, ask if she has any?"
"I'd rather not."
"That too much trouble?" A slow grin spreads across his face.
"I'm glad to see you. I'll cook something. I'll wash what you're wearing. What else can I do?"
"Nothing, Ma, nothing at all." But she doesn't believe him.
• • •
In the kitchen, taking the defrosted chicken from the fridge, she knows as if it's written on the wall that this time she's not to be spared. Well, okay, what more can happen? He'll want money. She'll go to the bank, take out a few hundred. He's her son, who else can she give it to? And she remembers that winter morning returning from Ava's, rushing him so she wouldn't be late for work. He became recalcitrant, moving ever more slowly. Finally, she told him she was leaving. He could get himself to school. "But I'll be late if I have to walk there." "Not my fault," she said, striding toward the door. "Ma," he called over and over, but she wouldn't turn around. That afternoon he disappeared. "I'd sure like a beer." He pokes his head in the kitchen.
"Go to the market."
"I'm hiding." His sullen words a cold fist in her belly.
She searches his face, the dark blue eyes with their long lashes, the beauty of them wasted like the rest of him. How could he be so stupid? "I'll go. I'll be back in a few minutes."
• • •
Once again she finds herself in the car heading toward the mall. Was it a bank? A robbery gone wrong, a teller wounded, blood on her son's hands, on his soul? Stop it, she tells herself. Until he shares what happened, she can't know. Years ago she made herself quit planning for his future, which hurt more than his demanding visits.
In the brightly lit market she hopes not to bump into Shelly, who works here. Any other time would be fine. Only two days ago they stood beside the colorful produce stands chatting. Shelly's a talker, said her youngest is in Iraq, which gives her nightmares. Poor Shelly.
She strides down the aisle, picks up a six-pack of Beck's and hurries to the cashier, no time to waste. Around her, people fill their carts as if today is just another day. She envies their indifference. Then consoles herself—no one really knows what goes on in another person's life.
• • •
The double-locked front door upsets her, makes her feel sneaky in her own house. He's right there waiting for a beer and follows her to the kitchen. She hands him one, puts the remaining bottles in the fridge. "Why not take a shower while I prepare dinner? Some of your clothes are in the dresser."
He twists off the cap, flips it in the sink. "Yeah. Good idea."
His narrow frame lopes easily out of the room. When he was little he'd curl up on her lap. The gentle weight of him against her breast, the grassy smell of his hair, imprints that never disappear. She begins breading the chicken the way he likes it.
Hearing the shower loud and certain, she switches on the small counter TV as she often does while preparing food. She surfs for news of robberies, murders, whatever. Nothing. Tomorrow's papers may enlighten her. Is that what she wants? Isn't it better not to know? A moment of uncertainty stills her: cook dinner, serve it, pretend everything's normal, then retreat to her bedroom. Or she could confront him. She takes a bottle of Beck's from the fridge and twists off the cap, the cold beer bitter in her throat.
• • •
He bounces down the stairs in a too-big pair of khakis and a faded black T-shirt she could've sworn she'd thrown away ages ago. His bare feet leave damp prints on the wood-slatted floor. A fringe of wet hair drips past his forehead.
"I bet the shower felt good." Some neutral ground has to be found.
"I forgot to close the curtain for a minute and got a little bit of water on the tiles. I threw down a towel." His voice matter-of-fact, but a challenge in his eyes, as if daring her to run up and fix the damage. The spilt juice, loose jar tops, left-out food, unlocked doors, half-open drawers. She tried to teach him, believed she could, but his habits never changed, and neither did her frustration.
"The floor will dry," she says crisply, and returns to the kitchen. Through the window she sees Bobby walking up the front steps carrying some boxes. Damn. She strides to the door to head him off.
"Hey sweetie, what's that you've got?"
He walks past her.
Tim salutes him. "Bobby, my man. You're a big guy now."
"Oh wow, I had no idea you were home."
Bobby deposits two boxes on the table. "One is a Scrabble game. My mom has two. The other is blueberry pie she brought home from the diner. Did your mother tell you she gave me your baseball mitt?"
"That's cool," Tim says.
"Want to play catch?"
"Not tonight. How's your mom?"
"She's out with Nick, her boyfriend."
"You like him?" Tim's voice deadpan.
"He's okay. His daughter's hilarious. She has a million funny stories. She won at Scrabble the other night and no one beats Mom. We have a marathon planned. The winner gets twenty dollars. I thought I'd practice with Dina."
"We can all play." Tim goes in the kitchen and returns with a beer. "Still too young for one of these, I guess."
"When is your mother expected home?" she asks.
He shrugs.
"Eat with us," Tim offers.
"I'm sure his mother has dinner for him."
"She can save it." Again Tim's voice gives nothing away. Why does he need Bobby here? She doesn't like the feel of it.
"She can save it," Bobby echoes Tim.
"Does she know you're here?"
"Where else would I be?" Bobby looks at her as if trying to figure out something.
She returns to the kitchen, dumps frozen broccoli in a saucepan of water, and waits for it to boil. A watched pot, Howie would've quipped. A man who liked his homilies, kitchen towels that read home sweet home, welcome mats, his-and-hers towels. She thought it a waste. They rarely had guests, not with her hospital shifts, but she saw no reason to squabble. Tim, however, wanted her to struggle, tried to engage her on a daily basis. She refused, had neither time nor energy. Sick people awaited her attention. Now she wonders if Tim needed her to fight. Children want to know they're important enough to stir up a ruckus. The water begins boiling. Early dinner, she thinks, then send Bobby home. An evening alone with her son, that's what she'll aim for, what she'll tell them both. She puts the ketchup bottle on the table.
• • •
As soon as they finish the game of Scrabble she suggests Bobby go home.
"Stay over. Why not? We're having fun, right, kiddo?" Tim speaks directly to Bobby.
"I'll call Mom."
"Where will he sleep?" she mumbles, confusion muddying her thinking.
"He'll share my room," Tim says.
Bobby looks at her, waiting for approval. The boy's not stupid.
"Sweetie, do me a favor, take the Scrabble up to Tim's room." She watches him run up the steps, then whispers, "What do you want with him?"
"Insurance policy. Don't worry, nothing will happen to him."
"I am worried." She stares at the hollows and planes of Tim's face, a replica of hers. "Why involve anybody else?"
"Bobby's a member of the family. He's the good boy." And Tim looks at her, mockingly.
"Tim, I'll have none of it. He's a neighbor's boy and should be sent home."
"In case of trouble?" An edge to his voice.
"Will there be any?"
"Depends." Is he toying with her?
"Let him go home, Tim. We can work out things without him."
"What things, Ma?"
"I don't know yet, but if you need to take someone, take me."
"That's a joke, right? You wouldn't know how to leave this place." She flashes on the times he begged her to take him somewhere, away, and always she had a reason—a good one, she believed—not to do so.
"Whatever happens, I'll help you. It'll be easier if it's just us."
"You may regret what you're saying." His eyes steady on her.
"I won't." The room seems darker though the lights are on.
"Okay. Listen . . . if my ride doesn't show, it could mean one of two things: He took the money and ran. Or he was picked up and gave me away. There'll be no way to know which." Tim speaks quickly and quietly.
If his ride doesn't show she'll have a felon under her roof. Still, he can't hang out here forever. It's the first place the police will look. She can't say so, can't have him believe she wants him to go even if she does. "We'll have to wait and see," her tone reasonable, even reassuring, though his face has gone a little blurry.
"Done," Bobby calls, bouncing down. Tim meets him at the foot of the stairs and in a low voice delivers some cockamamy story to send the boy home. Tim's good at that.
• • •
She wakes with a start. Squints to decipher the red digits on the new radio clock, nearly three. Dim voices reach her from downstairs. Did his partner arrive in the middle of the night? Her ears pick up the faint creak of the downstairs closet door. What does he want in there? A coat? Now Tim is climbing the stairs, maybe coming to say goodbye. No, he goes to his room. A minute later footsteps pad down again. Would he leave and not say a word? Let him. Yet she's out of bed peering through the slatted blinds at nothing more than velvet darkness. Slipping on a robe, she carefully takes the steps down.
He's on the couch, the TV volume low, the closet door ajar.
"Do you want a cup of hot chocolate?"
"I just finished the last beer. Go to sleep."
She sees the photo album he's tossed aside.
"I heard voices. I thought your partner showed up."
"I doubt he's coming. I shouldn't have left everything with him, too tempting," his face a screen of regret.
"What will you do?" Every visit ends with the same question.
"Maybe Canada . . . anywhere far from here." He gives a short laugh, almost a bark.
"You don't have a car."
"I'll switch buses till I get there. Can you dye my hair and cut it, too? Buy me a blazer, pants that fit, shoes, socks, and a pair of sunglasses. Can you do that?" The last words a childish plea.
She remembers a stormy night. Crashing thunder, lightning, wind rattling windows. It sounded like the roof would fly off. Tim toddled down, his eyes wide with terror. She picked him up, dragged an old sleeping bag into the living room along with a flashlight. She zipped them in, hunkered there till he fell asleep. There's nowhere to hide him now.
"Yes. I can do that. I'll take out cash too," she says. With whatever maternal influence she still has, she orders him to go to his room and get some sleep. He takes the stairs two at a time.
Switching off the TV, she replaces the photo album, shuts the closet door. She wonders if aiding his escape makes her a felon, too. It's against her nature to thwart authority. He's done something illegal; she's clothing him, giving him money. He's her son for heaven's sake. She'll shop and go to the bank early. If his partner shows up, the clothing and hair dye will be useful. Stretching out on the couch, she can't help feeling she's doing the right thing for the wrong reason.
If his father were alive, he'd pressure Tim to turn himself in and get him the best lawyer possible. She hoists herself off the couch, climbs upstairs.
"Tim, if I hire a lawyer, would you give yourself up?" The suggestion offered through his door. He doesn't respond.
"Are you asleep?"
Nothing.
"Tim?"
"Ma, that'd be easier for you than me, okay. So forget it." Again the childish voice, the boy who hated discomfort, feared difficulty. If Tim were caught red-handed on some video camera, jail would be inevitable. There's no way he'd survive in jail. With faint relief, she descends the steps.
• • •
The morning sun wakes her. Problems sometimes dissipate overnight, but the weight of Tim's needs is immediate. Tiptoeing upstairs, her back aching from the narrow couch, she listens at his door, hears nothing. In her room, she dresses quickly. She phones Bobby to say Tim isn't well and can he manage breakfast without her.
• • •
In the ATM vestibule of the still-shuttered bank, she averts her face from the video camera in the corner. Maybe his partner will show up. But maybe not. Tim should've taken his share of the money. What strange thoughts. She isn't used to all this intrigue. It's unreal, scary. Some people take risks for the fun of it—bungee jumping, mountain climbing, sky-diving—none of it anything she ever wanted to do.
The department store opens for business and she goes directly to menswear, no other customers in sight. That, too, feels eerie. Guessing at sizes, she chooses two pairs of pants, two shirts, a blazer, loafers, and socks. She piles the clothing on the counter. "My son won't let me shop for him anymore. They grow up and that's it," the cashier quips.
Taking the escalator to the basement pharmacy, she picks out a pair of wraparound sunglasses. Then she locates the hair dye, two bottles of brown-black color.
• • •
The blinds are drawn and the door double-locked no doubt. She hurries in as if she's the one they're after. Tim waits on the couch, his narrow body on guard.
"Want something to eat?"
"Do my hair first," his voice as tense as his face.
She follows him to the bathroom, drags in a stool, wraps a large towel around his shoulders. She cuts his hair, the back of his neck no longer boyish. A man shouldn't need his mother anymore. Maybe if he'd joined the army like other boys around here . . . There are too many ways to lose a son.
She mixes the dye with the solution, shakes it a few times, and applies it to his hair.
"I had an incident in the shower yesterday," she's surprised to hear herself say.
"Yeah?"
"I squirted a glob of shampoo in the palm of my hand, then rubbed it on my hair, but my hair was dry. I forgot to wet my hair. That's never happened before. It's frightening to forget the usual things."
"I once put salt in my coffee. It's not senility, Ma. It's preoccupation." His words are weirdly reassuring.
Waiting for the dye to take, she sweeps hair off the floor and wipes the steamy mirror, an eerie silence in the house. No radio. No TV. Outside noises are muffled, it's as if they've been sealed in.
"Ma, it's ready to wash off."
She'd leave it on another ten minutes, but he's too restless, fidgeting, in and out of the bathroom too many times to count.
She shampoos his hair and her hands are gentle on his scalp. The dye turns the sink black. Drying his head with a towel, she offers him a comb. The dark color pales his skin, but it suits him. "You do look different."
"Don't recognize me, huh?"
"You think I wouldn't know you?"
"Many ways to know me." He looks past her to the mirror.
She begins scrubbing the sink.
"If my partner does arrive it'll be by three this afternoon. That was the plan. If he's not here by three-thirty, drive me to a bus station near Montauk and stay with me until I pick up something going north. Did you get the money?"
• • •
She waits on the couch in her blue pantsuit and white turtleneck jersey. Montauk is two hours away. They could be stuck waiting in the car for god knows how long before a bus arrives. They should check a schedule. She's packed a small bag with a toothbrush and nightgown in case a motel becomes necessary. Tim once accused her of never venturing past anything familiar and now she's aiding and abetting a criminal. What's usual about that?
He walks slowly down the stairs, stopping to pose for her approval. He does look handsome in the new clothing. He could easily pass for a lawyer or a doctor, someone respected.
"Tell me one thing." She hands him three hundred dollars in twenties.
"One thing?"
"Was anyone hurt? Or shot? Or dead? Anything like that, I should know."
"Why? Will you love me less?"
"You're my boy, Tim."
"It was about money, that's it."
"I should open the blinds. They aren't usually shuttered at this time."
"Go ahead."
The afternoon sun no longer enters the room. A row of houses, outdoor chairs, closed garage doors, people at work, in school—nothing strange, except she's about to drive a fugitive to safety. He sits beside her on the couch. His loafers without a scuff, the crease in his pants sharp. It could be the last time she sees him. But it won't be. He'll return again, and again. Of this she's suddenly certain.
"I filled the gas tank," she says to say something.
"Good." But he's not listening, his mind elsewhere, planning lord knows what.
A black Honda pulls up in front of the house. Tim sprints upstairs and returns a moment later with a suitcase he found in the downstairs closet.
"Be careful," she mutters.
Words she ought to say slip away like time.
"You take care." He sounds excited he hasn't been left behind after all.
At the window she watches him slide into the passenger seat. Then he's gone. Just like that. A stab of disappointment takes her by surprise. She stares at the space where the car was. The afterimage contains more than the moment, but she blinks it away. She flashes on another memory: her husband's funeral, Tim slipping onto her lap, offering her his only candy bar.
We give each other what we can, she thinks, climbing the stairs to his room. The window is shut, the lights on, the bedding mussed, the closet door ajar, the chest drawers open. His dirty clothes are piled on a chair and a damp towel is on the floor next to an open magazine. Three empty beer bottles line the sill. Grease marks on the wall. A half-full chip bag sits near the bed. She'll air out the room, then tomorrow she'll sweep, vacuum, and change the sheets.
7
How We Know Before We Know
Wedding indeed. Rosalyn lifts one shoe from its box. What else could she tell Dina? At least she placed the escorting in the past where everything can be forgiven. More and more, she feels it's where it belongs. The long black skirt and blue silk tunic draped over a chair are waiting for her—whoever he is, he'll like it. It's such a chore dressing up when she'd rather order takeout and watch TV. Still, working for Annie has gone well for her, leaving her with a chunk of savings she'd never have accumulated from diner income. It's the money that bought her little villa . . . well, actually, a condo with terracotta floors and huge windows facing a terraced lawn.
Her eyes linger on the throw rugs bright as turquoise gems, the opalescent vase filled with daffodils, then slide to the sunburst clock on the wall. Jesus. Her father's waiting. He hates flowers, thinks they're a waste of money, promises to toss them when they're given to him.
• • •
The winter snows have damaged the road even more than last year. She drives cautiously past old houses, a few with tarps over their leaky roofs, others with cracked windows. This is where she grew up. The area depressed her then and still does now. Holidays were the worst, yet she'd look forward to Christmas every year as if it would be different. Her father would drink less. Her mother would find a gift that pleased Rosalyn. She and her cousins were always shooed in front of the TV, but her ear was on the adult-talk. Josie lost another baby; Marie Rose pregnant by god knows who; Artie can't get it up; Ron's temper is out of control; Tony's got no work again. After each tidbit, someone would sigh, "That's life, what can you do?" She hated their acceptance. Not that they'd listen to her, a mere girl, at least not until she was married with children, if then. They'd hear her out now, though. With wide eyes and slack jaws, yes they would.
She hoists the bag of food from the rear seat, feels a strain in her back. Yesterday . . . lugging boxes from the storeroom. She warned Murray, no more. If it's over two pounds, he carries it.
"Dad," she calls, shouldering open the door. Since the emphysema diagnosis he's remained dormant. His body will disintegrate. She's been over this with him. At least walk to the corner and back, she insisted. He won't hear it. When she phones her brother to complain, he's sympathetic, but rarely comes east. Neither of them is filled with affection for the man. How could they be? For too long after their mother died they had put up with him themselves. If she had the excuse of distance, she'd take it as well.
"Make sure to shut the door," his brusque voice a little wheezy.
"How about hello?"
He's broad-shouldered, with powerful, deeply veined hands, sheltered in his BarcaLounger the small, bomb-shaped oxygen container beside him. Not a trace of gray in his dark mane.
"We don't stand on formalities, you and me." As usual he shouts over the TV, which is on all the time. Does the man ever sleep?
"I brought frozen lunches and dinners. In and out, it's easy."
"You want me to pay you?"
Yes, why not? damn it. The man owns his house. He has a fireman's pension, doesn't spend a penny. "That's okay." How about thank you, she won't say, avoiding a lecture about duty or why have children.
"Can you stay?" He wants her to prepare and serve dinner.
But she's already in the narrow kitchen stuffing food in the freezer. The soiled towels heaped on the chair makes her wonder if the monthly cleaning service she hired is enough. When her mom was alive every room sparkled. After the cancer spread, her mother couldn't leave the bed, so Rosalyn had to mop, dust, whatever.
Peeling off the see-through covering on a turkey dinner, she places it in the microwave, an appliance her mother never owned. It's weird, so many years, yet, recently, thoughts of her mother spring to mind not just when she's here but also in the shower, supermarket, the oddest times. She recalls a story her mom told about living in the Bronx a few streets away from a Gypsy store. One winter evening, when her mother was seven, a Gypsy man scooped her up. Her mother screamed. He put her down and fled. The story was told as a warning to Rosalyn who often wandered away from the house. Her father, listening, muttered good riddance. To this day, she doesn't know if he meant his wife or Rosalyn.
She spoons hot food on a plate and places the meal on the TV tray in front of him.
"Do you think about Mom?"
"What's the point?"
"Memories, I don't know."
"Can't do a blessed thing to change the past. Today is what I have. You too. Make something of it. Where's your husband? Where's my grandchild?"
"Let's not, Dad."
"She's in her twenties now. She'd be a friend to me."
"You sure as hell didn't feel that way at the time," her voice rising above his.
"I was looking after your mother. It was never right. A child belongs with its parents. Period."
She watches him shovel food in his mouth. Damn him. What would he do if she died before him?
"I'll pick you up tomorrow for your doctor's appointment."
She hurries to the car, slides in, slams the door. How dare he mention the baby? The memory is hers, not his. He has no right to it, none at all. She remembers Carl Reese. Another lifetime. Someone told her . . . probably her father . . . Carl's in Iraq. Isn't he too old?
• • •
The hotel caters to conventions. Men wearing name tags wander in and out of the lounge but the bar isn't crowded. Burgundy-flocked paper darkens the walls. Several green-shaded lamps hang over the whiskey bottles and the indirect lighting casts a pinkish glow. She sits at a small square table nursing a glass of sparkling water. Dina should see her now, perfumed, coiffed, new shoes. Annie's message said his name is Jack Temple, a Londoner, carrying a newspaper. Arriving early gives her a chance to catch sight of her date before he sees her. If she gets a bad vibe she's out of there. It's happened once or twice. Annie, who runs the escort service, chooses carefully for her girls, and loves to hear next-day stories. Their phone calls mimic the confessional, with Annie as priest placing details in some universal order that undermines any thought of sin. Still, she can't help but wonder about the wives and girlfriends back home. When she says so, Annie swears it's genetic, that she's never met a man, gay or straight, who hasn't cheated on someone somewhere. Carl didn't cheat on her. Once the baby came, though, it was finished between them. Waiting for a strange man with Carl in her head. Too weird. It's her father's fault dredging him up.
• • •
A tall middle-aged man with a long, bony face and graying hair, carrying a newspaper stops at her table. He's dressed in an expensive-looking dark blazer over pale gray pants, gray shirt open at the neck. She pegs him as very English indeed.
"Rosalyn, I expect? Jack Temple." His bright green eyes carry the younger man he once was.
"Hi. Have a seat."
He seems pleased with what he sees. Why would someone like him need an escort?
"Here on business?" It's her job to help him relax, but he doesn't seem nervous.
"Yes, for a while."
"Let me guess . . . something to do with banks?"
"Not quite. I'm doing research at a lab on Long Island. I work for a pharmaceutical company."
"Impressive." Some men prattle on, which can be boring, but less wearing. It's hard to know with this one. Either way, she'd rather be home relaxing on her couch. Such thoughts aren't permitted. It's her job to be 100 percent present. She's learned the art of it, how to keep her distance and leave an impression of closeness.
The blue-white tablecloth, heavy linen napkins, crystal stemware, and elegant silverware are nothing at all like the diner, and nothing like what she grew up knowing. On the rare occasion her father took them out to dinner, but usually to some less than appetizing place. Here she is in for a sumptuous hotel meal, and not her first.
The waiter appears before she can settle in. He and Jack discuss the merits of Beaujolais or Sauvignon Blanc and Jack orders a bottle. Nodding his white-haired head, the waiter hurries off. She scans the other couples in the room, their intimacy, wondering as she always does if her status as an escort is apparent.
"May I say something about myself . . ."
"Of course." Her dates often attempt to define their goodness in the face of immorality.
"My wife has MS. There are limited hours we can spend together. We don't talk about my needs but she'd understand. A nurse cares for her. My son comes often, but he has his own life."
"That's a lot of information to tell someone you've just met."
"I want the woman who touches me to know something about me. It's less impersonal."
"And are you asking me to do the same?" Usually her dates couldn't care less.
"If you wish."
"I can't rattle off a bio." He's a stranger. It feels intrusive.
"Are you an actress, writer, a painter?"
"Why would you think so?"
"Creative women need to support themselves. And . . . well . . . you're very beautiful, radiant, really."
"This isn't the only work I do. I care for my sick dad, so I appreciate your situation. Shall we order?" She picks up the menu.
• • •
Except for an occasional headlight sweeping past, the road home is dark. Her mind replays the last hours. He was attentive, talkative. He told her about places he's visited, blue skies the color of her blouse, sunsets as tawny as Spanish wine. And middle age, how odd it feels to be there. He was an "up-by-the-bootstraps lad," worked his way through college. She found herself sharing snippets of her life—unusual—relating diner stories that had him laughing out loud. He was curious about her and easy to be with. The hours passed unnoticed, also unusual. Still, the faint embarrassment of exposure dogs her. He wants to see her again.
• • •
The breakfast rush is in full swing and the cacophony of sounds is jarring. Mila pours coffee with one hand, wipes surfaces with the other while trading words with customers. But that's Mila, her ability to juggle three things at once keeps Murray at a comfortable distance. Nick flips eggs, catches popping slices of toast, pulls plates out from the warmer. The distinct, watery slosh of the dishwasher surprises her. Murray asked Nick not to run it during busy hours, insisting the noise disturbed customers. Murray makes up things like that all the time, but isn't about to criticize Nick who's been pulling double shifts. They've all been covering for Bruce. Even if Bruce were ready to return, Murray's looking to hire someone "reliable." Changing into her work shoes, Rosalyn fights the urge to return home, to catch up on sleep.
Willy beckons her, his ancient arm in the air. A small man in a booth for four, Willy won't sit at a table because he doesn't want to reveal his skinny legs. They are two sticks. When does vanity end? She jots down his order, though Willy orders the same breakfast special every day. If she walks away without promising to return, he calls out, "Rosalyn, I need you." She fills his water glass and pats his arm. "I'll be right back."
Murray's standing at the counter. "Why the long face?" she quips, not expecting an answer.
"The whole thing . . . I don't get it . . . Sylvie leaves early, arrives home late. I have dinner alone when there's no reason for her to work. I don't like it. It's eating at me. What's the point of being married?"
"Talk to her. Tell her you're lonely." He won't. He'll never admit need. That feels familiar.
She places Willy's poached egg, wheat toast, small cereal box, and milk in front of him.
"Stay," he orders.
"For a minute." No doubt Murray's watching her. How he got Sylvie to marry him is the real question.
"You look lovely," Willy says.
"You say that every day."
"Sometimes I lie." He winks. "Did I mention . . . my sons are coming to visit? They're wonderful children, but it would kill me to move in with either one of them. At ninety, eating and sleeping are my last best functions. I need to do them on my own." His voice is thin, high, the testosterone long gone.
"I understand," she says sympathetically.
"I knew you would."
Why do people want to hang on so long? Are memories enough? Not that she'll ever see ninety. "How are you today?" she asks.
"My dear, the question is, will I make it here tomorrow?" He adds the third packet of sugar to his cereal, which he never finishes.
"And, will you?"
"Seems so, but my five senses are no longer intact. Tell me, does springtime still smell fresh? If so, it insists on love." They often have this conversation, which leads to his advice about her finding a companion. Usually it amuses her but today it's irritating and she doesn't respond, though Jack comes to mind. After sex, her dates want to sleep, happy to have her leave. Jack was different. He insisted on a post-midnight stroll along the dark flower-scented garden paths behind the hotel. Even if Jack were a free man . . . he's not.
"Did I say something to upset you?" Willy asks.
"Of course not." She gazes at his wizened face, the yellowish skin. His eyes, though, as black and shiny as patent leather. He's alone and as happy as his body allows. Something takes hold inside her, what, she can't exactly say, but it feels like a clutch, a squeeze against the future, a warning to do something now.
"I'll bring coffee in a minute," she calls over her shoulder, hurrying to the parking lot. Wedged between two cars, she takes the cell phone from her pocket, dials Annie. "It's Rosalyn," her voice low.
"How was last—"
"Fine, it was fine. It's not why I'm calling. How should I put this . . . I'm quitting," the words heavy in her mouth. "I'm getting too old for the routine. Or . . . maybe my day shift takes it out of me. I hate having to dress up when I feel like shit."
"Then . . . rest a few weeks." Annie's tone hesitant as if she's talking to someone ill. Her head does feel as if it's about to explode.
"No," she nearly shouts, shocking herself. Lowering her voice again, says, "It all feels . . . suddenly . . . beside the point."
"What point?" Annie sounds truly confused.
"I'm tired of meeting the needs of strangers." It's the best she can do.
"What about the money?"
"I have enough of everything but time." Where are these words coming from? She's not impulsive. "I've got customers waiting."
• • •
Switching on the car radio to interrupt the static in her brain, she pulls into the driveway. Her father walks slowly toward the car, portable oxygen canister in hand.
"I phoned you last night to pick up a six-pack today," he says, getting in.
"We'll do it on the way back from the doctor."
"Where were you?" his tone faintly accusatory.
"Out . . . on a date."
"Who's the boy?"
"Dad. I'm forty now."
For a moment he takes her in as if he might actually see her. Her hands tighten on the steering wheel, her head tense enough to crack.
"The food you brought yesterday was tainted."
"It was frozen."
"Kept me in the bathroom."
"Tell the doc, then."
"You don't care, do you?"
"Dad!"
"Be better for you if I was gone."
"You watch too many soap operas."
"Well, what else can I do?"
"I don't know. Invite more of your old pals to visit."
"They come when they can," his tone testy.
He's more protective of their feelings than hers. She says nothing more.
Accompanying him up the path to the doctor's office, she holds open the door. "I'll pick you up in an hour."
"What's the matter, got ants in your . . ." Hurrying back to the car, she drives to the beach.
With the windows rolled down, a balmy breeze, a hint of spring in it, the kind Willy can't smell anymore. Are there reasons for quitting other than the ones she told Annie, whose surprise and confusion mirrors her own. Forty's not old. Did something spook her? It's all so odd. She stares hard at the sand, water, the last shock of afternoon sun streaking purple and orange across the sky. Then she takes out her cell phone, calls the number Jack gave her.
• • •
Her hands are cold. She eyes a bottle of red wine on the side table, deciding whether to open it. A bottle of white wine chills in the fridge. She paces the living room like an anxious teen, then stops to look out the window. Jack's never been here in the weeks they've dated. She doesn't allow "dates" to come home with her. Hotel bar, restaurant, his room, then home, alone, that's been the routine with Jack, too. Now he's on his way here.
She watches his long legs precede his torso out of a town car. Watches him come up the drive, watches, too, as he takes in the landscape. Too late to change her mind, she opens the door. "Welcome to the villa."
His lips brush hers and he hands her flowers.
"Roses of all things . . . beautiful."
"Yes, they are."
She tosses out the brooding tulips, arranges the roses.
"Lovely villa," he says, looking around.
"I like it. A drink? Some music?"
"Let me." He riffles through her collection, stacks a few CDs, pours two glasses of wine, then sits beside her on the couch. He makes himself at home with ease.
"I'm quite glad you asked me over. I wondered, is the lady hiding a man in her closet." He grins. "By the way, my lab gets theater tickets. Let's take in a play. Also I want to explore the beaches here. People say they're more beautiful than the Riviera. Hard to believe."
"Sure, a play sounds great. I'm not sure about the Riviera, but late afternoons, the shore is wonderful."
"Perhaps this weekend then . . ."
"There's something I haven't told you." Ella's velvety tones float by. "It's the strangest thing . . . I quit the job."
"The diner?"
"No, the escort service."
"How come?"
"I don't know."
"When?"
"A while ago." She won't say the day after they met.
"Well, I hope I was responsible."
"No, at least I don't think so."
"I'm glad you did."
"Why?"
"You'll spend time only with me. It will be absolutely delightful." He folds her hand in his. "I couldn't be more fortunate." He pecks her cheek. "More wine?" He's up refilling their glasses.
• • •
In the shower, his large, firm hands slowly massage her soapy body. Two glasses of wine wait on the sink edge.
"You see," he shouts over the sound of the water. "It has nothing to do with getting clean. We should've done this before."
She laughs. A surprising lightness fills her.
"And also, I don't do all that many things well. This, however, I claim credit for success. Yes?"
"Yes," she shouts.
"The other bit I should share . . . well, my dear, it has to do with your outstanding body."
"Jack, I rarely believe the sentiments of excited men."
"Yes, indeed, you made it clear these last weeks. What can I say to make you trust my . . ." He stops his massage abruptly. "Let's get into bed. I'll warm it up for us."
He rinses off the soap and leaves first. She drapes herself in a large bath towel, follows his wet footsteps across the floor. The waning evening sunlight trickles through the shuttered blinds. She registers the stillness; the music has ended. She slides in beside him.
His arms go around her, his belly pressing her still-damp back, his mouth close to her ear.
"Rosalyn, dear. Something there in your breast."
"What?"
• • •
Locked in the diner bathroom, Mila drones on about the number of women who survive, customers who years later enjoy their burgers, women who . . . But she's only half listening. How we know before we know is the competing lyric in her head. It's why her mother's been visiting her thoughts. Why, too, the eerie sense of time. Even more crazy is the strange relief of no longer waiting. The doom she's carried since her mother's death has been born, the truth of it stark, almost energizing in its clarity.
Murray pounds on the door. "Hey! What's going on?"
"Be right out," Mila calls. "Hurry, tell me what the doctor said."
"It doesn't look promising, though the biopsy was inconclusive, which, he says, they often are. Tomorrow I visit the surgeon, the day after I go for a bone scan. Then . . . I don't know, a bunch of tests I guess."
"Ava and I will go with you. She, tomorrow, me the next day, Dina the day—"
"No . . ."
"Rosalyn, you'd certainly go with me." Mila cuffs her wrist. "So just let us do it."
"I do need help with my father. Could your daughter visit him two hours every afternoon for a while? She can put up his supper. See if he needs something at the store, whatever. I'll pay her eight an hour. I'll tell him I'm away on vacation. I don't want Murray to know either. Look at the way he's treating Bruce."
"Darla always needs money. It's a good time of day for her, after school, before gallivanting."
"Hey!" Murray bangs on the door.
"Coming." Mila's hand reaches for the knob.
"Go ahead, I need a minute."
In the silvering mirror a pale face greets her. She applies lipstick. There's distance between her and that person. Overwhelmed, is what it is . . . all the things to get done. Everything seems set out, no time for rumination, as if her body has sprung a leak. She glances at her watch and sees it's dinnertime. Not her usual shift, but Mila's in the kitchen helping Nick.
The restaurant's more crowded than usual, buzzing with impatient, hungry customers, arms beckoning, voices loud. Murray is greeting regulars as he fills water glasses and he eyes her as she steps through the door. Behind the counter, Ava serves one customer after another. About to wait on a table of four noisy people, the door chimes and Jack enters.
"God," she says low in her throat, on her way to stop him from taking another step inside. No one knows about that part of her life. "What are you doing?" she whispers harshly.
"You didn't call me last night. I won't have that." They stare at each other.
"We had no plan that I can remember," her tone far from welcoming.
"We don't need a plan. We've spent enough time together. You owed me a call," his tone controlled.
"That's a little proprietary."
"Look, it took me hours to find this damn place. I was worried. What happened at the doctor's?"
"I don't want to talk about it with you."
"Why not?"
"You're a married man with a life in London. You'll go back to that life."
"Well, that gets to the core . . . because Rosalyn, I'm here, with you, now. If you don't like me, if you find me a bore, if you would rather be with somebody else . . . say so." He's talking fast, nervously. "Otherwise, let's get out of here."
His eyes on her are wide, his mouth unsure, prepared for a verbal blow. He's vulnerable, like her. "I'll meet you as soon as I finish my shift."
"No. I'll find a table. I'm staying."
• • •
After the exam, the surgeon sits on the edge of an elegant leather-topped desk. His degrees descend the wall. He's tall, thin, with a head of curly blond hair, a smile to light the way. Is his demeanor part of the healing process? Does he offer it to each patient? No matter, he's too young, his life still beginning. She needs a doctor in his seventies who's seen it all.
He tells her about the statistics, studies, new treatments. How many of his patients have died of breast cancer? she wants to know. Not one of his statistics. He says, do this and the survival rate could be . . . Do that and . . . He says nothing will be known for sure until they stage the tumor. She's having trouble absorbing words—the door in her head is locked. It's too much information, she tells him. Mila, though, scribbles his words on index cards.
The afternoon sun does little to warm her body chilled by the A/C. Mila hands her the index cards, which she stuffs in her bag. The car isn't far, but they walk slowly, Mila's arm linking hers.
"What're you going to do? Breast off, chemo first, or . . . ?" Mila asks hesitantly.
"I don't know."
"It's an awful decision. Are you terrified?"
"I can't talk about it yet," she admits.
"Did you and Darla work out a deal?" Mila doesn't miss a beat.
"We did. She looks more like you every day except for the dark hair. Was her father dark?"
"He was dark all right. Who was the fine-looking man at the diner waiting for you?"
"I dated him for a few weeks. He's married."
"Now isn't the time to break up. You need as many with you as will stay. Sickness, divorce, birth, they take it out of you. Someone has to be there to empty the bucket. Should I drop you off or come in for a while?"
• • •
She kicks off her shoes, drops on the couch. The still-blank journal Jack bought her is on the table. Her feelings are muddled, alien, racing, her thoughts filled with the minutiae of things to do, stupid, unimportant bits and pieces taking up space in her head.
Every day, Jack asks how she's doing. Every day she says fine, her tone refusing talk about the possibility of malignant cells spreading like melting butter. He suggested a support group. She told him about Doris, one of her regulars, who attended a group to help grieve her son killed in Iraq. Doris quit after one meeting. She didn't want to hear how she'd feel a year from then. She wanted to make her own discoveries. To each her own journey, it's what she believes as well, and said as much to Jack.
Whatever her mother's journey, she didn't share it with Rosalyn, didn't talk about the disease, didn't reveal what was happening to her body. She never mentioned pain, disfigurement, or death. Then again, at seventeen, Rosalyn didn't want to hear, didn't want to take in the flat chest while her own breasts were flourishing. Her hands cup their fullness. She remembers Carl's head burrowing contentedly in the cleavage, his delight in their milk-laden heaviness. He was with her when the nun brought in the infant. She closed her eyes, lest the baby's face remain to haunt her. Now she wishes she had seen her, someone to hold on to.
The phone rings.
"It's me," Mila announces.
"You're home already?"
"Darla said your dad's cute and funny."
"You're kidding!"
"Swear to whatever. Ava and I hatched a plot to take you to dinner a few nights from now."
"I've had enough diner food."
"Funny, funny. The real stuff. Romano's, remember their salads? Even better, the wine they never tell you the name of . . . heaven. So warn your guy, he can't come. It's just us."
"Sounds great." She clicks off.
Cute and funny? Darla's gone to the wrong house.
• • •
It's a different exam room, small, cold, without windows. Lying on a narrow, padded table, a thin pillow beneath her head, wires attached to chest and legs, an EKG clicks faintly, recording her heartbeats. Yesterday, a whirring plate of light took three-dimensional pictures of her bones. Heart, bones, breasts, her body dissected for study; the tests, their definitions, the possible outcomes, layers of information have all taken their toll on her brain as well. Her voice has begun to echo in her head like a bad telephone connection; she senses herself watching herself, even when applying makeup. It's as if she's two people, one just a hair ahead or behind the other. Is this a heightened state or terror?
The technician removes the wires, wipes the gel off her body. The surgeon scrolls down the long sheet of paper, studying the EKG.
"Rosalyn, your heart is beautiful," he tells her, stuffing the graph in her file. "I'm sending the bloods to the lab. The bone scan hasn't come back yet. But let's look at the mammogram together. Get dressed, come to my office." He helps her off the table. The desire to hang on his arm, to stay in his sight at all times, is strong.
• • •
"I left work an hour ago to get here, but the traffic . . . listen, sorry I'm late, couldn't be helped," Jack bustles in, anticipating a scolding. But she hardly noticed the time. As usual he makes himself at home, uncorks wine, pours some in glasses, hands her one.
"I haven't done a thing about dinner," she mumbles more to herself.
"Low on the problem scale." He tugs her to sit beside him on the couch.
"Are you a problem solver as a scientist?" she asks, though concentrating is difficult.
"It's what they pay me for."
"Are you worth the money?"
"Absolutely."
"Tell me one of your great finds." She's trying.
"It'll sound like tooting my horn, is that how you say it?"
"Toot away," she orders.
"I discovered blending two types of old drugs produced a third that raised the number of white cells in the blood."
She stares at him. "Are you doing cancer research? Because that's too eerie."
"I never saw any reason to mention it before. It's where a great deal of the drug investigation is today." He slides an arm around her shoulders.
"You don't have to press me one place or another every time you mention cancer. I'm not that fragile." Actually, though, she's chilled.
"No you're not. In fact, your self-sufficiency is sometimes off-putting."
"Off-putting. That's very British. Aren't your countrywomen very self-sufficient?"
"In their public selves."
"I see." She wonders if his wife is clingy.
"I offended you when it wasn't my intent."
"Men want women to need them so they can feel strong and noble. But here's the thing . . . when women do lean on them, men feel suffocated."
"Wow. That's telling me."
"That wasn't my intent."
He laughs. "Touché. Nevertheless, you've seen the doctor again. What happened?"
"He showed me the mammogram. It's there, a white splat, not small, easy enough to see on the film. Also the staging came back. The surgery is being scheduled." She moves to the window. It's too dark to make out anything that isn't already familiar. He comes up behind her, nuzzles her neck.
"Your touches kind of scare me."
"That's simply terrible. What can I do?"
She wants to say, be cautious, because she's taking in the dimension of things, registering their very essence. She once read soldiers on the front lines create an impenetrable bubble to keep the world at a distance.
The phone rings.
She picks up the cordless. "Hello?"
"You said you were away."
"Dad?"
"Can't stand the sight of me anymore?" his voice explosive.
"Dad!"
"Lie to your father? Great! I actually thought Darla was my granddaughter, but she's only going on eighteen."
"Dad—"
"You've resented—"
"For craps sake, I have breast cancer." She hangs up. "Bastard," she mutters. "And you, too. Just go home."
"My sweet girl. I'm not about to honor your self-pity."
"Self-pity!"
He hands her the wine. "Drink up."
"I don't want it. And I don't want you here."
"Take a deep breath, my dear."
"I want you to leave."
He wraps his arms around her; his hard body a wall. "So you can be alone with your fears."
"So I can muster my strength."
"It's already there, in your eyes, determined jaw, set lips. Believe me, it would take an earthquake to undercut that."
"Why do you think you know me?"
"I don't. You won't let me. You won't share your dreams or your nightmares. Why didn't you tell your dad in the first place? Why must you carry the load by yourself?"
"And you'd like to take me to bed to prove your ability to comfort me, right?"
"I would, but not for that reason."
She gazes at him. Nothing in his expression mocks her. The accent makes him sound flip. "And the reason is . . . ?"
"I'm terribly smitten with you. I didn't want to be. It's why I hired someone instead of meeting a woman on my own. I thought hiring would alleviate better feelings." His voice so earnest it's almost comical.
"You talk funny."
He chuckles. "I'm going to cook dinner. Can I search your pantries?"
"Excuse me?"
• • •
She sets the table, her mind somewhere else. What if she fled? Stuffed the bad news in a corner of her brain the way the doctor stuffed the EKG in her file. What if she took off for California to walk the beaches? Or farther, Rome, Venice. Or maybe Turkey? She has the money. Spend it now. She looks out the window where things are as they were. That's the problem with fantasies. They change nothing.
He places a puffy salami omelet on the table, the garlic and onion smells palpable.
"Looks wonderful," she says, a bit sorry for her cutting words before.
"Now aren't you glad I stayed?" He holds out a chair for her.
"I won't be bribed." She sits across from him.
"Apparently. Yet it's exactly what I want to do. Cheer you."
"You're sweet."
"Not really." His expression clouds and she wonders if he's feeling guilty.
"Are you thinking about your wife?"
"Not thinking so much as worrying a bit. I spoke to the nurse this morning. My wife's been sleeping more. A bad sign."
"Do you like being surrounded by sick women?"
"What a thing to ask." He looks uneasy.
She shrugs. "Well, you are."
"I don't see you that way."
"What way?" She's no longer sure what they're talking about. Like those customers who insist on chatting. She provides trivial questions, and the answers don't matter.
"Like Lillian, incapacitated."
She wishes he hadn't said her name.
"Simply believe this. I'm here for you."
"But then you won't be."
"You're vulnerable. I'll continue to reassure you."
"Jack, that's condescending."
"Good! Sounds more like you."
"I'm in a very strange place," she admits.
"And I'm still drawn to you."
"Who knows what's going to happen to me."
"That's true about any of us," he says.
"You mean, today's what we have? Sounds like my dad."
He cuts the omelet, places some on her plate. She's not the least bit hungry but forks up a tiny piece because he's watching her. Ridiculously, Willy comes to mind. He still worries about what people will think.
"Things still matter," she muses.
He looks up. "What do you mean?"
"I'm surprised, is all."
"Crises propel us to odd places. A bit of an adventure . . ."
"That's inspiring, thank you."
"Adventures have no history, that's all."
"It's more complicated than an adventure," she says.
"Come now. You've heard about the best-laid plans . . ."
She nods, pushes away the barely touched food. "I'm really not hungry."
• • •
She drives to her dad's house. She hasn't spoken to him in a week. She considers leaving the bags of food in the driveway and taking off, but then finds herself with a shopping bag in each hand, walking up the scarred path. Fogged windows block anything inside. The house needs painting. Only the maple tree thrives, though no one ever cut back its branches.
"Dad?" She shuts the door behind her.
To her surprise he's in the kitchen.
"What're you doing?"
"Want coffee?" he asks.
"No." She wants out of here. Will resent any discussion about her body. And begins to stuff packets of frozen food in the freezer. He leans against the sink watching her. There's hardly room for the two of them.
"I hired Darla for the summer," he says gruffly.
"You what?"
"Going deaf?"
"You'll have to pay her."
"No kidding," he says.
"Did she agree?"
"She accepted." His eyes steady on her.
She wants to say you finally got off the chair. She wants to say it took the threat of death. She wants to say it's really too late. "Good, Dad. That'll be a help."
• • •
They've taken her street clothes, earrings, purse—anything that could identify her—and stowed them in some room she'll allegedly be wheeled to after recovery. Draped in a hospital robe, covered by a sheet, she's one of several bodies lined up between drawn curtains awaiting surgery. It's still possible, she isn't anesthetized yet. She could chance fate, shout, I changed my mind! Let me out of here! She makes no move, no sound, resignation heavier than the future.
Fingering the cold edges of the narrow gurney, A/C very high, no germs allowed, if she stays calm her teeth won't chatter. Breathing in deeply, she counts slowly on each exhale the way Dina taught her. It's no use. Her thoughts race, collide, refuse to remain long enough to read, as if there's something she must resolve. Dina has the keys to her condo and will take care of everything. Darla will deal with her dad. Ava and Mila drove her here. Ava didn't say much, though Mila went on about her daughter, how amazing it is that's she's grown, how worrisome, too, how she spends money like . . . Mila's chatter was more comforting than Ava's silence. They insisted on staying with her through admission, walked her down the long blue-carpeted corridor toward the heavy double doors leading to the area where a nurse took over. It surprised and scared her then when Ava suddenly hugged her so tight the breath was squished out of her.
Jack, too, on his last night here wrapped her so tightly she feared for her bones. Said over and over she was his godsend. How strange. He wanted to remain with her through surgery and then some, though his job at the lab was done. She wouldn't let him, didn't want him to see her in duress, wanted his image of her to be whole and beautiful. And, yes, she understood none of that mattered to him, but still, it's what she wanted. He wrote down a thousand phone numbers where he could be reached. He promised to stay in constant touch, made her promise to meet him in Europe when she recovered. Said if she didn't, he'd return to fetch her. She believes him.
Her doctor parts the curtains. He's in surgical garb, though his mouth remains uncovered. He smiles warmly; his warm hand squeezes hers. He alone understands what she's about to go through. He promised her a shot to relax her and leans over to inject her arm. He whispers two words she'd never say to herself, "think positive," though they both know truth will have its way.
8
About Time
"Mom, sit down."
"I'm making dinner." The don't-bother-me tone reserved for fussy customers, she's brought it home with her. Okay, she's overworked, working the diner kitchen . . . it's not her thing. Damn Murray. Rosalyn's illness, too . . . it frightens her—fear for those she loves.
"I can't talk to your ass, Mom."
"Darla!" She spins around. Without a bit of makeup, her daughter's a stunner, the contrast, dark hair, light skin. "Okay, what?"
"You're not going to be thrilled."
"Try me." The shag cut frames Darla's small face perfectly.
"I graduate in June."
"I know that." Adolescent nonsense. She reaches for an onion on top of the ancient fridge, notices the scratch marks on the door from a thousand magnets.
"In July, I'll be eighteen. I won't need your signature. It's May," her daughter recites.
She fishes for the missing knife buried under a pile of dishes in the cracked porcelain sink. Christ, the place could use some rehab. "Work the summer for Rosalyn's dad. Save money for the car's down payment. I can't—"
"Mom . . . Forget the car. I'm going to sign up."
She stares through the window at an identical clapboard house. A breeze flutters the short white curtains that need washing. "No you're not," she says softly, her gut cramping.
"It's the best way."
"To what? Die?" She sits across from her daughter.
"Don't be dramatic."
"It's out of the question, Darla." If she raises her voice, they'll fight. She'll lose. She takes a deep breath, tries not to sigh.
"If I sign up now, I get an extra thousand dollars."
"Money?" It's her fault, all her worrying out loud about it. She'll send Darla to her cousin in Arizona.
"You don't have any. I need a lot."
"They're not paying you to attend the opera."
"Mom, I'll be fine."
"You're only saying that because you're young and stupid."
"Thanks."
"It's a horrible choice. There are only downsides." Is this what women's liberation has brought? She needs a drink.
"On top of the thousand, there's a shitload of cash up front, so I could start a savings account. What am I going to do here? Work a few hours for Rosalyn's dad, a few hours more in some supermarket till I save enough to go to a third-rate community college? It's not how I see my future."
"Spend the summer with your cousin in Arizona. I'll scrape up the down payment for that jalopy you've been eyeing." Maybe Murray will let her work Rosalyn's shift as well.
"If you say no now, I'm going to sign up in July. So mull it over." Her daughter strides out of the kitchen.
She kills the stove flame and grabs a bottle of Johnnie Red from below the sink, a glass from the drain. She pours a few inches neat, sits on the couch, and drinks it down. The door slams. Out for the evening. The sigh that's been clogging her throat escapes. The girl's right about one thing—there's nothing special about living here. Darla could meet a guy and get pregnant. Her daughter's too smart for that. How smart is signing up, though?
It annoys the crap out of her that in a few weeks Darla won't need her approval to put her life on the line. Maybe it's true . . . what goes around . . . She devastated her parents when she eloped with Jimmy. But this is different. Darla could be maimed or killed. Christ, she has to do something to stop her. Times like these, a father would be helpful. Good god, it's been years since she had a thought like that.
She pours more scotch, looks around, but there's nothing worth selling. The room has darkened. She doesn't bother with the lamp. Her reflection's on the TV screen, a woman edging middle age with a daughter as old as she was then. The marks of time can't be hidden the way she's hidden Jimmy from Darla.
Sitting here will solve nothing and make her late for work. The overheated diner kitchen, that's what's waiting for her. One more day, Murray said, before a temp arrives and then back to her regular shifts, not that she loves them either. With glass in hand, she searches for a piece of paper, finds an index card on Darla's desk. Writes: Monday, my day off, we're going out to dinner. The pub you like. Don't make other plans. Off to work.
• • •
Darla walks the long route to Michelle's house. She needs to think. Her mother reacted as expected. The woman's scared of change. Why else would someone with her looks still be unmarried? She never gets a good answer to that question. It doesn't matter. She has no plan to follow in her mother's footsteps. College, law school, a job on Wall Street . . . she'll make a fortune and buy an apartment in Manhattan. Her mother will see she made the right decisions. The guys who've been in Iraq tell her the girls there do housework. Clean machinery, set up office stuff. They're not running around banging open doors with M16s or whatever they're called.
• • •
She rings the doorbell. Waits. "Damn," she mutters, just when you need someone. Not that she's crazy about being here, Michelle's dirty-mouthed brothers always in her face.
The upstairs window finally opens.
"Hi, let me in."
They traipse up the few steps to Michelle's room. "Where are your brothers?"
"Out with my dad."
An intact family she'd rather die than join. Michelle's father, a cop, never stops smiling. She can't trust someone who pretends everything's okay when his wife's messing around with whoever will have her.
Michelle, tall and broad-shouldered, stands in front of the window, her dark, wavy hair backlit by the evening sun. "What did your mother say?" Michelle has no patience for the finer feelings. She wants details. When Darla returns from a date, Michelle phones with clinical questions: What did his tongue taste like?
"Over her dead body. It's a first response."
"Did you tell her I'm signing up, too?"
"She'd accuse me of not making up my own mind."
"We need to go together."
She eyes the posters on the wall, no one she likes. "You're eighteen in June. I won't be able to sign up till late July."
"Work on your mother." Michelle rolls the squeaky desk chair back and forth.
"Like how?"
"The breakdown: you're depressed, no motivation, want to die. Refuse to get up for school."
"My mom thinks I'll get killed."
"Do you agree with her?" Michelle asks suspiciously.
"Of course not. I know she's negative. Have you spoken to any more of the guys who came back?"
"Ian said I was crazy, but he's been high since he came back." She laughs. "Let's smoke at the beach. My mother's car is outside. She won't be home till middle of the night."
• • •
They park in the empty lot. The beach won't be officially open until Memorial Day. Her bare feet tramp the damp sand. The sun has disappeared. They walk to the shore and sit, knees up, listening to the crash of waves. In the gray distance a ship cruises the horizon. Gritty wind blows in her face, her skin clammy. It's fine. She's open to the elements, but worries about how tough army training will be. She's not an athlete. Michelle can carry weight on her back. The thing to do is begin building muscle now. If she puts her mind to it she can do it.
"There's no guarantee we'll be sent to the same place for training," she says.
"Then we'll tell them the deal's off. They need us. They'll agree."
"You don't know that," her tone sullen.
"Did your mother say something you haven't told me?"
"That's not the point."
"Darla, we've been over points."
"What did your father say?" she asks.
"Women soldiers fuck up and complicate situations, then he laughed."
"And your mother?"
"Either she'd just had sex or was flying on chemicals. She looked at me like who was I, then said don't get raped. I told her I'd do my best."
"My mom's stubborn. It'll be hard to change her thinking. If worse comes to worst I'll wait till after my birthday. She can't stop me then." She doesn't say it'd be easier if there were two parents. Even if they both didn't want her to go, they'd have each other to bitch and moan to.
"The sooner we get out of this Long Island swamp the better," Michelle declares.
"I wonder what the desert will be like?"
"Check out National Geographic."
"I bet it has its own silence."
"There's a war going on."
"We'll get time off, sneak behind some dune, look at stars, smoke dope. I thought you brought some?"
"Coming right to you." Michelle digs a small plastic bag out of her purse, removes a joint, lights it, takes a drag, then passes it.
Inhaling deeply, she stares into the hazy nothingness. After the second hit, she sees a distant cloud drop behind the horizon.
• • •
Breakfast customers have cleared out, thank god. The lunch crowd will soon descend. Nick stacks salads in the big aluminum fridge; he's filled the bread bins. He's been here all night and must be exhausted. After an hour in the kitchen teaching a temp guy this and that, it became clear he's not a keeper. Damn. She slides onto a counter stool beside Ava, who's nursing a cup of coffee and thumbing through Newsweek. Murray hates his employees sitting even on break. His car keys dangle on the hook near the register. Why's he even here on a Sunday? The adjacent mirror reflects a swath of diner along with her sorrowful face.
She sighs loudly and Ava looks up.
"Darla wants to join . . . the army, the Marines, I don't know. Our conversation didn't get that far. Ever see the parents of dead soldiers on TV? It's beyond me how they continue to support the disaster. I'd never be that forgiving." She pulls napkins from the holder, then squeezes them back in.
"That's bad news." Ava folds away the paper. "Why?"
"The military can't get enough fools to volunteer so they're offering pots of gold. I can't compete with that." Would Ava lend her money? Christ, she's on the road to desperate.
"Are you two getting along?"
"Teenage girls and their moms, what's new? We're okay together. Our fights are like summer storms, they're over quickly." Long fingers of sun reach across the countertop, reminding her she'd rather be elsewhere.
"You have to stop her." Ava sounds alarmed, no doubt thinking of Bobby.
"How? Tie her up? Lock her in the bedroom?" She's read about parents who do such things. There's that woman who drove her kids into the water.
"Hell, my husband was killed in Iraq," Ava mutters, as if Mila didn't know.
For a moment they both stare into the mirror, silent. In the near distance trucks rumble on the highway.
"Should Nick speak to her, you know, a man, a vet," Ava offers.
"She'd ride my tail for telling you. The things you hope for . . . the girl's getting older . . . the two of us will talk . . . reason together. Think again." Not totally true. Darla is reasonable. In fact she's damned logical, which is why it's so difficult to win an argument. Her daughter will do well in life. But she has to be alive.
Nick, with gear in hand, ready to go home, comes around and whispers something to Ava, who nods. His hand brushes her cheek.
They watch him leave.
"It's good between you two," she says.
"I think so."
"Don't hurry it."
"What does that mean?"
"Enjoy each moment, I guess. Sounds corny, doesn't it?" But, actually, she means anything can happen and then what.
Murray comes up from the storeroom. So that's where he's been. "Ava? Write down toilet paper, cleaner, dozen rolls of towels, and, also, we're breaking glasses. That has to stop . . . another box water-size . . . they're damn expensive. Mila, nothing to do?"
If Rosalyn were here she'd remind Murray Ava's finished her shift.
Murray stacks some stray plates, dumps them in a bin under the counter with a crash, then takes a fistful of bills from the register, counts them, and slams the register shut. "Temp could use hands-on, Mila. He's alone in there. But maybe you're otherwise engaged."
"Sylvie better sleep with the guy . . ." she whispers to Ava, sliding off the stool to wipe wet silverware that would air-dry in a minute. Murray stuffs the wad of bills in the burlap bank bag. The thought of filling her pockets comes and goes.
• • •
The morning sun highlights the faded lime-color walls, water-stained ceiling, sagging beanbag chair. Darla sleeps tight. She perches on the edge of the too-thin mattress, thinks to stroke her daughter's hair, but touchy-feely is no longer a habit between them. There was a time Darla clung to her like an extra limb, her little arm circling Mila's leg as if she feared her mother would disappear.
"Time to get up, sweetie."
"Umm." Darla hugs the pillow, her painted-pink toenails bright against the graying sheet.
"You'll be late for school. Come on." Was she on the phone all night? No point asking, she needs calm between them.
"What time is it?" Darla mumbles.
"Seven-thirty."
Darla lets go of the pillow, swings her legs off the bed. "Why did you wait?"
"Relax. It's only seven."
"Damn! I always fall for your stupid trick."
"Because you're a great student. You'll get some kind of scholarship."
"Mom, I go to a less than mediocre high school. They don't even have a music department. They don't have any AP courses either. I'm not getting any scholarships without that kind of stuff. You just don't understand. I'll get financial aid, but it won't be from Yale or Harvard."
"Is that where you want to go?" It never crossed her mind.
"Eventually. To law school."
"So that's why you want to lose a limb in war."
"Jesus! You're so negative. It's a wonder I have any aspirations listening to you all my life."
"Darla, reconsider it. It's not a smart move. We'll find money somehow, somewhere."
"No we won't. I have no rich relatives or living grandparents. We manage, that's what we've always done. Big deal! Not only will the army bonus help, I'll have veteran's benefits. It'll make all the difference."
"I can't stop you after July, but waiting gives you a chance to change your mind. You sign up today, it's over." She wants to kick the wall.
"Let me get ready."
Reluctantly, she gets off the bed, shoves her hands in the back pockets of her jeans. "Did you see my note about dinner tonight?" But Darla has shut the bathroom door.
She listens for her daughter's tuneless voice to belt out a song in the shower. The staccato beat of water on plastic sends another message. Darla's pissed at her mother's refusal. Someday she'll understand. Maybe. She smells gas. The pilot light is out again. It could be the wind, so she shuts the window above the sink. Instantly, it's too hot. She's beginning to hate this kitchen. All kitchens. Opening the oven door, she strikes a match; it catches. She'll insist Darla apply to a good college. Then she'll find money somehow, at least for the first year. She remembers the costume jewelry, the silver watch, the wedding band her mother left her, the whole package worth a few hundred at most. That should pay for a day or two of college. Christ! Borrow from Rosalyn? She can't ask her now. There's Murray. She'd rather rob a gas station. Then Darla would have two parents in prison. Great!
She grabs a bowl, breaks two eggs and beats them so hard the table trembles.
• • •
"This pub . . . you never have to wait," she says, dropping the car keys in her bag.
"They have good Bloody Marys," Darla offers.
"When did you . . . ?"
"Oh, Mom, you're so predictable. I plan to have one tonight. Is that a problem?" Darla sashays ahead in a calf-length peasant skirt that swirls as she walks. She's built like her father, the long narrow body. Her deep, dusky voice is his as well.
"I'm refusing to argue with you."
"Okay then. Is there a particular occasion for this dinner?" Darla tosses back. "Someone new in your life? Changing jobs?"
"All of the above," she offers, enjoying a momentary lightness.
Beer and fish smells mingle. Netting hangs from the ceiling, a cardboard mermaid caught in one. Blue haze floats under fluorescent lights. Background music adds to the appreciable noise. In a high-backed booth with wooden benches, they open their menus. A woman and four antsy kids sit nearby.
The waiter, not much older than Darla, eyes her daughter appreciatively while scooping away two extra settings. He pours water and whips out his pad, stuff she's used to doing herself. They order drinks. She chooses fish and chips; Darla wants the clam and corn dinner.
"So, Mom?" Her daughter smirks.
"What?"
"You're not dropping fifty dollars here for nothing."
"Then what am I doing?"
"Buying my compliance?"
"Let's eat and enjoy, okay?"
"That'd be good."
The waiter returns quickly, the drinks festooned with celery and slices of lime. Again, he eyes Darla. "I hope she's old enough?"
"I'm her mother, you think I don't know her age? Christ!"
"Okay." He leaves. Darla grins.
"Good, Mom, that was believable. Anyway, I'm only weeks from legit."
"Wrong, sweetie. Legit here is twenty-one."
"That's as retrograde as everything in the burbs." Darla scans the room looking for proof.
"It's that bad living here?"
"Suffocating. People talk about anyone who's not like them."
"You find that everywhere."
"At least in the city there's shame."
She decides not to ask how she knows. "Is Michelle's family that way?"
"Her father is."
"Do you ever think about your father?" Her heart speeds up.
Darla stares at her. "What?"
She shrugs. "Just wondering."
"Mom, you don't wonder. What about my father?"
"Nope."
"You never mention him. He left you with a baby." Darla's eyes are steady on her. "How come you never remarried or even considered it?"
"You don't know what I considered." She tosses out the straw and drinks straight from the glass. The spiciness nearly chokes her. More likely it's the conversation. There are truths she's still afraid to tell this girl. That's the horror of a lie. You have to keep lying.
"Mom? Where are you?"
"Right here. The drink is dynamite."
"Answer my question."
"I never married because I was never divorced." She speaks low, as if the woman in the next booth were interested.
"There's a statute of limitations. All these years of abandonment . . . you don't know where he is . . . it's automatic . . ."
"You'll be a good lawyer."
"This isn't about me."
Yes it is, she thinks. And ponders ordering something stronger, a double scotch neat, but that'd be a giveaway.
"How come you don't date?"
"I went out with that Luke guy, remember? You didn't like him."
"Mom, that was ages ago. Don't tell me you stopped seeing him for my sake." Her daughter glares at her.
"Of course not. He turned out to be mean. Drank too much, too. I don't know. He didn't appeal to me. I'm having a hard time discussing boyfriends with my daughter."
"We're not discussing your habits in bed."
"Darla!" She finishes the drink and immediately wants another.
The waiter sets down their plates. Her food looks greasy, heavy, impossible to digest.
"Mom?"
"What now?"
"Are you a lesbian? I don't care, I'm just curious."
She stares at Darla, whose sudden childlike expression breaks her. "I don't know how to stop you from signing up. If there was a father around, he might change your mind."
"How would that work?" Darla sounds angry, but anything's better than that scrunched face. "My father would say, don't go, and I'd be scared to disobey him?"
The sudden clamor in the next booth is a relief. Water's wiped up, new napkins brought, children reseated.
"You still didn't answer my lesbian question." Darla's eyes on her again.
"I'm not a lesbian."
"Do you hate men because of how he treated you?"
"He treated me well."
"Oh what a consolation. A sweet guy who left you hanging."
"You're getting drunk."
"On one Bloody Mary? I doubt it, Mom. But it's good for us to chat like this. Not that I learn anything. You have a way of telling me zip, you know that?"
The girl's right. Mila, the queen of doublespeak, but it's no longer enough, not by a long shot. "Do you worry about what you don't know?"
"That's an interesting question. At times."
"I'm going to order another drink, then I'll tell you something." Maybe she's drunk.
"Uh-oh." Darla teases, though her eyes widen some.
She hails the waiter and orders a scotch neat. Then she chews on a piece of bread to put something in her stomach. "How's the food?"
"Fine."
Two of the kids climb noisily on and off the next bench. She can feel the vibrations. Someone should stop them.
"I bet those kids are a handful."
"Mom!" But she won't look at her daughter.
"You were like that, couldn't sit still. I can remember, I think you were five—"
"Mom!"
Her gorgeous girl will leave and maybe never return. What then? What now?
The waiter places the scotch in front of her. She takes a sip, the medicinal taste a reminder of the unpleasantness to come. And why is she doing this? Her cheeks are hot, her face flushed, a slight buzzing in her ears. She remembers like it was yesterday the two policemen at her door, the baby she didn't know how to comfort fidgeting in her arms.
"Mom, I'm waiting."
"Your father's in prison. He's been there sixteen years. We agreed you shouldn't know. That you should grow up without feeling stigmatized by his mistake." Each word a piece of flint cutting her throat.
Her daughter stares at her.
"My father's a criminal!" Darla's voice rises.
She nods. Her heart pounding now, she chokes down the scotch, which isn't helping, it's hard to breathe.
"You think lying about it makes me less the daughter of a criminal?" Darla pushes away her plate. "Did he murder someone?"
"There was a robbery. A person was shot. The law said anyone involved was guilty." Her robotic tone isn't helping. If she stops to take a breath she won't be able to go on.
"Did the man you call my father pull the trigger?"
"No. He was there, that's all."
"That's all! Jesus frigging god!"
"Darla!"
"You visit him on the sly?"
"I've never visited."
"Beautiful. You don't know where he's locked up, or it's just inconvenient?" her voice scary sweet.
"That's not important."
"What's important is how manipulative you are, lying to me all these years, laying it on me now to paralyze me. Think again, Mila, and think hard. Because you've given me even more reason to get the hell out of here."
Mila? She's been banished. "Your father—"
"Stop saying my father like I know him."
She takes a deep breath. "Jimmy was in the first Iraq war. It ruined him. He came back more restless than ever. He could hardly sit still, never mind keep a job." She can taste the bitterness. Even in bed he slept in fits and starts, except when he made love, the only time he could get out of his head. "He wanted money quick, just the way you do."
"Must be in the genes," Darla shouts, and bolts from the table. The woman in the next booth looks up.
• • •
Driving slowly, she rolls down the windows and scans each side of the road for Darla. Christ! What did she accomplish? Alienated her daughter . . . broke a promise. God knows what Darla thinks about any of it. And what did she expect from her daughter, a smile, a freaking hug? She opened the damned box, didn't she?
The heat in the car is suffocating; there isn't a breeze. It's hot like it was the last time she saw Jimmy when the A/C in the motel room was broken. He was afraid to open a window. They made a bed out of an empty dresser drawer and sat hunched over the baby talking softly. Jimmy's earnest expression, his hands pressing hers, his positive tone so certain. He knew the way for her to follow. He'd get to Florida, set up and send for them. If he was caught, ended up in jail, Darla must never know. He left money and took off. The baby woke up crying. She rocked her till morning and never did crack a window.
Her daughter's nowhere she can see, probably doubled back and called Michelle to pick her up. One thing is certain: Darla isn't on her way home. She doesn't want to go there, either. She heads for Sully's bar.
• • •
It's dark inside. A few regulars stare at the muted TV screen or maybe at their own ravaged faces in the mirror. She finds a small table in the rear. Fraying high school pennants decorate the wall, a faint yellowish light from the jukebox playing oldies. She hears the front door open and close but doesn't look up. She's tempted to cry but it won't do any good. The lying is over but not the anxiety, which fills her with cold, hard fear. She calls her home number. Useless. She leaves a message on her daughter's cell phone to contact her immediately. She scrolls down to Michelle's number, calls her. No answer. Leaves a message there, too. Where would Darla go this time of night?
The bartender, a tall, slim man in his fifties, waits impassively for her to decide. She glances at the soiled page that stands in for a menu, the smudgy print hard to decipher. Nothing she wants. She orders a double scotch neat, water on the side. Jimmy drank bourbon. She did, too, all those years ago. She liked so much of what he liked.
Her finger traces the table's gouged surface. Jimmy carved their initials in whatever tree trunk caught his fancy. It pleased her, the same as their long walks did, her hand in his the whole time, chatting about anything and everything. Old memories flickering again in her brain, it's her fault, saying his name aloud like he's part of her life.
The bartender places a sizable glass of scotch in front of her. Jimmy wrote her one letter from prison insisting she go on without him. That it was the only gift he could offer. She took it. Things happen to people every day; she knows that. Still, after his arrest, the separation was unbearable. He lived in her head, walked at her side, appeared wherever she went, at work, bars, laundromat, the supermarket. Holidays were hell. The pain was so intense something inside her finally switched off, released her. What exactly that was she never figured out. Explain that to her daughter.
How is it possible for someone who never knew her father to follow in his footsteps? It's eerie. She tries Darla's cell again. Damned gadgets go to voice mail after a few rings. How many times can she say call your mother? Darla often mentions clubs where they hang out, but did she listen to the names? Hopeless. Oh lord. When did he come in?
Murray, carrying a drink, strides toward her in dark slacks, white shirt. "Look who's here? Can I sit?" He doesn't wait for permission.
She nods anyway, helpless.
"How come you're here?" He's drinking neat like her.
"I could ask the same question."
"No one home yet." He sounds mellow, unusual for him.
"I fought with Darla, don't know where she ran to."
"These kids," he offers. "What about?"
"Wants to join the army." God, is she really sharing with Murray? He doesn't have a clue about her daughter. When Darla occasionally comes in for a meal, he doesn't chat with her, just notices the food she eats.
"That's crazy."
"I thought you liked the war."
"It's no place for women. Better stop her," his tone more dismissive than interested.
"After July she won't need my signature." The powerlessness of it all threatens to drown her.
"Tough, tough. Kids test us."
"You don't have children."
"Doesn't mean I'm stupid." He scowls at her.
Man's still her boss. "Of course not." She finishes the drink, which isn't doing a thing to calm her.
"The truth about kids is . . . they grow up."
"Murray, she'll be sent to a war zone. Growing older may not happen. Get it?" She can't help herself.
His hand covers hers.
She stares at him. "What's going on?" She removes her hand.
"I'm a little drunk, a lot lonely. Let me buy us a round." He holds up two fingers for the bartender to see. Then gazes at her as if she's the answer to a question he's been pondering. It doesn't take a shrink to know he wants her to probe his misery, help him unburden. Men like him expect that from a woman. An instant fantasy, bed down with him, then ask him to pay for Darla's education. The only thing more ludicrous would be doing it.
"One more and I'm off," she says. No doubt the man looked old when he was thirty. Some men are like that, wheelers and dealers, anxious about next steps while peering over their shoulder to see who's stalking them. It wears out the face. Jimmy was open, boyish, probably doesn't look much older now.
The bartender serves the drinks, places their empty glasses on a tray, then wipes the table with a rag and leaves.
After a quick sip, she reminds him, "Really . . . I have to go find Darla."
"Sure. Sure. But stay a few minutes. If something bad happens to me, Sylvie will be sorry."
"Why's that?" she mumbles. Because who cares. An unhappy man with a lot of money doesn't rouse her sympathy. One quarter of what he paid for the house would cover four years of college.
"The woman isn't being attentive the way it's supposed to be. She works late, sometimes even on weekends, but doesn't need to work at all. She cooks when she can, then freezes the damn food for me. I sit there alone with the dogs. The house is so big it echoes. Why did I even set it up? I don't know, Mila, I don't understand her. Women tend to be devious."
"Thanks, but I refuse the insult." He looks at her bewildered. She chooses to conserve her energy, which is dissipating with each drink.
"Maybe Sylvie's having an affair, but I don't think so. A man can smell that kind of thing. You know what I think? I think Sylvie doesn't know how to be married. No one taught her. Maybe we both need a few lessons," his tone gloomy.
"Well, amen to that," she says. "Murray, go home. Sylvie will be there by now. She'll worry."
"That's what I want," his tone defiant. "See—"
She'll trawl a few bars, find one of Darla's friends who can give her some place to search for her. Oh Jesus and Mary, the beach . . . she and Michelle always go there. At this time of night? Maniacs lurking in the shadows? Two beautiful young women walking alone in the dark?
"What I believe is—"
Christ! Panic has her by the throat. She grabs her purse.
"—sooner or later, everything has a solution," he says.
• • •
"Darla, I can't drive around all night. My father will have the force out. Stay at my house. We won't answer the phone."
"No." Not with Michelle's brothers parading around, she doesn't like sleeping with them in the next room, either, probably jerking off. "Drive me to Rosalyn's dad, the guy I work for. It's right down the road."
"This time of night?"
"He naps on and off all day and watches TV forever. I'll knock . . . just try it." She stares out the window seeing nothing. So what if this Jimmy guy fought in Iraq. That was a million years ago.
"Well, sorry for thinking you're nuts."
"It's around the corner, second house."
Michelle looks at her. "You can still change—"
"Wait to see if I get inside."
She runs up the old driveway, the blue and white TV screen flickers through the foggy window. She knocks hard at the door. Hears the slow plod of feet.
"Who's there?"
"It's me, Mr. Joseph, Darla." She waves at Michelle to leave.
He opens the door. "What are you doing here? Something happen? A guy chasing you?"
She smiles. Because what could the sick old man do about it. "I had a fight with my mom, and don't want to go home. I could sleep on the couch, do some work for you in the morning. You don't have to pay me. Sort of a trade-off." She hears the quickness of her words and wonders if he's gotten it all.
"Come in. Shut off the TV. Make some coffee."
She turns off the TV, goes in the kitchen. The coffee's already made. Old man probably forgot. This Jimmy guy could be old, too . . . more likely Mila's age. Who cares? It's all so stupid, the lies, the whatever. It probably made her mother feel powerful. Secrets can do that.
She's hyper enough but pours two cups, brings them in. The old man sits in his BarcaLounger but doesn't touch the coffee. She eyes the oxygen canister, the long tubes extending into his nose, the slow hiss of air, hypnotic. Dropping on the couch, she feels the springs. There's an unused bedroom. Later.
He removes the oxygen lines and hangs them over the chair arm. "What did you fight about?" His wheeze is more pronounced than usual. God, don't let him get worse while she's here.
"My mother revealed a long-held secret to stop me from joining the army. She figures if I'm going to die, I might as well know the truth."
"What secret?" He looks concerned. Sweet old man.
"My father's been in prison sixteen years. He didn't want me to see myself as the daughter of a criminal." Her mom's setting her up, that's what. Her mom wants her to talk to this Jimmy guy who went nuts in Iraq. Isn't going to happen.
"Well, maybe you're not."
"Mr. Joseph, they don't put people away all those years for no reason."
"Circumstances make people do stupid things, then they have to pay. Doesn't mean he's a horrible person. He could be. Doesn't mean he is."
"You're only saying that to make me feel better."
"I knew good people who did a lot of dumb things. Some ended up in jail, some in rehab, some in the ground. I've seen my share . . ." He gazes past her, his mouth slightly open.
"My mom's determined to keep me from signing up, but soon I'll be able to go without her permission. Thing is, the next few weeks with her will be hell. I wish I could leave tomorrow."
"I know a woman who had a baby when she was around your age. She gave it away, couldn't care for it properly, so she said. Your mother didn't do that." He takes a sip of coffee. "It's cold," he hands her the cup. For a moment she stares at it, then returns to the kitchen to make a fresh pot.
While it drips into the pot she checks the fridge. Stuff there for breakfast. The old man likes his food. She'll remind Mr. Joseph to replace the oxygen lines. The last thing she needs is for him to have a breathing attack. What if this Jimmy guy is sick like the old man, some disease or cancer? Shouldn't she know about this for her future? Everyone says she looks like her mom, still . . . how come this Jimmy guy wasn't curious about her? Prisoners always want contact with spouses and children. He sounds like a selfish prick. How could her mother love a guy like that? No wonder she doesn't go out much, scared of repeating the mess she made.
She carries in a steaming cup of coffee. His eyes are closed. She leaves the cup on his TV tray, tiptoes into the spare bedroom. She switches on one of the dim lamps. It must've been his daughter's room, though there's no sign of life anywhere. The bed's made, nothing on walls or dresser top except an old, grimy mirror. She could check the closets but she doesn't want to. Okay, she's surprised. Okay, shocked. Her mother gabs about every little thing. What else hasn't her mom told her? Who cares? Sooner than later she's out of here. Her mother will adjust. She'll have to. She checks her cell phone: five missed calls from her mother. Anyway, no one asked her mom to be a martyr. If she got pregnant, she'd have an abortion. There's also a message from Michelle. She listens. Her mother called the house and woke her father, who yelled up to ask where Darla was. "Obviously, I was fast asleep."
Tiptoeing out, she sees the old man is still napping. Should she reattach the oxygen? That's another thing. Stay home with her mom and then what? Become a nurse. Take care of sick old men. She's fond of Mr. Joseph. But he's not her future. And the hellhole he lives in isn't either. Maybe old people don't care where they live. That would never be her. Her house, too, is disgusting. It's clean, her mother keeps telling her. But what the hell does that mean? Clean? You can't call something so run-down clean. Jesus. These people fool themselves because they're either too lazy or too scared to change. Or worse, they don't know any better. It's depressing. Her cell phone vibrates and she takes it into the spare room, stares at it till it stops. Her mother's having a heart attack. She picks up the last message; Mila's walking the beach looking for her. She calls her mom's number.
"Darla?" her mother's breathless voice.
Who else, she wants to say. "I'm signing up day after my birthday. Period. Otherwise I'll live at Michelle's until its time to leave."
"I hear you."
"I want an agreement."
"I won't nag you. Come home. Where are you?"
"Rosalyn's dad's."
"Christ. You woke the old man?"
"He wasn't asleep. Mom, something else."
"Yes?"
"Why did you listen to this Jimmy guy?"
"I was young and scared and he sounded so sure."
"Are you sorry?"
"I don't let myself go there."
"All these years you didn't think about—"
"I did a lot at first."
"Were you heartbroken?"
"Let's talk more about it when we see each other. It's easier that way."
"Mom, easy isn't the way life is. Never mind. Just pick me up."
Mr. Joseph wheels the canister to the doorway. He's reattached the oxygen. Suddenly she feels like an intruder. He didn't offer this room. "My mom's coming to . . . I'm sorry if I . . ."
"Get me a six-pack on your way over tomorrow." He shuffles out.
She waits on the lawn, her back against the rough bark of a tree; light opening in the sky, the heat of the day beginning. Does her mother know where Mr. Joseph lives? Even if this Jimmy guy suddenly wants to see her, she'll refuse. What kind of father waits sixteen years to offer a hand? No kind she wants to know, though she'd probably get a lot of pointers from him about the desert.
• • •
Mila glances out the car window. Leaves hang precariously from tree limbs. In another week they'll fill the gutters. It's been a summer hot enough to wilt anyone's spirit. But now, a cool breeze ruffles her hair. The map's open on the seat beside her. She's terrible with directions, giving, taking, or following them. On long trips, her daughter was the guide. She's avoiding the highway, going through small towns different from her own. Huge houses that don't resemble each other remind her of the one Murray built. Her foot on the gas pedal is bare, though high-heeled shoes lie on the mat. Darla would get a chuckle seeing her in the sexy green dress she hasn't worn in years.
Darla's last e-mail said she might get a leave after basic, but "Mom, don't count on it." God knows she counts on nothing, trying to adjust is all. Less shopping, less cooking, no checking the clock to wonder where her daughter is, but none of it gives her an ounce of comfort. Dread dogs her even at the diner where she hoped to be too busy to notice. Morning, noon, night, she tells herself, what's done is done. No use. She can't accept the danger. Iraq? Afghanistan? It's where Darla's headed. Friends try to help. Ava offers words of support, Shelly, who should know, assures her that all will be fine, that time passes quickly. Even Sylvie sent a message. But, really, it's a crapshoot. The truth is no one has a clue what will happen, she least of all.
And truth is what's she's after. It's about time. He doesn't know she's coming. He'll recognize her, though. She's the best prize he ever won. He said so too many times to forget, his face lighting up whenever she came in view. She, too, always excited to see him, now as well. She's nervous, yes, but not scared. Years get used up; she can't fill them in for him, even if Jimmy does ask. Maybe he'll bow his head, reveal graying hair, or offer her that grin so close to sadness. Maybe he'll search for the girl in her that's no longer there.
9
Happiness Exists Somewhere
The news stuns her. Still seated on the exam table, legs dangling, she stares at the dove-gray suit hanging on the closed door as if it belonged to someone else, the person who walked in an hour ago. She can't go back to the office. A dental appointment, she lied. Her purse is beside a tray of instruments and she takes out the cell phone. The receptionist's voicemail picks up. "Hi, it's Sylvie, I'll be back a little later than expected. That is, if anyone asks. Thanks."
If she hadn't called in sick a few days ago she'd take the rest of the afternoon off. That, too, was a lie. She went to Liam's funeral. The entire service at the East Hampton church was bleak: a scattering of old people, the urn buried in an unmarked hole behind his house. Gone, all signs of him. Murray would've accompanied her, but she didn't want him to see her cry, didn't want to deal with the insistent questions that would follow.
He's already badgering her. In bed last night he wanted to know why she had to work? Why does she stay so late? Why can't she be home for dinner? Reasonable questions. She feigned sleep. He won't stop asking. She knows that. He's not an easy man to live with. He dotes on her but that's the problem. There's nothing concrete to hang her discontent on, except who he is.
Sliding off the table, she tears off the paper robe, stuffs it in the receptacle marked waste, and dresses. She needs to walk.
• • •
Fifth Avenue sometimes distracts her, even lifts her spirits. She allows herself to study the well-dressed people, stylish storefronts, soaring architecture, St. Patrick's Cathedral. She peers into the FAO Schwarz window of toys, decorated for Christmas, though it's weeks away. Grieving takes time, she reminds herself. Holidays don't help. But it's more than Liam. The sad eyes of a stuffed giraffe nearly as tall as the real thing stare at her. She turns away, surprised to see Shelly across the avenue, and finds herself striding toward her. Shelly's a woman of endurance, what she needs right now.
"Hi. What are you doing in the city?" Shelly's black coat does no justice to the lovely combination of her dark hair and light eyes. The woman should wear greens and purples.
"Hi to you. Bruce attends two days as an outpatient. I drive him in, walk around, then pick him up. I don't mind."
"How's he doing?"
"Up and about. They found medicine he can tolerate. Until now, the stuff they fed him made him a maniac. What are you doing here?"
"My office is nearby. A cup of coffee?"
"I wouldn't mind a glass of wine."
She leads them up a side street to an easy-to-miss Irish pub stuck between two restaurants. Inside there's no hint of afternoon light. The coziness suits her. No one would look for her here. People stand two deep at the bar, office workers, unemployed, lunchtime trysts, who knows?
They find a small table and a waitress appears dressed in slacks and a sweater. Maybe she's just helping out. Shelly orders a glass of house red. She asks for club soda.
"How's your youngest son doing?" What she wants to ask is how to go on when things are beyond control.
"In that horrible place." Shelly sighs. "The kids face death at every turn, not that I would say so to Mila. I can't pick up a newspaper or listen to the radio, forget TV. If Michael came home Bruce would recover faster. The doctor thinks Bruce has confused his own soldier past with Michael's war, which I could've told him weeks ago if he ever asked. I worry because so many reservists are being made to stay longer than their terms. Two months ago, my middle guy wrote to the National Guard that Michael has to come home for a family emergency. No dice."
"It is frightening. I'm so sorry." Shelly looks worn, her heart-shaped face thinner. "Have you eaten lunch?"
"Food isn't friendly lately. My oldest brings me takeout. Firstborns are worrywarts."
"You must've been a baby yourself when you had him."
"I swore I'd never go through the pain of it again, but the memory evaporates. It's the way of the world. Are you thinking of . . . ?"
Heat flames her cheeks. "Well . . . Murray's in his fifties."
"Unless they've given up their favorite pastime, age isn't a factor."
The waitress sets down their drinks. She's glad for the interruption. "Your children sound devoted. It must be comforting."
"I don't know. They have their own lives and a life takes time. If Bruce got sick like this with a bunch of youngsters in tow . . ." Shelly shakes her head.
"I'm sure now that Bruce is on the right medication he'll be on the mend shortly."
"He'd like to put in a few hours at the diner but full-time is still iffy." Shelly looks past her, clearly embarrassed to be asking.
Murray doesn't want Bruce coming back but she'll appeal to his sympathy bone. Bruce is a vet, son in Iraq. Murray likes feeling noble. He's always announcing how much food he sends to the shelter. "I'll talk to Murray about it. Nick's overwhelmed, I'm sure."
"That would be great. I hear you did a fine renovation job in the house. It must be lovely."
"Seems so." Is there nothing in her life greater than that damned house?
Shelly searches her face. "Is everything all right?"
"A dear friend died recently. It still grieves me." It was Liam who encouraged her to return to work, who said her unused energy was festering. The photo of Liam's dead son, what happened to it?
"You need to find something uplifting to get it off your mind. While Bruce was in the hospital I bought a print of women dancing in a circle. I hung it in my kitchen. I look at it and think happiness exists somewhere. Sounds silly, I know."
"God, no, Shelly. It's a wonderful thought. But . . ." she shrugs.
"What? You can tell me."
She gazes at Shelly, whose features blur slightly in the ashy darkness. How can she say she's pregnant with no idea what to do? Shelly will think her foolish for throwing away everything for a bit of pleasure. Is that what she did? Is that what she wanted to do? "My house feels too big," she says inanely.
"I can't say I know what that would be like."
Of course she wouldn't. They probably live in one of those tiny . . .
"I envy you," she admits. "You have a family who cares about you."
"Sylvie, everyone at the diner can see Murray's crazy about you. He'd do anything for you."
"Yes, he would," she agrees, letting the truth of it sink in to no avail.
• • •
At her desk, the computer open on the columns of numbers and names on the sales screen, but her mind refuses to focus, is everywhere but here. Another woman would be excited, perhaps even relieved. Not her; she's amazed, yes, but mostly scared and confused.
When her cell phone rings, her eyes flit to his corner office, vice president stenciled large. "Dinner tonight?" Harry's voice certain.
"Well . . ." she hesitates. "Can I get back to you?"
"Why?" he insists.
"I had tentative plans."
"Cancel them."
She doesn't answer.
"Buzz me." He clicks off.
There's nothing spontaneous about Harry. Most likely his wife called to say she'll be busy this evening. Across the wide expanse of desks separated by see-through walls of Plexiglas, salespeople talk into headphones while watching computer screens. Most here are starting careers. In the cafeteria and hallways they gab about ad agency profits and losses as if it were personal. The top echelon doesn't care a whit about any of them, the indifference palpable in Harry's anecdotes, which she counters. A spunky woman, he calls her.
It's no use, she can't concentrate, presses a few buttons and the document disappears. Endless white flakes drift across the snow-filled screensaver. Why choose a winter scene? She didn't; it was simply here, like Harry.
• • •
Harry waits at the cloakroom to check their coats. His elegant cocoa-brown suit fits smoothly across his broad shoulders. His stylishness first attracted her. They chatted a few times at the proverbial watercooler. He asked her to accompany him to account focus groups; wanted her input. She was flattered. In the chauffeured car maneuvering through crowded midtown, Harry didn't talk about work. He discussed foreign cinema, Italian films were his favorite, art shows in unexpected places. A man interested in museum exhibits, who'd been to the theater too many times to remember. Her acting background enthralled him. After their second "work" day, Harry asked her to dinner. She accepted easily, which surprises her still.
They're ushered to a table near a window of mullioned glass. Except for the warm coral blush of streetlamps, it's too dark to see much outside. Logs crackle in a nearby fireplace, the ambience seductive. People dine here late, usually after a cocktail party. They bustle in full of the cold outdoors, reluctant to give up their coats, impatient for whiskey to warm them. Most other nights, she frames the scene as if in a play, but she doesn't know what her role is now.
The menu is in his unblemished hands, manicured nails. She glances at his wedding band and hers.
"Shall I choose for us?" he asks as he does most evenings they're together.
"Why not?" she replies as usual, though not a bit hungry.
"I do like knowing you," he says softly, studying her face, which isn't as concentrated on his as he expects.
"Why's that?" she responds, also softly, while another scene plays in her head. I'm pregnant with your baby. Why didn't you use contraception? Because I had unprotected sex with my husband for a year, and nothing happened. I thought my fertility was gone. Likely story, he'll say. She glances at him. A man with four grown sons he'd never disappoint. He made that clear from date one.
"Your attitude makes me think. It's a challenge." He's trying to engage her. Who wants to bed down with a distant icy woman? Is that who she is tonight?
"My repertoire is endless," she offers through the fog of worry that threatens to dull her mind.
He smiles. Lovely teeth, strong chin, dark eyes that glitter. In the theater, he'd be the matinee idol but never Sir Laurence, whose talent reined over appearance.
"In my position, people tell me what I want to hear. You don't. That's what makes you refreshing."
"I try to be entertaining," she says lightly. Conversing about nothing feels almost beyond her.
"See, like that."
He can't see and he isn't curious either, not really. He's killing time till they're ready to leave for the corporate apartment, his to do with as he pleases. She suspects their encounters there are unlike anything he experiences at home.
The waiter pours a finger of wine in each glass, waits for Harry's nod of approval, then leaves. Harry is about to top hers but she places her hand over the rim. He glances at her but says nothing. She rarely has more than two drinks at any time, her mother's vacant alcoholic eyes never far away.
"There's something I can't figure out about you," he says.
"What could that be?" she asks, hoping miraculously he'll force the truth out of her.
"Are you ambitious?"
"Meaning am I after your job?" she teases, though disappointed.
"No, that would be stupid." He laughs. It's a nice sound, deep, warm.
"Which I'm not." She takes a sip of water. "Tell me," she probes gently, "why are we here, together?"
"Don't you think it's a little late in the day to be asking that?" His careful tone reminds her she's not delivering the correct lines.
"No, it's never too late." She shrugs, then fakes another smile.
"The brutal truth is I'm bored at home. And I find you appealing. I have from the beginning." He reaches across to stroke her hand. Two people married to others acting as if they're engaged in some dramatic first experience when passion's at its highest. Her interest in any of it tonight is less than nil.
"That doesn't sound so brutal," she says.
"Why the sudden curiosity?" He places the napkin on his lap. The man doesn't want to know.
"I'm interested in marriages. Why some go sour and some don't." No, Harry, I'm interested in how you'd react to my having a baby. I'm interested in knowing how disastrous your response would be. Interested, yes, but unable to put it to the test.
"My wife and I have been together twenty-eight years. Relationships plateau. Some get past that, other's teeter." He's reciting probably the same litany he gives all his women. Once she lied at an audition, said she was sixteen, and had to pretend for the duration of the play. But how long can she pretend not to be pregnant?
"An amount of time not to be sniffed at," she quips.
"You don't say much about your marriage . . ." He refills his wineglass.
"A bit stultifying," she offers. How to describe what she doesn't understand. Even if she did, it would feel disloyal to discuss it.
"You haven't been married that long," he reminds her.
"You can pepper me with questions in the boardroom, not here," she quips.
"In the boardroom I'd already know the answers."
She laughs. "So I hear." Rumor has it Harry's a hard man, that he fires people for first offenses. If she hints at her predicament he'll insist she deal with it immediately. She isn't ready, has no clue what ready would feel like.
Harry raises his hand and the waiter hurries over. He gives their order, pronouncing the French dishes with ease. He's comfortable in his skin, doesn't want to be anyone else. Onstage, she's been a mother, a wife, a lover. It's easy to play other people.
• • •
Her car, a dark shape beneath a sputtering lamp, is one of a few still at the station. She beeps open the door and slides in. On the road, an occasional light shimmers in the distance, the sense of the ocean near. She ratchets up the heat. Dead winter's on the way. Her refusal to go to the corporate apartment surprised him. She said dinner lasted longer than she expected. Harry didn't press her. He walked her to Penn Station, chatting easily. He kissed her cheek, his aloofness apparent. She didn't perform well tonight, didn't provide the pleasure the world owes him. He doesn't need her—she's a whim, a toy like the ones in the store window. Yes, the first weeks with him were exciting. Even now it's easy to imagine them in Bali, something they talked about. She sees them lying hand in hand on a sparkling white beach and she tells him she's pregnant, her voice languid, reassuring.
• • •
The dogs bark as she nears the door. Inside, they wrap themselves around her legs. Stroking their sleek bodies, she murmurs, "Hush," refusing to say their hateful names. Murray, in a thick terry cloth robe, watches like a pleased proprietor. He offers her a sip of his wine, which she declines. "Tired, I'm off to bed."
"I've been waiting up. I have something to say about us," he declares.
Is this the moment where everything changes? Has he had enough, wants out from this woman who fills so few of his needs. Oh she sleeps with him, but is she affectionate? Does she make him feel important? Or has someone told him about Harry? Is he about to send her away? She wants to say can't this wait till the weekend, that she doesn't like being blind sided, but Murray's withheld fury can be frightening. She follows him to the living room.
He sits too close on the ridiculously long sofa. She glances at the refurbished room, new chairs, lamps, and tables, paintings hung; the dogs are asleep in the corner; the silence broken by the sound of the surf, which pulses like an angry heart. Is now the time for truth? When he takes her hand, she feels only trepidation.
"I want to make you happy. But you're not behaving the way a wife should."
"Behaving? I'm not a child."
"Cut me a break, okay? I don't want to fight over words." He drops her hand. "I've decided we should go away for a week. I can't leave the diner longer than that. Lie on a beach, relax together." Oh god, he's climbed into her Harry fantasy. Is he toying with her too? Does he know more than he's letting on? Except it isn't like him to be subtle.
"And I don't like your job . . . it takes all your time . . . the hours . . . the nights . . ." The familiar lament goes on but she tunes out.
She sighs loudly. He stops talking and gazes at her.
"Having a beautiful house doesn't fulfill me the way it fulfills you. It's why I went back to work." The truth in pieces. It's the only path that seems possible.
"Why do you have to come home so late?" his voice plaintive, his expression sorrowful, deepening creases track the sides of his mouth. He can't fathom being the cause of her distress.
"A day or two . . . then I promise we'll have this talk. Now, though, I need to get some rest."
"We can't put it off longer than that." He strokes her hair. "Okay, go to bed."
She walks away quickly, praying he won't call her back. Coward! He turns off the lights, follows her, then slides in bed spooning her. He kisses the back of her neck, presses her shoulder to turn around. She does, easier than explaining why not.
• • •
She prepares the morning coffee, then leaves quickly to take her shower. When, finally, she hears his car backing out of the garage, dogs in the backseat, the relief floods her. She phones the office, leaves a message that she won't be in without saying why. Indecision churns her insides. Pulling on a pair of sweats and heavy socks, she finds her sneakers, and wraps a scarf around her neck. She grabs her old peacoat from the hall closet and stuffs gloves in her pocket.
• • •
Walking along the muddy shore, hair whipping her cheeks, spray dampening her face, she misses Liam again. He knew about Harry. She told him the affair was opening her mind to possibilities she'd forgotten existed. That Harry's hand might lead her out of her marriage. Liam thought she was searching for her feelings the way a painter seeks light.
She makes her way to where Liam had his lean-to. All traces of it are gone except for a vague shadow where the coal fire used to be. Up here the sand is drier. She sits and stares at the dusky horizon, the black water. A few gulls sweep over her still-discernible house. There's no hint of sun. The fast-moving clouds are dizzying. She never saw Liam work. He painted in the early morning but often he'd leave out a canvas for her to see. She'd marvel at his ability to see so many different scenes in the same place. He explained it's what made creating exciting.
Almost overnight, it seemed, he stopped painting and retreated to a chair on his patch of lawn. She brought whatever creature comforts she thought he needed, though he never asked for a thing. He wouldn't discuss what was ailing him. He hardly ate, barely talked. He took to his bed and stayed there. His last weeks remain painful to contemplate. He'd look around the room with faint wonder or study his hands as if they held some final secret.
Two of his sea paintings hang in her living room. Murray likes them because they prove where he lives. No doubt Harry would appreciate them as well.
Murray and Harry, each with his version of her. If she has an abortion, neither man would have to know. So what's stopping her? Nothing but a clump of cells moving toward recognition, that's all it is now, the doctor said. Soon, though, it'll resemble Casper the ghost in those ultrasound pictures women bring to the office. Except it'll be her ghost taking shape.
Dolls never attracted her, fake babies with silly big knees. A real baby will have needs she can't even fathom. She used to imagine being a mother different from her own. A ridiculous memory, as ridiculous as thinking she could care for a child by herself. Food, clothes, rent, sitters . . . her theater-days are over. And so too any chunks of money they used to bring in. Besides she's too old for anything but character roles and would have to wait eons between them.
The screams of wheeling gulls split the air. Wind bites her cheeks, sand lands on her lap. A coin of sun peeks through the clouds, then disappears, the entire scene bleak. If Liam were here he'd remind her that place reflects feelings. If she shared her ambivalence, he'd no doubt say there's such a fine line between delay and denial.
Hoisting herself off the sand, brushing herself off, she wraps the scarf around her head and trudges back to the house. If she dresses quickly, she'll make the afternoon train to Wantagh.
• • •
She pays the driver and watches the taxi head to the mall to pick up a fare. In the graying late afternoon light the neon sign flashes Murray's Diner. A silver-striped bus with polished aluminum siding. The long, wide windows clean as new morning. Murray's as fastidious about the diner's appearance as she is about her own. Dressed now in forest-green slacks, lime color sweater, makeup, she's ready for an audience.
Pushing open the door to the tinny sound of chimes, the sizzling smells of burgers and fries greet her along with the faint scent of barbecue sauce. Ava waves, eyebrows raised with questions. Why isn't Sylvie at work? Is something wrong? Did she quit? Q uestions only slightly offtrack. Once the affair ends, so will her job. Mila, wrapped in her puffy jacket, is ready to leave. The shift changes at four but everyone's putting in extra hours to cover for Rosalyn. The wall clock reads ten past five. She hangs her coat on one of the hooks.
"How are you?" Mila asks on her way out.
"Fine. And how's Darla?"
Mila shrugs, frowns. "Haven't heard bad news yet. Murray's in the storeroom, want me to get him?"
"No. I'll find him, thanks."
"I'm out of here, " Mila swings past, allowing in a burst of cold air.
"Coffee?" Ava calls.
"Tea, thanks." She sits on a stool, elbows on the counter. "You look wonderful." Ava's newly styled hair frames her slim face with soft curls.
"You think so?" Ava half turns to catch herself in the mirror, then leaves to set up two customers. It's not quite dinnertime, but half the tables are filled and there's a low buzz of voices.
Nick carries out a heavy tub of clean coffee mugs and places it under the counter. He looks a bit sleepy but handsome as ever; perhaps he just came in. He nods to her; she smiles, wonders if he opens up with Ava. Their romance upsets Murray, who fears that Nick will walk if he complains. Murray brings home each employee's transgression and wants her sympathy. It pisses him off when she defends the staff. He is holding open Rosalyn's job for another month, though he's sure she's not coming back. Even more surprising, he visited Rosalyn twice, once in the hospital, once at home. He said Dina was there helping out. She would've asked for particulars, but Murray's more about what's right or wrong than descriptions.
The leather booths, small, square tables, inlaid floor, low ceiling, Formica counter, familiar reflections in the gold-flecked mirror. At this very counter, she and Murray spoke for the first time ahead of a short courtship full of activity: bars, restaurants, sightseeing, Saturday nights at B&Bs in the Hamptons. Weekday evenings he'd show up outside her office full of enthusiasm about a new dinner place he'd discovered. He seemed indefatigable and it charmed her. He admired her, it was clear. He listed all the things about her that pleased him, and not just once. He was open about why he never married, how most women bored him. He told her what he wanted, what he was waiting for, drawing her into the simplicity of his confessions more than any chemical attraction.
What of that matters now? she wonders. Her face in the mirror offers no clues; her resolve dissipates by the second. She must get this scene over with. How else to go on? Leaving the comfort of other people's presence, she descends the storeroom steps slowly. The urge to turn back remains strong. Halfway down, the dogs leap up to greet her, nearly toppling her. "Easy, easy," she cautions. There are boxes piled against walls, supplies fill metal shelves that reach the low ceiling where a dim overhead bulb casts an eerie light.
Murray in rolled up shirtsleeves and an old cobbler's apron from who knows when stares at her as if she's a ghost.
"Didn't mean to shock you. I just wanted to—"
"Why aren't you at work?" he scolds. He fears surprises even more than she does.
"I took the day." She scans the room, which has no chairs and only two tiny windows. It's a cellar. A faint whiff of pesticide threatens to nauseate her.
"Why are you here?"
"Can we go somewhere else to talk?"
"Why?" his tone suspicious.
"I can't talk here." Either the room has contracted or claustrophobia has her by the throat.
"What do you want?" He's not going to make this easy.
"Murray . . . please," she begs. "Meet me somewhere."
He stands there as if his feet are nailed to the floor, wearing an angry anxious expression that asks why is she doing this to him?
"Murray? Where? Please?"
"Sully's," he says reluctantly. His eyes are steady on her.
"I'll be there." She turns away quickly lest he find a way to keep her in this dungeon.
The dogs follow her up the steps. "No. Stay."
"Rummy. Cheney. Down, now!" Murray shouts. His voice goes through her. The dogs obey.
Upstairs, a temp is wiping tables. Ava watches the woman with a mournful look, maybe thinking of Rosalyn.
Sylvie grabs her coat. "See you soon," she calls without turning. The door chimes shut behind her.
Snowflakes drift and whirl in the sudden wind, a purple tinge ahead of the darkening sky. Sounds of highway traffic fill her ears. The distant small houses with doors shut tight remind her there's no one available to help her. Why not simply disappear . . . woman last seen at diner . . . Pulling her coat close, she hurries toward the glittering shop lights of the mall.
• • •
The smell of spilled beer and old cigarette smoke lingers in the air, but it's warm inside. If Sully's were well lit all would be revealed, the warped floor, aging walls. Like an old theater, it's weathered many tales. She finds a table in the rear where early on she and Murray had drinks and spoke of past events. He described how badly he'd wanted to be a boxer, the hours he put in at the gym. But his hands were too small; no one would take him on; how awful the rejection. She told him about her mother's drinking problem, her father's suicide. He said simply, she'd turned out fine. What didn't she see then?
Her eyes travel the dark corridor leading to the door and his arrival. He'll explode, of course he will. He'll be enraged, incoherent, ask—no, demand—over and over why she took up with Harry as if any of her answers will register. He'll accuse her of being wanton, loose, a betrayer of everything good. Or is that what she thinks? He'll be devastated as well. Lord knows what it will do to him. Any other scene she'd know her cues, but not here, not now.
Her thoughts whirl dizzily, her throat tight, perhaps a shot of bourbon to warm the cold fear, or a martini for courage? The bartender is on his cell phone with his back to her. She's the only customer. The table holds a smudgy one-page menu she can't imagine consulting, and remembers Harry fingering last night's menu. It's finished with Harry, has to be, of that she's certain. What's weird is how little she cares.
The door bangs open and Murray rushes in, unzipped ski jacket flapping, woolen hat in hand; snow glancing his hair. His face scrunched tight, his eyes find her.
"I didn't order drinks," she says before he can sit. "I'll have club soda." He looks at her with alarm, then strides to the bar. Knocks hard on the counter to get the bartender's attention, then his eyes remain steady on her. She turns away, nothing on the wall but frayed pennants, old photos.
Murray slams down her soda, the spray wetting the table. He takes a long pull of scotch, then drops into a chair without taking off his jacket. His jaw is locked, his hands balled in fists, his shoulders hunched defensively. He's breathing hard. It's unnerving.
"Talk." His imperial tone, the one he uses with the dogs. Her stomach clenches. She takes a sip of soda. Truth in pieces, she reminds herself, first the pregnancy, then Harry, then she'll flee.
He leans his face close to hers, the smell of scotch on his breath mingling with aftershave. "Talk," he demands again.
"Murray, I'm pregnant, but it's not . . ."
He leaps up. "What the fuck . . . I thought you couldn't . . ." He's shouting.
"I thought so too."
He peers at her, eyes narrowing, face reddening.
"Murray, it's not—"
He grabs the back of her chair, slides it right and left, emitting little squeals or maybe he's choking. He's creating a scene; the bartender's watching. "Murray, stop, please."
"Okay, okay," his raw voice nearly breathless. "Oh baby, that's so marvelous. I'm happier even than when you said you'd marry me. I'm not happy, I'm thrilled. God, Sylvie. When did you find out?"
"Yesterday . . . but Murray, it's not . . ."
He clasps both her hands, kisses them. "How are you?"
"Good," she says, somewhat confused.
"Anything you want, baby, anything, just name it, it's yours for this great news." He's talking so fast, his certainty is overwhelming. Harry's response would look nothing like this.
"You really are happy," she murmurs.
"There's that empty room at the end of the foyer."
She knows where it is, knows where's he's going, and something snags in her throat, a pill too big to swallow.
"We'll make it into a nursery." He's still talking very loud.
She says nothing.
"We'll buy funny wallpaper, we'll hang those little musical toys. The dogs could be a problem. But they took to you real soon. I bet they'll protect the baby."
She says nothing.
He drains his glass. "Of course there's things I can't do. Diapering stuff. I don't know . . . maybe I can learn." He sounds faintly embarrassed. "I need a refill."
He hurries away. A deafening silence fills her head, the kind that occurs after an explosion. For a moment she can't remember where she is. Her eyes flick to the bright screen of the mute TV. Heavily equipped soldiers traipse through strange terrain, reminding her of Shelly who bought a painting of merry women, who refuses to be laid low by Bruce's condition, who makes the most of her situation. A customer enters and sits at the bar. He, too, leaves his coat on, wet with snow. Maybe he's staying for only one drink. She's not wearing boots, how will she manage the slippery outdoors? Her heart is pumping, her mind gone numb.
Murray's loud gleeful voice is offering to buy the man and bartender drinks. Her mother often offered her presents, old scarves or sweaters that stank of whiskey and cigarettes. She hated them. Once she wouldn't take the item. Her mother, angry, grabbed her arm, pushed her face close, and said in no uncertain terms, it was rude and unkind to refuse anyone's gift.
Is she giving Murray a gift or is he giving her one?
Whatever he's saying to the customer, she can't make out, but is sure he's boasting . . . a father . . . first time . . . never thought . . . His face is hidden but she imagines him grinning with flushed cheeks, the way he does before they make love. The customer lifts his glass in a toast. Did he turn to her?
The sad march of whiskey bottles across the back mirror leads to the door. It isn't far. She could run past Murray into the cold night. But what would she find there that isn't already here? Her eyes slide to the flickering yellow light of the jukebox. Maybe, somewhere, there's music.
10
The Things in Between
She's going crazy. Each morning, now, she talks to her image in the mirror, says, Rosalyn, life isn't half bad yet. Then she intones Sister Judi's words from long ago, everything can be gotten through—how did Sister know? Crazy, indeed, but so what? Spying a parking space, she pulls in, flips down the mirrored visor and checks her head scarf.
Dina's car pulls up beside hers.
They walk across the crowded mall, the heat of the day apparent in everyone's slow trudge.
"It's good not to have lunch at the diner. Murray's drone . . . should he sell, shouldn't he . . . who cares?" Dina asks no one in particular.
"He does sound serious," she murmurs.
"See what love can do?" Dina reminds her.
"All for Sylvie, right?"
"And the coming baby, don't forget."
"The baby . . . of course," she says quietly.
• • •
The café has A/C, thank heavens. It's crowded. Voices are loud; people seem indifferent to anyone hearing what they say. Jack, too, doesn't care. She has no idea where he is when he phones her—at work, a pub, in the street—only that he tries to probe her deepest thoughts, wants her to unburden.
The young, attractive hostess in a T-shirt, long skirt, and flip-flops leads them to a table, drops two menus, fills their water glasses. Beautiful is dangerous, her father would mutter when she was that age.
"Did you see her earrings . . . four hoops in one ear," Dina says, her lobes free of adornment.
"I'm having a glass of merlot with my sandwich."
"Okay, me too. Why not?" Dina agrees.
"Anything from your son?"
"If they caught him I'd know. Anyway, it hasn't been that long." Dina's dismissive tone surprises her.
"Are you worrying?" she asks.
"Thankful not to hear and ashamed to admit it. I can't bury my feelings anymore," Dina asserts.
"That's wisdom."
"Yes . . . compensation for the insults of aging and . . ." Dina stops. "Sorry."
"Don't be. You've nursed me through the whole rotten treatment. What would I have done without you?" Dina still shows up at ten every morning. Maybe she anticipates a time when Rosalyn won't be able to get out of bed. If so, she hasn't let on.
"I'm glad I had the skills to help," Dina says simply.
"Oh shit. Let's not talk about me. Your remarks about Tim . . . I understand. He's a problem that can't be solved easily."
"The guilt, the love . . . mixed together . . . that's hard too." Dina picks up the menu.
"Is there anything I can do?"
"Like what?" Again, Dina's faintly challenging tone surprises her. The truth is she's no longer curious about lives she needs to let go of.
"I don't know. It's what friends say," she offers.
"I suppose. I'm having a tuna melt." Dina closes the menu.
The waiter steps up to the table, wipes his hands on a stained towel tucked in his waist. Messy hair, sullen expression, clearly he'd rather be elsewhere. "We'll make your life easy. We'll both have the tuna melt with a glass of merlot."
"Did you return this morning's call from Jack?" Dina asks.
"I spoke to him three times this week, four last week."
"Are you avoiding him?" Dina searches her face.
"He's after me to go abroad."
"So?" Dina's chin lifts combatively.
"It's too far to travel." She doesn't say time has edges now. That there are things here she must do first.
"You're done with chemo for a while."
"I'm done with treatment, period. Anyway, Jack . . . it's complex."
"You mean he cares about you but since you're sick it's a waste of time?" Dina's serious eyes fasten on her.
"Don't be crude."
"Well, then, explain it better," Dina says matter-of-factly.
The waiter serves their drinks. "Foods still being prepared," he mumbles.
"Do you know how I met Jack?"
"At a bar, I thought."
"Well, that's true . . . I worked for an escort service. He paid for an arranged date with me."
"Oh." Her friend attempts to sound casual.
"I quit the service months ago," she assures her.
"Aren't those one-night stands," Dina's voice low.
"Jack was the guest who stayed. Why am I even telling you?"
"You wanted me to know," Dina says, her composure restored.
"Probably." Except it wasn't her intention. Lately, it's as if another person inside her decides what to say without her permission. It's what she fears, isn't it? Jack will hear what she isn't ready to share.
"Rosalyn, how you two met doesn't explain why you won't join him in Europe. Obviously he wants to show you around."
"You're one persistent lady. Let's say, I'm not in a vacation mood."
"What kind of mood is that?"
"Drop it. Please."
"For now." Dina takes a sip of merlot.
• • •
Glad to wave goodbye, she drives off. Dina's chatter about Jack felt intrusive. She doesn't want to think about him. He senses that but won't accept it. He can be endearing yet exasperating. She stuffs a pillow behind her lower back to ease the muscle spasm. She passes a row of refurbished houses with freshly painted porches and non-leaking roofs, a hard sun ignites the front lawns. Her father could move into one of the houses. She offered to arrange it months ago but he refused. It's been a few days since she saw him. It feels even more difficult to be with him. He stares at her like she might die in front of him, or else won't look at her at all. The man doesn't know how to be supportive. Simply doesn't. At least he approves of the high school student she hired to help him. She finds the boy aggressive.
She pulls into his driveway. Good, the student's car isn't there. She beeps to let her father know she's arrived. To her surprise, he walks out carrying the new oxygen container, which has a handle and resembles a thermos. "Let's go to the beach," he says, sliding in slowly.
"What?"
"Forgot where it is?"
She starts the car. "Why?"
"I want to be outdoors while it's warm."
The truth of that doesn't sit right, but she never could figure out how he thinks. She glances at his strong, craggy profile; he'll outlive her. She said as much to her brother, who reassured her that wasn't so in words that held no weight. The rest of the relatives are equally Pollyanna. No doubt family members need to believe what they will for their own comfort.
• • •
They sit on a boardwalk bench facing the water. A few clouds play hide-and-seek with the sun. Blankets, towels, umbrellas arrayed on the sand; lifeguards in high white chairs, whistles at the ready. Parents watch their children cavorting in the water, the noise of it all distant. She and her brother played here winter and summer, though her mom wouldn't allow them in the ocean even on the hottest days.
Lotion and salt air, she smells both, but feels outside, a witness. Yesterday, too, in the supermarket, she felt at the far end of a tunnel. Snippets of conversations reverberated in her head. It's as if what's said matters less than the things in between she must still uncover.
"Dad, do you know how to swim?"
He nods.
"Mom didn't. She was afraid of drowning."
"She was afraid of a lot of things."
"Parents pass on their quirks. I can't swim."
"Worse things have happened."
"When Mom took us here, she sat in a tiny canvas chair, her feet buried in the sand if it was warm. In the cold, we were all bundled up. She wore boots. You were never with us."
"The fire station didn't believe in time off for the beach."
"Or weekends?"
"What is this?"
"Just mulling stuff over, remembering . . . Mom made us wash our feet with the hose because you hated sand in the house. In winter, we had to leave our boots outside. It was a rule. She always wanted to please you."
"What do you want, Rosalyn?" She hears him breathing.
"Was Mom a happy person?"
"Who's happy?"
"It's a question I regret not asking her."
"That was a long time ago. Stop torturing yourself. And me."
He must've known what her mother felt; they lived together for god's sake. "Did you love Mom?"
"I was nineteen when we married." He lifts the oxygen container from the ground to the bench.
"Did you love us?" He spent more hours with his fire team than with them. The guys were his buddies, drinking mates, the ones he confided in if he confided at all.
"I supported my family, took care of all of you. What else can I say? What else do you expect?"
Is she stirring him up? Or will he switch on the TV as soon as he gets home to blot out the past hour? In third grade, she begged him to come talk to her class in his fireman's uniform. He refused, said, what for? She cried bitterly. Her mother whispered he was too shy; he wouldn't be comfortable. Comfort's what he always craved.
"I just want to understand you better," she says.
"Why?"
"Dad, you're exasperating."
He glances at her, then looks away. "Something I want to say . . ." his voice a hoarse whisper. "I have a bit of money. If there's a treatment out there your insurance won't cover, I'll pay for it."
"I have enough money."
"You never let me give you anything."
Is that true? She looks at him but he continues to face the water.
"Okay. Thanks, Dad. If I come up against that, I'll ask you for help." It's the best she can do.
"I meant what I said a while back about meeting my grandchild."
"Don't go there," her voice rising in desperation.
He inhales shakily. Then silence.
She could ask about the student helper or if his pals have visited, the ball games he loves, anything to break the silence. But, suddenly, she's weary of the ancient dance between parent and child. And she wonders, is it too soon to take him home? She mentioned a doctor's appointment at three, which is a lie, but the truth would be impossible to share.
He's hunched over on the bench, still staring ahead. Whatever he sees out there has captured his attention or is simply easier to look at.
• • •
From the driver's seat she watches him take small steps up the path. The maple tree in full leaf casts filigreed shadows, its thickly gnarled roots heaving the old lawn. She used to pray those roots would lift the house off its foundation so they'd have to move out. The prayer came back to her during the weeks of chemo. The intravenous bag was slowly deflating, her body exhausted, her mind, though, was wild with memories and fantasies. Faraway countries she'd visit, Zanzibar and Saint Kitts, names she heard somewhere but knew nothing about. Where's Zanzibar? She composed letters in her head to lots of people, but sent only one.
Her father reaches the door but doesn't turn or wave goodbye. She beeps to let him know she's leaving, glances at her watch. Nearly two. Arriving first is out of the question. If he's a no-show . . . but his terse phone message was explicit. Three p.m., Friendly Fishermen's Pub, Bridgton. He was never one for long phone conversations. What will be will be, she reminds herself, and refuses to give the next few hours form or content.
She knows the pub, which is dark in the afternoon and well lit in the evening. She ate there several times with Mila. Poor woman can barely talk about Darla's going to Afghanistan. Mila who will only step in a church to get out of the cold said she made a pact with God, promised not to complain if He brings her daughter back intact. On the other hand Mila talks nonstop about Jimmy, his gray hair, beard, handsome as ever, so recognizable, how each visit with him pleases her, his compliments about her youthful looks, how he loves seeing her and doesn't take his eyes off her. She even jabbers to Murray about him.
• • •
At a minute to three, she slides out of the car, walks up the back ramp, pushes open the heavy door, and enters near the bar. She scans two customers' faces, a muted TV screen, the bartender fiddling with a cranky A/C. Then she follows a long narrow corridor to the rear booths. She sees him in one, looking out the window. Is that buzz-cut marine, or army? Is he balding? His big shoulders the same as years ago, the chest broader, though. Wearing a white T-shirt, his muscular arms tanned, the short, flat fingers unchanged. She slips into the booth across from him. "Hey Carl."
"Rosalyn. Rosalyn. How the hell are you?" He grins; his wide black eyes no longer merry, his sun-weathered face creased. Years ago, his smooth skin was soft, no five-o'clock shadow either. Now the beginnings of a beard sprout under his chin.
"Not sure how to answer."
"Yeah, your letter said . . . sorry about . . ."
"Me too."
"Have a drink." His shot glass empty, his beer stein nearly drained, he hails the waiter, who looks no older than they were when they met. "Beer or bourbon?" the waiter asks, indifferent to her presence. Carl orders both and wine for her. "A very long time," he muses, taking her in.
She nods. He used to tease that she had two words for each one of his. Now she's strangely shy. "I heard you were in Iraq."
"Three stints. Reserves. I'm getting too old but I'd go again if they ask me."
"You don't look old."
"Yeah, friends never do."
"Are you working?"
"Helping my brother fix up a basement. I've only been back a few months. It feels forever. Can't fit in . . . it's like trying on an old jacket that won't button. Everything's too tight."
The waiter brings their drinks. Carl drains the last drops of beer and hands off the glass. "Last I heard you were at that diner."
"Who told you?"
"I can't remember. My memory's shot and so is my hearing, so talk loud."
"Was it awful?"
"Nothing good."
"Thank heavens you weren't injured."
"Not where you can see." He chuckles.
"But you want to go back?" Easiness creeping in between them.
"Funny, huh? The weird crap happens here. Bad sleep, jumping at sounds, drink like you know . . . but over there I know the sounds, sleep like a seal in the sun. Still drink like a . . . Hey, why talk about it? It comes out like a long whine." He downs the bourbon, takes a sip of the beer. Some of his color is from the booze. "How'd you find me?"
"I called some old friends . . . heard you were married." All this banter with someone she hasn't seen in too many years to count. Still, it's what one does, she supposes, except these questions are not what she's here for.
He gazes at her a moment. "Katie left me after the last tour, took a job in Atlanta. Couldn't get far enough away, I guess." He finishes the beer and looks around for the waiter. "Hey," he calls too loudly. "Another round."
The waiter frowns and the diner floats into her head. Murray barking at someone, customers impatient for service, Ava whirling from the counter to the tables. And Willy? She holds him there in his booth.
"Sorry about you and Katie. It's none of my business anyway."
His fingers drum the table, eager for his drinks. "You still have that terrific face. I'd know you anywhere."
Terrific face, unforgettable eyes . . . His husky voice comes back to her; so, too, the gentleness, how he cared for her. How could she forget? "That's sweet."
"And you, babe? Married a few times or still with the same lucky guy?"
"I never married. My aunts, my cousins, my father all point this out whenever they can. Not that any of their marriages are very inspiring either."
"Yeah, family. Best stay away from the bunch of them . . . or so I've learned. Anyway, if you never married I must've left an impression." He tries for boyishness, but sounds sarcastic. Jack's gentle, comforting, though persistent voice comes to mind. But Jack's older, hasn't been to war, his tragedy endured slowly over time, water on a rock.
"An impression . . . absolutely," she agrees. An impression she made sure to erase the way she managed to turn away from whatever needed tending. She won't let that happen again. She can't.
As soon as the waiter serves the drinks, Carl downs the shot of bourbon.
"You do that often?" She points to the empty glass.
"As often as I can. Don't worry. It takes a whole lot to knock me out. I'll be right back, going to visit a man about a—"
She watches him saunter to the restroom in his camouflage pants, a few bulging pockets with unbuttoned flaps. She remembers his tight jeans the night they broke up. She couldn't fit into hers. Her stomach was puffy and stretched. The stitches hadn't disappeared yet. Her breasts were still heavy. Her body felt unrecognizable, unlovable. His looked untouched, everything in place, without pain or scars.
It was a summer night and they took a blanket to the beach. Other couples were scattered about. They found a space to themselves. She remembers the water lapping gently at the shore, occasional laughter from somewhere nearby, a sky invaded by stars. He tried to kiss her. She turned away. He began stroking her hair. She told him to stop, said he made her want to vomit. He kept asking what's wrong, promising he could fix it. At first she didn't respond, then a stream of invective she's ashamed to recall burst from somewhere so deep inside and left her trembling. He drove her home without a word, didn't call again. It was what she wanted. She'd just given up her baby.
He slides in the booth and takes a long pull of the beer.
"I was remembering our breakup."
"Oh yeah?" He doesn't sound interested.
"I owe you an apology."
"Yeah . . . right . . . okay, I accept."
"You're not the least bit curious?"
"Hey, I figured out why you didn't want to see me anymore."
"Whew, that's a load off." She wipes her brow.
He chuckles. "Why are we here, babe?"
"I need to know about our daughter." There it is, the pussyfooting finished.
"What exactly?" She detects reluctance.
"I need information. Your sister's best friend took her." She can still taste the weird mixture of emptiness and relief.
"Rosalyn, Rosalyn . . . Why would she want you in her life now?" He finishes the beer and looks for the waiter.
"I made a will. I'm leaving her my savings, my condo. The attorney needs to be able to contact her when the time comes." Heat rushes to her cheeks as if she just spiked a fever. It's all she can do to sit still. But she can't take off, not yet.
He leans forward. The bourbon's worked itself into his face, sweating now. "I can't help you. I have nothing to do with her."
"She doesn't know you're her father?" her voice rising. Why would she? They gave her up. They have no rights. It's there again, the tunnel with him at the far end.
"The girl knows she's adopted. I hear the girl's mother filled her in some about us, how young we were . . . so on." His eyes are on her, but she's not sure what he's seeing. Suddenly his reticence, her powerlessness, the whole encounter pisses her off. And his drinking doesn't thrill her either.
"Think! Remember something for god's sake." She stares hard at him.
"Jesus where's the fucking waiter?" His hands open and close, his jaw slack. He used to be laid-back, pliable. Now he's brittle enough to break, and he knows it. Jack comes to mind again, his ability to withstand turmoil, his voice asking her to lean on him.
Two people talk softly in a nearby booth, their words hum past her ears. A few streaks of bright sun invade the room, land on the stained wooden floorboards. She takes a sip of the wine. It tastes old, rancid, vinegary, and she remembers the Chablis in her fridge.
"I'm going up there to get another drink. Fucking waiter." He grabs the table to steady himself. "Her name's Lacy Marino." Then sways toward the bar, his head bent beneath the mess of his life.
• • •
Home at last, the drive back was endless. Phone messages blink on the machine. Her flowers need fresh water. When they begin to wilt, she tosses them. These, though, meaty yellow roses, she bought yesterday. The girl's name repeats in her head as it has since he said it. A name matters. A name gives substance, rhythm, color—allows an image to form. Lacy as a grown woman, a combination of Carl and herself, dark eyes for sure, thick, wavy hair. Maybe she's petite like her or sturdy like Carl.
She finds herself in the bedroom closet pushing aside shoeboxes, scarves, purses on the too-high shelf till her fingers press the soft leather-like surface of an old photo album, which she retrieves. Dropping on the bed, she turns pages, a picture of her mom—who loved her children—reading the newspaper, which she did from back to front, wanting to fill her head with trivia before letting in bad news. Photos of friends she hasn't seen in years. And here's one of her at Lacy's age now. She's wearing jeans and a tank top, leaning against some guy's old jeep, holding back a curtain of hair to reveal what must've been new dangling earrings. She's grinning. When she was Lacy's age she flaunted her appeal yet worked hard, saved money, had lots of friends. Someone Lacy might approve of. Removing the photo carefully, she returns to the living room, slides it in the manila envelope on the coffee table. Then she picks up the cordless and calls Dina.
"It's me. There's something else I never told you."
"Hold on." She can almost hear Dina settling into her chair.
"Yes, Rosalyn, what is it?" her friend careful not to sound eager.
"When I was seventeen I had a baby . . ." she begins. It's a story she no longer wants to keep secret, the names and events unfurling easily. Lacy, Carl, the Manhattan foundling home where she lived for the final trimester, the nuns who cared for her, Carl's sister who took away the baby she never saw, her meeting with Carl today. She doesn't stop there but goes on about the envelope on the table, which contains her will, her father's address, other documents, and now a photo of her. "We'll never meet but Lacy will have something of me, of mine. Don't you see?"
"Rosalyn, you should meet your daughter," Dina declares.
Her eyes flit to the black-and-white print on the wall that resembles a Rorschach blot. "Well I hadn't thought."
"What's to think about?"
"Lacy, her family. I can't just walk into their lives and—" Again, saying her name feels gratifying, proprietary.
"She must be curious about you."
The phone at her ear, she pads across the gleaming terra-cotta floor, the aqua rug catching the last light of day, wondering what it would be like to actually see the girl. In the kitchen, she pours Chablis in a glass and takes a sip.
"Are you there?"
"I'm thinking. And drinking."
"I'll say one thing: do what pleases you, never mind the result."
"You speaking from experience?" She's aiming to tease her, but finds she's listening intently.
"I rarely did what pleased me, only what was necessary. After a while, I didn't know the difference."
It's not entirely true. Dina loved her work. Perhaps she's speaking of Tim, but it's too coded to get into now.
"Dina, I'm sorry."
"That's not the point. It's a lesson to share. Or at least to ponder."
"I hear you."
"It won't be that difficult to find her."
Suddenly the success of learning the girl's name slips away, the closure she sought reopened. "Listen, I can't talk about it right now."
"I didn't mean to—"
"I know. See you tomorrow." She clicks off. Though she's not sorry she told Dina—her secrets less amazing to others than she would have expected—a claw of anxiety tears at her.
The yellow-striped watering can sits on the counter. She fills it and steps out onto her small patio, scanning the darkening lawn, the starless sky. She hopes for a better night. The last two have been fitful. Sleep came late and only for several hours, kaleidoscopic images racing around her head leaving blurry smears.
She waters the spider plants, their tendrils nearly touching the ground, then goes in to look at her messages. There are three. Two from Jack asking her to call him, he misses her, he's worried, please phone him. One from Carl, whose slurry voice recites an address. She replays it twice to be certain.
• • •
Her car inches along in the early evening rush-hour traffic. The MapQ uest directions Bobby downloaded are on her lap. Reaching Bruckner Boulevard, the lanes fan out like fingers on a hand. It's bewildering till she spies the North Bronx exit sign underscored on the map. Then it's one bleak street after another toward Gun Hill Road. What a strange name. She's never been in the Bronx, though her mother often talked about her childhood there. She tries to recall the stories but her memory has become selective, permits only events involving her. She understands. She's rechewing experience, can taste it. It's made her talkative.
She chattered on and on the other night when Ava and Mila visited. She described her work as an escort, her dates that never mattered, except for Jack. Then Lacy, all about Lacy, how Carl wanted to marry, how her mother dissuaded her. How her father wants to know his granddaughter. How Dina suggested meeting her daughter. How what to do feels beyond her. Ava said little, refilling Rosalyn's wineglass till she was dizzy with words and drink. Mila, though, made comments all along . . . mothers and daughters, great stuff, unbeatable, deep love but not easy, even after a million years together, and warned that the girl might not jump for joy on hearing from Rosalyn.
Also on the phone with Jack, she gave him what he wanted, a dose of her life. Then she added her indecision about Lacy, her confusion about what to do next. To her surprise he didn't recommend a path, instead assured her she'd figure it out. But she hasn't. She thought about sending the girl a note. If no reply came, well, then it would be an answer of sorts. But she doesn't want an answer. What does she want? A reconciliation, a meeting, a sighting . . . she has no idea. If she were a private eye, she'd park the car near where Lacy lived, camera at hand, hat pulled low. She has no hat, no camera, no plan, and no clue what the next hours will bring, only the destination.
• • •
Her car scales the lengthy incline of Gun Hill Road. Six-story buildings climb the hill with her, gray brick façades, small-paned windows, some curtained, some not, others protected with bars. Worn stone steps lead to run-down entrance courtyards. Covered garbage cans cluster in front of locked alleys. Do poor people live here? She sees no boarded-up or half-gated shops, broken pavements or skinny kids sitting on stoops. Is it more or less impoverished than other places in the city? It's difficult to say.
Cars are parked on both sides of the street. There's an empty space in front of Lacy's building and she pulls in.
The summer's evening light is waning, the sky a mass of clouds. She turns off the A/C, rolls down the windows. A weak breeze crosses her shoulders, bare in a pale blue T-strap sundress; her head wrapped in a navy scarf. Choosing what to wear was a trip in itself, trying on and discarding one outfit after another. She felt she was preparing for an audition. Exasperated, Dina finally decided for her. How many times did she check her face in the mirror, thinner, longer, her eyes, thank god, no different. Her skin, though, not the least bit rosy.
If Lacy appears—perhaps on her way home from work—will their glances meet in some mystical recognition? More likely the girl will walk past her. How will she know it's her daughter? She's given Lacy a height, a weight, even color and style of dressing, created an image out of a name. What if Lacy doesn't resemble her or Carl? It happens. What image does Lacy have of her? The seventeen-year-old her adoptive mother glimpsed? Or one Lacy cobbled together from bits and pieces of overheard conversation. Whatever it is, it's not a woman without hair, sick, pale, in bad repair.
No doubt Carl told his sister that she plans to leave everything to Lacy. No doubt it's how he got an address from her. But did anyone tell the girl? Her watch reads seven. Could be Lacy won't get home till later. Could also be the girl's in her apartment and won't leave till morning. Does she sit here all night? She stares at the building. It's a sixth-floor apartment. Does she have the strength to climb up? Maybe there's an elevator? Doesn't matter. She'll get there, somehow, find 6J and ring the bell. The door will open. She'll introduce herself. Lacy could freak out. Then what? Answer correctly and win a daughter? Answer incorrectly, and . . . one thing she does know, a mother is more than the woman who birthed her.
And what is it she can add to her daughter's life? Who is this visit for? If she invades that life and then disappears forever—because that's what will happen—would that be fair? If she were going to be around for even a few years none of this would matter. They'd have time to know each other, time to compare likes and dislikes, disappointments as well as revelations. Without time, the girl will be left with sorrow, perhaps regret, what might have been, what isn't. How can she chance that? How can she impose this perverse need on her daughter? Dina's wrong. Consequences matter.
Something else tugs at her, something she didn't want to think about before driving here. Lacy could've found her if she'd wanted to. They were never more than an hour apart.
Once more she stares at the gray building, its prison-like façade, littered courtyard, forbidding back ally. Her clean, pretty condo comes to mind, with its small patio, its array of plants, glass-topped table, pillowed chairs, so comfortable, so inviting. All of it will belong to her daughter who can sit there with a glass of wine, a husband, maybe children. It's this she'll take away from being here. It's this she'll hold on to.
Carefully, she maneuvers the car out of the small space. The sky is darkening. Behind her, the streetlights brighten the pavement with an evening sun.
• • •
A jet flies low and loudly over the car as Dina pulls up in front of the departures building and pops the trunk. "Take your bag. I'll park and meet you back here."
With trepidation, she watches Dina drive away. The crowds make her nervous. People push past her. It's loud. Cars drive up to the curb nonstop. Redcaps hurry by, tugging trolleys of bags. Long lines form in front of outdoor check-in podiums. The every-which-way of it is confusing, how to get by, which direction to go? Figuring out anything here feels beyond her. A strong urge to be at home where it's quiet, predictable, assails her.
It'll be better on the plane, she assures herself. In her own seat, calm, maybe she'll sleep. She steps through the first set of automatic doors. A second set leads to the ticket counters, but she doesn't enter, fearing Dina won't find her. A spasm tightens her back. They occur more often now, but over-the-counter meds still help, though she has stronger stuff if she needs it. She fishes in her purse for the red pills, shakes out two, finds her water bottle and swallows them. Cars continue to pull up, unload, and drive away. Endless. Automatic doors open and close incessantly, hordes of people in and out. Where's Dina?
Jack's call this morning, he wanted to erase any last-minute concerns. Said he was aching to see her, had taken care of every little detail. Repeated some of the many places he'd show her. How gorgeous the weather was, the ocean something else. His tone was gentle, his words meant to reassure. She didn't say leaving home frightened her now.
Leaning against a wall, tempted to close her eyes, she goes over the checklist. House and car keys, her father's phone number and address to Dina. Done. Clearly marked envelope with will, deed, and financial stuff on coffee table should Dina need it in case she . . . Jesus, she's only going for two weeks. Could she be any more dramatic?
"Oh Dina," she exhales, "I thought I lost you." In her summery yellow dress and sandals, her friend looks cheery, youthful.
"Check in and we'll go sit somewhere. It's early."
The line is long and disorderly, baggage everywhere. People study their tickets, eye children, each other, emanating anticipation, annoyance. And what does she feel? The last time she flew was ten years ago, to Florida. Her newly divorced friend was anxious the entire trip about leaving her children. It wasn't much of a vacation, and nothing like what Jack has planned. Still, she's never flown across the ocean. She's never been unable to trust her body.
"Did I give you Jack's cell phone number?"
"And his e-mail address at work. Yes. Relax. I'm exceedingly efficient. I ran the ICU, remember?" Dina glances at her, then away.
From the ticket counter they weave through the crowd toward a small restaurant filled with travelers, tables crammed together, loud voices, scraping chairs. The bar is festooned in outer-space décor, silver stars shoot away from blue and red planets. They find a small table near a Plexiglas wall overlooking the tarmac, and order two cosmopolitans. Huge unmoving planes line up ready to take off, the weight and wingspans challenging the idea that they'll do so. The ground crew in orange jackets as bright as their wands of light that cut through the air in semaphore code. She watches it all, a moving tableau that her plane will soon join. Dusk is beginning to settle, the late sun falling somewhere in the sky.
The waitress sets down their drinks and hurries away. "Serving an airport crowd can't be easy. It makes the diner seem like a breeze. Mila gave me a note from Willy. He wrote, Miss you, always will. Then asked when I'm coming back. I'm not, you know. I haven't told Murray."
"I'm sure he figured it out."
"Ava says Murray actually wants to sell. Amazing, Murray no longer there. He's a fixture."
"More like a relic." Dina looks in her bag for a tissue, her expression troubled. "Something I'd like to ask you—it's none of my business and you can say so."
Afraid to hear anything disquieting, she wants only protective custody from those around her, cages of love. How silly is that? "Go ahead," she says, and takes a long pull of the drink. It's strong, tangy, cold. She decides she'll have another.
"Why did you decide against more treatment?"
She gazes at Dina, the small, round face, eyes that penetrate. She recalls the sleepless nights, anxious days weighing another round of chemo against the agony of side effects when no good outcome could be promised. Even now, the turmoil of that decision is easily resurrected.
"Jack's a cancer researcher. My doctors faxed him results after two rounds of chemo. He didn't want to mix doctor with boyfriend but I begged him. My mother had only snippets of information. She had to intuit on a daily basis. I told him that would send me bonkers way before the disease got me.
"He wasn't happy with what he saw. A pernicious cancer, invasive. The stats on treatment weren't promising. He said no one ever really knew how much time, that miracles happen . . . that word clarified things. Why spend the time I've got left throwing up, tasting mouth sores instead of food, becoming so weak I couldn't walk by myself. You know the rest of it."
"Yes. I can't imagine what I'd do," Dina says more to herself. "I think you're brave," her friend's eyes moist.
"Dina, it's okay—"
"Sorry, Rosalyn. It's not like me to be—"
"I know it's not. You're all steel wool, right?"
Dina smiles. "Not exactly, but after years in a hospital you grow some tough skin."
Her glass is nearly empty. "Let's have another."
"You do. I'm driving." Dina's eyes watch her.
Seeing the waitress zip from table to table, she decides to order at the bar, which is three deep with people whose raucousness tells her they're pain-free. Two young men smile, bow, and part to make room for her. She grins. Ordinary consideration feels extraordinarily reassuring these days.
"Damn, I forgot to throw out the flowers." She places her drink on the table. "They'll dry up and flake all over the . . . Christ, why am I still worrying about such unimportant crap? Is that crazy or what?"
Dina says nothing.
"Before you picked me up, I ran around the place, a bundle of indecision. Should I apply makeup at home, on the plane? Who cares . . . makeup for god's sake? I'm dealing with life and . . . Dina I expected to let go of trivia, though what that would feel like is beyond me. It does happen. I've heard it said often enough, but maybe closer to the end." Her throat tightens. She takes another pull of the drink.
Dina's hand slides across the table and squeezes hers. "I've been with lots of patients, and sick isn't dead. Yours is a warm body to touch and a mind to think and feel. You're alive, Rosalyn. Why shouldn't everything matter?"
"But the junk that fills my thoughts . . ." she shakes her head. "This morning . . . never mind."
"Oh go on, rattle away. Lord knows what goodies I'll hear."
"I was in the shower. The shapes on the curtain reminded me of Halloween when I was a kid, maybe eight. My mother cut out the face of my pumpkin. She made one eye round, the other square. God knows why. I wanted them to match. She tried to fix it, but only made it worse. I stamped my feet, crying I wanted another pumpkin. My father grabbed the damn thing, opened the door, and flung it out as far as he could. I was inconsolable, hated him for days. But this morning the memory struck me as funny, my father, in uniform, heavy boots tramping across the living room floor to throw out a little pumpkin. I laughed out loud. It's been that way recently. Events transformed by circumstance."
"Or time. As the years pass, I see things differently, too."
"Like what?"
"My husband's death. I didn't really grieve. I sucked it up as another glitch and soldiered on. Maybe that was necessary, but he deserved more. Thinking about it lately, the sadness is there, untouched. On the other hand, Tim's difficulties growing up . . . they don't seem as enormous now as they felt then."
"What do you do about any of it," she murmurs, watching the planes move silently down the runway. Flares light the way now, the darkness complete. People at the adjacent table noisily push back chairs, assemble their luggage and file out.
"Aren't we the serious ones?" Dina quips.
"You think?"
"You're about to take off for unknown parts. I think that's serious business." Dina glances at her watch.
"A house by the sea in a place called Mumbles. Sounds pretty grave. Have you been to Wales?"
Dina looks at her as if to say when would that have happened.
"Well, neither have I or a zillion other places. It's an adventure, isn't it?" The drinks have produced a comfortable buzz in her. She stands to gather her stuff. The waitress rushes over with the check. She leaves an outrageous tip.
• • •
They amble past food counters, bookstores, newsstands, kiosks selling all manner of things, luggage, watches, even jogging outfits. So many people going somewhere, it's as if two disconnected worlds exist, one here and now, another elsewhere. She is a bit drunk.
"The whole point of this uprooting trip is for you to have a marvelous time. Can you remember to do that?" Dina asks as they reach security.
She takes in her friend's face, the combative jaw sturdy as ever. "I'll try. But I'm a bit shaky," she admits.
"Who isn't?"
They hug for a long moment. Dina gently disengages. "I'll stay here till you get past security."
Several lines snake around awaiting passage through to the gates. Unlike the earlier ticket line, there's an air of expectancy, even gaiety. Women in jeans, shorts, flip-flops, men, too, in casual wear, all leaving, same as her.
Searching the large tote for her boarding pass, her fingers brush the address book. She promised to send postcards from everywhere. As the line moves forward, she loses sight of her friend, but knows Dina's there, watching, waiting. Her friends are like that, loyal, constant, strong-willed, and opinionated, like her. When she returns there'll be wine and laughter as they extract every detail of her trip. These women love stories.
She slings her bag onto the conveyor belt, tosses her shoes, purse, and sweater into plastic bins, all of which slide slowly through loose leather flaps into a tunnel to be scanned. It occurs to her she's done with scans. No more machines that click and buzz looking for hidden flaws, no more unwanted reports, no more uncertainty. She can drink as much as she wants, eat whatever pleases her, stay up late or sleep for hours, because why not? She can have or reject anything within reach. Everything out there is open to her, which is exciting yet weird.
Walking barefoot through the metal detector, a strange question pops into her head. Can people die happy even if they're not happy to die?
11
She was Definitely Here
"What is it?" She puts aside the fashion magazine someone left at the diner, the clothing nothing she'd waste her money on.
Nick is pacing with a beer bottle in hand. It's distracting.
He glances at her, his usually expressive eyes opaque, then walks out. She hears him in the kitchen. Is he worrying about his daughter? He said he was relieved Glory decided on the Peace Corps.
She picks up the magazine, flips pages looking for something besides skinny models with big lips. A window fan sends currents of hot air in her direction. A shower might help to cool her. He returns with another bottle of beer, though the one on the floor is half-full, and resumes pacing.
Ignore him, she thinks. If he wants her attention he'll have to say so; she isn't in the mood to read minds. It's too hot. The magazine is open on her lap, but her thoughts tick off chores she plans to accomplish. It's her day off. Her daily visit to Rosalyn, of course. She, Mila, and Dina follow a schedule of care with the help of Hospice. Rosalyn wants to be at home. Who can blame her. After Rosalyn, she'll drive . . .
He plops noisily on a chair and like a sullen adolescent moves the beer bottle back and forth on the dining table, leaving wet rings.
Glory pads in barefoot through the unlocked door in shorts, her T-strap top revealing glowing tanned skin. "Hi all." She registers her father, who's pacing again.
"How was the beach?" Ava asks.
"Hot and crowded, but fine. The three of us had fun in the water." When Nick's back is turned, Glory mouths, "What?"
She shrugs. "Where's Bobby?"
"Outside with Hamid." Glory told her Hamid's upset about her upcoming departure. They've been seeing each other a few months and plan to stay in touch. The girl's more open with her than Bobby is with Nick, whom he pegs as Glory's father and therefore off-limits.
Nick's at the window now, fingers drumming the wall. "Ask Hamid in for a beer," he says suddenly, eagerly, as if Hamid's presence will undo whatever's troubling him.
"Dad. How many times . . . he does not drink, it's against his—"
"Of course, of course. I can give him a soda."
"He's sandy, doesn't want to. Ava we're going to the antiwar vigil in Sag Harbor. Bobby's coming with us."
"He is?" Glory's invited him many times. He claims it bores him to stand there. It must be that he's taken a shine to Hamid. She remembers Bobby's difficulty letting go of Mark's friendship, which was more like a courtship.
"I hope he doesn't get restless . . ." But their eyes are on Nick, who trades his empty bottle for the one on the floor, then stares out the window again.
"Dad, I've raided the fridge. See . . ." Glory jiggles a bulging canvas tote.
"Sure, why not?" Nick doesn't give a glance, his foot tapping nervously. Whatever's out there has his full attention. She and Glory exchange a quick look.
"Are you feeling okay? You sound strange," Glory says.
"Don't I always? Next time tell Hamid to come in, sand or not."
"I'm not keeping him away, I promise. Dad, I'm leaving." No response.
"We'll see you later," she pats Glory's arm reassuringly because now she's worried, too. He's always responsive to Glory. What the hell's going on?
"Should I stay?" Glory whispers.
"No, it's better for us to be alone," she whispers back, not sure if that's true.
When the door closes, he says out of nowhere, "Glory's reliable."
"And right on the mark about you. You can't find a place for yourself. Are you anxious about something?"
"That's it," he agrees, then strides to the bedroom. She follows.
"About what?"
"About telling you," his high-pitched tone a cry.
A rush of adrenaline fires her system. "Me? What?"
"I'm getting a beer. Want one?"
"No. Take it easy with the stuff."
He lopes out in his bathing trunks and undershirt; his shoulders burdened by whatever she's about to hear. Her mind races through recent talks but nothing troublesome surfaces. An illness? He hasn't been to a doctor. Is he about to confess about an affair? Except their relationship is smooth, loving, Nick constant, loyal. He always wants to be with her.
Damn, she wishes they were at her house. The ambience at his less than reassuring. Though used to the warps and creaks of floors and doors here, it's less cared for than her own place. Her eyes sweep the yellowing walls of the bedroom, the tiny windows, paint cracks along the ceiling. Nothing attractive. Still, they've bonded in his saggy bed with its unmatched sheets and pillowcases. Their schedules are so erratic; to be alone with him she steals a few hours and comes here. Sometimes Nick's asleep. Always, though, as soon as she enters the bed his long limbs wrap her. The gentle strength of his hands is a constant surprise, so too the satisfying turbulence of their lovemaking. His almost childlike subsidence afterward follows her home, makes her feel safe, though she can't say why.
What's taking him so long? "Nick," she calls. Outside the window, the small lawn appears as thirsty as the white sky. The sprinkler is on though they're supposed to save water. It's been a week-long heat wave and the humidity has been merciless.
"I couldn't find a beer cold enough so I stuck a few in the freezer." He fiddles with the secondhand A/C in the window. He switches it on high and for a moment the whirring noise is everything.
"Let's talk in bed," he says.
"That's ridiculous."
"That's me, ridiculous," his serious tone is disconcerting.
He props up some pillows. "Get in first."
"I don't have to listen to you, Nick."
"Yes you do, for the next few minutes, then, maybe, never again." His large dark eyes beneath his thick brows stare at her unblinkingly.
"Christ, you're scaring me. You know I hate surprises and I hate being set up. You know that, Nick." She sits on the side of the bed, her feet planted on the floor.
He pulls up a chair. "It's hard to say everything because there's so much." And looks at her imploringly. Her stomach cramps.
"Are you sick?" Rosalyn's decline is never far from her thoughts.
He shakes his head. He holds both her hands as if he's sure she'll pull away at his first words. She won't. She'll listen to everything. It's what she does with Nick because sometimes it takes a while to figure out exactly what he wants.
"I have a plan that'll change our lives. For the better, is how I see it. I've thought through every angle. I don't find any flaws."
"Tell me."
"The diner's up for sale. We have to buy it."
Now it's her turn to stare. Has he gone off his meds? Where would they get that kind of money?
He lets go of her hands, begins pacing. "I know what you're thinking, but we can do it. If we get married, sell your house, move in here together, we'll raise enough cash for the down payment. The diner income will pay the mortgage and then some."
"Wow! And whoa," she says.
"Look at it any way you want, it's good." But he won't look at her.
Move in together? Marry? The diner? Sell her house? His words feel threatening. Words make things happen. They start wars.
"Nick, look at me. Did you expect me to say an immediate yes or no? You know me better than that."
"What scares you most?" His tone so earnest, she nearly relents to put him at ease.
"Well . . . buying the diner," she offers because she can't go near the rest. "That's big, Nick, too big for me to comprehend." The A/C isn't doing its job. The room is hot. They should open the window.
"Murray sells to some asshole, which is likely, we'll have to live with the consequences. The asshole will micromanage. They always do. Then what? Murray's no picnic, but he knows how far he can bug me. And what about you? The new guy will make you work things his way. And think about this." He begins pacing again. "New bosses bring in their own people. They fix up and sell. We might not have jobs. It might not stay a diner. We know the business . . . the customers. We'll change the name, spiff it up, place a few ads on the highway . . . that'll bring in more people, more profit," his arms gesturing, appealing. "Hard work isn't my problem. But here's the thing . . . No way can I look for a new job. No way. All that adjustment shit, can't do it. I don't want to." He's shaking his head.
She's stunned, can't remember the last time—if ever—he said as much at one time.
"Ava . . . you and me, we're good together. I couldn't . . . I wouldn't do this without you." He stops pacing, gazes at her with frightening intensity.
"You're generally so iffy about things," she murmurs, his sudden forcefulness unwelcome. He drops back on the chair, his body visibly deflating. But she can't tend to him now. A heavy band has wrapped her chest. She needs out of here, where she doesn't care, but not here. She needs air. "Let's talk later. I have to be at Rosalyn's at four. I'll see you after your shift. Okay? Lay off the beer, it's only three." She leans over to kiss him, breathes in his familiar peppery scent. His fingers circle her arm.
"You can't leave without giving me anything back."
"I need time, Nick. I can't get my mind around any of it. I can't even formulate the questions I know I have. We'll talk. Don't worry. We always do." She's beginning to sound hysterical.
He tugs her toward his lap. She knows where that'll lead, and pulls away.
• • •
On Sunrise Highway, cars whizzing past, A/C blasting, she presses the gas pedal but where she's headed remains a mystery. She lied. She's not due at Rosalyn's till five. She had to get away. Felt knocked over by a wave with nothing to grab onto, the old sensation that ebbed and flowed after her husband was killed. That, too, alarms her now.
Houses pass in a blur. She's seen them a million times before, old, worn, fractured windows, dirty aluminum siding, lawns too small to notice. Her house is better than that though not by much. Yet it's home. Her parents left her the place. Shouldn't that count for something? Rosalyn's gorgeous condo comes to mind, and so what? Okay, upheaval frightens her.
Her eyes land on a familiar exit sign. She heads toward the ramp leading off the highway. A few minutes and she's at the open wrought-iron gate. She drives through, turns onto a dirt path and parks. Her husband's grave is one hill over. During Bobby's first three years, she brought him here to visit on her husband's birthday, then Bobby began having bad dreams. They stopped coming. It was a relief.
Sparrows flit from tree to tree no doubt looking for some moisture. In the distance a few people attend a burial, fortunately their grief too far away to see. She remembers her husband's military funeral, the phony solemnity of uniformed strangers tending to the bereaved. She hated it. When they offered her the flag, she shook her head, wouldn't touch it, as if doing so would jeopardize the baby inside her. They handed the flag to his mother, too bent over in sorrow to see what she was accepting. It was a hot day like today. Everything dry, including her eyes, because she couldn't afford to cry, needed to conserve her strength for Bobby.
It isn't that she didn't love her husband, she did. When he proposed in his father's cluttered Ford, she said yes immediately. But at nineteen what did she know about anything? A lot less than now, and yes, she loves Nick, too, but it's different; she's different.
The thing is, she's been in charge of her life for years, with no one to answer to. She's made ends meet, god knows how, taken care of the house, her son, a job, made decisions on her own, big ones, little ones, daily. Why would she want to change any of it to buy a diner? They'll have to work even harder. They're not going to be much richer, either. Nick's drug bills alone are through the ceiling. He won't step into a VA hospital for free help, says he'd rather put a bullet in his head. Besides she's comfortable, even happy with the present arrangement. Why disturb that?
She flashes on her cop father. He'd analyze a homicide from every angle, yet each time he came up with a solution he'd shoot it down with another theory. It astonished her how many ways he could look at a situation. Nick can't do that; he's too impatient. It's not that he's going to undercut her response, but he won't get what there is to discuss. Either she agrees with his plan or she doesn't.
Sunshine rolls slowly down the hill leaving shadows . . . like a life. What's she doing here anyway? There are no friendly spirits, only sad memories. She drives back to the highway.
• • •
It's four-thirty when she pulls up in front of Rosalyn's condo, her head more muddled than it was an hour ago. A drink will help. Mila makes sure there's wine in Rosalyn's fridge. There's even leftover vodka, gin, and scotch from the night after Rosalyn returned from abroad. With a table filled with sinful food and drink, they partied hard. She, Dina, Mila, the four of them . . . drunk, wild, high on laughter, Mila snapping one picture after another with her cell phone. Rosalyn was like a kid who's been given everything she wanted for Christmas. She couldn't stop yakking about every unbelievable place Jack took her, banishing illness as only Rosalyn could.
• • •
The front door isn't locked. She finds Mila in the living room, looking half her age in a sleeveless shirt and shorts, hair in a ponytail. She takes in the surroundings as if for the first time. Teal-colored couch and chairs facing the plant-filled patio outside a large window, colorful rugs, black-and-white prints on the wall. Nick wouldn't care about any of it.
"Hospice hung another morphine drip. She's been mostly sleeping. Damn, it's unfair." Mila sighs deeply. "So energetic, now maybe a nod, a word . . ."
"I know. I was remembering her at our party. It wasn't that long ago."
"And at the diner, how she challenged Murray, telling him to give Sylvie room to breathe and didn't mean house space, either. Remember?"
"Yes. Always the odd yet correct take on a situation and not afraid to say so."
"Each day I come here it hits me again. There's no getting used to it. I was surprised when she sent Jack back to London. I wanted to say keep him around, who wants to die alone. Thing is, we do anyway, don't we, even if someone's in the room. Anyway, this is morbid. You're early. Is something the matter? Pale, too. Is it the heat?" Mila studies her a few seconds.
"Must be," she mumbles behind an urge to reveal everything and have Mila decide.
"I'll get my stuff." Mila gives her arm a squeeze.
"Wait. I need to talk." She sits on the couch.
"Oh lord, Ava, not bad news. I can't deal—"
"Nick wants to buy the diner."
"Great!" Mila claps. "You guys as bosses. No more Murray . . . hooray."
"Nick wants me to sell my house, use the money for the down payment, move in with him, marry, and—"
"Huge." Mila can't help grinning.
"It's such a big commitment, everything at once. How can I—"
Mila sits beside her. "Hold on a sec. If it doesn't work for you and Nick, you move out, find another place."
"It's not that easy."
"Nothing is."
"But I've been on my own so long."
"Yeah, change is horrible, but only for a short time." Mila still sounds gleeful.
"I don't want to sell my house."
"Don't. Sell his for hell's sake."
Why didn't that occur to her? He doesn't have a mortgage either.
"And listen to this . . . you don't have to marry him. Last I heard living together isn't against the law."
If only Mila's words could seep through the certainty of her resistance. But memory is in the way and trust comes hard to her.
"Do I want to change anything?" she blurts out.
"Hey, I'm no fan of unnecessary inconvenience, struggle for struggle's sake . . . all that bullshit, not when life's too ready to surprise you on its own. You didn't expect to be a widow. Rosalyn didn't expect to die young. I didn't bring up Darla to send her into danger?" Mila's face tenses, any mention of her daughter upsets her.
"True," she admits.
"The alternative, though, is some crazy attempt to hold it all in place. It can't be done. You know what I think, change isn't about taking chances. Uh-uh. It's about using time differently."
"I know."
"You can't know . . . it's all unknown. Did I believe I'd see my Jimmy again?"
"My Jimmy—" she repeats. My Nick, she wonders . . .
"The last great hurrah, Ava. We need it, even if—"
Mila's cell phone rings. She fishes it from her pocket. "Hello? Hello? Darla! Wow! Baby! Where are you, I mean right now? I'm with Ava at Ros . . . Hellohellohello . . . shit." She stares at the phone as if it will tell her something, then throws it on the couch. "Disconnected. Happens nine out of ten. I get to speak to the girl never, hardly. I can't stand it. It's like coitus interruptus. Not quite, but hearing her voice for two seconds, then gone, it's a frigging tease. Okay, I promised I wouldn't complain. But I can't get used to this crap, Ava. I just can't. Another six months in that terrible place, but who the hell knows, they could send her back again, they're not about to ask my permission. Bastards. I hate them. Hell and more hell. I need a drink."
"Are you on shift tonight?"
"Yeah. So what? Murray says one wrong word I'll let him have it." Mila seems to be channeling words Rosalyn would say. Is that what happens between friends?
She thinks to slip an arm around her friend's shoulders, but Mila's batting her eyes with the back of her hand to conceal tears.
"I'll check on Rosalyn," she says.
• • •
Light leaks through half-closed bedroom blinds. On the dresser, a vase with fresh lilies does little to mask medicinal smells. A portable commode, bedpan, oxygen tank crowd the space. Wipes, cotton balls, lotions, scented oils are on the bedside table. Soiled nightgowns are puddled in a corner. A silky robe drapes the chair, slippers beneath. Rosalyn's no longer walking.
Once upon a time Rosalyn spent hours shopping, choosing lamps, rugs, whatever doodads she envisioned would create a beautiful room. Now the lamps are gone, the rugs rolled up, furniture pushed to the wall to make room for the hospital bed, everything topsy-turvy to facilitate treatment. Material things suddenly made trivial, replaceable.
"Hey there." She perches on the side of the bed, strokes Rosalyn's cool, slim hand, the fingers splayed lightly on the coverlet. Rosalyn's wrists are the size of a child's, her body beneath the blanket shrinking back to where it began. Her hair, grown in some, is a dark halo around the thin white face, which becomes smoother each day as lines of definition disappear along with worries. Her eyes, though, remain luminous, feverish. Dina says morphine can do that.
Any other day she'd be applying moisturizer, adjusting pillows, fixing covers, chattering away, the whirlwind everyone says she is. But something unusual in Rosalyn's calm but distant expression warns her movement or noise will be distressing. She thinks to drape an arm around her friend, simply to be there, except that too might be disturbing, even painful.
She remembers the two of them out on the patio. Rosalyn half-reclined in a well-padded chair. It was early evening, the end of spring, not too hot or breezy, chips and dips and wine on the table, though Rosalyn took most of her sustenance through an IV hung on a stand beside her. Still talkative, though her voice weak, gravelly, her speech slowed, she described colorful dreams with real stories played out as if on a movie screen. Said with all the sleeping she does they kept her from being bored, and if they were chemical hallucinations, so be it. When the cell phone on the table rang, she handed it to Rosalyn. It was Jack. Rosalyn murmured a word or two, mostly listened, said "Me too" a few times before clicking off. Then turned to her and whispered love was a learning curve.
Rosalyn could do that, offer a usable truth in a few words. That night, long ago, at Murray's housewarming. Rosalyn traipsing the cold beach in high-heeled boots, so spry, so eager to be on the go, persuading her to leave the safety of the car to look at the stars, hear the waves. Rosalyn could do that too, insist on life.
Gently, she touches Rosalyn's forehead. It too feels cool. Asleep she looks young, vulnerable, far away, unwilling to be called awake. Tawny stripes of early evening sun slide across the blanket. She sits there till they disappear.
• • •
After tossing for hours, the sheets are hot, clammy. Her head filled to aching with Mila's words, Rosalyn's translucent face, Nick's earnest expression. She wants to be with him, of course she does. He loves her, relies on her. Pleasing him pleases her, yet her brain can't wrap itself around the future he proposes.
The chores she planned to do today didn't get done, that weighs on her too. Bobby needs another pair of summer pants, even if he won't take off those stupid jeans. And what about new sneakers, which he actually wants? Except none of that is important. Her friend is dying, her relationship with Nick threatened, her sense of order dissolving faster than ice on a grill. It's a test, a challenge to her resilience. All those years ago, everything crumbling, when going on seemed impossible, what did she do? Was it Bobby, his needs, his very being? Maybe. She sighs and switches on the lamp. It's after five.
Tiptoeing past her son's room, she considers two aspirin, anything to get another hour of rest. Instead she stands gazing out the living room window at the empty street of houses still shuttered against the day, wondering if somewhere others, too, are staring into the darkness edged now with pale blue light.
"Mom?"
"Oh honey, did I wake you?" How scrawny he looks in his long T-shirt, one he'll wear even during the day. The summer sun has whitened his hair and darkened his skin, her beautiful boy. After a few weeks indoors, the paleness he shares with her will return.
"Why are you up?" he wants to know.
"Why are you?" she teases.
"Can't you just answer?" one hand on his hip.
"I keep thinking of Rosalyn."
"Oh." He sits cross-legged in his faded TV chair, with it's food stains and god knows what else. He's offering her his company. She tries to read his face. But, really, what can the child know about illness? Nothing she hopes, ever.
"Want breakfast? It's getting light out."
"I'm starving." The boy eats like a logger and never gains weight.
"The whole deal? Pancakes and eggs?" she asks.
"Yup."
Life intrudes and that's encouraging. Children can do that.
Her arm around his shoulder, they traipse to the kitchen, where Bobby disentangles and plops on a chair. A surprising memory of the pristine, modern kitchen in Colorado comes to mind. If she and Mark had worked out together, there'd be no relationship with Nick. How strange.
"How would you feel if Nick moved in here?" Just saying so increases her adrenaline.
"Is he?" her cautious son, not willing to take a stand till she does.
"Honestly, I don't know. It's an ongoing discussion and how you feel about it matters."
"Would Glory come, too?" He studies the table as if the answer's written there.
"She'll be leaving in a month or so." Opening the fridge, she pulls out the eggs and milk, then pancake mix and syrup from the cabinet.
"She'll be here on vacations and holidays. Kids always come home for those," he informs her.
"Well . . . she may be too far away. But yes if she does she'll come here." She eyes the coffee, accepting the end of sleep.
"It's okay if Nick stays, except there have to be a few rules."
"Oh?"
"Nick can't use my computer or my TV chair or my bike or—"
The phone rings and the sound slices through her. She sprints to the living room and grabs it. "Yes?"
"It's over. Ava, she's gone, really gone."
Bobby follows her in.
"Dina, you shouldn't be alone."
"Hospice is here. Mila's on the way."
"Me too."
Her son looks at her.
"Rosalyn . . . it's over," she says. "Can you make some breakfast?"
He nods.
She ought to take a minute, talk about his feelings, reactions, what a mother's supposed to do. Later. She rushes to the bedroom, dresses quickly in shorts, shirt, flip-flops. Then phones Nick at the diner to tell him it's over. She can't bring herself to say Rosalyn's dead.
• • •
She steps outside. Her legs feel heavy as if the aches in Rosalyn's bones have landed in hers. The sky's heavy as well, and white, the sun pulsing somewhere far behind. The adjacent house is there same as yesterday, the dried-up lawn littered with sad little toys. Someone inside opens the blinds, someone alive. Anger surprises her throat.
In the car she rolls down the windows. The hot morning air, she needs it to breathe. She drives through streets too quiet by far. Noise, traffic, daily distractions could help. That's a joke . . . nothing helps. She knows that. The two baby-faced uniformed men who arrived to give her the news, who wanted to come in, sit with her, commune, she shut the door in their faces. Rude, yes, but weird things happen around death. Or maybe nothing that happens is weird. For days after, she of small appetite couldn't stop shoving food in her mouth. Chatted nonstop on the phone, but not about her husband. Friends tried to pry out her feelings but she would have none of it. Weeks after the funeral, leaving the obstetrician's office, she turned her ankle. Strangers had to help her, hugely pregnant, to a nearby bench. Waiting for a promised ice pack she began to cry, no, wail, and couldn't stop, her head screaming he's dead, gone, never again.
His death was a shock, yes, but Rosalyn's death she knew was coming, thought about it each day, tried to prepare herself for a world without her friend. So why does it feel sudden, cold and sharp? Why is it tearing away at something inside her she can't name but needs to hold on to? Why does she want to shout she'll never forget her? Some people pass through, not Rosalyn. She was definitely here.
12
Stop Here
The buzzing alarm wakes her. Reluctantly hoisting herself out of the warm bed, Ava reaches for her robe and slippers. She goes to the window. Snow again. The lawn and the shrubs are blanketed. Nick's car is gone. Her car, though, is shoveled out, rescued, the big shovel left lying in the driveway. She shuffles into the kitchen, rubbing her hands together for warmth. How sweet, he brewed the coffee. She pours a cup, sips at the hot liquid quickly, no time to linger. She scrambles two eggs, sticks the plate in the microwave for Bobby to reheat. She leaves jam, butter, milk, and cold cereal on the table, places two slices of bread on top of the toaster. She'll let him sleep. It's Saturday. The other morning he mumbled some criticism about Nick living here. She zeroed in with a bunch of questions but couldn't pin him down. He offered small grievances . . . Nick's odd sleeping hours, used his towel, TV on too loud . . . what he wanted was assurance that all would go on as before. She couldn't promise that.
She, too, is adjusting. Nick brought no furniture, a few bags of his and Glory's stuff, yet the house seems tighter. Or is it how she feels? What isn't in question is Nick's disdain for routine. Structure and expectations make him suffer, feel hemmed in, managed. He's explained this to her and she tries to understand, but it can be annoying. If the diner makes lots of money, she'll build an extension on the house. Fat chance. Nick would hate too much space, if it takes him too far away from her. He's grateful for her presence. And her? She's here with him, isn't she? That means something.
Pulling the list off the fridge, she grabs a pen from the cracked mug on the shelf, crosses out electrician, phone company, tires, adds baker, toothpaste, bulbs, laundry, Dina's birthday gift. Nick snickers at her sense of order. Well . . . tough. There's much to be done.
Another sip of coffee and she hurries to the bedroom to dress. Constitutionally unable to leave a mussed bed, she tightens the sheets, then tugs the yellow spread into place and remembers last night's pillow talk. Their voices low, her son a door away. Unexpected things can happen, she said. Would they have enough money? Someone could slip and fall, she warned. And what about the insurance policy, did he change the name? Does he realize that how people see, feel, talk about the diner will reflect not only on business, but on them as well? That whatever happens they're still accountable for paying their employees? He pulled her close to stop her chatter, murmured the diner isn't her house or her sole responsibility. "Yeah, I know, but . . ." the words muffled against him. "All we can do is our best," he declared, sounding more solemn than wise.
• • •
She pulls into the parking lot, relieved to see that the snowplow has come and gone. Slowing to flurries, the snow lands daintily on the windshield. She sits for a moment unwilling to begin the busy day. It's there, waiting, solid and snow-covered, undaunted by weather or change, a long bus with steamy windows, and as Nick likes to remind her, the only diner for miles. The other day Mila asked how it felt to be a proprietor. She had no idea. It's difficult to absorb the idea that the diner is theirs. Things need to be around awhile before she can claim them.
Enough musing, she chides herself, getting out of the car to hurry up the few steps. She pushes open the door and smells paint. Odors will not do. Also that ridiculous chime, they have to get rid of it. One more item for her list. The diner's been closed a few days, something Murray would never allow. How else would they get things done? Workers have installed indirect lighting, repainted inside and laid new floors. The wood tables with captain's chairs arrived yesterday. Nick would've left more of the old stuff intact, but she insisted. The room looks younger, inviting, warmer, which is a plus in this freeze.
Days ago workers removed the neon sign with Murray's name, and a new one needs to replace it. Baptism, christening, a party . . . it was Shelly's idea to have friends participate in renaming the diner. Leaving her boots at the door, she slips into shoes and hangs her coat on the new set of wooden pegs nearby. There are still a few hours before people arrive.
Murray, yakking loudly, hovers around Nick in the kitchen where she heads. He arrived so early. Damn. Rumor has it he's looking for a house in San Diego, maybe another restaurant there. Mila swears the man has more than enough to retire on. Still, what would he do at home except drive Sylvie crazy? Not her problem. Finding time for a hot bath is a problem.
The unusual clatter of dishes tells her Nick's trying to drown out Murray's prattle. She slides an arm around Nick's waist and whispers, "Thanks for shoveling out my car."
He doesn't respond, his face a tight mask, Murray's impinging on his space.
"—and always make sure there's enough toilet paper. It can turn off a customer like that." Murray snaps his fingers. "Also the paint smell. Do something about it."
"What can we do?" she says quickly, ready to wring her hands if needed.
He looks surprised to be asked, and describes some solvent spray that lifts off smells. The man does know the business. Wearing a black turtleneck sweater and black slacks instead of his usual rolled-up shirtsleeves and jeans, he appears faintly sinister.
"Great, that's a big help. Show me how the light panel works?"
He leads her behind the counter, teaches her what she's known for ages.
"How's the baby?" Really there's no time for chitchat.
"A tiger. He grabs my finger. The strength in him . . . a real toughie."
"I bet." She smiles, about to walk away.
"The dogs keep watch in front of the crib. You have to get my permission to—"
"Hello . . . Anyone? I need some help," Mila calls from the doorway, letting in the cold air.
Nick and the electrician hurry out to carry in Darla's wheelchair, too heavy for Mila to push up the snow-covered ramp; they set the chair down near a table.
"Hey, the conquering hero," Murray says.
"Shut up," Mila snaps, her face permanently tense. The woman has lost weight. Her hair's falling out, her eyes red-rimmed from lack of sleep. Dina suggested antidepressants. Mila shrugged her off, said there's no comfort to be had.
Mila's ragged, pain-filled voice calling to tell her of Darla's injury, she can't forget it. Did she receive a wire, a phone call, a man in uniform at her door, she never asked. Instead torn between grief for her friend and relief her son was intact, she drove fast to Mila's house and found her in bed, sobbing. Consoling words felt impossible, inadequate. She climbed in beside her and held her all night.
Embracing her friend now, she whispers, "I'm glad Darla came." She'd embrace Darla as well, but the girl's closed expression warns off hugs or questions. Darla takes in the new décor but says nothing. Her silky skin and thick dark hair remain, but her lovely full lips are pressed in a fixed line. Dressed in a down jacket, her useless legs in corduroy slacks. After weeks of pleading, threatening, cajoling, a zillion phone calls and VA visits, Mila managed to enroll Darla in a clinical trial for spinal nerve stimulation. If the trial succeeds, fingers crossed, Darla could someday get around with a walker or crutches. Maybe then, Mila hopes, her daughter will soften toward meeting her father.
When Willy opens the door, it gives her a start. It's been months since he was in here. He seems even tinier inside a long coat and fur hat, a scarf wrapped around several times. Last she saw him was Rosalyn's funeral where he kept muttering, "Not right." Lots of people attended, the church cool and cavernous but far from quiet, emotions flowing freely, including her own.
"I told Willy to come today." Mila leads him to a booth, begins to undress the old man.
• • •
Holding aloft a tray of hot finger foods, she carefully backs out of the kitchen, her friends' chatter loud and insistent. People who know each other. They've moved chairs into a tight little circle. Wet coats are piled high in one of the booths, the damp smell raising memories she has no time to decipher. The snow is coming down heavy now, layering tree limbs along the roadway, hushing traffic sounds. Inside, though, it's warm, safe, and promising, the indirect lighting softening people's faces. A red paper tablecloth covers the newly tiled countertop. Red and yellow balloons cling to the ceiling, trailing a broken spiderweb of strings.
"Ta-da," she announces, setting the tray on the counter near several champagne bottles. And remembers to add, "Bruce prepared these last night."
Nick, in new dark jeans and a navy crewneck sweater, pops the first cork to applause. His expression, focused, in charge, different from any she's seen before. She watches him pour generously into plastic cups. Murray would insist on glasses that could be washed and reused. But this is Nick, her Nick.
Bruce lifts his cup in a salute to Darla. "Glad you're home. My son's redeployed." Darla nods but says nothing. What can she say? I'm sure he'll be fine?
Bobby, seated beside Dina, sneaks glances at Darla, someone he knew before she enlisted, before the wheelchair, before he could ever imagine such damage. Well, okay. Forewarned is good.
Shelly weaves around chairs offering a tray of deviled eggs she prepared. "Ava, I told my oldest I'm coming in to help with the Sunday breakfast rush. He looked at me like I'd lost my marbles. You reach a certain age and these kids think you're finished. Think again, I didn't say."
"There's more hot food," Bruce lumbers toward the kitchen.
"I can get it," Murray says, but Bruce walks past him.
"A toast to Rosalyn," Willy's reedy voice insists.
"To our lady of the flowers," Dina chimes in, raising her drink high, dressed for a party in the long black skirt and blue tunic Rosalyn gave her.
"Rosalyn, dear Rosalyn," she murmurs, locking eyes with Mila. Both remembering, she's sure, the night of the funeral. The two of them plus Dina, sharing Rosalyn stories, laughing, crying, drinking at Sully's bar till the wee hours, none of them willing to go home alone with the loss.
At the counter, too near to where she's standing, Murray refills his drink and raises the cup. "To my old diner and now yours." He claps Nick on the shoulder. "To—"
"Hear, hear," Mila interrupts, sitting close to her daughter, whose jacket she's removed revealing Darla's slim torso in a deep purple Nehru shirt that could pass for festive.
"—the place where I met my wife," Murray continues, "where I spent most of my life, where all of—"
"A shrine will be built," Shelly says not too softly.
"When I opened the restaurant it was a nothing. If you could've seen the way—"
She takes a long drink of champagne, the lemony flavor the same as the one they shared after Nick closed on his house. They brought the bottle to bed, passed it back and forth till it was nearly empty. Trying to muffle their giggles, Bobby in the next room, they stared stupidly at TV sitcoms she can't remember a thing about now.
"—on the couch in the ladies room, I used to sleep there." Murray's voice drones on. "That's right. Once in a while I had company, before Sylvie." He turns to Nick, "And one time—"
"Enough," Nick says, his tone leaden. Grabbing a full bottle of champagne, he walks around refilling cups and returns to top hers as well.
Murray, watching, finishes his drink. "There's a lot you don't know—"
"Take a load off," Bruce orders, kicking out a chair, which Murray ignores.
"Look at all this new crap," Murray's arm sweeps the room. "It'll turn off old customers. Ask Willy. They're used to what was here. Too much alteration . . . What's with these lights? Armchairs? Pictures? It looks like a cocktail lounge. People come here for food, not entertainment. Next thing you know, there'll be some guitar player." He shakes his head, slides a hand across the counter, then again refills his cup.
"Murray, new management always makes—"
"What new management," he scolds her. "You guys have been here for years," his voice going up a few decibels.
"I like the way the place looks," Bobby speaks directly to Murray.
She flashes her son a grateful but warning smile.
Murray steps around a stool to get closer to Nick. "You can't hide in the kitchen anymore. Customers need to be chatted up, catered to, they—"
"Hey, Murray," Bruce growls, "give the man room."
"It takes more than kitchen savvy to make a restaurant work. You'll need to consult with other owners. I won't always be around."
Damn him. He's no longer the boss. How dare he take center stage? Rosalyn would get rid of him in a hand wave. With blood thrumming in her ears, she grabs a huge wooden spoon and bangs hard on the counter for quiet, which works faster than she expects. A tableau of faces turn to her and, for a dizzying moment, she's bewildered. Nick, too, seems to be waiting, but for what? Murray's eyes on her are challenging, his expression refusing to understand the moment. It's Bobby's expectant look that releases her. She hears herself declare loudly, "Names, everyone, I need names. And nothing more."
Bobby pops up. "Resurrection Diner."
"Great," she replaces the spoon, adrenaline high, and pulls a pad and pencil from her pocket to jot it down.
"Tiptoe In," Dina adds quickly.
"What kind of name is that?" Murray scoffs.
The door chimes. Chairs scrape and people turn to see. Hamid in a suit jacket over a cable-knit sweater, his hair covered in snow, rushes in. "So sorry to be late."
"No problem," Nick says gaily. Clearly, a welcome intrusion. "It's a party, not a meeting. Everyone, this is Hamid, Glory's good friend."
"Ha-mid," Murray draws out the name. Where you from, Iran?"
"Morocco."
"Oh yeah. Here in the US to—"
"Take some food," she interrupts.
"No, thank you." Hamid shrugs off his wet jacket, pulls up a chair beside Bobby, who looks pleased. "Glory e-mailed me two names." He waits.
"Yes?" pencil poised, her voice loud, ready to talk past Murray, who watches her warily.
"First name is Come Back Diner," Hamid pauses, no one says anything.
"Got it. Next?" her voice more normal though everyone's eyes are still on her.
"Welcome In Diner," Hamid offers with some assurance.
"That's as silly as Tip-toe In," Murray heckles.
"Hey—" Nick says evenly. "Everyone here participates equally. You want to give us a name or what?" He stares hard at Murray.
"Nick and Ava's," Murray replies sourly.
Mila rolls her eyes.
She writes it down. "Others?" she asks.
"A-One Diner," Mila raises her thumb.
"New Place," Willy points a bony finger at the additions.
"Fine Dine," Dina tries again.
Murray inches toward the door, god willing he'll slip out.
"Chow Down," Bruce offers.
"Eat Out," Mila declares.
"Food for Thought," the electrician perched on a stool suggests shyly.
"Second Chance," Shelly says loudly.
"Ace Diner." Bobby grins the way he does when he's playing Nintendo, aware this isn't a game but wanting to win.
She's writing now as fast as she can. When she looks up, her son's watching her. His admiration fills her with the kind of love only a child elicits. Her steady gaze embarrasses him. She turns away. Outside, a blanket of white as far as the eye can see. Bad weather curdled Murray's mood. It doesn't bother her.
"Hey, Ava?" Murray calls as if reading her mind. "What's your contribution?" He's leaning hard against the newly painted wall.
She hasn't thought about a name and admitted as much to Bobby yesterday, who took pains to explain that a name was important. A tag, he said, without a tag you're unknown. Lord help her, a lot rides on the diner being known. Her eyes scan her friends' waiting faces.
A name, she thinks. Give them a name, any name, but her mind blanks, wiped clean of names forevermore. She glances hopelessly at Nick, a name, she begs him silently, any name.
"Best Deal," he says softly, but loud enough for others to hear.
"It's Ava's turn," Murray shouts.
Darla's hand shoots up. "How about Stop Here?"
• • •
She sits at the bedroom window, watching the blue edge of dawn emerge. In the weeks since they closed on the diner she wakes at the same time each night. It's eerie. A scattering of illuminated snowflakes tumbles past the streetlamp, reluctant to hit the ground. The party's on her mind, though it ended hours ago. When Murray finally shut the door, she felt both relief and a chill of fear. The diner was theirs. In the twilight darkness people left in a group, Bobby went home with Dina. She and Nick watched the cars' red lights come on and then snake away. Alone, silent with their own thoughts, they tidied up some till hired cleaners arrived to finish the job.
Tomorrow the diner opens for business, the name, though, still up for grabs. All agreed Nick should make the decision, which delighted him. He's excited to be an owner. It's there in the way he palms the storeroom keys, responds patiently to salespeople who phone nonstop, checks and rechecks condiments, floors, appliances though no customer has yet been served. She's careful not to undercut his pleasure, careful to keep certain worries to herself. Because, really, it's not the best time to start a new venture. Look at how difficult it was to secure the mortgage, and what if they don't make enough to cover it? What if all the stuff they bought and still have to pay for doesn't result in more customers? What about her friends, now employees? They, too, must wonder, can she and Nick pull this off? If the diner fails, they'll be out of work. What then? These are the thoughts that visit before her day begins.
It's not that Nick's free of worry, he frets all the time, always has, about something happening to her, about Glory in Mali, whatever. Yet he sleeps, one arm up under the pillow. Amazing. Her eyes flick to her dead husband's portrait, a man who didn't fret. She offered to remove it. Nick said, no, he's gotten used to it, feels some affection for the guy.
Slipping out of the flannel robe, she climbs into bed. He curls around her back, warm, reassuring in its way.
"Couldn't sleep again?" she's surprised to hear him whisper.
"Umm."
"Want to say why?"
"Uh-uh."
"Nervous?"
"A bit," she admits.
"Me too. Things tank. We could live in a tent, no expenses, go from park to park."
"What about winter?"
"Problem," he agrees.
"We could go south."
"We could," his words low in his throat.
"There's Bobby—"
"Don't be real," he admonishes softly.
"That's a challenge," she whispers more to herself. The streetlamp flickers off and milky morning light brightens the sky.
Acknowledgments
My deep gratitude to Jane Lazarre for her talent, time, and unwavering eye on this book; to Jocelyn Lieu and Jan Clausen, whose attention to all on the page continue to impress me; to Tom Engelhardt for getting me started on the long journey here; to Denise C. for her insight and encouragement; and to Judi Brand, Elizabeth Strout, Barbara Schneider, Marsha Taubenhaus, Vickie Breitbart, Prue Glass, and Liz Gewirtzman for constant support and friendship.
Huge thanks to Dan Simon, Publisher extraordinaire, for his ongoing belief in this project; to my editor, Gabe Espinal, whose intelligence and easy ways made the process more than pleasant; to Gail Heimberg for her technical wizardry; to John Samuel Wiggins for his video; to Jesse Lichtenstein, Anne Rumberger, Elizabeth DeLong, and everyone at Seven Stories Press for their thoughtful and tender care in making this into a book. And deep and abiding appreciation to my agent, Melanie Jackson, a national treasure.
As always I remain grateful to my beloved, Charlie Wiggins, for his unfailing devotion, enthusiasm, and faith in what I do. And to the lights of my life, Georgina, Dónal, and Maya, you make it all matter.
About the Author
beverly gologorsky is the author of the acclaimed novel The Things We Do to Make it Home, originally published by Random House in 1999, reissued by Seven Stories in 2009, named a Notable Book by the New York Times, Best Fiction by Los Angeles Times, and a finalist for the Barnes and Noble Discover Great Writers Award. Her work has appeared in anthologies and magazines, including the New York Times, Newsweek, The Nation, and the LA Times. Former editor of two political journals, Viet-Report and Leviathan, she is acknowledged in the publication Feminists Who Changed America. She lives in New York and Maine.
About Seven Stories Press
seven stories press is an independent book publisher based in New York City. We publish works of the imagination by such writers as Nelson Algren, Russell Banks, Octavia E. Butler, Ani DiFranco, Assia Djebar, Ariel Dorfman, Coco Fusco, Barry Gifford, Martha Long, Luis Negrón, Hwang Sok-yong, Lee Stringer, and Kurt Vonnegut, to name a few, together with political titles by voices of conscience, including Subhankar Banerjee, the Boston Women's Health Collective, Noam Chomsky, Angela Y. Davis, Human Rights Watch, Derrick Jensen, Ralph Nader, Loretta Napoleoni, Gary Null, Greg Palast, Project Censored, Barbara Seaman, Alice Walker, Gary Webb, and Howard Zinn, among many others. Seven Stories Press believes publishers have a special responsibility to defend free speech and human rights, and to celebrate the gifts of the human imagination, wherever we can. In 2012 we launched Triangle Square books for young readers with strong social justice and narrative components, telling personal stories of courage and commitment. For additional information, visit www .sevenstories.com.
A Seven Stories Press Reading Group Guide
Stop Here
by Beverly Gologorsky
The following questions are suggested to enhance individual reading and invite group discussion regarding Beverly Gologorsky's Stop Here. We hope these questions provide additional topics for consideration and generate a stimulating dialogue with others.
For a complete listing of Seven Stories Press books featuring Reading Group Guides, please visit our website at www.sevenstories.com.
DISCUSSION QUESTIONS
1. War affects most of the characters, both directly and indirectly. How does each character react to the effects of war and what do their reactions say about their personalities?
2. Most of the characters in Stop Here are struggling to make ends meet and have limited opportunities because of their economic situations. How does each character respond to the stress of supporting themselves and their families? How does each character feel about their economic situation and how does that reflect their view of the world?
3. How does money affect the way the characters view Murray, the owner of the diner? How does it affect the way they view Sylvie, Murray's new wife?
4. What role does romantic love play in each character's life? Why do you think Ava is so hesitant about falling in love, first with Mark and then with Nick? How does Rosalyn use her illness as an excuse not to get close to Jack?
5. A prominent theme in this novel is the betrayals of the body—we see this through Dina and also Rosalyn. How do Dina and Rosalyn's reactions differ as their bodies start to fail them? What do they find comfort in to help them through these experiences?
6. What did the author accomplish by telling the story from multiple points of view? How did you feel when returning to a character after spending a few chapters away from him/her?
7. Many of the characters spend time fantasizing about what they thought their lives would look like or actions they would take if they could disregard the consequences. What role does fantasy play for each character and how does exploring their inner fantasies help you understand their characters?
8. Murray has never fought in a war and his thoughts on war differ greatly from other characters' opinions. How does his attitude compare with Bruce's, Liam's, and Ava's views on war and its place in society?
9. Do you think Sylvie loves Murray? What are her reasons for marrying him and how does her view of their relationship change when she gets pregnant and lets Murray believe the baby is his?
10. Shelly advises Sylvie to "Take what you can when you can where you can" (page 37). What do you think of this advice and how does this sentiment tie into the novel as a whole?
11. What is Sylvie searching for in her relationship with Liam? Does caring for him at the end of his life change anything for Sylvie?
12. What would you do if you were in Shelly's place, deciding between leaving Bruce and helping him through his debilitating depression?
13. Does Dina blame herself for her son Tim's actions? Do you understand her decision to help him escape and her refusal to ask him what he did? Do you think it is hard for her to admit that she prays that Tim stays away from her?
14. Do you understand Mila's reasons for keeping the truth about where Darla's father is secret? Do think she made the right decision in keeping the information from Darla? How might things have been different for Darla if she had grown up knowing her father was in prison?
15. The employees of Murray's diner feel like a family. What role in the family does each character play?
1. Contents
2. Usable Truths
3. Reunion
4. Imaginary Friends
5. The Way Things Work
6. In the Silence
7. Butter and Ketchup
8. How We Know Before We Know
9. About Time
10. Happiness Exists Somewhere
11. The Things in Between
12. She was Definitely Here
13. Stop Here
14. Acknowledgments
15. About the Author
16. About Seven Stories Press
17. A Seven Stories Press Reading Group Guide
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{"url":"https:\/\/plainmath.net\/algebra-i\/49689-y-equal-plus-determine-the-intercepts-of-the-quadratic-equation-in-both","text":"Sapewa\n\n2021-12-27\n\n$y={\\left(x+4\\right)}^{2}-7$\na) Determine the x-intercepts of the quadratic equation in both approximate and exact form\nb) Determine the vertex point.\nc) Determine the y-intercept.\nd) Graph the parabola.\n\nFasaniu\n\nExpert\n\nStep 1\nSolution:\u00a0$y={\\left(x+4\\right)}^{2}-7$\n${\\left(x+4\\right)}^{2}=y+7$\n1) By substituting\u00a0$y=0$, the x intercept can be achieved.\n${\\left(x+4\\right)}^{2}=0+7$\n${\\left(x+4\\right)}^{2}=7$\n$x+4=\u00b1\\sqrt{7}$\n$x=\u00b1\\sqrt{7}-4$\nBecause of this, x intercepts are \u00a0$\\sqrt{7}-4$\u00a0and\u00a0$-\\sqrt{7}-4$\n2) contrasting it with the fundamental parabola equation\n${\\left(x-h\\right)}^{2}=4a\\left(y-k\\right)$\n${\\left(x+4\\right)}^{2}=\\left(y+7\\right)$\n\nTherefore vertex is\nStep 2\n3) You can get the y intercept by changing\u00a0$x=0$\n$y={\\left(x+4\\right)}^{2}-7$\n$y={\\left(0+4\\right)}^{2}-7$\n$y=16-7=9$\n4)\u00a0${\\left(x-h\\right)}^{2}=4a\\left(y-k\\right)$\n${\\left(x+4\\right)}^{2}=\\left(y+7\\right)$\n$\\therefore 4a=1$\n$a=\\frac{1}{4}$\nTherefore four of the parabola is of\nAlso\u00a0$y={\\left(x+4\\right)}^{2}-7$\n$y={x}^{2}+16+8x-7$\n$y={x}^{2}+8x+9$\nCoefficient of\u00a0${x}^{2}=1\u21d2\\text{Parabola}$\u00a0opens up wards.\n\ntemzej9\n\nExpert\n\nStep 1\nUse binomial theorem ${\\left(a+b\\right)}^{2}={a}^{2}+2ab+{b}^{2}$ to expand ${\\left(x+4\\right)}^{2}$\n$y={x}^{2}+8x+16-7$\nSubtract 7 from 16 to get 9.\n$y={x}^{2}+8x+9$\nSwap sides so that all variable terms are on the left hand side.\n${x}^{2}+8x+9=y$\nSubtract y from both sides.\n${x}^{2}+8x+9-y=0$\nThis equation is in standard form: $a{x}^{2}+bx+c=0$\nSubstitute 1 for a, 8 for b, and $9-y$ for c in the quadratic formula, $\\frac{-b\u00b1\\sqrt{{b}^{2}-4ac}}{2a}$\n$x=\\frac{-8\u00b1\\sqrt{{8}^{2}-4\\left(9-y\\right)}}{2}$\nSquare 8.\n$x=\\frac{-8\u00b1\\sqrt{64-4\\left(9-y\\right)}}{2}$\nMultiply -4 times $9-y$.\n$x=\\frac{-8\u00b1\\sqrt{64+4y-36}}{2}$\nAdd 64 to $-36+4y$.\n$x=\\frac{-8\u00b1\\sqrt{4y+28}}{2}$\nTake the square root of $28+4y$.\n1) $x=\\frac{-8\u00b12\\sqrt{y+7}}{2}$\nNow solve the equation (1) when $\u00b1$ is plus. Add -8 to $2\\sqrt{7+y}$\n$x=\\frac{2\\sqrt{y+7}-8}{2}$\nDivide $-8+2\\sqrt{7+y}$ by 2\n$x=\\sqrt{y+7}-4$\nWhen $\u00b1$ is minus. Subtract $2\\sqrt{7+y}$ by\n$x=-\\sqrt{y+7}-4$\nThe equation is now solved\n$x=\\sqrt{y+7}-4$\n$x=-\\sqrt{y+7}-4$\n\nkarton\n\nExpert\n\nStep 1\na) Write the equation in standard form\n$-{x}^{2}-8x+y-9=0$\nStep 2\nSubstitute the values of a, b, and c into the quadratic formula\n$x=\\frac{-\\left(-8\\right)-\\sqrt{\\left(-8{\\right)}^{2}-4\\left(-1\\right)\\left(y-9\\right)}}{2\\left(-1\\right)}$ or $x=\\frac{-\\left(-8\\right)+\\sqrt{\\left(-8{\\right)}^{2}-4\\left(-1\\right)\\left(y-9\\right)}}{2\\left(-1\\right)}$\nStep 3\nSimplify the discriminant\n\nStep 4\nSimplify\n\nStep 5\nb) Let's focus on:\n$\\left(x+4{\\right)}^{2}-7$\nExpand the expression\n${x}^{2}+8x+16-7$\nStep 6\nGroup like terms together\n${x}^{2}+8x+\\left(16-7\\right)$\nStep 7\n${x}^{2}+8x+9$\nc) Parabola equation in polynomial form\nThe vertex ofan up-down facing parabola of the form $y=a{x}^{2}+bx+c$\u00a0is ${x}_{v}=-\\frac{b}{2a}$\nRewrite $y=\\left(x+4{\\right)}^{2}-7$ in the form $y=a{x}^{2}+bx+c$\n$y={x}^{2}+8x+9$\nThe parabola params are:\n$a=1,\\phantom{\\rule{1em}{0ex}}b=8,\\phantom{\\rule{1em}{0ex}}x=9$\n${x}_{v}=-\\frac{b}{2a}$\n${x}_{v}=-\\frac{8}{2\u00d71}$\nSimplify\n${x}_{v}=-4$\nPlug in ${x}_{v}=-4$ to find the ${y}_{v}$ value\n${y}_{v}=-7$\nTherefore the parabola vertex is\n(-4, -7)\nIf a<0, then the vertex is a maximum value\nIf Pa>0, then the vertex is a minimum value\nMinimum (-4,\u00a0-7)\nStep 8\n\nDo you have a similar question?","date":"2023-02-01 22:41:43","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 96, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8471927046775818, \"perplexity\": 2874.39861976632}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764499953.47\/warc\/CC-MAIN-20230201211725-20230202001725-00523.warc.gz\"}"} | null | null |
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DCHECK(ash::InstallAttributes::Get()->IsActiveDirectoryManaged());
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gaia::GaiaSource ActiveDirectoryAccountReconcilorDelegate::GetGaiaApiSource()
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const std::vector<gaia::ListedAccount>& gaia_accounts,
bool first_execution,
bool primary_has_error,
const gaia::MultiloginMode mode) const {
DCHECK_EQ(mode,
gaia::MultiloginMode::MULTILOGIN_UPDATE_COOKIE_ACCOUNTS_ORDER);
if (chrome_accounts.empty())
return chrome_accounts;
return ReorderChromeAccountsForReconcile(
chrome_accounts, GetFirstAccount(chrome_accounts, gaia_accounts),
gaia_accounts);
}
bool ActiveDirectoryAccountReconcilorDelegate::
ShouldAbortReconcileIfPrimaryHasError() const {
return false;
}
bool ActiveDirectoryAccountReconcilorDelegate::
ShouldRevokeTokensIfNoPrimaryAccount() const {
return false;
}
CoreAccountId ActiveDirectoryAccountReconcilorDelegate::GetFirstAccount(
const std::vector<CoreAccountId>& chrome_accounts,
const std::vector<gaia::ListedAccount>& gaia_accounts) const {
if (chrome_accounts.empty())
return CoreAccountId();
// Return first gaia account to preserve the account order in the cookie jar.
// (Gaia accounts which are NOT in chrome_accounts will be removed.) In case
// of first account mismatch, the cookie will be rebuilt and order of accounts
// will be changed.
if (!gaia_accounts.empty() &&
base::Contains(chrome_accounts, gaia_accounts[0].id)) {
return gaia_accounts[0].id;
}
// The cookie jar is empty or first Gaia account in the cookie jar is
// not present in Account Manager. Fall back to choosing the first account
// present in Account Manager as the first account for reconciliation.
return chrome_accounts[0];
}
} // namespace signin
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,604 |
\section{Introduction}
The interest in quantum field theory (henceforth QFT)
in classical curved space-times has
never ceased especially after the development of particle production
in cosmological
spacetimes, initiated by L. Parker in 1968 \cite{parker}, and
the discovery of black hole
evaporation by S. W. Hawking in 1974 \cite{hawking}.
Such a semiclassical treatment for gravity should hold in a region between
the Planck length and the Compton wavelength (and
certainly greater than this) of the matter field considered
and could be a useful route towards
understanding the principal ingredients for a theory
which unifies gravity and quantum mechanics.
All the theoretical motivations for studying QFT
in curved spacetimes are
corroborated by a possible application of the results to cosmology:
a spectrum of quantum fluctuations during inflation
could be related to the primordial spectrum
of cosmological perturbations \cite{k&tlinde}.
For the cosmological case particle production is related to the simple
and well studied case of a harmonic oscillator with time dependent
parameters. The time dependence of the metric leads to the appearance
of different vacua for differing times during the evolution of a free field.
When exact solutions to the field equations of motion are not
known, a systematic treatment based on the adiabatic approximation
can be found in the literature \cite{bd}. For this reason it
is a usual practice to
rescale fields by a time dependent factor
so as to obtain suitable expressions in order to
to implement an adiabatic expansion around a time independent
solution. This rescaling factor
is also related to the conformal weight of the field which leaves the
equations of motion invariant under a conformal trasformation in both
the metric and the field. Since the Robertson-Walker
(henceforth RW) metric is conformally related
to a Minkowski metric one has a correspondence between fields in
a RW metric and in flat spacetimes with time dependent masses.
For the case of massless conformally coupled fields the time dependent
parameters in the equations of motion disappear
and the theory in such a curved spacetime corresponds to a theory
in Minkowski spacetime.
The method of invariants \cite{lewis} allows one to exactly
quantize a harmonic oscillator with time dependent coefficients.
The application of this method in QFT
improves on the adiabatic approximation and
allows one to introduce a vacuum and a Fock space associated with the
quantum invariants.
The use of the method of quantum invariants \cite{lewis} has been
previously applied, within the context of cosmology, to a quantized
scalar field in a de Sitter space-time with a flat spatial section
\cite{gao0, desitter}. Such an approach also arises naturally within a
Born-Oppenheimer context, for the matter-gravity
system \cite{bv}, in a simple minisuperspace model when the semiclassical
limit is taken for gravity and fluctuations are neglected \cite{born}.
In the previously studied scalar case a massive scalar field with
a non-minimal coupling was examined \cite{desitter}.
It was found that the expectation value of the Hamiltonian of a Fourier
mode of the field grows
with time even in the massless conformally coupled case. However,
for a non-minimally coupled scalar field
the Hamiltonian density differs from the $00$ component of the
energy-momentum tensor, and the latter vanishes for zero mass and
conformal coupling.
The usual statement of no particle production in the massless
conformally coupled case should then be applied directly to the Einstein
equations: only the conformal anomaly is a source for the Einstein tensor
in this case. Further in \cite{desitter} it was also pointed out
that the usual conformal rescaling does not leave the
action invariant because of the presence of a boundary term.
It is then natural to re-examine, within the context of quantum invariants,
the results previously obtained for the electromagnetic field and spin
$1/2$ particles. This shall be done in the next two sections respectively.
In section 4 we discuss the conformal properties of the action considered,
while in section 5
we analyze in detail the problems arising from the quantization
of rescaled and non-rescaled fields.
Finally in section 6 we present our conclusions.
\section{Electromagnetic Field}
In this and the following sections we consider a RW line element
\begin{equation}
ds^2 = -d\tau^2 +a^2 (\tau) g_{ij} dx^i dx^j
\label{rwmetric}
\end{equation}
where $g_{ij}$ is the three metric for a flat three-space.
In the absence of charges the Lagrangian density for the electromagnetic
potential $A_\mu$ is given by:
\begin{eqnarray}
{\cal L}_{EM} &=& \sqrt{-g} \left[ -\frac{1}{4} F_{\mu \nu} F^{\mu \nu}
\right] \nonumber \\ &=& \frac{a^3}{4} \left[ + \frac{2}{a^2} F_{0j}^2
- \frac{1}{a^4} F_{ij}^2 \right]
\label{emlag}
\end{eqnarray}
where $F_{\mu \nu}\equiv \nabla_\mu A_\nu -\nabla_\nu A_\mu =
\partial _\mu A_\nu -\partial_\nu A_\mu $ (with $\nabla _\mu$ the covariant
derivative) because of the simmetry in the lower indices
of the Cristoffel connections.
On choosing the generalized Lorentz gauge $ \nabla _\mu A^{\mu} =0$ and
$ A_{0}=0$ (which is always possible in the source-free case)
and considering a RW metric, we obtain the so called radiation gauge
($\vec \nabla \cdot \vec A =0 $). On then substituting
$A_\mu = (A_0, \vec{A})$ in Eq. (\ref{emlag}) and using the radiation gauge
one has
\begin{eqnarray}
F_{0j} &=& \dot A_j \nonumber \\
F_{ij} &=& (\vec{\nabla} {\times} \vec{A})_{ij} \,,
\end{eqnarray}
where by the dot we denote a derivative with respect to the proper time
$\tau$. Let us
note that the vector potential $\vec{A}$ has been chosen to be the
covariant $A_{i}$,
and this is a {\em natural} choice if we wish to identify $F_{\mu \nu}$
with a two form (metric independent). Further the choice of $\vec{A}$
as a generalized coordinate with its subsequent quantization allows us
to use the method of invariants for all the Fourier components of
$\vec{A}$ \cite{gao} (this would not be possible for the case of the
identification with the controvariant form $A^{\mu}$).
The final expression one then obtains for the
electromagnetic
Lagrangian density is
\begin{equation}
{\cal L}_{EM} = \frac{1}{2} \left\{ a \dot{\vec{A}}^2 + \frac{1}{a}
\vec{A} \cdot \nabla^2 \vec{A} - \frac{1}{a} \vec{\nabla} \cdot
\left[ A_i \vec{\nabla} A_i - (\vec{A}\cdot \vec{\nabla})
\vec{A} \right] \right\} \,.
\end{equation}
We expand $\vec{A}$ as
\begin{equation}
\vec{A} = \frac{1}{\sqrt{V}} \sum_{\vec{k}\,,\lambda}
\left[ c_k^{(\lambda)} (\tau) \vec{\varepsilon}^{\,(\lambda)}
e^{i \vec{k} \cdot \vec{x}}
+ c_k^{(\lambda) *} (t)
\vec{\varepsilon}^{(\lambda) }
e^{-i \vec{k} \cdot \vec{x}} \right] \,,
\end{equation}
where $\lambda$ runs over the (two) polarization states and
$\vec{\varepsilon}^{\,(\lambda)}$ is a unit vector which satisfies
$\vec{k} \cdot \vec{\varepsilon}^{\,(\lambda)}=0$ and
$\vec{\varepsilon}^{\,(\lambda)} \cdot
\vec{\varepsilon}^{\,(\lambda')}=\delta_{\lambda\,\lambda'}$.
On further separating $c_k^{(\lambda)}$ into real and
imaginary parts
\begin{equation}
c_k^{(\lambda)} = \frac{1}{\sqrt{2}} \left( c_{k\,1}^{(\lambda)}
+ ic_{k\,2}^{(\lambda)} \right)
\end{equation}
the action becomes:
\begin{equation}
S = \sum_{k, i, \lambda} S^{(\lambda)}_{k \, i} =
\frac{1}{2}\sum_{k, i, \lambda} \int \left[ a \dot c_{k\,i}^{(\lambda)\,2}
- \frac{k^2}{a} c_{k\,i}^{(\lambda)\,2} \right] d\tau \,.
\end{equation}
Thus we see that the different modes $k, i, (\lambda)$ decouple and
one then obtains the following Hamiltonian for each mode:
\begin{equation}
H_{i\,k}^{(\lambda)} = \frac{1}{2}
\left( \frac{\pi_{i,k}^{(\lambda)\,2}}{a} + a
\omega_k^2 c_{k\,i}^{(\lambda)\,2} \right) \,,
\label{emham}
\end{equation}
where $\pi_{i,k}^{(\lambda)}= a \dot c_{k\,i}^{(\lambda)}$ and
$\omega_k^2 =k^2/a^2$. The classical equation of motion is:
\begin{equation}
\ddot{c}_{k\,i}^{(\lambda)} + \frac{\dot a}{a}\,
\dot{c}_{k\,i}^{(\lambda)} +
\omega_k^2 c_{k\,i}^{(\lambda)}= 0
\,
\label{emclass}
\end{equation}
and we proceed in analogy with the scalar field case \cite{desitter}.
On canonically quantizing, the Hamiltonian
in Eq. (\ref{emham}) can
be factorized as (henceforth we shall denote collectively
$\bf{k}, i, (\lambda)$ by $\sigma$ and retain the subscript $k$
only when relevant):
\begin{equation}
\hat{H}_{\sigma} = \hbar \omega_k \left(\,
\hat a_{\sigma}^{\dagger} \,
\hat a_{\sigma}
+ \frac{1}{2}\,\right)
\end{equation}
with
\begin{equation}
\begin{array}{c}
\hat{a}_{\sigma}
= \left( \frac{a \omega_k}{2 \hbar}
\right)^{\frac{1}{2}}
\left( \hat c_{\sigma} + i\,{\hat \pi_{\sigma} \over a\omega_k} \right) \,,
\\
\\
\hat{a}_{\sigma}^{\dagger} = \left( \frac{a \omega_k}{2 \hbar}
\right)^{\frac{1}{2}}
\left( \hat c_{\sigma} - i\,{\hat \pi_{\sigma} \over a\omega_k} \right)
\
\end{array}
\end{equation}
and $[\hat{a}_{\sigma}, \hat{a}_{\sigma}^{\dagger}]
=\delta_{\sigma\, \sigma'}$.
As we mentioned, a suitable method for the study of time dependent quantum
systems is that of invariants \cite{lewis}.
In particular a hermitian operator $\hat I$ which satisfies:
\begin{equation}
{\partial \hat I_{\sigma} (\tau) \over \partial\,\tau}
- \frac{i}{\hbar} [\hat I_{\sigma} (\tau), \hat H_{\sigma} (\tau)]=0 \,
\end{equation}
is an invariant.
The invariant $\hat I_{\sigma}$ has real, time independent,
eigenvalues
and
in our case, can be decomposed in terms of basic linear invariants
\cite{gao}:
\begin{equation}
\begin{array}{c}
\hat I_{b\, \sigma}(\tau) \equiv
e^{i\Theta_k(\tau)} \hat{b}_{\sigma} (\tau) \equiv { e^{i\Theta_k} \over
\sqrt{2\,\hbar}}\,
\left[{\hat c_{\sigma} \over \rho_k}+
i\,\left(\rho_k\,\hat \pi_{\sigma} - a \dot{\rho}_k \hat c_{\sigma} \right)\right] \,,
\\
\\
\hat I_{b\,\sigma}^{\dagger} (\tau) \equiv
e^{-i\Theta_k (\tau)} \hat{b}_\sigma(\tau) ^{\dagger} \equiv { e^{-i\Theta_k} \over
\sqrt{2\,\hbar}}\,\left[{\hat c_\sigma \over \rho_k}-
i\,\left(\rho_k\,\hat \pi_\sigma - a \dot{\rho}_k \hat c_\sigma
\right) \right]
\,, \label{inv}
\end{array}
\end{equation}
where $\rho_k(\tau)$ is real and satisfies \cite{lewis}:
\begin{equation}
\ddot{\rho}_k + \frac{\dot a}{a} \dot{\rho}_k
+ \omega^2_k \rho_k = \frac{1}{a^2 \rho_k^3}
\label{rho}
\end{equation}
with:
\begin{equation}
\Theta_k(\tau)=\int_{- \infty}^t
{d\tau'\over a(\tau')\,\rho_k^2(\tau')} \,.
\end{equation}
and:
\begin{equation}
[\hat{b}_\sigma, \hat{b}_{\sigma}^{\dagger}]
= \delta_{\sigma \sigma'} \, .
\end{equation}
Let us now note that the above equations (\ref{inv}) may be rewritten in terms
of the classical solutions $c_\sigma$ to Eq. (\ref{emclass}) through
$c=\rho_k e^{-i \Theta_k}$:
\begin{equation}
\hat I_{b \, \sigma} (\tau) = i (c^*_\sigma \hat \pi_\sigma -
\pi_{\sigma}^* \hat c_\sigma )
\end{equation}
and the quadratic, hermitian, adiabatic invariant originally
introduced in \cite{lewis} is given by:
\begin{equation}
\hat I_{\sigma} (\tau) =
\hbar \left(\hat b_\sigma^{\dagger} \, \hat b_\sigma + \frac{1}{2} \right)
= \frac{1}{2}
\left[ \frac{\hat c_\sigma^{2}}{\rho_k^2} + (\rho_k \hat \pi_\sigma - a
\dot{\rho}_k \hat c_\sigma)^2 \right]
\ ,
\end{equation}
The linearly independent solutions to the equation of motion
(\ref{emclass}) are the Bessel functions:
\begin{equation}
c = \eta^{1/2} \left\{ \begin{array}{c}
J_{1/2} (k\eta) \\
N_{1/2} (k\eta)
\end{array} \right.
\end{equation}
where $\eta$ is
the conformal time and the above solutions are true both for
de Sitter $a=-\frac{1}{H\eta}$ (H is the Hubble constant, $-\infty<\eta<0$)
and for power behaviour $a=\eta^p$ (with $p$ a positive real number).
The general solution to Eq. (\ref{rho})
can be written as a non-linear combination of the
solutions to the equations of motion as shown in \cite{desitter}:
\begin{equation}
\rho= \eta^{1/2} \left[ A J_{1/2}(k\eta) + B N_{1/2}(k\eta)
+ 2(AB-\frac{\pi^2}{4})
J_{1/2}(k\eta) N_{1/2}(k\eta) \right]^{\frac{1}{2}}
\end{equation}
where $A, B$ are real constants ($k$ independent because
of the spatial simmetry of RW spacetime).
On choosing $A=B=\pi/2$ one has
\begin{equation}
\rho = \frac{1}{\sqrt k}
\label{emfinal}
\end{equation}
which is independent of $\eta$. The choice $A=B$ is associated with
the adiabatic vacuum at early times
or the adiabatic vacuum for wavelengths $2 a \pi/k$ which
are well inside the Hubble radius $a/ \dot a = H^{-1} $
(see \cite{desitter}), i.e. the Bunch-Davies vacuum.
>From Eq. (\ref{emfinal}) we see that the annihilation operators
$\hat a$ and $\hat b$ coincide for all times which implies
\begin{equation}
\hat{H}_{k\,i}^{(\lambda)} = \omega_k \hat{I}_{k\,i}^{(\lambda)} \,.
\end{equation}
Hence for a (massless) photon
the initial adiabatic vacuum remains such for all times, leading
to a null photon production in RW spacetime, and its energy is redshifted
as expected for radiation.
We end by observing that if one relaxes the assumption of an adiabatic vacuum
at early times, i.e. allowes $A \ne B$, one has a
number of photons oscillating around the initial number, without
a net growth.
\section{Dirac Spinor Field}
For the case of a massive spin $\frac{1}{2}$ field $\Psi$ the
lagrangian density is:
\begin{equation}
{\cal L} = -\sqrt{-g} \left\{ \frac{i}{2} \left[ \bar{\Psi} \gamma^\mu
(\nabla_\mu \Psi) - (\nabla_\mu \bar{\Psi} )\gamma^\mu \Psi \right]
+ \mu \bar{\Psi} \Psi \right\}
\end{equation}
which in RW spacetimes can be rewritten as
\begin{eqnarray}
{\cal L} &=& a^3 \left[ \frac{i}{2} (\Psi^{\dagger} \dot \Psi -
\dot{\Psi}^{\dagger} \dot \Psi)- \Psi^\dagger \gamma_4 ({\vec{\gamma}\over{a}}
\cdot \vec{\nabla} + \mu) \Psi \right] \nonumber \\
&\equiv& a^3 \left[ \frac{i}{2} (\Psi^{\dagger} \dot \Psi -
\dot{\Psi}^{\dagger} \dot \Psi)- \Psi^\dagger M \Psi \right]
\end{eqnarray}
where $\mu$ is the inverse Compton wavelength of the spinor field and
we shall use the Pauli-Dirac representation for the $\gamma$ matrices.
On expanding
\begin{equation}
\Psi = \frac{1}{\sqrt{V}} \sum_{\bf k}
\Psi_{k} e^{i {\vec k} \cdot {\vec x}} \label{EFourier}
\end{equation}
one obtains an action:
\begin{eqnarray}
S &=& \sum_k S_k = \frac{1}{\sqrt{V}} \sum_k
\int d \tau a^3 \left[ \frac{i}{2} (\Psi^\dagger_k \dot \Psi_k -
\dot \Psi_k^\dagger \Psi_k) - \Psi_k^\dagger (\frac{i}{a} \gamma_4
\vec{\gamma} \cdot \vec{k} + \gamma_4 \mu) \Psi \right]
\nonumber \\
&\equiv& \frac{1}{\sqrt{V}} \sum_k
\int d \tau a^3 \left[ \frac{i}{2} (\Psi^\dagger_k \dot \Psi_k -
\dot \Psi_k^\dagger \Psi_k) - \Psi_k^\dagger M_k \Psi_k \right]
\end{eqnarray}
and again one can consider each Fourier mode separately and obtain
a single mode Hamitonian:
\begin{equation}
H_k = \psi_k^\dagger M_k \psi_k
\end{equation}
where $\psi_k = a^{3/2} \Psi _k$ and the matrix $M_k$ is:
\begin{equation}
M_k = \left( \begin{array}{cc}
\mu & \vec{\sigma} \cdot \frac{\vec{k}}{a} \\
\vec{\sigma} \cdot \frac{\vec{k}}{a} & -\mu
\end{array} \right) \,.
\label{fermmat}
\end{equation}
>From the Lagrangian density and eq.(\ref{EFourier}) one obtains a
classical equation of motion:
\begin{equation}
\left[ - i \frac{\partial}{\partial t} + i \gamma_4 \vec{\gamma}\cdot
\frac{\vec{k}}{a} + \gamma_4 \mu \right] w_k^{(r)} = 0
\end{equation}
which has been previously solved for a de Sitter space-time \cite{barut}
obtaining:
\begin{equation}
w_{1\,,\bf k}^{(r)} = \frac{1}{a^{1/2}} \left(
\begin{array}{c}
i Z_{\nu} \chi ^{(r)}\\
\frac{\vec \sigma \cdot \vec k}{k} Z_{\nu -1} \chi ^{(r)}
\end{array} \right) \mbox{ r=1,2}
\label{uno}
\end{equation}
and
\begin{equation}
w_{2\,,\bf k}^{(r)} = \frac{1}{a^{1/2}} \left(
\begin{array}{c}
- \frac{\vec\sigma \cdot \vec k}{k} Z_{\nu^* -1} \chi ^{(r)}\\
-i Z_{\nu^*} \chi ^{(r)}
\end{array} \right) \mbox{ r=3,4}
\label{due}
\end{equation}
where $r=1,3 (2,4)$ correspond to spin up (down) for the Pauli spinors
$\chi ^{(r)}$
and $Z_\nu (k |\eta|)$ are Bessel functions ($J_\nu$, $N_\nu$ or a
combination of them). Further $\nu=\frac{1}{2} - i\frac{\mu}{H}$
and we introduce a normalization factor $N_{a \, k}$ determined by
\begin{equation}
N_{a \, k}^2 w^{(r)\dagger}_{k} w^{(r')}_{k} = \delta_{r\, r'}
\end{equation}
It is now of interest to examine the adiabatic limit for which the
solutions (\ref{uno}) and (\ref{due}) reduce to the
usual static solutions:
\begin{equation}
u_{\bf k}^{(r)} = \left(
\begin{array}{c}
\chi ^{(r)}\\
\frac{\vec \sigma \cdot \vec k}{a(\omega_k+\mu)} \chi ^{(r)}
\end{array} \right) e^{-i \omega t} \quad r=1,2
\label{unobis}
\end{equation}
and
\begin{equation}
u_{\bf k}^{(r)} = \left(
\begin{array}{c}
- \frac{\vec \sigma \cdot \vec k}{a(\omega_k+\mu)} \chi ^{(r)}\\
\chi^{(r)}
\end{array} \right) e^{i \omega t} \quad r=3,4
\label{duebis}
\end{equation}
respectively, where $\omega=(\frac{k^2}{a^2} + \mu^2)^{1/2}$
and $r=1,3 (2,4)$ again refers to spin up (down) for the Pauli spinors.
As in the previous case we introduce a normalization factor $N_k$ determined by:
\begin{equation}
N_k^2 u_{\bf k}^{(r) \dagger} u_{\bf k}^{(r')}= \delta_{r\, r'} \, .
\end{equation}
We expect that the reduction to the usual static solutions occurs for very
early times ($\tau \rightarrow - \infty$)
or for wavelengths which are very small compared to the
de Sitter horizon $H^{-1}$. In such a limit ($-k\eta=\frac{k}{H} e^{-Ht}
\rightarrow \infty$) Eq. (\ref{unobis}) leads to
\begin{equation}
u_{\bf k}^{(r)} \rightarrow \frac{1}{\sqrt{2}} \left(
\begin{array}{c}
-\chi^{(r)} \\
\frac{\vec \sigma \cdot \vec k}{k} \chi^{(r)}
\end{array} \right) e^{i k \eta} \quad r=1,2
\end{equation}
which requires that one take the Hankel functions:
\begin{equation}
Z_\nu = H_\nu^{(1)}
\end{equation}
and a similar result only involving $H^{(2)}_{\nu ^*}$ is obtained from Eqs.
(\ref{due}) and (\ref{duebis}). Again, as in previous cases, we have a two
parameter
set of solutions: it is only by requiring agreement for a particular
(early time) limit with the adiabatic solutions that we constrain them.
We may quantize the system by postulating the usual canonical
anticommutation relations
\begin{equation}
\left\{ \hat \psi^\dagger_{\bf k},\hat \psi_{\bf k} \right\} = \hbar
\end{equation}
with all the other anticommutators being zero.
If we employ the adiabatic (static) solutions Eqs. (\ref{unobis}) and
(\ref{duebis}) one may expand
\begin{equation}
\hat \psi = \frac{1}{\sqrt V} \sum_{ k} e^{i {\vec k} \cdot {\vec
x}} \sum_r N_k u^{(r)}_{\bf k} \hat a_{\bf k}^{ (r)}
\end{equation}
with $\left\{\hat a^\dagger_{\bf k} , \hat a_k
\right\} = \hbar$ and on
substituting in the quantum Hamiltonian we obtain:
\begin{equation}
\hat H_{\bf k} =\hat {\psi}_k^\dagger M_k \hat {\psi}_k =
\sum_{r=1,2} \omega_k \hat a^{(r)\,\dagger}_{\bf k}
\hat a^{(r)}_{\bf k} - \sum_{r=3,4} \omega_k \hat
a^{(r)\,\dagger}_{\bf k}
\hat a^{(r)}_{\bf k}
\end{equation}
On then introducing a vacuum consisting of a "Dirac sea" of negative energy
states one has the usual interpretation of the destruction
of a negative energy as the creation of a positive energy antiparticle.
Thus through the usual replacements
\begin{eqnarray}
\hat a_{\bf k}^{(s)} = \hat a_{\bf k}^{(r)} \; ;
u_{\bf k}^{(s)} = u_{\bf k}^{(r)} \; \mbox{with r=s for $r=1,2$ }\nonumber \\
(-)^s c_{\bf k}^{(s)\,\dagger} = \hat a_{\bf k}^{(r)}
\; ;(-)^s v_{\bf k}^{(s)} = u_{-\bf k}^{(r)}\;
\mbox{with $s=1 (2)$ for $r=4(3)$}
\label{hole}
\end{eqnarray}
one obtains
\begin{equation}
\hat \psi = \frac{1}{\sqrt V} \sum_{{\bf k}\,s} N_k\left( e^{i {\vec k}
\cdot {\vec x}} \hat a_{\bf k}^{(s)} u_{\bf k}^{(s)}
+ e^{- i {\vec k}
\cdot {\vec x}} \hat c_{\bf k}^{(s) \, \dagger} v_{\bf k}^{(s)} \right)
\end{equation}
For this case, as in the previous section, one may also construct linear
invariants $\hat I$ by using Heisenberg fields and classical solutions through:
\begin{equation}
\hat I_{\bf k}^{(r)} = w_{\bf k}^{(r)\, \dagger} \hat \psi_{\bf k}
=N_{a \, k}^{-1}\hat b_{\bf k}^{(r)}
\end{equation}
where $\left\{ \hat b_{\bf k}^{(r)\, \dagger} ,\hat b_{\bf k}^{(r)}
\right\} = \hbar $
which, on using the anticommutation relations for $\hat \psi $ and the
classical equations of motions for $w$, satisfy:
\begin{equation}
{\partial \hat I_{\bf k}^{(r)} (t) \over \partial\,t}
- \frac{i}{\hbar} [\hat I_{\bf k}^{(r)} (t), \hat H_{\bf k} (t)]=0 \,.
\end{equation}
Correspondingly one may expand
\begin{equation}
\hat \psi =\frac{1}{\sqrt V} \sum_{\bf k} e^{i {\vec k} \cdot {\vec x}}
\sum_r N_{a \, k} w^{(r)}_{\bf k} \hat b_{\bf k}^{(r)}
\end{equation}
and again one may introduce a vacuum consisting of a 'sea' of the equivalent
of the negative energy states ($r=3,4$) with the usual interpretation of
a destruction operator as the creation of a 'hole'. Thus the relations
equivalent to eqs. (\ref{hole}) hold for the $b$ operators, where
the $d$ operators and the functions $y$ are introduced for $r=3,4$
(corresponding to $c$ and $w$),
obtaining
\begin{equation}
\hat \psi = \frac{1}{\sqrt V} \sum_{{\bf k}\,s}N_{a \, k}\left( e^{i {\vec k}
\cdot {\vec x}} \hat b_{\bf k}^{(s)} w_{\bf k}^{(s)}
+ e^{- i {\vec k}
\cdot {\vec x}} \hat d_{\bf k}^{(s) \, \dagger} y_{\bf k}^{(s)} \right)
\end{equation}
It is clear that the creation and destruction operators in the
two bases are related by a Bogoliubov transformation whose coefficients
may be easily determined;
thus for example the creation
of an invariant fermion quantum will correspond to a mixture
of the creation of a Dirac particle and the destruction of a
Dirac antiparticle. Further the quantum invariants allow
us to define an invariant vacuum
(the fermion equivalent of the Bunch-Davies vacuum \cite{bunchdavies}) by
\begin{equation}
b_{\bf k}^{(s)} |0\rangle_b = d_{\bf k}^{(s)} |0\rangle_b = 0
\end{equation}
We may now examine particle production from the vacuum during
a de Sitter expansion. In order to do this it will be sufficient
to consider the expectation value of the Hamiltonian with respect to the
above vacuum. On using the $\psi$ expansion in the
invariant basis it is straightforward to obtain
\begin{eqnarray}
\lim_{a \rightarrow \infty} { }_b\langle 0|\hat H_{\bf k} (t) |0\rangle_b
\!\!\!\!\! &=& \!\!\!\!\!
\lim_{a \rightarrow \infty} N_{a k}^2 \sum_{r=3,4} w_k^{(r)\dagger}M_k w_k^{(r)}
=\nonumber \\
\quad = 2\mu \left\{ \left| {\pi \left[ 1+i \cot {(\nu-1)\pi}\right]
\over{ \Gamma (\nu)^2}} \right| ^2 -1 \right\} & & \!\!\!\!\!\!\!\!\!\!\!\!
\left\{ \left| { \pi \left[ 1+i \cot {(\nu-1)\pi} \right]
\over{ \Gamma (\nu)^2}}\right| ^2 +1 \right\} ^{-1}
\label{limferm}
\end{eqnarray}
which is constant, that is the number of fermions does not increase in
time, and in particular we note that the RHS of Eq. (\ref{limferm})
is zero for $\mu = 0$ (actually there is a non-leading term ${\cal O} (k/a)$
corresponding to radiation).
Let us note that we have not concerned ourselves
with renormalizing the vacuum energy since we are just interested
in changes in it.
One may also consider the expectation value of the Hamiltonian
with respect to other states, such as coherent states. Such states, for
fermions, essentially consist of a superposition of zero and one fermion
states, because of the Pauli exclusion principle. It is straightforward to
verify that in this case also the expectation value of the Hamiltonian
is asymptotically constant in time and is of order ${\cal O} (k/a)$
for $\mu = 0$. This of course means that in the de Sitter expansion
the number of fermions in a given mode is asymptotically a constant.
In the next section we shall re-examine the results of this and the previous
section by examining the properties of the action rather than the
equations of motions under conformal transformations.
\section{Conformal Invariance}
Let us consider a general line element:
\begin{equation}
ds^2 = g_{\mu \nu} dx^\mu dx^\nu
\end{equation}
and the action of a massless conformally coupled scalar field:
\begin{equation}
{S}_{\Phi} = - \int d^4x \sqrt{-g} \left[
\frac{1}{2}
g^{\mu \nu} \nabla_{\mu} \Phi \nabla_{\nu} \Phi
+ \frac{1}{12} R {\Phi}^2 \right] \,.
\label{tilded}
\end{equation}
We may now perform a conformal trasformation:
\begin{eqnarray}
g_{\mu \nu} \rightarrow \tilde{g}_{\mu \nu} = \Omega^2 g_{\mu \nu}
\nonumber \\
\Phi \rightarrow \tilde \Phi = \Omega^{-1} \Phi
\label{conftras}
\end{eqnarray}
with $\Omega $ real, non-zero, continuous
and obtain for the change in the action
\begin{eqnarray}
\tilde{S}_{\tilde {\Phi}} - S_{\Phi} &=& \frac{1}{2} \int_V d^4 x \sqrt{-g}
\nabla^{\mu} \left( \Phi^2 \nabla_{\mu} \ln \Omega \right)
\nonumber \\
&=& \frac{1}{2} \int_{\Sigma(V)} d^3 \sigma \sqrt{-g}
n^{\mu} \Phi^2 \nabla_{\mu} \ln \Omega
\end{eqnarray}
where $n_\mu$ is a unit vector perpendicular to the three dimensional
surface $\Sigma$ containing V.
In particular for the RW metric (\ref{rwmetric}) the surface term becomes
\begin{equation}
-\frac{1}{2} \int_V d^3 x d\tau
{\partial \over{\partial \tau}} \left( \Phi^2 \dot a \right)
= - \frac{1}{2} \int_V d\eta d^3 x
{\partial \over{\partial \eta}} \left(\Phi^2 {1\over a}
{\partial a \over{\partial \eta}} \right)
\end{equation}
in agreement with our previous result \cite{desitter}. On the right
hand side we have introduced the conformal time $\eta$ with $a d\eta =
d\tau$ with which the metric (\ref{rwmetric}) becomes conformally flat:
\begin{equation}
ds^2 = a^2(\eta) (-d\eta^2 + d\vec{x}^2) \,.
\end{equation}
The electromagnetic field case is particularly simple: $A_\mu$
has zero conformal weight, that is it is unchanged
under conformal trasformations. This is also true for
$F_{\mu \nu}$ and for the action associated with it.
For the case of fermions, one also has that the massless spin $1/2$
Lagrangian density is invariant under conformal trasformations
with
\begin{equation}
\psi \rightarrow \tilde{\psi} = \Omega^{-3/2} \psi
\end{equation}
Thus for the case of the electromagnetic field and massless
fermions conformal invariance holds both for the action
and the equations of motion.
Accordingly
one then has that for a conformally flat metric the
flat space-times results are reproduced and there is no particle
production as the metric changes.
For a massless conformally coupled scalar field, on the other hand, conformal
invariance holds only for the equation of motion, while
the action and its
transformed version differ through the presence of a boundary (for
the RW metric a total derivative) term. Boundary terms,
which are classically associated with canonical transformations, do not
change
the equations of motion and conserved charges. However for a RW metric
the matter Hamiltonian is not a conserved charge
and hence the time evolution is changed.
Thus any attempt
to quantize employing Lagrangians obtained on neglecting
surface terms in RW metrics leads to questionable results, as also stated,
but for a different reason, in \cite{fulling}.
\section{Rescaled scalar field}
Let us further discuss the consequences of the presence of a (time derivative)
surface term in the scalar field action. As we have mentioned such a term
is classically associated with a canonical transformation and
quantum mechanically will correspond to a unitary transformation (as
is expected through the Poisson bracket - canonical commutator
correspondence, naturally on
neglecting eventual anomalies). On expanding a massive non-minimally
coupled scalar field in Fourier modes as in \cite{desitter}:
\begin{equation}
\Phi = \frac{1}{\sqrt{V}} \sum_{\bf k}
\left[ e^{i {\bf k} \cdot {\bf x}} \Phi_{k} (\tau) +
e^{- i {\bf k} \cdot {\bf x}} \Phi^*_{k} (\tau) \right]
\label{scalarFourier}
\end{equation}
and on separating real and imaginary parts:
\begin{equation}
\Phi_{k} (\tau) = \frac{1}{\sqrt{2}} \left( \phi_{k}^1 + i\phi_{k}^2
\right)
\end{equation}
one obtains an action for each mode ${\bf k}, i$
\begin{equation}
S_{\phi, k}^i = \frac{1}{2}
\int a^3\,d\tau \left( \dot{\phi}_k^{i\,2} - \omega_k^2 \phi_k^{i\,2}
\right)
\,,
\label{scalaraction}
\end{equation}
with $\omega_k^2 = \frac{k^2}{a^2} + \mu^2 +
\xi\,R$ and henceforth for semplicity we shall consider one mode $i, k$
and omit all such indices.
On rescaling $\phi = \zeta /a$ one immediately obtains the
corresponding action:
\begin{equation}
S_\zeta = \frac{1}{2}
\int a \,d\tau \left[ \dot{\zeta}^2 - (\omega^2 - \frac{1}{6} R)
\zeta^2
- {1\over a}\frac{d}{d\tau} \left(\dot a \zeta^2 \right)\right]
\,.
\label{reaction}
\end{equation}
From the above actions, Eqs. (\ref{scalaraction}) and (\ref{reaction})
one obtains the
following Hamiltonians (related
to evolution in proper time):
\begin{eqnarray}
H_\phi &=& \frac{1}{2a^3}
\pi_\phi^2 + \frac{a^3}{2} \omega^2 \phi^2
\label{ham_phi} \\
H_\zeta &=& \frac{\pi_{\zeta}^2}{2 a} +
\frac{a}{2} \omega^2 \zeta^2 + \frac{\dot{a}}{2a} (
\pi_{\zeta} \zeta + \zeta \pi_{\zeta} )
\label{ham_zeta} \,,
\end{eqnarray}
where $\pi_\phi = a^3 \dot \phi$, $\pi_{\zeta} = a \left(
\dot{\zeta} - \frac{\dot a}{a}
\zeta \right) = \pi_\phi/a$ , we have used $R=6 \left( {\ddot a \over a}
+ {{\dot a}^2 \over {a}^2} \right) $ and one has
\begin{equation}
H_\zeta = H_\phi + \frac{\dot a}{2a} (\phi \pi_{\phi} +
\pi_{\phi} \phi) \,.
\end{equation}
It is easy to verify that the trasformation from $\phi$ to $\zeta$
is canonical:
\begin{equation}
[\phi, \pi_\phi]_{PB} = [\zeta, \pi_\zeta]_{PB} = 1
\end{equation}
and one relates the Poisson brackets (PB) to commutators in order to
canonically quantize the two systems. For the system with Hamiltonian
(\ref{ham_phi}) one can introduce the following operator, which
factorizes the Hamiltonian,
\begin{equation}
\hat a_\phi = \left( \frac{a^3 \omega}{2 \hbar} \right)^{\frac{1}{2}}
\left( \hat \phi + i\,{\hat \pi_\phi \over a^3 \omega} \right) \,,
\end{equation}
and the operator related to the linear invariant
\begin{equation}
\hat b_\phi = \frac{1}{\sqrt{2 \hbar}}
\left[{\hat \phi \over \rho} +
i\,\left(\rho \,\hat \pi_\phi - a^3 \dot{\rho} \hat \phi \right)\right] \,,
\end{equation}
Similarly one may introduce the corresponding operators for the
Hamiltonian
(\ref{ham_zeta})
\begin{eqnarray}
\hat a_\zeta &=& \left( \frac{a \omega_D}{2 \hbar} \right)^{\frac{1}{2}}
\left( \hat \zeta + i\,{\hat \pi_\zeta \over a \omega_D} \right) \,,
\label{aoperator}
\\
\hat b_\zeta &=& \frac{1}{\sqrt{2 \hbar}}
\left\{ {\hat \zeta \over a \rho} +
i\,\left[ a \rho (\hat \pi_\zeta + \dot a \hat \zeta)
- (a \dot{\rho} + \dot a \rho) a \hat \zeta \right] \right\}\,,
\label{boperator}
\end{eqnarray}
where $\omega_D= (\omega^2 - \dot a^2/a^2)^{\frac{1}{2}}$ and in both
cases $\rho$ satisfies
\begin{equation}
\ddot{\rho} + 3 \frac{\dot a}{a} \dot{\rho}
+ \omega^2 \rho = \frac{1}{a^6 \rho^3}
\end{equation}
Of course the above operators satisfy
$[\hat a,\hat a^\dagger]=[\hat b,\hat b^\dagger]=1$ and
remarkably $\hat b_\phi$ and $\hat b_\zeta$ {\em coincide} implying that
they lead to the same vacuum and Fock space. Actually this
should not be surprising since the classical solutions to the
equation of motion for $\phi$ and $\zeta/a$ are the same.
On using the above operators one obtains:
\begin{eqnarray}
\hat I &=& \hbar (\hat b_\phi^\dagger \hat b_\phi + \frac{1}{2} ) =
\hbar (\hat b_\zeta^\dagger \hat b_\zeta + \frac{1}{2} ) \\
\hat H_\phi &=& \hbar \omega ( \hat a_\phi^\dagger \hat a_\phi + \frac{1}{2}) \\
\hat H_\zeta &=& \hbar \omega_D
( \hat a_\zeta^\dagger \hat a_\zeta + \frac{1}{2} ) \label{accazeta}
\end{eqnarray}
where $\hat I$ is an {\em invariant} and has time-independent eigenvalues,
while $\hat H_\phi$ and $\hat H_\zeta$ obviously do not and correspond to
the number of
$\phi$ and $\zeta$ quanta times their respective energies.
It is clear from Eq. (\ref{scalaraction}) that it is only the $\phi$
quanta
that have a
particle interpretation and the $\phi$ and $\zeta$ quanta are related through
a Bogoliubov transformation
\begin{equation}
\hat a_\phi = \frac{1}{2} \hat a_\zeta [ \frac{\omega^{1/2}}{\omega_D^{1/2}}
+ \frac{\omega_D^{1/2}}{\omega^{1/2}} ] + \frac{1}{2} \hat a_\zeta^\dagger
[\frac{\omega^{1/2}}{\omega_D^{1/2}}
- \frac{\omega_D^{1/2}}{\omega^{1/2}} ] \label{Bogtrans}
\end{equation}
corresponding to a squeezing \cite{schumaker}. Naturally the $\hat b$
and $\hat a$ are also related through a Bogoliubov transformation
and it is the $\hat b$ Fock space that describes the correct evolution.
Let us end by commenting that if we had omitted the surface term
in Eq. (\ref{reaction}) we would have obtained a third different
Hamiltonian (which, apart from overall scale factors, coincides with the 00
component of the scalar field energy-momentum tensor appearing in the
Einstein equations for $\xi =1/6 $)
\begin{equation}
\bar{H}_\zeta = \frac{1}{2 a} \bar{\pi}^2_\zeta + \frac{a}{2} ( \omega^2
- \frac{R}{6}) \zeta^2
\label{ham_skaz}
\end{equation}
where $\bar{\pi}_\zeta = a \dot \zeta = \pi_\zeta + \dot a \zeta $.
Again one may study the quantum system and introduce both linear invariant
operators and operators factorizing the Hamiltonian. It is straightforward
to verify that the former agree with Eq. (\ref{boperator}), that is one obtains
the same invariant vacuum and the Fock space. This is not surprising since the
equations of motion again are unchanged. For the quantum Hamiltonian,
on the other hand, one obtains:
\begin{equation}
\hat {\bar H}_{\zeta} = \hbar \bar \omega_D \left( \hat {\bar
a}_{\zeta}^{\dagger} \hat {\bar a}_{\zeta} +{1\over 2} \right)
\end{equation}
where $\bar {\omega}_D = ( \omega^2 - R/6 )^{1/2}$ and
\begin{equation}
\hat {\bar a}_{\zeta}^{\dagger} = {\left( {a\bar \omega_D
\over{2 \hbar}}\right)}^{1/2} \left[ \hat \zeta +{i\over{a\bar \omega _D}}
\hat{\bar \pi}_{\zeta} \right]
\end{equation}
which are related to the corresponding operators in Eq.(\ref{aoperator})
through a Bogolubov transformation:
\begin{equation}
\hat {\bar a}_{\zeta}= {1\over 2} \hat a_{\zeta} \left[ \left( 1+i{\dot a \over
{a \bar \omega_D}} \right) \left( {\bar \omega_D \over \omega_D}
\right)^{1\over 2}
+ \left( {\omega_D \over \bar \omega_D} \right)^{1\over 2} \right] +
{1\over 2} \hat a_{\zeta} ^{\dagger}
\left[ \left( 1+i {\dot a \over
{a \bar \omega_D}} \right) \left( {\bar \omega_D \over \omega_D}
\right)^{1\over 2}-\left( {\omega_D \over \bar \omega_D} \right)^{1\over 2}
\right]
\end{equation}
It is important to note the different spectrum obtained with respect to
Eqs. (\ref{accazeta}) and (\ref{Bogtrans}). Thus rescaling and omitting
the surface term obtained will modify the quantum system and the spectrum of
the Hamiltonian, leading to questionable results for some physical
quantities, while maintaining the invariant vacuum and Fock space which
corresponds to leaving the equations of motion unchanged.
For example we may
consider the invariant vacuum ($ \mid 0 \rangle _b$) expectation values of the
quantum Hamiltonians given by eqs.($70$) and eqs.($74$) for $\xi =0$ (thus we
do not have the presence of derivatives of $a$ in the matter Hamiltonian
- we return to this at the end of the next section) in a de Sitter space and
consider the
limit as $ a \longrightarrow \infty $. One obtains \cite{desitter}:
$$
{}_b \langle 0|\hat{H}_{\phi}|0 \rangle_b \simeq \frac{\hbar}{4} B \left(
\frac{\Gamma (\bar{\nu})}{\pi}
\right)^2 \left( \frac{k}{2H} \right)^{-2\bar{\nu}}
a^{2 \bar{\nu}} \left[ \mu ^2 + H^2 \left( \bar{\nu} - \frac{3}{2}
\right)^2 \right]
$$
$$
{}_b \langle 0|\hat{\bar{H}}_{\zeta}|0 \rangle _b \simeq \frac{\hbar}{4} B
\left(
\frac{\Gamma (\bar{\nu})}{\pi} \right)^2 \left( \frac{k}{2H} \right)^{-2\bar{\nu}}
a^{2\bar{\nu}} \left[ \mu ^2 -2H^2 + H^2 \left( \bar{\nu} - \frac{1}{2}
\right)^2 \right]
$$
where $B$ is a constant and $\bar{\nu}^2= 9/4 - \mu ^2/ H^2 $. We
immediately
note that the two expressions differ and in particular the first one vanishes
for $\mu =0$ (of course non-leading terms remain). Similar considerations
hold on considering expectation values
with respect to other states. A detailed discussion of the 'squeezing' effect
of the surface term for the $\xi=\mu=0$ case has been previously done
\cite{polarski}.
\section{Conclusions}
The method of invariants is a particularly suitable tool in order to
investigate quantum effects in time-dependent external fields.
On using a conserved
operator, one can find an invariant vacuum and an invariant Fock space,
which are preserved during the time evolution.
In this paper we have
used this tool
in order to investigate the relation between particle production
and the conformal properties of fields.
Motivated by the non conformal invariance of the action for the
scalar field case \cite{desitter} - shown in section 4 for a general
space-time -, we have analyzed the electromagnetic
and Dirac fields.
For the electromagnetic and the massless spinor fields the action
is conformally invariant, and we have verified that there is no particle
production in the Hamiltonian (which is equal to the energy density in
these cases). For the massive spinor case there is a non-trivial particle
production, but, because of the Pauli exclusion principle, the number of
particles in a given mode can never exceed one.
We have gone beyond our previous paper \cite{desitter}
and investigated, for cosmological spacetimes, some effects due to
the presence of the
surface term which violates the conformal invariance of the action in the
scalar field case with the following results:
1. We verified that the use of rescaled fields is classically a
canonical trasformation and a unitary trasformation quantum mechanically.
On retaining all terms in the action we used
the rescaled fields in order to canonically quantize the system. and
obtained the same invariant vacuum and the same invariant Fock space as
resulted on quantizing the original (non-rescaled) fields.
2. The fact that the energy density of a non-minimally coupled scalar
field (which is the source of the Einstein tensor) differs from the
canonically obtained Hamiltonians of the field and of the rescaled field
(which generate the correct time evolution for the two systems) implies that
the usual statement of no particle production - on neglecting
quantum anomalies - in the
massless conformally coupled scalar field should only be applied
to the source of Einstein tensor.Indeed as we pointed out in the previous
section apart from overall scale factors the
rescaled scalar field Hamiltonian, obtained on omitting surface terms,
coincides with the $00$ component of the scalar field energy momentum
tensor appearing in the Einstein equations for $\xi = 1/6 $.
3. The neglect of the surface term in the action for the
rescaled fields classically leads to
a third and different Hamiltonian (\ref{ham_skaz}), and so a different
time evolution.
On canonically quantizing this system, one however gets the
same invariant vacuum and Fock space corresponding to the fact that
the equations of motion are unchanged.
Let us end by noting that in all the above we have employed the usual
canonical formalism for the scalar field starting from the action and
obtaining the Hamiltonian describing time evolution. Time evolution
for matter can also be obtained from an initial action containing both
gravitation and matter leading to the Wheeler-De Witt equation which, of course,
does not contain time. On performing a Born-Oppenheimer decomposition
and considering the semiclassical limit for gravity one is led to the
Schrodinger (or Schwinger-Tomonaga ) equation for the evolution of matter
\cite{LakeTahoe}.
We feel it would be interesting to obtain a similar approach for the case
of a non-minimally
coupled scalar field. However there is an essential difficulty since a
non-minimal coupling leads to terms containing derivatives of the metric
appearing in the matter lagrangian. This will lead to momenta conjugate to the
metric appearing in the matter Hamiltonian in contrast with what is usually
assumed for a Born-Oppenheimer (or adiabatic) approach. Nonetheless we hope to
return to this.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 532 |
Home > All Blogs > Blog > The Inner Sanctum
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"A stellar sequel worth the wait." – Booklist 8/22/2013
Raccoons on a Mission! 7/1/2013
The SOURCE – June 13, 2013 6/14/2013
S&S Children's at BEA 2013! 5/13/2013
Pulseit: Read it, Heart it, Share it. 4/12/2013
Sharon M. Draper is at it again! 3/14/2013
2012 Winter Institute 1/13/2012
GOLIATH Audiobook 9/23/2011
Major Adult Authors Come to Teen! 9/19/2011
Perfect by Ellen Hopkins! 9/15/2011
The Man in the Moon – on sale today! 9/6/2011
The Unwanteds by Lisa McMann 9/1/2011
Exciting New Titles for Young Children 9/1/2011
New Teen Books Hit the Shelves 9/1/2011
THE SECRET WAR Audiobook 8/17/2011
Banned Books Week 8/16/2011
Michael Vey on sale today! 8/9/2011
The Smurfs Take on New York City 7/28/2011
Rave Review for Dreams of Significant Girls! 7/17/2011
New Teen Novel from Richard Paul Evans! 7/12/2011
Blood Red Road Trailer is here! 6/15/2011
Happy Dork Day!!!!!!!!! 6/7/2011
Summer Before Boys by Nora Raleigh Baskin 5/11/2011
The Beginning of a Summer of Firsts! 5/3/2011
Happy Book Birthday! 5/3/2011
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1. "A stellar sequel worth the wait." – Booklist
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By: Lucille, on 8/22/2013
Blog: The Inner Sanctum (Login to Add to MyJacketFlap)
In 2002, The House of the Scorpion by Nancy Farmer received the National Book Award, a Newbery Honor, and a Printz Honor–the only book to ever accomplish this feat! Now, over 10 years later, we are proud to bring you the highly anticipated sequel, The Lord of Opium, pubbing on September 3, 2013.
The Lord of Opium picks up right where The House of the Scorpion left off. Publisher's Weekly and Booklist have already given this book starred reviews, and the praise speaks for itself:
"A stellar sequel worth the wait."
"Lyrically written and filled with well-rounded, sometimes thorny characters, this superb novel is well worth the wait."
"The landscape of dystopian literature has changed significantly since the first book, but this sequel is still a cut above the rest."
— The Horn Book on September/October
The Lord of Opium by Nancy Farmer
2. Raccoons on a Mission!
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By: Lucille, on 7/1/2013
Meet Bingo and J'miah. They are going to save their beloved Sugar Man Swamp from a gator wrestler determined to turn it into an alligator-and-pony show AND from a herd of feral hogs, and they are going to discover ART on the way. For they are Scouts. They are best brothers. They are raccoons on a mission!
The True Blue Scouts of Sugar Man Swamp is the newest from Kathi Appelt, author of The Underneath, which was a Newbery Honor Book and a National Book Award finalist. Sugar Man is for a slightly younger audience than Kathi's previous novels–and is seriously funny! Laugh-out-loud-in-the-middle-of-class-reading-time funny. And it's a sensational read aloud. It has already received amazing acclaim with 3 starred reviews! (Kirkus, Booklist, and Publisher's Weekly.) It pubs next month on July 23!
3. The SOURCE – June 13, 2013
By: Brianne, on 6/14/2013
Highlights from this week's edition of THE SOURCE!
THE WHITE QUEEN, Philippa Gregory
9781476735481, Touchstone, $16.00
9781442342330, UAB, $29.95
The Starz drama series, THE WHITE QUEEN, was listed in Entertainment Weekly's "12 Shows You Can't Miss," calling it a great next pick for Game of Thrones fans! The series will air August 10. Watch the trailer and read about it here!
In anticipation of the show, we've repackaged Philippa Gregory's backlist titles: THE RED QUEEN, LADY OF THE RIVERS and KINGMAKER'S DAUGHTER. Look out for newest in the series, THE WHITE PRINCESS (9781451626094/
on sale July 23), and don't miss STORMBRINGERS (9780857077349), the second book in Philippa Gregory's teen series, ORDER OF DARKNESS.
BRUCE, Peter Ames Carlin
9781442357495, UAB, $29.95 On Sale September 17, 2013
Ridley Scott is producing a Bruce Springsteen documentary, "Springsteen & I," which hits theaters
June 22! Visit the website to find out more!
THE JOKER, Andrew Hudgins
9781476712710, S&S, $25.00
Elle Magazine review, June
Entertainment Weekly must list, June 14
You can download the full document here! The SOURCE June 13, 2013 6-6-2013
4. S&S Children's at BEA 2013!
It's that BEA time of year again!
Octavia Spencer, Cassandra Clare, Peter Brown, Brandon Mull, and Olivia… These are just some of the guests who will be joining us at BEA this year!
We're going to have lots of fun giveaways and have plenty to talk about with a fantastic fall 2013 line up. Come visit us anytime at Booth 2638, or visit our authors at their signings and panels.
Author Signings at the S&S Booth
Thursday, May 30 at 11:15 am
Peter Brown will be signing Creepy Carrots!
Friday, May 31 at 11 am
Octavia Spencer will be signing ARCs of Randi Rhodes, Ninja Detective: The Case of the Time-Capsule Bandit
Author Signings in the BEA Autographing Area
Abbi Glines will be signing Breathe
Table 5 from 12:00 — 12:30 pm
Susan Cooper will be signing Ghost Hawk
Matthew Van Fleet will be singing Lick
Table 10 from 1:00 — 1:30 pm
Jenny Han & Siobhan Vivian will be signing Fire with Fire
Table 8 from 2:00 — 2:30 pm
Neal Shusterman will be signing UnWholly
Cassandra Clare will be signing the movie tie-in edition of City of Bones
Brandon Mull will be signing A World without Heroes
Cat Patrick & Suzanne Young will be signing Just Like Fate
Table 12 from 11:30 — 12:00 pm
Patricia Polacco will be signing The Blessing Cup
*Times subject to change
Looking forward to seeing those of you who will be at BEA later this month! (And don't worry, we'll definitely save some goodies for those of you who will not be attending this year.)
5. Pulseit: Read it, Heart it, Share it.
April is the month of the Pulseit re-launch! For those of you unfamiliar with it, Pulseit is an online community for anyone ages 14 and older that loves teen books. It's where users can read free teen books & exclusive excerpts, like content on the site by hearting books, members, authors & other content, and share their reviews on books, ideas, and more. Plus – it's free to join & participate!
Here are some highlights of the new Pulseit site:
A new web address: The Pulseit site can now be found at www.pulseit.com
A brand new look: Pulseit will be updated with a new logo & look! See the image below for a sneak peek.
More content: At least two free reads a month with numerous excerpts.
Mobile: Books will now be readable on all devices: desktop computers, phones, and tablets.
For all readers: Pulseit membership, which has always been limited to teens between the ages of 14 and 18, will now be open to anyone ages 14 & up.
We also have bookmarks & postcards to help you promote Pulseit in your stores, libraries, and schools! Please send your request for bookmarks & postcards to [email protected]
6. Sharon M. Draper is at it again!
Sharon M. Draper is at it again, and this time with a page-turning novel for teens. PANIC is a riveting exploration of power: how quickly we can lose it — and how we can take it back.
Diamond knows not to get into a car with a stranger. But what if the stranger is well-dressed and handsome? On his way to meet his wife and daughter? And casting a movie that very night—a movie in need of a star dancer? What then? Then Diamond might make the wrong decision.
Sharon Draper's most recent book, OUT OF MY MIND, was on the New York Times bestseller list for over six months. It's also received phenomenal reviews. The Denver Post said, "If there's only one book teens and parents (and everyone else) can read this year, OUT OF MY MIND should be it."
We hope you're as excited as we are for PANIC to hit the shelves on this month!
Panic by Sharon M. Draper
7. 2012 Winter Institute
ABA's Seventh Annual Winter Institute in New Orleans is next week! S&S Children's Publishing will be attending and we look forward to seeing you there!
This year, T. R. Burns, author of Merits of Mischief, and Lisa M. Stasse, author of The Forsaken, will be at the Author Reception! We also have quite a number of galleys available for you in the Galley Room, so make sure you stop by to pick one up.
If you aren't able to attend, you can read more about the books by clicking the links below or by clicking to get a copy of the eGalley!
When You Were Mine
By: Rebecca Serle
On Sale: 5/1/12
A modern retelling of Shakespeare's classic Romeo and Juliet. This intensely romantic, tragic teen debut is narrated by Rosaline, the girl Romeo was supposed to love.
To read this eGalley click here
The Forsaken
By: Lisa M. Stasse
On Sale: 7/10/12
Combining elements of The Hunger Games, Lord of the Flies and Minority Report, The Forsaken imagines a world where criminals can be jailed without committing a crime. The first book in a riveting new dystopian trilogy.
Merits of Mischief: Bad Apple
By: T.R. Burns
An irreverent new middle grade series about a group of kids sent to a school to be trained as professional troublemakers!
Anyway*
By: Arthur Salm
Similar in tone to the Wimpy Kid series, Anyway* is peppered with humorous handwritten footnotes and doodles throughout and perfectly captures the viewpoint of a young boy doing his best to find his place in the world.
The Search for WondLa: A Hero for WondLa
By: Tony DiTerlizzi
The highly anticipated second novel in the New York Times bestselling The Search for WondLa series.
"An irresistible adventure. I can't wait to read the rest of the series."
-Rick Riordan, author of the Percy Jackson and the Olympians series
Heidi Heckelbeck Has a Secret
By: Wanda Coven
A brand-new young chapter-book series with witchy whimsy for readers between the ages of 5 and 7! Heidi Heckelbeck tells the story of a young girl with a bewitching secret!
Captain Awesome to the Rescue
By: Stan Kirby
With illustrations throughout, this series follows the adventures of eight-year-old Eugene McGillicudy¹s supersecret superhero alter ego: Captain Awesome.
1-2-3 Peas
By: Keith Baker
The peas are back in this new counting-themed book from Keith Baker, author of the Indie favorite and New York Times bestselling LMNO Peas!
Crafty Chloe
By: Kelly DiPucchio
Meet Chloe: a not-so-ordinary girl with extraordinary crafty talents! Fancy Nancy meets Martha Stewart in Crafty Chloe, the adorable DIY star of a new picture book series!
8. GOLIATH Audiobook
By: Lauren, on 9/23/2011
JacketFlap tags: audiobook, Audio, scott westerfeld, leviathan, goliath, alan cumming, Children, interview, Add a tag
It's finally here! The last installment of the LEVIATHAN trilogy, GOLIATH, is now available on CD and for download. Check out our exclusive interview with author Scott Westerfeld and narrator (and Emmy Award-nominee) Alan Cumming HERE!
9. Major Adult Authors Come to Teen!
JacketFlap tags: Samantha Van Leer, Children, Young Adult, Teen, Jodi Picoult, Philippa Gregory, Add a tag
Jodi Picoult, the internationally bestselling and award-winning author of numerous books, and her daughter Samantha Van Leer, have teamed up to write a new teen novel: Between the Lines, planned for release in the Summer 2012 season.
Picoult is the #1 New York Times bestselling author of several adult novels, including My Sister's Keeper, House Rules, Handle With Care, Change of Heart and Nineteen Minutes. Her daughter, Samantha Van Leer, is a junior in high school and pitched the idea of the book to Picoult, while she was in the middle of a book tour.
Between the Lines will be published jointly by Simon Pulse and Emily Bestler Books, and will be released on June 2012. This will be Van Leer's debut book, while it will serve as Picoult's debut teen book.
And another internationally bestselling adult author, Philippa Gregory, will also publish a series of young adult historical romance titles with Simon Pulse! Her first title is scheduled to release in Summer 2012.
Gregory is the #1 New York Times bestselling author of The Other Boleyn Girl and The Red Queen, with her Tudor series being adapted as a TV drama and The Other Boleyn Girl as a major film in 2008.
With this new deal, Gregory, and her historical and storytelling expertise, will be exposed to a new generation of young adult readers around the world.
10. Perfect by Ellen Hopkins!
JacketFlap tags: Children, Uncategorized, Poetry, Fiction, Young Adult, Teen, Ellen Hopkins, Perfect, Impulse, Add a tag
Perfect by New York Times bestselling author Ellen Hopkins is on sale now! A companion novel to the bestselling book Impulse, Perfect is a deeply empathetic and riveting portrait of teens striving past the possible to be perfect—and their quest to let perfection go and follow their hearts.
Ellen's new revamped site has a bunch of online assets include the video trailer, a video interview, an excerpt of the book, as well as video excerpts of the audiobook. Click to get to Ellen's new YA focused site: http://ellenhopkins.com/YoungAdult/perfect/
Also, Banned Books Week is coming up on 9/25! Visit our Banned Book Weeks site for PDFs of Ellen's manifesto and more! Visit: http://pages.simonandschuster.com/bannedbooksweek
11. The Man in the Moon – on sale today!
JacketFlap tags: Fall 2011, Guardians of Childhood, Children, Picture Book, William Joyce, Man in the Moon, Add a tag
William Joyce is a true luminary and creative spirit whose picture books are modern-day classics. More than a decade after his last picture book, Joyce returns with an amazing new book series: The Guardians of Childhood. The series will consist of 7 picture books and 7 chapter books, each focusing on an icon of childhood—the Man in the Moon, the Easter Bunny, Jack Frost, the Tooth Fairy, the Sandman and Santa Claus. In his first picture book adventure Joyce takes us to the moon with The Man in the Moon (Atheneum, 9781442430419) – on sale today!
The first chapter book in the series, Nicholas St. North and the Battle of the Nightmare King (Atheneum, 9781442430488), is on sale in October of this year.
The Man in the Moon has already received tons of praise. Publisher's Weekly gave it a starred review, calling it "a tale that's warm and fuzzy, swashbuckling, and dazzlingly inventive all at the same time" and Maurice Sendak calls it "a fabulous recapturing of an old, real fairy-tale world. Dark. Mysterious. Stunning!"
Joyce will also be all over national media this month – a list of when & where are all below!
XM Kids/ "Absolutely Mindy," 9/6
Westwood One Radio/ "First Light," 9/7
TV & Radio Satellite Tour, 9/7
CBS Radio/ "CBS Weekend Round-up," 9/19
FOX News Channel/ "FOX News Live," 9/25
We have a brand new website for the series which includes videos, introductions to the Guardians, a how-to guide for drawing a Moonbot and lots of downloadables including The Guardians Oath, trading cards, wallpapers, avatars, activities, and more! Click here for the website: http://theguardiansofchildhoodbooks.com/
12. The Unwanteds by Lisa McMann
JacketFlap tags: Unwanteds, Children, Middle-Grade, Lisa McMann, Add a tag
Lisa McMann, the New York Times bestselling author of the Wake Trilogy, has released her first middle-grade book, The Unwanteds. With a combination of middle-grade fantasy and dystopian themes, The Unwanteds is sure to appeal to many tween readers.
Every year in Quill, thirteen-year-olds are sorted into categories: the strong, intelligent Wanteds go to university, and the artistic Unwanteds are sent to their deaths.
Thirteen-year-old Alex tries his hardest to be stoic when his fate is announced as Unwanted, even while leaving behind his twin, Aaron, a Wanted. But, as secrets are revealed and transformations are made, Alex and Aaron's bond stretches and a threat arises that will pit brother against brother in an ultimate, magical battle.
The Unwanteds, 9781442407688, Aladdin, Hardcover
13. Exciting New Titles for Young Children
JacketFlap tags: Children, Picture Books, new, Add a tag
In Moo by Matthew Van Fleet, author of New York Times bestsellers Heads and Dog, the audience experiences interactive fun with his introduction to the habits and distinctive voices of seven species of barnyard residents. Toddlers will be delighted to try to identify and imitate every critter they see.
Visit http://authors.simonandschuster.com/Matthew-Van-Fleet/34583444 to learn more about Moo and more of Matthew Van Fleet's books.
Moo, 9781442435032, Paula Wiseman Books, Novelty
In the jazzy picture book Hootenanny! by Kimberly Ainsworth, sure to excite young readers, a hilarious group of owls sing and boogie their way up to the top of the Old Oak Tree for a party on Saturday night.
Visit http://books.simonandschuster.ca/Hootenanny!/Kimberly-Ainsworth/9781442422735 to read more about Hootenanny! and watch a video on the Hootenanny! Hop!
Hootenanny!, 9781442422735, Little Simon, Hardcover
Bill Martin Jr. and Lois Ehlert, the masterminds behind Chicka Chicka Boom Boom, have collaborated to make Ten Little Caterpillars. In this book young children will be able to explore this big world with the help of caterpillars, and who knows what they will find…maybe a butterfly?
Visit http://books.simonandschuster.com/Ten-Little-Caterpillars/Bill-Martin-Jr/9781442433854 to learn more about Ten Little Caterpillars and to read an excerpt.
Ten Little Caterpillars, 9781442433854, Beach Lane Books, Hardcover
14. New Teen Books Hit the Shelves
JacketFlap tags: Children, Fury, Teen, new, Vengeance, Dust & Decay, Add a tag
Elizabeth Miles' big teen debut, Fury, shows why sometimes sorry isn't enough…In Ascension, Maine, mistakes can be deadly. And three girls—three beautiful, mysterious girls—are here to choose who will pay.
Visit The Fury Series website at http://thefuryseries.com/ to learn more about the series and Elizabeth Miles.
Fury, 9781442422247, Simon Pulse, Hardcover
Dust & Decay by Jonathan Maberry, the exciting sequel to Rot & Ruin, brings Tom, Benny and Nix together and ready to leave their home forever in search for a better future. While it may seem easy, in the great Rot & Ruin everything wants to kill you.
Visit http://books.simonandschuster.com/Dust-Decay/Jonathan-Maberry/9781442402355 to read more about Dust & Decay and Rot & Ruin.
Dust & Decay, 9781442402355, Simon & Schuster Books for Young Readers, Hardcover
In Vengeance, Kate Brian brings her bestselling Private series to a shocking conclusion. For the many fans of the Private series, they will be sure to recognize what it means to go out with a bang.
Visit The Private Series Website at http://www.privatetheseries.com/ to learn more!
Vengeance 9781416984733, Simon & Schuster Books for Young Readers, Paperback
15. THE SECRET WAR Audiobook
THE SECRET WAR second installment in author Matt Myklusch's fast-paced, action-adventure Jack Blank series. Like JACK BLANK AND THE IMAGINE NATION, this audiobook is narrated by Tony Award-winner Norbert Leo Butz. Check out our video interview with Norbert about narrating audiobooks HERE.
16. Banned Books Week
JacketFlap tags: Children, Children's, Banned Books Week, Add a tag
Banned Books Week, set for September 24-October 1 this year, is soon approaching. Join Simon & Schuster as we celebrate people's freedom to read and enjoy books of all subjects, and spread the importance of the First Amendment.
Check out our web page http://pages.simonandschuster.com/bannedbooksweek where our favorite authors of challenged books, such as Lisa McMann and Ellen Hopkins, speak out against the banning of books.
You can also check out some of Simon & Schuster's challenged books like Wake by Lisa McMann, The Misfits by James Howe, Crank by Ellen Hopkins, and And Tango Makes Three by Peter Parnell and Justin Richardson.
"Censorship comes from a place of fear. We fear that which is different, what we cannot comprehend. Books gift us with understanding, empower us with knowledge. So, obviously, would-be censors have it all wrong."—Ellen Hopkins, author of frequently challenged book Crank
17. Michael Vey on sale today!
Michael Vey, the new teen novel from #1 New York Times bestselling author Richard Paul Evans is on sale today!
Richard is kicking off his tour with some major media, all happening today:
Glenn Beck's Nationally Syndicated Radio program—live in-studio
GBTV (Glenn Beck's syndicated internet show)—live in-studio
Radio Satellite Tour
Watch the trailer here: http://www.facebook.com/video/video.php?v=10150260801012129
Read the first 100 pages and get more assets here: http://pages.simonandschuster.com/michaelvey/
MICHAEL VEY
Written by Richard Paul Evans
Publication Date: 8/9/11 (laydown)
Simon Pulse/Mercury Ink
$17.99, Ages 12 up
18. The Smurfs Take on New York City
JacketFlap tags: The Smurfs, Children, Movies, Children's, Add a tag
They've been on television and in books, but on Friday, July 29th The Smurfs will be opening in movie theaters in 3D. They—and of course Gargamel too—will be coming to keep the audience laughing and on their toes as they are forced out of their village and find themselves in New York City.
But, for the audience that just can't keep themselves away from these little blue creatures, Simon & Schuster offers two lines of Smurfs book. The movie tie-ins include A Smurfin' Big Adventure!, Behold the Power of Gargamel!, and The Smurfs Movie Novelization. However, there is also the line of classic Smurf tales, which includes Meet Smurfette, Off to School!, and The Snow Giant, just to name a few.
The adventures of The Smurfs do not end in the movie theaters as families everywhere can sit back and enjoy classic and new tales.
19. Rave Review for Dreams of Significant Girls!
JacketFlap tags: Children, Fiction, Teen, New York Times Book Review, Christina Garcia, Dreams of Significant Girls, Add a tag
Dreams of Significant Girls by Christina Garcia (9781416979203), has just received this stellar review from The New York Times Book Review in the July 17th edition!
"Let's call them the original Gossip Girls. In Cristina Garcia's first young adult novel, three wealthy and adventurous ninth-grade girls from different worlds converge upon a Swiss boarding school for a summer of discovery…(DREAMS OF SIGNIFICANT GIRLS) takes you breathlessly and painfully back to the time when womanhood shimmered before you, always just out of reach, and you lunged for it, stupidly and bravely, with your first cigarette, your first kiss, your first swill of liquor, your first boy crush and your first girl crush."
20. New Teen Novel from Richard Paul Evans!
JacketFlap tags: Children, Fiction, Teen, August, 2011, Richard Paul Evans, Michael Vey, Add a tag
The first teen novel from bestselling author Richard Paul Evans, Michael Vey: The Prisoner of Cell 25 (9781451656503), is on sale 8/9/11!
There is a lot to say about this fantastic novel, but instead of hearing from me, I thought it would be better to let the author do the honors.
From Richard Paul Evans:
Four years ago, I wrote a novel called The Gift, the story of a young boy with the power to heal. A superhero. It was the most fun I'd had writing in years. About a year later an idea for a book began forming in my mind. The novel had everything I wanted to read when I was a boy. The main character, Michael Vey, was shy, small for his age, and moved around a lot, and he had Tourette's syndrome—all of which pretty much describes my childhood.
But Michael also had a special gift: He was born with electric powers.
I have already shared Michael Vey with over five hundred middle school students from three different schools, in three different cities. The response has been, pardon the pun, electric. Not since the The Christmas Box have I seen such an overwhelmingly positive reaction to one of my books.
On a personal level, I'm very pleased to have created a protagonist with Tourette's syndrome. I have TS. So does my son. In our test schools, the teachers reported that Michael Vey had helped students suffering with disabilities, from Tourette's to Asperger's, feel more accepted.
Like all my novels, Michael Vey, ultimately, is about hope. It's about the power and beauty each of us has inside, even when our outsides tell us different. It's a story, not about darkness, dysfunction, or dystopian worlds, but about light, loyalty, and friendship. In short, Michael Vey is the kind of message I want to deliver to today's youth.
You can read the eGalley by clicking this link: http://www.galleygrab.com/?asset_url=9781451661835
21. Blood Red Road Trailer is here!
Blood Red Road (9781442429987) which went on-sale 6-7 has received amazing reviews including the following stunner from the New York Times Book Review:
"Is it any surprise that Ridley Scott swiftly optioned the book? …what makes the story truly sing: Young's spare depictions of the struggle to survive and find companionship in a barren world that hardens hearts and minds. … Much of Young"s writing has an elemental power, unfolding across achingly barren land-scapes, full of blistering "hotwinds" and swirling clouds of orange dust. "
We also have a fantastic trailer–check it out at: http://www.youtube.com/watch?v=h734qfd2X1s
22. Happy Dork Day!!!!!!!!!
Happy Dork Day!!!!!!!!
Dork Diaries #3: Tales From a Not-So-Talented Pop Star is on-sale today.
Be sure to celebrate and— Let Your Inner Dork Shine Through!
23. Summer Before Boys by Nora Raleigh Baskin
JacketFlap tags: Children, Middle-Grade, Nora Raleigh Baskin, May 2011, Summer 2011, Summer Before Boys, Add a tag
SUMMER BEFORE BOYS (Simon & Schuster Books for Young Readers, 9781416986737) by award-winning author Nora Raleigh Baskin is now available!
Julia and Eliza are best friends, spending the summer together. Julia's mother is serving in the National Guard and Julia spends all of her time trying not to think about what could happen. So the girls lose themselves in their summer, hanging out at the resort where Eliza's father works. But when they meet a new boy, neither one of them is prepared for what it does to their friendship.
Nora Raleigh Baskin delivers a poignant look at the way a first crush can come between best friends and the importance of hanging on to the time you have as a kid before rushing into growing up.
Kirkus gave it a starred review, calling it "an extraordinary novel [that] explores the challenges faced by children whose parents have gone off to war … Baskin adeptly portrays Julia's ambivalence and anxiety in this thoughtful tale that artfully brings the war to the homefront."
24. The Beginning of a Summer of Firsts!
Summer of Firsts: Tween First in Series
Three books from our Summer of Firsts lists are on sale today!
KATIE AND THE CUPCAKE CURE and MIA IN THE MIX are the first & second book of a brand new tween series from Simon Spotlight! The Cupcake Diaries series is just the right dose of sweet for any tween!
Katie and the Cupcake Cure by Coco Simon
9781442422759, Simon Spotlight, Paperback
Mia in the Mix by Coco Simon
SEVEN SORCERERS by Caro King from Aladdin is the perfect fantasy read for any tween!
Seven Sorcerers by Caro King
9781442420427, Aladdin, Hardcover
And . . . THE GREAT HAMSTER MASSACRE by Katie Davies is an irresistibly funny mystery and Hanna Shaw's spot-on illustrations combine for a quirky, delightful read that is part detective tale, part diary, and altogether hilarious.
The Great Hamster Massacre by Katie Davies
9781442420625, Beach Lane Books, Hardcover
25. Happy Book Birthday!
Add/View Comments | Previous | Top |
MOONGLASS is the debut novel for Jessi Kirby!
Don't miss what Sarah Dessen calls "an incredible first novel."
Moonglass by Jessi Kirby
9781442416949, S&S BFYR, Hardcover
Also out tod ay is WHERE THINGS COME BACK by debut author John Corey-Whaley.
Kirkus Reviews calls it a "multilayered debut for sophisticated readers. Unexpected, thought-provoking storytelling."
Where Things Come Back by John Corey-Whaley
9781442413337, Atheneum, Hardcover
View Next 25 Posts | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,631 |
Karel Svoboda, né le à Prague et mort le à Jevany, est un compositeur tchèque de musique de film, , de musique pop et de comédie musicale.
Il a, entre autres, collaboré avec le chanteur Karel Gott et l'auteur de textes Jiří Štaidl.
Au milieu des années 1990, il compose l'intégralité des jingles de Česká televize, la télévision d'État tchèque.
Musique de films
Il a composé la musique des films suivants :
Každý mladý muž, 1965
Noc na Karlštejně, 1973
Tři oříšky pro Popelku, 1973
Jak se budí princezny, 1977
Což takhle dát si špenát, 1977
Smrt stopařek, 1979
Spievaj kovboj, 1981
Sůl nad zlato, 1982
Létající Čestmír, 1984
Big Man/Jack Clementi, 1987-1988
, 1988
Druhý dech, 1988
Kačenka a zase ta strašidla, 1992
Und keiner weint mir nach, 1996
Z pekla štěstí, 1999
Milenci a vrazi, 2004
Rodinná pouta, 2006
Karel Svoboda a également composé pour des séries télévisées :
Maya l'abeille (série télévisée), 1975 pour laquelle chante Karel Gott et dont la composition de Svoboda est reprise dans la plupart des langues.
Pinocchio (série télévisée d'animation, 1976), 1976
Nils Holgersson, 1980
Návštěvníci, 1983
Tao Tao (série télévisée), 1983-1984
Flash (série télévisée), 1993
Vic le Viking, dessin animé de la ZDF - ORF d'après les histoires de Runer Jonsson
Acteur
Karel Svoboda a joué dans les productions de ses amis :
Anna proletářka, 1953
Poslušně hlásím, 1958
Musicals
Dracula,
Monte Cristo
Golem
Liens externes
Site personnel
Compositeur tchèque de musique de film
Naissance en décembre 1938
Décès en janvier 2007
Suicide par arme à feu
Musicien suicidé
Suicide en Tchéquie
Décès à 68 ans | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,291 |
Archive for the 'History Repeats itself' Category
What's UP for Our Species?
"Skepticism is the chastity of the intellect, Santayana declared, and the metaphor is apt. The mind that seeks the deepest intellectual fulfillment does not give itself up to every passing idea." —(Richard Tarnas, COSMOS AND PSYCHE)
My last post engendered considerable response from readers who shared some very thoughtful views in a lengthy and rich conversation on a group Face Book page around the topic of the current climate crisis. The last word on the so-called "climate crisis," of course, has not been written, and may never be. I did receive a lead to my next book to read, COSMOS AND PSYCHE, which I am thoroughly enjoying. Thanks Don!
One of my followers, Marco, from Alojera, Canarias, Spain, shared these three perceptions:
1.That our environment is radically changing and requiring human behavior to adapt in ways that will challenge past behavior.
Yes, and "adapt" is the operative word for the purpose of the current post — as in evolve.
2. That aforementioned human behavior (particularly since the industrial revolution) has increasingly reflected a sense of identity separate and superior from the planet and its natural cycles; and that such separate, superior identity has been fertile ground for the emergence of aberrant, predatory, and ultimately self-defeating identity, convinced not of humankind belonging within a mutually interdependent ecosystem, but in an intrinsically fearful, competitive, and winner-takes-all matrix.
On the one hand, the industrial complex may be playing a remedial role in getting the oil off our water planet, two substances that don't mix well together. The late geology professor John Waskom, offered a rationale for the burning of oil, a foreign substance deposited on Earth by Venus when she was a comet some 3570 years ago — according to Dr. Immanuel Vlikovsky, author of Worlds In Collision, who was extremely knowledgeable in the texts of ancient peoples. Dr. Waskom's scenario was that beings with technological expertise were sent here by the Creators specifically to pump the oil out of the ground and dispose of it. On the other hand, I don't think the Creators had it in mind to turn the oil into plastic and other polluting and cancer causing chemicals. This has been the doings of "aberrant, predatory, and ultimately self-defeating" humans.
3. That as a result of the above, man is fearful of change, deeply convinced of his reliance on an unsustainable life-style. And unless something changes radically in man's psychological assumptions, the species will not survive this phase of environmental change.
Radical change is the only path to survival. The question is: Who is it that hopes to survive this "unsustainable life-style?" More to the point: "Who can survive it?" Perhaps those who have thoroughly established a more sustainable life-style in an isolated and insulated community can.
A more transcendent question is: Why limit our thinking to mere survival of our species? We can radically change our psychological assumptions. Why not think in terms of transcending the human condition altogether and creating a totally new and "greener" world on what seems to be emerging as a New Earth. God knows Gaia is doing her part to clear the way for a restoration to occur.
Behold, I create a new heaven and a new earth, for the old heaven and old earth are passed away; and there is no more sea.
Because the new heaven and new earth are one and the separate "sea" of fallen human consciousness that had been maintaining the old earth is no more. Human consciousness is transformed and transmuted into Divine Consciousness. This is how the New Earth appears: "Not by might nor by power, but by my Spirit saith the Lord of Hosts."
Marco added this quotation from Homer's The Odyssey, which speaks to the pathetic attitude of irresponsible humans.
Men are so quick to blame the gods: they say that we devise their misery. But they
themselves — in their depravity — design grief greater than the griefs that fate assigns.
Question: Is there really a "climate crisis? This reader's comment suggests otherwise:
I agree much more can be done to care for the planet, but panic is not good, and it's scaring children. Many say it's urgent, but 500! scientists and engineers just told the UN it's not an emergency, that climate policy is relying on inadequate models, so maybe some balance is required. Indeed, the Russian climate model says things are not as extreme . . . . Science is wonderful and very useful, but I believe science doesn't know everything and hasn't factored in all the variables . . . . [Go to this link for the whole story.]
https://www.washingtontimes.com/…/scientists-tell-un…/
My poet friend sent me these words of wisdom: "When there's a meme [an idea or behavior circulated on social media] being pushed at a mega scale it behooves one to ask why and for what purpose."
So I ask why and for what purpose. As I said in my previous post, I am feeling my way through all the reactive journalism and commentaries around the subject of climate change.
So, what is up for the Human Species?
All of this uproar over climate change poses a few questions in my own mind:
1. Are we facing the inevitable demise of our species, perhaps in the form of some sort of transmutation of our flesh bodies to a higher form of, say, the frequency of light?
2. This question gives rise to another: Is the earth's warming climate preparing the way by creating a vibrational caldron as a crucible in which this transmutation might occur "in the twinkling of an eye" and without too much suffering on the part of humans?
3. In other words, are we being asked by the Great Spirit Creator to let go of our attachment to these earthen forms, at least our identity with them, in order to cross over the alchemical threshold where physical materialization meets the spiritualizing realm, the abode of angels . . . if only in consciousness at first?
Transmutation of form follows transformation of consciousness, as consciousness is the matrix out of which form appears. In other words, there is a creative process to be stewarded here whereby all things are made new. Re-creation in consciousness is accomplished by Spirit as it finds expression through us in our living. My spiritual mentor articulated this principle forty years ago:
When the spiritual action is present the design in form appears. This is the principle which we have doubtless known for a long time but by which the restoration is made possible. Behold, I make all things new. As there is the means on earth by which the spiritual action may appear, then the substance in the realm of form leaps to reveal it. All this can occur very quickly when there is spiritual action and to the extent that there are people on the face of the earth who are willing to let it happen, who are no longer insisting that material substance should be held in the structured forms set up by fallen human nature. (Martin Cecil)
Lest We Forget Our Past
What's buried deep in the human psyche that's pushing up for conscious acknowledgment and remembrance? Are we awakening from our mass amnesia and remembering our traumatic past so that we do not repeat it?
In an effort to answer these questions, I would like to revisit a post from last year and excerpt the closing paragraphs. The post itself is a review of Graham Hancock's classic Fingerprints Of The Gods." Below is a link to the post itself.
EITHER WE EVOLVE OR WE PERISH
When human beings from around the globe, and from many different cultures, share a powerful and overwhelming intuition that a cataclysm is approaching, we are within our rights to ignore them. And when the voices of our distant ancestors, descending to us through myths and sacred architecture, speak to us of the physical obliteration of a great civilization in remote antiquity (and tell us that our own civilization is in jeopardy), we are entitled, if we wish, to stop our ears…
So it was, the Bible says, in the antediluvian world: "For in those days, before the Flood, people were eating, drinking, taking wives, taking husbands, right up to the moment that Noah went into the Ark, and they suspected nothing till the flood came and swept all away."
In the same manner it has been prophesied that the next global destruction will fall upon us suddenly "at an hour we do not suspect, like lightning striking in the east and flashing far into the west … The sun will be darkened, the moon will lose its brightness, the stars will fall from the sky and the powers of heaven will be shaken …."
What has happened before can happen again. What has been done before can be done again.
The thread that runs through the pages of Graham's book weaves a fabric upon which is painted a prophetic picture for our generation specifically that warns of impending disaster for our world and its population based simply on a cosmic clock ticking away the centuries and millennia over the past 12,500 years since a highly advanced civilization was all but wiped off the face of the earth by a cataclysm. That civilization was known as Atlantis and is now believed to be buried beneath a two-mile thick cap of ice. Today that island continent is called Antarctica. Buried with it are the historical records of what this civilization did that brought about its destruction in literally the blinking of an eye — like a thief in the night. It is likely we will never know the secret buried beneath those two miles of ice. An ancient lost civilization frozen in time and preserved till the day the ice melts and reveals its secrets. . . .
The survivors of the Deluge . . . left messages carved in stone which point to our day — when the sun rises on the Winter Solstice in the House of the constellation of Sagittarius at the South Gate of the Milky Way Galaxy between 1960 and 2040 AD — and when the pendulum of the cosmic clock strikes the hour when the hammer will likely fall again on the civilizations of man. It is also the dawning of the Age of Aquarius which guards the North Gate of the Milky Way Galaxy. As I wrote about in a previous post, this is also designated in Mayan cosmology as "the place of creation."
We ignore the myths of our ancient ancestors to our peril. The dawn of a New Day lightens the horizon. It's time to wake up and face the harvest of our sowing days as it spreads ugly before us. We've run into a concrete wall and there's no way to go forward — only up and over it. It's the way Gaia is going — up, up and away!
I believe that collectively — and restoration is a collective experience of transformation, as we are all in the same boat — we have the spiritual substance necessary for Spirit to work through in averting a cataclysmic demise of our species, and to turn the tide of self-destruction by climate collapse or nuclear war. I see this substance of love being generated by people the world over. My wife and I participated recently in a diverse group dialogue on the issue of racial justice. The atmosphere in the room was one of accord that we are one race, the human race. I came away with an uplifted sense that all is well. Unconquerable Life is prevailing, our differences adding color to its expression through us. We are beginning to truly love one another. Let's let this atmosphere increase and clarify to give Spirit a means for making all things new through Man restored.
I leave you with this timely video message: "If we do not wish to die, then we must evolve." Barbara Mor reads from The Great Cosmic Mother — Rediscovering the Religion of the Earth: https://youtu.be/oPtU2Rxbsjc
Until my next post,
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Survival of our species
Wheels Within Wheels: Breaking the Cycles of War, part 4
This will be my final post of this series. Enough already about ancient Roman wars repeating during the last zodiac cycle. It's time to move on to something a bit more NOW. This final post is about events that are at least current in our recent history.
As hard as the Cabal* tried to instigate wars, there inevitably appeared on the scene a hero whose actions averted a US attack on a Middle East country and a nuclear conflagration. Such was the case with Iran and Afghanistan. I will let David Wilcock tell how it all unfolded in this excerpt from his thoroughly researched and fascinating book The Synchronicity Key.
THE SKY IS NOT FALLING—ONLY OUR BLINDFOLDS ARE
Smart Enough Not to Take the Bait
I was very relieved to discover that the United States appears to be increasingly pulling away from the Roman wars that occurred in the Age of Aries. The potential triggers were still offered to the public, right on schedule, but we finally got smart enough not to take the bait. Masson believed that the period from 1988 to 1992 could be as dangerous as the Cuban Missile Crisis nuclear showdown, if not more so — but the proxywar with the USSR in Afghanistan did not turn into a world nuclear Armageddon.
We also saw the original, short-lived Gulf War in Iraq, known as Desert Storm, in the key risk period of 1988 to 1992. This war began on August 2, 1990, when Saddam Hussein invaded Kuwait. Aerial bombardment of Iraq began on January 17, 1991. Iraq declared a cease-fire only one hundred hours after the February 24, 1991, start of the ground campaign. Although many people died in the bombardment, Desert Storm was nowhere near the global disaster that Masson had feared it could have been if the zodiac cycle had exerted the full force of seemingly ancient Roman wars upon the destiny of the United States.
I graduated from high school in 1991, and that was also the last year I watched television on a regular basis. Since there was no Internet back then, I quickly made friends with books, because every time I turned on the television during the Gulf War, attractive female newscasters and unsmiling male anchors were interviewing "experts" who insisted the entire Middle East was about to light up like a huge, fiery cauldron of violence. Everyone was afraid that World War III was about to begin, fulfilling biblical prophecy — but the other Middle Eastern countries never took the bait. After all this time, they finally broke the old pattern and realized that the nemesis was trying to provoke them into having a violent reaction, in order to cause much greater harm — with vastly superior weapons.
American Heroes Block "Biblical Armageddon" in 2006
As we learned in the previous chapter, the period 1988 to 1992 in the Age of Pisces corresponds to Rome's second war with Macedonia — from 172 to 168 B.C. — in the Age of Aries. The next significant event was Rome's war against Lusitania, beginning in 154 B.C. When we look 2,160 years into the future, this brings us to the year 2006. Although it may seem, at first, that nothing significant happened in 2006, we did actually come very close to a disastrous war that same year.
On March 16, 2006, the United States formally declared war against Iran. The declaration appeared in that day's National Security Strategy announcement: "We may face no greater challenge from a single country than from Iran …. When the consequences of an attack with WMD are potentially so devastating, we cannot afford to stand idly by as grave dangers materialize."
This declaration also made it clear that the United States would use nuclear weapons to fight this conflict." This was very serious, and it appeared that a nuclear first strike against Iran was imminent. The Cabal* wanted their biblical Armageddon as soon as possible. Thankfully, people in key positions rose to stop the cycle of violence from playing out as it had in Rome during the Age of Aries. John Negroponte, the director of national intelligence, became the hero who directly confronted the nemesis when he told the press in April 2006 that it would be "a number of years off" before Iran would be "likely to have enough fissile material to assemble into or to put into a nuclear weapon—perhaps into the next decade."
A National Intelligence Estimate (NIE) that made the same conclusions about Iran was held up for more than a year by the Bush administration. It was finally released on December 4, 2007: "We judge that in fall 2003, Tehran halted its nuclear weapons program. Tehran has not restarted its nuclear weapons program as of mid-2007. We judge with high confidence that Iran will not be technically capable of producing and reprocessing enough plutonium for a weapon before about 20I5."
The very beginning of this ground-breaking report said, "This NIE does not assume that Iran intends to acquire nuclear weapons." The words "does not" were italicized in the original document. As of this writing in 20I3, it's now been seven years since Rome started fighting the Lusitanian War in the previous zodiac cycle, during the Age of Aries but we still haven't seen any overt aggression toward Iran. Nor are there any other major new wars. The United States does still maintain a presence in Afghanistan but has now largely pulled out of Iraq. I was quite relieved to discover that the old cycles finally appear to be breaking down.
If forgiveness is the key that will stop the Wheel of Karma from spinning, then it appears we are finally learning to love and accept one another. The nemesis can continue to harm us only if we fail to learn the lessons of forgiveness — and accept the temptation to turn against one another.
We Can Change the Outcome
It is very important to remember that we can change the outcome in these cycles. We are not trapped — and we do not need to keep repeating the same wars and atrocities again and again. As we learned earlier, we now have direct scientific proof that a small group of people can have a major effect on the behavior of the entire planet for the positive. Specifically, a group of seven thousand ordinary people was able to reduce worldwide terrorism by 72 percent–simply by meditating. They had similarly powerful effects in stopping wars, violent outbreaks, and loss of life. Fifty different scientific studies have validated this meditation effect. This proves that the cycles are not fixed. Wars will not keep repeating, right on schedule. We can change the outcome. The lesson is that if enough of us begin practicing peace in our lives, the ancient story finally achieves its purpose in bringing all of us face-to-face with the nemesis, so we can integrate our ego and learn not to blame one another for our feelings of pain, fear, and anger. We can finally master the lessons this pattern of archetypes is teaching us — and stop projecting our shadow onto others by making them into our nemeses.
We may then experience a stunning, worldwide curtain call, such as in a full breakdown of government, media, and financial secrecy. Based on the testimonies of multiple high-ranking witnesses I have personally interviewed, once the Cabal is fully exposed on the world stage, this will quickly lead to a disclosure of the advanced human relatives who have been assisting us-and were seen as gods in every ancient culture on earth. These people have been here all along but appear to have largely stepped behind the curtain since the rise of Islam in the 700's in order to allow us to become a modern society. Because they remain shielded from public view, each person has the freedom to accept or reject the idea of their presence. This premise of free will is very important in the Law of One series, but once we shift into fourth density, at the end of the twenty-five-thousand-year cycle, everything changes. Apparently, as we settle into this new reality, we will graduate into an entirely new time structure with surprisingly different rules.
I have often said that if reincarnation has been proven to be a scientific fact, how can we assume that we will just keep coming back and repeating the same lessons, lifetime after lifetime? Isn't there a point at which we learn the greatest teachings of the Hero's Journey — and are now ready to step into a higher level of our own human evolution? (Wilcock, THE SYNCHRONICITY KEY)
ASCENSION 2019
Our Earth and the human species are about to step into a higher level of evolution. Between now and my next series, make time and space available to view this presentation by David Wilcock on the spiritual implications of what's happening in our world today. I've been following David for several years and I am more and more convinced that he is indeed the reincarnation of Edgar Casey. His message here is for the entire human race — at least for those of us who are awake and available to hear it. The fact that you are following this blog indicates a heightened level of wakefulness. His is an urgent message that I know you will appreciate hearing.
*FOOTNOTE: The Federal Reserve: The Heart of the Cabal
Michel Helmer did not detect a cyclical connection between the Roman Empire and the United States before 1896. This was the year that big business bankers began seeing their plans realized with the election of the imperialist president McKinley. Under McKinley's reign, the Cabal soon began an unprecedented expansion of its power, using the United States as its new staging area. After many years of planning, the big bankers-such as the Rockefeller-Standard Oil dynasty in America and the Rothschild banking dynasty in Europe-pooled their resources to create the Federal Reserve in 1913. The creation of the Federal Reserve effectively overthrew the US Constitution. Harry V. Martin published the following research online in 1995, before so many others discovered the same facts. Now, even the corporate media is increasingly beginning to discuss this hidden truth. The number of people who know the real story of the Federal Reserve has skyrocketed since I first began researching this mystery in 1992. According to Martin, "Article I, Section 8, clause 5 of the United States Constitution provides that Congress shall have the power to coin money and regulate the value thereof, and of any foreign coins. But that is not the case. The United States government has no power to issue money, control the flow of money, or to even distribute it. That belongs to a private corporation, registered in the State of Delaware—the Federal Reserve Bank."
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The zodiac "wheel" cycle of 1,260 years is subdivided into smaller cycles: 630, 315, 260, 52 and as small as 13 years. In his controversial book The Synchronicity Key, lecturer and author David Wilcock studies repetitive events that fall within the zodiac cycle measured by the Mayan Great Year of 1,260 years connecting the Roman Empire in the Age of Aries with the modern world in the Age of Pisces. This zodiac cycle is accented by periods of war that mirror the many and various wars that took place during the reign of the Roman Empire. His purpose for studying this cycle and documenting his findings is to show how in our day, on this side of the zodiac cycle, the pattern of repeating wars has been broken and the pall of nuclear war that's been hanging over our heads for decades is no longer a threat.
My purpose in blogging about Wilcock's investigation is to bring his finding into the light of day for our more current awareness so that we can put to rest the background fears of nuclear annihilation that have kept us from moving forward with a revitalized sense of assurance that all is indeed well.
But this is not the only nor primary reason why all is well. The primary reason why all is well is because we are being watched over by "guardian angels" — in the form of "Watchers," according to David Wilcock's cosmology. But more about that in a future post.
In the previous post, we looked at two incidents where the imminent threat of nuclear war was defused. In this post I will present another war that mirrored a war in the Roman Empire on the far side of the zodiac cycle. I will again let David Wilcock tell the story from his investigative journalism in an excerpt from The Synchronicity Key.
VIETNAM, WATERGATE, AND THE FALL OF THE IRON CURTAIN
Rome went to war against Antiochus III in 192 B.C. Antiochus III was a king who ruled over Greater Syria and western Asia. Antiochus III invaded Greece with a ten-thousand-man army, triggering the Roman-Syrian War, which raged from 192 to 188 B.C.
When we advance this same time period ahead by 2,160 years, we have 1968 to 1972. This precisely corresponds to the key turning points of the Vietnam War—which also was a war in Asia.
The United States first began covert operations in North Vietnam in 1964. On August 2, 1964, three North Vietnamese PT boats fired on the USS Maddox. This led to the Gulf of Tonkin Resolution, in which Lyndon Baines Johnson, Kennedy's former vice president, gained permission to wage war against North Vietnam without a congressional declaration. The United States began bombing North Vietnam in 1965 while troop levels topped two hundred thousand. In 1967, Secretary of Defense Robert McNamara said the bombing raids were not effective enough to solve the problem and more needed to be done.
Then, in January 1968 — 2,160 years after the Asian king Antiochus III invaded Greece with ten thousand men, plunging him into an all-out war with the Roman Empire — North Vietnamese and Vietcong forces swept into South Vietnam. The Asian enemy attacked several cities, including the capital — similar to Antiochus' attack on Greece in the Age of Aries. This bold and daring military maneuver was called the Tet Offensive. Though this attack was repelled, it was a political and psychological victory, causing great questions about whether the United States was involved enough in the war. General William Westmoreland requested a doubling of the troop presence in February, calling for an additional 206,000 men. The idea of ordinary young men being drafted into military service suddenly became a very real and very terrifying prospect.
Then, on March 16, 1968, American soldiers massacred hundreds of innocent people in the village of My Lai. In 1969, when the incident became public knowledge, it caused shockwaves through the American political and military establishment as well as the general public. The American people had an opportunity to demand the end of the war right there, but the political will was not yet strong enough.
These three events—the Tet Offensive, the proposed doubling of the troop presence, and the My Lai massacre—dramatically increased the emotional impact of the war. The number of young men being drafted suddenly skyrocketed.
Again, these events all occurred in 1968–precisely 2,160 years after Rome went into full-scale war against the Asian king Antiochus III in 192 B.C.
A BITTER TASTE OF TREASON — Served on a Fifty-Two-Year-Old Platter
Furthermore, on March 17, 2013, it was revealed that presidential candidate Richard M. Nixon had deliberately sabotaged peace talks with Vietnam that same year, 1968. I happened to find this article through synchronicity, while looking for other information on Nixon to help flesh out this part of the book. This incredibly treasonous story was covered by Rachel Maddow on MSNBC, as well as in other media outlets, but I wouldn't have found it if I hadn't already been looking for links to Nixon and 1968. We now know that Nixon bribed the Vietnamese with promises that they would get a much better peace deal if they held off until he became president. This shocking betrayal of the American people guaranteed that the deadly Vietnam War would grow much, much larger-and far more profitable for the military-industrial complex.
This treasonous secret deal with the enemy provided fuel for much greater military power in Nixon's presidency. President Eisenhower had warned America about the growing, potentially "disastrous" menace of the military-industrial complex in his closing address on January 17, 196I. Eisenhower was another mentor figure who gave America a magic gift in this speech, which could ultimately be used to defeat the nemesis: "In the councils of government, we must guard against the acquisition of unwarranted influence, whether sought or unsought, by the military industrial complex. The potential for the disastrous rise of misplaced power exists-and will persist."
The president at the time, Lyndon Baines Johnson, was aware of Nixon's treasonous deal but said nothing. Nixon thereby sentenced hundreds of thousands of additional young men into the military draft, and tens of thousands more American soldiers into their deaths—when it was all entirely preventable. The audiotapes that proved Nixon did this were declassified by the LBJ Presidential Library in 2013. The story was published literally the day before this chapter of the book was being revisited for its final publication for the first time since 2010.
It is interesting that fifty-two years elapsed between Eisenhower's prophetic warning in January 1961 and the final exposure of Nixon's treason in 2013. The Maya strongly believed that history moved in fifty-two-year cycles, which were made up of four smaller cycles of thirteen years. People throughout Mesoamerica celebrated this "Sacred Round" cycle as "the Binding of the Years" and used it to help them understand past and future events." For example, the Spanish conqueror Hernan Cortes began destroying the Aztecs shortly after his initial, peaceful visit in November 1519 — causing the Aztec prophecy of "nine hells" of fifty-two years to begin. Five "Sacred Round" cycles of fifty-two years add up to the tzolkin cycle of 260 years, which was widely revered throughout Mesoamerica. Australian professor Robert Peden discovered that the 260-year tzolkin cycle is a perfect "common denominator" for all the orbits of the planets in our inner solar system. The period of 260 years is a sub cycle that divides perfectly into the exact length of every orbit within the inner solar system. It is quite astonishing that the "primitive" cultures of Mesoamerica were somehow able to discover this number.
They also constructed an estimated three hundred to five hundred pyramids out of huge stone blocks-which again suggests they may have had
access to advanced technology.
THE END OF THE ROMAN-SYRIAN WAR AND THE END OF VIETNAM
According to Helmer and Masson, Rome went to war with the Syrian king Antiochus III in 192 B.C.; 2,160 years later, in 1968, the Vietnam War dramatically expanded. (Remember that we now have absolute proof that Nixon bribed the Vietnamese government to extend the war.) The Roman-Syrian War lasted five years, finally ending in 188 B.C. If we advance 188 B.C. forward by 2,160 years, we arrive at 1972, which is the exact year that a cease-fire was negotiated by Henry Kissinger and Le Duc Tho.
John Lennon's song "Imagine," released in September 1971, now seems oddly prophetic as a foretelling of the defeat of the Cabal. In archetypal terms, "Imagine" was a harbinger of the Elixir of Immortality that would soon be seized — the promise of peace — once the dragon of the Cabal, and its military draft of ordinary young men, had been slain.
"I hope someday you'll join us; and the world will live as One."
The final peace treaty was signed with Vietnam and went into effect on January 27,1973. This led to the formal announcement of the end of the draft and the withdrawal of the last American troops from Vietnam.
[On April 30,1975, the North Vietnam Army tanks rolled through the gates of the Presidential Palace in Saigon headquarters, effectively ending the war, winning the war for Communist Russia — a huge loss for the USA, the nemesis defeating the hero in our Nation's "Hero's Journey."]
FURTHER BREAKS IN THE ZODIAC CYCLES OF WAR
There are a few other more recent breaks in the cycles of war as the United States in this age of Aquarius pulls away from the Roman wars that occupied the Age of Aries. These more recent incidences are the anti-terrorists wars in Afghanistan, Iraq and Syria, occasioned by the "9/11" attacks and destruction of the World Trade Center in New York, and the nuclear arms crisis of Iran, a crisis still unsettled now that President Trump has canceled the Iranian Nuclear Treaty. With the US pulling out of the INF treaty with the USSR yesterday, the nuclear arms race is likely to start up again. What insanity on the part of our leaders. I will present these events in my next two posts of this series. As always, I welcome your thoughts in the Comment section. Until my next post, fear not.
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Zodia Cycles
"Those who do not learn from history are doomed to repeat it."
(This saying is a paraphrase of the original words penned by George Santayana: "Those who cannot remember the past are condemned to repeat it.")
REMEMBERING OUR PAST
There have been several occasions when nuclear war has been averted in recent history. The most outstanding event occurred in 1983 when Stanislav Yevgrafovich Petrov, a lieutenant colonel on the Soviet Air Defense Forces, single-handedly saved the world from nuclear war by listening to his gut feeling about what turned out to be a nuclear false alarm. As the story is told by Wikipedia:
On 26 September 1983, three weeks after the Soviet military had shot down Korean Air Lines Flight 007, Petrov was the duty officer at the command center for the Oko nuclear early-warning system when the system reported that a missile had been launched from the United States, followed by up to five more. Petrov judged the reports to be a false alarm, and his decision to disobey orders, against Soviet military protocol, is credited with having prevented an erroneous retaliatory nuclear attack on the United States and its NATO allies that could have resulted in large-scale nuclear war. Investigation later confirmed that the Soviet satellite warning system had indeed malfunctioned.
(BREAKING NEWS: The US is withdrawing from an old Cold War nuclear arms treaty with Russia due to Russia's failure to comply with the NIF agreement. This may trigger another nuclear arms race.)
This story speaks to the insecurity inherent in modern technology's "Artificial Intelligence" when it involves vital information about threats to our national security. It also speaks to the presence of spirit guiding us in our decision making processes, working at times through our gut sensing, as it did with this Russian lieutenant colonel. This is a past event worthy of remembering.
THE CYCLES HAVE BEEN BROKEN
Looking back six decades to 1960 when the United States drew up plans to attack Cuba in what was actually a proxy war with the USSR, America's long-time nemesis, we see the war cycle broken by players on both sides of these global-threatening conflicts: the Korean war and the Cuban Missile Crisis. I'll let David Wilcock describe the details of these historical events in an excerpt from his controversial book The Synchronicity Key.
(Remember, we are looking at the influence of zodiac cycles on global events and how The Hero's Journey plays out in the lives of individuals and nations, as well as in the story of mankind's evolutionary journey. The zodiac cycle of 2,160 years Wilcock explores here brings to light the mirroring of the Roman Empire's warring years by our warring years this side of the zodiac cycle. This is as fascinating as it is educational for those of us who slept through history class — or were born in the post-war era, in which case you can learn what happened before your were born that got us where we are today.)
The Korean War–and a Chance for Peaceful Coexistence
The zodiac cycle gets slightly harder to follow once we see that the Second Punic War lasted until 201 B.C. [in the Age of Aries]–which is 1959 in our own Age of Pisces. Even though World War II seemingly ended in 1945 with Hitler and Japan's defeat, the United States immediately began fighting another Cabal-financed superpower–the USSR. The Cold War started directly after the end of World War II with the highly deadly nuclear arms race, threatening all life on earth. This significantly raised the stakes from the previous age, as no matter how much Rome attacked and defeated its neighbors, life on earth was never threatened in an overall sense.
The war against the USSR went hot from 1950 to 1953, when the United States battled the USSR's hidden ally, North Korea, in the Korean War. The Soviet Communists were supporting North Korea, while the United States was supporting the pro-Western regime of South Korea. The United States considered this a fight against global communism, and the threat of a much bigger war with the USSR or China was always looming. Five million soldiers and civilians were killed in this war. The Cold War continued escalating after the Korean conflict ended in 1953, until a remarkable breakthrough finally occurred. On September 25, 1959, Russian premier Nikita Khrushchev visited the United States to meet with President Eisenhower. This highly influential visit occurred exactly one zodiac cycle after the Second Punic War ended in 201 B.C. This was the first time in the entire history of the Cold War that a Soviet leader had visited the United States.
As revealed on Politico.com, in this groundbreaking summit, Khrushchev "denounced the 'excesses' of [communist] Stalinism and said he sought 'peaceful co-existence' with the United States …. In a joint communique, issued after two days of meetings, the leaders said…they believed that 'the question of general disarmament is the most important one facing the world today.'" Although the talks didn't last, this meeting was a key harbinger of a positive future, as it presented the people with a real opportunity to end the war, and achieve true, lasting peace.
Hardly anyone knew, at the time, that a very similar peace opportunity had occurred exactly 2,160 years before, when the second Punic War had ended in 201 B.C.
The Macedonian War
Rome's taste of peace was very short-lived, as the Macedonian War began the very next year, in 200 B.C. Macedonia is a very small country just north of Greece–and in our modern zodiac cycle, its equivalent may be Cuba…. In March of 1960, 2160 years after Rome attacked Macedonia in 200 B.C., the United States drew up plans to attack Cuba. The conflict between the United States and Cuba was another proxy war fought between the United States and the USSR. The Soviet Union had made economic and trade agreements with Fidel Castro, the prime minister of Cuba, in February 1960, and the United States felt the need to mount an immediate counterattack. Cuba was directly southeast of the continental United States, just ninety miles from the southern tip of Florida. This provided the USSR with a valuable strategic location from which to wage war–including the potential placement of first-strike nuclear weapons that would hit before the United States could effectively counter-attack.
On May I, 1960, the United States provocatively flew a U-2 spy plane over Russian airspace. The Soviets shot it down and captured the pilot, Gary Powers. The Cold War immediately restarted–precisely 2,160 years after Rome plunged into another war with Macedonia. Kennedy won the 1960 presidential election and was sworn in on January 20, 196I. In February 1961, very soon after his inauguration, Kennedy trusted his new, Cold War veteran advisers and authorized the CIA's plan to invade Cuba. Air strikes began on April 14, 1961, in B-26 bombers disguised as Cuban aircraft, but the planes were quickly identified as belonging to the United States.
Kennedy was embarrassed and canceled the next round of air strikes, but the land invasion went ahead on April 17, 1961, at the Bay of Pigs.
Twenty thousand Cuban troops were waiting for the invading force of about fourteen hundred Cuban exiles who had been trained by the United States. The battle quickly ended with 144 of the exiles killed and 1,189 captured. This was a great embarrassment for the United States and immediately threw its fledgling president into full-scale crisis. In terms of the ancient story, the failed Bay of Pigs invasion would be seen as another all-is-lost point for the United States–a nationwide dark night of the soul in its battle against the Soviet nemesis.
Predictably, the USSR then began beefing up Cuba with weapons including nukes. This caused the United States and USSR to come very close to full-scale nuclear war during the Cuban Missile Crisis, which reached its peak on October 22, 1962. On this day, Kennedy addressed the nation to present solid evidence that the Soviets had positioned nuclear weapons in Cuba. This was a tense third-act moment: The United States had regrouped, learned from the dark night of the soul in the Bay of Pigs disaster, and gained the strength to confront their own Soviet nemesis through its proxy state in Cuba. Out of a total of 1,436 B-52 bombers in the Strategic Air Command, fully one-eighth were sent airborne–to be ready to strike at a moment's notice. US military forces worldwide were placed at DEFCON 2, requiring an increase in force readiness. Twenty-three nuclear-armed B-52 bombers were placed in orbit within striking range of the Soviet Union as well.
After a series of terrifying moves in a very high-stakes chess game, Khrushchev admitted what the USSR had done and announced it would pull back as of October 28. All offensive weapons in Cuba would be dismantled and returned to the USSR. This was a great triumph moment for Kennedy and for America as a whole in the ancient Hero's Journey story line, and the dramatic events have triggered multiple movies, novels, and TV adaptations. The last US missiles were removed from Turkey on April 24, 1963, ending the conflict on both sides. In Rome, the Battle of Cynoscephalae ended in 197 B.C.–exactly 2,160 years before the Cuban Missile Crisis ended. The Battle of Cynoscephalae was the decisive turning point that led to Macedonia losing the war.
It's not always easy to figure out what our modern equivalents may be for the countries against which Rome waged war. The Age of Pisces equivalent of Macedonia appears to be Cuba, but the USSR was the real nemesis lurking behind the scenes all along. It seems that the great cycles of history describe an overall script of what types of events will happen, but various characters here on earth will shift in and out of those roles, ultimately depending upon how the people in each cycle respond. And, of course, not all events fit perfectly into these cycles. Multiple cycles can be intersecting and colliding at once, providing a push-pull of competing influences that we may not be able to map out without a great deal more information and computer power. Nonetheless, it is amazing to see how well the biggest events, and the most significant wars, all seem to reappear so precisely across the zodiac cycle.
The Macedonian War did not actually end at the battle of Cynoscephalae; it ended a year later, in I96 B.C. Although Kennedy was assassinated in November 22, I963, Khrushchev was ousted on October I4, I964–exactly 2,160 years after the Macedonian War finally ended for Rome. Although Khrushchev appeared to have been secretly working to negotiate peace treaties and alliances all along, he was still the primary figurehead for the USSR in worldly terms.
There may well be a vast wealth of other interlocking events that are not significant enough, in the Roman Empire, to be well documented as historical events–such as examples of music or theater that had as much of an effect as the Beatles would in the next cycle. There may well be a vast array of details that are repeating with equally stunning precision. This may all seem very hard to fathom, since we are used to thinking of time as completely linear. However, if we begin envisioning time as cyclical, moving through circular loops that are created by the celestial energy fields we move through as a planet, it makes more sense. Each time our planet reaches the same position in the circle, the events from previous rounds are much more likely to bleed through into our own reality-and repeat again.
Looking back at these warring years, we can see how close we came to a nuclear destruction of this civilization. The next opportunity to break the zodiac cycles of wars comes in 1964 with the Vietnam War–which I will write about in my next post. Until then,
David Wilcock, Ending Wars, Global Events, Healing Tones Blog, History Repeats itself, The Cabal, The Hero's Journey, The Nemesis, Zodiac Cycles & Wars
Cycles of War
Zodiac Cycles & Wars | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 311 |
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"redpajama_set_name": "RedPajamaC4"
} | 267 |
<component name="libraryTable">
<library name="Maven: io.dropwizard:dropwizard-jackson:1.1.3">
<CLASSES>
<root url="jar://$MAVEN_REPOSITORY$/io/dropwizard/dropwizard-jackson/1.1.3/dropwizard-jackson-1.1.3.jar!/" />
</CLASSES>
<JAVADOC>
<root url="jar://$MAVEN_REPOSITORY$/io/dropwizard/dropwizard-jackson/1.1.3/dropwizard-jackson-1.1.3-javadoc.jar!/" />
</JAVADOC>
<SOURCES>
<root url="jar://$MAVEN_REPOSITORY$/io/dropwizard/dropwizard-jackson/1.1.3/dropwizard-jackson-1.1.3-sources.jar!/" />
</SOURCES>
</library>
</component> | {
"redpajama_set_name": "RedPajamaGithub"
} | 9,077 |
package org.apache.camel.component.servicenow;
import java.time.LocalDate;
import java.time.LocalDateTime;
import java.time.LocalTime;
import java.time.format.DateTimeFormatter;
import java.util.HashMap;
import java.util.Map;
import com.fasterxml.jackson.annotation.JsonInclude;
import com.fasterxml.jackson.databind.DeserializationFeature;
import com.fasterxml.jackson.databind.ObjectMapper;
import com.fasterxml.jackson.databind.SerializationFeature;
import com.fasterxml.jackson.datatype.jdk8.Jdk8Module;
import com.fasterxml.jackson.datatype.jsr310.JavaTimeModule;
import com.fasterxml.jackson.datatype.jsr310.deser.LocalDateDeserializer;
import com.fasterxml.jackson.datatype.jsr310.deser.LocalDateTimeDeserializer;
import com.fasterxml.jackson.datatype.jsr310.deser.LocalTimeDeserializer;
import com.fasterxml.jackson.datatype.jsr310.ser.LocalDateSerializer;
import com.fasterxml.jackson.datatype.jsr310.ser.LocalDateTimeSerializer;
import com.fasterxml.jackson.datatype.jsr310.ser.LocalTimeSerializer;
import org.apache.camel.RuntimeCamelException;
import org.apache.camel.spi.Metadata;
import org.apache.camel.spi.UriParam;
import org.apache.camel.spi.UriParams;
import org.apache.camel.support.jsse.SSLContextParameters;
import org.apache.camel.util.ObjectHelper;
import org.apache.cxf.configuration.security.ProxyAuthorizationPolicy;
import org.apache.cxf.transports.http.configuration.HTTPClientPolicy;
@UriParams
public class ServiceNowConfiguration implements Cloneable {
@UriParam(label = "security", secret = true)
@Metadata(required = true)
private String userName;
@UriParam(label = "security", secret = true)
@Metadata(required = true)
private String password;
@UriParam(label = "security", secret = true)
private String oauthClientId;
@UriParam(label = "security", secret = true)
private String oauthClientSecret;
@UriParam(label = "security", secret = true)
private String oauthTokenUrl;
@UriParam(label = "security")
private String apiUrl;
@UriParam(label = "advanced")
private String apiVersion;
@UriParam
private String resource;
@UriParam
private String table;
@UriParam
private Boolean excludeReferenceLink = false;
@UriParam
private Boolean suppressAutoSysField = false;
@UriParam
private Boolean includeScores = false;
@UriParam
private Boolean includeAggregates = false;
@UriParam
private Boolean includeAvailableBreakdowns = false;
@UriParam
private Boolean includeAvailableAggregates = false;
@UriParam
private Boolean includeScoreNotes = false;
@UriParam
private Boolean topLevelOnly;
@UriParam
private Boolean favorites;
@UriParam(label = "advanced", defaultValue = "false")
private Boolean retrieveTargetRecordOnImport = false;
@UriParam
private Boolean key;
@UriParam
private Boolean target;
@UriParam(defaultValue = "true", enums = "false,true,all")
private String display = "true";
@UriParam(defaultValue = "10")
private Integer perPage = 10;
@UriParam(enums = "value,change,changeperc,gap,gapperc,duedate,name,order,default,group,indicator_group,frequency,target,date,trend,bullet,direction")
private String sortBy;
@UriParam(enums = "asc,desc")
private String sortDir;
@UriParam
private Boolean suppressPaginationHeader = false;
@UriParam(defaultValue = "false", enums = "false,true,all")
private String displayValue = "false";
@UriParam
private Boolean inputDisplayValue = false;
@UriParam(prefix = "model.", multiValue = true, javaType = "java.lang.String", description = "Defines both request and response models")
private transient Map<String, Class<?>> models; // field not in use as its a shortcut for both requestModels/responseModels
@UriParam(prefix = "request-model.", multiValue = true, javaType = "java.lang.String")
private Map<String, Class<?>> requestModels;
@UriParam(prefix = "response-model.", multiValue = true, javaType = "java.lang.String")
private Map<String, Class<?>> responseModels;
@UriParam(label = "advanced")
private ObjectMapper mapper;
@UriParam(defaultValue = "HELSINKI", enums = "FUJI,GENEVA,HELSINKI")
private ServiceNowRelease release = ServiceNowRelease.HELSINKI;
@UriParam(label = "security")
private SSLContextParameters sslContextParameters;
@UriParam(label = "advanced")
private HTTPClientPolicy httpClientPolicy;
@UriParam(label = "advanced")
private ProxyAuthorizationPolicy proxyAuthorizationPolicy;
@UriParam(label = "proxy")
private String proxyHost;
@UriParam(label = "proxy")
private Integer proxyPort;
@UriParam(label = "proxy,security")
private String proxyUserName;
@UriParam(label = "proxy,security")
private String proxyPassword;
@UriParam(label = "advanced", defaultValue = ServiceNowConstants.DEFAULT_DATE_FORMAT)
private String dateFormat = ServiceNowConstants.DEFAULT_DATE_FORMAT;
@UriParam(label = "advanced", defaultValue = ServiceNowConstants.DEFAULT_TIME_FORMAT)
private String timeFormat = ServiceNowConstants.DEFAULT_TIME_FORMAT;
@UriParam(label = "advanced", defaultValue = ServiceNowConstants.DEFAULT_DATE_TIME_FORMAT)
private String dateTimeFormat = ServiceNowConstants.DEFAULT_DATE_TIME_FORMAT;
public String getUserName() {
return userName;
}
public String getApiUrl() {
return apiUrl;
}
/**
* The ServiceNow REST API url
*/
public void setApiUrl(String apiUrl) {
this.apiUrl = apiUrl;
}
public boolean hasApiUrl() {
return apiUrl != null;
}
public String getApiVersion() {
return apiVersion;
}
/**
* The ServiceNow REST API version, default latest
*/
public void setApiVersion(String apiVersion) {
this.apiVersion = apiVersion;
}
/**
* ServiceNow user account name, MUST be provided
*/
public void setUserName(String userName) {
this.userName = userName;
}
public String getPassword() {
return password;
}
/**
* ServiceNow account password, MUST be provided
*/
public void setPassword(String password) {
this.password = password;
}
public String getOauthClientId() {
return oauthClientId;
}
/**
* OAuth2 ClientID
*/
public void setOauthClientId(String oauthClientId) {
this.oauthClientId = oauthClientId;
}
public String getOauthClientSecret() {
return oauthClientSecret;
}
/**
* OAuth2 ClientSecret
*/
public void setOauthClientSecret(String oauthClientSecret) {
this.oauthClientSecret = oauthClientSecret;
}
public String getOauthTokenUrl() {
return oauthTokenUrl;
}
public boolean hasOauthTokenUrl() {
return oauthTokenUrl != null;
}
/**
* OAuth token Url
*/
public void setOauthTokenUrl(String oauthTokenUrl) {
this.oauthTokenUrl = oauthTokenUrl;
}
public boolean hasBasicAuthentication() {
return ObjectHelper.isNotEmpty(userName)
&& ObjectHelper.isNotEmpty(password);
}
public boolean hasOAuthAuthentication() {
return ObjectHelper.isNotEmpty(userName)
&& ObjectHelper.isNotEmpty(password)
&& ObjectHelper.isNotEmpty(oauthClientId)
&& ObjectHelper.isNotEmpty(oauthClientSecret);
}
public String getResource() {
return resource;
}
/**
* The default resource, can be overridden by header CamelServiceNowResource
*/
public void setResource(String resource) {
this.resource = resource;
}
public String getTable() {
return table;
}
/**
* The default table, can be overridden by header CamelServiceNowTable
*/
public void setTable(String table) {
this.table = table;
}
public Boolean getExcludeReferenceLink() {
return excludeReferenceLink;
}
/**
* True to exclude Table API links for reference fields (default: false)
*/
public void setExcludeReferenceLink(Boolean excludeReferenceLink) {
this.excludeReferenceLink = excludeReferenceLink;
}
public Boolean getSuppressAutoSysField() {
return suppressAutoSysField;
}
/**
* True to suppress auto generation of system fields (default: false)
*/
public void setSuppressAutoSysField(Boolean suppressAutoSysField) {
this.suppressAutoSysField = suppressAutoSysField;
}
public Boolean getSuppressPaginationHeader() {
return suppressPaginationHeader;
}
/**
* Set this value to true to remove the Link header from the response. The
* Link header allows you to request additional pages of data when the number
* of records matching your query exceeds the query limit
*/
public void setSuppressPaginationHeader(Boolean suppressPaginationHeader) {
this.suppressPaginationHeader = suppressPaginationHeader;
}
public Boolean getIncludeScores() {
return includeScores;
}
/**
* Set this parameter to true to return all scores for a scorecard. If a value
* is not specified, this parameter defaults to false and returns only the most
* recent score value.
*/
public void setIncludeScores(Boolean includeScores) {
this.includeScores = includeScores;
}
public Boolean getIncludeAggregates() {
return includeAggregates;
}
/**
* Set this parameter to true to always return all available aggregates for
* an indicator, including when an aggregate has already been applied. If a
* value is not specified, this parameter defaults to false and returns no
* aggregates.
*/
public void setIncludeAggregates(Boolean includeAggregates) {
this.includeAggregates = includeAggregates;
}
public Boolean getIncludeAvailableBreakdowns() {
return includeAvailableBreakdowns;
}
/**
* Set this parameter to true to return all available breakdowns for an indicator.
* If a value is not specified, this parameter defaults to false and returns
* no breakdowns.
*/
public void setIncludeAvailableBreakdowns(Boolean includeAvailableBreakdowns) {
this.includeAvailableBreakdowns = includeAvailableBreakdowns;
}
public Boolean getIncludeAvailableAggregates() {
return includeAvailableAggregates;
}
/**
* Set this parameter to true to return all available aggregates for an indicator
* when no aggregate has been applied. If a value is not specified, this parameter
* defaults to false and returns no aggregates.
*/
public void setIncludeAvailableAggregates(Boolean includeAvailableAggregates) {
this.includeAvailableAggregates = includeAvailableAggregates;
}
public Boolean getIncludeScoreNotes() {
return includeScoreNotes;
}
/**
* Set this parameter to true to return all notes associated with the score.
* The note element contains the note text as well as the author and timestamp
* when the note was added.
*/
public void setIncludeScoreNotes(Boolean includeScoreNotes) {
this.includeScoreNotes = includeScoreNotes;
}
public Boolean getFavorites() {
return favorites;
}
/**
* Set this parameter to true to return only scorecards that are favorites of
* the querying user.
*/
public void setFavorites(Boolean favorites) {
this.favorites = favorites;
}
public Boolean getRetrieveTargetRecordOnImport() {
return retrieveTargetRecordOnImport;
}
/**
* Set this parameter to true to retrieve the target record when using import
* set api. The import set result is then replaced by the target record
*/
public void setRetrieveTargetRecordOnImport(Boolean retrieveTargetRecordOnImport) {
this.retrieveTargetRecordOnImport = retrieveTargetRecordOnImport;
}
public Boolean getKey() {
return key;
}
/**
* Set this parameter to true to return only scorecards for key indicators.
*/
public void setKey(Boolean key) {
this.key = key;
}
public Boolean getTarget() {
return target;
}
/**
* Set this parameter to true to return only scorecards that have a target.
*/
public void setTarget(Boolean target) {
this.target = target;
}
public String getDisplay() {
return display;
}
/**
* Set this parameter to true to return only scorecards where the indicator
* Display field is selected. Set this parameter to all to return scorecards
* with any Display field value. This parameter is true by default.
*/
public void setDisplay(String display) {
this.display = display;
}
public Integer getPerPage() {
return perPage;
}
/**
* Enter the maximum number of scorecards each query can return. By default
* this value is 10, and the maximum is 100.
*/
public void setPerPage(Integer perPage) {
this.perPage = perPage;
}
public String getSortBy() {
return sortBy;
}
/**
* Specify the value to use when sorting results. By default, queries sort
* records by value.
*/
public void setSortBy(String sortBy) {
this.sortBy = sortBy;
}
public String getSortDir() {
return sortDir;
}
/**
* Specify the sort direction, ascending or descending. By default, queries
* sort records in descending order. Use sysparm_sortdir=asc to sort in
* ascending order.
*/
public void setSortDir(String sortDir) {
this.sortDir = sortDir;
}
public String getDisplayValue() {
return displayValue;
}
/**
* Return the display value (true), actual value (false), or both (all) for
* reference fields (default: false)
*/
public void setDisplayValue(String displayValue) {
this.displayValue = displayValue;
}
public Boolean getInputDisplayValue() {
return inputDisplayValue;
}
/**
* True to set raw value of input fields (default: false)
*/
public void setInputDisplayValue(Boolean inputDisplayValue) {
this.inputDisplayValue = inputDisplayValue;
}
public Map<String, Class<?>> getRequestModels() {
return requestModels;
}
/**
* Sets Jackson's ObjectMapper to use for request/reply
*/
public void setMapper(ObjectMapper mapper) {
this.mapper = mapper;
}
public ObjectMapper getMapper() {
return mapper;
}
public ObjectMapper getOrCreateMapper() {
if (mapper == null) {
final DateTimeFormatter dateFormat = DateTimeFormatter.ofPattern(getDateFormat());
final DateTimeFormatter timeFormat = DateTimeFormatter.ofPattern(getTimeFormat());
final DateTimeFormatter dateTimeFormat = DateTimeFormatter.ofPattern(getDateTimeFormat());
this.mapper = new ObjectMapper()
.registerModule(new Jdk8Module())
.registerModule(new JavaTimeModule()
.addSerializer(LocalDate.class, new LocalDateSerializer(dateFormat))
.addDeserializer(LocalDate.class, new LocalDateDeserializer(dateFormat))
.addSerializer(LocalTime.class, new LocalTimeSerializer(timeFormat))
.addDeserializer(LocalTime.class, new LocalTimeDeserializer(timeFormat))
.addSerializer(LocalDateTime.class, new LocalDateTimeSerializer(dateTimeFormat))
.addDeserializer(LocalDateTime.class, new LocalDateTimeDeserializer(dateTimeFormat))
)
.configure(
SerializationFeature.WRITE_DATES_AS_TIMESTAMPS,
false)
.configure(
DeserializationFeature.FAIL_ON_UNKNOWN_PROPERTIES,
false)
.setSerializationInclusion(
JsonInclude.Include.NON_NULL
);
}
return mapper;
}
public boolean hasMapper() {
return mapper != null;
}
/**
* The ServiceNow release to target, default to Helsinki
*
* See https://docs.servicenow.com
*/
public void setRelease(ServiceNowRelease release) {
this.release = release;
}
public ServiceNowRelease getRelease() {
return release;
}
public Boolean getTopLevelOnly() {
return topLevelOnly;
}
/**
* Gets only those categories whose parent is a catalog.
*/
public void setTopLevelOnly(Boolean topLevelOnly) {
this.topLevelOnly = topLevelOnly;
}
public SSLContextParameters getSslContextParameters() {
return sslContextParameters;
}
/**
* To configure security using SSLContextParameters. See http://camel.apache.org/camel-configuration-utilities.html
*/
public void setSslContextParameters(SSLContextParameters sslContextParameters) {
this.sslContextParameters = sslContextParameters;
}
public HTTPClientPolicy getHttpClientPolicy() {
return httpClientPolicy;
}
/**
* To configure http-client
*/
public void setHttpClientPolicy(HTTPClientPolicy httpClientPolicy) {
this.httpClientPolicy = httpClientPolicy;
}
public ProxyAuthorizationPolicy getProxyAuthorizationPolicy() {
return proxyAuthorizationPolicy;
}
/**
* To configure proxy authentication
*/
public void setProxyAuthorizationPolicy(ProxyAuthorizationPolicy proxyAuthorizationPolicy) {
this.proxyAuthorizationPolicy = proxyAuthorizationPolicy;
}
public String getProxyHost() {
return proxyHost;
}
/**
* The proxy host name
*/
public void setProxyHost(String proxyHost) {
this.proxyHost = proxyHost;
}
public Integer getProxyPort() {
return proxyPort;
}
/**
* The proxy port number
*/
public void setProxyPort(Integer proxyPort) {
this.proxyPort = proxyPort;
}
public String getProxyUserName() {
return proxyUserName;
}
/**
* Username for proxy authentication
*/
public void setProxyUserName(String proxyUserName) {
this.proxyUserName = proxyUserName;
}
public String getProxyPassword() {
return proxyPassword;
}
/**
* Password for proxy authentication
*/
public void setProxyPassword(String proxyPassword) {
this.proxyPassword = proxyPassword;
}
public String getDateFormat() {
return dateFormat;
}
/**
* The date format used for Json serialization/deserialization
*/
public void setDateFormat(String dateFormat) {
this.dateFormat = dateFormat;
}
public String getTimeFormat() {
return timeFormat;
}
/**
* The time format used for Json serialization/deserialization
*/
public void setTimeFormat(String timeFormat) {
this.timeFormat = timeFormat;
}
public String getDateTimeFormat() {
return dateTimeFormat;
}
/**
* The date-time format used for Json serialization/deserialization
*/
public void setDateTimeFormat(String dateTimeFormat) {
this.dateTimeFormat = dateTimeFormat;
}
// *************************************************
//
// *************************************************
public void setModels(Map<String, Class<?>> models) {
setRequestModels(models);
setResponseModels(models);
}
public void addModel(String name, Class<?> type) {
addRequestModel(name, type);
addResponseModel(name, type);
}
// *************************************************
// Request model
// *************************************************
/**
* Defines the request model
*/
public void setRequestModels(Map<String, Class<?>> models) {
if (this.requestModels == null) {
this.requestModels = new HashMap<>();
}
this.requestModels.clear();
this.requestModels.putAll(models);
}
public void addRequestModel(String name, Class<?> type) {
if (this.requestModels == null) {
this.requestModels = new HashMap<>();
}
this.requestModels.put(name, type);
}
public Class<?> getRequestModel(String name) {
return getRequestModel(name, null);
}
public Class<?> getRequestModel(String name, Class<?> defaultType) {
Class<?> model = defaultType;
if (this.requestModels != null && this.requestModels.containsKey(name)) {
model = this.requestModels.get(name);
}
return model;
}
// *************************************************
// Response model
// *************************************************
/**
* Defines the response model
*/
public void setResponseModels(Map<String, Class<?>> models) {
if (this.responseModels == null) {
this.responseModels = new HashMap<>();
}
this.responseModels.putAll(models);
}
public void addResponseModel(String name, Class<?> type) {
if (this.responseModels == null) {
this.responseModels = new HashMap<>();
}
this.responseModels.clear();
this.responseModels.put(name, type);
}
public Class<?> getResponseModel(String name) {
return getResponseModel(name, null);
}
public Class<?> getResponseModel(String name, Class<?> defaultType) {
Class<?> model = defaultType;
if (this.responseModels != null && this.responseModels.containsKey(name)) {
model = this.responseModels.get(name);
}
return model;
}
// *************************************************
//
// *************************************************
public ServiceNowConfiguration copy() {
try {
return (ServiceNowConfiguration)super.clone();
} catch (CloneNotSupportedException e) {
throw new RuntimeCamelException(e);
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,946 |
Q: Why is the server getting an empty object when calling Meteor.method with a non-empty object? How do I call a Meteor method with a object as an argument, if this is at all possible?
Here is what I'm struggling with if it helps
I have the method:
'user.some.update.position'(coords) {
console.log('method: ', 'user.some.update.position');
console.log('this.userid: ', this.userId);
console.log('coords: ', coords);
check(coords, Object);
check(coords.latitude, Number);
check(coords.longitude, Number);
check(coords.accuracy, Number);
if (!this.userId)
throw new Meteor.Error('Not logged in', 'You are not logged in, please log in');
Meteor.users.update({
_id: this.userId
}, {
$set: {
coords,
lastUpdated: new Date()
}
});
return coords;
}
Which I want to call from the client like this:
> var coords = Geolocation.currentLocation().coords
undefined
> coords
Coordinates {latitude: 58.2441766, longitude: 8.376727899999999, altitude: null, accuracy: 25, altitudeAccuracy: null…}
> Meteor.call('user.some.update.position', coords, function(err, res) {if(err) console.log('err: ', err); if(res) console.log('res: ', res);})
undefined
VM7572:2 err: errorClass {error: 400, reason: "Match failed", details: undefined, message: "Match failed [400]", errorType: "Meteor.Error"}
But when I do that the server complains that coords is a empty object like this:
method: user.some.update.position
I20160220-15:49:34.226(1)? this.userid: nHqj3zaSWExRmqBZq
I20160220-15:49:34.226(1)? currentLocation: {}
I20160220-15:49:34.227(1)? Exception while invoking method 'user.some.update.position' Error: Match error: Expected number, got undefined
I20160220-15:49:34.227(1)? at Object.check (packages/check/match.js:33:1)
I20160220-15:49:34.228(1)? at [object Object]._meteorMeteor.Meteor.methods.user.some.update.position (server/methods/drivers.js:36:5)
I20160220-15:49:34.228(1)? at packages/check/match.js:103:1
I20160220-15:49:34.228(1)? at [object Object]._.extend.withValue (packages/meteor/dynamics_nodejs.js:56:1)
I20160220-15:49:34.228(1)? at Object.Match._failIfArgumentsAreNotAllChecked (packages/check/match.js:102:1)
I20160220-15:49:34.228(1)? at maybeAuditArgumentChecks (packages/ddp-server/livedata_server.js:1695:18)
I20160220-15:49:34.228(1)? at packages/ddp-server/livedata_server.js:708:19
I20160220-15:49:34.228(1)? at [object Object]._.extend.withValue (packages/meteor/dynamics_nodejs.js:56:1)
I20160220-15:49:34.228(1)? at packages/ddp-server/livedata_server.js:706:40
I20160220-15:49:34.229(1)? at [object Object]._.extend.withValue (packages/meteor/dynamics_nodejs.js:56:1)
I20160220-15:49:34.229(1)? Sanitized and reported to the client as: Match failed [400]
And the client complains:
err: errorClass {error: 400, reason: "Match failed", details: undefined, message: "Match failed [400]", errorType: "Meteor.Error"}
Edit: I am using ES6 and object destructuring.
A: It's clear from your error message that your object has nothing in it. please look at the 4th line of your error message, currentLocation contains no property. Sending a valid object will solve the problem.
A: what is $set: { coords, supposed to do? you can't do that. You need to take the content of that object apart and put it back together into the $set. Assuming that coords = {lat: 1234, lng: 2345} (or similar), you would do:
$set: {
lat: coords.lat,
lng: coords.lng,
...
or you can just add it as a sub-object;
$set: {
coords: coords,
A: Both the answers by Christian and Faysal contain valid information, I just want to expand on them a little:
The actual exception you're seeing, Error: Match error: Expected number, got undefined, is happening because of this line of code:
check(coords.latitude, Number);.
In your console log, you can see that coords is just an empty object:
I20160220-15:49:34.226(1)? currentLocation: {}
So when your check() method checks the currentLocation for coords.latitude, it throws the exception because coords.latitude is of type undefined, not of type Number like you said it would be in the check() statement.
Once you fix that, Christian pointed out that you'll get another error because of your update statement. MongoDB's $set requires you to pass in an object with the schema { $set: { <field1>: <value1>, ... } }. You are passing in: { $set: {}, lastUpdated: <A valid Date> }. Because that empty object doesn't match the field: value schema $set is expecting, it'll throw an exception. As he says, you'll need to either parse out that object into individual properties or pass it to a property itself.
A: The object that you are getting from the geolocation request implements the Coordinates interface. You cannot assume anything else about it.
In your case, the object is likely not serializable via EJSON, and therefore cannot be used as a Meteor method call argument as it is.
The method call routine calls EJSON.clone() and given the Coordinates object, it returns an empty object.
> EJSON.clone(coordinates)
Object {}
In Chrome's implementation, I assume that the properties are "owned" deeper in the prototype chain. EJSON relies on underscore's _.keys function when cloning, which in turn lists the object's own properties.
> coordinates.hasOwnProperty("latitude")
false
> "latitude" in coordinates
true
Since creating a custom EJSON type for it seems unrealistic, you can do as @ChristianFritz suggested or use underscore's _.pick.
> _.pick(coordinates, ['latitude', 'longitude', 'accuracy'])
Object {latitude: 12.1212, longitude: 34.3434, accuracy: 50}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 101 |
\section{Introduction}
Weak differential subordination of Banach space-valued martingales was recently discovered in the papers \cite{Y17FourUMD,Y17MartDec,Y17UMD^A,OY18} as a natural extension of differential subordination in the sense of Burkholder and Wang (see \cite{Wang,Burk84}) to infinite dimensions, and it has the following form: for a given Banach space $X$ an $X$-valued martingale $N$ is {\em weakly differentially subordinate} to an $X$-valued local martingale $M$ if a.s.\
\[
|\langle N_0, x^*\rangle| \leq |\langle M_0, x^*\rangle| \;\; \text{and}
\]
\[
[\langle N, x^*\rangle]_t - [\langle N, x^*\rangle]_s \leq [\langle M, x^*\rangle]_t - [\langle M, x^*\rangle]_s,\;\;\; 0\leq s\leq t,
\]
for any $x^*\in X^*$, where $[\,\cdot\,]$ is a {\em quadratic variation} of a martingale (see Subsection~\ref{subsec:qvandpdm}).
Weak differential subordination, especially if $X$ satisfies {\em the UMD property} (see Subsection \ref{subsec:UMD}), has several applications in Harmonic Analysis.
On the one hand, $L^p$-bounds for weakly differential subordinated {\em purely discontinuous} martingales imply estimates for $L^p$-norms of {\em L\'evy multipliers}. Namely, it was shown in \cite{Y17FourUMD} that if $T_m$ is a L\'evy multiplier (i.e.\ a Fourier multiplier generated by a L\'evy measure, see \cite{BB,BBB}), then by using weakly differential subordinated purely discontinuous martingales one gets that for any $1<p<\infty$ the $L^p$-norm of $T_m$ acting on $X$-valued functions is bounded by {\em the UMD constant} $\beta_{p, X}$ (which boundedness characterizes the UMD property, please see Subsection \ref{subsec:UMD}).
On the other hand, various bounds for weakly differential subordinated {\em orthogonal} martingales coincide with the same type of estimates for the {\em Hilbert transform} (see \cite{OY18} by Os\c{e}kowski and the author). Recall that two $X$-valued martingales $M$ and $N$ are orthogonal if a.s.\ for any $x^* \in X^*$
\[
\langle M_0, x^*\rangle\cdot\langle N_0, x^*\rangle=0 \;\; \text{and}\;\;
[\langle M, x^*\rangle,\langle N, x^*\rangle]_t=0,\;\;\; t\geq 0,
\]
where $[ \;\cdot\;, \,\cdot\;]$ is a {\em covariation} of two martingales (see Subsection \ref{subsec:qvandpdm}).
In particular, it was shown in \cite{OY18} that for any UMD Banach space $X$ and any $X$-valued orthogonal martingales $M$ and $N$ such that $N$ is weakly differentially subordinate to $M$ one has that for every $1<p<\infty$
\[
\mathbb E \|N_t\|^p \leq \hbar_{p, X}^p \mathbb E \|M_t\|^p,\;\;\; t\geq 0,
\]
where the sharp constant $\hbar_{p, X}$ is the norm of the Hilbert transform on $L^p(\mathbb R; X)$.
The goal of the present paper is to present sharp $L^p$ estimates for {\em strongly orthogonal} weakly differentially subordinated martingales. We call two $X$-valued martingales $M$ and $N$ strongly orthogonal if a.s.\ for any $x^*, y^* \in X^*$
\[
\langle M_0, x^*\rangle\cdot\langle N_0, y^*\rangle=0 \;\; \text{and}\;\;
[\langle M, x^*\rangle,\langle N, y^*\rangle]_t=0,\;\;\; t\geq 0.
\]
A classical example of strongly orthogonal martingales are stochastic integrals $\int \Phi\ud W$ and $\int \Phi\ud \widetilde W$, where $\Phi$ is $X$-valued elementary predictable, and $W$ and $\widetilde W$ are independent Brownian motions.
In the present paper we prove that for any strongly orthogonal weakly differentially subordinated martingales $M$ and $N$
\begin{equation}\label{eq:INTROSOWDSLpbounds}
\mathbb E \|N_t\|^p \leq \chi^p \mathbb E \|M_t\|^p,\;\;\; t\geq 0,\;\;1<p<\infty,
\end{equation}
where the sharp constant $\chi = \chi_{p, X}$ is within the range
\begin{equation}\label{eq:INTROchiestimates}
\max\{\sqrt{\beta_{p,X}}, \sqrt{\hbar_{p, X}}\} \leq \chi_{p, X} \leq \min\{\beta_{p, X}, \hbar_{p, X}\}.
\end{equation}
The main technique we used in order to prove \eqref{eq:INTROSOWDSLpbounds} is the {\em Bellman function method}. More specifically, we show that the following are equivalent
\begin{enumerate}[(A)]
\item \eqref{eq:INTROSOWDSLpbounds} holds for a constant $\chi>0$,
\item there exists $U^{SO}:X+iX \to \mathbb R$ such that $U^{SO}(x) \geq 0$ for any $x\in X$, $z \mapsto U^{SO}(x_0 + iy_0 + zx)$ in subharmonic in $z\in \mathbb C$ for any $x_0, y_0, x\in X$, and
\[
U^{SO}(x+iy) \leq \chi^p \|x\|^p - \|y\|^p,\;\;\;x,y \in X.
\]
\end{enumerate}
Notice that this method is not new while working with martingales with values in UMD Banach space. Namely, in \cite{Y17FourUMD} there was applied the {\em Burkholder function} $U:X \times X \to \mathbb R$ which first appeared in the paper \cite{Burk86} by Burkholder, and in \cite{OY18} there was used a {\em plurisubhirmonic function} $U_{\mathcal H}:X+iX\to \mathbb R$which first was constructed in the paper \cite{HKV03} by Hollenbeck, Kalton, and Verbitsky. The novelty of the present paper is in minimizing the necessary properties of the Bellman function. Namely, both $-U$ and $U_{\mathcal H}$ satisfy the property ${\rm (B)}$ outlined above (which makes the upper bound of \eqref{eq:INTROchiestimates} elementary).
In order to show the lower bounds of \eqref{eq:INTROchiestimates} and in order to characterize the least admissible cosntant $\chi_{p, X}$ we will need the example presented above. It turned out in Section \ref{sec:chipX} and \ref{sec:WDSofSOM} that the sharp constant $\chi_{p, X}$ is the smallest constant $\chi>0$ such that for any independent Brownian motions $W$ and $\widetilde W$ and for any elementary predictable $X$-valued $\Phi$ one has that
\[
\mathbb E \Bigl\|\int_0^{\infty}\Phi \ud \widetilde W \Bigr\|^p \leq \chi^p \mathbb E \Bigl\|\int_0^{\infty}\Phi \ud W \Bigr\|^p.
\]
Thus the desires lower bound of \eqref{eq:INTROchiestimates} follows from the well-known decoupling-type inequalities of Garling, see \cite{Gar85}.
\smallskip
Notice that if $X = \mathbb R$, then $\chi_{p, X} = \hbar_{p, X}$ (see Remark \ref{rem:-UandUHarediagplsh}). Nevertheless, it remains open whether this equality holds for a general UMD Banach space $X$. Moreover, if this is the case, then it proves a celebrated open problem about linear dependence of the constants $\beta_{p, X}$ and $\hbar_{p,X}$, see \cite[p.\ 48]{Bour84} and \cite{HNVW1,GM-SS,Y17FourUMD,OY18} (so far only a square dependence is known, see \eqref{eq:sqrtbetaleqhleqbeta^2}).
\medskip
\emph{Acknowledgment} --The author would like to thank Adam Os\c{e}kowski and Mark Veraar
for helpful comments. The author thanks Stefan Geiss for fruitful
discussions and for being the host while author's stay at
Jyv\"askyl\"a University where the present paper was written.
\section{Preliminaries}
Throughout the paper all Banach spaces are assumed to be over the scalar field $\mathbb R$ unless stated otherwise. We also assume that any filtration satisfies the usual conditions. In particular, any filtration is right-continuous, and thus all the local martingales exploited in the article have {\em c\`adl\`ag} versions (i.e.\ versions which are right continuous with left limits, see \cite{VerPhD,Y17FourUMD}). Furthermore, for any Banach space $X$, for any c\`adl\`ag process $A:\mathbb R_+ \times \Omega \to X$, and for any stopping time $\tau$ we define
\[
\Delta A_{\tau} := \lim_{\varepsilon\to 0} (A_{\tau} - A_{(\tau-\varepsilon)\vee 0}).
\]
\subsection{UMD Banach spaces}\label{subsec:UMD}
A Banach space $X$ is called {\it UMD} if for some (equivalently,
for all) $p \in (1,\infty)$ there exists a constant $\beta>0$ such
that for every $N \geq 1$, every martingale difference sequence
$(d_n)^N_{n=1}$ in $L^p(\Omega; X)$, and every $\{-1,1\}$-valued
sequence $(\varepsilon_n)^N_{n=1}$ we have
\begin{equation*}\label{eq:defofUMDconstant}
\Bigl(\mathbb E \Bigl\| \sum^N_{n=1} \varepsilon_n d_n\Bigr\|^p\Bigr )^{\frac
1p}
\leq \beta \Bigl(\mathbb E \Bigl \| \sum^N_{n=1}d_n\Bigr\|^p\Bigr )^{\frac 1p}.
\end{equation*}
The least admissible constant $\beta$ is denoted by $\beta_{p,X}$
and is called the {\it UMD$_p$~constant} or, in the case if the
value of $p$ is understood, the {\em UMD constant} of $X$. It is
well-known that UMD spaces obtain a large number of useful
properties, such as being reflexive. Examples of UMD spaces
include all finite dimensional spaces and the reflexive range of
$L^q$-, Besov, Sobolev, Schatten class, and Musielak--Orlicz
spaces. Example of spaces without the UMD property include all
nonreflexive Banach spaces, e.g.\ $L^1(0,1)$ or $C([0,1])$. We
refer to \cite{Burk01,HNVW1,Rubio86,Pis16} for details.
\subsection{Quadratic variation}\label{subsec:qvandpdm}
Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space with
a filtration $\mathbb F = (\mathcal F_t)_{t\geq 0}$ that satisfies
the usual conditions. Let $M:\mathbb R_+ \times \Omega \to \mathbb
R$ be a local martingale. We define a {\em quadratic variation} of
$M$ in the following way:
\begin{equation}\label{eq:defquadvar}
[M]_t := |M_0|^2 + \mathbb P-\lim_{{\rm mesh}\to 0}\sum_{n=1}^N \bigl|M(t_n)-M(t_{n-1})\bigr|^2,
\end{equation}
where the limit in probability is taken over partitions $0= t_0 <
\ldots < t_N = t$. Note that $[M]$ exists and is nondecreasing
a.s. The reader can find more on quadratic variations in
\cite{Kal,Prot,DM82}. For any martingales $M, N:\mathbb R_+ \times
\Omega \to \mathbb R$ we can define a {\em covariation}
$[M,N]:\mathbb R_+ \times \Omega \to \mathbb R$ as $[M,N] :=
\frac{1}{4}([M+N]-[M-N])$. Since $M$ and $N$ have c\`adl\`ag
versions, $[M,N]$ has a c\`adl\`ag version as well (see e.g.
\cite[Theorem I.4.47]{JS}).
\smallskip
A local martingale $M:\mathbb R_+ \times\Omega \to \mathbb R$ is
called {\em purely discontinuous} if $[M]$ is a.s.\ pure jump,
i.e.\ $[M]_t = \sum_{0\leq s \leq t}\Delta [M]_s$ a.s. Let $X$ be
a Banach space. Then an $X$-valued local martingale $M:\mathbb R_+
\times \Omega \to X$ is called {\em purely discontinuous} if
$\langle M, x^*\rangle$ is purely discontinuous for any $x^* \in
X^*$. Note that if $X$ is UMD, then any local martingale $M$ has a
unique decomposition into a sum of a continuous local martingale
$M^c$ with $M^c_0=0$ and a purely discontinuous local martingale
$M^d$ (see \cite{Y17GMY}). We refer to
\cite{Kal,JS,Y17FourUMD,Y17MartDec,Y17GMY} for details on purely
discontinuous martingales.
\subsection{Weak differential subordination of martingales}
Let $X$ be a Banach space.
Let $M,N:\mathbb R_+ \times \Omega \to X$ be local martingales. Then we say
that $N$ is {\em weakly differentially subordinate} to $M$ (we will denote this by $N \stackrel{w} \ll M$) if for each $x^* \in
X^*$ one has that $[\langle M,x^*\rangle]-[\langle N,x^*\rangle]$ is an a.s.\
nondecreasing function and $|\langle N_0,x^*\rangle|\leq |\langle
M_0,x^*\rangle|$~a.s.
The definition above first appeared in \cite{Y17FourUMD} as a natural
extension of differential subordination of real-valued
martingales. Later in \cite{Y17MartDec} there were obtained the
first $L^p$-estimated for weakly differentially subordinated
martingales, which have been significantly improved in \cite{OY18}
in the continuous-time case.
\subsection{Orthogonal martingales}
Let $M$ and $N$ be local martingales taking values in a given Banach
space $X$. Then $M$ and $N$ are said to be {\em orthogonal}, if
$\langle M_0, x^*\rangle\cdot \langle N_0, x^*\rangle =0 $ and
$[\langle M, x^*\rangle,\langle N, x^*\rangle] = 0$ almost surely
for all functionals $x^* \in X^*$.
\begin{remark}\label{rem:orth+wds}
Assume that $M$ and $N$ are local martingales taking values in some
Banach space $X$. If $M$ and $N$ are orthogonal and $N$ is weakly
differentially subordinate to $M$, then $N_0=0$ almost surely
(which follows immediately from the above definitions, see \cite{OY18}). Moreover,
under these assumptions, $N$ must have continuous trajectories
with probability $1$. Indeed, in such a case for any fixed $x^*\in
X^*$ the real-valued local martingales $\langle M, x^*\rangle$ and
$\langle N, x^*\rangle$ are orthogonal and we have $\langle N,
x^*\rangle \ll \langle M, x^*\rangle$. Therefore, $\langle N,
x^*\rangle$ has a continuous version for each $x^*\in X^*$ by
\cite[Lemma 3.1]{Os09a} (see also \cite[Lemma 1]{BanWang96}),
which in turn implies that $N$ is continuous since any $X$-valued local
martingale has a c\`adl\`ag version.
\end{remark}
\subsection{Stochastic integration}
For given Banach spaces $X$ and $Y$, the symbol $\mathcal{L}(X,Y)$
will denote the classes of all linear operators from $X$ to $Y$.
We will also use the notation $\mathcal{L}(X)=\mathcal{L}(X,X)$.
Suppose that $H$ is a Hilbert space. For each $h\in H$ and $x\in
X$, we denote by $h\otimes x$ the associated linear operator given
by $g\mapsto \langle g, h\rangle x$, $g\in H$. The process $\Phi:
\mathbb R_+ \times \Omega \to \mathcal L(H,X)$ is called
\textit{elementary predictable} with respect to the filtration
$\mathbb F = (\mathcal F_t)_{t \geq 0}$ if it is of the form
\begin{equation*}\label{eq:elempredict}
\Phi(t,\omega) = \sum_{k=1}^K\sum_{m=1}^M \mathbf 1_{(t_{k-1},t_k]\times B_{mk}}(t,\omega)
\sum_{n=1}^N h_n \otimes x_{kmn},\;\;\; t\geq 0, \omega \in
\Omega.
\end{equation*}
Here $0 \leq t_0 < \ldots < t_K <\infty$ is a finite increasing
sequence of nonegative numbers, the sets $B_{1k},\ldots,B_{Mk}$
belong to $\mathcal F_{t_{k-1}}$ for each $k = 1,\,2,\,\ldots,
K$, and the vectors $h_1,\ldots,h_N$ are assumed to be orthogonal.
Suppose further that $M$ is an adapted local martingale taking
values in $H$. Then the {\em stochastic integral} $ \int \Phi \ud
M:\mathbb R_+ \times \Omega \to X$ of $\Phi$ with respect to $M$
is defined by the formula
\begin{equation*}
\int_0^t\Phi \ud M = \sum_{k=1}^K\sum_{m=1}^M \mathbf 1_{B_{mk}}
\sum_{n=1}^N \langle(M(t_k\wedge t)- M(t_{k-1}\wedge t)),
h_n\rangle x_{kmn},\;\; t\geq 0.
\end{equation*}
\begin{remark}\label{rem:stochintgenPhi}
If both $X$ and $H$ are finite dimensional, then we may assume that $X$ is isomorphic to $\mathbb R^d$, and thus by \cite[Theorem 26.6 and 26.12]{Kal} we can extend the stochastic integration from elementary predictable processes to all the predictable processes $\Phi:\mathbb R_+ \times \Omega \to \mathcal L(H,X)$ with
$$
\mathbb E \Bigl( \sum_{i=1}^n\int_0^{\infty}\|\Phi h_i\|^2 \ud [\langle M, h_i\rangle]_s\Bigr)^{1/2}<\infty,
$$
where $n$ is the dimension of $H$ and $h_1,\ldots,h_n$ is an orthonormal basis of $H$. In fact, a similar characterization of stochastic integration can be shown for infinite dimensional $X$ and $H$ by using {\em $\gamma$-norms} (see \cite{NVW,Y18BDG,VY2016,Ver}).
\end{remark}
\subsection{Hilbert transform}\label{subsec:HT}
Let $X$ be a Banach space.
The {\em Hilbert transform} $\mathcal H_{X}$ is a singular integral
operator that maps a step function $f:\mathbb R\to X$ to the
function
\begin{equation*}\label{eq:defdefofRHT}
(\mathcal H_X f)(t):= \frac{1}{\pi}{\textnormal{p.v.}}\int_{\mathbb R}\frac{f(s)}{t-s}\ud s,\;\;\; t\in \mathbb R.
\end{equation*}
For any $1<p<\infty$ we denote the norm of $\mathcal H_{X}$ on
$L^p(\mathbb R; X)$ by $\hbar_{p, X}$. Note that due to
\cite{Burk83,Bour83} we have that $\hbar_{p, X} <\infty$ if and
only if $X$ is UMD. Moreover, due to \cite{Gar85,Bour83} we have
that for every $1<p<\infty$
\begin{equation}\label{eq:sqrtbetaleqhleqbeta^2}
\sqrt{\beta_{p,X}} \leq \hbar_{p, X} \leq \beta_{p, X}^2.
\end{equation}
\begin{remark}
Recently in \cite{OY18} it was shown that $\hbar_{p, X}$ is the smallest constant $\hbar$ such that there exists a {\em plurisubharmonic function} $U_{\mathcal H}:X+iX\to \mathbb R$ (i.e.\ $z\mapsto U_{\mathcal H}(x_0 + iy_0 + z(x+iy))$ is subharmonic in $z\in \mathbb C$ for any fixed $x_0, y_0, x, y\in X$) such that $U_{\mathcal H}(x)\geq 0$ for any $x\in X$ and $U_{\mathcal H}(x+iy) \leq \hbar^p \|x\|^p - \|y\|^p$ for all $x, y\in X$.
\end{remark}
\subsection{Bellman functions and function approximation}\label{subsec:funcapprox}
Let $X$ be a UMD Banach space, $1<p<\infty$. Throughout the paper
we will use different {\em Bellman functions}, i.e.\ functions
$u:X \times X \to \mathbb R$ which have certain appropriate
properties. Let us outline which functions we will use
\begin{itemize}
\item the Burkholder function $U:X \times X \to\mathbb R$ (see
e.g.\ \cite{HNVW1} and the proof of Corollary
\ref{cor:estimforchipX}),
\item a plurisubharmonic function $U_{\mathcal H}:X+iX \to \mathbb
R$ (see \cite{OY18} and Subsection~\ref{subsec:HT}),
\item a diagonally plurisubharmonic function $U^{SO}:X+iX \to
\mathbb R$ (see Section~\ref{sec:chipX}).
\end{itemize}
For all the Bellman functions named above we may assume that $X$
is finite dimensional and that the function is twice Fr\'echet
differentiable by an approximation argument exploited in
\cite{Y17MartDec,OY18,BO12}. We will not repeat this argument
here, but just shortly remind the reader the main steps.
\begin{itemize}
\item Since $X$ is UMD, it is reflexive, and by the Pettis
measurability theorem \cite[Theorem 1.1.20]{HNVW1} we may assume
that $X$ is separable. Thus $X^*$ is separable as well, and there
exist an increasing sequence $(Y_n)_{n\geq 1}$ of finite
dimensional subspaces of $X^*$ such that $X^* = \overline{\cup_n
Y_n}$. Let $P_n:Y_n \to X^*$ be the injection operator. In the
sequel we will need to show that $\mathbb E \|\eta\|^p \leq c_{p,
X}^p \mathbb E \|\xi\|^p $ for a certain pair of random variables $\xi, \eta \in
L^p(\Omega; X)$ and a certain constant $c_{p,
X}$. Since $\|P_n^* x\| \nearrow \|x\|$ monotonically as $n\to
\infty$ for any $x\in X$, by the monotone convergence theorem it
is sufficient to show that $\mathbb E \|P_n^*\eta\|^p \leq c_{p,
X}^p \mathbb E \|P_n^*\xi\|^p $ for any $n\geq 1$. Moreover, in fact we need to show that $\mathbb E \|P_n^*\eta\|^p \leq c_{p,
Y_n^*}^p \mathbb E \|P_n^*\xi\|^p $ since in our case $c_{p, X}$ equals either $\beta_{p, X}$, $\hbar_{p, X}$, or $\chi_{p, X}$ (see Section~\ref{sec:chipX} for the definition), and since all these constants can be represented as norms of operators having the same operators as their duals, so one has that analogously to \cite[Proposition 4.2.17]{HNVW1} $c_{p, X} = c_{p', X^*}$ (where $p' = p/(p-1)$), and in particular
\[
c_{p, Y_n^*} = c_{p', Y_n} \leq c_{p', X^*} = c_{p, X},
\]
Thus it is
sufficient to assume that $X$ is finite dimensional since both $P_n^* \xi$ and $P_n^* \eta$ have their values in a finite dimensional space $Y_n^*$.
\item Since $X$ is finite dimensional, for a Bellman function $u$ and
for any $\varepsilon>0$ we can define $u_{\varepsilon} := u *
\varepsilon^{-1}\phi(\varepsilon^{-1} \cdot)$, where $\phi:X \times X \to
\mathbb R_+$ is a $C^{\infty}$ function with a compact domain such
that $\int_{X \times X} \phi(x,y)\ud \lambda(x)\ud \lambda(y) = 1$
(here $\lambda$ is the {\em Lebesque measure} on $X$, see e.g.\
\cite[Remark 3.13]{Y17FourUMD} for the definition). Then
$u_{\varepsilon}$ preserves such properties of $u$ as convexity,
concavity, or subharmonicity on a linear subspace of $X \times X$,
and $u_{\varepsilon} \to u$ as $\varepsilon \to 0$ locally uniformly on $X
\times X$ due to continuity of $u$. Therefore by this approximation argument we may assume that $u$ is $C^{\infty}$.
\end{itemize}
\section{The $\chi_{p,X}$ constant}\label{sec:chipX}
Let $X$ be a Banach space, $1<p<\infty$. We define $\chi_{p,X}\in
[0,\infty]$ to be the least number $\chi>0$ such that for any independent
standard Brownian motions $W, \widetilde W:\mathbb R_+ \times
\Omega \to \mathbb R$ and for any elementary predictable with
respect to the filtration generated by both $W$ and $\widetilde W$
process $\Phi:\mathbb R_+ \times \Omega \to X$ one has that
\[
\mathbb E \Bigl\| \int_0^{\infty} \Phi \ud \widetilde W \Bigr\|^p
\leq \chi^p \mathbb E \Bigl\| \int_0^{\infty} \Phi \ud W
\Bigr\|^p.
\]
\begin{remark}
$\chi_{p,X}$ can be equivalently defined in the following way. Let $(\gamma_n)_{n\geq 1}$ and $(\tilde \gamma_n)_{n\geq 1}$ be sequences of independent standard Gaussian random variables, $\mathcal F_0 = \{\varnothing, \Omega\}$, and $\mathcal F_n = \sigma(\gamma_1, \tilde \gamma_1,\ldots,\gamma_n, \tilde \gamma_n)$ for $n\geq 1$. Then $\chi_{p, X}$ is the smallest $\chi>0$ such that for any $N\geq 1$ and any elementary step functions $v_0,\ldots,v_{N-1}:\Omega \to X$ with $v_n$ being $\mathcal F_n$-measurable for each $n =0, \ldots, N-1$, one has that
\begin{equation}\label{eq:defofchibygaussians}
\mathbb E\Bigl\|\sum_{n=1}^N\tilde\gamma_n v_{n-1}\Bigr\|^p \leq \chi^p \mathbb E\Bigl\|\sum_{n=1}^N\gamma_n v_{n-1}\Bigr\|^p.
\end{equation}
Indeed, one can represent the sums $\sum_{n=1}^N\gamma_n v_{n-1}$ and $\sum_{n=1}^N\tilde \gamma_n v_{n-1}$ as stochastic integrals with respect to independent Brownian motions $W$ and $\widetilde W$ by just letting $\gamma_n = W_n-W_{n-1}$ and $\tilde \gamma_n = \widetilde W_n-\widetilde W_{n-1}$. On the other hand, if $W$ and $\widetilde W$ are independent Brownian motions and if $\Phi$ is elementary predictable and defined by
\begin{equation*}
\Phi(t,\omega) = \sum_{k=1}^K\sum_{m=1}^M \mathbf 1_{(t_{k-1},t_k]\times B_{mk}}(t,\omega) x_{km},\;\;\; t\geq 0, \omega \in
\Omega.
\end{equation*}
where $0 \leq t_0 < \ldots < t_K <\infty$ is a finite increasing
sequence of nonnegative numbers and the sets $B_{1k},\ldots,B_{Mk}$
belong to $\mathcal F_{t_{k-1}}$ for each $k = 1,\,2,\,\ldots,
K$, then one can represent the stochastic integrals $\Phi \cdot W$ and $\Phi \cdot \widetilde W$ as the sums $\sum_{n=1}^N\gamma_n v_{n-1}$ and $\sum_{n=1}^N\tilde \gamma_n v_{n-1}$ in the following way
\[
\int_0^{\infty} \Phi \ud W = \sum_{k=1}^K\sum_{m=1}^M \mathbf 1_{B_{mk}}
(W(t_k)- W(t_{k-1})) x_{km} = \sum_{k=1}^K v_{k-1}
\gamma_k,
\]
\[
\int_0^{\infty} \Phi \ud \widetilde W = \sum_{k=1}^K\sum_{m=1}^M \mathbf 1_{B_{mk}}
(\widetilde W(t_k)-\widetilde W(t_{k-1})) x_{km} = \sum_{k=1}^K v_{k-1}
\tilde\gamma_k,
\]
where $\gamma_k = \tfrac{W(t_k)- W(t_{k-1})}{\sqrt{t_k-t_{k-1}}}$, $\tilde\gamma_k = \tfrac{\widetilde W(t_k)-\widetilde W(t_{k-1})}{\sqrt{t_k-t_{k-1}}}$, and
$v_{k-1} =\sqrt{t_k-t_{k-1}} \sum_{m=1}^M \mathbf 1_{B_{mk}} x_{km}.$
The martingale transform \eqref{eq:defofchibygaussians} appears while working with Volterra-type operators and stochastic shifts (see \cite{GY19}).
\end{remark}
Concerning the constant $\chi_{p,X}$ one can show the following
proposition. First we will define diagonally plurisubharmonic functions.
\begin{definition}
A function $F:X + iX \to \mathbb R$ is called {\em diagonally
plurisubharmonic} if $z \mapsto F(x_0 + iy_0 + zx)$ is subharmonic
in $z\in \mathbb C$ for any $x_0, y_0, x\in X$.
\end{definition}
\begin{proposition}\label{prop:chi<inftyiffexistsdiagplsfunc}
Let $X$ be a Banach space, $1<p<\infty$. Then the following are
equivalent
\begin{enumerate}[(i)]
\item $\chi_{p, X}<\infty$,
\item there exists a constant $\chi >0$ and a
diagonally plurisubharmonic $u:X+ iX \to \mathbb R$ such that
$u(x)\geq 0$ for any $x\in X$, $x\mapsto u(x + iy)$ is convex in
$x\in X$ for any $y\in X$, $y\mapsto u(x + iy)$ is concave in
$y\in X$ for any $x\in X$, and
\begin{equation}\label{eq:ubiggethenchipxp-yp}
u(x + iy) \leq \chi^p \|x\|^p - \|y\|^p,\;\;\; x,\; y\in X.
\end{equation}
\end{enumerate}
Moreover, if this is the case, then the smallest $\chi$ for which
such a function $u$ exists equals $\chi_{p, X}$.
\end{proposition}
\begin{proof}
We will prove both implications separately.
{\em $(i) \Rightarrow (ii)$.} In order to show this implication we
need to construct function $u$ for $\chi = \chi_{p, X}$. In this
case let us define the desired function $u$ to be as follows
\begin{equation}\label{eq:defofdiagplsu}
\begin{split}
u(x + iy) := \inf\Big\{&\chi_{p, X}^p \mathbb E\Bigl\|x +
\int_0^{\infty} \Phi \ud W \Bigr\|^p - \mathbb E\Bigl\|y +
\int_0^{\infty} \Phi \ud \widetilde W \Bigr\|^p:\\
& \Phi:\mathbb R_+ \times \Omega \to X \textnormal{elementary
predictable}\Big\},\;\;\;x, y\in X.
\end{split}
\end{equation}
First of all notice that $u$ is finite on $X+iX$. Indeed, one has
that for any elementary predictable $\Phi:\mathbb R_+ \times
\Omega \to X$ and for any $x, y\in X$ by the triangle inequality
\begin{align*}
\chi_{p, X}^p \mathbb E\Bigl\|x &+ \int_0^{\infty} \Phi \ud W
\Bigr\|^p - \mathbb E\Bigl\|y + \int_0^{\infty} \Phi \ud
\widetilde W
\Bigr\|^p\\
& \gtrsim_{p} \chi_{p, X}^p \mathbb E\Bigl\| \int_0^{\infty} \Phi
\ud W \Bigr\|^p - \mathbb E\Bigl\| \int_0^{\infty} \Phi \ud
\widetilde W
\Bigr\|^p -\chi_{p, X}^p\|x\|^p - \|y\|^p\geq -\chi_{p, X}^p\|x\|^p - \|y\|^p,
\end{align*}
where the latter holds by the definition of $\chi_{p, X}$.
Let us show that $u$ is continuous. For any $x, y, \tilde x,
\tilde y$ one has that by the triangle inequality
\begin{equation*}
\begin{split}
u(x+iy) =\inf\Big\{&\chi_{p, X}^p \mathbb E\Bigl\|x +
\int_0^{\infty} \Phi \ud W \Bigr\|^p - \mathbb E\Bigl\|y +
\int_0^{\infty} \Phi \ud \widetilde W \Bigr\|^p:\\
& \Phi:\mathbb R_+ \times \Omega \to X \textnormal{elementary
predictable}\Big\}\\
\lesssim_{p} \inf\Big\{&\chi_{p, X}^p \mathbb E\Bigl\|\tilde x +
\int_0^{\infty} \Phi \ud W \Bigr\|^p - \mathbb E\Bigl\| \tilde y +
\int_0^{\infty} \Phi \ud \widetilde W \Bigr\|^p:\\
& \Phi:\mathbb R_+ \times \Omega \to X \textnormal{elementary
predictable}\Big\} + \chi_{p, X}\|x-\tilde x\|^p + \|y-\tilde y\|^p\\
\leq u(\tilde x + &i\tilde y) + \chi_{p, X}\|x-\tilde x\|^p + \|y-\tilde
y\|^p,
\end{split}
\end{equation*}
so the continuity follows.
Now let us show that $u$ is diagonally plurisubharmonic. Fix $x_0,
y_0, x\in X$. We need to show that $z \mapsto u(x_0 + iy_0 + zx)$
is subharmonic in $z\in \mathbb C$. To this end we need to
prove that for any fixed $r>0$
\begin{equation}\label{eq:diagplshofu}
u(x_0 + iy_0) \leq \frac {1}{2\pi}\int_{0}^{2\pi} u(x_0 + iy_0 +
xre^{i\theta}) \ud \theta.
\end{equation}
Let $W, \widetilde W:\mathbb R_+ \times \Omega \to \mathbb R$ be
independent standard Brownian motions. Define a stopping time
$\tau$ in the following way
\begin{equation*}\label{eq:defoftauoutofcircle}
\tau := \inf \{t\geq 0: W_t^2 + \widetilde W_t^2 = r\}.
\end{equation*}
Fix $\varepsilon>0$. Note that since $u$ is continuous, there exist $\delta>0$
and a $\delta$-net $(a_n)_{n=1}^N = (x_n +
iy_n)_{n=1}^N$ of a compact set $A :=\{x_0 + iy_0 + xre^{i\theta}:
\theta \in [0,2\pi)\}\subset X + iX$ with
\begin{equation}\label{eq:controfanandepsassump}
|u(a) - u(a_n)| \leq \varepsilon \;\; \forall a\in A \;\; \text{such
that}\;\|a-a_n\| <\delta
\end{equation}
(here the norm on $A$ is assumed to be a usual norm on
$\mathbb C$ since $A$ can be represented as a circle on $\mathbb
C$). Let $B_t := W_{t+\tau} -W_{\tau}$, $\widetilde B_t :=
\widetilde W_{t+\tau} -\widetilde W_{\tau}$. Note that $B$ and
$\widetilde B$ are independent Brownian motions (see e.g.\ \cite[Theorem 13.11]{Kal}).
Therefore by the definition of $u$ for every $n=1,\ldots,N$ there
exists an elementary predictable with respect to the filtration
generated by $B$ and $\widetilde B$ process $\Phi_n:\mathbb R_+
\time \Omega \to X$ such that
\begin{equation}\label{eq:sharpguysforu(an)}
u(a_n) \geq \chi_{p, X}^p \mathbb E\Bigl\|x_n + \int_0^{\infty}
\Phi_n \ud B \Bigr\|^p - \mathbb E\Bigl\|y_n + \int_0^{\infty}
\Phi_n \ud \widetilde B \Bigr\|^p - \varepsilon.
\end{equation}
Now let us define a predictable with respect to the filtration
generated by $W$ and $\widetilde W$ process $\Phi$ in the
following way. $\Phi(t) = x$ if $t\leq \tau$ and $\Phi(t) =
\Phi_n(t-\tau)$ if $t> \tau$ and $a_n$ is the closest among the
set $(a_n)_{n=1}^N$ point to $x_0 + iy_0 + x(W_{\tau} +
i\widetilde W_{\tau})$. This is a predictable process
and since $\Phi$ takes values in a finite dimensional subspace of
$X$, it can be approximated by an elementary predictable process
(see Remark \ref{rem:stochintgenPhi}). Therefore we get that
\begin{align*}
u(x_0 + iy_0) &\leq \chi_{p, X}^p \mathbb E\Bigl\|x_0 +
\int_0^{\infty} \Phi \ud W \Bigr\|^p - \mathbb E\Bigl\|y_0 +
\int_0^{\infty} \Phi \ud \widetilde W \Bigr\|^p \\
&= \chi_{p, X}^p \mathbb E\Bigl\|x_0 + xW_{\tau} +
\int_0^{\infty} \Phi(t) \ud B_{t-\tau} \Bigr\|^p \\
&\quad\quad - \mathbb E\Bigl\|y_0 + x\widetilde W_{\tau} +
\int_0^{\infty} \Phi(t) \ud \widetilde
B_{t-\tau} \Bigr\|^p\\
&\stackrel{(i)}=\frac {1}{2\pi}\int_{0}^{2\pi} \chi_{p, X}^p
\mathbb E\Bigl\|x_0 + x \cos{\theta} + \int_0^{\infty}
\Phi_{n(\theta)}(t) \ud B_{t}
\Bigr\|^p\\
&\quad\quad - \mathbb E\Bigl\|y_0 + x \sin{\theta} +
\int_0^{\infty} \Phi_{n(\theta)}(t) \ud \widetilde B_{t} \Bigr\|^p
\ud
\theta\\
&\stackrel{(ii)}\leq \frac {1}{2\pi}\int_{0}^{2\pi} \chi_{p,
X}^p \mathbb E\Bigl\|x_{n(\theta)} + \int_0^{\infty}
\Phi_{n(\theta)}(t) \ud B_{t}
\Bigr\|^p\\
&\quad\quad - \mathbb E\Bigl\|y_{n(\theta)} + \int_0^{\infty}
\Phi_{n(\theta)}(t) \ud \widetilde B_{t} \Bigr\|^p \ud
\theta + c_p \delta\\
&\stackrel{(iii)}\leq \frac {1}{2\pi}\int_{0}^{2\pi}
u(a_{n(\theta)}) + \varepsilon \ud \theta +
c_p \delta\\
&\stackrel{(iv)}\leq \frac {1}{2\pi}\int_{0}^{2\pi} u(x_0 + iy_0 +
xre^{i\theta}) \ud \theta + c_p \delta + 2\varepsilon,
\end{align*}
where $n(\theta)$ is such $n$ that $a_{n}$ is the closest to $x_0
+ iy_0 + xre^{i\theta}$ among $(a_n)_{n=1}^N$, $(i)$ follows from
the definition of $\Phi$, $(ii)$ holds by the triangle inequality
and the fact that $(a_n)_{n=1}^N$ is a $\delta$-net of $A$ (where
the constant $c_p$ depends only on $p$), $(iii)$ holds by
\eqref{eq:sharpguysforu(an)}, and $(iv)$ holds by
\eqref{eq:controfanandepsassump}. Now if $\varepsilon\to 0$,
$\delta$ vanishes as well, and \eqref{eq:diagplshofu} follows.
Let us now show that $u(x)\geq 0$ for any $x\in X$. First notice
that $u$ is concave in the complex variable, i.e.\ $y\mapsto u(x +
iy)$ is concave in $y\in X$ for any $x\in X$, which follows
directly form the construction of $u$ in \eqref{eq:defofdiagplsu}.
Now one can show that $u$ is convex in the real variable, i.e.\
$x\mapsto u(x + iy)$ is convex in $x\in X$ for any $y\in X$, by
using the same argument as was used for plurisubharmonic functions
in \cite[Subsection 2.6]{OY18}. Next notice that $u$ is symmetric,
i.e.\ $u(x+iy) = u(-x-iy)$ for any $x, y\in X$. Thus $x\mapsto
u(x)$ is a symmetric convex function with $u(0)=0$, so it is
nonnegative.
{\em $(ii) \Rightarrow (i)$.} Let $u: X +iX \to \mathbb R$ be a
function from $(ii)$. We need to show that for any standard
Brownian motions $W, \widetilde W:\mathbb R_+ \times \Omega \to
\mathbb R$ and for any elementary predictable with respect to the
filtration generated by both $W$ and $\widetilde W$ process
$\Phi:\mathbb R_+ \times \Omega \to X$ one has that
\begin{equation}\label{eq:proofthatchifollfromdiagpls}
\mathbb E \Bigl\| \int_0^{\infty} \Phi \ud \widetilde W \Bigr\|^p
\leq \chi^p \mathbb E \Bigl\| \int_0^{\infty} \Phi \ud W
\Bigr\|^p.
\end{equation}
Since $\Phi$ is elementary predictable, it takes values in a
finite-dimensional subspace of $X$, so we may assume that $X$ is
finite-dimensional. Then by Subsection \ref{subsec:funcapprox} we can assume
that $u$ is twice differentiable on $X+ iX$ by a simple
convolution-type argument. Let $d<\infty$ be the dimension of $X$,
$(x_n)_{n=1}^d$ be the basis of $X$, $(x_n^*)_{n=1}^d$ be the {\em
corresponding dual basis} of $X^*$, i.e.\ a unique basis such that
$\langle x_n, x_m^*\rangle = \delta_{nm}$ for any $n,m=1,\ldots,d$
(see e.g.\ \cite{OY18,Y17MartDec,Y17FourUMD}). Then by It\^o's
formula \cite[Theorem 3.8]{Y17MartDec} and due to local
boundedness and twice differentiability of $u$ we have that (here
we define $M_t:=\int_0^t \Phi \ud W$ and $N_t:=\int_0^t \Phi \ud
\widetilde W$ for the convenience of the reader)
\begin{align}\label{eq:proofthatchiusingito}
\chi^p \mathbb E \Bigl\| \int_0^{\infty} \Phi \ud W \Bigr\|^p
- \mathbb E \Bigl\| \int_0^{\infty} \Phi \ud \widetilde W
\Bigr\|^p &\geq \mathbb E u\Bigl( \int_0^{\infty} \Phi \ud W
+i \int_0^{\infty} \Phi \ud \widetilde W \Bigr)\nonumber\\
& = \mathbb E u(M_0 + iN_0) + \mathbb E \int_0^{\infty} \bigl \langle \partial_x u( M_{t-} + iN_t), \ud
M_t\bigr\rangle\\
&\quad + \mathbb E \int_0^{\infty} \bigl\langle \partial_{ix} u( M_{t-} + iN_t),\ud N_t\bigr\rangle + \frac 12\mathbb E I,\nonumber
\end{align}
where
\[
I = \mathbb E \int_0^{\infty} \sum_{n,m=1}^d \Bigl(
\tfrac{\partial^2 u( M_{t-} + iN_t)}{\partial x_n x_m} +
\tfrac{\partial^2 u( M_{t-} + iN_t)}{\partial ix_n
ix_m}\Bigr)
\langle \Phi, x_n^*\rangle \cdot \langle \Phi, x_m^*\rangle \ud
t.
\]
First notice that $\mathbb E u(M_0 + iN_0) = \mathbb E u(0) = 0$
and analogously to \cite[proof of Theorem 3.18]{Y17FourUMD} both $\partial_x u( M_{t-} + iN_t)$ and $\partial_{ix} u( M_{t-} + iN_t)$ are stochastically integrable with respect to $M$ and $N$ respectively, so
$$
\mathbb E \int_0^{\infty} \bigl \langle \partial_x u( M_{t-} + iN_t), \ud
M_t\bigr\rangle + \mathbb E \int_0^{\infty} \bigl\langle \partial_{ix} u( M_{t-} + iN_t),\ud N_t\bigr\rangle= 0,
$$
where the latter holds since both stochastic integrals are
martingales which start in zero. Let us show that $\mathbb E I \geq 0$. Fix
$t\geq 0$ and $\omega \in \Omega$. By \cite[Lemma 3.7]{Y17MartDec}
we are free to choose any basis (and the corresponding dual
basis). In particular, we can assume that $x_1 = \Phi(t, \omega)$.
Then $\langle \Phi(t, \omega), x_n^*\rangle = \delta_{1n}$ for any $1\leq n\leq d$, so
(here we skip $(t, \omega)$ for the convenience of the reader)
\begin{multline*}
\sum_{n,m=1}^d \Bigl( \tfrac{\partial^2 u( M_{t-} +
iN_t)}{\partial x_n x_m} +
\tfrac{\partial^2 u( M_{t-} + iN_t)}{\partial ix_n
ix_m}\Bigr)
\langle \Phi, x_n^*\rangle \cdot \langle \Phi, x_m^*\rangle\\
= \tfrac{\partial^2 u( M_{t-} +
iN_t)}{\partial x_1^2} +
\tfrac{\partial^2 u( M_{t-} + iN_t)}{\partial ix_1^2} = \Delta u(M_{t-} + iN_t + z x_1)\big|_{z=0} \geq 0,
\end{multline*}
where $z\in \mathbb C$, and the latter inequality follows from the
diagonal plurisubharmonicity of $u$. Thus $\mathbb E I \geq 0$, and hence
\eqref{eq:proofthatchifollfromdiagpls} follows from
\eqref{eq:proofthatchiusingito}.
\end{proof}
\begin{remark}\label{rem:defofUSO}
Note that the maximum of any set of harmonic functions is harmonic
as well, so the maximum of any set of diagonally plurisubharmonic functions is
diagonally plurisubharmonic as well, and thus for any Banach space
$X$ and for any $1<p<\infty$ with $\chi_{p, X}<\infty$ we can
define an {\em optimal} diagonal plurisubharmonic function
$U^{SO}:X + iX \to \mathbb R$ as a supremum of all functions $u$
satisfying the conditions of Proposition
\ref{prop:chi<inftyiffexistsdiagplsfunc}$(ii)$.
Note that $U^{SO}$ coincides with the function $u$ defined by \eqref{eq:defofdiagplsu}. Indeed, let $U^{SO}$ be as defined above, $u$ be as in \eqref{eq:defofdiagplsu}. Then $U^{SO} \geq u$ by the definition of $U^{SO}$. Let us show that $U^{SO}(x+iy) \leq u(x+iy)$ for any $x, y\in X$. First fix independent Brownian motions $W$ and $\widetilde W$ and elementary predictable $\Phi:\mathbb R_+ \times \Omega \to X$. Then similarly to the It\^o argument from the proof of Proposition \ref{prop:chi<inftyiffexistsdiagplsfunc} one has that
\[
U^{SO} (x+iy) \leq \mathbb E U\Bigl(x+iy + \int_0^{\infty} \Phi \ud W + i \int_0^{\infty} \Phi \ud \widetilde W\Bigr).
\]
Thus
\begin{align*}
U^{SO} (x+iy)& \leq \inf\Bigl\{\mathbb E U\Bigl(x+iy + \int_0^{\infty} \Phi \ud W + i \int_0^{\infty} \Phi \ud \widetilde W\Bigr):\\
&\quad\quad\Phi \;\text{elementary predictable}\Bigr\} \leq u(x+iy),
\end{align*}
which implies the desired.
\end{remark}
As a corollary of Proposition \ref{prop:chi<inftyiffexistsdiagplsfunc} one can show the following upper and lower bounds for $\chi_{p, X}$. Recall that we define {\em decoupling constants} $\beta_{p,X}^{\gamma ,+}$ and $\beta_{p,X}^{\gamma ,-}$ to be the smallest possible $\beta^+$ and $\beta^-$ respectively for which
\[
\frac{1}{(\beta^-)^p}\mathbb E\Bigl\|\int_0^\infty \Phi \ud W\Bigr\|^p \leq \mathbb E\Bigl\|\int_0^\infty \Phi \ud \widetilde W\Bigr\|^p \leq (\beta^+)^p \mathbb E\Bigl\|\int_0^\infty \Phi \ud W\Bigr\|^p,
\]
where $W$ and $\widetilde{W}$ are independent standard Brownian motion, $\Phi:\mathbb R_+ \times \Omega \to X$ is elementary predictable which is independent of $\widetilde{W}$ (we refer the reader to \cite{Gar85,HNVW1,Ver07,MC,Geiss99,CG,OY18} for further details on decoupling constants).
\begin{corollary}\label{cor:estimforchipX}
Let $X$ be a Banach space, $1<p<\infty$. Then $\chi_{p,X}<\infty$
if and only if $X$ is a UMD Banach space. Moreover, if this is the
case, then
\begin{equation}\label{eq:uppandlowboundforchipX}
\max\Bigl\{\sqrt{\beta_{p, X}}, \sqrt{\hbar_{p,X}}\Bigr\} \stackrel{(i)}\leq \max\{\beta_{p, X}^{\gamma,+}, \beta_{p, X}^{\gamma, -}\} \stackrel{(ii)}\leq \chi_{p, X} \stackrel{(iii)}\leq \min\{\beta_{p, X}, \hbar_{p,X}\}.
\end{equation}
\end{corollary}
\begin{proof}
First we show \eqref{eq:uppandlowboundforchipX}, and then the
``iff'' statement will follow simultaneously. Let first show
$(iii)$ in \eqref{eq:uppandlowboundforchipX}. The fact that
$\chi_{p, X} \leq \hbar_{p,X}$ follows from \cite{OY18}, the
definition of $\chi_{p,X}$, and the fact that any two stochastic
integrals $\int \Phi \ud W$ and $\int \Phi \ud \widetilde W$ are
orthogonal martingales weakly differentially subordinate to each
other. The inequality $\chi_{p, X} \leq\beta_{p, X}$ can be proven
using a standard Burkholder function argument e.g.\ presented in
\cite{Y17FourUMD,Y17MartDec}. Indeed, if $\beta_{p, X}<\infty$,
then $X$ is a UMD Banach space, and their exists a {\em
zigzag-concave} function $U:X\times X \to \mathbb R$ (i.e.\ $z
\mapsto U(x+z, y+\alpha z)$ is concave in $z\in X$ for any $x,
y\in X$ and $\alpha \in [-1,1]$) such that $U(0,0)=0$ and
\[
U(x, y) \geq \|y\|^p - \beta_{p, X}^p\|x\|^p,\;\;\; x,y\in X.
\]
(This function is called {\em Burkholder}.) By a standard
convolution-type argument (see Subsection \ref{subsec:funcapprox}) we may
assume that $U$ is twice differentiable, and hence for any
independent standard Brownian motions $W$ and $\widetilde W$ and
for any elementary predictable $\Phi:\mathbb R_+ \times \Omega \to
X$ by It\^o's formula \cite[Theorem 3.8]{Y17MartDec} we have that
analogously to \eqref{eq:proofthatchiusingito} with denoting $M :=
\int \Phi \ud W$ and $ N := \int \Phi \ud \widetilde W$
\begin{align*}
\mathbb E \Bigl\| \int_0^{\infty} \Phi \ud \widetilde W \Bigr\|^p
- \beta_{p,X}^p \mathbb E \Bigl\| \int_0^{\infty} \Phi \ud W
\Bigr\|^p &\leq U\Bigl( \int_0^{\infty} \Phi \ud W , \int_0^{\infty} \Phi \ud \widetilde W\Bigr)\\
& = \frac12\int_0^{\infty} \tfrac{\partial^2 U(M_t, N_t)}{\partial (\Phi, 0)^2} + \tfrac{\partial^2 U(M_t, N_t)}{\partial (0,\Phi)^2} \ud t \\
&= \frac14\int_0^{\infty} \tfrac{\partial^2 U(M_t, N_t)}{\partial
(\Phi, \Phi)^2} + \tfrac{\partial^2 U(M_t, N_t)}{\partial
(\Phi,-\Phi)^2} \ud t \leq 0,
\end{align*}
where the latter inequality holds due to the zigzag-concavity of $U$ (so both $\tfrac{\partial^2 U(x, y)}{\partial (z, z)^2}$ and $\tfrac{\partial^2 U(x, y)}{\partial (z, -z)^2}$ and nonnegative for any $x,y, z\in X$). Thus $\chi_{p, X} \leq\beta_{p, X}$ holds true.
Now $(ii)$ of \eqref{eq:uppandlowboundforchipX} follows directly from the definitions of $\chi_{p, X}$, $\beta_{p, X}^{\gamma,+}$, and $\beta_{p, X}^{\gamma,-}$, while $(i)$ holds by \cite[p.\ 43 and Theorem 3]{Gar85}.
\end{proof}
\begin{remark}\label{rem:-UandUHarediagplsh}
Note that due to the latter proof for a Burkholder function $U$
one has that $-U$ is diagonal plurisubharmonic. Thus the proof of
$(iii)$ of \eqref{eq:uppandlowboundforchipX} has the following
form: {\em both $-U$ and $U_{\mathcal H}$ are diagonally
plurisubharmonic and thus satisfy the conditions of Proposition
\ref{prop:chi<inftyiffexistsdiagplsfunc}$(ii)$}, so the upper bound $(iii)$ of \eqref{eq:uppandlowboundforchipX} holds true.
We wish to notice that in the real-valued case functions $U^{SO}$ and $U_{\mathcal H}$ coincide since in this case there is no difference between plurisubharmonicity and diagonal plurisubharmonicity. Nevertheless, if the same holds for a general UMD Banach space, then $\hbar_{p, X} = \chi_{p, X}\leq \beta_{p, X}$, which would partly solve an open problem outlined in the introduction.
\end{remark}
\section{Weak differential subordination\\ of strongly orthogonal martingales}\label{sec:WDSofSOM}
Now we are ready to show the main result of the paper.
\begin{theorem}\label{thm:WDSforSOM}
Let $X$ be a UMD Banach space, $1<p<\infty$. Then for any strongly
orthogonal martingales $M,N:\mathbb R_+ \times \Omega \to X$ with
$N\stackrel{w} {\ll} M$ one has that
\[
\mathbb E \|N_t\|^p \leq \chi_{p, X}^p \mathbb E \|M_t\|^p, \;\;\;
t\geq 0.
\]
\end{theorem}
\begin{proof}
By Subsection \ref{subsec:funcapprox} we may assume that $X$ is
finite dimensional and that all the Bellman functions are smooth.
Due to \eqref{eq:ubiggethenchipxp-yp} we only need to show that
\begin{equation}\label{eq:proofofmainthmBellmanuse}
\mathbb E U^{SO}(M_t + iN_t) \geq 0,
\end{equation}
where $U^{SO}$ is as in Remark \ref{rem:defofUSO}. Let
$d\geq 0$ be the dimension of $X$. Since $N\stackrel{w} {\ll} M$
and since $M$ and $N$ are orthogonal, by \cite[Section 3]{OY18} we know
that after a proper time-change there exist a standard
$2d$-dimensional Brownian motion $W$ and predictable $\Phi,
\Psi:\mathbb R_+ \times \Omega \to \mathcal L(\mathbb R^{2d}, X)$
which are stochastically integrable with respect to $W$ such that
$N = \int \Psi \ud W$ and $M = M_0 + \int \Phi \ud W + M^d$, where
$M^d$ is purely discontinuous (see Subsection \ref{subsec:qvandpdm}). Moreover, as $M$ and $N$ are
strongly orthogonal, we have that for any $x^*, y^*\in X^*$ and
$t\geq 0$ by \cite[Theorem 26.6 and 26.13]{Kal}
\[
[\langle M, x^* \rangle,\langle N, y^* \rangle]_t = \int_0^t \bigl \langle \Phi^*(s)x^*, \Psi^*(s)y^*\bigr \rangle \ud s = 0.
\]
Therefore by the Lebesgue differentiation theorem $\langle \Phi^*x^*, \Psi^*y^*\bigr \rangle = 0$ a.e.\ on $\mathbb R_+ \times \Omega$. By choosing $(x^*, y^*)$ from a dense subset of $X^* \times X^*$ and using the fact that $(x^*, y^*) \mapsto \langle \Phi^*x^*, \Psi^*y^*\bigr \rangle$ is continuous on $X^* \times X^*$ on the whole $\mathbb R_+ \times \Omega$, one has
\begin{equation}\label{eq:Phix*Psix*=0}
\langle \Phi^*x^*, \Psi^*y^*\bigr \rangle = 0,\;\;\; x^*, y^* \in X^*,
\end{equation}
a.e.\ on $\mathbb R_+ \times \Omega$. Furthermore, by
\cite[Section 3]{OY18} we have that a.s.\ for any $0\leq s\leq t$ there
exists a skew-symmetric operator $A(s, \omega)\in \mathcal
L(\mathbb R^d)$ (i.e.\ $\langle Ah , h\rangle = 0$ for any $h\in
\mathbb R^d$) of norm at most one such that
\begin{equation}\label{eq:existofA}
\Psi(s, \omega) = \Phi(s,\omega) A (s,\omega).
\end{equation}
Now let us show \eqref{eq:proofofmainthmBellmanuse} using \eqref{eq:Phix*Psix*=0}. Let $(x_n)_{n=1}^d$ be a basis of $X$, $(x_n^*)_{n=1}^d$ be the corresponding dual basis of $X^*$. By It\^o's formula \cite[Theorem 3.8]{Y17MartDec} and smoothness of $U^{SO}$ we have that
\begin{align*}
\mathbb E U^{SO}(M_t + iN_t) = \mathbb E U^{SO}(M_0 + iN_0) + \mathbb E I_1 + \mathbb E I_2 + \frac 12 \mathbb E I_3,
\end{align*}
where
$$
I_1 = \int_0^t \langle \partial U^{SO} (M_{s-} + iN_s), \ud M_s + i\ud N_s \rangle,
$$
$$
I_2 = \sum_{0\leq s \leq t} \Delta U^{SO} (M_s + iN_s) - \langle \partial U^{SO} (M_{s-} + iN_s), \Delta M_s \rangle,
$$
and
\begin{align*}
I_3 &= \int_0^t \sum_{n, m=1}^d\tfrac{\partial^2 U^{SO} (M_{s-} + iN_s)}{\partial x_nx_m} \langle \Phi^*x_n^*,\Phi^* x_m^* \rangle\ud t\\
&\quad + 2\int_0^t \sum_{n, m=1}^d\tfrac{\partial^2 U^{SO} (M_{s-} + iN_s)}{\partial x_n ix_m} \langle \Phi^*x_n^*,\Psi^* x_m^* \rangle\ud t\\
&\quad\quad + \int_0^t \sum_{n, m=1}^d\tfrac{\partial^2 U^{SO} (M_{s-} + iN_s)}{\partial ix_n ix_m} \langle \Psi^*x_n^*,\Psi^* x_m^* \rangle\ud t.
\end{align*}
First notice that since $N_0=0$ and since $U^{SO}(x) \geq 0$ for any $x\in X$ we have that $\mathbb E U^{SO}(M_0 + iN_0) = \mathbb E U^{SO}(M_0) \geq 0$. Moreover, $\mathbb E I_1 = 0$ since this is a martingale that starts at zero (which follows similarly to the proof of Proposition \ref{prop:chi<inftyiffexistsdiagplsfunc}). Let us show that $I_2 \geq 0$ a.s. Note that $x \mapsto U^{SO}(x + iy)$ is convex in $x\in X$ for any $y\in X$ by Proposition \ref{prop:chi<inftyiffexistsdiagplsfunc}, so by the continuity of $N$ we have that for any $0\leq s \leq t$
\begin{align*}
U^{SO} (M_s + iN_s) \leq U^{SO} (M_{s-} + iN_s) + \langle \partial U^{SO} (M_{s-} + iN_s), \Delta M_s \rangle,
\end{align*}
and thus $I_2 \geq 0$ a.s.
Now we show that $I_3 \geq 0$ a.s. In order
to show this we need to prove that a.s.\ for every $0\leq s\leq t$
\begin{equation}\label{I3pointwisegeq0}
\begin{split}
\sum_{n, m=1}^d&\tfrac{\partial^2 U^{SO} (M_{s-} + iN_s)}{\partial
x_nx_m} \langle \Phi^*x_n^*,\Phi^* x_m^* \rangle\\
& \quad + \tfrac{\partial^2 U^{SO} (M_{s-} + iN_s)}{\partial x_n
ix_m}
\langle \Phi^*x_n^*,\Psi^* x_m^* \rangle\\
&\quad \quad+ \tfrac{\partial^2
U^{SO} (M_{s-} + iN_s)}{\partial ix_n ix_m} \langle
\Psi^*x_n^*,\Psi^* x_m^* \rangle \geq 0
\end{split}
\end{equation}
Fix $\omega \in \Omega$ and $0\leq s\leq t$ so that
\eqref{eq:Phix*Psix*=0} and \eqref{eq:existofA} hold true. Then the
expression on the left-hand side of \eqref{I3pointwisegeq0} gets
the following form
\begin{equation}\label{eq:I3simplerform}
\sum_{n, m=1}^d\tfrac{\partial^2 U^{SO} (M_{s-} + iN_s)}{\partial
x_nx_m} \langle \Phi^*x_n^*,\Phi^* x_m^* \rangle +
\tfrac{\partial^2 U^{SO} (M_{s-} + iN_s)}{\partial ix_n ix_m}
\langle \Psi^*x_n^*,\Psi^* x_m^* \rangle.
\end{equation}
Now analogously to \cite[Section 3]{OY18} the expression
\eqref{eq:I3simplerform} does not depend on the choice of the
basis $(x_n)_{n=1}^d$ or, equivalently, the choice of the basis
$(x_n^*)_{n=1}^d$ (since one can reconstruct the basis by its
corresponding dual basis, see \cite{OY18,Y17MartDec}). Moreover, by
\eqref{eq:existofA} for two symmetric nonnegative bilinear forms
$V, W :X^*\times X^* \to \mathbb R$ defined~by
\[
V(x^*, y^*) := \langle \Phi^*x^*,\Phi^* y^* \rangle,\;\; W(x^*,
y^*) := \langle \Psi^*x^*,\Psi^* y^* \rangle,\;\;\;\;\; x^*,
y^*\in X^*,
\]
we have that $V(x^*, x^*) = 0$ implies $W(x^*, x^*) =0$ for any
$x^*\in X^*$. Thus by \cite[Section 3]{OY18} there exist a basis
$(y_n^*)_{n=1}^d$ of $X^*$ with the corresponding dual basis
$(y_n)_{n=1}^d$ of $X$, a $[0,1]$-valued sequence
$(\lambda_n)_{n=1}^d$, and a number $0 \leq K \leq d$ such that
$V(y_n^*, y_m^*) = \delta_{nm} \mathbf 1_{m,n \leq K}$ and
$W(y_n^*, y_m^*) = \lambda_n\delta_{nm} \mathbf 1_{m,n \leq K}$
for any $m,n=1,\ldots,d$. Therefore by the discussion above we can
change the basis and get that the expression
\eqref{eq:I3simplerform} equals
\begin{equation}\label{eq:I3withnewbasis}
\begin{split}
\sum_{n, m=1}^d\tfrac{\partial^2 U^{SO} (M_{s-} + iN_s)}{\partial
y_ny_m}& \langle \Phi^*y_n^*,\Phi^* y_m^* \rangle +
\tfrac{\partial^2 U^{SO} (M_{s-} + iN_s)}{\partial iy_n iy_m}
\langle \Psi^*y_n^*,\Psi^* y_m^* \rangle\\
&= \sum_{n=1}^K\tfrac{\partial^2 U^{SO} (M_{s-} + iN_s)}{\partial
y_n^2} + \lambda_n\tfrac{\partial^2 U^{SO} (M_{s-} +
iN_s)}{\partial iy_n^2}.
\end{split}
\end{equation}
Since $y\mapsto U^{SO}(x+iy)$ is concave in $y\in X$ for any $x\in
X$, $\tfrac{\partial^2 U^{SO} (M_{s-} + iN_s)}{\partial iy_n^2}
\leq 0$, and hence due to the fact that $ 0\leq \lambda_n \leq 1$
we have that the latter expression of \eqref{eq:I3withnewbasis} is
bounded from below by (here $z\in \mathbb C$)
\begin{align*}
\sum_{n=1}^K\tfrac{\partial^2 U^{SO} (M_{s-} + iN_s)}{\partial
y_n^2} + \tfrac{\partial^2 U^{SO} (M_{s-} + iN_s)}{\partial iy_n^2} = \sum_{n=1}^K \Delta_z U^{SO} (M_{s-} + iN_s + zy_n)|_{z=0}
\geq 0,
\end{align*}
where the latter holds by the diagonal plurisubharmonicity of
$U^{SO}$. Therefore \eqref{I3pointwisegeq0} holds a.e.\ on
$\mathbb R_+ \times \Omega$, and thus $\mathbb EI_3 \geq 0$. This
completes the proof of \eqref{eq:proofofmainthmBellmanuse} and the
proof of the theorem.
\end{proof}
\bibliographystyle{plain}
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| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,472 |
Posted on March 28, 2019 by Kerri LeVanseler, Ph.D.
What does it take to find forgiveness? We may need forgiveness for ourselves or struggle with whether to forgive when people have wronged us in a way that has put a great strain on our hearts and our lives. The situations are unique to each individual's journey, with God as our guide for the actions we should take. Our behaviors when facing a decision and the practice of forgiveness have a critical role in our relationships in both the divine and human realms.
Forgiveness is important in many religious traditions and it is critical for our spiritual health and peace of mind. There is a Tibetan Buddist story about two monks (some updated versions change it to two prisoners of war). They had been in prison together where they were tortured by their captors. Years later, they meet again and the first one asks, "Have you forgiven them?" The other replies, "I will never forgive them! Never!" Then the first says, "Well, I guess they still have you in prison, don't they?" Forgiveness is a way to liberate your feelings about the worst of life's events.
In the Jewish culture, before Jesus, the steps to repentance and to earn forgiveness were often of a physical nature: an animal sacrificed as a sin offering, fasting, ripping of clothes, wearing sack cloth, an action by the High Priest. The act of killing an animal could symbolize the punishment and that justice was being extracted through the blood of the animal that takes the place of the one who sinned. Forgiveness was transactional. A sin against God required a debt to be paid back to God. After a sin against another person, the transgressor became obligated to the person they offended.
When Christians confess and ask God, with a repentant heart, sins are forgiven (1 John 1:9). Biblical lessons have taught that this became possible through the life and death of Jesus. The ultimate sacrificial act of love for humankind was God's gift to his creation when we were overwhelmed by sin.
God's plan through Jesus made it possible to be forgiven for the wrongs we have committed and the shame we feel. It isn't a physical transaction. It is a process that takes place in the heart, a re-connection to God. There is a joining to correct the break in our relationship that was caused by our sin. The teaching of Jesus before his crucifixion spoke of our responsibility to forgive others (Matthew 6:14-15). Seventy times seven times if that's what it takes (Matthew 18:22). We are not stuck waiting on the actions of others. Forgive as you have been forgiven (Ephesians 4:32).
When forgiveness is an exchange, an apology and justice is needed for the offense. But does an "I'm sorry" or a verdict of consequences really result in healing for the one who has been hurt? Evil people can act with disregard or hatred toward others, creating situations where repentance and restitution may be an impossibility. In the Hindu Dharma, the feminine forgiveness is granted when there has not been repentance, "which is higher and more noble than the masculine forgiveness granted only after there is repentance." Does it really take two willing parties for acts of forgiveness to be effective? I hope not!
Granting forgiveness when our pain and loss has been great, doesn't mean you forget. Forgiving doesn't mean we accept the wrongdoing as okay and it doesn't absolve the perpetrator of blame. We do not have to lose our story. There are lessons in it. Our experiences add to our wisdom. Learning to let go of the bitterness and anger, we can embrace the fullness of life and what awaits in our future. We let go so that we are not drawn into continual plans for seeking revenge or the feeling of being stuck as a victim without hope for the days ahead. When we forgive, we heal. Withholding forgiveness is like putting our peace of mind directly into the hands of someone who has already been shown to be untrustworthy. Isn't our peace more precious to us that we should take the actions necessary to remain in control of it?
For Muslims, forgiving means not carrying a grudge in your heart against that person. If you were given a chance to retaliate, you would choose not to and that you do not wish the other person evil, even secretly. If there is an inability to forgive, it may cause the person to repeatedly bring anger and bitterness into every relationship and new experience. If you cannot move on from the hurt, it is projected onto others. When praying for your own forgiveness to Allah, it is recommended that you use His names including "He who pardons," "He who forgives," and "The oft-forgiving" to reflect the attributes of God that apply.
To find forgiveness, pray about the situation. If you have offended someone, ask God to give you courage and humility to do what it takes to mend the relationship. If you've been wronged, don't sit and focus on your wounds and nurse hate for the person who hurt you. Pray for the persistence to get past the obstacles blocking your way forward. Pray for the person who offended you. This can be a very hard thing to do, but the deeper the pain they have caused is a sign of how far they have strayed from the path God had in mind for them. Pray that God's power will influence their future actions. Pray for God to give the grace to forgive them, "they don't know what they they've been doing" (Luke 23:34).
The acts of forgiveness takes a lot of work. Let us be able to show God that we are able to take the high road and that we trust God to know how justice is best served. In doing so, with help from God, we will be set free to experience peace and spiritual restoration that is beyond understanding. For that, we can give praise and thankfulness back to our big God.
Who Could Argue With Love?
Our All-Powerful, Loving God sends his only begotten son to be born of a virgin as a gift to mankind.
Even as a child Jesus loved to talk about God, amazing people with his understanding and his answers.
Parents brought their little children to Jesus so He could touch and bless them.
Jesus talks and socializes with people society judges and rejects. He befriends sinners.
Jesus taught that we shouldn't seek revenge or be a hypocrite, we should love our enemies, forgive and show mercy.
God shows how much he loved us by having Christ die for us, even though we were sinners.
God sent Jesus as a Prince of Peace, to show in the flesh how we are meant to love.
But through worldly ways, that type of love was rejected. Then and now, there are people who are driven by hate and violence. They argue and hurt others instead of showing love.
But in God's Kingdom — Love wins !!!! | {
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Once you have a Google account, raise a support ticket and we will add you to your University's Talis Aspire Reading Lists Google Analytics account.
You will receive notification that the account is linked and you are ready to start using.
Once you've received notification that your account is linked, you can access your Google Analytics by logging in at https://analytics.google.com. | {
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Research shows that increased interaction between parents and teachers/school results in higher achievement, and decreased behavioral challenges.
The Dallas Day School early childhood education center has a very active Parent Teacher Organization (PTO) that was formed to provide a communication forum for parents, families and staff. There are many opportunities to volunteer, whatever your talents, interests or time constraints. The PTO hosts general meetings and teacher appreciation lunches, organizes fundraisers and provides regular communication to parents and families. We encourage all parents to become an active participant in the Dallas Day School PTO.
Whether you have an hour a week or an hour a month, we have opportunities for you to participate and contribute to Dallas Day School. Here is a sample of many jobs – large and small.
To learn more about the Parent Teacher Organization or to get involved fill out the form below or email ddspto@dallasservices.org. | {
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\section{Introduction}
\label{sec:intro}
Graph Neural Networks (GNNs) are a widely used class of machine learning models. Since graphs occur naturally in several domains such as chemistry, biology, and medicine, GNNs have experienced widespread adoption.
Following a trend towards building more interpretable machine learning models, there have been numerous recent proposals to provide explanations for GNNs.
Most of the existing approaches provide post-hoc explanations starting from an already trained GNN to identify edges and node attributes that explain the model's prediction. However, as highlighted in \citet{faber2021comparing}, there might be some discrepancy between the ground-truth explanations and those attributed to the trained GNNs. Indeed, post-hoc explanations are often not able to faithfully represent the mechanisms of the original model \cite{rudin2018stop}. Unfortunately, the very definition of what constitutes a faithful explanation is still open to debate and there exist several competing positions on the matter. Recent work has also shown that post-hoc attribution methods are often not better than random baselines on the standard evaluation metrics for explanation accuracy and faithfulness~\cite{agarwal2022probing}.
A notable exception to a large number of post-hoc XAI methods is \textsc{ProtGNN}~\cite{zhang2021protgnn} which uses prototype-based learning.
A recently proposed alternative to post-hoc methods is the learning to explain (L2X) paradigm~\cite{chen2018learning}. The core difference to post-hoc methods is that the models are trained to, in the forward pass, discretely select a small subset of the input features as well as the parameters of a downstream model that uses only the selected features to make a prediction. The selected features are, therefore, faithful by design as they are the only ones used by the downstream model. Since the subset of features is sampled discretely, L2X requires a method for computing gradients of an expectation over a discrete probability distribution. \cite{chen2018learning} proposed a gradient estimator based on a relaxation of the discrete samples and tailored to the $k$-subset distribution.
With this work, we bring the L2X paradigm to graph representation learning. The important ingredient is a recently proposed method for computing gradients of an expectation over a complex exponential family distribution~\cite{niepert21imle}. The method facilitates approximate gradient backpropagation for models combining continuously differentiable GNNs with a black-box solver of combinatorial problems defined on graphs. Crucially, this allows us to learn to sample subgraphs with beneficial properties such as being connected and sparse. Contrary to prior work, this also creates a dependency between the random variables representing the presence of edges. The proposed framework \textsc{L2xGnn}\@\xspace, therefore, learns to select explanatory subgraph motifs and uses \emph{these and only these motifs} for its message-passing operations.
To the best of our knowledge, this is the first method for learning to explain GNNs.
The proposed framework is extensible as it can work with any optimization algorithm for graphs imposing properties on the sampled subgraphs.
We compare two different sampling strategies for obtaining sparse subgraph explanations resulting from two optimization problems on graphs: (1) the maximum-weight $k$-edge subgraph and (2) the maximum-weight $k$-edge connected subgraph problem.
We show empirically that \textsc{L2xGnn}\@\xspace when combined with a base GNN does not lose accuracy on several benchmark datasets. Moreover, we evaluate the explanations quantitatively and qualitatively. We also analyze the ability of \textsc{L2xGnn}\@\xspace to help in detecting shortcut learning which can be used for debugging the GNN.
Given the characteristics of the proposed method, our work improves model interpretability and increases the clarity of known black-box models as GNNs while maintaining competitive predictive capabilities.
\clearpage
\begin{figure*}[t!]
\centering
\includegraphics[width=0.7\textwidth]{figures/new/IMLE_framework_v3.pdf}
\caption{Workflow of the proposed approach.}
\label{fig:imle_workflow}
\end{figure*}
\section{Background}
\label{sec:background}
Let $\mathcal{G}(V, E)$ be a graph with $n=|V|$ the number of nodes. Let $\mathbf{X} \in \mathbb{R}^{n \times d}$ be the feature matrix that associates each node of the graph with a $d$-dimensional feature vector and let $\mathbf{A} \in \mathbb{R}^{n \times n}$ be the adjacency matrix. GNNs have three computations based on the message passing paradigm \cite{hamilton2017inductive} which is defined as
\begin{equation}
\label{eq:mpnn}
\mathbf{h}_i^{\ell} = \gamma \left(\mathbf{h}_i^{\ell-1},
\square_{j \in \mathcal{N}(v_i)} \phi \left(\mathbf{h}_i^{\ell-1}, \mathbf{h}_j^{\ell-1}, r_{ij} \right)\right),
\end{equation}
where $\gamma$, $\square$, and $\phi$ represent update, aggregation and message function respectively. \\
\textbf{Propagation step.} The message-passing network computes a message $m_{ij}^{\ell} = \phi(\mathbf{h}_i^{\ell-1}, \mathbf{h}_j^{\ell-1}, r_{ij})$ between every pair of nodes $(v_i, v_j)$. The function takes in input $v_i$'s and $v_j$'s representations $\mathbf{h}_i^{\ell-1}$ and $\mathbf{h}_j^{\ell-1}$ at the previous layer $\ell - 1$, and the relation $r_{ij}$ between the two nodes. \\
\textbf{Aggregation step.} For each node in the graph, the network performs an aggregation computation over the messages from $v_i$'s neighborhood $\mathcal{N}(v_i)$ to calculate an aggregated message $M_i^\ell = \square(\{m_{ij}^\ell \mid v_j \in \mathcal{N}(v_i)\})$. The definition of the aggregation function differs between methods \cite{hamilton2017inductive,velivckovic2018graph,xu2018powerful,duval2021graphsvx}. \\
\textbf{Update step.} Finally, the model non-linearly transforms the aggregated message $M_i^\ell$ and $v_i$'s representation from previous layer $\mathbf{h}_i^{\ell-1}$ to obtain $v_i$'s representation at layer $\ell$ as $\mathbf{h}_i^{\ell} = \gamma(M_i^\ell,\mathbf{h}_i^{\ell-1})$.
The final embedding for node $v_i$ after $L$ layers is $\mathbf{z}_i = \mathbf{h}_i^L$ and is used for node classification tasks. For graph classification, an additional readout function aggregates the node representations to obtain a graph representation $\mathbf{h}_G$. This function can be any permutation invariant function or a graph-level pooling function \cite{ying2018hierarchical,zhang2018end,lee2019self}. For Graph Isomorphism Networks (GINs) \cite{xu2018powerful}, for instance, the message passing operation for node $v_i$ is
\begin{equation}
\label{eq:gin_mp}
\mathbf{h}_i^{\ell} = \gamma^{\ell} \left(
\left(1 + \epsilon^{\ell} \right) \cdot \mathbf{h}_i^{\ell} +
\sum_{j \in \mathcal{N}(v_i)} \mathbf{h}_i^{\ell-1} \right),
\end{equation}
where $\gamma$ represents a multi-layer perceptron (MLP), and $\epsilon$ denotes a learnable parameter. We will write $\mathbf{H}_\ell = \textsc{Gnn}_\ell (\bm{A}, \mathbf{H}_{\ell-1})$ as a shorthand for the application of the $\ell^{\mbox{th}}$ layer of the GNN under consideration.
\section{Related Work}
There are several methods to explain the behavior of GNNs. Following \citet{yuan2020explainability}, explanatory methods for GNNs can be divided into several categories.
\noindent
\textbf{Gradient-based methods.} \cite{pope2019gradcam, baldassarre2019explainability, sanchez2020evaluating}. The main idea is to compute the gradients of the target prediction with respect to the corresponding input data. The larger the gradient values, the higher the importance of the input features.
\noindent
\textbf{Perturbation-based methods.} \cite{ying2019gnnexplainer, luo2020pgexplainer, funke2021zorro, loveland2021reliable, schlichtkrull2021interpreting, yuan2021subgraphx, perotti2022graphshap}. Here the objective is to study the models' output behavior under input perturbations. When the input is perturbed and we obtain an output comparable to the original one, we can conclude that the perturbed input information is not important for the current input. Inspired by causal inference methods, \cite{lin2021generative,lin2022orphicx,lucic2022cf,tan2022learning} attempt to provide explanations based on factual and counterfactual reasoning.
\noindent
\textbf{Surrogate methods.} \cite{huang2020graphlime, vu2020pgm, duval2021graphsvx, zhang2021relex, gui2022flowx}. First, these approaches generate a local dataset comprised of data points in the neighboring area of the input. The local dataset is assumed to be less complex and, consequently, can be analyzed through a simpler model. Then, a simple and interpretable surrogate model is used to capture local relationships that are used as explanations for the predictions of the original model.
\noindent
\textbf{Decomposition methods.} \cite{schwarzenberg2019layerwise, hu2020gcnlrp, schnake2020gnnlrp, feng2022degree}. These methods use decomposition rules to decompose the model predictions leading back to the input space. The prediction is considered as the target score. Then, starting from the output layer, the target score is decomposed at each preceding layer according to the decided decomposition rules. In this way, the initial target score is distributed among the neurons at every layer. Finally, the decomposed terms obtained at the input layer are associated to the input features and used as importance scores of the corresponding nodes and edges.
\noindent
\textbf{Model-level methods.} \cite{yuan2020xgnn}. Different from the instance-level methods above, these methods provide a general and high-level understanding of the models. In the context of GNNs, they aim at studying the input patterns that would lead to a certain target prediction. The generated explanations are general and provide a global understanding of the trained GNNs.
\noindent
\textbf{Prototype-based methods.} \cite{zhang2021protgnn} propose ProtGNN, a new explanatory method based on prototypes to provide \textit{built-in} explanations, overcoming the limitations of post-hoc techniques. The explanations are obtained following case-based reasoning, where new instances are compared with several learned \emph{prototypes}.
\noindent
\textbf{Concept-based methods.} \cite{magister2021gcexplainer} propose CGExplainer, a post-hoc explanatory methods for human-in-the-loop concept discovery. This concept representation learning method extracts concept-based explanations that allow the end-user to analyze predictions with a global view.
Additional works face the explainability problem from different perspectives as explanation supervision \cite{gao2021gnes} and motif-based generation \cite{yu2022motifexplainer}.
For a comprehensive discussion on methods to explain GNNs, we refer the reader to the survey \cite{yuan2020explainability}. Moreover, there have been related methods for learning the structure of graph neural networks~\cite{franceschi2019learning} and to sample subgraphs for subgraph aggregation methods~\cite{qian2022ordered} in a data-driven manner. These methods, however, are not concerned with the problem of explaining the behavior of graph neural networks.
\subsection{Limitations of Prior Work}
Existing XAI methods for GNNs have several limitations and can lead to inconsistencies~\cite{duval2021graphsvx,faber2021comparing}. We focus on the problem of identifying a subset of the edges as an explanation of the model's message-passing behavior. Hence, an explanation is equivalent to identifying a mask for the adjacency matrix of the original graph. Intuitively, an explanation can be \emph{accurate} and/or \emph{faithful}. It is accurate if it succeeds in identifying the edges in the \emph{input graph} responsible for the graph's class label. This property can, for example, be evaluated with synthetic data where the class label of a graph is determined by the presence or absence of a particular substructure. An explanation is faithful if the edges identified as the explanation \emph{cause} the prediction of the GNN on an input graph. Contrary to measuring accuracy, there is no consensus on evaluating faithfulness.
Recent work has proposed to measure unfaithfulness as the difference between the predictions of (1) the GNN on a perturbed adjacency matrix and (2) the GNN on the same perturbed adjacency matrix with edges removed by the explanation mask~\cite{pope2019gradcam,agarwal2022probing,agarwal2022explainable}.
We believe that this definition is problematic as the perturbation is typically implemented using a swap operation which replaces two existing edges $(a, b)$ and $(c, d)$ with two \emph{new} edges $(a, c)$ and $(b, d)$. Hence, these new edges are present in the unmasked adjacency matrix but not present in the masked one. It is, however, natural that the same GNN would predict highly different label distributions on these two graphs. For instance, consider a chemical compound where we remove and add new bonds. The resulting compounds and their properties can be chemically very different. Hence, contrary to prior work, we define a subgraph to be a faithful explanation, if it is a significantly smaller subgraph of the input graph and we know that \emph{only its structure} is used in the message-passing operations of equation~(\ref{eq:mpnn}).
\section{Learning to Explain Graph Neural Networks}
We propose a method that learns both (i) the parameters of a graph generative model and (ii) the parameters of a GNN operating on sparse subgraphs approximately sampled from said generative model in the forward pass. In line with prior work on learning to explain~\cite{chen2018learning}, the maximum probability subgraph is then used at test time to make the prediction and, therefore, serves as the faithful explanation. Since we aim to sample graphs with certain properties (e.g., connected subgraphs) we need a new approach to sampling and gradient estimation.
Contrary to prior work on edge masking \cite{schlichtkrull2021interpreting} which treats edges as independent binary random variables, we use a recently introduced method for backpropagating through optimization algorithms. This allows us to select subgraphs with specific properties and, therefore, to explicitly model dependencies between edge variables.
\subsection{Problem Statement and Framework}
We aim to jointly learn the parameters of a probability distribution over subgraphs \emph{with certain properties} and the parameters of a GNN operating on graphs sampled from said distribution in the context of the graph classification problem.
Here, the training data consists of a set of triples $\{(\mathbf{A}, \mathbf{X}, \mathbf{y})_j\}, j \in \{1, ..., N\}$, where $\mathbf{A}$ is an $n \times n$ binary adjacency matrix, $\mathbf{X} \in \mathbf{R}^{n \times d}$ a node attribute matrix with $d$ the number of node attributes, and $\mathbf{y}$ the target graph label. First, we have a learnable function $h_{\bm{v}}: \mathcal{A} \times \mathcal{X} \rightarrow \Theta$ where $\mathcal{A}$ is the set of all $n \times n$ adjacency matrices, $\mathcal{X}$ the set of all attribute matrices, $\bm{v}$ are the parameters of $h$, and $\Theta$ the set of possible edge parameter values. The function, which we refer to as the upstream model, maps the adjacency and attribute matrix to a matrix of edge weights $\bm{\theta} \in \mathbb{R}^{n \times x}$. Intuitively, $\bm{\theta}_{i,j}$ is the prior probability of edge $(i, j)$.
Next, we assume an algorithm $\mathtt{opt}: \Theta \rightarrow \mathcal{A}$ which returns the (approximate) solutions to an optimization problem on edge-weighted graphs. Examples of such optimization problems are the maximum-weight spanning tree or the maximum-weight $k$-edge connected subgraph problems. The optimization algorithm is treated as a black box. One can choose the optimization problem according to the application's requirements. We have found, for instance, that the connected subgraphs lead to better explanations in the domain of chemical compound classification. Contrary to prior work, the optimization problem creates a dependency between the binary variables modeling the edges.
For every binary adjacency matrix $\bm{Z} \in \mathcal{A}$, we write $\bm{Z} \in \mathcal{F}$ if and only if the adjacency matrix is a feasible solution (not necessarily an optimal one) of the chosen optimization problem.
We can now define a discrete exponential family distribution as
\begin{equation} \label{def-constrained-exp-family}
p(\bm{Z}; \bm{\theta}) = \left\lbrace
\begin{array}{ll}
\exp\left( \langle \bm{Z}, \bm{\theta} \rangle_{F} - B(\bm{\theta}) \right) & \text{if } \bm{Z} \in \mathcal{F}, \\
0 & \text{otherwise.}
\end{array}
\right.
\end{equation}
where $\langle \cdot, \cdot \rangle_{F}$ is the Frobenius inner product and $B(\bm{\theta})$ is the log-partition function defined as
$$B(\bm{\theta}) = \log\left(\sum_{\bm{A} \in \mathcal{F}} \exp \left( \langle \bm{Z}, \bm{\theta} \rangle_{F} \right)\right).$$
Hence, $p$ is a probability distribution over adjacency matrices that are feasible solutions to the optimization problem under consideration. Each feasible adjacency matrix's probability mass is proportional to the sum of its edge weights. For example, if the optimization problem is the maximum-weight $k$-edge connected subgraph problem, the distribution assigns a non-zero probability mass to all adjacency matrices of graphs that have $k$ edges and are connected.
Given an optimization problem, we would like to sample exactly from the above probability distribution $p(\bm{Z}; \bm{\theta})$. Unfortunately, this is intractable since computing the log-partition function is in general NP-hard. However, as in prior work~\cite{niepert21imle}, we can use perturb-and-MAP~\cite{Papandreou:2011} to \emph{approximately} sample from the above distribution as follows.
Let $\bm{\epsilon} \sim \rho(\bm{\epsilon})$ be a $n \times n$ matrix of appropriate random variables such as those following the Gumbel distribution. We can then \emph{approximately} sample an adjacency matrix $\bm{Z}$ from $p(\bm{Z}; \bm{\theta})$ by computing
$$\bm{Z} = \mathtt{opt}(\bm{\theta} + \bm{\epsilon}).$$
Hence, we can approximately sample by perturbing the edge weights (unnormalized probabilities) $\bm{\theta}$ and by applying the optimization algorithm to these perturbed weights.
In the final part of the model (the downstream model) we use the sampled $\bm{Z}$ as the input adjacency matrix to a message-passing neural network $f_{\bm{u}}: \mathcal{A} \times \mathcal{X} \rightarrow \mathcal{Y}$ computing $\hat{\bm{y}} = f_{\bm{u}}(\bm{Z}, \bm{X})$.
In summary, we have the following model architecture for training input data $(\bm{A}, \bm{X}, \bm{y})$:
\begin{align}
\label{eq:model_input}
\bm{\theta} & = h_{\bm{v}}(\bm{A}, \bm{X}) & \mbox{ with } \ \bm{A} \in \mathcal{A}, \bm{X} \in \mathcal{X}, \\
\label{eq:model_opt}
\bm{Z} & = \mathtt{opt}(\bm{\theta} + \bm{\epsilon}) & \mbox{ with } \ \bm{\epsilon} \sim \rho(\epsilon), \bm{\epsilon} \in \mathbb{R}^{n \times n}, \\
\label{eq:model_output}
\hat{\bm{y}} & = f_{\bm{u}}(\bm{Z}, \bm{X}) & \mbox{ with } \ \hat{\bm{y}} \in \mathcal{Y}, f_{\bm{u}}: \mathcal{A} \times \mathcal{X} \rightarrow \mathcal{Y}.
\end{align}
Figure~\ref{fig:imle_workflow} illustrates the architecture.
With $\bm{\omega} = (\bm{u},\bm{v})$ the learnable parameters of the model and the target variable $\bm{y}$ the loss is now defined as:
\begin{equation}
\label{eq:loss}
L(\bm{A}, \bm{X}, \bm{y};\bm{\omega}) = \mathbb{E}_{\bm{\epsilon} \sim \rho(\epsilon)}[\ell(f_{\bm{u}}(\bm{Z}, \bm{X}),\bm{y})],
\end{equation}
with $\bm{Z} = \mathtt{opt}(\bm{\theta} + \bm{\epsilon}) $, $\bm{\theta}=h_{\bm{v}}(\bm{A},\bm{X})$, and $\ell: \mathcal{Y} \times \mathcal{Y} \rightarrow \mathbb{R}^{+}$ a point-wise loss function.
The gradient of $L$ wrt $\bm{u}$ is
\begin{equation*}
\label{eq:gradient_u}
\nabla_{\bm{u}} L(\bm{A}, \bm{X}, \bm{y};\bm{\omega}) =
\mathbb{E}_{} [\partial_{\bm{u}}f_{\bm{u}}(\bm{Z}, \bm{X})^{\intercal}
\nabla_{\bm{y}} \ell (\hat{\bm{y}},\bm{y})]
\end{equation*}
which can be estimated by Monte-Carlo sampling. In contrast, the gradient of $L$ with respect to $\bm{v}$ is:
\begin{equation*}
\label{eq:gradient_v}
\nabla_{\bm{v}} L(\bm{A}, \bm{X}, \bm{y};\bm{\omega}) =
\partial_{\bm{v}}h_{\bm{v}}(\bm{A}, \bm{X})^{\intercal}
\nabla_{\bm{\theta}} L(\bm{A}, \bm{X}, \bm{y};\bm{\omega}),
\end{equation*}
where the challenge is to estimate $\nabla_{\bm{\theta}} L(\bm{A}, \bm{X}, \bm{y};\bm{\omega}) = \nabla_{\bm{\theta}}\mathbb{E}_{\bm{\epsilon} \sim \rho(\epsilon)}[\ell(f_{\bm{u}}(\bm{Z}, \bm{X}),\bm{y})]$ because $\bm{Z} = \mathtt{opt}(\bm{\theta} + \bm{\epsilon})$ is not continuously differentiable wrt $\bm{\theta}$. While it would be possible to use the score function estimator, its high variance typically makes it less competitive in practice~\cite{niepert21imle}.
\subsection{Implicit Maximum-Likelihood Learning}
The variant of \textsc{I-MLE}\@\xspace we use in this work estimates $\nabla_{\bm{\theta}} L(\bm{A}, \bm{X}, \bm{y};\bm{\omega})$ by implicitly creating a target distribution $q(\bm{Z}; \bm{\theta}')$ using perturbation-based implicit differentiation~\cite{domke2010implicit}. Here, the parameters $\bm{\theta}$ are moved in the direction of $-\nabla_{\bm{Z}} \ell (f_{\bm{u}}(\bm{A}, \bm{X}),\bm{y}))$, the negative gradient of the downstream loss with respect to the sampled adjacency matrix $\bm{Z}$, to construct $\bm{\theta}'$
\begin{equation}
\label{eq:target_distribution}
q(\bm{Z};\bm{\theta}^{\prime}) :=
p(\bm{Z};\bm{\theta} - \lambda \nabla_{\bm{Z}} \ell (f_{\bm{u}}(\bm{Z}, \bm{X}),\bm{y}))
\end{equation}
with $\bm{Z} = \mathtt{opt}(\bm{\theta} + \bm{\epsilon})$ and $\lambda > 0$ the strength of the perturbation. Intuitively, by moving the weights $\bm{\theta}$ into the direction of the negative gradients of $\bm{Z}$, the resulting distribution $q$ is more likely to generate samples with a lower downstream loss.
We approximate $\nabla_{\bm{\theta}} L(\bm{A}, \bm{X}, \bm{y};\bm{\omega})$ with a single-sample Monte Carlo estimate of the gradients of the KL divergence between $p$ and $q$:
\begin{equation}
\label{eq:approx_gradient}
\nabla_{\bm{\theta}} L(\bm{A}, \bm{X}, \bm{y};\bm{\omega}) \approx
\frac{1}{\lambda} \left(\mathtt{opt}(\bm{\theta} + \bm{\epsilon}) -
\mathtt{opt}(\bm{\theta}^{\prime} + \bm{\epsilon})\right).
\end{equation}
In other words, $\nabla_{\bm{\theta}} L(\bm{A}, \bm{X}, \bm{y};\bm{\omega})$ is approximated by the difference between an approximate sample from $p(\bm{Z};\bm{\theta})$ and an approximate sample from $q(\bm{Z};\bm{\theta}^{\prime})$. In this way we move the distribution $p(\bm{Z};\bm{\theta})$ closer to $q(\bm{Z};\bm{\theta}^{\prime})$. For further details on \textsc{I-MLE}\@\xspace we refer the reader to the original paper~\cite{niepert21imle}.
\begin{algorithm}[t]
\caption{Greedy algorithm $\mathtt{opt}_{\mathtt{con}}$ for the maximum-weight $k$-edge connected subgraph problem.}
\label{alg:cap}
\begin{algorithmic}
\Inputs{Input graph $\mathcal{G} = (V, E)$ \\
Number of edges $k$ \\ Edge weights $\bm{\theta}$ \\
}
\Initialize{
$e = \argmax_{e_{i,j}} \bm{\theta}_{i,j}$ \\
Set of selected edges $S = \{ e \}$ \\
Set of edges adjacent to selected edge $N = \mathcal{N}(e)$
}
\While{$|S| < k$ and $|N| > 0$}
\State $e = \argmax_{e_{i,j} \in N} \bm{\theta}_{i,j}$
\State $S = S \cup \{e\}$
\State $N = N \cup \mathcal{N}(e)$
\State $N = N - S$
\EndWhile
\State \textbf{Return:} Adjacency matrix $\bm{Z}$ of the subgraph induced by the set of selected edges $S$.
\end{algorithmic}
\end{algorithm}
\subsection{\textsc{L2xGnn}\@\xspace: Learning to Explain GNNs with \textsc{I-MLE}\@\xspace}
We now describe the class of \textsc{L2xGnn}\@\xspace models we use in the experiments. First, we need to define the function $h_{\bm{v}}(\bm{A}, \bm{X})$. Here we use a standard GNN (see equation~\ref{eq:mpnn}) to compute for every node $i$ and every layer $\ell$ the vector representation $\mathbf{h}_i^{\ell} = h_{\bm{v}}(\bm{A}, \bm{X})_{i,1:d}$.
We then compute the matrix of edge weights by taking the inner product between each pair of node embeddings. More formally, we compute $\bm{\theta}_{i,j} = \langle \mathbf{h}^{\ell}_i, \mathbf{h}_j^{\ell} \rangle$ for some fixed $\ell$. Typically, we choose $\ell=1$.
In this work, we sample the noise perturbations $\bm{\epsilon}$ from the sum of Gamma distribution~\cite{niepert21imle}.
Other noise distributions such as the Gumbel distribution are possible.
\begin{table*}[ht]
\caption{Prediction test accuracy for graph classification tasks. Standard deviations are over ten runs.}
\label{tab:accuracy}
\begin{center}
\begin{tabular}{lcccccc}
\toprule
\multicolumn{1}{c}{\multirow{2}{*}{Dataset}} & \multicolumn{3}{c}{GCN} & \multicolumn{3}{c}{GIN} \\ \cmidrule(r){2-4}\cmidrule(r){5-7}
\multicolumn{1}{c}{} & Original & \textsc{L2xGnn}\@\xspace $_{w/o_{con}}$ & \textsc{L2xGnn}\@\xspace & Original & \textsc{L2xGnn}\@\xspace $_{w/o_{con}}$ & \textsc{L2xGnn}\@\xspace \\
\midrule
DD & \textbf{0.720} \footnotesize$\pm$ 0.024 & 0.719 \footnotesize$\pm$ 0.031 & 0.719 \footnotesize$\pm$ 0.036
& 0.722 \footnotesize$\pm$ 0.027 & \textbf{0.739} \footnotesize$\pm$ 0.051 & 0.720 \footnotesize$\pm$ 0.030 \\
MUTAG & 0.734 \footnotesize$\pm$ 0.083 & 0.739 \footnotesize$\pm$ 0.111 & \textbf{0.745} \footnotesize$\pm$ 0.082
& \textbf{0.827} \footnotesize$\pm$ 0.051 & 0.814 \footnotesize$\pm$ 0.092 & 0.825 \footnotesize$\pm$ 0.078 \\
IMDB-B & 0.731 \footnotesize$\pm$ 0.032 & 0.660 \footnotesize$\pm$ 0.054 & \textbf{0.734} \footnotesize$\pm$ 0.047
& 0.721 \footnotesize$\pm$ 0.050 & 0.650 \footnotesize$\pm$ 0.050 & \textbf{0.724} \footnotesize$\pm$ 0.045 \\
IMDB-M & 0.500 \footnotesize$\pm$ 0.028 & \textbf{0.503} \footnotesize$\pm$ 0.032 & 0.490 \footnotesize$\pm$ 0.022
& \textbf{0.490} \footnotesize$\pm$ 0.047 & 0.488 \footnotesize$\pm$ 0.032 & 0.479 \footnotesize$\pm$ 0.035 \\
PROTEINS & 0.718 \footnotesize$\pm$ 0.044 & 0.711 \footnotesize$\pm$ 0.034 & \textbf{0.720} \footnotesize$\pm$ 0.053
& 0.708 \footnotesize$\pm$ 0.045 & 0.685 \footnotesize$\pm$ 0.029 & \textbf{0.709} \footnotesize$\pm$ 0.034 \\
YEAST & 0.881 \footnotesize$\pm$ 0.001 & \textbf{0.882} \footnotesize$\pm$ 0.002 & 0.881 \footnotesize$\pm$ 0.001
& \textbf{0.883} \footnotesize$\pm$ 0.001 & 0.882 \footnotesize$\pm$ 0.001 & 0.880 \footnotesize$\pm$ 0.002 \\
\bottomrule
\end{tabular}
\end{center}
\end{table*}
\subsubsection{Sampling constrained subgraphs.}
An advantage of the proposed method is its ability to integrate any graph optimization problem as long as there exists an algorithm $\mathtt{opt}$ for computing (approximate) solutions. In this work, we focus on two optimization problems: (1) The maximum-weight $k$-edge subgraph and (2) the maximum-weight $k$-edge connected subgraph problems. The former aims to find a maximum-weight subgraph with $k$ edges. The latter aims to find a \emph{connected} maximum-weight subgraph with $k$ edges. Other optimization problems are possible but we found that sparse and connected subgraphs provide a good efficiency-effectiveness trade-off.
Computing maximum weight $k$-edge subgraphs is highly efficient as we only need to select the $k$ edges with the maximum weights.
In order to compute \emph{connected} $k$-edge subgraphs we use a greedy approach. First, given a number $k$ of edges, we select a single edge $e_{i,j}$ with the highest weight $\bm{\theta}_{i,j}$ from the input graph. At every iteration of the algorithm, we select the next edge such that it (a) is connected to a previously selected edge and (b) has the maximum weight among all those connected edges. A more detailed description of the greedy algorithm is given in Algorithm \ref{alg:cap}.
Finally, we need to define the function $f_{\bm{u}}$ (the downstream function) of the proposed framework. Here, we again use a message-passing GNN that follows the update rule
\begin{equation}
\label{eq:mpnn-2}
\mathbf{h}_i^{\ell} = \gamma \left(\mathbf{h}_i^{\ell-1},
\square_{j \in \mathcal{N}(v_i)} \phi \left(\mathbf{h}_i^{\ell-1}, \mathbf{h}_j^{\ell-1}, r_{ij} \right)\right).
\end{equation}
The neighborhood structure $\mathcal{N}(\cdot)$, however, is defined through the output adjacency matrix $\bm{Z}$ of the optimization algorithm $\mathtt{opt}$
\begin{equation}
j \in \mathcal{N}(v_i) \Longleftrightarrow \bm{Z}_{i, j} = \bm{Z}_{j, i} = 1.
\end{equation}
Hence, if after the subgraph sampling, there exists a node $v_i$ which is an isolated node in the adjacency matrix $\bm{Z}$, that is, $\bm{Z}_{i,j} = \bm{Z}_{j,i} = 0 \ \forall j \in \{1, ..., n\}$, the embedding of the node will not be updated based on message passing steps with neighboring nodes. This means that, for \textit{isolated} nodes, the only information used in the downstream model is the one from the nodes themselves. Conceptually, $\bm{Z}$ works as a mask over the messages $m_{ij}^\ell$ computed at each layer $\ell$.
The adjacency matrix $\bm{Z}$ is then used in all subsequent layers of the GNN. In particular, for one layer $\ell$ we have
\begin{equation}
\label{eq:gnn_layer}
\mathbf{H}_\ell = \textsc{Gnn}_\ell (\bm{A} \odot \bm{Z}, \mathbf{H}_{\ell-1}),
\end{equation}
where $\odot$ is the Hadamard product.
Finally, the remaining part of the \textsc{L2xGnn}\@\xspace network for the graph classification is
\begin{equation}
\label{eq:classifier}
\mathbf{h}_{G} = \text{Pool}(\mathbf{H}_\ell) \qquad \hat{\bm{y}} = \eta(\mathbf{h}_G),
\end{equation}
where we use a global pooling operator to generate the (sub)graph representation $\mathbf{h}_G$ that will then be used by the MLP network $\eta(\cdot)$ to output a probability distribution $\hat{\bm{y}}$ over the class labels. Finally, a loss function is applied whose gradients are used to perform backpropagation. At test time, we use the maximum-probability subgraph for the explanation and prediction, that is, we do not perturb at test time.
\begin{table*}[t]\sffamily
\begin{tabular}{lC@{\hspace{2pt}}CC@{\hspace{2pt}}C@{}}
\toprule
& \multicolumn{2}{c}{GCN} & \multicolumn{2}{c}{GIN}\\
$w/_{con}$ &
\includegraphics[width=11em]{figures/new/Mutagenicity_GCN_graph_40.pdf} &
\includegraphics[width=11em]{figures/new/Mutagenicity_GCN_graph_97.pdf} &
\includegraphics[width=11em]{figures/new/Mutagenicity_GIN_graph_40.pdf} &
\includegraphics[width=11em]{figures/new/Mutagenicity_GIN_graph_97.pdf} \\
$w/o_{con}$ &
\includegraphics[width=11em]{figures/new/Mutagenicity_GCNdis_graph_40.pdf} &
\includegraphics[width=11em]{figures/new/Mutagenicity_GCNdis_graph_97.pdf} &
\includegraphics[width=11em]{figures/new/Mutagenicity_GINdis_graph_40.pdf} &
\includegraphics[width=11em]{figures/new/Mutagenicity_GINdis_graph_97.pdf} \\
\bottomrule
\end{tabular}
\captionof{figure}{Visualization of some of the subgraphs selected by \textsc{L2xGnn}\@\xspace for MUTAG$_0$ on the test set. The solid edges are the ones sampled by our approach. The first and second rows depict, respectively, some motifs with or without constraining the subgraphs to be connected. Blue, purple, red, and green nodes represent carbon (C), nitrogen (N), oxygen (O), and hydrogen (H) atoms respectively.}
\label{fig:subgraphs}
\end{table*}
\section{Experiments}
First, we evaluate the predictive performance of the model compared to baselines. Second, we qualitatively and quantitatively analyze the explanatory subgraphs for datasets for which we know the ground-truth motifs. Finally, we perform several ablation studies to investigate the effects of different model choices on the results.
\subsection{Datasets and Settings}
\paragraph{Datasets.} To understand the change in the predictive capabilities of the base models when integrating \textsc{L2xGnn}\@\xspace, we use six real-world datasets for graph classification tasks: MUTAG \cite{debnath1991structure}, PROTEINS \cite{borgwardt2005protein}, YEAST \cite{yan2008mining}, IMBD-BINARY, IMBD-MULTI \cite{yanardag2015deep}, and DD \cite{rossi2015network}. To quantitatively evaluate the quality of the explanations, we use datasets that include ground-truth edge masks. In particular, we use MUTAG$_0$ and BA2Motifs. MUTAG$_0$ is a dataset introduced in \citet{tan2022learning} which contains the benzene-NO$_2$ (i.e., a carbon ring with a nitro group (NO$_2$) attached) as the only discriminative motif between positive and negative labels. BA2Motifs is a synthetic dataset that was first introduced in \citet{luo2020pgexplainer}. The base graphs are Barabasi-Albert (BA) graphs. 50\% of the graphs are augmented with a \textit{house-motif} graphs, the rest with a \textit{5-node cycle motif}. The discriminative subgraph leading to different predictions is the motif attached to the BA graph.
\paragraph{Experimental settings.} To evaluate the quality of our approach, we use \textsc{L2xGnn}\@\xspace with several GNN base models including GCN \cite{kipf2016semi} and GIN \cite{xu2018powerful}. We compare the results when using the original model and when the same model is combined with our XAI method.
For model selection and evaluation, to fairly compare the methods, we follow a previously proposed protocol\footnote{\url{https://github.com/pyg-team/pytorch_geometric/tree/master/benchmark/kernel}}. The hyperparameter selection is performed via 10-fold cross-validation, where a training fold is randomly used as a validation set. The selection is performed for the number of layers $[1,2,3,4]$ and the number of hidden units $[16, 32, 64, 128]$ based on the validation accuracy. For a fair comparison with the backbone architectures, we select the best configuration for each dataset, and we integrate our approach into the best model. Instead of fixing a value $k$ for each input graph, we compute $k$ based on a ratio $R$ of edges to be kept. Once the hyperparameters of the default model are found, we select the best ratio $R$ from the set of values $[0.4, 0.5, 0.6, 0.7]$ based on the validation accuracy. We do not include extreme values for two reasons: (1) smaller values for $R$ lead to reduced predictive capabilities and do not lead to meaningful explanatory subgraphs; and (2) higher values would not remove enough edges compared to the original input. Finally, we choose the perturbation intensity $\lambda$ from the values $[10, 100, 1000]$.
\subsection{Empirical Results}
\paragraph{Graph Classification Comparison with Base GNNs.} Following the experimental procedure proposed in \citet{zhang2021protgnn}, Table \ref{tab:accuracy} lists the results of using \textsc{L2xGnn}\@\xspace with base GNN architectures for graph classification tasks. We observe that \textsc{L2xGnn}\@\xspace is competitive and sometimes even outperforms the base GNN models on the benchmark datasets. Note that the primary goal of this work is not to provide a better predictive model, but to provide faithful explanation masks while maintaining similar predictive capabilities.
\paragraph{Explanation Accuracy.}
We compare the proposed method with popular post-hoc explanation techniques including the GNN-Explainer \cite{ying2019gnnexplainer}, PGE-Explainer \cite{luo2020pgexplainer}, GradCAM \cite{pope2019gradcam} and SubgraphX \cite{yuan2021subgraphx}\footnote{Implementations taken from the \textsc{Dig} library \cite{liu2021dig}.}. We train a 3-layer GIN for $200$ epochs with hidden dimensions equal to $64$ and a learning rate equal to $0.001$. We save the best model according to the validation accuracy and we compare it with the post-hoc techniques. In our case, we integrate \textsc{L2xGnn}\@\xspace into the same architecture and learn the edge masking during training as described before. In Table \ref{tab:expl_accuracy}, we report the explanation accuracy evaluation with respect to the ground-truth motifs in comparison with post-hoc techniques. The explanation problem is formalized as a binary classification problem, where the edges belonging to the ground-truth motif are treated as positive labels. We observe that \textsc{L2xGnn}\@\xspace obtains better or the same results as the baseline GNN models. While for the post-hoc explanation techniques we cannot guarantee that the GNNs use exclusively the explanation subgraphs for the prediction \cite{yuan2020explainability}, our method, by providing \textit{faithful} explanations, overcomes this limitation. It is exactly the provided explanation that is used in the message-passing operations of \textsc{L2xGnn}\@\xspace.
\begin{table}[t]
\caption{Evaluation of explanations on synthetic graph classification datasets using a 3-layer GIN architecture.}
\label{tab:expl_accuracy}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tabular}{lcccccccc}
\toprule
\multicolumn{1}{c}{Dataset} & \multicolumn{4}{c}{BA-2MOTIFS} & \multicolumn{4}{c}{MUTAG$_{0}$}\\ \cmidrule(r){2-5}\cmidrule(r){6-9}
\multicolumn{1}{c}{} & Acc. & Pr. & Rec. & F$_1$ & Acc. & Pr. & Rec. & F$_1$ \\
\midrule
GNN-Exp. & .614 & .328 & .658 & .425 & .492 & .415 & .548 & .452 \\
GradCAM & .816 & .534 & .925 & .674 & .722 & .733 & .554 & .608\\
PGE-Exp. & .588 & .313 & .673 & .417 & .731 & .682 & .661 & .657\\
\textit{SubgraphX} & \textbf{.847} & \textbf{.640} & .920 & \textbf{.738} & .725 & \textbf{.765} & .497 & .583\\
\midrule
\textsc{L2xGnn}\@\xspace & .837 & .580 & .900 & .704 & \textbf{.794} & .706 & \textbf{.891} & \textbf{.762}\\
\textsc{L2xGnn}\@\xspace$_{w/o_{con}}$ & .818 & .549 & \textbf{.942} & .690 & .720 & .628 & .828 & .689 \\
\bottomrule
\end{tabular}}
\end{table}
\paragraph{Qualitative Evaluation of the Explanations.}
In Figure \ref{fig:subgraphs}, we present some of the subgraphs identified by \textsc{L2xGnn}\@\xspace when combined with two different base GNNs. Based on prior studies and chemical domain knowledge \cite{debnath1991structure,lin2021generative, tan2022learning}, carbon rings (the blue circles in the pictures) and $\text{NO}_2$ groups are known to be mutagenic. Interestingly, we can notice that, when using the information of connected subgraphs, the models are able to recognize a complete carbon ring with a $\text{NO}_2$ group in most of the cases. In some cases, the carbon ring is not complete, but the explanations are still helpful to understand which motifs are potentially important for the prediction. In the second row, we can observe the results of the sampling strategy where we do not require subgraphs to be connected. In this case, the carbon rings are not always identified. Instead, the $\text{NO}_2$ group is always considered important for the prediction. More generally, as also reported in \citet{yuan2021subgraphx}, studying connected subgraphs results in more natural motifs compared to the motifs obtained without the connectedness constraint.
\begin{table}[t]\sffamily
\resizebox{\columnwidth}{!}{%
\begin{tabular}{C@{\hspace{2pt}}CC@{\hspace{2pt}}C@{}}
\toprule
\multicolumn{2}{c}{GCN} & \multicolumn{2}{c}{GIN}\\
\midrule
\includegraphics[width=11em]{figures/new/ba_2motifs_GCN_graph_12.pdf} &
\includegraphics[width=11em]{figures/new/ba_2motifs_GCN_graph_82.pdf} &
\includegraphics[width=11em]{figures/new/ba_2motifs_GIN_graph_9.pdf} &
\includegraphics[width=11em]{figures/new/ba_2motifs_GIN_graph_2.pdf} \\
\bottomrule
\end{tabular}}
\captionof{figure}{Example of model analysis based on the generated explanations.}
\label{fig:shortcut_det}
\end{table}
\begin{table}[t]
\caption{Effect of the edge ratio on the prediction accuracy.}
\label{tab:ratio}
\centering
\resizebox{.92\columnwidth}{!}{%
\begin{tabular}{llllllllll}
\toprule
Ratio &
\multicolumn{1}{c}{0.2} & \multicolumn{1}{c}{0.3} & \multicolumn{1}{c}{0.4} & \multicolumn{1}{c}{0.5} & \multicolumn{1}{c}{0.6} & \multicolumn{1}{c}{0.7} & \multicolumn{1}{c}{0.8} & \multicolumn{1}{c}{0.9} \\
\midrule
& \multicolumn{8}{c}{MUTAG}\\
\midrule
GCN & .696 & .708 & .713 & .739 & .724 & .729 & .718 & .745 \\
GIN & .786 & .808 & .814 & .823 & .824 & .808 & .798 & .832 \\
\midrule
& \multicolumn{8}{c}{PROTEINS}\\
\midrule
GCN & .714 & .712 & .723 & .720 & .724 & .727 & .719 & .727 \\
GIN & .668 & .690 & .691 & .698 & .696 & .703 & .704 & .714 \\
\bottomrule
\end{tabular}}
\end{table}
\paragraph{Shortcut Learning Detection.} By generating \textit{faithful} subgraph explanations, our approach can be used to detect whether the predictive model is focusing on the expected features or if it is affected by shortcut learning. This is particularly important for GNNs, where seemingly small implementation differences can influence the learning process of the model \cite{schlichtkrull2021interpreting}. To this end, we use the BA2Motifs dataset \cite{luo2020pgexplainer}. We trained two different models, GCN and GIN, achieving a test accuracy of $0.67$ and $1.0$ respectively. Taking a closer look at the explanations of the first model, we observed that most of the correct predictions were (incorrectly) correlated with the cycle motif and that the explanations were similar to the ones reported in Figure~\ref{fig:shortcut_det}. The explanatory results show that the model is not learning the expected discriminative motifs and, consequently, the accuracy for the test set is poor. This insight can help users to change the configuration of the architecture or to use a different model (e.g., GIN). More generally, the results highlight that faithful explanations can facilitate model analysis and debugging.
\paragraph{Ablation Studies.} In Table~\ref{tab:accuracy}, we compare the two sampling strategies. From the results, the connected sampling is able to get better results than the non-connected counterpart on most datasets. In fact, the connectivity of subgraphs is essential to grasp the complete information about the important patterns, especially for chemical compound data where connected atoms are usually expected to create molecules or chemical groups. This aspect is also supported by the results obtained in Table \ref{tab:expl_accuracy}, where the connected strategy returns better explanations for the chemical dataset. Additionally, as previously mentioned, evaluating connected structures rather than just important edges looks more natural and intelligible. In Table \ref{tab:ratio}, we analyze the effect of the quantity of retained information on the prediction accuracy. A smaller ratio indicates that we retain fewer edges during training and, consequently, the resulting subgraphs are more sparse and, therefore, interpretable. As one can see, this affects the predictive capabilities only when $R$ is small. Starting from $R=0.5$, the ratio does not affect particularly the predictive capabilities of the model. In fact, for graph classification tasks, some of the information contained in the initial computational graph does not condition the prediction as the information may be redundant or noisy. For instance, considering the MUTAG dataset, we know that the initial graphs contain on average 20 edges. The discriminative motif benzene-$\text{NO}_2$, instead, contains around 9 edges, meaning that we ideally need 50\% of the original edges to obtain good results. This is in line with the findings of this analysis and the graph classification results previously reported in Table~\ref{tab:accuracy}.
\section{Conclusion and Limitations}
We propose \textsc{L2xGnn}\@\xspace\footnote{Code: \url{https://github.com/GiuseppeSerra93/L2XGNN}}, a framework that can be integrated into GNN architectures to learn to generate explanatory subgraphs which are exclusively used for the models' predictions. Our experimental findings demonstrate that the integration of \textsc{L2xGnn}\@\xspace with base GNNs does not affect the predictive capabilities of the model for graph classification tasks. Furthermore, according to the definition provided in the paper, the resulting explanations are \textit{faithful} since the retained information is the only one used by the model for prediction. Hence, differently from most of the common techniques, our explanations reveal the rationale of the GNNs and can also be used for model analysis and debugging.
A limitation of the approach is the reduced efficiency compared to baseline GNN models. Since we need to integrate an algorithm to compute (approximate) solutions to a combinatorial optimization problem, each forward-pass requires more time and resources. Moreover, depending on the choice of the optimization problem, we might not capture the structure of explanatory motifs required for the application under consideration.
\newpage
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,398 |
Congrats to Larry Lessig, Matt Haughey, Aaron Swartz, and all the many other folks at Creative Commons for their launch today! I see the Commons as a very ambitious undertaking, aiming both to make intellectual property rights more accessible to the people who produce content and to encourage those producers to allow greater reuse of their content. I'm anxious to see how people start using the licenses, and how developers integrate the licensing schemes into content production software (like Movable Type). | {
"redpajama_set_name": "RedPajamaC4"
} | 3,282 |
Home » Tyrannosaurus " Andrea" (PNSO) | Dinosaur Toy Blog
Tyrannosaurus " Andrea" (PNSO) | Dinosaur Toy Blog
Tyrannosaurus rex is like that actor typecast and boxed in a very particular role. Sometimes you become so good or famous for that image or role that it becomes hard for your adoring fans to picture you in anything else. In the case of Mr. Rex, it is playing the role of a villain; a bloodthirsty, cold-blooded killer always in search for its helpless victim. So ingrained is this image in our popular culture that it's hard to move away from it.Breaking away from one's typecast role or image when it does happen comes with high risk that could quickly torpedo one's career into oblivion if he or she doesn't have the strong popularity to endure any backlash.
While we have seen Tyrannosaurus rex model figures go through some changes over the years, the pose for the most part has remained the same: Striding with its mouth wide open in a silent eternal roar. So ubiquitous is this pose in the toy and model figure world that it has become predictable and yes, boring. This becomes even more of an issue for someone like me who's been in the collecting hobby for almost two decades. Finding a Tyrannosaurus rex that offers something different, something unique and exciting has proven to be elusive , which for me, have caused me to be even more indifferent to the many offerings now saturating the market. But once in a while, something unexpected does come out that rouse my interest.This model of the female Tyrannosaurus rex Andrea from PNSO is one of them.
Box and model, the mass produced final figure is pretty much true to the promo photos we saw.
As the most famous and lucrative dinosaur, companies can't put out enough figures of this iconic animal. So, it was not surprising that PNSO would release yet another figure of it right off the heels of their impressive Wilson in less than a year. Wilson (AKA Winter Wilson) burst into the scene with so much new to offer and surely stirred up some lively debate that earned him a place in the hall of most polarizing figures. Little did we know that in less than a year, all of these would be eclipsed by Andrea who in turn created even more of an "excitement" upon its release.
Two very different and unique pose that surely commands attention, its great to see a new pose.
Andrea can be considered both a groundbreaking as well as a pioneer for two reasons: the pose and for being a female. I will share my thoughts in a while, but these two attributes were so unconventional and unexpected that it immediately caused controversy as well as polarized fans and collectors. When it comes to creating controversial figures, PNSO surely have mastered the art for better or for worse.
Going the unconventional route, the prone pose is both exciting and unique making this model stand out from the crowd.
When it comes to PVC model figures of Tyrannosaurus rex (or most predatory theropod for that matter), very few have dared defy the old standard pose of striding/standing with mouth wide open. So, it was surprising to see that this female Tyrannosaurus named Andrea was given a prone or resting pose that went against the norm.This prone/resting pose is a risky choice especially for a huge carnivorous dinosaur ( for many, this means "not active"), but if you are going to take this huge of a risk, it's a smart move to give it to an animal that is so famous that it could afford it and there is no safer choice than Tyrannosaurus rex. Let's face it, it seems that no matter how inaccurate, ugly, or monstrous a figure of Tyrannosaurus rex is, it is always a huge seller, a true testament to its superstar status.
The only other sitting figure from a major brand.It seems this type of pose hasn't done so well sales wise that many companies are hesitant to risk investing on a figure sculpted this way.
The prone/resting pose is one that keeps on making it on the wish list for many collectors year after year, but it has remained elusive for any predatory carnivore dinosaur, it seems that companies are convinced it won't' sell (there may be some truth to this). I can only remember one other PVC (not resin, printed, or kits) figure in this pose, the small Daspletosaurus (guess what, another, but less famous Tyrannosaurus!)from CollectA that was released way back in 2013. Now CollectA is knows for going for the unconventional so it was no surprise that they did it, but it may have suffered in the sales department that no other models with this type of pose was produced after it, so, in this case the risk may not have paid off.
Like all carnivores, Tyrannosaurus rex probably spent most of it's time resting and not striding and roaring.
We had a glimpse of a prone posed T. rex in the artwork that accompanied Wilson V3, so there was speculation if we would see one eventually. When the figure was finally revealed, it was different from the silhouette image we got a glimpse of. Andrea's pose is more upright instead of completely down on the ground with her legs tucked in under her body very much like a bird. This pose is both exciting as well as refreshing, a welcome break from the typical. This pose also reminded us that Tyrannosaurus rex also did other things besides striding and having its mouth wide open roaring. Like most animals, it probably spent most of its time resting to conserve energy.
Gentlemen prefers blonde :Andrea with her suitors.
Andrea's pose has her looking relaxed, but at the same time alert and ready to spring to action when needed to. She has her head slightly turned to her side and close to the ground. Her huge body is touching the ground and her tiny arms tucked in and pulled close to her chest.Her strong legs are folded under her body and sticks a little bit to the side in a way that if a need arises, would allow her to quickly get up.
The Queen has arrived!
It is this leg's positioning, specifically the right one, that would earn a lot of criticism. At a quick glance, it looks okay, and perhaps not easily noticed by those not too familiar with dinosaur anatomy (or anatomy in general), but looking at it closer, one would notice how bent they are almost to the point of breaking especially for an animal as large as Tyrannosaurus rex.One thing that I really admire about PNSO models is in just how much attention is given on how the musculature, flesh, skin, and tendons all have a natural and organic feel to them especially those subtle ones that really captures movement as they follow the bones structure. So, it was a little surprising that they went with this acute angle of the front leg.
Many would equate the prone pose to being "boring", but far from it. It shows another side and activity that we rarely see in toy form.
It is understandable why the leg was sculpted in this position, again, for demonstrating how it would allow for an easier way to get up quickly. With such small useless arms and huge body, Tyrannosaurus rex couldn't push itself to get up quickly. We simply have nothing today that resembles this body plan, so it seems PNSO modeled this pose by studying how extant ratites (ostriches, emus) folded their legs under their body, not perfect, but it is the closest, I guess.
The back side showing the details and how the other leg is positioned.It is not as acutely angled as the front one but still looks painful.
The acute angle of the ankle bones looks painful and highly doubtful if the animal could fold it at such an angle especially with all that weight bearing down on it, but maybe it could on short durations, after all we also can bend our legs in some very uncomfortable position.Personally, I'm not too bothered by it at all as it may still fall within the realm of being possible, and knowing just how tricky it is to sculpt a folded leg and make it look natural, plus in time and at a short distance, one hardly notices it and becomes less of a distraction.
Feeding her young. The articulated mouth can open but not to its full extent since it's so low to the ground. A small raised pedestal would have been nice, but even without one, you can still position the mouth slightly open if you are still not tired of the open mouth pose.
Andrea's body is sculpted with the same high level of details that we have come to expect from PNSO. Wilson was heavily criticized for the large scales on his body, in Andrea, PNSO corrected that and sculpted the scales much smaller. This makes it hard to notice them especially from a distance, it is only upon closer inspection that one can appreciate the fine details of the scales. In many ways, the details on Andrea is much more impressive than Wilson since they are more complex. Unfortunately, there are still those who believe that she is less detailed or inferior in quality because of this.
The underside despite out of view is rich in details just like the rest of the body.
With this unique pose, how the muscles and skin fold look is very different from that of a standing one. PNSO did an exquisite job in really capturing all these subtle details of how muscles are bunched up when carrying some weight, you can see these especially on the stomach and tail region. In some parts you can almost feel the muscles roll from the pressure as well as the skin stretching. Even the underside is very detailed despite this side being mostly obscured from view unless you turn the figure. The long and powerful tail is slightly curved in a natural position that demonstrate the limitation and flexibility of such heavily muscled tail.
Andrea is slightly more robust than Wilson suggesting difference in sizes between the sexes (sexual dimorphism ). Unfortunately, there has been a lot of fat-shaming this figure on various blogs and social media.
The color is a shade lighter than Wilson with the exception of the head which shows darker markings. These subtle differences in color tones help differentiate the two and could be a result of sexual dimorphism. The head sculpt may look the same at a glance, but they are actually different. You can appreciate this by comparing the details on the head closely as well as the size, which in Wilson is slightly larger and maybe longer.
The head are different as you can see. The colors are much darker on her and the scales are also of different sizes.
For extinct animals, determining the sexes is challenging and often results in confusing one gender for another species. There are few rare instances such as embryonic fossil inside an adult or an individual caught and frozen in the act of giving birth that makes it easy to confirm the sexes, but for the majority it's still full of uncertainty.For Tyrannosaurus rex , there has been suggestion that they showed some form of sexual dimorphism with the females being slightly larger than the males. While this is still being debated, it offers a good and plausible possibility, and PNSO opted for this by making Andrea slightly larger than her mate Wilson, you can see this difference very well when viewed from the top.
A rare sight: a family of Tyrannosaurus rex!
It's rare to have a figure that is explicitly marketed as female especially amongst the big carnivores. While we see some mothers these are almost always herbivores, so it is refreshing to get a large predatory queen for a change. And what a queen she is. Her calm regal demeanor surely warrants attention and remind us that when it comes to Tyrannosaurus rex, it's not all ready for a bloody battle.
Yes, she can fish! I originally intended to pose her drinking, but I could't help adding the fish turning it into a fishing one instead.
Andrea's size has been fodder for many fat-shaming jokes and comments that has flooded the internet in various social media as well as blogs. It's hard not to think that this has something to do with her gender for, if you read blogs and forums, as well as various social media platforms, you know that sexism reared its ugly head. While Wilson created controversy and was heavily criticized , the words used are different than those thrown at Andrea.
PNSO's Tyrannosaurus rex Dynasty:The Past, the Present, and Future.
I have no doubt that despite some shortcomings, Andrea will go down in history as one of the most memorable figures long after her production date, something similar to the Battat Diplodocus. The difference between her and Wilson makes a convincing case for sexual dimorphism and the two really displays and complement each other.
What Child is this: What a scaled down Aaron may look like with Andrea, and with our little furry friend.
I have always wondered if Andrea was originally a Wilson prototype that was simply too good to be discarded, but seeing some Dioramas made by PNSO that featured a family of Tyrannosaurus rex, it seems like PNSO has always planned on creating a family portrait of the most famous dinosaur of all.Even the addition of Logan seems calculated and was a clever way to introduce a young one, and he does fit in very well with Andrea and Wilson making one truly unique, impressive and memorable display.
The Queen looking at her reflection.
For someone who is not really a big fan of tyrannosaurus rex, I was surprised by how many figures of it I actually reviewed! As I look back at all of them, I have covered a wide range from the vintage, the young, the fit, the party feathered, the old, the dead, and with Andrea finally a female, I realize that I have underestimated this animal and allowed myself to be distracted by the fatigue of having too many T. rexes. I have come to a new appreciation of just how interesting, complex, and fascinating this animal is despite suffering major overexposure. With this review, I am hesitant to say that I am done with this genus, but I would say that for now at least, I feel like I have covered a pretty diverse representation of this iconic animal in toy form.
The Royals!
Who knows, maybe, just maybe a fully grown up, fully feathered Aaron is just around the corner to tempt me to tackle this iconic dinosaur again, but until then, I'm happy with what I have covered so far. Well, that concludes our review, hope you like it. Until the next one, stay safe and healthy, Cheers!
Restyling This Mannequin- It's Not Pretty
La Union expands COVID-19 vaccination to children, college students | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,187 |
{"url":"https:\/\/chsasank.com\/classic_papers\/strength-of-weak-ties.html","text":"# The Strength of Weak Ties\n\nMark S. Granovetter | | 55 minutes to read.\n\nThis is a landmark paper in sociology. Idea is to use social networks to link micro and macro levels of sociology. Core idea of the paper is this: strong friendships and ties form cliques while weak ties act as \u2018bridges\u2019 that connect different cliques. Therefore, information from strong friendships\/ties is strongly correlated to your own, while weak ties give you information unknown to you. This makes weak ties crucial in explaining many phenomenon including diffusion of ideas, rumours and innovations, individual\u2019s network and even community cohesion and organization.\n\nYellow highlights\/annotations are my own. You can disable them.\n\n## Abstract\n\nAnalysis of social networks is suggested as a tool for linking micro and macro levels of sociological theory. The procedure is illustrated by elaboration of the macro implications of one aspect of small-scale interaction: the strength of dyadic ties. It is argued that the degree of overlap of two individuals\u2019 friendship networks varies directly with the strength of their tie to one another. The impact of this principle on diffusion of influence and information, mobility opportunity, and community organization is explored. Stress is laid on the cohesive power of weak ties. Most network models deal, implicitly, with strong ties, thus confining their applicability to small, well- defined groups. Emphasis on weak ties lends itself to discussion of relations between groups and to analysis of segments of social structure not easily defined in terms of primary groups.\n\n## Introduction\n\nA fundamental weakness of current sociological theory is that it does not relate micro-level interactions to macro-level patterns in any convincing way. Large-scale statistical, as well as qualitative, studies offer a good deal of insight into such macro phenomena as social mobility, community organization, and political structure. At the micro level, a large and increasing body of data and theory offers useful and illuminating ideas about what transpires within the confines of the small group. But how interaction in small groups aggregates to form large-scale patterns eludes us in most cases.\n\nI will argue, in this paper, that the analysis of processes in interpersonal networks provides the most fruitful micro-macro bridge. In one way or another, it is through these networks that small-scale interaction becomes translated into large-scale patterns, and that these, in turn, feed back into small groups.\n\nSociometry, the precursor of network analysis, has always been curiously peripheral\u2014invisible, really\u2014in sociological theory. This is partly because it has usually been studied and applied only as a branch of social psychology; it is also because of the inherent complexities of precise network analysis. We have had neither the theory nor the measurement and sampling techniques to move sociometry from the usual small-group level to that of larger structures. While a number of stimulating and suggestive studies have recently moved in this direction (Bott 1957; Mayer 1961; Milgram 1967; Boissevain 1968; Mitchell 1969), they do not treat structural issues in much theoretical detail. Studies which do so usually involve a level of technical complexity appropriate to such forbidding sources as the Bulletin of Mathematical Biophysics, where the original motivation for the study of networks was that of developing a theory of neural, rather than social, interaction (see the useful review of this literature by Coleman [1960]; also Rapoport [1963]).\n\nThe strategy of the present paper is to choose a rather limited aspect of small-scale interaction\u2014the strength of interpersonal ties\u2014and to show, in some detail, how the use of network analysis can relate this aspect to such varied macro phenomena as diffusion, social mobility, political organization, and social cohesion in general. While the analysis is essentially qualitative, a mathematically inclined reader will recognize the potential for models; mathematical arguments, leads, and references are suggested mostly in footnotes.\n\n## The Strength of Ties\n\nMost intuitive notions of the \u201cstrength\u201d of an interpersonal tie should be satisfied by the following definition: the strength of a tie is a (probably linear) combination of the amount of time, the emotional intensity, the intimacy (mutual confiding), and the reciprocal services which characterize the tie Ties discussed in this paper are assumed to be positive and symmetric; a comprehensive theory might require discussion of negative and\/or asymmetric ties, but this would add unnecessary complexity to the present, exploratory comments.. Each of these is somewhat independent of the other, though the set is obviously highly intracorrelated. Discussion of operational measures of and weights attaching to each of the four elements is postponed to future empirical studies. Some anthropologists suggest \u201cmultiplexity,\u201d that is, multiple contents in a relationship, as indicating a strong tie (Kapferer 1969, p. 213). While this may be accurate in some circumstances, ties with only one content or with diffuse content may be strong as well (Simmel 1950, pp. 317-29). The present definition would show most multiplex ties to be strong but also allow for other possibilities. It is sufficient for the present purpose if most of us can agree, on a rough intuitive basis, whether a given tie is strong, weak, or absent. Included in \u201cabsent\u201d are both the lack of any relationship and ties without substantial significance, such as a \u201cnodding\u201d relationship between people living on the same street, or the \u201ctie\u201d to the vendor from whom one customarily buys a morning newspaper. That two people \u201cknow\u201d each other by name need not move their relation out of this category if their interaction is negligible. In some contexts, however (disasters, for example), such \u201cnegligible\u201d ties might usefully be distinguished from the absence of one. This is an ambiguity caused by substitution, for convenience of exposition, of discrete values for an underlying continuous variable.\n\nConsider, now, any two arbitrarily selected individuals\u2014call them A and B\u2014and the set, S=C,D,E,\u2026, of all persons with ties to either or both of them. In Barnes\u2019s terminology, the union of their respective primary stars (1969, p. 58). The hypothesis which enables us to relate dyadic ties to larger structures is: the stronger the tie between A and B, the larger the proportion of individuals in S to whom they will both be tied, that is, connected by a weak or strong tie. This overlap in their friendship circles is predicted to be least when their tie is absent, most when it is strong, and intermediate when it is weak.\n\nThe proposed relationship results, first, from the tendency (by definition) of stronger ties to involve larger time commitments. If A-B and A-C ties exist, then the amount of time C spends with B depends (in part) on the amount A spends with B and C, respectively. (If the events \u201cA is with B\u201d and \u201cA is with C\u201d were independent, then the event \u201cC is with A and B\u201d would have probability equal to the product of their probabilities. For example, if A and B are together 60% of the time, and A and C 40%, then C, A, and B would be together 24% of the time. Such independence would be less likely after than before B and C became acquainted.) If C and B have no relationship, common strong ties to A will probably bring them into interaction and generate one. Implicit here is Homans\u2019s idea that \u201cthe more frequently persons interact with one another, the stronger their sentiments of friendship for one another are apt to be\u201d (1950, p. 133).\n\nThe hypothesis is made plausible also by empirical evidence that the stronger the tie connecting two individuals, the more similar they are, in various ways (Berscheid and Walster 1969, pp. 69-91; Bramel 1969, pp. 9-16; Brown 1965, pp. 71-90; Laumann 1968; Newcomb 1961, chap. 5; Precker 1952). Thus, if strong ties connect A to B and A to C, both C and B, being similar to A, are probably similar to one another, increasing the likelihood of a friendship once they have met. Applied in reverse, these two factors\u2014time and similarity\u2014indicate why weaker A-B and A-C ties make a C-B tie less likely than strong ones: C and B are less likely to interact and less likely to be compatible if they do.\n\nThe theory of cognitive balance, as formulated by Heider (1958) and especially by Newcomb (1961, pp. 4-23), also predicts this result. If strong ties A-B and A-C exist, and if B and C are aware of one another, anything short of a positive tie would introduce a \u201cpsychological strain\u201d into the situation since C will want his own feelings to be congruent with those of his good friend, A, and similarly, for B and his friend, A. Where the ties are weak, however, such consistency is psychologically less crucial. (On this point see also Homans [1950, p. 255] and Davis [1963, p. 448].)\n\nSome direct evidence for the basic hypothesis exists (Kapferer 1969, p. 229 n.; Laumann and Schuman 1967; Rapoport and Horvath 1961; Rapoport 1963). The models and experiments of Rapoport and his associates have been a major stimulus to this paper. In 1954 he commented on the \u201cwell-known fact that the likely contacts of two individuals who are closely acquainted tend to be more overlapping than those of two arbitrarily selected individuals\u201d (p. 75). His and Horvath\u2019s 1961 hypothesis is even closer to mine: \u201cone would expect the friendship relations, and therefore the overlap bias of the acquaintance circles, to become less tight with increasing numerical rank-order\u201d (p. 290). (i.e., best friend, second-best friend, third-best, etc.) Their development of this hypothesis, however, is quite different, substantively and mathematically, from mine (Rapoport 1953a, 1953b, 1954, 1963; Rapoport and Horvath 1961). This evidence is less comprehensive than one might hope. In addition, however, certain inferences from the hypothesis have received empirical support. Description of these inferences will suggest some of the substantive implications of the above argument.\n\n## Weak Ties in Diffusion Process\n\nTo derive implications for large networks of relations, it is necessary to frame the basic hypothesis more precisely. This can be done by investigating the possible triads consisting of strong, weak, or absent ties among A, B, and any arbitrarily chosen friend of either or both (i.e., some member of the set S, described above). A thorough mathematical model would do this in some detail, suggesting probabilities for various types. This analysis becomes rather involved, however, and it is sufficient for my purpose in this paper to say that the triad which is most unlikely to occur, under the hypothesis stated above, is that in which A and B are strongly linked, A has a strong tie to some friend C, but the tie between C and B is absent. This triad is shown in figure 1. To see the consequences of this assertion, I will exaggerate it in what follows by supposing that the triad shown never occurs\u2014that is, that the B-C tie is always present (whether weak or strong), given the other two strong ties. Whatever results are inferred from this supposition should tend to occur in the degree that the triad in question tends to be absent.\n\nSome evidence exists for this absence. Analyzing 651 sociograms, Davis (1970, p. 845) found that in 90% of them triads consisting of two mutual choices and one nonchoice occurred less than the expected random number of times. If we assume that mutual choice indicates a strong tie, this is strong evidence in the direction of my argument. This assumption is suggested by one of Davis\u2019s models (1970, p. 846) and made explicitly by Mazur (1971). It is not obvious, however. In a free-choice sociometric test or a fixed-choice one with a large number of choices, most strong ties would probably result in mutual choice, but some weak ones might as well. With a small, fixed number of choices, most mutual choices should be strong ties, but some strong ties might show up as asymmetric. For a general discussion of the biases introduced by sociometric procedures, see Holland and Leinhardt (1971b). Newcomb (1961, pp. 160- 65) reports that in triads consisting of dyads expressing mutual \u201chigh attraction,\u201d the configuration of three strong ties became increasingly frequent as people knew one another longer and better; the frequency of the triad pictured in figure 1 is not analyzed, but it is implied that processes of cognitive balance tended to eliminate it.\n\nThe significance of this triad\u2019s absence can be shown by using the concept of a \u201cbridge\u201d; this is a line in a network which provides the only path between two points (Harary, Norman, and Cartwright 1965, p. 198). Since, in general, each person has a great many contacts, a bridge between A and B provides the only route along which information or influence can flow from any contact of A to any contact of B, and, consequently, from anyone connected indirectly to A to anyone connected indirectly to B. Thus, in the study of diffusion, we can expect bridges to assume an important role.\n\nNow, if the stipulated triad is absent, it follows that, except under unlikely conditions, no strong tie is a bridge. Consider the strong tie A-B: if A has another strong tie to C, then forbidding the triad of figure 1 implies that a tie exists between C and B, so that the path A-C-B exists between A and B; hence, A-B is not a bridge. A strong tie can be a bridge, therefore, only if neither party to it has any other strong ties, unlikely in a social network of any size (though possible in a small group). Weak ties suffer no such restriction, though they are certainly not automatically bridges. What is important, rather, is that all bridges are weak ties.\n\nIn large networks it probably happens only rarely, in practice, that a specific tie provides the only path between two points. The bridging function may nevertheless be served locally. In figure 2a, for example, the tie A-B is not strictly a bridge, since one can construct the path A-E-J-B (and others). Yet, A-B is the shortest route to B for F, D, and C. This function is clearer in figure 2b. Here, A-B is, for C, D, and others, not only a local bridge to B, but, in most real instances of diffusion, a much more likely and efficient path. Harary et al. point out that \u201cthere may be a distance [length of path] beyond which it is not feasible for $u$ to communicate with $v$ because of costs or distortions entailed in each act of transmission. If $v$ does not lie within this critical distance, then he will not receive messages originating with $u$ (1965, p. 159). I will refer to a tie as a \u201clocal bridge of degree $n$\u201d if $n$ represents the shortest path between its two points (other than itself), and $n$ > 2. In figure 2a, A-B is a local bridge of degree 3, in 2b, of degree 13. As with bridges in a highway system, a local bridge in a social network will be more significant as a connection between two sectors to the extent that it is the only alternative for many people\u2014that is, as its degree increases. A bridge in the absolute sense is a local one of infinite degree. By the same logic used above, only weak ties may be local bridges.\n\nSuppose, now, that we adopt Davis\u2019s suggestion that \u201cin interpersonal flows of most any sort the probability that \u2018whatever it is\u2019 will flow from person $i$ to person $j$ is (a) directly proportional to the number of all-positive (friendship) paths connecting $i$ and $j$; and (b) inversely proportional to the length of such paths\u201d (1969, p. 549). Though this assumption seems plausible, it is by no means self-evident. Surprisingly little empirical evidence exists to support or refute it. The significance of weak ties, then, would be that those which are local bridges create more, and shorter, paths. Any given tie may, hypothetically, be removed from a network; the number of paths broken and the changes in average path length resulting between arbitrary pairs of points (with some limitation on length of path considered) can then be computed. The contention here is that removal of the average weak tie would do more \u201cdamage\u201d to transmission probabilities than would that of the average strong one. In a more comprehensive treatment it would be useful to consider to what extent a set of weak ties may be considered to have bridging functions. This generalization requires a long, complex discussion and is not attempted here (see Harary et al. 1965, pp. 211-16).\n\nIntuitively speaking, this means that whatever is to be diffused can reach a larger number of people, and traverse greater social distance (i.e., path length), We may define the \u201csocial distance\u201d between two individuals in a network as the number of lines in the shortest path from one to another. This is the same as the definition of \u201cdistance\u201d between points in graph theory (Harary et al. 1965, pp. 32-33, 138-41). The exact role of this quantity in diffusion and epidemic theory is discussed by Solomonoff and Rapoport (1951). when passed through weak ties rather than strong. If one tells a rumor to all his close friends, and they do likewise, many will hear the rumor a second and third time, since those linked by strong ties tend to share friends. If the motivation to spread the rumor is dampened a bit on each wave of retelling, then the rumor moving through strong ties is much more likely to be limited to a few cliques than that going via weak ones; bridges will not be crossed. If a damping effect is not specified, the whole population would hear the rumor after a sufficiently large number of retellings, since few real networks include totally self-contained cliques. The effective difference between using weak and strong ties, then, is one of people reached per unit of (ordinal) time. This could be called \u201cvelocity\u201d of transmission. I am indebted to Scott Feld for this point.\n\nSince sociologists and anthropologists have carried out many hundreds of diffusion studies\u2014Rogers\u2019s 1962 review dealt with 506\u2014one might suppose that the above claims could easily be put to test. But this is not so, for several reasons. To begin with, though most diffusion studies find that personal contacts are crucial, many undertake no sociometric investigation. (Rogers [1962] discusses this point.) When sociometric techniques are used, they tend to discourage the naming of those weakly tied to the respondent by sharply limiting the numbers of choices allowed. Hence, the proposed importance of weak ties in diffusion is not measured. Even when more sociometric information is collected there is almost never an attempt to directly retrace the exact interpersonal paths traversed by an (idea, rumor, or) innovation. More commonly, the time when each individual adopted the innovation is recorded, as is the number of sociometric choices he received from others in the study. Those receiving many choices are characterized as \u201ccentral,\u201d those with few as \u201cmarginal\u201d; this variable is then correlated with time of adoption and inferences made about what paths were probably followed by the innovation.\n\nOne point of controversy in diffusion studies can be related to my argument. Some have indicated that early innovators are marginal, that they \u201cunderconform to norms to such a degree that they are perceived as highly deviant\u201d (Rogers 1962, p. 197). Others (e.g., Coleman, Katz, and Menzel [1966] on the adoption of a new drug by doctors) find that those named more frequently adopt an innovation substantially earlier. Becker (1970) tries to resolve the question of whether early innovators are \u201ccentral\u201d or \u201cmarginal\u201d by referring to the \u201cperceived risks of adoption of a given innovation.\u201d His study of public health innovations shows that when a new program is thought relatively safe and uncontroversial (as with the drug of Coleman et al.), central figures lead in its adoption; otherwise, marginal ones do (p. 273). He explains the difference in terms of a greater desire of \u201ccentral\u201d figures to protect their professional reputation.\n\nKerckhoff, Back, and Miller (1965) reach a similar conclusion in a different type of study. A Southern textile plant had been swept by \u201chysterical contagion\u201d: a few, then more and more workers, claiming bites from a mysterious \u2018insect,\u2019 became nauseous, numb, and weak, leading to a plant shutdown. When the affected workers were asked to name their three best friends, many named one another, but the very earliest to be stricken were social isolates, receiving almost no choices. An explanation, compatible with Becker\u2019s, is offered: since the symptoms might be thought odd, early \u201cadopters\u201d were likely to be found among the marginal, those less subject to social pressures. Later, \u201cit is increasingly likely that some persons who are socially integrated will be affected. . . . The contagion enters social networks and is disseminated with increasing rapidity\u201d (p. 13). This is consistent with Rogers\u2019s comment that while the first adopters of innovations are marginal, the next group, \u201cearly adopters,\u201d \u201care a more integrated part of the local social system than the innovators\u201d (1962, p. 183).\n\n\u201cCentral\u201d and \u201cmarginal\u201d individuals may well be motivated as claimed; but if the marginal are genuinely so, it is difficult to see how they can ever spread innovations successfully. We may surmise that since the resistance to a risky or deviant activity is greater than to a safe or normal one, a larger number of people will have to be exposed to it and adopt it, in the early stages, before it will spread in a chain reaction. Individuals with many weak ties are, by my arguments, best placed to diffuse such a difficult innovation, since some of those ties will be local bridges. These individuals are what is often called, in organizational analysis, \u201cliaison persons,\u201d though their role here is different from the one usually discussed. (Cf. the concept in graph theory of a \u201ccut point\u201d\u2014one which, if removed from a graph, disconnects one part from another [Harary 1965].) In general, a bridge has one liaison person on each side, but the existence of a liaison person does not imply that of a bridge. For local bridges, the concept of local liaisons could be developed. In a more microscopically oriented discussion I would devote more time to the liaison role. For now, I only point out that, under the present assumptions, one can be a liaison between two network sectors only if all his ties into one or both are weak. An initially unpopular innovation spread by those with few weak ties is more likely to be confined to a few cliques, thus being stillborn and never finding its way into a diffusion study.\n\nThat the \u201cmarginal\u201d innovators of diffusion studies might actually be rich in weak ties is possible, given the usual sociometric technique, but in most cases this is purely speculative. Kerckhoff and Back, however, in a later more detailed analysis of the hysteria incident, indicate that besides asking about one\u2019s \u201cthree best friends,\u201d they also asked with whom workers ate, worked, shared car pools, etc. They report that five of the six workers earliest affected \u201care social isolates when friendship choices are used as the basis of analysis. Only 1 of the 6 is mentioned as a friend by anyone in our sample. This is made even more striking when we note that these 6 women are mentioned with considerable frequency when other bases for choice are used. In fact, they are chosen more frequently on a \u2018non-friendship\u2019 basis than are the women in any of the other categories\u201d (1968, p. 112).\n\nThis finding lends credence to the weak-tie argument, but is inconclusive. A somewhat different kind of diffusion study offers more direct support: the \u201csmall-world\u201d investigations of Milgram and his associates. The name of these studies stems from the typical comment of newly introduced individuals who discover some common acquaintance; this situation is generalized in an attempt to measure, for arbitrarily chosen pairs of individuals in the United States, how long a path of personal contacts would be needed to connect them. A booklet is given to randomly designated senders who are asked to forward it toward some named target person, via someone the sender knows personally who would be more likely than himself to know the target. The new recipient then advances the booklet similarly; eventually it reaches the target or someone fails to send it on. The proportion of such chains completed has ranged from 12% to 33% in different studies, and the number of links in completed chains has ranged from two to 10, averaging between five and eight (Milgram 1967; Travers and Milgram 1969; Korte and Milgram 1970).\n\nEach time someone forwards a booklet he also sends a postcard to the researchers, indicating, among other things, the relationship between himself and the next receiver. Two of the categories which can be chosen are \u201cfriend\u201d and \u2018acquaintance.\u201d I will assume that this corresponds to \u201cstrong\u201d and \u201cweak\u201d ties. In one of the studies, white senders were asked to forward the booklet to a target who was Negro. In such chains, a crucial point was the first sending of the booklet from a white to a Negro. In 50% of the instances where the white described this Negro as an \u201cacquaintance,\u201d the chain was ultimately completed; completion rate fell to 26%, however, when the white sent the booklet to a Negro \u201cfriend.\u201d (My computation, based on unpublished data kindly supplied by Charles Korte. See Korte [1967] and Korte and Milgram [1970].) Thus, weaker interracial ties can be seen to be more effective in bridging social distance.\n\nAnother relevant study, by Rapoport and Horvath (1961), is not exactly one of diffusion but is closely related in that it traces out paths along which diffusion could take place. They asked each individual in a Michigan junior high school (N = 851) to list his eight best friends in order of preference. Then, taking a number of random samples from the group (sample size, an arbitrary number, was nine), they traced out, for each sample, and averaged over all the samples, the total number of people reached by following along the network of first and second choices. That is, the first and second choices of each sample member were tabulated, then the first and second choices of these people were added in, etc., counting, at each remove, only names not previously chosen, and continuing until no new people were reached. The same procedure was followed using second and third choices, third and fourth, etc., up to seventh and eighth. (The theoretical connection of this tracing procedure to diffusion is discussed by Rapoport [1953a, 19530, and especially 1954].)\n\nThe smallest total number of people were reached through the networks generated by first and second choices\u2014presumably the strongest ties\u2014and the largest number through seventh and eighth choices. This corresponds to my assertion that more people can be reached through weak ties. A parameter in their mathematical model of the sociogram, designed to measure, approximately, the overlap of acquaintance circles, declined monotonically with increasing rank order of friends. This parameter, \u03b8, measures such overlap in the following sense: it is zero in a random net\u2014one in which individuals choose others at random\u2014and is one in a net made up entirely of cliques disconnected each from every other. Intermediate values of \u03b8, however, do not have a good intuitive interpretation in terms of individuals, but only with reference to the particular mathematical model defining the parameter; thus it does not correspond precisely to my arguments about friendship overlap.\n\n## Weak Ties in Egocentric Networks\n\nIn this section and the next, I want to discuss the general significance of the above findings and arguments at two levels: first that of individuals, then that of communities. These discussions make no pretense of being comprehensive; they are meant only to illustrate possible applications.\n\nIn recent years, a great deal of literature has appeared analyzing the impact on the behavior of individuals of the social networks in which they are imbedded. Some of the studies have emphasized the ways in which behavior is shaped and constrained by one\u2019s network (Bott 1957; Mayer 1961; Frankenberg 1965), others the ways in which individuals can manipulate these networks to achieve specific goals (Mayer 1966; Boissevain 1968; Kapferer 1969). Both facets are generally supposed to be affected by the structure of one\u2019s network. Bott argued that the crucial variable is that of whether one\u2019s friends tend to know one another (\u201ccloseknit\u201d network) or not (\u201cloose-knit\u201d network). Barnes makes this dichotomy into a continuous variable by counting the number of ties observed in the network formed by ego and his friends and dividing it by the ratio of possible ones; this then corresponds to what is often called network \u201cdensity\u201d (Barnes 1969; Tilly 1969). But if the crucial question is really whether ego\u2019s friends know each other, this measure should probably be computed after ego and his ties have been subtracted from the network; distortions caused by failure to do so will be especially great in small networks. It is important to note, also, that in nonegocentric networks, there is no simple correspondence between density and any \u201caverage\u201d measure of the extent to which the various egos have friends who know one another. \u201cDensity,\u201d as used here, should not be confused with the \u201caxone density\u201d of Rapoport\u2019s models\u2014the number of choices issuing from each node of a network.\n\nEpstein (1969) points out, however, that different parts of ego\u2019s network may have different density. He calls those with whom one \u201cinteracts most intensely and most regularly, and who are therefore also likely to come to know one another,\u201d the \u201ceffective network\u201d; the \u201cremainder constitute the extended network\u201d (pp. 110-11). This is close to saying, in my terms, that one\u2019s strong ties form a dense network, one\u2019s weak ties a less dense one. I would add that one\u2019s weak ties which are not local bridges might as well be counted with the strong ties, to maximize separation of the dense from the less dense network sectors.\n\nOne point on which there is no general agreement is whether ego\u2019s network should be treated as composed only of those to whom he is tied directly, or should include the contacts of his contacts, and\/or others. Analyses stressing encapsulation of an individual by his network tend to take the former position, those stressing manipulation of networks, the latter, since information or favors available through direct contacts may depend on who their contacts are. I would argue that by dividing ego\u2019s network into that part made up of strong and nonbridging weak ties on the one hand, and that of bridging weak ties on the other, both orientations can be dealt with. Ties in the former part should tend to be to people who not only know one another, but who also have few contacts not tied to ego as well. In the \u201cweak\u201d sector, however, not only will ego\u2019s contacts not be tied to one another, but they will be tied to individuals not tied to ego. Indirect contacts are thus typically reached through ties in this sector; such ties are then of importance not only in ego\u2019s manipulation of networks, but also in that they are the channels through which ideas, influences, or information socially distant from ego may reach him. The fewer indirect contacts one has the more encapsulated he will be in terms of knowledge of the world beyond his own friendship circle; thus, bridging weak ties (and the consequent indirect contacts) are important in both ways.\n\nI will develop this point empirically by citing some results from a labor-market study I have recently completed. Labor economists have long been aware that American blue-collar workers find out about new jobs more through personal contacts than by any other method. (Many studies are reviewed by Parnes 1954, chap. 5.) Recent studies suggest that this is also true for those in professional, technical, and managerial positions (Shapero, Howell, and Tombaugh 1965; Brown 1967; Granovetter 1970). My study of this question laid special emphasis on the nature of the tie between the job changer and the contact person who provided the necessary information.\n\nIn a random sample of recent professional, technical, and managerial job changers living in a Boston suburb, I asked those who found a new job through contacts how often they saw the contact around the time that he passed on job information to them. I will use this as a measure of tie strength. Although this corresponds only to the first of the four dimensions in my definition, supplementary anecdotal evidence from interviews makes it likely that, in this case, the entire definition is satisfied by this measure. At the time of research, it had not occurred to me that tie strength would be a useful variable. A natural a priori idea is that those with whom one has strong ties are more motivated to help with job information. Opposed to this greater motivation are the structural arguments I have been making: those to whom we are weakly tied are more likely to move in circles different from our own and will thus have access to information different from that which we receive.\n\nI have used the following categories for frequency of contact: often = at least twice a week; occasionally = more than once a year but less than twice a week; rarely = once a year or less. Of those finding a job through contacts, 16.7% reported that they saw their contact often at the time, 55.6% said occasionally, and 27.8% rarely (N = 54). The numbers reported are small because they represent a random subsample of 100, who were interviewed personally, of the total sample of 282. The personal interview allowed more detailed questioning. Comparisons between the mail sample and the interview sample on the large number of items which were put to both show almost no significant differences; this suggests that results observed in the smaller sample on those items put to it alone would not be much different in the mail sample. The skew is clearly to the weak end of the continuum, suggesting the primacy of structure over motivation.\n\nIn many cases, the contact was someone only marginally included in the current network of contacts, such as an old college friend or a former workmate or employer, with whom sporadic contact had been maintained (Granovetter 1970, pp. 76-80). Usually such ties had not even been very strong when first forged. For work-related ties, respondents almost invariably said that they never saw the person in a nonwork context. Often when I asked respondents whether a friend had told them about their current job, they said, \u201cNot a friend, an acquaintance.\u201d It was the frequency of this comment which suggested this section of the paper to me. Chance meetings or mutual friends operated to reactivate such ties. It is remarkable that people receive crucial information from individuals whose very existence they have forgotten. Donald Light has suggested to me an alternative reason to expect predominance of weak ties in transfer of job information. He reasons that most of any given person\u2019s ties are weak, so that we should expect, on a \u201crandom\u201d model, that most ties through which job information flows should be weak. Since baseline data on acquaintance networks are lacking, this objection remains inconclusive. Even if the premise were correct, however, one might still expect that greater motivation of close friends would overcome their being outnumbered. Different assumptions yield different \u201crandom\u201d models; it is not clear which one should be accepted as a starting point. One plausible such model would expect information to flow through ties in proportion to the time expended in interaction; this model would predict much more information via strong ties than one which merely counted all ties equally.\n\nThe usual dichotomy between \u201cformal\u201d or mass procedures and diffusion through personal contacts may thus be invalid in some cases where, instead, the former may be seen as a limiting case of long diffusion chains. This is especially likely where information of instrumental significance is involved. Such information is most valuable when earmarked for one person.\n\nFrom the individual\u2019s point of view, then, weak ties are an important resource in making possible mobility opportunity. Seen from a more macroscopic vantage, weak ties play a role in effecting social cohesion. When a man changes jobs, he is not only moving from one network of ties to another, but also establishing a link between these. Such a link is often of the same kind which facilitated his own movement. Especially within professional and technical specialties which are well defined and limited in size, this mobility sets up elaborate structures of bridging weak ties between the more coherent clusters that constitute operative networks in particular locations. Information and ideas thus flow more easily through the specialty, giving it some \u201csense of community,\u201d activated at meetings and conventions. Maintenance of weak ties may well be the most important consequence of such meetings.\n\n## Weak Ties and Community Organization\n\nThese comments about sense of community may remind us that in many cases it is desirable to deal with a unit of analysis larger than a single individual. I would like to develop my argument further by analyzing, in this section, why some communities organize for common goals easily and effectively whereas others seem unable to mobilize resources, even against dire threats. The Italian community of Boston\u2019s West End, for example, was unable to even form an organization to fight against the \u201curban renewal\u201d which ultimately destroyed it. This seems especially anomalous in view of Gans\u2019s description of West End social structure as cohesive (1962).\n\nVariations in culture and personality are often cited to explain such anomalies. Gans contrasts \u201clower\u201d-, \u201cworking\u201d-, and \u201cmiddle\u201d-class subcultures, concluding that only the last provides sufficient trust in leaders and practice in working toward common goals to enable formation of an effective organization. Thus, the working-class West End could not resist urban renewal (pp. 229-304). Yet, numerous well-documented cases show that some working-class communities have mobilized quite successfully against comparable or lesser threats (Dahl 1961, pp. 192-99; Keyes 1969; Davies 1966, chap. 4). This point was brought to my attention by Richard Wolfe. I would suggest, as a sharper analytical tool, examination of the network of ties comprising a community to see whether aspects of its structure might facilitate or block organization.\n\nImagine, to begin with, a community completely partitioned into cliques, such that each person is tied to every other in his clique and to none outside. Community organization would be severely inhibited. Leafletting, radio announcements, or other methods could insure that everyone was aware of some nascent organization; but studies of diffusion and mass communication have shown that people rarely act on mass-media information unless it is also transmitted through personal ties (Katz and Lazarsfeld 1955; Rogers 1962); otherwise one has no particular reason to think that an advertised product or an organization should be taken seriously. Enthusiasm for an organization in one clique, then, would not spread to others but would have to develop independently in each one to insure success.\n\nThe problem of trust is closely related. I would propose that whether a person trusts a given leader depends heavily on whether there exist intermediary personal contacts who can, from their own knowledge, assure him that the leader is trustworthy, and who can, if necessary, intercede with the leader or his lieutenants on his behalf. Trust in leaders is integrally related to the capacity to predict and affect their behavior. Leaders, for their part, have little motivation to be responsive or even trustworthy toward those to whom they have no direct or indirect connection. Thus, network fragmentation, by reducing drastically the number of paths from any leader to his potential followers, would inhibit trust in such leaders. This inhibition, furthermore, would not be entirely irrational.\n\nCould the West End\u2019s social structure really have been of this kind? Note first that while the structure hypothesized is, by definition, extremely fragmented, this is evident only at a macroscopic level\u2014from an \u201caerial view\u201d of the network. The local phenomenon is cohesion. (Davis [1967] also noted this paradox, in a related context.) An analyst studying such a group by participant observation might never see the extent of fragmentation, especially if the cliques were not earmarked by ethnic, cultural, or other visible differences. In the nature of participant observation, one is likely to get caught up in a fairly restricted circle; a few useful contacts are acquired and relied on for introduction to others. The \u201cproblem of entry into West End society was particularly vexing,\u201d Gans writes. But eventually, he and his wife \u201cwere welcomed by one of our neighbors and became friends with them. As a result they invited us to many of their evening gatherings and introduced us to other neighbors, relatives and friends. \u2026 As time went on\u2026 other West Enders . . . introduced me to relatives and friends, although most of the social gatherings at which I participated were those of our first contact and their circle\u201d (1962, pp. 340-41; emphasis supplied). Thus, his account of cohesive groups is not inconsistent with overall fragmentation.\n\nNow, suppose that all ties in the West End were either strong or absent, and that the triad of figure 1 did not occur. Then, for any ego, all his friends were friends of one another, and all their friends were ego\u2019s friends as well. Unless each person was strongly tied to all others in the community, network structure did indeed break down into the isolated cliques posited above. (In terms of Davis\u2019s mathematical treatment, the overall network was \u2018\u2018clusterable,\u201d\u2019 with unique clusters [1967, p. 186].) Since it is unlikely that anyone could sustain more than a few dozen strong ties, this would, in fact, have been the result.\n\nDid strong ties take up enough of the West Enders\u2019 social time to make this analysis even approximately applicable? Gans reported that \u201csociability is a routinized gathering of a relatively unchanging peer group of family members and friends that takes place several times a week.\u201d Some \u201cparticipate in informal cliques and in clubs made up of unrelated people\u2026 . In number, and in the amount of time devoted to them, however, these groups are much less important than the family circle\u201d (1962, pp. 74, 80). Moreover, two common sources of weak ties, formal organizations and work settings, did not provide them for the West End; organization membership was almost nil (pp. 104-7) and few worked within the area itself, so that ties formed at work were not relevant to the community (p. 122).\n\nNevertheless, in a community marked by geographic immobility and lifelong friendships (p. 19) it strains credulity to suppose that each person would not have known a great many others, so that there would have been some weak ties. The question is whether such ties were bridges. See Jane Jacobs\u2019s excellent, intuitive, discussion of bridging ties (\u201chop-skip links\u2019\u2019) in community organization (1961, chap. 6.) If none were, then the community would be fragmented in exactly the same way as described above, except that the cliques would then contain weak as well as strong ties. (This follows, again, from Davis\u2019s analysis of \u201cclusterability,\u201d with strong and weak ties called \u2018positive\u2019 and absent ones \u201cnegative\u201d [1967].) Such a pattern is made plausible by the lack of ways in the West End to develop weak ties other than by meeting friends of friends (where \u201cfriend\u201d includes relatives)\u2014in which case the new tie is automatically not a bridge. It is suggested, then, that for a community to have many weak ties which bridge, there must be several distinct ways or contexts in which people may form them. The case of Charlestown, a working-class community which successfully organized against the urban renewal plan of the same city (Boston) against which the West End was powerless, is instructive in this respect: unlike the West End, it had a rich organizational life, and most male residents worked within the area (Keyes 1969, chap. 4).\n\nIn the absence of actual network data, all this is speculation. The hard information needed to show either that the West End was fragmented or that communities which organized successfully were not, and that both patterns were due to the strategic role of weak ties, is not at hand and would not have been simple to collect. Nor has comparable information been collected in any context. But a theoretical framework has, at least, been suggested, with which one could not only carry out analyses post hoc, but also predict differential capacity of communities to act toward common goals. A rough principle with which to begin such an investigation might be: the more local bridges (per person?) in a community and the greater their degree, the more cohesive the community and the more capable of acting in concert. Study of the origins and nature (strength and content, for example) of such bridging ties would then offer unusual insight into the social dynamics of the community.\n\n## Micro and Macro Network Models\n\nUnlike most models of interpersonal networks, the one presented here is not meant primarily for application to small, face-to-face groups or to groups in confined institutional or organizational settings. Rather, it is meant for linkage of such small-scale levels with one another and with larger, more amorphous ones. This is why emphasis here has been placed more on weak ties than on strong. Weak ties are more likely to link members of different small groups than are strong ones, which tend to be concentrated within particular groups.\n\nFor this reason, my discussion does not lend itself to elucidation of the internal structure of small groups. This point can be made more clearly by contrasting the model of this paper to one with which it shares many similarities, that of James Davis, Paul Holland, and Samuel Leinhardt (hereafter, the DHL model) (Davis 1970; Davis and Leinhardt 1971; Holland and Leinhardt 1970, 1971a, 19715; Davis, Holland, and Leinhardt 1971; Leinhardt 1972). The authors, inspired by certain propositions in George Homans\u2019s The Human Group (1950), argue that \u201cthe central proposition in structural sociometry is this: Interpersonal choices tend to be transitive\u2014if P chooses O and O chooses X, then P is likely to choose X\u201d (Davis et al. 1971, p. 309). When this is true without exception, a sociogram can be divided into cliques in which every individual chooses every other; any asymmetric choices or nonchoices are between such cliques, and asymmetry, if present, runs only in one direction. A partial ordering of cliques may thus be inferred. If mutual choice implies equal, and assymmetric choice unequal, status, then this ordering reflects the stratification structure of the group (Holland and Leinhardt 1971a, pp. 107-14).\n\nOne immediate difference between this model and mine is that it is cast in terms of \u201cchoices\u201d rather than ties. Most sociometric tests ask people whom they like best or would prefer to do something with, rather than with whom they actually spend time. If transitivity is built more into our cognitive than our social structure, this method might overstate its prevalence. But since the DHL model could recast in terms of ties, this is not a conclusive difference.\n\nMore significant is the difference in the application of my argument to transitivity. Let P choose O and O choose X (or equivalently, let X choose O and O choose P): then I assert that transitivity\u2014P choosing X (or X, P)\u2014is most likely when both ties\u2014P-O and O-X\u2014are strong, least likely when both are weak, and of intermediate probability if one is strong and one weak. Transitivity, then, is claimed to be a function of the strength of ties, rather than a general feature of social structure.\n\nThe justification of this assertion is, in part, identical with that offered earlier for the triad designated A-B-C. In addition, it is important to point out here that the DHL model was designed for small groups, and with increasing size of the group considered the rationale for transitivity weakens. If P chooses O and O chooses X, P should choose X out of consistency; but if P does not know or barely knows X, nonchoice implies no inconsistency. For the logic of transitivity to apply, a group must be small enough so that any person knows enough about every other person to be able to decide whether to \u201cchoose\u201d him, and encounters him often enough that he feels the need for such a decision. Including weak ties in my model, then, lessens the expectation of transitivity and permits analysis of inter-group relationships and also of amorphous chunks of social structure which an analyst may ferret out as being of interest, but which are not easily defined in terms of face-to-face groups. Anthropologists have recently referred to such chunks as \u201cquasi-groups\u201d (Mayer 1966; Boissevain 1968).\n\nSince, as I have argued above, weak ties are poorly represented in sociograms, there is little in the DHL empirical studies\u2014which apply statistical tests to sociometric data\u2014to confirm or disconfirm my argument on transitivity. One finding does lend itself to speculation, however. Leinhardt (1972) shows that the sociograms of schoolchildren conform more and more closely to the transitive model as they become older, sixth graders being the oldest tested. He interprets this as reflecting cognitive development\u2014increasing capacity to make use of transitive logic. If my assertion is correct, an alternative possibility would be that children develop stronger ties with increasing age. This is consistent with some theories of child development (see especially Sullivan 1953, chap. 16) and would imply, on my argument, greater transitivity of structure. Some support for this explanation comes from Leinhardt\u2019s finding that proportion of choices which were mutual was positively correlated with both grade level and degree of transitivity. In these sociograms, with an average of only about four choices per child, it seems likely that most mutual choices reflected strong ties (see n. 7, above).\n\n## Conclusion\n\nThe major implication intended by this paper is that the personal experience of individuals is closely bound up with larger-scale aspects of social structure, well beyond the purview or control of particular individuals. Linkage of micro and macro levels is thus no luxury but of central importance to the development of sociological theory. Such linkage generates paradoxes: weak ties, often denounced as generative of alienation (Wirth 1938) are here seen as indispensable to individuals\u2019 opportunities and to their integration into communities; strong ties, breeding local cohesion, lead to overall fragmentation. Paradoxes are a welcome antidote to theories which explain everything all too neatly.\n\nThe model offered here is a very limited step in the linking of levels; it is a fragment of a theory. Treating only the strength of ties ignores, for instance, all the important issues involving their content. What is the relation between strength and degree of specialization of ties, or between strength and hierarchical structure? How can \u201cnegative\u201d ties be handled? Should tie strength be developed as a continuous variable? What is the developmental sequence of network structure over time?\n\nAs such questions are resolved, others will arise. Demography, coalition structure, and mobility are just a few of the variables which would be of special importance in developing micro-macro linkage with the help of network analysis; how these are related to the present discussion needs specification. My contribution here is mainly, then, exploratory and programmatic, its primary purpose being to generate interest in the proposed program of theory and research.\n\n## References\n\n\u2022 Barnes, J. A. 1969. \u201cNetworks and Political Process.\u201d In Social Networks in Urban Situations, edited by J. C. Mitchell. Manchester: Manchester University Press.\n\u2022 Becker, Marshall. 1970. \u201cSociometric Location and Innovativeness.\u201d American Socio- logical Review 35 (April): 267-82.\n\u2022 Berscheid, E., and E. Walster. 1969. Interpersonal Attraction. Reading, Mass.: Addison- Wesley.\n\u2022 Boissevain, J. 1968. \u201cThe Place of Non-Groups in the Social Sciences.\u201d Man 3 (December): 542-56.\n\u2022 Bott, Elizabeth. 1957. Family and Social Network. London: Tavistock.\n\u2022 Bramel, D. 1969. \u201cInterpersonal Attraction, Hostility and Perception.\u201d In Experimental Social Psychology, edited by Judson Mills. New York: Macmillan.\n\u2022 Brown, David. 1967. The Mobile Professors. Washington, D.C.: American Council on Education.\n\u2022 Brown, Roger. 1965. Social Psychology. New York: Free Press.\n\u2022 Coleman, J. S. 1960. \u201cThe Mathematical Study of Small Groups.\u201d In Mathematical Thinking in the Measurement of Behavior, edited by H. Solomon. Glencoe: Free Press.\n\u2022 Coleman, J. S., E. Katz, and H. Menzel. 1966. Medical Innovation: A Diffusion Study Indianapolis: Bobbs-Merrill.\n\u2022 Dahl, Robert. 1961. Who Governs? New Haven, Conn.: Yale University Press. Davies, J. C. 1966. Nezghborhood Groups and Urban Renewal. New York: Columbia University Press.\n\u2022 Davis, James A. 1963. \u201cStructural Balance, Mechanical Solidarity and Interpersonal Relations \u201d American Journal of Sociology 68 (January): 444-62.\n\u2022 \u2013. 1967. \u201cClustering and Structural Balance in Graphs.\u201d Human Relations 20\n\u2022 (May): 181-87.\n\u2022 \u2013 1969. \u201cSocial Structures and Cognitive Structures.\u201d In R. P. Abelson et al, Theories of Cognitive Consistency. Chicago: Rand McNally.\n\u2022 \u2013 1970. \u201cClustering and Hierarchy in Interpersonal Relations.\u201d American Sociological Review 35 (October): 843-52.\n\u2022 Davis, James A., P. Holland, and S. Leinhardt. 1971. \u201cComment.\u201d American Socio- logical Review 36 (April): 309-11.\n\u2022 Davis, James A., and S. Leinhardt. 1971. \u201cThe Structure of Positive Interpersonal Relations in Small Groups.\u201d In Sociological Theories in Progress. Vol. 2, edited by J. Berger, M. Zelditch, and B. Anderson. Boston: Houghton-Mifflin.\n\u2022 Epstein, A. 1969. \u201cThe Network and Urban Social Organization.\u201d In Social Networks in Urban Situations, edited by J. C. Mitchell. Manchester: Manchester University Press.\n\u2022 Frankenberg, R. 1965. Communities in Britain. Baltimore: Penguin.\n\u2022 Gans, Herbert. 1962. The Urban Villagers. New York: Free Press.\n\u2022 Granovetter, M. S. 1970. \u201cChanging Jobs: Channels of Mobility Information in a Suburban Community.\u201d Doctoral dissertation, Harvard University.\n\u2022 Harary, F. 1965. \u201cGraph Theory and Group Structure.\u201d In Readings in Mathematical Psychology. Vol. 2, edited by R. Luce, R. Bush, and E. Galanter. New York: Wiley. Harary, F., R. Norman, and D. Cartwright. 1965. Structural Models. New York: Wiley.\n\u2022 Heider, F. 1958. The Psychology of Interpersonal Relations. New York: Wiley. Holland, Paul, and S. Leinhardt. 1970. \u201cDetecting Structure in Sociometric Data.\u201d American Journal of Sociology 76 (November): 492-513.\n\u2022 \u2013 1971a. \u201cTransitivity in Structural Models of Small Groups.\u2019 Comparative\n\u2022 Group Studies 2:107-24.\n\u2022 \u2013 19716, \u201cMasking: The Structural Implications of Measurement Error in Sociometry.\u201d Mimeographed. Pittsburgh: Carnegie-Mellon University.\n\u2022 Homans, George. 1950. The Human Group. 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Poughkeepsie, N.Y.: Vassar College.\n\u2022 Korte, Charles, and Stanley Milgram. 1970. \u201cAcquaintance Networks between Racial Groups.\u201d Journal of Personality and Social Psychology 15 (June): 101-8.\n\u2022 Laumann, Edward. 1968. \u201cInterlocking and Radial Friendship Networks: A Cross-sectional Analysis.\u201d Mimeographed, Ann Arbor: University of Michigan.\n\u2022 Laumann, Edward, and H. Schuman. 1967. \u201cOpen and Closed Structures.\u201d Paper prepared for the 1967 ASA meeting. Mimeographed.\n\u2022 Leinhardt, Samuel. 1972. \u201cDevelopmental Change in the Sentiment Structure of childrens\u2019 Groups.\u201d American Sociological Review 37 (April): 202-12.\n\u2022 Mayer, Adrian. 1966. \u201cThe Significance of Quasi-Groups in the Study of Complex Societies.\u201d In The Social Anthropology of Complex Societies, edited by M. Banton. New York: Praeger.\n\u2022 Mayer, Phillip. 1961. Townsmen or Tribesmen? Capetown: Oxford.\n\u2022 Mazur, B. 1971. \u201cComment.\u201d American Sociological Review 36 (April): 308-9. Milgram, Stanley. 1967. \u201cThe Small-World Problem.\u201d Psychology Today 1 (May): 62-67.\n\u2022 Mitchell, J. Clyde. 1969. Social Networks in Urban Situations. Manchester: Manchester University Press.\n\u2022 Newcomb, T. M. 1961. The Acquaintance Process. New York: Holt, Rinehart & Winston.\n\u2022 Parnes, Herbert. 1954. Research on Labor Mobility. New York: Social Science Research Council.\n\u2022 Precker, Joseph. 1952. \u201cSimilarity of Valuings as a Factor in Selection of Peers and Near-Authority Figures.\u201d Journal of Abnormal and Social Psychology 47, suppl. (April): 406-14.\n\u2022 Rapoport, Anatol. 1953a. \u201cSpread of Information through a Population with Socio- Structural Bias. I. Assumption of Transitivity.\u201d Bulletin of Mathematical Biophysics 15 (December): 523-33.\n\u2022 \u2013 1953b. \u201cSpread of Information through a Population with Socio-Structural Bias. II. Various Models with Partial Transitivity.\u201d Bulletin of Mathematical Biophysics 15 (December): 535-46.\n\u2022 \u2013 1954. \u201cSpread of Information through a Population with Socio-Structural Bias. III. Suggested Experimental Procedures.\u201d Bulletin of Mathematical Biophysics 16 (March): 75-81.\n\u2022 \u2013 1963. \u201cMathematical Models of Social Interaction.\u201d In Handbook of Mathe- matical Psychology. Vol. 2, edited by R. Luce, R. Bush, and E. Galanter. New York: Wiley.\n\u2022 Rapoport, A., and W. Horvath. 1961. \u201cA Study of a Large Sociogram.\u201d Behavioral Science 6:279-91.\n\u2022 Rogers, Everett. 1962. Diffusion of Innovations. New York: Free Press.\n\u2022 Shapero, Albert, Richard Howell, and James Tombaugh. 1965. The Structure and Dynamics of the Defense R & D Industry. Menlo Park, Calif.: Stanford Research Institute.\n\u2022 Simmel, Georg. 1950. The Sociology of Georg Simmel. New York: Free Press. Solomonoff, Ray, and A. Rapoport. 1951. \u201cConnectivity of Random Nets.\u201d Bulletin of Mathematical Biophysics 13 (June): 107-17.\n\u2022 Sullivan, Harry Stack. 1953. The Interpersonal Theory of Psychiatry. New York: Norton.\n\u2022 Tilly, Charles. 1969. \u201cCommCuityn:Uirbatnizyati:on.\u201d Mimeographed. Ann Arbor: University of Michigan.\n\u2022 Travers, Jeffrey, and S. Milgram. 1969. \u201cAn Experimental Study of the \u2018Small-World\u2019 Problem.\u201d Sociometry 32 (December): 425-43.\n\u2022 Wirth, Louis. 1938. \u201cUrbanism as a Way of Life.\u201d American Journal of Sociology 44 (July): 1-24.","date":"2021-10-24 04:05:54","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5565111637115479, \"perplexity\": 2136.465068261677}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-43\/segments\/1634323585837.82\/warc\/CC-MAIN-20211024015104-20211024045104-00296.warc.gz\"}"} | null | null |
\section{Introduction}
Einstein's field equations can be deduced as equations of motion of the Einstein-Hilbert functional.
In the classical context one considers (Lorentzian or Riemannian) manifolds equipped only with the Levi-Civita connection.\medskip
\noindent
In the 1920s \'E.~Cartan investigated general orthogonal connections i.e.~connections which are compatible with the metric.
The difference of such a connection and the Levi-Civita connection is called torsion.
In his seminal articles \cite{Cartan23}, \cite{Cartan24} and \cite{Cartan25} Cartan observed that in general the torsion tensor splits into three components: the vectorial torsion, the totally anti-symmetric one and the one of Cartan type.
Taking the scalar curvature of orthogonal connections as the Langrangian one attains the Einstein-Cartan-Hilbert functional.
Its critial points are exactly Einstein manifolds, in particular the torsion of a critical point is zero.
(Physics' literature refers to this fact as the Palatini formalism.)\medskip
\noindent
Aiming for a unified theory of gravity and the other forces Chamseddine and Connes introduced the spectral action principle (\cite{ConnesChamseddine1}).
It states that any reasonable physical action is determined only by the spectrum of a Dirac operator.
Specifically, the Chamseddine-Connes spectral action comprises the Einstein-Hilbert action and the full bosonic part of the action of the Standard Model of Particle Physics, if one considers suitable twisted Dirac operators based on Levi-Civita connections.
It even predicts the correct Higgs potential necessary for the electro-weak symmetry breaking and allows to put constraints on the Higgs mass.
Into this framework orthogonal connections with totally anti-symmetric torsion have been incorporated in \cite{Torsion}, and by Iochum, Levy and Vassilevich in \cite{IochumLevy} for the purely gravitational action on manifolds with boundary.
Restricting to connections with totally anti-symmetric torsion was geometrically justified by the fact that the geodesics of such connections coincide with those of the Levi-Civita connection.\medskip
\noindent
Throughout the present article we consider closed manifolds, i.e. compact ones without boundary. For these we deal with the full class of orthogonal connections.
We review Cartan's classification and Einstein-Cartan theory in section~\ref{section1}, and we compute some curvature quantities in the case of totally anti-symmetric torsion in section~\ref{section2}.
In section~\ref{section3} we describe the Dirac operators constructed from orthogonal connections, and we notice that the vectorial component of the torsion has to be zero to assure that the Dirac operator is symmetric. This follows from a result by Friedrich and Sulanke (\cite{FriedrichSulanke}).
We show that the Cartan type component of the torsion has no effect on the Dirac operator (even pointwise) which provides another good reason to consider only anti-symmetric torsion.
\medskip
\noindent
In this setting many examples for commutative geometries in the sense of Connes' spectral triples (\cite{Connes96}) can be supplied.
Given the Reconstruction Theorem (\cite{Connes08}), we remark that in the even-dimensional case anti-symmetric torsion is reconstructable from the spectral data.
In four dimensions we calculate the purely gravitational part of the Chamseddine-Connes spectral action in some detail.
We find that some terms of the action given in \cite{Torsion} actually vanish, thus confirming the result by \cite{IochumLevy}.
For this action we derive the equations of motion in Theorem~\ref{Equation_of_Motion}.
One of them is a Proca equation for the torsion $3$-form which suggests an interpretation of the torsion as massive vector boson.
The set of critial points of the action, i.e.~the solutions of the equations of motion, contains all Einstein manifolds (with zero torsion).
Furthermore, in Lemmas~\ref{lemma_Robertson_Walker} and~\ref{warpedtorsiontorus} we exclude critical points which are warped products and carry special choices of non-zero torsion.
\medskip
\noindent
We tried to keep this text elementary and accessible, and we hope that it may also serve as an introduction for non-experts.
\medskip
\noindent
{\bf Acknoledgement:} The authors appreciate funding by the Deutsche Forschungsgemeinschaft, in particular by the SFB {\it Raum-Zeit-Materie}.
We would like to thank Christian B\"ar and Thomas Sch\"ucker for their support and helpful discussions.
\section{Orthogonal connections on Riemannian manifolds}\label{section1}
We consider an $n$-dimensional manifold $M$ equipped with some Riemannian metric $g$.
Let $\nabla^{g}$ denote the Levi-Civita connection on the tangent bundle.
For any affine connection $\nabla$ on the tangent bundle there exists a $(2,1)$-tensor field $A$ such that
\begin{equation}\label{def_Zusammenhang_mit_Torsion}
\nabla_X Y= \nabla^g_X Y +A(X,Y)
\end{equation}
for all vector fields $X,Y$.\medskip
\noindent
In this article we will require all connections $\nabla$ to be {\it orthogonal}, i.e.~for all vector fields $X,Y,Z$ one has
\begin{equation}\label{def_metrisch}
\partial_X \left\langle Y,Z \right\rangle=\left\langle \nabla_XY,Z\right\rangle+\left\langle Y,\nabla_XZ\right\rangle,
\end{equation}
where $\left\langle\cdot,\cdot\right\rangle$ denotes the scalar product given by the Riemannian metric $g$.
For any tangent vector $X$ one gets from (\ref{def_Zusammenhang_mit_Torsion}) and (\ref{def_metrisch}) that the endomorphism $A(X,\cdot)$ is skew-adjoint:
\begin{equation}\label{A_schiefadjungiert}
\left\langle A(X,Y), Z\right\rangle = -\left\langle Y,A(X, Z)\right\rangle.
\end{equation}
\noindent
Next, we want to express some curvature quantities for $\nabla$ in terms of $A$ and curvature quantities for $\nabla^g$.
To that end we fix some point $p\in M$, and we extend any tangent vectors $X,Y,Z,W\in T_p M$ to vector fields again denoted by $X,Y,Z,W$ being {\it synchronous} in $p$, which means
\begin{equation*}
\nabla^g_VX=\nabla^g_VY=\nabla^g_VZ=\nabla^g_VW=0 \quad\mbox{ for any tangent vector }V\in T_pM.
\end{equation*}
Furthermore, we choose a local orthogonal frame of vector fields $E_1,\ldots,E_n$ on a neighbourhood of $p$, all being synchronous in $p$.
Then the Lie bracket $[X,Y]=\nabla^g_XY-\nabla^g_YX=0$ vanishes in $p$, and synchronicity in $p$ implies
\begin{equation*}
\nabla_X\nabla_YZ
=\nabla^g_X\nabla^g_YZ
+\left(\nabla^g_X A \right)(Y,Z) + A\left(X,A(Y,Z)\right)
\end{equation*}
Hence, in $p$ the Riemann tensor of $\nabla$ reads as
\begin{align}
\label{Riemann_Torsion_Allgemein}
\operatorname{Riem}(X,Y)Z&=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ -\nabla_{[X,Y]}Z\nonumber \\
&=\operatorname{Riem}^g(X,Y)Z +\left(\nabla^g_X A \right)(Y,Z) -\left(\nabla^g_Y A \right)(X,Z)
+ A\left(X,A(Y,Z)\right)-A\left(Y,A(X,Z)\right)
\end{align}
where $\operatorname{Riem}^g$ denotes the Riemann tensor of $\nabla^g$.
We note that $\operatorname{Riem}(X,Y)Z$ is anti-symmetric in $X$ and $Y$.
And by differentiation of (\ref{A_schiefadjungiert}) we get that $(\nabla^g_{E_i} A )(E_j,\cdot)$ and $(\nabla^g_{E_j} A )(E_i,\cdot)$ are skew-adjoint, and therefore we have
\begin{equation}\label{Riemann_Torsion_antisymm_in_34_indices}
\left\langle\operatorname{Riem}(E_i,E_j)E_k,E_l\right\rangle = -
\left\langle\operatorname{Riem}(E_i,E_j)E_l,E_k\right\rangle.
\end{equation}
In general, $\operatorname{Riem}$ does not satisfy the Bianchi identity.
The Ricci curvature of $\nabla$ is defined as
\begin{equation*}
\operatorname{ric}(X,Y)=\operatorname{tr}\,\left( V\mapsto \operatorname{Riem}(V,X)Y\right),
\end{equation*}
by (\ref{Riemann_Torsion_Allgemein}) this can be expressed as
\begin{eqnarray}
\operatorname{ric}(X,Y)&=& \sum_{i=1}^n \left\langle \operatorname{Riem}(E_i,X)Y,E_i\right\rangle \nonumber\\
&=& \operatorname{ric}^g(X,Y)
+\sum_{i=1}^n \left(
\left\langle
\left(\nabla^g_{E_i}A \right)(X,Y),E_i
\right\rangle
-
\left\langle
\left(\nabla^g_{X}A \right)(E_i,Y),E_i
\right\rangle
\right)\nonumber\\
&&\qquad\qquad
+\sum_{i=1}^n\left(
-\left\langle
A(X,Y),A(E_i,E_i)
\right\rangle
+\left\langle
A(E_i,Y),A(X,E_i)
\right\rangle
\right)
\label{Ricci_Torsion_Allgemein}
\end{eqnarray}
where $\operatorname{ric}^g$ is the Ricci curvature of $\nabla^g$.
We have used that $A(E_i,\cdot)$ and $A(X,\cdot)$ are skew-adjoint.
\medskip
\noindent
One obtains the scalar curvature $R$ of $\nabla$ by taking yet another trace,
in $p$ it is given as $R=\sum_{j=1}^n \operatorname{ric}(E_j,E_j)$.
For the following calculation we use that $(\nabla^g_V A)(X,\cdot)$ is skew-adjoint for any tangent vectors $V,X$,
and we get:
\begin{eqnarray}
R&=&R^g+
\sum_{i,j=1}^n
\left(
\left\langle
\left(\nabla^g_{E_i}A \right)(E_j,E_j),E_i
\right\rangle
+
\left\langle
E_j, \left(\nabla^g_{E_j}A \right)(E_i,E_i)
\right\rangle
\right)
\nonumber\\
&& \qquad
+
\sum_{i,j=1}^n
\left(
-\left\langle A(E_j,E_j),A(E_i,E_i)
\right\rangle
+
\left\langle A(E_i,E_j),A(E_j,E_i)
\right\rangle
\right)\nonumber\\
&=& R^g+ 2\, \sum_{i,j=1}^n
\left\langle
\left(\nabla^g_{E_i}A \right)(E_j,E_j),E_i
\right\rangle
-
\big\|\sum_{i=1}^n A(E_i,E_i) \big\|^2
+
\sum_{i,j=1}^n
\left\langle A(E_i,E_j),A(E_j,E_i)
\right\rangle
\label{Scalar_Allgemein}
\end{eqnarray}
where $R^g$ denotes the scalar curvature of $\nabla^g$.\medskip
\noindent
The classification of orthogonal connections with torsions traces back to \cite[Chap.~VIII]{Cartan25}.
Here we adopt the notations of \cite[Chap.~3]{Tricerri} (see also \cite{AgricolaSrni}).
From (\ref{A_schiefadjungiert}) we know that the torsion tensor $A(X,\cdot)$ is skew-adjoint on the tangent space $T_pM$.
Any torsion tensor $A$ induces a $(3,0)$-tensor by setting
\begin{equation*}
A_{XYZ} =\left\langle A(X,Y),Z \right\rangle\qquad\mbox{ for any } X,Y,Z\in T_pM.
\end{equation*}
We define the space of all possible torsion tensors on $T_pM$ by
\begin{equation*}
\mathcal{T}(T_pM)=\left\{ A\in {\bigotimes}^3T^*_pM\; \big| \; A_{XYZ}=-A_{XZY}\quad\forall X,Y,Z\in T_pM \right\}.
\end{equation*}
This vector space carries a scalar product
\begin{eqnarray}
\left\langle A,A' \right\rangle= \sum_{i,j,k=1}^n A_{E_i E_j E_k} A'_{E_i E_j E_k},
\label{30tensorprod}
\end{eqnarray}
and the orthogonal group $O(T_pM)$ acts on $\mathcal{T}(T_pM)$ via $(\alpha A)_{XYZ}=A_{\alpha^{-1}(X)\alpha^{-1}(Y)\alpha^{-1}(Z)}$.\medskip
\noindent
For $A\in\mathcal{T}(T_pM)$ and $Z\in T_pM$ one denotes the trace over the first two entries by
\begin{equation}\label{definition_c12}
c_{12}(A)(Z)=\sum_{i=1}^n A_{E_i E_i Z}.
\end{equation}
The space of quadratic invariants on $\mathcal{T}(T_pM)$ with respect to the $O(T_pM)$-representation is spanned by the three quadratic forms
\begin{eqnarray}
\|A \|^2 &=& \langle A,A \rangle\,, \label{def_torsionsnorm}\\
\langle A,\widehat{A} \rangle &=& \sum_{i,j,k=1}^n A_{E_i E_j E_k}A_{E_j E_i E_k}\,, \label{def_torsionsnormtwisted}\\
\|c_{12}(A)\|^2 &=& \sum_{i,j,k=1}^n A_{E_i E_i E_k}A_{E_j E_j E_k}\,.\label{def_torsionsspurnorm}
\end{eqnarray}
Here $\widehat{A}$ denotes the $(3,0)$-tensor obtained from $A$ by interchanging the first two slots,
i.e.~$\widehat{A}_{XYZ} = A_{YXZ}$, for all tangent vectors $X,Y,Z$.
\begin{thm}\label{cartanklassifikation}
For $\dim(M)\ge 3$ one has the following decomposition of $\mathcal{T}(T_pM)$ into irreducible $O(T_pM)$-subrepresentations:
\[
\mathcal{T}(T_pM)\,=\,\mathcal{T}_1(T_pM)\,\oplus\, \mathcal{T}_2(T_pM)\,\oplus \,\mathcal{T}_3(T_pM).
\]
This decomposition is orthogonal with respect to $\langle\cdot,\cdot\rangle$, and it is given by
\begin{eqnarray*}
\mathcal{T}_1(T_pM)&=& \left\{ A\in \mathcal{T}(T_pM) \;\big|\; \exists V \mbox{ s.t. } \forall X,Y,Z:\, A_{XYZ}=
\langle X,Y \rangle \langle V,Z \rangle-\langle X,Z \rangle\langle V,Y \rangle\right\}, \\
\mathcal{T}_2(T_pM)&=& \left\{ A\in \mathcal{T}(T_pM) \;\big|\; \forall X,Y,Z:\,A_{XYZ}=-A_{YXZ}\right\}, \\
\mathcal{T}_3(T_pM)&=& \left\{ A\in \mathcal{T}(T_pM) \;\big|\; \forall X,Y,Z:\,A_{XYZ}+ A_{YZX}+ A_{ZXY}=0\mbox{ and } c_{12}(A)(Z)=0\right\}.
\end{eqnarray*}
\noindent
For $\dim(M) = 2$ the $O(T_pM)$-representation
\[
\mathcal{T}(T_pM)\,=\,\mathcal{T}_1(T_pM)
\]
is irreducible.{\hfill$\boxbox$}
\end{thm}
The above theorem is just Thm.~3.1 from \cite{Tricerri}.
The connections whose torsion tensor is contained in $\mathcal{T}_1(T_pM)\cong T_pM$ are called {\it vectorial}.
Those whose torsion tensor is in $\mathcal{T}_2(T_pM)={\bigwedge}^3T^*_pM$ are called {\it totally anti-symmetric},
and those with torsion tensor in $\mathcal{T}_3(T_pM)$ are called {\it of Cartan type}.\medskip
\noindent
We note that any Cartan type torsion tensor $A\in \mathcal{T}_3(T_pM)$ is trace-free in any pair of entries, i.e.~for any $Z$ one has
\[
\sum_{i=1}^n A_{E_i E_i Z}=0, \qquad
\sum_{i=1}^n A_{E_i Z E_i}=0, \qquad
\sum_{i=1}^n A_{Z E_i E_i}=0.
\]
The second equality holds as $A\in \mathcal{T}(T_pM)$, and the third one follows from the cyclic identity $A_{XYZ}+ A_{YZX}+ A_{ZXY}=0$.
\begin{rem}\label{spurnullfuerT2undT3}
The invariant quadratic form given in (\ref{def_torsionsspurnorm}) has the null space $\mathcal{T}_2(T_pM)\oplus\mathcal{T}_3(T_pM)$.
More precisely, one has $A\in \mathcal{T}_2(T_pM)\oplus\mathcal{T}_3(T_pM)$ if and only if
$c_{12}(A)(Z)=0$ for any $Z\in T_pM$.{\hfill$\boxbox$}
\end{rem}
\begin{rem}\label{orthogonalTiforotherproduct}
The decomposition given in Theorem~\ref{cartanklassifikation} is orthogonal with respect to the bilinear form given in (\ref{def_torsionsnormtwisted}), i.e.~for $\alpha,\beta\in\{1,2,3\}$, $\alpha\ne\beta$, and $A_\alpha\in \mathcal{T}_\alpha(T_pM)$, $A_\beta\in \mathcal{T}_\beta(T_pM)$ one gets $\langle A_\alpha, \widehat{A}_\beta \rangle=0$. {\hfill$\boxbox$}
\end{rem}
\noindent
Varying the base point $p\in M$, the decomposition in Theorem~\ref{cartanklassifikation} is parallel with respect to the Levi-Civita connection $\nabla^g$ (induced on $(3,0)$-tensor fields).
And from Theorem~\ref{cartanklassifikation} one gets immediately:
\begin{cor}\label{connectionswithtorsionsclassified}
For any orthogonal connection $\nabla$ on some Riemannian manifold of dimension $n\ge 3$
there exist a vector field $V$, a $3$-form $T$ and a $(3,0)$-tensor field $S$ with $S_p\in \mathcal{T}_3(T_pM)$ for any $p\in M$ such that $\nabla_XY=\nabla^g_XY+A(X,Y)$ takes the form
\[
A(X,Y)=\langle X,Y \rangle V- \langle V,Y \rangle X + T(X,Y,\cdot)^\sharp + S(X,Y,\cdot)^\sharp ,
\]
where $T(X,Y,\cdot)^\sharp$ and $S(X,Y,\cdot)^\sharp$ are the unique vectors with
\begin{eqnarray}
T(X,Y,Z)=\left\langle T(X,Y,\cdot)^\sharp,Z\right\rangle\mbox{ and }
S(X,Y,Z)=\left\langle S(X,Y,\cdot)^\sharp,Z\right\rangle\mbox{ for all }Z.
\label{sharp_def}
\end{eqnarray}
For any orthogonal connection these $V,T,S$ are unique.{\hfill$\boxbox$}
\end{cor}
\begin{lemma}
The scalar curvature of an orthogonal connection is given by
\[
R= R^g +2(n-1)\,\operatorname{div}^{\nabla^g}(V) -(n-1)(n-2)\,\|V\|^2 -\|T\|^2+\tfrac{1}{2}\,\|S\|^2
\]
with $V,T,S$ as in Corollary~\ref{connectionswithtorsionsclassified}, and $\operatorname{div}^{\nabla^g}(V)$ is the divergence of the vector field $V$ taken with respect to the Levi-Civita connection.
\label{Scalar_Norm}
\end{lemma}
\pf{ With the notations from (\ref{definition_c12})--(\ref{def_torsionsspurnorm}) we rewrite (\ref{Scalar_Allgemein}) as
\begin{equation}\label{eqn_R_general_morefancy}
R=R^g+2\,\sum_{i=1}^n c_{12}\left(\nabla^g_{E_i}A \right)(E_i) -\left\|c_{12}(A) \right\|^2 + \langle A, \widehat{A} \rangle.
\end{equation}
By Remark~\ref{spurnullfuerT2undT3} only the vectorial part of the torsion contributes to the $c_{12}$-terms, and therefore one gets
\begin{eqnarray}
\sum_{i=1}^n c_{12}\left(\nabla^g_{E_i}A \right)(E_i)&=& \sum_{i,j=1}^n
\left(
\langle E_j,E_j \rangle\, \langle \nabla^g_{E_i}V,E_i\rangle- \langle\nabla^g_{E_i}V,E_j \rangle\,\langle E_i,E_j\rangle
\right)\nonumber\\
&=& (n-1)\,\operatorname{div}^{\nabla^g}(V),\label{term_nablaspur}\\
\left\|c_{12}(A) \right\|^2 &=& \Big\|\sum_{j=1}^n c_{12}(A)(E_j)\,E_j\Big\|^2 \nonumber\\
&=& \Big\|\sum_{i,j=1}^n \left( \langle E_i,E_i \rangle\, \langle V,E_j\rangle\,E_j
- \langle V,E_i \rangle\,\langle E_i,E_j\rangle\,E_j
\right) \Big\|^2 \nonumber\\
&=& \Big\|\sum_{i=1}^n \left( V- \langle V,E_i \rangle\,E_i\right) \Big\|^2 \nonumber\\
&=& \|(n-1)\, V\|^2.\label{term_c12spur}
\end{eqnarray}
In order to compute the last term in (\ref{eqn_R_general_morefancy}) we decompose $A=A_1+A_2+A_3$ with $A_\alpha\in\mathcal{T}_\alpha(T_pM)$.
From Remark~\ref{orthogonalTiforotherproduct} we get
\[
\langle A ,\widehat{A} \rangle=\sum_{\alpha=1}^3\langle A_\alpha ,\widehat{A}_\alpha \rangle.
\]
For the vectorial part we get
\begin{eqnarray}
\langle A_1,\widehat{A}_1\rangle &=& \sum_{i,j,k=1}^n \big( \delta_{ij} \langle V,E_k\rangle -\delta_{ik}\langle V,E_j\rangle\big)\cdot
\big( \delta_{ji} \langle V,E_k\rangle -\delta_{jk}\langle V,E_i\rangle\big)
\nonumber\\
&=& \sum_{i,j,k=1}^n \big(\delta_{ij}\, \langle V,E_k\rangle^2 - \delta_{ij}\delta_{jk}\, \langle V,E_k\rangle \langle V,E_i\rangle
-\delta_{ik}\delta_{ji}\, \langle V,E_j\rangle \langle V,E_k\rangle
+\delta_{ik}\delta_{jk}\, \langle V,E_j\rangle \langle V,E_i\rangle
\big)
\nonumber\\
&=& (n-1)\,\|V\|^2
\end{eqnarray}
For the totally anti-symmetric part we get
\begin{equation}
\langle A_2,\widehat{A}_2\rangle \; = \sum_{i,j,k=1}^n T_{E_i E_j E_k}\,T_{E_j E_i E_k}
= -\sum_{i,j,k=1}^n T_{E_i E_j E_k}\,T_{E_i E_j E_k}
\; =\;-\|T\|^2.
\end{equation}
Finally, for the Cartan-type part we get
\begin{eqnarray}
\langle A_3,\widehat{A}_3\rangle &=& \sum_{i,j,k=1}^n S_{E_i E_j E_k}\,S_{E_j E_i E_k}\nonumber\\
&=& -\sum_{i,j,k=1}^n \big( S_{E_i E_j E_k}\,S_{´E_i E_k E_j}+S_{E_i E_j E_k}\,S_{E_k E_j E_i } \big) \label{cyclicschritt}\\
&=& \sum_{i,j,k=1}^n S_{E_i E_j E_k}\,S_{´E_i E_j E_k}- \sum_{i,j,k=1}^nS_{E_i E_k E_j}\,S_{E_k E_i E_j } \label{antisymmschritt}\\
&=& \tfrac12\,\|S\|^2, \label{cartantype-anteil}
\end{eqnarray}
where (\ref{cyclicschritt}) is due to the cyclic identity for $S$, (\ref{antisymmschritt}) follows from the anti-symmetry in the last two entries, and
$ \sum_{i,j,k=1}^nS_{E_i E_k E_j}\,S_{E_k E_i E_j }= \langle A_3,\widehat{A}_3\rangle $ implies (\ref{cartantype-anteil}).
Plugging (\ref{term_nablaspur})--(\ref{cartantype-anteil}) into (\ref{eqn_R_general_morefancy}) finishes the proof.
{\hfill$\boxbox$}
}
\begin{cor}\label{Einstein_Cartan_Hilbert}
Let $M$ be a closed manifold of dimension $n\ge 3$ with Riemannian metric $g$ and orthogonal connection $\nabla$.
Let $\operatorname{dvol}$ denote the Riemannian volume measure taken with respect to $g$.
Then the Einstein-Cartan-Hilbert functional is
\[
\int_M R \operatorname{dvol} = \int_M R^g \operatorname{dvol} -\,(n-1)(n-2)\,\int_M \|V\|^2\operatorname{dvol} -\int_M \|T\|^2 \operatorname{dvol}+\tfrac{1}{2}\,\int_M \|S\|^2\operatorname{dvol}.
\]
\end{cor}
\noindent
Considering variations over all Riemannian metrics, for which the volume $\operatorname{vol}_g(M)$ stays fixed, and all orthogonal connections (i.e.~over all torsion tensors), we get that $(M,g,\nabla)$ is a critical point of the Einstein-Cartan-Hilbert functional if and only if $(M,g)$ is an Einstein manifold and $\nabla=\nabla^g$ is the Levi-Civita connection (i.e.~$V\equiv 0$, $T\equiv 0$ and $S\equiv 0$).
\noindent
For an in depth treatment of the physical consequences of Einstein-Cartan-Hilbert theory
in Lorentzian geometry we refer to the classical review \cite{HHKN76} and the more
recent overview \cite{S02} and references therein.
\section{Curvature calculations in case of totally anti-symmetric torsion
in four dimensions}\label{section2}
Let us collect now some equalities involving curvature tensors and the
totally anti-symmetric torsion. To keep the main part of this paper as readable
as possible the proofs of the following theorems and lemmata have been
allocated to the appendix.
\noindent
We consider a $4$-dimensional manifold $M$ equipped with a Riemannian metric $g$.
Let $\nabla^{g}$ denote the Levi-Civita connection on the tangent bundle.
We fix some $3$-form $T$ on $M$ and some $s\in\mathbb{R}$, and we are studying the connection $\nabla$ which is given by
\begin{equation}
\label{vectortorsionconnection}
\nabla_X Y = \nabla^g_X Y + s \, T(X,Y,\cdot)^\sharp
\end{equation}
for any vector fields $X$ and $Y$ on $M$ and $T(X,Y,\cdot)^\sharp$ is defined as in (\ref{sharp_def}).
Hence $\nabla$ is an orthonormal connection with totally anti-symmetric torsion.
\subsection{Pointwise equalities}
In the following we want to express some curvature quantities for $\nabla$ in terms of $T$ and curvature quantities for $\nabla^g$. For the Riemann curvature
and the scalar curvature we will calculate the norms explicitly
in terms of $T$, the Levi-Civita connection and its curvatures.
The norm of the Ricci curvature is given in the appendix.
\medskip
\noindent
As in section~\ref{section1} we fix some point $p\in M$, and we extend any tangent vectors $X,Y,Z,W\in T_p M$ to vector fields again denoted by $X,Y,Z,W$ being synchronous in $p$.
Hence we obtain from (\ref{Riemann_Torsion_Allgemein}) the
Riemann curvature of $\nabla$
\begin{eqnarray}
\left\langle \operatorname{Riem}(X,Y)Z,W \right\rangle&=&\left\langle \operatorname{Riem}^g(X,Y)Z,W \right\rangle +
s\; \left( \left(\nabla^g_XT \right)(Y,Z,W)-\left(\nabla^g_YT \right)(X,Z,W)\right) \nonumber \\
&&\quad +s^2\,\left( T\left(X,T(Y,Z,\cdot)^\sharp,W \right) -T\left(Y,T(X,Z,\cdot)^\sharp,W \right) \right)
\nonumber\\
&=&\left\langle \operatorname{Riem}^g(X,Y)Z,W \right\rangle + s \; \left( \left(\nabla^g_XT \right)(Y,Z,W)-\left(\nabla^g_YT \right)(X,Z,W)\right) \nonumber \\
&&\quad +s^2\,\left(\left\langle T(X,Z,\cdot)^\sharp,T(Y,W,\cdot)^\sharp \right\rangle - \left\langle T(Y,Z,\cdot)^\sharp,T(X,W,\cdot)^\sharp\right\rangle
\right).
\label{Riemann_identity}
\end{eqnarray}
We used the identity
$T(X,T(Y,Z,\cdot)^\sharp,W)=-\left\langle T(X,W,\cdot)^\sharp,T(Y,Z;\cdot)^\sharp \right\rangle$, which follows from (\ref{sharp_def}).\medskip
\noindent
From (\ref{Ricci_Torsion_Allgemein}) we conclude that the Ricci curvature of
$\nabla$ for any orthonormal,
synchronous frame $E_1,\ldots,E_n$ defined on some neighbourhood of $p$ is
\begin{eqnarray}
\operatorname{ric}(X,Y)&=& \operatorname{ric}^g(X,Y) +s \;\sum_{i=1}^n \left(\nabla^g_{E_i}T \right)(X,Y,E_i)
-s^2\;\sum_{i=1}^n\left\langle T(E_i,X,\cdot)^\sharp,T(E_i,Y,\cdot)^\sharp \right\rangle.
\label{Ricci_identity}
\end{eqnarray}
This formula shows that in general the Ricci curvature is not symmetric in $X$ and $Y$.
\medskip
\noindent
From Lemma \ref{Scalar_Norm}
we get for the scalar curvature $R$ of $\nabla$ that
\begin{eqnarray}
R =R^g-s^2\, \left\|T \right\|^2.
\label{Scalar_identity}
\end{eqnarray}
\noindent
Next, we are aiming at finding an expression for the norm of the Riemann tensor.
As the vector fields $X,Y,Z,W$ are synchronous in $p$ we get for the differential $dT$ and the codifferential $\delta T$ of the $3$-form $T$:
\begin{align}
dT(X,Y,Z,W)&= \left(\nabla^g_X T\right)(Y,Z,W) -\left(\nabla^g_Y T\right)(X,Z,W)+\left(\nabla^g_Z T\right)(X,Y,W)-\left(\nabla^g_W T\right)(X,Y,Z),\label{exterior_differential}\\
\delta T(X,Y)&=-\sum_{i=1}^n\left(\nabla^g_{E_i} T\right)(X,Y,E_i).
\label{exterior_codifferential}
\end{align}
\noindent
We define the $(4,0)$-tensors $\operatorname{riem}$ and $\operatorname{riem}^g$ by
\begin{equation}\label{def_40riem}
\operatorname{riem}^{(g)}(X,Y,Z,W)=\left\langle \operatorname{Riem}^{(g)} (X,Y)Z,W \right\rangle.
\end{equation}
We decompose the Riemann tensor into its symmetric
and anti-symmetric component
\begin{eqnarray}
\operatorname{riem}(X,Y,Z,W) = \operatorname{riem}^S(X,Y,Z,W) + \operatorname{riem}^A(X,Y,Z,W).
\label{Riem_decomp}
\end{eqnarray}
The symmetric part of $\operatorname{riem}$ is
\begin{eqnarray}
\operatorname{riem}^S(X,Y,Z,W)=\tfrac{1}{2}\left(\operatorname{riem}(X,Y,Z,W)+ \operatorname{riem}(Z,W,X,Y)\right)
\nonumber\end{eqnarray}
and the anti-symmetric part of $\operatorname{riem}$ is then given by
\begin{eqnarray}
\operatorname{riem}^A(X,Y,Z,W)=\tfrac{1}{2}\left(\operatorname{riem}(X,Y,Z,W)- \operatorname{riem}(Z,W,X,Y)\right).\label{antisymm_Riemann}
\nonumber\end{eqnarray}
\noindent
Since $\operatorname{riem}^S$ and $\operatorname{riem}^A$ are orthogonal with respect to the scalar product of
$(4,0)$-tensors, as defined in (\ref{deftensorprod}), we find
\begin{eqnarray}
\|\operatorname{Riem} \|^2 = \| \operatorname{riem} \|^2 = \| \operatorname{riem}^S \|^2 + \| \operatorname{riem}^A \|^2
\label{Riemnormsqared}
\end{eqnarray}
We also decompose the Ricci curvature
into its symmetric and its anti-symmetric components
\begin{eqnarray}
\operatorname{ric}(X,Y) &=&\operatorname{ric}^S(X,Y) + \operatorname{ric}^A(X,Y),
\label{Ric_decomp}\\
\mbox{with }\quad
\operatorname{ric}^S(X,Y)&=&\tfrac{1}{2}\,\left(\operatorname{ric}(X,Y)+\operatorname{ric}(Y,X) \right)
\nonumber\\
\mbox{and }\quad
\operatorname{ric}^A(X,Y)&=&\tfrac{1}{2}\,\left(\operatorname{ric}(X,Y)-\operatorname{ric}(Y,X) \right).\nonumber
\end{eqnarray}
\noindent
Now we give an explicit formula
for $\|\operatorname{riem}\|^2$ in the case of $M$ being $4$-dimensional.
\begin{thm}\label{theoremriem}
Let $M$ be a $4$-dimensional manifold with Riemannian metric $g$ and
connection $\nabla$ as given in (\ref{vectortorsionconnection}).
Then the norm of the Riemann tensor of $\nabla$ is given by
\begin{eqnarray}
\left\|\operatorname{Riem} \right\|^2 = \left\|\operatorname{riem}^g \right\|^2 + \tfrac{1}{3} s^4\, \left\| T \right\|^4
+ \tfrac{1}{4} s^2 \, \left\|dT \right\|^2
- \tfrac{1}{3}s^2 \,R^g\,\left\|T\right\|^2
+4 s^2 \,\operatorname{B}(T) + \left\|\operatorname{riem}^A \right\|^2
\label{Ricnormsqared}
\end{eqnarray}
with
\begin{eqnarray}
\operatorname{B}(T) = \sum_{i,j,k} \operatorname{ric}^g(E_i,E_k)\,
\left\langle T(E_i,E_j,\cdot)^\sharp,T(E_j,E_k,\cdot)^\sharp \right\rangle
+\tfrac{1}{4}\,R^g\,\left\| T\right\|^2
\label{BBT}
\nonumber\end{eqnarray}
\end{thm}
\pf{See Appendix.}
\medskip
\noindent
We notice that the term $\operatorname{B}(T)$ couples the torsion to the Ricci curvature, and the term $\left\|\operatorname{riem}^A \right\|^2 $
is being computed in Lemma~\ref{riemAsquare}.
\subsection{Integral formulas}
In this section $(M,g)$ will be a closed, $4$-dimensional Riemannian manifold.
We will exploit the topological invariance of the Euler characteristic to
deduce integral formulas for $3$-forms defined on $M$.
\begin{Def}
Let $\nabla$ be an orthogonal connection on $M$
and let $\operatorname{Riem}_{ij}(X,Y):= \langle \operatorname{Riem}(E_i,E_j)X,Y\rangle$ be its
curvature $2$-form defined by equation (\ref{Riemann_Torsion_Allgemein}).
Define the $4$-form
\[
{\bf K} = \tfrac{1}{32 \pi^2 } \sum_{i,j,k,l=1}^4 \epsilon_{ijkl} \operatorname{Riem}_{ij}\wedge
\operatorname{Riem}_{kl},
\]
where $\epsilon_{ijkl}$ is the totally anti-symmetric tensor with normalisation
$\epsilon_{1234} = +1$.
\end{Def}
One obtains the classical result for the interplay between the topological invariant
Euler characteristic $\chi(M)$ and the curvature $2$-form of $\nabla$:
\begin{thm}
The Euler characteristic of $M$ is
\[
\chi(M) = \int_M {\bf K}.
\]
\end{thm}
\pf{
For a proof of this theorem we refer to \cite[Vol.~II, Chap.~XII, Thm.~5.1]{KN69}.
\hfill$\boxbox$
}
\medskip
\noindent
In four dimensions the Euler characteristic can be expressed in a particularly
convenient form in terms of squares of the Riemann, Ricci and
scalar curvature of $\nabla$.
\begin{thm}
Let $\nabla$ an orthogonal connection on $M$ and
let $\operatorname{riem}= \operatorname{riem}^S + \operatorname{riem}^A$, $\operatorname{ric}= \operatorname{ric}^S + \operatorname{ric}^A$ and $R$ be the
Riemann curvature, the
Ricci curvature and the scalar curvature of $\nabla$ decomposed into
their symmetric and anti-symmetric parts according to (\ref{Riem_decomp})
and (\ref{Ric_decomp}).
Then the Euler characteristic $\chi(M)$ is
\[
\chi(M) = \tfrac{1}{8 \pi^2} \, \int_M \left(R^2 - 4\, \| \operatorname{ric}^S \|^2
+ 4\, \| \operatorname{ric}^A \|^2 + \| \operatorname{riem}^S \|^2 - \| \operatorname{riem}^A \|^2 \right)\, dvol.
\]
\label{Eulerallg}
\end{thm}
\pf{See Appendix.}
\medskip
\noindent
The classical result for the Euler characteristic in terms of the curvatures of the Levi-Civita connection is due to Berger \cite{Berger},
and it follows immediately from the above theorem:
\begin{cor}\label{Eulerspezial}
Let $M$ be a $4$-dimensional manifold with Riemannian metric $g$ and
Levi-Civita connection $\nabla^g$.
Then the Euler characteristic $\chi(M)$ is given by
\[
\chi(M) = \tfrac{1}{8 \pi^2} \, \int_M \left((R^g)^2 - 4\, \| \operatorname{ric}^g \|^2
+ \| \operatorname{riem}^g \|^2 \right)\, dvol
\]
\end{cor}
\pf{
For $\nabla^g$ we have $\operatorname{riem}^S = \operatorname{riem}^g$, $\operatorname{ric}^S = \operatorname{ric}^g$ and hence $\operatorname{riem}^A\equiv 0$ and $\operatorname{ric}^A\equiv 0$.
\hfill$\boxbox$
}
\noindent
The fact that the Euler characteristic does not depend on the connection
allows us to deduce a useful integral formula for $3$-forms
on closed Riemannian $4$-manifolds.
\begin{lemma}\label{Eulervanish}
Let $M$ be a closed $4$-dimensional manifold with Riemannian metric $g$
and $T$ any $3$-form on $M$.
Let $R^g$ denote the scalar curvature of the Levi-Civita connection $\nabla^g$.
Then
\begin{eqnarray}
\int_M 4 \| \delta T \|^2 \, dvol
= \int_M \left( \tfrac{1}{3} \, R^g \| T \|^2 - \tfrac{1}{4} \|dT\|^2 + 4 B(T)
+ \frac{1}{s^2} \| \operatorname{riem}^A \|^2\right) \, dvol
\label{vanish}
\end{eqnarray}
with $\operatorname{B}(T)$ as defined in Theorem \ref{theoremriem} and $\operatorname{riem}^A$
is the anti-symmetric component of the Riemann curvature of $\nabla$ with
$s \,T$ as torsion $3$-form.
\label{intzero}
\end{lemma}
\pf{ See Appendix.}
\section{Dirac operators associated to orthogonal connections}\label{section3}
In this section we consider an $n$-dimensional oriented Riemannian manifold $(M,g)$ equipped with some spin structure.
Let $\nabla$ be an orthogonal connection given as in (\ref{def_Zusammenhang_mit_Torsion})--(\ref{A_schiefadjungiert}).
Then the connection $\nabla$ acting on vector fields induces a connection acting on spinor fields.
Next, we will briefly discuss the construction of this connection (compare \cite[p.~17f]{AgricolaSrni}, or see \cite[Chap.~II.4]{Lawson} for more details).
Again, we write $\nabla_XY=\nabla^g_XY+A(X,Y)$ with the Levi-Civita connection $\nabla^g$.
For any $X\in T_pM$ the endomorphism $A(X,\cdot)$ is skew-adjoint and hence it is an element of $\mathfrak{so}(T_pM)\cong\bigwedge^2T_pM$, we can express it as
\begin{equation}\label{torsionendo_in_terms_of_son}
A(X,\cdot) =\sum_{i< j} \alpha_{ij} \,E_i\wedge E_j.
\end{equation}
Here $E_i\wedge E_j$ is meant as the endomorphism of $ T_pM$ defined by $E_i\wedge E_j(Z)= \langle E_i,Z\rangle E_j-\langle E_j,Z\rangle E_i$.
For any $X\in T_pM$ one determines the coefficients in (\ref{torsionendo_in_terms_of_son}) by
\begin{equation}\label{determination_of_alphas}
\alpha_{ij}=\langle A(X,E_i), E_j \rangle= A_{X E_i E_j}.
\end{equation}
Each $E_i\wedge E_j$ lifts to $\frac12 E_i\cdot E_j$ in $\mathfrak{spin}(n)$, and the spinor connection induced by $\nabla$ is locally given by
\begin{equation}\label{spintorsionconnection}
\nabla_X\psi = \nabla^g_X\psi+\tfrac12 \sum_{i< j} \alpha_{ij} \, E_i\cdot E_j \cdot \psi
= \nabla^g_X\psi+\tfrac12 \sum_{i< j} A_{X E_i E_j} \, E_i\cdot E_j \cdot \psi.
\end{equation}
\begin{rem}
The connection given by (\ref{spintorsionconnection}) is compatible with the metric on spinors and with Clifford multiplication (see e.g.~\cite[Lemma~2.1]{AgricolaSrni}).
\end{rem}
\begin{rem}\label{spinorconnection_totallyanti}
For totally anti-symmetric torsion, given by a $3$-form $T$ as in Corollary~\ref{connectionswithtorsionsclassified}, one can rewrite (\ref{spintorsionconnection}) as
\[
\nabla_X \psi = \nabla^{g}_X \psi + \frac{1}{2} (X \lrcorner T) \cdot \psi,
\]
where $X \lrcorner T$ is the $2$-form defined by $X \lrcorner T(X,Z)=T(X,Y,Z).$
We recall that a $k$-form $\omega\in \bigwedge^kT_pM$, given as
$\omega=\sum\limits_{i_1,\ldots,i_k}\omega_{i_1,\ldots,i_k} \,{E_{i_1}}^\flat\wedge\ldots\wedge {E_{i_k}}^\flat$, acts on the spinor space as $\omega\cdot\psi= \sum\limits_{i_1,\ldots,i_k}\omega_{i_1,\ldots,i_k}\, E_{i_1}\cdot \ldots\cdot E_{i_k}\cdot \psi$.
\end{rem}
\begin{rem}
Not any connection on spinor fields is induced by an orthogonal connection on tangent vector fields.
For example, for the connection
$
\nabla_X\psi = \nabla^g_X\psi + X\cdot \psi
$
the endomorphism $\alpha$ of spinors defined by
\begin{equation}\label{tetraden}
\alpha(\psi)=\nabla_X\left(Y\cdot\psi\right)-Y\cdot\left(\nabla_X \psi\right)
\end{equation}
is given by multiplication by the Clifford element $\nabla_X^g Y+X\cdot Y-Y\cdot X$, which does not equal to the Clifford multiplication by any tangent vector. This consideration applies in any dimension $n\ge 2$.
\end{rem}
\begin{rem}
If we assume that for a spinor connection $\nabla$ for any vector fields $X,Y$ the endomorphism $\alpha$ defined in (\ref{tetraden}) is the Clifford multiplication by a tangent vector $V_{X,Y}$, i.e.~$\alpha(\psi)=V_{X,Y}\cdot \psi$ for all spinors $\psi$, then it can be shown than the assignment $\nabla_XY=V_{X,Y}$ defines an orthogonal connection on tangent vector fields such that the spinor connection is compatible with the Clifford multiplication.
In that case, physics literature occasionally refers to (\ref{tetraden}) as the {\it tetrad postulate}.
\end{rem}
\noindent
The Dirac operator associated to the spinor connection from (\ref{spintorsionconnection}) is defined as
\begin{eqnarray}
D\psi &=& \sum_{i=1}^n E_i\cdot \nabla_{E_i}\psi \nonumber\\
&=& D^g\psi +\tfrac12\sum_{i=1}^n \sum_{j<k} A_{E_i E_j E_k} \, E_i\cdot E_j \cdot E_k \cdot \psi
\nonumber\\
&=& D^g\psi +\tfrac14\sum_{i,j,k=1}^n A_{E_i E_j E_k} \, E_i\cdot E_j \cdot E_k \cdot \psi, \label{def_torsiondirac}
\end{eqnarray}
where $D^g$ is the Dirac operator induced by the Levi-Civita connection.\medskip
\noindent
The next theorem tells us when the Dirac operator $D$ is formally selfadjoint (i.e.~symmetric on the space of compactly supported smooth spinor fields as domain), it is provided as Satz 2 in \cite{FriedrichSulanke}:
\begin{thm}
The Dirac operator $D$ is formally selfadjoint if and only if the divergence of $\nabla$ coincides with the divergence of $\nabla^g$, i.e.~for any vector field $Z$ one has
\begin{equation}\label{divergenz_gleichheit}
\sum_{i=1}^n \langle \nabla_{E_i} Z,E_i \rangle =
\sum_{i=1}^n \langle \nabla^g_{E_i} Z,E_i\rangle
\end{equation}
in any point $p$ and for any orthonormal basis $E_1,\cdots, E_n$ of $T_pM$. {\hfill$\boxbox$}
\end{thm}
Taking the specific form of $\nabla_XY=\nabla^g_XY+A(X,Y)$ into account, we see that (\ref{divergenz_gleichheit}) is equivalent to
\[
c_{12}(A)(Z) = \sum_{i=1}^n \langle A(E_i,E_i),Z \rangle=-\sum_{i=1}^n \langle A(E_i,Z),E_i\rangle =0.
\]
Hence, we can conclude from Remark~\ref{spurnullfuerT2undT3}:
\begin{cor}\label{vectornull_dannselbstadjungiert}
The Dirac operator $D$ associated to an orthogonal connection is formally selfadjoint if and only if the $(3,0)$--torsion tensor $A$ does not have any vectorial compontent, i.e.~one has
\[
A_p\in\mathcal{T}_2(T_pM)\oplus\mathcal{T}_3(T_pM)
\]
in any point $p\in M$ {\hfill$\boxbox$}
\end{cor}
\noindent
The next lemma states that for the Dirac operator the Cartan type component of the torsion is invisible:
\begin{lemma}\label{Cartan_type_egal}
On a Riemannian spin manifold we consider some vector field $V$, some $3$-form $T$ and some $(3,0)$-tensor field $S$ with $S_p\in\mathcal{T}_3(T_pM)$ for any $p\in M$.
Let $\nabla_1$ and $\nabla_2$ be the orthogonal connections determined by
\begin{align*}
A_1(X,Y) & = \langle X,Y \rangle V- \langle V,Y \rangle X + T(X,Y,\cdot)^\sharp + S(X,Y,\cdot)^\sharp \quad\mbox{ and}\\
A_2(X,Y) & =\langle X,Y \rangle V- \langle V,Y \rangle X + T(X,Y,\cdot)^\sharp,
\end{align*}
respectively (compare Corollary~\ref{connectionswithtorsionsclassified}).
Denote the associated Dirac operators by $D_1$ and $D_2$.
Then, for any spinor field $\psi$ one has
\[
D_1 \psi =D_2\psi.
\]
\end{lemma}
{\pf
By (\ref{def_torsiondirac}) the difference of the two Dirac operators is
\begin{equation}\label{difference_Dirac_operators}
D_1\psi -D_2\psi =\tfrac14\sum_{i,j,k=1}^n S_{E_i E_j E_k} \, E_i\cdot E_j \cdot E_k \cdot \psi.
\end{equation}
We use the cyclic identity for $S$, the fact that $S$ is trace-free in any pair of entries and the Clifford relations $E_i\cdot E_j=-E_j\cdot E_i$ for $i\ne j$ as well, in order to obtain:
\begin{align*}
\sum_{i,j,k=1}^n S_{E_i E_j E_k} \, E_i\cdot E_j \cdot E_k
&= -\sum_{i,j,k=1}^n S_{E_j E_k E_i} \, E_i\cdot E_j \cdot E_k -\sum_{i,j,k=1}^n S_{E_k E_i E_j} \, E_i\cdot E_j \cdot E_k \\
&= -\sum_{i,j,k=1}^n S_{E_j E_k E_i} \, E_j\cdot E_k \cdot E_i -\sum_{i,j,k=1}^n S_{E_k E_i E_j} \, E_k\cdot E_i \cdot E_j \\
&= -2\,\sum_{i,j,k=1}^n S_{E_i E_j E_k} \, E_i\cdot E_j \cdot E_k.
\end{align*}
Therefore we get $\sum\limits_{i,j,k=1}^n S_{E_i E_j E_k} E_i\cdot E_j \cdot E_k=0$, and the right hand side of (\ref{difference_Dirac_operators}) is zero.
{\hfill$\boxbox$}
}\medskip
\noindent
One should note that the above lemma applies pointwise.
\begin{rem}
In the Lorentzian case it is known that torsion of Cartan type does not
contribute to the Dirac action under the integral \cite[Chap.~2.3]{S02}.
It is also known that the Dirac action is not real if the torsion has a
non-vanishing vectorial component \cite[Chap.~11.6]{GS87}.
Therefore only totally anti-symmetric torsion is considered to couple
to fermions reasonably.
\end{rem}
\noindent
The spinor connection in (\ref{spintorsionconnection}) is the connection which is induced by the tangent vector connection $\nabla$ given in (\ref{def_Zusammenhang_mit_Torsion}), hence one expects that their curvatures are related.
Let $(E_1,\ldots,E_n)$ be an arbitrary local orthonomal frame.
For $i,j$ the curvature endomorphism w.r.t.~this frame is defined as
\begin{eqnarray}
\Omega_{ij}\psi =\nabla_{E_i}\nabla_{E_j} \psi-\nabla_{E_j}\nabla_{E_i}\psi -\nabla_{[E_i,E_j]}\psi.
\nonumber\end{eqnarray}
These curvature endomorphisms for spinors can be naturally determined by the Riemann tensor for tangent vectors, compare with formula (4.37) in Theorem 4.15 of \cite[Chap.~II]{Lawson}.
\begin{lemma}
\label{curvaturetrace}
For the spinor connection $\nabla$ defined in (\ref{spintorsionconnection}) the curvature endomorphisms in $p$ are given by
\[
\Omega_{ij}\psi=\tfrac{1}{4}\,\sum_{a,b} \left\langle\operatorname{Riem}(E_i,E_j)E_a,E_b\right\rangle\,E_a\cdot E_b\cdot\psi
\]
with Riemann tensor as defined in (\ref{Riemann_Torsion_Allgemein}).
\hfill$\boxbox$
\end{lemma}
\begin{cor}\label{Omega-trace-Riem}
Let $\operatorname{tr}$ denote the trace over the spinor space over some footpoint $p$.
Then one has
\[
\sum_{i,j=1}^n\,\operatorname{tr}\left( \Omega_{ij}\Omega_{ij} \right)= -\tfrac{1}{8}\cdot 2^{[n/2]}\cdot\left\|\operatorname{Riem} \right\|^2
\]
where $\operatorname{Riem}$ is the Riemann tensor of the vector connection $\nabla$.
\end{cor}
\pf{
The spinor space has dimension $2^{[n/2]}$.
If $a\ne b$ and $c\ne d$ the Clifford relations imply
\[
\operatorname{tr}\left(E_a\cdot E_b\cdot E_c\cdot E_d \right) = 2^{[n/2]}\left(\delta_{bc}\delta_{ad}-\delta_{bd}\delta_{ac} \right).
\]
From Lemma~\ref{curvaturetrace} we derive
\begin{align*}
\sum_{i,j}\,\operatorname{tr}\left( \Omega_{ij}\Omega_{ij} \right)
&=\tfrac{1}{16}\,\sum_{i,j}\, \sum_{a\ne b}\, \sum_{c\ne d}
\left\langle\operatorname{Riem}(E_i,E_j)E_a,E_b\right\rangle\,\left\langle\operatorname{Riem}(E_i,E_j)E_c,E_d\right\rangle\,
\operatorname{tr}\left(E_a\cdot E_b\cdot E_c\cdot E_d \right)
\\
&=\tfrac{1}{16}\cdot 2^{[n/2]}\,\sum_{i,j}\, \sum_{a\ne b}\,\big(
\left\langle\operatorname{Riem}(E_i,E_j)E_a,E_b\right\rangle\,\left\langle\operatorname{Riem}(E_i,E_j)E_b,E_a\right\rangle\\
&\qquad\qquad\qquad\qquad\qquad
- \left\langle\operatorname{Riem}(E_i,E_j)E_a,E_b\right\rangle\,\left\langle\operatorname{Riem}(E_i,E_j)E_a,E_b\right\rangle
\big)\\
&=-\tfrac{1}{8}\cdot 2^{[n/2]}\,\sum_{i,j}\, \sum_{a\ne b}\,\left(\left\langle\operatorname{Riem}(E_i,E_j)E_a,E_b\right\rangle\right)^2
\\
&= -\tfrac{1}{8}\cdot 2^{[n/2]}\cdot\left\|\operatorname{Riem} \right\|^2,
\end{align*}
where we have used the anti-symmetry of $\operatorname{Riem}(E_i,E_j)E_a,E_b$ in the indices $a$ and $b$, which holds due to (\ref{Riemann_Torsion_antisymm_in_34_indices}).
\hfill$\boxbox$}
\section{Commutative geometries and the spectral action principle}\label{section4}
In this section we want to discuss torsion connections within the framework of Connes' noncommutative geometry (see \cite{Connes94}).
Let $M$ be a closed Riemannian manifold of dimension $n$ with some fixed spin structure.
We denote the algebra of smooth functions by $\mathcal{A}=C^\infty(M)$, and we denote the Hilbert space of square integrable spinor fields by $\mathcal{H}$.
The Dirac operator $D^g$ associated to the Levi-Civita connection is a selfadjoint operator in $\mathcal{H}$.
The triple $(\mathcal{A},\mathcal{H},D^g)$ forms a {\it canonical spectral triple}, and it satisfies all {\it axioms for commutative geometry} (see \cite{Connes96}, or \cite{CostaRica} for more details).\medskip
\noindent
Now, let $\nabla$ be an orthogonal connection on the tangent bundle of $M$, and let $D$ denote the associated Dirac operator.
By Corollary~\ref{vectornull_dannselbstadjungiert} we know that $D$ is symmetric if and only if the vectorial component of the torsion of $\nabla$ is zero.
In that case $D$ is the sum of a selfadjoint operator and a bounded symmetric one, and thus selfadjoint.
We notice that $D$ has the same principal symbol and the same Weyl asymptotics as $D^g$.
Furthermore, we note that any natural algebraic structure on the spinor space such as a real structure or the Clifford multiplication with the volume element are parallel with respect to any spinor connection which is induced by an orthogonal connection on the tangent bundle.
Therefore $D$ commutes or anti-commutes with such a structure exactly if $D^g$ does.
Following the details of the proof of \cite[Thm.~11.1]{CostaRica} we see that these observations suffice to verify all axioms for commutative geometry
and we conclude:
\begin{lemma}\label{commutative_geometry}
If the vectorial component of the torsion of $\nabla$ is zero,
the spectral triple $(\mathcal{A},\mathcal{H},D)$ satisfies the axioms for commutative geometry.
\hfill$\boxbox$
\end{lemma}
Connes' Reconstruction Theorem (conjectured in \cite{Connes96}, proved in \cite{Connes08}) states that, given a commutative geometry $(\mathcal{A},\mathcal{H},D)$, one can construct a differentiable spin manifold such that $\mathcal{A}$ coincides with the smooth functions on it.
Then, from the data $(\mathcal{A},\mathcal{H},D)$ one also gets a Riemannian metric and one obtains that $\mathcal{H}$ is isomorphic to the square integrable spinor fields (for some spin structure) (see \cite[Th\'eor\`eme 6]{Connes95}), and the natural Dirac operator we can always construct is the one induced by the Levi-Civita connection.\medskip
\noindent
In the above situation of Lemma~\ref{commutative_geometry}, one can algebraically recover the totally anti-symmetric torsion component from the spectral triple $(\mathcal{A},\mathcal{H},D)$ if the underlying manifold $M$ has even dimension.
This can be done by considering the endomorphism of spinors given as difference of $D$ and the Levi-Civita Dirac operator $D^g$, see (\ref{def_torsiondirac}).
In even dimensions the complex Clifford algebra and the space of endomorphisms of the spinor space are identical (see~\cite{Friedrich1997}, Proposition on p.~13), and hence the endomorphism $D-D^g$ can be uniquely determined as a $3$-form.
In odd dimensions this argument does not apply, as one easily sees by noticing that e.g.~in the $3$-dimensional case the volume form acts as multiple of the identity on the spinor space.\medskip
\noindent
By Lemma~\ref{Cartan_type_egal} the Cartan-type component of the torsion is invisible for the Dirac operator $D$, and therefore it cannot be recovered from $(\mathcal{A},\mathcal{H},D)$.
This can be interpreted as some sort of gauge freedom, which is schematically illustrated in Figure~\ref{figure_torsion_types}.
\begin{figure}[h]
\begin{pspicture}(-7,-2.1)(4,2.8)
\psline[fillcolor=lightgray,fillstyle=solid,linewidth=0,linecolor=lightgray](-1.3,-1)(-1.3,2)(4,2)(4,-1)
\psline[linewidth=0.03cm,linestyle=dotted](1.8,1.8)(0,0)
\psline[linewidth=.03cm,linecolor=blue](-1.2,-1)(-1.2,2)
\psline[linewidth=.03cm,linecolor=blue](-.6,-1)(-.6,-.7)
\psline[linewidth=.03cm,linecolor=blue](-.6,-.5)(-.6,2)
\psline[linewidth=.03cm,linecolor=blue](.6,-1)(.6,2)
\psline[linewidth=.03cm,linecolor=blue](1.2,-1)(1.2,2)
\psline[linewidth=.03cm,linecolor=blue](1.8,-1)(1.8,2)
\psline[linewidth=.03cm,linecolor=blue](2.4,-1)(2.4,2)
\psline[linewidth=.03cm,linecolor=blue](3,-1)(3,2)
\psline[linewidth=.03cm,linecolor=blue](3.6,-1)(3.6,2)
\psline[linewidth=0.05cm]{->}(-1.3,0)(4,0)
\psline[linewidth=0.05cm]{->}(0,-1)(0,2)
\psline[linewidth=0.05cm]{->}(0,0)(-1.5,-1.5)
\rput(3.8,-.25){$T$}
\rput(-.25,1.8){$S$}
\rput(-1.05,-1.4){$V$}
\end{pspicture}
\caption{Cartan-type component $S$ of torsion is invisible for Dirac operator (gauge freedom).}
\label{figure_torsion_types}
\end{figure}
\begin{rem}
For some even-dimensional compact Riemannian manifold $M$ with a given spin structure we fix $\mathcal{A}=C^\infty(M)$ and $\mathcal{H}$ the space of square integrable spinor fields.
The above considerations show that one has a family of Dirac operators parametrized by the $3$-forms $T\in \Omega^3(M)$ such that the associated spectral triples are pairwise disctinct commutative geometries.
Notice that all these Dirac operators are in the same $K$-homology class since they all have the same principal symbol.
We leave it open how big the class of first order operators $D$ in $\mathcal{H}$ is for which
$(\mathcal{A},\mathcal{H},D)$ forms a commutative geometry.
\end{rem}
\begin{rem}
For spectral triples of odd $KO$-dimension it has recently been
shown in \cite[Prop.~1.2]{SZ10} that one can modify the Dirac operator by adding a term induced
by a selfadjoint element of the algebra $\mathcal{A}$ and still finds the axioms of spectral triples satisfied.
In the case of $\mathcal{A}=C^\infty(M)$ and $KO$-dimensions 3 or 7 modulo 8, i.e.~$M$ is of dimension 3 or 7 modulo 8,
this modification is realised by adding a real-valued function
$\Phi \in \mathcal{A}$ to the Dirac operator, see \cite[Rem.~1.3]{SZ10}.
\end{rem}
\noindent
In the following we will only consider orthogonal connections $\nabla$ with zero vectorial component to ensure selfadjointness of the induced Dirac operator $D$.
For the computation of the Chamseddine-Connes spectral action (see \cite{ConnesChamseddine1}) we need the Seeley-deWitt coefficients $a_{2k}(D^2)$ of the heat trace asymptotics \cite{Gilkey95}
\begin{eqnarray}
{\rm Tr} \left( e^{-t \, D^2} \right) \sim \sum_{k\geq 0} t^{k- n/2} a_{2k} (D^2) \quad \mbox{ as }t\to 0.
\nonumber
\end{eqnarray}
\begin{prop}
The first two Seeley-deWitt coefficients are
\begin{align*}
a_0(D^2)&= \frac{1}{(4\pi)^{n/2}} \; 2^{[n/2]}\;\int_M dvol, \\
a_2(D^2)&= \frac{1}{(4\pi)^{n/2}} \; 2^{[n/2]}\;\int_M \left(\frac{3}{4}\|T\|^2-\frac{1}{12} R^g \right) dvol.
\end{align*}
\end{prop}
\pf{
By Lemma~\ref{Cartan_type_egal} we can assume without loss of generality that the Cartan-type component of the torsion vanishes.
The orthogonal connection $\nabla$ is given by
$
\nabla_XY= \nabla^g_XY +T(X,Y,\cdot)^\sharp.
$
Adapting \cite[Thm.~6.2]{AgricolaFriedrich} into our notation we get the Bochner formula
\begin{equation}\label{TorsionsBochner}
D^2=\Delta +\frac{3}{2}dT +\frac{1}{4}R^g -\frac{3}{4}\|T \|^2
\end{equation}
where $\Delta$ is the Laplacian associated to the spin connection
\begin{eqnarray}
\widetilde{\nabla}_X \psi = \nabla^{g}_X \psi + \frac{3}{2} (X \lrcorner T)
\cdot \psi,
\label{drittelspinconn}
\end{eqnarray}
which is induced (as in Remark~\ref{spinorconnection_totallyanti}) by the orthogonal connection
\begin{eqnarray}
\widetilde{\nabla}_XY= \nabla^g_XY +3T(X,Y,\cdot)^\sharp.
\label{drittelorthconn}
\end{eqnarray}
We notice that the trace of $dT$ taken over the spinor space is zero due to Clifford relations.
Inserting this into the general formulas for the Seeley-deWitt coefficients (see~\cite[Theorem~4.1.6]{Gilkey95}) the claim follows.
\hfill$\boxbox$}
\medskip
\noindent
If we consider the spectral action given by the $a_2(D^2)$ and variations with respect to the torsion $3$-form $T$ we find that $T=0$ is the only possibility for critical points.
Therefore this spectral action detects the Dirac operator induced by the Levi-Civita connection within the class of Dirac operators induced by orthogonal connections without vectorial torsion.
We note that this holds in any dimension.
This is in complete accordance with~\cite[Section~18.2]{ConnesMarcolli}.
\medskip
\noindent
The computation of $a_4(D^2)$ is more involved, we will give it only for $4$-dimensional manifolds.
In~\cite{IochumLevy} it has been noted that some terms given in~\cite{Torsion} vanish.
Similar results have been found before (compare~\cite{Goldthorpe}, \cite{Obukhov}, \cite{Grensing}).
The calculation given below is elementary, it takes place essentially in the tangent bundle and should therefore be easily accessible.
\begin{prop}
If $M$ is $4$-dimensional, the third Seeley-deWitt coefficient is
\begin{eqnarray}
a_4(D^2) = \tfrac{11}{720} \, \chi(M) - \tfrac{1}{320 \pi^2} \int_M \|C\|^2 \, dvol
- \tfrac{3}{32 \pi^2} \int_M \| \delta T \|^2 \, dvol,
\label{a4}
\end{eqnarray}
where $C$ is the Weyl curvature of $M$ (computed from the Levi-Civita connection).
\label{a4theorem}
\end{prop}
\pf{
We read (\ref{TorsionsBochner}) as $D^2=\Delta- E$ with potential $E = - \frac{3}{2}\, dT - \frac{1}{4}\, R^g + \frac{3}{4}\, \|T\|^2$.
From~\cite[Theorem~4.1.6,c)]{Gilkey95} we get
\[
a_4 (D^2) = \tfrac{1}{5760 \pi^2} \int_M \Big( \operatorname{tr} \big( 60 R^g E + 180 E^2 +
30 \,\sum_{i,j} \Omega_{ij} \Omega_{ij}\big)
+ 20\,( R^g)^2 - 8 \|\operatorname{ric}^g\|^2 + 8 \|\operatorname{Riem}^g\|^2 \Big) \, dvol\, ,
\]
where we have omitted the terms that integrate to zero over the closed manifold $M$ (Laplacians of functions).
The term $\Omega_{ij}$ is the curvature endomorphism for the spinor connection $\widetilde{\nabla}$.
The traces over the spinor space are
\begin{align*}
\operatorname{tr}(E)&=-R^g +3\,\|T\|^2\,,\\
\operatorname{tr}(E^2)&= \frac{1}{4}\,(R^g)^2-\frac{3}{2}\,R^g\,\|T\|^2+\frac{9}{4}\,\|T\|^4+\frac{9}{24}\|dT\|^2,
\end{align*}
where we note that $\operatorname{tr}(\omega^2)=\frac{1}{6} \,\|\omega\|^2$ for any $\omega\in\bigwedge^4$.
For the orthogonal connection $\widetilde{\nabla}$ on the tangent bundle we denote the Riemannian curvature by $\operatorname{Riem}$ and apply
Corollary~\ref{Omega-trace-Riem}.
Then we get
\begin{eqnarray}
a_4(D^2) &=& \tfrac{1}{16 \pi^2}\, \tfrac{1}{360} \,
\int_M \big( 5\,(R^g)^2 - 8 \,\left\| \operatorname{Ric}^g \right\|^2
+ 8 \,\left\| \operatorname{Riem}^g \right\|^2
- 15\, \left\| \operatorname{Riem} \right\|^2 \nonumber\\
&& \qquad \qquad \quad -90\, R^g\,\|T\|^2 +405\,\|T\|^4 +\tfrac{135}{2}
\left\|dT \right\|^2 \big) dvol
\nonumber \\
&=& \tfrac{1}{16 \pi^2}\,\tfrac{1}{360} \, \int_M \left( 5\,(R^g)^2 - 8\, \left\| \operatorname{Ric}^g \right\|^2
- 7\,\left\| \operatorname{Riem}^g \right\|^2 \right) dvol
\nonumber \\
&& \qquad - \tfrac{1}{16 \pi^2} \tfrac{1}{8}\,
\int_M \left( R^g \, \left\|T \right\|^2
-\tfrac{3}{4} \left\|dT \right\|^2 +12 \,\operatorname{B}(T) +\tfrac13\, \left\|\operatorname{riem}^A \right\|^2 \right) dvol
\nonumber\end{eqnarray}
by means of Proposition~\ref{theoremriem}.
With Lemma~\ref{Eulerspezial} we identify the first integral as the Euler characteristic plus the square of the Weyl curvature.
Lemma~\ref{Eulervanish} shows that the second integral equals
\[ \tfrac{1}{16 \pi^2} \tfrac{1}{8}\,
\int_M \left( R^g \, \left\|T \right\|^2
-\tfrac{3}{4} \left\|dT \right\|^2 +12 \,\operatorname{B}(T) +\tfrac13\, \left\|\operatorname{riem}^A \right\|^2 \right) dvol
= \tfrac{1}{16 \pi^2} \tfrac{3}{2} \int_M \| \delta T \|^2 \, dvol.
\]
This finishes the proof.
\hfill$\boxbox$}\medskip
\noindent
Next, we want to consider the Chamseddine-Connes spectral action (see \cite{ConnesChamseddine1}) for the Dirac operator $D$.
For $\Lambda >0$ it is defined as
\begin{eqnarray}
I_{CC} = {\rm Tr}\, F \left( \frac{D^2}{\Lambda^2} \right)
\nonumber
\end{eqnarray}
where ${\rm Tr}$ denotes the operator trace over $\mathcal{H}$ as before,
and $F:\mathbb{R}^+\to\mathbb{R}^+$ is a cut-off function with support in the interval $[0,+1]$ which is constant near the origin.
Using the heat trace asymptotics one gets an asymptotic expression for $I_{CC}$ as $\Lambda\to\infty$ (see \cite{ConnesChamseddine2} for details):
\begin{eqnarray}
I_{CC} = {\rm Tr} \, F \left( \frac{D^2}{\Lambda^2} \right) =
\Lambda^4 \, F_4 \, a_0 (D^2) + \Lambda^2 \, F_2 \, a_2(D^2)
+ \Lambda^0 \, F_0 \, a_4(D^2) + \operatorname{O}(\Lambda^{-\infty})
\label{specact}
\end{eqnarray}
with the first three moments of the cut-off function which are given
by
$F_4 = \int_0^\infty s \cdot F(s) \, ds$, $F_2 = \int_0^\infty F(s) \, ds$ and $F_0 = F(0)$.
Note that these moments are independent of the geometry of the manifold.
\medskip
\noindent
Now we want to deduce the equation of motion for $I_{CC}$.
In analogy to the Riemannian Einstein-Hilbert case we consider variations with respect to the metric and the torsion $3$-form while keeping the volume fixed.
Then $a_0(D^2)$ and $\chi(M)$ are constant and their variation vanishes.
In order to avoid further complications we neglect the contributions of the $\operatorname{O}(\Lambda^{-\infty})$-term.
Therefore we will consider the following action functional
\begin{eqnarray}
{\widetilde I}_{CC}=-\alpha \int_M R^g dvol -\beta \int_M \| C\|^2 dvol+\gamma_1 \int_M \| T\|^2 dvol -\gamma_2 \int_M \| \delta T\|^2 dvol,
\label{specacttruncated}
\end{eqnarray}
where $\alpha,\beta,\gamma_1,\gamma_2>0$.
\medskip
\noindent
Before we proceed let us briefly recall the standard scalar product on $k$-forms induced by a Riemannian metric $g$ (compare e.g.~with~\cite{Bleecker}).
Let $E_1,\ldots,E_n$ be an orthonormal basis of some tangent space $T_pM$.
Then the scalar product on $\bigwedge^k T^*_pM$ is uniquely determined by the requirement that $E^*_{i_1}\wedge\ldots\wedge E^*_{i_k}$, $i_1<\ldots<i_k$, form an orthonormal basis.
In local coordinates $(x^1,\ldots,x^n)$ it can be written as follows: for $\omega,\eta\in\bigwedge^k T^*_pM$ with
\[
\omega =\sum_{i_1<\ldots<i_k}\omega_{i_1\ldots i_k} dx^{i_1}\wedge\ldots\wedge dx^{i_k},\quad
\eta =\sum_{i_1<\ldots<i_k}\eta_{i_1\ldots i_k} dx^{i_1}\wedge\ldots\wedge dx^{i_k},
\]
where the coeffients $\omega_{i_1\ldots i_k}, \eta_{i_1\ldots i_k}$ are anti-symmetric in the indices $i_1\ldots i_k$,
the scalar product is then given as
\begin{eqnarray}\label{scalarproduct_forms}
\langle \omega,\eta\rangle_g = \frac{1}{k!} \sum_{\substack{i_1,\ldots ,i_k \\ j_1,\ldots ,j_k}} g^{i_1 j_1}\cdots g^{i_k j_k}\omega_{i_1\ldots i_k}\eta_{j_1\ldots j_k}.
\end{eqnarray}
Referring to the norm of $2$-forms and $3$-forms as used above we note that
$
\langle S,S\rangle_g = \tfrac12 \|S\|^2$,
$\langle T,T\rangle_g = \tfrac16 \|T\|^2$
for $S\in \bigwedge^2 T^*_pM$ and $T\in \bigwedge^3 T^*_pM$, compare (\ref{relationscalarprods}).
With respect to this scalar product the Hodge $*$-operator is an isometry, and on a $4$-manifold the $L^2$-adjoint of $d$ is $\delta=-*d*$ independently of the degree $k$ of the form.
For $3$-forms $\delta$ is given as in (\ref{exterior_codifferential}).\medskip
\noindent
Let $g$ be a Riemannian metric on $M$.
For any $k$-form $\eta$ on $M$ we define a symmetric $(2,0)$-tensor $g^\eta$
for $k \geq 2$ by
\begin{eqnarray}
g^\eta(X,Y)=\langle X\lrcorner \eta, Y \lrcorner \eta \rangle_g
\nonumber
\end{eqnarray}
and for $k=1$ by
\begin{eqnarray}
g^\eta(X,Y)= X\lrcorner \eta \cdot Y \lrcorner \eta = \eta(X) \, \eta(Y) \quad \mbox{ for any tangent vectors }X,Y.
\nonumber
\end{eqnarray}
For $(2,0)$-tensors $a$ and $h$ the natural scalar product defined in (\ref{deftensorprod}) reads as
\begin{eqnarray}\label{def_g-trace}
\langle a,h \rangle = \sum_{\substack{r,s\\i,j}} a_{ij} g^{ir} g^{js} h_{rs},
\end{eqnarray}
in local coordinates, as above.
\begin{lemma}
\label{variierealles}
Let $\left(g(t)\right)_t$ be a smooth family of Riemannian metrics on $M$ with $g(0)=g$ and ${\dot g}(0)=h$, and let $k\ge 1$.
Then for any $k$-form $\eta$ on $M$ we get
\[
\frac{d}{dt}\big|_{t=0} \langle \eta,\eta\rangle_{g(t)} = -\langle g^\eta , h\rangle.
\]
\end{lemma}
\pf{ In coordinates we write $\eta =\sum\limits_{i_1<\ldots<i_k}\eta_{i_1\ldots i_k} dx^{i_1}\wedge\ldots\wedge dx^{i_k}$ with
$\eta_{i_1\ldots i_k}$ anti-symmetric in the indices.
We recall that $\frac{d}{dt}\big|_{t=0} g^{ij}(t)=-\sum_{r,s} g^{ir}h_{rs}g^{js}$ and use (\ref{scalarproduct_forms}) to obtain
\begin{eqnarray*}
\frac{d}{dt}\big|_{t=0} \langle \eta,\eta\rangle_{g(t)}
&=& - \frac{1}{k!} \sum_{r,s}
\Big(
\sum_{\substack{i_1,\ldots ,i_k \\ j_1,\ldots ,j_k}} \eta_{i_1\ldots i_k} \eta_{j_1\ldots j_k}
\sum_{m=1}^k g^{i_1 j_1}\cdots {\widehat{g^{i_m j_m}}} \cdots g^{i_k j_k} g^{i_m r}g^{j_m s}
\Big)h_{rs}\\
&=& - \frac{1}{(k-1)!} \sum_{r,s}
\Big(
\sum_{\substack{i_1,\ldots ,i_k \\ j_1,\ldots ,j_k}} \eta_{i_1\ldots i_k} \eta_{j_1\ldots j_k}
g^{i_2 j_2}\cdots g^{i_k j_k}
\Big)
g^{i_1 r}g^{j_1 s}
h_{rs}\\
&=&
-\sum_{\substack{r,s\\i_1,j_1}} (g^\eta)_{i_1 j_1} g^{i_1 r}g^{j_1 s}
h_{rs}\\
&=& -\langle g^\eta , h\rangle,
\end{eqnarray*}
where we have used the total anti-symmetry of $\eta_{i_1\ldots i_k}$ in the indices for the second equality.
{\hfill$\boxbox$}}
\begin{thm}\label{Equation_of_Motion}
Any critical point $(M,g,T)$ of $ {\widetilde I}_{CC}$ satisfies
\begin{align}
0&= 3\gamma_1\, T -\gamma_2 \,d\delta T \label{variation_torsion} \\
0&= \alpha G^g - \beta B^g
+\gamma_1\big( -6 g^{*T}+\frac12 \|T\|^2 g\big)
-\gamma_2\big(- 2 g^{d*T}+\frac12 \|\delta T\|^2 g\big) \label{variation_metric}
\end{align}
where $G^g=\operatorname{ric}^g-\frac12 R^g \,g$ is the Einstein tensor of the metric $g$ and $B^g$ denotes its Bach tensor (for a definition see \cite[(4.77.)]{Besse}).
\end{thm}
\pf{
Let $(M,g,T)$ be a critial point.
We consider an arbitrary variation $T(t)$ of $3$-forms with $T(0)=T$ and ${\dot T}(0)=\tau$.
Then we have
\begin{align*}
0&= \frac{d}{dt}\big|_{t=0} \int_M \big(\gamma_1 \|T(t)\|^2 -\gamma_2 \|\delta T(t)\|^2 \big)dvol \\
&= \frac{d}{dt}\big|_{t=0} \int_M \big(6\gamma_1 \langle T(t),T(t)\rangle_g -2\gamma_2 \langle \delta T(t),\delta T(t)\rangle_g \big)dvol \\
&= \int_M \langle 3\gamma_1\, T -\gamma_2 \,d\delta T,4\tau \rangle_g dvol
\end{align*}
since $d$ is the adjoint of $\delta$.
As $\tau$ can be chosen arbitrarily we have established (\ref{variation_torsion}).\medskip
\noindent
Now we fix $T$ and consider an arbitrary variation $g(t)$ of Riemannian metrics with $g(0)=g$ and $\dot g(0)=h$.
In the following we label any object which depends on $g(t)$.
First we note that
$ \frac{d}{dt} |_{t=0} dvol_{g(t)} = \langle g,h\rangle dvol_g$
and by help of Lemma \ref{variierealles} we compute
\begin{align}
\frac{d}{dt}\big|_{t=0} \int_M\|T\|^2_{g(t)} dvol_{g(t)}
&= \frac{d}{dt}\big|_{t=0} \int_M 6\left\langle T,T \right\rangle_{g(t)}
dvol_{g(t)}\nonumber \\
&= \frac{d}{dt}\big|_{t=0} \int_M 6\left\langle *_{g(t)}\,T,*_{g(t)}\,T \right\rangle_{g(t)}
dvol_{g(t)}\nonumber \\
&= 3\int_M \left(\left\langle -2 g^{*_gT}
+ \left\langle*_g T , *_g T\right\rangle_g g,h \right\rangle
+4 \big\langle *_g T , \frac{d}{dt}\big|_{t=0} \left(*_{g(t)}\,T \right)\big\rangle_g
\right)
dvol_g\nonumber \\
&=\int_M \left(
\left\langle -6 g^{*_gT}
+ \frac12 \| T\|^2 \, g,h \right\rangle
+12 \big\langle *_g T , \frac{d}{dt}\big|_{t=0} \left(*_{g(t)}\,T \right)\big\rangle_g
\right)
dvol_g \label{dreck_1}
\end{align}
Now we calculate
\begin{align}
\frac{d}{dt}\big|_{t=0} \int_M\|\delta_{g(t)}T\|^2_{g(t)} dvol_{g(t)}
&= \frac{d}{dt}\big|_{t=0} \int_M 2\left\langle\delta_{g(t)} T,\delta_{g(t)} T \right\rangle_{g(t)}
dvol_{g(t)}\nonumber \\
&= \frac{d}{dt}\big|_{t=0} \int_M 2\left\langle d*_{g(t)} T,d*_{g(t)} T \right\rangle_{g(t)}
dvol_{g(t)}\nonumber \\
&= \int_M \Big(\left\langle -2 g^{d*_gT}
+ \left\langle d*_g T ,d *_g T\right\rangle_g g,h \right\rangle\nonumber \\
&\qquad\qquad\quad +4 \big\langle \delta_g d*_g T , \frac{d}{dt}\big|_{t=0} \left(*_{g(t)}\,T \right)\big\rangle_g
\Big)
dvol_g\nonumber \\
&= \int_M \Big(\left\langle -2 g^{d*_gT}
+ \frac12 \| \delta_g T \| \, g,h \right\rangle\nonumber \\
&\qquad\qquad\quad +12\,\frac{\gamma_1}{\gamma_2}\,\big\langle *_g T , \frac{d}{dt}\big|_{t=0} \left(*_{g(t)}\,T \right)\big\rangle_g
\Big)
dvol_g\label{dreck_2}
\end{align}
where we have inserted $3\gamma_1*_g\! T=\gamma_2\,\delta_g d *_g\! T$ which we obtained from (\ref{variation_torsion}).\medskip
\noindent
Finally \cite[Proposition~4.17]{Besse} and \cite[(4.77)]{Besse} tell us that
\begin{equation}
\frac{d}{dt}\big|_{t=0} \int_M \left(-\alpha R^{g(t)}-\beta \|C^{g(t)}\|^2_{g(t)}\right)
dvol_{g(t)}
=
\int_M\left\langle \alpha G^g-\beta B^g , h \right\rangle
dvol_g.\label{dreck_3}
\end{equation}
Combining (\ref{dreck_1}), (\ref{dreck_2}) and (\ref{dreck_3}) gives the assertion (\ref{variation_metric}).
{\hfill$\boxbox$}}
\begin{rem}
a) Equation (\ref{variation_torsion}) is a Proca equation for a $3$-form.
This suggests a physical interpretation of torsion as a massive vector boson.
This feature has been observed earlier in the Lorentzian context, see~\cite{Obukhov2}, and appears to be natural for dynamical Lagrangians of the torsion.\\
b) Equivalently, the Proca equation (\ref{variation_torsion}) can be expressed as
$\Delta T= \frac{\gamma_2}{3 \,\gamma_1} T $ under the condition that $dT=0$.
\end{rem}
\begin{rem}
Ricci flat manifolds $(M,g)$ with $T=0$ are critical points of ${\widetilde I}_{CC}$.
This follows from the fact that Ricci flat manifolds have vanishing Bach tensors (see \cite[Prop.~4.78]{Besse}).
\end{rem}
Finding solutions for the equations of motions (\ref{variation_torsion}) and (\ref{variation_metric}) with $T\ne 0$ is a challenge.
The following lemmas show that classes of warped products with special choices of $T$ can be excluded.
\begin{lemma}\label{lemma_Robertson_Walker}
Let $(N,h)$ be a compact oriented $3$-dimensional Riemannian manifold with constant curvature, and
let $f:S^1\to (0,\infty)$ be some smooth function on the circle $S^1= \mathbb{R}\slash \mathbb{Z}$.
Consider $M=S^1\times N$ equipped with the warped product metric $g=dt^2\oplus f(t)^2\,h$.
Let $\tau:M\to\mathbb{R}$ be a smooth function and set the $3$-form $T=\tau\cdot \pi^*dvol_{(N,h)}$ where $dvol_{(N,h)}$ is the Riemannian volume form of $(N,h)$ and $\pi:S^1\times N\to N$ is the canonical projection, and let this triple $(M,g,T)$ solve the equations of motions (\ref{variation_torsion}) and (\ref{variation_metric}). Then the torsion is zero: $T=0$.
\end{lemma}
\pf{
As $(N,h)$ is locally conformally flat, so is the warped product $(M,g)$.
Therefore the Bach tensor of $(M,g)$ vanishes, $B^g=0$.
In order to get the Einstein tensor $G^g$ we need to calculate the curvatures of $M$.
By $X$, $Y$, $Z$ we denote vectors tangent to the leaves $\{t\}\times N$, $\frac{\partial}{\partial t}$ is the unit normal vector field.
For the Levi-Connection connection we obain
\[
\nabla^g_X \tfrac{\partial}{\partial t}=\tfrac{\dot f(t)}{f(t)}\, X,\quad
\nabla^g_{\tfrac{\partial}{\partial t}} \tfrac{\partial}{\partial t} =0,\quad
\nabla^g_X Y =\nabla^h_X Y - \dot f(t) \,f(t)\,h(X,Y)\,\tfrac{\partial}{\partial t},
\]
where $X,Y$ are also considered as tangent vectors fields of $(N,h)$ and $\nabla^h$ is the corresponding Levi-Connection connection.
The Riemann tensor of $\nabla^g$ is given by
\begin{eqnarray*}
\operatorname{Riem}^g (X,Y)Z&=&\operatorname{Riem}^h (X,Y)Z +\left(\tfrac{\dot f(t)}{f(t)} \right)^2\cdot \left\{g(X,Z)Y- g(Y,Z)X \right\}\\
\operatorname{Riem}^g (X,\tfrac{\partial}{\partial t})Y&=& \tfrac{\ddot f(t)}{f(t)}\, g(X,Y)\, \tfrac{\partial}{\partial t}\\
\operatorname{Riem}^g (X,\tfrac{\partial}{\partial t})\tfrac{\partial}{\partial t}&=& -\tfrac{\ddot f(t)}{f(t)}\, X.
\end{eqnarray*}
From that we get the Ricci curvature and the scalar curvature
\begin{eqnarray*}
\operatorname{ric}^g &=&
\left(-3\tfrac{\ddot f(t)}{f(t)}\right)\,dt^2 \;\oplus\;\left(\operatorname{ric}^h -(\ddot f(t)\,f(t)+2(\dot f(t))^2)\,h \right)
\\
R^g &=&\tfrac{1}{f(t)^2}\left( R^h -6\ddot f(t)\,f(t) -6 (\dot f(t))^2 \right).
\end{eqnarray*}
As $(N,h)$ is assumed to have constant curvature there is a $\kappa\in\mathbb{R}$ such that
$\operatorname{ric}^h=2\kappa\,h$ and $R^h= 6\kappa$.
Hence the Einstein tensor of $(M,g)$ is
\begin{eqnarray}\label{warped_einstein}
G^g= \tfrac{3}{f(t)^2}\left( (\dot f(t))^2-\kappa \right) dt^2\;\oplus\;
\left( 2\ddot f(t)\,f(t)+ (\dot f(t))^2-\kappa \right)\, h.
\end{eqnarray}
Next, we consider normal coordinates $(x^1,x^2,x^3)$ of $N$ about $p$ such that $\tfrac{\partial}{\partial x^1}, \tfrac{\partial}{\partial x^2},\tfrac{\partial}{\partial x^3}$ forms an orthonormal basis of $T_pN$ with respect to $h$.
For the volume distortion $\sqrt{h}$, given by $\sqrt{h}(x)=\sqrt{\det(h_{ij}(x)}$, we get in $p$:
\[ \sqrt{h}=1,\quad \tfrac{\partial}{\partial x^i}\sqrt{h} =0,\quad\sum_{i=1}^3\tfrac{\partial^2}{(\partial x^i)^2}\sqrt{h}=-\tfrac{1}{3} R^h(p)= -\tfrac{1}{3}\kappa.
\]
Now we take the product chart $(t,x^1,x^2,x^3)$ of $M$.
In these coordinates the torsion $3$-form $T$ reads as $\tau(t,x^1,x^2,x^3)\sqrt{h}(x^1,x^2,x^3) dx^1\wedge dx^2\wedge dx^3$.
From (\ref{variation_torsion}) we get $dT=0$ which implies that $\tfrac{\partial \tau}{\partial t}\equiv 0$.\medskip
\noindent
As $dx^1,dx^2,dx^3$ is an orthonormal basis of $T^*_pN$ w.r.t.~$h$, we get that $dt, f\,dx^1, f\,dx^2, f\,dx^3$ forms an orthonormal basis of $T^*_{(t,p)}M$ w.r.t.~$g$.
Hence, we get
\begin{eqnarray}\label{tosionsnormquadrat}
\| T\|_g^2=6\langle T,T\rangle = \tfrac{6\tau^2}{f^6}.
\end{eqnarray}
In $(t,p)$ we get $*T= \tau(x^1,x^2,x^3)\tfrac{1}{f(t)^3} dt$ and thus
\begin{eqnarray}\label{gsternT}
g^{*T}= \tfrac{\tau^2}{f(t)^6}\,dt^2
\end{eqnarray}
Furthermore we obtain $d*T=\tfrac{1}{f(t)^3}\sum\limits_{i=1}^3\tfrac{\partial \tau}{\partial x^i} dx^i\wedge dt$, and therefore
\begin{eqnarray}
g^{d*T}(\tfrac{\partial }{\partial t}, \tfrac{\partial }{\partial t})&=& \tfrac{1}{f(t)^6}\,\|\operatorname{grad}^g\tau\|_g^2\nonumber\\
g^{d*T}(\tfrac{\partial }{\partial t},\tfrac{\partial }{\partial x^i}) &=& 0\nonumber\\
g^{d*T}(\tfrac{\partial }{\partial x^i},\tfrac{\partial }{\partial x^j})&=& \tfrac{1}{f(t)^6}\, d\tau(\tfrac{\partial }{\partial x^i})\cdot d\tau(\tfrac{\partial }{\partial x^j}).
\nonumber\end{eqnarray}
If we now assume that equation (\ref{variation_metric}) holds, we observe that after restricting the occuring $(2,0)$-tensors to $TN$ every ingredient is a multiple of $h$ except $g^{d*T}$. So (\ref{variation_metric}) can only hold if $d\tau=0$.
Therefore the function $\tau$ is constant, and $d*T=0$ and so $g^{d*T}=0$.
Then we decompose (\ref{variation_metric}) into its $dt^2$-component and its $h$-component:
\begin{eqnarray}
\tfrac{3\alpha}{f(t)^2}\left( \dot f(t))^2-\kappa\right)&=& \tfrac{6\gamma_1\tau^2}{f(t)^6}- \tfrac{6\gamma_1\tau^2}{2f(t)^6}\nonumber\\
\alpha\left( 2\ddot f(t)\,f(t)+ (\dot f(t))^2-\kappa \right)&=&- \tfrac{6\gamma_1\tau^2}{2f(t)^4}
\nonumber\end{eqnarray}
We divide the first equation by $3$, the second one by $(f(t)^2$, and substract the first from the second.
We get:
\[
\ddot f(t) = - \tfrac{\gamma_1\tau^2}{\alpha}\cdot \tfrac{1}{f(t)^5}.
\]
Since $f>0$ and $\alpha > 0$ we conclude that $\ddot f\geq 0$ and therefore $\dot f$ increases monotonically.
On the other hand $\dot f$ being a function on $S^1=\mathbb{R}\slash\mathbb{Z}$ is periodic.
Therefore $\dot f$ is constant and hence $\tau$ is zero.
{\hfill$\boxbox$}}
\begin{rem}
It should be noted that with the ansatz in Lemma~\ref{lemma_Robertson_Walker} one can obtain
restrictions on the geometry of $(N,h)$ pointwise from equation (\ref{variation_torsion}). Namely, in the normal coordinates from above one finds
\[
d\delta T = -\tau \sum_{i=1}^3 \tfrac{\partial^2}{(\partial x^i)^2}\sqrt{h}=\tfrac{1}{3} \kappa T.
\]
By (\ref{variation_torsion}) we have $\tfrac{1}{3} \kappa = \tfrac {3\gamma_1}{\gamma_2}>0$, therefore $(N,h)$ is a spherical space form.
\end{rem}
\begin{rem}
If we now consider formally the same equations of motions for Lorentzian manifolds, and if we admit also non-compact manifolds as solutions, the argument from
the proof of Lemma~\ref{lemma_Robertson_Walker} cannot discard the Robertson-Walker ansatz $M=\mathbb{R}\times N$ with $g= -dt^2\oplus f(t)^2\,h$ and
$T=\tau\cdot \pi^*dvol_{(N,h)}$ because
in the above proof the compactness of $S^1$ is essential.
\end{rem}
One could argue that in the above examples the torsion $T$ is not dynamical.
In the last example we consider a situation where torsion may be dynamical, and we show that the torsion vanishes by other reasons.
\begin{lemma}\label{warpedtorsiontorus}
Let $(N,h)$ be the flat torus $T^3=\mathbb{R}^3\slash\mathbb{Z}^3$, let $f:S^1\to (0,\infty)$ be some smooth function.
Consider $M=S^1\times N$ equipped with the warped product metric $g=dt^2\oplus f(t)^2\,h$.
Let $\tau:S^1\to (0,\infty)$ be a smooth function, and let
\[
T=\tau(t) (dx^1\wedge dx^2 +dx^2\wedge dx^3 + dx^3\wedge dx^1)\wedge dt.
\]
Assume that the triple $(M,g,T)$ solves the equations of motions (\ref{variation_torsion}) and (\ref{variation_metric}).
Then the torsion is zero: $T=0$.
\end{lemma}
\pf{
For the $T$ as above we find
\begin{eqnarray}
*T&=& \tfrac{\tau(t)}{f(t)}\left(dx^3+dx^1+dx^2\right), \nonumber\\
d*T&=&\tfrac{f(t)\dot\tau(t)-\tau(t)\dot f(t)}{f(t)^2} dt\wedge \left(dx^3+dx^1+dx^2\right).
\nonumber\end{eqnarray}
Furthermore we get $\| T\|^2=18\cdot (\tfrac{\tau(t)}{f(t)^2})^2$ and $\|\delta T\|^2= \tfrac{2}{f(t)^6}\cdot\left(f(t)\dot\tau(t)-\dot f(t)\tau(t)\right)^2$ and
\begin{eqnarray}
g^{*T}(\tfrac{\partial}{\partial t},\tfrac{\partial}{\partial t})&=& 0,\nonumber\\
g^{d*T}(\tfrac{\partial}{\partial t},\tfrac{\partial}{\partial t})&=& 3\cdot \tfrac{(f(t)\dot\tau(t)-\tau(t)\dot f(t))^2}{f(t)^6}.
\nonumber\end{eqnarray}
Now we insert twice $\tfrac{\partial}{\partial t}$ into the $(2,0)$-tensors of (\ref{variation_metric}) and use the specific form of the Einstein tensor (\ref{warped_einstein}). This yields
\[
0= 3\alpha \cdot \tfrac{\left( \dot f(t)\right)^2}{\left(f(t) \right)^2} + 9\gamma_1 \cdot \tfrac{\left(\tau(t)\right)^2}{\left(f(t)\right)^4}
+ 5\gamma_2 \cdot \tfrac{\left(f(t) \dot \tau(t) - \tau(t) \dot f(t)\right)^2}{\left(f(t)\right)^6}.
\]
Since $\alpha,\gamma_1,\gamma_2 > 0$ each summand is nonnegative, therefore each term vanishes individually, in particular the second one.
From this we conclude $\tau=0$.
{\hfill$\boxbox$}}
\begin{rem}
In the Lorentzian setting actions similar to $\widetilde{I}_{CC}$ have already been consider and some cosmological consequences for possible critical points with non-vanishing torsion have been discussed (see e.g.~\cite{HHKN76} or \cite{S02} and the references therein).
\end{rem}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,260 |
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Stockholm – film del 2013 diretto da Rodrigo Sorogoyen
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Stockholm – transatlantico svedese | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,622 |
@implementation CommentHeaderFooterView
- (instancetype)initWithReuseIdentifier:(NSString *)reuseIdentifier {
if (self = [super initWithReuseIdentifier:reuseIdentifier]) {
self.textLabel.textColor = [UIColor darkGrayColor];
self.contentView.backgroundColor = Color(206, 206, 206);
}
return self;
}
@end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 765 |
<!--公安视频 -->
<configuration>
<layer>http://[serverip]/arcgis/rest/services/putuo/pt_live/MapServer/3</layer>
<refreshrate>0</refreshrate>
<refreshneedquery>false</refreshneedquery>
<widgetvisible>false</widgetvisible>
<popup>
<title>{SXJMC}</title>
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<field name="ADD_PCSMC" alias="派出所:" visible="true" />
<field name="LG_ID" alias="编号:" visible="true"/>
<field name="DH" alias="电话:" visible="true"/>
<field name="FX"/>
<field name="picCount"/>
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<showzoomtobutton>false</showzoomtobutton>
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<button label="详细信息" action="showDevice" type="4" idfield="IPAndTDH" />
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<popupskinclass>CameraWithPic</popupskinclass>
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<?php
namespace Cardinity\Method\Payment;
use Cardinity\Method\ResultObject;
class AuthorizationInformation extends ResultObject
{
/** @type string URL where customer should be redirected to complete
* a payment authorization.
* Value assigned by Cardinity. */
private $url;
/** @type string Data which must be passed along with the customer being
* redirected.
* Value assigned by Cardinity.*/
private $data;
/**
* Gets the value of url.
* @return mixed
*/
public function getUrl()
{
return $this->url;
}
/**
* Sets the value of url.
* @param mixed $url the url
* @return void
*/
public function setUrl($url)
{
$this->url = $url;
}
/**
* Gets the value of data.
* @return mixed
*/
public function getData()
{
return $this->data;
}
/**
* Sets the value of data.
* @param mixed $data the data
* @return void
*/
public function setData($data)
{
$this->data = $data;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,107 |
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